Applied Fundamentals in Finance: Portfolio Management and Investments 3658410205, 9783658410209

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Springer Texts in Business and Economics

Enzo Mondello

Applied Fundamentals in Finance Portfolio Management and Investments

Springer Texts in Business and Economics

Springer Texts in Business and Economics (STBE) delivers high-quality instructional content for undergraduates and graduates in all areas of Business/Management Science and Economics. The series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual self-study. All texts are authored by established experts in their fields and offer a solid methodological background, often accompanied by problems and exercises.

Enzo Mondello

Applied Fundamentals in Finance Portfolio Management and Investments

Enzo Mondello CfBS Center for Business Studies AG Risch, Switzerland

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-658-41020-9 ISBN 978-3-658-41021-6 (eBook) https://doi.org/10.1007/978-3-658-41021-6 Translation from the German language edition: “Finance: Investments” by Enzo Mondello, # Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023. Published by Springer Fachmedien Wiesbaden. All Rights Reserved. # The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Gabler imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface

Finance or financial market theory deals with the raising and investment of capital on the financial markets. The importance of finance in investing, and therefore in financial analysis and in portfolio and risk management, is due to the fact that large fortunes can be both made and lost on the financial markets. The correct implementation of the concepts of finance makes it possible for profits to be made or for losses to be limited. Market participants should therefore understand the assumptions underlying the models and be aware of the limits of their application. This book is available in English and German. It is designed as a textbook and covers the concepts of finance that are relevant for making capital investments. The book is divided into four parts and consists of fifteen chapters. The first part covers portfolio management and consists of seven chapters. The various return and risk measures are introduced, followed by an examination of investment characteristics such as the properties of a return distribution, the efficient market hypothesis, and the relationship between market information efficiency and behavioural finance. Next, the construction of efficient risky portfolios that lie on the efficient frontier is demonstrated using historical return data. The optimal risky portfolio is obtained by combining the efficient frontier with the investor-specific indifference curves. The inclusion of the risk-free asset in the portfolio leads to the most efficient capital allocation line, on which the optimal portfolio is located. Next, the capital asset pricing model and the Fama–French model are presented—both models can be used to calculate the expected return on individual investments or portfolios. The first part ends with the portfolio management process, which focuses on the three phases of planning, execution, and feedback. This process ensures the systematic construction of a portfolio that is appropriate to the client’s needs. The second part of the book consists of three chapters and covers the analysis and valuation of equity securities. The dividend discount model, free cash flow models, such as the free cash flow to equity model, the free cash flow to firm model, and the adjusted present value model, and price and value multiples are presented. The equity valuation models are used in fundamental analysis to identify mispriced stocks. The third part is dedicated to bonds, which are the largest asset class worldwide. The two chapters on bonds include price and yield calculations as well as risk analysis by means of duration and convexity. The fourth part of the book covers derivatives and consists of three chapters. It describes the various uses, profit/loss calculation, and pricing of v

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Preface

derivatives. This part ends with a chapter on option strategies, which can be applied to modify the risk exposure of an asset or liability position. Furthermore, it is demonstrated how option strategies can be used to speculate on a predicted price direction and volatility of the underlying (e.g. a stock). The book is based on the following three principles. First, the concepts of finance are explained in an understandable way, presenting not only the theory but also the practical application. The practice-oriented character of the book is emphasised by the Microsoft Excel applications, which are found at the end of each chapter. Second, whenever possible, the concepts of finance are applied to concrete examples from the German and Swiss financial markets. Third, the book is largely modular, which means that the reader can also look up individual models such as the Markowitz model, the capital asset pricing model, the Fama-French model, or the Black-Scholes model. The fifteen chapters of the book each consist of an introduction, the learning content, a summary, problems and solutions, Microsoft Excel applications, as far as possible, and a bibliography. The stocks, indices, bonds, interest rates, currencies, and derivatives included in the application examples and exercises refer primarily to the German and Swiss financial markets. The motivation for writing books came from many years of teaching at universities, universities of applied sciences, and in the preparatory courses for the Chartered Financial Analyst (CFA®) at CfBS Center for Business Studies. I am very enthusiastic about transferring my finance knowledge into textbook form. The results of these efforts are the six books Applied Fundamentals in Finance (2023), Finance: Investments (2023), Corporate Finance (2022), Finance (2017), Aktienbewertung (2017), and Portfoliomanagement (2015). I would like to thank all those who have supported me in the preparation and writing of this book. In particular, I would like to thank my students who gave me valuable suggestions that contributed significantly to the successful completion of the book. The publication of this work marks the end of another book project for me. I hope that you enjoy reading the book as much as I enjoyed writing it. Risch, Switzerland January 2023

Enzo Mondello

About this Book

This book provides a comprehensive introduction to portfolio management and investments. It can be used at universities in lower/intermediate semesters as well as in continuing education courses. Furthermore, it is also suitable for practitioners who work in the fields of financial analysis and portfolio management or who aspire to such professional activity in the financial industry. The book has a high practical relevance. The portfolio and investment applications refer primarily to the German and Swiss financial markets and their implementations are also demonstrated in Microsoft Excel.

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Contents

Part I

Portfolio Management

1

Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Simple (Discrete) Investment Return . . . . . . . . . . . . . . . . . . . 1.3 Continuous Compounded Investment Return . . . . . . . . . . . . . . 1.4 Investment Return Over Several Periods . . . . . . . . . . . . . . . . . 1.5 Arithmetic Mean Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Geometric Mean Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Money-Weighted Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Real Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Expected Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 5 6 7 9 11 15 16 17 18 20 25

2

Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variance and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . 2.3 Average Return and Standard Deviation . . . . . . . . . . . . . . . . . 2.4 Downside Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Standard Normal Distribution Table . . . . . . . . . . . . . . . . . . Cumulative Probabilities for a Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative Probabilities for a Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 29 38 40 45 51 52 53 56 56 57 58

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3

Other Investment Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Properties of a Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . 3.3 Market Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Information Efficiency of Financial Markets . . . . . . . . 3.3.2 The Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Behavioural Finance and Market Efficiency . . . . . . . . 3.3.4 Market Liquidity and Trading Costs . . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 63 63 64 66 71 75 75 79 82 84 87 89 91 96 98

4

Efficient Risky Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Expected Return and Risk of a Two-Asset Portfolio . . . . . . . . 4.3 The Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Expected Return and Risk of a Portfolio Consisting of Many Risky Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Diversification Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 102 112

Optimal Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Concept of Risk Aversion . . . . . . . . . . . . . . . . . . . . . 5.2.2 Utility Theory and Indifference Curves . . . . . . . . . . . 5.3 The Optimal Risky Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Risk-Free Investment: Capital Allocation Line Model . . . . 5.5 Homogeneous Expectations: Capital Market Line Model . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 145 145 147 153 154 166 171 172 175 182 185

5

116 123 128 131 133 136 142

Contents

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6

Capital Asset Pricing Model and Fama-French Model . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Basics of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Calculation and Interpretation of the Beta . . . . . . . . . . 6.2.3 The Security Market Line . . . . . . . . . . . . . . . . . . . . . 6.2.4 Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Applications of the CAPM in Corporate Finance . . . . 6.3 Fama-French Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Risk Premiums for Size and Value . . . . . . . . . . . 6.3.2 Expected Rate of Return . . . . . . . . . . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 188 188 189 195 199 201 208 208 209 214 216 218 220 221 222

7

Portfolio Management Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Investment Objectives and Constraints . . . . . . . . . . . . 7.2.1.1 Risk Objectives . . . . . . . . . . . . . . . . . . . . . 7.2.1.2 Return Objectives . . . . . . . . . . . . . . . . . . . . 7.2.1.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Investment Policy Statement . . . . . . . . . . . . . . . . . . . 7.2.3 Capital Market Expectations . . . . . . . . . . . . . . . . . . . 7.2.4 Strategic Asset Allocation . . . . . . . . . . . . . . . . . . . . . 7.3 Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Monitoring the Investment Policy Statement . . . . . . . . 7.4.2 Monitoring Capital Market Expectations . . . . . . . . . . 7.4.3 Rebalancing the Portfolio . . . . . . . . . . . . . . . . . . . . . 7.4.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . 7.5 Performance Attribution of an Active Portfolio . . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 223 224 224 224 228 229 233 234 234 239 239 239 241 241 243 246 249 251 253 257

Part II 8

Equity Securities

Dividend Discount Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Fundamentals of Equity Valuation . . . . . . . . . . . . . . . . . . . . . 8.3 Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 One-Stage Dividend Discount Model . . . . . . . . . . . . . . . . . . .

261 261 262 265 268

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Contents

8.5 Two-Stage Dividend Discount Model . . . . . . . . . . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

10

.. .. .. .. ..

. . . . .

275 281 282 283 287

Free Cash Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Free Cash Flow to Equity Model . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Definition and Calculation of the FCFE . . . . . . . . . . . 9.2.3 Growth Rate of the FCFE . . . . . . . . . . . . . . . . . . . . . 9.2.4 One-Stage FCFE Model . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Two-Stage FCFE Model . . . . . . . . . . . . . . . . . . . . . . 9.3 Free Cash Flow to Firm Model . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Definition and Calculation of Free Cash Flow to Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Growth Rate of the FCFF . . . . . . . . . . . . . . . . . . . . . 9.3.3 One-Stage FCFF Model . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Comparison Between FCFE and FCFF Models . . . . . 9.4 Adjusted Present Value Model . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 290 290 290 291 294 297 301

Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Price-to-Earnings Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 P/E Ratio Based on Forecast Fundamentals . . . . . . . . 10.2.3 P/E Ratio Based on Comparable Companies . . . . . . . . 10.3 Price/Earnings-to-Growth Ratio . . . . . . . . . . . . . . . . . . . . . . . 10.4 Price-to-Book Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 P/B Ratio Based on Forecast Fundamentals . . . . . . . . 10.4.3 P/B Ratio Based on Comparable Companies . . . . . . . 10.5 Enterprise Value EBITDA Ratio . . . . . . . . . . . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327 327 329 329 332 335 338 342 342 345 347 350 355 356 358 362 363

301 303 304 310 310 316 318 319 325

Contents

Part III 11

12

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Bonds

Bond Price and Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Features of a Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Different Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Pricing of Fixed-Rate Bonds . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Pricing Fixed-Rate Bonds with a Fixed Risk-Adjusted Discount Rate . . . . . . . . . . . . . . . . . . . 11.4.2 Pricing Fixed-Rate Bonds with Risk-Adjusted Discount Rates That Correspond to the Timing of the Cash Flows . . . . . . . 11.4.3 Pricing of Zero-Coupon Bonds . . . . . . . . . . . . . . . . . 11.5 Pricing of Floating-Rate Notes . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Yield Measures for Fixed-Rate Bonds . . . . . . . . . . . . . . . . . . 11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367 367 368 371 374

383 387 389 394 400 402 404 408 410 411

Duration and Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Analysis of the Risk Factors . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Interest Rate Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Market Liquidity Risk . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Duration-Convexity Approach . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Modified Duration and Macaulay Duration . . . . . . . . . 12.4.2 Factors Affecting Duration and Price Volatility . . . . . 12.5 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413 413 413 413 415 417 419 421 423 423 427 430 437 440 442 443 445 447 447

375

xiv

Contents

Part IV

Derivatives

13

Futures, Forwards, and Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Use of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Futures and Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Futures Versus Forwards . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Profit and Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Leverage Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.4 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.6 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

451 451 452 454 454 457 459 460 464 466 469 471 473 474 477

14

Options: Basics and Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Basic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Profit and Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Intrinsic Value and Time Value . . . . . . . . . . . . . . . . . . . . . . . 14.5 Binomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 14.6 Black-Scholes Option Pricing Model . . . . . . . . . . . . . . . . . . 14.7 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Leverage Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9 Option Price Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.1 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.2 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.3 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.4 Rho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.5 Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.12 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

479 479 479 481 481 487 492 498 504 509 510 511 512 516 520 522 523 523 527 528 533 534 535

15

Option Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Synthetic Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Synthetic Call and Put Option . . . . . . . . . . . . . . . . . . . . . . . .

537 537 537 540

Contents

15.4

Covered Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Profit and Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Collar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Bull and Bear Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Bull Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Bear Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.3 Spread Strategy with Volatile Share Prices . . . . . . . . . 15.8 Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.1 Long Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.2 Short Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.3 Breakeven Share Price and Volatility . . . . . . . . . . . . . 15.9 Effects of Exercising Options on the Strategy . . . . . . . . . . . . . 15.10 Selection of the Option Strategy . . . . . . . . . . . . . . . . . . . . . . . 15.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.13 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Online Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

541 542 547 548 552 556 557 561 566 567 567 571 575 576 577 578 581 583 586 586

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

About the Author

Enzo Mondello is the founder, owner, and director of CfBS Center for Business Studies AG, which specialises in preparatory courses for CFA®, FRM®, CAIA®, and CMA since 2001. He has many years of teaching experience at universities, universities of applied sciences, and in CFA® preparation courses. Enzo Mondello, Dr. oec. publ., CFA, FRM, CAIA, studied business administration at the University of Zurich, where he graduated in 1995. In the same year, he obtained the diploma for the higher teaching profession in business subjects. From 1995 to 1998, he worked for PricewaterhouseCoopers in Zurich. During this time, he completed his PhD studies at the Department of Banking and Finance of the University of Zurich and received his PhD in 1999 with a dissertation on the subject of banking supervisory auditing of risk management and value at risk models. From 1999 to 2001, he was a lecturer in corporate finance and in banking and finance as well as project manager at the University of Applied Sciences Lucerne. He had teaching assignments, among others, at the University of Zurich and the University of Berne. Since 2001, as owner and managing director of CfBS Center for Business Studies AG, he provides live and online preparatory courses for the Chartered Financial Analyst (CFA®), Financial Risk Manager (FRM®), Chartered Alternative Investment Analyst (CAIA®), and Certified Management Accountant (CMA) certifications. From 2003 to 2011, he developed and directed the Master of Advanced Studies (MAS) in ‘Corporate Finance’ and ‘Banking and Finance (CFA® Track)’ at the University of Applied Sciences Northwestern Switzerland. He has published six books with Springer Gabler: Portfoliomanagement (second edition 2015), Aktienbewertung (second edition 2017), Finance (2017), Corporate Finance (2022), Finance: Investments (second edition 2023), and Applied Fundamentals in Finance (2023). In addition to his work as managing director of CfBS Center for Business Studies AG, he is currently also a lecturer in business administration at the University of St. Gallen, where he gives lectures on ‘Financial Risk Management’ and ‘Selected Finance Topics and their Application’ in the Master of Accounting and Corporate Finance. He is also a lecturer and subject leader for Bachelor and Master modules in banking and finance and corporate finance and for two Certified Advanced Studies (CAS) in ‘Portfolio and Wealth Management’ and ‘Asset Valuation’ at the Kalaidos University of Applied Sciences in Zurich. xvii

List of Abbreviations

ADR AG Ameribor APV BuBills CAPM CCP CD CDO CDS CEO CFA CHF CP DAX EBIT EBITDA EBT ECB e.g. EMIR Eq. ETF EUR Eurex EURIBOR EV EV/EBITDA EV/S EWMA FAUB FCFE

American depository receipt Aktiengesellschaft (public limited company) American Interbank Offered Rate Adjusted present value Non-interest-bearing Treasury bills of the Federal Republic of Germany Capital asset pricing model Central counterparty Certificate of deposit Collateralised debt obligation Credit default swap Chief executive officer Chartered financial analyst Swiss franc Commercial paper Deutscher Aktienindex (German stock index) Earnings before interest and taxes Earnings before interest, taxes, depreciation and amortisation Earnings before taxes European Central Bank Exempli gratia (for example) European Market Infrastructure Regulation Equation Exchange-traded fund Euro European Exchange European Interbank Offered Rate Enterprise value Enterprise value EBITDA ratio Enterprise value sales ratio Exponentially weighted moving average Fachausschuss für Unternehmensbewertung und Betriebswirtschaft Free cash flow to equity xix

xx

FCFF FFM FMIA FRN FTSE GARP GBP GICS HML iBobls iBunds ICB ICMA i.e. IFRS IPS IRR ISIN LIBOR ln LTM MiFID MSCI MVP MWR NTM NYSE OTC P/B P/C P/E PEG SARON SMB SMI SOFR SPI StGB S&P US USA USD US-GAAP

List of Abbreviations

Free cash flow to firm Fama-French model Financial Market Infrastructure Act Floating-rate note Financial Times Stock Exchange Growth at a reasonable price British pound Global Industry Classification System High minus low Inflation-indexed government bonds of the Federal Republic of Germany with original maturities of 5 years Inflation-indexed government bonds of the Federal Republic of Germany with original maturities more than 10 years Industrial Classification Benchmark International Capital Markets Association id est (that is) International Financial Reporting Standards Investment policy statement Internal rate of return International Securities Identification Number London Interbank Offered Rate Natural logarithm Last twelve months Markets in Financial Instruments Directive Morgan Stanley Capital Index Minimum variance portfolio Money-weighted rate of return Next twelve months New York Stock Exchange Over the counter Price-to-book ratio Price-to-cash-flow ratio Price-to-earnings ratio Price/earnings-to-growth ratio Swiss Average Rate Overnight Small minus big Swiss Market Index Secured overnight financing rate Swiss Performance Index Strafgesetzbuch (Swiss Penal Code) Standard & Poor’s United States (of America) United States of America US dollar US Generally Accepted Accounting Principles

List of Abbreviations

VAR VWAP WACC WpHG WTI YEN €STR

value at risk Volume-weighted average price Weighted average cost of capital Wertpapierhandelsgesetz (German Securities Trading Act) West Texas Intermediate Japanese yen Euro short-term rate

xxi

Part I Portfolio Management

1

Return

1.1

Introduction

The risk–return characteristics of individual assets play an important role in both portfolio construction and performance evaluation. In this chapter, the main focus is on returns. First, the periodic investment return is presented, which can be calculated as a simple (discrete) and continuous compounded return and for one or more periods. This is followed by an examination of the average returns of investments, which can be determined using the arithmetic or the geometric mean. While the performance of an investment can be assessed on the basis of the geometric mean return, the performance evaluation of an investor is carried out using the moneyweighted return. In addition, the asset return can be broken down into a nominal and real component. The expected return consists of the risk-free interest rate and a risk premium and can be estimated using, for example, historical returns or a prospective scenario analysis.

1.2

Simple (Discrete) Investment Return

Returns can be calculated either as a discrete return or as a continuous compounded return, and for one or more periods The holding period return represents the return from holding an asset for a specified period of time. The period can be 1 day, 1 week, 1 month, 1 year, 2 years, or some other time period. If the Mercedes-Benz Group stock was bought at a price per share of EUR 68.97 at the beginning of 2015 and later sold at a price per share of EUR 77.58 at the end of 2015 (see Table 1.1), the simple investment return is 12.484% [= (EUR 77.58 - EUR 68.97)/EUR 68.97]. Taking into account the dividend per share of EUR 2.45, the simple return is 16.036% [= (EUR 77.58 - EUR 68.97 + EUR 2.45)/ EUR 68.97]. In general, the simple periodic stock return (r), consisting of a price

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_1

3

4

1

Return

Table 1.1 Simple and continuous compounded returns of the Mercedes-Benz Group stock from 2008 to 2016 (Source: Refinitiv Eikon)

Year 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Share price at the end of December (in EUR) 66.50 26.70 37.23 50.73 33.92 41.32 62.90 68.97 77.58 70.72

Dividend per share (in EUR) 1.50 2.00 0.60 0.00 1.85 2.20 2.20 2.25 2.45 3.25

Simple return (in %)

Continuous compounded return (in %)

Year-end amount (in EUR)

-56.84 41.69 36.26 -29.49 28.30 57.55 13.23 16.04 -4.65

–84.03 34.85 30.94 –34.94 24.92 45.46 12.43 14.88 -4.76

28.70 40.67 55.41 39.07 50.13 78.98 89.43 103.77 98.95

return (capital gain/loss) and a dividend yield, is calculated with the following equation: r=

ðP1 - P0 Þ þ D ðP1 - P0 Þ D = þ = price return þ dividend yield, P0 P0 P0

ð1:1Þ

where P0 = price of the equity security at the beginning of the period, P1 = price of the equity security at the end of the period, and D = dividend per share. The example illustrates that the simple investment return of 16.036% is made up of two components. The first return component of 12.484% represents the price return and comprises the change in price during the period, set in relation to the price at the beginning of the period [(P1 - P0)/P0]. The second component is given by the dividend yield of 3.552%, where the dividend received during the period is divided by the share price at the beginning of the period (D/P0).1 The simple return illustrated with the example of the Mercedes-Benz Group stock can be calculated not only for equity securities but for all financial assets. For example, the simple return on a fixed-rate bond consists of the price return, which is the change in price divided by the price at the beginning of the period, and the coupon yield, which is the ratio between the coupon for the period and the bond price at the beginning of the period. In short, the simple return on any financial asset can be determined by dividing the sum of the capital gain or loss and the income by the price of the asset at the beginning of the period.

1

The simple investment return calculated in this way does not take into account any interest income from the reinvested dividend.

1.3 Continuous Compounded Investment Return

1.3

5

Continuous Compounded Investment Return

As an alternative to the simple return, it is also possible to calculate the continuous compounded return (log return) of an asset, which is used in a large number of financial market models, such as the Black-Scholes model.2 The continuous compounded return can be determined with the help of the simple return using a logarithmic transformation as follows: r s = lnð1 þ rÞ,

ð1:2Þ

where rs = continuous compounded or log return, r = simple return, and ln = natural logarithm. For example, if the Mercedes-Benz Group stock is bought at the beginning of 2015 at a price of EUR 68.97 and sold 1 year later at a price of EUR 77.58 after receiving a dividend of EUR 2.45, the continuous compounded return is 14.88% [= ln(1 + 0.1604)]. Table 1.1 presents the continuous compounded returns of the automobile stock from 2008 to 2016. By transforming Eq. (1.2), the simple or discrete return can be calculated from the continuous compounded return, since the inverse function of the natural logarithm is the exponential function, and therefore the following applies: ers = 1 þ r. Thus, using the continuous compounded return, the simple return can be determined as follows: r = ers - 1,

ð1:3Þ

where e = Euler’s number (e = 2.71828. . .), named after Leonhard Euler (1707-1783). If the continuous compounded return for the Mercedes-Benz Group stock is 14.88%, a simple return of 16.04% (=e0.1488 - 1) can be calculated from this. The continuous compounded and simple returns capture the same truth, and it is only the notation or formulaic representation of this truth that differs. An important point regarding the simple and the log return is that the smaller the simple return, the smaller the difference between the two return measures. This can be justified by the fact that the logarithm refers to the base of Euler’s number of e = 2.71828..., with the result that r ≈ rs holds. Consequently, for small return numbers, the simple and the continuous compounded returns are numerically approximately the same. In other words, if the simple return is small, it does not matter whether the

2

See Sect. 14.6.

6

1

Return

simple or the continuous compounded return is used. Table 1.1 illustrates that in 2016, for example, the two return figures are roughly equal, while in other years the difference between the two return figures is nevertheless substantial. The higher the simple return, the greater the difference.

1.4

Investment Return Over Several Periods

In many cases, the periodic investment return is calculated not only for one period but over a time horizon of several periods. On the basis of simple returns, the return over T periods can be determined as follows: rðT Þ = ½ð1 þ r 1 Þð1 þ r 2 Þ . . . ð1 þ r T Þ - 1,

ð1:4Þ

where r1 = simple return for period 1. If, for example, the Mercedes-Benz Group stock was purchased at the end of 2007 at a price of EUR 66.50 and held until the end of 2016, a return of 48.79% would result for the 9-year investment period if all dividends received are reinvested (for the simple annual returns of the automobile stock, see Table 1.1): rð9Þ = ½ð1 þ ð- 0:5684ÞÞ × ð1 þ 0:4169Þ × . . . × ð1 þ ð- 0:0465ÞÞ - 1 = 0:4879:

The multi-period return on an investment can also be calculated using continuous compounded returns, where the log returns are additive and not multiplicative like the simple returns: r s ðTÞ = r s,1 þ r s,2 þ . . . þ r s,T :

ð1:5Þ

For an investment period from the end of 2007 to the end of 2016, the 9-year continuous compounded return on Mercedes-Benz Group stock is 39.75% (for the continuous compounded annual returns, see Table 1.1): r s ð9Þ = - 0:8403 þ 0:3485 þ . . . þ ð- 0:0476Þ = 0:3975: If the Mercedes-Benz Group stock had been purchased at the end of 2007, held until the end of 2016 and all dividends received reinvested, the final value of this investment would be EUR 98.95. The last column in Table 1.1 presents the annual year-end values. The ending capital amount can be determined with either simple or continuous compounded returns. Using simple returns, the final value of EUR 98.95 can be calculated as follows:

1.5 Arithmetic Mean Return

7

EUR 66:50 × ð1 þ ð- 0:5684ÞÞ × ð1 þ 0:4169Þ × . . . × ð1 þ ð- 0:0465ÞÞ = EUR 66:50 × ð1 þ 0:4879Þ = EUR 98:95: Using log returns, the same final value is obtained as with simple returns:3 EUR 66:50 × eð- 0:8403 þ 0:3485 þ ... þ ð- 0:0476ÞÞ = EUR 66:50 × e0:3975 = EUR 98:96: In general, the final value of an investment with simple or continuous compounded returns over a time horizon of T periods can be determined with the following equations: FV = PVð1 þ r 1 Þð1 þ r 2 Þ . . . ð1 þ r T Þ = PV½1 þ rðT Þ

ð1:6Þ

or FV = PVeðrs,1

þ rs,2 þ ... þ rs,T Þ

= PVers ðTÞ ,

ð1:7Þ

where FV = final value of the investment, and PV = initial value of the investment. Thus, it does not matter whether simple or continuous compounded returns are used to calculate the final value of an asset.

1.5

Arithmetic Mean Return

Where investments have returns over several periods, it may be useful for purposes of comparison or understanding to determine an average return. The simplest approach to calculating the average return is the arithmetic mean return, in which the sum of the periodic returns is divided by the number of periods or returns. r =

1 r1 þ r2 þ . . . þ rT = T T

T

rt , t=1

where r = arithmetic mean return, rt = return for period t (t = 1 ... T ), and T = number of periods or returns.

3

The difference of EUR 0.01 is due to rounding of the annual returns.

ð1:8Þ

8

1

Return

Table 1.2 MSCI indices for China and Germany from 2007 to 2021 (Source: www.msci.com) Year 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Arithmetic mean Geometric mean Volatility (based on the arithmetic mean)

MSCI China Index annual return (in %) 66.24 -50.83 62.63 4.83 -18.24 23.10 3.96 8.26 -7.62 1.11 54.33 -18.75 23.66 29.67 -21.64 10.71 5.78 33.18

MSCI Germany Index annual return (in %) 35.93 -45.87 25.15 8.44 -18.08 30.90 31.37 -10.36 -1.89 2.75 27.70 -22.17 20.77 11.55 5.34 6.77 3.97 23.37

Table 1.2 presents the (simple) annual returns of the Chinese and German equity markets from 2007 to 2021, based on the Morgan Stanley Capital International (MSCI) index. The two MSCI indices are stated as a performance index in US dollars, excluding withholding taxes. In contrast to a price index, where only price changes are included in the return calculation, a performance index also takes into account dividends and other income components in addition to price changes. With 742 constituents, the MSCI China Index represents the country-specific large- and mid-cap equity market segments. It covers approximately 85% of the Chinese equity universe across A shares, H shares, B shares, red chips, P chips, and foreign listings (e.g. American depository receipts, ADRs). On the other hand, the MSCI Germany Index, with 61 constituents, measures the performance of the large- and mid-cap market segments and covers approximately 85% of the German equity market. An annual average return can be determined to assess the return performance over the 15-year period from 2007 to 2021. For example, the arithmetic mean of the MSCI China Index is 10.71% and can be calculated as follows: r =

66:24% þ ð- 50:83%Þ þ . . . þ ð- 21:64%Þ = 10:71%: 15

The arithmetic mean return of 10.71% represents an average return, which means that some annual returns are above and others below the arithmetic mean. On average, all returns equal 10.71%.

1.6 Geometric Mean Return

9

The arithmetic mean return cannot be used to determine the change in the invested capital over time. For example, if a financial asset is purchased at EUR 100 and the returns are 25% in the first year and -25% in the second year, the arithmetic mean return is 0% [= (25% + (-25%))/2]. The arithmetic mean return of 0% implies that after 2 years the investment amount of EUR 100 remains unchanged. In fact, after 1 year the amount invested increases by 25% to EUR 125 [= EUR 100 × (1 + 0.25)] and then falls by 25% to EUR 93.75 [= EUR 125 × (1 + (-0.25))] in the second year. Hence, the annual average return is not 0%, but negative, since the ending value of EUR 93.75 is below the initial value of EUR 100. The arithmetic mean is therefore not suitable for measuring the average rate of change of the invested capital over time; the geometric mean return should rather be used for this purpose.

1.6

Geometric Mean Return

The arithmetic mean return represents the average return achieved on an investment and assumes that the amount invested remains the same at the beginning of each period. In reality, the amount invested changes between periods due to the return earned (income and capital gains/losses). The geometric mean return considers the compounding effect of the returns, or the change in the initial investment amount due to the returns achieved. Assuming that the beginning capital does not change over the entire investment period, the geometric mean return is a better performance measure than the arithmetic mean return due to the compounding effect. The geometric mean return r G can be calculated with the following equation: 1=T

T

r G = ½ð1 þ r 1 Þð1 þ r 2 Þ . . . ð1 þ r T Þ1=T - 1 =

ð1 þ r t Þ

- 1:

ð1:9Þ

t=1

In line with Table 1.2, the geometric mean return for the equity market in China is 5.78% and can be determined as follows: rG = ½ð1 þ 0:6624Þ × ð1 þ ð- 0:5083ÞÞ × . . . × ð1 þ ð- 0:2164ÞÞ1=15 - 1 = 5:78%: In the previous example with a capital investment of EUR 100, the returns were 25% in the first year and -25% in the second year. The geometric mean return of this investment is -3.175% [=((1 + 0.25) × (1 + (-0.25)))1/2 - 1] and, in contrast to the arithmetic mean return of 0%, corresponds to the average annual rate of change of the invested capital. Accordingly, the final value of EUR 93.75 with the geometric mean return of -3.175 % can be calculated as follows:

10

1

Return

Table 1.3 Arithmetic versus geometric mean return (Source: Own illustration)

Year 0 1 2

Annual return (in %) 25 -25

Year-end amount (in EUR) 100.00 125.00 93.75

Year-end amount using arithmetic mean return of 0% (in EUR) 100.00 100.00 100.00

Year-end amount using geometric mean return of 3.175% (in EUR) 100.00 96.825 93.75

EUR 100 × ½1 þ ð- 0:03175Þ2 = EUR 93:75: Thus, the initial capital of EUR 100 falls in 2 years to an amount of EUR 93.75. Annually, the initial investment decreases by the geometric mean return of -3.175% (taking into account the compounding effect). Table 1.3 summarises the calculation of the year-end amount with the arithmetic and geometric mean return. Table 1.2 compares the arithmetic mean return with the geometric mean return for the two equity markets in China and Germany. The difference between the two returns of 4.93% (= 10.71% - 5.78%) for China is greater than that of 2.80% (= 6.77% - 3.97%) for Germany. This is due to the fact that the volatility of the Chinese equity market of 33.18% is higher than the 23.37% of the German stock market. All else being equal, higher (lower) volatility leads to a larger (smaller) difference between the arithmetic and geometric mean returns. Volatility can be viewed as the fluctuations or uncertainty of return data.4 When assessing the performance of highly volatile investments, such as Internet or technology stocks and private equity funds, the arithmetic mean return differs significantly from the geometric mean return, with the result that the change in the invested capital over time can only be tracked with the geometric mean return. With the geometric mean return, one obtains the final amount actually earned if the initial invested capital does not change. In the examples provided so far, the arithmetic mean return has always been larger than the geometric mean return. This is not a coincidence, since the arithmetic mean return is always greater than or at least equal to the geometric mean return. If there are no fluctuations in returns—that is, the returns are equal—both mean return measures are identical (e.g. 10% with annual return data of 10%). In all other cases, the arithmetic mean return exceeds the geometric mean return. The more volatile the return data, the greater the difference between the two mean returns.

4

For the calculation of volatility, see Sect. 2.2.

1.7 Money-Weighted Return

11

Table 1.4 Arithmetic versus geometric mean return (Source: Own illustration) Year 0 1 2 Arithmetic mean return Geometric mean return

1.7

Share price (in EUR) 100 200 100

Return (in %) 100 -50 25 0

Money-Weighted Return

Thus far, this chapter has dealt with the return on an investment. Another performance measure is the return of the investor, which is determined using the moneyweighted return. Depending on whether the chosen investment strategy is passive or active, the return of the asset and that of the investor may differ. The average return of an asset can be calculated using the arithmetic or geometric mean. The geometric mean return represents the periodic rate of change of the initial amount invested and thus captures the performance of a passive investment strategy in which the investor purchases an asset and holds it for a certain period of time (the buy-and-hold strategy). The investor does not make any investments other than reinvesting the income received from the asset. Not all market participants invest their money using a passive investment strategy. There are investors who, contrary to the findings of numerous empirical studies, adopt an active strategy and buy (sell) undervalued (overvalued) assets.5 The success of this strategy is determined by the timing of buying and selling. Depending on the right or wrong trading timing, the average return of the active investor is above or below the geometric mean return of the passive investor. The following example illustrates this relationship. An equity security is bought at a price of EUR 100. After 1 year, the share price rises to EUR 200 and falls to EUR 100 at the end of the following year. The simple returns are 100% [= (EUR 200 - EUR 100)/EUR 100] at the end of the first year and -50% [= (EUR 100 - EUR 200)/EUR 200] at the end of the second year. The arithmetic mean return is 25% [= (100% + (- 50%))/2]. By contrast, the geometric mean return is 0% [=((1 + 1) × (1 - 0.5))2 - 1]. The relevant data are presented in Table 1.4 below. A passive investor buys 10 shares at a unit price of EUR 100 and sells them in 2 years later at a unit price of EUR 100. To buy the 10 shares, a cash amount of EUR 1,000 is required. 2 years later, the investor receives cash proceeds from the sale of EUR 1000. If the present value of these cash flows is set at EUR 0, the internal rate of return (IRR), which corresponds to the discount rate, can be calculated using the following equation:

5

See Sect. 3.3.1.

12

1

- EUR 1000 þ

Return

EUR 0 EUR 1000 þ = EUR 0: ð1 þ IRRÞ1 ð1 þ IRRÞ2

Solving the equation for the IRR gives 0%.6 For the passive investor, the IRR is the same as the geometric mean return because they hold the shares over the 2-year period and do not make any further purchases or sales during this period. The IRR represents the investor’s money-weighted rate of return (MWR). In general, the MWR can be calculated by setting the present value of the cash flows at zero. In this case, the discount rate corresponds to the MWR return: T

CF0 þ t=1

CFt = 0, ð1 þ MWRÞt

ð1:10Þ

where CF0 = cash flow at the beginning of the first period, CFt = cash flow in period t (t = 1 ... T ), and T = number of periods. From the investor’s point of view, sums of money invested in an asset (or portfolio) represent an outgoing cash flow, while any sales proceeds during the investment period and the final amount of the asset (or portfolio) at the end of the investment horizon are to be regarded as incoming cash flows. In a passive strategy, the MWR (i.e. the investor’s return) and the geometric mean return (i.e. the asset’s return) are identical. In an active strategy, on the other hand, the investor’s performance can only be properly assessed using the MWR. For example, 10 shares are bought at a unit price of EUR 100. After 1 year, the investor buys another 15 shares at a price per unit of EUR 200. At the end of the second year, all 25 shares are sold at a unit price of EUR 100. The cash outflow at the time of purchase is EUR 1000 (= 10 × EUR 100). After 1 year, a further cash outflow of EUR 3000 (= 15 × EUR 200) takes place, while at the end of the second year, the cash proceeds from the sale are EUR 2500 (= 25 × EUR 100). The MWR of 32.06% can be calculated using the following equation:7

The IRR is typically calculated using a numerical method, with most financial calculators, as well as Microsoft Excel, employing Newton iterations. The Microsoft Excel applications at the end of the chapter demonstrate how to estimate the IRR. Another way to determine the IRR is to use the Texas Instrument BAII Plus calculator approved for the Chartered Financial Analyst (CFA®) exams as follows: [CF] 1000 [±] [ENTER] [#], 0 [ENTER] [#] [#], 1000 [ENTER] [#] [#]. Then press [IRR] and [CPT]. The expressions in brackets [ ] represent a key in the calculator. 7 The Texas Instrument BAII Plus calculator can be used to determine the MWR as follows: [CF] 1000 [± ] [ENTER] [#] [#], 3000 [±] [ENTER] [#] [# ], 2500 [ENTER] [#] [#]. Then press [IRR] and [CPT]. Before entering the figures, delete the figures from the previous example by using the keys [CF] [2nd] [CE/C]. 6

1.7 Money-Weighted Return

- EUR 1000 þ

13

- EUR 3000 EUR 2500 þ = EUR 0: ð1 þ MWRÞ1 ð1 þ MWRÞ2

The investor’s return of -32.06% is considerably worse than the geometric mean return of 0%, which reflects the asset return. This is due to the unfavourable timing of the stock purchase at the end of the first year. The additional 15 shares were bought at a relatively high price per share of EUR 200. At the end of the second year, the share price fell to EUR 100. The following example demonstrates the calculation of the MWR using the DAX 40 stock of Zalando. Example: Calculation of the Money-Weighted Return for the Stock of Zalando Below are the share prices and corresponding annual simple returns for the Zalando stock (an online platform for fashion and lifestyle) as at the end of December for each of the years from 2017 to 2020 (Source: Refinitiv Eikon): Year 2017 2018 2019 2020

Share price (in EUR) 44.12 22.44 45.47 91.06

Simple return (in %) -49.14 102.63 100.26

Between 2017 and 2020, Zalando shares did not pay any dividends. At the end of 2017, an investor buys 100 Zalando shares at a unit price of EUR 44.12. The following questions need to be answered: 1. What is the geometric mean return of the Zalando stock from the end of 2017 to 2020? 2. The investor buys a further 200 Zalando shares at the end of 2018 at a price per share of EUR 22.44. What is the MWR if the securities are sold at the end of 2020 at a price per share of EUR 91.06? 3. The investor sells 80 Zalando shares at a price per share of EUR 22.44 at the end of 2018. What is the MWR if the remaining securities are sold at the end of December 2020 at a price per share of EUR 91.06? Solution to 1 The geometric mean return is 27.32%: r G = ½ð1 þ ð - 0:4914ÞÞ × ð1 þ 1:0263Þ × ð1 þ 1:0026Þ1=3 - 1 = 27:32%:

(continued)

14

1

Return

Since the geometric mean return is the annual average investment return, it can also be calculated using the ending value and beginning value of the equity security as follows:8 rG =

EUR 91:06 EUR 44:12

1=3

- 1 = 27:32%:

If the investor pursues a passive strategy, they will achieve an average annual return of 27.32%. Solution to 2 The MWR is 55.23% and can be determined using the following equation: - EUR 4412 þ

- EUR 4488 EUR 0 EUR 27,318 þ þ = EUR 0: 1 2 ð1 þ MWRÞ ð1 þ MWRÞ ð1 þ MWRÞ3

The additional 200 Zalando shares were purchased at a favourable time when the price was relatively low. This explains why they perform better than the annual average investment return of 27.32%. The active strategy pays off for the investor. Solution to 3 The MWR is -9.22% and can be calculated using the following equation: - EUR 4412 þ

EUR 1795:20 EUR 0 EUR 1821:20 þ þ = EUR 0: ð1 þ MWRÞ1 ð1 þ MWRÞ2 ð1 þ MWRÞ3

The investor sold 80 Zalando shares at an unfavourable time when a relatively low price level prevailed. As a result, the annual average return of -9.22% falls short of the annual average investment return of 27.32%. The investor’s active strategy did not pay off. An investor’s MWR can be higher or lower than the geometric asset return. If the investor trades at the right time—that is, buys at lower prices and sells at higher ones—then the MWR exceeds the geometric investment return. However, should they choose the wrong timing for their trades—that is, buy at higher prices and sell at lower ones—their performance will be worse than the asset return. The disadvantage of the MWR is that it cannot be compared with the returns of other investors because the purchases (cash outflows) and sales (cash inflows) are different for each individual investor. To complete the discussion of investment returns, the real return and the expected return are examined below.

8

If the equation EUR 44:12 × ð1 þ rG Þ3 = EUR 91:06 is solved for r G , one arrives at 27.32%.

1.8 Real Rate of Return

1.8

15

Real Rate of Return

The nominal return on an investment consists of three components, namely the real risk-free interest rate for deferring consumption, inflation as compensation for lost purchasing power, and a risk premium for the risk of loss taken. The nominal simple investment return can therefore be calculated as follows: r = ð1 þ r F real Þð1 þ INFLÞð1 þ RPÞ - 1,

ð1:11Þ

where rF real = real risk-free rate, INFL = inflation rate, and RP = risk premium. By contrast, the real simple investment return is made up of the real risk-free interest rate and the risk premium: r Real = ð1 þ rF real Þð1 þ RPÞ - 1

ð1:12Þ

or r Real =

ð1 þ r Þ - 1: ð1 þ INFLÞ

ð1:13Þ

If inflation rates change over time, the use of real returns enables a performance comparison of the investment. Moreover, the inclusion of real returns is advantageous when returns are available in different currencies. This makes it possible to compare the returns of countries with different levels of inflation. The performance of an investment can be measured by the real after-tax return. This return measure represents compensation for the consumption deferred, the risk of loss taken, and the taxes paid. The after-tax real rate of return is a reliable benchmark for the investment decisions made by the investor. In portfolio theory, the after-tax real rate of return is generally not used because it is not possible to determine a uniform tax rate for all investors. For example, the amount of tax depends on the investor’s specific tax rate (e.g. through progression), the length of the investment period, and the tax effect of the investment (tax-free or normally taxed). Example: Calculation of the Real Return After Taxes An investor has earned a nominal return of 10% from an investment. The tax rate is 30%, while the inflation rate is 3%. What is the real return after tax? (continued)

16

1

Return

Solution First, the nominal after-tax return of 7% is calculated, since taxes are paid on the nominal amount: r After tax = 10% × ð1- 0:30Þ = 7%: If the inflation rate of 3% is taken into account, the real return after tax is 3.88%: r Real after tax =

1.9

1:07 - 1 = 3:88%: 1:03

Expected Return

Historical return is what an investor has actually earned in the past, whereas expected return is what an investor anticipates obtaining in the future. The expected return is a nominal return consisting of the real risk-free interest rate, the expected inflation rate, and the expected risk premium. The real risk-free interest rate is usually positive due to the deferral of consumption. In an inflationary environment, the expected inflation rate is also positive. If, on the other hand, there is deflation, the inflation rate is negative. Since market participants behave, on average, in a riskaverse manner, the expected risk premium is assumed to be positive. The greater the risk of loss, the higher the expected return. These relationships lead to the following equation for calculating the expected return: E ðr Þ = ð1 þ r F real Þ½1 þ E ðINFLÞ½1 þ EðRPÞ - 1,

ð1:14Þ

where E(INFL) = expected inflation rate, and E(RP) = expected risk premium. The average realised return (arithmetic or geometric mean) reflects the average return actually earned for a period in the past. Since an investment is risky, there is no certainty that the average return actually incurred will correspond to the expected return in the next period. If a sufficiently long time series is available, one can assume that the average historical return is a good indicator of the expected return. However, this assumption presupposes stable returns. Investment decisions made today depend on forecast future returns. Scenario analysis is the most frequently adopted approach to estimate expected returns. In this scheme, several possible scenarios are set out, and a probability as well as an expected return is assigned to each scenario. The expected return of an asset is equal to the sum of the probability-weighted scenario returns:

1.10

Summary

17

n

E ðr Þ =

pi E ðr i Þ,

ð1:15Þ

i=1

where pi = probability for the occurrence of scenario i, E(ri) = expected return for scenario i, and n = number of scenarios. In contrast to historical returns, expected returns are probabilistic in nature because they are not known with certainty. For example, it is assumed that the business cycle in the next period is equally likely to be distributed over each of the following four phases: economic boom, stagnation, recession, and depression. The expected returns of a cyclical equity security for the individual business cycle phases are 20%, 8%, 2%, and -12%, respectively. Thus, the expected return of the stock is 4.5% and can be calculated as follows: Eðr Þ = 0:25 × 20% þ 0:25 × 8% þ 0:25 × 2% þ 0:25 × ð- 12%Þ = 4:5%:

1.10

Summary

• The return on a financial asset consists of the income yield (dividends in the case of equity securities and coupons in the case of bonds) and the price return (capital gains/losses). The periodic investment return can be calculated over one or more periods. • The return on an investment can be determined with discrete or continuous compounding. The simple or discrete return is calculated over a period of time given by two single (or discrete) points in time, namely the beginning and end of the period. The continuous compounded or log return, on the other hand, refers to a return which is infinitely compounded during the investment period. • The simple and continuous compounded returns can also be estimated over several periods. The simple return over several periods is calculated multiplicatively from the simple period returns, while the log return over several periods is determined from the sum of the continuous compounded period returns. The return over several periods can be used to calculate the final value of an investment. It does not matter whether simple or log returns are used; the final value is the same in both cases. • The performance of different investments over several periods can be evaluated with the average return. Either the arithmetic or the geometric mean return is used for this purpose. The arithmetic mean return is easy to calculate and has known statistical properties. The geometric mean return, on the other hand, takes the compounding effect into account and is a better performance measure than the

18

•

•

• •

1

Return

arithmetic mean return if the amount invested does not change. However, if the amount of money invested in the portfolio shifts over the investment period, the MWR must be employed, which reflects the performance of an active strategy. Therefore, the return to investors should be determined utilising the MWR, while the asset return should be calculated with the geometric mean return. If the investor is following a passive strategy (i.e. a buy-and-hold strategy), the geometric mean return can be applied instead of the MWR, since both are equal. The real rate of return is calculated without including inflation as compensation for lost purchasing power. It consists of the real risk-free interest rate, which represents compensation for the deferral of consumption, and a risk premium. The real after-tax return is a reliable benchmark for the investment decisions made by an investor. Historical return is what an investor has actually earned in the past, whereas expected return is what an investor anticipates obtaining in the future. In contrast to historical returns, expected returns are probabilistic in nature because they are not known with certainty. The expected return is a nominal return consisting of the real risk-free interest rate, the expected inflation rate, and the expected risk premium. Assuming stable returns, the expected return can be calculated as the average of historical returns. However, this assumption often does not hold, and therefore, the expected return is usually estimated using scenario analysis.

1.11

Problems

1. An investor buys 200 shares at the beginning of the quarter at a price of EUR 25 per share. At the end of the quarter, they receive a dividend of EUR 5 per share, and the price of the equity security is EUR 30. What is the simple investment return for the quarter? 2. The following annual simple returns on a financial asset are available: Year 2019 2020 2021 2022

Annual simple return (in %) 12 24 35 10

a) What is the simple investment return over the 4-year period from 2019 to 2022? b) What is the continuous compounded investment return over the 4-year period from 2019 to 2022? c) At the beginning of 2019, the initial value of the asset was EUR 100. What is the final value of the asset using simple and continuous compounded returns? d) What is the annual arithmetic mean return using simple annual rates of return? e) What is the annual geometric mean return using simple annual rates of return?

1.11

Problems

19

3. The following simple returns of four mutual funds are available: Investment fund Delta Gamma Vega Rho

Time period after fund launch 78 days 136 days 18 weeks 14 months

Return since inception (in %) 3.52 4.58 4.81 20.44

Which of the four investment funds has the highest annual return? 4. The following simple returns are known for four asset classes: Asset class Equities Corporate bonds Money market debt claims of the Swiss Confederation (risk-free) Inflation

Return (in %) 9.5 4.4 2.1 1.3

a) What is the real rate of return on equities and corporate bonds? b) What is the risk premium on equities and corporate bonds? c) The tax rate is 30%. What are the real after-tax returns on equities and corporate bonds? 5. Presented below are the prices and corresponding annual simple returns for Twitter stock as at the end of December in each year from 2013 to 2016 (Source: Refinitiv Eikon): Year 2013 2014 2015 2016

Share price (in USD) 63.65 35.87 23.14 16.30

Simple return (in %) –43.64 –35.49 –29.56

Twitter Inc. has not paid any dividends since its initial public offering (IPO) in November 2013. At the end of 2013, an investor buys 100 Twitter shares at a price per share of USD 63.65. The following questions should be answered: a) What is the annual geometric mean return on Twitter stock from the end of 2013 through 2016? b) The investor buys a further 200 shares of Twitter Inc. at the end of 2014 at a price per share of USD 35.87. What is the MWR if the securities are sold at the end of 2016 at a price per share of USD 16.30?

20

1

Return

c) The investor sells 50 Twitter shares at a price per share of USD 35.87 at the end of 2014. What is the MWR if the remaining equity securities are sold at a price per share of USD 16.30 at the end of December 2016? 6. Presented below are the prices and corresponding annual returns for Yahoo! stock as at the end of December in each year from 2013 to 2016 (Source: Refinitiv Eikon): Year 2013 2014 2015 2016

Share price (in USD) 40.44 50.51 33.26 38.67

Simple return (in %) 24.90 –34.15 16.27

Yahoo! Inc. did not pay any dividends from 2013 to 2016. At the end of 2013, an investor buys 100 Yahoo! shares at a price per share of USD 40.44. The following questions should be answered: a) What is the geometric mean return on Yahoo! stock from the end of 2013 to 2016? b) The investor buys a further 200 shares of Yahoo! Inc. at the end of 2014 at a price per share of USD 50.51. What is the MWR if the securities are sold at the end of 2016 at a price per share of USD 38.67? c) The investor sells 50 Yahoo! shares at a price per share of USD 50.51 at the end of 2014. What is the MWR if the remaining securities are sold at a price per share of USD 38.67 at the end of December 2016?

1.12

Solutions

1. The simple investment return is 40% and can be calculated with the following equation:

r=

ðEUR 30 - EUR 25Þ þ EUR 5 = 40%: EUR 25

The price return is 20% [= (EUR 30 - EUR 25)/EUR 25] and the dividend yield is 20% (= EUR 5/EUR 25). Simple investment return = price return + dividend yield = 20% + 20% = 40% 2. a) The 4-year simple investment return is 3.42% and can be determined with the following equation:

1.12

Solutions

21

rð4Þ = ½ð1:12Þ × ð0:76Þ × ð1:35Þ × ð0:90Þ - 1 = 3:42%: b) The annual continuous compounded returns can be calculated as follows: rs, 2019 = lnð1:12Þ = 0:1133, rs, 2020 = lnð0:76Þ = - 0:2744, rs, 2021 = lnð1:35Þ = 0:3001, rs, 2022 = lnð0:90Þ = - 0:1054: The 4-year continuous compounded investment return is 3.36% and can be determined using the following equation: r s ð4Þ = 0:1133 þ ð- 0:2744Þ þ 0:3001 þ ð- 0:1054Þ = 3:36%: c) With the 4-year simple investment return of 3.42% the final value is EUR 103.42: FV = EUR 100 × 1:0342 = EUR 103:42: The same final value of EUR 103.42 can be calculated using the 4-year continuous compounded investment return of 3.36%: FV = EUR 100 × e0:0336 = EUR 103:42: Therefore, it makes no difference whether the final value of an asset is determined using the simple rate of return or the continuous compounded rate of return. d) The annual arithmetic mean return is 3.25% and can be determined with the following equation:

22

1

r=

Return

12% þ ð - 24%Þ þ 35% þ ð - 10%Þ = 3:25%: 4

e) The annual geometric mean return of 0.84% can be calculated with the following equation: r G = ½ð1:12Þ × ð0:76Þ × ð1:35Þ × ð0:90Þ1=4 - 1 = 0:84%: 3. The annual returns of the four investment funds can be calculated as follows: rDelta = ð1:0352Þ365=78 - 1 = 17:57%, r Gamma = ð1:0458Þ365=136 - 1 = 12:77%, rVega = ð1:0481Þ52=18 - 1 = 14:54%, r Rho = ð1:2044Þ12=14 - 1 = 17:28%: Delta’s mutual fund has the highest annual return of 17.57%.

4. a)

Real rate of return on equities =

Real rate of return on bonds =

1:095 - 1 = 8:1% 1:013 1:044 - 1 = 3:06% 1:013

b) Risk premium on equities =

Risk premium on bonds =

1:095 - 1 = 7:25% 1:021 1:044 - 1 = 2:25% 1:021

1.12

Solutions

23

c) Equities Nominal‐after tax return = 9:5% × ð1 - 0:30Þ = 5:28% Real‐after tax return =

1:0665 - 1 = 5:28% 1:013

Bonds Nominal‐after tax return = 4:4% × ð1 - 0:30Þ = 3:08% Real‐after tax return =

1:0308 - 1 = 1:76% 1:013

5. a) The geometric mean return is -36.50%: r G = ½ð1 þ ð - 0:4364ÞÞ × ð1 þ ð - 0:3549ÞÞ × ð1 þ ð - 0:2956ÞÞ1=3 - 1 = - 36:50%: Since the geometric mean return is the investment return, it can also be calculated using the final value and initial value of the equity security as follows:

rG =

USD 16:30 USD 63:65

1=3

- 1 = - 36:50%:

If the investor pursues a passive strategy, they will earn an annual negative return of 36.50%. b) The MWR is -34.37% and can be determined using the following equation: - USD 6365 þ

USD 4890 - USD 7174 USD 0 þ þ = USD 0: 1 2 ð1 þ MWRÞ ð1 þ MWRÞ3 ð1 þ MWRÞ

24

1

Return

Buying additional Twitter shares at a lower price than at the end of 2013 improves the negative investment return somewhat, although share prices continue to fall until the end of 2016. c) The MWR is –38.25% and can be calculated using the equation below: - USD 6365 þ

USD 815 USD 1793:50 USD 0 þ þ = USD 0: 1 2 ð1 þ MWRÞ ð1 þ MWRÞ3 ð1 þ MWRÞ

When measured against the passive strategy, there is a deterioration in performance.

6. a) The geometric mean return is -1.48%: r G = ½ð1 þ 0:2490Þ × ð1 þ ð - 0:3415ÞÞ × ð1 þ 0:1627Þ1=3 - 1 = - 1:48%: Since the geometric mean return corresponds to the annual average investment return, it can also be calculated using the ending value and beginning value of the equity security as follows:

rG =

USD 38:67 USD 40:44

1=3

- 1 = - 1:48%:

If the investor pursues a passive strategy, they will achieve an annual average return of -1.48%. b) The MWR is -8.34% and can be determined using the following equation: - USD 4044 þ

- USD 10,102 USD 0 USD 11,601 þ þ = USD 0: 1 2 ð1 þ MWRÞ ð1 þ MWRÞ ð1 þ MWRÞ3

Microsoft Excel Applications

25

The additional 200 Yahoo! shares were bought at an inopportune time when the price was relatively high. This explains why the performance is poorer than the annual average investment return of -1.48%. c) The MWR of 5.45% can be calculated using the equation below:

- USD 4044 þ

USD 1933:50 USD 2525:50 USD 0 þ þ = USD 0: 1 2 ð1 þ MWRÞ ð1 þ MWRÞ3 ð1 þ MWRÞ

The investor sold 50 Yahoo! shares at a favourable time when the price level was relatively high. Therefore, the annual average return of 5.45% exceeds the annual average investment return of -1.48%. The investor’s active strategy has paid off.

Microsoft Excel Applications • To calculate Euler’s number in Microsoft Excel, use the ‘EXP’ function. Euler’s number of 2.71828183. . . is obtained by entering = EXPð1Þ in a cell and pressing the Enter key. In the same way, any value can be found with the exponential function, which is calculated with the value ‘e’ (2.71828...) to the power of any number x: = EXPðxÞ: • The natural logarithm can be determined using the ‘LN’ function. For example, the natural logarithm of 1 can be determined by = LNð1Þ which is written in a cell and then confirmed with the Enter key. The result is 0. In general, the natural logarithm of a positive number x can be calculated as follows: = LNðxÞ:

26

1

Return

• Calculating the arithmetic and geometric mean returns in Excel is relatively simple. If, for example, 10 returns are listed in cells A1 to A10, the arithmetic mean return can be determined by entering the following in cell A11 = AVERAGEðA1:A10Þ and then pressing the Enter key. In order for the geometric mean to be determined type in cell A12 = GEOMEANð1 þ A1:A10Þ - 1 and then confirm with the key combination Ctrl+Shift+Enter. These three keys must be pressed simultaneously. If only the Enter key is pressed, an error message is displayed. Figure 1.1 demonstrates the calculation of the arithmetic and geometric mean returns for the MSCI Germany Index (for the return data, see Table 1.2). • In order to determine the IRR or the MWR, the cash movements into and out of the portfolio for an investment period of 3 years, for example, must first be entered in cells B1 to B4. Thereafter, the following expression can be typed in cell B5:

Fig. 1.1 Arithmetic and geometric mean return for the MSCI Germany Index (Source: Own illustration)

Microsoft Excel Applications

27

= IRRðB1:B4Þ, which is then confirmed with the Enter key.

2

Risk

2.1

Introduction

As with returns, there are different measures of risk, and it is difficult to find a general consensus on how to define risk. The perception of risk varies among market participants and depends, among other things, on the composition of the portfolio, the type of investor (private or institutional), and the investor’s attitude to risk. For a pension fund or insurance company, for example, the risk is that liabilities are not covered by assets. The risk of a mutual fund is characterised by the deviation of the portfolio’s return from a benchmark. A private investor, on the other hand, defines risk as the possibility of their investment decreasing in value as the result of a loss. Various risk measures are presented below. The variance or standard deviation as well as downside risk measures, such as the semi-standard deviation and the value at risk, are discussed.

2.2

Variance and Standard Deviation

Before describing the various quantitative approaches to calculating risk, a qualitative approach is first presented that allows an initial assessment of the risk of loss on investments. Figure 2.1 presents the monthly returns of the two stocks of MercedesBenz Group AG and Deutsche Bank AG for 2016. Figure 2.1 illustrates that the stock returns of Deutsche Bank vary more than those of Mercedes-Benz Group. The bank stock’s return spread is much wider, ranging from -32% to 15%, while the automobile stock has a return spread of -19% to 12%. Therefore, the bank stock appears riskier. The more the returns fluctuate over a period, the greater the uncertainty about future prices and returns, resulting in higher risk. A well-known statistical variable is the variance. It measures the average squared deviation of returns from the arithmetic mean return. If stable returns are assumed, # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_2

29

30

2 Risk

(Monthly returns) 20% 10%

Deutsche Bank AG Mercedes-Benz Group AG

0% -10% -20% -30% -40% Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec (Months for the year 2016) Fig. 2.1 Monthly stock returns of Mercedes-Benz Group AG and Deutsche Bank AG for 2016 (Source: Own illustration based on data from Refinitiv Eikon)

then the expected return can be determined using the arithmetic mean.1 A larger variance means a larger spread of returns and consequently a greater risk of loss. The variance (σ 2) of the population of return data can be calculated as follows: σ2 =

1 T

T

ðr t - μÞ2 ,

ð2:1Þ

t=1

where rt = return for period t, μ = arithmetic mean return (expected return) of the population, and T = number of periods or returns. Squaring is necessary for calculating the variance, as the deviations from the mean return can be positive or negative, which can lead to a variance of zero if the return deviations are added. On the other hand, squaring the return spreads produces positive values. The following example illustrates this relationship. The expected

1

See Sect. 1.9.

2.2 Variance and Standard Deviation

31

return on an investment is 10%. Returns of 20%, 10%, and 0% occur. The return deviations are therefore 10%, 0%, and -10%, resulting in a total deviation from the expected value of 0% [= 10% + 0% + (-10%)]. However, this does not correspond to the fluctuations around the expected value of + 10%, 0%, and -10%. If, on the other hand, the spreads are squared, positive values are obtained. Calculating a risk measure that uses squared deviations has several advantages. First, positive and negative return deviations do not cancel each other out because all variances are positive. Second, squaring the deviations leads to a stronger weight of large deviations. This is consistent with the concept of risk in that larger return fluctuations represent a higher risk of loss. Third, variance has statistical properties that lend themselves well to portfolio optimisation. The variance does not have the same unit as the return, which is expressed as a percentage. Therefore, the variance is converted into the standard deviation so that the same unit (per cent) is obtained as for the returns. The standard deviation (σ) of the population of return data can be determined by taking the root of the variance: σ=

1 T

T

ðr t - μ Þ2 :

ð2:2Þ

t=1

The standard deviation is referred to as volatility and is the most widely used risk measure in the financial markets. If a sample and not the population of return data is available, the variance or standard deviation is divided by T - 1 and not by T. This ~2 and sample standard deviation ðσ~Þ: results in the following sample variance σ σ~2 =

1 T -1

T

ðr t - r Þ2

ð2:3Þ

ðr t - r Þ2 ,

ð2:4Þ

t=1

and

σ~ =

1 T -1

T t=1

where r = arithmetic mean return (expected return) of sample. The sample variance is an approximation of the population variance. It is calculated using the mean of the sample rather than the mean of the entire population. The sample variance is divided by T - 1 and not by T, which ensures that the variance calculated on the basis of a sample is, on average (when random samples are drawn repeatedly), equal to the variance of the population. The individual squared return deviations ðrt - r Þ2 in both Eqs. (2.3) and (2.4) depend on the mean of the sample and not on the mean of the population. The mean of the sample is determined by the

32

2 Risk

individual return observations of the sample. If small return values are drawn at random, the mean also becomes small, as do the terms ðr t - r Þ2 . This effect is corrected for in the sample variance formula by dividing by T - 1 (rather than dividing by T ). This procedure makes it possible to make an unbiased estimate of the variance. Another aspect in the calculation of the variance or standard deviation is the mean return or expected return, which reflects the average return of the historical return data. If the standard deviation is determined with continuous compounded returns, the expected return can be calculated as the arithmetic mean of the continuous compounded returns, since these have the property of additivity.2 The standard deviation with continuous compounded returns can be calculated as follows: ~s = σ

1 T -1

T

ðr s,t - r s Þ2 ,

ð2:5Þ

t=1

where rs,t = continuous compounded return for period t, and r s = expected continuous compounded return (arithmetic mean) or r s =

1 T

T t=1

r s,t :

Furthermore, the choice of the historical time period and the frequency of the observed data are crucial for the estimation of volatility. There is a trade-off between the number of return observations (T) and the length of the historical time interval. If volatility is stationary over time, then the longest possible time period with a large number of returns should be taken to calculate a standard deviation with a low estimation error. However, in the case of non-stationary volatility, a long time period will result in a standard deviation that does not reflect the current risk of the investment. In this case, a rather short time period should be chosen, which on the one hand takes better account of a structural break in the data, but on the other produces a volatility measure that has a high estimation error due to the small number of return observations.3 The following example demonstrates the calculation of the volatility of the Mercedes-Benz Group stock using monthly returns for 2016.

2

See Sect. 1.4. As a guideline, volatility should not be calculated with less than 24 returns, as otherwise the statistical relevance of the risk measure is not given as the standard error is too high. 3

2.2 Variance and Standard Deviation

33

Example: Calculation of the Volatility of the Mercedes-Benz Group Stock Based on Monthly Returns for 2016 The following monthly prices, dividend per share, and simple returns are available for the Mercedes-Benz Group stock for 2016 (Source: Refinitiv Eikon): Month Beginning of January End of January End of February End of March End of April End of May End of June End of July End of August End of September End of October End of November End of December

Share price (in EUR) 77.58 64.16 63.10 67.37 64.19 61.39 53.52 60.82 62.08 62.71 64.91 62.76 70.72

Dividend per share (in EUR)

Simple return (in %)

–17.30 –1.65 6.77 0.10 –4.36 –12.82 13.64 2.07 1.01 3.51 –3.31 12.68

3.25

What is the annualised volatility of the continuous compounded and simple returns of the Mercedes-Benz Group stock? Solution First, the continuous compounded returns must be determined. The log return rs in the month of January of -18.99% can be calculated by solving the following equation according to the continuous compounded return rs: EUR 77:58 × ers × 1 = EUR 64:16 → rs = lnðEUR 64:16=EUR 77:58Þ = - 0:1899: Here, ln(EUR 64.16/EUR 77.58) corresponds to the value ln(1 + r) or is equal to ln[1 + (-0.1730)]. The sum of squared monthly return deviations of 0.0947 can be calculated as follows: Month Beginning of January End of January End of February End of March

Monthly continuous compounded return (rs,t)

Squared monthly return deviation ½ðr s,t - r s Þ2 

–0.1899 –0.0167 0.0655

0.0347 0.0002 0.0048

(continued)

34

2 Risk

Month End of April End of May End of June End of July End of August End of September End of October End of November End of December Total Average (expected return, r s )

Monthly continuous compounded return (rs,t) 0.0010 –0.0446 –0.1372 0.1279 0.0205 0.0101 0.0345 –0.0337 0.1194 –0.0432 –0.0036

Squared monthly return deviation ½ðr s,t - r s Þ2  0.0000 0.0017 0.0178 0.0173 0.0006 0.0002 0.0015 0.0009 0.0151 0.0947

The standard deviation of the monthly continuous compounded returns is 9.28%:4 σ~s =

1 T -1

T

ðr s,t - r s Þ2 = t=1

0:0947 = 0:0928: 12 - 1

Assuming that the returns occur independently (i.e. are not correlated) and thus follow a random walk, the annualised volatility of 32.15% can be calculated by multiplying the standard deviation of the monthly returns by the square root of 12 months as follows:5 (continued)

4

The Microsoft Excel applications at the end of the chapter demonstrate how to calculate the standard deviation. Another way to determine the standard deviation is to use the Texas Instrument BAII Plus calculator as follows: First, press [2nd] [Data] and use [2nd] [CE/C] to delete any numbers contained in a previous calculation. Then enter the continuous compounded returns as Xvalues: 0.1899 [ ±] [Enter] [#], 0.0167 [±] [Enter] [#], etc. The Y-values can be left at 1. After entering the returns, press the [2nd] [Stat] key combination. The display will show LIN. If not, LIN can be set by pressing [2nd] [Enter]. Tapping the [#] key displays the number of returns of 12, pressing [#] displays the expected return of -0.0036, and tapping [#] again displays the standard deviation of the sample of 0.0928. 5 See Sect. 3.3.2.

2.2 Variance and Standard Deviation

35

p ~s, annualised = 0:0928 × 12 = 0:3215: σ The annualised volatility based on continuous compounded returns of 32.15% can be converted into a standard deviation of simple returns of 37.92% as follows:6 ~ = eσ~s, annualised - 1 = e0:3215 - 1 = 0:3792: σ The annualised volatilities of 32.15% and 37.92% determined in the example are based on a very small data series of only 12 return observations. To obtain a longer data series and thereby reduce the standard error, daily share prices and returns over the past year can be used. Since 1 year consists of approximately 252 trading days, a volatility with a lower standard error can be determined from approximately 252 daily returns. A time period of 1 year is useful in that any structural breaks in the data can be better accounted for than with longer time intervals. In addition, different weights can be assigned to the return deviations. Recently occurred return deviations are given a higher weight and thus have a stronger influence on the calculation of volatility. The risk figure calculated in this way better reflects the current risk of loss of the equity security.7 The standard deviation is based on the statistical concept of the normal distribution— also known as the Gaussian distribution.8 The normal distribution is the most widely used distribution and has the following statistical properties: • All normal distributions are characterised by the same distribution shape. The distribution is bell-shaped and has only one ‘peak’ (single-peaked) in the middle of the distribution. The arithmetic mean, the median, and the mode are equal and are located in the middle of the distribution. • The normal distribution is symmetrically distributed around the mean. To the left and right of the mean, the distribution is a mirror image. • The normal distribution slopes slightly and asymptotically from the mean in both directions. The frequencies of the observations become increasingly smaller but

6

For the conversion from continuous to simple returns, see Sect. 1.3. For example, volatility can be estimated using the exponentially weighted moving average model (EWMA). Here, a decay factor that lies between 0 and 1 is used. This factor is responsible for assigning the weights which decrease the older the return observation is. In the model, the weights fall exponentially. 8 The normal distribution is a continuous distribution. The French mathematician Abraham de Moivre (1667-1754) introduced the normal distribution in 1733 when developing a version of the central limit theorem. The term ‘normal distribution’ was coined by the Göttingen mathematician and astronomer Carl Friedrich Gauss (1777-1827). Therefore, the term ‘Gaussian distribution’ is often used for this distribution in German-speaking countries. 7

36

2 Risk

(Density of probability) Probability density function

68.3%

95.5% 99.7% –28.20% –18.92% –9.64% 8.92% 18.20% 27.48% (Returns) Average or expected return of –0.36% Fig. 2.2 Expected return and standard deviation (Source: Own illustration)

never touch the X-axis. Consequently, the range is from minus infinity to plus infinity. • Of all the return observations, 68.3% are within plus/minus one standard deviation of the mean, 95.5% are within plus/minus two standard deviations of the mean, and 99.7% are within plus/minus three standard deviations of the mean. A monthly average or expected return of the Mercedes-Benz Group stock of 0.36% and a standard deviation of the monthly returns of 9.28% means that with a probability of 68.3% the returns fall within a range of -9.64% and 8.92%. With a probability of 95.5%, returns lie within a range of two standard deviations around the expected value, resulting in a return range of -18.92% and 18.20%. Furthermore, there is a 99.7% probability that returns will fall within three standard deviations of the expected value (i.e. between -28.20% and 27.48%). Figure 2.2 presents the relationship between expected return and standard deviation. Furthermore, the use of the normal distribution is supported by the fact that a distribution converges to a normal distribution given a sufficiently large number of independent, identically distributed random variables with finite variances. This approximation property is called the central limit theorem. Therefore, when the number of independent and identically distributed random returns is large, it is reasonable to assume a normal distribution even if the return distribution of a portfolio is not exactly normally distributed. In such a case, all statistical properties of the normal distribution can be taken for the return and risk analysis of an investment. The investment can be fully evaluated using the arithmetic mean

2.2 Variance and Standard Deviation

37

(expected return assuming stable returns) and the variance or standard deviation. All other higher central moments of the distribution, such as skewness and excess kurtosis, are zero. The normal distribution is a continuous one given by realisations of real numbers. For a continuous normally distributed random variable X, with arithmetic mean μ and variance σ 2 [X  Ν(μ, σ 2)], the probability distribution is described by the following density function (for -1 < x < +1): f ðxÞ = p

ðx - μÞ2 1 e - 2σ2 , 2 2πσ

ð2:6Þ

where x = certain value for the continuous random variable X, e = Euler’s number (e = 2.71828. . .), μ = arithmetic mean (equals median and modus) of X, and σ = standard deviation of X. The total area enclosed by the curve of the normal distribution—that is, the integral from -1 to +1—is always 1. The formula gives the probability that a random variable X takes on a value x. If the formula is integrated for a certain interval, the relevant area below the normal distribution is obtained, which corresponds to the probability sought. It is therefore possible to calculate the probability that the return will be above or below a value and within a certain range. For example, for Mercedes-Benz Group stock with a monthly average or expected return of -0.36% and a standard deviation of monthly returns of 9.28%, the probability is 71.82% that the monthly return will be less than 5%. Accordingly, the probability of the return being greater than 5% is 28.18%. The probability that the monthly returns will be in the range of -5% to 5% is 40.97%. The calculations can be made, for example, with a software solution such as Microsoft Excel.9 To determine the probability of a random variable, the arithmetic mean and the standard deviation of the distribution, as well as a target value for the random variable, are needed. The integral must then be determined using Eq. (2.6). This is a very laborious process without a software solution such as Microsoft Excel. However, the calculations can be simplified by transforming the normal distribution into a standard normal distribution with a mean of 0 and a variance or standard deviation of 1. This involves subtracting the arithmetic mean from the random variable and then dividing by the standard deviation. More precisely, if the random variable X follows a normal distribution with a mean of μ and a standard deviation of σ, then the random variable Z = (X - μ)/σ follows a standard normal distribution with a mean of 0 and a variance or standard deviation of 1. If μ = 0 and σ 2 = 1 are

9

See the Microsoft Excel applications at the end of this chapter.

38

2 Risk

used in Eq. (2.6), the following density function for the standard normal distribution results (for -1 < x < +1):10 x2 1 φðxÞ = p e - 2 , 2π

ð2:7Þ

where x = certain value for the continuous random variable Z. For example, for the Mercedes-Benz Group stock with a monthly average or expected return of -0.36% and a standard deviation of monthly returns of 9.28%, the probability of a monthly return of less than 5% can be calculated. For this purpose, the standard normal variable Z must first be determined: Z=

5% - ð- 0:36%Þ = 0:58: 9:28%

Using the standard normal distribution table (see Appendix), the standard normal variable of 0.58 yields a value of 0.7190. Thus, the probability that the monthly return will be less than 5% is 71.90%.11 Determining the probability using a standard normal distribution table is a simple alternative to a software solution such as Microsoft Excel.

2.3

Average Return and Standard Deviation

A low standard deviation implies that the fluctuations in returns around the average return are relatively small. By contrast, a higher standard deviation implies a wider spread of returns around the average return. The greater the spread of individual returns above and below the mean return, the greater the uncertainty about future returns. Consequently, high volatility or risk is considered ‘negative’ by most investors. A different view of volatility is presented below—one which not only represents uncertainty about future returns and thus a risk of loss but also has a negative impact on the final value of an investment. Table 2.1 presents six

The factor p12π ensures that the total area under the curve (and thus the integral from -1 to +1 as well) has an area of exactly 1. The ½ in the exponent of the e-function gives the normal distribution a unit variance (and thus a unit standard deviation as well). The axis of symmetry of the function is x = 0, where it also reaches its maximum value of p12π. The two reversal points are located at x = 1 and x = -1, respectively. 11 If the standard normal variable is rounded to three decimal places instead of two, this results in a value of 0.578, which is more accurate than 0.58. A value of 0.7183 [= 0.7157 + (0.7190 -0.7157) × 0.8] can be derived from the standard normal distribution table. Accordingly, the probability that the monthly return on Mercedes-Benz Group stock is less than 5% is 71.83%. With Microsoft Excel, the corresponding probability is 71.82%. 10

Year 1 2 3 4 5 6 7 8 9 10 r (in %) σ (in %) rG (in %) Final value (in EUR)

Investment A (return in %) 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.00 0.00 8.00 2159

Investment B (return in %) 12.0 4.0 12.0 4.0 12.0 4.0 12.0 4.0 12.0 4.0 8.00 4.00 7.93 2145

Investment C (return in %) 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 8.00 8.00 7.70 2100

Table 2.1 Risky investments with different volatilities (Source: Own illustration) Investment D (return in %) 20.0 –4.0 20.0 –4.0 20.0 –4.0 20.0 –4.0 20.0 –4.0 8.00 12.00 7.33 2029

Investment E (return in %) 24.0 –8.0 24.0 –8.0 24.0 –8.0 24.0 –8.0 24.0 –8.0 8.00 16.00 6.81 1933

Investment F (return in %) 32.0 –16.0 32.0 –16.0 32.0 –16.0 32.0 –16.0 32.0 –16.0 8.00 24.00 5.30 1676

2.3 Average Return and Standard Deviation 39

40

2 Risk

hypothetical investments, all of which have an arithmetic mean return of 8%, but with different levels of volatility. The volatility increases from investment A to investment F, which is indicated in the table with higher return fluctuations from left to right. By examining the table from left to right, one can see that the arithmetic mean return ðr Þ of 8% remains constant, while volatility (σ) increases from 0% to 24% and the geometric mean return ðrG Þ decreases from 8% to 5.3%. Thus, a riskier investment has a lower geometric mean return. The last row in the table presents the final value from all six assets for an investment of EUR 1000 at the beginning of the 10-year period. For example, the final value of investment B is EUR 2145 [=EUR 1.000 × (1.0793)10]. As the risk increases from investment A to investment F, the geometric mean return decreases, which has a negative effect on the final value of the investments. In other words, there is an inverse relationship between volatility and the ending value of investments. Therefore, higher volatility means not only a larger fluctuation of returns and thus a greater risk of loss, but also a smaller final value due to the lower geometric mean return.

2.4

Downside Risk

Intuitively, standard deviation is an attractive measure of risk. It is easy to calculate, is based on well-known statistical concepts such as the normal distribution, and is simple to interpret. As a measure of risk, standard deviation also has drawbacks. First, it measures the deviation of returns from the mean return and assumes that returns are normally distributed and independent. Both positive and negative deviations from the mean return occur equally. However, most return distributions are not normal and do not have a symmetric distribution. For example, two investments may have the same mean return and volatility, but different higher central moments of the distribution, such as skewness and kurtosis.12 There is also the question of whether investors define risk as deviation from a mean return or expected return. Many investors perceive risk as the failure to attain a certain targeted return, such as achieving the risk-free rate or another benchmark return. A pension fund, for example, must earn a certain rate of return on assets under management to avoid a shortfall and the need to increase contributions. Here, the risk of loss is understood as a negative deviation from the targeted rate of return (rather than a positive deviation). In addition, behavioural finance aspects need to be considered for private investors. Individuals have a high aversion to losses, that is, losses have a higher importance than gains. Therefore, one would have to assign more weight to negative deviations than to positive deviations when calculating volatility. Standard deviation as a measure of risk does not make this distinction.

12

For the skewness and kurtosis of the distribution, see Chap. 3.

2.4 Downside Risk

41

(Density of probability) Probability density function

Returns that are below the target return are included in the risk assessment.

Target return Average or expected Return

(Returns)

Fig. 2.3 Concept of downside risk (Source: Own illustration)

Thus, the risk assessment of an investment based on the standard deviation is critical and may lead to wrong conclusions. Downside risk measures such as the semi-standard deviation and the value at risk represent a further development of the standard deviation that takes the above points into account. In contrast to the standard deviation, the downside risk only considers negative deviations, that is, only returns that fall below a certain target return. Returns that occur above a target return are viewed as a profit opportunity rather than a risk of loss. Hence, the focus in risk management is on managing downside risk rather than standard deviation. Figure 2.3 presents the concept of downside risk. To calculate downside risk, negative return deviations are determined for those returns from a data series that are smaller than the target return, while deviations for returns above the target return are set to zero. Downside risk can therefore be calculated using the following equation: 1 T -1

Downside risk =

T t=1

where Xt = rt = return in the period t, r* = target return, and T = number of returns.

rt - r 0

if if

rt < r rt > r

,

X 2t ,

ð2:8Þ

42

2 Risk

The choice of target return has a significant impact on the amount of the estimated risk measure. The target return can be defined as expected return or arithmetic mean return, risk-free interest rate, zero per cent, or another benchmark return. If the arithmetic mean return is used as the target return in Eq. (2.8), the so-called semistandard deviation is calculated. The risk figure estimated in this way is proportional to the standard deviation and does not lead to greater insights into the risk of an investment or portfolio. If, on the other hand, a zero per cent target return is taken, the variability of negative returns (i.e. losses) is determined. This risk measure is suitable for risk-averse private investors. Furthermore, the target return can also be defined as a risk-free interest rate or a market index return. Institutional investors in particular, such as a mutual fund pursuing a passive equity strategy, can use an equity market index as a benchmark and measure the risk of loss as a negative deviation from the market index. Downside risk measures provide a better understanding of investment risk. However, several of these risk figures are based on the normal distribution. They express the risk of loss proportional to volatility and do not provide any additional risk information beyond that. It is also important to understand how they are determined because different assumptions make it difficult or impossible to compare downside risk measures. The following example demonstrates the calculation of the semi-standard deviation for the Mercedes-Benz Group stock. Example: Calculation of the Semi-Standard Deviation of the Mercedes-Benz Group Stock Using Monthly Returns for 2016 For the Mercedes-Benz Group stock the following monthly share prices, dividend per share, and simple returns are available for 2016 (Source: Refinitiv Eikon): Month Beginning of January End of January End of February End of March End of April End of May End of June End of July End of August End of September End of October End of November End of December

Share price (in EUR) 77.58 64.16 63.10 67.37 64.19 61.39 53.52 60.82 62.08 62.71 64.91 62.76 70.72

Dividend per share (in EUR)

3.25

Simple return (in %)

–17.30 –1.65 6.77 0.10 –4.36 –12.82 13.64 2.07 1.01 3.51 –3.31 12.68

2.4 Downside Risk

43

What is the semi-standard deviation of the continuous compounded and simple returns of Mercedes-Benz Group stock when the target return is given by the arithmetic mean return or expected return (assuming stable returns)? Solution The sum of squared monthly negative return deviations from the arithmetic mean return of -0.36% is 0.0553: Month Beginning of January End of January End of February End of March End of April End of May End of June End of July End of August End of September End of October End of November End of December Total Arithmetic mean or expected return, r s

Monthly continuous compounded return (rs,t)

Squared monthly negative return deviation X 2t

–0.1899 –0.0167 0.0655 0.0010 –0.0446 –0.1372 0.1279 0.0205 0.0101 0.0345 –0.0337 0.1194 –0.0432 –0.0036

0.0347 0.0002 0.0000 0.0000 0.0017 0.0178 0.0000 0.0000 0.0000 0.0000 0.0009 0.0000 0.0553

The semi-standard deviation of monthly continuous compounded returns is 7.09% and can be calculated as follows: Semi–standard deviationLog returns =

1 T -1

T t=1

X 2t =

0:0553 = 0:0709: 12 - 1

If independent returns are assumed, the annualised semi-standard deviation is 24.56%: p Semi–standard deviationLog returns annualised = 0:0709 × 12 = 0:2456: The annualised semi-standard deviation of continuous compounded returns of 24.56% can be converted to a semi-standard deviation of simple returns of 27.84% as follows: (continued)

44

2 Risk

Semi–standard deviationSimple returns annualised = e0:2456 - 1 = 0:2784: The semi-standard deviation measures the volatility below and not above the selected target return. If the risk-free monthly interest rate is approximately 0.17%, one would expect a higher volatility measure than the semi-standard deviation with a monthly target return of -0.36% (monthly average return of the Mercedes-Benz Group stock), since the return deviations are larger due to the higher target return of 0.17%. Table 2.2 compares the standard deviation with the semi-standard deviation of different target returns for the two equity securities of Mercedes-Benz Group and Deutsche Bank for 2016. The monthly average return of the stock (-0.36% for Mercedes-Benz Group and -2.22% for Deutsche Bank), the risk-free monthly interest rate of 0.17%, and zero per cent are used as target returns. Table 2.2 illustrates that the volatility measured with the standard deviation is greater for Deutsche Bank. The same conclusion is reached when the semi-standard deviations of the two equity securities are compared. In absolute terms, the negative return deviations of the bank stock are higher than the automobile stock. In relative terms, the share of the semi-standard deviation based on the arithmetic mean of the standard deviation is approximately 76% (= 24.56%/32.15%) for the MercedesBenz Group stock and approximately 78% (= 41.74%/53.28%) for the Deutsche Bank stock. Hence, the Deutsche Bank stock has a higher volatility below the average return. Furthermore, Table 2.2 illustrates that the lower the target return, the lower the semi-standard deviation, as the negative return deviations become correspondingly smaller. It should be noted that only semi-standard deviations of investments set with the same target return should be compared with each other. Another downside risk measure is the shortfall risk, which indicates the probability that a certain target return cannot be achieved within a specific investment horizon. The target return can be zero per cent, the risk-free interest rate, the market return, or another target figure. For example, for Mercedes-Benz Group stock, the monthly average return is -0.36% and the standard deviation of monthly returns is Table 2.2 Comparison of the standard deviation with different measures for the semi-standard deviation using the example of Mercedes-Benz Group AG and Deutsche Bank AG stocks for 2016 (based on monthly returns) Annualised risk measure Standard deviation Semi-standard deviation based on the arithmetic mean Semi-standard deviation based on the risk-free interest rate of 0.17% Semi-standard deviation based on the target return of zero per cent

Mercedes-Benz Group AG 32.15% 24.56% 25.52%

Deutsche Bank AG 53.28% 41.74% 46.38%

25.22%

46.04%

2.5 Value at Risk

45

9.28%.13 If the target return is defined as zero per cent, this results in a standard normal variable of 0.04:14 Z=

0% - ð- 0:36%Þ = 0:04: 9:28%

Using the standard normal distribution table in Appendix, a probability value of 51.60% can be identified. Accordingly, there is a probability of 51.60% that the monthly return will fall below zero per cent. This risk measure only indicates the probability of returns falling below the target return but not the possible amount of loss. A further development of the shortfall risk is the value at risk, which in addition to the probability also captures the possible loss amount during a specific time period.

2.5

Value at Risk

Value at risk (VAR) is a relatively new risk measure and became the most important risk measurement concept in the financial industry in the 1990s. It is used primarily to calculate expected portfolio losses resulting from market risk.15 VAR can also be employed to measure possible losses from credit risk and parts of operational risk.16 This section defines VAR and examines how to calculate VAR for an investment applying the variance-covariance method, as well as the issues arising using VAR as a risk measure. VAR is defined as the expected loss of an investment that can be incurred with a certain probability over a predefined time period under normal market conditions. This expected loss amount is characterised by the two parameters of probability and time period. VAR can be interpreted as both a maximum and a minimum expected loss figure. For example, the VAR of a portfolio can be described as follows: • Maximum VAR: ‘The value at risk of a portfolio is EUR 1 million for one day with a probability of 95%.’ This statement can be interpreted as follows: With a probability of 95%, the expected loss at the end of the next day is not greater than EUR 1 million.

13

See Sect. 2.2. A standard normal distribution is defined by a mean of 0 and a standard deviation of 1. A value (e.g. 0%) from a non-standard normal distribution can be transformed into a standard normal variable by subtracting the arithmetic mean from the value (e.g. 0%) and then dividing by the standard deviation. See Sect. 2.2. 15 Market risk is the risk of loss due to changes in equity prices, interest rates, foreign currencies, and commodity prices. For example, if an asset manager holds a bond portfolio, a rise in interest rates will cause the market value of the portfolio to fall. See Sect. 12.2.1. 16 The VAR calculation for credit risk and operational risk is more difficult than the VAR calculation for market risk, due to data availability, among other things. 14

46

2 Risk

(Density of probability)

Maximum VAR (95% VAR) Minimum VAR (5% VAR) 95% 5% VAR of EUR 1 million

(Market value changes of the portfolio)

Fig. 2.4 Maximum and minimum value at risk (Source: Own illustration)

• Minimum VAR: ‘The value at risk of a portfolio is EUR 1 million for one day with a probability of 5%.’ This means that with a probability of 5% the expected loss at the end of the next day is greater than EUR 1 million. It is not possible to predict the maximum loss with VAR. All that is known with absolute certainty is that one cannot lose more than the value of a portfolio consisting of long positions (without leverage). In particular, the VAR says nothing about possible loss incurred with a certain remaining probability. Figure 2.4 presents the maximum and minimum VAR under the assumption that the market value changes of the portfolio are normally distributed. Expressed in terms of probability theory, the VAR at a p % confidence level is the (1 - p%) quantile of the distribution of changes in the market value of the portfolio. For a VAR with a confidence level of 95%, the left quantile of the distribution is 5% (= 1 - 95%). The calculation of VAR depends on decisions relating to the probability level and the length of the time period. When measuring market risk, probabilities of 1%, 2.5%, or 5% are commonly used. A smaller probability (e.g. 1% versus 5%) results in a higher (more conservative) VAR value. The choice of time period also affects the amount of the expected loss. The longer the time interval chosen, the higher the VAR. A higher loss may be incurred over a longer period of time than over a shorter one. Financial institutions typically calculate the VAR of a trading portfolio with a time period of 1 day and/or 10 days, and with a probability of 1%, 2.5%, and/or 5%. To facilitate comparisons between different VAR measures, VAR is often expressed as a percentage (as a return measure) rather than an absolute expected

2.5 Value at Risk Table 2.3 Probability distribution of portfolio returns over a given time period (Source: Own illustration)

47

Portfolio return Less than -30% -30% to –25% -25% to –18% -18% to –10% - 10% to – 5% -5% to -2% -2% to 0% 0% to 2% 2% to 5% 5% to 10% 10% to 18% 18% to 25% 25% to 30% More than 30%

Probability 0.01 0.02 0.02 0.05 0.10 0.13 0.17 0.17 0.13 0.10 0.05 0.02 0.02 0.01 1.00

Cumulative probability 0.01 0.03 0.05 0.10 0.20 0.33 0.50 0.67 0.80 0.90 0.95 0.97 0.99 1.00

loss amount. The VAR can be determined from a probability distribution of portfolio returns. Taking the data from Table 2.3, the percentage VAR with a 5% probability is -18%. To obtain this percentage VAR, the probabilities are added until the cumulative probability is 5%. There is a 1% probability that the value of the portfolio will fall by at least -30%, a 2% probability that it will drop between -30% and 25%, and a 2% probability that the decline in value will be between -25% and 18%. Hence, the percentage 5% VAR is -18%. To calculate the absolute VAR, the 5% VAR of -18% should be multiplied by the market value of the portfolio. If the market value of the portfolio is assumed to be EUR 1 million, the absolute 5% VAR is - EUR 180,000 (= -0.18 × EUR 1 million). After defining the probability and the time period, the estimation method for the VAR must be specified. The three estimation methods are the analytical or variancecovariance method, and the historical and Monte Carlo simulations. The latter two VAR methods determine the changes in market value through revaluations of the portfolio. In the historical simulation method, the portfolio is revalued for scenarios that occurred in the past. Therefore, the simulation path is provided by the past. By contrast, the Monte Carlo simulation usually involves 10,000 runs that are produced with a random number generator to calculate the portfolio values or the market value changes derived from them for each individual run or scenario. The variance-covariance method is based on the assumption that the market value changes or the returns of the portfolio are normally distributed. The formulas for calculating the absolute and percentage VAR using this parametric approach17 are as follows: 17

A parametric approach assumes a certain distribution of changes in the market value of the portfolio when calculating the VAR (e.g. the normal distribution in the variance-covariance method).

48

2 Risk

VARAbsolute = Eðr ÞV þ Z α σV,

ð2:9Þ

VARIn% = E ðr Þ þ Z α σ,

ð2:10Þ

where E(r) = expected return of the portfolio, V = market value of the portfolio, Zα = standard normal variable at the left quantile α of the distribution, and σ = standard deviation of portfolio returns. A standard normal distribution is defined by a mean of 0 and a variance or standard deviation of 1. A value from a non-standard normal distribution can be transformed into a standard normal variable by subtracting the arithmetic mean from the value and then dividing by the standard deviation.18 In a standard normal distribution, 5% of all possible values are below the standard normal variable of 1.65. Therefore, a standard normal variable of -1.65 is taken for the 5% VAR. The VAR is often calculated with an expected return of 0% because a random change in the portfolio value is assumed (random walk).19 A randomly changed value has an expected value of approximately zero over a very short period of time, since the randomly occurring positive and negative changes in value cancel each other out. The absolute and percentage VAR can then be calculated using the following formulas: VARAbsolute = Z α σV,

ð2:11Þ

VARIn% = Z α σ:

ð2:12Þ

Example: Calculation of VAR A portfolio has a market value of EUR 1 million. The expected annual portfolio return is 10%, while the annualised standard deviation of portfolio returns is 30%. The following questions should be answered using the variance-covariance method: (continued)

18

See Sect. 2.4. In efficient markets, all information is contained in asset prices, and therefore, prices change only in response to new price-relevant information. Since price-relevant information occurs purely at random, asset prices follow a random walk. See Sect. 3.3.2. 19

2.5 Value at Risk

1. 2. 3. 4.

49

What is the absolute 5% VAR for 1 year? What is the absolute 1% VAR for 1 year? What is the absolute 5% VAR for 1 month? What is the absolute 1% VAR for 1 week? Solution to 1 The annual 5% VAR of - EUR 395,000 can be calculated as follows:

5%VARAbsolute = 0:10 × EUR 1 million þ ð- 1:65Þ × 0:30 × EUR 1 million = - EUR 395,000: A loss higher than EUR 395,000 is expected for one year with a probability of 5%. Solution to 2 In a standard normal distribution, 1% of all possible values are below the standard normal variable of -2.33. Therefore, the standard normal variable for the calculation of the 1% VAR is -2.33. This results in an annual 1% VAR of - EUR 599,000: 1%VARAbsolute = 0:10 × EUR 1 million þ ð- 2:33Þ × 0:30 × EUR 1 million = - EUR 599,000: A loss higher than EUR 599,000 is expected for one year with a probability of 1%. Solution to 3 First, the annual expected return and annualised volatility are converted into monthly values. It is important to note that the variance-covariance method is based on the assumption that the returns are normally distributed and occur independently of each other, and therefore, the expected return is proportional to time and the volatility is proportional to the root of time (the variance is proportional to time).20 Accordingly, the expected return and standard deviation of monthly returns can be calculated as follows: E ðr ÞMonthly =

0:10 = 0:0083, 12 months

σ Monthly = p

0:30 = 0:0866: 12 months

The monthly 5% VAR is - EUR 134,590: (continued)

20

See Sect. 3.3.2.

50

2 Risk

5%VARAbsolute = 0:0083 × EUR 1 million þ ð- 1:65Þ × 0:0866 × EUR 1 million = - EUR 134,590: A loss higher than EUR 134,590 is expected for one month with a probability of 5%. Solution to 4 First, the annual expected return and annualised standard deviation are converted into weekly values: E ðr ÞWeekly =

0:10 = 0:0019, 52 weeks

0:30 = 0:0416: σWeekly = p 52 weeks The weekly 1% VAR of - EUR 95,028 is calculated as follows: 1%VARAbsolute = 0:0019 × EUR 1 million þ ð- 2:33Þ × 0:0416 × EUR 1 million = - EUR 95,028: A loss higher than EUR 95,028 is expected for one week with a probability of 1%. If the market value changes of the portfolio are not normally distributed, then the variance-covariance method leads to a VAR that either overestimates or underestimates the portfolio risk. Therefore, revaluation approaches such as the historical simulation method or Monte Carlo simulation must be used to calculate VAR when distributions are not normal. Unlike the variance-covariance method, the revaluation approaches do not require the assumption of normal distribution. Furthermore, the VAR may violate the property of subadditivity, and therefore, the sum of two VAR positions may be smaller than the VAR of these two positions combined in a portfolio. The subadditivity property can be violated in the calculation of VAR by revaluation approaches that determine the VAR by examining the market value changes in the left quantile of the distribution. If, on the other hand, the variance-covariance method is used to estimate the VAR, the subadditivity property is fulfilled. Assuming a correlation coefficient of less than 1, the standard deviation of a portfolio is smaller than the sum of the weighted standard deviations of the individual investments.21 The VAR reflects the expected loss under normal market conditions. Losses from extreme market movements are not captured by the VAR. Therefore, in the context of risk management the VAR must always be stated together with a worst-case loss.

21

For the diversification effect, see Sect. 4.3.

2.6 Summary

51

Stress simulations allow the manager to analyse the portfolio under certain worstcase scenarios. Correlation among markets is very high in times of crisis because financial markets around the globe are closely intertwined, which leads to a significant risk of contagion. If, for example, the stock exchange in New York experiences large losses, other international stock exchanges such as Tokyo, London, Frankfurt, and Zurich may also be affected and suffer losses. Possible extreme scenarios must take such global contagion effects into account. The scenarios may consist of hypothetical events (e.g. a terrorist attack with biological weapons in a financial centre such as London or an attack by Russia in the Baltics) or past crises such as the October crash in 1987.22 In order to estimate an extreme portfolio loss, it is also feasible to change individual significant risk factors in the most unfavourable direction possible.

2.6

Summary

• The risk of an investment can be determined by a variety of risk measures such as variance or standard deviation, semi-standard deviation, shortfall risk, and VAR. • The standard deviation is calculated as the root of the average squared return deviations. The larger the return fluctuations around the mean, the larger the standard deviation and the greater the uncertainty about future returns. Furthermore, an investment with a higher standard deviation has a lower geometric mean return and therefore a lower final value. • The standard deviation captures both positive and negative deviations from the mean. However, positive deviations represent a chance of profit. Only negative deviations reflect a risk of loss. • Downside risk, on the other hand, takes into account only negative deviations from a target return and thus embodies the risk of loss better than standard deviation. Downside risk measures include semi-standard deviation, shortfall risk, and VAR. • In calculating the semi-standard deviation, the target return can be defined as the arithmetic mean return or expected return, zero per cent, risk-free interest rate, or the return of some other benchmark. • Shortfall risk is the probability that a specific target return cannot be achieved within a certain time period. A further development of this risk measure is the VAR, which captures not only the probability but also the possible amount of loss over a given time period.

22

On 19 October 1987 (‘Black Monday’), the Dow Jones index lost 22.6% of its value, the largest one-day percentage decline since World War I, when the New York Stock Exchange was closed for months after the war began and fell more than 24% when it reopened. Infected by the Dow Jones index, other stock markets around the world also plummeted. For example, on 19 October 1987, the then Swissindex lost 11.3% and on Tuesday a further 3.7% due to the staggered trading hours with the USA, while prices on Wall Street were already recovering slightly.

52

2 Risk

• The VAR is defined as the expected loss of an investment that can be incurred with a certain probability over a predefined time period under normal market conditions. This expected loss amount is characterised by two parameters, namely the probability and the time period. There are various methods of estimating VAR such as the analytical or variance-covariance method, the historical simulation method, and Monte Carlo simulation. For example, if the portfolio includes options, the market value changes of the portfolio are not normally distributed. The VAR of such a position is calculated using a revaluation approach such as the historical simulation method or Monte Carlo simulation. In addition, the VAR may violate the subadditivity property. Since the VAR is estimated under normal market conditions, it does not provide any information about the possible loss amount in the event of extreme market movements or a crash.

2.7

Problems

1. The following annual continuous compounded returns of an equity security are available (assumption: population of return data): Year 2019 2020 2021 2022

Annual log return (in %) 12 -24 35 -10

a) What is the standard deviation of the continuous compounded and simple returns? b) What is the semi-standard deviation of the continuous compounded returns with a target return of zero per cent? c) What is the shortfall risk with a target return of zero per cent? 2. A bank has published a daily VAR of EUR 25 million with a confidence level of 95% for the trading portfolio in its annual report. How can this risk figure be interpreted? 3. The following probability distribution of annual portfolio returns is available: Portfolio return Less than –25% -25% to –20% -20% to –12% -12% to –5% -5% to 0%

Probability 0.01 0.02 0.02 0.05 0.20 (continued)

2.8 Solutions

53

Portfolio return 0% to 5% 5% to 12% 12% to 16% 16% to 22% 22% to 25% More than 25%

Probability 0.15 0.15 0.15 0.10 0.10 0.05 1.00

The market value of the portfolio is EUR 25 million. a) What is the absolute 1% VAR for a period of 1 year? b) What is the absolute 5% VAR for a period of 1 year? 4. A portfolio has a market value of CHF 2 million. The expected annual portfolio return is 8%, while the annualised standard deviation of the portfolio returns is 25%. The portfolio returns occur independently of each other and are therefore not correlated. a) b) c) d)

What is the absolute 5% VAR for 1 year? What is the absolute 5% VAR for 1 month? What is the absolute 2.5% VAR for 1 week? What is the absolute 1% VAR for 1 day (252 trading days per year)?

2.8

Solutions

1. a) First, the expected continuous compounded return (arithmetic mean return) must be determined:

rs =

12% þ ð- 24%Þ þ 35% þ ð- 10%Þ = 3:25%: 4

The standard deviation of the continuous compounded returns is 22.38%:

σs =

1 × 4

ð0:12 - 0:0325Þ2 þð- 0:24 - 0:0325Þ2 þð0:35 - 0:0325Þ2 þð- 0:10 - 0:0325Þ2

= 0:2238:

The standard deviation of continuous compounded returns can be converted to a standard deviation of simple returns as follows:

54

2 Risk

σ = eσs - 1 = e0:2238 - 1 = 0:2508: b) With a target return of zero per cent, the semi-standard deviation of the continuous compounded returns is 13%: Semi–standard deviation =

1 × ð0 - 0Þ2 þ ð- 0:24 - 0Þ2 þ ð0 - 0Þ2 þ ð- 0:10 - 0Þ2 = 0:13: 4

c) With a target return of zero per cent, the standard normal variable is -0.15: Z=

0% - 3:25% = - 0:15: 22:38%

A probability value of 44.04% can be identified from the standard normal distribution table in Appendix. Thus, there is a probability of 44.04% that the target return of zero per cent will not be achieved. 2. There is a 95% probability that the bank will not lose more than EUR 25 million on the trading portfolio within 1 day (maximum VAR). There is a 5% probability that the financial institution will lose more than EUR 25 million on the trading portfolio within 1 day (minimum VAR). 3. a) 1%VARAbsolute = - 0:25 × EUR 25 million = - EUR 6:25 million: b) With a cumulative probability of 5%, the portfolio returns are between -20% and 12%. Therefore, the percentage VAR is -12%, while the absolute VAR is EUR 3 million. 5%VARAbsolute = - 0:12 × EUR 25 million = - EUR 3 million:

2.8 Solutions

55

4. a) 5%VARAbsolute = 0:08 × CHF 2 million þ ð- 1:65Þ × 0:25 × CHF 2 million = - CHF 665,000 A loss higher than CHF 665,000 is expected for one year with a probability of 5%. b) The monthly expected return and the standard deviation of monthly returns can be calculated as follows: E ðr ÞMonthly =

0:08 = 0:00667, 12 months

σ Monthly = p

0:25 = 0:07217: 12 months

The 5% VAR is—CHF 224,821: 5%VARAbsolute = 0:00667 × CHF 2 million þ ð- 1:65Þ × 0:07217 × CHF 2 million = - CHF 224,821: A loss higher than CHF 224,821 is expected for one month with a probability of 5%. c) The expected weekly return and the standard deviation of weekly returns can be calculated as follows: E ðr ÞWeekly =

0:08 = 0:00154, 52 weeks

0:25 = 0:03467: σ Weekly = p 52 weeks The 2.5% VAR is – CHF 132,826 and can be determined with the following equation:

56

2 Risk

2:5%VARAbsolute = 0:00154 × CHF 2 million þ ð- 1:96Þ × 0:03467 × CHF 2 million = - CHF 132,826: A loss higher than CHF 132,826 is expected for one week with a probability of 2.5%. d) The daily expected return and the standard deviation of daily returns can be determined as follows: E ðr ÞDaily =

0:08 = 0:00032, 252 trading days

σ Daily = p

0:25 = 0:01575: 252 trading days

The 1% VAR is – CHF 72,755: 1%VARAbsolute = 0:00032 × CHF 2 million þ ð- 2:33Þ × 0:01575 × CHF 2 million = - CHF 72,755: A loss higher than CHF 72,755 is expected for one day with a probability of 1%.

Appendix: Standard Normal Distribution Table Cumulative Probabilities for a Standard Normal Distribution PðZ ≤ xÞ = N ðxÞ for x ≥ 0 x

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881

0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910

0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939

0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967

0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995

0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023

0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051

0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078

0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106

0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133

(continued)

Appendix: Standard Normal Distribution Table

57

x

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00

0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 1.0000

0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987 0.9991 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 1.0000

0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987 0.9991 0.9994 0.9995 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.9991 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.9992 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990 0.9993 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

Cumulative Probabilities for a Standard Normal Distribution PðZ ≤ xÞ = N ðxÞ for x ≤ 0 x

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00

0.5000

0.4960

0.4920

0.4880

0.4840

0.4801

0.4761

0.4721

0.4681

0.09 0.4641

-0.10

0.4602

0.4562

0.4522

0.4483

0.4443

0.4404

0.4364

0.4325

0.4286

0.4247

-0.20

0.4207

0.4168

0.4129

0.4090

0.4052

0.4013

0.3974

0.3936

0.3897

0.3859

-0.30

0.3821

0.3783

0.3745

0.3707

0.3669

0.3632

0.3594

0.3557

0.3520

0.3483

-0.40

0.3446

0.3409

0.3372

0.3336

0.3300

0.3264

0.3228

0.3192

0.3156

0.3121

-0.50

0.3085

0.3050

0.3015

0.2981

0.2946

0.2912

0.2877

0.2843

0.2810

0.2776

-0.60

0.2743

0.2709

0.2676

0.2643

0.2611

0.2578

0.2546

0.2514

0.2483

0.2451

-0.70

0.2420

0.2389

0.2358

0.2327

0.2296

0.2266

0.2236

0.2206

0.2177

0.2148

(continued)

58

2 Risk

x

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-0.80

0.2119

0.2090

0.2061

0.2033

0.2005

0.1977

0.1949

0.1922

0.1894

0.1867

-0.90

0.1841

0.1814

0.1788

0.1762

0.1736

0.1711

0.1685

0.1660

0.1635

0.1611

-1.00

0.1587

0.1562

0.1539

0.1515

0.1492

0.1469

0.1446

0.1423

0.1401

0.1379

-1.10

0.1357

0.1335

0.1314

0.1292

0.1271

0.1251

0.1230

0.1210

0.1190

0.1170

-1.20

0.1151

0.1131

0.1112

0.1093

0.1075

0.1056

0.1038

0.1020

0.1003

0.0985

-1.30

0.0968

0.0951

0.0934

0.0918

0.0901

0.0885

0.0869

0.0853

0.0838

0.0823

-1.40

0.0808

0.0793

0.0778

0.0764

0.0749

0.0735

0.0721

0.0708

0.0694

0.0681

-1.50

0.0668

0.0655

0.0643

0.0630

0.0618

0.0606

0.0594

0.0582

0.0571

0.0559

-1.60

0.0548

0.0537

0.0526

0.0516

0.0505

0.0495

0.0485

0.0475

0.0465

0.0455

-1.70

0.0446

0.0436

0.0427

0.0418

0.0409

0.0401

0.0392

0.0384

0.0375

0.0367

-1.80

0.0359

0.0351

0.0344

0.0336

0.0329

0.0322

0.0314

0.0307

0.0301

0.0294

-1.90

0.0287

0.0281

0.0274

0.0268

0.0262

0.0256

0.0250

0.0244

0.0239

0.0233

-2.00

0.0228

0.0222

0.0217

0.0212

0.0207

0.0202

0.0197

0.0192

0.0188

0.0183

-2.10

0.0179

0.0174

0.0170

0.0166

0.0162

0.0158

0.0154

0.0150

0.0146

0.0143

-2.20

0.0139

0.0136

0.0132

0.0129

0.0125

0.0122

0.0119

0.0116

0.0113

0.0110

-2.30

0.0107

0.0104

0.0102

0.0099

0.0096

0.0094

0.0091

0.0089

0.0087

0.0084

-2.40

0.0082

0.0080

0.0078

0.0075

0.0073

0.0071

0.0069

0.0068

0.0066

0.0064

-2.50

0.0062

0.0060

0.0059

0.0057

0.0055

0.0054

0.0052

0.0051

0.0049

0.0048

-2.60

0.0047

0.0045

0.0044

0.0043

0.0041

0.0040

0.0039

0.0038

0.0037

0.0036

-2.70

0.0035

0.0034

0.0033

0.0032

0.0031

0.0030

0.0029

0.0028

0.0027

0.0026

-2.80

0.0026

0.0025

0.0024

0.0023

0.0023

0.0022

0.0021

0.0021

0.0020

0.0019

-2.90

0.0019

0.0018

0.0018

0.0017

0.0016

0.0016

0.0015

0.0015

0.0014

0.0014

-3.00

0.0013

0.0013

0.0013

0.0012

0.0012

0.0011

0.0011

0.0011

0.0010

0.0010

-3.10

0.0010

0.0009

0.0009

0.0009

0.0008

0.0008

0.0008

0.0008

0.0007

0.0007

-3.20

0.0007

0.0007

0.0006

0.0006

0.0006

0.0006

0.0006

0.0005

0.0005

0.0005

-3.30

0.0005

0.0005

0.0005

0.0004

0.0004

0.0004

0.0004

0.0004

0.0004

0.0003

-3.40

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0002

-3.50

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

-3.60

0.0002

0.0002

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

-3.70

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

-3.80

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

-3.90

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

-4.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Microsoft Excel Applications • To find the root of a number x in Excel, write the following expression in a cell and then press the Enter key: = SQRTðxÞ: • In order to calculate the variance and the standard deviation, the returns must first be inserted in cells A1 to A12.

Appendix: Standard Normal Distribution Table

59

The variance and standard deviation of a population where the sum of squared return variances is divided by T are obtained from the following expressions, which are entered in cells A13 and A14, for example:23 = VAR:PðA1:A12Þ, = STDEV:PðA1:A12Þ: The entry is then completed by pressing the Enter key. For the variance and standard deviation of a sample, the sum of the squared return variances is divided by T - 1. To calculate these two quantities, the following notations are written, for example, in cells A15 and A16 and then confirmed with the Enter key:24 = VAR:SðA1:A12Þ, = STDEV:SðA1:A12Þ: Figure 2.5 presents the calculation of the variance and standard deviation (population and sample) of the monthly continuous compounded returns of the Mercedes-Benz Group stock for 2016. • To determine the probabilities (areas) from a normal distribution, the arithmetic mean and the standard deviation are required. For example, the probability that the return r takes a value equal to or less than r0 can be calculated as follows: = NORM:DISTðr 0 ; r; σ; trueÞ: After entering, press the Enter key. The logical value ‘true’ or ‘false’ indicates which normal distribution function is used. ‘True’ stands for the normal cumulative distribution function and ‘false’ for the normal probability density function. The output of the normal cumulative distribution function corresponds to the area under the normal probability density function to the left of the selected threshold return. It represents the likelihood that a return is below a given threshold return.

23

In Excel 2007 and earlier versions, the notations for the population variance and the sample standard deviation are ‘varp’ and ‘stdevp’, respectively. 24 In Excel 2007 and earlier versions, the sample variance and sample standard deviation are calculated using the expressions ‘var’ and ‘stdev’, respectively.

60

2 Risk

Fig. 2.5 Variance and standard deviation (population and sample) of the monthly continuous compounded returns of the Mercedes-Benz Group stock for 2016 (Source: Own illustration)

• The probability that the return r has a value equal to or greater than r0 can be determined using the following notation: = 1 - NORM:DISTðr 0 ; r ; σ; trueÞ: The entry is completed by pressing the Enter key. • To calculate the probability that the return r is between r0 and r1 (where r0 < r1), enter the following expression into Microsoft Excel: = NORM:DISTðr 1 ; r ; σ; trueÞ - NORM:DISTðr 0 ; r ; σ; trueÞ and then confirm with the Enter key. • In order for the probability to be determined from a standard normal distribution, the standard normal variable must be calculated. For example, a distribution has an expected return of 7% and a standard deviation of returns of 30%. To determine the probability of obtaining a return of less than 10%, the standard normal variable of 0.1 [=(0.10 - 0.07)/0.30] must be calculated. To find the probability, enter the following expression into Excel: = NORM:S:DISTð0:10; trueÞ: After that, press the Enter key. Thus, the probability that the return is less than 10% is 53.98%. The logical value ‘true’ or ‘false’ indicates which standard normal

Appendix: Standard Normal Distribution Table

61

distribution function is utilised. ‘True’ stands for the standard normal cumulative distribution function and ‘false’ for the standard normal probability density function. • To calculate the semi-standard deviation with a target return equal to the arithmetic mean, for instance, the return data are to be entered in cells A1 to A12. In the free cell A13, the expected return or the mean value can be calculated with the function ‘=AVERAGE(A1:A12)’. Then, for example, the following formula is entered in cell B1 and concluded with the Enter key: = IFðA1 < $A$13; ðA1 - $A$13Þ^2; 0Þ: • The above formula is copied into cells B2 to B12. The dollar signs for cell A13 ensure that the value in this cell (i.e. the mean value) remains unchanged when copied. If the semi-standard deviation is to be determined for the sample of the return data, for example, the following expression can be entered in cell B13 and then confirmed with the Enter key: = SQRTðSUMðB1:B12Þ=ð12 - 1ÞÞ: • Figure 2.6 illustrates the calculation of the semi-standard deviation (sample) using the monthly continuous compounded returns of the Mercedes-Benz Group stock for 2016. The target return is given by the monthly average return (arithmetic mean).

Fig. 2.6 Semi-standard deviation of the monthly continuous compounded returns of the MercedesBenz Group stock for 2016 (Source: Own illustration)

3

Other Investment Characteristics

3.1

Introduction

The assumption that returns are normally distributed is very convenient, as only the first two moments of the distribution (i.e. the mean and the variance) are required to fully describe the return distribution. However, in the vast majority of cases, financial asset returns are not normally distributed, and higher central moments of the distribution such as skewness and kurtosis must therefore be considered. In addition, the market characteristics of investments are also important for their evaluation. The price of financial assets is affected by the information efficiency and liquidity of the markets. The latter has a significant impact on the level of trading costs. If the assumptions of normal distribution do not apply, and if market information efficiency and market liquidity are not given, higher central moments of the distribution such as skewness and kurtosis, as well as market characteristics, must be considered in order to be able to assess the investment.

3.2

Properties of a Distribution

3.2.1

Normal Distribution

The assumption of normal distribution is very tempting in view of the known statistical properties of the distribution. The mean and variance or standard deviation can be used to assess the investment. However, returns are often not normally distributed. On the one hand, the return distribution can be skewed—that is, the returns no longer occur symmetrically around the mean. On the other hand, the probability of extreme events may be higher or lower than in the case of the normal distribution. Empirically, major market movements in financial markets occur more frequently than would be expected on the basis of the normal distribution.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_3

63

64

3 Other Investment Characteristics

Fig. 3.1 Right- and leftskewed distributions (Source: Own illustration)

Mode Median

Right-skewed distribution

Mean

Left-skewed distribution

Mode Median Mean

3.2.2

Skewness

Skewness is the third central moment of a distribution. It measures the degree of symmetry of the returns around the mean. An important property of skewed distributions is that the mean (arithmetic mean or expected return if stable returns are assumed), the median (middle of the values), and the mode (most frequent value) are of different magnitudes, whereas these are the same in a normal distribution. The following relationship holds in the case of a right-skewed distribution: mean > median > mode. By contrast, in a left-skewed distribution the opposite relationship exists: mode > median > mean. In both cases, the centre of the distribution is given by the mode. Figure 3.1 presents a right-skewed and a left-skewed distribution. A skewness of zero means a symmetric distribution (mean = median = mode). In a right-skewed distribution, losses are small and frequent, while gains are extremely high and less frequent, with the result that the mean exceeds the median. A leftskewed distribution, on the other hand, is characterised by more frequent small gains and few extremely high losses, and the mean is therefore lower than the median. Such investments are not attractive.1

1

Market participants prefer investments with a high expected return (mean), low variance, positive skewness, and low kurtosis.

3.2 Properties of a Distribution

65

To calculate skewness, the average deviation of returns from the arithmetic mean or expected return to the power of three, is divided by the standard deviation of returns to the power of three. The sample skewness (also called sample relative skewness) can be determined as follows:2 T

T Ss = ð T - 1Þ ð T - 2Þ

t=1

ðr t - r Þ3 σ~3

,

ð3:1Þ

where T = number of return observations in the sample, rt = return for period t, r = arithmetic mean return (expected return), and σ = sample standard deviation of returns. The skewness of the distribution has no unity but can be both positive and negative. This is because the skewness (see Eq. 3.1) is calculated with the deviation of the returns from the mean return with an exponent of three, so that the sign of the deviation is preserved. The result is a numerical quantity for the extent of the distribution in one direction or the other. A normal distribution is symmetric and has a skewness of zero. A right-skewed distribution has a positive skewness, while a left-skewed distribution has a negative skewness. The skewness of the distribution can be interpreted by examining the numerator of the formula more closely. For example, in a right-skewed distribution, the mean exceeds the median because more than half of the return deviations from the mean are negative and less than half are positive. For the sum of the deviations to be positive, the negative return deviations must be small and frequent, and the positive return deviations must be less frequent and large. Thus, if skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations. Positive skewness also results when simple returns over several periods are used to construct the return distribution. For example, if an investment of EUR 100 has an average annual return of 40% and the annual return in the next 2 consecutive years is 60% (i.e. 20% above the average return of 40%), the final value of the investment in 2 years would be EUR 256 [= EUR 100 × (1.6)2], which corresponds to a return of 156% (= EUR 256/EUR 100 - 1) over 2 years. On the other hand, should the annual return over the next 2 years amount to 20% (i.e. 20% below the average return of 40%), the final value would be EUR 144 [= EUR 100 × (1.2)2], thus resulting in a 2-year return of 44% (= EUR 144/EUR 100 - 1). The average 2-year return equals 96% [= (1.4)2 - 1], and the 2-year return of 156% is therefore 60% higher than the 2-year average return, while the 2-year return of 44% is only 52% lower than the

2

The term T/[(T + 1)(T + 2)] corrects for a downward bias in the sample.

66

3 Other Investment Characteristics

average. Hence, the distribution of simple returns that span multiple periods is positively skewed, even though simple returns are normally distributed over one period.

3.2.3

Kurtosis

Kurtosis is the fourth central moment of the distribution. It measures the peakedness at the centre of the distribution and the tails at the two ends of the distribution. A normal distribution is called mesokurtic (meso from the Greek word for middle) and has a kurtosis of 3. A distribution which has a kurtosis greater than 3 is known as leptokurtic (lepto from the Greek word for slender). Such a distribution has more extreme observations than a normal distribution. Graphically, the distribution is steep in the middle (steep peaked) and has fatter tails than the normal distribution. If the distribution has a kurtosis of less than 3 it is called platykurtic (platy from the Greek word for broad). The returns are tightly clustered around the centre of the distribution. Graphically, the distribution is flat in the middle (flat-peaked) and the tails are thinner than the normal distribution. In financial market theory, the so-called excess kurtosis is often calculated and the number 3 is subtracted from the kurtosis. The calculation of the kurtosis involves the average deviation of the returns from the mean return to the power of four divided by the standard deviation of the returns to the power of four. The excess kurtosis of a sample can be determined with the following equation: T

T ðT þ 1Þ K E,s = ðT - 1ÞðT - 2ÞðT - 3Þ

t=1

ðr t - r Þ4 σ~4

-

3ðT - 1Þ2 : ðT - 2ÞðT - 3Þ

ð3:2Þ

The kurtosis is unit-free. A normal distribution has an excess kurtosis of zero. A leptokurtic distribution has an excess kurtosis greater than zero, and a platykurtic distribution has an excess kurtosis less than zero.3 Figure 3.2 presents the normal distribution compared to a leptokurtic and a platykurtic distribution with a positive and a negative excess kurtosis. The normal distribution has a skewness of zero and an excess kurtosis of zero. Therefore, it is defined by the first two moments of the distribution (mean and variance). Distributions that are not normally distributed have higher central moments such as skewness and kurtosis. Hence, it is important to add these two metrics to the analysis for distributions that deviate from the normal distribution. For example, empirical studies have demonstrated that in many cases the return distribution of equity securities has a positive excess kurtosis. If a return distribution is Some statistical software solutions, such as Microsoft Excel, use the excess kurtosis (kurtosis –3 for a very large sample) to calculate the kurtosis. For a normal distribution, the excess kurtosis is 0 (= 3 - 3).

3

3.2 Properties of a Distribution

67

Normal or mesokurtic distribution (excess kurtosis = 0) Leptokurtic distribution (excess kurtosis > 0)

Platykurtic distribution (excess kurtosis < 0)

(Returns) Fig. 3.2 Normal, leptokurtic, and platykurtic distributions with zero, positive, and negative excess kurtosis (Source: Own illustration)

leptokurtic and the normal distribution is used for the analysis, the probability of very high positive or very high negative returns is underestimated. Example: Calculation of the Expected Return, Standard Deviation, Skewness, and Excess Kurtosis of a Return Distribution The following 10 annual continuous compounded returns from 2012 to 2021 are given for a sample portfolio: Year 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021

Return (in %) 7.40 2.25 16.65 10.15 14.40 4.60 1.90 6.55 0.80 -6.50

1. What is the mean or expected return? 2. What is the standard deviation? 3. What is the skewness of the distribution? (continued)

68

3 Other Investment Characteristics

4. What is the excess kurtosis of the distribution? 5. How can the return distribution be interpreted using the calculated moments? Solution to 1 The mean or expected return of 5.82% can be calculated as the average (arithmetic mean) of the historical continuous compounded returns as follows: r=

7:4% þ 2:25% þ . . . þ ð - 6:50%Þ = 5:82%: 10

Solution to 2 Year 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Total

Return rt (in %) 7.40 2.25 16.65 10.15 14.40 4.60 1.90 6.55 0.80 -6.50

rt - r 1.58 -3.57 10.83 4.33 8.58 -1.22 -3.92 0.73 -5.02 -12.32

ðr t - r Þ2 2.496 12.745 117.289 18.749 73.616 1.488 15.366 0.533 25.2 151.782 419.264

The standard deviation of the annual returns is 6.825% and can be determined as follows: σ~ =

1 T -1

T

ðr t - r Þ2 = t=1

419:264 = 6:825%: 10 - 1

Solution to 3 Year 2012 2013 2014 2015 2016 2017 2018 2019 2020

Return rt (in %) 7.40 2.25 16.65 10.15 14.40 4.60 1.90 6.55 0.80

rt - r 1.58 -3.57 10.83 4.33 8.58 -1.22 -3.92 0.73 -5.02

ðr t - r Þ3 3.944 -45.499 1270.239 81.183 631.629 -1.816 -60.236 0.389 -126.506

(continued)

3.2 Properties of a Distribution

69

rt - r -12.32

Return rt (in %) -6.50

Year 2021 Total

ðr t - r Þ3 -1869.959 -116.632

The sample skewness of the distribution is 0.051 and can be calculated with the following equation: T

T Ss = ð T - 1Þ ð T - 2 Þ ×

t=1

ðr t - r Þ3 ~ σ

3

=

10 ð10 - 1Þ × ð10 - 2Þ

- 116:632 = - 0:051: 6:8253

The portfolio has five positive and five negative return deviations, respectively. Two large positive return deviations occurred in 2014 (10.83%) and 2016 (8.58%), which were approximately offset by the two negative return deviations in 2021 (-12.32%) and 2020 (-5.02%). Negative and positive deviations occur with equal frequency and approximately cancel each other out. This leads to the conclusion that the portfolio returns are approximately symmetric (slightly skewed to the left). Solution to 4 Year 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Total

Return rt (in %) 7.40 2.25 16.65 10.15 14.40 4.60 1.90 6.55 0.80 -6.50

rt - r 1.58 -3.57 10.83 4.33 8.58 -1.22 -3.92 0.73 -5.02 -12.32

ðr t - r Þ4 6.232 162.432 13,756.686 351.521 5419.374 2.215 236.126 0.284 635.06 23,037.897 43,607.827

The sample excess kurtosis of the distribution equals 0.047 and can be computed using the equation below: (continued)

70

3 Other Investment Characteristics

T

T ð T þ 1Þ K E,s = ðT - 1ÞðT - 2ÞðT - 3Þ =

t=1

ðr t - r Þ4 -

~4 σ

3ð T - 1Þ 2 ðT - 2ÞðT - 3Þ

10 × ð10 þ 1Þ 3 × ð10 - 1Þ2 43,607:827 × ð10 - 1Þ × ð10 - 2Þ × ð10 - 3Þ ð10 - 2Þ × ð10 - 3Þ 6:8254

= 0:047: An excess kurtosis of 0.047, which is close to 0, means that the distribution is approximately normal. Solution to 5 Both the skewness and excess kurtosis of the distribution are approximately 0. Thus, the distribution of the annual portfolio returns appears to be normally distributed over the 10-year period. With the help of the skewness and excess kurtosis of the distribution, it is possible to test the assumption of normal distribution on the basis of statistical tests. One of the most popular tests for normality is the Jarque-Bera test,4 which is a hypothesis test. The null hypothesis states that the actual returns are normally distributed. The alternative hypothesis assumes that the returns are not normally distributed. A normal distribution implies that the skewness and excess kurtosis of the sampling distribution are close to 0. The Jarque-Bera test converts the sample skewness and sample excess kurtosis into a statistical measure to test whether the calculated measure is significantly different from the expected value. The Jarque-Bera statistic is defined as follows:5 2

JB =

T 2 K E,s , S þ 4 6 s

ð3:3Þ

where T = number of return observations in the sample, Ss = sample skewness, and KE,s = sample excess kurtosis.

See Jarque and Bera (1987): ‘A test for normality of observations and regression residuals’, p. 163 ff. 5 In the formula, the sample skewness and the sample excess kurtosis are squared, which results in the Jarque-Bera statistic always being a positive number. If the skewness and excess kurtosis were not squared, for example, a negative skewness could cancel out a positive excess kurtosis in the formula, so that the Jarque-Bera statistic would falsely indicate a normal distribution. 4

3.2 Properties of a Distribution

71

The Jarque-Bera statistic is based on a chi-squared distribution with two degrees of freedom.6 If the returns are approximately normally distributed, the formula produces a low value for the Jarque–Bera statistic as a result of low skewness and excess kurtosis. In cases where the Jarque–Bera statistic calculated from a sample is greater than the critical value, the null hypothesis of normal distribution is rejected. The relevant critical value depends on the desired significance level for the test. For example, at significance levels of 5% and 1%, the critical values will be 5.99 and 9.21, respectively. The Jarque-Bera test is an asymptotic test that is not suitable for small samples. In the above example, it does not make sense to use the Jarque–Bera test due to the small sample of only 10 returns. The sample should contain at least 30 return observations (T ≥ 30). For example, assuming a sufficiently large sample of 250 returns whose distribution has a sample skewness of -0.051 and a sample excess kurtosis of 0.047, the Jarque-Bera statistic equals 0.13: JB =

0:0472 250 = 0:13: × ð - 0:051Þ2 þ 4 6

At a 5% significance level, the critical value is 5.99. Since the Jarque-Bera statistic of 0.13 is less than the critical value of 5.99, the null hypothesis that the returns are normally distributed cannot be rejected. Thus, the return distribution is normal.

3.2.4

Lognormal Distribution

An equity security is traded at a price of EUR 200 at the beginning of the year. At the end of the year, the share price falls to EUR 100, which corresponds to a return of 50% (= EUR 100/EUR 200 - 1). To get back to the initial price of EUR 200, a return of 100% (= EUR 200/EUR 100 - 1) is necessary. This means that if the return falls by x%, a return of more than x% is then required to get back to the initial value. The simple returns are therefore asymmetrical or positively skewed. If, on the other hand, the simple returns are converted in continuous compounded returns, a continuous compounded return of -69.31% [= ln(1 - 0.5)] is obtained for the price decline from EUR 200 to EUR 100 and a continuous compounded return of 69.31% [= ln(1 + 1)] for the price increase from EUR 100 to EUR 200. The fluctuations in returns are the same for a price decrease of EUR 100 and a subsequent price increase of EUR 100 and are therefore symmetrically distributed around the mean return of 0%. In other words, if continuous compounded returns are normally distributed, then simple returns are positively skewed. The following theorem applies in statistics: If a random variable X is lognormally distributed, then the random variable ln(X) is normally distributed. The following 6

The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics.

72 Fig. 3.3 Lognormal distribution (Source: Own illustration)

3 Other Investment Characteristics

(Density of probability)

0

(1+ r)

relationship holds between simple returns (r) and continuous compounded returns (rs): ð1 þ r Þ = ers . Hence, the continuous compounded return rs can be calculated using ln(1 + r).7 According to the theorem, if the values for 1+ r are lognormally distributed, then the values for ln(1 + r) are normally distributed. In other words, if 1 plus the simple returns are lognormally distributed, then the continuous compounded returns are normally distributed. In addition, the following relationship also holds – relevant, for example, in the case of the Black–Scholes model: If the continuous compounded returns are normally distributed, then the prices of the underlying asset are lognormally distributed [S1 = S0(1 + r)].8 A lognormal distribution is not symmetric but positively skewed. The distribution is defined only for positive values. Since the value of an investment (e.g. an equity security or a fixed-income security) cannot fall below zero, the maximum possible negative return is limited to -100%. It follows that (1 + r) cannot fall below 0. In financial market theory, therefore, consideration is given to the distribution of (1 + r) and not of r. Figure 3.3 presents the density function of the lognormal distribution. Since the price of an asset cannot fall below zero, it is usually assumed that 1 plus the simple returns of the asset are lognormally distributed. In the case of an equity security, for example, the simple returns lie in a range of values from -100% to +1; there are two reasons for this—first, negative share prices are not possible and, second, there is no upper share price limit. If continuous compounded returns are used instead of simple (discrete) returns, higher negative returns (in absolute terms) are obtained, which is consistent with the assumption of normal distribution, as the

7 8

See Sect. 1.3. See Sect. 14.6.

3.2 Properties of a Distribution

73

distribution has a range of return values from -1 to +1.9 Table 3.1 illustrates the relationship between simple returns and continuous compounded returns. Table 3.1 above indicates that negative simple returns are converted into higher negative log return values (in absolute terms) and positive simple returns into lower positive log return values. In contrast to simple returns, continuous compounded returns can fall below -100%, which is consistent with the normal distribution (or other distributions with strongly negative values). In financial market theory, continuous compounded returns are generally used for calculating the standard deviation and the beta of an equity security, as well as for econometric analyses. One of the reasons for this is that log returns, unlike simple returns, tend to be normally distributed. For example, if the Jarque-Bera test reveals that the continuous compounded returns of an investment are normally distributed, then 1 plus the simple returns must be lognormally distributed. Like the normal distribution, the lognormal distribution can be described by a density function. If a random variable ln(X) with an expected value of AM (arithmetic mean) and a standard deviation of SD is normally distributed, then X follows a lognormal distribution, given by the following probability density function, expected value, and variance: ½lnðxÞ - AM2 1 e - 2SD2 , f ðxÞ = p x 2πSD2

E ðX Þ = e

AMþSD2

2

,

2 2 VarðX Þ = eð2AMþSD Þ eSD - 1 ,

ð3:4Þ

ð3:5Þ ð3:6Þ

where x = certain value of the random variable X, e = base of the natural log function or Euler’s number (e = 2.71828. . .), and π = 3.14159. . . In Eqs. (3.5) and (3.6) the expected value and the variance of the lognormal distribution are given by the expected value (AM) and the variance (SD2) of the corresponding normal distribution. With the help of the density function (see Eq. 3.4), the probability that a random variable X takes a certain value x can be calculated. The expected value (AM) and the variance (SD2) from the corresponding normal distribution are used for this purpose. If 1 plus simple returns are lognormally distributed, the expected value and variance of the continuous compounded returns are used to calculate the probability; because if 1 plus simple returns are lognormally distributed, the continuous compounded returns are normally distributed. These 9

See Sect. 2.2.

Simple returns Continuous compounded returns

–99% –460.5%

–50% –69.3%

–10% –10.5% 0% 0%

10% 9.5%

Table 3.1 Logarithmic transformation of simple returns into continuous compounded returns (Source: Own illustration) 50% 40.5%

99% 68.8%

74 3 Other Investment Characteristics

3.3 Market Characteristics

75

calculations can be performed using Microsoft Excel, for example. For the portfolio in the previous example, given the annual continuous compounded returns for 2012 to 2021, the values for the first four moments of the distribution are as follows:10 Expected value (arithmetic mean) Standard deviation Skewness Excess kurtosis

5.82% 6.825% –0.051 0.047

Since the distribution of continuous compounded returns appears to be a normal distribution, 1 plus the simple returns are approximately lognormally distributed. Therefore, the probability that the simple return in the portfolio is less than or equal to 5% stands at 44.5%. This is the area to the left of 1 plus the 5% simple return under the lognormal distribution. Similarly, the probability that the simple return is greater than or equal to 10% can be determined. This probability is 29.3% and represents the area when the lognormal distribution is above 1 plus the simple return of 10%. The probability that the simple returns are located in an interval between 5% and 10% is 26.2%.11

3.3

Market Characteristics

3.3.1

Information Efficiency of Financial Markets

Market value is the price at which an asset can currently be bought or sold. By contrast, intrinsic value can be defined as the value that would be placed on an asset by investors if they had a complete understanding of its investment characteristics. For example, the relevant information to estimate the intrinsic value of an option-free bond with a cash flow model includes the coupon, the par value, the timing of coupon and principal payments, the other terms of the bond indenture such as the day-count convention for the accrued interest, an understanding of the default risk, other issue-specific characteristics, and market variables such as the term structure of interest rates and market liquidity.12 In a highly efficient market, market prices accurately reflect the intrinsic value. If the market is not efficient, the market price deviates from the intrinsic value, thus creating an opportunity to achieve an aboveaverage return. In an information-efficient market, asset prices react quickly to new information through the rational behaviour of investors. As a result, prices incorporate all past and present information. Consistent higher risk-adjusted returns are not possible in

10

See Sect. 3.2.3. See the Microsoft Excel applications at the end of this chapter for the calculation of probability with a lognormal distribution. 12 For the pricing of option-free bonds, see Chap. 11. 11

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3 Other Investment Characteristics

an efficient market.13 A passive investment strategy generates the same return as an active investment strategy, but has comparatively lower costs, such as transaction and analysis costs. On the other hand, if the market is information-inefficient, then asset prices do not contain all the price-relevant information; it is therefore possible to use an active investment strategy to exploit market inefficiencies and achieve above-average (abnormal) returns. In an information-inefficient market, an active investment strategy can lead to a higher return than a passive investment strategy. It is therefore important that investors understand the characteristics of informationefficient markets and are able to assess the degree of market efficiency. Fama (1970) distinguishes between three different degrees of market efficiency depending on the level of price-relevant information incorporated into the prices by the market participants:14 • Weak form • Semi-strong form • Strong form In the weak form of market information efficiency, prices incorporate all available historical information. Since current prices include all past price-relevant information, it is not possible to extrapolate historical prices and predict a price trend. Prices move only on the basis of new information and not on the basis of old information, because the latter has already been incorporated into prices by market participants. Technical analysis does examine past price movements to determine a future price trend or a buy or sell signal. However, if the market displays a weak form of information efficiency, above-average returns are not feasible with this strategy. Empirical studies have concluded that increased risk-adjusted returns cannot be achieved in developed markets on the basis of technical analysis.15 By contrast, in emerging markets, such as Hungary, Bangladesh, and Turkey, above-average returns are possible.16 In a market where the semi-strong form of information efficiency prevails, prices reflect not only past but also new public price-relevant information. Examples of publicly available information for stocks include data from published financial

13

A higher risk-adjusted or above-average return means that the return earned is greater than the return required based on the risk of the investment. The risk-adjusted required return can be calculated, for example, using a one-factor model such as the capital asset pricing model or a multifactor model such as the Fama–French model. The difference between the return earned and the required return calculated with a one- or multifactor model represents the above-average return or alpha. 14 See Fama (1970): ‘Efficient capital markets: a review of theory and empirical work’, p. 383 ff. 15 See, for example, Bessembinder and Chan (1998): ‘Market efficiency and the returns to technical analysis’, p. 5 ff., and Fifield et al. (2005): ‘An analysis of trading strategies in eleven European stock markets’, p. 531 ff. 16 See Fifield et al. (2005): ‘An analysis of trading strategies in eleven European stock markets’, p. 531 ff.

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77

statements and media releases (e.g. about dividends, operating cash flows, earnings, and changes in the company’s management and strategy), as well as financial market data (e.g. number of shares traded and beta of the equity security). In such a market, it is not possible to identify overvalued or undervalued securities based on publicly available information because this information has already been incorporated into share prices by market participants. An individual investor does not have access to any public information that another investor does not already possess. Therefore, abnormal returns cannot be achieved using publicly available data. When new public price-relevant information enters the market, prices change quickly to reflect the new level of information. If a company announces an unexpectedly high profit (earnings surprise), market participants react to this news, and the new information is therefore rapidly incorporated into the price. Thus, above-average profits based on such pricerelevant news are not possible in an information-efficient market of the semi-strong form. Most empirical studies consistently demonstrate that equity markets are semiinformationally efficient in developed countries; whereas this is fundamentally not the case in emerging markets.17 In an information-efficient market of the strong form, all price-relevant historical, public, and private information is incorporated into prices. By definition, an information-efficient market of the strong form is information-efficient in both the weak and semi-strong forms. It is not possible to earn above-average returns in such a market on the basis of private information. Prices reflect private information, such as management’s knowledge of its firm’s financial condition, which has not been publicly disseminated. In a market with strong information efficiency, private information such as insider knowledge does not lead to abnormal returns because this information has already been incorporated into prices. Empirical tests indicate that above-average returns can be obtained with private (i.e. non-public) information. Therefore, securities markets with a strong form of information efficiency do not exist.18 The degree of market efficiency is important for investors insofar as it influences the value of assets and thus reveals mispricing. The securities markets of developed countries basically exhibit the following characteristics with regard to their information efficiency: • Securities markets have a weak form of information efficiency. Thus, investors cannot generate above-average returns on the basis of historical prices and their extrapolation into the future.

See, for example, Gan et al. (2005): ‘Revisiting share market efficiency: Evidence from the New Zealand Australia, US and Japan stock indices’, p. 996 ff., and Raja et al. (2009): ‘Testing the semi-strong form efficiency of Indian stock market with respect to information content of stock split announcements—a study of IT industry’, p. 7 ff. 18 See, for example, Rozeff and Zaman (1988): ‘Market efficiency and insider trading: New evidence’, p. 25 ff. However, stock prices often reflect private information as well. See, for example, Meulbroek and Hart (1997): ‘The effect of illegal insider trading on takeover premia’, p. 51 ff. 17

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• Securities markets have a semi-strong form of information efficiency. In their investment decisions, investors and analysts must consider whether new public price-relevant information has already been incorporated into the asset price by market participants and how new price-relevant information affects the asset value. • Securities markets do not exhibit the strong form of information efficiency. Private information is usually not publicly available and thus cannot be known to all market participants. However, insider trading laws protect investors from market participants trading on the basis of private information (insider knowledge) and gaining an advantage. In the case of markets that exhibit the semi-strong form information efficiency, historical and new public price-relevant information cannot be used to earn aboveaverage returns. In such a market environment, a passive investment strategy is, in principle, more profitable than an active strategy. An empirical study by Malkiel (1995) demonstrates that, on average, mutual funds with an active strategy do not beat the overall market on a risk-adjusted basis.19 Actively managed mutual funds have, on average, the same return as the market before management fees and other expenses are added. However, if management fees and other expenses are deducted, the average return is lower. A similar result was reached in a study by Odean (1998b) for private investors using behavioural finance models that incorporate investor overconfidence. Overconfident investors will overestimate the value of their private information, causing them to trade too actively, and, consequently, to earn belowaverage returns.20 In principle, securities markets in developed countries exhibit the semi-strong form of information efficiency. Nevertheless, empirical studies have revealed that a number of market inefficiencies or anomalies exist that result in mispriced securities. Persistent market price anomalies are exceptions to market information efficiency. A price anomaly exists whenever the price of an asset is not justified by the pricerelevant information available. There are a variety of market price anomalies with equity securities that can be classified into three categories by the methods studied: time series (e.g. January effect), cross-sections (e.g. size and value effect, price-toearnings ratio, and price-to-book ratio), and others (e.g. earnings surprise and initial public offering).21 Identifying price anomalies is not straightforward. If the statistical methods used to identify market inefficiencies are corrected, most price anomalies disappear.

See Malkiel (1995): ‘Returns from investing in equity mutual funds 1971 to 1991’, p. 549 ff. See Odean (1998b): ‘Volume, volatility, price, and profit when all trades are above average age’, p. 1916. 21 See Mondello (2015): Portfoliomanagement: Theorie und Anwendungsbeispiele, p. 40 ff. 19 20

3.3 Market Characteristics

3.3.2

79

The Random Walk

In financial market theory, it is often assumed that the prices of assets such as an equity security follow a random walk. This assumption on price movements is consistent with the efficient market hypothesis, because all available price-relevant information is contained in asset prices, and therefore, prices move only as a result of new price-relevant information. Since new information emerges purely at random, asset prices follow a random walk. The random walk is illustrated below using a random experiment in which the current price of an equity security of EUR 100 changes by EUR 1 as a result of a coin toss. If the coin toss results in heads facing up, the share price rises by EUR 1. However, should tails be tossed, the share price falls by EUR 1. After four coin tosses, all share prices can be represented as follows (in EUR): 104 103 102

102 101

101 100

100

100

99

99 98

98 97 96

In the coin toss, the ups and downs (i.e. heads or tails) occur with equal probability, and therefore, the expected value of the share price is EUR 100. The variance, on the other hand, can be calculated with the following equation: N

σ2 = i=1

where

2

pi X i - X ,

ð3:7Þ

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3 Other Investment Characteristics

pi = probability that the value Xi occurs after a series of coin tosses. After a coin toss, the share prices stand at EUR 101 (heads) or EUR 99 (tails). The corresponding probability in each case is 50% that the coin toss will result in heads or tails. This leads to a variance of 1: σ 2 = 0:5 × ð101 - 100Þ2 þ 0:5 × ð99 - 100Þ2 = 1: p The standard deviation equals 1 = 1 . After two coin tosses, there is a probability of 50% (= 0.5 × 0.5 + 0.5 × 0.5) that the share price is EUR 100 since this value can be arrived at by two possibilities, namely heads–tails and tails–heads. The share prices of EUR 102 (heads–heads) and EUR 98 (tails–tails) can each be arrived at by one possibility, and the corresponding probability is therefore 25% (= 0.5 × 0.5). This results in a variance of 2: σ 2 = 0:25 × ð102 - 100Þ2 þ 0:5 × ð100 - 100Þ2 þ 0:25 × ð98 - 100Þ2 = 2: p The standard deviation for two coin tosses is 1.41 = 2 . The other values for the variance and the standard deviation (volatility) for a total of ten coin tosses are presented in Table 3.2. Table 3.2 indicates that the variance increases linearly with the number of tosses. By contrast, the standard deviation (volatility) increases with the root of the variance or the root of the number of coin tosses. If the coin tosses are replaced with time periods, the variance increases linearly with the time periods, and the standard deviation (volatility) increases with the root of the time periods. For example, assuming the standard deviation of daily stock returns is 1%, the annualised standard deviation of 15.87% for a random share price movement can be calculated as follows (1 year consists of 252 trading days): Table 3.2 Change in variance and standard deviation (volatility) for a random movement (Source: Own illustration)

Coin tosses 1 2 3 4 5 6 7 8 9 10

Variance 1.00 2.00 3.00a 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Standard deviation (volatility) 1.00 1.41 1.73b 2.00 2.24 2.45 2.65 2.83 3.00 3.16

σ = 0.125 × (103 - 100)2 + 0.375 × (101 - 100)2 + 0.375 × (99 100)2 + 0.125 × (97 - 100)2 = 3 p b σ = 3 = 1:73 a 2

3.3 Market Characteristics

81

p σ Annualised = 1% × 252 = 15:87%: In the random movement example above, past price movements have no effect on future prices. Hence, prices are not predictable. The assumption that asset prices follow a random movement is consistent with the efficient market hypothesis. This assumes that prices change only in response to new price-relevant information because all available information is already incorporated into asset prices. Since new information emerges purely at random, prices cannot be predicted and thus follow a random walk. If financial markets are information efficient, the autocorrelation of return time series, or the correlation of the returns of an asset over time, is zero. A zero autocorrelation means that returns of an asset are linearly independent over time. Therefore, it is important to examine the autocorrelation of return time series as this allows an assessment of whether prices are following a random movement.22 An autocorrelation of zero is consistent with the efficient market hypothesis. The autocorrelation of order k of a return time series with a constant expected value can be described as follows: AutocorrelationðkÞ =

E½ðr t - μÞðrt - k - μÞ , σt σt - k

ð3:8Þ

where rt = return of the asset at time t with expected value of μ and standard deviation σ t, and rt–k = return of the asset at time t - k with expected value of μ and standard deviation σ t–k. The formula for autocorrelation is the same as for the correlation coefficient,23 except that the numerator is not the covariance for the returns of two assets, but the returns of the same asset k periods apart. Accordingly, the range of values for the autocorrelation coefficient, like the correlation coefficient, is between -1 and 1, where 1 represents a perfect positive autocorrelation.

22

An informal approach to examining the autocorrelation of returns is to draw a scatter plot of returns rt versus returns rt–1. The scatter plot can be used to visually search for possible autocorrelation in the returns. If there is a positive autocorrelation, more return points are located in the northeast and southwest quadrants of the scatter plot where rt and rt–1 have the same sign. By contrast, when autocorrelation is negative, the return points are predominantly in the southeast and northwest quadrants. For zero autocorrelation, the return points occur uniformly in all four quadrants. A formal approach to determine the first-order autocorrelation of return time series is the Durbin-Watson test. See, for example, DeFusco et al. (2004): Quantitative Methods for Investment Analysis, p. 470 ff. 23 For the correlation coefficient, see Sect. 4.2.

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The autocorrelation of returns affects the relationship between the standard deviation and the time horizon. For example, if there is a positive autocorrelation, the standard deviation over t time periods is greater than the standard deviation multiplied by the square root of the number of time periods t. Accordingly, the volatility over several time periods is greater for a market with a clear return trend than for a market with an autocorrelation of zero (random movement). If the autocorrelation coefficient of returns is 1, the standard deviation over multiple time periods can be calculated by multiplying the standard deviation by the number of time periods t. If, on the other hand, the autocorrelation of the returns is negative (e.g. markets with a return reversal), the standard deviation over several time periods t is lower than the standard deviation multiplied by the square root of the number of time periods t. Example: Calculation of the Standard Deviation Over Several Time Periods with Different Autocorrelation An equity security has a standard deviation of daily returns of 1.2%. 1. What is the standard deviation of stock returns for a time period of 100 days if the returns are not correlated with each other (autocorrelation of returns of zero) or a random movement is assumed? 2. What is the maximum possible standard deviation for a time period of 100 days, assuming autocorrelation of returns? Solution to 1 The standard deviation of stock returns for a 100-day period is 12%, with an autocorrelation of zero: p σ 100 days = 1:2% × 100 = 12%: Solution to 2 With a coefficient for the autocorrelation of 1 (perfectly positive autocorrelation of the returns), the maximum possible standard deviation for a time period of 100 days is 120%: σ 100 days = 1:2% × 100 = 120%:

3.3.3

Behavioural Finance and Market Efficiency

The mindset and behaviour of market participants influence prices in financial markets. Behavioural finance theory attempts to explain whether individuals’

3.3 Market Characteristics

83

investment decisions are made rationally or irrationally.24 The theory focuses on the cognitive biases in investment decisions. The efficient market hypothesis, as well as most valuation models, assumes that the market is information efficient and that investors behave rationally. It is therefore useful to examine whether the behaviour of investors has an impact on market efficiency and asset prices. Most financial market theory models are based on the assumption that market participants are risk-averse. Risk aversion means that investors want to be adequately compensated for a higher risk of loss by a higher expected return. When risk is measured by standard deviation in the models, both positive and negative deviations from expected return lead to an increase in investment risk. In reality, however, individuals perceive the risk of an investment asymmetrically. They react more strongly to losses than to profits. Accordingly, they behave in a loss-averse rather than risk-averse manner because of their great fear of losses. When losses are incurred, an overreaction takes place, and positions are held in the hope that the price will recover. By contrast, profitable investments are often sold far too quickly. More time and energy are spent on avoiding losses than on achieving profits.25 Another behavioural characteristic is overconfidence in one’s own abilities when selecting investments. Overestimating one’s own understanding prevents one from processing the information appropriately.26 This is especially true for men and online traders.27 Overconfidence leads to wrong investment decisions and, consequently, in the case of mass behaviour, to mispricing in the markets. The gambler’s fallacy is the misconception that a random event becomes more likely the longer it has not occurred. This bias results in an incorrect assessment of probabilities and investment decisions since chance has no memory. The principle of mental accounting can be observed in investors who mentally divide their investments into two categories—those with profits and those with losses. This way of thinking prevents investments from being considered from a portfolio point of view and thus from a diversification perspective, which results in suboptimal investment decisions. Cognitive biases such as mental accounting and loss aversion can lead to herd behaviour or an information cascade. Social interaction and the resulting contagion

24 Behavioural finance deals with the behaviour of individuals in economic situations. It examines behaviours that contradict the model assumptions, such as risk aversion. 25 A number of studies reveal that individual investors frequently trade stocks too often, driving up commissions, and sell stocks with capital gains too early (prior to further price increases), while they hold onto stocks with capital losses too long (as the price continues to fall). See, for example, Barber and Odean (2000): ‘Trading is hazardous to your wealth: the common stock investment performance of individual investors’, p. 799 f., Odean (1998a): ‘Are investors reluctant to realize their losses?’, p. 1775 f., Shefrin and Statman (1985): ‘The disposition to sell winners too early and ride losers too long: theory and evidence’, p. 777 ff. 26 See, for example, Griffin and Tversky (1992): ‘The weighting of evidence and the determinants of confidence’, p. 411 ff. 27 See Barber and Odean (2001): ‘Boys will be boys: gender, overconfidence, and common stock investment’, p. 261 ff.

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of behavioural patterns are important factors in explaining price changes that cannot be attributed to new price-relevant information. Market-wide price anomalies can only arise from cognitive biases if many individuals adopt this unreasonable behaviour. Herd behaviour is characterised by investors who follow a price trend irrationally and are driven by emotions such as greed and fear of losses. For example, stocks are bought due to profit opportunities during a market bubble, while they are sold during a stock market crash for fear of losses. Accordingly, the actions of market participants do not have to be based on information processing. In an information cascade, on the other hand, investors buy and sell because of other investors who acted first. In this case, the investment decisions of others are observed and then imitated by investors independent of their own preferences. This behaviour can lead to directional stock returns and is consistent with the price anomaly from an overreaction by market participants. Whether the investment behaviour is rational due to the information cascade depends on the investors who bought or sold first. If they act rationally and are well informed, then the purchases or sales of the uninformed imitators can be equated with rational market behaviour. In such a case, the uninformed investors help to incorporate the relevant information into prices, which increases market efficiency. Behavioural finance theory is able to explain how markets function and how prices are determined. The question remains open as to whether the irrational behaviour of investors causes price anomalies. In principle, a distinction must be made between individual and social irrational behaviour. The former can be compensated for by the market, while the latter makes the market inefficient and removes prices from their intrinsic value. The assumption of market efficiency cannot be sustained if investors must act rationally in order for markets to be efficient. There are too many cognitive biases that result in irrational behaviour by individuals. On the other hand, the majority of empirical studies conclude that markets in developed countries are essentially informationally efficient in the semi-strong form. Hence, no above-average returns can be achieved with historical and new public information on a risk-adjusted basis.

3.3.4

Market Liquidity and Trading Costs

The prevailing market liquidity affects the value of assets. Financial assets with low market liquidity have a wide bid–ask spread. The bid–ask spread is the difference between the ask and bid prices. The larger this difference, the higher the trading costs. In a quote-driven market28, the bid and ask prices are provided by traders. For 28

Trading generally takes place in the following markets: quote-driven market, order-driven market, and broker market. In a quote-driven market, investors trade directly with traders, while in an order-driven market, trading takes place between investors (without intermediation of traders). In a broker market, on the other hand, the trader relies on a broker to find a counterparty to trade with.

3.3 Market Characteristics

85

the trader, the bid price represents the buying price for a given number of securities, while the ask price reflects the selling price. In this case, the ask price is higher than the bid price. From an investor’s perspective, executing a buy order at a low ask price is advantageous. A sell order, on the other hand, aims for a high bid price. Example: Bid–Ask Spread A portfolio manager of a bank submits a buy order for 500 shares to the bank’s own trading desk. The shares are traded in a quote-driven market. The market for this equity security consists of the three traders X, Y, and Z. The trading desk sees the following price quotes (in EUR) entered by the three traders on the screen at 09:23: • Trader X: bid of 55.85 for 300 shares and ask of 55.95 for 600 shares, • Trader Y: bid of 55.82 for 400 shares and ask of 55.98 for 300 shares, • Trader Z: bid of 55.80 for 300 shares and ask of 55.90 for 400 shares. The table below presents the bid prices from highest to lowest price, while the ask prices are listed from lowest to highest price. This order takes into account the best buy and sell price for the bank’s trading desk (highest bid or lowest ask price). Bid

Dealer X Y Z

Ask Time of the enter 09.22 09.21 09.18

Price (in EUR) 55.85 55.82 55.80

Quantity 300 400 300

Dealer Z X Y

Input of the enter 09.20 09.22 09.18

Price (in EUR) 55.90 55.95 55.98

Quantity 400 600 300

At what prices can the bank’s trading desk execute this order? Solution The bank’s trading desk buys 400 shares from trader Z at a unit price of EUR 55.90 and 100 shares from trader X at a unit price of EUR 55.95. In the example, the market bid–ask spread is EUR 0.05 and is equal to the difference between the lowest ask price of EUR 55.90 and the highest bid price of EUR 55.85. The market bid–ask spread is lower than the bid–ask spreads of the individual traders. Traders X, Y, and Z have spreads of EUR 0.10, EUR 0.16, and EUR 0.10, respectively, all of which are greater than the market bid–ask spread of EUR 0.05. If a stock trading at EUR 100 per share has a bid–ask spread of EUR 0.20, for example, the spread in relation to the price equals 0.2%. On the other hand, a stock with a price per share of EUR 10 and the same spread of EUR 0.20 has a bid–ask

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spread in relation to the price of 2%. Thus, the second equity security with a higher spread–price ratio of 2% has higher trading costs. Trading costs are made up of explicit and implicit costs. The explicit costs consist of direct trading costs such as broker commissions, taxes, and fees paid to the exchange. The trader or investor receives an invoice for these costs. By contrast, implicit trading costs have no invoice and include the following costs: • Bid–ask spread: An increase (decrease) in the bid–ask spread means, for example, that a stock can be bought at a higher (lower) ask price and sold at a lower (higher) bid price, resulting in higher (lower) implicit costs. • The effect of a trade order on the price of the transaction: For example, a trader splits the purchase of 800 bonds into two equal orders of 400 bonds each. The bid–ask spread is 98.355% to 98.675 %.29 The first order to buy 400 bonds is executed at an ask price of 98.675%. After this transaction, the bid–ask spread changes to 98.371% to 98.732%. The second order to buy 400 bonds is executed at an ask price of 98.732%. The first transaction caused a price increase in the market, and therefore, the second transaction was executed at a higher price of 0.057% (= 98.732% - 98.675%). If the par value of a bond is CHF 5,000, the result is higher transaction costs of CHF 1,140 (= CHF 5000 × 0.00057 × 400). • Opportunity costs: These costs result from an unexecuted buy or sell order. For example, a trader enters a buy limit order for bonds at 102.500% or better, good for 1 day, into the system. The market bid-ask spread at that time is 102.505% to 102.846%. After 1 day, the order has not been executed and the market bid-ask spread is 102.524% to 103.106%. The difference of 0.26% between the two market ask prices of 103.106% and 102.846% represents the opportunity cost of the unexecuted buy order. • Waiting costs: These costs arise because it is not possible to execute the buy or sell order during the desired time period. Due to the size of the order and the lack of market liquidity, the order can only be executed over a longer period of time. During this time, information enters the market that can influence the price, resulting in higher implied trading costs. One approach to estimate both the explicit and implicit trading costs is the volume-weighted average price (VWAP). The VWAP of a security is its average price during the trading day, calculated as the sum of the volume-weighted order prices. For example, shares of Delta AG are purchased at different times in the course of a trading day: 500 shares at a unit price of EUR 100, 600 shares at a unit price of EUR 101, and 900 shares at a unit price of EUR 103. The VWAP of EUR 101.65 can be determined as follows:

29

Bond prices are traded as a percentage of the par value. For instance, a bid price of 98.355% means a price equal to 98.355% of the par value. See Sect. 11.2.

3.4 Summary

87

VWAP 500 shares × EUR 100 þ 600 shares × EUR 101 þ 900 shares × EUR 103 2000 shares = EUR 101:65: =

If the 900 Delta shares are bought at a unit price of EUR 103, the implied trading costs amount to EUR 1215 [= (EUR 103 - EUR 101.65) × 900 shares]. The disadvantage of this approach is that the trader can coordinate the timing of his buy and sell orders with the level of the VWAP and thus influence the implied trading costs. Moreover, the calculated implied trading costs are not broken down further, and therefore, the impact of the trading order on the price of the transaction, the waiting costs, and the opportunity costs are not known. A better way to estimate the explicit and implicit trading costs is to adopt the implementation shortfall approach, which is less manipulable by traders and also reveals the origin of the implicit trading costs.30

3.4

Summary

• The assumption of normal distribution is very convenient because the distribution is completely defined by the mean and the variance. However, the returns on most financial assets, such as equity securities, are not normally distributed. In cases where returns are not normally distributed, the analysis has to be extended by including higher central moments of the distribution such as skewness and kurtosis. • The skewness of the distribution represents the third central moment of a distribution and measures the degree of symmetry of the returns around the mean. A normal distribution has a skewness of zero. The mean (arithmetic mean), median, and mode are equal. In a right-skewed distribution, losses are small and frequent, while gains are extremely high and less frequent, with the result that the mean exceeds the median and the mode. A left-skewed distribution, on the other hand, is characterised by more frequent small gains and few extremely high losses, and the mean is therefore lower than the median and the mode. Such investments are not attractive. • The kurtosis is the fourth central moment of a distribution. It measures the peakedness at the centre of the distribution and the tails at the two ends of the distribution. A normal distribution has a kurtosis of 3 and an excess kurtosis of 0. If a distribution has a kurtosis greater than 3 or a positive excess kurtosis, it is leptokurtic. Graphically, the distribution is steep in the middle (steep peaked) and has fatter tails at both ends of the distribution than a normal distribution. On the

30

For the implementation shortfall, see, for example, Mondello (2015): Portfoliomanagement: Theorie und Anwendungsbeispiele, p. 50 ff.

88

• •

•

• •

•

3 Other Investment Characteristics

other hand, if the kurtosis is less than 3 or the excess kurtosis is negative, the distribution is platykurtic. Graphically, the middle of the distribution is flat (flat peaked), and the tails are thinner than a normal distribution. Whether a normal distribution exists can be assessed by means of statistical tests such as the Jarque-Bera test, which includes the skewness and excess kurtosis of the sampling distribution. This is a hypothesis test. Since the price of financial assets cannot fall below zero and there is no upper price limit for equities, for example, financial market theory often works with a lognormal distribution, which represents a right-skewed distribution. The simple stock returns fall in a range of values from -100% to +1. If 1 plus the simple returns (1 + r) are lognormally distributed, the continuous compounded returns [ln(1 + r)] are normally distributed. Continuous compounded returns are generally used for calculating the standard deviation and the beta of equity securities, as well as for econometric analyses; one of the reasons for this being that they tend to be normally distributed, in contrast to simple returns. The degree of market information efficiency has an impact on the traded value of assets. Fama distinguishes three degrees of market efficiency: weak, semi-strong, and strong. The information efficiency of a market is weak if only historical information is incorporated into the asset price. Investors are unable to earn above-average returns when they extrapolate trends from historical prices. In a semi-strong form of market information efficiency, asset prices reflect both historical and new public information. Public information consists of annual price-relevant accounting and financial market data. If publicly available information is used, above-average returns are not possible in the semi-strong form of an information-efficient market. In the strong form of market information efficiency all historical, new public, and private price-relevant information is incorporated into prices by investors. In such a market, it is even impossible to achieve above-average returns with private information (insider knowledge). Empirical studies have demonstrated that securities markets of developed countries exhibit a semi-strong form of market efficiency, while securities markets with a weak form of information efficiency are the rule in emerging markets. Since the analysis of new public price-relevant information in a semi-strong information-efficient market does not lead to superior returns, a passive investment strategy is a better choice than an active strategy due to its lower costs. It is often assumed that the prices of financial assets follow a random walk. This assumption about price movements is consistent with the efficient market hypothesis because all available information is incorporated into asset prices by market participants, with the result that prices move only as a result of new price-relevant information. Since new information appears on a purely random basis, asset prices follow a random walk and thus cannot be predicted. If markets are information efficient, the autocorrelation of return time series, or the correlation of an asset’s returns over time, is zero. An autocorrelation of zero means that the returns of an asset are linearly independent over time and thus do not correlate with each other. Therefore, the standard deviation of one time period

3.5 Problems

89

(e.g. 1 week) can be converted into a standard deviation of multiple time periods (e.g. 52 weeks) by multiplying the standard deviation of one time period by the square root of the number of time periods. • The mindset and behaviour of investors influence prices in financial markets. Behavioural finance theory attempts to explain whether individuals’ investment decisions are rational or irrational. The focus is on the cognitive biases in investment decisions, such as loss aversion, overconfidence, gambler’s fallacy, and mental accounting. • Behavioural finance theory is able to explain how financial markets function and how prices are determined. The question remains open as to whether the irrational behaviour of investors generates price anomalies. In principle, a distinction must be made between individual and social irrational behaviour. The former can be compensated by the market, while the latter makes the market inefficient and removes prices from their intrinsic value. • The prevailing market liquidity affects the prices of assets. Trading costs are comprised of explicit and implicit costs. Implicit trading costs include the bid–ask spread, the effect of the trade order on the price of the transaction, waiting costs, and opportunity costs. A simple approach to estimating both explicit and implicit trading costs is the VWAP.

3.5

Problems

1. The table below presents the respective closing price of the Swiss Market Index (SMI) and the annual simple and continuous compounded returns from 2001 to 2017 (Source: Refinitv Eikon). Year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Closing price of the SMI 6417.84 4630.75 5487.81 5693.17 7583.93 8785.74 8484.46 5534.53 6545.91 6436.04 5936.23 6822.44 8202.98 8983.37 8818.09 8219.87 9381.87

Simple return (in %)

Continuous compounded return (in %)

-27.85 18.51 3.74 33.21 15.85 -3.43 -34.77 18.27 -1.68 -7.77 14.93 20.24 9.51 -1.84 - 6.78 14.14

-32.64 16.98 3.67 28.68 14.71 -3.49 -42.72 16.78 -1.69 -8.08 13.91 18.43 9.09 -1.86 -7.03 13.22

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3 Other Investment Characteristics

The analysis of the SMI return distribution is performed using the first four moments of the distribution: central tendency, dispersion, skewness, and kurtosis. a) To describe the central tendency of the return distribution, the arithmetic mean, the mode, and the median must be determined for the continuous compounded returns in the sample. What are the arithmetic mean and the median? b) To analyse the width or dispersion of the return distribution, the absolute average deviation and the standard deviation can be calculated. What is the standard deviation of the continuous compounded returns? c) To assess the extent to which the return distribution deviates from the normal distribution, the skewness and the kurtosis must be determined. What are the skewness and excess kurtosis for the SMI and how can these quantities be interpreted? 2. For a stock portfolio, the following values for the first four moments of the distribution are available from a time series of 252 continuous compounded daily returns: Arithmetic mean Standard deviation Skewness Excess kurtosis

1.41% 21.30% –0.52 –0.24

Using the Jarque-Bera test at a 5% significance level, determine whether the portfolio returns are normally distributed. 3. The following statements relate to the return distributions of investments: 1. Kurtosis represents the fourth central moment of the distribution. If the distribution is platykurtic, the kurtosis is negative. 2. Skewness is the third central moment of a distribution. A left-skewed distribution is characterised by many small gains and a few extremely high losses, resulting in the following relationship: mode > median > mean. 3. The normal distribution is completely described by the mean and the variance or standard deviation. All other higher central moments of the normal distribution such as skewness and excess kurtosis are zero. 4. In financial market theory, continuous compounded returns are generally used for calculating the standard deviation and the beta of an equity security, as well as for econometric analyses. One of the reasons for this is that continuous compounded returns tend to be normally distributed in contrast to simple returns. Indicate whether each of the above statements is true or false (with justification).

3.6 Solutions

91

4. The following statements relate to the efficient market hypothesis: 1. A financial market exhibits the semi-strong form of information efficiency when asset prices reflect all price-relevant historical and new public information. 2. Information-efficient markets in the semi-strong form are not necessarily information efficient in the weak form. 3. In a semi-strong information-efficient market, a passive investment strategy usually generates a higher return, after transaction and other costs, than an active investment strategy. 4. Fundamental analysis can be used to achieve above-average returns in a semistrong information-efficient market. 5. When a market exhibits weak information efficiency, an active investment strategy based on fundamental analysis can produce above-average returns. 6. An investment strategy that uses historical price charts to generate abnormal returns violates semi-strong information efficiency. Indicate whether each of the above statements is true or false (with justification). 5. Ms. Miller reads in the newspaper that Novartis has announced earnings that are above analysts’ consensus expectations (earnings surprise). Two days after this announcement, she calls her bank relationship manager and places a buy order for 100 Novartis shares. With this transaction she achieves an above-average return. Which form of the efficient market hypothesis is violated here? 6. The following shares of Gamma AG are sold at different times in the course of a trading day: Trading volume (number of shares) 400 800 1100 700

Transaction price per share (in EUR) 50.00 50.50 50.75 51.00

What are the VWAP of the trading day and the implied trading costs if a trader sold 800 Gamma shares at a price of EUR 50.50 per unit?

3.6

Solutions

1. a) The arithmetic mean return of the SMI is 2.37%:

r SMI =

-32:64% þ 16:98% þ . . . þ 13:22% = 2:37%: 16

92

3 Other Investment Characteristics

To determine the median return, the middle of the returns in the sample must be found. To do this, the continuous compounded returns are ranked from the highest to the lowest value: 28.68%, 18.43%, 16.98%, 16.78%, 14.71%, 13.91%, 13.22%, 9.09%, 3.67%, -1.69%, -1.86%, -3.49%, -7.03%, 8.08%, -32.64%, -42.72%. The median return lies in the middle, between 9.09% and 3.67%, and is equal to 6.38%:

Median return =

9:09% þ 3:67% = 6:38%: 2

The mode is the most frequently occurring return in a sample or distribution. In this example, it cannot be calculated because all returns are different. The following tables for standard deviation, skewness, and excess kurtosis were calculated with Microsoft Excel, and the figures rounded to two decimal places in each case, while the software calculates to several decimal places. b) Year 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Total

Continuous compounded return (in %) -32.64 16.98 3.67 28.68 14.71 -3.49 -42.72 16.78 -1.69 -8.08 13.91 18.43 9.09 -1.86 -7.03 13.22

rt - r -35.01 14.61 1.30 26.31 12.34 -5.86 -45.09 14.41 -4.06 -10.45 11.54 16.06 6.72 -4.23 -9.40 10.85

ðr t - r Þ2 1225.44 213.48 1.70 692.03 152.27 34.33 2033.38 207.75 16.51 109.28 133.27 257.86 45.13 17.87 88.27 117.78 5346.34

3.6 Solutions

93

The standard deviation of the SMI returns is 18.88%.

σ˜ =

1 T -1

T

5346:34 = 18:88%: 16 - 1

ðr t - rÞ2 = t=1

c) Skewness: Year 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Total

Continuous compounded return (in %) -32.64 16.98 3.67 28.68 14.71 -3.49 -42.72 16.78 -1.69 -8.08 13.91 18.43 9.09 -1.86 -7.03 13.22

T

Ss = ×

T ð T - 1 Þ ð T - 2Þ

t=1

ðr t - r Þ3 σ~3

=

rt - r -35.01 14.61 1.30 26.31 12.34 -5.86 -45.09 14.41 -4.06 -10.45 11.54 16.06 6.72 -4.23 -9.40 10.85

ðr t - r Þ3 -42,898.07 3119.20 2.22 18,204.80 1879.01 -201.17 -91,690.90 2994.31 -67.06 -1142.45 1538.52 4140.73 303.16 -75.52 -829.28 1278.16 -103,444.35

16 ð16 - 1Þ × ð16 - 2Þ

- 103,444:35 = - 1:17 ð18:88Þ3

The sample skewness of the return distribution is - 1.17. The annual returns of the SMI appear to be left-skewed from 2002 to 2017.

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3 Other Investment Characteristics

Excess kurtosis: Year 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Total

T

K E, s =

=

rt - r -35.01 14.61 1.30 26.31 12.34 -5.86 -45.09 14.41 -4.06 -10.45 11.54 16.06 6.72 -4.23 -9.40 10.85

Continuous compounded return (in %) -32.64 16.98 3.67 28.68 14.71 -3.49 -42.72 16.78 -1.69 -8.08 13.91 18.43 9.09 -1.86 -7.03 13.22

T ðT þ 1Þ ð T - 1Þ ð T - 2Þ ðT - 3Þ

t=1

ðr t - r Þ4 ~4 σ

-

ðr t - r Þ4 1,501,701.80 45,574.81 2.89 478,903.29 23,186.67 1178.70 4,134,614.30 43,158.18 272.43 11,943.09 17,761.08 66,491.99 2036.56 319.24 7791.19 13,871.22 6,348,807.43

3 ð T - 1Þ 2 ð T - 2Þ ð T - 3 Þ

16 × ð16 þ 1Þ 3 × ð16 - 1Þ2 6,348,807:43 × 4 ð16 - 2Þ × ð16 - 3Þ ð16 - 1Þ × ð16 - 2Þ × ð16 - 3Þ ð18:88Þ

= 1:27 The annual returns of the SMI have a positive excess kurtosis of 1.27 from 2002 to 2017 (leptokurtic). This means that there are more returns in the tails of the return distribution than in a normal distribution. Thus, the distribution of the annual SMI returns from 2002 to 2017 appears to be leftskewed with a skewness of -1.17 and leptokurtic with a positive excess kurtosis of 1.27. 2.

3.6 Solutions

95

The Jarque-Bera statistic is 11.96 and can be calculated as follows:

JB =

ð - 0:24Þ 252 × ð - 0:52Þ2 þ 6 4

2

= 11:96:

At a 5% significance level, the critical value is 5.99. Since the Jarque-Bera statistic of 11.96 is greater than the critical value of 5.99, the null hypothesis that the returns are normally distributed can be rejected. 3. 1. The first statement is false. In a platykurtic distribution, the kurtosis is less than 3 and cannot fall below 0. The excess kurtosis, on the other hand, is negative. 2. The second statement is true. In a left-skewed distribution, the centre of the distribution (i.e. the mode) is greater than the median and greater than the mean. Such a distribution includes many small gains and a few extremely large losses. 3. The third statement is true. The normal distribution is given by the first two moments, namely mean and variance. Other higher central moments such as skewness and excess kurtosis are zero. 4. The fourth statement is true. Simple returns cannot fall below -100% because negative asset prices (e.g. stock and bond prices) are not possible. By contrast, continuous compounded returns can take on higher negative values (in absolute terms), which is consistent with the assumption of the normal distribution (or indeed other distributions with strongly negative values). If 1 plus the simple returns (1 + r) are given by a lognormal distribution (e.g. equity securities with a range of values of the simple returns from -100% to +1), the continuous compounded returns [ln(1 + r)] are normally distributed. 4. 1. The first statement is true. In the semi-strong form of the efficient market hypothesis, asset prices reflect all price-relevant historical and new public information. Thus, using new private information (insider knowledge) can generate above-average returns. 2. The second statement is false. A semi-strong information-efficient market is also weakly information efficient. In addition to new public price-relevant information, prices also incorporate all historical price-relevant information. 3. The third statement is true. If financial markets are information efficient in the semi-strong form, it is not possible to earn abnormal returns with new public information. Therefore, it is difficult to compensate for the transaction and analysis costs of an active investment strategy with above-average returns. In such a market, a passive investment strategy is more profitable.

96

3 Other Investment Characteristics

4. The fourth statement is false. In fundamental analysis, public information is used (e.g. annual reports and press releases) to determine the intrinsic value of the financial asset. If the asset is undervalued (overvalued), it is bought (sold). In the semi-strong form of the efficient market hypothesis, prices include all public price-relevant information, and consequently abnormal returns are not possible with fundamental analysis. 5. The fifth statement is true. In a weakly information-efficient market, only historical information is incorporated into prices by investors. If new public pricerelevant information is analysed, an abnormal return can be earned. 6. The sixth statement is true. Not only the semi-strong but also the weak form of market efficiency is violated when above-average returns are achieved by analysing historical data. 5. If one achieves above-average returns with historical information, all three forms of the market information efficiency hypothesis (weak, semi-strong, and strong) are violated. 6. The VWAP is EUR 50.64 and can be calculated as follows:

VWAP =

400 shares × EUR 50þ800 shares × EUR 50:50þ1100 shares × EUR 50:75 þ700 shares × EUR 51

3000 shares

= EUR 50:64:

The trader receives proceeds from the sale of the 800 Gamma shares in the amount of EUR 40,400 (= 800 × EUR 50.50). Using the VWAP of the trading day, the sales proceeds amount to EUR 40,512 (= 800 × EUR 50.64). Thus, the implicit trading costs are EUR 112 (= EUR 40,512 - EUR 40,400).

Microsoft Excel Applications • To calculate the skewness of a return distribution (for a sample), first enter the returns in, for example, cells A1 to A12. After that, the following expression is written in an empty cell: = SKEWðA1:A12Þ

Microsoft Excel Applications

97

and then confirmed with the Enter key. • Microsoft Excel calculates the kurtosis using the sample excess kurtosis. If the returns are listed in cells A1 to A12, for example, the following expression can be entered in an empty cell: = KURTðA1:A12Þ: Then press the Enter key. • Until now, the individual statistical ratios such as mean, median, standard deviation, skewness, and excess kurtosis were determined separately. In Microsoft Excel, it is also possible to create a summary data output of these quantities. The following steps are required for this: 1. Click on ‘Data’ tab. 2. Click on ‘Data Analysis’. 3. This opens a window in which the function ‘Descriptive Statistics’ must be selected. 4. In the open dialog box, enter the returns in the ‘Input Range’ (e.g. A1:A12). Next, click on ‘Output Range’, in which a cell is defined (e.g. A13), where the summary data output is to appear. Finally, place a check mark in the ‘Summary Statistics’ and confirm with the OK key. • The mean and the standard deviation from the corresponding normal distribution are required for the calculation of probability in the case of a lognormal distribution. The simple returns (1 + r) from a lognormal distribution are to be converted into continuous compounded returns [ln(1 + r)] for the determination of the mean and the standard deviation, since these are normally distributed. If there are continuous compounded returns in cells A1 to A12, these can be used to calculate the mean (AM) and the standard deviation (SD) in cells A13 and A14. In the following Excel formulas, do not use the expressions 1 + r0, AM or SD but rather their current values. • In order to calculate the probability that a given value for the simple return r is less than or equal to r0 in a lognormal distribution, the function ‘lognorm.dist’ must be used as follows:31 = LOGNORM:DISTð1 þ r 0 ; AM; SD, trueÞ:

For compatibility with Excel 2007 and earlier also the function “lognormdist” may be used as follows: =lognormdist(1 + r0; AM; SD).

31

98

3 Other Investment Characteristics

Then press the Enter key. To calculate the cumulative lognormal distribution function the logical value ‘true’ must be used. If no value is entered, ‘false’ will be utilised by default, which is employed to determine the probability density function of the lognormal distribution. • The probability that the simple return r is greater than or equal to r0 in a lognormal distribution can be calculated in Microsoft Excel as follows: = 1 - LOGNORM:DISTð1 þ r 0 ; AM; SD; trueÞ: The formula must be completed by pressing the Enter key. • To determine the probability that, given a lognormal distribution, the simple return r lies between the two simple returns r0 and r1, where r0 < r1 enter the following expression in a cell: = LOGNORM:DISTð1 þ r 1 ; AM; SD; trueÞ - LOGNORM:DISTð1 þ r 0 ; AM; SD; trueÞ and then confirm with the Enter key.

References Barber, B.M., Odean, T.: Trading is hazardous to your wealth: the common stock investment performance of individual investors. J. Finance. 55(2), 773–806 (2000) Barber, B.M., Odean, T.: Boys will be boys: gender, overconfidence, and common stock investment. Q. J. Econ. 116(1), 261–292 (2001) Bessembinder, H., Chan, K.: Market efficiency and the returns to technical analysis. Financ. Manag. 27(2), 5–17 (1998) DeFusco, R.A., McLeavy, D.W., Pinto, J.E., Runkle, D.E.: Quantitative Methods for Investment Analysis, 2nd edn. CFA Institute, Charlottesville (2004) Fama, E.F.: Efficient capital markets: a review of theory and empirical work. J. Finance. 25(2), 383–417 (1970) Fifield, S.G., Power, D.M., Sinclair, C.D.: An analysis of trading strategies in eleven European stock markets. Eur. J. Finance. 11(6), 531–548 (2005) Gan, C., Lee, M., Hwa, A.Y., Zhang, J.: Revisiting share market efficiency: evidence from the New Zealand Australia, US and Japan stock indices. Am. J. Appl. Sci. 2(5), 996–1002 (2005) Griffin, D., Tversky, A.: The weighting of evidence and the determinants of confidence. Cogn. Psychol. 24(3), 411–435 (1992) Jarque, C.M., Bera, A.K.: A test for normality of observations and regression residuals. Int. Stat. Rev. 55(2), 163–172 (1987) Malkiel, B.G.: Returns from investing in equity mutual funds 1971 to 1991. J. Finance. 50(2), 549–572 (1995)

References

99

Meulbroek, L.K., Hart, C.: The effect of illegal insider trading on takeover premia. Eur. Finance Rev. 1, 51–80 (1997) Mondello, E.: Portfoliomanagement: Theorie und Anwendungsbeispiele, 2nd edn. Springer Fachmedien Wiesbaden, Wiesbaden (2015) Odean, T.: Are investors reluctant to realize their losses? J. Finance. 53(5), 1775–1798 (1998a) Odean, T.: Volume, volatility, price, and profit when all trades are above average age. J. Finance. 53(6), 1887–1934 (1998b) Raja, M., Sudhahar, J.C., Selvam, M.: Testing the semi-strong form efficiency of Indian stock market with respect to information content of stock split announcements - a study of IT industry. Int. Res. J. Finance Econ. 25, 7–20 (2009) Rozeff, M.S., Zaman, M.A.: Market efficiency and insider trading: new evidence. J. Bus. 61(1), 25–44 (1988) Shefrin, H., Statman, M.: The disposition to sell winners too early and ride losers too long: theory and evidence. J. Finance. 40(3), 777–790 (1985)

4

Efficient Risky Portfolios

4.1

Introduction

Rather than investing in a single asset, most investors put their money into a portfolio of assets. This raises the question of how to calculate the expected return and risk of a portfolio. Furthermore, it has to be determined which portfolios of risky assets are most efficient in terms of expected return and risk. Markowitz’s portfolio theory demonstrates how to construct the efficient frontier on which the most efficient risky portfolios lie with regard to expected return and risk.1 The efficient frontier is created from capital market data, which are used to estimate the expected return and standard deviation of the returns of individual assets, as well as the covariance or correlation coefficient between the returns of a pair of assets. Markowitz’s portfolio theory is based on the following assumptions: • The investment opportunities or assets are given by a distribution of expected returns accruing over a period. • Investors maximise expected utility over a period, with indifference curves (utility functions) characterised by decreasing marginal utility. • Investors perceive portfolio risk as fluctuations in expected returns, which are measured on the basis of the variance or standard deviation. • Investment decisions are based on expected return and risk, with the result that indifference curves are, among other things, a function of expected return and variance. • For a given level of risk, investors require a higher rather than a lower return; or for a given level of return, they choose the investment that has a lower (rather than a higher) level of risk. Accordingly, investors behave in a risk-averse manner.

See Markowitz (1952): ‘Portfolio selection’, p. 77 ff., and Markowitz (1959): Portfolio Selection: Efficient Diversification of Investments, p. 1 ff. 1

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_4

101

102

4.2

4

Efficient Risky Portfolios

Expected Return and Risk of a Two-Asset Portfolio

Portfolio construction is based on the fundamental notion of efficient diversification. Risky portfolios are comprised of two or more assets, and only a portfolio that has the lowest risk for a given expected return or the highest expected return for a given risk is efficient in terms of return expectation and risk. To determine the expected return and risk of a portfolio, the expected return of an individual asset must first be calculated. The expected return represents an expected value and may deviate from the actual return. It can be estimated using either past returns or a prospective scenario analysis. The expected return of an asset on the basis of historical values can be calculated as follows: E ðr Þ =

1 T

T

rt ,

ð4:1Þ

t=1

where T = number of periods or returns, and rt = return on a single asset for period t. The arithmetic mean of historical returns is a good approximation of the expected return only if returns are stable. Should this not be the case, then the expected return can be determined using a prospective scenario analysis. In such an analysis, the expected return represents the sum of the probability-weighted scenario returns and can be calculated with the following equation:2 n

E ðr Þ =

pi E ðr i Þ,

ð4:2Þ

i=1

where pi = probability for the occurrence of scenario i, E(ri) = expected return for scenario i, and n = number of scenarios. With this approach, relevant scenarios are identified and both the returns and the probabilities of occurrence of the individual scenarios are determined. By contrast, historical returns are based on time series of returns achieved in the past. A probability distribution is constructed from this limited data series. If a normal distribution exists, it is described by the expected return and the variance. Assuming that each return observation occurs with the same probability, the probability pi of

2

See Sect. 1.9.

4.2 Expected Return and Risk of a Two-Asset Portfolio

103

Eq. (4.2) can be replaced by 1/T for T return observations. Hence, the expected return can be determined as the arithmetic mean of the returns as indicated in Eq. (4.1). There are several ways to estimate volatility or fluctuations in asset returns.3 One of the most commonly used measures is the variance or standard deviation (volatility). Variance measures the average squared deviation of individual returns from the expected return. The standard deviation is the square root of the variance and thus represents a standardised version of the variance. Assuming probabilities and returns for each scenario, the variance of an asset can be calculated as follows: n

pi ½ r i - E ð r Þ  2 ,

σ2 =

ð4:3Þ

i=1

where E(ri) = expected return for scenario i calculated using Eq. (4.2). Example: Calculation of Expected Return and Risk of an Asset Using Prospective Scenario Analysis Financial analysts at a bank come to the conclusion that the business cycle in the next period is likely to be equally divided into the following four phases: boom, stagnation, recession, and depression. Moreover, they expect the returns of an equity security to follow the business cycle. The expected stock returns for each cycle are listed below: Business cycle Boom Stagnation Recession Depression

Expected stock return (in %) 30 15 5 –10

What is the expected return and the standard deviation of the returns for the equity security? Solution The expected return of the equity security is 10% and can be calculated as follows: Eðr Þ = 0:25 × 30% þ 0:25 × 15% þ 0:25 × 5% þ 0:25 × ð- 10%Þ = 10%: The standard deviation is 14.58% and can be determined with the following equation: (continued)

3

For the risk calculation of a single asset, see Chap. 2.

104

4

σ=

0:25 × ð0:30 - 0:10Þ2 þ0:25 × ð0:15 - 0:10Þ2 þ0:25 × ð0:05 - 0:10Þ2 þ0:25 × ð - 0:10 - 0:10Þ2

Efficient Risky Portfolios

= 14:58%:

Using historical returns that occur with an equal probability of 1/T, the variance and standard deviation of a sample can be calculated as follows: σ~2 =

σ~ =

1 T -1

1 T -1

T

½ r t - E ð r Þ 2 ,

ð4:4Þ

½r t - E ðr Þ2 ,

ð4:5Þ

t=1

T t=1

where E(r) = expected return calculated as the arithmetic mean of the historical returns from the sample (see Eq. 4.1), and T = number of periods or returns. The greater the variance or standard deviation, the greater the fluctuations of the individual returns around the expected return and the greater the uncertainty or risk of the asset. Table 4.1 presents the continuous compounded monthly returns and the expected return and standard deviation of Mercedes-Benz Group AG and Linde AG stocks, based on historical data relating to monthly prices from the end of July 2012 to the end of July 2017. The expected return of a portfolio consisting of two risky assets 1 and 2 can be calculated as the sum of the weighted expected returns of the two assets as follows: Table 4.1 Monthly returns of Mercedes-Benz Group AG and Linde AG stocks for monthly prices from the end of July 2012 to the end of July 2017 (Source: Refinitiv Eikon) Time period August 2012 September 2012 October 2012 ... May 2017 June 2017 July 2017 Expected return Standard deviation (volatility)

Monthly return of Mercedes-Benz Group stock (in %) –3.60 –3.48 –4.45 ... –5.70 –1.94 –6.65 0.64 7.53

Monthly return of Linde stock (in %) 3.41 6.75 –3.22 ... 2.99 –2.47 –2.44 0.48 5.55

4.2 Expected Return and Risk of a Two-Asset Portfolio

105

Eðr P Þ = w1 E ðr 1 Þ þ w2 E ðr 2 Þ,

ð4:6Þ

where w1 = market value weight of asset 1 in the portfolio, and E(r1) = expected return on asset 1 calculated using Eqs. (4.1) or (4.2). The sum of the asset weights is 1 (w1 + w2 = 1). It does not matter whether the value of the portfolio is EUR 1000 or EUR 100,000. Rather, the crucial factor is what proportion each asset makes up of the overall portfolio. For example, the expected monthly return on a portfolio consisting of 60% Mercedes-Benz Group shares and 40% Linde shares would be 0.576%: Eðr P Þ = 0:6 × 0:64% þ 0:4 × 0:48% = 0:576%: Portfolio risk is not calculated as the sum of the weighted standard deviations of the returns. Rather, in addition to the weights and the individual risk of the two assets (i.e. the standard deviation), portfolio risk also depends on the covariance or correlation, which measures the relationship between the returns of two assets. Covariance is a measure of the degree to which two variables ‘move together’ relative to their individual mean values over time. In portfolio analysis, a positive covariance means that, over a specified period of time, the majority of the return deviations of two assets move in the same direction. By contrast, a negative covariance indicates that the return deviations of a pair of assets move in the opposite direction.4 The magnitude of the covariance thus depends on the spreads of the returns and can be calculated on the basis of prospective scenarios as follows: n

cov1,2 =

pi ½r i,1 - E ðr 1 Þ½r i,2 - Eðr 2 Þ,

ð4:7Þ

i=1

where ri,1 = return on asset 1 for scenario i, and ri,2 = return on asset 2 for scenario i. Example: Calculation of Covariance with Prospective Scenario Analysis Financial analysts at a bank come to the conclusion that the business cycle in the next period is likely to be equally divided into the following four phases: boom, stagnation, recession, and depression. They expect the returns of stock (continued)

4

See Miller (2012): Mathematics and Statistics for Financial Risk Management, p. 56.

106

4

Efficient Risky Portfolios

A to follow the business cycle, while the returns of stock B are not affected by the economic cycle. The expected returns for the two securities are presented below: Business cycle Boom Stagnation Recession Depression

Return of stock A (in %) 30 15 5 –10

Return of stock B (in %) 10 –15 20 10

What is the covariance? Solution First, the expected return of the two securities must be calculated: E ðr A Þ = 0:25 × 30% þ 0:25 × 15% þ 0:25 × 5% þ 0:25 × ð- 10%Þ = 10%, E ðrB Þ = 0:25 × 10% þ 0:25 × ð- 15%Þ þ 0:25 × 20% þ 0:25 × 10% = 6:25%: The covariance is –0.004375 and can be determined as follows: covA,B = 0:25 × ð0:30 - 0:10Þ × ð0:10 - 0:0625Þ þ 0:25 × ð0:15 - 0:10Þ × ð- 0:15 - 0:0625Þ þ 0:25 × ð0:05 - 0:10Þ × ð0:20 - 0:0625Þ þ 0:25 × ð- 0:10 - 0:10Þ × ð0:10 - 0:0625Þ = - 0:004375: The covariance from a sample of historical returns can be calculated with the following equation: cov1,2 =

1 T -1

T

½r t,1 - E ðr 1 Þ½r t,2 - Eðr 2 Þ:

ð4:8Þ

t=1

The sample covariance is the average of the product of the return deviations of two assets from their expected (mean) sample returns. The unit of covariance is returns squared.5 The covariance is divided by T - 1 rather than by T to correct for the estimation error of the sample relative to the population of the data.6 The covariance is positive if the returns of asset 1 exceed their expected value whenever the returns of asset 2 also exceed their expected value, and if the returns of asset 1 are below their expected value whenever this is also the case with asset 2. In a scatter plot, most of the return deviation points for the two assets are located in the northeast and southwest quadrants, as illustrated in Fig. 4.1. If such a directional relationship can be established, the covariance is positive. By contrast, the covariance is negative whenever the returns of asset 1 are consistently above (below) their 5 6

If the random variables are returns, then the unit of covariance is returns squared. See Sect. 2.2.

4.2 Expected Return and Risk of a Two-Asset Portfolio

107

[rt, 1 – E(r1)] +

Negative covariance

Positive covariance

–

+ [rt, 2 – E(r2)]

Positive covariance

Negative covariance

– Fig. 4.1 Scatter plot for the covariance (Source: Own illustration)

expected value when the returns of asset 2 are below (above) their expected value. The majority of the return deviation points for the two assets are located in the northwest and southeast quadrants of the scatter plot. With a covariance of zero, there is no linear relationship between the return deviations of the two assets. In such a case, no conclusions can be drawn about the returns of asset 2 if the returns of asset 1 are above or below their expected value. The points of deviation in the returns of the two assets are evenly distributed in the scatter plot. For example, the covariance for the Mercedes-Benz Group and Linde stocks is 0.002049 (according to the historical return data in Table 4.1). A positive covariance means that the majority of the return deviations of two securities move in the same direction. However, the magnitude of the covariance is difficult to interpret because, like the variance, it embodies the square of the deviations. The covariance depends on the return variability of the two stocks. If the relationship between the returns of two assets is stable then, with a covariance of 0.002049 converging to 0, there is a weak positive relationship between the returns. On the other hand, if the returns are volatile—that is, they are far apart—the relationship may be strong or weak. To

108

4

Efficient Risky Portfolios

Table 4.2 Matrix for calculating the variance of a two-asset portfolio (Source: Own illustration) Asset 1

1 w21 σ 22

2 w1w2cov1, 2

2

w2w1cov2, 1

w22 σ 22

determine the strength of the relationship, the covariance must be standardised by including the standard deviation. This involves dividing the covariance by the standard deviation of the two securities. The standardised covariance or correlation coefficient (ρ1,2) can be calculated as follows: ρ1,2 =

cov1,2 , σ1σ2

ð4:9Þ

where σ 1 = standard deviation of returns on asset 1, and σ 2 = standard deviation of returns on asset 2. For the Mercedes-Benz Group and Linde stocks, the correlation coefficient would be 0.49 (according to the return data in Table 4.1): ρMercedes - Benz Group, Linde =

0:002049 = 0:49: 0:0753 × 0:0555

The sign of the correlation coefficient depends on the sign of the covariance since the standard deviations of the two securities are always positive. The correlation coefficient is a measure that is normalised to the range of values from 1 to -1. A correlation coefficient of 1 means that the returns move together in a completely linear manner. It indicates a perfect positive relationship between two return series such that when one stock’s return is above its mean, the other stock’s return will be above its mean by the comparable amount. A correlation coefficient of -1 indicates a perfect negative relationship and implies that the returns of two securities move in an entirely opposite direction. If the correlation coefficient has a value of 0, there is no linear relationship between the returns, and the returns of the two assets are therefore uncorrelated.7 To determine the risk of a two-asset portfolio, the portfolio variance can be divided into the individual risk factors. This produces a matrix consisting of the weighted variances and the weighted covariance of the two assets. This relationship is illustrated in Table 4.2. The sum of these risk components leads to the variance of the portfolio:

7

See Miller (2012): Mathematics and Statistics for Financial Risk Management, p. 56.

4.2 Expected Return and Risk of a Two-Asset Portfolio

σ 2P = w21 σ 22 þ w22 σ 22 þ w1 w2 cov1,2 þ w2 w1 cov2,1 = w21 σ 22 þ w22 σ 22 þ 2w1 w2 cov1,2 :

109

ð4:10Þ

The portfolio variance can be converted into the portfolio standard deviation using the root function: σP =

w21 σ 22 þ w22 σ 22 þ 2w1 w2 cov1,2 :

ð4:11Þ

The formula demonstrates that the standard deviation of portfolio returns is a function of the squared weighted variances of the individual assets and the weighted covariance of a pair of assets in the portfolio. Accordingly, the standard deviation of portfolio returns depends on the variances of the individual assets and the covariance of the two assets. For the given variances, a positive covariance leads to a higher standard deviation of the portfolio than a negative covariance. A portfolio consisting of 60% Mercedes-Benz Group shares and 40% Linde shares has a standard deviation of monthly returns of 5.93%: σP =

0:62 × 0:07532 þ 0:42 × 0:05552 þ 2 × 0:6 × 0:4 × 0:002049 = 5:93%:

The two stocks of Mercedes-Benz Group and Linde have a positive covariance of 0.002049. The diversification effect (i.e. the risk reduction) is higher with a negative than with a positive covariance, since the third term to the right of the equals sign in Eq. (4.11) (2w1w2cov1, 2) becomes negative with a negative covariance and thus results in a comparatively lower portfolio risk. The question that now arises is how high the positive covariance may be for a diversification effect still to exist. In answering this question, the covariance is an unsuitable risk measure to use as its level is difficult to interpret. Therefore, the covariance must be standardised and converted into the correlation coefficient. The correlation coefficient measures the degree of the relationship between the returns of two assets. If Eq. (4.9) is taken for the correlation coefficient and this equation is solved for the covariance, the following formula is obtained for the covariance: cov1,2 = ρ1,2 σ 1 σ 2 ,

ð4:12Þ

where ρ1,2 = correlation coefficient between the returns on asset 1 and asset 2. If the covariance (cov1,2) in Eq. (4.11) is replaced by the product of the correlation coefficient and the standard deviations of the individual assets (ρ1, 2σ 1σ 2), the portfolio risk can be determined with the following equation: σP =

w21 σ 22 þ w22 σ 22 þ 2w1 w2 ρ1,2 σ 1 σ 2 :

ð4:13Þ

110

4

Efficient Risky Portfolios

All else being equal, a higher correlation coefficient leads to a higher portfolio risk. Given a correlation coefficient of 1 (i.e. perfectly positively correlated returns), the calculation of portfolio risk is simplified as follows:8 σ P = w1 σ 1 þ w2 σ 2 :

ð4:14Þ

Hence, with a correlation coefficient of 1, the portfolio risk is equal to the sum of the weighted standard deviations of the individual assets. The construction of a portfolio with such assets does not lead to a reduction in the portfolio risk and thus there is no diversification effect. In general, the portfolio risk of N risky assets can be determined as the sum of the weighted standard deviations as follows: N

σP =

wi σ i ,

ð4:15Þ

i=1

where N = number of risky assets in the portfolio. Taking a correlation coefficient of 1 for the portfolio consisting of 60% Mercedes-Benz Group shares and 40% Linde shares and inserting it into Eq. (4.13) results in a value of 6.738% for the portfolio risk: σ P = 0:62 × 0:07532 þ 0:42 × 0:05552 þ 2 × 0:6 × 0:4 × 1 × 0:0753 × 0:0555 = 6:738%: If Eq. (4.14) is used to calculate the portfolio risk, the result is also 6.738%: σ P = 0:6 × 7:53% þ 0:4 × 5:55% = 6:738%: Equations (4.13) and (4.14) lead to the same portfolio risk of 6.738%. Hence, with a correlation coefficient of 1, the risk is additive. On the other hand, taking a correlation coefficient that is lower than 1 reduces the portfolio risk according to Eq. (4.13). Thus, there is always a diversification effect or a reduction in portfolio risk if the correlation coefficient between the returns of the two assets is less than 1. The lowest portfolio risk is obtained with a correlation coefficient of –1, that is, with perfectly negatively correlated returns of two assets. The variance and standard deviation of a portfolio consisting of a pair of assets with a correlation coefficient of -1 are calculated as follows:9

8

If the correlation coefficient is 1, then Eq. (4.13) can be written as portfolio variance in the following way: σ 2P = w21 σ 22 þ w22 σ 22 þ 2w1 w2 σ 1 σ 2 . The portfolio variance can be converted using the first binomial formula as follows: σ 2P = ðw1 σ 1 þ w2 σ 2 Þ2 . If the variance is converted into the standard deviation with the root function, Eq. (4.14) is obtained. 9 If the correlation coefficient is -1, the portfolio variance can be calculated with the following equation: σ 2P = w21 σ 22 þ w22 σ 22 - 2w1 w2 σ 1 σ 2 . The portfolio variance can then be transformed as

4.2 Expected Return and Risk of a Two-Asset Portfolio

111

σ 2P = ðw1 σ 1 - w2 σ 2 Þ2 ,

ð4:16Þ

σ P = jw1 σ 1 - w2 σ 2 j:

ð4:17Þ

With a correlation coefficient of -1, the risk of a two-asset portfolio can be completely eliminated, and Eq. (4.17) can be written as follows: 0 = w1σ 1 - w2σ 2. The weights of the two assets leading to a risk-free portfolio can be determined as follows:10 σ2 , σ1 þ σ2

ð4:18Þ

σ1 = 1 - w1 : σ1 þ σ2

ð4:19Þ

w1 = w2 =

Example: Expected Return of a Two-Asset Portfolio with a Risk of Zero Stock A has an expected return of 10% and a standard deviation of returns of 14.58%, while the expected return and standard deviation of returns of stock B are 6.25% and 12.93%, respectively. Assuming the returns of the two equity securities are completely negatively correlated, the correlation coefficient is 1. What is the expected return of the portfolio if the standard deviation of portfolio returns is 0%? Solution First, the weight of stock A for a two-asset portfolio with a risk of zero is calculated: wA =

σB 0:1293 = = 0:47: σ A þ σ B 0:1458 þ 0:1293

The weight of stock B equals 0.53 (= 1 - 0.47). Inserting the weights of the two securities of 47% and 53% in Eq. (4.13) leads to a portfolio risk of 0%: (continued)

follows: σ 2P = ðw1 σ 1 - w2 σ 2 Þ2 . The vertical bars in Eq. (4.17) mean that the portfolio risk cannot fall below zero. 10 The weight of asset 2 is w2 = 1 - w1, since the sum of the weights is 1. Substituting in Eq. (4.17) 1 - w1 for w2 and setting the equation equal to 0 gives w1σ 1 - (1 - w1)σ 2 = 0. If this equation is solved for w1, Eq. (4.18) results.

112

4

Efficient Risky Portfolios

(Expected return)

Mercedes-Benz Group 0.64% 0.576%

Portfolio Linde

0.48%

0% 0%

5.55% 7.53% 5.93% (Standard deviation of returns)

Fig. 4.2 Expected return and risk of Mercedes-Benz Group stock, Linde stock, and a portfolio comprising the two securities (Source: Own illustration)

σP =

0:472 ×0:14582 þ0:532 ×0:12932 þ2×0:47×0:53× ð -1Þ×0:1458×0:1293

= 0%: The expected return on this risk-free portfolio is 8.0125%: Eðr P Þ = 0:47 × 10% þ 0:53 × 6:25% = 8:0125%:

4.3

The Efficient Frontier

Figure 4.2 presents the expected returns and standard deviations of returns of Mercedes-Benz Group stock and Linde stock in a risk–return diagram. The Mercedes-Benz Group stock has a higher expected return and risk than the Linde

4.3 The Efficient Frontier

(Expected return)

113

Efficient frontier between MVP and Mercedes-Benz Group

MercedesBenz Group

0.64% 1

0.52% 0.48%

MVP

1‘

2 2‘ Linde

5.34% 5.55%

7.53% (Standard deviation of returns)

Fig. 4.3 Expected return and risk of different portfolio combinations with Linde and MercedesBenz Group stocks (Source: Own illustration)

stock.11 The figure also indicates an expected monthly return of 0.576% and a standard deviation of the monthly returns of 5.93% (with a correlation coefficient of 0.49) of the portfolio consisting of 60% Mercedes-Benz Group shares and 40% Linde shares. A portfolio can be constructed using any proportions (weights) of the two equity securities. Figure 4.3 presents a portfolio curve on which all possible portfolio combinations of the two stocks of Mercedes-Benz Group and Linde lie. These portfolios are based on the previously calculated correlation coefficient of 0.49. Portfolio 1 consists of 10% Linde shares and 90% Mercedes-Benz Group shares. This combination, which consists almost entirely of the automobile security, lies on the portfolio curve close to the risk–return point of the Mercedes-Benz Group stock.

11 Markowitz’s portfolio theory is based on the assumption that investors behave in a risk-averse manner. Due to the higher standard deviation of 7.53% (Linde: 5.55%), the Mercedes-Benz Group stock has a higher expected return than the Linde stock, namely 0.64% (Linde: 0.48%). See Table 4.1.

114

4

Efficient Risky Portfolios

A diversification effect exists whenever the correlation coefficient between the returns of two assets is less than 1. The correlation coefficient between the returns of the stocks of Mercedes-Benz Group and Linde is 0.49. If the straight line between the risk–return points of Linde and Mercedes-Benz Group is compared with the portfolio curve Linde–MVP–Mercedes-Benz Group, the diversification effect can be demonstrated. Assuming that the correlation coefficient between the returns of the two stocks is 1 instead of 0.49, then all combinations of the two securities are located on the straight line between the risk–return points of Linde and Mercedes-Benz Group. An example of this is portfolio 1' (with a correlation coefficient of 1), which consists of 10% Linde shares and 90% Mercedes-Benz Group shares. By contrast, all combinations of Linde and Mercedes-Benz Group stocks with a correlation coefficient of 0.49 lie on the portfolio curve Linde–MVP–Mercedes-Benz Group. A comparison of the two portfolios 1 and 1' in Fig. 4.3 reveals that both have the same expected portfolio return, while the standard deviation of portfolio 1 is lower due to the correlation coefficient of 0.49, which is less than 1. The risk–return point MVP in Fig. 4.3 represents the minimum variance portfolio. This portfolio has the lowest standard deviation of all possible combinations of the Linde and Mercedes-Benz Group stocks.12 The weight of an asset 1 in the minimum variance portfolio, which consists of assets 1 and 2, can be calculated using the following formula:13 w1 =

σ 21

σ 22 - cov1,2 : þ σ 22 - 2cov1,2

ð4:20Þ

The weight of Mercedes-Benz Group shares in the minimum variance portfolio of 22.17% can be determined as follows: wMercedes - Benz Group =

0:05552 - 0:002049 = 0:2217: 0:07532 þ 0:05552 - 2 × 0:002049

The sum of the weights is 1, and therefore, the weight of Linde shares equals 77.83% (= 1 - 0.2217). The minimum variance portfolio has an expected return of 0.52% and a standard deviation of returns of 5.34%: E ðr MVP Þ = 0:2217 × 0:64% þ 0:7783 × 0:48% = 0:52%,

12 This risk–return point should actually be called the minimum standard deviation portfolio, since the standard deviation, not the variance, is related to the return. See Roy (1952): ‘Safety first and the holding of assets’, p. 431 ff. 13 Equation (4.20) can be derived as follows: First, the formula for portfolio variance is taken σ 2P = w21 σ 21 þ w22 σ 22 þ 2w1 w2 ρ1,2 σ 1 σ 2 and the variable w2 is replaced by (1 - w1). Then the equation is derived according to w1 and this is set equal to 0. If the equation is solved for w1, Eq. (4.20) is obtained.

4.3 The Efficient Frontier

115

σ MVP = 0:22172 × 0:07532 þ 0:77832 × 0:05552 þ 2 × 0:2217 × 0:7783 × 0:002049 = 5:34%: An investor who invests in Linde and Mercedes-Benz Group stocks can achieve any risk–return point on the Linde–MVP–Mercedes-Benz Group portfolio curve by changing the weights of the two securities. However, it is not possible for them to achieve a risk–return point above or below the portfolio curve. A less risk-averse investor chooses investments with a high expected return and standard deviation, such as portfolio 1. On the other hand, a more risk-averse investor prefers an investment that has a lower standard deviation, such as portfolio 2. If an investor seeks the lowest possible risk from his investments, they select the minimum variance portfolio. Combinations of Linde and Mercedes-Benz Group stocks that result in a risk–return point on the portfolio curve below the minimum variance portfolio are not advantageous in terms of expected return and risk. Such portfolios have a lower expected return and a higher standard deviation than the minimum variance portfolio. The section of the curve between the risk–return points MVP and Mercedes-Benz Group is called the efficient frontier. Rational investors who behave in a risk-averse manner14 will only invest in portfolios that lie on the efficient frontier.15 For investments that are not perfectly positively or negatively correlated, the portion of the portfolio curve that lies above the minimum variance portfolio is concave. In Fig. 4.3, the concave part of the portfolio curve represents the efficient frontier (MVP–Mercedes-Benz Group). By contrast, the section of the portfolio curve below the minimum variance portfolio is convex (Linde–MVP).16 Figure 4.4 presents various portfolio curves for different correlations. The lower the correlation coefficient between the returns of two assets, the greater the diversification effect. The portfolio curve moves to the left along the X-axis due to the lower portfolio risk. In the extreme case, with a correlation coefficient of -1, a riskfree portfolio can be constructed. Equations (4.18) and (4.19) can be used to calculate the weights of the two assets in the portfolio. This case is rather theoretical, because in the financial markets most pairs of stocks have correlation coefficients that are at least greater than -1 and regularly even in the positive range.17

14

For the concept of risk aversion, see Sect. 5.2.1. Compared to portfolio 2, the portfolio 2' in Fig. 4.3 is not efficient and therefore does not lie on the efficient frontier because, for the same risk, the expected return of portfolio 2 is greater. 16 A concave curve is always created when a straight line passes through any two points below the curve. A convex curve, on the other hand, is characterised by a straight line between any two points that lie above the curve. A linear risk–return relationship between two assets (i.e. a correlation coefficient of -1 and 1) is both concave and convex. 17 See D’Ambrosio (1990): Portfolio Management Basics, pp. 2–21. 15

116

4

(Expected return)

Efficient Risky Portfolios

U = −1 U = −0.5 U=0 U = 0.5 U=1

0% 0%

(Standard deviation of returns)

Fig. 4.4 Portfolio curves with different correlation coefficients (Source: Own illustration)

4.4

Expected Return and Risk of a Portfolio Consisting of Many Risky Assets

Thus far, only a portfolio consisting of two risky assets has been considered. However, investors usually hold more than two assets in their portfolios. For example, a bank’s investment list may consist of 100 equity securities, from which a variety of portfolios can be formed by varying the number and weightings. In Fig. 4.5, risk–return point 1 may represent a portfolio of 50 securities, while risk– return point 2 consists of an investment combination of 70 securities. Risk–return point 3, on the other hand, contains a different portfolio consisting of 70 securities or the same 70 securities as in risk–return point 2, but with a different percentage mix. It is not possible to construct a portfolio with the 100 equity securities from the investment list whose expected return and standard deviation lie outside the area. If a portfolio contains only two securities, all possible combinations lie on a portfolio curve. On the other hand, if a portfolio is created with several risky assets, all possible portfolios are located within an area, as illustrated in Fig. 4.5. A riskaverse investor will select only those investments that lie on the concave curve between the minimum variance portfolio MVP and X. This portfolio curve represents the efficient frontier for a portfolio consisting of a large number of long risky assets. Each portfolio below the efficient frontier has either a lower expected return with the same risk or an equal expected return with higher risk than a portfolio

4.4 Expected Return and Risk of a Portfolio Consisting of Many Risky Assets

117

(Expected return) Efficient frontier between MVP and X

X

A B MVP

1

2

3

0% 0%

(Standard deviation of returns)

Fig. 4.5 Efficient frontier of portfolios consisting of long risky assets (Source: Own illustration)

on the efficient frontier. Portfolio B, for example, has a lower expected return than portfolio A, with the same risk. Therefore, a risk-averse investor prefers portfolio A, which lies on the efficient frontier. Those portfolios that have the maximum expected return for each level of risk lie on the efficient frontier. Conversely, only portfolios that have a minimum risk for each level of expected return are located on the efficient frontier. In the light of the diversification effect, it is reasonable to assume that the efficient frontier is composed of portfolios rather than individual assets. An exception is the end point of the efficient frontier for a portfolio consisting of long assets, which contains the asset with the highest expected return and risk.18 Investors can sell short securities that they do not own. In doing so they create short positions by borrowing securities from security lenders who are long holders. The short sellers then sell the borrowed securities to other traders. Short sellers close their positions by repurchasing the securities and returning them to the security lenders. If the securities drop in value, the short sellers make a profit because they repurchase the securities at lower prices than the prices at which they sold them.19 18

The highest risk–return point on the efficient frontier can be achieved in a long asset portfolio (i.e. without the possibility of taking short positions), buying the asset with the highest expected return and risk. Due to the risk aversion assumption, the asset with the highest expected return exhibits the highest risk. 19 For example, if an investor sells an equity security short for EUR 100 and buys the security on the market at a later date for EUR 90, the result is a profit of EUR 10. If a dividend of EUR 2 per share is paid during the open short position, the profit is reduced to EUR 8 (= EUR 10 - EUR 2).

118 Fig. 4.6 Efficient frontier of portfolios consisting of long and short positions (Source: Own illustration)

4

Efficient Risky Portfolios

(Expected return) No upper bound

Efficient frontier

MVP 0% 0%

(Standard deviation of returns)

On the other hand, if the securities rise in value, they will make a loss. It is also worthwhile to sell short securities when returns are expected to be positive, if the proceeds from the short sale can be used to buy assets with a higher expected return. Figure 4.6 illustrates the efficient frontier of portfolios with long and short positions; it has a concave shape, as in the case of a portfolio consisting only of long assets. The efficient frontier starts with the minimum variance portfolio and, in contrast to the efficient frontier of a long portfolio, does not have a specific end point. The lack of an upper bound on the efficient frontier is due to the possibility of combining long and short positions. Long and short positions can be used to construct portfolios with unlimited expected returns and risks. The following example demonstrates the impact of a long–short equity strategy compared to a long equity strategy on expected return and risk. An investor has EUR 100 that he wants to invest in shares of stocks A and B. Stock A has an expected return of 4% and a standard deviation of 20%, while the expected return and standard deviation of stock B are 12% and 30%, respectively. The correlation coefficient between the returns of the two securities is 0.5. The investor can buy stock B with the EUR 100 and achieve a return of 12% (long strategy). An alternative strategy is to sell shares of A short for EUR 1000 and buy shares of B for EUR 1100 (long-short strategy). The expected income from buying shares of stock B is EUR 132 (= EUR 1100 × 0.12), while the expected loss on shares of stock A is EUR 40 (= EUR 1000 × 0.04). Overall, the long–short strategy, excluding transaction costs, results in a profit of EUR 92 (= EUR 132 – EUR 40), which corresponds to an expected return of 92% on the invested capital of EUR 100. It should be noted that while the expected return has risen from 12% to

4.4 Expected Return and Risk of a Portfolio Consisting of Many Risky Assets

119

92%, the risk has increased from 30% to 287.92%.20 The expected return and the risk can be increased at will by raising the short and long weights. The calculation of the expected return and standard deviation of many assets turns out to be more complex than the two-asset portfolio. The expected return of a portfolio composed of N risky assets can be determined as the sum of the weighted returns as follows: N

E ðr P Þ =

wi E ðr i Þ,

ð4:21Þ

i=1

where N = number of assets in the portfolio, wi = market value weight of asset i in the portfolio, E(ri) = expected return on asset i, and N

wi = 1: i=1

Portfolio risk is not simply the sum of the weighted standard deviations of the individual assets, because the risk of loss is also influenced by the covariance or correlation coefficient. The standard deviation of a portfolio consisting of a large number of risky assets can be calculated using the following formula: N

σP =

w2i σ 2i þ 2 i=1

N -1

N

ð4:22Þ

wi wj covi,j , i = 1 j = iþ1

where covi,j = ρi,j σ i σ j : In this formula, long assets have a positive weight (w > 0), while short assets have a negative weight (w < 0). The sum of the weights is 1. The portfolio risk is equal to the sum of the weighted variances of the individual assets plus the sum of the weighted covariances between all pairs of assets in the portfolio. The representation in a matrix (see Table 4.3) assists in understanding the calculation of portfolio risk using Eq. (4.22). A portfolio contains N stocks. The

The weight of short position A is –1000% (= - EUR 1000/EUR 100), while the weight of long position B is 1100% (= EUR 1100/EUR 100). The sum of the weights is 100% (= -1000% +

20

1100%).

The

portfolio

risk

of

287.92%

can

be

calculated

as

ð - 10Þ2 × 0:22 þ 112 þ 0:32 þ 2 × ð - 10Þ × 11 × 0:5 × 0:2 × 0:3 = 287:92%:

follows:

σP =

120

4

Efficient Risky Portfolios

Table 4.3 Matrix for calculating the portfolio variance (Source: Own illustration) ...

Stock 1

1 w21 σ 21

2 w1w2cov1,2

3 w1w3cov1,3

N w1wNcov1,N

2 3

w2w1cov2,1 w3w1cov3,1

w22 σ 22 w3w2cov3,2

w2w3cov2,3 w23 σ 23

w2wNcov2,N w3wNcov3,N

... N

wNw1covN,1

wNw2covN,2

wNw3covN,3

w2N σ 2N

securities can be numbered from 1 to N on both the horizontal and vertical axes. Thus, an N × N = N2 matrix is obtained. The variances of the individual securities appear on the diagonal. For example, the variance of the first security in the portfolio is σ 21 . Each covariance of a pair of stocks appears twice in Table 4.3, once below and once above the diagonal. The number of diagonal terms (i.e. the variances) always corresponds to the sum of the N securities included in a portfolio. The covariance terms appear below and above the diagonal, and their number increases disproportionately compared to the variances. In total, the matrix contains N2 terms. The N variances appear on the diagonal. If the N variances are subtracted from the N2 terms N2 – N or N(N - 1) covariance terms are obtained. Since the same covariances lie above and below the diagonal, the number of covariances required is reduced to N(N - 1)/2. For example, calculating the risk of a portfolio consisting of 100 stocks requires 100 variances and 4950 covariances.21 This example reveals that it is primarily the covariances and not the variances that affect the amount of portfolio risk. In a portfolio of 100 securities, the risk is composed of 4950 covariances and only 100 variances. The primary risk driver is therefore the covariance or the correlation coefficient and not the variance. Table 4.4 illustrates how the number of covariances to be determined increases disproportionately with an increase in the number of securities in a portfolio. In order to construct the efficient frontier from N risky assets, minimum and maximum expected returns—that is, E(rmin) and E(rmax)—must first be determined.22 The next step is to calculate the weights of each asset in the portfolio that minimise the portfolio risk for all possible values of the expected returns between E(rmin) and E(rmax). Mathematically, the following objective function must be solved for the Z values (given expected portfolio returns) that lie between E(rmin) and E(rmax): The objective function for a given value of Z, namely to:

Number of covariances = 100 × (100 - 1)/2 = 4950. E(rmax) is unlimited for a portfolio with long and short assets. Consequently, the highest expected return must be chosen arbitrarily. By contrast, a portfolio consisting of long assets has a maximum expected return given by the riskiest asset with the highest expected return. 21 22

4.4 Expected Return and Risk of a Portfolio Consisting of Many Risky Assets

121

Table 4.4 Number of variances and covariances in a portfolio (Source: Own illustration) Number of securities in the portfolio 1 2 3 10 100 ... ... ... N

Number of terms in the matrix 1 4 9 100 10,000 ... ... ... N2

Number of variance terms (on the diagonal in the matrix) 1 2 3 10 100 ... ... ... N

N

minimise σ 2P by changing w =

i=1

w2i σ 2i þ 2

Number of covariance terms (below and above the diagonal in the matrix) 0 2 6 90 9900 ... ... ... N2 - N

N -1

N

wi wj ρi,j σ i σ j ,

i = 1 j = iþ1

is subject to the following constraints: N

E ðr P Þ =

wi E ðr i Þ = Z, i=1

and N

wi = 1:

ð4:23Þ

i=1

This optimisation problem determines the portfolio weights (w1, w2, w3, ..., wN) such that the variance of the returns is minimised for each given level of the expected portfolio return Z, where the sum of the weights equals 1. The weights define the portfolio with the lowest risk for the given return value Z. Thus, the varianceminimised portfolio is obtained for each expected return value Z. This portfolio corresponds to a risk–return point on the efficient frontier. If this is repeated for all possible values of Z, a series of risk–return points representing the entire efficient frontier are obtained. Equation (4.23) indicates the simplest case where the sum of the weights equals 1. Both long and short positions are allowed. If short positions are not permitted due to the investment policy, there is a further restriction, namely that the weights must be positive (wi ≥ 0, i = 1, ..., N ). The efficient frontier is constructed starting from the lowest to the highest expected return. The calculation of the optimal portfolio weight begins with the minimum expected return [E(rmin)] as the Z value, which is the smallest expected return value of all investments. The next step is to increase the Z-value by, say, 5 basis points (i.e. 0.0005) and solve Eq. (4.23) for the optimal portfolio weight. This step is repeated until the Z-value

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Efficient Risky Portfolios

(Expected return) 30% Efficient frontier 25% 20% 15% 10% 5% 0% 10%

15%

20%

25%

30%

(Standard deviation of returns) Fig. 4.7 Efficient frontier for the five DAX 40 stocks of Mercedes-Benz Group AG, Linde AG, Bayer AG, Siemens AG, and Adidas AG (Source: Own illustration based on data from Refinitiv Eikon)

with the maximum expected return [Z = E(rmax)] is obtained. Typically, this optimisation problem is solved with the aid of a computer program that relies on algorithms containing the objective function and constraints of Eq. (4.23) and processes them into systems of equations using the Lagrange method.23 Figure 4.7 presents the efficient frontier for the five DAX 40 stocks of MercedesBenz Group AG, Linde AG, Bayer AG, Siemens AG, and Adidas AG, which was created using Microsoft Excel. The efficient frontier based on annual expected returns was calculated with Eq. (4.23) and includes both long and short equity positions. The data used for this calculation are based on continuous compounded returns for monthly shares prices from the end of July 2012 to the end of July 2017 (Source: Refinitiv Eikon; since the beginning of March 2023, Linde shares have no longer been listed in the DAX 40). The construction of the efficient frontier is

23

Risk–return optimisation methods that are restricted to long positions use the Kuhn–Tucker approach in addition to the Lagrange method in order to process the additional constraint of positive asset weights (wi ≥ 0) in the algorithms. For possible algorithms for calculating the efficient frontier, see Markowitz (1959): Portfolio Selection: Efficient Diversification of Investments, p. 309 ff. In the development of portfolio theory, the main merit of Markowitz is the construction of the efficient frontier on the basis of expected returns, standard deviations, and correlation coefficients (diversification), as well as the calculation of the efficient frontier with algorithms.

4.5 Diversification Effect

123

described in the Microsoft Excel applications presented at the end of the chapter. Here, the portfolio variance for each target return is minimised in accordance with the Lagrange method. Excel can only be employed to build the efficient frontier for a small number of securities. Moreover, it is somewhat laborious to determine the inverse of the variance-covariance matrix using Excel.

4.5

Diversification Effect

Prices of equity securities change as a result of company- and market-specific factors. Company-specific factors influencing the share price include, for example, a new chief executive officer (CEO), the launch of a new product or service, the development of better technology by competitors in the market, damage to a company’s reputation due to criminal activity or unethical behaviour by management, and so forth. These are idiosyncratic factors that emerge from the company (or industry) and affect the share price. In addition, there are several factors that are unrelated to the company. These include macroeconomic events such as unexpected changes in interest rates, inflation rate, and growth rate of gross domestic product, or political events such as presidential and parliamentary elections. These crosseconomic (market-related) factors affect the share price of all companies at the same time. The majority of the share prices move in the same direction as the equity market. These idiosyncratic factors influencing the share price are usually referred to as unsystematic risk factors. By contrast, the cross-economic events that are exogenous and cannot be attributed to the company or industry are known as systematic risk factors (market risk). The price volatility of an investment is thus affected by both unsystematic and systematic risk factors. In portfolio theory, this division of overall risk into unsystematic and systematic parts is used to explain the diversification effect resulting from portfolio construction. As investors are generally risk-averse, they are interested in reducing risk, preferably without reducing return. The diversification effect of a portfolio consisting of long assets is best illustrated by the following example. A portfolio contains equity securities that all have the same average variance ðσ 2 Þ and covariance ðcovÞ. In addition, all securities have the same weight in the portfolio. Under these assumptions—following the matrix for calculating the portfolio variance (see Table 4.3)—all terms on the diagonal (the variances) and the terms above and below the diagonal (the covariances) are equal. Accordingly, the variance of the portfolio is calculated from the sum of the terms in the matrix as follows: σ 2P = N

1 1 σ 2 þ N ðN- 1Þ 2 cov: 2 N N

ð4:24Þ

The portfolio variance consists of the sum of the weighted variances and covariances. Dividing the N variances by the number of terms in the matrix of N2 yields the percentage of variances in the portfolio. The percentage of covariances, on

124

4

Efficient Risky Portfolios

the other hand, is calculated by the number of covariances of N(N – 1) divided by the number of terms in the matrix of N2. Multiplying the terms in Eq. (4.24), the following equation for the portfolio variance is obtained: σ 2P =

1 2 N -1 σ þ cov: N N

ð4:25Þ

Equation (4.25) expresses the variance of this particular portfolio as the weighted sum of the average variance and covariance. If the number of assets (N ) in the portfolio is increased towards infinity, the portfolio variance tends towards the average covariance.24 This example leads to the following conclusion: The variances of the individual assets can be eliminated by diversification, while the term of the covariance remains and thus cannot be eliminated. Hence, the portfolio variance can be represented as follows: σ2P = cov þ ðσ2P - covÞ or Total portfolio risk = systematic or non - diversifiable risk þ unsystematic or diversifiable risk

ð4:26Þ

The average covariance ðcovÞ is the risk that remains in a long asset portfolio. This risk of loss is called systematic or non-diversifiable risk. By contrast, unsystematic risk or diversifiable risk σ 2P - cov can be eliminated in a large enough portfolio through diversification. Figure 4.8 illustrates the relationship between the portfolio variance and the number of long stocks in a portfolio, given a standard deviation of the individual stocks of 20% and an average covariance of 0.012. For an investor who owns a well-diversified portfolio, it is not important how high the risk (variance) of an individual asset is. Rather, what matters is by how much a new asset reduces portfolio risk, or how much that purchase contributes to diversification. An investor’s risk in such a case can be defined as the contribution of the new asset to the risk of the overall portfolio and depends on the covariance or correlation of the asset to the other assets in the portfolio. A central question in portfolio management deals with the number of risky assets required for a well-diversified portfolio. To answer this question, the average covariance in Eq. (4.25) is replaced by the product of the average correlation coefficient and the average variance,25 leading to the following equation for the portfolio variance:

If N tends to infinity, then the first term of Eq. (4.25) of (1/N )σ 2 converges to 0, while the second term of ððN- 1Þ=N Þ cov approaches the average covariance. Consequently, for a large number of assets, the portfolio variance is equal to the average covariance. 25 The covariance of two random variables is the correlation coefficient multiplied by the standard deviations of the two variables. The assumption is that all securities have the same standard deviation of returns, and the average covariance with the correlation coefficient is therefore calculated as follows: cov = ρσ 2 . 24

4.5 Diversification Effect

125

(Variance of portfolio returns) 0.045 VP2

0.04 0.035 Company-specific risk, unsystematic risk or diversifiable risk

0.03 0.025 0.02

cov

0.015 0.01

Market risk, systematic risk or non-diversifiable risk

0.005 0 0

5

10

15

20

25

30

35

40

(Number of stocks) Fig. 4.8 Relationship between portfolio variance and the number of long stocks in a portfolio. (This figure is only valid under the assumption that the risk is not additive or that the correlation coefficient between equity securities is below 1 and thus a diversification effect can be achieved) (Source: Own illustration)

σ 2P =

1 2 N -1 σ þ ρσ 2 , N N

ð4:27Þ

where σ = all assets have the same standard deviation of the returns, and ρ = same correlation coefficient between pairs of assets. When the above formula is multiplied out, the following equation for portfolio variance results:26 σ 2P = σ 2

1-ρ þρ : N

ð4:28Þ

If the portfolio consists of only one stock, the portfolio variance corresponds to the average variance of the individual security ðσ 2 Þ. However, increasing the number σP = σ2

26 2

1þðN - 1Þρ N

= σ2

1þNρ - ρ N

= σ2

1-ρ N

2 þ Nρ N =σ

1-ρ N

þρ

126

4

Efficient Risky Portfolios

of equity securities (N ) results in a lower portfolio variance. For example, if an average correlation coefficient of 0.3 and a standard deviation of returns of 20% are assumed, which is the same for all stocks, then the portfolio variance equals 0.04 for 1 stock, 0.026 for 2 stocks, 0.0148 for 10 stocks, 0.0129 for 30 stocks, 0.01228 for 100 stocks, and 0.01203 for 1000 stocks. This example demonstrates that a large part of the diversification effect can be achieved with 30 securities. If the number of equity securities in the portfolio increases from 30 to 1000, the portfolio variance can only be reduced by approximately 6.8%, whereas the reduction potential of the portfolio variance with a portfolio of 30 stocks (compared to one stock) is approximately 67.8%. Furthermore, the example reveals that the portfolio variance tends towards the average covariance of 0.012 (= 0.3 × 0.22) when the number of equity securities increases. The number of assets required to diversify a portfolio depends on the average correlation coefficient between the returns of a pair of assets. The higher the correlation, the more assets are required for a desired diversification effect in the portfolio. Empirical studies reveal—depending on the study period and the prevailing average correlation and volatility in the equity markets—that between 20 and 50 stocks are sufficient to achieve optimal diversification.27 In addition, it should be noted that diversifying a portfolio does not come free of charge. When buying securities, trading costs28 are incurred, which must be set off against the benefit gained from diversification. Example: Diversification Effect In an equity market, the average standard deviation of stock returns is 20%, while the average covariance between stocks is 0.016. 1. What is the respective portfolio variance for 1, 30, and 1000 stocks? 2. What conclusions on the diversification effect can be drawn from question 1? Solution to 1 First, the average correlation coefficient of 0.4 is calculated with the average standard deviation and covariance as follows: ρ=

cov 0:016 = = 0:4: 0:20 × 0:20 σ2

Accordingly, Eq. (4.28) can be used to determine the portfolio variance for 1, 30, and 1000 stocks as follows: (continued)

27 28

See DeFusco et al. (2004): Quantitative Methods for Investment Analysis, p. 607 ff. See Sect. 3.3.4.

4.5 Diversification Effect

σ2P, 1 stock = 0:22 × σ2P, 30 stocks = 0:22 × σ2P, 1000 stocks = 0:22 ×

127

1 - 0:4 þ 0:4 = 0:04, 1 1 - 0:4 þ 0:4 = 0:0168, 30 1 - 0:4 þ 0:4 = 0:01602: 1000

Solution to 2 If the number of equity securities in the portfolio increases by 29 (from 1 to 30 stocks), the portfolio variance falls by more than half (58%). A further increase to 1000 stocks leads only to a reduction in the portfolio variance of approximately 4.6%. If the number of equity securities increases, the portfolio variance tends towards the average covariance of 0.016. A large part of the diversification effect is achieved with 30 stocks. This number is sufficient if the average equity volatility in a market is 20% and the average correlation coefficient between pairs of stocks is 0.4. For example, Evans and Archer (1968) conducted a study on diversification and calculated the standard deviation for portfolios of increasing numbers of stocks up to 20. The results indicated a large initial impact wherein the major diversification benefits were achieved rather quickly. Specifically, approximately 90% of the maximum diversification benefit was realised from portfolios of 12 to 18 stocks.29 A later study conducted by Statman (1987) compared the diversification benefits of lower portfolio risk to the added transaction costs with more securities. The number of stocks should be increased in a portfolio as long as the marginal benefits obtained through risk reduction exceed the marginal costs in the form of transaction costs. The study concluded that a well-diversified portfolio of randomly chosen stocks must consist of at least 30 stocks for a borrowing investor and 40 stocks for a lending investor.30 Portfolio construction and hence diversification are not limited to the domestic capital market (e.g. the German or Swiss equity market). If a portfolio is created with equity securities from several countries, the diversification effect can be increased. The risk of a well-diversified portfolio of international stocks is lower than a welldiversified portfolio of stocks from a single country. For example, if an investor owns a well-diversified portfolio of German stocks, they are exposed to the See Evans and Archer (1968): ‘Diversification and the reduction of dispersion: an empirical analysis’, p. 761 ff. 30 See Statman (1987): ‘How many stocks make a diversified portfolio?’, p. 353 ff. A borrowing investor borrows money at a certain interest rate to invest more than 100% in a stock portfolio. By contrast, a lending investor allocates his money to a risk-free asset and a stock portfolio. 29

128

4

Efficient Risky Portfolios

systematic risk of the German equity market. By contrast, a portfolio of US stocks is exposed to the systematic risk of the US equity market. The factors influencing the German equity market are not perfectly positively correlated with those of the US equity market. Put differently, market-related events that have an impact on the German equity market do not play a role in the USA and vice versa. If the investor now buys US equities, they can reduce the risk of the German stock portfolio. If the portfolio consists of global equities, one is still exposed to systematic risk. The complete elimination of risk is not possible due to systematic factors such as international crises and conflicts, the oil price, and the gold price. Diversification into equities from one market can reduce the overall risk of a portfolio by eliminating the unsystematic risk. Furthermore, systematic risk can be reduced by diversification into the global equity market.31 Another important aspect of risk reduction is diversification into different asset classes. In addition to equities, a well-diversified portfolio includes bonds, real estate, commodities (e.g. gold), and so forth.32 Table 4.5 presents correlations among major US asset classes and international stocks based on historical annual returns between 1970 and 2020. The highest correlation coefficient of 0.72 can be found between US large-cap stocks and US small-cap stocks, whereas the correlation coefficient between US large-cap stocks and international stocks is slightly lower at 0.67. Despite the high correlation, there is still a diversification effect as the correlation coefficients are less than 1. The correlation coefficient between the US small-cap stocks and international stocks is lower at 0.52. However, the correlation coefficients between the equity and bond asset classes are lower—for example, 0.04 between US large-cap stocks and US long-term Treasury bonds, and -0.13 between US small-cap stocks and US long-term Treasury bonds. Thus, the highest correlation occurs between domestic equity securities (US large- and small-cap stocks). Adding international stocks to the US equity portfolio will further reduce the risk. An additional reduction in risk is obtained when corporate and Treasury bonds and Treasury bills are included in the equity portfolio. Harry Markowitz illustrated the diversification effect with the phrase, ’Don’t put all your eggs in one basket.’ Indeed, if all one’s eggs are in the same basket and it falls, the eggs will break. If each egg is placed in a separate basket, one is better protected—if the basket falls, one loses only one egg and not all of them. In the context of portfolio theory, the individual assets represent the baskets and the money invested in each asset represents the eggs.

4.6

Summary

• The expected portfolio return is the sum of the weighted individual asset returns. However, the calculation of the portfolio variance is more complicated as it depends on the variances of the individual assets, on the covariance or correlation

31 32

See Spremann (2000): Portfoliomanagement, p. 155 f. See, for example, Zisler (1990): Real Estate Portfolio Management, pp. 10–52.

Asset classes International Stocks US large-cap stocks US small-cap stocks US long-term corporate bonds US long-term Treasury bonds US Treasury bills US inflation 1.00 0.72 0.27 0.04

0.03 -0.12

-0.11

0.03 -0.05

US large-cap stocks

International stocks 1.00 0.67 0.52 0.06

0.04 0.06

-0.13

1.00 0.09

US small-cap stocks

0.04 -0.31

0.89

1.00

US long-term corporate bonds

0.08 -0.26

1.00

US long-term Treasury bonds

1.00 0.70

US Treasury bills

1.00

US inflation

Table 4.5 Correlations between US stocks, US bonds, and international stocks between 1970 and 2020 (Source: Ibbotson and Harrington 2021: Stocks, Bonds, Bills, and Inflation® (SBBI®) - 2021 Summary Edition, p. 198)

4.6 Summary 129

130

•

• •

• •

•

•

•

4

Efficient Risky Portfolios

coefficient between the returns of pairs of assets, and on the weights of each asset in the portfolio. Covariance measures the relationship between the return deviations of two assets. A positive (negative) covariance means that the return deviations of two assets move in the same (opposite) direction for a specific period of time. However, the magnitude of the covariance is difficult to interpret because, like the variance, it embodies the square of the return deviations. Therefore, the correlation coefficient is calculated by dividing the covariance by the corresponding standard deviations of the two assets. The correlation coefficient ranges from 1 to -1. A correlation coefficient of 1 means that the returns move completely in the same direction, while a correlation coefficient of -1 implies that the returns move in a completely opposite direction. If the correlation coefficient has a value of 0, there is no linear relationship between the returns. In this case, the returns of two assets are uncorrelated. There is always a diversification effect in a portfolio if the correlation coefficient between the returns of two assets is less than 1. In a two-asset portfolio, the risk can be entirely eliminated if two assets with a correlation coefficient of -1 are used for portfolio construction. However, this case is rather theoretical since in the financial markets most pairs of stocks have correlation coefficients that are at least greater than -1 and regularly even in the positive range. The efficient frontier for a two-asset portfolio is given by the part of the portfolio curve that passes through the minimum variance portfolio and the asset with the highest expected return. The efficient frontier is concave. With regard to portfolios constructed with several risky assets, those portfolios with the highest expected return and the lowest standard deviation lie on the efficient frontier. The calculation of the efficient frontier is an optimisation problem. The portfolio weights are determined such that the portfolio variances for the respective expected returns are minimised, where the sum of the weights equals 1. Usually, this optimisation problem is solved with a computer program that relies on algorithms. The primary risk driver in a portfolio of N risky assets is the covariance or correlation coefficient, not the variance of the individual assets. For example, if a portfolio consists of 100 stocks, then the portfolio variance (ignoring weights) is calculated as the sum of 100 variances and 9900 covariances. In other words, the higher the number of risky assets in a portfolio, the less relevant the standalone risk of each asset (i.e. the variance). Rather, what matters is how much each asset contributes to the overall risk of the portfolio. This can be measured with the covariance or correlation coefficient. In a well-diversified portfolio consisting of long equity securities, the variances of the individual stocks can be eliminated but not the covariances. Therefore, the market risk (systematic risk) remains in the equity portfolio, while the companyspecific risk of loss (unsystematic risk) is removed. Empirical studies reveal that, depending on the average volatility and correlation in the equity markets, between 20 and 50 stocks are sufficient to achieve optimal diversification of a long equity portfolio.

4.7 Problems

131

• Diversification into equities from one market can reduce the overall risk of a portfolio by eliminating the unsystematic risk. Furthermore, systematic risk can be reduced by diversification in the global equity market. • Another important aspect of risk reduction is diversification into different asset classes. In addition to equities, a well-diversified portfolio contains bonds, real estate, commodities (e.g. gold), and so forth.

4.7

Problems

1. A stock X has an expected return of 10% and a standard deviation of returns of 8%. By contrast, stock Y has an expected return of 16% and a standard deviation of returns of 25%. a) What is the expected return on a portfolio consisting of 40% stock X and 60% stock Y? b) What is the standard deviation of the portfolio if the correlation coefficient between the returns of securities X and Y is 0.3? 2. An investor owns 100 shares of company A and 400 shares of company B. Stock A trades on the market at a price per share of EUR 50, while stock B’s price per share is EUR 25. The expected return and the standard deviation of returns of stock A are 14% and 10%, respectively. By contrast, stock B has an expected return of 20% and a standard deviation of returns of 22%. The correlation coefficient between the returns of the two stocks is 0.28. a) What is the expected return and standard deviation of the portfolio? b) The investor sells 300 shares of stock B. What is the expected return and standard deviation of the new portfolio? 3. The stocks of companies Z and X each have an expected return of 15% and a standard deviation of returns of 30%. The correlation coefficient between the returns of the two stocks is -1. The returns of the two stocks are therefore completely negatively correlated. What is the expected return on the equity portfolio if the standard deviation of portfolio returns is 0%? 4. A portfolio consisting of equally weighted stocks has an identical correlation coefficient of 0.4 between all pairs of stocks. Each stock has the same variance of 0.0625. What is the standard deviation of the equity portfolio if the portfolio consists of 30 stocks, 100 stocks, and an unlimited number of stocks? 5. A portfolio contains the following three stocks: Stock A B C

Expected return (in %) 12 10 15

Standard deviation (in %) 25 30 35

The following correlation matrix is given for the three stocks: Stock A B C

A 1.00 0.40 0.60

B 0.40 1.00 0.80

C 0.60 0.80 1.00

132

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Efficient Risky Portfolios

The market values of stocks A, B, and C are EUR 40,000, EUR 30,000, and EUR 30,000, respectively. What is the expected return and standard deviation of the equity portfolio? 6. The following statements relate to portfolios of risky assets: 1. For a long equity portfolio with a correlation coefficient of 1 between all pairs of stocks, firm-specific risk can be eliminated by increasing the number of equity securities in the portfolio. 2. In an equally weighted portfolio, where all stocks have the same average variance and all pairs of stocks have the same average covariance, the portfolio risk tends towards the average covariance as the number of stocks increases. 3. The systematic risk of a well-diversified portfolio consisting of domestic equities can be eliminated by buying foreign equities and expanding the equity portfolio to include other asset classes such as bonds, real estate, and commodities. 4. In a portfolio consisting of two long stocks, risk can be completely eliminated by finding two stocks with a correlation coefficient of zero. 5. The efficient frontier of a long equity portfolio is convex, with higher risk resulting in higher expected return. 6. Only risky equity portfolios that have the highest expected return for a given level of risk lie on the efficient frontier created from long stocks. Indicate whether each of the above statements is true or false (with justification). 7. Evaluating the stocks of companies A and B, an investor expects the following returns, depending on the state of the economy: Business cycle Boom Stagnation Recession

Probability (in %) 30 60 10

Expected return of stock A (in %) 16.5 10.2 -3.5

Expected return of stock B (in %) 8.5 8.2 5.0

a) What are the expected returns of stocks A and B? b) What are the standard deviations of the returns of stocks A and B? c) What are the covariance and the correlation coefficient between the returns of stocks A and B? d) What is the expected return and the standard deviation of a portfolio 40% of which comprises shares of A and 60% shares of B?

4.8 Solutions

4.8

133

Solutions

1. a) E ðr P Þ = 0:4 × 10% þ 0:6 × 16% = 13:6% b) The portfolio’s standard deviation of 16.25% can be calculated as follows: σP =

0:42 × 0:082 þ 0:62 × 0:252 þ 2 × 0:4 × 0:6 × 0:3 × 0:08 × 0:25 = 16:25%:

2. a)

wA =

100 × EUR 50 = 0:333 ð100 × EUR 50Þ þ ð400 × EUR 25Þ

wB =

400 × EUR 25 = 0:667 ð100 × EUR 50Þ þ ð400 × EUR 25Þ

E ðr P Þ = 0:333 × 14% þ 0:667 × 20% = 18% σ P = 0:3332 × 0:102 þ 0:6672 × 0:222 þ 2 × 0:333 × 0:667 × 0:28 × 0:10 × 0:22 = 15:93% The expected return of the portfolio is 18%, while the standard deviation of the portfolio returns is 15.93%. b) wA =

100 × EUR 50 = 0:667 ð100 × EUR 50Þ þ ð100 × EUR 25Þ

wB =

100 × EUR 25 = 0:333 ð100 × EUR 50Þ þ ð100 × EUR 25Þ

E ðr P Þ = 0:667 × 14% þ 0:333 × 20% = 16%

134

4

Efficient Risky Portfolios

σ P = 0:6672 × 0:102 þ 0:3332 × 0:222 þ 2 × 0:667 × 0:333 × 0:28 × 0:10 × 0:22 = 11:20% Selling shares of B results in both a lower expected portfolio return of 16% (versus 18%) and a lower portfolio risk of 11.20% (versus 15.93%). This is because stock B has a higher expected return and standard deviation than stock A. 3. First, the weight of stock Z is calculated as follows: wZ =

σX 0:30 = = 0:5: σ Z þ σ X 0:30 þ 0:30

The weight of stock X is also 0.5 (= 1 - 0.5). Using the weights of the two stocks of 0.5 each, the portfolio risk is 0%:

σP =

0:52 × 0:32 þ 0:52 × 0:32 þ 2 × 0:5 × 0:5 × ð - 1Þ × 0:3 × 0:3 = 0%:

The expected portfolio return is 15%: E ðrP Þ = 0:5 × 15% þ 0:5 × 15% = 15%: 4. Equation (4.28) can be used to calculate the standard deviation of the equally weighted portfolio for the different number of stocks:

σ 30 stocks =

0:0625 ×

1 - 0:4 þ 0:4 = 16:20%, 30

σ 100 stocks =

0:0625 ×

1 - 0:4 þ 0:4 = 15:93%: 100

With an infinite number of stocks in the portfolio, the first term in the bracketed expression to the right of the first equals sign tends to 0, resulting in the lowest standard deviation of the portfolio of 15.81%: p σ Infinite number of stocks = 0:0625 × 0:4 = 15:81%:

4.8 Solutions

135

The ratio of the portfolio standard deviation consisting of 30 stocks and the portfolio standard deviation with an infinite number of stocks is 1.0247: 16:20% σ 30 stocks = = 1:0247: σInfinite number of stocks 15:81% The portfolio standard deviation calculated with 30 stocks is approximately 102.5% of the portfolio standard deviation determined with an infinite number of stocks. This example demonstrates that a very good diversification of the portfolio can be achieved with 30 stocks if the average correlation coefficient between pairs of stocks is 0.4 and the average volatility in the equity market is 25%. 5. E ðrP Þ = 0:4 × 12% þ 0:3 × 10% þ 0:3 × 15% = 12:3% σP =

0:42 × 0:252 þ0:32 × 0:32 þ0:32 × 0:352 þ2 × 0:4 × 0:3 × 0:4 × 0:25 × 0:3 þ2 × 0:4 × 0:3 × 0:6 × 0:25 × 0:35þ2 × 0:3 × 0:3 × 0:8 × 0:3 × 0:35

= 25:307%

The expected return of the portfolio is 12.3%, while the portfolio risk is 25.31%. 6. 1. The first statement is false. If the correlation coefficient is 1, there is no diversification effect. 2. The second statement is true. The risk of such a portfolio tends towards the average covariance as the number of risky assets increases. 3. The third statement is false. Systematic risk can be reduced, but not eliminated, with foreign equities and other asset classes. 4. The fourth statement is false. The risk of a two-stock portfolio can only be completely eliminated if one finds two securities with a correlation coefficient of -1. 5. The fifth statement is false. The efficient frontier is concave. 6. The sixth statement is false. The upper bound of the efficient frontier constructed from long equity securities is given by the security with the highest expected return and risk. 7. a) E ðrA Þ = 0:3 × 16:5% þ 0:6 × 10:2% þ 0:1 × ð- 3:5%Þ = 10:72%

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Efficient Risky Portfolios

Eðr B Þ = 0:3 × 8:5% þ 0:6 × 8:2% þ 0:1 × 5:0% = 7:97% b) 0:3× ð0:165-0:1072Þ2 þ0:6× ð0:102-0:1072Þ2 þ0:1× ð -0:035-0:1072Þ2

σA =

= 5:51% σ=

0:3× ð0:085-0:0797Þ2 þ 0:6× ð0:082-0:0797Þ2 þ 0:1× ð0:05-0:0797Þ2 =1%

c) covA,B = 0:3 × ð0:165 - 0:1072Þ × ð0:085 - 0:0797Þ þ 0:6 × ð0:102 - 0:1072Þ × ð0:082 - 0:0797Þ þ 0:1 × ð- 0:035 - 0:1072Þ × ð0:05 - 0:0797Þ = 0:00050706 ρA,B =

covA,B 0:00050706 = = 0:92 σAσB 0:0551 × 0:01

d) Eðr P Þ = 0:4 × 10:72% þ 0:6 × 7:97% = 9:07% σP =

0:42 × 0:05512 þ 0:62 × 0:012 þ 2 × 0:4 × 0:6 × 0:00050706 = 2:766%

The expected portfolio return is 9.07%, while the portfolio risk is 2.766%.

Microsoft Excel Applications • To calculate the covariance (population), first enter the returns of the first asset, for example in cells A1 to A12, and the returns of the second asset in cells B1 to B12. Then write the following expression in a free cell: = COVARðA1:A12; B1:B12Þ and confirm with the Enter key. • The correlation coefficient (sample) can be calculated with the same returns as above using the following notation in a free cell:

Microsoft Excel Applications

137

= CORRELðA1:A12; B1:B12Þ: The formula expression must be completed by pressing the Enter key. • The following part sets out how the efficient frontier for the five DAX 40 stocks of Mercedes-Benz Group AG, Linde AG, Siemens AG, Bayer AG, and Adidas AG can be constructed using Microsoft Excel. For this purpose, the monthly continuous compounded returns for the monthly share prices from the end of June 2012 to the end of July 2017 are used. For example, the continuous compound returns of the Mercedes-Benz Group stock can be recorded in cells A3 to A62, and the continuous compound returns of the Linde stock in cells B3 to B62, and so on. • The next step is to calculate the monthly expected returns for all five stocks with the ‘AVERAGE’ function and the standard deviations and variances of the monthly returns with the ‘STDEV.S’ and ‘VAR.S’ functions, respectively. Since the efficient frontier is usually constructed on the basis of annual returns, the expected return and variance must be converted to annual values by multiplying each by 12 months. The standard deviation can be annualised by multiplying by the square root of 12 months. • The next step is to determine the covariances between the different pairs of stocks. To do this, the correlation coefficients are first estimated using the Excel function ‘CORREL’. Figure 4.9 presents the variance-correlation matrix in which the function for calculating the correlation coefficient between the returns of the Mercedes-Benz Group and Linde stocks is illustrated. • The covariance is the correlation coefficient between the returns of two securities multiplied by their standard deviations. This leads to the variance-covariance matrix. The expected returns can then be combined into a return vector. To do this, it is sufficient to arrange the respective return values in a column one below the other. Figure 4.10 presents the return vector together with the variancecovariance matrix. • For a given target return, the minimum variance of a portfolio can be calculated using the constants A, B, and C as follows: σ 2P = A - 2BμP þ CμP2 AC - B2:

ð4:29Þ

• The constant A is the product of the transposed return vector (μt), the inverted variance-covariance matrix (∑-1), and the simple return vector (μ):

138

4

Efficient Risky Portfolios

Fig. 4.9 Variance-correlation matrix (Source: Own illustration)

Fig. 4.10 Return vector and variance-covariance matrix (Source: Own illustration)

A = μt

-1

μ:

ð4:30Þ

• In order to transpose the return vector, as many cells as there are assets in the portfolio must first be marked in a row. Figure 4.11 illustrates how these cells are assigned the corresponding Excel function ‘TRANSPOSE’ and the return vector

Microsoft Excel Applications

139

Fig. 4.11 Transposition of the return vector (Source: Own illustration)

to be transposed is specified as the data range. The function should be concluded with the key combination Ctrl+Shift+Enter so that Excel recognises the matrix calculation as such. • Next, the variance-covariance matrix must be inverted. For this purpose, a free cell range is marked, which has the same size as the corresponding matrix. To invert the matrix, this cell range must be assigned the function ‘MINVERSE’, and the variance-covariance matrix must be defined for the data range. Again, the data input must be confirmed with the key combination Ctrl+Shift+Enter. This procedure is illustrated in Fig. 4.12. • After these calculations, the constant A of 0.942781 can be determined. The Excel function for the multiplication of matrices is ‘MMULT’. Vectors and matrices must always be multiplied in the correct order. Therefore, the transposed return vector should first be multiplied by the inverted variance-covariance matrix, and only then should the resulting vector be multiplied by the return vector. This procedure is illustrated in Fig. 4.13. • The next step is to determine the constant B of 4.569518 and the constant C of 40.700664 using the following formulas: B = 1t and

-1

μ

ð4:31Þ

140

4

Efficient Risky Portfolios

Fig. 4.12 Inversion of the variance-covariance matrix (Source: Own illustration)

Fig. 4.13 Calculation of the constant A (Source: Own illustration)

C = 1t

-1

1:

ð4:32Þ

• The procedure is almost identical to the calculation of constant A. It is only necessary to replace the return vector with the 1s vector at the corresponding points. Figure 4.14 illustrates this for constant B and Fig. 4.15 for constant C.

Microsoft Excel Applications

141

Fig. 4.14 Calculation of the constant B (Source: Own illustration)

Fig. 4.15 Calculation of the constant C (Source: Own illustration)

• In order for the efficient frontier to be displayed graphically in Excel, a series of target returns is selected. A corresponding number of target returns are defined, for example, at intervals of 0.5%. The next step is to store the corresponding formula for the minimum portfolio variance in each cell of another column. To do this, the variables of the formula should be related to the corresponding cells containing the constants A, B, and C, and the respective target return. It is sufficient to enter the formula in the first cell and to mark the cell reference to

142

4

Efficient Risky Portfolios

Fig. 4.16 Variance and standard deviation of the most efficient portfolios with selected target return (Source: Own illustration)

the constants A, B, and C with dollar signs, as indicated in Fig. 4.16. In this way, the formula can be copied into all further cells since, as intended, it is only the cell reference for the target return that changes. • In a further column, the portfolio variance determined in each case should be converted into the standard deviation using the root function. Thereafter, the efficient frontier can be created as a diagram. Excel offers numerous options for this. For example, a scatter plot with connected data points can be selected via the ‘Insert’ tab. If the standard deviation is defined for the X-values and the target return for the Y-values, this results in the familiar representation of the efficient frontier (see Fig. 4.7).

References D’Ambrosio, C.A.: Portfolio management basics. In: Maginn, J.L., Tuttle, D.L. (eds.) Managing Investment Portfolios: A Dynamic Process, 2nd edn. Wiley, Boston (1990) DeFusco, R.A., McLeavy, D.W., Pinto, J.E., Runkle, D.E.: Quantitative Methods for Investment Analysis, 2nd edn. Charlottesville, CFA Institute (2004) Evans, J.L., Archer, S.H.: Diversification and the reduction of dispersion: an empirical analysis. J. Finance. 23(5), 761–767 (1968) Ibbotson, R.G., Harrington, J.P.: Stocks, Bonds, Bills, and Inflation® (SBBI®) – 2021 Summary Edition. CFA Institute, Charlottesville (2021) Markowitz, H.: Portfolio selection. J. Finance. 7(1), 77–91 (1952) Markowitz, H.: Portfolio Selection: Efficient Diversification of Investments. Wiley, New York (1959) Miller, M.B.: Mathematics and Statistics for Financial Risk Management. Wiley, Hoboken, NJ (2012)

References

143

Roy, A.D.: Safety first and the holding of assets. Econometrica. 20(3), 431–439 (1952) Spremann, K.: Portfolio Management. Oldenbourg, Wien (2000) Statman, M.: How many stocks make a diversified portfolio? J. Financ. Quant. Anal. 22(3), 353–363 (1987) Zisler, R.C.: Real Estate portfolio management. In: Maginn, J.L., Tuttle, D.L. (eds.) Managing Investment Portfolios: A Dynamic Process, 2nd edn. Wiley, Boston (1990)

5

Optimal Portfolio

5.1

Introduction

In addition to the return–risk characteristics of the assets, the investor’s attitude to risk must also be taken into account in order to achieve the investment objectives. This chapter demonstrates how the efficient frontier is combined with the investorspecific indifference curves to arrive at the optimal risky portfolio. The efficient frontier is constructed using capital market data with the expected return and standard deviation of returns on individual assets and the covariance or correlation coefficient between returns of pairs of assets. Indifference curves, by contrast, measure the benefit the investor gains from holding the portfolio. In calculating the benefit or utility, relevant factors include the degree of risk aversion of an individual investor, in addition to the expected return and the risk. The point of contact between the efficient frontier and the highest possible investor-specific indifference curve represents the optimal portfolio of risky assets. If the risk-free asset is included in the portfolio construction, the optimal portfolio lies on the most efficient capital allocation line. Assuming that market participants have identical (homogeneous) capital market expectations, then all investors invest in the same portfolio of risky assets or market portfolio. All investment combinations of the riskfree asset and the market portfolio lie on the capital market line.

5.2

Risk Aversion

5.2.1

Concept of Risk Aversion

The degree of risk aversion an investor has depends on their behaviour in uncertain situations. For example, an investor has the following two alternatives: (1) they receive EUR 100 safely, or (2) they agree to play a game with a 50% probability of receiving EUR 200 or EUR 0 in each case. The expected value in both cases, under # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_5

145

146

5

Optimal Portfolio

certainty and uncertainty, is EUR 100.1 The investor has a total of three options: to choose the game, to take the EUR 100, or to be indifferent about the two possibilities. The behaviour of the investor makes it possible to classify their risk behaviour, although it should be noted that this is an illustrative example. As a rule, a single decision is not sufficient to conclusively determine the risk behaviour of an investor. A risk-seeking investor selects the game. The game has an uncertain outcome, but it has the same expected value as the safe choice of EUR 100. A risk-seeking investor would also accept an expected value of less than EUR 100 (e.g. EUR 75) as long as they have the possibility of receiving more than the guaranteed EUR 100. Risk-taking behaviour can be observed, for example, in lottery games or in casinos. For example, people buy lottery tickets even though the expected price is lower than the price paid for the tickets. A risk-neutral investor is indifferent about the two choices. Risk neutrality means that only the expected return matters to an investor, while risk is irrelevant. Therefore, investments with a higher expected return are preferred, regardless of their risk. Risk-neutral behaviour can be observed when the investment constitutes only a small part of an investor’s total wealth. For example, a very wealthy investor is indifferent when it comes to choosing between the guaranteed payout of EUR 100 or gambling. A risk-averse investor, on the other hand, opts for the guaranteed payout because they are not willing to take the risk of not receiving a payout in the end. Depending on the degree of risk aversion, such an investor will even accept a payout of EUR 80 instead of the EUR 100 expected from the game. Basically, risk-averse investors tend to make investments with a lower risk and a guaranteed return. They prefer to invest in assets that have a lower risk of loss for the same expected return or a higher expected return for the same risk. By contrast, a risk-neutral investor maximises return regardless of risk, while a risk-seeking investor maximises both return and risk. Numerous empirical studies using historical data series relating to investments indicate that there is a positive relationship between return and risk.2 The higher the risk, the higher the return on investments. These positive risk premiums imply that investors behave in a risk-averse manner. For example, one indication of risk-averse behaviour is the purchase of insurance. The purchase of insurance, such as car or health insurance, represents a hedge against future risks such as a car accident or illness. The insurance premium paid protects against the future uncertainty that a large amount of money will be needed to pay for such an event. In the bond market, investors expect a higher yield to maturity for issuers or bonds with higher credit risk. This is another indication that market participants are behaving in a risk-averse

1

The expected value under uncertainty is calculated as the sum of the probability-weighted payouts as follows: 0.5 × EUR 200 + 0.5 × EUR 0 = EUR 100. 2 See Reilly and Brown (2000): Investment Analysis and Portfolio Management, p. 259.

5.2 Risk Aversion

147

manner. Based on these observations, financial market theory generally assumes that investors are risk-averse and thus demand a higher return for taking a higher risk.

5.2.2

Utility Theory and Indifference Curves

A risk-averse investor prefers the guaranteed payout of EUR 100 to the uncertainty of the expected payout of EUR 100 resulting from the game. Therefore, the utility or satisfaction from the guaranteed payout is higher for such an investor. In general, utility is a measure of the relative satisfaction derived from the consumption of goods and services.3 If this microeconomic definition is applied to investing, then the utility of an investor arises from holding different portfolios. Investors have various preferences in terms of return and risk. Hence, the ranking of investments differs among risk-averse investors. All risk-averse investors prefer the guaranteed payout of EUR 100 to the game. If, for example, the guaranteed payout is only EUR 60—that is, below the expected value of the gamble of EUR 100—then it is no longer clear whether all risk-averse investors would prefer the guaranteed payout to the gamble. The ranking of investments or the investor’s utility is higher when the expected return is greater and lower when the risk is higher. If the risk increases with the expected return, the classification of the investments is no longer clear. The expected payout of the game is EUR 100, while the guaranteed payout is EUR 60. The game has a higher expected return and risk than the guaranteed payout. The trade-off between expected return and risk can be measured by utility functions. Portfolios with a higher expected return lead to a higher utility for the investor, while portfolios with a higher risk result in a lower investor’s utility. The following utility function takes into account the expected return and the variance to measure the utility (U ) of single assets or portfolios:4 U = E ðr Þ -

1 2 Aσ , 2

ð5:1Þ

where E(r) = expected return, σ 2 = variance, and A = degree of risk aversion (risk aversion coefficient). According to the formula above, a higher expected return results in a greater utility. A higher risk, on the other hand, leads to a lower investor’s utility. These properties of the utility function reflect the concept of risk aversion. The extent of the

3

See Gwartney et al. (2000): Economics, Private and Public Choice, p. 912. For the calculation of the utility, the expected return and the standard deviation of the returns must be entered into the formula in decimal places and not as a percentage. 4

148

5

Optimal Portfolio

influence of the variance on the utility depends on the degree of risk aversion of the investor. The degree of risk aversion represents the additional expected return an investor requires to accept an additional unit of risk. Higher risk aversion is captured by a higher coefficient A in Eq. (5.1). Investors who have a high degree of risk aversion give more weight to the risk of an investment, which results in a utility that is lower than that of less risk-averse investors. Risk aversion and risk tolerance have an opposite relationship. High (low) risk aversion implies low (high) risk tolerance. An investor’s risk aversion or tolerance can be assessed as below average (low), average (medium), or above average (high) based on the investor’s willingness and ability to bear losses. Risk aversion can be determined through an interview or questionnaire in which the investor provides information about their investment and risk preferences.5 In order to define an approximate benchmark, a risk aversion coefficient A of 6–8 is chosen for high or above-average risk aversion; a medium or average risk aversion is given by a coefficient A of 3 to 5; and an investor with a low or below-average risk aversion has a coefficient A of 1 to 2. Equation (5.1) can only be used to rank different assets or portfolios. The investor’s satisfaction cannot be measured with the utility function. For example, a utility that is twice as high (e.g. a utility of 4 compared to 2) does not mean that an investor is twice as satisfied with a utility of 4 compared to a utility of 2. Rather, the utility function indicates that an investment with a higher utility (e.g. 4) is preferred to an investment with a lower utility (say 2). Investors choose the investment with the higher utility. Moreover, the utility functions of different investors cannot be compared or aggregated for the market as a whole. Utility functions are essentially investor-specific. For risk-averse investors, the risk aversion coefficient is between 1 and 8. An increase in risk leads to a lower utility due to the positive risk aversion coefficient (A > 0). In the case of a risk-neutral investor, however, the risk aversion coefficient is 0 (A = 0). Thus, a change in risk does not affect utility. By contrast, a risk-seeking investor has a coefficient that is less than 0 or negative (A < 0). Accordingly, a higher risk results in a greater utility. It should be noted that a risk-free investment (σ 2 = 0) produces the same utility for all three types of investors. The higher the expected return, the greater the utility. Example: Calculation of Utility An investment has an expected return of 10% and a standard deviation of returns of 20%. The risk aversion coefficient for an investor with average risk aversion is 3. (continued)

5

For an example of a standardised risk assessment questionnaire, see Sect. 7.2.1.1.

5.2 Risk Aversion

149

1. What is the utility of this investment for the investor with a risk aversion coefficient of 3? 2. What is the minimum risk-free return required to obtain the same utility from the investment as in question 1? Solution to 1 The utility of this investment for the investor is 0.04 and can be calculated as follows: U = 0:1 - 0:5 × 3 × 0:22 = 0:04: Solution to 2 A risk-free investment has a risk of 0 (σ = 0). Therefore, the second term to the right of the equals sign is 0 (= 0.5 × 3 × 02). To achieve the same utility of 0.04, the risk-free return must be 4%. A risky investment with an expected return of 10% and a standard deviation of 20% has the same utility as a riskfree investment with a return of 4% if the investor’s risk aversion coefficient is 3. Indifference curves represent risk–return combinations of portfolios that have the same utility for a given investor. An investor is indifferent if the portfolios lie on the same indifference curve because they have the same utility. Indifference curves are therefore defined in terms of the trade-off between expected return and risk.6 Figure 5.1 presents an indifference curve with a risk aversion coefficient of 3, where all portfolios have the same utility of 0.04. Starting from risk–return point A, a higher standard deviation results in a lower utility. This lower utility must be compensated by a higher expected return. Risk–return point B, which is also on the indifference curve, has a higher standard deviation and a higher expected return than A. Investments with the same utility have a higher expected return and higher risk than investments with lower expected return and risk. Indifference curves are continuous between all risk–return points because there are an infinite number of portfolios that have the same utility for an investor. Indifference curves can be created with the risk aversion coefficient, the expected return, and the variance. For example, an investor has a risk aversion coefficient of 3 and has invested their money in a risk-free asset earning a return of 4%. According to Eq. (5.1), the utility of this investment is 0.04 (= 0.04 - 0.5 × 3 × 02). To construct the indifference curve, the expected returns of risky investments must be determined, given a utility of 0.04 and the respective risk level. If Eq. (5.1) is solved for the expected return, the following equation is obtained:

6

See Reilly and Brown (2000): Investment Analysis and Portfolio Management, p. 230.

150

5

Optimal Portfolio

(Expected return) 20%

Indifference curve with A = 3 and U = 0.04

18% 16% 14% 12%

B

E(rB) 10% 8%

E(rA)

A

6% 4% 2% 0% 0%

10%

σA

20%

30%

σB (Standard deviation of returns)

Fig. 5.1 Indifference curve (Source: Own illustration)

1 E ðr Þ = U þ Aσ 2 : 2

ð5:2Þ

For a standard deviation of 1% (σ = 0.01), the expected return is 4.015%,7 while for a risk of 2% (σ = 0.02) it is 4.06%. If this calculation is repeated for a large number of increasing standard deviations, the corresponding return values are obtained. If these values are plotted in a risk–return diagram and the risk–return points are connected, this results in the indifference curve presented in Fig. 5.1, with a utility of 0.04. Figure 5.2 presents indifference curves with different levels of utility. An investor is indifferent about the risk–return points A and B, since they lie on the same indifference curve and thus generate the same utility for the investor. A comparison between risk–return point C, which lies on indifference curve 2, and risk–return point B reveals that both investments have the same standard deviation, but B’s expected return is higher. Risk-averse investors prefer portfolio B because the

7

E ðr Þ = 0:04 þ 12 × 3 × 0:012 = 0:04015

5.2 Risk Aversion Fig. 5.2 Indifference curves with different utility levels (Source: Own illustration)

151

(Expected return)

Greater utility Medium 1 B

E(rB) E(rA) E(rC)

A

C

utility 2 3 Smaller utility Indifference curves

0% 0%

σA σ B = σ C

(Standard deviation of returns)

expected return is greater, for the same level of risk. Hence, indifference curve 1 has a higher utility than curve 2. The utility of risk-averse investors increases as expected return rises and standard deviation falls. In a risk–return diagram, this relationship can be represented by a northwestward shift of the indifference curve. Portfolios that lie on a higher indifference curve have a higher expected return for any given unit of risk and therefore result in a greater utility for the investor. The indifference curves are convex because there is a diminishing marginal utility between the increase in risk and the expected return. If the risk increases, risk-averse investors demand a higher return. In this case, the expected return rises disproportionately, making the indifference curve steeper. For an additional unit of risk, investors require an above-average increase in return. An increase in risk aversion leads to a steeper indifference curve because the investor accepts a higher risk only if they receive a higher return. Figure 5.3 presents indifference curves for different degrees of risk aversion. The indifference curves are very steep for highly risk-averse investors. For less risk-averse investors, on the other hand, the indifference curves are flatter because they demand a lower return when risk increases. Risk-seeking investors have indifference curves with a negative slope. In the case of the indifference curve 5 in Fig. 5.3, for example, there is a negative relationship between expected return and standard deviation. Along indifference curve 5, a risk-seeking investor’s utility remains unchanged as expected return falls (rises) and standard deviation rises (falls). By contrast, risk-neutral investors have flat indifference curves because a change in risk has no effect on the investor’s utility.

152

5

High risk aversion (A from 6 – 8)

(Expected return)

1

Average risk aversion (A from 3 – 5) 3

2

4

Optimal Portfolio

Low risk aversion (A from 1 – 2 )

Risk neutral (A = 0)

5 Risk-seeking (A < 0)

0% 0%

(Standard deviation of returns)

Fig. 5.3 Indifference curves for investors with different risk behaviour (Source: Own illustration)

Example: Calculation of the Utility for Different Investments A portfolio manager holds four investments that have the following expected returns and standard deviations of returns: Investment 1 2 3 4

Expected return (in %) 8 12 16 20

Standard deviation (in %) 24 30 36 42

1. Which investment does an average risk-averse investor with a risk aversion coefficient of 5 select? 2. Which investment does a below-average risk-averse investor with a risk aversion coefficient of 2 select? 3. Which investment does a risk-neutral investor select? 4. Which investment does a risk-seeking investor select? Solutions to 1 and 2 The utility is calculated for each individual investment. For example, for an average risk-averse investor with a risk aversion coefficient of 5, the utility of the first investment is calculated as follows: (continued)

5.3 The Optimal Risky Portfolio

U = 0:08 -

153

1 × 5 × 0:242 = - 0:064: 2

If these calculations are repeated for all investments with risk aversion coefficients of 2 and 5, the following utility values are obtained: Investment 1 2 3 4

Expected return (in %) 8 12 16 20

Standard deviation (in %) 24 30 36 42

Utility A=2 0.0224 0.0300 0.0304 0.0236

Utility A=5 -0.0640 -0.1050 -0.1640 -0.2410

The average risk-averse investor with a risk aversion coefficient of 5 selects investment 1, while the below-average risk-averse investor with a coefficient of 2 selects investment 3. Solution to 3 A risk-neutral investor does not consider risk in their investment decision. Therefore, they select investment 4, which has the highest expected return of 20%. Solution to 4 A risk-seeking investor favours investments with a high expected return and a high standard deviation of returns because they have a negative risk aversion coefficient. Investment 4 has the greatest utility for such an investor.

5.3

The Optimal Risky Portfolio

The efficient frontier is constructed using capital market data such as the expected return, the standard deviation of returns, and the covariance or correlation coefficient between the returns of two assets. The most efficient portfolio of risky assets in terms of expected return and risk lies on this curve. Efficient portfolios have the highest expected return for a given level of risk, or the lowest risk for a given return expectation. The construction of the efficient frontier assumes that investors behave in a risk-averse manner. The indifference curves, on the other hand, reveal the investor’s utility from investments as a function of the investor’s risk behaviour. They are defined in terms of a trade-off between the expected return and the risk. The optimal portfolio of risky assets is obtained if the efficient frontier is combined with the investor-specific indifference curves. Figure 5.4 illustrates the optimal risky portfolio for two investors with different degrees of risk aversion. Investor A, who is more risk averse, has steeper indifference curves A1, A2, and A3. For an additional unit of risk, investor A demands a

154

5

Fig. 5.4 Optimal risky portfolio (Source: Own illustration)

B6 B5

(Expected return)

B4 X

E(rX )

E(rP )

Optimal Portfolio

Efficient frontier P A3 A2 A1

0% 0%

σP

σX

(Standard deviation of returns)

higher return than investor B, who has less steep indifference curves B4, B5, and B6. The optimal risky portfolio lies on the efficient frontier and has the highest possible utility for an investor. Therefore, it lies on the point of contact between the efficient frontier and the indifference curve with the highest attainable utility. Investor A selects portfolio P on the efficient frontier where indifference curve A2 touches the efficient frontier. Investor B, on the other hand, who is less risk averse, chooses portfolio X, which has a higher expected return and a higher risk than portfolio P. The indifference curves A3 and B6 cannot be achieved because no portfolios can be formed for these utility functions.

5.4

The Risk-Free Investment: Capital Allocation Line Model

A risky investment has uncertain future returns. This uncertainty can be measured using the standard deviation. By contrast, the expected return of a short-term riskfree investment is certain. In order for an investment to be risk-free and thus yield a safe return, it must not have credit risk, interest rate risk, inflation risk, or reinvestment risk. Government bonds with an impeccable creditworthiness, such as bonds issued by the Federal Republic of Germany and the Swiss Confederation, have little to no credit risk. They are considered a safe investment in normal times. Government bonds are exposed to the risk of price changes due to interest rate changes if the investment period does not correspond to the lifetime of the bond. If interest rates rise, bond prices fall. Interest rate risk increases with longer bond maturities. A rise in inflation leads to a higher nominal interest rate and a corresponding fall in the price of the bond. There are government bonds with a prime credit rating that provide inflation protection. Inflation-protected government bonds are issued by the US Treasury and the Federal Republic of Germany, for example. Reinvestment risk is another risk of a coupon-paying bond. For example, if the bond matures after

5.4 The Risk-Free Investment: Capital Allocation Line Model

155

10 years and the coupons are paid annually, then the annual coupons received during the life of the bond must be invested. In this case, the interest rate for investing the coupons in the future is not known today, and therefore, the yield of the bond cannot be determined with certainty. The reinvestment risk can be eliminated with zerocoupon securities because these securities pay no coupons, and there is consequently no reinvestment risk. For example, non-interest-bearing treasury bills of the Federal Republic of Germany and money market book claims of the Swiss Confederation have no coupons and therefore no reinvestment risk. In principle, an investment can be described as risk-free if it has no credit risk, has no interest rate risk, is protected against inflation, and has no reinvestment risk. Typically, short-term risk-free government securities—such as non-interest-bearing treasury bills of the Federal Republic of Germany with maturities of 12 months and money market book claims of the Swiss Confederation with maturities of 3, 6, and 12 months—are considered risk-free investments. Due to the short terms to maturity—a maximum of 1 year—the interest rate risk is relatively low. They generally have no credit risk and the short-term maturity usually coincides with the investment period specified by the investor. These securities of the money market segment are issued at a discount price (below the par value of 100%) in the case of a positive interest rate level and are repaid at the par value of 100% on the maturity date. If the time to maturity of these risk-free securities is congruent with the investment period, the investor can achieve a known return (provided inflation is negligible over this period). The risk, or standard deviation of returns, of a risk-free investment is 0% (σ = 0) because the expected return over the life of the investment is certain. The expected return of a portfolio [E(rTP)] consisting of a risk-free investment and a portfolio of risky assets can be determined as follows: E ðr TP Þ = wF r F þ wP Eðr P Þ,

ð5:3Þ

where wF = market value weight of the risk-free investment in the total portfolio, wP = market value weight of the risky portfolio in the total portfolio, rF = risk-free return, and E(rP) = expected return of the risky portfolio. The variance of a portfolio consisting of the two investments in the risk-free asset and in the portfolio of risky assets can be calculated using the following equation: σ 2TP = w2F σ 2F þ w2P σ 2P þ 2wF wP ρF,P σ F σ P ,

ð5:4Þ

where σ F = standard deviation of the returns of the risk-free asset, σ P = standard deviation of the returns of the risky portfolio, and ρF, P = correlation coefficient between the returns of the risk-free asset and the portfolio of risky assets.

156

5

(Expected return)

Most efficient capital allocation line

Tangent portfolio (TAP)

Efficient frontier

Er E(r1) E(r2 )

rF

Optimal Portfolio

Capital allocation line for investments rF and X

1 X 2

0% 0% σ1 = σ2

σTAP

(Standard deviation of returns)

Fig. 5.5 Capital allocation line (Source: Own illustration)

The standard deviation of the returns of a short-term risk-free asset is 0% (σ F = 0). Accordingly, the first and third terms of Eq. (5.4) to the right of the equals sign are omitted,8 leading to the following formula for the variance of the portfolio: 2 = w2P σ 2P : σ TP

ð5:5Þ

Therefore, the standard deviation of the portfolio’s returns is: σ TP =

w2P σ 2P = wP σ P :

ð5:6Þ

The standard deviation of the portfolio consisting of a combination of the riskfree asset and a portfolio of risky assets is the weighted standard deviation of the risky portfolio. Hence, the inclusion of a risk-free asset in a portfolio of risky assets changes its risk–return characteristics. The expected return and the standard deviation of such a portfolio can be calculated with linear equations (see Eqs. 5.3 and 5.6). If this relationship is transferred to a risk–return diagram, the expected return and the risk of the portfolio can be depicted with a straight line between the risk-free asset and the portfolio of risky assets. Figure 5.5 presents the most efficient capital allocation line in terms of expected return and risk. It represents a combination of a risk-free investment and a portfolio of risky assets lying on the efficient frontier. The most efficient capital allocation line can be drawn as a tangent, starting from the risk-free return rF to the

8

The correlation coefficient between the returns of the risk-free investment and the portfolio of risky assets is 0, because the risk-free investment has no fluctuations in returns, while the portfolio of risky investments has volatile returns.

5.4 The Risk-Free Investment: Capital Allocation Line Model

157

Most efficient capital allocation line

(Expected return) (Y)

Tangent portfolio (TAP)

Efficient frontier

Er r rF a

σ

∆Y ∆X

0

Er σ

0% 0%

σ

(Standard deviation of returns) (X)

Fig. 5.6 Determination of the expected return of a portfolio using the most efficient capital allocation line (Source: Own illustration)

efficient frontier. The resulting portfolio of risky assets on the efficient frontier represents the tangent portfolio. The risk–return point 1 lies on the most efficient capital allocation line and consists of a combination of the risk-free asset and the tangent portfolio. Portfolio 2, on the other hand, is located on a less efficient capital allocation line and is made up of the risk-free asset and the portfolio of risky assets X, which lies on the efficient frontier. Both investment combinations 1 and 2 have the same risk, but the expected return of portfolio 1, which is on the most efficient capital allocation line, is higher. Risk-averse investors select portfolio 1 because it has a higher expected return than portfolio 2 for the same risk. The capital allocation line is a straight line and thus given by the following linear function: Y = a þ bX,

ð5:7Þ

where a = intercept of the straight line, and b = slope of the straight line, calculated by dividing the change in the dependent variable Y by the change in the independent variable X (b = ΔY/ΔX). Figure 5.6 demonstrates that the intercept of the most efficient capital allocation line (a) corresponds to the risk-free return rF. The slope of the straight line (b) can be determined by the difference between the expected return of the tangent portfolio

158

5

Optimal Portfolio

and the risk-free return divided by the standard deviation of the returns of the tangent portfolio. This leads to the following formula for calculating the expected return of a portfolio which is composed of a risk-free asset and the tangent portfolio of risky assets: Eðr TAP Þ - r F σ TP , σ TAP

E ðr TP Þ = r F þ

ð5:8Þ

where E(rTAP) = expected return of the tangent portfolio, σ TAP = standard deviation of the returns of the tangent portfolio, and σ TP = standard deviation of the returns of the total portfolio (σ TP = wTAPσ TAP). The slope of the most efficient capital allocation line is given by the Sharpe ratio of the tangent portfolio. The Sharpe ratio indicates the proportion of return above the risk-free rate for one unit of total risk taken; in other words, how much the excess return increases when the standard deviation increases by one unit (e.g. 1%). The most efficient capital allocation line has a higher slope—or Sharpe ratio—than any other capital allocation line between the risk-free asset and any risky portfolio on the efficient frontier. To determine the tangent portfolio, an optimisation problem must be solved. The objective function is to maximise the Sharpe ratio, while the constraint must be satisfied that the sum of the weights equals 1: The objective function, namely to: maximise the Sharpe ratio by changing w =

E ðrTAP - r F Þ σ TAP

ð5:9Þ

is subject to the following constraint: N

wi = 1: i=1

The solution to the optimisation problem is presented in the section on the Microsoft Excel applications listed at the end of the chapter, using the example of the five DAX 40 stocks of Mercedes-Benz Group AG, Linde AG, Siemens AG, Bayer AG, and Adidas AG. The 60 monthly returns employed for the calculation are based on the monthly prices of the five DAX 40 stocks from the end of July 2012 to the end of July 2017. If a portfolio consists of two risky assets 1 and 2, the weight of asset 1 in the tangent portfolio can be determined with the following equation:9 9

The equation for the optimal weight of asset 1 can be derived as follows: The expected return

w1E(r1) + (1 - w1)E(r2) and standard deviation

w21 σ21 þ ð1 - w1 Þ2 σ22 þ 2w1 ð1 - w1 Þρ1,2 σ1 σ2 are

5.4 The Risk-Free Investment: Capital Allocation Line Model

w1 =

159

½E ðr 1 Þ - r F σ22 - ½E ðr 2 Þ - r F cov1,2 : ð5:10Þ ½Eðr 1 Þ - r F σ22 þ ½E ðr 2 Þ - r F σ21 - ½Eðr 1 Þ - r F þ Eðr 2 Þ - r F cov1,2

Example: Tangent Portfolio with Two Risky Assets A portfolio consists of the two stocks A and B. The portfolio manager expects the following returns and standard deviations for the two securities: Stock A B

Expected return (in %) 10 20

Standard deviation (in %) 30 50

The correlation coefficient between the returns of the two equity securities is 0.01. The risk-free return is 2%. 1. What are the weights of the two stocks A and B in the tangent portfolio? 2. What is the expected return and risk of the tangent portfolio? 3. What is the Sharpe ratio of the tangent portfolio or the slope of the most efficient capital allocation line? Solution to 1 The covariance between the returns of the two stocks A and B can be determined as follows: covA,B = ρA,B σ A σ B = 0:01 × 0:3 × 0:5 = 0:0015: The weight of security A in the tangent portfolio can be calculated as 55.1% with the following equation: wA =

ð0:10 - 0:02Þ × 0:502 - ð0:20 - 0:02Þ × 0:0015 ð0:10 - 0:02Þ × 0:502 þð0:20 - 0:02Þ × 0:302 - ð0:10 - 0:02þ0:20 - 0:02Þ × 0:0015

= 0:551:

The weight of equity security B in the tangent portfolio is therefore 44.9%: wB = 1 - 0:551 = 0:449:

(continued)

substituted into the Sharpe ratio equation for the tangent portfolio. Then the first derivate of the Sharpe ratio with respect to a change in the weight of investment 1 (w1) is calculated, set equal to 0 and solved according to w1.

160

5

Optimal Portfolio

Thus, 55.1% of the tangent portfolio consists of stock A, and 44.9% of stock B. Solution to 2 The expected return of 14.49% and the standard deviation of returns of 28.01% of the tangent portfolio can be determined as follows: EðrTAP Þ = 0:551 × 10% þ 0:449 × 20% = 14:49%, σ TAP = 0:5512 × 0:302 þ 0:4492 × 0:502 þ 2 × 0:551 × 0:449 × 0:0015 = 28:01%: Solution to 3 The slope of the most efficient capital allocation line is given by the Sharpe ratio of the tangent portfolio and is 0.446: SRTAP =

14:49% - 2% = 0:446: 28:01%

The tangent portfolio lies on the most efficient capital allocation line. The slope of this capital allocation line is higher than the slope of any other capital allocation line between another risky portfolio on the efficient frontier and the risk-free asset. If the investor borrows money at the risk-free rate, they can invest more than 100% of their capital in the tangent portfolio. In such a case, the optimal portfolio is located to the right of the tangent portfolio on the most efficient capital allocation line. The leveraged portfolio has a higher expected return than the tangent portfolio but also a higher risk. Example: Expected Return and Risk of a Portfolio on the Most Efficient Capital Allocation Line A portfolio manager has constructed the efficient frontier with an investment list of 100 equity securities. They construct a portfolio for two clients, each worth EUR 100,000, consisting of a risk-free investment and an equity portfolio, which is formed from the investment list. To provide the most efficient equity portfolio for their clients, they determine the tangent portfolio that lies on the most efficient capital allocation line. The tangent portfolio has an expected return of 10% and a standard deviation of returns of 30%. The interest rate for risk-free investments is 3%. (continued)

5.4 The Risk-Free Investment: Capital Allocation Line Model

161

1. For the first client, the portfolio manager decides to invest EUR 60,000 in a risk-free asset and EUR 40,000 in the tangent portfolio. What is the expected return and risk of this portfolio? 2. The second client is less risk-averse and has expressed a desire to achieve an expected return of more than 10%. To meet this return requirement, the portfolio manager borrows EUR 40,000 at the risk-free rate and invests EUR 140,000 in the tangent portfolio. What is the expected return and risk of this portfolio? Solution to 1 The expected return on the portfolio is 5.8%: EðrTP Þ = 0:60 × 3% þ 0:40 × 10% = 5:8%: The standard deviation of portfolio returns is 12% and can be calculated as follows: σ TP = 0:40 × 30% = 12%: Equation (5.8) can also be used to determine the expected portfolio return of 5.8%: E ðr TP Þ = 3% þ

10% - 3% × 12% = 5:8%: 30%

The portfolio with a weight of 60% in the risk-free asset and a weight of 40% in the tangent portfolio has an expected return of 5.8% and a risk of 12%. Solution to 2 To invest 140% of EUR 100,000 in the tangent portfolio, 40% of EUR 100,000 must be borrowed at the risk-free interest rate. The expected return of 12.8% can be calculated as follows, taking into account the interest cost of borrowing money: E ðr TP Þ = 1:40 × 10% þ ð- 0:40Þ × 3% = 12:8%: The risk of this leveraged portfolio is 42% and can be determined as follows: σ TP = 1:40 × 30% = 42%: The expected portfolio return of 12.8 % can also be estimated with Eq. (5.8): (continued)

162

5

E ðr TP Þ = 3% þ

Optimal Portfolio

10% - 3% × 42% = 12:8%: 30%

The leveraged portfolio has an expected return of 12.8% and a risk of 42%. The expected return of 12.8% is higher than the 10% return required by the client. However, this investment has a risk of 42%, while the tangent portfolio, whose expected return is 10%, has a return volatility of only 30%. Figure 5.7 illustrates the relationship between these investments. The selection of the optimal portfolio on the most efficient capital allocation line represents a trade-off decision between expected return and risk. Less risk-averse investors hold a higher percentage of the tangent portfolio in the optimal portfolio. By contrast, investors with higher risk aversion prefer a smaller amount to be allocated to the tangent portfolio and accordingly invest more of the total portfolio in the risk-free asset. Maximum utility can be achieved for the investor through the optimal allocation of capital between the tangent portfolio and risk-free investment. The optimal or utility-maximising weight of the tangent portfolio of risky assets in the overall portfolio can be calculated as follows:10 wTAP =

Eðr TAP Þ - r F , Aσ 2TAP

ð5:11Þ

where wTAP = optimal weight of the tangent portfolio in the overall portfolio that maximises the investor’s utility. (In financial market theory, the symbol ‘*’ is usually used to represent the optimum.) The above formula demonstrates that the optimal allocation to the tangent portfolio of risky assets is proportional to the risk premium (excess portfolio return over the risk-free interest rate) and inversely proportional to the investor’s risk aversion and the portfolio risk.

The utility maximisation problem can be written as follows: Max U = E ðr TAP Þ - 12 Aσ 2TAP . If the expected tangent portfolio return E(rTAP) is replaced by rF + wTAP [E(rTAP) - rF] and the tangent portfolio variance σ 2TAP by w2TAP σ 2TAP , the following equation for the maximisation of utility is obtained: Max U = rF þ wTAP ½E ðr TAP Þ - r F  - 12 Aw2TAP σ 2TAP : To solve the maximisation problem, the first derivative with respect to a change in wTAP must be calculated and then set to 0: 0 = E ðr TAP Þ - r F - wTAP Aσ 2TAP : Solving this equation for wTAP gives Eq. (5.11). See Mondello (2015): Portfoliomanagement: Theorie und Anwendungsbeispiele, p. 154 ff. 10

5.4 The Risk-Free Investment: Capital Allocation Line Model

(Expected return)

Tangent portfolio (TAP)

E(r2) = 12.8%

163

Most efficient capital allocation line 2 Efficient frontier

E(rTAP) = 10% 1 E(r1) = 5.8% rF

0% 0% σ1 = 12%

σTAP = 30% σ2 = 42%

(Standard deviation of returns)

Fig. 5.7 Expected return and risk of portfolios on the most efficient capital allocation line. (To make the graph clearer, the expected return on the Y-axis and the standard deviation on the X-axis are scaled differently) (Source: Own illustration)

Example: Calculation of Capital Allocation In line with the previous example, the expected return and standard deviation of returns of the tangent portfolio are 10% and 30%, respectively. The risk-free return is 3%. The investor has a risk aversion coefficient of 2. 1. What are the weights of the tangent portfolio and the risk-free asset that make up the optimal portfolio when the investor’s utility is maximised? 2. What is the expected return and risk of the optimal portfolio? 3. What is the Sharpe ratio of the optimal portfolio? Solution to 1 The optimal weight of the tangent portfolio in the overall portfolio is 38.89% and can be calculated as follows: wTAP =

0:10 - 0:03 = 0:3889: 2 × 0:302

Hence, 38.89% of the optimal portfolio consists of the tangent portfolio, and 61.11% of it consists of the risk-free asset. This portfolio provides the investor with the highest utility. (continued)

164

5

Optimal Portfolio

Solution to 2 The expected return of the optimal portfolio is 5.72%: E ðr OP Þ = 0:3889 × 10% þ 0:6111 × 3% = 5:72%: The risk of the optimal portfolio is 11.67% and can be determined as follows: σ OP = 0:3889 × 30% = 11:67%: Figure 5.8 illustrates the optimal portfolio with an expected return of 5.72% and a standard deviation of returns of 11.67%, which lies on the most efficient capital allocation line and on the highest achievable indifference curve. Solution to 3 The Sharpe ratio of the optimal portfolio is 0.2331: SROP =

5:72% - 3% = 0:2331: 11:67%

The Sharpe ratio corresponds to the slope of the capital allocation line. The steeper the slope, the higher the increase in the difference between the expected return and the risk-free interest rate (risk premium) when risk increases. Portfolios that lie on the most efficient capital allocation line have a higher Sharpe ratio than all other possible investments and are therefore more efficient in terms of expected return and risk.

(Expected return)

E(rTAP) = 10% E(rOP) = 5.72%

Optimal portfolio (OP)

Most efficient capital allocation line

Tangentportfolio (TAP)

Efficient frontier

Indifference curves

rF

0% 0% σTAP = 30% σOP = 11.67%

(Standard deviation of returns)

Fig. 5.8 Optimal portfolio on the most efficient capital allocation line. (To make the graph clearer, the expected return on the Y-axis and the standard deviation on the X-axis are scaled differently) (Source: Own illustration)

5.4 The Risk-Free Investment: Capital Allocation Line Model

165

A new risky asset is added to an existing portfolio only if this new asset combination has a higher risk-adjusted return than the existing portfolio. The inclusion of a new asset in a portfolio is optimal if the following condition is met:11 Eðr new Þ - r F > σnew

EðrP Þ - r F ρnew, P , σP

ð5:12Þ

where E(rnew) = expected return of new asset, σ new = standard deviation of returns of new asset, and ρnew, P = correlation coefficient between the returns of the new asset and the portfolio. To obtain a higher Sharpe ratio by adding a new risky asset to a portfolio, the Sharpe ratio of the new asset must be greater than the product of the Sharpe ratio of the portfolio and the correlation coefficient between the returns of the new asset and the portfolio. The new portfolio lies on a superior efficient frontier, resulting in a tangent portfolio that has a higher Sharpe ratio. Equation (5.12) only demonstrates that it is possible to improve the efficient frontier by including a new asset in the portfolio. However, the formula says nothing about the required weighting of the new asset in the portfolio. The optimal portfolio weights leading to the new efficient frontier can be determined by the risk–return optimisation procedure according to Eq. (4.23).12 Example: Adding an Asset Class to an Existing Portfolio A portfolio manager of a Swiss pension fund has invested in Swiss equities, Swiss bonds, real estate in Switzerland, and global equities. The portfolio has a Sharpe ratio of 0.30. The investment committee of the pension fund is considering adding one of the following two asset classes to the portfolio: • Private equity: estimated Sharpe ratio of 0.12 and estimated correlation to existing portfolio of 0.45. • Hedge funds: estimated Sharpe ratio of 0.35 and estimated correlation to existing portfolio of 0.70. (continued)

11

The expected return of a new asset can be calculated based on the risk–return trade-off with the

existing portfolio as follows: E ðr new Þ = r F þ ½E ðr P- r F 

σ new ρnew,P σP

. If the risk-adjusted return of the

portfolio increases, then the new asset should be added to the portfolio. Equation (5.12) is obtained if the condition is rewritten using the Sharpe ratio on both sides of the equation. See Elton et al. (1987): ‘Professionally managed, publicly traded commodity funds’, p. 198. 12 See Sect. 4.4.

166

5

Optimal Portfolio

Which of the two asset classes must the investment committee include in the existing portfolio in order to increase its Sharpe ratio? Solution Private equity: The Sharpe ratio of the existing portfolio multiplied by the correlation coefficient between private equity and the existing portfolio results in a value of 0.135 (= 0.30 × 0.45). As the estimated private equity Sharpe ratio of 0.12 is lower than 0.135, the investment committee will not include this asset class in the existing portfolio. Hedge funds: The product of the Sharpe ratio of the existing portfolio and the correlation coefficient between hedge funds and the existing portfolio is 0.21 (= 0.30 × 0.70). As the estimated hedge fund Sharpe ratio of 0.35 is higher than 0.21, the investment committee will add this asset class to the existing portfolio.

5.5

Homogeneous Expectations: Capital Market Line Model

In line with the assumption of homogeneous (identical) capital market expectations, all investors construct the same efficient frontier and thus select the same portfolio of risky assets on the efficient frontier in the presence of a risk-free investment. Since all investors invest in the same risky portfolio, this is the market portfolio. In theory, the market portfolio includes all risky assets such as equity securities, fixed-income securities, real estate, commodities, and even human capital. The problem with this definition of the market portfolio is that not all assets are tradable. For example, the Federal Chancellery in Berlin and the Federal Parliament Building in Berne are assets that cannot be traded. Furthermore, stocks are traded in China, for example, which cannot be purchased by foreign investors. Therefore, the market portfolio must contain, as far as possible, all assets that are tradable and can be invested in. Such a market portfolio cannot be replicated in a single index due to the large number of tradable and investable assets worldwide. For this reason, a specific equity index with a high market capitalisation is often used as an approximation of the market portfolio. For example, the S&P 500 represents approximately 80% of the equity market capitalisation in the USA and approximately 32% of the global equity market.13 The SMI reflects approximately 85%–90% of the total capitalisation of the Swiss equity market, while the DAX 40 represents approximately 80% of the equity market capitalisation in Germany. In the following, the market portfolio is approximated by an equity index that can be easily expanded with other tradable investments.

13 The S&P 500 (Standard & Poor’s 500) comprises the 500 stocks with the highest market capitalisation in the US equity market.

5.5 Homogeneous Expectations: Capital Market Line Model

(Expected return) (Y) E(rOP)

167

Indifference curve Optimal portfolio (OP)

Market portfolio (MP)

Capital market line

Efficient frontier

E(rMP) ΔY = E(rMP) – rF rF

a = rF Δ Y E(rMP) – rF b= = ΔX σMP

a ΔX = σ MP – 0 0% 0%

σMP

σOP

(Standard deviation of returns) (X)

Fig. 5.9 Capital market line (Source: Own illustration)

If a risk-free asset is combined with the market portfolio, the most efficient investment combination in terms of expected return and risk lies on the capital market line.14 Figure 5.9 presents the capital market line, which can be drawn as a tangent, starting from the risk-free return rF to the efficient frontier. If the investorspecific indifference curves are incorporated into the risk–return diagram, the optimal portfolio consisting of the risk-free investment and the market portfolio is obtained. An optimal portfolio on the capital market line with a higher expected return and standard deviation than the market portfolio can be achieved only by borrowing money at the risk-free rate and investing more than 100% in the market portfolio. Since the capital market line is a straight line, the expected return can be determined as the dependent variable (Y ) with a linear function Y = a + bX. The intercept (a) in Fig. 5.9 is given by the risk-free return (rF), while the slope (b) represents the Sharpe ratio of the market portfolio [(E(rMP) - rF)/σ MP]. Therefore, the expected return of the optimal portfolio comprising the risk-free asset and the market portfolio can be calculated as follows:

14

See Tobin (1958): ‘Liquidity preference as behavior towards risk’, p. 65 ff.

168

5

E ðr OP Þ = r F þ

Eðr MP Þ - r F σ OP , σ MP

Optimal Portfolio

ð5:13Þ

where σ OP = wMP σ MP : Passive investors allocate their money on the capital market line depending on their risk aversion. The weight of the market portfolio in the optimal portfolio can be calculated with Eq. (5.11) wMP = ðE ðr MP Þ - r F Þ=Aσ 2MP . Example: Calculation of Capital Allocation, Expected Return, and Risk in the Capital Market Line Model An institutional investor constructs a portfolio consisting of non-interestbearing treasury bills of the Federal Republic of Germany (BuBills) and an exchange-traded fund (ETF) on the DAX 40. The treasury bills have a return of 2%, while the expected return and standard deviation of the DAX 40 ETF are 15% and 30%, respectively. The investor has a risk aversion coefficient of 3. 1. What proportions of the market portfolio (DAX 40 ETF) and the risk-free asset (BuBills) make up the optimal portfolio if the investor's utility is maximised? 2. What are the expected return and standard deviation of returns of the optimal portfolio on the capital market line? 3. What is the investor’s utility? Solution to 1 According to Eq. (5.11), the weight of the market portfolio in the optimal portfolio is 48.15%: wMP =

0:15 - 0:02 = 0:4815: 3 × 0:302

The investor with a risk aversion coefficient of 3 constructs a portfolio, 48.15% of which consists of ETFs on the DAX 40 and 51.85% of BuBills. This portfolio has the highest utility for the passive investor. Solution to 2 The expected return and standard deviation of returns of the optimal portfolio can be determined as follows: (continued)

5.5 Homogeneous Expectations: Capital Market Line Model

169

E ðr OP Þ = 0:4815 × 15% þ 0:5185 × 2% = 8:26%, σ OP = 0:4815 × 30% = 14:45%: The expected return of the optimal portfolio of 8.26% can also be calculated with the following equation: Eðr OP Þ = 2% þ

15% - 2% × 14:45% = 8:26%: 30%

Solution to 3 The utility of the institutional investor is 0.05128: U = 0:0826 - 0:5 × 3 × 0:14452 = 0:05128: The assumption that market participants can borrow money at the same risk-free rate as a government is not realistic. Rather, investors usually have to pay a higher interest rate than the government for borrowing money. With different interest rates for investing and borrowing, the capital market line turns into a kinked line. The slope of the capital market line between the risk–return point of the risk-free asset and the market portfolio is higher [E(rMP - rF)/σ MP] than the slope of the line to the right of the market portfolio [E(rMP - rB)/σ MP] because the interest rate for borrowing money (rB) is higher than that for investing money (rF). Figure 5.10 illustrates the kinked capital market line under the assumption that the tangent portfolio (equal to market portfolio) for investing and borrowing money at the risk-free rate is identical. The expected return of the optimal portfolio where the investor borrows money at the risk-free interest rate can be calculated using the following equation: E ðrOP Þ = r B þ

Eðr MP Þ - r B σ OP , σ MP

ð5:14Þ

where rB = risk-free interest rate for borrowing money. If there are two risk-free interest rates for investing and borrowing money, two tangent portfolios on the efficient frontier are obtained. In Fig. 5.11, L represents the portfolio of risky assets that investors buy when they invest at the risk-free rate. The tangent portfolio B, on the other hand, indicates the risky portfolio when borrowing money at the risk-free interest rate. The market portfolio must lie on the efficient frontier between the two tangent portfolios L and B. This is because investors hold only risky portfolios that are on the efficient frontier between L and B. The expected return of the market portfolio is the average expected returns from the two portfolios

170

5

(Expected return)

E(rMP)

Market portfolio (MP) Investing money at the risk-free rate

Optimal Portfolio

Kinked capital market line Efficient frontier Borrowing money at the risk-free rate

rB rF

0% σMP

0%

(Standard deviation of returns)

Fig. 5.10 Kinked capital market line with identical tangent portfolio for investing and borrowing money at different risk-free interest rates (Source: Own illustration)

(Expected return)

Market portfolio (MP)

G

B

Kinked capital market line

Efficient frontier

L

E(rMP)

rB rF

0% 0%

σMP

(Standard deviation of returns)

Fig. 5.11 Kinked capital market line with two tangent portfolios for investing and borrowing money at different risk-free interest rates (Source: Own illustration)

5.6 Summary

171

L and B and all other risky portfolios that lie on the efficient frontier between L and B. The market portfolio must therefore be on the efficient frontier between the two tangent portfolios L and B. Figure 5.11 presents the kinked capital market line under the realistic assumption that the tangent portfolios for investing and borrowing money are not equal. The kinked capital market line passes through the risk–return points rF – L – MP – B - G.

5.6

Summary

• In financial market theory, it is usually assumed that investors behave in a riskaverse manner and demand a higher expected return for a higher risk. An investor’s utility function reflects this trade-off between expected return and risk. A higher expected return results in a higher utility, while a higher risk leads to a lower utility for a risk-averse investor. The more risk-averse an investor is, the higher the risk aversion coefficient used in the utility function. Hence, investors with higher risk aversion give more weight to risk, which lowers utility. • Utility is represented by indifference curves in a risk–return diagram. An investor is indifferent to investments that lie on the same indifference curve, as these have the same utility. The more (less) risk-averse the investor, the steeper (flatter) the indifference curves. • The point of contact between the efficient frontier and the highest achievable investor-specific indifference curve represents the optimal portfolio of risky assets. The efficient frontier is determined with capital market data using the expected return, standard deviation, and covariance or correlation coefficient, while the indifference curves are constructed with the degree of risk aversion of the investor in addition to the expected return and risk. A highly risk-averse investor will select an optimal portfolio of risky assets on the efficient frontier, with a lower expected return and lower risk than a less risk-averse investor. • If the investment universe is expanded by the possibility of investing and borrowing at the risk-free rate, the most efficient portfolio in terms of expected return and risk between the risk–return point of a risk-free asset and an efficient risky portfolio (tangent portfolio) lies on the most efficient capital allocation line. Since all investors have heterogeneous expectations about the expected return and standard deviation of single assets and the correlation coefficient of pairs of assets, they construct different efficient frontiers, resulting in different efficient capital allocation lines. • It is beneficial to add a new asset to a portfolio if the Sharpe ratio of the new asset is greater than the Sharpe ratio of the current portfolio multiplied by the correlation coefficient between the new asset and the portfolio. The inclusion of the new asset leads to an increase in the risk-adjusted portfolio return. • If all investors have homogeneous capital market expectations they calculate the same expected returns, standard deviations, and correlation coefficients and consequently construct the same efficient frontier. With the inclusion of the risk-free asset, they invest in the same tangent portfolio of risky assets or market

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portfolio. The market portfolio lies on the capital market line, which can be drawn as a tangent, starting from the risk–return point of the risk-free asset to the efficient frontier. • Depending on their risk tolerance, investors can select their utility-maximising portfolio on the capital market line. If they are more risk-averse, they will acquire risk-free investments in addition to the market portfolio. Less risk-averse investors, on the other hand, will borrow additional money at the risk-free rate and invest it in the market portfolio, thus constructing a leveraged portfolio. • If there are two risk-free interest rates for investing and borrowing, two tangent portfolios on the efficient frontier are obtained. The market portfolio lies on the efficient frontier between the two tangent portfolios. The expected return of the market portfolio is the average expected returns from the two tangent portfolios and all other risky portfolios that lie on the efficient frontier between these two portfolios. The capital market line is kinked as a result of the two tangent portfolios and the market portfolio located between them.

5.7

Problems

1. The expected return of the tangent portfolio is 12%, while its standard deviation is 20%. Investing and borrowing money is done at the risk-free interest rate of 4%. a) What is the expected return on a portfolio with a standard deviation of returns of 30% that lies on the most efficient capital allocation line? b) Which parts of the tangent portfolio and the risk-free position does this portfolio comprise? 2. The expected market return and the risk-free interest rate are 8% and 2%, respectively. An investor with assets of EUR 1.8 million wants to earn a return of 10% on a portfolio consisting of a risk-free position and the market portfolio. How much money does the investor need to borrow at the risk-free rate to achieve their target return of 10%? 3. A portfolio manager of a pension fund evaluates three investment funds: a riskless money market fund with an expected return of 2%, an equity fund, and a corporate bond fund. The expected returns and standard deviations of the two risky funds are given as follows: Fund Equity fund Bond fund

Expected return (in %) 15 10

Standard deviation (in %) 28 14

The correlation coefficient between the returns of the two risky funds is 0.2. a) What are the weights of the two risky funds in the minimum variance portfolio and what are the expected return and risk of this portfolio? b) What are the weights of the two risky funds in the tangent portfolio and what are the expected return and risk of this portfolio?

5.7 Problems

173

c) What is the slope of the most efficient capital allocation line passing through the tangent portfolio? d) The portfolio manager is targeting an expected return of 11% with an investment on the most efficient capital allocation line. What is the standard deviation of the returns of this portfolio and what are the weights of the three funds that make up this portfolio? 4. A portfolio manager expects the following returns and standard deviations of returns for the following four investments: Investment 1 2 3 4

Expected return (in %) 9 10 12 9

Standard deviation (in %) 4 16 30 50

The following questions must be answered using the utility function U = Eðr Þ - 0:5 A σ 2 : a) Which of the four investments does a risk-neutral investor select? b) Which of the four investments does a risk-seeking investor with a risk aversion coefficient of -3 select? c) Which of the four investments does an average risk-averse investor with a risk aversion coefficient of 3 prefer? d) Which of the four investments does an above-average risk-averse investor with a risk aversion coefficient of 6 select? 5. The tangent portfolio of risky assets, which lies on the most efficient capital allocation line, has an expected return of 12% and a standard deviation of returns of 30%. The yield on non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills) is 2%. The investor has a risk aversion coefficient of 4. a) What are the weights of the tangent portfolio and the BuBills in the client’s optimal portfolio that lies on the most efficient capital allocation line? b) What are the expected return and standard deviation of returns of the optimal portfolio? c) What is the Sharpe ratio of the optimal portfolio or the slope of the most efficient capital allocation line? 6. The expected returns and standard deviations of returns are given for the following investments: Investment A B

Expected return (in %) 10.0 15.0

Standard deviation (in %) 10.0 30.0 (continued)

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(Expected return) 25%

Optimal Portfolio

C Most efficient capital allocation line

20%

TAP B

15% D

A 10% 5%

E

0% 0%

10%

20%

30% 40% (Standard deviation of returns)

Fig. 5.12 Investments in a risk–return diagram (Source: Own illustration) Investment C D (gold) E (risk-free asset) TAP (tangent portfolio)

Expected return (in %) 24.0 10.8 3.0 18.2

Standard deviation (in %) 30.0 30.0 0.0 21.8

Investments A and C lie on the most efficient capital allocation line and consist of the risk-free asset and the tangential portfolio. Investment B is composed of risky assets, while D is a gold investment. Figure 5.12 presents these investments in a risk–return diagram. The following questions must be answered: a) Which investments are not selected by a rational risk-averse investor? b) Which of these investments will a risk-neutral investor choose? c) Why is gold not on the efficient frontier, even though many rational investors hold it in a diversified portfolio? d) What is the utility of holding the tangent portfolio for an average risk-averse investor with a risk aversion coefficient of 4?

5.8 Solutions

175

e) What are the weights of the risk-free asset and the tangent portfolio in the optimal portfolio on the most efficient capital allocation line for a risk-averse investor with a risk aversion coefficient of 4 and what are the expected return and risk of this optimal portfolio? f) What is the slope of the most efficient capital allocation line?

5.8

Solutions

1. a) E ðr P Þ = 4% þ

12% - 4% × 30% = 16% 20%

b) E ðr P Þ = ð1- wTAP Þr F þ wTAP E ðr TAP Þ, where wTAP = weight of the tangent portfolio in the overall portfolio, E(rTAP) = expected return of the tangent portfolio, and rF = risk-free return. Inserting 16% for the expected portfolio return, 4% for the risk-free return, and 12% for the expected return of the tangent portfolio in the above formula results in the following equation: 16% = ð1- wTAP Þ × 4% þ wTAP × 12%: Multiplying out the right side of the equals sign leads to the equation below: 16% = 4% - 4% × wTAP þ 12% × wTAP = 4% þ 8% × wTAP : The next step is to subtract 4% from both sides of the equals sign: 12% = 8% × wTAP : If both sides of the equals sign are divided by 8%, the weight of the tangent portfolio is obtained:

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wTAP =

Optimal Portfolio

12% = 1:5: 8%

The weight of the tangent portfolio in the new portfolio is 150%. In order to invest 150% of the available capital in the tangent portfolio, 50% of the needed capital must be borrowed at the risk-free interest rate.

2. Since the expected portfolio return of 10% is higher than the expected market return of 8%, the investor has to borrow money and pay interest on it. Thus, the expected portfolio return can be determined as follows: Eðr P Þ = 10% = wMP × 8% þ ð1- wMP Þ × 2%: Solving this equation for the weight of the market portfolio leads to a weight of 1 1/3: wMP = 1 1=3: The weight of the risk-free short position in the total portfolio is -1/3 (= 1 - 1 1/3). Accordingly, the investor must borrow EUR 600,000 (= 1/3 × EUR 1,800,000) at the risk-free interest rate. 3. a) The weight of the equity fund in the minimum variance portfolio of the two risky funds can be calculated using the following equation:

wEquity fund =

σ2Bond fund - covEquity fund, Bond fund : σ2Equity fund þ σ2Bond fund - 2covEquity fund, Bond fund

The covariance of the two funds is 0.00784 and can be determined as follows: covEquity fund, Bond fund = 0:2 × 0:28 × 0:14 = 0:00784:

5.8 Solutions

177

If the covariance and the variances of the corresponding funds are incorporated into the above formula, a weight of 14.29% is obtained for the equity fund in the minimum variance portfolio:

wEquity fund =

0:142 - 0:00784 = 0:1429: 0:28 þ 0:142 - 2 × 0:00784 2

The weight of the bond fund in the minimum variance portfolio is 85.71% (= 1 0.1429). The expected return and standard deviation of the minimum variance portfolio (MVP) can be determined as follows: Eðr MVP Þ = 0:1429 × 15% þ 0:8571 × 10% = 10:71%, σ MVP = 0:14292 × 0:282 þ 0:85712 × 0:142 þ 2 × 0:1429 × 0:8571 × 0:00784 = 13:39%: b) The point of contact between the capital allocation line with the highest Sharpe ratio and the efficient frontier represents the optimal portfolio of risky assets (tangent portfolio). For a two-asset portfolio, the weight of asset 1 (equity fund in the example) can be determined using the following equation: w1 =

½E ðr1 Þ - rF σ 22 - ½E ðr 2 Þ - r F cov1,2 : ½Eðr 1 Þ - r F σ 22 þ ½Eðr 2 Þ - r F σ 21 - ½E ðr 1 Þ - r F þ Eðr 2 Þ - r F cov1,2

The weight of the equity fund is 26.78% and can be calculated as follows:

wEquity fund =

ð0:15 - 0:02Þ × 0:142 - ð0:10 - 0:02Þ × 0:00784 ð0:15 - 0:02Þ × 0:142 þð0:10 - 0:02Þ × 0:282 - ð0:15 - 0:02þ0:10 - 0:02Þ × 0:00784

The weight of the bond fund is 73.22%: wBond fund = 1 - 0:2678 = 0:7322:

= 0:2678:

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The expected return and standard deviation of the tangent portfolio comprising the two funds can be determined as follows: EðrTAP Þ = 0:2678 × 15% þ 0:7322 × 10% = 11:34%, σ TAP = 0:26782 × 0:282 þ 0:73222 × 0:142 þ 2 × 0:2678 × 0:7322 × 0:00784 = 13:86%: c) The slope of the most efficient capital allocation line is given by the Sharpe ratio of the tangent portfolio and is 0.6739:

SRTAP =

11:34% - 2% = 0:6739: 13:86%

The slope of the most efficient capital allocation line is higher than the slope of any other capital allocation line between a portfolio on the efficient frontier and the risk-free asset. d) The expected return of a portfolio that lies on the most efficient capital allocation line can be calculated as follows:

EðrP Þ = r F þ

Eðr OP Þ - r F σOP → 11% = 2% þ 0:6739 × σP : σOP

The portfolio risk of 13.36% is obtained by solving the equation according to σ P: σP =

11% - 2% = 13:36%: 0:6739

The expected return of the portfolio consists of the sum of the weighted expected returns. The portfolio is made up of the risk-free money market fund and the two risky funds (tangent portfolio).

5.8 Solutions

179

Eðr P Þ = wF r F þ ð1- wF ÞEðr TAP Þ 11% = wF × 2% þ ð1- wF Þ × 11:34% wF = 0:0364 wTAP = 1 - 0:0364 = 0:9636 The weights of the two risky funds in the tangent portfolio can be calculated as follows: wEquity fund = 0:2678 × 0:9636 = 0:2581, wBond fund = 0:7322 × 0:9636 = 0:7055: The portfolio with an expected return of 11%, which lies on the most efficient capital allocation line, is made up of the three funds in the following proportions: Weight of risk-free money market fund + Weight of equity fund + Weight of bond fund = Total weight

3.64% +25.81% +70.55% =100.00%

4. a) A risk-neutral investor has a risk aversion coefficient of 0 (A = 0). The investor’s utility is therefore given by the expected return [U = E(r)]. Investment 3 has the highest utility for a risk-neutral investor as it has the highest expected return of 12%. b) U 1 = 0:09 - 0:5 × ð- 3Þ × 0:042 = 0:0924 U 2 = 0:10 - 0:5 × ð- 3Þ × 0:162 = 0:1384 U 3 = 0:12 - 0:5 × ð- 3Þ × 0:302 = 0:255 U 4 = 0:09 - 0:5 × ð- 3Þ × 0:502 = 0:465

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Optimal Portfolio

Investment 4 has the highest utility of 0.465 for a risk-seeking investor with a risk aversion coefficient of -3. c) U 1 = 0:09 - 0:5 × 3 × 0:042 = 0:0876 U 2 = 0:10 - 0:5 × 3 × 0:162 = 0:0616 U 3 = 0:12 - 0:5 × 3 × 0:302 = - 0:015 U 4 = 0:09 - 0:5 × 3 × 0:502 = - 0:285 Investment 1 provides the highest utility of 0.0876 for an average risk-averse investor with a risk aversion coefficient of 3. d) U 1 = 0:09 - 0:5 × 6 × 0:042 = 0:0852 U 2 = 0:10 - 0:5 × 6 × 0:162 = 0:0232 U 3 = 0:12 - 0:5 × 6 × 0:302 = - 0:150 U 4 = 0:09 - 0:5 × 6 × 0:502 = - 0:660 Investment 1 has the highest utility of 0.0852 for an above-average risk-averse investor with a risk aversion coefficient of 6. 5. a) The optimal weight of the tangent portfolio in the overall portfolio is 27.78% and can be calculated as follows:

w2TAP =

E ðr TAP Þ - r F 0:12 - 0:02 = = 0:2778: Aσ 2TAP 4 × 0:302

Accordingly, 27.78% of the optimal portfolio consists of the tangent portfolio and 72.22% of BuBills (risk-free investment). This portfolio has the highest utility for the investor with a risk aversion coefficient of 4. b)

5.8 Solutions

181

E ðr OP Þ = 0:2778 × 12% þ 0:7222 × 2% = 4:78% σ OP = 0:2778 × 30% = 8:33% c) The Sharpe ratio of the optimal portfolio is 0.334:

SROP =

4:78% - 2% = 0:334: 8:33%

6. a) Investments B and D are not selected by a rational risk-averse investor because investment C has a higher expected return for the same risk. Portfolio C lies on the most efficient capital allocation line and consists of a short risk-free asset position (borrowing money at the risk-free interest rate) and the tangent portfolio. b) Portfolio C is best suited for a risk-neutral investor because this investment on the most efficient capital allocation line has the highest expected return. Risk is not an issue. c) A portfolio consisting of 100% gold is not held by investors because gold allows us to diversify the portfolio well due to the low correlation coefficient with other asset classes such as equities, bonds, real estate, and private equity. Gold has a high risk of 30% and a relatively low expected return of 10.8%. Accordingly, gold is less efficient in terms of expected return and risk than, for example, Portfolio C, which lies on the most efficient capital allocation line and has a higher expected return for the same risk. d) The utility of the tangent portfolio for an average risk-averse investor with a risk aversion coefficient of 4 is 0.0870: U = 0:182 - 0:5 × 4 × 0:2182 = 0:0870: e)

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Optimal Portfolio

The weight of the tangent portfolio in the optimal portfolio is 79.96%: wTAP =

E ðr TAP Þ - r F 0:182 - 0:03 = = 0:7996: Aσ 2TAP 4 × 0:2182

Thus, the optimal portfolio consists of 79.96% of the tangent portfolio and 20.04% of the risk-free asset. This portfolio has the highest utility for the investor with a risk aversion coefficient of 4. The expected return of the optimal portfolio is 15.15% and can be calculated as follows: E ðrOP Þ = 0:7996 × 18:2% þ 0:2004 × 3% = 15:15%: The risk of the optimal portfolio is 17.43% and can be determined as follows: σ OP = 0:7996 × 21:8% = 17:43%: f) The slope of the most efficient capital allocation line corresponds to the Sharpe ratio of the optimal portfolio and is 0.697:

SROP =

15:15% - 3% = 0:697: 17:43%

Alternatively, the slope of the most efficient capital allocation line can be determined at the risk–return point of the tangent portfolio as follows:

Slope =

18:2% - 3% = 0:697: 21:8% - 0%

Microsoft Excel Applications This section demonstrates how Microsoft Excel can be used to determine the weights of the individual investments in the tangent portfolio for the most efficient capital allocation line. The most efficient capital allocation line has the highest slope, or the highest Sharpe ratio, of all possible capital allocation lines that can be determined by

Microsoft Excel Applications

183

combining the risk-free asset with a portfolio of risky assets on the efficient frontier. Determining the optimal tangent portfolio is an optimisation problem in which the objective function consists of maximising the Sharpe ratio while satisfying the constraint that the sum of the weights equals 1. If no short positions are allowed in the portfolio, the additional constraint that the weights must be positive is required. This optimisation problem is presented using the five DAX 40 stocks of MercedesBenz Group AG, Linde AG, Siemens AG, Bayer AG, and Adidas AG for the monthly share prices from the end of July 2012 to the end of July 2017 (Source: Refinitiv Eikon; since the beginning of March 2023, Linde shares have no longer been listed in the DAX 40). The input parameters required for this consist of the expected return and the variance of the five equity securities, as well as the covariances between returns of two securities. The optimisation is performed with Excel Solver so that a numerical (and not an analytical) solution is found. Excel Solver allows a large number of constraints to be entered in the dialog box. Three steps are necessary to determine the maximum Sharpe ratio and the optimal composition of the tangent portfolio. First, all input parameters—such as the expected returns, standard deviations and variances of the individual securities, the covariances between two securities, and the risk-free return—must be entered. Second, the expected return and standard deviation as well as the Sharpe ratio of the portfolio must be calculated using the input parameters. In the third and final step, Excel Solver is used to calculate the optimal portfolio allocation (i.e. the optimal weighting of each security), expected return, standard deviation, and Sharpe ratio of the optimal tangent portfolio. Figure 5.13

Fig. 5.13 Composition and Sharpe ratio of the optimal tangent portfolio consisting of the five DAX 40 stocks of Mercedes-Benz Group AG, Linde AG, Siemens AG, Bayer AG, and Adidas AG from the end of July 2012 to the end of July 2017 (Source: Own illustration based on data from Refinitiv Eikon)

184

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Optimal Portfolio

presents the calculation of the optimal tangent portfolio of the five DAX 40 stocks in Microsoft Excel. The numbers not highlighted in a colour reflect the input parameters, while the numbers in the cells highlighted in a colour contain the calculations. Cell H19 (risk of the optimal tangent portfolio) and column J (portfolio allocation, expected return, and Sharpe ratio of the optimal tangent portfolio) indicate the output range. The following steps must be carried out in order to determine the optimal tangent portfolio with the maximum Sharpe ratio: • First step: Entering the input parameters (monthly values) – The risk-free return of 0.001 should be inserted in cell C4. – The expected returns of the five DAX 40 stocks should be inserted in cells C8 to G8. – The standard deviations of the five DAX 40 stocks should be inserted in cells C11 to G11. – The variance-covariance matrix should be inserted in cells C14 to G18. – The weights should be inserted in cells J5 to J9, where the sum of the weights is 1, so that the calculations can be initialised with Excel Solver. • Second step: Calculations – The expected portfolio return in cell J14 can be determined as follows: =J5*C8+J6*D8+J7*E8+J8*F8+J9*G8. – To calculate the standard deviation of the portfolio returns in cell H19, the variances in cells C19 to G19 must first be estimated. The variance in cell C19 is calculated as follows: =J5*sumproduct(J5:J9;C14:C18). For cell D19, enter the following expression: =J6*sumproduct(J5:J9;D14:D18). The calculations for the variances in cells E19 to G19 are performed analogously. – Cell H19 contains the standard deviation of portfolio returns, which comprises the variances from cells C19 to G19: =sum(C19:G19)^0.5. – The Sharpe ratio in cell J17 is obtained as follows: =(J14-C4)/H19. • Third step: Excel Solver – Set objective (i.e. Sharpe ratio): J17 – to: Max – by changing variable cells: J5:J9, subject to the constraints: J10 = 1 and J5:J9 >= 0. In order for the optimisation to be initialised with Excel Solver, the weights should first be entered, where the sum of the weights equals 1. A practical rule of thumb is that the initial weights of the individual investments are 1/N. Here, N represents the number of assets in the portfolio, so that the portfolio consists of equally weighted securities. In the example, the portfolio is composed of five DAX 40 stocks, each with an initial weight of 20%. Excel Solver is activated via the ‘Data’ tab. ‘Solver’ should then be selected (on the right of the toolbar). A dialogue box opens in which five entries are required. First, to set the objective the Sharpe ratio should be entered, which is located in cell ‘J17’. Then click on ‘Max’. For the changing variable cells, enter the output range of

References

185

the optimal portfolio allocation, which is ‘J5:J9’. Finally, the constraints of the objective function (maximising the Sharpe ratio) should be recorded. The first constraint is that the sum of the weights equals 1. To enter this constraint, click on ‘Add’. The next step is to fill in the three windows in the dialogue box. In the first cell reference window, type in ‘J10’. In the second cell select ‘=’, and in the constraint window type in ‘1’. After the entry has been made, press ‘OK’. The second constraint is that the weights must be positive in order to avoid short positions in the optimal tangent portfolio. Once again, click on ‘Add’ in the dialogue box for the constraints. Type ‘J5:J9’ in the cell reference window. Then select ‘>=’. Enter ‘0’ in the constraint window and confirm with ‘OK’. Click on ‘Solve’ in the dialogue box. If the solution is to be used, this must be confirmed by selecting ‘OK’. The maximum Sharpe ratio is 0.26, while the Mercedes-Benz Group and Linde stocks each have a weight of 0%, the weight of the Siemens stock is 15.1%, the Bayer stock 10.9%, and the Adidas stock 74.0%. Thus, the optimal tangent portfolio, which lies on the most efficient capital allocation line, is composed of only three long equity securities, and the sum of their weights is 1.

References Elton, E.J., Gruber, M.J., Rentzler, J.C.: Professionally managed, publicly traded commodity funds. J. Bus. 60(2), 175–199 (1987) Gwartney, J.D., Stroup, R.L., Sobel, R.S.: Economics, Private and Public Choice, 9th edn. SouthWestern Cengage Learning, Mason (2000) Mondello, E.: Portfoliomanagement: Theorie und Anwendungsbeispiele, 2nd edn. Springer, Wiesbaden (2015) Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 6th edn. South-Western Cengage Learning, Jefferson City (2000) Tobin, J.: Liquidity preference as behavior towards risk. Rev. Econ. Stud. 25(2), 65–86 (1958)

6

Capital Asset Pricing Model and FamaFrench Model

6.1

Introduction

Investors want to be compensated for taking higher risk by a higher expected return. This raises the question of the level of return compensation. In financial market theory, this question is answered by one-factor and multifactor models which determine the expected return of a single asset or a portfolio of assets with one or more systematic risk factors. The most widely used model is probably the capital asset pricing model (CAPM). With this one-factor model, which is typically applied to equity securities, the expected return of a stock or stock portfolio is calculated by adding a risk premium to the risk-free rate. The former is the product of the expected equity market risk premium and the beta of the investment. The higher (lower) the systematic risk or market risk of the investment, the higher (lower) the beta and hence the expected return. However, empirical studies on equity securities demonstrate that stock returns are correlated not only with equity market returns but also with other factors. Two of these risk factors are the size of the firm (measured by market capitalisation) and the book-to-price ratio, which were captured by Eugène Fama and Kenneth French in a multifactor model. The Fama–French model (FFM) is a three-factor model that explains expected returns in terms of risk premiums and the corresponding betas for market, size, and value. Both the CAPM and the FFM are based on the assumption that investors are compensated by a premium when they assume systematic risk. Hence, only systematic risk is relevant to valuation. These two models differ in how systematic risk is measured. In the CAPM, the systematic risk is given by the market portfolio, whereas the FFM uses size and value as systematic risk factors in addition to the market portfolio. This chapter examines these two models.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_6

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6.2

Capital Asset Pricing Model

6.2.1

Basics of the Model

The CAPM represents one of the most important innovations in financial market theory.1 The model is straightforward and intuitive in its comprehensibility and application, as only one systematic risk factor is used to determine the expected return of an investment. This factor consists of the product of the expected market risk premium and the beta of the investment. The expected CAPM return can be calculated as follows: Eðr i Þ = r F þ MRPβi ,

ð6:1Þ

where E(ri) = expected return on investment i, rF = risk-free interest rate, MRP = expected market risk premium, and βi = beta of the investment i. The riskier the investment, the higher the beta, and hence the higher the expected return. The relationship between the expected return and the beta (risk) is linear. The CAPM assumes that the expected return of an investment depends solely on systematic risk or beta and not on total risk or standard deviation. Investors can diversify their portfolios in such a way that unsystematic or firm-specific risk is no longer a factor in calculating returns.2 For example, two investments with identical beta have the same expected return because they have the same market risk. Like other models, the CAPM is based on simplifying assumptions and largely ignores the complexity that characterises financial markets. These assumptions make it possible to gain initial insights into the pricing of assets. The assumptions of the CAPM are as follows: • • • •

1

Investors behave in a rational and risk-averse manner and maximise their utility. Markets are frictionless and there are no transaction costs or taxes. All investors plan for the same investment period. All investors have homogeneous (the same) capital market expectations, and therefore, the strong form of information efficiency holds in financial markets

Markowitz’s 1952 portfolio model laid the foundation for modern portfolio theory. Some 12 years later, the theory was further developed into the CAPM by Wiliam Sharpe, John Lintner, and Jan Mossin. See Sharpe (1964): ‘Capital asset prices: a theory of market equilibrium under conditions of risk’, p. 425 ff; Lintner (1965): ‘The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets’, p. 13 ff; Mossin (1966): ‘Equilibrium in a capital asset market’, p. 768 ff. 2 See Sect. 4.5.

6.2 Capital Asset Pricing Model

189

and the optimal risky portfolio is given by the market portfolio of the capital market line model. • All assets are infinitely divisible and tradable. • Investors are price takers, so no individual investor can change prices in the market by buying and selling an asset. The objective of these assumptions is to define an investor who selects a particular portfolio that is efficient in terms of expected return and risk. Market inefficiencies arising from operational (transaction costs, taxes, etc.) and informational inefficiencies are excluded from the CAPM. Although some of these assumptions are unrealistic, their removal leads to only small changes in the explanatory power of the model.3 The CAPM has gained acceptance in practice despite its partially unrealistic assumptions. Although the CAPM can, in principle, be used to determine expected returns for any asset, it is usually used only for equities. Therefore, the focus of the following discussion is on equities.

6.2.2

Calculation and Interpretation of the Beta

Beta represents the measure of risk in the CAPM. To calculate beta, a simple linear regression analysis is performed in which a regression is carried out between the returns of the equity security as the dependent variable and the returns of the equity market as the independent variable. The regression equation is as follows: r i,t = αi þ βi r M,t þ εi,t ,

ð6:2Þ

where ri,t = return of stock i for time period t, αi = constant of the regression equation (intercept of the regression line), βi = regression coefficient for the equity market return of the regression equation (slope of the regression line) or beta of stock i, rM,t = return of the equity market for time period t, and εi, t = error term of residual returns of stock i for time period t. The slope of the regression line reflects the beta of the equity security, which measures the change in the stock return with respect to a change in the stock market return. Accordingly, beta is a measure of the market risk or systematic risk of an equity security. Figure 6.1 illustrates the estimation of beta using simple linear regression analysis.

3

For the relaxing of assumptions in the CAPM and their impact on the model, see, for example, Reilly and Brown (2000): Investment Analysis and Portfolio Management, p. 314 ff.

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(Returns of stock ri) Regression line Δ ri Δ rM

Slope

∆ri ∆rM

beta

ri,1 εi,1 (Returns of stock market rM)

ri',1

Fig. 6.1 Estimation of the historical beta (Source: Own illustration)

The regression line is determined using the least squares method. This method minimises the vertical distances between the observed returns (ri,t) and the corresponding values on the regression line r 0i,t , or the residual deviations (εi, t). T t=1

T

ε2i,t =

t=1

ðr i,t - r 0i,t Þ → minimise 2

ð6:3Þ

The slope of the regression line reflects the beta of the equity security and can be calculated with the following formula.4 βi =

covðr i , r M Þ , σ 2M

ð6:4Þ

where

4

According to the least squares method the regression line runs through the arithmetic mean of the X-values X and the arithmetic mean of the Y-values Y . The X-value corresponds to the independent variable (rM), while the Y value reflects the dependent variable (ri). The function of the regression line is: Y′ = a + bX. The regression coefficient b can be calculated as follows: ½ðX - X ÞðY - Y Þ covX,Y = σ2 : b= 2 X ðX - X Þ

6.2 Capital Asset Pricing Model

191

cov(ri, rM) = covariance between the returns of stock i and the returns of the equity market, and σ 2M = variance of equity market returns. The beta can be determined not only with the covariance but also with the correlation coefficient. The correlation coefficient is given by the standardised covariance:5 ρi,M =

covðr i , r M Þ , σiσM

ð6:5Þ

where σ i = standard deviation of the returns of stock i, and σ M = standard deviation of the equity market returns. If the equation is solved for covariance, cov(ri, rM) = ρi, M σ i σ M is obtained. Substituting the expression for the covariance into Eq. (6.4) yields the following equation for the beta of a stock i: βi =

ρi,M σ i σ M ρi,M σ i = : σM σ 2M

ð6:6Þ

Thus, the beta of an equity security can be calculated either with the covariance (see Eq. 6.4) or with the correlation coefficient (see Eq. 6.6). Both equations for calculating the beta indicate that beta is a sensitivity measure. It measures how much the stock return changes when the stock market return changes. For example, if the beta of an equity security is 1.2 and the stock market return increases by 2%, then the return of the security will increase by 2.4% (= 1.2 × 2%). The beta reflects the systematic risk of the stock, or that portion of the risk that cannot be eliminated through diversification. A positive beta implies that stock returns move in the same direction as the stock market returns. A negative beta, on the other hand, means that returns of the equity security move in the opposite direction from the stock market. A risk-free investment has a beta of 0 because the covariance with other investments or with the market will be 0. By contrast, the beta of the market is 1 (βM = 1). Replacing the standard deviation of the equity security (σ i) with that of the market (σ M) in the numerator of Eq. (6.6), and taking into account that the correlation of the market with itself is 1 (ρM, M = 1), a beta of 1 results for the market: βi =

5

See Sect. 4.2.

ρi,M σ i σ M ρM,M σ M = = 1: σM σ 2M

ð6:7Þ

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6 Capital Asset Pricing Model and Fama–French Model

The market’s beta of 1 can also be explained by the fact that the average of all the betas of the equity securities traded in the market is 1. A majority of traded securities have a positive beta because their returns move in the same direction as the overall stock market. Stocks with a negative beta tend to be the exception. Example: Calculation of Beta The volatility (standard deviation) of the market is 25%. An analyst wants to calculate the beta for the following investments: 1. Non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills). 2. Gold with a volatility of 30% and a correlation coefficient to the overall market of -0.4. 3. Stock with a volatility of 45% and a correlation coefficient to the overall market of 0.7. Solution to 1 The non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills) are short-term risk-free government securities. The return on these investments is known, and therefore, the standard deviation and the correlation coefficient to the overall market are 0. Hence, the beta of BuBills is also 0. Solution to 2 The beta of gold is -0.48 and can be calculated as follows: βGold =

- 0:4 × 30% = - 0:48: 25%

For example, if the overall market falls by 5%, then the return on gold rises by 2.4% [= (-0.48) × (-5%)]. Solution to 3 The beta of the equity security is 1.26: βEquity security =

0:7 × 45% = 1:26: 25%

If the stock market rises by, say, 2%, the stock return increases by 2.52% (= 1.26 × 2%). Figure 6.2 presents the regression line for the Mercedes-Benz Group stock, which has a slope or beta of 1.455. The simple linear regression analysis is based on 60 monthly continuous compounded returns for the monthly prices of the

6.2 Capital Asset Pricing Model

193

(Returns of the Mercedes-Benz Group stock) 20% Regression line 15% 10% 5%

0% -20%

-15%

-10%

-5%

0% -5% -10% -15% -20%

5%

10%

15% 20% (Returns of the DAX)

Regression stasc: • Constant (intercept): −0.0077 • Beta (slope): 1.455 • R2 = 0.6935 • Standard error (SEE) = 0.0412 • Standard error beta = 0.1270 • t-stasc beta = 11.46 • t-stasc constant = −1.42

Fig. 6.2 Beta of the Mercedes-Benz Group stock (Source: Own illustration based on data from Refinitiv Eikon)

automobile stock and the DAX 30 from the end of July 2012 to the end of July 2017 (a time period of 5 years; Source: Refinitiv Eikon).6 Since the DAX 30 (now DAX 40) is a performance index, the returns of the Mercedes-Benz Group stock include dividends in addition to capital gains and losses. However, if the equity index is a price index (e.g. the SMI), the stock returns without dividends should be used in the regression analysis. This ensures that the returns employed for the regression are consistent. The following three decisions must be made in a regression: 1. Length of the time period for the regression 2. Periodicity of returns 3. Market index 1. The longer the data series, the more returns are available for the regression. This leads to a smaller standard error. However, the risk situation of a company can also change (e.g. due to a change in the business model, an acquisition, or higher financial risk due to a higher debt ratio), with the result that a long time period no longer accurately reflects the current risk. Therefore, a period of 5 years is usually

6

The coefficient of determination is 0.6935. Accordingly, the return changes in the DAX 30 explain approximately 69% of the return changes in the Mercedes-Benz Group stock. The slope of the regression line has a t-statistic of 11.46. The critical t-value at T - 2 or at around 60 degrees of freedom and a significance level of 5% is approximately 2. Therefore, the slope is statistically significant at the 5% significance level.

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6 Capital Asset Pricing Model and Fama–French Model

used for the regression of the return data, which represents a compromise between a time period that is too long and one that is too short. 2. Stock returns are available on an annual, monthly, weekly, daily, and intra-day basis. Daily or intra-day returns increase the number of observations in the regression, but this can lead to a wrong beta because there are certain days or hours when the security is not traded. In particular, small firms may be affected by a stock not trading if daily returns are employed in the regression. Annual returns are not used because there are not sufficient return observations with a time horizon of 5 years for a statistically significant analysis. Hence, weekly or monthly returns should be utilised in the regression. In practice, a regression over a 5-year period with 60 monthly returns is often used. The CAPM is based on the assumption that all investors have homogeneous capital market expectations and thus select the same risky portfolio, which is given by the market portfolio of the capital market line model. The market portfolio includes by definition all tradable risky assets. However, such a market index does not exist, and therefore, the stock market index of the home country in which the stock is traded and is also part of the index is usually selected. For example, the Financial Times Stock Exchange (FTSE) is used for UK shares, the Nikkei for Japanese shares, the SMI or Swiss Performance Index (SPI) for Swiss shares, and the New York Stock Exchange Composite (NYSE) or the Standard & Poor’s 500 (S&P 500) for US shares. The beta of an equity security moves towards its expected value of 1 in the long run. This empirical observation can be explained by the fact that the company will build a diversified product line over time and is thus exposed to the influences of the overall stock market. Moreover, the average beta of all equity securities in the market will be 1. Therefore, the best estimate of the expected beta represents the average value of 1. A formula commonly used to adjust the historical beta against the expected long-term value of 1 is the following:7 βi, Adjusted = a þ bβi ,

ð6:8Þ

where a = 0.333, b = 0.667, and βi = beta of the stock i from the regression with historical return data. For example, the beta of the Mercedes-Benz Group stock from the simple linear regression analysis of 1.455 can be converted to an adjusted beta of 1.303 (= 0.333 + 0.667 × 1.455) by means of the equation above. Although in practice the adjusted beta is often calculated with the coefficients a = 0.333 and b = 0.667, it has not been empirically proved that these two coefficients are the best values for correcting the 7

See Blume (1971): ‘On the assessment of risk’, p. 8 ff.

6.2 Capital Asset Pricing Model

195

(Expected return) (Y)

Security market line

M

E rM ∆Y

E rM

rF

Slope a

∆X

∆Y ∆X

E rM 1–0

E rM

1–0

0% 0

βM = 1

(Beta) (X)

Fig. 6.3 Security market line (Source: Own illustration)

beta towards 1. The correction of the beta to its expected value of 1 is primarily relevant for the determination of a long-term expected return, as is the case, for example, for the calculation of the cost of equity in corporate finance or of the discount rate in a cash flow model to determine the intrinsic value of a stock.8

6.2.3

The Security Market Line

The security market line (SML) is a graphical representation of the CAPM, with beta as a measure of risk on the X-axis and expected return on the Y-axis. The relationship between risk and expected return is linear in the CAPM. The SML can be drawn once one is able to define two risk–return points in the diagram. One of the assumptions in the CAPM is that investors can lend and borrow money at an identical risk-free rate. This leads to the first risk–return point of rF in Figure 6.3. Another assumption of the CAPM is that investors have homogeneous capital market expectations and therefore invest in the same optimal portfolio of risky assets, which is the market portfolio. The risk–return point of the market portfolio M is given by its beta of 1 and its expected return. The SML passes through the two risk–return points rF and M and is illustrated in Fig. 6.3. The SML is a straight line given by the equation Y = a + bX. The dependent variable Y represents the expected return of asset i, while the independent variable X reflects the beta of asset i. In Fig. 6.3, the intercept (a) is the risk-free interest rate 8

See Sect. 6.2.5.

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6 Capital Asset Pricing Model and Fama–French Model

(rF), while the slope of the SML (b) corresponds to the difference between the expected return of the market portfolio and the risk-free interest rate. Thus, in the CAPM the expected return of an investment i can be calculated as follows: Eðr i Þ = r F þ ½E ðr M Þ - r F βi :

ð6:9Þ

The CAPM demonstrates that the primary determinant of an investment’s expected return is beta, or the measure of how strongly the investment’s returns correlate with market returns. There is a positive relationship between expected return and beta. Equity securities with a beta greater than 1 have an expected return that is higher than that of the stock market. By contrast, equity securities with a beta of less than 1 have a lower expected return than the stock market. If the security has a negative beta, this can lead to an expected return that is below the risk-free interest rate. The capital market line model and the capital allocation line model make it possible to determine the expected return and risk of an efficient portfolio consisting of the risk-free position and a portfolio of risky assets lying on the efficient frontier. The CAPM, on the other hand, can be applied to calculate the expected return and risk not only of portfolios but also of individual assets. Another important difference is in the risk figure used. The capital allocation line model and the capital market line model define risk in terms of standard deviation, which is a measure of total risk. The portfolios of risky assets in the two models that lie on the efficient frontier are well diversified, and the total risk is therefore equal to the non-diversifiable systematic risk. By contrast, risk in the CAPM is calculated with beta, which is a measure of market risk and not of total risk. Example: Calculation of Expected Return with the CAPM The market portfolio has an expected return of 10% and a standard deviation of returns of 25%. The risk-free interest rate is 2%. 1. An equity security has a standard deviation of returns of 30%. The correlation coefficient between the security and the stock market returns is 0. What is the expected return of the equity security using the CAPM? 2. Another equity security has a standard deviation of returns of 30%, with a correlation coefficient between security and stock market returns of 0.6. What is the expected return of the stock using the CAPM? Solution to 1 The beta of the equity security is 0 and can be calculated as follows: βEquity security =

0 × 30% = 0: 25%

The expected return on the stock is 2%: (continued)

6.2 Capital Asset Pricing Model

197

E r Equity security = 2% þ ð10% - 2%Þ × 0 = 2%: Since the stock has no market risk, the risk premium (i.e. the market risk premium multiplied by beta) will be 0%. The expected CAPM return of the security is thus equal to the risk-free interest rate of 2%. Solution to 2 The beta of the equity security is 0.72 and can be determined as follows: βEquity security =

0:6 × 30% = 0:72: 25%

The expected CAPM return of the stock is 7.76%: E rEquity security = 2% þ ð10% - 2%Þ × 0:72 = 7:76%: The stock’s beta of 0.72 is lower than the beta of the market portfolio of 1, and therefore, the stock’s expected CAPM return of 7.76% is less than the expected stock market return of 10%. The CAPM can be applied to calculate the expected return of individual stocks as well as of a portfolio of stocks. If there is an equity portfolio, the expected portfolio return can be determined using the following equation: E ðrP Þ = r F þ ½Eðr M Þ - r F βP ,

ð6:10Þ

where βP = beta of the equity portfolio. The beta of the stock portfolio is the sum of the weighted betas of the individual securities in the portfolio:9 N

βP =

wi βi ,

ð6:11Þ

i=1

where

9

The expected return of a two-asset portfolio can be calculated using the following formula: E(rP) = w1E(r1) + w2E(r2). Substituting into the equation for E(r1) and E(r2) the expected CAPM return of E(ri) = rF + [E(rM) - rF]βi results in the following equation for the expected return of the portfolio: E(rP) = w1rF + w1β1 [E(rM) - rF] + w2β2 [E(rM) - rF] = rF + (w1β1 + w2β2) [E(rM) rF]. The equation demonstrates that the beta of the two-asset portfolio is the sum of the weighted betas of the two assets (βP = w1β1 + w2β2).

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6 Capital Asset Pricing Model and Fama–French Model

wi = weight of stock i in the portfolio based on market values, βi = beta of stock i, and N

wi = 1: i=1

Example: Expected Return and Beta of a Portfolio A portfolio manager has invested EUR 0.5 million in BuBills, which have an expected return of 1%. In addition, the portfolio consists of exchange traded funds on the HDAX (the market portfolio) with a market value of EUR 1 million. The expected return and standard deviation of the ETFs on the HDAX are 10% and 30%, respectively. The portfolio also contains MercedesBenz Group shares with a beta of 1.455. The market value of the MercedesBenz Group shares is EUR 1 million. What is the expected return of this portfolio using the CAPM? Solution First, the portfolio beta has to be calculated. The weight of the BuBills in the portfolio is 20% (= EUR 0.5 million/EUR 2.5 million), while the market portfolio and the position of Mercedes-Benz Group shares each have a weight of 40% (= EUR 1 million/EUR 2.5 million). The BuBills have a beta of 0, whereas the ETFs on the HDAX have a beta of 1. The portfolio beta of 0.982 can be determined as follows: βP = 0:2 × 0 þ 0:4 × 1 þ 0:4 × 1:455 = 0:982: The expected CAPM return of the portfolio is 9.838%: EðrP Þ = 1% þ ð10% - 1%Þ × 0:982 = 9:838%: Alternatively, the expected portfolio return of 9.838% can be estimated by first calculating the expected CAPM returns of the individual assets. In a second step, the expected portfolio return is obtained by adding the weighted single asset returns. Eðr BuBills Þ = 1% þ ð10% - 1%Þ × 0 = 1% EðrETF HDAX Þ = 1% þ ð10% - 1%Þ × 1 = 10% E r Mercedes - Benz Group = 1% þ ð10% - 1%Þ × 1:455 = 14:095% E ðrP Þ = 0:2 × 1% þ 0:4 × 10% þ 0:4 × 14:095% = 9:838%

6.2 Capital Asset Pricing Model

6.2.4

199

Equilibrium Model

The CAPM is based on the assumption that investors have homogeneous capital market expectations and behave in a rational and risk-averse manner that maximises utility. These assumptions lead to all investors calculating identical values for the individual assets and consequently constructing the same optimal risky portfolio, that is, the market portfolio. If the expected cash flows of an asset are discounted by the expected return, the price (intrinsic value) can be determined. If all investors have the same expectations regarding the future cash flows and the required return, they arrive at the same intrinsic value. The CAPM’s assumption of strong market information efficiency (homogeneous expectations) and operational market efficiency (i.e. no transaction costs) means that all assets are correctly valued and therefore lie on the SML. In equilibrium, all individual assets and portfolios lie on the SML. An investment such as an equity security is undervalued if its market price is lower than its intrinsic value. The share price is too low because the expected return for a given beta is too high relative to the required CAPM return. By contrast, a stock is overvalued if the expected return for a given beta is below the SML. When equity securities are undervalued, market participants buy them, causing the price to rise and the expected return to fall. The price correction in the market takes place until the mispricing disappears. This equilibrium in the model is achieved when the expected return is on the SML. On the other hand, if a stock is overvalued, it is sold. This leads to a lower price, and as a result, the expected return increases. Once equilibrium has been restored by the sales, the expected return of the equity security lies on the SML. This relationship is illustrated in Fig. 6.4. (Expected return) Security market line

A E(rA) E(rA, CAPM) M

Positive alpha, security is undervalued

E(rM) E(rB, CAPM)

Negative alpha, security is overvalued

E(rB) rF

B

0

βB = 0.8 βM =1 βA = 1.4

Fig. 6.4 CAPM as an equilibrium model (Source: Own illustration)

(Beta)

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6 Capital Asset Pricing Model and Fama–French Model

In Fig. 6.4, security A is undervalued because the expected return of the asset is higher than that which can be determined using the CAPM. The alpha—given by the vertical distance between the risk–return point A or B and the SML—is positive in the case of an undervalued asset, while an overvalued asset has a negative alpha. The portfolio manager can determine the expected return of a stock by its projected capital gain or loss and dividend yield. The alpha of an investment is calculated by deducting the required CAPM return from the investors’ expected return: ðP - P0 Þ D1 þ  - ½rF þ ðEðr M Þ - r F Þβ, Alpha = ½ 1 P0 P0

ð6:12Þ

where P0 = share price at the beginning of the period, P1 = estimated share price (intrinsic value) at the end of the period using valuation models, and D1 = expected dividend per share at the end of the period. If investors buy the undervalued stock A, its price increases. If a higher price for P0 is incorporated into the above equation, the expected return falls, resulting in a lower alpha. This buying process lasts until the alpha is 0% or the security is priced correctly, that is, market price = intrinsic value. By contrast, an overvalued stock is sold so that the price P0 decreases, leading to a higher expected return. This selling process continues until the expected return settles back on the SML or the alpha is 0%. Thus, the CAPM is an equilibrium model. This equilibrium exists in an efficient market in which, among other things, all available information is processed in the prices and therefore there are no mispricings. In a market that is not efficient, however, there are undervalued and overvalued assets. If an investor is able to identify mispriced assets, they can achieve an excess or abnormal return (alpha). The condition for this is that equilibrium is restored after the mispricing has been identified. Example: Determining Overvalued and Undervalued Equity Securities with the CAPM On 18 August 2017, a sell-side analyst examines the two equity securities of Mercedes-Benz Group AG and Linde AG applying the CAPM. Using fundamental analysis, they determine the following intrinsic share values and dividends per share in 1 year’s time for the two securities: Stock Mercedes-Benz Group AG Linde AG

Current share price (P0) EUR 67.10

Intrinsic share value in 1 year’s time (P1) EUR 70.05

Dividend per share in 1 year’s time (D1) EUR 3.45

EUR 165.05

EUR 187.45

EUR 3.95

The analyst employs simple linear regression analysis to calculate the betas of the two securities. The beta of Mercedes-Benz Group stock is 1.455, while (continued)

6.2 Capital Asset Pricing Model

201

the beta of Linde stock is 1.1. Non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills) with a time to maturity of 1 year have a return of -0.2%. The DAX is expected to yield 8%. 1. What is the alpha of the Mercedes-Benz Group stock and what is the investment recommendation? 2. What is the alpha of the Linde stock and what is the investment recommendation? Solution to 1 The alpha of the Mercedes-Benz Group stock is –2.19%: AlphaMercedes - Benz Group =

ðEUR 70:05 - EUR 67:10Þ þ EUR 3:45 EUR 67:10

- ½- 0:002 þ ð0:08 - ð- 0:002ÞÞ × 1:455 = -2:19%: Therefore, the automobile stock appears to be overvalued, and the sell-side analyst will issue a sell recommendation. Solution to 2 The alpha of the Linde stock is 7.14% and can be calculated as follows: AlphaLinde =

ðEUR 187:45 - EUR 165:05Þ þ EUR 3:95 EUR 165:05

- ½- 0:002 þ ð0:08 - ð-0:002ÞÞ × 1:1 = 7:14%: Due to the positive alpha, the security appears to be undervalued. The sellside analyst will issue a buy recommendation.

6.2.5

Applications of the CAPM in Corporate Finance

The weighted average cost of capital (WACC) is an important measure in corporate finance. The WACC is used, among other things, to determine the increase in firm value, and thus in shareholder value, to assess the profitability of investment projects, and to set the optimal capital structure. The weighted average cost of capital represents the expected return of the capital holders (i.e. debt and equity providers) and is made up of a risk-free interest rate and a risk premium. The risk premium reflects a return compensation for the investment and financing risk of the company. The higher the risk from the investment projects and from the capital structure, the

202

6 Capital Asset Pricing Model and Fama–French Model

higher the expected return of the capital providers or the WACC.10 The weighted average cost of capital is the sum of the weighted cost components for interestbearing debt and equity capital and can be calculated follows: WACC = wD cD ð1- t Þ þ wE cE ,

ð6:13Þ

where wD = market value weight of interest-bearing debt, wE = market value weight of equity, cD = cost of debt (expected return of debt holders), t = marginal income tax rate, and cE = cost of equity (expected return of equity holders). The cost of equity can be determined by various methods, one of which is the CAPM. If the cost of equity is estimated with this capital market model, a long-term risk-free interest rate and a long-term expected market risk premium must be specified; since in corporate finance the WACC and thus the cost of equity are used, for example, to discount expected cash flows of an investment project or to determine a target capital structure. This is in contrast to portfolio theory, where the expected CAPM return is measured over a shorter period of time (e.g. 1 year) in order to calculate the alpha.11 Here, the valuation parameters consist of the risk-free interest rate and the expected market return for the corresponding short-term period. In accordance with the CAPM, the cost of equity can be calculated using the following formula: cE = r F þ MRPβEquity security,

ð6:14Þ

where rF = long-term risk-free interest rate, MRP = long-term expected market risk premium, and βEquity security = beta of the stock. The beta of the stock can be estimated with a simple linear regression analysis in which a regression is performed between the stock returns and the stock market returns. The slope of the regression line corresponds to the beta of the stock. To run this regression, it is common to use 60 monthly continuous compounded returns over a 5-year period on the one hand, and a broad equity index on the other. Then the beta from the regression is corrected for mean reversion, that is towards its long-term expected value of 1.12

10

See Mondello (2022): Corporate Finance: Theorie und Anwendungsbeispiele, p. 154 ff. See Sect. 6.2.4. 12 See Sect. 6.2.2. 11

6.2 Capital Asset Pricing Model

203

Since the application of the cost of equity in corporate finance is forward-looking, a long-term risk-free interest rate is taken. For this purpose, the yield to maturity of a prime government bond, such as those of the Federal Republic of Germany and the Swiss Confederation with a time to maturity of 10–15 years, is suitable. Compared to government bonds with longer maturities (e.g. 30 years), a maturity of 10 years has the advantage that these securities are more liquid and the yield curve is usually relatively flat from the 10th year onwards. However, the interest rate differences in 2017 for 10- and 30-year bonds of the Federal Republic of Germany are large in relative terms (0.4% versus 1.1%), and therefore, the yield to maturity of 30-year German government bonds is used for the long-term risk-free interest rate in the following. Another means of estimating the discount rate is to employ annual risk-free interest rates that match the timing of the cash flows over the expected useful lifetime of the investment project. For example, if an equity-financed project has a useful lifetime of 10 years, the expected annual cash flows are discounted at different costof-equity rates, where the risk-free rates are based on the yield to maturity of prime government bonds and maturities of 1 year, 2 years, and so on. This approach is particularly suitable when assessing investment projects that have a limited useful life. For the calculation of the equity value under the going concern assumption, riskfree interest rates that match the timing of the expected free cash flows to equity are not practicable because the forecasted free cash flows must be discounted over an infinitely long period of time, and prime government bonds often have a maximum maturity of 30 years. The market risk premium is the difference between the stock market return and the risk-free interest rate. It reflects the excess return of the stock market over a riskfree investment. The expected market risk premium can be determined as an average of historical return data, implicitly from an equity valuation model, or using a macroeconomic model.13 A common approach is to estimate an expected market risk premium using historical return data. This involves taking an average of the annual stock market returns and subtracting the average of the annual risk-free interest rates. In order to estimate the historical market risk premium, the following four variables must be determined: 1. 2. 3. 4.

Equity index Length of the return time series Method for averaging the return data Risk-free interest rate term

1. A representative equity index of the country in which the stock is traded is usually chosen. This is a well-diversified and market-capitalised equity index. This index should be as broad as possible and not dominated by individual securities with large market capitalisations.

13

See Mondello (2017a): Aktienbewertung: Theorie und Anwendungsbeispiele, p. 61 ff.

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6 Capital Asset Pricing Model and Fama–French Model

2. The choice of the time period over which the returns are determined has a significant influence on the level of the market risk premium. The principle applies that a longer time period or a higher number of return observations results in a smaller standard error.14 For this reason, a long observation period is usually selected, covering annual returns of more than 100 years. 3. The average value of the market return and the risk-free interest rate can be determined using the arithmetic mean or the geometric mean. When calculating the arithmetic mean return, it is assumed that the capital invested at the beginning of the period remains unchanged and therefore does not increase or decrease with the annual return. By contrast, the geometric mean return takes into account the compounding effect on the amount invested at the beginning of the period. If it is assumed that historical returns are not correlated with each other and thus follow a random movement, the arithmetic mean return represents an unbiased estimator. However, empirical studies demonstrate that stock returns exhibit negative autocorrelation over time,15 and therefore, the use of arithmetic mean tends to produce a risk premium that is too high. Moreover, in corporate finance, a long-term expected return of the equity providers (cost of equity) has to be estimated, which also argues for the use of the geometric mean return as this takes into account the compounding effect over several periods. Hence, the geometric mean is the better method for averaging when calculating the historical market risk premium. Due to the different advantages and disadvantages of the two methods, it is also possible to apply a weighted average of the arithmetic and the geometric mean. In this case, the weighting of the geometric mean return increases proportionally with the length of the return time series.16 4. Risk-free interest rates of short- or long-term prime government bonds can be used to calculate the historical market risk premium. In the presence of a normal yield curve, short-term government securities lead to a higher market risk premium than long-term risk-free securities. Since the applications in corporate finance are forward-looking, a long-term expected market risk premium should be determined when calculating the cost of equity. For this purpose, risk-free interest rates of long-term government securities of 10 years or longer are the better choice. Table 6.1 presents the historical market risk premiums from 1900 to 2018 for a large number of developed countries. The data are from Dimson, Marsh, and Staunton and are included in the Credit Suisse Global Investment Returns Yearbook 2019 by the Credit Suisse Research Institute. For the USA, risk premiums are measured against Treasury bond yields. For all other countries, corresponding yields

See Merton (1980): ‘On estimating the expected return on the market: an exploratory investigation’, p. 323 ff. 15 See Fama and French (1988): ‘Permanent and temporary components of stock prices’, p. 246 ff., Drobetz and Wegmann (2002): ‘Mean reversion on global stock markets’, p. 230 ff. 16 See Blume (1974): ‘Unbiased estimators of long-run expected rates of return’, p. 634 ff. 14

6.2 Capital Asset Pricing Model

205

of long-term interest rate instruments, such as yields to maturity of government bonds, are taken. The yield time series used for Germany excludes 1922 and 1923, which were years of hyperinflation. Germany, for example, has a historical market risk premium of 4.8% measured with the geometric mean, while the arithmetic mean p is 8.2%. The corresponding standard error of the estimate is 2.6% = 28:2%= 117 , despite the very long data series. At a confidence interval of 95%, the range of returns for the historical market risk premium of 8.2% measured with the arithmetic mean lies between 3.1% and 13.3% (= 8.2% ± 1.96 × 2.6%). For Switzerland, the historical market risk premium calculated with the geometric mean is 2.1%, while the arithmetic mean is 3.6%. Assuming a confidence interval of 95%, the range of the historical market risk premium of 3.6% based on the arithmetic mean lies between 0.5% and 6.7% (= 3.6% ± 1.96 × 1.6%). Example: Calculation of the Cost of Equity for Mercedes-Benz Group The following data is available as of early August 2017: • The beta of the Mercedes-Benz Group stock is 1.455 and was estimated using a simple linear regression analysis with 60 monthly log returns for the monthly share prices from the end of July 2012 to the end of July 2017 (see Fig. 6.2). • The historical market risk premium for Germany is 4.8%. • The yield to maturity of 30-year bonds issued by the Federal Republic of Germany is 1.1%. According to the CAPM, what is the cost of equity for Mercedes-Benz Group? Solution Since the cost of equity reflects a long-term measure, the historical beta of the automobile stock is first corrected towards its long-term expected value of 1, resulting in an adjusted beta of 1.303: βAdjusted = 0:333 þ 0:667 × 1:455 = 1:303: The car manufacturer’s cost of equity is 7.35% and can be calculated as follows: cE = 1:1% þ 4:8% × 1:303 = 7:35%: There are also reasons that warrant a higher market risk premium than the historical market risk premium. For example, a higher market risk premium can be justified by the occurrence of extreme events such as the recent global financial crisis, with falling share prices in 2008 and 2009, and the subsequent European debt crisis, as well as the increased correlation of global markets (due to increasing

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Table 6.1 Country-specific market risk premiums (1900 to 2018) (Source: Credit Suisse Research Institute 2019: Credit Suisse Global Investment Returns Yearbook 2019, p. 32.) Return from stock market - Return from long-term bonds Country/ Geometric mean Arithmetic mean region (in %) (in %) Australia 4.9 6.5 Austria 2.7 21.1 Belgium 2.1 4.1 Canada 3.3 4.9 Denmark 3.3 4.9 Finland 5.1 8.6 France 3.0 5.3 Germany 4.8 8.2 Ireland 2.5 4.5 Italy 3.1 6.4 Japan 4.9 8.9 Netherlands 3.2 5.5 New Zealand 4.0 5.5 Norway 2.5 5.4 Portugal 5.1 9.2 South Africa 5.1 6.8 Spain 1.6 3.6 Sweden 3.0 5.2 Switzerland 2.1 3.6 United 3.5 4.9 Kingdom USA 4.3 6.4 Europe 2.9 4.2 World ex 2.6 3.7 USA World 3.0 4.3

Standard error (in %) 1.7 14.0 1.9 1.7 1.7 2.7 2.1 2.6 1.8 2.7 2.9 2.0 1.6 2.5 2.9 1.8 1.9 1.9 1.6 1.6

Standard deviation (in %) 18.0 150.9 20.8 18.1 18.5 29.7 22.5 28.2 19.8 29.1 32.2 22.1 17.6 27.3 31.3 19.4 20.5 21.2 17.4 17.0

1.9 1.4 1.3

20.6 15.7 14.5

1.4

15.3

integration). In a letter dated 19 September 2012, for example, the FAUB (Fachausschuss für Unternehmensbewertung und Betriebswirtschaft) in Germany proposed a higher market risk premium in the range of 5.5% to 7%. The higher market risk premium in Germany was essentially justified by a change in risk tolerance.17 On 25 October 2019, the FAUB issued a new cost of capital recommendation for Germany, as the yield curve of German government bonds derived using the Svensson method is in negative territory for almost the entire term of 30 years.18 The long-term risk-free interest rate relevant for the valuation is effectively 0% for 17

See FAUB (2012): Hinweise des FAUB zur Berücksichtigung der Finanzmarktkrise bei der Ermittlung des Kapitalisierungszinssatzes in der Unternehmensbewertung, p. 2. 18 For the Svensson method, see Svensson (1994): Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, p. 1 ff.

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the first time and threatens to become negative in the foreseeable future due to the expansive monetary policy of the European Central Bank (ECB). Therefore, the previous recommendation for the market risk premium was increased from a range of 5.5% to 7% to a range of 6% to 8%.19 With an expected market risk premium of 7% (instead of 4,8%), the Mercedes-Benz Group’s cost of equity rises from 7.35% to 10.22%: cE = 1:1% þ 7% × 1:303 = 10:22%: To determine the weighted average cost of capital, the cost of debt and the market value weights for debt and equity are required in addition to the cost of equity. The cost of debt corresponds to the expected return of the debt holders, which consists of the risk-free interest rate and a risk premium reflecting the credit risk of the company: cD = r F þ CP,

ð6:15Þ

where rF = long-term risk-free interest rate, and CP = credit risk premium. If the risk-free rate rises, the cost of debt increases. Similarly, there is a positive relationship between credit risk and the cost of debt because the cost of borrowing money increases with a higher probability of default. The cost of debt can be estimated using several methods.20 If the company has long-term bonds with a high market liquidity outstanding, the yield to maturity of the bonds is equal to the cost of debt.21 Another approach is based on the company’s external rating, which is assigned by rating agencies such as Standard & Poor’s, Moody’s, and Fitch. The company rating can be applied to determine the cost of debt by taking the yield to maturity of long-term bonds with the same rating and a high market liquidity. For example, according to Standard & Poor’s, Mercedes-Benz Group’s long-term rating is A-. Assuming a credit risk premium for this rating of 1.13% and a long-term risk-free rate of 1.1% leads to a cost of debt of 2.23%: cD = 1:1% þ 1:13% = 2:23%: If the company does not have an external rating, the cost of debt can be determined either by the interest rate of recent borrowings or by a synthetic rating established using financial ratios (e.g. interest coverage ratio).22 In the WACC formula (see Eq. 6.13), the cost of debt is multiplied by 1 minus the marginal income tax rate due to the tax deductibility of the interest payments. Since

19

See https://www.idw.de/idw/idw-aktuell/neue-kapitalkostenempfehlungen-des-faub/120158 See Courtois et al. (2008): Cost of Capital, p. 135 ff. 21 For the yield to maturity of bonds, see Sect. 10.4.1. 22 See Mondello (2022): Corporate Finance: Theorie und Anwendungsbeispiele, p. 210 f. 20

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the application of the WACC in corporate finance is forward-looking, the weights of debt and equity are based on market values rather than on book values; alternatively—and a better option—a target capital structure provided by the company can be used, which in some cases can be found on the company’s website under the section dealing with investor relations. Example: Calculation of the Weighted Average Cost of Capital for Mercedes-Benz Group Mercedes-Benz Group’s cost of equity is 10.22%, and its cost of debt is 2.23%. The assumed target capital structure consists of 65% debt and 35% equity, while the marginal income tax rate for Germany is 30%. What is the weighted average cost of capital for the Mercedes-Benz Group? Solution The weighted average cost of capital is 4.59% and can be calculated as follows: WACC = 0:65 × 2:23% × ð1- 0:3Þ þ 0:35 × 10:22% = 4:59%:

6.3

Fama-French Model

6.3.1

The Risk Premiums for Size and Value

The CAPM is based on a theoretically sound model that assumes rational, riskaverse, and well-diversified investors who maximise utility and all have the same capital market expectations. Hence, the expected return depends only on systematic risk or market risk and is measured by the beta. Empirically, the CAPM has been confirmed but also rejected by many studies. For example, Fama and French (1992) published a study that questioned the CAPM.23 This study concluded that there was no statistically significant relationship between expected return and beta for US stocks from 1963 to 1990. Rather, the average returns of US stocks are explained on the basis of other factors, such as firm size, financial leverage, price-to-earnings ratio, and book-to-price ratio. In this regard, the size of the firm and the book-to-price ratio are the dominant explanatory variables. Given the importance of Fama and French’s study, several empirical studies were conducted after 1992. Some of these studies supported Fama and French’s results,24 while others concluded that there is a statistically significant relationship between beta and expected return.25 See Fama and French (1992): ‘The cross section of expected stock returns’, p. 427 ff. See, for example, Dennis et al. (1995): ‘The effects of rebalancing on size and book-to-market ratio portfolio returns’, p. 47 ff. 25 See, for example, Kothari et al. (1995): ‘Another look at the cross section of expected stock returns’, p. 185 ff. 23 24

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The results of empirical studies on the CAPM are therefore not uniform. Rather, empirical data for a large number of countries and over different time periods demonstrate that the size of a company (i.e. market capitalisation) and the bookto-price ratio have an influence on expected returns. For example, there is a negative relationship between market capitalisation and returns. In the long run, the stock returns of companies with a small market capitalisation are higher than those with a large market capitalisation. Furthermore, there is a positive relationship between the book-to-price ratio and stock returns. In the long run, the returns of stocks with a large book-to-price ratio are higher than those of stocks with a small book-to-price ratio. The ratio between the book value and the share price provides an indication of whether the security is correctly valued. A high (low) book-to-price ratio is an indicator that the stock is undervalued (overvalued). Equity securities with a high book-to-price ratio are called value stocks because the price is too low in relation to the book value and they are therefore comparatively cheap. By contrast, securities with a low book-to-price ratio are referred to as growth stocks. The high price of the security is due to the positive growth prospects. In the long run, the returns of cheap stocks (value stocks) are higher than those of expensive stocks (growth stocks). The two risk premiums for size (market capitalisation) and value (book-to-price ratio) play an important role in explaining stock returns. For example, small companies are likely to be less well diversified than large ones and thus not necessarily able to withstand large economic downturns. If the ratio of book value to share price increases, the equity security appears relatively cheap. However, the reason for the share price decline may be a deterioration in the company’s performance. Buying value stocks means betting on an economic recovery of the company and thus on an increase in the price of the securities. The higher risk of stocks with a small market capitalisation and a higher book-to-price ratio justifies a higher expected return. This explanation assumes an efficient capital market in which market participants demand a higher return on investments with higher risk. By contrast, the higher returns of stocks with a small market capitalisation and value bias can also be explained by an inefficient market where price anomalies are present. The following discussion is based on the first explanatory approach, which assumes an efficient capital market in which an increase in risk is compensated with a higher return. According to the CAPM, equity securities with a higher (lower) market risk have a higher (lower) expected return. Moreover, empirical studies demonstrate that, in the long run, returns are higher for securities with a small rather than a large market capitalisation and for securities with a high rather than a low book-to-price ratio. When these findings are combined, stock returns can be explained by the market risk premium and the risk premiums for size and value. This three-factor model for market, size, and value is known as the Fama–French model (FFM).

6.3.2

Expected Rate of Return

The three-factor model developed by Fama and French (1996) explains equity returns using the stock market risk premium (RM), the market capitalisation of the

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firm’s stocks (SMB), and the book-to-price ratio (HML). For this purpose, the excess returns over the risk-free return of a stock or a stock portfolio (Ri,t = ri, t - rF) are regressed against the excess returns of the stock market, size, and value by means of a multiple linear regression analysis. The resulting regression equation is as follows:26 Ri,t = αi þ βi,M RM, t þ βi,SMB SMBt þ βi,HML HMLt þ εi,t ,

ð6:16Þ

where RM = return difference between a market-weighted equity index and a risk-free asset; this risk factor corresponds to the market risk premium in the CAPM; SMB = return difference between three portfolios containing stocks with small market capitalisation and three portfolios consisting of stocks with large market capitalisation; this risk factor for company size thus represents an excess return for small-cap stocks versus large-cap stocks (small minus big); HML = return difference between two portfolios containing stocks with high bookto-price ratios and two portfolios comprising stocks with low book-to-price ratios; equity securities with a high book-to-price ratio (or a low price-to-book ratio) have a value bias, while equity securities with a low book-to-price ratio (or a high price-to-book ratio) have a growth bias; this risk factor for value thus reflects an excess return for value stocks versus growth stocks (high minus low); αi = constant (intercept) of the regression equation (fixed return contribution from unsystematic risk); βi = regression coefficients or betas for market (RM), size (SMB), and value (HML); and εi, t = error term of residual returns of asset i for period t (return contribution from unsystematic risk). The market, size, and value betas can be estimated for a single stock or for a stock portfolio using multiple linear regression analysis. As with the CAPM, the regression is typically performed using 60 monthly continuous compounded returns over a 5-year period. Therefore, monthly returns are required for the risk-free interest rate and for the risk premiums for market, size, and value, which are the same for all stocks. Assuming a constant and an error term from unsystematic risk of zero (αi = 0 and εi, t = 0), the return in the FFM can be determined as follows: r i,t = r F þ βi,M RM,t þ βi,SMB SMBt þ βi,HML HMLt :

ð6:17Þ

The return is made up of the risk-free interest rate and a risk premium given by the three systematic risk factors of stock market (RM), size (SMB), and value (HML). The three systematic risk factors can be viewed as the average return of a long–short

26

See Fama and French (1996): ‘Multifactor explanations of asset pricing anomalies’, p. 55 ff.

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portfolio with a net investment of zero. The RM factor represents a short position in riskless assets and a long position in the market portfolio. Thus, money is borrowed at the risk-free rate and invested in a broad equity index (market portfolio). The SMB factor reflects the average return of a short position in large-cap stocks, with the cash inflow from the short sale invested in small-cap stocks. The HML factor, on the other hand, represents the average return from a short position in equity securities with a low book-to-price ratio and a long position in equity securities with a high book-toprice ratio purchased with the funds from the short sale position. In addition to the excess market return over the risk-free rate, in the FFM equity returns are explained by the two other systematic risk factors of size and value. Therefore, the beta for the market risk premium is not identical to the beta in the CAPM. The systematic risk factors in the model can be divided into the following two groups: • A systematic risk factor for the stock market (RM) which, like the CAPM, reflects the risk specific to the equity market or market risk. • Two systematic risk factors for size (SMB) and value (HML) which describe fundamental characteristics of the company. Example: Expected Return Based on the CAPM and the FFM Using the Stock of Adidas AG A portfolio manager wants to determine the expected return of the Adidas stock applying the CAPM and the FFM. A simple linear regression analysis between the monthly excess returns of the security and those of the market from the end of June 2011 to the end of June 2016 leads to the following regression equation for the market model:27 RAdidas,t = 0:009 þ 0:801 × RM,t þ εAdidas,t : The beta is 0.801 and is statistically significant at a significance level of 5%, with a t-statistic of 5.14.28 The constant or intercept of 0.009, on the other (continued)

27

In the market model a regression takes place between the excess returns over the risk-free rate of the stock and those of the stock market. The regression equation represents a return compensation for the company-specific risk (unsystematic risk) and the market risk (systematic risk). The CAPM is a special case of the market model where firm-specific risk is zero (αi = 0 and εi, t = 0). Assuming that the effect of the risk-free rate in calculating beta is small, beta can also be determined by regressing the stock and stock market returns (and not the excess returns over the risk-free rate). See Sect. 6.2.2 and Mondello (2017b): Finance: Theorie und Anwendungsbeispiele, p. 143 ff. The data for the market’s monthly excess returns were obtained from the website of the Humboldt University in Berlin and refer to Germany. See https://www.wiwi.hu-berlin.de/de/professuren/bwl/bb/daten/ fama-french-factors-germany/fama-french-factors-for-germany. 28 The critical t-value at T - 2 or at approximately 60 degrees of freedom and a significance level of 5 % is approximately 2.

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6 Capital Asset Pricing Model and Fama–French Model

hand, has a t-statistic of 1.16 and is thus not relevant for calculating the excess return of the stock. The adjusted coefficient of determination of the regression is 0.30. Since the CAPM only provides a return compensation for systematic risk, the simple linear regression is run again. For this purpose, the intercept is set to 0, which leads to the following regression equation: RAdidas,t = 0 þ 0:825 × RM,t þ εAdidas,t : The beta of 0.825 has a t-statistic of 5.33 and is therefore statistically significant at a significance level of 5%. The adjusted coefficient of determination of the regression is 0.31. The regression between the monthly excess stock returns and the monthly returns of the three systematic risk factors of market (RM), size (SMB), and value (HML) from the end of June 2011 to the end of June 2016 results in the following regression equation for the FFM (multiple linear regression analysis):29 RAdidas,t = 0:008 þ 0:962 × RM,t þ 0:329 × SMBt - 0:679 × HMLt þ εAdidas,t : The adjusted coefficient of determination of the regression is 0.35. The intercept with a t-statistic of 1.08 is not statistically significant at a significance level of 5%. The regression coefficients (or betas) are statistically significant at a level of 5% for the systematic risk factors of market and value, with a tstatistic of 4.36 and -2.25, respectively. By contrast, the t-statistic for the regression coefficient of the systematic risk factor size is 0.98. Hence, the beta for the size factor is not statistically significant at a significance level of 5%. This indicates multicollinearity. Multicollinearity occurs when two or more independent variables are highly correlated with each other, such that the regression coefficients estimated with the regression are inaccurate. Therefore, the multiple linear regression is run again with the two systematic risk factors of market and value since they have regression coefficients that are statistically significant. The regression equation is as follows: RAdidas,t = 0:010 þ 0:803 × RM,t - 0:685 × HMLt þ εAdidas,t : The adjusted coefficient of determination of the regression is 0.35. The two regression coefficients for market and value are statistically significant at a significance level of 5%, with t-statistics of 5.33 and -2.27, respectively, while the intercept is statistically not significant at a significance level of 5%, (continued) 29 The monthly return data for the risk factors market, size, and value were obtained from the website of the Humboldt University in Berlin and refer to Germany. See https://www.wiwi.huberlin.de/de/professuren/bwl/bb/daten/fama-french-factors-germany/fama-french-factors-forgermany.

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with a t-statistic of 1.28. Since the FFM is applied to estimate the expected return using the systematic risk factors, the multiple linear regression is performed again. For this purpose, the intercept is set to 0, similar to the CAPM, which leads to the following regression equation: RAdidas,t = 0 þ 0:828 × RM,t - 0:671 × HMLt þ εAdidas,t : The adjusted coefficient of determination of the regression is 0.35. The two regression coefficients are statistically significant at a significance level of 5%, with t-statistics of 5.52 and -2.21, respectively. The coefficient of determination or the explanatory power of the regression is greater than in the CAPM because the FFM uses several systematic risk factors to calculate returns. The annual risk-free interest rate is -0.2%. The market risk premium is 6.16%, while the risk premiums for size and value are -1.81% and 4.07%, respectively.30 What is the expected return of the Adidas stock according to the CAPM and the FFM? Solution The expected CAPM return of the Adidas stock is 4.882% and can be calculated as follows: E ðr Adidas Þ = - 0:2% þ 0:825 × 6:16% = 4:882%: In accordance with the FFM, the expected return of 2.17% can be determined taking the two systematic risk factors of stock market (RM) and value (HML), using the equation below: E ðr Adidas Þ = - 0:2% þ 0:828 × 6:16% - 0:671 × 4:07% = 2:170%: The beta for value is negative. Thus, the Adidas stock is a growth stock. The return compensation for the systematic risk is 2.37%. (=0.828 × 6.16 % - 0.671 × 4.07%). If the negative risk-free interest rate of 0.2% is added to the risk premium of 2.37%, the expected return on the Adidas stock is 2.17% using the FFM. The return component from the unsystematic or diversifiable risk is zero (αi = 0 and εi, t = 0). The negative risk premium for size of –1.81% for Germany contradicts the assumption of market efficiency stated by Fama and French, because equity securities with a small market capitalisation are riskier than securities with a large market capitalisation, and market participants demand a higher return as a result.

30

These risk premiums apply to Germany and represent an annualised average of the corresponding monthly risk premiums from July 1958 to June 2016. See https://www.wiwi.hu-berlin.de/de/ professuren/bwl/bb/daten/fama-french-factors-germany/fama-french-factors-for-germany

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Table 6.2 Assessment of betas for size (SMB) and value (HML) (Source: Own illustration) Beta Size (SMB) Value (HML)

Positive Small market capitalisation Value

Negative Large market capitalisation Growth

It is not unusual, as the Adidas stock example demonstrates, for the beta to be negative for the value (and the size). By contrast, negative betas for the market are the exception because stock returns are usually positively correlated with the stock market. Table 6.2 provides guidelines for judging a single stock or stock portfolio by its betas for size (SMB) and value (HML). Moreover, the empirical work of Fama and French suggests that the risk premiums for size and value are generally positive. For example, relative to Europe, the risk premiums for size and value are 0.98% and 4.27%, respectively (based on annual risk premiums over the period 1991-2016).31 Davis et al. (2000) empirically tested the FFM in their study.32 They concluded that the intercept from the multiple linear regression analysis is small and basically not statistically significant. The coefficient of determination for the US stock portfolios analysed is above 0.90. Furthermore, the regression coefficients (i.e. betas) for the two systematic risk factors—size and value—are statistically significant with high t-statistics. These results indicate that the systematic risk factors in the model explain stock portfolio returns well. One possible interpretation of these empirical results is that size and value are complementary to the CAPM in capturing returns. This reasoning is consistent with the FFM and assumes that size and value are systematic risk factors. The higher risk of stocks with small market capitalisation and high book-to-price ratio is compensated by a higher return in an efficient market. Another interpretation is that these risk premiums for value and size are due to the irrational behaviour of investors (behavioural bias).

6.4

Summary

• The CAPM is one of the most important innovations in financial market theory. The model assumes a linear relationship between risk and expected return. Investors are only compensated for systematic risk with a return because they can eliminate the unsystematic or idiosyncratic risk in a portfolio. Therefore, the expected return is a function of the systematic risk of the investment, which is measured with the beta. Although the CAPM can, in principle, be used to determine expected returns for any asset, it is usually used only for equities. • The CAPM is based on the following assumptions: (1) Investors behave in a rational and risk-averse manner and maximise their utility. (2) Markets are

31

See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. No corresponding data are available for Germany and Switzerland. 32 See Davis et al. (2000): ‘Characteristics, covariances, and average returns, 1929 to 1997’, p. 389 ff.

6.4 Summary

•

•

•

•

215

frictionless and there are no transaction costs or taxes. (3) All investors plan for the same investment period. (4) All investors have homogeneous capital market expectations. (5) All investments are infinitely divisible and tradable. (6) Investors are price takers. The expected return in the CAPM is estimated by adding a premium for market risk to the risk-free return. The risk premium can be calculated by multiplying the stock market risk premium by the beta of the equity security. Beta is a sensitivity measure and indicates how much the stock return will change if the stock market return moves. This measure of risk is determined by performing a regression between stock and stock market returns (simple linear regression analysis). The beta corresponds to the slope of the regression line. The CAPM is an equilibrium model. All investments that are correctly valued lie on the SML. If stocks are undervalued or overvalued, in a beta–return diagram they are located above or below the SML, respectively. Undervalued (overvalued) stocks are bought (sold) by market participants, resulting in a higher (lower) price and a lower (higher) expected return. This buying (selling) process continues until the expected return of the stock has fallen (risen) to the corresponding beta–return point on the SML. If the market is efficient, all investments lie on the SML because, among other things, all available information has been processed into prices, with the result that excess returns are not possible or alpha is zero. Therefore, the CAPM is based on the assumption that capital markets are informationally efficient in the strong form, as well as operationally efficient (i.e. no transaction costs and taxes). The CAPM is applied in corporate finance to estimate the cost of equity, which is a component of the weighted average cost of capital. In contrast to portfolio management, a long-term risk-free interest rate is taken instead of a short-term one, which usually corresponds to the yield to maturity of prime government bonds with a maturity of 10 or more years. In addition, the expected market risk premium can be estimated using historical return data, a macroeconomic model, or an equity valuation model with current stock market data. The historical market risk premium is the difference between the average of stock market returns and risk-free returns. This risk premium is based on a broad equity index, a long time series of annual returns (more than 100 years), the geometric mean for averaging the returns, and a long-term risk-free interest rate of, say, 10 years. The predictions of the CAPM are that the market portfolio is efficient in terms of expected return and risk and that all assets are correctly valued and therefore lie on the SML. Despite the mixed results of empirical tests, the CAPM is widely used in practice. The division of risk into a systematic part and an unsystematic part is consistent with a logical way of thinking in portfolio management. Furthermore, the CAPM assumes that the market portfolio is efficient. This assumption can be observed by evaluating the performance of passive investment strategies. A market index portfolio has, on average, a higher return after deducting transaction and management costs than an active investment strategy. Therefore, a one-factor model with an expected alpha of zero represents a realistic investment decision model in an efficient market.

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• Return anomalies can be observed in the market that cannot be explained by the CAPM. These include the size effect and the value effect. The size and value return compensations are either due to market inefficiencies or represent systematic risk factors in an efficient market that can be captured in a multifactor model. • For example, there is a negative relationship between market capitalisation and returns. In the long run, stock returns are higher for companies with small rather than large market capitalisation. In addition, there is a positive relationship between the book-to-price ratio and stock returns. In the long run, the returns of equity securities with a high rather than low book-to-price ratio are higher. • The FFM is a three-factor model that includes risk premiums for market (same as CAPM), size, and value. This involves a regression between the excess returns over the risk-free rate of a single stock or a stock portfolio and the returns of the risk premiums for the market, size, and value (multiple linear regression analysis). The regression coefficients in the regression equation represent the betas for market, size, and value. A positive (negative) beta for size is an indication that the equity security is a small- (large-) cap stock and therefore has a higher (lower) risk of loss, which should be compensated with a higher (lower) required return. A positive (negative) beta for value means that it is a stock with a value bias (growth bias). A too low (too high) price of a value stock (growth stock) can be explained in an efficient market by a lack of (high) profitability of the company, and therefore as a result of the higher (lower) systematic risk, a higher (lower) return is demanded by market participants. Typically, risk premiums are positive for size and value. Like the market risk premium in the CAPM, they can be determined using historical return data.

6.5

Problems

1. The risk-free interest rate is 2%. The shares of Rho AG have a beta of 1.2. Investors expect a return of 14%, which coincides with that of the CAPM. a) What is the market risk premium? b) The shares of Vega AG have a beta of 0.8. What is the expected return according to the CAPM? c) An investor has invested EUR 100,000 in shares of Rho AG and Vega AG. The portfolio has a beta of 1.1. What are the weights of the two equity securities in the portfolio? What is the expected return of this stock portfolio? 2. The covariance between the stock Z and the stock market is 0.0455. The variance of stock market returns is 0.0785. The market risk premium is 5.5%, while the riskfree rate is 1.5%. What is the expected return of stock Z in accordance with the CAPM?

6.5 Problems

217

3. A financial analyst is examining the stocks of Delta AG and Kappa AG. He has compiled the following data for the two equity securities: Stock Delta AG Kappa AG

Expected return of the financial analyst 10.0% 5.7%

Beta 1.3 0.9

The risk-free interest rate is 1.5%, while the market risk premium is 5.5%. Are the stocks of Delta AG and Kappa AG overvalued or undervalued if the analyst uses the CAPM for his investment decisions? 4. A portfolio manager of an asset management firm has compiled the following data for the stocks of Gamma AG and Vega AG:

Beta Covariance with HDAX

Gamma AG 1.4 Not known

Vega AG Not known 0.06

The analysis department of the asset management firm provides the portfolio manager for the HDAX an expected return of 8% and a standard deviation of 20%. The return on BuBills with a maturity of 1 year is 2%. The portfolio manager constructs a portfolio with a market value of EUR 200,000. He invests EUR 60,000 in Gamma shares and EUR 140,000 in Vega shares. What is the expected return of the portfolio? 5. According to the FFM, the following betas and risk premiums exist for a stock: Risk factor Market risk premium (RM) Size (SMB) Value (HML)

Beta 1.3 -0.6 0.3

The risk-free interest rate is 2%. a) What is the expected return in accordance with the FFM? b) To which investment style can the stock be assigned?

Risk premium 5.0% 2.2% 4.8%

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Solutions

1. a) If the CAPM return equation is solved for the expected market return, a value of 12% is obtained:

E ðr M Þ =

Eðr Rho Þ - r F 14% - 2% þ rF = þ 2% = 12%: β 1:2

The market risk premium is therefore 10%: E ðr M Þ - r F = 12% - 2% = 10%: b) The expected return of the Vega stock is 10% and can be calculated as follows: E rVega = 2% þ ð12% - 2%Þ × 0:8 = 10%: c) The portfolio beta can be determined with the following formula: βP = wRho βRho þ ð1- wRho ÞβVega : Inserting the betas in the equation, the following formula is obtained: 1:1 = wRho × 1:2 þ ð1- wRho Þ × 0:8: Solving for the weight of Rho (wRho) leads to a value of 0.75. Thus, the weights of the Rho and Vega securities in the portfolio are 75% and 25%, respectively. An expected portfolio return of 13% can be calculated as follows: Eðr P Þ = 2% þ ð12% - 2%Þ × 1:1 = 13% or

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219

E ðr P Þ = 0:75 × 14% þ 0:25 × 10% = 13%: 2. The beta of stock Z is 0.58 and can be determined with the equation below:

βZ =

covZ,M 0:0455 = = 0:58: 0:0785 σ 2M

The expected CAPM return of stock Z is 4.69% and can be calculated as follows: E ðrZ Þ = 1:5% þ 5:5% × 0:58 = 4:69%: 3. Stock Delta Kappa

Expected CAPM return 8.65% (= 1.5% + 5.5% × 1.3) 6.45% (= 1.5% + 5.5% × 0.9)

Expected return of the analyst 10.0%

Overvalued or undervalued Undervalued

5.7%

Overvalued

The Delta stock has a positive alpha of 1.35% (= 10% - 8.65%) and is therefore undervalued. By contrast, the Kappa stock is overvalued, with a negative alpha of 0.75% (= 5.7% - 6.45%). 4. EðrGamma Þ = 2% þ ð8% - 2%Þ × 1:4 = 10:4% βVega =

covVega, HDAX 0:06 = = 1:5 σ2HDAX 0:202

E r Vega = 2% þ ð8% - 2%Þ × 1:5 = 11% EðrP Þ = 0:3 × 10:4% þ 0:7 × 11% = 10:82% The expected portfolio return of 10.82% can also be calculated as follows: βP = 0:3 × 1:4 þ 0:7 × 1:5 = 1:47,

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Eðr P Þ = 2% þ ð8% - 2%Þ × 1:47 = 10:82%: 5. a) According to the FFM, the expected stock return is 8.62% and can be calculated as follows: E r Equity security = 2% þ 1:3 × 5% þ ð- 0:6Þ × 2:2% þ 0:3 × 4:8% = 8:62%: b) The beta for the company size (SMB) is negative, while the beta for the value (HML) is positive. Accordingly, a stock with large market capitalisation and value bias is present. The beta for the market risk premium of 1.3 means that the stock has a higher market risk than the stock market, which has a beta of 1. The equity security therefore reflects the investment style of large market capitalisation with a value bias.

Microsoft Excel Applications • Performing a linear regression analysis is straightforward in Microsoft Excel, although there are more comprehensive and better developed software solutions on the market. For example, there are 60 return observations for each of three time series in cells A1 to A60, B1 to B60, and C1 to C60. The first column A contains the monthly return values for the dependent variable, while the other two columns B and C include the corresponding values for the two independent variables. To open the dialog box, first click on the ‘Data’ tab and then select ‘Data analysis’. Select ‘Regression’ from the analysis functions that appear and then confirm with ‘OK’. The dialogue box for the regression will now appear. • Simple linear regression with one independent variable can be performed as follows: – In the regression dialogue box, the monthly returns of the dependent variables (A1:A60) should be entered at the ‘Input Y range’. – Then click on the ‘Input X range’ and enter the monthly returns of the independent variables (B1:B60). – Finally, click on ‘Output options’ and specify a free cell in the window provided, where the output of the regression should appear. Note that the output spans several cells (9 columns and 18 rows). To display the output press ‘OK’ in the dialogue box. • Multiple linear regression with more than one independent variable is performed in the same way as simple linear regression, except for a single step. With the ‘Input X range’, all independent variables are entered, as opposed to only one

References

221

independent variable. For example, with two independent variables, cells B1 to B60 and C1 to C60 should be entered (i.e. B1:C60). • The regression coefficient of a simple linear regression analysis (e.g. the beta of a stock in the CAPM) can be calculated in Excel using a shortcut. The following expression is written in a free cell: = LINESTðA1:A60; B1:B60Þ and then confirmed with the Enter key. • Similarly, the regression coefficients of a multiple linear regression analysis can be calculated using the ‘LINEST’ function. For example, cells A1 to A60 contain the return observations for the dependent variable, while cells B1 to B60 and C1 to C60 contain the corresponding data for the two independent variables. To determine the regression coefficients of the two independent variables, mark the two free cells B61 and C61. Then enter the following expression: = LINESTðA1:A60; B1:C60Þ and finish by pressing the key combination Ctrl+Shift+Enter at the same time. It should be noted that Microsoft Excel displays the regression coefficients in reverse order. Therefore, the regression coefficient for the independent variable of column C (B) is in cell B61 (C61). • To create the regression line from the simple linear regression analysis in the scatter plot, mark the returns of the independent and dependent variables with the right mouse button. Then click on the ‘Insert’ tab and thereafter on ‘Insert scatter (X, Y)’. Select the appropriate scatter plot in the diagram layouts and click on ‘Chart elements’, which can be found on the plus button (top right of the diagram), and select ‘Trendline’.

References Blume, M.E.: On the assessment of risk. J. Finance. 26(1), 1–10 (1971) Blume, M.E.: Unbiased estimators of long-run expected rates of return. J. Am. Stat. Assoc. 69(347), 634–638 (1974) Credit Suisse: Global Investment Returns Yearbook 2019. Credit Suisse Research Institute, Zurich (2019) Courtois, Y., Lai, G.C., Peterson Drake, P.: Cost of capital. In: Clayman, M.R., Fridson, M.S., Troughton, G.H. (eds.) Corporate Finance: A Practical Approach, pp. 127–169. Wiley, Hoboken, NJ (2008) Davis, J.L., Fama, E.F., French, K.R.: Characteristics, covariances, and average returns, 1929 to 1997. J. Finance. 55(1), 389–406 (2000)

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Dennis, P., Perfect, S., Snow, K., Wiles, K.: The effects of rebalancing on size and book-to-market ratio portfolio returns. Financial Anal. J. 51(3), 47–57 (1995) Drobetz, W., Wegmann, P.: Mean reversion on global stock markets. Swiss J. Econ. Stat. (SJES). 138(3), 215–239 (2002) Fama, E.F., French, K.R.: Permanent and temporary components of stock prices. J. Polit. Econ. 96(2), 246–273 (1988) Fama, E.F., French, K.R.: The cross section of expected stock returns. J. Finance. 47(2), 427–465 (1992) Fama, E.F., French, K.R.: Multifactor explanations of asset pricing anomalies. J. Finance. 51(1), 55–84 (1996) IDW: FAUB: Hinweise des FAUB zur Berücksichtigung der Finanzmarktkrise bei der Ermittlung des Kapitalisierungszinssatzes in der Unternehmensbewertung, pp. 1–2. Springer (2012) Kothari, S.P., Shanken, J., Sloan, R.G.: Another look at the cross section of expected stock returns. J. Finance. 50(2), 185–224 (1995) Lintner, J.: The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat. 47(1), 13–37 (1965) Merton, R.C.: On estimating the expected return on the market: an exploratory investigation. J. Financ. Econ. 8(4), 323–361 (1980) Mondello, E.: Aktienbewertung: Theorie und Anwendungsbeispiele, 2nd edn. Springer, Wiesbaden (2017a) Mondello, E.: Finance: Theorie und Anwendungsbeispiele. Springer, Wiesbaden (2017b) Mondello, E.: Corporate Finance: Theorie und Anwendungsbeispiele. Springer, Wiesbaden (2022) Mossin, J.: Equilibrium in a capital asset market. Econometrica. 34(4), 768–783 (1966) Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 6th edn. South-Western Cengage Learning, Jefferson City (2000) Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Finance. 19(3), 425–442 (1964) Svensson, L.E.: Estimating and Interpreting Forward Interest Rates: Sweden 1992–1994. National Bureau of Economic Research Working Paper Series 4871 (1994)

Online Sources Fama, French: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html (2021). Accessed 18 December 2021 FAUB: Neue Kapitalkostenempfehlungen des FAUB. https://www.idw.de/idw/idw-aktuell/neuekapitalkostenempfehlungen-des-faub/120158 (2019). Accessed 25 March 2021 Humbold-Universität zu Berlin: https://www.wiwi.hu-berlin.de/de/professuren/bwl/bb/daten/famafrench-factors-germany/fama-french-factors-for-germany (2021). Accessed 18 December 2021

7

Portfolio Management Process

7.1

Introduction

The portfolio management process consists of various steps and ensures the systematic construction of a portfolio that is appropriate to the client’s needs. The process consists of three phases, namely planning, execution, and feedback. In the planning phase, the financial markets and investor needs are analysed, the long-term investment policy is formulated, and the strategic asset allocation is determined. In the execution phase, the portfolio is created and the assets required by the investment policy are purchased. The feedback phase concludes the process and includes monitoring of the investment policy and capital market expectations, as well as rebalancing and performance evaluation of the portfolio.1 The long-term investment policy statement (IPS) is the core of the portfolio management process. It is a document that contains the investor’s return objectives and risk tolerance, as well as existing investment constraints such as time horizon, required liquidity, tax situation, legal and regulatory factors, and unique circumstances. The expected return and risk of assets such as equities and bonds are estimated based on long-term capital market expectations. The investment policy, together with the long-term capital market expectations, forms the starting point for determining strategic asset allocation. This approach to establishing the strategic asset allocation ensures that the investor’s long-term return and risk objectives can be achieved, while taking the investment restrictions into account.2 Figure 7.1 presents the portfolio management process, which consists of the three phases—planning, execution, and feedback—described below.

1 2

See Maginn and Tuttle 1990: ‘The Portfolio Management Process and Its Dynamics’, p. 1-1 ff. See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 40 ff.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_7

223

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Portfolio Management Process

1. Planning Investment objectives and constraints: Investment policy statement

Long-term capital market expectations

Strategic asset allocation 2. Execution Construction of optimal portfolio 3. Feedback Monitoring of capital market data

Monitoring and evaluation of portfolio

Attainment of investment objectives Performance mesurement

Fig. 7.1 Portfolio management process (Source: Based on Maginn and Tuttle 1990: ‘The Portfolio Management Process and Its Dynamics’, p. 1–4)

7.2

Planning

7.2.1

Investment Objectives and Constraints

The first step is to determine the investment objectives and constraints for an investor. Investment objectives include risk and return. Internal restrictions consist of liquidity requirements, the investment horizon, and investor-specific circumstances; while external restrictions on investment policy are caused by the tax situation and the legal framework. In this context, liquidity and the investment horizon, in particular, influence the ability to bear risk and consequently the return and risk objectives.3

7.2.1.1 Risk Objectives The risk policy formulated in the IPS influences the level of the expected portfolio return. Higher risk implies a higher expected return and vice versa. The risk

3

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 42 ff.

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225

objectives reflect the investor’s risk tolerance and define the desired level of portfolio risk. Risk can be measured in absolute or relative terms. The absolute risk relates, for example, of the objective of losing no more than 6% of the invested capital in the next 12 months. This loss figure can be estimated with the standard deviation or variance or with the value at risk. By contrast, a relative risk measure represents a return deviation of the portfolio relative to a benchmark. It can be calculated with the standard deviation of the return differences between the portfolio and the benchmark. For institutional clients such as pension funds, payment obligations to pension plan participants reflect the benchmark. The risk objective is to minimise the probability that the pension fund will have a shortfall (i.e. the pension obligation exceeds the fair value of pension plan assets) or will not be able to fulfil its payment promise. An investor’s risk tolerance is characterised by their ability to bear potential losses and by their willingness to take risk (risk attitude).4 Above-average (belowaverage) risk tolerance is indicated by high (low) ability to accept risk and high (low) risk willingness. The ability to absorb losses depends on factors such as investment horizon, expected income, and amount of net worth. For example, an investor with an investment horizon of 25 years has a higher ability to bear risk than an investor with an investment horizon of only 3 years; 25 years is a longer period in which to recoup any losses than 3 years. Risk tolerance is affected by the investor’s psychology and current circumstances. Risk attitude is influenced by psychological factors such as personality type, self-confidence, and propensity for independent thinking. There is no universal method to measure risk tolerance. Risk tolerance can be evaluated by talking to the client or by using a standardised questionnaire.5 Table 7.1 presents an example of an analysis of risk attitude by means of a standardised assessment questionnaire, which employs primarily psychological factors (statements 2–5).6 The four answers (a), (b), (c), and (d) are assigned 1, 2, 3, and 4 points, respectively. If the points are added together, a statement can be made about the investor’s willingness to take risk. The lowest score of 5 means that the investor has a very low-risk attitude. By contrast, the maximum score of 20 indicates a very high willingness to assume risk. In assessing risk tolerance, the ability to bear risk and risk attitude is examined. If there is a discrepancy between ability and willingness, the investor must be educated with regard to the risks of loss and opportunities for profit. If risk willingness is above average and ability to bear risk is below average, the ability to accept risk is the limiting factor, and risk tolerance is classified as below average. If, on the other hand, the willingness to take risk is below average and the ability to bear risk is

4

See Bodie and Merton 2000: Finance, p. 322. See Kaiser 1990: ‘Individual Investors’, p. 3–12. 6 See Grable and Joo 2004: ‘Environmental and biopsychosocial factors associated with financial risk tolerance’, p. 73 f. 5

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Table 7.1 Assessment sheet for risk attitude (Source: Based on Grable and Joo 2004: ‘Environmental and biopsychosocial factors associated with financial risk tolerance’, p. 73 f) Statement 1. Investing money is a complex activity that I have no idea about.

2. I feel better when my money is in the bank account and not invested in the stock market.

3. When I think about the word ‘risk’, the word ‘loss’ comes to mind.

4. To make money in stocks and bonds, you need luck.

5. Safety is more important than profit (return) when investing.

Answer (a) Agree in full. (b) Agree in substance. (c) Disagree in substance. (d) Do not agree at all. (a) Agree in full. (b) Agree in substance. (c) Disagree in substance. (d) Do not agree at all. (a) Agree in full. (b) Agree in substance. (c) Disagree in substance. (d) Do not agree at all. (a) Agree in full. (b) Agree in substance. (c) Disagree in substance. (d) Do not agree at all. (a) Agree in full. (b) Agree in substance. (c) Disagree in substance. (d) Do not agree at all.

above average, the client must be informed of possible lost profits. For example, a wealthy investor with a 25-year time horizon has a high ability to accept risk. However, if he has a low willingness to take risk, there would be a conflict. The limiting factor is characterised by the willingness to assume risk, and the risk tolerance is classified as above average. If there are no discrepancies between ability to accept risk and risk willingness, the classification of risk tolerance is relatively simple. Table 7.2 illustrates how risk tolerance can be assessed.

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227

Table 7.2 Assessment of risk tolerance based on ability to bear risk and willingness to take risk (Source: Own illustration) Ability to bear risk Above average Below average Below average Above average

Willingness to take risk Above average Below average Above average Below average

Risk tolerance Above average Below average Below average Above average

Example: Determining Risk Tolerance A portfolio manager assesses the risk tolerance of two new clients, Peter Miller and Christian Franck. Peter Miller is 40 years old and works as a coffee trader for a commodity trading company in Germany. He is married and has two children, aged 10 and 14. The portfolio manager compiles the following relevant factors for assessing risk tolerance: • Miller’s annual income is EUR 300,000, which covers considerably more than the family’s living expenses. • The mortgage on the house is paid off and the family has savings of EUR 1.5 million. • Miller considers his job to be relatively secure. • He is well versed in financial matters and believes that stocks generate positive long-term returns. • Miller expects his savings to be sufficient to retire at age 55. • Following the risk assessment sheet (see Table 7.1), Miller does not agree at all that he ‘feels better if the money is invested in the bank account and not in the stock market’ and that ‘safety is more important than profit’. The second client, Christian Franck, is self-employed and owns a small restaurant in Cologne, Germany. He is 50 years old, divorced, and has four children aged between 10 and 16. The portfolio manager compiles the following relevant factors for assessing risk tolerance: • Franck’s average annual income is EUR 50,000 and is subject to large fluctuations. • The alimony payments for his divorced wife and the four children amount to EUR 20,000 per year. • There is a mortgage of EUR 80,000 on the apartment and he has savings of EUR 30,000. • He is well versed in financial matters and believes that equities generate positive returns over the long term. (continued)

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• Franck expects that a large part of the EUR 30,000 in savings will be needed to finance his children’s living and study costs at university. • Following the risk assessment sheet (see Table 7.1), Franck does not agree at all that ‘you need luck to make money with stocks and bonds’ and that ‘risk equals loss’.

1. What is the assessment of Miller’s risk tolerance? 2. How does the assessment of Franck’s risk tolerance turn out? Solution to 1 Miller has a high income that covers far more than the family’s living expenses. He has a high net worth (a paid-off house and savings of EUR 1.5 million), has a relatively secure job, and has a long-time horizon until his planned retirement in 15 years. These facts suggest that Miller’s ability to bear risk is above average. The responses to the standardised questionnaire indicate an above-average willingness to take risk. Miller’s risk tolerance can therefore be classified as above average. Solution to 2 Franck has a relatively low and volatile income. In addition, he must make mortgage repayments and annual alimony payments of EUR 20,000. The time horizon until he must provide financial support for his children’s studies is relatively short, ranging from 2 to 8 years. The financial situation and the relatively short investment horizon until he must make support payments for the children’s university education suggest a low ability to accept risk. By contrast, responses to the standardised questionnaire suggest an above-average willingness to take risk. There is a conflict between below-average ability to bear risk and above-average risk willingness, and therefore his risk tolerance must be classified as below average. The portfolio manager has to inform Franck about his financial situation and, contrary to his risk appetite, propose a portfolio that is relatively low-risk.

7.2.1.2 Return Objectives An investor’s desired return must be consistent with the risk objectives. The targeted return can be realistic or unrealistic. Therefore, the portfolio manager must match high-return requests with the client’s risk tolerance and expected capital market data. Unlike a desired return, which can be diminished if there is a mismatch with the risk objective, high required returns represent a potential conflict between return and risk objectives in an IPS.

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The total return is the sum of the price return and the income return of the assets included in the portfolio (e.g. for equities, return on capital gains/losses, dividends, and income on reinvested dividends). It can be defined either as an absolute value (e.g. 10%) or as a relative value, that is, portfolio return versus benchmark return (e.g. benchmark return + 2%). A distinction can also be made between a nominal and a real return and between a return before and after tax.7 The return required to meet the investment objectives must be achieved, on average, over the investment period. For example, a private investor needs a certain geometric average annual return to obtain an investment amount at the end of employment that is sufficient for retirement. Assume a married couple needs an amount of EUR 1.5 million in 20 years, including expected inflation, to financially secure their retirement. Their current assets are EUR 0.5 million. To reach the desired final amount of EUR 1.5 million, the couple needs to earn an annual aftertax return of 5.65% [= (EUR 1,500,000/EUR 500,000)1/20 - 1]. If the tax rate is 30%, the required pre-tax return is 8.07% [= 5.65%/(1 - 0.30)]. Another example is an investor who needs a sufficient portfolio return in retirement to cover his expenses. For this purpose, the retiree must earn, for example, a 5% after-tax return on his invested capital to cover living expenses. The 5% required rate of return should be understood in real and after-tax terms. If the expected annual inflation rate is 2% and the tax rate is 35%, the nominal pre-tax return is 10.77% [= (5% + 2%)/ (1 - 0.35)]. A pension fund, for example, must generate an average return from its investment portfolio in order to pay pensions to current and future retirees. If the required return is not achieved on average, a permanent underfunding arises, and the pension fund must be restructured by means of contributions or lowering the pension benefit payments.

7.2.1.3 Constraints Liquidity The IPS should list the circumstances that lead to cash withdrawals from the portfolio. For a private wealth management client, for example, these might be payments to purchase a new house or for continuing education. For a pension fund, by contrast, liquidity needs consist of monthly pension payments and expected lump-sum withdrawals. Liquidity requirements should be set in such a way that both planned and unplanned expenditures can be covered. Accordingly, the portfolio must contain cash or cash equivalents that can be quickly converted into cash. There is a liquidity risk if the investments can only be sold at a discount due to a lack of market liquidity. The portfolio manager controls security selection, but not the investor’s liquidity needs. Therefore, liquidity management has to be done through investment selection.8 For example, if the portfolio’s assets and income are high, less liquid investments can be held. Another factor is the price risk of assets. In the event of a 7 8

See Sect. 1.8. See Kaiser 1990: ‘Individual Investors’, p. 3–21 f.

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market collapse, the market liquidity of assets with a higher price risk usually falls. When the timing of an investor’s potential payment shortfall correlates with a market downturn, security selection should move towards less price-sensitive assets. In order to anticipate future liquidity needs, both liquidity and price risk should be considered, and a portion of the portfolio held in liquid and less price-sensitive assets. Investment horizon The investment period depends on the investment objectives and ranges from short to long term. For example, a multi-stage investment period may consist of a combination of a short-term and a long-term period. In the short term, an investor may need to pay for the living expenses of his children studying at university abroad, while the long-term focus is on financial security after retirement. In general, the longer the investment horizon, the higher the ability to bear risk due to accumulated savings and investment income. Long-term earned income from employment also allows for higher portfolio risk. For a short-term investor, cash and cash equivalents are a safe investment. By contrast, an investor with a long-term investment horizon is exposed to reinvestment risk when holding cash and cash equivalents. Unique circumstances The specific circumstances of an investor may influence portfolio allocation.9 A client may have reservations about certain investments due to religious and ethical considerations. For example, an Islamic investor who follows Sharia (Islamic law) will not invest in casino stocks and bonds because Sharia prohibits gambling and lending money at interest. For ethical reasons, one can avoid investing in companies that are involved in the arms or tobacco industries or contribute to high levels of environmental pollution. Another unique circumstance is that an entrepreneur may forbid the portfolio manager to buy stocks of competitors. Taxes The tax burden varies from investor to investor. In many countries, pension funds do not pay any tax on investment income. Private investors, on the other hand, usually have different tax rates for investment income such as dividends and interest and for capital gains. Typically, the tax rate is higher for income than for capital gains. Investment income is taxed as soon as it is earned. By contrast, only realised capital gains are taxable. Unrealised capital gains are generally not subject to capital gains tax, resulting in a tax benefit (time value effect of tax payment). In Switzerland, dividend distributions are subject to withholding tax of 35% on the gross distribution amount, regardless of the legal status of the shareholder. As a result of the withholding tax deducted from dividend income, dividend recipients in Switzerland are required to declare investment income received in their tax returns.

9

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 51.

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231

Overpayments of withholding tax are offset or refunded. For example, if an investor owns 400 shares that pay a dividend per share of CHF 2, the gross dividend amounts to CHF 800. After deduction of withholding tax, the investor receives a net dividend of CHF 520. For tax purposes, the gross dividend of CHF 800 is added to the taxable income. The withholding tax paid of CHF 280 is offset against the tax liability or refunded. Capital gains from the sale of privately managed securities are exempt from tax in Switzerland. However, if purchases and sales of securities are made in a manner that goes beyond the simple management of private assets, capital gains are taxed as income from self-employment. The principle applied is that capital gains are taxable in the case of professional stock trading. Furthermore, Switzerland has a wealth tax on the ownership of domestic and foreign securities.10 In Germany, dividends and private capital gains are subject to withholding tax. The flat rate of withholding tax is 25%. In addition, a solidarity surcharge of 5.5% is levied on the withholding tax, resulting in a total tax burden of 26.375% (= 25% × 1.055). If applicable, church tax is added to this. Private shareholders receive income from equity securities after deduction of the withholding tax, which is paid to the tax office. Capital income does not have to be declared in the income tax return and is therefore not subject to the investor-specific income tax rate.11 The portfolio must be aligned with the investor’s tax situation. For example, a taxable investor should hold a portfolio focused on capital gains if these are taxed at a lower rate than investment income. By contrast, it is immaterial whether a tax-exempt investor, such as a pension fund, holds investments with capital gains or income. Legal and Regulatory Factors Legal and regulatory constraints that limit investment activities must be listed in the IPS. In many countries, there is a legal framework for pension funds that regulates portfolio allocation. For example, a maximum allocation may be prescribed in the case of risky investments such as equities. In terms of legislation on insider trading, an individual who has confidential, price-sensitive information about a particular company (i.e. an insider) may not use this knowledge to buy or sell securities in order to gain a pecuniary benefit. In Switzerland, for example, sales of securities made in advance of an earnings warning have been covered by the Swiss Penal Code since 2008.12

10

See Mondello 2022: Corporate Finance: Theorie und Anwendungsbeispiele, p. 957. See Perridon et al. 2017: Finanzwirtschaft der Unternehmung, p. 410 f. 12 In Switzerland, insider offences are regulated in the Swiss Penal Code (Art. 161 StGB). In Germany, the prohibition of insider trading is listed in the Securities Trading Act (§ 14 WpHG). 11

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Example: Investment Policy Statement Urs Meier is 50 years old, single, and a self-employed management consultant. He owns assets worth CHF 2.5 million, the majority of which is invested in small-cap stocks. Over the past 4 years, his portfolio has generated an average annual total return of 18%. He hopes that the portfolio will continue to generate a similarly high return. He considers his own risk tolerance to be average. When a friend points out to him that the risk of small-cap stocks is high, he is surprised by this statement. Meier would like to retire at the age of 65. His current income by far exceeds the cost of living. Upon retirement, he plans to sell his consulting business for CHF 1 million. The income is subject to a tax rate of 35%. What return and risk objectives and constraints should a long-term IPS include for Meier? Solution Risk objectives: Meier has an above-average risk tolerance because both his ability to bear risk and risk willingness are high. His ability to bear risk is high due to the relatively large asset base of CHF 2.5 million, the long investment period, the high income to cover living costs, and the low liquidity needs. The concentration of his assets in equities with a low market capitalisation and the desire for a high return indicate a high willingness to take risks. Return objectives: Meier’s financial situation—characterised by a long investment period, relatively large assets, high income, low liquidity needs, and above-average risk tolerance—leads to the conclusion that a high total return should be sought, consistent with long-term capital market expectations. However, a desired return of 18% is unrealistic. Liquidity: Liquidity requirements are low. Investment horizon: Meier has a long investment horizon since he is only 50 years old. The investment horizon consists of two phases. The first phase lasts approximately 15 years and ends when he stops working, while the second phase extends from retirement until death (approximately 15–20 years). Taxes: Meier pays a 35% tax rate on income. Tax considerations should therefore play a significant role in his investment decisions. However, capital gains from the sale of privately held securities are exempt from tax in Switzerland.

7.2 Planning

7.2.2

233

Investment Policy Statement

Once the return and risk objectives and the constraints have been defined, the longterm investment policy can be formulated. This policy document serves as the basis for all investment decisions.13 The IPS usually includes the following content: • A brief description of the investor • The reason for determining the investment policy • The duties and responsibilities (including fiduciary duties) of all parties involved in the investment decision (e.g. investor, investment manager, any investment committee, and custodian bank) • A report on the investment objectives and constraints • A schedule for reviewing the investment performance and policy • Performance measures and benchmarks to evaluate investment performance • Investment strategies and investment styles (e.g. value or growth bias in equities) • Guidelines for portfolio reallocation based on the feedback phase The IPS and, in particular, the risk/return objectives and constraints, together with the long-term capital market expectations, constitute the starting point for establishing strategic asset allocation. The strategic asset allocation can also be integrated into the IPS, which is regularly the case in portfolio management practice. However, in the following discussion the strategic asset allocation is considered a result of the IPS and is therefore not part of the IPS. The investment strategy is developed in the planning phase and describes the procedure to be followed for investment analysis and security selection. It forms the basis for investment decisions and identifies which strategies can be used to achieve the risk and return objectives of the IPS. Investment strategies can generally be divided into passive, active, and semi-active strategies. In a passive investment strategy, the composition of the portfolio remains the same, regardless of new capital market expectations. This strategy can be implemented by indexing to an equity index, for example. If the stock portfolio is indexed to the DAX 40, for example, the portfolio will only change if the stock composition of the DAX 40 changes. Should a new stock be included in the DAX 40, a revised stock portfolio that tracks the changed equity index must be constructed. New capital market expectations do not lead to a reallocation of assets in the portfolio. In addition, a passive investment strategy can be implemented through a buy-and-hold strategy. For example, a portfolio manager can buy bonds and hold them until maturity. In an active investment strategy a portfolio manager reacts to new capital market expectations and rebalances the portfolio accordingly. The performance of the portfolio is compared with a benchmark of the same investment style. In order to achieve an excess return or a positive alpha (return above the benchmark), the

13

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 52.

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portfolio weights of undervalued (overvalued) securities are increased (reduced) in relation to the benchmark. The fact that the weights of the assets in the portfolio deviate from those of the benchmark is the result of the manager’s own capital market expectations, which differ from the overall expectations of the market participants. If the manager is correct in his analysis on average and makes the corresponding investment decisions, he will achieve a return above the benchmark or a positive alpha and the active strategy will pay off.14 The semi-active investment strategy, or enhanced index strategy, is a hybrid form of the passive and the active strategy. The objective is to generate positive alpha while controlling or minimising active risk relative to the benchmark. In the case of bonds, for example, the portfolio manager can increase (decrease) the portfolio weights of undervalued (overvalued) bonds relative to the benchmark, while controlling interest rate risk and equating the duration of the bond portfolio to that of the benchmark.

7.2.3

Capital Market Expectations

The portfolio manager’s long-term capital market expectations lead to return and risk estimates for the individual asset classes, which, in addition to the investment objectives and constraints, are relevant for determining the strategic asset allocation.15 The portfolio created in this way is efficient in terms of expected return and risk, and therefore has the highest expected return for the given risk.16 For each individual asset class, the capital market data consist of the expected return, the standard deviation of returns and the corresponding correlation coefficients (or covariances) between the returns of two asset classes. The expected return can be split into a risk-free interest rate and a risk premium for the asset class in question. Expected returns, standard deviations, and correlation coefficients can be determined using, for example, historical return data or returns based on scenario analysis.17

7.2.4

Strategic Asset Allocation

The final phase of the planning process involves determining the strategic asset allocation, which reflects the long-term portfolio exposure to systematic risks set out in the investment policy. The asset allocation decision is a critical component of the portfolio management process and has an important influence on investment

14

See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 1109 ff. See Diermeier 1990: ‘Capital Market Expectations: The Macro Forecasts’, p. 5-2. 16 See Sect. 4.4. 17 See Sect. 4.2 and Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 70 ff. 15

7.2 Planning

235

performance.18 In general, the following four decisions have to be made when creating an investment strategy:19 • • • •

What asset classes should be considered for investment? What target weights should be assigned to each eligible asset class? What are the allowable allocation ranges for each asset class? What specific assets should be purchased for the portfolio?

Long-term capital market expectations are combined with the investor’s risk tolerance to determine the target weights of the individual asset classes within the portfolio with a view to meeting the investment objectives of the IPS and dealing with its constraints. Target weights are defined within a maximum and minimum range. If the asset classes in the portfolio deviate from this predefined range, a correction is made by buying and selling assets. An asset class consists of a group of investments with similar characteristics. Asset class selection is an important decision that has a lasting impact on the expected return and risk of a portfolio. It is made through asset classes that are permitted by the IPS referred to earlier. Before possible asset classes are chosen, the criteria for defining them must first be established: • Investments within an asset class should be as homogeneous as possible and should have similar characteristics. If equities are defined as an asset class, then no bonds should be included, for example. • Asset classes should be independent of each other. Overlapping asset classes reduce the effectiveness of controlling risks in strategic asset allocation and lead to problems in estimating expected returns. For example, for a Swiss investor, the ‘domestic equities’ asset class should be combined with the ‘European equities ex Switzerland’ asset class and not with the ‘European equities’ asset class because the latter also includes Swiss equities. • Asset classes should be diversified in order to reduce the systematic risk of the portfolio. They should not have high correlations with each other, which would make them redundant since they would duplicate risk exposure. • All the chosen asset classes should, together, represent a large proportion of the world’s investable assets. A portfolio that meets this characteristic is more efficient in terms of expected return and risk. • An asset class should constitute a significant component of the portfolio without affecting the liquidity of the asset mix. Corrections in the strategic asset allocation

18

Several studies have examined the effect of the strategic asset allocation on investment performance. Studies for pension funds and mutual funds found that approximately 90% of a fund’s returns over time can be explained by its strategic asset allocation. See, for example, Ibbotson and Kaplan 2000: ‘Does asset allocation policy explain 40, 90, or 100 percent of performance?’, p. 26 ff. 19 See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 53.

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should not result in price movements of the investments included in the corresponding asset class and should not cause high transaction costs. Traditional asset classes consist of the following categories, each of which can be broken down further: • Domestic equities: This asset class can be categorised further by breaking down the market capitalisation of equities into small, medium, and large market capitalisation. • Domestic bonds: The maturity of the bonds can be applied to divide this asset class into medium-term and long-term domestic bonds. Another feature is inflation protection, which allows for a subdivision into nominal and inflation-linked bonds. The real value preservation of a portfolio is an important objective in portfolio management. • Foreign equities: This asset class can be further broken down into developed and emerging equity markets. • Foreign bonds: A further separation into developed and emerging bond markets is possible. • Real estate: Today, this asset class is often listed under alternative investments rather than traditional investments.20 Alternative investments consist of real estate, private equity, commodities, and hedge funds. They should be managed as separate asset classes because they are not homogeneous and therefore have different characteristics. • Cash and cash equivalents. When investing in assets denominated in a foreign currency, the return and risk of a portfolio are affected by currency fluctuations. If long positions in a foreign currency are held in the portfolio and the foreign currency depreciates against the investor’s domestic currency, this results in a loss. A gain, and therefore a positive contribution to the portfolio return, occurs when the foreign currency appreciates against the domestic currency. Empirical studies demonstrate that currency risk, measured by the standard deviation of exchange rate changes, is about half as large as equity market risk. By contrast, currency risk is often larger than local currency bond price risk (approximately twice as large).21 Furthermore, in times of crisis, correlations between capital markets around the world increase, leading to higher portfolio risk. It is also important to take into account special aspects of emerging markets such as the limited free float of shares, ownership constraints on foreign

20

In the early 1990s, real estate was considered a traditional asset class, along with equities and bonds. Today, real estate is referred to as an alternative asset class, along with hedge funds, private equity, and commodities. 21 See Solnik and McLeavey 2004: International Investments, p. 471 ff.

7.2 Planning

Investor: Net worth and risk attitude

237

Investor: Risk tolerance function

Investor: Risik tolerance

Optimiser

Capital market conditions

Prediction procedure

Strategic asset allocation (investor’s asset mix)

Returns

Expected returns, risks, and correlations

Fig. 7.2 Key steps in determining strategic asset allocation (Source: Based on Sharpe 1990: ‘Asset Allocation’, p. 7–20)

investors, the quality of corporate information, and asset returns that are not normally distributed.22 Figure 7.2 illustrates the most important steps for determining strategic asset allocation.23 Net worth and risk attitude influence an investor’s risk tolerance. Risk tolerance or the degree of risk aversion can be taken to define utility functions.24 Capital market analysis makes it possible to obtain information on, among other things, current and historical asset prices, which can be used to estimate expected returns, standard deviations, and correlation coefficients. Various techniques, such as historical returns, returns based on scenario analysis, or

22

When the optimal portfolio or strategic asset allocation is constructed using the Markowitz model (mean variance method), it is assumed that returns are normally distributed. See Sect. 4.4. This is an unrealistic assumption for investments in emerging markets. 23 See Sharpe 1990: ‘Asset Allocation’, p. 7–18 ff. 24 See Sect. 5.2.2.

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regression analysis with historical returns applying the market model can be used to calculate expected returns, standard deviations, and correlation coefficients.25 The investor’s risk tolerance and the estimated expected returns, risks, and correlations are incorporated into an optimisation procedure to identify the optimal portfolio or strategic asset allocation. The point of contact between the efficient frontier and the investor-specific indifference curve with the highest achievable utility represents the optimal portfolio of risky assets.26 With the inclusion of the risk-free asset, the most efficient portfolio in terms of expected return and risk lies on the most efficient capital allocation line.27 The construction of the efficient frontier is crucial in determining strategic asset allocation. It can be generated using various methods such as the Markowitz model, the market model, simulation models, or the Black– Litterman model.28 The returns from the strategic asset allocation that occur in one period influence the portfolio value at the beginning of the next period. Figure 7.2 illustrates this relationship with a feedback loop between the returns and the net worth of the investor. The returns earned from the portfolio are compared to capital market conditions for the purpose of testing the prediction procedure, which is also marked with a feedback loop. The loops in the figure indicate that these processes are continuous. Thus, the decisions and portfolio results in one period affect the decisions in the following period. The rectangles drawn with a plain line in Fig. 7.2 change from one period to the next. By contrast, the rectangles drawn with a bold line remain unchanged because they are decision rules or procedures. For example, the investor’s risk tolerance may change while the risk tolerance function used to measure is given and remains unchanged. Similarly, the forecast expected returns, risks, and correlations change periodically, while the prediction procedure remains the same. The optimal strategic asset allocation may also change, while the optimisation procedure, once chosen, usually remains the same. The process illustrated in Fig. 7.2 can be applied to both strategic and tactical asset allocation. With the tactical asset allocation, the weights of the asset classes are temporarily adjusted, based on short-term capital market expectations, in order to enhance the performance of the portfolio. Overvalued assets are sold and undervalued assets are purchased with the funds received from the sale. Tactical asset allocation is guided by short-term capital market expectations, while strategic asset allocation is driven by long-term capital market expectations. Changes in an investor’s net worth or attitude to risk play a subordinate role in tactical asset allocation.29

25

See Mondello 2015: Portfoliomanagement: Theorie und Anwendungsbeispiele, p. 198 ff. See Sect. 5.3. 27 See Sect. 5.4. 28 See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 143 ff. For the BlackLitterman model see Black and Litterman 1992: ‘Global portfolio optimization’, p. 28 ff. 29 See Sharpe 1990: ‘Asset Allocation’, p. 7–24 f. 26

7.4 Feedback

7.3

239

Execution

In the execution phase, a manager combines investment strategies with capital market expectations to select assets for the portfolio. In a financial institution, analysts prepare an investment list that the portfolio manager uses to buy the securities. The purchase itself, however, takes place in the trading department. If the investor’s situation or long-term capital market expectations change, the composition of the portfolio is revised. The execution phase interacts constantly with the feedback phase. Implementing strategic asset allocation is as important as selecting assets. High transaction costs reduce investment performance. Trading costs consist of explicit and implicit costs. Explicit costs include broker commissions, exchange fees, and taxes. Implicit costs, on the other hand, comprise the bid-ask spread, the effect of a trade order on the price of the transaction, opportunity costs, and waiting costs.30

7.4

Feedback

The feedback phase is an important part of the portfolio management process. It consists of the monitoring of the IPS and capital market expectations, the possible rebalancing of the portfolio, and the performance evaluation, which are described below.

7.4.1

Monitoring the Investment Policy Statement

A client’s needs and circumstances change over time. Hence, the portfolio manager must identify the client’s needs and, if necessary, anticipate events that may lead to a change in those needs. Regular client meetings at which a client’s needs, circumstances, and investment objectives can be clarified are suitable for this purpose. If significant changes are found, the IPS must be revised and the portfolio adjusted accordingly. In private wealth management, reviews usually take place on a semi-annual or quarterly basis. By contrast, for institutional investors, reviews are conducted as part of the annual asset allocation assessment. A review of the IPS will examine changes in the following areas: 1. 2. 3. 4. 5.

30

Circumstances and net worth Liquidity requirements Investment horizon Taxes Unique circumstances

See Sect. 3.3.4.

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Portfolio Management Process

1. Changes in the circumstances of private wealth management clients occur as a result of events such as a change in employment, marriage, and/or the birth of children, which may affect income, spending, exposure, and preferences to risk. A client’s net worth is a key component of the IPS. Wealth is a measure of financial success achieved and has an impact on future investment planning. Changes in net worth arise from the client’s saving and spending behaviour, investment performance, and events such as gifts and inheritances. According to utility theory, an increase in wealth leads to higher risk tolerance and therefore a higher expected return. However, portfolio managers should only incorporate material and permanent changes in wealth when determining risk tolerance. Temporary wealth shifts should not lead to a change in risk tolerance. 2. Changing liquidity needs of private wealth management clients arise from events such as job loss, illness, court judgements, retirement, divorce, death of a spouse, or property purchase. If significant cash withdrawals from the portfolio are likely, liquid investments should be held. Furthermore, if money is to be spent in the short term, investing in liquid assets with low price risk, such as money market securities, is advisable. 3. The investment horizon shortens the older a person becomes. In order to be able to cover planned expenses during retirement, the risks in the portfolio should be reduced, and less should be invested in equities and more in coupon-paying prime bonds, which guarantee a constant return. Moreover, the price risk of bonds with a high credit quality is lower than that of equities. Many clients have multi-stage investment horizons. For example, a working person is in an accumulation phase that lasts until retirement. During this phase, wealth is accumulated through savings and investments. In retirement, the assets are spent and ultimately inherited. Typically, changes in investment policy are necessary when one phase ends (e.g. due to retirement or the sale of a family business) and a new one begins. Although the phases are generally predictable, abrupt changes also occur from time to time. For example, the death of a working spouse may trigger an adjustment of the IPS. The portfolio manager must therefore keep an eye on changes in the investment horizon and prepare for them by proposing solutions in the IPS. 4. For taxable investors, all investment decisions must be made on an after-tax basis. Therefore, the manager must create a portfolio that takes into account the client’s current and future tax situation. The investment horizon and reallocations of the portfolio are important factors in this process. Monitoring a client’s tax situation may, for example, involve the following actions: • The realisation of profits is shifted from a year with a high tax burden to a year with a low tax burden. • Expenses are increased in a year with a high tax burden. • At the end of the year, losses are realised in order to reduce the realised gains for the year and hence the taxes.

7.4 Feedback

241

• Assets with high unrealised gains should be used for charitable donations, as donations to charitable institutions can be given preferential tax treatment. 5. If specific circumstances change, the IPS must be adjusted accordingly. For example, an investor may hold a large position in a particular stock that they received as a salary component from their work as a management board member. After the vesting period has expired, it may make sense to sell the stock position due to the high concentration of risk in the portfolio and buy new investments with the proceeds of the sale.

7.4.2

Monitoring Capital Market Expectations

In addition to monitoring investment policy, long-term capital market expectations must also be reviewed. The economy moves from phases of expansion to contraction, which are characterised by different features. The financial markets are closely linked to the state of the economy as a whole and its future development, which have a significant influence on return expectations and risks of the asset classes and individual investments. In this context, the portfolio manager must monitor for example changes in expected returns, risks and correlations of asset classes, the market cycle, monetary policy, the yield curve, and inflation. The expected returns, risks, and correlations of asset classes may change in such a way that the strategic asset allocation no longer meets the client’s investment objectives and therefore needs to be adjusted. The market cycle and valuation levels should also be monitored in order to form an opinion on the short-term risks and return opportunities in the financial markets. As a result, tactical adjustments to asset allocation can be made. Furthermore, the monetary policy of central banks has an important influence on bond and equity markets. For example, if central banks buy government securities on the financial markets, their prices rise while interest rates fall. A lower interest rate level leads to bonds being a less attractive asset class, and thus changes in asset allocation become necessary.

7.4.3

Rebalancing the Portfolio

Monitoring a portfolio is a continuous process. In this process, events and trends are observed that have an influence on the future price development of assets and asset classes and their suitability for achieving the investment objectives. This may lead to changes in the IPS or to the substitution of assets in the portfolio. Furthermore, if the weights of asset classes deviate from the target weights of the strategic asset allocation due to price changes of the assets contained therein, reallocations must be made in the portfolio to bring the weights back to their target levels. A portfolio reallocation may also be necessary if the investment objectives or the constraints of the IPS change. Moreover, changes in long-term capital market expectations can also result in corrections to the strategic asset allocation.

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Reallocations of a portfolio also take place in tactical asset allocation. Portfolio reallocations resulting from changes in the value of investments are described below. When rebalancing a portfolio there is a trade-off between benefits and costs. In a case where the strategic asset allocation is given by an optimal portfolio that provides the investor with the greatest possible utility, any deviation from the optimum means a loss of benefit. The necessary rebalancing of a fixed asset mix requires, on the one hand, selling assets that are increasing in value and, on the other hand, buying assets that are decreasing in value, which is equivalent to a contrarian investment strategy. Since this strategy provides liquidity in the market, a positive return can be expected, which increases the benefit. The costs of portfolio rebalancing include transaction costs and taxes. As assets are sold with price appreciation when the portfolio is readjusted, capital gains are realised, which must be taxed accordingly. The two main rebalancing strategies are: 1. Calendar strategy 2. Percentage-of-portfolio strategy 1. In the calendar strategy, the portfolio is periodically adjusted to the target weightings defined in the strategic asset allocation. Rebalancing can take place monthly, quarterly, semi-annually, or annually. If, for example, the target weights for three asset classes are 40%, 30%, and 30%, the weights are periodically (e.g. every 3 months) moved back to the targets. The advantages of this rebalancing strategy are that it is simple and does not require continuous monitoring of the portfolio. However, the frequency of the reviews must be aligned with the volatility of the asset classes in the portfolio so that their weights do not deviate too much from the target weights within a specified period. Moreover, it is not primarily the market but the specified period that is relevant for adjusting the weights. 2. In the percentage-of-portfolio strategy, a rebalancing of the portfolio to the target weights of the strategic asset allocation occurs when the weight of an asset class falls outside a specified range. For example, if the target weight of an asset class is 40% and the range is set at 35% to 45%, a correction of the weight takes place whenever the weight of the asset class falls outside the specified range of 35% to 45%. For the rebalancing of the entire portfolio to the target weights, it is sufficient if an asset class drops outside the range. In contrast to the calendar strategy, the rebalancing of asset classes is not carried out periodically, but when the predefined ranges are exceeded or fallen below. Therefore, the percentage-of-portfolio strategy allows for tighter control of deviations from target levels and is thus closely linked to the market performance of the asset classes. For the strategy to be implemented efficiently, a short review frequency is required. The greatest precision can be achieved with a daily review.

7.4 Feedback

7.4.4

243

Performance Evaluation

Portfolio performance must be evaluated periodically in order to assess the achievement of investment objectives and the skills of the portfolio manager. The manager’s skills are assessed through performance measurement and attribution and on the basis of a track record of realised returns in the past. Performance measurement involves determining the portfolio’s return. Risk-adjusted return measures such as the Sharpe ratio and the information ratio can be used. The Sharpe ratio sets the portfolio return above the risk-free rate in relation to the total portfolio risk (standard deviation) and measures the risky part of the portfolio return for a unit of total risk:31 SR =

rP - rF , σP

ð7:1Þ

where rP = return of the portfolio, rF = risk-free interest rate, and σ P = standard deviation of portfolio returns. The risk-adjusted return of an active strategy can be calculated by the information ratio, which consists of the quotient between the active return and the active risk. The active return is the difference between the returns of the portfolio and the benchmark. The active risk, on the other hand, is the standard deviation of the active returns, that is, the standard deviation between the returns of the portfolio and the benchmark of the same investment style. The information ratio measures the manager’s excess return for a unit of active risk and is determined as follows: IR =

rP - rB r = A, σA σA

ð7:2Þ

where rP = return of the portfolio, rB = return of the benchmark, rA = active return (rP - rB), and σ A = standard deviation of active returns. The excess or active return in the numerator of Eq. (7.2) represents the manager’s ability to use their talent and information to generate a portfolio return that differs from that of the benchmark against which the performance is being measured. By contrast, the denominator measures the amount of active (unsystematic) risk that the manager incurred in pursuit of excess returns. Reasonable information ratio levels

See Sharpe 1966a: ‘Mutual fund performance’, p. 119 ff. For a more recent interpretation of this risk-adjusted return measure, also see Sharpe 1966b: ‘The Sharpe ratio’, p. 49 ff.

31

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should range from 0.5 to 1, with a portfolio manager having an information ratio of 0.5 being good and one with an information ratio of 1 being exceptional.32 Example: Sharpe Ratio and Information Ratio A portfolio manager examines four investment funds with an active investment strategy for large-cap German equities for their client. The benchmark is the DAX 40, which has a return of 6% and a standard deviation of returns (volatility) of 20%. The portfolio manager has compiled the following data for the four investment funds: Investment fund A B C D

Return 8.2%

9.8%

Volatility 25% 30% 21% 24%

Active risk 2.8% 2.4% 2.0%

Information ratio 0.9 1.2 1.1

The portfolio manager wants to select the investment fund with the highest information ratio that meets the following return objectives in the client’s IPS: • The active return must be at least 2.5%. • The Sharpe ratio must be not less than 0.3. The risk-free interest rate is 2%. Which investment fund has the highest information ratio and meets the investment guidelines contained in the IPS? Solution Investment fund A: r A = 8:2% - 6% = 2:2% SR =

8:2% - 2% = 0:248 25%

Investment fund A does not comply with the investment guidelines, which require an active return of at least 2.5% and a Sharpe ratio of 0.3. Investment fund B: rA = 1:2 × 2:8% = 3:36%

(continued)

32

See Grinold and Kahn 2000: Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk, p. 109 ff.

7.4 Feedback

245

3:36% = rP - 6% → rP = 9:36% SR =

9:36% - 2% = 0:245 30%

The Sharpe ratio of 0.245 is too low in relation to the required risk-adjusted return of 0.3. Investment fund C: r A = 1:1 × 2:4% = 2:64% 2:64% = rP - 6% → rP = 8:64% SR =

8:64% - 2% = 0:316 21%

Investment fund C complies with the investment guidelines and, in addition to a Sharpe ratio of 0.316, has an information ratio of 1.1. Investment fund D: SR =

9:8% - 2% = 0:325 24%

rA = 9:8% - 6% = 3:8% IR =

3:8% = 1:9 2%

Investment fund D meets the return objectives of the IPS and has the highest information ratio of 1.9. The portfolio manager will select this fund for the client. Performance attribution makes it possible to evaluate the sources of the returns. When assessing the performance of the active manager, the return of the portfolio is compared with that of the benchmark of the same investment style. In the case of an active investment strategy, if there is an excess return of the portfolio relative to the benchmark, the portfolio manager has added value. In portfolio management practice, returns are often broken down into three components to assess performance: (1) strategic asset allocation, (2) market timing reflecting tactical versus strategic asset allocation, and (3) security selection, which is the manager’s ability to choose individual securities with superior returns within an asset class.

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7

7.5

Portfolio Management Process

Performance Attribution of an Active Portfolio

Comparing the return of a portfolio with that of the benchmark, the manager’s added value or active return can be calculated as follows: rA = rP - rB :

ð7:3Þ

Since the portfolio return is the sum of the weighted returns of the individual assets, the formula for determining the active return can be written as follows: N

N

rA =

wPi r i i=1

wBi r i ,

ð7:4Þ

i=1

where wPi = weight of asset i in the portfolio, wBi = weight of asset i in the benchmark portfolio, ri = return of asset i, and N = number of assets in the portfolio or in the benchmark portfolio. The portfolio manager’s added value is equal to the sum of the difference in the weights of the individual assets between the portfolio and the benchmark multiplied by the return of the corresponding asset: N

rA =

ðwPi- wBi Þr i :

ð7:5Þ

i=1

If the above equation is rearranged, the following formula to calculate the active return is obtained:33 N

rA =

ðwPi- wBi Þðr i- r B Þ:

ð7:6Þ

i=1

Equation (7.6) represents the simplest equation for performance attribution. The added value by the portfolio manager is due to two return components: (1) different weights of the individual assets in the portfolio and the benchmark and (2) returns of the individual assets compared to the total return of the benchmark.34 Table 7.3

33

The sum of the weights is 1, and therefore the difference between the sum of the weights in the N

portfolio and the benchmark is 0: i=1

ðwPi- wBi Þ = 0: Multiplying the difference in the weights by

the benchmark return gives 0: N i=1 34

ðwPi- wBi Þr B = r B

N i=1

ðwPi- wBi Þ = 0.

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 1130.

7.5 Performance Attribution of an Active Portfolio Table 7.3 Active return: different weights and returns of portfolio and benchmark (Source: Own illustration)

wPi - wBi Positive Negative Positive Negative

247 ri - rB Positive Positive Negative Negative

Active return Positive Negative Negative Positive

illustrates the relationship between the different weights and returns of the portfolio and the benchmark. A portfolio manager adds value by overweighting (underweighting) assets with above-average (below-average) performance relative to the benchmark. A more detailed and informative performance attribution breaks the active portfolio return down into the following three components: (1) pure sector allocation,35 (2) security selection within sector, and (3) sector allocation/security selection interaction. N

rA

ðActive returnÞ

=

N

ðwPi - wBi Þðr Bi - r B Þ þ i=1

ðPure sector allocationÞ

wBi ðr Pi - r Bi Þ i=1 ðSecurity selection within sectorÞ

N

þ

ðwPi - wBi Þðr Pi - r Bi Þ i=1 ðSector allocation=security selection interactionÞ

ð7:7Þ

The ‘pure sector allocation’ is the difference between the sector allocations (weights) of the portfolio and the benchmark. The assumption is that the manager holds the same assets in the portfolio with the same weights within a sector as the benchmark. Hence, relative performance is explained by the manager’s decisions regarding the different sector weights. The ‘security selection within sector’ is calculated as the sum of the benchmark weighted return differences between the returns of the portfolio and the benchmark for each sector. The manager assigns each sector in the portfolio the same weighting as the benchmark. However, he holds different asset weights within the sector. As a result, relative performance is due to the manager’s asset selection. The ‘sector allocation/security selection interaction’ determines the joint effect of sector allocation and security selection on relative performance. It is the sum of the product of the weight differences and the return differences between the portfolio and the benchmark for each sector. If the manager decides to increase the allocation to a particular asset, the weight of the sector within the portfolio will also increase if no other asset in the sector is reduced. Therefore, security selection drives sector weights. The consequence of this relationship is that in performance attribution 35 For example, the SPI (Swiss Performance Index) is divided into the following sectors or industries (sector weights as at 29 October 2021): healthcare (34%), consumer goods (26%), financials (17%), industrials (16%), basic materials (3%), oil and gas (0%), services (1%), telecommunications (1%), technology (2%), and utilities (0%).

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practice, the ‘sector allocation/security selection interaction’ is often integrated with ‘security selection within the sector’. Example: Performance Attribution A portfolio manager invests in small and medium-sized companies whose stocks are primarily listed in Switzerland. The benchmark of the active investment strategy is the SPI EXTRA®, which includes all SPI stocks except the SMI (Swiss blue chips index). The following performance data are available for the SPI EXTRA® benchmark and the active portfolio:

Sector Industry Finance Consumer goods & services Othera Total

Weight of the SPI EXTRA® (in %) 33.0 29.5 19.3

Sector return of the SPI EXTRA® (in %) 5 -5 8

18.2 100.0

12

Weight of the portfolio (in %) 20.2 39.4 29.4

11.0 100.0

Sector return of the portfolio (in %) 6 8 10

8

a

As at December 31, 2011, the other sectors of the SPI EXTRA® consisted of health care (9.8%), technology (3.8%), basic materials (3.7%), utilities (0.8%), and oil and gas (0.1%)

1. What is the active return of the portfolio? 2. What are the returns attributable to ‘pure sector allocation’, ‘security selection within sector’, and ‘sector allocation/security selection interaction’ that make up the active return of the portfolio? Solution to 1 r P = 0:202 × 6% þ 0:394 × 8% þ 0:294 × 10% þ 0:11 × 8% = 8:184% rB = 0:33 × 5% þ 0:295 × ð- 5%Þ þ 0:193 × 8% þ 0:182 × 12% = 3:903% rA = 8:184% - 3:903% = 4:281% Solution to 2 Pure sector allocation = ð0:202 - 0:33Þ × ð0:05 - 0:03903Þ þ ð0:394 - 0:295Þ × ð - 0:05 - 0:03903Þ þ ð0:294 - 0:193Þ × ð0:08 - 0:03903Þ þ ð0:11 - 0:182Þ × ð0:12 - 0:03903Þ = - 0:0119

(continued)

7.6 Summary

249

Security selection within sector =0:33×ð0:06-0:05Þ þ 0:295 ×ð0:08-ð-0:05ÞÞ þ 0:193×ð0:10-0:08Þ þ 0:182×ð0:08-0:12Þ=0:03823 Sector allocation=security selection interaction = ð0:202 - 0:33Þ × ð0:06 - 0:05Þ þ ð0:394 - 0:295Þ × ð0:08 - ð- 0:05ÞÞ þ ð0:294 - 0:193Þ × ð0:10 - 0:08Þ þ ð0:11 - 0:182Þ × ð0:08 - 0:12Þ = 0:01649 The performance attribution can be represented as follows: Benchmark return

3.903% Pure sector allocation return Security selection within sector return Sector allocation/security selection interaction return

Active portfolio return Portfolio return

- 1.191% + 3.823% + 1.649% = 4.281%

+ 4.281% = 8.148%

The deviations of the portfolio’s sector weights from the benchmark lead to a negative return contribution of 1.191%. The active portfolio return of 4.281% can primarily be explained by the portfolio manager’s security selection skills of 3.823%.

7.6

Summary

• The portfolio management process consists of three phases: planning, execution, and feedback. It represents a systematic approach which makes it possible to construct a portfolio that meets the investor’s needs. In the planning phase, the capital markets and the investor’s exigencies are analysed, the IPS is defined, and the strategic asset allocation is determined. In the execution phase, the portfolio is created and the assets required by the IPS are purchased. Feedback involves monitoring, rebalancing, and evaluating the portfolio. It ensures that the objectives required by the IPS are achieved. • The long-term IPS forms the core of the portfolio management process. This document sets out the investor’s risk tolerance and return objectives, as well as the existing constraints such as the investment horizon, liquidity requirements, tax situation, legal and regulatory factors, and unique circumstances. The IPS and the long-term capital market expectations form the starting point for specifying

250

•

•

• •

•

• •

•

• •

7

Portfolio Management Process

the strategic asset allocation, which reflects the long-term target portfolio exposure to systematic risks. For the assessment of risk tolerance, the ability to bear risk and risk willingness are examined. If there is a discrepancy between ability and willingness, the investor must be educated regarding the risks of loss and the opportunities for profit. The portfolio manager must reconcile high desired returns with the client’s ability to bear risk and with expected capital market data. In contrast to a desired return, which can be reduced if there is a discrepancy with the risk objective, high returns required to achieve an investment goal represent a potential conflict between return and risk objectives in the IPS. The feedback phase consists of monitoring the IPS and capital market expectations, any rebalancing of the portfolio, and performance evaluation. Once the investment objectives relating to return and risk have been defined, the constraints must be specified in the IPS. The liquidity requirements should be determined in such a way that unplanned expenses can be covered in addition to planned expenses. The investment horizon can be short to long term and may comprise several stages. Taxes represent a further restriction. Therefore, the portfolio must be aligned with the investor’s tax situation. Legal and regulatory factors play an important role, especially for institutional investors such as pension plans. The final phase of the planning process involves defining the strategic asset allocation, which represents the investor’s target long-term portfolio exposure to systematic risks. The long-term capital market expectations are combined with the objectives and constraints of the IPS to determine the target weights of the individual asset classes within the portfolio. An asset class consists of a group of investments with similar characteristics. The selection of asset classes is an important decision that has a lasting impact on the return and risk of a portfolio. When monitoring the IPS, changes in circumstances and wealth, liquidity requirements, investment horizon, taxes, and unique circumstances should be examined. If significant changes are identified, the IPS must be revised and the portfolio adjusted accordingly. In addition to monitoring the IPS, long-term capital market expectations must also be reviewed. In doing so, the portfolio manager should monitor, for example, changes in the expected returns, risks, and correlations of asset classes, the market cycle, monetary policy, the yield curve, and inflation. Portfolio reallocations are made due to changes in investment objectives or constraints in the IPS, changes in capital market expectations, deviations of asset classes from the strategic asset allocation, and tactical asset allocation. The adjustment of the asset class weights to the target weights of the strategic asset allocation not only provides the investor with a benefit but also incurs costs. A deviation from the optimal portfolio represents a loss of benefit for the investor, which can be offset by rebalancing the portfolio. This involves costs such as transaction costs and taxes.

7.7 Problems

251

• If the asset class weights deviate from the target weights of the strategic asset allocation, rebalancing is carried out using either the calendar strategy or the percentage-of-portfolio strategy. With the calendar strategy, the rebalancing of the portfolio takes place periodically. The disadvantage of this simple rebalancing strategy is that there is no relationship with market performance. With the percentage-of-portfolio strategy, on the other hand, an asset class is adjusted if it falls outside a certain range. Thus, unlike the calendar strategy, there is closer monitoring of deviations from the strategic asset allocation, which is also market-related. • Portfolio performance must be evaluated periodically in order to assess the achievement of investment objectives and the skills of the portfolio manager. The manager’s skills are judged on the measurement and attribution of performance and on a track record of returns realised in the past. • In performance measurement, the return of the portfolio is determined. Riskadjusted return measures such as the Sharpe ratio and the information ratio can be used for this purpose. By contrast, performance attribution is used to identify and calculate the sources of return. When assessing the performance of an active manager, the return of the portfolio is compared with that of the benchmark of the same investment style. If there is an excess return of the portfolio relative to the benchmark, the portfolio manager has earned an active return (added value). In the practice of portfolio management, performance is often evaluated by breaking down returns into the following three components: (1) strategic asset allocation, (2) market timing (tactical versus strategic asset allocation), and (3) security selection (the manager’s ability to select individual securities within the asset class).

7.7

Problems

1. An investment advisor gathers the following data for a new client: • Age: 40. • Family: married but divorce proceedings are pending; three children aged 14, 16, and 18. • Hobbies: paragliding. • Profession: derivatives trader at an investment bank. • Net worth: EUR 1 million in a non-diversified portfolio and a house worth EUR 400,000. • Planned retirement: at the age of 65. • Current income: cost of living slightly exceeds current income. • Future expected expenses: education costs of EUR 20,000 per year per child; the capital settlement expected from the divorce for the wife amounts to half of the net worth. What risk objectives and constraints, such as liquidity and investment horizon, should a long-term IPS for the client include?

252

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Portfolio Management Process

2. Sabine and Klaus Miller recently moved from Frankfurt am Main to Paris and have temporarily rented an apartment in Paris. Klaus Miller is 45 years old and works as a partner at the consulting firm Delta SA. He earns EUR 180,000 before taxes. Sabine Miller is 38 years old and, as a housewife and mother, takes care of their two children aged 3 and 5 full-time. She recently inherited EUR 1 million after taxes from her parents. In addition, Mr. and Mrs. Miller own the following assets: • Cash of EUR 20,000. • Portfolio of equities and bonds worth EUR 200,000. • Equity securities worth EUR 250,000 in the consulting company Delta SA. The value of Delta shares has risen sharply in recent years due to the company’s good earnings situation. Klaus Miller is confident that the value of equity securities will continue to rise in the future. The Miller family requires a down payment of EUR 200,000 for the purchase of an apartment. The family’s annual living expenses amount to EUR 120,000. Sabine and Klaus Miller would like to accumulate sufficient funds to retire in 20 years. They also want to finance their two children’s university studies in England. In addition, the following information is known about their attitude to investing: • They describe the volatility of their equity and bond portfolios as too high and do not wish to lose more than 10% of portfolio value in a given year. • They do not want stocks and bonds of weapons industry companies in their portfolio. A portfolio manager has calculated that a net worth of EUR 2.5 million will be needed in 20 years to finance the children’s education costs and retirement. The tax rate on the working income and investment earnings including capital gains is 35%. a) What risk and return objectives should be included in a long-term IPS for the Miller family? (Assumption for the return calculation: inflation for the cost-ofliving and that for earned income cancel each other out; the risk-adjusted interest rate is 4%.) b) What constraints such as investment horizon, liquidity, taxes, and unique circumstances should be listed in a long-term IPS for the Miller family? 3. The following annual returns are available for a portfolio with an active strategy and the corresponding benchmark of the same investment style: Year 2017 2018

Portfolio return (in %) 4.0 8.0

Benchmark return (in %) 6.0 3.0 (continued)

7.8 Solutions

253

Year 2019 2020 2021

Portfolio return (in %) 5.5 2.0 6.5

Benchmark return (in %) 7.5 4.0 2.0

What is the average annual information ratio from 2017 to 2021 (assumption: sample of return data)? 4. A portfolio manager invests in Swiss equities and uses the SPI as a benchmark. The relevant performance data for the SPI benchmark and the active portfolio are presented in the following table. Sector Health Consumer goods Finance Othera Total

Weight of the SPI (in %) 32.5 26.8

Sector return of the SPI (in %) 6 8

Weight of the portfolio (in %) 20.0 40.0

Sector return of the portfolio (in %) 9 7

18.4 22.3 100.0

-10 8

20.0 20.0 100.0

4 9

a

As at December 31, 2011, the other sectors of the SPI consisted of health care (9.8%), technology (3.8%), basic materials (3.7%), utilities (0.8%), and oil and gas (0.1%)

a) What is the active return of the portfolio? b) What are the return components of ‘pure sector allocation’, ‘security selection within sector’ and ‘sector allocation/security selection interaction’ that make up the active return of the portfolio?

7.8

Solutions

1. Risk objectives: The client’s professional occupation (derivatives trader), hobbies (paragliding), and non-diversified portfolio indicate a high willingness to take risk. The current ability to assume risk is rather low due to the expected capital settlement for the wife (half of the net worth of EUR 1.4 million), the future education costs of EUR 20,000 per year per child, and the cost of living above the income. The discrepancy between above-average willingness and below-average ability to bear risk leads to a below-average risk tolerance. Liquidity: The expected liquidity needs are high due to the impending divorce and education costs for the children. Likewise, the family’s cost of living exceeds current income, resulting in immediate liquidity needs.

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Investment horizon: The client’s investment horizon is long because he is only 40 years old. The investment horizon consists of two phases. The first phase lasts 25 years and ends upon retirement. The second phase starts with retirement at the age of 65 and ends with death (approximately 15–20 years). 2. a) Risk objectives: The couple’s risk willingness can be classified as below average due to their dissatisfaction with the high portfolio volatility and their desire not to lose more than 10% of their portfolio value in 1 year. Their ability to bear risk can be described as average. The net worth and the long investment horizon indicate an above-average ability to accept risk, while the annual living costs of EUR 120,000 exceed earned income after taxes of EUR 117,000 (= EUR 180,000 × 0.65). The difference of EUR 3000 must be covered by the expected portfolio return, which reduces the ability to bear risk. Risk tolerance can best be defined as below average due to the below-average willingness to take risk and average ability to bear risk. Return objectives: The return target includes growth of the portfolio value to finance the future education costs of the children and retirement in 20 years. In addition, the living expenses of EUR 3000 per year, which are not covered by earned income, must be taken into account. The present value of the annual net expenditure of EUR 40,771 can be determined as follows: 20

Present value of annual net expenditure = t=1

EUR 3000 = EUR 40,771: ð1:04Þt

The current asset value of EUR 1,229,229 is calculated as follows (in EUR): Present value of annual net expenditure + Inheritance + Portfolio of equities and bonds + Shares of the consulting company Delta SA + Cash - Down payment on the purchase of a flat = Current asset value

- 40,771 + 1,000,000 + 200,000 + 250,000 + 20,000 - 200,000 = 1,229,229

7.8 Solutions

255

The pre-tax return is 3.613% and can be calculated as follows:

Return before taxes =

EUR 2,500,000 EUR 1,229,229

1=20

- 1 = 3:613%:

To achieve an asset value of EUR 2.5 million in 20 years, an annual return after taxes of 5.558% is required:

Return after taxes =

3:613% = 5:558%: 1 - 0:35

There is no need to adjust the return for inflation, since it is assumed that cost-ofliving inflation and earned income inflation cancel each other out. b) Investment horizon: The investment horizon consists of two phases. The first phase lasts 20 years and is characterised by Klaus Miller’s employment and the university education costs of the children. The second phase begins with retirement and ends with the death of the couple. Liquidity: The immediate liquidity needs include the down payment of EUR 200,000 on the purchase of the apartment. Taxes: The 35% tax rate applies to both earned income and investment income. Unique circumstances: The shareholding of EUR 250,000 in the consulting company Delta SA corresponds to approximately 20% of the current net worth of EUR 1,229,229. This concentration of risk should be reduced in future by a better diversification of the assets. The desire not to hold weapons industry stocks and bonds in the portfolio is an investment restriction. 3. Average portfolio return =

4% þ 8% þ 5:5% þ 2% þ 6:5% = 5:2% 5

256

7

Average benchmark return =

Portfolio Management Process

6% þ 3% þ 7:5% þ 4% þ 2% = 4:5% 5

The return differentials between the portfolio and the benchmark or the active returns are: -2%, 5%, -2%, -2%, and 4.5%. Average active return =

ð - 2%Þ þ 5% þ ð - 2%Þ þ ð - 2%Þ þ 4:5% = 0:7% 5

Standard deviation of active returns =

1 × 5-1

ð - 0:02 - 0:007Þ2 þð0:05 - 0:007Þ2 þð - 0:02 - 0:007Þ2 þð - 0:02 - 0:007Þ2 þð0:045 - 0:007Þ2

= 0:037014

The information ratio of 0.189 can be calculated as follows:

IR =

5:2% - 4:5% = 0:189: 3:7014%

The portfolio’s active return of 0.7% is due to the manager’s decisions to change the weights of the assets in the portfolio relative to the benchmark and to select the mispriced stocks. The positive information ratio is 0.189. In the manager’s evaluations, an information ratio greater than 0.5 is considered ‘good’. 4. a) rP = 0:2 × 9% þ 0:4 × 7% þ 0:2 × 4% þ 0:2 × 9% = 7:2% rB = 0:325 × 6% þ 0:268 × 8% þ 0:184 × ð- 10%Þ þ 0:223 × 8% = 4:038% rA = 7:2% - 4:038% = 3:162% b) Pure sector allocation = ð0:20 - 0:325Þ × ð0:06 - 0:04038Þ þ ð0:40 - 0:268Þ × ð0:08 - 0:04038Þ þ ð0:20 - 0:184Þ × ð- 0:10 - 0:04038Þ þ ð0:20 - 0:223Þ × ð0:08 - 0:04038Þ = - 0:00038

References

257

Security selection within sector = 0:325 × ð0:09 - 0:06Þ þ 0:268 × ð0:07 - 0:08Þ þ 0:184 × ð0:04 - ð- 0:10ÞÞ þ 0:223 × ð0:09 - 0:08Þ = 0:03506 Sector allocation=security selection interaction = ð0:20 - 0:325Þ × ð0:09 - 0:06Þ þ ð0:40 - 0:268Þ × ð0:07 - 0:08Þ þ ð0:20 - 0:184Þ × ð0:04 - ð- 0:10ÞÞ þ ð0:20 - 0:223Þ × ð0:09 - 0:08Þ = - 0:00306 The performance attribution can be represented as follows: Benchmark return

4.038% Pure sector allocation return Security selection within sector return Sector allocation/security selection interaction return

Active portfolio return Portfolio return

- 0.038% + 3.506% - 0.306% = 3.162%

+ 3.162% = 7.200%

The active portfolio return of 3.162% is due primarily to the portfolio manager’s security selection skills (3.506%).

References Black, F., Litterman, R.: Global portfolio optimization. Financial Anal. J. 48(5), 28–43 (1992) Bodie, Z., Merton, R.C.: Finance. Upper Saddle River (2000) Diermeier, J.J.: Capital market expectations: the macro forecasts. In: Maginn, J.L., Tuttle, D.L. (eds.) Managing Investment Portfolios: A Dynamic Process, 2nd edn, pp. 5-1–5-77, Boston, NY (1990) Grable, J.E., Joo, S.H.: Environmental and biopsychosocial factors associated with financial risk tolerance. Financial Counsell. Plann. 15(1), 73–82 (2004) Grinold, R.C., Kahn, R.N.: Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk, 2nd edn, New York (2000) Ibbotson, R.G., Kaplan, P.D.: Does asset allocation policy explain 40, 90, or 100 percent of performance? Financial Anal. J. 56(1), 26–33 (2000) Kaiser, R.W.: Individual investors. In: Maginn, J.L., Tuttle, D.L. (eds.) Managing Investment Portfolios: A Dynamic Process, 2nd edn, pp. 3-1–3-46, Boston, NY (1990) Perridon, L., Steiner, M., Rathgeber, A.: Finanzwirtschaft der Unternehmung, 17th edn, Munich (2017) Maginn, J.L., Tuttle, D.L.: The portfolio management process and its dynamics. In: Maginn, J.L., Tuttle, D.L. (eds.) Managing Investment Portfolios: A Dynamic Process, 2nd edn, pp. 1-1–1-11, Boston, NY (1990) Mondello, E.: Portfoliomanagement: Theorie und Anwendungsbeispiele, 2nd edn, Wiesbaden (2015) Mondello, E.: Finance: Theorie und Anwendungsbeispiele, Wiesbaden (2017) Mondello, E.: Corporate Finance: Theorie und Anwendungsbeispiele, Wiesbaden (2022)

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Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 7th edn, Mason (2003) Sharpe, W.F.: Mutual fund performance. J. Bus. 39(1), 119–138 (1966a) Sharpe, W.F.: The Sharpe ratio. J. Portfolio Manag. 21(1), 49–59 (1966b) Sharpe, W.F.: Asset allocation. In: Maginn, J.L., Tuttle, D.L. (eds.) Managing Investment Portfolios: A Dynamic Process, 2nd edn, pp. 7-1–7-71, Boston, NY (1990) Solnik, B., McLeavey, D.: International Investments, 5th edn, Boston, NY (2004)

Part II Equity Securities

8

Dividend Discount Model

8.1

Introduction

Equity analysis is carried out by means of fundamental and/or technical analysis. Fundamental analysis examines the factors influencing the share price that are relevant to valuation. For this purpose, information on the overall economy, the industry, and the company are analysed. Central to fundamental analysis is a valuation model on the basis of which the intrinsic value is calculated. To arrive at an investment decision, the intrinsic value determined with the valuation model is compared with the market price. If the intrinsic value exceeds (falls below) the market price, the equity security appears to be undervalued (overvalued). By contrast, the technical analysis works with data on the share price and trading volume in order to be able to predict the future price movement of the security. Thus, the investment decision in technical analysis is based on the direction of the predicted price change, which is determined with the help of chart images and patterns. Benjamin Graham (1894-1976), a US economist and investor, is considered the founder of fundamental analysis and value investing. He held the view that a stock should be bought if its price is below its fundamental value. To estimate the fundamental value of a stock, he used various ratios such as the price-to-earnings ratio, the price-to-book ratio, the dividend yield, and the earnings growth. In 1934, Benjamin Graham, together with David Dodd, published the book Security Analysis, which contains the basic principles of fundamental analysis and value-based stock selection.1 The book is still considered the fundamental work for value investors today. In 1949, the first edition of The Intelligent Investor was published, a popular scientific version of Security Analysis, which became a bestseller and was reprinted several times.2 Graham also contributed to the development of the Chartered

1 2

See Graham and Dodd 1934: Security Analysis, p. 1 ff. See Graham 1949: The Intelligent Investor: A Book of Practical Counsel, p. 1 ff.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_8

261

262

8 Dividend Discount Model

Financial Analyst (CFA®) certification, which is the global standard in investment analysis and portfolio management today. The valuation models applied in fundamental analysis can be divided into absolute and relative models. The absolute models can be grouped into cash flow models and value-added models. Cash flow models include the dividend discount model, the free cash flow to equity model, and the free cash flow to firm model.3 Value-added models consist of models such as the residual income valuation model, in which earnings after the cost of debt and equity are discounted. To calculate the intrinsic share value, the present value of these residual earnings per share is added to the book value per share.4 Relative valuation models, on the other hand, consist of multiples such as the price-to-earnings ratio and the enterprise value-to-EBITDA ratio. They are compared for the subject company with the corresponding multiples of a peer group in order to evaluate whether the stock is correctly valued relative to a benchmark.5 This chapter presents the dividend discount model, which, like the other cash flow models, is used to calculate the intrinsic share value under the going concern assumption. According to the model, the intrinsic value equals the present value of the future dividends that investors can expect when buying stocks. For this purpose, future dividends are discounted with the expected return of the shareholders.

8.2

Fundamentals of Equity Valuation

The price an investor is willing to pay for an equity security depends on its expected cash flows. Since dividends will be received in the future, the time value effect must be taken into account. An amount of money received in the future is worth less than the same amount of money today. In order to estimate the intrinsic share value, the dividends paid in the future must therefore be discounted to the valuation date. Hence, in a dividend discount model, the valuation parameters consist of the expected cash flows and the discount rate. The latter represents the expected return of the shareholders. The intrinsic value is obtained by discounting the dividends with the expected return. Investors buying the share at this price achieve the investment return they require for the risk taken. This valuation principle underlying cash flowbased models was first described by John Williams in the late 1930s.6 Equity ownership often results in dividend income. If the stock is sold after a certain period (e.g. 5 years), in addition to the last dividend the investor receives the proceeds from the sale of the equity security. Accordingly, the intrinsic share value (P0) over a certain investment period T can be determined as follows:

For the free cash flow models, see Chap. 9. For the residual income valuation model, see, for example, Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 354 ff. 5 For the multiples, see Chap. 10. 6 See Williams 1938: The Theory of Investment Value, p. 55 ff. 3 4

8.2 Fundamentals of Equity Valuation

P0 =

263

D1 D2 D3 D þ PT þ þ þ ... þ T , ½1 þ EðrÞT ½1 þ EðrÞ1 ½1 þ EðrÞ2 ½1 þ EðrÞ3

ð8:1Þ

where D1 = expected dividend per share for the period 1, E(r) = expected return of the shareholders or discount rate, and PT = expected share price at the end of the investment period T. The formula demonstrates the calculation of the intrinsic value of an equity security over a finite investment horizon. This value results both from the discounted dividends during the investment period and from the present value of the sales proceeds at the end of the investment period. The expected sale price can be calculated by discounting the future dividends accruing after the end of the investment horizon T over an infinite period of time: PT =

DTþ1 DTþ2 DTþ3 þ þ þ ... ½1 þ E ðr ÞTþ1 ½1 þ Eðr ÞTþ2 ½1 þ Eðr ÞTþ3

ð8:2Þ

If the valuation is carried out under the going concern principle, the dividends are discounted over an infinite period of time because the stock (like the company) has an unlimited life. The intrinsic share value can be estimated with the basic formula of the dividend discount model as follows: P0 =

1 t=1

Dt : ½1 þ E ðr Þt

ð8:3Þ

For a risk-free coupon-paying bond, the valuation parameters are known. The cash flows consist of the fixed coupons which are paid periodically (e.g. every 6 months or every year) and the principal value at maturity, while the expected return is given by the risk-free interest rate. By contrast, for equities the valuation parameters such as future cash flows and expected return are not known in advance and have to be estimated. The expected cash flows can be determined using growth assumptions. The risk is integrated into the expected return through the risk premium, which represents a return compensation for the business and financial risk of the company. Furthermore, the calculation of the present value must take into account that dividends will be paid over an infinitely long period of time, since the duration of the company’s activity is essentially not limited. Figure 8.1 illustrates the basic principles of equity valuation with the dividend discount model and the resulting valuation issues. The valuation principle with a cash flow model is relatively simple. Nevertheless, the application of the model poses a challenge when it comes to selecting an appropriate valuation model and estimating the valuation parameters. The following steps are required to determine the intrinsic share value with a cash flow model:

264

8 Dividend Discount Model

Dividend per share1

Dividend per share2

Dividend per share3

1

2

3

0

Intrinsic share value

…

Dividend per sharef f (Years)

Dividend per share

Valuaon issues: • Future dividends • Expected return • Unlimited life of .the share (going concern) Fig. 8.1 Calculation of the intrinsic share value with the dividend discount model (Source: Own illustration)

• Estimation of expected cash flows such as dividends or free cash flows. • Calculation of the risk-adjusted discount rate or the expected return. • Selection of a valuation model based on the assumed growth assumptions and the specific valuation situation. Example: Calculation of the Intrinsic Share Value in the Event of a Company Liquidation in Four Years A company plans to cease operations in 4 years. An annual dividend per share of CHF 5 is expected in each of the first 3 years. Dividends are paid out at the end of the respective years. At the end of the 4-year period, a liquidating dividend per share of CHF 100 is issued. The expected return is 10%. What is the intrinsic share value? Solution The intrinsic share value is CHF 80.74 and can be calculated using the dividend discount model as follows: P0 =

CHF 5 CHF 5 CHF 5 CHF 100 þ þ þ = CHF 80:74: ð1:10Þ1 ð1:10Þ2 ð1:10Þ3 ð1:10Þ4

8.3 Growth Rate

265

The various methods for estimating growth rates and the one- and two-stage dividend discount models are described below. The expected return is usually determined using the CAPM, which has already been discussed in Chap. 6.7

8.3

Growth Rate

The estimated growth rates used to predict future earnings or dividends are a critical valuation parameter in a dividend discount model. Earnings or dividend growth rates can typically be determined using the following three methods:8 1. Statistical forecasting models 2. Company fundamentals 3. Consensus forecasts from analysts Statistical forecasting models rely on historical company earnings to calculate the growth rate. The historical growth rate can be estimated by averaging past percentage changes in earnings applying the arithmetic or geometric mean. In addition, regression models or time series analyses can be used. Statistical forecasting models are appropriate for mature companies whose average future growth rate is the same as in the recent past. However, these models are not suitable for high-growth companies because past growth is not a good indicator of future growth. The growth rate can also be estimated on the basis of fundamental data relating to the company. The endogenous or fundamental growth rate reflects a growth rate that the company can sustain for a given return on equity and dividend policy without issuing new equity. The fundamental growth rate (g) can be calculated as follows: g = bROE,

ð8:4Þ

where b = retention rate of earnings, and ROE = return on equity.

7

See Sect. 6.2. In addition to the CAPM, the expected return can also be estimated with the Fama– French model or with other multifactor models such as the Carhart model and the Pastor– Stambaugh model. However, these models are not suitable for calculating the expected return of an individual stock due to the statistical difficulties such as multicollinearity associated with multiple linear regression analysis. See Sect. 6.3. Build-up methods can also be used. Here, the expected return is estimated with the risk-free rate plus a series of risk premiums, which are partly estimated subjectively. See, for example, Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 109 ff. 8 See Damodaran 2012: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, p. 272 ff.

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8 Dividend Discount Model

The higher the retention rate of the earnings and the return on equity, the higher the fundamental growth rate. The retention rate is obtained by subtracting the earnings payout ratio from 1. The payout ratio can be calculated by dividing the dividend by the net income. The return on equity is obtained by dividing the net income by the shareholder’s equity at the beginning of the period, or by dividing it by the average shareholder’s equity.9 In the calculations below, the equity at the beginning of the period is used to determine the return on equity. This leads to the following formula for calculating the fundamental growth rate:10 g = 1-

Dividends Net income : × Net income Equity

ð8:5Þ

The formula indicates that a low earnings payout ratio results in a high growth rate since a higher proportion of net income is retained and is thus available to finance the company’s growth. This relationship can be clarified if the expression to the right of the equals sign is multiplied out:11 g=

Net income - Dividends : Equity

ð8:6Þ

Thus, the fundamental growth rate represents the growth rate of equity, that is, retained earnings divided by the equity at the beginning of the period. Return on equity can be further broken down into the following three financial ratios of net profit margin, total asset turnover, and financial leverage:12 ROE =

Total assets Net income Revenues × × : Total assets Revenues Equity

ð8:7Þ

Net income divided by revenues reflects the company’s net profit margin. It measures the amount of profit for a unit of revenues earned. The higher the net profit margin, the higher the company’s profitability and hence the higher the return on equity. The second ratio—revenues divided by total assets—provides the total asset turnover. A total asset turnover of 1 means that the company generates revenues of EUR 1 for invested assets of EUR 1. The greater the ratio, the higher the operating efficiency of the company and consequently the higher the return on equity. The financial leverage or the equity multiplier increases if equity is replaced with debt. Thus, a higher level of indebtedness results in a higher return on equity. Equation (8.7) is known as the DuPont model13 and when inserted instead of the

Average equity = (Equity at the beginning of the period + Equity at the end of the period)/2. See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 349. 11 income × Net income Net Income - Dividends g = NetEquity - Dividends . Net Income × Equity = Equity 9

10

See Peterson Drake 2008: ‘Financial Statement Analysis’, p. 341. The DuPont model (i.e. the breakdown of return on equity into net profit margin, total asset turnover, and financial leverage) was developed by E.I. du Pont de Nemours & Company.

12 13

8.3 Growth Rate

267

return on equity in Eq. (8.4), leads to the following formula for calculating the fundamental growth rate: g= ×

Net income - Dividends Net income Revenues × × Equity Revenues Total assets Total assets : Equity

ð8:8Þ

Accordingly, the fundamental growth rate is a function of the retention rate, net profit margin, total asset turnover, and financial leverage. The net profit margin and total asset turnover reflect the contribution to return on equity from investment decisions. The retention rate and financial leverage, by contrast, are a result of the financing decisions. The level of the growth rate, therefore, depends on the company’s investment and financing policy. Example: Calculation of the Fundamental Growth Rate for the Stock of Mercedes-Benz Group AG The following data are available for the Mercedes-Benz Group for 2016 (in EUR million):14 Net income (profit attributable to shareholders of Mercedes-Benz Group AG)a Shareholders’ equity at the beginning of the yeara Shareholders’ equity at the end of the yeara Revenues Total assets at the beginning of the year Total assets at the end of the year Dividends a

8526 53,561 57,950 153,261 217,166 242,988 3477

Without non-controlling interests (minority interests)

What is the fundamental growth rate that can be used to estimate future dividends? Solution The earnings retention rate of 59.22% can be determined as follows: b=

EUR 8526 million - EUR 3477 million = 0:5922: EUR 8526 million

The return on equity is 15.92% and can be calculated as follows: (continued)

14

See Mercedes-Benz Group 2017: Geschäftsbericht 2016, p. 216 ff.

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8 Dividend Discount Model

ROE = ×

EUR 8526 million EUR 153,261 million × EUR 217,166 million EUR 153,261 million

EUR 217,166 million = 15:92% EUR 53,561 million

or ROE =

EUR 8526 million = 15:92% EUR 53,561 million

The earnings retention rate is 59.22%, while the return on equity is 15.92%. This leads to a fundamental growth rate of 9.43%: g = 0:5922 × 15:92% = 9:43%: If the company’s growth is above 9.43%, capital must be raised on the financial market to finance the excess growth. Accordingly, the company can finance a sustainable growth of 9.43% through retained earnings. Future dividends can also be estimated using analysts’ consensus earnings forecasts. For example, financial information service providers such as Thomson Reuters and Bloomberg provide consensus forecasts with earnings growth rates ranging from 1 to 5 years. Analysts’ earnings estimates are based on statistical forecasting models and company fundamentals, among other factors.

8.4

One-Stage Dividend Discount Model

Earnings can either be distributed in the form of dividends or reinvested in the company. They constitute the most important value driver of a stock. In principle, dividend discount models can be used to value equity securities in the following situations: • The company pays dividends, and a historical data series is available to estimate future dividends. • Management has established a dividend policy that is both understandable and related to the profitability and the value of the company. This implies a constant earnings payout ratio so that dividends reflect the profitability of the company. If earnings increase (decrease), dividends increase (decrease). • The dividends and the free cash flows to equity are approximately the same. For example, the stocks of Mercedes-Benz Group AG and Linde AG can be valued using the dividend discount model, as the two companies pay dividends and have an

8.4 One-Stage Dividend Discount Model

269

approximately constant earnings payout ratio.15 Therefore, the following valuation examples are described using these two securities. Furthermore, the dividend discount model is suitable for determining the intrinsic value of equity securities in the case of minority shareholders, as they do not control the distribution of free cash flows. The one-stage dividend discount model, also known as the Gordon–Growth model, is based on the assumption that dividends grow at a constant rate over an infinite time period.16 The expected dividend per share in the next period can be calculated by multiplying the current dividend per share by 1 plus the constant growth rate: D1 = D0 ð1 þ gÞ,

ð8:9Þ

where D1 = dividend per share for the next period, D0 = dividend per share for the current period, and g = constant growth rate. For example, if the dividend per share is EUR 5 and there is a constant growth rate of 5%, then the dividend per share will be EUR 5.25 [= EUR 5 × (1.05)1] in 1 year. In 2 years the dividend per share will be EUR 5.513 [= EUR 5 × (1.05)2], in 3 years EUR 5.788 [= EUR 5 × (1.05)3], and so forth. The following formula is used to calculate the intrinsic share value (going concern principle): P0 =

D0 ð1 þ gÞ1 D0 ð1 þ gÞ2 D0 ð1 þ gÞT þ þ . . . þ þ ...: ½1 þ EðrÞT ½1 þ EðrÞ1 ½1 þ EðrÞ2

ð8:10Þ

This equation displays a geometric series because each term in the formula is equal to the previous term multiplied by a constant. The constant is (1 + g)/[1 + E(r)]. This equation for calculating the intrinsic share value can be expressed compactly as follows:17 P0 =

D0 ð1 þ gÞ D1 = , E ðr Þ - g E ðr Þ - g

ð8:11Þ

where E(r) > g. For example, applying the Gordon–Growth model to a stock with an expected dividend per share of EUR 5 (D1 = EUR 5) in the next period, a long-term expected 15

For example, Mercedes-Benz Group has a payout ratio in the years 2011–2016 that ranges between 35% and 41%. By contrast, Linde’s payout ratio ranges between 41% and 57% in a period from 2012 to 2016. 16 See Gordon 1962: The Investment, Financing, and Valuation of the Corporation, p. 1 ff. 17 The first term of an unbounded geometric series is equal to a and the growth factor is n, where | n| < 1. The sum of a + an + an2 + ... is a/(1 - n). If for a = D1/[1 + E(r)] and for n = (1 + g)/[1 + E(r)] is substituted, one arrives at Eq. (8.11) or the Gordon-Growth model.

270

8 Dividend Discount Model

return of 10%, and a perpetual constant growth rate of 3% results in an intrinsic share value of EUR 71.43: P0 =

EUR 5 = EUR 71:43: 0:10 - 0:03

To calculate the intrinsic share value, the expected return must be greater than the growth rate [E(r) > g]. If the expected return is equal to the growth rate [E(r) = g], dividends increase at the same rate at which they are discounted. This leads to an infinitely high equity value, which is equal to the sum of all undiscounted future dividends. If the expected return is smaller than the growth rate [E(r) < g], the model yields a negative equity value. Unlimited and negative equity values do not make sense, and an expected return equal to or below the growth rate is therefore not appropriate. The Gordon–Growth model can also be applied to calculate the intrinsic share value of companies that have a negative growth rate. A negative growth rate implies that the share value is decreasing.18 The higher the negative growth rate is estimated, the lower the intrinsic share value. The premise of long-term negative growth is justified for companies that are exposed to a sustained decline in demand for their products and services due to technological progress or social upheaval. The Gordon–Growth model is based on the going concern assumption, and dividends are therefore incurred over an infinite time period. Accordingly, the expected return and the growth rate must be long-term estimates. The reliability of the valuation model depends primarily on the expected return and the long-term growth rate, as the dividend of the next period can usually be estimated with sufficient accuracy. In this context, the equity value determined on the basis of the model reacts very sensitively to changes in the expected return and the growth rate. Small changes in these two valuation parameters can lead to a relatively large change in the calculated share value. For this reason, sensitivity analysis should be carried out to reveal the extent to which the value of the security moves with a change in the two parameters. In particular, a sensitivity analysis is justified if there is uncertainty about the level of the valuation parameters. If, for example, the dividend per share in the next period is EUR 5.25 (D1 = EUR 5.25), the expected return is 10%, and the long-term growth rate is 5%, the intrinsic share value is EUR 105 [= EUR 5.25/ (0.10 - 0.05)]. Assuming that the expected return and the growth rate each change by 0.5% upwards and downwards, the one-stage dividend discount model yields the following equity values: E(r) = 9.5% E(r) = 10.0% E(r) = 10.5%

18

g = 4.5% EUR 105.00 EUR 95.45 EUR 87.50

g = 5% EUR 116.67 EUR 105.00 EUR 95.45

g = 5.5% EUR 131.25 EUR 116.67 EUR 105.00

In the Gordon-Growth model, when the growth rate is negative, the numerator decreases [D0(1 + g)] and the denominator increases [E(r) - g], with the result that the intrinsic value of the stock is lower with a negative growth rate than with a positive one.

8.4 One-Stage Dividend Discount Model

271

(Intrinsic share value) EUR 300 EUR 250 EUR 200 EUR 150 EUR 100 EUR 50 EUR 0 14%

12%

10%

8%

6%

4%

2%

(Expected return – growth rate) Fig. 8.2 Exponential increase in intrinsic share value with convergence of expected return and growth rate (Source: Own illustration)

The sensitivity analysis reveals that even small changes in the expected return and growth rate have a significant impact on the share value. The calculated equity values range from EUR 87.50 to EUR 131.25. The lowest share value of EUR 87.50 is due to the largest difference between the expected return and the growth rate of 6%. Conversely, the highest equity value of EUR 131.25 results from the lowest difference between the two valuation parameters of 4%. The difference between the highest and lowest equity value is approximately 50%, while the deviation from the base scenario of EUR 105 is approximately 25% upwards and approximately 17% downwards. If the difference between the expected return and the growth rate decreases, the calculated share value increases exponentially, as illustrated in Fig. 8.2. The expected return and the growth rate can take on different values depending on the method chosen. For example, the CAPM or the build-up method can be applied to determine the expected return, while a historical or fundamental approach can be adopted to estimate the growth rate. Therefore, the intrinsic share value must be calculated using a range of estimated valuation parameters to ensure that the share value falls within a range.

272

8 Dividend Discount Model

The level of the constant growth rate in the one-stage dividend discount model should be lower than or equal to the growth rate of the gross domestic product (GDP) of the country in which a company conducts its main business activities. No company, no matter how successful, can forever achieve a higher growth rate than that of the economy as a whole. The sustainable fundamental growth rate is calculated by taking the difference between 1 and the long-term earnings payout ratio multiplied by the long-term return on equity. Thus, the constant earnings payout ratio δ in the model can be determined as follows:19 δ=1-

g , ROE

ð8:12Þ

where g = fundamental growth rate, and ROE = long-term return on equity. Since the Gordon–Growth model assumes perpetual constant growth, it is suitable for companies in the maturity phase of their life cycle with a well-established dividend policy that have a growth rate similar to the economy as a whole. Possible examples are companies in the energy supply and food retail sectors. For example, if the expected return is calculated with the CAPM, the beta should be close to 1. Depending on the firm’s systematic risk, a beta between 0.8 and 1.2 can be chosen. This range reflects the adjusted beta of most listed companies.20 A beta of 1 means that the stock or the company has the same risk as the overall market, which is a realistic assumption for a mature firm operating in a saturated market. Accordingly, the Gordon–Growth model is suitable for stable and mature companies that pay sufficiently high dividends and do not build up excessive cash reserves—with the result that dividends and free cash flows to equity are approximately equal—and have a beta of approximately 1.21 If the market price and the intrinsic value of the equity security are the same, or if one assumes informationally and operationally efficient capital markets, important information can be gained about the factors relevant to valuation, such as the growth rate and the expected return. Thus, the one-stage dividend discount model [P0 = D1/ (E(r) - g)] can be rearranged in accordance with the growth rate, which leads to the following equation for the implied growth rate:

19 The fundamental growth rate can be calculated with the payout ratio as follows: g = (1 - δ) ROE. If this equation is rearranged, the payout ratio δ is obtained. 20 For example, 63% of DAX 30 stocks and 70% of SMI stocks have an adjusted beta of between 0.8 and 1.2 (Source: Refinitiv Eikon and own calculations; as at January 2015). 21 See Damodaran 2012: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, p. 324.

8.4 One-Stage Dividend Discount Model

273

gImplied = E ðr Þ -

D1 : P0

ð8:13Þ

Hence, the implied growth rate can be determined by subtracting the expected dividend yield from the expected return. For example, the Mercedes-Benz Group stock is trading at a price per share of EUR 71.18 at the beginning of 2017. The expected dividend per share for the year is EUR 3.48 (Source: Thomson Reuters consensus forecast), while the long-term expected CAPM return of the car manufacturer is 10.22%.22 The implied growth rate of 5.33% can be calculated as follows: gImplied = 0:1022 -

EUR 3:48 = 0:0533: EUR 71:18

The Mercedes-Benz Group stock is valued on the market as if it possessed a longterm dividend growth rate of 5.33%. Implied growth rates estimated in this way can be used for stock selection. If the equity analysis leads to the conclusion that the long-term growth rate is higher (lower) than 5.33%, the security appears undervalued (overvalued) and should therefore be bought (sold). The Gordon– Growth model can also be used to estimate the implied expected return under the assumption of market efficiency: E ðr ÞImplied =

D1 þ g: P0

ð8:14Þ

The expected shareholders’ return is made up of the expected dividend yield and the growth rate. The latter reflects the growth rate of the equity or stock price.23 The Linde stock is valued below using the one-stage dividend discount model. The focus is on the step-by-step application of the valuation model, with the calculations based on plausible assumptions which can be further refined with more detailed analysis. Example: Valuation of the Linde Stock with the One-Stage Dividend Discount Model The following data are available for the listed stock of Linde AG at the beginning of 2017 (Source: Refinitiv Eikon): Dividend per share (for 2016) Traded share price

EUR 3.70 EUR 165.74

It is assumed that the company has the same long-term nominal growth as the German economy as a whole. Long-term real GDP growth rate in Germany (continued)

22 23

See Sect. 6.2.5. See Sect. 8.3.

274

8 Dividend Discount Model

is estimated at 1.2% per year, while the long-term inflation rate is anticipated to be 2.4% per year (Source: IMF—World Economic Outlook). The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%. The expected market risk premium for Germany is 7%. A long-term beta of 1 is assumed for Linde shares. 1. What is the intrinsic share value of Linde using the Gordon–Growth model? 2. Is the stock correctly valued according to the Gordon–Growth model? 3. What is the implied growth rate? Solution to 1 The expected CAPM return is 8.1% and can be calculated as follows: Eðr Þ = 1:1% þ 7% × 1 = 8:1%: The nominal GDP growth rate of 3.63% is estimated using the compounding effect with the real GDP growth rate of 1.2% and the expected long-term inflation rate of 2.4% as follows: g = ð1 þ 0:012Þ × ð1 þ 0:024Þ - 1 = 3:63%: If 8.1% is taken for the expected return, 3.63% for the constant growth rate, and EUR 3.70 for the last dividend paid in the one-stage dividend discount model, the intrinsic share value is EUR 85.78: P0 =

EUR 3:70 × 1:0363 = EUR 85:78: 0:081 - 0:0363

Solution to 2 Based on the Gordon–Growth model, Linde shares are overvalued by approximately 93%, as the intrinsic value of EUR 85.78 is lower than the traded share price of EUR 165.74. It should be mentioned at this point that the calculation of the intrinsic value using the one-stage dividend discount model is a conservative approach that assumes that Linde has the same annual growth rate as the economy as a whole of 3.63%. However, the financial information service provider Thomson Reuters indicates a consensus forecast of the annual growth rate for the next 3 years of 9.43%. A perpetual constant growth rate of 3.63% per annum is therefore too low for Linde. Thus, the assumption of perpetual constant growth is not appropriate. Furthermore, the company is in a position to pay out higher dividends, as the free cash flows to equity are greater than the dividends. A higher growth rate, at least over a limited time period, and higher cash flows lead to a higher intrinsic share value. (continued)

8.5 Two-Stage Dividend Discount Model

275

Solution to 3 The implied growth rate can be calculated using the Gordon–Growth model as follows: EUR 165:74 =

EUR 3:70 × ð1 þ gÞ → g = 5:7395%: 0:081 - g

The market price of the stock of EUR 165.74, which is higher than the intrinsic value, implies a higher perpetual constant growth rate of 5.7395% (instead of 3.63%). A higher growth rate is justified by the consensus forecast of 9.43% per year for the next 3 years. A multi-stage valuation model is more appropriate for a high-growth company like Linde.

8.5

Two-Stage Dividend Discount Model

For most companies, the assumption of eternal constant growth does not apply because they go through several growth phases in the course of their life cycle. Multi-stage valuation models make it possible to incorporate different growth phases. The following life cycle growth phases can be distinguished: start-up, rapid expansion, high growth, mature growth, and decline.24 The multi-stage valuation models are based on the assumption that over a limited period of time the company exhibits an earnings trend that deviates from the longterm average. After a certain number of years, the company returns to the long-term growth path. Two growth phases are distinguished in the two-stage dividend discount model. In the first phase, which is limited in time (e.g. 5–8 years), dividends increase at a certain growth rate. In the subsequent second phase, perpetual constant growth is assumed. In most cases, the growth rate in the first period is greater than in the second constant phase. There are also companies which, due to business and/or financial problems, exhibit low or negative growth in the first stage, which is replaced by higher constant growth in the second stage after the difficulties have been overcome. In the two-stage dividend discount model, the transition from the first to the second growth phase is characterised by an abrupt decline in the growth rate to a lower long-term level. In this process, the high growth rate at the end of the first stage—of, for example, 15%—falls abruptly to a lower constant growth rate of, for example, 4%. The intrinsic share value can be calculated on the basis of the two-stage dividend discount model as follows:

24

See Koller et al. 2010: Valuation: Measuring and Managing the Value of Companies, p. 89 ff.

276

8 Dividend Discount Model

T

P0 =

Dt þ ½1 þ Eðr 1 Þt

t=1 ðShort - term high growthÞ

PT ½1 þ Eðr1 ÞT

,

ð8:15Þ

ðLong - term perpetual growthÞ

where Dt = expected dividend per share in period t, E(r1) = expected return of shareholders in the first growth phase, and PT = terminal share value at the end of period T (intrinsic share value at the end of the first growth phase). The first term to the right of the equals sign reflects the intrinsic share value for the first growth stage. The second term to the right of the equals sign contains the terminal value of the share (PT) at the end of the first or beginning of the second growth stage, which is discounted to the valuation date with the expected return of the first growth stage. The terminal value can be determined either with a price multiple, such as the price-to-earnings ratio or the price-to-book ratio, or with a one-stage dividend discount model as follows: PT =

DTþ1 , E ð r 2 Þ - gT

ð8:16Þ

where DT+1 = expected dividend per share at the end of the first period of the second perpetual growth phase, E(r2) = expected return of shareholders in the second perpetual growth phase, and gT = perpetual constant growth rate. In calculating the terminal value, a constant growth rate gT is used, which should not exceed the long-term nominal growth rate of the economy as a whole. Furthermore, the earnings payout ratio must be consistent with the estimated growth rate. High (low) growth implies a low (high) earnings payout ratio. If growth is high, earnings are reinvested and not distributed to shareholders; whereas if growth is low, less earnings are reinvested and higher dividends are paid as a result. A drop of the growth rate in the second phase leads to an increase of the earnings payout ratio. The new payout ratio in the perpetual growth phase can be determined with 1 - gT/ROET (see Eq. 8.12). Figure 8.3 illustrates the relationship between the growth rate and the earnings payout ratio in the two-stage dividend discount model. As a rule, the expected shareholders’ return in the first growth stage is greater than in the second perpetual growth stage due to the higher risk. For example, high-growth companies have a relatively high beta as a result of the greater risk. It is not uncommon for high-growth companies such as internet, telecommunications or biotechnology companies to have a beta of approximately 2. In the second

8.5 Two-Stage Dividend Discount Model

277

(Growth rate) Lower earnings payout ratio

g Higher earnings payout ratio

gT

(Years)

Phase with high constant growth

Phase with perpetual constant growth

Fig. 8.3 Course of the growth rate and change in the earnings payout ratio in the two-stage dividend discount model (Source: Own illustration)

perpetual growth phase, the risk of the company usually decreases. For this stage, it is appropriate to assume a beta of approximately 1 (range between 0.8 and 1.2), as the company is in the maturity phase and thus bears the same risk as the economy as a whole.25 Similarly, the return on equity is greater in the high-growth phase than in the perpetual constant growth phase because the investment projects are more profitable in the first phase. To determine the return on equity in the second perpetual growth phase, the industry average return on equity or the company’s expected cost of equity in that period can be used as a benchmark. In the following example, the Mercedes-Benz Group stock is valued using the two-stage dividend discount model. The focus is on the step-by-step application of the valuation model, where calculations are based on plausible assumptions that can be further refined with a more detailed analysis.

25

Most listed stocks have an adjusted beta that lies in a range between 0.8 and 1.2. In addition, there are companies (e.g. in the commodities industry) that have a beta that is far below 1. For such companies, the low beta can be left in the valuation. If, on the other hand, diversification into other business areas is assumed in the maturity phase, a higher beta (e.g. 0.8) should be selected for these equity securities.

278

8 Dividend Discount Model

Example: Valuation of the Mercedes-Benz Group Stock with the Two-Stage Dividend Discount Model The following data are available for the listed stock of Mercedes-Benz Group AG at the beginning of 2017 (Source: Refinitiv Eikon): Historical beta Dividend per share (for 2016) Traded share price Return on equity (for 2016) Earnings payout ratio (for 2016)

1.52 EUR 3.25 EUR 71.18 15.3% 40.78%

It is assumed that earnings and dividends will increase annually over the next 5 years at the 2016 fundamental growth rate of 9.06%. The growth rate of 9.06% constitutes the product of the retention rate of 0.5922 (= 1 - 0.4078) and the return on equity of 15.3%. In the second perpetual growth phase, which begins after 5 years, the company exhibits a perpetual constant growth of 3.63% per year, which corresponds to the expected nominal growth rate of the GDP in Germany. Furthermore, it is assumed that at the beginning of the second growth stage the return on equity falls from 15.3% to 7%. It is also supposed that the adjusted beta of the Mercedes-Benz Group stock in the second perpetual growth phase is 1.2. The yield to maturity of 5-year and 30-year bonds of the Federal Republic of Germany are -0.4% and 1.1%, respectively, while the expected market risk premium for Germany is 7%. What is the intrinsic value of the Mercedes-Benz Group share according to the two-stage dividend discount model and is the security correctly valued? Solution The expected CAPM return for the first 5-year growth phase is 10.24%: E ðr 1 Þ = - 0:4% þ 7% × 1:52 = 10:24%: The next step is to calculate the annual dividends per share at a growth rate of 9.06% for the first 5-year growth phase: D1 = EUR 3:25 × ð1:0906Þ1 = EUR 3:54, D2 = EUR 3:25 × ð1:0906Þ2 = EUR 3:87, D3 = EUR 3:25 × ð1:0906Þ3 = EUR 4:22, D4 = EUR 3:25 × ð1:0906Þ4 = EUR 4:60,

(continued)

8.5 Two-Stage Dividend Discount Model

279

D5 = EUR 3:25 × ð1:0906Þ5 = EUR 5:01: In order to determine the dividend per share at the end of the first year of the second perpetual growth period, the new earnings payout ratio must first be estimated: δSecond perpetual growth phase = 1 -

0:0363 = 0:4814: 0:07

With a payout ratio of 40.78%, the earnings per share for 2016 are EUR 7.97 (= EUR 3.25/0.4078). Earnings per share in the sixth year are EUR 12.74: EPS6 = EUR 7:97 × ð1:0906Þ5 × 1:0363 = EUR 12:74: The dividend per share at the end of the first year of the second perpetual growth phase is EUR 6.13 (= EUR 12.74 × 0.4814). The expected return in the second growth stage is calculated with a beta of 1.2 and is 9.5% (= 1.1% + 7% × 1.2). The terminal value of the share in 5 years is EUR 104.43: P5 =

EUR 6:13 = EUR 104:43: 0:095 - 0:0363

At the valuation date, the intrinsic value of the Mercedes-Benz Group share is EUR 79.88 and can be calculated as follows: P0 = þ

EUR 3:54 EUR 3:87 EUR 4:22 EUR 4:60 þ þ þ ð1:1024Þ1 ð1:1024Þ2 ð1:1024Þ3 ð1:1024Þ4 EUR 5:01 þ EUR 104:43 = EUR 79:88: ð1:1024Þ5

Based on the assumptions made and the valuation model applied, the Mercedes-Benz Group stock appears undervalued by approximately 11%, as the intrinsic value of EUR 79.88 is above the market price of EUR 71.18. Figure 8.4 presents the calculation of the intrinsic value using the two-stage dividend discount model. A significant disadvantage of multi-stage cash flow models is that the terminal value makes up a large part of the intrinsic value calculated. In the example at hand, the terminal value at the valuation date is EUR 64.14 [= EUR 104.43/(1.1024)5] and accounts for 80.3% of the share’s intrinsic value of EUR 79.88. Hence, only 19.7% of the intrinsic value is attributable to the present value of the dividends in the 5-year high-growth period. The two-stage dividend discount model is justified in all cases where the company goes through an exceptionally high growth phase for a few years before growth

280

8 Dividend Discount Model

P5

EUR 3.54

EUR 3.87

EUR 6.13 0.095– 0.0363

EUR 4.22

EUR 4.60

EUR 5.01

EUR 6.13

3

4

5

6

f 0

1

2

Perpetual constant growth period

High growth period

P0

EUR 3.54 1.1024 1

EUR 3.87 1.1024 2 1.1024

EUR 4.22 1.1024 3

(Years)

EUR 4.60 1.1024 4

5

Fig. 8.4 Calculation of the intrinsic value of the Mercedes-Benz Group stock using the two-stage dividend discount model (Source: Own illustration)

settles down to a constant level. Extraordinarily high growth phases can occur due to a patent, a copyright, high legal or infrastructure-related barriers to entry, an advantage as the first provider of a product or service, or some other factor that contributes to a leading market position. The high growth observed during this period is usually not sustainable due to factors such as a patent expiring or competitors entering the market and capturing market share. At the end of the high-growth period, the growth rate falls back to a level equal to that of the economy as a whole. The assumption that the high growth rate drops abruptly to a lower constant level after a certain number of years represents a limitation of the valuation model, as there is no transition phase between the high growth phase and the perpetual constant growth phase. Therefore, the valuation model is suitable for companies whose growth rates in the first high-growth phase are not significantly higher than those in the second perpetual growth phase; for example, growth rates of 7–12% in the first phase, which then fall abruptly to a constant rate of, say, 2–5%. For companies that have an exceptionally high growth rate of, say, 30%, the assumption of an abrupt drop to a much lower constant rate is usually not realistic. Rather, a scenario that assumes a gradual decline in the growth rate would be appropriate. The valuation principle underlying the dividend discount model is simple and logical. No more should be paid for an equity security than the present value of the cash flows that can be expected in the future. Nevertheless, the implementation of the model is anything but simple, as various assumptions have to be made with regard to the calculation of the expected return and the dividend growth rate.

8.6 Summary

281

The objective of fundamental analysis is to identify undervalued or overvalued securities in order to earn an excess return (alpha). Therefore, various empirical studies have investigated whether mispriced equity securities can be identified with the application of the dividend discount model. The majority of the results of these studies indicate that excess returns (before tax) are achieved in the long run with the valuation model.26 However, the interpretation of these results is difficult in that it is not clear whether the excess returns are due to the identification of mispriced securities with the valuation model or to known price anomalies in the market, which can be determined with the price-to-earnings ratio and the dividend yield.27 An empirical study by Jacobs and Levy (1988) demonstrates that stock selection using a low price-to-earnings ratio results in an average return contribution of 0.92% per quarter, while stock selection applying the dividend discount model leads to an average return contribution of only 0.06% per quarter.28 These results are an indication that the majority of the excess returns captured with the cash flow models in the long run can be explained by the price-to-earnings ratio and/or the dividend yield.

8.6

Summary

• Equity securities can be valued under the going concern assumption using cash flow models, added-value models (e.g. residual income valuation models), and multiples. • When valuing equity securities with dividend discount models, the future dividends and the expected shareholders’ return have to be estimated. The choice of valuation model—that is, one-stage or multi-stage dividend discount model— depends on the assumed dividend growth pattern. • Dividend discount models can be used for equity valuation if the following conditions exist: (1) The company pays dividends and a data series of historical dividends is available. (2) The dividend policy is such that there is a correlation between profitability and dividends paid (constant earnings payout ratio). (3) The dividends and the free cash flows to equity are approximately the same. Furthermore, the dividend discount model can be applied to value equity securities that are held by minority shareholders, as they do not control the distribution of the free cash flows. See, for example, Sorensen and Williamson 1985: ‘Some evidence on the value of the dividend discount models’, p. 60 ff. 27 Studies on market information efficiency conclude that buying equity securities with a comparatively low price-to-earnings ratio makes it possible to earn excess returns in the long run. Likewise, abnormal returns can be achieved with equity securities that have a comparatively high dividend yield. The dividend discount model can be used to identify undervalued (overvalued) stocks that have a low (high) price-to-earnings ratio and a high (low) dividend yield. Hence, the observed price anomalies in the market are consistent with the results of the valuation model. 28 See Jacobs and Levy 1988: ‘On the value of ‘value’, p. 47 ff. 26

282

8 Dividend Discount Model

• Expected dividends can be estimated using growth rates. Statistical forecasting models based on historical earnings and company fundamentals can be applied to determine growth rates; moreover, consensus forecasts from analysts can also be used. • The one-stage dividend discount model or Gordon–Growth model assumes perpetual dividend growth and a constant earnings payout ratio. In order to apply the model, the expected return must be higher than the growth rate. The intrinsic value calculated with the model reacts very sensitively to changes in the two valuation parameters expected return and growth rate. The valuation model is suitable for companies in the maturity phase of their life cycle that operate in a saturated market. • The two-stage dividend discount model is based on the assumption that an initial period of high growth is followed by a perpetual growth phase. After the first growth stage, the growth rate falls abruptly to a lower constant level and remains there in perpetuity. The terminal value at the beginning of the second perpetual growth stage can be calculated either with a one-stage dividend discount model or with a price multiple. In many cases, the terminal value exceeds 75% of the calculated intrinsic value. The valuation model is suitable for companies that experience high growth due to a patent, high legal or infrastructure barriers to entry, or an advantage as a first mover of a product. The high growth that occurs during this period is usually unsustainable, and after the strong growth period ends, the earnings growth rate falls to a lower constant level. • Equity securities with high dividends and a low price-to-earnings ratio are most likely to be undervalued according to the dividend discount model, because high dividends or earnings lead to a high intrinsic share value. Empirical studies indicate that excess returns can be achieved over the long run with the valuation model. However, these abnormal returns are largely due to price anomalies such as low price-to-earnings ratio and high dividend yield.

8.7

Problems

1. Delta AG has earnings per share of EUR 6 and a dividend per share of EUR 3.60 in the year just ended. The earnings payout ratio and the return on equity of 8% are expected to remain constant in the future. The historical beta of the stock is 0.8. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%, while the expected market risk premium for Germany is 7%. What is the intrinsic share value of Delta according to the one-stage dividend discount model? 2. An analyst has compiled the following data on a mature company that has a well-established dividend policy and operates in a saturated market:

8.8 Solutions

283

Long-term earnings retention rate Long-term return on equity Historical beta of the stock Earnings per share in the year just ended Traded share price

35% 10% 1 EUR 4.50 EUR 60

The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%, while the expected market risk premium for Germany is 7%. What is the intrinsic share value using a one-stage dividend discount model? Is the security correctly valued and what is the expected alpha if the price correction occurs over a 1-year period? 3. At the beginning of 2017, an investor wants to buy Mercedes-Benz Group shares, which paid a dividend per share of EUR 3.25 in 2016. They assume an annual earnings growth rate of 10% and a constant payout ratio for the next 2 years. They also expect a dividend yield of 5% in 1 year. The historical beta of the stock is 1.52. The yield on non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills) with a maturity of 1 year is -0.2%. The expected market risk premium for Germany is 7%. At what share price is the investor willing to buy the Mercedes-Benz Group stock if they sell the security in 1 year? (Assumption: The dividend is paid out at the end of the investment period.) 4. Gamma AG paid a dividend per share of EUR 2.50 in the year just ended. An analyst expects an annual earnings growth rate of 10% for the next 2 years. During these 2 years, the earnings payout ratio remains constant. After 2 years, the annual growth rate falls abruptly to a level of 3.63%, which corresponds to the long-term nominal growth rate of the GDP. Furthermore, the return on equity also drops from 16% in the first growth stage to 8% in the second perpetual growth stage. The expected return of the shareholders is 9% in the first 2 years. The expected return of the shareholders is 7% for the perpetual growth stage. What is the intrinsic share value of Gamma according to the two-stage dividend discount model?

8.8

Solutions

1. The earnings payout ratio is 60% (= EUR 3.60/EUR 6). The long-term fundamental growth rate of 3.2% can be calculated as follows: g = ð1- 0:6Þ × 8% = 3:2%:

284

8 Dividend Discount Model

To estimate the expected CAPM return, the adjusted beta must first be determined: βAdjusted = 0:333 þ 0:667 × 0:8 = 0:867: The expected CAPM return is 7.169%: Eðr Þ = 1:1% þ 7% × 0:867 = 7:169%: The intrinsic share value of Delta is EUR 93.61 and can be estimated with a one-stage dividend discount model as follows:

P0 =

EUR 3:60 × 1:032 = EUR 93:61: 0:07169 - 0:032

2. The long-term fundamental growth rate is 3.5%: g = 0:35 × 10% = 3:5%: The expected CAPM return of 8.1% can be determined as follows: Eðr Þ = 1:1% þ 7% × 1 = 8:1%: The earnings payout ratio is 65% (= 1 - 0.35) and the dividend per share is EUR 2.925 (= 0.65 × EUR 4.50) in the year just ended. The intrinsic share value is EUR 65.81 and can be calculated with the following equation:

P0 =

EUR 2:925 × 1:035 = EUR 65:81: 0:081 - 0:035

The equity security appears undervalued because the intrinsic value of EUR 65.81 is above the market price of EUR 60. If market participants buy the undervalued security and the market price of EUR 60 moves to the intrinsic value of EUR 65.81 over a 1-year period, an excess return (alpha) of 6.63% is achieved:

8.8 Solutions

Alpha =

285

ðEUR 65:81 - EUR 60Þ þ EUR 2:925 × 1:035 - 0:081 = 6:63%: EUR 60

3. The historical beta of the Mercedes-Benz Group stock should not be adjusted for the mean reversion, since an expected return over a period of 1 year is to be determined. The expected CAPM return is 10.44%: Eðr Þ = - 0:2% þ 7% × 1:52 = 10:44%: The dividend per share in 2 years is EUR 3.93: D2 = EUR 3:25 × ð1:10Þ2 = EUR 3:93: The expected dividend yield in 1 year of 5% equals the dividend per share in 2 years divided by the expected share price in 1 year: 5% =

EUR 3:93 : P1

Therefore, the expected share price in 1 year of EUR 78.60 can be calculated as follows: P1 =

EUR 3:93 = EUR 78:60: 0:05

The expected cash flows over a period of 1 year consist of the dividend and the proceeds from the sale of the share. If these cash flows are discounted at the beginning of the investment period, the price that the investor is willing to pay for the stock is obtained (assumption: the dividend is paid out at the end of the investment period):

P0 =

EUR 3:25 × 1:1 EUR 78:60 þ = EUR 74:41: ð1:1044Þ1 ð1:1044Þ1

4. First, the dividends per share during the 2-year high-growth phase must be calculated:

286

8 Dividend Discount Model

D1 = EUR 2:50 × ð1:10Þ1 = EUR 2:75, D2 = EUR 2:50 × ð1:10Þ2 = EUR 3:025: The earnings payout ratio is 37.5% in the first growth stage and can be determined as follows: δFirst growth stage = 1 -

0:10 = 0:375: 0:16

Therefore, the earnings per share in the current period are EUR 6.667 (= EUR 2.50/ 0.375). The earnings per share at the end of the first year of the second perpetual growth stage can be calculated as follows: EPS3 = EUR 6:667 × ð1:10Þ3 × 1:0363 = EUR 8:36: In the second perpetual growth phase, the earnings payout ratio increases from 37.5% to 54.6% because growth and profitability decline:

δSecond growth stage = 1 -

0:0363 = 0:546: 0:08

The dividend per share at the end of the first year of the second perpetual growth stage of EUR 4.565 can be determined by multiplying the earnings per share of EUR 8.36 by the earnings payout ratio of 54.6%: D3 = EUR 8:36 × 0:546 = EUR 4:565: The terminal value of the share at the beginning of the second perpetual growth phase is EUR 135.46:

P2 =

EUR 4:565 = EUR 135:46: 0:07 - 0:0363

The intrinsic share value of Gamma is EUR 119.08 and can be calculated as follows using the two-stage dividend discount model:

References

287

P0 =

EUR 2:75 EUR 3:025 þ EUR 135:46 þ = EUR 119:08: ð1:09Þ1 ð1:09Þ2

References Damodaran, A.: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, 3rd edn, Hoboken (2012) Gordon, M.J.: The Investment, Financing, and Valuation of the Corporation, Homewood (1962) Graham, B.: The Intelligent Investor: A Book of Practical Counsel, New York (1949) Graham, B., Dodd, D.L.: Security Analysis, New York (1934) Jacobs, B.I., Levy, K.N.: On the value of ‘value’. Financial Anal. J. 44(4), 47–62 (1988) Koller, T., Goedhart, M., Wessels, D.: Valuation: Measuring and Managing the Value of Companies, 5th edn, Hoboken (2010) Mercedes-Benz Group: Geschäftsbericht 2016. Stuttgart (2017) Mondello, E.: Aktienbewertung: Theorie und Anwendungsbeispiele, 2nd edn, Wiesbaden (2017) Peterson Drake, P.: Financial statement analysis. In: Clayman, M.R., Fridson, M.S., Troughton, G.H. (eds.) Corporate Finance: A Practical Approach, pp. 311–366 (2008) Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 7th edn, Mason (2003) Sorensen, E.H., Williamson, D.A.: Some evidence on the value of the dividend discount models. Financial Anal. J. 41(6), 60–69 (1985) Williams, J.B.: The Theory of Investment Value, Cambridge (1938)

9

Free Cash Flow Models

9.1

Introduction

Cash flow models include not only the dividend discount model but also free cash flow models, in which free cash flows are discounted at the expected rate of return instead of dividends. Free cash flows are the operating cash flows generated by the company less the net capital expenditures that are required for its operating activities. They can be determined either after payment of the debt provider claims (free cash flows to equity) or before these claims are paid (free cash flows to firm). The advantage of these valuation models is that they are conceptually sound and suitable for most equity valuation applications. In particular, they can be used in the following cases: • The company does not pay dividends. • The company pays dividends, but they differ significantly from the free cash flows that can be distributed to equity investors. • Free cash flows are related to profitability and thus to the company’s value creation. • The equity valuation is carried out from the perspective of a controlling interest, as the majority shareholder has control over the distribution of free cash flows. The free cash flow to equity model and the free cash flow to firm model are presented below. The adjusted present value model, which is a further development of the free cash flow to firm model, is also described.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_9

289

290

9

9.2

Free Cash Flow to Equity Model

9.2.1

Overview

Free Cash Flow Models

The dividend discount model is based on the assumption that only the future dividends distributed to shareholders are relevant for valuation. By contrast, the free cash flow to equity (FCFE) model presumes that all cash potentially distributable to shareholders—that is, FCFE—is used for valuation. Only in the rarest of cases are FCFE paid out in full as dividends, with the result that the dividend discount model and the FCFE model usually produce different results for the intrinsic share value. Unlike dividends, free cash flows must first be calculated using the company’s existing financial information, which requires an understanding of the method used to calculate the free cash flows and the financial accounting. In order to estimate future cash flows, the growth rate must also be determined. Therefore, the following discussion starts with the calculation of the free cash flows to equity and the fundamental growth rate. It goes on to demonstrate how the intrinsic share value can be estimated by applying a one- and two-stage free cash flow to equity model.

9.2.2

Definition and Calculation of the FCFE

To gain a better understanding of how FCFE are determined, the calculation of the company’s net income is presented first. The left column of Table 9.1 presents a profit and loss statement. It starts with the revenues that a company has earned during a certain period (e.g. 1 year). Operating expenses as well as depreciation and amortisation are deducted from the revenues, resulting in the operating income (earnings before interest and taxes, EBIT). The interest expense is subtracted from the operating income, which leads to the earnings before taxes (EBT). After deducting the income tax expense, the net income is obtained. The right-hand column of Table 9.1 demonstrates the calculation method of the FCFE. First, non-cash expenses such as depreciation and amortisation have to be added to net income. Then, investments in non-cash working capital (accounts receivable plus inventories less accounts payable), which represent cash outflows Table 9.1 Net income versus free cash flow to equity (Source: Own illustration) Revenues - Operating expenses - Depreciation and amortisation = Operating income (EBIT) - Interest expense = Earnings before taxes (EBT) - Income tax expense = Net income

Net income + Depreciation and amortisation - Investments in non-cash working capital - Net capital expenditures + Net increase in interest-bearing debt capital = Free cash flow to equity

9.2 Free Cash Flow to Equity Model

291

for daily operations, must be deducted. Other cash outflows are net capital expenditures. They include net investments (investments less disinvestments) in property, plant and equipment such as machinery and equipment, in intangible assets such as software, patents, and trademarks, in other companies and in financial assets. Investments in non-cash working capital and net capital expenditures are cash outflows and are therefore no longer available for distribution to equity investors. Finally, the net increase in interest-bearing debt capital must be added. Raising debt capital in a period increases the FCFE, while repayment of debt capital represents a cash outflow and thus reduces the freely available cash flow that can be paid out to shareholders.

9.2.3

Growth Rate of the FCFE

The expected FCFE growth rate can be determined either as a historical growth rate or as a fundamental growth rate. The latter is based on fundamental data of the company and consists of the product of the reinvestment rate of equity and the return on equity:1 g = I E ROE,

ð9:1Þ

where IE = reinvestment rate of equity, and ROE = return on equity. The equity reinvestment rate measures the percentage of equity in after-tax profit that is invested in the business. It can be calculated by dividing the equity capital reinvested in the operating assets by the after-tax operating income as follows: IE =

EquityReinvested Net income - IncomeCash × ð1 - Tax rateÞ

ð9:2Þ

where EquityReinvested = equity reinvested in the operating activities, and IncomeCash = income from non-operating assets such as cash and cash equivalents.

1

See Damodaran 2012: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, p. 285. The fundamental growth rate of dividends or earnings is calculated using retained earnings, assuming that the only source of equity is retained earnings. See Sect. 8.3. However, a company can increase its growth not only with internal funds, but also with external funds, by issuing new equity. Thus, the growth rate must be a measure that makes it possible to estimate how much equity a firm reinvests in its operations in the form of net capital expenditures and investments in non-cash working capital.

292

9

Free Cash Flow Models

The equity capital reinvested in the operations consists of the net capital expenditures (Capex) minus depreciation and amortisation (Depr) plus the investments in non-cash working capital (I WC) and minus the net increase in interest-bearing debt capital (ΔDebt): EquityReinvested = Capex - Depr þ I WC - ΔDebt:

ð9:3Þ

In the denominator of Eq. (9.2), after-tax income from non-operating cash and cash equivalents must be deducted from net income, because future growth of the business depends on the after-tax operating income and not on total net income. Hence, income from additionally built-up cash—cash that is invested in marketable securities, for example, and is thus not necessary for operations—does not play a role in estimating the FCFE growth rate. The return on equity must also be adjusted for the non-operating components. On the one hand, the tax-adjusted income from cash and cash equivalents is subtracted from the net income and, on the other hand, the book value of cash and cash equivalents is deducted from the book value of the equity. The return on equity calculated in this way reflects the return of the equity capital invested in operational assets without cash and cash equivalents: ROE =

Net income - IncomeCash × ð1 - Tax rateÞ , EquityBegining - CashBeginning

ð9:4Þ

where EquityBeginning = book value of equity at the beginning of the period, and CashBeginning = book value of cash and cash equivalents not required for operations at the beginning of the period. Higher FCFE growth can be explained by a higher reinvestment rate and/or a higher return on equity. However, higher growth does not necessarily mean that the equity value increases. For example, although a higher reinvestment rate leads to higher growth, free cash flows decrease in the face of higher net capital expenditures and investments in non-cash working capital. A higher return on equity results in an increase in growth, but if the higher return is earned on riskier investments, the cost of equity increases. Therefore, an increase in the reinvestment rate and the return on equity does not only have a positive effect on the equity value. Example: Calculation of the Fundamental Free Cash Flow to Equity Growth Rate The following information is available for Delta AG (in EUR million): Net income Book value of equity at the beginning of the year Book value of equity at the end of the year

1800 15,000 16,200

(continued)

9.2 Free Cash Flow to Equity Model

Net capital expenditures Depreciation and amortisation Investments in non-cash working capital Borrowing of interest-bearing debt Repayment of interest-bearing debt Book value of cash and cash equivalents at the beginning of the yeara Book value of cash and cash equivalents at the end of the yeara Income from cash and cash equivalentsa a

293

4000 2000 500 5000 4000 1000 1200 40

Assumption: not operationally necessary

The income tax rate is 30%. What is the fundamental FCFE growth rate of Delta AG? Solution The reinvested equity and the corresponding reinvestment rate can be determined as follows: EquityReinvested = Capex - Depr þ I WC - ΔDebt = EUR 4000 million - EUR 2000 million þ EUR 500 million - ðEUR 5000 million - EUR 4000 millionÞ = EUR 1500 million, IE = =

EquityReinvested Net income - IncomeCash × ð1 - Tax rateÞ EUR 1500 million = 0:8465: EUR 1800 million - EUR 40 million × ð1 - 0:3Þ

The reinvestment rate of equity is 84.65%. The return on equity of 12.66% can be calculated as follows: ROE =

EUR 1800 million - EUR 40 million × ð1 - 0:3Þ = 12:66%: EUR 15,000 million - EUR 1000 million

The fundamental FCFE growth rate of 10.72% can be determined by multiplying the equity reinvestment rate by the return on equity: g = 0:8465 × 12:66% = 10:72%: Assuming that it is a representative year for the equity reinvested in operating assets and after-tax operating income and that the relationship between historical free cash flows and fundamentals remains the same in the near future, the calculated growth rate of 10.72% can be used to estimate the FCFE in the first growth stage in a multi-stage FCFE model.

294

9.2.4

9

Free Cash Flow Models

One-Stage FCFE Model

In the case of a one-stage FCFE model, a perpetual constant growth of the FCFE is assumed. The FCFE in each period is equal to the FCFE from the period just ended multiplied by 1 plus the constant growth rate. The intrinsic value of the equity can be calculated with a constant growth model as follows [E(r) > g]:2 V E,

0

=

FCFE0 ð1 þ gÞ FCFE1 = , E ðr Þ - g E ðr Þ - g

ð9:5Þ

where VE, 0 = intrinsic equity value at the valuation date, E(r) = expected return of the shareholders, and g = perpetual constant growth rate of free cash flows to equity. The amount of cash and cash equivalents not needed for operations and the market value of other non-operating assets not required for operations (if material and financed with equity or built-up cash reserves) are added to the intrinsic equity value calculated at the time of valuation, since future FCFE will only be generated with the operating assets. To determine the intrinsic share value, the adjusted intrinsic equity value is divided by the number of outstanding shares:3 P0 =

V E, 0 þ Cash0 þ NOA0 , Shares0

ð9:6Þ

where Cash0 = value of cash and cash equivalents not required for operations at the time of valuation, NOA0 = market value of non-operating assets (financed with equity or built-up cash reserves) at the valuation date, and Shares0 = number of shares outstanding at the valuation date. The constant growth rate of the FCFE should not exceed the nominal growth rate of the GDP of the country in which the company conducts its operations. The assumption of perpetual constant growth is appropriate for mature companies that operate in a saturated market and therefore have the same average growth as the overall economy.4 For mature companies with perpetual constant growth, the difference between investments in property, plant and equipment and depreciation is not excessive because the need for expansion investments is rather low due to relatively 2

For the derivation, see Sect. 8.4. If a mature company pays out all FCFE as dividends, the one-stage FCFE model produces the same intrinsic share value as the Gordon-Growth model (one-stage dividend discount model). 3 See Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 243. 4 See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 83.

9.2 Free Cash Flow to Equity Model

295

low growth. In addition, the company risk mirrors the average risk of the economy as a whole, and the beta of the stock is therefore close to 1. The reinvestment rate of equity (reinvested equity divided by after-tax operating income) of a mature company can be estimated, for example, using an average equity reinvestment rate of all mature companies in the same industry or using fundamental data of the company. When using fundamental data, the equity reinvestment rate can be calculated by rearranging Eq. (9.1) as follows: IE =

g : ROE

ð9:7Þ

For example, if the constant growth rate is 3.6% and the return on equity is 8%, 45% of the taxed profit is invested in fixed assets (net capital expenditures) and non-cash working capital. Accordingly, the ratio between the FCFE and the operating profit after tax is 55%. This leads to the following formula for calculating the FCFE:5 FCFE = NIð1- I E Þ,

ð9:8Þ

where NI = net income. If, for example, the net income is EUR 10 million and the reinvestment rate of equity is 45%, the FCFE will be EUR 5.5 million [= EUR 10 million × (1 - 0.45)].6 Example: Calculation of the Intrinsic Share Value Using the One-Stage FCFE Model The following data is available for Gamma AG (in EUR million):

Net income Depreciation and amortisation Cash and cash equivalentsa Income from cash and cash equivalentsa Equity Investments in non-cash working capital Net capital expenditures Net increase in interest-bearing debt capital a

2023 5000 3200 7900 160 72,000 2000 8000 4700

2022 4800 3000 7700 150 69,000 1800 7000 3400

Assumption: not operationally necessary

(continued)

FCFE = NI [1 - (Capex - Depr + I WC - ΔDebt)/NI] = NI - Capex + Depr - I WC + ΔDebt It is assumed that the non-cash expenses and revenues consist only of depreciation and amortisation, as well as the change in non-cash working capital. See Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 244. 5 6

296

9

Free Cash Flow Models

The company is assumed to have perpetual constant growth based on 2023 fundamentals. The income tax rate is 30%. As at the end of 2023, a total of 500 million shares are outstanding. The long-term beta of Gamma’s stock is 1. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%, while the expected market risk premium for Germany is 7%. What is the intrinsic share value of Gamma according to the one-stage FCFE model at the end of 2023? Solution To determine the fundamental growth rate, the return on equity of 7.97% adjusted for non-operating cash is calculated first: ROE =

EUR 5000 million - EUR 160 million × ð1 - 0:3Þ = 7:97%: EUR 69,000 million - EUR 7700 million

The reinvested equity is EUR 2100 million (= EUR 8000 million - EUR 3200 million + EUR 2000 million - EUR 4700 million). The reinvestment rate of the equity capital is 0.4296 and can be calculated as follows: IE =

EUR 2100 million = 0:4296: EUR 5000 million - EUR 160 million × ð1 - 0:3Þ

The fundamental FCFE growth rate is 3.4%: g = 0:4296 × 7:97% = 3:42%: The FCFE of EUR 2788 million can be determined with the net income of EUR 5000 million, the tax-adjusted income from non-operating cash and cash equivalents of EUR 112 million [= EUR 160 million × (1 - 0.3)],7 and the equity reinvestment rate of 0.4296 as follows: FCFE2023 = ½EUR 5000 million - EUR 160 million × ð1- 0:3Þ × ð1- 0:4296Þ = EUR 2788 million: Alternatively, the FCFE can also be calculated as follows (in EUR million): Net income - Tax-adjusted income from non-operating cash and cash equivalents [160 x (1 - 0.3)]. + Depreciation and amortisation - Investments in non-cash working capital - Net capital expenditures + Net increase in interest-bearing debt capital = FCFE

7

5000 - 112 + 3200 - 2000 - 8000 + 4700 = 2788

Since the FCFE are generated from the operating assets and it is assumed in the example that the cash and cash equivalents are not required for operations, the tax-adjusted income from the cash and cash equivalents must be deducted from net income.

9.2 Free Cash Flow to Equity Model

297

The expected CAPM return of 8.1% can be calculated as follows: Eðr Þ = 1:1% þ 7% × 1 = 8:1%: The one-stage FCFE model produces an intrinsic equity value of EUR 61,610 million: V E,

2023

=

EUR 2788 million × 1:0342 = EUR 61,610 million: 0:081 - 0:0342

The cash and cash equivalents of EUR 7900 million should be added to the calculated intrinsic equity value of EUR 61,610 million, which leads to a value of EUR 69,510 million. If this amount is divided by the 500 million outstanding shares, the intrinsic share value of EUR 139.02 is obtained: P2023 =

9.2.5

EUR 69,510 million = EUR 139:02: 500 million shares

Two-Stage FCFE Model

The basic variant of the two-stage FCFE model assumes a high constant growth of the FCFE for a first time period of, for example, 3–10 years, which is followed by a second period with a perpetual constant growth rate of the FCFE. With the two-stage model, the intrinsic value of the equity can be calculated as follows: T

V E, 0 =

FCFEt þ ½1 þ Eðr 1 Þt

t=1 ðShort - term high growthÞ

V E, T ½1 þ Eðr 1 ÞT

,

ð9:9Þ

ðLong - term perpetual growthÞ

where FCFEt = free cash flow to equity in period t, E(r1) = expected return of shareholders in the first high-growth phase, and VE, T = terminal value of equity at the end of period T (intrinsic value of equity at the end of the first growth phase). To determine the intrinsic share value, the calculated equity value is divided by the number of outstanding shares. The terminal value of the equity can be calculated with a price multiple or with a one-stage FCFE model.8 With a price multiple, for

8

See Bodie et al. 2009: Investments, p. 613.

298

9

Free Cash Flow Models

example, the terminal value can be determined by multiplying the leading price-toearnings ratio by the earnings per share of the next year. With a one-stage FCFE model, on the other hand, the terminal value can be estimated with the following equation under the assumption of perpetual constant growth [E(r2) > gT]: V E,

T

=

FCFETþ1 , E ð r 2 Þ - gT

ð9:10Þ

where FCFET+1 = free cash flow to equity at the end of the first period of the second perpetual growth phase, E(r2) = expected return of shareholders in the second perpetual growth phase, and gT = perpetual constant growth rate. For the calculation of the terminal value, it is important that the FCFE are aligned with the stability assumption of a company in the maturity phase.9 Thus, the need for expansion investments, given by the difference between investments in property, plant and equipment and depreciation, is smaller in the second perpetual growth phase than in the first high growth period. The expansion investments for the perpetual growth phase can be estimated either with an average investment value of mature companies in the industry or with fundamental data of the company (IE = g/ROE). Furthermore, a beta close to 1 (range between 0.8 and 1.2) should be chosen, as the company is subject to a similar risk as the economy as a whole. The debt-to-equity ratio also needs to be adjusted. For example, a company may have a high level of debt at the beginning of the growth phase, which is gradually reduced to an industry-typical level over time.10 The terminal value usually accounts for a high percentage of the total equity value. Proportions of 75% and higher are common for public companies with listed stocks. Young growth companies with negative FCFE in the first growth phase have a terminal value that even exceeds 100% of the calculated equity value.11 The high share price of such companies is explained by the high terminal value which is a reflection of positive long-term capital market expectations on the part of investors. Dividends and share buybacks play only a subordinate role for such stocks. Moreover, the terminal value reacts very sensitively to changes in valuation parameters such as the growth rate and expected return. The impact of the valuation parameters on the calculated share value can be assessed with the assistance of a sensitivity analysis. If the valuation is based on overly optimistic forecasts (e.g. as a result of a

9

See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 83. See Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 251. 11 See Koller et al. 2010: Valuation: Measuring and Managing the Value of Companies, p. 214. 10

9.2 Free Cash Flow to Equity Model

299

bull market), a scenario analysis can be used to adjust the excessively high terminal value downwards. The following example demonstrates the calculation of the intrinsic share value with the two-stage FCFE model for Mercedes-Benz Group AG, where the future FCFE are estimated by means of the fundamental growth rate. The example is based on plausible assumptions that can be refined with a more detailed analysis. Example: Valuation of the Mercedes-Benz Group Stock with the Two-Stage FCFE Model The following data from the 2016 annual report are available for the Mercedes-Benz Group (in EUR million):12

Net income (profit attributable to shareholders of Mercedes-Benz Group AG)a Depreciation and amortisation Cash and cash equivalentsb Equity (attributable to shareholders of Mercedes-Benz Group AG)c Change in non-cash working capitald Net capital expenditures Increase in interest-bearing debt Value of non-operating assets (including marketable securities and equity investments valued using the equity method)e

2016 7821

2015 7999

5478 10,981 57,950 10,281 14,666 15,763 14,846

5384 9936 53,561 14,654 9722 12,464 11,906

a

Net income from operations after taxes and interest and without non-controlling interests (minority interests) b Assumption: not required for operations c Excluding non-controlling interests (minority interests) d Including other non-cash assets and liabilities e Assumptions: (1) Income from this position is not included in net income from operations. (2) The position is financed with equity capital or with built-up cash reserves

As at the end of December 2016, Mercedes-Benz Group has 1070 million shares outstanding. The shares are traded at a price of EUR 70.71. The historical beta of the stock is 1.52. The yields to maturity of 5-year and 30-year bonds of the Federal Republic of Germany are -0.4% and 1.1%, respectively, while the expected market risk premium for Germany is 7%. FCFE are assumed to increase over the next 5 years based on 2016 fundamentals before annual growth drops abruptly to a constant level of 3.63%, which corresponds to the annual long-term nominal growth of Germany’s GDP. In the second perpetual growth stage, the adjusted beta of (continued)

12

See Mercedes-Benz Group 2017: Geschäftsbericht 2016, p. 216 ff.

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Free Cash Flow Models

the stock is 1.2 and the return on equity is 8%. What is the intrinsic share value of Mercedes-Benz Group according to the two-stage FCFE model? Solution The equity reinvestment rate of 0.4739 can be determined as follows: IE =

EUR 14,666 million - EUR 5478 million þEUR 10,281 million - EUR 15,763 million

EUR 7821 million

= 0:4739:

The return on equity is 24.66%: ROE =

EUR 7821 million EUR 53,561 million - EUR 9936 million - EUR 11,906 million

= 24:66%: The fundamental growth rate of 11.69% is obtained by multiplying the reinvestment rate of equity by the return on equity: g = 0:4793 × 24:66% = 11:69%: The FCFE for 2016 can be calculated as follows: FCFE2016 = EUR 7821 million × ð1- 0:4739Þ = EUR 4115 million: The FCFE increases by 11.69% annually in the first high-growth period, resulting in the following FCFE values for the first 5 years: FCFE2017 = EUR 4115 million × ð1:1169Þ1 = EUR 4596 million, FCFE2018 = EUR 4115 million × ð1:1169Þ2 = EUR 5133 million, FCFE2019 = EUR 4115 million × ð1:1169Þ3 = EUR 5733 million, FCFE2020 = EUR 4115 million × ð1:1169Þ4 = EUR 6404 million, FCFE2021 = EUR 4115 million × ð1:1169Þ5 = EUR 7152 million: The expected CAPM return in the first growth phase is 10.24% (= -0.4% + 7% × 1.52), while in the second perpetual growth phase the expected CAPM return is 9.5% (= 1.1% + 7% × 1.2). In the second perpetual growth phase, the equity reinvestment rate decreases from 0.4739 to 0.4538: (continued)

9.3 Free Cash Flow to Firm Model

IE =

301

g 3:63% = = 0:4538: ROE 8%

The FCFE at the end of the first year of the second perpetual growth stage is EUR 7694 million: FCFE2022 = EUR 7821 million × ð1:1169Þ5 × 1:0363 × ð1- 0:4538Þ = EUR 7694 million: The terminal value at the beginning of the second perpetual growth phase can be determined as follows: V E, 2021 =

EUR 7694 million = EUR 131, 073 million: 0:095 - 0:0363

The intrinsic equity value is EUR 101,905 million and is equal to the present value of all future FCFE: V E, 2016 = þ

EUR 4596 million EUR 5133 million EUR 5733 million þ þ ð1:1024Þ1 ð1:1024Þ2 ð1:1024Þ3

EUR 6404 million EUR 7152 million þ EUR 131, 073 million þ ð1:1024Þ4 ð1:1024Þ5

= EUR 101, 905 million: With the cash and cash equivalents of EUR 10,981 million and the value of the non-operating assets of EUR 14,846 million, the adjusted intrinsic equity value is EUR 127,732 million. If this value is divided by 1070 million shares outstanding, the result is an intrinsic share value of EUR 119.38. Compared to the traded share price of EUR 70.71, the security is undervalued by approximately 41% based on the calculations and the assumptions made in the example.

9.3

Free Cash Flow to Firm Model

9.3.1

Definition and Calculation of Free Cash Flow to Firm

Free cash flow to firm (FCFF) refers to the cash earned in a certain period that can be potentially distributed to the debt and equity providers. In order to calculate the FCFF, the FCFE must be adjusted for the claims of the debt capital providers. For

302

9

Free Cash Flow Models

Table 9.2 Calculation of the free cash flow to firm (Source: Own illustration) Variant 1 Net income + Depreciation and amortisation - Investments in non-cash working capital - Net capital expenditures + Net increase in interest-bearing debt capital = Free cash flow to equity + Interest expense x (1 - tax rate) - Net increase in interest-bearing debt capital = Free cash flow to firm

Variant 2 Net income + Interest expense x (1 - tax rate) + Depreciation and amortisation - Investments in non-cash working capital - Net capital expenditures = Free cash flow to firm

this purpose, the tax-adjusted interest expense on debt is added to the FCFE and the net increase in interest-bearing debt is subtracted. Alternatively, the FCFF can be determined by starting with the net income from operations. In this case, the tax-adjusted interest expense on debt must be added to the net income. This equals the operating income multiplied by 1 minus the income tax rate [EBIT × (1 - tax rate)] or the operating income after tax.13 Next, non-cash expenses such as depreciation and amortisation are added, and investments in non-cash working capital and net capital expenditures are subtracted as these are needed for the company’s activities and are therefore not available to the providers of capital. Table 9.2 presents the calculation of the FCFF, assuming that depreciation and amortisation (as well as the change in non-cash working capital) are the only non-cash expenses and revenues. The FCFF can be distributed to debt providers as interest payments and repayments of outstanding debt capital and to equity providers as dividends and as part of a share buyback programme. Free cash flows to firm can be considered as debt-free cash flows because they are not affected by interest payments and net borrowing. By contrast, FCFE depend on the capital structure of the company. Table 9.3 presents the calculation of FCFE and FCFF for three different capital structure scenarios. The company can borrow money at an interest rate of 4%. The income tax rate is 30%. The figures in the table illustrate that when the debt ratio increases, interest expense increases and after-tax income decreases, leading to a decline in FCFE. On the other hand, FCFF remain unchanged. However, FCFE do not fall by the full amount of interest expense, but only by the tax-adjusted interest expense [interest expense × (1 - income tax rate], because interest on debt is tax deductible. The last line in Table 9.3 presents the FCFF, which amount to EUR 27,000 regardless of the debt ratio. Thus, the FCFF are debt-adjusted free cash flows that reflect the free cash flows of an unlevered or debt-free company.

13

Net income from operations + interest expense × (1 - tax rate) = EBIT × (1 - tax rate).

9.3 Free Cash Flow to Firm Model

303

Table 9.3 FCFE and FCFF for different capital structure scenarios (Source: Own illustration) 0% debt ratio (in EUR) Capital structure Debt capital Equity capital Total capital Profit and loss statement Revenues Operating expenses Depreciation and amortisation Operating income Interest expense Operating income before taxes Income tax expense Net income Calculation of free cash flows Net income Depreciation and amortisation Investments in non-cash working capital Net capital expenditures FCFE Tax-adjusted interest expense FCFF

9.3.2

40% debt ratio (in EUR)

60% debt ratio (in EUR)

0 100,000 100,000

40,000 60,000 100,000

60,000 40,000 100,000

200,000 -120,000 -20,000 60,000 0 60,000 -18,000 42,000

200,000 -120,000 -20,000 60,000 -1600 58,400 -17,520 40,880

200,000 -120,000 -20,000 60,000 -2400 57,600 -17,280 40,320

42,000 20,000 -5000

40,880 20,000 -5000

40,320 20,000 -5000

-30,000 27,000 0 27,000

-30,000 25,880 1120 27,000

-30,000 25,320 1680 27,000

Growth Rate of the FCFF

The growth rate of FCFF can be calculated using historical FCFF or as a fundamental (endogenous) growth rate. The fundamental growth rate of FCFF refers to the growth rate of after-tax operating income [EBIT × (1 - income tax rate)], as this income from operations is available to all capital providers. By contrast, the FCFE growth rate is based on net income generated from operating activities.14 The fundamental growth rate of operating income after tax can be calculated by multiplying the reinvestment rate of total capital (IC) by the return on capital (ROC) adjusted for non-operating cash and cash equivalents as follows:15 g = I C ROC,

ð9:11Þ

where

14

See Sect. 9.2.3. See Damodaran 2012: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, p. 289. 15

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9

Free Cash Flow Models

IC = proportion of the after-tax operating income that is reinvested in fixed assets (net capital expenditures) and non-cash working capital (where t = income tax rate): (Capex - Depr + I WC)/[EBIT(1 - t)], and ROC = return on capital invested: EBIT(1 - t)/(EquityBeginning + DebtBeginning CashBeginning). The fundamental growth rate in operating income is therefore given by the product of the capital reinvestment rate and the return on the capital invested. The former measures the proportion of the after-tax operating income that is invested in fixed assets (net capital expenditures) and non-cash working capital, while the latter measures the profitability of these reinvestments. Accordingly, the growth rate can be increased through an increased capital reinvestment rate and/or through an improved return on capital resulting from investments in operational assets. Higher growth does not necessarily mean that the value of the company increases. For example, a higher capital reinvestment rate contributes to higher growth, but at the same time FCFF fall as a result of the investments in non-cash working capital and net capital expenditures. If higher returns on capital are earned in riskier business areas, the cost of capital can rise despite an improved growth rate, which has a negative impact on the value of the company. For companies with capital reinvestment rates of more than 100%, part of the growth is due to acquisitions. For the calculation of the long-term fundamental growth rate, cash expenditures for mergers and acquisitions should be excluded from net capital expenditures. A mature company operating in a saturated market does not exhibit high perpetual external growth. Similarly, the capital reinvestment rate must be adjusted for a decrease in non-cash working capital, as working capital cannot drop below zero in the long run. For this purpose, investments in working capital can be normalised or average values of comparable companies can be taken.

9.3.3

One-Stage FCFF Model

Since FCFF are available to all providers of capital, they are discounted with the expected return of the capital providers or with the weighted average cost of capital (WACC) for the calculation of the enterprise value. Depending on the expected cash flow pattern, one-stage or multi-stage models can be used for equity valuation. If FCFF are expected to increase at a constant growth rate over an infinite time period, the enterprise value at the valuation date can be determined on the basis of a one-stage model as follows (WACC > g): EV0 = where

FCFF0 ð1 þ gÞ FCFF1 = , WACC - g WACC - g

ð9:12Þ

9.3 Free Cash Flow to Firm Model

305

EV0 = enterprise value at the valuation date, FCFF1 = free cash flow to firm for period 1, WACC = weighted average cost of capital, and g = perpetual constant growth rate of FCFF. The WACC can be calculated with the following equation:16 WACC = wD cD ð1- t Þ þ wE cE ,

ð9:13Þ

where wD = market value weight of interest-bearing debt, wE = market value weight of equity, cD = cost of debt (expected return of debt providers), t = marginal income tax rate, and cE = cost of equity (expected return of equity providers). The company valuation with the FCFF model refers to expected FCFF which will be earned from the operating activities. If the company owns significant non-operating assets, such as non-operating cash, real estate, land reserves, or financial investments in fixed-income securities or in equity securities of non-operating subsidiaries or of other companies, these must be added to the calculated enterprise value:17 V 0 = EV0 þ NOA0 ,

ð9:14Þ

where NOA0 = market value of non-operating assets (including non-operating cash and cash equivalents) at the valuation date. As a general rule, assets that do not produce operating cash flows and are therefore not used to determine FCFF are added to the enterprise value. Non-operating assets are to be included in the valuation at market value (and not at book value). In order to estimate the intrinsic equity value, the market value of the interestbearing debt capital (VD,0) and the market value of non-controlling interests (Min0)—that is, the claims of the minority shareholders—must be subtracted from the enterprise value (WACC > g):18

16

See Sect. 6.2.5. See Koller et al. 2010: Valuation: Measuring and Managing the Value of Companies, p. 275. 18 See Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 280. 17

306

9

V E, 0 =

Free Cash Flow Models

FCFF1 þ NOA0 - V D, 0 - Min0 : WACC - g

ð9:15Þ

The interest-bearing debt consists of current and non-current debt capital. This includes corporate bonds, interest-bearing shareholder loans, bank loans, and liabilities to affiliated companies and companies in which a participating interest is held or to which bills payable exist, as well as liabilities from financial and operating leases. In addition, pension liabilities, which reflect obligations from a company’s pension plan, should also be allocated to interest-bearing liabilities if there is no matching pension asset value in the balance sheet.19 If the interest-bearing financial liabilities (e.g. bonds) are traded on a liquid market, the intrinsic value equals the traded price. However, most financial liabilities are not traded on financial markets, and the intrinsic value can therefore be determined by discounting the expected cash flows.20 The cash flows of interest-bearing liabilities consist of interest and principal payments, which are discounted with the expected return of the debt providers (cost of debt) for the calculation of the intrinsic value. In valuation practice, the book value is often taken for debt capital that is not traded on a market, since the information relevant to the valuation regarding the cash flows and the discount rate is usually not available.21 The calculation of the enterprise value with the one-stage FCFF model assumes that the long-term growth rate of FCFF is not greater than the growth rate of the economy as a whole. Furthermore, the capital reinvestment rate and the growth rate of the FCFF must be consistent with the company’s fundamentals. Thus, the following relationship must hold between the capital reinvestment rate, the estimated growth rate, and return on capital: IC =

g : ROC

ð9:16Þ

For the calculation of the capital reinvestment rate of a company in the maturity phase, investments in non-cash working capital and net capital expenditures, which are necessary in a saturated market, must be taken into account. In doing so, one cannot assume a perpetual constant decrease in working capital; otherwise working capital would fall below zero at some point in the future. When determining the WACC, it is important to use a beta of approximately 1 (range between 0.8 and 1.2) for estimating the cost of equity with the CAPM. Typically, companies in the maturity phase refinance themselves with more debt

19

See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 27. 20 See Koller et al. 2010: Valuation: Measuring and Managing the Value of Companies, p. 282. 21 See Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 281.

9.3 Free Cash Flow to Firm Model

307

than equity, since equity is repaid to the company owners due to the lack of profitable investment projects. The value calculated applying the one-stage FCFF model reacts very sensitively to small changes in the expected growth rate and the WACC. The difference between the WACC and the long-term growth rate is lower than with the one-stage FCFE model because the cost of capital is lower than the cost of equity. Moreover, the growth rate should not be higher than the long-term nominal GDP growth rate. Example: Calculation of the Intrinsic Share Value With the One-Stage FCFF Model The following balance sheet and income statement information is available for Omega AG (in EUR million): Balance sheet Cash and cash equivalentsa Accounts receivables Inventories Property, plant and equipment Other fixed assets Total assets Accounts payables Current interest-bearing financial liabilities Long-term interest-bearing financial liabilities Total liabilities Equity Total liabilities and equity a

1000

Profit and loss statement Revenues Cost of goods sold Other operating expenses Depreciation and amortisation Operating income (EBIT) Interest expense Operating income before taxes Income tax expense

8000

Net income

1036

Other relevant information Net capital expenditures Investments in non-cash working capital

3400 400

2100 3600 2300 15,000 3000 26,000 3000

12,000 14,000 26,000

22,000 12,000 5000 3000 2000 520 1480 444

Assumption: not operationally necessary

At the end of the previous year, cash and cash equivalents amounted to EUR 2000 million, liabilities to EUR 11,000 million, and equity to EUR 13,600 million. Depreciation and amortisation (and the change in non-cash working capital) are the only non-cash expenses and revenues. The historical beta of the stock is 1.2. The credit risk premium for the calculation of the cost of debt is 2.3%. Furthermore, it is assumed that the optimal capital structure corresponds to the ratio between the book values of interest-bearing debt and equity. There are a total of 500 million shares outstanding. The income tax rate is 30%. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%, while the expected market risk premium for Germany is 7%. What is the intrinsic share value of Omega according to the one-stage (continued)

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9

Free Cash Flow Models

FCFF model if perpetual constant growth is based on the company’s fundamentals? Solution In order to determine the fundamental growth rate of FCFF, the reinvestment rate of total capital must first be calculated: IC =

EUR 3400 million - EUR 3000 million þ EUR 400 million = 0:5714: EUR 2000 million × ð1 - 0:3Þ

The return on capital is calculated with the operating income (EBIT) and with the book values of the debt and equity capital, as well as with the cash not required for operations of the previous year, as follows: ROC =

EUR 2000 million × ð1 - 0:3Þ EUR 11,000 million þ EUR 13,600 million - EUR 2000 million

= 6:19%: The fundamental growth rate is 3.54%: g = 0:5714 × 6:19% = 3:54%: The FCFF of EUR 600 million can be determined as follows: FCFF0 = EUR 2000 million × ð1 - 0:3Þ þ EUR 3000 million - EUR 3400 million - EUR 400 million = EUR 600 million or FCFF0 = EUR 2000 million × ð1- 0:3Þ × ð1- 0:5714Þ = EUR 600 million: The cost of debt is 3.4% and is made up of the long-term risk-free interest rate of 1.1% and the credit risk premium of 2.3%: cD = 1:1% þ 2:3% = 3:4%: The adjusted beta of 1.13 can be calculated from the historical beta as follows: βAdjusted = 0:333 þ 0:667 × 1:2 = 1:13: The cost of equity is 9.01% and can be estimated with the CAPM using the following equation: cE = 1:1% þ 7% × 1:13 = 9:01%:

(continued)

9.3 Free Cash Flow to Firm Model

309

The weights for interest-bearing debt and equity can be determined as follows: wD =

EUR 9000 million = 39:13%, EUR 9000 million þ EUR 14,000 million

wE =

EUR 14, 000 million = 60:87%: EUR 9000 million þ EUR 14, 000 million

The WACC is 6.42% and can be determined using the calculated weights and the cost components for interest-bearing debt and equity as follows: WACC = 0:3913 × 3:4% × ð1- 0:3Þ þ 0:6087 × 9:01% = 6:42%: The one-stage FCFF model results in an enterprise value of EUR 21,571 million: EV0 =

EUR 600 million × 1:0354 = EUR 21,571 million: 0:0642 - 0:0354

If the market value of the interest-bearing debt of EUR 9000 million is subtracted from the enterprise value, and the value of cash and cash equivalents not required for operations of EUR 2100 million is added, one arrives at an intrinsic equity value of EUR 14,671 million: V E, 0 = EUR 21, 571 million - EUR 9000 million þ EUR 2100 million = EUR 14, 671 million: The intrinsic share value of Omega is EUR 29.34: P0 =

EUR 14,671 million = EUR 29:34: 500 million shares

In order to determine the enterprise value, among other things, the WACC, which consists of the sum of the capital-weighted cost components for equity and debt capital, must be calculated. The weights required to calculate the WACC are preferably based on a target capital structure disclosed by the company. If no such information is available, the weights can be estimated using the current capital structure with the market values of the interest-bearing debt and equity. To calculate the market value of the equity, the share price is needed, which in turn is calculated in the valuation model by discounting the FCFF. Consequently, there is a circularity problem in the calculations as the share price is applied as a parameter for determining the WACC, which is then used as a discount rate in the company valuation. One possible solution is to take the share price estimated by means of the FCFF model to

310

9

Free Cash Flow Models

calculate the WACC in turn. The intrinsic share value is calculated using the new WACC. If these calculations are repeated several times, the share price applied for the weighting converges with the calculated share price.22 Furthermore, the weights for debt and equity capital can also be determined using the average capital structure of comparable companies. This approach assumes that the average capital structure of the peer firms corresponds to the target capital structure of the subject company.23

9.3.4

Comparison Between FCFE and FCFF Models

With the dividend discount model and the FCFE model (net valuation method), the intrinsic equity value is calculated directly, whereas with the FCFF model (gross valuation method), the enterprise value is determined first. To calculate the intrinsic equity value with the FCFF model, the market value of the non-operating assets is added to the estimated enterprise value, while the market value of the interestbearing debt capital is subtracted from the enterprise value. Calculating the intrinsic equity value with the FCFF model is more cumbersome than with the dividend discount model or the FCFE model. Nevertheless, the gross valuation method is used for a company with an unstable capital structure and a high debt ratio, since no changes in interest-bearing debt have to be estimated when determining FCFF (in contrast to FCFE). In the FCFF model, however, the weight of interest-bearing debt and the cost of debt must be determined in order to calculate the WACC.24 Table 9.4 compares the FCFE model with the FCFF model.

9.4

Adjusted Present Value Model

The adjusted present value (APV) approach25 can be applied to value companies that have a fixed repayment schedule for debt capital (e.g. leveraged acquisitions such as a management leveraged buyout). According to the APV model, the value of the company is the sum of two components of value, one derived from the company’s operations and the other from its capital structure. The first component reflects the value of the unlevered company, where the FCFF are discounted with the cost of capital for a debt-free company. In this way, the company value resulting from the operating activity can be estimated. The second component of value reflects the benefit that arises from debt financing through tax savings, since interest payments

See Schwetzler and Darijtschuk 1999: ‘Unternehmensbewertung mit Hilfe der DCF-Methode – eine Anmerkung zum “Zirkularitätsproblem”’, p. 295 ff. 23 See Courtois et al. 2008: Cost of Capital, p. 131. 24 See Barker 2001: Determining Value: Valuation Models and Financial Statements, p. 197. 25 See Myers 1974: ‘Interactions of corporate financing and investment decisions - implications for capital budgeting’, p. 4. 22

9.4 Adjusted Present Value Model

311

Table 9.4 FCFE model versus FCFF model (Source: Own illustration) Definition of cash flows Investment assumptions

Expected growth

Cash and cash equivalents

Discount rate Application

FCFE model FCFE are available to equity providers.

FCFF model FCFF are available to debt and equity providers.

The net capital expenditures adjusted for depreciation/amortisation and the investments in non-cash working capital less the net increase in debt borrowings reflect the financing of the operating assets with equity. Growth refers to operating assets financed with equity. It can be determined either with historical data or with fundamental data. Income from the non-operating cash and cash equivalents is not part of the FCFE. Therefore, the value of the non-operating cash and cash equivalents must be added to the calculated equity value. Expected return of equity providers (cost of equity). If a stable future capital structure is assumed, the future changes in debt capital and thus the FCFE can be well estimated. The intrinsic equity value can be determined directly with the FCFE model (net valuation method).

The difference between net capital expenditures and depreciation/ amortisation, as well as investments in non-cash working capital, increase the operating assets. These investments are financed with debt and equity. Growth relates to operating assets financed with total capital. The estimation procedures are the same as for the FCFE model. Income from the non-operating cash and cash equivalents is not part of the FCFF. Therefore, the value of the non-operating cash and cash equivalents must be added to the calculated enterprise value.a Expected return of total capital providers (WACC). If changes in the capital structure are to be expected and there is a high debt ratio, it is comparatively easy to forecast the FCFF. However, in order to calculate the WACC, the future capital structure and the cost of debt must be estimated.

a

Since FCFF are calculated on the basis of the operating income (EBIT) and income from non-operating cash and cash equivalents is not included in the EBIT, there is no need to adjust FCFF for this non-operating income. By contrast, FCFE is determined starting with net income, which includes income from non-operating cash and cash equivalents. Therefore, the after-tax income from the cash and cash equivalents not required for operations must be removed from net income in order to calculate FCFE

on debt are tax deductible. Thus, the enterprise value can be calculated with the APV model as follows: EV0 =

FCFF1 FCFFT þ EVT þ ... þ þ PVðTSÞ, 1 ð1 þ cu ÞT ð1 þ cu Þ

ð9:17Þ

where FCFF1 = free cash flow to firm for the period 1, cu = cost of capital of an unlevered (debt-free) company, EVT = terminal value of the unlevered enterprise value at the end of period T, and PV(TS) = present value of the interest-related tax savings.

312

9

Free Cash Flow Models

In practice, the APV model is often implemented only with the tax benefit of debt capital, which is given by the last term to the right of the equals sign; that is, by the present value of the interest-related tax savings or PV(TS). The cost of debt capital in the form of expected insolvency costs is usually not considered.26 The tax benefit of debt results from the tax deductibility of the interest payments. The annual tax savings are determined by multiplying the income tax rate by the annual interest payments, which leads to the following formula for the company valuation with the APV model: EV0 = þ

tc V FCFF1 FCFFT þ EVT þ ... þ þ D D, 01 þ . . . T 1 ð1 þ cu Þ ð1 þ cu Þ ð1 þ cD Þ

tcD V D, T - 1 þ FVTS, T , ð1 þ cD ÞT

ð9:18Þ

where t = income tax rate, cD = interest rate for debt or cost of debt, VD = value of interest-bearing debt capital, and FVTS, T = final value of the tax savings for the debt capital at the end of period T. In the formula, it is assumed that the interest rate for debt capital and the cost of debt (expected return of the debt capital providers) are the same and that the interest payments therefore equal the product of the cost of debt and the value of the debt capital. Furthermore, if one assumes that the amount of debt capital remains unchanged in the future, then the annual tax savings are constant if the income tax rate and the cost of debt capital remain the same. This leads to the following equation for calculating the present value of the interest-related tax savings: PVðTSÞ =

tcD V D = tV D : cD

ð9:19Þ

In Eq. (9.18), the unlevered enterprise value is given by the first half of the terms to the right of the equals sign, where the FCFF are discounted with the debt-free cost of capital. This is the value of a company that is entirely financed with equity and therefore reflects the company value on the basis of operating activities that are not influenced by the capital structure or the level of indebtedness. In order to determine the cost of capital used for discounting the FCFF, the beta must first be adjusted for the systematic financial risk, since beta is a measure for the overall systematic risk of the company. The total risk consists of the risk of loss arising from the business activity and the capital structure. The former is affected by the industry in which the company operates and is thus linked to corporate activity. The latter risk of loss

26 For the integration of expected insolvency costs into the APV model, see, for example, Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 315 ff.

9.4 Adjusted Present Value Model

313

results from the company’s level of indebtedness and increases with a higher debt ratio. The beta adjusted for financial risk or the unlevered (or asset) beta can be calculated under certain conditions, which will not be discussed in detail here, as follows:27 βUnlevered =

βEquity security , ½1 þ ð1 - t ÞðV D =V E Þ

ð9:20Þ

where βEquity security = beta of the equity security, t = income tax rate, and VD/VE = debt-to-equity ratio (gearing ratio). Since the denominator of the term to the right of the equals sign is greater than or equal to 1, the unlevered beta is always lower than the beta of the stock (levered beta). The unlevered beta includes only the systematic risk from business activity, while the beta of the equity security accounts for the total systematic risk (i.e. including financial risk). In order to calculate the cost of equity of a debt-free company, the unlevered beta must be used to determine the cost of equity with the CAPM: cE = r F þ MRPβUnlevered ,

ð9:21Þ

where rF= long-term risk-free interest rate, and MRP = expected market risk premium. The APV model, like the FCFF model, is a gross valuation method in which the enterprise value is determined first. In order to estimate the intrinsic share value, the market value of the interest-bearing debt capital must be subtracted from the calculated enterprise value and then divided by the number of outstanding shares. Example: Calculation of Enterprise Value Using the APV Model for a Debt-Financed Acquisition The unlisted Fitness Chain AG will be bought by a private equity company at the end of year t for a price of EUR 100 million, with EUR 90 million of the purchase price being financed with debt. The pre-tax cost of debt is 6%. The (continued)

See Hamada 1972: ‘The effect of the firm’s capital structure on the systematic risk of common stock’, p. 435 ff. For the derivation, see also Mondello 2022: Corporate Finance: Theorie und Anwendungsbeispiele, p. 184 ff. 27

314

9

Free Cash Flow Models

debt repayment plan requires annual repayments of EUR 14 million over the next 5 years. It is assumed that the remaining debt of EUR 20 million will be used to finance the business activity and will therefore not be repaid. In year t, the operating income (EBIT) is EUR 4.5 million and the revenues of the Fitness Chain AG are EUR 30 million. The return on capital is 12%. The company is expected to grow at a constant rate of 3.4% per year in perpetuity. The Fitness Chain AG has no debt capital prior to the acquisition. The beta of comparable fitness companies is 1.2. The average debt-to-equity ratio of comparable companies is 0.5. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%, while the expected market risk premium for Germany is 7%. What is the enterprise value of the Fitness Chain AG at the end of year t according to the APV model? Solution First, the unlevered value of the company should be determined. To do this, the beta adjusted for financial risk, using data relating to the peer companies, is calculated: βUnlevered =

1:2 = 0:889: ½1 þ ð1 - 0:3Þ × 0:5

The cost of capital of the unlevered company is 7.323%: cU = 1:1% þ 7% × 0:889 = 7:323%: The capital or equity reinvestment rate of 0.283 can be calculated with the perpetual constant growth rate of 3.4% and the return on capital of 12% as follows: IC =

3:4% = 0:283: 12%

The FCFF of EUR 2.25855 million for the year t can be determined as follows using the tax-adjusted EBIT and the reinvestment rate: FCFFt = EUR 4:5 million × ð1- 0:3Þ × ð1- 0:283Þ = EUR 2:25855 million: According to the one-stage model, the unlevered enterprise value is EUR 59.529 million: EVUnlevered =

EUR 2:25855 million × 1:034 = EUR 59:529 million 0:07323 - 0:034

The present values of the annual tax savings in the next 5 years and the perpetual annual tax savings thereafter can be determined as follows (in EUR million): Year t+1

Debt capital at the beginning of the year 90

Interest payment 5.40a

Tax saving 1.620b

Present value of the tax savings 1.528c

(continued)

9.4 Adjusted Present Value Model

Year t+2 t+3 t+4 t+5 Forever Total

Debt capital at the beginning of the year 76 62 48 34 20

315

Interest payment 4.56 3.72 2.88 2.04 1.20

Tax saving 1.368 1.116 0.864 0.612 0.360

Present value of the tax savings 1.218 0.937 0.684 0.457 4.484 9.308

90 × 0.06 = 5.40 5.40 × 0.3 = 1.62 c 1:620 1:06 = 1:528 a

b

To calculate the present value of the perpetual tax savings from the sixth year (t + 6) of EUR 4.484 million, the perpetual annual tax savings of EUR 0.36 million (= EUR 20 million × 0.06 × 0.3) must first be determined. If the perpetual annual tax saving of EUR 0.36 million is discounted at a discount rate of 6%, the present value of the tax savings in 5 years of EUR 6 million (= EUR 0.36 million/0.06) is obtained. To arrive at the present value of the tax benefit of EUR 4.484 million, the EUR 6 million must be discounted at 6% over 5 years [= EUR 6 million/(1.06)5]. The total present value of the tax savings is EUR 9.308 million. According to the APV model, the enterprise value of Fitness Chain AG is EUR 113.844 million: Unlevered enterprise value + Present value of tax savings = Enterprise value

EUR 59.529 million + EUR 9.308 million = EUR 68.837 million

Based on the APV model, the price of EUR 100 million paid by the private equity firm for Fitness Chain AG is too high. Due to the different way in which debt is incorporated into the calculation of enterprise value, the APV and FCFF models do not necessarily produce the same value. In the APV model, the interest-related tax savings are determined with the assistance of the available debt capital, whereas in the FCFF model the tax benefit is calculated using the target capital structure. For example, to meet the target capital structure, a high-growth company will need to borrow capital in the future. The resulting tax benefit is already included in the calculated enterprise value. If no changes in the debt ratio are expected in the future, the FCFF model is easier to apply because, in contrast to the APV model, no amount of debt capital needs to be

316

9

Free Cash Flow Models

estimated in the future. If, on the other hand, changes in the capital structure are expected—as in the case of acquisitions with an excessive amount of debt capital and a negotiated repayment plan for the debt capital—the APV model is the more appropriate valuation model.28

9.5

Summary

• In contrast to the dividend discount model, the FCFE model considers not only the dividends but all freely available cash flows that the company can pay out to its shareholders. • Free cash flow models can be used in the following cases in particular: (1) The company does not pay dividends. (2) The dividends differ significantly from the FCFE. (3) There is a correlation between the profitability of the company and the free cash flows. (4) equity valuation is from the perspective of a controlling interest. • The FCFE reflect funds that are potentially distributable to the equity providers. Assuming that depreciation and amortisation, as well as the change in non-cash working capital, are the only non-cash expenses and revenues, FCFE can be calculated as follows: Net income + depreciation and amortisation - investments in non-cash working capital - net capital expenditures + net increase in interestbearing debt. • Depending on the expected free cash flow patterns, one-stage or multi-stage valuation models can be used for equity valuation. • The one-stage FCFE model is suitable for the equity valuation of mature companies operating in a saturated market that have a long-term growth rate that does not exceed that of the economy as a whole. The intrinsic equity value is determined by dividing the FCFE of the next year by the difference between the expected return of the shareholders and the long-term growth rate. Adding the value of non-operating cash and of other non-operating assets (financed with equity or built-up cash reserves) to the intrinsic equity value and dividing this value by the number of outstanding shares gives the intrinsic share value. The growth rate applied in the calculations should not be greater than the long-term nominal growth of the economy as a whole or of the GDP. Furthermore, the difference between investments in property, plant and equipment and depreciation, that is, expansion investments, must be consistent with future growth. • The basic variant of the two-stage FCFE model assumes high constant growth in the first growth stage, which is followed by a second perpetual growth stage with a lower constant growth rate. The terminal value at the beginning of the second perpetual growth phase can be estimated with a one-stage FCFE model. The forecast FCFE must be consistent with the stability assumption of a company in the maturity phase. For example, the need for expansion investments is lower than

28

See Parrino 2013: ‘Choosing the Right Valuation Approach’, p. 267.

9.5 Summary

•

•

• •

•

•

317

in the first high-growth stage and the beta of the stock should be close to 1 (range between 0.8 and 1.2). The FCFF represent the funds that are potentially distributable to the providers of debt and equity capital. They can be calculated, for example, from the FCFE or from the operating income. Assuming that depreciation and amortisation, as well as the change in non-cash working capital, are the only non-cash expenses and revenues, FCFF can be determined as follows: EBIT × (1 - income tax rate) + depreciation and amortisation - investments in non-cash working capital - net capital expenditures. The enterprise value is determined by discounting the FCFF by the WACC. In order to estimate the intrinsic equity value, the market value of the non-operating assets, including cash and cash equivalents not required for operations, is added to the enterprise value. Then the market value of the interest-bearing debt capital and the value of non-controlling interests (minority interests) are subtracted. Finally, to obtain the intrinsic share value the calculated equity value is divided by the number of outstanding shares. In the FCFF model, the tax benefit of debt capital is included in the discount rate (WACC) and not in the FCFF, as no assumptions about future changes in debt capital are needed to calculate the FCFF. The FCFF model is suitable for companies with negative FCFE and/or unstable capital structure and high debt ratio because no assumptions about the change in debt capital are required to estimate future FCFF. However, assumptions about debt capital, such as the level of the cost of debt and the market value of interestbearing debt (or the target capital structure), are necessary to determine the WACC. If changes in debt capital are expected in the future, such as with a fixed repayment schedule for debt capital, the enterprise value can be calculated using the APV model. The enterprise value is arrived at by adding the value of the unlevered company and the present value of the interest-related tax savings. An important valuation issue that affects all cash flow models is the calculation of the terminal value, which reflects the intrinsic value at the beginning of the last perpetual constant growth stage. In many cases, the terminal value accounts for more than 75% of the intrinsic value. It is therefore important that this value is carefully estimated. If the terminal value is determined using a one-stage model, it is worth noting that it is very sensitive to small changes in the expected return and the growth rate. A sensitivity analysis can be applied to track the changes. In addition, the terminal value can also be estimated by means of a price or value multiple. Again, it is essential to understand by how much the terminal value changes when the value or the type of the multiple changes. Therefore, it is not surprising that the use of spreadsheets is widespread in valuation practice.

318

9.6

9

Free Cash Flow Models

Problems

1. Kappa AG is a mature company operating in the machinery industry. In the year t, net income is EUR 60 million. The equity capital at the beginning of the year t is EUR 600 million. The cash and cash equivalents not required for operations are EUR 200 million at the beginning of the year t and EUR 250 million at the end of the year t. The interest income from the cash and cash equivalents is EUR 20 million. The cost of equity is 12%, while the income tax rate is 30%. A total of 10 million shares are outstanding. What is the intrinsic share value of Kappa AG using the one-stage FCFE model, assuming a long-term growth rate of 3.6%? (The valuation date is the end of year t.) 2. The following data are available for Vega AG for the year t (in EUR million): Net income Depreciation and amortisation Income from cash and cash equivalents Cash and cash equivalents at the end of the year ta Equity at the end of the year t Investments in non-cash working capital Net capital expenditures Increase in interest-bearing debt capital a

1200 600 60 2000 14,000 400 2800 1800

Assumption: not operationally necessary

At the end of the previous year (i.e. end of year t - 1), cash and cash equivalents amount to EUR 1900 million and equity to EUR 13,200 million. The income tax rate is 30%. The historical beta of the stock is 1.3. The yield to maturity of 3-year and 30-year bonds of the Federal Republic of Germany are -0.6% and 1.1%, respectively. The expected market risk premium for Germany is 7%. FCFE increase over the next 3 years based on the fundamental growth rate of year t. After this first growth stage, growth falls abruptly to a perpetual constant level of 3.63%, which corresponds to the long-term nominal growth of the GDP in Germany. In the second growth stage, a beta of 1 is applied. In addition, a return on equity of 8% is expected. Vega has 500 million shares outstanding. What is the intrinsic share value of Vega according to the two-stage FCFE model? (The valuation date is the end of year t.) 3. The following data are available for Rho AG for the year t (in EUR million): Operating income (EBIT) Depreciation and amortisation Investments in non-cash working capital Net capital expenditures Book value of equity at beginning of year t

300 120 30 150 1050 (continued)

9.7 Solutions

319

Book value of equity at the end of the year t Book value of interest-bearing debt at the beginning of the year t Book value of interest-bearing debt at the end of the year t

1100 700 735

Depreciation and amortisation, as well as the change in non-cash working capital, are the only non-cash expenses and revenues. Cash and cash equivalents not required for operations amount to EUR 45 million at the beginning of the year t and EUR 50 million at the end of the year t. The market value of interest-bearing debt is EUR 750 million. The income tax rate is 30%. The WACC is 7.8%. At the end of year t, a total of 10 million shares are outstanding. What is the intrinsic share value of Rho using the one-stage FCFF model if perpetual constant growth is based on the company’s fundamentals of year t? (The valuation date is the end of year t). 4. Gamma AG is a company in the watch industry. In year t, its operating income (EBIT) is EUR 6 million and its revenues are EUR 40 million. Future company profits will be taxed at an income tax rate of 30%. The return on capital is 10%. The market value of the interest-bearing debt is EUR 65 million. It is assumed that the debt capital will remain at the same level in the future. The cost of debt is 5%. The company is also expected to grow at a constant rate of 3.6% per year in perpetuity. A total of 1 million shares are outstanding. The average adjusted beta of comparable companies in the watch industry is 1.2. The average debt-to-equity ratio of comparable companies is 0.5. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%, while the expected market risk premium for Germany is 7%. What is the intrinsic share value of Gamma according to the APV model? (The valuation date is the end of year t.)

9.7

Solutions

1. The return on equity adjusted for cash and cash equivalents not required for operations can be determined with the following equation:

ROE =

EUR 60 million - EUR 20 million × ð1 - 0:3Þ = 11:5%: EUR 600 million - EUR 200 million

The equity reinvestment rate of 0.313 can be calculated using the long-term growth rate of 3.6% and the return on equity of 11.5%, adjusted for non-operating cash and cash equivalents, as follows:

320

9

IE =

Free Cash Flow Models

g 3:6% = = 0:313: ROE 11:5%

The FCFE amount to EUR 31.602 million: FCFEt = ½EUR 60 million - EUR 20 million × ð1- 0:3Þ × ð1- 0:313Þ = EUR 31:602 million: The intrinsic equity value of EUR 389.758 million can be determined using the one-stage FCFE model as follows:

V E, t =

EUR 31:602 million × 1:036 = EUR 389:758 million: 0:12 - 0:036

With cash and cash equivalents not required for operations, the intrinsic adjusted equity value is EUR 639.758 million (= EUR 389.758 million + EUR 250 million). Dividing this value by the number of outstanding shares results in an intrinsic share value of EUR 63.98:

Pt =

EUR 639:758 million = EUR 63:98: 10 million shares

2. The return on equity adjusted for cash and cash equivalents not required for operations is 10.25%:

ROE =

EUR 1200 million - EUR 60 million × ð1 - 0:3Þ = 10:25%: EUR 13,200 million - EUR 1900 million

The reinvestment rate of equity is 0.6908: EUR 2800 million - EUR 600 million þ EUR 400 million - EUR 1800 million EUR 1200 million - EUR 60 million × ð1 - 0:3Þ = 0:6908

IE =

9.7 Solutions

321

The fundamental growth rate for year t of 7.08% can be calculated as follows: g = 0:6908 × 10:25% = 7:08%: The FCFE for year t can be determined with the following equation: FCFEt = ½EUR 1200 million - EUR 60 million × ð1- 0:3Þ × ð1- 0:6908Þ = EUR 358:05 million: The FCFE increase by 7.08% annually in the first high-growth stage, resulting in the following FCFE for the next 3 years: FCFEtþ1 = EUR 358:05 million × ð1:0708Þ1 = EUR 383:40 million, FCFEtþ2 = EUR 358:05 million × ð1:0708Þ2 = EUR 410:54 million, FCFEtþ3 = EUR 358:05 million × ð1:0708Þ3 = EUR 439:61 million: In the first high-growth stage, the expected CAPM return is 8.5%: E ðr Þ1st growth stage = - 0:6% þ 7% × 1:3 = 8:5%: In the second perpetual growth stage, the expected CAPM return is 8.1%: E ðr Þ2nd growth stage = 1:1% þ 7% × 1 = 8:1%: In the second perpetual growth phase, the reinvestment rate of equity decreases from 0.6908 to 0.4538:

IE =

3:63% = 0:4538: 8%

The FCFE of the first year of the second perpetual growth stage is EUR 804.77 million:

322

9

Free Cash Flow Models

FCFEtþ4 = ½EUR 1200 million - EUR 60 million × ð1- 0:3Þ × ð1:0708Þ3 × 1:0363 × ð1- 0:4538Þ = EUR 804:77 million: The terminal value at the beginning of the second perpetual growth stage of EUR 18,004 million can be determined with the one-stage FCFE model as follows:

V E, tþ3 =

EUR 804:77 million = EUR 18, 004 million: 0:081 - 0:0363

The intrinsic equity value is EUR 15,142 million:

V E, t = þ

EUR 383:40 million EUR 410:54 million þ ð1:085Þ1 ð1:085Þ2

EUR 439:61 million þ EUR 18, 004 million = EUR 15, 142 million: ð1:085Þ3

With the amount of cash and cash equivalents not required for operations, the intrinsic adjusted equity value is EUR 17,142 million (= EUR 15,142 million + EUR 2000 million). Dividing this value by the number of outstanding shares leads to an intrinsic share value of EUR 34.28:

Pt =

EUR 17,142 million = EUR 34:28: 500 million shares

3. The capital reinvestment rate of 28.57% can be calculated as follows:

IC =

EUR 150 million - EUR 120 million þ EUR 30 million = 0:2857: EUR 300 million × ð1 - 0:3Þ

The return on capital is 12.32%:

ROC =

EUR 300 million × ð1 - 0:3Þ = 12:32%: EUR 700 million þ EUR 1050 million - EUR 45 million

The fundamental FCFF growth rate of 3.52% is obtained by multiplying the capital reinvestment rate by the return on capital:

9.7 Solutions

323

g = 0:2857 × 12:32% = 3:52%: The FCFF of EUR 150 million can be calculated by adding depreciation and amortisation to the after-tax operating income and subtracting from this value the investments in non-cash working capital and net capital expenditures: FCFFt = EUR 300 million × ð1- 0:3Þ þ EUR 120 million - EUR 30 million - EUR 150 million = EUR 150 million: Alternatively, the FCFF of EUR 150 million can be determined with the operating income after tax and the capital reinvestment rate as follows: FCFFt = ½EUR 300 million × ð1- 0:3Þ × ð1- 0:2857Þ = EUR 150 million: According to the one-stage FCFF model, the enterprise value is EUR 3628.04 million:

EVt =

EUR 150 million × 1:0352 = EUR 3628:04 million: 0:078 - 0:0352

The value of the non-operating cash and cash equivalents of EUR 50 million must be added to the enterprise value of EUR 3628.04 million, and the market value of the interest-bearing debt of EUR 750 million must be deducted from it, resulting in an intrinsic equity value of EUR 2928.04 million. To determine the intrinsic share value of Rho of EUR 292.80, the intrinsic equity value must be divided by the number of outstanding shares:

Pt =

EUR 2928:04 million = EUR 292:80: 10 million shares

4. The unlevered beta is 0.889 and can be calculated as follows: βUnlevered =

1:2 = 0:889: ½1 þ ð1 - 0:3Þ × 0:5

324

9

Free Cash Flow Models

The cost of capital of the debt-free company is 7.323%: cU = 1:1% þ 7% × 0:889 = 7:323%: The capital or equity reinvestment rate of 0.36 can be calculated with the perpetual constant growth rate of 3.6% and the return on capital of 10% as follows:

IC =

g 3:6% = 0:36: = ROC 10%

The FCFF of EUR 2.688 million for the year t can be determined with the following equation: FCFFt = EUR 6 million × ð1- 0:3Þ × ð1- 0:36Þ = EUR 2:688 million: Applying a one-stage model leads to an unlevered enterprise value of EUR 74.799 million:

EVUnlevered,t =

EUR 2:688 million × 1:036 = EUR 74:799 million: 0:07323 - 0:036

The present value of the interest-related tax savings of EUR 19.5 million can be determined as follows: PVðTSÞt = 0:3 × EUR 65 million = EUR 19:5 million: According to the APV model, the enterprise value consists of the unlevered enterprise value and the present value of the interest-related tax savings, resulting in an enterprise value of EUR 94.299 million: EVt = EUR 74:799 million þ EUR 19:5 million = EUR 94:299 million: The intrinsic equity value of EUR 29.299 million can be determined by subtracting the market value of the interest-bearing debt of EUR 65 million from the enterprise value of EUR 94.299 million. To calculate the intrinsic share value

References

325

of EUR 29.30, the intrinsic equity value must be divided by the number of outstanding shares:

Pt =

EUR 29:299 million = EUR 29:30: 1 million shares

References Barker, R.: Determining value: valuation models and financial statements, Harlow (2001) Bodie, Z., Kane, A., Marcus, A.J.: Investments, 8th edn, New York (2009) Courtois, Y., Lai, G.C., Peterson Drake, P.: Cost of capital. In: Clayman, M.R., Fridson, M.S., Troughton, G.H. (eds.) Corporate finance: a practical approach, pp. 127–169, Hoboken (2008) Damodaran, A.: Investment valuation: tools and techniques for determining the value of any asset, 3rd edn, Hoboken (2012) Frykman, D., Tolleryd, J.: Corporate valuation: an easy guide to measuring value, Harlow (2003) Hamada, R.S.: The effect of the firm’s capital structure on the systematic risk of common stocks. J. Finance. 27(2), 435–452 (1972) Koller, T., Goedhart, M., Wessels, D.: Valuation: measuring and managing the value of companies, 5th edn, Hoboken (2010) Mercedes-Benz Group: Geschäftsbericht 2016. Stuttgart (2017) Mondello, E.: Aktienbewertung: Theorie und Anwendungsbeispiele, 2nd edn, Wiesbaden (2017) Mondello, E.: Corporate Finance: Theorie und Anwendungsbeispiele, Wiesbaden (2022) Myers, S.C.: Interactions of corporate financing and investment decisions – implications for capital budgeting. J. Finance. 29(1), 1–25 (1974) Parrino, R.: Choosing the right valuation approach. In: Larrabee, D.T., Voss, J.A. (eds.) Valuation techniques, pp. 259–278, Hoboken (2013) Schwetzler, B., Darijtschuk, N.: Unternehmensbewertung mit Hilfe der DCF-Methode – eine Anmerkung zum “Zirkularitätsproblem”. Zeitschrift für Betriebswirtschaft. 69(3), 295–318 (1999)

Multiples

10.1

10

Introduction

The intrinsic value of an equity security is determined on the basis of a cash flow model using the growth rate of the cash flows and the expected return. Relative valuation analysis, on the other hand, assesses the value of an equity security against a benchmark employing a multiple. This approach makes it possible to examine whether the security is valued correctly relative to the stocks of comparable companies. The fundamental economic principle of the comparables method is based on the law of one price, according to which two identical assets are traded at the same price.1 Essentially, a distinction is made between price multiples and value multiples. With a price multiple, the price of an equity security is set in relation to a financial variable that has a significant influence on the share price. The variable chosen for this purpose is, for example, the earnings or the book value per share. The intuition behind value multiples is similar. Investors evaluate the market value of an entire company relative to the amount of earnings before interest, taxes, depreciation and amortisation (EBITDA), sales, operating cash flow, or free cash flow to firm. Thus, the enterprise value is considered in relation to a financial variable that affects its value. The price and value multiples can be used to determine whether the stock is correctly valued in the market. The use of multiples in equity valuation assumes that capital markets are informationally efficient. However, the efficient market hypothesis applies only to the peer group. By contrast, an inefficient capital market is assumed for the company to be valued. Only then it is possible to identify a mispriced stock with the assistance of peer companies that are correctly valued. Moreover, the inefficiency in this

1

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 388.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_10

327

328

10 Multiples

Fig. 10.1 Common industry multiples (Source: Based on Hasler 2013: Aktien richtig bewerten, p. 286)

sub-sector of the capital market must gradually disappear so that an alpha can be achieved by detecting the mispriced stock by means of multiples. A multiple reflects the price an investor has to pay to buy a certain valuation unit of the company such as earnings. At the beginning of equity valuation, the accepted multiples consisted only of the price-to-earnings ratio (P/E) and the price-to-book ratio (P/B).2 In the meantime, the number of multiples has increased considerably. In addition to the P/E ratio and the P/B ratio, the price/earnings-to-growth ratio (PEG) and the price-to-cash-flow ratio (P/C), as well as enterprise value-based multiples such as the enterprise value EBITDA ratio (EV/EBITDA) and enterprise value sales ratio (EV/S), among others, are used today. Figure 10.1 presents an overview of the most common industry-specific multiples. Multiples can also be calculated with non-financial variables if they affect the share price. For mobile telecommunications, for example, the number of subscribers is defined as the relevant performance indicator. For hospitals and hotels, the relevant performance indicator is the number of beds, while for social media companies, it is the number of registered users. 2

Graham and Dodd described the P/E ratio and the P/B ratio in equity valuation as early as 1934. See Graham and Dodd 1934: Security Analysis, p. 351 ff.

10.2

Price-to-Earnings Ratio

329

The following section examines the price-to-earnings ratio, the price/earnings-togrowth ratio, and the price-to-book ratio as examples of price multiples. It goes on to discuss the enterprise value EBITDA ratio, which belongs to the value multiples.

10.2

Price-to-Earnings Ratio

10.2.1 Definition The share price is arrived at by dividing the market value of equity by the number of outstanding shares. For example, a two-for-one stock split doubles the number of shares, which halves the share price. Since the price of an equity security is affected by the number of outstanding shares, the share prices of different companies are not easily comparable. Nevertheless, in order to be able to compare the share prices of similar companies, the prices must be standardised or brought to a comparable basis with the assistance of a multiple. For example, the P/E ratio can be calculated by considering the price of a share in relation to the earnings per share: P Share price = , E Annual EPS

ð10:1Þ

where EPS = earnings per share. The P/E ratio is the most widely used price multiple.3 The numerator of the multiple contains the market value of the equity per share or the traded share price. The denominator, on the other hand, includes the earnings per share, which reflects the profitability of the equity as the earnings per share can be used together with the book value per share to calculate the return on equity. While the determination of the numerator, the traded share price, does not pose any problems—at least for listed companies—the setting of the denominator is a challenge. Various earnings measures can be used, such as the earnings of the past business year, the earnings of the past 12 months, or the expected future earnings. For valuation purposes, diluted recurring earnings are usually used. The following are the three main variants of the P/E ratio: • The current P/E ratio, which is based on the earnings of the past financial year (or the most recently published annual result) • The trailing P/E ratio, which is based on the earnings of the last 12 months (LTM or last 12 months)

3 A survey conducted by Bank of America Merrill Lynch in 2012 indicates that 81% of institutional investors took the P/E ratio into account in their equity analysis. Thus, the P/E ratio is the most utilised valuation indicator. See Bank of America Merrill Lynch 2012: Annual Institutional Factor Survey, p. 18.

330

10 Multiples

• The forward P/E ratio, which refers to the expected earnings of the next 12 months (NTM or next 12 months) Share prices are not determined by past data, but by expectations. Therefore, whenever possible, the forward P/E ratio is preferable to the trailing P/E ratio. The basic idea behind using multiples is that the price of an equity security should not be evaluated in isolation. Therefore, the share price must be considered in relation to another variable to determine how much one is willing to pay for a unit of, say, earnings or book value. For example, a P/E ratio of 15 means that 15 units of a currency (e.g. EUR 15) are needed to buy one currency unit of earnings (e.g. EUR 1).4 This standardisation makes it possible to compare share prices with each other. If one stock has a P/E ratio of 20 and another stock has a ratio of 15, then one pays more for one unit of earnings (EUR 20 instead of EUR 15 for a profit of EUR 1). If the two securities are comparable—that is, they have similar growth, risk, and expected cash flows—the stock with the P/E ratio of 15 is relatively undervalued compared to the stock with the higher price multiple of 20. Relative valuation analysis can be used to determine whether the equity security is correctly valued in relation to stocks of one or a group of similar companies. The assumption of this analysis is that the stocks of the peer companies in the market are correctly valued on average. One of the advantages of the P/E ratio is that the company’s profitability as the main driver of the share price is included in the multiple in the form of earnings per share. Moreover, the P/E ratio is an accepted valuation indicator among market participants when buying and selling stocks and is accordingly widely used. Empirical studies conclude that the relative level of the P/E ratio has an influence on the long-term average stock return.5 The disadvantages of using the P/E ratio are due primarily to the characteristics of earnings. For example, earnings per share can be zero, negative, or very small compared to the share price, with the result that the P/E ratio is not meaningful or makes no economic sense. The amount of earnings reported is affected by the interpretation and application of the accounting standards used. Earnings per share can also be very volatile, which is regularly the case, especially with high-growth and high-risk companies. Furthermore, different levels of gearing are not explicitly taken into account in the P/E ratio, since in the price multiple the numerator consists of the share price and the denominator of earnings after taxes and interest on debt per share.

4

Another interpretation of a P/E ratio of 15 is that it takes 15 years to pay back the price paid for the stock with the earnings. Accordingly, a lower P/E ratio is preferable to a higher P/E ratio because the share price paid for one unit of earnings is lower and the payback period is shorter. 5 Equity securities with a value bias are characterised, among other things, by a below-average P/E ratio and have a consistently higher risk-adjusted return over longer periods of time than stocks with a high P/E ratio (stocks with a growth bias). See Fama and French 1998: ‘Value versus growth: the international evidence’, p. 1975 ff.

10.2

Price-to-Earnings Ratio

331

Example: Comparables Method Delta stock trades at a price per share of EUR 50 and has had earnings per share of EUR 2.50 for the last 12 months. The comparable company Gamma, which has similar growth, risk, and expected cash flows, has a trailing P/E ratio of 23. 1. Is the Delta stock correctly valued relative to the Gamma stock? 2. What is the share price of Delta if it is assumed that the Gamma stock is correctly valued or that both equity securities are trading at the same trailing P/E ratio of 23? Solution to 1 The trailing P/E ratio of the Delta stock is 20 and can be calculated as follows: Trailing P=E of Delta =

EUR 50 = 20: EUR 2:50

Compared to the Gamma stock’s trailing P/E ratio of 23, the Delta stock’s lower trailing P/E ratio of 20 leads to the conclusion that the Delta stock is undervalued compared to the stock of the benchmark company Gamma. Delta’s share price should be higher so that both securities trade at the same P/E ratio of 23. Solution to 2 Assuming that the trailing P/E ratio of Delta is equal to that of Gamma and that the Gamma stock is correctly valued, an intrinsic value of EUR 57.50 can be calculated for the Delta shares: PDelta = 23 × EUR 2:50 = EUR 57:50: If the calculated share price of EUR 57.50 is compared with the traded share price of EUR 50, the Delta stock is again undervalued. The example demonstrates that a comparison of the multiples, just like the calculation of the intrinsic value, leads to the same conclusion, namely that the Delta stock is undervalued. A multiple can also be determined using forecast fundamentals of the company, such as the expected growth rate, risk, and projected cash flows, which significantly influence the firm value. They can be converted into a multiple by means of a cash flow model. Thus, one can determine the intrinsic equity value by applying a cash flow model and then convert it into a price multiple by dividing the intrinsic equity value by the expected earnings. If, for example, the intrinsic equity value is EUR 45 million and the expected earnings are EUR 3 million, the result is a forward P/E ratio of 15. Comparing the price multiple calculated in this way with the P/E ratio of

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Mulple e.g. price-to-earnings rao (P/E)

Method based on forecast fundamentals

Comparables method

Comparable companies: P/E of A = 10 P/E of B = 13 P/E of C = 16 ø P/E = (10 + 13 + 16)/3 = 13 Intrinsic share value = jusfied P/E x EPS

Intrinsic share value = ø P/E x EPS

Is the equity security correctly valued? - Comparison between intrinsic share value and traded share price or - between jusfied P/E and P/E of the stock

Is the equity security correctly valued? - Comparison between intrinsic share value and traded share price or - between ø P/E and P/E of the stock

Fig. 10.2 Calculation of the price-to-earnings ratio and intrinsic share value with the method based on forecast fundamentals and the comparables method (Source: Own illustration)

the equity security (traded share price divided by the expected earnings per share), it is possible to determine whether the security is correctly valued in the market. If the forward P/E ratio calculated with the cash flow model is higher (lower) than the forward P/E ratio observable in the market, the stock is undervalued (overvalued). In addition, the intrinsic share value can also be estimated by multiplying the forward P/E ratio of 15 by the expected earnings per share of the company. A comparison with the traded share price makes it possible to assess whether the security is correctly valued. Figure 10.2 contrasts the two methods for determining multiples, that is, the method based on forecast fundamentals and comparables method, which are described in more detail below.

10.2.2 P/E Ratio Based on Forecast Fundamentals By calculating the P/E ratio using a cash flow model, it is possible to determine the share price that must be paid for one unit of earnings. This incorporates the forecast fundamentals of the company, such as profitability, growth, and risk.6 The one-stage

6

See Barker 2001: Determining Value: Valuation Models and Financial Statements, p. 54 ff.

10.2

Price-to-Earnings Ratio

333

dividend discount model (Gordon–Growth model) is applied below to calculate the justified price multiple. A one-stage model is appropriate for mature companies in a saturated market. For growth stocks, multiples should be determined using a multistage dividend discount model.7 In a one-stage dividend discount model, the intrinsic share value can be calculated as follows [E(r) > g]:8 P0 =

D0 ð1 þ gÞ D1 = , E ðr Þ - g E ðr Þ - g

ð10:2Þ

where D1 = expected dividend per share in period 1, E(r) = expected return of shareholders, and g = perpetual constant growth rate of dividends. Dividing both sides of the equals sign by the expected earnings per share for the next year (E1) yields the justified forward (or leading) P/E ratio [E(r) > g]: D =E 1-b P0 , = 1 1 = E 1 E ðr Þ - g E ðr Þ - g

ð10:3Þ

where b = retention rate of earnings or 1 - b = payout ratio of earnings. The valuation model can also be used to determine the justified trailing P/E ratio by dividing Eq. (10.2) on both sides of the equals sign by the last 12 months’ earnings per share [E(r) > g]: D ð1 þ gÞ=E 0 ð1 - bÞð1 þ gÞ P0 = 0 = : E0 E ðr Þ - g E ðr Þ - g

ð10:4Þ

The justified forward and trailing P/E ratios have a positive relationship with the earnings payout ratio and growth rate and a negative relationship with the expected return, which reflects the risk of the company. The influence of one of these three fundamental factors on the level of the multiple is examined by changing one factor while all other factors remain unchanged. A higher intrinsic share value and thus a higher P/E ratio, for example, is obtained either with a higher earnings payout ratio, a higher growth rate or a lower expected return.9

7

For the application of the two-stage dividend discount model for growth stocks, see, for example, Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 453 ff. 8 See Sect. 8.4. 9 In this analysis, it should be noted that a higher earnings payout ratio results in a lower earnings retention rate, and thus a lower growth rate (g = b ROE). Therefore, this statement is only valid if

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For a given growth rate, the higher (lower) the earnings payout ratio, the higher (lower) the P/E ratio. This relationship can be explained by the fact that companies with low investment needs have a higher earnings payout ratio and hence a higher P/E ratio than firms with high capital expenditures. Furthermore, the lower the difference between the expected return and the growth rate, the higher the P/E ratio. If this difference is negative, the result is a negative P/E ratio, which does not make economic sense. Example: Calculation of the Justified Trailing P/E Ratio and of the Intrinsic Share Value Using the Deutsche Telekom Stock The following information is available for the listed Deutsche Telekom stock at the beginning of 2022 (Source: Refinitiv Eikon): Dividend per share (for 2021) Earnings per share (for 2021) Share price Historical beta

EUR 0.64 EUR 0.87 EUR 16.55 0.70

Based on the 2021 fundamentals, it is assumed the company will grow in perpetuity. The return on equity is 10.6%. The yield to maturity on 30-year bonds of the Federal Republic of Germany is 1.1%. The expected market risk premium for Germany is 7%. What is the justified trailing P/E ratio and the intrinsic share value of Deutsche Telekom? Solution In order to estimate the justified trailing P/E ratio, the adjusted beta of the stock must first be calculated, followed by the expected CAPM return: βAdjusted = 0:333 þ 0:667 × 0:70 = 0:80, Eðr Þ = 1:1% þ 7% × 0:80 = 6:70%: The earnings payout ratio is 0.7356 (= EUR 0.64/EUR 0.87). The fundamental growth rate is 2.80% and can be determined as follows: g = ð1- 0:7356Þ × 10:6% = 2:80%: The justified trailing P/E ratio of 19.39 can be calculated as follows: (continued)

one fundamental factor is changed and the other fundamental factors remain the same. For the fundamental earnings growth rate, see Sect. 8.3.

10.2

Price-to-Earnings Ratio

335

P0 0:7356 × 1:028 = = 19:39: E0 0:067 - 0:028 The intrinsic share value of EUR 16.87 is arrived at by multiplying the justified trailing P/E ratio of 19.39 by the earnings per share of EUR 0.87: P0 = 19:39 × EUR 0:87 = EUR 16:87: The intrinsic share value of EUR 16.87 is higher than the traded share price of EUR 16.55. The Deutsche Telekom stock, therefore, appears to be undervalued by approximately 2%. The same conclusion is reached when the trailing P/E ratio of Deutsche Telekom stock of 19.02 (= EUR 16.55/ EUR 0.87) is compared with the justified trailing P/E ratio of 19.39.

10.2.3 P/E Ratio Based on Comparable Companies The current P/E ratio of the equity security can be compared with the P/E ratio of stocks from benchmark companies to determine whether the security is correctly valued. For this purpose, the benchmark (or peer) firms must be identified.10 Common valuation practice defines a benchmark company as a company that operates in the same industry or sector. This is not the most appropriate approach for identifying peer companies as it does not take into account differences in fundamentals. A benchmark company must have expected cash flows, growth prospects, and risk that are similar to those of the firm being valued. A stock has the same price as the stock of an identical peer company if the forecast fundamentals such as cash flows, growth rate, and expected return are the same. This definition does not make reference to the affiliation of equity security to an industry or sector. Accordingly, a machinery industry stock can be compared to equity securities in other industries, such as health care, as long as the expected cash flows, growth prospect, and risk are the same. Nevertheless, analysts use comparable companies that operate in the same industry or sector as a benchmark. The implicit assumption here is that stocks in the same industry sector have the same risk, growth, and cash flow patterns and are therefore comparable to each other.11 If there are a large number of peer companies, other criteria such as similar company size (market capitalisation) or similar revenues are usually used in order to reduce the number of comparable firms. If a company is compared with the market leader in the industry, care should be taken that this security can be traded at a valuation premium due to its outstanding 10

See Pinto et al. 2010: Equity Asset Valuation, p. 279. See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 51.

11

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10 Multiples

market position. In addition, the number of comparable companies in a specific country such as Germany or Switzerland may be relatively small or, in extreme cases, no corresponding company may be listed on the home stock exchange. In such a case, foreign companies are also included in the peer group. As a result of different accounting standards, such as IFRS, US-GAAP, and other national accounting standards, the valuation-related figures such as earnings have to be adjusted so that the multiples are comparable. Standardised classification systems such as the Global Industry Classification System (GICS) from Standard & Poor’s and MSCI Barra12 or the Industrial Classification Benchmark (ICB) from the Dow Jones and FTSE13 are used to determine the peer group. This has the advantage that there is no need for a subjective evaluation of the peer group to which stocks belong. The price multiple is calculated both for the company being valued and for each comparable company. An average value or benchmark multiple is then determined. To evaluate an individual equity security, the P/E ratio of the stock is compared with the benchmark P/E ratio. In addition, it is assessed whether the forecast fundamentals such as growth, risk, and cash flows make it possible to explain the difference between the two price multiples. For example, if a stock has a P/E ratio of 12 and the P/E ratio of the peer companies is 18, the stock may still be correctly valued if the difference can be explained by the forecast fundamentals such as lower growth and/or higher risk than the benchmark companies. In the event that the differences between the two multiples cannot be explained by the fundamentals, the stock appears undervalued relative to the benchmark.14 The following example demonstrates how the stock of Mercedes-Benz Group is evaluated using the comparables method with the P/E ratio. Example: Relative Valuation Analysis of the Mercedes-Benz Group Stock Based on the Comparables Method Using the Price-to-Earnings Ratio For the ‘automobile manufacturers’ (GICS) sub-sector, the following global peer companies are given in order of market capitalisation (excluding Tesla Inc. due to negative trailing P/E ratio) with the corresponding trailing P/E ratios, expected annual earnings growth rates for the next 4 years, and betas as at the end of October 2017 (Source: Refinitiv Eikon): Company Toyota Motor Corp Volkswagen Mercedes-Benz Group General Motors Co BMW

Trailing P/E ratio 11.2 8.7 7.4 7.2 7.4

Earnings growth rate 6.3% 33.3% 3.8% -2.3% 2.3%

Beta 1.16 1.57 1.54 1.59 1.43

(continued) 12

See http://www.msci.com/products/indices/sector/gics. See https://www.ftserussell.com/data/industry-classification-benchmark-icb. 14 See Martin 2013: ‘Traditional Equity Valuation Methods’, p. 164 ff. 13

10.2

Price-to-Earnings Ratio

Company SAIC Motor Corp Honda Motor Co Nissan Motor Co Audi Hyundai Renault Fiat Chrysler Automobiles Suzuki Motor Corp Peugeot SA Arithmetic mean Median

337

Trailing P/E ratio 11.1 9.5 6.4 12.5 9.7 5.3 7.9 14.0 11.1 9.2 9.1

Earnings growth rate 7.9% 4.5% 1.3% -29.3% 8.4% 7.7% 23.1% 8.9% 2.7% 5.6% 5.4%

Beta 0.70 1.29 1.08 0.40 1.03 1.81 1.36 1.08 1.79 1.27 1.33

The following questions must be answered: 1. Is the arithmetic mean or the median the better method to calculate, for example, the benchmark value for the trailing P/E ratio? 2. Is the Mercedes-Benz Group stock correctly valued compared to the benchmark? (The median should be taken for the analysis.) Solution to 1 The median represents the middle of all P/E values, with half of the values below and half above the median. Unlike the arithmetic mean, the median is not affected by outliers in the data, and therefore the median is more suitable for determining the benchmark value. However, there are no outliers in the present example. Therefore, the median and the arithmetic mean are close to each other and both can be used for the relative value analysis. Solution to 2 Without taking differences in fundamentals into account, the MercedesBenz Group stock is undervalued, as the P/E ratio of 7.4 is lower than the benchmark P/E ratio of 9.1. If the differences in fundamentals are included in the analysis, the stock no longer appears undervalued, since the earnings growth rate of 3.8% is lower than the median of 5.4% and the beta of the automobile stock of 1.54 is higher than the median of 1.33. A lower earnings growth rate and higher risk imply a lower share price, and therefore the conclusion that the security is undervalued cannot be justified. To assess the value of the stock, an average of the stock’s past P/E ratios can also be used as a benchmark, as long as the company’s fundamentals have not changed significantly over time. The relative valuation method applied in this way assumes

338

10 Multiples

that the P/E ratio of the stock converges to its own historical average.15 However, a company may have undergone such a transformation process that a comparison between the current and historical multiples does not make sense. For example, two decades ago, European telecommunications companies were local monopolies with a single business segment. Today, they offer a variety of products and services both domestically and abroad. Finally, the P/E ratio of the stock can be related to the P/E ratio of an equity index (e.g. DAX 40 for large-cap German stocks or SMI for largecap Swiss stocks). The relative P/E ratio calculated in this way enables an assessment of whether the equity security is correctly valued compared to the market as a whole.16

10.3

Price/Earnings-to-Growth Ratio

A multiple can be adjusted by its most important fundamental factor. This factor can be determined by running a regression between the multiple as the dependent variable and all fundamental factors as the independent variables. The fundamental factor that statistically best explains the multiple (i.e. has the highest t-statistic) is the dominant variable. Studies conclude that the expected earnings growth rate has the highest impact on the P/E ratio.17 The price/earnings-to-growth ratio (PEG) is the P/E ratio divided by the expected earnings growth rate multiplied by 100: PEG =

P=E , g × 100

ð10:5Þ

where P/E = price-to-earnings ratio, and g = expected earnings growth rate. For example, if the P/E ratio is 15 and the expected earnings growth rate is 5%, the result is a PEG ratio of 3 [= 15/(0.05 × 100)]. The PEG ratio reflects the P/E ratio of the stock for one percentage point of expected earnings. The price multiple calculated in this way can be compared with multiples of companies in the same industry. Equity securities with a low PEG ratio are more attractive than securities with a high PEG ratio, all else being equal. Such stocks have a low P/E ratio and a high expected earnings growth rate. Accordingly, the PEG ratio can be used to combine an investment strategy that is based on a value bias (undervalued stock due to a low P/E ratio) with an investment strategy that is based on a growth bias. As an 15

See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 48. 16 See Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 474 ff. 17 See, for example, Fairfield 1994: ‘P/E, P/B and the present value of future dividends’, p. 30.

10.3

Price/Earnings-to-Growth Ratio

339

indicator of an attractive equity security, a PEG ratio of 1 or less than 1 is often used in valuation practice because the P/E ratio is lower than the expected earnings growth rate multiplied by 100.18 This equity strategy is known as growth at a reasonable price (GARP). The PEG ratio should be calculated with the trailing P/E ratio (or with the current P/E ratio), which takes into account the earnings of the last 12 months, and not with the forward P/E ratio; otherwise, the earnings for the next period are counted twice, which leads to a PEG ratio that is too low. The PEG ratio should be calculated consistently and in the same way for all peer companies and the company to be valued. For example, the same time period for the expected earnings growth rate (e.g. 3 or 5 years) should be applied for all stocks. The growth rates should also be obtained from the same data source. For example, analysts’ consensus forecasts of future earnings can be found in databases such as Bloomberg. Alternatively, growth rates can be estimated using historical data or company’s fundamentals.19 Example: Relative Valuation Analysis of the Mercedes-Benz Group Stock Based on the Comparables Method Using the Price/Earnings-to-Growth Ratio For the ‘automobile manufacturers’ sub-sector (GICS), the following global peer companies are given in order of market capitalisation (excluding Tesla Inc. due to negative trailing P/E ratio) with the corresponding trailing P/E ratios, expected annual earnings growth rates for the next 4 years, PEG ratios, and betas as at the end of October 2017 (Source: Refinitiv Eikon): Company Toyota Motor Corp Volkswagen Mercedes-Benz Group General Motors Co BMW SAIC Motor Corp Honda Motor Co Nissan Motor Co Audi Hyundai Renault Fiat Chrysler Automobiles

Trailing P/E ratio 11.2 8.7 7.4

Expected earnings growth rate 6.3% 33.3% 3.8%

PEG ratio 1.8 0.3 1.9

Beta 1.16 1.57 1.54

7.2 7.4 11.1 9.5 6.4 12.5 9.7 5.3 7.9

-2.3% 2.3% 7.9% 4.5% 1.3% -29.3% 8.4% 7.7% 23.1%

n/a 3.2 1.4 2.1 4.9 n/a 1.2 0.7 0.3

1.59 1.43 0.70 1.29 1.08 0.40 1.03 1.81 1.36

(continued) 18 19

See, for example, Glenn 2011: How to Value Shares and Outperform the Market, p. 35. See Sect. 8.3.

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Company Suzuki Motor Corp Peugeot SA Arithmetic mean Median

Trailing P/E ratio 14.0 11.1 9.2 9.1

Expected earnings growth rate 8.9% 2.7% 5.6% 5.4%

PEG ratio 1.6 4.1 2.0 1.7

Beta 1.08 1.79 1.27 1.33

Is the Mercedes-Benz Group stock correctly valued based on the comparables method using the PEG ratio? (The median should be taken for the analysis.) Solution Compared to the benchmark P/E ratio of 9.1, the Mercedes-Benz Group stock is undervalued, with a P/E ratio of 7.4. This conclusion assumes that all stocks have the same expected earnings growth rate, cash flow pattern, and risk. The PEG ratio can be used to incorporate the expected earnings growth rate into the analysis. The P/E ratio of the Mercedes-Benz Group stock adjusted for the expected earnings growth rate can be calculated with the benchmark PEG ratio of 1.7 and the expected earnings growth rate of the automobile stock of 3.8% as follows: Adjusted P/E ratio of the Mercedes-Benz Group stock = 1.7×3.8 = 6.5. If the adjusted P/E ratio of 6.5 is compared with the traded P/E ratio of the Mercedes-Benz Group stock of 7.4, the security no longer appears undervalued but overvalued. However, this conclusion applies only if all equity securities have the same risk. Moreover, this analysis assumes a linear relationship between the P/E ratio and the expected earnings growth rate. A cash flow model can be used to identify the company’s fundamentals that have an impact on the PEG ratio. The intrinsic share value can be estimated with a one-stage dividend discount model as follows [E(r) > g]:20 P0 =

D 0 ð 1 þ gÞ : E ðr Þ - g

The dividend per share (D0) is equal to the earnings per share (E0) multiplied by the earnings payout ratio or by 1 minus the earnings retention rate (1 - b), which leads to the following price equation [E(r) > g]: P0 =

20

See Sect. 8.4.

E 0 ð 1 - b Þ ð 1 þ gÞ : E ðr Þ - g

10.3

Price/Earnings-to-Growth Ratio

341

If both sides of the equals sign are divided by the earnings per share (E0) and then by the expected earnings growth rate (g) multiplied by 100, the following formula for the justified PEG ratio is obtained: PEG =

ð 1 - bÞ × ð 1 þ g Þ : g × 100 × ½Eðr Þ - g

ð10:6Þ

The net impact on the price multiple depends on the corresponding level of the growth rate. However, the expected growth rate cannot exceed the expected return, as a negative P/E ratio makes no economic sense.21 Moreover, there is a positive relationship of the multiple with the earnings payout ratio, while the relationship with the expected return or risk (beta) is negative (if all other factors remain unchanged). Example: Calculation of the Justified Price/Earnings-to-Growth Ratio Using the Deutsche Telekom AG Stock The following information is available for the listed Deutsche Telekom stock at the beginning of 2022 (Source: Refinitiv Eikon): Dividend per share (for 2021) Earnings per share (for 2021) Share price Historical beta

EUR 0.64 EUR 0.87 EUR 16.55 0.70

The company is assumed to grow in perpetuity based on its 2021 fundamentals. The return on equity is 10.6%. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%. The expected market risk premium for Germany is 7%. What is the justified PEG ratio of the Deutsche Telekom stock and is the stock correctly valued? Solution The adjusted beta of 0.80 and the expected CAPM return of 6.70% can be calculated as follows: βAdjusted = 0:333 þ 0:667 × 0:70 = 0:80, Eðr Þ = 1:1% þ 7% × 0:80 = 6:70%: The earnings payout ratio is 0.7356 (= EUR 0.64/EUR 0.87), while the fundamental earnings growth rate is 2.8% [= (1 - 0.7356) × 10.6%]. The justified PEG ratio of 6.92 can be determined with the following equation: (continued)

21

See Sect. 10.2.1.

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10 Multiples

PEG =

0:7356 × 1:028 = 6:92: 0:028 × 100 × ð0:067 - 0:028Þ

If the stock’s trailing P/E ratio of 19.02 (= EUR 16.55/EUR 0.87) is divided by the long-term expected earnings growth rate of 2.8% multiplied by 100, the result is a PEG ratio of 6.79. Accordingly, the Deutsche Telekom stock appears to be undervalued based on the PEG ratio.

10.4

Price-to-Book Ratio

10.4.1 Definition The price-to-book (P/B) ratio is an important price multiple and is widely used in valuation practice.22 To calculate the P/B ratio, the traded share price is divided by the book value per share: P Share price = , B Book value per share

ð10:7Þ

where Book value per share =

Book value of equity : Number of shares outstanding

The P/B ratio is a price multiple because the share price is divided by the book value per share, which is an equity measure. The term ‘book value’ in accounting refers to the value of assets and liabilities in the balance sheet. The book value of equity can be obtained by deducting the book value of liabilities from the book value of assets. In order to calculate the price multiple, any non-controlling interests (minority interests) must be removed from the book value of the equity because the equity securities are held by the shareholders of the parent company. The book value of the equity can then be divided by the number of outstanding shares. For example, at the end of December 2021, Mercedes-Benz Group AG had an equity book value (excluding non-controlling interests) of EUR 71,951 million and outstanding shares of 1069.837 million, resulting in a book value of EUR 67.25 per share:23

22

A survey by Bank of America Merrill Lynch in 2012 indicates that 53% of the institutional investors surveyed use the P/B ratio in equity analysis. See Bank of America Merrill Lynch 2012: Annual Institutional Factor Survey, p. 18. 23 See Mercedes-Benz Group 2022: Annual Report 2021, p. 182 ff.

10.4

Price-to-Book Ratio

343

Book value per share =

EUR 71,951 million = EUR 67:25: 1069:837 million shares

The Mercedes-Benz Group stock trades at a price per share of EUR 67.59 at the end of December 2021, leading to a P/B ratio of 1.01: P EUR 67:59 = = 1:01: B EUR 67:25 If the company has different classes of shares—for example, ordinary shares and preference shares—the price of each class of shares may be different and it is not clear how the book value of equity should be divided between the different classes of shares. Nevertheless, a P/B ratio can be determined for all share classes by dividing the total market value of equity—that is, the number of shares in the different categories multiplied by the corresponding share price—by the book value of equity: P Market value of equity = : B Book value of equity

ð10:8Þ

For example, if the profitability of the invested assets increases, the market value of the assets or equity increases (the corresponding book values, however, remain the same), which results in a higher P/B ratio. Thus, there is a positive relationship between earnings or profitability and the P/B ratio. The relationship between the P/B ratio and profitability can also be illustrated with the following equation by transforming the price multiple as follows: P0 P0 = B0 E1

E1 = ðP0 =E1 ÞROE, B0

ð10:9Þ

where B0 = book value per share at the beginning of the period, E1 = earnings per share in period 1, and ROE = return on equity. The equation indicates that the P/B ratio is positively related to the leading P/E ratio and the return on equity. Furthermore, the equation illustrates that the only difference in fundamental factors between the P/B ratio and the P/E ratio is the return on equity. In principle, companies with a higher return on equity can be expected to trade at a higher P/B ratio. If companies generate a return on equity above the cost of equity, they are likely to be valued at a P/B ratio well above 1. A non-profitable company with a return on equity below the cost of equity has a P/B ratio of less than 1. In particular, companies in economic difficulties or in the maturity phase of their life cycle are often no longer able to earn the cost of equity capital. In such cases, the share price falls below the book value per share, if no economic recovery is expected.

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The book value of equity is positive under the going concern assumption, and therefore the P/B ratio, unlike the P/E ratio, can be used even if earnings per share are zero or negative. Furthermore, the book value per share is a more stable measure than the earnings per share. For example, if earnings are very volatile, the P/B ratio is more meaningful than the P/E ratio. For companies in the financial industry, such as banks and insurance companies, the book value and the market value of assets are approximately equal because the assets consist mainly of liquid assets. Accordingly, if the P/B ratio deviates significantly from 1, this is an indication of an incorrect valuation of the stock. Finally, empirical studies have concluded that the P/B ratio can be used to explain long-term average returns of equity securities.24 One argument against the use of the P/B ratio is that book values, as well as earnings, are affected by the estimates made when applying the accounting and valuation principles. If peer companies and the company to be valued use different accounting standards (e.g. IFRS and US GAAP), the P/B ratios of the stocks are no longer comparable. For example, depending on the accounting standards used, companies may capitalise development costs on the balance sheet or recognise them as an expense on the income statement. Capitalising these costs on the balance sheet results in a higher book value of equity and thus a lower P/B ratio. Accordingly, adjustments must be made to ensure the comparability of the P/B ratio. In addition, intangibles such as human capital, corporate reputation, comparative advantages, and customer relationships are not included in the book value of equity and are therefore not considered in the relative valuation analysis. Hence, the P/B ratio is not suitable for companies that depend primarily on human capital. These include, for example, software companies, investment banks, and real estate developers. The highest P/B ratios are observed in industries in which the most valuable assets are not recognised on the balance sheet. In the software or biotechnology industry, if development and research costs are expensed (i.e. not capitalised), the book value tends to be too low or the P/B ratio too high. Similarly, the book value of equity is too low for branded companies since a significant portion of their earnings is due to the internally generated brand name, which is not listed on the balance sheet. These companies have high returns on equity due to the rather low book value of equity and are traded on the market with a P/B ratio that is well above average. Applying the P/B ratio for equity valuation in such cases results in a mispricing.25

24 25

See, for example, Fama and French 1992: ‘The cross-section of expected stock returns’, p. 427 ff. See Martin 2013: ‘Traditional Equity Valuation Methods’, p. 157.

10.4

Price-to-Book Ratio

345

10.4.2 P/B Ratio Based on Forecast Fundamentals The P/B ratio, like the P/E ratio, can be calculated using forecast fundamentals of the company. Following the one-stage dividend discount model (Gordon-Growth model), the intrinsic share value can be determined as follows [E(r) > g]:26 P0 =

D1 : E ðr Þ - g

ð10:10Þ

Replacing the expected dividend per share (D1) by the product of the expected earnings per share (E1) and the payout ratio (1 - b) yields the following price equation for the one-stage dividend discount model [E(r) > g]: P0 =

E 1 ð 1 - bÞ , E ðr Þ - g

ð10:11Þ

where b = earnings retention rate, and (1 - b) = earnings payout ratio. The return on equity (ROE) is equal to the expected earnings per share (E1) divided by the book value per share at the valuation date (B0). If this equation, ROE = E1/B0, is solved for the expected earnings per share, the result is E1 = ROE B0. Replacing the earnings per share in Eq. (10.11) by the product of the return on equity and the book value per share and dividing both sides of the equals sign by the book value per share leads to the justified trailing P/B ratio [E(r) > g]: P0 ROE ð1 - bÞ = : B0 E ðr Þ - g

ð10:12Þ

Accordingly, the P/B ratio is positively related to the return on equity, the earnings payout ratio, and the expected earnings growth rate. Its relationship to the expected return or risk, on the other hand, is negative. If the payout ratio of 1 - b is replaced by 1 - g/ROE in Eq. (10.12),27 the justified trailing P/B ratio can be calculated as follows [E(r) > g and ROE > g]:28 P0 ROE - g = : B0 E ðr Þ - g

26

ð10:13Þ

See Sect. 8.4. The fundamental earnings growth rate can be determined as follows: g = (1 - δ) ROE. If this equation is solved for the earnings payout ratio (δ), δ = 1 - g/ROE is obtained. g ROEð1 - ROE Þ ROE - g ROE ð1 - bÞ 28 P0 = EðrÞ - g B0 = E ðr Þ - g = E ðr Þ - g 27

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10 Multiples

The equation indicates that the P/B ratio of a company with a perpetual constant earnings growth rate—in other words, a company in the maturity phase of its life cycle—increases when the return on equity rises or the expected return falls. If the return on equity is greater (less) than the expected return, the share price exceeds (falls below) the book value per share.29 This relationship can be demonstrated more clearly if the earnings growth rate is set equal to zero (g = 0): P0/B0 = ROE/E(r). Example: Calculation of the Justified Price-to-Book Ratio Using the Deutsche Telekom AG Stock The following information is available for the listed Deutsche Telekom stock at the beginning of 2022 (Source: Refinitiv Eikon): Dividend per share (for 2021) Earnings per share (for 2021) Share price Book value per share Historical beta

EUR 0.64 EUR 0.87 EUR 16.55 EUR 8.56 0.70

The company is assumed to grow in perpetuity based on its 2021 fundamentals. The return on equity is 10.6%, and the yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%. The expected market risk premium for Germany is 7%. What is the justified trailing P/B ratio of the Deutsche Telekom stock and is the equity security correctly valued? Solution The earnings payout ratio is 0.7356 and can be calculated as follows: δ=

EUR 0:64 = 0:7356: EUR 0:87

The fundamental earnings growth rate is 2.8%: g = ð1- 0:7356Þ × 10:6% = 2:8%: The adjusted beta of the stock of 0.80, and the expected CAPM return of 6.70% can be determined as follows: βAdjusted = 0:333 þ 0:667 × 0:70 = 0:80, Eðr Þ = 1:1% þ 7% × 0:80 = 6:70%: The justified trailing P/B ratio of the Deutsche Telekom stock is to 2 and can be calculated with one of the following two equations: (continued) 29

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 392.

10.4

Price-to-Book Ratio

347

P0 ROE ð1 - bÞ 0:106 × 0:7356 = = = 2:0 B0 0:067 - 0:028 E ðr Þ - g or P0 ROE - g 0:106 - 0:028 = = = 2:0: 0:067 - 0:028 B0 E ðr Þ - g Due to the calculated P/B ratio of 2, Deutsche Telekom shares must trade at a price that is significantly higher than the book value per share because the return on equity of 10.6% exceeds the expected return of 6.7%. For example, at the beginning of 2022, the Deutsche Telekom share price is EUR 16.55, while the book value per share is EUR 8.56. The justified P/B ratio of 2 is higher than the traded P/B ratio of Deutsche Telekom shares of 1.9 (= EUR 16.55/EUR 8.56). Consequently, equity security appears undervalued. If the return on equity and the expected return of the shareholders are the same [ROE = E(r)], the company achieves a return exactly equal to the expected return, and the P/B ratio of the stock is 1. If the return on equity exceeds the expected return [ROE > E(r)], the P/B ratio is above 1. In the opposite case [ROE < E(r)], the result is a P/B ratio of less than 1. Figure 10.3 presents the P/B ratio for the Deutsche Telekom stock for different levels of the difference between the return on equity and the expected return. The figure illustrates that a decrease in the difference between the return on equity and the expected return results in a lower P/B ratio. If the difference is zero, the P/B ratio is 1.

10.4.3 P/B Ratio Based on Comparable Companies Using the comparables method in relative valuation analysis, the average P/B ratio is first calculated from a benchmark group. Then the P/B ratio of the stock to be valued is compared with the benchmark P/B ratio to determine whether the security is correctly priced. Any differences in fundamentals are assessed on subjective judgement. The return on equity is the most important fundamental factor because the share price is significantly affected by the profitability of the company. Other fundamental factors are the expected earnings growth rate, the earnings payout ratio, and the expected return or risk. The following example demonstrates the application of the P/B ratio based on the comparables method, using the Mercedes-Benz Group stock.

348

10 Multiples

(P/B rao)

8 7 6 5 4 3 2 1 0 12.0% 10.5% 9.0%

7.5%

6.0%

4.5%

3.0%

1.5%

0.0%

-1.5%

(Return on equity − expected return) Fig. 10.3 Price-to-book ratio and difference between return on equity and expected return (Source: Own illustration)

Example: Relative Valuation Analysis of the Mercedes-Benz Group Stock Based on the Comparables Method Using the Price-to-Book Ratio For the ‘automobile manufacturers’ sub-sector (GICS), the following global peer companies are given in order of market capitalisation with the corresponding P/B ratios, expected annual earnings growth rates for the next 4 years, returns on equity, and betas as at the end of October 2017 (Source: Refinitiv Eikon): Company Toyota Motor Corp Volkswagen Mercedes-Benz Group General Motors Co BMW Tesla Inc. SAIC Motor Corp Honda Motor Co Nissan Motor Co Audi Hyundai Renault

P/B ratio 1.1 0.7 1.2 1.5 1.1 11.3 1.8 0.8 0.9 1.1 0.6 0.8

Expected earnings growth rate 6.3% 33.3% 3.8% -2.3% 2.3% 114.4% 7.9% 4.5% 1.3% -29.3% 8.4% 7.7%

Return on equity 11.1% 8.5% 16.4% 21.8% 16.2% -20.1% 17.2% 9.3% 14.3% 9.5% 6.3% 14.1%

Beta 1.16 1.57 1.54 1.59 1.43 0.98 0.70 1.29 1.08 0.40 1.03 1.81

(continued)

10.4

Price-to-Book Ratio

Company Fiat Chrysler Automobiles Suzuki Motor Corp Peugeot SA Arithmetic mean Median

349

P/B ratio 1.4

Expected earnings growth rate 23.1%

Return on equity 15.4%

Beta 1.36

2.2 1.3 1.9 1.1

8.9% 2.7% 12.9% 6.3%

17.8% 13.0% 11.4% 14.1%

1.08 1.79 1.25 1.29

Is the Mercedes-Benz Group stock correctly valued based on the comparables method using the P/B ratio? (The median should be taken for this analysis.) Solution Compared to the median of 1.1, the Mercedes-Benz Group stock appears slightly overvalued, with a P/B ratio of 1.2. The return on equity of 16.4%, which is higher than the median of 14.1%, justifies the higher share valuation. However, the lower expected earnings growth rate of 3.8% versus the median of 6.3% and the higher beta of 1.54 versus the median of 1.29 suggest a lower valuation, with the result that the equity security appears overvalued. A clear judgement as to whether the Mercedes-Benz Group stock is overvalued cannot be made on the basis of the differences in fundamentals between the stock and the benchmark. If, on the other hand, only the most important fundamental factor, namely the return on equity, is taken into account, the Mercedes-Benz Group stock appears to be correctly valued. Given the positive relationship between the P/B ratio and the return on equity, it is not surprising that stocks with a high (low) return on equity trade at a high (low) P/B ratio.30 Accordingly, equity securities with a comparatively high (low) P/B ratio and low (high) return on equity should attract investors’ attention. For this purpose, stocks can be classified in a two-dimensional matrix consisting of the two criteria of P/B ratio and the difference between the return on equity and the expected return. This makes it possible to identify stocks that are mispriced in the market because the level of the P/B ratio does not correspond to the fundamental factors or the difference between the return on equity and the expected return. Figure 10.4 presents the two-dimensional matrix that makes it possible to identify mispriced stocks. The four quadrants of the matrix can each be defined by the median of the P/B ratio and the difference between the return on equity and the expected return.31

30

Empirical studies demonstrate the positive correlation between P/B ratio and return on equity. See, for example, Fairfield 1994: ‘P/E, P/B and the present value of future dividends’, p. 30. 31 For an example with companies from the car manufacturers sub-sector, see Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 491 ff.

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10 Multiples

(P/B rao) Equity security appears correctly valued

Equity security appears overvalued

• High P/B rao • High P/B rao • Low difference between return • High difference between return on equity and expected return on equity and expected return

Median

Equity security appears correctly valued

Equity security appears undervalued

• Low P/B rao • Low P/B rao • Low difference between return • High difference between return on equity and expected return on equity and expected return

0 0

Median (Return on equity – expected return)

Fig. 10.4 Matrix for identifying mispriced equity securities using the price-to-book ratio and the difference between the return on equity and the expected return (Source: Based on Damodaran 2012: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, p. 524)

10.5

Enterprise Value EBITDA Ratio

Enterprise value-based multiples place the value of the operating company in relation to a quantity such as EBITDA and EBIT, which is attributable to all providers of capital. Price multiples, on the other hand, divide the share price by, for example, the earnings per share or the book value per share, all of which are quantities allocable to the equity providers.32 The equity-based price multiples are influenced by the company’s debt level. Thus, the P/E ratio may increase with a higher debt-to-equity ratio as a result of a higher return on equity and a consequent higher earnings growth rate. By contrast, the value multiples have the advantage that their level is not affected by the debt-to-equity ratio, with the result that companies with different gearing ratios can be compared with each other. Therefore, value multiples are suitable for assessing the value of a business model.33

32

See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 47. 33 See Koller et al. 2010: Valuation: Measuring and Managing the Value of Companies, p. 314.

10.5

Enterprise Value EBITDA Ratio

351

The enterprise value EBITDA multiple (EV/EBITDA) measures the relationship between the total market value of the company, adjusted for cash and cash equivalents, and the operating result (EBIT) before deduction of depreciation and amortisation. The consistency between the numerator and the denominator is given in the multiple, as the total enterprise value consisting of the market value of debt and equity is divided by an earnings figure before interest on debt. For example, the value multiple can be calculated on a forward basis as follows: Forward EV=EBITDA ratio =

EV0 , EBITDA1

ð10:14Þ

where EV0 = enterprise value = market value of equity + market value of debt + non-controlling interests - cash and cash equivalents. Cash and cash equivalents (e.g. marketable securities with maturities of less than 1 year) are deducted from the enterprise value because the interest income from cash and cash equivalents is not included in EBITDA. If this adjustment is not made, the multiple will be too high. Alternatively, it can be argued that in the case of an acquisition only the net price paid for the company should be taken into account. After the acquisition, the acquirer gains access to cash and cash equivalents that can be used to repay part of the purchase price (for example, part of the debt capital that was needed to finance the acquisition can be repaid). Analogous to this consideration, market values are taken for the debt and equity in the numerator of the value multiple, since the acquirer pays the market value and not the book value when repaying the debt. For example, if bonds are outstanding, they must be purchased at market price upon repayment. If the debt capital is not traded on the market (such as a loan from a bank), the book values from the balance sheet can be used for the valuation. Only the interest-bearing debt capital is considered. Non-interest-bearing liabilities such as trade payables or guarantee provisions are not included in the calculation of the enterprise value. The market value of equity corresponds to the market capitalisation of all share categories (e.g. ordinary and preference shares) on the valuation date. Non-controlling interests are included in equity in the consolidated balance sheet under IFRS. They arise in the case of majority shareholdings of less than 100%. Since non-controlling interests are not included in the share price, they must be added separately to the market value of the equity in order to determine the enterprise value. EBITDA is the corporate result before interest, income taxes, depreciation, and amortisation. This figure eliminates distortions caused by different capital intensities and different income tax rates. EBITDA (as well as EBIT) reflect the earnings power from the company’s business activities. For the calculation of the value multiple, the

352

10 Multiples

EBITDA reported in the income statement is adjusted for extraordinary and non-recurring income and expense items.34 Example: Calculation of the Enterprise Value EBITDA Ratio The following information from the annual financial statements as of the end of December 2022 is available for Vega AG, a fictitious company operating in the steel industry (in EUR million): Balance sheet Cash and cash equivalents Accounts receivable Inventories Other assets Current assets Property, plant and equipment Financial assets (no shareholdings) Intangible assets Fixed assets Total assets Accounts payable Short-term interest-bearing financial liabilities Guarantee provisions Long-term interest-bearing financial liabilities Total liabilities Share capital Additional paid-in capital Retained earnings Non-controlling interests Total equity Total liabilities and equity

2022 2108 3630 1112 7 6857 11,173 9788 5000 25,961 32,818 2806 1370 2768 13,216 20,160 1200 2658 8340 460 12,658 32,818

Vega’s share capital of EUR 1200 million consists of 800 million ordinary shares with a par value of EUR 1 and 400 million non-voting preference shares with a par value of EUR 1. The preference shares carry an interim dividend per share of EUR 0.05. As of the end of December 2022, the ordinary shares are traded at a price of EUR 20, while the preference shares have a price of EUR 15. For the year 2022, EBITDA amount to EUR 5962 million. (continued)

34

See Koller et al. 2010: Valuation: Measuring and Managing the Value of Companies, p. 317.

10.5

Enterprise Value EBITDA Ratio

353

1. What is Vega’s trailing EV/EBITDA multiple? 2. Is the Vega stock correctly valued if the trailing EV/EBITDA ratio of the peer companies is 6.5? Solution to 1 The enterprise value of EUR 34,938 million can be calculated as follows: Market value of ordinary shares (800 million shares × EUR 20) + Market value of preference shares (400 million shares × EUR 15) + Book value of current financial liabilities + Book value of non-current financial liabilities + Non-controlling interests - Cash and cash equivalents = Enterprise value

EUR 16,000 million + EUR 6000 million + EUR 1370 million + EUR 13,216 million + EUR 460 million - EUR 2108 million = EUR 34,938 million

The trailing EV/EBITDA ratio is 5.86: EV0 EUR 34,938 million = = 5:86: EUR 5962 million EBITDA0 Solution to 2 The Vega stock appears undervalued due to the lower EV/EBITDA ratio of 5.86 (versus 6.5). The EV/EBITDA ratio is more suitable than the P/E ratio for comparing companies with different debt ratios because EBITDA, in contrast to net income, is a measure of earnings before deducting interest on borrowed capital and is therefore not influenced by the capital structure. Furthermore, EBITDA is often positive when the company’s net income is negative. The use of the EBITDA in a multiple is carried out, in particular, in those valuation cases in which the lower earnings levels such as EBIT and net income are negative. Furthermore, the application of different depreciation methods across different companies does not affect EBITDA, while the comparability of EBIT and net income is affected. Since depreciation and amortisation are added to operating income (EBIT), the EV/EBITDA ratio is suitable for capital-intensive industries such as steel, utilities, and telecommunications, where major infrastructure investments are required. Companies in such industries have high depreciation charges. For example, mobile communications providers require significant capital expenditures in the expansion and maintenance of network infrastructure. The use of the EV/EBITDA ratio is more appropriate for such companies than the P/E ratio due to the capital-intensive investments and the long-term orientation of the business model.

354

10 Multiples

A disadvantage of the EV/EBITDA ratio is that the free cash flows to firm have a stronger link to equity valuation than EBITDA. Only when depreciation and investments in fixed assets and net working capital cancel each other out are EBITDA and free cash flows to firm approximately equal. The justified forward EV/EBITDA ratio can be calculated as follows [WACC > g].35 1 ð1 - t Þ ð1 - t Þ - Depr EV0 EBITDA1 = EBITDA1 WACC - g

EI1 EBITDA1

,

ð10:15Þ

where t = income tax rate, Depr = depreciation and amortisation of fixed assets, and EI = expansion investments including investments in net working capital (EI = Capex - Depr + I NWC). The formula demonstrates that the EV/EBITDA multiple is influenced by the following fundamental factors:36 • Income tax rate: With an increase (decrease) in the tax rate, the value multiple falls (rises), all else being equal. • Depreciation and amortisation: The higher (lower) the level of depreciation and amortisation in EBITDA, the lower (higher) the EV-based multiple, all else being equal. • Expansion investments: A larger (smaller) share of investments in net working capital and fixed assets less depreciation relative to EBITDA leads to a lower (higher) EV/EBITDA ratio, all else being equal. • Weighted average cost of capital: A higher (lower) cost of capital results in a lower (higher) value multiple, all else being equal. • Growth rate: If the expected growth rate increases (decreases) due to, say, a higher (lower) return on assets, the EV/EBITDA ratio increases (decreases), all else being equal. Companies with a higher level of depreciation and amortisation in EBITDA trade at a lower EV/EBITDA ratio than companies whose depreciation and amortisation are of lesser importance. The same applies to companies with high capital expenditures compared to EBITDA. Accordingly, equity securities of capital- and depreciation-intensive industries, such as telecommunications, trade at a lower EV/EBITDA multiple than the stocks of less capital- and depreciation-intensive 35

For the derivation of the formula, see Mondello 2017: Aktienbewertung: Theorie und Anwendungsbeispiele, p. 528 ff. 36 See Frykman and Tolleryd 2003: Corporate Valuation: An Easy Guide to Measuring Value, p. 58.

10.6

Summary

355

industries (e.g. technology). Since individual industries have different value multiples, EV-based multiples of the same industry (and not of different industries) should be compared.

10.6

Summary

• Multiples can be categorised into price and value multiples. In the case of price multiples, both the numerator and the denominator have equity-related variables. The share price is set in relation to the earnings per share, the book value per share, or the free cash flows to equity per share. By contrast, value multiples consist of total capital-related variables. The enterprise value is divided by the EBIT, the EBITDA, the free cash flows to firm or by the sales. • Company’s fundamentals such as growth, expected cash flows, and risk influence the value of an equity security. Based on a one- or multi-stage cash flow model, the corresponding multiple for a mature or high-growth company can be derived. The intrinsic share value can be calculated if the multiple (e.g. P/E ratio) is multiplied by a company-specific variable that corresponds to the denominator of the multiple (e.g. earnings per share). A comparison with the traded share price makes it possible to assess whether the security is correctly valued. The multiple derived from a cash flow model can also be compared with the traded multiple of the stock in order to identify any mispricing. • Multiples are usually applied with the comparables method. The analysis begins with two basic decisions, which relate to selecting the multiple and determining the comparable companies. The multiple is calculated for the company being valued and for the peer companies, subsequent to which an average (or median) value is calculated. In order to assess the share price, the multiples of the stock and the benchmark are compared. If the two multiples are apart, a subjective judgement can be made as to whether differences in fundamentals such as growth, expected cash flows, and risk can explain this discrepancy. If it is concluded that the differences in fundamentals do not justify the difference between the two multiples, the equity security is mispriced. In the event that the multiple of the security is higher (lower) than the corresponding benchmark multiple, the stock appears overvalued (undervalued). • The P/E ratio can be calculated both with the earnings per share of the last 12 months (trailing P/E ratio) and with the expected next year’s earnings per share (forward P/E ratio). The earnings per share are adjusted for non-recurring expenses and revenues. The P/E ratio is the most widely used multiple since earnings reflect the profitability of the company and thus represent the most important value driver of an equity security. However, reported after-tax earnings are distorted by the application and interpretation of the accounting standards used. Moreover, the earnings can be very volatile as well as negative. • The P/B ratio is widely used in equity valuation. Unlike earnings, the book value of equity is less volatile and positive under the going-concern assumption. With very volatile and possibly negative after-tax earnings, the P/B ratio is more

356

10 Multiples

meaningful than the P/E ratio. The P/B ratio is positively related to return on equity, earnings growth, and payout ratio. By contrast, the ratio is negatively affected by the expected return. • Enterprise value-based multiples place the enterprise value in relation to a total capital-related variable such as EBIT, EBITDA, and sales. The enterprise value consists of the market value of equity and interest-bearing debt capital less cash and cash equivalents (including marketable securities with a term of less than 1 year). • The EV/EBITDA multiple measures the ratio between the operating enterprise value and the operating income (EBIT) before deduction of depreciation and amortisation. Unlike the P/E ratio or the P/B ratio, this value multiple can be used to compare companies with different levels of debt-to-equity ratios. • The expected growth rate, which depends on the return on total capital, among other things, has a positive effect on the EV/EBITDA ratio. On the other hand, the income tax rate, the share of depreciation and amortisation in EBITDA, expansion investments measured against EBITDA, and the WACC have a negative impact on the level of the multiple.

10.7

Problems

1. The following statements can be made on the use of multiples: 1. Assuming falling after-tax earnings in next year, the forward P/E ratio is greater than the trailing P/E ratio if the share price remains unchanged. 2. A value multiple consists of the ratio between the enterprise value and the operating income after interest and taxes. 3. High positive P/E ratios or outliers (negative P/E ratios are not included in the analysis) lead to a right-skewed distribution of the multiples. Therefore, the median and not the arithmetic mean should be used for averaging the multiples. 4. A permanent decrease in the return on equity leads to a decrease in the P/B ratio. 5. An equity security trading at a price below book value per share is undervalued. 6. The relationship between the enterprise value EBITDA multiple and investments in expansion projects and growth rate is positive. Indicate whether each of the above statements is true or false (with justification).

10.7

Problems

357

2. An analyst examines the valuation of equity securities in the beverage industry. The benchmark companies have a forward P/E ratio of 20, which corresponds to the median of all positive P/E ratios of the peer companies. Earnings per share of CHF 4 are expected for the stock of Spring Water AG in the next year. The analyst calculates an intrinsic share value of CHF 80 (= 20 × CHF 4). A comparison with the traded share price of CHF 105 indicates that the share price is too high. Therefore, the security appears overvalued. a) Why can the conclusion that equity security is overvalued be wrong? b) What additional information about the equity security being valued and the benchmark is needed to support the conclusion that the security is overvalued? 3. The following information is available for the listed HeidelbergCement AG stock as of the end of December 2016 (Source: Refinitiv Eikon): Dividend per share (for 2016) Earnings per share (for 2016) Share price Book value per share Historical beta

EUR 1.60 EUR 3.66 EUR 88.63 EUR 81.11 1.09

It is assumed that the company will grow in perpetuity based on 2016 fundamentals. The company’s return on equity is 4.6%, while the earnings retention rate is 56%. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%. The expected market risk premium for Germany is 7%. a) What is the justified trailing P/E ratio of the HeidelbergCement stock and is the security correctly valued? b) What is the justified trailing P/B ratio of the HeidelbergCement stock and is the security correctly valued? 4. The following information is available for the listed Linde stock at the beginning of 2017 (Source: Refinitiv Eikon): Dividend per share (for 2016) Earnings per share (for 2016) Share price Book value per share Historical beta

EUR 3.70 EUR 6.50 EUR 165.74 EUR 78.52 0.89

The company is assumed to grow in perpetuity based on its 2016 fundamentals. The return on equity is 8.9%. The yield to maturity of 30-year bonds of the Federal Republic of Germany is 1.1%. The expected market risk premium for Germany is 7%. a) What is the justified P/E ratio of the Linde stock and is the security correctly valued?

358

10 Multiples

b) What is the justified PEG ratio of the Linde stock and is the security correctly valued? c) What is the justified P/B ratio of the Linde stock and is the security correctly valued? 5. For the ‘airlines’ sub-sector (GICS), the following global peer companies are given in order of market capitalisation with the corresponding trailing P/E ratios, expected annual earnings growth rates for the next 4 years, and betas as at the end of October 2017 (Source: Refinitiv Eikon): Company Delta Air Lines Southwest Airlines. American Airlines Group Ryanair Holdings Air China United Continental Holdings International Consolidated Airlines Deutsche Lufthansa ANA Holdings China Eastern Airlines Japan Airlines China Southern Airlines Arithmetic mean Median

Trailing P/E ratio 10.1 15.7 12.0 13.5 17.5 8.5 8.1 7.2 10.6 17.8 8.2 18.3 12.3 11.3

Earnings growth rate 7.4% 15.9% 5.6% 8.0% 9.8% 8.2% 16.2% 1.1% 11.2% 14.4% -5.9% 11.9% 8.7% 9.0%

Beta 1.28 1.32 0.98 1.01 1.44 1.01 0.85 0.82 0.63 1.06 0.54 1.40 1.03 1.01

Is the Deutsche Lufthansa correctly valued based on the comparables method? (The median should be taken for this analysis.)

10.8

Solutions

1. 1. The first statement is true. The relationship between the P/E ratio and earnings per share is negative. Since lower after-tax earnings are expected in the future, the forward P/E ratio is greater than the trailing P/E ratio. 2. The second statement is false. In a value multiple, the numerator comprises the enterprise value (market value of equity and interest bearing-debt less cash and cash equivalents), while the denominator includes a total capital-related variable, not an equity-related variable. After-tax earnings represent a quantity that is related to equity. By contrast, EBIT and EBITDA are total capital-related variables that can be allocated to all capital providers (and not just equity providers).

10.8

Solutions

359

3. The third statement is true. Since only positive P/E ratios are included in the relative valuation analysis, a right-skewed distribution results due to outliers. To solve the problem of outliers in the data, the median can be used as an average value instead of the arithmetic mean. 4. The fourth statement is true. A decline in the return on equity has a negative impact on the level of the P/B ratio. 5. The fifth statement is false. If the return on equity is above the expected return of the shareholders, then a stock with a price-to-book ratio of less than 1 appears undervalued. If, on the other hand, the return on equity is below the expected shareholders’ return, the security is correctly valued. 6. The sixth statement is false. The relationship between the EV/EBITDA multiple and investments in expansion projects is negative. Only the relationship with the growth rate is positive. 2. a) The conclusion that the stock is overvalued may be wrong for the following reasons: The benchmark stocks of the beverage industry sector are undervalued. The average value of the forward P/E ratio of 20 is therefore too low, and as a result, the calculated intrinsic share value of Spring Water of CHF 80 is too low. The fundamental factors consisting of earnings growth rate, earnings payout ratio, and risk are different for the stock and the benchmark. For example, the Spring Water stock may be correctly valued if the expected earnings growth rate is higher and the risk is lower than the corresponding average values of the benchmark companies. Accordingly, in order to calculate the intrinsic share value, the average forward P/E ratio of 20 must be adjusted upwards. b) The conclusion that the Spring Water stock is overvalued can be maintained if: • The peer companies are correctly valued on average. • There are no significant differences in fundamentals between the stock being valued and the benchmark. 3. a) The fundamental earnings growth rate is 2.58%: g = 0:56 × 4:6% = 2:58%:

360

10 Multiples

The adjusted beta of the stock of 1.06, and the expected CAPM return of 8.52% can be calculated as follows: βAdjusted = 0:333 þ 0:667 × 1:09 = 1:06, Eðr Þ = 1:1% þ 7% × 1:06 = 8:52%: The justified trailing P/E ratio of the HeidelbergCement stock is 7.6 and can be determined with the following equation: P0 0:44 × 1:0258 = = 7:6: E 0 0:0852 - 0:0258 The trailing P/E ratio is 24.2 (= EUR 88.63/EUR 3.66). Therefore, the HeidelbergCement stock appears overvalued. b) The justified trailing P/B ratio of the HeidelbergCement stock is 0.34 and can be calculated with one of the two equations below: P0 ROEð1 - bÞ 0:046 × ð1 - 0:56Þ = = = 0:34 0:0852 - 0:0258 B0 E ðr Þ - g or P0 ROE - g 0:046 - 0:0258 = = = 0:34: B0 0:0852 - 0:0258 E ðr Þ - g The justified P/B ratio of 0.34 is below the traded P/B ratio of the HeidelbergCement stock of 1.09 (= EUR 88.63/EUR 81.11), with the result that the equity security appears overvalued. 4. a) The adjusted beta of 0.93 and the expected CAPM return of 5.84% can be calculated as follows:

10.8

Solutions

361

βAdjusted = 0:333 þ 0:667 × 0:89 = 0:93, Eðr Þ = 1:1% þ 7% × 0:93 = 7:61%: The earnings payout ratio is 0.5692 (= EUR 3.70/EUR 6.50), while the fundamental earnings growth rate is 3.83% [= (1 - 0.5692) × 8.9%]. The justified trailing P/E ratio of 15.63 can be calculated as follows: P0 0:5692 × 1:0383 = = 15:63: E 0 0:0761 - 0:0383 The intrinsic share value of EUR 101.60 is arrived at by multiplying the justified trailing P/E ratio of 15.63 by the earnings per share of EUR 6.50: P0 = 15:63 × EUR 6:50 = EUR 101:60: The intrinsic share value of EUR 101.60 is lower than the traded share price of EUR 165.74. The Linde stock, therefore, appears to be overvalued. The same conclusion is reached when the trailing P/E ratio of Linde stock of 25.5 (= EUR 165.74/ EUR 6.50) is compared with the justified trailing P/E ratio of 15.63. b) The justified PEG ratio of 4.08 can be determined with the following equation:

PEG =

0:5692 × 1:0383 = 4:08: 0:0383 × 100 × ð0:0761 - 0:0383Þ

If the stock’s trailing P/E ratio of 25.5 (= EUR 165.74/EUR 6.50) is divided by the long-term expected earnings growth rate of 3.83% multiplied by 100, the result is a PEG ratio of 6.66. Accordingly, Linde stock appears to be overvalued based on the PEG ratio. c)

362

10 Multiples

The justified trailing P/B ratio of the Linde stock is to 1.34: 0:089 × 0:5692 P0 ROE ð1 - bÞ = = = 1:34 B0 0:0761 - 0:0383 E ðr Þ - g or P0 ROE - g 0:089 - 0:0383 = = = 1:34: B0 0:0761 - 0:0383 E ðr Þ - g Due to the calculated P/B ratio of 1.34, Linde shares must trade at a price that is higher than the book value per share because the return on equity of 8.9% exceeds the expected return of 7.61%. For example, at the beginning of 2017, the Linde share price is EUR 165.74, while the book value per share is EUR 78.52. The justified P/B ratio of 1.34 is lower than the traded P/B ratio of Linde shares of 2.1 (= EUR 165.74/EUR 78.52), and therefore the equity security appears overvalued. 5. As at the end of October 2017, the equity security of Deutsche Lufthansa has a trailing P/E ratio of 7.2, which is lower than the benchmark median of 11.3. Therefore, the security appears undervalued. However, an earnings growth rate of 1.1% which is lower than the benchmark growth rate of 9% suggests that the lower valuation is justified. When the stock’s beta of 0.82 is included in the analysis, which is below the median of 1.01, a higher valuation appears appropriate, supporting the conclusion from the P/E comparison that the airline stock is undervalued. As the inclusion of the differences in fundamentals in the relative valuation analysis leads to different results, the conclusion from the comparison of the P/E ratios cannot be confirmed that the Deutsche Lufthansa stock is undervalued. Therefore, a further analysis using additional multiples and valuation models is necessary.

References Bank of America Merrill Lynch: Annual Institutional Factor Survey, United States, pp. 1-27 (2012) Barker, R.: Determining value: valuation models and financial statements, Harlow (2001) Damodaran, A.: Investment valuation: tools and techniques for determining the value of any asset, 3rd edn, Hoboken (2012)

References

363

Fairfield, P.M.: P/E, P/B and the present value of future dividends. Financial Anal. J. 50(4), 23–31 (1994) Fama, E.F., French, K.R.: The cross-section of expected stock returns. J. Finance. 47(2), 427–465 (1992) Fama, E.F., French, K.R.: Value versus growth: the international evidence. J. Finance. 53(6), 1975–1999 (1998) Frykman, D., Tolleryd, J.: Corporate valuation: an easy guide to measuring value, Harlow (2003) Glenn, M.: How to value shares and outperform the market. Petersfield Hampshire (2011) Graham, B., Dodd, D.L.: Security analysis, New York (1934) Hasler, P.T.: Aktien richtig bewerten, Berlin (2013) Koller, T., Goedhart, M., Wessels, D.: Valuation: measuring and managing the value of companies, 5th edn, Hoboken (2010) Martin, T.A.: Traditional equity valuation methods. In: Larrabee, D.T., Voss, J.A. (eds.) Valuation techniques, pp. 155–176, Hoboken (2013) Mercedes-Benz Group: Annual Report 2021, Stuttgart (2022) Mondello, E.: Aktienbewertung: Theorie und Anwendungsbeispiele, 2nd edn, Wiesbaden (2017) Pinto, J.E., Henry, E., Robinson, T.R., Stowe, J.D.: Equity asset valuation, 2nd edn, Hoboken (2010) Reilly, F.K., Brown, K.C.: Investment analysis and portfolio management, 7th edn, Mason (2003)

Online Sources FTSE Russell: Industry Classification Benchmark (ICB). https://www.ftserussell.com/data/ industry-classification-benchmark-icb. Accessed 30 June 2022 MSCI: The Global Industry Classification Standard (GICS®). http://www.msci.com/products/ indices/sector/gics (2022). Accessed 30 June 2022

Part III Bonds

Bond Price and Yield

11.1

11

Introduction

A fixed-income security represents a financial obligation of an entity or issuer that promises to pay a specified sum of money at predefined future dates. In the case of a debt obligation, the issuer is called the borrower or debtor, while the investor who purchases a fixed-income security is known as the lender or creditor. The promised payments of the issuer consist of interest payments and the repayment of the amount borrowed. Examples include option-free bonds, bonds with embedded options (e.g. callable bonds and convertible bonds), mortgage-backed securities, and assetbacked securities.1 The terms ‘fixed-income security’ and ‘bond’ are used interchangeably in the following two chapters. Bonds belong to the traditional asset classes together with equity securities. These are interest-bearing securities that are securitised and thus tradable. The global bond market is large and diverse and offers important investment opportunities, especially for institutional investors. Its importance can also be seen in the fact that bond markets are larger than equity markets worldwide. In contrast to equity securities, the majority of fixed-income securities are traded over the counter. An understanding of bonds is useful as these securities increase the universe of investments available to create a diversified portfolio. The issuer of a bond is the borrower, while the investors act as lenders and purchase the security. By issuing the bond, the issuer receives the purchase price/ issue proceeds from the investors and, in return, agrees to make fixed or variable interest payments (coupons) during the life of the security. The principal or par value of the bond is redeemed either during or usually at the end of the bond’s time to maturity.

1

See Fabozzi 2000: Fixed-income Analysis for the Chartered Financial Analyst® Program, p. 4.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_11

367

368

11

Bond Price and Yield

The influence of the bond market on the rest of the financial market and on the real economy is great, as the global financial crisis of 2008 and the resulting European debt crisis have revealed. The global financial crisis was triggered in 2007 by losses on securitised US subprime mortgages—collateralised debt obligations (CDOs)—which led to a loss of confidence and an earthquake in the global financial market. The subsequent debt crisis in the euro area has increased the credit risk and therefore the interest costs for borrowing in some European countries, which the European Central Bank (ECB) continues to control with an expansionary monetary policy (as of October 2022). These distortions highlight the risks that can emerge from the bond market and thus from the financial system. The capital market is represented by long-term securities such as bonds and stocks, as well as loans. By contrast, the money market includes short-term securities and loans with an original time to maturity of less than 1 year. Figure 11.1 presents an overview of the financial market which comprises the capital market and the money market. Derivatives are based on underlying assets of the capital market and the money market.2 The chapter first deals with the basics and the various types of bonds, followed by the pricing of option-free bonds with the cash flow model. In addition to the pricing of fixed-rate bonds, the pricing of floating-rate bonds is also examined. The chapter concludes with a discussion of the three yield measures of current yield, yield to maturity, and total return, which allow investors to assess the return on their investment in fixed-rate option-free bonds.

11.2

Basic Features of a Bond

The basic features of a bond can be outlined in terms of the issuer, the time to maturity, the par or principal value, the coupon, and the currency. Thus, bonds can be issued by public institutions, financial institutions, such as banks and insurance companies, and non-financial corporations. The time to maturity specifies the date or the number of years before a bond matures. It is important for the analysis of the debt instrument as it indicates over which period the coupons and the par value are paid. The time to maturity of a fixed-income security can range from overnight to more than 30 years. Securities with an original time to maturity of less than 1 year are called money market instruments. In Germany, these include non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills) and in Switzerland money market book claims of the Swiss Confederation. If, on the other hand, the securities have a time to maturity that is longer than 1 year, they are known as capital market instruments. The par value represents the debt amount to be repaid by the issuer. The bond is usually repaid at 100% of the par value on the maturity date (plain vanilla bond) or during the time to maturity of the security. The par value is

2

Alternative investments, i.e., real estate, private equity, hedge funds, and commodities, are not included in Fig. 11.1.

11.2

Basic Features of a Bond

369

Financial market

Capital market (long-term)

Money market (short-term)

Securitised (e.g., commercial papers, BuBills, and Treasury-bills)

Not securitised (e.g., money market deposits, short-term loans, repos)

Securitised: Bonds (fixedincome securities)

Not securitised: Loans

Securitised: Equity securities

Derivatives (e.g., Treasury bill futures)

Derivatives (e.g., 3months EURIBOR futures)

Derivatives (e.g., bond futures)

Derivatives (e.g., credit default swaps)

Derivatives (e.g., equity futures, options, and swaps)

Fig. 11.1 Financial market (Source: Own illustration)

also used to determine the coupon payment. Bond prices are quoted as a percentage of the par value. For example, if the par value is EUR 1000 and the bond is trading at a price of 108%, the bond price is EUR 1080 (= 1.08 × EUR 1000). The coupon is expressed as a percentage of the par value. For example, if the coupon rate is 3% and the par value is EUR 1000, the coupon is EUR 30 (= 0.03 × EUR 1000). Coupons can be paid annually, semi-annually, quarterly, or even monthly. For example, coupons on bonds issued by the Federal Republic of Germany and the Swiss Confederation are payable annually. By contrast, US Treasury bonds pay coupons semi-annually. A coupon can be either fixed or variable. A variable coupon is based on a reference interest rate such as EURIBOR or €STR for the euro, LIBOR, SOFR,

370

11

Bond Price and Yield

and Ameribor for the US dollar, and SARON for the Swiss franc.3 The coupon is reset at the beginning of each interest period, and the payment is made at the end of each interest period. Finally, bonds can be issued in any currency, although a large number of bonds worldwide are issued in either US dollars or euros.4 Bonds can have different types of collateral and be secured or unsecured. Secured (senior) bonds are backed by a legal claim on some specified property of the issuer in the case of default. This includes, for example, mortgage loans, items of property, receivables from companies, or other bonds. These assets are used to ensure that the payment obligation to investors is met, so that coupon and principal payments can continue to be made even in the event of insolvency. On the other hand, unsecured bonds (debentures) are backed only by the promise of the issuer to pay coupon and principal on a timely basis. Therefore, they are secured by the general credit of the issuer. In the event of a default, the issuer is liable with all their assets and business activities. Issuing bonds in the form of a physical certificate is no longer appropriate today due to digitalisation and high costs. Today, they are usually stored in electronic form at the depositories. A global certificate (collective certificate) can be issued, which replaces multiple individual certificates and is deposited in a securities clearing and deposit bank. The individual bondholders are entered electronically in a securities account and thus hold shares in the global certificate. This form of registration is no longer necessary in today’s world either. Rather, it is sufficient if the electronic entry is made in a debt register. In this context, reference is made to a book-entry right, since no certificate is issued. In the case of bonds issued by the Federal Republic of Germany, for example, an electronic entry is made in the Federal Debt Register. Issuing uncertificated securities can save costs and improve security against loss. The transfer of ownership of a security takes place in the secondary market and often does not occur through physical delivery. Thus, a change of ownership of the bond is entered electronically in the securities account. When transferring ownership, a distinction must be made between bearer and registered securities. Bonds are usually bearer securities, where possession is sufficient to be able to assert the securitised claims arising from the security. The owner is not recorded by name, and therefore possession and ownership amount to the same thing. Securities can be transferred quickly and relatively easily in the form of bearer bonds. Bearer securities are usually marked with an identification number such as an International

3

LIBOR (London Interbank Offered Rate) was an important reference interest rate in the money market and served as a base rate for construction of loans up to complex derivative transactions. Due to manipulation of LIBOR by the banks involved in determining the interest rate, it was discontinued at the end of 2021, and a new, more reliable system was introduced. For the euro area, EURIBOR and €STR (Euro Short-Term Rate) are the relevant reference rates. In Switzerland, the LIBOR was replaced by SARON (Swiss Average Rate Overnight). However, LIBOR will continue to be provided for the US dollar until the end of June 2023. Alternatives for the US dollar LIBOR are the SOFR (Secured Overnight Finance Rate) and the Ameribor (American Interbank Offered Rate). 4 See Bank for International Settlements 2015: BIS Quarterly Review December 2015, A 10.

11.3

Different Types of Bonds

371

Bonds

Fixed-rate bonds

Floating-rate notes

Special types of bonds

• Convertible bonds • Bonds with warrants • Dual-currency bonds • Index-linked bonds • Step-up bonds Fig. 11.2 Overview of the different types of bonds (Source: Own illustration)

Securities Identification Number (ISIN). In the case of registered securities, the issuer maintains records of owners, and the interest and principal are paid directly to them. This makes the transfer of the security more difficult, as the new owner must be registered.

11.3

Different Types of Bonds

Bonds consist of fixed-rate bonds, floating rate notes, zero-coupon bonds, and special types of bonds such as convertible bonds, bonds with warrants, dualcurrency bonds, index-linked bonds (e.g. inflation-linked securities), and step-up bonds. Figure 11.2 presents an overview of the various types of bonds. Most bonds have a fixed coupon. For example, as of November 2021, long-term fixed-rate bonds (without zero-coupon bonds) issued by entities located in the euro area accounted for about 71.5% of outstanding debt securities. The share of long-term floating-rate debt securities, on the other hand, was only 16%. The remaining securities were divided between short-term debt securities with a market share of 7.6% and zero-coupon securities (and revaluation effects) with a share of 4.9%.5 The investor pays a price when buying a coupon bond. In return, they receive coupon payments from the issuer, which are paid at the end of each interest period. In addition, at the maturity date of the bond, the investor is repaid the amount invested, which in most cases corresponds to the par value of the security, so that the 5

See http://sdw.ecb.europa.eu/reports.do?node=1000002757

372

11

(Cash flows in EUR) 1000

Bond Price and Yield

Coupon EUR 50 + principal EUR 1000

Coupon Coupon EUR 50 EUR 50

Coupon EUR 50

Coupon EUR 50

3

4

0 0

1

2

5 (Years)

-1000 Purchase price EUR 1000 Fig. 11.3 Cash flow pattern of a fixed-rate plain vanilla bond (Source: Own illustration)

bond is repaid at 100% of its principal value. Such a fixed-rate bond is also called a plain vanilla bond. Figure 11.3 presents the cash flow pattern of a fixed-rate plain vanilla bond with a time to maturity of 5 years, an annual coupon rate of 5%, and a purchase price of 100%. The par value is EUR 1000. The coupon rate is always expressed as an annual interest rate. The coupon is calculated by multiplying the coupon rate by the par value. In the case of a bond with semi-annual interest payments and a coupon rate of 5%, a coupon rate of 2.5% of the par value is paid every 6 months. The purchase price of the bond is usually approximately 100% of the par value, as the amount of the coupon is determined based on the yield to maturity at the time of issue.6 The purchase price is usually paid two business days after the close of trading (settlement date). Zero-coupon bonds have fixed coupon rate of zero per cent. Hence, they pay no coupon and are therefore issued at a deep discount price. On the maturity date of the bonds, the investor is paid the par value. Thus, the interest earned at the maturity date is given by the difference between the par value and the price paid for the security. For example, if an investor purchases a zero-coupon bond for 65% and holds it until the maturity date, the interest earned is 35% (= 100% - 65%). Floating-rate notes do not have a fixed coupon. Rather, the coupon payment resets periodically according to some reference rate. The most common floating-rate

6

See Mondello 2022: Corporate Finance: Theorie und Anwendungsbeispiele, p. 766.

11.3

Different Types of Bonds

373

notes are money market floaters, where the reference rate is based on a money market rate such as EURIBOR for the euro or Compounded SARON for the Swiss franc. The term of the reference interest rate is usually 3 or 6 months. The coupon is reset on each interest payment date. A spread is added to the reference rate. The spread is also known as the quoted margin and is expressed in basis points.7 It is fixed at the time of issue and usually remains constant over the entire life of the floater. The amount of the quoted margin depends on the creditworthiness of the issuer. It can also be negative if the rating of the reference rate is worse than that of the issuer (with equal market liquidity). If the 6-month EURIBOR is 0.5% on the coupon reset date and the quoted margin is -20 basis points, the coupon rate is 0.3% [= 0.5% + (-0.2%)]. This is the interest rate that the issuer agrees to pay on the next coupon date 6 months hence. Bonds can also contain embedded options such as a call option or a conversion option. For example, a callable bond gives the issuer the right to redeem all or part of a bond issue before the scheduled maturity date. The call option embedded in the bond protects the issuer from falling interest rates. If interest rates fall sufficiently below the issue’s coupon rate or the credit quality of the borrower improves, the issuer will replace the callable bond with another issue with a lower coupon rate. In this way, the borrower can reduce their capital costs. The right of the issuer to call back the bond is a disadvantage to the bondholder since proceeds received must be reinvested in the market at a lower rate. In the case of a convertible bond, the investor, rather than the issuer, owns the embedded option and therefore the right to convert the bond into a specified number of shares of the issuer. The bondholder will exercise the conversion option if the share price increases. Even without conversion, they can participate in the price upside potential of the stock due to the increase in the price of the convertible bond. Issuing a convertible bond is also advantageous for a corporation. For example, it can reduce interest costs because the conversion option is paid for by the bondholder in the form of a lower coupon and/or a higher price. A bond with warrants is a fixed-income security with the right to purchase equity securities of the issuer, thereby providing a capital gain if the price of the stock rises. The subscription or purchase right is securitised in the warrant. The subscription ratio, the subscription price, and the subscription period are determined when the warrant is issued. The issuer can reduce interest costs by issuing a warrant bond because the subscription option is paid for by the investor in the form of a lower coupon and/or a higher price. The warrant can be separated from the warrant bond after issuance, so that the bond trades as an ex-warrant (and no longer as a cum warrant). Unlike a convertible bond, the bond remains in place when the option is exercised. Dual-currency bonds have different currencies for interest and redemption payments. Thus, the issue and coupon payments are made in one currency (e.g. Swiss franc) and the principal payment at maturity in another currency

7 In the fixed-income market, changes or differences in interest rates are referred to in terms of basis points. One basis point is defined as 0.01% or 0.0001. Therefore, 100 basis points are equal to 1%.

374

11

Bond Price and Yield

(e.g. euro). Therefore, if a Swiss company needs to finance a long-term investment project in Italy that will take a few years to become profitable, for example, it could issue a Swiss franc/euro dual-currency bond. The coupon payments in Swiss francs can be made from the company’s cash flows earned in Switzerland, whereas the principal can be repaid in euros, utilising the cash flows generated in Italy once the investment project becomes profitable. Inflation-indexed bonds (linkers) grant the investor protection against inflation. The coupon and/or the par value are adjusted with an inflation index. The index is usually a consumer price index. If the inflation index increases during the life of the bond, the coupon and/or principal value increase. The extent of inflation protection depends on the underlying inflation index. If this is a consumer price index, for example, one is only protected against a price increase of the goods contained therein, but not of other goods. Governments play an important role in issuing inflation-linked bonds. For example, the Federal Republic of Germany issues inflation-indexed government bonds with original maturities of 5 years (iBobls) and more than 10 years (iBunds). These are capital-indexed bonds for which the interest payment is determined with the help of a fixed coupon rate that is multiplied by the inflation-adjusted par value. In this way, both interest and principal payments are corrected with an inflation index. In the case of step-up bonds, the fixed or variable coupon rate increases over time. This grants the investor protection against rising interest rates. If the coupon is only increased once during the time to maturity of the bond, it is called a single-step-up bond. In cases where the coupon increases several times, it is referred to as a multistep-up bond. If the kind of step-up bond chosen is a floating-rate note, the quoted margin increases by a fixed percentage.

11.4

Pricing of Fixed-Rate Bonds

The pricing of an option-free fixed-rate bond is carried out with the cash flow model. To calculate the bond price, the expected cash flows, consisting of a series of coupon payments and the repayment of the full principal value at maturity, are discounted with the expected return. If the bond price determined in this way is paid and the bond is held until maturity, the expected return used for discounting the cash flows in the valuation model is earned. The expected return or the risk-adjusted discount rate is made up of a risk-free rate and a risk premium. Future cash flows can be discounted either with a fixed expected return or, as is common in valuation practice, with a series of expected returns that correspond to the timing of the cash flows.

11.4

Pricing of Fixed-Rate Bonds

375

11.4.1 Pricing Fixed-Rate Bonds with a Fixed Risk-Adjusted Discount Rate With a fixed risk-adjusted discount rate, it is possible to calculate the price of an option-free fixed-rate bond on a specified coupon date, where the principal value is paid on the maturity date (plain vanilla bond), as follows:8 B0 =

C C C þ PV þ þ ... þ , 1 2 ½1 þ EðrÞT ½1 þ EðrÞ ½1 þ EðrÞ

ð11:1Þ

where B0 = price of the fixed-rate (plain vanilla) bond on a coupon date, C = coupon payment per period, PV = principal or par value paid at maturity, E(r) = expected (required) return or risk-adjusted discount rate, and T = number of years or evenly spaced periods to maturity. The most important difference between equity and bond valuation with a cash flow model is that with option-free fixed-rate bonds the future cash flows do not have to be estimated under uncertainty, but are known. For example, if the annual coupon rate is 3%, the time to maturity of the bond is 5 years, and the risk-adjusted discount rate is 4%, the bond price is 95.548%:9 B0 =

3% 3% 3% 3% 103% þ þ þ þ = 95:548%: ð1:04Þ1 ð1:04Þ2 ð1:04Þ3 ð1:04Þ4 ð1:04Þ5

If the plain vanilla bond is traded on the market at the calculated bond price of 95.548%, it is a discount bond because the bond price is lower than 100%. With a discount bond, the coupon rate is lower than the expected return. For example, if the par value is EUR 1000, the price is EUR 955.48 (= 0.95548 × EUR 1000). With a premium bond, by contrast, the coupon rate is higher than the expected return. If, for example, the annual coupon rate is 5%, the time to maturity of the bond is 5 years, and the risk-adjusted discount rate is 4%, a bond price of 104.452% results, which is thus greater than 100%:

8

See Fabozzi, F. J. 2000: Fixed-income Analysis for the Chartered Financial Analyst® Program, p. 154. 9 The Microsoft Excel applications at the end of the chapter demonstrate how to calculate the bond price in Excel. Another way to determine the bond price is to use the Texas Instrument BAII Plus calculator approved for CFA® exams. The bond price can be calculated by entering 5 [N] for the time to maturity, 3 [PMT] for the coupon, 100 [FV] for the par value, and 4 [I/Y] for the expected return. Then press [CPT] and [PV]. The expression in brackets [ ] represents a key in the calculator.

376

11

B0 =

Bond Price and Yield

5% 5% 5% 5% 105% þ þ þ þ = 104:452%: 1 2 3 4 ð1:04Þ ð1:04Þ ð1:04Þ ð1:04Þ ð1:04Þ5

A plain vanilla bond trades at a price of 100% when the coupon rate and the expected return are the same. For example, if both the annual coupon rate and the risk-adjusted discount rate are 4%, the 5-year bond is priced at 100%: B0 =

4% 4% 4% 4% 104% þ þ þ þ = 100%: ð1:04Þ1 ð1:04Þ2 ð1:04Þ3 ð1:04Þ4 ð1:04Þ5

Whether the debt security is a discount bond, premium bond, or par bond depends on the difference between the coupon rate and the expected return and can be summarised as follows:10 • Coupon rate < expected return: discount bond (bond price below par value). • Coupon rate > expected return: premium bond (bond price above par value). • Coupon rate = expected return: par bond (bond price equal to par value). The risk-adjusted discount rate in the cash flow model corresponds to the expected return, which represents a return compensation of the investors for the risk assumed with the bond. The expected return consists of the nominal risk-free rate and a risk premium: Eðr Þ = r F þ RP,

ð11:2Þ

where rF= nominal risk-free interest rate, and RP = risk premium. The nominal risk-free rate consists of the real risk-free rate and the expected inflation rate. The risk premium, on the other hand, reflects a return compensation for the credit risk of the issuer and the market liquidity risk of the security.11 Credit risk is the risk of loss resulting from the issuer failing to make full and timely payments of the agreed coupon and/or principal, which can lead to a default in payment or even insolvency of the debtor. The market liquidity risk reflects the liquidity of the bond on the market. A bond has low market liquidity and thus higher risk if it has low trading volume, resulting in a wide bid-ask price spread. Purchases are made at a higher ask price and sales at a lower bid price than in the case of a liquid bond. With a fixed-rate bond, the coupon remains constant, while the expected return changes over the bond’s time to maturity. If, for example, the risk-free rate, the credit

10 11

See Buckley et al. 1998: Corporate Finance Europe, p. 115 f. See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 748 f.

11.4

Pricing of Fixed-Rate Bonds

377

(Bond price) 160% 140% B 120%

Convexity effect: B

B 100% B

80% Convex price curve

60% 40% 20% 0% 0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

(Expected return or risk-adjusted discount rate) Fig. 11.4 Price curve for 20-year option-free fixed-rate bond with an annual coupon rate of 3% (Source: Own illustration)

risk, and/or the market liquidity risk increase (decrease), the bond price decreases (increases). Hence, there is an inverse relationship between the bond price and the expected return. All else being equal, an increase in interest rates and/or in the risk premium leads to a higher required return on the part of the investors. Since the bond’s cash flows are fixed, a higher expected return only occurs if the bond price falls. In short, the coupon and the principal remain the same over the time to maturity of the bond, while the expected return, and thus the bond price, vary due to a change in interest rates and/or in the risk premium. Figure 11.4 presents the inverse price-yield relationship for a 20-year fixed-rate bond with an annual coupon rate of 3%. It also demonstrates that the price curve is convex. For example, if the expected return changes by ±1%, the price increase is greater than the price decrease. The figure illustrates this positive convexity effect based on an expected return of 3% and a price of 100%. A drop in the expected return to 2% results in a price of 116.351%, which is equivalent to a price increase of 16.351%. However, an increase in the expected return from 3% to 4% causes the bond price to fall by 13.590%—from 100% to 86.410%. The comparatively higher price increase is due to the convex price curve. The higher the convexity of a bond, the more attractive the security is for market participants, because with an equal downward and upward change in the expected return, the price increase is higher and the price decrease is lower.

378

11

Bond Price and Yield

If a fixed risk-adjusted discount rate is taken as the expected return for discounting the coupons and the par value, the expected return corresponds to the yield to maturity. The yield to maturity of a corporate bond results from the yield to maturity of a risk-free government bond with the same time to maturity and currency plus a risk premium as compensation for the credit and market liquidity risk. The financial information provider Bloomberg coined the term ‘G-spread’ to refer to the risk premium. Thus, the yield to maturity of a corporate bond can be calculated as follows: YTMCorporate bond = YTMGovernment bond þ Gspread,

ð11:3Þ

where YTMCorporate bond = yield to maturity of a corporate bond, and YTMGovernment bond = yield to maturity of a risk-free government bond with the same time to maturity and currency as the corporate bond. If, for example, the yields to maturity of a 10-year corporate bond and a 10-year risk-free government bond are 3% and 2.3%, respectively, the G-spread is 0.7% or 70 basis points (= 3% - 2.3%). In the event that no risk-free government bond with the same time to maturity can be found on the market, the yields to maturity of two risk-free government bonds with a longer and shorter time to maturity than the corporate bond can be interpolated linearly (T2 > Tt > T1):12 YTMGovernment bond,

Tt

= YTMGovernment bond,

 ðYTMGovernment bond,

T2

T1

þ

- YTMGovernment bond,

Tt - T1 T2 - T1 T 1 Þ,

ð11:4Þ

where YTMGovernment bond, T t = linearly interpolated yield to maturity of the risk-free government bond with time to maturity Tt, YTMGovernment bond, T 1 = yield to maturity of the risk-free government bond with time to maturity T1, and YTMGovernment bond, T 2 = yield to maturity of the risk-free government bond with time to maturity T2. For example, if the yields to maturity of 9-year and 11-year risk-free government bonds are 2% and 2.6%, respectively, a linearly interpolated yield to maturity for the 10-year risk-free government bond of 2.3% is obtained: YTMGovernment bond,

12

10 years

= 2% þ

10 years - 9 years × ð2:6% - 2%Þ = 2:3%: 11 years - 9 years

See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 525.

11.4

Pricing of Fixed-Rate Bonds

379

Example: Pricing of the Mercedes-Benz Group AG 2% 2019/2031 Bond on a Coupon Date The following information is available for the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Rating:

Mercedes-Benz Group AG DE000A2TR083 Corporate bond EUR EUR 1000 EUR 750 million 27 February 2019 Actual/actual ICMA 2% fixed, annual Every year on 27 February 27 February 2031 A3 (according to Moody’s as at 13 December 2019)

The following information is available for 27 February 2021: • The interpolated yield to maturity of bonds from the Federal Republic of Germany with the same time to maturity as the 2% Mercedes-Benz Group bond is -0.252%. • The G-spread is 97.2 basis points. What is the price of the Mercedes-Benz Group bond on the coupon date of 27 February 2021? Solution The time to maturity of the bond is 10 years. The yield to maturity is 0.720% (= -0.252% + 0.972%). The price of the Mercedes-Benz Group bond is 112.307% and can be calculated as follows: B0, MercedesBenz Group =

2% 2% 102% þ þ ... þ ð1:00720Þ1 ð1:00720Þ2 ð1:00720Þ10

= 112:307%: For the calculation of the yield to maturity, the swap rate can be taken as a benchmark rate in addition to the risk-free rate (yield to maturity of the risk-free government bond). For this purpose, a risk premium, which is referred to as the I-spread or interpolated spread by the financial information provider Bloomberg, is added to the average swap rate (average of bid and ask swap rates) with the same time to maturity and currency as the corporate bond:13 13

See Ho et al. 2015: ‘The Term Structure and Interest Rate Dynamics’, p. 500.

380

11

Bond Price and Yield

YTMCorporate bond = Swap rate þ Ispread,

ð11:5Þ

where Swap rate = swap rate with same time to maturity and currency as corporate bond. In the event that no swap rate with the same time to maturity as the corporate bond can be found, two swap rates with a longer and shorter maturity than the corporate bond are interpolated linearly. The I-spread can be positive or negative. This depends on whether the credit and market liquidity risk of the bond is higher or lower than that of the swap rate determined with the interest rates of the interbank market or the floating side of the interest rate swaps (e.g. EURIBOR for the euro or SARON for the Swiss franc). For example, on 27 February 2021, the 10-year swap rate based on the 6-month EURIBOR is 0.045%, while the I-spread for the 2% Mercedes-Benz Group bond is 67.5 basis points (Source: Refinitiv Eikon). The positive I-spread is due to the fact that the credit and/or market liquidity risk of the Mercedes-Benz Group bond are higher than the swap rate. The yield to maturity of the Mercedes-Benz Group bond is 0.720% and can be calculated adding the I-spread of 67.5 basis points to the 10-year swap rate of 0.045%. If a bond is not purchased on a coupon date but between two coupon dates, the bond price paid, or full price, consists of the quoted (traded) bond price and the accrued interest. The quoted bond price is also known as the clean or flat price. Thus, the full price can be determined as follows: B0, Full = B0, Clean þ AI,

ð11:6Þ

where B0, Full = full price, B0, Clean = clean price, and AI = accrued interest. The clean price is usually quoted by bond dealers in the market and is calculated by deducting the accrued interest from the full price. If a trade occurs, the full price paid on the settlement date is obtained by adding the accrued interest to the clean price on the trading day. In contrast to the trading day, the settlement date is the day on which the buyer pays for the bond and the seller delivers the security. The settlement date is typically specified as the trading day plus two business days (t + 2). The accrued interest is the interest earned by the seller between the date on which the last coupon payment was made and the settlement date. By paying the full price, the buyer compensates the seller for the accrued interest. The buyer recovers the accrued interest when the next coupon payment is received. On a coupon date, the accrued interest is zero, and the full price and the clean price are therefore equal. The accrued interest can be calculated with the following formula: AI = C where

t , n

ð11:7Þ

11.4

Pricing of Fixed-Rate Bonds

381

C = coupon for the period n, t = number of days from the last coupon payment date to the settlement date, n = number of days in the coupon period, and t/n = fraction of the coupon period that has elapsed since the last coupon payment. The full price of a plain vanilla bond between two coupon dates can be determined by discounting the expected cash flows with a fixed risk-adjusted discount rate or yield to maturity as follows:14 B0, Full =

C C C þ PV þ þ ... þ ð11:8Þ t , 1 - t=n 2 - t=n ð1 þ YTMÞ ð1 þ YTMÞ ð1 þ YTMÞT - n

where T = number of years (evenly spaced periods to maturity) from the beginning of the coupon period in which the settlement date falls until the maturity date. Example: Pricing of the Mercedes-Benz Group AG 2% 2019/2031 Bond between Two Coupon Dates The following information is available for the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

Mercedes-Benz Group AG DE000A2TR083 Corporate bond EUR EUR 1000 EUR 750 million 27 February 2019 Actual/actual ICMA 2% fixed, annual Every year on 27 February 27 February 2031 2 business days after close of trading A3 (according to Moody’s as at 13 December 2019)

The Mercedes-Benz Group bond will be purchased on Friday, 12 March 2021. The settlement date is Tuesday, 16 March 2021. The yield to maturity of the corporate bond is 0.6229%. The following questions must be answered: 1. What is the full price of the Mercedes-Benz Group bond on the settlement date of 16 March 2021? (continued) 14

See Adams and Smith 2015: ‘Introduction to Fixed-Income Valuation’, p. 106.

382

11

Bond Price and Yield

2. What is the accrued interest? 3. What is the clean price of the Mercedes-Benz Group bond on the settlement date of 16 March 2021? Solution to 1 In order to calculate the days from the beginning of the coupon period to the settlement date, the day-count convention actual/actual should be used. For this purpose, the current days per month and the current days per year are to be calculated. There are a total of 17 days from the beginning of the coupon period to the settlement date (1 day in February and 16 days in March). The year or the coupon period consists of 365 days. Thus, the fractional coupon period of t/n is 0.047 years (= 17 days/365 days). Figure 11.5 presents the periods relevant for discounting the cash flows. The full price of the MercedesBenz Group bond of 113.344% can be determined as follows: B0, Full = þ

2% 2% þ þ ... 1 - 0:047 ð1:006229Þ ð1:006229Þ2 - 0:047

102% = 113:344%: ð1:006229Þ10 - 0:047

Solution to 2 The accrued interest reflects the pro rata coupon to which the seller of the bond is entitled. There are a total of 17 days from the last coupon date to the settlement date, resulting in accrued interest of 0.093%: AI = 2% ×

17 days = 0:093%: 365 days

Solution to 3 The clean price of 113.251% is calculated by subtracting the accrued interest from the full price:15 B0, Clean = 113:344% - 0:093% = 113:251%:

15 The Microsoft Excel applications at the end of the chapter demonstrate how the clean price of the bond can be calculated in Excel. Another way to determine the clean price is to use the CFA® approved calculator Texas Instrument BAII Plus. The clean price can be calculated with the bond function [2nd] [BOND] by entering 3.1621 [Enter] [#] for the settlement date, 2 [Enter] [#] for the coupon, 2.2731 [Enter] [#] for the maturity date, 100 [Enter] [#] for the par value, ACT [#] for the day-count convention, 1/Y [#] for the coupon frequency and 0.6229 [Enter] [#] for the yield to maturity. The accrued interest is displayed when the [#] key is pressed. The expression in brackets [ ] represents a key in the calculator.

11.4

Pricing of Fixed-Rate Bonds

383

9.953 years 4.953 years 3.953 years 2.953 years 1.953 years 0.953 years

27 Feb 2021

2%

2%

2%

2%

2%

102%

27 Feb 2022

27 Feb 2023

27 Feb 2024

27 Feb 2025

27 Feb 2026

27 Feb 2031

Settlement date 16 March 2021 Fig. 11.5 Periods for discounting cash flows (Source: Own illustration)

11.4.2 Pricing Fixed-Rate Bonds with Risk-Adjusted Discount Rates That Correspond to the Timing of the Cash Flows The price of a bond is given by the present value of its expected cash flows consisting of coupon and principal payments. Instead of using a fixed discount rate, the cash flows can also be discounted employing discount rates that match the timing of the expected cash flows. For this purpose, the spot rate curve or the swap rate curve can be defined as a benchmark curve. In valuation practice, the swap rate curve is typically taken because the swap market has more maturities than the government bond market for building the benchmark curve. Moreover, the swap market is usually more liquid. Since the swap rate curve is constructed from interbank market interest rates such as EURIBOR or €STR for the euro and SARON for the Swiss franc, the swap market reflects the credit risk of the interbank market for loans, either overnight with €STR or SARON or based on maturities varying from 1 week to 12 months with EURIBOR. The overnight rates are a good approximation of the risk-free interest rate and are available over a period of up to 1 year as compounded average rates.

384

11

Bond Price and Yield

The short end (i.e. the beginning) of the swap rate curve is created with money market rates such as the EURIBOR or €STR for the euro or the SARON for Swiss francs, which have maturities of up to 1 year. Short-term money market futures such as the 3-month EURIBOR futures can also be used for this purpose. Swap rates are the standard instrument for building up the interbank curve (swap rate curve) especially for longer maturities. The choice of instruments depends on their liquidity. The principle applies that the higher the liquidity, the more suitable the instrument is for creating the swap rate curve. It therefore seems appropriate to construct it from different instruments. Money market rates are used for the short end of the swap rate curve, then forward rate agreements and/or money market futures, and finally swap rates for longer maturities. The swap rate curve constructed in this way is transformed into the zero-coupon swap rate curve using the bootstrapping method. In doing so, the reinvestment risk is eliminated from the swap rates.16 However, the bootstrapping method is applied only if no overnight interest rates such as €STR or SARON were used to calculate the swap rates. The risk-adjusted discount rates can be determined by taking the rates from the spot rate curve or the swap rate curve and adding a risk premium, which is referred to as the Z-spread by the financial information provider Bloomberg. Figure 11.6 presents the EURIBOR zero-coupon swap rate curve for 8 November 2017. The (Interest rate) 2.5%

Discount rate curve

2.0%

Z-spread of 50 basis points

1.5% 1.0%

EURIBOR zero-coupon swap rate curve

0.5% 0.0% -0.5% 0

10

20

30

40

50 (Years)

Fig. 11.6 EURIBOR zero-coupon swap rate curve as at 8 November 2017 and discount rate curve with a Z-spread of 50 basis points (Source: Own illustration based on data from Refinitiv Eikon)

16

See Hull 2006: Options, Futures, and Other Derivatives, p. 160 ff.

11.4

Pricing of Fixed-Rate Bonds

385

discount rate curve is obtained by adding the Z-spread to the zero-coupon swap rate curve. The figure presents a positive Z-spread of 50 basis points. The ‘intrinsic’ value of the bond is obtained by discounting the expected cash flows of a corporate bond with the rates from the discount rate curve that correspond to the timing of the expected cash flows. The following example illustrates the price calculation of the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031, using riskadjusted discount rates that correspond to the timing of the bond’s cash flows. Example: Pricing of the Mercedes-Benz Group AG 2% 2019/2031 Bond between Two Coupon Dates with Risk-Adjusted Discount Rates that Correspond to the Timing of the Expected Cash Flows The following information is available for the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

Mercedes-Benz Group AG DE000A2TR083 Corporate bond EUR EUR 1000 EUR 750 million 27 February 2019 Actual/actual ICMA 2% fixed, annual Every year on 27 February 27 February 2031 2 business days after close of trading A3 (according to Moody’s as at 13 December 2019)

The Mercedes-Benz Group bond will be purchased on Friday, 12 March 2021. The settlement date is Tuesday, 16 March 2021. The Z-spread is 60.71 basis points. The linearly interpolated EURIBOR zero-coupon swap rates are as follows (Source: Refinitiv Eikon and own calculations): • • • • • • • • • •

0.953-year swap rate: -0.5080% 1.953-year swap rate: -0.4845% 2.953-year swap rate: -0.4433% 3.953-year swap rate: -0.3857% 4.953-year swap rate: -0.3191% 5.953-year swap rate: -0.2473% 6.953-year swap rate: -0.1739% 7.935-year swap rate: -0.1017% 8.953-year swap rate: -0.0031% 9.953-year swap rate: 0.0364% (continued)

386

11

Bond Price and Yield

What are the full price and the clean price of the Mercedes-Benz Group bond on the settlement date of 16 March 2021? Solution In order to calculate the full price of the corporate bond, the risk-adjusted discount rates that correspond to the timing of the expected cash flows must first be calculated: Zero-coupon swap rate –0.5080% –0.4845% –0.4433% –0.3857% -0.3191% -0.2473% –0.1739% -0.1017% -0.0031% 0.0364%

Z-spread 0.6071% 0.6071% 0.6071% 0.6071% 0.6071% 0.6071% 0.6071% 0.6071% 0.6071% 0.6071%

Discount rate 0.0991% 0.1226% 0.1638% 0.2214% 0.2880% 0.3598% 0.4332% 0.5054% 0.6040% 0.6435%

The full price of the Mercedes-Benz Group bond is 113.344% and can be determined as follows: B0, Full =

2% 2% 102% þ þ ... þ ð1:000991Þ0:953 ð1:001226Þ1:953 ð1:006435Þ9:953

= 113:344%: The accrued interest is 0.093%: AI = 2% ×

17 days = 0:093%: 365 days

The clean price of the Mercedes-Benz Group bond is 113.251% and can be calculated as follows: B0, Clean = 113:344% - 0:093% = 113:251%: The ‘intrinsic’ value of an option-free fixed-rate bond is determined in valuation practice using discount rates that correspond to the timing of the expected cash flows rather than a fixed discount rate. Only in the rare case where the benchmark curve is flat are the fixed discount rate and the discount rates that match the timing of the future cash flows equal. However, the benchmark curve is usually upward sloping, with the result that long-term rates are higher than short-term rates. In such a case,

11.4

Pricing of Fixed-Rate Bonds

387

the bond price is calculated using the maturity-related rates from the discount rate curve.

11.4.3 Pricing of Zero-Coupon Bonds The price of zero-coupon bonds can also be determined using the cash flow model. However, unlike coupon bonds, zero-coupon debt securities do not pay a coupon. Thus, the cash flows of the security consist only of the principal paid on the maturity date, and therefore the price can be calculated by discounting the principal with the expected return or yield to maturity as follows: B0 =

PV , ½1 þ E ðr ÞT

ð11:9Þ

where PV = par value of the zero-coupon bond, E(r) = expected return, and T = number of years or evenly spaced periods to maturity. The expected return consists of a benchmark rate, such as the swap rate with the same time to maturity and currency as the corporate bond, and a risk premium (Z-spread) which reflects a return compensation for credit risk and market liquidity risk. In order to determine the swap rate, a linear interpolation with a longer and a shorter swap rate is required (T2 > Tt > T1):17 Swap rateT t = Swap rateT 1 þ

Tt - T1 T2 - T1

Swap rateT 2 - Swap rateT 1

ð11:10Þ

where Swap rateT t = linearly interpolated swap rate with maturity Tt, Swap rateT 1 = swap rate with shorter maturity of T1, and Swap rateT 2 = swap rate with longer maturity of T2. The price of the zero-coupon bond trades at a price below 100% due to the lack of a coupon. Since the security pays no coupon, no interest accrues, and the clean price and the full price are therefore identical. The following example illustrates the price calculation of the Mercedes-Benz Group 0% bond with maturity date of 8 February 2024.

17

See O’Kane and Sen 2005: ‘Credit spreads explained’, p. 65.

388

11

Bond Price and Yield

Example: Pricing of the Mercedes-Benz Group AG 0% 2019/2024 Bond The following information is available for the Mercedes-Benz Group AG 0% bond, maturing on 8 February 2024 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

Mercedes-Benz Group AG DE000A2YNZV0 Corporate bond EUR EUR 1000 EUR 750 million 8 August 2019 Actual/actual ICMA 0% fixed, annual Annually on 8 February each year 8 February 2024 2 business days after close of trading A3 (according to Moody’s)

The Mercedes-Benz Group 0% bond will be purchased on Friday, 12 March 2021. The settlement date is Tuesday, 16 March 2021. The Z-spread is 46.08 basis points. The EURIBOR zero-coupon swap rates are as follows (Source: Refinitiv Eikon): • 2-year swap rate: -0.4841% • 3-year swap rate: -0.4429% What is the price of the Mercedes-Benz Group 0% bond on the settlement date of 16 March 2021? Solution The time to maturity of the bond is 2.9014 years (= 2 years + 329 days/365 days). The 2.9014-year linearly interpolated EURIBOR zero-coupon swap rate of -0.4470% can be determined as follows: Swap rate2:9014 years = - 0:4841% þ

2:9014 years - 2 years 3 years - 2 years

× ½ - 0:4429% - ð - 0:4841%Þ = - 0:4470%: The expected return or risk-adjusted discount rate is 0.0138% (= 0.4470% + 0.4608%). The price of the Mercedes-Benz Group 0% bond is 99.960% and can be calculated as follows: B0 =

100% = 99:960%: ð1:000138Þ2:9014

11.5

Pricing of Floating-Rate Notes

11.5

389

Pricing of Floating-Rate Notes

In the case of floating-rate notes (FRNs)—also known as floaters—the coupon rate is reset on each interest date for the next period. Therefore, the coupon rate is made up of the reset reference interest rate (e.g. EURIBOR for the euro) and the fixed quoted margin. The coupon to be paid at the end of the interest period is thus known at the beginning of the interest period. However, the coupons following the first period are unknown on the valuation date because the level of the reference rate for the further coupon payments is determined only at the beginning of the corresponding periods. In order to estimate the future coupons nevertheless, forward interest rates are calculated using the yield curve for the reference rate applicable on the valuation date.18 For example, the forward EURIBOR, which starts in 6 months for 6 months, can be estimated using the 6-month and the 12-month EURIBOR. The ending value of an amount of money invested at the 12-month EURIBOR is equal to the ending value of the same amount of capital invested first at the 6-month EURIBOR and then at the forward EURIBOR starting in 6 months and ending after 6 months. The final value of the two strategies (spot market strategy and combined spot/forward market strategy) must be the same in an arbitrage-free market, since with both strategies the capital is invested at the EURIBOR over 1 year. This relationship leads to the following formula (t2 > t1): BVð1 þ r t2 Þt2 = BVð1 þ r t2 Þt1 ð1 þ FRt1,t2 Þðt2 - t1Þ ,

ð11:11Þ

where BV = beginning value of the capital invested (or borrowed), rt1 = reference rate for period t1 of the combined spot and forward market strategy, rt2 = reference rate for period t2 of the spot market strategy, and FRt1,t2 = forward rate starting at the end of period t1 and with a maturity of t2 - t1. If the above equation is solved for FRt1,t2, the following formula is obtained for calculating the forward rate: ð1 þ rt2 Þt2 FRt1,t2 = ð1 þ rt1 Þt1

1=ðt2 - t1Þ

- 1:

ð11:12Þ

For example, if the 6-month and the 12-month EURIBOR are 0.4% and 0.6%, respectively, the forward rate that starts to run in 6 months and lasts 6 months thereafter is 0.8%: FR6,12 =

18

ð1:006Þ12=12 ð1:004Þ6=12

1=ð12=12 - 6=12Þ

- 1 = 0:8%:

See Fabozzi 1993: Fixed-income Mathematics: Analytical and Statistical Techniques, p. 90.

390

11

Bond Price and Yield

Assuming the quoted margin is 150 basis points, the coupon rate for the second semi-annual coupon period will be 2.3% (= 0.8% + 1.5%). The cash flows of the floating-rate note are made up of the coupons determined with the forward rates and the quoted margin, as well as the principal on the maturity date. The principal on a floater is typically not amortised during the time to maturity, but is redeemed in full at maturity. The cash flows of the security are discounted with risk-adjusted discount rates that correspond to the timing of the expected cash flows. The individual discount rates consist of the zero-coupon rates of the reference rate matching the timing of the cash flows plus a risk premium, which is referred to as the discounted margin. If, for example, the discount margin is 120 basis points, the discount rate for the second coupon of 2.3% is 1.8% (= 0.6% + 1.2%). The following example illustrates the pricing of the floating-rate note of Mercedes-Benz Group that matures on 12 January 2019. Example: Pricing of the Mercedes-Benz Group AG 2016/2019 Floating-Rate Note The following information is available for the floating-rate note of MercedesBenz Group AG, maturing on 12 January 2019 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Issue price: Coupon:

Interest dates: Start of interest run: Maturity date:

Mercedes-Benz Group AG DE000A169GZ7 Floating-rate note EUR EUR 100,000 EUR 1.25 billion 12 January 2016 100% Reference interest rate: 3-month EURIBOR Quoted margin: 53 basis points Payment: quarterly Interest fixing date: 2 business days before the start of the period Day-count convention: actual/360 Every 12 January, 12 April, 12 July, and 12 October 12 January 2016 12 January 2019

The trading occurs on Wednesday 15 November 2017, making the settlement date Friday 17 November 2017. The 3-month EURIBOR at the start of the current coupon period (i.e. 12 October 2017) is -0.329%. The linearly interpolated EURIBOR zero-coupon swap rates relevant for the calculation of the coupons and the discount rates are as follows (Source: Refinitiv Eikon and own calculations): (continued)

11.5

Pricing of Floating-Rate Notes

• • • • •

391

0.156-year swap rate: -0.407% 0.406-year swap rate: -0.345% 0.658-year swap rate: -0.310% 0.914-year swap rate: -0.305% 1.169-year swap rate: -0.303%

The discount margin is 18 basis points. What are the full price and the clean price of the Mercedes-Benz Group floating-rate note on the settlement date of 17 November 2017? Solution The first coupon, which is paid on 12 January 2018 and consists of the 3-month EURIBOR at the beginning of the coupon period on 12 October 2017 and the quoted margin, is 0.201% (= -0.329% + 0.53%). In order to determine the other coupon rates, the forward EURIBOR, with a time to maturity after the respective expiration dates of 3 months, must first be determined: ð1 - 0:00345Þ0:406 FR0:156,0:406 = ð1 - 0:00407Þ0:156

FR0:406,0:658 =

ð1 - 0:00310Þ0:658 ð1 - 0:00345Þ0:406

ð1 - 0:00305Þ0:914 FR0:658,0:914 = ð1 - 0:00310Þ0:658

FR0:914,1:169 =

ð1 - 0:00303Þ1:196 ð1 - 0:00305Þ0:914

1=ð0:406 - 0:156Þ

- 1 = - 0:306%,

1=ð0:658 - 0:406Þ

- 1 = - 0:254%,

1=ð0:914 - 0:658Þ

- 1 = - 0:292%,

1=ð1:169 - 0:914Þ

- 1 = - 0:296%:

The quarterly coupon rates consist of the forward EURIBOR plus the quoted margin of 0.53%. They can be calculated for the corresponding interest periods as follows: Interest period 12 January to 12 April 2018 12 April to 12 July 2018 12 July to 12 October 2018 12 October 2018 to 12 January 2019

Reference forward rate -0.306% -0.254% -0.292% -0.296%

Quoted margin 0.53% 0.53% 0.53% 0.53%

Coupon rate 0.224% 0.276% 0.238% 0.234%

(continued)

392

11

Bond Price and Yield

The coupon rates are annualised rates, but the coupons are paid quarterly; therefore, they must be converted to quarterly coupon rates using the EURIBOR day-count convention actual/360: Interest period 12 October 2017 to 12 January 2018

Quarterly coupon rate 0:201% ×

92 days 360 days

= 0:0514%

12 January to 12 April 2018

0:224% ×

90 days 360 days

= 0:0560%

12 April to 12 July 2018

0:276% ×

91 days 360 days

= 0:0698%

12 July to 12 October 2018

0:238% ×

92 days 360 days

= 0:0608%

12 October 2018 to 12 January 2019

0:234% ×

92 days 360 days

= 0:0598%

The risk-adjusted discount rates that match the timing of the expected cash flows consist of the corresponding EURIBOR zero-coupon swap rates and the fixed discount margin of 18 basis points: • • • • •

Discount rate for 0.156-year: -0.407% + 0.18% = -0.227% Discount rate for 0.406-years: -0.345% + 0.18% = -0.165% Discount rate for 0.658-years: -0.310% + 0.18% = -0.13% Discount rate for 0.914-years: -0.305% + 0.18% = -0.125% Discount rate for 1.169-years: -0.303% + 0.18% = -0.123%

The full price of the Mercedes-Benz Group floating-rate note is 100.442% and can be calculated as follows: B0, Full = þ

0:0698% 0:0514% 0:0560% þ þ 0:406 0:156 ð1 - 0:00165Þ ð1 - 0:00227Þ ð1 - 0:0013Þ0:658

0:0608% 100:0598% þ = 100:442%: 0:914 ð1 - 0:00125Þ ð1 - 0:00123Þ1:169

Figure 11.7 presents the periods relevant for discounting the cash flows. There are a total of 36 days from the beginning of the interest period on 12 October 2017 to the settlement date on 17 November 2017. Accordingly, the accrued interest is 0.020%: AI = 0:201% ×

36 days = 0:020%: 360 days

The clean price is 100.422% and can be determined as follows: B0, Clean = 100:442% - 0:020% = 100:422%:

11.5

Pricing of Floating-Rate Notes

393

1.169 years (= 421 days/360 days) 0.914 years (= 329 days/360 days) 0.658 years (= 237 days/360 days) 0.406 years (= 146 days/360 days) 0.156 years (= 56 days/360 days)

12.10. 2017

0.0514%

0.056%

0.0698%

0.0608%

100.0598%

12.1. 2018

12.4. 2018

12.7. 2018

12.10. 2018

12.1. 2019

Settlement date 17.11.2017 Fig. 11.7 Periods for discounting cash flows (Source: Own illustration)

Changes in interest rates only affect the price of the floating-rate note during a coupon period. If the reference interest rate rises (falls), the price of the floater decreases (increases). This interest rate risk exists only until the next coupon date since the reference rate is reset to its actual value at the beginning of the next interest period. If the quoted margin is equal to the discount margin and the yield curve for the reference rate is flat, the coupon rate and the expected return are the same and the price of the floater is 100%. However, a decrease (increase) in the reference interest rate results in a lower (higher) coupon rate, which has an impact on the return of the floater. If the reference interest rate falls (rises), then the coupon payments decrease (increase), while the price hardly changes. By contrast, the price of a fixed-rate bond increases (decreases) if the reference interest rate decreases (increases). However, the coupon remains unchanged. Credit risk is the most important risk factor for the price of a floating-rate note. If the credit risk increases, the discount margin—and thus the expected return—rise, which results in a price decline. On the other hand, a reduction in credit risk leads to a lower discount margin. The price of the floater increases as a result of the lower expected return. Changes in market liquidity also affect the discount margin and therefore the price of a floating-rate note. A lower (higher) market liquidity of the security leads to a higher (lower) discount margin, with the result that the price decreases (increases). With respect to changes in credit risk and market liquidity risk, the price of a floater reacts in the same way as the price of a fixed-rate bond.19

19

See Adams and Smith 2015: ‘Introduction to Fixed-Income Valuation’, p. 119 f.

394

11.6

11

Bond Price and Yield

Yield Measures for Fixed-Rate Bonds

The yield is a way of assessing the profitability or return of a bond. If an investor purchases a fixed-income security, one or more of the following sources of return can be expected:20 • Coupons • Capital gain or loss at maturity or from sale before maturity • Income from the reinvested interim cash flows such as coupons A yield measure for fixed-rate bonds must take all return components into account. The coupon is probably the most obvious part of the yield. A capital gain (capital loss) on an option-free bond occurs when the par value at maturity or the sale price of the security before maturity is greater (less) than the purchase price. The following yield measures are commonly used in the market for option-free fixed-rate bonds:21 1. Current yield 2. Yield to maturity 3. Total return (1) When calculating the current yield, the annual coupon is divided by the clean price of the bond (i.e. excluding accrued interest):22 Current yield =

C , B0, Clean

ð11:13Þ

where C = annual coupon, and B0, Clean = clean price of the bond. For example, if the coupon rate is 5% and the traded price of the bond is 104%, the current yield is 4.81% (= 5%/104%). The current yield will be higher (lower) than the coupon rate when the security sells at a discount (premium). If the bond sells at par, the current yield and the coupon rate are the same. The current yield can be determined relatively easily. However, the yield measure does not include the capital gain or loss and the income from reinvested coupons. It is only the coupon that is covered by the return figure.

20

See Fabozzi 2000: Fixed-income Analysis for the Chartered Financial Analyst® Program, p. 186. 21 See Adams and Smith 2015: ‘Introduction to Fixed-Income Valuation’, p. 116. For floating-rate notes, the discount margin can be used as a yield measure. See Sect. 11.5. 22 See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 735.

11.6

Yield Measures for Fixed-Rate Bonds

395

(2) The yield to maturity is probably the most widely used bond performance measure. It can be calculated between two coupon dates from the full price of the bond (i.e. with accrued interest) as follows:23 B0, Full =

C C C þ PV þ þ ... þ t , ð11:14Þ 1 - t=n 2 - t=n ð1 þ YTMÞ ð1 þ YTMÞ ð1 þ YTMÞT - n

where YTM = yield to maturity. There is no accrued interest on a coupon date, with the result that the full price is equal to the clean price and the yield to maturity can be determined by applying the following price equation: B0, Full =

C þ PV C C þ þ ... þ : ð1 þ YTMÞT ð1 þ YTMÞ1 ð1 þ YTMÞ2

ð11:15Þ

For example, if a fixed-rate bond with an annual coupon rate of 3% and a time to maturity of 5 years trades at a price of 95.548% on a coupon date, the yield to maturity can be estimated using the price equation below: 95:548% = þ

3% 3% 3% 3% þ þ þ 1 2 3 ð1 þ YTMÞ ð1 þ YTMÞ ð1 þ YTMÞ4 ð1 þ YTMÞ

103% : ð1 þ YTMÞ5

The yield to maturity of the 5-year 3% bond is 4%.24 Buying the security at a price of 95.548% and holding it until maturity in 5 years’ time will result in an average annual yield of 4%, taking into account the compounding effect. There are a number of countries, such as the USA, whose government and corporate bonds pay a semi-annual coupon. In the case of semi-annual coupons, the bond price can be calculated with the following equation on an interest date:25

23

For the price equation, see Sect. 11.4.1. The Microsoft Excel applications at the end of the chapter demonstrate how to calculate the yield to maturity in Excel. Another way to determine the yield to maturity is to use the Texas Instrument BAII Plus calculator approved for CFA® exams. The yield to maturity can be determined by entering 5 [N] for the time to maturity, 3 [PMT] for the coupon, 100 [FV] for the par value and 95.548 [±] [PV] for the price. Then press [CPT] and [I/Y]. The expression in brackets [ ] represents a key in the calculator. 25 See Brigham and Houston 2004: Fundamentals of Financial Management, p. 281. 24

396

11

B0, Full =

ð1

C 2 1 þ YTM 2 Þ

þ

ð1

C 2 2 þ YTM 2 Þ

þ ... þ

C 2

Bond Price and Yield

þ PV

ð1 þ YTM 2 Þ

T

ð11:16Þ

,

where T = number of evenly spaced half-year periods to maturity. The cash flows of the bond are discounted at the semi-annual yield to maturity (i.e. YTM/2). The yield to maturity for the 5-year 3% bond at a price of 95.509% can be determined with the following price equation: 95:509% =

1:5% ð1 þ

YTM 1 2 Þ

þ

1:5% ð1 þ

YTM 2 2 Þ

þ

1:5% ð1 þ

YTM 3 2 Þ

þ ... þ

101:5% ð1 þ YTM 2 Þ

10

:

The discount rate or the semi-annual maturity yield (i.e. YTM/2) is 2%.26 The annualised yield to maturity is therefore 4% (= 2 × 2%). Annualising the semiannual yield to maturity by a factor of 2 is consistent with market convention and is referred to as the bond equivalent yield. Taking the compounding effect into account, an annualised yield to maturity of 4.04% can be calculated: ð1 þ YTMÞ1 = 1 þ

YTM 2

2

→ YTM = ð1:02Þ2 - 1 = 0:0404:

The yield to maturity annualised in this way is known as the effective annual yield. It incorporates the compounding effect and is thus higher than the bond equivalent yield of 4%.27 In contrast to the current yield, the yield to maturity captures all return components. Nevertheless, it has two major disadvantages. First, the bond is assumed to be held to maturity, and second, the coupons are invested at the yield to maturity, although this may differ from the current reinvestment rate.28 Therefore, if the yield to maturity of a bond is 6%, for example, in order to earn that return the coupons received must be reinvested at an interest rate of 6%. To illustrate this reinvestment rate assumption underlying the yield to maturity, the following equation is introduced, which can be used to determine the return on any financial asset, such as equities and fixed-income securities: BVð1 þ ReturnÞT = FV,

ð11:17Þ

where

26

The Microsoft Excel applications at the end of the chapter demonstrate how to calculate the yield to maturity in Excel. 27 See Fabozzi 2007: Fixed-income Analysis, p. 122. 28 See Brealey and Myers 1996: Principles of Corporate Finance, p. 648.

11.6

Yield Measures for Fixed-Rate Bonds

397

BV = beginning value of the investment, FV = final value of the investment, and T = number of evenly spaced periods (e.g. years) over the investment period. Solving the above equation according to the return leads to the following equation: Return =

FV BV

1=T

- 1:

ð11:18Þ

The purchase price or initial value is 95.548% for the 5-year bond with an annual coupon rate of 3% and a yield to maturity of 4%. The final value of this bond investment, on the other hand, comprises the security’s principal (100% of par value) paid at maturity and the final value of the reinvested coupons. The latter can be determined by investing the 3% coupon in 1 year for 4 years at a yield to maturity of 4%, the 3% coupon in 2 years for 3 years at a yield to maturity of 4%, and so on: FV of reinvested coupons = 3% × ð1:04Þ4 þ 3% × ð1:04Þ3 þ 3% × ð1:04Þ2 þ 3% × ð1:04Þ1 þ 3% × ð1:04Þ0 = 16:249%: The final value is therefore 116.249% (= 100% + 16.249%). The annual average return of the 5-year bond investment with an initial value of 95.548% and a final value of 116.249% is 4% and can be calculated as follows: Return =

116:249% 95:548%

1=5

- 1 = 4%:

The return calculation illustrates the assumptions underlying the yield to maturity, namely that the bond is held until maturity and that the coupons are reinvested at the yield to maturity. (3) When calculating the total return, the assumptions of the yield to maturity are no longer necessary. Hence, using Eq. (11.18), the return can be calculated for any investment period and not only for the entire maturity of the bond. Furthermore, a reinvestment rate can be taken for the coupons at which they can actually be invested on the market. The total return reflects the yield that the investor expects to realise from the three sources of return consisting of the coupons, capital gain/loss, and income on the reinvested coupons.29 The following example illustrates the calculation of the current yield, yield to maturity, and total return using the Mercedes-Benz Group 2% bond that matures on 27 February 2031.

29 See Fabozzi 2000: Fixed-income Analysis for the Chartered Financial Analyst® Program, p. 189.

398

11

Bond Price and Yield

Example: Calculation of the Current Yield, the Yield to Maturity, and the Total Return of the Mercedes-Benz Group AG 2% 2019/2031 Bond The following information is available for the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

Mercedes-Benz Group AG DE000A2TR083 Corporate bond EUR EUR 1000 EUR 750 million 27 February 2019 Actual/actual ICMA 2% fixed, annual Every year on 27 February 27 February 2031 2 business days after close of trading A3 (according to Moody’s as at 13 December 2019)

The Mercedes-Benz Group bond will be purchased on Friday, 12 March 2021 at an ask price of 113.251%. The bond will be sold later on a coupon date of 27 February 2025. A yield to maturity of 2% is expected on this date due to rising interest rates. The coupons can be invested at a market rate of 0.5%. What are the current yield, the yield to maturity, and the total return? Solution The current yield of the Mercedes-Benz Group bond is 1.77%, which is calculated by dividing the annual coupon by the clean price: Current yield =

2% = 0:0177: 113:251%

The settlement date is Tuesday, 16 March 2021. There are a total of 17 days from the last coupon date of 27 February 2021 to the settlement date of 16 March 2021. Hence, the accrued interest is 0.093%: AI = 2% ×

17 days = 0:093%: 365 days

The full price is 113.344% and can be determined as follows: B0, Full = 113:251% þ 0:093% = 113:344%:

(continued)

11.6

Yield Measures for Fixed-Rate Bonds

399

There are a total of 348 days or 0.953 years (= 348 days/365 days) from the settlement date of 16 March 2021 to the next coupon date of 27 February 2022. The yield to maturity of 0.6229% can be calculated using the following price equation:30 113:344% =

2% 2% 102% þ þ ... þ : 0:953 1:953 ð1 þ YTMÞ ð1 þ YTMÞ ð1 þ YTMÞ9:953

The initial value of the investment comprises the purchase price of the bond of 113.344%. The final value, which is calculated below, is made up of the expected sale price of the Mercedes-Benz Group bond on 27 February 2025 and the final value of the reinvested coupons. On 27 February 2025, the debt security has a remaining time to maturity of 6 years. Accordingly, the expected sale price of 100% can be determined using the expected yield to maturity of 2% as follows: B27 February 2025 =

2% 2% 102% þ þ ... þ = 100%: ð1:02Þ1 ð1:02Þ2 ð1:02Þ6

The coupons can be invested at an interest rate of 0.5%. The first coupon of 2% will be received on 27 February 2022 and can be invested over 3 years at an interest rate of 0.5% until the sale date of 27 February 2025. The second coupon is paid the following year on 27 February 2023 and can be invested over 2 years at 0.5% and so on, resulting in a final value of reinvested coupons of 8.060%: FV of reinvested coupons = 2% × ð1:005Þ3 þ 2% × ð1:005Þ2 þ 2% × ð1:005Þ1 þ 2% × ð1:005Þ0 = 8:060%: The final value of the bond investment is 108.060% and can be determined as follows: Expected sale price of the bond on 27 February 2025 + Final value of the reinvested coupons = Final value of the bond investment

100.000% + 8.060% = 108.060%

The investment period lasts 3.953 years. Thus, the total return of the Mercedes-Benz Group bond is -1.2%: (continued)

30

The Microsoft Excel applications at the end of the chapter demonstrate how to calculate the yield to maturity in Excel.

400

11

Total return =

108:060% 113:344%

Bond Price and Yield

1=3:953

- 1 = - 0:012:

The total return of -1.2% is lower than the yield to maturity of 0.6229% at the time of purchase of the Mercedes-Benz Group bond for two reasons: first, the coupons are invested at a slightly lower interest rate of 0.5%; and second, the expected yield to maturity at the time of sale of the bond is higher at 2%. Hence, the bond is sold at a price of 100%, which is lower than the purchase price of 113.344%. If the yield to maturity had remained unchanged at 0.6229%, the selling price would have been higher, namely 108.085%, which would have had a positive effect on the total return. Thus, the rising yield forecast due to expected interest rate increases has a negative impact on the total return.

11.7

Summary

• Bonds are interest-bearing securities that are securitised and thus tradable. The issuer of a bond is the borrower, while the investors act as lenders and purchase the security. By issuing the bond, the issuer receives the purchase price/issue proceeds from the investors and in return agrees to make fixed or variable interest payments (coupons) during the life of the security. The principal or par value of the bond is redeemed either during or, usually, at the end of the bond’s time to maturity. • Bonds are characterised by basic features such as the issuer, the time to maturity, the par value, the coupon, and the currency. • Bonds are usually bearer securities, where possession is sufficient to be able to assert securitised claims arising from the security. The owner is not recorded by name, with the result that possession and ownership amount to the same thing. In this form, they can be transferred quickly and relatively easily. • The pricing of option-free fixed-rate bonds and floating-rate notes is carried out with a cash flow model. In order to calculate the price, the expected cash flows consisting of coupon payments and the principal payment at maturity are discounted with the expected return. • The expected return used to discount the cash flows is made up of a benchmark rate and a risk premium. The benchmark rate is usually the swap rate derived from the floating side of an interest rate swap (e.g. EURIBOR or €STR for the euro and SARON for the Swiss franc), rather than the spot rate, because the swap market has more maturities than the primary market of prime government bonds and is also very liquid. Moreover, market participants such as banks can refinance at the interest rate of the floating side of the swap. If the benchmark is given by the swap

11.7

•

•

•

•

•

•

Summary

401

rate, the risk premium represents a return compensation for the bond’s higher or lower credit risk and market liquidity risk. Therefore, the risk premium can be positive or negative. Accordingly, the risk-adjusted discount rates consist of the swap rates that correspond to the timing of the bond’s cash flows plus a risk premium known as the Z-spread. The risk factors of an option-free fixed-rate corporate bond consist of the interest rate risk, the credit risk, and the market liquidity risk. The relationship between these risk factors and the bond price is negative. For example, if interest rates increase, the expected return rises, which causes the bond price to fall. In the event that the credit risk or the market liquidity risk increases, the expected return rises, with the result that the bond price falls. The price curve of an option-free fixed-rate bond is convex. If the expected return changes by an equal amount upwards and downwards (e.g. ±1%), the price increase is higher than the price decrease. Market participants, therefore, find a bond with a higher positive convexity more attractive. They are therefore willing to pay a higher price for such a security, which has a negative impact on the yield. If the full price of a fixed-rate bond or a floating-rate note is determined between two coupon dates, accrued interest must be added to the quoted bond price (clean price). The accrued interest represents the interest earned by the seller between the date on which the last coupon payment was made and the settlement date. By paying the full price, the buyer compensates the seller for the accrued interest. The buyer recovers the accrued interest when the next coupon payment is received. On a coupon date, the accrued interest is zero, and the full price and the clean price are therefore equal. Accrued interest is determined using the day-count convention specified in the issuing prospectus, such as actual/360 or actual/actual. A zero-coupon bond pays the par value on the maturity date. There are no coupons. Accordingly, the price results from the discounted par value. A swap rate that corresponds to the timing of the principal payment at maturity plus a risk premium (Z-spread) is used as the risk-adjusted discount rate. Since no coupons are paid, the clean price and the full price are the same. In the case of a floating-rate note, the coupon consists of the reference interest rate (e.g. EURIBOR for the euro and Compounded SARON for the Swiss franc) and a fixed risk premium, known as the quoted margin. Since the coupon rate is reset on each interest date, the future coupons are not known at the time of valuation. Therefore, forward rates are determined for the reference rate, which start on the respective interest dates and have the same maturity as the coupons. To estimate future coupon rates, the fixed quoted margin is added to the forward rates of the reference rate. The full price of the floating-rate note can be determined by discounting the coupons with the swap rates that correspond to the timing of the security’s cash flows plus a fixed risk premium, referred to as the discounted margin. Deducting the accrued interest from the full price leads to the clean price. The return components of an option-free fixed-rate bond comprise the coupon, the capital gain or loss, and the income from the reinvested coupons. The current

402

11

Bond Price and Yield

yield, the yield to maturity, and the total return are widely used yield measures in the bond market. • With the current yield, the annual coupon is divided by the clean price. The return measure calculated in this way only captures the coupon. The other return sources such as capital gain or loss and income from reinvested coupons are not included in the return figure. • The yield to maturity can be calculated by incorporating the full price, the coupons, and the par value of the bond in the price equation and proceeding to solve the equation according to the fixed risk-adjusted discount rate. The yield to maturity assumes that the bond is held to maturity and that the coupons are reinvested at the yield to maturity (and not at the prevailing market rate). • Unlike the yield to maturity, the total return allows the yield to be estimated for any investment period. In addition, the coupons can be reinvested at an interest rate available on the market. In a nutshell, the total return reflects the yield that the investor expects to realise from all three sources of return, namely coupon, capital gain/loss, and income on the reinvested coupons.

11.8

Problems

1. The following information is available for the Mercedes-Benz Group AG 1.4% bond, maturing on 12 January 2024 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

Mercedes-Benz Group AG DE000A169G15 Corporate bond EUR EUR 1000 EUR 500 million 12 January 2016 actual/actual ICMA 1.400% fixed, annual Annually on 12 January 12 January 2024 2 business days after close of trading A2 (according to Moody’s as at 3 February 2017)

The Mercedes-Benz Group bond will be purchased on Friday, 24 November 2017. The settlement date is Tuesday, 28 November 2017. The Z-spread is 10 basis points. The linearly interpolated EURIBOR zero-coupon swap rates are as follows (Source: Refinitiv Eikon and own calculations): • 0.123-year swap rate: -0.400% • 1.123-year swap rate: -0.308% • 2.123-year swap rate: -0.178%

11.8

• • • •

Problems

403

3.123-year swap rate: -0.055% 4.123-year swap rate: 0.086% 5.123-year swap rate: 0.225% 6.123-year swap rate: 0.359%

a) What is the full price of the Mercedes-Benz Group bond on the settlement date of 28 November 2017? b) What is the clean price of the Mercedes-Benz Group bond on the settlement date of 28 November 2017? 2. The following information is available for the NRW Bank’s 0% bond, maturing on 30 June 2022 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

NRW Bank DE000NWB13U1 Corporate bond EUR EUR 1000 EUR 10 million 17 August 2010 30/360 0% fixed, annual Annually on 30 June each year 30 June 2022 2 business days after close of trading Aa1 (according to Moody’s as at 7 March 2014)

The 0% bond of NRW Bank will be purchased on Friday, 24 November 2017. The settlement date is Tuesday, 28 November 2017. The Z-spread is -12 basis points. The EURIBOR zero-coupon swap rates on 24 November 2017 are as follows (Source: Refinitiv Eikon): • 4.5-year swap rate: 0.139% • 4.75-year swap rate: 0.174% What is the price of NRW Bank’s 0% bond on the settlement date of 28 November 2017? 3. A 4-year floating-rate note trades at 97% on the coupon date. The coupon is paid annually and consists of the 12-month EURIBOR and a quoted margin of 80 basis points. The current 12-month EURIBOR is 1%. The day-count convention is 30/360. It is assumed that the coupon periods are evenly spaced and that the EURIBOR zero-coupon swap rate curve is flat. What is the discount margin of the floating-rate note?

404

11

Bond Price and Yield

4. The following information is available for the Linde Finance B. V. 2% bond, maturing on 18 April 2023 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date:

Linde Finance B. V. DE000A1R07P5 Corporate bond EUR EUR 1000 EUR 650 million 18 April 2013 Actual/actual ICMA 2% fixed, annual Every year on 18 April 18 April 2023 2 business days after close of trading

The Linde Finance B. V. bond is purchased on Friday 24 November 2017 at an ask price of 110.145%. The bond will be sold later on a coupon date of 18 April 2020. A yield to maturity of 1% is expected on this date due to rising interest rates. The coupons can be invested at a market rate of 0.25%. What are the current yield, the yield to maturity, and the total return of the Linde Finance B. V. bond?

11.9

Solutions

1. a) There are a total of 45 days or 0.123 years (= 45 days/365 days) from the settlement date of 28 November 2017 to the next coupon date of 12 January 2018. The time to maturity of the bond is therefore 6.123 years. In order to calculate the full price of the corporate bond, the risk-adjusted discount rates that correspond to the timing of the bond’s cash flows must first be calculated: Zero-coupon swap rate -0.400% -0.308% -0.178% -0.055% 0.086% 0.225% 0.359%

Z-Spread 0.100% 0.100% 0.100% 0.100% 0.100% 0.100% 0.100%

Discount rate -0.300% -0.208% -0.078% 0.045% 0.186% 0.325% 0.459%

The full price of the Mercedes-Benz Group bond is 106.967% and can be determined as follows:

11.9

Solutions

B0, Full =

405

1:4% 1:4% 101:4% þ þ ... þ ð1 - 0:003Þ0:123 ð1 - 0:00208Þ1:123 ð1 þ 0:00459Þ6:123

= 106:967%: b) There are a total of 320 days from the last coupon date of 12 January 2017 to the settlement date of 28 November 2017. Hence, the accrued interest is 1.227%:

AI = 1:4% ×

320 days = 1:227%: 365 days

The clean price of 105.740% can be calculated by subtracting the accrued interest from the full price: B0, Clean = 106:967% - 1:227% = 105:740%: 2. There are a total of 212 days, or 0.589 years (= 212 days/360 days) from the settlement date of 28 November 2017 to the next coupon date of 30 June 2018, using the day-count convention 30/360. The 4.589-years linearly interpolated EURIBOR zero-coupon swap rate of 0.151% can be determined as follows:

Swap rate4:589 years = 0:139% þ

4:589 years - 4:5 years × ½0:174% - 0:139% 4:75 years - 4:5 years

= 0:151%: The expected return or risk-adjusted discount rate is 0.031% [= 0.151% + (0.12%)]. The price of the zero-coupon bond is 99.858% and can be calculated as follows:

B0 =

100% = 99:858%: ð1:00031Þ4:589

3. Since the EURIBOR zero-coupon swap rate curve is flat, the spot rates and the forward rates are equal. Thus, the coupon rate for all interest periods is 1.8%

406

11

Bond Price and Yield

(= 1% + 0.8%). The expected return or risk-adjusted discount rate is made up of the EURIBOR zero-coupon swap rate of 1% for each maturity and the discount margin. The price equation of the 4-year floating-rate note can be expressed as follows:

97% = þ

1:8% 1:8% 1:8% þ þ ð1 þ 0:01 þ DMÞ1 ð1 þ 0:01 þ DMÞ2 ð1 þ 0:01 þ DMÞ3

101:8% : ð1 þ 0:01 þ DMÞ4

Solving the price equation according to the expected return (i.e. 1% + DM), results in 2.6%. The discount margin is therefore 160 basis points or 1.6%: DM = 2:6% - 1% = 1:6%: Since the discount margin of 160 basis points is higher than the quoted margin of 80 basis points, the price of the floater of 97% is below the par value of 100% (discount floater). 4. The current yield of the Linde Finance B. V. bond is 1.82% and can be calculated as follows:

Current yield =

2% = 0:0182: 110:145%

The settlement date is Tuesday, 28 November 2017. There are a total of 224 days from the last coupon date of 18 April 2017 to the settlement date of 28 November 2017. Hence, the accrued interest is 1.227%:

AI = 2% ×

224 days = 1:227%: 365 days

The full price is 111.372%: B0, Full = 110:145% þ 1:227% = 111:372%:

Solutions

407

There are a total of 141 days, or 0.386 years (= 141 days/365 days), from the settlement date of 28 November 2017 to the next coupon date of 18 April 2018. The yield to maturity of 0.1099% can be determined with the following price equation:31

111:372% = þ

2% 2% 2% þ þ þ ... 0:386 1:386 ð1 þ YTMÞ ð1 þ YTMÞ2:386 ð1 þ YTMÞ

102% : ð1 þ YTMÞ5:386

The expected sale price on 18 April 2020 can be calculated with the projected yield to maturity of 1% as follows:

B18 April 2020 =

2% 2% 102% þ þ = 102:941%: 1 2 ð1:01Þ ð1:01Þ ð1:01Þ3

The final value of the reinvested coupons, which will be received annually on 18 April for each of the years 2018, 2019, and 2020, can be calculated using the reinvestment rate of 0.25% as follows: FV of reinvested coupons = 2% × ð1:0025Þ2 þ 2% × ð1:0025Þ1 þ 2% × ð1:0025Þ0 = 6:015%: The final value of 108.956% is made up of the expected sales price of 102.941% and the final value of the reinvested coupons of 6.015%. The purchase price of the Linde Finance B. V. bond is 111.372%. The total return is therefore -0.915%:

Total return =

31

108:956% 111:372%

1=2:386

- 1 = - 0:00915:

The Microsoft Excel applications at the end of the chapter demonstrate how to calculate the yield to maturity in Excel.

408

11

Bond Price and Yield

Microsoft Excel Applications • In Excel, the price of an option-free fixed-rate bond on a coupon date can be calculated with the function ‘NPV’. For example, the cash flows of the MercedesBenz Group bond with a coupon of 2% and a time to maturity of 10 years can be entered in cells A2 to A11. Cell A11 contains the par value as well as the last coupon. The discount rate (in decimal places) of 0.0072 should be entered in cell A1. Then the following expression should be written in an empty cell: = NPVðA1; A2:A11Þ: Then press the Enter key. The price calculation is presented in Fig. 11.8. • The price of an option-free fixed-rate bond can also be determined between two coupon dates. For this purpose, the ‘PRICE’ function in Excel can be used to calculate the clean price of a bond. The ‘PRICE’ function is defined as follows: PRICE(settlement date; maturity date; annualised coupon rate; annualised yield to maturity; 100 or par value in %; frequency of coupon payments; basis).

Fig. 11.8 Calculation of the price and the yield to maturity using the example of the MercedesBenz Group AG 2% 2019/2031 bond on the coupon date of 27 February 2021 and between two coupon dates on 16 March 2021 (Source: Own illustration)

Microsoft Excel Applications

409

• The frequency for coupon payments is 1 for annual, 2 for semi-annual, and 4 for quarterly. The basis is specified for the different day-count conventions as follows: – – – –

Basis 0 (or no specification): 30/360 Basis 1: actual/actual Basis 2: actual/360 Basis 3: actual/365

• If, for example, the clean price of the Mercedes-Benz Group AG 2% bond with a maturity date of 27 February 2031 and a yield to maturity of 0.6229% is to be determined on the settlement date of 16 March 2021, then the following should be entered: in cell B1 the settlement date of 16.03.2021, in cell B2 the maturity date of 27.02.2031, in cell B3 the annual coupon rate of 2%, in cell B4 the annualised yield to maturity of 0.6229%, in cell B5 the par value without the percentage sign of 100, in cell B6 the coupon frequency of 1, and in cell B7 the basis of 1. The following expression can be written in an empty cell: = PriceðB1; B2; B3; B4; B5; B6; B7Þ: Then press the Enter key. The price calculation is presented in Fig. 11.8. • The ‘PRICE’ function can also be used to determine the price on a coupon date without having to enter all cash flows of the bond separately, as is the case with the ‘NPV’ function. For example, if the time to maturity of the bond is 20 years, 01.01.2020 can be used for the settlement date and 01.01.2040 for the maturity date. • In order to determine the yield to maturity of the Mercedes-Benz Group bond with an annual coupon of 2% and a time to maturity of 10 years on a coupon date, the price of the bond must be written with a minus sign (-112.307) in cell C1, for example. Then the cash flows can be entered in cells C2 to C11. The yield to maturity can be determined with the function ‘IRR’ (internal rate of return), and this step is concluded by pressing the Enter key (see Fig. 11.8): = IRRðC1:C11Þ: • If, for example, the coupons are paid semi-annually, the ‘IRR’ function is applied to obtain the semi-annual yield to maturity, which is then annualised. Using the bond equivalent yield, the annualised yield to maturity can be calculated by multiplying the semi-annual yield to maturity by two. • If the yield to maturity for an option-free fixed-rate bond is to be determined between two coupon dates, the ‘YIELD’ function can be used, which is defined as follows:

410

11

Bond Price and Yield

YIELD(settlement date; maturity date; annualised coupon rate; price of the bond in % of the par value of 100; 100 or par value in %; frequency of coupon payments; basis). • For the Mercedes-Benz Group AG 2% bond, enter the settlement date of 16.03.2021 in cell D1, the maturity date of 27.02.2031 in cell D2, the annual coupon rate of 2% in cell D3, the clean price of 113.250 in cell D4, the par value of 100 in cell D5, the coupon frequency of 1 in cell D6, and the basis of 1 in cell D7. The following expression should be written in an empty cell: = YieldðD1; D2; D3; D4; D5; D6; D7Þ: Then press the Enter key. The calculation of the yield to maturity is presented in Fig. 11.8. • The yield to maturity calculated with the ‘YIELD’ function is the annualised yield to maturity according to the bond equivalent yield convention. If one is on a coupon date and the bond has a time to maturity of 20 years, for example, two fictitious dates can again be used for the settlement date and the maturity date (e.g. 01.01.2020 and 01.01.2040).

References Adams, J.F., Smith, D.J.: Introduction to fixed-income valuation. In: Petitt, B.S., Pinto, J.E., Pirie, W.L. (eds.) Fixed-income analysis, 3rd edn, pp. 91–150, Hoboken (2015) Bank for International Settlements 2015: BIS Quarterly Review December 2015, Basel (2015) Brealey, R.A., Myers, S.C.: Principles of corporate finance, 3rd edn, New York (1996) Brigham, E.F., Houston, J.F.: Fundamentals of financial management, 10th edn, Mason (2004) Buckley, A., Ross, S.A., Westerfield, R.W., Jaffe, J.F.: Corporate Finance Europe, London (1998) Fabozzi, F.J.: Fixed-income mathematics: analytical and statistical techniques, 2nd edn, Chicago (1993) Fabozzi, F.J.: Fixed-income analysis for the Chartered Financial Analyst® program, New Hope (2000) Fabozzi, F.J.: Fixed-income analysis, 2nd edn, Hoboken (2007) Ho, T.S., Lee, S.B., Wilcox, S.E.: The term structure and interest rate dynamics. In: Petitt, B.S., Pinto, J.E., Pirie, W.L. (eds.) Fixed-income analysis, 3rd edn, pp. 473–527, Hoboken (2015) Hull, J.C.: Options, futures, and other derivatives, 6th edn, Upper Saddle River (2006) Mondello, E.: Finance: Theorie und Anwendungsbeispiele, Wiesbaden (2017) Mondello, E.: Corporate Finance: Theorie und Anwendungsbeispiele, Wiesbaden (2022) O’Kane, D., Sen, S.: Credit spreads explained. J. Credit Risk. 1(2), 61–78 (2005) Reilly, F.K., Brown, K.C.: Investment analysis and portfolio management, 7th edn, Mason (2003)

References

411

Online Sources European Central Bank: Euro Area Securities Issue Statistics – Table Debt Securities by Original Maturity and Currency. http://sdw.ecb.europa.eu/reports.do?node=1000002757. Accessed 30 June 2022

Duration and Convexity

12.1

12

Introduction

The risk of a bond is analysed using the sensitivity measures of modified duration and modified convexity. These risk measures can be used to assess how much the bond price changes when the expected return (or the yield to maturity) moves. Duration improves with convexity in view of the fact that the relationship between price and yield to maturity of a fixed-rate bond is not linear. The relationship between the bond price and the expected return is negative. If the expected return increases (decreases), the price of the bond decreases (increases). The changes in the yield to maturity result from a change in the benchmark rate and the risk premium. Therefore, this chapter starts with an analysis of the risk factors relevant for valuation. This is followed by a presentation of the duration-convexity approach, which can be derived from the second-order Taylor series expansion. Next, Macaulay duration, modified duration, and modified convexity are described. Finally, the chapter examines applications of duration and convexity in portfolio management, which can be used for tactical asset allocation, investment strategies, such as the immunisation strategy, and for exploiting and hedging predicted interest rate and credit risk changes.

12.2

Analysis of the Risk Factors

12.2.1 Overview For example, purchasing a 5-year corporate bond with the intention of selling it after 1 year exposes the buyer to price risk. Should the expected return increase in 1 year, the bond price will decrease, resulting in a capital loss if everything else remains the same. In order to analyse the price risk, the risk factors have to be identified. To do

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_12

413

414

12

Duration and Convexity

this, the expected return is decomposed into its components, namely the benchmark rate and the risk premium: E ðr Þ = BR þ RP,

ð12:1Þ

where BR = benchmark rate, and RP = risk premium. When calculating the price of an option-free bond, the swap rate is usually taken as the benchmark rate.1 Nevertheless, the swap rate—derived from an interest rate for term deposits up to 12 months such as EURIBOR—includes a return compensation for the credit risk of the interbank market, and therefore the risk-free interest rate is more suitable as a benchmark rate for the risk analysis. €STR and SARON are better approximations of the risk-free interest rate than EURIBOR because they are overnight rates. If interest rates rise, the benchmark rate, and thus the expected return, increase, leading to a decline in the price of the bond. This risk of loss is referred to as interest rate risk.2 By contrast, the risk premium represents a return compensation for the credit risk and the market liquidity risk of the option-free bond. If the issuer fails to make full and timely payments of interest and/or principal according to the term of the fixed-income security or if this risk increases, the risk premium and thus the expected return rise, causing the bond price to fall. This risk of loss is called credit risk.3 Market liquidity risk, on the other hand, is related to the bond’s liquidity in the market. The lower the market liquidity, the higher the required risk premium or the expected return, and therefore the purchased bond can be sold at a lower price.4 Figure 12.1 presents the risk factors of an option-free bond based on the expected return. The three most important risk factors of an option-free fixed-rate bond are described in more detail below, namely the interest rate risk, the credit risk, and the market liquidity risk. For floating-rate notes, the interest rate risk is less relevant because the bond price has a price of 100% at each coupon date if the credit risk has not changed. On each coupon date, assuming a flat term structure, the coupon rate and the expected return are equal, and the price is therefore 100%. Consequently, the bond price can only change as a result of an interest rate change between two coupon dates. Therefore, with floating-rate notes, the risk of loss arises primarily from credit risk.5

1

See Sect. 11.4.2. See Tuckman and Serrat 2012: Fixed Income Securities: Tools for Today’s Markets, p. 118. 3 See Gootkind 2015: ‘Fundamentals of Credit Analysis’, p. 212. 4 See Fabozzi 2007: Fixed Income Analysis, p. 32. 5 See Sect. 11.5. 2

12.2

Analysis of the Risk Factors

Expected return

=

Risk-free interest rate

415

+

Risk premium

Interest rate risk

Credit risk

Market liquidity risk

Fig. 12.1 Risk factors of an option-free bond (Source: Own illustration)

12.2.2 Interest Rate Risk Prime government bonds from developed countries such as Germany, Switzerland, and the USA have almost no credit risk. Furthermore, market liquidity, especially in the primary market, is also very high. Nevertheless, such bonds are exposed to interest rate risk. Changes in interest rates affect not only the income on reinvested coupons, but also the price of the bond if it sold before it matures. A rise in interest rates as a result of a restrictive monetary policy, for example, causes the bond price to fall and the income from reinvested coupons to rise because the received coupons can be invested at a higher interest rate. Hence, the price risk and the reinvestment risk have an opposite effect on the return.6 For example, if an investor buys a bond on one coupon date and sells it on the next coupon date, he is only exposed to price risk. Price risk becomes more significant when the investment period is relatively short compared to the bond’s time to maturity. By contrast, the reinvestment risk of the coupons has a greater effect on the return when the investment horizon is relatively long compared to the bond’s time to maturity. For example, an investor buys a 5-year fixed-rate bond with an annual coupon rate of 3%. The yield to maturity is 3%, and the purchase price is therefore 100%. If the bond is sold in 1 year and the yield to maturity has increased from 3% to 4% due to an increase in interest rates, the bond can be sold at a price of 96.370%: B0 =

6

3% 3% 3% 103% þ þ þ = 96:370%: 1 2 3 ð1:04Þ ð1:04Þ ð1:04Þ ð1:04Þ4

See Adams and Smith 2015: ‘Understanding Fixed-Income Risk and Return’, p. 160 f.

416

12

Duration and Convexity

The final value of the investment of 99.370% is made up of the sales price of 96.370% and the coupon of 3%. Thus, the total return is -0.63% and can be calculated as follows: Total return =

99:370% - 1 = - 0:0063: 100%

The debt security is sold in 1 year, and therefore the investor is only exposed to the price risk; there is no reinvestment risk. In the event that the bond is sold in 4 years and the yield to maturity has already risen from 3% to 4% after 1 year, a selling price of 99.038% results for the bond in 4 years: B0 =

103% = 99:038%: ð1:04Þ1

Assuming that the coupons can be invested at the higher yield to maturity of 4%, the final value of the reinvested coupons is 12.739%: FV of reinvested coupons = 3% × ð1:04Þ3 þ 3% × ð1:04Þ2 þ 3% × ð1:04Þ1 þ 3% × ð1:04Þ0 = 12:739%: The final value in 4 years is 111.777% (= 99.038% + 12.739%). The annual total return is therefore 2.82%: Total return =

111:777% 100%

1=4

- 1 = 0:0282:

With a 4-year investment horizon, the return contribution from price risk is negative because the bond is sold at a price of 99.038%, which is lower than the purchase price of 100%. On the other hand, the contribution of the reinvestment risk to the return is positive because the coupons can be invested at a higher interest rate of 4%. Thus, the price risk and the reinvestment risk have an opposite effect on the return. With a long (short) investment horizon the effect of the reinvestment risk on the return is greater (smaller) than the price risk. In a buy-and-hold strategy, if the bond is bought at a price of 100% at the time of issue and held until maturity, the price risk is irrelevant. The bondholder receives the par value of 100% at maturity. However, they are exposed to the reinvestment risk. Should interest rates fall, the coupons can be invested at a lower interest rate, which reduces the income from the reinvested coupons and thus the return.

12.2

Analysis of the Risk Factors

417

12.2.3 Credit Risk Credit risk is the risk that the bond issuer fails to make the contractually agreed-upon interest and principal payments, which can lead to a delay in payment or even to the insolvency of the issuer. Not all bonds are equally affected by credit risk. For example, it is fairly certain that governments of developed countries with good credit ratings and companies with a large stock market capitalisation will meet their contractual payment obligations. By contrast, government bonds of developing countries or debt securities of companies with a poor credit rating are much riskier. The largest default on government bonds to date, involving more than USD 100 billion, took place in Argentina in December 2001. As a result of a severe recession lasting more than 3 years, the Argentine government stopped making debt payments at that time. The probability of such possible credit events is assessed by rating agencies and determined in the form of ratings. The most important rating agencies worldwide are Standard & Poor’s, Fitch Ratings, and Moody’s Investors Service. They assign a rating based on their assessment of the creditworthiness and probability of default of a debtor or a specific bond. Each agency uses its own system. The subdivision of issuers and bonds into different rating classes is based on the US school grades A, B, C, and so forth. Table 12.1 presents the long-term ratings ranked from highest to lowest of S&P, Fitch, and Moody’s.7 AAA for S&P and Fitch and Aaa for Moody’s reflect the best rating classification. Such a rating does not mean that default is not possible. Rather, the top rating reflects an extremely low probability of default. Other bonds with a ‘high-quality grade’ and thus minimal credit risk have a rating of AA or Aa. Bonds with a low credit risk have an A rating, while bonds with a BBB or Baa rating have a medium credit risk. Bonds with a BBB- or Baa3 rating or better are called investment grade. Bonds with a rating of BB+ or Ba1 rating or worse have a speculative credit character and a higher probability of default. They are classified as non-investment grade or speculative grade (also called junk or high yield). This is significant in that market participants such as pension plans may have investment restrictions that prohibit investment in non-investment grade debt securities. The worst credit rating receives a D rating from S&P and Fitch and a C rating from Moody’s, which means that the issuer is in default. In addition, Fitch distinguishes between a restricted default (RD) and a full default (D). In the former type of default, insolvency proceedings have not yet been initiated and business activities are continuing, whereas in the case of a full default either insolvency proceedings have been initiated or business activities have been discontinued. Moreover, rating agencies will typically provide a positive, stable, or negative outlook on their determined ratings. They may also disclose other indicators on the potential direction of their ratings under

7

The rating agencies also provide ratings on short-term debt instruments, such as commercial papers and bank deposits. However, the scales they use for the ranking are different from those used for the long-term ranking. For example, Fitch uses F-1, F-2, and F-3 for prime short-term debt investments.

418

12

Duration and Convexity

Table 12.1 Long-term rating classification of the three largest rating agencies worldwide (Source: Own illustration) Investment grade

Non-investment grade or speculative grade

S&P AAA

Fitch AAA

Moody’s Aaa

AA+ AA AAA+

AA+ AA AAA+

Aa1 Aa2 Aa3 A1

A ABBB+

A ABBB+

A2 A3 Baa1

BBB BBBBB+ BB BBB+

BBB BBBBB+ BB BBB+

Baa2 Baa3 Ba1 Ba2 Ba3 B1

B BCCC+

B BCCC+

B2 B3 Caa1

CCC CCCCC C D

CCC CCCCC C RD/D

Caa2 Caa3 Ca C

Credit risk High-quality grade: minimal credit risk

Upper-medium grade: low credit risk

Low-medium grade: medium credit risk

Speculative grade: high credit risk

Speculative grade: very high credit risk

Default

certain circumstances such as ‘on review for a downgrade’ or ‘on credit watch for an upgrade’. In addition, the ratings are supported by extensive written commentary, financial analysis of the rated debtor, and summary industry statistics. In general, issuers of bonds rated as investment grade have better access to the capital market and can borrow at lower interest costs than those rated as non-investment grade or speculative grade. Rating agencies are usually criticised for not downgrading a credit rating quickly enough when the fundamental factors of a company suddenly deteriorate sharply. Rating agencies have been repeatedly observed to carry out a downgrade from investment grade to non-investment grade only when the critical situation of the company is already known to the public. For example, the bank Lehman Brothers was downgraded to non-investment grade in 2008 only a few days before it filed for bankruptcy. In addition to the credit ratings provided by the rating agencies, credit risk can be assessed with credit default swaps (CDSs), which are credit derivatives in which the

12.2

Analysis of the Risk Factors

419

Table 12.2 Assessment of creditworthiness based on ratings and CDS spreads (Source: https:// derivate.bnpparibas.com/service/ueber-uns/bonitat-und-credit-default-swaps) Bank HSBC DZ Bank UBS BNP Paribas Bank of America Commerzbank Deutsche Bank Credit Suisse Unicredit Royal Bank of Scotland

S&P Rating AAAAA+ A AAABBB+ BBB BBB-

S&P Outlook Stable Stable Stable Stable Stable Negative Negative Stable Stable Stable

CDS spread in basis points 22.50 74.88 22.32 25.85 43.51 53.38 73.86 54.79 65.53 82.16

buyer pays the seller a CDS premium periodically (typically every quarter) for default protection. The CDS spread reflects a market-based measure of credit risk as CDSs are traded on the market. If the market assesses the probability of default as high (low), the CDS spread will rise (fall). Table 12.2 presents the S&P rating and the premium (price) in basis points of 5-year CDSs for banks as at 1 December 2017. For example, DZ Bank is rated AA- by S&P, while the CDS premium of 74.88 basis points is relatively high and corresponds more to a BBB rating.

12.2.4 Market Liquidity Risk A bond is illiquid if it has a low trading volume and therefore the price has a wide bid–ask spread. The purchase occurs at a higher ask price and the sale at a lower bid price than in the case of a liquid bond. This risk of loss is called the market liquidity risk.8 The market liquidity of a bond is influenced by the following factors:9 • Issuer: Bonds from issuers that are widely known have a higher demand and acceptance in the market. In addition, larger issuers with regular bond issues usually undertake price management and actively quote prices. A risk of loss exists if the trading volume, and thus the market liquidity, are negatively affected by a possible downgrade of the credit rating. This is particularly true in a market environment in which the risk aversion of investors increases. • Issue volume: Typically, liquidity increases (decreases) with a higher (lower) issue volume because the bonds are sold among a larger (smaller) number of market participants who can trade the bond on the market. With small issues, even small turnovers are sufficient to move the price considerably. Price jumps can also

8

See Fabozzi 2007: Fixed Income Analysis, p. 32. See Diwald 2012: Anleihen verstehen: Grundlagen verzinslicher Wertpapiere und weiterführende Produkte, p. 215 ff.

9

420

•

•

•

• • •

•

12

Duration and Convexity

occur. For many investors it is important that the bond is liquid so that they can sell the position at any time without a price discount. Therefore, they are willing to pay a higher price for liquid bonds, which is reflected in a lower yield. This is particularly important in times of crisis, when market participants invest not only in safe but also in liquid securities. Age of issue: Newly issued fixed-income securities are typically more liquid because previously issued debt securities are held in portfolios primarily by institutional investors and are not available for trading purposes. Moreover, new issues are more likely to track current market conditions, such as current coupon rates, and are therefore also more likely to be traded. Benchmark bond: A bond can be a benchmark for other debt securities if it has a large volume and liquidity and is also widely spread among market participants. This criterion is often met by newly issued prime government bonds (e.g. bonds from the Federal Republic of Germany). Credit rating: When the rating is downgraded below a certain level (e.g. from investment grade to speculative grade), certain investors such as mutual funds with a conservative investment policy or pension funds are no longer allowed to purchase or hold the bond. This has a negative effect on market liquidity. Bonds with and without embedded options: Plain vanilla bonds are usually more liquid than debt securities with embedded options (e.g. convertible bonds) because they have higher demand and are easier to value. Market making: The market making activity of dealers and brokers can improve market liquidity in the predominantly over-the-counter trading of bonds. Bonds that are listed are not necessarily liquid. Market environment: A favourable market environment in which supply and demand are high has a positive effect on turnover and market liquidity. If risk aversion among investors decreases, more risky debt securities are bought, which improves market liquidity. On the other hand, if risk aversion increases, risky debt securities are sold. The money is then invested in safe securities such as prime government bonds. Issuer refinancing needs: If an issuer has an urgent refinancing need and the market has only low turnover or is not sufficiently absorbent, the issue can only be placed on the market at a lower price.

12.3

Duration-Convexity Approach

12.3

421

Duration-Convexity Approach

The price change of an option-free fixed-rate bond due to a change in the yield to maturity can be approximated by using the Taylor series expansion with a secondorder approximation. After a change in the yield to maturity the bond price can be approximated over a short time period as follows:10 B1 ≈ B0 þ

dB 1 d2 B ΔYTM þ ΔYTM2 , dYTM 2 dYTM2

ð12:2Þ

where B0 = bond price before change in yield to maturity, B1 = bond price after change in yield to maturity, and ΔYTM = change in yield to maturity. The change in the bond price can therefore be approximated as follows: B1 - B0 ≈

dB 1 d2 B ΔYTM þ ΔYTM2 : dYTM 2 dYTM2

ð12:3Þ

dB/dYTM represents the first derivative of the bond price with respect to an infinitesimal change in the yield to maturity. The second derivative is given in the formula by d2B/dYTM2. If the first and second derivatives are each divided by the bond price B0, the modified duration (MDUR) and the modified convexity (MCONV) are obtained, respectively: MDUR = -

1 dB , B0 dYTM

ð12:4Þ

MCONV =

1 d2 B : B0 dYTM2

ð12:5Þ

Multiplying the modified duration and the modified convexity each by the bond price B0 and inserting them into Eq. (12.3) in the place of the first and second derivatives, one can approximate the change in the bond price due to a change in the yield to maturity with the Taylor series expansion of the second order as follows: 1 ΔB ≈ ð- MDURÞ B0 ΔYTM þ ðMCONVÞ B0 ΔYTM2 , 2

10

ð12:6Þ

For the derivation of the Taylor series expansion, see, for example, Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 961 f.

422

12

Duration and Convexity

where ΔB = B1 - B0 :

Example: Calculation of the Price Change of an Option-Free Fixed-Rate Bond Using the Taylor Series Expansion with a Second-Order Approximation An option-free fixed-rate bond with a time to maturity of 5 years and an annual coupon rate of 3% has a yield to maturity of 4%. The modified duration of the bond is 4.528, while the modified convexity is 25.598. According to the second-order Taylor series expansion, what are the absolute change in price and the price of the bond if interest rates fall by 75 basis points along the yield curve (parallel downward shift in interest rates)? Solution The price of the 5-year 3% bond can be calculated as follows: B0 =

3% 3% 3% 3% 103% þ þ þ þ = 95:548%: 1 2 3 4 ð1:04Þ ð1:04Þ ð1:04Þ ð1:04Þ ð1:04Þ5

The approximate absolute price change of the bond is 3.314% and can be determined using the second-order Taylor series expansion as follows: ΔB ≈ ð- 4:528Þ × 95:548% × ð- 0:0075Þ þ 0:5 × 25:598 × 95:548% × ð - 0:0075Þ2 = 3:314%: According to the duration-convexity approach the price of the bond after the interest rate change is equal to 98.862%: B1 = B0 þ ΔB = 95:548% þ 3:314% = 98:862%: If the price of the fixed-income security is determined by applying the cash flow model with a yield to maturity of 3.25% (= 4% - 0.75%), a price of 98.863% is obtained: B1 =

3% 3% 3% 3% 103% þ þ þ þ 1 2 3 4 ð1:0325Þ ð1:0325Þ ð1:0325Þ ð1:0325Þ ð1:0325Þ5

= 98:863%: The price of 98.863% calculated with the cash flow model is very close to the price of 98.862% determined with the Taylor series expansion. Hence, the second-order Taylor series expansion provides a very good approximation of the price change.

12.4

Duration

12.4

423

Duration

The modified duration of a bond measures the sensitivity of the full price (including accrued interest) to a change in the yield to maturity or, more generally, if interest rate risk is the focus of the analysis, to a change in the risk-free interest rate. Consequently, all other risk factors such as credit and market liquidity risk are left unchanged. Furthermore, it is assumed that the time to maturity, and as a result the accrued interest, remain the same, which means that a change in the full price reflects a change in the clean price. Hence, the modified duration makes it possible to estimate an immediate change in the bond price due to a change in the yield to maturity.11 The modified duration and the Macaulay duration are described in more detail below, prior to the discussion of the modified convexity and the applications of these measures in portfolio management.

12.4.1 Modified Duration and Macaulay Duration With the Taylor series expansion of the first order, the approximate absolute price change of a bond can be calculated as follows: ΔB ≈ ð- MDURÞB0 ΔYTM:

ð12:7Þ

Solving the formula according to the modified duration leads to the following equation: MDUR ≈ -

ΔB=B0 Δ%B =, ΔYTM ΔYTM

ð12:8Þ

where Δ%B = percentage change in the price of the bond. For example, if the modified duration is 5 and the yield to maturity increases by 1%, the bond price falls by approximately 5%. The modified duration thus demonstrates by how much the price of the bond changes in percentage terms when the yield to maturity moves. Like the beta of an equity security, it is a sensitivity measure. The beta measures how much the return of the stock changes when the return of the equity market moves.12 According to the cash flow model, the price equation of an option-free fixed-rate bond on a coupon date can be stated as follows:

11 12

See Adams and Smith 2015: ‘Understanding Fixed-Income Risk and Return’, p. 162 f. See Sect. 6.2.2.

424

12 T

B0 =

Duration and Convexity

CFt ð1 þ YTMÞ - t ,

ð12:9Þ

t=1

where CFt = cash flows of the fixed-rate bond comprising coupon payments for the periods t = 1 to T and principal payment at maturity T. In risk analysis, it is important to understand how much the price of the bond will change if the yield to maturity moves. For this purpose, the first derivative of the bond price with respect to an infinitesimal (infinitely small) change in the yield to maturity is calculated as follows:13 ΔB = ΔYTM

T

ð- t ÞCFt ð1 þ YTMÞ - ðtþ1Þ = -

t=1

1 ð1 þ YTMÞ

tCFt  : ð1 þ YTMÞt

T t=1

ð12:10Þ

The modified duration of the option-free fixed-rate bond is obtained by dividing the first derivative, or Eq. (12.10), by the price of the bond B0: T

tCFt ð1 þ YTMÞt ΔB=ΔYTM 1 t=1 =: B0 B0 ð1 þ YTMÞ

ð12:11Þ

tCFt ð1 þ YTMÞt The term to the right of the equals sign is called the Macaulay B0 duration, which is named after the Canadian economist Frederick Macaulay.14 If the YTM , the modified duration Macaulay duration (MacDUR) is divided by 1 þ m results: T t=1

MDUR =

MacDUR , YTM 1þ m

ð12:12Þ

where

13

In the following explanations, the limit notation is not used. Therefore, the following relationship dB ΔB applies: dYTM = lim ΔYTM = - MDUR B0 : ΔYTM → 0

14

See Macaulay 1938: Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856, p. 1 ff.

12.4

Duration

425

m = number of coupons per year (e.g. m = 1 for a bond with an annual coupon and m = 2 for a bond with a coupon paid semi-annually). The Macaulay duration is equal to the present value of the time-weighted cash flows divided by the present value of the cash flows or the bond price. It can be determined on a coupon date for a bond with an annual coupon as follows:

MacDUR ¼

T t=1

tCFt ð1 þ YTMÞt B0

T × ðC þ PVÞ 2×C 1×C þ þ ... þ ð1 þ YTMÞT ð1 þ YTMÞ1 ð1 þ YTMÞ2 ¼ : C C þ PV C þ þ . . . þ ð1 þ YTMÞT ð1 þ YTMÞ1 ð1 þ YTMÞ2

ð12:13Þ

where C = coupon payment for periods t = 1 to T, PV = principal or par value paid at maturity T, and T = number of years or evenly spaced periods to maturity. The Macaulay duration is the weighted average time period of the bond’s cash flows. Thus, Macaulay duration is a measure of the average life of the bond and is expressed in years. For example, if the price of a 5-year bond with an annual coupon rate of 3% and a yield to maturity of 4% is 95.548%, the Macaulay duration of 4.709 can be determined as follows: 1 × 3% 2 × 3% 3 × 3% 4 × 3% 5 × 103% þ þ þ þ ð1:04Þ1 ð1:04Þ2 ð1:04Þ3 ð1:04Þ4 ð1:04Þ5 = 4:709: MacDUR = 95:548% The Macaulay duration, or weighted average time period of the cash flows, is 4.709 years. It is less than the time to maturity of the bond of 5 years because a fixedrate bond pays coupons before the maturity date. The modified duration of 4.528 results from dividing the Macaulay duration by 1 plus the yield to maturity: MDUR =

4:709 = 4:528: 1:04

In contrast to the Macaulay duration, the modified duration is a sensitivity measure. For example, if the yield to maturity falls by 1%, the bond price rises by approximately 4.528%: Δ%B ≈ ð- 4:528Þ × ð- 0:01Þ = 0:04528:

426

12

Duration and Convexity

Example: Calculation of the Macaulay Duration and the Modified Duration Using the Mercedes-Benz Group AG 2% 2019/2031 Bond The following information is available for the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

Mercedes-Benz Group AG DE000A2TR083 Corporate bond EUR EUR 1000 EUR 750 million 27 February 2019 Actual/actual ICMA 2% fixed, annual Every year on 27 February 27 February 2031 2 business days after close of trading A3 (according to Moody’s as at 13 December 2019)

The yield to maturity is 0.6394% on 27 February 2021. What are the Macaulay duration and the modified duration of the Mercedes-Benz Group bond? Solution The time to maturity of the bond is 10 years. The full price of the MercedesBenz Group bond of 113.1395% can be determined as follows on the coupon date of 27 February 2021: B0 =

2% 2% 102% þ þ ... þ = 113:1395%: 1 2 ð1:006394Þ ð1:006394Þ ð1:006394Þ10

The Macaulay duration of 9.223 can be found by dividing the present value of the time-weighted cash flows by the full price of the bond: 2 × 2% 10 × 102% 1 × 2% þ þ ... þ ð1:006394Þ10 ð1:006394Þ1 ð1:006394Þ2 MacDUR = = 9:223: 113:1395% The modified duration of the Mercedes-Benz Group bond is 9.164 and can be calculated as follows: MDUR =

9:223 = 9:164: 1:006394

12.4

Duration

427

Table 12.3 Calculation of the Macaulay duration (Source: Own illustration)

Period (t) 1 2 3 4 5 Total

Cash flow (CFt) 3 3 3 3 103

Present value of cash flows CFt ð1 þ YTMÞt 2.885 2.774 2.667 2.564 84.658 95.548

Weight CFt ð1 þ YTMÞt B0

Period × weight CFt ð1 þ YTMÞt t B0

0.030 0.029 0.028 0.027 0.886 1.000

0.030 0.058 0.084 0.107 4.430 4.709

12.4.2 Factors Affecting Duration and Price Volatility Modified duration is used to estimate the price change of a bond when the yield to maturity moves. The higher (lower) the modified duration, the higher (lower) the price change for a given change in the yield to maturity. Therefore, the modified duration can be applied to assess the price volatility of the bond when risk factors such as the interest rate level, the creditworthiness of the issuer, and/or the market liquidity of the bond change. The yield to maturity is a function of these risk factors.15 The price volatility of an option-free fixed-rate bond can be evaluated with the Macaulay duration because the relationship between the Macaulay duration and the modified duration is positive (see Eq. 12.12). For this purpose, the Macaulay duration can be defined by the following equation, where the present value of the time-weighted cash flows per period is divided by the bond price (on a coupon date): C C C þ PV 1 2 ð1 þ YTMÞ ð1 þ YTMÞ ð1 þ YTMÞT þ 2× þ ... þ T × : MacDUR = 1 × B0 B0 B0 ð12:14Þ The Macaulay duration can be interpreted, as already mentioned, as the weighted average time period of the cash flows. The weights are given by the present value of each cash flow divided by the bond price.16 Table 12.3 presents the calculation of the Macaulay duration of 4.709 for the 5-year bond with an annual coupon rate of 3% and a yield to maturity of 4% using Eq. (12.14). The first column indicates years 1–5 on which the cash flows of the bond are paid out. The second column presents the annual cash flows of the bond. The third column indicates the present value of each cash flow discounted at the 4% yield 15 16

See Sect. 12.2.1. See Fabozzi 1993: Fixed Income Mathematics: Analytical and Statistical Techniques, p. 158.

428

12

Duration and Convexity

Table 12.4 Impact of a change in bond’s time to maturity, coupon rate, and yield to maturity on Macaulay duration (Source: Own illustration) Par value Time to maturity Coupon rate Coupon frequency Yield to maturity Price Macaulay duration

Reference bond 100% 5 years 3% Annual 4% 95.548% 4.709

Bond 2 100% 6 years 3% Annual 4% 94.758% 5.566

Bond 3 100% 5 years 2% Annual 4% 91.096% 4.797

Bond 4 100% 5 years 3% Annual 3% 100% 4.717

to maturity and the sum of these present values, which equals the bond price of 95.548%. The fourth column presents the present value of each cash flow divided by the bond price or the weights, which add up to 1. The last column indicates the product of the first and fourth columns, or the time periods of the cash flows multiplied by the corresponding weight. The sum of these numbers gives the Macaulay duration of 4.709. With the Macaulay duration, the weighted average time period of the bond’s cash flows is determined by taking into account both the different timing of cash flows and the amount of cash flows relative to the bond price. For the 5-year bond with an annual coupon rate of 3%, the cash flow at the end of each year for the first 4 years is 3%, whereas the cash flow at maturity of 103% is much larger. As a result, the Macaulay duration of 4.709 years is less than the time to maturity of the bond of 5 years. Furthermore, the Macaulay duration is relatively close to the time to maturity of 5 years because the largest cash flow of 103% occurs on the maturity date. Table 12.3 indicates that the weight of the cash flow at the end of the first year is 3% and contributes only 0.64% (= 0.03/4.709) to the Macaulay duration of 4.709. By contrast, 88.6% of the bond price is recovered at maturity, contributing 94.08% (= 4.430/4.709) to the Macaulay duration. The Macaulay duration of a zero-coupon bond can be determined from Eq. (12.14) as follows: PV ð1 þ YTMÞT MacDUR0% bond = T B0

= T:

ð12:15Þ

The present value of the principal paid at maturity [PV/(1 + YTM)T] equals the bond price B0, and therefore the Macaulay duration of a zero-coupon debt security corresponds to its time to maturity T. For example, in the case of a 5-year zerocoupon bond, the weighted average time period of the cash flows is 5 years because the principal is paid out only on the bond’s maturity date. Macaulay duration is positively related to time to maturity. The longer (shorter) the time to maturity, the higher (lower) the Macaulay duration and thus the modified duration, resulting in a higher (lower) price change when the yield to maturity moves. Conversely, the relationship between Macaulay duration and the coupon

12.4

Duration

429

Table 12.5 Non-linear relationship between the bond’s time to maturity and Macaulay duration (Source: Own illustration) Time to maturity of bond in years Macaulay duration

5 4.709

10 8.723

20 14.909

30 19.104

40 21.845

50 23.581

rate, as well as the yield to maturity, is negative. To illustrate this relationship, Table 12.4 again lists the 5-year bond with an annual coupon rate of 3% and a yield to maturity of 4%. This bond is taken as the reference instrument so that it is possible to examine how the Macaulay duration varies when the time to maturity (bond 2), the coupon rate (bond 3), and the yield to maturity (bond 4) are changed. The table indicates that the Macaulay duration increases from 4.709 to 5.566 for a 1-year increase in time to maturity. The Macaulay duration increases from 4.709 to 4.797 when the coupon rate decreases from 3% to 2%. Finally, the Macaulay duration increases from 4.709 to 4.717 when the yield to maturity falls by 1%. The positive relationship between Macaulay duration and time to maturity is obvious. The longer (shorter) the time to maturity, the longer (shorter) the weighted average time period of the cash flows, all other factors remaining equal. However, this relationship is not linear. Table 12.5 indicates for the 5-year bond with a coupon rate of 3% and a yield to maturity of 4% that as the time to maturity gradually increases, the Macaulay duration increases at a lower rate. If, for example, the bond’s time to maturity increases by a factor of 10—for example, from 5 to 50 years—the Macaulay duration increases only by a factor of approximately 5, from 4.709 to 23.581. As the bond’s time to maturity increases, the cash flows are discounted over a longer period of time, resulting in a lower present value for later cash flows. Therefore, the Macaulay duration increases at a lower rate than the bond’s time to maturity. This relationship applies to almost all coupon-paying bonds.17 For zerocoupon bonds, however, the Macaulay duration and the time to maturity are equal, and therefore both increase at the same rate. There is a negative relationship between the Macaulay duration and the coupon rate. This relationship can be explained by the fact that a higher coupon rate leads to relatively higher present values of the cash flows (weights) before maturity. Hence, the weight of the cash flow at maturity is lower, implying a decrease in the Macaulay duration. Finally, the relationship between the Macaulay duration and the yield to maturity is also negative because, as the yield to maturity increases, cash flows closer in time receive a larger weight and cash flows further away in time receive a smaller weight due to the discounting effect, resulting in a lower Macaulay duration.18

17 Discount bonds with a long time to maturity and a coupon rate far below the yield to maturity are an exception. With such bonds, when the time to maturity shortens, the Macaulay duration can first rise before it starts to fall. See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 567 ff. 18 See Adams and Smith 2015: ‘Understanding Fixed-Income Risk and Return’, p. 176.

430

12

Duration and Convexity

Example: Assessing the Price Volatility of Option-Free Bonds A portfolio consists of the following four option-free fixed-rate bonds: Bond Bond A Bond B Bond C Bond D

Time to maturity 20 years 20 years 10 years 9 years

Coupon rate (paid annually) 0% 5.5% 6.0% 6.5%

Yield to maturity 5.2% 5.5% 5.5% 5.7%

How can the price volatility of the four bonds be assessed? Solution The 20-year zero-coupon bond has the highest duration or price volatility because it is the security with the longest time to maturity and the lowest coupon rate and yield to maturity. Bond B has the second-highest price volatility. Like the zero-coupon security, it has the longest time to maturity of 20 years, but has a higher coupon rate and a higher yield to maturity. Bond C has a lower duration than the first two securities due to its shorter time to maturity of 10 years and higher coupon rate. Bond D has the shortest time to maturity of 9 years and the highest coupon rate and yield to maturity, and therefore the duration or price volatility is the lowest. If the modified durations of the four bonds are calculated, the following values are obtained, which confirm the results of the qualitative analysis: • • • •

12.5

Bond A: Modified duration = 19.011 (= 20/1.052) Bond B: Modified duration = 11.950 Bond C: Modified duration = 7.438 Bond D: Modified duration = 6.760

Convexity

Modified duration is a sensitivity measure which is used to determine the linear relationship between the bond price and the yield. However, this relationship is not linear, because the price curve is convex.19 Hence, modified duration is an appropriate price sensitivity measure only for small yield changes. For example, for the 5-year bond with an annual coupon rate of 3%, a price of 95.548%, and a modified duration of 4.528, an increase in the yield to maturity from 4% to 5% results in a price change according to the first-order Taylor series expansion of approximately 4.326%:

19

See Sect. 11.4.1.

12.5

Convexity

431

Table 12.6 Price change of the 5-year 3% bond calculated with the duration approach, the duration-convexity approach, and the cash flow model (Source: Own illustration) Yield to maturity 2.5% 3.0% 3.3% 3.5% 3.7% 3.9% 4.0%

Price change using duration approach (first-order Taylor series expansion) 6.490% 4.326% 3.028% 2.163% 1.298% 0.433% Not applicable

Price change using durationconvexity approach (second-order Taylor series expansion) 6.765% 4.449% 3.088% 2.194% 1.309% 0.434% Not applicable

4.1% 4.3% 4.5% 4.7% 5.0% 5.5%

-0.433% -1.298% -2.163% -3.028% -4.326% -6.490%

-0.431% -1.287% -2.133% -2.969% -4.204% -6.214%

Price change using cash flow model 6.775% 4.452% 3.090% 2.194% 1.309% 0.434% Not applicable -0.431% -1.287% -2.133% -2.970% -4.207% -6.224%

Δ%B ≈ ð- 4:528Þ × 95:548% × 0:01 = - 4:326%: The bond price after the change in yield to maturity is therefore 91.222%: B1 = B0 þ ΔB = 95:548% þ ð- 4:326%Þ = 91:222%: Based on the cash flow model, the price after the 1% increase in the yield to maturity would be 91.341%: B1 =

3% 3% 3% 3% 103% þ þ þ þ = 91:341%: 1 2 3 4 ð1:05Þ ð1:05Þ ð1:05Þ ð1:05Þ ð1:05Þ5

The example demonstrates that in the case of an increase in the yield to maturity, the price decline is overestimated using the Taylor series expansion of the first order, which is based on modified duration. Instead of decreasing, as predicted by the firstorder Taylor series expansion, by 4.326%, the bond price falls by only 4.207% (= 95.548% - 91.341%). A better approximation of the price change is obtained if, in addition to the modified duration, the modified convexity of the bond of 25.598 is used, because the relationship between the bond price and the yield to maturity is not linear. Applying the duration-convexity approach (Taylor series expansion of the second order) results in a price change of approximately -4.204%, which is very close to the actual price movement of -4.207%: ΔB ≈ ð- 4:528Þ × 95:548% × 0:01 þ 0:5 × 25:598 × 95:548% × ð0:01Þ2 = - 4:204%: Table 12.6 presents the price change of the 5-year 3% bond with a yield to maturity of 4% for a stepwise equal change in the yield to maturity downwards and

432

12

Duration and Convexity

(Price) 115%

Convexity adjustment of 1 MCONV B ∆YTM 2

110% 105% 100% B0 95% 90%

Price curve

85% 80%

Tangent line at the point B0 and YTM0

75% 0%

1%

2%

3%

4% YTM0

5%

6%

7%

8%

(Yield to maturity)

Fig. 12.2 Adjustment of modified duration with modified convexity (Source: Own illustration)

upwards using the Taylor expansion series of the first order and the second order, as well as the cash flow model. In general, the modified duration is a good approximation for small changes in yield, but not for larger yield changes. Furthermore, Table 12.6 illustrates that the duration approach overestimates the price decrease of the bond when the yield to maturity increases. Adopting the duration-convexity approach leads to a better approximation of the actual price change measured with the cash flow model. For this purpose, a convexity component of 0.5 MCONV B0 (ΔYTM)2 must be added to the first-order Taylor series expansion of (-MDUR) B0 ΔYTM. By contrast, the duration approach underestimates the price increase of the bond when the yield to maturity falls. In order to calculate a better approximation of the actual price increase, the duration-convexity approach must be applied, in which the convexity component of 0.5 MCONV B0 (ΔYTM)2 is added to the first-order Taylor series expansion of (-MDUR) B0 ΔYTM. The figures presented in Table 12.6 also indicate that due to the convex price curve, the price increase exceeds the price decrease when the yield to maturity moves evenly downwards and upwards. Furthermore, the modified convexity of an option-free bond is positive. The convexity adjustment [i.e. 0.5 MCONV B0 (ΔYTM)2] is added to the first term of the Taylor series expansion to estimate a more accurate price change. This additional term is positive as the yield to maturity is squared. The error in the calculation of the price change caused by the duration is corrected by the positive convexity term. Figure 12.2 presents the price curve of the 5-year 3% bond. If the price change is calculated using the modified duration, it can be represented with a tangent line at the price point B0 of 95.548% and at the yield to maturity point YTM0 of 4%. At this price-yield point, the slope of the tangent line and the price curve are equal. Thus, with

12.5

Convexity

433

the duration approach the change in the bond price along the tangent line is estimated. The tangent line captures the linear relationship between the bond price and the yield to maturity. The duration approach is an appropriate means of calculating the bond price movement caused by small yield changes, because there is little difference between the tangent line and the price curve line if the yield changes are small. However, the priceyield relationship is not linear, which means that for a larger yield change, a convexity adjustment of 0.5 MCONV B0 (ΔYTM)2 must be added to the first-order Taylor expansion to obtain a better approximation of the price change. Since large interest rate changes over short time periods hardly ever occur, the modified duration is in most cases suitable for evaluating the price volatility of a bond.20 The size of the convexity adjustment depends on the price curve. The more convex the price curve, the greater the difference between the bond price on the curve line and the price calculated with the modified duration on the tangent line. The convexity is positively affected by the time to maturity and negatively by the coupon rate and the yield to maturity. In other words, modified duration is a good approximation of the price volatility of a bond when the bond’s time to maturity is short and the coupon rate and the yield to maturity are high. Another factor that has an impact on convexity is the dispersion of the cash flows, which is the degree to which coupon and principal payments are spread over time. Bonds with the same duration but higher dispersion of cash flows have a greater convexity.21 A bond with a higher rather than lower positive convexity will experience a higher price increase when the yield falls and a lower price decrease when the yield rises. Therefore, investors are willing to pay a higher price for an option-free debt security with a greater positive convexity. However, a higher price has a negative impact on the yield to maturity. In other words, the benefit of a high positive convexity is paid for at a higher price, which lowers the yield on the bond investment. As Nobel prize-winning economist Milton Friedman said, ‘There is no such thing as a free lunch.’22 The modified convexity is the second derivative of the option-free fixed-rate bond price with respect to a change in the yield to maturity divided by the bond price.23 The second derivative is equal to the first derivative with respect to a change in the yield and can be calculated on a coupon date as follows:24

20

See Fabozzi 2007: Fixed Income Analysis, p. 180. See Adams and Smith 2015: ‘Understanding Fixed-Income Risk and Return’, p. 192. 22 Milton Friedman was born in 1912 and died in 2006. He was awarded the Nobel Prize in Economics in 1976. 23 See Sect. 12.3. 24 See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 586. 21

Δ2 B = ΔYTM2 T

= t=1

T

- ðt þ 1Þð- t ÞCFt ð1 þ YTMÞ - ðtþ1Þ - 1 =

€t = 1

ðt 2 þ t ÞCFt 1 = tþ2 ð1 þ YTMÞ2 ð1 þ YTMÞ

T t=1

T t=1

ðt 2 þ t ÞCFt ð1 þ YTMÞt

t 2 þ t CFt ð1 þ YTMÞ - ðtþ2Þ

434

12

Δ2 B 1 = ΔYTM2 ð1 þ YTMÞ2

T t=1

Duration and Convexity

ðt 2 þ t ÞCFt , ð1 þ YTMÞt

ð12:16Þ

where CFt = cash flows of the fixed-rate bond consisting of coupon payments for the periods t = 1 to T and principal payment at maturity T. The modified convexity can be determined by dividing the second derivative by the bond price:

MCONV =

1 ð1 þ YTMÞ2

T t=1

ðt 2 þ tÞCFt ð1 þ YTMÞt : B0

ð12:17Þ

To calculate the modified convexity of the 5-year 3% bond with a yield to maturity of 4% and a price of 95.548%, the value of the formula term from ðt 2 þ tÞCFt of 2645.458 must first be determined: Eq. (12.17) Tt = 1 ð1 þ YTMÞt

Period (year t) 1 2 3 4 5 Total

2

t +t 2 6 12 20 30

CFt 3 3 3 3 103

2

(t + t)CFt 6 18 36 60 3090 3210

ðt 2 þ tÞCFt ð1 þ YTMÞt 5.769 16.642 32.004 51.288 2539.755 2645.458

The modified convexity of the 5-year 3% bond is 25.598: MCONV =

2645:458 1 = 25:598: × 2 95:548 ð1:04Þ

Example: Calculation of the Modified Convexity and Approximate Price Change with the Duration-Convexity Approach for the Mercedes-Benz Group AG 2% 2019/2031 Bond The following information is available for the Mercedes-Benz Group AG 2% bond, maturing on 27 February 2031 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency:

Mercedes-Benz Group AG DE000A2TR083 Corporate bond EUR

(continued)

12.5

Convexity

435

Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date: Settlement date: Rating:

EUR 1000 EUR 750 million 27 February 2019 Actual/actual ICMA 2% fixed, annual Every year on 27 February 27 February 2031 2 business days after close of trading A3 (according to Moody’s as at 13 December 2019)

The yield to maturity is 0.6394% on 27 February 2021. The modified duration of the Mercedes-Benz Group bond is 9.164. 1. What is the modified convexity of the Mercedes-Benz Group bond? 2. What is the bond price with the duration-convexity approach and the cash flow model when the yield to maturity rises from 0.6394% to 1.5%? Solution to 1 The time to maturity of the bond is 10 years. The full price of the MercedesBenz Group bond is 113.140% and can be calculated as follows on the coupon date of 27 February 2021: B0 =

2% 2% 102% þ þ ... þ = 113:140%: 1 2 ð1:006394Þ ð1:006394Þ ð1:006394Þ10

To determine the modified convexity of the Mercedes-Benz Group bond, ðt 2 þ tÞCFt of 11,158.4375 must first be the value of the formula term Tt = 1 ð1 þ YTMÞt computed:

Period (year t) 1 2 3 4 5 6 7 8 9 10 Total

2

t +t 2 6 12 20 30 42 56 72 90 110

CFt 2 2 2 2 2 2 2 2 2 102

2

(t + t)CFt 4 12 24 40 60 84 112 144 180 11,220 11,880

ðt 2 þ tÞCFt ð1 þ YTMÞt 3.9746 11.8480 23.5455 38.9931 58.1181 80.8483 107.1129 136.8416 169.9653 10,527.1902 11,158.4375

(continued)

436

12

Duration and Convexity

The modified convexity of the 10-year Mercedes-Benz Group 2% bond is 97.376: MCONV =

1 11,158:4375 × = 97:376: 113:140 ð1:006394Þ2

Solution to 2 The approximate price change of the Mercedes-Benz Group bond of 8.515% can be determined with the duration-convexity approach as follows: ΔB ≈ ð- 9:164Þ × 113:140% × 0:008606 þ 0:5 × 97:376 × 113:140% × ð0:008606Þ2 = - 8:515%: According to the duration-convexity approach, the price of the bond after the yield change is 104.625%: B1 = B0 þ ΔB = 113:140% þ ð- 8:515%Þ = 104:625%: If the price of the Mercedes-Benz Group bond is determined using the cash flow model, the price is 104.611%: B1 =

2% 2% 102% þ þ ... þ = 104:611%: ð1:015Þ1 ð1:015Þ2 ð1:015Þ10

The price of 104.611% calculated with the cash flow model is very close to the price of 104.625% obtained with the Taylor series expansion of the second order. Hence, the duration-convexity approach makes it possible to estimate a very good approximation of the price change. For an option-free fixed-rate bond, the same modified duration and modified convexity measures apply for a change in the benchmark rate as for a change in the risk premium. A shift in the benchmark rate can result from changes in the expected inflation rate or the real interest rate. By contrast, a shift in the risk premium can arise from a change in credit risk of the issuer or in the market liquidity of the bond.25 Thus, for an option-free fixed-rate bond the ‘inflation duration’, the ‘real rate duration’, the ‘credit spread duration’, and the ‘market liquidity duration’ are all the same number. Modified duration is a sensitivity measure that makes it possible to

25

See Sect. 12.2.1.

12.6

Applications

437

determine how much the bond price will move (percentage wise) if the expected inflation rate, the real interest rate, the credit spread, and the market liquidity spread were to change by a certain amount.26 For example, the yield to maturity of a corporate bond is 4.5%. It consists of a risk-free benchmark yield of 2.1% and a risk premium of 2.4%. Two percent of the risk premium can be attributed to credit risk and the remaining 0.4% to market liquidity risk. If the modified duration, which is the same as the credit spread duration, is 5 and one expects the credit spread to rise from 2% to 2.8%, the bond price falls by approximately 4%: Δ%B ≈ ð- 5Þ × 0:8% = - 4%:

12.6

Applications

Duration and convexity are important tools in the management of a bond portfolio. They are used in the following portfolio management applications: 1. Tactical asset allocation 2. Exploiting interest rate expectations 3. Immunisation strategy 1. Tactical asset allocation: The bond portion of a portfolio may deviate from the weighting specified in the strategic asset allocation over a short period of time. If the portfolio manager expects rising interest rates and share prices, for example, they can reduce the bond portion and at the same time increase the equity portion in the portfolio. The bond allocation in the portfolio can be reduced synthetically with short bond futures. The number of bond futures required for this can be determined with a hedge ratio, which consists of the difference between the target duration of 0 and the modified duration of the bond position divided by the implied modified duration of the bond futures: Hedge ratio =

0 - MDURB , MDURF

ð12:18Þ

where MDURB = modified duration of bond portion in the portfolio, and MDURF = implied modified duration of bond futures (modified duration of the futures’ underlying bond).

26

See Adams and Smith 2015: ‘Understanding Fixed-Income Risk and Return’, p. 201.

438

12

Duration and Convexity

Assuming the modified duration of the bond position is 10 and the implied modified duration of a Euro-Bund futures contract is 8, the hedge ratio is -1.25 [= (0–10)/ 8]. To determine the number of Euro-Bund futures, the hedge ratio is multiplied by the market value of the bond position and divided by the value of one bond futures contract:27

NF =

0 - MDURB B , MDURF Fq

ð12:19Þ

where NF = number of bond futures, B = market value of bond portion in the portfolio, F = quoted (traded) price of bond futures, and q = contract size of bond futures (e.g. for a Euro-Bund futures contract EUR 100,000).

For example, if the market value of the bond position is EUR 15 million and the Euro-Bund futures contract trades at a price of 150%, a total of 125 short EuroBund futures contracts are necessary to temporarily eliminate the interest rate risk exposure arising from the bonds in the portfolio: N F = -1:25 ×

EUR 15, 000, 000 1:5 × EUR 100, 000

= -125:

The share of the diversified equity position in the portfolio can be synthetically increased with long equity index futures. Derivatives are suitable for tactical asset allocation because they have lower transaction costs than spot market instruments such as bonds and equities. Moreover, implementation is faster because with derivatives the systematic risk exposures are traded and not the individual assets in the portfolio. In addition, the portfolio manager’s investment strategy is not disrupted because the assets still remain in the portfolio. With derivatives, only the asset allocation is synthetically changed. 2. Exploiting interest rate expectations: The portfolio manager can use their own interest rate expectations to achieve a higher return by adjusting the modified duration of the bond portfolio with the help of interest rate derivatives such as bond futures. Should the portfolio manager expect interest rates to fall (rise),

27

The yield beta, which is the change in the yield of the bond position divided by the change in the yield of the bond futures’ underlying government bond, is assumed to be 1. See, for example, Chance 2003: Analysis of Derivatives for the CFA® Program, p. 350.

12.6

Applications

439

they can increase (decrease) the modified duration of the bond portfolio with long (short) bond futures.28 For example, if the modified duration of the bond portfolio with a market value of EUR 15 million is 10 and the portfolio manager forecasts falling interest rates, they can temporarily increase the modified duration of the portfolio to 12 using bond futures. Taking Euro-Bund futures with an implied modified duration of 8 that are traded at a price of 150%, a total of 25 long Euro-Bund futures contracts are required:

NF =

12 - 10 8

×

EUR 15, 000, 000 1:5 × EUR 100, 000

= 25:

Should the portfolio manager’s expectations be correct and interest rates fall by 1%, for example, the market value of the bond portfolio will increase by 12% instead of 10%. The portfolio manager can also achieve the higher modified duration of 12 by buying bonds with a longer time to maturity, lower coupon rate, and/or lower yield to maturity, because such bonds have a higher modified duration. However, this strategy is more expensive than employing exchange-traded bond futures. 3. Immunisation strategy: The objective of an immunisation strategy is that the cash flows from the bond portfolio can cover one or more payments from an obligation. Thus, an immunisation can be applied to structure a bond portfolio designed to fund a single liability or multiple liabilities. For example, pension plans immunise their bond portfolio by, among other things, equating the modified duration of the bond portfolio and the duration of the pension obligation payments. If, for example, a single liability must be funded, in addition to equating the durations, the market value of the bond portfolio is set equal to the present value of the single future liability payment for a given interest rate change29 ð- MDURB Þ B0 Δi = ð- MDURL Þ L0 Δi, where MDURB = modified duration of the bond portfolio, B0 = market value of the bond portfolio, MDURL = duration of a single future liability payment, L0 = present value of the single future liability payment, and Δi = change in interest rates.

28 29

See Fong and Guin 2015b: ‘Fixed-Income Portfolio Management – Part II’, p. 596. See Fabozzi 2007: Fixed Income Analysis, p. 546.

ð12:20Þ

440

12

Duration and Convexity

The equation demonstrates that the bond portfolio can be immunised by equating the durations when the market value of the bond portfolio and the present value of the future liability payment are the same. The duration of the liability payment is given by its time to maturity, which is equal to the Macaulay duration in the case of only one payment (analogous to the zero-coupon bond). Should the objective of the immunisation strategy be to generate a bond portfolio value that is higher than the present value of the liability payments, convexity must be considered in addition to equating the durations. In doing so, the modified convexity of the bond portfolio must exceed that of the liability payment. In the event of an increase in interest rates, the market value of the bond portfolio drops less than the present value of the liability payment. If, on the other hand, interest rates fall, the increase in the market value of the bond portfolio exceeds the increase in the present value of the liability payment. In both interest rate scenarios, the immunisation strategy produces an asset surplus.30

12.7

Summary

• The price of an option-free bond is a function of the yield to maturity. As the yield to maturity rises (falls), the bond price decreases (increases). The yield to maturity consists of the nominal risk-free interest rate and a risk premium, which, in the case of an option-free bond, is a return compensation for the credit risk of the issuer and market liquidity risk of the bond. The holder of a bond is exposed to interest rate risk. If interest rates rise (fall), the bond price decreases (increases). However, with a relatively long investment horizon, the coupons can be reinvested at a higher (lower) interest rate, which has a positive (negative) effect on the total return. Thus, the interest rate risk has an impact on the price risk and the reinvestment risk. Credit risk is the risk that the bond issuer fails to make the contractually agreed interest and principal payments, which can lead to a delay in payment or even to the insolvency of the issuer. The market liquidity risk, on the other hand, is related to the liquidity of the bond in the market. Should the liquidity of the bond in the market diminish, the bid–ask spread widens. Hence, investors can sell the bond at a lower bid price and buy the security at a higher ask price. • Unlike fixed-rate bonds, floating-rate notes are less exposed to interest rate risk because, all else being equal, the price can only change between two coupon dates as a result of a change in interest rates. The dominant risk factor for a floating-rate note is credit risk. • Investors can assess the credit risk of the issuer or of bonds with the help of credit ratings assigned by rating agencies such as Standard & Poor’s, Fitch, and Moody’s. The credit risk can also be evaluated through CDSs, which, in contrast to credit ratings, provide a time and market-based credit risk measure in the form

30

See Fong and Guin 2015a: ‘Fixed-Income Portfolio Management – Part I’, p. 567 f.

12.7

•

•

•

•

•

•

•

Summary

441

of a CDS spread. The size of the CDS spread depends on the creditworthiness of the issuer. The higher (lower) the probability of default and the loss given default, the higher (lower) the CDS spread. The price change of an option-free fixed-rate bond can be estimated with the second-order Taylor series expansion. The modified duration and the modified convexity are required for this. The modified duration is obtained by dividing the first derivative of the bond price with respect to a change in the yield to maturity by the bond price. It captures the linear price-yield relationship. Modified convexity, on the other hand, measures the non-linear price-yield relationship and is determined by dividing the second derivative of the bond price with respect to a change in the yield to maturity by the bond price. Modified duration is a sensitivity measure. For example, a modified duration of 5 means that the bond price increases (decreases) by approximately 5% if the yield to maturity falls (rises) by 1%. In the vast majority of cases, modified duration is a reasonable approximation and is therefore widely used to assess the price volatility of an option-free bond. Instead, the Macaulay duration can be interpreted as the weighted average time period of the bond’s cash flows. The higher (lower) the modified duration, the higher (lower) the price change, and thus the price volatility of the debt security, for a given change in the yield to maturity. The duration of an option-free fixed-rate bond increases (decreases) with a longer (shorter) time to maturity. By contrast, the duration falls (rises) with a higher (lower) coupon rate and a higher (lower) yield to maturity. If the price change of a bond is measured only with the modified duration or with the Taylor series expansion of the first order, the price increase is underestimated when the yield to maturity decreases, and the price decrease is overestimated when the yield to maturity increases. This error becomes greater the larger the change in the yield to maturity and the higher the modified convexity of the bond. Therefore, especially in the case of larger yield movements, modified convexity must be included in the calculation of the price change in addition to modified duration. In doing so, a convexity adjustment of 0.5 MCONV B0 (ΔYTM)2 must be added to the first term of the Taylor series expansion of (-MDUR) B0 ΔYTM. The modified convexity of an option-free fixed-rate bond is always positive. The modified convexity increases (decreases) with a longer (shorter) time to maturity. On the other hand, it falls (rises) with a higher (lower) coupon rate and a higher (lower) yield to maturity. Bonds with the same duration, but with higher dispersion of cash flows have a greater convexity. Modified duration is a sensitivity measure that makes it possible to determine how much the bond price will change if the expected inflation rate, the real interest rate, the credit spread, and the market liquidity spread were to change by a certain amount. Modified duration is used in portfolio management for tactical asset allocation, for exploiting forecast interest rate and credit spread changes, and for the immunisation strategy. Modified convexity is applied in an immunisation strategy to achieve an asset surplus, with the result that the market value of the bond portfolio exceeds the present value of the liability payment.

442

12.8

12

Duration and Convexity

Problems

1. The following information is available for the Mercedes-Benz Group AG 0.75% bond, maturing on 11 May 2023 (Source: Refinitiv Eikon): Issuer: ISIN: Type: Currency: Denomination: Issue volume: Issue date: Day-count convention: Coupon: Interest date: Maturity date:

Mercedes-Benz Group AG DE000A169NB4 Corporate bond EUR EUR 1000 EUR 750 million 11 May 2016 Actual/actual ICMA 0.75% fixed, annually Every year on 11 May 11 May 2023

The yield to maturity is 0.5074% on 11 May 2017. The following questions are to be answered: a) What are the Macaulay duration and the modified duration? b) What is the modified convexity? c) How much does the price of the Mercedes-Benz Group bond change using the duration-convexity approach if the yield to maturity increases by 0.5%? 2. A portfolio is made up of the following four option-free bonds: Bond Zero-coupon bond A Fixed-rate bond B Fixed-rate bond C Floating-rate note D

Time to Maturity 10 years 10 years 7 years 15 years

Coupon rate (paid annually) 0% 3.5% 4% 2.5%

Yield to Maturity 3.5% 3.8% 4.2% 2.5%

How can the price volatility of the four bonds be assessed?

3. The following statements relate to the modified duration and modified convexity of option-free fixed-rate bonds: 1. The lower the yield to maturity, the higher the modified duration and modified convexity. 2. If the price change of a bond is only measured with the modified duration or with the Taylor series expansion of the first order, the price increase is overestimated in the case of a decrease in the yield to maturity and the price decrease is underestimated in the case of an increase in the yield to maturity. 3. With the modified duration, the price change of a bond with a high modified convexity can be calculated with a high degree of accuracy. 4. Bonds with a long time to maturity have a high modified duration and modified convexity.

12.9

Solutions

443

5. In the case of a zero-coupon bond, the modified duration and the time to maturity are equal. Indicate whether each of the above statements is true or false (with justification). 4. A portfolio of corporate bonds has a market value of EUR 80 million and a modified duration of 12. The portfolio manager expects interest rates to rise in the coming weeks as a result of monetary policy measures by the European Central Bank. Therefore, they want to reduce the modified duration of the bond portfolio to 4. For this purpose, they intend to use Euro-Bund futures, which have a quoted price of 160% and an implied modified duration of 8. How many Euro-Bund futures contracts are needed to reduce the modified duration of the bond portfolio from 12 to 4?

12.9

Solutions

1. a) The time to maturity of the bond is 6 years. The price of the Mercedes-Benz Group bond of 101.430% can be calculated as follows on the coupon date 11 May 2017: B0 =

0:75% 0:75% 0:75% 0:75% 0:75% þ þ þ þ ð1:005074Þ1 ð1:005074Þ2 ð1:005074Þ3 ð1:005074Þ4 ð1:005074Þ5 þ

100:75% = 101:430%: ð1:005074Þ6

The Macaulay duration of 5.891 corresponds to the weighted average time period of the cash flows or average life of the bond and can be determined using the following table (rounded to three decimal places, but further calculated with unrounded figures):

Period (t) 1 2 3 4 5 6 Total

Cash flow (CFt) 0.75 0.75 0.75 0.75 0.75 100.75

Present value of cash flows CFt ð1 þ YTMÞt 0.746 0.742 0.739 0.735 0.731 97.737 101.430

Weight CFt ð1 þ YTMÞt B0

Period x weight CFt ð1 þ YTMÞt t B0

0.007 0.007 0.007 0.007 0.007 0.964 1.000

0.007 0.015 0.022 0.029 0.036 5.782 5.891

444

12

Duration and Convexity

The modified duration of the Mercedes-Benz Group bond is 5.86: MDUR =

5:891 = 5:86: 1:005074

b) To determine the modified convexity of the 6-year 0.75% bond with a yield to ðt 2 þ tÞCFt T of maturity of 0.5074%, the value of the formula term t=1 ð1 þ YTMÞt 4156.381 must first be calculated:

Period (t) 1 2 3 4 5 6 Total

2

t +t 2 6 12 20 30 42

CFt 0.75 0.75 0.75 0.75 0.75 100.75

2

(t + t)CFt 1.50 4.50 9.00 15.00 22.50 4231.50 4284.00

ðt 2 þ tÞCFt ð1 þ YTMÞt 1.492 4.455 8.864 14.699 21.938 4104.933 4156.381

The modified convexity of the 6-year 0.75% bond of Mercedes-Benz Group AG is 40.565 and can be calculated as follows: MCONV =

1 4156:381 × = 40:565: 2 101:430 ð1:005074Þ

c) If the yield to maturity increases by 0.5%, the price of the Mercedes-Benz Group bond decreases by approximately 2.92% using the duration-convexity approach: ΔB ≈ ð- 5:86Þ × 101:430% × 0:005 þ 0:5 × 40:565 × 101:430% × ð0:005Þ2 = - 2:92%: 2. Zero-coupon bond A has the highest modified duration or price volatility because it has the same or longer time to maturity as the other two fixed-rate bonds B and C, as well as the lowest coupon rate and yield to maturity. Fixedrate bond B has the same 10-year maturity as the zero-coupon bond. However, the coupon rate and the yield to maturity are higher. Hence, it has a lower duration. Compared to bonds A and B, fixed-rate bond C has a shorter time to maturity of 7 years and a higher coupon rate and yield to maturity. Thus, the modified duration and consequently the price volatility is lower. The lowest duration can be assigned to the floating-rate note D because the bond price can only change between two coupon dates as a result of an interest rate change. If the modified durations of the four bonds are calculated, the following values are obtained, which confirm the results of the qualitative analysis: • Zero-coupon bond A: Modified duration = 10/1.035 = 9.662 • Fixed-rate bond B: Modified duration = 8.273

Microsoft Excel Applications

445

• Fixed-rate bond C: Modified duration = 5.985 • Floating-rate note D: Modified duration = 1/1.025 = 0.976 3. 1. The first statement is true. A lower yield to maturity results in a higher duration and convexity, all else being equal. 2. The second statement is false. If the price change of the bond is only measured with the modified duration, the price increase is underestimated when the yield to maturity decreases, and the price decrease is overestimated when the yield to maturity increases. 3. The third statement is false. When the bond has a high convexity, the price change should be determined with the modified duration as well as the modified convexity. 4. The fourth statement is true. The relationship between time to maturity and duration and convexity is positive. 5. The fifth statement is false. The Macaulay duration and not the modified duration equals the time to maturity of the zero-coupon bond. 4. The contract value of Euro-Bund futures is EUR 160,000 (= 1.6 x EUR 100,000). To reduce the modified duration of the bond portfolio from 12 to 4, a total of 500 short Euro-Bund futures contracts are required: NF =

4 - 12 8

×

EUR 80, 000, 000 = -500: EUR 160, 000

If interest rates rise by 1% in the next few weeks, for example, this leads to a percentage decline in the bond portfolio of approximately only 4% instead of 12%. Unless the portfolio manager offsets the short futures position with a long futures position on the same underlying asset and expiration date, this durationbased hedging strategy remains in place until the expiration date of the EuroBund futures. The maximum time to expiration of Euro-Bund futures is 9 months.

Microsoft Excel Applications • The calculation of the duration in Excel is similar to the determination of the bond price and the yield to maturity. First, the following should be entered: in cell A1 the settlement date, in cell A2 the maturity date, in cell A3 the coupon rate (in per cent), in cell A4 the yield to maturity (in percent), in cell A5 the frequency of coupon payments, and in cell A6 the basis of the corresponding day-count convention. Next, the following expression should be written in an empty cell to calculate the Macaulay duration: = DURATION ðA1; A2; A3; A4; A5; A6Þ:

446

12

Duration and Convexity

Fig. 12.3 Calculation of the Macaulay duration and the modified duration using the MercedesBenz Group AG 2% 2019/2031 bond on the coupon date of 27 February 2021 (Source: Own illustration)

Then press the Enter key. • To determine the modified duration, use the ‘MDURATION’ function and after entering the function in an empty cell confirm with the Enter key:

= MDURATION ðA1; A2; A3; A4; A5; A6Þ: Figure 12.3 presents the calculation of the Macaulay duration and the modified duration for the Mercedes-Benz Group AG 2% 2019/2031 bond on the coupon date 27 February 2021. • In Excel, both the settlement date and the yield to maturity are required to determine the duration. If the bond is on a coupon date, two fictitious dates can be taken for this, which reflect the time to maturity of the bond. • A function for calculating the modified convexity does not exist in Excel. One possibility is to calculate the approximated modified convexity with the following formula:31 MCONVApproximated =

31

B - þ Bþ - 2B0 , B0 ðΔYTMÞ2

See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 589 ff.

ð12:21Þ

References

447

where B- = full price of the bond after a decrease in the yield to maturity, B+ = full price of the bond after an increase in the yield to maturity, B0 = full price of the bond before the change in yield to maturity, and ΔYTM = change in yield to maturity (in decimal places). • The prices of the bond after a change in the yield to maturity (e.g. by 1%) can be calculated using the ‘Price’ function in Excel and inserted into the equation above.

References Adams, J.F., Smith, D.J.: Understanding fixed-income risk and return. In: Petitt, B.S., Pinto, J.E., Pirie, W.L. (eds.) Fixed Income Analysis, 3rd edn, pp. 153–209. Wiley, Hoboken, NJ (2015) Chance, D.M.: Analysis of Derivatives for the CFA® Program. Association for Investment Management and Research, Charlottesville, VA (2003) Diwald, H.: Anleihen verstehen: Grundlagen verzinslicher Wertpapiere und weiterführende Produkte. Deutscher Taschenbuch Verlag, München (2012) Fabozzi, F.J.: Fixed Income Analysis, 2nd edn. Wiley, Hoboken, NJ (2007) Fabozzi, F.J.: Fixed Income Mathematics: Analytical and Statistical Techniques, 2nd edn. Irwin, Chicago et al. (1993) Fong, H.G., Guin, L.D.: Fixed-income portfolio management – part I. In: Petitt, B.S., Pinto, J.E., Pirie, W.L. (eds.) Fixed Income Analysis, 3rd edn, pp. 531–584. Wiley, Hoboken, NJ (2015a) Fong, H.G., Guin, L.D.: Fixed-income portfolio management – part II. In: Petitt, B.S., Pinto, J.E., Pirie, W.L. (eds.) Fixed Income Analysis, 3rd edn, pp. 585–632. Wiley, Hoboken, NJ (2015b) Gootkind, C.L.: Fundamentals of credit analysis. In: Petitt, B.S., Pinto, J.E., Pirie, W.L. (eds.) Fixed Income Analysis, 3rd edn, pp. 211–278. Wiley, Hoboken, NJ (2015) Macaulay, F.R.: Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. National Bureau of Economic Research, New York (1938) Mondello, E.: Finance: Theorie und Anwendungsbeispiele. Springer Fachmedien, Wiesbaden (2017) Tuckman, B., Serrat, A.: Fixed Income Securities: Tools for Today’s Markets, 3rd edn. Wiley, Hoboken, NJ (2012)

Online Sources BNP Paribas: Bonität und Credit Defaults Swaps. https://derivate.bnpparibas.com/service/ueberuns/bonitat-und-credit-default-swaps. Accessed on 1 December 2017

Part IV Derivatives

Futures, Forwards, and Swaps

13.1

13

Introduction

A derivative is a contract between two parties whose value is derived from the price of an underlying asset. Depending on the underlying asset, one speaks of financial, commodity, and credit derivatives, among others. The underlying of a financial derivative refers to a financial instrument such as an equity security or a fixedincome security, a currency or a financial indicator such as an interest rate, equity, or bond index. For commodity derivatives, the underlying is given by a commodity— for example, gold, silver, oil, or wheat. By contrast, credit derivatives are based on a credit event, which can occur, for example, in the form of a debtor default on a bond or loan. Derivatives can also be divided into forward commitments and contingent claims. A forward commitment is a contractual agreement between two parties, which involves an obligation to buy or sell an underlying asset at the expiration date of the contract and at a price that is specified at the start of the agreement. Examples are futures and forwards, which differ in that the former are traded on an exchange and the latter over the counter. Swaps also belong to the category of forward commitments. The parties to a swap contract enter into an obligation to exchange a series of cash flows at specific points in time for a predetermined period of time. By contrast, a contingent claim grants the buyer the right to buy (call option) or to sell (put option) the underlying asset at the agreed strike price either during or at the end of the option life. The option seller, on the other hand, has the obligation to sell (call option) or buy (put option) the underlying asset at the strike price. Derivatives can also be classified on the basis of their profit/loss profile. Forward commitments have a linear profit/loss profile, while contingent claims have a nonlinear profit/loss pattern due to the option right and the option premium paid for it. Figure 13.1 presents a possible classification of derivative instruments. The chapter begins with the use of derivatives, which includes risk hedging, risktaking (speculation and trading), and the exploitation of price differences (arbitrage). # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_13

451

452

13

Futures, Forwards, and Swaps

Derivatives

Forward commitments

Examples

Exchangetraded • Financial futures • Commodity futures

Contingent claims

Over-thecounter

Exchangetraded

• Forwards • Swaps

• Options • Options on futures

Over-thecounter • Options • Swaptions • Exotic options

Underlying assets

Equity securities/equity indices Interest rates/fixed-income securities Currencies Commodities Loans Natural events/weather Real estate Fig. 13.1 Classification of derivative instruments (Source: Own illustration)

After differentiating between futures and forwards, the calculation of profit/loss, price, and value is explained. This is followed by a discussion of how forwards and futures can be applied to hedge risky positions. The chapter ends with an examination of interest rate swaps.

13.2

Use of Derivatives

With financial derivatives, a risk position, for example in stocks or bonds, can be entered into or restructured quickly and cost-effectively according to expectations and risk preferences. Essentially, the following three motives for the use of derivatives can be distinguished:1

1

See, for example, Eurex 2007: Aktien- und Aktienindexderivate: Handelsstrategien, p. 23.

13.2

Use of Derivatives

453

1. Risk hedging 2. Risk-taking (speculation, trading) 3. Exploiting price differences (arbitrage) 1. Risk hedging: In risk hedging, the price risk of, for example, an equity position is eliminated or reduced by a derivative with an opposite profit/loss profile. The price risk of the equity position can be completely eliminated (perfect hedge) or only partially reduced. Furthermore, risk hedging can be divided into a portfolio hedge, in which the risk of loss of an existing asset or liability position (or long or short securities) is hedged, and an anticipatory hedge, in which the price risk of a planned long or short risk position is eliminated. 2. Risk-taking: The risk is assumed by market participants who, based on price expectations, do not establish a risk position using an asset of the spot market but with a derivative of that asset. This is often both simpler and more cost-effective. For example, it is easier to enter into a short equity position with short futures contracts than by selling short an equity security. This speculative risk-taking (expectation of falling prices) is diametrically opposed to hedging. In hedging, the risk of loss of an asset or liability is hedged with a derivative. In speculation, the risk position to be hedged is not owned. Rather, with the help of derivatives, an attempt is made to earn a profit based on expected price changes of the underlying asset. A risk position is built up for this purpose. Hedging and speculation are often seen as complementary activities. The assumption here is that risk hedging requires market participants, that is, speculators, to take the risk. However, this is not always the case. For example, market participants who hedge the risk in their positions can trade with each other. An airline can protect against rising jet fuel prices with long crude oil futures, while a crude oil producer can hedge falling prices with short crude oil futures.2 Likewise, speculators often trade with each other as a result of different price expectations. Hence, a counterparty with an opposite risk exposure or with different price expectations is needed to hedge and bear risk.3 3. Exploiting price differences: When price differences exist for the same financial asset in different markets, a risk-free profit can be achieved by simultaneously buying and selling the asset in two different markets.4 Such a long–short transaction is called arbitrage. If the markets are efficient, no arbitrage opportunities exist. For example, should a stock be traded on one market at EUR 100 and at the same time on another market at EUR 99, a risk-free profit of EUR 1 can be earned with an arbitrage transaction (without taking transaction costs into account). For this purpose, the stock is sold on the more expensive market at EUR 100 and simultaneously bought on the cheaper market at EUR 99. Therefore, the cash

2 The prices of crude oil and jet fuel are highly correlated, and therefore crude oil futures can be used to hedge the price risk of jet fuel. 3 See Chance 2003: Analysis of Derivatives for the CFA® Program, p. 14. 4 See Kolb 2000: Futures, Options, and Swaps, p. 7.

454

13

Futures, Forwards, and Swaps

proceeds from the sale can be used to finance the purchase of the security. The remaining difference represents the risk-free arbitrage profit. With the long–short position the price risk of the equity security is eliminated, and it therefore does not matter whether the share price rises or falls. Since there is no price risk, the profit made from the arbitrage strategy is risk-free. If many market participants carry out this arbitrage strategy, the two prices of the stock converge to each other and the arbitrage opportunity disappears. The security trades at a single price in the two markets. Accordingly, the principle that no arbitrage opportunity exists is often referred to as the law of one price. Arbitrage activities are generally carried out by investment banks, as they have the appropriate technical equipment, very quick access to the relevant markets, and low transaction costs.5 In a cash-and-carry and reverse cash-and-carry arbitrage, unjustified price differences between the traded futures price and the futures price calculated with the cost-of-carry model, which is based on the costs of the spot market, are exploited. With the help of arbitrage, a price equilibrium is established on the spot and futures markets. This arbitrage principle is an important concept for the pricing of derivatives.6

13.3

Futures and Forwards

13.3.1 Futures Versus Forwards Futures and forwards are traded on the derivatives markets, where the closing and fulfilment of the contract are separated in time. The price, the underlying asset, and the time to expiration of the derivative are determined when the contract has been agreed. The purchase and sale of the underlying, on the other hand, takes place at the expiration date of the contract.7 Futures are traded on a derivatives exchange such as the Eurex (European Exchange). Trading takes place electronically and anonymously. The contract features such as the underlying asset, the contract size, and the time to expiration are specified by the derivatives exchange. Thus, futures contracts are standardised, with the result that the exchange, rather than the individual parties, sets the terms and conditions, with the exception of the price. For example, the DAX futures traded on the Eurex have standardised contract features that relate to the underlying asset (DAX 40), the contract size of EUR 25 per index point, and the maximum time to expiration of 9 months with the three nearest quarterly months of the March, June, September, and December cycle. DAX futures expire on the third Friday of the corresponding expiration month and are cash settled. The standardisation of the derivatives leads to homogeneity, which increases liquidity on the markets. The

5

See Bösch 2014: Derivate: Verstehen, anwenden und bewerten, p. 10. For cash-and-carry and reverse cash-and-carry arbitrage, see Sect. 13.3.4. 7 See Reilly and Brown 2006: Investment Analysis and Portfolio Management, p. 808. 6

13.3

Futures and Forwards

455

derivatives exchange acts as the contractual partner for the buyer and the seller and guarantees to each party the performance of the other party, through a mechanism known as the clearinghouse. This largely eliminates the settlement risk (credit risk). In the case of futures, a security deposit must be made with the derivatives exchange in the form of an initial margin. In addition, there is a variation margin which includes the profit or loss realised at the end of each trading day. This procedure is known as daily settlement or marking to market. The daily profit/loss is calculated on the basis of the difference between the closing futures prices at the end of the trading day and the previous trading day. A profit is credited and a loss is debited to the margin account. Once the margin account falls below the maintenance margin (e.g. 70% or 80% of the initial margin) due to realised losses, the investor receives a margin call. This is a request for payment to restore the margin account to the original initial margin level. Should the investor not comply with the payment request, the position will be closed out. The main purpose of the initial margin, the marking to market, and the margin call, if any, is to reduce the probability of a default of the holder of the futures on the expiration date of the contract. Unlike futures, forwards are traded over the counter (OTC). Trading takes place on electronic trading platforms such as TradeWeb and Bloomberg or through direct contact between market participants (e.g. by telephone). The forward contract’s terms and conditions are not specified by an exchange, but are negotiated individually between the contracting parties. Therefore, forwards are customised contracts. In response to the 2008 financial crisis, the G-20 countries decided at a 2009 summit meeting in Pittsburgh to reduce the systematic risk of the OTC markets in order to make OTC trading more transparent and safer. One of the key reform points is that OTC derivatives are to be classified into ‘centrally cleared’ and ‘bilaterally traded’ instruments. The centrally cleared OTC derivatives must be cleared through a central counterparty (CCP), so that both contracting parties to the forwards have the CCP as a counterparty. This way, the credit risk is largely eliminated, analogous to futures. In the case of bilaterally traded OTC derivatives, the two contracting parties still face each other as counterparties. Regardless of the classification as centrally cleared or bilaterally traded, both contracting parties must deposit collateral in the form of an initial margin. Moreover, there is a daily profit/loss settlement which, in contrast to futures, is not realised. Thus, any profit serves as collateral against a possible default of the counterparty. The margin account consists of the security deposits (initial margin) and the daily unrealised profits/losses (variation margin). The adoption of the new regulations in the European Union has taken place in terms of the Markets in Financial Instruments Directive II (MiFID II) and the European Market Infrastructure Regulation (EMIR). The corresponding regulations in Switzerland are found in the Financial Market Infrastructure Act (FMIA). As a result of these regulations, which reduce the systematic risk of OTC derivatives, the differences between futures and forwards have decreased considerably. Table 13.1 compares futures with forwards. Futures and forwards can be held until the expiration date or closed out (offset) prior to expiration. Unlike forwards, futures are not usually intended to be held until the expiration date. They are typically offset with an opposite futures contract that

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Table 13.1 Comparison of futures and forwards (Source: Own illustration) Features Trade Contract’s terms and conditions

Futures Futures are traded on a derivatives exchange. The contract’s terms and conditions are specified by the derivatives exchange and are thus standardised.

Market liquidity

The market liquidity is typically high.

Maturity

The contract term tends to be short-term, although medium-term maturities also exist in some cases. The clearinghouse is exposed to credit risk.

Credit risk

Margins Cash flow

Initial margin and variation margin There is a daily settlement (marking to market). The profits/losses are realised at the end of each trading day.

Contract fulfilment

Only a small percentage of the contracts are fulfilled. In most cases, the position is closed out before the end of the contract.

Forwards Forwards are traded over the counter. The contract’s terms and conditions are negotiated individually between the parties, often using certain marketable features. Forwards are customised. Market liquidity varies according to market segment and product type. Only for certain derivatives such as interest rate options, interest rate swaps, and currency forwards is liquidity relatively high. The maturity spectrum is very broad and ranges between short-, medium- and long-term. For centrally cleared OTC derivatives, the clearinghouse is exposed to credit risk. In the case of bilaterally traded OTC derivatives, however, the credit risk is assumed by the owners of the long and short forward contracts, with an obligation to provide collateral. Initial margin and variation margin There is a daily settlement which results in an adjustment of the collateral so that no realised profits/ losses arise. The fulfilment of the contract is often intentional.

has the same underlying asset and the same expiration date. For example, if the holder of 10 long DAX futures has contracts expiring in September 2023, they must enter into 10 short DAX futures contracts that also expire in September 2023 in order to offset the long DAX futures. In the event that the futures or forward position is not closed out prior to expiration, either physical delivery of the underlying asset or cash settlement will take place. Cash settlement is used for equity index futures because the physical delivery of an equity index is difficult and expensive. For DAX and other equity index futures, cash settlement takes place with the realised daily profit/ loss of the last trading day.

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13.3.2 Profit and Loss A long position in a futures/forward contract is an obligation to buy an underlying asset at a later date and at a price agreed today. A short position in a futures/forward contract, on the other hand, represents a corresponding obligation to sell. If, at the expiration date of the forward contract, the spot price of the underlying assets exceeds the forward price agreed when the contract was entered into, the owner of the long position realises a profit and the owner of the short position incurs a loss. The holder of the long futures/forward contract has made a profit because they can acquire the underlying asset at a lower forward price than the prevailing spot price. By contrast, the holder of the short futures/forward contract incurs a loss. They must sell the underlying asset at a forward price below the spot price. If they were to sell the underlying directly on the spot market instead of on the futures/forward market, the cash proceeds would be greater. However, should the spot price of the underlying asset fall below the contractually agreed forward price on the expiration date of the futures/forward, the owner of the long futures/forward position incurs a loss, while the owner of the short futures/forward position realises a profit. The loss on the long position is because the underlying has to be bought at a forward price that is higher than the spot price. For the short position, the obligation to sell at a higher price, results in a profit. The profit/loss of futures/forward contracts can be determined at expiration as follows (without taking into account interest on the margin account of a futures contract): Profit=loss of long futures=forward = ST - F 0 ,

ð13:1Þ

Profit=loss of short futures=forward = - ðST - F 0 Þ,

ð13:2Þ

where ST = spot price of the underlying asset at the expiration date T of the futures/forward contract, and F0 = agreed forward price at inception of futures/forward contract. For example, a long forward on the stock of Mercedes-Benz Group AG has a forward price of EUR 70. If the automobile stock is traded at a price of EUR 73 on the expiration date of the forward contract, the profit is EUR 3 (= EUR 73 EUR 70). On the other hand, with a share price of EUR 67, a loss of EUR 3 (= EUR 67 - EUR 70) occurs. Thus, a share price above (below) the forward price leads to a profit (loss). By contrast, a short forward contract produces a loss (profit) if the share price exceeds (falls below) the forward price. If, for example, the share price is EUR 73, holding a short forward contract leads to a loss of EUR 3 [= (EUR 73 - EUR 70)]. A share price of EUR 67 results in a profit of EUR 3 [= (EUR 67 - EUR 70)]. Figure 13.2 presents on the expiration date the profit/loss profile of the holder of the long and short futures/forward contract on the MercedesBenz Group stock. The profit/loss of both long and short positions are symmetric, or

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(Profit/ loss in EUR) 70 60 50

Short futures/forward

Long futures/forward

40 30 20 10 0 -10 0 -20 -30 -40 -50

10 20 30 40 50 60 70 80 90 100 110 120 130 140 (Futures/forward price Future/ or price of underlying forward price asset on expiration date at inception of contract in EUR) of contract (F0)

-60 -70 Fig. 13.2 Profit/loss diagram of a long and short futures/forward contract on the expiration date (Source: Own illustration)

two-sided, around the agreed futures/forward price. The maximum possible profit of a long futures/forward contract on an equity security is unlimited, since there is no upper price limit for a stock. However, the potential loss is limited to the agreed futures/forward price since the share price cannot fall below zero. With a short position, the maximum profit is limited to the futures/forward price, while the maximum loss is unlimited. Figure 13.2 demonstrates that the profits/losses of the long and short positions are mirror images of each other. Thus, futures/forward contracts are zero-sum games.8 Market participants who forecast rising prices for the underlying asset can make a profit with a long futures/forward contract if their price expectations are met. On the other hand, if they expect the price of the underlying asset to fall, they can enter into a corresponding short futures/forward contract.

8

See Reilly and Brown 2006: Investment Analysis and Portfolio Management, p. 819.

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13.3.3 Leverage Effect Another important property of derivatives is the leverage effect. With a derivative, a much smaller amount of money is required to enter into the risk exposure of the underlying asset and thus to participate in its price movements. For example, a futures/forward contract requires an initial margin that is much lower than the price of the underlying asset. Since the holder of a futures/forward position participates fully in the price movements of the underlying, they earn a much higher or lower return than if they owned the underlying. This leverage effect on the return is illustrated below using DAX futures. At the end of the trading day on 21 December 2017, an investor entered into a long DAX futures contract expiring in March 2018 (FDAX Mar 18) at a price of 13,098 index points. The contract value per index point is EUR 25. Accordingly, the investor has an obligation to buy the DAX 30 at a price of EUR 327,450 (= 13,098 index points × EUR 25 per index point) on the expiration date of the futures (i.e. on the third Friday of the expiration month March 2018). On the Eurex, the required additional margin or initial margin for a DAX futures contract is EUR 24,560 (on 21 December 2017). If, for example, the DAX futures price rises to 13,300 index points at the end of the next trading day on 22 December 2017, the result is a profit of EUR 5050: Profit long DAX futures = ð13, 300 index points - 13, 098 index pointsÞ × EUR 25 per index point = EUR 5050: With respect to the deposited additional margin of EUR 24,560, the daily return for the holder of the long DAX futures is 20.56%: Return of long DAX futures =

EUR 5050 = 20:56%: EUR 24, 560

The DAX 30 closes at 13,110 index points on 21 December 2017. Assuming that the DAX 30 rises by 202 index points at the end of the next trading day on 22 December 2017, in line with the futures price, the return on the underlying equity index would be 1.54%: Return of DAX 30 =

202 index points = 1:54%: 13,110 index points

The return of the DAX futures is 13.4 times higher (= 20.56%/1.54%) than that of the DAX 30. This results in a return leverage of 13.4. If, on the other hand, the DAX 30 falls by 1.54%, the loss is 13.4 times higher in the case of the long DAX futures. The leverage effect is due to the fact that with derivatives it is possible to fully participate in the price movements of the underlying with a comparatively lower capital commitment. Thus, derivatives trading offers not only great opportunities for profit, but also great risks of loss. Table 13.2 compares the two

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Table 13.2 Leverage effect for the DAX futures (Source: Own illustration) DAX 30 DAX futures End of trading day 21 December 2017 Price 13,110 13,098 1st price scenario at the end of the trading day 22 December 2017 Price 13,312 13,300 13, 312 - 13, 110 ð13, 300 - 13, 098Þ × EUR 25 Return = 1:54% = 20:56% 13, 110 EUR 24, 560 2nd price scenario at the end of the trading day 22 December 2017 Price 12,908 12,896 12, 908 - 13, 110 ð12, 896 - 13, 098Þ × EUR 25 Return = - 1:54% = - 20:56% 13, 110 EUR 24, 560

price scenarios of the DAX futures and the DAX 30, as well as the effect on the returns.

13.3.4 Pricing The price of futures/forwards can be determined using the replication approach of the cost-of-carry model, where the costs in the futures/forward market are replicated by the costs in the spot market. For example, should the investment objective be to own 100 barrels of crude oil (one barrel contains 159 litres of crude oil) in 1 year, the barrels can be purchased either on the spot market or on the futures market. If the crude oil barrels are purchased on the spot market, the spot price has to be paid. Financing the purchase with money borrowed incurs interest costs. Opportunity costs arise when buying with one’s own money as the money cannot be invested elsewhere. In addition, crude oil has to be held for a period of 1 year, which leads to storage, insurance, and transportation costs. If commodities have a convenience yield, which is the benefit that arises from its physical possession, this must be deducted from the costs of the spot market transaction.9 A convenience yield for heating oil, for example, occurs when holding heating oil today provides a greater benefit than holding this commodity position at a future date. For example, the demand for heating oil is greater in the winter months than in the summer months, which means that the heating oil from a spot market strategy can be sold at a high

9 A convenience yield is the benefit of having the commodity available to use. It can also be viewed as a measure of how much a buyer would pay to circumvent the inconvenience of constantly ordering new quantities of the commodity and be exposed to the risk that the supply of the material will not arrive when needed. Convenience yields vary with the level of inventories. Lower inventories lead to a higher convenience yield, as commodity consumers will pay a higher price to secure adequate supplies to operate their businesses. Hence, the convenience yield is an economic benefit and not a monetary benefit. See Black et al. 2012: CAIA Level II: Advanced Core Topics in Alternative Investments, p. 303.

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461

price in the winter and bought at a lower price in the summer. The profit thus achieved reduces the costs of the spot market strategy. If the 100 barrels of crude oil are not bought on the spot market but on the futures market, a futures price must be agreed today that will be paid for the 100 barrels of crude oil in one year’s time. With both the spot market strategy and the forward market strategy, the investment goal of having 100 barrels of crude oil in 1 year can be achieved. Therefore, the cost in the forward market, that is, the forward price, must be the same as the cost in the spot market, which is the spot price of the underlying asset plus the financing and holding costs minus any convenience yield. According to the cost-of-carry model the price of a commodity futures/forward contract can be calculated as follows: F 0 = S0 eðrs þ c

- yÞT

ð13:3Þ

,

where F0 = price of commodity futures/forward at inception of the contract, S0 = spot price of the underlying commodity, e = Euler’s number (2.71828 . . .), rs = continuous compounded interest rate per annum for a time period of T years, c = continuous compounded rate for the commodity’s carrying costs per annum, y = continuous compounded convenience yield per annum, and T = time to expiration of the futures/forward expressed in years. The carrying costs and the convenience yield cannot be observed directly on the market and must therefore be estimated separately. The convenience yield depends largely on the expectations of market participants about the future availability of the commodity. The higher the probability of a shortage, the greater the benefit from holding the commodity in the inventory and thus the convenience yield.10 Example: Pricing of a West Texas Intermediate Oil Futures Contract In December 2017, a barrel of West Texas Intermediate (WTI) crude oil is trading at USD 58.36. The continuous compounded rate for the carrying costs is 0.3%, while the continuous compounded convenience yield is 5.3%. The 1-year continuous compounded LIBOR for the US dollar is 2%. What is the price of a 1-year WTI oil futures contract? Solution According to the cost-of-carry model, the price of the WTI oil futures is USD 56.64: F 0 = USD 58:36 × eð0:02 þ 0:003

10

- 0:053Þ × 1

= USD 56:64:

See Chance 2003: Analysis of Derivatives for the CFA® Program, p. 112.

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In the example, the convenience yield of crude oil exceeds the financing and carrying costs (rs + c < y), which leads to the futures price of USD 56.64 falling below the commodity’s spot price of USD 58.36 (F < S). A futures market with such a price situation is said to be in backwardation. If, on the other hand, the financing and carrying costs are greater than the convenience yield (rs + c > y), the commodity futures price will be above the spot price of the underlying (F > S). Such a market situation is called contango.11 It is also important to note that over time the futures price will converge to the price of the underlying asset. The futures price and the underlying price are equal on the expiration date of the derivative. When the underlying is a financial asset such as an equity security or a fixedincome security, the carrying costs can be disregarded because their amount is insignificant. However, unlike commodities, financial assets generate income that reduces the costs of the spot market transaction. These include dividends in the case of stocks or an equity index, coupons in the case of bonds, and interest income in the case of a foreign currency. Thus, the price of a financial futures/forward contract can be determined according to the cost-of-carry model as follows: F 0 = S0 eðrs

- dÞT

ð13:4Þ

,

where d = continuous compounded income yield per annum (e.g. dividend yield of an equity security). Example: Pricing of a DAX Forward and an SMI Forward The DAX 40 (performance index) closes at 13,094.54 index points on 4 March 2022. An expected continuous compounded dividend yield of 2.38% is assumed for the DAX 40. The 12-months continuous compounded EURIBOR is -0.359%. By contrast, the SMI (price index) is traded at 11,344.90 index points on 4 March 2022. The expected continuous compounded dividend yield is estimated at 2.92%. The 12-months Compounded SARON is -0.715%. What is the price of the DAX forward and the SMI forward, each with a time to expiration of 12 months? Solution The DAX 40 is a performance index where dividends are reinvested in the equity index. Since the dividends are already included in the equity index, no dividends need to be deducted from the financing costs when calculating the forward price. The DAX forward price of 13,047.61 index points can therefore be calculated with the cost-of-carry model as follows: (continued)

11

See Reilly and Brown 2006: Investment Analysis and Portfolio Management, p. 857 f.

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F 0, DAX = 13, 094:54 × e - 0:00359 × 12=12 = 13, 047:61: In contrast to the DAX forward, the dividends must be subtracted from the costs of the spot market transaction for the SMI forward because the SMI is a price index and thus does not contain any dividends. The SMI forward price of 10,939.92 index points can be determined by using the cost-of-carry model as follows: F 0, SMI = 11, 344:90 × eð- 0:00715

- 0:0292Þ × 12=12

= 10, 939:92:

The cost-of-carry model is based on the assumption that if the traded futures price deviates from the cost-of-carry model price, the market participants carry out arbitrage transactions until the market price matches the model price and the costs of the spot market and the futures market are equal again. For example, an investor has purchased equity futures with a time to expiration of 6 months. The underlying stock is traded at a price per share of EUR 100 and the continuously compounded interest rate is 1%. According to the cost-of-carry-model the equity futures price is EUR 100.50: F 0 = EUR 100 × e0:01 × 6=12 = EUR 100:50: If the equity futures trades at a price of EUR 101, it is overvalued. Hence, market participants will sell the overvalued futures at a price of EUR 101 (short futures) and buy the underlying equity security at a price of EUR 100. To finance the purchase of the security, market participants will borrow money at an interest rate of 1%. On the expiration date of the futures, the stock purchased 6 months ago is sold at the agreed forward price of EUR 101. In addition, the borrowed money, including interest totalling EUR 100.50 (=EUR 100 × e0.01 × 6/12), is repaid. The resulting risk-free arbitrage profit is EUR 0.50 (= EUR 101 - EUR 100.50). This so-called cash-andcarry arbitrage (long underlying and short futures) is carried out until the mispricing on the market has disappeared. If, on the other hand, the equity futures trades at a price of EUR 100, it is undervalued compared to the cost-of-carry model price of EUR 100.50. Investors can earn a risk-free arbitrage profit by buying the undervalued futures (long futures) and simultaneously selling the underlying equity security short at a price of EUR 100. The proceeds from the short sale are invested at an interest rate of 1% over 6 months. On the expiration date of the futures, the market participants receive the invested cash proceeds including interest of EUR 100.50 (=EUR 100 × e0.01 × 6/12). At the same time, they buy the stock through the long futures at the agreed futures price of EUR 100 and fulfil the delivery obligation from the short sale with the purchased stock. This strategy is referred to as reverse cashand-carry arbitrage (short underlying and long futures) and results in a risk-free arbitrage profit of EUR 0.50 (= EUR 100.50 - EUR 100). The market participants

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will carry out these arbitrage transactions until the traded futures price matches the model price. Arbitrage is an important concept for the price calculation of futures/forwards with the cost-of-carry model. Without the possibility of cash-and-carry and reverse cash-and-carry arbitrage, the market price deviates from the cost-of-carry model price.12

13.3.5 Valuation Unlike futures, forwards are not marked to market. Rather, the profits/losses accumulate with the result that either an asset position (unrealised gain) or a liability position (unrealised loss) is created during the lifetime of the forward contract. The profit/loss is typically not realised until the expiration date of the forward. By contrast, futures have a value of zero at the end of each trading day because the profit/loss is realised, and as a result a profit is credited and a loss is debited daily to the margin account. Hence, the value calculation is only relevant for forwards, but not for futures. The valuation of forwards is explained below using an equity forward, whose calculations are carried out using months and not days for the sake of simplicity. For example, a forward contract on the Mercedes-Benz Group stock has a time to expiration of 9 months, lasting from 3 January to 3 October 2017. The automobile stock trades at a price of EUR 72.20 per share on 3 January 2017. The dividend payment of EUR 3.25 per share will take place on 3 April 2017. The continuous compounded EURIBOR for 3 and 9 months are -0.313% and -0.140%, respectively. The forward price at inception can be calculated with the cost-of-carry model. The costs of the spot market transaction are made up of the share price and the financing costs minus the dividend. Accordingly, the forward price of EUR 68.88 can be determined as follows: F 0 = EUR 72:20 -

EUR 3:25 × e - 0:00140 × 9=12 = EUR 68:88: e-0:00313 × 3=12

If an investor enters into a long equity forward, they have the obligation on 3 January 2017 to purchase the Mercedes-Benz Group stock on 3 October 2017 at a price per share of EUR 68.88. The value of this long forward contract at the time it is entered into is zero and changes during the lifetime of the forward due to changes in the share price and interest rates. After 6 months, that is, on 3 July 2017, the automobile stock trades at a price per share of EUR 64.17. The continuously compounded 3-month EURIBOR is -0.331%. Since there are no dividends during

12

For the price calculation of futures in imperfect markets, which are characterised, for example, by transaction costs, different interest rates for borrowing and lending money, and restrictions on short sales, see Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 769 ff.

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465

the remaining time to expiration of the derivative, the forward price on 3 July 2017 is EUR 64.12 and can be calculated using the share price and the financing costs as follows: F t = EUR 64:17 × e-0:00331 × 3=12 = EUR 64:12: The holder of the long forward incurs a loss of EUR 4.76 (= EUR 64.12 EUR 68.88). This loss arises because of the obligation to buy the Mercedes-Benz Group stock at the agreed forward price of EUR 68.88. If the investor enters into the same long forward contract on 3 July 2017, the forward price is lower at EUR 64.12. Thus, the higher agreed forward price for the automobile stock results in a loss of EUR 4.76. As this loss will only be realised on the expiration date of the forward, it has to be discounted with the 3-month EURIBOR to account for the time value of money effect at the valuation date of 3 July 2017, which leads to a negative value of the long equity forward of EUR 4.76: V t,

Long

=

-EUR 4:76 = -EUR 4:76: e-0:00331 × 3=12

Accordingly, the holder of the long forward has a liability position (accumulated unrealised loss). For the holder of the short forward, on the other hand, the profit on 3 July 2017 is EUR 4.76 [= -(EUR 64.12 - EUR 68.88)]. A profit is achieved because they have the obligation to sell the stock at the agreed forward price of EUR 68.88 in 3 months, which is higher than using a 3-month forward with a price of EUR 64.12 on 3 July 2017. The profit of EUR 4.76 can only be realised on the expiration date of the forward, and therefore the value of the short forward of EUR 4.76 can be determined by discounting the profit at the valuation date: V t,

Short

=

EUR 4:76 e-0:00331 × 3=12

= EUR 4:76:

The example demonstrates that the value of a forward is the discounted profit/loss at the valuation date. In general, the values of a long and a short forward can be calculated with the following formulas: V t,

Long

=

Ft - F0 er s ð T - t Þ

ð13:5Þ

Ft - F0 , ers ðT - tÞ

ð13:6Þ

and V t, Short = where

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F0 = forward price at inception, Ft = forward price at valuation date t, rs = continuous compounded interest rate per annum, and T - t = time to expiration of the forward expressed in years. In the case of forwards, a distinction is made between price and value. The forward price indicates the price at which the underlying asset can be bought or sold at the expiration date of the derivative. By contrast, the value reflects the accumulated unrealised profit/loss during the lifetime of the forward. At inception of the forward, the value is zero. For other financial instruments (with the exception of swaps), there is no difference between price and value. For example, the price and value are the same for stocks, bonds, and options.

13.3.6 Hedging The price risk of an asset or liability can be managed with derivatives such as futures and forwards. For this purpose, the price risk can be completely or only partially eliminated. Essentially, the following two risk hedging situations can be distinguished:13 1. Hedging an existing asset or liability position 2. Anticipatory hedge 1. Hedging an existing asset or liability position: The price risk of an asset or liability can be controlled with derivatives such as futures and forwards. For example, an investor owns 500 shares in Siemens AG, which are trading at a price per share of EUR 132.95 on 2 May 2017. They expect the share price to fall over the next 6 months. Therefore, they fully hedge the price risk of the long equity position with a 6-month short equity forward (perfect hedge). If the share price falls as predicted, the investor suffers a loss on the long equity position, which is offset by the gain on the short forward position. In order to determine the forward price when the contract was entered into on 2 May 2017, the continuous compounded 6-month EURIBOR of -0.248% is required, in addition to the share price of EUR 132.95. The forward price of EUR 132.785 can be calculated with the cost-of-carry model as follows: F 0 = EUR 132:95 × e-0:00248 × 6=12 = EUR 132:785:

13 See Rudolph and Schäfer 2010: Derivative Finanzmarktinstrumente: Eine anwendungsbezogene Einführung in Märkte, Strategien und Bewertung, p. 33.

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Thus, the investor has entered into the obligation to sell 500 Siemens shares on 2 November 2017 at a price per share of EUR 132.785. On the expiration date of the derivative, the stock is traded at a price per share of EUR 124.50, resulting in a profit on the short forward contract of EUR 4142.50: Profit short forward = -500 × ðEUR 124:50 - EUR 132:785Þ = EUR 4142:50: On the long equity position, the investor loses EUR 4225: Loss long equity = 500 × ðEUR 124:50 - EUR 132:95Þ = -EUR 4225: The net loss from the hedging strategy is EUR 82.50 (= EUR 4142.50 EUR 4225). Therefore, the investor achieves a 6-month return of -0.124%: Return6 months =

-EUR 82:50 = -0:124%: 500 × EUR 132:95

The annualised return is therefore -0.248%, which corresponds to the 6-month EURIBOR at the beginning of the hedging strategy: Return12 months = e-0:00124 × 2 - 1 = -0:248%: The example demonstrates that the price risk is completely eliminated and the return achieved is equal to EURIBOR or the financing interest rate. Thus, at the beginning of the hedging strategy, it is already known what the return on the hedged position will be. Since the EURIBOR is negative, the return on the hedging strategy is also negative. With this hedge, not only the loss potential but also the profit potential of the equity position is eliminated. Therefore, this hedging strategy makes sense when investors expect falling (rising) prices for a long (short) equity position. 2. Anticipatory hedge: In the case of an anticipatory hedge, the price risk of a planned purchase or sale of a financial asset can be removed with long or short futures/forwards. Such a hedge is appropriate when an investor knows they will have to purchase or sell a certain asset in the future and want to lock in a price now. If the intention is to buy (sell) a stock in the future, for example, one is exposed to the risk of rising (falling) prices. This price risk can be hedged with a long (short) equity futures/forward position.14

14

See Hull 2006: Options, Futures, and Other Derivatives, p. 48 f.

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For example, an investor wants to buy 800 Lufthansa shares in 6 months. They expect that the share price will rise in the next 6 months. To eliminate the price risk, they enter into a 6-month long equity forward on 1 June 2017. The Lufthansa stock trades at a price per share of EUR 17.29 on 1 June 2017. The continuous compounded 6-month EURIBOR is -0.254%. The forward price of EUR 17.268 can be determined with the cost-of-carry model as follows: F 0 = EUR 17:29 × e - 0:00254 × 6=12 = EUR 17:268: Accordingly, the investor has the obligation to buy 800 shares of Lufthansa AG at a price per share of EUR 17.268 at the forward’s expiration date of 1 December 2017. On this day, the stock of the airline company is traded at a price per share of EUR 29.07. The profit of the long forward contract is EUR 9441.60 and can be calculated as follows: Profit long forward = 800 × ðEUR 29:07 - EUR 17:268Þ = EUR 9441:60: On 1 December 2017, the investor buys 800 Lufthansa shares for EUR 23,256 (= 800 × EUR 29.07). The stock purchase is EUR 9424 [= EUR 23,256 - (800 × EUR 17.29)] more expensive than on 1 June 2017. This loss of EUR 9424 is offset against the EUR 9441.60 profit from the long forward, resulting in a net profit from the hedging strategy of EUR 17.60 (= EUR 9441.60 - EUR 9424). The 6-month return of the strategy is 0.127%: Return6 months =

EUR 17:60 = 0:127%: 800 × EUR 17:29

The annualised hedge return is therefore 0.254%: Return12 months = e0:00127 × 2 - 1 = 0:254%: Hence, the investor achieves a return of 0.254% with the anticipatory hedge, which is positive in this example because the 6-month EURIBOR of -0.254% is negative. The return of the hedging strategy is given by the financing interest rate and is therefore already known at the beginning of the hedge. If the share price falls, the investor can buy the security at a later date at a lower price. However, this profit is absorbed by the loss on the long forward. The cost of the hedging strategy is equal to the financing interest rate. With an anticipatory hedge, both a potential gain and loss on the later purchase of the stock are eliminated.

13.4

Swaps

13.4

469

Swaps

A swap is an agreement between two parties who commit to exchange a series of future cash flows. Like futures and forwards, they are forward commitments and can be considered as a portfolio (or series) of forwards. Examples include interest rate swaps, currency swaps, equity swaps, and non-financial swaps such as commodity swaps, credit default swaps, and total return swaps. Interest rate swaps are described below. They are referred to as plain vanilla swaps and are probably the most common derivative transaction in the global financial system.15 In an interest rate swap both contracting parties exchange a fixed interest payment for a floating interest payment in the same currency over the same time period. There is no exchange of the notional principal. The notional principal is only used to compute the interest payments. The fixed interest payments are calculated by multiplying the notional principal by the swap rate, while the floating interest payments are determined by multiplying the notional principal by a reference interest rate in the money market such as the EURIBOR or Compounded €STR for the euro or the Compounded SARON for the Swiss franc. Since the reference interest rate changes from one interest period to the next, the floating interest payment must be recalculated for each period. The interest payments can be, and nearly always are, netted. The party owing the greater amount pays the net difference, which significantly reduces the credit risk.16 The fixed and floating payments can be calculated for the respective interest periods as follows:17 Fixed payment = Notional princpal × Swap rate ×

t , 365 days

Floating payment = Notional princpal × Reference rate ×

t , 360 days

ð13:7Þ ð13:8Þ

where t = days of the interest period. For the sake of simplicity, months rather than days are used to calculate the interest payments below. For example, there is an interest rate swap with a time to maturity (or tenor) of 10 years and a notional principal of EUR 20 million. The fixed and floating interest payments are made semi-annually, at the beginning of January and July. At the beginning of January 2018, the 10-year swap rate for the euro is 0.866%, while the 6-month EURIBOR is -0.271% (Source: Refinitiv Eikon). The

15

See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 705. See Watsham 1998: Futures and Options in Risk Management, p. 464 ff. 17 The day count convention for EURIBOR, €STR, and SARON is actual/360, whereas the swap rate is usually quoted as actual/365 or 30/360. 16

470

13

Futures, Forwards, and Swaps

fixed and floating payments can be determined for the first semi-annual interest period as follows: Fixed payment = EUR 20, 000, 000 × 0:00866 ×

6 months = EUR 86, 600, 12 months

6 months Floating payment = EUR 20, 000, 000 × ð - 0:00271Þ × 12 months = - EUR 27, 100: The fixed and floating interest payments are calculated for the next 10 years at the beginning of each semi-annual interest period and paid at the end of the corresponding interest period. Since the cash flows refer to the same currency, namely the euro, they are netted. Thus, the counterparty with the fixed interest payment pays EUR 113,700 [= EUR 86,600 - (-EUR 27,100)]. As long as the 6-month EURIBOR is negative, the floating leg of the swap receives not only the fixed but also the floating interest payments. The counterparty paying the swap rate and receiving the floating rate is called the payer, while the party receiving the swap rate and paying the floating rate is known as the receiver. Accordingly, the terms ‘payer’ and ‘receiver’ refer to the fixed interest payments. The terms ‘long’ and ‘short’, which are used for buying and selling futures/forwards and options, are not applied in the case of swaps. The swap rate is offered by financial institutions at a bid and ask rate (buying and selling rate), with the bid–ask spread usually being between 1 and 4 basis points, reflecting the high market liquidity of interest rate swaps. The swap rate is the average of the bid and ask rates. At the beginning of January 2018, the 10-year swap rate for the euro is quoted at a bid rate of 0.860% and an ask rate of 0.866% (Source: Refinitiv Eikon). The bid–ask spread is 0.6 basis points, while the swap rate is 0.863% [(= 0.860% + 0.866%)/2]. Market participants enter into a payer swap at the ask rate and a receiver swap at the bid rate with a financial institution. Trading usually takes place through electronic trading platforms. Borrowers can use interest rate swaps to manage their exposure to rising interest rates and convert a floating-rate loan to a fixed-rate loan. For example, at the beginning of January 2018, a company has a floating-rate note outstanding with a time to maturity of 10 years and a par value of EUR 100 million. The coupon rate of the floater consists of the 6-month EURIBOR plus a quoted margin of 75 basis points. Since the company expects rising interest rates, and thus higher capital costs, it decides to enter into a 10-year payer swap with a notional principal of EUR 100 million in early January 2018. Interest payments are made semi-annually. The swap rate is 0.866%, while the floating rate is given by the 6-month EURIBOR. The floating-rate note is transformed into a fixed-rate debt position with the payer swap, which has a fixed annual interest rate of 1.616%, regardless of the interest rate level:

13.5

Summary

471

Balance Sheet Assets

Risk: rising interest rates

Liabilities and Equity

Floating-rate note

EURIBOR + 0.75%

Payer Swap: Pays swap rate

0.866 %

Receives EURIBOR

EURIBOR

Fig. 13.3 Transformation of an issued floating-rate note into a fixed-rate debt position using a payer swap. (Source: Own illustration) Floating-rate note: pays coupon rate Payer swap: pays swap rate Payer swap: receives EURIBOR Net interest rate (net interest cost)

EURIBOR + 0.75% + 0.866% - EURIBOR = 1.616%

With the help of the payer swap, the floating-rate note is converted into a fixedrate debt position with a fixed annual interest cost of 1.616%. Figure 13.3 presents the interest payments of the floating-rate note and the payer swap.

13.5

Summary

• A derivative is a contract between two parties whose value is derived from the price of an underlying asset. Depending on the underlying asset, a distinction is made between financial, commodity, and credit derivatives, among others. • Derivatives can also be divided into forward commitments and contingent claims. A forward commitment is an obligation to buy or sell a certain number or quantity of an underlying asset at an agreed price at a later date. By contrast, a contingent claim grants the buyer the right to buy (call option) or to sell (put option) the underlying asset at the agreed strike price either during or at the end of the option life. The option seller, on the other hand, has the obligation to sell (call option) or buy (put option) the underlying asset at the strike price. He must fulfil this obligation if the holder of the call or put exercises their buy or sell option.

472

13

Futures, Forwards, and Swaps

• Derivatives are used by market participants to hedge the price risk of an asset or liability (hedging), assume risk (speculation, trading), and exploit price differences (arbitrage). • Derivatives are traded on an exchange as well as over the counter. Exchangetraded derivatives include futures and options, which are traded electronically and anonymously. The contract features such as the underlying, the contract size, and the time to expiration are specified by the derivatives exchange. Thus, a futures contract is standardised so that the exchange, rather than the individual parties, sets the terms and conditions, with the exception of the price. The derivatives exchange acts as the contractual partner for the buyer and the seller and guarantees to each party the performance of the other party, through a mechanism known as the clearinghouse. This largely eliminates the settlement risk (credit risk). A security deposit must be made with the derivatives exchange in the form of an initial margin. In addition, there is a variation margin which includes the profit or loss realised at the end of each trading day. This procedure is known as daily settlement or marking to market. Derivatives traded over the counter include forwards, swaps, and options, whose contractual features are not standardised but are negotiated individually by the contracting parties. Trading takes place through electronic trading platforms or through direct contacts between market participants. Over-the-counter derivatives are classified into centrally cleared and bilaterally traded instruments. Regardless of the classification, a security deposit in the form of an initial margin must be paid. In addition, daily unrealised profits/losses are calculated (variation margin). • With a long futures/forward contract, a profit (loss) is made when the futures/ forward price rises (falls). With a short futures/forward contract it is the other way round. A profit (loss) is incurred if the futures/forward price falls (rises). Futures/ forward contracts are zero-sum games because the profits/losses of the long and short positions are mirror images of each other. • With a derivative, a much smaller amount of money is required to enter into the risk exposure of the underlying asset and thus to participate in its price movements. Therefore, the return on derivatives is many times higher or lower than the return on the underlying asset. This is known as leverage. • The price of a futures/forward contract can be estimated by applying the cost-ofcarry model. With this replication approach the costs of the futures market (i.e. the forward price) are determined using the costs of the spot market. In the case of a commodity futures/forward contract, the costs of the spot market consist of the spot price of the underlying commodity plus the financing and carrying costs minus a possible benefit of holding the commodity in the inventory (convenience yield). In the case of a financial futures/forward contract, the costs of the spot market comprise the spot price of the underlying financial asset plus the financing costs less any income from the financial asset. If the market price of the futures/ forward deviates from the cost-of-carry model price, investors can earn a risk-free profit with an arbitrage strategy. In the event that the futures/forward is overvalued (undervalued), they will carry out a cash-and-carry arbitrage (reverse cash-and-carry arbitrage) until the market price has moved to the model price. In

13.6

•

•

•

•

Problems

473

this equilibrium price the costs of the futures/forward market and the costs of the spot market are equal. The derivative is correctly priced. The value calculation is only relevant for forwards, but not for futures because, unlike futures, they are not marked to market. An accumulated unrealised profit (loss) represents an asset (liability) position. The value of a forward can be calculated by discounting the profit/loss at the valuation date. At the inception of a forward the value is zero. In the case of a hedge, a distinction is made between hedging an existing asset or liability position and an anticipatory hedge. In hedging an existing position, the price risk of an asset is eliminated with short futures/forwards and of a liability with long futures/forwards. The return is already known at the beginning of the hedge and equals the financing interest rate. On the other hand, an anticipatory hedge removes the price risk of a planned purchase or sale of a financial asset with long or short futures/forwards. Such a hedge is appropriate when an investor knows they will have to purchase or sell a certain asset in the future and want to lock in a price now. The return of the hedging strategy is equal to the financing interest rate. Both a potential gain and loss on the later purchase or sale of the asset are eliminated. A swap is an agreement between two parties who commit to exchange a series of future cash flows. For example, in an interest rate swap a fixed interest payment is exchanged for a floating interest payment in the same currency over the same time period. There is no exchange of the notional principal. The notional principal is only used to determine the interest payments. The fixed interest payments are calculated by multiplying the notional principal by the swap rate, while the floating interest payments are determined by multiplying the notional principal by a reference interest rate in the money market such as the EURIBOR or Compounded €STR for the euro, or the Compounded SARON for the Swiss franc. The interest payments can be, and nearly always are, netted. The holder of a payer swap pays the swap rate and receives the floating rate, while the holder of a receiver swap receives the swap rate and pays the floating rate. Accordingly, the terms ‘payer’ and ‘receiver’ refer to the fixed interest payments. If interest rates are expected to rise in the future, a floating-rate loan or note can be converted into a fixed-rate debt position using a payer swap. In such a case, the floating interest payments of the loan or note and the swap cancel each other out, leaving a fixed interest rate exposure.

13.6

Problems

1. An investor owns 5000 shares in Lufthansa AG. The stock of the airline company is trading at EUR 17.41 on 1 June 2017. The continuous compounded 3-month EURIBOR is -0.329%. What is the profit/loss on a 3-month short forward, expiring on 1 September 2017, on the 5000 Lufthansa shares if the share price is EUR 18.68 and the continuous compounded 1-month EURIBOR is -0.371% on 1 August 2017?

474

13

Futures, Forwards, and Swaps

2. An investor enters into a long forward contract on 1000 shares of Bayer AG with a time to expiration of 9 months on 1 February 2017. The Bayer stock trades at a price per share of EUR 102.87 on 1 February 2017. The dividend per share of EUR 2.70 will be distributed at the beginning of May 2017 after the annual shareholder meeting. The continuous compounded 3-month and 9-month EURIBOR are -0.318% and -0.140%, respectively. a) At what forward price can Bayer AG stock be bought in 9 months? b) The Bayer stock trades at a price per share of EUR 107.45 on 1 August 2017. The continuous compounded 3-month EURIBOR is -0.331%. What is the value of the long forward on 1 August 2017? c) The Bayer stock trades at a price per share of EUR 114.30 on the expiration date of the forward (i.e. 1 November 2017). What is the value of the long forward on 1 November 2017? 3. An institutional investor holds a long equity position consisting of 2600 shares of Siemens AG and a short equity position consisting of 3000 shares of MercedesBenz Group AG on 1 June 2017. The investor expects falling prices for the Siemens stock and rising prices for the automobile stock in the coming months. On 1 June 2017, the Siemens share is trading at a price of EUR 127.05, while the Mercedes-Benz Group share is trading at a price of EUR 65.37. As the institutional investor will hold the two equity positions for a period of 5 months, they want to hedge the price risk using 5-month forwards with an expiration date on 1 November 2017. The continuous compounded 5-month EURIBOR is 0.280%. a) The Siemens stock is trading at a price per share of EUR 125.10 on 1 November 2017. What is the forward price at the inception of the contract on 1 June 2017 and the profit/loss from the hedging strategy on 1 November 2017? b) The Mercedes-Benz Group stock is trading at a price per share of EUR 72.93 on 1 November 2017. What is the forward price at the inception of the contract on 1 June 2017 and the profit/loss from the hedging strategy on 1 November 2017? 4. An interest rate swap with a tenor of 10 years and a notional principal of EUR 50 million has a swap rate of 1%. Interest payments are made quarterly. The 3-month EURIBOR at the beginning of the interest period is 0.25%. What is the interest payment at the end of the interest period for the holder of the payer swap and the receiver swap?

13.7

Solutions

1. In order to calculate the profit/loss of the short forward, the forward prices on 1 June and 1 August 2017 must first be determined:

13.7

Solutions

475

F 0 = EUR 17:41 × e - 0:00329 × 3=12 = EUR 17:40, F t = EUR 18:68 × e - 0:00371 × 1=12 = EUR 18:67: The short forward produces a loss of EUR 6350 on 1 August 2017, which can be calculated as follows: Loss short forward = -5000 × ðEUR 18:67 - EUR 17:40Þ = -EUR 6350: 2. a) The forward price of EUR 100.063 can be calculated with the cost-of-carry model as follows: F 0 = EUR 102:87 -

EUR 2:70 × e - 0:0014 × 9=12 = EUR 100:063: e - 0:00318 × 3=12

b) In order to determine the value of the long forward, the forward price at the valuation date of 1 August 2017 must first be calculated: F t = EUR 107:45 × e - 0:00331 × 3=12 = EUR 107:361: The value of the long forward is EUR 7304.04 and can be determined by discounting the profit on the valuation date: V t, Long =

1000 × ðEUR 107:361 - EUR 100:063Þ = EUR 7304:04: e - 0:00331 × 3=12

c) The value of the long forward on the expiration date equals the profit of EUR 14,237: V T, Long = 1000 × ðEUR 114:30 - EUR 100:063Þ = EUR 14, 237: 3. a) The forward price on the Siemens stock of EUR 126.902 can be calculated with the cost-of-carry model as follows: F 0 = EUR 127:05 × e - 0:00280 × 5=12 = EUR 126:902: The long equity position should be hedged with a short equity forward. At the expiration date of the short forward on 1 November 2017, a profit of EUR 4685.20 is realised: Profit short forward = - 2600 × ðEUR 125:10 - EUR 126:902Þ = EUR 4685:20:

476

13

Futures, Forwards, and Swaps

The loss on the long equity position is EUR 5070 [= 2600 × (EUR 125.10 EUR 127.05)]. Thus, the net loss from the hedging strategy is EUR 384.80 (= EUR 4685.20 - EUR 5070), resulting in a 5-month return of -0.1165%: Return5 months =

- EUR 384:80 = - 0:1165%: 2600 × EUR 127:05

The annualised hedge return is therefore -0.279% (=e-0.001165 × 12/5 - 1), which corresponds to the 5-month EURIBOR. b) The short equity position can be hedged with a long equity forward. The forward price on the Mercedes-Benz Group stock is EUR 65.294: F 0 = EUR 65:37 × e - 0:00280 × 5=12 = EUR 65:294: On the expiration date of the long equity forward, a profit of EUR 22,908 is realised: Profit long forward = 3000 × ðEUR 72:93 - EUR 65:294Þ = EUR 22, 908: The short equity position produces a loss of EUR 22,680 [= -3000 × (EUR 72.93 - EUR 65.37)]. Hence, the net profit of the hedging strategy is EUR 228 (= EUR 22,908 - EUR 22,680). The 5-month return of the strategy is 0.1163% and can be calculated as follows: Return5 months =

EUR 228 = 0:1163%: 3000 × EUR 65:37

The annualised return of the hedge is 0.279% (=e0.001163 × corresponds to the 5-month EURIBOR for a short position.

12/5

- 1) and

4. The holder of the payer swap pays the swap rate and receives the EURIBOR. Accordingly, they owe the fixed interest payment of EUR 125,000: Fixed payment = EUR 50, 000, 000 × 0:01 ×

3 months = EUR 125, 000: 12 months

By contrast, the receiver swap holder pays the floating interest rate, resulting in a payment of EUR 31,250: Floating payment = EUR 50, 000, 000 × 0:0025 ×

3 months = EUR 31, 250: 12 months

The interest payments are netted so that the holder of the payer swap makes a payment of EUR 93,750 (= EUR 125,000 - EUR 31,250) to the receiver swap holder at the end of the interest period.

References

477

References Black, K.H., Chambers, D.R., Kazemi, H.: CAIA Level II: Advanced Core Topics in Alternative Investments, 2nd edn. Wiley, Hoboken (2012) Bösch, M.: Derivate: Verstehen, anwenden und bewerten, 3rd edn. Auflage, München (2014) Chance, D.M.: Analysis of Derivatives for the CFA® Program. Association for Investment Management and Research, Charlottesville (2003) Eurex: Aktien- und Aktienindexderivate: Handelsstrategien, Eschborn, Zürich (2007) Hull, J.C.: Options, Futures, and Other Derivatives, 6th edn. Prentice Hall, Upper Saddle River (2006) Kolb, R.W.: Futures, Options, and Swaps, 3rd edn. Blackwell, Malden (2000) Mondello, E.: Finance: Theorie und Anwendungsbeispiele. Springer Fachmedien, Wiesbaden (2017) Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 8th edn. Thompson South-Western, Mason (2006) Rudolph, B., Schäfer, K.: Derivative Finanzmarktinstrumente: Eine anwendungsbezogene Einführung in Märkte, Strategien und Bewertung, 2nd edn. Springer, Berlin, Heidelberg (2010) Watsham, T.J.: Futures and Options in Risk Management, 2nd edn, London et al (1998)

Options: Basics and Valuation

14.1

14

Introduction

Options, like futures and forwards, are derivative instruments that provide the opportunity to buy or sell an underlying asset with a specific expiration date. However, a long option gives the holder the right, not the obligation, to buy (call) or sell (put) an underlying asset. On the other hand, the holder of the short option or the option seller has the obligation to fulfil the option buyer’s right to buy or sell. For the right to purchase or sell an underlying asset, an option premium is paid from the buyer to the seller of the option. By contrast, futures and forwards involve no cash upfront payment. The basic characteristics and the profit and loss calculation of call and put options are discussed below. This is followed by an examination of option pricing using the one-stage and two-stage binomial model, the Black–Scholes model, and put–call parity. The leverage is then described, which reflects the return leverage of options against the underlying asset. The chapter ends with the option price sensitivities. They allow to examine how much the option price changes when a risk factor (e.g. price or price volatility of underlying) moves. Options on individual stocks, also called equity options, are among the most popular and are subsequently used to illustrate the profit and loss calculation, the pricing, the leverage effect, and the option price sensitivities.

14.2

Basic Characteristics

A call option grants the holder the right to buy a certain quantity of an underlying asset during or at the end of the option life at a price agreed in advance. The holder of a put option, on the other hand, has the right to sell a certain quantity of an underlying asset during or at the end of the option life at a fixed price. The price at which the owner of the option can purchase or sell the underlying asset is known as # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_14

479

480

14

Options: Basics and Valuation

the strike price or exercise price. By contrast, the seller of the option has the commitment to sell (call) or to buy (put) the underlying asset. Since the seller must wait to see whether the buyer exercises his call or put, he is also known as the writer. For this commitment, the option seller receives an option premium from the option buyer. An option that can be exercised at any time during its time to expiration is called an American option. If the option can be exercised only on the expiration date, it is referred to as a European option. The terms ‘American’ and ‘European’ do not refer to the trading location or the origin of the option, but only to the exercise modality. When the expiration date arrives, an option that is not exercised simply expires. The options are named after the underlying asset. If, for example, the underlying assets refer to an equity security, currency, interest rate, or gold, one speaks of an equity option, currency option, interest rate option, or gold option. Essentially, four basic positions can be distinguished in an option transaction, namely the purchase and sale of a call and the purchase and sale of a put. The purchase of an option corresponds to a long position, while the sale of the option represents a short position. What happens at exercise depends on whether the option is a call or a put. If the owner of the long call (put) exercises their right, they pay (receive) the strike price and receive (deliver) the underlying or receive an equivalent cash settlement. On the other side of the transaction stands the owner of the short call (put) who receives (pays) the strike price from (to) the holder of the long call (put) and delivers (receives) the underlying, or alternatively, pays an equivalent cash settlement. Therefore, when the option is exercised, either a physical delivery of the underlying asset or a cash settlement takes place. In the case of a cash settlement, the option holder exercising the call receives the difference between the market value of the underlying asset and the strike price from the option seller in cash. If the option holder exercises a put, they receive the difference between the strike price and the market price of the underlying in cash. In the case of physical delivery, the period between the last trading day and the delivery day of the underlying depends on the delivery practices for a corresponding spot market transaction. For options on securities, the period is usually 2 business days. This gives the seller of a call the opportunity to buy the underlying asset on the spot market in order to be able to fulfil their delivery obligations on time. Figure 14.1 presents the four basic option positions. The profit and loss characteristics associated with the long and short option positions are different.1 Options do not have to be held until the expiration date, but can be closed out early like futures and forwards. Exchange-traded options can be offset before the expiration date by entering into an opposite option position. For example, a long call requires a short call with the same underlying asset, strike price, and time to expiration. The derivatives exchange (e.g. Eurex) acts as counterparty to an exchange-traded option. If the option is not closed out, it can be exercised by the buyer in the event of an advantageous price scenario. European options can only be

1

See Sect. 14.3.

14.3

Profit and Loss

481

Long option The buyer of the option pays the option premium and receives the right to buy or sell the underlying asset.

Short option The seller of the option receives the option premium and has the commitment to sell or buy the underying asset.

Call (option to buy)

The buyer of the call option has the right to purchase the underlying asset.

The seller of the call option (writer) has the obligation to sell the underlying asset if the owner of the long call exercises his right.

Put (option to sell)

The buyer of the put The seller of the put option has the right to option (writer) has the sell the underlying asset. obligation to buy the underlying asset if the owner of the long put exercises his right.

Contract position

Type of option

Fig. 14.1 Four basic option positions (Source: Own illustration)

exercised on the expiration date, while American options can be exercised at any time during their lifetime.

14.3

Profit and Loss

14.3.1 Call Option The profit and loss calculation will be presented below using a call equity option. For example, a long call option has a strike price of EUR 100. The price of the call is EUR 10. If the share price is EUR 95 on the expiration date of the call option, the holder of the long call will not exercise the option, because the equity security can be bought on the market at a price lower than the agreed strike price of EUR 100. The call option therefore expires worthless. Since the holder of the long call option has paid the premium of EUR 10, a loss of EUR 10 is incurred. The maximum possible loss is limited to the option premium paid. It does not matter how far below the strike price the share price is traded on the expiration date of the call. The loss of EUR 10 is reduced or also turned into a profit if the share price exceeds the strike price of EUR 100. In this case, the call option is always exercised (without taking transaction costs into account). If, for example, the share price is EUR 105, the purchase of the

482

14

(Profit/ loss in EUR) 50 40

Options: Basics and Valuation

Long call Profit is limited to the option premium received of EUR 10.

Unlimited potential for profit

30 20 10 0 -10

0

20

40

60

80

100

120

140

160

180

(Share price in EUR) -20 -30 -40 -50

Loss is limited to the option premium paid of EUR 10.

Breakeven share price of EUR 110 Unlimited potential for loss

Short call

Fig. 14.2 Profit/loss diagram of a long and a short equity call option on the expiration date (Source: Own illustration)

stock using the call is worthwhile because the security, which has a price of EUR 105 on the market, can be bought at a price of EUR 100. The gain of EUR 5 generated by exercising the option is reduced by the option premium paid of EUR 10, resulting in a net loss of EUR 5. Without exercising the call option, the loss would be higher, namely EUR 10. One enters the profit area if the share price exceeds the exercise price plus the option premium paid. If the share price is EUR 120, exercising the option results in a gain of EUR 20, because the security can be bought at the exercise price of EUR 100, but has a value of EUR 120. The gain from the exercise of EUR 20 is reduced by the option premium paid of EUR 10, leaving a net profit of EUR 10. The maximum possible profit of the long call is unlimited, as there is no upper limit for the share price, while the loss is limited to the option premium paid. The short call has the opposite profit/loss pattern to the long call. Should the share price be below the strike price on the expiration date, the call is not exercised by the option buyer, with the result that the profit equals the option premium received, which thus represents the maximum possible profit. If, on the other hand, the share price exceeds the strike price, the call is exercised. A loss is incurred by the option seller whenever the difference between the share price and the exercise price exceeds the option premium received. Since there is no upper limit for the share price, the maximum possible loss is unlimited. Figure 14.2 presents the asymmetric profit/loss pattern of a long and a short equity call option with a strike price of EUR 100 and an

14.3

Profit and Loss

483

option premium of EUR 10 on the expiration date. The figure also indicates that no new money is created with options. There is only a redistribution of wealth between the two contracting parties of the option. In order to calculate the net profit/loss of a long call and a short call, the gain from the exercise or the corresponding option value on the expiration date must first be calculated. The call option is exercised on the expiration date if the share price exceeds the strike price. In this case, the value of the option consists of the share price minus the strike price. However, should the share price be below the exercise price, the call option expires worthless. The value of the call is zero. Accordingly, the value at expiration VT of a long call and a short call can be determined as follows: V T, Long call = max ð0, ST - XÞ

ð14:1Þ

V T, Short call = - max ð0, ST - XÞ,

ð14:2Þ

and

where ST = price of the underlying asset at the option expiration date T, and X = strike (exercise) price. The profit/loss of a long call option can be calculated by deducting the premium paid from the call value at expiration: Profit=lossLong call = V T, Long call - c0 ,

ð14:3Þ

where c0 = premium paid to purchase the call option. The profit/loss of a short call option equals the call value at expiration plus the premium received from selling the call: Profit=lossShort call = V T, Short call þ c0 ,

ð14:4Þ

where c0 = premium received from the sale of the call option. The breakeven point for a long call is reached when the share price at expiration exceeds the exercise price by the amount of the premium paid to recover the cost of the premium. With a short call, the loss begins when the option premium received is no longer sufficient to cover the loss from the exercise of the option (ST - X). Thus, for a long call and a short call, the breakeven share price at expiration equals the exercise price plus the option premium paid or received:

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Table 14.1 Maximum profit, maximum loss, breakeven share price, and profit/loss of a long call and a short call at expiration (Source: Own illustration) Call Long call Short call

Maximum profit Unlimited c0 (premium received)

Maximum loss c0 (premium paid) Unlimited

Breakeven share price X + c0 X + c0

Breakeven ST = X þ c0 :

Profit/loss VT, Long call - c0 or max(0, ST - X) - c0 VT, Short call + c0 or – max (0, ST - X) + c0

ð14:5Þ

For a call option with a strike price of EUR 100 and a premium of EUR 10, the breakeven share price at expiration is EUR 110 (= EUR 100 + EUR 10). Table 14.1 presents the maximum profit, maximum loss, breakeven share price, and profit/loss of a long call and a short call at expiration. The moneyness of options describes the relationship between the price of the underlying and the exercise price. A distinction is made between options that are in the money, at the money, and out of the money. A call option is in the money if the price of the underlying asset is higher than the strike price. In this price scenario it is worthwhile to exercise the call (without considering transaction costs). For at-themoney options, the price of the underlying assets is equal to the strike price. At-themoney call options are not exercised, as transaction costs always lead to a loss. For an out-of-the-money call option, the price of the underlying is below the exercise price. Out-of-the-money call options are not exercised because the underlying can be bought more cheaply on the spot market. Example: Profit/Loss Calculation of a Call Option A European call option on the Mercedes-Benz Group stock has a strike price of EUR 68. The premium for the call option is EUR 5, while the automobile stock is trading at EUR 70. 1. What is the profit/loss for the call buyer if the share price is either EUR 80 or EUR 60 at expiration of the option? 2. What is the profit/loss for the call seller if the share price is either EUR 80 or EUR 60 at expiration of the option? 3. What is the maximum profit and loss for the buyer and the seller of the call option? 4. What is the breakeven share price for the buyer and the seller of the call option at expiration? (continued)

14.3

Profit and Loss

485

Solution to 1 At a share price of EUR 80 at expiration, the call option ends in the money and is exercised. A profit of EUR 7 can be determined as follows for the holder of the long call option: V T, Long call = max ðEUR 0, EUR 80 - EUR 68Þ = EUR 12, ProfitLong call = EUR 12 - EUR 5 = EUR 7: At a share price of EUR 60 at expiration, the option expires out of the money and is not exercised. Accordingly, the holder of the long call option incurs a loss of EUR 5, which is the same as the option premium paid: V T, Long call = max ðEUR 0, EUR 60 - EUR 68Þ = EUR 0, LossLong call = EUR 0 - EUR 5 = - EUR 5: Solution to 2 At a share price of EUR 80 at expiration, the call option ends in the money and is exercised by the holder of the long call option. A loss of EUR 7 for the holder of the short call option can be determined as follows: V T, Short call = - max ðEUR 0, EUR 80 - EUR 68Þ = - EUR 12, LossShort call = - EUR 12 þ EUR 5 = - EUR 7: At a share price of EUR 60 at expiration, the option expires out of the money and is therefore not exercised, resulting in a profit of EUR 5, which corresponds to the premium received from the sale of the call: V T, Short call = - max ðEUR 0, EUR 60 - EUR 68Þ = EUR 0, ProfitShort call = EUR 0 þ EUR 5 = EUR 5: The example demonstrates that options, like futures/forwards, are a zerosum game. The profit (loss) of one party is equal to the loss (profit) of the other party. As a result, no new money is created, but the money is merely redistributed among the contracting parties of the option. Solution to 3 The maximum profit for the call buyer option is unlimited because there is no upper limit for the share price. By contrast, the loss is limited to the paid option premium of EUR 5. The maximum profit for the seller of the call option is limited to the option premium received of EUR 5, while the potential loss is unlimited as there is no upper price limit for the share price. (continued)

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14

Options: Basics and Valuation

Solution to 4 The breakeven share price of the buyer and the seller of the call option at expiration is EUR 73 and can be calculated as follows: Breakeven ST = EUR 68 þ EUR 5 = EUR 73: The holder of the long call option makes a profit if the share price at expiration exceeds the breakeven share price of EUR 73. A share price of EUR 73 results in a call value at expiration of EUR 5 [= max(EUR 0, EUR 73 - EUR 68)], and thus after deducting the paid option premium of EUR 5, the profit/loss is EUR 0 (= EUR 5 - EUR 5). The profit area for the holder of the short call option begins when the share price at expiration falls below the breakeven share price of EUR 73. At a share price of EUR 73, there is neither a profit nor a loss, since the loss from exercising the option of EUR 5 [= - max(EUR 0, EUR 73 - EUR 68)] is offset by the option premium received of EUR 5. The holder of a long call can only make a profit on the expiration date of the option if the share price is higher than the strike price by more than the option premium paid. Therefore, they speculate on rising prices of the underlying. The higher the price increase of the underlying, the higher the profit. If the price of the underlying falls despite a predicted price rise, they cannot lose more than the option premium paid. Furthermore, in the case of a long call, the purchase date of the underlying is postponed to the exercise date, so that the funds not tied up in the underlying can be invested in the money market until the option has been exercised. In a positive interest rate environment, an interest income accrues that can be used to finance the purchase of the call. If, for example, the premium for an equity call is EUR 5 and the underlying stock is traded at a price per share of EUR 100, an interest rate of 5% is sufficient for a 1-year call to finance the purchase of the option completely with the interest income. Unlike the case of a share purchase, buying a call allows a profit to be made if the share price rises and a loss to be avoided if the share price falls. The risk for the holder of the long call consists not only of falling prices of the underlying, but also of the limited option life. Thus, the opportunity to make a profit because of rising underlying prices is limited in time. The call buyer must therefore correctly predict not only the right price forecast but also the period of the price increase. Furthermore, the choice of the exercise price plays an important role in the purchase of a call, as it has an influence on the amount of the option premium paid. A higher (lower) strike price will result in a lower (higher) option price, all else being equal. However, a higher exercise price increases the breakeven share price, and the share price increase must therefore be correspondingly large for a profit to be made on the call option. Therefore, when choosing the strike price, there is a trade-off between the maximum loss, which is limited to the option premium paid and is

14.3

Profit and Loss

487

smaller the higher the strike price, and the extent of the price increase required to reach breakeven. The opportunities and risks for the call buyer are exactly opposite for the call seller. Thus, the call option seller profits if the underlying price remains below the strike price, as in this price scenario the option is not exercised by the holder of the long call. Since the buyer’s call right is limited in time, price increases of the underlying asset after the expiration date are meaningless. As a result, time works in favour of the option seller. In addition, the choice of strike price has an impact on the likelihood of an option being exercised. The higher the strike price chosen, the lower the probability of an option exercise by the call buyer. However, the cash proceeds from the sale of the call also decrease because a higher strike price results in a lower option premium.

14.3.2 Put Option The put buyer exercises the option to sell the underlying asset once the price of the underlying is lower than the strike price. If, for example, the put price is EUR 10, the exercise price EUR 100, and the share price EUR 95, the gain from exercising the option is EUR 5 (= EUR 100 - EUR 95). The equity security worth EUR 95 in the market can be sold at a price of EUR 100 through the option, which leads to an exercise gain of EUR 5. This gain is offset against the paid option premium of EUR 10, resulting in a net loss of EUR 5 (= EUR 5 - EUR 10), which is lower than the paid premium of EUR 10. Thus, exercising the put option is worthwhile. Should the share price at expiration of the long put be EUR 80, the option is again exercised because the stock can be sold through the option at a higher price of EUR 100 than on the spot market at a price of EUR 80. The gain from exercising the option is EUR 20 (= EUR 100 - EUR 80). If the paid option premium of EUR 10 is deducted from this gain, the result is a net profit of EUR 10 (= EUR 20 - EUR 10). The greater the difference between the strike price and the price of the underlying less the option premium paid, the higher the profit. However, the share price cannot fall below zero, so the maximum possible profit is the exercise price minus the option premium paid. If, on the other hand, the share price on the expiration date of the option is EUR 120, the put option is not exercised because the underlying stock can be sold on the spot market at a higher price of EUR 120 than through the option at the strike price of EUR 100. Hence, the loss is limited to the paid option premium of EUR 10. In the case of a short put option, a loss results from the exercise of the option on the expiration date if the share price falls below the strike price, as the underlying must be bought by the option seller at a price above the market value. The net profit/ loss is equal to the exercise loss reduced by the option premium received from the sale of the option. Should the share price exceed the exercise price, the put is not exercised by the option buyer. In this case, the profit is made up of the option premium received, which at the same time represents the maximum possible profit. Figure 14.3 presents the asymmetric profit/loss pattern of a long and a short equity

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14

(Profit/ loss in EUR) 50

Options: Basics and Valuation

Profit is limited to EUR 90

Long put

Profit is limited to the option premium received of EUR 10

40 30 20 10 0 -10 -20 -30

0

20

40

80

100

120

140

160

180

(Share price in EUR) Loss is limited to EUR 90

-40 -50

60

Breakeven share price of EUR 90 Loss is limited to the option premium paid of EUR 10

Short put

Fig. 14.3 Profit/loss diagram of a long and a short equity put option on the expiration date (Source: Own illustration)

put option with a strike price of EUR 100 and an option premium of EUR 10 on the expiration date. In order to determine the net profit/loss of a long put and a short put, the gain from the exercise or the corresponding option values on the expiration date VT must first be calculated: V T, Long put = max ð0, X - ST Þ

ð14:6Þ

V T, Short put = - max ð0, X - ST Þ:

ð14:7Þ

and

The profit/loss of a long put can be calculated by subtracting the option premium paid from the put value at expiration: Profit=lossLong put = V T, Long put - p0 , where

ð14:8Þ

14.3

Profit and Loss

489

Table 14.2 Maximum profit, maximum loss, breakeven share price, and profit/loss of a long put and a short put at expiration (Source: Own illustration) Put Long put Short put

Maximum profit X - p0 p0 (premium received)

Maximum loss p0 (premium paid) X - p0

Breakeven share price X - p0 X - p0

Profit/loss VT, Long put - p0 or max(0, X ST) - p0 VT, Short put + p0 or - max (0, X - S T) + p 0

p0 = premium paid to purchase the put option. The profit/loss of a short put equals the put value at expiration plus the option premium received: Profit=lossShort put = V T, Short put þ p0 ,

ð14:9Þ

where p0 = premium received from the sale of the put option. For a long put, breakeven is reached when the share price falls below the strike price by the amount of the option premium paid to recover the cost of the premium. The gain from the exercise (X - ST) is offset by the paid option premium. The profit area begins with a share price below the breakeven share price. On the other hand, for a short put option, breakeven is reached when the share price is below the strike price by the amount of the option premium received. The loss from the exercise (X ST) is absorbed by the option premium received from the sale of the option. The profit area begins with a share price above the breakeven share price. Accordingly, the breakeven share price at expiration is the same for a long put and a short put and can be calculated as follows: Breakeven ST = X - p0 :

ð14:10Þ

For a put option with a strike price of EUR 100 and a premium of EUR 10, the breakeven share price at expiration is EUR 90 (= EUR 100 - EUR 10). Table 14.2 presents the maximum profit, maximum loss, breakeven share price, and profit/loss of a long put and a short put at expiration. A put option is in the money if the underlying price falls below the strike price. In such a price scenario, exercising the put is advantageous without taking transaction costs into account. For at-the-money options, the price of the underlying is equal to the strike price. An at-the-money put option is not exercised because transaction costs always result in a loss. For an out-of-the-money put option, the underlying price exceeds the strike price. Such options are not exercised because the underlying can be sold at a higher price on the spot market.

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Options: Basics and Valuation

Example: Profit/Loss Calculation of a Put Option A European put option on the stock of Lufthansa AG has a strike price of EUR 20. The premium for the put option is EUR 3, while the Lufthansa stock trades at a price per share of EUR 18. 1. What is the profit/loss for the buyer of the put option at expiration if the share price either rises to EUR 23 or falls to EUR 13? 2. What is the profit/loss for the seller of the put option if the share price on the expiration date of the option is either EUR 23 or EUR 13? 3. What is the maximum profit and loss for the buyer and the seller of the put option? 4. What is the breakeven share price at expiration for the buyer and the seller of the put option? Solution to 1 At a share price of EUR 23 at expiration, the put option expires worthless and is not exercised. The put value at expiration of EUR 0 can be calculated as follows: V T, Long put = max ðEUR 0, EUR 20 - EUR 23Þ = EUR 0: The loss is limited to the paid option premium of EUR 3: LossLong put = EUR 0 - EUR 3 = - EUR 3: At a share price of EUR 13, the put option expires in the money and is exercised. The put value at expiration is EUR 7 and can be determined as follows: V T, Long put = max ðEUR 0, EUR 20 - EUR 13Þ = EUR 7: To calculate the profit/loss, the option premium paid must be deducted from the put value at expiration, leading to a profit of EUR 4: ProfitLong put = EUR 7 - EUR 3 = EUR 4: Solution to 2 At a share price of EUR 23 at expiration, the put option expires worthless, resulting in a profit in the amount of the option premium received of EUR 3: V T, Short put = - max ðEUR 0, EUR 20 - EUR 23Þ = EUR 0, ProfitShort put = EUR 0 þ EUR 3 = EUR 3:

(continued)

14.3

Profit and Loss

491

At a share price of EUR 13 at expiration, the put option ends in the money and is exercised by the option buyer. The loss for the seller of the put option is EUR 4 and can be determined as follows: V T, Short put = - max ðEUR 0, EUR 20 - EUR 13Þ = - EUR 7, LossShort put = - EUR 7 þ EUR 3 = - EUR 4: Solution to 3 The maximum profit for the buyer of the put option is capped at the strike price less the premium paid and totals EUR 17 (= EUR 20 - EUR 3). The maximum loss is limited to the paid option price of EUR 3. Conversely, the maximum profit for the seller of the put option is limited to the premium received of EUR 3, while the maximum loss of EUR 17 is made up of the strike price minus the premium received. Solution to 4 The breakeven share price at expiration is EUR 17 for the buyer and the seller of the put option and can be determined as follows: Breakeven ST = EUR 20 - EUR 3 = EUR 17: The buyer of the put option makes a profit at expiration if the share price falls below the breakeven share price of EUR 17. A share price of EUR 17 results in a put value at expiration of EUR 3 [= max(EUR 0, EUR 20 EUR 17)] and hence, after deducting the paid option premium of EUR 3, a profit/loss of EUR 0 (= EUR 3 - EUR 3). The profit area for the seller of the put option begins when the share price exceeds the breakeven price of EUR 17. At a share price of EUR 17, there is neither a profit nor a loss because the loss from exercising the option of EUR 3 [=-max(EUR 0, EUR 20 EUR 17)] is offset by the option premium received of EUR 3. As with the call buyer, the limited time to expiration also poses a risk for the put buyer. The holder of the long put must properly anticipate the price decrease of the underlying and correctly forecast the period of the price drop. If a price decrease of the underlying occurs only after the expiration date of the option, the buyer of the put loses the option premium paid despite the correct price forecast. The amount of the option premium paid depends, among other things, on the selected exercise price. The lower (higher) the strike price, the lower (higher) the option price because the buyer of the put can only sell the underlying at a lower (higher) strike price. Accordingly, by choosing the exercise price, the holder of the long put decides on the amount of the option premium to be paid and, as a result, on their maximum loss. In addition, the level of the strike price influences the breakeven share price. For example, a lower exercise price results in a lower option premium, on the one hand,

492

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Options: Basics and Valuation

and a lower breakeven share price on the other, with the result that a higher price decline of the underlying must take place in order to break even. The opportunities and risks of the seller are exactly the opposite of those of the buyer. Since the time to expiration is limited, time works for the put seller. If the price of the underlying does not fall below the strike price, the put option is not exercised by the holder of the long put and expires worthless. In addition, the choice of strike price influences the put seller’s risk of loss. The higher the selected strike price, the higher the proceeds (option premium) from the sale of the put. However, the probability of a loss increases with a higher strike price.

14.4

Intrinsic Value and Time Value

The option price consists of an intrinsic value and a time value. It is easiest to determine on the expiration date because at that time the time value is zero and the price of the option is given by the intrinsic value, which corresponds to the option value at expiration or the exercised value of the option. For example, the price of a call option at expiration or the intrinsic value is either zero or, if positive, the difference between the price of the underlying and the exercise price: cT = maxð0, ST- X Þ,

ð14:11Þ

where cT = call price at expiration T. On the expiration date, the price or the intrinsic value of a put option is the higher of zero or the difference between the strike price and the underlying price: pT = maxð0, X- ST Þ,

ð14:12Þ

where pT = put price at expiration T. During the life of an option, its price consists not only of an intrinsic value but also of a time value. In-the-money options have a positive intrinsic value in addition to a time value, while the price of at-the-money or out-of-the-money options is made up exclusively of the time value. The intrinsic value is zero. For example, if an inthe-money equity call option with a strike price of EUR 100 trades at a price of EUR 12 and the share price is EUR 108, the intrinsic value of the call option is EUR 8 [= max(EUR 0, EUR 108 - EUR 100)]. The intrinsic value reflects the gain if the option is exercised today. The difference of EUR 4 between the option price of EUR 12 and the intrinsic value of EUR 8 is the time value. Investors are willing to pay this value due to the chance that an amount exceeding the intrinsic value can be earned with the option. When the option expires, this opportunity no longer exists,

14.4

Intrinsic Value and Time Value

493

and therefore the option price is given by the intrinsic value. However, during the life of the option, its price is made up of an intrinsic value and a time value: Option price = intrinsic value þ time value:

ð14:13Þ

Table 14.3 presents an extract of the American call options on the Mercedes-Benz Group stock traded on the Eurex with an expiration date in June 2018 (DAI Jun 2018) for 29 December 2017. On 29 December 2017, the Mercedes-Benz Group stock is quoted at a closing share price of EUR 70.80. For example, an in-the-money call option with a strike price of EUR 68 is traded at a price of EUR 4.40 (ask price).2 The intrinsic value is EUR 2.80 [= max(EUR 0, EUR 70.80 - EUR 68)], while the time value is EUR 1.60 (= EUR 4.40 - EUR 2.80). The table also indicates that out-of-the-money (as well as at-the-money) options are quoted at a price consisting of the time value only. The intrinsic value of such options is accordingly EUR 0. Furthermore, the table indicates that the time value of the option decreases the further it moves out of the money and in the money. Options that are at the money have the greatest time value, as they have the greatest chance of moving in the money. The intrinsic value depends on the price of the underlying and the strike price, while the time value is influenced by the volatility of the underlying price, the time to expiration of the option, the risk-free interest rate, and the income of the underlying (e.g. dividends in the case of equity securities or interest income in the case of foreign currencies). Consequently, there are several risk factors that affect the price of an option, which are presented in Fig. 14.4. The option price (OP) is a function of several risk factors and can be stated as follows for an equity option: OP = f ðS, X, σ, T, r Fs , DÞ,

ð14:14Þ

where S = stock price, X = strike (exercise) price, σ = price volatility of the stock, T = time to expiration of the option (in years), rFs = nominal risk-free interest rate, and D = expected dividend per share. The stock price, the risk-free rate, and the dividend are observable market prices, while the strike price and the option’s time to expiration are defined by the contract itself. Hence, the only risk factor which is neither observable nor written in the contract is the volatility.3 It can be estimated as historical volatility by calculating the 2

The purchase of an option takes place at the ask price, while the sale of the option occurs at the bid price. Like all other financial instruments, options are quoted at a bid and ask price. 3 See Reilly and Brown 2006: Investment Analysis and Portfolio Management, p. 913.

Strike price Call price Intrinsic value Time value

56.00 15.04 14.80 0.24

60.00 11.09 10.80 0.29

64.00 7.47 6.80 0.67

66.00 5.85 4.80 1.05

68.00 4.40 2.80 1.60

72.00 2.23 0.00 2.23

76.00 1.08 0.00 1.08

80.00 0.56 0.00 0.56

84.00 0.33 0.00 0.33

88.00 0.20 0.00 0.20

Table 14.3 Eurex call options on Mercedes-Benz Group stock (in EUR) (Source: www.eurex.com/ex-en/markets/equ/opt/Mercedes-Benz-Group-2884628)

494 14 Options: Basics and Valuation

14.4

Intrinsic Value and Time Value

Option price

=

495

+

Intrinsic value

Time value

Risk factors

Call

Put

max(0, S – X)

max(0, X – S)

• Price volatility of the underlying • Time to expiration • Risk-free interest rate • Income of underlying asset

Risk factors • Price of underlying asset • Strike price Fig. 14.4 Option price and risk factors (Source: Own illustration)

standard deviation of continuously compounded stock price returns.4 The advantage of historical volatility is that it is easy to calculate, and no assumption of market efficiency for the stock is required. However, its disadvantage is that stable returns are assumed, that is, stock price behaviour in the future will continue as it has in the past. An alternative to historical volatility is implied volatility, which is determined in accordance with an option pricing model using the traded option price. To identify the implied volatility, an iterative procedure such as the Newton–Raphson method is applied.5 Although implied volatility is the same volatility forecast that investors use to set the option price, it is assumed that the option is priced correctly in the market. A study conducted by Beckers (1981) concludes that implied volatility is more suitable than historical volatility for forecasting future stock price movements.6 A change in a risk factor results in a change in the price of an option. The risk factors have the following positive or negative relationship to the price of an equity option (if all other risk factors remain the same): • The price of a call option increases when the price of the underlying stock increases. If the call option is out of the money, then the probability that the option will expire in the money on the expiration date increases with the share

4

See Sect. 2.2. See, for example, Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 958 ff. 6 See Beckers 1981: ‘Standard deviations implied in option prices as predictors of future stock price variability’, p. 363 ff. 5

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Options: Basics and Valuation

price. If the call option is already in the money, the intrinsic value of the call option increases with the share price. Conversely, the price of a put option rises as the price of the underlying falls. If the put option is out of the money, the probability that the put option will expire in the money increases with the decline in the share price. When the put option is already in the money, the intrinsic value increases as the share price decreases. Thus, there is a positive relationship between the call price and the underlying price, while this relationship is negative for a put option. • A higher strike price has a negative price effect on a call because the holder of the long call can purchase the equity security at a higher price. As a result, the call becomes less valuable with a higher strike price. By contrast, a lower strike price has a positive price effect on the call, as the holder of the long call can buy the equity security at a lower price. With a put option, on the other hand, the relationship between the strike price and the put price is positive. A higher (lower) strike price leads to a higher (lower) put price. The holder of the long put can sell the underlying at a higher (lower) price if the strike price is higher (lower), which has a positive (negative) effect on the put price. • The volatility of the stock price can be interpreted as a measure of price uncertainty. A higher price volatility means that the chance that the equity security will do very well or very poorly increases. For the shareholder, a higher and a lower share price tend to offset each other. However, this is not the case for the holder of an option. The buyer of a call benefits from the price increase of the underlying, but the loss is limited to the option premium paid in the event of a price drop of the underlying. Similarly, the buyer of a put benefits from a decrease in the price of the underlying, but in the event of a price increase the downside risk is limited to the option premium paid. Hence, if the price volatility rises, the chance that a profit can be made with a long call and a long put increases. Accordingly, there is a positive relationship between volatility and option price. At-the-money options with a long time to expiration have the highest price sensitivity to changes in price volatility of the underlying. • If the option has a longer time to expiration, the chance of a higher profit is higher because the underlying price can move in a favourable direction over a longer period of time. This positive relationship between option price and the option’s time to expiration applies to American options, but not in every case to European options. If, for example, there are two European put options with expiration dates in 1 month and in 1 year, respectively, and the price of the underlying falls to zero, the maximum possible profit for the holder of the long put is reached. The put option with a time to expiration of 1 month can be exercised earlier, and therefore the maximum profit can be realised earlier than in the case of the put with a time to expiration of 1 year. Hence, European put options with a shorter maturity may have a higher price. This is not the case with American put options, which can be exercised at any time. Moreover, in the case of European call options, there is no clear relationship between time to expiration and price. Consider two European call options on an equity security with expiration dates in 1 month and in 3 months and a large dividend payment expected in 9 weeks. The payment of the dividend

14.4

Intrinsic Value and Time Value

497

Table 14.4 Risk factors and price of equity options (Source: Own illustration) Risk factors Share price S: rises (falls) Strike price X: rises (falls) Volatility σ: rises (falls) Time to expiration T: rises (falls)a Risk-free interest rate rFs: rises (falls) Dividend D: rises (falls) a

Call price Rises (falls) Falls (rises) Rises (falls) Rises (falls) Rises (falls) Falls (rises)

Put price Falls (rises) Rises (falls) Rises (falls) Rises (falls) Falls (rises) Rises (falls)

For European options, there is no clear relationship between time to expiration and option price

leads to a lower share price in 9 weeks, with the result that the price of the call option with the shorter maturity is not affected by this dividend payment, while the price of the call option with a longer time to expiration of 3 months decreases. Therefore, the price of the short-life call could be worth more than the price of the long-life call. • If the risk-free interest rate rises, the price of a call option increases, while the price of a put option falls. This relationship is not obvious and can be explained as follows. The buyer of a call option has the right to purchase the underlying asset. The money is not tied up in the underlying and can therefore be invested at the risk-free interest rate. If interest rates rise (fall), the income from the interestbearing investment increases (decreases), which makes holding a call option more (less) valuable. The buyer of a put option, on the other hand, has the right to sell the underlying asset. They receive the proceeds from the sale of the underlying only at the time the option is exercised. When interest rates rise, they lose more interest while waiting to sell the underlying through the put. Thus, the opportunity cost of waiting is higher (lower) when interest rates are higher (lower), which makes holding a put option less (more) valuable. • A dividend payment causes the share price to fall. A lower share price results in a lower call price and a higher put price. Dividends are paid out during the lifetime of the option, and therefore they affect the time value and not the intrinsic value of the option. Table 14.4 presents the relationship between the risk factors and the price of equity call and put options when only one risk factor changes and all other risk factors remain the same. On the expiration date of the option, the option price is given by the intrinsic value, which can be determined by the share price and the strike price. The time value is zero. By contrast, the option price before expiration consists of the intrinsic value and the time value, the latter depending on volatility, the time to expiration, the risk-free interest rate and, in the case of an equity option, on any future dividend. Therefore, the pricing of options before expiration is not straightforward and is carried out using valuation models such as the binomial model and the Black-

498

14

Options: Basics and Valuation

Scholes model, which are described below for European equity options.7 The critical variable in pricing options is the volatility because it is not directly observable either in the option contract or in the market. Thus, it must be estimated separately as historical volatility or implied volatility.

14.5

Binomial Option Pricing Model

The binomial option pricing model is a discrete-time option pricing model in which the underlying price either rises or falls at the end of each time period. The time to expiration of the option is divided into different periods (e.g. days, weeks, or months) with the result that time and hence the price of the underlying change gradually. The Cox–Ross–Rubinstein option valuation model is presented below.8 The factors resulting in an up move and a down move of the underlying price can be determined as follows for one-period in the binomial tree: u = eσ

p Δt

ð14:15Þ

,

and d = e-σ

p Δt

=

1 , u

ð14:16Þ

where σ = annualised standard deviation of daily log returns of the underlying (historical volatility) or annualised implied volatility, and Δt = one period in the binomial tree (time interval expressed in years). For example, on 16 November 2022, the Mercedes-Benz Group stock trades at a price per share of EUR 61.15. The annualised implied volatility of the share price is 36% (Source: Refinitiv Eikon). Over a period of 6 months, the factors resulting in an up and down move of the share price are 1.290 and 0.775, respectively, and can be calculated as follows: u = e0:36 ×

p

d = e - 0:36 ×

6=12

p

= 1:290,

6=12

= 0:775:

Hence, at the end of the 6-month period the share price rises from EUR 61.15 to EUR 78.88 (= EUR 61.15 × 1.290) or falls from EUR 61.15 to EUR 47.39 7

For a more detailed description of option valuation models, see, for example, Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 938 ff. 8 See Cox et al. 1979: ‘Option pricing: a simplified approach’, p. 249.

14.5

Binomial Option Pricing Model

499

Su = EUR 78.88

S0 = EUR 61.15

Sd = EUR 47.39 Year 0

Year 0.5

Fig. 14.5 One-period binomial tree for Mercedes-Benz Group stock (Source: Own illustration)

(= EUR 61.15 × 0.775). Figure 14.5 presents the share prices for the 6-month period using a one-period binomial tree. Consider a 6-month call option on the Mercedes-Benz Group stock with a strike price of EUR 60. First, the call prices on the expiration date must be calculated. After an upward movement, the share price is EUR 78.88, which leads to a value of the call option on the expiration date of EUR 18.88 [= max(EUR 0, EUR 78.88 - EUR 60)]. In the event of a downward movement of the share price the call option expires worthless because the share price of EUR 47.39 is below the strike price of EUR 60 [max(EUR 0, EUR 47.39 - EUR 60) = EUR 0]. The next step is to determine the risk-neutral probability of an up move πu with the following formula:9 πu =

erFs Δt - d , u-d

ð14:17Þ

where rFs = continuously compounded risk-free interest rate, d = factor for a down move of the underlying price, and u = factor for an up move of the underlying price.

9

The expected return of the underlying asset in a risk-neutral world is given by the risk-free interest rate rFs. The expected underlying price at the end of a period (Δt) is S0 erFs Δt and equals the sum of the probability-weighted share prices at the end of the period, which is given as follows: π uS0u + (1 π u)S0d. If the following equation is solved for the risk-neutral probability of an upward movement (π u), Eq. (14.17) is obtained for Δt = 1: S0 erFs Δt = π u S0 u þ ð1- π u ÞS0 d:

500

14

Options: Basics and Valuation

Since the sum of the risk-neutral probabilities is 1, the risk-neutral probability of a down move πd can be calculated as follows: πd = 1 - πu :

ð14:18Þ

The continuously compounded 6-month risk-free interest rate approximated by the 6-month EURIBOR is 2.311% (Source: Refinitiv Eikon). Accordingly, the riskneutral probabilities of an up and down move can be determined as follows: πu =

e0:02311 × 6=12 - 0:775 = 0:459, 1:290 - 0:775 π d = 1 - 0:459 = 0:541:

The expected call price at the beginning of the period can be estimated by discounting the sum of the probability-weighted option prices at the end of the period with the risk-free interest rate as follows: c0 =

π u cu þ π d cd , erFs Δt

ð14:19Þ

where cu = call price after an up move of the underlying price, and cd = call price after a down move of the underlying price. Thus, the call price on the Mercedes-Benz Group stock is EUR 8.57: c0 =

0:459 × EUR 18:88 þ 0:541 × EUR 0 = EUR 8:57: e0:02311 × 6=12

Figure 14.6 presents the call price calculation applying a one-period binomial model. In this example, the call price has been estimated on the basis of one period of 6 months. A more accurate option value is obtained by increasing the number of periods in the binomial tree. For example, days can be taken for each period instead of a single 6-month period, which results in a more realistic option price. In the following example, a two-period binomial model is used to again calculate the price of the 6-month call option on the Mercedes-Benz Group stock with a strike price of EUR 60. In accordance with the Cox, Ross, and Rubinstein model the factors resulting in an up and down move of the share price over a 3-month period must be determined first: u = e0:36 ×

p

3=12

= 1:197,

14.5

Binomial Option Pricing Model

501

Su = EUR 78.88 cu = EUR 18.88 πu = 0,459

S0 = EUR 61.15 c0 = EUR 8.57

c

π c

c

πd = 0,541

e

Sd = EUR 47.39 cd = EUR 0 Year 0

Year 0.5

Fig. 14.6 One-period binomial tree for Mercedes-Benz Group stock and call option (Source: Own illustration)

d = e - 0:36 ×

p

3=12

= 0:835:

The share price at the end of the first 3-month period moves either up to EUR 73.20 (= EUR 61.15 × 1.197) or down to EUR 51.06 (= EUR 61.15 × 0.835). In the second 3-month period, the share price after an upward movement at the end of the first period either rises to EUR 87.62 (= EUR 73.20 × 1.197) or falls to EUR 61.12 (= EUR 73.20 × 0.835). The price of the security after a downward movement at the end of the first period either increases to EUR 61.12 (= EUR 51.06 × 1.197) or decreases to EUR 42.64 (= EUR 51.06 × 0.835) in the next period. Figure 14.7 presents the movement of the share price in a two-period binomial tree. The value of the call at expiration after two up movements of the share price is EUR 27.62 [= max(EUR 0, EUR 87.62 – EUR 60)], while after an up-down and a down-up move of the share price the value of the call is EUR 1.12 [= max(EUR 0, EUR 61.12 – EUR 60)]. After two down movements of the underlying price the call expires worthless [max(EUR 0, EUR 42.64 – EUR 60) = EUR 0]. The risk-neutral probabilities of an up and down move over a 3-month period can be determined as follows, if the continuously compounded 3-month risk-free interest rate approximated by the 3-month EURIBOR is 1.803% (Source: Refinitiv Eikon): πu =

e0:01803 × 3=12 - 0:835 = 0:468, 1:197 - 0:835 π d = 1 - 0:468 = 0:532:

502

14

Options: Basics and Valuation

Suu = EUR 87.62

Su = EUR 73.20 Sud = EUR 61.12 S0 = EUR 61.15 Sdu = EUR 61.12 Sd = EUR 51.06

Sdd = EUR 42.64 Year 0

Year 0.25

Year 0.5

Fig. 14.7 Two-period binomial tree for Mercedes-Benz Group stock (Source: Own illustration)

There are two call prices for an up move and a down move of the share price at the end of the first 3-month period, which are estimated using backward induction. To calculate, for example, the call price after an upward movement of the share price, the probability-weighted call prices after two up moves and after an up-down move must be discounted with the risk-free rate over a period of 3 months. Therefore, the call price after an up movement and after a down movement of the underlying price can be estimated applying the equation from the one-period model as follows: π u cuu þ π d cud erFs Δt

ð14:20Þ

π u cdu þ π d cdd , erFs Δt

ð14:21Þ

cu = and cd = where

cuu = call price at expiration after two up moves of the underlying price, cud = call price at expiration after up-down move of the underlying price, cdu = call price at expiration after down-up move of the underlying price, and cdd = call price at expiration after two down moves of the underlying price.

14.5

Binomial Option Pricing Model

503

Suu = EUR 87.62 cuu = EUR 27.62 S u = EUR 73.20 cu = EUR 13.46 Sud = EUR 61.12 cud = EUR 1.12 Sdu = EUR 61.12 cdu = EUR 1.12

S0 = EUR 61.15 c0 = EUR 6.55

Sd = EUR 51.06 cd = EUR 0.52 Sdd = EUR 42.64 cdd = EUR 0 Year 0

Year 0.25

Year 0.5

Fig. 14.8 Two-period binomial tree for Mercedes-Benz Group share prices and call prices (Source: Own illustration)

The call prices after the first upward and downward movement of the share price are EUR 13.46 and EUR 0.52, respectively: cu =

0:468 × EUR 27:62 þ 0:532 × EUR 1:12 = EUR 13:46, e0:01803 × 3=12

cd =

0:468 × EUR 1:12 þ 0:532 × EUR 0 = EUR 0:52: e0:01803 × 3=12

The expected call price of EUR 6.55 can be calculated as follows: c0 =

0:468 × EUR 13:46 þ 0:532 × EUR 0:52 = EUR 6:55: e0:01803 × 3=12

The call price determined with the two-period binomial model of EUR 6.55 is more accurate than the price of EUR 8.57 estimated with the one-period binomial model. If, for example, a 150-period binomial model is used, the calculated call price with a 6-month risk-free rate of 2.311% is EUR 7.05. Figure 14.8 presents the calculated share and call prices along the two-period binomial tree.

504

14

Options: Basics and Valuation

The binomial model is sufficiently flexible to be applied not only for the price calculation of European options, but also for the pricing of American options.10 The price calculation of a European option with the binomial model can be summarised in the following steps: • A plot of the binomial tree with the share prices is required, where the share prices at the end of each period are calculated with the upward and downward factors. • At the end of the binomial tree—that is, on the expiration date of the option—the intrinsic option values [call price = max(0, ST - X) and put price = max(0, X ST)] are to be calculated. • Using backward induction, the option prices are determined starting from the end of the period in the binomial tree by multiplying the risk-neutral probabilities of an upward or downward movement by the corresponding option prices at the end of the respective period and then discounting them with the risk-free interest rate.

14.6

Black-Scholes Option Pricing Model

The Black-Scholes model, also known as the Black-Scholes-Merton model, dates back to 1973.11 This is the first and probably best-known pricing model for European options. The Black–Scholes model enabled market participants to calculate the option price for the first time, which contributed significantly to the fact that option trading increased significantly in subsequent years. Moreover, this model has been pivotal for the growth and success of financial engineering in the last 40 years. In contrast to the binomial model, the Black–Scholes model assumes that share prices follow a geometric lognormal diffusion process which is continuous in time and not discrete in time and therefore makes it possible to depict price movements more realistically. In addition, the Black–Scholes model can only be applied to European options. American options—with the exception of American call options on non-dividend paying stocks—cannot be priced with the model. For pricing American options, the best approach is to use the binomial model with a large number of time periods.12 The assumptions of the Black–Scholes model can be listed as follows:

10

See, for example, Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 949 ff. The uncertainty associated with the early exercise of American options makes it impossible to derive a closed-form solution such as the Black-Scholes model. 11 See Black and Scholes 1972: ‘The valuation of option contracts and a test of market efficiency’, p. 399 ff., Black and Scholes 1973: ‘The pricing of options and corporate liabilities’, p. 637 ff. and Merton 1973: ‘Theory of rational option pricing’, p. 141 ff. Myron Scholes and Robert Merton received the Nobel Prize for Economics in 1997 for their work on option pricing. Fischer Black died in 1995. 12 See, for example, Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 949 ff.

14.6

Black–Scholes Option Pricing Model

505

• The simple returns of the underlying asset follow a geometric Brownian motion (or Wiener process), and the prices of the underlying are log normally distributed. The lognormal distribution ensures that the price of the underlying cannot fall below zero. The continuous compounded returns of the underlying, by contrast, are normally distributed. • The risk-free interest rate is constant and the same for all maturities. It does not follow any random walk. • Money can be borrowed or invested at the risk-free interest rate. • The price volatility of the underlying is constant and does not change over time. It is assumed that the volatility is known at all times. However, in reality, the volatility is not known and must be estimated. • There are no cash flows on the underlying (e.g. no dividends in the case of stocks). • Short selling of securities and the full use of proceeds is permitted. • There are no taxes or transaction costs. Some of these assumptions can be relaxed. For example, the model can be adjusted to calculate the price of European options on dividend-paying stocks. The formulas for calculating the price of a call and put option with the Black–Scholes model are as follows: c0 = S0 N ðd1 Þ - Xe - rFs T N ðd 2 Þ

ð14:22Þ

p0 = Xe - rFs T ½1- N ðd 2 Þ - S0 ½1- N ðd 1 Þ,

ð14:23Þ

and

where d1 = d2 =

lnðS0 =X Þ þ ðrFs þ σ 2 =2ÞT p , σ T

p lnðS0 =X Þ þ ðr Fs - σ 2 =2ÞT p = d1 - σ T , σ T

N(d1) = area under the probability density function of the standard normal distribution from -1 to d1, N(d2) = area under the probability density function of the standard normal distribution from -1 to d2, σ = price volatility of the underlying, which is an annualised standard deviation of the continuously compounded returns on the underlying, r Fs = continuously compounded risk-free interest rate, and T =time to expiration (in years), equal to number of days to expiration divided by 365 days.

506

14

(Density of probability)

Options: Basics and Valuation

N(0, 1) N(d1)

0

d1

(Standard normal variable)

Fig. 14.9 Area N(d1) under the probability density function of the standard normal distribution (Source: Own illustration)

The Black–Scholes option pricing equations include the underlying price, strike price, volatility, risk-free interest rate, and time to expiration as risk factors which have an influence on the intrinsic value and time value of an option. For example, the pricing equation for the call option (see Eq. 14.22) consists of an intrinsic value component S0 - Xe - rFs T , where the strike price deducted from the underlying price is discounted to the valuation date—since a European option can only be exercised on the expiration date—and a time value component given by the standard normal distribution areas N(d1) and N(d2). N(d1) and N(d2) represents the area under the standard normal distribution up to the standard normal variables d1 and d2 and thus indicates the probability that a standard normally distributed random variable assumes a value less than or equal to d1 and d2, respectively. Figure 14.9 presents the area N(d1) under the probability density function of the standard normal distribution. The call price calculated with the Black–Scholes model is thus the difference between the current underlying price and the discounted strike price, with both components of the intrinsic value weighted by the probabilities N(d1) and N(d2) from the standard normal distribution. N(d1) is the delta of the call, while N(d2) indicates the probability that the call option will expire in the money. Hence, the price calculation of the call option in the Black–Scholes model can be interpreted as the underlying price multiplied by the delta, minus the discounted strike price, multiplied by the probability that the option will be exercised.

14.6

Black–Scholes Option Pricing Model

507

(Density of probability) N(0, 1)

1– N(d1)

0

d1

(Standard normal variable)

Fig. 14.10 Area 1 – N(d1) under the probability density function of the standard normal distribution. (Source: Own illustration)

Analogous to the call price, the price of a put option can be interpreted as the present value of the strike price multiplied by the probability that the option will expire in the money [1 – N(d2)], minus the underlying price multiplied by the delta of the put option [1 – N(d1)]. Figure 14.10 presents the area of 1 – N(d1) under the probability density function of the standard normal distribution. Example: Calculation of the Call and Put Price with the BlackScholes Model The Mercedes-Benz Group stock trades at a price per share of EUR 61.15 on 16 November 2022. The annualised implied volatility of the stock is 36%. The continuously compounded 6-month risk-free interest rate is 2.311% (Source: Refinitiv Eikon). 1. What is the price of a European call option with a strike price of EUR 60 and a time to expiration of 6 months according to the Black-Scholes model? 2. What is the price of a European put option with a strike price of EUR 60 and a time to expiration of 6 months based on the Black-Scholes model? (continued)

508

14

Options: Basics and Valuation

Solution to 1 The risk factors for determining the option price are: S0 = 61.15, X = 60, σ = 0.36, r Fs = 0.02311, and T = 6/12 = 0.5. The standard normal variables d1 and d2 can be calculated as follows: d1 =

lnð61:15=60Þ þ 0:02311 þ 0:362 =2 × 0:5 p = 0:2473, 0:36 × 0:5 p d2 = 0:2473 - 0:36 × 0:5 = - 0:00726:

The areas of the standard normal distribution N(d1) and N(d2) can be determined either from a standard normal distribution table (see Appendix in Chap. 2) or using the Microsoft Excel function ‘NORMSDIST’. The latter yields values for N(d1) of 0.5977 and N(d2) of 0.4971. The call price of EUR 7.066 can be calculated as follows: c0 = EUR 61:15 × 0:5977 - EUR 60 × e - 0:02311 × 0:5 × 0:4971 = EUR 7:066: Solution to 2 The areas of the standard normal distribution of 1 - N(d1) and 1 - N(d2) can be determined as follows: 1 - N ðd 1 Þ = 1 - 0:5977 = 0:4023, 1 - N ðd 2 Þ = 1 - 0:4971 = 0:5029: The put price is EUR 5.227: p0 = EUR 60 × e - 0:02311 × 0:5 × 0:5029 - EUR 61:15 × 0:4023 = EUR 5:227: The Black-Scholes model assumes no dividends are paid during the life of the option. The model was generalised by Merton to allow for a constant continuous dividend yield.13 The corresponding dividend-adjusted option pricing formulas for a call and put are as follows: c0 = S0 e - qT N ðd 1 Þ - Xe - rFs T N ðd2 Þ

ð14:24Þ

p0 = Xe - rFs T ½1- N ðd 2 Þ - S0 e - qT ½1- N ðd 1 Þ,

ð14:25Þ

and

where 13

See Merton 1973: ‘Theory of rational option pricing’, p. 141 ff.

14.7

Put–Call Parity

509

q = constant continuous dividend yield, ln ðS0 =X Þ þ ðr Fs - q þ σ2 =2ÞT p d1 = , and σ T p ln ðS0 =X Þ þ ðr Fs - q - σ2 =2ÞT p d2 = = d1 - σ T : σ T The dividend yield reduces the stock price because it is assumed that on the ex-dividend date the stock price falls by the full amount of the dividend paid. Furthermore, the formula assumes that the amount and timing of the dividends during the life of an option can be predicted with certainty. However, in the real world, neither dividend yield nor volatility is known with certainty, and empirical evidence suggests that both vary stochastically over time. Models that incorporate these stochastic variations have been developed and are used in practice.14

14.7

Put-Call Parity

There is a relationship between the call price, the put price, and the price of the underlying, which is captured by the put–call parity. The parity ensures that the prices of these three financial instruments do not move independently of each other. The put–call parity assumes that the call and the put are European options and that they have the same underlying, the same strike price, and the same time to expiration. If these assumptions hold, the put–call parity can be stated as follows: c0 þ Xe - rFs T = p0 þ S0 :

ð14:26Þ

According to the put–call parity, the value of a fiduciary call ðc0 þ Xe - rFs T Þ equals the value of a protective put ( p0 + S0). The fiduciary call consists of a long call and a long risk-free zero-coupon bond maturing on the option expiration date, with a face value of X equal to the strike price of the option. The protective put consists of a long put and a long equity security. If the parity is violated, a risk-free arbitrage profit can be made, and therefore market participants ensure that parity is restored between the fiduciary call and the protective put. The put–call parity can be derived on the expiration date of the call and the put. If the share price exceeds the strike price (ST > X) at the expiration date of the two options, the long call option expires in the money and has a value of ST - X. The price of the zero-coupon bond at maturity is X. Accordingly, the value of the fiduciary call is ST (= ST - X + X). The protective put has the same value of ST because the long put expires worthless and the long stock has a value of ST. If, on the other hand, the share price falls below the strike price (ST < X) when the two options

14 For option pricing models using stochastic volatility models, see, for example, Hull 2006: Options, Futures, and Other Derivatives, p. 566 ff.

510

14

Table 14.5 Derivation of the put-call parity (Source: Own illustration)

Strategies Fiduciary call Long call Long 0% bond Total Protective put Long put Long equity Total

Options: Basics and Valuation

Current value

Value at expiration T ST < X ST > X

c0 Xe - rFs T c0 þ Xe - rFs T

ST - X X ST

0 X X

p0 S0 p0 + S0

0 ST ST

X - ST ST X

expire, the fiduciary call has a value of X because the call option expires worthless and the value of the zero-coupon bond at maturity is X. The value of the protective put is the result of the long put expiring in the money and having a value of X - ST, and the stock trading at a price of ST. As a result, the value of the protective put is also X (= X - ST + ST). Table 14.5 presents the derivation of the put–call parity.15 One of the situations in which put–call parity can be applied is when no market price is available for one of the two options.16 For example, the price of the call option on the Mercedes-Benz Group stock with a strike price of EUR 60 and a time to expiration of 6 months can be calculated using the put price of EUR 5.227 as follows, if the price of the automobile stock is traded at a price of EUR 61.15 on 16 November 2022 and the continuously compounded 6-month risk-free interest rate is 2.311%: c0 = EUR 5:227 þ EUR 61:15 - EUR 60 × e - 0:02311 × 0:5 = EUR 7:066:

14.8

Leverage Effect

Like futures/forwards, options can generate high positive returns but also high negative returns due to the capital commitment being lower than the price paid for the underlying. For example, if the price of the Mercedes-Benz Group stock increases from EUR 61.15 to EUR 66.15, the call price increases from EUR 7.066 to EUR 10.344. Accordingly, the call price increases by EUR 3.278 when the share price increases by EUR 5. The option price increases by 46.39% (= EUR 3.278/ EUR 7.066), while the share price increases by only 8.18% (= EUR 5/EUR 61.15). Thus, there is a leverage of 5.7 with the call option, which means that one earns 5.7 times more (46.39% compared to 8.18%) with a long call option than with the underlying asset if the share price increases by EUR 5. Leverage can generally be calculated with the following formula: 15 16

See Watsham 1998: Futures and Options in Risk Management, p. 127. For the construction of a synthetic all and put option with the put–call parity, see Sect. 15.3.

14.9

Option Price Sensitivities

511

Leverage =

ΔOP=OP0 , ΔS=S0

ð14:27Þ

where ΔOP = change in the option price in the event of a change in the underlying price, OP0 = option price before change in the underlying price, ΔS = change in the price of the underlying, and S0 = price of the underlying before change. If, for example, the share price falls by EUR 5 from EUR 61.15 to EUR 56.15, the call price drops by EUR 2.665 from EUR 7.066 to EUR 4.401. The leverage of 4.6 can be determined as follows: Leverage =

- EUR 2:665 =EUR 7:066 = 4:6: - EUR 5 =EUR 61:15

Accordingly, the negative return of the long call option is 4.6 times higher than that of the long equity position if the share price falls by EUR 5. The leverage effect illustrates that options are very risky financial instruments, as they offer not only great opportunities for profit, but also great risks of loss. In addition to speculation, options can also be used to hedge the price risk of an existing long or short position of the underlying. The common risk hedging strategies for a long equity position are the protective put, the covered call, and the collar, which are described in the next chapter.17

14.9

Option Price Sensitivities

The five risk factors of an option price are underlying price, strike price, volatility, risk-free rate, and time to expiration.18 Option price sensitivities make it possible to examine how much the option price changes when a risk factor moves, if all other risk factors remain unchanged. They are also called Greeks because they are often referred to by Greek names. The following option price sensitivities for equity options are examined in this section: • Delta: sensitivity of the option price to a change in the price of the underlying. • Gamma: measure of how much delta changes in response to a move in the price of the underlying. • Vega: sensitivity of the option price to a change in volatility. • Rho: sensitivity of the option price to a change in the risk-free rate.

17 18

See Chap. 15. See Sect. 14.4.

512

14

Options: Basics and Valuation

• Theta: rate at which the time value of the option decays as the expiration date approaches.

14.9.1 Delta Changes in the price of the underlying result in a change in the option price. For example, if the share price of an out-of-the-money call option increases, the probability that the call option will expire in the money on the expiration date increases with the share price, with the result that the call price rises. If the call is already in the money, then the option price or its intrinsic value rises with the share price. Thus, there is a positive relationship between the call price and the share price. The extent to which the option price changes with a move in the underlying price is measured by the delta, which reflects the linear relationship between the option price and the share price. In other words, delta is the first partial derivative of the call price with respect to the underlying price. The delta of a call option can generally be defined as follows: DeltaCall =

Δc , ΔS

ð14:28Þ

where Δc = change in the call price, and ΔS = change in the price of the underlying. For example, if one takes a call option on a non-dividend-paying stock with risk factors of S0 = 42, X = 40, σ = 0.30, rFs = 0.03, and T = 0.75, the result is a price curve with a positive slope, as presented in Fig. 14.11. The delta of the call option corresponds to the rate of change of the option price for a very small (infinitesimal) change in the underlying price. It is the slope of the price curve at a given call price and share price. In Fig. 14.11, with a share price of EUR 42 and a call price of EUR 5.793, the delta is 0.66. This means that if the share price rises from EUR 42 to EUR 42.20, the call price tends to increase by EUR 0.132 [= 0.66 × (EUR 42.20 - EUR 42)] to EUR 5.925. Thus, the change in the call price can be determined using the Taylor series expansion with a first-order approximation, assuming a linear relationship between the call price and the price of the underlying asset: Δc ≈ DeltaCall × ΔS:

ð14:29Þ

For a call option which is deep in the money, the slope of the price curve or the delta is 1. A change in the share price of EUR 1 leads to a change in the price of the call option of EUR 1. The call price is given by the intrinsic value, while the time value approaches zero. This is presented in Fig. 14.11 by the dashed line (intrinsic value), which exhibits practically the same shape as the price curve for call options that are deep in the money. If, on the other hand, the call option is deep out of the

14.9

Option Price Sensitivities

513

(Call price in EUR)

Tangent with slope of 0.66

45 40 35 30 25 20 15 10 5.793 5 0

Delta of 1

Price curve Delta of 0

0

10

Delta of 0.66

20

30

40 42

50

60

70

80

(Share price in EUR)

Fig. 14.11 Price curve of a call option (Source: Own illustration)

money, the delta is close to 0. If the share price moves, the call price remains unchanged because the chance of earning money with the option is still low. The intrinsic value is zero for out-of-the-money options, and therefore the price consists of the time value only. For at-the-money options, the delta is approximately 0.5. Options that are at the money and close to expiration have a very unstable delta. If the call expires in the money, the delta is 1. By contrast, the delta is 0 should the option expire out of the money. The value range of the delta is between 0 and 1. The delta of the put option corresponds to the rate of change of the option price for a very small (infinitesimal) change in the underlying price. The price of a put rises when the price of the underlying falls, as the probability that the option will expire in the money on the expiration date increases. Therefore, the relationship between the put price and the share price is negative. The delta of a put option, which reflects the linear price relationship between the put option and the underlying asset, can generally be defined as follows: DeltaPut = where Δp = change in the put price, and ΔS = change in the price of the underlying.

Δp , ΔS

ð14:30Þ

514

14

(Put price in EUR) 45 40

Options: Basics and Valuation

Delta of –1

35 Price curve

30 25

Delta of –0.34

20

Delta of 0

15 10 5 2.90 0 0

10

20

30

40 42

50

60 70 80 (Share price in EUR)

Fig. 14.12 Price curve of a put option (Source: Own illustration)

If, for example, there is a put option on a non-dividend-paying stock with risk factors of S0 = 42, X = 40, σ = 0.3, rFs = 0.03, and T = 0.75, a price curve will result as presented in Fig. 14.12. The slope of the price curve reflects the delta for a given put price and share price. With a put price of EUR 2.90 and a share price of EUR 42, the delta is -0.34, which is indicated in the figure by a tangent at this put-stock-price point. The negative slope of the price curve implies a negative delta, which lies between -1 and 0. Furthermore, the figure indicates that a deep in-the-money put option has a delta of -1. For example, if the share price falls (rises) by EUR 1, the put price rises (falls) by EUR 1. A deep out-of-the-money put has a delta of 0. The intrinsic value of the option is zero, and therefore the put price is given by the time value. An at-themoney put has a delta of approximately -0.5. At-the-money put options that are close to expiration have an unstable delta. If the put option expires in the money, the delta is -1, otherwise it is 0. The value range of the delta is between -1 and 0. In Fig. 14.12, with a share price of EUR 42 and a put price of EUR 2.90, the delta is - 0.34. This means that if the share price falls from EUR 42 to EUR 41.80, the put price tends to increase by EUR 0.068 [= (-0.34) × (EUR 41.80 - EUR 42)] to EUR 2.968. Thus, the change in the put price can be determined using the Taylor series expansion with a first-order approximation, assuming a linear relationship between the put price and the price of the underlying asset:

14.9

Option Price Sensitivities

515

Δp ≈ DeltaPut × ΔS:

ð14:31Þ

Delta is an important risk measure as it defines the sensitivity of the option price to a change in the price of the underlying. Traders, in particular options dealers, use delta to construct hedges to eliminate the linear price risk of their long and short option positions.19 To implement a delta hedge on equity options, the delta of the option position must be offset by the delta of the underlying equity position, which leads to the following equation: N × DeltaEquity security þ F × DeltaOption = 0,

ð14:32Þ

where N = number of shares of the equity security, DeltaEquity security = delta of the equity security, F = number of options, and DeltaOption = delta of the option. Thus, the delta-hedged option position has a delta of zero. Solving the above equation according to the number of shares leads to the following equation for the delta hedge: N= -

DeltaOption × F: DeltaEquity security

ð14:33Þ

Suppose the price of an equity call option with a delta of 0.66 is EUR 5.793 and the price of the underlying equity security is EUR 42. The latter has a delta of 1. An investor holds 40 call option contracts to buy 4000 shares. To hedge the delta risk of the long call position a short position of 2640 shares is required: N= -

0:66 × 4000 = - 2640: 1

The delta risk of the long call position is therefore eliminated by selling short 2640 shares. The gain (loss) of the call position would then tend to be offset by the loss (gain) of the equity position. If, for example, the share price goes up by EUR 1, the value of the long call position increases in accordance with the delta by EUR 2640 (= 0.66 × 4000 calls × EUR 1). However, the increase in share price will produce a loss on the short equity position of EUR 2640 (= -2640 shares × EUR 1). Thus, the gain on the long call position is exactly offset by the loss on the short equity position. As the share price rises (falls), the delta of the call option also goes up (goes down), because the slope of the price curve increases (decreases) with a higher (lower) share price (see Fig. 14.11). A higher or lower delta requires that the number

19

See Figlewski 1990: ‘Theoretical Valuation Models’, p. 105.

516

14

Options: Basics and Valuation

of shares must be adjusted so that the position remains delta neutral. For example, if the share price rises from EUR 42 to EUR 44, the delta of the call option increases from 0.66 to 0.72, which leads to the following number of short shares to maintain the option position delta neutral: N= -

0:72 × 4000 = - 2880: 1

The value of the long call position increases by EUR 5760 (= 0.72 × 4000 calls × EUR 2), while the short equity position will produce a loss of EUR 5760 (= -2880 shares × EUR 2). The example demonstrates that a delta-neutral hedge is a dynamic hedging strategy. In contrast to a hedge with forwards or futures, with options the delta or hedge ratio changes with a change in the underlying price, and therefore the number of shares must be periodically adjusted in order to remain hedged, otherwise full protection against the delta risk persists for a short period of time only.

14.9.2 Gamma The gamma of an option is the rate of change of the delta of the option with respect to a movement in the underlying asset price. Hence, gamma measures the change in delta for a one-unit change in the price of the underlying asset.20 The gamma of a call and a put option can generally be stated as follows: GammaCall or put =

ΔDelta : ΔS

ð14:34Þ

In other words, gamma is the second partial derivative of the option price with respect to the underlying price. A high gamma results from a large change in the delta. This is the case for options that are at the money and close to expiration. If the option expires in the money, the delta is 1 for a call and -1 for a put. However, should the option expire out of the money, the delta will be 0. Thus, a delta hedge will work poorly for options that are at the money and close to the expiration date. For such options gamma is large.21 Therefore, frequent adjustments are required to keep the option position delta neutral. By contrast, options that are deep in the money or out of the money have small gamma because price changes in the underlying asset result in small changes in the delta. Adjustments need to be made only relatively infrequently to keep the option position delta neutral.22 The relationship between option price and underlying price is not linear. As a result, in addition to the delta, the gamma must also be considered in order to approximate the option price change for price movements of the underlying over a

20

See Watsham 1998: Futures and Options in Risk Management, p. 121. See Figlewski 1990: ‘Theoretical Valuation Models’, p. 111. 22 See Hull 2012: Risk Management and Financial Institutions, p. 145. 21

14.9

Option Price Sensitivities

517

(Call price in EUR) 7

1 2

6 5 4 1 2

3 2 1 0 34

35

36

37

38

39

40 S0

41

42

43

44

45

46

(Share price in EUR)

Fig. 14.13 Delta and gamma of a call option (Source: Own illustration)

short time period using the Taylor series expansion with a second-order approximation: ΔOP ≈ Delta × ΔS þ

1 × Gamma × ΔS2 , 2

ð14:35Þ

where ΔOP = change in option price. If the first term of the Taylor series expansion of Delta × ΔS is taken to measure the price change of a call option, the price increase is underestimated when the share price increases. Accordingly, the second term of the Taylor series expansion of 0.5 × Gamma × ΔS2 must be added in order to obtain a better estimate of the option price, which lies on the price curve. If, on the other hand, the share price decreases, one overestimates the price decrease of the call option with the term Delta × ΔS. To obtain a better estimate of the call price, which is located on the price curve, the second term of 0.5 × Gamma × ΔS2 has to be added again. Figure 14.13 presents this relationship for an at-the-money European equity call option with risk factors of S0 = 40, X = 40, σ = 0.3, rFs = 0.02, and T = 0.25, where the share price changes by plus/minus EUR 3. The price adjustment of the call option is made in each share price scenario by the gamma, which is positive for long call options, since the second

518

14

(Put price in EUR) 7 6

Options: Basics and Valuation

1 2

5 4

1 2

3 2 1 0 34

35

36

37

38

39

40 S0

41

42

43

44

45

46

(Share price in EUR)

Fig. 14.14 Delta and gamma of a put option (Source: Own illustration)

term must be added to the first term of the Taylor Series expansion in the case of rising and falling share prices. The gamma of a short call option, on the other hand, is negative because a short option has a profit/loss pattern opposite to that of a long option. By hedging the delta risk of a long call option, the call price movement of Delta × ΔS can be eliminated. In this case, a profit is made due to the positive gamma when the underlying price moves (see Fig. 14.13). A positive gamma should not be hedged because it reflects a profit opportunity and not a risk of loss. Conversely, a short call option has a negative gamma, which leads to a loss if the underlying price changes. This risk of loss can be removed with a gamma hedge.23 When measuring the price change of a put option with the first term of the Taylor series expansion of Delta × ΔS, the price increase of the put option is underestimated if the share price falls; therefore, the term of 0.5 × Gamma × ΔS2 must be added to obtain the option price that lies on the price curve. Should the share price increase, the price decrease of the put option is overestimated by the first term of the Taylor series expansion of Delta × ΔS. To correct the price of the put option, the term of 0.5 × Gamma × ΔS2 must be added again. Figure 14.14 presents this relationship

23

See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 980 f.

14.9

Option Price Sensitivities

Table 14.6 Delta and gamma of long and short options (Source: Own illustration)

Options Long call Short call Long put Short put

519

Delta Positive Negative Negative Positive

Gamma Positive Negative Positive Negative

using an at-the-money European equity put option with risk factors of S0 = 40, X = 40, σ = 0.3, r Fs = 0.02, and T = 0.25, where the share price changes by plus/minus EUR 3. The gamma is positive for a long put which represents a profit opportunity. When the underlying price moves, the positive gamma leads to a profit. Conversely, a short put option has a negative gamma. If the price of the underlying asset changes, a loss is made as a result of the negative gamma. Table 14.6 presents the delta and gamma of long and short options. Long options have a positive gamma, while short options have a negative gamma. A positive gamma represents a profit opportunity, whereas a negative gamma reflects a risk of loss that can be hedged in an option portfolio.24 Example: Delta and Gamma An options trader holds 10 short European call contracts on shares of Mercedes-Benz Group AG. One call contract refers to 100 shares. The calls have a strike price of EUR 67 and a time to expiration of 279 days. The Mercedes-Benz Group stock is traded at a closing price per share of EUR 63.75 on 25 March 2022. The annualised volatility of the automobile stock is 34.6%, while the dividend per share is EUR 5. The risk-free interest rate is -0.45%. According to the Black–Scholes model, the call price is EUR 4.14. The option has a delta of 0.387 and a gamma of 0.021. 1. How many Mercedes-Benz Group shares are required to hedge the delta risk of the short call position? 2. What is the profit/loss of the delta-hedged short call position according to Taylor series expansion with a second-order approximation if the share price rises by EUR 2? Solutions to 1 The delta of short calls is negative. One short call option has a delta of 0.387, while 1000 short call options have a delta of -387 (= -1000 × 0.387). This means that if the share price rises (falls) by EUR 1, a loss (profit) of (continued)

24

See Figlewski 1990: ‘Theoretical Valuation Models’, p. 111.

520

14

Options: Basics and Valuation

EUR 387 is incurred. To hedge the delta risk, a total of 387 Mercedes-Benz Group shares should be bought: N= -

- 0:387 × 1000 = 387: 1

Solutions to 2 The gamma of the short call position is negative and is -21 (= -1000 × 0.021). In accordance with the Taylor series expansion with a second-order approximation, an increase in the share price by EUR 2 leads to a loss on the short call contracts of approximately EUR 816: Δc ≈ ð - 387Þ × EUR 2 þ 0:5 × ð - 21Þ × ðEUR 2Þ2 = - EUR 816: The delta of the hedged call option position is 0, and therefore a loss of approximately EUR 42 is incurred: Δc ≈ 0 × EUR 2 þ 0:5 × ð - 21Þ × ðEUR 2Þ2 = - EUR 42: The larger the negative gamma, the greater the loss of the delta-neutral option position when the share price rises or falls. The gamma risk can be hedged by adding options on the same underlying asset.25

14.9.3 Vega The relationship between option price and price volatility of the underlying is referred to as vega. Although vega is considered an option Greek, it is the name of a star in the constellation Lyra, not a letter, and its origin is Latin. Vega is positive for long call and put options, meaning that if the price volatility of the underlying increases (decreases), both call and put prices rise (fall). However, vega is negative for short options.26 Vega is the first partial derivative of the call price or put price with respect to volatility. The vega of a call and a put can generally be defined as follows: VegaCall or put =

25 26

ΔOP , Δσ

See Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 980 f. See Figlewski 1990: ‘Theoretical Valuation Models’, p. 112.

ð14:36Þ

14.9

Option Price Sensitivities

521

(Vega) 0.16 0.14 0.12 3 month until expiration date

0.1 1 year until expiration date

0.08 0.06 0.04 0.02 0 0

10

20

30

40 X

50

60

70

80

90

(Share price in EUR)

Fig. 14.15 Relationship between vega, underlying price, and time to expiration (Source: Own illustration)

where ΔOP = change in the option price (call price or put price), and Δσ = change in the price volatility of the underlying. The vega of a long option is always positive. An increase in volatility leads to an increase in the option price, resulting in a profit. By contrast, a short option has a negative vega. If volatility increases, the option price rises, causing a loss.27 Figure 14.15 presents both the relationship between the vega and the underlying price and the relationship between vega and time to expiration. There are two options with regard to time to expiration, namely 3 months and 1 year, which have risk factors of X = 40, σ = 0.3, and r Fs = 0.02. The share price varies. The graph indicates that atthe-money options have the highest vega. Option prices react very sensitively to changes in volatility. Options that are deep in the money and deep out of the money have a low rate of price change to movements in volatility because the time value of these options is low. The chance of earning a higher profit does not change with a shift in volatility. Therefore, the vega for options that are deep in the money or deep out of the money tends to zero. In addition, the time to expiration influences the time

27

See Hull 2012: Risk Management and Financial Institutions, p. 146 f.

522 Fig. 14.16 Relationship between the price of an equity option and the risk-free rate (Source: Own illustration)

14

Options: Basics and Valuation

(Call price) 8 6 4 2 0 0

2

4

6

8

10

(Risk-free interest rate in %) (Put price) 4 3 2 1 0 0

2

4

6

8

10

(Risk-free interest rate in %) value of an option. Thus, an option with a longer time to expiration has a higher vega than an option with a shorter lifetime, all else being equal.

14.9.4 Rho Rho represents the rate of change of the option price with respect to an interest rate change. It is positive for a long call option. If the risk-free interest rate rises, the call price increases. Conversely, rho is negative for a long put option, as an increase in the risk-free interest rate causes a decrease in the put price, since the put holder loses interest while waiting until option expiration to receive the strike price. For both calls and puts, the longer the time to expiration, the larger is the effect of interest rate changes on the option price. Moreover, in-the-money options have a greater rho because the strike price is more likely to be paid.28

28

See Figlewski 1990: ‘Theoretical Valuation Models’, p. 112.

14.10

Summary

523

Equity options react relatively weakly to interest rate changes, as presented in Fig. 14.16 for the call and put options with the risk factors of S0 = 42, X = 40, σ = 0.3, and T = 0.75. The risk-free interest rate varies. However, if the underlying is a fixed-rate bond, there is a strong relationship between the option price and the riskfree interest rate as the latter also has an influence on the level of the bond price.29

14.9.5 Theta The theta of the option corresponds to the rate of change of the option price over time. The time to expiration of the option is constantly moving towards zero. As the option approaches the expiration date, the option price moves to the intrinsic value, while the time value decreases. The rate of change of the time value decay can be measured by the theta.30 The theta is generally negative because the option price decreases with the passage of time. American options always have a negative theta, whereas European put options can have a positive theta in certain cases.31 In general, the longer the time to expiration of the option, the higher the option price. Figure 14.17 presents the value decay of the call and put option with the risk factors of S0 = 42, X = 40, σ = 0.3, and rFs = 0.03. The expiration time varies. The theta becomes more negative as the expiration date approaches if the option is at the money. Essentially, the same occurs with in-the-money and out-of-themoney options, except that the theta becomes less negative when the expiration date is imminent.32 Unlike the other option price sensitivities, the theta does not represent a risk factor, because the time course is known. Therefore, unexpected changes cannot occur, and it makes no sense to hedge an option against the passage of time.33 Nevertheless, the theta is regarded as an important option price sensitivity measure. In the case of a delta-neutral portfolio, for example, the theta can be used as an approximation of the gamma.34

14.10 Summary • A call option grants the holder the right to buy a certain quantity of an underlying asset during or at the end of the option life at a fixed strike price. By contrast, the holder of a put option has the right to sell a certain quantity of an underlying asset during or at the end of the option life at a predefined strike price. • The seller of the option has the obligation to sell (call) or to buy (put) the underlying asset in the event that the option is exercised. Since the seller must See Chance 2003: Analysis of Derivatives for the CFA® Program, p. 221 and Sect. 12.2. See Figlewski 1990: ‘Theoretical Valuation Models’, p. 111. 31 See Sect. 14.4. 32 See Watsham 1998: Futures and Options in Risk Management, p. 124. 33 See Hull 2012: Risk Management and Financial Institutions, p. 148. 34 See Hull 2006: Options, Futures, and Other Derivatives, p. 359. 29 30

524 Fig. 14.17 Relationship between option price and time to expiration (Source: Own illustration)

14

Options: Basics and Valuation

(Call price) 8 6 4 2 0 0

0.2

0.4

0.6

0.8

(Time to expiration in years) (Put price) 4 3 2 1 0 0

0.2

0.4

0.6

0.8

(Time to expiration in years) wait to see whether the buyer exercises their call or put, they are also known as the writer. For this commitment, the option seller receives an option premium from the option buyer. • An option that can be exercised at any time during its time to expiration is called an American option. If the option can be exercised only on the expiration date, it is referred to as a European option. • Four basic positions can be distinguished in an option transaction, namely the purchase and sale of a call and the purchase and sale of a put. Buying (selling) an option is equivalent to a long (short) position. • The profit/loss pattern of an option is asymmetric. Thus, with a long call or a long put, no more than the option price paid can be lost, while the maximum possible profit with a long equity call is unlimited and with a long put is limited by the strike price minus the option premium paid. By contrast, for a short call or a short put, the maximum profit is limited to the option premium received. The loss potential is unlimited for a short equity call and limited to the difference between the strike price and the option premium received for a short put, because the price of the underlying cannot fall below zero. The profit/loss pattern of a long and a short option (call or put) is a mirror image. Consequently, options are a zero-sum game. The profit (loss) of one party is equal to the loss (profit) of the other party.

14.10

•

•

•

•

• • • •

• • •

Summary

525

As a result, no new money is created, but money is merely redistributed among the contracting parties of the option. For an in-the-money call option, the intrinsic value is the difference between the underlying price and the strike price. If the call option is at the money or out of the money, it is not exercised because this would result in a loss. The intrinsic value is therefore zero. Before expiration, the price of an at-the-money or out-of-the money call is given by the time value, whereas the price of an in-the-money call consists of an intrinsic value and a time value. The intrinsic value of an in-the-money put option equals the strike price minus the underlying price and thus corresponds to the gain from exercising the option. If the put option is at the money or out of the money, the intrinsic value is zero, and therefore the option price before expiration consists of the time value only. Conversely, the price of an in-the-money put consists of an intrinsic value and a time value. The time value reflects the chance that the option will move in the money. Therefore, time value is greatest for options that are at the money. The further the option moves out the money and in the money, the more the time value declines. The intrinsic value depends on the price of the underlying and the strike price, while the time value is influenced by the price volatility of the underlying, the time to expiration of the option, the risk-free interest rate, and the income of the underlying (e.g. dividends in the case of equity securities or interest income in the case of foreign currencies). There is a positive relationship between the call price and the underlying price, while this relationship is negative for a put option. A higher exercise price leads to a lower call price, and vice versa. With a put option, on the other hand, the relationship between the strike price and the put price is positive. The higher (lower) the volatility, the higher (lower) the option price. Options that are at the money and have a long time to expiration have the greatest price sensitivity to changes in volatility. A longer (shorter) time to expiration results in a higher (lower) option price. However, this relationship applies only to American options. Short-life European options can have a higher option price than those with a longer time to expiration. This is the case, for example, with European put options if the underlying price falls close to zero or to zero. The relationship between the risk-free interest rate and the option price is positive for a call option and negative for a put option. A change in the dividend has an impact on the share price. A higher (lower) dividend leads to a lower (higher) share price, which has a negative (positive) effect on the call price and a positive (negative) effect on the put price. The prices of options can be calculated with the binomial model and the Black– Scholes model. In contrast to the Black–Scholes model, the binomial model is more flexible and can be used for the pricing not only of European but also of American options.

526

14

Options: Basics and Valuation

• The binomial model is a discrete-time option pricing model. It specifies a successive path of future prices of the underlying, with the price either rising or falling at the end of each period. At the end of the binomial tree which corresponds to the expiration date of the option, the intrinsic value of the option is determined with the underlying price and the strike price. The time value of the option is zero. Using backward induction, the option price is calculated for each period in the binomial tree. At each node in the tree, the option price is determined by discounting the two probability-weighted option prices of the previous period with the risk-free rate. This risk-neutral valuation is carried out with the riskneutral probabilities and the risk-free interest rate. • The Black–Scholes model can be used to estimate the prices of European options only. The price of a call option with the Black-Scholes model can be interpreted as the underlying price multiplied by the delta, minus the discounted strike price, multiplied by the probability that the option will be exercised. The price of a put option, on the other hand, is equal to the present value of the strike price multiplied by the probability that the put option will expire in the money, minus the underlying price multiplied by the delta of the put. • The put–call parity represents a parity between a fiduciary call and a protective put and not between a call and a put. It applies only to European call and put options that refer to the same underlying and have the same strike price and time to expiration. The fiduciary call consists of a long call option and a long risk-free zero-coupon bond, while the protective put consists of a long put option and a long equity security. One of the situations in which the put–call parity can be used is if there is no market price for one of the two options (call or put). • Like futures/forwards, options are exposed to a leverage effect due to a capital commitment that is significantly lower than the purchase of the underlying, which can lead to both large positive returns and large negative returns. • The delta of the option corresponds to the rate of change of the option price with respect to a movement of the underlying price. The delta of a call option is between 0 and 1. Put options, on the other hand, have a negative delta, ranging from -1 to 0, due to the negative relationship between option price and underlying price. Long calls and short puts have a positive delta, while short calls and long puts have a negative delta. • The delta risk of equity options is eliminated by selling or buying the underlying equity security. A delta hedge is a dynamic hedge as the delta changes due to the change in the underlying price and the time to expiration of the option. Therefore, in contrast to a hedge strategy with forwards and futures, the delta-hedged option position must be continuously adjusted in order to remain delta neutral. • The gamma of an option is the rate of change of the option’s delta relative to a movement in the underlying price. It is largest for options that are at the money and close to expiration, as such options have a very unstable delta. The gamma is positive for long options and negative for short options. • An option’s vega reflects the rate of change of the option price with respect to a movement in the price volatility of the underlying. For long options, the vega is positive because the option price rises with an increase in volatility. Conversely,

14.11

Problems

527

short options have a negative vega. The vega is largest for options that are at the money and have a long time to expiration. • The rho of the option represents the rate of change of the option price relative to an interest rate change. This option Greek is positive for a long call and a short put and negative for a short call and a long put. Unlike interest rate options, equity options react relatively weakly to interest rate movements. • The theta of the option corresponds to the rate of change of the option price over time. The time to expiration of the option is constantly moving towards zero. Like gamma, theta has the highest value (in absolute terms) when the option is at the money and close to expiration. Such options have a high chance of expiring in the money.

14.11 Problems 1. A DAX call option expiring in March 2018 (ODAX Mar 18) with a strike price of 12,800 index points is listed on the Eurex on 29 December 2018 at a bid price of 385.30 index points and an ask price of 409 index points. The contract value per index point is EUR 5. The following questions must be answered: a) What is the profit/loss of the long DAX call option if the underlying equity index is at 13,300 and at 12,300 index points when the option expires? b) What is the profit/loss of the short DAX call option if the underlying equity index at expiration of the option is at 13,300 and 12,300 index points? c) What is the maximum profit and loss of the long DAX call and the short DAX call at expiration? d) What is the breakeven price of the equity index of the long DAX call and the short DAX call at expiration? 2. Deutsche Lufthansa AG stock trades at a price per share of EUR 7.42 on 21 November 2022. The annualised implied volatility of the equity security is 43%. The continuously compounded 3-month risk-free interest rate is 1.8%. What is the price of a put option with a strike price of EUR 7 and a time to expiration of 3 months according to the Cox–Ross–Rubinstein model (one-period binomial model)? 3. Bayer AG stock trades at a price per share of EUR 113.20 at the end of June 2017. The annualised implied volatility of the stock is 18%. The continuously compounded 6-month risk-free rate is -0.7%. a) What is the price of a European call option with a strike price of EUR 110 and a time to expiration of 6 months based on the Black–Scholes model? b) What is the price of a European put option with a strike price of EUR 110 and a time to expiration of 6 months based on the Black-Scholes model? 4. A European call option on Bayer AG stock with a strike price of EUR 110 and a time to expiration of 6 months has a market price of EUR 7.18 at the end of June 2017. The Bayer stock trades at a price per share of EUR 113.20 at the end of June 2017. The continuously compounded 6-month risk-free interest rate is -0.7%. What is the price of the put option according to put–call parity?

528

14

Options: Basics and Valuation

5. Nestlé S.A. stock trades at a price per share of CHF 121.04 at the end of the trading day of 18 March 2022. The annualised implied volatility of the stock is 23%. The 6-month and 12-month Compounded SARON is -0.714%. What is the price of a Nestlé call option with a strike price of CHF 120 and a time to expiration of 1 year using the Cox–Ross–Rubinstein model (two-period binomial model)? 6. An options trader holds 30 long European put contracts on shares of Nestlé S.A. One put contract refers to 100 shares. The puts have a strike price of CHF 120 and a time to expiration of 281 days. The Nestlé stock trades at a closing price per share of CHF 119.20 on 25 March 2022. The annualised implied volatility of the stock is 15%, while the dividend per share is CHF 2.80. The risk-free interest rate is -0.71%. According to the Black–Scholes model, the put price is CHF 8.54. The option has a delta of -0.583 and a gamma of 0.026. a) How many Nestlé shares are required to hedge the delta risk of the long put contracts? b) What is the profit/loss of the delta-hedged long put position according to Taylor series expansion with a second-order approximation if the Nestlé share price rises by CHF 5?

14.12 Solutions 1. a) In order to calculate the option value at expiration, the index price and the exercise price must each be multiplied by EUR 5. With a DAX price of 13,300 index points, the value at expiration for the long call option is EUR 2500: VT, Long call = max ðEUR 0, EUR 66, 500 - EUR 64, 000Þ = EUR 2500: The option premium for the holder of the long call corresponds to the ask price of 409 index points quoted on the Eurex, with the result that the option premium paid is EUR 2045 (= 409 index points × EUR 5). The profit of EUR 455 can be determined as follows: ProfitLong call = EUR 2500 - EUR 2045 = EUR 455: At a DAX price of 12,300 index points, the call option expires out of the money and is not exercised. The option value at expiration is EUR 0: V T, Long call = max ðEUR 0, EUR 61, 500 - EUR 64, 000Þ = EUR 0: The loss of the long DAX call is equal to the paid option premium of EUR 2045.

14.12

Solutions

529

b) At a DAX price of 13,300 index points, the holder of the long call will exercise the option, resulting in an exercise loss of EUR 2500 for the holder of the short call: V T, Short call = - max ðEUR 0, EUR 66, 500 - EUR 64, 000Þ = - EUR 2500: The loss is reduced by the option premium received of EUR 1926.50 (= 385.3 index points × EUR 5), resulting in a loss of EUR 573.50: LossShort call = - EUR 2500 þ EUR 1926:50 = - EUR 573:50: If the DAX is at 12,300 index points, the call option expires out of the money and is not exercised by the call buyer. The option value on the expiration date is EUR 0. Accordingly, the profit for the call seller consists of the option premium received of EUR 1926.50. c) The maximum profit for the call buyer is unlimited as there is no upper price limit for the DAX. The maximum loss is limited to the paid option premium of EUR 2045. Conversely, the maximum profit for the call seller is given by the premium received of EUR 1926.50, while the maximum loss is unlimited. d) For the long call holder, the DAX breakeven price of EUR 66,045 can be calculated as follows on the expiration date: BreakevenST , Long call = EUR 64, 000 þ EUR 2045 = EUR 66, 045: For the holder of the short call, the breakeven price of the DAX is EUR 65,926.50 on the expiration date: BreakevenST , Short call = EUR 64, 000 þ EUR 1926:50 = EUR 65, 926:50: 2. First, it is necessary to determine the factors for an up move and a down move, as well as the risk-neutral probabilities of an upward and downward movement of the share price: u = e0:43 ×

p

d = e - 0:43 × πu =

3=12

p

= 1:240,

3=12

= 0:807,

e0:018 × 3=12 - 0:807 = 0:456, 1:240 - 0:807 π d = 1 - 0:456 = 0:544:

The share prices after an upward and downward movement can be calculated as follows:

530

14

Options: Basics and Valuation

Su = EUR 7:42 × 1:240 = EUR 9:20, Sd = EUR 7:42 × 0:807 = EUR 5:99: On the expiration date of the option, put prices after an up move and a down move of the share price are EUR 0 and EUR 3.30, respectively: pu = max ðEUR 0, EUR 7 - EUR 9:20Þ = EUR 0, pd = max ðEUR 0, EUR 7 - EUR 5:99Þ = EUR 1:01: The expected put price of EUR 0.55 can be determined by discounting the riskneutral probability-weighted option prices after an up and down movement on the expiration date with the risk-free interest rate: p0 =

0:456 × EUR 0 þ 0:544 × EUR 1:01 = EUR 0:55: e0:018 × 3=12

3. a) The risk factors for calculating the option price are: S0 = 113.20, X = 110, σ = 0.18, r Fs = -0.007, and T = 6/12 = 0.5. The standard normal variables d1 and d2 can be determined with the equations below: d1 =

lnð113:20=110Þ þ - 0:007 þ 0:182 =2 × 0:5 p = 0:2615, 0:18 × 0:5 p d2 = 0:2615 - 0:18 × 0:5 = 0:1342:

The Excel function “NORMSDIST” produces values for N(d1) of 0.6031 and N(d2) of 0.5534. The call price of EUR 7.18 can be calculated as follows: c0 = EUR 113:20 × 0:6031 - EUR 110 × e - ð - 0:007Þ × 0:5 × 0:5534 = EUR 7:18: b) The areas of the standard normal distribution of 1 – N(d1) and 1 – N(d2) can be determined as follows: 1 - N ðd 1 Þ = 1 - 0:6031 = 0:3969, 1 - N ðd 2 Þ = 1 - 0:5534 = 0:4466: The put price is EUR 4.37:

14.12

Solutions

531

p0 = EUR 110 × e - ð - 0:007Þ × 0:5 × 0:4466 - EUR 113:20 × 0:3969 = EUR 4:37: 4. The price of a European put option on the Bayer stock of EUR 4.37 can be calculated using the put–call parity as follows: p0 = EUR 7:18 þ EUR 110 × e - ð - 0:007Þ × 0:5 - EUR 113:20 = EUR 4:37: 5. First, it is necessary to determine the factors of an up move and a down move, as well as the risk-neutral probabilities of an upward and downward movement of the share price: u = e0:23 ×

p 0:5

d = e - 0:23 × πu =

p

= 1:1766,

0:5

= 0:8499,

e - 0:00714 × 0:5 - 0:8499 = 0:4485, 1:1766 - 0:8499 π d = 1 - 0:4485 = 0:5515:

The share prices after an upward and downward movement at the end of the first 6-month period can be estimated as follows: Su = CHF 121:04 × 1:1766 = CHF 142:42, Sd = CHF 121:04 × 0:8499 = CHF 102:87: The share prices after a further upward and downward movement at the end of the second half-year period can be calculated as follows: Suu = CHF 121:04 × ð1:1766Þ2 = CHF 167:57, Sud = CHF 121:04 × 1:1766 × 0:8499 = CHF 121:04, Sdu = CHF 121:04 × 0:8499 × 1:1766 = CHF 121:04, Sdd = CHF 121:04 × ð0:8499Þ2 = CHF 87:43: The following call prices result on the expiration date of the option: cuu = maxðCHF 0, CHF 167:57 - CHF 120Þ = CHF 47:57, cud = maxðCHF 0, CHF 121:04 - CHF 120Þ = CHF 1:04, cdu = maxðCHF 0, CHF 121:04 - CHF 120Þ = CHF 1:04,

532

14

Options: Basics and Valuation

cdd = maxðCHF 0, CHF 87:43 - CHF 120Þ = CHF 0: The call prices at the end of the first 6-month period can be determined as follows after an upward and downward movement of the share price: cu =

0:4485 × CHF 47:57 þ 0:5515 × CHF 1:04 = CHF 21:99, e - 0:00714 × 0:5

cd =

0:4485 × CHF 1:04 þ 0:5515 × CHF 0 = CHF 0:468: e - 0:00714 × 0:5

The expected call price with the two-period binomial model is CHF 10.16: c0 =

0:4485 × CHF 21:99 þ 0:5515 × CHF 0:468 = CHF 10:16: e - 0:00714 × 0:5

A more accurate call price of CHF 11.17 is obtained by applying a binomial model consisting of 150 periods, as well as the Black-Scholes model. 6. a) One long put option has a delta of -0.583, while 3000 long put options have a delta of -1749 [= 3000 × (- 0.583)]. This means that if the share price rises (falls) by CHF 1, a loss (profit) of CHF 1749 is incurred (realised). A total of 1749 Nestlé shares need to be bought in order to hedge the delta risk: - 0:583 × 3000 = 1749: N= 1 b) The gamma of the long put position is positive and is 78 (= 3000 × 0.026). According to the Taylor series expansion with a second-order approximation, an increase of CHF 5 in the share price leads to a loss on the long put contracts of approximately CHF 7770: Δp ≈ ð- 1749Þ × CHF 5 þ 0:5 × 78 × ðCHF 5Þ2 = - CHF 7770: The delta of the hedged put option position is 0, resulting in a profit of approximately CHF 975: Δp ≈ 0 × CHF 5 þ 0:5 × 78 × ðCHF 5Þ2 = CHF 975: The larger the positive gamma, the greater the profit when the share price rises or falls. Thus, there is no gamma risk to be hedged.

Microsoft Excel Applications

533

Microsoft Excel Applications • In order to determine the area under a probability density function of the standard normal distribution, the standard normal variable is required. For example, the probability that a value is equal to or less than the standard normal variable Z can be calculated as follows: = NORMSDISTðZÞ: Then press the Enter key. The following demonstrates how Excel can be used to calculate the price of a call and a put with the Black–Scholes model. For example, in cells B1 to B5, the risk factors such as the share price, the strike price, the volatility, the time to expiration in years, and the risk-free interest rate should be entered. The formula for estimating the standard normal variable d1 should be typed in cell B7: = ðLNðB1=B2Þ þ ðB5 þ B3^2=2Þ B4Þ=ðB3 B4^0:5Þ: • Next, the equation for determining the standard normal variable d2 should be inserted in cell B8: = B7 - B3 B40:5 : • In cells B9 and B10, the areas under the probability density function of the standard normal distribution N(d1) and N(d2) can be estimated using =NORMSDIST(B7) and =NORMSDIST(B8), respectively. In cells B11 and B12, 1 - N(d1) and 1 - N(d2) should be entered. The discount factor of e - rFs T should be typed in cell B13 with the formula expression =EXP(-B5*B4). To calculate the call price and the put price the following equations should be entered in cells B14 and B15: = B1 B9 - B2 B13 B10 and = B2 B13 B12 - B1 B11:

534

14

Options: Basics and Valuation

Fig. 14.18 Black-Scholes model in Excel (Source: Own illustration)

Figure 14.18 presents how to apply the Black–Scholes model in Excel to calculate the call and put price with a share price of 100, a strike price of 90, a volatility of 30%, a time to expiration of 0.75 years, and an interest rate of 1%.

References Beckers, S.: Standard deviations implied in option prices as predictors of future stock price variability. J. Bank. Financ. 5(3), 363–381 (1981) Black, F., Scholes, M.: The valuation of option contracts and a test of market efficiency. J. Financ. 27(2), 399–417 (1972) Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973) Chance, D.M.: Analysis of Derivatives for the CFA® Program. Association for Investment Management and Research, Charlottesville (2003) Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econ. 7(3), 229–263 (1979) Figlewski, S.: Theoretical valuation models. In: Figlewski, S., Silber, W.L., Subrahmanyam, M.G. (eds.) Financial Options: From Theory to Practice, pp. 77–134. McGraw-Hill, New York (1990) Hull, J.C.: Options, Futures, and Other Derivatives, 6th edn. Prentice Hall, Upper Saddle River, NJ (2006) Hull, J.C.: Risk Management and Financial Institutions, 3rd edn. Wiley, Hoboken (2012) Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141–183 (1973) Mondello, E.: Finance: Theorie und Anwendungsbeispiele. Springer, Wiesbaden (2017) Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 8th edn. Cengage Learning, Mason (2006) Watsham, T.J.: Futures and Options in Risk Management, 2nd edn. Thomson Learning, London et al (1998)

References

535

Online Sources European Exchange: Eurex Markets. https://www.eurex.com/ex-en/markets/equ/opt/MercedesBenz-Group-2884628. Accessed on 29 December 2017.

Option Strategies

15.1

15

Introduction

Calls and puts can be used in a variety of ways. They allow market participants to modify a risk position or implement an investment strategy. Some option strategies have been designed to make a profit if a certain market condition occurs and are therefore purely speculative in nature. Other strategies, however, are defensive and allow protection against an unfavourable market development. This chapter describes the use of options in typical investment situations. The chapter begins with put–call parity, which can be applied to create synthetic long– short positions of an equity security, a call option, and a put option. It goes on to explain how the risk exposure of a long equity position can be modified with a covered call and a protective put strategy and in which situations the respective strategy is appropriate. Another option strategy associated with hedging the price risk exposure of a long equity security is a collar, where a price floor and a price ceiling are set on the underlying asset. If the share price falls outside these price limits, there is no further loss or gain. The chapter ends with option strategies that can be constructed from a combination of calls and puts with different strike prices— strategies such as bull and bear spreads, as well as the straddle.

15.2

Synthetic Equity

Derivatives can be used to synthetically create a specific financial instrument with a desired risk exposure. Calls and puts can be used for this purpose, in addition to other derivatives such as futures, forwards, or swaps. Options have the advantage that they have an asymmetrical profit/loss pattern. This means that the underlying can be protected against an unfavourable price movement, but it is still possible to benefit from the price movement of the underlying in the other direction. This is not feasible

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6_15

537

538

15

Option Strategies

Table 15.1 Equivalence between long equity and synthetic long equity (Source: Own illustration) Transaction Long equity Buy equity security Synthetic long equity Buy call Sell put Buy 0%-bond Total

Current value

Value at expiration T ST < X

ST > X

S0

ST

ST

c0 -p0 Xe - rFs T c0 - p0 þ Xe - rFs T

0 -(X - ST) X ST

ST - X 0 X ST

with futures, forwards, and swaps since these derivatives have a symmetrical profit/ loss pattern.1 In many cases, there are several possible combinations that can be used to create a specific risk exposure. For example, a long equity security can be replicated synthetically by applying put–call parity, where the value of a fiduciary call is equal to that of a protective put ðc0 þ Xe - rFs T = p0 þ S0 Þ.2 If the put–call parity is solved for the long equity position, the following equation is obtained:3 S0 = c0 þ Xe - rFs T - p0 :

ð15:1Þ

Therefore, the long equity security can be synthetically constructed with a European long call (c0), a long risk-free zero-coupon bond ðXe - rFs T Þ, and a European short put (-p0). Table 15.1 indicates that a long equity and a synthetically produced long equity are equivalents. Both have the same value at expiration of ST. If at expiration T of the two options the share price is below the strike price, ST < X, the put option of the synthetic stock expires in the money and there is an exercise loss of - (X - ST) for the short put. By contrast, the long call option expires worthless, while the price of the long zero-coupon bond is given at maturity by its par value of X. The value of the synthetic long equity is ST [= -(X - ST) + X] and thus corresponds to the price of the long stock of ST. If, on the other hand, the share price exceeds the strike price, ST > X, on the expiration date of the two options, the call option expires in the money, and an exercise gain of ST - X results. The short put option expires worthless, while the long zero-coupon bond has a value of X. Accordingly, the value of the synthetic long equity is again ST (= ST - X + X). A synthetic short equity can be created according to put–call parity as follows: - S0 = - c0 - Xe - rFs T þ p0 :

ð15:2Þ

See Chance 2003: Analysis of Derivatives for the CFA® Program, p. 412. See Sect. 14.7. 3 A plus sign in the equation indicates a long position, while a minus sign indicates a short position. 1 2

15.2

Synthetic Equity

539

Table 15.2 Equivalence between short equity and synthetic short equity (Source: Own illustration) Transaction Short equity Short equity security Synthetic short equity Sell call Buy put Short 0%-bond Total

Current value

Value at expiration T ST > X ST < X

-S0

-ST

-ST

-c0 p0 - Xe - rFs T - c0 þ p0 - Xe - rFs T

0 (X - ST) -X -ST

-(ST - X) 0 -X -ST

Thus, a short stock can be synthetically constructed with a European short call (-c0), a short risk-free zero-coupon bond ð- Xe - rFs T Þ, and a European long put ( p0). Table 15.2 presents the equivalence between a short equity and a synthetically produced short equity. Both have the same value on the expiration date of the two options of -ST. If the share price falls below the strike price on the expiration date ST < X, the long put expires in the money and produces an exercise gain of X - ST. By contrast, the short call option expires worthless. The short zero-coupon bond has a value of -X,4 and therefore the value of the synthetic short equity is -ST [= (X ST) - X]. If the share price exceeds the strike price on the expiration date of the two options ST > X, the short call option incurs an exercise loss of -(ST - X). The long put expires worthless, while the short zero-coupon bond has a value of -X. Accordingly, the value of the synthetic short equity is again -ST [= -(ST - X) - X] and is therefore equal to the price of the short equity position of -ST. A synthetic long equity position can also be constructed with futures/forwards. For example, the risk exposure of the long stock can be replicated by entering into a long equity futures contract. Since long futures represent an obligation to buy the underlying asset at contract expiration, no cash payment is made when entering into the futures contract. Thus, over the lifetime of the futures the money can be invested in a risk-free zero-coupon debt security. Accordingly, a long equity security can be synthetically created with a long equity futures contract and a long risk-free zerocoupon bond: S0 = F 0 þ Xe - rFs T ,

ð15:3Þ

where S0 = price of equity security, F0 = price of equity futures/forward, and

4

The zero-coupon bond is bought at the par value of X on the maturity date so that the bond position from the short sale can be closed.

540

15

Option Strategies

Xe - rFs T = price of risk-free zero-coupon debt security, e.g. non-interest-bearing treasury bills of the Federal Republic of Germany (BuBills) or money market book claims of the Swiss Confederation.

15.3

Synthetic Call and Put Option

When the US Chicago Board Options Exchange opened in 1973, only calls were traded—no puts. Regulators at the time were concerned about allowing financial instruments such as puts to speculate on prices in a falling market environment. However, market participants can also synthetically create a long put option with a long call option, a short stock, and a long risk-free zero-coupon debt security and thus speculate on falling market prices. Synthetic put options were widely used among investors at the time, with the result that regulators approved a pilot project in 1975 involving put options on 25 different stocks. The exchange-traded puts did not cause any significant market distortions, and derivatives exchanges were therefore able to gradually expand their listed put options. Nevertheless, it is important to know how to replicate a long put synthetically, as mispriced options can be more expensive than a synthetic long put option. A synthetic long put option can be created on the basis of put–call parity as follows: p0 = c0 - S0 þ Xe - rFs T :

ð15:4Þ

Thus, a synthetic long put can be constructed with a long call option, a short equity security, and a long risk-free zero-coupon debt security. Table 15.3 indicates that a long put and a synthetic long put are equivalents. Both the put and the synthetic put lead to the same value at expiration. If at expiration T the share price is below the strike price, ST < X, the long call option expires worthless. The short equity security has a price of -ST, while the long zero-coupon debt security has a value of X, reflecting the par value at maturity. Accordingly, the value of the synthetic long put option on the expiration date is X - ST (= 0 - ST + X) and is equal to the value of the long put of X - ST, which expires in the money. If, on the other hand, the share price exceeds the strike price at the expiration date T, ST > X, Table 15.3 Equivalence between long put and synthetic long put (Source: Own illustration) Transaction Long put Buy put Synthetic long put Buy call Short equity security Buy 0%-bond Total

Current value

Value at expiration T ST > X ST < X

p0

X - ST

0

c0 -S0 Xe - rFs T c0 - S0 þ Xe - rFs T

0 -ST X X - ST

ST - X -ST X 0

15.4

Covered Call

541

Table 15.4 Equivalence between long call and synthetic long call (Source: Own illustration) Transaction Long call Buy call Synthetic long call Buy put Buy equity security Short 0%-bond Total

Current value

Value at expiration T ST > X ST < X

c0

0

ST - X

p0 S0 - Xe - rFs T p0 þ S0 - Xe - rFs T

X - ST ST -X 0

0 ST -X ST - X

the long call expires in the money and produces an exercise gain of ST - X. Since the values of the short stock and the long zero-coupon debt security are –ST and X, respectively, the synthetic put results in a value of 0 [= (ST - X) - ST + X], which corresponds to the value of the out-of-the-money long put option of 0. A synthetic long call option can be created using put–call parity as follows: c0 = p0 þ S0 - Xe - rFs T :

ð15:5Þ

Therefore, a synthetic long call option consists of a long put, a long equity security, and a short risk-free zero-coupon debt security. Table 15.4 illustrates that a long call option and a synthetic long call option are equivalents. Both have the same value at expiration. If at expiration T the share price is below the strike price, ST < X, the long put of the synthetic call expires in the money, and an exercise gain of X - ST arises. The long equity security has a value of ST, while the short zerocoupon debt security has a value of -X. Hence, the value of the synthetic long call option on the expiration date is 0 [= (X - ST) + ST - X] and is equal to the value of the long call, which expires worthless. If, on the other hand, the share price exceeds the strike price at expiration T, ST > X, the long put expires out of the money and has a value of 0. Thus, the synthetic call consists of the long equity security with a price of ST and the short zero-coupon bond with a value of -X, which is equal to the value of the long call option at expiration of ST - X.

15.4

Covered Call

A covered call strategy is an option strategy commonly used by both private and institutional investors. It is a relatively conservative strategy, but also one of the most misunderstood strategies. A covered call is a position where an investor owns the underlying and sells a call. If the underlying price exceeds the strike price on the expiration date of the option, the holder of the long call option will buy the underlying at the strike price, and therefore the holder of the short call option must deliver the underlying asset at the strike price. As compensation for the sales obligation, the seller of the call receives from the buyer of the call an upfront option

542

15

Option Strategies

premium at inception of the contract.5 In the following section, the underlying asset is an equity security.

15.4.1 Profit and Loss In order to calculate the profit/loss of the covered call strategy on the expiration date of the call option, the value of the strategy at expiration must first be determined, which equals the share price minus the intrinsic value of the call option: V T = ST - maxð0, ST- X Þ,

ð15:6Þ

where VT = value at expiration of the covered call strategy. If the share price exceeds the strike price, the intrinsic value of the call option represents an exercise loss for the holder of the short call option. To calculate the value at expiration of the covered call strategy the loss must be deducted from the share price. In other words, the profit from the increase in the share price above the strike price belongs to the buyer of the call, which means that the holder of the covered call cannot make any profits from share price increases above the strike price. The profit/loss of the covered call results from the difference between the value at expiration and the costs at the inception of the strategy, which consist of the price paid for the share minus the premium received from the sale of the call. Thus, the profit/loss at expiration can be determined as follows: Profit=loss = V T - S0 þ c0 :

ð15:7Þ

For example, an investor buys a Mercedes-Benz Group share on 12 January 2018 at a price of EUR 74.25. In order to earn an additional income, they sell an out-of-the-money European call option expiring in March 2018 with a strike price of EUR 76 at a price of EUR 1.41. Table 15.5 presents the prices listed on the Eurex for European call and put options on the Mercedes-Benz Group stock for the expiration months of January, February, and March 2018 and exercise prices of EUR 68, EUR 72, EUR 74, EUR 76, and EUR 80. The options expire on the last Friday of the respective expiration month. The prices are quoted on the Eurex for one option, although the contract size consists of 100 options. If the price of the automobile share is EUR 80 on the expiration date of the call option, the option expires in the money and is exercised by the holder of the long call. The value of the covered call strategy at expiration of EUR 76 and the profit of EUR 3.16 can be calculated as follows: 5

See Yates and Kopprasch 1980: ‘Writing call options: profits and risks’, p. 74 ff.

15.4

Covered Call

543

Table 15.5 Prices for European call and put options with different expiration dates and strike prices on the Mercedes-Benz Group stock (Source: www.eurex.com/ex-en/markets/equ/opt/ Mercedes-Benz-Group-2884628) Calls (in EUR) January February 6.21 6.41 2.30 2.98 0.73 1.74 0.14 0.92 0.02 0.22

March 6.65 3.47 2.38 1.41 0.49

Strike price (in EUR) 68 72 74 76 80

Puts (in EUR) January February 0.03 0.24 0.13 0.83 0.56 1.58 1.96 2.76 5.85 6.07

March 0.50 1.33 2.14 3.27 6.35

V T = EUR 80 - max ðEUR 0, EUR 80 - EUR 76Þ = EUR 76, Profit = EUR 76 - EUR 74:25 þ EUR 1:41 = EUR 3:16: However, the contract size on the Eurex consists of 100 options, and therefore the covered call strategy can be implemented with the purchase of 100 Mercedes-Benz Group shares and the simultaneous sale of a call contract. Therefore, the value at expiration is EUR 7600 and the profit is EUR 316: V T = 100 × EUR 80 - 100 × ½max ðEUR 0, EUR 80 - EUR 76Þ = EUR 7600, Profit = EUR 7600 - 100 × EUR 74:25 þ 100 × EUR 1:41 = EUR 316: Figure 15.1 presents the profit/loss diagram of the covered call on the expiration date of the short call, in which the strategy is displayed with a long Mercedes-Benz Group share and a short call option. The diagram indicates, among other things, that regardless of the increase in the share price, the maximum profit of the covered call is EUR 3.16. With a share price of EUR 74.25 and a strike price of EUR 76, the maximum profit from the share price increase is EUR 1.75, since the automobile stock is sold through the short call option at a price of EUR 76. Consequently, the maximum share price gain consists of the difference between the strike price and the share price at the inception of the strategy. In addition, the investor receives proceeds from the sale of the call option of EUR 1.41, which means that the maximum profit is EUR 3.16 [= (EUR 76 - EUR 74.25) + EUR 1.41]. The maximum profit at expiration can therefore be determined with the following formula: Maximum profit = X - S0 þ c0 :

ð15:8Þ

If the share price falls, a loss on the long equity security occurs, which is reduced by the premium received from the sale of the call option. Should the call expire out of the money, the long call option is not exercised by its holder. The maximum loss of the covered call strategy occurs when the share price falls to zero. In this price scenario, the value of the strategy on the expiration date of the call option is zero. Subtracting the initial value of the strategy of S0 – c0 from the value at expiration

544

15

(Profit/loss in EUR) 20

Long Mercedes-Benz Group share

Short call

Option Strategies

Maximum profit from covered call of EUR 3.16

10 0 0

10

20

30

40

50

60

70

80

-10

90 100 110 120 (Share price in EUR)

-20 Covered call -30

Breakeven share price of covered call of EUR 72.84

-40 -50 -60 -70

Maximum loss from covered call of EUR 72.84

-80 Fig. 15.1 Profit/loss of the covered call strategy at expiration (Source: Own illustration)

leads to the maximum loss at expiration, which can be calculated with the following formula: Maximum loss = S0 - c0 :

ð15:9Þ

In the event that the price of the automobile share falls to EUR 0, the value of the covered call strategy at expiration is EUR 0 [= EUR 0 - max(EUR 0, EUR 0 EUR 76)]. Accordingly, the maximum loss of the strategy at expiration is EUR 72.84 (= EUR 74.25 - EUR 1.41). Thus, the maximum possible loss of the long stock of EUR 74.25 is reduced by the option premium received of EUR 1.41, with the result that a maximum loss of EUR 72.84 remains. The breakeven share price at expiration represents the breakeven point of the strategy where there is neither a profit nor a loss. The breakeven share price is obtained if the share price falls by the amount of the option premium received. At this share price, the loss of the equity security is offset by the option premium received from the call sale, with the result that there is neither a profit nor a loss. In the example, the breakeven point is at a share price of EUR 72.84 (= EUR 74.25 -

15.4

Covered Call

545

Table 15.6 Value, profit/loss, maximum profit, maximum loss, and breakeven share price of the covered call strategy at expiration (Source: Own illustration) Covered call Value at expiration VT Profit/loss at expiration Maximum profit at expiration Maximum loss at expiration Breakeven share price at expiration ST

Formula ST - max (0, ST - X) VT - S0 + c0 X - S0 + c 0 S0 - c 0 S0 - c 0

EUR 1.41). In general, the breakeven share price at expiration can be determined with the following equation: Breakeven share price ST = S0 - c0 :

ð15:10Þ

Table 15.6 summarises the presented formulas of the covered call strategy at expiration for the value, profit/loss, maximum profit, maximum loss, and breakeven share price. With a covered call, the risk of loss of a long equity position can be reduced by selling call options. Hence, the loss resulting from a decrease in the share price is reduced by the premium received from the sale of the call option. In return, the profit potential of the strategy is limited. If call options are sold without owning the underlying, the risk of loss is unlimited, since the price of an equity security has no upper limit. On the other hand, the profit potential is limited to the option premium received.6 Example: Covered Call Strategy A portfolio manager owns 10,000 shares of Mercedes-Benz Group trading at a price per share of EUR 74.25 on 12 January 2018. To earn income, they sell 10,000 out-of-the-money European call options (or 100 call option contracts) on the Eurex with a strike price of EUR 80 and expiration month of September 2018. A call option is traded at a price of EUR 1.50. 1. What is the profit/loss of the covered call strategy if the price of the Mercedes-Benz Group share on the expiration date of the call is EUR 85 and EUR 65? 2. What is the maximum profit, maximum loss, and breakeven share price of the strategy at expiration? Solution to 1 With a share price of EUR 85, the value of the strategy at expiration is EUR 800,000: (continued) 6

See Sect. 14.3.1.

546

15

Option Strategies

V T = 10, 000 × EUR 85 - 10, 000 × ½max ðEUR 0, EUR 85 - EUR 80Þ = EUR 800, 000: The profit/loss can be determined by subtracting from the value at expiration of EUR 800,000 the value of the equity position at the beginning of EUR 742,500 (= 10,000 × EUR 74.25) and adding the premium received from the sale of the call options of EUR 15,000 (= 10,000 × EUR 1.50). Therefore, the profit at expiration is EUR 72,500: Profit = EUR 800, 000 - EUR 742, 500 þ EUR 15, 000 = EUR 72, 500: At a share price of EUR 65, the value of the strategy at expiration is EUR 650,000: V T = 10, 000 × EUR 65 - 10, 000 × ½max ðEUR 0, EUR 65 - EUR 80Þ = EUR 650, 000: The loss of EUR 77,500 on the expiration date of the call can be determined as follows: Loss = EUR 650, 000 - EUR 742, 500 þ EUR 15, 000 = - EUR 77, 500: Without the sale of the call options, the loss would be EUR 92,500 (= EUR 650,000 - EUR 742,500). Thus, the income from the sale of the call options reduces the loss of EUR 92,500 to EUR 77,500 (= EUR 92,500 EUR 15,000). Solution to 2 The maximum profit, maximum loss, and breakeven share price at expiration can be determined as follows: Maximum profit = X - S0 þ c0 = 10, 000 × ðEUR 80 - EUR 74:25 þ EUR 1:50Þ = EUR 72, 500, Maximum loss = S0 - c0 = 10, 000 × ðEUR 74:25 - EUR 1:50Þ = EUR 727, 500, Breakeven share price ST = S0 - c0 = EUR 74:25 - EUR 1:50 = EUR 72:75: The covered call strategy is a more conservative strategy than buying an equity security because the risk of loss from a share price decline is reduced by the option premium received. By contrast, profits are foregone in the event of a share price increase above the strike price. The level of the selected exercise price has an influence on the extent of the loss if the share price falls and on the maximum profit if the share price rises. The higher (lower) the strike price, the higher (lower) the

15.4

Covered Call

547

maximum possible capital gain on the stock and the lower (higher) the option premium received. There is no actual decision rule, and therefore the selection of strike price for the call option depends on the risk preferences of the investors in the market. Most covered call strategies are realised with exchange-traded options, which means that the holder of the covered short call has the option to close out the option position before expiration by entering into a long call with the same underlying, strike price, and expiration date. If, for example, the price of the Mercedes-Benz Group share falls after the covered call has been implemented, the call price also falls. Provided the investor assumes that the decline in the price of the automobile stock is temporary, they can offset the short call by buying a call option at a lower price and thus make a profit. In the event that the share price recovers, they can sell a covered call option again.

15.4.2 Objectives The most common motivation for writing covered call options is to earn an income. When a call option is sold, all profits from the share price increase above the strike price are forfeited. As compensation, the seller of the covered call receives the option premium. A higher option premium can be earned by selling call options with a longer time to expiration. However, the risk of exercising the option increases because the probability is greater with a longer time to expiration that the option will move in the money and that the holder of the long call will therefore exercise their option to buy the underlying equity security. In this context, share price expectations also play an important role. Provided that the share price does not rise significantly during the life of the option, a covered call is an attractive strategy. The proceeds from the sale of the option help to improve the return on the equity position.7 The covered call strategy can also be used to reduce an overweighted equity position in a portfolio. For example, a portfolio manager holds 50,000 MercedesBenz Group shares in the portfolio. On 12 January 2018, the automobile stock is traded at a price per share of EUR 74.25. The portfolio manager expects the share price to remain relatively stable over the next few weeks. Therefore, they sell 200 inthe-money call option contracts expiring in February 2018 with a strike price of EUR 72 at a price per call of EUR 2.98 (see Table 15.5). The proceeds from the call sale amount to EUR 59,600 (= 200 × 100 × EUR 2.98). Since the portfolio manager assumes a relatively stable share price, there is a high probability that the in-themoney call options will be exercised on the expiration date. If the call options expire in the money, they can reduce the excessive weight of Mercedes-Benz Group shares in the portfolio as planned. In doing so, the shares are sold at a unit price of EUR 74.98, which is made up of the exercise price of EUR 72 and the option premium of EUR 2.98 received. They could also have sold the shares at a unit price

7

See Figlewski 1990: ‘Basic Price Relationships and Basic Trading Strategies’, p. 58.

548

15

Option Strategies

of EUR 74.25. However, the portfolio manager achieves higher sales proceeds of EUR 74.98 per share through the covered call sale, which is equivalent to a return of 0.98% (= EUR 74.98/EUR 74.25 –1) or additional proceeds of EUR 14,600 [= 20,000 × (EUR 74.98 - EUR 74.25)]. The option price consists of an intrinsic value and a time value. In the example, the intrinsic value is EUR 2.25 (= EUR 74.25 EUR 72), while the time value is EUR 0.73 (= EUR 2.98 - EUR 2.25). Compared to the sale of the shares at a unit price of EUR 74.25, the holder of the covered short call realises the time value of the sold call options of EUR 14,600 (= 20,000 × EUR 0.73), which increases the sales proceeds of the shares and thus represents additional proceeds. If, against the expectations of the portfolio manager, the share price rises, the covered call strategy causes an opportunity loss, since the sales proceeds would have been greater without the covered call strategy. Another possible use of the covered call strategy is to sell covered call options on an undervalued equity security at an exercise price that is close to the intrinsic value of the security. For example, if financial analysts at a bank assume that the intrinsic value of the Mercedes-Benz Group share is EUR 76 when the security is traded at a price per share of EUR 74.25, the trading department of the financial institution can sell a covered call. In a case where the bank holds 40,000 Mercedes-Benz Group shares in its trading portfolio, 400 call option contracts with an expiration date of February 2018 and a strike price of EUR 76 can be sold at a price per call of EUR 0.92 (see Table 15.5), resulting in sales proceeds of EUR 36,800 (= 400 × 100 × EUR 0.92). If the call options expire in the money, the Mercedes-Benz Group shares are sold at the strike price or the intrinsic value of EUR 76, which means that an additional return of 1.24% (= EUR 0.92/EUR 74.25) arises. Should the call options expire out of the money, the bank can again sell covered calls. With this strategy, there is a risk that unexpected negative company news or a general stock market decline will result in a sharp decrease in the share price. However, the loss of the equity position is reduced by the premium received from the sale of the calls. Conversely, the share price can rise significantly as a result of unexpected good company news. If the share price exceeds the strike price, there is an opportunity loss because the shares are sold at the lower strike price.

15.5

Protective Put

The holder of an equity security is exposed to the risk of a price decrease. A long put option, which moves in the money if the share price drops below the strike price, offers protection against the risk of loss. In this case, the loss from the share price decline below the strike price is offset by the exercise gain of the long put. However, an option premium must be paid for this protection. In the event of an increase in the share price, however, the price upside potential of the security remains, but is reduced by the option premium paid. This strategy is referred to as protective put. By contrast, with the covered call strategy a premium is received from the sale of the call option, but in return the profit potential of the underlying is given up and the loss potential is reduced only by the option premium received.

15.5

Protective Put

549

The objective of a protective put is, on the one hand, to hedge the price risk of a long equity position with a long put and, on the other hand, to still participate in the price upside potential of the long stock. In order to calculate the profit/loss of the protective put strategy at expiration, the value at expiration must first be determined, which is given by the share price and the intrinsic value of the long put: V T = ST þ maxð0, X- ST Þ:

ð15:11Þ

The profit/loss at expiration results from the value at expiration minus the costs at the inception of the strategy, which consist of the share price and the purchase price of the put option: Profit=loss = V T - S0 - p0 :

ð15:12Þ

An investor owns a Mercedes-Benz Group share, which is traded at a price of EUR 74.25 on 12 January 2018. In order to protect against a share price decline, they buy an exchange-traded European put option expiring in March 2018 with a strike price of EUR 74. The put option is traded on the Eurex at a price of EUR 2.14 (see Table 15.5). If the price of the automobile stock falls to EUR 68 on the expiration date of the put option, the value at expiration is EUR 74: V T = EUR 68 þ max ðEUR 0, EUR 74 - EUR 68Þ = EUR 74: The loss of the protective put of EUR 2.39 arises from the value at expiration of EUR 74 less the value at the inception of the strategy, which is made up of the price of the Mercedes-Benz Group share of EUR 74.25 and the price paid for the put option of EUR 2.14: Loss = EUR 74 - EUR 74:25 - EUR 2:14 = - EUR 2:39: The loss caused by the falling share price of EUR 6.25 (= EUR 68 - EUR 74.25) is partially offset by the exercise gain of the long put option of EUR 6 [= max (EUR 0, EUR 74 - EUR 68)]. Furthermore, the option premium paid of EUR 2.14 must be considered, resulting in a strategy loss of EUR 2.39 (= - EUR 6.25 + EUR 6 - EUR 2.14). Figure 15.2 presents the profit/loss diagram of the protective put on the expiration date. If the share price falls below the strike price, the put option expires in the money. The loss of the equity security from a further price decrease is offset by the exercise gain of the long put option, and therefore the maximum possible loss is limited. The maximum loss of the protective put strategy at expiration equals the price decrease of the share to the strike price plus the premium paid for the long put: Maximum loss = S0 - X þ p0 :

ð15:13Þ

In the example, the maximum possible loss at expiration is EUR 2.39 (= EUR 74.25 - EUR 74 + EUR 2.14). By contrast, the maximum profit is unlimited because the share price has no upper limit:

550

15

(Profit/loss in EUR)

Maximum profit of protective put is unlimited

Long put

60 50 40 30 20

Option Strategies

Long Mercedes-Benz Group share

Maximum loss of protective put of EUR 2.39

10

Protective put

0 -10 0

10

20

30

40

50

60

70

80

-20 -30

90 100 110 120 (Share price at expiration in EUR)

-40 Breakeven share price of protective put of EUR 76.39

-50 -60 -70 -80

Fig. 15.2 Profit/loss diagram of the protective put strategy at expiration (Source: Own illustration) Table 15.7 Value, profit/loss, maximum loss, maximum profit, and breakeven share price of the protective put strategy at expiration (Source: Own illustration) Protective put Value at expiration VT Profit/loss at expiration Maximum loss at expiration Maximum profit at expiration Breakeven share price at expiration ST

Maximum profit = 1:

Formula ST + max (0, X - ST) VT - S0 - p0 S0 - X + p0 1 S0 + p0

ð15:14Þ

In order to break even, the share price must rise by a sufficiently large amount so that the capital gain of the share offsets the option premium paid. Therefore, the breakeven share price of the strategy at expiration can be calculated as follows: Breakeven share price ST = S0 þ p0 :

ð15:15Þ

In the example, the breakeven share price is EUR 76.39 (= EUR 74.25 + EUR 2.14). Table 15.7 presents a compact overview of the formulas

15.5

Protective Put

551

that can be used to calculate the value, profit/loss, maximum loss, maximum profit, and breakeven share price at expiration for a protective put strategy. Example: Protective Put Strategy A portfolio manager holds 60,000 Mercedes-Benz Group shares trading at a price per share of EUR 74.25 on 12 January 2018. To hedge the price risk of the equity position, they buy 600 European put option contracts on the Eurex, expiring in March 2018 with a strike price of EUR 76 at a price per put of EUR 3.27 (see Table 15.5). 1. What is the profit/loss of the protective put strategy at expiration if the price of the Mercedes-Benz Group share is EUR 68? 2. What is the maximum profit, maximum loss, and breakeven share price of the protective put strategy at expiration? Solution to 1 In order to calculate the profit/loss at expiration, the value of the protective put strategy at expiration must first be calculated: V T = 60, 000 × EUR 68 þ 60, 000 × ½max ðEUR 0, EUR 76 - EUR 68Þ = EUR 4, 560, 000: The loss of EUR 91,200 can be determined by subtracting the beginning costs of the strategy from the strategy value at expiration: Loss = EUR 4, 560, 000 - 60, 000 × ðEUR 74:25 þ EUR 3:27Þ = - EUR 91, 200: Solution to 2 The maximum profit is unlimited, while the maximum loss at expiration of EUR 91,200 can be calculated as follows: Maximum loss = S0 - X þ p0 = 60, 000 × ðEUR 74:25 - EUR 76 þ EUR 3:27Þ = EUR 91, 200: The breakeven share price at expiration is EUR 77.52: Breakeven share price ST = S0 þ p0 = EUR 74:25 þ EUR 3:27 = EUR 77:52: The costs of the protective put strategy depend on the chosen strike price of the put option, on the one hand, and on the time to expiration of the option on the other. Hedging the price risk of a long equity security with an out-of-the-money put option with a low strike price is cheaper than an at-the-money or in-the-money put option with a high strike price. However, there is also less protection against the risk of loss,

552

15

Option Strategies

as the share price decline must be higher for the long put option with the low strike price to move in the money. A protective put strategy is often compared to an insurance policy. The option premium can be put on a par with the insurance premium. The lower the exercise price, the lower the option premium as well as the price protection. Transferred to an insurance policy, a low premium means a higher self-retention for the policyholder. Furthermore, options with a longer time to expiration are more expensive because there is a chance that the option will move in the money or even further in the money over a longer period of time. If the investor wants longer protection against a falling share price, put options with a longer time to expiration should be chosen, which are accordingly more expensive. If, for example, a portfolio manager forecasts that the share price will fall sharply as a result of a corporate event (e.g. negative earnings surprise), they can hedge the long equity position with a long put. Once the event occurs and the share price falls as a result, the protection is no longer necessary; the portfolio manager can therefore close out the long put option before expiration and thus recover part of the option premium paid from the proceeds of the sale. In addition, the corporate event can lead to an increase in stock price volatility, which has a positive effect on the put price and thus on the sales proceeds. Buying long put options on an ongoing basis to protect the long equity position against possible losses is a very expensive strategy that can destroy most of the longterm profit of an otherwise good investment. Therefore, a protective put strategy should be used in those situations where a fall in stock prices is expected during a certain time period.

15.6

Collar

In a protective put strategy, the downside risk of the stock is eliminated with the purchase of a put option. The costs of this hedging strategy may be financed with the sale of a call option whose strike price is higher than the share price and the strike price of the long put. Such an option strategy is known as a collar.8 In the case of a collar, the long equity position has a price floor that corresponds to the strike price of the long put. Should the share price fall below the strike price of the put, no further losses are incurred because the profit of the long put absorbs the loss of the long equity security. On the other hand, the upper price limit is given by the strike price of the short call. If the share price rises above the strike price, the short call incurs a loss that is offset by the profit of the long equity security. If the option premiums of the put and the call are equal, this strategy is referred to as a zero-cost collar.9 In order to calculate the profit/loss of the collar at expiration, the value of the strategy at expiration must first be determined, which equals the share price plus the intrinsic value of the put and minus the intrinsic value of the call (X1 < X2): 8 9

See Watsham 1998: Futures and Options in Risk Management, p. 177. See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 892.

15.6

Collar

553

V T = ST þ maxð0, X 1- ST Þ - maxð0, ST- X 2 Þ,

ð15:16Þ

where X1 = strike price of the put, and X2 = strike price of the call. The beginning costs of the strategy are made up of the share price paid plus the premium paid for buying the put minus the premium received from selling the call. The profit/loss of the collar at expiration is obtained by subtracting the beginning costs from the value of the strategy at expiration: Profit=loss = V T - S0 - p0 þ c0 :

ð15:17Þ

For example, the Mercedes-Benz Group stock is traded at a price per share of EUR 74.25 on 12 January 2018. To implement a collar strategy, a long put option expiring in March 2018 with a strike price of EUR 68 is used, which is traded on the Eurex at a price of EUR 0.50 (see Table 15.5). In addition, a call option with the same expiration date in March 2018 and a strike price of EUR 80 is employed. It is traded on the Eurex at a price of EUR 0.49 (see Table 15.5). If the price of the automobile share falls to EUR 65 on the expiration date of the two options, the strategy value is EUR 68: V T = EUR 65 þ max ðEUR 0, EUR 68 - EUR 65Þ - max ðEUR 0, EUR 65 - EUR 80Þ = EUR 68: The loss of the long stock below the strike price of EUR 3 is offset by the intrinsic value (exercise gain) of the long put of EUR 3, while the short call option expires worthless. Thus, the price floor of EUR 68 or the strike price of the long put represents the strategy value at expiration. The loss of EUR 6.26 at expiration can be determined as follows: Loss = EUR 68 - EUR 74:25 - EUR 0:50 þ EUR 0:49 = - EUR 6:26: The loss of EUR 6.26 consists of the share price decline to the strike price of the put of EUR 6.25 (= EUR 68 - EUR 74.25) and the net option premium paid of EUR 0.01 (= - EUR 0.50 + EUR 0.49). Figure 15.3 presents the profit/loss diagram of the collar strategy on the expiration date of the two options. It does not matter how far the share price falls below the strike price; the maximum possible loss of the collar is EUR 6.26. In general, the maximum loss at expiration can be determined with the following formula: Maximum loss = X 1 - S0 - p0 þ c0 ,

ð15:18Þ

554

15

(Profit/loss in EUR)

Option Strategies

Maximum profit of EUR 5.74

7 6 5 4

Collar

3 2 1 0 -1 0

10

20

30

40

50

60

70

80

90 100 110 120 130 140 (Share price at expiration in EUR)

-2 -3

Maximum loss of EUR 6.26

-4 -5 -6

Breakeven share price of EUR 74.26

-7 Fig. 15.3 Profit/loss diagram of the collar strategy at expiration (Source: Own illustration)

where X1 = strike price of the put. If, on the other hand, the price of the Mercedes-Benz Group share rises to EUR 83 when the two options expire, a profit at expiration of EUR 5.74 is realised: VT = EUR 83 þ max ðEUR 0, EUR 68 - EUR 83Þ - max ðEUR 0, EUR 83 - EUR 80Þ = EUR 80, Profit = EUR 80 - EUR 74:25 - EUR 0:50 þ EUR 0:49 = EUR 5:74: The profit at expiration of EUR 5.74 consists of the share price increase up to the strike price of the call of EUR 5.75 (= EUR 80 - EUR 74.25) minus the net option premium paid of EUR 0.01. The profit of EUR 5.74 is at the same time the maximum possible profit of the strategy, because the share price increase is limited to the strike price of the call. In general, the maximum profit of the collar at expiration can be determined with the following formula: Maximum profit = X 2 - S0 - p0 þ c0 ,

ð15:19Þ

15.6

Collar

555

Table 15.8 Value, profit/loss, maximum loss, maximum profit, and breakeven share price of the collar strategy at expiration (Source: Own illustration) Collar Value at expiration VT Profit/loss at expiration Maximum loss at expiration Maximum profit at expiration Breakeven share price at expiration ST

Formula ST + max (0, X1 - ST) - max (0, ST - X2) VT - S0 - p0 + c0 X1 - S0 - p0 + c0 X2 - S0 - p0 + c0 S0 + p0 - c0

where X2 = strike price of the call. The breakeven share price at expiration equals the share price at the inception of the strategy plus the premium paid for the put and minus the premium received for the call: Breakeven share price ST = S0 þ p0 - c0 :

ð15:20Þ

In the example, the breakeven share price is EUR 74.26 (= EUR 74.25 + EUR 0.50 - EUR 0.49). Table 15.8 summarises the formulas that can be applied to calculate for the collar strategy the value, profit/loss, maximum loss, maximum profit, and breakeven share price at expiration. Example: Collar Strategy A portfolio manager holds an equity position consisting of 20,000 shares of Siemens AG. The Siemens share is traded at a price of EUR 122.50 on 15 January 2018. They want to hedge the downside risk of the equity position and buy 200 European put option contracts on the Eurex with an expiration date of June 2018 and an exercise price of EUR 110, at a price per put of EUR 2.47. In order to partially finance the purchase of the put options, the portfolio manager sells 200 European call option contracts on the Eurex expiring in June 2018 with an exercise price of EUR 130 at a price per call of EUR 1.79. 1. What is the profit/loss of the collar strategy at expiration if the share price of the Siemens stock is EUR 100? 2. What is the maximum profit, maximum loss, and breakeven share price of the collar strategy at expiration? Solution to 1 The value of the collar strategy at expiration is EUR 2,200,000: (continued)

556

15

Option Strategies

V T = 20, 000 × EUR 100 þ 20, 000 × ½max ðEUR 0, EUR 110 - EUR 100Þ - ½max ðEUR 0, EUR 100 - EUR 130Þ = EUR 2, 200, 000: The loss at expiration is EUR 263,600 and can be calculated as follows: Loss = EUR 2, 200, 000 - 20, 000 × ðEUR 122:50 þ EUR 2:47 - EUR 1:79Þ = - EUR 263, 600: Solution to 2 The maximum profit, maximum loss, and breakeven share price at expiration can be determined as follows: Maximum profit = X 2 - S0 - p0 þ c0 = 20, 000 × ðEUR 130 - EUR 122:50 - EUR 2:47 þ EUR 1:79Þ = EUR 136, 400, Maximum loss = X 1 - S0 - p0 þ c0 = 20, 000 × ðEUR 110 - EUR 122:50 - EUR 2:47 þ EUR 1:79Þ = - EUR 263, 600, Breakeven share price ST = S0 þ p0 - c0 = EUR 122:50 þ EUR 2:47 - EUR 1:79 = EUR 123:18: With a collar strategy, an upper and a lower price limit of the stock are set. Therefore, this strategy makes it possible to reduce the risk of a loss, while the profit potential is capped in return. A collar is appropriate if the investor expects the share price to fall. In this case, they can finance the purchase of the put option partly or entirely with the proceeds from the sale of the call option. In contrast to a protective put, however, they give up the unlimited upside price potential of the equity security.

15.7

Bull and Bear Spreads

A spread is an option strategy that involves buying and selling call and put options that are based on the same underlying asset but have different strike prices or times to expiration.10 If the options differ by time to expiration, the spread is called a time spread. This strategy exploits differences in the price volatility of the underlying as perceived by investors in the market. By contrast, money spreads refer to options

10

See Black 1975: ‘Fact and fantasy in the use of options’, p. 39.

15.7

Bull and Bear Spreads

557

with different strike prices. These include bull and bear spreads. In these spread strategies, an option with a specific expiration date and strike price is bought and another option with the same expiration but different strike price is sold. Both options have the same underlying asset. The term ‘spread’ is used because the value of the strategy at expiration is based on the difference, or spread, between the two strike prices.11 Bull and bear spreads are examined below.

15.7.1 Bull Spread With a bull spread, investors earn money when the price of the underlying rises by a predicted amount.12 The strategy can be implemented with either calls or puts. If calls are used, a call option with a lower strike price (X1) is bought and another call option with a higher strike price (X2) is sold at the same time. The value of the bull call spread strategy at expiration is equal to the intrinsic value of the long call option minus the intrinsic value of the short call option (X1 < X2): V T = maxð0, ST- X 1 Þ - maxð0, ST- X 2 Þ,

ð15:21Þ

where X1 = strike price of the long call option, and X2 = strike price of the short call option. The call option with the lower strike price is more expensive than a call with a higher strike price. Since a call with a lower strike price is bought and a call with a higher strike price is sold, the option premium paid is greater than the option premium received, and therefore the implementation of the bull call spread strategy causes a cash outflow. The profit/loss on the expiration date of the two options is the value of the strategy at expiration minus the net option premium paid: Profit=loss = V T - c1 þ c2 ,

ð15:22Þ

where c1 = price of long call option with lower strike price X1, and c2 = price of short call option with higher strike price X2. The profit/loss of the strategy at expiration can also be determined with the following formula by subtracting the profit/loss of the short call from the profit/ loss of the long call:

11 12

See Chance 2003: Analysis of Derivatives for the CFA® Program, p. 430. See Figlewski 1990: ‘Basic Price Relationships and Basic Trading Strategies’, p. 63.

558

15

Option Strategies

(Profit/loss in EUR) 8 7 6 5 4 3 2 1 0 -1 50 -2 -3 -4 -5 -6 -7 -8

Maximum profit of EUR 4.48

Bull call spread

55

60

65

70

75

X1 Maximum loss of EUR 1.52

80

85

90 95 100 (Share price at expiration in EUR)

X2

Breakeven share price of EUR 75.52

Fig. 15.4 Profit/loss diagram of the bull call spread strategy at expiration (Source: Own illustration)

Profit=loss = maxð0, ST- X 1 Þ - c1 - maxð0, ST- X 2 Þ þ c2 :

ð15:23Þ

For example, on 12 January 2018, an investor expects that the price of the Mercedes-Benz Group share will rise from EUR 74.25 to approximately EUR 80 at the expiration date of the February equity options contracts on the Eurex. Therefore, they buy a European long call option expiring in February 2018 with a strike price of EUR 74. The call premium is EUR 1.74 (see Table 15.5). To reduce the cost of the long call, they sell a European call option expiring in February 2018 with a strike price of EUR 80. The premium received from the sale of the call is EUR 0.22 (see Table 15.5). As a result, the net option premium paid for the bull call spread strategy is EUR 1.52 (= EUR 1.74 - EUR 0.22). Figure 15.4 presents the profit/loss diagram of the bull call spread strategy at expiration. The combined long/short call option position limits the loss, on the one hand, and the profit on the other. If, for example, the price of the automobile share is EUR 82 on the expiration date of the two options, the value of the strategy and the profit at expiration will be EUR 6 and EUR 4.48, respectively:

15.7

Bull and Bear Spreads

559

V T = max ðEUR 0, EUR 82 - EUR 74Þ - max ðEUR 0, EUR 82 - EUR 80Þ = EUR 6, Profit = EUR 6 - EUR 1:74 þ EUR 0:22 = EUR 4:48: If the share price falls below the strike price X1 on the expiration date, both call options expire worthless. The value of the strategy at expiration is zero, while the loss is equal to the net option premium paid. Hence, the maximum loss at expiration can be determined with the following equation: Maximum loss = c1 - c2 :

ð15:24Þ

In the example, the maximum possible loss at expiration is EUR 1.52 (= EUR 1.74 - EUR 0.22). In the event that the share price on the expiration date is above the strike price of the short call of X2, both calls expire in the money. Above the strike price of X2, the profit of the long call and the loss of the short call cancel each other out. Should the share price at expiration be greater than X1 but less than X2, the long call option expires in the money and the short call option expires out of the money. The short call remains out of the money until the share price does not reach the strike price X2. Accordingly, the maximum profit at expiration can be calculated by subtracting the net option premium paid from the difference between the strike prices X2 and X1: Maximum profit = X 2 - X 1 - c1 þ c2 :

ð15:25Þ

In the example, the maximum possible profit at expiration for the investor is EUR 4.48 (= EUR 80 - EUR 74 - EUR 1.74 + EUR 0.22). The profit area is reached when the share price exceeds the strike price X1 of the long call by the amount of the net option premium paid. Thus, the breakeven share price at expiration can be determined as follows: Breakeven share price ST = X 1 þ c1 - c2 :

ð15:26Þ

For the investor in the example, this results in a breakeven share price of EUR 75.52 (= EUR 74 + EUR 1.74 - EUR 0.22). Accordingly, the share price must increase by 1.71% to reach breakeven. Should the investor, on the other hand, choose a long call with the same expiration date and a strike price of EUR 76, the option premium is lower and is EUR 0.92 (see Table 15.5). However, the breakeven share price increases to EUR 76.70 (= EUR 76 + EUR 0.92 - EUR 0.22). Thus, the share price must increase by 3.3% in order to break even. There is no rule for the choice of the exercise price. Rather, the level of the selected strike price depends on the investor’s risk preferences. Table 15.9 summarises the formulas for calculating the strategy value, profit/loss, maximum loss, maximum profit, and breakeven share price at expiration.

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Table 15.9 Value, profit/loss, maximum loss, maximum profit, and break-even share price of the bull call spread strategy at expiration (Source: Own illustration) Bull call spread Value at expiration VT Profit/loss at expiration Maximum loss at expiration Maximum profit at expiration Breakeven share price at expiration ST

Formula max(0, ST - X1) - max (0, ST - X2) VT - c1 + c2 c1 - c2 X2 - X1 - c1 + c2 X1 + c1 - c2

Example: Bull Call Spread Strategy An investor expects the share price of Bayer AG stock to reach EUR 115 in the next few weeks and then to remain at this price level. The Bayer stock is trading at a price per share of EUR 103.50 on 17 January 2018. The investor wants to use a bull call spread to implement his price expectation. Therefore, they buy 100 European long call options contracts on the Eurex expiring in March 2018 with an exercise price of EUR 100 at a price per call of EUR 5.87. At the same time, they sell 100 European call options contracts with the same expiration date and an exercise price of EUR 115 at a price per call of EUR 0.66. 1. What is the profit/loss of the bull call spread strategy at expiration if the Bayer share price is EUR 95 and EUR 120? 2. What is the maximum profit, maximum loss, and breakeven share price of the bull call spread strategy at expiration? Solution to 1 With a Bayer share price of EUR 95, the value and the loss of the bull call spread strategy at expiration can be calculated as follows: V T = 10, 000 × ½max ðEUR 0, EUR 95 - EUR 100Þ - max ðEUR 0, EUR 95 - EUR 115Þ = EUR 0, Loss = EUR 0 - 10, 000 × ðEUR 5:87 - EUR 0:66Þ = - EUR 52, 100: Since both call options expire worthless, there is a loss to the extent of the net option premium paid of EUR 52,100. If the share price is EUR 120 at expiration of the two call options, the value and the profit are EUR 150,000 and EUR 97,900, respectively: (continued)

15.7

Bull and Bear Spreads

561

V T = 10, 000 × ½max ðEUR 0, EUR 120 - EUR 100Þ - max ðEUR 0, EUR 120 - EUR 115Þ = EUR 150, 000, Profit = EUR 150, 000 - 10, 000 × ðEUR 5:87 - EUR 0:66Þ = EUR 97, 900: Solution to 2 The maximum profit, maximum loss, and breakeven share price on the expiration date of the two options can be determined as follows: Maximum profit = X 2 - X 1 - c1 þ c2 = 10, 000 × ðEUR 115 - EUR 100 - EUR 5:87 þ EUR 0:66Þ = EUR 97, 900, Maximum loss = c1 - c2 = 10, 000 × ðEUR 5:87 - EUR 0:66Þ = EUR 52, 100, Breakeven share price ST = X 1 þ c1 - c2 = EUR 100 þ EUR 5:87 - EUR 0:66 = EUR 105:21: The profit/loss pattern at expiration of a bull spread is similar to that of a collar. Both the profit and loss potential are limited. The profit of the strategy arises from price movements of the underlying between the strike prices of the two options. Bull call spreads are used by investors who expect price increases of the underlying up to the strike price of the short call. Conversely, bear spreads are employed by investors who assume that the price of the underlying will fall by a predicted amount. Thus, spreads are bets that the price of the underlying will move by a certain amount in one direction or the other.13

15.7.2 Bear Spread If a call with a lower strike price is sold and another call with a higher strike price is bought, the opposite results of a bull spread are achieved. The profit/loss graph at expiration is completely reversed with the result that the profit is on the downside and the loss is on the upside. This strategy is called a bear call spread and can be applied when the share price is predicted to fall by a certain amount. However, it is more intuitive to use puts rather than calls for a bear spread since puts increase in value when the share price goes down. This involves buying a put with a higher strike price and selling another put with the same expiration date and underlying at a lower strike price. When implementing this strategy, it is assumed that the share

13

See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 1000.

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15

Option Strategies

price does not fall below the strike price of the short put.14 The value at expiration of the bear put spread strategy can be calculated by subtracting the intrinsic value of the short put from the intrinsic value of the long put (X1 < X2): V T = maxð0, X 2- ST Þ - maxð0, X 1- ST Þ,

ð15:27Þ

where X1 = strike price of the short put option, and X2 = strike price of the long put option. The higher the strike price, the higher the put price. Therefore, the premium paid for the long put exceeds that for the short put, with the result that a net option premium is paid for implementing the strategy. The profit/loss of the bear put spread strategy at expiration can be calculated by deducting the net option premium paid from the value of the strategy at expiration: Profit=loss = V T - p2 þ p1 ,

ð15:28Þ

where p1 = price of short put option with lower strike price X1, and p2 = price of long put option with higher strike price X2. Alternatively, the profit/loss at expiration of the bear put spread strategy can be determined with the following formula, in which the profit/loss of the short put is subtracted from the profit/loss of the long put: Profit=loss = maxð0, X 2- ST Þ - p2 - maxð0, X 1- ST Þ þ p1 :

ð15:29Þ

For example, should an investor expect on January 12, 2018 that the price of the Mercedes-Benz Group share will fall from EUR 74.25 to approximately EUR 68 at the expiration date of the February equity option contracts on the Eurex, they can buy a European put option expiring in February 2018 with a strike price of EUR 76 at a price per put of EUR 2.76 and simultaneously sell a European put option expiring in February 2018 with a strike price of EUR 68 at a price per put of EUR 0.24 (see Table 15.5). The net option premium paid is EUR 2.52 (= EUR 2.76 - EUR 0.24). The costs of the strategy are lower than a long put, but the investor gives up the profit opportunity of a share price decline below the strike price of the short put. Thus, both the profit and the loss are limited. Figure 15.5 presents the profit/loss diagram of the bear put spread strategy at expiration. If, for example, the price of the automobile share on the expiration date of the two options is EUR 65, the value and the profit of the strategy at expiration are EUR 8 and EUR 5.48, respectively: 14

See Hull 2006: Options, Futures, and Other Derivatives, p. 227 f.

15.7

Bull and Bear Spreads

(Profit/loss in EUR) 8

563

Maximum profit of EUR 5.48 Bear put spread

6 4 2 0 50

55

60

65

70

75

-2 -4 -6

X1

80 85 90 95 100 (Share price at expiration in EUR)

X2 Breakeven share price of EUR 73.48

Maximum loss of EUR 2.52

-8 Fig. 15.5 Profit/loss diagram of the bear put spread strategy at expiration (Source: Own illustration)

V T = max ðEUR 0, EUR 76 - EUR 65Þ - max ðEUR 0, EUR 68 - EUR 65Þ = EUR 8, Profit = EUR 8 - EUR 2:76 þ EUR 0:24 = EUR 5:48: Should the share price rise above the strike price X2 on the expiration date, both put options expire worthless. The value of the bear put spread strategy at expiration is zero, while the loss is given by the net option premium paid. Hence, the maximum loss at expiration can be determined as follows: Maximum loss = p2 - p1 :

ð15:30Þ

In the example, the maximum possible loss is EUR 2.52 (= EUR 2.76 EUR 0.24). If the share price on the expiration date is below the strike price of the short put of X1, both puts expire in the money. Below the strike price of X1, the profit of the long put and the loss of the short put cancel each other out. Should the share price be less than X2 but still greater than X1, the long put option expires in the

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Table 15.10 Value, profit/loss, maximum loss, maximum profit, and breakeven share price of the bear put spread strategy at expiration (Source: Own illustration) Bear put spread Value at expiration VT Profit/loss at expiration Maximum loss at expiration Maximum profit at expiration Breakeven share price at expiration ST

Formula max(0, X2 - ST) - max (0, X1 - ST) VT - p2 + p1 p2 - p1 X2 - X1 - p2 + p1 X2 - p2 + p1

money, while the short put option expires out of the money. The short put remains out of the money until the share price does not fall below the strike price X1. Hence, the maximum profit at expiration can be determined by subtracting the net option premium paid from the difference between the two strike prices X2 and X1: Maximum profit = X 2 - X 1 - p2 þ p1 :

ð15:31Þ

In the example, the maximum possible profit for the investor is EUR 5.48 (= EUR 76 - EUR 68 - EUR 2.76 + EUR 0.24). The profit area is reached when the share price falls below the strike price X2 of the long put by the amount of the net option premium paid. At this share price, the profit of the long put is offset by the net option premium paid for the strategy. The breakeven share price at expiration can be calculated as follows: Breakeven share price ST = X 2 - p2 þ p1 :

ð15:32Þ

In the example, the breakeven share price is EUR 73.48 (= EUR 76 EUR 2.76 + EUR 0.24). Thus, the share price must fall by 1.04% in order to break even. If, on the other hand, the investor chooses a long put with the same expiration date but with a lower strike price of EUR 74, the option premium is cheaper at EUR 1.58 (see Table 15.5). However, the breakeven share price at expiration falls to EUR 72.66 (= EUR 74 - EUR 1.58 + EUR 0.24). Therefore, the share price must drop by 2.14% in order to break even. The amount of the selected strike price depends on the investor’s risk preferences, analogous to the bull spread. Table 15.10 presents the formulas for calculating the value, profit/loss, maximum loss, maximum profit, and breakeven share price of the bear put strategy at expiration. Example: Bear Put Spread Strategy The stock of Siemens AG is traded at a price per share of EUR 122.70 on 18 January 2018. An investor expects the Siemens stock to drop to a share price of approximately EUR 110 at the expiration date of the March equity options contracts traded on the Eurex. The investor wants to implement their price forecast with a bear put spread. Therefore, they buy 200 European long (continued)

15.7

Bull and Bear Spreads

565

put options contracts on the Eurex expiring in March 2018 with an exercise price of EUR 125 at a price per put of EUR 7.55. They simultaneously sell 200 European put options contracts with the same expiration date and an exercise price of EUR 110 at a price per put of EUR 0.95. 1. What is the profit/loss of the bear put spread strategy at expiration if the price of the Siemens share is EUR 105 and EUR 130? 2. What is the maximum profit, maximum loss, and breakeven share price of the bear put strategy at expiration? Solution to 1 With a Siemens share price of EUR 105, the value and the profit of the bear put spread strategy at expiration can be determined as follows: V T = 20, 000 × ½max ðEUR 0, EUR 125 - EUR 105Þ - 20, 000 × ½max ðEUR 0, EUR 110 - EUR 105Þ = EUR 300, 000, Profit = EUR 300, 000 - 20, 000 × ðEUR 7:55 - EUR 0:95Þ = EUR 168, 000: If the share price is EUR 130, the value and the loss of the strategy at expiration are EUR 0 and EUR 132,000, respectively: V T = 20, 000 × ½max ðEUR 0, EUR 125 - EUR 130Þ - 20, 000 × ½max ðEUR 0, EUR 110 - EUR 130Þ = EUR 0, Loss = EUR 0 - 20, 000 × ðEUR 7:55 - EUR 0:95Þ = - EUR 132, 000: As both put options expire worthless, there is a loss to the extent of the net option premium paid of EUR 132,000. Solution to 2 The maximum profit, maximum loss, and breakeven share price on the expiration date of the two options can be calculated as follows: Maximum profit = X 2 - X 1 - p2 þ p1 = 20, 000 × ðEUR 125 - EUR 110 - EUR 7:55 þ EUR 0:95Þ = EUR 168, 000, Maximum loss = p2 - p1 = 20, 000 × ðEUR 7:55 - EUR 0:95Þ = EUR 132, 000, Breakeven share price ST = X 2 - p2 þ p1 = EUR 125 - EUR 7:55 þ EUR 0:95 = EUR 118:40:

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Option Strategies

15.7.3 Spread Strategy with Volatile Share Prices A spread strategy is suitable for a volatile equity security in a trend market.15 For example, on 12 January 2018 an investor buys a European call option on the Mercedes-Benz Group stock expiring in March 2018, with a strike price of EUR 76, at a price per call of EUR 1.41 (see Table 15.5). On this day, the Mercedes-Benz Group share is trading at a price of EUR 74.25. Assuming that the share price of the automobile stock rises to EUR 83 on 18 January 2018, the investor believes that the share price will not increase any further and they therefore sell a European call option expiring in March 2018, with an exercise price of EUR 84, at a price per call of EUR 1.85. The investor therefore owns a 76/84 bull call spread expiring in March 2018. Five days later, that is, on 23 January 2018, the MercedesBenz Group share price falls to EUR 75. The call with a strike price of EUR 76 trades at a price of EUR 1.71, while the call with a strike price of EUR 84 trades at a price of EUR 0.14. The investor decides to close out the short call with a strike price of EUR 84. Thus, they pay a premium of EUR 0.14 (long call) to offset the short call position. However, they sold the call option 5 days ago at a price of EUR 1.85, and they therefore make a profit of EUR 1.71 (= EUR 1.85 – EUR 0.14). They still own the long call option with a strike price of EUR 76. After another 5 days, that is, on 28 January 2018, the share price rises to EUR 84. The price of the call option expiring in March 2018 with a strike price of EUR 84 is now EUR 2.15. The investor sells the call option at this price, thus restoring the 76/84 bull call spread expiring in March 2018. In total, the outgoing cash is EUR 1.55 (= EUR 1.41 + EUR 0.14) and the incoming cash is EUR 4 (= EUR 1.85 + EUR 2.15), resulting in a profit of EUR 2.45. The option purchases and sales can be summarised as follows: Date 12 January 2018 18 January 2018 23 January 2018 28 January 2018 Total Profit

Transaction Buy call expiring in March 2018 with X = EUR 76 Sell call expiring in March 2018 with X = EUR 84 Buy call expiring in March 2018 with X = EUR 84 Sell call expiring in March 2018 with X = EUR 84

Outgoing cash (in EUR) 1.41

Incoming cash (in EUR)

1.85 0.14 2.15 1.55 2.45

4.00

Spreads can be used not only to earn a profit from expectations of rising and falling prices, but also to profit from changes in volatility levels. There is no certainty that a predicted share price will be reached or that the price of a stock will remain volatile. Nevertheless, knowledgeable investors can make profits at a relatively low

15

See Figlewski 1990: ‘Basic Price Relationships and Basic Trading Strategies’, p. 64.

15.8

Straddle

567

risk with a spread strategy if they correctly anticipate not only the price direction but also the price volatility of the underlying.

15.8

Straddle

15.8.1 Long Straddle A straddle differs from the option strategies presented so far because it is neither bullish nor bearish. A long call is used when the price of the underlying is expected to rise. A long put, on the other hand, is entered into when it is assumed that the price of the underlying will fall. If the investor does not have a clear opinion about the price direction of the underlying and expects the price volatility of the underlying to increase, a long straddle can be applied.16 In this case, the investor buys a call and a put with the same exercise price and expiration date on the same underlying. With this strategy, a profit is made when the price of the underlying asset moves up and down by more than the option premiums paid. The higher the price movement in one direction or the other, the higher the profit. High volatility therefore increases the profit of the option strategy. However, the costs of the long straddle are relatively high because a call and a put option must be purchased to implement the strategy. The costs from the option purchase are not reduced by the sale of an option, as is the case with a collar or a spread. Hence, a significant price movement of the underlying asset must take place for a profit to be made.17 The options chosen for the strategy usually have a strike price that is close to the current price of the underlying. If the underlying price exceeds or falls below the strike price at the expiration date of the two options, either the call or the put expires in the money. The value of the long straddle at expiration is determined by adding the intrinsic value of the long put to the intrinsic value of the long call: V T = maxð0, ST- X Þ þ maxð0, X- ST Þ:

ð15:33Þ

If the underlying price is below the strike price, ST < X, on the expiration date of the two long options, the call option expires worthless, and therefore the value of the straddle is equal to the intrinsic value of the put of X - ST. On the other hand, if the underlying price exceeds the strike price, ST > X, when both options expire, the put option expires out of the money and the call option in the money. In this price scenario, the value of the straddle is given by the intrinsic value of the call of ST - X. To calculate the profit/loss on the expiration date of the two options, the costs of the long straddle—that is, the premiums paid for the call and the put—must be deducted from the value of the strategy at expiration:

16 17

See Figlewski 1990: ‘Basic Price Relationships and Basic Trading Strategies’, p. 66. See Hull 2006: Options, Futures, and Other Derivatives, p. 234 f.

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15

Option Strategies

Profit=loss = V T - c0 - p0 :

ð15:34Þ

For example, the Mercedes-Benz Group stock is traded at a price per share of EUR 74.25 on 12 January 2018. An investor expects that stock price volatility will increase. Therefore, they buy a long call and a long put with an expiration date of March 2018 and a strike price of EUR 74. The premiums paid for the call and the put amount to EUR 2.38 and EUR 2.14, respectively (see Table 15.5). Thus, the costs of the long straddle amount to EUR 4.52. If the share price of the automobile stock rises to EUR 90 on the expiration date of the two options, the value of the strategy at expiration is EUR 16: V T = max ðEUR 0, EUR 90 - EUR 74Þ þ max ðEUR 0, EUR 74 - EUR 90Þ = EUR 16: The profit of EUR 11.48 is obtained by deducting the costs for the purchase of the call and the put from the value at expiration: Profit = EUR 16 - EUR 2:38 - EUR 2:14 = EUR 11:48: If, on the other hand, the price of the Mercedes-Benz Group share falls to EUR 58 on the expiration date of the two options, the value and profit of the long straddle are EUR 16 and EUR 11.48, respectively: V T = max ðEUR 0, EUR 58 - EUR 74Þ þ max ðEUR 0, EUR 74 - EUR 58Þ = EUR 16, Profit = EUR 16 - EUR 2:38 - EUR 2:14 = EUR 11:48: In the example, the share price moves by the same amount of plus/minus EUR 16 around the strike price of EUR 74, resulting in a profit of EUR 11.48 regardless of the direction of the price change. Thus, the profit/loss pattern of the long straddle is symmetric. Figure 15.6 presents the profit/loss diagram of the long straddle strategy at expiration. With a long call, the maximum profit is unlimited, as there is no upper price limit for the stock. If the share price rises, the maximum profit of the long straddle is unlimited: Maximum profit in the event of a share price increase = 1:

ð15:35Þ

Conversely, the maximum profit of a long put is limited because the share price cannot fall below zero. As a result, the maximum possible profit in the case of a share price drop is given by the strike price minus the option premiums paid for the call and the put:

15.8

Straddle

569

(Profit/loss in EUR) 80

Long straddle

70 60 50 40 30 20 10 0 0 -10

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 (Share price at expiration in EUR) X

Fig. 15.6 Profit/loss diagram of the long straddle strategy at expiration (Source: Own illustration)

Maximum profit in the event of a share price decrease = X - c0 - p0 :

ð15:36Þ

In the example, the maximum profit in the presence of a drop in the share price is EUR 69.48 (= EUR 74 - EUR 2.38 - EUR 2.14). The maximum loss of the long straddle occurs if, on the expiration date of both options, the share price is equal to the strike price. In this price scenario, both the call and the put expire worthless. Accordingly, the maximum loss of the strategy consists of the option premiums paid: Maximum loss = c0 þ p0 :

ð15:37Þ

In the example, the maximum loss is EUR 4.52 (= EUR 2.38 + EUR 2.14). In a long straddle, there are two breakeven share prices at expiration. In order to break even, the share price must rise more than the option premiums paid, with the result that the first breakeven share price is given by the strike price plus the option premiums paid for the two options. By contrast, when the share price falls, breakeven is reached if the drop in price exceeds the option premiums paid. Thus, the second breakeven share price equals the strike price minus the costs for the two

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Option Strategies

Table 15.11 Value, profit/loss, maximum profit, maximum loss, and breakeven share prices of the long straddle strategy at expiration (Source: Own illustration) Long Straddle Value at expiration VT Profit/loss at expiration Maximum profit in the event of a share price increase at expiration Maximum profit in the event of a share price decline at expiration Maximum loss at expiration Breakeven share price at expiration ST

Formula max(0, ST - X) + max (0, X S T) VT - c0 - p0 1 X - c0 - p0 c 0 + p0 X ± (c0 + p0)

options. In general, the formula for calculating the breakeven share price at expiration is as follows: Breakeven share price ST = X ± ðc0 þ p0 Þ:

ð15:38Þ

In the example, the breakeven share prices of the long straddle amount to EUR 78.52 (= EUR 74 + EUR 2.38 + EUR 2.14) and EUR 69.48 (= EUR 74 EUR 2.38 - EUR 2.14). Should the share price at expiration of the strategy be between the two breakeven share prices of EUR 69.48 and EUR 78.52, the long straddle does not produce a profit. The price of the Mercedes-Benz Group share must rise by more than 5.75% or fall by more than 6.42% during the lifetime of the options for the strategy to realise a profit. Should the investor assume that such a price movement is unlikely, they can enter into a short straddle.18 Table 15.11 summarises the formulas for calculating the value, profit/loss, maximum profit, maximum loss, and breakeven share prices of the long straddle at expiration. Buying two options instead of one increases the initial cost. To make a profit from the investment, stock price movements must be greater than buying one option with a price forecast of the security in one direction. Looked at this way, a long straddle is a volatility play as prices must move strongly one way or the other.19 A long straddle can be considered when important corporate events occur, such as the announcement of a company’s quarterly earnings. If the level of the company’s earnings is uncertain due to the course of business, the stock price volatility increases. This leads to rising option prices on the market, and therefore the implementation of the strategy becomes more expensive. Consequently, using a straddle while waiting for an event that is already anticipated by market participants is not a promising strategy. A long straddle is only a viable strategy if the investor expects volatility to be higher than the market consensus. If they are correct in their volatility assessment and the

18 19

See Sect. 15.8.2. See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 995.

15.8

Straddle

571

other market participants are wrong on average, it is possible to earn a profit with a long straddle.

15.8.2 Short Straddle A short straddle can be implemented by selling a call and a put with identical time to expiration and exercise price on the same underlying asset. In the event of large price movements or increased price volatility of the underlying, losses are incurred because either the call or the put moves in the money. A profit is made if the price of the underlying changes only slightly or price volatility decreases. The value of the short straddle strategy at expiration is made up of the intrinsic value of the short call and the short put: V T = - maxð0, ST- X Þ - maxð0, X- ST Þ:

ð15:39Þ

The profit/loss of the strategy at expiration is obtained by adding the premiums received from the sale of the call and the put to the value at expiration: Profit=loss = V T þ c0 þ p0 :

ð15:40Þ

If the underlying price and the strike price are the same on the expiration date of the two options, neither the call nor the put expires in the money, and the options are therefore not exercised. In this price scenario, the maximum possible profit of the short straddle at expiration is achieved and can therefore be calculated as follows: Maximum profit = c0 þ p0 :

ð15:41Þ

If the underlying price exceeds the strike price, the call option moves in the money. The maximum loss is unlimited on the short call if the underlying is an equity security because the share price has no upper limit: Maximum loss in the event of a share price increase = 1:

ð15:42Þ

Conversely, if the share price drops below the strike price, the short put option moves in the money. Since the share price cannot fall below zero, the maximum possible loss at expiration equals the strike price less the premiums received from the sale of the options: Maximum loss in the event of a share price decrease = X - c0 - p0 :

ð15:43Þ

The long and short straddle have the same breakeven share prices. A short straddle results in a profit as long as the share price neither falls below nor exceeds the strike price to the extent of the premiums received from the option sale. In general, the breakeven share price at expiration can be determined with the following equation:

572

15

Breakeven share price ST = X ± ðc0 þ p0 Þ:

Option Strategies

ð15:44Þ

Table 15.12 summarises the formulas for calculating the value, profit/loss, maximum profit, maximum loss, and breakeven share prices of the short straddle at expiration. Figure 15.7 presents the profit/loss diagram of the short straddle at expiration resulting from the short call and the short put on the Mercedes-Benz Table 15.12 Value, profit/loss, maximum profit, maximum loss, and breakeven share prices of the short straddle strategy at expiration (Source: Own illustration) Short straddle Value at expiration VT Profit/loss at expiration Maximum profit at expiration Maximum loss in the event of a share price increase at expiration Maximum loss in the event of a share price decrease at expiration Breakeven share price at expiration ST

(Profit/loss in EUR)

Formula - max (0, ST - X) - max (0, X ST) VT + c0 + p0 c0 + p0 1 X - c0 - p0 X ± (c0 + p0)

Short straddle

10 0 0 -10 -20

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 (Share price X at expiration in EUR)

-30 -40 -50 -60 -70 -80 Fig. 15.7 Profit/loss diagram of the short straddle strategy at expiration (Source: Own illustration)

15.8

Straddle

573

Group stock with expiration in March 2018 and the strike price of EUR 74 of the previous example. Example: Straddle Strategy The Linde AG stock is traded at a price per share of EUR 203.70 on 19 January 2018. An investor expects price volatility in Linde shares to increase over the next 9 weeks. They therefore buy 100 American call and 100 American put option contracts on the Eurex with an expiration date of March 2018 and an exercise price of EUR 200. The option premiums paid are EUR 11.31 for the call and EUR 7.31 for the put. 1. What is the profit/loss of the long straddle strategy at expiration if the price of Linde share is EUR 180 and EUR 220? 2. What is the maximum profit, maximum loss, and breakeven share price of the long straddle strategy at expiration? 3. What is the maximum profit, maximum loss, and breakeven share price of the short straddle strategy at expiration? Solution to 1 With a share price of EUR 180, the profit of the long straddle strategy at expiration of EUR 13,800 can be calculated as follows: V T = 10, 000 × ½max ðEUR 0, EUR 180 - EUR 200Þ þ 10, 000 × ½max ðEUR 0, EUR 200 - EUR 180Þ = EUR 200, 000, Profit = EUR 200, 000 - 10, 000 × ðEUR 11:31 þ EUR 7:31Þ = EUR 13, 800: If the share price is EUR 220, the profit on the expiration date of the two options is EUR 13,800: V T = 10, 000 × ½max ðEUR 0, EUR 220 - EUR 200Þ þ 10, 000 × ½max ðEUR 0, EUR 200 - EUR 220Þ = EUR 200, 000, Profit = EUR 200, 000 - 10, 000 × ðEUR 11:31 þ EUR7:31Þ = EUR 13, 800: The profit/loss pattern of a straddle is symmetrical. Solution to 2 The maximum profit, maximum loss, and breakeven share prices of the long straddle at expiration can be determined as follows: (continued)

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Maximum profit in the event of a share price increase = 1, Maximum profit in the event of a share price decrease = X - c0 - p0 = 10, 000 × ðEUR 200 - EUR 11:31 - EUR 7:31Þ = EUR 1, 813, 800, Maximum loss = c0 þ p0 = 10, 000 × ðEUR 11:31 þ EUR 7:31Þ = EUR 186, 200, 1st breakeven share price ST = X þ ðc0 þ p0 Þ = EUR 200 þ EUR 11:31 þ EUR 7:31 = EUR 218:62, 2nd breakeven share price ST = X - ðc0 þ p0 Þ = EUR 200 - EUR 11:31 - EUR 7:31 = EUR 181:38: Solution to 3 The maximum profit, maximum loss, and breakeven share prices of the short straddle at expiration can be calculated as follows: Maximum profit = c0 þ p0 = 10, 000 × ðEUR 11:31 þ EUR 7:31Þ = EUR 186, 200, Maximum loss in the event of a share price increase = 1, Maximum loss in the event of a share price decrease = X - c0 - p0 = 10, 000 × ðEUR 200 - EUR 11:31 - EUR 7:31Þ = EUR 1, 813, 800, 1st breakeven share price ST = X þ ðc0 þ p0 Þ = EUR 200 þ EUR 11:31 þ EUR 7:31 = EUR 218:62, 2nd breakeven share price ST = X - ðc0 þ p0 Þ = EUR 200 - EUR 11:31 - EUR 7:31 = EUR 181:38: The short straddle has the opposite profit/loss pattern to the long straddle. This is demonstrated in the maximum profit and loss. However, the breakeven share prices are equal. As already mentioned, long straddles are used by investors who expect an increased price volatility of the stock but do not know in which direction the price will move. If there is nevertheless a perception of the price direction, a long call or a long put can be added to the straddle. Buying a call option alongside the long straddle is called a strap. The share price is expected to move significantly, with a price increase considered more likely than a decrease. However, combining a long put with a long straddle is referred to as a strip. Again, the share price is expected to move substantially, with a price decrease being more likely than an increase. If the expected price direction occurs and price volatility rises, the profit increases

15.8

Straddle

575

accordingly.20 Another form of a straddle is a strangle. In a long strangle, a put and a call are bought with the same time to expiration and on the same underlying, but with different strike prices. The put has the lower strike price. Strike prices are often chosen so that both options are out of the money, which makes the options cheaper.21 The profit/loss pattern on the expiration date looks similar to that of a long straddle. However, the profit/loss line is flat between the two strike prices, because if the share price is between the two strike prices, both the put and the call expire out of the money and the loss is made up of the option premiums paid. Accordingly, a long strangle results in a profit when there is a large movement of the share price in one direction or the other and the price volatility increases. In contrast to a long straddle, the share price movement must be greater for a profit to be made. A short strangle is implemented by selling a put and a call with the same time to expiration and on the same underlying, but with different strike prices. This strategy might seem slightly less risky than a short straddle because the maximum profit occurs over a broader share price range.22

15.8.3 Breakeven Share Price and Volatility In order to assess the option strategy, investors often prepare a profit/loss chart. This chart can be used to evaluate the profit/loss of the strategy at different share prices. Moreover, option pricing theory can be applied to better understand the breakeven share price points in the chart. The option price depends on the expected price volatility of the underlying. The higher (lower) the volatility, the higher (lower) the option price. The implied volatility of an option can be calculated with the traded option price and an option pricing model such as the Black–Scholes model for European options.23 Options traders use implied volatility as a kind of option currency. For example, an option price of EUR 4 cannot be directly compared with prices of other options. Rather, one must examine whether the options are in the money, at the money, or out of the money and how long the corresponding times to expiration are. By contrast, implied volatility is a measure that enables a direct comparison with other options. A high (low) implied volatility warrants a higher (lower) option price.24 For example, the options on the Mercedes-Benz Group stock have an annualised implied volatility of 18%. The automobile share is traded at a price of EUR 74.25 on 12 January 2018. The long straddle is implemented with the assistance of a long call and a long put with an expiration date of March 2018 and a strike price of EUR 74.

20

See Hull 2006: Options, Futures, and Other Derivatives, p. 236. See Reilly and Brown 2003: Investment Analysis and Portfolio Management, p. 998. 22 See Figlewski 1990: ‘Basic Price Relationships and Basic Trading Strategies’, p. 68 ff. 23 For the calculation of implied volatility, see, for example, Mondello 2017: Finance: Theorie und Anwendungsbeispiele, p. 958 ff. 24 See Chance 2003: Analysis of Derivatives for the CFA® Program, p. 225 f. 21

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The premiums paid for the call and the put are EUR 2.38 and EUR 2.14, respectively (see Table 15.5). In order for the two breakeven share prices to be reached, the price of the Mercedes-Benz Group share must move up by 5.75% [= (EUR 74 + EUR 2.38 + EUR 2.14)/EUR 74.25 –1] and down by 6.42% [= (EUR 74 - EUR 2.38 - EUR 2.14)/EUR 74.25 –1]. The two options expire in 63 days. Excluding the weekends, there are 45 trading days until the expiration date. One year consists of approximately 252 trading days. Thus, the 45-day volatility of 5.75% required for a share price increase to break even can be annualised as follows, assuming that the stock price returns are uncorrelated and therefore, follow a random walk:25 σ Annualised = 0:0575 ×

252 days = 0:1361: 45 days

In the event of a share price decline, a 45-day volatility of 6.42% is necessary to break even. The annualised volatility is 15.19% and can be calculated with the equation below: σ Annualised = 0:0642 ×

252 days = 0:1519: 45 days

Since the implied volatility of 18% is greater than the volatilities required for a profit in the case of an increase and a decrease in the share price of 13.61% and 15.19%, it is worth implementing the long straddle strategy. This is because the 18% implied volatility can be expected to cause the price movements to be sufficiently large for the share price to exceed or fall below the breakeven share price on the expiration date. Long straddles and strangles are usually established shortly before an expected increase in stock price volatility; for example, around the time that certain pricesensitive public information is announced, such as economic data (e.g. inflation) or a corporate event (e.g. quarterly earnings, a court verdict, or a merger). It is important to consider that the implied volatility may have risen substantially in anticipation of greater volatility surrounding the announcement. In such cases, a short straddle may be profitable if the volatility falls after the announcement.26

15.9

Effects of Exercising Options on the Strategy

The seller of an option has the obligation to deliver (call) or receive (put) the underlying asset at the exercise price. In the case of American options, they have no control over whether or when the option is exercised. The effects of an exercise

25 26

See Sect. 3.3.2. See Watsham 1998: Futures and Options in Risk Management, p. 197.

15.10

Selection of the Option Strategy

577

can be considerable for the option seller, especially if it occurs unexpectedly and they are not prepared for it. For example, an investor implements a bull call spread strategy by buying an American equity call option with a strike price of EUR 80 and simultaneously selling an American call option with the same time to expiration and the same underlying at a strike price of EUR 90. Four weeks before the expiration date of the two options, the price of the underlying share rises to EUR 93. The strategy has reached its maximum value of EUR 10. The holder of the long American call with a strike price of EUR 90 decides to exercise the option and buy the underlying share at a price of EUR 90. If the investor in the bull call spread does not own the underlying, they must buy the stock on the market at a price per share of EUR 93. If the option contract consists of 100 options, a total of EUR 9300 (= 100 × EUR 93) is required. Alternatively, the investor in the bull call spread can exercise the long call option and buy the shares needed to fulfil the obligation of the short call at an exercise price of EUR 80. In doing so, the investor loses the time value of the call, since this option, instead of being exercised, could have been sold. If, for example, the price of the call option with an exercise price of EUR 80 is trading at a price of EUR 13.80, the intrinsic value is EUR 13 [= max(EUR 0, EUR 93 - EUR 80)] and the time value is EUR 0.80 (= EUR 13.80 – EUR 13). Exercising the call option, the investor acquires the share at a price of EUR 80, which has a market value of EUR 93, thus realising an exercise gain of EUR 13, which corresponds to the intrinsic value of the call. By contrast, the sale of the call option results in proceeds of EUR 13.80, which are higher than the earned exercise gain of EUR 13. If the investor buys the share on the market at a price of EUR 93 after closing out the call option, they pay a net price of EUR 79.20 (= EUR 93 EUR 13.80). Hence, it is cheaper for the investor to sell the call (i.e. close out the long call) and buy the share at the market price than to exercise the call option. Therefore, the investor should think about the implications of exercising an option before implementing an option strategy that includes short options.

15.10 Selection of the Option Strategy The risk of loss and the profit opportunity arising from derivatives depend on how they are used. For example, derivatives can be combined with other assets such as equity securities and fixed-income securities to obtain a desired risk profile. Examples include the covered call, the protective put, and the collar. Derivatives can also assist an investor or a portfolio manager to react quickly to changing market conditions or to respond to client requests. Regardless of their use, derivatives should always be employed with a clearly defined investment objective. Every trading transaction begins with a market and product analysis. If options are used in the trading strategy, it is not sufficient to predict only the price direction of the market and the price of the asset; price volatility must also be taken into account. For example, with a long call option on an equity security, it is not sufficient for the share price to rise in order to earn a profit. The expected price increase of the

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Table 15.13 Selection of option strategy based on expected price direction and price volatility (Source: Own illustration) Expected price direction

Expected price volatility

Rise Stable Fall

Strongly downwards (bearish view) Long puts Long puts and short calls Short calls

Moderate (neutral view) Long straddle Spreads Short straddle

Strongly upwards (bullish view) Long calls Long calls and short puts Short puts

security must exceed the option premium paid. The profit must be sufficiently high to compensate for the loss of time value of the call. The buyer of a straddle assumes an increase in price volatility of the underlying asset. It is not important whether the price of the underlying rises or falls. Thus, the holder of the long straddle is neutral with regard to the price direction. By contrast, the seller of a straddle expects a decrease in price volatility. Like the holder of the long straddle, they are neutral with regard to the price direction. In a spread strategy, an average volatility and an estimated up or down price movement of the underlying within a certain range are assumed. Hence, the underlying market is not clearly trending upward or downward. With a bullish view, a bull call spread may be used, and with a bearish view, a bear put spread. Furthermore, spreads make it possible to reduce total costs because they are constructed by buying one option and selling another. On the other hand, if a strong price increase and high price volatility are predicted, a call can be bought with this bullish view. The combination of a long call with a short put is appropriate in the case of a strong price increase and average price volatility. With this strategy, in the event of a price increase, a profit is made on the long call, the purchase of which is partially or entirely financed by the sale of the put. If a strong price decline and a high price volatility is forecast, a put can be bought in order to implement this bearish view. The combination of a long put with a short call is appropriate in the case of a strong price decline and average price volatility. Table 15.13 presents the selection of the option strategy, which depends on the expected price direction and volatility. The implemented strategy can be quickly adjusted at any time due to changing expectations. The covered call, the protective put, and the collar are not included in the table, as these are risk management strategies in which the risk exposure of a long underlying asset (e.g. an equity security) is changed with options.

15.11 Summary • A synthetic long equity security can be created according to put–call parity by entering into a long call, a long risk-free zero-coupon bond (lending money at the risk-free interest rate), and a short put. For this purpose, European options are used that have the same underlying, strike price, and time to expiration.

15.11

Summary

579

• A synthetic short equity security can be constructed with a short call, a short riskfree zero-coupon bond (borrowing money at the risk-free interest rate), and a long put. • A synthetic put consists of a long call, a short equity security, and a long risk-free zero-coupon bond. By contrast, a synthetic call is made up of a long put, a long equity security, and a short risk-free zero-coupon bond. • A covered call is a combination of a long equity security and a short call on that security. Writing covered calls makes it possible to earn income, to sell an overweighted equity position in the portfolio, or to sell shares when they reach a target price. • The maximum profit of a covered call at expiration is given by the difference between the strike price and the share price at the inception of the strategy plus the option premium received from the sale of the call. The maximum possible loss at expiration is incurred if the share price falls to zero. In this price scenario, the loss is equal to the beginning share price minus the option premium received. Thus, the risk of loss of the long equity security is reduced by the option premium collected from the sale of the call. However, the profit potential of the strategy is limited. In contrast to a long equity security, no profit is made if the share price rises above the strike price of the short call. The breakeven share price at expiration equals the share price at the inception of the strategy minus the option premium received. • A protective put is a long equity security combined with a long put on that security. This strategy provides the long equity position with loss protection. Consistently holding a protective put position is expensive and not optimal. Rather, this strategy should be applied when falling share prices are expected over a certain period of time. The maximum loss of the protective put at expiration is limited, while the maximum profit is unlimited because the share price has no upper limit. Unlike the case of a long equity security, the profit is reduced by the put premium paid. In order to reach breakeven at expiration of the strategy, the share price must rise to the extent of the option premium paid. • A collar is made up of a long equity security, a long put, and a short call. The strike price of the put is lower than that of the call. With the collar, the long stock has a lower and upper price limit, which are given by the strike prices of the put and the call. This limits the risk of loss of the equity security, but also the potential for profit. Unlike the case of a protective put, the unlimited profit potential of the stock is given up. However, the purchase of the put option to protect the downside risk of the underlying is partially or fully financed by the sale of the call option. This strategy is used when a price decline of the share is assumed. • The maximum loss of a collar at expiration is given by the difference between the strike price of the put and the share price at the inception of the strategy minus the net option premium paid. The maximum profit at expiration is also limited and equals the difference between the strike price of the call and the share price at the inception of the strategy minus the net option premium paid. Breakeven at expiration is reached when the share price rises to the extent of the net option premium paid.

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• An investor who buys an equity call option has an unlimited maximum profit, while the maximum loss is limited to the option premium paid. If they also sell a call option with a higher exercise price, the long call is converted into a bull call spread. This spread strategy limits the profit potential and reduces the costs compared to the long call strategy. Bull call spreads are used by investors who expect price increases up to the strike price of the short call. • The profit/loss pattern of a bull call spread is similar to that of a collar. Both the profit and loss potential are limited. If the share price falls below the strike price of the long call on the expiration date, both call options expire worthless, with the result that the maximum loss at expiration is equal to the net option premium paid. The maximum possible profit at expiration, on the other hand, occurs if the share price exceeds the strike price of the short call. It equals the difference between the two strike prices minus the net option premium paid. Breakeven is reached when the share price at expiration exceeds the strike price of the long call by the amount of the net option premium paid. • In a bear put spread, a put with a higher strike price is bought and another put with the same time to expiration and underlying is sold at a lower strike price. Investors do not expect the share price to fall below the strike price of the short put option. If the share price rises above the strike price of the long put on the expiration date, both options expire worthless, with the result that the maximum loss of the strategy corresponds to the net option premium paid. The maximum profit at expiration occurs when the share price falls below the strike price of the short put option and can be calculated by subtracting the net option premium paid from the difference between the two strike prices of the long and short put. The profit area is reached when the share price at expiration falls below the strike price of the long put to the extent of the net option premium paid. • A long straddle consists of a long call and a long put with the same strike price. Moreover, both options have the same time to expiration and are based on the same underlying asset. The risk of a long straddle is limited to the premiums paid to buy the call and the put. Both options expire worthless if the share price and the strike price are the same on the expiration date. At any other share price at expiration, either the call or the put expires in the money, and the value of the strategy is given by the intrinsic value of one of the two options. The profit area is entered when the share price deviates from the strike price by more than the option premiums paid. Both options lose their time value at expiration, and it therefore takes a significant movement in the price of the underlying to break even. A long straddle is used by investors when an increase in price volatility is expected. • A short straddle consists of a short call and a short put. Both options have the same strike price, the same time to expiration, and the same underlying. If the share price and the strike price are equal on the expiration date, both options expire worthless, and therefore the maximum possible profit consists of the premiums received from the sale of the call and the put. The risk of loss is caused by upward and downward movements in the share price. If the share price rises, the short call option moves in the money. Since the share has no upper price limit,

15.12

Problems

581

the maximum possible loss is unlimited. If, on the other hand, the share price falls, the short put option expires in the money. In this price scenario, the maximum loss is given by the strike price minus the premiums received from the option sale. The breakeven share prices at expiration are the same for the long and short straddle. Investors use a short straddle when they expect price volatility to fall. • If the option’s implied volatility is greater (less) than the volatility needed to break even, the long (short) straddle is worth implementing. • The choice of option strategy depends on the expected price direction and volatility of the underlying asset.

15.12 Problems 1. Peter Mueller, CFA, is a portfolio manager at an asset management company that manages the money of wealthy private clients. Alois Hofmann is a client of the asset management company. His portfolio is invested in equities. The investment policy is aggressive. Alois Hofmann’s cash reserves have declined sharply as a result of the high costs of living. Due to an unexpectedly high cash outlay of EUR 30,000, he contacts Peter Mueller, asking him to provide the required amount of money relatively quickly. The recently revised investment policy for Alois Hofmann’s portfolio allows the use of long options and the sale of covered options. Peter Mueller’s asset management company expects a stable, slightly falling equity market for the next 6 months. Alois Hofmann’s portfolio includes 8000 shares of Siemens AG, which were recently added. On 19 January 2018, the Siemens share is trading at a price of EUR 123.70. Peter Mueller is considering the use of European options traded on the Eurex and expiring in March 2018, which are listed below (in EUR): Call 3.79 1.81 0.80

Exercise price 120 125 130

Put 3.33 6.37 1.36

Source: https://www.eurex.com/ex-de/maerkte/equ/opt/Siemens-953546

Which option strategy should Peter Mueller recommend to the client?

2. Maximilian Wegmann, CFA, is a portfolio manager at an asset management firm. Among other things, he manages a position consisting of 20,000 shares in Linde AG for a wealthy private client. Linde shares are trading at EUR 200.60 on 22 January 2018. The Linde Group will publish its corporate results in 1 week. Maximilian Wegmann expects an increase in earnings, which will, however, be lower than the market consensus. Therefore, he wants to protect the long equity position against the risk of a price decline. The costs required for this hedge should be kept to a minimum. He is considering the use of American

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options on Linde shares traded on the Eurex and expiring in February 2018, which are listed below (in EUR): Call 15.90 8.61 3.84

Exercise price 190 200 210

Put 1.85 4.56 9.81

Source: https://www.eurex.com/ex-en/markets/equ/opt/Linde-945916

Which option strategy should Maximilian Wegmann use?

3. Peter Meier, CFA, is the managing director of an investment foundation. He is evaluating the purchase of a new equity position. However, he considers the share price to be too high. Since he intends to buy 100,000 shares, a price difference of a few euros on a share matters a lot. The equity security is listed at a price per share of EUR 100. Peter Meier is willing to pay a price of EUR 97.50 for the share. A put option with a time to expiration of 30 days and a strike price of EUR 102 is traded at a price of EUR 4.57. How can the put option be used to achieve Peter Meier’s target purchase price? 4. Claudia Berger, CFA, has been following the stock of Gamma AG for several months. The company is involved in a legal dispute, and Claudia Berger is considering the use of a long straddle. The Gamma share is traded at a price of EUR 50. She is considering the use of at-the-money call and put options with a time to expiration of 30 days, which are traded at a price of EUR 1.72 and EUR 1.70, respectively. After the close of trading, there is news that a verdict on the legal dispute will be announced in the course of the next day. Claudia Berger expects the share price to move up or down by 10% after the court decision. When the market opens the next day, she finds that the prices for the call and the put have risen to EUR 3.50 and EUR 3.48, respectively, while the stock is still trading at a price of EUR 50. What impact will these new option prices have on the planned long straddle strategy? 5. Sarah Baumeister, CFA, expects that the price of an equity security will rise from EUR 80 to EUR 87 in the next 30 days due to an unexpected event. Therefore, she is considering the use of call options with a time to expiration of 30 days. The price of a 30-day call option with a strike price of EUR 80 is EUR 2.74. On the other hand, the price of a 30-day call option with a strike price of EUR 87 is EUR 0.63. Which option strategy is suitable based on Sarah Baumeister’s price expectation? 6. Heinz Meier, CFA, is a portfolio manager at Rock Investment AG. He expects that the share price of Alpha AG could either fall or rise substantially due to a forthcoming court case verdict that might be positive or negative for the company. The Alpha share is currently trading at a price of CHF 31 (prior to the announcement of the court decision). Meier does not have any Alpha shares in his portfolio at present. Nevertheless, he would like to profit from the price

15.13

Solutions

583

movements of the share by implementing a long straddle. Meier has compiled the following data for the European options on the Alpha shares:

Price Strike price Time to expiration Annualised implied volatility Dividend yield Delta Gamma

Call option Not available CHF 30 60 days 28% 0% 0.635 0.107

Put option CHF 0.93 CHF 30 60 days 28% 0% -0.365 0.107

a) What is the price of the European call option according to put–call parity if the continuous compounded risk-free interest rate is 0.25%? b) Is the implementation of the long straddle strategy profitable assuming 60 days correspond to 43 trading days and 1 year consists of 252 trading days?

15.13 Solutions 1. In order to earn income, Mueller will recommend the sale of covered call options in accordance with the investment policy. One option contract comprises 100 options on the Eurex. The 8000 Siemens shares enable the sale of 80 call option contracts. The proceeds from the sale of the covered call options with a strike price of EUR 125 and EUR 130 are too low to finance the required amount of EUR 30,000. Therefore, the portfolio manager decides to sell the in-the-money call options with a strike price of EUR 120, resulting in proceeds of EUR 30,320 (= 80 × 100 × EUR 3.79). Since these call options are in the money, there is a high risk of an option exercise on the expiration date. However, based on the market forecast made by the asset management firm, this risk appears to be reasonable. 2. Since hedging costs are to be kept to a minimum, Wegmann should first consider selling covered call options. For example, if he sells in-the-money call options with a strike price of EUR 190, he receives a premium of EUR 15.90. The breakeven share price at expiration is EUR 184.70 (= EUR 200.60 EUR 15.90). If the Linde share price falls below EUR 184.70, a loss is incurred. On the other hand, if he sells a call option that is only slightly in the money, with a strike price of EUR 200, the breakeven share price is EUR 191.99 (= EUR 200.60 - EUR 8.61). In the case of a sale of covered call options, the risk of loss of the equity security is reduced by the option premium received, but

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in return the profit potential of the security is given up. If Wegmann’s forecasts are wrong and the company’s earnings are higher than expected by the market, any profit from a possible price increase is foregone. An alternative to the covered call is the protective put. For example, Wegmann can buy the put option with a strike price of EUR 190 with the intention of selling it shortly after the publication of the company’s earnings. If Wegmann’s earnings expectations are met, the Linde share price is likely to fall, causing the price of the put to rise. The profit from the sale of the put can be used to offset the loss on the long equity position. If the company’s published earnings are in line with the market consensus, the put can be sold at a price close to the purchase price. However, if the company’s published earnings are better than expected by the market, the share price will probably increase, which will lead to a decrease in the put price. In this situation, the hedge is no longer needed and the put can be sold to recover part of the purchase price. 3. In-the-money put options can be sold in order to reach the purchase price target of the share. If the share price is above the strike price on the expiration date, the put options are not exercised. Conversely, when the share price is below the strike price, the puts are exercised and the shares must be bought at the strike price. Meier sells 1000 put option contracts, resulting in proceeds of EUR 457,000 (= 1000 × 100 × EUR 4.57). The short put position, if exercised, represents the obligation to buy 100,000 shares at a price per share of EUR 102. For example, if the share price on the expiration date of the put option is EUR 100, the put option expires in the money and is exercised. Meier must buy 100,000 shares at a unit price of EUR 102. This incurs costs of EUR 10,200,000 (= 100,000 × EUR 102). Deducting from this the premium of EUR 457,000 received from the option sale, results in an effective purchase price for the shares of EUR 9,743,000 (= EUR 10,200,000 - EUR 457,000). Converted to one share, the effective share price paid is EUR 97.43 (= EUR 9,743,000/100,000 shares), which is lower than the target share price of EUR 97.50. For example, if the share price is EUR 105 on the expiration date of the put, the option expires out of the money and is not exercised. Meier can keep the premium from the option sale of EUR 457,000, but he cannot buy the equity security at the desired price of EUR 97.50. Compared to the share price of EUR 100 at the inception of the strategy, a possible purchase at a price of EUR 105 results in additional costs of EUR 43,000 [= 100,000 × (EUR 105 - EUR 100) EUR 457,000], which means that the effective share price to be paid is EUR 100.43. 4. Berger expects a share price movement of plus/minus 10%. The forecast price movement must be sufficiently high to recover the premiums paid for the long straddle. Prior to the announcement of the court verdict, the costs of the planned option strategy amount to EUR 3.42 (= EUR 1.72 + EUR 1.70). If the share price changes by 10% or EUR 5, the costs of the strategy can be more than covered, making the long straddle profitable. However, the news of the imminent court decision has led to an increase in implied volatility and thus in option prices. The cost of the long straddle is now EUR 6.98 (= EUR 3.50 + EUR 3.48). In order for

Solutions

585

the breakeven share price to be reached at expiration, a higher share price movement of approximately 14% (= EUR 6.98/EUR 50) is necessary. With a projected price change of 10%, the long straddle is no longer attractive. 5. If Burmeister buys a call with a strike price of EUR 80 and is correct in her price expectation, she makes a profit at expiration of EUR 4.26 [= max(EUR 0, EUR 87 - EUR 80) - EUR 2.74]. Alternatively, she can use an 80/87 bull call spread by buying a call option with a strike price of EUR 80 and simultaneously selling a call option with a strike price of EUR 87. The profit of the spread strategy at expiration is EUR 4.89 (= EUR 87 - EUR 80 EUR 2.74 + EUR 0.63). Thus, the bull call spread is the more attractive strategy. 6. a) CHF 30 c0 = CHF 0:93 þ CHF 31 60 days = CHF 1:9423 0:0025 × ð365 daysÞ e b) For breakeven share prices to be reached, the share price must move 6.03% upwards and 12.48% downwards: ðCHF 30 þ CHF 0:93 þ CHF 1:94Þ=CHF 31 - 1 = 6:03% ðCHF 30 - CHF 0:93 - CHF 1:94Þ=CHF 31 - 1 = - 12:48% An annualised volatility of 14.60% is required for the increase in the share price to move into the profit area: σAnnualised = 6:03% ×

252 days = 14:60%: 43 days

An annualised volatility of 30.21% is needed for the decrease in the share price to enter into the profit area: σAnnualised = 12:48% ×

252 days = 30:21%: 43 days

The long straddle strategy is only profitable in the event of a rise in the share price (i.e. if the court decision is positive for the company), because the implied volatility of 28% is then greater than the volatility of 14.60% required for the profit.

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References Black, F.: Fact and fantasy in the use of options. Financ. Anal. J. 31(4), 36–41., 61–72 (1975) Chance, D.M.: Analysis of Derivatives for the CFA® Program. Association for Investment Management and Research, Charlottesville, VA (2003) Mondello, E.: Finance: Theorie und Anwendungsbeispiele. Springer, Wiesbaden (2017) Figlewski, S.: Basic price relationships and basic trading strategies. In: Figlewski, S., Silber, W.L., Subrahmanyam, M.G. (eds.) Financial Options: From Theory to Practice, pp. 20–76. McGrawHill, New York (1990) Hull, J.C.: Options, Futures, and Other Derivatives, 6th edn. Prentice Hall, Upper Saddle River (2006) Reilly, F.K., Brown, K.C.: Investment Analysis and Portfolio Management, 7th edn. SouthWestern/Thomson Learning, Mason, OH (2003) Watsham, T.J.: Futures and Options in Risk Management, 2nd edn. International Thomson Business Press, London et al (1998) Yates, J.W., Kopprasch, R.W.: Writing covered call options: profits and risks. J. Portf. Manag. 7(1), 74–79 (1980)

Online Sources European Exchange: Eurex Markets. https://www.eurex.com/ex-en/markets/equ/opt/Linde-94591 6. Accessed on 22 January 2018a. European Exchange: Eurex Markets. https://www.eurex.com/ex-en/markets/equ/opt/MercedesBenz-Group-2884628. Accessed on 12 January 2018. European Exchange: Eurex Markets. https://www.eurex.com/ex-de/maerkte/equ/opt/Siemens-953 546. Accessed on 12 January 2018b.

Index

A Adjusted present value (APV) model, 310 Alpha, 200 Alternative assets, 236 American option, 480 Anticipatory hedge, 467 Asset classes, 235 Autocorrelation of return time series, 81

B Bear put spread, 562 Bear spread, 561 Behavioural finance, 82 Beta, 189 Beta for size, 214 Beta for value, 214 Bid-ask spread, 84, 419 Binomial option pricing model, 498 Black-Scholes option pricing model, 504 Bond equivalent yield, 396 Bonds with warrants, 373 Bull call spread, 557 Bull spread, 557

C Calendar rebalancing strategy, 242 Callable bonds, 373 Call option, 479, 481–487 Capital allocation line, 156 Capital asset pricing model (CAPM), 188 Capital market, 368 Cash-and-carry arbitrage, 463 Clean price of a bond, 380 Collar, 552–556 Commodity futures/forward, 461 Comparables method, 332

Contingent claim, 451 Continuous compounded return, 5 Convenience yield, 460 Convertible bonds, 373 Convexity, 377 Correlation coefficient, 108 Cost-of-carry model, 460 Cost of debt, 207 Cost of equity, 202 Covariance, 105 Covered call, 541–548 Cox–Ross–Rubinstein option pricing model, 498 Credit default swaps, 418 Credit risk, 376, 393, 417 Current price-to-earnings ratio, 329 Current yield, 394

D Delta, 512–516 Delta hedge on options, 515 Delta of a call option, 512 Delta of a put option, 513 Density function of the normal distribution, 37 Density function of the standard normal distribution, 38 Discount bonds, 375 Discount margin, 390 Diversification effect, 114, 123–128 Downside risk, 40–45 Dual-currency bonds, 373 DuPont model, 266 Duration-convexity approach, 421–422

E Effective annual yield, 396

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 E. Mondello, Applied Fundamentals in Finance, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-41021-6

587

588

Index

Efficient frontier, 112–115, 120, 153 Enterprise value EBITDA ratio, 350–355 European option, 480 Excess kurtosis, 66 Expected return, 16

Intrinsic value of options, 492 Investment policy statement, xx, 233 Investment ratio of equity, 291 Investment ratio of total capital, 303 I-spread, 379

F Fama–French model, 208–214 Fiduciary call, 509 Financial futures/forward, 462 Financial leverage, 266 Financial market, 368 Fixed-rate bonds, 374 Floating-rate notes (FRNs), xx, 372, 389 Forward commitment, 451 Forward price-to-earnings ratio, 330 Forward rate, 389 Forwards, 455 Free cash flow to equity (FCFE) Model, 290–301 Free cash flow to firm (FCFF) model, xx, 301–311 Full price of a bond, 380 Fundamental analysis, 261 Futures, 454

J Jarque-Bera test, 70 Justified forward enterprise value EBITDA ratio, 354 Justified forward (or leading) P/E ratio, 333 Justified price/earnings-to-growth ratio, 341 Justified trailing price-to-book ratio, 345

G Gamma, 516 Geometric mean return, 9 Gordon–Growth model, 269 Growth rate of the free cash flows to equity, 291 Growth rate of the free cash flows to firm, 303 G-spread, 378

M Macaulay duration, 424, 427 Maintenance margin, 455 Margin call, 455 Market liquidity risk, 376, 419–420 Market portfolio, 166 Market price anomalies, 78 Market risk premium, 203 Method based on forecast fundamentals, 332 Minimum variance portfolio, 114 Modified convexity, 421, 430–437 Modified duration, 421 Money market, 368 Moneyness of call options, 484 Moneyness of put options, 489 Money spread, 556 Money-weighted return, 11 Monte Carlo simulation, 47

H Hedging an existing long or short position, 466

I Idiosyncratic factors, 123 Immunisation strategy, 439 Implied expected return, 273 Implied growth rate, 273 Indifference curve, 149 Inflation-indexed bonds, 374 Information ratio, 243 Initial margin, 455 Interest rate risk, 415–416 Interest rate swap, 469 Internal rate of return, 11

K Kinked capital market line, 169 Kurtosis, 66

L Last squares method, 190 Leverage effect of futures, 459 Leverage effect of options, 510–511 Lognormal distribution, 71–75

N Net profit margin, 266 Nominal return, 15 Nominal risk-free rate, 376 Normal distribution, 35

Index O One-stage dividend discount model, 268–275, 333 One-stage free cash flow to equity model, 294 One-stage free cash flow to firm model, 304 Optimal risky portfolio, 153–154 Option price sensitivities (option Greeks), 511

P Payer swap, 470 Percentage-of-portfolio rebalancing strategy, 242 Performance index, 8 Plain vanilla bond, 372 Portfolio performance attribution, 246–249 Portfolio performance measurement, 243 Portfolio risk, 119 Price/earnings-to-growth ratio, 338–342 Price index, 8 Price multiple, 327 Price risk of a bond, 415 Price-to-book (P/B) ratio, 342 Price-to-book (P/B) ratio based on comparable companies, 347–349 Price-to-book (P/B) ratio based on forecast fundamentals, 345–347 Price-to-earnings (P/E) ratio, 329–338 Price-to-earnings (P/E) ratio based on comparable companies, 335–338 Price-to-earnings (P/E) ratio based on forecast fundamentals, 332–335 Protective put, 509, 548–552 Put-call parity, 509, 538 Put option, 479, 487–492

Q Quoted margin, 373, 389

R Random walk, 79 Real return, 15 Receiver swap, 470 Reinvestment risk of a bond, 415 Return on equity, 292 Return-risk optimization, 121 Reverse cash-and-carry arbitrage, 463 Rho, 522 Risk-adjusted discount rate for bonds, 376 Risk aversion, 145, 148 Risk factors of options, 493

589 Risk-free investment, 154 Risk-neutral probability, 499 Risk tolerance, 148, 225

S Scenario analysis, 16 Secured bonds, 370 Security market line (SML), 195 Semi-standard deviation, 42 Semi-strong form of market information efficiency, 76 Sensitivity analysis, 270 Settlement date, 380 Sharpe ratio, 243 Shortfall risk, 44 Short selling, 117 Simple linear regression analysis, 189 Simple (discrete) return, 3 Skewness, 64 Standard normal variable, 38 Statistical forecasting models, 265 Step-up bonds, 374 Straddle, 567 Strangle, 575 Strap, 574 Strip, 574 Strong form of market information efficiency, 77 Subadditivity, 50 Swap, 469 Swap rate, 470 Swap rate curve, 383 Synthetic long call option, 541 Synthetic long equity security, 538 Synthetic long put option, 540 Synthetic short equity security, 538 Systematic risk, 123, 124

T Tactical asset allocation, 238, 437 Taylor series expansion, 421 Taylor series expansion with a second-order approximation, 517 Technical analysis, 261 Terminal value of the stock, 276 Theta, 523 Time spread, 556 Time value of options, 492 Total asset turnover, 266 Total return, 397 Traditional assets, 236

590 Trailing price-to-earnings ratio, 329 Two-stage dividend discount model, 275–281 Two-stage free cash flow to equity model, 297

U Unlevered beta, 313 Unsecured bonds, 370 Unsystematic risk, 123, 124 Utility function, 147

V Value at risk, 45 Value multiple, 327 Variance, 29 Variance-covariance method, 47 Variation margin, 455, 472 Vega, 520

Index Volatility, 31, 496 Volume-weighted average price (VWAP), 86

W Weak form of market information efficiency, 76 Weighted average cost of capital (WACC), 201, 305 Withholding tax, 230

Y Yield to maturity, 378, 395

Z Zero-cost collar, 552 Zero-coupon bonds, 372, 387 Z-spread, 384