FIXED INCOME ANALYTICS : bonds in high and low interest rate environments. [2 ed.] 9783030471576, 3030471578

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Wolfgang Marty

Fixed Income Analytics

Bonds in High and Low Interest Rate Environments Second Edition

Fixed Income Analytics

Wolfgang Marty

Fixed Income Analytics Bonds in High and Low Interest Rate Environments Second Edition

Wolfgang Marty Zurich, Switzerland

ISBN 978-3-030-47157-6 ISBN 978-3-030-47158-3 https://doi.org/10.1007/978-3-030-47158-3

(eBook)

# Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

In light of an investment environment characterized by low yields and new regulatory capital regimes, it has become increasingly demanding for investors to achieve sustainable returns. Particularly, fixed income investments are called into question. There is a solution. Since the foundation of AgaNola a decade ago, we have put our interest into convertibles, and at this point, we want to thank our clients for having supported us also in challenging times—particularly when convertible bonds were considered at most a niche investment. Unjustly! For being a hybrid, convertible bonds offer the “best of both worlds,” the benefits of an equity with the advantages of a corporate bond. AgaNola is considered a leading provider in this asset class, and to date, convertible bonds remain the core competence of us as a specialized asset manager. As we consider increasingly popular convertible bonds a living and dynamic universe, we are placing great importance on research and the exploration of the nature of this asset class. As an internationally renowned expert in the fixed income and bond field, Dr. Wolfgang Marty has contributed valuable insights to our work— making the bridge from theory to portfolio management. AgaNola is committed to continue to support his fundamental research. We wish Wolfgang Marty lots of success with his latest book. AgaNola AG Pfäffikon, Switzerland

Stefan Hiestand

v

Foreword

Compared to other asset classes, fixed income investments are routinely considered as a relatively well-understood, transparent, and (above all) safe investment. The notions of yield, duration, and convexity are referred to confidently and resolutely in the context of single bonds as well as bond portfolios, and the effects of interest rates are generally believed to be well understood. At the same time, we live in a world where the amount of private, corporate, and sovereign debt is steadily increasing and where postcrisis stimuli continue to affect and distort investor behavior and markets in an unprecedented way. And that is even before we start contemplating the enormous uncertainties introduced by negative interest rates. In his book, Dr. Wolfgang Marty covers and expands on classic fixed income theory and terminology with a clarity and transparency that is rare to be found in a world where computerization of accepted facts often is the norm. Wolfgang highlights obvious but commonly unknown conflicts that can be observed, for example, when applying standard theory outside its default setting or when migrating from single to multiple bond portfolios. He also includes the effects of negative interest rates into standard theory. Wolfgang’s book makes highly informative reading for anyone exposed to fixed income concepts, be it as a portfolio manager or as an investor, and it shows that often we understand less than we think when studying bond or bond portfolio holdings purely based on their commonly accepted key metrics; Wolfgang encourages to ask questions. Anyone building automated software would benefit from familiarity with the model discrepancies highlighted as it is to everyone’s disadvantage if we find these too deeply rooted in commonly and widely applied tools. In summary, Wolfgang’s book makes interesting reading for the fixed income novice as well as the seasoned practitioner. Record Currency Management Windsor, UK

Jan Hendrik Witte

vii

Preface

Computers have become more and more powerful and often are an invaluable aid. But there is a considerable disadvantage: often, the output of a computer program is difficult to understand, and the end user may be swamped by data. In addition, computers solve problems in many dimensions, and, as human beings, we struggle thinking in more than a few dimensions. To provide a sound background of understanding to anyone working in fixed income, we intend to illustrate here the essential basic calculations, followed by easy to understand examples. The reporting of return and risk figure is paramount in the asset management industry, and the portfolio manager is often rewarded on performance figures. The first motivation for the here presented material were the findings of a working group of the Swiss Bond Commission (OKS), where we compared the yield for a fixed income benchmark portfolio calculated by different software providers: we found different yields for the same portfolio and the same underlying time periods. The following questions are obvious: How can a regulating body accept ambiguous figures? Should there not be a standard? An additional complication is linearization, often the first step in analyzing a bond portfolio. The yield of the bonds in a bond portfolio is routinely added to report the yield of the total bond portfolio, and different durations of bonds in the portfolio are simply added to indicate the duration of a bond portfolio. We found that linearization works well for a flat yield curve, but the more the yield deviates from a flat curve, the more the resulting figures become questionable. Also, historically, interest rates have been positive. In the present market conditions, however, interest rates are close to zero or even slightly negative. We find ourselves confronted with several questions: Does the notion of duration still make sense in this new environment? And which formulae can be applied for interest rates equal or very close to zero? How do discount factors behave? In the following, we attempt to include negative interest in our considerations. For instance, in the world of convertibles, yield to maturities can easily be negative and is not problematic. We describe the here presented material in three ways. Firstly, we use words and sentences, in order to give an introduction into in the notions, definitions, ideas, and concepts. Secondly, we introduce equations. Thirdly, we also use tables and figures in order to make the outputs of our numerical calculations accessible.

ix

Preface for Second Edition

The first edition has been published in 2017. In the meantime, I had different opportunities for representing the material of the book. I realized that Fixed Income Analytics is not limited to topics that are known in the community. I could also establish new work relationships. In particular, I would like to mention Moorad Choudry. His book Analysing and Interpreting the Yield Curve was of special interest for me. We had interesting discussions and I contributed a chapter on negative interest rate in the second edition of his book which appeared in 2019. Special thanks go to Martin Hillebrand and his team working at ESM (European Stability Mechanism). I could discuss my recent research results. As I am a member of the Fixed Income Index Commission at the SIX, Fixed Income Analytics could be used for enhancing the analytics of the SBI (Swiss Bond Index). The ongoing conversations with Andreas Henke are inspiring and constructive. For instance, the topic “Macaulay duration” is revised and expanded. In particular, negative rates are also considered. I am grateful to Ralf Seschek who updated and refined his sections on the comparison of the normal distribution with the data observed in the market. A chapter on multicurrency portfolio has been added. We start with some basics notions of hedging currency exposure followed by an exposition of the be Karnosky–Singer attribution. My first book Portfolio Analytics describes the main concepts of performance attribution. In the second edition of Fixed Income Analytics, we focus on the currency effect of a multicurrency portfolio. Many thanks to Karl Weber. He provided all charts for Chap. 10. They can be updated continually. Pfaeffikon SZ, Switzerland November 24, 2019

Wolfgang Marty

xi

Acknowledgments

This book is based on several presentations, courses, and seminars held in Europe and the Middle East. The presented material is based on a compilation of notes and presentations. Presenting fixed income is a unique experiment and I am grateful for the many feedbacks from the audience. The initial motivation for the book was a seminar held at the education center of the SIX Swiss Exchange. I became aware that many issues in fixed income need to be restudied and revised; moreover, I did not find satisfying answers to my questions in the pertinent literature. The SIX Swiss Exchange Bond Advisory Group was an excellent platform for analyzing open issues. Furthermore, the working group “Portfolio Analytics” of the Swiss Bond Commission was instrumental for the research activities. In particular my thanks go to Geraldine Haldi, Dominik Studer, and Jan Witte. They revised part of the manuscript and provided helpful comments. The European Bond Commission (EBC) was very important for my professional development. The members of the EBC Executive Committee Chris Golden and Christian Schelling gave me continuing support for my activities, and the EBC sessions throughout Europe yielded important ideas for the book. At the moment I am focusing on convertibles. My thanks go to Marco Turinello and Lukas Buxtorf for introducing me into the analytics of convertibles. The last chapter of the book is dedicated to convertibles. The book was written over several years, and I am grateful to my present employer AgaNola for the opportunity to complete this book.

xiii

Conventions

This book consists of eight chapters. The chapters are divided into sections. (1.2.3) denotes formula (3) in Sect. 1.2. If we refer to formula (2) in Sect. 1.2, we only write (2); otherwise we use the full reference (1.2.2). Within the chapters, definitions, assumptions, theorems, and examples are numerated continually, e.g., Theorem 2.1 refers to Theorem 1 in Chapter 2. Square brackets [ ] contain references. The details of the references are given at the end of each chapter.

xv

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

The Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Return Over a Time Unit . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

5 5 7 12

3

The Flat Yield Curve Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Description of a Straight Bond . . . . . . . . . . . . . . . . . . . . 3.2 Yield Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Durations and Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

19 19 27 34 37

4

The Internal Rate of Return for a Bond Portfolio . . . . . . . . . . . . . 4.1 The Direct Yield of the Portfolio . . . . . . . . . . . . . . . . . . . . . . 4.2 Different Approximation Scheme for the Internal Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Macaulay Duration Approximation Versus Modified Duration Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

39 39

.

56

. . .

66 74 86

5

The Term Structure of Interest Rate . . . . . . . . . . . . . . . . . . . . . . . 5.1 Spot Rate and the Forward Rate . . . . . . . . . . . . . . . . . . . . . . . 5.2 Discrete Forward Rate and the Instantaneous Forward Curve . . 5.3 Spot Rate and Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Effective Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 87 . 88 . 91 . 94 . 110 . 111

6

Spread Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Interest Rate Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Rating Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Composite Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Optionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 117 126 128 131 xvii

xviii

Contents

7

Different Fixed Income Instruments . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Segmentation of the Yield Curve . . . . . . . . . . . . . . . . . . . . . . . 7.2 Floating Rate Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Interest Rate Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Asset Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 134 136 141 143

8

Fixed-Income Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Definition and Fundamental Properties . . . . . . . . . . . . . . . . . . 8.2 Constructing a Fixed Income Benchmark . . . . . . . . . . . . . . . . 8.3 Recent Developments in the Benchmark Industry . . . . . . . . . . 8.4 Fixed Income as Asset Class . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Equity Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Fixed Income Indices . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Hedged Fixed Income Indices . . . . . . . . . . . . . . . . . . 8.4.4 Commodity Index . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Balanced Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Portfolio Selection and Modern Portfolio Theory . . . . 8.5.2 Pitfalls of Modern Portfolio Theory . . . . . . . . . . . . . . 8.5.3 Modern Portfolio Theory Assumptions Versus Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Hypothesis Testing of the MPT Assumptions . . . . . . . 8.5.5 Visual Comparison of Idealized and Real Data . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

145 145 146 148 149 150 152 154 154 155 156 157

. . . .

157 157 159 163

9

Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Basics Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Embedded Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Hedging Proprieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Modelling the Price of a Convertible . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 171 174 175 178

10

Multi-Currency Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Currency Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Karnosky-Singer-Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Return Attribution for a Multi-Currency Portfolio . . . . 10.3.2 Extension of the BHB to Multi-Currency Portfolio . . . 10.4 Two Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

179 179 182 186 186 190 195 200

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Closed Formula for the Geometrical Series . . . . . . . . . . Appendix B: Landau Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Application of the Landau Symbol to the Taylor Series . Appendix D: Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

203 203 205 205 206

Contents

Appendix E: Integral, Riemann Sum . . . . . . . . . . . . . . . . . . . . . . . . Appendix F: Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix G: Macaulay Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix H: Explicit Formula of Condition (3.3.6) . . . . . . . . . . . . . . Appendix I: A Closed Formula for Convexity . . . . . . . . . . . . . . . . . . Appendix J: Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix K: See Also [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix L: Derivation of Black Scholes Differential Equation . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

. . . . . . . . .

208 209 209 215 217 219 220 222 224

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

About the Author

Wolfgang Marty is senior investment strategist at AgaNola, Pfaeffikon SZ, Switzerland. Between 1998 and 2015, he was working with Credit Suisse. He joined Credit Suisse Asset Management in 1998 as head product engineer. He specializes in performance attribution, portfolio optimization, and fixed income in general. Prior to joining Credit Suisse Asset Management, Marty worked for UBS AG in London, Chicago, and Zurich. He started his career as an assistant for applied mathematics at the Swiss Federal Institute of Technology. Marty holds a university degree in mathematics from the Swiss Federal Institute of Technology in Zurich and a doctorate from the University of Zurich. He chairs the method and measure subcommittee of the European Bond Commission (EBC) and is president of the Swiss Bond Commission (OKS). Furthermore, he is a member of the Fixed Income Index Commission at the SIX Swiss Exchange and a member of the Index Team that monitors the Liquid Swiss Index (LSI).

xxi

1

Introduction

Abstract

A fixed-income security is a financial obligation of an entity that promises to pay a specified sum of money at specified future dates. The entity can be a government, a company, or an individual and is called an issuer. The investor lends a specified amount of money to the issuer. A bond is a legal engagement between the issuer and an investor. A fixed-income security is a financial obligation of an entity that promises to pay a specified sum of money at specified future dates. The entity can be a government, a company, or an individual and is called an issuer. The investor lends a specified amount of money to the issuer. A bond is a legal engagement between the issuer and an investor. A bond is a fixed-income instrument and has usually a finite live. Periodic future cash flows from the issuer to the investor are called the coupon of the bond. Coupons are unaffected by market movements for the live of the bond and reflect the notion “fixed income.” As depicted in Fig. 1.1, a straight bond or a coupon paying bond is a bond that pays a coupon periodically and pays back at the end of its live the money that was originally invested. For precise definitions and analytics, we refer to Chap. 3. The bond markets have grown tremendously, and today there is a large universe of organizations that issues bonds. Together with equities, bonds are the two major traditional asset classes in financial markets. There are much different bonds than equities. For instance, there were 5447 shares traded and admitted to trading on the EU regulated market (mifiddatabase.esma.europa.eu), and TRAX has data for 300,000 government bonds, corporate bonds, medium-term notes, and private derivative issues (xtrakter.com).

# Springer Nature Switzerland AG 2020 W. Marty, Fixed Income Analytics, https://doi.org/10.1007/978-3-030-47158-3_1

1

2

1 Introduction cash flows

Original investment coupon

coupon

coupon

coupon

Original Investment + coupon

time

Fig. 1.1 Straight bonds

Outstanding U.S. Bond Market Debt 18.000,0 16.000,0 14.000,0 12.000,0 Municipal

10.000,0

Treasury 8.000,0

Mortgage related

6.000,0

Corporate

4.000,0 2.000,0 0,0 1980

1985

1990

1995

2000

2005

2010

2015

Fig. 1.2 The development of the US bond market

The time to maturity and the coupon are fixed at the issuance of a bond and are thus called static data or reference data, whereas the market price is determined by the trading activity and is thus called market data. Unlike equities, every bond has potentially special and unique features. A company has one or two kinds of equities but many different bonds. Bond markets are very fragmented. Figure 1.2 (see www.sifma.org/research/statistics.aspx) shows the development of the four most important segments of the US Bond Market. Ever since interest rates began to climb in the late 1960s, the appeal for fixed-income instrument has increased. This is due to the fact that interest levels were competitive with other instruments, and at the same time, the market rates began to fluctuate widely, providing investors with attractive capital gain opportunities emphasizing

1

Introduction

3

that fixed income is not necessarily fixed income. Only for the buy and hold investor, i.e., the investor who keeps the bond till maturity, cash flows are fixed. The here presented material gives a comprehensive introduction to fixed-income analytics. Some of the topics are: • The transition from a single bond to portfolio of bonds is examined. We investigate the nonlinearity of income since just adding characteristics of individual bonds yields in general wrong results for the overall portfolio. • We consider market-relevant values for interest rates and examine different shape of the yield curve. In particular, we discuss negative interest rates. • We introduce the main ideas for assessing the credit quality of a bond. We compile different definitions of the default of a bond. • We describe the construction of an income benchmark and give an overview of different benchmark providers. We now provide more detail about the different chapters of this book. Chapter 2 describes the time value of money. This chapter contains the building blocks of a fixed-income instrument. We introduce the concept of an interest rate. We stress specifically that throughout this book and all its results, we treat negative and positive interest rates with generality (rather than favoring positive interest rates as has been so common in the literature until now). In Chap. 3, the flat yield curve concept is explained, i.e., every cash flow is discounted by the same interest rate. This does not mean that the yield curve is flat. If all bonds have the same yield, the yield curve is said to be flat. We discuss deviation of the flat yield curve . The yield to maturity is a well-established measurement for indicating a bond’s future yield. It is derived from the coupon, the nominal value, and the term to maturity of the bond. Portfolio analysis frequently refers to the “yield.” The question is which yield? In the following, we will not focus on a single bond. Rather, we will examine the ex ante yield of an entire bond portfolio, i.e., exclusively future cash flows are factored into the calculation. The equation for yield to maturity will be generalized to derive an equation for the bond portfolio (internal rate of return). This equation is not solved exactly by the programs offered by most software providers; instead, it is considered in combination with the yields to maturity of the individual bonds. In Chap. 4, we speak about the transition from yield curve to spot curves and spot curves to forward curves (see Fig. 1.3). Figure 1.3 refers to a specific time and does not say anything about the dynamic of the curve. Actual prices are measured in the marketplace, and yield, spot, and forward curve are in general calculated or computed. Duration is a risk measure of bonds and bond portfolios. Here, we assess the durations in the context of a bond and a portfolio of bonds. Effective duration versus durations based on the flat yield concept is discussed. Modified duration is used for a sensitivity analysis of a bond portfolio. The different durations we introduced tackle the interest risk and the yield curve risk. The duration is the fulcrum of a bond and can be compared to an equilibrium in physics.

4 Fig. 1.3 Different interest rate term structures

1 Introduction

forward rate

spot rate

yield to maturities In Chap. 5, we depart from the assumption that a straight bond is riskless. We consider credit markets. The credit quality of a bond is described by different spreads. We introduce the normal spread and the Z-spread and give the definition of default of a bond from S&P, Moodys, and Fitch. More recent developments of credit markets are described. We illustrate some figures from a transition matrix and discuss composition ratings followed by the description of call and put features of a bond. In Chap. 6, we start with float rate notes. Unlike fixed coupons, floating rates are tied to the short end of the yield curve . We give an introduction in the analytics of floating rate notes. We then proceed with the interest rate swap, which exchanges the liability of two counterparties. Interest swap markets are important for steering the duration of a bond portfolio. In the last section of the chapter, asset swaps are described. Starting point in Chap. 7 are the basic characteristics of a benchmark. An overview of different benchmark providers is given. We describe benchmarks from different asset classes and discuss benchmarks for a balanced portfolio. We give more recent developments in the benchmark industry. In Chap. 8, we give an introduction into convertible bonds. Convertible is corporate bond with an option on the stock of the issuing company. Convertibles can behave like a bond as well as a stock. We compile the most important notions describing a convertible. Difficulties of pricing a convertible are discussed.

2

The Time Value of Money

Abstract

In this chapter, we introduce the basic notions and methods for assessing fixedincome instruments. The subject of this chapter is the connection between time and the value of money.

2.1

The Return Over a Time Unit

Return measurement always relates to a time span, i.e., it matters whether you earn a specific amount of money over a day or a month. Therefore, return measurement has to be relative to a unit time period. In finance, the most prominent examples are a day, a month, or a year. In Fig. 2.1 we see a unit time period and a partition into four time spans of the same length. With a beginning value BV and a yearly or annular interest r, we write EV1 ¼ BVð1 þ rÞ

ð2:1:1Þ

for the ending value EV1. The underlying assumptions of (1) are that: • We hold the beginning value over 1 year. • There is no interest payment and no cash flow during the year. Example 2.1 We consider for BV a Coupon C of an annual paying bond. Then (1) expresses the ending value EV1 after 1 year. In the European bond market, coupons are usually paid yearly. The index 1 in EV1 says that there is no cash flow during the year and EV1. Next, we assume that one half of the interest is pay out in the middle of the year, which gives # Springer Nature Switzerland AG 2020 W. Marty, Fixed Income Analytics, https://doi.org/10.1007/978-3-030-47158-3_2

5

6

2

Fig. 2.1 The time unit

t1 = 0.25

The Time Value of Money

t3 = 0.75 t

t0 = 0

t2 = 0.5

t4 = 1

  h
i
 r r r2 EV2 ¼ BV  1 þ : 1þ ¼ BV 1 þ r þ : 2 2 4 Here, we have a reinvestment assumption about the middle of the year: we assume that the money received is reinvested with the same interest rate r. We observe that EV2 > EV1, and we proceed by iterating and taking the limit:
 1 n , n ¼ 1, 2, 3, . . . EV ¼ BVn!1 lim 1 þ n The question is whether the sequence EVn is bounded or unbounded. The answer is that the sequence is convergent since from calculus we know that
 1 n ¼e lim 1 þ n!1 n with e ¼ 2:71828 18284 5905: From calculus we also have 


  r r n 1 nr 1 n lim 1 þ ¼ lim 1 þ ¼ lim 1þ ¼ er : n!1 n!1 n!1 n n n Hence, when compounding with an infinitely small compounding interval, the continuous compounding expression becomes EV1 ¼ BVer : Example 2.2 For r ¼ 0.05 (¼5% annually) and BV ¼ $100 we get in decimals. EV2 ¼ $105.06250 (semi-annual). EV4 ¼ $105.09453 (quarterly). EV100 ¼ $105.1257960. EV1000 ¼ $105.1269782. EV10000 ¼ $105.1270965. EV100000 ¼ $105.1271083. EV1 ¼ $105.1271109 (continuous).

2.2 Discount Factors

7

Definition 2.1 The return AERðnÞ ¼

EVn  BV , n ¼ 1, 2, 3, . . . BV

is called the annual effective rate. Remark 2.1 For discrete compounding, we have AERðnÞ ¼


 EVn  BV EVn r n ¼  1, n ¼ 1, 2, . . . : 1¼ 1þ BV n BV

and for continuous compounding, we have with n ! 1 AER ¼ er  1: Example 2.3 We consider a semiannual bond with face value F 1 year before maturing. Furthermore, we assume there are two coupons, i.e., we get C/2 in the middle of the year and C/2 at the end of the year. By using continuous compounding and prevailing interest r1 and r2, we find P¼

2.2


 C r21 C e þ F þ er 2 : 2 2

Discount Factors

The time value of money concept is concerned with the relationship between cash flow C occurring on different dates. If C > 0 or C < 0, the investor has an inflow or outflow, resp., in his or her portfolio. The cash flow can occur at arbitrary different dates. A simple time pattern is depicted in Fig. 2.1. In Fig. 2.2, we introduce N time knots between the time knot t0 and tN, where the time t is the independent variable. We specify N (not necessarily equidistant) knots on the time axis with corresponding times tk and denote them by tk , 0  k  N:

Fig. 2.2 The time axis

ð2:2:1Þ

Equidistant knots t t 0 = 0 t1

t2

tk

tN–1

tN = T

8

2

The Time Value of Money

By assuming t0 ¼ 0, t0 is the present or for short t0 is now. However, in principle, t0 can be in the past (t0 < 0) or in the future (t0 > 0). For illustration purposes, we use years as units. Then, for equidistant knots of annual cash flows between t0 and tN, we have tk ¼ k, 0  k  N:

ð2:2:2Þ

For two equidistant knots over 1 year, we have N ¼ 2, and the time knots are marked by 1 t1 ¼ , 2 t2 ¼ 1: Definition 2.2 The discount factor function or for short the discount factor d(r(t 2 tk), t, t k) with an annual discount rate function r(t  tk) > 1, k ¼ 0,. . ., N, at arbitrary time tk 2 R1 for arbitrary t 2 R1, is defined by dðrðt  tk Þ, t, tk Þ ¼

1 , ð1 þ rðt  tk ÞÞðttk Þ

ð2:2:3aÞ

and for equidistant knots tj ¼ j with rj ¼ r(tj) and tk ¼ 0, the abbreviation      dj r j ¼ d r t j , t j , 0 ¼ 

1 t 1 þ rj j

ð2:2:3bÞ

is often used. We see that in (3), $1 is discounted by the discount factor d(r, t, tk). We consider in the following the more general form by considering a cash flow C and a beginning value BV: BVðC, rðt  tk Þ, t, tk Þ ¼ C Cdðr, t, tk Þ ð1 þ rðt  tk ÞÞðttk Þ :

ð2:2:4Þ

Example 2.4 We choose N ¼ 4 in (1, 2) with a cash flow $3 in t ¼ tN ¼ 4. With t0 ¼ 0 and r (t) ¼ r ¼ 5% in (3), we have for the beginning value BV with (4) BVð$3, 2%, 2, 0Þ ¼

$2 $2 ¼ ¼ $1:567052: ð1 þ rÞ4 ð1 þ 0:05Þ4

2.2 Discount Factors

9

Fig. 2.3 Discount factor ex post and ex ante

3.00 2.50 2.00

r = 0.5 r=0

1.50

r = –0.5

1.00

–6.00

–4.00

–2.00

0.50 0.00

2.00

4.00

6.00

In Fig. 2.3, we assume N ¼ 10 and show the discount factor for the interest rates r ¼ 0.05, r ¼ 0, and r ¼ 0.05 between the times t0 ¼ 5 (ex post) and t10 ¼ 5 (ex ante). We see that the behavior of the discount factors is different for positive and negative discount factors. Remark 2.2 From Eq. (2.1.1), we have with C ¼ EV after one time unit C ¼ BV ð1 þ rÞ: On the interval r 2 (1, 0), we see that value is destroyed, i.e., C < BV, and for r ¼ 1, we have complete loss, i.e., C ¼ 0. The following lemma summarizes some fundamental properties about discount factors: Lemma 2.1 In (4) we have under the assumption C > 0: (a) For fixed r 2 R1 with r > 1 and t 2 R1 with t > 0, BV(C, r, t, tk) is a monotonically increasing linear function of C, i.e., BVðλC, r, t, tk Þ ¼ λBVðC, r, t, tk Þ, λ 2 R1 :

ð2:2:5Þ

(5) says that by changing the cash flow by a fixed factor, the value at present is multiplied by the same factor. (b) For fixed C 2 R1 and t 2 R1 with t > 0, BV(C, r, t, tk) is a monotonically decreasing function of r. The higher the interest, the less worth is the money at present. (c) For C 2 R1 and for t 2 R1, BV(C, r, t, tk) is for a fixed r 2 R1:

10

2

The Time Value of Money

• With r > 0 monotonically decreasing • With r ¼ 0 constant • With 1 < r < 0 monotonically increasing function of t (d) The series of the discount factor

dn ¼

C , n ¼ 1, 2, 3, . . . ð1 þ r Þn

are: • For r > 0, monotonically decreasing and converging with limit 0. • For r ¼ 1, the series is constant with dn ¼ 1, n ¼ 1, 2, 3, . . ., • For 1 < r < 0, the series dn is diverging for n ! 1 . Proof From (4), we have 1 1 1 ¼ , t ðttk Þ ð1 þ rÞ ð1 þ rÞtk ð1 þ rÞ and by assuming r > 1, we have 1 > 0, ð1 þ rÞtk i.e., in order to show monotonicity, it is enough to consider BVðC, r, t, 0Þ ¼

C : ð1 þ r Þt

The assertions a and b follow from the partial derivatives ∂BV 1 ¼ > 0, ∂C ð1 þ r Þt ∂BV ¼ Ct ð1 þ rÞτ1 < 0: ∂r The assertion c follows also from the partial derivative and the hypothesis that the coupon is positive. We have to distinguish the following cases:

2.2 Discount Factors

11

• For r > 0, ∂BV ¼ Cet ∂t

ln ð1þrÞ

ð ln ð1 þ rÞÞ < 0:

• For r ¼ 0, ∂BV ¼ 0, ∂t • For 1 < r < 0, ∂BV ¼ C et ∂t

ln ð1þrÞ

ð ln ð1 þ rÞÞ > 0:

The assertion d follows from induction with respect to n. □ Lemma 2.1 discusses the monotonicity of the discount factors. We assumed three independent variables, C, r, and t. In the following lemma, we change the three variables simultaneously, and we see that there is no monotonicity. Lemma 2.2 For 100C ¼ t (1  t  10) and C ¼ r, the function defined in (4) has a global maximum for C ¼ 7.259173%, and we have BVðC, C, C=100Þ ¼ 4:364739: Proof By assumption, we have: BVðC, r, tÞ ¼

C C ¼ : ð1 þ rÞt ð1 þ CÞ100C

We use the product rule for the derivative

dBV dC

1 ð1 þ CÞ100C 1 1 dexpð100C ¼ þ C þC ¼ 100C 100C dC dC ð1 þ CÞ ð1 þ C Þ dð100C ln ð1 þ CÞÞ 1 ¼ þ Cexpð100C ln ð1þCÞÞ dC ð1 þ CÞ100C

 1 100C ¼ : 1 þ C 100 ln ð1 þ CÞ þ 1þC ð1 þ CÞ100C

The condition

d

ln ð1þCÞÞ

12

2

Fig. 2.4 Global maximum

The Time Value of Money

5 4

BV

3 function values

2

first Derivative second derivative

1 0 1

2 3

4 5

–1

6 7

8 9 10 11

C

∂BV ¼0 ∂C is the same as 1  100 ln C




1þ

 C 100C  ¼ 0: 100 1þC

Figure 2.4 shows this function, and a numerical method calculates the values stated in the lemma, which completes the proof. □ We investigate the behavior of the discount factors in more detail in Example 4.7 (Chap. 4).

2.3

Annuities

In this section, we consider multiple cash flows. We start with the following definition: Definition 2.3 An annuity is a finite set of level sequential cash flows at equidistant knots (2.2.2). An ordinary annuity has a first cash flow one period from the present, i.e., in the time point t1 ¼ 1. An annuity due has a first cash flow immediately, i.e., at t0 ¼ 0. A perpetuity or a perpetual annuity is a set of level never-ending sequential cash flows. Lemma 2.3 A closed formula for the beginning value BVor of an ordinary annuity in the time span between t0 ¼ 0 and tN is, for 1 < r < 0 or r > 0,

2.3 Annuities

13

! C 1 , 1 BVor ¼ r ð1 þ r ÞN

ð2:3:1aÞ

and for the ending value EVor we find EVor ¼


 C ð1 þ r ÞN  1 : r

ð2:3:1bÞ

For r ¼ 0, we have BVor ¼ EVo ¼ N: A closed formula for an annuity due for the beginning value BVdue in the time span t0 ¼ 0 and tN is, for 1 < r < 0 or r > 0, BVdue

! C 1 1þr ¼ , r ð1 þ r ÞN

ð2:3:2aÞ

and for the ending value EVdue, we find EVdue ¼


 C ð1 þ rÞNþ1  1 : r

ð2:3:2bÞ

For r ¼ 0, we have BVdue ¼ N þ 1: A closed formula for the value PBVor of perpetual ordinary annuity in t0 ¼ 0 is, for 1 < r < 0 or r > 0, PBVor ¼

C : r

ð2:3:3aÞ

A closed formula for the value PBVdue of an perpetual annuity due in t0, t0 ¼ 0, denoted by PBVdue, is, for 1 < r < 0 or r > 0, PBVdue ¼

Cð1 þ rÞ : r

ð2:3:3bÞ

Proof We use the closed formula of a geometric series. For details see Appendix A. Remark 2.3 EV and BV are related by

□

14

2

EV ¼

The Time Value of Money

BV , ð1 þ r ÞN

ð2:3:4Þ

and for N ! 1 and r > 0, we have EV ¼ 0: Example 2.5 For N ¼ 1 in (1a), we have BVor ¼



 C 1 C 1þr1 C ¼ ¼ 1 : r 1þr r 1þr 1þr

For N ¼ 1 in (2a), we have

BVdue



 C ð1 þ r Þ2  1 Cð2r þ r2 Þ C 1 C ¼ ¼ 1þr : ¼ ¼Cþ r 1þr 1þr r ð1 þ r Þ r ð1 þ r Þ

Example 2.6 With C ¼ $150,000 and r ¼ 3%, we have by (3a) $150, 000 ¼ $5, 000, 000, 3% i.e., for an annual income of $150,000, the capital of $5,000,000 is needed. The following lemmas decompose the balance at each point of time of a cash flow into the cash flow and the accumulated interest rate. Lemma 2.4 (Repayment of Mortgage) We assume that a BV and an interest r > 0 are given. The periodic payment of ordinary annuity is Cor ¼ BV

r 1 1  ð1þr ÞN

,

ð2:3:5aÞ

and for an annuity due, we have Cdue ¼ BV

r : ð1 þ rÞ  ð1þr1ÞNþ1

Starting with an initial value BV, we consider the iteration

ð2:3:5bÞ

2.3 Annuities

15

Bnþ1 ¼ Bn þ

C : ð1 þ rÞnþ1

ð2:3:6Þ

C With B1 ¼ 1þr and (6) with n ¼ 2,. . .,N, we have for an ordinary annuity (1a)

BN ¼ Bord ¼ BV:

ð2:3:7aÞ

With B0 ¼ C and (6) with n ¼ 1,. . .,N, we have for an annuity due (2a) BN ¼ Bdue ¼ BV:

ð2:3:7bÞ

We decompose the annuity by the part that is due to the interest rate in the last period and the part which is due to the amortizing part C ¼ r Bn þ ðC  r Bn Þ: Proof We consider the partial sum Bn ¼

n X k¼1

C : ð1 þ rÞk

Then, for n ¼ N, we have (7a) based on (5) and Lemma 2.2. We consider the partial sum Bn ¼

n X k¼0

C ð1 þ r Þk

with B0 ¼ C, and the proof (7b) follows like for the proof for (7a).

□

Lemma 2.5 (Accumulation of Capital) We assume that an EV and an interest rate r > 0 are given. The periodic payment of ordinary annuity is r , ð1 þ r ÞN  1

ð2:3:8aÞ

r : 1 ð1 þ rÞNþ1  1þr

ð2:3:8bÞ

Cor ¼ EV and for an annuity due, we have Cdue ¼ EV We consider the iteration

16

2

The Time Value of Money

Enþ1 ¼ ð1 þ rÞnþ1 C þ En :

ð2:3:9Þ

With E1 ¼ (1 + r) C and (9) with n ¼ 2,. . .,N, we have for an ordinary annuity (1b) Eor ¼ EN ¼ EV:

ð2:3:10aÞ

With E0 ¼ C and (9) with n ¼ 1,. . .,N, we have for an annuity due in (2b) Edue ¼ EN ¼ EV:

ð2:3:10bÞ

We decompose the annuity by the part which is due to increase of the balance minus the interest rate payment in last period: C ¼ ðC þ r E n Þ  r E n : Proof We consider the partial sum En ¼

n X

ð1 þ rÞk C:

k¼1

Then, for n ¼ N, we have (10a) based on (5) Lemma 2.3. We consider the partial sum En ¼

n X

ð1 þ rÞk C:

k¼0

with E0 ¼ C. The proof (10b) follows like the proof for (10a).

□

Example 2.7 We consider a fixed-rate mortgage such that the payments are equal. We consider a mortgage of $100,000 with a mortgage rate of 8.125% over 10 years with monthly payments. The investor pays off the mortgage completely in equal installments. We have C ¼ BV ¼

r ð1 þ r ÞN ¼ $742:50: ð1 þ r ÞN  1

The amortization schedule is in Table 2.1.

2.3 Annuities

17

Table 2.1 Amortization schedule

Month 1 2 ⋮ 359 360

Beginning of month mortgage balance 100,000 99,934 ⋮ 1470.05 737.50

Mortgage payment 742.50 742.50 ⋮ 742.50 742.50

Interest 677.08 676.64 ⋮ 14.88 4.99

Scheduled principal repayment 65.41 65.86 ⋮ 727.54 737.50

End of month mortgage balance 99,934.59 99,868.73 ⋮ 737.50 0.000

3

The Flat Yield Curve Concept

Abstract

Starting point are basic notions for assessing a single bonds. We define and discuss the flat yield concept. The non-linearity of the price and the yield of the Bond is examined. We discuss the Macaulay duration of a single bond in detail, including negative interest rates. We consider the transition from a single bond to a bond portfolio.

3.1

The Description of a Straight Bond

The financial Market consists of the credit market, the capital market, and the money market. The bond market is part of the capital market. The financial industry distinguishes traditional and alternative Investments. Fixed income instruments are traditional investments. We start with the following definitions. Definition 3.1 A straight bond with price P will pay back the original investment at its maturity date T and will pay a specified amount of interest on specific dates periodically. A straight bond is the most basic of debt investments. It is also known as a plain vanilla or bullet bond. The cash flows illustrated in Fig. 1.1 are referring to a straight bond. Example 3.1 (Description of a Bond Universe) Most of the Bonds in the Swiss Bond Market are straight bonds. Definition 3.2 The face value F of a bond is the amount repaid to the investor when the bond matures. The face value is also called the par value of a bond or the principal, stated or maturity value of a bond. # Springer Nature Switzerland AG 2020 W. Marty, Fixed Income Analytics, https://doi.org/10.1007/978-3-030-47158-3_3

19

20

3 The Flat Yield Curve Concept

Fig. 3.1 Cash flows

F+CN IP,P

C1

C2

Ck

CN–1 t

t0 = 0

t1

t2

tk

tN–1

tN = T

Definition 3.3 Coupon C is a term used for each interest payment made to the bond holder. We distinguish between registered and unregistered Bonds. A bearer bond is unregistered and the investor is anonymous. Whoever physically holds the paper on which the bond is issued owns the bond. Recovery of the value of a bearer bond in the event of its loss, theft, or destruction is usually impossible. The collection of the coupon is the task of the investor. Often, the bank collects the coupon payment on behalf of the investor. If the issuer of the bond kept a record of the investor, we speak of a registered bond. The issuer of the bond sends the coupon payments to the investor. Figure 3.1 shows on the horizontal axis the specific dates and the corresponding cash flows denoted with the coupons and the face value. Generally, a fixed income instrument is a series of cash flows of coupons and a face value. A straight bond is the starting point for studying fixed income instruments. Definition 3.4 The time to maturity tm = t - tN is the remaining lifetime of the bond. Remark 3.1 In Fig. 3.1, we assume that t0 is the origin, i.e., t0 ¼ 0 on the time axes. Moreover, in the following, we assume that tk, k ¼ 1,. . . .,N are the times of the analysis. Definition 3.5 A zero coupon bond is a bond which does not pay interest before the maturity date. A straight bond can be considered as a series of zero coupon Bonds. Bonds have therefore a so-called linear structure. Definition 3.6 The flat rate concept assumes that each coupon and the face value of a specific bond are discounted by the same interest rate. We consider an annual paying bond with N coupons C and face value F. Referring to knots (2.1.2), we specify N equidistant knots on the time axis with corresponding time tk denote them tk ¼ k, 0  k  N: We consider the price P of a bond that is valid at issuance:

3.1 The Description of a Straight Bond

21

Pðt0 Þ ¼ Pð0Þ ¼

N X j¼1

C F þ : ð1 þ r Þj ð1 þ r ÞN

The price P of a bond that is valid just after the payment of a coupon is Pðtk Þ ¼ PðkÞ ¼

N X j¼kþ1

C F þ : ð1 þ rÞjk ð1 þ rÞN

ð3:1:1aÞ

The formula is only valid at the day the coupon is paid. We consider the remaining coupons of the bond at time tk C , j ¼ k þ 1, . . . , n: ð1 þ r Þj We extend to any time and to the period before the coupon the next coupon for the coupon at time tk+j   C tkþj t , t 2 tkþj1 , tkþj , j ¼ 1, . . . , N: ð1 þ r Þ

ð3:1:1bÞ

We proceed with the following definition: Definition 3.7 (Invoice Price) A bond pricing with invoice Price (IP) quotes the price of a bond that includes the present values of all future cash flows incurring including the interest accruing until the next coupon payment. Remark 3.2 P and IP as a function depend on the variables r, C, F, tN ¼ N, and the time and the frequency of the coupon payments. In this book, we only consider annual buying bonds and, therefore, the frequency is 1 and C, F are kept constant. In the following, we consider one point in time. Therefore we suppress these arguments. Remark 3.3 The invoice price is also called the dirty or full price. For k ¼ 0,. . .,N - 1, we have with (1) IPðtÞ ¼

N X 1 C F þ , t 2 ½tk , tkþ1 : tkþ1 ‐t j ð1 þ r Þ ð1 þ r ÞN j¼kþ1 ð1 þ rÞ

By using the closed formula, we have

ð3:1:2aÞ

22

3 The Flat Yield Curve Concept

price (P) = 100

+

Invoice price

today

accrued interest

clean price

α

Coupon (C) Coupon (C)

Coupon (C)

Face (F) + Coupon (C)

Coupon (C)

–

Time

Fig. 3.2 The invoice and the clean price

IPðtÞ ¼

1 ð1 þ rÞtkþ1 ‐t C  r

1 1 þ r‐ ð1 þ rÞN

! ‐ k‐1

! þ

F ð1 þ rÞN‐k‐1

2 ½tk , tkþ1 :

,t ð3:1:2bÞ

Then we have, for t to tk+1, IP ðtk Þ  IPþ ð tk Þ ¼ C, k ¼ 1, . . . , N  1: In Fig. 3.2, we observe a Zig-Zag line which represents the invoice price of a bond as introduced in Definition 3.7. We distinguish between days where a coupon is paid and days where no coupon is paid. We require the following definition. Definition 3.8 (Accrued Interest) Accrued interest is a accounting method for measuring the interest rate that is either payable or receivable and has been recognized but not yet paid or received. It occurs as a result of the difference in timing of cash flows and the measurement of these cash flows. If we assume periodic coupon payments C, then, for k ¼ 1,. . . ., N - 1, the accrued interest AIk is defined by AIk ¼ ðt  tk1 Þ C, t 2 ½tk1 , tk Þ: Remark 3.4 For k ¼ 1,. . . ., N - 1, we have no AI, but the coupon is paid. Remark 3.5 By referring to Fig. 3.2, we introduce

ð3:1:3aÞ

3.1 The Description of a Straight Bond

23

Accrued interest

C 360

xx.xx.xx

xx+1.xx.xx xx+2.xx.xx

Time [days]

Fig. 3.3 Accrued interest Table 3.1 Calculation of the accrued interest

20.08.XX

01.1.XX

30.08.XX September October November December 25.01.XX

Number of days 11 30 30 30 30 24 155

ακ ¼ 1  ðt  tk1 Þ C, k ¼ 0, . . . , N  1, and AIκ ¼ ð1  ακ ÞC, k ¼ 0, . . . , N  1, t 2 ½tk1 , tk Þ

ð3:1:3bÞ

is the same as (3a). As accrued interest is calculated daily, we have to change from the unit year to the unit days, and we obtain a step function (Fig. 3.3). If the month is calculated with 30 days, the accrued interest is horizontal and stays the same. In February, the accrued interest changes vertically, and accrued interested is cumulated. The International Capital Market Association (ICMA) recommends in its Rule 251 that the number of days accrued should be calculated as the difference between the date of the last payment inclusive (or the date from which the coupon is due, for a new issue) up until, but not including, the value date of the transaction. Example 3.2 (Day Counting) We consider a bond that pays a coupon at 20.08.xx and we assume that the value date of the transaction is 25.01.xx. Assuming that the month is calculating with 30 days, Table 3.1 gives the number of days.

24

3 The Flat Yield Curve Concept

Definition 3.9 (Clean Price) The price of a coupon bond not including any accrued interest is called clean price and is denoted by P. Remark 3.6 The Flat or simple price is the same a clean Price. We find that Clean price ¼ Invoice Price  Accrued Interest, i.e., by (3), * 1 PðtÞ ¼ ð1 þ rÞtk

t

C r

+ ð1 þ rÞ 

1 ð1 þ r ÞN

tk

! þ

F ð1 þ r ÞN

tk

 ð1  αÞC, t 2 ½k  1, k: The price of the bond is based on the evaluation of all cash flows. In mathematical terms expressed, this means that the bond has a linear structure. For continuous compounding, we have Pð0Þ ¼

N X

Cejr þ FeTr :

ð3:1:4Þ

j¼1

If we spread the Coupon over the time t 2 [tk  1, tk], i.e., if we consider continuous compounding by starting an equal distant sample of the interval tj to tj +1 (see Appendix E), then

lim

M!1

M X m¼1

C M

m ð1 þ rÞððtj þMÞÞ

Ztjþ1 ¼ tj

C dt ð1 þ r Þt

and we have P ð 0Þ ¼

tjþ1 N1 Z X j¼1

tj

C F ¼ t dt þ ð1 þ r Þm ð1 þ r Þ

ZtN 0

C F t dt þ ð1 þ rÞm : ð1 þ r Þ

We see that by continuous compounding we have no accrued interest, i.e., the invoice price is equal to the flat price. Theorem 3.1 We consider the recursion

3.1 The Description of a Straight Bond

25

P1 ¼

FþC 1þr

ð3:1:5aÞ

and Pnþ1 ¼

Pn þ C , 1  n  N  1: 1þr

ð3:1:5bÞ

At the times knot defined by (2.1.1), the following holds with P1 ¼ F ¼ 100 and n ¼ 1, 2, 3,. . . .,N - 1. (a) If r < C, then Pn + 1 > Pn, (b) if r ¼ C, then Pn ¼ 100, (c) and if r > C, then Pn + 1 < Pn. Proof For (a) and (b), this follows as the closed formula for Pn is Pn ¼

C r

 1

1 ð1 þ rÞn

 þ

F , 1  n  N: ð1 þ r Þn

For the difference, we then have !

Pnþ1  Pn ¼

C r

  F C 1  1  r ð1 þ r Þn ð1 þ rÞnþ1     1 C 1  F  1 : r 1þr ð1 þ r Þn

1 1 ð1 þ rÞnþ1 þ

F ¼ ð1 þ r Þn

þ

Then, for Cr < 1 and Cr > 1, the sequence is increasing and decreasing, respectively, and for C ¼ r, the difference vanishes. Therefore, the assertion (c) is shown. □ Example 3.3 We consider a face value F ¼ 100. Then, with C ¼ 2% and r ¼ 4%, we have P10 ¼ 124:012, P20 ¼ 159:556, and, with C ¼ 4% and r ¼ 2%, we have P10 ¼ 78:100, P20 ¼ 51:405: Definition 3.10 If the bond price is P ¼ 100, then the bond price is said to be at par. If the Bond Price P is less than 100, then we have a discount Bond. If the bond Price P is over 100, then we have a premium Bond.

26

3 The Flat Yield Curve Concept

Corollary 3.1 We consider an annual paying bond with price P, yields r, and Coupon C with C > 0, C 2 R1. At the times of knots as defined in (2.2.1), the following holds for n ¼ 1, 2, 3,. . . .,N - 1. (a) If 0 < r < C, then Pn < 100, (b) if r ¼ C, then Pn ¼ 100, (c) and if r > C or  1 < r < 0 (negative interest), then Pn > 100. Proof We consider the recursion (5) and we prove the corollary by induction with respect to n. For n ¼ 1, we distinguish the following cases. ð aÞ P 1 ¼

FþC FþC < ¼ 100, 1þr 1þC

ð b Þ P1 ¼

FþC FþC ¼ ¼ 100, 1þr 1þC

ðcÞ P1 ¼

FþC FþC > ¼ 100: 1þr 1þC

Assuming that the assertion is true for n, we consider n ¼ 1, 2, 3,. . . .,N - 1 as Pnþ1 ¼

Pn þ C 1þr

(a) Case Pn < 100: as F ¼ 100, this follows by Pn < F by (5) Pnþ1
100: as F ¼ 100, we have Pn > F by (5) and Pnþ1 >

FþC > 100: 1þr

Example 3.4 A treasury bill (or for short T-bill) is a zero coupon money market instrument.

□

3.2 Yield Measures

3.2

27

Yield Measures

As depicted in Fig. 3.4, we consider a bond portfolio with cash-flow at fixed equidistant time points tk ¼ k, k ¼ 1,. . . .,N ¼ T. Definition 3.11 (Constituents of a Bond Portfolio) For the Price Pj of the Bond j, 1  j  n, with time of maturities 1  Tj  Tn and with cash-flows Cj,k and face values Fj, we have the price of a bond as a function of r as Pj ðrÞ ¼

Tj‐1 X k¼0

Cj,k ð1 þ r Þ

k

þ

CTj þ FTj ð1 þ rÞTj

, 8rj 2 R1 :

ð3:2:1Þ

Without loss of generality, we now assume that the bond is sorted in ascending order, i.e., 1  Tj  Tj + 1  Tn, 1  j  j + 1  n. Figure 3.5 shows the maturity profile of the portfolio. Definition 3.12 (Yield to Maturity) Assuming that the price of the bond is given, the yield to maturity (YTM) rj of a bond with price Pj is the solution of (1) Tj1  X CTj þ FTj Cj,k Pj rj ¼ T :  k þ  1 þ rj j k¼1 1 þ rj

ð3:2:2Þ

In the following, the assumptions are that • it is assumed that all Coupons are paid, i.e., there are no defaults, • the investor holds the bond until maturity, • we are looking forward, i.e., we consider the cash flow in the future. Fig. 3.4 Equidistant knots over unit intervals

IP,P

C1

Ck

CN–1

F+CN t

t0 = 0

Fig. 3.5 The maturity profile

t1

tk

tN–1

tN = T

Bond n j

Cj,k

Time k

Tn

28

3 The Flat Yield Curve Concept

• The yield to maturity is the solution of this equation written down here, which says that the cash flows in the future discounted to today equal to the price paid in the market. The principle is based on an arbitrage relationship, i.e., a condition which avoids a situation with a profit without risk. • It is not clear how yield to maturity is added for different bon • It is not clear whether the solution is unique. For the last two points, there is research being conducted in [1, 2]. We assume that a portfolio with n bonds is ordered with decreasing times to maturity. We assume that in this portfolio there are Nj of Bond j, and n is the number of bonds that have a cash flow in time tk ¼ k, 1  k  Tj. Then the portfolio value Po is ! Tj n n X X X Nj Cj,k Nj Fj PoðrÞ ¼ Nj Pj ðrÞ ¼ þ : ð3:2:3Þ k ð1 þ r ÞT j k¼1 ð1 þ rÞ j¼1 j¼1 A solution of (3) is called the true yield or the internal rate of return. The internal rate of rate is a solution of a transcendental equation. In the Sect. 3.3, we examine different methods for approximating solutions of (3). Remark 3.7 In this section and the following section, we consider the flat rate concept. This, however, does not mean that the yield curve is flat. In the following, we consider a portfolio consisting of only one bond with a given price. We denote the yield to maturity with YTM. Example 3.5 (YTM of a Zero Coupon Bond) The price of zero coupon bond is P¼

F , t 2 R1 : ð1 þ r Þt

By solving with respect to r, we find rffiffiffi t F  1: YTM ¼ P Definition 3.13 The yield to maturity of a zero coupon bond is called the spot rate.

3.2 Yield Measures Table 3.2 Pricing of a bond

29

r (%) 2 3 4

1 2.9412 2.9126 2.8846

2 2.8835 2.8278 2.7737

3 97.0592 94.2596 91.5666

Total 102.8839 100.0000 97.2249

Example 3.6 (YTM in the Last Period) As can be seen in the proof of Theorem 2.1, the price of a Bond in the last period is P¼

FþC , 1þr

and therefore we have YTM ¼

FþC  1: P

Example 3.7 We consider 3 bonds that have a coupon of 3% with YTM’s 2%, 3% or 4% and time to maturity 3 years. In Table 3.2 the cash flow analysis can be seen. The second column (t ¼ 0) of Table 3.2 shows the price of the different bonds. In this example, the YTD is given. In practice, the bond is given, and the yield to maturity has to be computed. We illustrate the general principle that if the YTD is below the coupon, then the price is above the par value (premium bond). If the YTD is equal to the coupon, then we have a par bond. And that if YTD is beyond the coupon, the price is below the par value (discount bond). Example 3.8 We want to determine whether the yield of a semi-annual 6% 15 year bond with face value $100 selling at $84.25 is 7.2%, 7.6%, or 7.8%. We compute the present value PVC of the cash flows of the cash by using the formula C PVC ¼ 2

1

1

ð1þ2r Þ , r n

and the PVF of the Face Value F PVF ¼ F 

1 n : 1 þ 2r

The price P of the Bond is the P ¼ PVC þ PVF : Table 3.3 below shows the computed values. We see that for the price $84.25 we have YTM ¼ 7.8%. With YTD ¼ 6.0%, (1) yields P ¼ 100%.

30 Table 3.3 Different yield of maturity

3 The Flat Yield Curve Concept

YTM 0.072 0.074 0.078

Cash flow 54.4913 53.8191 52.5118

Face value 34.6105 33.6231 31.7346

PV of the bond 89.1017 87.4422 84.2465

Definition 3.14 (Current Yield) The current yield or direct yield of a bond j is defined by DYdir,j ¼

Cj : Pj

ð3:2:4Þ

Theorem 3.2 We consider an annual paying bond with price P, yield to maturity r, and coupon C. Then, at the times of knots defined in (2.1.2), the following holds. (a) If P > 100, then C > DY > YTD, (b) if P ¼ 100, then C ¼ DY ¼ YTD; (c) and if P < 100, then C < DY < YTD. Proof As in Theorem 3.1, we consider P1 ¼ Pnþ1 ¼

FþC , 1þr

Pn þ C , 1  n  N  1: 1þr

Then we have P ¼ PN, and we proof the theorem by induction. For N ¼ 1, we have P1 ¼

FþC , 1þr

and thus FþC  1 ¼ r ¼ YTM, P1 and hence FþC  1 ¼ r ¼ YTM, P1 i.e.,

3.2 Yield Measures

31

F  1 þ DY ¼ YTM: P1

ð3:2:5Þ

For a discount bond, the assertion follows from PF1 > 1, for a par bond we have ¼ 1, and for a premium bonds PF1 < 1. We assume that the assertion is true for n and consider

F P1

Pnþ1 ¼

Pn þ C : 1þr

We find by (3) that Pn  1 þ DY ¼ YTM: Pnþ1 The assertion now follows from Theorem 3.1. □ Remark 3.8 The direct yield of a zero coupon bond is 0, which makes little sense. Definition 3.15 Par yield or par rate denotes the coupon rate for which the price of a bond is equal to its nominal value (or par value). In the following, we illustrate Theorem 3.2 and the concept of IRR. Example 3.9 (Discussion of the Cash Flows for a Bond with Two Cash Flows) We consider a Bond with 2 years to maturity and investigate the reinvested rate. By (3.1.1a), we have P¼

C FþC þ : 1 þ r ð1 þ rÞ2

By using the discount factor (2.2.3b) d¼

1 , 1þr

we find that P ¼ C d þ ðF þ CÞ d2 : Hence, 0 ¼ ðF þ CÞ d2 þ C d  P: The solutions are

ð3:2:6Þ

32

3 The Flat Yield Curve Concept 2.00

1.50

1.00

0.50 C=0

Price funcon

C>0

0.00 –2.00

–1.50

–1.00

–0.50

0.00

0.50

1.00

1.50

2.00

C 0, then r > 0 (see Figs. 3.7 and 3.8). Thus, the property b. in Theorem 3.2 is extended to the real numbers. We proceed by assuming that the rate after time 1 is fixed and invest the return. We call this return modified internal rate of return. We consider 2 cases. (a) C > 0 (outflow) The cash flow is discounted to the time t ¼ 0 and P¼

C FþC þ , 1 þ r 0 ð1 þ r Þ2

and thus sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FþC  1: r¼ C P  1þr 0 (b) C < 0 (inflow) The cash flow is discounted to the time t ¼ 2

ð3:2:7Þ

34

3 The Flat Yield Curve Concept

Table 3.4 Modified rate of return

Spot 0 1 2 3 4 5 6 7 8 9 10

Cash flow 5 0.051315 0.051041 0.050773 0.050510 0.050252 0.050000 0.049752 0.049510 0.049272 0.049038 0.048809

Cash flow 5 0.024695 0.029320 0.033925 0.038509 0.043072 0.047616 0.052141 0.056646 0.061131 0.065598 0.070047

ð1 þ rÞ2 P ¼ ð1 þ r0 Þ C þ ðF þ CÞ, i.e., with positive coefficient, we have ð1 þ rÞ2 P  ð1 þ r0 Þ C ¼ F þ C, and then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F þ C þ ð1 þ r0 ÞC r¼  1: P

ð3:2:8Þ

In the numerical example, we chose C ¼ 5% and C ¼ 5% and an internal rate of r ¼ 5%. In Column 2 and 3 of Table 3.4, we see the modified internal rate. The return for an out flow varies more than for an inflow. For outflows, the return goes with the spot rates, for out flow it is the opposite.

3.3

Durations and Convexity

Frederik Macaulay originally began in 1938 to investigate the impact of interest rates on the movements of bonds. For instance, solely looking at the time to maturity does suffice for this purpose. A zero coupon bond and a bond with a coupon with the same time to maturity might have a different behaviour under different rate scenarios. The basic idea of duration is the amalgamation of different characteristics of a straight bond. It helps to manage the risk of bonds and bond portfolios. There are different duration concepts. We start by the duration concept that is based on three independent variables, namely the time to maturity, the coupon and the price or equivalently the time to maturity, the coupon and the yield to maturity.

3.3 Durations and Convexity

35

Definition 3.16 (Macaulay Duration) The Macaulay duration DiMac ðrÞ of a Bond with N Coupons C, Face Value F, the fraction α between coupon payment and an annual yield to maturity r measures the average life of Bond j and is defined as the weighted average of the time to the next cash flows, namely N P

DjMac ðrÞ

¼

j¼1

ðj‐1 þ αÞ ð1þrCÞj‐1þα þ ðN‐1 þ αÞ N P

C j‐1þα

j¼1

ð1 þ rÞ

þ ð1

F ð1þrÞN‐1þα

:

ð3:3:1Þ

F þ rÞN‐1þα

In the nominator, there is the time weighted by the discounted cash flows and is a continuous function (for details see Appendix G). In the denominator we have the invoice price of the bond. As described in Sect. 3.1 (see for example Fig. 3.2) the invoice price dropped when a coupon is due. Thus the Macaulay Duration goes up when a coupon is due. The crucial property of the Macaulay Duration is as follows. At each time, one can evaluate the value a bond. If yields are changing, as of today, the value of the bond is changing due to the changing the reinvestment values of the coupons and the price of the bond changes. Under some assumptions, for the Macaulay Duration, these effects are compensating. For a zero bond the Macaulay duration is equal to the number of years to maturity. This can be seen algebraically from the formula (1), that is we have Lemma 3.1 The Macaulay Duration of a Zero Coupon DiMac ðrÞ ¼ N‐1 þ α: Remark 3.9 A closed formula for the Macaulay Duration is found in Appendix G. Remark 3.10 Furthermore, we see that the unit of Macaulay duration is time. In the following, we want to illustrate the Macaulay Duration. The following formula is the value at issuance of the bond: N P

 Djmac rj ¼

k

k¼1 N P k¼1

C

ð1þrj Þ

k

C k ð1þrj Þ

þN þ

F

ð1þrj Þ

F N ð1þrj Þ

N

¼

ð3:3:2Þ

36

3 The Flat Yield Curve Concept

Table 3.5 Calculating the Macaulay duration

r (%) 2 3 4

Time weighted price 0

Time weighted coupons 1

Time weighted coupon 2

Time weighted coupons + face 3

299.8858 291.3470 283.1318

2.9412 2.9126 2.8846

5.7670 5.6556 5.5473

291.1776 282.7788 274.6999

N P k¼1

k

C F N k þ N ð1þrj Þ ð1þrj Þ  : P rj

The formula is only applicable to times when there is a coupon payment. Furthermore, we see that the unit of Macaulay duration is time. Remark 3.11 The Macaulay duration is invariant under the multiplication by a discount factor and thus (2) is the same as N P

k C k‐N þ N F  ð1þrj Þ Djmac rj ¼ k¼1N : P C þ F k‐N k¼1 ð1þrj Þ Example 3.10 (Continuation of Example 3.7) We consider 3 bonds with 3 years to maturity each and with 3% coupons and yields 2, 3 and 4. Table 3.1 shows the denominator in (2) and the following Table 3.5 shows the nominator in (2) By dividing the numerator by the denominator, the Macaulay durations are 2.9148 2.9135 2.9121 Remark 3.12 The Macaulay duration can be calculated in two ways. Either by evaluating the series, or there is a closed formula (see Appendix G). We note that the Macaulay Duration is less than the time to maturity. In Fig. 3.9, we have a graphical illustration of the Macaulay duration. We see the fulcrum or the equilibrium of the bond, i.e., 50% of the weight is on either side. We see that the scale is at balance. The black area is the discount value of the cash flows.

References

37

Fig. 3.9 Macaulay duration

me now

Duraon (Equilibrium)

References 1. Wolfgang M (2017) Fixed income analytics, 1st edn. Springer, Cham 2. Yuri S, Wolfgang M (2011) Properties of the IRR equation with regards to the ambiguity of calculating the rate of return and a maximum number of solutions. J Perform Meas 15 (3):302–310

4

The Internal Rate of Return for a Bond Portfolio

Abstract

We consider a portfolio of bonds and discuss different yields. We examine the internal rate of return of a bond portfolio and its approximations used in the industry. Our analysis is based on power series. The here presented material is illustrated by numerical examples.

4.1

The Direct Yield of the Portfolio

Based on (3.2.3), we start with the net asset value NAV of a portfolio defined by NAVðrÞ ¼ PB0 

Tj n X X Nj Cj j¼1

¼

n X j¼1

k¼1

ð1 þ rÞk

þ

Nj Fj

!

ð1 þ rÞTj

! Tj n X   X Nj Cj Nj Fj Nj Pj rj  þ , k ð1 þ rÞTj k¼1 ð1 þ rÞ j¼1

ð4:1:1aÞ

for r > 1. With the abbreviations   Pj ¼ Pj rj , j ¼ 1, . . . :, n, and Pj ðrÞ ¼

Tj X k¼1

Cj ð1 þ r Þ

k

þ

Fj ð1 þ rÞTj

,

we have

# Springer Nature Switzerland AG 2020 W. Marty, Fixed Income Analytics, https://doi.org/10.1007/978-3-030-47158-3_4

39

40

4 The Internal Rate of Return for a Bond Portfolio

Fig. 4.1 A realization of a zero of the IRR equation

NAV(r) 1.00 0.50 –0.60

–0.40

–0.20

0.00 0.00 –0.50

0.20

0.40

0.60

–1.00 –1.50 –2.00 –2.50

NAVðrÞ ¼

n X

Nj Pj 

j¼1

n X

Nj Pj ðrÞ:

ð4:1:1bÞ

j¼1

A solution IR of (1), i.e., a real number IR 2 R1 that satisfies NAVðIRÞ ¼ 0, is called an internal rate of return of the portfolio and (1) is then called the equation of the internal rate of return equation. Remark 4.1 In [1], it is shown that (1) has in general multiple solutions. However, in practical situations, there exists only one unique business relevant solution. Thus, in this section, we assume that (1) has at least one real solution IR with ∂ NAVðIRÞ 6¼ 0 ∂ IR and by considering NAVðIRÞ ¼ 0 we can assume without loss of generality that ∂ NAVðIRÞ > 0, ∂ IR i.e., NAV(r) is locally monotonically increasing as seen in Fig. 4.1. Example 4.1 We consider a portfolio of two zero bonds with prices

4.1 The Direct Yield of the Portfolio

41

P1 ðr1 Þ ¼ P2 ð r 2 Þ ¼

F , 1 þ r1

F , ð1 þ r2 Þ2

and choose r1 ¼ 0.05, r2 ¼ 0.025, and F ¼ 1. The condition NAV(r) ¼ 0 leads to solving the quadratic equation for the IRR and with the chosen values we find IR1 ¼ 0:03335918, IR2 ¼ 1:508203: Figure 4.1 shows the NAV(r) function in the neighborhood of IR1. IR2 is not considered in (1) as the discount factor is negative (see also [2]). We consider a bond portfolio with n bonds that have the yield to maturities YTMs r1, . . .,rj, . . .,rn with prices P1, . . .,Pj, . . .,Pn and each position consists of N1, . . ., Nj, . . .,Nn units. For the solution IR, we propose the approximations n X

rnom ¼

wj r j ,

ð4:1:2aÞ

j¼1

where wj ¼

Nj , n P Ni

ð4:1:2bÞ

i¼1

and rlin ¼

n X

b j rj , w

ð4:1:3aÞ

j¼1

where bj ¼ w

Nj  Pj : n P Ni  Pi

ð4:1:3bÞ

i¼1

Following Definition 3.14, we also consider the direct yield r dir of a bond portfolio by denoting by Cj, 1  j  n, the coupons of the bonds with the prices Pj, 1  j  n, rdir ¼

N1 C1 þ N2 C2 þ : . . . þ Nn Cn : N1 P1 þ N2 P2 þ : . . . þ Nn Pn

ð4:1:4Þ

42

4 The Internal Rate of Return for a Bond Portfolio

Lemma 4.1 (Linearity of the Direct Yield) If we consider a portfolio of n bonds (see Definition 3.11) with coupons Cj, 1  j  n with Nj, 1  j  n units, then the direct yield is linear, i.e., the direct yield of a portfolio is the market price weighted direct yield of the individual bonds of the portfolio. Proof By (3b) and (4), we have by using the definition (3.2.4) rdir ¼

N1 C1 N C N C þ n2 2 þ :...... þ n n n n P P P Ni Pi Ni Pi Ni Pi i¼1

i¼1

i¼1

N P N C N P N C N P N C ¼ n 1 1 1 1 þ n 2 2 2 2 þ :... þ n n n n n P P P Ni Pi  N1 P1 Ni Pi  N2 P2 Ni Pi  Nn Pn i¼1

b1 ¼w

i¼1

i¼1

N1 C1 N C N C b 2 2 2 þ :: . . . þ w bn n n ¼ þw N1 P1 N2 P2 Nn Pn

n X

wj rdir,j :

j¼1

□ In the following, lemma assumes a flat yield curve, i.e., the YTM is the same for all bonds in a specific bond universe. The flat yield concept introduced in Definition 3.6, however, refers to a specific bond. Different discount factors are in general applied to different bonds. The assumption of the following lemma is thus more restrictive that the flat yield concept. The solution IR of (1) is the equal to the YTM. Lemma 4.2 (More Than the Flat Yield Concept) We consider (1) and assume a flat yield curve, i.e., every bond portfolio with n bonds (see Definition 3.11) satisfies with the same yield to maturity, i.e., YTM ¼ rj

ð4:1:5Þ

with Nj, 1  j  n units. Then we have IR ¼ YTM, i.e., NAVðIRÞ ¼ NAVðYTMÞ ¼ 0: Proof We use (1)

ð4:1:6Þ

4.1 The Direct Yield of the Portfolio

NAVðrÞ ¼

n X j¼1

43

! Tj n X   X Nj Cj Nj Fj : Nj Pj rj  þ k ð1 þ r ÞT j k¼1 ð1 þ rÞ j¼1

By using (5), we have NAVðrÞ ¼

n X

Nj Pj ðYTMÞ 

j¼1

Tj n X X Nj Cj j¼1

k¼1

ð1 þ rÞk

þ

Nj Fj ð1 þ rÞTj

! ,

thus YTM is a solution of (1) and the lemma is thus shown.

□

In the following, we discuss the approximations (2), (3), and (4) of the solution IR of (1). The following lemma shows an easy situation where the weights of (3b) are reducing to (2b). Lemma 4.3 We consider (1) and assume that the portfolio consists of n par bonds (see Definition 3.15) with YTMs rj with Nj, 1  j  n units that mature in 1 year. It follows that rnom and rlin are a solution of (1) IR ¼ rnom , IR ¼ rlin , NAVðrnom Þ ¼ 0, NAVðrlin Þ ¼ 0: Proof For a bond with 1 year to run, we have   Fj þ Cj P rj ¼ : 1 þ rj We use the assumptions Pj ¼ 1, Fj ¼ Pj with Cj ¼ rj, and consider the Eq. (1) for the IRR NAVðrÞ ¼

n X j¼1

    n n n X X X Nj Cj þ Fj Nj r j þ 1 ¼ Nj  Nj  : 1þr 1þr j¼1 j¼1 j¼1

Thus, by solving (1b), we have ð1 þ r Þ

n X j¼1

and hence

Nj ¼

n X j¼1

  Nj r j þ 1 ,

44

4 The Internal Rate of Return for a Bond Portfolio

r

n X

Nj ¼

j¼1

n X

Nj r j :

j¼1

By solving for r, we find (2), and as Pj ¼ 1, we have (3).

□

The following Lemma 4.4 is a first generalization of Lemma 4.3. We assume a credit portfolio, i.e., a portfolio with different par yields with the same time to maturity. Lemma 4.4 (Same Time to Maturity) We consider (1) and assume that the bonds have the same time to maturity T ¼ 1, 2, 3,. . ., i.e., Tj ¼ T, 1  j  n and distinguish two cases. (a) (par bonds) The n bond are all par bonds, i.e., the coupon is equal to the YTM, i.e., Cj ¼ rj. Then (2) yields a solution of (1), i.e., rnom ¼ IR with NAVðIRÞ ¼ NAVðrnom Þ ¼ 0,

ð4:1:7aÞ

and (3) yields a solution of (1), i.e., rlin ¼ IR and NAVðIRÞ ¼ NAVðrlin Þ ¼ 0:

ð4:1:7bÞ

(b) (par bonds except one bond) The bonds are par bonds except one bond, i.e., there exist a m, 1  m  n such that C1 ¼ r1 , . . . , Cm1 ¼ rm1 , Cm 6¼ rm , Cmþ1 ¼ rmþ1 , . . . , Cn ¼ rn :

ð4:1:8Þ

We find that the direct yield satisfies the equation Nm Pm þ

n X

Ni 

i¼1 i6¼m

n X Nj j¼1

Cj r

¼ 0:

ð4:1:9Þ

For the approximation of rdir, we have NAVðrdir Þ ¼ Nm R1m ,

ð4:1:10aÞ

where R1m ¼ R1 ðPm , Fm , rdir , TÞ ¼ with

Fm  Pm ð1 þ rdir ÞT

ð4:1:10bÞ

4.1 The Direct Yield of the Portfolio

bj ¼ w

45

Nj Nm Pm þ

ð4:1:11aÞ

n P

Ni

i¼1 i6¼m

and bm ¼ w

Nm Pm : n P Nm Pm þ Ni

ð4:1:11bÞ

i¼1 i6¼m

In (3), we find that the direct yield approximation rdir is related to rlin by rdir ¼ rlin þ

R2 ðCm , Fm , rm , Pm Þ , n P Nm Pm þ Ni

ð4:1:11cÞ

i¼1 i6¼m

where R2m ¼ R2 ðCm , Fm , rm , rm , TÞ ¼ Nm

Cm  rm Fm : ð1 þ r m ÞT

ð4:1:11dÞ

For the approximation properties of the error terms, we are assuming rdir > 0 for T ! 1, which gives Rim ! 0,

i ¼ 1, 2,

ð4:1:12aÞ

Rim ! 0,

i ¼ 1,

ð4:1:12bÞ

and for rm ! Cm 2:

Proof We consider (1) with Tj ¼ T and apply the closed formula (A1.1) of Appendix A for 1jm

P j ðr Þ ¼

  1 Cj 1  ð1þr T Þ r

þ

Fj ð1 þ r ÞT

:

ð4:1:13Þ

46

4 The Internal Rate of Return for a Bond Portfolio

We substitute in (1b)

NAVðrÞ ¼

n X

Nj Pj 

j¼1

n X

0 @

  1 Nj Cj 1  ð1þr T Þ r

j¼1

1 þ

Nj Fj A :

ð1 þ r ÞT

Thus, NAVðrÞ ¼

n  X j¼1

Nj Cj Nj Pj  r



" # n  X Nj Cj 1  : ð4:1:14Þ Nj Fj  r ð1 þ rÞT j¼1

We start with case (a). By assumption, we have Pj ¼ Fj ¼ 1 and rj ¼ Cj, and hence NAVðrÞ ¼

n X j¼1

" # n n n X X X Nj rj Nj r j 1  : Nj  Nj  r r ð1 þ rÞT j¼1 j¼1 j¼1

By solving the first two parts of the sum and the expression in the parentheses of this equation for r, we find r

n X

Nj 

j¼1

n X

Nj rj ¼ 0,

j¼1

i.e., NAVðIRRÞ ¼ NAVðrnom Þ ¼ NAVðrlin Þ ¼ 0: We proceed with case (b) and focus on the assertion formulated in (8). From (13), we have for j ¼ m

Pm ¼

 Cm 1  ð1þr1

 T

mÞ

rm

þ

Fm : ð1 þ r m ÞT

This is the same as Cm ¼ rm  Pm þ

1 ðCm  Fm  rm Þ: ð1 þ r m ÞT

ð4:1:15Þ

By extracting the index m, we have, with (14), Pj ¼ Fj ¼ 1 for j 6¼ m, and

4.1 The Direct Yield of the Portfolio

47

 n  Nj Cj Nm Cm X NAVðrÞ ¼ Nm Pm  Nj  þ r r j¼0 j6¼m

3

2

7 6 6 7 n  X 7 6 N C 1 N C j j m m 7: 6  Nj  þ T 6Nm Fm  7 r r ð1 þ r Þ 6 7 j¼0 5 4 j6¼m

By adding and subtracting the price with the units on the right-hand side, this is the same as NAVðrÞ ¼ Nm Pm 

 n  Nj Cj Nm Cm X Nj  þ r r j¼0 j6¼m

3

2

6 7 n  6 Nj Cj 7 1 Nm Cm X 7 6  þ Nm Pm  Nm Pm þ Nm Fm  Nj  7: T6 r r 7 6 ð1 þ rÞ 4 5 j¼0

ð4:1:16Þ

j6¼m

We neglect the difference Nm Fm  Nm Pm of the right-hand side of the equation and consider the equation by using the assumption that, for j 6¼ m, we have Pj ¼ Fj ¼ 1 and NAVapp ðrÞ ¼ Nm Pm 

 n  Nj Cj Nm Cm X Nj  þ r r j¼0 j6¼m

3

2

7 6 7 6   n 6 Nj Cj 7 1 Nm Cm X 7 6  Nj  þ 7: 6Nm Pm  r r 7 ð1 þ r ÞT 6 7 6 j¼0 5 4 j6¼m

We see that

ð4:1:17Þ

48

4 The Internal Rate of Return for a Bond Portfolio

NAVapp ðrÞ ¼ 0 is the same as r Nm Pm  Nm Cm þ

n  X

 rNj  Nj Cj ¼ 0,

j¼0 j6¼m

i.e., the direct yield n P

rdir ¼

Nj Cj

j¼0

Nm Pm þ

n P

ð4:1:18Þ Ni

i¼1 i6¼m

satisfies (7) and (15). Furthermore, we have by (14) Fm  Pm , ð1 þ r ÞT

NAVðrÞ  NAVapp ðrÞ ¼ Nm and thus NAVðrdir Þ ¼ Nm

Fm  Pm : ð1 þ rdir ÞT

We have shown (8) and we proceed to the relation of rlin and rdir. By using the assumption (6) of the lemma together with (13) and (16), we have n P

Nj

rj þ Nm  Pm  rm þ ð1þr1

mÞ

T

ðCm  Fm  rm Þ

j¼0

rdir ¼

j6¼m

Nm Pm

n P

: Ni

i¼1 i6¼m

Hence, with R2m ¼ R2 ðCm , Fm , rm , TÞ ¼ Nm we have (9)

Cm  rm Fm , ð1 þ r m ÞT

4.1 The Direct Yield of the Portfolio

rdir ¼ rlin þ

49

R2 ðCm , Fm , rm , Pm Þ : n P Nm Pm þ Ni i¼1 i6¼m

The approximation properties (10) follow from case (a) and from the explicit expression for the error term. □ Remark 4.2 There is only a minor difference between case (a) and (b). In (a), we assume that all bonds are par bonds, whereas in (b) we assume that all bonds are par except one. In the following example, we consider a portfolio with three bonds. Applying the above lemma, we see that the IRR reduces to the YTM of a bond of the portfolio. Example 4.2 [Lemma 4.2 Case (a)] We consider a portfolio that consists of 3 units of different par bonds with face value Fj ¼ F, j ¼ 1, 2, 3, i.e., C1 ¼ r1 , C2 ¼ r2 , C3 ¼ r3

ð4:1:19aÞ

with the same time to maturity and with the yield to maturities r1 ¼ r2  α, r3 ¼ r2 þ α, α 2 R1 :

ð4:1:19bÞ

We evaluate the assumption of the example by (1a) and find first by the par assumption that 3 X   Pj rj ¼ 3P2 ðr2 Þ: j¼1

Furthermore, we have by (19) for 1  j  n C1 C α ¼ 2 , ð1 þ r Þj ð1 þ r Þj C3 C þα ¼ 2 : j ð1 þ r Þj ð1 þ r Þ For the portfolio P with PðrÞ ¼ P1 ðrÞ þ P2 ðrÞ þ P3 ðrÞ, we have

ð4:1:20aÞ

50

4 The Internal Rate of Return for a Bond Portfolio

3PðrÞ ¼

n X j¼1

3C2 3F þ n: ð1 þ r Þj ð1 þ r Þ

ð4:1:20bÞ

Consequently, we have by (1a) and (20) P2 ðr2 Þ ¼ PðrÞ:

ð4:1:21Þ

Then, we have by (19b) IR ¼ r2 : By evaluation of (2), we have wj ¼

1 , j ¼ 1, 2, 3, . . . , 3

and thus rnom ¼

n X rj : 3 j¼1

By using (19b), we have rnom ¼ r2 , and hence IR ¼ rnom : As we assume par bonds, we find with the evaluation of (3) IR ¼ rlin : The following example illustrates Lemma 4.6. Example 4.3 (Lemma 4.6 [Case (a) and (b)] with Two Bonds of Same Time of Maturity) We consider two bonds with Tj ¼ T ¼ 2, j ¼ 1, 2. The price of the first bond is P1 ðrÞ ¼

2 X j¼1

C1 F1 þ j ð1 þ r Þ ð1 þ r Þ2

and is par, i.e., r ¼ C1. We choose F1 ¼ 1 and C1 ¼ 8%. The price of the second bond is

4.1 The Direct Yield of the Portfolio Table 4.1 Approximation error of the yield to maturity

51

C2 0.00000000 2.00000000 4.00000000 6.00000000 8.00000000 10.00000000 12.00000000 14.00000000 16.00000000

P2 ðrÞ ¼

2 X j¼1

NAV(dir), R1 2 13.11224095 9.65281093 6.32190018 3.10775779 0.00000000 3.01058057 5.93217499 8.77209282 11.53687628

R2 2 6.85871056 5.14403292 3.42935528 1.71467764 0.00000000 1.71467764 3.42935528 5.14403292 6.85871056

C2 F2 þ : ð1 þ rÞj ð1 þ rÞ2

We choose F2 ¼ 1 and a different value for the coupon C2. In addition, we use the price P2 such that YTM ¼ 0.08. We consider the bond portfolio PðrÞ ¼ P1 ðrÞ þ P2 ðrÞ: We look at the different approximations of the IRR. As both the yields to maturity are rk ¼ 0:08, k ¼ 1, 2,

ð4:1:22aÞ

IR ¼ rnom ¼ rlin ¼ 0:08:

ð4:1:22bÞ

we have

Comparing rnom and rlin, the numerical value shows that the direct yield only approximates the IR. As the time of maturity is short, the approximation of the direct yield is rather bad as it does not reflect time value and assume time to maturity infinity (compare Theorem 3.7). In Table 4.1, we show the error if substituting the direct yield instead of IR in (1), i.e. R21 . For the cash flow C ¼ 8%, we expand the above Eq. (22b) with IRR ¼ rnom ¼ rlin ¼ rdir ¼ 0:08: Introducing the error terms (10a) and (11d) for par bonds, we see that R11 ¼ 0 and ¼ 0: Furthermore, R22 indicates the error from the par. We see that we have discount and premium bonds for C different from C ¼ 8%. The following example shows that the Lemma 4.3 cannot be generalized to different time to maturities. R12

Example 4.4 (Different Time to Maturity) We consider two par bonds with time to maturities T1 ¼ 1 und T2 ¼ 2. We have

52

4 The Internal Rate of Return for a Bond Portfolio

Table 4.2 Par bond

C2 0.0500000 0.0600000 0.0700000 0.0800000 0.0900000 0.1000000 0.1100000 0.1200000 0.1300000 0.1400000 0.1500000

IRR 0.050000 0.056606 0.063199 0.069778 0.076344 0.082897 0.089437 0.095965 0.102481 0.108984 0.115475

P1 ðrÞ ¼

rlin 0.050000 0.055000 0.060000 0.065000 0.070000 0.075000 0.080000 0.085000 0.090000 0.095000 0.100000

IRR–rlin 0.000000 0.001606 0.003199 0.004778 0.006344 0.007897 0.009437 0.010965 0.012481 0.013984 0.015475

F1 þ C1 1þr

with r ¼ 0.05, C1 ¼ r, and F1 ¼ 1, and P2 ðrÞ ¼

2 X j¼1

C2 F þ C2 þ 2 : j ð1 þ r Þ ð1 þ r Þ2

with r ¼ C2 and F2 ¼ 1. We find that rnom ¼ rlin ¼ rdir :

ð4:1:23Þ

However, from Table 4.2, we see that all measurement deviates from IR, and the difference is increasinfg with increasing cash flow. Thus, Lemma 4.3 is not valid. We proceed with a generalization of Lemma 4.3. Theorem 4.1 (Same Time to Maturity) We consider (1) and assume that the bonds have the same time to maturity, i.e., Tj ¼ T, 1  j  n, and the first m bonds are not par. C1 6¼ r1 , . . . , Cm1 6¼ rm1 , Cm 6¼ rm , Cmþ1 ¼ rmþ1 , . . . , Cn ¼ rn :

ð4:1:24Þ

We find that the direct yield satisfies the equation m X j¼1

Nj Pj

þ

n X

Ni 

j¼mþ1

For the approximation of rdir, we have

n X Nj Cj ¼ 0: r j¼1

ð4:1:25Þ

4.1 The Direct Yield of the Portfolio

53

NAVðrdir Þ ¼

m X

ð4:1:26aÞ

R1j ,

j¼1

where   R1j ¼ R1 Pj , Fj , rdir , T ¼ Nj

Fj  Pj

:

ð4:1:26bÞ

, j ¼ 1, 2, . . . , m,

ð4:1:27aÞ

ð1 þ rdir ÞT

With bj ¼ w

m P

Nj Pj Ni Pi þ

i¼1

n P

Ni

i¼mþ1

and bj ¼ w

m P i¼1

Nj Ni Pi þ

n P

, j ¼ m þ 1, 2, . . . , n,

ð4:1:27bÞ

Ni

i¼mþ1

in (3), we find that the direct yield approximation rdir is related to rlin by m P

rdir ¼ rlin þ

i¼1 n P

  R2 Cj , Fj , rj , Pj Ni Pi þ

n P

i¼1

i¼1

i6¼m

i6¼m

,

ð4:1:27cÞ

Ni

where   Cj  rj Fj R2i ¼ R2 Ci , Fj , ri , T ¼ Nj  T : 1 þ rj

ð4:1:27dÞ

For the approximation properties of the error terms, we are assuming rdir > 0 for T ! 1 and for j ¼ 1,. . .,m we get Rij ! 0, i ¼ 1, 2,

ð4:1:28aÞ

Rij ! 0, i ¼ 1, 2:

ð4:1:28bÞ

and for rj ! Cj

Proof By extracting the indices j ¼ 1, . . ., m, we have, with (12) and

54

4 The Internal Rate of Return for a Bond Portfolio

Pj ¼ Fj ¼ 1, j ¼ m þ 1, 2, . . . , n,    m  n X X Nj Cj Nj Cj Nj Pj  Nj  þ r r j¼0 j¼mþ1 "   # m  n X X Nj Cj Nj Cj 1  Nj Fj  Nj  þ : r r ð1 þ rÞT j¼1 j¼mþ1

NAVðrÞ ¼

By adding and subtracting the price with the units on the right-side hand, this is the same as NAVðrÞ ¼

m  X

   n X Nj Cj Nj Cj Nj  þ r r j¼1 j¼mþ1 "   # m  n X X Nj Cj Nj C j 1 Nj Pj  Nj Pj þ Nj Fj  Nj  : þ  r r ð1 þ rÞT j¼1 j¼mþ1 Nj P j 

ð4:1:29Þ

We neglect the difference Nj Fj  Nj Pj on the right-hand side (22) of the equation and consider the equation by using the assumption that for j 6¼ m we have Pj ¼ Fj ¼ 1.    m  n X X Nj Cj Nj Cj Nj Pj  Nj  þ r r j¼1 j¼mþ1 "   # m  n X X Nj Cj Nj Cj 1  Nj Pj  Nj  þ : r r ð1 þ rÞT j¼1 j¼mþ1

NAVapp ðrÞ ¼

We see that NAVapp ðrÞ ¼ 0 is the same as m  X j¼1

i.e., the direct yield

n X    rNj Pj  Nj Cj þ r Nj  Nj Cj ¼ 0, j¼mþ1

ð4:1:30Þ

4.1 The Direct Yield of the Portfolio

55 n P

rdir ¼

Nj Cj

j¼0 m P i¼0

ð4:1:31Þ

n P

Ni Pi þ

Ni

i¼mþ1

satisfies (23) and (28). Furthermore, we have by (27) NAVðrÞ  NAVapp ðrÞ ¼

m X

Nj

j¼1

Fj  Pj ð1 þ r ÞT

,

and thus NAVðrdir Þ ¼

m X

Nj

j¼1

Fj  Pj ð1 þ rdir ÞT

:

We have shown (24) and we proceed to the relation of rlin and rdir. By using the assumption (22) of the lemma together with (13) and (29), we have m P

rdir ¼

Nj  Pj  rj þ

i¼1

1 T ð1þrj Þ

m P



n  P Cj  Fj  rj þ Nj rj j¼mþ1

Ni Pi þ

i¼1

n P

:

Ni

i¼mþ1

Hence, we have (25) rdir ¼ rlin þ

m X j¼1

Cj  rj Fj Nj  T : 1 þ rj

The approximation properties (26) follow from case (a) and from the explicit expression for the error term. □ Definition 4.1 We denote with Rdir the residual value of rdir, i.e., the value that results by evaluating (1) by rdir instead of the solution IR. Remark 4.3 The residual Rdir in Lemma 4.3 is (8), and in Theorem 4.1 is based on (24) Rdir ¼

m X

  R1 Pj , Fj , rdir , T :

j¼1

We will investigate the residual value of the rnom and rlin.

56

4 The Internal Rate of Return for a Bond Portfolio

Corollary 4.1 For a portfolio consisting solely of perpetual bonds, the yield to maturity is the same as the direct and the solution of the (1) is equal to (2) and (3). Proof The assertion follows from the asymptotic behavior stipulated in Theorem 4.1 and Lemma 4.1. □ The above Theorem 4.1 gives explicit expressions for the direct yield, the connection between the linear approximation and the direct yield, and the residual value evaluated in the NAV Eq. (1a)

4.2

Different Approximation Scheme for the Internal Rate of Return

In this paragraph we start with the analysis of rnom and rlin defined in (4.1.2) and (4.1.3). The following Lemma 4.5 investigates the first order approximation and second approximation of the discount factor for the solution of internal rate of return (independent variable) versus the discount factor of the yield to maturity of a bond (see also Fig. 4.2). We use in the following extensively the Landau symbol. This indicates the magnitude of rest of Taylor series and asses the goodness of the approximation. The precise definition is in Appendix B. We derive the following identities: Lemma 4.5 We consider in (4.1.1) the discount factors 

1 1þr

k

and consider the following cases: (a) For k ¼ 1 and rj, j ¼ 1,. . ., n with rj 6¼  1, and for r 6¼  1, we have

Fig. 4.2 The increment of the IRR equation

1+r

1 1+rj

4.2 Different Approximation Scheme for the Internal Rate of Return

57

1 1  ¼ 1þr 1 þ rj               2  1  r  rj þ r  rj r þ O r3 þ O r2 rj þ O r rj þ O rj 3 : ð4:2:1Þ (b) For k ¼ 2, 3,. . . and rj, j ¼ 1,. . ., n with rj 6¼ 1, and for r 6¼, 1 we have        kð k þ 1Þ kð k  1Þ 1 1 r rj ¼  k 1  k r  r j þ r  r j 2 2 ð1 þ r Þk 1 þ rj            2 þ O r3 þ O r2 rj þ O r rj þ O rj 3 : ð4:2:2Þ Proof We consider case (a) and start with the reformulation that 1 1þr

¼

1 1 ¼ ¼ 1 þ r  rj þ rj 1 þ rj þ r  rj 

1 ¼ 1 þ rj

1   r  rj 1 þ rj 1þ 1 þ rj !  2  3   r  rj r  rj r  rj 1 þ  E1 r, rj , 1 þ rj 1 þ rj 1 þ rj 

ð4:2:3aÞ

where   E1 r, rj ¼

1 rr : 1 þ 1þrjj

ð4:2:3bÞ

We proceed by using  2   1 ¼ 1  r j þ r j E2 r j , 1 þ rj

ð4:2:4aÞ

where   E2 r j ¼ Thus

1 , 1 þ rj

ð4:2:4bÞ

58

4 The Internal Rate of Return for a Bond Portfolio

D  2  E r  rj  ¼ r  rj 1  rj þ rj E2 rj 1 þ rj With (3), we have by ordering the powers of r  rj        2   r  rj 2  1  r  rj rj E2 rj rrj ¼ 1  r  r j þ r  rj rj þ 1 þ r 1 þ 1þrj j  3   r  rj þ E1 r, rj : 1 þ rj With the O-Symbol, we have by (4)       r  rj 2 1 rr ¼ 1  r  rj þ r  rj r j þ 1 þ rj 1 þ 1þrjj           2 3 þ O r3 þ O r2 rj þ O r rj þ O rj      2   2  2 ¼ 1  r  rj þ r  rj rj þ r  rj 1  rj þ rj E2 rj            2 3 þ O r3 þ O r2 rj þ O r rj þ O rj      2     ¼ 1  r  rj þ r  rj r j þ r  rj þ O r3 þ O r2 rj        2 3 þ O r rj þ O rj : Thus, we have 1 1þr

       1  1  r  r j þ r  rj rj þ r  rj 2 ¼ 1 þ rj      3     2 þ O rj 3 þO r þ O r2 rj þ O r rj                1  1  r  r j þ r  rj r þ O r3 þ O r 2 rj þ O r rj 2 þ O rj 3 , ¼ 1 þ rj

and we find (1). We proceed with k ¼ 2,. . ., j ¼ 1,. . .,n, and 1 1 1 1 ¼ k ¼  k ¼  k , k  rr ð1 þ rÞk 1 þ r  rj þ rj 1 þ rj þ r  rj 1 þ 1þrjj 1 þ rj and, with (3) and (4), we get, by applying the binomial series, that

4.2 Different Approximation Scheme for the Internal Rate of Return



1 rr

1 þ 1þrjj

59

!k     r  rj r  rj 2 r  rj 3   1 þ  E1 r, rj 1 þ rj 1 þ rj 1 þ rj

k ¼

!p     r  rj r  rj 2 r  rj 3   ¼ þ  E1 r, rj  1 þ rj 1 þ rj 1 þ rj p p¼0 !     r  rj r  rj 2 r  rj 3   ¼1k  þ E1 r, rj 1 þ rj 1 þ rj 1 þ rj !2       ! r  rj 2 r  rj 3   r  rj 3 k ðk  1 Þ r  r j þ  þ E1 r, rj þO : 2 1 þ rj 1 þ rj 1 þ rj 1 þ rj k X

k

!

By arranging the linear and quadratic terms, we find, by using (3), 

    ! r  r j kð k þ 1Þ r  r j 2 r  rj 3 þO k ¼ 1  k 1 þ r þ 2 1 þ rj 1 þ rj rrj j

1

1 þ 1þrj

   2   ¼ 1  k r  r j 1  r j þ r j E2 r j  2   2  2 r  rj 3 kð k þ 1Þ  r  r j 1  r j þ r j E2 r j þO 2 1 þ rj       kð k þ 1Þ 2 r  rj ¼ 1  k r  rj þ k r  rj rj þ 2            2 þ O r3 þ O r2 rj þ O r r j þ O rj 3       kð k þ 1Þ kð k  1Þ ¼ 1  k r  rj þ r  rj r rj 2 2            2  3 þ O r3 þ O r2 rj þ O r r j þ O rj , þ

and thus we have        kð k þ 1Þ kð k  1Þ 1 1 r rj ¼ k 1  k r  r j þ r  r j 2 2 ð1 þ r Þk 1 þ rj             2 3 þ O r3 þ O r2 rj þ O r rj þ O rj , i.e., we have obtain (2).

□

From Lemma 4.5, we have the following corollaries. Corollary 4.2 is a preparation for Theorem 4.2, whereas Corollary 4.3 is a preparation for Theorem 4.3. Corollary 4.2 (Linear Approximation) For k ¼ 2, 3,. . . and rj, j ¼ 1,. . ., n, with rj 2 R1, rj 6¼  1, and r 2 R1, r 6¼  1, we have

60

4 The Internal Rate of Return for a Bond Portfolio

     1 1 ¼ k 1  r  r j þ O r  r j : k ð1 þ r Þ 1 þ rj Corollary 4.3 (Quadratic Approximation) For k ¼ 1, 2, 3,. . . and rj, j ¼ 1,. . ., n, with rj 2 R1, rj 6¼  1, and r 2 R1, r 6¼  1, we have         2   1 1 2 ¼ 1  k r  r : þ O r þ O r r þ O rj   j j k k ð1 þ r Þ 1 þ rj Definition 4.2 An interval I δ, δ > 0, δ 2 R 1 of Cj, j ¼ 1,..,n, as depicted in Fig. 4.3 is defined by the following set:

  Iδ ¼ x 2 Cj  δ, Cj þ δ : The following theorem characterizes the residual of the linear approximations. The part of the portfolio that matures in 1 year has a quadratic error term, and the rest of the portfolio has a linear error term. Theorem 4.2 (Linear Approximation) We consider a portfolio with n bonds and yield to maturities r1,. . .,rj,. . .,rn, 1  j  n, that have time to maturities Tj ¼ 1, 1  j  m, and Tj > 1, Tj 2 N, m þ 1  j  n: By substituting rnom, rlin, resp., in (4.1.1) instead of IR, we have the following expression for the residual value: ! ! ! n X     Nj Fj þ Nj Cj  N j C j      r  rj r NAVðrÞ ¼ r  rj r þ 1 þ rj 1 þ rj j¼1 j¼mþ1 ! Tj n X X    Nj Cj Nj Fj þ  T j ð k  1Þ r  r j  k þ  1 þ rj k¼2 1 þ rj j¼mþ1     2   2 þ O r þ O r rj þ O rj m X

ð4:2:5Þ

Cj

Fig. 4.3 An interval of the real axis

δ>0

4.2 Different Approximation Scheme for the Internal Rate of Return

61

and there exists an δ > 0 and a Iδ(Cj), 1  j  m, such for all YTM r1,. . .,rj,. . .,rn, in this interval we have (5) for the residual NAV(rnom), NAV(rlin), resp. More precisely, by substituting rnom, rlin in (4.1.1) instead of IR, we find with (5) the deviation from O when using the approximations rnom, rlin instead of IR. Thus, for a portfolio that have bonds with 1 year to maturity (k ¼ 1, m ¼ n, resp.), we have quadratic approximation otherwise we have only linear approximation (k > 1, m > n, resp.). Proof We start by assuming that the YTM are par, i.e.,     rj ¼ Cj , P rj ¼ F rj , 1  j  n:

ð4:2:6Þ

With (4.1.1a) we find by the hypothesis (6) NAVðrÞ ¼

n X

Tj n X   X Nj Fj rj 

j¼1

j¼1

k¼1

Nj Cj ð1 þ r Þk

þ

Nj Fj ð1 þ r ÞT j

! :

We consider three parts. The first part are the bonds maturing in 1 year, and the second are the coupons of the bonds that have maturity longer than 1 year and are due in 1 year. The third part are the cash flows (i.e., coupons and face value) that are due after the first year.  n n X X   Nj Cj Nj Fj Nj Cj Nj Fj rj þ þ þ 1 þ r 1 þ r 1 þ r j¼1 j¼1 j¼mþ1 j¼mþ1 !  Tj m m  X X X N j Cj Nj Fj Nj Cj Nj Fj  þ þ ¼ k 1 þ rj 1 þ r j ð1 þ rÞTj j¼mþ1 k¼2 ð1 þ rÞ j¼1     n X Nj Cj N j Fj Nj Cj Nj Cj þ þ þ  1 þ r 1 þ r 1 þ rj 1 þ r j¼mþ1 ! !! T Tj j n X N j Cj X X Nj Fj N j Cj N j Fj  T   k þ   k þ  Tj 1 þ rj j 1 þ rj k¼2 1 þ rj k¼2 1 þ rj j¼mþ1     m  n X X Nj Cj N j Cj Nj Fj N j Fj Nj Cj N j Cj ¼ þ þ    1 þ rj 1 þ r 1 þ rj 1 þ r 1 þ rj 1 þ r j¼1 j¼mþ1 ! ! Tj n X N j Cj X N j Cj Nj Fj N j Fj þ þ :  k þ Tj þ ð1 þ rÞTj ð1 þ rÞk 1 þ rj j¼mþ1 k¼2 1 þ rj

NAVðrÞ ¼

m X

m   X Nj Fj rj 



We use Lemma 4.5 in case 1 (k ¼ 1) and Corollary 4.2. As Corollary connects the discount factors of the internal rate of return and the yield of maturity, we find by canceling the terms of order zero

62

4 The Internal Rate of Return for a Bond Portfolio

  m  X     Nj Fj þ Cj   NAVðrÞ ¼  r  rj þ r  rj rj 1 þ rj j¼1   m X Nj Cj        r  rj þ r  rj rj þ 1 þ rj j¼1     3     2 þ O r þ O r2 rj þ O r rj þ O rj 3 ! Tj n X X    Nj Cj Nj Fj þ Tj k r  rj  k þ  1 þ rj 1 þ rj j¼mþ1 k¼2         2 þ O r2 þ O r rj þ O rj : By neglecting the cubic terms, we get.   m X     Nj Fj þ Cj   NAVðrÞ ¼  r  rj þ r  rj rj 1 þ r j j¼1   m X Nj Cj        r  rj þ r  rj rj þ 1 þ rj j¼1 ! Tj n X X    Nj Cj Nj Fj þ Tj  r  rj  k þ  1 þ rj 1 þ rj j¼mþ1 k¼2 ! Tj n X X    Nj Cj Nj Fj þ Tj ðk  1Þ r  rj  k þ  1 þ rj 1 þ rj j¼mþ1 k¼2         2 þ O r2 þ O r rj þ O r j : We consider    m  m  X  X  Nj Fj þ Cj   Nj Cj   NAVapp ðrÞ ¼  r  rj þ  r  rj 1 þ rj 1 þ rj j¼1 j¼1 ! Tj n X X    Nj Cj Nj Fj þ Tj  r  rj :  k þ  1 þ rj 1 þ rj j¼mþ1 k¼2 By solving for r, we find (4.1.2) and (4.1.3) under the assumption (6). As the Landau symbols are valid on an interval, we can perturb to a not par yield rnom. The same applies for rnom. Thus, we have the residual value (5) for all bonds in the neighborhood of par bond. □ Analogously in (4.1.1), we denote with DjMac ðrÞ the Macaulay duration and with modified duration of bond j with yield of maturity rj. We proceed by the

DjMod ðrÞ the

4.2 Different Approximation Scheme for the Internal Rate of Return

63

following two proposals for the approximation of the IR, denoted by rmac and rmod; more specially we consider rmac ¼

n X

vj r j ,

ð4:2:7aÞ

j¼1

and with the abbreviation in (4.1.1b), we consider   Nj Pj Djmac rj vj ¼ n  , P Ni Pi Dimac rj

ð4:2:7bÞ

i¼1

and, similarly, for rmod ¼

n X

e vj r j

ð4:2:8aÞ

j¼1

and again with the abbreviation in (4.1.1b)   Nj Pj Djmod rj e vj ¼ n  : P Ni Pi Dimod rj

ð4:2:8bÞ

i¼1

Lemma 4.6 For a portfolio consisting of n bonds (see Definition 3.11) with yield to maturity r1,. . .,rj,. . .,rn and with Nj, 1  j  n, units that matures in 1 year, it follows that IR ¼ rlin ¼ rmac : And assuming in addition that the yield to maturity are the same, i.e., YTM ¼ rj , 1  j  n, we have IR ¼ rmod : Proof For a bond with 1 year to run, we have   Fj þ Cj P rj ¼ : 1 þ rj

64

4 The Internal Rate of Return for a Bond Portfolio

We use the assumptions P ¼ 1, Fj ¼ Pj with Cj ¼ rj, and consider the equation for the IR rate of return (4.1.1) n X j¼0 n X

  n X Nj Fj þ Cj , Nj Pj ¼ 1þr j¼0

Nj Pj ð1 þ rÞ ¼

j¼0

n X

  Nj Pj 1 þ rj :

j¼0

By solving for r, we find (4.1.3).

□

Lemma 4.7 We assume a flat curve, i.e., for a portfolio consisting of n bonds (see Definition 3.9) with yield to maturities r1,. . .,rj,. . .,rn, with Nj, 1  j  n units, we assume YTM ¼ rj , 1  j  n: Then there follows IR ¼ rnom ¼ rlin ¼ rmac ¼ rmod : Proof As the weights in (4.1.2), (4.1.3), (7), and (8) add up to 1, and all YTM in the portfolio are the same we have YTM ¼ rnom ¼ rlin ¼ rmac ¼ rmod , □

which entails the assertion by Lemma 4.1.

We proceed by analyzing rmac and rmod. In the following theorem, we investigate the residuals of rmac and rmod. Theorem 4.3 (Approximation Macaulay Duration) We consider a portfolio with n Bonds and yield to maturities r1 > 0,. . .,rj > 0,. . ., rn > 0, 1  j  n, that have time to maturities Tj  1, Tj 2 N, 1  j  n. Then we have.     kð k  1Þ kð k þ 1Þ r r r  r þ  k mac j j 2 2 1 þ rj j¼1 k¼1        Tn X   Tj Tj  1 Tj Tj þ 1 Nj Fj þ rj þ r rmac  rj  T 2 2 1 þ rj j j¼1       2    þ O r3mac þ O r2mac rj þ O rmac rj þ O rj 3 ,

NAVðrmac Þ ¼

Tn n X X

Nj Cj



ð4:2:9Þ

4.2 Different Approximation Scheme for the Internal Rate of Return

65

and there exists a δ > 0 and a Iδ (Cj), 1  j  n, such that, for all YTM r1,. . .,rj,. . .,rn in this interval we have (9) for the residual NAV(rmac). More precisely, by substituting rmac in (9), we find the deviation from O when using the approximation rmac instead of IRR. Proof We start by assuming that the YTM are par, i.e., rj ¼ Cj , Pj ¼ Fj , 1  j  n: With (1a), we find ! Tj n X   X Nj Cj Nj Fj NAVðrÞ ¼ Nj Pj rj  þ k ð1 þ rÞTj k¼1 ð1 þ rÞ j¼1 j¼1 ! !! Tj Tj n X X X Nj Cj Nj Fj Nj Cj Nj Fj ¼ þ T   k þ  k ð1 þ rÞTj 1 þ rj j k¼1 1 þ rj k¼1 ð1 þ rÞ j¼1 * T + j n X X Nj Cj Nj Cj Nj Fj Nj Fj ¼ : þ Tj   k  k ð 1 þ rÞTj ð 1 þ r Þ 1 þ rj k¼1 1 þ rj j¼1 n X

We use Lemma 4.5 and find by canceling the terms of 0th order *

* T ++ j n X X   Nj Cj   Nj Fj   NAVðrÞ ¼ T Tj r  rj  k k r  r j þ  1 þ rj j k¼1 1 þ rj j¼1 * T  j n X X  kð k  1Þ Nj Cj  kð k  1Þ  rj þ r  k r  rj 2 2 k¼1 1 þ rj j¼1       Tj Tj  1 Tj Tj þ 1 Nj Fj  rj þ r þ T r  r j 2 2 1 þ rj j          E 2 þO r3 þ O r2 rj þ O r rj þ O rj 3 : ð4:2:10Þ We consider the approximation * T + j n X X    Nj Cj   Nj Fj  NAVapp ðrÞ ¼ T Tj r  rj  k k r  rj þ  1 þ rj j k¼1 1 þ rj j¼1 of (10). By solving for r, we find (7), and we have the residual (9). As the Landau symbols are valid on the interval, we can perturb the not par yield. Thus, we have the residual value (9) for all bonds in the neighborhood of the par bonds. □

66

4 The Internal Rate of Return for a Bond Portfolio

4.3

Macaulay Duration Approximation Versus Modified Duration Approximation

In the following theorem, we show that the modified duration yields a worse approximation than the Macaulay duration of the IRR. Theorem 4.4 (Macaulay Duration Versus Modified Duration) We consider a portfolio consisting of n bonds that have YTMs with time to maturities 1 ¼ T 1  . . .  Tj  . . .  TN, Tj 2 N, 1  j  N. We assume a flat curve, i.e., YTM ¼ rj , 1  j  n:

ð4:3:1Þ

Then we have a solution IR of (4.1.1) with IR ¼ YTM with rmac ¼ IR, rmod ¼ IR and there exists a δ 2 R 1, δ > 0, and an interval Iδ ¼ (δ, δ) such that IR 2 Iδ

ð4:3:2Þ

r1 , . . . :, r j , . . . :, rn

ð4:3:3Þ

and the YTM

are in Iδ with 0 < rmod  rmac  IR: More specifically, if (1) is satisfied, we have rmod ¼ rmac ¼ IR,

ð4:3:4Þ

and if there exists an index j, 1  j  n, such that rj 6¼ IR,

ð4:3:5Þ

0 < rmod < rmac < IR:

ð4:3:6Þ

then we have

Proof We start by assuming that the YTM are par, i.e.

4.3 Macaulay Duration Approximation Versus Modified Duration Approximation

67

rj ¼ Cj , Pj ¼ Fj ¼ 1, 1  j  n: Following Remark 4.1, there exists a δ1 2 R j ¼ 1,. . .,n, in

1,

0 < δ1 < δ such that, for rj,

Iδ1 ¼ ðδ1 , δ1 Þ, we have (a) for (2) and (3), we have IR 2 Iδ1 (b) NAV(r), r 2 (δ1, δ1) is monotonically decreasing (c) IR is the only solution of (1) in Iδ1 We first consider (1). Then, from (1), we have NAVðIRÞ ¼ 0 and as the weights (4.2.7) and (4.2.8) sum up to one we have rmac ¼ IR and rmod ¼ IR and thus NAVðrmac Þ ¼ 0, NAVðrmod Þ ¼ 0: We proceed with the assertion (6) and by (4.1.1), we have * T + * T + j j n n X X X X Nj Cj Nj Nj Cj Nj NAVðrÞ ¼ þ T   k þ  k ð1 þ rÞTj 1 þ rj j k¼1 1 þ rj k¼1 ð1 þ rÞ j¼1 j¼1 * T + * T + j j n n X X X X Nj Cj Nj Cj Nj Nj ¼ þ  k   k  ð1 þ rÞTj ð1 þ rÞk k¼1 1 þ rj k¼1 1 þ rj j¼1 j¼1 From Lemma 4.5 we have for r ¼ IR       kð k þ 1Þ kð k  1Þ 1 1 IR  rj ¼ k 1  k IR  rj þ IR  rj 2 2 ð1 þ IRÞk 1 þ rj           2 þO IR3 þ O IR2 rj þ O IR rj þ O rj 3 This the same as

68

4 The Internal Rate of Return for a Bond Portfolio

     1 1 ¼ k 1  k IR  rj ð1 þ IRÞ  k IR  rj IR k ð1 þ IRÞ 1 þ rj         kðk þ 1Þ kðk  1Þ       2 þ IR  rj þ O rj 3 , IR rj þ O IR3 þ O IR2 rj þ O IR rj 2 2

thus     2 kðk  1Þ  1 1 IR  r ¼ 1  k IR  r ð 1 þ IR Þ þ  k j j 2 ð1 þ IRÞk 1 þ rj           2 þ O IR3 þ O IR2 rj þ O IR rj þ O rj 3 : We sum up and find *

n X

*

Tj X

  Nj Cj   Nj Fj   NAVðIRÞ ¼ Tj Tj IR  rj  k k IR  rj ð1 þ IRÞ þ  1 þ rj 1 þ rj k¼1 j¼1 * T  2 j n X X  Nj Cj kðk  1Þ  þ IR  rj  k 2 k¼1 1 þ rj j¼1    +  2     Tj Tj  1  Nj Fj IR  r O IR3 þ O IR2 rj þ  Tj j 2 1 þ rj      E 2 ¼ 0: þ O rj 3 þO IR rj

!!+

We neglect the cubic terms and define the NAV1(r) !+!+   Nj Cj   Nj Fj   NAV1 ðrÞ ¼ , T Tj r  rj   k k r  r j ð1 þ r Þ  1 þ rj j 1 þ rj k¼1 j¼1 * T      + j n X X  2  2 Tj Tj  1  Nj Cj Nj Fj kðk  1Þ  r  rj  r  rj : þ T  k 2 2 1 þ rj j k¼1 1 þ rj j¼1 n X

*

Tj X

Then there exists a δ2 2 R1, δ2 > 0 with 0 < δ2 < δ1 lδ2 ¼ ðδ2 , δ2 Þ, and a IR0 in Iδ2 such that NAVðIR0 Þ ¼ 0 and as the quadratic error are positive we have *

n X j¼1

 Nj Cj    k k IR0  rj ð1 þ IR0 Þ 1 þ rj

!

 Nj Fj   þ T Tj IR0  rj ð1 þ IR0 Þ0: 1 þ rj j

The there exists a δ3 2 R1, δ3 > 0 with O < δ3 < δ3 with

+ > 0:

4.3 Macaulay Duration Approximation Versus Modified Duration Approximation

69

Iδ3 ¼ ðδ3 , δ3 Þ, such that *

*

n X

Tj X k¼1

j¼1

    Nj Fj   Tj Tj IR  rj  k k IR  rj þ  1 þ rj 1 þ rj Nj Cj

!+ > 0:

We consider gð r Þ ¼ a 1 r  b1 where a1 ¼

n X

n X     Nj Djmac rj , b1 ¼ Nj Djmac rj rj :

j¼1

j¼1

We have g(IR) > 0 and we look at the line between g(0) ¼ b1 and g(IR), and as g(0) < 0, we have 0 < rmac < IR: We consider hð r Þ ¼ a 2 r  b2 where a2 ¼

n X

  Nj Djmod rj

¼

j¼1

b2 ¼

n X j¼1

  Nj Djmod rj rj

n X j¼1

¼

n X j¼1

    n X   Djmac rj rj j Nj ¼ Nj Dmac rj 1  1 þ rj 1 þ rj j¼1

    n X   Djmac rj rj j Nj r ¼ Nj Dmac rj 1  r: 1 þ rj j 1 þ rj j j¼1

We look at the line between the differences gðrmac Þ  hðrmac Þ > 0 and the intersection of the vertical axis gð 0Þ ¼ b1 > b2 ¼ hð 0Þ and we conclude (8). As the Landau symbols are valid on the interval, we can perturb to intervals of the par yields. The proof is thus completed. □

70

4 The Internal Rate of Return for a Bond Portfolio

Example 4.5 We choose the parameter of the Example 4.1. Although we consider two zero bond and not par bond we find with rmac ¼ 0:03333640, and rmod ¼ 0:03320328, and 0 < rmod < rmac < IR the statement (6) of Theorem 4.4. We do not discuss the magnitude of the deviation from a flat curve and par bonds. Theorem 4.4 is only valid locally, i.e., our claims assume that there exists an open interval around the flat curve. Example 4.6 We consider two par bonds with prices P1 ðr1 Þ ¼ P 2 ðr 2 Þ ¼

F þ C1 , 1 þ r1

C2 F þ C2 þ , 1 þ r 2 ð1 þ r 2 Þ2

with C1 ¼ 1%, C2 ¼ 9%, r1 ¼ C1, r2 ¼ C2, and find IRR ¼ 0:06279 rmac ¼ rmod ¼

D1mac r1 þ D2mac r2 1 0:01 þ 1:91 0:09 ¼ 0:06258, ¼ 1 þ 1:91 D1mac þ D2mac

D1mod r1 þ D2mod r2 0:9900 0:01 þ 1:7591 0:09 ¼ 0:06119: ¼ 0:9900 þ 1:7591 D1mod þ D2mod

Thus we have 0 < rmod < rmac < IR: In Fig. 4.4 we see the difference of the NAV and the line g. Example 4.7 We consider two zero bonds with prices P1 ðr1 Þ ¼

F , 1 þ r1

4.3 Macaulay Duration Approximation Versus Modified Duration Approximation Fig. 4.4 Approximation of the NAV

71

2.E–03 0.E+00 0.00 –2.E–03

0.02

0.04

0.06

–4.E–03 –6.E–03 –8.E–03 –1.E–02 –1.E–02

P2 ðr2 Þ ¼

F , ð1 þ r2 Þ2

and choose r1 ¼ 0.00 and F ¼ 1 with r2 ¼ k 10%, k ¼ 0, 1, . . . , 8: We find, for (1a), NAVðrÞ ¼

1 1 1 1  þ  : 1 þ r1 ð1 þ r2 Þ2 1 þ r ð1 þ rÞ2

For the residual in r1, we find NAVðr1 Þ ¼

1 1  , ð1 þ r 2 Þ2 ð1 þ r 1 Þ2

and for the residual in r1 we find NAVðr2 Þ ¼

1 1  : 1 þ r1 1 þ r2

Based on (38), we have rmac ¼

2 X

vj r j

j¼1

with     j 1 Pj Djmac rj 1þrj Dmac rj ¼ 2   : vj ¼ 2 P P 1 i i Pi Dimac ðri Þ D ð r Þ mac i 1þri i¼1

i¼1

0.08

0.10

72

4 The Internal Rate of Return for a Bond Portfolio

 1 1þr1

rmac ¼ 1 1þr1

þ2



1 1þr2

2  r 1 þ

2 1 1þr1

1 1þr2

þ2



2

1 1þr2

2  r2 :

For the approximation by the modified duration, we have 

rmod ¼ 

1 1þr1

 3 1 2 1þr 2  3  r1 þ  2  3  r2 : 1 1 1 þ 2 1þr2 þ 2 1þr1 1þr2

1 1þr1 2

2

From (1), we have P1 ðr1 Þ þ P2 ðr2 Þ ¼

F F  : 2 ð1 þ r Þ ð1 þ r Þ2

And, using the price function, we find F F F F þ ¼ þ : 2 2 2 ð1 þ rÞ ð1 þ r Þ2 ð1 þ r 1 Þ ð1 þ r 2 Þ The IR can be explicitly calculated by solving a quadratic Eq. (23). In Fig. 4.5, we show the difference IR  rmac , IR  rmod : As shown in Theorem 4.4, we see that rmac and rmod underestimate the IR and rmac is a better approximation than rmod.

Fig. 4.5 Approximation of two zero bonds

0.14 0.12

Erorr versus IRR

0.10 0.08 Mac

0.06

Mod 0.04 0.02 0.00 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.02 Yield of the zero bond

4.3 Macaulay Duration Approximation Versus Modified Duration Approximation

73

Although we have not par bonds in this example, rmod  rmac

ð3:4:48Þ

can be shown explicitly. With a1 ¼ 1 + 2, b1 ¼ r1 + 2 r2 a2 ¼

1 2 þ , ð1 þ r1 Þ2 ð1 þ r2 Þ2

b2 ¼

r1 2r2 þ : 2 ð1 þ r 1 Þ ð1 þ r 2 Þ2

(48) is the same as b2 b1  a2 a1 thus a 1 b2  a 2 b1 a 1 b2  a 2 b1

3r1 6r2 þ 2 ð1 þ r1 Þ ð1 þ r 2 Þ2 r1 2r1 2r2 4r2     2 2 2 ð1 þ r 1 Þ ð1 þ r 2 Þ ð1 þ r 1 Þ ð1 þ r 2 Þ2 2r1 2r2 2r1 2r2 ¼ þ   2 2 2 ð1 þ r1 Þ ð1 þ r 2 Þ ð1 þ r 2 Þ ð1 þ r 1 Þ2 ¼

¼ 2

ðr1  r2 Þ2 : ð1 þ r 2 Þð1 þ r 2 Þ

The equality holds if and only if r1 ¼ r2. Remark 4.4 (The Approximation of IRR By Macaulay Weight Versus Modified Duration Weights) We give an intuitive explanation why the approximation of the internal rate of return by Macaulay duration is superior to modified duration. In Theorem 4.1, we have decomposed the bond portfolio into zero bonds. For a zero bond, the Macaulay duration is equal to the time to maturity, and the linear terms vanish and the NAV consists only of quadratic terms. This process is unique. But by substituting, the modified duration linear term does not vanish, and the NAV is bigger.

74

4 The Internal Rate of Return for a Bond Portfolio

4.4

Numerical Illustrations

We refer to (3.3.2) and consider the Macaulay duration at issuance or just after the coupon payment. The Macaulay duration Dmac(r) is then defined by N P

Dmac ðrÞ ¼

C ð1þ rÞk

k

k¼1 N P

þ

C ð1þrÞk

k¼1

þN

F ð1þrÞN

:

ð4:4:1aÞ

F ð1þrÞN

This is the same as N P

C ð1þrÞk

k

k¼1

Dmac ðrÞ ¼

þN

F ð1þ rÞN

ð4:4:1bÞ

PðrÞ

where P(r) is the price of the bond. The value of the Macaulay duration depends on the coupon, the time to maturity, and interest rate. Mostly the yield to maturity for a bond is substituted for the interest rate. This is consistent with the formulae introduced in (2), (3), and (39) as they reduce to the yield to maturity for a bond portfolio that reduces to a single bond. Here we consider a bond portfolio and recall (3.3.5). The Macaulay duration of a portfolio is defined by

DPo mac ðrÞ

¼

¼

n P

Tj P

j¼1

k¼1

k

Nj Cj ð1 þ r Þk

n P

Tj P Nj Cj

j¼1

k¼1 ð1

n P

Tj P

j¼1

k¼1

k

þ r Þk

þ Tj þ

Nj Cj

Nj Fj

!

ð1 þ rÞTj ! Nj Fj

ð1 þ r ÞT j ! Nj Fj

þ Tj ð1 þ rÞTj ð1 þ r Þk PoV

:

We discuss different possibilities for the interest rates (see Fig. 4.6). In the commercial software, the yield to maturity approach is mostly used

DPo Mac

¼

n P

Tj P

j¼1

k¼1

! k

Nj C j

ð1þrj Þ

þ Tj

k

n P

Tj P

N j Cj

j¼1

k¼1

ð1þrj Þ

k

þ

Nj Fj

ð1þrj Þ j ! : T

Nj Fj

ð1þrj Þ

Tj

4.4 Numerical Illustrations

75 r1,r2,.......,rn Yield to maturity (YTM)

YTM approach

IRR (Internal rate of rate)

r = rtrue

r = rdur

r = rlin

d1,d2,.......,dn

durtrue

durdur

durlin

Dur = ∑nk=1 wkdk

Fig. 4.6 Different calculation of the Macaulay duration

This approach is based in the flat yield concept. The yield to maturity of each individual bond is substituted. The theoretical drawback is that we use different interest rates at the same time, and that is the main caveat of the flat yield concept. If the yield is flat or nearly flat, the approach is acceptable. We propose to use the IR or an approximation thereof and present some numerical experiment in the following section. We proceed by compiling some analytic expression for calculating the Macaulay duration. In the following lemma, we discuss the price function relating to different interest rate. Lemma 4.8 (Price Approximation of a Bond and Bond Portfolio) We denote with rapp one of the approximations rnom (see (4.1.2)), rlin (see (4.1.3)), rdir (see (4.1.4)), rmac (see (7)), or rmod (see (8)). Then we have with Δrapp ¼ rapp  IR,

ð4:4:2Þ

and the discrete version DDmod [see (3.3.6) and (3.3.8)] for a bond with price P(r)     2   1 1 ¼   1  DDmod rapp Δrapp þ o Δrapp PðIRÞ P rapp

ð4:4:3aÞ

and for a bond portfolio Po(r)     2  1 1 ¼   1  DDmod rapp Δrapp þ O Δrapp : PoðIRÞ Po rapp Proof By the right-hand side of (3a), we have

ð4:4:3bÞ

76

4 The Internal Rate of Return for a Bond Portfolio

1 1   ¼   PðIRÞ P rapp  P rapp þ PðIRÞ ¼

1  :   Pðrapp ÞPðIRÞ P rapp 1  P r ð app Þ

By using the discrete version of the modified duration (see Definition 3.18), we have by (2) DDmod ðr, ΔrÞ ¼ 

ΔPðrÞ Δr

PðrÞ

,

and, by using the abbreviation (1), we find the assertion (3a) of the lemma. The assertion (3b) follows analogously. □ The following theorem discusses the different duration when changing the level. The following theorem decomposes the error between the IR and the approximation rapp when evaluating the Macaulay duration of the bond. The linear term has two parts. The first term stems from evaluating the denominator and the quadratic cross terms, and the second term comes from the price approximation in the nominator. The corresponding result for a bond portfolio is in Theorem 4.6. Theorem 4.5 (Approximation Macaulay Duration of Bond) We denote with rapp one of the approximation rnom (see (4.1.2)), rlin (see (4.1.3)), rdir (see (4.1.4)), rmac (see (4.2.7)) or rmod (see (4.2.8)). Then we have     1 Dmac rapp  Dmac ðIRÞ ¼ fK Δrapp Papp 1       2   o   þ K2 Δrapp • 1 þ DDmod • Δrapp þ O rapp þ O rapp IR þ O IR2 ð4:4:4aÞ where Δrapp is defined by (2) and K1 ¼

N X k¼1



k2 C 1 þ rapp

k þ 

N2 F 1 þ rapp

N

ð4:4:4bÞ

and K2 ¼

N X k¼1

! 

kC 1 þ rapp

Proof By (1) and Lemma 4.8, we have

k þ 

NF 1 þ rapp

N

DDmod :

ð4:4:4cÞ

4.4 Numerical Illustrations

77

! N X 1 kC NF Dmac rapp  Dmac ðIRÞ ¼   N  k þ  P rapp 1 þ rapp k¼1 1 þ r app ! ! N N X X 1 kC NF 1 kC NF þ  ¼    k þ  k PðIRÞ k¼1 ð1 þ IRÞk ð1 þ IRÞN P rapp 1 þ rapp k¼1 1 þ rapp ! N   X   2  1 kC NF 1  DDmod rapp Δrapp þ O Δrapp ¼    k þ P rapp ð1 þ IRÞN k¼1 1 þ r app ! N   X   2  1 kC kC þ 1  DDmod rapp Δrapp þ O Δrapp  PðIRÞ k¼1 ð1 þ IRÞk ð1 þ IRÞN !     2   NF NF þ 1  DDmod rapp Δrapp þ O Δrapp N  ð1 þ IRÞN 1 þ rapp 



ð4:4:5Þ By Corollary 4.2 we have       1 1 2 ¼ N 1  k IR  rapp þ O rapp k ð1 þ IRÞ 1 þ rapp       1 þ O rapp • IRÞ þ O IR2 ¼  N 1 þ k rapp  IR 1 þ rapp        2 þ O rapp • IR þ O IR2 þ O rapp and with (5) we find N X   1 kC kC Dmac rapp  Dmac ðIRÞ ¼    k   k P rapp 1 þ rapp 1 þ rapp k¼1         2 • 1  kΔrapp þ O rapp þ O rapp • IR þ O IR2

   2  NF Þþ • 1 þ DDmod rapp Δrapp þ O Δrapp N 1 þ rapp         NF 2 þ O rapp • IR þ O IR2  N • 1  NΔrapp þ O rapp 1 þ rapp     2   • 1 þ DDmod rapp Δrapp þ O Δrapp Þ :

By arranging the terms and by canceling the term 0ter order

78

4 The Internal Rate of Return for a Bond Portfolio

N X      1 kC Dmac rapp  Dmac ðIRÞ ¼    k • kΔrapp þ DDmod rapp Δrapp P rapp 1 þ rapp k¼1        2   2 þ kDDmod rapp Δrapp þ O rapp þ O rapp • IR þ O IR2    NF þ N • NΔrapp þDDmod rapp Δrapp 1 þ rapp        2   2 þ N • DDmod rapp Δrapp þ O rapp þ O rapp • IR þ O IR2

□

which yields the assertion of the theorem.

Theorem 4.6 (Macaulay Duration of a Bond Portfolio) We denote with rapp one of the approximations rnom (see (4.1.2)), rlin (see (4.1.3)), rdir (see (4.4.4)), rmac (see (4.2.7)) or rmod (see (4.2.8). Then we have with (4.2.8b)   Po DPo mac rapp  Dmac ðIRÞ ¼

    1  K1 Δrapp þ K2 Δrapp • 1 þ DDPmod • Δrapp Poapp

    2    þ O Δrapp þ O rapp IR þ O IR2 g where K1 ¼

n X

tj X

j¼1

k¼1



k2 N j C j 1 þ rapp

k þ 

N2j Fj 1 þ rapp

! N

and K2 ¼

Tj n X X j¼1

k¼1



kNj Cj 1 þ rapp

k þ 

Nj Fj 1 þ rapp

Proof We apply the proof of Theorem 4.1 to a portfolio.

!! N

DDPo mod

□

We consider again with examples that constituent of two bonds. First we consider two bonds that fit into the framework for Lemma 4.3 and use again the data from Example 3.18. Then we apply (4.1.18) to the bonds consider in Example 4.3 and compare the different approximations to IR. In Example 4.9 we consider zero bonds and investigate the error of IR when evaluating (4.1.18) and (4.1.19). Example 4.8 (Continuation of Example 4.3) We again use the date from Example 4.5. With (4.2.7) and (4.2.8), we find by using r2 ¼ 8%, and then we have

4.4 Numerical Illustrations

79

0.0002 0.0001 0.0000 6.0% 6.5% 7.0% 7.5% 8.0% 8.5% 9.0% 9.5% 10.0% Error linear Error Duration Error Nomial

–0.0001 –0.0002 –0.0003 –0.0004 –0.0005

Fig. 4.7 Different approximation of the IRR

IR ¼ rmod ¼ rmac ¼ 8%: We consider r2 ¼ 8%  α0:5%, α ¼ 4, 3, 2, 1, 0: And with the coupons in Table 3.5, we have the full problem we have tackled in Lemma 4.3. We see in Fig. 4.7 that the approximation with Macaulay duration formulated in (38) is best for approximating IR. The following example shows the easiest portfolio of two bonds with different times to maturity. Example 4.9 (Continuation of Example 4.4) We again use the date from Example 4.6. By referring to Table 4.2 and (4.1.23), we see that the linear approach and the direct yields the same result which is different from the numerical value for IR. Figure 4.8 shows that the quadratic approach with the Macaulay approximation (4.2.7) is much better than all linear measurement introduced here. Example 4.10 (Zero Bonds, Continuation of Example 4.5) As in Example 4.7, we consider two zero bonds with price P1 ð r 1 Þ ¼

F F , P ðr Þ ¼ : 1 þ r1 2 2 ð1 þ r 2 Þ2

and choose F ¼ 1 and r1 ¼ 5%. We vary over r2 and we see that the approximations (2), (3), (38), and (39) are different, except we have a flat yield curve consisting of entry r1 and r2 (compare Lemma 4.7). Table 4.3 illustrates Theorem 4.2–4.5. In Fig. 4.9 we illustrate Corollary 4.2 and 4.3 with the second bond by the quadratic approximation

80

4 The Internal Rate of Return for a Bond Portfolio 0.2% 0.0% 1

2

3

4

5

6

7

8

9

10 11

–0.2% –0.4% –0.6% IRR - rlin

–0.8%

IRR - rmac

–1.0% –1.2% –1.4% –1.6% –1.8%

Fig. 4.8 Linear versus quadratic Table 4.3 IRR and its approximation r2 (%) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

IRR 0.0333592 0.0367271 0.0400751 0.0434032 0.0467115 0.0500000 0.0532689 0.0565182 0.0597480 0.0629585 0.0661496

IRR-rnom 0.0041408 0.0032729 0.0024249 0.0015968 0.0007885 0.0000000 0.0007689 0.0015182 0.0022480 0.0029585 0.0036496

IRR-rlin 0.0041445 0.0033245 0.0024999 0.0016709 0.0008376 0.0000000 0.0008417 0.0016875 0.0025372 0.0033908 0.0042482

IRR-rmac 0.0000225 0.0000145 0.0000082 0.0000037 0.0000009 0.0000000 0.0000009 0.0000037 0.0000085 0.0000151 0.0000237

IRR-rmod 0.0001559 0.0001000 0.0000564 0.0000251 0.0000063 0.0000000 0.0000063 0.0000253 0.0000570 0.0001015 0.0001588

0.02 0.01 0.00

2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5%

–0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07

Fig. 4.9 Approximation of discount factor

quadratic error linear error

4.4 Numerical Illustrations

81

1 ¼ 1  2r2 þ Oððr2 ÞÞ2 , ð1 þ r 2 Þ2 and the linear approximation 1 ¼ 1  r2 þ Oðr2 Þ: ð1 þ r 2 Þ2

Definition 4.3 We consider a yield curveconsidering the following yield r1 > 0, . . . , rj > 0, . . . , rn > 0, 1  j  n and the time points t1 > 0, . . . , tj > 0, . . . , tn > 0, 1  j  Tn : We discuss three cases: (a) If r1 ¼ r2 , . . . rj ¼ rjþ1 . . . , rn1 ¼ rn , 1  j  n the yield curve is said to be flat. (b) If r1 > r2 , . . . rj > rjþ1 . . . , rn1 > rn , 1  j  n the yield curve is said to be normal. (c) If r1 < r2 , . . . rj < rjþ1 . . . , rn1 < rn , 1  j  n the yield curve is said to be inverted. Example 4.11 (Continued Example 4.2) We consider again three par bonds with the same time to maturity Ti ¼ 3. The portfolios can be considered as a credit portfolios with bonds that have the same time to maturity. In Portfolio 1 (α ¼ 0), all bonds have the same credit quality. In Portfolios 2 and 3 (α ¼ 2, α ¼ 2), we assume two different portfolios with different credit. In Table 4.4 we assume three portfolios with different par yield.

82

4 The Internal Rate of Return for a Bond Portfolio

Table 4.4 Different credit qualities

Portfolio 1 2 3

Table 4.5 Different approximation of par bonds

(1) + (2) (%) 6.0000 6.0000 6.0000

Table 4.6 Different yield of maturity

Scenario 1 2 3

Table 4.7 Approximation of the IRR

(1) (%) 2.0000 4.00000 7.33333

r1 (%) 6 4 2

r2 (%) 6 6 6

(38) (%) 6.00000 5.97581 5.90317

r1 (%) 2 2 2

(2) (%) 2.00000 3.92464 6.86353

r3 (%) 6 8 10

(IRR) (%) 6.00000 6.00000 6.00000

r2 (%) 2 4 8

(38) (%) 2.00000 3.92379 6.85795

r3 (%) 2 6 12

(IRR) (%) 2.00000 3.94907 7.01624

The IRR in the Table 4.5 follows from (21). We see that the Macaulay duration approach (38) yields a wrong result. Example 4.12 (Not Par, Same Time to Maturity) We consider three bonds with coupon 2% and the time to maturity T ¼ 3. As exposed in Table 4.6 we assume first a flat curve and then two choices with discount bonds. Table 4.7 shows the results of our calculation. We see that (38) can be a better and a worse approximation of the IIR than (2). Example 4.13 (Different Time to Maturity) We consider three bonds with Coupon C ¼ 6% with N1 ¼ 1, N2 ¼ 1, N3 ¼ 1. Furthermore we assume time to maturities T1 ¼ 1, T1 ¼ 2, T1 ¼ 3 and three yield scenarios r1, r2, r3 as specified in the Table 4.8. Table 4.9 shows the results of our calculation. We see that (2) is a much better approximation of the IRR that (1).

4.4 Numerical Illustrations

83

Table 4.8 Yield scenario

Scenario 1 2 3

Table 4.9 Flat versus non-flat curve

(1) (%) 6.0000 6.0000 6.0000

r1 (%) 6 0 12

r2 (%) 6 6 6

(2) (%) 6.00000 7.58253 3.75359

r3 (%) 6 12 0

(IRR) (%) 6.00000 7.58107 3.76000

yield spread (normal) 10.00% 9.50% 9.00% IRR Nom lin MacDur

8.50% 8.00% 7.50% 7.00% 9

10 11 12 13 yield long maturity bond

14

Fig. 4.10 Approximation of IR for increasing yields

Example 4.14 (Yield Spread, Different Time to Maturities) We consider three bonds with Coupon C ¼ 9% with Nj ¼ 1, j ¼ 1, 2, 3, and the time to maturities are T1 ¼ 4, T2 ¼ 9, T3 ¼ 14. We assume with r2 ¼ 9% and considered: 1. r1 ¼ r2  α, r3 ¼ r2 + α (normal yield curve 2. r1 ¼ r2 + α, r3 ¼ r2  α, (inverted yield curve) We chose α 2 N between 1%  α  5%. The Figs. 4.10 and 4.11 shows the difference of the different approximation for IR. We see that the duration approximations (38) yield the best approximation. Figures 4.12 and 4.13 are the accompanying Macaulay duration calculations.

84

4 The Internal Rate of Return for a Bond Portfolio yield spread (inverse) 10.0% 9.5% 9.0% 8.5% 8.0% IRR Nom lin MacDur

7.5% 7.0% 6.5% 6.0% 5.5% 5.0% yield long maturity bond

Fig. 4.11 Approximation of IR for decreasing yields

Macaulay Duration in years (normal) 6.40 6.20 6.00 Durshort DurIRR Dur yield nom duryield lin durMac

5.80 5.60 5.40 5.20 5.00 9

10

11

12

13

14

yield long maturity bond

Fig. 4.12 The Macaulay duration calculation for increasing yields

Example 4.15 (Yield Spread, Same Time to Maturities) We consider three bonds with Coupon C ¼ 9%, with Ni ¼ 1, i ¼ 1, 2, 3. We assume r1 ¼ 2%, r2 ¼ 9%, r3 ¼ 16% and vary over the time to maturities of the portfolio T 2 N and 10  T  15. Figures 4.14 and 4.15 show the difference of the different approximations of IR and the accompanying Macaulay duration calculations. The approximation error of the IRR is a linear function of the time to maturity. The linear

4.4 Numerical Illustrations

85

Macaulay Duration in years (inverse) 7.40 7.20 7.00 6.80 Durshort DurIRR Dur yield nom duryield lin durMac

6.60 6.40 6.20 6.00 5.80 5.60 9

10

11

12

13

14

yield long maturity bond

Fig. 4.13 Macaulay duration calculation for decreasing yields

Yield spread 1400 bps

Approximation for the IRR

10%

9% IRR Nom LIN MacDur

8%

7%

6%

5%

10

11

12 13 Time to Maturity

14

15

Fig. 4.14 Same time to maturity of the bonds

approximation is the best approximation in which is in line with Examples 4.4 and 4.11. Remark 4.5 As introduced in Definition 3.6, the flat yield curve concept assumes that every cash flow of a single bond is discounted with the same yield, but we know that we have

86

4 The Internal Rate of Return for a Bond Portfolio Macaulay Duration in years 10.0 9.5 9.0 8.5

Dur IRR Dur yield nom Dur yield lin Dur Mac Dur short

8.0 7.5 7.0 6.5 6.0

10

11

12 13 Time to Maturity

14

15

Fig. 4.15 Different duration measures

different yield for different time to maturities and coupons at the same time are discounted with different yields. In a riskless world, this makes little sense. We need a model that derives from the observed yield to maturity the fair spot for any time. Remark 4.6 Credit bond can also be tackled with the consideration on the yield to maturities given in this chapter. However, with the duration introduced so far does not reflect credit risk. If yields are increasing because of a possible credit event-like default, we would say that the duration is diminishing, and we have less risk which is a wrong conclusion in our context.

References 1. Wolfgang M (2015) Portfolio analytics, 2nd edn. Springer, Cham 2. Markowitz HM (1952) Portfolio selection. J Financ 7:77–91. https://doi.org/10.1111/j.15406261.1952.tb01525.x

5

The Term Structure of Interest Rate

Abstract

We depart from the flat yield concept. The flat yield concept allows different discount factors for different bonds although cash flow occurs at the same time in the future. The concept of time value of money does not allow this situation and spot curves avoid this deficiency. We discuss the transition from yield curves to spot curves and spot curves to forward curves. In this chapter, we depart from the flat yield concept (see Definition 3.6) as discussed in Chap. 3. The flat yield concept allows different discount factors for different bonds although cash flow occurs at the same time in the future. The concept of time value of money does not allow this situation and spot curves avoid this deficiency. Figure 5.1 shows the backbone of this chapter. We discuss the transition from yield curve to spot curves and spot curves to forward curves. Figure 5.1 refers to a specific time and does not say anything about the dynamic of the curve. Starting point is a set of bonds of similar quality and the accompanying market price. The set of yield is the input to a scatterplot showing the time versus to yield. The term yield curve suggests that we can find a yield to maturity for any time. Yield curve modelling then refers to the transition for the scatter plot to a curve. It is often said that the spot rates are the basis of a specific bond universe. Spot rates have a wide area of application like for instance • Scenario analysis for a bond portfolio • Rich/cheap analysis of the price of a bond paid in market • Price of a recently issue bond The forward rate gives an indication of future interest rates. The material here discussed is also extensively discussed in the literature (See e.g. [1, 2]). We confine our exposition to some basic ideas and concepts. # Springer Nature Switzerland AG 2020 W. Marty, Fixed Income Analytics, https://doi.org/10.1007/978-3-030-47158-3_5

87

88

5

The Term Structure of Interest Rate

Fig. 5.1 Different curves based on market data

forward rate curve

Spot rate curve

yield curves

Actual Bond Prices

5.1

Spot Rate and the Forward Rate

In this section, we discuss the basics concept of the term structure of the interest rate. We consider a partition of the time axis with unit year t0 ¼ 0, : . . . , tk < tk1 , . . . , tN ¼ T

ð5:1:1aÞ

of the interval [0, T], and k1 hk

¼ tk  tk1 , k ¼ 1, . . . , N

ð5:1:1bÞ

is the time span between tk and tk-1. Definition 5.1 The yield to maturity of a Zero Coupon Bond with time to maturity t 2 R1 is denoted by st and is called the interest zero rate or simply the zero rate for time t 2 R1, t 2 [0, T]. In the following, we assume that the zero rates s(t), t 2 R1, are given for any time t 2 R1, t 2 [0, T]. In Definition 5.1, it is assumed that the interest rate starts at t ¼ 0. Definition 5.2 An annual interest rate k-1fk that starts with a time t > 0 over the time span [tk-1, tk] in the future is called a forward interest rate or simply a forward rate. Remark 5.1 The notation k-1fk, k ¼ 1,. . .,N, implicitly assumes that the forward rate is constant and that there is no compounding in the time period k-1tk. We consider a spot rate curve s(tk), k ¼ 1,. . .,N. Forward rates are derived from the spot rates. They are indicators of future interest rate implied from a no arbitrage condition which states that an investor that first invests $1 in the period 1 and then

5.1 Spot Rate and the Forward Rate Fig. 5.2 Forward rate

89 1

0

S1

2

1f2

S2

reinvests in period 2 must receive the same amount that as an investor that invests $1 over both periods (see Fig. 5.1.2). The spot rates s1 and s2 with s1 < s2 and the forward rate 1f2 are related by ð1 þ s2 Þ2 ¼ ð1 þ s1 Þ  ð1þ1 f 2 Þ:

ð5:1:2Þ

Interest rates are always referring to a time span. In (2), it is assumed that the period 1 and period 2 have the same length. In most cases, the interest rate are quoted annually and the underlying period is years. The calculation of the right side of (2) reflects an investor that receives interest after period 1 and reinvests in period 2. We speak of compounding, more specially we have the following definition: Definition 5.3 Compounding is the reinvestment of the income to earn more income in the subsequent periods. If the income and the gains are retained within the investment vehicle or reinvested, they will accumulate and contribute to the starting balance for each subsequent period’s income calculation. Example 5.1 We assume that the unit is years with the spot rate s1 ¼ 2.0000% for the first year and the spot rate s2 ¼ 2.5000% for the first 2 years. Referring to (1), we have N ¼ 2 and t0 ¼ 0, t1 ¼ 1, t2 ¼ 2. The forward rate between the end of the first year and the end of the second year is then 1f2

¼

ð1 þ s2 Þ2  1: ð1 þ s1 Þ

ð5:1:3Þ

The numerical value is 1f 2

¼ 3:0025%:

Because the spot curve is upward sloping, we see that the forward rate is above the two spot rates. We generalize (2) by considering a fraction of the underlying base unit. Here the base unit is not equal to the validity of the interest rate. From (1) and (2), we have ð1 þ h  s2 Þ2 ¼ ð1 þ h  s1 Þ  ð1 þ h0 f 1 Þ:

90

5

The Term Structure of Interest Rate

Example 5.2 We assume that the unit is years with the spot rate s1 ¼ 2.0000% for the first half year and the spot rate s2 ¼ 2.5000% for the second half year. Referring to (1), we have N ¼ 2 and t0 ¼ 0, t1 ¼ 0.5, t2 ¼ 1. Based on (3) the numerical value is 1f 2

¼ 3:0012%:

By iterating (3) for k ¼ 2,. . .,N, the term structure of the spot rate are then related to the forward rate by ð1 þ h  sk Þk ¼ ð1 þ h  s1 Þ  ð1 þ h1 f 2 Þ  . . .  ð1 þ hk1 f k Þ, 1  k  N, and with s1 ¼ 0f1 we have ð1 þ h  sk Þk ¼ ð1 þ h0 f 1 Þ  ð1 þ h1 f 2 Þ  . . . . . . :  ð1 þ hk1 f k Þ:

ð5:1:4aÞ

We see that the spot rates are the geometrical compounded means of the forward rates. We compound twice
k
2
2
2 h h h h 1 þ  sk ¼ 1 þ 0 f 1  1 þ 1 f 2  . . . . . . :  1 þ k1 f k 2 2 2 2 and by iterating we find for continuous compounding ehksk ¼ eh0 f 1  eh1 f 2 . . . ehk‐1 f k

ð5:1:4bÞ

Definition 5.4 The yield to maturity of a Zero Coupon Bond at issuance time tB > 0, with time to maturity tE 2 R1, with tB < tE, is called the forward yield and is denoted by BsE, and the forward rate yield curve is a plot of the forward rate against the term to maturity. By using the notation in Definition 5.4, (4b) is the same as ð1 þ h0 sk Þk ¼ ð1 þ h0 f 1 Þ  ð1 þ h1 f 2 Þ  . . . . . . :  ð1 þ hk1 f k Þ: The term structure of the forward yield after the first period  j   1 þ h1 sj ¼ ð1 þ h1 f 2 Þ  . . . . . . :  1 þ hj1 f j , 1  j  N,

ð5:1:5aÞ

and, more generally after the k-th period  k   1 þ hk sj ¼ ð1 þ hk f kþ1 Þ   1 þ hj1 f j , 0  k < j  N:

ð5:1:5bÞ

We see that there is a forward rate at every time point tk in the future and an accompanying forward yield curve. This leads to the following remark.

5.2 Discrete Forward Rate and the Instantaneous Forward Curve

91

Remark 5.2 In an optimisation tool, the economist is considering the forward curve at his time horizon and forecasts against this forward yield curve.

5.2

Discrete Forward Rate and the Instantaneous Forward Curve

Based on the definition of the discount factor d in (2.1.3), we define the discount factor dk starting with by t0 ¼ 0. dk ¼ dk ðrðtk Þ, tk Þ ¼

1 : ð1 þ sðtk ÞÞtk

Thus, we have by (5.1.5)      1 1 þ 0 h1  0 f 1 : 1 þ 1 h2  1 f 2 : . . . . . . :: 1 þ k‐1 hk  k‐1 f k dk    , ¼ 1 1 þ 0 h1  0 f 1 : 1 þ 1 h2  1 f 2 : . . . . . . :: 1 þ k‐2 hk‐1  k‐2 f k‐1 d k

1

i.e., we have 1 dk 1 dk

 1 ¼ k1 hk  k1 f k : 1

We find the marginal increase of the discount factor over the time is dk

1

 dk ¼ k‐1 hk  k‐1 f k : dk

We define the instantaneous forward rate f(t), t 2 [0, T], by f ðtÞ ¼ 

∂dðtÞ ∂t

dðtÞ

:

ð5:2:1Þ

A function is called continuous if the function has no jumps, i.e., if the originals of a function are closed together, the images are also close to together. A step function is a function that is piecewise constant. For a typical example we refer to (4) as we have f ðtÞ¼k1 f k , t 2 ½tk1 , t, k ¼ 1, . . . ::, N: The precise definition of a continuous function is in [3].

92

5

The Term Structure of Interest Rate

Theorem 5.1 Assume that the forward continuous rate f(t) of the interval [0, T] are annual. Then, for tB, tE with tE > tB and tB 2 [0, T], tE 2 [0, T], and continuous compounding forward rates, we have for the spot rate s(t) 0 1 0 1 ZtE ZtE 0 d ð t Þ 1@ 1 t sðt Þ ‐ f ðτÞdτA ¼ @tB sðtB Þ ‐ sðtE Þ ¼ dτA, ð5:2:2aÞ tE B B tE dð t Þ tB

tB

and the effective spot return efs is RtA efs ¼ etB

f ðtÞdt

 1:

ð5:2:2bÞ

Proof We consider (5.1.4b) with tB ¼ t0, tE ¼ tN etE sE ðtÞtB sB ðtÞ ¼ e0 t1 f 1 ðtÞþ:...þk tk1 f k

ðtÞþ:...þN1 tN f N ðtÞ

:

ð5:2:3Þ

We introduce two step functions. With mk ðf Þ ¼ min f ðtÞ, Mk ðf Þ ¼ max f ðtÞ, t2½tk , tkþ1 

t2½tk , tkþ1 

we approximate N X

ZtB k f kþ1

mk ðf Þ 

k¼1

N X

f ðτÞdτ 

k f kþ1

M k ðf Þ

k¼1

tA

by a lower, upper, resp. approximation of f. By applying the rules of the logarithm, we find N X

ZtB k f kþ1

mk ðf Þ 

k¼1

N X

f ðτÞdτ 

k f kþ1

M k ðf Þ

k¼1

tA

by considering the definition of the integral (see Appendix E).

□

Corollary 5.1 Assume that the forward rates f(t) of the interval [0, T] are annual and continuous. Then, for continuous compounding forward rates, we have for the spot rate s(t) SðTÞ ¼

1 T

ZT f ðτÞdτ ¼ 0

and the effective spot return efs is

1 T

ZT

∂dðtÞ ∂t

dðtÞ 0

dτ

5.2 Discrete Forward Rate and the Instantaneous Forward Curve

RT efs ¼ e 0

f ðtÞdt

93

 1:

Proof The proof follows from Theorem 5.1 by tB ¼ 0, tE ¼ T.

□

Example 5.3 (Annual Versus Continuous Compounding) We assume 4 equidistant knots (1) with T ¼ 3, and the unit of the time axis is years. We consider an initial investment of BV ¼ $100, and the spot curve is 0 s1

¼ 2%on ½0, 1Þ, 1 f 2 ¼ 3%on ½1, 2Þ and2 f 3 ¼ 4%on ½2, 3Þ:

By using continuous compounding, the ending value EV1 is EV1 ¼ BV  e0:09 ¼ BV  e0:02  e0:03  e0:04 : The numerical value is EV1 ¼ $1:094174, and annually we have EV3 ¼ ð1 þ 0:02Þ  ð1 þ 0:03Þ  ð1 þ 0:04Þ: The numerical value is EV3 ¼ $1:092624: Nelson-Siegel and its extensions is a very popular method for fitting yield curves. It is widely discussed in the literature (see for instance [1, 2]). We only use it here for illustrating Theorem 5.1. Example 5.4 (Nelson-Siegel) Starting point of the Nelson-Siegel model is a functional form of the annualized forward curve with 3 parameters β1 2 R1, β2 2 R1Z and β3 2 R1 h i t t t f ðtÞ ¼ β1 þ β2 eλ  β3 eλ , λ 2 R1 : λ Thus, we have f ð 0 Þ ¼ β1 þ β 2 : We consider the indefinite Integral

94

5

Zy β1

Zy 1 dx þ β2

Zy

x

e dx þ β3

The Term Structure of Interest Rate

xex dx ¼ β0 þ β1 y  ðβ2 þ β3 Þey  β3 yey

and define the annualized spot curve by sðyÞ ¼

β0 þ β1 y  ðβ2 þ β3 Þey  β3 yey : y

ð5:2:4Þ

By choosing β0 ¼ β 2 þ β 3 , the singularity is removable and we have lim sðyÞ ¼

y!0

β1 y þ ðβ2 þ β3 Þð1  e‐y Þ‐β3 ye‐y ¼ β 1 ‐ β3 : y

Thus, we have s ð y Þ ¼ β1 þ

ðβ2 þ β3 Þð1‐e‐y Þ ‐β3 e‐y : y

We consider the time scaling t y¼ , λ And the spot rate (4), by exchanging the variable t by y, is 1  e λ t

sðtÞ ¼ β1 þ ðβ2 þ β3 Þ

t λ

 β3 eλ , λ 2 R1 , t

ð5:2:5Þ

and we have s ð 0 Þ ¼ β1 þ β 3 :

5.3

Spot Rate and Yield Curve

Definition 5.5 The notion term structure refers to the distribution of any rates along the time axes. The term structure of the par yield rates is called the par yield rate curve. The par bond rate curve is representative for the value of a coupon the bond universe in paying at each point of the axis. Starting with a given spot curve s(t), we derive in this section a par yield curve.

5.3 Spot Rate and Yield Curve

95

Theorem 5.2 We assume that the spot curve s(t) and the discount factor d(t) are continuous and expressed in years. The par yield rpar of a Bond with time to maturity tN ¼ T at t ¼ 0 is then given by discrete compounding rpar ðTÞ ¼

1 ‐ dN ðTÞ N P d ðtk Þ

ð5:3:1aÞ

k¼1

and by continuous compounding rpar ðTÞ ¼

1 ‐ es N T : N P s k k e

ð5:3:1bÞ

k¼1

Proof We start by (3.1.1) by evaluation at t0 ¼ 0: Pð0Þ ¼

N X j¼1

C 1 þ : j ð1 þ rÞ ð1 þ rÞN

By using the assumption that the spot rate are given by Pð0Þ ¼

N X j¼1



C 1 ,  j þ ð1 þ sðtN ÞÞN 1 þ s tj

for a bond at par, we have Pðt0 Þ ¼ 1 and C ¼ rpar, and thus, for annual compounding, we have 1¼

N X j¼1



rpar 1 ,  j þ ð1 þ sðt N ÞÞN 1 þ s tj

And, by continuous compounding and (3.1.4), we have 1¼

N X

rpar e‐jsðtj Þ þ e‐NsðtN Þ ,

j¼1

which yields (1). □ The Government Bond Markets of some developed countries like for e.g. US, UK, Japan or Switzerland are the least likely to default. These governments will pay a minimum of borrowing costs. The US government is and has been the most important bond market globally. It has been best populated in

96

5

The Term Structure of Interest Rate

term of time to maturities of individually bonds. Often it is said that these markets are riskless. An investor who holds the bond until maturity has no interest rate risk, because he gets his investment back at the time of maturity. However, an investor that trades its positions in the portfolio is exposed to the interest market risk. We proceed with a general definition of market risk: Definition 5.6 The term market risk reflects the possibility that an investor experiences loss due to factors affecting the overall performance of financial markets. Remark 5.3 Examples for market risk are natural disaster, recession, political turmoil, changes in interest rates and terrorist attacks. Remark 5.4 The interest rate market risk is the market risk affecting the yield curve, regardless of the risk originating from specific issuer of fixed income instrument. The inflation and money market policy of central banks are typical market risks for interest rates. There are also theories that try to explain the behaviour of the interest rates (see for e.g. [1]). The market price of a bond is exposed to interest rate market risk, the coupon payment and the face value, however, are untouched. Thus, in economic research, market prices are important for assessing interest rate market. The spot rate curve can be used as a benchmark for pricing bonds. This type of rate curve can be built from on-the-run treasuries. In Theorem 5.2, we derive the par yield curve from the spot rate. As bond prices are traded in the market place, the question is rather: how can yields to maturities be transformed in spot rates.

Lemma 5.1 (Linear Structure of the Bond) A bond is equal to a series of zero coupon bonds. Proof Follow from the Definition 5.7 and the price formula of the invoice price (3.1.2). □ We measure the Bond Price in the market and we assume that we know the term and condition of the Bond. With Bootstrapping, we describe the transition with which the spot curve and the forwards rate are calculated from the Bond Price and the Coupon. Bootstrapping makes the very restrictive assumption that we have a price at each point. We assume that the Price PN and the coupon CN and time tN of the bond is given. We proceed iteratively and consider a bond that matures in 1 year, P1 ¼ i.e.,

C1 þ F1 , 1 þ s1

ð5:3:2aÞ

5.3 Spot Rate and Yield Curve

97

s1 ¼

C 1 þ F1  1: P1

ð5:3:2bÞ

For two periods, we have P2 ¼

C2 C þ 100 þ 2 , 1 þ s1 ð1 þ s2 Þ2

i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 þ 100 s2 ¼  1, C2 P2  1þs 1 PN ¼

CN CN C þ FN þ ...... þ þ N : 1 þ s1 ð1 þ sN1 ÞN1 ð1 þ sN ÞN

We proceed with illustrating that the only yield curve that is equal to the spot curve is the par yield. The following example shows that two par bonds with different par yields exemplify the general case, i.e., the yield of the maturity in the first interval is equal to the spot rate, and the second spot rate is numerically different to the par yield. Example 5.5 We consider a par bond with 1 year maturity with a coupon C1 ¼ 10% and a par bond with 2 year maturity with a coupon C2 ¼ 10%. For the spot curve we have s1 ¼

F1 þ C1 ‐1 ¼ 10% P1

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffi F þ C2 ‐ s2 ¼ C2 P2 ‐ 1þs 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 100% þ 10% ‐ 1 ¼ 1:1 100% ‐ 10% 1:1

1¼

1 ¼ 0:1, s2 ¼ 10%,

Thus, the 2-year spot is equal to the yield to maturity. In the following, we illustrate that if the par yield of the 2-year par yield is different to 1-year par yield, then the 2 year spot yield is different to the par yield of the 2 year bond. We consider par yields between 0% and 20%, and Fig. 5.3 shows the par yield of the 2 year bond versus the difference of the par yield 2 year spot and the 2 year spot rate. If the coupon is zero, we find s2 ¼ 0. We see that the spot rate is bigger than the par yield if

98

5

The Term Structure of Interest Rate

Fig. 5.3 Par versus spot

the par yield is smaller than s1, and that the spot rate is smaller than the par yield if the par yield is bigger than s1. The par yields are averaging the spot rates. The question is whether a par bond can be replaced by a bond with the same yield of maturity leaving the spot rate unchanged. The following example shows that this is in general not the case. Example 5.6 We consider a par Bond with 1 year maturity with a coupon C1 ¼ 10% and a par Bond with 2 year maturity with a coupon C2. We distinguish 2 cases. (a) C2 ¼ 10%. By changing the C2 and the price of the bond with 2 year to maturity such the yield to maturity is equal to 10%, an analysis shows that s2 ¼ C2 ¼ 10%. (b) C2 ¼ 15 % + α % , α 2 Z,  20  α  20. The following Fig. 5.4 shows the spot rate s2 as a function of the coupon C2. We keep the yield to maturity 15% constant, i.e., we consider non par yield. If the coupon is zero we have a second zero coupon. The analysis of the formulae shows that the difference is an almost linear relationship, and the par yield is unique in the sense that the spot rate changes by exchanging a bond with the same time to maturity and the same yield to maturity. The following formula computes recursively the spot rate sn vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Ck þ Fk sk ¼ u  1, 1  k  N: k u kP 1 t 1 Pk  Ck ð1þsn Þn n¼1

This leads to the following Theorem.

5.3 Spot Rate and Yield Curve

99

16.0% 15.8%

S2

15.6% 15.4% 15.2% 15.0% 14.8% 14.6% -5%

0%

5%

10%

15%

20%

25%

30%

35%

C2

Fig. 5.4 Same yield to maturity versus different coupons

Theorem 5.3 (Bootstrapping) We consider a Portfolio that consists of a series of par straight bonds Pk with face Fk and coupons Ck that have time to maturities Tk ¼ k, 1  k  N and coupon payments at Tk ¼ j, 1  j  k, 1  k  N: We consider a flat curve with interest rate r 2 R1 and assume that Pk > Ck

k‐1 X n¼1

1 ‐1, 1  k  N: ð1 þ sn Þn

Then, there exits an Interval Iδ(rr) such that, for each rk 2 R1 in Iδ(rk) with price Pk(rk), starting with s1 ¼ r1, the spot rates sk, 2  k  N, are recursively given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Ck þ Fk sk ¼ u  1, 1  k  N: ð5:3:3Þ k u k‐1 P t 1 Pk ‐Ck n ð1þsn Þ n¼1

Proof We consider a series of Bonds Bk, 2  k  N, with Prices and Face Value, resp., as PN ¼ 100%, FN ¼ 100%, N ¼ 1, . . . :, resp:, and Coupons

ð5:3:4aÞ

100

5

The Term Structure of Interest Rate

1  r ¼ C, 1  k  N:

ð5:3:4bÞ

Then the yield of maturities rN of the Bonds BN satisfy ð aÞ r N ¼

C , N ¼ 1, . . . : 100

ðbÞ SN ¼

C , N ¼ 1, . . . : 100

We pursue a proof by induction. For N ¼ 1, the assertion follows from (1). By hypothesis of the theorem, we start by PN ¼ 100, N ¼ 1, . . . : (a) By considering the equation for the yield of the maturity r, we have PNþ1 ðrÞ ¼

CNþ1 C C þ FNþ1 þ Nþ1 þ . . . þ Nþ1 Nþ1 ¼ 1 þ r ð1 þ r Þ2 ð1 þ rÞ

! 1 CNþ1 CNþ1 þ FNþ1 CNþ1 þ þ ... þ : 1þr 1þr ð1 þ r ÞN By substituting r ¼ rN and using (4), PNþ1 ðrN Þ ¼

1 ðC þ PðrN ÞÞ, N ¼ 1, . . . :, 1 þ rN

i.e., PNþ1 ðrN Þ ¼

1 ðC þ PðrN ÞÞ, 1 þ rN

and defining rNþ1 ¼ rN ¼

C , 100

we find that rN + 1 satisfies PNþ1 ðrNþ1 Þ ¼

1 ðC þ PðrNþ1 ÞÞ ¼ 100, 1 þ rNþ1

i.e., rN + 1 is the yield of maturity of PN + 1. (b) We start by (3), (4), and we use the induction assumption

5.3 Spot Rate and Yield Curve

sNþ1

101

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u CþF u  1, 1  k  N: ¼ Nþ1 u N P t C PNþ1  ð1þCÞk k¼1

Following (a), rN+1 is the yield of maturity from PN+1 PNþ1 ðrNþ1 Þ 

C C C þ þ ... þ ¼ 1 þ rNþ1 ð1 þ rNþ1 Þ2 ð1 þ rNþ1 ÞN

ð5:3:5Þ

CþF : ð1 þ rNþ1 ÞNþ1 Thus, we find sNþ1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u CþF u ¼ Nþ1  1, 1  n  N, u N P t C PNþ1  ð1þCÞk k¼1

and thus, we have rN + 1 ¼ rN ¼ C and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CþF sNþ1 ¼ Nþ1 FþC  1, 1  n  N, ð1þCÞNþ1

i.e., sNþ1 ¼ C, 1  n  N: Remark 5.5 As exposed in [4], we consider 0 1 þ r1 0 B 0 B : B B A ¼ B rj : B : : @ rn :

0 0

0 0

1 þ rj

0

: :

0 :

Then, the system of linear equations Ad ¼ P for the discount factors

0 0

□

1

0

P1

1

C B C C B C C B C C C 0 C, P ¼ B B Pj C: C B C 0 A @ A Pn 1 þ rn

102 Table 5.1 Calculation by bootstrapping

5 Time tk Spot rate Time tk Spot rate

t¼1 0.0100 t¼6 0.0640

0

d1

The Term Structure of Interest Rate

t¼2 0.0201 t¼7 0.0760

t¼3 0.0304 t¼8 0.0910

t¼4 0.0411 t¼9 0.1074

t¼5 0.0522 t ¼ 10 0.1270

1

B C B C B C C d¼B B dj C B C @ A dj is the same as evaluating the recursive formula (3). Remark 5.6 The yield to maturity of a Bond is not used in (3) and can be calculated by price, coupon, and time to maturity of the Bond. The spot curve allows pricing a Bond Universe consistently. Example 5.7 We consider Bonds where the coupon is equal the yield to maturity rk ¼

Ck ¼ k, 1  k  10, 100

ð5:3:6Þ

i.e., according to Lemma 5.1, we have Pk ¼ 100. The values in Table 5.1 are calculated by formula (3). The ending spot rate is above the yield to maturity. In addition, the difference between consecutive spot rates is increasing. In general, the spot curve has to be modelled from many bonds. The problem is underdetermined. The spot curve is then estimated by a few numbers of parameters and a functional form of the term functional. However, we have only points in time. For Fig. 5.5, we expand the spot rate to the time axis between 0 and 10 years by a piecewise constant function. The values for any time are in Table 5.2. In mathematical terms, the function is not continuous and has jumps at the time t ¼ 1,. . .,10. The function is called left sided continuous. The Bootstrapping has the following two properties. 1. It makes the assumption that we have a price at each Coupon payment. 2. There are no parameter specification like in the Nelson-Siegel.

5.3 Spot Rate and Yield Curve

103

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0

2

4

6

8

10

Fig. 5.5 Spot rates for 1–10 years to maturity Table 5.2 Step function 0