Fixed Income Investing: A Classic in a Time of Increased Uncertainty 3030879216, 9783030879211

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Thomas Poufinas

Fixed Income Investing A Classic in a Time of Increased Uncertainty

Fixed Income Investing

Thomas Poufinas

Fixed Income Investing A Classic in a Time of Increased Uncertainty

Thomas Poufinas Department of Economics Democritus University of Thrace Komotini, Greece

ISBN 978-3-030-87921-1 ISBN 978-3-030-87922-8 (eBook) https://doi.org/10.1007/978-3-030-87922-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Palgrave Macmillan imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Angelina and Katherine

Preface

For those of us who were active in the investment world in the turn of the millennium one of the biggest potential showstoppers seemed to be the millennium bug. However, anticipation at the time prevented any big disruptions and only minor issues were reported. Having entered the third decade of the very same millennium (2020–2021) we could not have expected that a pandemic would prevail in almost all financial, economic, social and political activities. A pandemic sounded too remote in the past and too unlikely to occur in the future. It seems that if it were discussed—in the financial services industry—then that should have been only between insurers and reinsurers when they drafted their treaties. At the same time, whenever a market crash or crisis was discussed, the reference point was this of 1929 as any more recent crisis or crash seemed to have been more easily surpassed. Yet, it was seven or eight years after the turn of the millennium (2007–2008) that a prolonged financial and sovereign crisis emerged. In some countries the effects were managed in a relatively short period of time, whereas in others it took more time to recover—if they ever did. It seems though that since then central banks and governments were willing to provide support to the economies of the countries at a macro level, translated to assistance to enterprises and households at a micro level. For more than ten years the central banks, the FED and the ECB as an example, have been providing ample support to the financial markets and the economies through purchase programs that pertained primarily to

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PREFACE

fixed income securities. The so called quantitative easing programs were not limited to government bonds, but extended gradually to corporate bonds, ETFs and covered bonds. Furthermore, following the unprecedented impact of the slowdown or shutdown due to the pandemic, it even included fallen angels. This was the ‘medicine’ used since the 2007– 2008 crisis several times and was put at work even more intensively since the beginning of 2020. This policy/approach has kept interest rates low and in some cases even negative. Fixed income investments have always been of interest to the investors; they were considered as safe havens when the stock markets had negative years, as they provided a steady, positive yield. Prior to the crises they were perceived to be moving at opposite directions from stocks, thus offering sufficient diversification. Furthermore, they were the preferred investment means for insurance companies and pension schemes, as they could be used to accumulate the benefits promised to the insured/beneficiaries and the pensioners respectively. However, at the present time (end of 2020— beginning of 2021) bond yields are considerably low, and in some cases government bond yields are even negative. More specifically all of the Eurozone countries post negative yields for at least one of their sovereign issued and some of them record negative yields for all of their issues, such as Germany and the Netherlands for example. These two countries have 30-year bonds with negative yields to maturity, whereas Switzerland’s 50year bond issue has negative yield too. In the USA yields are still positive but at their lowest levels for the last forty years. This realization makes the study of fixed income investments more relevant than ever. In a low or even negative interest rate environment, with many challenges still emerging, it is important to understand the mechanics of fixed income instruments, their use as investment vehicles, their role as financing tools, the risks they entail, their association with the crisis, their link with the overall debt of a country, as well as the applicable regulatory framework. In this book, combining the market and academic experience of more than two decades, we try to present the full spectrum of topics relevant to fixed income investments. Starting from the very basic notions related to bonds and bond valuation to the more advanced concepts relevant to interest rate and credit derivatives and from the (recent) regulatory framework to the (more traditional) risk management techniques. The writing of this book has been influenced by my own exposure to the field (as a practitioner at the beginning and as an academic in the end) on one hand

PREFACE

ix

and by the books I have studied all these years on the other hand. These books helped me not only understand these topics but also apply them in practice. Up to date academic research and papers also contributed, but I feel that the fundamentals were established by classical editions, such as Investments, by Bodie, Kane & Marcus (1996); Investment Science, by Luenberger (1998); Options, Futures and Other Derivatives, by Hull (1997, 2005, 2012); and, The Fundamentals of Risk Measurement, by Marrison (2002). This book is structured in the following manner. Chapter 1 is an introduction to the basic notions of investments in general and bonds in particular. Chapter 2 addresses the main concepts related to bonds and their valuation. Chapter 3 discusses the term structure of interest rates. Chapter 4 offers the principles of fixed income portfolio management. Chapter 5 analyzes interest rate derivatives, whereas Chapter 6 focuses on credit derivatives. Chapter 7 presents the bond markets. Chapter 8 describes bond funds. Chapter 9 explains the risks associated with fixed income investments and how they are managed. Chapter 10 attempts to capture the impact of crises and the reaction to them on bonds and bond yields. Chapter 11 investigates the link between bond issuance and the overall debt of a country. Chapter 12 compares bonds with stocks. Chapter 13 shows how fixed income instruments can be used for hedging, speculation and arbitrage. Finally, Chapter 14 sketches the impact of the applicable (current and upcoming) regulatory framework. Hopefully the book will become a useful guide—next to established books—for students, academics and practitioners/investors in understanding fixed income investing as a classic but also in a time of increased uncertainty. Komotini, Greece

Thomas Poufinas

References Bodie, Z., Kane, A. & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. Hull, J. C. (1997). Options, Futures and Other Derivatives (3rd ed.). Prentice Hall International, Inc. Hull, J. C. (2005). Options, Futures and Other Derivatives (5th ed.). Prentice Hall International, Inc.

x

PREFACE

Hull, J. C. (2012). Options, Futures and Other Derivatives (8th ed.). Prentice Hall International, Inc. Luenberger, D. G. (1998). Investment Science. Oxford University Press. Marrison, C. (2002, July 18). The Fundamentals of Risk Measurement (1st ed.). McGraw-Hill Education.

Acknowledgements

Writing this book on fixed income investing proved to be a very pleasant and productive experience. It was an opportunity to put on paper the market and teaching experience I have gained on the field over the past years. The success of this undertaking would not have been possible without the support—active or passive—of some people. First, many thanks are due to my two co-authors Nick Apergis and Giusy Chesini for their contribution in two chapters (10 and 11) of the book. These chapters are based on recent original research that we prepared for another project with Palgrave Macmillan. Tackling the topic was in the initial proposal of the book, as the link of fixed income securities with debt and crisis could not have been missed from a book on fixed income. Having though recent evidence on the topic could not have been omitted either from the relevant chapters. Although I really enjoyed the time invested in the writing of the book, I feel it was occasionally taken from tasks or people that expected some other deliverable from me. For this I am thankful to my family for their understanding and support, as well as to my colleagues, students and staff who accepted a potential last minute submission, delivery or notice due to this effort. Last but not least, my gratitude is in order to the members of Palgrave Macmillan who believed in the proposal, accepted its publication and

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ACKNOWLEDGEMENTS

guided me until the submission of the chapters. With the order met—in most cases remotely, I am thankful to Tula Weis, Lucy Kidwell, Ashwini Elango, Susan Westendorf and Shukkanthy Siva for their trust, patience and advice.

Contents

1

1

Introduction

2

Bonds

47

3

Term Structure of Interest Rates

99

4

Fixed Income Portfolio Management

169

5

Interest Rate Derivatives

265

6

Credit Derivatives

349

7

Bond Markets

369

8

Bond Funds

407

9

Risks and Risk Management

455

10

Bonds and Crises

535

11

Bonds and Debt

557

12

Bonds versus Stocks

571

13

Hedging, Speculation and Arbitrage

621

14

Bonds and Regulation

649

Index

695

xiii

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 2.1 Fig. 2.2 Fig. 2.3

Fig. 2.4

Fig. 2.5

Fig. 2.6

Fig. 2.7

Risk-return ratio (Source Created by the author) Risk taken per type of investment (Source Created by the author) APP amount per period implemented (Source Created by the author with data assembled from the ECB [2021]) Example of a bearer bond in paper format (Source Created by the author) Spread as a function of time to maturity per credit rating class (Source Created by the author) Average cumulative issuer-weighted global default rates (%), 1970–2010—part I (Source Created by the author with data assembled from Moody’s [2011]) Average cumulative issuer-weighted global default rates (%), 1970–2010—part II (Source Created by the author with data assembled from Moody’s [2011]) Global corporate average cumulative default rates (%) by rating modifier (1981–2018)—part I (Source Created by the author with data assembled from S&P [2019]) Global corporate average cumulative default rates (%) by rating modifier (1981–2018)—part II (Source Created by the author with data assembled from S&P [2019]) Global corporate average annual default rates by (%) rating modifier (1981–2018)—part I (Source Created by the author based on author’s calculations with data from Standard and Poor’s [S&P, 2019])

7 19 37 50 59

60

60

61

62

62

xv

xvi

LIST OF FIGURES

Fig. 2.8

Fig. 2.9 Fig. 2.10 Fig. 3.1 Fig. 3.2

Fig. 3.3

Fig. 3.4

Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8

Fig. 3.9 Fig. 3.10 Fig. 3.11

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4

Global corporate average annual default rates (%) by rating modifier (1981–2018)—part II (Source Created by the author based on author’s calculations with data from Standard and Poor’s [S&P, 2019]) Bond valuation (Source Created by the author) Relationship between price and bond yield (Source Created by the author) Term structure of interest rates (Source Created by the author with data assembled from Statista [2021]) Two investment alternatives for a two year horizon: one 2-year period versus two 1-year rolling periods (Source Created by the author) Three investment alternatives for a three year horizon: one 2-year & one 1-year periods versus one 3-year period versus one 1-year & one 2-year periods (Source Created by the author) Four investment alternatives for a three year horizon: one 2-year & one 1-year periods versus one 3-year period versus one 1-year & one 2-year periods versus three one-year rolling periods (Source Created by the author) Zero-coupon yield curve for 6 years (Source Created by the author) Types of yield curves (Source Created by the author) Term structure of interest rates for 3 years (Source Created by the author) Interest rate mean reversion property (Source Created by the author with information assembled from Hull [1997]) Trinomial tree standard branching (Source Created by the author) Basic trinomial tree (Source Created by the author) Trinomial tree branching methods (Source Created by the author with information assembled from Hull [1997]) Price sensitivity to interest rate changes (Source Created by the author) Duration as a function of time to maturity, coupon and interest rate (Source Created by the author) Using duration to calculate the change in the bond price (Source Created by the author) Duration of a zero-coupon bond (Source Created by the author)

63 69 76 106

107

110

114 120 121 129

148 154 156

156 175 182 199 199

LIST OF FIGURES

Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. Fig. Fig. Fig. Fig. Fig.

5.12 5.13 5.14 5.15 5.16 5.17

Fig. 6.1

Duration of a coupon-bearing bond (Source Created by the author) Duration at the coupon payment time instant (Source Created by the author) Duration shift after the coupon payment instant (Source Created by the author) Convexity (Source Created by the author) Bank assets and liabilities (Source Created by the author) Pension and insurance assets and liabilities (Source Created by the author) Cash flow matching for a pure endowment portfolio (Source Created by the author) Schematic representation of an interest rate swap (Source Created by the author) Using a swap to transform a liability (Source Created by the author) Using a swap to transform a liability—An example (Source Created by the author) Using a swap to transform an asset (Source Created by the author) Using a swap to transform an asset—An example (Source Created by the author) Construction of a swap (Source Created by the author) Construction of a swap—An example (Source Created by the author) The role of intermediaries in swaps (Source Created by the author) Example of a currency swap (Source Created by the author) Trinomial interest rate tree for the pricing of options (Source Created by the author) Trinomial interest rate tree—An example (Source Created by the author) MBS (Source Created by the author) CMO (Source Created by the author) ABS (Source Created by the author) CDO (Source Created by the author) CDO Tranches (Source Created by the author) Borrowing interest rate for a floating rate loan with a cap or floor (Source Created by the author with information assembled from Hull [1997]) CDS (Source Created by the author)

xvii

200 200 201 208 235 239 248 285 286 286 287 288 289 289 291 296 305 310 318 323 324 326 328

333 353

xviii

LIST OF FIGURES

Fig. 6.2 Fig. 6.3 Fig. 7.1

Fig. 8.1

Fig. 8.2

Fig. 8.3

Fig. 8.4

Fig. 8.5

Fig. 8.6

Fig. 8.7

Fig. 8.8

Fig. 8.9

Fig. 8.10

Fig. 8.11

Total return swap (Source Created by the author) CDO (Source Created by the author) Government bond yield to maturity (%) for all available maturities for the USA, the UK, Germany, France and Italy (Source Created by the author with data assembled from World Government Bonds (2021)) Net assets of worldwide regulated open-end funds (in trillion USD) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) Net assets of worldwide regulated open-end funds (percent of Total) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) AUM of bond and money market mutual funds (in billion USD) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) Number of bond and money market mutual funds (Source Created by the author with data assembled from the Investment Company Institute (2020b)) AUM by type of bond mutual funds (in billion USD) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) AUM by type of bond mutual funds (in percent) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) Number of bond mutual funds by type (Source Created by the author with data assembled from the Investment Company Institute (2020b)) Number of bond mutual funds by type (in percent) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) AUM of bond ETFs by major bond type investment (Source Created by the author with data assembled from the ETF Database (2020)) Number of bond ETFs by major bond type investment (Source Created by the author with data assembled from the ETF Database (2020)) Average dividend yield of bond ETFs by major bond type investment (in percent) (Source Created by the author with data assembled from the ETF Database (2020))

359 362

386

418

418

420

421

422

423

423

424

434

434

435

LIST OF FIGURES

Fig. 8.12

Fig. 8.13

Fig. 8.14

Fig. 8.15

Fig. 8.16

Fig. 8.17

Fig. 8.18

Fig. 8.19

Fig. 8.20

Fig. 8.21

Fig. 8.22 Fig. 8.23

Fig. 8.24

AUM of bond ETFs by major bond duration investment (Source Created by the author with data assembled from the ETF Database (2020)) Number of bond ETFs by major bond duration investment (Source Created by the author with data assembled from the ETF Database (2020)) Average dividend yield of bond ETFs by major bond duration investment (in percent) (Source Created by the author with data assembled from the ETF Database (2020)) Comparison of private debt with other asset categories—Part I (Source Created by the author with information assembled from PRI (2019)) Comparison of private debt with other asset categories—Part II (Source Created by the author with information assembled from PRI (2019)) Risk-return profiles for various investment strategies (Source Created by the author with information assembled from PRI (2019), IHS Markit (2017) and NN Investment Partners (2017)) Relationship between bank and non-bank loans for leveraged loans (Source Created by the author with data assembled from the IMF (2020)) Evolution of the AUM by type of alternative investment (Source Created by the author with data assembled from PRI (2019) and from Preqin (2018b)) Private debt analysis by investor type (Source Created by the author with data assembled from PRI (2019) and from Preqin (2018a)) Expectations of private debt funds for sectors that will attract their highest levels of investment (Source Created by the author with data assembled from PRI (2019) and from Intertrust (2018)) Private debt funds evolution (Source Created by the author with data assembled from Preqin (2020a)) Private debt fund evolution by region (in USD) (Source Created by the author with data assembled from PitchBook (2020)) Private debt fund evolution by region (in number) (Source Created by the author with data assembled from PitchBook (2020))

xix

436

436

437

440

440

441

444

445

446

447 447

448

448

xx

LIST OF FIGURES

Fig. 8.25

Fig. 9.1 Fig. 9.2

Fig. 9.3

Fig. 9.4

Fig. 9.5

Fig. 9.6 Fig. 9.7

Fig. 9.8

Fig. 9.9

Fig. 9.10

Fig. 9.11 Fig. 9.12 Fig. 9.13 Fig. 9.14

Private debt fund comparative performance (Source Created by the author with data assembled from PitchBook (2020)) 1−p Value at Risk (Source Created by the author with information assembled from Marrison [2002]) Comparison of the VaR calculation methods—Part I and Part II (Source Created by the author with information assembled from Marrison [2002]) Average corporate debt recovery rates measured by post-default trading prices (1982–2010) (Source Created by the author with data assembled from Moody’s [2011]) Average corporate debt recovery rates measured by ultimate recoveries (1987–2010) (Source Created by the author with data assembled from Moody’s [2011]) One-year global corporate transition rates (% – 2018) (Source Created by the author with data assembled from S&P [2019]) Credit Default Swap (Source Created by the Author) Global Corporate Average Cumulative Default Rates (%) by Rating Modifier (1981–2018)—Part I (Source Created by the author with data assembled from S&P [2019]) Global Corporate Average Cumulative Default Rates (%) by Rating Modifier (1981–2018)—Part II (Source Created by the author with data assembled from S&P [2019]) Global Corporate Average Annual Default Rates by (%) Rating Modifier (1981–2018)—Part I (Source Created by the author based on author’s calculations with data from Standard and Poor’s [S&P, 2019]) Global Corporate Average Annual Default Rates (%) by Rating Modifier (1981–2018)—Part II (Source Created by the author based on author’s calculations with data from Standard and Poor’s [S&P, 2019]) Annual corporate rating changes (Source Created by the author with data assembled from S&P [2019]) Number of issuers (as of 1/1) (Source Created by the author with data assembled from S&P [2019]) Downgrade/upgrade ratio (Source Created by the author with data assembled from S&P [2019]) Rate change statistics (Source Created by the author with data assembled from S&P [2019])

449 465

473

511

511

517 519

520

520

521

521 522 522 523 523

LIST OF FIGURES

Fig. 9.15 Fig. 10.1

Fig. 10.2

Fig. 10.3

Fig. 11.1

Fig. 11.2

Fig. 11.3

Fig. 11.4

Downgrade/upgrade ratio statistics (Source Created by the author with data assembled from S&P [2019]) Percentage of national government bonds held by domestic banks (Source Created by the authors with data assembled from the Bruegel Datasets [2020]) Percentage of national government bonds held by non-residents (Source Created by the authors with data assembled from the Bruegel Datasets [2020]) Percentage of national government bonds held by central banks (Source Created by the authors with data assembled from the Bruegel Datasets [2020]) Trend of gross consolidated general debt (in millions) (Source Created by the authors with data assembled from the ECB Statistical Data Warehouse [2021]) General government gross debt (% GDP) (Source Created by the authors with data assembled from the International Monetary Fund, World Economic Outlook Database, October 2020) Gross general debt broken down into financial instruments for 12 Eurozone countries (average %) (Note: 2020: until Q2. Source: Created by the Authors with Data Assembled from the ECB Statistical Data Warehouse [2021]) Non-resident investors in country’s sovereign debt (%) (Source Created by the authors with data assembled from the Bruegel Datasets [2020])

xxi

524

547

548

550

562

563

564

566

List of Tables

Table Table Table Table Table

2.1 2.2 3.1 3.2 3.3

Table 3.4 Table Table Table Table

3.5 3.6 4.1 4.2

Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9

Credit ratings (long-term) by agency Yield to call versus yield to maturity Spot, forward and short rates Spot, forward and short rates for a 3-year time interval Spot, forward and short rates with semi-annual compounding Spot, forward and short rates with continuous compounding Zero-coupon yield curve for 6 years Interest rate term structure models Price change of a zero-coupon bond (initial r = 4%) Price change of a coupon-bearing bond (c = 4%, initial r = 4%) Price change of a zero-coupon bond (initial r = 8%) Price change of a coupon-bearing bond (c = 4%, initial r = 8%) Duration of a zero-coupon bond (initial r = 4%) Duration of a coupon-bearing bond (c = 4%, initial r = 4%) Duration of a zero-coupon bond (initial r = 8%) Duration of a coupon-bearing bond (c = 4%, initial r = 8%) Price change of a zero-coupon bond (initial r = 4%) with duration

56 80 113 114 116 118 119 146 173 173 173 174 179 179 180 180 184

xxiii

xxiv

LIST OF TABLES

Table 4.10 Table 4.11 Table 4.12 Table 4.13

Table 4.14

Table 4.15

Table 4.16

Table 4.17 Table 4.18 Table 4.19 Table 4.20 Table 4.21 Table 4.22 Table 4.23 Table 4.24 Table 4.25 Table 4.26 Table 4.27 Table 4.28

Price change of a coupon-bearing bond (c = 4%, initial r = 4%) with duration Price change of a zero-coupon bond (initial r = 8%) with duration Price change of a coupon-bearing bond (c = 4%, initial r = 8%) with duration Deviation of the percentage price change of a zero-coupon bond (initial r = 4%) with the use of duration from the actual percentage price change Deviation of the percentage price change of a coupon-bearing bond (c = 4%, initial r = 4%) with the use of duration from the actual percentage price change Deviation of the percentage price change of a zero-coupon bond (initial r = 8%) with the use of duration from the actual percentage price change Deviation of the percentage price change of a coupon-bearing bond (c = 4%, initial r = 8%) with the use of duration from the actual percentage price change Price change of a zero-coupon bond (initial r = 4%) Price change of a coupon-bearing bond (c = 4%, initial r = 4%) Price change of a zero-coupon bond (initial r = 8%) Price change of a coupon-bearing bond (c = 4%, initial r = 8%) Duration of a zero-coupon bond (initial r = 4%) Duration of a coupon-bearing bond (c = 4%, initial r = 4%) Duration of a zero-coupon bond (initial r = 8%) Duration of a coupon-bearing bond (c = 4%, initial r = 8%) Price change of a zero-coupon bond (initial r = 4%) with duration Price change of a coupon-bearing bond (c = 4%, initial r = 4%) with duration Price change of a zero-coupon bond (initial r = 8%) with duration Price change of a coupon-bearing bond (c = 4%, initial r = 8%) with duration

185 185 185

186

187

187

187 188 188 188 189 189 189 189 190 190 190 191 191

LIST OF TABLES

Table 4.29

Table 4.30

Table 4.31

Table 4.32

Table 4.33 Table 4.34 Table 4.35 Table 4.36 Table 4.37 Table 4.38 Table 4.39 Table 4.40 Table 4.41

Table 4.42

Table 4.43

Deviation of the percentage price change of a zero-coupon bond (initial r = 4%) with the use of duration from the actual percentage price change Deviation of the percentage price change of a coupon-bearing bond (c = 4%, initial r = 4%) with the use of duration from the actual percentage price change Deviation of the percentage price change of a zero-coupon bond (initial r = 8%) with the use of duration from the actual percentage price change Deviation of the percentage price change of a coupon-bearing bond (c = 4%, initial r = 8%) with the use of duration from the actual percentage price change Convexity of a zero-coupon bond (initial r = 4%) Convexity of a coupon-bearing bond (c = 4%, initial r = 4%) Convexity of a zero-coupon bond (initial r = 8%) Convexity of a coupon-bearing bond (c = 4%, initial r = 8%) Price change of a zero-coupon bond (initial r = 4%) with duration and convexity Price change of a coupon-bearing bond (c = 4%, initial r = 4%) with duration and convexity Price change of a zero-coupon bond (initial r = 8%) with duration and convexity Price change of a coupon-bearing bond (c = 4%, initial r = 8%) with duration and convexity Deviation of the percentage price change of a zero-coupon bond (initial r = 4%) with the use of duration and convexity from the actual percentage price change Deviation of the percentage price change of coupon-bearing bond (c = 4%, initial r = 4%) with the use of duration and convexity from the actual percentage price change Deviation of the percentage price change of a zero-coupon bond (initial r = 8%) with the use of duration and convexity from the actual percentage price change

xxv

191

192

192

192 209 209 209 209 211 212 212 212

214

214

214

xxvi

LIST OF TABLES

Table 4.44

Table 4.45 Table 4.46 Table 4.47 Table 4.48 Table 4.49

Table 4.50

Table 4.51

Table 4.52

Table 4.53

Table 4.54

Table 4.55 Table 4.56 Table 4.57

Deviation of the percentage price change of coupon-bearing bond (c = 4%, initial r = 8%) with the use of duration and convexity from the actual percentage price change Price change of a zero-coupon bond (initial r = 4%) with duration and convexity Price change of a coupon-bearing bond (c = 4%, initial r = 4%) with duration and convexity Price change of a zero-coupon bond (initial r = 8%) with duration and convexity Price change of a coupon-bearing bond (c = 4%, initial r = 8%) with duration and convexity Deviation of the percentage price change of a zero-coupon bond (initial r = 4%) with the use of duration and convexity from the actual percentage price change Deviation of the percentage price change of coupon-bearing bond (c = 4%, initial r = 4%) with the use of duration and convexity from the actual percentage price change Deviation of the percentage price change of a zero-coupon bond (initial r = 8%) with the use of duration and convexity from the actual percentage price change Deviation of the percentage price change of a coupon-bearing bond (c = 4%, initial r = 8%) with the use of duration and convexity from the actual percentage price change Time to maturity of a zero-coupon bond or coupon-bearing bond (c = 4%, initial r = 4% or r = 8%) Weighted average maturity of a zero-coupon bond or a coupon-bearing bond (c = 4%, initial r = 4% or r = 8%) Weighted average cash flow of a zero-coupon bond (initial r = 4% or r = 8%) Weighted average cash flow of a coupon-bearing bond (c = 4%, initial r = 4% or r = 8%) Price value of a basis point and percentage price change of a basis point of a zero-coupon bond (initial r = 4%)

215 215 215 216 216

216

217

217

217

224

225 226 227 227

LIST OF TABLES

Table 4.58

Table 4.59 Table 4.60

Table 4.61 Table 4.62 Table 4.63 Table 4.64 Table Table Table Table Table Table Table

4.65 4.66 4.67 5.1 5.2 5.3 7.1

Table 7.2 Table 7.3 Table Table Table Table Table Table Table Table Table Table

8.1 8.2 9.1 9.2 9.3 9.4 9.5 10.1 10.2 10.3

Price value of a basis point and percentage price change of a basis point of a coupon-bearing bond (c = 4%, initial r = 4%) Price value of a basis point and percentage price change of a basis point of a zero-coupon bond (initial r = 8%) Price value of a basis point and percentage price change of a basis point of a coupon-bearing bond (c = 4%, initial r = 8%) Yield value of a price change of -10 Euro of a zero-coupon bond (initial r = 4%) Yield value of a price change of -10 Euro of a coupon-bearing bond (c = 4%, initial r = 4%) Yield value of a price change of -10 Euro of a zero-coupon bond (initial r = 8%) Yield value of a price change of -10 Euro of a coupon-bearing bond (c = 4%, initial r = 8%) Cellular approach Bond cash flows and duration (r = 6%) Bond cash flows and duration (r = 5%) Candidate bonds for delivery under a futures contract Cash flow per candidate bond for delivery Swap cash flows Government bond yield to maturity (%) for all available maturities Number of bond indices—example (as published in the Wall Street Journal ) Example of bond indices (as published in the Wall Street Journal ) Bond ETF metrics by major bond type investment Bond ETF metrics by major bond duration investment Properties of VaR Calculation Approaches Historical VaR Calculation Monte Carlo VaR Calculation Expected loss from default Cumulative and annual default Overview of the variables Sovereign European bond yield estimates (all bonds) Sovereign European bond yield estimates (all bonds)-Alternative definitions of the external competitiveness variable—The role of the financial and sovereign debt crises

xxvii

228 228

228 229 229 229 230 234 243 244 278 278 287 388 400 401 433 435 469 477 480 507 509 538 539

541

xxviii

LIST OF TABLES

Table 10.4 Table 10.5

Table 10.6 Table Table Table Table

12.1 12.2 14.1 14.2

Sovereign European bond yield estimates—The role of the issue currency Sovereign European bond yield estimates (all bonds)—Alternative definitions of the external competitiveness variable—The role of the QE programmes Sovereign European bond yield estimates—North vs. South bond issues Common versus preferred shares Stocks versus bonds Market structure under MifID II/MifIR Obligations of SIs compared to Trading Venues and Non-SIs

542

544 545 576 602 683 684

CHAPTER 1

Introduction

The growth of the bond market is of particular importance for any economy as it achieves the reduction of borrowing costs for both businesses and the public sector. The former are looking for an alternative to bank lending way to raise debt, hopefully at a lower cost. The latter are trying to more efficiently manage their debt levels. This growth has been accelerated during the last decade which has been characterized by an unprecedented rise of debt at all levels—public and private, sovereign and corporate or even individual. This was the outcome of (the support provided by the central banks due to) initially the financial crisis (2007– 2008), which was succeeded by the debt crisis (2009—depending on the country) and finally of the pandemic (2020). However, a few decades ago, the bond market was primarily composed of only plain vanilla issues, the valuation of which was not difficult. Rising interest rates and fluctuations in the late 1970s and early 1980s have led to the emergence of new (at the time) types of bonds such as zero-coupon bonds and floating rate notes (FRN’s) or bonds with builtin purchase or sale options (call or put options). Such bonds proved though to be extremely useful even in the 1990s for issuers—especially countries/governments—who/which faced high but anticipated declining future yields. They assisted such countries in securing the viability and affordability of their debt as the improvement of their financials allowed the reduction of their borrowing cost. In the case of Europe, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_1

1

2

T. POUFINAS

this could have been the outcome of convergence process as part of the EMU (European Monetary Union), which led to the adoption of the Euro as the common currency of certain countries. Since then, financial engineering has manufactured some interesting structured notes, for example by stripping the coupons from the body of the bond and creating two streams of flows, by linking the coupon to the inflation or to any other index or yield/interest rate spread. In the first example, two streams of flows are created, giving rise to a (synthetically created) zero coupon bond and a (synthetically created) annuity. In the second, the coupon payment is related to a specific price index—usually the Consumer Price Index (CPI)—and the bonds are known as indexlinked bonds. In the third, it can be any other index, such as a stock index, in which case the coupon is paid when the index remains above or below or at a certain level or crosses a certain level from above or from below. In the fourth, any spread of yields or interest rate term structures— such as swap spreads—may be used to determine the periodic payment of interest. No matter what the variant created was, the principles remain the same; a periodic flow of income (which could be null) and/or a final payment (normally the face value). This is not only a function of bonds; it governs pretty much every fixed income investment, the most characteristic representatives of which are bonds. The last decade or so the scenery has changed; after the 2007–2008 sovereign and financial crises interest rates and bond yields globally have dropped to unprecedented levels, reaching negative levels (in most of the euro-zone countries for example) for rather long maturities (in some cases 10–15 years). This creates a paradox to the eyes of a reader. Are we paying to lend the issuer? Why is there still interest in fixed income investments and in particular bonds? To answer that, one needs to consider at the same time one more question; what else? Fixed income instruments, including bonds, are one type of investments made available to investors. They are usually considered of bearing lower risk compared for example to equity and as such they are (perceived to be) more suitable for relatively risk averse investors. One can argue that this was not exactly what we experienced during the latest crisis in 2008 and the years immediately following it. Yet though, this is still the case; overall, fixed income instruments are exhibiting lower risk levels versus equity (not to mention derivatives). Consequently, investors use fixed income investments for the part of their capital for which they

1

INTRODUCTION

3

want to bear lower levels of risk, even the ones that produce negative yields or interest rates. This is probably due to their superior quality and hence such a move—especially in distressed times—is known as “flying to quality”. Furthermore, one needs to take into account the global environment. In most of the developed world inflation has shrunk and the levels of CPI change have in many cases turned negative (as was the case in several member states of the European Union and the euro-zone in particular). Consequently, the interest rates that were offering compensation for the purchasing power lost due to inflation could afford being negative; the net result, i.e. the real interest rate (defined approximately as the difference between the nominal interest rate and the inflation rate) was most likely positive. Finally, one of the interested parties, in addition to the issuer and the investor, is the regulator. Central Banks had a key role to play; they injected capital in the market(s) so as to smoothen the consequences of the crisis by purchasing bonds and several other types of fixed income issues. This capital injection, known also as quantitative easing, targeted in providing the necessary liquidity that would keep the financing tap open, so that the crisis would not harm the entrepreneurial activity and the ability of the countries to repay their debt. The amounts were enormous, which also drove interest rates to negative territories. This realization raises one more question; for how long? Although any attempt to forecast interest rates can be notoriously bad, the consensus indicates that interest rates are not expected to go higher for quite some time in the foreseeable future (short- to mid- term). This was the anticipation in 2019 and this continues to be the anticipation in 2020 and the beginning of 2021, as with the start of the new decade humanity has to confront one more challenge; this of a pandemic. The root cause is not debt-related this time; however it is forecasted that it will impact the overall economic activity, as a consequence of which the economic growth will suffer (at least for 2020). To prevent the worse from happening, countries and central banks have mobilized again funds to support the economies and the markets. Quantitative easing will continue and thus interest rates will remain at relatively low levels. This statement, whether it will prove correct or not, brings one more question; will it ever change? We cannot tell, as according to some experts we have entered a new era, triggered by a virus, the durability and behavior of which is not well known. Consequently, our routine lifestyle

4

T. POUFINAS

could change. A change in the direction of the interest rates could be seen if the change of the production levels and our purchasing attitude result in higher inflation levels, which in their turn could lift interest rates higher. This discussion of course is not over; but it indicates that further digging into fixed income investments is more relevant than ever. Fixed income investments are in the spot light, potentially along with other types of securities. It seems though that they have not left the front stage, especially after the global crisis that commenced in 2007–2008, as debt amounts and defaults have proved to be a hard puzzle for the investors, both at corporate and government level. In this chapter we will try to explain the basic notions pertaining to fixed income investments and in particular bonds. We also give an introduction of the basic investment concepts, the different asset classes, the markets, as well as the participants involved. We explain the basic characteristics of bonds and stocks, as well as their basic functions both as investment instruments and means of financing. We present the bond issuance process and the roles of the parties involved. This helps the reader understand the broader environment in which bonds are placed. However, as we progress in the book we will add into our quiver more and more of the knowledge that has been accumulated through the years with regards to fixed income investing, reaching to the most recent developments and trends.

1.1

Introductory Notions

As fixed income securities in general and bonds in particular are investment, as well as financing vehicles/means we first attempt to place them in the entire investment universe and investment process. 1.1.1

Basic Steps in the Investment Process

The basic steps undertaken in the investment process, even if nothing else is considered, are primarily the following two: 1. Security analysis : This step has to do with the decision as to whether a securityis properly valued/priced. This is important as an investor needs to know if the amount he or she will disburse to acquire

1

INTRODUCTION

5

the security reflects its true/fair value. The intention is to identify mispriced securities, i.e. securities whose market price differs from their estimated (or perceived) value. Of course the latter can be subject to questioning, as it is computed based on a theoretical model that does not necessarily capture the market forces. 2. Portfolio management : This step pertains to the combination of the securities that have been selected after the analysis that has been performed according to the first step in a portfolio. Such a portfolio is compiled in order to match the needs of the investor and/or reflect his or her preferences, part of which is his or her risk-appetite and risk tolerance. It furthermore entails the monitoring of the portfolio and the evaluation/measurement of its performance—most frequently against the assumed risk. In the portfolio construction process, a portfolio manager attempts to find the best relationship between risk and return. A top-down analysis of the strategies followed by portfolio managers starts with the most general details of portfolio construction and ends with the most specific ones. This analysis leads to three decisions that are to be made: a. Capital allocation decision: This decision reflects the choice of the part of the total portfolio that will be placed in safe(r) but low yielding securities and the part that will be placed in securities with higher yield but with (higher assumed) risk. Of course, one could argue that the judgment of whether the expected return and the assumed risk are high or low can be very subjective and may be based on perceptions. The success of the investment process is to involve tools and methods that allow the quantification of risk and return as well as the perceptions/preferences of the investor and thus facilitate the separation between risky and non-risky assets and asset classes. b. Asset allocation decision: This decision pertains to the choice of the broad categories of risky assets at which the amount of money that has been decided to be invested in risky assets (according to the capital allocation decision), will be directed. These categories are/may be stocks, bonds, real estate, and many more. This choice is also made in a way that captures the risk and return preferences of the investor in a quantifiable manner.

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T. POUFINAS

c. Security selection decision: This decision addresses the selection of the specific securities that will comprise the portfolio from each of the asset categories/classes (as chosen via the asset allocation decision process). This means that specific stocks, bonds and securities from the other classes are chosen for inclusion in the investment portfolio. The selection process takes into account the targets of the investor as well as his or her preferred management type (passive or active) (Bodie et al., 1996). As the notions of risk and return have already been mentioned several times, we offer next their definition. 1.1.2

Return

The return of an investment, a portfolio or an asset is the ratio of the value of the investment at maturity or at the time of sale of the asset or closing of the position held on the asset to the value of the investment at its inception. The rate of return is the ratio of the change in the value of the investment at maturity or at the time of sale of the asset or closing of the position held on the asset from the value of the investment at its inception, to the value of the investment at its inception (Bodie et al., 1996). 1.1.3

Risk

The risk borne by an investment, a portfolio or an asset is the possibility that the return on an investment may be different from the expected return. In reality, if an investor has purchased the investment, portfolio or asset it refers to the possibility that the return on the investment is inferior to the expected one, as risk is ultimately the possibility of losing part or all of the initial investment (Bodie et al., 1996). It is usually measured by the standard deviation (which measures the discrepancy/distance of the actual return from the expected return). Globally though, risk describes the uncertainty of the investment outcome. This is due to the fact that the potential excess or lower return versus the expected return constitutes a difference from the perceived known outcome. Even the positive return can lead to a loss if for example the investor has short-sold the asset.

Expected Return (Mean)

1

Low Assumed Risk Low Expected Return

INTRODUCTION

7

High Assumed Risk High Expected Return

Assumed Risk (Standard Deviation) Fig. 1.1 Risk-return ratio (Source Created by the author)

1.1.4

Risk-Return Ratio

The return on the selected investment/asset/portfolio is directly dependent on the risk taken. A common misconception is that higher risk means higher return. The correct wording, however, is that the higher the assumed risk is, the higher the expected return becomes. Of course there is no guarantee. Higher risk can mean higher losses as well as higher profits (Bodie et al., 1996). As a matter of fact, (rational) investors attempt to maximize the expected return versus the risk they assume or minimize the risk they assume against the return they expect. They thus focus on the quotient of return over risk or risk over return, often referred to as the risk-return ratio. Figure 1.1 shows the risk-return ratio. 1.1.5

Benchmark

The evaluation of an investment should always be done in comparison to the return of a “reference point”, known as a benchmark, which has comparable characteristics to the investment or portfolio of interest. This means that the portfolio and the benchmark have the same (or similar) composition and therefore it is justified to compare the performance of the examined portfolio with that of the benchmark (portfolio) (Bodie et al., 1996). For example, the return of a fixed income portfolio that invests in a European market or in a set of European bonds or fixed income instruments cannot be compared with that of the DJIA index or the performance of a T-Bill. Furthermore, it cannot be compared with a bond

8

T. POUFINAS

index that invests in US bonds or European bonds that are different from the positions of the portfolio (in terms of specific assets and/or weights). 1.1.6

Differentiation

The aforementioned discussion revealed so far that investments carry risk. A natural question is how to reduce, minimize or even annihilate this risk—if possible. A well known approach is this of differentiation. Differentiation can be viewed as a risk management technique that allows a wide range of investments to be balanced (combined) in a portfolio so as to minimize the impact that each individual asset can have on the portfolio performance (Bodie et al., 1996). It pretty much reflects the well-known proverb that prompts not to put all your eggs in one basket. 1.1.7

Types of Assets

There are two broad types of assets; real assets and financial assets. 1. Real assets are/can be (among others) land/plant, buildings/property, equipment/machinery (used to produce goods), knowledge, and workers/employees/human resources who have some experience to use the former. Real assets generate income. 2. Financial assets are/can be stocks and bonds (among others). Financial assets distribute income to investors (Bodie et al., 1996). In this book we study fixed income financial assets only, with some reference to other financial asset classes, as necessary, for comparison purposes. 1.1.8

Participants of the Financial System

So far we focused on the processes/steps/decisions involved in the investment function as well as the means, i.e. the assets/securities/tools employed in order to perform investments. However, what is missing so far is the human and/or entity factor that drives the investment process in a way that directs capital and or resources from the investors/employees/workers to the investees/companies/government

1

INTRODUCTION

9

and delivers return/income in return. The participants of the financial system and their roles can be summarized in the following three categories/sectors (Bodie et al., 1996): 1. The household sector: i. It makes decisions about work, education/training, retirement planning and savings or consumption. ii. Households need investments with a relatively high rate of return and low to medium risk. 2. The business sector: i. It needs to raise money in order to finance its investments in real assets, such as factories/plant, equipment/machinery, technology, etc. ii. There are two main types of financing: borrowing from banks or reaching households by issuing bonds and/or obtaining coowners by issuing shares of stock. 3. The government sector: i. It also needs to raise money to finance its expenses. This is done through taxes or lending—that is, there is no issuance of shares of stock. ii. An important role of the government is to enforce the regulations for the operation of the economic/financial environment.

1.2 1.2.1

Bonds

What Is a Bond?

A bond issue is a loan with one borrower but usually more than one lender. The loan is broken into several pieces, each one of which bears the same value. A bond is a security under which the issuer (or debtor or borrower) promises to pay the buyer (or investor) the initial loan amount plus interest. A bond is briefly described or determined by four parameters: (a) the face/par value, i.e. the nominal amount of the loan; (b) the bond issuer, i.e. the borrower; (c) the coupon (rate), i.e. the interest (rate) paid by/charged at the loan; and (d) the maturity date, i.e. the date that marks the expiration of the loan.

10

T. POUFINAS

1.2.2

Bond Indenture

A bond indenture is the legal document that contains all the details of the bond issue. It is the contract that is drafted between the issuer/debtor and the bondholder/investor (similar to a contract/agreement signed between a borrower and a lender). It can contain restrictions that apply to the issuing party in order to protect the interests of the investing/lending party. They pertain to the issuance of additional debt, the payment of dividends, the existence of collateral or the set up of a sinking fund. These restrictions are also known as protective or restrictive covenants. They are normally seen by the investors as safety nets and as such they make the bond issue more attractive (Bodie et al., 1996). The first restriction may regulate the issuance of further debt by requiring that any potential additional debt is subordinated or junior to the existing debt (subordination clause). The second may put restrictions on the dividends paid to the shareholders so as to make sure that there enough assets to pay the bondholders first. The third provisions for the retention of an asset—the collateral—that the bondholders will receive in case the issuing entity defaults. The fourth allows for the repurchase of a portion of the outstanding bonds prior to their maturity date each year, either at a market price or at a predetermined price as specified in the sinking fund provision, so as to relieve the burden of the payment that is to be made at the maturity of the bond. 1.2.3

Bond Categories

The term bonds (Treasury Bonds & Treasury Notes) usually refers to the securities that last more than a year. Issues with maturities equal to or less than a year are classified as Treasury Bills (or T-Bills). Depending on the issuer bonds are divided into (Bodie et al., 1996): a. Government bonds b. Corporate bonds c. Bonds of supranational organizations (supranational(s), such as the World Bank, the European Investment Bank—E.I.B., the European Bank for Reconstruction and Development—E.B.R.D.) d. Municipal bonds e. Special purpose vehicles (SPVs).

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INTRODUCTION

11

Usually the issuer of the bond with a very good name (which presumably reflects its capacity to repay debt) does not give to the bond buyers some additional safety/security (premium). Bond issues that enjoy the benefit of the security offered by collateral are called secured bonds. The first issues without security are called debentures while the ones that follow are called subordinated debentures . In some cases the payment of the coupon or the capital is linked to the course of an investment or is covered by a collateral or pledge. Some examples are (Bodie et al., 1996): 1. Revenue Bonds : Interest and capital are paid from the income derived from an investment that is financed by the bond issue. 2. Mortgages: The money from the issue is used to buy land and buildings that also serve as a collateral or pledge. 3. Income Bonds : The coupon is paid only if there have been profits from the investment of the capital that has been collected from the issue. Depending on how the coupon is set and paid, the bonds are divided into (Bodie et al., 1996): 1. Annuities : They have a fixed coupon rate which is paid at regular time intervals, usually every year or semester. 1.1 Sometimes they are referred to as fixed rate bonds, because their coupon is fixed. 1.2 The term annuity may be used when the borrowed capital (nominal amount) is not repaid at maturity and only interim (coupon) payments are made in the meantime. 2. Zero-Coupon: Interest payments are made on the bond maturity date. 2.1 In contrast, bonds that carry a positive coupon are called couponbearing bonds. 3. Floating Rate Notes —FRN’s : The coupon rate changes each period according to the issuance terms.

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1.2.4

History of Bonds

Bonds have been used for more than 300 years in a form that resembles to their present one. However, according to some they are dated back to 2400 B.C. The use of bonds has grown significantly over the years and these days they are one of the main security types used by governments and companies to secure financing. The first bond in history seems to date back to 2400 B.C. in the form of a stone that has been discovered at Nippur, in Mesopotamia at that time, in Iraq at present-time. This bond guaranteed the payment of grain by the principal and the surety bond guaranteed reimbursement in case the principal did not meet his or her payment obligation. The currency at that time period was corn (Cummars, 2014). The second significant finding is dated in the 1100s when the city of Venice issued bonds in order to secure funds for its wars. The market evolved in the city during the fourteenth century. The Venetians could purchase and trade government securities that were essentially perpetuities with a predetermined rate (Cummars, 2014). Later on, in 1693, it was the Bank of England that issued the first ever government bond in order to finance its war against France. It was a mix of lottery and annuity bonds. The example seems to have been followed by other governments in Europe who issued bonds to fund wars and government spending (Cummars, 2014). In the US the first government bond issuance appears during the Revolutionary War; it was used to collect the money needed to finance the war. They had the form of Treasury loan certificates. Individuals purchased more than USD27 million. The first US Treasury bonds were issued to finance World War I; they were initially called Liberty Bonds and they date back to 1917. The initial issue was of USD5 billion at 3.5% interest rate (Cummars, 2014). Since then bonds were used more and more by governments in the US and around the world. The same trend was observed also in the corporate world; companies issued also bonds to finance their operations. Although at the beginning it was primarily an investment-grade firm market during the late 1970s and through the 80s non-investment grade public companies were given the green light to issue bonds. It was in 1977 that Bear Stearns underwrote the first new issue of non-investment grade bond in decades (Cummars, 2014).

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INTRODUCTION

13

It was in the 1980s that the first derivatives on debt appeared with the creation of collateralized debt obligations. These products became more popular after the year 2000 and the relevant market grew a lot (Cummars, 2014). It extended to mortgage-backed securities and other asset-backed securities. These derivative products—although bearing a superior credit rating—carried increased risk in some of their tranches. Moreover, they were significantly hot during the housing crisis, as the non-payment of loans impacted the holders of such fixed income instruments. They are considered to have contributed to the financial crisis. Along with these fixed income derivatives, bonds—in particular noninvestment grade—and stocks were hit. The Federal Reserve implemented a quantitative easing approach that entailed the purchase of fixed income securities, including mortgage backed securities in order to stop the financial crisis. The same paradigm was followed by central banks worldwide, which launched similar quantitative easing programs. This trend continued over the years until it was reinforced at present times as the economic activity has been hit by the pandemic. Central banks resumed their purchase programs and extended them to include fallen angels (bonds whose credit rating has deteriorated from investment to noninvestment grade) and ETFs in the US, as well as non-investment grade government bonds in Europe.

1.3

Stock

Stocks (or shares of stock or equity), unlike bonds, do not comprise an obligation by the issuer to the holder, but constitute a title deed to a “piece” of the company that issued them. Their main forms are: common stock and preferred stock. The terms stock and share are sometimes used interchangeably. It is however useful to know how they are distinguished. Stocks are securities that indicate ownership of a part of one company (or more companies). A share indicates ownership of a portion of a particular company. It is the smallest denomination of a company’s stock. 1.3.1

Common Stock

Common (shares of) stock indicate(s) ownership in a business. Each common (share of) stock entitles its holder to one vote right. They may pay dividends. Non-payment of a dividend is not a default.

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T. POUFINAS

Holders of common (shares of) stock are the last beneficiaries of the assets or income of the company (residual claim) especially in case of liquidation of its assets. Also, the maximum loss that a shareholder can incur in case of a company going bankrupt is his or her initial investment (limited liability). That is, there is no claim/requirement on his or her personal assets (Bodie et al., 1996). 1.3.2

Preferred Stock

Preferred (shares of) stock has (have) characteristics similar to those of both stocks and bonds. The feature that makes them look like a bond is that they promise their owners a dividend every year. In this sense they look like bonds of infinite time to maturity, known as perpetuities . Failure to pay the dividend does not constitute a default and in this respect they resemble stocks (Bodie et al., 1996). Usually the dividend is cumulative, i.e. the holders of preferred stock receive the dividend that has not been paid before the holders of common shares. There are preferred (shares of) stock(s) with or without voting rights . Example 1.1 To understand how bonds are placed in the spectrum of assets but also financing tools, let us assume that we want to set up a new company. Assume that its first product will be chocolate bars. We need capital to set up a chocolate factory (buy the land, build the plant, buy or lease the equipment, hire staff, etc.). Our initial capital does not suffice to launch our venture. To complement our own funds we decide to borrow money from a bank. As we have no prior credit history, i.e. we did not have a loan in the past, banks do not know whether we can repay our loan. As a result the interest rate they charge is not low. As a matter of fact, before granting the loan they need to be convinced that we will have a solid plan of what to do with the money; i.e. the need to see our business plan. Assume that we do have a good business plan and all the banks that we apply for a loan approve our application. If all of them offer us comparable terms, including the same expiration date, then we will choose the bank with the lowest interest rate. Let it be 8% per annum (the percentage is only for illustrative purposes and does not necessarily reflect reality).

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After 8 years we want to extend our activity to produce flavored chocolate bars. All these years we have been properly paying our debt and gradually we have had a small profit. However, the new product line requires an additional investment that we cannot afford. We therefore pay again a visit to each of the banks to apply for a loan. All others being equal we select the bank with the lowest interest rate. Assume that this time it is 6.5%. The interest rate is lower because we now have a credit history, our brand is on the radar and we repay our loan. After another 8 years we want to extend again our activity and produce more cocoa and chocolate related products. Assume that we have been meeting our debt obligations successfully and that our brand recognition has significantly increased. Although we posted profits, our own funds do not suffice. We apply for a bank loan again and this time we can access an interest rate of 5%—as before the minimum of the interest rates offered to us by the different banks. However, we feel that this interest rate is high. We are willing to exploit other financing solutions. We have two alternative routes; one is to issue a bond and the other is to issue new (shares of) stock. Issuing a bond means to essentially borrow funds from the broader investment audience. It means that we will divide the loan into smaller pieces, each with a certain nominal value and we will distribute them to several lenders/creditors. We now become the bond issuer and our lenders become the bondholders or investors. The bond can be made available for private placement—to specific, usually institutional investors, or for public placement—to all potential investors. In all cases, as this is the first time we address investors other than the bank, the investors need to have an assessment of our ability to make the payments promised by the bond. Such an assessment is offered by the rating agencies. These agencies offer a credit rating which indicates the probability of default, i.e. the probability that a certain issuer will miss one or more payments completely or deviate from the payment schedule. The investors then can compare the rate of return they receive from a bond with the rate of return of other bonds (or investments) that carry the same credit rating. This means that they expect that bonds that have been issued by different issuers with the same credit rating need to have (almost) equal rate of return (or yield). Assume that we address one of these rating agencies and we receive a credit rating for which the anticipated yield is 4%; then we will most likely opt to issue a bond instead of relying once more to direct bank lending.

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However, if we receive a lower credit rating or if the credit rating we are assigned indicates an expected yield of 5.5%, then we may revert back to borrowing from the bank or exploit the issuance of (shares of) stock, a process that will be explained shortly. Suppose that the credit rating we receive justifies a yield of 4%. Then we issue the bond. As a bond is a form of loan, we maintain ownership of our company. After another 8 years we wish to extend our activities to other countries. Assume we have been making the bond or other loan payments in a timely manner and as promised. We have been profitable all these years. This is a totally new undertaking though and requires a significant additional capital. Our own funds are not enough. Once and again we have the same choices in order to finance these activities; borrow from a bank, issue a bond or issue stock. We feel we should not increase our debt and thus we decide to issue new shares of stock in order to attract new investors. This can take place either through private placement, i.e. with funds coming from specific, usually institutional investors or through public offering, i.e. with funds coming from all potential investors. The latter can be achieved by having the shares of our company listed in an organized stock exchange. Assume we pursue the second alternative. We issue new stock that is made available only to new investors. This time we have no interest rate charged, as this is not debt financing any more. There is no commitment to pay a specific rate of return. However, the investors that have purchased our shares of stock—the shareholders, are co-owners of our company and expect part of the profits that our operations will deliver. Consequently, we have no obligations similar to debt, but instead we have let part of our company go to the hands of other investors. In summary, we realize that a bond is on one hand a financing tool for the issuing firm that constitutes a form of debt and on the other hand an investment vehicle for the investor that offers a promised stream of cash flows during the life of the bond. In contrast, stock is on one side a financing tool for the issuing firm that constitutes a form of ownership and on the other side an investment vehicle for the investor that offers part of the earnings of the company—when it records profits—during the life of the company (usually assumed ad infinitum). We further elaborate on the differences between stocks and bonds in Chapter 12.

1

1.4

INTRODUCTION

17

Markets

Markets are divided into 4 categories based on the assets traded in them; the money market(s), the fixed income capital market(s) (or bond market(s)), the equity market(s) and the derivative market(s) (Bodie et al., 1996). 1.4.1

Money Markets

In money market(s), fixed income assets with a time to maturity of less than one year are traded. It could also be considered a subcategory of the fixed income securities (capital) market. It usually comprises securities that are: – – – –

short-term, accessible to private investors, with high liquidity, with great marketability.

These include: – – – – – –

Treasury Bills, Certificates of Deposit, Commercial Paper, Banker’s Acceptances, Eurodollars, Repos and Reverses. 1.4.2

Fixed Income Capital Market (or Bond Market)

In fixed income capital market(s) or bond market(s), fixed income securities with a time to maturity of more than one year are traded. It usually consists of securities that are: – – – –

longer lasting; not always accessible to private investors; sometimes with limited liquidity; sometimes with reduced marketability.

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These include: – – – – – –

Treasury Notes and Bonds (Government Bonds) Federal Agency Debt (Bonds of Government (State) Organizations) Supranational Bonds Municipal Bonds Corporate Bonds Mortgage-Backed Securities and other Asset-Backed Securities 1.4.3

Equity Markets

In equity markets the shares of stock of companies are traded. These are organized stock markets designed to ensure the liquidity and transparency of transactions. As a large number of corporate stocks are traded in these markets, the general movement of transactions is monitored daily through one or more indices, which usually consist of stocks of companies that are considered representative of the whole market. 1.4.4

Derivative Markets

This is where the derivatives are traded. These markets are distinguished in futures markets and options markets. 1.4.4.1 Futures Markets These are the markets where the futures contracts are traded. Futures are derivative contracts, according to which each of the two counterparties has the obligation to deliver or receive on a specific future date the underlying value (underlying asset) at a price and other characteristics specified at the time of entry into the contract. There is no cost (premium) as a futures contract constitutes an obligation. 1.4.4.2 Options Markets These are the markets where options are listed. Options are derivative contracts, according to which each of the two counterparties has the right to sell (put option) or buy (call option) to (European) or up to (American) a specific future date the underlying value (underlying asset) at a price and other features specified at the time of entry into the contract. There is a premium as an option is a right and not an obligation.

1

INTRODUCTION

19

High Risk

Derivative Markets

Stock Markets

Bond Markets

Money Markets

Low Risk

Fig. 1.2 Risk taken per type of investment (Source Created by the author)

Money markets are perceived as the least risky markets, followed by the bond markets, then the equity markets and finally the derivative markets. Figure 1.2 shows the risk taken by the investor who chooses each of the above investment vehicles (or markets). We present the particulars of the bond markets, including the money markets in Chapter 7. We elaborate on interest rate derivatives in Chapter 5.

1.5

Primary and Secondary Markets 1.5.1

Primary Market

The primary market is the market in which investment banks issue new securities (bonds, stocks, etc.) to the investing public. It pertains to the

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initial disposition of securities, which in the case of the government/state could be done directly, i.e. without the intermediation of an investment bank. There are two types of primary issues (for stocks and bonds) (Bodie et al., 1996): a. Initial Public Offerings (IPO’s), where securities are made available to the investment audience for the first time. b. Seasoned New Issues , where securities that have already been issued are reissued—also known as issue re-opening. Another distinction between primary market issues has to do with the targeted audience and also has two types (Bodie et al., 1996): a. Public Offering, in which securities are made available to the broader investment audience; the amount of the issue is usually large. It is addressed to a large number of investors. It then trades on the secondary market. The approval of the regulatory authorities— usually the capital markets commission of the country where the issue takes place and/or it is made available (SEC in the USA) may be required. b. Private Placement , in which securities are made available to a few— usually institutional—investors. The issues are usually small(er) and they do not address the general investing public. Most of the times, the securities are held until their maturity date. The securities do not have high liquidity and marketability, but they offer superior returns, possibly at a greater risk. It is worth mentioning that private debt (and private equity) is considered among the most popular investment vehicles within the alternative investments the last few years. This is due to the fact that interest rates and yields of good quality corporate or government bonds are extremely low or even negative. Institutional investors, especially the ones with longer investment horizons, include private debt in their portfolios in the quest of a higher performance, even at a higher risk. Pension schemes are among these institutional investors, as they can afford waiting for longer periods, due to the long-term nature of their liabilities; at the same time they need positive returns to preserve or grow the contributions made by their

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21

members, as this is money that they will receive at retirement. To make sure that the issuing entities will not default they scrutinize their finances and operations, as a result of which they may have big stakes. The steps followed for public offering are (Bodie et al., 1996): 1. The new bonds (or stocks) are made available through underwriting performed by investment banks. 2. Investment banks usually set up underwriting syndicates to make new bonds available. 3. Some of the banks are heading the offer, called the lead underwriters . 4. A preliminary prospectus is drafted to describe the terms of the issue and the prospects of the issuing entity. It is submitted for approval to the capital markets commission or the responsible supervising authority. 5. Finally, after the relevant approvals have been granted and the required adjustments have been made, the prospectus is drawn up. The selling price of the securities to the investing public is announced. This is a common practice for relatively large issues so as to provide access to a larger number of potential investors. Investment banks advise the issuers on the terms at which they should try to sell their securities. There are two methods followed in underwriting securities (Bodie et al., 1996): a. The firm commitment agreement, in which the investment banks— underwriters buy the securities from the issuer and undertake to sell them to the investing public. The sale of the securities from the issuer to the underwriters is done at a lower price than they are made available to the public (the difference is the spread, which is the remunaration of the underwriters). Underwriters take the risk of not selling the entire issue. b. The best efforts agreement, in which investment banks—underwriters agree to help the issuer sell its securities to the public. The underwriters do not buy the securities. Instead they act as intermediaries. They do not run the risk of holding to unsold securities.

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The issuers can select the underwriters with one of the following two ways (Bodie et al., 1996): a. Negotiations , where the spread is negotiated and the underwriter may receive some of the securities of the issuer as compensation. b. Competitive bidding , where the candidate underwriters are invited to submit their offers. 1.5.2

Secondary Market

The secondary market is the market in which the purchases and sales of securities that have already been issued take place. Its operation is very important as it allows investors to trade securities fast, without affecting the price of securities so much, due to the liquidity it offers. At the same time it offers transparency, standardization and safety as transactions are under the supervision of the competent authorities. Secondary markets include (Bodie et al., 1996): a. Organized stock exchanges , where stocks and bonds are typically listed and traded. b. Over-The-Counter (OTC) markets , which are non-organized markets, where stocks and bonds are bought and sold. Dealers (and brokers) offer prices to buy or sell securities and hope to get the spread as compensation. The OTC trading of listed securities is known as the Third Market. c. The direct trading between two counterparties , which takes place without the intermediation of a broker. It is used by large institutional investors that want to avoid transaction costs. This market, also known as the Fourth Market, flourished after the introduction of electronic trading systems (such as Posit, Instinet).

1.6

Issuers

The most common bond issuers (and their issues) are (Bodie et al., 1996): – Governments/States (Notes, Bonds, T-Bills): they issue bonds in order to raise funds to pay debt, meet obligations and finance

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projects. The issues are named by their time to maturity, with TBills having maturity dates of less than or equal to 1 year, Notes having maturities from 1 to 10 years and Bonds having maturities of more than 10 years. • Index-Linked Bonds: these are bonds that are primarily issued by governments whose interest income paid is connected with a specific price index, such as the consumer price index (CPI). These bonds are used to obtain protection from the changes in the associated index, such as the CPI, whose change is inflation. Their coupon payments in this way are adjusted for inflation and thus a known real rate of return is ensured. – Municipalities and Regions (Bonds): they issue bonds in order to finance projects, such the construction of schools, libraries, roads, bridges, public transit, parks and pathways, and the funding of police and/or fire departments, community centers and waste management. – Supranational entities (Bonds): they issue bonds in order to fund their operations. Supranational bonds resemble to government bonds; in contrast supranational entities are not based in a specific country; they rather have members located in several countries. – Corporations—Firms (Bonds and Shares): they issue bonds in order to finance their existing operations, new operations, products or projects, etc. – Government Agencies (Bonds): they issue bonds to finance their own activities. – Mortgage-Backed Securities, Asset Backed Securities and SPVs (Bonds/Derivatives): they pool together a set of mortgages on real estate properties or other assets—depending on the type and they are securitized in a fixed income security. However, as they are structured on an underlying asset they are also considered as derivatives.

1.7

Intermediaries

Intermediaries bring issuers and consumers/investors together. In this way they do not need to know/contact each other directly. They offer convenience and access to markets that certain investors could not have

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managed on their own. The main types of intermediaries include (Bodie et al., 1996): – Investment Bank(er)s—Primary Market: they advise the issuer on the pricing of the securities to be issued, on the market conditions, on the appropriate interest rates, etc. When a security is made available for the first time in the market, they usually perform the marketing-/distribution-related functions. They can also assist issuers in structuring securities with special characteristics. – Dealers and Brokers—Secondary Market: they act as links between buyers and sellers of goods and assets, so they do not need to search for them. Brokers assist buyers and sellers in finding the good or asset they wish. There needs to be some activity for brokers to be present. When the activity is even higher dealers arise. Their difference from, brokers is that they purchase assets for their own inventory and trade for their own accounts. The dealers make their profit from the difference between the price they sell an asset from their inventory and the price they buy that asset for their inventory. This is the bid-ask spread. – Mutual Funds: they usually collect small amounts of money from individual/retail/private investors and allocate them to a bigger portfolio, giving shares or units in return. The size of the portfolio allows it to invest to a wide range of securities—which in the context of this book are fixed income securities. In this way smaller investors can gain access to investments that they could not have accessed differently. The mutual funds may make available a wide range of investment strategies, which are standardized though. Moreover, they can benefit from the diversification that the mutual fund can achieve, either within a broad investment universe (e.g. through a global balanced fund) or within a specific thematic investment universe (e.g. emerging market sovereign bond fund). Mutual funds charge a fee and/or publish different purchase/acquisition and redemption prices. – Exchange Traded Funds: these are variants of mutual funds that are traded in a stock exchange. – Investment companies: they operate in a way similar to mutual funds; however, they address usually larger investors, with specific targets with the portfolios they construct. In contrast to mutual funds they can be tailor made or can combine similar preferences of different

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investors. They are thus most often not available to small retail investors. Investment companies charge a fee and quite often an over-performance fee. – Insurance Companies: they usually offer access to a well diversified portfolio via an insurance policy. This portfolio is formed by investing the premia that the insured or policyholders pay. The portfolio that underlies this policy is earmarked to deliver a certain amount or annuity at the maturity of the policy to the policyholder or to the beneficiaries (if different from the policyholder) in case a certain event occurs. The investment risk of the portfolio can be born either by the insurer or by the insured. In the former case, especially when there is an interest rate guarantee the portfolio invests in fixed income securities. In the latter case, the insured can select a portfolio that is usually made up from a range of mutual funds. Insurance policies can be either individual or group. Insurance companies provision for a profit margin in their premium estimation. – Pension Funds: pension funds operate in a similar manner to insurance companies. In this case the payments made by the participants of the pension fund are called contributions. Nonetheless, they have a very specific scope; to match a certain retirement income; this income is usually a percentage of the income that the pensioner earned during his or her active employment years. The amount can be offered as lump sum or as an annuity or both. Pension funds may offer guarantees; alternatively they may guarantee no amount, but still attempt to optimize the return against the risk in the investment portfolio. Guarantees were possible in the past through fixed income investments (mainly government bonds) that offered returns higher than the guaranteed interest rates. Nowadays, as interest rates have turned negative or close to zero, these guarantees are no longer feasible. Pension funds, whether they offer guarantees (normally from the past) or not still invest heavily in fixed income. Nevertheless, the fixed income investments they pursue are primarily corporate bonds or private debt. – Commercial Banks: they usually raise funds by taking deposits and lend money through loans. The former is a liability, as the bank essentially borrows money from the depositors for which it pays interest. The later is an asset as the bank practically purchases the debt/loan expecting to receive interest. The difference (spread)

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between the interest rates paid to the depositors and charged to the borrowers is the profit of the bank. An extensive presentation of bonds funds takes place in Chapter 8.

1.8

The Role of Central Banks

When talking about fixed income investing, fixed income markets and fixed income securities one cannot miss the role of central banks. Central banks traditionally pursue three goals; (i) to maintain the price stability; (ii) to maintain financial stability; and (iii) to support the financing needs of a country in times of crisis (Goodhart, 2010). The last role may have been the most eminent during the last decade; however, before that it was not necessarily the prevailing function of a central bank. Nevertheless the 2008 financial crisis and the turbulence it created promoted even more the perception of a central bank as a lender of last resort. This means that it can provide funds in the economy of a country when the commercial banks cannot do so. In such a way it secures that the banking system will not fail. The first role is probably the most well-known routine function. It has to be seen in comparison with the monetary regime. The purpose is to maintain the price stability taking into account the currency-related objectives (e.g. pegged exchange rate, convergence targets) or inflationrelated objectives. The second has to do with the promotion of financial development. The third, in normal times is linked with the appropriate use of the financial resources of a country (Goodhart, 2010). A central bank acts also as the regulatory authority of the monetary policy of a country, being the only entity that has the empowerment to print notes and coins. At the same time it is the supervisory authority for banks and other financial or insurance institutions—depending on the country. One can thus say that a central bank operates both at macro and micro level. The former is practiced when it focuses on price stability, and thus on regulating the level of inflation, by controlling the money supply; it thus exercises its monetary policy. The central bank can achieve that via its open market operations that either increase or decrease liquidity which impacts inflation and interest rates. The latter is realized when it focuses on the financial stability by securing that commercial banks maintain the reserves required to ensure their solvency or by lending them (at the

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discount rate) the funds they may need in order to meet their liabilities to their clients. Because of the diversity and significance of the roles and functions that are executed under the auspices of a central bank, it is usually independent from the government; its policy is also uninfluenced from the governmental fiscal policy. Furthermore, it remains divested from commercial banking activities. The first central bank seems to have been founded in 1668 in Sweden. The Swedish Riksbank was a joint stock enterprise whose scope was to lend funds to the government and operate as a clearing house for commercial activities. In 1694 the Bank of England was established. It had a similar structure and objective with the Riksbank. The Bank of France was established by Napoleon in 1800 so as (i) to deal with the monetary issues that emerged as a result of the hyperinflation that appeared during the French Revolution; and (ii) to help the government financing. The Federal Reserve was created in 1913, with the mandate (i) to offer a uniform and elastic currency aiming at accommodating the seasonal, cyclical and secular shifts observed in the economy; and (ii) to operate as a lender of last resort (Bordo, 2007; Goodhart, 2010). The European Central Bank (ECB) and the European System of Central Banks (ESCB) were established in 1998. In order to better illustrate the role of central banks we present the structure and the functions of the FED and the ECB. 1.8.1

FED

The Fed (Federal Reserve System—FRS) is the central bank of the USA. Its main function in the bond market is to set and manage the monetary policy. This includes controlling the money supply and/or interest rates. The Fed was created by the Congress in 1913 to help promote a safe and sound monetary and financial system for the United States (Fed, 2016). 1.8.1.1 Structure of the Fed The Federal Reserve System consists of two parts (The Federal Reserve System, 2021): the Board of Governors in Washington, DC and the 12 regional (district) Federal Reserve Banks and 24 Branches. These are located in Boston, New York, Philadelphia, Cleveland, Richmond, Atlanta, Chicago, St. Louis, Minneapolis, Kansas City, Dallas, and San Francisco.

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The Federal Reserve has three key entities; the Federal Reserve Board of Governors, the Federal Reserve Banks and the Federal Open Market Committee (FOMC). They make decisions that help promote the health of the economy and the stability of the financial system of the US. The Fed Board consists of 7 members appointed by the President of the United States for 14 years of service. The years of service are divided so that a new governor is appointed every 2 years. One of them is appointed by the President of the USA as Chair of the Board of Governors for 4 years. The Chair of the Board has significant power to conduct monetary policy. The FED system consists of 12 regions (districts), each of which has its local Federal Reserve Bank and Branches. These Federal Reserve Banks are privately owned by commercial banks that are members of the FED in the region. Each local bank has a chairperson on the Federal Open Market Committee (FOMC). The FOMC is the monetary policy making body and as such pursues the monetary policy. The voting members of the FOMC are the 7 governors, the president of the local Federal Reserve of New York and on a rolling basis 4 of the presidents of the local Federal Reserve banks. The FOMC has 8 regular meetings per annum with the goal of setting the guidelines for the monetary policy, which in their turn draft the directions for the growth rate of money supply and the level of interest rates. Other significant entities that contribute to the Federal Reserve Functions are the depository institutions and the advisory councils (FED, 2021b). • The depository institutions are the banks, thrifts and credit unions. They offer transaction or checking accounts to the public, and may maintain own accounts at their local Federal Reserve Banks. They have reserve requirements in line with their checking accounts balances. If a depository institution has a balance in excess of the required one, then it may lend this excess to other depository institutions so that they can match their own requirements. This affects interest rates, asset prices, exchanges rates, wealth and as a result the total demand in the economy. The FOMC sets a target for the federal fund rates and pursues this target via open market operations. • There are four advisory councils that assist and advise the Board on matters of public policy.

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– The Federal Advisory Council (FAC) consists of 12 representatives of the banking industry. It meets with the Board 4 times per annum. Its members serve three one-year terms. Each Reserve Bank chooses one representative of its District to FAC on an annual basis. – The Community Depository Institutions Advisory Council (CDIAC) collects information and opinions from thrift institutions, i.e. savings and loans institutions, as well as mutual savings banks and credit unions as well as community banks. When it was initially created by the Board of Governors it had in its scope only the first two. It offers to the Board views about the economy globally and specific matters, such as lending specifically. – The Model Validation Council offers expert and independent input relevant to the evaluation of the models employed in the stress tests performed by banking institutions. It was launched in 2012 following the Dodd-Frank Wall Street Reform and Consumer Protection Act with the aims at bettering the quality of—and consequently increasing the confidence in—the stress tests. – The Community Advisory Council (CAC) monitors the economic conditions and financial services needs of consumers and communities, primarily the low- and moderate-income ones. It was established by the Board in 2015 to complement FAC and CDIAC, so as to properly represent the third stakeholder, next to the Fed and the depository institutions, the consumers. It meets semiannually with members of the Board. It has 15 members with interchanging/rolling three-year terms selected by a Board by public nomination. Advisory committees operate also at the level of the Federal Reserve Banks. The main ones are these that offer advice on agricultural, small business and labor issues (two times a year). 1.8.1.2 Functions of the FED The FED performs five main functions in order to secure the effective operation of the economy of the US and the public interest. As such it (FED, 2021b)

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• Conducts the monetary policy of the country in order to maximize employment, secure price stability and moderate long-term interest rates. • Promotes the stability of the financial system and focuses on containing systemic risks by actively monitoring and engaging the influencing determinants in and out of the country. • Promotes the safety and solvency of individual financial institutions and follows their effect on the global financial system. • Fosters the safety and efficiency of the payment and settlement system via services to banks and the government of the country that ease the transactional and payment activity in USD. • Promotes the protection of the consumer and the development of the community via supervision, study of the consumer and community rising needs and concerns, and implementation of consumer-related legislation and regulation, with an overall focus to the consumer. 1.8.1.3 Monetary Policy Objectives The goals of the monetary policy are i. the projection of economic growth, ii. the full employment, and iii. the low inflation. Economic growth is desirable because it means an increase in the wealth of the state. Full-time employment is desirable for two reasons: a. more employees means more production, and b. competent people have the right to equal employment opportunities. Low inflation is desirable because it gives incentives to work and increase production. These goals are often competitive and often make it difficult for the Fed to make choices. It is e.g. difficult to have both full employment and low inflation. Keeping inflation low requires a tight monetary policy, keeping money supply growth low and raising interest rates. Low money

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supply and high interest rates mean a reduction in investment, income and employment. A compromise policy is usually followed. 1.8.1.4 Conducting Monetary Policy There are typically 3 tools for conducting monetary policy: a. change in the reserve requirements b. change in the interest rates for bank borrowing c. open market operations. The first two bring about major and inflexible policy changes. The third is the most commonly used and is used effectively for small adjustments or policy modifications. Open market transactions involve the purchase and sale by the FED of securities (US Treasuries) on the open market. – If the Fed buys bonds from an individual, then the seller’s commercial bank account increases at the sale price. Since the money in the account can be spent, the money supply increases and interest rates fall. – If the Fed sells bonds to the individual, his or her account is reduced. The money supply is declining and interest rates are rising. – If the Fed buys securities from a commercial bank, then the bank balance to the Fed increases. The amount above the required reserves can be used to give loans to customers and thus increase the money supply. The Fed has unlimited purchasing power, as it can generate balances in any amount. The Fed pursues a monetary policy by amassing a large portfolio of US Treasuries from which it earns interest. Part of this is used to cover expenses and the rest is returned to Treasury. 1.8.1.5 Quantitative Easing (QE) The FED has also exercised unconventional monetary policy actions when needed and in particular during the last almost one and a half decade. This became widely known—although it was not the first time used—as quantitative easing (QE).

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Quantitative easing refers to a particular form of monetary policy, according to which a central bank purchases bonds—mainly government bonds—and/or other financial assets in order to increase liquidity and consequently support the economic activity. It is used on an ad hoc basis, when traditional monetary policy cannot be efficient. Through these purchase programs money injection is performed, with the purchase of even riskier than usual assets, at a large scale and through a pre-defined period of time. The balance sheet of the FED expanded and contracted over the years, depending on the stage at which the economy was. The 2007–2008 financial crisis and the recession that followed it led to a significant increase of the total assets. From a total of USD870 billion in August 2007 it reached USD4.5 trillion at the end of 2014–early 2015. From October 2017 to August 2019 the total assets declined to reach USD3.8 trillion. Since then the pandemic hit and as a result in March 2020 the FED announced an envelope of USD700 billion to be disposed for asset purchases in order to support the liquidity and assist in weathering the consequences of the pandemic. Since then additional asset purchases have been announced and pursued as the pandemic is not over yet. At the end of 2020 (and mid-January 2021 as well) the total assets have increased to approximately USD7.4 trillion (FED, 2021a). Consequently, the FED has implemented overall a total of 4 QE rounds: • The first round started in November 2008 with the purchase of USD600 billion of MBSs. The purchase program kept stopping and restarting in line with the performance of the economy, leading to a total asset amount of approximately USD2.4 trillion (FED, 2008). • The second round was launched in November 2010 and included the purchase of an additional USD600 billion of Treasury securities (FED, 2010). • The third round was announced in September 2012 with an openend purchase of USD40 billion MBSs per month. The amount was more than doubled to USD85 billion per month in December 2012 (FED, 2012). • The fourth round commenced in September 2019 but was strongly enforced in March 2020 as a result of the pandemic with the announcement of USD700 billion of asset purchases (FED, 2020).

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It is worth noting that in June 2013 the FED announced a tapering of some of its QE policies provided that there will be positive developments in the economy. These purchases stopped at the end of 2014 (FED, 2013). 1.8.2

ECB

The European Central Bank is responsible for maintaining price stability in the Eurozone. It was founded on June 1, 1998 and is one of the newest central banks in the world. The following discussion has been taken from the ECB (2021) and the Bank of International Settlements—BIS (2021). 1.8.2.1 Objective of the European System of Central Banks The primary objective of the European System of Central Banks (ESCB), as set out in the Statute of the European System of Central Banks and of the European Central Bank, is to maintain price stability. In line with this primary target, the ESCB aims to foster the general economic policies in the Community in order to facilitate the fulfillment of the targets of the Community. To succeed in this goal the ESCB operates in line with the principle of an open market economy with free competition so as to secure the optimization of resource allocation. 1.8.2.2 The Eurosystem Monetary Policy Strategy As stated above the main objective of the ESCB is the maintenance of price stability. To facilitate the success of this goal the Treaty establishing the European Community attempt to safeguard a balanced approach by offering to the ESCB and the Eurosystem a significant degree of institutional independence, yet in parallel requires the honoring of transparency and democratic control. On 13 October 1998, the Governing Council of the European Central Bank (ECB) announced the Eurosystem’s stability-oriented monetary policy strategy, so as to properly orchestrate the monetary policy decisions in the Third Stage of Economic and Monetary Union (EMU). Its main components are 1. A quantitative definition of price stability and 2. The two pillars that are employed in order to reach this objective, namely:

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i. the leading role of money ii. the rigorous evaluation of the price movements expectations and the associated volatility risks in the euro zone. 1.8.2.3 The Operational Framework of the Eurosystem The Statute clearly defines the monetary functions and operations of the Eurosystem. In line with these definitions, the European Monetary Institute has determined the functional framework that will enable a single monetary policy. The Governing Council of the ECB a. Took the final decisions on the operational framework in the second half of 1998. b. Has maintained the authority to proceed with certain modifications of the instruments and procedures as described in the following subsections. 1.8.2.4 Monetary Policy Instruments and Procedures The functional framework consists of a number of instruments. The Eurosystem performs open market operations, offers standing facilities and requires credit institutions to maintain minimum reserves in accounts with national central banks (NCBs) in the euro area. Furthermore, due to the extraordinary conditions that emerged and prevailed since the last financial crisis, the ECB has put in force also non-standard monetary policy measures since 2009 in order to support the standard operations of the Eurosystem. These are the asset purchase programs (APP) and, recently, the pandemic emergency purchase program (PEPP). 1.8.2.5 Open Market Operations An important role in Eurosystem’s monetary policy is played by the open market operations used by the Eurosystem to influence interest rates, manage the market liquidity levels and set the direction of the monetary policy. The Eurosystem has five types of instruments for performing open market operations. The most important instrument is reversible transactions (applied under repurchase agreements or secured loans). The Eurosystem may also use outright transactions, debt securities, currency swaps and acceptance of time deposits. Open market operations are carried out by the ECB, which also decides on the instruments to be used,

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as well as on the terms and conditions for the execution of such operations. Open market operations can be conducted through formal tenders, express tenders or bilateral procedures. Depending on their purpose, frequency and procedures used, Eurosystem open market operations fall into the following four categories: – The main refinancing operations are regular reverse liquidity transactions, which take place once a week and last for one week. • They are conducted by the NCBs through standard auctions and according to a predetermined schedule. • They offer the main volume of the refinancing in the financial sector and as such they have a key contribution in meeting the objectives of the open market transactions of the Eurosystem. – The longer-term refinancing operations are reverse liquidity transactions that take place once a month and last for three months. • They are conducted by the NCBs through standard auctions and according to a predetermined schedule. • They offer additional, longer-term refinancing to counterparties. • The Eurosystem’s rule is not to send messages to the market through these transactions and, thus, it usually acts as a “recipient” (i.e. accepts the interest rates offered by the counterparties). – The smoothing (or fine-tuning) operations of short-term liquidity fluctuations are carried out in special cases, with the aim of both managing the market liquidity levels and influencing interest rates, in particular so as to smooth out the effects of unexpected liquidity fluctuations on interest rates. • They are usually carried out by the NCBs through fast tenders or bilateral procedures. The Governing Council of the ECB decides whether, in exceptional circumstances, short-term liquidity smoothing operations, through bilateral procedures, may be carried out by the ECB itself.

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• They are carried out mainly as reverse transactions, but can also be executed as outright transactions, currency swaps and acceptance of time (term) deposits so as to serve the specific targets set by the operations. – In addition, the Eurosystem can carry out structural operations through the issuance of securities, reverse transactions and outright transactions. • They are carried out by the NCBs through formal auctions when they are performed as reverse transactions or via the bond issuance. They are executed through bilateral procedures when they are performed as outright transactions. • They take place in order to adjust the Eurosystem’s structural position relevant to the financial sector when deemed necessary by the ECB (on a regular or extraordinary basis). 1.8.2.6 Standing Facilities Standing facilities provide and absorb overnight liquidity, set the monetary policy direction and put a bound on overnight market interest rates. There are two options to the eligible counterparties, managed by the NCBs: a. The marginal lending facility, which offers overnight liquidity by the NCBs with the appropriate collateral. It serves as an upper limit of the overnight market rate. b. The deposit facility, which allows for overnight deposits with the NCBs. It serves as lower limit of the overnight market rate. 1.8.2.7 Minimum Reserves The Governing Council of the ECB has decided to implement the minimum reserve system as an integral part of the monetary policy operating framework in the Third Stage. It seeks to stabilize money market interest rates, create (or widen) a structural liquidity deficit and, possibly, contribute to controlling monetary expansion. The minimum reserve is directly dependent on the balance sheet items of the interested financial institution. The latter can use the average approach, i.e. the requirement is met (or not met) on the basis of the daily average of the reserves held by the institution during a monthly maintenance period. The reserves

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bear the average interest rate of the main refinancing operations of the Eurosystem over the period held. 1.8.2.8 Asset Purchase Programs As mentioned, the APP is part of a package of non-standard monetary policy measures that the ECB has put forward to whether the crises. It was launched in mid-2014 in line with the ECB objective of safeguarding price stability. It was deployed in four directions: • • • •

The The The The

corporate sector purchase program (CSPP) public sector purchase program (PSPP) asset-backed securities purchase program (ABSPP) third covered bond purchase program (CBPP3)

The APPs spread in the period between October 2014 and December 2018 with a monthly purchase pace of (on average) as indicated in Fig. 1.3. In the following period, i.e. between January 2019 and October 2019, the maturities were reinvested, so as to keep each APP at the December 2018 levels. On 12 September 2019 the ECB Governing Council launch APP Amount

Amount in Billion Euro

90 80 70 60 50 40 30 20 10 0 March 2015 March 2016

April 2016 - March April 2017 2017 December 2017

January 2018 September 2018

October 2018 December 2018

Period

Fig. 1.3 APP amount per period implemented (Source Created by the author with data assembled from the ECB [2021])

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anew the APP at a monthly pace of e20 billion (starting 1 November 2019) without indicating a specific termination date other than only before raising its interest rates. The pandemic has not yet indicated that such a move is anticipated. 1.8.2.9 Pandemic Emergency Purchase Program The PEPP, similar to the APP, is a non-standard monetary policy measure launched in March 2020 to help weather the threats to the monetary policy transmission mechanism and the euro as a result of the pandemic. The PEPP is an asset purchase program of temporary nature and addresses private and public sector securities. The initial amount of e750 billion has been increased two times in 2020, to reach e1,850 billion as a result of the persistence of the pandemic. Asset eligibility remains the same with the APP with the extension towards the inclusion of Greek Government bonds that have been excluded so far, being non-investment grade. The residual maturity of eligible public sector securities starts from 70 days and reaches 30 years and 364 days. Their participation takes place as per the Eurosystem capital key of the NCBs. Purchases (will) take place in line with the market evolutions and so as to avoid harming the effort to mitigate the declining trend of the inflation, caused by the pandemic. This approach is expected to facilitate the monetary policy transmission. The remaining maturity of eligible non-financial commercial paper under the CSPP (as well as the PEPP) was extended to at least 28 days from the 6-month maturity threshold. The PEPP is expected to continue until the maximum date between the time that the Governing Council decides that the pandemic crisis has finished and the end of March 2022. The maturing securities will be reinvested at minimum until the end of 2023. The objective is that the PEPP portfolio will be unwound in a way that does not endanger the desired monetary policy. The participating securities can be lent in order to further increase liquidity and candidate collaterals. 1.8.2.10 Counterparties The monetary policy framework of the Eurosystem is designed in such a way that secures that a series of counterparties will be eligible to participate. However, access to the standing facilities and open market operations is limited to institutions to which the minimum reserve requirement applies. In addition the Eurosystem may select a limited

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number of counterparties to participate in operations to normalize shortterm liquidity fluctuations. In the case of outright transactions, no prior restrictions are imposed on counterparties. Currency swaps are carried out using factors that are actively involved in the foreign exchange market. 1.8.2.11 Eligible Assets According to the Statute, all Eurosystem credit transactions must exhibit adequate security provided by assets that are eligible to serve as collateral. The Eurosystem operations can thus be based on a wide range of assets. There are two categories of eligible assets, mainly for reasons related to the Eurosystem itself: “tier one” and “tier two”. The former comprises of marketable securities that meet eligibility criteria that are common to the entire euro area as defined by the ECB. The latter comprises of additional assets, marketable and non-marketable, that are important for the national financial markets and banking systems and whose eligibility criteria are set by the NCBs and are subject to ECB approval. The two levels of asset quality—as captured by the tiers—are not distinguished in terms of their eligibility to participate in the different Eurosystem monetary policy operations (except that tier two assets are not usually considered in outright transactions). Both the Eurosystem monetary policy and overnight credit operations apply the same eligibility criteria for deeming whether assets qualify for collateral. In addition, Eurosystem counterparties can choose to employ eligible assets on a cross-border basis, i.e. they can borrow from the central bank of the Member State in which they are established, mobilizing assets located in another Member State (ECB, 1998). 1.8.2.12

Organization of the European System of Central Banks (ESCB) The European System of Central Banks (ESCB) is made of the European Central Bank (ECB) and the national central banks (NCBs) of the 27 Member States of the European Union. The term “Eurosystem” refers to the ECB and the NCBs of the 19 Member States that have adopted the euro. However, NCBs in non-euro area Member States are still members of the ESCB with the ability to pursue their national monetary policy without participating though in the decision process relevant to the single currency monetary policy. As stated earlier, the primary objective of the European System of Central Banks (ESCB), as set out in the Statute of the European System of Central Banks and of the European Central Bank, is to maintain price

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stability. In line with this primary target, the ESCB aims to foster the general economic policies in the Community in order to facilitate the fulfillment of the targets of the Community. The key tasks of the Eurosystem are: i. to formulate and implement the monetary policy of the euro area; ii. to carry out foreign exchange transactions; iii. to hold and manage the official foreign reserves of the Member States; and iv. to promote the smooth operation of payment systems. In addition, the Eurosystem v. contributes to the application by the competent authorities of the policies relevant to prudential supervision of the credit institutions as well as the stability of the financial system; vi. pursues an advisory role to the Community and to national authorities on matters within its capacity; vii. collects the necessary statistical information either from the competent national authorities or directly from economic operators, in order to carry out the tasks of the ESCB and the ECB (with the assistance of the NCBs) (Bank of Greece, 2021). Intra—Eurosystem decisions are taken by the Governing Council and the Executive Board, being the ECB’s decision—making bodies. However, as not all Member States have adopted the euro yet, the General Council will serve as the third decision making body. The Governing Council The Governing Council is composed of all the members of the Executive Board and the governors of the NCBs of the Member States that have adopted the euro. Its main duties are: i. to set out the guidelines and take the steps needed in order to carry out the tasks assigned to the Eurosystem; ii. to formulate the euro area monetary policy, incorporating—where necessary—actions on intermediate monetary targets, key interest rates and reserve supply; and

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INTRODUCTION

41

iii. to draw up the guidelines needed for their realization. The Executive Board The Executive Board consists of the President, the Vice-President and four other members, selected from distinguished personalities with proven expertise in the fields of monetary policy or banking. Their appointment requires the agreement of all interested parties; namely the Governments of the Member States at the level of Heads of State or Government, the Council of the EU, the European Parliament and the Governing Council of the ECB (or the Council of the European Monetary Institute (EMI) for the first appointments). The main tasks of the Executive Board are: i. to implement monetary policy in the directions set by the Governing Council of the ECB and guide NCBs accordingly; and ii. to perform in line with the empowerment it received by the Governing Council of the ECB. The General Council consists of the President, the Vice-President and the Governors of the NCBs of all 27 Member States. The General Council shall carry out the duties that the ECB took over from the EMI and which shall continue to be carried out during the Third Stage of Economic and Monetary Union (EMU) due to the existence of one or more Member States that have not adopted Euro yet. The General Council also contributes to: i. the advisory functions performed by the ECB; ii. the collection of statistical data/input; iii. the compilation of the annual reports published by the ECB; iv. the establishment of the guidelines that will drive the standardization of accounting and reporting of the NCB operations; v. the assumption of measures—in addition to the ones provisioned in the Treaty—to determine the allocation key for subscription to the capital of the ECB; vi. the introduction of the employment terms of the staff of the ECB; and vii. the planning needed in order to lock the exchange rates of the currencies of the Member States that have not joined the euro.

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The Eurosystem is independent. As such none of the ECB, the NCBs, or any member of their decision-making bodies may request or receive directions from any other body when they execute their duties concerning the Eurosystem, and vice versa. I.e. EU institutions or bodies, as well as the governments of the Member States, may not attempt to distract the members of the decision-making bodies of the ECB or the NCBs in the performance of their duties. In order to ensure the term of office of the NCB Governors and the members of the Executive Committee, the Statute provides for the following measures: i. the term of office of the Governors is at least five years and is renewable; ii. the term of office of the members of the Executive Committee is at least eight years and is non-renewable (note that a partial renewal system was used to form the first Executive Committee, with the exception of the Chair, to ensure continuity), and iii. the dismissal of a member from his or her duties is possible only if incapacity or serious misconduct are detected. In this case, the Court of Justice of the EU shall have jurisdiction to settle any dispute. The ECB’s capital amounts to EUR10,825,007,069.61 (as per December 29, 2020). NCBs are the only registered shareholders and holders of ECB capital. The subscription in the capital is made on the basis of the allocation key determined according to the respective shares of the EU Member States in the GDP and the population of the European Union. In addition, the NCBs of the Member States that adopted the euro transferred to the ECB foreign exchange reserves. The amount transferred by each NCB was calculated in line with its shareholding fraction in the subscribed capital of the ECB; in return each NCB was credited by the ECB with a claim in euro equivalent to the amount it transferred.

Exercises Exercise 1 Select an existing company or country that wishes to issue a five-year bond.

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A. Explain what steps it must take in order to issue the bond. B. In which market will it be traded after its issuance? C. When would the issuance be considered successful? D. If the Central Bank wishes to pursue a loose monetary policy, then what steps should it take regarding these bonds? Exercise 2 Insurance companies are NOT allowed to buy bonds because they have a high risk. A. True B. False Explain your answer. Exercise 3 Zero coupon bonds pay only the face value at maturity and make no interim payment. A. True B. False Explain your answer. Exercise 4 Municipalities are not allowed to issue bonds. Only countries-states can. A. True B. False Explain your answer. Exercise 5 Which of the following is correct?

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I. If the Central Bank buys bonds from an individual, then interest rates go up. II. If the Central Bank sells bonds to an individual, then interest rates go up. III. If the Central Bank buys securities from a commercial bank, then the money supply increases. IV. If the Central Bank sells securities to a commercial bank, then the money supply increases. A. I, IV B. I, III C. II, III D. II, IV Explain your answer. Exercise 6 a. What other issuers could issue bonds in addition to the ones listed in this chapter? b. Review the literature for such potential cases. Exercise 7 Compare the role of Central Banks traditionally and their role over the last 15 years. a. What similarities and differences do you observe? b. Where are they attributed? c. Explain the evolution of their policies and their determinants. Exercise 8 Compare the role of the Fed with the ECB. a. What similarities or differences do you observe? b. Where are they attributed? c. Explain the evolution of their policies and their determinants.

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d. Can you judge—if possible—whether one of the two has been more efficient/effective in tackling the recent crises (financial, sovereign and pandemic)? Exercise 9 Look at the policies of other central banks, e.g. the Bank of England, the Bank of Japan, etc. a. Repeat exercise 7. b. Repeat exercise 8. c. What can you infer? (Hint: you may need to do some research of your own for these central banks). Exercise 10 Study the PEPP. Research the economic outlook for the EU and the Eurozone. a. Do you think it is sufficient? b. What else should be taken into account? c. Can it be applied uniformly across the countries or should idiosyncrasies of the countries be taken more into account? (Hint: you may need to do some research of your own for these topics).

References Bank of Greece. (2021). Eurosystem. https://www.bankofgreece.gr/en/thebank/eurosystem-and-euro-area/eurosystem. Accessed: January 2021. BIS. (2021). Eurosystem. https://www.bis.org/mc/currency_areas/xm.htm. Accessed: January 2021. Bodie, Z., Kane, A., & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. Bordo, M. D. (2007). A brief history of central banks. Federal Reserve Bank of Cleveland. https://www.clevelandfed.org/en/newsroom-and-events/ publications/economic-commentary/economic-commentary-archives/2007-

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economic-commentaries/ec-20071201-a-brief-history-of-central-banks.aspx. Accessed: January 2021. Cummans, J. (2014, October 1). A brief history of bond investing, bond education center. BondFunds.Com. http://bondfunds.com/education/a-brief-his tory-of-bond-investing/. Accessed: January 2021 ECB. (1998, October 23). Eligible assets. https://www.ecb.europa.eu/press/pr/ date/1998/html/pr981023.en.html. Accessed: January 2021. ECB. (2021). Monetary policy. https://www.ecb.europa.eu/. Accessed: January 2021. FED. (2008, November 25). Federal Reserve announces it will initiate a program to purchase the direct obligations of housing-related government-sponsored enterprises and mortgage-backed securities backed by Fannie Mae, Freddie Mac, and Ginnie Mae. Press Release. https://www.federalreserve.gov/. Accessed: January 2021. FED. (2010, November 3). Minutes of the Federal Open Market Committee— Summary of economic projections. https://www.federalreserve.gov/. Accessed: January 2021. FED. (2012, October 24). Federal Reserve issues FOMC statement. Press Release. https://www.federalreserve.gov/. Accessed: January 2021. FED. (2013, June 19). Transcript of Chairman Bernanke’s Press Conference. Media Center. https://www.federalreserve.gov/. Accessed: January 2021. FED. (2016, October). Federal Reserve Board: What is the FED? https:// www.federalreserve.gov/mediacenter/files/what-is-the-fed-transcript-2016. pdf. Accessed: January 2021. FED. (2020, March 15). Federal Reserve issues FOMC statement. Press Release. https://www.federalreserve.gov/. Accessed: January 2021. FED. (2021a). Credit and liquidity programs and the balance sheet. https:// www.federalreserve.gov/. Accessed: January 2021. FED. (2021b). Purposes and functions. https://www.federalreserve.gov/. Accessed: January 2021. Goodhart, C. A. E. (2010). The changing role of central banks (BIS Working Papers, No. 326). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=171 7776. Accessed: January 2021. The Federal Reserve System. (2021). https://federalreserveonline.org/. Accessed: January 2021.

CHAPTER 2

Bonds

Bonds are the most well known representatives of fixed income investments. They are loans with one borrower and usually more than one lender. The borrower is known as the issuer, whereas the lenders are known as bond holders or debt holders or creditors or simply investors. Bonds are issued by governments, states, municipalities and companies among others (as is also mentioned in Chapter 1) to borrow money so as to finance projects or operations. Governments (federal and regional/local) need to fund the building of infrastructure or to refinance their debt. The taxes collected are not enough—most of the times—so bonds are a good and popular way to raise money. Corporations need to fund their growth or running operations at an affordable cost; bank loans may not be enough or may come at a relatively high cost. Consequently, both governments and companies turn to institutional and individual investors in order to raise capital through lending. The interest paid is on one hand the cost of the issuer and on the other hand the return of the investor. Compared to bank loans, bonds may be made available to the broader investing public. At the same time, when they are listed in organized exchanges, they are very easily trade-able and can be bought and sold at the will of the interested investors. As such they are suitable for investment portfolios of many types of investors (institutional or individual). In addition, they come at varying levels of assumed risk, which determine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_2

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the expected return. Consequently, they match several risk-return profiles and have become quite popular over the years. As in most instances (until at least the 2008 crisis) they behaved in a way opposite to stocks, they were good diversification means when included in investor portfolios. When people feared a drop in the stock markets the retreated to the safety of bonds and when they were optimistic about the performance of the stock markets they jumped on to stocks. The same switch of preferred asset class took place depending on the outlook investors had on the projected track of interest rates. However, this has changed over the last decade, most likely due to the intervention of the central banks; they provided liquidity through their quantitative easing (QE) programs, which led both equity and bond markets to higher and higher levels as the investors realized that no matter what the bad news may have been, the central banks would offer support. When the going went tough, in most instances they were leaving from both markets. As a result there were several instances that bonds and stocks moved towards the same direction. It therefore makes sense to try to understand on one hand how bonds are structured as debt (fixed income) instruments (i.e. their characteristics and pricing) and on the other hand how they can be used as investment means within a portfolio (i.e. the fluctuations of their prices and the factors that cause it). In this chapter we present the characteristics of bonds, as well as the risks that come from bonds; the default risk and the market risk. We explain how a bond is priced and how its yield is determined. We elaborate on the different types of bonds. After having read this chapter the reader will be in place to realize how a bond behaves as an investment vehicle on its own.

2.1

Basic Bond Characteristics, Concepts and Notions 2.1.1

Basic Bond Characteristics

A bond is a debt issue in which the debtor-borrower issues or sells a debt to the lender-investor. This debt is often referred to as IOU (I owe you), with the difference that the bond represents usually a more formal debt instrument. The relevant/underlying agreement determines the payments

2

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49

to be made by the issuing/borrowing party to the investing/lending party and the time instants at which these payments will be made. The main difference between a standard loan and a bond is that the loan is most of the times an agreement between one borrower and one lender, whereas the bond is an agreement between one borrower (the issuer) and several lenders (the bondholders or the investors). A bond resembles a single payment loan with or without intermediate interest payments. A single payment loan is a loan that requires that the entire principal is paid on the maturity date of the loan as a lump sum. There are two variants; one that requires that interest is paid at the maturity date of the loan along with the principal and another that demands the periodic payment of interest (e.g. monthly, quarterly, semi-annually or annually). A bond can be issued with or without intermediate interest payments. In the previous framework, a standard coupon-bearing bond provisions for intermediate payments; i.e. the issuer has the obligation to make periodic payments (usually on a semi-annual or annual basis) to the holder of the bond until it matures. These payments reflect the interest, which in the case of bonds is referred to as the coupon. On top of these periodic (interest) payments, the issuer is obliged to make one last payment to the holder of the bond at the maturity date of the bond; namely the face value (or par value) of the bond, which essentially represents the loan principal amount. The standard bond quotation releases the coupon rate and not the coupon amount; this is defined as the coupon payment divided by the face value. It practically stands for the interest rate (in comparison to a standard loan). The coupon needs to be high enough to attract investors. A coupon-bearing bond is essentially a single payment loan with intermediate interest payments. Furthermore, there can be bonds that make no coupon payments. These are known as zero-coupon bonds , as essentially the coupon rate is zero. Investors receive only the face value (nominal value) at maturity but no intermediate payments. In other words they receive only the equivalent of the loan principal amount but no interest. There is no coupon in this case to attract investors and so the prices are well below the face value. The investor’s income comes from this very difference. A zero-coupon bond is practically a single payment loan without intermediate interest payments. Bonds in the past had a paper/physical form/format and were bearer bonds, i.e. contained no owner information. They issued coupon

50

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payments or repaid the principal amount to whoever was in physical possession of the bond paper/certificate (as it had physical form). The certificate had perforated coupons (at the side and/or at the bottom) corresponding to the interest payments that were to be made by the bond (Fig. 2.1). The owner of the bond had to simply cut the coupons and present them to the issuer so as to collect the interest. Until today, this explains why the interest payments made by the bond are referred to as coupons, even though most of the bonds are in electronic and not in paper form/format.

Issuer Face Value Issue Date Maturity Date Coupon Rate Coupon Frequency

Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount

Company ABC123 EUR 1,000 January 21, 2022 EUR 20 Company ABC123 EUR 1,000 January 21, 2024 EUR 20 Company ABC123 EUR 1,000 January 21, 2026 EUR 20 Company ABC123 EUR 1,000 January 21, 2028 EUR 20 Company ABC123 EUR 1,000 January 21, 2030 EUR 20

Company ABC123 EUR 1,000 January 21, 2021 January 21, 2031 2% Annual

Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount Issuer Face Value Coupon Payment Date Coupon Amount

Company ABC123 EUR 1,000 January 21, 2023 EUR 20 Company ABC123 EUR 1,000 January 21, 2025 EUR 20 Company ABC123 EUR 1,000 January 21, 2027 EUR 20 Company ABC123 EUR 1,000 January 21, 2029 EUR 20 Company ABC123 EUR 1,000 January 21, 2031 EUR 20

Fig. 2.1 Example of a bearer bond in paper format (Source Created by the author)

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These days, in addition to being in electronic form/format, most bonds are registered, i.e. the details of the owner (such as name, address, etc.) are archived to ensure that the bond coupon payments are made to the correct investor. Both combinations are (theoretically) possible; i.e. physical/paper or electronic/non-physical format bonds can be registered, with electronically registered bonds being the only available form in certain countries. This allows the payment of the coupon periodically and the face value on the maturity date at a designated bank account of the owner without his or her intervention. Registered bonds exhibit additional security as they make payments only to the registered owners. To that extend a bond does not make payments to any party that presents a bond certificate without being a registered owner. At the same time, if a registered bond (in physical/paper form/format) is destroyed or lost or stolen, then it can be easily replaced as the particulars of the owner have been archived. Bearer bonds do not offer this safety, as they make payments to whoever is in physical possession of the bond certificate. In addition they may be destroyed, lost or stolen without having the possibility to replace them as the particulars of the owners are not logged. 2.1.2

Basic Bond Concepts and Notions

Further to the definitions provided above, there is a series of concepts relevant to bonds. The basic notions associated with bonds, and in particular with the different types of bonds are (Bodie, 1996): 1. Government Bonds or Treasury Bonds and Notes: These are the bonds that are issued by a central government. They aim at assisting the different nations at collecting funds—other than taxes—to finance their activities and the benefits offered to their citizens. 2. Corporate Bonds : These are the bonds that are issued by companies/firms again in order to finance their activities/operations. 3. Accrued Interest : The prices displayed on trading monitors/screens are not the cash prices at which the transaction takes place. The accrued interest must be calculated on (top of) the purchase price. It is the part of the coupon that is paid to the seller by the buyer so that he or she receives the portion of the interest that corresponds to the time period prior to the transaction. It is

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estimated from the time that the last coupon prior to the transaction is paid and before the next coupon payment is made. As this calculation takes place every time a bond changes hands, investors are properly compensated for the period they hold the bond; on one hand when they sell the bond they receive interest since the last coupon payment whereas on the other hand they have already paid interest since the last coupon payment when they purchased the bond (if for example both transactions take place between the same two coupon payments). This gives the invoice price. 4. Quoted Price or Clean Price: This is the price that can be seen on a trading system or listed in an organized exchange. It does not account for the accrued interest. 5. Cash Price or Dirty Price: This is the price that accounts also for the accrued interest and is computed as the sum of the quoted price and the accrued interest. 6. Floating Rate Notes (FRNs): In these bonds the coupon is not preset but is rather linked to the current course of market interest rates. It is usually the sum of a reference variable interest rate and a fixed yield spread. The yield spread, i.e. the additional reward offered for any potential incremental risk born by the investor, is usually unchanged for the whole life of the bond. It does not change due to a potential change in the company’s ability to repay its debt. Such an improvement or worsening is thus reflected in the bond price. 7. Callable Bonds —Call Provisions : They give the possibility/right/option to the issuer to repurchase/recall the bond at a predetermined call price, at a predetermined date (or dates) before the bond matures. The bond is referred to as a callable bond and is the combination of a bond (sold by the issuer) with a call option (acquired by the issuer). The issuer has (a long position on) the call option and thus the right to purchase the bond back. It is exercised by the issuer if the interest rates fall, and the coupon becomes comparatively much higher than them, as in this case lower coupons can be achieved with a new issue. There is usually a protection period during which the call option (deferred callable bonds ) is not exercised. Callable bonds offer either higher coupons or lower prices compared to the plain vanilla bonds; this is the compensation of the investor for the risk he or she bears that the bond is recalled prior to its maturity date. It is at the same time

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the implicit cost to the issuer for having acquired the option to recall the bond prior to its maturity date. 8. Puttable Bonds : These are bonds that give the option/right/possibility to the investor/bondholder to sell the bond if he or she wants before it matures at a predetermined (put) price at a predetermined date (or dates). Puttable bonds are a combination of a bond and a put option, i.e. an option to sell the bond prior to its maturity date. The investor has (a long position on) the bond and the option, whereas the issuer has sold them to him or her (who then has a short position). The investor keeps the bond if it gives a coupon higher than the yields/interest rates available in the market for the same level of risk. He or she sells it if the coupon is lower than the yields/interest rates offered in the market for comparable risk. As puttable bonds offer to the investor the right to sell back the bond before it matures, they come at a lower coupon or higher price compared to plain vanilla bonds. This is the compensation to the issuer for bearing the risk of having to repurchase the bond before it matures. At the same time it is the cost to the investor for acquiring the put option 9. Convertible Bonds /Convertibles: These are bonds with an option/right given to the investors to convert/exchange each bond with a predetermined number of common (shares of) stock of the issuing company. The number of shares corresponding to each bond (to be converted) is referred to as the conversion ratio. The current value of these shares (corresponding to each bond to be converted) is known as the market conversion value. The amount by which the market conversion value exceeds the actual bond value is defined as the conversion premium. Convertibles have a lower coupon or higher price compared to plain vanilla bonds as they give the right to the investors to access the stocks of the company, potentially benefiting from the conversion premium. This is the compensation of the issuer for receiving back the bond before it matures. At the same time it is the cost to the investor for having been given the conversion right. 10. Preferred Stock: It is considered equity and not debt. However, it does have some common features with debt instruments/bonds. Namely, it promises a flow of dividends. So in a way it resembles a fixed income security. But the inability to pay the promised cash

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flows does not constitute a default. It looks like perpetuity, as the dividend payments are assumed to be made ad infinitum. Additional concepts and notions relevant to bonds and fixed income instruments are presented in the sections and chapters that follow and in particular in the fixed income/bond market chapter. A comparison of bonds with stocks is made in the relevant chapter.

2.2

Bond-Related Risks

In Chapter 1, where we outlined the basic steps of the investing process, it became apparent that investors attempt to maximize the risk-return ratio. Fixed income instruments in general and bonds in particular are considered as bearing less risk compared to stocks; however, they still constitute a form of debt. As such, although they promise steady and specific cash flows, this does not mean that the investor is secured. There is always the risk that the issuer will default on its obligation! Perhaps influenced by government bonds, we believe that this is not the case. But it happens in corporate bonds, and the recent turbulent financial environment revealed that it could happen to government bonds as well. If the issuer goes bankrupt then the investor will not receive the promised income. This is referred to as default risk. As mentioned, in the latest financial crisis of 2007–2008 even the security of government bonds was questioned; several European countries, especially of the South, experienced difficulty in making certain coupon or face value payments. Some of these countries had to receive support from the European Union (EU), European Central Bank (ECB) and/or the International Monetary Fund (IMF) to avoid default. Specific examples were Greece, Portugal, Italy, Spain, Ireland and Cyprus; the first five gave rise to acronyms such as GIPSI or PIGS or PIIGS depending on the combination of countries and order of initial letters. Greece and Cyprus seemed to be the most distressed, with Greece restructuring its debt towards the private sector by replacing the at-the-time available bonds and applying a haircut on the amount of debt. In our example in Chapter 1 we indicated that the risk borne by the investors when they buy our bond is remunerated by the interest rate/yield they expect as their return. It is thus clear that the default risk associated with a bond determines the level of the interest rate (depicted by the coupon) that the investor/bondholder expects to receive

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for committing his or her capital to fund the issuing entity. Another determinant of this return is the overall level of interest rates in the market; this poses another risk, often referred to as interest rate risk. Consequently there are primarily two sources of risk associated with a bond; the default risk and the interest rate risk. We study them both in this book; however in this chapter we focus on the basic notions of default risk. 2.2.1

Ratings and Rating Agencies

In the example we used in Chapter 1 we explained that our capacity to repay our debt is scored by specializing firms so that investors can compare our promised return with similar/comparable other issuers. The default risk is thus rated by firms, known as (credit) rating agencies , such as Moody’s Investor Services (Moody’s, 2021), Standard & Poor’s Corporation (S&P Global, 2021) and Fitch (Fitch Ratings, 2020b). The rating agencies provide information on the companies’ finances and the quality of government, corporate, municipal and other issuer bond issues. This quality is measured with a score/grade/rate, known as credit rating . Evaluating the ability of a debtor to repay his or her debt is a sine qua non for any type of loan. Banks have been performing that since their origins for individuals and companies; however this process is known as credit scoring and the numerical evaluation of the creditworthiness of the borrower is known as credit score. Rating agencies usually have bigger issuers under their scope that go beyond the application of a bank loan. The best grades that can be assigned by rating agencies are AAA (S&P, Fitch) and Aaa (Moody’s) and the grade can go down to D or C (depending on the agency). The scales/ratings/categories used and their descriptions are shown in Table 2.1. There are also subdivisions of the above scale, called rating modifiers, in order to better capture the creditworthiness of bonds (and their issuing entities). S&P and Fitch use ± and Moody’s uses 1, 2, 3 as suffixes next to the various categorizations. S&P adds these suffixes (±) to all rates from AA to CCC, whereas Fitch adds modifiers (±) to all rates from AA to B. Finally, Moody’s uses suffixes (1, 2, and 3) for all rates from Aa to Caa. S&P uses the Selective Default (SD) rating when the issuer has selectively defaulted on a specific issue or class of issues but continues to meet other financial obligations by making payments in a timely manner. In a

S&P

AAA

AA

A

BBB

BB

B

CCC

Aaa

Aa

A

Baa

Ba

B

Caa

CCC

B

BB

BBB

A

AA

AAA

Fitch The highest rating. Extremely high ability to meet financial commitments, i.e. make interest and principal payments Very high ability to meet financial commitments, i.e. make interest and principal payments High ability to meet financial commitments, i.e. make interest and principal payments—somewhat more vulnerable to adverse economic conditions than the higher ratings Adequate ability to meet financial commitments, i.e. make interest and principal payments—it is even more vulnerable to adverse economic conditions Some ability to meet financial commitments, i.e. make interest and principal payments and potential protective characteristics—adverse economic conditions may render this ability inadequate Somewhat lower but sufficient current ability to meet financial commitments, i.e. make interest and principal payments and potential protective characteristics—in adverse economic conditions more likely not to have the ability to meet financial commitments Lower and vulnerable current ability to meet financial commitments, i.e. make interest and principal payments and potential protective characteristics—adverse economic conditions may impair this ability to meet financial commitments

Description

Credit ratings (long-term) by agency

Moody’s

Table 2.1

Substantial credit risk

Highly speculative

Speculative

Good credit quality

High credit quality

Very high credit quality

(Of the) Highest credit quality

Quality

56 T. POUFINAS

D

Quality

Very low and highly vulnerable current ability to meet Very high level of credit risk financial commitments, i.e. make interest and principal payments and potential protective characteristics—default appears highly probable or almost a certainty Very low and very highly vulnerable current ability to meet Near default financial commitments, i.e. make interest and principal payments and potential protective characteristics—a default or default like process may have already commenced and the ability to meet obligations may be irrevocably impaired, with potentially lower seniority in recovery Already in default, financial commitments are not met, i.e. (At) Default interest and principal payments are not made on the due date and potentially a bankruptcy process of any type or similar action has or is taking place or business has stopped

Description

Source Created by the author with information assembled from Moody’s (2021), S&P (2021) and Fitch (2020b)

D

C

C

C

CC

CC

Ca

Fitch

S&P

Moody’s

2 BONDS

57

58

T. POUFINAS

similar manner Fitch employs the term Restricted Default (RD) when the issuer has experienced a selective payment default on a specific class of debt, as well as (among others) an uncured payment default, a distressed debt exchange, the uncured expiry of any grace period, as well as the extension of multiple waivers, without though having entered into a bankruptcy process of any type or having stopped operating. Bonds that have a grade above BBB (S&P, Fitch) or Baa (Moody’s) are considered as investment grade. This qualifies them as suitable for investment by investors that have minimum bond quality requirements such as pension schemes and insurance companies. Bonds with a lower grade are characterized as non-investment grade or speculative grade or junk bonds . These bonds have in many occasions been excluded from candidate investments for pension schemes and insurers; however, the quest for higher performance has led such institutional investors to consider non-investment grade bonds for their portfolios. Rating agencies rate issues and issuers for their long term obligations as per the aforementioned scales. Moreover, they rate both obligations and issuers in the short-term. The particulars can be found in Moody’s (2021), S&P (2021) and Fitch (2020b). As became clear in Chapter 1, but also in the previous discussion, investors expect a compensation for the incremental risk they are willing to undertake. The additional performance that an investor expects due to the risk of default on the issuer is also known as (credit) spread. Therefore, the bond yield is the risk-free interest rate increased by the spread: y = r f + spread.

(2.1)

The higher (the best) the credit rating the lower the spread, as the risk undertaken by the investor is lower. The spread is probably anticipated to be higher as the time to maturity of the bond increases. This is due to the fact that the owner of the bond is exposed to the credit risk stemming from the issuer for a longer period of time. Furthermore, he or she has committed capital for a longer period of time and has thus given away the benefit of liquidity. The spread per credit rating as function of the time to maturity is illustrated in Fig. 2.2 (we will talk about the interest rate term structure in Chapter 3). The credit ratings are not carved in stone or written in blood; they may change. Their shift reflects a change in the ability of issuers to repay their debt. An improvement, i.e. a shift upwards, is known as an upgrade,

2

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59

Spread over risk free BBB

A AA AAA

Maturity Fig. 2.2 Spread as a function of time to maturity per credit rating class (Source Created by the author)

whereas deterioration, i.e. a move downwards, is called a downgrade. The change to the credit rating immediately preceding or succeeding the assigned one is referred to as one notch. For example, a move from a credit rating of A to A+ (as per the S&P scale) is an upgrade by one notch. Example 2.1 The average cumulative (issuer-weighted) default rates for the years 1970– 2010 as posted by Moody’s are shown in Figs. 2.3 and 2.4. Figure 2.3 shows the rate of default for companies starting with a specific credit rating today for the next few years. For example, a company with a credit rating A has a chance of 0.055% to default within a year and 0.177% to default within two years, It has a chance of 0.122% (=0.177% − 0.055%) to default between years 1 and 2. A company that has an initial credit rating of Baa has a chance of 0.181% to default on its obligation by the end of the first year, 0.510% by the end of the second year and so on. Sometimes this is mentioned as probability of default; however caution is recommended when this term is used, as the aforementioned percentages are frequencies estimated from historical data out of a monitored set of companies. This will be further discussed below.

60

T. POUFINAS

Cumulative Default Rates per Credit Rating

90% 80% 70%

Percent

60%

Aaa Aa

50%

A 40%

Baa

30%

Ba B

20%

Caa-C 10% 0% 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Year

Fig. 2.3 Average cumulative issuer-weighted global default rates (%), 1970– 2010—part I (Source Created by the author with data assembled from Moody’s [2011])

Cumulative Default Rates per Grade Class

60%

50%

Percent

40%

30%

Investment grade

20%

All rated

Speculative grade

10%

0% 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

Year

Fig. 2.4 Average cumulative issuer-weighted global default rates (%), 1970– 2010—part II (Source Created by the author with data assembled from Moody’s [2011])

2

61

BONDS

The ratings as posted by Standard and Poor’s—using also the rating modifiers are depicted in Figs. 2.5 and 2.6. We can use the cumulative default rates to calculate the annual default rates; they are illustrated in Figs. 2.7 and 2.8. Comparing the two sources, i.e. Moody’s and Standard and Poor’s we see that the results can be quite close or apart, depending on the number of years and the credit quality. The observed differences are potentially attributed to the fact that there is a difference of almost 10 years in the periods captured and the bond universe that has been rated. An investor needs to be careful when employing the credit ratings in order to make decisions. There are three points of attention; (i) the relatively qualitative nature of the grades; (ii) the indicatively quantitative nature of the grades; and (iii) the corporate nature of rating agencies. These grades have comparative and not absolute value as they are assigned based on the study of a few thousand companies and other issuing entities. In other words, they are used so as to know whether comparable issuers—in terms of creditworthiness—offer similar returns. Cumulative Default Rates by Rating Modifier 60% AAA

50%

AA+ AA AA-

Percent

40%

A+ A A-

30%

BBB+ BBB BBB-

20%

BB+ BB BB-

10%

B+ B B-

0%

CCC/C

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Year

Fig. 2.5 Global corporate average cumulative default rates (%) by rating modifier (1981–2018)—part I (Source Created by the author with data assembled from S&P [2019])

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T. POUFINAS

Fig. 2.6 Global corporate average cumulative default rates (%) by rating modifier (1981–2018)—part II (Source Created by the author with data assembled from S&P [2019])

Annual Default Rates by Rating Modifier

30%

AAA AA+

25%

AA AAA+

Percent

20%

A A-

15%

BBB+ BBB BBB-

10%

BB+ BB

5%

BBB+

0%

B

1

2

3

4

5

6

7

8

Year

9

10

11

12

13

14

15

BCCC/C

Fig. 2.7 Global corporate average annual default rates by (%) rating modifier (1981–2018)—part I (Source Created by the author based on author’s calculations with data from Standard and Poor’s [S&P, 2019])

2

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63

Annual Default Rates per Grade Class

4% 4% 3%

Percent

3% 2%

Investment grade SpeculaƟve grade All rated

2% 1% 1% 0%

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Year

Fig. 2.8 Global corporate average annual default rates (%) by rating modifier (1981–2018)—part II (Source Created by the author based on author’s calculations with data from Standard and Poor’s [S&P, 2019])

They are also used to rank companies or issuers with regards to their ability to repay debt. As such they can be a valuable risk management tool. But they have primarily a qualitative nature. Next to them there is usually a default frequency (or default rate) assigned, which is estimated from the number of companies that defaulted out of the evaluated sample, during a specific period of time, let’s say a year. These default rates are sometimes associated with or even mentioned as probabilities of default. Attention is needed when the term probability of default is used in the context of credit ratings; a company (or any other issuer) will either default or not default within one year. This is better described by a Bernoulli (binary) probability distribution; with let’s say a probability of default p. In order to properly estimate the probability of default of a specific issuer within one year we need to know the value of p. This is unique and idiosyncratic for each entity. It may be approximated with the default frequency associated with a credit rating but it is not exactly it. Consequently, credit ratings have some quantitative nature as depicted by default rates per credit rating. However, their computation is based on historical default frequencies and they thus constitute estimates of the actual probability of default of a specific entity; they are thus good mainly for capturing the global default trend.

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T. POUFINAS

Finally, one should not ignore that rating agencies are corporations, run by people and populated by people. Although, as in all professions, people may do their best to deliver the optimal result in the optimal time span (which is usually considered the shortest possible), this may not always be feasible. Furthermore, updates of the credit ratings may be necessary, when the creditworthiness of the issuers change. The timing may not be the desired one; sometimes it is a bit too early or too late. Besides that, the ratings assigned may not be as accurate, for several reasons, such as the quality of the available data and information, or because of rapid developments inside or outside the underlying entity; or simply because of the human nature. Example 2.2 The COVID-19 pandemic that hit the world at the end of 2019 and was fully deployed in 2020 is anticipated to have unprecedented impact not only to human lives and lifestyles but also to the economic activity. The International Monetary Fund (IMF) World Economic Outlook (International Monetary Fund, 2020) foresees a drop of 3% in the global economy (in the baseline scenario), which is at the area of 6.1% for the advanced economies, 5.9% for the USA and 7.5% for the countries of the euro-zone. This is expected to trigger a series of defaults as well as a deterioration of the creditworthiness of bond issuers all over the world. The central banks, such as the Fed and the ECB, backed by the US and EU governments respectively, have announced that they will provide once more the required liquidity; however, certain down grades or even defaults are inevitable; the question is how many will there be and for how long. According to the International Monetary Fund World Economic Outlook (International Monetary Fund, 2020), in the baseline scenario 2021 is expected to be a year of growth; however, this is subject to a successful weathering of the pandemic. If true, downgrades will most likely be followed by upgrades. Indicative of this climate is the Fitch Non-Rating Commentary (Fitch Ratings, 2020a) stating that foresees additional downgrades to noninvestment grade—more than observed in the precious years—due to the worsening of the relevant (sovereign) credit indicators caused by the pandemic and the drop in oil prices. Since the middle of the 1990s a total of 27 downgrades to non-investment grade were recorded compared to 31 upgrades to investment grade.

2

2.2.2

BONDS

65

Factors Affecting the Credit Rating

But how are the credit ratings estimated? What are the factors/parameters that determine them? There are several factors/determinants of the risk/creditworthiness of a bond (issuer). These are primarily (financial) ratios that are globally used to evaluate the solvency and/or performance of any company. The credit ratings produced by the rating agencies take into account these ratios among other parameters. The main ones are (Bodie et al., 1996): 1. Coverage Ratios : These are ratios of corporate earnings (profits) to different types of fixed costs. They reflect the capacity of a company to repay its debt and overall financial obligations. The main ones are: times-interest-earned (TIE) ratio = earnings before interest (payments) and taxes EBIT = , interest obligations (or expense) interest expense (2.2) fixed-charge coverage ratio (FCCR) = EBIT + fixed charges before taxes (FCBT) EBIT + FCBT = . all cash obligations interest expense + FCBT (2.3) The first indicates the capacity of a company to cover its debt obligations, whereas the second shows the capacity of a company to meet all its fixed-charge obligations. Low ratios may imply difficulties in meeting obligations, potentially originating from low cash flows. 2. Cash Flow-to-Debt Ratio: This ratio indicates the ability of a company (as well as the time it would take) to pay off all of its debt if it used all of its operating cash flow for debt repayment. cash flow-to-debt ratio =

total cash flow . outstanding debt

(2.4)

Alternatively, the cash flow from operations is used in the numerator. This ratio acts as a metric of the overall health of a company and resembles coverage ratios. A high ratio indicates that a company is

66

T. POUFINAS

capable to repay its debt, and is thus better placed to take on more debt if necessary. 3. Leverage Ratios : These are ratios that show the proportional contribution of debt to the financing of an entity, as well as its capacity to meet its financial obligations. The most representative leverage ratio is calculated as the ratio of debt to equity and is given by: leverage ratio =

debt . equity

(2.5)

A high ratio is indicative of high debt levels compared to the equity of the issuing entity, which could prohibit it from meeting its debt obligations. 4. Liquidity Ratios : These are ratios that reflect the ability of a company to cover its short term liabilities/obligations. The main ones are: current ratio = quick ratio =

current assets , current liabilities

current assets ex inventory(ies) . current liabilities

(2.6) (2.7)

The difference between these two ratios is that the second one reflects the capacity of a company to cover its short-term liabilities with assets that can be immediately converted into cash. The former is also referred to as working capital ratio, whereas the latter is also referred to as acid-test ratio. Liquidity ratios measure the ability of the evaluated firm to pay the upcoming payables due with the cash it collects. Low liquidity ratios indicate potential difficulties in meeting short term liabilities. 5. Profitability Ratios : These are ratios that depict the earnings/profitability of a company versus its assets or equity. The most well known ones are: return on assets (ROA) =

earnings before interest and taxes (EBIT) , total assets (2.8)

return on equity (ROE) =

earnings before interest and taxes (EBIT) . equity (2.9)

2

BONDS

67

Both indices can be defined with net income in the numerator which is produced if interest and taxes are added to EBIT. Their difference is the inclusion or exclusion of debt in the denominator, which indicates leverage. They indicate how well the capital of the company is managed. Furthermore, they serve as investment performance metrics. They show the financial strength of the company and as such higher ratios are preferred by investors. They are also indicative of the ability of a company to borrow money as its earnings reflect its ability to repay them. Bond ratings are directly dependent on the financial data available. Investors are monitoring potential changes in the corporate bond ratings. So the prices (usually) change while anticipating for the changes in the ratings to happen and not after the changes are made. Usually downgrades are accompanied by unusual returns on the company’s stock. Several studies have focused on whether financial ratios can be used to predict default risk. One such method is to give each company a score (called the Z -score) based on its financial characteristics (Altman, 1968). If the score is above a certain level the company is considered creditworthy. If the score is below this level then there is a risk of bankruptcy in the future. The separating equation (discriminant function) developed by Altman (1968) is: retained earnings working capital + 0.014 · total assets total assets market value of equity EBIT + 0.006 · + 0.033 · total assets book value of total debt sales . (2.10) + 0.999 · total assets

Z = 0.012 ·

Since then several approaches have been used in order to predict bankruptcy. Nowadays, the evolution of information technology (and artificial intelligence) has facilitated the use of machine learning techniques for the prediction of bankruptcy. Such a presentation lies below the scope of this book. An overview of the applicable methods may be found for example in Qu et al. (2019).

68

T. POUFINAS

2.3

Bond Valuation

Apparently, the most interesting part of a bond is its pricing. This has to do with the fact that bonds are traded in the secondary market(s). Therefore, as opposed to bank loans bonds can change hands, i.e. they can be bought or sold. Consequently, it is of interest to know the price that the buyer has to pay to the seller. As with any stream of cash flows, the value of a bond (at a certain point of time) is computed as the present value of the cash flows received from it. These cash flows are the coupons and the face value paid at maturity— for a coupon-bearing bond—or only the face value paid at maturity—for a zero-coupon bond. More precisely, c1 c N + FV c2 + ··· + + (1 + r ) (1 + r )2 (1 + r ) N N cn FV = + , n=1 (1 + r )n (1 + r ) N

PV0 =

(2.11)

where, cn r FV N PV0

is is is is is

the the the the the

coupon paid by the bond at time n, interest rate used for discounting, face value of the bond, time to maturity of the bond, present value of the bond.

The interest rate r is the interest rate that corresponds to the same level of risk as the one born by the bond. We sometimes refer to it as the return of an investment of equivalent level of risk. It is usually calculated as the risk-free rate r f augmented with the spread that compensates the investor for the risk he or she is exposed to by holding to the bond. It could be given by the equation r = r f + spread.

(2.12)

In fixed rate bonds the coupon is the same in all periods; however in floating rate notes it changes at the beginning of each period. In zerocoupon bonds the coupon is equal to zero and the present value equation

2

BONDS

69

changes to PV0 =

FV . (1 + r ) N

(2.13)

In Eq. (2.11), although we did not explicitly say it, one could have assumed that the coupon is annual; it could be semi-annual or of any other frequency. In that case the coupon rate is divided with the number of payments that occur in a year—a.k.a. the frequency of the coupon payment and the interest rate used for discounting has to be the equivalent with the same compounding frequency, e.g. semi-annual. In that case N is not years, but the number of periods until the maturity of the bond. This means that if the coupon is paid on a semi-annual basis, then N would equal twice the number of years until the maturity date of the bond The present value calculation for a bond maturing (for example) in 4 years and making annual coupon payments can be pictorially seen in Fig. 2.9. A coupon-bearing bond can be seen as a set or a portfolio of zero coupon bonds. Each of these zero-coupon bonds matures on each of the coupon-payment dates of the coupon-bearing bond and its face value is equal to the cash flow paid by the coupon-bearing bond at that time Present value of bond payments

0

c/(1+r)

1

2

c

3

c

c

c/(1+r)2 c/(1+r)

3

(c+FV)/(1+r)

4

Discounting of bond payments

Fig. 2.9 Bond valuation (Source Created by the author)

4

Time

c+ FV

Bond payments

70

T. POUFINAS

instant. It is therefore equal to the coupon (amount), except for the last one that is equal to the sum of the coupon and the face value. Example 2.3 Let us consider a bond that matures in 6 years, with a coupon rate of 4% paid annually and a face value per bond of 1,000 Euro. If the discount rate is 2% per annum with annual compounding, then the value of the bond today is estimated as 40 40 + 1000 + ··· + (1 + 0.02) (1 + 0.02)6 6 40 1, 000 = + = 1, 112.03. n=1 (1 + 0.02)n (1 + 0.02)6

PV0 =

(2.14)

If the coupon of 4% is paid semi-annually and the interest rate is 2% per annum with semi-annual compounding, then the coupon payment would be 40/2 = 20 Euro per semester and the discounting would use 2%/2 = 1%. There are this time 12 payments during the life of the bond. The first eleven are of 20 Euro, whereas the last one is the sum of the last coupon and the face value, i.e. 1,020 Euro. The value of the bond is given by 20 + 1000 20 + ··· + (1 + 0.01) (1 + 0.01)12 12 40 1, 000 = + = 1, 112.55. n n=1 (1 + 0.01) (1 + 0.01)12

PV0 =

2.3.1

(2.15)

The Relation Between Price and Present Value

But is the value of the bond, calculated as the present value of the payments made by the bond in Eq. (2.11) for coupon-bearing bonds—or Eq. (2.13) for zero coupon bonds—the price of the bond? We sometimes refer to it as the theoretical price or fair price, since this is the price we expect to observe in the market. However, the market may think differently, at least instantly; because as soon as investors realize that there is a difference then they take positions that would equate the present value and the market price of the bond.

2

BONDS

71

One could even debate whether the terms value and price could be used interchangeably. For the sake of good order we explain the difference. The term value most of the times refers to a (monetary) figure assigned to a product (or service) based on a theoretical method or model. The term price refers to the money that the seller receives in the market for selling a product or service and the buyer pays for buying it. In our context, the value assigned to the bond is the present value that was employed in Eq. (2.11) to put a monetary indication to the bond. Nonetheless it is restricted by the way Eq. (2.11) receives its input and produces its output. The input is the coupon (rate), the time to maturity, the interest rate and the face value. The output is the present value. One potential restriction is that the interest rate seems to be flat (although we will lift this restriction later). The price is determined by the will of the buyer and the seller to come to an agreement and they may believe that a different amount of money needs to be exchanged. In an organized market the difference between the (present) value and the price is usually instantaneous as investors will try to take advantage of it to make a riskless profit (known as arbitrage). We will explain why this cannot be the case later in this chapter when we define the notion of yield to maturity. 2.3.2

Bid Price and Ask Price

One more point of attention is that usually in an organized market we observe two prices; the bid price and the ask price (or offer price). The former is the price that the intermediary (dealer) is willing to buy the bond and the latter is the price at which he or she is willing to sell it. The bid price is lower than the ask price; the difference, known as bid-ask spread or bid-offer spread is the compensation of the intermediary. This is a transaction cost that alters the price compared to the one assigned through the present value formula of Eq. (2.11). When we follow the price of a bond in our screen through an electronic trading system we see two prices; the price at which we buy (the ask price, which is higher) and the price at which we sell (the bid price, which is lower). The bid-ask spread is usually lower for bonds that exhibit high liquidity and marketability compared to the bonds that have low liquidity and marketability.

72

T. POUFINAS

Example 2.4 If the bond of Example 2.3 was listed in an organized exchange, then we would see a bid price of 1,111.93 Euro and an ask price of 1,112.13 Euro compared to a single price of 1,112.03 Euro. Following the previous conversation this indicates that the broker is willing to buy the bond from us at 1,111.93 Euro and he or she is willing to sell it to us at 1,112.13 Euro. The difference of the two is 0.20 Euro and is the bid-ask spread.

2.3.3

The Relation Between Quoted Price and Cash Price

There is one more distinction of bond prices; this of a quoted price and a cash price which was addressed in Sect. 2.1.2, where the relevant definitions were given. Recall that the quoted price or clean price is the price that can be seen on a trading system or listed in an organized exchange. It does not account for the accrued interest. The cash or dirty price is the price that accounts also for the accrued interest and is computed as the sum of the quoted price and the accrued interest. The accrued interest is the part of the coupon that is paid to the seller by the buyer so that he or she receives the portion of the interest that corresponds to the time period prior to the transaction. The relation connecting the two of them is Pcash = Pquoted + AI,

(2.16)

where P cash P quoted AI

denotes the cash bond price denotes the quoted bond price denotes the accrued interest since the last coupon payment.

Example 2.5 Assume that it is March 5, 2019 and we are interested in a bond with a coupon rate of 4% and semi-annual payments. The bonds face value is 100 Euro. The quoted bond price is 95.50 Euro. The accrued interest is calculated on an actual/actual basis. The last coupon was paid on January 5, 2019. This means that 59 days of interest have accrued for the following

2

BONDS

73

coupon, payable on July 5, 2020. This means that cash price = 95.50 +

59 · 2 = 96.15. 181

(2.17)

If the face value per bond had been 1,000 Euro, then the cash price per bond would have been 961.52 Euro. 2.3.4

The Relation Between Price and Face Value

From Eq. (2.11) one can immediately infer that the present value of the bond, which if no other assumption is made equals the bond price, is a decreasing function of the interest rate, as it is essentially a rational function of the interest rate. Moreover, we observe that – If c = r, then PV0 = FV. – If c > r, then PV0 > FV. – If c < r, then PV0 < FV. These are interpreted as follows – If the interest rate is equal to the coupon (rate), then the bond price will be equal to the face value. This may not be difficult to intuitively understand. If the bond gives as a coupon the interest rate of the alternative investment of the same degree of risk, then its price will be equal to its face value, i.e. the amount that the investor lends to the issuer. The former is appropriately compensated by the coupon payments and thus is willing to pay the loan value, i.e. the face value. – If the interest rate is lower than the coupon (rate), then since the price is a decreasing function of the interest rate it will be higher than the face value. This may be intuitively understood by the fact that since the coupon is higher than the performance (interest rate) of the alternative investment of the same degree of risk, investors will want to invest in the bond so its price will be (driven) higher than the face value. – If the interest rate is higher than the coupon, then since the price is a decreasing function of the interest rate it will be lower than the face value. This may be intuitively understood by the fact that since the coupon is lower than the performance of the alternative investment

74

T. POUFINAS

of the same degree of risk, then investors will want to invest in the alternative investment, so its price will be (driven) lower than the nominal value, so that it becomes attractive. When the price of a bond (which if not stated otherwise shall be equal to its present value) is higher than the face value, we say that the bond trades at a premium and it is then called a premium bond. In contrast, when the bond price is less than the face value, then we say that the bond trades at a discount and it is then called a discount bond.

2.4

Bond Yields

But what really matters to the investor is the return, known as yield, of the investment in the bond. There are several ways to measure bond yields. We need measures that account for the current income as well as the increase or decrease in the price of the bond during its lifetime. The main measures are presented below. 2.4.1

Yield to Maturity (YTM)

It measures the total return (coupon plus capital gains or losses) of the investment. In other words, it is the real return for the investor (from the moment of purchase) under the assumptions that: a. the bond will be held until maturity and b. all coupons will be reinvested at the same rate, i.e. YTM. It is practically the internal rate of return (IRR) of the investment in a bond. To calculate the YTM we solve the equation that gives us the value of the bond with respect to YTM. It is practically Eq. (2.11) for coupon-bearing or (2.13) for zero-coupon bonds, though this time we know the purchase price and we are looking for the yield to maturity, usually denoted with y. I.e., if we know the price P 0 , then we can solve for y: P0 =

c2 c1 c N + FV + + ··· + ⇒ y = .... (1 + y) (1 + y)2 (1 + y) N

(2.18)

2

BONDS

75

Equation (2.18) is usually solved with numerical methods when N > 2 for a coupon bearing bond. If N = 2 it can be solved analytically. Of course there is an explicit solution of Eq. (2.13) for a zero-coupon bond: P0 =

FV ⇒y= (1 + y) N



FV P0

1

N

− 1.

(2.19)

Example 2.6 Take for example the first bond of Example 2.3 and assume that its price in the market is 1,054.17 Euro. Then 1, 054.17 =

40 40 + 1,000 40 + + ··· + ⇒ y = 3%. (2.20) (1 + y) (1 + y)2 (1 + y)6

The ϒTM is essentially the discount rate, unless there is a reason that the price of the bond differs from the value we calculated. So if P 0 denotes the price of the bond in the market, and PV0 the present value (theoretical price) of the bond we calculated, then the following hold true: • If P 0 = PV0 , then y = r. • If P 0 > PV0 , then y < r. • If P 0 < PV0 , then y > r. In the previous y indicates the yield to maturity and r the interest rate. We note that the bond price is a decreasing function of the YTM, as it a rational function of the YTM. Its graph is depicted by the curve in Fig. 2.10. When we follow the price of a bond in an electronic trading system, we see two prices, the price we buy (the highest—the ask price) and the price we sell (the lowest—the bid price). Next to them we see two YTMs, one corresponding to the bid price and another corresponding to the ask price. The first is obviously higher than the second (due to the inverse relationship connecting the price with the YTM. The yield to maturity makes 3 implicit assumptions, namely that i. the bond is held to maturity;

76

T. POUFINAS

Price

Yield Fig. 2.10 Relationship between price and bond yield (Source Created by the author)

ii. the interest rate is flat (and equal to the yield to maturity); and iii. the coupons are invested at the yield to maturity (which is the same with the interest rate). We will address (i) later in this section, when we discuss the notion of holding period return. We will revisit (ii) in Chapter 3. We will tackle (iii) in this section with the introduction of realized compound yield. 2.4.2

Effective Yield (EY)

The effective (annual) yield of a bond takes into account the compounding frequency. It measures the actual return for bonds that pay coupon more than once a year. For a bond that makes coupon payments n times a year, this yield is calculated as follows:     n EY = 1 + y n − 1, (2.21)

2

BONDS

77

where y is the yield to maturity and n is the number of times that a coupon is paid within one year. Example 2.7 Consider for example the second bond of Example 2.3 that makes semiannual coupon payments. Assume that its price is 1,054.54 Euro. Then 20 + 1000 20 + ··· + (1 + y/2) (1 + y/2)12 = 1, 054.54 ⇒ y/2 = 1.5% ⇒ y = 3%.

PV0 =

(2.22)

However, one could assume that the interest is earned with semi-annual compounding, thus y(2) = (1 + y/2)2 − 1 = (1.015)2 − 1 = 1.030225 − 1 = 3.0225% ≈ 3.02%,

(2.23)

where y denotes the yield to maturity per annum and y (2) the effective yield per annum with semi-annual compounding. The effective annual yield gives the compounded interest rate. 2.4.3

Current Yield (CY)

Current yield accounts only for the part of the return that is linked to the interest (coupon) payments. No capital gains or losses are considered. Curreny Yield = Annual Coupon/Bond Clean Price.

(2.24)

It should be noted that for bonds traded at a discount, YTM is always higher than the current yield. Similarly, for bonds traded in premium, YTM is always lower than the current yield. Example 2.8 Let us consider the first bond of Example 2.3 and assume that its price in the market is 1,054.17 Euro. Its coupon is 4% paid annually. Consequently, the current yield is given by CY =

40 = 0.0379 = 3.79%. 1, 054.17

(2.25)

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T. POUFINAS

2.4.4

Realized Compound Yield (RCY)

The realized yield is the actual return the investors earn when all coupons are reinvested. The yield to maturity (being the IRR of the investment to the bond) makes the implicit assumption that all coupons are reinvested in the yield to maturity. However, this may not be the case, as when the investors receive the coupons from the issuer the interest rates available for reinvestment in the market may be totally different. As a matter of fact an investor may choose not to reinvest the coupon at all or invest it at a non-fixed income asset. The realized compound yield is given by  RCY =

VN P0

1

N

− 1,

(2.26)

where V N is the final value accumulated from the investment of the bond at maturity and P 0 is the purchase price. V N comprises of the face value and the coupons paid along with the return earned on each coupon for the time it was invested (Terregrossa & Moy, 2011). If the coupon is annual for example, then the coupon paid at time n is reinvested for N–n years. Example 2.9 Let us consider once and again the first bond of Example 2.3 with a price in the market equal to 1,054.17 Euro. The yield to maturity is 3% (as per Example 2.6). Assume initially that coupons are reinvested at 3%. Each coupon is reinvested for N–n years, where N = 6 is the maturity date of the bond and n is the year that the coupon is paid. Then, the value V N that has been accumulated at the maturity of the bond is the sum of the coupons reinvested at 3% and the face value paid at maturity: VN = 40 · (1 + 0.03)5 + 40 · (1 + 0.03)4 + · · · + 1.040 = 1, 258.74. (2.27) As a result the realized compound yield is given by  RCY =

1, 258.74 1, 054.17

1 6

− 1 = 3%,

(2.28)

i.e. it equals the yield to maturity. This is probably anticipated as this is precisely one of the implicit assumptions of the yield to maturity (and as

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a matter of fact of IRR); namely that coupons are reinvested at the yield to maturity. If we now we assume that coupons are reinvested at 2%, then VN = 40 · (1 + 0.02)5 + 40 · (1 + 0.02)4 + · · · + 1.040 = 1, 252.32 (2.29) and  RCY =

1, 252.32 1, 054.17

1 6

− 1 = 2.91%,

(2.30)

which is lower than 3%, something that was probably expected as coupons were reinvested at an interest rate lower than 3%. Interest rates were assumed to be flat, even though different from the yield to maturity. In Chapter 3 we will lift this assumption and we will revisit the reinvestment of coupons at interest rates that are not necessarily fixed over time. 2.4.5

Simple Yield to Maturity (SY)

It is also known as Japanese Yield. The calculation of simple yield takes into account the current yield as well as any capital gains or losses. It is calculated as follows: SY =

c+

FV−P N

P

,

(2.31)

where c is the coupon, FV is the face value of the bond, P is the (clean) price of the bond at the time of the purchase and N counts the remaining life of the bond in years. It is calculated if we divide the total number of days until the maturity of the bond by 365. Some authors define as simple (or approximate) yield the aforementioned fraction with (P + FV)/2 instead of P at the denominator. Example 2.10 Take for example the first bond of Example 2.3 and assume that its price in the market is 1,054.17 Euro. Then SY =

40 + 1,000−1,054.17 6 = 0.0294 = 2.94% 1, 054.17

(2.32)

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T. POUFINAS

If we were to use the alternative definition, then the simple yield would equal 3.02%. 2.4.6

Yield to Call

As mentioned earlier the YTM calculation incorporates the implicit assumption that the bond will be held until its maturity date. However, if the bond is callable, then the issuer may repurchase it before its maturity date from the investor. As said in Sect. 2.1.2 above, these are essentially bonds that have a built-in call option in which the issuer has a long position (purchase position). In these bonds, the issuer has the option to repurchase the bond at a predetermined price, on a predetermined date before the maturity date. It is exercised by the issuer if the interest rates fall and the coupon is much higher. There is usually a protection period during which the call option is not exercised (deferred callable bonds ). In this case, in addition to yield to maturity, yield to call is also of interest to the investor. It is calculated in a way similar to the YTM (i.e. by solving the bond present value equation for the yield) but the time to call replaces the time to maturity and the value at which the call is exercised replaces the face value. Example 2.11 Let us consider a bond that matures in 20 years and pays an annual coupon of 4% with a face value equal to 1,000 Euro. Assume that the option can be called at a (strike) price of 1,100 Euro in 10 years. If the bond price is currently 1,148.77 Euro, then the yield to maturity is 3% and the yield to call is 3.11% (rounded to two decimals) (with the corresponding data depicted in Table 2.2). Table 2.2 Yield to call versus yield to maturity Coupon Number of payments Final payment Price

Yield to call

Yield to maturity

40 10 1,100 1,148.77

40 20 1,000 1,148.77

Source Created by the author

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Most callable bonds are issued with an initial protection period. They cannot be called during this period. Bonds in deep discount (i.e. whose price is significantly below par) are implicitly protected. Even if interest rates fall sharply, their price will be below the call price. Bonds in premium (i.e. whose prices are significantly higher than par) sold near call prices are likely to be called if interest rates fall further. 2.4.7

Yield to Put

Yield to put is symmetric to yield to call for puttable bonds. It is the yield on a bond that is calculated until the (first) date that it is permissible to sell the bond back to the issuer instead of the time to maturity. 2.4.8

Yield to Worst

Yield to worst (YTW) is the worst possible yield that an investor can earn on a bond assuming that the terms of the contract are honored and the issuer does not default. It is usually referenced when a bond has provisions that give the right to the issuer to retire it earlier than its maturity date, most commonly when the bond is callable. It depicts the worst possible yield at the earliest retirement date, as provisioned by the bond. Hence the yield to worst is the lower of the yield to call and yield to maturity.

2.5

Holding Period Return

Investors are not obliged to hold bonds until they mature. They may keep them for a shorter period of time that matches their investment horizon or performance needs and/or expectations. As a result, their return may be different from the yield to maturity when they hold the bond for a time interval shorter than the time to maturity. We need to estimate the return achieved if a bond is sold prior to its maturity. Let us consider a bond that expires in N years, pays an annual coupon c and has a yield to maturity y = r. We notice that its value at time 0 is: P0 =

c c c c + FV + + + ··· . 2 3 1+r (1 + r ) (1 + r ) (1 + r ) N

(2.33)

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T. POUFINAS

Its price at time instant 1 is: P1 =

c c c + FV + + ··· . 2 1+r (1 + r ) (1 + r ) N −1

(2.34)

Its price at time instant 2 is: P2 =

c c + FV . + ··· 1+r (1 + r ) N −2

(2.35)

Its price at time instant k (where k = 0, 1, 2,…, N ) is: Pk =

c + FV c + ··· + . 1+r (1 + r ) N −k

(2.36)

Its performance if the investor keeps it for 1 year is: HPR1 =

P1 + c − P0 , P0

(2.37)

since the investor has also received the coupon paid at the end of year 1. HPR stands for holding period return, which is the return achieved from the investment in the bond for the period it was held. The index indicates the years (periods) that the bond was held. But we notice that P1 + c = c +

c c c + FV + + ··· = P0 · (1 + r ). (2.38) 2 1+r (1 + r ) (1 + r ) N −1

So we get that: HPR1 =

P1 + c − P0 P0 · (1 + r ) − P0 = = r = (1 + r ) − 1. P0 P0

(2.39)

That is, the return of the bond held for 1 year is exactly the yield to maturity of the bond. If the investor holds the bond for two years, then HPR2 =

P2 + c + c · (1 + r ) − P0 , P0

(2.40)

since the investor has also received the coupon paid at the end of the second year (on top of the coupon paid at the end of the first year). The

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first year coupon is reinvested for 1 year. We assume that the reinvestment rate is the fixed interest rate. This is a strong hypothesis, which if removed the result changes. We will revisit it in Chapter 3, where we allow interest rates to change over time. We find that: P2 + c + c · (1 + r ) = c · (1 + r ) + c + = P0 · (1 + r )2 .

c c + FV c + ··· + 2 1+r (1 + r ) (1 + r ) N −1 (2.41)

So we get that: HPR2 =

P2 + c + c · (1 + r ) − P0 P0 · (1 + r )2 − P0 = = (1 + r )2 − 1. P0 P0 (2.42)

That is, the annual return of holding the bond for 2 years is exactly the yield of maturity of the bond, since (1 + r )2 − 1.

(2.43)

is the return on investment of 1 Euro for 2 years with annual compounding and we have assumed that y = r. If the investor holds the bond for k years, then, HPRk =

Pk + c + c · (1 + r ) + · · · c · (1 + r )k−1 − P0 . P0

(2.44)

since the investor has also received the coupon paid at the end of k years. The coupon paid at the end of the first year is reinvested for k − 1 years, the coupon paid at the end of the second year for k − 2 years, the coupon paid at the end of k − 1 year for 1 year, while the last coupon is not reinvested. We consider the reinvestment rate to be the fixed interest rate (used for discounting and is equal to the yield to maturity), which, as we have said, is a strong assumption that we will lift in Chapter 3. We find that: Pk + c + c · (1 + r ) + · · · c · (1 + r )k−1 = c + FV c + ··· c · (1 + r )k−1 + · · · c + = P0 · (1 + r )k . 1+r (1 + r ) N −1

(2.45)

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T. POUFINAS

So we get that: Pk + c + c · (1 + r ) + · · · c · (1 + r )k−1 − P0 P0 P0 · (1 + r )k − P0 = = (1 + r )k − 1. P0

HPRk =

(2.46)

That is, the annual bond return for holding it for k years is exactly the yield to maturity of the bond, since (1 + r )k − 1

(2.47)

is the investment of 1 Euro for k years compounded annually and we have assumed that y = r. Example 2.12 Let us consider for one more time the first bond of Example 2.3 with a price in the market equal to 1,054.17 Euro. The yield to maturity is 3% (as per Example 2.6). An investor purchases the bond today for P 0 and sells it after 3 years for P 3 . Assume that coupons are reinvested at 3%, which is equal to the yield to maturity. Then the holding period return for the 3 year period is HPR3 = (1 + 0.03)3 − 1 = 0.0927 = 9.27% .

(2.48)

We note that this is the return for a 3 year period; its corresponding annualized return is 3\%. If we had applied the explicit formula, then we would have received P3 + c + c · (1 + r ) + c · (1 + r )3−1 − P0 P0 1, 028.29 + 40 + 40 · (1.03) + 40 · (1.03)2 − 1, 054.17 = 1, 054.17 1, 151.92 − 1, 054.17 = 0.0927 = 9.27%, (2.49) = 1, 054.17

HPR3 =

which is identical to the return found with the simplified formula. Although we will address interest rates that are not flat in Chapter 3, let us attempt to derive the formula of holding period return when the

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reinvestment rate is flat but different from the yield to maturity. In such a case the simplified equations do not hold true. Consequently, if r  is the reinvestment rate, then the holding period return for k years becomes: HPRk =

Pk + c + c · (1 + r  ) + · · · c · (1 + r  )k−1 − P0 . P0

(2.50)

However, this expression cannot be simplified further, as it is not necessarily true that the reinvestment rate r  is equal to the yield to maturity or interest rate r at time t = 0. Example 2.13 Let us consider the bond of Example 2.12. The bond is purchased again today and is sold three years from now. Let us assume though that the reinvestment rate this time becomes 2%. The holding period return is given by P3 + c + c · (1 + r  ) + c · (1 + r  )3−1 − P0 P0 1, 028.29 + 40 + 40 · (1.02) + 40 · (1.02)2 − 1, 054.17 = 1, 054.17 1, 150.70 − 1, 054.17 = 0.0916 = 9.16%. (2.51) = 1, 054.17

HPR3 =

The difference from the result of Eq. (2.49) in Example 2.12 is due to the difference of the coupon reinvestment rate of 2% from the bond yield to maturity of 3%. 2.5.1

The Relation Between Price and Present Value—Revisited

We stated earlier that a potential difference between the present value and the price of the bond would lead to arbitrage opportunities. After having defined the yield to maturity we are in position to explain why. Furthermore, we assume that there are no arbitrage opportunities, as if there were, then all investors would try to take advantage of them and thus they would cease to exist. To understand how this would work, assume that the bond price, denoted by P 0 can be different from the present value of the bond, denoted by PV0 . There are two cases; namely that

86

T. POUFINAS

P 0 < PV0 or that P 0 > PV0 . If we show that none of these cases can hold true, then it is only the equality that could hold true. Recall that we are in a fixed interest rate environment, thus interest rates do not change. Without loss of generality we assume that the bond is zero-coupon. Assume for example that P0 < PV0 ;

(2.52)

then y > r due to the reciprocal relation between the bond price/present value and the interest rate. An investor can at • t =0 – Borrow P 0 at an interest rate r until the maturity date of the bond N . – Purchase the bond for P 0 . • t = N (the maturity date of the bond) – Collect FV = P0 ∗ (1 + y) N > P0 ∗ (1 + r ) N . – Repay the loan with interest of P0 ∗ (1 + r ) N . – Post a profit of P0 ∗ (1 + y) N − P0 ∗ (1 + r ) N > 0. This means that there are arbitrage opportunities, which we assumed cannot be the case. Assume now that P0 > PV0 ;

(2.53)

then y < r due to the reciprocal relation between the bond price/present value and the interest rate. An investor can at • t =0 – Short-sell the bond (i.e. borrow the bond and sell it) for P 0 . – Invest the proceeds at an interest rate r until the maturity date of the bond N . • t = N (the maturity date of the bond) – Collect P0 ∗ (1 + r ) N > P0 ∗ (1 + y) N = FV. – Purchase the bond for FV and return it to its owner (practically pay FV to the owner).

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– Post a profit of P0 ∗ (1 + r ) N − P0 ∗ (1 + y) N > 0. This means that there are arbitrage opportunities, which we assumed cannot be the case. As a result, none of these inequalities can hold true and thus P0 = PV0 .

(2.54)

A similar proof can be given when the bond pays a periodic coupon. In that case though the investor reinvests the periodic payments received in the first case and is obliged to pay them to the owner of the bond in the second case.

2.6

Bond Price Movements

Bond prices do not remain constant; they move. For a certain bond, we realize by observing Eq. (2.11) that if its coupon is fixed, then it is the passage of time and a potential move of the interest rates that can lead to a price change. Even if we did not rely on this remark though, we understand that as bonds are traded, their prices move as a result of the change of the preferences of the investors, as well as of their risk appetite. The behavior of the investors determines the demand and supply in the capital markets and this holds true for the fixed income markets too. Example 2.14 Let us consider the bond of Example 2.3, i.e. a bond that matures in 6 years, with a coupon rate of 4% paid annually and a face value per title of 1,000 Euro. If the discount rate is 2% per annum with annual compounding, then the value of the bond today is estimated as 1,112.03 as per Eq. (2.14). Using Eq. (2.11) we can see that if the interest rate becomes 3% i.e. increases, then the price changes to up

40 + 1000 40 + ··· + (1 + 0.03) (1 + 0.03)6 6 40 1, 000 = + = 1, 054.17. n=1 (1 + 0.03)n (1 + 0.03)6

PV0 =

(2.55)

88

T. POUFINAS

If the interest rate moves to 1%, i.e. decreases, then the resulting price is 40 + 1000 40 + ··· + (1 + 0.01) (1 + 0.01)6 6 40 1, 000 = + = 1, 173.86. n=1 (1 + 0.01)n (1 + 0.01)6

PVdown = 0

(2.56)

The percentage change for the interest rate increase is DPVup =

1, 054.17 − 1, 112.03 = −5.20%. 1, 112.03

(2.57)

The percentage change for the interest rate decrease is DPVdown =

1, 173.86 − 1, 112.03 = −5.56%. 1, 112.03

(2.58)

We observe that the interest rate increase and the interest rate decrease do not lead to symmetric value decrease and increase respectively. Looking at Eq. (2.11) we observe that an increase in interest rates means a fall in the price of bonds and vice versa. In other words, bond prices are reciprocal to interest rates. This relationship is very important for the calculation of the YTM or bond prices depending on which of the two is known. It helps managers in the calculations they perform in order to determine the positions they need to take. However, it is very important to note that for a bond, an increase in interest rates results in a fall in the price, which is less than the increase in price that results if the interest rate is reduced by the same percentage. This means that the curve that graphs the relationship between bond prices and interest rates is a convex curve. Therefore, a gradual increase in the interest rate means a gradually smaller reduction in the price of the bond. Consequently, the curve becomes flatter and flatter. These concepts will be better understood after the presentation of the concepts of duration and convexity in Chapter 4 that follows. In that framework, we will also note that the longer the time to maturity of the bond, the more sensitive its price is to changes in interest rates. At this point this may be intuitively understood by the fact that the longer the maturation of a bond, the longer the investor has locked his or her money at a certain interest rate (the coupon). This implies a bigger loss in a possible increase in interest rates.

2

2.7

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89

Government Bonds

A government bond is the bond issued by the central government of a country. A government issues bonds in order to raise funds to pay debt, meet obligations and finance projects, such as infrastructure, etc. The term bond is used to refer to government debt issues with a maturity longer than one year. Maturities shorter than one year are typically considered as money market instruments. A government bond is considered to be of the lowest risk among the debt issues of a country, compared for example to a corporate bond or a local government bond that has been issued in the same country. This is due to the fact that it is backed by the government, which can repay its debt more easily than the companies or the municipalities of the country. A government can achieve that because it can issue money and collect taxes and therefore its creditworthiness is higher. The lower risk though results in lower interest being paid by a government bond compared to interest paid by any other the debt instrument that has been issued in the same country by a company or a local government. Since the latest financial crisis (2008) several governments issue bonds with zero or negative yield to maturity. These bonds are primarily used for safe keeping, wealth preservation and portfolio diversification. However, a potential disadvantage is that they may lead to an overall loss in the purchasing power, if inflation becomes higher than their rate of return. We discuss the particulars of government bonds, the different types of government bonds and the government bond markets in Chapter 7.

2.8

Corporate Bonds

A corporate bond is a bond issued by a company. For a company a bond is one of the ways it has to finance its existing operations, new operations, products or projects, etc. Usually it refers to debt issues with a maturity longer than one year. A corporate bond tends to have a shorter maturity date than the corresponding government bonds. It offers a higher yield—at least when compared to a government bond with similar characteristics issued by the country of its domiciliation. Companies have issued bonds more actively the last ten years due on one hand to the lower interest rate environment

90

T. POUFINAS

that followed the 2008 financial crisis, and on the other hand due to the reduction of the traditional loans offered by banks. The yield to maturity of a corporate bond is often compared with the yield to maturity of the corresponding government bond of the country at which the company domiciles. The difference is known as the credit spread. A corporate bond is perceived to be of higher risk than the corresponding government bond with the same characteristics. The spread is the compensation for this additional risk. A corporate bond is though perceived to be less risky than the stock of the same company. The relevant discussion takes place in Chapter 12. We present the particulars of corporate bonds, the different types of corporate bonds and the corporate bond markets in Chapter 7.

2.9

Other Bond Types

Government bonds and corporate bonds are only a subset the available types of bonds issued. Other popular types of bonds include municipal bonds and mortgage-backed securities. In addition, debt issues of maturity dates less than a year attract the interest of the investors. These securities can also be issued by governments, companies, local governments and other institutions. They are considered as money market instruments. 2.9.1

Municipal Bonds

A municipal bond is a bond issued by a local government, such as region, state, city, etc. A municipal bond is used primarily to finance an infrastructure project, such as the construction of a school, a library, a road, a bridge, a park, etc. It can also be used to provide funding of the police and/or fire department, of a community center, etc. (Corporate Finance Institute, 2021). A municipal bond, also known as muni, may enjoy preferential tax treatment for the interest payments it makes, as is the case for the US for federal, and most of the state and local taxes. The capital gains though that an investor may incur are subject to the applicable taxes. This tax exemption makes a municipal bond quite attractive for investors that are subject to a high tax rate. There can be municipal bonds that are not tax free. As a result of this tax exemption the yield of a municipal bond may be lower than the yield of a comparable bond of the same risk.

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We illustrate the particulars of municipal bonds, the different types of municipal bonds and the municipal bond markets in Chapter 7. 2.9.2

Mortgage-Backed Securities

A mortgage-backed security (or mortgage-backed bond)—MBS is a fixed income (debt) security that is created by a collection/pool/basket of mortgages or is secured by a collection/pool/basket of mortgages (Bodie et al., 1996). It is created when a pool of mortgages on real estate properties is packaged and then sold to investors. This process is called securitization. Such a package originates from the entity that provided the loans, such as a bank. Through a MBS the bank essentially sells its claim to the payments (interest and principal) from the mortgages; such payments take place during the repayment period of the loan. The entity that originated the mortgage continues to service it; however, it passes the interest and capital payments to the holder of the mortgage-backed security (or bond). There is a series of similar securities that can be created; the most well known are asset-backed securities, collateralized mortgage obligations and collateralized debt obligations. They are considered as derivative products. We elaborate on the particulars of these securities in Chapter 5. The relevant markets are described in Chapter 7. 2.9.3

Money Market Instruments

A money market security is a short-term debt issue of high quality whose maturity is usually up to one year or even less. A money market security is considered to be of the highest degree of safety; as a result it offers a relatively low rate of return. A money market security can be issued by the same types of entities that issue a bond, such a government, a company, a bank, a municipality, etc. However, not all entities within a category can or do issue such short-term instruments. The main types of money market securities are treasury bills, certificates of deposit, commercial paper, bankers’ acceptances, Eurodollars and repurchase agreements. The particulars and the characteristics of the money markets and their instruments are presented in Chapter 7.

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T. POUFINAS

Exercises Exercise 1 Let us consider a bond with a face value of EUR100 with a maturity of 2 years and an annual coupon at rate of 6% per annum. The market interest rate (for the same level of risk) is flat and equal to 4% per annum. a. What is the present value (or price) of the bond? b. What is the yield-to-maturity of the bond? c. Compare the present value with the face value. Explain their relation. Exercise 2 Let us consider a bond with a nominal value of EUR100 with a maturity of 2 years and an annual coupon at a rate of 6% per annum. a. If the bond price is EUR104 what is the yield-to-maturity of the bond? b. Compare the yield-to-maturity with the coupon rate. Explain their relation. c. For what interest rate is the price equal to the face value? Exercise 3 Let us consider a bond with a face value of EUR100 with a maturity of 2 years and an annual coupon at a rate of 6% per annum. The market interest rate (for the same level of risk) is flat and equal to 4% per annum. a. An investor holds the bond until maturity and invests the coupons at the market interest rate (which is equal to the yield to maturity). What is the holding period return of the investor? b. An investor holds the bond until maturity and invests the coupons at an interest rate of 2% per annum. What is the holding period return of the investor?

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93

Exercise 4 The higher the discount rate is, the lower the bond price becomes. A. True B. False Explain your answer. Exercise 5 The calculation of the price of a bond with a CCC credit rating can be done by discounting the cash flows it pays at the risk-free rate. A. True B. False Explain your answer. Exercise 6 If the face value of the bond is EUR1,000 and the coupon is higher than the discount rate at some point in time, then the price of the bond is: A. Greater than EUR1,000. B. Less than EUR1,000. C. Equal to EUR1,000. D. We cannot answer this question because we do not know the interest rate and the coupon. Explain you answer. Exercise 7 Consider a zero-coupon bond, with a face value of EUR1,000 that expires in 10 years. The price of the bond today is EUR500. Then, its yield to maturity is:

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T. POUFINAS

A. −50% B. 5% C. 10% D. 7.18% Explain your answer. Exercise 8 A bond that matures in 3 years has a face value of EUR1,000. It makes annual coupon payments at a rate of 3% per annum. The discount rate is 4% per annum with annual compounding. There are no transaction fees or bid-ask spread. The bond trades at EUR980. a. Find the present value of the bond. b. Find the yield to maturity that corresponds to the price at which the bond trades. c. Are there any opportunities to lock into riskless profit (arbitrage)? d. How would you exploit them? e. What will happen to the bond price and the bond yield soon after you (as well as the other investors) take advantage of potential such opportunities? f. What could be the reasons for such a difference between the bond present value and the bond market price? Exercise 9 Let us consider a bond with a face value of EUR1,000 with a maturity of 4 years that makes semi-annual coupon payments at a coupon rate of 6% per annum. The market interest rate (for the same level of risk) is flat and equal to 4% per annum with semiannual compounding. a. What is the present value (or price) of the bond? b. What is the yield-to-maturity of the bond? c. Compare the present value with the face value. Explain their relation.

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95

Exercise 10 Let us consider a bond with a face value of EUR1,000 with a maturity of 4 years that makes semi-annual coupon payments at a coupon rate of 6% per annum. a. If the bond price is EUR104 what is the yield-to-maturity of the bond? b. Compare the yield-to-maturity with the coupon rate. Explain their relation. c. For what interest rate is the price equal to the face value? Exercise 11 Let us consider a bond with a nominal value of EUR1,000 with a maturity of 2 years that makes semi-annual coupon payments at a coupon rate of 6% per annum. The market interest rate (for the same level of risk) is flat and equal to 4% per annum with semiannual compounding. a. An investor holds the bond until maturity and invests the coupons at the market interest rate (which is equal to the yield to maturity). What is the holding period return of the investor? b. An investor holds the bond until maturity and invests the coupons at an interest rate of 2% per annum with semiannual compounding. What is the holding period return of the investor? Exercise 12 Assume that it is February 22, 2021 and we are interested in a bond with a coupon rate of 6% and semi-annual payments. The face value of the bonds is 100 Euro. The quoted bond price is 97.00 Euro. The accrued interest is calculated on an actual/actual basis. The last coupon was paid in January 12, 2021. What is the cash bond price? Exercise 13 Investigate the credit ratings that have been announced over the years by the credit rating agencies. What do you observe? Have the relative frequencies increased or decreased over the years? Is there any link

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between the increase and decrease of these frequencies and potential external/global and not idiosyncratic events? Exercise 14 A zero-coupon bond has a face value of EUR1,000 and matures in 10 years. The discount rate is 5% per annum with annual compounding. a. What is the present value of the bond? b. If the bond price today is EUR500, then what is the yield to maturity of the bond? Exercise 15 Repeat exercise 14 with a. Semiannual compounding. b. Continuous compounding.

References Altman, E. I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The Journal of Finance, 23(4), 589–609. Bodie, Z., Kane, A., & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. Corporate Finance Institute. (2021). Municipal Bond. https://corporatefinanc einstitute.com/resources/knowledge/trading-investing/municipal-bond/. Accessed: January 2021. Fitch Ratings. (2020a, April 20). Further sovereign downgrades to sub-investment grade probable in 2020. Fitch Wire. https://www.fitchratings.com/research/ sovereigns/further-sovereign-downgrades-to-sub-investment-grade-probablein-2020-20-04-2020. Accessed: June 2020. Fitch Ratings. (2020b, June 11). Rating definitions. Special Report. https:// www.fitchratings.com/research/fund-asset-managers/rating-definitions-1106-2020. Accessed: January 2021. International Monetary Fund. (2020, April). IMF World Economic Outlook: The great lockdown. Washington, DC. https://www.imf.org/en/Publications/ WEO/Issues/2020/04/14/weo-april-2020. Accessed: June 2020.

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Moody’s. (2011, February 28). Corporate default and recovery rates, 1920–2010. Global Corporate Finance. Moody’s Investor Service. http://efinance.org. cn/cn/FEben/Corporate%20Default%20and%20Recovery%20Rates,19202010.pdf. Accessed: January 2021. Moody’s. (2021, January 26). Rating symbols and definitions. Moody’s Investor Service. https://www.moodys.com/researchdocumentcontentpage. aspx?docid=PBC_79004. Accessed: January 2021. Qu, Y., Quan, P., Lei, M., & Shi, Y. (2019). Review of bankruptcy prediction using machine learning and deep learning techniques. Procedia Computer Science, 162, 895–899. S&P Global. (2019). Default, transition and recovery: 2018 annual global corporate default and rating transitions study. Ratings Direct. www.spratings.com. Accessed: 04 April 2020. S&P Global. (2021, January 5). S&P global ratings definitions. https://www.sta ndardandpoors.com/en_US/web/guest/article/-/view/sourceId/504352. Accessed: January 2021. Terregrossa, R., & Moy, R. L. (2011). Computing realized compound yield with a financial calculator: A note. American Journal of Business Education (AJBE), 4(6), 11–14.

CHAPTER 3

Term Structure of Interest Rates

Until now, we have considered interest rates to be constant/fixed for all time periods or time instants. Nevertheless, we know that in practice this is not the case. The simplest example is a term/time deposit in a bank, where the higher the time until maturity, the higher the interest rate (usually). The same is true for the yield to maturity of bonds. It is not the same for all maturity dates, even if the issuer is the same, such as the government of a country. It is thus evident, from these simple realizations that interest rates most likely differ with the time horizon of the investment/investor. However, interest rates change also with the passage of time. For example, the interest rate of an one-month term deposit today is different from the one-month term deposit rate one year ago (last year) and will be most likely different from the corresponding interest rate one year from now (next year). Which raises one obvious question; how can we tell anything about the interest rates that will apply in the future? In our example of the onemonth term deposit, we know what the current interest rate is and what the interest rate one year ago was. What will the one-month term deposit be next year? It seems that future interest rates are unknown; as a matter of fact not only they are unknown but they are uncertain. As such they are better described as random variables.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_3

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We can thus infer that interest rates depend on time, as well as other parameters that influence their level. They depend on the time instant that the investment commences and on the time instant that the investments matures. As a result, interest rates commencing at any point of time are functions of the time to maturity. As opposed to bonds (or stocks or any other asset) for which at any point of time we have only one single value (or price) assigned, interest rates at any point of time constitute a function, whose value depends on the maturity of the investment. The term that is used to describe this behavior of interest rates is interest rate term structure. Consequently, a series of questions arises; how can we know the interest rates, depending on how long the investment lasts? What is it that affects their level and how can we determine/find them? What can we tell about future interest rates? In other words how can we capture the determinants of the interest rates and how can we potentially model their evolution/movements? In this chapter we analyze the behavior of interest rates, which are far from being constant, as is often assumed—at least for educational purposes. Interest rates are a source of uncertainty and any attempt to forecast them can be notoriously bad. We however deploy the theories of the interest rate term structure, explain its behavior and examine a series of interest rate models. This chapter assists the reader in comprehending the way interest rates move.

3.1

Definition

The interest rate term structure or the term structure of interest rates is (a function that depicts) the relationship between interest rates and different terms, i.e. time instants or maturities. It is usually known by its graph which elaborates this relationship. As became apparent in Chapter 2, interest rates and bond yields (to maturity) are interrelated. As a matter of fact, when interest rates were constant, they coincided with the yields to maturity of the bonds under discussion, provided that their prices where equal to their (present) values. It is therefore a natural question to ask what is the overall relation of the term structure and bond yields and how can one lead to the other. An additional inquiry is how the term structure can be used to price bonds and other fixed income securities or instruments.

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To elaborate the relationship of interest rates with time we present a hypothetical but real life example. The interest rates are a bit on the highend for the current period of time (January, 2021); however using lower (or even negative) interest rates would (probably) create unnecessary difficulty in understanding the relevant notions. Having said that one should not ignore that interest rates are (and have been) at very low levels (or zero/negative) as we will later illustrate, and according to the Fed (CBS, 2021) and the ECB (ECB, 2021) this will be maintained for as long as the pandemic tantalizes the economies. This stance is confirmed even if some inflationary trends are evident down the road. Example 3.1 Assume that an individual investor walks into a bank branch (or calls) to discuss a term deposit for an amount of 100,000 Euro. Prepared for a series of alternatives the individual asks the personal banking officer what his or her alternatives are: Alternative 1 Individual investor: If I deposited today 100,000 Euro for 1 month, would you be able to offer me an interest rate? Personal banking officer: Of course (as this is practically what a bank does)! The interest rate for a term deposit of 100,000 Euro maturing in 1 month would be 1% per annum (looking at a list of interest rates for term deposits with different maturity dates). Explanation: This is exactly what a bank does for a term deposit. The personal banking officer has already available (most likely via the dealing room or private banking department of the bank) a list of interest rates per maturity date, usually for 1, 2, 3, 6, 12 or even 18 months. He or she practically has the term structure of interest rates for the short-term time intervals. Alternative 2 Individual investor: If I deposited today 100,000 Euro for 2 months, would you be able to offer me an interest rate? Personal banking officer: Naturally (again, as this is practically what a bank does)! The interest rate for a term deposit of 100,000 Euro maturing in 2 months would be 2% per annum. Explanation: This is the same as in Alternative 1. The only difference is that the term deposit matures in 2 months. Alternative 3

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Individual investor: Assume now that we sign an agreement that I will deposit 100,000 Euro in 1 month, for 1 month, i.e. maturing in two months; then would you be able to offer me an interest rate today? Personal banking officer: This is a bit more complicated. Let me call our dealing room… I am glad to inform you (after exchanging the relevant information with the dealing room staff) that in that case the interest rate for a term deposit of 100,000 Euro commencing 1 month and maturing in 2 months would be 3% per annum. This is conditional on signing the agreement now. Explanation: This is a bit trickier than Alternatives 1 and 2. One may wonder how the bank can come up today with an interest rate that commences 1 month from today and matures 2 months from today. The answer lies in one very important detail; namely that the agreement is signed today. As will become evident in this chapter these rates are derived with a concrete mathematical formula from the rates of Alternatives 1 and 2. Hence, there is no guessing or forecasting. Alternative 4 Individual investor: Assume now that I come in 1 month from today with 100,000 Euro for a term deposit starting at that time and maturing one month later, i.e. 2 months from today. We sign no agreement today. Are you able to indicate what the interest rate will be? Personal banking officer: I am afraid that as long as there is no agreement, we will have to look at the 1-month interest rates for your term deposit at that time. Explanation: This is far more complicated that Alternative 3 and as a matter of fact quite different from Alternative 3. There is no commitment today. As a result the interest rate sought is the future 1-month term deposit rate for a term deposit that starts in 1 month. This is not the same as Alternative 3. The interest rates 1 month from now are totally unknown; and as mentioned earlier, they are uncertain, i.e. random variables. As such any potential estimate of these future interest rates entails interest rate forecasting, as it practically replaces a random variable with a number! Following this example, we realize that interest rates are not constant; in some cases they are not only unknown but even uncertain. They depend on the time instant t 0 on which the interest rate is observed or agreed, the time instant t 1 on which the (investment underlying the)

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interest rate starts, the time instant t 2 on which the (investment underlying the) interest rate ends and a variable ω that reflects uncertainty. We thus infer that interest rates are functions of the form r : [0, +∞) × [0, +∞) × [0, +∞) ×  → ,

r = r (t0 , t1 , t2 , ω), (3.1)

where  is the set of events (sample space) out of which ω takes its values and  is the set of real numbers. Linking this with the Alternatives of Example 3.1 we realize that essentially Alternatives 1 and 2 have as a common characteristic that t 1 = t 0 = 0, i.e. the time instant on which the interest rate is agreed (or observed) coincides with the time instant it commences, which in both cases is the present time. Moreover, these interest rates are known with certainty; hence do not depend on ω. As a result, both these interest rates are given by a simplification of Eq. (3.1) r = r (0, 0, t2 ) = r (t2 ) =: st2

(3.2)

and depend only on their maturity date. In Alternative 1 t 2 = 1/12 and in Alternative 2, t 2 = 2/12. These interest rates are called spot rates . We denote them with s T , (s for spot), where T is the maturity of the (underlying investment of the) interest rate. The current spot rates are rates that are agreed (observed) today, start today, and mature on a specific, known future date. Going now to Alternative 3, we see that t 0 = 0 but t 1 = 1/12 and t 2 = 2/12. The difference is that the time that the interest rate is agreed is different from the time instant at which it starts. This interest rate is also known with certainty (although it is possibly not obvious how it was derived) and therefore there is no dependence on ω. As a result, the interest rate is given by r = r (0, t1 , t2 ) = r (t1 , t2 ) =: f t1 , t2

(3.3)

and depends on the starting and ending dates of the (underlying investment of the) interest rate. In Alternative 3 t 1 = 1/12 and t 2 = 2/12. These interest rates are called forward rates . We denote them with f T1 , T2 (f for forward), where T 1 is the starting and T 2 is the ending date

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of the (underlying investment of the) interest rate. The current forward rates are rates that are agreed (observed) today, start on a specific, known future date and mature on a specific, known posterior date. When the forward rates cover one single period of time (one month in our example) they are called short rates . As we will realize later, short rates are key in constructing or even modeling the term structure. Turning to Alternative 4 we realize that t 0 = 1/12, t 1 = 1/12 and t 2 = 2/12. This means that the interest rate is observed on the same day as its starting date; however, this is a date in the future. Such an interest rate cannot be known with certainty and depends on ω. The interest rate is expressed by the general form of Eq. (3.1) r = r (t1 , t1 , t2 , ω) =: st1 , t2 (ω)

(3.4)

and depends on the future agreement, starting and ending dates. In Alternative 4, t 0 = t 1 = 1/12 and t 2 = 2/12. These interest rates are called future spot rates . We denote them with sT1 ,T2 (ω) or st,T (ω) (again s for spot) where T 1 (or t ) is the agreement and starting date and T 2 (or T ) is the ending date of the (underlying investment of the) interest rate. There is clear dependence on ω as these interest rates are not known with certainty, which makes them random variables. The future spot rates are rates that are agreed (observed) and start on a specific, known future date and mature on a specific, known posterior date. It is natural to wonder what the relation between the forward rate and the future spot rate of the same time interval is, i.e. how do f T1 , T2 and sT1 , T2 compare. The comparison is not immediate as the former is a (known) number, whereas the latter is an (unknown) random variable. It would make sense though to calculate the expected (mean) value of the random variable, i.e. the expected value of the future spot rate eT1 , T2 := E(sT1 , T2 ).

(3.5)

This is called the expected future spot rate. The notation eT1 ,T2 or et,T is similar to the one we used before; it denotes the expected future spot rate (e for expected) that is agreed and starts on T 1 (or t ) and matures on T 2 (or T ). The expected future spot rate is a number and can therefore be compared to the forward rate that covers the same interval. We will investigate the relationship between the forward rate and the expected future spot rate when we present the theories of the term structure.

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If t (T 1 ) is fixed and T (T 2 ) is allowed to vary, then (st,T )T ∈[0,+∞) is a random process st,T : [0, +∞) ×  → .

(3.6)

We will model interest rates later in this chapter. However, it is worth observing that when t = 0, then these are the current spot rates, which are known with certainty. Recall that we started this section by defining the term structure of interest rates as the (function that depicts) the relationship between interest rates and different terms, i.e. time instants or maturities. When we are referring to the present time, then it is practically the maturity date that varies. Consequently, the term structure of interest rates practically refers to the spot rates. It is usually the graph of this relationship/function that comes to the minds of people when they thing of the term structure. As a matter of fact this is nothing but the graphical representation of spot rates as a function of the time (to maturity), i.e. s : [0, +∞) → .

(3.7)

The term structure of interest rates is graphed on a chart that has time to maturity on the horizontal axis and the spot interest rates on the vertical axis. If an investor is interested in finding the spot rate for a specific maturity date, then he or she has to look for the interest rate that corresponds to this date on the graph. An example is given in Fig. 3.1 for the US Treasury yield curve (as of January 2021). There are two natural questions; (i) what is the relation between spot and forward rates; and (ii) how is the term structure constructed?

3.2

Relation Between Spot and Forward Rates

Looking at Example 3.1 and in particular Alternative 3, one may inquire how could the bank offer an interest rate for a term deposit that starts on a future date? As explained earlier, the answer lies in the signing of the agreement today. As we will show, the forward interest rates are derived from the spot rates.

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Fig. 3.1 Term structure of interest rates (Source Created by the author with data assembled from Statista [2021])

3.2.1

Annual Compounding

To illustrate, let us assume without loss of generality that the time instants are whole years, i.e. 1, 2, 3…N (for N a natural number) and that compounding is annual (Luenberger [1998] or Bodie et al. [1996]). This restriction will be lifted later on; any compounding frequency, including annual compounding will be considered. For starters, let N = 2. Following the rationale of Example 3.1, let T 1 = 1 and T 2 = 2. An investor that has a time horizon of 2 years has two alternatives with certain outcome available; either invest his or her capital for 2 years directly at the 2-year spot rate s 2 or invest for 1 year at the 1-year spot rate s 1 and then reinvest the amount available at the end of the first year at the 1-year forward rate f 1, 2 (between years 1 and 2). The investor has a third alternative, which however has an uncertain outcome; invest for 1 year at the 1-year spot rate and reinvest at the end of the first year for one more year at the 1-year spot rate that will be available at that time. That spot rate is at the present time a random variable; hence the investor is exposed to reinvestment risk.

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0

Fig. 3.2 Two investment alternatives for a two year horizon: one 2-year period versus two 1-year rolling periods (Source Created by the author)

1

2

s1 1

(1+s2)2

s2

1

107

f1,2 1+s1

(1+s1)(1+f1,2)

The first alternative will yield (for each monetary unit invested): 1 → (1 + s2 )2 .

(3.8)

The second alternative will deliver (for each monetary unit invested): 1 → (1 + s1 ) · (1 + f 1, 2 ).

(3.9)

The alternatives with certain outcome are depicted in Fig. 3.2: These two outcomes must be equal; otherwise there would be arbitrage opportunities. Therefore, (1 + s2 )2 = (1 + s1 ) · (1 + f 1, 2 ) ⇒ (1 + f 1, 2 ) =

(1 + s2 )2 (1 + s2 )2 ⇒ f 1, 2 = − 1. (1 + s1 ) (1 + s1 )

(3.10)

We can thus infer that forward rates can be indeed derived from the spot rates. This explains how the bank could come up with an interest rate in Alternative 3; knowing the spot rates for 1 month and 2 months the forward rate between months 1 and 2 can be calculated from the spot rates. We will see how this calculation takes place when we generalize the compounding frequency. Let us explain why there would be arbitrage opportunities if the outcomes of the investments of expressions (3.8) and (3.9) were not equal. Assume that (1 + s2 )2 < (1 + s1 ) · (1 + f 1, 2 ). An investor can perform arbitrage as follows: • At time t = 0

(3.11)

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– Borrow 1 monetary unit for 2 years at s 2 . – Invest it for 1 year at the spot rate s 1 and agree to reinvest the outcome at the end of year 1 for one more year at the forward rate f 1, 2 . • At time t = 2 – The outcome of the investment is (1 + s1 ) · (1 + f 1, 2 ). – The investor owes (1 + s2 )2 . – As the former is bigger than the latter (if inequality (3.11) held true), the investor has a profit of (1 + s1 ) · (1 + f 1, 2 ) − (1 + s2 )2 > 0 without risk and without any initial disbursement, i.e. has performed arbitrage. Consequently, inequality (3.11) cannot hold true. We can reject the reverse inequality in a similar way. Assume that (1 + s2 )2 > (1 + s1 ) · (1 + f 1, 2 ).

(3.12)

An investor can exploit arbitrage opportunities as follows: • At time t = 0 – Borrow 1 monetary unit for 1 year at s 1 and agree to refinance his loan (with the accumulated interest) for one more year at the end of year one at the forward rate f 1, 2 . – Invest this 1 monetary unit for 2 years at s 2 . • At time t = 2 – The outcome of the investment is (1 + s2 )2 . – The investor owes (1 + s1 ) · (1 + f 1, 2 ). – As the former is bigger than the latter (if inequality (3.12) held true), the investor has a profit of (1 + s2 )2 − (1 + s1 ) · (1 + f 1, 2 ) > 0 without risk and without any initial disbursement, i.e. has performed arbitrage. As a result, inequality (3.12) cannot hold true. Having rejected inequalities (3.11) and (3.12) we conclude that the first equation of (3.10) must be true and thus forward rates are indeed determined by spot rates according to the last equality of (3.10) We will use this argument in the

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following so as to establish the remaining relationships between spot and forward rates. Even though arbitrage opportunities may be available momentarily, investors will immediately try to take advantage of them and thus equilibrium will recur. In our discussion, as is standard in these elaborations, we assumed there are no costs associated with these transactions and that investors can borrow and lend money at the same rate; in practice this is not the case as transaction costs, fees or even taxes may apply and borrowing and lending rates are not necessarily equal. Let us now continue our illustration by adding one more year. We consider an investor that has a 3-year horizon. The investor now has 3 alternatives with certain outcome; invest his or her capital for 3 years at the 3-year spot rate s 3 or invest it for 2 years at the 2-year spot rate s 2 and agree to reinvest the outcome at the 1-year forward rate f 2, 3 between years 2 and 3 or invest it for 1 year at the 1-year spot rate s 1 and agree to reinvest the outcome at the 2-year forward rate f 1, 3 between years 1 and 3. Of course, he or she can opt to invest for 1 year at the 1-year spot rate s1 and agree to reinvest the outcome at the 1-year forward rate f 1, 2 between years 1 and 2 and further reinvest the outcome at the 1-year forward rate f 2, 3 between years 2 and 3. This is equivalent to investing his or her capital for 2 years at the 2-year spot rate s 2 and agreeing to reinvest the outcome at the 1-year forward rate f 2, 3 between years 2 and 3 because of Eq. (3.10) that was previously established. Furthermore, there are more alternative combinations that are uncertain. If the investor does not wish to “lock” a reinvestment rate at the end of year 2 or year 1 respectively, then he or she can wait and reinvest at the spot rate that will be available for 1 year (between years 2 and 3) or for 2 years (between years 1 and 3) or even 2 consecutive years (between year 1 and 2 and between year 2 and 3). However in all these cases he or she is exposed to the reinvestment risk as the reinvestment rate is uncertain, being a random variable. These cases will be examined later when we discuss the theories of the term structure. The first alternative will deliver (for each monetary unit invested): 1 → (1 + s3 )3 .

(3.13)

The second alternative will deliver (for each monetary unit invested): 1 → (1 + s2 )2 · (1 + f 2, 3 ).

(3.14)

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The third alternative in a similar manner will give (for each monetary unit invested) 1 → (1 + s1 ) · (1 + f 1, 3 )2 .

(3.15)

These alternatives are illustrated in Fig. 3.3: All these investment alternatives must produce the same result as otherwise there would be arbitrage opportunities and as explained earlier we assume that such opportunities are not available; as even if there were, investors would immediately take advantage of them and would thus vanish. Equating the first and the second outcomes, we see that (1 + s3 )3 = (1 + s2 )2 · (1 + f 2, 3 ) ⇒ (1 + f 2, 3 ) =

(1 + s3 )3 (1 + s3 )3 ⇒ f = − 1. 2, 3 (1 + s2 )2 (1 + s2 )2

(3.16)

This resembles a lot to (3.10) as it again generates a one-year forward rate; this time between years 2 and 3. Equating the first and the third, we observe that (1 + s3 )3 = (1 + s1 ) · (1 + f 1, 3 )2 ⇒ 1  (1 + s3 )3 (1 + s3 )3 2 2 ⇒ f 1, 3 = (1 + f 1, 3 ) = −1 (1 + s1 ) (1 + s1 )

0

1

2 s2

3 f2,3

(1+s2)

1

(3.17)

2

(1+s2)2(1+f2,3)

s3 (1+s3)3

1 s1 1

f1,3 1+s1

(1+s1)(1+f1,3)2

Fig. 3.3 Three investment alternatives for a three year horizon: one 2-year & one 1-year periods versus one 3-year period versus one 1-year & one 2-year periods (Source Created by the author)

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which is similar to the previous equations that deliver the forward rates, but also differs in the sense that it gives a forward rate for two years instead of one. Evidently, the aforementioned equations can be combined in all possible ways to deliver additional relations between spot and forward rates. We can therefore see that combining (3.10) and (3.16) that (1 + s3 )3 = (1 + s2 )2 · (1 + f 2, 3 ) = (1 + s1 ) · (1 + f 1, 2 ) · (1 + f 2, 3 ). (3.18) This is one relation that will prove to be quite useful in deriving interest rates or pricing bonds as it links the spot rates with only annual forward rates, i.e. the (annual) short rates, which we will discuss later in this section. We are now ready to generalize. If an investor has an investment horizon of n years, then he or she has n alternatives similar to the ones we described earlier for 2-year and 3-year investment horizons. For each intermediate year m (m = 1, 2…n) he or she can invest at the m-year spot rate s m and agree to reinvest the outcome at the (n−m)-year forward rate f m, n . Of course, when m = n the interest rate is the n-year spot rate and there is no reinvestment. All investment alternatives must yield the same result, as otherwise there would be arbitrage opportunities. Consequently, for any m the following equation must hold true (see also Luenberger, 1998): (1 + sn )n = (1 + sm )m · (1 + f m, n )(n−m) ⇒ (1 + sn )n (1 + f m, n )(n−m) = ⇒ (1 + sm )m  1  (1 + sn )n (n−m) f m, n = −1 (1 + sm )m

(3.19)

for all forward rates for years m = 1, 2…n. In the previous demonstration, we could have allowed m to take n−1 as its maximum value; however in this case we would need to separately consider the n-year direct investment alternative. When m = n (3.19) stops trivially at the first equation which becomes a tautology. Some authors (see for example Luenberger, 1998) allow for m to equal 0. When m = 0, then the forward rates at the initial time are nothing but the spot

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rates, i.e. f 0, n = s n . Sometimes this notation is used in order to have a uniform notation for all rates, spot and forward, with the understanding that forward rates at the initial time instant are identical to the spot rates. One remark is that all forward rates until any given value of m can be determined from the spot rates up to the time instant m; hence their values do not change when considering longer time horizons. Furthermore, generalizing Eq. (3.18) we derive that (1 + sn )n = (1 + sn−1 )n−1 · (1 + f n−1, n ) = (1 + s1 ) · (1 + f 1, 2 ) · (1 + f 2, 3 ) · · · (1 + f n−1, n )

(3.20)

which expresses spot rates as a function of the annual forward rates, i.e. the (annual) short rates. It shows that the investment outcome for a period of n years at the n-year spot rate is the same with the outcome achieved if the investment is rolled over annually at the agreed annual forward (short) rate; i.e. each year for n years with n−1 rollovers on top of the initial investment, from year 1 to year 2, from year 2 to year 3, and finally from year n−1 to year n. Short rates are denoted with r (following Luenberger [1998] or Hull [1997]). One may wonder why we revert to our very first symbolism of interest rates (recall that we used r to denote interest rates in the previous chapters) for short rates. The reason will become apparent when we realize that when we want to develop models of the term structure, i.e. models that capture the movement of interest rates over time, it is the short rate that is being modeled. We will therefore use the notation (see for example Luenberger, 1998) rm−1 := f m−1, m

(3.21)

to denote the short rate that is applied to the 1-year period that starts at time m−1 and ends at time m, for m = 1, 2…n−1. Equation (3.20) becomes (1 + sn )n = (1 + r0 ) · (1 + r1 ) · (1 + r2 ) · · · (1 + rn−1 ).

(3.22)

When m = 1, then the short rate coincides with the first year spot rate as r0 = f 0, 1 = s1 .

(3.23)

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Table 3.1 Spot, forward and short rates Forward rates Spot rates r0 = f 0, 1 = s1 r1 = f 1, 2 r2 = f 2, 3 . . . rn−2 = f n−2, n−1 rn−1 = f n−1, n

Short rates

f 0, 2 = s2

f 0, 3 = s3

···

f 1, 3 f 2, 4 . . . f n−2, n

f 1, 4 f 2, 5

··· ···

f 0, n−2 = sn−2 f 1, n−1 f 2, n

f 0, n−1 = sn−1 f 1, n

f 0, n = sn

Source Created by the author

We can now put all interest rates in a tabular form, hoping that it will help grasping the relationship among spot, forward and short interest rates pictorially (see also Luenberger, 1998) (Table 3.1). forward (with spot and We can see that for n years there are n·(n+1) 2 short) rates. Example 3.2 To elaborate, let us assume that our time span is 3 years and that compounding is annual. There are therefore 3 spot rates, s 1 , s 2 and s 3 ; 3 forward rates, f 1, 2 , f 1, 3 and f 2, 3 ; and 3 short rates r 0 , r 1 and r 2 . Assume that the spot rates are 1%, 2% and 3%. Then the forward rates are given by f 1, 2 = f 2, 3 = 

f 1, 3

1.022 (1 + s2 )2 −1= − 1 = 0.0301 = 3.01%, (1 + s1 ) 1.01

(3.24)

(1 + s3 )3 (1.03)3 − 1 = − 1 = 0.0503 = 5.03%, (1 + s2 )2 (1.02)2

(3.25)

(1 + s3 )3 = (1 + s1 )

 21



1.033 −1= 1.01

 21

− 1 = 0.0401 = 4.01%.

They can be presented also in a tabular form (Table 3.2).

(3.26)

114

T. POUFINAS

Table 3.2 Spot, forward and short rates for a 3-year time interval

Forward rates Spot rates Short rates

1.00% 3.01% 5.03%

2.00% 4.01%

3.00%

Source Created by the author

0

1

2 2.00% 2 (1.02)

1

3 5.03% 2

(1.02) (1.0503)

3.00% (1.03)3

1 1.00% 1 1

4.01%

(1.01)(1.0401)2 1.01 1.00% 3.01% 5.03% 1.01 (1.01)(1.0301) (1.01)(1.0301)(1.0503)

Fig. 3.4 Four investment alternatives for a three year horizon: one 2-year & one 1-year periods versus one 3-year period versus one 1-year & one 2-year periods versus three one-year rolling periods (Source Created by the author)

They are also illustrated in Fig. 3.4. 3.2.2

Multi-Period Compounding Per Year

Assume now that the compounding frequency is more than once per year. Often enough it can be twice (2 times) a year, called semi-annual compounding, 4 times a year, called quarterly compounding, 12 times a year, called monthly compounding. Although not very common, it could be 365 times a year, called daily compounding. In annual compounding the compounding period is equal to 1; in semi-annual compounding the compounding period is equal to 1/2; in quarterly compounding the compounding period is equal to 1/4; in monthly compounding the compounding period is equal to 1/12; in daily compounding the compounding period is equal to 1/365.

3

TERM STRUCTURE OF INTEREST RATES

115

Let us denote in general the compounding frequency with the Greek letter ν and the compounding period with the Greek letter π. This means that π = 1/ν and in this case there are ν compounding periods per year. The term structure, spot, forward and short rates can thus be defined when the compounding period (or frequency) is different from 1 per year. All illustrations, proofs and notations change in order to reflect the ν-period compounding per year and period replaces year. Therefore, spot, forward and spot rates cover integral multiples of the compounding period π = 1/ν. All time instants considered need not be years; they can be time instants corresponding to such multiples of the compounding period. To avoid potential misunderstanding we will denote the compounding frequency with a superscript (ν) at the spot, forward and short rates. The corre(ν) (ν) sponding notation will thus be sT for the spot rate, f T1 , T2 for the

forward rate and r T(ν) for the short rate; in all cases T , T 1 and T 2 are integral multiples of π = 1/ν. Formulas (3.8) to (3.23) hold true with indices showing periods instead of years. One point of attention though is that the interest rates have to be divided by the compounding frequency (ν). This is necessary as interest rates are always denoted (and quoted) as percentages per annum (yearly) no matter what the compounding frequency is. To illustrate, we realize that Eq. (3.19) becomes ⎧ ⎫  1 ⎨ (1 + s (ν) /ν)n (n−m) ⎬ n (ν) − 1 , (3.27) f m, n =ν· ⎩ (1 + sm(ν) /ν)m ⎭ where m, n are integral multiples of the compounding frequency ν. We observe that ν multiplies the quantity that was previously expressing the forward rate f m, n as the latter is also compounded ν times a year. Example 3.3 To illustrate, let us assume that ν = 2 or π = 1/2, i.e. that compounding is semi-annual. If we consider a time span of 2 years, i.e. 4 semesters, then there are 4 spot rates applicable in that period, depending on the (2) (2) (2) (2) , s2/2 , s3/2 and s4/2 . If there is time horizon of the investor; namely s1/2 no confusion and it is clear that the compounding is semiannual, then the denominator may be dropped, understanding that the spot rates start at

116

T. POUFINAS

Table 3.3 Spot, forward and short rates with semi-annual compounding Forward rates Spot rates Short rates

(2)

(2)

(2)

(2)

(2)

= f 0, 1/2 = s1/2 (2) (2) r1 = f 1/2, 2/2

f 0, 2/2 = s2/2 (2) f 1/2, 3/2

r2

f 2/2, 4/2

r0

(2)

(2) = f 2/2, 3/2 (2) (2) r3 = f 3/2, 4/2

(2)

(2)

(2)

f 0, 3/2 = s3/2 (2) f 1/2, 4/2

(2)

(2)

f 0, 4/2 = s4/2

Source Created by the author

the present time instant and end in 1, 2, 3 and 4 semesters respectively. The forward and short rate notations are derived in a similar manner and are mapped in Table 3.3. Again, the denominator 2 can be dropped when it is clear that compounding is semiannual and there is no fear of confusion. Example 3.4 Revisiting Example 3.1 and assuming that compounding is monthly, we are in place to explain how the bank could come up with the forward rate between the first and the second month. We drop the denominator, i.e. 12, from the subscript as it is evident that the compounding is monthly, in order to familiarize ourselves with this notation as well; the indices become 1 and 2 instead of 1/12 and 2/12. The forward rate is given by the equation     (12) (1 + s2 /12)2 1.0016672 (12) − 1 = 0.0300 = 3%. − 1 = 12 · f 1, 2 = 12 · (12) 1.000833 (1 + s1 /12) (3.28) Consequently, the bank does not have to guess in order to offer a rate for a term deposit of 100,000 Euro that starts in 1 month, ends in 2 months (i.e. one month later) and is agreed today. As explained, the key feature of this deposit is that it is agreed today; the interest rate that applies is nothing but the 1-month forward rate starting 1 month from today and ending 2 months from today.

3

3.2.3

TERM STRUCTURE OF INTEREST RATES

117

Continuous Compounding

Finally, we look at continuous compounding, which occurs when each time instant is a compounding instant. The outcome of such an investment is the limit of the multi-period compounding when the frequency ν converges to infinity (ν → ∞). Indeed if the spot interest rate is fixed for a certain maturity date and only the compounding frequency increases, then lim (1 + sT /v)ν·T = esT ·T according to the definition of e. ν→∞

The time instants under consideration can be any positive real number and Eqs. (3.19) and (3.27) change to esn ·n = esm ·m · e fm, n ·(n−m) ⇒ esn ·n = esm ·m+ f m, n ·(n−m) ⇒ sn · n = sm · m + f m, n · (n − m) ⇒ s n · n − sm · m f m, n = n−m

(3.29)

for all m, n positive real numbers and 0 < m < n. When n = m, (3.29) stops at the first equation, as it is a tautology. The expression of the forward rate receives its simplest possible form with continuous compounding. As it can be used for any time instant it is extremely useful when modeling the term structure and when pricing derivatives. Example 3.4 To elaborate, let us repeat Example 3.2 with compounding changing to continuous instead of annual. There are therefore 3 spot rates, s 1 , s 2 and s 3 ; 3 forward rates, f 1, 2 , f 1, 3 and f 2, 3 ; and 3 short rates r 0 , r 1 and r 2 . The spot rates are once and again 1%, 2% and 3%. Then the forward rates are given by f 1, 2 =

2% · 2 − 1% · 1 s2 · 2 − s1 · 1 = = 3% 2−1 2−1

(3.30)

f 2, 3 =

3% · 3 − 2% · 2 s 3 · 3 − s3 · 3 = = 5% 2−1 3−2

(3.31)

f 1, 3 =

3% · 3 − 1% · 1 s 3 · 3 − s1 · 1 = = 4%. 3−1 3−1

(3.32)

118

T. POUFINAS

Table 3.4 Spot, forward and short rates with continuous compounding

Forward rates Spot rates Short rates

1.00% 3.00% 5.00%

2.00% 4.00%

3.00%

Source Created by the author

We observe that they come up to be whole number percentages (no decimals as with annual compounding). They can also be shown in a tabular form (Table 3.4).

3.3 Relation of the Term Structure with the Bond Yields After having defined the term structure and having established the relation between spot, forward and short rates, we are in place to establish their relation with the bond yields, which are essentially also interest rates; as a matter of fact when interest rates were considered to be flat, they were identical to the interest rates that corresponded to an investment that had the same risk with the bond and the same time to maturity. In order to establish this relation we consider now a zero-coupon bond maturing in N years. For simplicity we will assume that compounding is annual. Combining Eqs. (2.13) and (2.19) of Chapter 2 we derive that FV FV = PV0 = P0 = ⇒ sN = y = N (1 + s N ) (1 + y) N



FV P0

1

N

− 1, (3.33)

where s N is the spot rate that matures at time N (of an investment bearing the same risk with the bond) and y is the yield to maturity of the bond. The realization that they are equal means that the spot rates are essentially the yields to maturity of zero-coupon bonds that mature on the same date. Consequently, if for each time instant T there was a zero coupon bond with the same maturity date, then we would have derived all spot rates and could have constructed the term structure corresponding to these spot rates; of course all bonds should have the same level of risk.

3

119

TERM STRUCTURE OF INTEREST RATES

Table 3.5 Zero-coupon yield curve for 6 years Maturity

1

2

3

4

5

6

Zero-coupon bond price Zero-coupon yield Spot rate

990 1.01% 1.01%

970 1.53% 1.53%

950 1.72% 1.72%

930 1.83% 1.83%

910 1.90% 1.90%

890 1.96% 1.96%

Source Created by the author

As a result of this the term structure is also known as zero-coupon yield curve; the two are used interchangeably. Example 3.5 Assume that there are available 6 zero-coupon bonds maturing in 1, 2, 3, 4, 5, 6 years respectively, issued by the same party (e.g. a government), hence all carrying the same credit rating. The bonds have a face value of 1,000 Euro and trade at 990, 970, 950, 930, 910 and 890 Euro respectively. The yield to maturity of each bond is estimated by Eq. (2.19) of Chapter 2 (or (3.33) above) and is found to be 1.01%, 1.53%, 1.72%, 1.83%, 1.90% and 1.96% respectively. Consequently, the spot rates are s1 = 1.01%, s2 = 1.53%, s3 = 1.72%, s4 = 1.83%, s5 = 1.90% and s6 = 1.96%.

(3.34)

This can be seen in a tabular form (Table 3.5). It can also be mapped in a Cartesian diagram (Fig. 3.5). As we will see later, to produce the entire yield curve we simply connect the points of the aforementioned chart, most frequently with linear segments, essentially performing (linear) interpolation. This approach paves the path for bond valuation as well as the construction of the yield curve when interest rates are not flat/horizontal as was the case in Chapter 2, which we discuss next. 3.3.1

Types of Yield Curves

There are four main types of yield curves depending on their shape, which reflects the relationship of short-, mid- and long-term interest rates (yields). These are (Fig. 3.5):

120

T. POUFINAS

Zero-coupon yield curve

2.50%

Spot Rates

2.00% 1.50% 1.00% 0.50% 0.00%

0

1

2

3

4

5

6

7

Maturity

Fig. 3.5 Zero-coupon yield curve for 6 years (Source Created by the author)

• The flat or horizontal yield curve, whose graph is a horizontal line, indicating that interest rates (yields) remain fixed for all maturities. This is the simplest assumption and we have used that when pricing bonds in Chapter 2. It indicates that investors expect the same compensation for all maturities, thus they do not consider time to maturity as a risk factor that justifies higher returns. • The normal yield curve, which is an upward sloping yield curve, indicating that as the time to maturity increases the interest rate increases. Therefore long-term interest rates are higher than midterm interest rates, which are higher than short-term interest rates. This is usually a concave function of time, with the curve becoming asymptotically flat, as interest rates tend to stabilize after a long period of time. This phenomenon is called ‘flattening’. This is the most common shape and depicts the belief that investors anticipate higher returns for longer maturities. • The inverted yield curve, which is a downward slopping curve, indicating that as the time to maturity increases the interest rate decreases. Therefore short-term interest rates are higher than midterm interest rates, which are higher than long-term interest rates. This is usually a convex function of time, with the curve becoming asymptotically flat, as interest rates tend to stabilize after a long period of time. Such a shape indicates the anticipation of a drop of long-term interest rates.

3

TERM STRUCTURE OF INTEREST RATES

121

Yield Normal

Flat Humped Inverted

Time

Fig. 3.6 Types of yield curves (Source Created by the author)

• The humped yield curve, which is a bell-shaped curve, reflecting an increase in the medium maturities and a drop in the short and the long maturities. Therefore mid-term interest rates are higher than both the short-term and the long-term interest rates. Such a bellshaped curve is not frequently observed; it is indicative of a potential distress of the issuer in the mid-term, which is expected to smoothen in the long-term, as was the case of Greece during the financial crisis (Fig. 3.6).

3.4

Bond Valuation, Pricing and Return 3.4.1

Bond Valuation

3.4.1.1 Annual Compounding With the understanding that interest rates are not necessarily fixed and that they are better described as functions of time, depicted by the term structure, the question of how to value bonds becomes valid again. The value of a bond at any given point of time is calculated as the present value of the payments it makes—coupons (if any) and face value (at maturity); this time though the discounting takes place with the spot rate that applies at each time instant that a bond payment occurs. Therefore, Eq. (2.11) of Chapter 2 changes to PV0 =

c2 c1 c N + FV + + ··· + (1 + s1 ) (1 + s2 )2 (1 + s N ) N

122

T. POUFINAS

=

N  n=1

cn FV + , n (1 + sn ) (1 + s N ) N

(3.35)

where, cn : sn : FV: N: PV0 :

is is is is is

the the the the the

coupon paid by the bond at time n, interest rate used for discounting, face value of the bond, time to maturity of the bond, present value of the bond.

Once and again the spot rates correspond to the level of risk of the bond. They are given by the respective risk free rate with the addition of the appropriate (for the level of risk) spread on top of it. This involves three term structures; the one that corresponds to the risk of the bond, the risk-free and their difference, which is the spread. Hence Eq. (2.12) of Chapter 2 becomes sn = r f, n + spreadn .

(3.36)

We kept the notation of the risk-free rate as it was, with the addition of the time index trusting that there will be no confusion by not introducing s instead of r (as we did for spot rates globally) and that r f immediately prompts to the risk-free (spot) rate. Equation (3.35) for zero-coupon bonds is given by PV0 =

FV . (1 + s N ) N

(3.37)

In the aforementioned equations the compounding and the coupon payment frequency were assumed to be annual; however, they could have been of any frequency, with the spot rates and the coupons divided with the corresponding number. If this had been the case, then N would have denoted periods instead of years. At the limit, continuous compounding may be used.

3

TERM STRUCTURE OF INTEREST RATES

123

Example 3.6 We will apply the aforementioned approach on the bond of Example 2.3 of Chapter 2. The bond (issue) matures in 6 years, with a coupon rate of 4% paid annually and a face value per bond of 1,000 Euro. The discount rate is given by the term structure of Example 3.5 above that was derived as a zero-coupon yield curve (shown in Table 3.5). Spot rates are per annum with annual compounding. The value of the bond today is estimated as PV0 =

40 40 40 + + 2 (1.0101) (1.0153) (1.0172)3 40 40 40 + 1000 + + + = 1, 115.60 (1.0183)4 (1.0196)6 (1.0190)5

(3.38)

Apparently the (present) value of the bond is different from the one of Example 2.3 in Chapter 2 as the spot rates used for discounting the payments made by the bond are different. The interest rate used in Example 2.3 of Chapter 2 was a flat 2% whereas in this example the interest rates start from about 1% at year 1 and increase until they reach a level of a little bit below 2%. 3.4.1.2 Multi-Period Compounding Per Year Bond pricing formula (3.35) can be easily adapted to reflect multi-period compounding per year. If ν denotes the compounding frequency, then coupon is paid ν times a year and interest is compounded ν times a year. Equation (3.35) becomes PV0 =

N  n=1

cn /ν FV + , n (1 + sn /ν) (1 + s N /ν) N

(3.39)

where n = 1…N counts the compounding periods (instead of years). 3.4.1.3 Continuous Compounding In a similar manner formula (3.35) can be adapted for continuous compounding. One can readily see that it changes to PV0 =

N  n=1

cn · e−sn ·n + FV · e−s N ·N .

(3.40)

124

T. POUFINAS

It is clear that when compounding is continuous the dates on which the bond makes a payment do not have to be integers. I.e. for any set of time instants t 0 < t 1 < … < t N the price (or present value) of a bond is PV0 =

N 

cn · e−sn ·tn + FV · e−s N ·t N .

(3.41)

n=1

Example 3.7 We repeat Example 3.6 with continuous compounding to see that P = 40 · e−0.0101·1 + 40 · e−0.0153·2 + 40 · e−0.0172·3 + 40 · e−0.0183·4 + 40 · e−0.0190·5 + 1, 040 · e−0.0196·6 = 1, 114.46.

(3.42)

The (small) difference in the price of −1.14 Euro or −0.10% is due to the difference in the compounding frequency. 3.4.2

Bond Pricing and Yield to Maturity

The use of a term structure that is not flat raises one question. Can the yield to maturity be calculated? And if yes what is its added value—being a fixed rate—if interest rates are not fixed? The answer lies within the definition of the yield to maturity; it was defined as the internal rate of return of the investment to the bond. It therefore indicates what would have been the return from the investment to the bond if it had been held to maturity and if all coupons (for coupon bearing bonds) had been invested in that rate. The yield to maturity can be calculated as before, no matter what the term structure is, and is a measure used to compare the return of bonds incorporating their risk. An investor needs to know if the investment to the bond offers a return that compensates him or her for the risk he or she is exposed to. The present value (or price) of the bond is not (necessarily such) a good metric for comparing bonds as it is an absolute amount that depends on the face value, the coupon, the time to maturity, as well as the term structure used for discounting the cash flows of the bond. The yield to maturity (as well as any other of the yield measures that were presented in Chapter 2) though is a percentage that accounts for all that, but

3

TERM STRUCTURE OF INTEREST RATES

125

normalizes the outcome against the price and thus makes the comparison feasible. Investors expect to have (about) the same return from bonds that have (about) the same risk level and (about) the same remaining time to maturity even though their initial coupon, initial time to maturity and price are different. Example 3.8 To understand that, consider for example a firm that has an unchanged credit rating for the last 10 years and the outlook is that it will remain the same in the foreseeable future. The firm has issued a 20-year bond ten years ago. It thus has a remaining time to maturity of 10 years. A second firm exhibits identical characteristics in terms of credit rating and lines of business and is perceived by the investors (and the rating agencies) as having the same exact level of risk. The second firm issues a 10-year bond today. The term structure is apparently very different; only by chance it could have been close to the term structure 10 years ago. The price of the bond of the second firm is most likely quite different from the price of the bond of the first firm, as even if they have the same face value the level of coupon can be quite different. However, their yield to maturity is expected to be equal or very close. Example 3.9 Assume now that the price of the bond in Example 3.6 is indeed 1,115.60 Euro. The yield to maturity of the bond of Example 3.6 is found by solving Eq. (2.18) of Chapter 2. As a result, we employ numerical methods to find that 40 40 40 + + + (1 + y) (1 + y)2 (1 + y)3 40 40 40 + 1000 + + = 1, 115.60 4 5 (1 + y) (1 + y)6 (1 + y) ⇒ y = 1.94%

(3.43)

If the price of the bond had been 1,054.17, then the YTM would have been 3% as was found with the use of Eq. (3.20) in Example 2.6 of Chapter 2. This indicates that the calculation of the YTM does not depend on the term structure.

126

T. POUFINAS

The yield to maturity is a fixed percentage (in this example 1.94%) as opposed to the spot rates and is probably never realized in practice, as it assumes that the investor holds the bond until its maturity date and that coupons are reinvested at the bond (or at a rate equal to the YTM). As explained the primary use of the YTM is in the comparison of the returns of the bonds given their level of risk. For the same level of risk and maturity date investors most likely privilege the bond with the highest yield to maturity. 3.4.3

Bond Return

The yield to maturity has among its hypothesis that the investor holds the bond until its maturity. In that case the return from the bond—modulo the reinvestment risk—is the yield to maturity which is the internal rate of return of the bond. When the bond is not held until its maturity date, the return from the bond is better described from the holding period return. Introducing a term structure that is not flat influences also the holding period return of the bond. Recall that the first calculations we performed in Sect. 2.5 assumed that the coupons are reinvested at a constant interest rate which was equal to the yield to maturity and led to the simplified formula (2.46). Then, we allowed the reinvestment rate to be different from the YTM, however still fixed and known (as shown in equation (2.50)). In reality, the interest rates at which coupons are reinvested are unknown and uncertain, unless the investor “locks” them with the use of forward rates (through forward rate agreements). What is also uncertain a priori is the price of the bond at the end of the holding period. Consequently, the holding period return is a random variable as well, or at best can be calculated under the assumption that the coupons earned are reinvested at the corresponding forward rate and that the selling price of the bond is agreed, potentially derived with the use of forward rates for discounting. The relevant formula becomes HPRsk =

Pk + c + c · (1 + s1, k ) + · · · c · (1 + sk−1, k )k−1 − P0 , P0

(3.44)

3

TERM STRUCTURE OF INTEREST RATES

127

where s i, k , i = 1…k−1 denotes the future spot rate that starts at time i and ends at time k. We introduced the superscript s in HPR to denote the dependence from a non-horizontal term structure. Of course the holding period return becomes known a posteriori, i.e. when the investor has sold the bond and has reinvested the coupons. It simply reflects though the performance achieved and provides no forward looking estimate of the anticipated future return from holding the bond.

3.5

Construction of the Term Structure

Investors are interested in constructing the term structure applicable to the pricing of bonds. We saw that the term structure (for a certain credit rating or risk level) is the zero-coupon yield curve constructed from a set of zero-coupon bonds with maturity date equal to each time instant of the time horizon of interest. However, having a zero-coupon bond with a maturity date T , for each positive real number T is not happening in reality. There may not even be zero-coupon bonds with maturity dates on positive integer time instants. What we may have in practice is a series of bonds, of the same credit quality (often of the same issuer and more often sovereign bonds), with finitely many maturity dates in the future. These maturity dates may not even be positive integer time instants as bonds may mature at any point of time in the future. Luckily enough we can use these bonds to derive certain points of the yield curve; namely, the ones that correspond to their maturity dates. We use interpolation (most frequently linear) to fill in the interest rates for the time intervals between these time instants. To illustrate, let us consider an example. Example 3.10 There are 3 bonds available, all with a face value of 1,000 Euro. The first is zero-coupon and matures in 1 year; it trades at a price of 980.40 Euro. The second pays an annual coupon of 3%, matures in 2 years and has a price of 1,009.80 Euro. The third is zero-coupon, matures in 3 years and has a price of 920.50 Euro. We can use these bonds to construct specific points of the term structure.

128

T. POUFINAS

We know that the YTM of the first bond is nothing but the spot rate of the first year. More precisely, 980.40 =

1, 000 1, 000 ⇒ s1 = − 1 ⇒ s1 = 2.00%. 1 + s1 980.40

(3.45)

To compute the price of the second bond we need the spot rates of the first two years. We have just found the spot rate of the first year, hence we can use it to solve for the spot rate of the second year. Its price satisfies the equation 1, 030 1, 030 30 30 + + ⇒ 1, 009.80 = ⇒ 2 1 + s1 (1 + s2 ) 1.02 (1 + s2 )2 1, 030 1, 030 ⇒ (1 + s2 )2 = 1, 009.80 − 29.41 = = 1.051 ⇒ (1 + s2 )2 980.39 1 + s2 = 1.0511/2 ⇒ s2 = 1.0250 − 1 ⇒ s2 = 2.50% (3.46) 1, 009.80 =

as s 1 has been found from Eq. (3.45) Finally, the third spot rate is found from the third bond as

1, 000 1, 000 1/3 920.50 = ⇒ s = − 1 ⇒ s3 = 2.80%. (3.47) 3 (1 + s3 )3 920.50 The term structure can be drawn if we use linear interpolation to complete the intermediate intervals (Fig. 3.7). This method of constructing the term structure is known as bootstrap method (Hull, 1997). From the term structure as constructed above we can derive the forward rates with the use of Eq. (3.27) to see that f 1, 2 = 3.00%, f 1, 3 = 3.20% and f 2, 3 = 3.40%. The short rates are r 0 = 2.00%, r 1 = 3.00% and r 2 = 3.20%. Example 3.11 A bond that is issued today pays a coupon of 4% and matures in 3 years. It has the same credit rating with the bonds of Example 3.10. Its price is given by PV0 =

40 40 1, 040 + + = 1, 034.61. 2 1.020 1.025 1.0283

(3.48)

3

TERM STRUCTURE OF INTEREST RATES

129

Term structure 3.00%

Interest Rate

2.50% 2.00% 1.50% 1.00% 0.50% 0.00%

1

2

3

Maturity

Fig. 3.7 Term structure of interest rates for 3 years (Source Created by the author)

When interest rates are not at integral time instants continuous compounding may be preferred as it does not depend on the compounding period. Otherwise, simple compounding is used between two compounding instants. Furthermore, simple compounding is used— which is equivalent to linear interpolation—for payments made by the available bonds in time instants that are intermediate to the compounding time instants. Globally, the construction of the term structure involves several bonds, whose payments occur at different time instants, which are not necessarily all integral multiples of the same period. However, we can always follow the aforementioned interpolation approach and express all interest rates as linear functions of the integral interest rates. For simplicity and without loss of generality we assume annual compounding and annual coupon payments that take place on compounding time instants. If N is the longest maturity among the available bonds, then we need N bonds to construct the term structure until year N . We thus derive a N × N

130

T. POUFINAS

system of equations C N1 C11 C21 + + ··· + 2 1 + s1 (1 + s2 ) (1 + s N ) N C N2 C12 C22 P02 = + + · · · + 1 + s1 (1 + s2 )2 (1 + s N ) N .. . P01 =

P0N =

(3.49)

C NN C1N C2N + + · · · + 1 + s1 (1 + s2 )2 (1 + s N ) N

where for j, n = 1…N. j

Cn : sn : N: j P0 :

is is is is

the the the the

payment made by bond j at time n, spot interest rate of time n used for discounting, time to maturity of the longest maturing bond, present value of bond j.

If a bond j has a maturity date N j that is shorter than N , then the payments made by that bond after its maturity date are obviously equal to zero. The payments made by the bonds are their coupons (unless a bond is zero-coupon), except for the last one that equals its coupon plus its face value. One issue that has been brought forward is that the system of equations (3.49) may not hold exactly, and may need to be corrected with an error term. This has been attributed to taxes and call options embedded to the bond (Bodie et al., 1996).

3.6 The Moves and the Uncertainty of (the Term Structure of) Interest Rates So far we have assumed that interest rates are known with certainty, even when they are not fixed. As a result, investment strategies with the same maturity date should yield the same returns, provided they have the same level of risk. Otherwise there would be arbitrage opportunities (as explained earlier in this chapter); when such opportunities appear the market players immediately take advantage of them and thus equilibrium

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is reached. Of course, one needs to consider potential taxes and expenses or differences between the lending and borrowing rates that make arbitrage opportunities more difficult to exploit (which we will not take into account in the analysis that follows). In our discussion above though, we explained that interest rates—especially in the future—may not be known with certainty. We referred earlier to future spot rates, which were not known with certainty. In order to illustrate the behavior of interest rates under uncertainty, before we study the relevant models of the term structure below, we assume without loss of generality that compounding is annual. Moreover, we will consider only one-year periods; i.e. we look at the outcome of our investments only at the annual anniversaries since their inception and not in the interim. Without loss of generality we consider a total horizon of two years; nevertheless our findings can be readily generalized to any investment horizon or number of years. We consider two routes; one that employs investments in (zerocoupon) bonds, following the presentation of Bodie et al. (1996), and a second that is based on our term-deposit investments, as introduced in the previous sections. When investment takes place via (zero-coupon) bonds, then if we consider that we have a two-year horizon, then as investors we have two alternative choices: A. Buy a zero-coupon bond that expires in two years (which is the analogue of the two-year term deposit discussed in Sect. 3.2.1 above). B. Buy a zero-coupon bond that expires in one year and repeat it, i.e. buy a second zero-coupon bond that expires in one year, at the end of the first year (which is the analogue of the two rolling one-year term deposits discussed in Sect. 3.2.1 above). Then, assuming that the future spot rates are equal to the forward rates, if interest rates are deterministic we have with certainty (Bodie et al., 1996): (1 + s1 ) · (1 + f 1, 2 ) = (1 + s2 )2 = (1 + r0 ) · (1 + r1 ),

(3.50)

where r 0 = s 1 is the short rate of the first year, which thus equals to the spot rate of the first year; r 1 = f 1, 2 is the short rate of the second year and is therefore equal to the forward rate between years 1 and 2; and

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s 2 is the spot rate of the second year. This is not new according to our previous analysis. But what if we do not know the interest rate for the second year with certainty? Then the investor would be exposed to reinvestment risk, which results from the interest rate risk, as the interest rate risk in year two is not known. As a matter of fact this interest rate is nothing else than the future spot rate for the second year. This future spot rate, denoted by s 1, 2 , is a random variable—as seen from today. The natural metric we can use in order to estimate a potential value is its expected value; we denoted it by e 1, 2 = E (s 1, 2 ) earlier in this chapter. If in the previous equation we replace r 1 by s 1, 2 then this equation will not hold true anymore as the left hand side will be a random variable, whereas the right hand side will be a known number. The next best try would be to replace r 1 with e 1, 2 , i.e. the expected future spot rate. The question is though if the expectation of the investor (or the market) is indeed such that makes this equation hold true. In other words, why should (1 + r0 ) · (1 + e1,2 ) = (1 + s2 )2 ,

(3.51)

(1 + r0 ) · (1 + e1,2 ) > (1 + s2 )2

(3.52)

(1 + r0 ) · (1 + e1,2 ) < (1 + s2 )2

(3.53)

and not have

or not have

holding true? If (3.52) or (3.53) hold true instead of (3.51) then there are no arbitrage opportunities, as the last two equations do not have agreed/locked interest rates –as were the forward rates—but expected rates instead. To demonstrate we proceed with a numerical example. Example 3.12 Using the data of Example 3.2 we have that r 0 = s 1 = 1%, s 2 = 2%, r 1 = f 1, 2 = 3.01%. If e 1, 2 had been equal to f 1, 2 , then Eq. (3.51) would have held true as it becomes the equation we have used to derive the

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forward rates. Namely, (1 + r0 )(1 + e1,2 ) = (1 + s1 )(1 + f 1,2 ) = (1 + s2 )2 .

(3.54)

If the valuation of the bonds was based on the expected interest rates, then the one-year zero-coupon bond would be sold for 1, 000/1.01 = 990.10

(3.55)

1, 000/(1.01 · 1.0301) = 1, 000/(1.02)2 = 961.17.

(3.56)

and the two-year for

Assume now that we have an investor with a short-term horizon. Considering that the short-term horizon refers to one year and the long-term horizon refers to two years, he or she can buy the one-year zero-coupon bond that will definitely expire in 1 year at 1,000 Euro and lock in a certain yield of 1%. Alternatively he or she can buy the two-year zero-coupon bond. Then, • If he or she insists on the one year horizon, then he or she has to sell the two-year bond. However, there is a risk, as the selling price one year from now is not known. We can only determine it provided his or her expectation comes true and as matter of fact if it is equal to the forward rate. • If he or she is willing to switch to a two year horizon, then he or she is exposed to a dilemma of which risk to assume: – Invest for one year and then reinvest the proceeds at the end of the first year. Then he or she is exposed to the reinvestment risk; the interest rate for year two is not known. Thus, if it is equal to the expected one, which in its turn coincides with the forward rate, then the outcome is known. If it differs, then the final outcome could be inferior or superior to the anticipated one. – Invest for two years directly. In this case, as he or she purchases the two year bond, he or she locks a return equal to the yield to maturity, since the bond is zero-coupon and thus—provided there is no default—he or she will collect the face value on its maturity date. However, having left the comfort and the

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freedom of choice of the one year horizon, he or she runs the risk of making a lower return than if he or she had invested for one year and then had reinvested the proceeds. To illustrate the outcomes of these choices, we realize that the expected return for the first year is also 1%. The second year has an expected yield of 3.01%, following our assumption that the expected rate is equal to the forward rate. This means an expected price for the bond at the end of the first year of 1, 000/(1.0301) = 970.78.

(3.57)

The HPR for the first year is also 1%. But—as we explained above—no matter what the choice of the investor is, the two-year return is at risk. If the interest rate exceeds (is inferior to) the expectations and is more (less) than 3.01%, then the price will be below (above) 970.78 Euro. In any case, the investor wants an incentive or compensation to buy the twoyear bond that gives him or her the same HPR1 for the first year, which is no better than that of the safe/risk-free one-year rate. The only reason to do this is to have a higher expected return, which would drive the two-year bond price under 961.17 Euro. To understand and explain the potential outcomes, we will consider alternative routes that justify the inequalities (3.52) and (3.53) instead of the equality (3.51) above. These pertain to the investment horizon of the investors. We analyze two scenarios; what if all (or most) investors had a short-term horizon and what if all (or most) investors had a long-term horizon. What if all (or most) investors had a short-term horizon? If all (or most) investors had (or have) a short-term horizon and were willing to buy the two-year bond if its price fell, let’s say to 951.75 Euro (instead of 961.17 Euro), then this would give an expected holding period return of 2% as estimated in equation (3.58): EHPR1 = 970.78/951.75 − 1 = 1.02 − 1 = 0.02 = 2%.

(3.58)

The risk premium is 1% over the 1% one-year interest rate (as the exepcted one-year holding period return was found to be 2%). This is the compensation to the investor for the risk he or she took because of the uncertainty. In this case though the yield to maturity of the two-year

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zero-coupon bond becomes (Bodie et al., 1996) y2 = s2 = (1, 000/951.75)1/2 − 1 = 0.025 = 2.50%.

(3.59)

and  = f 1,2

(1 + s2 )2 (1.025)2 −1= − 1 = 0.0403 = 4.03%. (1 + s1 ) 1.01

(3.60)

That is, the forward rate is higher than the expected short rate of 3.01%. This is not paradoxical because the forward rate is the interest rate that should apply in the second year so that short-term and long-term investments (via the one-year and two-year bonds respectively) produce the same result so as not to have arbitrage opportunities, but without taking risk into account. If we consider the associated risk then short-term investors will not want to invest in the long-term (two-year) bond except if its expected return is higher than that of the short-term (one-year) bond. A risk-averse investor would not invest in the long-term (two-year) bond unless the actual forward rate is superior to his or her expected return (Bodie et al., 1996), i.e. e1,2 < f 1, 2 .

(3.61)

This is because the lower the expected spot rate for the second year, the higher the expected return on the two-year bond. We could have shown that in an alternative way, without the use of bonds, but term-deposits instead, which was the approach followed in the previous sections. An investor that has a one-year horizon needs a higher two-year return in order to be convinced to switch to the longerterm horizon. This means that the two-year interest rate should result in a return that exceeds the combined rolling one-year investment returns, incorporating the expectation of the investor for the second year, i.e. (1 + s1 ) · (1 + e1, 2 ) < (1 + s2 )2 .

(3.62)

The right hand side can be replaced by its equivalent expression that involves the one-year spot rate and the one-year forward rate. It therefore becomes (1 + s1 ) · (1 + e1, 2 ) < (1 + s2 )2 = (1 + s1 ) · (1 + f 1, 2 ) ⇒ e1, 2 < f 1, 2 . (3.63)

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An interpretation of this finding is that the forward rate between years one and two should exceed the expected future spot rate of the investor for the same period, in order for him or her to be convinced to switch from the short-term horizon to the long-term horizon. The difference between the expected future spot rate and the forward rate is the liquidity premium, i.e. l1, 2 = f 1, 2 − e1, 2 > 0.

(3.64)

Following our first demonstration, it compensates investors for the uncertainty with which they will sell their bond at the end of the year (Bodie et al., 1996). According to our second elaboration, it compensates investors for staying longer than their investment horizon on the same investment and thus they give away the benefits of liquidity that the short-term horizon offers. Indeed, the investor that holds onto one year investments has the advantage of liquidity as he or she receives the proceeds of the one-year investment at the end of the first year and he or she can choose then how to invest them. The opposite assumption could hold true; i.e. that all (or most) investors had a long-term horizon. If investors have a long-term horizon then they consider long-term bonds to be safer. Buying at 961.17 Euro a two-year zero-coupon bond secures a yield to maturity of 2% per annum and delivers a face value of 1,000 Euro at the maturity of the bond. Instead, the investor can buy a one-year zero-coupon bond and at the end of the first year use the outcome of the first to buy another one-year zero-coupon bond. The return of the investor in two years will be 961.17 · (1.01) · (1 + s1, 2 ),

(3.65)

which is an amount that is not known with certainty though as the future spot rate is a random variable. The interest rate at which the returns are the same, i.e. this amount equals the face value of 1,000 Euro is the forward rate of 3.01% The expected return is 961.17 · (1.01) · (1 + e1, 2 ).

(3.66)

If the expected future spot rate is equal to the forward rate, then the expected yield from the two one-year rollover investments will be equal to the known yield of the two-year zero-coupon bond (Bodie et al., 1996).

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This is true if the investor is indifferent to the uncertainty that this sequence of one-year investments has. However, recall that the long-term investor prefers the safety of a known yield. He or she is therefore willing to invest in short-term (one-year) bonds only if the expected return is higher than that of the long-term (two-year) bond. Or put differently— so as to employ again our term-deposit approach—he or she needs the total outcome of the two consecutive one year investments to exceed his or her safe two-year return. That is (for both approaches) (1 + s1 ) · (1 + e1, 2 ) > (1 + s2 )2 .

(3.67)

In our example above we know that s1 = 1% and s2 = 2%. Nevertheless, we proceed with the general case to see that as the right hand side can be replaced by its equivalent expression that uses the one-year spot rate and the one-year forward rate, it becomes (1 + s1 ) · (1 + e1, 2 ) > (1 + s2 )2 = (1 + s1 ) · (1 + f 1, 2 ) ⇒ e1, 2 > f 1, 2 . (3.68) The difference is again the liquidity premium, i.e. l1,2 = f 1,2 − e1,2 < 0.

(3.69)

If for example e 1, 2 had been equal to 4.01% then the liquidity premium would have been −1%. Whether or not forward rates are equal to the expected future spot (or short) rates depends on how willing the investor is to assume the risk of interest rate changes or to invest in bonds (or other fixed income instruments) that do not meet his or her investment horizon (Bodie et al., 1996).

3.7

Approaches and Theories of the Term Structure Evolution Over Time

But how is the interest rate term structure formed/determined? And how can we estimate future spot interest rates? These are now random variables. What would be an estimate of their expected price? Are forward rates good estimates of the expected future spot rates or not? We note that the answer to this question may be different, depending on the theory we adopt for the interest rate term structure.

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There are four main approaches/theories that attempt to explain the term structure (Bodie et al., 1996; Luenberger, 1998). These are the expectation hypothesis theory, the liquidity preference theory, the market segmentation theory and the preferred habitat theory. 3.7.1

The Expectation Hypothesis Theory

This is the easiest to comprehend (and possibly accept) theory concerning the interest rate term structure. It claims that the forward rate is equal to the expected (according to the market view) future spot (or short) rates for a period of time and the liquidity premium is zero, i.e. f 1,2 = e1,2 .

(3.70)

This is in line with the first potential case we examined in the previous section, i.e. that forward rates capture the expectations of investors about future spot rates. 3.7.2

The Liquidity Preference Theory

Proponents of this theory believe that most investors in the market are short-term investors and thus the forward rate is higher than the expected future spot (or short) interest rate over a period of time, i.e. f 1,2 > e1,2 .

(3.71)

We illustrated that in the previous section, when we tried to capture what would make an investor with a short-term horizon adopt a long-term horizon. The difference of the two is the liquidity premium. 3.7.3

The Market Segmentation Theory

According to this theory, long-term bonds are traded in a different market than short-term bonds. Each of these markets independently reaches equilibrium. The transactions and interests of long-term lenders and borrowers set interest rates on long-term bonds. Likewise, the transactions and interests of short-term investors and issuers affect short-term interest rates, regardless of the transactions and interests of the long-term ones.

3

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The Preferred Habitat Theory

According to this theory, investors have a preference for certain maturities and need a premium to be convinced to change. The long-term and short-term markets are not so strictly separated to prevent investors from changing their horizons when they deem that the premium justifies the move.

3.8

Formation of the Term Structure

We have seen that when future short rates are known with certainty, then 1 + y N = 1 + s N = [(1 + r0 )(1 + r1 ) · · · (1 + r N −1 )]1/N ,

(3.72)

1 + y N = 1 + s N = [(1 + s1 )(1 + f 1, 2 ) · · · (1 + f N −1, N )]1/N ,

(3.73)

or

where yN : sN : s 1: r n −1 : f n −1, n : N:

is the yield to maturity of a zero-coupon bond maturing in N years, is the N -year spot rate, is the first year spot rate, is the n-year short rate for n = 1…N , is the n-year annual forward rate, is the investment horizon.

When future spot (or short) rates are uncertain, then the aforementioned equation changes to the anticipated 1 + y N = 1 + s N = [(1 + s1 )(1 + e1,2 ) · · · (1 + e N −1, N )]1/N ,

(3.74)

where e n−1,n is the n-year expected future spot rate (annual in our case) and s 1 is the first year spot rate. However, we do not know if this equation will necessarily hold true. If the expectation hypothesis theory holds, then these expected future spot rates are equal to the corresponding forward rates and the equation is indeed valid.

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These formulas show that there is a relationship between yields of bonds with different maturities and forward interest rates. We use it to analyze the yield curve (Bodie et al., 1996). If the yield curve is increasing, then f n, n+1 is higher than y n (or s n ). So on any maturity date n the yield curve is an increasing function of time if the forward rate for the next period exceeds the yield at that maturity date. The new forward rate is higher than the geometric average of previously observed forward rates. This can be intuitively understood as it needs to be high enough to close the gap from the previous yield. But what makes the forward rate higher? Recall that: f n,n+1 = en,n+1 + ln,n+1 .

(3.75)

Liquidity premium is the necessary incentive for investors to hold a bond with a maturity that does not meet their investment horizon. Liquidity premium is not necessarily positive, unless one accepts the liquidity preference theory. If most investors are long-term then the liquidity premium is negative (as we realized earlier). This equation indicates that there are two possible explanations for the forward rate to increase. Investors anticipate: i. Either that interest rates will increase, ii. Or a high liquidity premium in order to invest in long(er)-term bonds. It is not correct to say that if the yield curve is upward slopping then investors trust that interest rates will at some future point of time rise; liquidity premium may have a significant contribution in the determination of interest rates. Assume for example, that we have a flat expected annual spot (or short) rate of 1% and a fixed liquidity premium of 1%; then all forward rates are equal to 2% as evedenced by equation (3.75). The yield curve is upward slopping and commences at 1% for one-year bonds and asymptotically reaches 2% for long(er)-term bonds as more forward rates of 2% are added. Consequently if increases in future interest rates are expected, then this results in an upward yield curve. Nevertheless an upward yield curve does not mean that interest rates will be higher in the future. As a result, the very existence/acceptance of the liquidity premium makes it hard to make any inferences with regards to the moves of the

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interest rates by using the yield curve (only). However, it is very useful for an investor to evaluate the expected sentiment of the market as well as his or her own anticipations. As a matter of fact contrasting the two may impact his or her stance/position on the course of interest rates and his or her respective investment decisions. One assumption would be that liquidity premium is flat. The expected market rates are then given by the difference of the liquidity premium from the forward rate. But this method would hardly work as: i. It is very difficult to get an approximation of the liquidity premium. This is done by comparing the forward rates with the realized future spot (or short) rates and estimating their differences. The latter are affected by a variety of economic developments/circumstances/factors. ii. It is not entirely correct to assume that liquidity premium is flat. Relevant studies indicate a large variation in the interest rate risk for bonds of different maturities (Startz [1982] as per Bodie et al. [1996]). The conventional ‘assumption’ that the yield curve is rising (normal yield curve) especially for short maturities is probably reflecting the view that long-term bonds have a positive liquidity premium. Thus we interpret an inverted yield curve as an indication that interest rates are expected to fall. If the term premium, i.e. the difference between the yields of longterm and short-term bonds is generally positive, then expected interest rate decreases result in a declining yield curve (Bodie et al., 1996).

3.9

Determinants of the Interest Rates

After this presentation and before moving to the modeling of interest rates and their term structure one may ask several questions. Two come to mind. The first one is what are really the factors that determine the level (or the change) of interest rates? The second—which is definitely connected—is what makes interest rates fall or rise? The latter is absolutely relevant to the current period as interest rates are at very low or even negative levels and it is a question what will make them rise again. For starters, there are two constituents that are worth investigating (Bodie et al., 1996):

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i. Real interest rates and ii. Inflation. The nominal interest rate, i.e. the rate that is announced by a bank or issuer has two components: real interest rate and inflation; the latter is rather a compensation for the reduction of the purchasing power of money. Consequently, the following equation holds true (Bodie et al., 1996): 1 + r = (1 + ρ) · (1 + π ) ⇒ r ≈ ρ + π.

(3.76)

In the afore mentioned equation r: ρ: π:

is the nominal interest rate, is the real interest rate, is the inflation rate.

The second approximate equation in (3.76) is derived as a result of the assumption that the product ρ·π is small compared to (the order of) ρ and π. The equation linking the real and the nominal interest rate with the use of inflation can be used both for actual and expected rates. We can thus infer that a change in nominal interest rates is due to either a change in real interest rates or a change in inflation. A distinction is needed between the two cases as they are combined with different economic circumstances (Bodie et al., 1996). High real interest rates can mean (Bodie et al., 1996): i. Fast growing economy, ii. Increased budget deficits and iii. Tight monetary policy. High inflation can mean (Bodie et al., 1996): i. Fast growing economy, ii. Increased money supply and iii. Potential shocks related to the supply.

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These determinants may affect the levels and evolution of interest rates and fixed income investments in several ways. For example, the disruption observed in the supply and the logistics chain due to the pandemic is expected to increase inflation. However, it is not clear whether this inflationary trend will persist or will retreat. It very much depends on the course of the pandemic. As a consequence, central banks are not prepared yet to stop their quantitative easing programs, until they are certain that the economies will be able to weather the effects of the pandemic. At the time of this writing there is no sign that the pandemic is over; hence the support offered by the central banks is not expected to stop. However, if they decided to end the purchase programs, then interest rate increases are likely to occur. Following this present time example (end of 2020—beginning 2021) we will try at this point to make a log of the determinants of the interest rates, in line with our discussion in these first three chapters. Although the list may not be exhaustive, these are (see also Analyst Prep, 2020; Bean, 2017; European Parliament, 1999): • The demand and supply of money; this is evidenced by the stance of the central banks, as well as the behavior of the players of the financial system, i.e. households, enterprises and governments. Households normally supply money, whereas enterprises and governments demand money. When the central banks want to increase the supply they purchase fixed income securities; their prices rise and thus the interest rates drop. When they want to decrease liquidity they do the opposite and interest rates rise. The liquidity preference theory also supports this factor. When investors privilege liquidity, then they demand cash over interest-bearing longer term securities. They sell the latter and thus bond prices fall, leading to interest rate rise. When the supply of money increases, then investors will buy bonds; their prices will rise, leading to an interest rate drop. • The risk associated with the money borrowed/lent or invested. We have seen that the interest rate is the sum of the risk-free rate and a spread that compensates the investor for the risk he or she bears for lending funds to an issuer or debtor. The rating agencies have a role in assessing the default risk, which determines the spread. The creditworthiness of the borrower is thus significant in determining the applicable interest rate. The seniority of the bond/fixed income

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•

• •

•

•

• •

T. POUFINAS

security in the capital structure of the company is also important for the level of the relevant interest rate. The inflation or inflationary expectations or in other words the potential loss of the purchasing power of money. As we explained earlier in this section, investors anticipate compensation for inflation; hence the nominal interest rate embeds the real interest rate as well as inflation. The maturity date of the bond or fixed income security or loan; in other words the time horizon of the investment or the length of time for which the money has been lent. The currency or the exchange rate between the domestic and the foreign currency (i.e. the FX rate) can also affect the interest rate. Competing FX rates may lead to selling domestic securities to invest in foreign securities; the selling of the domestic securities will result in a price drop hence an interest rate increase. The opposite will happen for the foreign interest rates. The liquidity of financial markets, as well as of the fixed income securities of interest, affects the bid-yield and offer/ask-yield and thus the level of interest rates. For example a bond with low liquidity will have a lower bid price (and thus higher bid-yield) and a higher ask price (and thus lower ask-yield). The absence or presence of embedded options or guarantees (such as put or call provisions) have an impact on the yields of corporate bonds (or any other fixed income security) and thus to the interest rates. The overhead costs of the intermediaries, such as banks, also affect the level of interest rates; such costs need to be covered as well by the interest rate required in loans or bond issues. The tax rates that are applicable on interest payments may also affect the level of interest rates, as investors demand interest rates that suffice to pay also the tax expenditure.

Nonetheless, we attempt an identification of the determinants and a quantification of their contribution via econometric models in Chapters 10 and 11 that follow.

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Stochastic Interest Rate Models

In this section we try to present the most well-known models that describe the movement of the interest rate term structure and to use them to price bonds or fixed income instruments. In Chapter 5 we use them to price interest rate options. These models are more complex than those that describe the movement of a stock or a stock index, as they deal with the movement of an entire curve and not just a variable. Over time, the interest rate of each maturity in the curve changes, as does the shape of the curve. When the stock price moves it draws a path. When the term structure of interest rates moves it draws a surface. In modeling interest rates a question that comes to mind is which rate to model, as the term structure incorporates one interest rate per maturity date, which also changes as the time passes. Furthermore, there are additional rates, such as the forward rates and the spot rates. Our guide here is the modeling of the stock prices. We thus want an interest rate that is uniquely defined at each point of time and can generate the entire term structure. This is due to the fact that at each time instant the stock price receives one value only (even if as seen from today this is a random variable), whereas the term structure at each time instant is a curve (i.e. a set of infinitely many interest rates, one per maturity date). This role is played by the short rate (with an extension of its definition). Interest rate models capture the move of the short rate (adapted for each point of time instead of a time interval) in a way that will be explained in this section. Before extending the definition of the short rate, we distinguish the models based on two criteria: time continuity and position of the initial term term structure in the model. With regards to the first criterion, we distinguish two types of models: • Continuous-times models. • Discrete-time models. In the first case we are trying to model the movement of the interest rate curve in a way similar to a geometric Brownian motion (or with variants and extensions thereof), while in the second with a methodology similar to that of the binomial trees (or with variants and extensions thereof). As a matter of fact, in the modeling of interest rates we employ trinomial trees, as will be explained in the relevant section that follows. These two are the most popular approaches used—at least for educational purposes—for the

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modeling of the move of stocks and stock indices and as such we extend their application to the modeling of the move of interest rates. With regards to the second criterion, we distinguish two basic types of models (Hull, 1997): • The equilibrium models, in which the initial interest rate term structure is an output. • The no-arbitrage models, in which the initial interest rate term structure is used as input. In summary (Table 3.6). In the aforementioned presentation of the various models we focused on the short rate; this indicates that we limit our presentation to one source of risk/uncertainty. The models that replicate the movement of interest rates based on one risk factor are called one-factor models. A one-factor model encompasses that (i) all interest rates move in the same direction over a short period of time, but not necessarily by the same magnitude and (ii) the term structure may change with time. The difference from the corresponding modeling of stock price movements is eminent; in the stochastic process that captures the stock price moves the interest rate used for discounting is usually assumed to be constant. This is not the case here, as it is the interest rate that is being modeled. Our analysis focuses on one-factor models only. We need to mention though that multi-factor models (e.g. two- and three- factor models) have been developed. Examples of two-factor models are these of Longstaff and Schwartz (1992) and Hull and White (1994b). Chen has developed a three-factor model (1996). Multi-factor models have the advantage of Table 3.6 Interest rate term structure models

Model

Discrete-time

Continuous-time

Equilibrium

Trinomial Tree Initial term structure output

No-arbitrage

Trinomial Tree Initial term structure input

Geometric Brownian Motion Initial term structure output Geometric Brownian Motion Initial term structure an input

Source Created by the author

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replicating more closely and consistently the movements of the interest rates than one-factor models. However we feel they lie beyond the scope of this book; as long as the reader has been acquainted with one-factor models, he or she can easily comprehend multi-factor models by referring to the previous sources. In modeling interest rate movements we need to carefully account for the properties of interest rates. As said, the aforementioned approaches constitute extensions of similar approaches used for stocks. Yet, there is dissimilarity between interest rates and stock prices; namely interest rates tend to return to a long-term average with the passage of time. This property of interest rates is called mean reversion property (Hull, 1997). It seems that this property has been violated the last decade or so as the Quantitative Easing (QE) programs of the Central Banks aiming to provide liquidity to the market(s) have led the interest rates to unprecedented low levels, even at the negative territory. The trigger was economic—financial until 2019; at the turn of the new decade the root cause is the recent pandemic that has pushed back the productive activity and along with it the growth of the economies around the globe. Consequently, interest rates are anticipated to remain at low levels (even negative in some countries) for an indefinite period of time—until at least there are signs of recovery (CBS, 2021; ECB, 2021) (Fig. 3.8). The mean reversion property claims that when the short interest rate is low then it tends to return to a higher level, when it is high it tends to return to a lower level. This property is reasonable, as when interest rates are high there is less demand for debt capital, so they drop. When interest rates are low, then there is a greater demand for debt capital and so interest rates go up. As ample liquidity is provided the last years through the central bank QE programs this argument has not been confirmed; interest rates remain low. In each of the models that we present we make a note of whether it captures the mean reversion property. We still present models that do not exhibit mean reversion due to the simplicity in their presentation and application (at least for educational purposes) on one hand, and as they are natural extensions of the models used to reflect the movement of stock prices on the other hand. 3.10.1

Continuous-Time Term Structure Models

We first present continuous-time models. In these models we consider all time instants in the time interval of interest. As mentioned earlier the

148

T. POUFINAS

Interest Rate Mean Reversion Interest Rate

High interest rates tend to drop

Reversion Level

Low interest rates tend to rise Time Fig. 3.8 Interest rate mean reversion property (Source Created by the author with information assembled from Hull [1997])

models we present follow the movement of the short rate. The choice of the short rate is not arbitrary; recall in the discussion in the previous sections, and in particular as per equation (3.22) that the spot rates can be derived from the short rates. We only need to define the short rate in the context of a time instant instead of a time interval. The short rate r at time t is the interest rate for an infinitesimal period of time at time t. It is essentially the short rate that was defined earlier in Sect. 3.2; however the period of time it covers is infinitesimally small in contrast with the compounding period that we used earlier. This can be probably explained by the fact that on one hand we use continuous compounding and on the other hand we want to model the movement of interest rates in a way similar that is done for equities; thus we need a very small change of the interest rate. This short rate is known as the instantaneous short rate. Let, in line with the way we constructed the interest rate term structure, r (t0, t1 , t2 , ω) denote the interest rate that is observed at time t 0 , starts at time t 1 and matures at time t 2 , with ω symbolizing the possible

3

TERM STRUCTURE OF INTEREST RATES

149

events. The instantaneous short rate at a given time t is defined as: r (t) := r (t, ω) = lim r (t, t, t + t, ω). t→0

(3.77)

This interest rate is not known with certainty and is thus a random variable. Even if we manage to model the course followed by the short rates, we still need to find a way to produce the term structure from them. To do that recall that earlier in this chapter, when the interest rates were deterministic (i.e. known with certainty), we constructed the term structure as the yield curve of zero coupon bonds. In other words, we realized that a spot rate is equal to the yield to maturity of a zero-coupon bond with the same maturity. This is the route that we follow in the case of stochastic interest rates. The present value at time t of a zero-coupon bond that matures at time T and has a face value of 1 monetary unit (e.g. e 1 or $ 1 payable at T ) is (Hull, 1997): T

Pt = Eˆ t [e−r (T −t) ],

(3.78)

where r denotes the average short rate in the interval [t, T ], given by 1 · r= T−t

T r (τ )dτ .

(3.79)

t

Please note that r is still a random variable, as it is the average of the short rate over time. The symbol Eˆ t stands for the expected value at time t in a risk-neutral world. The rationale behind the validity of equation (3.78) is that the price of the zero coupon bond at time t is nothing but its present value. As the interest rate is stochastic, i.e. not known with certainty, then the expected value needs to be introduced so as to produce a numeral and not a random variable. For T a fixed maturity date the aforementioned bond price is essentially what we have so far denoted by P t . However, as we want both the maturity date T , as well as the starting date t to vary we introduce the notation with the subscripts. Let now sTt denote the spot interest rate with continuous compounding at time t for the time interval [t, T ]. If we are at time 0, then this is the future spot rate st, T which is a random variable. However, when we are at time t, then it is known with certainty

150

T. POUFINAS

and this is why we introduced a different notation (with the beginning and the ending date as superscript and subscript respectively). Furthermore, in line with the notation of the bond price we want both t and T to vary, thus we wish to distinguish the notation from this of the future spot rate. Using the standard equation for the price/present value of a zero coupon bond, but with the newly introduced notation we receive that (Hull, 1997): Pt = e−sT (T −t) t

T

1 ln P(t, T ) T −t

(3.81)

1 ln Eˆ t [e−r (T −t) ]. T −t

(3.82)

⇔ sTt = − ⇔ sTt = −

(3.80)

This equation shows that we can construct the interest rate term structure at any given point of time from r provided we know its risk-neutral stochastic process. So if we have somehow modeled the process for r, then we can generate the initial interest rate term structure and its evolution in the future. 3.10.1.1 Equilibrium Models Equilibrium models attempt to capture/replicate/reproduce the move of the short rate (i.e. the short-term risk-free interest rate) and then do the same for the move of bond prices. As the initial interest rate term structure is an output in these models their input is determined by (hypotheses made on) key (macro)economic variables. As said, we employ only onefactor models. Usually in a risk-free world the instantaneous short rate is given by an Itô process of the form: dr = μ(r )dt + σ (r )dz.

(3.83)

In such a model both μ() and σ () are considered functions of the short rate r and not of time. The most well-known such models are the Rendleman and Bartter model (1980), the Vasicek model (1977), and the Cox, Ingersoll and Ross —CIR model (1985). Their particulars are (Hull, 1997):

3

TERM STRUCTURE OF INTEREST RATES

Model

μ(r )

σ (r )

Rendleman and Bartter Vasicek Cox, Ingersoll, Ross

μ·r a · (β − r ) a · (β − r )

σ ·r σ √ σ· r

151

Mean-reversion No Yes Yes

All models can be used to price zero coupon bonds and then produce from them the term structure from equation (3.82) above. Vasicek (1977) and Cox et al. (1985) proved that (3.78) can be used to value zerocoupon bonds as (with the notation of Hull adapted to ours [1997]): T

Pt = T At e−r (t)·T Bt ,

(3.84)

where Model Vasicek

T Bt

T At

a = 0

1−e−a(T −t) a

  2 β−σ 2 /2) σ 2 B(t, T )2 exp (B(t, T )−(T −t))(a − 2 4a

a=0

T −t

CIR

2(eγ (T −t) −1) (γ +α)(eγ (T −t) −1)+2γ

a

exp[σ 2 (T − t)3 /6] 

2γ e(a+γ )(T −t)/2 (γ +a)(eγ (T −t) −1)+2γ )  γ = a 2 + 2σ 2

2aβ/σ 2 ,

We note that • For the model of Rendleman and Bartter we could use the expression derived for Vasicek’s model with β = 0 and μ = −α. • In the model of Cox, Ingersoll and Ross interest rates are always non-negative, whereas in the other two models the replicated interest rates could take negative values. From (3.84) and (3.81) we can derive the term structure as: sTt = −

1 1 · ln T At + · T Bt · r (t). T −t T −t

(3.85)

Therefore, the full interest rate term structure can be produced as a function of r(t ) if α, β, σ are known. sTt is a linear function of r(t ). The latter

152

T. POUFINAS

determines the interest rate term structure at time t and the way by which it is generated. 3.10.1.2 No-arbitrage Models The equilibrium models are easy to build and comprehend, however their output does not immediately match the initial interest rate term structure. With careful selection of the parameters they can roughly match the initial interest rate curve. But usually the approach exhibits errors. In contrast, no-arbitrage models are constructed in such a way that they are fully aligned with the initial interest rate term structure. In this section we will study one factor no-arbitrage models. The instantaneous short rate suffices for the construction of equilibrium models. However, we need to define the instantaneous forward rate for the development of no-arbitrage models. In a way similar to the introduction of a slightly different notation above for the spot rate, we use a slightly different notation for the forward rate. The symbol f Tt1, T2 is identical to f T 1 , T2 when t = 0. When t > 0, then f Tt1 , T2 is a random variable as seen from today. However, when we are at time t it is not a random variable any more. We introduce this notation as we want to allow t (as well as T 1 and T 2 ) to vary. The instantaneous forward rate is defined in a way similar to the instantaneous short rate as: t f (t, T ) := f (t, T, ω) = lim r (t, T, T + T, ω) = lim f T, T +T . T →0

T →0

(3.86)

Please note that time t is not necessarily equal to 0. When interest rates were known with certainty, the distinction was not probably as important. In the case of models of the term structure though it is the initial term structure that is known with certainty, but all other future term structures are not. In other words, at time t = 0 we know r (0), T P0 and f (0, T ) provided we know the initial term structure. This is not the case for any other time t at which we are required to model the term structure. In the case of no-arbitrage models μ becomes also a function of time. Hence Eq. (3.83) becomes dr = μ(r, t)dt + σ (r )dz.

(3.87)

We present two no-arbitrage models; the Ho and Lee model (1986) and the Hull and White model (1990). Their particulars are (Hull, 1997):

3

Model

μ(r, t)

Ho and Lee Hull and White

TERM STRUCTURE OF INTEREST RATES

σ (r )

θ (t)

θ (t)

σ

(θ (t) − ar )

σ

∂ f (0, t)/∂t + σ 2 2 ∂ f (0, t)/∂t + a f (0, t) + σ2a (1 − e−2at )

153

Mean-reversion No Yes

We observe that • Both models have the initial term structure as input via the instantaneous forward rate at time 0. • The model of Ho and Lee is a specific case of the Hull and White model for α = 0; the model of Hull and White is a generalization of the model of Ho and Lee that incorporates the mean reversion property. • The model of Hull and White is an extension of Vasicek’s model that fits the initial term structure. The price of a zero-coupon bond is given by (3.84) with (Hull, 1997): Model

T Bt

lnT At

Ho and Lee

T −t

ln T P 0 − (T − t) ∂tt 0 − 21 σ 2 t (T − t)2 t 0 P ∂ ln P ln T P 0 − B(t, T ) ∂tt 0 − 13 σ 2 (e−aT − e−at )2 (e2at − 1)

Hull and White

P

1−e−a(T −t) a

t 0

∂ ln P

4a

The term structure is then produced by Eq. (3.85) 3.10.2

Discrete-Time Term-Structure Models

We next present discrete-time models. In these models we partition the time intervals into smaller subintervals of certain length/step. Having done that we note that interest rate trees are traditionally used in order to reproduce/replicate/capture the movements of interest rates in discrete-time. They pretty much reflect what the aforementioned stochastic processes generated for the short rate in continuous-time. They are used in a way similar to stocks, where their application is probably more straightforward. Thus, in order to build an interest rate tree we divide (partition) the time horizon of interest into smaller subintervals

154

T. POUFINAS

of step t. Then the interest rates depicted by the tree are the interest rates with continuous compounding for a period t. The construction of the tree replicates the stochastic continuous-time process for the instantaneous short rate. Nonetheless, trees used to model interest rates exhibit an important difference versus trees that map stock price moves; the latter assume that interest rate remains unchanged at each node for discounting purposes, whereas the former do not, as the interest rate used at each node for discounting is different. This can be explained by the fact that interest rates used for discounting cannot be assumed to be fixed, as it is exactly interest rates that are being modeled. The modeling of the short rate movement is done with a trinomial tree and not a binomial one, which allows for more potential interest rate moves, and better captures the mean reversion property. 3.10.2.1 Standard or Basic Trinomial Tree A standard or basic (plain vanilla) trinomial tree is constructed by first dividing the time horizon interval into smaller sub-intervals of equal length t. If T is the time horizon of the investor, which could be the maturity of a bond, of an obligation or of a derivative, then for a partition of N sub-intervals the time period t is given by t = T /N or vice-versa the time to maturity T satisfies the relation T = t ✕ N . The simplest tree has N = 1 (Fig. 3.9) and it is the building component that is repeated at each node for N > 1 (Fig. 3.9).

Fig. 3.9 Trinomial tree standard branching (Source Created by the author)

0

T

3

TERM STRUCTURE OF INTEREST RATES

155

The trinomial tree starts from the initial short rate and then provisions for 3 potential moves; up, middle (straight) and a down. Each of them has its own probability and is represented by a tree branch (often referred to as standard branching, Fig. 3.9). At each node of the tree (until the time horizon has been reached) all three moves are possible and there are three branches that start from this node that indicate these moves. We assume that the interest rate trees are recombining in the sense that an up move followed by a down move leads to the same node as two middle (straight) moves; the same holds true for a down move followed by an up move. Similarly, an up move followed by a middle (straight) move leads to the same node as a middle (straight) move followed by an up move and so on so forth. A basic interest rate tree is deployed with such a standard branching over the desired number of periods. We illustrate the approach with a 2step recombining trinomial tree with standard branching. However, the methodology can be replicated for any number of steps as presented for a 2-step tree (Fig. 3.10). As shown in Fig. 3.10, We denote by, t: pu : pm : pd : ru i d j : rm :

The time interval between any two steps. The probability of an up movement. The probability of a middle movement. The probability of a down movement. The interest rate that corresponds to i movements up and j movements down. The interest rate that corresponds to i movements up and j = i movements down. This is the middle level rate and is equal to r0 .

It is important to note that although u i d j is an index and not a multiplier the powers annihilate each other as d = u −1 . This means for example that an up move (i = 1) followed by a down move (j = 1) leads to r0 ≡ rm which is the initial rate; an up move (i = 1) followed by a middle (horizontal) move (j = 0) leads to ru . 3.10.2.2 General Trinomial Tree—The Model of Hull and White Standard branching does not reflect the mean reversion property; it seems that interest rates can go as high or as low as possible by increasing the

156

T. POUFINAS

ru 2

ru

ru

r0

r0

rd

rd

pu

r0

pm pd

rd 2

0

Fig. 3.10

Δt

T=2Δt

Basic trinomial tree (Source Created by the author)

number of steps. Hull and White (1993) have come up with variants of the standard branching so as to allow the representation of the mean reversion property (Hull, 1997) (Fig. 3.11). Branching 1

0

Branching 2

T

0

Branching 3

T

0

T

Fig. 3.11 Trinomial tree branching methods (Source Created by the author with information assembled from Hull [1997])

3

TERM STRUCTURE OF INTEREST RATES

157

Branching 1 is symmetric and is the standard branching. Branching 2 is (kind of) upward skewed and is used to capture the mean reversion property for low interest rates, while branching 3 is (kind of) downward skewed and is used to capture the mean reversion property for high interest rates. Branching 2 has three branches; middle (straight), one up and two up. Branching 3 has also three branches but in the opposite direction; middle (straight), one down and two down. We will use the terms branching 1 - symmetric, branching 2 - upward skewed and branching 3 - downward skewed interchangeably. We will try to illustrate the process of constructing interest rate trees of Hull and White (1994a) as presented in Hull (1997). This process is divided into two steps. It is based on the Hull and White model. The first step is to construct a symmetric tree for the interest rate (variable) ρ which is initially 0 and follows the process: dρ = −aρdt + σ dz.

(3.88)

This process is symmetric around ρ = 0. The difference ρ(t + t) − ρ(t) is a random variable that follows a normal distribution. If terms of order greater than t are ignored, then the expected value of ρ(t + t) − ρ(t) is −aρ(t)t and its volatility is σ 2 t. The distance between the interest rates of the tree is ρ and is taken equal to: √ (3.89) ρ = σ 3t. This choice according to Hull and White (1994a) is indicated by theoretical work in numerical methods in order to minimize the error of the approximation. First we construct a tree for ρ where the nodes are equidistant from each other with respect to ρ and t. What we need to consider is the branching method that should be applied to each node and the probability of getting to one node starting from another at the previous time step. Let (κ, λ) denote the node that is reached at t = κt and ρ = λρ. The branching chosen must ensure positive probabilities. For a > 0 we change from symmetric branching (1) to downward skewed branching (3) for λ quite large and positive, while we change to upward skewed branching (2) for λ large enough and negative. We

158

T. POUFINAS

define as λmax and λmin respectively the values of λ for which we change branching. According to Hull and White (1994a), the probabilities are always positive if we set  (3.90) λmax = [0.184 at] + 1, λmin = −λmax .

(3.91)

In equation (3.90) the notation [x] denotes the integral (or integer) part of a number x. In order to estimate pu , pm , pd , we realize that they need to agree with the expected change of ρ in the next period of time t, as well as the volatility of this change. The probabilities must also add to 1. We thus obtain 3 equations with 3 unknowns. If we have symmetric branching (1) then (Hull, 1997): pu ρ − pd ρ = −a jρt,

(3.92)

pu r 2 + pd r 2 = σ 2 t + a 2 j 2 ρ 2 t 2 ,

(3.93)

pu + pm + pd = 1.

(3.94)

We have similar systems of equations for branching 2 and branching 3. The solution of these systems yields the probability functions as (Hull, 1997): Branching pu pm pd

Symmetric (1)

Upward skewed (2)

a 2 λ2 t 2 −aλt

1 6 + 2 2 − a 2 λ2 t 2 3 1 a 2 λ2 t 2 +aλt 6 + 2

a 2 λ2 t 2 +aλt

1 6 + 2 − 13 − a 2 λ2 t 2 − 2aλt a 2 λ2 t 2 +3aλt 7 6 + 2

Downward skewed (3) 7 + a 2 λ2 t 2 −3aλt 6 2 − 13 − a 2 λ2 t 2 + 2aλt 1 + a 2 λ2 t 2 −aλt 6 2

The 2nd step is to convert the tree to a tree with respect to r. We set: r˜ (t) := r (t) − ρ(t),

(3.95)

to receive by differentiation and equations (3.87) and (3.88) the differential equation: d r˜ (t) = [θ (t) − ar˜ (t)]dt.

(3.96)

3

TERM STRUCTURE OF INTEREST RATES

159

The solution of this equation is (Hull, 1997): ⎤ ⎡ t σ2 r˜ (t) = e−at ⎣r (0) + eaτ θ (τ )dτ ⎦ = f (0, t) + 2 (1 − e−at )2 . (3.97) 2a 0

Using (3.97) we can build a tree for r from the tree already constructed for ρ, since from (3.95) r = ρ + r˜ . The tree is constructed if we set the interest rate at time κt equal to: r (κt) = ρ(κt) + r˜ (κt).

(3.98)

However, the tree that is constructed in such a way is not necessarily aligned with the initial interest rate term structure since r˜ that was found in a continuous time framework, is now used in a discrete time framework. To overcome this problem we can calculate r˜ with an iterative procedure, so that the initial term structure is consistently fitted. This process for each κ allows the calculation of r˜κ (i.e. the difference of ρ from r at time κΔt ). by going backwards until time 0 via all possible paths that lead to time κΔt taking into account the probabilities of each path that links the node at time 0 with all the nodes at time κΔt. However, this process is cumbersome as it involves several paths as κ increases and could even be confusing. We therefore follow the simplified approach presented by Hull (1997) that at each node defines a zero-coupon bond that pays its face value only if reaching this particular node. Estimating the present value of these zero-coupon bonds can make the process leaner and the equations easier to understand. We therefore define r˜κ := r (κt) − ρ(κt),

(3.99)

and P˜κ, λ to be the present value of a zero-coupon bond that pays a face value of 1 monetary unit (e.g. either e 1 or $ 1) if we reach the node (κ, λ) and 0 in all other cases. r˜κ and P˜κ, λ will be calculated by an inductive procedure that we will describe immediately below so that the initial term structure is fitted. Note here that the use of P˜κ, λ is auxiliary, as we mentioned earlier, so that for educational purposes we do not convey large and perplex expressions. The more familiar reader may not need them. We will show both approaches for the step that we will analytically present.

160

T. POUFINAS

Suppose that st0 is the initial spot rate maturing at time t, thus replicating the initial interest rate term structure with continuous compounding and which is known (according to the notation we introduced earlier in the chapter). In what follows we drop the superscript when we are at time 0 as it refers to the known with certainty present-time term structure (provided there is no risk of confusion). We note that 0 = st , r (0) = r˜0 = st

(3.100)

must hold true, since ρ(0) = 0. That is, the interest rate should be equal to the original interest rate for the time period t. It is chosen so as to give the correct price to a zero-coupon bond maturing at time t, as interest rates were seen to be essentially the yields to maturity of zerocoupon bonds (of the same risk and maturity date as the interest rates). This sets the starting position/starting node of the tree. Obviously, by its definition P˜0, 0 = 1.

(3.101)

Next, we recall that we have 3 branches that end in 3 nodes with the probabilities as given above. It follows that: P˜1, 1 = pu e−˜r0 t ,

(3.102)

P˜1, 0 = pm e−˜r0 t ,

(3.103)

P˜1, −1 = pd e−˜r0 t .

(3.104)

r˜1 is chosen so as to give the matching price to a zero-coupon bond that matures at time 2t. The price of the bond at the top node is e−(˜r1 +ρ)t , at the middle node e−(˜r1 )t and at the bottom node e−(˜r1 −ρ)t . To understand this, we must recall that the interest rate of each node refers to the corresponding time period t ahead of it. So a bond that matures at time 2t, may take the 3 above prices in the immediately preceding time period, as shown by the respective interest rates at the relevant nodes. We also recall that the interest rate step is ρ, as given by (3.89) Then we look for the value at the initial node. This can be found by discounting with r (0) = r˜0 given the probability of reaching each of

3

TERM STRUCTURE OF INTEREST RATES

161

the three nodes and adding or making use of the intermediate quantities P˜1, λ , λ = −1, 0, 1, as found in (3.102), (3.103), and (3.104). The two approaches are equivalent. At the same time, however, by using the initial interest rate term structure, we should obtain the same present value. The two discounting alternatives give as value to the original node: t P0

= P˜1, 1 e−(˜r1 +ρ)t + P˜1, 0 e−˜r1 t + P˜1, −1 e−(˜r1 −ρ)t = pu e−˜r0 t e−(˜r1 +ρ)t + pm e−˜r0 t e−˜r1 t + pd e−˜r0 t e−(˜r1 −ρ)t = e−s2t ·2t .

(3.105)

The third expression in the aforementioned equation is derived from the replacement of the P˜1, λ ’s by their equal quantities, as expressed in Eqs. (3.102), (3.103) and (3.104). This comes as no surprise as the price as seen at time 0 is precisely the sum of the present values of one monetary unit weighted by the probability of ending at each of the corresponding 0 is the spot rate of the initial term nodes. Please note that s2t = s2t structure and thus we drop the superscript (as indicated earlier). The only unknown of this equation is r˜1 , which is easily given as:   P˜1, 1 e−ρt + P˜1, 0 + P˜1, −1 eρt 1 ln r˜1 = t e−s2t ·2t   pu e−(˜r0 +ρ)t + pm e−˜r0 t + pd e−(˜r0 −ρ)t 1 ln = . (3.106) t e−s2t ·2t The second expression of (3.106) is justified in a similar manner to the third expression of Eq. (3.105). Using this we calculate r at the 3 nodes of the tree from (3.98) at time t. One can see that we do not really need to employ P˜κ, λ , as we stated earlier, as long as we are familiar with what they represent. We continue by calculating P˜2, 2 , P˜2, 1 , P˜2, 0 , P˜2, −1 , P˜2, −2 , which in turn lead to the calculation of r˜2 , which leads to the calculation of interest rate r at the five nodes at time 2t through equation (3.98). Then we repeat for time 3t, and so on and so forth. The general formulas are as follows. Assume that the P˜κ, λ ’s have been calculated for all κ ≤ ξ , ξ ≥ 0. We want to estimate r˜ξ so that the tree correctly values a zero-coupon bond maturing at the moment (ξ + 1)t.

162

T. POUFINAS

The interest rate at the node (ξ, λ) is r˜ξ + λρ. The price of a zerocoupon bond maturing at that moment is: (ξ +1)t P0

=

κξ 

P˜ξ, λ e−(˜rξ +λρ)t = e−s(ξ +1)t ·(ξ +1)t .

(3.107)

λ=−κξ

The solution of the above equation for r˜ξ gives: ⎡ κ ⎤ ξ  −λρt ˜ Pξ, λ e ⎢ ⎥ ⎥ λ=−κξ 1 ⎢ ⎢ ⎥, ln⎢ −s r˜ξ = ⎥ ·(ξ +1)t (ξ +1)τ t ⎣ e ⎦

(3.108)

where κ ξ denotes the number of nodes at time ξ Δt. If the branching had always been the standard (symmetric) branching 1, then κ ξ would be equal to ξ. However, it is less than ξ when the non-standard (upward skewed and downward skewed) branching alternatives 2 and 3 are used. 0 Please note that s(ξ +1)t = s(ξ +1)t is the spot rate of the initial term structure and thus we drop the superscript (as indicated earlier). Equation (3.108) is nothing but a general form of (3.106), which supports our statement that the auxiliary zero-coupon bonds are not necessarily needed to find the interest rates. Then the P˜κ, λ ’s for κ = ξ + 1 are calculated by:  P˜ξ +1, λ = (3.109) P˜ξ,  · pr (, λ)e−(˜rξ +ρ)t , 

where pr (, λ) is the probability of moving from node (ξ, ) to node (ξ + 1, λ) and the sum extends to all the values of  for which this probability is positive. It is worth noting here that this process can be extended to other models of the form (Hull, 1997): d f (r ) = [θ (t) − a f (r )]dt + σ dz. The relevant steps are similar to the above.

(3.110)

3

TERM STRUCTURE OF INTEREST RATES

163

Exercises Exercise 1 An investor knows the particulars of three bonds and wishes to find the interest rate term structure. The bonds are: • A zero coupon bond with a maturity of 1 year which sells at a price of EUR 980.4. • A bond with a maturity of 2 years, annual coupon payments at a coupon rate of 3% with a price of EUR 1,009.8. • A zero coupon bond maturing in 3 years with a price of EUR 915.2. All bonds have a face value of EUR 1,000. a. What are the spot, forward and short interest rates for the first 3 years? b. Draw the graph of the interest rate term structure. c. What is the price of a bond that expires in 3 years, has a face value of EUR 1,000 and pays an annual coupon at a coupon rate of 4%? Exercise 2 Assume that there is a zero coupon bond with a maturity of 1 year at a price of EUR 99, a zero coupon bond with a maturity of 2 years at a price of EUR 96 and a coupon-bearing bond that pays coupons annually at a rate of 4% per annum with a maturity of 3 years and a price of EUR 101. All bonds have a face value of EUR 100. a. What are the spot, forward and short interest rates for the first 3 years? b. Draw the graph of the interest rate term structure on a maturity date—interest rate diagram using linear interpolation/extrapolation where/as necessary. c. What is the price of a bond that expires in 3 years, has a face value of EUR 100 and pays an annual coupon at a coupon rate of 3% per annum?

164

T. POUFINAS

Exercise 3 The future spot rates for a period of time are equal to the corresponding forward rates. This statement: A. Is B. Is C. Is D. Is

always correct. never correct. correct if we accept the Expectations Hypothesis Theory. correct if we accept the Liquidity Preference Theory.

Justify your answer. Exercise 4 The future spot rates for a period of time are lower than the corresponding forward rates. This sentence: A. Is always correct. B. Is never correct. C. Is correct if we accept the Expectations Hypothesis Theory. D. It is correct if we accept the Liquidity Preference Theory. Justify your answer. Exercise 5 The interest rate term structure includes: A. All spot, forward and short rates. B. Only spot rates. C. Only forward rates. D. Only short rates. Justify you answer.

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TERM STRUCTURE OF INTEREST RATES

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Exercise 6 Consider a zero-coupon bond, with a face value of EUR 1,000 that matures in 2 years. The one-year spot rate is 2% and the two-year spot rate is 4%. The price of the bond today is: A. EUR B. EUR C. EUR D. EUR

961.17 1,000 924.56 942.68

Justify you answer. Exercise 7 Let us assume that a = 0.08 and b = 0.08 in Vasicek’s model and in the Cox, Ingersoll, Ross model. Suppose that in both models, the initial short rate is 4% and the initial standard deviation of the short rate is 1%. Use both models to price a zero-coupon bond that matures in 8 years. Compare the outputs of the two models. What do you observe? Exercise 8 Indicate which models of the interest rate curve you know. a. What are their similarities and what are their differences? b. Outline how they are used to value bonds. c. What is the difficulty? Exercise 9 Let us consider a bond that matures in 4 years, has a face value of EUR 1,000 and makes annual coupon payments at a coupon rate of 4% per annum. The spot rate for a one-year period is 2%, for a two-year period 3%, for a three-year period 3.5% and for a four-year period is 3.8%. a. Find the present value of the bond. b. What is the yield to maturity of the bond if the bond price is equal to the present value you calculated in question (a)?

166

T. POUFINAS

c. What conclusions can you draw about potential weaknesses of the yield to maturity? d. If interest rates are not fixed, i.e. the term structure is not horizontal, then what is the added value of the yield to maturity? Exercise 10 Let us assume that an investor believes that the liquidity preference theory holds true. He or she wants to take a bet on the evolution of the future spot rates so as to potentially gain from their move. a. What positions could he or she take in order to exploit such potential opportunities? b. What risk is he or she exposed to?

References Analyst Prep. (2020, January 31). Determinants of interest rates. https://analys tprep.com/study-notes/actuarial-exams/soa/fm-financial-mathematics/det erminants-of-interest-rates/. Accessed: January 2021. Bean, M. A. (2017). Determinants of interest rates. Education and Examination Committee. Society of Actuaries. Financial Mathematics Study Note, FM-26-17. https://www.soa.org/globalassets/assets/Files/Edu/2017/fmdeterminants-interest-rates.pdf. Accessed: January 2021. Bodie, Z., Kane, A., & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. CBS. (2021, April 12). Fed Chair Jerome Powell tells 60 Minutes America is going back to work. 60 minutes. CBS News. https://www.cbsnews.com/ news/60-minutes-jerome-powell-federal-reserve-economy-update-202104-11/. Accessed: April 2021. Chen, L. (1996). A three-factor model of the term structure of interest rates. In Interest rate dynamics, derivatives pricing, and risk management. Lecture Notes in Economics and Mathematical Systems (Vol. 435). Springer. https:// doi.org/10.1007/978-3-642-46825-4_1 Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407. ECB—European Central Bank. (2021, April 22). Monetary policy decisions. Press Release. https://www.ecb.europa.eu/press/pr/date/2021/html/ecb. mp210422~f075ebe1f0.en.html. Accessed: April 2021.

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European Parliament. (1999, November). The determination of interest rates. Economic Affairs Series, ECON 116 EN (PE 168.283). https://www.europarl. europa.eu/workingpapers/econ/116/116_en.htm. Accessed: January 2021. Ho, T. S., & Lee, S. B. (1986). Term structure movements and pricing interest rate contingent claims. The Journal of Finance, 41(5), 1011–1029. Hull, J. C. (1997). Options, futures and other derivatives (3rd ed.). Prentice Hall International, Inc. Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. The Review of Financial Studies, 3(4), 573–592. Hull, J., & White, A. (1993). One-factor interest-rate models and the valuation of interest-rate derivative securities. Journal of Financial and Quantitative Analysis, 28(2), 235–254. Hull, J., & White, A. (1994a). Numerical procedures for implementing term structure models I: Single-factor models. Journal of Derivatives, 2(1), 7–16. Hull, J. C., & White, A. D. (1994b). Numerical procedures for implementing term structure models II: Two-factor models. The Journal of Derivatives, 2(2), 37–48. Longstaff, F. A., & Schwartz, E. S. (1992). Interest rate volatility and the term structure: A two-factor general equilibrium model. The Journal of Finance, 47 (4), 1259–1282. Luenberger, D. G. (1998). Investment science. Oxford University Press. Rendleman, R. J., & Bartter, B. J. (1980). The pricing of options on debt securities. Journal of Financial and Quantitative Analysis, 15(1), 11–24. Startz, R. (1982). Do forecast errors or term premia really make the difference between long and short rates? Journal of Financial Economics, 10(3), 323– 329. Statista. (2021). Treasury yield curve in the United States as of January 2021. https://www.statista.com/statistics/1058454/yield-curve-usa/. Accessed: February 2021. Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188. World Government Bonds. (2021). United States Government Bonds— Yields Curve. http://www.worldgovernmentbonds.com/country/united-sta tes/. Accessed: February 2021.

CHAPTER 4

Fixed Income Portfolio Management

So far we have mainly studied the properties of bonds as debt instruments individually as well as the mechanics of their pricing. We have realized that interest rates are not necessarily fixed and may change not only with their maturity date, but also with the passage of time. At each time instant an investor faces a new term structure. As a result the price of bonds may change as time passes and time to maturity decreases, coupons are paid and interest rates change. This means that an investor is exposed to the risk of the change of the price of the bond, which is known as market risk. As the price of the bond depends on the level of interest rates, this type of market risk has two sources; the change in the risk-free interest rate, referred to as interest rate risk and the change in the spread, referred to as spread risk. Sometimes the broader term interest rate risk is used to address this type of risk when the two components of interest rate, i.e. the risk-free rate and the spread, are not considered separately. One could say that there is a third source of risk; namely that the issuer may fail to repay the debt, i.e. to default, known as credit risk. This will be dealt separately in Chapters 6 and 9. Knowing the change to the bond price that a change in interest rate may bring is important to investors so that they can immediately know their profit or loss from the move.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_4

169

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T. POUFINAS

Furthermore, bonds rarely are the single investments of an investor. Most of the times they are part of a bigger portfolio, consisting of more than one bond issue or fixed income instruments, among several asset classes. Consequently, it is of interest to know how the entire bond (or fixed income) portfolio (or sub-portfolio) will be affected by potential changes in the interest rates. There are two interested parties; the asset managers and the asset owners. The former construct and manage the fixed income portfolios; the latter invest their capital to the fixed income portfolios that the asset managers offer them—unless they have the size, the access and the know-how to do it on their own. Asset managers follow certain approaches in order to set up and to manage the portfolios. Most of the times, either they build the portfolios in advance and then try to distribute them to the asset owners that have a (risk) profile that is aligned with the characteristics of the fund or they try to shape a portfolio that matches the exact needs of one single asset owner or several asset owners that have similar investment targets. As managers try to meet either the preset targets of the portfolio or the specific ones dictated by the asset owners, they implement strategies that attempt to achieve specific goals driven by the underlying market or the liabilities of the asset owners or both. In most cases they try not only to meet these goals but exceed them. In this chapter we enter into the particulars of fixed income portfolio management, looking at bonds within a portfolio and not individually. We realize that interest rate risk is the driver of portfolio management and present the notions of duration and convexity as means to measure and manage this risk. We present the two different approaches for managing a fixed income portfolio, the passive and active management. This chapter introduces the reader into the details of fixed income portfolio management.

4.1

Interest Rate Risk

As became apparent in Chapter 3, interest rates change as the time to maturity of the underlying investment varies and as time passes. In Chapter 2 we realized that the price is a decreasing function of the interest rate. When the interest rate increases, then the price drops and when the interest rate decreases, then the price rises. As interest rates go down or

4

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171

up, prices go up or down (respectively) and they produce profits or losses for the investors that hold them. Consequently, from the moment that an investor acquires a bond he or she is exposed to a price change that leads to a profit or loss, known as market risk. In the case of bonds this change emanates from the change of the interest rate. This is broadly called interest rate risk and is considered as a category of market risk. The level of a given interest rate can change either because the risk-free rate changed or because the spread changed. The latter is known as spread risk and differs from the default risk of the issuer which is called credit risk or counterparty risk. The impact of interest rate changes is even more important for bond or globally fixed income portfolios. As interest rates shift, the price of each fixed income security goes up or down, and this results in a change to the total value of the portfolio. Portfolio managers and investors follow this change of value as it determines the performance of the portfolio. The sensitivity of the value change affects the portfolio performance and drives to a certain extend the selection of the securities that will be included in the portfolio. With this observation in mind managers and investors need to know and monitor the determinants of the sensitivity of the price/value change of a fixed income security to interest rate shifts. 4.1.1

Price Sensitivity to Interest Rate Moves

To investigate the sensitivity of the bond price to the changes of the interest rates we will initially assume that the term structure is horizontal and later on we will lift this restriction. Equation (2.1) of Chapter 2 indicates that the price (which is assumed to be equal to the present value of the bond) is a decreasing function of the interest rate r PV0 = P0 = =

N  n=1

c2 c1 c N + FV + + ··· + (1 + r ) (1 + r )2 (1 + r ) N

cn FV + . (1 + r )n (1 + r ) N

(4.1)

The function that measures the price change to the interest rate change is the first derivative of the price P (we drop for simplicity the subscript

172

T. POUFINAS

0) as a function of the interest rate r c1 c2 c N + FV dP = −1 · −2· − ··· − N · 2 3 dr (1 + r ) (1 + r ) (1 + r ) N +1   c1 1 c2 c N + FV · 1· =− −2· − ··· − N · 1+r (1 + r )1 (1 + r )2 (1 + r ) N  N   1 cn FV · =− n· +N· . (4.2) 1+r (1 + r )n (1 + r ) N n=1

If the bond pays no coupon, i.e. it is a zero-coupon bond, then Eq. (4.2) becomes FV FV dP 1 = −N · ·N· =− , dr (1 + r ) N +1 1+r (1 + r ) N

(4.3)

which is a much simpler formula. The negative sign in both formulas verifies what we already knew; namely, that as interest rates rise the bond price drops. Equations (4.2) and (4.3) indicate that the price change to the interest rate change depends on the time to maturity, the coupon and the interest rate (or yield to maturity). We will try to understand with an example how each of them impacts the price change. Example 4.1 Let us consider six bonds all of which have a face value of 1,000 Euro. Three bonds pay no coupon and the other three pay a coupon of 4%. There are three different maturity dates; 1 year, 10 years and 20 years. For each maturity date there is a zero-coupon and a coupon-bearing bond with that maturity. We assume initially that the interest rate is 4% for all maturity dates and then becomes 3% and 5% and we map the prices as well as the price changes for all interest rates. This is shown in Tables 4.1 and 4.2 respectively (see also Bodie et al., 1996). Then, to comprehend the effect of the level of the interest rate we assume that initially the interest rate is 8% and then shifts to 9% and 7%. The results are depicted in Tables 4.3 and 4.4 for the zero-coupon and the coupon-bearing bonds respectively. We observe that

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Table 4.1 Price change of a zero-coupon bond (initial r = 4%) Time to maturity

Interest rate

4% 5% % change 3% % change

1

10

20

961.54 952.38 −0.95% 970.87 0.97%

675.56 613.91 −9.13% 744.09 10.14%

456.39 376.89 −17.42% 553.68 21.32%

Source Created by the author

Table 4.2 Price change of a coupon-bearing bond (c = 4%, initial r = 4%) Time to maturity

Interest rate

4% 5% % change 3% % change

1

10

20

1,000.00 990.48 −0.95% 1,009.71 0.97%

1,000.00 922.78 −7.72% 1,085.30 8.53%

1,000.00 875.38 −12.46% 1,148.77 14.88%

Source Created by the author

Table 4.3 Price change of a zero-coupon bond (initial r = 8%) Time to maturity

Interest rate

8% 9% % change 7% % change

1

10

20

925.93 917.43 −0.92% 934.58 0.93%

463.19 422.41 −8.80% 508.35 9.75%

214.55 178.43 −16.83% 258.42 20.45%

Source Created by the author

a. Both for the zero-coupon and coupon-bearing bond, the longer the maturity the bigger the change in absolute value (all other things being equal). This is probably anticipated as both involve higher

174

T. POUFINAS

Table 4.4 Price change of a coupon-bearing bond (c = 4%, initial r = 8%) Time to maturity

Interest rate

8% 9% % change 7% % change

1

10

20

962.96 954.13 −0.92% 971.96 0.93%

731.60 679.12 −7.17% 789.29 7.89%

607.27 543.57 −10.49% 682.18 12.33%

Source Created by the author

powers of the discounting factor 1 + r as the maturity of the bond increases. This can be seen in all 4 tables. b. The bigger the coupon, the smaller the change in absolute value (again all other things being equal). This can be seen by comparing Table 4.1 with Table 4.2 and Table 4.3 with Table 4.4. Coupon bearing bonds post smaller changes in absolute value than the corresponding zero-coupon bonds with the same maturity dates. This difference can be intuitively explained by the fact that a zero-coupon bond resembles/behaves like a “longer maturing” bond than a coupon-bearing bond of the same maturity. We claim that as the former makes only one payment on its maturity date, whereas the latter starts making payments well before its maturity date. Each such payment has its own maturity date. We explained in Chapter 2 that a coupon bearing bond resembles a portfolio of zero-coupon bonds, each of which matures on a coupon payment date. Hence, the coupon-bearing bond maturity can be seen as the mean of the maturities of its payments. It thus has a different “effective maturity”. We note that the change is the same for the zero-coupon and coupon-bearing bonds that mature in one year. However, a couponbearing bond with annual coupon payments that matures in one year makes not intermediate payments and thus behaves identically with a zero-coupon bond that matures in one year’s time. c. The bigger the initial interest rate the smaller the change in absolute value (all other things being equal). Contrasting Table 4.1 with Table 4.3 (for zero-coupon bonds) and Table 4.2 with Table 4.4 (for coupon-bearing bonds) we realize that when the initial interest rate is 4% its shift by ±1 percentage point leads to a bigger price

4

Coupon L

H Maturity

FIXED INCOME PORTFOLIO MANAGEMENT

Medium Sensitivity

High Sensitivity

Low Sensitivity

Medium Sensitivity

175

Interest Rate L

H L

H

Fig. 4.1 Price sensitivity to interest rate changes (Source Created by the author)

change (again in absolute value) than when the initial interest rate is 8% and is shifted by ±1% for all maturities. d. The interest rate increase does not lead to a symmetric change of the bond price in absolute value. Looking at all 4 tables we comprehend that an interest rate increase results in a price drop and an interest rate decrease results in a price increase which however differ in absolute value for each of the bonds, no matter what the initial interest rate is. As a matter of fact in all the bonds we consider and for both initial interest rates the increase of the interest rate causes a bigger (absolute) change than an equal decrease of the interest rate. This indicates that the change of the price change needs to be explored; this is practically the second derivative of the price as a function of the interest rate and will be introduced later in this chapter. Pictorially the sensitivity of the price to the interest rate is portrayed in Fig. 4.1. This analysis leads to one natural question: how to measure the impact of interest rate changes on the bond price. In the following sections we will present the applicable metrics that can be used to reflect how interest rate moves affect/cause price changes. 4.1.2

Duration

Duration is considered as the best single measure of the price change that results from the interest rate change, i.e. a bond (price) risk measure. This means that duration can be used on its own and give a somehow reliable indication of the sensitivity or volatility of the price of the bond as a function of the interest rate.

176

T. POUFINAS

Duration is defined with the use of the first derivative that was previously generated in Eq. (4.2), by essentially finding the relative or proportional price change to the initial price; which is practically what is of interest to the investor, as he or she needs to know the percentage price change. The first derivative produces the absolute change; however in bonds, as prices are most of the times referred to (or rather quoted) in terms of a face value of 100, i.e. out of 100, a normalization of the price change with regards to the price is necessary so as to find a comparable result. Continuing from Eq. (4.2) we find that   N  1  1 cn FV 1 dP =− · n· +N· . (4.4) P dr 1+r P (1 + r )n (1 + r ) N n=1

The quantity in the curly brackets is defined as the duration or Macaulay duration of the bond, named after Frederick Macaulay, who introduced the notion in 1938 (Macaulay, 1938) and is thus given by    N 1  cn FV n· +N· , (4.5) D := P (1 + r )n (1 + r ) N n=1

whereas the expression with the discounting factor included is defined as the modified duration of the bond 1 · Dm := 1+r



  N 1  1 dP cn FV D =− · . n· +N· = n N P (1 + r ) (1 + r ) 1+r P dr n=1

When a bond is zero-coupon, then Eq. (4.5) becomes 

 (1 + r ) N FV FV 1 = N· ·N· D= = N. N P (1 + r ) FV (1 + r ) N

(4.6)

(4.7)

The second equality of (4.7) above is derived by replacing P with its equal expression, using Eq. (2.13) of Chapter 2. Equation (4.7) reveals that for zero-coupon bonds duration is equal to the time to maturity of the bond. We will study the properties of duration, as well as its use as a measure of the price change of a bond.

4

4.1.3

FIXED INCOME PORTFOLIO MANAGEMENT

177

Properties of Duration

Duration, as does the bond price, depends on the interest rate (which is essentially the bond yield, as the interest rate may not be flat), the coupon and the time to maturity of the bond. We rewrite Eq. (4.5) to see that   N  N  1  cn FV Cn /P D= n· + N · n· , (4.8) = n N P (1 + r ) (1 + r ) (1 + r )n n=1

n=1

where C n is the payment made by the bond at time n, i.e. the coupon for all time instants prior to the maturity date and the coupon plus the face value at maturity. We realize that Eq. (4.8) can again be rewritten as D=

N  n=1

 Cn /P = n · wn , n (1 + r ) N

n·

(4.9)

n=1

where wn =

Cn /P Cn /(1 + r )n , = (1 + r )n P

n = 1 . . . N.

(4.10)

Summing up w n , for n = 1…N , we notice that N  n=1

wn =

  1 c1 1 c2 c N + FV = · · P = 1. + + · · · + P (1 + r )1 (1 + r )2 (1 + r ) N P (4.11)

As a result, these quotients, which multiply the time instants on which a payment is made, act as weights of these time instants. These weights are the present values of the payments made by the bond—normalized or adjusted as percentage of the price of the bond, as practically they are the components of the price of the bond. This means that the duration of a bond is the weighted sum of the time instants on which payments are made. These time instants are measured in years. The weights are pure numbers as C n and P are measured in monetary units, hence their ratio is a pure number, and the interest rate is a pure number. Hence, duration is measured in years. This explains the use of the term duration and

178

T. POUFINAS

unveils one important property of the duration; it pretty much measures time and more precisely the average time that the investor has to wait in order to recuperate the money he or she paid to purchase the bond—in present value terms. The fact that duration equals the time to maturity for zero-coupon bonds verifies this interpretation. A zero-coupon bond makes no payments before its maturity date and the investor has to wait until its maturity to collect the face value of the fund, whose present value is practically the price he or she paid to acquire it. Moreover, as the time to maturity is measured in years, it justifies that duration is also measured in years, thus it expresses time. Proposition 4.1 The duration of the bond is always smaller or equal to its time to maturity. Moreover, it is equal to its time to maturity when the bond is zero-coupon and (strictly) less than the time to maturity when the bond is coupon-bearing. Proof Equation (4.7) elaborates that for zero-coupon bonds the result already holds true. For coupon-coupon bearing bonds, we start with Eq. (4.5) and realize that all time instants before the time to maturity of the bond are less than its maturity date N (provided N > 1)  N    cn FV 1 · n· +N· D= P (1 + r )n (1 + r ) N n=1  N   1 cn FV < · N· +N· P (1 + r )n (1 + r ) N n=1  N   1 cn FV = ·N· + P (1 + r )n (1 + r ) N n=1

1 ·N·P=N = P ⇒D0 Pcash Pcash

(4.46)

as per Eq. (4.39) or the first equation of (4.45), which completes the proof of this part. Q.E.D. b. We use Eqs. (4.41) and (4.39) to see that (see also CIIA, 2004)

(1 + r )t · D= Pcash

N  n=1

N 

(n − t) · Cn n=1 ⇒D= N (1 + r )n 

(n−t)·Cn (1+r )n

n=1

Cn (1+r )n

,

(4.47)

204

T. POUFINAS

when P cash is replaced by its equal expression in (4.39). We observe that the duration is a function of t. We take the derivative of D with respect to t to find that  d

dD = dt

N 

n=1

 (n−t)·Cn (1+r )n

dt N  n=1

−

n=1

=

Cn (1+r )n

N 

N  n=1

Cn (1+r )n

= −1 < 0,

(4.48)

Cn (1+r )n

as the denominator is a constant function with respect to t. Consequently, the duration D is a decreasing function of t; moreover Eq. (4.48) shows that it decreases as a linear function of time t between two coupon payment dates, which is the desired result of this part. Q.E.D. 4.1.8

Portfolio Duration

As mentioned, we are interested in bonds not only as investment vehicles individually but primarily as parts of bigger portfolios. Investors form portfolios consisting of several bonds. It thus makes sense to calculate the duration of a bond portfolio. As a matter of fact the added value of duration is primarily spotted in its use as a measure of the portfolio sensitivity in interest rate shifts. Determining the price shift for each bond with the use of full non-linear pricing (with the use of the present value formula) could be tedious and time-consuming. Duration provides a quick calculation which for small interest rate moves provides relatively accurate results. Proposition 4.4 The duration of portfolio is given by Dport =

I 

wi · Di ,

(4.49)

i=1

where wi :

is the weight of bond i in the portfolio (in market value terms),

Di :

is the duration of bond i in the portfolio,

4

I:

FIXED INCOME PORTFOLIO MANAGEMENT

205

is the number of bonds in the portfolio.

Proof Let us consider a portfolio consisting of I bonds. We will assume that all bonds mature at the maturity date of the longest maturing bond. All payments past their actual maturity date are obviously zero. Without loss of generality we will assume that all payments made by the bonds in the portfolio take place on the same coupon payment dates. If not, then we take the union of the payment dates and whenever a bond does not make an actual payment, then the amount is zero. The value of the portfolio is (see also Luenberger, 1998) Pport =

I 

xi · Pi .

(4.50)

i=1

The payment that is made by all the bonds in the portfolio at time n is port

Cn

:=

I 

xi · Cni .

(4.51)

i=1

In Eqs. (4.50) and (4.51) for i = 1…I and n = 1…N Cni : xi : N: Pi :

is the payment made by bond i at time n, is the number of the minimum denomination securities (bonds) held from bond issue i, is the time to maturity of the longest maturing bond, is the price (present value) of bond i.

The contribution of bond i to the value of the portfolio is wi :=

xi · Pi . Pport

(4.52)

As (4.50) holds true, I  i=1

wi =

I  Pport xi · Pi = = 1. Pport Pport i=1

(4.53)

206

T. POUFINAS

The duration of the portfolio is given by Dport = Dport = Dport = Dport =

1

N 

Pport

n=1

1

I 

Pport

i=1

I 

1

i=1

Pport

I 

I n·

· Cni ⇒ (1 + r )n i=1 x i

xi · Pi ·

N 1  Cni · n· ⇒ Pi (1 + r )n n=1

(4.54)

· xi · Pi · Di ⇒

wi · Di

i=1

which is the desired result. Q.E.D. 4.1.9

Duration for Multi-Period Compounding

If there are more than one compounding periods per year—let us say ν, then the duration is given by (see also Luenberger, 1998)  D=

  N N  1 n cn Cn /P n FV · · +N· , = P ν (1 + r/ν)n (1 + r ) N ν (1 + r/ν)n n=1

n=1

(4.55)

where n = 1…N measures the number of compounding periods. The modified duration is given by Dm =

D . 1 + r/ν

(4.56)

4

4.1.10

FIXED INCOME PORTFOLIO MANAGEMENT

207

Duration for Continuous Compounding

When compounding is continuous, then the formula for duration changes to D=

N 1  · n · Cn · e−r ·n . P

(4.57)

n=1

We observe that as the 1/(1 + r) factor is not present anymore, modified duration coincides with duration, i.e. Dm = D.

4.1.11

(4.58)

Convexity

From our argumentation it is clear that the graphic representation of the bond price with respect to its yield to maturity (or interest rate) is a convex curve. In addition, the use of duration as a measure of the change in the price of the bond with respect to the change the yield to maturity (or interest rate) is good only for small changes in the yields. Furthermore, the duration on its own is not a comparative measure of the volatility of the price of two different bonds. If two bonds have the same duration and the same yield then a change in the yield will not affect both bonds the same. The bond with the highest curvature is less affected, in the sense that it will have a higher price as interest rates rise or fall (Fig. 4.8). As can be seen in Fig. 4.8, bonds A and B have the same price at a certain interest rate (yield) level. When the interest rate moves the use of duration assigns to both bonds the same price, which in reality is not true. When yields change slightly, then the prices of both bonds A and B will change by about equal percentages, hence such an approximation may be acceptable. However, when yields change significantly, for example increase, then the prices of both bonds will fall, but the price of B will fall more than that of A. To overpass this deficiency of duration we introduce convexity, which is defined with the use of the second derivative of the bond price with

208

T. POUFINAS

Bond Price

PA*=PB*

Bond A

PA**>PB**

Bond B

PD**

Tangent (DuraƟon) y*

y**

Yield

Fig. 4.8 Convexity (Source Created by the author)

respect to the interest rate (or yield to maturity) Conv =

N  1 d2 P 1 1 Cn · = · n · (n + 1) · , P dr 2 P (1 + r )2 (1 + r )n

(4.59)

n=1

where C n is the payment made by the bond at time n, n = 1…N . Convexity measures the rate of change of the slope of the price-yield (or price-interest rate) curve with respect to the yield (or interest rate) change and changes with time. It complements duration as a bond (price) risk measure. Convexity is beneficial to the investor as (all other things being equal) it always has a positive price effect for increasing as well as decreasing prices. In addition, the higher the convexity, the higher the positive price effect is. Example 4.6 We calculate the convexity for the 6 bonds of Example 4.1 with the use of Eq. (4.59) for interest rate r = 4% and r = 8%. The results are posted in Tables 4.33, 4.34, 4.35, and 4.36. We observe that in a way similar to duration, for plain vanilla bonds a. Convexity increases as the time to maturity of the bond increases (all other things being equal), as evidenced by all four Tables 4.33–4.36.

4

FIXED INCOME PORTFOLIO MANAGEMENT

Table 4.33 Convexity of a zero-coupon bond (initial r = 4%)

209

Time to maturity

Convexity

1

10

20

1.85

101.70

388.31

1

10

20

1.85

80.75

240.68

1

10

20

1.71

94.31

360.08

1

10

20

1.71

71.22

183.92

Source Created by the author

Table 4.34 Convexity of a coupon-bearing bond (c = 4%, initial r = 4%)

Time to maturity

Convexity

Source Created by the author

Table 4.35 Convexity of a zero-coupon bond (initial r = 8%)

Time to maturity

Convexity

Source Created by the author

Table 4.36 Convexity of a coupon-bearing bond (c = 4%, initial r = 8%)

Time to maturity

Convexity

Source Created by the author

b. As the coupon increases the convexity decreases (again all other things being equal), as the comparison of Table 4.33 with Table 4.34 and Table 4.35 with Table 4.36 reveals. Coupon-bearing bonds have a smaller convexity than the corresponding zero-coupon bonds with the same maturity dates—except when the bond matures in one year. When the maturity date of the coupon-bearing bond is 1, then it makes no intermediate payments and thus behaves in a way similar to a zero-coupon bond.

210

T. POUFINAS

c. The bigger the interest rate is the smaller the convexity of the bond becomes (all other things being equal), as can be seen by contrasting Tables 4.33 and 4.35 (for zero-coupon bonds), as well as Tables 4.34 and 4.36 (for coupon bearing bonds). 4.1.12

Uses of Convexity

Convexity can be used to produce more accurate results of a bond price change when the interest rate changes. Employing the Taylor series of the bond price as a function of the interest rate r, we observe that 1 d2 P 1 dP · · r + · · r 2 + · · · ⇒ 1! dr 2! dr 2 P 1 1 dP 1 1 d2 P · r 2 + · · · ⇒ = · · · r + · · P 1! P dr 2! P dr 2 P 1 dP 1 1 d2 P ≈ · · r + · · · r 2 P P dr 2 P dr 2

P(r + r ) = P(r ) +

(4.60)

where the first implication is derived by subtracting P (r) from P (r + r) and dividing both sides by P = P (r), whereas the second leads to an approximate equality for r small if we drop terms of power equal to or higher than 3, them being very small. We recall though that the multiplier of r is nothing else than minus the modified duration—D m and the multiplier of r 2 is the convexity Conv. Equation (4.60) thus becomes P 1 ≈ −Dm · r + · Conv · r 2 ⇔ P 2 D 1 P ≈− · r + · Conv · r 2 ⇔ P 1+r 2 1 P ≈ −Dm · P · r + · Conv · P · r 2 ⇔ 2 1 D · P · r + · Conv · P · r 2 , P ≈ − 1+r 2

(4.61)

where the second and fourth equations are derived by replacing modified duration with its equal expression that incorporates duration. Equation (4.61) provides an improvement of the approximation produced with

4

FIXED INCOME PORTFOLIO MANAGEMENT

211

the use of duration only, as it corrects for the curvature of the actual bond price against the linear approximation generated by duration and modified duration in Eqs. (4.14) and (4.16) respectively. Equation (4.17) still holds true, i.e. P  ≈ P + P

(4.62)

with P given by (4.61). Equations (4.61) explain the use of convexity along with duration in order to approximate the bond price change that results from an interest rate change. They hold true, as was the case when duration only was used, provided that i. There is a parallel shift of the term structure by r. ii. The shift r is very small. iii. The interest rate change r is instantaneous. iv. The term structure is horizontal. Recall, that we will lift the last condition later on. Example 4.7 We repeat the calculations of Example 4.3 with the use of duration and convexity to find the approximate bond prices. Tables 4.9–4.12 change to Tables 4.37, 4.38, 4.39, and 4.40. This time we apply Eqs. (4.61) and (4.62) to derive the price change as well as the new price. Table 4.37 Price change of a zero-coupon bond (initial r = 4%) with duration and convexity Time to maturity

Interest rate

4% 5% % change 3% % change

Source Created by the author

1

10

20

961.54 952.38 −0.95% 970.87 0.97%

675.56 614.04 −9.11% 743.96 10.12%

456.39 377.48 −17.29% 553.01 21.17%

212

T. POUFINAS

Table 4.38 Price change of a coupon-bearing bond (c = 4%, initial r = 4%) with duration and convexity Time to maturity

Interest rate

4% 5% % change 3% % change

1

10

20

1,000.00 990.48 −0.95% 1,009.71 0.97%

1,000.00 922.93 −7.71% 1,085.15 8.51%

1,000.00 876.13 −12.39% 1,147.94 14.79%

Source Created by the author

Table 4.39 Price change of a zero-coupon bond (initial r = 8%) with duration and convexity Time to maturity

Interest rate

8% 9% % change 7% % change

1

10

20

925.93 917.43 −0.92% 934.58 0.93%

463.19 422.49 −8.79% 508.27 9.73%

214.55 178.68 −16.72% 258.14 20.32%

Source Created by the author

Table 4.40 Price change of a coupon-bearing bond (c = 4%, initial r = 8%) with duration and convexity Time to maturity

Interest rate

8% 9% % change 7% % change

Source Created by the author

1

10

20

962.96 954.13 −0.92% 971.96 0.93%

731.60 679.21 −7.16% 789.20 7.87%

607.27 543.90 −10.44% 681.81 12.27%

4

FIXED INCOME PORTFOLIO MANAGEMENT

213

Tables 4.37–4.40 exhibit results similar to the results that were derived with the use of duration only, as was seen in Tables 4.9–4.12. However the use of the convexity on top of the duration secures more accurate outcomes than the use of duration only. More precisely a. The output of Eqs. (4.61) and (4.62) is in line with the results of Tables 4.1–4.4, as well as Tables 4.9–4.12 in terms of the direction and the magnitude of the price change when the interest rate changes. When the interest rate increases the price drops and vice versa. b. Contrary to the changes delivered with the use of duration only, the price changes that are calculated with the use of convexity are not symmetric. This is in line with the actual price change that is derived when the present value bond pricing formula is employed. This is the merit of the use of convexity; the linear pricing of the duration formula is (positively) adjusted for curvature. c. The use of duration and convexity underestimates the price change when interest rates increase or decrease (in absolute values) compared with the price change that is produced with the use of the present value bond pricing formula (at least in this example). d. The deviation increases (in absolute value) as the time to maturity increases, as the coupon decreases and as the interest rate decreases. e. The deviation is much smaller in this example (less than 1% in all cases) when convexity is introduced, even though the interest rate change of ±1 percentage point is not a small change. This finding confirms the improvement of the approximation with convexity. This is further illustrated in the following example. Example 4.8 We calculate the deviation of the price change with and without the use of duration and convexity for the bonds of Example 4.7 as we did in Example 4.3. The results are displayed in Tables 4.41, 4.42, 4.43 and 4.44. The results—as in the case of the use of duration only—are in line with point (d) and (e) above. Although the deviation is much smaller when convexity is inserted, we investigate once more what would have happened if the interest rate change had been smaller.

214

T. POUFINAS

Table 4.41 Deviation of the percentage price change of a zero-coupon bond (initial r = 4%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 4 to 5% From 4 to 3%

1

10

20

−0.01% −0.01%

−0.21% −0.20%

−0.74% −0.68%

Source Created by the author

Table 4.42 Deviation of the percentage price change of coupon-bearing bond (c = 4%, initial r = 4%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 4 to 5% From 4 to 3%

1

10

20

−0.01% −0.01%

−0.19% −0.18%

−0.60% −0.56%

Source Created by the author

Table 4.43 Deviation of the percentage price change of a zero-coupon bond (initial r = 8%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 8 to 9% From 8 to 7%

1

10

20

−0.01% −0.01%

−0.19% −0.18%

−0.69% −0.63%

Source Created by the author

Example 4.9 We repeat Examples 4.7 and 4.8 with an interest rate change of ±0.10 percentage points. The findings are shown in Tables 4.45, 4.46, 4.47, 4.48, 4.49, 4.50, 4.51, and 4.52.

4

215

FIXED INCOME PORTFOLIO MANAGEMENT

Table 4.44 Deviation of the percentage price change of coupon-bearing bond (c = 4%, initial r = 8%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 8 to 9% From 8 to 7%

1

10

20

−0.01% −0.01%

−0.17% −0.17%

−0.52% −0.49%

Source Created by the author

Table 4.45 Price change of a zero-coupon bond (initial r = 4%) with duration and convexity Time to maturity

Interest rate

4% 4.10% % change 3.90% % change

1

10

20

961.54 960.61 −0.10% 962.46 0.10%

675.56 669.10 −0.96% 682.09 0.97%

456.39 447.70 −1.90% 465.25 1.94%

Source Created by the author

Table 4.46 Price change of a coupon-bearing bond (c = 4%, initial r = 4%) with duration and convexity Time to maturity

Interest rate

4% 4.10% % change 3.90% % change

Source Created by the author

1

10

20

1,000.00 999.04 −0.10% 1,000.96 0.10%

1,000.00 991.93 −0.81% 1,008.15 0.82%

1,000.00 986.53 −1.35% 1,013.71 1.37%

216

T. POUFINAS

Table 4.47 Price change of a zero-coupon bond (initial r = 8%) with duration and convexity Time to maturity

Interest rate

8% 8.10% % change 7.90% % change

1

10

20

925.93 925.07 −0.09% 926.78 0.09%

463.19 458.93 −0.92% 467.50 0.93%

214.55 210.61 −1.83% 218.56 1.87%

Source Created by the author

Table 4.48 Price change of a coupon-bearing bond (c = 4%, initial r = 8%) with duration and convexity Time to maturity

Interest rate

8% 8.10% % change 7.90% % change

1

10

20

962.96 962.07 −0.09% 963.86 0.09%

731.60 726.12 −0.75% 737.12 0.76%

607.27 600.43 −1.13% 614.23 1.14%

Source Created by the author

Table 4.49 Deviation of the percentage price change of a zero-coupon bond (initial r = 4%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 4 to 4.10% From 4 to 3.90%

Source Created by the author

1

10

0.00% 0.00%

0.00% 0.00%

20 −0.01% −0.01%

4

217

FIXED INCOME PORTFOLIO MANAGEMENT

Table 4.50 Deviation of the percentage price change of coupon-bearing bond (c = 4%, initial r = 4%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 4 to 4.10% From 4 to 3.90%

1

10

0.00% 0.00%

0.00% 0.00%

20 −0.01% −0.01%

Source Created by the author

Table 4.51 Deviation of the percentage price change of a zero-coupon bond (initial r = 8%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 8 to 8.10% From 8 to 7.90%

1

10

0.00% 0.00%

0.00% 0.00%

20 −0.01% −0.01%

Source Created by the author

Table 4.52 Deviation of the percentage price change of a coupon-bearing bond (c = 4%, initial r = 8%) with the use of duration and convexity from the actual percentage price change Time to maturity

Deviation

From 8 to 9% From 8 to 7%

1

10

0.00% 0.00%

0.00% 0.00%

20 −0.01% −0.01%

Source Created by the author

Comparing Tables 4.45–4.52 with Tables 4.37–4.44 we observe that a. The sensitivity of the price to the interest rate change is much smaller with the use of convexity and duration in magnitude but

218

T. POUFINAS

depends again on the same parameters; the time to maturity, the interest rate and the coupon rate. b. The deviation of the price change with the use of duration and convexity compared with the price change with the use of the present value formula is much smaller when the interest rate change becomes smaller—and in this example it is 0 (with at least threedecimal-digit accuracy). Consequently, we realize that the use of the duration and convexity secures much more accurate results when the interest rate move is smaller. This is probably expected as the estimate of the price with the use of duration and convexity according to formulas (4.61) is based on Taylor’s polynomial approximation. Its application requires a small r. 4.1.13

Convexity Between Coupon Payment Dates

The convexity between two coupon payment dates can be derived in a way similar to the one the corresponding duration was found. We differentiate Eq. (4.40) with respect to the interest rate r to see that for a time instant t between 0 and 1 N (1 + r )t  (n − t) · (n − t + 1) · Cn d 2 Pcash = · . dr 2 (1 + r )2 (1 + r )n

(4.63)

n=1

Consequently, Conv =

1 Pcash

·

N d 2 Pcash (1 + r )t 1  (n − t) · (n − t + 1) · Cn · = · . dr 2 (1 + r )2 P (1 + r )n n=1

4.1.14

(4.64)

Portfolio Convexity

The convexity of a portfolio of bonds (or fixed income securities in general) is given by a formula that is analogous to that of the duration of the portfolio.

4

FIXED INCOME PORTFOLIO MANAGEMENT

219

Proposition 4.5 The convexity of portfolio is given by Convport =

I 

wi · Convi ,

(4.65)

i=1

where wi :

is the weight of bond i in the portfolio (in market value terms),

Convi : I:

is the convexity of bond i in the portfolio,

is the number of bonds in the portfolio.

Proof The proof is similar to the one given for the bond duration and can be repeated with duration replaced by convexity. Q.E.D.

4.1.15

Duration for Non-Horizontal Yield Curve

We are now in place to generalize the use of duration for non-horizontal term structures. This is an important step, as we realized that interest rates are not constant and depend on the maturity of the investment. Consequently, it is important to have a bond (price) risk measure even when the term structure is not flat. We thus move from a specific fixed (flat) yield framework to a general term structure framework. However, the shift of the term structure can only be parallel. We distinguish two extensions; one for discrete-time compounding and another for continuous-time compounding (Luenberger, 1998). As the term structure is not horizontal any more, we denote the parallel shift by l. The first three assumptions of the use of duration still hold true, i.e. i. There is a parallel shift of the term structure by l. ii. The shift l is very small. iii. The interest rate change l is instantaneous.

220

T. POUFINAS

4.1.15.1 Quasi-Modified Duration Without loss of generality we will assume that the compounding is annual; the generalization for multi-period compounding within one year is relatively easy as soon as the relevant formulas have been established. The price P (l ) of a bond for a parallel term structure shift by l is (see also Luenberger, 1998) P(l) = PV0 (l) =

N  n=1

Cn , (1 + sn + l)n

(4.66)

where Cn : sn : N: PV0 :

is is is is

the the the the

payment made by the bond at time n, (current) spot interest rate used for discounting, time to maturity of the bond, present value of the bond.

We differentiate with respect to l to find that  N  d P(0) d P(l)  Cn ≡ = − n· ,  dl dl l=0 (1 + sn )n+1

(4.67)

n=1

which prompts to modified duration as expressed by Eq. (4.6). To reach a similar expression, we need to drop the negative sign and divide by the bond price. We thus define the quasi-modified duration as Dqm := −

N d P(0) 1  1 Cn · =− · n· , P(0) dl P (1 + sn )n+1

(4.68)

n=1

for P (0) = P the current price of the bond. We remark that a. The quasi-modified duration reflects the bond price change/sensitivity with respect to a parallel shift in the term structure. b. The quasi-modified duration is still measured in time.

4

FIXED INCOME PORTFOLIO MANAGEMENT

221

c. Reaching an expression equivalent to duration would require pulling out a common factor, which is not possible when the term structure is not horizontal. When the interest rate was constant, then there was a common factor of 1/(1 + r), which led to the definition of duration. Such a common factor does not exist this time. d. As a common factor cannot be pulled out, the sum in Eq. (4.68) is not the weighted sum of the time instants on which a payment is made by the bond. If compounding had not been annual, bet there had been ν compounding periods per year, then Eq. (4.68) would have changed to Dqm := −

N 1 n Cn 1 d P(0) · = · · P dl P ν (1 + sn(ν) /ν)n+1 n=1

(4.69)

with n this time counting compounding periods instead of years. Example 4.10 We consider the bond of Example 3.6, with the term structure of Example 3.5. More precisely, the bond matures in 6 years, with a coupon rate of 4% paid annually and a face value per minimum denomination bond (security) of 1,000 Euro. The spot rates are given by s1 = 1.01%, s2 = 1.53%, s3 = 1.72%, s4 = 1.83%, s5 = 1.90% and s6 = 1.96%.

(4.70)

The price of the bond has been found to be 1,115.60 Euro. The quasimodified duration is  40 1 40 40 Dqm = · 1· +2· +3· 1, 115.60 (1.0101)2 (1.0153)3 (1.0172)4  40 40 40 + 1000 = 5.38 (4.71) +4 · + 5 · + 6 · (1.0190)6 (1.0196)7 (1.0183)5 as ν = 1.

222

T. POUFINAS

4.1.15.2 Fisher-Weil Duration In a similar manner we derive a bond (price) risk measure when compounding is continuous. As before the price P (l ) of a bond for a parallel term structure shift by l is (see also Luenberger, 1998) P(l) =

N 

Cn · e−(sn +l)·n .

(4.72)

n=1

We take the first derivative with respect to l to find that  N  d P(l)  d P(0) ≡ = − n · Cn · e−sn ·n . dl dl l=0

(4.73)

n=1

We define the Fisher-Weil duration as DFW := −

N d P(0) 1  1 · = · n · Cn · e−sn ·n P(0) dl P

(4.74)

n=1

for P (0) = P the current price of the bond. We see that a. The Fisher-Weil duration is a measure the bond price change/sensitivity with respect to a parallel shift in the term structure. b. The Fisher-Weil duration is still measured in time. c. The Fisher-Weil duration is the weighted sum of the time instants on which a payment is made by the bond as N  n=1

(Cn /P) · e−sn ·n =

N 1  1 · · P = 1. Cn · e−sn ·n = P P

(4.75)

n=1

The roots of Fisher-Weil duration can be found in 1971 (Fisher & Weil, 1971).

4

223

FIXED INCOME PORTFOLIO MANAGEMENT

Example 4.11 We repeat Example 4.10 with continuous compounding to see that P=

N 

Cn · e−sn ·n = 40 · e−0.0101·1 + 40 · e−0.0153·2 + 40 · e−0.0172·3

n=1

+ 40 · e−0.0183·4 + 40 · e−0.0190·5 + 1, 040 · e−0.0196·6 = 1, 114.46 (4.76) and DFW =

 1 · 1 · 40 · e−0.0101·1 + 2 · 40 · e−0.0153·2 1, 114.46 + 3 · 40 · e−0.0172·3 + 4 · 40 · e−0.0183·4  +5 · 40 · e−0.0190·5 + 6 · 1, 040 · e−0.0196·6 = 5.48 .

4.1.16

(4.77)

Other Measures

There are more bond price volatility measures that are used instead of duration; however, the output they produce when they are used to estimate the bond price change is not as accurate as the one delivered with the use of duration (and convexity). For the sake of completeness we present them here. These are, the time to maturity, the average weighted maturity, the weighted average cash flow, the price value of a basis point and the yield value of a price change. 4.1.16.1 Time to Maturity The time to maturity is probably the easiest metric to use. It is the number of years remaining until the bond matures. As we have seen the bond price sensitivity increases with its maturity date. The time to maturity is not a very good proxy of the bond price risk, because i. It ignores the (potential) payments made by the bond before its maturity date, which can lead to mistakes in the risk assessment process.

224

T. POUFINAS

Table 4.53 Time to maturity of a zero-coupon bond or coupon-bearing bond (c = 4%, initial r = 4% or r = 8%)

Time to maturity

Time to maturity

1

10

20

1.00

10.00

20.00

Source Created by the author

ii. It silently assumes that there is a linear relationship between time to maturity and price volatility, which we know is not the case because of the curvature of the bond price with respect to the interest rate. Example 4.12 Let us consider the bonds of example 4.1. We realize that the time to maturity is equal to the maturity date of the bond, no matter what the coupon or the interest rate is. I.e. for the zero-coupon and the couponbearing bonds maturing in 1, 10 or 20 years the time to maturity is 1, 10 or 20 years respectively, regardless of whether the initial interest rate is 4% or 8% (Table 4.53). 4.1.16.2 Weighted Average Maturity The weighted average maturity (WAM) or weighted average life (WAL) is the weighted average of the principal repayment dates. It does not account for coupon payments, but only for principal repayments. It is given by N  Pr n · n, WAM := Pr Total n=1

where N: Prn : PrTotal :

is the maturity date of the bond or bond portfolio, is the principal repayment at time n, is the total principal to be repaid.

The weighted average maturity is still a not very good proxy as

(4.78)

4

Table 4.54 Weighted average maturity of a zero-coupon bond or a coupon-bearing bond (c = 4%, initial r = 4% or r = 8%)

FIXED INCOME PORTFOLIO MANAGEMENT

225

Time to maturity

Weighted average maturity

1

10

20

1.00

10.00

20.00

Source Created by the author

i. For plain vanilla bonds it equals to the time to maturity; it is an improved measure for sinking funds and mortgage backed securities. ii. It ignores coupons and thus does not consider the full impact of the bond cash flow distribution on its risk. As per (i) all zero-coupon or coupon-bearing bonds have a weighted average maturity equal to their time to maturity as they make one single principal payment on their maturity date WAM :=

Pr n FV · N = N. ·N = Pr Total FV

(4.79)

Example 4.13 The bonds of Example 4.1 have a weighted average maturity equal to their time to maturity no matter what their coupon or interest rate is (Table 4.54). Example 4.14 Consider an obligation that makes a principal repayment of 100 Euro per year for the next 10 years. The total principal paid is 1,000 and the weighted average maturity is found by applying Eq. (4.78) WAM =

10  100 · n = 5.5. 1, 000

(4.80)

n=1

4.1.16.3 Weighted Average Cash Flow The weighted average cash flow (WACF) is the weighted average of the cash flow payment dates. It resembles the weighted average maturity, with

226

T. POUFINAS

the exception that it considers all the cash flows, including coupons and principal repayments. It is calculated by WACF :=

N  CFn · n, CFTotal

(4.81)

n=1

where N: CFn : CFTotal :

is the maturity date of the bond, is the cash flow paid at time n, is the total cash flow amount to be repaid.

For zero-coupon bonds Eq. (4.79) becomes WACF :=

CF N FV · N = N, ·N = CFTotal FV

(4.82)

i.e. it equals the maturity date and the duration of the bond. Although an improved metric, the weighted average cash flow still has one drawback; cash flow payments are considered on a nominal and not on a present value basis. It is as if interest rates are set equal to 0% in a present value framework. Example 4.15 For the bonds of Example 4.1 we find that the zero coupon bonds have weighted average cash flow equal to the maturity date, which is equal to duration, as evidenced by Eq. (4.82). For the remaining bonds (Tables 4.55 and 4.56).

Table 4.55 Weighted average cash flow of a zero-coupon bond (initial r = 4% or r = 8%)

Time to maturity

Weighted average cash flow Source Created by the author

1

10

20

1.00

10.00

20.00

4

Table 4.56 Weighted average cash flow of a coupon-bearing bond (c = 4%, initial r = 4% or r = 8%)

FIXED INCOME PORTFOLIO MANAGEMENT

227

Time to maturity

Weighted average cash flow

1

10

20

1.00

8.71

15.78

Source Created by the author

4.1.16.4 Price Value of a Basis Point The price value of a basis point (PVBP) measures the change in the price of a bond when the bond yield changes by 1 bp (0.01% or 0.0001) in absolute value terms. For small changes in yield such as 1 bp the percentage change in the bond price is about the same regardless of whether the yield increases or decreases. Observe that the price value of a basis point measures the volatility of price change in terms of monetary units (e.g. Euro or dollars). If we divide it by the initial price then we get the percentage price change for a 1-basis-pont (PPCBP) shift in the term structure (or yield curve). Example 4.16 We use the bonds of Example 4.1 and apply an interest rate shift of ±1 bps to find the corresponding price value of a basis point in Tables 4.57, 4.58, 4.59 and 4.60. We observe that a. The properties of a parallel shift of a flat yield curve remain the same as the ones observed in Example 4.1. b. Even though we use full nonlinear pricing a 1 basis point increase leads to the same price value basis point as a 1 basis point decrease. Table 4.57 Price value of a basis point and percentage price change of a basis point of a zero-coupon bond (initial r = 4%)

Time to maturity

PVBP PPCBP

1

10

20

0.09 0.01%

0.65 0.10%

0.88 0.19%

Source Created by the author

228

T. POUFINAS

Table 4.58 Price value of a basis point and percentage price change of a basis point of a coupon-bearing bond (c = 4%, initial r = 4%)

Time to maturity

PVBP PPCBP

1

10

20

0.10 0.01%

0.81 0.08%

1.36 0.14%

1

10

20

0.09 0.01%

0.43 0.09%

0.40 0.19%

1

10

20

0.09 0.01%

0.55 0.08%

0.69 0.11%

Source Created by the author

Table 4.59 Price value of a basis point and percentage price change of a basis point of a zero-coupon bond (initial r = 8%)

Time to maturity

PVBP PPCBP

Source Created by the author

Table 4.60 Price value of a basis point and percentage price change of a basis point of a coupon-bearing bond (c = 4%, initial r = 8%)

Time to maturity

PVBP PPCBP

Source Created by the author

Some authors calculate the price value of a basis point by finding the price with a 1 basis point increase, the price with a 1 basis point decrease, take their difference and divide by 2. As a ±1 bps shift is very small the outcome of this calculation for the bonds of Example 4.16 is identical since a 1 bp increase produces almost the same price change in absolute terms with a 1 bp decrease. Some other authors use modified duration to derive an approximation. They apply Eq. (4.16) for r = 0.01%. Again, as anticipated, as a 1 bp shift is very small the output is very close to the one derived by our definition. As a matter of fact this can be readily verified for the bonds of Example 4.16.

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4.1.16.5 Yield Value of a Price Change The yield value of a price change (YVPC) measures the change in the bond yield from a certain price change. To find it we calculate the bond yield when a specific price change is applied. The yield value of a price change is the difference between the new yield and the initial yield. The smaller the yield value of a price change is, the greater the price volatility becomes, as it would take a smaller change in yield to result in a specific price change. Example 4.17 We calculate the yield value of a price change for the bonds of Example 4.1. We consider a 10 Euro price decrease. The results are illustrated in Tables 4.61, 4.62, 4.63 and 4.64.

Table 4.61 Yield value of a price change of -10 Euro of a zero-coupon bond (initial r = 4%)

Time to maturity

YVPC

1

10

20

1.09300%

0.15521%

0.11527%

1

10

20

1.05050%

0.12405%

0.07406%

1

10

20

1.17913%

0.23598%

0.25805%

Source Created by the author

Table 4.62 Yield value of a price change of -10 Euro of a coupon-bearing bond (c = 4%, initial r = 4%)

Time to maturity

YVPC

Source Created by the author

Table 4.63 Yield value of a price change of -10 Euro of a zero-coupon bond (initial r = 8%)

Time to maturity

YVPC

Source Created by the author

230

T. POUFINAS

Table 4.64 Yield value of a price change of -10 Euro of a coupon-bearing bond (c = 4%, initial r = 8%)

Time to maturity

YVPC

1

10

20

1.13331%

0.18342%

0.14675%

Source Created by the author

4.2

Passive Bond Portfolio Management

Passive bond portfolio management considers the prices of securities in the market as fair and more or less correctly determined. It does not attempt to outperform the market by exploiting some information or knowledge. It tries to maintain a certain balance between risk and return given the market opportunities. Managers simply try to monitor, manage and mitigate the risk assumed by their portfolio. Passive Bond Management usually comes in 3 forms (following Bodie et al., 1996): i. Bond Indexing ii. Immunization iii. Cash Flow Matching 4.2.1

Bond Indexing

This strategy seeks to copy the allocation and performance of a bond index. In other words, the portfolio that will be formed based on this index will have the same risk—return profile with the portfolio underlying the index. Bond portfolio managers that pursue this strategy follow indices that are determined within their investment mandate(s), which means that they try to replicate the index as closely as they can in terms of composition and thus in terms of risk and return. It is common though that their investment mandate is not limited to one single index but rather prescribes a blend of indices, with a certain weight assigned to each index. The index or indices used depend on the types of fixed income instruments that the managers wish to include in their portfolios, in line with the targets and risk –return profiles of their customers—investors. There

4

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is a wide set of indices available, provided by a series of investment firms or banks. Some of them follow (the list is not exhaustive). The bond indices that are mainly used in the USA are: • • • •

(Bank of America) Merrill Lynch Domestic Master (Barclays) Lehman Brothers US Treasury Index The Capital Markets Bond Index Citi US Broad Investment-Grade Bond Index (USBIG).

These indices contain a few thousand (in some cases even each) of government, corporate, mortgage-backed and Yankee bonds with maturities of more than one year. Well known global bond indices are: • (Bank of America) Merrill Lynch Global Bond Index • Bloomberg Barclays Global Aggregate Bond Index • Citi World Broad Investment-Grade Bond Index (WorldBIG). Some government bond indices are: • • • •

Barclays Inflation-Linked Euro Government Bond Index Citi World Government Bond Index (WGBI) FTSE UK Gilts Index Series J.P. Morgan Government Bond Index.

Representative emerging markets indices are: • J.P. Morgan Emerging Markets Bond Index • Citi Emerging Markets Broad Bond Index (EMUSDBBI). Typical high-yield bond indices are: • • • • • •

(Bank of America) Merrill Lynch High-Yield Master II Barclays High-Yield Index Bear Stearns High-Yield Index Citi US High-Yield Market Index (Credit Suisse) First Boston High-Yield II Index S&P US Issued High-Yield Corporate Bond Index.

232

T. POUFINAS

4.2.1.1 Potential Problems of Bond Indexing There are some issues that could hinder bond portfolio (or fund) managers from replicating the index fully or accurately. Some of them are a. Each index contains a big number of different bond issues (even a few thousand in some cases). Even if the different bond issues were fewer, the size of the fund may not be large enough to allow it to include all possible issues with the weights that they contribute to the index. b. Some bonds may not have the required liquidity, making trading difficult for managers. Lack of liquidity affects prices so that they are (potentially) no longer fair. c. Bonds are removed from the indices as they mature or their maturity date approaches (usually when it become less than one year). Managers will need to adjust or rebalance their portfolios. d. New bonds are issued and are included in the indices. In this case, too, managers must include these bonds in their portfolios. So they have to deal with the phenomenon of a continuous change of the indices by rebalancing their portfolios so that they follow the indices with precision. e. Bonds may pay a coupon which must be reinvested. This makes the replication of the index by a manager even more difficult. Consequently, passive bond portfolio management does not mean that the manager does not act!!! 4.2.1.2 Potential Solutions to Bond-Indexing Problems The usual solution to the above issues that may arise is to follow the index as closely as possible. To achieve this, the cellular approach may be applied (Bodie et al., 1996). According to this, the bonds in the market of focus—and as a result of the index which is considered to be a good replica of the market—are categorized according to different criteria into different subcategories. Bonds belonging to the same category (cell) are considered relatively similar (and potentially can be used interchangeably in terms of investability). The percentages of the bonds corresponding to each category (cell) are calculated. The manager then constructs a bond portfolio by selecting representative bonds from each category (cell),

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matching in this way (with some approximation) the bond market/index of interest. He or she picks—most likely—bonds that are the most liquid. The resulting portfolio may not be identical to the index; however the main characteristics are maintained. These usually include the breakdown of the maturity dates, the coupon rates paid by the bonds, the credit quality (or default risk) of the issuers, the industry in which the issuers is active, etc. Since the resulting portfolio is as close as possible to the underlying index—at least with regards to the aforementioned attributes—it is anticipated that its performance will also be very close to that of the index. A measure of how well the above technique (or any other index-linked investment strategy in general) works is the tracking error. Depending on the convention, the tracking error is defined either as the difference between the (percentage) performance of the portfolio and the (percentage) performance of its underlying index or the standard deviation of this difference. We will be using the latter. The bigger the portfolio the better it can mimic/follow the index and therefore the smaller the tracking error. Example 4.18 Assume that a bond portfolio manager wishes to apply the cellular approach to his or her bond universe, which consists of global government bonds only, valued in the same currency. He or she considers all possible credit ratings and all possible maturity dates and comes up with the following hive (percentages are arbitrary) (Table 4.65): The manager then will select out of each cell the bonds that he or she considers the most representative; most likely the ones that exhibit the highest liquidity or that potentially offer the highest yield. The latter though already constitutes active bond portfolio management that will be visited in Sect. 4.3. 4.2.2

Immunization

Portfolio immunization is a different approach that can be followed in order to construct a bond portfolio. Instead of following an underlying index portfolio immunization dictates that the manager immunizes/protects his or her portfolio under management from interest rate risk as a whole.

234

T. POUFINAS

Table 4.65 Cellular approach Time to maturity (in years)

Credit rating

AAA AA A BBB BB B CCC CC C D

30

1.1% 2.2% 3.3% 4.4% 5.5% 4.5% 3.4% 0.0% 0.0% 0.0%

0.2% 0.4% 0.7% 0.9% 1.1% 0.9% 0.7% 0.0% 0.0% 0.0%

0.3% 0.6% 0.8% 1.1% 1.4% 1.1% 0.9% 0.0% 0.0% 0.0%

0.6% 1.1% 1.7% 2.2% 2.8% 2.3% 1.7% 0.0% 0.0% 0.0%

1.7% 3.3% 5.0% 6.6% 8.3% 6.8% 5.1% 0.0% 0.0% 0.0%

0.3% 0.6% 0.8% 1.1% 1.4% 1.1% 0.9% 0.0% 0.0% 0.0%

0.2% 0.4% 0.7% 0.9% 1.1% 0.9% 0.7% 0.0% 0.0% 0.0%

0.2% 0.4% 0.7% 0.9% 1.1% 0.9% 0.7% 0.0% 0.0% 0.0%

1.1% 2.2% 3.3% 4.4% 5.5% 4.5% 3.4% 0.0% 0.0% 0.0%

Source Created by the author

To understand how this approach works and what the potentially different strategies that can be followed are we distinguish two cases or types of investors (see also Bodie et al., 1996; Luenberger,1998): i. Financial institutions, such as banks, that have a short-term (or shorter-term) horizon whose liability is to make payments at present time and are thus interested in securing the net worth or market value or present value of their fixed income portfolio from interest rate fluctuations. ii. Financial institutions, such as pension funds or insurance companies that have a long-term (or longer-term) horizon whose liability is to make payments in the future and are therefore interested in safeguarding the future value of their fixed income portfolio from interest rate fluctuations. In both cases the source of risk is the same; namely the interest rate fluctuations. Interest rate fluctuations change the values (both present and future) of the portfolios and thus impact the ability of the financial institutions to meet their short-term or long-term liabilities. Consequently, both types of financial institutions are interested in methods that will allow them to monitor, manage and mitigate this risk. With the right choice of maturities in their portfolios they can reduce or eliminate the

4

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risk from interest rate moves. With the term immunization we describe any investment approach that can be employed to protect a position from changes in interest rates. 4.2.3

Net Worth Immunization

Many financial institutions, such as banks, wish to match their assets with their liabilities (see also Bodie et al., 1996). Such a match does not happen automatically and is desired both in terms of payment/cash flow amounts and in terms of payment/cash flow time instants. The liabilities of the banks (that pursue traditional banking activities such as offering savings accounts and loans) are primarily customer deposits, the majority of which are short-term and therefore have a low duration. The assets of the banks are mainly loans—commercial, consumer or mortgage loans— which are mid-term or long-term and therefore have a longer duration than deposits. As a result, they are more sensitive to changes in interest rates than deposits (Fig. 4.9). Figure 4.9 depicts the loans and the deposits of the bank, which are its assets and liabilities respectively. It gathers all loans into a single payment that is to be made on its duration and all liabilities into a single payment that is also to be made on its duration. In reality this is not the case; loans as well as deposits spread over time. However, loans tend to have much longer maturities than deposits. Loans and deposits behave like fixed income securities as they make streams of payments that are determined from an interest rate that is applied to a nominal value (the outstanding loan amount or the deposit amount respectively). Loans (Assets)

DL

DA

Deposits (LiabiliƟes) Fig. 4.9 Bank assets and liabilities (Source Created by the author)

Time

236

T. POUFINAS

If interest rates rise, then banks experience a drop in their net worth as the (present) value of their assets (loans) falls more than the (present) value of their liabilities (deposits), since the former tend to have longer maturities than the latter. To tackle this mismatch between the payment times of their assets and liabilities banks apply an approach which is known as asset liability management . It pretty much tries to match the assets with the liabilities. In the context of duration, the methods employed are often referred to as gap management , as they aim to reduce/close the gap between the duration of assets and liabilities. To illustrate, we will assume (without loss of generality) that the loans and the deposits offer the same flat interest rate. The bank wants at minimum to have the present value of its assets, equal to the present value of its liabilities: PV A = PV L .

(4.83)

However, at the same time, the duration of its assets is higher than the duration of its liabilities: D A > DL .

(4.84)

If interest rates increase, let us say by a small parallel shift r then both the present value of assets and the present value of liabilities will decrease. However, the present value of assets will decrease more than the present value of liabilities as: PV A ≈ −

1 1 · D A · PV A · r < − · D L · PV L · r ≈ PV L 1+r 1+r (4.85)

as a result of (4.83) and (4.84) and the fact that r is positive (due to the interest rate increase). Consequently, the present value of the assets will be lower than the present value of the liabilities of the bank after the interest rate shift (denoted with a “prime”, following the notation of [4.62]) 



PVA ≈ PV A + PV A < PV L + PV L ≈ PV L .

(4.86)

As a result, the bank will not have the necessary assets to match its liabilities. To tackle that the bank attempts to close the duration gap and thus

4

237

FIXED INCOME PORTFOLIO MANAGEMENT

set the duration of its assets equal to the duration of its liabilities, i.e. D A = DL .

(4.87)

If the bank achieves that, then combined with (4.83), a small increase r in the interest rates will result in the change in assets being (almost) equal to the change in liabilities PV A ≈ −

1 1 · D A · PV A · r = − · D L · PV L · r ≈ PV L . 1+r 1+r (4.88)

The latter yields that 



PVA ≈ PV A + PV A ≈ PV L + PV L ≈ PV L ,

(4.89)

which means that roughly the assets continue to match the liabilities of the bank. Achieving at minimum the equality of the present value of the assets with the present value of the liabilities and the equality of the duration of the assets with the duration of the liabilities is the objective of net worth immunization. The focus is in the short-term (net worth) as the bank liabilities (deposits) are short-term. This determines the liability driven investment (LDI) strategy that the bank has to follow with respect to its assets. Banks therefore try to reduce the duration of their loans and increase the duration of their deposits. One way to achieve lower duration of the assets is through adjustable rate mortgages. The value of these loans is affected less than the fixed rate loans, since their interest rates are linked to some index of the reference/underlying interest rate. As a matter of fact in this way they transfer part of the interest rate risk to the borrowers. On the liability side, one way to increase/extend the duration of the liabilities is through long(er)-term time deposits, such as CDs (Certificates of Deposit). At the same time, banks may issue bonds with longer maturities and convert some of deposits of their customers to bond holdings. These actions reduce the duration gap. Financial engineering has given additional tools that can help in this direction; mortgage backed securities are structured products—considered as derivatives—that allow banks to securitize their mortgage portfolio and offer it to investors, thus collect the required cash earlier than

238

T. POUFINAS

the maturity dates of the mortgage loans. The same approach may be applied to any loan portfolio. These types of securities will be examined in Chapter 5. In this way the bank tries to immunize its overall position from interest rate movements. Its assets are approximately equal to its liabilities (in terms of present value). If their durations are also equal then any change in interest rates will affect both of them equally. Its total net worth will not change. In summary, net worth immunization requires a portfolio with a total duration of zero. This may be achieved if the assets and the liabilities have the same size and duration. 4.2.4

Target Date Immunization

Pension funds and insurance companies differ from banks in that their liabilities are primarily further into the future. Consequently, they are interested more in being able to make future payments than in their net worth at present times. Pension funds for example have assumed the obligation to deliver specific/promised benefits to employees when they retire and therefore must ensure that the necessary assets are available so that the promised payments are made in the future. As interest rates change, so does the value of the assets earmarked to meet the liabilities. The same holds true for the rate of return of these fixed income assets. The asset manager must therefore immunize the value of his or her portfolio from interest rate shifts at a future/target date that is determined by the dates on which the liabilities/promised benefits are to be paid The manager has to deal with two types or two consequences of interest rate risk, which are neutralized/offsetting each other though: i. Price risk ii. Reinvestment rate risk Rising interest rates are causing capital losses, as the (present) value of bonds and fixed income portfolios drops. However, the revenue from the reinvestment of coupons will increase. The suitable choice of the duration of the portfolio secures that each outcome offsets/annihilates the other. If the duration of the portfolio is chosen to be equal to the target date (horizon) of the investor then the accumulated value of the portfolio at the selected target date will not be affected by interest rate moves.

4

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This means that for a target date (horizon) equal to the duration of the portfolio, the price risk and the reinvestment risk offset one another! Pension funds and (mainly life) insurance organizations face this type of problem. The benefits that are promised to the insured or beneficiaries are the liabilities and the contributions or premia collected are the assets. Contrary to the banks, the liabilities are long term—as benefits are paid after a certain number of years (except maybe for property and casualty insurance) and assets are short term—as contributions or premia are paid either as lump sum or periodically, but in most cases before the benefits are paid. This is definitely true for pensions and pension-type insurance products such as pure endowments, annuities, etc. Consequently, in this case, liabilities tend to be more sensitive to interest rate shifts compared to assets (Fig. 4.10). Figure 4.10 graphically presents the assets and the liabilities of a pension scheme or a life insurance company; the contributions or premia are the assets and have been aggregated to one single amount payable on its (and their) duration and the benefits are the liabilities which have also been summed up into an amount payable on its (and their) duration. This is a simplified pictorial representation which facilitates the understanding of the issue that pension funds and insurers face. Normally benefits as well as contributions or premia spread over time. Benefits usually have longer maturities than contributions or premia, as the latter need to be paid so that the policy is valid. Contributions or premia and benefits resemble to cash flows made by fixed income securities, as they are either lump sum or periodic. In the former case they look like zero-coupon bonds; in the latter they are similar to annuities. Premia (Assets)

DA

DL

Time

Benefits (Liabilies) Fig. 4.10 author)

Pension and insurance assets and liabilities (Source Created by the

240

T. POUFINAS

The problem that insurers face is the drop of interest rates as the (future or present) value of their assets (premia) will increase less than the (future or present) value of their liabilities (benefits), since the former have shorter maturities than the latter. They also apply asset liability management to surpass this mismatch between the assets and the liabilities. As bankers, they try to reduce the gap between the duration of assets and liabilities. Their problem though is the opposite of the one that the bankers face; it is the interest rate drop that is not wanted. To show how pension funds and insurance companies put asset liability management at work, we assume (without loss of generality) that contributions or premia and benefits earn or promise the same flat interest rate. The pension fund or insurance company desires to have the future (or equivalently the present) value of its assets equal to the future (or equivalently the present) value of its liabilities: FV A = FV L ⇔ PV A = PV L .

(4.90)

As mentioned earlier though the duration of the assets is lower than the duration of the liabilities: D A < DL .

(4.91)

If interest rates drop, for example by a small parallel shift r, then both the present value of assets and the present value of liabilities will increase. As a matter of fact the same will apply to the corresponding future values. As the duration of the assets is lower than the duration of the liabilities, the present value of the assets will increase less than the present value of the liabilities as: PV A ≈ −

1 1 · D A · PV A · r < − · D L · PV L · r ≈ PV L 1+r 1+r (4.92)

as a result of (4.90) and (4.91) and the fact that r is negative (as there is an interest rate decrease). As a result, the present value of the assets will be lower than the present value of the liabilities of the pension fund or the insurance company (denoted with a “prime” following the notation of [4.62] and [4.86]) 



PVA ≈ PV A + PV A < PV L + PV L ≈ PV L .

(4.93)

4

FIXED INCOME PORTFOLIO MANAGEMENT

241

The same will hold true also for the future values of the assets and the liabilities, as they are derived from the present values with the multiplication by a compounding factor of (1 + r + r)N , where N is the date that the liability must be met. In our simplified approach this would be DL . If this happens then the pension fund or the insurer will not have the necessary assets to meet its or his liabilities. To circumvent this, the pension fund or the insurer tries to close the duration gap by equalizing the duration of its or his assets and liabilities, i.e. D A = DL .

(4.94)

If this is achieved, then combined with (4.90), a small decrease r in the interest rates will result in a change in the assets that will be (almost) equal with the change in liabilities PV A ≈ −

1 1 · D A · PV A · r = − · D L · PV L · r ≈ PV L . 1+r 1+r (4.95)

The latter gives that 



PVA ≈ PV A + PV A ≈ PV L + PV L ≈ PV L ,

(4.96)

which means that the assets continue to match the liabilities of the pension fund or the insurance company. The same holds true for the future values, as they are derived from the present values with the multiplication by a compounding factor of (1 + r + r)N , where N is the date that the liability must be met. In our simplified approach, as mentioned earlier, this would be D L . Securing at least the equality of the future (or present) value of the assets with the future (or present) value of the liabilities and the equality of the duration of the assets with the duration of the liabilities is the objective of target date immunization. The focus is in the long-term (target date) as the pension fund or insurance company liabilities (pensions or benefits) are long-term. This determines the liability driven investment (LDI) strategy that the pension scheme or the insurer has to follow with respect to its assets. Pension funds and insurance companies try to increase the duration of their assets. To achieve that they invest the premia in most of the cases

242

T. POUFINAS

to fixed income securities so as to increase their duration. Equities, real estate and other asset classes are eligible investments according to their investment mandates. At the same time they try to shorten the period of interest rate guarantees or abolish interest rate guarantees completely. Consequently, there is a shift from defined benefit to defined contribution pension schemes and from traditional insurance products to unit linked. These are some of the choices that can help reduce the duration gap. Derivatives may also be used to lock interest rate levels. They will be presented in Chapter 5. Following these approaches the pension fund or the insurance company tries to immunize its overall position from interest rate movements. Its assets are approximately equal to its liabilities. If their durations are equal then any change in interest rates will affect both equally. Its total future or present value will not change. This may be achieved if the assets and the liabilities have the same size and duration. Example 4.19 An insurance company just opened its doors, let one customer in and closed its doors. It sold to the customer a policy that guaranteed an interest rate of 6% on the invested amount, on a single premium pension product that pays its benefit as a lump sum amount in 10 years. The amount is paid to the insured if he or she is alive at the end of the ten years or to his or her beneficiaries if he or she is not. The insured pays to the insurer an amount of 10,000.00 Euro. After the subtraction of policy and management fees, which are paid up front, an amount of 9,070.50 Euro is invested. Consequently, the insurer has a liability of 16, 243.89 = 9, 070.50 × (1.06)10

(4.97)

in ten years. The portfolio manager of the insurer has found a bond in the market that has 14 years remaining until its maturity and pays an annual coupon of 5% (which was most likely the level of interest rates at its issuance date, if we assume all interest rates being equal at a certain point of time). This is the asset of the insurer. Apparently the present value of the asset is equal to the present value of the liability. But is the duration of the asset equal to the duration of the liability? As a matter of fact, as the insurer wishes to achieve target

4

FIXED INCOME PORTFOLIO MANAGEMENT

243

date immunization the future value of the asset (along with the reinvested coupons) has to equal the future value of the liability. Is this indeed the case? As the bond matures after the liability is due, the manager will have to sell the bond at the end of the 10th year. At the same time he or she will have to reinvest all coupons at the available interest rate, which was assumed to be the promised rate of 6%. The relevant calculations are mapped in Table 4.66. We first observe that the duration of the asset is indeed close to 10, as is the duration of the liability. The coupons that are collected up to year 10 are reinvested at an interest rate of 6%. As the bond is sold at the end of year 10, the coupons that are payable after year 10, as well as the face value are discounted to produce the selling price of the bond at the end of the 10th year. Adding the reinvested coupons as well as the bond selling price yields a lump sum of 16,243.89 Euro; this is precisely the amount of the liability. Table 4.66 Bond cash flows and duration (r = 6%) Year

Cashflow

1 2 3 4 5 6 7 8 9 10 11 12 13 14

500 500 500 500 500 500 500 500 500 500 500 500 500 10,500

Discounting factor

Discounted cashflow

Weighted

Reinvested

r

0.94 0.89 0.84 0.79 0.75 0.70 0.67 0.63 0.59 0.56 0.53 0.50 0.47 0.44 Price

471.70 445.00 419.81 396.05 373.63 352.48 332.53 313.71 295.95 279.20 263.39 248.48 234.42 4,644.16 9,070.50 Duration

471.70 890.00 1,259.43 1,584.19 1,868.15 2,114.88 2,327.70 2,509.65 2,663.54 2,791.97 2,897.33 2,981.82 3,047.45 65,018.24 92,426.05 10.19 Future value

844.74 796.92 751.82 709.26 669.11 631.24 595.51 561.80 530.00 10,153.49

6%

Source Created by the author

16,243.89

244

T. POUFINAS

But is the portfolio immunized? What will happen if the interest rate drops? Assume that the interest rate drops to 5% immediately (after the bond is purchased). What happens is that the coupons are reinvested at a lower rate; however the selling price of the bond will increase as discounting will take place with a lower interest rate. The new calculations are shown in Table 4.67. We observe that the value of the investment at the end of the 10th year, at which time the bond is sold, is 16,288.95 Euro >16,243.89 Euro. This means that the asset still covers the liability. Example 4.20 An insurance company has issued a policy that promises the amount of 1,000,000.00 Euro in 3 years. It offers an interest rate guarantee of 4%. This is the liability of the insurance company. It collects today the present value of this amount. The portfolio manager is able to find in the market two bonds that mature in 2 and 4 years respectively, without a coupon, with a nominal value of 100 Euro each. These two bonds are the assets Table 4.67 Bond cash flows and duration (r = 5%) Year

Cashflow

1 2 3 4 5 6 7 8 9 10 11 12 13 14

500 500 500 500 500 500 500 500 500 500 500 500 500 10,500

Discounting factor

Discounted cashflow

Weighted

Reinvested

r

0.95 0.91 0.86 0.82 0.78 0.75 0.71 0.68 0.64 0.61 0.58 0.56 0.53 0.51 Price

476.2 453.5 431.9 411.4 391.8 373.1 355.3 338.4 322.3 307.0 292.3 278.4 265.2 5,303.2 10,000.0 Duration

476.2 907.0 1,295.8 1,645.4 1,958.8 2,238.6 2,487.4 2,707.4 2,900.7 3,069.6 3,215.7 3,341.0 3,447.1 74,245.0 103,935.7 10.39 Future value

775.7 738.7 703.6 670.0 638.1 607.8 578.8 551.3 525.0 10,500.0

5%

Source Created by the author

16,288.95

4

FIXED INCOME PORTFOLIO MANAGEMENT

245

of the insurance company. The interest rate is fixed and equal to 4%. The portfolio manager needs to answer the following questions: a. What are the present value and the duration of the liability and bonds? b. How can we build a portfolio of the two bonds that immunizes the obligation? c. What number of minimum denomination securities/bonds (of face value 100 Euro) must be obtained from each bond (issue)? d. If the interest rate becomes 3% does the bond portfolio still cover the liability? Answer: a. The present values of the liability and the bonds (which are actually the prices of the bonds) are PV L =

1, 000, 000 = 888, 996.36 (1.04)3

(4.98)

P01 = PV10 =

100 = 92.46 (1.04)2

(4.99)

P02 = PV20 =

100 = 85.48 (1.04)4

(4.100)

As the liability makes no interim payments, it resembles a zero-coupon bond. Therefore its duration is equal to its maturity date. The same holds true for the bonds that are zero coupon. Therefore DL = 3

(4.101)

D1 = 2

(4.102)

D2 = 4.

(4.103)

b. The portfolio will be immunized if the present value of the assets is equal to the present value of the liabilities and the duration of the

246

T. POUFINAS

assets is equal to the duration of the liabilities. The former is definitely the case as the portfolio manager can invest only the present value of the liability: PV A = PV L = 888, 996.36.

(4.104)

To achieve equal asset and liability durations the portfolio manager needs to split this amount so that the weighted average of the duration of the bonds is equal to the duration of the liability. If he or she invests a proportion of w 1 = w in the first bond, then he or she will invest w 2 = 1 − w in the second bond. The following equality must then hold true: w1 · D1 + w2 · D2 = D A = D L ⇔ w · 2 + (1 − w) · 4 = 3 ⇔ w = 0.5, (4.105) which was probably expected as 3 is in the middle of 2 and 4. This shows that the portfolio manager has to place an amount of 444, 498.18 = 88, 996.36/ 2

(4.106)

to each of the bonds. iii. To find how many minimum denomination securities/bonds he or she needs to purchase, he or she simply has to divide the amount invested in each of the bonds by the corresponding price. Let us denote by x 1 and x 2 the number of minimum denomination securities/bonds (of face value 100 Euro) for each of the bonds. Then x1 =

444, 498.18 = 4, 807.69 92.46

(4.107)

x2 =

444, 498.18 = 5, 200.00. 85.48

(4.108)

One can readily see that for this number of minimum denomination securities the present value of the portfolio equals the present value of the liability: PV A = x1 · P01 + x2 · P02

4

FIXED INCOME PORTFOLIO MANAGEMENT

= 4, 807.69 · 92.46 + 5, 200.00 · 85.48 = 888, 996.36.

247

(4.109)

iv. If the interest rate drops to 3%, then the present value of the liability as well as the prices of the bonds will increase. However, the number of minimum denomination securities/bonds per bond (issue) remains the same. As a result 1, 000, 000 = 915, 141.66 (1.03)3



PV L =

(4.110)





100 = 94.26 (1.03)2

(4.111)





100 = 88.85. (1.03)4

(4.112)

P01 = PV01 = P02 = PV02 = The value of the portfolio changes to 





PV A = x0, 1 · P01 + x2 · P02 = 4, 807.69 · 94.26 + 5, 200.00 · 88.85 = 915, 184.38,

(4.113)

which slightly exceeds the liability. As a result the immunization strategy pursued by the insurance company worked in this case.

4.2.5

Cash Flow Matching

Why look for a portfolio with duration equal to that of the liability? Wouldn’t it be a better alternative to find a zero-coupon bond that gives an amount equal to the liability when we need it? Unfortunately, it is not certain that such a bond exists. If we manage to find such a bond then we have secured the portfolio immunization. This strategy is called cash flow matching . As liabilities though usually extend over different time periods, cash flow matching is required over more than one period. This strategy is called dedication strategy. It is achieved by choosing zero-coupon bonds or coupon-bearing bonds which give total cash flows that cover exactly the liabilities when they occur. This eliminates the interest rate risk. No rebalancing is required. The portfolio provides the necessary liquidity to

248

T. POUFINAS

Cash Flow Matching

500

Amount (Euro)

400 300 200 100 0 -100 1

4

7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

-200 -300 -400 -500

Time (Years)

Fig. 4.11 Cash flow matching for a pure endowment portfolio (Source Created by the author)

meet liabilities independently of the course of interest rates (see also Bodie et al., 1996). Example 4.21 An insurance company has mapped its pure endowment portfolio liabilities over the next 50 years, assuming no new inflows. It wishes to perform cash flow matching. To achieve that it needs to find assets whose combined cash flows will match exactly the liability cash flows. The graphical representation resembles to a fishbone (Fig. 4.11). Following such a dedication strategy is not simple. It could be done by structured products, but allows no flexibility when rebalancing is necessary. In addition, there are new insured and thus new inflows or policy surrenders and deaths that result in additional outflows. As a result, cash flow matching is also dynamic. 4.2.5.1 Potential Problems of Cash Flow Matching This strategy is not always so easy to implement. Managers usually try to immunize their portfolio with bonds they deem mispriced, even if they do not want to bet on interest rates. It is a characteristic of human nature to look for ways to beat the market, even if passive management is exploited.

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The limitations set by the dedication strategy in the choice of bonds make sometimes its execution with mispriced bonds only unfeasible, although it would result in a higher portfolio return. Many times in practice its execution is impossible. A pension fund that has to pay an amount to current and future retirees ad infinitum will have to buy bonds with maturity dates that are hundreds of years in the future. There are no such bonds. Dedication is not possible; immunization is though! Example 4.22 Assume that the liability of an insurer is essentially perpetuity. If the interest rate is 5.26%, the duration of the obligation is 1.0526 / 0.0526 = 20 years. By buying a zero-coupon with a maturity of 20 years and an amount equal to the liability the portfolio manager has immunized the liability. However, finding bonds for every single year so as to implement a dedication strategy is not feasible.

4.2.6

Concerns Relevant to Immunization

There is a series of concerns relevant to immunization. Besides, immunization is based on the notion of duration, which although it offers a way to measure and mitigate interest rate risk, it is still an approximation. The main concerns about immunization follow. 1. The elaboration of immunization that took place in the previous paragraphs assumes a horizontal yield curve. This is consistent with the definition of duration, which has been primarily given for a constant yield curve. If the yield curve is not flat then we need to use the quasi-modified or the Fisher-Weil duration. 2. Even if this change is made, the immunization we have described protects the portfolio only for parallel shifts of the yield curve. Generalizations in the definition of duration have been proposed to solve the problem. But they have not increased the effectiveness of the models. 3. Immunization does not work in an inflationary environment. It produces results in nominal amounts. It does not encounter for liabilities projected in the future.

250

T. POUFINAS

4. Many investors find an immunized portfolio extremely conservative. It does not offer opportunities for increased performance.

4.3

Active Bond Portfolio Management

Active bond portfolio management as opposed to passive bond portfolio management tries to achieve above-average returns by taking risks. It is not generally accepted that the market values the different securities correctly. As bond prices depend on interest rates their potential mispricing may be due to the course of interest rates. Hence, active portfolio management is the quest of mispriced securities or evolution/forecast of interest rates. As a result there is an effort to outperform the market. Any information or knowledge about interest rate movements and therefore bond prices is investigated. Entire industries or specific bonds that are considered to be mispriced by the market are examined. Active bond portfolio management takes place most of the times in two main forms (Bodie, et al., 1996): a. Interest Rate Forecasting b. Identification of Mispriced Bonds These two constitute the two potential sources of over-performance and thus of excess gain in the bond market. 4.3.1

Interest Rate Forecasting

In this form of active bond portfolio management, the portfolio manager (or the affiliated analyst) attempts to forecast the move of the interest rate term structures in the future. We use plural, because he or she is normally interested in the entire spectrum of interest rates in the fixed income markets. There are several models he or she can employ in order to model the move of future interest rates. However, forecasting interest rates is not a trivial task and it can easily fail; one does not have to predict a value only—as is the case with stocks, but an entire yield curve, as became apparent in the analysis of Chapter 3. Besides the difficulty surrounding interest rate forecasting, there are some simple rules to follow. If the portfolio manager expects an interest

4

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251

rate decrease, then he or she increases the duration of his or her portfolio, as the prices of bonds with longer durations are expected to increase more. Vice versa, if he or she anticipates an interest rate increase, then he or she decreases the duration of his or her portfolio, as the prices of the bonds with shorter durations are expected to drop less. 4.3.2

Identification of Mispriced Bonds

In the second form of active bond portfolio management, the portfolio manager (or his affiliated analyst) attempts to find mispriced securities in the entire spectrum of the fixed income markets. In other words, he or she focuses on securities with a credit spread (or default premium) that is higher than justified by the credit rating or the creditworthiness of the issuer. A high spread compared to peer-issuers may indicate an underpriced bond that if purchased will offer an excess return when its spread reverts to the comparable level. These techniques outperform only if the incumbent analyst has access to information or has gained knowledge or exhibits intuition that is superior to that of those of the market. If everyone knows that interest rates will fall then there is no possibility to make (excess) profit. A potential fall has been priced in, in the sense that bonds with a longer duration are more expensive, since the future interest rates of specific periods (future short rates) are expected to fall. As a result, when performing active bond portfolio management, there are two important things that need to be kept in mind (Bodie et al., 1996): a. It is the differential information that is valued and not just the information that is common. b. Interest rate forecasts have not been very successful overall. 4.3.3

Taxonomy of Active Bond Portfolio Management Strategies

Following the approach of Homer and Liebowitz (1972, 2013) as presented in Bodie et al. (1996) we realize that they had provided taxonomy of the active management strategies in 4 types of bond swaps. They are used to explain the choices that managers that pursue active management make and measure the potential gain (or loss). These swaps are the

252

T. POUFINAS

i. Substitution swap ii. Intermarket spread swap iii. Rate anticipation swap iv. Pure yield pickup swap. According to Homer and Liebowitz (1972, 2013) there is a variety of other swaps, including tax swaps (in order to achieve a beneficial tax treatment), multibond swaps, swaps for improved liquidity, for improved quality, to increase or reduce volatility, etc. The portfolio managers count on a realignment of values to take place over a period of time in order to improve their rate of return. This period is called workout time or workout period of the swap (Homer & Liebovitz, 1972, 2013). 4.3.3.1 Substitution Swap This is the swap of a bond for another bond that is almost identical in terms of its characteristics. The bonds participating in the swap must have the same coupon, the same maturity date, the same credit quality, etc. This swap is based on the view that the two bonds are temporarily relatively mispriced by the market and their price (or rather yield) differential offers excess profit opportunities. Example 4.23 Assume that two corporate bonds, issued by companies A and B have the same characteristics and the bond of A has a yield to maturity of 3%, whereas the bond of B has a yield to maturity of 3.10%. If the bonds have the same credit rating there is no reason for this differential. Consequently the bond with the higher yield is more attractive. An active portfolio manager swaps the bond of company A for the bond of company B anticipating that the gap in the yield to maturity will close, offering him or her an excess return.

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4.3.3.2 Intermarket Spread Swap This is the swap of a bond of one industry for the bond of another industry. It takes place when we trust that the yield spread between the two industries is temporarily disrupted. Example 4.24 Assume that the yield spread between a 10-year government bond and a 10-year corporate bond is now 1.5%; however, the historical spread is 1.0%. An active portfolio manager may consider swaping the government bond for the corporate bond. If the spread returns to its historical levels, then he or she will experience a higher performance compared to the government bond. We need to be cautious in exploiting this type of swap. We need to be certain that the yield spread is indeed abnormal. If for example the credit spread of corporate bonds increases in the market because it discounts a recession, then it does not mean that the corporate bonds are more attractive than the government bonds. 4.3.3.3 Rate Anticipation Swap This swap entails interest rate forecasting. If investors trust that interest rates will increase, then they swap to shorter duration bonds. If they tust that interest rates will decrease, then they swap to longer duration bonds. Example 4.25 Assume that an investor anticipates an interest rate decrease. He can then swap a 5-year government bond for a 20-year government bond. They both have the same level of credit risk; however the second bond has a longer duration, as a result of which its price will increase more if interest rates drop. 4.3.3.4 Pure Yield Pickup Swap An investor that exploits this type of swap is not anticipating a higher yield because the bond is mispriced in the market. He or she opts to increase his or her yield by investing in bonds with higher yield. In several cases this comes with a longer time to maturity; consequently the portfolio

254

T. POUFINAS

manager attempts to benefit from a term premium. At the same time he or she is exposed to the interest rate risk of this strategy. Example 4.26 Assume that a government bond that matures in 1 year has a yield to maturity of 0.5%. A government bond that matures in 20 years has a yield to maturity of 2.5%. An investor may swap the short maturity bond with the long maturity bond in order to take advantage of the higher yield to maturity. If the yield curve does not move upwards while he or she is invested in the (longer duration) bond, then he or she will enjoy a higher return; if it moves upwards then he or she will post a bigger loss.

4.3.4

Horizon Analysis

This is a form of interest rate forecasting (see also Bodie et al., 1996). More precisely the investor: 1. Chooses his or her investment horizon, which indicates the holding period of the bond. 2. Projects/forecasts the yield curve at the end of the holding period. 3. Finds the value of all bonds that are candidate for investment with the use of the forecasted yield curve at the end of the holding period. 4. Calculates the value of the bond at the beginning of the investment period. 5. Estimates potential capital gains or losses. 6. Computes the income produced by the reinvestment of coupons. 7. Produces the holding period return. 8. Invests in the bond(s) that exhibit(s) the maximum holding period return.

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Example 4.27 A portfolio manager has an investment horizon of 5 years. He or she wants to assess two candidate bonds through horizon analysis. The first bond has a maturity of 20 years and pays an annual coupon of 4%. The second bond has a maturity of 15 years and pays an annual coupon of 4.25%. Both bonds have a face value of 1,000 Euro. The interest rate is flat 3% per annum. The portfolio manager assumes that the interest rate will remain the same for the next 5 years; hence all coupons will be reinvested at 3%. He or she predicts that after year 5 and for the remaining of the life of the longest maturing bond it will drop to 2.5%. He or she computes the prices of the bonds: P01 =

20 

40 1.000 + = 1, 148.77 (1.03)n (1.03)15

(4.114)

15  42.5 1.000 + = 1, 149.22. n (1.03) (1.03)15

(4.115)

n=1

P02 =

n=1

The prices are very close to each other. Without performing horizon analysis he or she cannot make a choice as the two bonds have the same yield. The former has a longer maturity date, whereas the latter has a bigger coupon. He or she calculates the prices of the two bonds at the end of year 5 to find: P51 =

15  n=1

P52 =

10  n=1

40 1.000 + = 1, 185.72 (1.025)n (1.025)15

(4.116)

42.5 1.000 + = 1, 153.16. (1.025)n (1.025)10

(4.117)

He or she then estimates the value of the reinvested coupons (denoted with RIC) for each of the bonds to find that RIC15 =

5  n=1

40 · (1.03)5−n = 212.37

(4.118)

256

T. POUFINAS

RIC25 =

5 

42.5 · (1.03)5−n = 225.64.

(4.119)

n=1

The holding period return of each bond is then found to be HPR15 =

(P51 − P01 ) + R I C51

P01 (1, 185.72 − 1, 148.77) + 212.37 = 21.70% = 1, 148.77

HPR25 =

(4.120)

(P52 − P02 ) + R I C52

P02 (1, 153.16 − 1, 149.22) + 225.64 = 19.98%. = 1, 149.22

(4.121)

The portfolio manager, all other things being equal, will opt for the first bond as it offers a higher holding period return, based on his or her interest rate projections.

4.4

Combination of Active and Passive Bond Portfolio Management

It is in the human nature to believe that we can always beat any potential benchmark. As a result, even if we feel safer with a passive bond portfolio management, we are inclined to try an active bond portfolio management. But is there a safety net? Is there a way to revert to a passive bond portfolio management that will deliver the desired outcome in case the active bond portfolio management does not succeed? Contingent immunization does exactly that (see also Leibowitz and Weinberger, 1982; Bodie et al., 1996). The portfolio manager can start with an active bond portfolio management and at the same time he or she can set a floor that if reached he or she switches to a passive bond portfolio strategy. If this threshold is not reached, then he or she can continue with the active management until the end of his or her investment horizon. Example 4.29

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257

Assume that the portfolio manager has a portfolio of 100 million Euro. He or she has a target of delivering 121.90 million Euro in 10 years, which is equivalent to an interest rate guarantee of 2% per annum, with annual compounding. The portfolio manager can find a zero coupon bond that yields a 3% per annum; he or she can thus immunize the portfolio and “lock” an amount of 134.39 million Euro (=100 × (1.03)10 ). He or she feels that he or she can achieve an even higher return. He realizes, that at a 3% yield he or she needs 90.70 million Euro (=121.90/(1.03)10 ) and not 100 million Euro to meet his or her target. So he or she can exploit an active management strategy and if at a certain point of time t he or she ever reaches a level of 121.90/(1 + r )(10−t) ,

(4.122)

where r is the market interest rate and 10-t is the remaining time to maturity, then he or she can switch to a passive management strategy so as to secure the promised amount of 121.90 million Euro. He or she practically applies an immunization strategy—contingent on the outcome—that will safeguard the minimum promised/accepted return.

Exercises Exercise 1 Consider a bond with a nominal value of e 1,000 with a maturity of 3 years and a coupon rate of 4%. The market interest rate is fixed and equal to 3%. a. Calculate the bond price using the formula of the present value. b. Calculate the duration of the bond. c. Calculate the convexity of the bond. The interest rate increases by 1%. iv. What is the new price of the bond with the formula of the present value? v. What is the new bond price using duration? vi. How do the results of (d) and (e) compare?

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T. POUFINAS

vii. What is the new bond price using duration and convexity? viii. How do the results of (d), (e) and (g) compare? The interest rate decreases by 1%. i. Repeat questions (d) to (h). ii. Compare the findings with those of (d) to (h). The interest rate increases by 0.1%. k. Repeat questions (d) to (h). l. Compare the findings with those of (d) to (h). The interest rate decreases by 0.1%. k. Repeat questions (d) to (h). l. Compare the findings with those of (j) and (k). Exercise 2 A government bond has a date of issue 15/11/2020 and an expiration date of 15/11/2023. Its coupon is annual and at a rate of 3% per annum. An investor intends to buy EUR 100,000 in face value on 15/11/2020. a. How much will he or she pay if the 3-year interest rate is fixed at 3% per annum? b. How much will he or she pay if the 3-year interest rate is fixed at 4% per annum? c. How much will he or she pay if the 3-year interest rate is fixed at 2% per annum? d. What is the yield-to-maturity in each of the cases? e. What is the duration of the bond in each case (calculate the duration for an interest rate of 3%, 4% and 2%)? What do you notice about the changes in duration relative to the changes in interest rates? f. Assume that the 3-year interest rate is flat at 3% per annum. Use duration to observe what happens if the interest rate changes by + 1%. How does it compare with the result of question (b)? g. Calculate Convexity (for a flat 3-year interest rate of 3% per annum). How does the result of question (f) change if you use both duration and convexity?

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h. Assume that the 3-year interest rate is flat at 3% per annum. Use duration to observe what happens if the interest rate changes by − 1%. How does it compare with the result of question (c)? i. Calculate Convexity (for a flat 3-year interest rate of 3% per annum). How does the result of question (h) change if you use both duration and convexity? Exercise 3 An insurance company has an obligation of EUR 1,000,000 in 3 years and wants to invest enough money today to meet this future obligation. The investment must be such as to protect the company against interest rate risk. The interest rate is fixed and equal to 4% per annum. The company decides to invest in two bonds B1 and B2. B1 matures in 4 years and pays an annual coupon with a coupon rate of 2% per annum. B2 matures in 2 years and pays an annual coupon with a coupon rate of 4% per annum. Both B1 and B2 have a nominal/face value of EUR 100. a. What are the prices (present values) of the bonds and what is the present value of the liability? b. What are the durations of the bonds and the liability? c. Which portfolio investing in B1 and B2 immunizes against parallel movements of the interest rate curve? Find the number of each of the bonds B1 and B2 that the investor has to buy in order to achieve immunization. d. If the interest rate becomes 3%, does this portfolio still cover the liability? Exercise 4 The larger the bond coupon, the more volatile the bond price is to interest rate changes. A. True B. False Elaborate on your answer.

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Exercise 5 The higher the interest rate, the more volatile the bond price is to changes in interest rates. A. True B. False Elaborate on your answer. Exercise 6 The longer the maturity date of the bond, the more volatile the bond is to changes in interest rates. A. True B. False Elaborate on your answer. Exercise 7 Consider a zero-coupon bond, with a face value of EUR 1,000 that expires in 10 years. The duration of the bond is: A. 0 B. 5 C. 10 D. Cannot be answered. Elaborate on your answer. Exercise 8 In passive bond portfolio management the manager does not act at all. A. True B. False

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Justify your answer. Exercise 9 In active bond portfolio management, the manager accepts the valuation of bonds from the market as correct. A. True B. False Justify your answer. Exercise 10 In the market you can find two bonds, each with a maturity date of 4 years and a face value of EUR 1,000. The first makes annual coupon payments of 4% per annum and the second makes annual coupon payments of 6% per annum. The discount rate for both is 5%. You decide to split your wealth between the two bonds. a. Find the present value of each of the bonds. b. Find the duration of each of the bonds. c. For each potential allocation of your wealth to the two bonds find the present value of the resulting portfolio. d. For each potential allocation of your wealth to the two bonds find the duration of the resulting portfolio. e. What conclusions can you draw about the present value and the duration? Exercise 11 Repeat exercise 10 assuming that you invested 40% in the first bond and 60% in the second bond. a. Find the yield to maturity of the portfolio you created. b. How does it compare to the weighted average of the yields to maturity of the individual bonds? c. Comment.

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Exercise 12 In exercise 3 above consider that one year has lapsed, hence there are two more years remaining until the maturity of the obligation. a. Answer the questions of exercise 3 anew. b. How should the company rebalance the portfolio in order to immunize the portfolio anew? c. Explain the reallocation of funds as necessary. Exercise 13 In exercise 3 and exercise 12 above assume that 2 years have lapsed. a. What situation is the company confronted with in terms of immunizing its portfolio? b. If in the market there is a T-Bill maturing in 6 months with a face value of EUR 100, then how can the company rebalance its portfolio in order to immunize it anew? Exercise 14 An insurance company knows that its obligations on annual basis for the next five years will be EUR 5 million each year. In the market, there is a T-Bill that matures in 1-year and a 5-year coupon-bearing bond that has a coupon rate of 5%. The company aims at securing cash flow matching. If it can find a financial institution that can strip the coupons from the bond, then how could it succeed in completely matching the cash flows? Exercise 15 An investment manager wishes to follow a benchmark that comprises of 3 bonds that mature in 5, 7 and 10 years. Their percentage participation in the index is 30%, 30% and 40% respectively. They all have face value of EUR 1,000. The bonds make annual coupon payments at coupon rates of 3%, 4% and 5% respectively. Assume that the discount rates are given in the following table:

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263

Year

1

2

3

4

5

6

7

8

9

10

Rate

2.2%

2.4%

2.6%

2.8%

3%

3.2%

3.4%

3.6%

3.8%

4%

The investor anticipates a drop in the interest rates and so he or she decides to overweight the longer maturing bonds. He or she allocates his or her wealth at 10%, 40% and 50% in the three bonds respectively. a. What is his or her performance against the benchmark if the interest rate term structure parallel shifts by −1%? b. What is his or her performance against the benchmark if the interest rate term structure parallel shifts by +1%? c. What safety nets can he or she put in order to revert to the benchmark in case his or her strategy/expectations prove wrong?

References Bodie, Z., Kane, A., & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. CIIA. (2004). Fixed income valuation and analysis. Course Manual. Fisher, L., & Weil, R. L. (1971). Coping with the risk of interest-rate fluctuations: Returns to bondholders from naive and optimal strategies. The Journal of Business, 44(4), 408–431. Homer, S., & Leibowitz, M. L. (1972). Inside the Yield Book. Original Edition as Part III of Inside the Yield Book—The classic that created the science of bond analysis (3rd ed.). Wiley/Bloomberg Press. Homer, S., & Leibowitz, M. L. with Bova, A. & Kogelman, S. (2013). Inside the Yield Book—The classic that created the science of bond analysis (3rd ed.). Wiley/Bloomberg Press. Leibowitz, M. L., & Weinberger, A. (1982). Contingent immunization—Part I: Risk control procedures. Financial Analysts Journal, 38(6), 17–31. Luenberger, D. G. (1998). Investment science. Oxford University Press. Macaulay, F. R. (1938). The concept of long term interest rates, Chapter II in Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the United States since 1856. Publications of the National Bureau of Economic Research, Inc. Number 33, 24–53. https://www.nber. org/chapters/c6342.pdf. Accessed: 06 June 2020.

CHAPTER 5

Interest Rate Derivatives

Interest rate derivatives are financial products whose underlying asset/instrument/value is a fixed income security or instrument that depends on interest rates either directly or indirectly. Accordingly, the payments they make depend on interest rates. As such, the focus is primarily in their valuation, as well as in their use. We have realized that interest rates depend on a series of parameters (variables). Consequently, the pricing (valuation) of interest rate derivatives is anticipated to be more demanding than the pricing (valuation) of derivatives on equity, FX or indices. Interest rate derivatives are used also for hedging interest rate risk. As a result they are important risk management tools. The valuation of these derivatives is not as straightforward as this of other derivatives because of the more complex nature of interest rates compared to other underlying assets/instruments. More specifically, as became apparent in Chapter 3 (Hull, 1997): 1. The stochastic processes, along with the associated probability distribution that interest rates follow are more complex than the stochastic process and the probability distribution followed by the price of a stock or an index or an FX rate. 2. The valuation of a derivative on a stock or an index or a foreign exchange rate is based on the modeling of its price movement. In the case of interest rates, the question arises as to which interest rate © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_5

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should be modeled, since in essence we have an interest rate term structure and not a single price. While the movement of a stock price (or index or FX rate) draws a path, the movement of an interest rate term structure draws a surface. 3. The individual interest rates on the term structure experience their own specific volatility, which is an input to the pricing model, which causes an additional complexity. 4. In interest rate derivatives, interest rates are the underlying instrument (directly or indirectly) and the input in the discounting factors at the same time. Hence they cannot be assumed to be fixed for the life of the derivative, as is usually the case for derivatives on stock, indices or FX rates. In this chapter we present some of the most well-known interest rate derivatives and the valuation methods most commonly used. These interest rates are repos, forward rate agreements, interest rate futures, interest rate swaps, interest rate options, embedded bond options, Mortgage Backed Securities, Collateralized Mortgage Obligations, Asset Backed Securities, swaptions and caps and floors. We study their characteristics; we price them and set the grounds for their use as tools for hedging interest rate risk, in particular by institutional investors, such as social security funds, pension schemes, insurers and bankers. After having read this chapter the reader is expected to have a very good understanding of how the interest rate derivatives operate. It is interesting to note that according to some (Luenberger, 1998) even a bond is a derivative on the interest rate as its value depends on the interest rate itself. We will not include bonds in the interest rate derivatives chapter; however, their particulars and valuation have been presented in Chapter 2.

5.1

Repos

A repo (acronym for repurchase agreement ) is an agreement between one counterparty (a security holder) and a second counterparty, where the former agrees to sell the security to the latter and to buy it back at a predetermined higher price, on a predetermined future date. For the first counterparty this is a loan. The interest earned from the loan is derived from the difference between the repurchase price and the selling price. This interest is known as the repo rate.

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The maturity of the repo may be from one day to a few months or even year(s). The former is called overnight repo and the latter is known as term repo. The risk at which the counterparties are exposed is limited; if the security holder (the selling party) does not repurchase the security sold to the second counterparty (the buying party), then the second counterparty gets to keep the security and if the second counterparty does not sell the security then the (ex) security holder gets to keep the cash (loan) (Hull, 1997). To find the repo rate, we realize that simple discounting is used for a short period of time. Assuming a 365 day count convention (it could be 360) we see that    Prepurchase n  365 ⇔ rr = , (5.1) −1 · Prepurchase = Pselling 1 + rr 365 Pselling n where P repurchase : P selling : n: rr:

is the repurchase price is the selling price is the number of days between the sale and the repurchase is the repo rate.

When the latter equation is used, the repo rate found is sometimes referred to as the implied repo rate. Example 5.1 Assume that a repo on a zero-coupon bond maturing in 7 days is agreed between two counterparties. The selling party will receive 1 million Euro and will repurchase the bond after 7 days for 1,000,383.56 Euro. The (implied) repo rate is found by   365 1, 000, 383.56 −1 · = 2.00%. (5.2) rr = 1, 000, 000.00 7 Several term deposits are backed by repos. As a matter of fact a repo resembles a collateralized deposit (ICMA, 2019) as it essentially offers an

268

T. POUFINAS

interest rate that is materialized though the sale and repurchase of the underlying security. In reality, a repo contractually is just that: the sale and the repurchase of a (fixed income) security. This property determines the very nature of the transaction for the counterparties. Namely, the buying counterparty has a temporary use of the security and the selling counterparty has a temporary use of the cash received from the sale. At the beginning the security is sold, but it has to be repurchased at the end of the agreement. Hence, the security and the cash return to their initial/original owners. There is a debate on whether a repo is a derivative or not—in which case it classifies as a money market instrument (Faure, 2011). There are arguments for both opinions. To substantiate the opinion according to which a repo is a derivative, we realize that the repo is an agreement, and as such a separate instrument, on an underlying security to sell the security at its initiation and buy it back on its maturity date. The repurchase price reflects the selling price and the repo rate. The applicable rate is the price of money during the life of the agreement. It can be traded in the money market. These features indicate that a repo is not very different from a futures (or forward) contract, which is an agreement, and as such a separate instrument, on an underlying security to buy or sell the security at a predetermined price on its maturity date. The futures price is the future value of the spot price, calculated with the use of the appropriate/relevant interest rate that is applicable for the remaining life of the futures contract (potentially reduced by the accrued income). However, as mentioned earlier, the repo resembles a collateralized deposit. As such the buying counterparty focuses primarily on the interest he or she will receive from the repo; he or she is not paying attention to the mechanics of the repo, as he or she is interested in investing a certain amount of money for a specific period of time and earn an interest rate (the repo rate). However, for the seller of the repo there is a bigger resemblance with a derivative. One may also say that a futures contract—which was used above in order to explain why a repo can be considered a derivative—is used primarily for hedging; however it can also be used as an investment vehicle to gain access/exposure in a market, by taking a long position on a futures contract written on the underlying security of interest to the investor. At the same time, a repo—by its very nature—could be used for hedging the outcome of an interest rate derivative, as it allows locking an

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interest rate for a predetermined period of time. Such derivatives could be forward rate agreements and (the floating interest rate of) swaps (Faure, 2011). 5.1.1

Reverse Repo

A reverse repo (or reverse repurchase agreement ) is an agreement according to which one counterparty agrees to purchase a security from a second counterparty at the present time and sell it back at a predetermined future date at a predetermined higher price. For the second (selling) counterparty this is a repo, as it sells the security and agrees to buy it back at a higher price, which according to the definition given above is nothing but a repurchase agreement.

5.2

Interest Rate Forward Contracts 5.2.1

Forward Rate Agreements

A forward rate agreement (FRA) is a forward contract (to be formally defined below) whose underlying asset (value) is an interest rate, and its delivery price agreed to be paid on its delivery date is an interest rate covering a period after the delivery date. In essence (and this is why we present it before forward contracts), two counterparties agree on the interest rate that will be valid in a future period of time and on a specific principal. The FRA is settled in cash at the commencement of the relevant period (Hull, 1997). Let the (notional) principal (face value) be FV (for example 100 Euro or 100 USD) and an interest rate R K (i.e. the equivalent of the delivery price of the corresponding forward contract) be agreed for the period between T 1 and T 2 on this amount. Let s 1 be the spot interest rate for the time interval [0, T 1 ] and s 2 the spot rate for the interval [0, T 2 ]. According to the FRA one counterparty (the one with the long position) pays FV (e.g. 100 Euro or 100 USD) at time T 1 and receives FV · e R K (T2 −T1 )

(5.3)

270

T. POUFINAS

at time T 2 . The FRA is an agreement to receive the interest rate R K (which as will be seen when forward contracts are discussed is the equivalent of the delivery price). The agreement is made at t = 0 and is honored at t = T 1 (which in the context of forward contracts is the delivery date). In order to find the value of the FRA at time t = 0 we discount the cash flows of the counterparty with the spot interest rates (Hull, 1997): PV0 = FV · e R K (T2 −T1 ) e−s2 T2 − FV · e−s1 T1 .

(5.4)

As there is no outflow (or inflow) at time t = 0, the value of the FRA must be zero (because no counterparty makes or receives any payment). This yields that R K · (T2 − T1 ) − s2 T2 = −s1 T1 ⇒ R K =

s2 T2 − s1 T1 = f T1 ,T2 . T2 − T1

(5.5)

This is nothing but the forward rate with continuous compounding between times T 1 and T 2 . As the FRA is settled in cash, the actual interest rate R at time T 1 for the period [T 1 , T 2 ] is compared with the agreed interest rate R K . The cash settlement amount at time T 1 is calculated from the relevant cash flows as: −FV + FV · e R K (T2 −T1 ) e−R(T2 −T1 ) .

(5.6)

The interest rate R is nothing but the spot interest rate at time T 1 . When seen at time t = 0 (i.e. today) it is the future spot rate sT1 , T2 and it is a random variable (as explained in Chapter 3). When time T 1 arrives R becomes the at-the-time spot rate for the time interval [0, T 2 −T 1 ], i.e. R = sTT21 .We us the superscript T 1 to denote that this is the spot rate (maturing at time T 2 ) when the time instant T 1 becomes the present time. We keep the notation R though so as to immediately compare it with the interest rate of the FRA. The FRA obtains a value until its maturity though, as interest rates change. If we consider a time instant t in the time interval [0, T 1 ], then the value of the FRA becomes PVt = FV · e R K (T2 −T1 ) e−s2 (T2 −t) − FV · e−s1 (T1 −t) , t

t

(5.7)

where s1t and s2t are the spot rates with maturities T 1 and T 2 respectively but observed at time t (instead at time 0).

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The forward rate with continuous compounding at time t is given by s2t (T2 − t) − s1t (T1 − t) ⇔ f Tt1 ,T1 (T2 − T1 ) + s1t (T1 − t) T2 − T1 = s2t (T2 − t) (5.8)

f Tt1 ,T2 =

where f Tt1 ,T2 is the forward rate with starting at time T 1 and ending at time T 2 but observed at time t (instead at time 0). We combine it with the equation that gives the value of the FRA at time t to see that PVt = [−FV + FVe R K (T2 −T1 ) e

− f Tt

1 ,T2

(T2 −T1 )

]e−s1 (T1 −t) . t

(5.9)

This is the same as Eq. (5.6)—in present value terms—with R = f Tt1 ,T2 . This means that the value of the FRA can be found by assuming that the at-the-time current forward rates materialize (Hull, 1997). 5.2.2

Forward Contract on a Bond

A forward contract is an agreement between two counterparties, to buy or sell an underlying asset on a predetermined future date at a predetermined price. The predetermined future date is called delivery date and the predetermined price is called delivery price. It is an over-the-counter (OTC) derivative (Hull, 1997). A forward contract on a bond provisions the physical delivery of a specific bond on the delivery date for the delivery price. The party with the long position has the obligation to buy the bond and the party with the short position has the obligation to sell the bond according to the terms of the forward contract. Proposition 5.1 Let us consider a forward contract on a bond or any other fixed income security that pays known cash income during the life of the forward contract. The delivery date of the forward contract is on time T 1 and the maturity of the bond is on time T 2 > T 1 . Let K be the delivery price, r be the risk-free rate, P 0 denote the price of the bond and I 0 denote the present value of the coupons (or other cash income) paid in the interval [0, T 1 ]. Then the value of the forward contract at time t = 0 becomes (Hull, 1997). f 0 = (P0 − I0 ) − K e−r T1 .

(5.10)

272

T. POUFINAS

Proof To prove that this equation holds true, we realize that it is equivalent to f 0 + K e−r T1 = P0 − I0 .

(5.11)

The left hand side resembles a portfolio consisting of one long forward contract and cash equal to the present value of K, i.e. K e−r T1 . We will denote this portfolio with f . The right hand side resembles a portfolio with one long bond and a loan equal to the present value of the coupons paid by the bond, i.e. I 0 . We will denote this portfolio with B . Cash in both portfolios is invested at the risk-free rate r. We realize that on the delivery date of the forward contract, the cash has grown to become K e−r T1 er T1 = K , as cash was invested at the riskfree rate. This amount is used to buy the bond according to the terms of the forward contract. Hence the value of the portfolio f on the delivery date is equal to the value of the bond  f (T1 ) = PT1 .

(5.12)

At the same time, assuming that the coupons collected are reinvested, their value on the delivery date has become I0 er T1 . This is exactly the outstanding loan amount. As a result the income from the reinvested coupons can be used to repay the loan. Therefore, the value of the portfolio B is equal to the value of the bond  B (T1 ) = PT1 .

(5.13)

We realize that the value of the two portfolios is equal on the delivery date of the forward contract. Provided there are no arbitrage opportunities (because if there were investors would immediately take advantage of them and equilibrium would be established), the value of the two portfolios has to be the same at any point of time, i.e.  f (t) =  B (t).

(5.14)

For t = 0 this is Eq. (5.11), which means that Eq. (5.10) must hold true. Q.E.D. As a matter of fact our elaboration did not depend on the time instant at which the forward contract was valued. It holds true for any time instant.

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Corollary 5.1 The value of a forward contract on a bond at time t is given by f t = (Pt − It ) − K e−r (T1 −t) .

(5.15)

In the aforementioned equations when there is no risk of confusing the delivery date of the forward contract with the maturity date of the bond the index may be dropped from the maturity date of the forward contract. Equation (5.15) becomes f t = (Pt − It ) − K e−r (T −t) .

5.3

(5.16)

Interest Rate Futures

A futures contract is nothing else than a forward contract that is traded on an organized exchange. As the futures contract is traded, the focus is on the futures price. Futures contracts can be written on a single bold issue or on a basket of bonds, which is actually the most common in the bond futures markets. In the latter the counterparty with the short position can choose among a series of bonds which one he or she will deliver upon expiration of the contract. 5.3.1

Futures Contract on a Single Bond Issue

Assume we have a futures contract on a bond (or fixed income security) that provides known cash income. The maturity of the futures contract is on time T 1 and the maturity of the bond on time T 2 > T 1 . Proposition 5.2 Let r be the risk-free rate, P 0 denote the price of the bond and I 0 denote the present value of the coupons (or other cash income) paid in the interval [0, T 1 ]. Then the futures price at time t = 0 becomes (Hull, 1997). F0 = (P0 − I0 )er T1 . Proof

(5.17)

274

T. POUFINAS

In order to prove this equality we need to show that none of the alternative inequalities can hold true. Assume that F0 > (P0 − I0 )er T1

(5.18)

was true. An investor could at t = 0 • Take a short position on the futures contract. • Borrow P 0 to buy the bond (or fixed income security). • Collect the coupons (or other cash payments) during the life of the futures contract; their present value is I 0 . • Reinvest the coupons until the maturity of the futures contract, i.e. time T 1 . At time t = T 1 the investor can • Sell the bond for F 0 according to the terms of the futures contract. • Collect the (future) value of the reinvested coupons; it has become I0 er T1 . • Repay the amount due under the loan (principal and interest), which has grown to P0 er T1 . • Enjoy a profit of F0 + I0 er T1 − P0 er T1 = F0 − (P0 − I0 )er T1 > 0.

(5.19)

This is a riskless profit, with no initial investment on behalf of the investor, which is an arbitrage. Assuming there are no arbitrage opportunities, because if there were investors would immediately take advantage of them, this inequality cannot hold true. To show that the opposite inequality cannot hold true as well, let us assume that it did, i.e. F0 < (P0 − I0 )er T1 . An investor could at time t = 0 • Take a long position on the futures contract. • Short-sell the bond (or fixed income security) for P 0 .

(5.20)

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• Invest the proceeds for interest rate r until the maturity of the futures contract. • Put aside the present value of the coupons (or other cash income) paid during the life of the futures contract as they are due to the owner of the bond; their present value is I 0 . At time t = T 1 the investor can • Buy the bond for F 0 according to the terms of the futures contract. • Return the bond to its owner. • Repay the value of the coupons owed to the owner of the bond, which has become I0 er T1 . • Collect the invested amount of the proceeds, which has grown to P0 er T1 . • Enjoy a profit of −F0 − I0 er T1 + P0 er T1 = −F0 + (P0 − I0 )er T1 > 0.

(5.21)

This means that there is an arbitrage opportunity, which we assumed is not the case. Consequently this inequality cannot hold true as well. Hence the initial equality is validated. Q.E.D. Corollary 5.2 In a similar manner we calculate the futures price at any time instant t as Ft = (Pt − It )er (T1 −t) .

(5.22)

When there is no need to distinguish between the maturity of the futures contract and the maturity of the bond we drop the index from the maturity date of the futures contract and the above equations become F0 = (P0 − I0 )er T ,

(5.23)

Ft = (Pt − It )er (T −t) .

(5.24)

5.3.2

Bond Futures

In reality a bond futures contract is written on a basket of bonds. There is physical delivery. This means that the party with the short position can

276

T. POUFINAS

choose from a set of bonds which one to deliver at the maturity of the futures contract. The bonds in the basket have certain quality, maturity and call-ability criteria; their credit rating has to be of specific grades, their maturity has to be longer than a number of years and they cannot be called for a certain number of years. However, there can only be one futures price at the derivatives exchange, called the quoted futures price. A natural question though is how can this be linked to the set of bonds that are candidates for delivery? How is the amount paid (received) by the party with the long (short) position affected by the bond delivered? First of all, we recall that for bonds we have also defined the quoted or clean price and the cash or dirty price. In Chapter 2 we realized that Pcash = Pquoted + AI,

(5.25)

where P cash: P quoted: AI:

denotes the cash bond price denotes the quoted bond price denotes the accrued interest since the last coupon payment.

As a result, we understand that a similar arrangement most likely holds for the futures contract. However, in this case there is only one quoted futures price but more than one candidate bonds. To convert the quoted futures price to a price suitable for the chosen bond a specific parameter, called conversion factor (cf), is used. It converts the quoted futures price to the quoted price applicable to the delivery; the latter is the product of the quoted futures price and the conversion factor. We need also to account for the accrued interest. The equation thus becomes: Fcash = Fquoted x cfbond

delivered

+ AI,

(5.26)

where F cash: F quoted: cfbond delivered:

denotes the cash futures price denotes the quoted futures price denotes the conversion factor of the bond to be delivered.

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The party with the short position thus buys the bond he or she chooses to deliver at the bond market and delivers it under the terms of the futures contract. He or she thus pays the cash bond price as given by Eq. (5.25) and receives the cash futures price as depicted by Eq. (5.26). His or her net outcome from the transactions is Net Outcomeshort =Fquoted x cfbond

delivered

− Pquoted .

(5.27)

The party with the short price wants to maximize this amount or minimize the opposite (Hull, 1997), i.e. −Net Outcomeshort = Pquoted − Fquoted x cfbond

delivered .

(5.28)

The bond for which the maximum of the former or the minimum of the latter is achieved is known as cheapest-to-deliver bond (CTD). This is found by computing the aforementioned differences for all the candidate bonds. If  denotes the number of bonds that are candidate for delivery then the cheapest-to-deliver is the bond for which j

j

CTD CTD × cfCTD − Pquoted = max (Fquoted × cf j − Pquoted ) Fquoted j=1...

(5.29)

or equivalently j

j

CTD CTD − Fquoted × cfCTD = min (Pquoted − Fquoted × cf j ). Pquoted j=1...

(5.30)

Example 5.2 Assume that two entities have entered a bond futures contract maturing in 6 months with a quoted futures price of 94.00 (out of a face value of 100.00). The bonds that are candidate for delivery have the following characteristics (Table 5.1): To find the cheapest to deliver bond, we need to solve (5.29) or (5.30). Using Eq. (5.28) for the three bonds we see that (Table 5.2): The cheapest to deliver bond is the second one. The question that is naturally brought up is how do we find the quoted futures price? The answer is not immediate as the choices of the short party with regards to the bond to be delivered and the time to be delivered are not necessarily known and thus cannot be easily valued (Hull,

278

T. POUFINAS

Table 5.1 Candidate bonds for delivery under a futures contract

Bond 1 2 3

Quoted price

Conversion factor

101.75 110.50 115.25

1.0456 1.1678 1.2123

Source Created by the author

Table 5.2 Cash flow per candidate bond for delivery

Bond

Cash flow

1 2 3

3.4636 0.7268 1.2938

Source Created by the author

1997). If we assume that the cheapest-to-deliver bond and the delivery date are known then we may proceed as follows: • We find the quoted price P quoted of the bond to be delivered. • The cash price (P cash ) of the bond to be delivered can be calculated from its quoted price (P quoted ) and the accrued interest (AI) in line with Eq. (5.25) above. • The cash futures price becomes Fcash,0 = (Pcash,0 − I0 )er T .

(5.31)

• For a conversion factor cf the quoted futures price becomes Fquoted,0 =

Fcash,0 − AI . cf

(5.32)

Example 5.3 An interest rate futures contract expires in 1 year and the holder of the short position can choose from a series of bonds which one he or she will deliver at its expiration. The cheapest-to-deliver bond (with a face value 100 Euros) has a conversion factor of 1.4 and during the

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life of the futures contract it pays a coupon of 4% (per year) semiannually in 4 and 10 months. It pre-existed 14 months before the start of the futures contract (and therefore paid coupons before) and matures 7 years and 10 months after the maturity of the futures contract. Today’s price (quoted) is 103. The discount rate is fixed at 3% with continuous compounding. What is the quoted futures price? Answer: The accrued interest at time = 0 is the part of the coupon that is payable in 4 months from the start of the futures contract. Therefore, it has accrued for 2 months and is equal to AI0 = 2 ·

2 = 0.67. 6

(5.33)

The cash bond price is Pcash,0 = Pquoted,0 + AI0 = 103 + 0.67 = 103.67.

(5.34)

The present value of the coupons that are paid during the life of the futures contract is 4

10

I0 = 2e−0.03 12 + 2e−0.03 12 = 3.93.

(5.35)

The cash futures price is given by Fcash,0 = (103.67 − 3.93)e0.03·1 = 102.77.

(5.36)

The accrued interest at the maturity of the futures contract is the part of the coupon that is payable 16 months after the start of the futures contract, i.e. 4 months after its maturity. It accrues for two months until the maturity of the futures contract: AI1 = 2 ·

2 = 0.67. 6

(5.37)

The quoted futures price is thus given by Fquoted,0 =

102.77 − 0.67 = 72.93. 1.4

(5.38)

The size of a bond futures contracts is commonly for 100,000 Euro in the various Euro-zone derivatives exchanges. The same holds true in the US, in which case it is equal to $100,000.

280

T. POUFINAS

5.3.3

T-Bill Futures

T-Bills have usually a shorter life than bonds and notes and are considered as discount instruments as they pay no coupons; the investor receives only the face value at maturity. We often refer to them with their yield. Let us consider a futures contract on a T-Bill. The futures contract has a maturity date of T 1 and the T-Bill a maturity date of T 2 > T 1 . If the face value of the T-Bill is FV (e.g. 100 Euro or USD), then—keeping the same notation as in bond futures and FRAs—its present value is (Hull, 1997) PV0 = FVe−s2 T2 .

(5.39)

As there is no income paid on the T-Bill, the futures price becomes F0 = FVe−s2 T2 es1 T1 = FVe−(s2 T2 −s1 T1 ) ⇒ F0 = FVe− f T1 ,T2 (T2 −T1 ) , (5.40) as per Eq. (5.8). An interpretation of the last equality is that the futures price of a T-Bill is the one corresponding to an interest rate on the delivery date (essentially the at the time spot rate) equal to the current forward rate for the time interval [T 1 , T 2 ]. The price quote of an n-day T-Bill in the US is done by its discount rate (dr), assuming an actual/360 day count convention. For a T-Bill with a cash price of P cash this becomes (Hull, 1997) Pquoted =

360 (FV − Pcash ) = dr. n

(5.41)

For a T-Bill with cash futures price F cash the T-Bill futures price quote is given by: 360 (FV − Fcash ), n

(5.42)

n (FV − Fquoted ). 360

(5.43)

Fquoted = FV − which is equivalent to Fcash = FV −

In reality, one 90-day T-Bill futures contract is for the delivery of 1 million monetary units (i.e. Euro or USD) of T-Bills as FV = 100. Therefore the

5

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INTEREST RATE DERIVATIVES

aforementioned equation for the contract cash futures price (F contract becomes Fcontract cash = 10, 000[FV − 0.25(FV − Fquoted )].

cash )

(5.44)

As futures prices are marked to market, the amount that is paid or received by each of the counterparties is equal to the change in the contract cash price. To elaborate, assume that there a change in the quoted futures price by 1bps (=0.01%). Then, as FV = 100 Fcontract cash = 10, 000 · 0.25 · 0.01 = 25.

(5.45)

Example 5.4 Assume that the quoted price of a 90-day futures contract on a T-Bill is 94.00. Then the corresponding cash price is Fcash = 100 −

90 (100 − 94) = 98.5. 360

(5.46)

The contract price is Fcontract cash = 10, 000[100 − 0.25(100 − 94)] = 985, 000.

5.3.4

(5.47)

Eurodollar Futures

A Eurodollar is a dollar deposited in a bank (US or non-US) outside the US. The interest rate that applies when one bank deposits a Eurodollar to another bank is referred to as the Eurodollar interest rate (Hull, 1997). This interest rate is also referred to as the London Interbank Offer Rate or LIBOR. LIBOR is a broadly used benchmark for setting interest rates on floating-rate loans (mortgages, corporate, etc.). It will be gradually replaced (from January 1st, 2022 to July 1st, 2023 - depending on the reference period) by other interest rates. Such rates are the EURIBOR, the Secured Overnight Financing Rate (SOFR), the Sterling Overnight Index Average (SONIA), and the Tokyo Overnight Average Rate (TONA) for EUR-, USD-, GBP- and JPY-denominated loans respectively.

282

T. POUFINAS

The formula used to find the amount paid under a Eurodollar Futures contract is the same as the one for T-Bill Futures. This means that Eqs. (5.42), (5.43) and (5.44) hold exactly as written above. There are differences between the T-Bill and Eurodollar futures contracts. Namely, the price of a T-Bill futures contract converges at maturity to the price of the underlying T-Bill. If the contract is held to maturity then it is delivered from the party with the short position to the party with the long position. In contrast, a Eurodollar futures contract has no physical delivery but rather involves a cash settlement. The corresponding marked to market futures price is set equal to   T (4) (5.48) Fcontract cash, T = 10, 000 100 − 0.25 · s90−day , T (4)

with s90−day being the at the time (T ) quoted 90-day Eurodollar spot rate compounded quarterly. (Hull, 1997). Recall that in the case of a TBill that was a discount rate. Consequently, the difference between the two futures contracts may be summarized in the fact that a T-Bill futures contract is written on T-Bill (or discount rate), whereas a Eurodollar futures contract is written on an interest rate (Hull, 1997). Example 5.5 Assume that the quoted price of a 90-day Eurodollar futures contract is 94.00. Then the corresponding contract futures cash price is Fcontract cash = 10, 000[100 − 0.25(100 − 94)] = 985, 000.

5.3.5

(5.49)

Duration Hedging

We saw earlier, in Chapter 4, that portfolio managers can apply passive bond portfolio management with the use of duration; in that case they tried to match the duration of assets with the duration of liabilities. This approach was called portfolio immunization. Its target was to protect the portfolio from interest rate moves. Futures contracts can also be used to protect fixed income portfolios from interest rate moves; thus hedging interest rate risk. The question is how many contracts are needed for such hedging strategies to be

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283

implemented. Duration can be used to find the number of contracts required. We assume continuous compounding. Let as denote with : D : F: DF :

the the the the

value of the fixed income portfolio being hedged. duration of the fixed income portfolio being hedged. futures price of the futures contract used. duration of the asset underlying the futures contract.

Using Eqs. (4.16) and (4.57) of Chapter 4 we get that for a small shift r of the interest rate (or y of the yield curve)  ≈ −D r

(5.50)

F ≈ −FD F r.

(5.51)

and

If H is the number of futures contracts used to hedge against the interest rate move, then a new portfolio  =  − HF.

(5.52)

is formed with the inclusion of the futures contract. As we wish to hedge the initial portfolio from the interest rate movements, this means that the portfolio with the futures contract should not change in value for (small) shifts in the interest rate. This means that  =  − H F = 0 ⇒ −D r + HFD F r = 0 D . ⇒H= FD F

(5.53)

This number of contracts is called the duration-based hedge ratio. The duration of the fixed income portfolio along with the futures contract becomes equal to zero (Hull, 1997).

284

T. POUFINAS

Example 5.6 It is now July. An investor has invested 10 million Euro in a government bond portfolio. He or she wants to hedge against interest rate moves until year-end and for that he or she uses a futures contract maturing in December (understanding that his or her portfolio may remain unhedged for a few days). The futures price is 94. The duration of the portfolio is 7.5 years and the duration of the cheapest-to-deliver bond is 6.8 years. One futures contract is for the delivery of 100,000 Euro of face value of bonds. The number of contracts is H=

10, 000, 000 · 7.5 = 117.32. 100, 000 · (94/100) · 6.8

(5.54)

This is usually rounded to the nearest whole number, i.e. 117 contracts short. This means that the portfolio is slightly underhedged.

5.4

Interest Rate Swaps

A swap is an agreement according to which two counterparties exchange predetermined cash flows (or cash flows with a predetermined pattern), at predetermined future dates. A swap differs from a forward contract in that it involves a series of payments in more than one future dates, whereas a forward contract involves only one such payment on the delivery date. The value of these cash flow exchanges is based on the price of the underlying products. The payments made by one party to the other take place either directly or not, in which case there may be a financial institution as an intermediary. Counterparties assume financial risk once they agree to such a cash flow exchange. The simplest and most common swap involves the exchange of cash flows between two investors A and B with two different interest rates, one of which is fixed and the other of which is floating. This is called an interest rate swap. The cash flows are calculated as the interest payments (cash flows)—for the fixed and floating interest rates—on a nominal principal for a certain number of years (or periods). The nominal principal is not exchanged. An interest rate swap is based on a single/common currency for both parties (see also Hull, 1997) (Fig. 5.1).

5

INTEREST RATE DERIVATIVES

285

fixed A

B floating

Fig. 5.1 Schematic representation of an interest rate swap (Source Created by the author)

The motivation for entering a swap may be the existing assets or the liabilities of the counterparties. Suppose that individual/investor/party A wants to borrow at a floating interest rate and individual/investor/party B at a fixed rate (i.e. they practically act as investees). If, today, A can borrow cheaper at a fixed interest rate while B can borrow cheaper at a floating rate, then they can do the following: • They both borrow the same principal at a fixed rate (investor A) and at a floating interest rate (investor B). • They enter into a swap, according to which A pays B the loan interest of B, and B pays A the loan interest of A. • In this way, they have both borrowed with favorable terms. The swap has to do with the exchange of cash flows between A and B and not with the lending that each of them has done. A symmetric rationale can be followed when the two individuals/investors/parties want to lend and not to borrow (in which case they act indeed as investors). This exchange, however, carries risks. The first risk is the default on the agreement (credit risk); the second is the interest rate risk, primarily born by the counterparty that pays floating (in this case counterparty A). 5.4.1

Main Reasons for Entering a Swap

There are a number of reasons for which investors would consider entering a swap. They do it to transform a liability or an asset or because they realize that they have a competitive advantage in one of the interest rates and they wish to exploit it (Hull, 1997). Assume that investor A has a fixed interest rate liability and investor B has a floating rate liability. They enter a swap where A pays B the floating rate and receives from B the fixed rate (Fig. 5.2). The combined cash

286

T. POUFINAS fixed

fixed A

B floating

floating

Fig. 5.2 Using a swap to transform a liability (Source Created by the author)

flows result in A paying a floating rate and in B paying fixed (see also Hull, 1997). Example 5.7 Suppose that A borrows at a fixed interest rate of 6.2% and B borrows at a floating interest rate that is Euribor +0.6%. They have both borrowed the same principal amount. They enter a swap through which A pays to B Euribor and receives from B 6%. Thus, their net cash flows become: Payments A : −6.2% − Euribor + 6% = − (Euribor + 0.2%), (5.55) Payments B : − (Euribor + 0.6%) − 6% + Euribor = −6.6%. (5.56) The final result is for A to pay Euribor +0.2%, which is floating and for B to pay 6.6% which is fixed. Figure 5.2 becomes in this case (Fig. 5.3). Suppose that the nominal principal is 100 million Euro. If the swap makes annual payments and a posteriori we know the level of the Euribor rate, then we can compile the table of the cash flows exchanged according to the terms of the swap. The Euribor rate is depicted directly in Table 5.3. Amounts are in million Euro. Assume now that investor A has a floating rate asset and investor B has a fixed rate asset. They enter a swap where A pays B the floating rate and receives from B the fixed rate (see Fig. 5.4). The combined cash flows 6%

6.2% A

B Euribor

Euribor+0.6%

Fig. 5.3 Using a swap to transform a liability—An example (Source Created by the author)

5

INTEREST RATE DERIVATIVES

287

Table 5.3 Swap cash flows Date

Euribor

Floating rate

0 1 2 3 4 Total

5.20% 5.80% 6.30% 6.50% 6.60%

5.20% 5.80% 6.30% 6.50% 6.60%

Fixed rate 6% 6% 6% 6% 6%

A to B

5.2 5.8 6.3 6.5

B to A

Net A

Net B

6 6 6 6

0.8 0.2 −0.3 −0.5 0.2

−0.8 −0.2 0.3 0.5 −0.2

Source Created by the author

fixed A floating

fixed B

floating

Fig. 5.4 Using a swap to transform an asset (Source Created by the author)

result in A receiving fixed and B receiving floating (see also Hull, 1997). Example 5.8 Suppose that A holds an asset that pays a floating interest rate of Euribor−0.2% and B holds an asset that pays a fixed interest rate of 5.7%. They enter a swap through which A pays to B Euribor and receives 6%. Thus, their net rates become: Payments A : (Euribor + 0.2%) + 6% − Euribor = 6.2%,

(5.57)

Payments B : 5.7% + Euribor − 6% = Euribor − 0.3%.

(5.58)

The final result is for A to receive 6.2% which is fixed and for B to receive Euribor-0.3%, which is floating. Figure 5.4 becomes in this example (Fig. 5.5). The table for this swap is identical to Table 5.3 of Example 5.7 provided the swap has the same terms. There are times when one of the two investors has a comparative advantage in borrowing at one of the two interest rates (fixed or floating) but is interested in borrowing at the other. The two investors then borrow at the interest rates at which they are comparatively “better” and then enter into a swap so as to lock a joint benefit, by achieving a lower

288

T. POUFINAS 6% A Euribor-0.2%

5.70% B

Euribor

Fig. 5.5 Using a swap to transform an asset—An example (Source Created by the author)

borrowing rate than if they had borrowed directly. This is called the comparative advantage argument (Hull, 1997). It is better understood through an example. Example 5.9 Consider two counterparties A and B who can borrow at the interest rates shown in the table below. Counterparty A B

Fixed rate 6.05% 7.25%

Floating rate 6-month Euribor+0.25% 6-month Euribor+0.95%

We observe that in both interest rate markets (fixed and floating) A can borrow at a lower interest rate than B. Possibly this is due to the better creditworthiness of A compared to B. In the fixed rate market B pays a higher interest rate by 1.2 percentage points, while in the floating rate market pays a higher interest rate by 0.7 percentage points. So A is much better in the fixed rate market and less good in the floating rate market. We then say that A has a comparative advantage in the fixed interest rate market, while B has a comparative advantage in the floating interest rate market. Assume that B wants to borrow at a fixed rate and A at a floating rate. Looking however at the interest rates they can access, A borrows at a fixed interest rate of 6.05% and B borrows at a floating rate of Euribor +0.95% and they enter a swap where A pays a fixed rate and receives a floating rate. How could A and B construct the swap so as to benefit and achieve lower borrowing costs than if they had borrowed directly? The way to answer the question is this (Hull, 1997):

5

289

INTEREST RATE DERIVATIVES

• We take the interest rate differences that can be achieved in each of the two markets. These are 1.2% in the fixed rate market and 0.7% in the floating rate market. • We calculate the absolute value of their difference, i.e. 0.5% (=1.2% −0.7%). • We find half of the difference, i.e. 0.25% (= 0.5% / 2). • This is the benefit that each of the counterparties would like to have in relation to direct borrowing from the market they were interested in. The above means that combining the initial borrowing along with the swap, A in the example should succeed in borrowing at Euribor and B at 7%. To construct the swap we set Euribor as the floating interest rate and then find the fixed interest rate. For A to succeed in borrowing at Euribor and B at 7%, the swap fixed interest rate must be 6.05%. To find that, let x be the fixed rate of the swap. Then it will look like (Fig. 5.6). The combined payments result in A paying: −6.05% + Euribor + x = −Euribor ⇒ x = 6.05%

(5.59)

and in B paying: −(Euribor + 0.95% ) − x + Euribor = −7% ⇒ x = 6.05% .

(5.60)

We observe that both equations yield the same solution as the fixed rate of the swap. The swap then becomes (Fig. 5.7) 6.05%

x A

B Euribor

Euribor+0.95%

Fig. 5.6 Construction of a swap (Source Created by the author)

6.05%

6.05% A

B Euribor

Euribor+0.95%

Fig. 5.7 Construction of a swap—An example (Source Created by the author)

290

T. POUFINAS

This is only one of the swaps that can be constructed. There are infinitely many swaps, as for each choice of a floating interest rate there is a different fixed interest rate that fulfils the comparative advantage argument. In this example we chose Euribor as the floating rate, which made all calculations much easier. We could have chosen Euribor + spread (spread could be negative) and this would change the fixed rate accordingly. 5.4.2

Swap Financial Intermediation

There are times when counterparties may not know each other so as to enter a swap. Also they may not be interested in entering a swap at the same time. Finally, they may not be able to manage a swap. In these cases the swap is made possible through a financial institution (e.g. a bank) that acts as an intermediary. For this role the financial institution receives a fee, which is usually charged to both counterparties by reducing the interest rate they receive or increasing the interest rate they pay. This practically creates two swaps; one between the financial institution and company A and another between the financial institution and company B. The positions of the financial institution are opposite. The financial institution is obviously interested in taking a position on both swaps at the same time in order to offset the positions between them. If it does not succeed then it warehouses the first swap it enters until it finds the second one that will offset it. This is called warehousing (Hull, 1997). In this case it is exposed to interest rate risk until it manages to find the second swap. It is up to the financial institution to hedge the interest rate risk until it enters the second swap or assume it without hedging. In any case, the financial institution acting as an intermediary is exposed to credit risk. Example 5.10 Assume that the floating interest rate of the asset held by counterparty A is Euribor −0.2% and the fixed interest rate of the asset held by counterparty B is 5.7%. They enter a swap each through a bank that charges a total of 0.03% divided equally among them. Through the swap, A pays Euribor to the bank and receives 5.985%. B receives Euribor from the bank and

5 5.985% A

6.015% B

Euribor-0.2%

Euribor

291

INTEREST RATE DERIVATIVES 5.70% B Euribor

Fig. 5.8 The role of intermediaries in swaps (Source Created by the author)

pays 6.015%. Thus, their total payments become: Payments A : (Euribor + 0.2%) + 5.985% − Euribor = 6.185%, (5.61) Payments B : 5.7% + Euribor − 6.015% = Euribor − 0.3015%. (5.62) The net cash flows result in A receiving a fixed rate of 6.185% and B receiving a floating rate of Euribor−0.3015%. The bank receives 6.015% from B and pays 5.985% to A. It receives Euribor from A and pays Euribor to B. Finally it has a margin of 0.03% left. This is graphically shown below (Fig. 5.8). 5.4.3

Interest Rate Swap Valuation

The valuation of an interest rate swap is done by observing that if the nominal value (principal) is added to the last exchange of payments, then the cash flows are similar to the cash flows of a bond. The cash flows up to the penultimate correspond to coupons and the last one to the last coupon plus the face value. One of the bonds has a fixed interest rate, while the other has a floating rate. As a result counterparty A, who receives a fixed interest rate and pays a floating interest rate is like having a long position on a fixed rate bond and a short position on a floating rate bond. The opposite holds true for counterparty B (see also Hull, 1997). Proposition 5.3 The value of the swap at time 0, let it be S0 , is given by the relation: S0 = Pfixed − Pfloating ,

(5.63)

where Pfixed and Pfloating are the present values of the fixed and floating rate bonds mentioned above. Proof

292

T. POUFINAS

The proof is trivial from the above arguments. Q.E.D. If FV is the face value, c fixed the payments (in amount) corresponding to the fixed interest rate and c floating the first payment (in amount) corresponding to the floating interest rate (since this is the only known one), then (Hull, 1997) N 

cfixed e−si ·τi + FVe−s N ·τ N ,

(5.64)

Pfloating = cfloating e−s1 ·τ1 + FVe−s1 ·τ1 .

(5.65)

Pfixed =

i=1

By si we denote the spot interest rates with maturity dates τi , i = 1 . . . N . The latter are the times at which payments are exchanged, with N indicating the number of the exchanges. The first equation is just the present value of a bond that pays a fixed coupon, calculated with continuous compounding. The second is the present value of a floating rate bond. The price (present value) of this bond immediately after the payment of a coupon becomes immediately (and instantaneously) equal to the face value (principal) FV. Such a bond behaves like a rolling zero-coupon bond portfolio, each of which expires at the time of a swap payment, as the coupon is reset on each anniversary (year or semester—for annual or semi-annual coupons respectively). With this observation we understand that we practically only need the first coupon to calculate the present value of a floating rate bond. The above equations, if properly modified, are valid for any time and so we can calculate the value of the swap at any time in its life. Since the entry in a swap does not require an initial payment (premium), we understand that its initial value should be zero. This determines the fixed interest rate of the swap and/or the floating rate spread. Example 5.11 Let us consider a swap, according to which the bank is going to receive 4% (annualized) every semester and pay the semiannual Euribor on a nominal principal of 100 million Euro. The remaining duration of the agreement is 15 months. We know that the spot rates used for discounting are 5%, 5.25% and 5.5% for 3, 9 and 15 months, respectively, and that the halfyear Euribor was 5.1% 3 months ago.

5

INTEREST RATE DERIVATIVES

293

Let us first look at the cash flows: cfixed = 0.04/2 × 100 = 2 million Euro and therefore: Pfixed = 2e−0.05× 0.25 + 2e−0.0525× 0.75 + 102e−0.055×1.25 = 99.12. (5.66) Also, the amount that is expected to be paid in three months is: cfloating = 0.051/2 × 100 = 2.55 million Euro, because the semiannual LIBOR was 5.1% three months ago. Therefore: Pfloating = (2.55 + 100)e−0.05 × 0.25 = 101.28.

(5.67)

The value of the agreement is: 99.12 − 101.28 = −2.16 million.

(5.68)

A negative sign means that the bank is losing money. The reason is that it receives a fixed rate of 4%, while it is charged with interest rates higher than 5%. A swap, in a way similar to a forward contract, has no initial payments by any of the counterparties. This means that its value at initiation has to be zero. As soon as the maturity date and the payment exchange dates of the swap have been set, the only parameter/variable that can possibly be determined so that the value of the swap is zero, is the fixed rate of the swap (and/or the spread of the floating rate). By combining Eqs. (5.63), (5.64) and (5.65) we get that S0 = Pfixed − Pfloating = 0 ⇒ N 

cfixed e−si ·τi + FVe−s N ·τ N − cfloating e−s1 ·τ1 + FVe−s1 ·τ1 = 0 ⇒

i=1 N 

cfixed e−si ·τi + FVe−s N ·τ N = cfloating e−s1 ·τ1 + FVe−s1 ·τ1 ⇒ (5.69)

i=1

cfixed ·

N 

e−si ·τi + FVe−s N ·τ N = cfloating e−s1 ·τ1 + FVe−s1 ·τ1 ⇒

i=1

cfixed =

cfloating e−s1 ·τ1 + FVe−s1 ·τ1 − FVe−s N ·τ N N  i=1

e−si ·τi

.

294

T. POUFINAS

5.4.4

Currency Swaps

Currency swaps are essentially interest rate swaps as again there is an exchange of cash flows calculated with two different interest rates. Their difference from the plain vanilla interest rate swaps is that the interest rates apply to (notional) principals in different currencies and they are both fixed (i.e. there is no floating interest rate). The reasons for which investors enter currency swaps are the same with those for which they enter interest rate swaps. So, they get into a currency swap (Hull, 1997): • To convert a liability that is in one currency into a liability that is in another currency (e.g. a loan). • To convert an asset that is in one currency into an asset that is in another currency (e.g. a bond). • Because one of the two investors has a comparative advantage in one of the two interest rates in the different currencies. The particulars of currency swaps are similar to those of interest rate swaps. The role of intermediaries is exactly the same as in interest rate swaps. However, it is important to note that in currency swaps intermediaries are also exposed to foreign exchange risk (FX risk) in addition to credit risk and interest rate risk (the latter if the intermediary has not entered both swaps at the same time). The intermediating financial institution may decide to hedge or keep the FX and or the interest risk (the latter while the swap remains warehoused). We illustrate with the use of an example. Example 5.12 Counterparties A and B can borrow at the following USD (dollar) and GBP (sterling) rates: Counterparty

USD

GBP

A B

4.0% 6.0%

5.8% 6.2%

5

INTEREST RATE DERIVATIVES

295

A has a comparative advantage in USD markets and B in GBP markets, as A is much better at borrowing in USD than in GBP; B is less bad in the GBP market. The interest rate differences are 2% and 0.4% for the USD and GBP loans respectively. Counterparty A wants to borrow in GBP and counterparty B in USD. Instead, A borrows in USD and B in GBP. Through a bank, they enter into a swap, each with the bank, in order to take advantage of the comparative advantage they have in the opposite interest rates at which they want to borrow. If the bank did not exist, they would have benefited 1.6 percentage points in total or 0.8 percentage points each. However, the bank charges 0.4 percentage points in total, equally to both companies, so 0.2 percentage points to each. Therefore the benefit for each is 0.6 percentage points. A swap with the bank has the following characteristics: A receives 4% in USD and pays 5.2% in GBP. The converse holds true for the bank. The net result for A (modulo the difference between the two currencies) is: Payment A : −4% − 5.2% + 4% = −5.2%,

(5.70)

which is better by 0.6% compared to 5.8%. B’s swap with the bank has the following characteristics: B receives 6.2% in GBP and pays 5.4% in USDs. The converse holds true for the bank. The net result for B (modulo the difference between the two currencies) is: Payment B : −6.2% − 5.4% + 6.2% = −5.4%,

(5.71)

which is better by 0.6% compared to 6%. As for the bank, it has taken positions in two swaps. The net result for the bank (modulo the difference between the two currencies) is: Bank payment : − 4% + 5.2% − 6.2% + 5.4% = 0.4%,

which is its fee. This is graphically shown below (Fig. 5.9).

(5.72)

296

T. POUFINAS $ 4%

$ 4% A

$ 5.4% Bank

£ 5.2%

B £ 6.2%

£ 6.2%

Fig. 5.9 Example of a currency swap (Source Created by the author)

The pricing of currency swaps follows the same approach we used for pricing the plain vanilla interest rate swaps. Each of the counterparties is long one bond and is short another bond. Both bonds pay a fixed coupon, however in different currencies. Proposition 5.4 If FX (1 unit of foreign currency = FX units of domestic currency) denotes the spot exchange rate, P foreign and P domestic the values of the bonds in foreign and domestic currency respectively, then the value of the swap is given by the equation (Hull, 1997): S0 = FX · Pforeign − Pdomestic .

(5.73)

Proof The proof is trivial from the aforementioned discussion. Q.E.D. Example 5.12 Yields in the US and Japan are 5% and 2%, respectively. A US bank has entered into a swap, under which it will receive 3% in yen and pay 4% in dollars each year. The nominal amounts are $ 10 million and ¥ 1,000 million. The current exchange rate is S = ¥ 110 / $ 1, while the remaining term of the agreement is two years. We have (Hull, 1997): Pdomestic = 0.4e−0.05×1 + 10.4e−0.05×2 = $9.79 million,

(5.74)

Bforeign = 30e−0.02 × 1 + 1030e−0.02 × 2 =  1019.02 million.

(5.75)

Therefore, the value of the swap today is: S0 = FXBforeign − Bdomestic = 1019.02 / 110 − 9.79 = − $ 0.53 million.

(5.76)

5

INTEREST RATE DERIVATIVES

297

The value for the counterparty of the bank is the exact opposite, i.e. $ 0.53 million, which translates to 0.53 × 110 =  57.97 million.

5.5

(5.77)

Interest Rate Options

Interest rate options are options written on fixed income instruments, i.e. assets or other tools whose value depends on an interest rate. The most common interest rate options are probably the bond options. A bond option is an option to buy or sell a bond at a certain price (the strike or exercise price) at or until a predetermined future date (the maturity or expiration date). Bond options can be European or American. The former are exercised only on the maturity date and the latter can be exercised at any time until the maturity date. The pricing of bond options (or interest rate options in general) depends very much on the approach followed for the pricing of the underlying bond, which in turn depends on the assumption made on the interest rate model. As such, the simplest pricing method assumes a deterministic interest rate, whereas the more advanced techniques employ stochastic interest rate models. In this section we focus primarily on the pricing of European bond options and draft the approach for American bond options. We assume that the reader is familiar with discrete-time and continuous-time option pricing techniques as we apply them in the valuation of bond (interest rate) options. When interest rates are deterministic (non-stochastic, i.e. known with certainty), we use Black’s model (1976) to price (value) European bond options. When interest rates are stochastic, then we use the approach recommended by Jamshidian (1989) combined with the interest rate models of Vasicek (1977), Ho and Lee (1986) and Hull and White (1990, 1993, 1994) to price (value) European bond options. We sketch the pricing of American bond options with the use of the tree building approach that is based on the model of Hull and White.

298

T. POUFINAS

5.5.1

Interest Rate Option Pricing with the Use of Black’s Model

The simplest approach and the closest to the infamous Black–Scholes formula, which is used to price European stock options, is the one of Black (1976), which is used for pricing options on futures (Hull, 1997), provided interest rates are fixed and known with certainty. If the bond price at the maturity of the option is assumed to follow the lognormal distribution, then Black’s model can be applied to value the option, using a forward or a futures √ contract on the bond with price F . The volatility σ is defined so that σ T is the standard deviation of the logarithm of the bond price at the maturity of the option (Hull, 1997). Recall that F is calculated from the spot price of the bond (let it be P ) as F = (P − I )esT T ,

(5.78)

where I is the present value of the coupons paid by the bond throughout the life of the contract and s T is the current spot rate with continuous compounding that matures at T . To value a European interest rate option with the use of Black’s model we first assume that we have a European call option written on a bond whose price is P. We subsequently realize that we can proceed with a European put option on a bond in a similar manner. We are interested in the valuation of the option at the present time t = 0. It is apparent that the bond has a maturity date that is bigger than the expiration date of the option. We set as Hull, 1997): K: T 1:

sT1 : F: σ:

the exercise price of the call (or put) option. the expiration date of the call (or put) option (we use the subscript 1 as T2 is used to denote the maturity date of the bond or any other date after the maturity of the option, i.e. with T1 T 1 . Attention is needed in the interpretation of the exercise price. There are two conventions used in practice; that the exercise price is a cash price and that it is a quoted price. As in the valuation models we produce cash prices, as became evident in the valuation of interest rate futures contracts, the exercise price has to be a cash price at all times. Therefore depending on the case (Hull, 1997):

300

T. POUFINAS

• If it is given as a cash price, then it remains as is. • If it is given as a quoted price, then is has to be converted to a cash price by adding to it the accrued interest. Example 5.12 A European call option that expires in 1 year is written on a bond that expires in 9 years and 6 months. The bond pays an annual coupon of 4%; the first coupon payment is made in 6 months from today and the last at the maturity of the bond. The cash strike price is e 1,000 and the cash bond price is e 950. The semiannual discount rate is 3.0% per annum and the annual discount rate is 3.5% per annum with continuous compounding. The bond price volatility is 10%. In order to calculate the option price we first calculate the present value of the coupon paid during the lifetime of the option. This is I = 40e−0.03×(1/2) = 39.40.

(5.85)

The futures price is therefore: F = (950 − 39.40)e0.035×1 = 943.03.

(5.86)

Using Black’s model we now find that the price of the option is: c = e−sT1 ·T1 · [F N (d1 ) − X N (d2 )] = e−0.035×1 · [943.03N (d1 ) − 1, 000N (d2 )] = 16.14.

5.5.2

(5.87)

Bond Option Pricing with Continuous-Time Models

When interest rates are stochastic and not deterministic we can still price European interest rate or bond options. To do that we employ the interest rate models that were presented in Chapter 3 instead of Black’s model. The valuation of bond options is due to Jamshidian (1989) and we apply it to Vasicek’s model, to the Ho & Lee model and to the Hull & White model. The global pricing formula is the same, except for one of its parameters that depends on volatility. Furthermore, we first present the

5

INTEREST RATE DERIVATIVES

301

valuation of zero-coupon bond options and then the valuation of couponbearing bond options, as a coupon-bearing bond is essentially a portfolio of zero-coupon bonds. 5.5.2.1 Zero-Coupon Bond Options Zero-coupon bond options can be valued using all three models (Vasicek, Ho & Lee, Hull & White models) as presented in Chapter 3. The value of a call option that is written on a zero-coupon bond is for all models equal to (Hull, 1997; Jamshidian, 1989): c0E, L = FV · T2 P0 · N (d1 ) − K · T1 P0 · N (d2 ),

(5.88)

whereas the value of a put option is equal to p0E, L = K · T1 P0 · N (−d2 ) − FV · T2 P0 · N (−d1 ).

(5.89)

where d1 =

ln

F V ·T2 P0 K ·T1 P0

+

σP

σ P2 2

,

d2 = d1 − σ P ,

(5.90) (5.91)

and FV: T 2: K: T 1:

is the principal of the bond (face value) is the maturity of the bond is the strike price of the option the expiration date of the call option.

The value for σ P per model is

1 − e−2aT1 σ Vasicek σ P = [1 − e−a(T2 −T1 ) ] a 2a Ho and Lee σ P = σ (T2 − T1 ) T1

(5.92) (5.93)

302

T. POUFINAS

1 − e−2aT1 σ 1 − e−a(T2 −T1 ) Hull and White σ P = a 2a

(5.94)

Theses equations hold also for any time instant t in the interval [0, T 1 ]. 5.5.2.2 Coupon-Bearing Bond Options The valuation of European options on a bond that pays a coupon is made with the realization that the latter can be considered as a portfolio of zero coupon bonds. The option is then analyzed as a portfolio of options written on the respective zero coupon bonds, which from the previous paragraph we know how to value. The sum of the values of these options gives us the value of the original option. This method is also attributed to Jamshidian (1989) and is outlined below (Hull, 1997). Suppose we have an option with a strike price of K and a maturity date of T 1 . We assume that the bond provides a total of  cash flows after the maturity of the option, let them be Ci , i = 1 . . . . We use a different symbol for the number of payments made by the bond as in the context of option pricing we use N to denote the standard normal distribution function. These are essentially the coupons and the final payment of the face value. These payments shall be made at time instants τi > T1 , i = 1 . . . . The maturity date of the bond T 2 is essentially equal to τ . We define as: rK: Ki :

the short rate r at time T 1 that equalizes the price of the couponbearing bond with the strike price. the present value at time T 1 of a zero-coupon bond with face value of one monetary unit (either e 1 or $ 1) payable at its maturity date τi > T1 , when r = r K .

As bond prices are functions of the interest rate, r K can be found by solving the bond price equation with respect to r using a numerical method. This is not different from the yield-to-maturity calculation back in Chapters 2 and 3. Recall that with the notation of Chapter 3 τi PT1 is the price at time T 1 of a zero-coupon bond, whose face value is equal to 1 monetary unit (e.g. 1 Euro or 1 USD), payable at its maturity dateτi > T1 . As a couponbearing bond is essentially a portfolio of zero coupon bonds, each with a face value equal to the cash flow payments of the bond, this makes the

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303

price (value) of the coupon-bearing bond at time T 1 equal to: PT1 =

 

Ci · τi PT1 .

(5.95)

i=1

The payoff of the option is therefore given by



max PT1 − K , 0 = max

  

 Ci · τi PT1 − K , 0 .

(5.96)

i=1

We will try to decompose the call option on the coupon-bearing bond to a portfolio of call options on the zero-coupon bonds that comprise it. Equation (3.85) of Chapter 3 indicates that all spot interest rates are increasing functions of the short rate r. Therefore, all bond prices are decreasing functions of r. This means that the value of the couponbearing bond at time T 1 exceeds the strike price K if and only if r < r K . In this case the option must be exercised. In addition, the zero-coupon bond with maturityτi > T1 is worth more than Ci K i at time T 1 if and only if r < r K . Consequently, the European call option on the couponbearing bond will be exercised if and only if all European call options with strike price Ci K i , written on the zero-coupon bonds that comprise the coupon bearing bond, are exercised. This implies that the option on the coupon-bearing bond can be decomposed to a portfolio of options on the zero-coupon bonds. This yields that the payoff from the option is (Hull, 1997):  

Ci max

τi PT1

 − Ki , 0 .

(5.97)

i=1

As a result the value of the option on the coupon-bearing bond is the sum of the values of  options written on the zero-coupon bonds. Its value, using (5.97) and (5.84) becomes: c0E

=

 

Ci · c0E, i

i=1

=

  i=1

Ci · [F V · τι P0 · N (d1i ) − K i · T1 P0 · N (d2i )],

(5.98)

304

T. POUFINAS

where c0E, i are the values of the options on zero-coupon bonds for each cash flow Ci , as shown in (5.88). In Eq. (5.98) we inserted a superscript in d 1 and d 2 as they depend on the time instants τ i . Similarly we obtain the valuation equations for put options. 5.5.3

Bond Option Pricing with Discrete-Time Models

The most popular discrete-time (numerical) method for the pricing of European and American options makes use of trees. For European and American options on stocks the binary tree pricing of Cox, Ross and Rubinstein (Cox et al., 1979) is the most well known approach, at least for educational purposes. Binomial (or binary) trees are initially used to model the move of the underlying equity and then the options written on the stock are priced. The modeling of interest rates is done with the use of trinomial trees (as presented in Chapter 3), in order to capture the mean reversion property. As soon as the possible interest rates have been mapped on the tree then the prices of the underlying bond (or fixed income instrument) are calculated and finally the values of the derivative on the nodes are computed. We start as usual with the terminal nodes, i.e. at the maturity of the option and then we go backwards to find the value of the option on each node, until we find its value at time t = 0. 5.5.3.1 Pricing Interest Rate Options with a Trinomial Tree We illustrate the approach with a 2-step recombining trinomial tree with standard branching. However, the methodology can be replicated for any number of steps as presented. The trinomial tree in Fig. 5.10 does not distinguish in its notation between European and American options (or call and put options, although c usually refers to call options) so that is not too populated. We denote by t: pu : pm : pd : ru i d j :

the time interval between any two steps. the probability of an up movement. the probability of a middle movement. the probability of a down movement. the interest rate that corresponds to i movements up and j movements down.

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305

ru 2

pu

r0

pm

c0

pd

ru

cu22 ru

cu1

cu2

r0

r0

cm1

cm2

rd

rd

cd1

cd2 rd 2

cd22 0

Δt

T=2Δt

Fig. 5.10 Trinomial interest rate tree for the pricing of options (Source Created by the author)

rm : cuE,i dkj : cmE, k : cuA,i dkj : cmA, k :

the interest rate that corresponds to i movements up and j = i movements down. This is the middle level rate and is equal to r0 . the European call option value that corresponds to i movements up and j movements down at time step k. the European call option value that corresponds to i movements up and j = i movements down at time step k. This is the middle level value of the derivative. the American call option value that corresponds to i movements up and j movements down at time step k. the American call option value that corresponds to i movements up and j = i movements down at time step k. This is the middle level value of the derivative.

306

T. POUFINAS

It is important to note that although u i d j is an index and not a multiplier the powers annihilate each other as if d = u −1 . This means for example that an up move (i = 1) followed by a down move (j = 1) leads to r0 ≡ rm which is the initial rate; an up move (i = 1) followed by a middle (horizontal) move (j = 0) leads to ru . For the derivative though we also distinguish the time step k as it is not necessarily true that landing at the same level of the recombining trinomial tree would yield the same option value. For example an up move (i = 1) followed by a down move (j = 1) leads to r0 ≡ rm ; however it is E, 2 E, 0 = cmE, 2 will be equal to cud = cmE, 0 ≡ c0 , not necessarily true that cud A, 2 A, 0 or that cud = cmA, 2 will be equal to cud = cmA, 0 ≡ c0 . Please note that as the tree is recombining for the interest rate it is not necessary to have a superscript for the time step at the interest rate. If it were not (as we will see later), then the distinction would be necessary. A similar rationale can be applied to trees with more steps. Assume that we have an option (call type on the interest rate) that pays at the end of the second step, i.e. T = 2t, a payoff of max[FV(r − R K ); 0],

(5.99)

where r is the t-period rate and R K is the strike price (in terms of the interest rate). Usually FV is taken equal to 100 monetary units (e.g. Euro or USD) as it immediately converts percentage rates to amounts; however any face value FV can be used instead. There are 5 potential outcomes one for each value of the interest rate, namely ru 2 , ru , r , rd and rd 2 . Assuming first that the option is European (although they are exactly the same at the terminal step also for American), then these are cuE,2 2 = max[FV(ru 2 − R K ); 0],

(5.100)

cuE, 2 = max[FV(ru − R K ); 0],

(5.101)

cmE, 2 = max[FV(r0 − R K ); 0],

(5.102)

cdE, 2 = max[FV(rd − R K ); 0],

(5.103)

cdE,2 2 = max[FV(rd 2 − R K ); 0].

(5.104)

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INTEREST RATE DERIVATIVES

307

Going one step back, i.e. at t = t, the value of the call option at each node is calculated by discounting the expected value of the option as seen from that node, which corresponds to the expected value of the option values that are at the end of the branches that start from this node. To elaborate, let us consider the node at which the interest rate is ru . Then, the value of the derivative at that node is the discounted expected value as calculated from its values at the nodes of time step 2 at which the interest rate is ru 2 , ru , r0 . The interest rate used for discounting is the t-period rate when at the node at which the interest rate is ru . The outcome becomes: cuE, 1 = ( pu · cuE,2 2 + pm · cuE, 2 + pd · cmE, 2 ) · e−ru ·t ,

(5.105)

cmE, 1 = ( pu × cuE, 2 + pm × cmE, 2 + pd × cdE, 2 ) × e−r0 ×t ,

(5.106)

cdE, 1 = ( pu · cmE, 2 + pm · cdE, 2 + pd · cdE,2 2 ) · e−rd ·t .

(5.107)

At the initial node, at time t=0 the value of the option becomes cmE, 0 ≡ c0E = ( pu · cuE, 1 + pm · cmE, 1 + pd · cdE, 1 ) · e−r0 ·t .

(5.108)

If the option was American, then, the equations at the terminal time step would be identical, i.e. cuA,2 2 = max[FV(ru 2 − R K ); 0],

(5.109)

cuA, 2 = max[FV(ru − R K ); 0],

(5.110)

cmA, 2 = max[F V (r0 − R K ); 0],

(5.111)

cdA, 2 = max[FV(rd − R K ); 0],

(5.112)

cdA,2 2 = max[FV(rd 2 − R K ); 0].

(5.113)

At each of the middle steps though, which in our illustrative case is for t = t, we need to check whether early exercise is optimal. If we were for example at the node at which the interest rate is ru , then we would have

308

T. POUFINAS

to compare the discounted expected value with the payoff of the early exercise, which would have been max[F V (ru − R K ); 0].

(5.114)

The investor would then choose the maximum between the discounted expected value and the payoff of the early exercise. This would hold true for all nodes of this time step; thus cuA, 1 = max{( pu · cuA,2 2 + pm · cuA, 2 + pd · cmA, 2 ) · e−ru ·t ; max[FV(ru − R K ); 0]},

(5.115)

cmA, 1 = max{( pu · cuA, 2 + pm · cmA, 2 + pd · cdA, 2 ) · e−r0 ·t ; max[FV(r0 − R K ); 0]}, (5.116) cdA, 1 = max{( pu · cmA, 2 + pm · cdA, 2 + pd · cdA,2 2 ) · e−rd ·t ; max[FV(rd − R K ); 0]}.

(5.117)

The value of the option at the initial node remains unchanged, i.e. cmA, 0 ≡ c0A = ( pu · cuA, 1 + pm · cmA, 1 + pd · cdA, 1 ) · e−r0 ·t .

(5.118)

The same rationale may be repeated for European and American interest rate put options. Consequently, we now have an approach to price American interest rate options. Example 5.14 Let as assume now that we build a two step tree for which the up, middle and down probabilities are 0.25, 0.5 and 0.25 respectively. Let T = 2, thus t = 1. Assume that R K = 4.1%. The initial interest rate is r 0 = 4% and each up or down move is by 1 percentage point. Assume that the face value is 100. There is a derivative (call option) on the interest rate that has a payoff at maturity of max[100(r − 4.1%); 0].

(5.119)

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INTEREST RATE DERIVATIVES

309

By applying the aforementioned equations we derive that at the terminal time step: cuE,2 2 = max[100(6% − 4.1%); 0] = 1.9 = cuA,2 2 ,

(5.120)

cuE, 2 = max[100(5% − 4.1%); 0] = 0.9 = cuA, 2 ,

(5.121)

cmE, 2 = max[100(4% − 4.1%); 0] = 0 = cmA, 2 ,

(5.122)

cdE, 2 = max[100(3% − 4.1%); 0] = 0 = cdA, 2 ,

(5.123)

cdE,2 2 = max[100(2% − 4.1%); 0] = 0 = cdA,2 2 .

(5.124)

At the first time step for the European option we see that cuE, 1 = (0.25 · 1.9 + 0.5 · 0.9 + 0.25 · 0) · e−0.05·1 = 0.88,

(5.125)

cmE, 1 = (0.25 · 0.9 + 0.5 · 0 + 0.25 · 0) · e−0.04·1 = 0.22,

(5.126)

cdE, 1 = (0.25 · 0 + 0.5 · 0 + 0.25 · 0) · e−0.03·1 = 0.

(5.127)

The equations for the American option change to cuA, 1 = max{(0.25 · 1.9 + 0.5 · 0.9 + 0.25 · 0) · e−0.05·1 ; max[100(5% − 4.1%); 0]} = 0.90,

(5.128)

cmA, 1 = max{(0.25 · 0.9 + 0.5 · 0 + 0.25 · 0) · e−0.04·1 ; max[100(4% − 4.1%); 0]} = 0.22, cdA, 1 = max{(0.25 · 0 + 0.5 · 0 + 0.25 · 0) · e−0.03·1 ; max[100(3% − 4.1%); 0]} = 0.

(5.129)

(5.130)

We observe that the American option privileges the early exercise at the top node of the first time step. This affects the value of the option also at the initial time step, as the European call option has a value of cmE, 0 ≡ c0E = (0.25 · 0.88 + 0.5 · 0.22 + 0.25 · 0) · e−0.04·1

310

T. POUFINAS

= 0.315,

(5.131)

whereas the American call option has a value of cmA, 0 ≡ c0A = (0.25 · 0.90 + 0.5 · 0.22 + 0.25 · 0) · e−0.04·1 = 0.320.

(5.132)

Apparently the American call option has a slightly higher price. We used 3 decimals to reveal this difference, as the two premia are the same at two decimals. As the amounts involved are usually much higher than the principal of 100 we used, the difference would be more apparent. The aforementioned interest rates and option values are shown in Fig. 5.11. The interest rates are above each node and the option prices are below each node. The European call option prices are at the left and the American call prices are at the right.

Fig. 5.11 Trinomial interest rate tree—An example (Source Created by the author)

5

INTEREST RATE DERIVATIVES

311

5.5.3.2

Pricing Bond Options with the Hull & White Trinomial Tree A variation of the aforementioned approach would be to use the trinomial trees constructed in Chapter 3 for the Hull & White model. The approach is pretty much the same; however the branching is non-standard and the probabilities of up, middle and down movement are endogenously produced, whereas in the example we presented in the previous subsection the probabilities where exogenously defined. Attention is needed only in the construction of the interest rate or bond trinomial tree according to the particulars of the approach from the initial term structure. The pricing of the option is done as follows (Hull, 1997): 1. The initial zero-coupon yield curve is considered. 2. Interest rates corresponding to maturities that are interim points of the yield curve are filled with linear interpolation. 3. The instantaneous short rate r(t ) can be derived from Eq. (3.85) of Chapter 3 as r (t) =

t · t + lnt+t At st+t t+t Bt

,

(5.133)

t is the spot interest rate for the period t, when at time t. where st+t

4. The term structure at a node of the Hull & White tree is calculated with the use Eq. (3.85) of Chapter 3 after the instantaneous short rate has been produced from the previous Eq. (5.133). 5. The bond is priced on each node. 6. Following the procedure described in the previous subsection the option is priced.

5.6

Bonds with Embedded Options

There are bonds that have embedded options. Of course, the options are not separated from the bonds, but the distinction helps us to understand the operating mechanism of these bonds. Bonds with embedded options may be viewed as a portfolio of an underlying bullet (plain vanilla) bond, i.e. a bond that cannot be redeemed before their maturity and an option

312

T. POUFINAS

(on it) to buy or sell it. These bonds and their corresponding derivatives are examined below. The most common examples are callable bonds and puttable bonds which were defined in Chapter 2. We also attempt to price these bonds. 5.6.1

Callable Bonds

A callable bond allows the issuer to repurchase the bond early, i.e. before it matures at a predetermined price at a predetermined time or time instants in the future. It therefore behaves like a bond that has an embedded call option. The holder of such a bond is like having a long position on the bond and a short position on a call option on the bond. The issuer has a short position on the bond and a long position on a call option. The exercise price of the call option is the price that the issuer has agreed to pay to buy back the bond (Hull, 1997). Although the bond cannot be separated from the call option its price will look like Pactual = Pbullet − c,

(5.134)

where Pactual : Pbullet : c:

the callable bond price. the price of an equivalent non-callable bond. the value of the call.

Callable bonds cannot be repurchased until a certain point of time since their issuance has elapsed. Then, the price at which the bond is repurchased drops with the passage of time. An investor who has acquired a callable bond faces the risk of early repurchase from the issuer; he or she thus expects a higher yield compared to a plain vanilla bond. This is also explained by the fact that he or she has a short position on a call option, so he or she has received a premium for that. A higher yield than a non-callable bond is reflected in either a lower purchase price or a higher coupon. Both lead to a higher yield (see also Hull, 1997). The issuer exercises the right to call the bond when the bond price is higher than the exercise price. It is caused by falling interest rates, so it has an interest in buying the bonds early as he or she may be able to borrow at a lower interest rate.

5

5.6.2

INTEREST RATE DERIVATIVES

313

Puttable Bonds

A puttable bond gives the holder the right to sell it back to the issuer earlier than the maturity date of the bond at a predetermined price and on a predetermined date or dates in the future. It therefore behaves like a bond that has an embedded put option. The holder of this bond is like having a long position on the bond and a long position on a put option. The issuer has a short position on the bond and a short position on a put option respectively. Although the bond cannot be separated from the call option its price will look like Pactual = Pbullet + p, Pactual : Pbullet : p:

(5.135)

the putable bond price the price of an equivalent non-puttable bond the value of the put.

An investor who has acquired a puttable bond has substantially less risk, as he or she has opportunities to sell the bond especially if there is a fall in its price. This means that such a bond has a lower yield than a bond that does not have the option to be redeemed earlier than its maturity date. This can be seen from the fact that the holder of the puttable bond has actually paid a premium for its purchase. The lower yield compared to a plain vanilla bond is reflected in either a higher price or in a lower coupon. Both lead to lower yields (see also Hull, 1997). The investor exercises the option to sell the bond early if the bond price falls below the exercise price. This happens when interest rates go up. He or she can therefore gain access to a higher yielding bond, which means that he or she has an interest in redeeming the bond early. 5.6.3

Option-Adjusted Spread

For the valuation of bonds or MBSs that have embedded options we use the option-adjusted spread (OAS). This is a measure of excess return, over the yield of a risk-free bond (e.g. government bond), incorporating the options embedded in the bond—hence the term spread. To calculate the OAS, we first calculate the value of the interest rate derivative (bond with embedded options in our case) with the embedded option using the zero-coupon risk-free (or government) curve/term

314

T. POUFINAS

structure. This is inserted to the valuation model used. This value is compared to the market price of the interest rate derivative (bond with embedded options). The process is repeated until we find the parallel shift of the interest rate curve that equates the bond value estimated by the model to its market price. This parallel shift is the OAS. To understand how it works we consider a callable bond. Initially we will not inquire the method we use to model the interest rates. Whatever the model is, assume that the price of the bond for the appropriate (risk(0) free, let us say government) yield curve is P0 > Pactual . We need to move the yield curve up by a parallel shift so that the price matches the actual price. Depending on the model used this may involve an iterative numerical method. Assume that we apply a parallel shift of λ(1) on the risk-free yield curve, where the superscript denotes the step of the iterative (0) (1) process. Suppose that the new price satisfies P0 > Pactual > P0 . Then, the new parallel shift to be applied to the yield curve is given by linearly interpolating: λ(2) = λ(1) ·

(0)

P0 − Pactual (0)

(1)

P0 − P0

.

(5.136)

We apply the new parallel shift to the yield curve and then re-price the bond. Assume that the new price is closer but still not exactly equal to the actual price P0(0) > Pactual > P0(2) > P0(1) . We linearly interpolate once more to find: λ(3) = λ(2) ·

(0)

P0 − Pactual (0)

(2)

P0 − P0

.

(5.137)

We repeat the process until the price is equal to the actual price. If this happens after m steps, then the option adjusted spread is OAS = λ(m) .

(5.138)

The option adjusted spread shows the incremental yield (spread) over the underlying yield curve that is due to the (embedded) call option (see also Hull, 1997 for a numerical example). We will now sketch the way to find the value of the option. Assume that we have a callable bond on a coupon bearing bond that matures on T 2 and makes annual coupon payments. The bond can be called on a

5

INTEREST RATE DERIVATIVES

315

single date T 1 < T 2 at par. A bond that is callable on a single date only is referred to as discretely callable. We will assume for simplicity that T 2 −T 1 is an integer number of years (or compounding periods in general). The embedded option behaves like a European call option on the bond. Let the bond be rated A. As such it carries both a default and a call risk. Its actual price in the market is known and we are interested in the yield spread and the OAS of the bond. The steps we follow are: 1. We determine the risk-free benchmark rates. This is the interest rate term structure or yield curve for the given credit rating and is constructed from the bonds that are available in the market. 2. We model the interest rates. This can be done even with the plain vanilla trinomial tree we presented in Chapter 3. 3. We value (price) the bond on the tree ignoring the call provision (i.e. as a bullet bond). To do that, we recall that a coupon-bearing bond behaves like a portfolio of zero-coupon bonds. We thus value on a separate tree each of the cash flows paid by the bond and then we add them. This gives a price of the bond which we denote by Pb . This is higher than the actual price of the bond Pactual , as it has not accounted yet for the call provision. 4. We measure the option adjusted spread. We apply a parallel shift and the aforementioned OAS approach to find the spread that results in the actual price; we denote it by λb . However, we still have not accounted for the call provision. At some of the terminal nodes the price of the bond will be higher than the strike price (K ) and thus the call option will be exercised. We reprice the bond and find that the new price that takes into account the call option, denoted by Pc is lower than the actual price. This is a result of the exercise that sets the value of the bond at K at the nodes at which the option is exercised (from a value that was higher than K, hence the exercise). In order to exactly match the actual price we need to reduce the spread to a level λc < λb . This is the spread that accounts for the call provision and sets the bond price equal to the actual price.

316

T. POUFINAS

If it were possible to buy the bond at Pc < Pactual , then it would provide an incremental return equavalent to that of the bullet. This means that for the same price as the bullet bond the callable bond provides a lower expected return; for the same OAS λb it would have to be valued at a lower price. 5. We value the option. We now use the tree to value the option using risk-free rates plus λc . Let c denote the value of the call. Then the value of the bullet would be

Pbullet = Pactual + c.

(5.139)

The yield to maturity of the bullet bond is referred to as the option-free yield. It is compared with the yield to maturity of the benchmark yield curve. The difference of the two is known as the option-free yield spread. I.e. option − free spread = yoption−free − ybenchmark .

(5.140)

Sometimes the option value is measured in terms of basis points (yields) as the difference between the yield to maturity of the specific bond issue and the option-free yield spread. I.e. option value in bps = ybond − yoption

5.7

- free .

(5.141)

Mortgage Backed Securities

Mortgage-backed securities, also known as MBS, are securitized mortgages. They are constructed by financial institutions when they securitize all or part of their mortgage portfolio—which is usually mortgaged to the property. These loans are placed in a pool and this debt is divided into pieces that constitute securities. These securities are available to investors and are known as mortgage-backed securities (Hull, 1997). An investor, for each such security in his or her possession, receives the part of the capital and interest cash flows that correspond to the security (as a percentage of the total pool) and which result from the mortgage loans of the pool.

5

INTEREST RATE DERIVATIVES

317

The borrower usually has the right to repay his or her loan early. So he or she has an American-type call option to repay the loan to the lender at face value (outstanding balance). Early repayments take place made for various reasons, the most common of which are (Hull, 1997): • The decrease of interest rates, triggering the owner—borrower to refinance the loan at a lower interest rate. • The sale of the property (house). An important component in the valuation of an MBS is the prepayment function. This is the function that describes the expected prepayments (or early repayments) in the loan portfolio that makes up the MBS at a given time and depends on the interest rate term structure and other related variables. It also depends on the behavior of the borrowers and that is what it tries to capture. The prepayment function is not reliable in predicting the prepayment of an individual mortgage. A mortgage will either be prepaid (early repaid) or not. However, the accuracy of the prediction can be improved by pooling together a set of similar mortgage loans, as the law of large numbers applies (Hull, 1997). Prepayments are not only due to interest rate movements. However, prepayments are more likely to be made in a low interest rate environment as borrowers can borrow at a lower interest rate. This means that investors in an MBS will lose part of the cash flows, the one that comes from interest. They therefore demand a higher return for this “risk”, as compensation for the prepayment options they have “written”/sold/granted. Another risk is that the borrower is unable to repay his loan (default). This means that he or she cannot pay one or more installments. In the US, mortgage loans are guaranteed by government agencies such as the Government National Mortgage Association (GNMA) or the Federal National Mortgage Association (FNMA), thus protecting investors from a potential default of the borrower (Hull, 1997) (Fig. 5.12). These MBSs give to all investors the same cash flows and thus they are all exposed to the same prepayment (or early repayment) risk and receive the same return. They are also referred to as pass-throughs because of this mechanism of their operation, i.e. to pass the payments of the borrowers to the security holders (Hull, 1997). Each of these securities

318

T. POUFINAS

Mortgage 1

MBS 1

Mortgage 2

MBS 2

Mortgage 3

MBS 3

Pool

Mortgage M

Fig. 5.12

SPV

MBS I

MBS (Source Created by the author)

at any point of time pays an amount that is the sum of three components: the interest on the outstanding principal; the amortization on the outstanding principal and the prepayments. The latter can be due to the ordinary prepayment or due to the default of the borrower in which a potential payment is made by the insurer. There are three main triggers of prepayment (CIIA, 2004) • Mobility: they pertain primarily to demographic, i.e. human, social and natural causes and are not related to economic variables. Such reasons include death, divorce, inheritance, sale, relocation, etc. These triggers are expected to occur at a relatively constant (or at least predictable) rate over time. Their cumulative volume reflects (the passage of) time, also referred to as seasoning. • Refinancing: they are related to the partial or full repayment of the mortgage loan mainly to take advantage of potentially lower interest rates on a new mortgage or to take a new, possible larger loan if there is additional collateral available. These reasons are primarily related to economic variables. They reflect primarily the interest rate level. • Default: they are partially due to personal events and partially due to macroeconomic events. To a certain extend they are expected in a portfolio of mortgage loans. As a mortgage loan has the property as collateral a default may create a problem only if the value of the real estate is lower than the loan and interest rates are rising so refinancing would not help. If there is insurance, then it kicks in

5

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319

and prepays the relevant amount to the MBS investors. They reflect mainly the real estate prices. 5.7.1

Interest Only and Principal Only MBS

This is an MBS type where interest payments are separated from capital payments. All capital payments go to securities called principal only (PO) and all interest payments go to another group called interest only (IO). These MBSs are also known as stripped (Hull, 1997). As the prepayments increase, the POs acquire more value and the IOs less. The opposite happens when prepayments are reduced. In POs a total capital is returned to the investor. If there is default protection then only the time is unknown. A high rate of prepayments leads to early repayment of the principal which is good for the PO holder. A low rate of prepayments leads to a slow repayment of the principal and thus a reduction in the return of the POs. On the contrary, for IOs the cash flows are not certain. As the prepayment rate increases, the cash flows received from the IO investor decrease (Hull, 1997). 5.7.2

MBS Cash Flows and Valuation

We are interested in finding the cash flows of an MBS at each time t. If prepayment was not an option, then the cash flows would be similar to those of an annuity as they would depend purely on the installments that are paid by the borrowers. However, prepayments affect these cash flows significantly. The prepayment pattern is to a great extend behavioral; however, there have been attempts to standardize it or forecast it. We will proceed as if we knew it and we will explore the standard approach later. We assume that we know the initial mortgage balance MB 0 and that payments take place on a monthly basis. We denote by (CIIA, 2004) for month m = 1…M. CFIm : NIPm : SPRm : UPRm :

The total cash flow to the investor. The monthly interest payment net of servicing and other fees. The scheduled principal payment for month m. The forecasted unscheduled principal prepayments in month m.

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T. POUFINAS

GIPm : MBm : MSPm : SMMm : MTPm : SF m : ir: sf: M: m:

The gross interest (coupon) payments—including fees. The mortgage balance at time m. The monthly scheduled payment made by borrowers. The standard (or other) monthly mortality rate, i.e. the prepayment rate. The monthly total payment made by borrowers. The monthly service fee amount. The monthly interest rate—denoted ir so as to distinguish it from the term structure. The monthly service (and other) fee rate. The time to maturity in months The number of months since the initiation of the mortgage.

The cash flows that are paid to the MBS investors are (CIIA, 2004) CFIm = SPRm + UPRm + NIPm .

(5.142)

However, the payments made by the borrowers are a bit higher as the service fee has not been subtracted yet. Therefore MTPm = GIPm + SPRm + UPRm .

(5.143)

The interest (coupon) payments are GIPm = ri · MBm−1 .

(5.144)

SFm = sf · MBm−1 .

(5.145)

The service fee is

The interest payment net of service fee is NIPm = (ir − sf) · MBm−1 .

(5.146)

The monthly scheduled payment made by the borrowers includes the accrued interest for the month (coupon) and the amortization on the mortgage balance and is thus given by MSPm = MBm ·

ir · (1 + ir) M−m+1 . (1 + ir) M−m+1 − 1

(5.147)

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The scheduled principal repayments are SPRm = MSPm − GIPm .

(5.148)

We need though to account for the forecasted unscheduled principal repayments. These are forecasted at a rate SMM m and are thus calculated by UPRm = SMMm · (MBm−1 − SPRm ).

(5.149)

The total repayments are SPRm + UPRm ,

(5.150)

which is the amount that reduces the mortgage balance, i.e. MBm − MBm−1 = SPRm + UPRm .

(5.151)

Using (5.142) and (5.146) we see that the cash flow to the MBS investors becomes: CFIm = SPRm + UPRm + NIPm = SPRm + UPRm + GIPm − SFm .

(5.152)

As far as the prepayment rate is concerned, we accept that it follows the Public Securities Association (PSA) standard prepayment benchmark. The PSA model assumes that prepayments are low for new mortgages but increase with the passage of time until it reaches a maximum. The monthly rate is 0.2% for the first month; it increases by 0.2% each month for the next 30 months until it attains a rate of 6% per annum. It remains at this level for the remaining of the life of the MBS (CIIA, 2004). We define as Conditional Prepayment Rate (CPR) the annual prepayment rate; the adjective conditional determines that the rate is conditional on the remaining mortgage balance. Consequently, the PSA benchmark can be expressed as  m · 6% m ≤ 30 months . (5.153) CPRm = 30 CPR = 6% m > 30 months This benchmark is referred to as 100% PSA or 100 PSA. If slower or faster prepayment rates are assumed for a certain MBS, then the notation

322

T. POUFINAS

changes accordingly. For example 200 PSA means a prepayment rate of 2 times the CPR of the PSA benchmark. 50 PSA means 0.5 times the CPR of the PSA benchmark and so on and so forth (CIIA, 2004). Recall that in the previous discussion we used the single monthly mortality rate. According to the PSA benchmark, a CPR of 6% means that 6% of the mortgage balance is prepaid per year. To go from the annual to the monthly churn rate we observe that (1 − CPRm ) = (1 − SMMm )12 ⇒ SMMm = 1 − (1 − CPRm )1/12 .

5.8

(5.154)

Collateralized Mortgage Obligations

The pass-through mortgage-backed securities expose all investors to the same type and level of risk; namely, they are all exposed to the same level of default (credit) risk and the same level of prepayment (or early repayment) risk. Furthermore, all investors receive the same return (Hull, 1997). To allow for differentiation in the risk profile of the investors a structured product called collateralized mortgage obligation (CMO) has been developed. A CMO provisions for more security types, which essentially divide/group the investors in groups or classes or tranches (from the French word slice). The CMO directs the prepayments to the different classes according preset rules (Hull, 1997). For example, there could be 3 groups (or tranches) A, B, C in such a way that all repayments (early and scheduled) are distributed to A until its investors are repaid. They are then distributed to B until its investors are repaid and then to C. A-investors have a higher risk of early repayment. The securities of A have shorter maturity than the securities of B, which in turn have shorter maturity than the securities of C. This is also due to the fact that besides prepayments, A-investors receive usually also all principal amortization and the interest on its outstanding principal. B receives only the corresponding interest. It receives no principal until A has received all its capital and matures. Similarly, C receives only interest until group B matures. Other arrangements could be possible; however this type of a CMO is considered to be the most common and is sometimes referred to as sequential CMO or sequential-pay CMO; it consists of three of four tranches that mature sequentially.

5

INTEREST RATE DERIVATIVES

Mortgage 1

323

Tranche A

Mortgage 2 Tranche B

Mortgage 3

Pool

SPV Tranche C

Residual (Z)

Mortgage M

Fig. 5.13

CMO (Source Created by the author)

Some CMOs may have a Z-tranche; in some case they may have no Ztranches (hence in parentheses in Fig. 5.13). It is the tranche that receives the lowest ranking or seniority. The Z-tranche receives no interest or cash flow until the more senior tranches have been paid off. The interest that would correspond to the Z-tranche is used to pay off the principal of the more senior tranches. It functions as a cushion to the risks that the other tranches face; namely prepayment risk and extension risk. As it receives no payments until all the other tranches have been retired, it behaves like a zero-coupon bond and as such it sometimes called an accrual tranche or Z-bond. It can take decades (20 years is not uncommon) before the Z-tranche receives any payments. It has increased volatility as interest rates change over time and the mortgage pool is affected not only by interest rates but by prepayments, refinancing and defaults. As such of course it protects the upper tranches. The Z-tranche investors have a lower reinvestment risk as they receive no payments until all other tranches mature. Their interest accrues and due to its long time to maturity (and thus duration) it can be a suitable asset to match long term liabilities.

5.9

Asset Backed Securities

An asset-backed security (ABS) is a security backed by assets different from real estate (mortgage). In this way it resembles MBS. The backing assets can be loans (except for mortgage loans), credit card receivables,

324

T. POUFINAS

leases, etc. that normally produce interest or other forms of income. The structure of an asset-backed security is similar to that of an MBS or rather a CMO; a pool is created so as to include these assets and payments are made to investors via securities (bonds). These securities are separated in tranches of different risk (and thus expected return) so as to address the risk profiles of the investors. The highest rated tranche (bond) could be rated as high as AAA and constitutes the senior tranche. This is offered the lowest promised return as it has the lowest risk among the tranches. An interim tranche, the mezzanine tranche, follows that is rated lower (for example BBB). This tranche has a somehow higher promised return as it bears higher risk. There could be more tranches. There is also a last residual tranche that is not rated that absorbs the first losses. This is referred to as the toxic waste or equity tranche. It bears the highest possible risk and thus it has a significantly higher expected return. As such it is most of the times retained by the issuing financial institution and is not made available to investors (Hull, 2012; Marrison, 2002) (Fig. 5.14). The aforementioned pictorial representation depicts the ABS in a decreasing order of risk borne by the various tranches. Placing the senior tranche at the top and the equity tranche at the bottom would indicate an increasing order of risk; this is the approach selected for the other derivative products structured through securitization, such as CMOs and CDOs. An ABS overall operates like these structured products. However, we chose both representations so that the reader is exposed in both directions of risk-exposure and seniority. Asset 1

Equity or Toxic Waste

Asset 2 Asset 3

Pool

Senior or AAA Tranche

Asset N

Fig. 5.14

SPV

Mezzanine or BBB Tranche

ABS (Source Created by the author)

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325

In order to separate the credit risk originating from the issuing financial institution from the credit risk of the ABS a special purpose vehicle (SPV) is established. The role of the SPV is to transfer the payments from the pool to the tranches and thus to the investors. The SPV is endowed with assets of the highest quality, very often of the highest possible credit rating. As a result it could be that the SPV has a better credit rating than the issuing financial institution (e.g. bank) that created the ABS. This ensures that even if the financial institution defaults, the payments of the ABS will not be disrupted. The SPV separates the credit risk of the issuing financial institution from the credit risk stemming from the asset pool. However, the ABS still carries the risk that the payments to the investors are not made in full. This risk depends (i) on the level of over-collateralization, i.e. the excess of the asset pool value over the security value; and (ii) on the volatility of the asset value (Marrison, 2002). Example 5.15 An example would be that the equity (toxic waste) tranche absorbs the first 5% and the mezzanine tranche the second 15% (totaling 20%). The senior tranche is left with the remaining 80%. Thus if losses exceed 5% then the equity tranche loses all of its principal, if they exceed 20% then the mezzanine tranche loses all of its principal and the senior tranche bears the residual losses (on top of 20%). In a way similar to the development of collateralized mortgage obligations from mortgage backed securities, a series of collateralized debt obligations have been developed. Their aim is to create more tranches, so that investors can find the tranche appropriate to their risk profile and appetite, in order to gain access to the desired return. Their names vary and are known as collateralized bond obligations (CBOs), where the backing assets are bonds; collateralized debt obligations (CDOs), where the backing asset can be any type of debt—but according to some bonds only; and collateralized loan obligations, where the backing assets are loans. 5.9.1

Collateralized Debt Obligations

A collateralized debt obligation (CDO) is a structured product (a derivative) that is backed by a pool of loans, bonds and other assets. CDOs

326

T. POUFINAS

do not specialize in one type of debt. They are similar in structure to a collateralized mortgage obligation (CMO). However, CDOs represent different types of debt and credit risk. As a matter of fact collateralized bond obligations (CBOs) and collateralized loan obligations (CLOs), as well as CMOs are specific types of CDOs. It is evident that a CDO can incorporate different asset classes. A general simplified definition is that a CDO is a securitization of corporate obligations. CDOs may securitize (or re-securitize) among others commercial loans, corporate bonds, ABSs, and emerging market debt. Even tranches of CDOs may be (and have been) re-securitized into CDOs of CDOs (Fig. 5.15). CDOs issue multiple classes of equity and debt in a way similar to CMOs. The classes are organized in tranches with varying levels of seniority in terms of default and repayment. These can be the equity tranche also referred to as junior subordinated notes, preferred stock or income notes, is the lowest tranche. Some other times the junior tranche is differentiated from the equity tranche. The equity tranche comes last in receiving payments and as in the case of CMOs it absorbs the first losses or delays in payments. This function of the equity tranche makes the more senior debt tranches of better credit quality, as it receives payments only after the debt tranche claims have received their cash flows. The same holds true for subordinated CDO debt tranches; they offer protection to more senior tranches by absorbing the losses before them and thus offer to investors a higher coupon as a compensation for the risk Asset 1

Senior (AAA/ AA)

Asset 2

Senior (A)

Asset 3

Pool

SPV

Mezzanine (BBB) Junior (BB) Equity (NR)

Asset N

Fig. 5.15

CDO (Source Created by the author)

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INTEREST RATE DERIVATIVES

327

they are exposed to. Sometimes coupon payments may be deferred until the payments from the pool suffice. In a way similar to CMOs the higher the credit rating of the tranche is, the lower the risk becomes; thus the lower the promised interest (or coupon) paid by the tranche (or bond). In case of a default the senior tranches are paid first from the collateralized pool of assets, followed by mezzanine and junior according their credit ratings. CDOs may offer additional features some of which include (Fig. 5.16): • They may draw money from the investors with delay as needed when the assets are purchased over time. • The leverage of the CDO may be adjusted with the introduction of a revolving tranche. • The same seniority tranche may be comprised of separate fixed and floating rate sub-tranches. • Debt tranches are sometimes protected by bond insurance. • The seniority can also be created synthetically outside the CDO structure with the use of a credit default swap. 5.9.2

Collateralized Bond Obligations

A collateralized bond obligation (CBO) is a structured product (a derivative) that is backed by a pool of junk bonds, i.e. bonds that are not investment grade and thus their credit rating is below BBB. The pool created includes a wide range of credit qualities and from a series of different issuers that allows the creation of tranches some of which (or even all except for the toxic waste) are investment grade. The latter is feasible due to the diversification described in the pool of bonds. In addition it is usually over-collateralized. It is therefore considered that the probability that all the bonds in the pool default is quite smaller than the probability of default of each individual bond. Each tranche bears a different level of risk. As in CMOs and CDOs the top tranche has the highest quality and the lowest risk and as such has a priority in receiving payments; however, it pays the lowest interest rate. The middle tranche(s) bear(s) higher risk and follow(s) the top tranche in receiving payments; it (they) therefore promise(s) a higher interest rate than the top tranche. Finally, the bottom tranche bears the highest risk and thus promises the highest return. As in all other similar structured

328

T. POUFINAS

Tranches

Seniority

Rating

A-1 Floating Rate Revolving Facility

A2 Fixed Rate Tranche

Senior

AAA/AA

B-1 Floating Rate Tranche

B-2 Fixed Rate Tranche

Senior

A

C Fixed or Floating Rate Tranche

Mezzanine

BBB

D Fixed or Floating Rate Tranche

Junior

BB

Equity Most Subordinate Tranche

Equity

NR

Fig. 5.16

CDO Tranches (Source Created by the author)

products the bottom tranche absorbs the first losses and the top part the last, if they surpass the intermediate tranche(s). 5.9.3

Collateralized Loan Obligations

A collateralized loan obligation (CLO) is similar to a CMO, however the assets being securitized are loans and most of the times corporate loans (with low credit ratings) or loans taken out by private equity firms to pursue leverage buy outs. A CLO is structured in tranches just as the CDOs and CMOs, with the same risk and return characteristics and priority in repayment. MBSs, CMOs, ABSs CDOs, CBOs and CLOs may be used as part of ALM strategies as they were described in Chapter 4. They can transfer

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329

long-term assets from the banks to investors, thus facilitating the achievement of immunization. Furthermore, they can allow for regulatory capital reduction.

5.10

Swaptions

In this section we focus on European swap options (swaptions ). These are options on interest rate swaps. Under a swaption the counterparty with the long position has the right (but not the obligation) to enter into a predetermined interest rate swap at a predetermined time in the future (as European option). The underlying value of the swaption is the fixed interest rate of a swap (Hull, 1997). Swaptions can be used by a financial institution or company that knows it will have to make payments calculated with a floating interest rate at a future point of time and wants to exchange it for a fixed rate. At the maturity of the swaption it should compare the fixed interest rate of the swap it finds in the market with the fixed interest rate it secures from the swaption and choose the lower one (Hull, 1997). Another (symmetrical) case is that of a financial institution or company (usually an insurance company) that has guaranteed a fixed interest rate, but estimates that the assets it has invested will not deliver that interest rate. These assets have a floating return. Such a company therefore acquires a swaption according to which it receives the fixed interest rate of the swap and pays the floating one. If at the maturity of the swaption it can find in the market a swap that pays a higher interest rate than that of the swaption then it chooses it. Otherwise it exercises the swaption and enters the corresponding swap. The swaption has a premium but gives the right to the counterparty with the long position not to exercise it if it can find a swap with a fixed interest rate lower or higher (depending on whether it pays fixed or receives fixed rate) than that of the swaption. Alternatively, if the financial institution or company does not wish to pay a premium, it may enter into a forward swap. Their advantage over swaptions is that they do not have an initial outflow (premium), but they are an obligation and thus commit the company to enter the specific swap (Hull, 1997). Even if the company finds a swap with a better interest rate, it cannot take advantage of it.

330

T. POUFINAS

5.10.1

Valuation of Swaptions

Our presentation so far with regards to the pricing of options assumes that the underlying asset follows a lognormal distribution. The same holds true in the valuation of European swaptions at which the underlying instrument is swap interest rate that is available at the maturity date of the option. We consider a swaption according to which the investor has the right to pay a fixed interest rate R K and receive a floating rate through a swap that lasts  years and begins in T years. Payments are made at a frequency of ν per year on a face value (principal) FV. Suppose that the fixed interest rate of the swap at the maturity of the swaption is R. This is what we denoted in the interest rate swap section above as c fixed . We introduced though the notation R K as an analogue to the strike price K. The swap interest rate is a random variable as seen from today; hence the notation R is more representative compared to c fixed that denotes a known fixed interest rate. Both R K and R are compounded ν times a year. The investor compares these two interest rates and chooses the lower one (Hull, 1997). A swap is a series of payment exchanges. The payoff of the swaption for each of these payments is: FV max(R − R K , 0), ν

(5.155)

for payments exchanged ν times a year, at times τi = T +i/ν, i = 1 . . . ν. This resembles the payment of a long call written on the interest rate R with an exercise price R K . Black’s model indicates that the value at the current time of such an option (and therefore the value of the relevant payment) is: FV −si ·τi e [FN(d1 ) − R K N (d2 )], ν

(5.156)

where F is the forward rate of the swap (forward swap rate) corresponding to R, s i is the spot rate with maturity τ i , with continuous compounding and  ln(F R X ) + σ 2 T /2 d1 = , (157a) √ σ T √ d2 = d1 − σ T . (157b)

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331

The total value of the swaption is given as the sum of the values of (5.156) for all payment exchanges and is calculated as (Hull, 1997): swaptionc0 =

m  FV i=1

ν

e−si ·τi [F N (d1 ) − R K N (d2 )]

(5.158)

or swaptionc0

 m   FV −si ·τi · = e · [FN(d1 ) − R K N (d2 )]. ν

(5.159)

i=1

Similarly, a swaption that gives to the counterparty with the long position the right to receive a fixed interest rate R K has a payoff from the swaption of: FV max(R K − R, 0). ν

(5.160)

This resembles a long put option written on R. The value of the swaption is calculated with the use of Black’s model to become (Hull, 1997): p swaption0

  m  FV −si ·τi = e · [R K N (−d2 ) − FN(−d1 )]. · ν

(5.161)

i=1

Example 5.16 Consider a swaption that entitles the counterparty with the long position to pay 5% on a 3-year swap that starts in 2 years and receive a floating rate. Swap payment exchanges are annual and are made at a nominal value of e 100. The swap volatility is 20%. For the purposes of the example, we assume that the floating interest rate term structure is horizontal at the level of 4.50% per year with continuous compounding. We observe that the present value of swap payments (since ν = 1) is: 100(e−0.045×3 + e−0.045×4 + e−0.045×5 ) = 250.75.

(5.162)

The interest rate of 4.50% with continuous compounding has an equivalent interest rate of 4.60% with annual compounding. So the forward

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T. POUFINAS

interest rate is also 4.60%. Therefore we have F = 0.046, R K = 0.05, T = 2 and σ = 0.2. By applying (5.155) we get that the value of the swaption is e 1.

5.11

Caps and Floors 5.11.1

Caps

Interest rate caps are derivatives that offer protection (upper bound) against interest rate hikes on a floating-rate loan. This protects the borrower from interest rates rising above a certain level—called the cap rate. They are usually traded in over-the-counter markets (OTC markets) (Hull, 1997). It is clear that such an “upper bound” in the level of the interest rate is not for free and therefore results to a charge that has to be paid by the borrower. This charge is either distributed periodically, embedded in the interest rate of the loan, when the protection is offered as a (packaged) feature of the loan by the financial institution that offers it or it is collected in advance as a lump sum, when the protection is offered by a third party (Hull, 1997). The introduction of a cap changes the effective interest rate of the loan to the minimum of the current interest rate and the cap rate. This is shown schematically in Fig. 5.17. A cap can be seen as a portfolio of interest rate options. To see that, let R K be the cap rate and FV denote the face value (principal). If τ is the period between two installments, then interest payments are made at times τi = iτ , i = 1 . . .  during the life of the cap. Then the borrower based on the floating interest rate R i determined at the moment iτ makes the following payment at time (i + 1)τ: τ FVRi

if

Ri < R K

(163a)

τ FVR K

if

Ri > R K .

(163b)

This is equal to: τ FV min(Ri , R K ).

(5.164)

The interest owed by the borrower in one installment is however τ F V Ri ,

(5.165)

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INTEREST RATE DERIVATIVES

333

Interest rate cap & floor Interest Rate

Floor rate Cap rate

Floating

Floored Capped

Time

Fig. 5.17 Borrowing interest rate for a floating rate loan with a cap or floor (Source Created by the author with information assembled from Hull [1997])

if this protection does not exist. Combining (5.164) and (5.165) we get that the interest paid is: Ii+1 = τ FVRi − τ FV max(Ri − R K , 0).

(5.166)

However, as this amount is paid, the cash flows for the borrower become: −Ii+1 = −τ FVRi + τ FV max(Ri − R X , 0).

(5.167)

The latter relationship indicates that interest payment can be broken down into a short position on the loan and a long position on a call option on the interest rate. The call option is written on the interest rate R i that is observed at the time instant iτ and is paid at the time instant (i + 1)τ. The exercise price is nothing but the cap rate R K . At the same time the lender has a long position on the loan and a short position on the call option. The mechanics of a loan with a cap rate indicate that the borrower must pay the amount determined by Eq. (5.167) and the lender to “return” the payoff of the option. The latter is positive if the lending rate exceeds the cap rate. For this right the lender “receives” a fee. There are therefore  such options called caplets . The cap is a portfolio of caplets. They are usually made so that the interest rate of the first

334

T. POUFINAS

period, as defined at the time of the disbursement of the loan (t = 0) does not lead to a payment at the time instant τ. 5.11.1.1 Valuation of Caps The caplet corresponding to the rate observed at time iτ has a payoff at time (i + 1)τ equal to (Hull, 1997): τ FV max(Ri − R K , 0).

(5.168)

If the interest rate R i follows a lognormal distribution with volatility measure σ i , then using Black’s model we obtain the value of the caplet at the present time as: capleti+1 := τ FVe−si+1 ·(i+1)τ [Fi N (d i1 ) − R K N (d2i )], 0

(5.169)

where s i+1 is the spot interest rate with continuous compounding maturing at time (i + 1)τ and F i is the forward rate for the time interval [iτ , (i + 1)τ ]. In addition,   ln(Fi R K ) + (σi2 2)iτ d1i = , (170a) √ σi iτ √ (170b) d2i = d1i − σi iτ . R K and F i have a compounding period of τ as opposed to the spot rates s i+1 that are continuously compounded. The total value of the cap is the sum of the values of the caplets that make it up and with the use of (5.169) is given by the relation: cap0 : =

−1 

capleti+1 0

i=0

= τ FV

−1 

e−sk+1 ·(i+1)τ [Fi N (d1i ) − R K N (d2i )].

(5.171)

i=0

5.11.2

Floors

By analogy, interest rate floors are derivatives that offer protection (lower bound) against interest rate droops on a floating-rate loan. The floor

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335

protects the lender against falling interest rates. This level is called the floor rate (Hull, 1997). This is shown schematically in Fig. 5.17. It behaves like a portfolio of put options written on the lending rate. They are usually written by the borrower and sold to the lender. Obviously their existence reduces the cost of borrowing for the borrower and this is reflected in a lower interest rate. 5.11.2.1 Valuation of Floors The value for the corresponding floorlet is derived also from Black’s model and becomes: := τ FVe−si+1 ·(i+1)τ [R K N (−d2i ) − Fi N (−d1i )]. floorleti+1 o

(5.172)

In a similar manner, the corresponding floor has a value of: floor0 : =

−1 

floorleti+1 0

i=0

= τ FV

−1 

e−sik+1 ·(i+1)τ [R K N (−d2i ) − Fi N (−d1i )].

(5.173)

i=0

In the valuation of caps and floors two approaches can be followed with regards to the volatility measures (Hull, 1997): • The volatility per caplet (floorlet) is different, in which case volatilities are known as forward forward volatilities . • All caplets (floorlets) have the same volatility for a specific maturity date of the cap (floor), but may change when the maturity date changes, in which case they are known as flat volatilities . 5.11.3

Collars

An interest rate collar is a derivative that sets both an upper and lower bound at the level of the borrowing interest rate. It behaves as a portfolio of a long cap and a short floor. It is usually made so that the net charge is zero, by equating the premium of the cap to the premium of the floor (Hull, 1997).

336

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An investor that has a long cap and a short floor experiences a net payment of τ FV max(Ri − R K , 0) − τ FV max(R K − Ri , 0) = τ FV(Ri − R K ).

(5.174)

at each time instant τ i . This resembles the payments made by a counterparty of a swap that makes payments at a fixed rate R K on a nominal principal of FV with period τ and receives payments with floating rate. Therefore the following equation must hold true: cap0 = floor0 + S0 ,

(5.175)

where cap 0 is the value of the cap, floor 0 is the value of the floor and S 0 is the value of the swap. Example 5.17 Let us consider a caplet that has an upper bound of 10% on a loan of e 100,000 for 3 months starting in 1 year. The compounding frequency is quarterly. Suppose that the forward interest rate for the quarter starting in 1 year is 8.75% with quarterly compounding. The current fifteenmonth interest rate is 8% per annum with continuous compounding. The volatility of the caplet quarterly interest rate is 20% per annum. We are looking for the value of the caplet. From the data of the exercise we have that: FV = 100, 000, R K = 0.10, Fi = 0.0875, si+1 = 0.08, τ = 0.25, iτ = 1, σi = 0.2.

(5.176)

Substituting at (5.169), using (5.170a) and (5.170b) we find that the value of the caplet is: capleti+1 = 63.67. 0

(5.177)

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Exercises Exercise 1 A forward contract is written on EUR 100,000 of face value of the corporate bond of company ABC. It has a delivery date of 1 year and a delivery price of EUR 95 per EUR 100 of face value. The bond price is currently EUR 93 per EUR 100 of face value (which is the minimum bond denomination) and makes two coupon payments of EUR 2 each (per EUR 100 of face value) in 3 months from today and in 9 months from today. The interest rate is fixed and equal to 4% per annum with continuous compounding. a. What is the value of forward contract today (long position)? b. Are there any arbitrage opportunities? If yes, show how an investor can take advantage of them (for this part, consider that there is the possibility of entering/exiting a forward contract after its start with the possible payment/collection of the value of the forward). c. If an investor agreed to enter the forward contract today and demanded that the value of the contract is zero, then what would be the delivery price that he or she would accept (long position)? d. What is the value of the forward contract after 6 months in questions (a) and (b) if the bond price at that time is EUR 94 (long position)? Exercise 2 An interest rate futures contract matures in 1 year and the counterparty with the short position can choose from a series of bonds which one to deliver at the maturity of the contract. One of the bonds that is a candidate for delivery, has a face value of EUR 100, a conversion factor of 1.4 and during the life of the futures contract makes two semi-annual coupon payments in 4 and 10 months with a coupon rate of 4% per annum. It was issued 14 months before the start of the futures contract (and therefore paid coupons before) and matures 7 years and 10 months after the maturity of the futures contract. The discount rate is fixed at 3% per annum with continuous compounding. a. What is the quoted bond price? b. What is the quoted futures price?

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c. The counterparty with the short position finds an alternative bond (with a face value of EUR 100) with conversion factor 1.2, the same coupon payment time instants, i.e. semiannual coupon payments in 4 and 10 months; the coupon rate of this bond is 6% per annum. It was issued 2 years and 2 months before the starting date of the futures contract and matures 11 years and 10 months after the maturity date of the futures contract. Which bond would he or she prefer to deliver? Exercise 3 An investor takes a short position in an interest rate futures contract expiring in 1 year from today. He or she has the choice to deliver one of the 3 bonds shown in the table below. All 3 bonds have a face value of EUR 100 and their maturity date is given in years and months from today. The discount rate is fixed and equal to 5% per annum with continuous compounding. Bond

Maturity

Coupon

1 2 3

1 year and 3 months 2 years and 6 months 3 years and 9 months

3% annual 4% annual 0%

Quoted price

Conversion factor 1.500

a. Find the quoted price of each bond at the starting date of the futures contract (today). b. Use the 1st bond to find the quoted futures price. c. For which conversion factors of the other bonds (i.e. 2nd and 3rd) could they all qualify as the cheapest to deliver? d. If the quoted futures price was EUR 100, are there any arbitrage opportunities? If so, how could you take advantage of them using the 1st bond? Exercise 4 An investor takes a short position in a 1-year futures contract that is written on bond index when the index is at 300 points and the interest

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rate is 4% per annum with continuous compounding. The multiplier of this contract is EUR 5. a. What is the initial (monetary) value of the futures contract? b. If after 6 months the price of the index is at 280 points and the interest rate has remained the same, then what is the (monetary) value of the futures contract? c. If the investor closes his or her position at that moment, what is his or her profit or loss? Exercise 5 Two banks A and B have access to both floating and fixed interest rate loans as follows (interest rates are with annual compounding and per annum): Bank

Fixed interest rate

Floating interest rate

A B

5% 6%

Libor% Libor+0.4%

A wants to borrow with floating interest rate and B with fixed interest rate. However, they observe that A has a comparative advantage in the fixed interest rate loans and B in the floating interest rate loans. So A borrows at a fixed interest rate and B at a floating interest rate and they enter a swap. a. Construct a swap using the comparative advantage argument. Suppose that the swap is for 3 years and with an annual exchange of payments in a notional principal amount of EUR 1,000,000. The discount rate for the cash flows is 4% per annum with continuous compounding. b. If the Libor of the 1st year is 5.5%, what is the value of the swap for each of the banks? c. If the banks wanted to enter a swap with a zero initial value (i.e. fair swap) what should be the fixed interest rate of the swap (the Libor of the 1st year remains 5.5%).

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d. Continuing question (b), if the Libor of the 2nd year becomes 5% what is the value of the swap at the beginning of the 2nd year? e. If bank A defaults at the beginning of the 2nd year (after making the 1st payment) what is the loss (if any) for bank B? f. If bank B defaults at the beginning of the 2nd year (after making the 1st payment) what is the loss (if any) for bank A? Exercise 6 Two banks A and B have access to both floating and fixed interest rate loans as follows (interest rates are with annual compounding and per annum): Bank

Fixed interest rate

Floating interest rate

A B

4% 5%

Libor+0.2% Libor+0.6%

A wants to borrow with floating interest rate and B with fixed interest rate. However, they observe that A has a comparative advantage in the fixed interest rate loans and B in the floating interest rate loans. So A borrows at a fixed interest rate and B at a floating interest rate and they enter a swap. a. Construct a swap using the comparative advantage argument. b. Suppose the swap matures in 3 years and with an annual exchange of payments on a notional principal amount of EUR 100,000. Assume that the Libor for the 1st year is 3% per annum, for the 2nd year 4% per annum and for the 3rd year 5% per annum. Show the payments that A makes to B, the payments that A receives from B and the net cash flow of each of A and B for each year. c. What are the total profit or loss for each of A and B for the entire life of the swap? Exercise 7 Let us consider a bond with a current price of EUR 99 and a maturity date of 9 months from today on which a forward contract, a European

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call option and a European put option are written. All three expire (or mature) in half-a-year and have an exercise—delivery or strike—price of EUR 120. The bond makes no coupon payments during the life of the derivatives. The premium of the call option is EUR 20 while the premium of the put option is EUR 30. a. What is the profit or loss for the counterparty with the long position in each of the above derivative products at maturity? (Hint: the general equation is needed.) b. Show the result graphically on the maturity date. c. Answer (a) and (b) for the short position holder. d. What are the similarities and differences of these derivatives? e. If the bond price at maturity is EUR 130, what is the profit or loss for each of the above derivatives and for each position? Exercise 8 A bank has introduced am innovative loan, which provides protection to the borrower against rising interest rates by setting a cap of 8% per annum with annual compounding on the borrowing rate (in case of rising interest rates). The loan matures in 3 years and the payments are annual. The discount rates for the 1st, 2nd and 3rd year are 4%, 5% and 6% per annum respectively with continuous compounding, while the respective annual forward rates for each year are 4%, 6% and 8% per annum with continuous compounding. The volatility is 10% per annum. a. What option has the bank actually offered to the borrower? b. What is the cost (premium) of this option if the loan amount is EUR 100,000? c. How much would the loan installment increase if this cost was evenly apportioned? d. How does the real borrowing interest rate change for the borrower? Exercise 9 An insurance company is going to acquire an option that gives it the right to enter into a swap in which it receives 4% per annum to cover an interest rate guarantee obligation to its policyholders. The option expires

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in 1 year, while the swap has annual payments in a notional principal amount of EUR 100 million and lasts 2 years after the expiration of the option. The floating interest rate is the LIBOR and its term structure is horizontal and equal to 5% with continuous compounding. This is the yield curve that the company uses for discounting cash flows. The volatility for the swap rate is 10% per annum. a. What type of option does the insurance company have? b. What is its value? c. When will the company exercise it and when not? d. If the company is able to find in the market a swap whose fixed interest rate is 5% per annum, then which one will it prefer? e. What are the reasons for a company to enter such an option? What is its loss if it does not exercise it? Exercise 10 A bank has granted its subsidiary insurance company an option for a fee— premium that gives the holder the right to enter into a swap in which the bank will pay 3%. The insurance company uses it to cover an interest rate guarantee it has offered to its policyholders. The option expires in 1 year, while the swap has annual payments in a notional principal of EUR 100 million and matures in 3 years. The floating interest rate is the LIBOR and its term structure is horizontal and equal to 2% with continuous compounding. This is the yield curve that the company uses for discounting cash flows. The volatility for the swap rate is 20% per annum. a. What type of option has the bank granted? What is its position? b. How much will it charge the insurance company? c. When does the bank face the exercise of the option and when not? If there is a swap in the market whose fixed interest rate is 4%, will the option be exercised? d. What is the benefit for the bank if it provides such an option? What is its loss if exercised? What is the loss for the insurance company if it exercises it?

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Exercise 11 Assume that the Euribor yield curve is fixed at 3% per annum with annual compounding. A derivative entitles the counterparty with the long position to pay 2.8% per annum on a three-year swap that starts in 2 years and to receive Euribor. Payments are made annually. The volatility of the swap rate is 10% per annum and the notional principal amount is EUR 100 million. a. What type of derivative is this? b. What is its value? Exercise 12 A one-year European put option is written on a 2-year bond with semiannual coupon payments. The bond pays coupon in 6 months, 12 months and so on from today which at a rate of 4% per annum. The strike cash price is EUR 100 and the current cash price of the bond is EUR 95. The 6-month discount rate is 3.5% and the 12-month discount rate is 4.5% per annum with continuous compounding. The volatility of the bond price is 10%. a. What is the value of the option? b. How will your answer change if the afore mentioned strike price was not cash byt quoted price? Exercise 13 Indicate which models of the interest rate curve you know. a. What are their similarities and what are their differences? b. Outline how they are used to value derivatives on bonds. c. What is the difficulty? Exercise 14 Use Black’s model to find the value of a European put option that matures in 1 year, written on a 5-year bond. Assume that the present value (price)

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of the bond is EUR 115, the strike price of the option is EUR100, the one-year interest rate is 5% per annum, the volatility measure of the bond price is 4% per annum and the present value of the coupons to be paid during the lifetime of the option is EUR 5. Exercise 15 A one-year European call option is written on a 3-year bond with semiannual coupon payments. The bond pays coupon in 6, 12, etc. months from today at a coupon rate of 5% per annum. The strike cash price is EUR 1,000 and the current cash bond price is 950 Euros. The six-month discount rate is 4.5% and the 12-month discount rate is 5.5% per annum with continuous compounding. The volatility of the bond price is 20%. a. What is the value of the option? b. How will your answer change if the afore mentioned strike price was not cash byt quoted price? Exercise 16 a. Construct a 2-step trinomial tree to evaluate a European call option on an interest rate, in which each step is for a period of one year and the probabilities of an up, flat and down movement of the interest rate are 20%, 50% and 30% respectively. The initial interest rate is 9% and in each step the rise is by +2 percentage points, while the fall by −2 percentage points. The strike price of the call Option is 10% and the nominal value on which it is calculated is EUR 100. b. How does the result change if the option is a European put option? Exercise 17 Suppose you want to use Vasicek’s model with α = 0.1, b = 0.1 and σ = 0.02. The initial price of the instantaneous short rate (at time t = 0) is 10% per annum with continuous compounding. a. Find the value of a European call option today (at time t = 0) which expires in 1 year and is written on a zero-coupon bond that expires in 2 years and has a face value of EUR 100.

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b. What is the value of the option if it is a European put option? c. Does the put-call parity apply in this case? Exercise 18 For all the bonds of Chapter 3 assume that you have a European call option and a European put option on the bonds that expires in 3 months and definitely prior to the maturity of the bonds. Assume that the strike price is equal to the face value. a. Apply all option pricing models of this chapter to price the options. You are free to choose the parameters needed in the models. b. Compare the output of the different the different models. Exercise19 Assume that the Euribor yield curve is fixed at 2% per annum. A swaption gives the holder the right to receive 1.9% on a three-year swap that starts in 2 years. Payments are made annually. The volatility of the swap rate is 20% per annum and the notional principal amount is EUR 100 million. What is the value of the swaption (with the use of Black’s model)? Exercise 20 A company knows that it will receive EUR 10 million in 6 months. It wishes to invest this amount upon receipt for another 180 days. It wants to secure/lock an interest rate guarantee. a. What position should it take in interest rate options to hedge the risk of interest rate changes between now and the end of 6 months? b. What other derivative products could it use to secure the same guarantee? c. Compare the alternatives of the company and explain the advantages and disadvantages of each of them.

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Exercise 21 A company wishes to have an upper limit (cap) on its lending rate of 4%. It borrows at the 6-month Euribor. The principal amount is EUR 10 million. On the day of interest rate reset the 6-month Euribor is 5% per annum. a. What derivative will it use? b. What payment will be made under this derivative? c. When/in which cases will the payment take place? Exercise 22 Analyze and compare all models of the interest rate term structure that were developed in this chapter? In which categories are they classified? a. What are their similarities and what are their differences? b. Outline how they are used to value bonds and derivatives on bonds. c. What is the difficulty? Exercise 23 Use Vasicek’s model to price a European call option if you know that a = 0.05, b = 0.04 and σ = 0.01 and the initial value of the short rate is 2%.The European call option expires in 1 year and is written on a 5-year zero-coupon bond with a face value of EUR 1,000 and a strike price of EUR 900. Exercise 24 How does the answer to the previous Exercise (23) change if the option is a European put optio with the same characteristics? Use your answer in both exercises to examine whether the put-call parity is valid. What do you observe? Exercise 25 Repeat exercises 23 and 24 using Hull and White’s model.

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a. What is the main difference? b. What part of the data is not needed in this model? c. How do your findings compare? d. Where are the differences attributed?

References Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1–2), 167–179. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407. Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7 (3), 229–263. CIIA. (2004). Fixed Income Valuation and Analysis. Course Manual. Faure, P. (2011). Is the repo a derivative?: Opinion. African Review of Economics and Finance, 2(2), 194–203. Ho, T. S., & Lee, S. B. (1986). Term structure movements and pricing interest rate contingent claims. The Journal of Finance, 41(5), 1011–1029. Hull, J. C. (1997). Options, futures and other derivatives (3rd ed.). Prentice Hall International, Inc. Hull, J. C. (2012). Options, futures and other derivatives (8th ed.). Prentice Hall International, Inc. Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. The Review of Financial Studies, 3(4), 573–592. Hull, J., & White, A. (1993). One-factor interest-rate models and the valuation of interest-rate derivative securities. Journal of Financial and Quantitative Analysis, 28(2), 235–254. Hull, J., & White, A. (1994). Numerical procedures for implementing term structure models I: Single-factor models. Journal of Derivatives, 2(1), 7–16. ICMA. (2019). Frequently asked questions or Repo. Updated January 2019. https://www.icmagroup.org/assets/documents/Regulatory/Repo/RepoFAQs-January-2019.pdf. Accessed: August 2020. Jamshidian, F. (1989). An exact bond option formula. The Journal of Finance, 44(1), 205–209. Luenberger, D. G. (1998). Investment science. Oxford University Press. Marrison, C. (2002, July 18). The fundamentals of risk measurement (1st ed.). McGraw-Hill Education. Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188.

CHAPTER 6

Credit Derivatives

Credit derivatives are contracts whose value depends on the reliability— creditworthiness of an issuer or institution. This can be a company or a state. These derivatives have gained prominence in recent years as they were written in debt issues of issuers who could not meet their obligations and have thus protected their buyers. However, there have been cases where some of them have led to losses due to the decline of the underlying market instead of yielding the expected—comparatively higher—returns. These contracts can be used on the one hand to protect their holders from changes in the creditworthiness of issuers, on the other hand to give them access to higher returns than they could achieve through simple fixed income issues. One should not confuse credit derivatives with interest rate derivatives. They are both written on fixed income securities; however they are triggered by different events. An interest rate derivative may be exercised when an interest rate or a price that depends on an interest rate moves to a certain direction. A credit derivative is exercised when a credit event takes place. The term credit event refers to the sudden and substantial negative change to the capacity of a borrower to accommodate its payment obligations. There are six credit event categories as per the International Swaps and derivatives Association (ISDA); (1) bankruptcy; (2) obligation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_6

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acceleration; (3) obligation default; (4) payment default; (5) repudiation/moratorium; and (6) debt restructuring. Bankruptcy, payment default and debt restructuring are the most frequently observed credit events (Corporate Finance Institute, 2020; ISDA, 2019a, 2019b). Bankruptcy is defined as the legal process that follows the inability of an entity (individual, organization, state/government) to pay back their outstanding debt. The bankruptcy request is normally filed by the debtor; however, sometimes it can be filed by the creditor. A bankrupt entity is not solvent anymore and most likely has ceased meeting any potential regulatory solvency requirements ahead of time. Companies can go bankrupt for a number of reasons; most likely due to a combination of causes. The recent financial crisis (2007–2008) as well as the crisis caused by the pandemic have both led several companies to file for bankruptcy. The reason was that they faced adverse market conditions that proved to be not favorable for their products or services and as a result they generated revenues that were significantly less than their projections and/or obligations/liabilities. Such companies were not in position to repay their outstanding debts to their creditors and in some cases not even their routine daily operating expenses. Especially in the crisis due to the pandemic they were not even operational for extended periods of times and the support programs provided by the different governments did not suffice to keep them alive. When these conditions prevailed, the companies had to file for bankruptcy (Corporate Finance Institute, 2020). Nevertheless, one has to investigate if it was really the crisis that caused the bankruptcy or the company was already facing structural issues and was not going to make it anyway. In other words, it is important to identify what the root cause of the bankruptcy was. Payment default is defined as the inability of an entity (individual, organization, state/government) to make one or more payments on their outstanding debt. Repeated payment defaults could be a prelude to or indications of upcoming bankruptcy. However, the two are different. A bankruptcy means that the debtor will not be able to pay the creditor the full amount due. A payment default means that the debtor will not be able to pay the creditor in a timely manner. Payment defaults are also due to several reasons and in most cases they are the result of a mix of causes. During the latest (2007–2008) financial crisis, as well as during the pandemic several companies were not in position to make their scheduled payments, without though going bankrupt. The relief measures introduced by the governments prevented bankruptcy and

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even rendered payment delays to some extent acceptable. Companies with volatile or fragile revenues experienced even more decreased earnings— due to market and/or global conditions—and as a result they failed to make interest payments to their creditors when they are due (Corporate Finance Institute, 2020). In a way similar to bankruptcy it is important to distinguish whether the default is due to the global or market conditions or the company would not be able to make the payments any way in the long run due to inherent problems. Debt restructuring is defined as the change to the terms of the debt of an entity (individual, organization, state/government), in favor of the debtor as a result of which it becomes less beneficial for the creditors, who most likely have to loosen up one or more of the conditions of the loan. It is usually the outcome of the inability of the debtor to make the payments—either in full or in a timely manner. The most common forms of debt restructuring are the reduction of the loan/debt outstanding principal amount (also known as haircut), the decrease of the coupon or interest rate, the postponement of due payments, the extension of the loan/debt maturity date or the change in the order of payment priority. A debt restructuring may take place for example when a company faces liquidity issues and cannot be certain that it will service its debt. During the recent (2007–2008) financial and the subsequent (2009) sovereign crisis there were cases of companies (or even governments/countries) that had to restructure their debt so that it remains viable. It usually involved an extension of the maturity date, a lowering of the coupon rate, and in some cases even a haircut. If a company has issued for example a 10year bond with a face value of 1,000,000 Euro and a coupon payment of 5%, then it can restructure its debt by issuing a 10-year (or 20-year) bond with the same face value and a coupon payment of 2% (Corporate Finance Institute, 2020). In a way similar to bankruptcy and payment default a cautious investor has to examine whether it was the adverse conditions that led to debt restructuring or the debt was not viable anyway. The point of reference in determining when a credit event has occurred is the International Swaps and Derivatives Association (ISDA). It is a trade institution that has been established by more than 800 participants from almost 60 countries around the world. ISDA has developed a standardized contract for derivatives transactions—known as the ISDA Master Agreement. Since 1992 it is used as template for entering into derivatives contracts, facilitating standardization. The participants include

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individuals and entities that work with over the counter (OTC) derivatives, i.e. dealers, service providers, and other end-derivatives-users. ISDA oversees the standards and the language used when referring to derivatives transactions—known as the financial products markup language (FpML) (Corporate Finance Institute, 2021). In this chapter we present all types of credit derivatives, such as credit default swaps, total return swaps and credit spread options. As in the previous chapter we investigate how they can be used in order to hedge the credit risk that is generated by fixed income instruments that have been included in investor portfolios. After having read this chapter the reader is expected to have a very good understanding of how the credit derivatives work.

6.1

Credit Default Swaps

Credit default swaps (CDS) are contracts that protect their holder against the default risk of an issuer/entity (e.g. a company or a government). In this respect they somehow offer insurance protection. The entity against which protection is offered is the reference entity and its default is defined (and mentioned) as a credit event. The buyer of this insurance protection acquires—for a fee—the right to sell a specific bond issued by the reference entity at its face value in case of a credit event. This bond, which is the underlying asset, is the reference obligation and the total face value of the bond underlying the swap is the notional principal of the swap. The seller of the swap is offering this insurance protection against the default of the reference entity—for a fee (Hull, 2005). The buyer of the CDS makes periodic payments—similar to the periodic payments of insurance premium—to the seller until either the CDS matures or a credit event occurs, whichever comes first. If a credit event takes place prior to the maturity of the CDS, then (Hull, 2005) a. The buyer makes one last accrual payment to the seller if payments are made in arrears; no such payment is made if the payments are made in advance. b. The CDS can be settled, depending on its terms, by: • Physical delivery: the buyer delivers the bond to the seller and collects its face value.

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Periodic payments ProtecƟon Buyer

ProtecƟon Seller Payment in case of default

Fig. 6.1 CDS (Source Created by the author)

• Cash settlement: the seller pays the buyer an amount of (100 − Z )% × FV, where FV: is the face value (notional principal) of the reference obligation. Z: is the mid-price of the bond after a predetermined number of days from the credit event offered by a sample of traders. The buyer can collect the residual amount (Z % × FV) by selling the bond in the market. Figure 6.1 illustrates the cash flows exchanged in a CDS. A CDS resembles a life insurance protection policy where default is the analogue of death/mortality—if it can happen at most one time or disability/illness—if it can happen more than one times. Normally a default is an event that happens once during the life of a reference obligation. Credit derivatives are active credit risk management tools. They can thus exchange their exposure to the risk of one entity for the risk of another entity. This is done by buying a CDS written on the bond of one issuer and selling a CDS written on the bond of another issuer. 6.1.1

CDS Valuation

As mentioned, a CDS is similar to a life insurance protection policy. To value a CDS we mimic the pricing of such a policy. The probability of death is replaced by the probability of default. To do that (following Hull [2005]) we assume that: • Its face value/notional principal is 1 monetary unit (e.g. 1 Euro or 1 USD). • Default events, interest rates and recovery rates are mutually independent.

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• The claim in case of default is the notional principal plus accrued interest. • The default can occur at distinct times τ1 , τ2 , . . . τI . • The periodic payments from the buyer to the seller take place at the end of the period (annuity immediate) In the valuation we use the following definitions: T: prτι : ERR: (ν)

at|s :

pdt : PR: PR0 : prnd T : AIt :

The maturity date of the CDS in years. The risk-neutral probability of default at time τi . The expected recovery rate for the reference obligation in a risk—neutral world—and is independent of the default. The present value of a series of payments that pays 1 monetary unit (e.g. e1 or $1) at payment times from time 0 to time t, with a term structure denoted by s (spot rate), under the assumption of a payment frequency of ν times a year, i.e. of amount 1/ν per payment. We may assume without loss of generality that ν = 1 and drop the superscript. The payment date immediately preceding t. The payment per monetary unit made at each of the above times from the buyer to the seller. The value of PR that makes the value of the CDS equal to zero. The risk-neutral probability that a credit event will not occur in the life of the CDS. The accrued interest of the reference obligation at time t as a percentage of the notional principal.

It follows from the previous definitions that: prnd T =1−

I 

prτι .

(6.1)

i=1

Payments are made from the buyer to the seller either until a credit event occurs or until the CDS natures, whichever occurs first. This means that

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the present value of the payments of the buyer to the seller is given by: PVbuyer, 0 : = PR ·

I    aτ¯i |s + (τi − pdτi ) · (1 + sτi )−τi prτi i=1

+ PR · prnd T · aT¯ |s .

(6.2)

If a credit event occurs at time τi i = 1 . . . I then the risk-neutral expected value of the reference obligation as a percentage of the notional principal is: EVτi := (1 + AIτi ) · ERR,

(6.3)

and thus the risk-neutral payoff made by the CDS is the difference: EPτi : = 1 − EVτi = 1 − (1 + AIτi ) · ERR.

(6.4)

The present value of the total expected payment from the CDS is given as the sum of the above payments multiplied by the corresponding probability: PVseller, 0 :=

I    1 − (1 + AIτi ) · ERR · prτi · (1 + sτi )−τi .

(6.5)

i=1

We conclude that the value of the CDS for the buyer becomes: CDS0 : = PVseller, 0 − PVbuyer, 0 =

I    1 − (1 + AIτi ) · ERR · prτi · (1 + sτi )−τi i=1

− PR ·

I    aτ¯i |s + (τi − pdτi ) · (1 + sτi )−τi prτi

i=1 + PR · prnd T

· aT¯ |s .

(6.6)

The value of the CDS at the time it commences should be zero as there is no payment by any counterparty. The CDS spread is the value of PR

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that makes the value of the CDS zero. It is given by:  I  −τi i=1 1 − (1 + AIτi ) · ERR · prτi · (1 + sτi ) . PR0 = I   −τi prτi + prnd i=1 aτ i |s + (τi − pdτi · (1 + sτi ) T · aT |s

(6.7)

The variable PR0 is also known as credit default swap spread or CDS spread. It is the payment made each year, as a percentage of the notional principal, for a new CDS. The valuation method can be generalized to capture defaults that can occur at any time. If prt denotes the risk-neutral probability density function at time t Eq. (6.7) that computes the CDS spread becomes: T PR0 =  T 0

0

[1 − (1 + AIt ) · ERR] · prt · (1 + st )−t dt

[at|s + (t − pdt · (1 + st )−t ]prt dt + prnd T · aT |s

.

(6.8)

Example 6.1 Let us consider a simple CDS that expires in 5 years. Payments are made every six months. The reference entity is a bond that matures in 5 years and pays a semi-annual coupon of 6% per annum. If default occurs then this is done immediately before the coupon payment and is made at the end of year 1, 2, 3, 4, 5. The probability of default for these years is respectively 2.50%, 2.75%, 3.00%, 3.25% and 3.50%. The recovery rate is 60%. The risk-free interest rate is 3% with semi-annual compounding. We wish to calculate the CDS spread. The CDS spread is given by formula (6.7). From the aforementioned information we have: pr1 = 0.025, pr2 = 0.0275, pr3 = 0.030, pr4 = 0.0325, pr5 = 0.035 & prnd 5 = 0.85 AIτi = 0.03 & τi − pdτi = 0, ERR = 0.60.

∀i

(6.9) (6.10) (6.11)

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We estimate that: (2)

(2)

(2)

(2)

(2)

(1 + s1 /2)2 = 0.9707, (1 + s2 /2)4 = 0.9422, (1 + s3 /2)6 = 0.9145, (1 + s4 /2)8 = 0.8877, (1 + s5 /2)10 = 0.8617,

(6.12)

a (2) = 0.9779, a (2) = 1.9272, a (2) = 2.8486, a (2) = 3.7430, a (2) = 4.6111. 1|s

2|s

3|s

4|s

5|s

(6.13)

Substituting in (6.7) we get: PR0 = 0.05219/4.36537 = 0.011955 = 1.1955% ≈ 1.20%.

(6.14)

This means that if the payments are semi-annual, then half of this spread will be paid every six months, i.e. 0.5978% ≈ 0.60%. 6.1.2

Binary CDS

A binary CDS is similar to a regular CDS except that the payment made by the seller to the buyer is a fixed amount. Consequently, the expected recovery rate impacts the likelihood of default, but does not impact the payoff to the buyer. Therefore in a binary CDS the CDS spread has increased sensitivity to the recovery rate (Hull, 2005). 6.1.3

Basket CDS

The basket CDS is a CDS written on a pool of underlying securities rather than on a single security. Hence, it offers protection from a basket of reference entities. There are two types of baskets: • In the first type the seller pays the buyer when any of the reference entities defaults and is called an add-up basket CDS. It is essentially similar to a portfolio of individual CDSs written on each individual reference entity. • In the second type the seller pays the buyer only once, as soon as there is one reference entity that defaults. When the seller pays the buyer the CDS expires and thus none of the counterparties makes any additional payments. This is called a first-to-default basket CDS (Hull, 2005).

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We will not enter into the details of the valuation of non-plain vanilla CDSs. However, simulation methods may be used (such as Monte Carlo) to generate a number of scenarios capturing the potential default (Hull, 2005). 6.1.4

CDS Risks

The risks of a CDS are apparently related with the default of the reference entity (or entities). The seller bears this risk. The seller also bears the risk that the buyer will not be able to make the periodic payments. However, this is a relatively small amount and if the buyer stops making payments then the swap expires; the buyer simply has no default protection any more. The buyer though is also exposed to a risk; the counterparty (or credit) risk stemming from the seller. This risk is present to the buyer only if the seller defaults before the default of the reference entity. If the latter happens, then the buyer should find and buy a new CDS, from a new seller, to maintain protection against possible default by the reference entity—provided insurance is still sought. The new CDS has a maturity date of T -τ, where T is the expiration date of the first CDS and τ is the time at which the seller defaulted. The buyer posts a loss if the payments expensed for the replacing CDS are higher than the payments of the defaulted CDS. Higher new payments are anticipated to happen due to a potential deterioration of the reference entity’s creditworthiness in the time interval [0, τ ]. Consequently, the seller default risk depends on the potential increase of the probability of default of the reference entity and the correlation between the two credit events, i.e. the default of the reference entity and the default of the seller. The estimation of the impact of this risk can be done with the use of numerical methods, such as Monte Carlo simulation. The scenarios ran capture the possibility of default of either the reference entity or the seller. We will not present though the particulars. An elaboration can be found in Hull and White (2001) or Hull (2005).

6.2

Total Return Swaps

A total return swap (TRS) is a swap that exchanges all the proceeds (total return) of an asset with a floating interest rate plus a spread. The total return from the reference asset (e.g. a bond) includes any periodic

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payments (e.g. coupons or interest) as well as any other return (profit or loss) that is produced during the life of the swap. Total return swaps are sources of financing. An investor interested in buying an asset—let us say a bond—agrees with a financial institution that the latter buys the bond and receives a floating interest rate plus a spread from the former while it repays the bond yield. The combination of the resulting cash flows for the investor is equivalent to these of a floating rate loan (plus the spread) used to finance the purchase of the bond. However, the owner of the bond is not the investor but the financial institution. The risks involved are similar to a plain vanilla CDS; the financial institution owns the asset (assume again a bond) until the swap matures. As such it is exposed to the credit risk of the bond issuer. It is also exposed to the counterparty (credit) risk of the investor. This risk is though lower compared to an outright loan to the investor. Assuming that there is no risk counterparty risk for any of the two counterparties, then the value of the swap for the investor at any given time is the difference between the value of the actual underlying bond and the value of a similar floating rate (plus spread) bond of the same face value (the latter is subtracted from the former). It is the exact opposite for the financial institution (the former is subtracted from the latter). The value of the swap at inception should be zero for each of the counterparties as there is no payment by any of them. The spread that is added on top of the floating interest rate compensates the financial institution for the investor default risk. The financial institution posts a loss only if the underlying asset experiences a decreased value when the investor defaults. The spread therefore depends on the investor’s creditworthiness or credit rating, the bond issuer’s creditworthiness or credit rating and their correlation. In a total return swap the investor is often referred to as the receiver, whereas the financial institution as the payer (Fig. 6.2). There are a couple of variants to the standard TRS: Total asset return Payer

Receiver FloaƟng rate + spread

Fig. 6.2 Total return swap (Source Created by the author)

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• Instead of paying cash for the change in the value of the underlying bond the bond is physically delivered; at maturity the financial institution delivers the bond and receives its face value. • Instead of a single payment of the change of value at maturity, there are periodic payments. Example 6.2 Suppose an investor enters a TRS with a financial institution to receive the total proceeds of a bond with a 2% coupon and pay an interest rate equal to EURIBOR plus 1%. The face value is 500 Million Euro and the expiration of the agreement is in 4 years. The payments are: • On the coupon payment dates, the counterparties exchange payments in a way similar to a standard interest rate swap. – The financial institution pays the coupons (at 2%) received from the bond investment, i.e. 10 Million Euro. – The investormakes payments that are calculated with an interest rate of Euribor + 1% on 500 Million Euro. This resembles to the coupon payments of a floating rate note. If Euribor had been 0.8% at the beginning of the year, then he or she would have made a payment of 9 Million Euro = (0.8% + 1%) × 500 Million Euro. • At the maturity of the swap there is a payment for the change of the value of the bond. – If the value of the bond is higher by 8% for example, then the financial institution pays 40 Million Euro at the end of 4 years. – If the value of the bond falls 8%, then the investor pays 40 million Euros at the end of 4 years. • If the bond issuer defaults then the swap expires and the investor pays 500 million minus the market value of the bond.

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Collateralized Debt Obligations

Collateralized debt obligations (CDO’s) are derivatives similar to collateralized mortgage obligations except that the underlying securities are other-than-mortgage debt securities. Mortgage loans were the constituent securities of the underlying pool of CMOs. At CMO’s the risk of early repayment is packaged, while at CDO’s the credit risk is packaged. We introduced CDOs in Chapter 5 among interest rate derivatives so as to capture that aspect of their operation. For completeness we complement the presentation in this chapter, so as to elaborate their credit structure. In a similar way to CMO’s, classes/categories of securities known as tranches are created from a portfolio of corporate bonds or bank loans. • The first tranche includes x1 % of the total face value (or notional principal) of the underlying portfolio and absorbs the first x1 % of the losses from default (e.g. non-payment of an installment or nonrepayment of the loan). • The second part includes the next x2 % of the total face value (or notional principal) of the underlying portfolio and absorbs the next x2 % of the losses from default (e.g. non-payment of an installment or non-repayment of the loan). • Continuing similarly, the n-th tranche includes x M % of the total face value (or notional principal) of the underlying portfolio and absorbs the last x M % (and therefore the rest) of the losses from default (e.g. non-payment of an installment or non-repayment of the loan). Obviously the sum of xi , i = 1 . . . M is equal to one. The last tranche has the best credit rating, as there is a relatively small probability that defaults will be so many that the losses they will result will affect it. The penultimate has a lower credit rating, but still a relatively good one. “Going up” to the first tranches the credit rating deteriorates as the risk of loss due to default increases. In fact, it may be higher than that of the original portfolio. In the lower tranches the risk may be lower than in the original portfolio. The yields (yield to maturity) of the tranches are respectively: y M < y M−1 < · · · < y2 < y1 ,

(6.16)

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Tranche 1 1st x1% of loss Yield y1

Bond 1 Bond 2 Bond 3

Trust

Bond N

Tranche 2 2nd x2% of loss Yield y2

Tranche M Mth xM% of loss Yield yM

Average Yield y

Fig. 6.3 CDO (Source Created by the author)

thus reflecting that the lower the return the lower the level of risk. These are shown schematically in Fig. 6.3. The first tranche in particular is known as “toxic waste” and is usually not marketed but is rather kept by the writer of the CDO due to the very high risk it poses. The remaining tranches are available to investors, depending on the level of risk they are willing to take. A CDO allows the creation of high quality debt issues from medium or low quality debt issues. The risk borne by the buyers of each of the tranches depends on the correlation between the defaults of the participating issuers. High rating is the result of low correlations.

6.4

Credit Spread Options

Credit spread options are options whose underlying asset (instrument) is either a credit spread or an asset dependent on credit spread (Hull, 2005). This is similar to an interest rate option; however recall that an interest rate usually corresponds to the risk of the investment that is dependent on the creditworthiness of the investee. The interest rate (or yield) is decomposed to the sum of the risk-free rate and the spread. The spread is the part of the interest rate that compensates the investor for the credit risk he or she is willing to assume. Credit spread options focus on the second component of interest rate, i.e. the spread. They expire not only on their predetermined maturity date but also when the issuer of the underlying asset defaults.

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There is a third alternative of the use of the term credit spread option; it refers to a strategy or contract that involves a long position in one option along with a short position in a similar option. The two options are of the same type (i.e. either both call options or both put options), they have an identical maturity date, but they are written with different strike prices. Sometimes this strategy is referred to as credit spread (i.e. the word option is dropped). The use of the term credit spread in this case is justified by the fact that the proceeds of the strategy are credited to the account of one of the counterparties as he or she posts an initial inflow as a result of the difference in the premia of the long and short positions. Alternatively, the term is used, because it transfers credit risk from one counterparty to the other (Frederick, 2019). 6.4.1

Credit Spread Options Written on the Spread

When the option is written on the credit spread, then the payoff is a function of the change in spread. Depending on whether the option is a European put or a European call credit spread option it is given by: cTE = D max(spreadT − spread K , 0),

(6.17)

pTE = D max(spread K − spreadT , 0).

(6.18)

In the above equations spreadT is the spread at the maturity date of the option, spread K is the strike price of the spread and D is the duration employed to convert the spread into monetary units (amount) (Hull, 2005). These payoffs are similar to those of interest rates options; interest rate has been replaced with spread. Black’s model can be used for the valuation of these options under the hypothesis that the future spread—with the condition that there will be no default—follows a lognormal distribution. This condition leads to a modification of Black’s model so that it is multiplied by the probability that there will be no default until the maturity date of the option (Hull, 2005). 6.4.2

Credit Spread Options Written on the Credit Asset

When the option is written on an asset that depends on a credit spread the payoff is a function of the change in the price of the asset. Depending

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on whether the option is a European call or European put option the payoff is given by: cTE = max(PT − K , 0),

(6.19)

pTE = max(K − PT , 0).

(6.20)

In this case PT is the value of the underlying asset at the maturity of the option and K is the exercise price. Black’s model can be used for the valuation of these options under the hypothesis that the price of the underlying asset—on the condition that there will be no default—follows a lognormal distribution. This condition leads to a modification of Black’s model so that it is multiplied by the probability that there will be no default until the maturity date of the option (Hull, 2005). 6.4.3

Credit Spread Options Strategy

There are two main types of credit spread strategies (Frederick, 2019): • A credit put spread that is created when a short position is taken on a put option on an underlying debt security with a strike price and a long position is taken on a put option on the same underlying security with the same maturity date but with a lower strike price. As the premium of a put option is an increasing function of the strike price, the premium of the short position is higher than the premium of the long position, resulting in an initial inflow for the seller. This is a bullish position as the investor that pursues it anticipates that the price of the underlying asset will increase and in any case remain above the higher strike price so that the options mature without being exercised. The income produced by the difference of the premia will then be his or her profit. • A credit call spread that is created when a short position is taken on a call option on an underlying debt security with a strike price and a long position is taken on a call option on the same underlying security with the same maturity date but with a higher strike price. As the premium of a call option is a decreasing function of the strike price, the premium of the short position is higher than the premium of the long position, resulting in an initial inflow for the

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seller. This is a bearish position as the investor that pursues it anticipates that the price of the underlying asset will decrease and in any case remain below the lower strike price so that the options mature without being exercised. The income produced by the difference of the premia will then be his or her profit. Credit spread option strategies (Frederick, 2019): • Set a stop-loss for the seller compared to uncovered positions. More specifically – In the case of a credit put spread if the price of the underlying asset drops below the higher strike price then the seller will be obliged to purchase it from the buyer; however he or she has some downward protection as he or she can sell the asset at least at the lower strike price as he or she has a long position on the put option with the lower strike price. – In the case of a credit call spread if the price of the underlying asset increases more than the lower strike price then the seller will be obliged sell it to the buyer; however he or she has some upward protection as he or she can buy the asset at most at the higher strike price as he or she has a long position on the call option with the higher strike price. • Provide a lower profit opportunity than the corresponding uncovered positions (i.e. short put or short call) for the seller as he or she collects the higher premium but disburses the lower premium. This is done so that the seller obtains the aforementioned stop-loss safety net. However they cost less than the corresponding uncovered positions (i.e. short put or short call) for the buyer. • They require a lower margin compared to the corresponding uncovered positions (i.e. short put or short call) as the exposure of the seller is lower. The pricing of credit spreads is pretty much straightforward. It is the difference of the premia. More specifically the initial inflow for the seller for the credit put spread (CPS) and credit call spread (CCS) respectively is: E E − pLow , premiumCPS = pHigh

(6.21)

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E E premiumCCS = cLow − cHigh .

(6.22)

A strategy that combines a credit spread option and a CDS provides protection to against both the increase of the spread and the default of the issuer (Hull, 2005).

Exercises Exercise 1 A credit default swap (CDS) is agreed between a buyer and a seller. The premium that the buyer agrees to pay to the seller is 50 basis points, which is paid on an annual basis. The CDS expires in 6 years. The notional principal amount is EUR100 million and the CDS is settled in cash. The bond issuer defaults in 4 years and 6 months. The price of the reference bond when the credit event occurs (or immediately after) is estimated at 30% of its face value. What are the cash flows and their payment times for both the seller and the buyer of the CDS? Exercise 2 Explain the mechanics of a first-to-default CDS. Explain how its value changes (monotonicity) as the correlation between the reference entities in the basket changes. Exercise 3 An asset swap is an agreement for the exchange of a series of cash flows, as they arise from one asset—security with another series of cash flows. One such case is the exchange of coupons of a bond with a floating interest rate plus/minus a spread. Compare the asset swap with the total return swap. What are their similarities and what are their differences? Exercise 4 Let us consider a CDS that matures in 5 years. Payments are made on a semi-annual basis. The reference obligation is a bond that pays a semiannual coupon of 6% per year. We consider that default can occur at times 1 year, 2 years, 3 years, 4 years and 5 years. The corresponding default

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probabilities are 2%, 3%, 10%, 20% and 30% respectively in these years. If default occurs, then it will be just before a bond coupon payment. The recovery rate is 40%. We assume that the risk-free interest rate curve is horizontal at 4% per annum with continuous compounding. a. What is the CDS spread? b. What would be the CDS spread if the derivative was a binary CDS? Exercise 5 Let us consider a CDS on a reference entity that matures in 4 years. If default occurs then it can be only at times 1 year, 2 years, 3 years and 4 years. As a result there is no accrued interest for the reference entity at the time of a (potential) default. The default probabilities are 2.25%, 2.5%, 2.75% and 3% for each of these years. The recovery rate is 40%. Payments are made on a semi-annual basis. The risk-free zero-coupon yield curve is flat at 4%. a. What is the CDS spread? b. What would be the CDS spread if the derivative was a binary CDS?

References Corporate Finance Institute. (2020). Credit event. https://corporatefinanceinsti tute.com/resources/knowledge/finance/credit-event/. Accessed: December 2020. Corporate Finance Institute. (2021). International Swaps and Derivatives Association (ISDA). https://corporatefinanceinstitute.com/resources/knowle dge/trading-investing/international-swaps-and-derivatives-association-isda/. Accessed: January 2021. Frederick, R. (2019, June 19). Reducing risk with a credit spread options strategy. Charles Schwab. https://www.schwab.com/resource-center/ins ights/content/reducing-risk-with-credit-spread-options-strategy-0. Accessed: April 2021. Hull, J. C. (2005). Options, futures and other derivatives (5th ed.). Prentice Hall International, Inc. Hull, J. C., & White, A. D. (2001). Valuing credit default swaps II: Modeling default correlations. The Journal of Derivatives, 8(3), 12–21.

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ISDA. (2019a, March 6). Proposed Amendments to the 2014 ISDA Credit Derivatives Definitions Relating to Narrowly Tailored Credit Events. https:// www.isda.org/2019/03/06/proposed-amendments-to-the-2014-isda-creditderivatives-definitions-relating-to-narrowly-tailored-credit-events/. Accessed: December 2020. ISDA. (2019b, July 15). 2019 Narrowly Tailored Credit Event Supplement to the 2014 ISDA Credit Derivatives Definitions. https://www.isda.org/book/ 2019-narrowly-tailored-credit-event-supplement-to-the-2014-isda-credit-der ivatives-definitions/. Accessed: December 2020.

CHAPTER 7

Bond Markets

Already in Chapter 1 we gave a prelude of the bond markets and the main functions of the primary and secondary markets. We also identified the interested parties such as the investors, the issuers and the intermediaries. In Chapter 2 we briefly mentioned some of the bond types and their main characteristics. However, we still have to understand which the bond markets are and what the products traded there are. Furthermore, the bond markets have evolved through the years and even though traditional fixed income securities have always attracted interest, there have been other products that have come to the forefront. Fixed income products have increased significantly not only as far as the total outstanding face value or market value is concerned but also in terms of the different types of securities as well as the types of markets. They have therefore attracted the attention of the regulators that attempt to set up a more standardized and rigorous framework for the operation of bond markets, similar to that of the stock markets. At the same time, the last ten years the bond markets—both corporate and sovereign—have experienced the presence of the central banks, which have provided liquidity in the markets to support the economies and the Euro. These bond purchase programs, known also as quantitative easing, resulted in a significant drop in the interest rates. As a matter of fact all countries of the Eurozone have at least one issue with a negative yield and

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_7

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there are countries whose issues have negative yield for all maturities. This practically means that investors pay the issuing countries to lend them! The bond purchase programs have been extended to corporate bonds, covered bonds and recently to ETFs as well as fallen angels, i.e. corporate bonds that have lost their investment grade as a result of the pandemic and its consequences to the economies. The concentration of fixed income securities in the balance sheets of the central banks has raised one interesting question. Is this the end of the bonds markets when there are central banks that own more than 50% of the debt of a country? The reason behind this question stems primarily from the fact that as central banks purchase fixed income securities at large scale, they are taking away the yield opportunities from bond investors. Even though this may not be the end of the bond markets, it signals one think; that the bond markets are changing and traditional fixed income investing does not work the way it did. The concern stated about the end of the bond markets—even more about the potential death of the bonds markets (Michele, 2020)—is not unjustified. However, the cause may be identified. The pandemic has led to an extended pause of the economic activity worldwide, which now requires unprecedented amounts of money in order first to survive and later to resume. This support that has been offered by the central banks, the governments as well as the European Union was necessary so that financing is affordable and controllable. The risks faced by the economies globally are similar to the financial crisis they had to weather back in 2008. The remedy is similar to what was offered then as well as in similar occasions in the past; support and liquidity. So, is this the end of the bond markets? The answer is most likely no. However, (slightly paraphrasing R.E.M.) this may be the end of the (bond) world as we know it (R.E.M., 1987). Investors need to evolve along with the era they are living in. They need to look at the bond issues that are capable of attracting interest anew. More precisely, they need to identify issues that offer higher yield, well with higher risk, and potentially not as high a return in absolute terms as it used to be. As we are living in a low interest rate environment, even the issues that carry higher risk post lower rates of return and even shrunk spreads over the low risk or risk-free fixed income securities. Potential candidates include peripheral European countries that have had a harder time during the previous crisis—and may still have, such as the European South. These are members of the Eurozone, so there are

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limited chances that they (will) fail (again). At the same time investment grade corporate bonds, municipal bonds and securitizations (in Europe and in the US) could be considered. There are also issues that are not in the scope of the purchase programs that have been launched by the central banks. These are non-investment grade corporate issues that come from companies with strong balance sheets, that have been not affected by the shut down or that can get through it and have maintained their access to capital. Some of these issuers may have a non-investment grade just because their country of domiciliation does not enjoy a high investment grade. Private debt is a potential direction; however it may not be suitable for all investors, as it exhibits higher risk. Having said that, investors need to be cautious; no matter what their investment choice is, the prolonged distress conditions could lead to defaults and downgrades. It may be optimal to invest through professionally managed investment portfolios. After all we should not forget why we consider bonds in the investment portfolios; they provide diversification against stocks and they tend to have a positive and comparatively more predictable performance when even good stocks record negative performance (Parish, 2014). One could very well claim that this has not always been the case the last few years, as both stock and bond prices were going up no matter what. It may be true; however the fundamental principles are still there. Following the aforementioned discussion, in this chapter we look at the fixed income markets, with a fresher look, addressing also contemporary trends. We do not ignore though traditional issues, starting from the money market and going to the bond market. We present the different products that are traded in these markets and explain the way they operate. This chapter exposes the reader to the details of the markets at which bonds are traded.

7.1

Contemporary Trends in the Bond Markets

Unlike stock markets, the bond markets are mainly over-the-counter (OTC) markets and as such they do not always exhibit the same transparency and liquidity or standardization. There is no centralized bond exchange and thus there is no real time visibility of all bond trades as is the case for stock trades. Although several steps are being taken with the more recent regulatory frameworks in Europe, the US but also globally, there are still areas of improvement. This has led to the introduction of

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fixed income ETFs that enjoy significant growth as they are listed in a stock exchange. Bond markets tend to see more mediocre variance than the stock markets do. Volatility is there though and as became apparent in the previous chapters it is primarily due to the interest rate and/ or spread movements. Bond markets seem to be more “stable” than stock markets. As a matter of fact, although we compare bonds with stocks in Chapter 12, we may observe that the bond of a company may be much less volatile than the stock of the very same company. The reason behind that is that a bond is practically a loan that has been split to several lenders. As such, it bears a contractual obligation to make a series of payments at certain future times; coupon (interest during its life) and face value (at maturity) payments. Not delivering on any of these promises constitutes a credit event which may lead to bankruptcy. It is not the same as not making a dividend payment at common shares of stock. Bond holders have a priority over shareholders at good times and at bad times; if the company goes bankrupt, they recover part of their capital due via the assets of the company. In addition, the terms of a bond are known well ahead of time. As a result of the additional security offered by the priority of the bond holders in receiving payments and the knowledge of the terms of the issue bonds exhibit a lower volatility than stocks—even if they are issued by the same firm (Vanguard, 2021). 7.1.1

The Size of the Bond Markets

The size of the bond markets has grown considerably over the last years. According to the International Capital Markets Association— ICMA (ICMA, 2020) the overall size of the global bond markets face value in USD is calculated to circa USD 128.3 Trillion. Sovereigns, Supranational and Agencies (SSA) bonds amount to USD 87.5 Trillion or 68%, whereas corporate bonds come up to USD 40.9 Trillion or 32% (as per August, 2020). The US comes first in the SSA bond markets with USD 22.4 Trillion and China comes second with USD 19.8 Trillion. The two of them add to 62% of the SSA bonds. Sovereigns add up to USD 63.7 Trillion, representing 73% of the SSA bonds (as per August, 2020). The total European (European Economic Area including Switzerland) SSA bond market is estimated to a combined equivalent nominal value of approximately EUR 12.2 Trillion (as per May, 2020).

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The US comes first also in the global corporate issues with USD 10.9 Trillion and China comes second with USD 7.4 Trillion. They add up to 45% of the corporate bonds. The corporate issues of financial institutions represent 53% or USD 21.5 Trillion (as per August, 2020). The total European (European Economic Area including Switzerland) investment grade corporate bond market is estimated to about EUR 5.65 Trillion in face value. 7.1.2

The Modernization of the Bond Markets

The bond markets have been known of not necessarily exhibiting the same transparency, ease of access and liquidity as the stock markets. However, since the last financial crisis in 2008, several steps have been taken towards the modernization of the bond markets. This has been facilitated or even driven by a series of factors. First, the low interest rate environment and the reduced ability of banks to provide the corporate loans that the market needed, directed several firms to public or private debt issues. Second, the regulatory framework evolved not only in the US and Europe, but also globally. Third, an array of products, such as ETFs, came to the spot light as means of fixed income investing, as they demonstrated increased transparency and liquidity as well as stock exchange listing. Finally, ITC advances assisted in the digitalization (or electronification) of the transactions and the establishment of electronic trading platforms. The changes experienced in these areas are gradually transforming the bond markets from over-the-counter, relatively illiquid and nontransparent to more standardized, more liquid and more transparent. Even if the focus was initially on SSA and investment grade corporate bonds, gradually even non-investment grade issues are added in the radar, with the organized exchanges in the world gaining a bigger role. 7.1.3 7.1.3.1

The Drivers of the Bond Market Growth

The Low Interest Rates and the Increase of Corporate Bond Issues The drop of the interest rates, mainly as a result of the support provided by the central banks, especially in Europe and the US, made bond issuance more affordable. Combined with the unwillingness of the banks to provide business loans to certain firms, as a result of the distressed

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loans they have accumulated in their portfolios during and after the crisis, corporate issues were in some cases a one-way path. Issuers with an investment grade credit rating enjoyed significantly lower interest rates, following the drop of the sovereign “risk-free” rates and the shrinking of spreads. However, even new entrants, as well as issuers whose rating was non-investment grade (high-yield) benefited too. Even if they were borrowing at higher rates than the corresponding investment grade firms, their cost of debt capital was at unprecedented low levels. The same held true for unrated firms. This increase in corporate bond issues was further assisted by the more active participation of exchanges around the world, which hosted some of the new issues. This provided transparency and a sense of security to the investors, knowing that some of the issues were not even investment grade. The credit risk is still there; however, the perception of increased transparency and liquidity reduced resistance. But above all, in a low or even negative interest rate environment, investors, both institutional and individual, saw opportunities for a positive rate of return. The safety net provided by the central banks, even inadvertently, led to euphoria and risk did not and still does not hinder the placement of funds in such investments. Pension schemes considered sub-investment grade corporate bonds, as well as unrated or private fixed income securities, as the only means to deliver returns that could match the promise they have made to the workers, pensioners and beneficiaries. Because, even if these instruments have lower credit ratings they still provide steady cash flows that allow appropriate cash flow planning and matching with the liabilities of the pension schemes. Over the years the lower-rated BBB segment of the bond market was expanded, as on one hand it is still investment grade and on the other hand offers somehow higher returns. In Europe, the euro-denominated investment grade debt is scratching the Euro 3.0 trillion threshold, after having reached the area of 2.8 trillion Euro by September 2020 (Weinberg et al., 2020). 7.1.3.2 The Role of Regulation Regulation over the last few years and in particular since the 2008 financial crisis has played an important role in the direction of the growth and the modernization of the bond markets. Besides the potential ongoing discussion and concern about the need for such an extensive regulatory framework both in the US and in Europe, the regulatory reforms have

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definitely affected the bond market organization and set up, transparency and liquidity, as well as the new product development. First, the increased capital requirements for the traditional fixed income market players, i.e. bankers, insurers and intermediaries, as well as other participants led to increased cost for attracting financing. The latter led to a drop of the risk that such players were eager to maintain; their role changed into primarily bringing together selling counterparties with buying counterparties (Prager et al., 2018). Second, the increased transparency requirements and investor protection rules may have been initially been perceived as showstoppers as they potentially increase operational costs. However, in the long run, they increase the loyalty and the retention of (retail) customers to the compliant bond market participants. The relevant legislation for the global financial markets is Basel III, complemented in Europe by MiFID II and in the US by the Volcker Rule in particular and the Dodd-Frank Wall Street Reform and Consumer Protection Act in general, with the repealing amendments performed by the Trump administration. The Basel Committee on Banking Supervision (BCBS) has set rules that have been recently updated (Basel III ) with regard to the global regulatory standards on bank capital adequacy and liquidity. The updated capital requirements led to increased and better-rated capital, a more rigorous liquidity structure and a capital-to-asset ratio dependent on the economic conditions. In Europe, the Markets in Financial Instruments Directive (MiFID) II has as its scope to secure transparency. To achieve that it targets at establishing a solid framework for (most of) the fixed income instruments and transfer (a material portion of) the OTC activity to trading venues that are regulated. In the US the Dodd-Frank Wall Street Reform and Consumer Protection Act introduced changes in the financial regulation and affected all federal financial regulatory agencies and almost all aspects of the financial services industry. The Vocker Rule, a provision of the act, restricted banks from entering in a series of proprietary trading transactions/ positions and being affiliated with certain types of funds (hedge and private equity). The Trump administration rolled back a significant part of the Act through the Economic Growth, Regulatory Relief and Consumer Protection Act (Prager et al., 2018).

376

T. POUFINAS

In Europe, Solvency II introduced rules similar to Basel III for insurers, resulting in similar consequences. Furthermore, the European Commission has adopted the regulation on PRIIPs (packaged retail investment and insurance products), which obliges all parties that produce and/or sell investment products to provide investors with key information documents (KIDs). The regulation aims at ensuring that the retail investors who select PRIIPs enjoy the necessary transparency in order to make a prudent and well-informed decision. The regulation affects all packaged, publicly available financial products that invest in underlying assets, among which fixed income securities, in order to provide a return to the investor, who is though in this way exposed to risk. The regulation practically applies to the entire spectrum of packaged retail investment and insurance products that are made available in the European Union. 7.1.3.3 The Importance of New Products During the last decade, since the 2008 financial crisis new(er) products were developed or were standardized and thus became more broadly available. Product development captured interest rate and credit derivatives (such as interest rate futures and options, interest rate and credit default swaps, as well as covered bonds) and exchange traded funds. Starting with derivatives, we observe that except for the plain vanilla bonds, counterparties entered a range of interest rate and credit derivatives, all of which were presented in the previous two chapters. The interest rate derivative markets were overall more solidly built, standardized and transparent, as to a great extend they were organized by exchanges. However, (the majority of) credit derivatives were tailor made, bilateral and were primarily arranged between intermediaries and counterparties. As a result of the 2008 financial crisis, credit derivates came several times to the spotlight as investors were trying to hedge against the potential default of enterprises or even countries/ governments to whose debt they were exposed. Gradually, credit default swaps were homogenized and standardized; nevertheless the rest of the credit derivatives remained tailor made and sometimes difficult to comprehend (Prager et al., 2018). With time, and especially during the years that followed the crisis, standardization in the credit derivatives was further enhanced via trading for example through the Credit Default Swap Index (CDX) or iTraxx. CDX is a CDS index that is based on a basket of single issuer CDSs, issued by North American or emerging market companies. iTraxx is a set of indices

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for the CDS market in Europe, Australia and Asia. Next to the standardization of CDS, the same held true for Total Return Swaps (TRS) that were based on an index. Both derivatives have been presented in Chapter 6. Standardized derivatives gave the opportunity to fixed income investors to shift from the plain vanilla bonds to standardized derivatives so as to access this type of exposure—provided this was compatible with their investment mandates and policies. At the same time, even OTC derivatives started being centrally cleared which helped in increasing confidence on one hand and containing counterparty risk on the other hand. Bond Exchange Traded Funds (ETFs ) offered a reliable choice in fixed income investing as they were listed in exchanges and at the same time they constituted standardized and mandated fixed income portfolios with no specific maturity date or prescribed investment horizon. The former allowed for liquidity and accessibility by institutional as well as individual investors. The latter offered a diversified portfolio, reduced trading and potentially embedded hedging—if subscribed by the investment mandate/policy of the ETF. ETFs have gained ground during the last few years and are no longer viewed as the passive-management relative of the mutual fund family. They have gained their own identity and are the preferred choice of several retail investors, investment/ portfolio managers, insurance companies and pension schemes. They are being considered not only as components of bigger bond portfolios but also as replacements of fixed income securities and index derivatives (Prager et al., 2018). When it comes to the implementation of fixed income strategies, they are used for portfolio completion purposes, for tactical asset allocation, portfolio rebalancing and liquidity management. They may also be used to shift exposure between the underlying bond market and the ETF by capitalizing on the creation/redemption mechanism of the latter. Their growing popularity is attributed to the merits they have introduced in fixed income portfolio management, such as liquidity, operational simplicity and speed of execution (Prager et al., 2018). The rise of the ETFs led to the development of derivatives on these ETFs, such as swaps and options. Moreover, it has triggered the creation of indices that the ETFs follow. ETFs can invest only in investment grade or only in high yield bonds depending on the targeted investors (Prager et al., 2018). We discuss ETFs in more detail in Chapter 8.

378

T. POUFINAS

Another fixed income vehicle that gained grounds primarily in the European Union is that of the covered bond. It is a debt issue originated by a financial/credit institution that pools together a set of loans and then resells them—usually through a different entity/financial institution. They are engineered in a way similar to mortgage-backed and asset-backed securities. Covered bonds are as such considered derivative instruments. They may also include public sector loans and mortgage loans. The individual loans remain with the issuing institution and they provide an additional protection to the covered bondholders. They are thus part of what is known as the double-resource protection. If the issuer defaults, then the covered bondholder has two layers of protection. The first is offered by the loans of the pool, whereas as the second is provided by the rest of the assets of the issuer (European Commission, 2020b). The investors may still receive the interest and principal payments from the underlying assets of the covered bonds even if the issuer becomes insolvent. Because of this two-layer protection they receive a very high credit rating (AAA). Covered bonds are popular in Europe and have recently started growing also in the US. By issuing bonds financial/ credit institutions can release capital—in a period that it has become scarce—and allocate it in other operations, such as offering new mortgage loans to potential homeowners. They can also be used to fund infrastructure projects, alleviating in such a way the burden from local/ regional or central governments. It seems that this “new” product engineering that is based on pools of assets and/or indices will be the prevailing one in a low interest rate environment. No matter whether such a product is considered as a tool in the spot or in the derivatives market, it is an important constituent in the fixed income portfolio formation process. These products have already room for providing more granularity and refinement through the focus in sub-indices or specific themes, geographies, etc. This trend is expected to be assisted by the increased application of the technological advances in the trading and clearing operations (Prager et al., 2018). 7.1.3.4 The Contribution of ITC Advances The bond market has evolved and is expected to evolve even more as information technology and communication (ITC) tools are used as enablers in the fixed income market transactions and trading.

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Not too long ago, bond trading has been taking place in peripheral, over the counter markets at which investors transacted directly with intermediaries. Trading used to be two-sided and verbal. Request for Quote (RFQ) venues were gradually introduced initially for more liquid bonds and transparency was not at the level of stocks, even when the Trade Reporting and Compliance Engine (TRACE) was introduced in the US. Especially after the crisis most asset classes were traded electronically and this has been permitted—but may have been even pushed forward— by non-traditional traders, such as non-bank trading firms and ETF market makers. This resulted in the growth of electronic transactional activity globally, which led to the rise of trading protocols and platforms. As matter of fact the increased use of technology forced traditional players, such as bankers and other intermediaries to embed it in their standard operations (Prager et al., 2018). This trend is often referred to as electronification and has been observed both in the US and Europe. As a matter of fact, the regulatory framework gradually pushes for equity like approaches, especially in Europe (MiFID II). The latter already has the majority (60%) of its secondary market transactions through electronic trading venues (Weinberg et al., 2020). It is not only electronic execution though that has increased. The full trading process automation is expanding. As a result of technological advances: • The interested parties can trade bonds and entire portfolios with the use of algorithms, they are able to price systematically and redesign the entire process with the use of ITC. • An increasing number of bonds are being traded through a fully automated process, without the intervention of a human trader. • Firms can source and aggregate liquidity, as well as to quote and respond to RFQs automatically. • Order and execution management systems (OMS/ EMS) can operate jointly with even additional functionalities. The former is employed so as to build portfolios, manage orders, and control compliance and operations; the latter is utilized in order to aggregate data, support decision making, route orders, and execute orders in an automated manner. • Smart order routing (SOR) was strengthened (A-Team, 2020).

380

T. POUFINAS

Electronification progresses in Europe as the regulatory framework (MiFID II) attempts to introduce a homogeneous trading structure. The combination of the technological advances with the push of the regulatory authorities (with MiFID II at hand) results in increased transparency. The latter is expected to further grow as there is a wish to disseminate almost real time post-trade data in fixed income markets via Approved Publication Agreements (APAs). This could be achieved with the use of a technology provider—enabler that would produce a unique APA record from the separate APA records, incorporating even the TRACE data reporting (in the US)—as per the vision of the European Securities and Markets Authority (ESMA) (Weinberg et al., 2020). This trend is anticipated to expand more so as to include more and more issues in Europe and in the US as well as in the remaining bond markets. It could apply to single issues, bond ETFs, as well as derivatives. The expansion of electronification offers to market participants several opportunities, such as the ability to deliver excess returns (alpha), perform tailor made or larger scale transactions, develop additional capabilities and optimize fixed income portfolio management (Weinberg et al., 2020). Besides the impact electronification has in the market per se it is expected to affect the technology providers as well. Synergies, mergers and acquisitions are most likely in the agenda in order to accommodate the requirements of the bond markets. In addition, it attracts but also permits the emergence of non-bank electronic liquidity providers (ELPs) (A-Team, 2020). Technological advances are anticipated to further allow the use of algorithmic pricing models, the transition to digital processes and assist the establishment of pan-European bond market (Weinberg et al., 2020). As discussed later in the book, in Chapter 14, the Capital Markets Union (CMU) is one of the targets that have been set by the European Union. Furthermore, there are two additional benefits from the process automation and digitization. First, the accumulation of data allows for increased market intelligence. The analysis of the data will enable well informed trades that will optimize the execution of transactions involving fixed income securities. Second, the ease of participation in the fixed income markets, as a result of the use of ITC is anticipated to improve liquidity and pricing. All in all, the electronification—as well as the introduction of FinTech—in the fixed income markets, affects them in four possible ways (ICMA, 2017):

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i. Processes become (more) efficient and direct (straight-through). ii. Liquidity can be (effectively) sourced. iii. Regulatory compliance can be established (it is even as a prerequisite). iv. Data can be managed (effectively). The aforementioned attributes characterize the market structure and their degree of implementation depends very much on the market (primary, secondary, repo) as well as on the fixed income security traded (government, corporate, investment grade, high yield, etc.). Furthermore, they are expected to gradually become more intense, due to the development of more demanding regulatory frameworks, such as MiFID II in Europe. The pursue and introduction of innovation is already in the agenda of the countries and in particular the European Union in the post-pandemic era as we illustrate in Chapter 14. The green and digital transition that is being pushed forward, as a way to recuperate the ground lost due to the pandemic and secure a sustainable growth, will further foster such technological advances and will further contribute to the modernization of the bond markets.

7.2

Traditional Notions of the Bond Markets

Although we have started our presentation with the latest developments and trends in the bonds markets, we still need to complete the presentation of the bond markets with regards to their traditional concepts. The term bond market is used to describe the set of transactions and issues of bonds and other fixed income or debt securities. It is also referred to as debt market or fixed income market. The traditional role of a bond market is on one hand to bring together bond issuers that wish to raise capital and investors that are willing to finance their activities through bonds, and on the other hand to facilitate the purchase and sale of bonds that have already been issued. The various bond markets offer to the investors a wide range of possibilities, with varying flexibility and tax benefits. In Europe, there is an effort to achieve integration also at the bond market front. The common currency assists a lot; the regulation attempts to complete this task. Transacting has become easier and fairer, due to the novelties introduced in the

382

T. POUFINAS

bond markets, as explained in the first section of this chapter. Furthermore, newer issues in several countries are listed in order to increase transparency and the sense of security. In the chapters of the book we present the different types of fixed income securities and derivatives, as well as the particulars of the markets they constitute. In Chapter 1 we presented the main issuers, as well as the main players of the bond markets. We also explained the difference between the primary and secondary market. In this chapter we give more of a global overview of the main and traditional bond markets. 7.2.1

Government Bond Market

The government bond market is the framework under which bonds issued by the central governments of the various countries around the world are traded. The governments issue bonds in order to raise funds to pay their debt, meet their obligations and finance their projects, such as infrastructure, etc. Usually they refer to debt issues with a maturity longer than one year. Maturities shorter than one year are typically considered as money market instruments. Government bonds are considered to be as bearing the lowest risk among the debt issues of a country, i.e. when compared to corporate bonds or local government bonds of the same country as they are backed by the government, which normally has increased capacity to collect the necessary proceeds to repay, issue money and enjoy higher creditworthiness. Consequently, the government bond markets are potentially exhibiting the lowest volatility among the bond markets. Because of the reduced risk the interest paid by government bonds is lower than the debt instruments that have been issued in the same country by companies or local governments. Since the latest financial crisis (2008) several government bonds have zero or negative yield to maturity. Their merit is though that they can be used for safe keeping, i.e. for capital preservation. Their potential disadvantage is that they may lead to an overall loss in the purchasing power, especially if inflation exceeds their rate of return. At issuance their coupon rate is set at a percentage almost equal to the yield of maturity that corresponds to their credit rating, as implied from the relevant yield curve. This brings their initial price at or near their face value (par). Sometimes it is deliberately set at such a level that leads to a bond price slightly less than par. In such a way there is a small capital

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gain if held until maturity, at which time the bond will pay the face value provided there is not default in the mean time. In addition, it supports the expectation that the bond price will rise until it reaches (or even exceeds) its face value. Although this is not guaranteed in the short-term (or at any point of time before the maturity date of the bond), it still creates a (psychological) comfort and motivation to the investor. To understand the size and characteristics of the government bond market we present the particulars of the US, German, French, Italian and UK bond market, all of which exhibit significant size. In addition, we exhibit the yield to maturity of the US, Eorozone, UK and Japanese bond markets for comparison purposes. In the US the bonds traded are the Treasury Notes, with maturities from one to ten years and the Treasury Bonds with longer maturities, ranging usually from ten to thirty years. The two of them pay coupon on a semi-annual basis. The yield to maturity that is disclosed is computed with the use of simple interest as two times the semiannual yield (i.e. on an annual percentage rate (APR) and not on an effective annual yield basis). This is often referred to as the bond equivalent yield (Bodie et al., 1996). The US government issues maturities up to 30 years. The credit rating of the country is AA + (as per S&P). The yield to maturity of the 10year US Government Note was at 0.91% on December 31st, 2020 and at 1.02% on January 27, 2021. The 5-year CDS quotation was 11.30, thus implying a probability of default of 0.19% (World Government Bonds, 2021). A specific type of US Government Bond that can be met in the government bond market is the Treasury STRIP (Separate Trading of Registered Interest and Principal Security). A STRIP is created when the coupons of the bond are taken apart from the principal of the bond, referred to as coupon stripping. This stripped bond is thus a zero coupon bond and is sold to investors at a discount (for positive interest rates) as seen in Chapter 2. The coupons that have been stripped are made available as distinct investments. The investor of a Treasury STRIP receives the full face value at maturity and thus makes a profit equal to the difference of the purchase price from the face value. STRIPs can be created from any US Treasury issue with a maturity of greater than or equal to 10 years. Investors can purchase them from intermediaries but not from the government. STRIPs are among the preferred choices of funds that have to meet long-term liabilities and need to perform asset liability matching or portfolio immunization. These

384

T. POUFINAS

include pension funds and individual pension (retirement) accounts (Zipf, 1997). Original issue discount securities are bonds whose price is less than their face value at the time they are first issued. This can happen if the coupon rate is determined to be significantly less than the yield to maturity that corresponds to this bond when issued. As a result these bonds have a price at issuance that is materially lower than their face value (Zipf, 1997; Nasdaq, 2021). This discount at the time of issuance is known as the original issue discount (OID). Zero-coupon bonds are typical examples of OID securities, provided that interest rates are positive. As a matter of fact, all other things being equal, they offer the largest OID among discount bonds, which is expected to attract investors. Besides the US government, there are certain federal government agencies in the US that can issue bonds to secure funding for their own operations. These bonds are known as Federal (Government) Agency Bonds (or Debt). Their role is to complement the (potentially insufficient) private credit provided to specific sectors of the economy. It is primarily directed to home mortgages and farm credit. When an agency is owned by the government then its debt is considered to be of the same quality as the government debt (i.e. practically risk-free). The Government National Mortgage Association (GNMA or Ginnie Mae) is such an example. Alternatively, an agency may be backed (sponsored) by the federal government. The debt repayment in this case is not explicitly guaranteed by the federal government; however it is implicitly perceived that the government provides protection from default. Thus, their debt issues are considered to be almost of the same quality, which results in a spread very close to zero. Examples of such federally sponsored agencies are the Federal Home Loan Bank (FHLB), the Federal National Mortgage Association (FNMA or Fannie Mae), the Federal Home Loan Mortgage Corporation (FHLMC or Freddie Mac) and the farm credit agencies (Bodie et al., 1996). The coupon is usually fixed and payable on a semi-annual basis. They minimum investment is normally of USD 10,000 and increases at a step of USD 5,000 (USD 25,000 for GNMA). The FHLB directs the proceeds to savings and loan institutions so that they offer home mortgage loans, whereas Freddie Mac and Ginnie Mac to the mortgage market so as to offer liquidity (especially before they launched mortgage-backed securities) (Bodie et al., 1996).

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In Europe, most government bond issues have annual coupon payments. Although the risk that the investor is exposed to is driven primarily by the creditworthiness of the issuing country, there is some implicit security for the member states of the European Union (EU), as well as for the countries that share the Euro. Practically the entire Eurozone or EU may have to go bankrupt for the invested capital to be lost. The recent history showed that for countries that were challenged more during the latest sovereign crisis (especially Greece, Cyprus and Portugal) a lending mechanism was activated to secure funding at affordable levels. Even in the case of Greece that a haircut was applied for the private sector investors, the performance of the bonds that replaced the bonds that were withdrawn was such that the money lost due to the haircut has been recuperated in less than ten years time. The biggest economies in the EU are those of Germany, France and Italy. Until recently it was also this of the UK, which however stepped out of the EU as of January 1st, 2021. In the Eurozone the yield to maturity of a government bonds is benchmarked with that of the German Bund—and in particular the 10-year issue. The German bond market is thus characterized by a comparable stability, potentially reflecting the path that its economy has followed. Hence the yield to maturity of a government bond is often referenced with its spread over the German Bund. The German government issues maturities up to 30 years. They represent a claim on the debt register of the country rather than the purchase of a certificate (Geneve Invest, 2021). The credit rating of the country is AAA (as per S&P). The yield to maturity of the 10-year German Government Bond was at –0.57% on December 31st, 2021 and at –0.53% on January 27, 2021. The 5-year CDS quotation is 11.00, thus implying a probability of default of 0.18% (World Government Bonds, 2021). France’s government bonds are known as OATs (Obligations assimilables du Trésor) and cover maturities up to 50 years. The credit rating of the country is AA (as per S&P). The yield to maturity of the 10-year French Government Bond was at –0.34% on December 31st, 2021 and at –0.29% on January 27, 2021. The 5-year CDS quotation is 16.50, thus implying a probability of default of 0.28% (World Government Bonds, 2021). Italy’s bond market is one of the biggets, which however exhibited the years increased volatility as a result of the concern around the capacity of the country to repay its debt (Geneve Invest, 2021). It issues maturities

386

T. POUFINAS

up to 50 years. The credit rating of the country is BBB (as per S&P). The yield to maturity of the 10-year Italian Government Bond was at 0.52% on December 31st, 2021 and at 0.61% on January 27, 2021. The 5-year CDS quotation is 99.50, thus implying a probability of default of 1.66% (World Government Bonds, 2021). The UK government bonds are known as Gilts. They come in two types; with fixed rates and indexed-linked. They can be acquired either via the U.K. Debt Management Office or (authorized) brokers (Geneve Invest, 2021). The UK issues maturities up to 50 years. The credit rating of the country is AA (as per S&P). The yield to maturity of the 10-year UK Government Bond was at 0.20% on December 31st, 2021 and at 0.27% on January 27, 2021. The 5-year CDS quotation is 17.28 thus implying a probability of default of 0.29% (World Government Bonds, 2021). The yield to maturity for all different government bond issues (maturities) of the aforementioned five countries is presented in Fig. 7.1 below.

Fig. 7.1 Government bond yield to maturity (%) for all available maturities for the USA, the UK, Germany, France and Italy (Source Created by the author with data assembled from World Government Bonds (2021))

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The yield to maturity for all different government bond issues (maturities) of these five countries along with the yield to maturity of the remaining countries of the Eurozone, as well as Japan is depicted in Table 7.1 that follows. Figure 7.1 and Table 7.1 illustrate also the yield to maturity for maturities less than one year, which are usually considered money market instruments. Money markets are discussed in Sect. 7.4 below. 7.2.2

Corporate Bond Market

The corporate bond market is the venue in which the bonds issued by companies around the globe are traded. Companies issue bonds in order to finance their existing operations, new operations, products or projects, repay existing debt, acquire other companies, etc. Usually they refer to debt issues with a maturity longer than one year. However, corporate bonds tend to have overall shorter maturities dates compared to government bonds and higher yields—at least when compared to government bonds issued by the country of their domiciliation. The corporate bond market has grown significantly the last ten years as the access of companies to them was facilitated by the lower interest rate environment that followed the 2008 financial crisis, along with the shortage of funds offered through traditional bank loans. Their yield to maturity is often referred to with the spread over the yield to maturity of the corresponding government bond, issued by the country at which they domicile. This is also known as the credit spread. Corporate bonds are perceived to be of higher risk than the corresponding government bonds, hence the yield spread over them. However, corporate bonds are perceived to be safer than the stocks issued by the same company. The relevant discussion takes place in Chapter 12. Corporate bonds resemble to government bonds; however the issuer is a company and not a government. They usually pay coupons semiannually in the US, but the frequency of coupon payment can be different, such as annual in Europe. They differ from the government bonds in the level of risk at which they expose investors. Depending on the security they offer they are distinguished to secured bonds and unsecured bonds. The former are backed by specific assets upon which investors can draw in case the issuing company goes bankrupt. The latter have no such collateral and are known as debentures. Subordinated debentures have a lower priority in drawing on the collateral of the issuing company in case it goes bankrupt (Bodie et al., 1996).

0.053 0.081 0.084

0.091 0.119 0.175

−0.133 −0.139 −0.11 −0.097 −0.058 −0.002 0.066 0.125 0.213 0.268 0.373

1.799 Netherlands −0.631 −0.632

0.531 Malta −0.471 −0.455

Lithuania

1.781

Latvia

−0.582

1.508

0.146 0.248 0.429

−0.321 −0.186 −0.108

0.774 0.806 0.834 0.746 0.658 Ireland

1.587

1.134

1.061

−0.124

−0.361

0.484

1.021

0.713

0.413

−0.475 −0.517 −0.443 −0.41 −0.403 −0.365 −0.307 −0.148 −0.012 0.285 0.346 0.406 0.475 0.609

−0.631 −0.648 −0.623 −0.631 −0.619 −0.686 −0.709 −0.663 −0.632 −0.573 −0.534 −0.447 −0.368 −0.294

−0.634 −0.643 −0.632 −0.655 −0.729 −0.773 −0.77 −0.733 −0.731 −0.69 −0.647 −0.588 −0.528

−0.539

Portugal

−0.011 0.068 0.134 0.217

−0.159

−0.688 −0.696 −0.688 −0.686 −0.68 −0.62 −0.605 −0.555 −0.487 −0.416

Austria

−0.007 −0.053 −0.038

Italy

1m 3m 6m 9m 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y 12y 13y 15y 18y 20y 25y 30y 40y 50y Maturity 1m 3m

France

USA

UK

Maturity

Germany

Government bond yield to maturity (%) for all available maturities

Table 7.1

Slovakia

0.159

−0.138

−0.621 −0.618 −0.628 −0.635 −0.62 −0.681 −0.71 −0.695 −0.672 −0.644 −0.546 −0.537 −0.433 −0.362

Belgium

Slovenia

0.238

0.064

0.021

−0.225

0.022

Cyprus

Spain −0.59 −0.534

−0.37

Finland

Japan −0.187 −0.091

1.017 1.139

0.946

0.668

0.025

−0.512 −0.283 −0.257

Greece

388 T. POUFINAS

0.716

0.394

0.369

−0.509 lePara> −0.668 −0.619 −0.545 −0.443 −0.369 −0.257 −0.17 −0.103 0.021

−0.526

Austria

Source Created by the author with data assembled from World Government Bonds (2021)

0.333

−0.22 −0.217 −0.154

0.89 1.071

0.221

−0.005

−0.01 −0.415

0.55

−0.229

−0.71 −0.749 −0.745 −0.72 −0.691 −0.646 −0.601 −0.543 −0.483

−0.645

Italy

−0.069

−0.235

−0.343

−0.69 0 −0.667 −0.641 −0.595 −0.532 −0.471 −0.402 −0.317

−0.432

France

−0.577

Germany

6m 9m 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y 12y 13y 15y 18y 20y 25y 30y 40y 50y

USA

UK

Maturity

0.479

0.251

−0.08

−0.308

−0.557 −0.532 −0.46

−0.62 −0.66

27.679

Belgium

0.291 0.384

0.101

−0.167

−0.389 −0.322

−0.564

−0.812 −0.5 −0.562

Cyprus

0.657 0.729 0.906

0.399

−0.505 −0.513 −0.512 −0.533 −0.538 −0.491 −0.449 −0.372 −0.225 −0.165 −0.069 0.075

Finland

0.65 0.681

0.443

0.267

−0.097 −0.209 −0.134 −0.135 −0.13 −0.13 −0.115 −0.11 −0.085 −0.05 −0.005 0.035

Greece

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There can also be guaranteed bonds; these are bonds whose payments are guaranteed by another party. This decreases the default risk as the guarantor honors the terms of the bond in case the issuer cannot. There is still some risk left as the guarantor may also not be able to make the payments provisioned by the indenture (Corporate Finance Institute, 2021a). The corporate bond markets – in a way similar to equity markets— are categorized according to the industry at which the issuing company belongs. The basic categories are public utilities, transportations, industrials, banks and finance companies, and international issues. The basic industries can be broken down to subsectors (Corporate Finance Institute, 2021a). Specific types of bonds may be found in the corporate bond markets; these are among others mortgage bonds, collateral trust bonds and equipment trust certificates. The former are bonds that have mortgages as collateral and are known as mortgage-backed securities. In case the bond cannot make the promised payments, the property can be disposed of so that the proceeds are used to make these payments. These bonds are considered as derivatives and have been presented in Chapter 5. The second are similar to mortgage bonds but collateral comes from securities of companies different from the issuing company. They collateralize such securities (stocks, bonds or other investments) when issuing bonds. The third are usually related to lease of equipment. These bonds are issued to finance the purchase of equipment that is then leased to the interested party. At expiration, the equipment passes to the ownership of the interested party (Corporate Finance Institute, 2021a). Bonds traded in the corporate bond market often come with embedded options, as discussed also in Chapter 2. Callable bonds give the right to the issuing company to buy back (or repay early) the bond from the bond bearer on or until a predetermined future date at a predetermined price. Puttable bonds give the right to the bond bearer to sell the bond back to the issuing company on or until a predetermined future date at a predetermined price. Convertible bonds give the right to the bond bearer to convert each bond to a predetermined number of shares of stock on or until a predetermined future date (Bodie et al., 1996). The risks present in the corporate bond market fall in five categories; (i) the market risk, which in the case of bonds translates to the interest rate risk, which affects all bonds and leads to a price drop when interest rates increase; (ii) the default or counterparty or credit risk of the company, i.e.

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its inability to pay the promised coupon or face value amounts as scheduled; (iii) the early redemption or call risk that is generated by the fact that certain corporate bonds have a call option embedded, which gives the right to the issuer to redeem it before its expiration date, especially if interest rates drop; (iv) the inflation risk, which may lead to a decline in the purchasing power of the money received from the coupons and the face value of the bond; and (v) the liquidity risk, i.e. the (in)ability to sell (or buy) a bond at the appropriate/ desired price and time, as corporate bonds do not necessarily enjoy the liquidity of government bonds, especially if not traded in an organized exchange; (Securities and Exchange Commission, 2013). The liquidity risk is of concern to the market participants as it makes the buying and selling of corporate bonds harder, which in turn results into higher costs for the issuing entities as well as the investors (European Commission, 2020a). As the yield to maturity of the government bonds in the Eurozone and the UK are negative for the shorter durations and yields in the US have also dropped significantly, the corporate bond markets have attracted the interest of the investors, including the low ranks of the investment grade bonds (BBB) and the high ranks of the non-investment grade corporate bonds (BB). Improving the functioning of the corporate bonds markets is important for the sustainability of the increasing trend in the corporate bond issues that has been observed as a result of the decreasing interest rates, as well as the corporate bond purchases pursued by the central banks, in an environment full of challenges and changes. The corporate bond market is quite important for the European companies as corporate bonds are viewed not only as financing tools for the issuing firm but as a means to achieve growth and increase employment. In addition, they are perceived as important investment vehicles due to their positive yield versus the zero or negative return of the bank accounts and government bond issues. Securing these attributes at a pan-European level is among the objectives of the capital markets union (CMU) pursued by the European Commission (European Commission, 2020a). 7.2.3

Municipal Bond Market

The municipal bond market is the place in which bonds issued by the local governments (regions, states, cities) and other government-owned

392

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bodies (school districts, airports, seaports) are traded. The proceeds from these bonds are used primarily to finance infrastructure projects. Examples are the construction of schools, libraries, roads, bridges, public transit, parks and pathways, and the funding of police and/or fire departments, of community centers and waste management (Corporate Finance Institute, 2021b). These bonds are also referred to as munis and they may enjoy preferential tax treatment as far as the interest income is concerned, as is the case in the US (for federal, as well as most of state and local taxes). The capital gains though that may be realized either at maturity or if sold before maturity are subject to the applicable taxes. This tax exemption makes the municipal bond market quite attractive for investors who fall in a high tax bracket. However, there can be municipal bonds that are not tax free. In the municipal bond market there are two main types of municipal bonds based on the source of its interest and principal payments; the general obligation bonds and the revenue bonds. The former are issued by government-owned bodies and are backed by the taxing capacity of the issuing entity. They are not secured by the revenues generated from a specific government-run project. The latter are issued by government-owned bodies to finance specific projects and are supported by the proceeds generated from that project or the municipal organization that runs it. It can be in the form of tolls or taxes (e.g. sales tax, property tax, fuel tax, etc.) or general funds. These bonds may be issued by ports, airports, turnpikes, hospitals, etc. (Bodie et al., 1996; Corporate Finance Institute, 2021b). The municipal bond market is considered as a low risk market in comparison to the corporate bond market. However, it is considered as riskier than the government bond markets. The risks that the municipal bond market entails can be summarized in six categories; (i) the market risk, which in the case of bonds stems from the interest rate risk (inherent in all types of bonds), which leads bond prices to drop when interest rates increase; (ii) the default risk or counterparty risk or credit risk of the issuer as municipal bonds are not guaranteed by the central/ federal government (it is not unheard that a local government has defaulted on a bond); (iii) the early redemption risk that stems from the fact that several municipal bonds carry call provisions that allow the issuing entity to repurchase the bond before its maturity date, especially when interest rates decline; (iv) the inflation risk that is caused from the reduced purchasing power of

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the bond cash flows; (v) the liquidity risk, as the municipal bonds are not as liquid as the bonds issued by other parties; and (vi) the fluctuation of the yield due to the potential variation of the revenue generated from the projects financed or the sources earmarked to back the bond— present primarily in revenue bonds. Especially with regards to the risks of revenue bonds, we realize that even the default risk they carry may be higher compared to that of general obligation bonds, as the projects or the sources that produce the proceeds used to make the interest payments and principal repayment may not generate the required revenue to secure such payments. This is due among others reasons to potential changes in the consumer behavior or the appearance of a decline in the economic activity (Bodie et al., 1996; Corporate Finance Institute, 2021b). One particular type of securities traded in the municipal bond markets is that of the industrial development bond (IDB) or the industrial revenue bond (IRB). IRBs are issued by local government entities in order to finance the activities of a private company. Through IRBs a private company acquires access to the (tax-exempt) borrowing that the local government can achieve (Bodie, 1996). It capitalizes on the borrowing capacity of the local government that may not have been able to obtain otherwise. The borrowed funds are used mainly to build a factory or other facility that will be operated by the private company. The rationale behind such a bond lies in the wish of the local government to attract a business activity, create job positions and opportunities in the area. It has to prove though that the project financed through an IRB will deliver public benefits. An IRB is a variety of private activity bonds (PAB), which are issued by the local government to finance particular projects of private entities. The reasoning behind the issuance of PABs is the same with that if IRBs, i.e. to return value to the local society as whole. However, they cover a wider range of projects, such as the construction of hospitals or airports. The tax exempt status that the municipal bond market/issue may enjoy, results in lower returns versus the corresponding taxable bond markets/issues. Consequently, to make the comparison, one needs to find the equivalent taxable yield of the municipal bond. To find that, we realize that if ϕ is the tax rate, then the yield yt of a taxable bond and the yield yt e m of a tax-exempt municipal bond have to satisfy the following Eq. 7.1: yt × (1 − ϕ) = ytem ⇔ yt = ytem /(1 − ϕ)

(7.1)

394

T. POUFINAS

The left hand side of the first equation explains that the yield of a taxable bond is essentially reduced by the tax rate (Bodie et al., 1996). 7.2.4

Mortgage-Backed Security Market

The mortgage-backed security market or generally the asset-backed security market experienced significant growth the last decades. The market includes also collateralized mortgage obligations and collateralized debt obligations. A mortgage-backed security (or mortgage-backed bond) is the outcome of the securitization of a basket of mortgages. It is therefore an asset that implies direct ownership of a basket of mortgages or an asset that is backed by a basket of mortgages. They are created when a pool of mortgages on real estate properties is packaged and then sold in the secondary market. These packages are originated from the entities that provided the loans, such as banks. They essentially sell their receivables (interest and principal) from the mortgages; such payments take place during the repayment period of the loan. The entity that originated the mortgage continues to service it; however, it transfers the cash flows it collects to the holder of the mortgage-backed security (or bond). Recall, that this type of mortgage-backed securities is known as pass-through, since it simply passes the payments to the purchaser of the security with no further distinction (Bodie et al., 1996). The growth of the mortgage-backed security market that originated in 1970 by the Government National Mortgage Association was assisted by the issuance of similar securities by other entities, gradually including banks and other lending institutions. It extended to securities that were backed by other asset classes. This created an even bigger market of assetbacked securities. When tranches were created, pretty much addressing different investor risk profiles, the market grew even more to incorporate collateralized mortgage obligations and collateralized debt obligations instead of pass-through securities only. Even if there is a guarantee that the payments will be received there is no guarantee that the rate of return will be fixed. If the interest rates drop, then the MBS investors will receive lower payments. The risks that the investors active in these markets are exposed to are similar to the risks presented in the other markets. Nevertheless there is one additional risk; this of the early repayment (or prepayment) of the mortgage loan. This risk is generated by the option that the borrowers have to completely

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pay off the outstanding principal of their mortgages before their maturity date. This resembles to a call option on the loan. Homeowners may choose to exercise it when interest rates drop and they can find a loan with lower interest payments and use it to refinance their mortgage by prepaying the existing loan. These prepayments are directed to the passthrough investors, who in such a way miss future interest payments or potential capital gains from the drop of the interest rates (Bodie et al., 1996). All these products, which are classified as derivative products, have been dully explained in Chapter 5. 7.2.5

Money Market

The money market is the sub-market of the fixed-income market in which short-term and highly liquid fixed income issues are traded. These securities are of high quality and their maturity is usually up to one year or even less. These markets involve both large denominations and large volume trades at a wholesale level between institutional investors and financial institutions or traders. At the same time though they can be accessible to retail investors, who can invest through money market funds or money market accounts (Bodie et al., 1996; Corporate Finance Institute, 2021c). The money market is considered to be of the highest degree of safety; as a result it offers relatively low rates of return. As a matter of fact, due to the low or even negative interest rate environment, several money market funds post negative rates of return in Europe for the last eight to ten years. The money market is a significant constituent of the fixed income market ecosystem. It provides short-term financing to governments, financial institutions and large creditworthy firms. They can thus secure the desired liquidity. This liquidity comes from the excess liquidity of investors who can place their capital in the money market for interest although these days it is at zero or even negative levels and is thus primarily used for safe-keeping. The central banks can use it to regulate— influence the flow of liquidity in the financial system (Corporate Finance Institute, 2021c). The money market is used to implement central bank policies, finance trade, assist entrepreneurial growth and facilitate commercial bank operations. As far its first function is concerned, through the money market the central banks can roll out their policy making strategy and secure the

396

T. POUFINAS

health of the financial playing field. One such action is the development of the appropriate (short-term) interest rate policy, which is important for the determination of the desired conditions in the banking sector. Moreover, the central banks can affect the market subsectors and implement their global monetary policies (Corporate Finance Institute, 2021c). When it comes to its second function, it secures the necessary (shortterm) funds to the domestic and global traders, which are the required “oil” to keep the activity running. It facilitates the payment of goods and services, as well as the financing of specific activities such as agriculture and smaller sectors (Corporate Finance Institute, 2021c). The third function focuses also on short-term lending of enterprises as a means to obtain working capital. The latter is necessary to cover potential lack of cash that is needed to meet short-term outflows. Shortterm debt, in the form of commercial paper for example, can facilitate the seamless operation of enterprises and can be a prelude to the long-term debt that can be drawn from the capital markets. The two combined can ensure that the growth of companies is not interrupted; in particular the money market covers the short-term end (Corporate Finance Institute, 2021c). Commercial banks may also use the money market to regulate their short-term liquidity needs. On one hand they can invest their excess liquidity and earn interest. The money market instruments are easily convertible to cash to cover potentially increased or unexpected withdrawals by savers. On the other hand they can borrow from the money market to cover possible short-term liquidity needs to substitute central bank lending, potentially at a lower interest rate (Corporate Finance Institute, 2021c). The most frequently used money market instruments are treasury bills, certificates of deposit, commercial paper, bankers’ acceptances, Eurodollars, repurchase (and reverse repurchase) agreements and tax anticipation notes (Bodie et al., 1996). 7.2.5.1 Treasury Bills Treasury bills or T-bills are fixed income securities with maturities of less than or equal to one year that are issued by the US and other governments. They have maturities of 1, 3, 6, 9 and 12 months (1 year) expressed usually in days or weeks depending on the issuing country. For example 13-week or 91-day (for the 3-month), 26-week or 182-day (for the 6-month) or 52-week (for the 12-month) T-bills.

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Treasury bills pay no coupon and are offered at a discount from their stated face value payable at maturity. The difference of the two constitutes the profit of the investor and is the base for the evaluation of the discount yield. They are traded pretty much by all wholesale (intermediaries, bankers, pension schemes, and insurers) and retail parties (individual investors) (Corporate Finance Institute, 2021c). As can be seen in Table 1 the relevant rates of return are negative for all the countries of the Eurozone except for Cyprus and Slovakia. They are also negative in the UK and Japan but not in the USA. 7.2.5.2 Certificates of Deposit A certificate of deposit or CD is essentially a time (or term) deposit agreed with a credit institution. Time deposit rates remain constant for the deposit period. Early withdrawal is not possible and there is usually a penalty in case it happens. Their maturity dates are normally up to a year but can be found with maturity dates from 18 months to 5 years. They can be issued in any denominations usually with a minimum of USD/EUR 100,000. They are issued by a commercial bank but can be purchased also through brokerage firms. Certificates of Deposit enjoy the insurance provided by the Federal Deposit Insurance Corporation (FDIC) (Bodie et al., 1996; Corporate Finance Institute, 2021c; Zipf 1997). 7.2.5.3 Commercial Paper Commercial paper is practically a short-term debt security that has been issued by a company. It is usually unsecured and it is used to cover shortterm payables. It is usually issued only by companies with high credit rating and so the risk is perceived to be low. Their maturity is usually between one and nine months (270 days). They are usually issued in multiples of USD/EUR 100,000 (Bodie et al., 1996; Corporate Finance Institute, 2021c). These notes are often supported by a bank line of credit upon which the borrower can draw—if needed—to repay the debt when it expires. Individual investors can access commercial paper through money market funds. It is issued at a discount; the difference between the market price and the face value represents the earnings of the holder (Corporate Finance Institute, 2021c; Bodie et al., 1996).

398

T. POUFINAS

7.2.5.4 Bankers’ Acceptances Bankers’ acceptances are short term notes (or debt) that are issued by a company but are guaranteed by a bank. It is initiated with the order of a company to a bank to pay a specific amount of money on a future date. They thus resemble to a post-dated check. By endorsing the order the bank becomes liable for the final payment to the bearer of the acceptance. They expire usually in one to six months. They can be traded at the secondary markets. They are also sold at a discount—similar to T-Bills. The difference between the selling price and the face value of the payment order is the profit to the investor. They are often used in international trade to facilitate the transactions of a trader whose creditworthiness is not known to its counterparty (Bodie et al., 1996; Corporate Finance Institute, 2021c). 7.2.5.5 Eurodollars Eurodollars are deposits in USD at banks or bank branches that are not located in the US. As such they are not subject to the Federal Reserve regulations. Although they are referred to as Eurodollars, these accounts need not be in Europe or in the Eurozone. However, this is where these deposits initially took place. Such deposits are used by wholesale investors (money market funds, banks, etc.) as they tend to slightly over-perform US government bonds. They normally involve large amounts for maturities up to six months. 7.2.5.6 Repurchase and Reverse Repurchase Agreements A repurchase agreement or a repo is a short-term loan that involves the selling of a security with an agreement to repurchase it at a higher price at a later predetermined date. This difference in price is the interest earned. They are usually overnight but can reach maturities of up to 30 days or more. These variants are known as term repos. The central banks (such as the Federal Reserve) use repos in order to regulate the supply of money as well as the reserves held by commercial banks. The reverse repo is the purchase of a security with an agreement to resell it at a higher price in a predetermined future date. It is symmetric to the repo, seen from the angle of the buying counterparty (Bodie et al., 1996; Corporate Finance Institute, 2021c). We have discussed repos in detail in Chapter 5.

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7.2.5.7 Tax Anticipation Notes A tax anticipation note (TAN) is a short-term debt issue used by a local or regional government (municipality, region, state, city, etc.) to fund a short-term project or present operations. A TAN is repaid with taxes that are paid during its life. Local or regional governments rely on TANs to secure short-term financing for projects (road construction, building repairs, etc.) before the needed tax money is available. In this way the funded undertaking is not delayed. They have maturities of less than or equal to one year and they thus bear a low interest rate. The interest generated by TANs may be tax exempt, in which case the interest rate becomes even lower (Zipf, 1997). 7.2.6

Bond Indices

Bond indices are used to track the bond markets in a way similar to the one that stock indices track the stock markets. Bond indices are representative of a bond market composition and at the same time a measure of a bond market performance. Bond indices may reflect the behavior of a an entire bond market, but may also be employed to compare markets in terms of composition and performancewithin a region, industry sector or other asset category. As explained in Chapter 4, they have a significant role in bond portfolio management; this of a benchmark, as the performance of fixed income portfolios can be compared against indices with similar compositions, which most likely they already track. Indices are used by all bond market participants, such as asset managers, asset owners, central bankers, regulators, etc. in order to follow the market developments, as well as to pursue active and passive bond portfolio management. In active bond portfolio management the managers try to perform better than the relevant market; the index is used in order to assess the potential (over)performance. In passive portfolio management the managers simply track the index, which in its turn follows the market—hopefully with a zero or close to zero tracking error—that the managers wish to copy. Such index tracking funds can be good proxies of the markets they follow—when they exhibit a zero or near zero tracking error and thus give the opportunity to mimic the index by investing in such funds. The specificities of the bond markets—especially when compared to the stock markets—that we presented above, such as the bilateral trade via a dealer instead of public trading, the reduced liquidity and transparency,

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the OTC instead of on-an-exchange transactions, the several bond issues by the same issuer compared to one only (common) stock, have made the introduction of bond indices more challenging versus stock indices. The introduction of bond index funds in the 1980’s by Vanguard was succeeded by the development of indices that provide coverage for practically any bond market. Investing in indices through bond index tracking funds lifts the difficulties attributed to the bond markets and allows for increased risk management and thus lower uncertainty. This can help investors succeed a more efficient risk-return relation (IHS Markit, 2018). In Chapter 4 we presented representative bond indices. Many more exist. A comprehensive list can be found in ETF Database (2021). In one day (January 28, 2021) a list of 57 indices was published in the Wall Street Journal : (Table 7.2). One can see that there is a wide array of indices, each with its own particulars, reflecting the bond sub-market captured such as the issuer type (government, municipal, corporate, mortgage-backed, etc.); the credit quality (investment and non-investment grade); the geography (region i.e. global, country, emerging, etc.); and the purpose (green, social, etc.) (Table 7.3). When following bond indices an investor needs to properly evaluate its composition, as the biggest issues normally weigh more. This implies that these companies may have more debt than the remaining companies, which is not necessarily as good as having high market capitalization Table 7.2 Number of bond indices—example (as published in the Wall Street Journal ) Provider

Broad market

Bloomberg 2 Barclays ICE Data Services S&P Dow Jones J.P. Morgan Total 2

Government bonds

US agency

US corporate

4

5 5

High yield

Tax-exempt

Mortgage backed 4

5

2

5

3

13

9 9

4

12

5

18

7

Source Created by the author with information assembled from The Wall Street Journal (2021)

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Table 7.3 Example of bond indices (as published in the Wall Street Journal ) Broad market

Government bonds

High yield bonds

Bloomberg Barclays U.S. Government/Credit U.S. Aggregate

J.P. Morgan Global Government Canada EMU France Germany Japan Netherlands U.K. Emerging Markets

ICE Data Services High Yield Constrained Triple-C-rated (CCC) High Yield 100 Europe High Yield Constrained Global High Yield Constrained

US agency

US corporate

Tax-exempt

Mortgage-backed

Bloomberg Barclays

Bloomberg Barclays

U.S. Agency 10–20 years

U.S. Corporate Intermediate

S&P Dow Jones Indices National AMT-Free California AMT-Free

20-plus years

Long-term

New York AMT-Free

Yankee

Double-A-rated (AA) Triple-B-rated (Baa)

Short Term National AMT-Free 1–5 Year National AMT-Free Intermediate Term National AMT-Free Short Term California AMT-Free Short Term New York AMT-Free Intermediate Term California AMT-Free Intermediate Term New York AMT-Free 7–12 Year National AMT-Free 12–22 Year National AMT-Free

Bloomberg Barclays Mortgage-Backed Ginnie Mae (GNMA) Freddie Mac (FHLMC) Fannie Mae (FNMA) ICE Data Services Ginnie Mae (GNMA) Fannie Mae (FNMA) Freddie Mac (FHLMC)

S&P Dow Jones U.S. Issued High Yield U.S. Issued Investment Grade ICE Data Services 1–10 Year Maturities 10 + Year Maturities Corporate Master

(continued)

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Table 7.3 (continued) US agency

US corporate

Tax-exempt

High Yield

Long Term National AMT-Free ICE Data Services Muni Master 7–12 years 12–22 years 22-plus years Bond Buyer 6% Muni

Yankee Bonds

Mortgage-backed

Source Created by the author with information assembled from The Wall Street Journal (2021)

(when looking at stocks). In contrast, these bonds, i.e. the biggest issues may exhibit more liquidity than smaller issues (IHS Markit, 2018). As we have noted in several points in this book, the liquidity of the bond market and thus of the issues that comprise an index may be low. However, indices are still valuable tools in bond portfolio management, as explained also in Chapter 4.

Exercises Exercise 1 a. In what ways are the bond markets different from the equity markets? b. What is the perceived problematic around the bond markets? c. What are the efforts that the regulators pursue? d. How can ITC assist? e. What issues are difficult to overpass? Exercise 2 a. Explain the differences between the different money market instruments/markets. Look at the way they operate today. b. What market/ instruments would be more suitable for a retail investor?

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c. What market/ instruments would be more suitable for an institutional investor? d. What performance/return do they post these days (2021)? Why? Exercise 3 a. What is the contribution of electronification in the bonds markets? b. What is the contribution of the existing and upcoming regulation in the bonds markets? c. What changes/advances are there still to come? Exercise 4 Compare the yield to maturity of the government bond issues in the Eurozone, the US, the UK and Japan. a. What do you observe? b. How are the differences explained? c. What choices does an investor have to overcome the low or negative yields? (Hint: you can refer to other chapters as well). Exercise 5 Repeat exercise 4 for corporate bonds and municipal bondswhere available. You may need to do some research on your own to find the relevant yields. a. What do you observe for the different countries? b. How do the corporate bond yields compare with the government bond yields of the same country?

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References A-Team. (2020, November 30). Electronification in fixed income—Where is it headed? A-Team Insight. https://a-teaminsight.com/electronification-infixed-income-where-is-it-headed/?brand=ati. Accessed: January 2021. Bodie, Z., Kane, A., & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. Corporate Finance Institute. (2021a). Corporate bonds: Debt issued by corporations. https://corporatefinanceinstitute.com/resources/knowledge/finance/ corporate-bonds/. Accessed: January 2021. Corporate Finance Institute. (2021b). Municipal bond. https://corporatefinanc einstitute.com/resources/knowledge/trading-investing/municipal-bond/. Accessed: January 2021. Corporate Finance Institute. (2021c). Money Market: Market for lending and borrowing for short-term cash flow needs. https://corporatefinanceinsti tute.com/resources/knowledge/trading-investing/what-is-money-market/. Accessed: January 2021. European Commission. (2020a). Corporate bonds. https://ec.europa.eu/info/ business-economy-euro/banking-and-finance/financial-markets/corporatebonds_en. Accessed: January 2021. European Commission. (2020b). Covered bonds. https://ec.europa.eu/info/bus iness-economy-euro/banking-and-finance/financial-supervision-and-risk-man agement/managing-risks-banks-and-financial-institutions/covered-bonds_en. Accessed: January 2021. ETF Database. (2021). List of bond/fixed income indexes. https://etfdb.com/ind exes/bondfixed-income/. Accessed: January 2021. Geneve Invest. (2021). Investment guidance for making money in the European bond market. https://www.geneveinvest.com/european-bond-market/. Accessed: January 2021. HIS Markit. (2018). What are bond indices? iBoxx indices. 258506022-CW0718. https://cdn.ihsmarkit.com/www/pdf/What-Are-Bond-Indices.pdf. Accessed: January 2021. ICMA. (2017). Market electronification and FinTech. www.icmagroup.org›assets ›Market-Infrastructure. Accessed: January 2021. ICMA. (2020). Bond market size. https://www.icmagroup.org/Regulatory-Pol icy-and-Market-Practice/Secondary-Markets/bond-market-size/. Accessed: January 2021. Michele, B. (2020, April 23). Is the bond market dead? JB Morgan asset management. https://am.jpmorgan.com/it/en/asset-management/instituti onal/insights/portfolio-insights/fixed-income/fixed-income-perspectives/isthe-bond-market-dead/. Accessed: January 2021. Nasdaq. (2021). Discount securities. https://www.nasdaq.com/glossary/d/dis count-securities. Accessed: January 2021.

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Parish, A. (2014). Are bonds dead? Positioning your portfolio for today’s markets. Research & Portfolio Strategy. Sage Advisory Services. https://www.etf.com/ sites/default/files/3gms_arebondsdead.pdf. Accessed: January 2021. Prager, R., Pybus, B., Pachatouridi, V., Veiner, D., Laipply, S., & Koay, H. S. (2018). The next generation bond market. BlackRock. https://www.sec. gov/spotlight/fixed-income-advisory-committee/blackrock-next-generationbond-market-fimsa-011118.pdf. Accessed: January 2021. R.E.M. (1987). It’s the end of the world. https://www.google.com/search?q=r. e.m.+this+is+the+end+of+the+world&oq=R.E.M.+This+is+the+end+of&aqs= chrome.1.69i57j0i22i30.13215j0j7&sourceid=chrome&ie=UTF-8. Accessed: April 2021. Securities and Exchange Commission. (2013).What are corporate bonds? Investor Bulletin, SEC Pub. No 149 (6/13). https://www.sec.gov/files/ib_corporate bonds.pdf. Accessed: January 2021. The Wall Street Journal. (2021, January 28). Markets Tracking bond benchmarks, Thursday. https://www.wsj.com/market-data/bonds/benchmarks. Accessed: January 2021. Vanguard. (2021). Bond markets. https://investor.vanguard.com/investing/por tfolio-management/bond-market. Accessed: January 2021. Weinberg, C., Cohen, S., Claringbull, A., Cohen, S., Olson, B., Pachatouridi, V., Blackman, N., & Sethi, N. (2020). The modernization of the European bond market. BlackRock. https://www.blackrock.com/dk/formidler/litera ture/whitepaper/modernisation-of-the-bond-market-en-emea-prof-whitep aper.pdf. Accessed: January, 2021. World Government Bonds. (2021). Yields curve. http://www.worldgovernment bonds.com. Accessed: January 2021. Zipf, R. (1997). How the bond market works (2nd ed.). New York Institute of Finance.

CHAPTER 8

Bond Funds

After having been exposed to all the particulars of fixed income securities and derivatives a natural question is how one could gain access to a portfolio of fixed income assets. Institutional investors or investors with a significant wealth could attempt to build such a portfolio on their own, by selecting the appropriate strategy, as described in Chapter 4. Individual— and in particular retail—investors may not have the ability to do that, as building a (well diversified) portfolio that matches their needs may require the purchase of a significant number of different issues. This may require a capital that exceeds their available budget or wealth. Both institutional and individual investors may face though the same dilemma; will they manage such a portfolio themselves or will they reach out to a professional asset manager? The answer to this question is important as even if they have (or feel they have) sufficient monetary resources, they may not have the required skill to build and manage such a fixed income portfolio. It could be that they are confident enough to construct a portfolio that invests in listed securities, but they are not as keen with non-listed securities. In both cases they may want to exploit the portfolio management solutions offered by professionals and/or firms that offer fixed income asset management services or products. Gaining access to fixed income portfolios can take place in two ways. One is to have a professional asset manager create a customized portfolio that meets the needs of the investor; the other is to purchase a portion © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_8

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(usually measured in shares or units) of an existing fixed income portfolio or fund. There is a wide array of fixed income funds that are distributed to institutional and individual investors; from mutual funds and exchange traded funds to private debt funds. They differ not only into the underlying mechanics but also in the liquidity and the risk at which the investor is exposed. The rise of the crisis in 2007–2008 led several investors to bond or money market funds, which in some cases posted negative performance as a result of the negative interest rates, so as to protect their wealth from the risks the other asset classes posted, or even the fear of exits from the common currency in the Eurozone. The prolonged period of negative (or low) yields has driven several—primarily institutional—investors to private debt funds which could offer higher returns, with higher assumed risk; the latter was somehow mitigated versus outright private debt investments due to the number of issues and issuers at which the private debt fund invested. In this chapter we discuss the composition, structure, characteristics, similarities and differences of the various types of bond (fixed income) funds available to the investors, such as mutual funds, ETFs and private debt funds. We investigate for which investor profile each of these funds is suitable and what investor goals they can be used to match. This chapter helps the reader understand the different types of bond funds.

8.1 8.1.1

Bond Mutual Funds Bond Mutual Fund Definition

A bond mutual fund is an investment company or a financial vehicle that collects money from several investors, thus creating a pool, and invests it in bonds or other fixed income securities. This basket of security holdings comprises an investment portfolio of bonds or other debt instruments, such as government, municipal, corporate bonds, as well as mortgage backed securities (Investment Company Institute, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Bond mutual funds target at securing a steady flow of revenues with low to medium risk. Mutual funds are distributed in shares (or units). This is the smallest piece of a mutual fund that is made available to investors. As each share portrays a part of the fund it gives access to (and ownership of) a portion of the entire portfolio. Consequently, an investor invests to a bond mutual

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fund is actually purchasing a piece of the portfolio equal to the value of the shares he or she owns. In this way he or she practically acquires a diversified portfolio that could not have constructed by him or herself. This is due to the fact that he or she would not have as an individual (or even institutional) investor the required wealth (capital) or the expertise to buy all the securities that the bond mutual fund consists of. Investors thus turn to bond mutual funds because they enjoy professional management, diversification, affordability, liquidity, as well as potential income and tax benefits (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Professional management secures that expert managers and analysts have the knowhow and the equipment to research and analyze the market as well as the individual securities and the creditworthiness of the issuers in order to make well-informed investment decisions (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Diversification ensures that not all eggs are put in the same basket. This means that the bond fund invests its money in a wide variety of bonds in terms of maturity, issuer or even industry and sector, as well as region (geography) and credit quality, depending of course in the orientation of the fund (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Mutual funds allow investors to purchase and sell their shares/units more easily than individual bond/fixed income securities. They thus enjoy increased liquidity compared to outright bond issues. The subscription and the redemption to bond/fixed income mutual funds may involve entrance or exit fees respectively (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). As most bond mutual funds have a rather low minimum investment amount, they are affordable to most investors, even individual investors that have a limited share of their wallet to dispose. This further unveils the merit of diversification as (primarily retail) investors acquire a bond portfolio that may not have been otherwise able to do so (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Several bond mutual funds pay a periodic income (monthly, quarterly, etc. depending on the fund); the amount is not fixed but depends on the streams of payments that are made by the bonds in the portfolio as well as the market conditions. Investors that privilege such periodic income flows can opt for dividend paying mutual funds. Investors that do not wish to have such a periodic income may prefer to invest in mutual funds that

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reinvest the dividend by default (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Certain bond mutual funds (municipal bond funds) offer tax-free income. Depending on the tax bracket of the investor, it may be more beneficial to invest in a tax-free municipal bond fund instead of a taxable bond fund, even though the (tax-free) municipal bond (fund) yields may be lower than the (taxable) bond (fund) yields (Fidelity, 2020a, 2020b; US Securities and Exchange Commission, 2020a). Bond mutual funds are considered as portfolios that generate income; sometimes they are referred to as income portfolios. In a (currently) negative interest rate environment achieving a positive income flow though is not trivial (as per January 2021). 8.1.2

Types of Bond Mutual Funds

There are several types of bond mutual funds. They are primarily characterized by the bond categories they are built from. Taxonomy could be made in terms of the expected return and the risk they bear for the investor, in line with his or her risk-return profile. This gives rise to money market, investment-grade and high-yield bond funds. Additional types may be considered in terms of the issuer (central or local government, industry or sector related) or the geography (region, country, continent or international). 8.1.2.1 Money Market Funds Investors that want (almost) risk-free bonds that are highly liquid and carry very low risk may invest in money market funds. They consist of T-Bills or bonds with maturity normally less than a year; they thus behave in a way similar to T-Bills. Historically these funds had some low positive yield; however, these days they are in the negative territory due to the negative interest rate environment. Money market funds could be considered as a separate category since they may invest in cash, repos and other instruments that are equivalent to cash. 8.1.2.2 Investment-Grade Bond Funds Investors that are willing to undertake somewhat higher risk in order to achieve a higher return may choose to invest in bond funds that consist of investment grade bonds. These are typically issued by governments (e.g. the U.S. government), government agencies, corporations and/or

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municipalities. These bonds have a credit rating that is BBB- or higher (according to the S&P credit rating scale, S&P Global 2019). These funds may fall into subcategories depending on the specific bond type in which they invest: • Government (or sovereign) bond funds, which invest in bonds issued and guaranteed by a government such as the U.S. government or a Eurozone country. In this case of the US they primarily include Treasury bonds and Treasury bills. They may also contain structured fixed income securities that enjoy the support of the government, such as mortgage-backed securities or asset-backed securities or bonds issued by entities such as Fannie Mae and Freddie Mac. In the Eurozone they may invest in covered bonds. The low or negative yield paid by most high quality sovereign bonds has led the return of government bond funds the last years to rather low territories. These funds exhibit rather low credit risk though (Fidelity, 2020b; Corporate Finance Institute, 2021a). • Inflation-protected bond funds, which in the case of the US invest to Treasury Inflation Protected Securities, known as TIPS. These are bonds that offer protection against inflation. They function in such a way that their face value changes so as to mirror the Consumer Price Index (CPI). Eurozone countries issue HICP-linked notes, where HICP stands for Harmonized Index of Consumer Prices. They are RPI-linked in the UK, i.e. the Retail Price Index (RPI) (Fidelity, 2020b; PIMCO, 2016). • Mortgage-backed bond funds, which invest in mortgage-backed securities. These instruments may carry higher risk than government bonds. As a result, investors that are willing to consider these funds take on more risk in order to achieve a potentially higher return (Fidelity, 2020b). • Corporate bond funds, which invest in bonds issued by enterprises. These bonds carry more risk but are expected to deliver higher return than government bonds. These funds may exhibit higher volatility than the other funds of this type (Fidelity, 2020b). 8.1.2.3 High-Yield Bond Funds These funds invest in non-investment grade bonds, i.e. bonds with a credit rating of BB + and below (according to the S&P credit rating

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scale, S&P Global, 2019). As such they are expected to achieve higher returns than the investment-grade bond funds at a higher risk though; their level of volatility is also anticipated to be higher (Fidelity, 2020b; Corporate Finance Institute, 2021a). These funds (as well as their underlying bonds) become popular in low interest rate environments as the one we are currently experiencing. Consequently, they are good for the part of the portfolio that the candidate investor is willing to take on higher risk. In addition, they may be used in order to increase the diversification of the portfolio and thus improve the overall risk-return profile. Increased caution is required when investing in such funds; the issuing entities may not be as solid and the regulatory framework of their country of origin may not be as demanding, thus resulting in a higher probability of loss (Fidelity, 2020b). A finer subcategory is the high-yield bank loan funds that invest primarily in sub-investment grade floating rate loans; these are sometimes called leveraged loans. Their coupons are equal to a benchmark reference rate, such as the LIBOR (London Interbank Offer Rate) plus a spread that is the compensation of the investor for the incremental risk. These bond funds enjoy a level of higher protection both in terms of credit as well as interest rate risk, as on one hand loans are senior to bonds and on the other hand coupons increase as interest rates increase (Corporate Finance Institute, 2021a). 8.1.2.4 Multi-Sector Bond Funds These funds invest in a blend of bond types in terms of issuer, credit quality, average maturity and average duration. They thus mix all the previously mentioned bond types (government, corporate, etc.). They are considered as more suitable for investors that wish to have the highest possible diversification through one fund (Corporate Finance Institute, 2021a; Fidelity, 2020b). Multi-sector bond funds may be offered with varying time horizons, such as short-term or long-term. The former invest in shorter term government or investment-grade corporate bonds. They enjoy lower volatility but are expected to have lower performance. The latter invest in longer maturity issues, which have longer durations and may thus be more volatile to the interest rate shifts (Corporate Finance Institute, 2021a; Fidelity, 2020b).

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8.1.2.5 Municipal Bond Funds These funds invest in bonds issued by municipalities. Their prime feature is that the income flows generated by their holdings are typically taxexempt (in the U.S.). They constitute good choices for investors that fall into higher tax brackets and are in favor of investments that offer tax advantages even though they achieve lower yields. Tax advantages can become higher or lower. Usually the coupon payments that are made from the proceeds of the funded projects are federal-tax exempt; they may also be state and local-tax exempt based on the state of residence of the individual investor. In contrast, when municipal bond holdings are disposed of by the mutual fund, then the revenue distributed to the investors may be subject to tax (Alternative Minimum Tax). Furthermore, the potential capital gain that is the outcome of the sale of municipal bond fund shares is subject to federal and/or state taxes (Corporate Finance Institute, 2021a; Fidelity, 2020b). 8.1.2.6 International Bond Funds These funds invest in domestic and foreign (to the country of interest) government and corporate bonds. They are also known as global bond funds. Sometimes international funds refer to funds that invest in countries other than the home country, whereas global funds refer to funds that invest in bonds issued at foreign as well as the home country (Corporate Finance Institute, 2021a; Fidelity, 2020b). This mix of different originating countries could allow for increased diversification, as well as potentially higher return, especially when the foreign issuers offer higher returns (most likely through the assumption of higher risk). In addition, these funds assist the investors in better managing interest rate risk as well as the overall economic risk of one country or region, by placing their capital in a broader range of countries or regions (Corporate Finance Institute, 2021a; Fidelity, 2020b). 8.1.2.7 Other Types of Bond Funds There are other types of bond funds whose classification is based on other characteristics. The main characteristics are • Time horizon or maturity, according to which bond funds are broken down with respect to the time to maturity of the underlying debt instruments or for the repayment of the principal.

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– In the former this can range from a few weeks (ultra short-term funds ) or a few months (short-term funds ) to a big number— let’s say 30—years (long-term funds ). There are funds that can change between short-term and long-term fixed income securities by investing in the entire spectrum of bonds and money market assets with no limit on the expiration date or asset class (dynamic funds ). – In the latter, also known as fixed maturity funds , the fund has a specific expiration date that can range from one month to a few (e.g. 5) years. The fund manager invests in fixed income securities with a maturity date equal to that of the fund. As these securities are held to maturity the interest rate risk is mitigated. If they bare coupon, then it is still present in the form of reinvestment risk. • Sector or industry, according to which bond funds invest in fixedincome securities of one specific sector or industry. As holdings are contained within one sector (which normally has a smaller number of securities than the market), the concentration risk increases the total risk; of course the expectation is that they will post higher returns. • Geography, according to which bond funds invest in fixed-income securities of a specific country or region. – Emerging market bond funds invest in emerging or developing country debt. Investors anticipate higher returns by clearly assuming higher risk. 8.1.3

The Mechanics of Bond Mutual Funds

Investors choose bond mutual funds instead of picking individual stocks as an efficient way of accessing a diversified fixed income portfolio. Bond mutual funds (except for specific exceptions) do not have a fixed expiration date on which the net asset value is repaid; hence the value of the investment varies over time. The debt instruments held by a bond mutual fund make interest payments on their coupon payment anniversary (typically annually or semi-annually). Bond mutual funds pay interest periodically (e.g. monthly or quarterly) in their turn by combining the coupon payments of the various debt instruments included in the portfolio. Therefore, the periodic interest flows to the fund holder are not fixed.

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The bonds that constitute the mutual fund are purchased and disposed of by the portfolio manager in line with the prevailing market conditions, the investment mandate/instructions or the benchmark of the mutual fund. They are seldom held until their expiration date - unless the type of the fund dictates so (as is the case in fixed maturity funds). As explained in the previous subsection bond funds may choose to invest in different types of bonds, which characterize the bond fund. However, the bond fund mandate may not be as narrow, limiting it only to a specific bond type. Broader mandates give the flexibility to investment managers to pick up the specific bond types up to a certain percentage (let us say 65%) and they are allowed to include other bond types in the remaining part (in our case 35%). 8.1.3.1 Pricing As a bond mutual fund is a portfolio that is sold to several investors in shares (or units) a share price needs to be determined. This is called net asset value (NAV) and is the value of all the securities in the portfolio divided by the number of shares. On a daily basis the NAV is found by dividing the total value of the assets under management by the total number of shares outstanding on that date: Net Asset Valuet =

Assets Under Managementt Number of outstanding sharest

(8.1)

The NAV is used in order to estimate the performance of the mutual fund as well as the purchase (buying or ask or offer) and redemption (selling or bid) prices. It could be that at the numerator there are liabilities (and/or expenses) that need to be subtracted from the assets of the mutual fund. In this case the numerator needs to change to Assets-Liabilities-Expenses (Corporate Finance Institute, 2021b). The purchase or offer price is calculated if the sales charge (or fee or load) is added to the NAV. This is also known as front-end load. As a result O f f er price = NAV + sales load

(8.2)

This is the price at which an investor can purchase a share of an open-end mutual fund on a daily basis.

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When a shareholder of a mutual fund wishes to sell his or her shares, then he or she sells them back to the company. The mutual fund management company redeems the shares by buying them back within a certain number of days. The redemption or bid price is equal to the NAV if there are no redemption charges. Otherwise redemption charges may be subtracted. They are also known as back-end loads or back-end sales charges. The bid price is thus the difference of this load from the NAV: Bid price = NAV − r edemption load

(8.3)

The contingent deferred sales charge and the exit fee are forms of back-end loads. The former depends on the holding period and usually decreases over time and finally may drop to zero (after a period of 5–10 years) (Fidelity, 2020c; U.S. Securities and Exchange Commission, 2020b). The difference between the two prices is essentially the compensation of the distributor. It is known as bid-offer spread: Bid − Offer Spread = Offer price − Bid price Bid − Offer Spread (%) =

Offer price−−Bid price NAV

(8.4) (8.5)

8.1.3.2 Shareholder Fees There is a series of fees that may apply on top of the front—and backloads. Some of them are (Fidelity, 2020c; US Securities and Exchange Commission, 2020b): • Short-term redemption fees, which may be charged by some funds in order to cover the costs related to the short-term trading of the shares of the fund for periods ranging from less than 30 days to less than 180 days. • Short-term trading fees, which may be applied to funds that have no transaction fees for selling or exchanging shares within a certain number of days from the purchase date (e.g. 60 days). • Transactions fees, which resemble to intermediary commissions and are charged mainly by no-load funds when purchasing (or even selling—depending on the fund) shares of the fund.

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• Purchase fees, which are charged in order to cover costs, related to the purchases and are paid to the fund and not to the intermediary (hence they are different from the front-end load). • Exchange fees, which are charged for transferring from one fund to another fund (usually of the same family). • Account fees, which are charged to cover the expenses required for the account maintenance, often enforced when the amount in the account falls below a predetermined minimum threshold. 8.1.3.3 Operating Expenses There are some (annual) operating expenses charged by the fund. These are (Fidelity, 2020c; US Securities and Exchange Commission, 2020b): • Management fees, which are the charges associated with the remuneration of the fund managers and investment advisors who have been assigned the management of the fund, along with any other admin expenses that have not been separately provisioned. • Distribution fees, which include costs associated with the promotion and distribution of the funds; the former includes among others advertising, preparing and shipping promotional material, whereas the latter refers to the intermediary remuneration. • Other expenses, which cover all other costs, such as certain customer service and support functions offered among others by custodian, legal, accounting, and other admin divisions of the mutual fund management company. • Expense Ratio (also known as Management Expense Ratio—MER or Total Annual Fund Operating Expenses), which estimated as the sum of all operating (including management) expenses, divided by the (average net) assets of the fund (in percent). 8.1.4

The Bond Mutual Fund Market

Bond mutual funds comprise a significant part of the mutual fund market worldwide. The net assets of the regulated open-end bond funds worldwide rose to $11.8 trillion or 21% in 2019—out of a total of almost $55 trillion. The net assets of the corresponding money market funds rose to $6.9 trillion or 13% in 2019 (Figs. 8.1 and 8.2, Investment Company Institute, 2020b).

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60 50 40 30 20 10 0 2010

2011

Equity

2012

2013

2014

Mixed/Other

2015

Bond

2016

2017

2018

Money Market

2019 Total

Fig. 8.1 Net assets of worldwide regulated open-end funds (in trillion USD) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% 2010

2011 Equity

2012

2013

2014

Mixed/Other

2015 Bond

2016

2017

2018

2019

Money market

Fig. 8.2 Net assets of worldwide regulated open-end funds (percent of Total) (Source Created by the author with data assembled from the Investment Company Institute (2020b))

Both bond and money market funds experienced their pick in 2019 in terms of assets under management for the period under examination, i.e. years 2010–2019. However, when looking at percentages, bond funds

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seem to maintain comparable popularity, whereas money market funds seem to become less popular in favor of equity funds (Fig. 8.2). Overall, equity funds had for the entire period 2010 to 2019 the highest assets under management, with bond and mixed funds following, being close to each other, exchanging their position depending on the year (Fig. 8.1). Regulated open-end funds do not include only mutual funds though. They are defined as “collective investment pools that are substantively regulated, open-end investment funds ” (Investment Company Institute, 2020b). Open-end funds are defined as funds that make available new and cancel existing shares or units as demanded by their (candidate new or already existing shareholders). These funds are regulated. As such, specific disclosure requirements apply; the organization needs to be set up in certain forms (e.g. corporations or trusts); assets are kept by a custodian; minimum capital requirements are defined; asset valuation follows certain rules; certain investment practices are enforced (e.g. limited leverage, qualifying assets and portfolio diversification). It seems however that the reporting is not uniform across the different geographic regions • In the US regulated funds incorporate on top of open-end funds (i.e. mutual funds and ETFs) unit investment trusts and closed-end funds. • In Europe regulated funds contain UCITS (Undertakings for Collective Investment in Transferable Securities), i.e. ETFs, money market funds and other similar fund types, as well as alternative investment funds (AIFs). • There are countries, at which regulation has been extended to institutional funds (addressing only certain institutional investors), guaranteed or protected funds and open-end real estate funds (Investment Company Institute, 2020b). In the analysis that follows, the regulated open-end funds include mutual funds, ETFs, and institutional funds. One of the fund categories/types considered is this of mixed/other funds; these funds include balanced/mixed funds, guaranteed/protected funds, real estate funds, as well as other funds, i.e. the fund categories described in the third bullet, thus providing the largest possible set of funds (Investment Company Institute, 2020b).

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In the USA the total assets under management of bond mutual funds in 2019 came up to $4,704.33 billion or 22.09% of the entire mutual fund market; $3,890.23 billion (or 18.27%) were in taxable bond funds and $814.10 billion (or 3.82%) in municipal bond funds (Fig. 8.3). There were 2,156 bond mutual funds or 27.14% of the total number of mutual funds; 1,602 funds (or 20.16%) invested in taxable bonds and 554 (or 6.97%) in municipal bonds (Fig. 8.4). At the same time the assets under management of money market (MM) funds amounted to $3,632.00 billion or 17.06% of the total mutual fund assets under management; $3,494.38 billion (or 16.41%) were invested in taxable bond funds, whereas $137.62 billion (or 0.65%) in tax-exempt bond mutual funds (Fig. 8.3). There were 364 money market mutual funds or 4.58% of the total number of mutual funds; 284 (or 3.57%) invested in taxable assets and 80 (or 1.01%) invested in tax-exempt assets (Fig. 8.4). For the period under examination (years 2000–2019) the pick with regards to the assets under management seems to be in the year 2019 for taxable and municipal bond funds. The same holds true for taxable money market funds, which had seen a drop from the previous high of 2008, most likely attributed to the financial crisis. That was the year

Bond Taxable

Bond Municipal

MM Taxable

2018

2019

2017

2016

2015

2014

2013

2012

2011

2010

2009

2007

2008

2006

2005

2004

2003

2002

2001

2000

4,500 4,000 3,500 3,000 2,500 2,000 1,500 1,000 500 0

MM Tax-exempt

Fig. 8.3 AUM of bond and money market mutual funds (in billion USD) (Source Created by the author with data assembled from the Investment Company Institute (2020b))

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1,800 1,600 1,400 1,200 1,000 800 600 400 200 0

Bond Taxable

Bond Municipal

MM Taxable

MM Tax-exempt

Fig. 8.4 Number of bond and money market mutual funds (Source Created by the author with data assembled from the Investment Company Institute (2020b))

that the tax-exempt money market funds experienced their highest assets under management. Taxable bond mutual funds posted their maximum percentage as a category of mutual funds in 2012 (21.52%), whereas municipal bond funds recorded their maximum percentage in 2002 (5.17%). Both taxable and tax-exempt money market funds exhibited their highest percentage in 2008 (34.71 and 5.13% respectively), as they were possibly among the favorite choices of the investors amid the financial crisis (Fig. 8.4). They were potentially considered safer than longer or even short-term assets, including bank deposits. Even if an issuer or a bank defaulted the money market funds had two competitive advantages; they were diversified portfolios and were easily liquidable. Hence, they could be transferred to any custodian (even cross-border) in case of default and easily converted to other asset classes when the turbulence would be over. Bond funds followed a clearly increasing trend, lagging money market funds at almost the first half of the period, but surpassing them at the second half. More precisely taxable bond funds exceeded taxable money market funds from 2012 and onwards and municipal bonds funds surpassed non-taxable money market funds from 2009 and on (Fig. 8.3). This is probably due to the fact that the financial (and the subsequent sovereign) crisis of 2008 had been somehow overcome. Money market funds had started posting very low or even negative returns, hence fixed

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income investors moved back to longer maturities; at the same time, trust to (certain) sovereigns has been reinstated. The highest number of taxable bond mutual funds was recorded in 2018 (1.618), whereas the highest for municipal mutual funds in the year 2000 (870) and has been declining ever since. Both taxable and tax-exempt money market mutual funds posted their maximum in the year 2000 (703 and 334 funds). The number of taxable bond mutual funds had its highest percentage in 2019 (20.16%), whereas the number of municipal bond funds picked in 2000 (10.70%). The same holds true for taxable and tax-exempt money market mutual funds (8.64 and 4.11% respectively) as evidenced in Fig. 8.4. As a matter of fact the number of taxable bond funds has always been higher than the number of municipal bond funds, which was higher than the number of taxable money market funds, which in its turn was higher than the number of non-taxable money market funds. When it comes to the different types of bond mutual funds the breakdown in 2019 was as follows (Figs. 8.5, 8.6, 8.7 and 8.8): 2,500 2,000 1,500 1,000 500 0

Investment grade

High yield

World

MulƟsector

State muni

NaƟonal muni

Government

Fig. 8.5 AUM by type of bond mutual funds (in billion USD) (Source Created by the author with data assembled from the Investment Company Institute (2020b))

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60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 20002001200220032004200520062007200820092010201120122013201420152016201720182019 Investment grade

High yield

World

MulƟsector

State muni

NaƟonal muni

Government

Fig. 8.6 AUM by type of bond mutual funds (in percent) (Source Created by the author with data assembled from the Investment Company Institute (2020b)) 700 600 500 400 300 200 100 0

Investment grade

High yield

World

MulƟsector

State muni

NaƟonal muni

Government

Fig. 8.7 Number of bond mutual funds by type (Source Created by the author with data assembled from the Investment Company Institute (2020b))

• investment grade had $2,159.31 billion in AUM (or 45.90% of total bond fund AUM) and 593 mutual funds (or 27.50% of the total bond mutual funds);

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35.00% 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00%

Investment grade

High yield

World

MulƟsector

State muni

NaƟonal muni

Government

Fig. 8.8 Number of bond mutual funds by type (in percent) (Source Created by the author with data assembled from the Investment Company Institute (2020b))

• high yield posted $337.16 billion in AUM (or 7.17% of total bond fund AUM) and 254 mutual funds (or 11.78% of the total bond mutual funds); • world reported $548.16 billion in AUM (or 11.65% of total bond fund AUM) and 348 mutual funds (or 16.14% of the total bond mutual funds); • government recorded $347 billion in AUM (or 7.38% of total bond fund AUM) and 193 mutual funds (or 8.95% of the total bond mutual funds); • multi-sector had $498.58 billion in AUM (or 10.60% of total bond fund AUM) and 214 mutual funds (or 9.93% of the total bond mutual funds); • state municipal posted $183.64 billion in AUM (or 3.90% of total bond fund AUM) and 285 mutual funds (or 13.22% of the total bond mutual funds); and • national municipal reported $630.46 billion in AUM (or 13.40% of total bond fund AUM) and 269 mutual funds (or 12.48% of the total bond mutual funds). The highest value of the assets under management per type of fund (Fig. 8.5) was posted in.

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• 2019 for the investment grade bond mutual funds ($2,159.31 billion); • 2013 for the high-yield bond mutual funds ($417.12 billion); • 2019 for the world bond mutual funds ($548.16 billion); • 2019 for the government bond mutual funds ($347.00 billion); • 2019 for the multi-sector bond mutual funds ($498.58 billion); • 2019 for the state municipal bond mutual funds ($183.64 billion); • 2019 for the national municipal bond mutual funds ($630.46 billion). Moreover, investment grade bonds posted for the entire period (2000– 2019) the highest assets under management with the other types following and exchanging positions depending on the year. All of them had an overall increasing trend within the period, with the high-yield bond funds showing some swings after the year 2013, in which it recorded its maximum (Fig. 8.5). The highest value of the assets under management in percent per type of fund (Fig. 8.6) was posted in • • • • • • •

2011 2004 2014 2002 2019 2000 2000

for for for for for for for

the the the the the the the

investment grade bond mutual funds (48.02%); high-yield bond mutual funds (12.78%); world bond mutual funds (13.50%); government bond mutual funds (19.25%); multi-sector bond mutual funds (10.60%); state municipal bond mutual funds (16.14%); national municipal bond mutual funds (17.93%).

Although all types of funds showed an increase in their assets under management, with the exception of high-yield bond funds, it seems that they all lost grounds since their pick (as a percent of total bond mutual funds) in favor of multi-sector bond mutual funds; the later recorded their maximum percentage in 2019. This is possibly interpreted by the search for diversification and/or benefit from all possible fixed income classes as interest rates are at record low levels (or even negative). The highest number per type of fund (Fig. 8.7) was recorded in • 2016 for the investment grade bond mutual funds (621); • 2019 for the high-yield bond mutual funds (254);

426

• • • • •

T. POUFINAS

2015 2000 2019 2000 2000

and 2016 for the world bond mutual funds (370); for the government bond mutual funds (322); for the multi-sector bond mutual funds (214); for the state municipal bond mutual funds (588); for the national municipal bond mutual funds (282).

Investment grade bond funds had the highest number of all types of funds after 2001 for the period of interest (2000–2019). State muni bond funds showed the highest drop in the period, losing their top position and gradually ranking third, after world funds since 2014. The number of the remaining types of funds followed different trends. Government, state muni and national muni ended in 2019 at levels lower than the ones they had in 2000. The remaining funds ended at higher levels. The highest number of bond mutual funds as a percent per type (Fig. 8.8) was reported in • • • • • • •

2010 2019 2015 2000 2019 2000 2000

for for for for for for for

the the the the the the the

investment grade bond mutual funds (30.83%); high-yield bond mutual funds (11.78%); world bond mutual funds (17.19%); government bond mutual funds (14.43%); multi-sector bond mutual funds (9.93%); state municipal bond mutual funds (26.34%); national municipal bond mutual funds (12.63%).

When it comes to the number of funds almost all types showed a decrease from their maximum values in the number of funds as well as in popularity among bond mutual types except for high-yield and multi-sector bond mutual funds, which recorded their maximum number and percentage in 2019. For multi-sector funds this is in line with the pick in the assets under management. This is not the case for high-yield bonds whose number moved in the opposite direction of their AUM. This is probably attributed to the fact that investors seek higher yields in a low interest rate environment so more funds were launched to attract them; however, the preference to these funds have not been reflected in their AUM yet.

8

8.1.5

BOND FUNDS

427

Comparison of Bond Mutual Funds with Bonds

Even though bond funds are a collection of individual bonds, there are still differences between them. These differences pertain to the principal, income, diversification and liquidity as is probably anticipated. An individual bond promises to return the face value of the investor at the maturity or call date. Even if the bond price varies over time, the investor can hold on to the bond until its expiration (or call) date on which he or she collects the face value. This promise is not kept when the issuer defaults. A bond mutual fund does not promise to pay back the invested amount principal, although it attempts to increase it. Its price also changes on a daily basis. An individual fixed-rate bond promises to pay a known set of interim cash flows (interest or coupon) on certain dates, usually annually or semi-annually. Bond mutual funds pay interest that varies as the underlying portfolio may change over time. Moreover, the cash flows from the mutual fund to the fund holder are periodic (e.g. monthly) so as to aggregate all the payments that are made during the period by the bonds in the underlying portfolio. With a relatively low initial investment amount bond mutual funds offer access to a diversified portfolio. This is not feasible with individual bonds. If an investor wanted to build such a diversified portfolio him or herself, then he or she would need a much more substantial initial capital. Bond mutual funds can be sold at any time at the NAV more easily compared to an individual bond. The latter may not be as liquid. This may increase the time it takes to sell it, as well as the price at which it will be sold, making the sale more expensive. 8.1.6

Concerns About Bond Mutual Funds

Despite the advantages that bond mutual funds have for the investors, there are some points of attention. These pertain to expenses, taxes and risks. As became apparent earlier bond funds are subject to a series of expenses, which consume part of the investor’s capital or performance. As a result, the return of the fund needs to exceed a certain threshold before even the investor breaks even. The dividends distributed by bond mutual funds are usually taxable; the same holds true for any capital gains. Bond funds are exposed to a number of risks such as market-related (interest

428

T. POUFINAS

rate, principal and call, as well as reinvestment and inflation) and credit risks (Fidelity, 2020a). The most apparent form of market risk is interest rate risk, which pertains to the drop (increase) of the bond price as interest rates increase (decrease). Recall that the longer the duration of the bond, the higher its volatility to interest rate moves is. As a result a bond fund with longer duration will have a more volatile unit price compared to a shorter duration bond fund as interest rates increase or decrease. This fluctuation has an impact on the periodic cash flows made by the bond fund (Fidelity, 2020a). Interest rate risk is also inherent in what is known as the reinvestment risk; as the proceeds of a bond need to be reinvested there is no guarantee that this will be at the same interest rate as at the time of purchase. The reinvestment interest rate can be higher or lower, thus causing an uncertainty over the value of the bond portfolio underlying the fund. Furthermore, the interest rate movements influence the price of the bonds and therefore of the fund. The principal of an investor is at risk when investing in bond mutual funds. The purchase and redemption of bond mutual fund shares takes place at the NAV (plus or minus any applicable charges). If the redemption price is lower than the purchase price, then they investor may forfeit part or all of his or her initial principal (Fidelity, 2020a). If the fund invests in callable bonds then it is exposed to the risk of the bond being called, thus missing some of the interest payments made by the bond being called. The fund will then most likely be forced to reinvest at a lower interest rate, as the bond was probably called because the issuer realized that borrowing at a lower interest rate was feasible (Fidelity, 2020a). Reinvestment risk is present also as part of the call risk; the payoff received as a result of the exercise of the call option by the issuer has to be reinvested by the fund at an interest rate (yield) that is not known with certainty. Inflation risk stems from the fact that the real return of the bond fund is different from the nominal one as (positive) inflation consumes part of the performance at it reduces the purchasing power of the future cash flows that the fund delivers to the investor. Inflation impacts the periodic payments made by the bonds—hence the mutual fund—as well as the call price paid by a callable bond or the face value paid by a bond at maturity (if held until then by the fund)—hence the redemption value (Fidelity, 2020a).

8

BOND FUNDS

429

Bond mutual funds are clearly exposed to credit risk as their underlying portfolio consists of bonds whose issuers may default on their promises. Would this happen, a part of the portfolio value will be lost, modulo any potential redemptions. If the fund invests partially or fully in lower quality bonds, such as non-investment grade or high-yield bonds, then this risk is higher than the risk borne by bond funds that invest only in investment-grade bonds. Furthermore, they are exposed to the possibility of a downgrade which will reduce the bond and thus the fund price (Fidelity, 2020a). Of course these risks are better mitigated in a bond mutual fund in contrast with individual bond holdings. Bond funds invest in an array of bonds; thus default (credit) risk and the call risk are somehow diluted when the bond being affected is part of a (much) bigger portfolio. The latter can be lowered with the use of credit derivatives. The inflation risk can be reduced with the use of inflation protected issues. The interest rate risk can be contained with the use of interest rate derivatives or appropriate selection of the duration of the underlying portfolio, in line with the interest rate movement expectations.

8.2

Bond ETFs (Exchange Traded Funds) 8.2.1

Bond ETF Definition

A bond or fixed income ETF (Exchange Traded Fund) is an exchange traded fund that invests in bonds only. It offers exposure to a portfolio (basket) of bonds. Bond or fixed income ETFs cover the entire fixed income market; from high quality government debt to non-investment grade emerging market debt. The holding periods can be long-term or short-term. Bond ETFs select the securities in which they invest as any other professionally managed fixed income portfolio; the portfolio manager needs to determine the type of exposure he or she wants. This includes the types of bonds, the credit ratings and the interest rate risk, as well as the geographic exposure, i.e. US, UK, euro-zone, emerging markets, etc. (ETF, 2020; State Street, 2018). 8.2.2

Types of Bond ETFs

Bond ETFs are usually categorized as follows (ETF, 2020):

430

T. POUFINAS

• Sovereign ETFs, which invest in debt instruments issued by governments, such the US, the UK, countries of the Eurozone, etc. • Corporate ETFs, which invest in debt instruments issued by companies. • Municipal ETFs, which invest debt instruments issued by municipalities (local governments). • Broad Market ETFs, which invest both in sovereign and corporate fixed income securities 8.2.3

The Mechanics of ETFs

Bond ETFs pursue passive bond management. They track fixed income indices. Most such indices are market value weighted (i.e. total outstanding debt issuance); however, some could be based on credit ratings, liquidity or currency denomination (ETF, 2020). Bond ETFs are traded in major stock exchanges. This offers additional liquidity versus corporate bonds; one could say that the liquidity of sovereign ETFs is comparable with that of government bonds. Bond ETFs also assist in securing market stability and transparency, especially when bond markets are under pressure. Bond ETFs trade in organized stock exchanges intraday, whereas individual bonds are (still materially) sold over the counter by intermediaries. It may not be easy for investors to buy or sell a bond at an attractive price (or at any price). As bond ETFs trade on an organized exchange, they may overpass this problematic. Consequently, investors have increased transparency and ease in transacting in the organized (stock) exchange compared to the (OTC) bond market. Bond ETFs exhibit superior liquidity over bond mutual funds, which post a daily price—usually at closure. This is important when markets are under pressure; there is always a price at which the buyers and sellers can transact. They may not like the price but it is available. Bond ETFs offer periodic income through periodic (monthly) dividend payments. If there are capital gains, then they are often disbursed on an annual basis (annual dividend). To explain the rationale behind this payment pattern, recall that a bond pays its coupon annually or semi-annually. A bond ETF is a portfolio of bonds with different maturity dates. As such their coupon payments are spread over time. ETFs pay interest periodically (monthly) within the year that comprise of the

8

BOND FUNDS

431

coupon payments of the bonds that had a coupon payment in that month. As a result the interest paid at each period (month) may be different. Individual bonds have a specific maturity date. In contrast, the ETF portfolio is rebalanced and its positions are not left to expire. Instead they are purchased and disposed of so that they always fulfill the duration requirements of the fund or as their maturity date approaches. The ETF manager has to consistently, narrowly and carefully follow the underlying benchmark/index. This is not always trivial as the bond market liquidity (especially corporate bond market) is limited compared to the stock market. Securities that have the required liquidity need to be selected. Furthermore, indices rebalance when a bond is issued or when a bond matures. They may also rebalance to adjust for the market weight (on a daily basis). The manager of the bond ETF has to follow that by adjusting the composition of the underlying portfolio accordingly. As this is not always feasible, since certain bond issues may exhibit decreased liquidity, managers use the cellular approach described in Chapter 3. They therefore select a representative subset of bonds that are included in the index and can successfully proxy it. These are usually the issues with the highest outstanding amount, which at the same time exhibit the highest liquidity. Consequently, the bond ETF is subject to tracking error. This tends to be smaller for government bond ETFs compared to corporate bond ETFs. 8.2.4

Bond ETF Market

The bond ETF market is rather new compared to the other bond funds and grew significantly over the last 18 years. Its origins go back to 2002 with the launch of the first fixed income ETF. In 2008 its assets under management (AUM) were $48 billion or approximately 1.9% of the total fixed income funds AUM globally. They made for 0.2% of the available for investment fixed income amount globally, comprising of investment and non-investment grade issues denominated in a wide range of currencies. At the end of the first semester of 2018 the fixed income ETFs AUM had reached $800 billion or 10.2% of the fund market globally. They made up for 1.5% of the available for investment fixed income amount. Fixed income ETF markets still accounted though for a thin slice in their corresponding sub-markets. Consequently, they do not significantly affect the market prices of the fixed income securities at which they invest (State Street, 2018).

432

T. POUFINAS

On August 31, 2020 there were 411 US-listed bond ETFs, with $988.4 billion of assets under management (ETF Database, 2020). These bond ETFs are classified based on the bond category (Table 8.1, Figs. 8.9, 8.10 and 8.11), as well as the bond duration (Table 8.2, Figs. 8.12, 8.13 and 8.14) at which they are mostly exposed. They are monitored through a series of metrics, such as their assets under management, the number of ETFs per category, the return, the fund flow, the expense and the dividend paid. As of the above reference date we observe that • Total Bond Market ETFs rank first in terms of assets under management, with approximately $298 billion, followed by Investment Grade Corporate ETFs with approximately $201 billion and Treasuries ETFs with approximately $165 billion. • Total Bond Market comes first also with regards to the number of ETFs, with 88 ETFs, followed by Treasuries with 51 ETFs and Investment Grade Corporate with 50 ETFs. • Convertible ETFs come first in terms of the Average 3-month return, posting a return of 15.32%, followed by Build America with 5.89% and Emerging Markets with 4.76%. • Total Bond Market recorded the highest three month flow of approximately $25 billion, followed by Investment Grade Corporate with approximately $24 billion and TIPS with approximately $9 billion. • Investment Grade Corporate ETFs seem to charge the lowest average expense of 9bps, followed by Target Maturity Date Corporate Bonds with 10bps and Money Market with 10 bps. • Junk pay the highest dividend with an average of 5.18%. Target Maturity Date Junk Bonds come next with 4.77%. The third position is held by Emerging Markets with an average dividend of 4.37%. The afore mentioned findings indicate that Total Bond ETFs dominate (as of the reference date) the Bond ETF market in terms of assets under management, number and flows (for the last 3 months). Convertible ETFs dominated in terms of return and Junk ETFs prevailed in terms of dividend yield. However both categories bear substantial expense ratios (compared to the other categories) in the area of 40 bps. When the duration exposure is used we realize that

8

BOND FUNDS

433

Table 8.1 Bond ETF metrics by major bond type investment Bond

AUM

#

Avg. 3-Month fund 3-Month flow return (%)

Bank loans Build America California munis China bonds Convertible Emerging markets Floating rate bonds Floating rate treasury High yield munis International corporate International treasury Investment grade corporate Junk Money market Mortgagebacked Municipal bonds New York munis Target maturity date corporate bonds Target maturity date Junk bonds Target maturity date munis TIPS Total bond market Treasuries U.S. agency

$7,594,835,133 $2,022,375,245 $2,123,606,527 $19,147,309 $6,544,343,005 $26,962,715,233

7 1 3 2 3 21

1.49 5.89 1.58 3.69 15.32 4.76

$9,266,481,380

4

$1,895,440,516

Avg. Avg. expense dividend ratio Yield (%) (%)

$55,493,596 $594,238,759 $87,425,715 $4,443,941 $400,282,087 $969,686,722

0.70 0.28 0.27 0.60 0.39 0.39

4.26 3.26 2.02 4.24 2.28 4.37

0.86

−$175,090,303

0.19

1.91

2

0.03

−$307,804,130

0.15

1.10

$4,063,602,448

2

2.78

$124,810,602

0.35

4.07

$19,714,221,972

11

0.94

$3,775,005,816

0.17

2.00

$2,454,846,295

5

4.61

$220,671,729

0.34

0.73

$200,684,445,158 50

2.42

$23,580,854,878

0.09

2.65

$74,421,708,764 $1,462,195,678 $44,149,793,165

48 1 13

1.95 0.03 0.28

$4,574,646,535 −$98,813,160 $2,537,079,562

0.42 0.10 0.15

5.18 2.11 2.26

$46,524,111,365

28

1.62

$4,057,191,682

0.14

2.06

$570,032,320

2

1.42

$31,018,276

0.26

2.13

$15,991,965,071

22

1.03

$1,420,904,864

0.10

2.57

$2,502,142,493

12

1.19

$43,263,564

0.42

4.77

$1,711,498,387

18

0.60

$147.193.360

0.18

1.35

$54,305,325,944 16 $297,899,339,816 88

3.93 1.55

$8,827,106,069 $24,526,827,037

0.14 0.11

1.12 2.40

$164,654,293,181 51 $863,524,564 1

1.28 0.85

−$8,525,973,759 $150,999,226

0.13 0.20

1.41 1.66

Source Created by the author with data assembled from the ETF Database (2020)

434

T. POUFINAS

350.00

300.00

Billion USD

250.00

200.00

150.00

100.00

50.00

.00

Fig. 8.9 AUM of bond ETFs by major bond type investment (Source Created by the author with data assembled from the ETF Database (2020)) 100 90 80

Number

70 60 50 40 30 20 10 0

Fig. 8.10 Number of bond ETFs by major bond type investment (Source Created by the author with data assembled from the ETF Database (2020))

8

BOND FUNDS

435

6.000% 5.000%

Percent

4.000% 3.000% 2.000% 1.000% .000%

Fig. 8.11 Average dividend yield of bond ETFs by major bond type investment (in percent) (Source Created by the author with data assembled from the ETF Database (2020))

Table 8.2 Bond ETF metrics by major bond duration investment Bond duration

AUM

#

All-term Intermediateterm Long-term Short-term Ultra short-term Zero duration

$554,599,840,505 207 $181,778,552,234 94

2.26 2.11

$49,192,713,867 $13,230,700,167

0.15 0.21

2.48 3.17

$90,984,030,187 $93,776,976,766 $66,355,530,559 $907,060,718

2.47 0.52 0.29 1.80

$5,846,768,254 $2,326,814,297 −$3,545,122,063 −$30,411,857

0.17 0.11 0.16 0.33

2.30 1.81 1.56 3.70

38 38 23 11

Avg. 3- 3-Month fund month flow return (%)

Avg. Avg. expense dividend ratio yield (%) (%)

Source Created by the author with data assembled from the ETF Database (2020)

• The All-Term ETFs exhibit the highest assets under management, reaching almost $555 billion. Intermediate-Term ETFs come next with approximately $182 billion, followed by Short-Term with almost $94 billion.

436

T. POUFINAS 600

Billion USD

500 400 300 200 100 0

Fig. 8.12 AUM of bond ETFs by major bond duration investment (Source Created by the author with data assembled from the ETF Database (2020)) 250

Number

200 150 100 50 0

Fig. 8.13 Number of bond ETFs by major bond duration investment (Source Created by the author with data assembled from the ETF Database (2020))

8

BOND FUNDS

437

4.000% 3.5000%

Percent

3.000% 2.5000% 2.000% 1.5000% 1.000% .5000% .000%

Fig. 8.14 Average dividend yield of bond ETFs by major bond duration investment (in percent) (Source Created by the author with data assembled from the ETF Database (2020))

• The same ranking holds true with regards to the number of ETFs. • Long-Term ETFs post the highest average 3-month performance of 2.47%, followed by All-Term with 2.26% and Intermediate Term with 2.11%. • The All-Term ETFs recorded the top 3-month flow of approximately $49 billion, followed by Intermediate-Term with almost $13 billion and Long-Term with circa $6 billion. • Short-Term ETFs charge the lowest average expenses of 11 bps, followed by All-Term with 15 bps and Ultra Short-Term with 16 bps. • Zero Duration ETFs pay the highest average dividend of 3.70%.Intermediate-Term come next with 3.17%, followed by AllTerm with 2.48%. We conclude that All-Time ETFs come first in terms of assets under management, number of ETFs and flows (for the last 3 months). LongTerm ETFs were the first in terms of return and Zero Duration ETFs in terms of dividend. Long-Term ETFs charge medium expenses (17 bps), whereas Zero-Duration ETFs charge the highest (33 bps).

438

T. POUFINAS

8.2.5

Comparison of Bond ETFs with Bond Mutual Funds

Bond ETFs and mutual funds have similarities but also differences. They are summarized below: • Trading venue: As mentioned, bond ETFs trade at an organized (stock) exchange and thus there is always a price at which a transaction can take place, at any point of time during the trading hours. Bond mutual funds can be purchased or redeemed only once the daily NAV has been calculated. • Management style: Bond ETFs are more appropriate choices for investors that prefer passive management. Investors that favor active management are probably better off with a mutual fund that matches their needs. • Trading frequency: Bond ETFs are probably more suitable for investors that intend to buy and sell frequently. Both fund types can be used to match longer investment horizons. • Transparency: Bond ETFs offer increased transparency as their positions are visible at any point of time. Mutual funds offer this visibility at a lower frequency. • Sellability: Although bond ETFs do offer intraday liquidity and a price during the trading hours at which a transaction can take place, there could be times that selling (or buying) is not feasible. Mutual fund shares (or units) can always be redeemed (or purchased) via the mutual fund management company. 8.2.6

Concerns About Bond ETFs

Although bond ETFs offer access to a basket of bonds and as such there is increased diversification, liquidity and an available price to buy or sell there are still some concerns around them. Individual bonds pay the face value and as a result the principal of the investor at maturity; as bond ETFs do not mature there is no such payment. As such the initial investment is not guaranteed. An interest rate increase results in a decrease of both ETF and bond prices. This is interest rate risk; as bonds have a specific maturity date this interest rate risk can be contained. Managing such a risk tends to be harder for an ETF.

8

8.3 8.3.1

BOND FUNDS

439

Private Debt Funds Private Debt Definition

Private debt is an alternative investment that pertains to debt issues that are not issued or traded publicly (they are not traded on regulated stock markets) and are not required to be rated by a credit rating agency. They are often used to finance a private company, while providing investors with a steady stream of revenue. Private debt has recently only qualified as an asset category. It is considered as the youngest private capital asset class (Preqin, 2020b; PRI, 2019). The term is used to describe a spectrum of relevant investments. It usually refers to non-bank credit which is not broadly available in the marketplace. Private means that the investment vehicle employed is not publicly available. Thus both listed—public and private—unlisted companies can pursue private debt financing. Private debt belongs in the same debt family as alternative debt or alternative credit. It is often referred to as direct lending, private lending and private credit. Private debt holders practically substitute for bank in providing credit to entities such as companies or project financing vehicles. Examples are the development of businesses, infrastructure or real estate development (PRI, 2019; World Economic Forum, 2015). Note here that some investors and authors consider private debt part of private equity. However, as it has its own special characteristics, we treat it separately. From this, however, the reader can conclude that its history is essentially parallel to that of private equity. It has of course its own advantages and disadvantages when compared to private equity and/or public debt. 8.3.2

Positioning Private Debt Among Other Asset Categories

It is worth seeing how private debt is positioned among other asset categories both by investors—as an investment vehicle and by business—as a form of financing. This comparison is shown in Figs. 8.15 and 8.16. Also of interest is its benchmarking in terms of risk and return. This is depicted in Fig. 8.17.

440

T. POUFINAS

Public Debt

Public Equity

Liquidity

H

Private Equity L

Private Debt Income

Growth Objective

Fig. 8.15 Comparison of private debt with other asset categories—Part I (Source Created by the author with information assembled from PRI (2019))

H

Control

Private Debt Public Debt

Private Equity

Public Equity

L Limited

Unlimited Holding period

Fig. 8.16 Comparison of private debt with other asset categories—Part II (Source Created by the author with information assembled from PRI (2019))

8.3.3

Private Debt Types

As explained, private debt constitutes credit provided primarily from institutional investors (insurers and pension or other funds) except for bankers. Private debt issues exhibit decreased liquidity compared to public

8

H

Expected Return

Junior Subordinated Second Lien Mezzanine Senior Senior Debt

Unitranche

BOND FUNDS

441

Equity Private Equity

Listed Equity High Yield Debt

Bonds M M

H Assumed Risk (Standard Deviation)

Fig. 8.17 Risk-return profiles for various investment strategies (Source Created by the author with information assembled from PRI (2019), IHS Markit (2017) and NN Investment Partners (2017))

corporate issues. They became popular initially in the United Kingdom and the United States, where they were already employed especially for development financing and acquisitions. However, the quest for higher returns in a low or even negative interest rate environment has put private debt in the radar of the investors and entrepreneurs of several countries. Fund managers generally specialize in specific market segments; the extended list includes senior debt, mezzanine debt, credit opportunities, distressed debt, infrastructure debt, real-estate debt, special structures and venture debt (Golding, 2019; PitchBook, 2019). Emphasis is given primarily on the first four. The following terminology pertaining to the private debt types is standard in the relevant market and comes mainly from Golding (2019) for the first four and Pitchbook (2019) for the remaining. 8.3.3.1 Senior Debt Senior debt describes the loans that enjoy the highest security (and are thus secured) used mainly for acquisitions and development. Their yield comes primarily from interest flows.

442

T. POUFINAS

8.3.3.2 Mezzanine Debt (Intermediate Debt) Mezzanine debt is financing that is positioned between debt and equities. It is mainly used for acquisitions and development and is often of lower priority than bank debt. The yield comes from interest flows and equity warrants. 8.3.3.3 Credit Opportunities Credit opportunities refer to broad spectrum of financing schemes; from the refinancing of firms that stopped receiving public financing to secondary transactions. 8.3.3.4 Distressed Debt Distressed debt refers to financing through the acquisition of senior secured loans in the secondary market with a discount as the borrowing entities are experiencing financial hardship. 8.3.3.5 Infrastructure Debt This is debt employed for new and existing infrastructure management with long-term horizons due to the longer lifespan/duration of the relevant assets. 8.3.3.6 Real Estate Debt Real estate debt entails usually outright loans to finance real estate purchases, as well as securitized real estate lending that takes place in the secondary market. 8.3.3.7 Special Situations These are debt or structured equity investments that aim at financing the acquisition of (control of) a company that usually experiences financial hardship, encompassing the trading in the secondary market of (perceived) mispriced distressed debt. 8.3.3.8 Venture Debt (Business Debt) This is debt that is offered to firms that already enjoy venture capital financing so that entrepreneurs will not have to give away an additional portion of their enterprise.

8

8.3.4

BOND FUNDS

443

Reasons to Invest in Private Debt (Funds)

The main reasons for investing in private debt (funds) can be summarized as follows (Golding, 2019): 1. They entail (positive) spreads compared to government, corporate and high yield bonds. 2. They have low correlation with the traditional assets which facilitates differentiation. 3. They can be used for risk reduction as they exhibit comparatively reduced volatility with stable returns even at different market cycles. 4. They exhibit a proven track record and employ several alternative investment strategies. 8.3.5

The Private Debt Market

Private debt makes up for a significant share of private markets, in the area of 10–15% of all assets under management. It has attracted the interest of investees and investors; several private medium-sized companies have issued private debt. Investors have turned to private debt more aggressively as a result of the low interest rate environment. Furthermore, they perceive it as a diversifying position. The global market has grown significantly as a result of both supply and demand factors (PRI, 2019), as can be seen in Figs. 8.18 and 8.19. The main supply factors are that (PRI, 2019) • Banks offer a very limited number of loans (especially to smaller private companies) as a result of stricter regulatory capital requirements in the EU. • Outright loans (direct lending) kicked in to fill this gap offering to investors returns that are higher than government bond yields or corporate bond yields and in some cases index linked interest. The main demand factors are that (PRI, 2019) • Investors try to achieve excess returns in an extremely low interest rate environment.

444

T. POUFINAS

Fig. 8.18 Relationship between bank and non-bank loans for leveraged loans (Source Created by the author with data assembled from the IMF (2020))

• Investors try to secure differentiation against their positions in traditional asset classes. • Investors and investees are looking for floating interest rate debt so as to mitigate interest rate risk (depending on the expectations of each side). • Investees turned to private debt as a means of financing after the turnaround of the economies, following the 2007–2008 crisis. The total volume of private debt assets managed by institutional investors was approximately $638 billion worldwide in 2017 and reached a record high of $812 billion in 2019. This is more approximately 4 times the volume of assets under management in 2007. There are mainly two private debt markets worldwide, in the US and in the EU, with the former being larger and more mature than the latter (PRI (2019); Preqin (2018a)). Private debt funds attracted about $107 billion in new capital in 2017 worldwide, broken down at $67 billion in the US, $33 billion in Europe and $6 billion in Asia. In other countries (Germany and Nordic countries) - banks continue to have the lead in providing loans for historical and/

8

BOND FUNDS

445

Fig. 8.19 Evolution of the AUM by type of alternative investment (Source Created by the author with data assembled from PRI (2019) and from Preqin (2018b))

or regulatory reasons. After an increase the new capital in private debt returned to comparable levels in 2019. In terms of AUM private debt ranked in 2017 below private equity, hedge funds and real estate, but ahead of infrastructure and natural resources in a $10.31 trillion AUM alternative investment industry. The number of active investors reached 4,155 and the number of fund managers increased to 1,764 (PRI (2019); Preqin (2018a)). The most active players are pension funds, foundations, endowments and insurers (Fig. 8.20) (PRI (2019); Preqin (2018a)). The sectors that are expected to attract investments in the next 12– 24 months (as per the 4th quarter of 2017) are real assets such as infrastructure and commercial real estate, with private equity financing coming up next (Fig. 8.21) (PRI (2019); Intertrust (2018)).

446

T. POUFINAS

Wealth Manager 6%

Private Sector Pension Fund 16%

Family Office 8%

Other 14%

Public Pension Fund 13%

Fund of Funds Manager 6%

Foundation 13%

Asset Manager 6% Insurance Company 9%

Endowment Plan 9%

Fig. 8.20 Private debt analysis by investor type (Source Created by the author with data assembled from PRI (2019) and from Preqin (2018a))

The private debt funds have increased over the years, with a drop since 2017 that seemed to have been the best year (Fig. 8.22). At the beginning 436 funds are raising capital, aiming at $192 billion. The funds raised are primarily in North America, followed by Europe, both in monetary terms and number of funds (Figs. 8.23 and 8.24). Private debt comes of course with risk; investors need to be cautious of that, especially during financial crises that the capacity of the borrowers to repay debt may be affected. However, in the long term private debt seems to be offering returns that are comparable with other asset classes (Fig. 8.25). 8.3.6

Comparison of Private Debt with Private Equity

Private debt offers lending to a company so as to finance existing or new operations and infrastructure. This loan is often collateralized (backed) by an asset of the borrowing firm, such as real estate. Private debt funds

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60%

Percent

50% 40% 30% 20% 10% 0%

Fig. 8.21 Expectations of private debt funds for sectors that will attract their highest levels of investment (Source Created by the author with data assembled from PRI (2019) and from Intertrust (2018)) Number of Funds

Total Capital Targeted (Billion USD)

446

436 399 354

261

265 191

107

118

128

65

66

68

2011

2012

2013

118

121

169

192 168

118

54

2014

2015

2016

2017

2018

2019

2020

Fig. 8.22 Private debt funds evolution (Source Created by the author with data assembled from Preqin (2020a))

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$160.00 $140.00 $120.00 $100.00 $80.00 $60.00 $40.00 $20.00 $.00 2009

2010

2011

2012

North America

2013

2014

2015

2016

Rest of World

Europe

2017

2018

2019

Total

Fig. 8.23 Private debt fund evolution by region (in USD) (Source Created by the author with data assembled from PitchBook (2020)) 180 160 140 120 100 80 60 40 20 0 2009

2010

2011

2012

North America

2013 Europe

2014

2015

2016

Rest of World

2017

2018

2019

Total

Fig. 8.24 Private debt fund evolution by region (in number) (Source Created by the author with data assembled from PitchBook (2020))

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S&P 500

Russell 3000

Russell 2000 Growth

Morningstar US Real Assets

Bloomberg Barclays US Corporate High Yield 17% 16% 15% 14% 13%

11% 10% 9% 9%

9% 8%

12% 11%

8% 7%

7% 6% 5%

5% 5%

4% 3% 2%

1-Year

3-Year

5-Year

10-Year

Fig. 8.25 Private debt fund comparative performance (Source Created by the author with data assembled from PitchBook (2020))

do not target at ownership, whereas private equity funds do (partially or fully). Private debt funds can sometimes be open-end funds, while private equity funds are often closed-end. Private debt fund returns come from the interest on loans, while private equity fund returns come from the rise in the value of the investee companies (Prestige Funds, 2019). 8.3.7

Concerns About Private Debt

Despite the attractiveness of private debt, there are some concerns that stem mainly from the risks involved in investing in it. They can be summarized as follows (Golding, 2019): • There is no guarantee that a specific performance (monetary or percentage) will be achieved. Past returns do not guarantee the future returns. • Non-managing minority shareholders have no or limited effect over the fund manager.

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• The potential use of leverage may improve performance, but leads to higher chances of posting losses. • The market values may be volatile due to macroeconomic and/ or market variable/ determinant movements including market interest rates. • They are not regulated usually and provide minimal or no protection. • Investors are exposed to the tax and regulatory risks emanating from the funds as well as their underlying investments. • Investors are exposed to losses that can be as high as their full investment.

Exercises Exercise 1 Look for an example of private debt financing in the international market. A. Describe how it worked for investors. B. Describe how it worked for the company and why it used this form of financing. C. Describe the final outcome—if it is known, i.e. whether it was considered successful or unsuccessful. D. Explain if you would do something different both on the part of the company and on the part of the investors. Exercise 2 Investments in government bonds are alternative investments. A. True B. False Explain your answer.

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Exercise 3 The reasons to invest in private debt are: I. The II. The III. The IV. The

attractive, stable profit margins low correlation with other categories of assets risk reduction listing of a company in the stock market

A. All of the above B. None of the above C. I, II, III D. II, III, IV Justify your answer. Exercise 4 Types of private debt are: I. Mezzanine debt II. Distressed debt III. Government bonds A. All of the above B. II C. I, II D. None of the above Explain your answer. Exercise 5 There is absolutely no concern for investing in private debt. It is better than any other form of investment. A. True B. False

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Explain your answer.

References Corporate Finance Institute. (2021a). Debt/Bond Fund. https://corporatefinanc einstitute.com/resources/knowledge/trading-investing/debt-bond-fund/. Accessed: January 2021. Corporate Finance Institute. (2021b). Net asset value. https://corporatefin anceinstitute.com/resources/knowledge/finance/net-asset-value/. Accessed: January 2021. ETF. (2020). Fixed income investing ETFs: The basics. https://www.etf.com/etfeducation-center/etf-basics/fixed-income-investing-etfs-the-basics. Accessed: September 2020. ETF database. (2020). https://etfdb.com/. Accessed: September 2020. Fidelity. (2020a). What are bond funds? https://www.fidelity.com/learningcenter/investment-products/mutual-funds/what-are-bond-funds. Accessed: September 2020. Fidelity. (2020b). Types of bond funds. https://www.fidelity.com/learningcenter/investment-products/mutual-funds/types-of-bond-funds. Accessed: September 2020. Fidelity. (2020c). Mutual fund fees and expenses. https://www.fidelity.com/ learning-center/investment-products/mutual-funds/fees-expenses. Accessed: September 2020. Golding. (2019). Private bebt—Security and returns from private lending. https://www.goldingcapital.com/en/investors/private-debt. Accessed: September 2019. IHS Markit. (2017). The rise of private debt. https://ihsmarkit.com/researchanalysis/07082017-In-My-Opinion-The-Rise-of-Private-Debt.html. Accessed: September 2019. IMF—International Monetary Fund. (2020). Global financial stability report: Markets in the time of COVID-19, April 2020. https://www.imf.org/en/ Publications/GFSR/Issues/2020/04/14/global-financial-stability-reportapril-2020. Accessed: February 2021. Intertrust. (2018). Changing tides: Global private debt market in 2018. https:// www.intertrustgroup.com/wp-content/uploads/2021/01/Intertrust-Pri vate-Debt-Report-2018.pdf. Accessed: September 2019. Investment Company Institute. (2020a). Trends in mutual fund investing: July 2020. www.ici.org. Accessed: September 2020. Investment Company Institute. (2020b). 2020 investment company fact book: A review of trends and activities in the investment company industry, 60th edition. https://ici.org/research/stats/factbook. Accessed: September 2020.

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NN Investment Partners. (2017). Alternative credit and its asset classes: A guide to understanding the complex universe of private debt assets. https://assets.ctf assets.net/y4nxuejkhx03/3PYG54Iy5WG5QCV7Efyl5x/f4c28cf7c2da5cbfe6 89a38bbc968f7d/NN_IP_Guidebook_to_Alternative_Credit.pdf Accessed: September 2020. PIMCO. (2016). Understanding investing: Inflation-linked bonds. https://glo bal.pimco.com/en-gbl/resources/education/understanding-inflation-linkedbonds. Accessed: January 2021. PitchBook. (2019). What is private debt? https://pitchbook.com/blog/what-isprivate-debt. Accessed: September 2019. PitchBook. (2020). Global private debt report. https://pitchbook.com/news/ reports/h2-2019-global-private-debt-report. Accessed: September 2020. Prestige Funds. (2019). Private debt. https://www.prestigefunds.com/know/pri vate-debt/. Accessed: September 2019. Preqin. (2018a). Private debt spotlight, 3(3), (March). https://www.pre qin.com/insights/research/reports/private-debt-spotlight-march-2018. Accessed: September 2019. Preqin. (2018b). 2018 Preqin global private debt report—Sample pages. https:// docs.preqin.com/samples/2018-Preqin-Global-Private-Debt-Report-SamplePages.pdf. Accessed: September 2019. Preqin. (2020a). 2019 Private debt fundraising & deals update. https://www. preqin.com/insights/research/factsheets/2019-private-debt-fundraisingdeals-update. Accessed: September 2020. Preqin. (2020b). 2020 Preqin global private debt report. https://www.pre qin.com/insights/global-reports/2020-preqin-global-private-debt-report. Accessed: September 2020. Principles for Responsible Investment—PRI. (2019). Private debt overview. https://www.unpri.org/private-debt/an-overview-of-private-debt/4057.art icle. Accessed: September 2019. S&P Global. (2019). Default, transition and recovery: 2018 Annual global corporate default and rating transitions study. Ratings Direct. www.spratings. com. Accessed: April 4, 2020. State Street. (2018). Fixed income ETFs fact vs. Fiction. https://www.ssga. com/library-content/story/fixed-income/etf/apac/fi-fact-vs-fiction-en.pdf. Accessed: September 2020. U.S. Securities and Exchange Commission. (2020a). Mutual funds. https:// www.investor.gov/introduction-investing/investing-basics/investment-pro ducts/mutual-funds-and-exchange-traded-1. Accessed: September 2020. U.S. Securities and Exchange Commission. (2020b). Mutual fund fees and expenses. https://www.investor.gov/introduction-investing/investing-basics/ glossary/mutual-fund-fees-and-expenses. Accessed: September 2020.

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World Economic Forum—WEF. (2015). Alternative investments 2020—An introduction to alternative investments. http://www3.weforum.org/docs/ WEF_Alternative_Investments_2020_An_Introduction_to_AI.pdf. Accessed: September 2019.

CHAPTER 9

Risks and Risk Management

The discussion that took place in the previous chapter indicated that fixed income investing exposes investors to risks. This probably comes as no surprise, as all types of investments bear risks. However, in the minds of some people bonds and in particular sovereign bonds are considered as risk-free investments, at least in terms of making the promised payments. The recent history, and in particular the 2008 crisis, reminded us that governments may not be able to honor their obligations. Enterprises may not be able to make all the payments of the bonds they issue as well; and in some cases the perception is that they may default easier than governments. The chance of an issuer not making all promised payments is described by the term counterparty or credit risk. Investors may acquire credit protection through the suitable credit derivatives. Even if certain issuers show limited credit risk, there are other forms of risk to which bond investors are exposed to. The most apparent one is interest rate risk, which stems from the change of the interest rates. We explained in chapter 4 how a shift of the interest rate (term structure) may affect the price of a bond. Even if an investor has no intention of selling a coupon-bearing bond and considers that the quality of the bond exposes him or her to limited counterparty risk, he or she is still exposed to reinvestment risk, which is a form of interest rate risk. This is due to the fact that he or she cannot secure (unless he or she undertakes a position in the appropriate derivative) that the interest rate at which © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_9

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the bond coupons will be reinvested will be identical to the one that he or she enjoys at the initiation of his or her investment. Moreover, if the investor marks-to-market his or her portfolio or does not pursue a buy-and-hold strategy, then he or she is vulnerable to the bond price changes that are caused by the interest rate changes – unless he or she has acquired protection through the appropriate derivatives. Interest rate risk has two sources; the risk-free interest rate and the spread. Both of them may change and cause the bond price to move; the former as a result of a shift of the risk-free interest rate term structure and the latter as an outcome of a change in the credit rating of an issuer or of the overall reward that investors require for undertaking the specific level of credit risk. Liquidity—or rather the lack of it—is an additional source of risk that may be present in bonds, especially corporate bonds. (Il)liquidity risk pertains to the inability that an investor may face in acquiring or giving away a bond or other fixed income security. It is in particular the latter that is considered mostly unpleasant or undesired. Especially when an investor pursues a forced sale, he or she may realize that selling a bond may not be feasible; or if accomplished, it could result either in a very long time interval required to complete the sale or in a significant loss to finish it off within a specified period. Analogous situations may arise when attempting to buy a bond, when required to follow a benchmark or fill in an order; the time required may be too long or the purchase price may be too high. Corporate bonds or private debt issues may exhibit increased illiquidity. Although this may result in decreased volatility, as the bond price does not frequently change, it could lead to inability to sell the corresponding security, i.e. expose the investor to liquidity risk. Investors seem to expose themselves more and more to liquidity risk as in the quest of performance they turn to corporate issues and private debt. The illiquidity of the former is further intensified by the quantitative easing or corporate bond purchase programs applied by the central banks in order to support the economies and the markets in an era of a pandemic. Such programs were enforced also in the recent past; the trigger may have been different, but the outcome was similar. Such actions have been and are pursued by the ECB, the Bank of England (ICMA, 2020), and the FED (Federal Reserve, 2020a). The latter even extended the corporate sector purchases to ETFs containing corporate issues (New York Fed, 2020) as well as to fallen angels (Federal Reserve, 2020b), i.e. bonds that moved

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from investment grade to non-investment grade ratings as a result of the strain they endured due to the pandemic. Currency may put at stake the value of a bond portfolio as its fluctuations could result in changes in the fixed income security prices. Currency or FX (foreign exchange) risk is present when a portfolio contains positions that are valued at a currency different from the one that the portfolio is valued. Consequently, its value may change when there are shifts in the exchange rate of the portfolio and the instrument valuation currencies. Investors that do not consider foreign exchange as a source of income but see it only as a source or risk can protect against it with the use of the appropriate derivatives. Investors, insurers and bankers though make their profits by assuming risks. Consequently, it is to their interest to measure and manage these risks so that their assumption is well-informed and prudent. As a matter of fact two questions are relevant; namely, “Is the reward worth taking the risk?” and “What is the level of reward that would make the risk worth taking?” Answering these questions is of the utmost importance in risk management. It determines also the means that investors may use to mitigate the risks that come from fixed income instruments or absorb the subsequent losses would they occur. The former refer to management actions as well as potential derivatives or other protection means that can be used. The latter relate to the capital that can act as cushion in case the actions enforced do not prevent the risk from happening. One should not forget that besides bonds and other fixed income securities it is also interest rate derivatives that bear interest rate risk. This chapter analyzes deeper the risks that are generated from fixed income instruments, such as interest rate risk, credit risk and other potential risks, such as liquidity risk, and offers ways to manage them. It introduces measures such as Value-at-Risk and explains how they can be used to measure and manage the relevant risks. After having read this chapter the reader will be able to understand better the sources of risks which appear in a bond portfolio, as well as the ways to measure and manage them.

9.1

Interest Rate Risk

Interest rate risk is a form of market risk, i.e. the risk that is due to changes in the values of traded instruments. As mentioned it is distinguished in risk-free interest rate risk and credit spread risk, depending on

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its source. The former stems from changes in the risk-free interest rate term structure; the latter emanates from moves in the credit spread. There is a series of approaches that can be used to quantify the impact of interest rate related risk factors, i.e. risk-free interest rate and credit spread risks, to the value of fixed income securities/bonds. The most frequently used ones are sensitivity analysis, stress testing, scenario testing and value at risk (Marrison, 2002). In using them, one can recall that there are two ways to measure the price change of a bond that results from an interest rate shift; an analytical (non-linear) pricing, with the use of the present value formula that gives the price of a bond or an approximate (linear) pricing with the use of duration. The applicable regulatory framework for financial institutions and insurance organizations provisions for periodic, as well as ad hoc, stresses or scenarios in order to investigate the viability of these entities in adverse economic conditions. This is performed (also) by testing the adequacy of their capital to absorb the resulting losses. 9.1.1

Sensitivity Analysis

Sensitivity analysis is an approach that depicts the expected change in the value of a fixed income portfolio when a small change (perturbation) in any of the risk factors under investigation takes place, i.e. the risk free interest rate and the credit spread. The idea of sensitivity is based on the very definition of the first derivative of a function, which is essentially a measure of its change with respect to a perturbation of its argument (variable). For a bond (fixed income) portfolio with value  B that depends on a risk factor φ (usually risk-free rate or spread), sensitivity can be measured with one of the following three expressions (Marrison, 2002): ∂:= %B :=

B (φ+ε)−B (φ) ∂B = lim ∂φ ε→0 ε

(9.1)

B B (φ + ε) − B (φ) B = = , φ ε ε

(9.2)

B (φ + ε) = B (φ) + ε × ∂,

(9.3)

where ε or φ is a small shift of the factor φ. Equations (9.1), (9.2), and (9.3) are equivalent, considering that for a small change of φ the first derivative is approximately equal to the ratio of the change of the

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portfolio value over the change of the factor, i.e. ∂ ≈ %B for a small ε. Equation (9.3) is derived from the other two with ∂ being essentially the first derivative or its ratio of changes approximation. In the case of bonds, there are two risk factors; hence we are looking for the differential of the bond portfolio value B : B =

∂B ∂B s, r f + ∂r f ∂s

(9.4)

where the interest rate (or yield) is given by r = r f + s = y,

(9.5)

for rf s

the risk-free interest rate. the credit spread.

For bonds, the first derivative (Eq. 9.1) practically leads to the use of duration, as the latter is defined via the first derivative of the bond price (value) with respect to interest rate – as seen in Chapter 4. A point of attention is that duration can be applied by definition only when the interest rate change is small and remains the same for all time instants in order to find the price change (Eq. 9.2) or the new price (Eq. 9.3). Consequently, sensitivity analysis uses a linear approximation to assess the risk. 9.1.2

Stress Testing

Sensitivity analysis works for small shifts of the risk-free rate or the spread. If any of these factors experiences a bigger change – as could be in the case of a crisis, then the linear approximation employed by sensitivity analysis will not depict the change of the value of the fixed income portfolio as accurately. To overcome this limitation we may employ stress testing for bigger changes in the risk factors. This approach does not rely on (linear) approximations; instead it uses the actual analytical present value (hence non-linear) bond pricing formula to calculate the value of the fixed income portfolio and the loss that results from an interest rate (risk-free rate or spread) shift. As the plain vanilla present value formula is used and the interest rate (risk-free rate or spread) shifts are standardized, stress

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testing is relatively straightforward to apply and comprehend (Marrison, 2002). Example 9.1 One can easily understand the following findings/statements: • An interest rate increase of 2%, will lead to a portfolio loss of EUR10 million. • An interest rate increase of 4%, will lead to a portfolio loss of EUR 18 million.

Example 9.2 Interest rate changes applied in stress testing could be set at −10%, −5%, 5%, 10% of the level of interest rate (or risk-free interest rate or spread). Alternatively a parallel shift of −10 bps, −5 bps, 5 bps, 10 bps, etc. may be applied. When performing stress tests we need to decide a priori whether there are any factors that should shift en bloc. This is often referred to as “blocking” and facilitates the evaluation of the resulting loss. For example we could assume that all Eurozone or all European Union interest rates move at the same time and by the same percentage rather than separately for each country. The same could be applied at exchange rates, which we will visit in Sect. 9.3 of this chapter. The drawback of blocking is that there is no visibility any more on the loss that would occur if interest rates have shifted by a different percentage. Blocking may counterbalance gains and losses that occur from the different interest rates that have been shifted en bloc (Marrison, 2002). Please note that blocking could be applied to the underlying risk-free interest rates or to the credit spreads. The former could be the same for all bonds anyway. For example, in the case of the Eurozone government bonds a common risk-free interest rate may be used. 9.1.2.1 Stress Testing Steps There are certain steps that we usually follow in order to perform stress testing. The main ones are (Marrison, 2002):

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1. Select the risk factors (interest rate, spread, currency, etc.) that determine the value of the fixed income portfolio under investigation. 2. Choose the factors that move en bloc and those that do not (e.g. all Eurozone interest rates for a bank located outside Europe). 3. Choose the shift that will be applied per factor (e.g. perturb by 2–3 or 4–6 times the standard deviation or a standard percentage shift of the interest rate term structure). 4. Find the new value of the fixed income securities and thus of the portfolio with the use of the analytical and not approximate formulas (e.g. for bonds use the present value formula and not duration). 5. Record the change in the value of the portfolio as a result of the change in the risk factor (e.g. risk-free rate or spread), with emphasis on the loss generated. 6. Repeat on a periodic basis, potentially updating the stresses tested. 9.1.2.2 Stress Testing Limitations Stress testing exhibits a series of limitations. The main ones are (Marrison, 2002): 1. The outputs are not ranked in terms of the significance they may have for the investor. 2. There is no probability assigned to each of the shifts applied. 3. Considering that certain factors change en bloc whereas some other factors do not is equivalent to assuming that factors have a correlation of 1 or 0 respectively. This is not necessarily the case as the correlation of any two factors may take in reality a value between −1 and 1. In addition, this could cover up losses that would have been posted if some factors had not changed identically. Example 9.3 Assume that a US bank holds a long position in the German Bund and a short position in Greek Government Bonds. If their yields are blocked together then when different interest rates are perturbed it could be that gains from one position counterbalance the losses of the other, which would possibly mean that there is no risk. However, if we unbundle them,

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then the result could be very different; of course our concern is about posting higher overall losses.

9.1.3

Scenario Testing

Scenario testing provides an approach to address the concerns of stress testing. It resembles to stress testing as they both use the explicit valuation (or pricing) formulas to calculate the new value of the fixed income portfolio and the resulting loss that are generated by risk factor shifts. In this case though the shifts are not standardized; they are rather determined in a way that reflects the idiosyncrasies of the portfolio under valuation. To capture that, a set of scenarios that are considered/perceived worst for the portfolio are tested. These scenarios can be single-digit in number and may be generated by the risk officers and the investment managers or replicate previous crises that could (or are feared to) cause a material loss in the fixed income portfolio. When testing the previous crises, the risk factor moves that took place during the crisis are projected into present terms, and their impact in the portfolio value is studied (Marrison, 2002). Examples of such scenarios may be (the list is not inclusive but rather indicative): • The 1987 Black Monday, when a 20% decrease occurred in the US within 1 day. • The Russian financial crisis in 2008. • The Lehman Brothers default in 2008. • The Greek financial crisis in 2010. • The oil prices drop in May 2010 • The Greek financial crisis in 2015. • Brexit in 2016. When using the current portfolio, along with the opinion of the investment managers or risk officers (or any other party with expertise on the field) the scenarios that are expected to result in the highest potential losses are tested. For each such scenario the change caused in the portfolio value by the move of the risk factors (which has to be determined first) is estimated (Marrison, 2002). Examples of such expert derived scenarios could be (the list is indicative):

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Equities down by 10% (or more or even a crash). A volatility increase by 20% (or more). US 10 yr treasury yields up by 100 bps (or more). Interest rates up by 100bps (or more). US Curve steepening. Euro redenomination risk. Euro up 10% vs. USD.

9.1.3.1 Scenario Testing Steps There are certain steps that we usually follow in order to perform scenario testing. These are (Marrison, 2002): 1. Select an array of adverse scenarios that could distress the fixed income markets. 2. Select the risk factors (interest rate, spread, currency, etc.) that determine the value of the fixed income portfolio under investigation and assess the shift caused to each of the risk factors per scenario tested. 3. Estimate changes in each risk factor based on the crisis scenarios identified (based on expert opinion and historical data). 4. Find the new value of the fixed income securities and thus of the portfolio with the use of the analytical and not approximate formulas (e.g. for bonds use the present value formula and not duration). 5. Record the change in the value of the portfolio as a result of the change in the risk factor (e.g. risk-free rate or spread), with emphasis on the loss generated. 6. Repeat on a periodic basis, potentially updating the scenarios tested. 9.1.3.2 Scenario Testing Limitations Scenario testing exhibits a series of limitations. The main ones are (Marrison, 2002): 1. The tests may require a significant amount of time. 2. The number of the scenarios tested is relatively small. 3. The choice of scenarios and the shift of the risk factors are custommade. 4. The scenarios may be proposed by the portfolio managers, in which case there may be a bias.

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5. There is no probability assigned to each of the scenarios tested. 9.1.4

Value at Risk

The previous three approaches fail to capture the probability of occurrence of a scenario or a risk factor shift. In order to properly encompass the probability of a change in a risk factor and the implied change in the portfolio value we employ a measure called Value at Risk (VaR). It reflects the expected loss in the fixed income portfolio value as a consequence of extreme risk factor changes. It is possibly the best risk metric that can be used on its own in order to measure the loss that is produced by such changes (Marrison, 2002). Investors (and risk officers) are interested in extreme losses, i.e. losses that have a small probability p (let’s say 1%) of occurring during a trading day. This means that on average the portfolio will lose at least the amount of VaR on p times 250 days per year (assuming that there are approximately 250 trading days in a calendar year). This translates to 2–3 days per annum for p equal to 1% (as 1% times 250 equals 2.5, which lies between 2 and 3). For p equal to 2.5% then 6–7 days per annum on average the portfolio will lose at least the amount of VaR (as 2.5% times 250 equals 6.25). VaR can be calculated for each fixed income security and portfolio, as well as for any financial institution (e.g. a bank, an insurance company, a pension fund or an investment firm) (Marrison, 2002). In calculating VaR we assume that the risk factor changes follow a normal distribution. As a consequence, 1− p VaRT :=z p

× σT ,

(9.6)

where zp

the z-score that corresponds to probability p.

We formally produce the equation that gives zp in Eq. (9.6) that follows when we elaborate on the general definition of VaR (later in this chapter). Recall though from the particulars of the standard normal distribution that if p is equal to 1% (or 2.5%), then zp is equal to 2.32 (or 1.96 respectively); hence, there is a probability of 1% (or 2.5%) that the portfolio will lose more than 2.32 (or 1.96 respectively) standard deviations.

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Probability

probability p

Portfolio Value 1-pVaR=zpσ

Fig. 9.1 1−p Value at Risk (Source Created by the author with information assembled from Marrison [2002])

VaR is defined for any investment horizon T. When T is equal to one day then VaR is often referred to as DEaR (Daily Earnings at Risk) (Fig. 9.1), i.e. VaR1

- day :

= DEaR

(9.7)

As the loss from the bond price change is due to a move of the corresponding interest rate (yield), the VaR of a fixed income portfolio  B that corresponds to the interest rate move that has a small probability p of occurring is given approximately by (Marrisson, 2002):   dB × r, (9.8) 1− p V a R ≈ − dr where r

denotes the daily interest rate move that has a probability p of occurring.

Assuming that the interest rate changes follow a normal distribution (Marrison, 2002), Eq. (9.8) becomes (Marrison, 2002)   dB × z p × σr , Va R ≈ − (9.9) 1− p dr where

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σr

denotes the volatility (standard deviation) of the daily interest rate move.

Using Eq. (4.16) of Chapter 4, Eq. (9.8) becomes 1− p Va R

≈ (Dm × B ) × r ,

(9.10)

and thus Eq. (9.9) changes into 1− p Va R

≈ (Dm × B ) × z p × σr

(9.11)

The product of modified duration with the bond portfolio value is frequently referred to as money duration (or dollar duration), let it be MoD i.e. MoD =Dm × B

(9.12)

Equations (9.8) and (9.11) may be expressed as ≈ (MoD) × r

(9.13)

≈ (MoD) × z p × σr

(9.14)

1− p Va R

and 1− p Va R

respectively. This means that the 99% VaR (or 97.5% VaR) of a bond portfolio can be estimated by multiplying the money duration of the portfolio by 2.32 (or 1.96) standard deviations (see also Marrison, 2002). Example 9.4 Let us consider a fixed income portfolio consisting of a single bond, whose duration is 10 years, its current price is EUR 10,000 and the daily standard deviation of the actual level of interest rates is 0.1 percentage points. Assume that changes in the interest rate are normally distributed and that the entire interest rate term structure shifts in a parallel manner. This means that we can employ duration in order to estimate the change in the bond price. Consequently, the 99% and 97.5% Value at Risk is given by 99%

VAR ≈ $10,000 × 10 × 0.1% × 2.32 = EUR 232,

(9.15.1)

9

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VAR ≈ $10,000 × 10 × 0.1% × 1.96 = EUR 196

467

(9.15.2)

The difference is attributed to the fact that as the probability p decreases the level of loss is harder to reach; however it becomes higher. As mentioned earlier, VaR is calculated over any time horizon; however in our previous discussion we primarily referred on the loss that the fixed income portfolio may experience on a day. We can employ Eq. (9.7) to move from the daily VaR to a horizon T VaR (Marrison, 2002): √ √ (9.16) Va R T =V a R 1-day · T =D Ea R T This equation can be readily produced by the properties of the standard deviation provided that certain assumptions hold true: • The shifts in the applicable risk factors follow a normal distribution. • VaR 1-day remains unchanged for the entire time interval [0, T]. • The daily shifts of any two consecutive days are independent random variables (i.e. there is no serial correlation). 9.1.4.1 VaR Limitations Attention needs to be paid to the fact that VaR does not correspond to the highest loss that the fixed income portfolio may post. Instead, it reflects losses that are expected to be recorded a few times within a year. In this respect it captures big losses that may occur a small number of times (p times 250) a year but does not capture gigantic losses that may occur one time every ten years. Consequently, it is not identical to the capital that serves as a cushion to absorb potential losses; nevertheless, it may be used to derive this capital. Furthermore, in order to still account for extremely high losses stress testing or scenario testing are more suitable approaches (Marrison, 2002). In this respect, as we will see in the VaR calculation approaches, VaR relies to a certain extend (depending on the method) on historical data. Consequently, as the interest is to depict less frequent but adverse events (ideally even catastrophic), the inclusion or omission of such events may increase or decrease VaR even if the composition and/or the risk profile of the portfolio remains unchanged. To illustrate, one can observe that the inclusion of older data could lead to the inclusion of older crises in the VaR calculation that impact it negatively without reflecting the actual

468

T. POUFINAS

risk level of the portfolio, when a significant amount of time has passed since the crisis. The same holds true when entering a crisis after a period of euphoria; the older data tend to impact VaR positively. 9.1.5

VaR Calculation

Of course the natural next question is how to calculate VaR. Contrary to the order of presentation we followed for the other three approaches we dedicate a separate section to the calculation of VaR. This is due to the fact that VaR is the measure that encompasses not only the potential loss but also the associated probability. There are 3 methods that are commonly used in order to calculate VaR: parametric VaR, historical simulation and Monte Carlo simulation. All three methods share some common characteristics and limitations. More precisely, for the case of fixed income securities, instruments and portfolio all approaches: (i) employ the same risk factors (interest rates – risk-free and spread, FX rates, forward prices and implied volatilities); (ii) rely on historical data to derive the probability distributions; (iii) take for granted that the correlation of any two risk factors remains unchanged; and (iv) group/aggregate/accumulate payments (cash flows) on specific time instants—often referred to as binning or grouping or mapping (Marrison, 2002). The particulars are presented below in a tabular form (Table 9.1): Among the aforementioned attributes, it is probably binning that needs to be elaborated more; we do that with the use of an example. Payments are usually clustered at 3, 6, 12, and18 months and 2, 5, 7, 10, 15 and 20 years (and may extend to 25, 30, 35, 40 and 50 years); these are the most common maturities of T-Bills and bonds/notes. The effort is to maintain the present values and durations (and consequently the estimated stand-alone VaR amounts) of the initial payments (Marrison, 2002; EIOPA, 2021). Example 9.5 Let us consider a zero-coupon bond that pays EUR 100 in 6 years. This can be mapped as a zero-coupon bond that pays EUR 75 in 5 years and a zero coupon bond that pays EUR 25 in 10 years. Assume an interest rate of 5%. Observe that the present values (prices) as well as the durations of the single bond and the two bonds that were produced as a result of mapping are close to each other. Indeed, calculating the present

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Table 9.1 Properties of VaR Calculation Approaches Description

Advantage (A)/ Disadvantage (D)

Application

Risk factors

Select a relatively small number of risk factors in order to value all assets in the portfolio

A: Smaller amount of data compared to historical price data D: Potentially reduced accuracy

Probability distributions

Historical data is used to derive probability distributions, variances and correlations

A: Availability of data and ease in calculations D: Uncertainty of how much back in time to go and inclusion (or negligence) of less frequent adverse events

Correlation

Remains the same for any two risk factors

A: Simplicity in calculations D: Past relations do not guarantee future relations

Binning

Payments are grouped/ aggregated/ accumulated into specific time instants

A: Simplicity in calculations D: The initial payment time instants are ignored

Use a few points of the interest rate term-structure to find the value of a much bigger number of securities Inclusion (omission) of older data could lead to the inclusion (omission) of older crises in the VaR calculation that impact it negatively (positively) without reflecting the actual risk level of the portfolio Factors with correlation of 0 (or 1) in the past will have a correlation of (0 or 1) in the future The frequent cash flows (coupon or face value) of a broad bond portfolio are replaced with consolidated payments on fewer instants

Source Created by the Author with information assembled from Marrison [2002]

values Single

PV0

PVBin 0 =

=

100 = 74.62, 1.056

75 25 + = 74.11, 1.055 1.0510

(9.17) (9.18)

470

T. POUFINAS

we realize that they differ only by a small amount, corresponding to a percentage difference of 0.69%. Estimating the durations as Single

= 6,

(9.19)

  75 1 25 =5.99, · 5· + 10 · 74.11 1.0510 1.055

(9.20)

D0 DBin 0 =

we see that they are even closer (almost equal). As a result the VaR of the single zero-coupon bond will be almost identical with the VaR of the portfolio that consists of the two zero-coupon bonds that were generated through binning. To find the exact equivalent cash flow amounts (bond face values) when performing binning, we practically need to solve a system of equations. More precisely Single

PV0 Single

D0

Bin,1

= D0

Bin,1

= PV0 Bin,1

×

PV0 Bin,1

PV0

Bin,2

+ PV0

Bin,2

Bin,2

+ PV0

+ D0

,

(9.21) Bin,2

×

PV0 Bin,1

PV0

Bin,2

+ PV0

(9.22)

This is a system of equations with two unknowns; the present values of the two cash flow amounts (bond face values) that are equivalent to the initial cash flow (bond face value). The denominator of the second equation is nothing but the present value of the initial amount, which is given by the first equation of our system. Please note that the subscript 0 is not necessarily needed in the notation of duration; however we use it so as to make clear that this process could be repeated at any other time. As soon as we have found the present values of the two equivalent separate bonds we calculate their face values as Bin,1

= PV0

Bin,2

= PV0

FVT1 FVT2

Bin,1

× (1+r )T1 ,

(9.23)

Bin,2

× (1+r )T2 ,

(9.24)

where T1 and T2 are the maturity dates of the two bonds or the times at which cash flow payments are grouped; r denotes the interest rate

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which does not have to be flat. If we use a non-horizontal term structure though, then we need to employ the appropriate variants of duration as discussed in Chapter 4. Of course, as our bonds are zero-coupon, the outcome will not change. Furthermore, compounding does not have to be annual; the aforementioned equations can change as discussed in Chapters 2, , 3 and 4 to account for different compounding frequencies. All three methods are assessed in terms of the computation speed, the non-linearity capturability, the non-normality capturability and the historical data independence. The first and the last are probably self-explanatory. The second reflects the capacity of the approach to capture the nonlinearity of the (relations between) fixed income security prices and/or risk factors (and their changes). This is important in the case of fixed income instruments as if a method uses duration, then it misses the nonlinearity of the present value formula (in case of coupon-bearing bonds for example). This is the case when parametric VaR is employed. If the present value formula is used, then the method expresses non-linearity, as is the case for historical and Monte Carlo simulations. The third pertains to the capacity of the technique to capture the non-normality of the distributions followed by fixed income security prices and/or risk factors (and their changes). Parametric and Monte Carlo VaR assume that the change of a risk factor (e.g. the interest rate) follows a normal distribution; historical VaR does not. The attributes of the different approaches are summarized pictorially in the following figures 9.2a and 9.2b (Marrison, 2002). 9.1.5.1 Parametric VaR Parametric VaR - and hence the name – hypothesizes that the probability distribution followed by the change of the applicable risk factors, i.e. the risk-free rate, the spread, etc., is normal. Under parametric VaR, the price change of a fixed income security is a linear function of the change in the risk factor (risk-free rate or spread) that caused it. If the fixed income security was a bond, then this linear relation is expressed by duration (Marrison, 2002). The steps for the calculation of parametric VaR are (Marrison, 2002): 1. Identify the risk factors that affect the value of the fixed income portfolio under investigation – usually the risk-free rate, the spread and the FX rate.

472

T. POUFINAS

2. Estimate the first derivative of each fixed income security in the portfolio to each risk factor that has been identified as relevant – being a measure of sensitivity. 3. Calculate the standard deviation of the changes and the correlations of any two of them with the use of risk-factor historical data. 4. Compute the standard deviation of the (change in) the fixed income portfolio value as the product of the first derivatives with the standard deviations and/or all correlations where applicable. 5. Approximate the 1−p VaR as z p times the standard deviation of the value of the fixed income portfolio under the hypothesis that the loss follows a normal distribution. Parametric VaR is faster than Historical and Monte Carlo Simulation (Fig. 9.2a). VaR Contribution can be analytically found, as will be explained later in this chapter. Nevertheless it presents certain limitations, as parametric VaR fails to sufficiently capture non-linearity (Fig. 9.2b). It fails to capture non-normality (Fig. 9.2b) and thus events that have a low frequency of occurring (and are at the tail of the distribution). Recall that we faced a similar concern when we examined the appropriateness of the normal distribution to replicate the change of interest rates when we modeled the move of interest rates in Chapter 3. To overcome this constrain investors may opt to use historical simulation, which is discussed next. Furthermore, it considers the correlations between risk factors as unchanged with the passage of time (Marrison, 2002). To understand the calculation of parametric VaR for a fixed income portfolio, we start with an example on a zero-coupon bond. Example 9.6 Let us consider a zero-coupon Euro-denominated government bond held by a financial institution domiciled at a country-member of the Eurozone. If it matures in T years and the interest rate is r, then its price is given by (we drop the index 0 from the bond price at time t = 0) P=

FV (1+r )T

,

(9.25)

where FV is the face value of the bond. The sensitivity of the price (value) of the bond to changes in interest rates is measured by the first derivative,

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a

Computation speed

H Parametric VaR M

Historical VaR Monte Carlo VaR

L L

M Historical data independence

H

b Non-linearity capturability

H Monte Carlo VaR

Historical VaR

M

Parametric VaR L L

M Non-normality capturability

H

Fig. 9.2 Comparison of the VaR calculation methods—Part I and Part II (Source Created by the author with information assembled from Marrison [2002])

which is duration dollars or money duration, i.e. ∂ P −T × FV = ≡ ∂r ∂r (1+r )T + 1

(9.26)

474

T. POUFINAS

The change in the value is thus approximated by the sensitivity multiplied by the change in interest rates: P=∂r × r

(9.27)

σ P2 =(∂r × σr )2 ⇒ σ P =|∂r | × σr ,

(9.28)

This means that

and thus 1− p Va R

=z p × |∂r | × σr

(9.29)

The absolute value is introduced because ∂r is negative. For a bond face value of EUR 100 and a maturity of 6 years (as in Example 9.6), an interest rate of 5% and a standard deviation of the interest rate of 0.5% the aforementioned equations yield that: P =

100 = 74.62, 1.056

(9.30)

∂ P −6 × 100 = = −426.41 ≡ ∂r , ∂r (1.05)7

(9.31)

σ P = 426.41 × 0.5% = 2.13,

(9.32)

99% VaR

= 2.32 × 426.41 × 0.5% = 4.95

(9.33)

9.1.5.2 Historical VaR Historical VaR is probably the easiest of the three methods to apply and comprehend. Its competitive disadvantage versus parametric VaR is that it is slower (Fig. 9.2a). On the other hand though, it does not employ a specific probability distribution - and in particular the normal distribution (Fig. 9.2b). Furthermore, it prices the fixed income instruments analytically and not with linear approximations (Fig. 9.2b). However, it relies heavily and explicitly on historical data (Fig. 9.2a); thus, the inclusion or omission of extreme events (such as crises) may significantly impact its output, even though the risk profile of the fixed income portfolio may have remained unaltered. It uses historical data for the desired number of

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days (Marrison, 2002). We usually go so many days back in time as our time horizon is. The steps to estimate historical VaR are (Marrison, 2002): 1. For a time horizon of H trading days we employ historical data of the previous H days. 2. We compute the daily percentage change for each risk factor (e.g. risk-free rate or spread) for each of these H days. 3. We multiply the present day risk factor values with each of these daily percentage changes to produce H scenarios for the risk factor values of the following day. 4. We value the fixed income portfolio for each of these scenarios with the use of analytical formulas, i.e. the present value formula for bonds and not the linear approximation produced by duration. 5. We rank the losses in a decreasing order (highest loss ranks first, lowest loss ranks last). 6. We estimate the 1-p VaR as the loss that corresponds to the p-times-H worst scenario. Example 9.7 Assume that our time horizon is one year or 250 trading days (approximately). Recall that when calculating the 99% VaR (or 97.5% VaR), we are essentially finding the loss that we could reach in 2–3 (or 6–7) trading days in a year. This is due to the fact that 1% (or 2.5%) of 250 days is 2.5 (or 6.25) days, i.e. between 2 and 3 (or 6 and 7) days. The 3rd (or 7th) worst day is therefore the 99% VaR (or 97.5% VaR). Let us work once and again with our zero-coupon Euro-denominated bond that pays 100 Euro in 6 years. The interest rate today is taken to be 5%. We wish to calculate the Historical VaR of the bond. To do that we realize that 5% is the 6-year yield to maturity (or interest rate) as taken from the interest-rate term structure (zero-coupon yield curve) that applies to the credit rating of the bond. We observe the 6-year interest rate for each day over the chosen period of time. For the sake of this example we produce 250 interest rates ourselves through a pseudorandom number generator. We then calculate the percentage change of the interest rate per day for 250 days (i.e. the trading days in one calendar

476

T. POUFINAS

year) (see also Marrison, 2002): rd =

rd+1 −rd rd

(9.34)

We then apply each percentage change to 5% (the current 6-year yield, denoted by r 0 ) and produce 250 interest rate (yield) scenarios. Please note that in Table 9.2 below we start from today, hence each prior day is counted going back in time and thus carries a negative sign.   (9.35) rdhistorical =r0 · 1+r−250+(d - 1) Each scenario (historical day) d, for d = 1…250, corresponds to the (250−(d−1))th day before today. For each scenario we estimate the bond price (value really as it is not the market price), as Pdhistorical = 

100

6 1+rdhistorical

(9.36)

After that we find the bond price (value) change by taking its difference from today’s price (Table 9.2). Pd =Pdhistorical −P0

(9.37)

To estimate 99% VaR (or 97.5% VaR) we look for the 3rd (or 7th) worst day in terms of value change of the bond. To find it we rank the losses in a decreasing order rank (or the price change in an increasing order). As can be seen from Table 9.2 this corresponds to a loss of 3.72 Euro (or 3.51 Euro respectively). 9.1.5.3 Monte Carlo VaR Monte Carlo VaR is calculated via a series of scenarios that (pseudo) randomly produce (the change of) the risk factors (e.g. risk-free rate and spread) and from this the subsequent fixed income portfolio value (change). Monte Carlo VaR employs analytical valuation formulas (e.g. the present value for bonds) and not linear approximations (e.g. the duration for bonds) – resembling to historical VaR. It therefore depicts non-linearity (Fig. 9.2b). It can deliver though – by construction - any desired number of scenarios. However, this leads in slower computation (Fig. 9.2a). According to Monte Carlo VaR the risk factors follow a

9

Table 9.2 Historical VaR Calculation

Source Created by the Author

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477

478

T. POUFINAS

normal or lognormal distribution; hence it fails to capture non-normality (Fig. 9.2b). More specifically, for one risk factor, such as the interest rate (change), the hypothesis is that it follows a normal distribution; for more than one risk factors, that they move by a multivariate normal distribution, as does parametric VaR (Marrison, 2002). The steps to estimate Monte Carlo VaR are similar to those of historical VaR, with the exception that historical data are replaced with draws from a standard normal distribution: 1. We use a pseudo-random number generator that produces draws from a normal distribution with mean 0 and standard deviation 1. 2. We compute H (percentage change) scenarios for each risk factor (e.g. risk-free rate or spread). 3. We multiply the present day risk factor values with each of these daily percentage changes to produce H scenarios for the risk factor values of the following day. 4. We value the fixed income portfolio for each of these scenarios with the use of analytical formulas, i.e. the present value formula for bonds and not the linear approximation produced by duration. 5. We rank the losses in a decreasing order (highest loss ranks first, lowest loss ranks last). 6. We estimate the 1-p VaR as the loss that corresponds to the p-times-H worst scenario or p-th percentile worst loss. Example 9.8 We use for one more time our zero-coupon Euro-denominated bond that pays 100 Euro in 6 years. The interest rate today is taken to be 5%. We wish to calculate the Monte Carlo VaR of the bond with the use of 1,000 scenarios. We work in a way similar to historical VaR in order to produce 1,000 scenarios for the interest rate change. However, in this case we use a pseudo-random number generator that produces draws from a normal distribution with mean 0 and standard deviation 1. Then we produce 1,000 scenarios of interest rates with the use of the formula r =r0 +σr · z=5%+0.5% · z, z ∼ N(0; 1)

(9.38)

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We then calculate the price of the bond with each of these interest rates with the use of formula (36) and estimate the price change through Eq. (9.37) versus the initial price of 74.62 Euro. The difference is that the interest rates of the 1,000 scenarios are derived via Eq. (9.38) and not from historical rates. In this case the superscript ‘historical’ may need to be replaced with ‘Monte Carlo’ to reflect the method change; the number of days (250) needs to be replaced by the number of scenarios as well. The 99% VaR (or 97.5% VaR) is chosen to be the 1-percentile (or 2.5 percentile) worst loss. For 1,000 scenarios this is the 10th (or 25th) worst outcome. If 10,000 scenarios are run then the 99% VaR (or 97.5% VaR) would be the 100th (or 250th) worst outcome. We find that the 10th (or 25th) worst scenario yields a loss of 4.77 Euro (or 4.04 Euro), which is the 99% VaR (or 97.5% VaR), as revealed in Table 9.3.

9.2

Foreign Exchange Risk

Foreign exchange risk is a common type of risk in fixed income securities. Even a deposit in a foreign currency constitutes a fixed income investment that is exposed to foreign exchange risk. Let alone a bond that is issued in a foreign currency relative to the valuation currency of the investor. As a result it is important to quantify the incremental loss it may cause next to the interest rate as a risk factor. We employ once and again VaR and attempt to estimate it when foreign exchange risk is present. 9.2.1

Parametric VaR

Example 9.9 (Example 9.6 Continued) Assume now that the bond of Example 9.6 is owned by a US-based financial institution. As the bank values its portfolio in dollars it is exposed to foreign exchange risk on top of the interest rate risk. The price of the bond in US dollars is given by (we drop the subscript 0 that corresponds to the present time t = 0) (see also Marrison, 2002): P $= S × P e = S ×

FV (1+r )T

(9.39)

480

T. POUFINAS

Table 9.3 Monte Carlo VaR Calculation Scenario 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Interest Rate (%) 4,93% 4,70% 5,07% 4,98% 4,96% 5,57% 5,26% 5,64% 5,13% 5,78% 4,81% 4,49% 5,08% 5,02% 5,20% 3,94% 4,45% 5,34% 5,95% 5,31% 4,95% 5,14% 5,36% 4,55% 4,54% 5,10% 5,33% 5,22% 5,69% 4,87% 4,78% 5,20% 5,05% 4,79% 5,19% 4,68% 5,71% 5,62% 5,77% 6,02% 5,23% 5,01% 5,09% 4,84% 4,36% 5,46% 4,66% 5,20% 4,08% 4,74% 4,37% 5,06% 4,43% 4,99% 5,09% 4,77% 4,06% 4,34% 6,17% 4,96% 4,12% 4,78% 4,41% 4,67% 4,58% 4,42% 5,27% 4,38% 4,67% 4,75% 4,74% 4,66% 5,37% 5,05% 4,71% 5,13% 4,50% 4,82% 6,04% 5,44% 5,52% 5,55% 3,63% 5,38% 4,78% 5,10% 5,23% 5,05% 5,23% 4,42% 5,69% 4,67% 6,01% 3,42% 4,24% 4,19% 5,19% 4,85% 5,44% 5,04%

Scenario Price Change Scenario Price (€) (€) 74,90 0,28 101 75,92 1,30 102 74,32 -0,30 103 74,72 0,10 104 74,78 0,15 105 72,23 -2,39 106 73,52 -1,10 107 71,94 -2,68 108 74,05 -0,57 109 71,38 -3,24 110 75,42 0,80 111 76,82 2,19 112 74,26 -0,36 113 74,54 -0,08 114 73,78 -0,84 115 79,29 4,67 116 77,01 2,39 117 73,20 -1,42 118 70,68 -3,94 119 73,32 -1,30 120 74,84 0,22 121 74,01 -0,61 122 73,10 -1,52 123 76,58 1,96 124 76,62 2,00 125 74,20 -0,42 126 73,25 -1,37 127 73,69 -0,93 128 71,73 -2,89 129 75,18 0,56 130 75,58 0,96 131 73,79 -0,84 132 74,40 -0,22 133 75,54 0,92 134 73,80 -0,82 135 75,99 1,37 136 71,67 -2,95 137 72,05 -2,58 138 71,44 -3,18 139 70,42 -4,20 140 73,64 -0,98 141 74,56 -0,06 142 74,22 -0,40 143 75,32 0,70 144 77,41 2,79 145 72,71 -1,91 146 76,08 1,46 147 73,79 -0,83 148 78,65 4,02 149 75,72 1,10 150 77,37 2,75 151 74,37 -0,25 152 77,09 2,47 153 74,67 0,04 154 74,25 -0,37 155 75,62 1,00 156 78,77 4,15 157 77,51 2,89 158 69,82 -4,81 159 74,80 0,18 160 78,47 3,85 161 75,59 0,97 162 77,17 2,55 163 76,05 1,43 164 76,42 1,80 165 77,16 2,54 166 73,47 -1,15 167 77,34 2,72 168 76,03 1,41 169 75,72 1,10 170 75,75 1,13 171 76,07 1,45 172 73,06 -1,56 173 74,43 -0,20 174 75,86 1,24 175 74,09 -0,53 176 76,80 2,18 177 75,40 0,77 178 70,36 -4,27 179 72,76 -1,86 180 72,46 -2,16 181 72,33 -2,29 182 80,75 6,12 183 73,02 -1,60 184 75,57 0,95 185 74,19 -0,44 186 73,65 -0,97 187 74,41 -0,21 188 73,67 -0,95 189 77,17 2,54 190 71,73 -2,89 191 76,06 1,43 192 70,44 -4,18 193 81,72 7,10 194 77,93 3,31 195 78,18 3,56 196 73,81 -0,81 197 75,26 0,64 198 72,76 -1,86 199 74,45 -0,17 200

Interest Rate (%) 4,93% 5,66% 5,45% 4,38% 5,45% 4,95% 4,75% 4,34% 4,51% 5,58% 5,91% 5,68% 4,95% 4,60% 4,58% 5,27% 4,93% 5,02% 5,85% 5,07% 5,08% 4,86% 5,29% 5,45% 4,91% 5,58% 5,27% 4,90% 4,62% 5,72% 5,22% 5,17% 4,20% 4,44% 5,17% 4,33% 4,82% 4,02% 5,40% 4,70% 5,98% 4,94% 5,65% 4,45% 5,50% 5,27% 4,70% 5,21% 4,78% 5,85% 4,90% 4,71% 5,85% 5,54% 4,41% 4,96% 5,23% 5,58% 4,95% 4,44% 4,66% 4,53% 4,35% 5,92% 5,56% 4,95% 5,40% 5,27% 4,87% 6,03% 4,67% 5,16% 4,79% 4,91% 5,17% 4,64% 5,00% 4,40% 5,71% 4,54% 4,43% 5,91% 5,43% 4,37% 5,22% 5,21% 4,31% 5,20% 4,55% 5,84% 4,30% 5,73% 4,74% 4,37% 5,01% 4,71% 4,75% 4,14% 4,74% 4,72%

Scenario Price Change Scenario Price (€) (€) 74,94 0,32 201 71,85 -2,77 202 72,73 -1,89 203 77,31 2,69 204 72,72 -1,90 205 74,82 0,20 206 75,70 1,08 207 77,50 2,88 208 76,72 2,10 209 72,21 -2,41 210 70,84 -3,78 211 71,77 -2,85 212 74,83 0,21 213 76,35 1,73 214 76,43 1,81 215 73,48 -1,14 216 74,90 0,28 217 74,53 -0,09 218 71,09 -3,53 219 74,32 -0,30 220 74,27 -0,35 221 75,22 0,60 222 73,40 -1,23 223 72,73 -1,89 224 75,03 0,40 225 72,18 -2,44 226 73,48 -1,14 227 75,04 0,42 228 76,25 1,63 229 71,64 -2,98 230 73,70 -0,92 231 73,92 -0,70 232 78,13 3,51 233 77,07 2,45 234 73,91 -0,71 235 77,56 2,94 236 75,40 0,78 237 78,92 4,30 238 72,96 -1,67 239 75,91 1,29 240 70,58 -4,04 241 74,87 0,25 242 71,92 -2,70 243 77,03 2,41 244 72,52 -2,10 245 73,49 -1,13 246 75,92 1,30 247 73,75 -0,87 248 75,57 0,95 249 71,10 -3,52 250 75,05 0,43 251 75,88 1,25 252 71,11 -3,51 253 72,36 -2,26 254 77,20 2,58 255 74,80 0,18 256 73,63 -0,99 257 72,22 -2,41 258 74,85 0,23 259 77,05 2,43 260 76,09 1,46 261 76,64 2,02 262 77,46 2,83 263 70,82 -3,80 264 72,30 -2,32 265 74,83 0,21 266 72,93 -1,69 267 73,49 -1,13 268 75,17 0,55 269 70,38 -4,24 270 76,06 1,44 271 73,96 -0,67 272 75,50 0,88 273 75,02 0,40 274 73,89 -0,73 275 76,19 1,57 276 74,62 0,00 277 77,24 2,62 278 71,68 -2,94 279 76,64 2,01 280 77,08 2,46 281 70,87 -3,75 282 72,82 -1,80 283 77,35 2,73 284 73,69 -0,93 285 73,71 -0,91 286 77,65 3,03 287 73,78 -0,84 288 76,56 1,94 289 71,14 -3,48 290 77,67 3,05 291 71,57 -3,05 292 75,74 1,12 293 77,35 2,73 294 74,57 -0,05 295 75,88 1,26 296 75,68 1,06 297 78,39 3,77 298 75,75 1,12 299 75,82 1,20 300

Interest Rate (%) 5,19% 5,16% 4,72% 4,76% 5,43% 5,18% 5,16% 4,22% 5,66% 4,47% 5,55% 5,10% 5,00% 4,71% 3,44% 4,22% 5,26% 5,30% 4,24% 5,36% 5,13% 4,26% 4,79% 5,27% 5,34% 4,87% 4,94% 4,67% 4,25% 4,26% 4,48% 4,78% 5,41% 5,88% 4,67% 4,70% 4,30% 5,58% 4,21% 5,18% 4,75% 5,19% 4,79% 5,27% 4,94% 5,50% 5,11% 5,04% 4,40% 5,56% 4,51% 4,83% 5,74% 3,76% 4,68% 5,47% 4,76% 4,24% 5,05% 6,05% 4,96% 4,67% 4,59% 4,31% 5,33% 5,24% 4,45% 5,52% 5,46% 4,68% 4,93% 4,52% 4,73% 4,57% 4,86% 4,43% 4,63% 4,93% 4,09% 4,44% 5,07% 5,12% 4,90% 4,74% 5,74% 4,88% 5,42% 5,45% 4,38% 5,61% 4,90% 5,67% 5,24% 5,41% 5,63% 4,75% 5,71% 4,72% 4,28% 4,73%

Scenario Price Change Scenario Price (€) (€) 73,81 -0,81 301 73,94 -0,68 302 75,83 1,20 303 75,67 1,05 304 72,80 -1,82 305 73,86 -0,76 306 73,95 -0,67 307 78,05 3,42 308 71,85 -2,77 309 76,93 2,31 310 72,33 -2,29 311 74,20 -0,42 312 74,60 -0,02 313 75,85 1,23 314 81,64 7,01 315 78,04 3,42 316 73,53 -1,09 317 73,34 -1,28 318 77,92 3,30 319 73,10 -1,53 320 74,06 -0,56 321 77,87 3,25 322 75,53 0,91 323 73,47 -1,15 324 73,18 -1,44 325 75,19 0,57 326 74,90 0,28 327 76,06 1,44 328 77,88 3,26 329 77,85 3,23 330 76,89 2,27 331 75,58 0,96 332 72,89 -1,73 333 70,98 -3,64 334 76,06 1,44 335 75,90 1,27 336 77,69 3,07 337 72,21 -2,41 338 78,08 3,46 339 73,88 -0,74 340 75,68 1,06 341 73,81 -0,81 342 75,53 0,91 343 73,48 -1,14 344 74,86 0,24 345 72,54 -2,08 346 74,17 -0,45 347 74,43 -0,19 348 77,24 2,62 349 72,28 -2,34 350 76,74 2,12 351 75,34 0,72 352 71,55 -3,07 353 80,14 5,52 354 76,02 1,40 355 72,63 -1,99 356 75,64 1,02 357 77,97 3,35 358 74,41 -0,22 359 70,29 -4,33 360 74,81 0,18 361 76,04 1,41 362 76,38 1,76 363 77,64 3,02 364 73,23 -1,39 365 73,60 -1,02 366 76,99 2,37 367 72,44 -2,18 368 72,70 -1,92 369 76,00 1,38 370 74,90 0,28 371 76,71 2,09 372 75,80 1,18 373 76,49 1,87 374 75,23 0,61 375 77,10 2,48 376 76,23 1,61 377 74,92 0,30 378 78,61 3,98 379 77,07 2,45 380 74,32 -0,30 381 74,11 -0,51 382 75,06 0,44 383 75,75 1,12 384 71,53 -3,09 385 75,14 0,52 386 72,88 -1,75 387 72,71 -1,91 388 77,31 2,69 389 72,06 -2,56 390 75,07 0,44 391 71,84 -2,78 392 73,59 -1,03 393 72,91 -1,71 394 71,98 -2,64 395 75,69 1,07 396 71,65 -2,97 397 75,81 1,18 398 77,76 3,13 399 75,76 1,14 400

Interest Rate (%) 5,00% 4,45% 4,32% 4,58% 4,72% 4,00% 4,84% 4,74% 4,81% 4,58% 5,43% 5,21% 5,61% 4,98% 4,33% 5,41% 4,96% 4,54% 5,92% 4,69% 5,73% 4,98% 5,19% 4,75% 5,55% 5,03% 5,05% 5,30% 4,35% 4,78% 4,78% 5,63% 5,32% 5,89% 4,65% 5,26% 4,31% 4,87% 4,77% 4,60% 5,37% 5,62% 4,90% 4,77% 5,01% 5,17% 4,32% 4,31% 5,70% 5,80% 4,29% 4,73% 4,61% 4,95% 5,01% 5,90% 5,62% 5,90% 5,12% 4,92% 4,62% 4,74% 5,00% 5,00% 5,40% 5,44% 5,55% 5,03% 5,26% 3,80% 4,82% 4,34% 5,41% 5,59% 5,33% 4,29% 4,30% 4,77% 5,60% 4,61% 4,88% 5,09% 5,02% 4,65% 5,06% 4,56% 5,22% 4,93% 4,93% 4,56% 4,11% 5,90% 5,26% 3,95% 4,99% 4,71% 4,89% 5,27% 5,40% 4,67%

Scenario Price Change Scenario Price (€) (€) 74,61 -0,01 401 76,99 2,37 402 77,57 2,95 403 76,42 1,80 404 75,84 1,22 405 79,01 4,39 406 75,30 0,68 407 75,74 1,12 408 75,42 0,80 409 76,42 1,80 410 72,80 -1,82 411 73,75 -0,87 412 72,07 -2,55 413 74,70 0,08 414 77,56 2,94 415 72,92 -1,71 416 74,79 0,17 417 76,64 2,01 418 70,80 -3,82 419 75,96 1,34 420 71,59 -3,03 421 74,73 0,10 422 73,80 -0,82 423 75,68 1,05 424 72,30 -2,32 425 74,50 -0,12 426 74,39 -0,23 427 73,37 -1,25 428 77,45 2,83 429 0,94 430 75,57 75,58 0,96 431 72,00 -2,62 432 73,29 -1,33 433 70,93 -3,69 434 76,12 1,50 435 73,53 -1,09 436 77,65 3,03 437 75,17 0,55 438 75,62 0,99 439 76,33 1,71 440 73,08 -1,54 441 72,02 -2,60 442 75,03 0,41 443 75,59 0,97 444 74,57 -0,05 445 73,91 -0,72 446 77,57 2,95 447 77,63 3,00 448 71,73 -2,90 449 71,31 -3,31 450 77,71 3,09 451 75,77 1,15 452 76,29 1,67 453 74,84 0,22 454 74,57 -0,05 455 70,90 -3,72 456 72,05 -2,57 457 70,89 -3,73 458 74,11 -0,51 459 74,96 0,34 460 76,28 1,65 461 75,76 1,14 462 74,60 -0,02 463 74,61 -0,01 464 72,93 -1,69 465 72,76 -1,86 466 72,30 -2,32 467 74,48 -0,15 468 73,53 -1,09 469 79,95 5,32 470 75,41 0,79 471 77,48 2,86 472 72,89 -1,73 473 72,14 -2,48 474 73,23 -1,39 475 77,70 3,08 476 77,68 3,06 477 75,61 0,99 478 72,13 -2,49 479 76,30 1,68 480 75,12 0,50 481 74,23 -0,39 482 74,55 -0,07 483 76,15 1,53 484 74,37 -0,25 485 76,52 1,89 486 73,70 -0,92 487 74,93 0,31 488 74,93 0,31 489 76,53 1,91 490 78,52 3,90 491 70,91 -3,71 492 73,54 -1,09 493 79,26 4,64 494 74,66 0,04 495 75,88 1,26 496 75,08 0,46 497 73,47 -1,16 498 72,94 -1,68 499 76,03 1,40 500

Interest Rate (%) 4,85% 4,85% 5,60% 4,62% 4,79% 5,22% 5,42% 4,04% 4,53% 4,94% 4,88% 4,22% 5,33% 3,90% 4,30% 4,58% 5,18% 4,91% 5,30% 6,09% 4,21% 5,48% 5,75% 4,46% 5,47% 4,42% 3,93% 5,09% 4,94% 4,62% 5,54% 4,41% 4,40% 5,26% 4,59% 5,46% 5,16% 4,79% 5,33% 5,03% 5,08% 4,72% 5,54% 4,71% 5,68% 4,67% 5,92% 4,74% 5,93% 4,41% 4,21% 4,88% 4,87% 5,62% 5,63% 4,87% 5,39% 4,02% 5,13% 4,92% 3,96% 4,63% 5,42% 4,68% 4,78% 5,34% 4,72% 5,09% 4,59% 5,52% 4,56% 4,97% 4,69% 4,17% 4,24% 5,50% 5,25% 5,17% 4,61% 5,57% 4,65% 5,52% 5,42% 5,19% 4,79% 5,08% 4,94% 4,99% 5,62% 5,56% 4,97% 4,93% 5,00% 5,18% 5,32% 4,95% 4,77% 4,97% 5,29% 4,54%

Scenario Price Change Price (€) (€) 75,28 0,66 75,27 0,64 72,13 -2,50 76,26 1,64 75,52 0,90 73,71 -0,92 72,87 -1,75 78,84 4,21 76,65 2,03 74,89 0,27 75,15 0,53 78,04 3,42 73,24 -1,38 79,47 4,85 77,68 3,06 76,44 1,82 73,85 -0,77 75,00 0,38 73,35 -1,28 70,14 -4,48 78,09 3,47 72,59 -2,03 71,51 -3,12 76,99 2,37 72,66 -1,96 77,14 2,52 79,35 4,72 74,25 -0,38 74,86 0,24 76,24 1,62 72,34 -2,28 77,18 2,56 77,25 2,63 73,50 -1,12 76,40 1,78 72,69 -1,93 73,93 -0,69 75,53 0,91 73,22 -1,40 74,50 -0,12 74,27 -0,35 75,82 1,20 72,35 -2,28 75,85 1,23 71,77 -2,85 76,05 1,43 70,80 -3,82 75,73 1,11 70,77 -3,85 77,20 2,58 78,09 3,47 75,13 0,51 75,18 0,56 72,04 -2,58 72,00 -2,62 75,18 0,56 72,96 -1,66 78,96 4,34 74,07 -0,55 74,96 0,34 79,23 4,61 76,23 1,61 72,87 -1,75 76,00 1,38 75,55 0,93 73,19 -1,43 75,81 1,19 74,25 -0,37 76,38 1,76 72,42 -2,20 76,52 1,90 74,75 0,13 75,98 1,36 78,27 3,64 77,93 3,30 72,51 -2,11 73,57 -1,06 73,89 -0,73 76,31 1,69 72,24 -2,38 76,14 1,52 72,46 -2,16 72,86 -1,76 73,81 -0,81 75,54 0,92 74,26 -0,36 74,86 0,24 74,65 0,03 72,01 -2,61 72,27 -2,35 74,73 0,11 74,90 0,28 74,63 0,01 73,85 -0,77 73,28 -1,34 74,84 0,22 75,62 0,99 74,74 0,12 73,42 -1,20 76,61 1,99

Source Created by the Author

(continued)

9

RISKS AND RISK MANAGEMENT

481

Table 9.3 (continued) Scenario 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600

Interest Rate (%) 4,27% 5,39% 4,20% 5,29% 5,13% 5,34% 4,92% 6,19% 5,47% 5,07% 4,93% 5,09% 5,50% 5,47% 5,55% 5,74% 5,45% 5,09% 4,96% 5,36% 5,38% 5,66% 5,45% 5,43% 4,65% 5,82% 4,93% 4,76% 4,09% 5,42% 5,70% 4,99% 5,28% 6,03% 4,64% 4,65% 5,31% 3,56% 5,69% 5,69% 5,48% 5,28% 4,37% 4,55% 4,78% 4,59% 4,51% 4,95% 5,55% 4,82% 5,52% 4,59% 5,60% 4,48% 5,33% 4,62% 4,63% 5,26% 5,52% 4,19% 4,95% 5,17% 5,19% 5,35% 5,35% 5,66% 5,47% 5,11% 5,35% 4,65% 4,90% 5,10% 4,82% 4,80% 4,69% 5,32% 5,09% 5,54% 5,59% 5,10% 4,64% 5,29% 4,84% 5,33% 5,25% 4,48% 5,22% 5,47% 5,17% 4,83% 5,40% 4,87% 5,40% 5,28% 4,75% 4,97% 4,58% 4,93% 5,36% 5,39%

Scenario Price Change Scenario Price (€) (€) 77,81 3,19 601 72,99 -1,63 602 78,15 3,53 603 73,41 -1,21 604 74,09 -0,53 605 73,17 -1,45 606 74,98 0,36 607 69,76 -4,86 608 72,66 -1,96 609 74,34 -0,28 610 74,91 0,29 611 74,24 -0,38 612 72,52 -2,10 613 72,64 -1,98 614 72,31 -2,31 615 71,55 -3,07 616 72,73 -1,89 617 74,22 -0,40 618 74,78 0,15 619 73,10 -1,52 620 73,04 -1,58 621 71,88 -2,74 622 72,72 -1,90 623 72,82 -1,80 624 76,15 1,53 625 71,22 -3,40 626 74,92 0,29 627 75,65 1,02 628 78,63 4,01 629 72,84 -1,78 630 71,71 -2,91 631 74,67 0,05 632 73,42 -1,20 633 70,39 -4,23 634 76,20 1,57 635 76,14 1,52 636 73,32 -1,30 637 81,08 6,46 638 71,73 -2,89 639 71,74 -2,88 640 72,61 -2,01 641 73,43 -1,19 642 77,37 2,75 643 76,59 1,96 644 75,57 0,95 645 76,41 1,79 646 76,76 2,13 647 74,82 0,20 648 72,33 -2,29 649 75,40 0,78 650 72,43 -2,19 651 76,40 1,78 652 72,09 -2,53 653 76,86 2,24 654 73,24 -1,38 655 76,26 1,64 656 76,20 1,58 657 73,54 -1,08 658 72,44 -2,18 659 78,16 3,53 660 74,83 0,21 661 73,91 -0,71 662 73,82 -0,81 663 73,15 -1,47 664 73,14 -1,48 665 71,86 -2,76 666 72,67 -1,95 667 74,14 -0,48 668 73,15 -1,47 669 76,13 1,51 670 75,06 0,44 671 74,22 -0,40 672 75,40 0,77 673 75,49 0,87 674 75,97 1,35 675 73,29 -1,33 676 74,24 -0,39 677 72,35 -2,27 678 72,15 -2,47 679 74,19 -0,43 680 76,19 1,57 681 73,39 -1,23 682 75,33 0,71 683 73,25 -1,37 684 73,56 -1,06 685 76,86 2,24 686 73,68 -0,94 687 72,65 -1,97 688 73,91 -0,72 689 75,34 0,72 690 72,93 -1,69 691 75,16 0,54 692 72,95 -1,67 693 73,43 -1,19 694 75,70 1,08 695 74,75 0,13 696 76,42 1,80 697 74,91 0,28 698 73,09 -1,54 699 72,99 -1,63 700

Interest Rate (%) 5,70% 5,15% 4,89% 4,30% 4,84% 5,16% 5,13% 5,22% 4,64% 6,04% 4,68% 5,63% 4,98% 5,07% 5,21% 5,03% 5,56% 5,44% 4,34% 5,07% 5,05% 4,62% 4,48% 5,10% 4,85% 5,23% 5,57% 4,96% 4,75% 5,68% 4,42% 4,82% 5,84% 4,43% 5,05% 4,50% 5,50% 4,72% 4,83% 5,30% 5,32% 5,18% 5,39% 5,66% 4,77% 5,34% 5,05% 4,59% 4,85% 4,94% 4,32% 5,16% 4,35% 5,33% 4,47% 5,03% 4,49% 4,36% 5,28% 4,52% 4,30% 5,51% 5,36% 4,81% 5,61% 4,52% 5,77% 4,26% 5,41% 6,35% 5,16% 4,52% 4,99% 4,91% 5,41% 5,01% 4,86% 4,98% 4,96% 4,50% 6,38% 5,28% 4,73% 5,09% 4,53% 4,43% 5,14% 4,54% 5,61% 5,61% 5,06% 5,48% 5,60% 6,04% 4,07% 5,22% 5,33% 4,35% 5,13% 4,96%

Scenario Price Change Scenario Price (€) (€) 71,71 -2,91 701 73,99 -0,63 702 75,10 0,48 703 77,67 3,04 704 75,29 0,66 705 73,94 -0,68 706 74,05 -0,57 707 73,71 -0,91 708 76,19 1,57 709 70,32 -4,30 710 76,01 1,39 711 72,01 -2,62 712 74,72 0,10 713 74,31 -0,31 714 73,72 -0,91 715 74,48 -0,14 716 72,27 -2,36 717 72,76 -1,86 718 77,48 2,86 719 74,32 -0,30 720 74,42 -0,20 721 76,26 1,64 722 76,88 2,25 723 74,21 -0,41 724 75,26 0,64 725 73,64 -0,98 726 72,25 -2,37 727 74,80 0,18 728 75,70 1,08 729 71,77 -2,85 730 77,16 2,54 731 75,39 0,77 732 71,16 -3,47 733 77,12 2,50 734 74,41 -0,22 735 76,77 2,15 736 72,54 -2,08 737 75,83 1,21 738 75,36 0,74 739 73,36 -1,26 740 73,29 -1,33 741 73,86 -0,76 742 72,98 -1,64 743 71,88 -2,74 744 75,62 1,00 745 73,18 -1,44 746 74,39 -0,23 747 76,41 1,79 748 75,29 0,66 749 74,88 0,26 750 77,60 2,98 751 73,95 -0,67 752 77,45 2,82 753 73,24 -1,38 754 76,94 2,32 755 74,49 -0,14 756 76,85 2,23 757 77,43 2,81 758 73,43 -1,19 759 76,71 2,09 760 77,70 3,07 761 72,47 -2,15 762 73,10 -1,52 763 75,42 0,80 764 72,06 -2,57 765 76,71 2,09 766 71,43 -3,20 767 77,84 3,21 768 72,89 -1,74 769 69,10 -5,52 770 73,93 -0,69 771 76,70 2,08 772 74,67 0,05 773 75,02 0,40 774 72,92 -1,71 775 74,59 -0,03 776 75,24 0,62 777 74,71 0,09 778 74,78 0,16 779 76,77 2,15 780 69,01 -5,62 781 73,46 -1,17 782 75,78 1,16 783 74,22 -0,40 784 76,66 2,04 785 77,11 2,49 786 74,02 -0,60 787 76,59 1,97 788 72,06 -2,56 789 72,07 -2,55 790 74,38 -0,25 791 72,59 -2,03 792 72,11 -2,51 793 70,33 -4,30 794 78,71 4,09 795 73,71 -0,91 796 73,23 -1,40 797 77,47 2,85 798 74,07 -0,56 799 74,78 0,16 800

Source Created by the Author

Interest Rate (%) 4,25% 4,52% 5,32% 4,63% 3,87% 5,60% 4,54% 5,22% 4,93% 5,60% 4,43% 5,58% 5,18% 4,99% 5,01% 4,00% 5,63% 5,77% 5,61% 4,41% 5,68% 5,41% 4,38% 4,71% 4,85% 4,25% 4,68% 4,86% 5,38% 4,37% 5,07% 4,77% 4,51% 4,16% 4,03% 5,26% 5,41% 4,75% 4,98% 4,83% 4,15% 6,01% 5,38% 4,77% 4,69% 4,86% 5,16% 5,28% 5,44% 4,18% 5,86% 4,13% 4,57% 4,80% 4,21% 5,49% 4,50% 4,83% 4,56% 5,71% 6,30% 6,19% 5,07% 5,57% 4,02% 5,16% 4,48% 5,25% 5,31% 4,06% 4,72% 4,71% 4,18% 4,70% 5,10% 4,63% 5,99% 4,88% 4,79% 4,53% 5,31% 5,53% 4,45% 5,67% 4,98% 5,33% 5,22% 4,91% 4,94% 5,10% 4,55% 5,17% 5,59% 5,10% 5,17% 4,87% 4,42% 5,23% 4,98% 4,38%

Scenario Price Change Scenario Price (€) (€) 77,92 3,30 801 76,71 2,09 802 73,27 -1,35 803 76,20 1,58 804 79,63 5,00 805 72,12 -2,50 806 76,62 2,00 807 73,71 -0,91 808 74,93 0,31 809 72,10 -2,52 810 77,10 2,48 811 72,20 -2,42 812 73,85 -0,78 813 74,68 0,06 814 74,58 -0,04 815 79,04 4,42 816 71,99 -2,63 817 71,43 -3,19 818 72,06 -2,56 819 77,18 2,55 820 71,77 -2,85 821 72,91 -1,71 822 77,30 2,68 823 75,88 1,26 824 75,27 0,65 825 77,92 3,30 826 76,02 1,40 827 75,23 0,61 828 73,01 -1,61 829 77,38 2,76 830 74,32 -0,30 831 75,60 0,98 832 76,75 2,13 833 78,28 3,66 834 78,88 4,26 835 73,51 -1,11 836 72,89 -1,73 837 75,72 1,09 838 74,71 0,09 839 75,36 0,74 840 78,35 3,73 841 70,44 -4,18 842 73,01 -1,61 843 75,61 0,99 844 75,98 1,36 845 75,23 0,61 846 73,93 -0,70 847 73,45 -1,17 848 72,79 -1,83 849 78,20 3,57 850 71,06 -3,56 851 78,42 3,80 852 76,47 1,85 853 75,50 0,87 854 78,09 3,47 855 72,56 -2,06 856 76,78 2,16 857 75,34 0,72 858 76,52 1,90 859 71,67 -2,95 860 69,32 -5,30 861 69,73 -4,89 862 74,30 -0,32 863 72,26 -2,36 864 78,94 4,32 865 73,93 -0,69 866 76,86 2,24 867 73,57 -1,05 868 73,31 -1,32 869 78,77 4,15 870 75,83 1,21 871 75,86 1,23 872 78,23 3,61 873 75,92 1,30 874 74,19 -0,43 875 76,23 1,61 876 70,55 -4,07 877 75,13 0,51 878 75,53 0,91 879 76,65 2,03 880 73,30 -1,32 881 72,39 -2,23 882 77,02 2,40 883 71,85 -2,78 884 74,70 0,08 885 73,22 -1,40 886 73,67 -0,95 887 75,03 0,41 888 74,86 0,24 889 74,20 -0,42 890 76,57 1,95 891 73,92 -0,70 892 72,14 -2,48 893 74,19 -0,43 894 73,89 -0,73 895 75,19 0,57 896 77,16 2,54 897 73,65 -0,97 898 74,73 0,10 899 77,30 2,68 900

Interest Rate (%) 4,93% 4,71% 4,57% 4,47% 4,82% 4,65% 5,30% 4,44% 4,60% 4,56% 4,94% 5,07% 5,04% 5,77% 5,44% 3,90% 5,93% 5,35% 4,62% 4,61% 4,64% 4,91% 4,42% 5,40% 5,38% 5,62% 4,60% 4,33% 4,44% 4,89% 5,87% 5,19% 4,60% 4,46% 5,07% 4,81% 4,79% 5,61% 4,84% 4,89% 5,56% 5,26% 5,32% 5,54% 5,99% 5,59% 5,39% 4,83% 4,56% 4,86% 5,36% 5,57% 4,86% 4,71% 4,34% 5,65% 4,82% 5,45% 5,82% 5,17% 3,42% 4,95% 5,57% 5,46% 6,21% 5,33% 4,75% 4,87% 5,25% 4,91% 4,76% 4,69% 4,53% 5,14% 4,80% 5,51% 4,38% 4,75% 4,41% 5,16% 4,87% 5,16% 4,97% 3,82% 5,07% 4,33% 5,30% 5,16% 4,93% 5,31% 5,38% 5,38% 5,09% 4,51% 5,64% 6,16% 5,29% 4,85% 5,05% 6,01%

Scenario Price Change Scenario Price (€) (€) 74,94 0,32 901 75,86 1,24 902 76,48 1,85 903 76,94 2,32 904 75,38 0,75 905 76,12 1,50 906 73,36 -1,26 907 77,06 2,44 908 76,37 1,75 909 76,54 1,92 910 74,89 0,27 911 74,33 -0,29 912 74,44 -0,18 913 71,41 -3,21 914 72,76 -1,86 915 79,50 4,88 916 70,77 -3,85 917 73,15 -1,47 918 76,26 1,64 919 76,30 1,68 920 76,17 1,55 921 75,02 0,40 922 77,16 2,54 923 72,92 -1,70 924 73,03 -1,59 925 72,05 -2,57 926 76,35 1,73 927 77,56 2,94 928 77,06 2,44 929 75,09 0,46 930 71,01 -3,61 931 73,80 -0,82 932 76,36 1,74 933 76,96 2,33 934 74,32 -0,31 935 75,43 0,81 936 75,51 0,89 937 72,06 -2,56 938 75,30 0,68 939 75,09 0,47 940 72,30 -2,33 941 73,54 -1,08 942 73,26 -1,37 943 72,36 -2,26 944 70,52 -4,10 945 72,14 -2,48 946 72,97 -1,65 947 75,35 0,73 948 76,51 1,89 949 75,24 0,61 950 73,10 -1,52 951 72,22 -2,40 952 75,22 0,59 953 75,86 1,23 954 77,50 2,88 955 71,91 -2,72 956 75,41 0,78 957 72,75 -1,87 958 71,20 -3,42 959 73,89 -0,73 960 81,72 7,10 961 74,82 0,20 962 72,25 -2,37 963 72,68 -1,94 964 69,65 -4,97 965 73,23 -1,39 966 75,69 1,07 967 75,18 0,55 968 73,58 -1,05 969 75,00 0,37 970 75,66 1,04 971 75,95 1,33 972 76,65 2,03 973 74,01 -0,61 974 75,50 0,88 975 72,47 -2,16 976 77,33 2,71 977 75,71 1,09 978 77,19 2,57 979 73,94 -0,68 980 75,19 0,57 981 73,95 -0,67 982 74,73 0,11 983 79,87 5,25 984 74,31 -0,31 985 77,55 2,93 986 73,36 -1,26 987 73,94 -0,69 988 74,91 0,29 989 73,32 -1,30 990 73,00 -1,62 991 73,04 -1,59 992 74,25 -0,37 993 76,75 2,13 994 71,97 -2,65 995 69,85 -4,77 996 73,41 -1,21 997 75,26 0,64 998 74,41 -0,21 999 70,44 -4,18 1000

Interest Rate (%) 5,24% 4,60% 5,11% 5,44% 5,07% 4,74% 5,35% 4,66% 5,05% 5,31% 4,28% 5,31% 4,59% 4,55% 4,87% 4,86% 5,44% 5,20% 4,49% 4,18% 5,58% 4,42% 4,79% 5,56% 5,33% 5,31% 5,57% 5,32% 3,83% 4,01% 5,52% 5,95% 4,58% 5,11% 5,18% 5,86% 4,73% 4,88% 4,83% 5,26% 5,58% 4,63% 4,32% 5,83% 5,48% 3,99% 4,49% 5,19% 5,30% 5,11% 5,85% 4,21% 5,38% 5,04% 4,93% 5,41% 4,88% 5,29% 5,09% 6,19% 4,72% 5,05% 4,56% 4,83% 4,42% 5,88% 5,59% 4,71% 5,47% 5,47% 5,76% 4,46% 5,13% 5,13% 4,52% 3,98% 5,59% 5,39% 5,85% 4,11% 4,65% 5,18% 4,22% 5,42% 5,38% 5,41% 4,76% 4,79% 5,57% 5,12% 4,94% 4,89% 6,14% 5,35% 5,20% 4,76% 4,88% 4,56% 4,80% 5,02%

Scenario Price Change Price (€) (€) 73,62 -1,00 76,37 1,75 74,15 -0,47 72,78 -1,84 74,31 -0,31 75,73 1,11 73,14 -1,48 76,10 1,47 74,43 -0,19 73,29 -1,33 77,77 3,15 73,30 -1,32 76,38 1,76 76,57 1,95 75,17 0,55 75,23 0,61 72,77 -1,86 73,78 -0,84 76,85 2,23 78,23 3,60 72,18 -2,44 77,14 2,52 75,52 0,90 72,26 -2,36 73,23 -1,39 73,33 -1,29 72,24 -2,38 73,27 -1,35 79,81 5,19 79,00 4,38 72,43 -2,19 70,68 -3,94 76,42 1,80 74,14 -0,48 73,87 -0,75 71,07 -3,55 75,78 1,16 75,14 0,51 75,36 0,74 73,52 -1,10 72,18 -2,44 76,23 1,61 77,57 2,95 71,16 -3,46 72,60 -2,03 79,06 4,44 76,85 2,23 73,84 -0,78 73,37 -1,25 74,17 -0,46 71,08 -3,54 78,08 3,46 73,04 -1,58 74,44 -0,18 74,91 0,29 72,90 -1,72 75,14 0,52 73,41 -1,21 74,25 -0,37 69,74 -4,89 75,83 1,21 74,40 -0,22 76,53 1,90 75,33 0,71 77,13 2,50 70,98 -3,64 72,13 -2,49 75,88 1,26 72,64 -1,98 72,63 -1,99 71,48 -3,14 76,98 2,36 74,06 -0,56 74,07 -0,55 76,70 2,08 79,14 4,52 72,17 -2,45 72,96 -1,66 71,09 -3,53 78,52 3,90 76,12 1,50 73,85 -0,77 78,02 3,40 72,87 -1,76 73,04 -1,58 72,88 -1,74 75,64 1,01 75,50 0,88 72,26 -2,36 74,12 -0,50 74,88 0,25 75,10 0,47 69,96 -4,67 73,16 -1,46 73,77 -0,85 75,66 1,04 75,15 0,53 76,54 1,92 75,50 0,87 74,52 -0,10

482

T. POUFINAS

where S is the foreign exchange rate, P $ the price in US dollars and P e the price in Euro (as has been found in Eq. [9.21] above). The change in value due to changes in interest rates is given by a variation of Eq. (9.26) so as to include the foreign exchange rate: −T × FV ∂ P$ =S × ≡ ∂r$ ∂r (1+r )T +1

(9.40)

Consequently, Eq. (9.27) becomes P $ =∂r$ × r

(9.41)

We still need to account for the change in the price due to the change in the foreign exchange rate. The price change as a result of the change in the foreign exchange rate is depicted by the first partial derivative of the price in US dollars as a function of S. This means that FV ∂ P$ = ≡ ∂s , ∂ S (1+r )T

(9.42)

and thus the price change is approximated by P $ =∂ S × S

(9.43)

The total change in the bond price (value) as a result of a change in both risk factors (the interest rate and the FX rate) is this approximated by the sum of the changes as P $ =∂ S × S+∂r$ × r

(9.44)

The variance σ P $ of the bond price is given by 2    σ P2 $ =(∂ S · σ S )2 + ∂r$ · σr +2 · ρ S,r · (∂ S · σ S ) · ∂r$ · σr ,

(9.45)

where ρ S,r is the correlation coefficient of S and r . The aforementioned equation gives the variance of a random variable that is the sum of two random variables; in our case these are the change in interest rate and the change in the FX rate.

9

RISKS AND RISK MANAGEMENT

483

Consequently the 1−p VaR is given by  2    $ $ 2 1− p Va R =z p · σ P $ =z p · (∂ S · σ S ) + ∂r · σr +2 · ρ S,r · (∂ S · σ S ) · ∂r · σr (9.46) For the bond of example 9.7 we will assume that the Euro – USD exchange rate is 1.17 (i.e. 1 Euro = 1.17 USD - on November 1, 2020). Let us assume that the volatility of the exchange rate is 0.01 and the correlation between interest rates and the foreign exchange rates is −0.5. We then see that P $ = S × P e =1.17 ×

FV (1+r )T

=1.17 × 74.62=87.31,

FV ∂ P$ = = 74.62 ≡ ∂ S , ∂ S (1+r )T −T · FV ∂ P$ = 1.17 × = −1.17 × 426.41 = 498.90 ≡ ∂r$ , ∂r (1+r )T +1

(9.47) (9.48) (9.49)

We plug them in Eq. (9.45) to derive that σ P2 $ =8.64 ⇒ σ P $ =2.94

(9.50)

and thus for p = 1% or p = 2.5% 99% Va R

=2.32 · σ P $ =6.82,

(9.51.a)

=1.96 · σ P $ =5.76

(9.51.b)

97.5% Va R

Once and again we realize that the smaller the probability the higher the potential loss that corresponds to it – as depicted by VaR. A natural next question is what happens if we have a portfolio, i.e. we hold more than one positions. In this case the sensitivity of the portfolio to each risk factor is the sum of the sensitivities of the individual positions. We illustrate initially with an example. Example 9.10

484

T. POUFINAS

Let us consider once again the zero-coupon Euro-denominated bond of our previous examples and in addition cash position of 10 Euros. This is not unusual in practice as investors tend to hold a part of their fixed income portfolios in cash. The present value of the portfolio  B is then (see also Marrison, 2002) $B =S × P e +S × C e =1.17 × 74.62+1.17 × 10=87.31+11.7=99.0, (9.52) where C e is the cash position. The sensitivity of the portfolio to the interest rate remains as before since the cash position does not depend on the interest rate. However, the sensitivity of the value of the portfolio to changes in the foreign exchange rate differs as ∂S =

∂$B FV = + C e =74.62+10=84.62 ∂ S (1+r )T

(9.53)

We replace the new sensitivity in Eq. (9.45) to see that 2 σ =9.05 B

(9.54)

and thus for p = 1% 99%

VaR = 2.32 · σ B = 6.98

(9.55)

The 99% VaR is slightly higher as a result of the additional position in cash. The 97.5% VaR becomes 5.89, which is also slightly higher due to the cash position added to the bond.

9.2.2

Monte Carlo VaR

Monte Carlo VaR can be a bit more challenging to implement when we have more than one risk factor. At this point, assuming that we have only foreign exchange and interest rate risk, i.e. two factors, we present an approach that could generalize the single-factor model. In line with the single-factor considerations, the assumption is that the (changes of the) two risk factors follow bivariate normal distribution. We proceed as in the single-factor Monte Carlo with the drawing of numbers, so as to produce a pseudo-random generating process that will deliver the

9

485

RISKS AND RISK MANAGEMENT

required scenarios (Marrison, 2002). For each scenario the steps can be summarized as follows: • For the first risk factor, i.e. foreign exchange (denoted by S) – We draw a random number z S from a standard normal distribution – We multiply z S by the standard deviation σ S to generate the change of the first risk factor. • For the second risk factor, i.e. the interest rate (denoted by r) – We multiply z S by the correlation coefficient ρ S,r – We draw a second random number z r from a standard normal distribution, which is independent from the first one – We multiply z r by the square root of 1 minus the square of the correlation coefficient – We add the two to generate a random number that has a standard deviation of 1 and whose correlation with the first factor is the given correlation ρ S,r – We multiply this random number by the standard deviation σ r , to create the change of the second risk factor (with the desired standard deviation). The aforementioned steps lead to the following equations (see also Marrison, 2002): S =σ S · z S , z S ∼ N(0,1)

(9.56)

  2 , zr ∼ N(0,1) r =σr · z S · ρ S,r +zr · 1−ρ S,r

(9.57)

Example 9.11 Continuing our previous example, 9.10 that is based on the data of example 9.6 we can see that the aforementioned equations become: S = 0.01 · z S , z S ∼ N(0,1)   √ r = 0.005 −0.5 · z S +zr · 1−0.25 , zr ∼ N(0,1)

(9.58) (9.59)

486

T. POUFINAS

9.3

Economic Capital

Having entered the analysis of risks inherent in a fixed income portfolio, we could not omit a (brief) introduction to the economic capital. Following Marrison (2002), economic capital is the (shareholder’s) equity (or net value, i.e. the difference of liabilities from assets) that the entity under investigation must secure at the beginning of the year so that the default probability (reflecting the desired credit rating) during the year is limited/small. If there is no such target, then the capital can be in line with the regulatory capital requirements. In the case of insurance organizations for example, the regulatory framework, known as Solvency II, requires the maintenance of capital that covers for risks that have a 0.5% probability, i.e. once in two hundred years, of occurring. The target credit rating could result in a higher or lower economic capital; nevertheless, the regulatory capital requirement, known as Solvency Capital Requirement (SCR) has to be met. The economic capital allows the entity to perform its daily activities, execute its business plan and reach or preserve its targeted credit rating. Shareholders expect/ require a return on the money they contribute to the capital of the company. The performance on the capital of the shareholders that accounts for the risk they are exposed to, is often referred to as risk-adjusted return on capital (RAROC). It is defined as the expected net risk-adjusted profit divided by the economic capital requirement. Shareholders often set a minimum requirement/ threshold for RAROC. This is known as the hurdle rate (HR). This is the threshold that determines that viability of an investment or portfolio in risk terms. RAROC is a percentage. When the return is measured in monetary terms, then it is often referred to as shareholder value added (SVA). It is the difference of required profitability to meet the hurdle rate (i.e. the product of the hurdle rate with the economic capital) from the actual or expected profitability (Marrison, 2002). Economic capital is not exactly the same as VaR. To connect the two we realize that 1-pVaR measures the loss that the entity can have with a probability of p over a day. To find the related economic capital we need to convert this percentage to the probability of default implied by the target credit rating and move from 1 day to 1 year. For the fixed income portfolio of the entity we assume that losses are normally distributed. Depending on the desired credit rating of the entity, its economic capital

9

RISKS AND RISK MANAGEMENT

487

(E cr ) should be given by ECcr =acr · σ1year =acr ·

√ 250 · σ1day ,

(9.60)

where a cr corresponds to the credit rating that the company wishes to have. Recall that 1−p VaR is given by 1− p VaR

= z p · σ1day ⇒ σ1day = 1− p VaR/z p

Replacing in (9.60) we get that √ ECcr = (acr · 250/z p ) · 1− p VaR

9.4

(9.61)

(9.62)

VaR Generalization 9.4.1

Parametric VaR

The aforementioned examples bring forward one natural question; how do the calculations change when we have more than one risk factors or more than one positions? We study the changes in the subsections that follow. 9.4.1.1 More Than Two Factors When more than one risk factors are present, but still there is one position, i.e. one fixed income security that is exposed to several risk factors, the steps of example 9.10 above are used to see that the generalization of Eq. (9.44) applies. Indeed, assuming that the fixed income position bears ℵ risk factors we see that P = ∂1 × φ1 + ∂2 × φ2 + · · · ∂ℵ × φℵ

(9.63)

which yields that the standard deviation of the change in value of the position is given by σ P2 = (∂1 · σ1 )2 + (∂2 · σ2 )2 + 2 · ρ1,2 · (∂1 · σ1 ) · (∂2 · σ2 )+ ···+ (∂ℵ−1 · σℵ−1 )2 + (∂ℵ · σℵ )2 + 2 · ρℵ−1,ℵ · (∂ℵ−1 · σℵ−1 ) · (∂ℵ · σℵ ) (9.64)

488

T. POUFINAS

This implies that (see also Marrison, 2002)

 ℵ ℵ

 2 VaR = z · σ = z · (∂ · σ ) + 2 · ρi, j · (∂i · σi ) · (∂ j · σ j ) 1− p p P p i i i=1

i< j

(9.65) or that



  ℵ ℵ ℵ ℵ     ρi,j · (∂i · σi ) · (∂j · σj ) = zp · σi,j · ∂i · ∂j 1− p VaR = zp · i=1 j=1

i=1 j=1

(9.66) In the above we denote by ℵ ϕi ∂i σi ρi, j σi, j

the number of risk factors the i-th factor the derivative (sensitivity) of the position with respect to ϕi i.e. ∂ P/∂ϕi the standard deviation of the change of the i-th factor the correlation of the changes of factors i and j The covariance of the changes of factors i and j

The above can be represented with the use of matrix notation. If ∂ denotes the vector of sensitivities and V the variance (-covariance) matrix, then (see also Marrison, 2002)  VaR = z · (9.67) ∂ · · ∂ T 1− p p Where ∂ := [∂1 , ∂2 , · · · , ∂ℵ ] And

⎡

σ12 σ1,2 ⎢ σ2,1 σ 2 2 ⎢

:= ⎢ . .. ⎣ .. . σℵ,1 σℵ,2

⎤ · · · σ1,ℵ · · · σ2,ℵ ⎥ ⎥ . . .. ⎥ . . ⎦ · · · σℵ2

(9.68)

(9.69)

9

∂ T ran

RISKS AND RISK MANAGEMENT

489

denotes the transpose of ∂

Example 9.12 We revisit example 9.10 to see that the matrix notation yields that (see also Marrison) ∂ = [74.62 ∂ T ran =

− 498.90], 

74.62 498.90

(9.70)

 (9.71)

and 

0.0001 −0.000025

= −0.000025 0.000025



The matrix multiplication delivers the same result as before, i.e.  · · ∂ Tran = 6.82 99% VaR = 2.32 · ∂

(9.72)

(9.73)

9.4.1.2 More Positions The natural next extension is to consider multiple securities, i.e. a portfolio, along with the multiple factors. The sensitivity vector for the portfolio is the sum of the sensitivity vectors of the securities as the first derivative of the sum of functions is the sum of their first derivatives. If we have I securities, then the sensitivity vectors become (Marrison, 2002)   ∂ 1 = ∂11 , ∂21 , · · · , ∂ℵ1   ∂ 2 = ∂12 , ∂22 , · · · , ∂ℵ2 (9.74) .. .   ∂ I = ∂1I , ∂2I , · · · , ∂ℵI

490

T. POUFINAS

which for the portfolio (Π B ) yields that ⎡ ⎤ I I I

j j j ∂1 , ∂2 , . . . , ∂ℵ ⎦ ∂ B = ∂ 1 + ∂ 2 + · · · + ∂ I = ⎣ j=1

Consequently the portfolio 1− p VaRB

j=1

(9.75)

j=1

1-p VaR

is given by T ran = z p · ∂ B · · ∂  B

(9.76)

The variance (-covariance) matrix remains as is provided we have the same ℵ factors as before. Example 9.13 With the introduction of the aforementioned generalized approach example 9.11 can be revisited to give that ∂ Bond = [74.62

− 498.90]

(9.77)

and ∂ Cash = [10

(9.78)

0]

This gives that ∂ B = ∂ Bond + ∂ Cash = [84.62 Therefore 99% VaRB

= 2.32 ·

97.5% VaRB

− 498.90]

(9.79)

T ran = 6.98, ∂ B · · ∂  B

(9.80)

T ran = 5.89 ∂ B · · ∂  B

(9.81)



= 1.96 ·

In the previous analysis we have that Pr(Z < −z p ) = Pr(Z > z p ) = p,

(9.82)

z p = −1 (1 − p),

(9.83)

or

for

9

Z zp Φ

RISKS AND RISK MANAGEMENT

491

a random variable that follows the standard normal distribution N (0, 1) the level of loss that corresponds to a small probability p the cumulative standard normal distribution function.

9.4.2

Monte Carlo VaR with More Than Two Factors

We saw earlier how to set up Monte Carlo simulation for two factors. For more than one risk factor, the hypothesis is once and again, that the risk factors move by a multivariate normal distribution. However, the implementation is a bit more challenging as we need to properly capture the correlations. To do that, we apply matrix decomposition techniques to the variance matrix. The two most common approaches are the Cholesky decomposition and the eigendecomposition (or spectral decomposition) (Marrison, 2002). We simply draft how they work. The Cholesky decomposition expresses the variance matrix as the product of an upper triangular matrix U (or equivalently a lower triangular matrix L) with its transpose. This means that

= U T ran · U

(9.84)

Then we consider independent random draws of the standard normal distribution; these draws are as many as our risk factors are. Therefore, for ℵ risk factors this can be expressed with a vector z = [z 1 , z 2 , . . . z ℵ ]

(9.85)

If we multiply U by z then we get a new vector of factors, let it be z , that preserves the correlations as the initial ones. These outcomes are used essentially to replicate the changes of our initial factors. If φ denotes the vector of the factor changes, then this is expressed as: φ = z = z · U

(9.86)

or ⎡

u 11 ⎢ 0 ⎢ [φ1 φ2 . . . φℵ ] = [z 1 z 2 . . . z ℵ ] · ⎢ . ⎣ .. 0

u 12 u 22 .. . 0

··· ··· .. .

⎤ u 1ℵ u 2ℵ ⎥ ⎥ .. ⎥ . ⎦

0 u ℵℵ

(9.87)

492

T. POUFINAS

After that we proceed as before to produce the scenarios for the Monte Carlo simulation. The Cholesky decomposition requires that the covariance matrix is positive definite, i.e. that all eigenvalues are positive. To tackle this limitation we employ the eigendecomposition. This approach overcomes the positive definiteness restriction. It provides an expression for the covariance matrix as a multiple of a diagonal matrix, a second matrix and its transpose, which is also its inverse, as follows:

= Q T ran ·  · Q

(9.88)

I = Q T ran · Q

(9.89)

where I is the identity matrix and Λ is the diagonal matrix whose diagonal has the eigenvalues, i.e. ⎡

λ1 ⎢0 ⎢ =⎢ . ⎣ ..

0 λ2 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(9.90)

0 0 0 λℵ We then produce the changes of our factors as with the Cholesky decomposition; however we use the property that

= Q T ran ·  · Q = (1/2 · Q)T ran · (1/2 · Q)

(9.91)

to replace the matrix U by the product of Λ1/2 with Q . We then derive the new vector of changes of the risk factors as √ √ √ ⎤ q11 λ1 q12 λ1 · · · q1ℵ λ1 ⎢ q21 √λ2 q22 √λ2 · · · q2ℵ √λ2 ⎥ ⎥ ⎢ [φ1 φ2 . . . φℵ ] = [z 1 z 2 . . . z ℵ ] · ⎢ ⎥ .. .. .. .. ⎦ ⎣ . . . . √ √ √ qℵ1 λℵ qℵ2 λℵ · · · qℵℵ λℵ (9.92) ⎡

Beyond that point we proceed as before to perform the desired scenarios of the Monte Carlo simulation and the consequent estimation of VaR.

9

9.5

RISKS AND RISK MANAGEMENT

493

VaR Contribution

So far we managed to calculate the VaR of a portfolio of fixed income securities that is exposed to a series of risk factors. At the same time we may of course estimate the VaR of each of these securities individually. This is often referred to as stand-alone VaR and is denoted by S-VaR or SVaR. The stand-alone VaR for a fixed income security (or risk factor) disregards the rest of the securities (or risk factors) of the investor portfolio. The total VaR is calculated by using all the fixed income securities and risk factors of the portfolio (or investor). We can readily infer that the stand-alone VaR has two major drawbacks. Namely, (i) the sum of the individual security S-VaRs is not equal to the total fixed income portfolio VaR; and (ii) the individual security S-VaR does not take into account the (correlations with the) other securities of the portfolio (Marrison, 2002). To see that, if we had a fixed income (or bond) portfolio Π B that consisted of I fixed income securities (or depended on ℵ risk factors), let them be B 1 , B 2 , … BI (or φ1 , φ2 , … φℵ respectively) then the VaR of the portfolio would have to account not only for the stand-alone VaR of the two securities (or risk factors) but also for their joint risk as measured by the correlation. With the use if Eq. (9.45) it may be readily seen that (Marrison, 2002):

 I I   ρ B j ,B SVaR B j SVaR B VaRB = j=1 =1

 I I 

 2 = SVaR B j + 2 ρ B j ,B SVaR B j SVaR B j=1

or VaRB

(9.93)

j VaRB where ζj ζ B  ζ B > VaRB

denotes the loss that comes from security j, denotes the loss of the entire portfolio  B , denotes the condition that the portfolio loss exceeds VaR.

4. The VaR contribution of security j is defined as: VaRC j = VaRB × %C j

(9.120)

9

9.5.3

RISKS AND RISK MANAGEMENT

499

Economic Capital Revisited

Having defined the VaR contribution we have a way to allocate economic capital at a fixed income security (or sub-portfolio) level. The equation that gives the allocated (economic) capital (AEC ) at security j is (Marrison, 2002): AEC j = ECB ·

VaRC j VaRB

(9.121)

This allows also the calculation of RAROC and SVA at a security (or sub-portfolio) level as (Marrison, 2002): RAROC j =

NI j AEC j

(9.122)

where NI j is the net income produced by the security (or sub-portfolio). The shareholder value added is calculated by (Marrison, 2002): j

SVA j = NI j − AEC j × HRhd

(9.123)

where HRhd

denotes the hurdle rate for an investment horizon in days hd.

9.6

Liquidity Risk

As even publically traded (especially corporate) bonds have reduced liquidity one can easily understand that liquidity is an important source of risk for a fixed income portfolio. As such it can contribute to the portfolio loss and thus it needs to be considered when estimating VaR and Economic Capital. Fixed income portfolios are exposed to the risk that a fixed income security may not be disposed of (or purchased) at the desired pace (i.e. quickly enough) at a (perceived) fair (or without affecting its) price. This may occur when the fixed income security has a reduced marketability, i.e. a small number of investors show interest in transacting on it. This is referred to as trading liquidity risk (Marrison, 2002). If this occurs, then losses may be posted as (Marrison, 2002):

500

T. POUFINAS

• The investor (or portfolio manager) needs to pressingly sell one or more fixed income securities in order to use the proceeds so as to cover a liability. An example would be a fixed income mutual fund with increased redemption applications as a result of a sharp upward move in the interest rates. The portfolio manager may be forced to accept a sale price that is much lower than its (perceived) fair price. • The investor (or portfolio manager) intends to sell a fixed income security that is generating a loss or so as to prevent a (further) loss but realizes that this cannot be completed immediately; it rather takes more time (days) to dispose of the security at a (perceived) fair (or without affecting its) price. This delay may result in additional losses while he or she waits for the sale to be concluded. If in the aforementioned cases the portfolio manager is obliged to transact, then he or she has to accept either a (much) lower price so as to conclude the sale quickly or a (much) longer execution time so as to make the sale at (or close to) the (perceived) fair price. Both of them can lead to great losses; the former may lead to an instant high loss, whereas the latter may lead to a cumulative high loss as a sum of smaller losses that the manager took over an extended period of time (see also Marrison, 2002). But why do portfolio managers select illiquid fixed income securities? Unless it is part of a benchmark they follow or investment mandate they need to honor, it is primarily done because illiquid instruments are expected to have superior performance over liquid securities with comparatively lower volatility. This anticipated excess return though is nothing more than the compensation to the investor for the higher (il)liquidity risk he or she is willing to accept (see also Marrison, 2002). Private debt issues are examples of (il)liquidity risk; they do not trade in organized exchanges, they may not be priced on a daily basis and unless a credit event occurs they may exhibit low volatility. Corporate bond issues are also representative cases; some of them are not as liquid, even though they may be listed in organized exchanges. As explained earlier, this may be due to their small size, along with the fact that they may have participated in central bank purchase programs. Consequently, the (il)liquidity risk needs to be measured so that the investors have no incentive to buy them thinking that there will be no charge (in terms of Economic Capital) by the regulator and as a result by their financial institution or insurance organization for this additional risk.

9

RISKS AND RISK MANAGEMENT

501

The liquidity risk can be measured in (at least three) possible ways; the loss due to a forced sale, the close-out time, and the bid-ask spread. We study them in the following subsections, following the approach of Marrison (2002). 9.6.1

Forced Sale

The first one involves the sale of a fixed income security and the measurement of the loss recorded for immediately concluding the sale or the time it takes to make the sale at (or close to) its (perceived) fair price. This metric though is not as reliable, since on one hand it interferes with the portfolio management process and on the other hand it does not necessarily proof-test the portfolio in real adverse states (Marrison, 2002). 9.6.2

Close-Out Time

The second method counts the number of days that are needed to sell a fixed income security. The close-out time is the time needed to reduce the holding of a fixed income security to a size that will cause no further losses to the fixed income security. It is therefore the time needed either to sell or hedge the fixed income security. The the close-out time in days (CoD) is given by CoD :=

MV ,  × Vol

(9.124)

where,  MV Vol

denotes the % of the daily volume that can be disposed of without materially impacting the price of the security, denotes the (market) value of the entire fixed income security holding, denotes the daily volume of the security.

The close-out time metric can be used to define the close-out adjusted VaR, which incorporates the measurement of (il)liquidity risk. It assumes that it takes CoD days to sell an illiquid fixed income security compared to an equivalent fixed income security that would be liquid and can thus be sold in one day. It therefore considers that the position will be held

502

T. POUFINAS

for CoD days and on the last day it will be fully sold. As a result, the VaR of such an illiquid security (denoted by LVaR as it adjusts for liquidity) is the VaR that the security would have if it could be sold in one day times the square root of CoD (Marrison, 2002), i.e. √ (9.125) LVaR = VaRCoD = CoD · VaR1−day This holds true under the (implicit) assumption that the loss over CoD days will be the sum of the losses per day for CoD days, which are independent and identically distributed (IID). To see why this holds true, we reason that the loss of an investor that remains exposed to an illiquid security for a period of CoD days is the sum of the losses that may occur on each individual day. The daily as well as the cumulative CoD-day period losses are of course random variables. As a result the CoD-day period loss is given by (Marrison, 2002) ZCoD =

CoD

ζd ,

(9.126)

d=1

where Z CoD is the cumulative loss for an exposure period of CoD days and ζ d is the loss on day d, for d = 1…CoD. As these losses are IID the variance of the cumulative loss is equal to the sum of the individual variances (Marrison, 2002): σ Z2CoD =

CoD

σζ2d

(9.127)

d=1

As these variances are all equal, we derive that σ Z2CoD = CoD · σζ21

(9.128)

or that σ Z CoD = The

1-p VaR

√

CoD·σζ1

(9.129)

is thus given by

1− p VaR CoD

√ √ = z p · σ Z CoD = z p · CoD · σζ1 = CoD · z p · σζ1 √ = CoD · 1− p VaR1−day , (9.130)

9

RISKS AND RISK MANAGEMENT

503

as was announced in Eq. (9.125) above. A variation of the previous estimation uses the hypothesis that the fixed income security is sold gradually over the CoD-day interval. As a matter of fact the reduction of the holding (and thus of its variance) is linear until it is completely sold out. This modifies Eq. (9.128) to σ Z2CoD

=

CoD

σζ2d

=

σζ21

·

CoD−1 

CoD − d 2

d=1

CoD

d=0

≡ σζ21 S2CoD ,

2 where SCoD is the quantity in the brackets. The calculated as 1− p VaR CoD

1-p VaR

(9.131)

is therefore

= z p · σ Z CoD = z p · σζ1 · SCoD = SCoD · z p · σζ1 = SCoD · 1− p VaR1−day (9.132)

Clearly, the linearly adjusted VaR is smaller than the square-root-of-CoD adjusted VaR. To see that observe that for CoD > 1 2 SCoD =

CoD−1

 d=0

CoD − d CoD

2
Every time the companies with which the company under evaluation is compared change, its estimated value also changes. – Is the evaluation correct compared to other companies? ==> If the performance of other companies is significantly different from the average, then the average performance is not a representative performance. • They do not take into account changes in the stock market. – It is possible that an important event has affected prices upwards or downwards in the last year and it is estimated that this change will be maintained as it is justified. • Different ratios can produce stock price estimates that are far apart. – The solution lies in the use of the weighted average of the valuations generated by the various multiples.

12

BONDS VERSUS STOCKS

595

• They do not take into account the prospects of the company, but are based on data from the current and previous years. After all, how do we value the stock when the market ratios give contradictory results? The answer is given by the third bullet above: we use a weighted average of the outputs delivered by the various ratios employed. That is, P/E

P/BV

(P)val =w P/E · (P)val + w P/BV · (P)val P/I V

+ w P/I V · (P)val

P/C V

+ w P/C V · (P)val

(12.40)

where wi (P)ival

denotes the weight for each of the P/E, P/BV, P/V, and P/CV ratios. denotes the value estimate generated with the use of each of the P/E, P/BV, P/IV, P/CV ratios.

The criticism that could be exercised here is that the weighting is subjective. Example 12.11 Suppose that for a company the stock valuation produced via the various multiples and the weights used by an investor are: Ratio

Stock value

Weight

P/E P/BV P/CV

10 15 18

50% 40% 10%

Then the value attributed to this company per share is: (P)val = 10 · 0.5 + 15 · 0.4 + 18 · 0.1 = 12.8

(12.41)

596

T. POUFINAS

12.3 12.3.1

Stock Returns Holding Period Return

As in the case of bonds it makes sense to ask what the holding period return is for an investor that decides to hold the bond for a specific number of years. Adapting the generic holding period return Eq. (12.38) of Chapter 2 to reflect the fact that stocks pay dividends instead of coupons that are paid by the bonds we get H P RH =

PH + D H + D H −1 · (1 + r ) + · · · D1 · (1 + r ) H −1 − P0 P0 (12.42)

where H is the investment horizon of the investor. Example 12.12 An investor decides to purchase the stock of Example 12.4 at time t = 0 and to sell it at time t = 3, i.e. he or she has an investment horizon equal to 3 years. He or she invests dividends at an interest rate of 2% per annum with annual compounding. What is the (expected) holding period return from this position on the stock? Answer We assume that the expectations of Example 12.4 are realized. This means that P 3 = V 3 and P 0 = V 0 . Otherwise, the findings reflect an expected holding period return. According to Eqs. (12.1) and (12.14) the stock is purchased for 17.13 Euro and is sold at 21 Euro. As dividends are invested at an interest rate of r = 2%, we use Eq. (12.15) to find that the 3-year holding period return becomes P3 + D3 + D2 · (1 + r ) + D1 · (1 + r )3−1 − P0 P0 21 + 2 + 1, 5 · (1.02) + 1 · (1.02)2 − 17.13 = 17.13 8.4437 = 0.4930 = 49.30% = 17.13

H P R3 =

(12.43)

12

BONDS VERSUS STOCKS

597

The annualized holding period return is found if we solve the equation (1 + ρ)3 =1 + 0.4930 ⇒ 1 + ρ =(1.4930)1/3 ⇒ ρ = 0.1429 = 14.29%

12.3.2

(12.44)

Capital Asset Pricing Model - CAPM

We still need to explain how to find k, the expected or required return of the common equity shareholders. A potential route is offered by the capital asset pricing model as developed by Fama and French (1992, 1993). The CAPM states that the expected return of a risky asset or a portfolio i is given by   (12.45) r i − r f = βi r M − r f where βi =

σi σi M = ρi M 2 σM σM

(12.46)

The latter ratio is known as the (market) beta of the asset or the portfolio. In the previous equation the following notation is used: ri rM rf σiM σi σM ρiM

denotes the expected return of the risky asset denotes the expected return of the market portfolio denotes the risk-free rate enotes the covariance of the returns of the asset and the market portfolio denotes the standard deviation of the return of the asset denotes the standard deviation of the return of the market portfolio denotes the correlation of the returns of the asset and the market portfolio

CAPM states that the excess return of the risky asset over the risk-free rate is proportional to the excess return of the market portfolio over the risk-free rate. The coefficient of this proportionality is the (market) beta.

598

T. POUFINAS

When it comes to our stock valuation approach, we employ CAPM to estimate k. As a matter of fact k is nothing else than r i . This means that   k = r f + βi r M − r f (12.47) Example 12.13 A firm recorded earnings of 2 Million Euro for the fiscal year that just ended. The firm has only common equity that consists of 1 Million shares of stock. It distributed as dividend 50% of the earnings. The return on equity is 10%. The return of the stock index of the stock exchange in which the stock is listed is 20%. The risk-free rate is 4%. The beta of the stock is 0.5. What is the value of a share of stock? Answer To answer this question we combine the approach used in the previous examples, along with the use of CAPM to estimate the expected return for the shareholders. The latter, with the use of Eq. (12.20), yields that   k = r f + βi r M − r f = 0.04 + 0.5 · (0.2 − 0.04) = 0.12 = 12% (12.48) The dividend that has just been distributed per share according to Eq. (12.17) equals D0 = E 0 × d =

2, 000, 000 · 0.5 = 2 · 0.5 = 1 1, 000, 000

(12.49)

The dividend growth rate according to Eq. (12.18) is g = (1 − d) · R O E = 0.5 · 0.10 = 0.05 = 5%

(12.50)

Using now Eq. (12.6) we infer that V0 =

1 · (1.05) 1.05 D0 · (1 + g) = = = 15 k−g 0.12 − 0.05 0.07

12.3.3

(12.51)

Dividend Yield

The dividend yield is for stocks the equivalent of current yield for bonds. It is the financial ratio that measures the percentage that a company pays

12

BONDS VERSUS STOCKS

599

out in dividends on an annual basis relative to its stock price (or market value). It is estimated as Dividend Yield =

Annual Dividend per Share Stock Price per Share

(12.52)

Example 12.14 A company is listed in a stock exchange and trades for a stock price of 20 Euro. It makes during the year four quarterly dividend payments of 0.25 Euro each. The dividend yield ratio is calculated as DY =

1 0.25 + 0.25 + 0.25 + 0.25 = = 5% 20 20

(12.53)

This means that the investor that owns shares of the company will earn 5% on these shares in the form of dividends. The dividend yield, as is the case with all financial ratios, can be used for comparison purposes in the stock selection process among stocks of the same industry and across years. Attention however is in order for two main reasons; • An increasing dividend yield may be due to a declining price and not an increasing dividend. • An increasing dividend yield due to an increasing dividend may indicate that little or no part of the earnings is reinvested so as to produce new earnings and allow the company to grow. Normally investors would select the company with the higher dividend yield. However, even this criterion needs to be carefully applied; a fast growing, relatively young company may elect to pay a low or no dividend at all and instead reinvest the money into the company in order to achieve future growth. At the same time, a mature company could post a relatively high dividend yield as it may lack future growth potential. This could work as an incentive for investors to hold the stock of the company instead of selling it so as to buy the stock of high-growth competitors.

600

T. POUFINAS

The dividend yield can be thus used in order to indicate if a stock aligns with their investment strategy and annual yield objective. Example 12.15 Two companies, ABC and DEF are active in the same industry. The first one is in the market for 50 years, whereas the second entered the market 5 years ago. They are both listed in the same exchange. ABC trades for 20 Euro per share and distributed a dividend of 1 Euro per annum per share. DEF trades for 22 Euro and just paid out a dividend of 0.55 Euro per annum per share. Which one would you recommend buying based on the dividend yield? Answer The dividend yield of company ABC is DY ABC =

1 = 5% 20

(12.54)

The dividend yield of company DEF is DY D E F =

0.55 = 2.5% 22

(12.55)

We realize that company ABC has twice the dividend yield of company DEF. Moreover, as ABC is more years in the market than DEF this difference seems to be sustainable, at least in the near future. Hence, ABC constitutes a safer choice compared to DEF, with this criterion. Dividend yield is the return metric that focuses on dividend and not capital gains that come through the stock price appreciation. The pros of focusing on dividend yields relate to the fact that investors that choose firms with high dividend yields are more likely to reinvest the dividends to the company. This will drive its price even higher. This leads to compounding gains in the long run. The cons of focusing on high dividend yields have to do with the possibility that companies that distribute high dividends may do that at the expense of future growth. In other words, it limits the future earnings generating capacity of the company so that it delivers more and more earnings. Reinvesting earnings may result in lower dividends but may lead to higher earnings, thus higher stock price; in this case investors can realize capital gains as the price of their stock increases.

12

12.4

BONDS VERSUS STOCKS

601

Comparison of Bonds and Stocks

The previous discussion unveiled the differences and similarities of bonds and stocks. Investors often have to choose between a stock and a bond; most often though, they try to combine stocks and bonds in one portfolio. An interesting question that can be asked to (or is faced by) an investor - assuming that he or she considers providing capital for a company – is: Would you invest to the bond or the stock of a company? Sometimes investors think that a company is not solvent enough to service its debt obligations and hence refrain from lending it or purchasing its bonds. Yet, at the same time, they are willing to buy its stock and thus become co-owners. This may seem as a paradox. However, sometimes investors feel that they can invest in the stocks of a company in the short term and benefit from an increase of its price plus potential dividend payments. Such a short-term profit may not be feasible by investing in a corporate bond (see for example Corporate Finance Institute, 2021a). A comparison of the main characteristics of bonds and stocks is demonstrated in Table 12.2. 12.4.1

Selecting Between a Stock and a Bond with the Use of HPR

We offer a potential selection process between the stock and the bond of a company, with a caveat though. Example 12.16 Let us consider now an investor who has a horizon of 3 years. He or she thinks of financing a company and wishes to choose between the stocks and the bonds of the company. The candidate bond expires in 4 years, has a face value of 100 Euro and pays an annual coupon of 5%. The discount rate is 6%. The common stock just paid dividend of 1 Euro per share. The dividend is expected to increase at a rate of 5% per year. The required rate of return by the investors is 15%. The investor wishes to choose between the bond and the stock with the use of the holding period return. (a) What is the bond price today? (b) What will the price of the bond be in 3 years?

602

T. POUFINAS

Table 12.2 Stocks versus bonds Feature

Stocks

Bonds

Instrument type Holder ownership Type of financing Voting Rights Face Value Periodic payments Maturity Comparative Advantage Rate of Return Source of returns

Equity Yes—Owner Company Ownership Interest Yes No—except for preferred Dividends Perpetuity Voting rights Risk-free + risk premium (In line with) Company Earnings

(Perceived) Risk (Potential) Return Guarantee/Promise Issuers

High No

Debt No—Lender Debt No Yes Interest—Coupons Finite or Perpetuity Priority in repayment Risk-free + spread (Interest and face value) To be repaid even without Earnings Low-Moderate Yes (except if default)

Companies

Convertibility Callability

No—except for preferred Yes

Companies, Governments, Financial Institutions, Municipalities Yes (if provisioned) Yes

Source Created by the Author

(c) If the investor invests the coupons at an interest rate of 2%, what is the 3-year holding period return? (d) What is the annualized holding period return? (e) What is the share price today? (f) What will the share price be in 3 years? (g) If the investor invests dividends at an interest rate of 2%, what is the 3-year holding period return? (h) What is the annualized holding period return? (i) Which of the two will the investor choose – the bond or the stock? (j) What does this approach not take into account? Answer

12

BONDS VERSUS STOCKS

603

(a) The bond price is nothing than the present value of the bond as calculated by Eq. (2.1) of Chapter 2. It gives that P0 = P V0 =

5 5 5 105 + + + = 96.5348 (1.06) (1.06)2 (1.06)3 (1.06)4 (12.56)

(b) We use the same formula for the remaining year after the 3rd year to see that P3 =

105 = 99.0566 (1.06)1

(12.57)

(c) Eq. (2.38) of Chapter 2 can be used to calculate the 3-year holding period return as P3 + c + c · (1 + r ) + c · (1 + r )3−1 − P0 P0 99.0566 + 5 + 5 · (1.02) + 5 · (1.02)2 − 96.5348 = 96.5348 =0.1846 = 18.46% (12.58)

H P R3 =

(d) The annualized return is given by (1 + ρ)3 =1 + 0.1846 ⇒ 1 + ρ =(1.1846)1/3 ⇒ ρ = 0.0581 = 5.81%

(12.59)

(e) We use Gordon’s formula (12.6) to see that for the stock the price today is given by S0 = V0 =

1 · (1.05) = 10.5 0.15 − 0.05

(12.60)

(f) We use Gordon’s formula (12.6) to see that for the stock the price 3 years from today is given by S3 = V3 =

1 · (1.05)3 · (1.05) D3 · (1.05) = = 12.155 0.15 − 0.05 0.15 − 0.05

(12.61)

(g) The 3-year holding period return is given by Eq. (12.26) as H P R3 =

S3 + D3 + D2 · (1 + r ) + D1 · (1 + r )3−1 − S0 P0

604

T. POUFINAS

12.155 + 1.1576 + 1.1025 · (1.02) + 1.05 · (1.02)2 − 10.5 10.5 =0.4790 = 47.90% (12.62)

=

The dividends have been found with the use of Eqs. (12.3) taking into account that D 0 = 1. (h) The annualized 3-year holding period return is given by (1 + ρ)3 =1 + 0.4790 ⇒ 1 + ρ =(1.4790)1/3 ⇒ ρ = 0.1394 = 13.94%

(12.63)

(i) The investor would choose the stock as it has a higher holding period return. (j) This approach does not account for risk. It could be that stock has a higher risk, hence the higher holding period return. The investor needs to adjust for this risk, with a potential use of the volatility of the two securities. In addition, the investor may wish to use a utility function in order to rank his or her potential choices, including all portfolios that he or she can construct with the bond and the stock.

12.4.2

Selecting Between a Stock and a Bond with the Use of Current Yield and Dividend Yield

Investors can also compare the current yield of a bond with the dividend yield of a stock in order to choose between a stock and a bond. However, this approach does not account for the risk undertaken as well. Example 12.17 A company is listed in a stock exchange and trades for a stock price of 20 Euro. It makes during the year total dividend payments of 1 Euro. The same company has issued a bond that matures in 5 years and makes annual coupon payments of 2.2%. Its market bond price is 1,100 Euro for a face value of 1,000 Euro. An investor considers that he or she is exposed at the same level of risk with both the bond and the stock of the company and he or she thus compares the current yield with the dividend yield. Which of the two would he choose to invest in? Answer

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The annual coupon paid by the company is 2.2% × 1, 000 Euro = 22 Euro

(12.64)

Therefore, the current yield of the bond is given by CY =

22 = 0, 02 = 2% 1, 100

(12.65)

The dividend yield of the stock is given by DY =

1 = 0, 05 = 5% 20

(12.66)

If the investor wishes to choose between the stock and the bond with the use of the current and dividend yields then he or she is better off with the stock. His or her rationale is possibly explained by the fact that since he or she is putting his or her money at the same company, he or she is better off being an owner than a lender. According to the exercise, he or she feels exposed to the same level of risk from the bond and the stock of the company and thus aims for the highest yield.

12.4.3

Selecting Between a Stock and a Bond with the Use of Utility Functions

Investors can use a utility function to rank all potential investments and choose the one that maximizes their utility function. A utility function assigns a score to each and every investment in such a way that usually privileges return and penalizes risk. We used the word ‘usually’ because the last statement holds true mainly for risk averse investors; i.e. investors that expect higher return to assume higher risk. There are several functions that can be used as utility functions. They need to exhibit two main characteristics for being representative of the behavior of a rational, risk averse investor. We would expect that if the final wealth of the investor increases then the utility function ranks higher the investment with the higher final wealth, i.e. to be an increasing function. If its first derivative exists then it should be positive. Between two investments with the same final expected wealth, one of which generates this final wealth with certainty the investor should prefer

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the investment that produces the final wealth with certainty. That is, the utility function should rank higher the investment with the more certain outcome. Such a function should be concave. If its second derivative exists then it should be negative. An investor (and a function) that exhibits this characteristic is called risk averse. The risk aversion of the investor is measured with the Arrow–Pratt absolute risk aversion coefficient (Bodie, 1996; Luenberger, 1998): A(x) = −

U  (x) U  (x)

(12.67)

Two utility functions are said to be equivalent if they give the same ranking to all possible alternative investments, even if they do not give them the same grade. We can easily find that if U is a utility function then V = aU + b for a > 0, b is consistently a utility function equivalent to U. But what is important is that if U and V are two equivalent utility functions then they are associated with a linear transformation of this form. The risk aversion coefficient is the same for all equivalent utility functions due to the normalization achieved by its denominator. The main utility functions are presented below (Bodie, 1996; Luenberger, 1998): a) Linear: it is given by equation U (x) = x

(12.68)

An investor with this utility function evaluates investments based on the amount of expected final wealth ignoring the risk taken. For this reason this the investor is characterized as risk neutral. This is also shown by the fact that A(x) = 0

(12.69)

b) Exponential: it is given by equation U (x) = −e−ax

(12.70)

for a fixed number a>0. In this case we observe that A(x) = a

(12.71)

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i.e. the risk aversion coefficient is constant. c) Logarithmic: it is given by equation U (x) = ln(x)

(12.72)

and is therefore defined when x is positive (i.e. x> 0). In this case we observe that the risk aversion coefficient is A(x) = 1/ x,

(12.73)

i.e. the investor’s aversion to risk decreases as the level of wealth increases. In other words, investors who have this utility function are willing to take a bigger risk as they become more financially secure. d) Power: it is given by equation, U (x) = bx b ,

(12.74)

for a fixed number b ≤ 1 and b = 0. e) Quadratic: it is given by equation, U (x) = x − bx 2 ,

(12.75)

for a fixed number b > 0. This function is increasing when x < 1/(2b). A generalization of this utility function is given by U (x) = ax − 21 bx 2 ,

(12.76)

with a > 0 and b > 0 fixed numbers. The function is increasing for x < a/b. Example 12.18 Assume that the utility function is given by Eq. (12.75) above. Suppose that all the random variables we are interested in belong to this set. Then for a portfolio with final wealth W1 , what we are looking for is the maximum of: E[U (W1 )] = E(aW1 − 21 bW12 ) = aE(W1 ) − 21 bE(W12 ) ⇒ E[U (W1 )] = aE(W1 ) − 21 b[E(W1 )]2 − 21 bvar(W1 )

(12.77)

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T. POUFINAS

for all the portfolios that the investor can construct. W1 is not known with certainty and is therefore a random variable. In contrast, the initial wealth of the investor, W0 , is known with certainty. The utility function is “applied” to the final wealth of the investor and gives us the utility U (W1 ) of the investor for each investment. But this is also a random variable and therefore does not yet constitute a “score” for the investment. For this we consider its expected value, i.e. E[U (W1 )]. With this approach a real number is assigned to each investment and the investor’s problem is to choose the investment that maximizes the expected value of the utility function, i.e. to find ℘

max E[U (W1 )], ℘

(12.78)

where ℘ runs across all the possible portfolios that can be built from the available assets (with or without risk). This problem can be equivalently converted to a problem that uses the expected percentage return of the investments instead of the final monetary value of the portfolio or the final wealth of the investor in monetary terms. To do that we assume that the investor has a horizon of 1 term (e.g. 1 year) to see that W1 = W0 · (1 + r ),

(12.79)

where r is the return of the investment portfolio at which W 0 was placed. The return r is also a random variable. Assuming, without loss of generality that the initial wealth is one monetary unit, i.e. that W0 = 1, then W1 = 1 + r

(12.80)

The maximization problem we need to solve changes to max E[U (1 + r℘ )] ℘

(12.81)

If we accept the mean–variance analysis of Harry Markowitz (1956), then investors choose among investments by using a utility function of the form U (σ, r ) = r −

1 · A · σ 2, 2

(12.82)

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where r σ A

denotes the expected return of the investment denotes the standard deviation (or volatility) of the return of the investment denotes the risk aversion coefficient of the investor – normally an integer.

For a risk-averse investor A is greater than zero; for a risk-neutral investor A is equal to zero; and, for a risk-lover A is less than zero. Example 12.19 Assume that an investor wants to invest in a company and he or she tries to assess whether it is better to be an owner or a lender, i.e. buy the bond or the stock. From historical data he or she has estimated an expected annual return of 5% for the bond and 10% for the stock. The corresponding standard deviation of the return is 10% for the bond and 20% for the stock. (a) If the risk aversion coefficient of the investor is 2 would he or she prefer the bond or the stock? (b) If the risk aversion coefficient of the investor is 3 would he or she prefer the bond or the stock? (c) If the risk aversion coefficient of the investor is 4 would he or she prefer the bond or the stock? Answer For all possible values of the risk aversion coefficient we use Eq. (12.82) to estimate the utility function of the investor. Substituting for the three different values of A and the expected return as well as the standard deviation as given in the exercise we get the following results:

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A

Utility of bond

Utility of stock

2 3 4

0.040 0.035 0.030

0.060 0.040 0.020

We therefore see that (a) For A = 2 the utility of the stock (0.060) is higher than the utility of bond (0.040), hence the investor would invest in the stock of the company. A = 2 indicates a relatively low risk aversion; hence this particular investor does not penalize risk enough to refrain from investing in the stock. (b) For A = 3 the same holds true, even though the two utilities are closer as the utility of the stock (0.040) is less than before and the utility of the bond (0.035) is higher than before. (c) For A = 4 the utility of the stock (0.020) is lower than the utility of the bond (0.030), hence the investor would invest in the bond of the company. A = 4 indicates a higher risk aversion; hence this particular investor penalizes risk more and thus prefers the bond over the stock. In general, we can use this or any utility function to find the optimal mix between the stock and the bond if the investor wishes to split his or her wealth among the two. Let rB σB rS σS ρ y 1−y

denote the expected return of the bond denote the standard deviation (or volatility) of the return of the bond denote the expected return of the stock denote the standard deviation (or volatility) of the return of the stock denote the correlation coefficient of the returns of the bond and stock denote the portion (percentage) of the wealth invested in the bond denote the portion (percentage) of the wealth invested in the stock

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denote the expected return of the portfolio that invests in the bond and the stock denote the standard deviation (or volatility) of the return of the portfolio

rP σP

Recall that r P = y · r B + (1 − y) · r S

(12.83)

and σP2 = y 2 · σB2 + (1 − y)2 · σS2 + 2 · y · (1 − y) · ρ · σB · σs

(12.84)

If we substitute in Eq. (12.82), we have that U (σ, r ) = r − −

1 · A · σ 2 = [y · r B + (1 − y) · r S ] 2

1 · A · [y 2 · σB2 + (1 − y)2 · σS2 + 2 · y · (1 − y) · ρ · σB · σs ] =: h(y) 2 (12.85)

This means that this is a function of y. To maximize it, we need to find its maximum in terms of y. Its first derivative becomes: 1 · A · [2 · y · σB2 − 2 · (1 − y) · σS2 + 2 · (1 − 2y) · ρ · σB · σs ] 2 = (r B − r S ) − A · [y · σB2 − ·(1 − y) · σS2 + (1 − 2y) · ρ · σB · σs ] h  (y) = (r B − r S ) −

= (r B − r S ) − A · [y · (σB2 + σS2 − 2ρ · σB · σs ) − σS2 + ρ · σB · σs ]

(12.86)

This is a linear equation of y and can be solved for y to find the critical point. What is of interest is to confirm that at this critical point the function h has a maximum. To confirm that we calculate its second derivative: h  (y) = −A · (σB2 + σS2 − 2ρ · σB · σs )

(12.87)

ρ ≤ 1 ⇒ −ρ ≥ −1

(12.88)

Observe that as

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T. POUFINAS

Hence as the standard deviations and the correlation are nonnegative, we derive that σB2 + σS2 − 2ρ · σB · σs ≥ σB2 + σS2 − 2 · σB · σs = (σB − σS )2 ≥ 0 (12.89) Therefore, for different standard deviations or ρ strictly less than 1 the latter is strictly positive and thus for A > 0 h  (y) < 0

(12.90)

Consequently, the critical point is a maximum. The value of y for which the maximum is achieved, let it be y* indicates the optimal split of the wealth of the investor between the stock and the bond of the company. If the standard deviations had been by chance equal, and the correlation had been equal to 1, then the investor would have obviously invested all his or her wealth in the asset with the highest return as Eq. (12.82) becomes U (σ, r ) = r −

1 1 · A · σ 2 = [y · r B + (1 − y) · r S ] − · A · σ 2 =: h(y) 2 2 (12.91)

where σ = σB = σ S

12.5

(12.92)

Company Financing

As we have already explained bonds, as well as stocks are not only investment securities but primarily financing vehicles for the companies that issue them. Companies may receive capital in both ways; i.e. receive both debt and equity capital. Debt can be in the form of a bank loan or via a bond issue. Equity capital can be through common equity or preferred equity. It can also be through retained earnings. Recall that the return that an investor achieves through a security is the cost of capital (or financing cost) for the issuing company (or entity). The cost of capital is defined as the percentage of return that a company must earn from its investments in order to maintain its value & price. It is nothing more than the percentage to be returned to the investors who chose to invest their money in it.

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It is clear that a company ensures its operation by attracting funds from individual or institutional investors who are interested in placing them in it. The company must have a clear picture of the disposal of the funds it raises in order to optimize their use. It is clear that while investors investing capital are interested in optimizing the expected return on expected risk, the company is trying to optimize the capital mix it raises. This is because part of it comes from lending (through bonds for example), another part from equity (e.g. from retained earnings), a third part from the issue of preferred shares and one last part from the issue of common shares. To these, we may need to add non-controlling interests if applicable. The weighted average cost of each investment instrument is the cost of capital of the company. The company analyzes the budgeted capital investments to make the appropriate decision. Practically, the respective return for each asset class it uses comes from the use of the Net Present Value (NPV) and the Internal Rate of Return (IRR). The company is interested in determining the optimal capital structure. By achieving this, the cost of capital is minimized and the value of the company is maximized The components of the cost of capital are summarized as follows: i. Cost of debt capital (or borrowed funds)—B This is the interest rate of the loan or the yield to maturity of the bond. ii. Cost of preferred equity—PE This is calculated with the fixed dividend version of Gordon’s formula as kP E =

DP E , PP E − I C P E

where k PE D PE P PE IC PE

denotes denotes denotes denotes

the the the the

required return on preferred shares fixed dividend per share share price issuing cost per share

iii. Cost of common equity from retained earnings—RE

(12.93)

614

T. POUFINAS

This is calculated either with the use of CAPM as   kRE = r f + β · r M − r f

(12.94)

or with the use of Gordon’s formula as (where RE replaces PE in the index and 0 denotes the present time t = 0) kRE =

D0 · (1 + g) +g PR E, 0

(12.95)

iv. Cost of common equity from stock issuance—CE This is calculated as before, except for the consideration of the issuing cost (where CE replaces PE in the index and 0 denotes the present time t = 0), i.e. kC E =

D0 · (1 + g) +g PC E, 0 − I CC E

(12.96)

The Weighted Average Cost of Capital (WACC) is the weighted average of each cost component of each source of financing. The weights are the participation rates of each source of capital: W ACC = w B · k B · (1 − φ) + w P E · k P E + w R E · k R E + wC E · kC E (12.97) where w i is the weight of source i and ϕ denotes the tax rate of the company. Usually, there is a tax deduction for interest due, which explains the multiplication of the cost of debt capital component by 1−φ.

Exercises Exercise 1 Company ABC uses equity for its financing. The company is listed in the stock exchange of its country of domiciliation. The company is considered mature and thus the growth rate of its dividends is considered fixed. It is known that the company just posted a profit of EUR 2 per share, of which 50% is distributed as a dividend. The dividend growth rate is 6%. The required/ expected rate of return on the stock is 16%. (a) What is the share price today?

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615

(b) What is the share price 3 years from today? (c) An investor has a horizon of 3 years and assumes that if he or she invests in the stock he or she will sell it in 3 years. He or she intends to invest the dividends he or she will receive from the share at the risk-free interest rate, which is 1%. What is the holding period return of the investor? Exercise 2 Company ABC uses equity for its financing. The company is listed in the stock exchange of its country of domiciliation. The company is considered mature and thus the growth rate of its dividends is considered fixed. It is known that the company just posted a profit of EUR 2 per share, of which 50% is distributed as a dividend. The dividend growth rate is 6%. If the share price today is EUR 17.67, what is the required/ expected rate of return? Exercise 3 Company ABC uses equity for its financing. The company is listed in the stock exchange of its country of domiciliation. The company is considered mature and thus the growth rate of its dividends is considered fixed. It is known that the company just posted a profit of EUR 2 per share, of which 50% is distributed as a dividend. If the share price today is 13.5 Euros, then what is the dividend growth rate? Exercise 4 Company ABC uses equity for its financing. The company is listed in the stock exchange of its country of domiciliation. The company is considered mature and thus the growth rate of its dividends is considered fixed. It is known that the company just posted a profit of EUR 2 per share. The dividend growth rate is 6%. The required/ expected rate of return on the stock is 16%. If the share price today is EUR 12.72, then what is the part of the profits that is paid out as dividend?

616

T. POUFINAS

Exercise 5 Company ABC uses equity for its financing. The company is listed in the stock exchange of its country of domiciliation. The company is considered mature and thus the growth rate of its dividends is considered fixed. It is known that the company just posted a profit of EUR 2 per share, of which 50% is distributed as a dividend. The dividend growth rate is 6%. The required/ expected rate of return on the stock is 16%. (a) If the share price on the stock exchange was EUR 15 today, would you buy it? If not, what position could you take in order to take a bet on the move of the stock. (b) For what required/ expected rate of return would you be willing to buy the stock at EUR 15? Exercise 6 Company ABC uses bonds and stock for its financing. • It has issued a bond with a nominal value of EUR 100 with 5 years remaining until its expiration with a 5% fixed coupon. The discount rate is 4% for the bond. • The company’s share is listed in the stock exchange. The company is considered mature and thus the growth rate of its dividends can be assumed to be fixed. The company just posted profit of EUR 2 per share, of which 60% is distributed as a dividend. The dividend growth rate is 8%. The required return on the share is 18%. • An investor has a horizon of 3 years and assumes that if he or she invests either in the bond or in the share he or she will sell it at the end of years. He or she intends to invest the coupons he or she will receive from the bond or the dividends he or she will receive from the share at the risk-free interest rate, which is 1%. (a) What is the price of the bond today and in 3 years? (b) What is the holding period return for the bond for the 3 years? (c) What is the share price today and in 3 years? (d) What is the holding period return of the stock for the 3 years?

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(e) Which asset will the investor choose if he or she uses the holding period return for the 3 years as a selection criterion? What does this approach not take into account? Exercise 7 An investor is considering two alternative investments: buying a bond or buying a stock. The bond matures in 4-years and has an annual coupon rate of 3%. Its face value is EUR 1,000. Its price at that time is also EUR 1,000. The stock just gave a dividend of EUR 10 per share and has a dividend growth rate of 10% per year. Its price at that time is EUR 220. The investor believes that the market prices both securities fairly. (a) What is the yield to maturity of the bond? (b) What is the expected rate of return on the stock? (c) If the investor wants to choose between the two by comparing the previous two rates of return, which investment will he or she choose? Exercise 8 Suppose that your investment universe consists of two risky assets, a bond and a stock and one risk-free asset. The bond and the stock have an expected return of 10% and 15% respectively and a standard deviation of 20% and 25% respectively. The return on the risk-free asset is 5%. The correlation coefficient between the bond and the stock is 0.5. (a) Find the expected return and standard deviation of a portfolio that invests 40% in the bond and 60% in the stock. (b) An investor has a utility function U (σ, r ) = r −

1 Aσ 2 2

The risk aversion factor A is 4. He or she wants to allocate his or her wealth between the portfolio of question (a) and the risk-free asset by maximizing his or her utility function.

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T. POUFINAS

(i) How should he or she distribute his or her wealth between the portfolio and the risk-free asset? (ii) What is the expected return and standard deviation of the final portfolio? Exercise 9 An investor has EUR 100,000 available to invest. (I) He or she is offered to acquire a share with a price of EUR 100, for one year, which with a probability of 60% will rise to EUR 140 while with a probability of 40% it will fall to EUR 80. (a) What is the expected return E(r) of the investor (for the amount of EUR 100,000) in amount and in percentage? (b) What is the standard deviation σ of the expected return (for the amount of EUR 100,000) in amount and in percentage? (II) Alternatively the investor can place his or her money in an annual T-Bill, which is considered to be risk-free, with a face value of EUR 100 which gives an annual yield of 2%. (c) What is the expected return E(r) of the investor (for the amount of EUR 100,000) in amount and percentage? (d) What is the standard deviation of the return σ (for the amount of EUR 100,000) in amount and percentage? (III) The utility function of the investor is given by the formula: U(σ , E(r))=E(r)−σ 2 (e) Will the investor prefer the stock or the T-Bill? Exercise 10 An investor has EUR 1,000,000 available for investment. (I) He or she is offered to acquire a share with a price of EUR 10, for one year, which with a 50% probability will rise to EUR 14, while with a 50% probability will fall to EUR 6.

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(a) What is the expected profit E(r) of the investor (for the amount of EUR 1,000,000) in amount and percentage? (b) What is the standard deviation σ of the expected profit (for the amount of e 1,000,000) in amount and percentage? (II) Alternatively, the investor can place his or her money in a tenyear bond with a face value of EUR 100, which pays an annual coupon at a rate of 5% per annum. The price of the bond at the given time is equal to its face value. It is estimated that, for one year, the bond price has a 70% probability of rising to EUR 105 and a 30% probability of falling to EUR 95. (c) What is the expected profit E(r) of the investor (for the amount of EUR 1,000,000) in amount and percentage? (d) What is the standard deviation σ of the expected profit (for the amount of EUR 1,000,000) in amount and percentage? (III) The utility function of the investor is given by the formula: U(σ , E(r))=E(r)−0.02σ 2 (e) Will the investor prefer the stock or the bond?

References Bodie, Z., Kane, A., & Marcus, A. J. (1996). Investments (3rd ed.). The McGraw Hill Companies, Inc. Chamberlain, L., & Hay, W. W. (1931). Investment and speculation. Studies of Modern Movements and Basic Principles (p. 55). Henry Holt and Company, 1931. https://www.maggs.com/investment-and-speculation-studies-of-mod ern-movements-and-basic-principles_231247.htm. Accessed: January 2021. Corporate Finance Institute. (2021a). Bonds vs. stocks. https://corporatefinanc einstitute.com/resources/knowledge/trading-investing/bonds-vs-stocks/. Accessed: January 2021. Corporate Finance Institute. (2021b). Common vs. preferred shares. https:// corporatefinanceinstitute.com/resources/knowledge/finance/common-vspreferred-shares/. Accessed: January 2021. Federal Reserve. (2020). Secondary market corporate credit facility, April 9, 2020. https://www.federalreserve.gov/newsevents/pressreleases/files/mon etary20200409a2.pdf. Accessed: 28 October 2020.

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Fama, E. F., & French, K. R. (1992). The cross-section of expected stock returns. The Journal of Finance, XLVII (2), 427–465. Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33: 3–56. Luenberger, D. G. (1998). Investment science. Oxford University Press. Markowitz, H. (1956). The optimization of a quadratic function subject to linear constraints. Naval Research Logistics Quarterly, 3(1–2), 111–133. Siegel, J. J. (1998). Stocks for the long run—The definitive guide to financial market returns and long-term investment strategies (2nd ed.). McGraw-Hill.

CHAPTER 13

Hedging, Speculation and Arbitrage

Derivatives offer to the interested investors, traders and other participants of the financial markets several opportunities when used on their own or along with the underlying asset. The use of the derivatives with or without the corresponding asset allows them to pursue a series of alternative targets with regards to their investment output. Investors can pursue three different, basic (or plain vanilla) and common strategies, with the use of the derivative and the underlying fixed income security; hedging, speculation and arbitrage. The same holds true of course for all asset classes and the underlying asset category may not be significant for the description or implementation of the strategy. This distinction classifies investors and traders in three categories; hedgers, speculators and arbitrageurs. All strategies are implemented with a specific investment horizon, as specified by the corresponding investors. Hedgers are the investors that implement hedging. They aim at lifting volatility from their final outcome; they wish to make their (investment) result or total payoff as certain as possible and somehow immunize it from the movement of the variables that may affect it. If possible they wish to “lock” it. They are not interested in outperforming the markets; they simply want to manage the risk and secure their portfolio, cash flows, disbursements or collections. They take a position both in the derivatives and the spot markets.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_13

621

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Arbitrageurs are the investors that attempt to identify arbitrage opportunities. They act in a similar manner to hedgers. They aim at “locking” a result, but this time a profit. As a matter of fact they wish to make a profit without potentially using any initial capital. They can do so by taking a position both in the derivates and the spot markets to capitalize in a potential imbalance they have identified between the two markets. Such an imbalance may be due to a potential mispricing of the derivative versus the underlying asset. Like hedgers, arbitragers achieve in managing their (market) risk, by securing a profit. Speculators—the investors that pursue speculation, on the other hand, do not take positions in both markets; they simply take a bet on the evolution of the derivative (or underlying asset). They do so by attempting to forecast the development of one or more of the variables that affect the price of the derivative and take a position that will deliver a profit. However, contrary to hedgers and arbitragers, they are exposed to risk; their investment outcome is not “locked”. If their prediction of the direction of the variable movement proves to be correct, then they make a profit, whose magnitude may vary. If their prediction on the other hand proves to be wrong, then they may suffer a loss. In this chapter we explain how hedging, speculation and arbitrage can be done, what the conditions to achieve it are and what the risks that may appear are. This chapter exposes the reader to the strategies that may be exploited to achieve hedging, speculation and arbitrage.

13.1

Hedging

As explained, hedging is a strategy that is followed by investors that wish to have an investment outcome as certain as possible with the use of a derivative and an underlying security. They do so by taking the opposite positions in the derivative and in the spot markets. This means that if they take a long position on the (fixed income) security, then they will take a short position on the derivative and vice versa; that if they take a short position on the (fixed income) security, then they will take a long position on the derivative. The former is called short hedge, whereas the latter is called long hedge. Hedging is applied for a specific investment horizon. As a result the derivative selected needs to have a matching or nearest maturity date. Sometimes, for liquidity purposes, the investors may choose to select shorter derivative expiration dates and then roll the hedge forward.

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Hedging simply renders the result as certain as possible. It is primarily a risk management approach. Investors may have been better off without hedging. However, they are in place to determine it only at the end of their investment horizon. For example, the holder of the long position in the derivative, i.e. the counter party that will buy the underlying asset through the derivative, could have had profited if he or she had purchased the underlying asset in the spot market at the end of his or her investment horizon, if its spot price was lower than the delivery or strike price determined by the derivative. In a similar manner, the counterparty with the short position, i.e. the party that will sell the underlying asset as agreed by the derivative contract, could have had benefited more, if he or she had sold the underlying asset in the spot market, if its spot price was higher than the delivery or strike price determined by the derivative. Of course derivatives do provide such freedom; recall that the counterparties are bound to such an agreement in forward contracts. In that case they are “locking” the price that they will pay or receive for the underlying (fixed income) security. Nonetheless, options protect investors against the movements of the price of the underlying asset that are adverse to them depending on whether they are set to buy or sell. Options give to the counterparty with the long position the right to decide whether he or she will transact through the option. Consequently, if an investor uses an option instead of a forward to hedge, then he or she may decide not to exercise the option if the price movement was favorable to him or her. Recall though that options have a premium that is paid at the time the position is taken by the counterparty with the long position. Options offer protection and demand a premium; they thus offer some sort of insurance protection. Forward (and futures) contracts on the other hand carry no initial cost and are thus obligations. We present futures and forward contracts first. 13.1.1

Using Futures or Forward Contracts to Hedge

13.1.1.1 Optimal Hedge Ratio Hedging with the use of a futures or forward contract aims at minimizing the risk of the position of the investor. If this risk is fully hedged then the hedge is called perfect. Sometimes, especially when using futures contracts to hedge, the fixed income asset to be hedged and the underlying asset of the future contract may be different. The ratio of the size of the position

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in futures contracts to the size of the exposure in the spot market is called the hedge ratio. When the fixed income security underlying the futures contract is the same as the one that is hedged, then the hedge ratio is 1. However, when the two securities are different then the hedge ratio may not be 1. A hedger is interested in calculating the optimal size of the position to be taken in the futures or forward contracts versus his or her position in the spot market. Optimal hedge ratio is the hedge ratio that ensures the minimum risk from both positions. To find it, we assume that we measure the risk of a portfolio of assets and derivatives by its variance or standard deviation. We denote by P: P: F: F : σ P : σ F : ρ P, F : hr: hropt :

the spot value of the fixed income portfolio being hedged. the change in the spot value of the portfolio during the hedging horizon. the futures price of the futures contract used. the change in the futures price during the hedging horizon. the volatility of P. the volatility of F . the correlation of P and F . the hedge ratio. the optimal hedge ratio.

Determining the optimal hedge ratio resembles to the Markowitz meanvariance analysis. Let us first assume that the investor is long the portfolio and thus takes a short position in the futures contract. We let  denote the portfolio of the fixed income portfolio and the futures contract (Hull, 1997):  = P − hr · F.

(13.1)

 = P − hr · F.

(13.2)

The change in its value is

The volatility (or variance) of the portfolio is 2 2 2 σ = σP + hr2 · σF − 2 · hr · ρP, F · σP · σF .

(13.3)

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We want to minimize the volatility of the portfolio as a function of hr. To do that we take the first derivative with respect to h and set it equal to zero to find that 2 0 =2 · hr · σF − 2 · ρP, σP = ρP, F · . σF

· σP · σF ⇒ hr

F

(13.4)

As the second derivative of the variance of the portfolio with respect to h is 2 0 = 2 · σF >0

(13.5)

we can conclude that when hr is given by Eq. (13.4), then the variance of the total portfolio is minimum. We thus say that this is the optimal hedge ratio, i.e. hropt = ρP, F ·

σP . σF

(13.6)

We can use the optimal hedge ratio in order to find the optimal number of contracts—where applicable. More precisely, if Q P: Q F: N opt :

denotes the size of the portfolio to be hedged, denotes the size of the futures contract, denotes the number of futures contracts used for hedging,

then the number of futures contracts required is N opt = hr opt ·

QP . QF

(13.7)

Example 13.1 Let as consider an investor that has decided to invest 10,000,000 Euro in a single corporate bond issue. He or she wishes to hedge against the risk of a fall in the price of the bond (or rise in the interest rates) for a year. To do that he or she decides to use a futures contract, that matures in one year and is written on the 10-year government bond of the country of domiciliation of the issuer. The standard deviation of the change of the

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bond price is 0.16; the standard deviation of the change of the futures price is 0.18; the correlation coefficient of the two is 0.9. If one futures contract is written on a bond face value of 100,000 Euro, then how many contracts will the investor need to hedge his or her position? Answer The optimal hedge ratio is calculated as hr opt = ρP, F ·

σP 0.16 = 0.8. = 0.9 · σF 0.18

(13.8)

Consequently, the number of contracts needed is N opt = hr opt ·

QP 10, 000, 000 = 80. = 0.8 · QF 100, 000

(13.9)

This means that the investor will have to take a short position in 80 contracts in order to hedge his or her investment in the corporate bond issue. 13.1.1.2 Duration-Based Hedging In Chapter 5 we presented duration-based hedging; we briefly present it here for completeness of the presentation of hedging the risks that stem from variables that determine the price of fixed income instruments. One—potentially the primary—such variable is the interest rate. As mentioned also in Chapter 5, futures contracts can be used to protect fixed income portfolios from interest rate moves, i.e. hedge against interest rate risk. The question is how many contracts are needed for such hedging strategies to be implemented. The approach that uses duration in order to find the number of contracts required to pursue hedging against interest rate is called duration-based hedging. Recall that we assume continuous compounding and we denote with (Hull, 1997). P: DP : F: DF : N: r: y:

the value of the fixed income portfolio being hedged. the duration of the fixed income portfolio being hedged. the futures price of the futures contract used. the duration of the asset underlying the futures contract. the number of futures contracts used to hedge against the interest rate moves. a small shift of the interest rate = y a small shift of the yield curve = r

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We use once and again Eqs. (4.16) and (4.58) of Chapter 4 to see that P ≈ −PD P r

(13.10)

F ≈ −FD F r.

(13.11)

and

Assuming we are long the fixed income portfolio, we take a short position in the futures contracts in order to protect the portfolio from the interest rate moves. If we create a new portfolio with a short position in N futures contracts (N to be found) then the value of the new portfolio would be P − NF.

(13.12)

A (small) shift of the interest rate should have no impact in the value of the portfolio, i.e. P − N F = 0 ⇒ −PD P r + NFD F r PD P =0⇒N = . FD F

(13.13)

This number of contracts is called the duration-based hedge ratio (Hull, 1997). The duration of the fixed income portfolio along with the futures contract becomes equal to zero. We repeat the example of Chapter 5, in order to illustrate the use of futures contracts in hedging. Example 13.2 It is now July. An investor has invested 10 million Euro in a government bond portfolio. He or she wants to hedge against interest rate moves until year-end and for that he or she uses a futures contract maturing in December (understanding that his or her portfolio may remain unhedged for a few days). The futures price is 94. The duration of the portfolio is 7.5 years and the duration of the cheapest-to-deliver bond is 6.8 years. One futures contract is for the delivery of 100,000 Euro face value of bonds. The number of contracts is N=

10, 000, 000 · 7.5 = 117.32. 100, 000 · (94/100) · 6.8

(13.14)

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T. POUFINAS

This is usually rounded to the nearest whole number, i.e. 117 contracts short. This means that the portfolio is slightly underhedged. Example 13.3 A company issues a 5-year bond with a 3% coupon, which is the current interest rate corresponding to its credit rating. The minimum bond denomination, i.e. each bond note—if it had a physical form—has a face value of 1,000 Euro. The amount the firm wants to raise is 100 Million Euro. The placement of the bond in the primary market is planned in 3 months from today, when the 5 years until its expiration will start counting. The company estimates that the full issue will be disposed to the investors. However, the management of the firm worries that the pandemic will lead to inflationary pressures and thus interest rates will increase in the next quarter, which will result in receiving less than 100 Million Euro. The management addresses a bank to enter a forward (or) futures contract for the sale of the bonds at their nominal value. But the bank believes just the opposite; i.e., that interest rates will fall, as a result of the support provided by the Central Bank. The firm therefore agrees to enter into a forward (or futures) contract for the purchase of the bond with a delivery price equal to the face value of the bond. They agree that each contract concerns the delivery of 100 bond notes at their face value. As the company does not want to forfeit the benefit of a potential drop in the interest rates, it decides to protect in this way 50% of the issue. As the futures contract is written on the bond itself the hedge ratio is considered to be equal to 1. a. What is the position of the issuing company per forward (or futures) contract and what is the position of the bank? How many bond notes and what amount does it concern? What is the name of the strategy implemented by the company? b. What is the number of the contracts required to implement its strategy? What is the risk that the company remains exposed to? c. After 3 months the interest rate corresponding to the credit rating of the company has moved to 4.13%. How much did the company raise through the primary bond market and what amount through the forward (or futures) contracts? How does it compare to the amount

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it wanted to raise in total? What is the corresponding percentage of improvement or deterioration? How does it compare to the amount it would raise if it did not implement this strategy? What is the corresponding percentage of improvement or deterioration? d. Was the choice of the company to implement this strategy a right one? Did it manage to protect the amount it wanted to raise? What would be the result if the interest rate had fallen instead of having risen? Illustrate with a case/example with the data of the exercise. Answer a. The company has taken a short position for 100 bond notes per forward (or futures) contract at 1,000 Euros per bond note, i.e. for 100 × 1,000 = 100,000 Euro. The bank has taken a long position for the exact same number of notes and the same amount. The strategy implemented by the company is a short hedge since the bond has not been issued at the present time and therefore has a positive position in the spot market. So a short position is required in the forward (or futures) contract to hedge the risk. This strategy is called short hedge. b. The total number of securities is the ratio of the total issuance amount and the face value of each note. It is therefore calculated as: 100, 000, 000/1, 000 = 100, 000.

(13.15)

To calculate the number of contracts we take into account that the hedge ratio is 1, i.e. hr = 1. The number of contracts required takes into account the amount to be hedged which is 50% × 100,000 × 1,000 = 50,000,000 Euro, the size of the contract which is 100 × 1,000 = 100,000 Euro and the hedge ratio which is equal to 1. Therefore, the number of contracts is: N = 1 X (50, 000, 000 /100, 000) = 500.

(13.16)

The company remains exposed to the risk of price change for the part of the securities (issue) that it has not hedged.

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T. POUFINAS

c. If the interest rate remained constant and equal to the coupon, i.e. 3% then the selling price of each bond would be: P0 =

30 30 30 1, 030 30 + + + = 1, 000. + 1.03 1.032 1.033 1.034 1.035

(13.17)

since the bond coupon in amount is 1,000 × 3% = 30. The selling price of each bond note in three months, when the bond is issued in the primary market, but also the expiration of the forward (futures) contract becomes: P0 =

30 30 30 30 1, 030 + + + + = 950. 2 3 4 1.0413 1.0413 1.0413 1.0413 1.04135 (13.18)

From the forward contract the firm will receive: 500 × 100 × 1,000 = 50,000,000.

(13.19)

This amount, however, corresponds to 500 × 100 = 40,000 notes. The remaining 50,000 notes were made available in the primary market for 950 Euros each, i.e.: 50,000 × 950 = 47,500,000.

(13.20)

Therefore the total amount raised by the company is the sum of the two previous amounts, namely: 50,000,000 + 47,500,000 = 97,500,000.

(13.21)

The amount is 2,500,000 Euro less than the amount of 100,000,000 Euro it wanted to raise or 2,500,000/100,000,000 = 2.5% lower. If it had not implemented the short hedge, then it would have received: 100,000 × 950 = 95,000,000.

(13.22)

Therefore the amount it received is by 2.5 Million Euro higher with the short hedge it applied. In percentage it amounts to 2,500,000/95,000,000 = 2.63% higher than if it had not implemented the hedging strategy.

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d. From the result we can judge that the company’s decision to carry out a hedging strategy was correct as it managed to make a profit in the futures market. With the use of a forward (or futures) contract it managed to hedge the selling price for the part of the issue it chose to hedge. The company chose to pursue a partial hedge as it wanted to benefit from a potential fall or no move in interest rates, which would mean an increase or no move in the selling price above or from the nominal value of 1,000 Euro. If the opposite had happened, that is, if there had been a fall in interest rates, i.e. an increase in the bond price above the face value of 1,000 Euro then it would have benefited from the securities that were made available directly to the primary market, but would have suffered a loss from the part of the securities it decided to hedge through the forward (or futures) contracts, which in any case would have sold at their nominal value. If, for example, the interest rate had fallen to 1.94%, then the market price of the bond in the primary market in 3 months would have been: P0 =

30 30 30 1, 030 30 + + + = 1, 050. + 1.0194 1.01942 1.01943 1.01944 1.01945 (13.23)

From the forward (or futures) contract it would have received: 500 × 100 × 1,000 = 50,000,000.

(13.24)

This amount, however, corresponds to 500 × 100 = 50,000 notes. The remaining 50,000 notes were made available in the primary market for 1,050 Euro each, i.e.: 50,000 × 1,050 = 52,500,000.

(13.25)

Therefore the total amount raised by the company is the sum of the two previous amounts, namely: 50,000,000 + 52,500,000 = 102,500,000.

(13.26)

The amount is 2,500,000 Euro higher than the amount of 100,000,000 it wanted to raise or 2,500,000/100,000,000 = 2.5% higher. If it had not implemented a short hedge, then it would have received: 100,000 × 1.050 = 105,000,000.

(13.27)

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T. POUFINAS

Therefore the amount it received is by 2.5 Million Euro lower with the short hedge it applied. In percentage it amounts to 2,500,000/105,000,000 = 2.38% lower than if it did not implemented the hedging strategy. 13.1.2

Using Options to Hedge

We demonstrate the use of options to hedge with an example. Example 13.4 Let us consider an investor who has just invested 980,000 Euro in a bond issue that matures in 10 years time, whose minimum denomination note has a face value of 1,000 Euro and sells for 980 Euro. He or she has thus purchased 1,000 such notes. The investor wishes to protect his or her portfolio for the next 12 months against 950,000 Euro. He or she could hedge with the use of futures contracts that trade on the bond. However, he or she does not want to miss the opportunity of a potential price increase (or interest rate drop). He or she therefore takes a long position in a European put option to sell the bond for 950 Euro per 1,000 Euro of face value note (or minimum denomination bond). Assume that each option is written on 100 such notes (i.e. a face value of 100,000 Euro). The investor needs 10 such contracts in order to hedge his or her portfolio. The option has a premium of 10 Euro per note, thus a total cost of 1,000 Euro (=10 Euro × 100 notes) per contract or 10,000 Euro (=1,000 Euro × 10 contracts) for the entire hedge. If on the maturity date of the option the bond price is above 950 Euro, then the option will not be exercised and will expire worthless. The investor has forfeited the 10,000 Euro premium he or she has paid for the put option. The value of his or her bond will be higher than 950,000 Euro and the total worth of his or her portfolio—incorporating the cost of the option—will be at least 940,000 Euro. If on the maturity date of the option the bond price is below 950 Euro, then the investor can sell the entire bond position for 950,000 Euro. After subtracting the cost of the option the total amount drops to 940,000 Euro. We realize that—modulo the cost of the option—the value of the bond portfolio of the investor is greater than or equal to 950,000 Euro at all

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times. It is greater than or equal to 940,000 Euro if the cost of the option is considered. This strategy is called protective put.

13.2

Speculation

Contrary to the use of futures and forward contracts or options for minimizing or potentially lifting risk, investors can use the very same derivatives to assume risk. This is done by taking a bet in the move of a certain market or asset via a derivative instead of taking directly a comparable bet on the underlying asset. 13.2.1

Using Forward or Futures Contracts to Speculate

We show how an investor can speculate with the use of forward or futures contracts with an example. Example 13.5 Let us consider an investor that wishes to bet on the drop of the interest rates or an increase in the bond prices. He or she has two alternatives; the first is to purchase a bond in the spot market and wait for the interest rates to drop; the second is to take a long position on a futures contract that is written on the bond. Assume that his or her investment horizon is 3 months, the bond price has a face value of 1,000 Euro, its price is currently at 950 Euro and it pays no coupon during the life of the futures contract. The futures contract matures in 3 months, its size is for a face value of 100,000 Euro and its price is F0 = 950 · e0.05·0,25 = 961.95

(13.28)

per 1,000 Euro of face value, for an interest rate of 5%. The investor can either bet directly on the spot market or via the derivatives market. • Let as first assume that he or she buys the bond at 950 Euro. For a face value of 100,000 Euro he or she pays 95,000 Euro. – If he or she proves to be right then interest rates will drop and the bond price will go up. Let’s say that its price at the end of

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3 months is 970 Euro. Then he or she makes a profit of 2,000 Euro = (970 − 950) × 100 Euro.

(13.29)

– If he or she proves to be wrong, then if interest rates rise and the bond price drops, let’s say to 930 Euro, then he or she will post a loss of 2,000 Euro = (930 − 950) × 100 Euro.

(13.30)

• If he or she takes a comparable position in the futures market, then there are also two potential outcomes. – In the first case, when the bond price goes up at the end of 3 months the futures price coincides with the spot price and he or she thus makes a profit of (970 − 961.95) × 100 = 805.05 Euro.

(13.31)

– In the second case, when the bond price drops, as the spot price coincides with the futures price, he or she makes a loss of

(930 − 961.95) × 100 = −3,194.95 Euro.

(13.32)

The futures price outcome appears to be worse in both scenarios; however there are two main differences. • First, in the purchase of the bond the investor makes an initial disbursement of 95,000 Euro, whereas in the case of the futures contact he or she makes no initial disbursement. If he or she did not have the 95,000 Euro, then he or she would have to borrow it and thus carry the burden of the interest. As a matter of fact, assuming that the interest rate is the same, i.e. 5% flat, then he or she would owe Loan0.25 = 95,000·e0.05·0.25 = 96,194.95.

(13.33)

This means that his or her final profit or loss would have to account for interest and is thus

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– In the case that the bond price rises to 970 equal to a profit of 97,000 − 96,194.95 = 805.05 Euro.

(13.34)

– In the case that the bond price drops to 930 equal to a loss of 96,194.95 − 97, 000 = −3,194.95 Euro.

(13.35)

We observe that it equals the profit or loss of the alternative bet in the futures market. We would obtain a similar result if we had considered that he or she already was in possession of the 95,000 Euro but instead of investing it at 5% for 3 months he or she decided to buy the bond. Similarly, we can view the cash flows from the side of the investor that chose to take the bet in the futures market. Taking a long position in the futures contract requires no initial payment. Hence, he or she could have invested the 95,000 Euro, provided he already had this amount, for 3 months at 5%. This would earn him or her an interest amount of Interest0.25 = 95, 000 · e0.05·0.25 − 95,000 = 1,194.95. (13.36) Adding this amount to the profit or loss he or she encountered from the futures contract we realize that it becomes – When the bond price rises to 970 Euro, equal to a profit of 805.05 + 1,194.95 = 2,000 Euro.

(13.37)

– When the bond price drops to 930 Euro, equal to a loss of −3,194.95 + 1,194.95 = −2,000 Euro.

(13.38)

• Consequently, it is equal to the profit or loss of an equivalent bet in the spot market. • Second, taking a position in the futures market requires the deposit of a “guarantee”, called margin. This amount is deposited in the margin account, it increases or decreases on a daily basis, depending

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on the increase or decrease of the exposure. This amount is much smaller from the value of the underlying asset. However, when the position is closed, the amount is returned to the investor. As a result, it does not alter the outcome of the bet with the use of a futures contract. Following this discussion a natural question rises; why take the bet through a futures contract and not by purchasing the bond directly? There are two straightforward answers. • The first is that the purchase of the bond requires a cash outflow, whereas the position in the futures contract does not. The initial margin amount, along with any potential add on is returned to the investor when the position is closed. • The second is that the use of futures contracts allows also taking a bet on the drop of the price, by taking a short position. One could say that this can be achieved also with short-selling. This is true; however it entails borrowing the bond and then returning it to its owner at the end of the short-sell. Nonetheless, short selling is done via a derivative (lending/borrowing contract) in some exchanges. Finally, one needs to note that both positions can be closed before the end of the horizon of the investor so as to contain potential losses or benefit from any profit that can be realized earlier. 13.2.2

Using Options to Speculate

We use an example to show how an investor can speculate with the use of options. Example 13.6 Let us consider the same exact set up with Example 13.5. However, let us assume that the investor has the choice to speculate also with the use of a European call option that matures (or that expires) in 3 months time from today. The strike price of the option is 965 Euro. We make the hypothesis that the option is also for the delivery of a bond face value of 100,000 Euro. As he or she wishes to bet on the increase of the bond price (or equivalently the drop of the interest rate) he or she can take a

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long position on a European call option. We assume that the volatility of the futures contract that is written on the bond is 0.18 = 18%. This time the investor can use also the European call option to speculate. We use Black’s Model to find the value of the call option. This is given by Eq. (5.75) of Chapter 5—with the help of Eqs. (5.77) and (5.78). More precisely, d1 =

d2 =

ln(961.95/965) + (0.182 /2) · 0.25 ln(F/ X ) + (σ 2 /2)T = = 0.0098 √ √ σ T 0.18 · 0.25 (13.39) √ √ ln(F/ X ) − (σ 2 /2)(T ) = d1 − σ T = 0.0098 − 0.18 · 0.25 = −0.0802. √ σ T (13.40)

Therefore, the premium of the option at t = 0 is c0 = e−r T [FN(d1 ) − XN(d2 )] ⇒ c0 = 100 · e−0.05·0.25 · [961.65 · N (0.0098) − 965 · N (−0.0802)] ⇒ c0 = 100 · 39.60 = 3, 960.04 (13.41) The investor faces two scenarios at the maturity of the option: • If the bond price is greater than or equal to 965 Euro, then he or she will exercise the option. If the bond price was as before 970 Euro, then he or she exercises the option for a payoff of 97,000 − 96,500 = 500 Euro.

(13.42)

Taking into account the premium of the option this leads to a loss of −3,960.04 + 500 = −3,460.04 Euro.

(13.43)

• If the bond price is less than 965 Euro, then he or she will not exercise the option and it will expire worthless. He or she will record a loss of an amount equal to the premium, i.e. −3,960.04 Euro.

(13.44)

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We observe that the final output with regards to the option is a bit worse in all these cases compared both with the outright position at the spot market or the long position at the futures market. This is due to the premium that the investor has to pay. • With the option he or she starts earning a profit only if the bond price ends higher than the strike price plus the premium of the option, i.e. for a bond price higher than 965 + 39.60 = 1,004.60 Euro.

(13.45)

• In a similar manner, he is better off at the drop of the bond price only if the bond price ends less than the current price minus the premium of the option 950 − 39.60 = 910.40 Euro.

(13.46)

In the aforementioned discussion we need to consider some additional factors. • As before, the impact of investing the amount of 95,000 Euro which is the initial investment cost for the purchase of the bond has to be considered. The loss of the investor that chooses to take a long position on the call option is reduced by the interest earned on this amount (potentially less the premium paid). • Depending on the position in the option, there could be a margin amount that needs to be deposited at the margin account. As explained before, this amount is returned upon the position closeout. • Taking a position in the option offers protection as the loss is limited at the level of the premium—in case the markets move in the opposite direction. This is not the case in the outright bond position. • In all cases, the position can be closed prior to the end of the investment horizon in order to reduce losses or realize any gain. In the options market, this can be done by collecting the premium at the time of the position closeout, as the investor will take a short position.

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639

Arbitrage

Using Forward or Futures Contracts to Exploit Arbitrage Opportunities

We illustrate the use of a forward contract to perform arbitrage with an example. Example 13.7 Let us consider a forward contract that is written on a 100 securities (minimum denomination notes) of a bond with a face value of 1,000 Euro. The forward contract has a delivery date of 12 months (1 year) and a delivery cash price of 1,000 per bond security. The bond matures in 10 years and 9 months. Its cash price today is 920 Euro and is expected to pay a coupon of 4% semi-annually, i.e. of 20 Euro, in 3 months and 9 months from today. The interest rate is assumed to be flat at 3% with continuous compounding. a. What is the value of the forward contract today for the long position? b. Are there any arbitrage opportunities? If yes, how can an investor take advantage of them? c. If an investor had been willing to enter the forward contract today, then what would have been the delivery price that he or she would have accepted? d. What is the value of the forward contract (with the initial delivery price) six months later if the bond cash price at the time is 960 Euro? Answer a. In order to find the value of the forward contract, we use Eq. (5.10) of Chapter 5 to see that f 0 = (P0 − I0 ) − Ke−r T1 ,

(13.47)

where the delivery date of the forward contract is on time T 1 and the maturity of the bond on time T 2 > T 1 ; K denotes the delivery price, r is the risk-free rate, P 0 denotes the price of the bond and

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I 0 denotes the present value of the coupons (or other cash income) paid in the interval [0, T 1 ]. As the bond is expected to make two coupon payments of 40 Euro each during the life of the forward contract their present value I 0 is calculated as (per security) I0 = 20 · e−0,03·0.25 + 20 · e−0,03·0.75 = 39.41.

(13.48)

Replacing for P 0 = 920, I 0 = 39.41, K = 1,000, r = 3%, and T 1 = 1 year and multiplying by the number of securities (100) we receive that f 0 = 100 · [(920 − 39.41) − 1,000e−.0.03·1 ] = −8,985.11. (13.49) This is the value of the forward contract when this contract is written. b. Such a contract offers arbitrage opportunities for the short position holder, as it has a negative value. More precisely an investor can pursue the following strategy. At time t = 0 the investor • Takes a short position on the forward contract to deliver 100 securities (minimum denomination bond notes) for 1,000 Euro each. • Borrows 920 Euro per security (note), i.e. a total of 92,000 Euro, for 1 year at a rate of 3%. • Buys 100 securities (notes) of the bond for 920 Euro per security (bond note), i.e. a total of 92,000 Euro. At time t = 1/4 = 0.25 (3 months) the investor • Receives a coupon payment of 20 Euro per security (note), i.e. a total of 2,000 for the 100 securities (notes). • Invests the coupon until the delivery date, i.e. for 9 months (0.75 years) at a rate of 3%. At time t = 3/4 = 0.75 (9 months) the investor • Receives a coupon payment of 20 Euro per security (note), i.e. a total of 2,000 for the 100 securities (notes). • Invests the coupon until the delivery date, i.e. for 3 months (0.25 years) at a rate of 3%.

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At time t = 1 year (12 months) the investor • Sells the bond under the terms of the forward contract for 1,000 Euro per security, i.e. a total of 100,000 Euro for the 100 securities (notes). • Receives the proceeds of the reinvestment of the coupons he or she received, i.e. I1 := (20 · e0.03·0.75 + 20 · e0.03·0.25 ) = 40, 61

(13.50)

per security (note), or a total of 4,060.57 Euro for the 100 bond securities (notes). • Repays the loan, which including taxes has grown to the amount of Loan1 := 920 · e0,03·1 = 948.02

(13.51)

per security (note), or a total of 94,801.82 Euro for the 100 bond securities (notes). • ➜ Records a profit of Profit1 = 100,000 + 4,060.57 − 94,801.82 = 9,258.75 Euro. (13.52) c. In order for a counterparty to enter a forward contract it would have to be fair. This means that its value would have to be zero. Solving for the delivery price we find that 0 = f 0 = (P0 − I0 ) − K e−r T1 ⇒ K = (P0 − I0 ) · er T1 ⇒ K = 100 · (920 − 39.41) · e0.03·1 = 90,741.25.

(13.53)

d. Six months from today there is only one coupon to be paid during the remaining lifetime of the forward contract. Its present value at the time is (per security) I0.5 = 20 · e−0.03·0.25 = 19.85.

(13.54)

As per Eq. (5.15) of Chapter 5, this yields that the value of the forward contract at that time becomes f 0.5 = (P0.5 − I0.5 ) − K e−0.03·0.25

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= 100 · (960 − 19.85) − 1, 000 · e−0.03·0.25 = −4,496.25. (13.55) One can see that no matter what the value of the forward contract is at time t = 0, if the prices evolve in the indicated manner, then the forward contract will have a negative value for the party with the long position and, of course, a positive value for the party with the short position. The party with the short position could perform arbitrage by mimicking the aforementioned steps. However, this is not a strategy that could have been planned from the inception of the contract if its value at time t = 0 had been 0. If the party with the short position had not been already in possession of the bond to be delivered, then he or she could have borrowed an amount equal to its spot price for six months, in order to purchase the bond and wait until the delivery date to deliver the bond under the terms of the forward contract and repay the loan. He or she would still have posted a profit of 4,564.20 Euro, as he or she had done before. We need to note two things as a result of this example: 1. We used the cash bond price and the cash delivery price, as the value of the forward contract holds true with cash prices. If any of the prices had been the quoted price (either the bond price or the delivery price), then we would have had to convert it to the corresponding cash price by adding the accrued interest. 2. If the value of the forward contract had been positive, then it would have been the counterparty with the long position that would have been able to take advantage of the arbitrage opportunities. To do so he or she At time t = 0 would • Short-sell the bond (i.e. would borrow it and sell it). • Invest the proceeds at the given interest rate until the delivery date. At the delivery date t = 1 year would • Collect the invested amount with interest • Use the investment outcome to purchase the bond under the terms of the forward contract.

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• Return the bond. • Pay the future value of the coupons paid during the life of the forward to the owner of the bond. • ➜ Post a profit from the difference of the delivery price and the future value of the coupons from the investment outcome. 13.3.2

Using Options to Exploit Arbitrage Opportunities

We demonstrate the employment of options to exploit arbitrage opportunities with the use of an example. Example 13.8 Let us assume that a European call option is written on a bond with face value of 1,000 Euro. The bond matures in 10 years from today, has a cash price today of 980 Euro and makes annual coupon payments. The option matures in three months from today and has a cash strike price of 970 Euro. The bond makes no coupon payments during the life of the option. The premium of the option is 20 Euro. The interest rate is 5% with continuous compounding. An investor can profit from arbitrage opportunities as follows: At t = 0 he or she • Sells the bond short for 980 Euro. • Takes a long position on the call option and pays the premium of 20 Euro. • Invests the difference of 960 Euro for 3 months at 5%. At t = 0.25 (3 months). –If the bond price is higher than (or equal to) the strike price of 970 Euro, then the call option can be exercised. The investor • Will receive from the investment an amount of Investment0.25 := 960 · e0,05·0.25 = 972.08 • Will exercise the call option and purchase the bond for 970. • Will return the bond.

(13.56)

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• Will post a profit of 972.08 − 970 = 2.08 > 0.

(13.57)

–If the bond price is lower than the strike price of 970 Euro, then the call option will not be exercised. The investor • Will receive from the investment an amount of Investment0.25 := 960 · e0,05·0.25 = 972.08

(13.58)

• Will buy the bond from the bond market for a price PT < 970 Euro. • Will return the bond. • Will post a profit of 972.08 − PT > 972.08 − 970 = 2.08 > 0.

(13.59)

Thus the investor can lock into a profit of at least 2.08 Euro in all cases. In the aforementioned discussion and examples, please note that it was implicitly assumed that there are no transaction costs or taxes. The latter can wipe out any potential arbitrage opportunities. Whenever though such arbitrage opportunities are observed, then they can last for a small period of time. This is due to the fact that investors will attempt to profit from them and thus the price of the call option will increase (as investors will be taking long position), whereas the price of the bond will drop (as investors will be short selling it), until equilibrium is reached.

Exercises Exercise 1 An investor wishes to invest 1,000,000 Euro in a single corporate bond issue. He or she wishes to hedge against the risk of a price decrease (or interest rate increase) for 3 months. To do that he or she decides to use a futures contract, that matures in 3 months and is written on the 5year government bond of the country of domiciliation of the issuer. The standard deviation of the change of the bond price is 0.2; the standard

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deviation of the change of the futures price is 0.15; the correlation coefficient of the two is 0.6. If one futures contract is written on a bond face value of 100,000 Euro, then how many contracts will the investor need to hedge his or her position? Exercise 2 It is now January. An investor has invested 1 million Euro in a government bond portfolio. He or she wants to hedge against interest rate moves until the end of the 1st semester of the year and for that he or she uses a futures contract maturing in June (understanding that his portfolio may remain un-hedged for a few days). The futures price is 96. The duration of the portfolio is 5 years and the duration of the cheapest-to-deliver bond is 6 years. One futures contract is for the delivery of 100,000 Euro of face value of bonds. a .What position should the investor take on the futures contract? b .How many contracts does he or she need to hedge the portfolio? c .Consider a scenario of portfolio value at the end of the semester and explain how the hedge worked. Exercise 3 Repeat Example 13.3 from the perspective of a buyer of the bond and not the issuing company. What are your findings? Exercise 4 Let us consider a forward contract that is written on a bond with a face value of 1,000 Euro. The forward contract has a delivery date of 6 months (1/2 year) and a delivery cash price of 1,000 Euro per bond security. The bond matures in 7 years and 8 months. Its cash price today is 940 Euro and is expected to pay a coupon of 6% semi-annually, i.e. of 30 Euro, in 4 months and 8 months from today. The interest rate is assumed to be flat at 4% with continuous compounding. a. What is the value of the forward contract today for the long position?

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b. Are there any arbitrage opportunities? If yes, how can an investor take advantage of them? c. If an investor was willing to enter the forward contract today, then what would be the delivery price that he or she would accept? d. What is the value of the forward contract (with the initial delivery price) five months later if the bond cash price at the time is 980 Euro? Exercise 5 Let us assume that a European call option is written on a bond with face value of 1,000 Euro. The bond matures in 5 years from today, has a cash price today of 970 Euro and makes annual coupon payments. The option matures in four months from today and has a cash strike price of 960 Euro. The bond makes no coupon payments during the life of the option. The premium of the option is 20 Euro. The interest rate is 4% with continuous compounding. a. Are there any arbitrage opportunities? b. How can an investor take advantage of it? Exercise 6 Repeat Exercise 5 for a European put option instead of a European call option. Exercise 7 Repeat Example 13.5 with the consideration that the investor wishes to bet on the increase of the interest rates or the decrease of the bond prices. Exercise 8 Repeat Example 13.6 with the consideration that the investor wishes to bet on the increase of the interest rates or the decrease of the bond prices.

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Exercise 9 Find a real example of a bond or any other fixed income security and a derivative written on it and investigate whether there is a possibility to speculate. How would you pursue it? Exercise 10 Find a real example of a bond or any other fixed income security and a derivative written on it and investigate whether there are arbitrage opportunities. a. How would you exploit it? b. What are additional factors that you need to consider that potentially do not allow the exploitation of such opportunities?

Reference Hull, J. C. (1997). Options, futures and other derivatives (3rd ed.). Prentice Hall International, Inc.

CHAPTER 14

Bonds and Regulation

Bonds and other fixed income instruments are securities that are issued in the primary markets and are then publicly traded in the secondary fixed income markets or are held in the private markets when they constitute private debt. No matter what their form or the market they are traded in is, they are subject to a set of laws and regulations. The financial and debt crisis of the previous decade unveiled the importance of the applicable law and regulatory framework. Investors suddenly realized that a default is not unimaginable and that even a country— member state of the European Union could face it. The question of what are their rights when a company or a state cannot honor its obligations became pressing. The importance of the law under which a bond was issued emerged. At the same time concerns were raised as to whether all bond issues, especially corporate or structured were suitable for all investors; the protection of investors, the reassurance that fixed income markets are transparent, fair and efficient, and the prevention from systemic risk came to the forefront. All of the sudden fixed income investments and in particular bonds (even government bonds) did not seem to be any more the safe haven they were perceived to have been. The significance of an efficient regulatory framework that would safeguard the integrity of the market and the interests of the investors was realized.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8_14

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Regulation sometimes is thought as restrictive; however, when in place, it fosters the structured liberalization of the market, it ensures the employment of high quality staff in the financial services, it introduces high ethical standards, it promotes high business standards and secures that firms are prudently run. Rigorous regulation increases the confidence of all market players as it fosters controls and checks, efficient risk management processes, transparency, market discipline, self-regulation and self-assessment as well as compliance. Financial institutions and enterprises find value in these principles as when they operate within such a regulatory framework they can gain the trust of the investors, who at the same time feel more confident, precisely because such principles are in place. Good regulation stems from the ability of the regulatory/supervisory authorities to first act as advisors to the supervised parties by offering guidance before they act as auditors—if necessary. But even in this case, when the actions taken are constructive and corrective—if needed—but not necessarily punitive for inadvertent and non-crucial slips, regulators could be seen as partners. To that end effective regulation can be achieved if the regulatory/supervisory authorities are empowered to properly monitor their underlying markets and are populated with the suitable staff that has the necessary skills to carry out these tasks. This is important especially in countries at which there are more than one supervisory authorities that need to cooperate. Regulation both in the USA and in the European Union has been increasing the last years so as to set level playing field among investors, issuers and regulatory authorities and exchanges. Consequently, bond issuance and pricing may be affected in particular when it comes to corporate bonds, convertible bonds, covered bonds, as well as other fixed income instruments. In this chapter we try to present the regulation that applies to bonds and this is the added value to the reader.

14.1

Capital Markets Union

The Capital Markets Union (CMU) is the undertaking that the European Union (EU) pursues in order to create a single transaction environment (or framework)—a single market—for capital. It aims at providing benefits to all stakeholders; the enterprises, the individuals/households and the member states. It is thus anticipated to appropriately streamline the

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available funds to enterprises, offer increased choices for savers—investors, and strengthen the economy of the EU as a whole—and as a result the economies of the member states (European Commission, 2020a). The CMU aims at facilitating the flow of capital (investments and savings) across the EU, creating a level playing field, in a similar manner that European legislation has done in other areas, such as banking and insurance, with the introduction of a common regulatory framework. The objective is that individuals (citizens—investors/households), enterprises and states will all benefit from this uninterrupted flow of capital. As a matter of fact it is seen as lever that will lift the EU economy from the crisis and assist it weather the challenges that emerged from the experience of the pandemic (European Commission, 2020a). Bonds, and in particular corporate, covered, structured and green fixed income securities, as well as alternative fixed income investments fall— among other financing tools—within the scope of the CMU. This is brought forward even more when these instruments are used for the financing of SMEs or are used as investments vehicles by individuals. They are considered as means that can assist the growth of SMEs as well as of the economies. Furthermore, they are viewed as assets, which, within the framework of a single market for capital, can facilitate the development of green and sustainable sectors of the economy, technology, as well as the achievement of adequate pensions for retirees (European Commission, 2020a). 14.1.1

The History of CMU

The origins of the CMU may be found at the very creation of the European Economic Community (EEC) or Common Market with the Treaty of Rome back in 1958. The idea was further reinforced with the establishment of the Single Market in 1986, with the Single European Act and the free movement of capital through the Maastricht treaty in 1992. It was intensified in 1999 with the Financial Service Action Plan and the introduction of the Euro. However, a fully operational single market is not in place yet. The more recent history has a series of milestones starting with the 2015 Action Plan that incorporated CMU as a top priority and set the grounds for its establishment. In 2016 the European Commission (EC) followed the progress and set out the next steps to accelerate the completion of CMU. In 2017 the EC implemented further steps towards the

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achievement of CMU via the mid-term review. All of the adopted actions have been completed by 2019. In 2020 the new action plan on the CMU was introduced (European Commission, 2020a). 14.1.2

The Benefits of the CMU

The introduction of the common currency, the euro, was probably the single most important step towards the economic union of the member states of the EU. However, the market-related institutions operate still to a great extend independently. As a result the same holds true largely for the market players. Although several steps have been taken towards a single market via the convergence of banking and insurance, as well as investor protection regulation and legislation the capital markets are still not fully integrated. Consequently, the individuals and the enterprises cannot enjoy yet the advantages of seamless investing and funding respectively. The CMU is anticipated to deliver increased reliability, efficiency and depth in both. The experience of pension funds, which seem to be the biggest investors in the different asset classes, including alternative investments, teaches us that investing in the real economy can have significant contribution to growth and can relieve the states from the burden they have to carry (Poufinas & Kouskouna, 2017). Moreover, the alternative investments constitute an important source of funding for SMEs and they unveil the role of the stock exchanges and other institutions when they take place via the appropriate funds, such as ETFs (Poufinas & Polychronou, 2018). The return to growth is imperative in order to exit the distress that the pandemic has caused. The capital injected by the central bank(s) may have worked as a cushion to prevent from further decline of personal and business incomes but it will not be enough to secure growth. It will be the market(s) that are expected to be steam engine that will drive growth anew. Investing and financing will have to be streamlined through it (them) and having a single market in place will allow for the optimal use of capital for the recovery and sustainability of growth. All stakeholders can benefit from the CMU; the individuals/households, the enterprises, the society, the member states and the EU as a whole. As explained above, the individuals have the opportunity to optimize the use of their current wealth through a wide array of savings and investments so as to achieve their future wealth targets. The

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enterprises, and in particular the SMEs, can attract capital that will help them grow and be able to stand up at a pan-European or even global level (European Commission, 2020a). The society will enjoy the advantages of the CMU via (i) the green and digital transition, which will be facilitated by the funds that will be directed to companies that foster it; (ii) the improvement of the pension adequacy at retirement for the ageing populations; and (iii) the attraction of funds for the implementation of long-term projects that will promote innovation, research and development for new technologies (and not only) (European Commission, 2020a). Finally, the member states and the EU as a whole will gain from the CMU as it will enable (i) increased resilience against adverse economic conditions and distressed environments; (ii) integrated to a single EU and thus stronger capital markets; (iii) higher employment and economic growth for the member states and the EU as a whole; (iv) completion of the economic and monetary union; and (v) global strengthening of the euro (European Commission, 2020a). 14.1.3

What Has Been Done so Far Towards the Establishment of the CMU?

The legislative proposals that have been put forward so far by the EC in order to build a CMU are (European Commission, 2020b): 1. The establishment of a simple, transparent and standardized securitization that aims at broadening the investment possibilities and intensifying lending capabilities towards the European individuals/households and enterprises. 2. The introduction of uniform prospectus regulation that intents to enable especially small and medium enterprises (SMEs) to enter the financial markets. 3. The adoption of a series of measures addressing collective investment funds for the formation of the appropriate regulatory framework (such as the setting up of a European Venture Capital Fund Regulation (EuVECA) and a European Social Entrepreneurship Funds Regulation (EuSEF)) so as to boost venture capital and social investment in the European Union and enable the distribution of collective investment funds at a cross-border level by

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lifting unnecessary requirements/restrictions and homogenizing local practices. 4. The launch of a Pan-European Personal Pension Product (PEPP) that will allow citizens to accumulate their retirement savings in a seamless manner across the EU member states. 5. The use of covered bonds as a means to provide capital to the banks with a long-term horizon and backing to the real economy. 6. The employment of crowdfunding as a source of financing for start-ups, simultaneously protecting investors from the associated risks. 7. The review of the prudential rules for the investment firms in order to set uniform standards for the larger, systemic financial institutions and potentially simplified standards for smaller firms—through the application of a proportionality principle. 8. The adoption of rules that foster preventive restructuring and second-chance offering for the rescue of candid business persons that went bankrupt or are facing financial strain. 9. The advancement of more proportionate rules that will allow SMEs to enter the capital markets—known as SMEs growth markets— while safeguarding the interests of market participants (investors, intermediaries, operators, issuers, etc.). 10. The removal of uncertainties pertaining to the ownership of claims when they are set across the (borders of the) member states by specifying which national law applies. 11. The proposal to further reinforce the supervision of EU financial institutions so as to better contain or prevent money-laundering and terrorist financing. 12. The amendment of the European market infrastructure regulation (EMIR) in order to secure that the supervision exercised by the European Union over central counterparties (both EU and nonEU) is robust and effective. 14.1.4

Challenges and the New CMU Action Plan

There are still many things that need to be put in place in order to enjoy a single market. One could say that this is the case for any other dimension the European Commission has established common rules. Preparation was necessary before uniform norms could be applied. The roots of the

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challenges are pretty much the same and they are related to the historical, the customary and the cultural differences of the member states; as such they need to be respected so that the single market processes are accepted. The aforementioned determinants can be identified in several aspects of the legislation that has been (or still is) applicable to the member states such as the tax enforcement, the supervision implementation or the insolvency framework. As with the introduction of any other joint European undertaking, the ingredients of the success are similar; attentiveness, commitment and support—primarily political—are needed to overcome potential showstoppers (European Commission, 2020a). The response of the EC to the aforementioned challenges is a new action plan. It paves the route towards a single capital market. The path towards CMU is taken step by step so as to secure that all hurdles that impede capital from moving freely are gradually lifted. The stakeholders in this journey are the European Commission, the national authorities, as well as the market operators (European Commission, 2020a). There are 3 main pillars in this undertaking (quoted from European Commission, 2020a): I. To “support a green, digital, inclusive and resilient economic recovery by making financing more accessible to European companies.” II. To “make the EU a safer place for the individuals to save and invest long-term.” III. To “integrate the national capital markets into a genuine single market .” The new CMU actions per pillar are presented below (European Commission, 2020a): Pillar I: Actions addressing primarily SMEs 1. Launch an EU platform (European Single Access Point) where investors can find all the firm specific financial and sustainability particulars. 2. Facilitate the entrance of small and innovative enterprises in the public markets through leaner listing procedures so as to exploit the funds that are made available in these markets.

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3. Direct additional funding, aiming at longer investment horizons, to enterprises and infrastructure undertakings, privileging the ones that emphasize on smart, sustainable and inclusive growth. 4. Invite bankers and insurers to place their capital in equity and other long-term securities/asset classes. 5. Highlight the benefits of establishing a condition that provides access to alternative financing for enterprises whose loan request has not been approved. 6. Back the lending of European enterprises and primarily SMEs by bettering the securitization process and market. Pillar II: Actions addressing primarily retail investors 7. Ameliorate the level of financial literacy through a European-wide framework that fosters financial competence and stimulates member states to advance financial education and responsible investing. 8. Establish (or rather bring back) confidence for retail investors by rationalizing the amount of the provided information and improving the level of the offered financial advice. 9. Assist member states to better the adequacy of their pensions. Pillar III: Actions addressing primarily the establishment of a single market 10. Decrease cross-border investment-related costs through easier taxapplication processes. 11. Encourage further the alignment of the insolvency frameworks in the different EU countries. 12. Facilitate shareholders to secure their rights at a cross-border level. 13. Increase the offer of settlement services at a cross-border level within the EU. 14. Build a consolidated data-source to incorporate the trading conditions across all EU trading places in order to encourage competition. 15. Enhance investment protection and enable more actively investments at a cross-border level. 16. Enrich the single rulebook applicable to capital markets (and financial institutions) and support the convergence of the supervisory frameworks and practices of the different member states.

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CMU and Bonds

The capital markets union envisaged by the European Commission could not have left out the fixed incomes markets. As becomes evident in Sect. 14.3 the European regulation via MiFID II/MiFIR has taken several steps towards increasing the transparency and liquidity of the fixed income markets. This has also become evident from Chapter 7, where we discussed bond markets. Furthermore, as will be seen in Sect. 14.2, green bonds offer a tool that can finance the green and digital transition that is in the recovery agenda of the EU Green Deal, in order to ensure a sustainable growth. But what has already been done in the direction of capital markets union and bonds/fixed income instruments? What are the recommended next steps? There are two main fixed-income-related pillars exploited by the EU in the deployment of the capital markets union; these are covered bonds and the SME growth markets. More precisely, the legislative reforms that have been adopted in 2019, provision, among others, for a harmonized framework for EU covered bonds and rules promoting access to SME growth markets. 14.1.5.1 Covered Bonds Covered bonds are financial instruments that a credit institution can issue, which offer increased safety for the investors as they are supported by a set (pool) of (earmarked) assets, such as mortgages or public debt. The covered bond investors have a preferential claim on these assets in case the issuing entity (most often a bank) defaults on its obligation. In this respect—depending on the asset backing them—they resemble to mortgage-backed or asset-backed securities (European Council, 2019). The harmonized regulatory framework that has been set forth determines • An array of rules for common (harmonized) product requirements and the supervision applicable to covered bonds, thus making sure that the investors enjoy a great degree of protection. • A common definition and structural features for being qualified as a covered bond and thus be able to use the EU covered bond label which grants the comparative advantage of the preferential

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prudential treatment to covered bonds under the capital requirement regulation. • A stable source of capital for credit institutions, which in their turn can use it on one hand to lend individuals/households and business with better terms and on the other hand offer less risky alternative investments to investors (European Council, 2019). Covered bonds are of such importance for the European capital markets that they have been part of the ECB purchase programs, as has been discussed in Chapter 1. As becomes clear (also from the next section), they constitute a vehicle thought which the banks can secure capital and subsequently provide additional lending to SMEs. 14.1.5.2 SME Financing When it comes to SME financing, the European Council realizes that the small and medium enterprises need to exploit novel sources of capital, especially when the traditional financing sources (e.g. credit institutions) cannot provide them with the funds needed for their operations. The emphasis is given in the SME growth markets entrance requirements, introducing a proportionality to the SME growth market issuers, thus permitting easier access, without lifting though the market integrity and investor protection prerequisites (European Council, 2019). The relevant legislation realizes that it is the intention of the CMU to facilitate the access of SMEs to public markets with the issuance of bonds (and stock) and in this way lessen their dependence on bank lending through diversified market-based sources of funding. Furthermore, it admits that the bond private placement process, which may be a first step for non-bank financing for SMEs, can be rather burdensome and as a result dis-incentivize the issuing enterprise as well as the interested investors. In order to make the private placement of bonds more attractive it loosens the insider information disclosure requirements to the existence of an adequate non-disclosure agreement (European Parliament, 2019). Under the new CMU action plan, and in alignment with the (the need for a) green, digital, inclusive and resilient economic recovery, a series of measures have been brought forward in order to facilitate the access of companies to financing. As we have noted, the first CMU action plan already focuses on the funding of SMEs via bonds and furthermore

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via the issuance of private debt. The second (proposed) CMU plan realizes that bonds (and private equity) have had an increasing share in the company financing, complementing bank lending. This was partially due to the actions implemented via the first CMU plan. However, the use of certain forms of funding seems still to be under-represented. It is thus the aim of the proposed measures to assist companies in exploiting all available sources of capital. These efforts need to address not only the companies that need financing, but also the remaining stakeholders and in particular the investors and the markets/intermediaries. The proposed CMU action plan attempts to make the EU an even safer place for individuals to save and invest in the long term. Furthermore, it aims at integrating national capital markets into a genuine single market. The second CMU action plan consists of 3 objectives (or objective areas) and a total of 16 targets (or targeted action areas) in each of which the European Commission has put one or more specific actions indicating how to achieve the target and a time line for their completion. We highlight these targets (or targeted action areas) for the sake of completeness; however, we comment only on the targets/actions that address the financing of SMEs in connection with fixed income explicitly. The targets/actions that have been set forward are to: 1. Increase visibility of enterprises to investors at a cross-border level. 2. Facilitate entrance to public markets. 3. Promote instruments that foster long-term investment horizons. 4. Support institutional investors to pursue more long-term and equity funding. 5. Guide SMEs in considering alternative providers of financing. 6. Assist credit institutions (mainly banks) in offering more loans targeting the real economy. 7. Enhance the financial literacy of individuals. 8. (Re)Establish the confidence of retail investors in capital markets. 9. Assist citizens toward and during their retirement. 10. Ease the tax levels applied to cross-border investment. 11. Increase the predictability of the results of insolvency processes that pertain to cross-border investments. 12. Accommodate the participation of the shareholders. 13. Deploy settlement services at a cross-border level. 14. Set up a consolidated tape.

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15. Strengthen investment protection and accommodation. 16. Enhance supervision. Targets/actions 1–6 contribute to the objective of supporting a green, digital, inclusive and resilient economic recovery by facilitating the access of enterprises to financing. Targets/actions 7–9 aim at fulfilling the objective of making the European Union safer for individual savings and investments. Finally, targets/actions 10–16 pursue the success of the objective of establishing a true single market through the integration of the national capital markets. Apparently, although all objectives and targeted action areas are expected to contribute in the growth and strengthening of the bond/fixed income markets, it is primarily the first objective with its six actions that explicitly addresses fixed income investments. More precisely, action 2 aims at promoting and diversifying the access of small and innovative companies to funding, and to this direction the European Commission will purse the simplification of rules for the listing of SMEs in public markets. This is expected to homogenize the SME definition across financial legislation, potentially simplify the market abuse status quo and propose transitional provisions for enterprises that issue securities for the first time on regulated and SME growth markets. Actions 3 and 4 aim at supporting long-term investing and thus facilitating financing by all potential sources of capital; long-term investment funds, bankers and insurers. This is anticipated to take place during the review or implementation of the applicable regulations, i.e. the European long-term investment fund (ELTIF) regulation for long-term investment funds, Solvency II for insurance companies and Basel III for banks. The initiatives undertaken via these actions are anticipated to assist not only long-term SME equity placements (that is explicitly mentioned), but also debt investing. Action 5 paves the path for referring SMEs to alternative sources of capital when their credit application has been rejected by one or more banks. Its feasibility needs to be confirmed; however it can very well mean that SMEs can turn not only to equity finance (that is explicitly mentioned), but also to debt providers. Action 6 intends to alleviate the loan burden from the banks via the strengthening of the securitization market and framework so that they are able to offer sustainable and stable financing to the real economy, with emphasis on SMEs and the green transition. The review will include

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simple, transparent and standardized (STS) securitization, as well as nonSTS securitization. The implication in the fixed income markets is implicit; however, it means that structured securities will be made available so that SME lending further progresses (European Commission, 2020f, 2020g).

14.2

Green Bonds

One can readily see that the green (and digital) transition is high in the agenda of the EU globally but also specifically within the CMU effort. The EU endeavor in this direction has been branded as the Green Deal. Bonds—in this case known as green bonds—have a crucial role to play in it, as the funds to support Europe’s Green Deal are expected to be both public and private. 14.2.1

The EU Green Deal

To understand the support that regulation will provide to green bond issuance we note that the EU Green Deal has been put by the European Commission at the center of the European Recovery Strategy (Wacker et al., 2020). The proposal is that the new money that will be raised will be invested in order to (exact quote from the European Commission press release, European Commission, 2020h, pp. 1–2). • “Support the member states with investments and reforms .” • “Kick-start the EU economy through incentives to private investments .” • “Address the lessons of the crisis .” All pillars incorporate a series of projects that aim at facilitating the envisaged green (and digital) transition of the economies of the member states and of the EU as a whole. A significant portion of the Next Generation EU fund (that amounts to EUR750 billion) will be streamlined to the EU countries so as to expedite the green transition not only of the countries themselves but also of the enterprises, fostering climate neutrality, rural area support, structural changes towards a greener economy, and a series of goals that are aligned “with the new biodiversity and Farm to Fork strategies ” (European Commission, 2020h, p. 2).

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An important component of the Green Deal is the Climate Law, which entails a legal commitment for the EU to achieve climate neutrality by 2050. Other significant components are strategies and actions towards the supply of clean, affordable and secure energy, biodiversity, zero pollution, circular economy and sustainable food production. To pursue these targets a series of financial and economic reforms in the public and private sectors have been provisioned in the EU Green Deal. As far as the former are concerned they focus on “(i) the Sustainable Europe Investment Plan and (ii) the Renewed Strategy on Sustainable Finance.” The latter is concentrated on “(i) the rapid decarbonization of energy systems; (ii) the innovation in sustainable industry; (iii) the large-scale renovation of existing buildings; (iv) the development of cleaner public and private transport; and (v) the progress towards sustainable food systems ” (European Commission, 2020c, p. 9). In that direction, the Technical Expert Group formatted its report in March 2020 on the recommended EU Taxonomy. The taxonomy is expected to assist “investors, companies, issuers and project promoters ” to follow a route towards “the transition to a low-carbon, resilient and resource-efficient economy”. The taxonomy calls for “performance thresholds, the ‘technical screening criteria’, for economic activities which” (European Commission, 2020c, p. 2): • Have a considerable contribution “to one of six environmental objectives: – – – – – –

Climate change mitigation Climate change adaptation Sustainable and protection of water and marine resources Transition to a circular economy Pollution prevention and control Protection and restoration of biodiversity and ecosystems .”

• “Do no significant harm (DNSH) to the other five”—where applicable. • “Meet minimum safeguards, such as the OECD Guidelines on Multinational Enterprises and the UN Guiding Principles on Business and Human Rights.”

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These performance thresholds are expected to assist all interested parties (enterprises, investees/issuers and project promoters) to obtain green financing in order to better their environmental accomplishments and distinguish among their operations the ones that are already friendly to the environment. This is anticipated to drive the growth of industries with a low carbon footprint and at the same time will lead to the decarbonization of industries with high carbon footprint (European Commission, 2020c). Although the taxonomy preceded the Green Deal, it is anticipated to facilitate the implementation of the reforms pursued in the framework of the Green Deal. The key environmental objectives of the two are aligned. The EU taxonomy is an important step towards sustainable finance and is expected to have an effect on investors and issuers. 14.2.1.1 Other Initiatives One can note that EU Green Deal is probably the epitome of similar trends and initiatives around the globe. Moreover, it is probably the formal expression of existing requests that come from the heart of the society and relate to the environmental sustainability. Governments seem to be urged to embed climate change policy into the financing that will be streamlined to assist the economic recovery from the pandemic. Three representative cases are these of the UK, Canada and The Netherlands. In the UK, the Climate Change Committee (CCC), an independent adviser to the government, with the purpose of tackling the climate change, recommended, with a letter to the Prime Minister, that the effective climate policy plays a part in the recovery from the pandemic. The six principles brought forward (Climate Change Committee, 2020) in order to succeed in a resilient recovery are: 1. Support the economic recovery via climate investments. 2. Encourage positive long-term behaviors via new social norms. 3. Address the broader ‘resilience deficit’ on climate change via policies that will help mitigate the climate change risks. 4. Insert fairness as a fundamental principle by sharing the benefits of the actions on climate change. 5. Secure that the recovery does not result in increased climate risk or greenhouse gas emissions by making sure that the support provided to carbon-intensive sectors is balanced by counter-actions on climate

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change and that the new investments can endure against the climate change. 6. Employ fiscal changes to reinforce stimuli that will assist in reducing emissions by appropriately adjusting the relevant pricing (e.g. carbon) or tax policies so as to reach a level of zero-net emissions. Also in the UK, 206 leading UK business, investors and business networks, addressed an open letter to the Prime Minister, standing by the Government in its effort to face the recent pandemic crisis, but also urging it to offer a clear vision for exiting the crisis that is in accordance with the social, environmental and climate goals of the country. The letter states that an ambitious low carbon growth and environmental improvement plan can assist in successfully confronting the economic and social concerns that the UK is facing and can strengthen the economy so as to tackle future challenges, such as the ones that are associated to climate change. The letter indicates that the efforts to save and revive the economy must be in accordance with the net-zero-emissions target by 2050 (Lovell, 2020). In Canada, a Large Employer Emergency Financing Facility (LEEFF) was established in order to provide bridge financing to Canada’s large companies which, due to the pandemic, cannot meet their financing needs with conventional means. These are companies with annual revenues in excess of CAD 300 million. However, one important requirement of the support program is that the “recipient companies would be required to commit to publish annual climate-related disclosure reports consistent with the Financial Stability Board’s Task Force on Climate-related Financial Disclosures , including how their future operations will support environmental sustainability and national climate goals ” (Trudeau, 2020). In The Netherlands, 170 academics/scientists from 8 different Dutch universities signed a manifesto with a list of five proposals aiming at implementing a model that will allow post-pandemic development and will assist in weathering the pandemic, as well as future social and environmental crises. Two of the five suggestions are aligned with environmental targets. More specifically, they propose an agricultural transformation in the direction of regenerative agriculture based on biodiversity conservation and a move from the current potentially unnecessary and costly consumption and travel habits to reduced and thus more sustainable consumption and travel levels (Leiden University, 2020).

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Consequently, the EU Green Deal, along with the EU taxonomy, as well as other similar initiatives—some of which were mentioned above, is anticipated to have an impact on the global investing and financing activity, including fixed income investing. It will challenge the traditional bond markets and we trust it will gradually and eventually prevail in the relevant regulation. 14.2.2

What Are Green Bonds?

Green bonds are fixed income instruments the proceeds of which are used for the sole purpose of financing or refinancing, in part or in full, new and/or existing Green Projects that qualify and are aligned with the Green Bond Principles (ICMA, 2018). The Green Bond Principles and the Green Projects are presented in Sects. 14.2.4 and 14.2.5. Green bonds are issued by parties (issuers) that pursue projects that have proven contribution to environmental sustainability. They are appropriate fixed income securities for gaining access to long-term project financing. They are also suitable for responsible—sustainable investors, who are cautious of the use of the funds they provide (Wacker et al., 2020). The major types of green projects, which can as such be funded by green bonds, are renewable energy, green buildings, clean transportation, as well as water and wastewater management. The green bond principles incorporate the (disclosure of the) specific use and management of proceeds, the evaluation and the selection (process) of the project, the periodic reporting, as well as the voluntary verification by a third party (Wacker et al., 2020). A green bond resembles to a plain vanilla bond of the issuing entity and carries the same credit rating or default risk. As a result it offers comparable yield and overall return. Issuers though exploit the capital offered by sustainable investors, who grow rapidly, have long-term horizons, and enjoy the publicity gained by their sustainable investments or business (Wacker et al., 2020). 14.2.3

Why Are Green Bonds Becoming Popular?

The previous discussion unveiled the rationale behind the use of Green Bonds; they facilitate the flow of investments toward undertakings that are considered sustainable and climate friendly and are thus in line with the

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corresponding objectives of the interested parties (Wacker et al., 2020). Consequently, they are perceived as being used for a good cause and this leads to increased popularity. The first green bonds were issued more than a decade ago by investment banks; the intention was to support the Sustainable Development Goals (SDGs) of the United Nations. They focused primarily on achieving the climate-related targets (Wacker et al., 2020). In the course of time, Environmental, Social and Governance (ESG) emerged in the investment—corporate world, which brought such targets higher in the agenda. The green bond market aims at enabling issuers that implement environmental sustainability projects to gain access to the capital offered by sustainable investors (Wacker et al., 2020). To accelerate that and to attract investors a (first) set of Green Bond Principles (GPBs) was presented by the International Capital Market Association (ICMA, 2018). These guidelines were voluntary in nature. Their introduction though created the framework that triggered increased green bond issuance. Moreover, it is recommended that they are subjected to an external review. The review process aims at confirming the alignment of the bond or the bond program with the GBPs. 14.2.4

What Are the Green Bond Principles?

The Green Bond Principles (GBPs) constitute a set of guidelines that appropriate the issuance process of green bonds in order to promote transparency and disclosure so as to secure the integrity of the green bond market. They are voluntary in nature. They address all the stakeholders of that market; the issuers, the investors and the underwriters (ICMA, 2018). The issuers have a clear array of rules that they can (voluntarily) follow so as to issue a green bond that captures the trust of the investors. The information that needs to be disclosed has to be accurate and privilege transparency and integrity. The investors are assisted by the GBPs as they receive all the necessary information to assess the effect of the investments performed through green bonds to the environment. They appreciate the transparency as they can in their turn select investments that may have a true impact in their own areas of interest. The underwriters—and along with them banks and placement agents—are facilitated on setting

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the market standards, taking it one step further with the required disclosures and securing the transparency that will further boost the relevant transactions (ICMA, 2018). The Green Bond Principles define (i) the use of proceeds; (ii) the process for project evaluation and selection; (iii) the management of proceeds; and (iv) the corresponding reporting (as per ICMA, 2018, p. 3). A brief summary of each of them is given in the next paragraphs. 14.2.4.1 Use of Proceeds The use of proceeds concerns the very nature of green bonds. Green bonds do aim at employing the proceeds to fund Green Projects. The latter have to be clearly stated in the bond indenture that accompanies the issue. They need to offer specific benefits to the environment—ideally quantifiable. There needs to be a clear separation between the portions that aim at financing and refinancing, with the particulars of the refinanced green projects appropriately disclosed. The candidate projects cover a wide range of environmental goals such as “climate change mitigation, climate change adaptation, natural resource conservation, biodiversity conservation and pollution prevention and control ” (ICMA, 2018, p. 3). The eligible green projects are depicted in Sect. 14.2.5 that follows. 14.2.4.2 Process for Project Evaluation and Selection The issuing entity has to disclose the particulars of the green project that was selected to be funded through the proceeds of the green bond to be launched. More specifically, the issuer has to let the investors know the environmental sustainability targets to be met; the process by which the project was qualified as green—assessed as falling within one of the green project categories; and, the applicable eligibility criteria, exclusion criteria and risk management processes—should environmental or social risks connected with the project arise. Additional information contains the connection of the selected project with the overall strategy and policy of the issuer with regards to environmental sustainability and the green standards or certifications that have been followed in the project selection process. An external review is suggested in order to foster increased transparency (ICMA, 2018). The latter is described in Sect. 14.2.6 below. 14.2.4.3 Management of Proceeds The net proceeds of the green bonds have to be appropriately monitored by the issuer in a separate sub-account or sub-portfolio. Furthermore, this

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has to be verified by the issuer through a formal, appropriate, and internal lending and investment process for green projects. For the time interval that the green bond has not been repaid, the balance of the net proceeds has to be adjusted periodically so as to correspond to the eligible green projects that were funded during that interval of time. The issuer also has to inform the investors on the particulars of the investment of the unallocated balance. The recommendation is to have the internal monitoring and fund allocation processes audited by an external auditor or other third party as this enhances transparency and inspires trust (ICMA, 2018). 14.2.4.4 Reporting Reporting—at least on an annual basis—should inform the interested audience on the use of proceeds and any significant evolution. The projects that have drawn funds from the green bond, their (brief) description, the amount they drew, and their expected effect have to be included in the report. In case there are confidentiality restrictions, a generic presentation is suggested. The inclusion of qualitative and quantitative (key) performance indicators, along with the disclosure of the relevant assumptions and methodology, or any impact measurement is recommended. There are voluntary guidelines for impact reporting in certain areas (energy efficiency, renewable energy, water, and wastewater projects) that can be used by issuers to address the market players. Also, the illustration of the alignment of the green bond with the GBPs is proposed (ICMA, 2018). 14.2.5

What Are the Eligible Green Projects?

The main eligible Green Project categories are (quoted verbatim from ICMA, 2018, p. 4; Wacker et al., 2020, slide 10): • • • • • • • •

Renewable energy Energy efficiency Pollution prevention and control Environmentally sustainable management of living natural resources and land use Terrestrial and aquatic biodiversity conservation Clean transportation Sustainable water and wastewater management Climate change adaptation

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• Eco-efficient and/or circular economy adapted products, production technologies and processes • Green buildings The particulars of the aforementioned categories can be found in ICMA (2018). The list is not exclusive and there may be other projects characterized as green. They may also cover related and supporting expenditures such as R&D. The list however describes the majority of the types of the eligible projects. Furthermore, they do not intend to restrict the interested parties in identifying the environmental benefits they wish to quote. Taxonomies, like the one the EU published, are in place, and can serve as a compass to help stakeholders identify projects that qualify as green. 14.2.6

What Is the External Review Process?

In order to verify and safeguard the alignment of the green bond issuance or program with the GBPs the performance of an external review is suggested. Such an external review may be carried by independent consultants and/or institutions that specialize in environmental sustainability and/or other fields relevant to the launch of a green bond (e.g. the establishment of the green bond framework, the underlying assets, the procedures, or the reporting). The reviews are grouped into the following types: (i) second party opinion; (ii) verification; (iii) certification; and (iv) green bond scoring/rating (ICMA, 2018). They are highlighted in the following paragraphs. 14.2.6.1 Second Party Opinion This is an opinion issued by a party (institution) that is independent from the issuer and has proven know-how on environmental issues (sustainability). The opinion evaluates essentially the issuer for its alignment with the GBPs; its strategy, objectives, policy and processes, as well as the attributes of the projects to be funded in terms of environmental sustainability. The party that offers the second opinion has to be also independent from the party that potentially advised the issuer on the set up of its green bond framework; if not then appropriate Chinese walls are in order to secure the impartiality of the opinion (ICMA, 2018).

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14.2.6.2 Verification This is a statement of satisfaction of certain criteria that is also produced by a party independent from the issuer. The criteria may be connected with the business processes or environmental standards. The verification pertains to the assessment of the characteristics of the underlying assets in terms of environmental sustainability, the internal monitoring approach of the use of proceeds, the fund placement, the statement of environmental impact, as well as the reporting in line with the GBPs (ICMA, 2018). 14.2.6.3 Certification This is a confirmation that key features, such as the green bond or green bond framework or use of proceeds matches a prescribed external green standard or label. A qualified, accredited third party produces the certification after validating the consistency of the green bond with a set of criteria that need to be met (ICMA, 2018). 14.2.6.4 Green Bond Scoring/Rating This is a grade that assesses an important characteristic/aspect such as the green bond (itself) or green bond framework or use of proceeds according to an established scoring/rating methodology. It is performed by a qualified third party, such as a specialized research company or a rating agency. The evaluation process is based on specific metrics or benchmarks, such as the environmental performance, the process relative to the GBPs, etc. Although the scoring produced is different from the credit rating assigned by a rating agency, it can be the case that the latter depicts also environmental risks—if significant (ICMA, 2018). The external review may be full—covering all four pillars of the GBPs, or partial—covering only certain dimensions of the green bond or green bond framework. It can be that confidentiality restrictions apply though. In any case the disclosure of external reviews is suggested, as it fosters transparency and can increase investor confidence. At the same time, the external reviewers are recommended to publish their credentials and know-how as well as the scope of the reviews performed. 14.2.7

The Green Bond Market

The green bond market has reached USD760 billion as per May 28, 2020 (according to Wacker et al., 2020) and is anticipated to reach USD1 trillion as of 2021 (Barbiroglio, 2020). NN Investment Partners (2020)

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subscribes to this point of view estimating EUR1 trillion by year-end 2021 and even forecasts an amount of EUR2 trillion by year-end 2023. As per the Climate Bond Initiative data, the market has reached already USD867.8 billion by the end of H1 2020. Europe represented more than half of the total (55%) for the first time. The issuance in the first half of 2020 retreated to below half of 2019 levels in every geographical region, except for Latin America that remained at comparable levels due to the continuing issuance form Chile (Climate Bonds Initiative, 2020). This may have been the result of the pandemic. As per the climate bond initiative data, the green bond market has already exceeded the USD1 trillion threshold by year-end 2020 primarily supported by government backed entities, whereas the private sector did not advance. This is also attributed to the pandemic as public sector exhibits higher resilience versus the private sector (Climate Bonds Initiative, 2021). The main issuers are utilities, banks and public sector agencies. They are primarily denominated in Euro and US Dollar. They are investable along with other bonds and can be part of well diversified portfolios (Wacker et al., 2020). The overall green bond market at the end of 2019 exhibited (Almeida, 2020): • A cumulative issuance of USD754 billion, since their inception in 2007. • A total of 5,931 deals. • An overall of 927 issuers. • A strong triad of issuing countries: USA with USD171.5 billion; China with USD107.3 billion; and France with USD86.7 billion. • A milestone of USD100 billion of Certified Climate Bonds. The new green bond activity in 2019 can be summarized as follows (Almeida, 2020): • • • • •

An issuance of USD258.9 billion versus USD171.2 billion in 2018. An array of 1,802 deals versus 1,591 in 2018. A block of 506 issuers compared to 347 in 2018. A group of 291 new issuers compared to 204 in 2018. A set of 8 new countries: Russia, Saudi Arabia, Ukraine, Ecuador, Greece, Kenya, Panama and Barbados.

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The regional breakdown indicates that (Almeida, 2020) • As far as the green bond markets are concerned, Europe had 25, Asia–Pacific had 18, Latin America had 11, Africa had 6 and North America had 2. • When it comes to issuers, Asia–Pacific was represented by 345 issuers, Europe by 269, North America by 167, Latin America by 47 and Africa by 16; there were also 11 Supranational issuers. • With regards to the amount issued (change since 2018 in parentheses), Europe came first with USD307.4 billion (+74%), North America came second with USD190.4 billion (+46%), closely followed by Asia–Pacific with USD183.6 billion (+29%), succeeded by Supranational organizations with USD79.4 billion (+9%), followed by Latin America with USD12.9 billion (+216%) and Africa with USD2.7 billion (+495%). The activity seems to be concentrated in the top 10 countries (Almeida, 2020). Namely, USA, China, France, Germany, The Netherlands, Sweden, Japan, Canada, Italy and Spain accumulate: • • • •

352 issuers, which is 69.6% of the total. 1540 Deals, which constitutes 85.5% of the total. USD184.3 billion, which amounts for 71.2% of the total. 49% of the amount change since 2018.

The highlights of the particulars by country show that (Almeida, 2020): • The leading issuers (with the change since last year in parentheses) remain pretty much the same, i.e. USA with USD51.3 billion (+44%); China with USD31.3 billion (+1%) and France with USD30.1 billion (+113%), followed by Germany with USD18.7 billion (+144%) and The Netherlands with USD15.1 billion (+105%) in the top 5 positions. – The first 10 positions were completed with Sweden with USD10.3 billion (+66%), Japan with USD7.2 billion (+73%), Canada with USD7.0 billion (+63%), Italy with USD6.8 billion (+128%) and Spain with USD6.5 billion (+3%).

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• Almost the same leading countries are observed in terms of the number of issuers; USA came first with 105 issuers (+59%), China follows with 79 issuers (+14%), Japan came third with 47 issuers (+43%), followed by Sweden with 40 issuers (+43%) and France with 19 issuers (+58%). – The Netherlands followed with 15 issuers; Canada came next with 14 issuers; Germany succeeded with 12 issuers; Spain posted 11 issuers; and finally Italy had 10 issuers. • The number of deals for these countries amounted to 1,128 deals for the USA, 106 deals for Sweden, 99 deals for China, 66 deals for Japan, 54 deals for France, 25 deals for Germany, 17 deals for The Netherlands, Spain and Canada, and 11 deals for Italy. The breakdown by issuer shows (with the change since last year in parentheses) that (Almeida, 2020): • The issuers with the highest issuance amounts were Fannie Mae (USA) with USD22.8 billion (+13%), KfW (Germany) with USD9.0 billion (+375%), the Dutch State Treasury with USD6.7 billion (first issue), the Republic of France with USD6.6 billion (+9%) and ICBC (China) with 5.9 billion (+154%). • The issuers with the highest number of green bonds were Fannie Mae (USA) with 991 deals (−10%), Vasakronan (Sweden) with 29 deals (+93%), IFC (Supranational) with 16 deals (+78%), SNFC (France) with 12 deals (0 in 2018) and ADB (Supranational) with 11 deals (+57%). The details of the deals, external review and use of proceeds indicate that (Almeida, 2020): • The average deal size was USD144 million compared to USD108 million in 2018. The median deal size was USD31 million versus USD21 million in 2018. 10% of the deals were in excess of USD500 million against 6% in 2018. – Fannie Mae seems to be an important contributor; the aforementioned figures (2018 in parentheses) if Fannie Mae is excluded change to an average deal size of USD291 million

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(USD208 million); a median deal size of USD119 million (USD122 million); and 21% (20%) of the deals in amounts higher than USD500 million. • The majority of the deals and the amount issued was in hard currency (2018 in parentheses); 83% (87%) of the deals were in hard currency; 83% (80%) of the amount was issued in hard currency; and 34 (31) currencies in total were used. – Again, if Fannie Mae is excluded the number of deals in hard currency becomes 63% (59%); and the amount issued in hard currency changes to 81% (77%). • External review is gaining grounds; 92% (95% in 2017) of the deals received an external review; 86% (89% in 2017) of the volume underwent an external review; and 17% (14% in 2018) of the volume received a CBI Certification. – If Fannie Mae is not considered, then the percentage of deals that got an external review was 81% (83% in 2017); and the volume that obtained an external review was 85% (88% in 2017). • All use of proceeds categories posted an amount increase since 2018. – Energy, buildings and transport accounted for USD80 billion out of the additional USD88 billion that flew in during the year. Their percentage contribution amounts to 82%, with energy exhibiting a 32%, buildings 30% and transport 20%. – Water, waste and land had a smaller growth. Their total contribution was 15%, broken down to 9% for water and 3% for each of water and land use. – Industry, ICT and unallocated adaptation and resilience contribute the remaining 3%, with 1% each. As evidenced from the aforementioned amounts, the increase in 2019 compared to 2018 in some countries is impressive, reaching double digit figures. This is representative of the dynamics of green bond issuance, especially in Europe, that overall seems to have taken the lead. There are two main drivers of this observed increase; the rising demand coming from the investor size, who wish to make their portfolios greener on one hand and the rising supply of green bonds from issuers in the

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EU on the other hand. The latter is anticipated to grow even more as a result of the EU Green Bond Standard implementation in 2021. The undertaking enhances the transparency and reporting in the (sustainable) fixed income market, which is an objective of the EU regulator globally, as we will see in the next section, and in the agenda of the CMU. As such, it is expected that the aforementioned may be adopted as a global standard. In addition, as mentioned earlier, the EU dedicates a significant portion (more than 30%) of the EUR750 billion recovery fund to projects whose financing will come from green debt (NN Investment Partners, 2020). A similar trend is observed in the US, where issuance is though driven by large corporations, which in this way indicate the adoption of more responsible and sustainable practices. The green bond issuance is also becoming popular in Asia, backed by the ambition of China to achieve carbon neutrality by 2060 (NN Investment Partners, 2020). A third important determinant of the green bond market is the green bond performance, which seems to exceed that of their traditional peers. According to NN Investment Partners (2020), the Bloomberg Barclays MSCI Euro Green Bond Index posted a return of 3.2%per annum in the time period January 2016–August 2020. This was 0.70% higher than the Bloomberg Barclays MSCI Euro Aggregate Index. Moreover, green bonds recorded a superior performance in every year, except for 2017. Furthermore, as mentioned earlier, the projection for the years to come is that the green bond market will grow significantly. The biggest activity is anticipated in financial institutions, governments and overall climaterelated issuers. Moreover, a widening of the spectrum is foreseen, so as to incorporate sustainability and social bonds, due to the post-pandemic economy revival efforts, which is linked to (a green and digital) transition (as per the Green Deal). Brown sectors, that are still a necessity, such as aviation, steel and cement, will attract focus, as they are at the epicenter of this transition, especially when broader ESG criteria are taken into account. In addition, a further homogenization of the taxonomies is expected, so as to facilitate all stakeholders in following a well-defined set of guidelines and disclosures (Almeida, 2020). 14.2.8

Green Bonds Versus Plain Vanilla Bonds

A green bond is essentially a regular bond with embedded environmental benefits. As such it shares with a plain vanilla bond the following characteristics (Wacker et al., 2020):

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• It carries a comparable claim against the issuer in case of default/insolvency. • It is assigned an equivalent credit rating. • It gives the same yield and risk premium. • It is included in bond indices. However, it exhibits some additional features, in line with the GBPs (Wacker et al., 2020): • The green project to be financed and the environmental sustainability objectives to be met have to be disclosed to the investors. • The use of proceeds is dedicated to specific green projects, which need to offer specific benefits to the environment—ideally quantifiable. • There is a distinct, strict tracing of the proceeds. • There is a reporting requirement for the time period that the bond is outstanding. • There is a requirement for an external review that covers one or more of the following topics: second party opinion; verification; certification; and green bond scoring/rating. The issuing companies seem to undertake an additional expense in order to launch a green bond, even if—at least for the time being—the yields seem to be the same with plain vanilla bonds. The rationale behind such a decision lies within a series of reasons (Wacker et al., 2020): • Green bonds seem to appeal to investors that are more cautious about the environment and the society and wish to see more value returning to them. These investors are becoming more and more and the ESG investing criteria are facilitating this increase. Furthermore, they have longer-term horizons. This may be beneficial for the issuing party, as it provides stability. • Green bonds require increased transparency, monitoring and reporting. The issuing companies may have these requirements work for them; they can promote their climate-friendly culture, commitment and actions, which, as said before, is valued not only by the investors but by the society as a whole.

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• Green bonds are anticipated to enjoy a boost as a result of the EU and other country determination to advance with regards to climaterelated and overall environmental targets, especially as a means to accelerate the recovery from the consequences of the pandemic. 14.2.9

Concerns Around Green Bonds

In the previous subsections we noted that the simultaneous rise in the demand and supply of green bonds has made them rather fashionable and popular; and vice versa the rise in their popularity among issuers and investors has led to both increased demand and supply. The biggest concern around green bond issuance is greenwashing. The term was introduced in 1986 by the environmentalist Jay Westerveld and in the case of green bonds it refers to the practice of channeling proceeds from green bonds towards projects or activities having negligible or even negative environmental benefits. The first use of the term was not for green bonds though. It was triggered by a practice followed by hotels to put notices/cards in the (bath)rooms recommending the reuse of the towels as such a habit would assist in saving the environment. However, Westerveld observed that at the same time there was zero or negligible attempt—except for a few cases—to contain the unnecessary use of energy. He thus expressed the opinion that the motivation behind this push/crusade, which has been labeled green, was in fact higher financial gain. Greenwashing is a source of material reputational risk for green bond holders (and issuers). It may appall environmentally and socially conscious investors that base their investments on the value returned to the environment or the society or employ environmental, social and governance (ESG) criteria. According to the analysis performed by NN Investment Partners (2020) around 15% of the green bonds are related to disputed approaches that are against environmental norms. This means that the issuers may indeed direct the collected funds to finance environmental sustainability projects; however they still have operations that negatively affect other areas. The GBPs aim precisely at securing that the green bonds to be launched, as well as their issuers, will not relate to such practices and will finance genuinely green projects.

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14.3

MifID, MifID II/MiFIR and Bond Markets

When discussing the bond (and fixed income) markets in Chapter 7, we identified their key trends and elaborated on the contribution of the US and EU regulators. This means, that although the bond markets set the trends to a great extend on their own, it is sometimes the intervention of the regulatory authorities that expedites the setting of the trend or the implementation of the indicated directions. Such directions are related to the overall organization of the bond markets, and in particular the liquidity, transparency and electronification—among others. The EU regulator has been defining the framework for the last 15 years or so for the European financial markets via MifID at the beginning and MiFID II/MiFIR a couple of years ago. The impact of the EU regulation on bond markets is worth exploring as—besides some concerns—it facilitates the transition of bond markets to a more contemporary era. The objective is that bond markets and their stakeholders (investors, issuers, intermediaries and operators) are inspired the same confidence with stock markets and their stakeholders. This is made possible to a great extend by establishing a uniform, level playing field across all European Union countries via a common legislation and regulatory framework. Emphasis is given in disclosures and transparency which was one of the major differences observed between bond and stock markets. We present some if the basic directions of this EU regulation that applies to bond markets. 14.3.1

MiFID

MiFID stands for the Markets in Financial Instruments Directive. It has come into force in November 2007 across the EU member states. It is among the fundamental principles of the regulation that the EU has implemented for the financial markets with the aim of rendering them more competitive. To do so, it fosters the development of a single market for investment-related operations, including all kinds of associated services, and of a homogeneous framework for investor protection from the risks that are inherent in financial instruments. The objective has been (and still is) to secure the efficiency, resilience and transparency of the EU financial markets (ESMA, 2020). As such MiFID has addressed the (European Commission, 2004; European Securities and Markets Authority—ESMA, 2020):

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• authorization requirements, conditions and procedures applicable to investment operations and services offered; • investor protection by establishing conduct of business and organizational rules for investment-related operations and providers; • transparency requirements via reporting and record keeping of the transactions so as to secure the integrity of the markets and prevent market abuse; • protection of the operators and the retail investors through a series of measures that ensure their fair treatment; • appointment of competent authorities in the member states so as to achieve a harmonized supervision and procedures relevant to the market players and operations; • prudential assessment pertaining to the acquisition, capital and/or voting rights of investment firms; • rules on the organization and operation of investment firms; and • requirements on the (access of) trading of financial instruments; It has thus set the rules for the provision of investment services by banks and investment firms, as well as the operation of traditional stock exchanges and alternative trading venues. Besides the added value of MiFID in the aforementioned directions, there were points for improvement, which were probably highlighted during the 2008 financial crisis. This led to a proposal for its revision, which emerged to a revised Directive and a new Regulation. Their shaping took almost two years (starting 2011) before the European Parliament and the Council of the European Union accepted them. MiFID remained in force until January 2, 2018, when it was succeeded by MiFID II/MiFIR. 14.3.2

MiFID II/MiFIR

MiFID II may be seen as a reinforced evolution of MiFID. Its goal is to strengthen the rules on security markets (European Commission, 2020i) by • making sure that organized trading occurs on trading platforms that are regulated;

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• launching requirements in order to regulate algorithmic and high frequency trading; • ameliorating the transparency and supervision of financial markets— with emphasis on financial and commodity derivatives markets; and • intensifying investor protection and advancing the terms under which the relevant business is conducted—addressing also the competition relevant to the trading and clearing functions. In addition MiFID II introduces requirements relevant to the organization and conduct of business of the stakeholders in the financial markets which aim at fortifying investor protection. MiFIR determines the requirements (European Commission, 2020i) pertaining to • disclosing to the public the trading data/information; • disclosing to the regulatory and supervisory authorities the transaction data/information; • compulsory derivatives trading on organized venues; • ensuring increased completion by eliminating potential obstacles between trading venues and clearing service providers; and • specifying supervisory measures for financial instruments and derivatives contracts. MiFID II and MifIR are anticipated to improve the way the financial markets across the member states of the EU operate. Furthermore, it introduces a level playing field. More specifically the transition from MifID to MiFID II/MifIR is expected to (ESMA, 2020): • secure that the markets within the EU are safer, fairer, and more efficient; • help achieve more transparency for all stakeholders; • enable the accessibility of increased information via the introduction of new reporting requirements and tests and the limitation of opaque markets and OTC trading; • enforce a rigorous set of organizational requirements applicable to investment firms and trading venues with the adoption of rules that regulate high-frequency trading;

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• allow for more competition through provisions that govern “the non-discriminatory access to central counterparties (CCPs), trading venues and benchmarks ”; • fortify the protection offered to investors via a series of initiatives, such as – the implementation of new rules on product governance and independent investment advice; – the enhancement of the requirements relevant to structured deposits; – the advancement of the rules that govern several aspects and parties of the investment services, such as the accountability of the managing entities, the information and reporting released to investors, the cross-selling practices deployed, the incentives offered, the remuneration of the involved staff and the best execution mandate. 14.3.3

Impact on Bond Markets

The impact of MifID II/MiFIR on the bond or the broader fixed income markets stems from the overall objective of the revised directive and the new regulation to make the markets more robust on one hand and from the specific reference to non-equity instruments with emphasis on the transparency of the corresponding markets. The approach attempts to a great extend to treat the fixed income securities and markets in a way similar to equity markets (equity-like). This may be justified by the fact that equity markets have already tackled issues that pertained to transparency, liquidity and the use of new technologies. However, this is also a point that the participants in the fixed income markets make that the Directive and the Regulation could have done differently (AFM, 2020). The main goals the MiFID II sets for fixed income markets are (ICMA, 2017): • Establish a price discovery process in the fixed income markets. • Promote the use of trading venues for fixed income trading, instead of the use of OTC trading. Such trading venues are Regulated Markets (RMs), Organized Trading Facilities (OTFs) and Multilateral Trading Facilities (MTFs).

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• Increase transparency in OTC trading environments with the introduction of the Systemic Internalizer (SI) regime. • Secure the availability of reference data for fixed income securities. 14.3.4

Market Structure

The market structure model promoted by the regulation is deployed through four pillars (as can be seen in the second bullet above); the Regulated Markets (RMs), the Organized Trading Facilities (OTFs), the Multilateral Trading Facilities (MTFs) and the Systemic Internalisers (SIs). Their particulars in short are (ICMA, 2017): • A Regulated Market (RM) constitutes “a multilateral system that is operated and/or managed by a market operator” (ICMA, 2017, p. 6), whose role is to connect or facilitate the connection of multiple third parties who wish to buy or sell financial securities and has been licensed by the competent authorities. • A Multilateral Trading Facility (MTF) is “a multilateral system that is operated by an investment firm or market operator” (ICMA, 2017, p. 6), with the purpose of connecting multiple third parties who wish to buy or sell financial securities in general. • An Organized Trading Facility (OTF) is “a multilateral system other than an RM or an MTF, in which multiple third parties ” (ICMA, 2017, p. 6), who wish to buy or sell bonds, structured products and/or derivatives are connected. In contrast to RMs and MTFs, OTFs can exercise discretion in the order execution, but are still obliged to comply with the pre-trade transparency and best execution requirements. • A Systemic Internaliser (SI) is “an investment firm” that uses proprietary capital “by executing client orders outside a trading venue” (ICMA, 2017, p.6) and is not a trading venue itself. Some additional restrictions that are enforced in the applicable market model are that (ICMA, 2017, p. 6): • RMs and MTFs are not permitted to execute client dealing on own account or to participate in matched principal trading.

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• OTFs may use their proprietary capital when not taking part in matched principal trading only for illiquid government fixed income securities. • OTFs and SIs cannot co-exist in the same legal body and cannot be brought together in order to facilitate the interaction of orders or quotes (Table 14.1). Fixed income instruments fall within the non-equity asset classes. From the aforementioned stakeholders the SIs are potentially worth explaining as their introduction in fixed income markets aims at securing a convergence of OTC trading with the organized trading. As a result, the SIs—a concept traditionally met in equity markets—have within their scope the fulfillment of pre- and post-trade reporting obligations. According to the formal definition provided by MiFID II (European Commission, 2014a), the SIs are “investment firms, which on an organized, frequent, systematic and substantial basis, deal on own account when executing client orders outside a regulated market (RM), multilateral trading facility (MTF ) or organized trading facility (OTF ) (together referred to as Trading Venues) without operating a multilateral system” (Table 14.1). Practically, the SI obligations are similar to Trading Venue obligations. Only Non-SI obligations differ. A summary can be seen in Table 14.2. MifID II/MiFIR explain the particulars of frequent, systematic and substantial—so that an investment firm qualifies for an SI, as well as Table 14.1 Market structure under MifID II/MifIR

Financial Instruments Transaction execution Proprietary capital Matched principal trading

Equity Non-Equity Discretionary Non-discretionary Not permitted Permitted Not permitted Permitted

RM

MTF

X X

X X

X X

X X

X

X

OTF

SI

X X

X X X

X with exceptions X X client consent

Source Created by the author with information assembled from ICMA (2017)

X exceptional cases

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Table 14.2 Obligations of SIs compared to Trading Venues and Non-SIs Transparency obligation

Trading Venue

SI

Non-SI

Pre-trade information provision Post-trade information provision Best execution data provision Reference data provision Post-trade reporting

A (Non-investment firm) A

A (Investment firm)

N/A

A

A

A

A

A

A

A

N/A

A

A

A: selling only

Note A: Applicable, N/A: Not Applicable Source Created by the Author with Information Assembled from ICMA (2017)

relevant matters such as the determination, the requirements and the discretion (European Commission, 2014a, 2014b; ICMA, 2017). 14.3.5

Transparency

One of the targets of MiFID II/MifIR is to secure transparency across fixed income markets as well, also with the use of electronic means. This is sought before and after the trade. The spectrum of the asset classes that pre-trade requirements cover has been broadened to include non-equity securities, such as bonds, structured products, derivatives and emission allowances. Our interest in this book is in fixed income securities and (the respective) derivatives. This widening of rules to govern also fixed income markets implies that Regulated Markets, Multilateral Trade Facilities and Organized Trade Facilities are obliged to publish electronically to the public the current prices (bid and ask/offer) as well as the depth of the market/orders placed. Certain thresholds are enforced as a result of which the publication requirement of these data may be waived when: • The orders placed are large in scale (LIS). • The request for quote (RFQ) and voice systems volumes exceed the size-specific to the instrument (SSTI) threshold. • The traded securities are illiquid.

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Table 14.2 depicts the pre-trade transparency enforcement; it is not applicable only to non-systemic internalisers, whereas for trading venues and systemic internalisers it is applicable for non-investment firms and investment firms respectively (ICMA, 2017). There are also post-trade requirements that demand the reporting/publication of the post-trade data in near real-time, i.e. within 15 minutes, via a trading venue or an Approved Publication Arrangement (APA). The targeted audience is the broader investment public in this case as well. It can be seen in Table 14.2, the post-trade requirements are applicable horizontally to all types of providers. An APA is an entity (person) that is authorized to publish trade reports on behalf of investment firms. The data recorded are usually the time and date of the trade, the time of the publication, the unique identification number for the security, known as ISIN (International Securities Identification Number), the name of the security, the price and the currency at which it is quoted, the volume and the type of trade. MifID II/MiFIR explain the details of the pre- and post-trade transparency/reporting requirements, along with the thresholds and waivers (European Commission, 2014a, 2014b; ICMA, 2017). 14.3.6

Best Execution Data Provision

The trading venues and the SIs, as well as the execution venues, including the market makers and other intermediaries that have been assigned the task of providing liquidity to the market, must publish (to the broader investment public/audience) the data that pertain to the quality of the execution of the transactions that took place within their operation, free of charge and annually at a minimum (on a quarterly basis for execution venues and no later than 3 months after the end of each quarter). The published data are the price of the trades, the applicable costs, the speed and the likelihood of execution per security. Furthermore, the published data are separated (in amount and nature) per trading environment (i.e. system, mode and platform), so as to better depict the achieved quality of execution. In addition, it is required that they are made available to the public electronically, downloadable and machine/computer-readable. Investment firms that execute client orders must provide to the public

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• their first 5 execution venues by trading volume at which their client orders were executed in the previous year; and • the quality of the execution, on an annual basis and per asset class, – as a percentage of the total volumes that the firm executed and – as a percentage of the number of the orders that the firm executed per each of these asset classes. The reported data are separated between retail and institutional/professional clients (ICMA, 2017). Furthermore, the investment firms must inform the public in a different publication about the first 5 execution venues (in terms of volume) at which they executed securities financing transactions, i.e. repurchase agreements, security lending, sell-buy backs or buy-sell backs, and margin lending. They must expressly show the asset classes whose number of executed orders was very small. They must also release a summary of the analysis and the conclusions with regards to the particulars of the execution (i.e. quality and venues per security class) (ICMA, 2017). 14.3.7

Third Country Trading Venues

Obviously there can be trades in a non-EU country (also referred to as a third country) trading venue. As such, there may be a distinction on the transparency requirements that apply to such a country. If the posttrade publications are not comparable, then the trades must be published in the European Union through an APA. If the post-trade transparency requirements are comparable though, then the supplementary publication in the EU is waived. Furthermore, on one hand third-country trading venues cannot ask consideration directly from the EU member-state national competent authorities; on the other hand though, EU investment firms that trade in non-EU countries must notify their competent authorities (ICMA, 2017). 14.3.8

Costs and Charges Disclosure

Transparency rules could not miss the applicable costs and charges to the clients (existing and prospective). MifID II requires investment firms

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to disclose to their clients all associated costs and charges pertaining to investment services—incorporating orders (before and after the services are offered). Examples include fees of all types concerning all involved parties, such as intermediary, subscription and redemption (payable to the fund manager), and platform (if any) fees. They extend to all other potential costs and charges relevant to the transaction itself that determine the cash price such as taxes of all types (e.g. stamp duty, transaction taxes, etc.), foreign exchange fees, price spread etc. (ICMA, 2017). 14.3.9

Research Unbundling Rules

One of the challenges to the investment community that MiFID II introduced was the research unbundling rules. The rationale behind the unbundling reforms of the revised Directive was to ensure that portfolio managers represent in the best possible manner the interests of their clients and that their investment decisions remain uninfluenced by third parties. It therefore provisions, among others, that investment firms are not allowed to receive research with no charge when offering portfolio management or independent advisory services. The said research will have to be paid either on their expense or via a designated Research Payment Account (RPA). The latter would be financed by their customers through either a commission sharing agreement (CSA) or a direct debit. In fixed income in particular, research cannot be covered by commissions, which means that the investment firm will either bear the expense or will directly charge its customers via an RPA. If a firm opts for an RPA, it is then required to set in advance an annual budget earmarked for research and disclose the research costs to the customer. Consequently, the unbundling reform requires portfolio managers to refrain from bundling research costs into transaction fees; they are asked instead to pay for research separately from order execution fees. In their turn they will either have to pay the research themselves or debit their customers in a transparent manner (ICMA, 2017). 14.3.10

Securities Financing Transactions

Securities financing transactions (SFTs) give to investors (and firms) the possibility to employ assets, such as the stocks or bonds they possess, in

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order to receive financing for their operations. They refer to any transaction that securities can be used to borrow cash or vice versa and thus resemble to collateralized loans. To better understand that, we observe that in these transactions, the ownership of the underlying security changes in return for cash, which also changes ownership. These changes are temporary, as in the maturity of the SFT the ownership changes back to its initial status and the counterparties end up with their initial holdings. However there is a small fee that is added or subtracted depending on the nature of the transaction (European Commission, 2016; London Stock Exchange Group, 2018). Examples of securities financing transactions are (European Commission, 2016): • a repurchase agreement, i.e. the sale of a security with a repurchase agreement in a specific future date at a specific higher price, thus producing a yield/compensation for making use of the cash; • a security lending (or borrowing) agreement for a fee, which involves a guarantee offered by the borrower in the form of securities or cash; • a sell-buy back or buy-sell back agreement; • a margin lending agreement. SFTs are also in the scope of MiFID II/MiFIR, except for the cases they are explicitly exempted. However, the financial crisis in 2008 unveiled certain risks that emanated from STFs that were not properly anticipated by regulators and supervisors primarily due to the lack of data. The EU in response adopted the securities financing transactions regulation (SFTR) to increase transparency. The regulation requires that (European Commission, 2016): • STFs are reported in trade repositories (central databases), with the exception of those taking place via central banks. • Investment firms disclose to investors in their periodic reports and pre-investment documents the information that is relevant to the use of SFTs. • The minimum transparency requirements (e.g. risk notification and customer accordance), pertaining to the reuse of collateral, are fulfilled.

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MiFID II does not allow the conclusion of title transfer collateral arrangements (TTCAs, such as repos, securities lending) with retail (individual) investors (clients). However it does allow TTCAs with non-retail (professional) investors (clients). SFTs have to be included in the publications of the execution venue particulars but not in the publications of the data relating to the quality of execution of the transactions. SFTs are not in the scope of pre- and post-trade transparency rules (European Commission, 2017; ICMA, 2017). The pandemic has triggered a discussion within the European Commission with regards to the regulatory framework in connection with the capital markets recovery. The European Commission announced in July 2020 a Capital Markets Recovery Package, which reviews and proposes specific amendments to MiFID II, the capital requirements regulation, the securitization regulation, and the prospectus regulation. In the proposed amendments in MiFID II, emphasis is given in the product governance of bonds, according to which proportionality to the requirements is recommended, when plain vanilla bond transactions with eligible counterparties are considered. There is a proposal of lifting the product governance requirements in particular for simple corporate bonds with investor protective features, such as make-whole clauses, i.e. provisions that allow the issuer to retire the bond early by paying off the remaining debt earlier than its maturity. This is expected to facilitate the distribution of corporate bonds to a wider audience, as these products are generally perceived as safe and simple and eligible also for retail clients (individual investors) (European Commission, 2020d). Furthermore, as securitization falls within the fixed income instrument umbrella, thus in the scope of this book, we note that the European Commission has put forward a proposal to amend the general framework for securitization and promote a specific framework for simple, transparent and standardized (STS) securitization in order to assist the recovery from the pandemic. The rationale behind this review and proposal is to upgrade the means that the banks have in order to maintain and if possible increase their lending ability towards the participants of the real economy and more specifically SMEs. Securitization is an important facilitator of this process as on one hand it helps banks release capital (as they convert loans to marketable instruments), which can be used to provide additional loans, and on the other hand it addresses a bigger set of investors that could provide capital towards the recovery of the economy.

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Under the prism of simple, transparent and standardized securitization, the European Commission proposes a series of requirements that address each of the constituents of the approach; i.e. requirements relating to simplicity, requirements relating to transparency and requirements relating to standardization. Furthermore, it sets forth requirements relating to the credit protection agreement, the third party verification agent and the synthetic excess spread. The particulars can be found in the relevant European Commission documentation (European Commission, 2020e).

Exercises Exercise 1 Investigate whether there are initiatives similar to the CMU around the world—even at a smaller scale. What do you observe? Exercise 2 Investigate the equivalent of MifID in other countries/markets, such as the US. a. What are the similarities and differences? b. Could there be global agreements? c. What would be the advantages or disadvantages of such agreements? Exercise 3 Consider a country or company that has issued both a regular bond and a green bond. a. What are the differences? b. What are the similarities? c. Is their yield to maturity comparable? d. Is the overall performance comparable?

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Exercise 4 a. What is the impact of the applicable regulatory frameworks in the bond markets? b. How does it affect bond issuance? Exercise 5 Can you investigate, with the potential use of an econometric model, if the introduction of the applicable regulation has impacted the yield, volume, and other key parameters of a bond issue across the various countries? (Hint: this exercise requires the use of econometric models and the access to the necessary data.)

References AFM. (2020, August 27). A review of MiFID II and MiFIR—Impact on the fixed income and derivative markets. https://www.afm.nl. Accessed: 8 December 2020. Almeida, M. (2020, July). Global Green Bond State of the Market 2019, Climate Bonds Initiative. https://www.climatebonds.net/system/tdf/reports/cbi_ sotm_2019_vol1_04d.pdf?file=1&type=node&id=47577&force=0. Accessed: 8 December 2020. Barbiroglio, E. (2020). Green bond market will reach $1 trillion with German new issuance. Forbes. https://www.forbes.com/sites/emanuelabarbiroglio/ 2020/09/02/green-bond-market-will-reach-1-trillion-with-german-new-iss uance/. Accessed: 8 December 2020. Climate Bonds Initiative. (2020). Interactive data platform—As of H1 2020. https://www.climatebonds.net/market/data/. Accessed: 8 December 2020. Climate Bonds Initiative. (2021). Interactive data platform—As of end 2020. https://www.climatebonds.net/market/data/. Accessed: 18 June 2021. Climate Change Committee. (2020, May 6). Building a resilient recovery from the COVID-19 crisis. https://www.theccc.org.uk/publication/letter-bui lding-a-resilient-recovery-from-the-covid-19-crisis-to-prime-minister-boris-joh nson/. Accessed: 8 December 2020. ESMA. (2020). MIFID II . https://www.esma.europa.eu/policy-rules/mifid-iiand-mifir. Accessed: 8 December 2020. European Commission. (2004). Markets in financial instruments (MiFID) and investment services—Summary of: Directive 2004/39/EC on markets in

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financial instruments. https://eur-lex.europa.eu/legal-content/EN/LSU/? uri=celex:32004L0039. Accessed: 8 December 2020. European Commission. (2014a). Directive 2014/65/EU of the European Parliament and of the Council of 15 May 2014 on markets in financial instruments and amending Directive 2002/92/EC and Directive 2011/61/EU. Official Journal of the European Union L 173/349, 12.6.2014. https://eur-lex.europa.eu/legal-content/EN/TXT/ PDF/?uri=CELEX:32014L0065&from=EN. Accessed: January 2021. European Commission. (2014b). Regulation (EU) No 600/2014 of the European Parliament and of the Council of 15 May 2014 on markets in financial instruments and amending Regulation (EU) No 648/2012. Official Journal of the European Union L 173/84, 12.6.2014. https://eur-lex. europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32014R0600&fro m=EN. Accessed: January 2021. European Commission. (2016). Securities financing transactions (SFTs). https://ec.europa.eu/info/business-economy-euro/banking-and-finance/fin ancial-markets/post-trade-services/securities-financing-transactions-sfts_en. Accessed: January 2021. European Commission. (2017). Commission Delegated Directive (EU) 2017/593 of 7 April 2016. Official Journal of the European Union, L 87/500, 31.3.2017. https://eur-lex.europa.eu/legal-content/EN/TXT/ PDF/?uri=CELEX:32017L0593&from=EN. Accessed: January 2021. European Commission. (2020a). Factsheet: What is the capital markets union? https://ec.europa.eu/info/sites/info/files/business_economy_euro/gro wth_and_investment/documents/200924-capital-markets-union-action-planfactsheet_en.pdf. Accessed: 8 December 2020. European Commission. (2020b). Legislative measures taken so far to build a CMU . https://ec.europa.eu/info/business-economy-euro/growth-and-inv estment/capital-markets-union/legislative-measures-taken-so-far-build-cmu. Accessed: 8 December 2020. European Commission. (2020c). Taxonomy: Final report of the Technical Expert Group on Sustainable Finance. Financing a Sustainable European Economy— Technical Report. https://ec.europa.eu/info/sites/info/files/business_eco nomy_euro/banking_and_finance/documents/200309-sustainable-financeteg-final-report-taxonomy_en.pdf. Accessed: 8 December 2020 European Commission. (2020d). Proposal for a Directive of the European Parliament and of the Council amending Directive 2014/65/EU as regards information requirements, product governance and position limits to help the recovery from the COVID-19 pandemic, 2020/0152 (COD). https://ec.europa.eu/ finance/docs/law/200724-mifid-review-proposal_en.pdf. Accessed: January 2021.

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European Commission. (2020e). Proposal for a Regulation of the European Parliament and of the Council amending Regulation (EU) 2017/2402 laying down a general framework for securitization and creating a specific framework for simple, transparent and standardized securitization to help the recovery from the COVID-19 pandemic, 2020/0151 (COD). https://ec.europa.eu/finance/ docs/law/200724-securitisation-review-proposal_en.pdf. Accessed: January 2021. European Commission. (2020f). Communication from the Commission to the European Parliament, the Council, the European Economic and Social Committee and the Committee of the Regions—A Capital Markets Union for people and businesses-new action plan, COM(2020) 590 final. https://eurlex.europa.eu/resource.html?uri=cellar:61042990-fe46-11ea-b44f-01aa75ed7 1a1.0001.02/DOC_1&format=PDF. Accessed: January 2021. European Commission. (2020g). Annex to the Communication from the Commission to the European Parliament, the Council, the European Economic and Social Committee and the Committee of the Regions—A Capital Markets Union for people and businesses—New action plan, COM(2020) 590 final Annex. https://eur-lex.europa.eu/resource.html?uri=cellar:61042990-fe4611ea-b44f-01aa75ed71a1.0001.02/DOC_2&format=PDF. Accessed: January 2021. European Commission. (2020h, May 27). Europe’s moment: Repair and prepare for the next generation. Press release. https://ec.europa.eu/commission/pre sscorner/detail/en/ip_20_940. Accessed: 6 November 2020. European Commission. (2020i). Investment services and regulated markets— Markets in financial instruments directive (MiFID): EU laws aimed at making financial markets more efficient, resilient and transparent, and at strengthening the protection of investors. https://ec.europa.eu/info/businesseconomy-euro/banking-and-finance/financial-markets/securities-markets/inv estment-services-and-regulated-markets-markets-financial-instruments-direct ive-mifid_en. Accessed: 6 November 2020. European Council. (2019). Capital Market Union: Council adopts legislative reforms. https://www.consilium.europa.eu/en/press/press-releases/2019/ 11/08/capital-markets-union-council-adopts-legislative-reforms/. Accessed: January 2021. European Parliament. (2019). Regulation of the European Parliament and of the Council amending Directive 2014/65/EU and Regulations (EU) No 596/2014 and (EU) 2017/1129 as regards the promotion of the use of SME growth markets, 2018/0165 (COD). https://data.consilium.europa.eu/doc/ document/PE-89–2019-INIT/en/pdf. Accessed: January 2021. ICMA. (2017, August). MiFID II/MiFIR and fixed income. https://www.icm agroup.org/Regulatory-Policy-and-Market-Practice/Secondary-Markets/sec

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ondary-market-practices-committee-smpc-and-related-working-groups/mifidii-r-working-group/. Accessed: January 2021. ICMA. (2018, June). Green Bond Principles: Voluntary process guidelines for issuing green bonds. The Green Bond Principles. https://www.icmagroup. org/assets/documents/Regulatory/Green-Bonds/Green-Bonds-PrinciplesJune-2018-270520.pdf. Accessed: January 2021. Leiden University. (2020, April 14). 170 scientists sign manifesto—Planning for Post-Corona: Five proposals to craft a radically more sustainable and equal world. https://www.universiteitleiden.nl/en/news/2020/04/170-scientistssign-manifesto. Accessed: 8 December 2020. London Stock Exchange Group. (2018). Securities Financing Transactions Regulation (SFTR). https://www.lseg.com/markets-products-and-services/ post-trade-services/unavista/regulation/securities-financing-transactions-reg ulation. Accessed: January 2021. Lovell, J. (2020, June 3). Build Back Better—Open Letter to the Prime Minister. Hillbreak. https://www.hillbreak.com/build-back-better-open-let ter-to-the-prime-minister/. Accessed: 8 December 2020. NN Investment Partners. (2020). Global green bond market set to hit EUR 2 trillion in three years, says NN Investment Partners. https://www.nnip.com/ en-INT/professional/insights/global-green-bond-market-set-to-hit-eur-2-tri llion-in-three-years-says-nn-ip. Accessed: 8 December 2020. Poufinas, T., & Kouskouna, E. (2017). On the split of social security contributions between funded and pay-as-you-go pension schemes; contribution to growth. In The Greek debt crisis—In quest of growth in times of austerity (pp. 129–152). Palgrave Macmillan. Poufinas, T., & Polychronou, M. (2018). Alternative investments as a financing tool for small and medium enterprises. Bulletin of Applied Economics, 5(2), 13–44. Trudeau, J. (2020, May 11). Prime Minister announces additional support for businesses to help save Canadian jobs. Justin Trudeau, Prime Minister of Canada. https://pm.gc.ca/en/news/news-releases/2020/05/11/primeminister-announces-additional-support-businesses-help-save. Accessed: 8 December 2020. Wacker, T., Bolliger, M., & Seimen, M. (2020). Green bond slide pack. UBS. https://www.ubs.com/. Accessed: 6 November 2020.

Index

A account fees, 417 accrued interest (AI), 51, 52, 72, 201, 276, 278, 279, 300, 320, 354, 510, 514, 642 actions addressing primarily retail investor, 656 actions addressing primarily SMEs, 655 actions addressing primarily the establishment of a single market, 656 activating downgrades, 516, 517 active bond portfolio management horizon analysis, 255 identification of mispriced bonds, 250, 251 interest rate forecasting, 102, 250 active bond portfolio management strategies, 251 taxonomy of, 251 address the lessons of the crisis, 661 AFM, 681

agreement, 10, 21, 41, 48, 49, 71, 102, 104, 105, 266, 268–271, 284, 285, 292, 293, 296, 360, 396, 398, 623, 658, 688, 690 repurchase, 34, 266, 269, 688 algorithmic bond trading, 379 allocated (economic) capital, 499 allocation, 6, 33, 41, 42, 230, 377, 668 capital, 5 All-Term ETFs, 435, 437 Almeida, M., 671–673, 675 ALM risk, 524–526 Altavilla, C., 546, 552 alternative approaches for calculating the probability of default, 514 American options on bonds, 304 price, 304 amount, 3–5, 9, 11, 20, 24, 25, 31, 32, 38, 42, 49, 50, 54, 70, 71, 73, 101, 106, 124, 136, 180, 205, 226, 235, 239, 242–244, 246, 247, 249, 257, 268–270, 272, 274–277, 281, 282, 286,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Poufinas, Fixed Income Investing, https://doi.org/10.1007/978-3-030-87922-8

695

696

INDEX

292, 293, 296, 306, 310, 318–321, 333, 350, 351, 353, 354, 357, 358, 363, 370, 372, 391, 398, 409, 417, 420, 427, 431, 463, 464, 468–470, 504, 506, 507, 509, 510, 512, 513, 546, 553, 557, 558, 561, 562, 564, 566, 583, 585, 586, 591, 606, 628–632, 635–638, 641, 642, 656, 661, 668, 671–674, 685 notional principal, 518 analysis, 5, 131, 132, 146, 175, 250, 380, 419, 446, 486, 490, 514, 530, 540, 542, 559, 561, 565, 590, 608, 624, 677 security, 4, 686 sensitivity, 458, 459, 530 Analyst Prep, 143 annual compounding, 70, 77, 83, 87, 106, 114, 118, 121, 123, 129, 257, 331, 579, 580, 596 annuity(ies), 2, 11, 12, 25, 193, 196, 239, 319, 354 Apergis, N., 537–539, 541, 542, 544, 545, 559 approach, 8, 13, 33, 36, 38, 67, 119, 123, 129, 137, 138, 145, 147, 152, 155, 159, 161, 170, 232–236, 238, 241, 282, 296–298, 304, 308, 311, 315, 319, 324, 335, 379, 431, 458, 459, 462, 464, 467–469, 471, 484, 490–492, 501, 503, 505, 516, 525, 526, 529, 530, 573, 577, 578, 584, 587, 589, 598, 604, 608, 626, 670, 677, 681, 690 cellular, 232, 233, 431 approved publication arrangement (APA), 380, 685, 686 arbitrage

forward contracts, 272, 639, 640, 642 futures contracts, 639 options, 643 arbitrageur, 621, 622 Aristei, D., 537 ask, 24, 72, 75, 100, 141, 415, 505, 559, 596, 684, 686 ask price, 71, 72, 75, 144, 505 asset backed securities (ABS), 23, 266, 323–325 asset-backed securities purchase program (ABSPP), 37, 550 asset class, 4, 5, 8, 48, 170, 242, 326, 379, 394, 408, 414, 421, 439, 444, 446, 526, 536, 550, 571, 613, 621, 652, 656, 683, 684, 686 asset liability management (ALM), 236, 240, 328, 506, 525, 526, 530 asset liability matching, 383 asset purchase programs (APPs), 34, 37, 38, 550 assets, 5, 8–10, 14, 17, 23, 24, 32, 39, 66, 235–242, 244, 245, 248, 265, 282, 285, 297, 323–325, 327–329, 372, 376, 378, 387, 407, 414, 415, 417–421, 424– 426, 432, 435, 437, 442–445, 469, 486, 515, 525, 526, 537, 561, 573, 587, 591, 608, 624, 651, 657, 669, 670, 687 types of, 8 A-Team, 379, 380 Ávila, F., 527, 528 B back-end load, 416 Bangia, A., 505 bank, 1, 14–16, 25, 26, 28, 31, 47, 51, 55, 68, 91, 99, 101, 102,

INDEX

105, 107, 116, 142, 235–238, 281, 290–293, 295–297, 325, 361, 375, 387, 391, 397, 398, 412, 421, 439, 442, 444, 464, 479, 506, 516, 525, 526, 549, 552, 612, 628, 629, 657–659 commercial, 25, 26, 28, 31, 395–398 bankers’ acceptances, 91, 396, 398 Bank of America Merrill Lynch Domestic Master, 231 Merrill Lynch Global Bond Index, 231 Merrill Lynch High-Yield Master II, 231 Bank of England, 12, 27, 456 Bank of International Settlements (BIS), 33 bankruptcy, 57, 58, 67, 349–351, 372 Barbiroglio, E., 670 Barclays, 231, 401 Lehman Brothers US Treasury Index, 231 Barclays High-Yield Index, 231 Barclays Inflation-Linked Euro Government Bond Ind, 231 Bartter, 151 Basel Committee on Banking Supervision (BCBS), 375 Basel III, 375, 376, 660 basic trinomial tree, 156 basket CDS, 357 first-to-default, 357 Bean, M.A., 143 bearer, 49–51, 390, 398 Bear Stearns, 12 Bear Stearns High-Yield Index, 231 Beirne, J., 540 benchmark, 7, 256, 315, 316, 321, 322, 385, 399, 412, 415, 431, 456, 500, 536, 670, 681

697

benefits of the CMU, 652 best effort, 21 best execution data provision, 685 MifID, MifID II/MiFIR, 685 bid, 415, 505, 684 bid-ask spread, 71, 501, 504, 505 bid-offer spread, 71, 416 bid price, 71, 72, 75, 144, 416, 505 binary CDS, 357 binomial trees, 145 Black, F., 298, 299 Blackman, N., 374, 379, 380 Black Monday, 1987, 462 Black’s model, 297–300, 330, 331, 334, 335, 363, 364, 637 BlackRock, 405 Bloomberg Barclays Global Aggregate Bond Index, 231 Bloomberg Barclays MSCI Euro Green Bond Index, 675 Bodie, Z., 6–11, 14, 17, 20–22, 24, 51, 65, 91, 106, 130, 131, 135–138, 140–142, 172, 230, 232, 234, 235, 248, 251, 254, 256, 383, 384, 387, 390, 392–398, 606 Bolliger, M., 661, 665, 666, 668, 670, 671, 675, 676 bond categories, 10, 410 bond characteristics, 48 bond coupon-bearing, 69, 74, 172–174, 178–181, 193, 197, 199, 200, 202, 224, 301–303, 455, 471 bond ETFs (Exchange Traded Funds) comparison of bond mutual funds, 427 concerns, 438 definition, 429 market, 431 types, 429 bond fund(s)

698

INDEX

high-yield, 410, 411, 425 international, 406, 413 investment-grade, 410, 412 multi-sector, 412 municipal, 410, 413, 420–422 other types of, 413 bond futures, 273, 275, 277, 279, 280 bondholder, 10, 15, 49, 53, 54, 179, 180, 184, 199, 378, 510, 512, 573–575 bond income, 7 bond indenture, 10, 667 bond index/bond indexing, 8, 230, 400 problems of, 232 bond indices, 231, 399–401, 676 bond issue, 9–11, 55, 89, 90, 144, 170, 232, 316, 352, 370, 385–387, 391, 400, 431, 545, 546, 612, 632, 649 bond issuer, 9, 15, 22, 64, 359, 360, 381, 518 bond market growth, 373 drivers of the, 373 bond market(s) contemporary trends, 371 contemporary trends in the, 371 corporate, 90, 369, 373, 387, 390–392, 431 government, 89, 382, 383, 392, 549 green, 666, 670–672, 675 MifID, MifID II/MiFIR and, 678, 681 MifID, MifID II/ MiFIR impact on, 681 modernization of the, 373, 374, 381 municipal, 91, 391–393 size of the, 372 traditional notions of the, 381

bond maturity, 11, 507 bond mutual fund(s) comparison of bonds, 54, 427 comparison with bond ETFs, 438 concerns about, 427 definition, 408 high-yield, 425, 426 international, 410 investment-grade, 12, 412 market, 414, 417 mechanics of, 414 money market, 371, 397, 398, 408, 410, 417–422 multi-sector, 425, 426 municipal, 422 operating expenses, 417 pricing, 415 shareholder fees, 416 types of, 410, 422 bond options, 297, 300, 304, 311 embedded, 130, 311, 390 bond portfolio management, 402 active, 233, 250, 251, 256, 399 passive, 230, 232, 250, 256, 282, 399 bond price movements, 87 bond pricing, 123, 213, 459 bond return, 84, 126 bond(s) American Options, 304, 309 and crises, 535, 546 and stocks, comparison of, 601 callable, 52, 80, 81, 312, 314, 316, 390, 428 CMU and, 657, 659 comparison with bond mutual funds, 54, 427 convertible, 53, 390, 650 corporate, 10, 18, 20, 25, 51, 54, 67, 89, 90, 144, 252, 253, 391, 408, 411–413, 430, 431,

INDEX

433, 443, 456, 536, 558, 601, 650, 689 coupon-bearing, 11, 49, 68–70, 172–174, 178–181, 184, 185, 190, 193, 197, 199, 209, 224, 225, 227, 247, 301–303, 315, 455, 471, 510 covered, 370, 376, 378, 411, 536, 550, 650, 654, 657, 658 and debt, 557 deferred callable, 52, 80 European options, 302 exchange traded funds (ETFs), 376, 377, 408, 429–433, 435 floating rate notes (FRN’s), 1, 11, 52, 68, 360 forward contract on a, 271, 273 government, 10, 12, 13, 20, 23, 25, 32, 38, 51, 54, 89, 90, 231, 233, 253, 254, 284, 313, 382, 385–387, 391, 398, 400, 401, 411, 425, 426, 430, 431, 443, 460, 472, 536, 537, 546–548, 550, 551, 559, 561, 625, 627, 649 green, 657, 661, 665–671, 673–677 index-linked, 2, 23, 386, 443 industrial development, 393 industrial revenue, 393 issued by the bank, 510 junk, 58, 327, 432, 433 maturity, 254 of supranational organizations, 10 options, 1, 53, 80, 297, 300, 301, 304, 311 puttable, 53, 81, 312, 313, 390 regulation, 381, 649 revenue, 11, 392, 393 speculative grade, 58 supranational, 23

699

supranational organizations, 10, 18, 23 types, 90, 369, 411, 412, 415, 434, 435 valuation, 68, 119, 121 without coupon, 244 zero-coupon, 1, 49, 68, 69, 74, 75, 86, 118, 119, 122, 127, 130, 131, 133, 135, 136, 139, 149, 151, 153, 159–162, 172–174, 176, 178–181, 184–192, 199, 209–212, 214–217, 224–229, 239, 245, 247, 267, 292, 301–304, 315, 323, 384, 468, 470, 472, 507, 508, 510 bonds and regulation capital markets union (CMU), 380, 391, 657 Green bonds, 657, 661, 665 MifID, MifID II/MiFIR and bond markets, 678 bond yield(s) impact of crisis on the, 537 the impact of public and private debt to, 559 the impact of QE on the, 542 Bordo, M.D., 27 Bova, A., 251, 252 Brexit, 462 broad market ETFs, 430 brokers, 22, 24, 72, 386 Broner, F., 562 Brownian motion, 145, 299 business, 1, 13, 14, 29, 57, 125, 373, 393, 439, 486, 552, 577, 650, 652, 654, 658, 664, 665, 670, 679, 680 debt, 442 risk, 529 sector, 9

700

INDEX

buyer, 9, 11, 24, 51, 68, 71, 72, 352–355, 357, 358, 362, 365, 402, 430, 505, 515, 518, 591 protection, 349 C Caggiano, G., 565 Calabrese, R., 528 calculation, 52, 79, 80, 88, 107, 125, 159, 161, 204, 228, 302, 468, 584, 586, 589, 591 VaR, 467–469, 473, 477, 499 callable bond(s), 52, 81, 312, 314, 316, 390, 428 deferred, 52, 80 call provisions, 52, 144, 315, 392 capital, 2, 3, 8, 11, 14, 16, 21, 38, 41, 42, 47, 55, 58, 66, 67, 74, 77, 79, 87, 90, 91, 106, 109, 144, 147, 170, 180, 238, 254, 316, 319, 322, 329, 371, 372, 375, 378, 381, 382, 385, 392, 395, 396, 407, 409, 413, 419, 427, 430, 439, 442–446, 457, 458, 467, 486, 525, 526, 528, 530, 531, 546, 553, 560, 571, 579, 590, 600, 601, 612–614, 622, 650–660, 665, 666, 679, 682, 689 allocation, 5 capital asset pricing model (CAPM), 597 The Capital Markets Bond Index, 231 capital market union (CMU) benefits of the, 652 establishment of the, 380, 651, 653 history of, 651 caplet, 333–336 cap(s), 332–336 interest rate, 332, 333, 335 rate, 332, 333 valuation of, 334, 335

cash flow(s), 16, 54, 65, 68, 69, 124, 195, 198, 199, 225, 226, 235, 239, 244, 247, 248, 270, 278, 284–287, 291, 293, 294, 302, 304, 315–317, 319–321, 323, 326, 333, 353, 359, 374, 393, 394, 427, 428, 468–470, 526, 590, 594, 621, 635 matching, 230, 247, 248, 525, 526, 530 MBS, 317 cash flow-to-debt ratio, 65 cash futures price, 277–281 cash price, 51, 52, 72, 73, 201, 203, 278, 280–282, 299, 300, 639, 642, 643, 687 CDS Valuation, 353 cellular approach, 232–234, 431 central bank(s), 1, 3, 13, 26, 27, 32, 33, 39, 48, 64, 143, 147, 369–371, 373, 374, 391, 395, 396, 398, 456, 500, 535, 536, 543, 545, 546, 549–551, 557–561, 566, 567, 571, 652, 688 the role of, 26, 27 certificates of deposit (CDs), 17, 91, 237, 396, 397 certification, 667, 669, 670, 674, 676 Chair, 28, 42 challenges, 3, 385, 391, 553, 651, 655, 664, 665 new CMU action plan, 654 Chamberlain, L., 573 change, 3, 4, 6, 23, 52, 58, 59, 64, 67, 68, 86–88, 99, 112, 115, 117, 139, 141, 142, 146, 148, 157, 158, 169–176, 183–186, 188–191, 199, 207, 208, 211–213, 215, 216, 227, 229, 232, 234, 237, 238, 241, 242, 249, 281, 283, 290, 323, 335,

INDEX

349, 351, 360, 363, 414, 427, 455–459, 461–464, 466, 471, 472, 474–476, 478, 479, 482, 485, 487, 488, 523, 526, 594, 624–626, 663, 664, 672, 673 percentage price, 176, 186, 187, 191, 192, 214–217, 227, 228 price, 87, 171–176, 181, 183–186, 190, 191, 198, 210–213, 215, 216, 218, 220, 222, 223, 227–230, 456, 458, 459, 465, 471, 476, 479, 482, 629 changes in the competitive environment, 529 Chernobai, A.S., 529 Chesini, G., xi, 537–539, 541, 542, 544, 545, 559 Cholesky decomposition, 491, 492 Citi Emerging Markets Broad Bond Index (EMUSDBBI), 231 Citi US Broad Investment-Grade Bond Index (USBIG), 231 Citi World Broad Investment-Grade Bond Index (WorldBIG), 231 Citi World Government Bond Index (WGBI), 231 clean, 79, 201, 662 clean price, 52, 72, 77, 276 clean transportation, 665, 668 climate bond initiative, 671 climate change adaptation, 662, 667, 668 Climate Change Committee (CCC), 55, 56, 234, 401, 663 climate change mitigation, 662, 667 Climate Law, 662 climate-related financial disclosures, 664 close-out time, 501, 503, 504 CMU and bonds, 657 collar(s), 335 interest rate, 335

701

collateralized bond obligations (CBOs), 325–328 collateralized debt obligations (CDOs), 13, 91, 322, 324–328, 361, 362, 394 collateralized loan obligations (CLOs), 325, 326, 328 collateralized mortgage obligations (CMOs), 91, 266, 322–328, 361, 394 collateral(s), 10, 11, 36, 38, 39, 318, 387, 390, 546, 688, 689 commercial bank, 25, 26, 28, 31, 395–398 commercial loans, 326 commercial paper, 38, 91, 396, 397 commission sharing agreement (CSA), 687 common market, 651 common stock, 13, 573–575, 577–580, 582 Community Advisory Council (CAC), 29 Community Depository Institutions Advisory Council (CDIAC), 29 company(ies) financing, 444, 601, 612, 658, 659 insurance, 25, 58, 234, 238–242, 244, 245, 247, 248, 329, 377, 464, 525, 526, 660 investment, 24, 25, 408 comparison of bond ETFs with bond mutual funds, 438 comparison of bond mutual funds with bonds, 427 comparison of bonds and stocks, 601 comparison of private debt with private equity, 446 competitive bidding, 22 composition of debt, 563, 564 composition of debt and sustainability, 561

702

INDEX

compounding, 69, 76, 106, 107, 113–118, 122–124, 129, 131, 148, 206, 207, 219–222, 241, 315, 334, 336, 356, 471, 508, 600 concentration, 370, 528 concentration risk, 414, 524, 526, 527 credit, 526, 528 concerns about bond ETFs, 438 concerns about bond mutual funds, 427 concerns about private debt, 449 concerns around green bonds, 677 conditional prepayment rate (CPR), 321 conducting monetary policy, 31 Consiglio, A., 553 construction, 23, 90, 119, 152, 154, 289, 311, 392, 393, 399, 476 portfolio, 5, 527 construction of the term structure, 127, 129 consumer price index (CPI), 2, 3, 23, 411 contemporary trends in the bond markets, 371 contingent immunization, 256 continuous compounding, 117, 118, 122–124, 129, 148, 149, 154, 160, 223, 270, 271, 279, 283, 292, 298, 300, 330, 331, 334, 336, 506, 626, 639, 643 duration for, 207 contribution, 20, 25, 35, 66, 140, 144, 205, 239, 240, 242, 495, 498, 652, 662, 665, 674, 678 value at risk (VaR), 530 contribution of ICT advances, 674 conversion factor, 276, 278 conversion premium, 53 conversion ratio, 53

convertible bond, 53, 390, 650 convertible ETFs, 432 convexity, 88, 170, 207–219, 223, 530 uses of, 198, 210, 211, 213 convexity between coupon payment dates, 218 corporate bond issue(s), 373, 374, 391, 500, 625, 626 corporate bond market, 90, 373, 387, 390–392, 431 corporate bonds, 10, 18, 25, 51, 54, 67, 89, 90, 144, 252, 253, 326, 361, 370–374, 382, 387, 391, 408, 411–413, 430, 431, 433, 456, 536, 558, 601, 650, 689 corporate ETFs, 430, 432 Corporate Finance Institute, 90, 350– 352, 390, 392, 393, 395–398, 411–413, 415, 525, 526, 573, 601 corporate sector purchase program (CSPP), 37, 38, 550 Corsetti, G., 548 cost of borrowed funds, 613 cost of common equity from retained earnings, 613 cost of common equity from stock issuance, 614 cost of debt capital, 374, 613, 614 cost of preferred equity, 613 costs and charges disclosure, 686 MifID, MifID II/MiFIR, 686 Council of the European Monetary Institute (EMI), 41 counterparties, 18, 22, 35, 36, 38, 39, 267–269, 271, 281, 284, 285, 288–290, 293, 294, 296, 357, 359, 360, 363, 375, 376, 515, 518, 623, 654, 688, 689 counterparty risk, 171, 359, 377, 455

INDEX

coupon, 2, 9, 11, 23, 49–54, 68–85, 87, 88, 118, 121–130, 178–182, 186, 193, 195, 200–205, 209, 213, 218, 221, 224–226, 232, 233, 238, 242–244, 252, 254, 255, 271–276, 279, 280, 291, 292, 296, 298, 300, 302, 312, 313, 320, 326, 327, 351, 356, 360, 372, 382–385, 387, 391, 397, 412–414, 427, 430, 431, 456, 469, 507, 510, 572, 575–577, 596, 604, 605, 628, 630, 633, 639–641, 643 coupon-bearing bond(s), 11, 49, 68–70, 75, 124, 172–174, 178– 181, 184, 185, 187–193, 197, 199, 200, 202, 209, 210, 212, 214–217, 224, 225, 228–230, 247, 301–303, 314, 315, 455, 471, 510 covariance matrix, 492 coverage ratio, 65 covered bond purchase program (CBPP3), 37, 550 covered bonds, 370, 376, 378, 411, 536, 650, 654, 657, 658 COVID-19, 64, 551, 552, 563, 567 Cox, Ingersoll, Ross model, 150 credit, 14, 15, 28, 29, 34, 39, 40, 42, 55, 56, 58, 61, 64, 90, 127, 233, 251–253, 322, 326, 327, 349–355, 358, 359, 361–363, 365, 372, 376, 378, 384, 397, 400, 412, 429, 439, 440, 455, 458–460, 500, 509, 510, 513, 515, 518, 519, 524, 526–528, 543, 552, 572, 657–660, 690 opportunities, 441, 442 credit-card receivables, 323 credit concentration risk, 526, 528 credit default swap index (CDX), 376 credit default swaps (CDSs)

703

basket, 357, 376 binary, 357 credit derivatives basket CDS, 357 binary CDS, 357 collateralized debt obligations (CDOs), 13 credit default swaps (CDSs), 352, 376, 518 credit spread options, 352 total return swaps (TRS), 352, 518 credit opportunities, 441 credit rating, 13, 15, 16, 55, 56, 58, 59, 61, 63–65, 119, 125, 127, 128, 233, 234, 251, 252, 276, 315, 325, 327, 328, 359, 361, 374, 378, 382, 383, 385, 386, 397, 411, 429, 430, 439, 456, 475, 486, 487, 510, 513, 517, 519, 526, 527, 531, 536, 572, 628, 665, 670, 676 factors affecting the, 65 credit risk expected losses in case of default, 506 probability of default, 524 protection, 353, 516, 518, 678 reduction, 516 credit spread option(s), 352, 362–366 Credit Suisse, 231 First Boston High-Yield II Index, 231 crisis/crises COVID pandemic, 37, 446, 462, 467, 469, 474, 535, 536, 538, 540, 541, 549, 552, 557, 559, 560, 563, 565–567, 571, 664 financial, 1, 13, 26, 32, 34, 54, 89, 90, 121, 350, 370, 373, 374, 376, 382, 387, 420, 421, 462, 538, 540, 546, 549, 552, 559, 562, 565, 679, 688

704

INDEX

sovereign, 351, 385, 535, 537, 540, 546–549, 558–560, 563, 565 Cummars, 12, 13 currency, 2, 12, 26, 27, 39, 144, 233, 294, 296, 381, 408, 430, 457, 461, 463, 479, 525, 526, 536, 537, 542, 546, 564, 652, 674, 685 swaps, 34, 36, 39, 294 currency swaps, 34, 36, 39, 294, 296 current yield (CY), 77, 79, 598, 604, 605 selecting between a stock and a bond, 601, 605 D Dabrowski, M., 563 date, 11, 12, 14, 18, 38, 52, 53, 57, 80, 81, 103–105, 118, 149, 150, 174, 238, 239, 241, 242, 266, 269–273, 278, 280, 284, 287, 297, 298, 301, 313, 315, 354, 358, 390, 391, 398, 414–416, 427, 432, 525, 537, 639, 640, 642, 685, 688 maturity, 9, 10, 20, 23, 49, 51–53, 69, 78, 80, 81, 86, 89, 90, 99, 101, 103, 105, 117, 118, 126, 127, 130, 133, 140, 144, 145, 149, 169, 172, 174, 177–181, 200, 201, 205, 209, 223–226, 232, 233, 238, 245, 249, 252, 255, 268, 273, 275, 280, 292, 293, 297–299, 302, 313, 330, 335, 351, 354, 358, 362–364, 377, 383, 392, 395, 397, 414, 430, 431, 438, 470, 510, 512, 513, 515, 576, 622, 632 dealers, 22, 24, 71, 352, 399 debt business, 442 composition of, 561, 563, 564

distressed, 58, 441, 442 Federal Agency, 18 infrastructure, 441, 442 intermediate, 442 junior, 10 mezzanine, 441, 442 outstanding, 350, 351, 430, 564 real estate, 442 restructuring, 54, 350, 351 senior, 326, 441 subordinated, 10 sustainability of, 548, 553, 558, 561, 562, 567 venture, 441, 442 debt restructuring, 350, 351 debt sustainability, 553, 567 decision, 4–6, 8, 9, 28, 34, 39, 40, 42, 61, 376, 379, 409, 531, 543, 575, 586, 613, 631, 676, 687 capital allocation, 5 decomposition, 491 Cholesky, 491, 492 eigen-value, 492 default expected losses, 506, 507, 508, 514 loss given, 514 obligation, 350 payment, 58, 350, 351 probability of, 15, 59, 63, 327, 353, 354, 356, 358, 383, 385, 386, 486, 506–510, 512–514, 524 rate, 59, 60, 63, 519, 522, 524 restricted, 58 risk-neutral probability of, 354 risk of, 48, 54, 55, 58, 67, 143, 171, 233, 352, 358, 359, 390, 392, 393, 506, 537, 559, 665 selective, 55 deferred callable bonds , 52, 80 definition bond ETF, 429

INDEX

bond mutual fund, 408 private debt, 439 demand of money, 143 derivative(s) credit, 349, 352, 353, 376, 429, 455, 515, 518, 530 interest rate, 19, 265, 266, 268, 299, 349, 361, 429, 457 market(s), 17–19, 376, 378, 633, 680 determinants of the interest rates, 100, 141, 143 diagonal matrix, 492 Diebold, F., 505 differentiation, 8, 322, 443, 444 digitalization, 373 direct trading, 22 dirty, 201 dirty price, 52, 72, 276 discounted dividend model (DDM), 577, 578 distressed debt, 58, 441, 442 distribution fees, 417 diversification, 24, 48, 89, 327, 371, 409, 412, 413, 419, 425, 427, 438, 546 dividends, 10, 13, 53, 427, 572, 574–576, 578, 580–582, 590, 596, 599, 600, 602, 604 stock valuation based on, 577 dividend yield, 432, 435, 437, 598–600, 604, 605 selecting between a stock and a bond, 604 Dodd-Frank Wall Street Reform and Consumer Protection Act, 29, 375 do no significant harm (DNSH), 662 Dow Jones Industrial Average (DJIA), 7 Dow Jones Industrial Average (DJIA) Index, 7

705

downgrades, 59, 64, 67, 371, 429, 517, 519, 522, 536 activating, 516, 517 triggering, 516, 517 drawbacks, 226, 493 of scenario testing, 462, 463 of stress testing, 460, 461 drivers of the bond market growth, 373 duration between coupon payment dates, 200 dollars, 473 Fisher-Weil, 222, 249 for continuous compounding, 207 for multi-period compounding, 206 for non-horizontal yield curve, 219 hedging, 282, 626 impact of coupon payment on, 202 impact of time lapse on, 202 interpretation of, 198 Macaulay, 176 properties of, 176, 177 quasi-modified, 220, 221, 249 uses of, 182 duration-based hedge ratio, 283, 627 duration-based hedging, 626 dynamic funds, 414

E eco-efficient and/or circular economy adapted products, production technologies and processes, 669 economic and monetary union (EMU), 2, 33, 41, 401, 565, 653 economic capital, 486, 494, 499, 500, 504, 531 Economic Growth, Regulatory Relief and Consumer Protection Act, 375 The Economic Times, 558

706

INDEX

effective yield (EY), 76, 77 eigen-value decomposition, 492 electronic form, 51 electronic liquidity providers (ELPs), 380 electronification, 373, 379, 380, 678 eligible assets, 39, 536 eligible green projects, 667, 668 embedded bond options, 130, 144, 266, 313 energy efficiency, 668 Entropy Concentration Index (ECI), 529 environmentally sustainable management of living natural resources and land use, 668 environmental, social and governance (ESG), 666, 675–677 equilibrium models Cox, Ingersoll, Ross, 150 one factor, 152 Rendleman & Bartter, 150, 151 Vasicek, 150, 151, 153, 301 equity, 2, 13, 48, 53, 66, 265, 304, 324–326, 379, 419, 442, 486, 536, 575–577, 579–584, 589–591, 597, 598, 602, 612, 613, 656, 659, 660, 681, 683 financial ratios, 585 market(s), 17–19, 390, 681, 683 private, 20, 328, 375, 439, 445, 449, 659 Erce, A., 549 establishment of the CMU, 653 estimating g, 582 ETF database, 400, 432–437 Eurodollar futures, 281, 282 Eurodollars, 17, 91, 281, 282, 396, 398 Euro Interbank Offered Rate (Euribor), 286–292, 360

European Bank for Reconstruction and Development (EBRD), 10 European Banking Union (EBU), 549 European Central Bank (ECB), 27, 33–37, 39–42, 54, 64, 101, 147, 456, 536, 538, 542, 543, 549–553, 560, 563, 567, 658 Governing Council of the, 33–36, 41 European Commission (EC), 376, 378, 391, 651–655, 657, 659–663, 678–680, 683–685, 688–690 European Economic Community (EEC), 651 European Investment Bank (EIB), 10 European long-term investment fund (ELTIF), 660 European market infrastructure regulation (EMIR), 654 European Monetary Institute (EMI), 34, 41 European Options on Bonds, 302 price, 302 European Parliament, 41, 143, 553, 658, 679 European Recovery Strategy, 661 European Securities and Markets Authority (ESMA), 380, 678, 680 European swap options, 330 valuation of, 302, 330 European System of Central Banks (ESCB), 27, 33, 39, 40 objective of the, 33, 39 organization of the European Union (EU), 3, 39, 42, 54, 64, 370, 376, 378, 380, 381, 385, 443, 444, 460, 530, 558, 561, 567, 649–657, 659–663, 665, 669, 675, 677–680, 686, 688

INDEX

Green Deal, 657, 661–663, 665 European Venture Capital Fund Regulation (EuVECA), 653 Eurosystem, 33–40, 42, 549, 550 the operational framework of the, 34 Eurosystem monetary policy strategy, 33 euro-zone, 3, 64, 279, 429, 472, 536 euro-zone countries, 2 exchange fees, 417, 687 exchange traded funds (ETFs) All-Term, 435, 437 bond, 377, 408, 430–432 broad market, 430 convertible, 432 corporate, 430, 432 investment grade corporate, 432 long-term, 437 mechanics of, 430 municipal, 430 short-term, 437 sovereign, 430 total bond market, 432, 433 zero duration, 437 Executive Committee, 42 Chair, 42 expectation hypothesis theory, 138, 139 expected future spot rate, 104, 132, 136, 137, 139 expected losses in case of default, 506 expected recovery rate, 354, 357, 513 expense ratio, 417, 432 expenses, 9, 31, 131, 358, 415, 417, 427, 432, 437, 558, 600, 676, 687 other, 417 exploit arbitrage opportunities using forward contracts to, 639 using futures contracts to, 639 using options to, 643

707

external review process, 669 extrapolation, 163

F Fabozzi, F.J., 529 face value, 2, 49, 51, 54, 68–71, 73, 74, 78–80, 87, 119, 121–125, 127, 130, 133, 136, 149, 159, 172, 176–179, 193, 195, 201, 221, 243, 246, 255, 269, 277, 278, 280, 284, 291, 292, 301, 302, 306, 308, 317, 330, 332, 351–353, 359–361, 369, 372, 373, 382–384, 391, 397, 398, 411, 427, 428, 438, 469, 470, 472, 474, 507, 510, 514, 518, 572, 576, 577, 601, 602, 604, 626–629, 631–633, 636, 639, 643 factor(s), 8, 39, 48, 141, 143, 146, 174, 176, 181, 207, 221, 241, 266, 373, 443, 459–461, 469, 475, 484, 485, 488–492, 495, 529, 543, 638 affecting the credit rating, 65 conversion, 276, 278 failure risk, 529 fallen angels, 13, 370, 456, 536, 561, 571 Fama, E.F., 597 Fastenrath, F., 567 Federal Advisory Council (FAC), 29 Federal Agency Debt, 18 Federal Deposit Insurance Corporation (FDIC), 397 Federal Home Loan Bank (FHLB), 384 Federal Home Loan Mortgage Corporation (FHLMC/Freddie Mac), 384, 401, 411

708

INDEX

Federal National Mortgage Association (FNMA/Fannie Mae), 317, 384, 401, 411, 673, 674 Federal Open Market Committee (FOMC), 28 Federal Reserve (FED), 13, 27, 28, 31–33, 398, 456, 536, 572 functions of, 27, 29 structure of, 27 Federal Reserve System (FRS), 27 fees account, 417 distribution, 417 exchange, 417, 687 management, 242, 417 purchase, 417 transactions, 416, 687 Fidelity, 409–413, 416, 417, 428, 429 financial crisis, 1, 13, 26, 32, 34, 54, 89, 90, 121, 350, 370, 373, 374, 376, 382, 387, 420, 421, 462, 538, 540, 546, 549, 552, 559, 562, 565, 679, 688 financial intermediation, 290 swap, 290 financial products markup language (FpML), 352 financial ratios, 67, 527, 585, 599 equity, 585 market, 585 Financial Service Action Plan, 651 Financial Stability Board’s Task Force, 664 financial system, 27, 28, 30, 40, 143, 546, 552 participants of, 8, 9 financing, 3, 4, 9, 12, 14–16, 26, 27, 66, 359, 370, 375, 391, 395, 396, 399, 439, 441, 442, 445, 535, 551, 553, 557, 558, 560, 571, 572, 602, 614, 651, 652,

654, 658–660, 663–665, 667, 675, 688 company, 14, 444, 579, 601, 612, 659 small and medium enterprise (SMEs), 651, 653, 658–660 FinTech, 380 firm commitment, 21 first-to-default basket CDS, 357 Fisher, L., 222 Fisher-Weil duration, 222, 249 Fitch, 55–58 Fitch Ratings, 55, 64 fixed income, 3, 4, 7, 8, 13, 17, 25, 26, 48, 54, 87, 91, 170, 171, 234, 238, 250, 251, 268, 282, 283, 349, 369–373, 375, 377, 378, 380, 381, 395, 399, 407, 408, 414, 422, 425, 429–431, 455, 458, 459, 461–467, 472, 474–476, 478, 484, 486, 493, 494, 498–500, 519, 524, 528, 529, 535, 571, 622–624, 626, 627, 649, 657, 659–661, 665, 675, 678, 681, 683, 684, 687 capital market, 17 instrument, 2, 7, 13, 54, 137, 145, 170, 230, 297, 304, 352, 375, 457, 471, 474, 494, 496, 526, 537, 546, 626, 649, 650, 657, 665, 683, 689 investment, 2, 4, 25, 47, 479, 529–531, 572, 594, 649, 651, 660 portfolio management, 170, 377, 380, 525 flat volatilities, 335 floating rate notes (FRN’s), 1, 11, 52, 68 floorlet, 335 floor(s), 256, 266, 333–336 interest rate, 334

INDEX

rate, 335 valuation of, 335 forced sale, 456, 501 foreign exchange (FX), 144, 265, 266, 294, 457, 468, 471, 482 foreign exchange risk (FX risk), 294, 479 Monte Carlo VaR, 471, 476, 478, 484 parametric VaR, 468, 471, 474 formula, 71, 84, 102, 115, 123, 126, 140, 161, 172, 186, 190, 192, 204, 207, 213, 218, 220, 282, 298, 300, 356, 458, 459, 461–463, 471, 475, 476, 478, 479, 525, 575, 577, 603 Gordon’s, 578, 603, 613, 614 forward contract(s) arbitrage, 272, 642 bond, 271–273 hedging, 623 interest rate, 269, 270 speculation, 633 forward forward volatilities , 335 forward rate agreements (FRA), 126, 266, 269–271 forward rate(s), 103–118, 126, 128, 131–141, 145, 152, 153, 270, 271, 280, 330, 334 Fratscher, M., 540 Fratzscher, M., 547 Freitakas, E., 527, 528 French, 322, 383, 546 French Government Bond, 385 French, K.R., 597 front-end load, 417 FTSE UK Gilts Index Series, 231 functions of the FED, 27, 29 fund(s) bond, 24, 408–415, 417, 418, 420–422, 426–429, 431 dynamic, 414

709

exchange traded, 24, 376, 408, 429 high-yield bond, 410, 411, 425 international bond, 413 investment-grade bond, 410, 429 long-term, 414 maturity, 414, 415 money market, 395, 397, 398, 408, 410, 417, 419–422 multi-sector bond, 412 municipal bond, 10, 90, 91, 371, 392, 393, 410, 413, 420, 422, 433 mutual, 24, 25, 377, 408, 409, 413, 415–417, 419–428, 438, 500 other types of bond, 413 pension, 25, 234, 238–242, 249, 384, 445, 464, 652 private debt, 408, 443, 444, 446–449 short-term, 414 ultra short-term, 437 futures bond, 273, 275, 277, 279, 280 Eurodollar, 281, 282 interest rate, 99, 100, 102, 140, 250, 251 T-Bill, 280, 282 futures contracts arbitrage, 275, 642 hedging, 268, 282, 625, 627 on a single bond, 273 speculation, 573 futures markets, 18, 631, 634, 635, 638 future spot rate, 104, 127, 131, 132, 136, 138, 149, 150, 270 futures price, 268, 273, 275, 276, 280–284, 299, 300, 512, 624, 626, 627, 634 cash, 271–273, 276–281 quoted, 276–279, 281

710

INDEX

G General Council President, 41 Vice-President, 41 General Trinomial Tree, 155 geometric Brownian motion, 145, 146, 299 German bund, 385, 461, 553 German Government Bond, 385 Gilt, 386 Gini coefficient (G), 528 Golding, 441, 449 Goodhart, C.A.E., 26, 27 Gordon, 578 Gordon’s formula, 578, 580, 603, 613, 614 Governing Council of the ECB, 34–36, 41 government, 4, 8, 9, 12, 20, 27, 30, 41, 51, 55, 89–91, 99, 119, 231, 313, 314, 317, 350–352, 381–385, 387, 391–393, 399, 400, 408, 410–413, 424, 426, 429, 443, 508, 527, 537, 546, 547, 557–559, 561–565, 567, 663, 664, 671 government bond market, 89, 382, 383, 392, 549 government bonds, 10, 12, 13, 20, 23, 32, 51, 54, 89, 90, 231, 233, 253, 254, 284, 313, 382, 385–387, 391, 398, 400, 401, 411, 425, 426, 430, 431, 443, 460, 472, 536, 537, 546–548, 550, 551, 559, 561, 625, 627, 649 Government National Mortgage Association (GNMA/Ginnie Mae), 317, 384, 394, 401 government sector, 9, 561 governors of the NCBs, 40, 41 Greco, L., 565

Greek financial crisis, 462 green bond market, 666, 670–672, 675 Green Bond Principles (GBPs), 294, 295, 665–670, 676, 677 green bond(s) becoming popular, 665, 675 concerns, 667, 677 market, 666, 670–672, 675 principles, 665–667 scoring/rating, 669, 670, 676 versus plain vanilla bonds, 675 What are, 665, 666 green buildings, 665, 669 Green Deal, 561, 661–663, 675 green projects, 665, 667, 668, 676, 677 eligible, 667, 668 greenwashing, 677 Gross, P., 567 guarantees, 7, 12, 25, 144, 242, 244, 257, 317, 329, 383, 384, 390, 392, 394, 398, 411, 419, 428, 438, 469, 516, 574, 575, 635, 688 collaterals, 516 H Hall and Tideman Index (THI), 528 Hay, W.W., 573 hedge using forward contracts to, 623 using futures contracts to, 623 using options to, 632 hedger, 621, 622, 624 hedge ratio, 624, 628, 629 duration-based, 283, 627 hedging duration-based, 626 forward contracts, 623 futures contracts, 268, 282, 625–627

INDEX

options, 623, 632 Herfindahl-Hirschman Index (HHI), 527, 528 high-yield bond funds, 410, 411, 425, 426, 429 historical VaR, 471, 474–478 history of CMU, 651 Ho & Lee model, 152, 300 holding period return (HPR), 76, 81, 82, 84, 85, 126, 127, 134, 254, 256, 596, 601, 603, 604 selecting between a stock and a bond, 601 Homer, S., 251, 252 horizon analysis, 254, 255 Ho, T.S., 153, 297, 301 household, 9, 143, 557–559, 650–653, 658 Hull & White model, 152, 157, 300, 311 Hull, J., 146, 153, 156–158, 297, 301, 311 human mistakes on trades, 529 I ICMA, 267 ICT advances contribution of, 378 identification of mispriced bonds, 250, 251 IHS Markit, 400, 441 immunization contingent, 256 net worth, 235, 237, 238, 525 target date, 238, 241, 243 impact of coupon payment on duration, 202 of crisis on the bond yield, 537 of public and private debt to bond yields, 559 of QE on the bond yield, 542

711

of time lapse on duration, 202 on stakeholders, 543 implications, 210, 661 policymaking, 560 importance of new products, 376 income, 2, 8, 9, 11, 14, 23, 25, 29, 31, 49, 54, 74, 254, 268, 271–273, 275, 280, 324, 326, 364, 365, 392, 395, 409, 410, 413, 414, 427, 430, 457, 575, 640, 652, 657 bond(s), 11 net, 67, 499 required, 407 indenture, 390 bond, 10, 667 independent and identically distributed (IID), 502 index, 2, 23, 82, 122, 145, 155, 230, 232, 233, 237, 265, 266, 275, 299, 306, 376, 377, 399, 402, 431, 472, 528, 538, 587, 592, 598 bond, 7, 230, 273, 400 Dow Jones Industrial Average (DJIA), 7 index-linked bonds, 2, 23 industrial development bond (IDB), 393 industrial revenue bond (IRB), 393 inflation, 2–4, 23, 26, 30, 38, 89, 142, 144, 382, 411, 428, 429 information and communication technology (ICT), 373, 378–380, 674 infrastructure debt, 441, 442 Ingersoll, J.E., 151 initial public offerings (IPOs), 20 instantaneous short rate, 148–150, 152, 154, 311

712

INDEX

institutional investor, 15, 16, 20, 22, 58, 266, 395, 407, 419, 440, 444, 613, 659 insurance companies, 25, 58, 234, 238, 240, 241, 377, 660 interest, 2, 7, 9, 23, 25, 29, 31, 47, 49–52, 56, 57, 67, 68, 72, 77, 80, 86, 89–91, 108, 123, 127, 143, 144, 147, 153, 170, 176, 233, 266, 268, 274, 278, 284, 285, 294, 312, 313, 316–320, 322–324, 327, 332, 333, 351, 359, 369, 370, 372, 378, 382, 383, 391–396, 398, 414, 426–428, 430, 431, 439, 441–443, 449, 457, 467, 475, 499, 513, 543, 558, 562, 565, 602, 611, 614, 634, 635, 638, 642, 666, 684 accrued, 51, 52, 72, 201, 276, 278, 279, 300, 320, 354, 510, 514, 642 interest only (IO), 319 interest only MBS, 319 interest rate cap, 332 interest rate collar, 335 interest rate derivatives asset backed securities, 266 bond futures, 280 callable bond(s), 52, 81, 312, 314, 428 collateralized bond obligation(s), 327 collateralized debt obligation(s), 327, 361 collateralized loan obligation(s), 327 collateralized mortgage obligation(s), 266, 361 currency swap(s), 36 embedded bond option(s), 266 Eurodollar futures, 281, 282

forward contract(s), 269, 633 forward rate agreement(s), 266 futures, 99, 140, 250, 251 interest rate option(s), 145, 266, 297, 298, 308, 362 interest rate swap(s), 266, 284, 285, 291, 294, 296, 329, 330, 360, 515 mortgage backed security(ies), 225, 237, 266, 316 option(s), 266, 297–299, 329, 633 puttable bond, 313 repos, 266, 267 swap(s), 253, 266, 269, 289–291, 294–296, 329, 330 swaption(s), 266, 329, 330 T-Bill Futures, 280 interest rate floor, 334 interest rate forecasting, 102, 250, 253, 254 interest rate forward contracts, 269 interest rate futures, 266, 273, 299, 376 interest rate models, 100, 145, 297, 300 interest rate moves, 88, 154, 171, 175, 192, 204, 207, 218, 235, 238, 282–284, 349, 428, 465, 466, 525, 575, 626, 627 interest rate options, 145, 266, 297, 298, 304, 308, 362 interest rate risk scenario testing, 458, 463 sensitivity analysis, 459 stress testing, 458–460 value at risk, 458, 465, 530 interest rate(s) cap, 332, 333, 335 collar, 335 determinants of the, 100, 141, 143 floor, 334

INDEX

low, 83, 157, 317, 370, 373, 378, 399, 412, 426, 443 nominal, 3, 142, 144, 284 risk, 53–55, 68, 73, 118, 120, 132, 134, 137, 141, 146, 169–171, 175, 233, 234, 237, 238, 247, 249, 254, 265, 266, 282, 285, 290, 294, 390, 392, 412–414, 428, 429, 438, 444, 455–457, 479, 484, 495, 525, 626 swaps, 284, 285, 294, 329, 330, 360, 515 valuation, 291 structure, 2, 58, 100, 137, 138, 145, 146, 148, 150–152, 159–161, 250, 266, 315, 317, 331, 456, 458, 461, 466 trees, 153, 155, 157 uncertainty, 131 intermarket spread swap, 252, 253 intermediary(ies), 21, 23, 24, 71, 144, 284, 290, 291, 294, 369, 375, 376, 379, 383, 397, 416, 417, 430, 654, 659, 678, 685, 687 intermediate debt, 442 international bond funds, 413 International Capital Markets Association (ICMA), 372, 380, 456, 665–670, 681–687, 689 International Monetary Fund (IMF), 54, 64, 444 International Swaps and derivatives Association (ISDA), 349–352 interpolation, 119, 127–129, 311 interpretation of duration, 198 investment bank(er), 19–21, 24, 666 investment companies, 24, 25 Investment Company Institute, 408, 417–424 investment-grade bond funds, 410, 412 investment grade corporate ETFs, 432

713

investment process, 4, 5, 8, 531, 668 invoice price, 52 issuer(s), 3, 9, 10, 13, 15, 21, 22, 24, 47, 49, 50, 52–55, 58, 63, 65, 73, 78, 80, 81, 99, 121, 127, 142, 143, 169, 171, 251, 312, 313, 349, 352, 353, 362, 378, 387, 390, 391, 400, 409, 410, 412, 421, 427, 428, 455, 456, 506, 510, 513, 524, 526–528, 553, 557, 572, 574, 625, 667–670, 673, 676, 689 bond, 9, 15, 22, 64, 359, 360, 381, 518 issuers of borrowers of the same kind of credit, the same aim of the credit or source of repayment, 527 issuers or borrowers from economically related countries, 527 issuers or borrowers from the same geographical region, 527 Issuers or borrowers from the same sector of the economy, 527 issuers or borrowers of the same country, 89, 382, 527 Issuers or borrowers of the same rating especially of lower credit quality, 527 issuers or borrowers that have received credit in the same (foreign) currency, 479, 527 issuers or borrowers that use the same collateral or types of collateral, 527 issue(s) bond, 9–11, 55, 89, 90, 144, 170, 205, 232, 316, 352, 370, 373, 374, 385–387, 391, 400, 431, 545, 546, 612, 632, 649 corporate bond, 373, 374, 391, 500, 625, 626

714

INDEX

iTraxx, 376

J J.P. Morgan Emerging Markets Bond Index, 231 J.P. Morgan Government Bond Index, 231 junior debt, 10 junk bond, 58, 327, 432, 433

K Kane, A., 6–11, 14, 17, 20–22, 24, 51, 65, 91, 106, 130, 131, 135–138, 140–142, 172, 230, 232, 234, 235, 248, 251, 254, 256, 383, 384, 387, 390, 392–398 Kick-start the EU economy through incentives to private investments, 661 Koay, H., 375–379 Kogelman, S., 251, 252 Kouskouna, E., 652 Kwon, S., 540, 559

L Laipply, S., 375–379 Large Employer Emergency Financing Facility (LEEFF), 664 large in scale (LIS) orders, 684 lead underwriters, 21 Lee, L.B., 153, 297, 301 Lehman Brothers, 462, 538, 559 Leibowitz, M.L., 251 Leiden University, 664 lender, 9, 10, 15, 26, 27, 47–49, 138, 317, 333, 335, 372, 506, 572, 605, 609 leverage ratio, 66 Liebowitz, M.L., 252

liquidity adjusted VaR (LVaR), 502, 505 liquidity preference theory, 138, 140, 143 liquidity ratio, 66 liquidity risk bid-ask spread, 505 close-out time, 501 forced sale, 456 load back-end, 416 front-end, 415, 417 redemption, 416 sales, 415 London Interbank Offer Rate (LIBOR), 281, 293, 412 London Stock Exchange Group, 688 long position, 52, 53, 80, 268, 269, 271, 274, 282, 291, 312, 313, 329, 331, 333, 363–365, 461, 622, 623, 629, 632, 633, 635, 637, 638, 642 long-term ETFs, 437 long-term funds, 414 long-term investments, 135, 659, 660 López-Gallo, F., 527, 528 loss given default (LGD), 514 Lovell, J., 664 low interest rates, 83, 157, 317, 370, 373, 378, 399, 412, 426, 443 Luenberger, D.G., 106, 111–113, 138, 205, 206, 219, 220, 222, 234, 266, 606 M Maastricht treaty, 562, 651 Macaulay duration, 176 Macaulay, F.R., 176 management, 6, 23, 170, 233, 236, 248, 251, 256, 257, 377, 392, 407, 409, 415–421, 424–426, 430, 432, 435, 437, 438,

INDEX

442–444, 457, 525, 526, 530, 531, 565–567, 628, 665, 668 portfolio, 5, 170, 250, 399, 407, 501, 687 risk, 8, 63, 265, 353, 400, 457, 529–531, 567, 623, 650, 667 management expense ratio (MER), 417 management fees, 242, 417 management of proceeds, 665, 667 Marcus, A.J., 6–11, 14, 17, 20–22, 24, 51, 91, 106, 130, 131, 135–138, 140–142, 172, 230, 232, 234, 235, 248, 251, 254, 256, 383, 384, 387, 390, 392–398, 606 market/equity financial ratios, 585 market conversion value, 53 market financial ratios, 585 market(s) bond, 604 bond ETF, 431 bond mutual fund, 417, 420, 422 green bond, 666, 671, 672, 675 mortgage-backed security, 91, 394 over-the-counter (OTC), 22, 271, 332, 371, 373 private debt, 443, 444, 649 segmentation theory, 138 Markets in Financial Instruments Directive II (MiFID II), 375, 379–381, 679–681, 683, 686, 687, 689 Markets in Financial Instruments Directive (MiFID), 678–680 Markets in Financial Instruments Regulation (MiFIR), 657, 680 market structure MifID, MifID II/MiFIR, 683 Markowitz, H., 608 Márquez, J., 527, 528

715

Marrison, C., 324, 325, 458, 460– 469, 471–473, 475, 476, 478, 479, 484–486, 488, 489, 491, 493, 494, 498–502, 504, 505, 526, 529 Martelli, D., 537 Martin, A., 562 maturity, 6, 10, 11, 25, 38, 49, 68, 69, 74, 78, 79, 81, 83, 89, 91, 99, 100, 103, 105, 118, 119, 121, 124, 129, 136, 140, 145, 149, 154, 172–174, 177, 178, 193, 197, 199, 202, 219, 223, 242, 249, 255, 267, 270, 273–276, 279, 280, 282, 297, 299–301, 303, 304, 308, 322, 329, 330, 352, 360, 372, 382, 383, 385, 387–389, 392, 395, 397, 409, 410, 412–414, 427, 428, 438, 474, 507–509, 526, 536, 549, 557, 561, 564, 565, 576, 602, 637, 639, 688, 689 bond, 254 date, 9, 10, 20, 49, 51–53, 69, 78, 80, 81, 86, 89, 90, 99, 101, 103, 105, 117, 118, 126, 127, 130, 140, 144, 145, 149, 169, 172, 174, 177–181, 200, 205, 209, 223, 225, 226, 232, 233, 238, 245, 249, 252, 255, 268, 273, 275, 280, 292, 293, 297–299, 302, 313, 330, 335, 351, 362–364, 377, 383, 392, 395, 397, 414, 430, 431, 438, 470, 510, 512, 513, 515, 576, 622, 632 funds, 414, 415 option, 53, 298, 299, 302, 311, 364, 515 MBS cash flows, 319 MBS valuation, 313, 317, 319

716

INDEX

mean reversion property, 147, 148, 153–157, 304 mechanics of bond mutual funds, 268, 333, 408, 414 operating expenses, 417 pricing, 169 shareholder fees, 416 mechanics of ETFs, 430 member states, 3, 39–42, 385, 540, 543, 553, 560, 649–656, 661, 678–680 Merler, S., 546, 566 Merrill Lynch, 231 mezzanine debt, 441 Michele, B., 370 MifID, MifID II/MiFIR best execution data provision, 685 costs and charges disclosure, 686 market structure, 381, 682, 683 research unbundling rules, 687 securities financing transactions, 686–688 third country trading venues, 686 transparency, 375, 678, 684, 686 MifID, MifID II/MiFIR and bond markets, 678 MifID, MifID II/MiFIR Impact on bond markets, 681 minimum reserves, 34, 36, 38 mistakes on the implementation of legal/regulatory framework, 530 mistakes or misconduct by staff, 42 model Black’s, 297–300, 331, 334, 335, 363, 364, 637 Cox, Ingersoll, Ross, 150 Ho & Lee, 152, 153, 300 Hull & White, 152, 153, 157, 300, 311 Rendleman & Bartter, 150, 151 Vasicek, 150, 151, 153, 300 Model Validation Council, 29

modernization of the bond markets, 373, 374, 381 monetary policy instruments and procedures, 34 monetary policy objectives, 30 monetary policy strategy, 33 Eurosystem, 33 money demand of, 143 market funds, 395, 410, 418–421 market instruments, 89–91, 268, 382, 387, 396 supply of, 143, 398 money market(s) bankers’ acceptances, 91 certificates of deposit, 91 commercial paper, 91, 396, 397 Eurodollars, 91 funds, 395, 410, 418–422 instruments, 89–91, 268, 382, 387, 396 repurchase and reverse repurchase agreements, 398 tax anticipation notes, 396, 399 treasury bills, 91 Monte Carlo Value at Risk Contribution, 498 Monte Carlo VaR, 471, 476, 478, 480, 484, 491 FX Risk, 468 Moody’s, 55–58, 60, 61, 510, 511 Moody’s Investor Services, 55 Morgan, J.P., 400, 401 mortgage backed securities (MBS), 13, 23, 32, 90, 91, 225, 237, 316–325, 328, 384, 390, 394, 408, 411 mortgage-backed security market, 394 Moy, R.L., 78 multilateral trading facilities (MTFs), 681–683 multi-period compounding, 117

INDEX

duration for, 206 multi-period compounding per year, 114, 123, 220 multi-sector bond funds, 412, 426 muni, 90, 402, 426 municipal bond(s), 11, 18, 90, 91, 371, 392, 393 funds, 410, 413, 420–422 market, 91, 391–393 municipal ETFs, 430 municipalities, 47, 89, 411, 413, 430 mutual fund(s) bond, 408–410, 414, 415, 421–430 high-yield bond, 425, 426, 429 international bond, 413, 429 investment-grade bond, 429 money market Bond, 408, 420–422 multi-sector bond, 412 municipal bond, 410, 413, 420–422, 425, 426 other types bond, 413

N Nasdaq, 384 national central bank (NCB), 34–36, 38–42, 551 negotiations, 22 net asset value (NAV), 414–416, 427, 428, 438, 590 net income, 67, 499 netting, 516 net worth immunization, 235, 237, 238 new CMU action plan, 654, 658 challenges, 654 new products, 15, 375 importance of, 376 New York Fed, 456 Next Generation EU fund, 661 NN Investment Partners, 441, 670, 675, 677

717

no-arbitrage models, 146, 152 Ho & Lee, 152 Hull & White, 152 nominal interest rate, 3, 142, 144 nominal value, 15, 49, 74, 235, 244, 291, 331, 372, 628, 631 non-horizontal yield curve, 219 duration for, 219 normal distribution, 157, 302, 464, 465, 467, 471, 472, 474, 478, 484, 485, 491 notional principal, 352–356, 361, 518 O Objective of the European System of Central Banks, 33 obligation, 12, 13, 15, 16, 18, 22, 49, 55, 57–59, 65, 66, 89, 154, 225, 238, 249, 259, 271, 329, 341, 349, 350, 382, 392, 393, 455, 506, 508, 515, 558, 567, 572, 601, 623, 649, 657, 683 acceleration, 350 default, 350 reference, 352–355, 518 Obligation assimilables du Trésor (OAT), 385 OECD Guidelines on Multinational Enterprises, 662 offer price, 22, 71, 415 oil prices drop, 462 one-factor models, 146, 147, 150 open market operations, 26, 28, 31, 34, 35, 38 operating expenses, 350, 417 of bond mutual funds, 417 operating risk/operations risk, 529 the operational framework of the Eurosystem, 38 opportunities, 30, 85–87, 94, 107–111, 130–132, 135, 230, 250, 252, 272, 274, 313, 338,

718

INDEX

370, 374, 380, 393, 621, 622, 642–644 credit, 442 optimal hedge ratio, 623–626 option-adjusted spread (OAS), 313–316 option(s) American, 304, 309 arbitrage, 643 credit spread, 352, 362–365 European, 302, 309, 515 hedging, 632 interest rate, 145, 297, 298, 304, 308, 362 markets, 18, 638 maturity, 53, 80, 298, 299, 302, 304, 311, 364, 637 on swap, 302, 329 speculation, 633 options on bonds, 302, 304 price, 304 order and execution management systems (OMS/EMS), 379 organized stock exchanges, 16, 22, 430 organized trading facilities (OTFs), 681–683 original issue discount (OID), 384 other asset categories, Positioning private debt among, 439, 440 other bond types, 90, 415 other expenses, 417 other risks ALM, 524 concentration risk, 414, 524, 526 operating risk, 529 other types of bond funds, 413 outright monetary transactions (OMT), 549 outstanding debt, 350, 430, 564 overnight repo, 267

over-the-counter (OTC), 22, 332, 352, 371, 375, 377, 400, 430, 680–683 P Pachatouridi, V., 375–379 packaged retail investment and insurance products (PRIIPs), 376 pandemic, 1, 3, 13, 32, 38, 64, 101, 147, 350, 370, 381, 456, 457, 536, 551–553, 558, 560, 561, 563, 571, 628, 651, 652, 663, 664, 671, 677, 689 pandemic emergency purchase program (PEPP), 34, 38, 551, 552 Pan-European Personal Pension Product (PEPP), 654 Panjer, H.H., 529 paper/physical form, 49 parametric value at risk contribution, 494 parametric VaR, 468, 471, 472, 474, 478, 487, 495 FX risk, 468, 479 Parish, A., 371 participants of the financial system, 8, 9 par value, 9, 49, 573, 574 passive bond portfolio management bond indexing, 232 cash flow matching, 230 immunization, 230, 233, 249 payment default, 58, 350, 351 pension fund, 25, 234, 238–242, 249, 384, 445, 652 percentage price change, 176, 186, 187, 191, 192, 214–217, 227, 228 performance, 5, 7, 20, 25, 32, 42, 48, 58, 65, 67, 73, 81, 82, 127, 171, 230, 233, 250, 253, 263, 371,

INDEX

385, 399, 408, 412, 415, 427, 428, 437, 449, 450, 456, 486, 500, 529, 535, 571, 594, 663, 668–670, 675 portfolio, 8, 171 periodic payment of interest, 2, 49 perpetuities, 12, 14, 54, 193, 195, 196, 249, 573, 576, 602 Pisani-Ferry, J., 546, 566 PitchBook, 441 plain vanilla bonds, 52, 53, 208, 225, 312, 313, 376, 377, 665, 675, 676, 689 versus green bonds, 665, 675 policymaking, 28, 395 implications, 560 pollution prevention and control, 662, 667, 668 Polychronou, M., 652 Porro, F., 528 portfolio construction, 5, 527 duration, 204 management, 5, 170, 250, 399, 407, 501, 687 performance, 8, 171 position, 6, 36, 80, 85, 145, 160, 235, 238, 242, 276, 290, 350, 364, 365, 419, 426, 432, 443, 455, 461, 484, 487, 488, 498, 501, 504, 515, 537, 538, 543, 596, 621–624, 626, 628, 629, 632, 634–636, 638 long, 52, 53, 80, 268, 269, 271, 274, 282, 291, 312, 313, 329, 331, 333, 363–365, 461, 622, 623, 629, 632, 633, 635, 636, 638, 639, 642 short, 53, 271, 273–275, 277, 278, 282, 291, 312, 313, 333, 363, 364, 461, 622–624, 626, 627, 629, 636, 638, 640, 642

719

positioning private debt among other asset categories, 439 Poufinas, T., 652 Prager, R., 375–379 preferred habitat theory, 138, 139 preferred stock, 13, 14, 53, 326, 573–575 without voting rights, 14 with voting rights, 14, 573 preliminary prospectus, 21 premium, 11, 18, 25, 74, 81, 134, 136–141, 242, 251, 254, 292, 312, 313, 329, 335, 341, 342, 352, 364, 365, 515, 559, 565, 576, 602, 623, 632, 637, 638, 643, 646, 676 conversion, 53 Preqin, 439, 444, 445 present value, 68, 70, 71, 73–75, 80, 85, 86, 121, 122, 124, 130, 149, 150, 159, 161, 171, 177, 178, 186, 190, 198, 204, 205, 213, 218, 220, 226, 234, 236–238, 240–242, 244–247, 271–275, 279, 280, 291, 292, 298, 300, 302, 331, 354, 355, 458, 459, 461, 463, 468, 470, 471, 475, 476, 478, 484, 506, 508, 510, 512, 575, 577, 603, 640, 641 President, 28, 41 price/pricing, 24, 48, 68, 111, 117, 120, 121, 127, 186, 204, 213, 227, 265, 266, 296–298, 300, 302, 304, 305, 311, 330, 353, 365, 380, 458, 462, 515, 557, 558, 650, 664 ask, 71, 72, 75, 144, 505 bid, 71, 72, 75, 144, 416, 505 cash, 51, 72, 73, 201, 203, 278, 280–282, 299, 300, 639, 642, 643, 687 cash futures, 276–281

720

INDEX

clean, 52, 72 dirty, 52, 72, 276 invoice, 52 offer, 22, 71, 415 of bond mutual funds, 297 purchase, 51, 74, 78, 312, 383, 428, 456, 574, 580 quoted, 52, 72, 201, 276, 278, 281, 282, 299, 300, 504, 505, 642 quoted futures, 276–279, 281 redemption, 24, 415, 428 risk, 223, 238, 239 sensitivity, 171, 175, 223 price American Options on Bonds, 304 price change, 87, 171–176, 181, 183– 186, 188–191, 198, 210–213, 215, 216, 218, 220, 222, 223, 227–230, 456, 458, 459, 465, 471, 476, 479, 482, 629 percentage, 176, 186, 187, 191, 192, 214–217, 228 price European Options on Bonds, 302 price-to-earnings (P/E), 585–589, 591, 593–595 ratio, 585 price value of a basis point (PVBP), 223, 227, 228 primary market(s), 19, 20, 24, 628, 630, 631, 649 principal, 12, 49, 50, 56, 57, 91, 224–226, 269, 274, 284–286, 291, 292, 301, 310, 318, 319, 321–323, 325, 330, 332, 336, 378, 383, 392–395, 413, 427, 428, 438, 514, 550, 682, 683 notional, 269, 294, 352–356, 361, 518 principal only (PO) MBS, 319 principles, 2, 371, 650, 663

green bond, 665, 666 Principles for Responsible Investment (PRI), 439, 443–445 private debt business, 439, 442, 443 comparison of with private equity, 446 concerns, 449 credit opportunities, 441 definition, 439 distressed, 442 funds, 439 infrastructure, 442, 446 intermediate, 442 market, 443, 444 mezzanine, 327, 442 positioning among other asset categories, 439 real estate, 439, 445, 446 real estate debt, 442 reasons to invest in, 443 senior, 441 special situations, 442 venture, 442 private debt types, 408, 440 business debt, 442 credit opportunities, 441 distressed debt, 58 infrastructure debt, 442 intermediate debt, 442 mezzanine debt, 441 real estate debt, 442 senior debt, 326, 441 special situations, 442 venture debt, 443 private equity, 20, 328, 375, 439, 445, 449, 659 comparison of private debt with, 446 private placement, 15, 16, 20, 658 probability of default bond without coupon, 244

INDEX

coupon-bearing bonds, 11 recovery rate, 357, 512, 514 risk-neutral, 354 probability of default (PD), 15, 59, 63, 327, 353, 356, 358, 383, 385, 386, 486, 508, 509, 512–514, 524 alternative approaches for calculating the, 514 problems of bond indexing, 232 process(ing), 2, 4, 6, 16, 39, 55, 57, 58, 91, 105, 146, 150, 154, 157, 159, 162, 223, 265, 314, 350, 378–380, 470, 484, 501, 525, 527, 529, 567, 586, 591, 599, 601, 656, 658, 665–667, 670, 681, 689 for project evaluation and selection, 667 risk, 529 profitability ratio, 66 properties of duration, 176, 177 property, 8, 147, 178, 239, 268, 316, 318, 390, 392, 492, 591 mean reversion, 147, 148, 153–157, 304 prospectus, 21, 653, 689 protection, 23, 52, 80, 81, 319, 326, 332–334, 341, 352, 353, 357, 358, 365, 366, 375, 378, 384, 411, 412, 450, 456, 457, 518, 623, 638, 649, 652, 656–658, 660, 679–681, 690 and restoration of biodiversity and ecosystems, 662 buyer, 349 protection from credit risk activating/triggering downgrades, 517 credit derivatives, 349, 455, 518 credit risk reduction, 516 guarantees - collaterals, 516

721

netting, 516 protection seller, 352 provisions, 10, 41, 49, 81, 155, 271, 322, 375, 390, 417, 458, 518, 552, 567, 589, 657, 660, 662, 679, 681, 687, 689 call, 52, 144, 315, 392 public sector purchase program (PSPP), 37, 550 public securities association (PSA), 321, 322 standard prepayment benchmark, 321 purchase fees, 417 purchase price, 51, 74, 78, 312, 383, 428, 456, 574, 580 purchase program asset, 34, 37, 38, 550 asset-backed securities, 13, 91, 324, 378, 394, 411, 657 corporate sector, 37, 38, 550 public sector, 37, 550 third covered bond, 37 pure yield pickup swap, 252, 253 puttable bond(s), 53, 81, 312, 313, 390 Q quantitative easing (QE), 31–33, 48, 147, 536, 538, 543, 544, 559, 560 quasi-modified duration, 220, 221 quoted futures price, 276–279, 281 quoted price, 52, 72, 201, 276, 278, 281, 282, 299, 300, 504, 505, 642 R Rachev, S.T., 529 rate anticipation swap, 252, 253

722

INDEX

cap, 332, 333 default, 59–61, 63, 519, 522, 524 expected future spot, 104, 132, 136–139 expected recovery, 354, 357, 513 floor, 335 forward, 103–105, 107–109, 111–113, 115–117, 126, 131–136, 138–141, 145, 152, 153, 270, 271, 280, 330, 334 future spot, 104, 127, 131, 132, 136–139, 141, 270 instantaneous short, 148–150, 152, 154, 311 recovery, 353, 356, 506, 510–513 repo, 266–268 short, 104, 111–118, 128, 131, 135, 139, 145, 146, 148–150, 153–155, 251, 302, 303 spot, 103, 105–109, 111–119, 121–123, 126, 128, 131, 132, 135, 137, 139, 140, 145, 148, 149, 152, 160–162, 221, 269, 270, 280, 282, 292, 298, 330, 354 rating agencies, 15, 55, 58, 61, 64, 65, 125, 143, 519 rating modifier, 55, 61–63, 520, 521 ratio cash flow-to-debt, 65 conversion, 53 coverage, 65 expense, 417, 432 leverage, 66 liquidity, 66 management expense, 417 price-to-earnings, 585–589, 591, 594 profitability, 66 real estate debt, 442 realized compound yield (RCY), 76, 78

reasons to invest in private debt, 455 recovery rate, 353, 356, 357, 506, 510–513 expected, 354, 357, 513 redemption load, 416 redemption price, 24, 428, 574 reference obligation, 352–355, 518 regions, 28, 90, 391, 399, 400, 409, 410, 413, 414, 419, 448, 517, 519, 671 registered, 42, 51 regulated markets (RMs), 681–684 regulation, 9, 30, 374–376, 398, 419, 530, 549, 552, 649, 650, 653, 657, 658, 660, 661, 665, 678, 679, 681, 682, 688, 689 bonds, 381, 649, 678 role of, 374 reinvestment rate risk, 238 related issuers/borrowers, 675 relationship, 5, 75, 76, 88, 100, 101, 104, 105, 109, 113, 119, 140, 224, 333, 444, 531, 546 between price and face value, 73 between price and present value, 85 between quoted price and cash price, 72 between sovereign bonds’ holders and crises, 546 between spot and forward rates, 105 of the term structure with the bond yields, 118 Rendleman & Bartter Model, 150, 151 Rendleman, R.J., 151 renewable energy, 665, 668 renewed strategy on sustainable finance, 662 reporting, 41, 380, 419, 665, 667–670, 675, 676, 679–681, 683, 685

INDEX

repo(s) overnight, 267 reverse, 269, 398 term, 267, 398 repudiation/moratorium, 350 repurchase agreement(s), 34, 91, 266, 269, 398, 686, 688 request for quotation (RFQ), 379, 684 research unbundling rules, 687 MifID, MifID II/MiFIR, 687 residual claim, 14 restricted default (RD), 58 retail checking accounts, 28 retail savings accounts, 235 return, 5–7, 9, 15, 16, 20, 24, 25, 31, 42, 47, 48, 54, 55, 61, 67, 68, 74, 76–78, 81–84, 86, 89, 91, 120, 121, 124–127, 130, 133–137, 147, 230, 238, 249–254, 257, 268, 275, 313, 316, 317, 319, 322, 324, 325, 327, 329, 333, 349, 358, 362, 370, 374, 376, 380, 382, 391, 393–395, 397, 408, 410–414, 421, 427, 428, 432, 437, 439, 441, 443, 445, 446, 449, 486, 500, 525, 561, 573–575, 577, 579–584, 597, 598, 600, 603, 605, 608–613, 636, 638, 643, 652, 665, 675, 677, 688 revenue bond(s), 11, 392, 393 reverse repo, 269, 398 reverse repurchase agreements, 269 review process, 666 external, 669 risk ALM, 524, 525 business, 529 concentration, 414, 526, 527 counterparty, 171, 359, 377, 455

723

credit, 56–58, 169, 171, 253, 285, 290, 294, 325, 326, 352, 353, 359, 361–363, 374, 390, 411, 428, 429, 455–457, 506, 515, 516, 524, 561, 572 failure, 529 foreign exchange (FX), 457, 479, 485 interest rate, 3, 55, 58, 120, 132–134, 143, 169, 171, 223, 234, 238, 285, 290, 294, 390, 392, 412–414, 428, 429, 438, 444, 455–457, 479, 484, 495, 626 management, 8, 63, 265, 400, 455, 457, 529–531, 567, 623, 650, 667 neutral probability, 356 operating, 529 price, 223, 239 processing, 529 reinvestment rate, 238 return ratio, 7 spread, 169, 171, 457, 458 risk-adjusted return on capital (RAROC), 486, 499 risk factor, 120, 458, 459, 461–464, 467–469, 471, 472, 475, 476, 478, 479, 482–485, 487, 491–496 risk management, 8, 63, 265, 400, 455, 457, 529–531, 567, 623, 650, 667 risk of default, 58, 506, 559 risks other, 524 the role of central banks, 26, 27 role of regulation, 374 Ross, S.A., 151 Russian financial crisis, 462

724

INDEX

S S&P, 55–59, 383, 385, 386, 411, 517, 519, 524 S&P Global, 55, 411, 412 S&P US Issued High-Yield Corporate Bond Index, 231 sales load, 415 scenario testing, 462, 463, 467 steps to conduct, 463 Schuermann, T., 505 Schwan, M., 567 seasoned new issues, 20 secondary market(s), 19, 20, 22, 24, 68, 369, 379, 382, 394, 398, 442, 549 second party opinion, 669, 676 sector business, 9 government, 9, 561 household, 9 Securities and Exchange Commission (SEC), 20, 391 securities financing transactions (SFTs), 687–689 MifID, MifID II/MiFIR, 686 Securities Markets Programme (SMP), 549 security analysis, 4 security(ies), 4–6, 8–13, 17, 19–22, 24, 31, 34, 36, 38, 39, 51, 54, 91, 143, 144, 171, 205, 221, 230, 238, 245–247, 250, 251, 266–269, 316, 317, 319, 322–325, 357, 361, 364, 369, 372, 374, 381, 382, 384, 385, 387, 390, 394, 395, 397, 398, 407–409, 414, 415, 429, 431, 441, 456, 469, 489, 493, 497, 499–502, 504, 505, 515, 526, 542, 545, 549, 551, 552, 561, 564, 565, 604, 612, 622–624, 629, 631, 639–641, 656, 660,

661, 679, 682, 684–686, 688, 689 fixed income, 4, 13, 17, 23–26, 53, 100, 143, 144, 171, 218, 235, 239, 242, 265, 271, 273, 274, 349, 369, 370, 374, 376, 377, 380–382, 396, 407, 408, 411, 414, 430, 431, 456, 457, 461, 463, 464, 468, 471, 472, 479, 487, 493–495, 498–501, 503–505, 526, 536, 537, 621, 624, 651, 665, 681–684 mortgage backed, 13, 225, 237, 316, 317, 319, 321, 323, 325, 408 Seimen, M., 661, 665, 666, 668, 670, 671, 675, 676 selecting between a stock and a bond with the use of current yield and dividend yield, 604 selecting between a stock and a bond with the use of HPR, 601 selecting between a stock and a bond with the use of utility functions, 605 selective default (SD), 55 seller, 24, 31, 51, 68, 71, 72, 268, 352–355, 357, 358, 364, 365, 430, 505, 518 protection, 352, 358 senior debt, 326 sensitivity, 171, 175, 181, 190, 204, 217, 220, 222, 357, 458, 472, 474, 483, 484, 488, 495 analysis, 458, 459, 530 vector, 489, 495 separate trading of registered interest and principal security (STRIP), 383 sequential CMO, 322 sequential-pay CMO, 322 shareholder fees, 416

INDEX

shareholder value added (SVA), 486, 499 shares, 9, 13–16, 385, 408, 409, 415, 443, 468, 527, 547, 550, 572, 575, 577, 579–593, 659 short position, 53, 271, 273–275, 277, 278, 282, 291, 312, 313, 333, 363, 364, 622–624, 626, 627, 629, 636, 638, 640, 642 short rate, 104, 111–118, 131, 135, 139, 145, 146, 148–150, 153–155, 251 instantaneous, 148, 150, 152–154, 311 short-term, 17, 35, 39, 58, 66, 91, 101, 120, 121, 133–136, 138, 141, 150, 234, 235, 237, 383, 395–399, 412, 414, 416, 421, 429, 435, 564, 565, 601 ETFs, 437 funds, 414 redemption fees, 416 trading fees, 416 Siegel, J.J., 573 Simonelli, S., 546 simple, transparent and standardized (STS), 653, 661, 689, 690 simple yield to maturity (SY), 79 Single European Act, 651 single issuer/borrower, 9, 10, 26, 47–49, 55, 138, 143, 237, 317–320, 332, 333, 335, 349, 376, 394, 397, 446, 506, 510, 527, 528, 688 single market, 650–652, 654, 655, 659, 660, 678 Single Resolution Mechanism (SRM), 549 sinking fund, 10 situations special, 442

725

size, 24, 170, 232, 238, 242, 279, 372, 383, 500, 501, 538, 549, 550, 562, 565, 623–625, 629, 633, 673, 674 of bond markets, 372 size-specific to the instrument (SSTI), 684 Skridulyte, R., 527, 528 Small and Medium Enterprise financing, 658 Small and Medium Enterprise (SME), 657, 658, 660, 661 smart order routing (SOR), 379 solvency, 26, 30, 65, 350, 546, 547, 564 Solvency Capital Requirement (SCR), 486 Solvency II, 376, 486, 660 sovereign and financial crises, 2 sovereign bonds’ holders and crises, relationship between, 546 sovereign crisis, 351, 385, 535, 537, 540, 546–549, 558–560, 563, 565 sovereign ETFs, 429, 430 Sovereigns, Supranational and Agencies (SSA), 372, 373 special purpose vehicles (SPVs), 10, 23, 325 special situations, 442 speculation forward contracts, 633 futures contracts, 633 options, 636, 637 speculative grade bond, 58 speculator, 621, 622 spot and forward rates, 105, 111 spot rate(s), 103, 105, 107–109, 111– 113, 115, 118, 119, 121–123, 126, 128, 131, 140, 145, 152, 160–162, 221, 269, 270, 280, 292, 298, 330

726

INDEX

expected future, 104, 132, 136–139 future, 104, 127, 131, 132, 136, 149, 150, 270 spread bid-offer, 71, 416 option-adjusted, 313, 315 risk, 169, 171, 457 swap, 2, 356 stakeholders, impact on, 543 stand-alone VaR (SVaR/S-VaR), 468, 493, 495 Standard & Poor’s Corporation, 55 standard normal distribution N(0,1), 491 standard or basic trinomial tree, 154 standard prepayment benchmark -public securities association (PSA), 321 standard trinomial tree, 154 standing facilities, 34, 36, 38 Startz, R., 141 states, 20, 30, 41, 90, 349–351, 392, 399, 413, 424, 426, 527, 561, 567, 649 State Street, 429, 431 steps to conduct scenario testing, 463 steps to conduct stress testing, 460 STFs, 688 stock and bond selecting between with the use of current yield and dividend yield, 604 selecting between with the use of HPR, 601 selecting between with the use of utility functions, 605 stock returns capital asset pricing model (CAPM), 597, 598 dividend yield, 598–600, 604, 605 holding period return (HPR), 596, 604

stock(s), 2, 4–6, 8, 9, 13–16, 18–22, 27, 48, 53, 54, 67, 90, 100, 145–147, 153, 154, 250, 265, 266, 298, 369, 371–373, 379, 387, 390, 399, 400, 402, 414, 430, 438, 439, 526, 538, 571–573, 575–577, 580–596, 598–601, 603–605, 609, 610, 612, 658, 678, 687 common, 13, 573–575, 578, 579 exchange, 24, 372, 373, 430, 580, 581, 583, 584, 589, 593, 598, 599, 604, 652, 679 preferred, 13, 14, 53, 326, 573–575 versus bonds, 602 stock valuation dividends, 577 Gordon’s formula, 578, 603 market/equity financial ratios, 585 stress testing, 459–462, 467 drawbacks of, 460 steps to conduct, 460 Stroughair, J., 505 structure of the FED, 27 subordinated debentures, 11, 387 subordinated debt, 10 substitution swap, 252 supply of money, 143, 398 support the member states with investments and reforms, 661 supranational and agencies (SSA), 372 supranational bonds, 23 supranational organizations, 672 bonds, 23 sustainability, 391, 549, 558, 561, 567, 652, 655, 663–667, 669, 670, 675–677 sustainable and protection of water and marine resources, 662 sustainable development goals (SDGs), 666

INDEX

Sustainable Europe Investment Plan, 662 sustainable water and wastewater management, 668 swap(s) credit default, 327, 352, 356, 376, 518, 519 currency, 34, 36, 39, 294, 296 financial intermediation, 290 interest rate, 330 intermarket spread, 253 option, 329 pure yield pickup, 252, 253 rate anticipation, 252, 253 spread, 2 substitution, 252 total return, 352, 358, 359 swaption(s), 266, 329–332 systemic internalizer (SI), 682, 683

T target date immunization, 238, 241, 242 tax anticipation note (TAN), 396, 399 tax exemption, 90, 392 taxonomy, 251, 410, 662, 663, 665 taxonomy of active bond portfolio management strategies intermarket spread swap, 253 pure yield pickup swap, 253 rate anticipation swap, 253 substitution swap, 252 T-Bill futures, 280, 282 term repo, 267, 398 term structure construction of the, 127, 129 explanation of the, 101 of interest rates, 100, 101, 105, 106, 129 theories of, 104, 109, 137 Terregrossa, R., 78

727

terrestrial and aquatic biodiversity conservation, 668 testing, 458, 462, 530 stress, 459–462, 467, 530 third country trading venues - MifID, MifID II/MiFIR, 686 third covered bond purchase program (CBPP3), 37 Third Stage of Economic and Monetary Union, 33, 41 time to maturity, 14, 17, 23, 58, 59, 68, 71, 80, 81, 88, 100, 105, 118, 120, 122, 124, 125, 130, 154, 169, 170, 172–174, 176–182, 184–192, 201, 202, 205, 208, 209, 211–218, 220, 223–230, 234, 253, 257, 320, 323, 413 Total Annual Fund Operating Expenses, 417 total bond market ETFs, 432 total return swap(s), 358–360, 377, 518 toxic waste, 324, 325, 327, 362 traded bonds, 74, 77, 371, 383, 390 Trade Reporting and Compliance Engine (TRACE), 379, 380 trading, 22, 51, 52, 71, 72, 75, 232, 373, 375–380, 399, 416, 438, 442, 464, 475, 499, 589, 593, 656, 679–683, 685, 686 direct, 22 traditional notions of the bond markets, 381 Trampusch, C., 567 tranche, 323–328, 361, 362 transactions fees, 416 transfer collateral arrangements (TTCAs), 689 transition to a circular economy, 662 transparency, 18, 22, 33, 371, 373– 376, 379, 380, 382, 399, 430,

728

INDEX

438, 650, 657, 666–668, 670, 675, 676, 678–682, 684–686, 688–690 MifID, MifID II/MiFIR, 375 treasury bills/T-bills, 10, 17, 22, 23, 280, 396, 397, 410, 411, 468, 537, 546 treasury bonds, 12, 383, 411 treasury notes, 18, 383 Treasury STRIP (Separate Trading of Registered Interest and Principal Security), 383 Treaty of Rome, 651 triggering downgrades, 516, 517 trinomial tree basic, 154 general, 155 standard, 154 Trudeau, J., 664 trust, 253, 390, 419, 422, 531, 650, 665, 666, 668 types, 1, 3, 4, 9, 12, 20, 24, 34, 47, 48, 51, 65, 89–91, 145, 146, 230, 234, 238, 251, 322, 326, 352, 357, 364, 369, 375, 382, 386, 390, 392, 408, 410, 412, 415, 419, 422, 425, 426, 429, 438, 455, 527, 531, 536, 573, 665, 669, 685, 687 of assets, 8 of bond ETFs, 429 of bond mutual funds, 410 private debt, 440 of yield curves, 119 U U.K. Debt Management Office, 386 UK government bonds, 386 ultra short-term funds, 414 underwriting syndicates, 21 UN Guiding Principles on Business and Human Rights, 662

use of proceeds, 667, 668, 670, 673, 674, 676 uses of convexity, 210 uses of duration, 182 using forward contracts to exploit arbitrage opportunities, 639 using forward contracts to hedge, 623 using forward contracts to speculate, 633 using futures contracts to exploit arbitrage opportunities, 639 using futures contracts to hedge, 283, 623, 626, 629, 631 using futures contracts to speculate, 633 using options to exploit arbitrage opportunities, 639, 643 using options to hedge, 632 using options to speculate, 636 US Securities and Exchange Commission, 408–410, 416, 417 utility functions, 604–610 selecting between a stock and a bond, 605 selecting between a stock and a bond with the use of, 605 V valuation bond, 68, 119, 121 caps, 334, 335 credit default swaps (CDSs), 353 European swap options, 302, 330 floors, 335 interest rate swap, 291 MBS, 314, 317 stock, 575, 595, 598 value face, 2, 49, 51, 54, 68–71, 73, 74, 78–80, 87, 119, 121–125, 127, 130, 133, 136, 149, 159, 172, 176–179, 193, 195, 201, 221,

INDEX

243, 245, 255, 277, 278, 280, 284, 291, 292, 302, 306, 308, 330, 332, 351, 352, 359–361, 369, 372, 373, 382–384, 391, 397, 398, 411, 427, 428, 438, 470, 472, 474, 507, 510, 518, 572, 577, 601, 602, 604, 626–629, 631–633, 639 market conversion, 53 nominal, 15, 74, 235, 244, 291, 331, 372, 628, 631 par, 9, 49, 573, 574 value at risk contribution, 494 Monte Carlo, 492, 498 parametric, 494 value at risk (VaR), 458, 464, 466 historical, 471, 474–476, 478 liquidity adjusted, 502, 505 Monte Carlo, 471, 472, 476, 479, 484, 492 parametric, 468, 471, 472, 474, 487, 495 stand-alone, 468, 493, 495 Vanguard, 372, 400 VaR calculation historical, 467, 498 Monte Carlo, 476, 498 parametric, 472 VaR contribution, 472, 493–499 Monte Carlo, 472, 498 parametric, 494 VaR generalization general probability of loss, 464, 468 Monte Carlo with more than two factors, 491 more positions, 489 more than two factors, 487 parametric VaR, 487 Vasicek model, 150, 151, 153, 300 Vasicek, O., 151, 297, 301 Veiner, D., 375–379 Ventura, J., 562

729

venture debt, 441 verification, 665, 669, 670, 676, 690 Vice-President, 41 volatilities, 34, 157, 158, 175, 207, 223, 224, 229, 252, 266, 298, 300, 323, 325, 331, 334–336, 372, 382, 385, 411, 412, 428, 456, 463, 466, 468, 483, 500, 505, 538, 543, 565, 572, 604, 609–611, 621, 624, 625, 637 flat, 335 forward forward, 335 Volcker Rule, 375 voting rights, 572–574, 576, 602, 679

W Wacker, T., 661, 665, 666, 668, 670, 671, 675, 676 the Wall Street Journal, 400, 401 weighted average cash flow, 223, 225–227 Weighted Average Cost of Capital (WACC), 614 weighted average life, 224 weighted average maturity (WAM), 224, 225 Weil, R.L., 222 Weinberg, C., 374, 379, 380 White, A., 146, 153, 156–158, 297, 301, 311 Will, F., 540 without voting rights, 14 with voting rights, 14 Wolswijk, G., 565 World Bank, 10 World Economic Forum (WEF), 439 World Government Bonds, 383, 385, 386 write an option, 515 writer, 362, 515

730

INDEX

Y yield curves, 119–121, 123, 127, 140, 141, 149, 227, 249, 250, 254, 283, 311, 314–316, 382, 475, 558, 626 yield(s), 1–3, 5, 15, 16, 20, 48, 52–54, 58, 74, 76–81, 83, 89, 90, 100, 107, 111, 118–120, 124, 130, 133, 134, 136, 137, 140, 141, 144, 158, 177, 207, 208, 227, 229, 233, 237, 243, 252, 253, 255, 257, 270, 280, 289, 296, 303, 306, 312–314, 316, 359, 361, 362, 369, 370, 377, 381–383, 387, 391, 393, 394, 397, 408, 410, 411, 413, 428, 441–443, 459, 461, 463, 465, 474, 476, 479, 487, 489, 490, 494, 495, 505, 535–546, 549, 559–561, 565, 571, 582, 584, 585, 598, 600, 605, 641, 665, 676, 688 dividend, 432, 433, 435, 437, 598–600, 605 yield to call, 80, 81

yield to maturity (YTM), 71, 74–85, 88–90, 99, 118, 119, 124–126, 128, 133, 134, 136, 139, 149, 172, 193, 207, 208, 252, 254, 302, 316, 361, 382–388, 391, 475, 507, 508, 557–559, 613 yield to put, 81 yield to worst (YTW), 81 yield value of a price change, 223, 229

Z Zenios, S., 553 zero-coupon bond(s), 1, 11, 49, 68, 69, 74, 75, 118, 119, 120, 123, 127, 131, 133, 134, 136, 151, 153, 159–162, 172–174, 176, 178–181, 184, 189, 199, 209, 224–227, 229, 239, 245, 247, 249, 267, 292, 301–303, 311, 313, 315, 323, 468, 470–472, 475, 478, 484, 506, 507, 509 zero duration ETFs, 437 Zipf, R., 384, 397 Z-tranche, 323