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With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and market- ing print and electronic products and services for our customers’ profes- sional and personal knowledge and understanding.com Bond Math The Theory Behind the Formulas Second Edition DONALD J.com Cover image: abstract © aleksandarvelasevic/iStock.com Cover design: Wiley Copyright © 2014 by Donald J. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. The First Edition of Bond Math was published by John Wiley & Sons, Inc.
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Interest rates—Mathematical models. Zero coupon securities.63'2301519—dc23 2014018633 Printed in the United States of America.com To my students www.com Contents Preface to the Second Edition xi Preface to the First Edition xiii CHAPTER 1 Money Market Interest Rates 1 Interest Rates in Textbook Theory 2 Money Market Add-On Rates 3 Money Market Discount Rates 6 Two Cash Flows, Many Money Market Rates 9 A History Lesson on Money Market Certificates 12 Periodicity Conversions 13 Treasury Bill Auction Results 15 The Future: Hourly Interest Rates? 19 Conclusion 21 CHAPTER 2 Zero-Coupon Bonds 23 The Story of TIGRS, CATS, LIONS, and STRIPS 24 Yields to Maturity on Zero-Coupon Bonds 27 Horizon Yields and Holding-Period Rates of Return 30 Changes in Bond Prices and Yields 32 Credit Spreads and the Implied Probability of Default 34 Conclusion 38 CHAPTER 3 Prices and Yields on Coupon Bonds 39 Market Demand and Supply 40 Bond Prices and Yields to Maturity in a World of No Arbitrage 44 Some Other Yield Statistics 48 Horizon Yields 52 Some Uses of Yield-to-Maturity Statistics 53 vii www.com viii CONTENTS Implied Probability of Default on Coupon Bonds 55 Bond Pricing between Coupon Dates 56 A Real Corporate Bond 59 Conclusion 63 CHAPTER 4 Bond Taxation 65 Basic Bond Taxation 66 Market Discount Bonds 68 A Real Market Discount Corporate Bond 70 Premium Bonds 74 Original Issue Discount Bonds 77 Municipal Bonds 79 Conclusion 82 CHAPTER 5 Yield Curves 83 An Intuitive Forward Curve 84 Classic Theories of the Term Structure of Interest Rates 87 Accurate Implied Forward Rates 91 Money Market Implied Forward Rates 93 Calculating and Using Implied Spot (Zero-Coupon) Rates 96 More Applications for the Implied Spot and Forward Curves 99 Discount Factors 105 Conclusion 109 CHAPTER 6 Duration and Convexity 111 Yield Duration and Convexity Relationships 112 Yield Duration 115 The Relationship between Yield Duration and Maturity 118 Yield Convexity 121 Bloomberg Yield Duration and Convexity 125 Curve Duration and Convexity 129 Conclusion 138 CHAPTER 7 Floaters and Linkers 139 Floating-Rate Notes in General 140 A Simple Floater Valuation Model 141 A Somewhat More Complex Floater Valuation Model 146 www.com Contents ix An Actual Floater 149 Inflation-Indexed Bonds: C-Linkers and P-Linkers 157 Linker Taxation 162 Linker Duration 165 Conclusion 171 CHAPTER 8 Interest Rate Swaps 173 Pricing an Interest Rate Swap 174 Interest Rate Forwards and Futures 178 Inferring the Forward Curve 181 Valuing an Interest Rate Swap 185 Interest Rate Swap Duration 188 Collateralized Swaps 192 Traditional LIBOR Discounting 193 OIS Discounting 196 The LIBOR Forward Curve for OIS Discounting 198 Conclusion 202 CHAPTER 9 Bond Portfolios 205 Bond Portfolio Statistics in Theory 205 Bond Portfolio Statistics in Practice 208 A Real Bond Portfolio 213 Thoughts on Bond Portfolio Statistics 223 Conclusion 225 CHAPTER 10 Bond Strategies 227 Acting on a Rate View 228 An Interest Rate Swap Overlay Strategy 233 Classic Immunization Theory 237 Immunization Implementation Issues 242 Liability-Driven Investing 245 Closing Thoughts: Target-Duration Bond Funds 246 Technical Appendix 249 Acronyms 267 Bibliographic Notes 269 www.com x CONTENTS About the Author 275 Acknowledgments 277 About the Companion Website 279 Index 281 www.com Preface to the Second Edition I am pleased to present the second edition of Bond Math. I’m sure my edi- tors at Wiley will disagree but I’m more impressed with who reads the book rather than how many. I’ve been very happy with reader responses to the first edition.
Best of all, based on the book, I was invited by CFA Institute to write two new readings on Fixed‐Income Valuation and Risk and Return for the Chartered Financial Analyst® Level I curriculum. I was joined in that endeavor by James Adams, with whom I’ve been writing a series of articles on corporate finance applications of derivatives to hedge interest rate risk. One of the changes to the second edition of this book is to align the notation and terminology used in Bond Math with the CFA Institute readings. Also, I have added the simple model to value floating‐rate notes that is used in the Fixed‐Income Valuation reading.
One of my objectives is to explain the math behind numbers presented on commonly used Bloomberg pages, primarily the Yield and Spread Analysis page for bonds. Bloomberg has changed the format of this page since the first edition, so it is timely to update the examples. I like the new format—the page is less “busy,” as a graphic designer might say. In Chapters 3 and 6 I show the formulas that generate the various risk and return statistics for fixed‐income bonds, that is, yield to maturity, modified duration, and convexity, included on that page.
But still there are some Bloomberg numbers that I think are misleading and unreliable. You see in Chapter 4 that Bloomberg makes a curi- ous assumption for some bonds to get the projected after‐tax rate of return, namely, that current U. tax law does not apply to the investor. Also, you see in Chapter 7 that Bloomberg shows some hard‐to‐understand (and therefore use) modified duration results for a floating‐rate note.
Chapter 8 is significantly revised from the first edition. I now include discussion of how the financial crisis of 2007 to 2009 has changed deriva- tives valuation. The traditional method to value interest rate swaps, which I use in the first edition, is called LIBOR discounting. The idea is that LIBOR is a workable and reasonable proxy for the interbank “risk‐free” interest rate.
The financial crisis revealed the flaws in that assumption. Nowadays, OIS discounting is the standard. Rates on overnight indexed swaps are now used to generate the discount factors to value derivatives. You see that with xi www.com xii PREFACE TO THE SECOND EDITION OIS discounting, care must be taken in valuing a swap as a combination of fixed‐rate and floating‐rate bonds, as you might have learned in a deriva- tives textbook.
A second edition of Bond Math has been on my wish list. Next on the list is to have it translated from American to British financial English and use examples of U. gilts instead of U. The title of the transla- tion would have to be Bond Maths.com Preface to the First Edition T his book could be titled Applied Bond Math or, perhaps, Practical Bond Math.
Those who do serious research on fixed‐income securities and markets know that this subject matter goes far beyond the mathematics cov- ered herein. Those who are interested in discussions about “pricing kernels” and “stochastic discount rates” will have to look elsewhere. My target audi- ence is those who work in the finance industry (or aspire to), know what a Bloomberg page is, and in the course of the day might hear or use terms such as “yield to maturity,” “forward curve,” and “modified duration.” My objective in Bond Math is to explain the theory and assump- tions that lie behind the commonly used statistics regarding the risk and return on bonds. I show many of the formulas that are used to calculate yield and duration statistics and, in the Technical Appendix, their formal derivations.
But I do not expect a reader to actually use the formulas or do the calculations. There is much to be gained by recognizing that “there exists an equation” and becoming more comfortable using a number that is taken from a Bloomberg page, knowing that the result could have been obtained using a bond math formula. This book is based on my 25 years of experience teaching this material to graduate students and finance professionals. For that, I thank the many deans, department chairs, and program directors at the Boston University School of Management who have allowed me to continue teaching fixed‐ income courses over the years.
I thank Euromoney Training in New York and Hong Kong for organizing four‐day intensive courses for me all over the world. I thank training coordinators at Chase Manhattan Bank (and its heritage banks, Manufacturers Hanover and Chemical), Lehman Brothers, and the Bank of Boston for paying me handsomely to teach their employees on so many occasions in so many interesting venues. Bond math has been very, very good to me. The title of this book emanates from an eponymous two‐day course I taught many years ago at the old Manny Hanny.
(Okay, I admit that I have always wanted to use the word “eponymous”; now I can cross that off my bucket list.) I thank Keith Brown of the University of Texas at Austin, who co‐designed and co‐taught many of those executive training courses, xiii www.com xiv PREFACE TO THE FIRST EDITION for emphasizing the value of relating the formulas to results reported on Bloomberg.