Fixed IncomeSecuritiesTools for Todays Markets
Second Edition
BRUCE TUCKMAN
John Wiley & Sons, Inc.
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Additional Praise for Fixed Income Securities:
Tools for Todays Markets, 2nd Edition
In my opinion, this edition of Tuckmans book has no match in terms ofclarity, accessibility and applicability to todays bond markets.
Vineer Bhansali, Ph.D.Executive Vice PresidentHead of Portfolio AnalyticsPIMCO
Tuckmans book is a must for the bookshelf of anyone interested in theconcepts of fixed income markets and their application. Throughout thebook, the basic concepts are illustrated with numerical examples that makethem easier to apply from a practical perspective.
Marti G. SubrahmanyamCharles E. Merrill Professor of Finance, Economics
and International BusinessStern School of Business, New York University
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John Wiley & Sons
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Fixed IncomeSecuritiesTools for Todays Markets
Second Edition
BRUCE TUCKMAN
John Wiley & Sons, Inc.
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Copyright 2002 by Bruce Tuckman. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means, electronic, mechanical, photocopying,recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the1976 United States Copyright Act, without either the prior written permission of thePublisher, or authorization through payment of the appropriate per-copy fee to theCopyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests tothe Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, e-mail: [email protected].
Limit of Liability/Disclaimer of Warranty: While the publisher and author have usedtheir best efforts in preparing this book, they make no representations or warrantieswith respect to the accuracy or completeness of the contents of this book andspecifically disclaim any implied warranties of merchantability or fitness for a particularpurpose. No warranty may be created or extended by sales representatives or writtensales materials. The advice and strategies contained herein may not be suitable for yoursituation. You should consult with a professional where appropriate. Neither thepublisher nor author shall be liable for any loss of profit or any other commercialdamages, including but not limited to special, incidental, consequential, or otherdamages.
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Library of Congress Cataloging-in-Publication Data:Tuckman, Bruce.
Fixed income securities : tools for todays market / Bruce Tuckman.2nd ed.
p. cm.(Wiley finance series)ISBN 0-471-06317-7 (cloth)ISBN 0-471-06322-3 (paperback)
1. Fixed income securities. I. Title. II. Series.HG4650 .T83 2002332.63'2044dc21 2002005425
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
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http://www.copyright.com
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CONTENTS
INTRODUCTION xiii
ACKNOWLEDGMENTS xv
PART ONEThe Relative Pricing of Fixed Income Securities with Fixed Cash Flows 1
CHAPTER 1Bond Prices, Discount Factors, and Arbitrage 3
The Time Value of Money 3Treasury Bond Quotations 4Discount Factors 6The Law of One Price 8Arbitrage and the Law of One Price 10Treasury STRIPS 12APPENDIX 1A
Deriving the Replicating Portfolio 17APPENDIX 1B
APPLICATION: Treasury Triplets and High Coupon Bonds 18
CHAPTER 2Bond Prices, Spot Rates, and Forward Rates 23
Semiannual Compounding 23Spot Rates 25Forward Rates 28Maturity and Bond Price 32Maturity and Bond Return 34Treasury STRIPS, Continued 37
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APPENDIX 2AThe Relation between Spot and Forward Rates and the Slope of the Term Structure 38
CHAPTER 3Yield-to-Maturity 41
Definition and Interpretation 41Yield-to-Maturity and Spot Rates 46Yield-to-Maturity and Relative Value: The Coupon Effect 50Yield-to-Maturity and Realized Return 51
CHAPTER 4Generalizations and Curve Fitting 53
Accrued Interest 53Compounding Conventions 56Yield and Compounding Conventions 59Bad Days 60Introduction to Curve Fitting 61Piecewise Cubics 69APPLICATION: Fitting the Term Structure in the U.S. Treasury Market
on February 15, 2001 71TRADING CASE STUDY: A 7s-8s-9s Butterfly 77APPENDIX 4A
Continuous Compounding 83APPENDIX 4B
A Simple Cubic Spline 85
PART TWOMeasures of Price Sensitivity and Hedging 87
CHAPTER 5One-Factor Measures of Price Sensitivity 89
DV01 91A Hedging Example, Part I: Hedging a Call Option 95
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Duration 98Convexity 101A Hedging Example, Part II: A Short Convexity Position 103Estimating Price Changes and Returns with DV01, Duration, and Convexity 105Convexity in the Investment and Asset-Liability Management Contexts 108Measuring the Price Sensitivity of Portfolios 109A Hedging Example, Part III: The Negative Convexity of Callable Bonds 111
CHAPTER 6Measures of Price Sensitivity Based on Parallel Yield Shifts 115
Yield-Based DV01 115Modified and Macaulay Duration 119Zero Coupon Bonds and a Reinterpretation of Duration 120Par Bonds and Perpetuities 122Duration, DV01, Maturity, and Coupon: A Graphical Analysis 124Duration, DV01, and Yield 127Yield-Based Convexity 127Yield-Based Convexity of Zero Coupon Bonds 128The Barbell versus the Bullet 129
CHAPTER 7Key Rate and Bucket Exposures 133
Key Rate Shifts 134Key Rate 01s and Key Rate Durations 135Hedging with Key Rate Exposures 137Choosing Key Rates 140Bucket Shifts and Exposures 142Immunization 147Multi-Factor Exposures and Risk Management 147
CHAPTER 8Regression-Based Hedging 149
Volatility-Weighted Hedging 150One-Variable Regression-Based Hedging 153Two-Variable Regression-Based Hedging 158TRADING CASE STUDY: The Pricing of the 20-Year U.S. Treasury Sector 161A Comment on Level Regressions 166
Contents vii
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PART THREETerm Structure Models 169
CHAPTER 9The Science of Term Structure Models 171
Rate and Price Trees 171Arbitrage Pricing of Derivatives 174Risk-Neutral Pricing 177Arbitrage Pricing in a Multi-Period Setting 179Example: Pricing a CMT Swap 185Reducing the Time Step 187Fixed Income versus Equity Derivatives 190
CHAPTER 10The Short-Rate Process and the Shape of the Term Structure 193
Expectations 194Volatility and Convexity 196Risk Premium 201A Mathematical Description of Expectations, Convexity, and
Risk Premium 206APPLICATION: Expectations, Convexity, and Risk Premium in the
U.S. Treasury Market on February 15, 2001 212APPENDIX 10A
Proofs of Equations (10.19) and (10.25) 214
CHAPTER 11The Art of Term Structure Models: Drift 219
Normally Distributed Rates, Zero Drift: Model 1 219Drift and Risk Premium: Model 2 225Time-Dependent Drift: The Ho-Lee Model 228Desirability of Fitting to the Term Structure 229Mean Reversion: The Vasicek (1977) Model 232
CHAPTER 12The Art of Term Structure Models: Volatility and Distribution 245
Time-Dependent Volatility: Model 3 245Volatility as a Function of the Short Rate: The Cox-Ingersoll-Ross
and Lognormal Models 248
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Tree for the Original Salomon Brothers Model 251A Lognormal Model with Mean Reversion: The Black-Karasinski Model 253Selected List of One-Factor Term Structure Models 255APPENDIX 12A
Closed-Form Solutions for Spot Rates 257
CHAPTER 13Multi-Factor Term Structure Models 259
Motivation from Principal Components 259A Two-Factor Model 263Tree Implementation 265Properties of the Two-Factor Model 269Other Two-Factor and Multi-Factor Modeling Approaches 274APPENDIX 13A
Closed-Form Solution for Spot Rates
