Interest Rate Risk Modeling The Fixed Income Valuation Course SANJAY K. NAWALKHA GLORIA M. SOTO NATALIA A. BELIAEVA John Wiley & Sons, Inc. Interest Rate Risk Modeling Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic prod- ucts and services for our customers’ professional and personal knowledge and understanding. The Wiley Finance series contains books written specifically for finance and in- vestment professionals as well as sophisticated individual investors and their finan- cial advisors. Book topics range from portfolio management to e-commerce, risk management, financial engineering, valuation and financial instrument analysis, as well as much more. For a list of available titles, please visit our web site at www.WileyFinance.com. Interest Rate Risk Modeling The Fixed Income Valuation Course SANJAY K. NAWALKHA GLORIA M. SOTO NATALIA A. BELIAEVA John Wiley & Sons, Inc. Copyright © 2005 by Sanjay K. Nawalkha, Gloria M. Soto, and Natalia A. Beliaeva. 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, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. 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For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Nawalkha, Sanjay K. Interest rate risk modeling : the fixed income valuation course / Sanjay K. Nawalkha, Gloria M. Soto, Natalia A. Beliaeva. p. cm.—(Wiley finance series) Includes bibliographical references and index. ISBN-13 978-0-471-42724-7 (cloth / cd-rom) ISBN-10 0-471-42724-1 (cloth / cd-rom) 1. Interest rate risk—Mathematical models. 2. Bonds—Valuation—Mathematical models. 3. Fixed-income securities—Valuation—Mathematical models. I. Title: Fixed income valuation course. II. Soto, Gloria M. III. Beliaeva, Natalia A. IV. Title. V. Series. HG6024.5.N39 2005 332.63′23—dc22 2005000048 Printed in the United States of America. 10 987654321 To our parents —Natalia and Sanjay To J. Alberto —Gloria Preface his is the first book of the trilogy on a fixed-income valuation course Tby Wiley finance covering the following three areas of fixed-income valuation: 1. Interest rate risk modeling 2. Ter m structure modeling 3. Credit risk modeling Unlike other books in fixed-income valuation, which are either too rig- orous but mathematically demanding, or easy-to-read but lacking in impor- tant details, our goal is to provide readability with sufficient rigor. In the first book, we give a basic introduction to various fixed-income securities and their derivatives. The principal focus of this book is on measuring and managing interest rate risk arising from general nonparallel rate changes in the term structure of interest rates. This book covers five types of interest rate risk models in the fixed-income literature. These models can be applied in a variety of contexts by financial institutions ranging from commercial banks to fixed-income hedge funds. These institutions can design and exe- cute strategies that range from simplest duration-based hedging to the more sophisticated immunization or speculative yield-curve programs, based on multiple risk measures with off-balance sheet positions in swaps, interest rate options, and interest rate futures. The five interest rate risk models covered in this book are the duration and convexity models in Chapter 2, M-Absolute/M-Square models in Chap- ter 4, duration vector model in Chapter 5, key rate duration model in Chap- ter 9, and principal component duration model in Chapter 10. Applications using some of these models are given for regular bonds in Chapters 2,4,5,9, and 10; Treasury futures and Eurodollar futures in Chapter 6; bond options and callable bonds in Chapter 7; forward rate agreements, interest rate op- tions, swaps, and swaptions in Chapter 8; mortgage securities in Chapter 10; and default-prone corporate bonds in Chapter 11. Chapter 3 also shows how to estimate the term structure of interest rates from a cross-section of bond prices using the Nelson-Siegel exponential vii viii PREFACE model and the McCulloch’s cubic spline model. The interest rate options, such as caps, floors, collars, and swaptions in Chapter 8 are priced using the LIBOR market models of Jamshidian and others. The default-prone zero- coupon bonds in Chapter 11 are priced using the models of Merton and Nawalkha-Shimko et al., while default-prone coupon bonds are priced using the first passage probability models of Longstaff and Schwartz, and Collin- Dufresne and Goldstein. All three books of the trilogy come with software in a user-friendly excel/VBA format that covers a variety of models in the three respective areas. The software is organized to correspond with the models covered in different chapters, so it can be used as a powerful supplement in the learning process. Using the software for the current book, the user could, for example, design a multiple factor hedging strategy using the three key rate durations or using the three principal component durations. The user could solve for the no- tional amounts corresponding to interest rate swaps of different maturities to protect against the height, slope, and curvature shifts in the yield curve using a three-element duration vector model. The user could pick from a variety of multiple factor hedging and speculative strategies, such as immunization, bond index replication, and speculative yield-curve strategies, using a variety of interest rate contingent claims, such as regular bonds, bond options, Trea- sury futures (on T-bills, T-notes, and T-bonds), Eurodollar futures, forward rate agreements, interest rate options (e.g., caps, floors, and collars), swaps, swaptions, and default-prone corporate bonds. Finally, based on Craig Holden’s excel program, the software for Chapter 3 also demonstrates a ped- agogically useful term structure “movie” using monthly zero-coupon rates as well as forward rates over the period from 1946 to 1991. After reading chapters on given topics from these books, the reader should be able to follow the examples and be ready to apply these models without searching for missing details from other sources (as we often did while writing this book). Though many of our programs require coding in advanced scientific languages, such as C, C++, the final output is always presented in user-friendly excel/VBA spreadsheets. These spreadsheets allow the readers with basic excel skills to instantly play with these models. This book will be useful to both fixed-income practitioners, as well as graduate and advanced undergraduate students in an introductory course in fixed-income valuation. Since this book is a part of the trilogy, it is integrated both conceptually and in terms of the mathematical notation, with the next two books to fol- low. This implies low cost to the user in reading the next two books, espe- cially for practitioners who do not have the luxury of taking fixed-income courses. The second book on term structure modeling covers various term structure models from the basic Vasicek/CIR models to the more advanced Preface ix quadratic, HJM, and LIBOR market models. The third book covers both the structural and reduced-form models on credit risk as well as valuation of credit derivatives. Va r i ous aspects of this trilogy on the fixed-income valuation course, in- cluding the book descriptions, software details, and future updates are available on the web site www.fixedincomerisk.com. SANJAY K. NAWA LKHA GLORIA M. SOTO NATALIA A. BELIAEVA Acknowledgments e would like to thank the many individuals who helped with this book Wproject, some in small ways, others in substantial ways, including Christopher Schwarz, Aixin Ma, Hossein Kazemi, Bing Liang, Nelson Lacey, Sanjiv Das, Huston McCulloch, Hyuna Park, Saira Latif, and Ying Li.