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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Finance Editions
- Sprache: Englisch
A bond calculation quick reference, complete with context and application insights
Bond Math is a quick and easy resource that puts the intricacies of bond calculations into a clear and logical order. This simple, readable guide provides a handy reference, teaching the reader how to think about the essentials of bond math. Much more than just a book of formulas, the emphasis is on how to think about bonds and the associated math, with plenty of examples, anecdotes, and thought-provoking insights that sometimes run counter to conventional wisdom. This updated second edition includes popular Bloomberg pages used in fixed-income analysis, including the Yield and Spread Analysis page, plus a companion website complete with an Online Workbook of multiple choice questions and answers and spreadsheet exercises. Detailed coverage of key calculations, including thorough explanations, provide practical guidance to working bond professionals.
The bond market is the largest and most liquid in the world, encompassing everything from Treasuries and investment grade corporate paper to municipals and junk bonds, trading over $900 billion daily in the U.S. alone. Bond Math is a guide to the inevitable calculations involved in managing bonds, with expert insight on the portfolios and investment strategies that puts the math in perspective. Clear and concise without sacrificing detail, this book helps readers to:
- Delineate the characteristics of different types of debt securities
- Calculate implied forward and spot rates and discount factors
- Work with rates of return, yield statistics, and interest rate swaps
- Understand duration-based risk measures, and more
Memorizing formulas is one thing, but really learning how to mentally approach the math behind bonds is something else entirely. This approach places calculations in context, and enables easier transition from theory to application. For the bond professional seeking a quick math reference, Bond Math provides that and so much more.
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Seitenzahl: 445
Veröffentlichungsjahr: 2014
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Bond Math
The Theory Behind the Formulas
Second Edition
DONALD J. SMITH
Cover image: abstract © aleksandarvelasevic/iStock.com Cover design: Wiley
Copyright © 2014 by Donald J. Smith. 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. in 2011.
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. Requests to the 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, or online at www.wiley.com/go/permissions.
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Library of Congress Cataloging-in-Publication Data:
Smith, Donald J., 1947- Bond math : the theory behind the formulas / Donald J. Smith. — Second edition. pages cm. — (Wiley finance) Includes bibliographical references and index. ISBN 978-1-118-86632-0 (hardback); 978-1-118-86629-0 (ebk); 978-1-118-86636-8 (ebk) 1. Bonds—Mathematical models. 2. Interest rates—Mathematical models. 3. Zero coupon securities. I. Title. HG4651.S57 2014 332.63′2301519—dc23
2014018633
To my students
Contents
Preface to the Second Edition
Preface to the First Edition
CHAPTER 1 Money Market Interest Rates
Interest Rates in Textbook Theory
Money Market Add-On Rates
Money Market Discount Rates
Two Cash Flows, Many Money Market Rates
A History Lesson on Money Market Certificates
Periodicity Conversions
Treasury Bill Auction Results
The Future: Hourly Interest Rates?
Conclusion
CHAPTER 2 Zero-Coupon Bonds
The Story of TIGRS, CATS, LIONS, and STRIPS
Yields to Maturity on Zero-Coupon Bonds
Horizon Yields and Holding-Period Rates of Return
Changes in Bond Prices and Yields
Credit Spreads and the Implied Probability of Default
Conclusion
CHAPTER 3 Prices and Yields on Coupon Bonds
Market Demand and Supply
Bond Prices and Yields to Maturity in a World of No Arbitrage
Some Other Yield Statistics
Horizon Yields
Some Uses of Yield-to-Maturity Statistics
Implied Probability of Default on Coupon Bonds
Bond Pricing between Coupon Dates
A Real Corporate Bond
Conclusion
CHAPTER 4 Bond Taxation
Basic Bond Taxation
Market Discount Bonds
A Real Market Discount Corporate Bond
Premium Bonds
Original Issue Discount Bonds
Municipal Bonds
Conclusion
CHAPTER 5 Yield Curves
An Intuitive Forward Curve
Classic Theories of the Term Structure of Interest Rates
Accurate Implied Forward Rates
Money Market Implied Forward Rates
Calculating and Using Implied Spot (Zero-Coupon) Rates
More Applications for the Implied Spot and Forward Curves
Discount Factors
Conclusion
CHAPTER 6 Duration and Convexity
Yield Duration and Convexity Relationships
Yield Duration
The Relationship between Yield Duration and Maturity
Yield Convexity
Bloomberg Yield Duration and Convexity
Curve Duration and Convexity
Conclusion
CHAPTER 7 Floaters and Linkers
Floating-Rate Notes in General
A Simple Floater Valuation Model
A Somewhat More Complex Floater Valuation Model
An Actual Floater
Inflation-Indexed Bonds: C-Linkers and P-Linkers
Linker Taxation
Linker Duration
Conclusion
CHAPTER 8 Interest Rate Swaps
Pricing an Interest Rate Swap
Interest Rate Forwards and Futures
Inferring the Forward Curve
Valuing an Interest Rate Swap
Interest Rate Swap Duration
Collateralized Swaps
Traditional LIBOR Discounting
OIS Discounting
The LIBOR Forward Curve for OIS Discounting
Conclusion
CHAPTER 9 Bond Portfolios
Bond Portfolio Statistics in Theory
Bond Portfolio Statistics in Practice
A Real Bond Portfolio
Thoughts on Bond Portfolio Statistics
Conclusion
CHAPTER 10 Bond Strategies
Acting on a Rate View
An Interest Rate Swap Overlay Strategy
Classic Immunization Theory
Immunization Implementation Issues
Liability-Driven Investing
Closing Thoughts: Target-Duration Bond Funds
Technical Appendix
Chapter 1 Money Market Interest Rates
Chapter 3 Prices and Yields on Coupon Bonds
Chapter 6 Duration and Convexity
Chapter 7 Floaters and Linkers
Chapter 9 Bond Portfolios
Acronyms
Bibliographic Notes
Chapter 1 Money Market Interest Rates
Chapter 2 Zero-Coupon Bonds
Chapter 3 Prices and Yields on Coupon Bonds
Chapter 4 Bond Taxation
Chapter 5 Yield Curves
Chapter 6 Duration and Convexity
Chapter 7 Floaters and Linkers
Chapter 8 Interest Rate Swaps
Chapter 9 Bond Portfolios
Chapter 10 Bond Strategies
About the Author
Acknowledgments
About the Companion Website
Index
End User License Agreement
List of Tables
Chapter 1
TABLE 1.1
TABLE 1.2
Chapter 2
TABLE 2.1
Chapter 3
TABLE 3.1
TABLE 3.2
Chapter 4
TABLE 4.1
Chapter 5
TABLE 5.1
Chapter 7
TABLE 7.1
TABLE 7.2
TABLE 7.3
TABLE 7.4
TABLE 7.5
TABLE 7.6
TABLE 7.7
Chapter 8
TABLE 8.1
TABLE 8.2
Chapter 9
TABLE 9.1
TABLE 9.2
TABLE 9.3
Chapter 10
TABLE 10.1
List of Illustrations
Chapter 2
FIGURE 2.1
Monthly Averages of Daily 10-Year U.S. Treasury Note Yields from April 1953 to May 2010
FIGURE 2.2
Constant-Yield Price Trajectory, 10-Year, Zero-Coupon Corporate Bond Priced to Yield 5.174% (s.a.)
FIGURE 2.3
Probability Distribution for Rates of Return on a Corporate Bond to a Buy-and-Hold Investor
Chapter 3
FIGURE 3.1
Exchange Diagram
FIGURE 3.2A
The Interest Rate Diagram
FIGURE 3.2B
The Bond Price Diagram
FIGURE 3.3A
The Interest Rate Diagram
FIGURE 3.3B
The Bond Price Diagram
FIGURE 3.4
Data vs. Information
FIGURE 3.5
Bloomberg Yield and Spread Analysis Page for the IBM 8 3/8% Bond Due 11/01/2019
Chapter 4
FIGURE 4.1
Bloomberg Yield and Spread Analysis Page, AAPL 3.85% Bond Due 5/04/2043, Assuming 43.40% Ordinary Income Tax Rate and 23.80% Capital Gains Rate
FIGURE 4.2
Bloomberg Descriptive Page, AAPL 3.85% Bond Due 5/04/2043
FIGURE 4.3
Bloomberg Yield and Spread Analysis Page, IBM 8.375% Bond Due 11/01/2019, Assuming 43.40% Ordinary Income Tax Rate and 23.80% Capital Gains Rate
Chapter 5
FIGURE 5.1
The LIBOR Hair Chart
FIGURE 5.2
Summary of Implied Forward Rate Calculations
Chapter 6
FIGURE 6.1
Relationships between Macaulay Duration and Maturity
FIGURE 6.2
Bloomberg Yield and Spread Analysis Page (YA), AAPL 3.85% Bond Due May 4, 2043
FIGURE 6.3
Bloomberg Yield and Spread Analysis Page, 6% Fannie Mae Callable Bond
FIGURE 6.4
Bloomberg Option-Adjusted Spread Analysis Page (OAS1), Fannie Mae Callable Bond
FIGURE 6.5
Bloomberg Yield Analysis Page (YA), Treasury P-STRIPS Due May 15, 2042.
Chapter 7
FIGURE 7.1
Macaulay Duration of a Floating-Rate Note
FIGURE 7.2
Bloomberg Description Page, Citigroup Global Markets Floating-Rate Note
FIGURE 7.3
Bloomberg Yield and Spread Analysis Page (YAS), Citigroup Global Markets Floating-Rate Note, Settlement on March 28, 2014
FIGURE 7.4
Bloomberg Yield and Spread Analysis Page (YAS), Citigroup Global Markets Floating-Rate Note, Settlement on April 17, 2014
FIGURE 7.5
Bloomberg Floater Analysis Page (YASN), Citigroup Global Markets Floating-Rate Note, Settlement on April 17, 2014
Chapter 8
FIGURE 8.1
Two-Year, Quarterly Net Settlement, Interest Rate Swap 3.40% Fixed versus 3-Month LIBOR
Chapter 9
FIGURE 9.1
Bloomberg Yield and Spread Analysis Page, 0.375% Treasury Note Due 2/15/2016
FIGURE 9.2
Bloomberg Yield and Spread Analysis Page, 2% Treasury Note Due 2/15/2023
FIGURE 9.3
Bloomberg Yield and Spread Analysis Page, 3.125% Treasury Bond Due 2/15/2043
FIGURE 9.4
Bloomberg Yield and Spread Analysis Page, 0% Treasury P-STRIPS Due 2/15/2043
Chapter 10
FIGURE 10.1
Parallel Upward and Downward Yield Curve Shifts
FIGURE 10.2
Steepening and Flattening Yield Curve Shifts
FIGURE 10.3
Shape-Changing Yield Curve Shifts
FIGURE 10.4
Immunization as Zero Replication
Guide
Cover
Table of Contents
Preface
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Preface to the Second Edition
I am pleased to present the second edition of Bond Math. I’m sure my editors 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 curious assumption for some bonds to get the projected after-tax rate of return, namely, that current U.S. 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 derivatives 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 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 derivatives 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.K. gilts instead of U.S. Treasuries. The title of the translation would have to be Bond Maths.
Preface to the First Edition
This 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 covered herein. Those who are interested in discussions about “pricing kernels” and “stochastic discount rates” will have to look elsewhere. My target audience 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 assumptions 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, for emphasizing the value of relating the formulas to results reported on Bloomberg. I have found that users of “black box” technologies find comfort in knowing how those bond numbers are calculated, which ones are useful, which ones are essentially meaningless, and which ones are just wrong.
Our journey through applied and practical bond math starts in the money market, where we have to deal with anachronisms like discount rates and a 360-day year. A key point in Chapter 1 is that knowing the periodicity of an annual interest rate (i.e., the assumed number of periods in the year) is critical. Converting from one periodicity to another—for instance, from quarterly to semiannual—is a core bond math calculation that I use throughout the book. Money market rates can be deceiving because they are not intuitive and do not follow classic time-value-of-money principles taught in introductory finance courses. You have to know what you are doing to play with T-bills, commercial paper, and bankers acceptances.
Chapters 2 and 3 go deep into calculating prices and yields, first on zero-coupon bonds to get the ideas out for a simple security like U.S. Treasury STRIPS (i.e., just two cash flows) and then on coupon bonds for which coupon reinvestment is an issue. The yield to maturity on a bond is a summary statistic about its cash flows—it’s important to know the assumptions that underlie this widely quoted measure of an investor’s rate of return and what to do when those assumptions are untenable. I decipher Bloomberg’s Yield Analysis page for a typical corporate bond, showing the math behind “street convention,” “U.S. government equivalent,” and “true” yields. The problem is distinguishing between yields that are pure data (and can be overlooked) and those that provide information useful in making a decision about the bond.
Chapter 4 continues the exploration of rate-of-return measures on an after-tax basis for corporate, Treasury, and municipal bonds. Like all tax matters, this necessarily gets technical and complicated. Taxation, at least in the U.S., depends on when the bond was issued (there were significant changes in the 1980s and 1990s), at what issuance price (there are different rules for original issue discount bonds), and whether a bond issued at (or close to) par value is later purchased at a premium or discount. Given the inevitability of taxes, this is important stuff—and it is stuff on which Bloomberg sometimes reports a misleading result, at least for U.S. investors.
Yield curve analysis, in Chapter 5, is arguably the most important topic in the book. There are many practical applications arising from bootstrapped implied zero-coupon (or spot) rates and implied forward rates—identifying arbitrage opportunities, obtaining discount factors to get present values, calculating spreads, and pricing and valuing derivatives. However, the operative assumption in this analysis is “no arbitrage”—that is, transactions costs and counterparty credit risk are sufficiently small so that trading eliminates any arbitrage opportunity. Therefore, while mathematically elegant, yield curve analysis is best applied to Treasury securities and LIBOR-based interest rate derivatives for which the no-arbitrage assumption is reasonable.
Duration and convexity, the subject of Chapter 6, is the most mathematical topic in this book. These statistics, which in classic form measure the sensitivity of the bond price to a change in its yield to maturity, can be derived with algebra and calculus. Those details are relegated to the Technical Appendix. Another version of the risk statistics measures the sensitivity of the bond price to a shift in the entire Treasury yield curve. I call the former yield and the latter curve duration and convexity and demonstrate where and how they are presented on Bloomberg pages.
Chapters 7 and 8 examine floating-rate notes (floaters), inflation-indexed bonds (linkers), and interest rate swaps. The idea is to use the bond math toolkit—periodicity conversions, bond valuation, after-tax rates of return, implied spot rates, implied forward rates, and duration and convexity—to examine securities other than traditional fixed-rate and zero-coupon bonds. In particular, I look for circumstances of negative duration, meaning market value and interest rates are positively correlated. That’s an obvious feature for one type of interest rate swap but a real oddity for a floater and a linker.
Understanding the risk and return characteristics for an individual bond is easy compared to a portfolio of bonds. In Chapter 9, I show different ways of getting summary statistics. One is to treat the portfolio as a big bundle of cash flow and derive its yield, duration, and convexity is if it were just a single bond with many variable payments. While that is theoretically correct, in practice portfolio statistics are calculated as weighted averages of those for the constituent bonds. Some statistics can be aggregated in this manner and provide reasonable estimates of the “true” values, depending on how the weights are calculated and on the shape of the yield curve.
Chapter 10 is on bond strategies. If your hope is that I’ll show you how to get rich by trading bonds, you’ll be disappointed. My focus is on how the bond math tools and the various risk and return statistics that we can calculate for individual bonds and portfolios can facilitate either aggressive or passive investment strategies. I’ll discuss derivative overlays, immunization, and liability-driven investing and conclude with a request that the finance industry create target-duration bond funds.
I’d like to thank my Wiley editors for allowing me to deviate from their usual publishing standards so that I can use in this book acronyms, italics, and notation as I prefer. Now let’s get started in the money market.
