Fixed-Income Securities and Derivatives Handbook E-Book

Moorad Choudhry

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Bloomberg Professional
• Sprache: Englisch

Seitenzahl: 629

Veröffentlichungsjahr: 2010

Beschreibung

The definitive guide to fixed-come securities-revised to reflect today's dynamic financial environment The Second Edition of the Fixed-Income Securities and Derivatives Handbook offers a completely updated and revised look at an important area of today's financial world. In addition to providing an accessible description of the main elements of the debt market, concentrating on the instruments used and their applications, this edition takes into account the effect of the recent financial crisis on fixed income securities and derivatives. As timely as it is timeless, the Second Edition of the Fixed-Income Securities and Derivatives Handbook includes a wealth of new material on such topics as covered and convertible bonds, swaps, synthetic securitization, and bond portfolio management, as well as discussions regarding new regulatory twists and the evolving derivatives market. * Offers a more detailed look at the basic principles of securitization and an updated chapter on collateralized debt obligations * Covers bond mathematics, pricing and yield analytics, and term structure models * Includes a new chapter on credit analysis and the different metrics used to measure bond-relative value * Contains illustrative case studies and real-world examples of the topics touched upon throughout the book Written in a straightforward and accessible style, Moorad Choudhry's new book offers the ideal mix of practical tips and academic theory within this important field.

Leseprobe

Table of Contents

Praise

Title Page

Copyright Page

Foreword

PREFACE

A Word on the Mathematics

Acknowledgments

PART ONE - INTRODUCTION TO BONDS

CHAPTER 1 - The Bond Instrument

The Time Value of Money

Bond Pricing and Yield: The Traditional Approach

Accrued Interest

CHAPTER 2 - Bond Instruments and Interest Rate Risk

Duration, Modified Duration, and Convexity

CHAPTER 3 - Bond Pricing and Spot and Forward Rates

Zero-Coupon Bonds

Coupon Bonds

Bond Price in Continuous Time

Forward Rates

Term Structure Hypotheses

CHAPTER 4 - Interest Rate Modeling

Basic Concepts

One-Factor Term-Structure Models

Further One-Factor Term-Structure Models

Two-Factor Interest Rate Models

Choosing a Term-Structure Model

CHAPTER 5 - Fitting the Yield Curve

Yield Curve Smoothing

Smoothing Techniques

Cubic Polynomials

Non-Parametric Methods

Comparing Curves

Cubic Spline Methodology

The Hypothesis

Practical Approach

A Look at Forward Rates

Conclusion

PART TWO - SELECTED CASH AND DERIVATIVE INSTRUMENTS

CHAPTER 6 - Forwards and Futures Valuation

Forwards and Futures

Forward-Spot Parity

The Basis and Implied Repo Rate

CHAPTER 7 - Swaps

Interest Rate Swaps

Generic Swap Valuation

Non-Plain Vanilla Interest Rate Swaps

Swaptions

Interest Rate Swap Applications

CHAPTER 8 - Options

Option Basics

Option Instruments

Option Pricing: Setting the Scene

Option Pricing

The Black-Scholes Option Model

Other Option Models

CHAPTER 9 - Measuring Option Risk

Option Price Behavior

The Greeks

The Option Smile

Caps and Floors

CHAPTER 10 - Credit Derivatives

Credit Risk

Credit Risk and Credit Derivatives

Credit Derivative Instruments

Investment Applications

Credit Derivatives and Relative Value Trading

Credit-Derivative Pricing

CHAPTER 11 - The Analysis of Bonds with Embedded Options

Understanding Option Elements Embedded in a Bond

Basic Options Features

Option Valuation

The Call Provision

The Binomial Tree of Short-Term Interest Rates

Arbitrage-Free Pricing

Options Pricing

Risk-Neutral Pricing

Recombining and Nonrecombining Trees

Pricing Callable Bonds

Price and Yield Sensitivity

Measuring Bond Yield Spreads

CHAPTER 12 - Option-Adjusted Spread Analysis

Introduction

A Theoretical Framework

The Methodology in Practice

CHAPTER 13 - Convertible Bonds

Basic Features

Advantages of Issuing and Holding Convertibles

Convertible Bond Valuation

Pricing Spreadsheet

CHAPTER 14 - Inflation-Indexed Bonds

Basic Concepts

Index-Linked Bond Cash Flows and Yields

Analysis of Real Interest Rates

CHAPTER 15 - Securitization and Asset-Backed Securities

The Concept of Securitization

The Process of Securitization

Securitizing Mortgages

Cash Flow Patterns

ABS Structures: A Primer on Performance Metrics and Test Measures

Securitization: Features of the 2007-2009 Financial Crisis

CHAPTER 16 - Collateralized Debt Obligations

CDO Structures

Motivation Behind CDO Issuance

Analysis and Evaluation

Expected Loss

CDO Market Overview Since 2005

PART THREE - SELECTED MARKET TRADING CONSIDERATIONS

CHAPTER 17 - The Yield Curve, Bond Yield, and Spot Rates

Practical Uses of Redemption Yield and Duration

Illustrating Bond Yield Using a Microsoft Excel Spreadsheet

Implied Spot Rates and Market Zero-Coupon Yields

Implied Spot Yields and Zero-Coupon Bond Yields

Determining Strip Values

Strips Market Anomalies

Strips Trading Strategy

CHAPTER 18 - Approaches to Trading

Futures Trading

Yield Curves and Relative Value

Characterizing the Complete Term Structure

Hedging Bond Positions

Summary of the Derivation of the Optimum-Hedge Equation

CHAPTER 19 - Credit Analysis and Relative Value Measurement

Credit Ratings

Credit Analysis

Industry-Specific Analysis

The Art of Credit Analysis

Bond Spreads and Relative Value

APPENDIX I - The Black-Scholes Model in Microsoft Excel

APPENDIX II - Iterative Formula Spreadsheet

APPENDIX III - Pricing Spreadsheet

REFERENCES

ABOUT THE AUTHOR

INDEX

PRAISE FOR

Fixed-Income Securities and Derivatives Handbook Second Edition

“The best book about fixed income securities out there.”

—Patrick Y. Shim, Financial Advisor CG Investment Group Wells Fargo Advisors LLC

“A truly excellent resource and recommended reading for anyone with an interest or involvement in the bond and fixed income markets.”

—Mohamoud Dualeh, Service Officer Abu Dhabi Commercial Bank, UAE

“Professor Choudhry has a brilliant knack for taking complex subjects and writing about them in an accessible and reader-friendly manner. A valuable reference work on fixed income securities and their related derivatives, and a particularly excellent introduction to credit derivatives.”

—Rod Pienaar, Executive Director Prime Services, UBS AG, London

“A comprehensive yet succinct coverage of fixed income debt market instruments, which will be instructive for investors as well as analysts and bankers.”

—Zhuoshi Liu, Analyst Barrie & Hibbert, Edinburgh

“A work of excellence!”

—Abukar Ali FX Analytics, Bloomberg L.P., London

Copyright © 2010, 2005 by Moorad Choudhry. 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, [phone], fax [phone], 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, [phone], fax [phone], or online at http://www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

The information herein is taken from sources believed to be reliable. However, it is intended for purposes of information and education only and is not guaranteed by CME Group Inc. or any of its subsidiaries as to accuracy, completeness, nor any trading result and does not constitute trading advice or constitute a solicitation of the purchase or sale of any futures or options.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com.

ISBN-13 978-1-576-60334-5

“What’s the secret, Sean?”“Buy cheap, sell dear . . . !”

FOREWORD

I remember the first time I met Moorad Choudhry—a man, it seems, who is very well known in the banking community. It was late evening on a rainy day in London at the Dorchester Hotel, where Moody’s Investors Service was hosting a cocktail party in connection with its annual asset-backed commercial paper conference. I was invited by my law firm to mingle with bankers, raters, accountants, and other lawyers on a day of panel workshops covering the short-term money markets.

Standing with some attendees that I knew fairly well, I noticed a wizened, older gentleman with grey hair and a classic English pinstripe suit peering into our group, studying me. A few minutes later, an immaculately dressed lady with dark hair and glasses walked by, paused, and did the same. A former colleague of mine noticed this, and, laughing, said that at least three other people had similar reactions to my presence. Notwithstanding my hearty laugh back, I quickly examined my clothing to make sure nothing was untoward.

Having moved to the other side of the room, I noticed that the attention I was receiving continued unabated. I asked the banker with whom I was speaking if he also observed this, and he chuckled knowingly, explaining that people must have been mistaking me for Moorad Choudhry, then Head of Treasury at KBC Financial Products. He pointed to a flat-screen TV that was showing replays of different panel presentations from the conference. To my surprise, sitting on the panel was a strikingly handsome, charismatic and commanding gentleman of South Asian descent. My description is certainly accurate, but note that were I to describe Moorad in any other way, it would be demeaning to me—we do look alike!

About four years later, I got to know the man behind the reputation—first as a client and then as a dear friend. Speaking professionally, Moorad is a challenging and, at times, daunting client—having earned the respect of my mentor, Jim Croke, years before I ever started working with Moorad. He always demands the best work product, which is not easy to provide given his desire to create new, innovative and dynamic structured credit products. His drive is fostered by an academic zeal to find market efficiencies where others have failed to look, but his approach to deals is equally tempered by an overriding principle that his reputation as a scrupulous businessman must not be tarnished. He reminds me in many ways of the mythical English cricketer of old, who would rather lose a match than dishonor the rules of the game.

Personally, Moorad has always been steadfast and loyal to his friends and family—and, dare I say it, lawyers; he is an affable, easy-going and charismatic person, with a deep desire to understand people and the motivations that keep them going. His enthusiasm and intensity are contagious. Indeed, there are very few people in the world today that can make dinner conversation entertaining and lively when the topic is the mechanics of derivatives. It is not surprising that Moorad is a Visiting Professor at the Department of Economics at London Metropolitan University, not to mention a Research Fellow at the ICMA Centre at Reading University.

Somehow, in the midst of all of these responsibilities, Moorad still finds the time to be a prolific author. He has written numerous books on the markets and is considered by many in both academia and the business world as a go-to person for financial insight. His reach in literary media is far and wide—once, an incredulous colleague called me from the US on a Sunday afternoon just to tell me that he had come across a letter in The Economist that was from “that Moorad Choudhry guy.”

Moorad’s passion for and understanding of the markets is transposed in his written work. His writing is clear and concise, with easily accessible text, setting him apart from many of his academic peers; his ability to convey ideas stems from a wealth of professional knowledge and experience. While Moorad certainly provides an excellent overview of the interplay of market forces, a careful reader will also discern discrete and valuable insights, showing that Moorad can be granular in his approach to highlight important points of market intelligence. He is careful to incorporate current market trends to keep his materials fresh and vibrant, and he regularly invites submissions and comment from pre-eminent market participants to enhance the practical value of his work.

Needless to say, I think you will enjoy reading The Fixed-Income Securities & Derivatives Handbook. If you are reading it for the first time, I encourage you to soak up as much as you can, as Moorad certainly provides you with a helpful background to an integral part of the debt capital markets. If you have already read this book, however, I suggest you keep it as a reference. Having this book on hand is like having Moorad Choudhry available at your beck and call for market advice, and that, Dear Reader, is a valuable thing to have.

Sharad A. Samy, Esq. General Counsel Aladdin Capital Holdings LLC Stamford, CT March 25, 2010

PREFACE

The banking and liquidity crisis of 2007-2009, which led to global recession and the first examples of deflation in developed economies since the Second World War, had a profound effect on financial markets. Its aftermath is an opportune time to update a textbook on fixed-income securities and derivatives, as we can review the impact that the crisis had on the instruments concerned.

The primary objective behind writing the first edition of this book was to produce a practitioner-viewpoint accessible description of the main elements of the markets, concentrating on the instruments used and their applications. The book was intended to be a proper “handbook,” that is, one that was both succinct and concise, and dispensed with extraneous supporting material.

Part One, “Introduction to Bonds,” covers bond mathematics, including pricing and yield analytics. This includes modified duration and convexity. Chapters also cover the concept of spot (zero-coupon) and forward rates, and the rates implied by market bond prices and yields; yield-curve fitting techniques; an account of spline fitting using regression techniques; and an introductory discussion of term structure models.

Part Two, “Cash and Derivative Instruments and Analysis,” has an analysis of various instruments including callable bonds that feature embedded options. There is a discussion of securitization and the impact on the market of the financial crisis. Other chapters cover U.S. Treasury TIPS securities, and the use and applications of credit derivatives.

Finally, Part Three, “Selected Market Analysis and Trading Considerations,” covers the practical uses of redemption yield and duration as well as trading techniques based on the author’s personal experiences at a primary dealer’s desk.

As well as revising and updating the existing chapters, new material in this second edition includes the following:

• A practical demonstration of cubic spline methodology, useful in constructing yield curves.

• An update on developments in the credit derivative market, which takes in the development of the iTraxx and CD-X indices, credit default swap trading strategies, and the 2009 “big bang” in the CDS market. There is also an accessible look at CDS pricing principles.

• A new chapter on convertible bonds.

• Description, in the chapter on inflation-linked bonds, of inflation-indexed derivatives.

• A more detailed look at the basic principles of securitization, and an update of the chapter on CDOs.

• A new chapter on credit analysis and the different metrics used to measure bond relative value.

As with the first edition, use is made of case studies and real-world examples to help illustrate key concepts.

Comments on the text are most welcome and should be sent to the author in care of Bloomberg Press.

A Word on the Mathematics

This is what I wrote in the Preface to the first edition:

Financial subjects such as the debt capital markets are essentially quantitative disciplines, and as such it is not possible to describe them, let alone analyze them, without a certain amount of numerical input. To maintain accessibility of this book, the level of mathematics used has been limited; as a result many topics could not be reviewed in full detail. There are very few derivations, for example, and fewer proofs. This has not, in the author’s opinion, impaired the analysis as the reader is still left with an understanding of the techniques required in the context of market instruments.

Actually, if the credit crunch taught us anything, it was that the muchhyped “quant” was overrated and had been given a prominence on the trading floor that far outweighed his or her value-added to the business. Even the expression “quantitative finance” is redundant: finance is by definition a quantitative subject, so adding the spurious “quantitative” to it is a bit like saying “wet swimming” or “aerial flying.” The “quants” and their physics disciplines and mind-crunching VaR models didn’t predict the credit crunch, didn’t stop the credit crunch, and didn’t come anywhere near estimating accurately its anticipated and actual losses. In fact, an overemphasis on mathematical models and black boxes only served to help remove focus from basic but important issues such as the value of sound credit analysis, loan origination standards, and liquidity (funding) risk management. So in hindsight, a lack of emphasis on the mathematics was not necessarily a bad thing. In any case, there certainly isn’t any complex math in this second edition! If anything, the contrary. And as for VaR: after the credit crunch it was noted that the banks that had adopted it didn’t do any better than the banks that had continued to use good old-fashioned modified duration for their market risk management. As a measurement methodology it remains in widespread use, but it should never, in line with all methods, be viewed as being a complete measure.

Acknowledgments

Thanks to my main man Stuart Turner.

Thanks also to Eric Subliskey at JPMorgan Securities, Jim Croke and Sharad Samy at Orrick, and Khurram Butt for being all-round top chaps and gentlemen to boot.

Thanks to Courty Gates at Clearwater Analytics and Philip Fernandez for pointing out all errors in the first edition, which have now been corrected.

PART ONE

INTRODUCTION TO BONDS

Part One describes fixed-income market analysis and the basic concepts relating to bond instruments. The analytic building blocks are generic and thus applicable to any market. The analysis is simplest when applied to plain vanilla default-free bonds; as the instruments analyzed become more complex, additional techniques and assumptions are required.

The first two chapters of this section discuss bond pricing and yields, moving on to an explanation of such traditional interest rate risk measures as modified duration and convexity, followed by a discussion of floating-rate notes (FRNs). Chapter 3 looks at spot and forward rates, the derivation of such rates from market yields, and the yield curve.

Yield-curve analysis and the modeling of the term structure of interest rates are among the most heavily researched areas of financial economics. The treatment here has been kept as concise as possible, at just two chapters. The References section at the end of the book directs interested readers to accessible and readable resources that provide more detail.

CHAPTER 1

The Bond Instrument

Bonds are the basic ingredient of the U.S. debt-capital market, which is the cornerstone of the U.S. economy. All evening television news programs include a slot during which the newscaster informs viewers where the main stock market indexes closed that day and where key foreign exchange rates ended up. Financial sections of most newspapers also indicate at what yield the Treasury long bond closed. This coverage reflects the fact that bond prices are affected directly by economic and political events, and yield levels on certain government bonds are fundamental economic indicators. The yield level on the U.S. Treasury long bond, for instance, mirrors the market’s view on U.S. interest rates, inflation, public-sector debt, and economic growth.

The media report the bond yield level because it is so important to the country’s economy—as important as the level of the equity market and more relevant as an indicator of the health and direction of the economy. Because of the size and crucial nature of the debt markets, a large number of market participants, ranging from bond issuers to bond investors and associated intermediaries, are interested in analyzing them. This chapter introduces the building blocks of the analysis.

Bonds are debt instruments that represent cash flows payable during a specified time period. They are essentially loans. The cash flows they represent are the interest payments on the loan and the loan redemption. Unlike commercial bank loans, however, bonds are tradable in a secondary market. Bonds are commonly referred to as fixed-income instruments. This term goes back to a time when bonds paid fixed coupons each year. Today that is not necessarily the case. Asset-backed bonds, for instance, are issued in a number of tranches—related securities from the same issuer—each of which pays a different fixed or floating coupon. Nevertheless, this is still commonly referred to as the fixed-income market.

In the past, bond analysis was frequently limited to calculating gross redemption yield, or yield to maturity. Today basic bond math involves different concepts and calculations. These are described in several of the references for Chapter 3, such as Ingersoll (1987), Shiller (1990), Neftci (1996), Jarrow (1996), Van Deventer (1997), and Sundaresan (1997). This chapter reviews the basic elements. Bond pricing, together with the academic approach to it and a review of the term structure of interest rates, are discussed in depth in Chapter 3.

In the analysis that follows, bonds are assumed to be default free. This means there is no possibility that the interest payments and principal repayment will not be made. Such an assumption is entirely reasonable for government bonds such as U.S. Treasuries and U.K. gilt-edged securities. It is less so when you are dealing with the debt of corporate and lower-rated sovereign borrowers. The valuation and analysis of bonds carrying default risk, however, are based on those of default-free government securities. Essentially, the yield investors demand from borrowers whose credit standing is not risk-free is the yield on government securities plus some credit risk premium.

The Time Value of Money

Bond prices are expressed “per 100 nominal”—that is, as a percentage of the bond’s face value. (The convention in certain markets is to quote a price per 1,000 nominal, but this is rare.) For example, if the price of a U.S. dollar-denominated bond is quoted as 98.00, this means that for every $100 of the bond’s face value, a buyer would pay $98. The principles of pricing in the bond market are the same as those in other financial markets: the price of a financial instrument is equal to the sum of the present values of all the future cash flows from the instrument. The interest rate used to derive the present value of the cash flows, known as the discount rate, is key, since it reflects where the bond is trading and how its return is perceived by the market. All the factors that identify the bond—including the nature of the issuer, the maturity date, the coupon, and the currency in which it was issued—influence the bond’s discount rate. Comparable bonds have similar discount rates. The following sections explain the traditional approach to bond pricing for plain vanilla instruments, making certain assumptions to keep the analysis simple. After that, a more formal analysis is presented.

Basic Features and Definitions

One of the key identifying features of a bond is its issuer, the entity that is borrowing funds by issuing the bond in the market. Issuers generally fall into one of four categories: governments and their agencies; local governments, or municipal authorities; supranational bodies, such as the World Bank; and corporations. Within the municipal and corporate markets there are a wide range of issuers that differ in their ability to make the interest payments on their debt and repay the full loan. An issuer’s ability to make these payments is identified by its credit rating.

Another key feature of a bond is its term to maturity: the number of years over which the issuer has promised to meet the conditions of the debt obligation. The practice in the bond market is to refer to the term to maturity of a bond simply as its maturity or term. Bonds are debt capital market securities and therefore have maturities longer than one year. This differentiates them from money market securities. Bonds also have more intricate cash flow patterns than money market securities, which usually have just one cash flow at maturity. As a result, bonds are more complex to price than money market instruments, and their prices are more sensitive to changes in the general level of interest rates.

A bond’s term to maturity is crucial because it indicates the period during which the bondholder can expect to receive coupon payments and the number of years before the principal is paid back. The principal of a bond—also referred to as its redemption value, maturity value, par value, or face value—is the amount that the issuer agrees to repay the bondholder on the maturity, or redemption, date, when the debt ceases to exist and the issuer redeems the bond. The coupon rate, or nominal rate, is the interest rate that the issuer agrees to pay during the bond’s term. The annual interest payment made to bondholders is the bond’s coupon. The cash amount of the coupon is the coupon rate multiplied by the principal of the bond. For example, a bond with a coupon rate of 8 percent and a principal of $1,000 will pay an annual cash amount of $80.

A bond’s term to maturity also influences the volatility of its price. All else being equal, the longer the term to maturity of a bond, the greater its price volatility.

There are a large variety of bonds. The most common type is the plain vanilla, otherwise known as the straight, conventional, or bullet bond. A plain vanilla bond pays a regular—annual or semiannual—fixed interest payment over a fixed term. All other types of bonds are variations on this theme.

In the United States, all bonds make periodic coupon payments except for one type: the zero-coupon. Zero-coupon bonds do not pay any coupon. Instead, investors buy them at a discount to face value and redeem them at par. Interest on the bond is thus paid at maturity, realized as the difference between the principal value and the discounted purchase price.

Floating-rate bonds, often referred to as floating-rate notes (FRNs), also exist. The coupon rates of these bonds are reset periodically according to a predetermined benchmark, such as 3-month or 6-month LIBOR (London interbank offered rate). LIBOR is the official benchmark rate at which commercial banks will lend funds to other banks in the interbank market. It is an average of the offered rates posted by all the main commercial banks and is reported by the British Bankers Association at 11.00 hours each business day. For this reason, FRNs typically trade more like money market instruments than like conventional bonds.

A bond issue may include a provision that gives either the bondholder or the issuer the option to take some action with respect to the other party. The most common type of option embedded in a bond is a call feature. This grants the issuer the right to “call” the bond by repaying the debt, fully or partially, on designated dates before the maturity date. A put provision gives bondholders the right to sell the issue back to the issuer at par on designated dates before the maturity date. A convertible bond contains a provision giving bondholders the right to exchange the issue for a specified number of stock shares, or equity, in the issuing company. The presence of embedded options makes the valuation of such bonds more complicated than that of plain vanilla bonds.

Present Value and Discounting

Since fixed-income instruments are essentially collections of cash flows, it is useful to begin by reviewing two key concepts of cash flow analysis: discounting and present value. Understanding these concepts is essential. In the following discussion, the interest rates cited are assumed to be the market-determined rates.

Financial arithmetic demonstrates that the value of $1 received today is not the same as that of $1 received in the future. Assuming an interest rate of 10 percent a year, a choice between receiving $1 in a year and receiving the same amount today is really a choice between having $1 a year from now and having $1 plus $0.10—the interest on $1 for one year at 10 percent per annum.

The market convention is to quote annualized interest rates: the rate corresponding to the amount of interest that would be earned if the investment term were one year. Consider a three-month deposit of $100 in a bank earning a rate of 6 percent a year. The annual interest gain would be $6. The interest earned for the ninety days of the deposit is proportional to that gain, as calculated below:

Note that, in the sterling markets, the number of days in the year is taken to be 365, but most other markets—including the dollar and euro markets—use a 360-day year. (These conventions are discussed more fully later in the chapter.)

This computation assumes that the interest payments made during the investment term are reinvested at an interest rate equal to the first year’s rate. That is why the example stated that the 6 percent rate was fixed for three years. Compounding obviously results in higher returns than those earned with simple interest.

Now consider a deposit of $100 for one year, still at a rate of 6 percent but compounded quarterly. Again assuming that the interest payments will be reinvested at the initial interest rate of 6 percent, the total return at the end of the year will be

The terminal value for quarterly compounding is thus about $0.13 more than that for annual compounded interest.

In general, if compounding takes place m times per year, then at the end of n years, mn interest payments will have been made, and the future value of the principal is computed using the formula (1.4).

(1.4)

As the example above illustrates, more frequent compounding results in higher total returns. FIGURE 1.1 shows the interest rate factors corresponding to different frequencies of compounding on a base rate of 6 percent a year.

FIGURE 1.1Impact of Compounding

(1.5)

As compounding becomes continuous and m and hence w approach infinity, the expression in the square brackets in (1.5) approaches the mathematical constant e (the base of natural logarithmic functions), which is equal to approximately 2.718281.

Substituting e into (1.5) gives us

(1.6)

In (1.6) ern is the exponential function of rn. It represents the continuously compounded interest rate factor. To compute this factor for an interest rate of 6 percent over a term of one year, set r to 6 percent and n to 1, giving

The convention in both wholesale and personal, or retail, markets is to quote an annual interest rate, whatever the term of the investment, whether it be overnight or 10 years. Lenders wishing to earn interest at the rate quoted have to place their funds on deposit for one year. For example, if you open a bank account that pays 3.5 percent interest and close it after six months, the interest you actually earn will be equal to 1.75 percent of your deposit. The actual return on a three-year building society bond that pays a 6.75 percent fixed rate compounded annually is 21.65 percent. The quoted rate is the annual one-year equivalent. An overnight deposit in the wholesale, or interbank, market is still quoted as an annual rate, even though interest is earned for only one day.

Quoting annualized rates allows deposits and loans of different maturities and involving different instruments to be compared. Be careful when comparing interest rates for products that have different payment frequencies. As shown in the earlier examples, the actual interest earned on a deposit paying 6 percent semiannually will be greater than on one paying 6 percent annually. The convention in the money markets is to quote the applicable interest rate taking into account payment frequency.

(1.7)

Equation (1.7) applies to investments earning annual interest payments, giving the present value of a known future sum.

To calculate the present value of an investment, you must prorate the interest that would be earned for a whole year over the number of days in the investment period, as was done in (1.2). The result is equation (1.8).

(1.8)

When interest is compounded more than once a year, the formula for calculating present value is modified, as it was in (1.4). The result is shown in equation (1.9).

(1.9)

For example, the present value of $100 to be received at the end of five years, assuming an interest rate of 5 percent, with quarterly compounding is

Say the 1-month (30-day) interest rate is 5.25 percent and the 2-month (60-day) rate is 5.75 percent. If a customer wishes to borrow money for 40 days, the bank can calculate the required rate using straight line interpolation as follows: the difference between 30 and 40 is one-third that between 30 and 60, so the increase from the 30-day to the 40-day rate is assumed to be one-third the increase between the 30-day and the 60-day rates, giving the following computation

What about the interest rate for a period that is shorter or longer than the two whose rates are known, rather than lying between them? What if the customer in the example above wished to borrow money for 64 days? In this case, the interbank desk would extrapolate from the relationship between the known 1-month and 2-month rates, again assuming a uniform rate of change in the interest rates along the maturity spectrum. So given the 1-month rate of 5.25 percent and the 2-month rate of 5.75 percent, the 64-day rate would be

Just as future and present value can be derived from one another, given an investment period and interest rate, so can the interest rate for a period be calculated given a present and a future value. The basic equation is merely rearranged again to solve for r. This, as discussed later, is known as the investment’s yield.

Discount Factors

For instance, the five-year discount factor for a rate of 6 percent compounded annually is

The set of discount factors for every period from one day to 30 years and longer is termed the discount function. Since the following discussion is in terms of PV, discount factors may be used to value any financial instrument that generates future cash flows. For example, the present value of an instrument generating a cash flow of $103.50 payable at the end of six months would be determined as follows, given a six-month discount factor of 0.98756:

Discount factors can also be used to calculate the future value of a present investment by inverting the formula. In the example above, the six-month discount factor of 0.98756 signifies that $1 receivable in six months has a present value of $0.98756. By the same reasoning, $1 today would in six months be worth

It is possible to derive discount factors from current bond prices. This process can be illustrated using the set of hypothetical bonds, all assumed to have semiannual coupons, that are shown in FIGURE 1.2, together with their prices.

FIGURE 1.2Hypothetical Set of Bonds and Bond Prices

The first bond in Figure 1.2 matures in precisely six months. Its final cash flow will be $103.50, comprising the final coupon payment of $3.50 and the redemption payment of $100. The price, or present value, of this bond is $101.65. Using this, the six-month discount factor may be calculated as follows:

Using this six-month discount factor, the one-year factor can be derived from the second bond in Figure 1.2, the 8 percent due 2001. This bond pays a coupon of $4 in six months and, in one year, makes a payment of $104, consisting of another $4 coupon payment plus $100 return of principal.

The price of the one-year bond is $101.89. As with the 6-month bond, the price is also its present value, equal to the sum of the present values of its total cash flows. This relationship can be expressed in the following equation:

The value of d0.5 is known to be 0.98213. That leaves d1 as the only unknown in the equation, which may be rearranged to solve for it:

The same procedure can be repeated for the remaining two bonds, using the discount factors derived in the previous steps to derive the set of discount factors in FIGURE 1.3. These factors may also be graphed as a continuous function, as shown in FIGURE 1.4.

FIGURE 1.3Discount Factors Calculated Using BootstrappingTechnique

FIGURE 1.4Hypothetical Discount Function

This technique of calculating discount factors, known as bootstrapping, is conceptually neat, but may not work so well in practice. Problems arise when you do not have a set of bonds that mature at precise six-month intervals. Liquidity issues connected with individual bonds can also cause complications. This is true because the price of the bond, which is still the sum of the present values of the cash flows, may reflect liquidity considerations (e.g., hard to buy or sell the bond, difficult to find) that do not reflect the market as a whole but peculiarities of that specific bond. The approach, however, is still worth knowing.

Note that the discount factors in Figure 1.3 decrease as the bond’s maturity increases. This makes intuitive sense, since the present value of something to be received in the future diminishes the further in the future the date of receipt lies.

Bond Pricing and Yield: The Traditional Approach

The discount rate used to derive the present value of a bond’s cash flows is the interest rate that the bondholders require as compensation for the risk of lending their money to the issuer. The yield investors require on a bond depends on a number of political and economic factors, including what other bonds in the same class are yielding. Yield is always quoted as an annualized interest rate. This means that the rate used to discount the cash flows of a bond paying semiannual coupons is exactly half the bond’s yield.

Bond Pricing

Bonds in the U.S. domestic market—as opposed to international securities denominated in U.S. dollars, such as USD Eurobonds—usually pay semi-annual coupons. Such bonds may be priced using the expression in (1.13), which is a modification of (1.12) allowing for twice-yearly discounting.

(1.13)

Note that 2N is now the power to which the discount factor is raised. This is because a bond that pays a semiannual coupon makes two interest payments a year. It might therefore be convenient to replace the number of years to maturity with the number of interest periods, which could be represented by the variable n, resulting in formula (1.14).

(1.14)

This formula calculates the fair price on a coupon payment date, so there is no accrued interest incorporated into the price. Accrued interest is an accounting convention that treats coupon interest as accruing every day a bond is held; this accrued amount is added to the discounted present value of the bond (the clean price) to obtain the market value of the bond, known as the dirty price. The price calculation is made as of the bond’s settlement date, the date on which it actually changes hands after being traded. For a new bond issue, the settlement date is the day when the investors take delivery of the bond and the issuer receives payment. The settlement date for a bond traded in the secondary market—the market where bonds are bought and sold after they are first issued—is the day the buyer transfers payment to the seller of the bond and the seller transfers the bond to the buyer.

Different markets have different settlement conventions. U.S. Treasuries, for example, normally settle on “T + 1”: one business day after the trade date; T. Eurobonds, on the other hand, settle on T + 3. The term value date is sometimes used in place of settlement date; however, the two terms are not strictly synonymous. A settlement date can fall only on a business day; a bond traded on a Friday, therefore, will settle on a Monday. A value date, in contrast, can sometimes fall on a non-business day—when accrued interest is being calculated, for example.

Equation (1.14) assumes an even number of coupon payment dates remaining before maturity. If there are an odd number, the formula is modified as shown in (1.15).

(1.15)

Another assumption embodied in the standard formula is that the bond is traded for settlement on a day that is precisely one interest period before the next coupon payment. If the trade takes place between coupon dates, the formula is modified. This is done by adjusting the exponent for the discount factor using ratio i, shown in (1.16).

(1.16)

The denominator of this ratio is the number of calendar days between the last coupon date and the next one. This figure depends on the day-count convention (see below) used for that particular bond. Using i, the price formula is modified as (1.17) (for annual-coupon-paying bonds; for bonds with semiannual coupons, r/2 replaces r).

(1.17)

where the variables C, M, n, and r are as before

As noted above, the bond market includes securities, known as zero-coupon bonds, or strips, that do not pay coupons. These are priced by setting C to 0 in the pricing equation. The only cash flow is the maturity payment, resulting in formula (1.18)

(1.18)

where M and r are as before and N is the number of years to maturity.

Note that, even though these bonds pay no actual coupons, their prices and yields must be calculated on the basis of quasi-coupon periods, which are based on the interest periods of bonds denominated in the same currency. A U.S. dollar or a sterling five-year zero-coupon bond, for example, would be assumed to cover 10 quasi-coupon periods, and the price equation would accordingly be modified as (1.19).

(1.19)

It is clear from the bond price formula that a bond’s yield and its price are closely related. Specifically, the price moves in the opposite direction from the yield. This is because a bond’s price is the net present value of its cash flows; if the discount rate—that is, the yield required by investors—increases, the present values of the cash flows decrease. In the same way, if the required yield decreases, the price of the bond rises. The relationship between a bond’s price and any required yield level is illustrated by the graph in FIGURE 1.5, which plots the yield against the corresponding price to form a convex curve.

Bond Yield

The discussion so far has involved calculating the price of a bond given its yield. This procedure can be reversed to find a bond’s yield when its price is known. This is equivalent to calculating the bond’s internal rate of return, or IRR, also known as its yield to maturity or gross redemption yield (also yield to workout). These are among the various measures used in the markets to estimate the return generated from holding a bond.

Summary of the Price/Yield Relationship

• At issue, if a bond is priced at par, its coupon will equal the yield that the market requires, reflecting factors such as the bond’s term to maturity, the issuer’s credit rating, and the yield on current bonds of comparable quality.

FIGURE 1.5The Price/Yield Relationship

• If the required yield rises above the coupon rate, the bond price will decrease.

• If the required yield falls below the coupon rate, the bond price will increase.

In most markets, bonds are traded on the basis of their prices. Because different bonds can generate different and complicated cash flow patterns, however, they are generally compared in terms of their yields. For example, market makers usually quote two-way prices at which they will buy or sell particular bonds, but it is the yield at which the bonds are trading that is important to the market makers’ customers. This is because a bond’s price does not tell buyers anything useful about what they are getting. Remember that in any market a number of bonds exist with different issuers, coupons, and terms to maturity. It is their yields that are compared, not their prices.

The yield on any investment is the discount rate that will make the present value of its cash flows equal its initial cost or price. Mathematically, an investment’s yield, represented by r, is the interest rate that satisfies the bond price equation, repeated here as (1.20).

(1.20)

Other types of yield measure, however, are used in the market for different purposes. The simplest is the current yield, also know as the flat, interest, or running yield. These are computed by formula (1.21).

(1.21)

where rc is the current yield.

In this equation the percentage for C is not expressed as a decimal. Current yield ignores any capital gain or loss that might arise from holding and trading a bond and does not consider the time value of money. It calculates the coupon income as a proportion of the price paid for the bond. For this to be an accurate representation of return, the bond would have to be more like an annuity than a fixed-term instrument.

Current yield is useful as a “rough and ready” interest rate calculation; it is often used to estimate the cost of or profit from holding a bond for a short term. For example, if short-term interest rates, such as the one-week or three-month, are higher than the current yield, holding the bond is said to involve a running cost. This is also known as negative carry or negative funding. The concept is used by bond traders, market makers, and leveraged investors, but it is useful for all market practitioners, since it represents the investor’s short-term cost of holding or funding a bond. The yield to maturity (YTM)—or, as it is known in sterling markets, gross redemption yield—is the most frequently used measure of bond return. Yield to maturity takes into account the pattern of coupon payments, the bond’s term to maturity, and the capital gain (or loss) arising over the remaining life of the bond. The bond price formula shows the relationship between these elements and demonstrates their importance in determining price. The YTM calculation discounts the cash flows to maturity, employing the concept of the time value of money.

As noted earlier, the formula for calculating YTM is essentially that for calculating the price of a bond, repeated as (1.12). (For the YTM of bonds with semiannual coupon, the formula must be modified, as in (1.13).) Note, though, that this equation has two variables, the price P and yield r. It cannot, therefore, be rearranged to solve for yield r explicitly. In fact, the only way to solve for the yield is to use numerical iteration. This involves estimating a value for r and calculating the price associated with it. If the calculated price is higher than the bond’s current price, the estimate for r is lower than the actual yield, so it must be raised. This process of calculation and adjustment up or down is repeated until the estimates converge on a level that generates the bond’s current price.

To differentiate redemption yield from other yield and interest rate measures described in this book, it will be referred to as rm. Note that this section is concerned with the gross redemption yield, the yield that results from payment of coupons without deduction of any withholding tax. The net redemption yield is what will be received if the bond is traded in a market where bonds pay coupon net, without withholding tax. It is obtained by multiplying the coupon rate C by (1 - marginal tax rate). The net redemption yield is always lower than the gross redemption yield.

The key assumption behind the YTM calculation has already been discussed—that the redemption yield rm remains stable for the entire life of the bond, so that all coupons are reinvested at this same rate. The assumption is unrealistic, however. It can be predicted with virtual certainty that the interest rates paid by instruments with maturities equal to those of the bond at each coupon date will differ from rm at some point, at least, during the life of the bond. In practice, however, investors require a rate of return that is equivalent to the price that they are paying for a bond, and the redemption yield is as good a measurement as any.

A more accurate approach might be the one used to price interest rate swaps: to calculate the present values of future cash flows using discount rates determined by the markets’ view on where interest rates will be at those points. These expected rates are known as forward interest rates. Forward rates, however, are implied, and a YTM derived using them is as speculative as one calculated using the conventional formula. This is because the real market interest rate at any time is invariably different from the one implied earlier in the forward markets. So a YTM calculation made using forward rates would not equal the yield actually realized either. The zero-coupon rate, it will be demonstrated later, is the true interest rate for any term to maturity. Still, despite the limitations imposed by its underlying assumptions, the YTM is the main measure of return used in the markets.

Calculating the redemption yield of bonds that pay semiannual coupons involves the semiannual discounting of those payments. This approach is appropriate for most U.S. bonds and U.K. gilts. Government bonds in most of continental Europe and most Eurobonds, however, pay annual coupon payments. The appropriate method of calculating their redemption yields is to use annual discounting. The two yield measures are not directly comparable.

It is possible to make a Eurobond directly comparable with a U.K. gilt by using semiannual discounting of the former’s annual coupon payments or using annual discounting of the latter’s semiannual payments. The formulas for the semiannual and annual calculations appeared as (1.13) and (1.12), respectively, and are repeated here as (1.22) and (1.23).

(1.22)

(1.23)

Consider a bond with a dirty price—including the accrued interest the seller is entitled to receive—of $97.89, a coupon of 6 percent, and five years to maturity. FIGURE 1.6 shows the gross redemption yields this bond would have under the different yield-calculation conventions.

These figures demonstrate the impact that the coupon-payment and discounting frequencies have on a bond’s redemption yield calculation. Specifically, increasing the frequency of discounting lowers the calculated yield, while increasing the frequency of payments raises it. When comparing yields for bonds that trade in markets with different conventions, it is important to convert all the yields to the same calculation basis.

FIGURE 1.6Yield and Payment Frequency

It might seem that doubling a semiannual yield figure would produce the annualized equivalent; the real result, however, is an underestimate of the true annualized yield. This is because of the multiplicative effects of discounting. The correct procedure for converting semiannual and quarterly yields into annualized ones is shown in (1.24).

EXAMPLE:Comparing Yields to Maturity

A U.S. Treasury paying semiannual coupons, with a maturity of 10 years, has a quoted yield of 4.89 percent. A European government bond with a similar maturity is quoted at a yield of 4.96 percent. Which bond has the higher yield to maturity in practice?

The effective annual yield of the Treasury is

Comparing the securities using the same calculation basis reveals that the European government bond does indeed have the higher yield.

b. Formulas for converting between semiannual and annual yields

c. Formulas for converting between quarterly and annual yields where rmq, rms, and rma are, respectively, the quarterly, semiannually, and annually discounted yields to maturity.

The market convention is sometimes simply to double the semiannual yield to obtain the annualized yields, despite the fact that this produces an inaccurate result. It is only acceptable to do this for rough calculations. An annualized yield obtained in this manner is known as a bond equivalent yield. It was noted earlier that the one disadvantage of the YTM measure is that its calculation incorporates the unrealistic assumption that each coupon payment, as it becomes due, is reinvested at the rate rm. Another disadvantage is that it does not deal with the situation in which investors do not hold their bonds to maturity. In these cases, the redemption yield will not be as great. Investors might therefore be interested in other measures of return, such as the equivalent zero-coupon yield, considered a true yield.

To review, the redemption yield measure assumes that

• the bond is held to maturity

• all coupons during the bond’s life are reinvested at the same (redemption yield) rate

Given these assumptions, the YTM can be viewed as an expected or anticipated yield. It is closest to reality when an investor buys a bond on first issue and holds it to maturity. Even then, however, the actual realized yield at maturity would be different from the YTM because of the unrealistic nature of the second assumption. It is clearly unlikely that all the coupons of any but the shortest-maturity bond will be reinvested at the same rate. As noted earlier, market interest rates are in a state of constant flux, and this would affect money reinvestment rates. Therefore, although yield to maturity is the main market measure of bond levels, it is not a true interest rate. This is an important point. Chapter 2 will explore the concept of a true interest rate.

Another problem with YTM is that it discounts a bond’s coupons at the yield specific to that bond. It thus cannot serve as an accurate basis for comparing bonds. Consider a two-year and a five-year bond. These securities will invariably have different YTMs. Accordingly, the coupon cash flows they generate in two years’ time will be discounted at different rates (assuming the yield curve is not flat). This is clearly not correct. The present value calculated today of a cash flow occurring in two years’ time should be the same whether that cash flow is generated by a short- or a long-dated bond.

Floating Rate Notes

Floating rate notes (FRNs) are bonds that have variable rates of interest; the coupon rate is linked to a specified index and changes periodically as the index changes. An FRN is usually issued with a coupon that pays a fixed spread over a reference index; for example, the coupon may be 50 basis points over the six-month interbank rate. Since the value for the reference benchmark index is not known, it is not possible to calculate the redemption yield for an FRN. The FRN market in countries such as the United States and United Kingdom is large and well-developed; floating-rate bonds are particularly popular with short-term investors and financial institutions such as banks.

The rate against which the FRN coupon is set is known as the reference rate. In the United States market, FRNs frequently set their coupons in line with the Treasury bill rate. The spread over the reference note is called the index spread. The index spread is the number of basis points over the reference rate; in a few cases the index spread is negative, so it is subtracted from the reference rate.