Introduction to Fixed Income Analytics E-Book

Frank J. Fabozzi

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 684

Veröffentlichungsjahr: 2010

Beschreibung

A comprehensive introduction to the key concepts of fixed income analytics

The First Edition of Introduction to Fixed Income Analytics skillfully covered the fundamentals of this discipline and was the first book to feature Bloomberg screens in examples and illustrations. Since publication over eight years ago, the markets have experienced cathartic change.

That's why authors Frank Fabozzi and Steven Mann have returned with a fully updated Second Edition. This reliable resource reflects current economic conditions, and offers additional chapters on relative value analysis, value-at-risk measures and information on instruments like TIPS (treasury inflation protected securities).

• Offers insights into value-at-risk, relative value measures, convertible bond analysis, and much more
• Includes updated charts and descriptions using Bloomberg screens
• Covers important analytical concepts used by portfolio managers

Understanding fixed-income analytics is essential in today's dynamic financial environment. The Second Edition of Introduction to Fixed Income Analytics will help you build a solid foundation in this field.

Leseprobe

Table of Contents

Title Page

Copyright Page

Dedication

Preface

About the Authors

CHAPTER 1 - Time Value of Money

FUTURE VALUE OF A SINGLE CASH FLOW

PRESENT VALUE OF A SINGLE CASH FLOW

COMPOUNDING/DISCOUNTING WHEN INTEREST IS PAID MORE THAN ANNUALLY

FUTURE AND PRESENT VALUES OF AN ORDINARY ANNUITY

YIELD (INTERNAL RATE OF RETURN)

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

APPENDIX: COMPOUNDING AND DISCOUNTING IN CONTINUOUS TIME

QUESTIONS

CHAPTER 2 - Yield Curve Analysis

A BOND IS A PACKAGE OF ZERO-COUPON INSTRUMENTS

THEORETICAL SPOT RATES

FORWARD RATES

DYNAMICS OF THE YIELD CURVE

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 3 - Day Count Conventions and Accrued Interest

DAY COUNT CONVENTIONS

COMPUTING THE ACCRUED INTEREST

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 4 - Valuation of Option-Free Bonds

GENERAL PRINCIPLES OF VALUATION

DETERMINING A BOND’S VALUE

THE PRICE/DISCOUNT RATE RELATIONSHIP

TIME PATH OF BOND

VALUING A ZERO-COUPON BOND

VALUING A BOND BETWEEN COUPON PAYMENTS

TRADITIONAL APPROACH TO VALUATION

THE ARBITRAGE-FREE VALUATION APPROACH

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 5 - Yield Measures

SOURCES OF RETURN

TRADITIONAL YIELD MEASURES

YIELD TO CALL

YIELD TO PUT

YIELD TO WORST

CASH FLOW YIELD

PORTFOLIO YIELD MEASURES

YIELD MEASURES FOR U.S. TREASURY BILLS

YIELD SPREAD MEASURES RELATIVE TO A SPOT RATE CURVE

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

APPENDIX: MATHEMATICS OF THE INTERNAL RATE OF RETURN

QUESTIONS

CHAPTER 6 - Analysis of Floating Rate Securities

GENERAL FEATURES OF FLOATERS

VALUING A RISKY FLOATER

VALUATION OF FLOATERS WITH EMBEDDED OPTIONS

MARGIN MEASURES

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 7 - Valuation of Bonds with Embedded Options

OVERVIEW OF THE VALUATION OF BONDS WITH EMBEDDED OPTIONS

OPTION-ADJUSTED SPREAD AND OPTION COST

LATTICE MODEL

BINOMIAL MODEL

ILLUSTRATION

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 8 - Cash Flow for Mortgage-Backed Securities and Amortizing ...

CASH FLOW OF MORTGAGE-BACKED SECURITIES

AMORTIZING ASSET-BACKED SECURITIES

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 9 - Valuation of Mortgage-Backed and Asset-Backed Securities

STATIC CASH FLOW YIELD ANALYSIS

MONTE CARLO SIMULATION/OAS

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 10 - Analysis of Convertible Bonds

GENERAL CHARACTERISTICS OF CONVERTIBLE BONDS

TOOLS FOR ANALYZING CONVERTIBLES

CALL AND PUT FEATURES

CONVERTIBLE BOND ARBITRAGE

OTHER TYPES OF CONVERTIBLES

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 11 - Total Return

COMPUTING THE TOTAL RETURN

OAS-TOTAL RETURN

TOTAL RETURN TO MATURITY

TOTAL RETURN FOR A MORTGAGE-BACKED SECURITY

PORTFOLIO TOTAL RETURN

TOTAL RETURN ANALYSIS FOR MULTIPLE SCENARIOS

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 12 - Measuring Interest Rate Risk

THE FULL VALUATION APPROACH

PRICE VOLATILITY CHARACTERISTICS OF BONDS

DURATION

OTHER DURATION MEASURES

CONVEXITY

PRICE VALUE OF A BASIS POINT

THE IMPORTANCE OF YIELD VOLATILITY

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 13 - Value-at-Risk Measure and Extensions

VALUE-AT-RISK

CONDITIONAL VALUE-AT-RISK

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 14 - Analysis of Inflation-Protected Bonds

BREAKEVEN INFLATION RATE

VALUATION OF TIPS

MEASURING INTEREST RATE RISK

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 15 - The Tools of Relative Value Analysis

HOW PORTFOLIO MANAGERS ADD VALUE

YIELD SPREADS OVER SWAP AND TREASURY CURVES

ASSET SWAPS

CREDIT DEFAULT SWAPS

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 16 - Analysis of Interest Rate Swaps

DESCRIPTION OF AN INTEREST RATE SWAP

INTERPRETING A SWAP POSITION

TERMINOLOGY, CONVENTIONS, AND MARKET QUOTES

VALUING INTEREST RATE SWAPS

PRIMARY DETERMINANTS OF SWAP SPREADS

DOLLAR DURATION OF A SWAP

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

CHAPTER 17 - Estimating Yield Volatility

HISTORICAL VOLATILITY

IMPLIED VOLATILITY

FORECASTING YIELD VOLATILITY

CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

QUESTIONS

Index

The Frank J. Fabozzi Series

Fixed Income Securities, Second Edition by Frank J. Fabozzi

Focus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. Abate

Handbook of Global Fixed Income Calculations by Dragomir Krgin

Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi

Real Options and Option-Embedded Securities by William T. Moore

Capital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. Fabozzi

The Exchange-Traded Funds Manual by Gary L. Gastineau

Professional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi

Investing in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia Pilarinu

Handbook of Alternative Assets by Mark J. P. Anson

The Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

The Handbook of Financial Instruments edited by Frank J. Fabozzi

Collateralized Debt Obligations: Structures and Analysis by Laurie S. Goodman and Frank J. Fabozzi

Interest Rate, Term Structure, and Valuation Modeling edited by Frank J. Fabozzi

Investment Performance Measurement by Bruce J. Feibel

The Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. Fabozzi

The Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. Markowitz

Foundations of Economic Value Added, Second Edition by James L. Grant

Financial Management and Analysis, Second Edition by Frank J. Fabozzi and Pamela P. Peterson

Measuring and Controlling Interest Rate and Credit Risk, Second Edition by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

Professional Perspectives on Fixed Income Portfolio Management, Volume 4 edited by Frank J. Fabozzi

The Handbook of European Fixed Income Securities edited by Frank J. Fabozzi and Moorad Choudhry

The Handbook of European Structured Financial Products edited by Frank J. Fabozzi and Moorad Choudhry

The Mathematics of Financial Modeling and Investment Management by Sergio M. Focardi and Frank J. Fabozzi

Short Selling: Strategies, Risks, and Rewards edited by Frank J. Fabozzi

The Real Estate Investment Handbook by G. Timothy Haight and Daniel Singer

Market Neutral Strategies edited by Bruce I. Jacobs and Kenneth N. Levy

Securities Finance: Securities Lending and Repurchase Agreements edited by Frank J. Fabozzi and Steven V. Mann

Fat-Tailed and Skewed Asset Return Distributions by Svetlozar T. Rachev, Christian Menn, and Frank J. Fabozzi

Financial Modeling of the Equity Market: From CAPM to Cointegration by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm

Advanced Bond Portfolio Management: Best Practices in Modeling and Strategies edited by Frank J. Fabozzi, Lionel Martellini, and Philippe Priaulet

Analysis of Financial Statements, Second Edition by Pamela P. Peterson and Frank J. Fabozzi

Collateralized Debt Obligations: Structures and Analysis, Second Edition by Douglas J. Lucas, Laurie S. Goodman, and Frank J. Fabozzi

Handbook of Alternative Assets, Second Edition by Mark J. P. Anson

Introduction to Structured Finance by Frank J. Fabozzi, Henry A. Davis, and Moorad Choudhry

Financial Econometrics by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo Jasic

Developments in Collateralized Debt Obligations: New Products and Insights by Douglas J. Lucas, Laurie S. Goodman, Frank J. Fabozzi, and Rebecca J. Manning

Robust Portfolio Optimization and Management by Frank J. Fabozzi, Peter N. Kolm, Dessislava A. Pachamanova, and Sergio M. Focardi

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimizations by Svetlozar T. Rachev, Stogan V. Stoyanov, and Frank J. Fabozzi

How to Select Investment Managers and Evaluate Performance by G. Timothy Haight, Stephen O. Morrell, and Glenn E. Ross

Bayesian Methods in Finance by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. Fabozzi

Structured Products and Related Credit Derivatives by Brian P. Lancaster, Glenn M. Schultz, and Frank J. Fabozzi

Quantitative Equity Investing: Techniques and Strategies by Frank J. Fabozzi, CFA, Sergio M. Focardi, Petter N. Kolm

Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Fabozzi, Frank J.

Includes index.

ISBN 978-0-470-57213-9 (cloth); 978-0-470-92209-5 (ebk); 978-0-470-92210-1 (ebk)

1. Fixed-income securities. 2. Fixed income securities—Mathematics. 3. Rate of return. 4. Risk management. I. Mann, Steven V. II. Title.

HG4650.F335 2010 332.63’2—dc22

2010026721

FJFTo my wife Donna and my children Patricia, Karly, and Francesco

SVMTo my wife, Mary - TDA

Preface

Participants in the fixed income market are inundated with terms and concepts in both the popular press and, more typically, in research reports and professional journal articles. Making life more difficult for professionals in this market sector is the fact that for some important analytical concepts, the same concept is referred to in different ways by different dealer firms and asset management firms. The purpose of this book is to describe the key analytical concepts used in the fixed income market and illustrate how they are computed. The book is not only intended for professionals but also newcomers to the field. It is for this reason that we provide end of chapter questions.

Although market professionals often want a walk through demonstration of how a metric is computed, once they are comfortable with the concept and its computation, professionals then rely on vendors of analytical systems. Probably the most popular system relied upon by fixed income professionals is the Bloomberg System. For this reason, every chapter ties in the analytical concepts that are available on Bloomberg and walks the reader through the relevant Bloomberg screens. We want to thank Bloomberg Financial for granting us permission to reproduce the screens that we used in our exhibits.

We begin the book with an explanation of the most basic concept in finance: the time value of money. In Chapter 2, we describe yield curve analysis, discussing the importance of spot rates and forward rates. The fixed income market has adopted various conventions for determining the number of days when computing accrued interest when trades are settled. These market conventions are the subject of Chapter 3.

The basics of bond valuation are covered in Chapter 4. Our focus in this chapter is on option-free bonds (i.e., bonds that are not callable, putable or convertible) and that have a fixed coupon rate. Yield measures for bonds are covered in Chapter 5.

The analysis of floating rate securities and bonds whose coupon interest is linked to some inflation measure are the subjects of Chapters 6 and 14, respectively. Bonds with embedded options are the subjects of Chapters 7, 9, and 10. Chapter 7 explains how to analyze callable and putable agency and corporate bonds. All residential mortgage-backed securities and certain asset-backed securities grant borrowers a prepayment option and, therefore, these securities have an embedded call option. Chapter 9 explains how these bonds are valued. For those readers unfamiliar with mortgage-backed and asset-backed securities, Chapter 8 explains them and how their cash flows are estimated. Convertible bond valuation is the subject of Chapter 10.

While one often hears about yield measures, portfolio managers are assessed based on their performance, which is measured in terms of total return. Chapter 11 demonstrates the calculation of this measure for individual bonds and portfolios.

A key analytical concept for quantifying and controlling the interest risk of a portfolio or trading position is duration and convexity. These measures of interest rate risk are explained in Chapter 12. One of the limitations of these two measures for use in portfolio risk management is that they assume that if interest rates change, the interest rate for all maturities change by the same amount. This is known as the parallel yield curve shift assumption. An analytical framework for assessing how a portfolio’s value might change if this assumption is relaxed is the calculation of a portfolio’s key rate durations, which is also explained in Chapter 12.

There are other measures used frequently for quantifying a portfolio’s risk exposure. The most popular one is the value-at-risk (VaR) metric. In Chapter 13 we explain not only the reason for the popularity of this metric and alternatives methodologies for calculating it, but the severe limitations of this measure. We explain a superior metric for quantifying risk, conditional VaR.

The approach to bond valuation described in the earlier chapters of the book are based on the discounted cash flow framework. Another approach to valuing bonds for inclusion in a portfolio or positioning for a trade is relative valuation. When properly interpreted, the tools of relative value analysis offer investors some clues about how similar bonds are currently priced in the market on a relative basis. Relative value analysis is the subject of Chapter 15.

An important derivative instrument in the fixed income market for controlling risk is an interest rate swap and is the subject of Chapter 16. After describing a swap’s counterparties, risk-return profile, and economic interpretation, we illustrate how to value it.

As explained in several chapters, a key input into a valuation model is the expected interest rate volatility or expected yield volatility. How this measure is estimated is covered in Chapter 17.

We would like to thank Kimberly Bradshaw for her editorial assistance and Megan Orem for her patience in typesetting this book.

Frank J. Fabozzi Steven V. Mann

About the Authors

Frank J. Fabozzi, Ph.D., CFA, CPA is Professor in the Practice of Finance in the Yale School of Management. Prior to joining the Yale faculty, he was a Visiting Professor of Finance in the Sloan School at MIT. Professor Fabozzi is a Fellow of the International Center for Finance at Yale University and on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University. He is the editor of the Journal of Portfolio Management and an associate editor of the Journal of Fixed Income and the Journal of Structured Finance. He is a trustee for the BlackRock family of closed-end funds. In 2002, Professor Fabozzi was inducted into the Fixed Income Analysts Society’s Hall of Fame and is the 2007 recipient of the C. Stewart Sheppard Award given by the CFA Institute. He has authored numerous books in investment management and structured finance. Professor Fabozzi earned a doctorate in economics from the City University of New York in 1972 and earned the designation of Chartered Financial Analyst and Certified Public Accountant.

Steven V. Mann, Ph.D., is Professor of Finance and Chair at the Moore School of Business, University of South Carolina. He has co-authored four books, edited two books, and written numerous articles in the area of investments, primarily fixed income securities and derivatives. Professor Mann is an accomplished teacher winning over 20 awards for excellence in teaching. He also works as a consultant to investment banks and commercial banks, and has conducted training programs for financial institutions throughout the United States. Professor Mann also serves as an expert witness in legal proceedings on matters involving fixed income securities.

CHAPTER 1

Time Value of Money

A security is a package of cash flows. The cash flows are delivered across time with varying degrees of uncertainty. To value a security, we must determine how much this package of cash flows is worth today. This process employs a fundamental finance principle—the time value of money. Simply stated, one dollar today is worth more than one dollar to be received in the future. The reason is that the money has a time value. One dollar today can be invested, start earning interest immediately, and grow to a larger amount in the future. Conversely, one dollar to be received one year from today is worth less than one dollar delivered today. This is true because an individual can invest an amount of money less than one dollar today and at some interest rate it will grow to one dollar in a year’s time.

The purpose of this chapter is to introduce the fundamental principles of future value (i.e., compounding cash flows) and present value (i.e., discounting cash flows). These principles will be employed in every chapter in the remainder of the book. To be sure, no matter how complicated the security’s cash flows become (e.g., bonds with embedded options, interest rate swaps, etc.), determining how much they are worth today involves taking present values. In addition, we introduce the concept of yield, which is a measure of potential return and explain how to compute the yield on any investment.

FUTURE VALUE OF A SINGLE CASH FLOW

Suppose an individual invests $100 at 5% compounded annually for three years. We call the $100 invested the original principal and denote it as P. In this example, the annual interest rate is 5% and is the compensation the investor receives for giving up the use of his or her money for one year’s time. Intuitively, the interest rate is a bribe offered to induce an individual to postpone their consumption of one dollar until some time in the future. If interest is compounded annually, this means that interest is paid for use of the money only once per year.

So the question at hand is how much $100 will be worth at the end of three years if it earns interest at 5% compounded annually?

In words, if an individual invests $100 that earns 5% compounded annually, at the end of one year the amount invested will grow to $105 (i.e., the original principal of $100 plus $5 interest).

Note that during the second year, we earn $5.25 in interest rather than $5 because we are earning interest on our interest from the first year. This example illustrates an important point about how securities’ returns work; returns reproduce multiplicatively rather than additively.

The future value of $100 invested for three years earning 5% interest compounded annually is $115.7625.

The general formula for the future value of a single cash flow N years in the future given an interest rate i is

(1.1)

As we will see in due course, the longer the investment, the more dramatic the impact of even relatively small changes in interest rates on future values.

PRESENT VALUE OF A SINGLE CASH FLOW

The present value of a single cash flow asks the opposite question. Namely, how much is a single cash flow to be received in the future worth today given a particular interest rate? Suppose the interest rate is 10%, how much is $161.05 to be received five years hence worth today? This question can be easily visualized on the time line presented below:

Alternatively, given the interest rate is 10%, how much would one have to invest today to have $161.05 in five years? The process is called “discounting” because as long as interest rates are positive, the amount invested (the present value) will be less than $161.05 (the future value) because of the time value of money.1

In order to answer the question of how much we would have to invest today at 10% to have $161.05 in five years, we must solve for P

So, the present value of $161.05 delivered five years hence at 10% is $100.

It is easy to see that the mathematics conform to our intuition. When we calculate a future value, we ask how much will the dollars invested today be worth in the future given a particular interest rate. So, the mathematics of future value involve multiplication by a value greater than one (i.e., making things bigger). Correspondingly, when we find present values, we ask how much a future amount of dollars is worth today given a particular interest rate. Thus, the mathematics of present value involve division by a value greater than one (i.e., making things smaller).

The general formula for the present value (PV) of a single cash flow N years in the future given an interest rate i is

(1.2)

Note that we have replaced P with PV. In addition, PV does not have a subscript because we assume it is the value at time 0 (i.e., today).

It is instructive to write the expression for the present value of a single cash flow as follows

The term in brackets is equal to the present value of one dollar to be received N years hence given interest rate i and is often called a discount factor . The present value of a single cash flow is the product of the cash flow to be received (FVN) and the discount factor. Essentially, the discount factor is today’s value of one dollar that is expected to be delivered at some time in the future given a particular interest rate. An analogy will illustrate the point.

Suppose a U.S. investor receives cash payments of $200,000, ¥500,000, and £600,000. How much does the investor receive? We cannot simply add up the cash flows since the three cash flows are denominated in different currencies. In order to determine how much the investor receives, we would convert the three cash flows into a common currency (say, U.S. dollars) using currency exchange rates. Similarly, we cannot value cash flows to be received at different dates in the future merely by taking their sum. The expected cash flows are delivered at different times and are denominated in different “currencies” (Year 1 dollars, Year 2 dollars, etc.). We use discount factors just like exchange rates to convert cash flows to be received across time into a “common currency” called the present value (i.e., Year 0 dollars).

To illustrate this, we return to the last example—what is the present value of $161.05 to be received five years from today given that the interest rate is 10%? The present value can be written as

One dollar to be received in five years is worth $0.6209 today given the interest rate is 10%. We expect to receive $161.05 Year 5 dollars each worth 0.6209 dollars today. The present value is $100, which is the quantity ($161.05) multiplied by the price per unit ($0.6209).

As can be easily seen from the present value expression, the discount factor depends on two things. First, holding the interest rate constant, the longer the time until the cash flow is to be received, the lower the discount factor. To illustrate this, suppose we have $100 to be received 10 years from now and the interest rate is 10%. What is the present value?

Now suppose the cash flow is to be received 20 years hence instead, all else the same. What is the present value?

The discount factor falls 0.3855 to 0.1486. This is simply the time value of money at work. The present value is lower the farther into the future the cash flow will be received.

Why this occurs is apparent from looking at the present value equation. The numerator remains the same and is being divided by a larger number in the denominator as one plus the discount rate is being raised to ever higher powers. This is an important property of the present value: for a given interest rate, the farther into the future a cash flow is received, the lower its present value. Simply put, as cash flows move away from the present, they are worth less to us today. Intuitively, we can invest an even smaller amount now ($14.86) today and it will have more time to grow (20 years versus 10 years) to be equal in size to the payment to be received, $100.

The second factor driving the discount factor is the level of the interest rate. Specifically, holding the time to receipt constant, the discount factor is inversely related to the interest rate. Suppose, once again, we have $100 to be received 10 years from now at 10%. From our previous calculations, we know that the present value is $38.55. Now suppose everything is the same except that the interest rate is 12%. What is the present value when the interest rate increases?

As the interest rate rises from 10% to 12%, the present value of $100 to be received 10 years from today falls from $38.55 to $32.20. The reasoning is equally straightforward. If the amount invested compounds at a faster rate (12% versus 10%), we can invest a smaller amount now ($32.20 versus $38.55) and still have $100 after 10 years.

The relationship between the present value of a single cash flow ($100 to be received 10 years hence) and the level of the interest rate is presented in Exhibit 1.1. For now, there are two things to note about present value/interest rate relationship depicted in the exhibit. First, the relationship is downward sloping. This is simply the inverse relationship between present values and interest rates at work. Second, the relationship is a curve rather than a straight line. In fact, the shape of the curve in Exhibit 1.1 is referred to as convex. By convex, it simply means the curve is “bowed in” relative to the origin.

This second observation raises two questions about the convex or curved shape of the present value/interest rate relationship. First, why is it curved? Second, what is the significance of the curvature? The answer to the first question is mathematical. The answer lies in the denominator of the present value formula. Since we are raising one plus the discount rate to powers greater than one, it should not be surprising that the relationship between the present value and the interest rate is not linear. The answer to the second question requires an entire chapter. Specifically, as we see in Chapter 12, this convexity or bowed shape has implications for the price volatility of a bond when interest rates change. What is important to understand at this point is that the relationship is not linear.

EXHIBIT 1.1 PV/Interest Rate Relationship

COMPOUNDING/DISCOUNTING WHEN INTEREST IS PAID MORE THAN ANNUALLY

An investment may pay interest more frequently than once per year (e.g., semiannually, quarterly, monthly, weekly). If an investment pays interest compounded semiannually, then interest is added to the principal twice a year. To account for this, the future value and present value computations presented above require two simple modifications. First, the annual interest rate is adjusted by dividing by the number of times that interest is paid per year. The adjusted interest rate is called a periodic interest rate. Second, the number of years, N, is replaced with the number of periods, n, which is found by multiplying the number of years by the number of times that interest is paid per year.

Future Value of a Single Cash Flow with More Frequent Compounding

This future value is larger than if interest were compounded annually. With annual compounding, the future value would be $694,746.34. The higher future value when interest is paid semiannually reflects the fact that the interest is being added to principal more frequently, which in turn earns interest sooner.

Present Value of a Single Cash Flow Using Periodic Interest Rates

We must also adjust our present value expression to account for more frequent compounding. The same two adjustments are required. First, like before, we must convert the annual interest rate into a periodic interest rate. Second, we need to convert the number of years until the cash flow is to be received into the appropriate number of periods that matches the compounding frequency.

Plugging this information into the present value expression gives us:

This present value is smaller than if interest were compounded annually. With annual compounding, the present value would be $46,319.35. The lower value when interest is paid semiannually means that for a given annual interest rate we can invest a smaller amount today and still have $100,000 in 10 years with more frequent compounding.

Plugging this information into the present value expression given by equation (1.4) gives

FUTURE AND PRESENT VALUES OF AN ORDINARY ANNUITY

Most securities promise to deliver more than one cash flow. As such, most of the time when we make future/present value calculations, we are working with multiple cash flows. The simplest package of cash flows is called an annuity. An annuity is a series of payments of fixed amounts for a specified number of periods. The specific type of annuity we are dealing with in our applications is an ordinary annuity. The adjective “ordinary” tells us that the annuity payments come at the end of the period and the first payment is one period from now.

Future Value of an Ordinary Annuity

Suppose an investor expects to receive $100 at the end of each of the next three years and the relevant interest rate is 5% compounded annually. This annuity can be visualized on the time line presented below:

What is the future value of this annuity at the end of year 3? Of course, one way to determine this amount is to find the future value of each payment as of the end of year 3 and simply add them up. The first $100 payment will earn 5% interest for two years while the second $100 payment will earn 5% for one year. The third $100 payment is already at the end of the year (i.e., denominated in year 3 dollars) so no adjustment is necessary. Mathematically, the summation of the future values of these three cash flows can be written as:

So, if the investor receives $100 at the end of each of the next three years and can reinvest the cash flows at 5% compounded annually, then at the end of three years the investment will have grown to $315.25.

This expression can be rewritten as follows by factoring out the $100 annuity payment:

$100[(1.05)2 + (1.05)1 + (1.05)0]

The term in brackets is the future value of an ordinary annuity of $1 per year. Multiplying the future value of an ordinary annuity of $1 by the annuity payment produces the future value of an ordinary annuity.

This value agrees with our earlier calculation.

Future Value of an Ordinary Annuity when Payments Occur More Than Once per Year

The future value of an ordinary annuity can be easily generalized to handle situations in which payments are made more than one time per year. For example, instead of assuming an investor receives and then reinvests $100 per year for three years, starting one year from now, suppose that the investor receives $50 every six months for three years, starting six months from now.

The value in brackets is the future value of an ordinary annuity of $1 per period.

Let us return to the example above and assume an annuity of $50 for six semiannual periods. The number line would appear as follows:

Although the total of the cash payments received by the investor over three years is $300 in both examples, the future value is higher ($319.39) when the cash flows are $50 every six months for six periods rather than $100 a year for three years ($315.25). This is true because of the more frequent reinvestment of the payments received by the investor.

Present Value of an Annuity

The coupon payments of a fixed rate bond are an ordinary annuity. Accordingly, in order to value a bond, we must be able to find the present value of an annuity. In this section, we turn our attention to this operation. Suppose we have an ordinary annuity of $300 for three years. These cash flows are pictured on the time line below:

Suppose that the relevant interest rate is 12% compounded annually. What is the present value of this annuity? Of course, we can take the present value of each cash flow individually and then sum them up. The present value is $720.57. To see this, we employ the present value of a single cash flow as follows:

We can rewrite the summation of these present values horizontally as shown below:

This expression can be rewritten by factoring out the $300 annuity payment as follows:

The term in brackets is the present value of an ordinary annuity of $1 for three years at 12%.

This value agrees with our earlier calculation.

Present Value of an Ordinary Annuity when Payments Occur More Than Once per Year

The present value of an ordinary annuity can be generalized to deal with cash payments that occur more frequently than one time per year. For example, instead of assuming an investor receives $300 per year for three years, starting one year from now, suppose instead that the investor receives $150 every six months for three years, starting six months from now.

The value in brackets is the present value of an ordinary annuity of $1 per period.

Let us return to the example above and assume an annuity of $150 for 6 semiannual periods. The time line would appear as follows:

Although the total cash payments received by the investor over three years are $900 in both examples, the present value is higher ($737.60) when the cash flows are $150 every six months for six periods rather than $300 a year for three years ($720.57). This result makes sense because half the cash flows are six months closer when they are received semiannually so their present value should be higher.

Present Value of a Perpetual Annuity

The reason equation (1.9) is so simple can be found in equation (1.8), which is the general formula for the present value of an ordinary annuity of $1 per period. As the number of periods n gets very large, the numerator of the term in brackets in equation (1.8) collapses to 1 because the term 1/(1 + i)n approaches zero producing equation (1.9), which is the present value of the perpetual annuity formula.

Let’s use equation (1.9) to find the present value of a perpetual annuity. Suppose a financial instrument promises to pay $350 per year in perpetuity. The investor requires an annual interest rate of 7% from this investment. What is the present value of this package of cash flows?

Present Value of a Package of Cash Flows with Unequal Interest Rates

To this point in our discussion, we have used the same interest rate to compute present values regardless of when the cash flows were to be delivered in the future. This will not generally be the case in practice. As we see in Chapter 2, the interest rates used to compute present values will depend on, among other things, the shape of the Treasury yield curve. Each cash flow will be discounted back to the present using a unique interest rate. Accordingly, the present value of a package of cash flows is the sum of the present values of each individual cash flow that comprises the package where each present value is computed using a unique interest rate.

As an illustration of this process, consider a 4-year 9% coupon bond with a $1,000 maturity value. Assume, for simplicity, the bond delivers coupon interest payments annually. The bond’s cash flows and required interest rates are shown below:

Years from NowAnnual Cash Payments (in dollars)Required Interest Rate (%)1$906.072906.173906.7041,0906.88

The present value of each cash flow is determined using the appropriate interest rate as shown below:

The present value of the cash flows is $1,074.07777.

Since the process of discounting cash flows with multiple interest rates is so important to our work in later chapters, let’s work through another example. We demonstrate how to find the present value of the fixed rate payments in an interest rate swap. As explained in Chapter 13, in an interest rate swap, two counterparties agree to exchange periodic interest payments. The dollar amount of the interest payments exchanged is based on some notional principal. The dollar amount each counterparty pays to the other is the agreed-upon periodic interest rate multiplied by the notional principal.

To illustrate an interest rate swap, suppose that for the next five years party A agrees to pay party B 10% per year, while party B agrees to pay party A 6-month LIBOR (the reference rate). Party A is a fixed rate payer/ floating rate receiver, while party B is a floating rate payer/fixed rate receiver. Assume the notional principal is $50 million, and that payments are exchanged every six months for the next five years. This means that every six months, party A (the fixed rate payer/floating rate receiver) will pay party B $2.5 million (10% × $50 million × 0.5). The amount that party B (floating rate payer/fixed rate receiver) will be 6-month LIBOR × $50 million × 0.5. For example, if 6-month LIBOR is 7%, party B will pay party A $1.75 (7% × $50 million × 0.5). Note that we multiply by 0.5 because one-half year’s interest is being paid.2

Let’s compute the present value of the fixed rate payments made by party A. As we see in Chapter 2, every cash flow should be discounted using its own interest rate. These interest rates are determined using Eurodollar futures contracts as described in Chapter 13. For now, we take the interest rates as given. The interest rate swap’s fixed rate payments and required semiannual interest rates are shown below:

Periods from NowSemiannual Fixed Rate Payments (in millions of dollars)Required Semiannual Interest Rate (%)1$2.53.0022.53.1532.53.2042.53.3052.53.3862.53.4272.53.4582.53.5092.53.53102.53.54

The present value of this interest rate swap’s fixed rate payments using the appropriate semiannual interest rates is shown below:3

The present value of the fixed rate payments in this interest rate swap is $20.87249 million.

YIELD (INTERNAL RATE OF RETURN)

Yield is a measure of potential return from an investment over a stated time horizon. We discuss several yield measures for both fixed rate and floating rate securities (e.g., yield-to-maturity, yield-to-call, discounted margin, etc.) in later chapters. In this section, we explain how to compute the yield on any investment.

Computing the Yield on Any Investment

The individual terms summed to produce the price are the present values of the cash flow. The yield calculated from the expression above is also termed the internal rate of return.

There is no closed-form expression for determining an investment’s yield given its price (except for investments with only one cash flow). The yield is, therefore, found by an iterative process. The objective is to find the interest rate that will make the present value of the cash flows equal to the price. The procedure is as follows:

Step 1 Select an interest rate.

Step 2 Compute the present value of each cash flow by using the interest rate selected in Step 1.

Step 3 Total the present value of the cash flows found in Step 2.

Step 4 Compare the total present value found in Step 3 with the price of the investment. Then, if the present value of the cash flows found in Step 3 is equal to the price of the investment, the interest rate selected in Step 1 is the yield. If the total present value of the cash flows found in Step 3 is more than the price of the investment, the interest rate selected is not the yield. Go back to Step 1 and use a higher interest rate. If the total present value of the cash flows found in Step 3 is less than the price of the investment, the interest rate used is not the yield. Go back to Step 1 and use a lower interest rate.

We illustrate how these steps are implemented.

Suppose a financial instrument offers the following annual payments for the next five years as displayed in Exhibit 1.2.

Suppose that the price of this financial instrument is $1,084.25. What is the yield or internal rate of return offered by this financial instrument?

EXHIBIT 1.2 Cash Flows from a Financial Instrument

Years from NowAnnual Cash Payments (in dollars)1$8028038048051,080

EXHIBIT 1.3 Present Value at 5%

Years from NowAnnual Cash Payments (in dollars)Present Value of Cash Flow at 5%1$80$76.190528072.562438069.107048065.816251,080846.2083Total Present Value$1,129.88

EXHIBIT 1.4 Present Value at 7%

Years from NowAnnual Cash Payments (in dollars)Present Value of Cash Flow at 7%1$80$74.766428069.875138065.303848061.031651,080770.0251Total Present Value$1,041.00

To compute the yield, we must compute the total present value of these cash flows using different interest rates until we find the one that makes the present value of the cash flows equal to $1,084.25 (the price). Suppose 5% is selected, the calculation is presented in Exhibit 1.3.

The present value using a 5% interest rate exceeds the price of $1,084.25, so a higher interest rate must be tried. If a 7% interest rate is utilized, the present value is $1,041.00 as seen in Exhibit 1.4.

At 7%, the total present value of the cash flows is less than the price of $1,084.25. Accordingly, the present value must be computed with a lower interest rate. The present value at 6% is presented in Exhibit 1.5.

The present value of the cash flows at 6% is equal to the price of the financial instrument when a 6% interest rate is used. Therefore, the yield is 6%.

It is important to bear in mind that the yield computed using equation (1.11) is now the yield for the period. If the cash flows are delivered semiannually, the yield is a semiannual yield. If the cash flows are delivered quarterly, the yield is a quarterly yield, and so forth. The annual rate is determined by multiplying the yield for the period by the number of periods per year (m).

EXHIBIT 1.5 Present Value at 6%

Years from NowAnnual Cash Payments (in dollars)Present Value of Cash Flow at 6%1$80$75.471728071.199738067.169548063.367551,080807.0388Total Present Value$1,041.00

EXHIBIT 1.6 Yield Calculation with Semiannual Cash Flows

Annual Interest Rate (%)Semiannual Interest Rate (%)Total Present Value ($)63.01,035.1073.51,000.0084.0966.3494.5934.04

As an illustration, suppose an investor is considering the purchase of a financial instrument that promises to deliver the following semiannual cash flows:

• Eight payments of $35 every six months for four years

• $1,000 eight semiannual periods from now

Suppose the price of this financial instrument is $934.04. What yield is this financial instrument offering? The yield is calculated via the iterative procedure explained before and the results are summarized in Exhibit 1.6.

When a semiannual rate interest rate of 4.5% is used to compute the total present value of the cash flows, the total present value is equal to the price of $934.04. Therefore, the semiannual yield is 4.5%. Doubling this yield gives an annual yield of 9%.

Yield Calculation When There is Only One Cash Flow

We can solve this expression for y by first dividing both sides by $4,139.25:

The yield on this investment is therefore 6.5%

As an illustration, suppose that a security can be purchased for $71,298.62 today and promises to pay $100,000 five years hence. What is the yield? The answer is 7% and the calculation is detailed below:

Annualizing Yields

Up to this point in our discussion, we have converted periodic interest rates (semiannual, quarterly, monthly, etc.) into annual interest rates by simply multiplying the periodic rate by the frequency of payments per year. For example, we converted a semiannual rate into an annual rate by multiplying it by 2. Similarly, we converted an annual rate into a semiannual rate by dividing it by 2.

Interest is $102.50 on a $1,000 investment and the yield is 10.25% ($102.50/$1,000). The 10.25% is called the effective annual yield.

We can reverse the process and compute the periodic interest rate that will produce a given annual interest rate. For example, suppose we need to know what semiannual interest rate would produce an effective annual yield of 8%. The following formula is employed:

(1.14)

CONCEPTS PRESENTED IN THIS CHAPTER (IN ORDER OF PRESENTATION)

Time value of money Original principal Discount factor Periodic interest rate Annuity Ordinary annuity Future value of an ordinary annuity of $1 per year Present value of an ordinary annuity of $1 per period Perpetual annuity Yield Internal rate of return Effective annual yield

APPENDIX: COMPOUNDING AND DISCOUNTING IN CONTINUOUS TIME

Most valuation models of derivative instruments (futures/forwards, options, swaps, caps, floors) utilize continuous compounding and discounting. Thus, in this section, we develop these important ideas. As we see, although the mathematics are somewhat more involved, the basic principles we have learned to this point are exactly the same.

Normally, when computing present and future values, we assume that interest is added to the principal once each period, where the period may be one year, a month, a day, etc. Consider an extreme example: the future value of $100 one year hence, given a 100% interest rate and annual compounding is $200. This amount represents the present value ($100) plus the interest earned over the year ($100), which is added to the principal at the end of the year.

If the other factors remain unchanged, increasing the frequency with which interest is added to the principal (e.g., semiannually, quarterly, monthly, etc.) increases the future value. The future value of $100 one year hence, given a 100% rate and semiannual compounding is $225. Two steps are required to arrive at this amount. At the end of the first six months, the original $100 grows to $150, which represents the original principal ($100) plus the interest earned ($50) over the first six months at a periodic rate of 50%. The periodic rate is simply the annual rate (100%) divided by two, which is the number of times that interest is paid per year. During the second six months, although the account is still earning interest at a periodic rate of 50%, the principal is now $150. Accordingly, an additional $75 interest is added at the end of the period, bringing the total to $225. We earn $25 more in interest in the second six months (as opposed to the first six months) because our interest is also earning interest at a periodic rate of 50%.

So it goes with compound interest. The sooner interest is added to the principal, the sooner interest is earned on a larger balance at the same periodic rate. Therefore, it is not surprising that as annual periods are divided into even smaller increments of time (e.g., quarterly, monthly, daily, etc.), the future value of our $100 at the end of one year continues to grow.

Exhibit A1 depicts what happens to the future value of $100 one year hence given a 100% interest rate as we increase the number of times per year interest is added to the principal. The vertical axis measures the future value at year end; the horizontal axis measures the frequency of compounding per year. The “1” on the horizontal axis is annual compounding, the “2” semiannual compounding, and so forth to “8760,” which represents compounding interest every hour.

EXHIBIT A1 Future Value of $100 at 100%

The rate of increase in the future value is decreasing as we move from annual compounding ($200) to weekly compounding ($269.26) to hourly compounding ($271.81). As it turns out, no matter how frequently the interest is added to our account (every minute, every second, ...), the future value of $100 one year hence at 100% interest can be no more than $271.83. The amount $271.83 is the future value of $100 at 100% if interest is added to our balance continuously; interest is added to our account at literally each instant of time rather than once per period. The future value of $271.83 is the highest possible, given an interest rate of 100%. A future value computed when interest is compounded continuously represents a natural upper bound, similar to the speed of light.

This exercise usually engenders two questions. First, why is there an upper bound? Second, why is the upper bound $271.83? We consider each in turn.

Let’s answer the first question by appealing to an analogy. Suppose you are going to fill a bathtub with water. You turn the faucet a quarter turn to the left and water begins to pour into the bathtub. This is analogous to how interest is added to the principal when interest is compounded continuously—the water tumbles out in a continuous stream. Suppose you are going to fill the bathtub for four minutes. Even though the water is coming out of the faucet in a continuous stream, the amount of water in the bathtub will only reach a certain level. The only way we can get more water in the bathtub in a given amount of time is to increase the water pressure. Similarly, if we invest $100 for one year, the only way we can achieve a higher future value than $271.83 is to increase the interest rate above 100%.4

The answer to the second question requires a brief mathematical interlude. The future value of $1 when interest is compounded more than once per year is given by (1 + i/m)m where i is the annual interest rate and m is the frequency of compounding. When i is 100% and as m goes to infinity (i.e., continuous compounding), the future value of $1 converges to 2.71828... This number, which is denoted by the letter e in honor of the famous Swiss mathematician Euler, is one of the most important numbers in mathematics. Among its many attributes, e is the base of natural logarithms (i.e., the natural logarithm of e is one).5