IT training simulation and optimization in finance modeling with MATLAB, risk, or VBA pachamanova fabozzi 2010 10 05

They review current simulation and optimization methodologies—along with the available software—and proceed with portfolio risk management, modeling of random processes, pricing of fi na

With Simulation and Optimization in Finance and its

companion Web site, authors Dessislava Pachamanova and Frank Fabozzi explain the application of these tools for both fi nancial professionals and academics in this fi eld.Divided into fi ve comprehensive parts, this reliable guide provides an accessible introduction to the simulation and optimization techniques most widely used in fi nance, while offering fundamental background information on the fi nancial concepts surrounding these techniques

In addition, the authors use simulation and optimization

as a means to clarify diffi cult concepts in traditional risk models in fi nance, and explain how to build fi nancial models with certain software They review current simulation and optimization methodologies—along with the available software—and proceed with portfolio risk management, modeling of random processes, pricing of

fi nancial derivatives, and capital budgeting applications.Designed for practitioners and students, this book:

• Contains a unique combination of fi nance theory and rigorous mathematical modeling emphasizing

a hands-on approach through implementation with software

• Highlights both classical applications and more recent developments such as pricing of mortgage-backed securities

• Includes models and code in both based software (@RISK, Solver, and VBA) and mathematical modeling software (MATLAB)

spreadsheet-• Incorporates a companion Web site containing ancillary materials, including the models and code used in the book, appendices with introductions to the software, and practice sections

• And much more

( c o n t i n u e d o n b a c k f l a p )

DESSISLAVA A PACHAMANOVA, P H D, is an

Associate Professor of Operations Research at Babson

College where she holds the Zwerling Term Chair

She has published a number of articles in operations

research, fi nance, and engineering journals, and

co-authored the Wiley title Robust Portfolio Optimization

and Management Pachamanova’s academic research is

supplemented by consulting and previous work in the

fi nancial industry, including projects with quantitative

strategy groups at WestLB and Goldman Sachs She

holds an AB in mathematics from Princeton University

and a PhD in operations research from the Sloan School of

Management at MIT

FRANK J FABOZZI, P H D, CFA, CPA, is Professor

in the Practice of Finance and Becton Fellow at the

Yale School of Management and Editor of the Journal

of Portfolio Management He is an Affi liated Professor

at the University of Karlsruhe’s Institute of Statistics,

Econometrics, and Mathematical Finance and is on the

Advisory Council for the Department of Operations

Research and Financial Engineering at Princeton

University He earned a doctorate in economics from the

City University of New York

Jacket Image: © Getty Images

Engaging and accessible, this book and its companion Web site provide an introduction to the simulation and optimization techniques most widely used in

fi nance, while, at the same time, offering essential information on the fi nancial concepts surrounding these applications.

This practical guide is divided into fi ve informative parts:

• Part I, Fundamental Concepts, provides insights on the most important

issues in fi nance, simulation, optimization, and optimization under uncertainty

• Part II, Portfolio Optimization and Risk Measures, reviews the theory

and practice of equity and fi xed income portfolio management, from classical frameworks to recent advances in the theory of risk measurement

• Part III, Asset Pricing Models, discusses classical static and dynamic models

for asset pricing, such as factor models and different types of random walks

• Part IV, Derivative Pricing and Use, introduces important types of fi nancial

derivatives, shows how their value can be determined by simulation, and discusses how derivatives can be employed for portfolio risk management and return enhancement purposes

• Part V, Capital Budgeting Decisions, reviews capital budgeting decision

models, including real options, and discusses applications of simulation and optimization in capital budgeting under uncertainty

Supplemented with models and code in both spreadsheet-based software (@RISK,

Solver, and VBA) and mathematical modeling software (MATLAB), Simulation and Optimization in Finance is a well-rounded guide to a dynamic discipline.

Modeling with MATLAB,

@RISK, or VBA

vi

Simulation and Optimization in

Finance

i

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

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

The Handbook of Commodity Investing by Frank J Fabozzi, Roland F ¨uss, and Dieter G Kaiser

The Handbook of Municipal Bonds edited by Sylvan G Feldstein and Frank J Fabozzi

Subprime Mortgage Credit Derivatives by Laurie S Goodman, Shumin Li, Douglas J Lucas, Thomas A Zimmerman,

and Frank J Fabozzi

Introduction to Securitization by Frank J Fabozzi and Vinod Kothari

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

Handbook of Finance: Volume I: Financial Markets and Instruments edited by Frank J Fabozzi

Handbook of Finance: Volume II: Financial Management and Asset Management edited by Frank J Fabozzi

Handbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools edited by Frank J Fabozzi

Finance: Capital Markets, Financial Management, and Investment Management by Frank J Fabozzi and Pamela

Peterson-Drake

Active Private Equity Real Estate Strategy edited by David J Lynn

Foundations and Applications of the Time Value of Money by Pamela Peterson-Drake and Frank J Fabozzi

Leveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives by Stephen Antczak,

Douglas Lucas, and Frank J Fabozzi

Modern Financial Systems: Theory and Applications by Edwin Neave

Institutional Investment Management: Equity and Bond Portfolio Strategies and Applications by Frank J Fabozzi

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

Simulation and Optimization in Finance: Modeling with MATLAB, @RISK, or VBA by Dessislava A Pachamanova and

Frank J Fabozzi

ii

Simulation and Optimization in

Copyright
C 2010 by John Wiley & Sons, Inc 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

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the accuracy or completeness of the contents of this book and specifically disclaim any implied

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visit our Web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Pachamanova, Dessislava A.

Simulation and optimization in finance : modeling with MATLAB, @RISK, or VBA /

Dessislava A Pachamanova, Frank J Fabozzi.

p cm – (Frank J Fabozzi series ; 173) Includes index.

ISBN 978-0-470-37189-3 (cloth); 978-0-470-88211-5 (ebk);

Dessislava A Pachamanova

To my husband, Christian, and my children,

Anna and Coleman Frank J Fabozzi

To my wife, Donna, and my children, Patricia,

Karly, and Francesco

v

vi

Basic Theory of Interest; Asset Classes; Basic TradingTerminology; Calculating Rate of Return; Valuation;

Important Concepts in Fixed Income; Summary; Notes

CHAPTER 3

Random Variables, Probability Distributions, and

What is a Probability Distribution?; BernoulliProbability Distribution and Probability MassFunctions; Binomial Probability Distribution andDiscrete Distributions; Normal Distribution andProbability Density Functions; Concept of CumulativeProbability; Describing Distributions; Brief Overview

of Some Important Probability Distributions;

Dependence Between Two Random Variables:

Covariance and Correlation; Sums of RandomVariables; Joint Probability Distributions andConditional Probability; From Probability Theory toStatistical Measurement: Probability Distributions andSampling; Summary; Software Hints; Notes

vii

CHAPTER 4

Monte Carlo Simulation: A Simple Example; Why UseSimulation?; Important Questions in SimulationModeling; Random Number Generation; Summary;

Software Hints; Notes

CHAPTER 5

Optimization Formulations; Important Types ofOptimization Problems; Optimization ProblemFormulation Examples; Optimization Algorithms;

Optimization Duality; Multistage Optimization;

Optimization Software; Summary; Software Hints; Notes

CHAPTER 6

Dynamic Programming; Stochastic Programming;

Robust Optimization; Summary; Notes

PART TWO

Portfolio Optimization and Risk Measures

CHAPTER 7

The Case for Diversification; The ClassicalMean-Variance Optimization Framework; EfficientFrontiers; Alternative Formulations of the ClassicalMean-Variance Optimization Problem; The CapitalMarket Line; Expected Utility Theory; Summary;

Software Hints; Notes

CHAPTER 8

Classes of Risk Measures; Value-At-Risk; ConditionalValue-At-Risk and the Concept of Coherent RiskMeasures; Summary; Software Hints; Notes

CHAPTER 9

The Investment Process; Portfolio ConstraintsCommonly Used in Practice; Benchmark Exposure andTracking Error Minimization; Incorporating

Transaction Costs; Incorporating Taxes; MultiaccountOptimization; Robust Parameter Estimation; PortfolioResampling; Robust Portfolio Optimization; Summary;

Software Hints; Notes

CHAPTER 10

Measuring Bond Portfolio Risk; The Spectrum of BondPortfolio Management Strategies; Liability-DrivenStrategies; Summary; Notes

Binomial Trees; Arithmetic Random Walks; GeometricRandom Walks; Mean Reversion; Advanced RandomWalk Models; Stochastic Processes; Summary;

Software Hints; Notes

Software Hints; Notes

CHAPTER 14

Computing Option Prices with Crude Monte CarloSimulation; Variance Reduction Techniques;

Quasirandom Number Sequences; More SimulationApplication Examples; Summary; Software Hints; Notes

CHAPTER 15

Types of Asset-Backed Securities; Mortgage-BackedSecurities: Important Terminology; Types of RMBSStructures; Pricing RMBS by Simulation; UsingSimulation to Estimate Sensitivity of RMBS Prices toDifferent Factors; Structuring RMBS Deals UsingDynamic Programming; Summary; Notes

CHAPTER 16

Using Derivatives in Equity Portfolio Management;

Using Derivatives in Bond Portfolio Management;

Using Futures to Implement an Asset AllocationDecision; Measuring Portfolio Risk When the PortfolioContains Derivatives; Summary; Notes

PART FIVE

Capital Budgeting Decisions

CHAPTER 17

Classifying Investment Projects; Investment Decisionsand Wealth Maximization; Evaluating Project Risk;

Case Study; Managing Portfolios of Projects; Summary;

Software Hints; Notes

CHAPTER 18

Types of Real Options; Real Options and FinancialOptions; New View of NPV; Option to Expand;

Option to Abandon; More Real Options Examples;

Estimation of Inputs for Real Option ValuationModels; Summary; Software Hints; Notes

Simulation and Optimization in Finance: Modeling with MATLAB,

@RISK, or VBA is an introduction to two quantitative modeling tools—

simulation and optimization—and their applications in financial risk

man-agement In addition to laying a solid theoretical foundation and discussing

the practical implications of applying simulation and optimization

tech-niques, the book uses simulation and optimization as a means to clarify

difficult concepts in traditional risk models in finance, and explains how to

build financial models with software The book covers a wide range of

ap-plications and is written in a theoretically rigorous way, which will make it

of interest to both practitioners and academics It can be used as a self-study

aid by finance practitioners and students who have some fundamental

back-ground in calculus and statistics, or as a textbook in finance and quantitative

methods courses In addition, this book is accompanied by a web site where

readers can go to download an array of supplementary materials Please

see the “Companion Web Site” section toward the end of this Preface for

more details

C E N T R A L T H E M E S

Simulation and Optimization in Finance contains 18 chapters in five parts.

Part One, Fundamental Concepts, provides background on the most

impor-tant finance, simulation, optimization, and optimization under uncertainty

concepts that are necessary to understand the financial applications in later

parts of the book Part Two, Portfolio Optimization and Risk Measures,

reviews the theory and practice of equity and fixed income portfolio

man-agement, from classical frameworks, such as mean-variance optimization,

to recent advances in the theory of risk measurement, such as value-at-risk

and conditional value-at-risk estimation Part Three, Asset Pricing Models,

discusses classical static and dynamic models for asset pricing, such as factor

models and different types of random walks Part Four, Derivative Pricing

and Use, introduces important types of financial derivatives, shows how

their value can be determined by simulation, reviews advanced simulation

xi

methods for efficient implementation of pricing algorithms, and discusses

how derivatives can be employed for portfolio risk management and return

enhancement purposes Part Five, Capital Budgeting Decisions, reviews

cap-ital budgeting decision models, including real options, and discusses

applica-tions of simulation and optimization in capital budgeting under uncertainty

It is important to note that there often are multiple numerical methods

that can be used to handle a particular problem in finance Many of the

topics listed here, especially asset and derivative pricing models, however,

have traditionally been out of reach for readers without advanced degrees in

mathematics because understanding the theory behind the models and the

advanced methods for modeling requires years of training Simulation and

optimization formulations provide a framework within which very

challeng-ing concepts can be explained through simple visualization and hands-on

implementation, which makes the material accessible to readers with little

background in advanced mathematics

S O F T W A R E

In our experience, teaching and learning cannot be effective without

exam-ples and hands-on implementation Most of the chapters in this book have

“Software Hints” sections that explain how to use the applications under

discussion The examples themselves are posted on the companion web site

discussed later in the Preface

In Simulation and Optimization in Finance, we assume basic

familiar-ity with spreadsheets and Microsoft Excel, and use two different platforms

to implement concepts and algorithms: the Palisade Decision Tools Suite

and other Excel-based software (@RISK1, Solver2, VBA3), and MATLAB4

Readers do not need to learn both; they can choose one or the other,

depend-ing on their level of familiarity and comfort with spreadsheet programs and

their add-ins versus programming environments such as MATLAB

Specifi-cally, users with finance and social science backgrounds typically prefer an

Excel-based implementation, whereas users with engineering and

quanti-tative backgrounds prefer MATLAB Some tasks and implementations are

easier in one environment than in the other, and students who have used this

book in the form of lecture notes in the past have felt they benefitted from

learning about both platforms Basic introductions to the software used in

the book are provided in Appendices B through D, which can be accessed at

the companion web site

Although Excel and other programs are used extensively in this book,

we were wary of turning it into a software tutorial Our goal was to

com-bine concepts and tools for implementing them in an effective manner

without necessarily covering every aspect of working in a specific software

environment

We have, of course, attempted to implement all examples correctly

That said, the code is provided “as is” and is intended only to illustrate

the concepts in this book Readers who use the code for financial decision

making are doing so at their own risk For full information on the terms

of use of the code, please see the licensing information in each file on the

companion web site

The following web sites provide useful information about Palisade

De-cision Tools Suite and MATLAB Readers can download trial versions or

purchase the software

Palisade Decision Tools Suite, http://www.palisade.com

MATLAB, http://www.mathworks.com

T E A C H I N G

Simulation and Optimization in Finance: Modeling with MATLAB, @RISK,

or VBA covers finance and applied quantitative methods theory, as well as

a wide range of applications It can be used as a textbook for upper-level

undergraduate or lower-level graduate (such as MBA or Master’s) courses

in applied quantitative methods, operations research, decision sciences, or

financial engineering, finance courses in derivatives, investments or

corpo-rate finance with an emphasis on modeling, or as a supplement in a special

topics course in quantitative methods or finance In addition, the book can

be used as a self-study aid by students, or serve as a reference for student

projects

The book assumes that the reader has no background in finance or

ad-vanced quantitative methods except for basic calculus and statistics Most

quantitative concepts necessary for understanding the notation or

applica-tions are introduced and explained in endnotes, software hints, and online

appendices This makes the book suitable for readers with a wide range of

backgrounds and particularly so as a textbook for classes with mixed

audi-ences (such as engineering and business students) In fact, the idea for this

book project matured after years of searching for an appropriate text for a

course with a mixed audience that needed a good reference for both finance

and quantitative methods topics

Every chapter follows the same basic outline The concepts are

intro-duced in the main body of the chapter, and illustrations are provided At

the end of each chapter, there is a summary that contains the most

impor-tant discussion points A Software Hints section provides instructions and

code for implementing the examples in the chapter with both Excel-based

software and MATLAB

On the companion web site, there are practice sections for selected

chapters These sections feature examples that complement those found

in their respective chapters Some practice sections contain cases as well

The cases are more in-depth exercises that focus on a particular practical

application not necessarily covered in the chapter, but possible to address

with the tools introduced in that chapter

We recommend that before proceeding with the main body of this book,

readers consult the four appendices on the companion web site, namely

Appendix A, Basic Linear Algebra Concepts; Appendix B, Introduction to

@RISK; Appendix C, Introduction to MATLAB; and Appendix D,

Intro-duction to Visual Basic for Applications They provide background on basic

mathematical and programming concepts that enable readers to understand

the implementation and the code provided in the Software Hints sections

The chapters that introduce fundamental concepts all contain code that

can be found on the companion web site Some more advanced chapters do

not; the idea is that at that point students are sufficiently familiar with the

applications and models to put together examples on their own based on the

code provided in previous chapters The material in the advanced chapters

can be used also as templates for student course projects

A typical course may start with the material in Chapters 2 through 6

It can then cover the material in Chapters 7 through 9, which focus on

applications of optimization for single-period optimal portfolio allocation

and risk management The course then proceeds with Chapters 11 through

14, which introduce static and dynamic asset pricing models through

sim-ulation as well as derivative pricing by simsim-ulation, and ends with Chapters

17 and 18, which discuss applications of simulation and optimization in

capital budgeting Chapters 10, 15, and 16 represent good assignments for

final projects because they use concepts similar to other chapters, but in a

different context and without as much implementation detail

Depending on the nature of the course, only some of Chapters 2 through

6 will need to be covered explicitly; but the information in these chapters is

useful in case the instructor would like to assign the chapters as reading for

students who lack some of the necessary background for the course

C O M P A N I O N W E B S I T E

Additional material for Simulation and Optimization in Finance can be

downloaded by visiting www.wiley.com/go/pachamanova Please log in to

the web site using this password: finance123 The files on this companion

web site are organized in the following folders: Appendices, Code, and

Practice The Appendices directory contains Appendix A through D The

Practice directory contains practice problems and cases indexed by chapter

(Practice problems are present for Chapters 4–16, 18, and Appendix D, as a

bonus to the content in the book Please note, however, that only problems

are offered without solutions.) The Code directory has Excel and MATLAB

subdirectories that contain files for use with the corresponding software

The latter files are referenced in the main body of the book and the Software

Hints sections for selected chapters

The companion web site is a great resource for readers interested in

actually implementing the concepts in the book Such readers should begin

by reading the applicable appendix on the companion web site with

infor-mation about the software they intend to use, then read the main body of a

chapter, the chapter’s Software Hints, and, finally, the Excel model files or

MATLAB code in the code directory on the companion web site

N O T E S

1 An Excel add-in for simulation.

2 An Excel add-in for optimization that comes standard with Excel.

3 Visual Basic for Applications—a programming language that can be

used to automate tasks in Excel

4 A programming environment for mathematical and engineering

appli-cations that provides users with tools for number array manipulation,

statistical estimation, simulation, optimization, and others

About the Authors

Dessislava A Pachamanova is an Associate Professor of Operations

Re-search at Babson College where she holds the Zwerling Term Chair Her

research interests lie in the areas of portfolio risk management, simulation,

high-performance optimization, and financial engineering She has published

a number of articles in operations research, finance, and engineering

jour-nals, and coauthored the Wiley title Robust Portfolio Optimization and

Management (2007) Dessislava’s academic research is supplemented by

consulting and previous work in the financial industry, including projects

with quantitative strategy groups at WestLB and Goldman Sachs She holds

an AB in mathematics from Princeton University and a PhD in operations

research from the Sloan School of Management at MIT

Frank J Fabozzi is Professor in the Practice of Finance in the School of

Management at Yale University Prior to joining the Yale faculty, he was

a Visiting Professor of Finance in the Sloan School at MIT Frank is a

Fel-low 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

Port-folio Management and an associate editor of the Journal of Fixed Income.

He earned a doctorate in economics from the City University of New York

in 1972 In 2002 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 earned the designation of Chartered Financial

Ana-lyst and Certified Public Accountant He has authored and edited numerous

books in finance

xvi

In writing a book that covers such a wide range of topics in simulation,

optimization, and finance, we were fortunate to have received valuable

help from a number of individuals The following people have commented

on chapters or sections of chapters or provided helpful references and

intro-ductions:

Anthony Corr, Brett McElwee, and Max Capetta of Continuum Capital

Management

Nalan Gulpinar of the University of Warwick Business School

Craig Stephenson of Babson College

Hugh Crowther of Crowther Investment, LLC

Bruce Collins of Western Connecticut State University

Pamela Drake of James Madison University

Zack Coburn implemented the VBA code for the Software Hints

sec-tions in Chapters 7 and 14 Christian Hicks helped with writing and testing

some of the VBA code in the book, such as the VBA implementation of the

American option pricing model with least squares in Chapter 14 Professor

Mark Potter of Babson College allowed us to modify his case, “Reebok

International: Strategic Asset Allocation,” for use as an example in Chapter

17, and some of the ideas are based on case spreadsheet models further

de-veloped by Kathy Hevert and Richard Bliss of Babson College Some of the

cases and examples in the book are based on ideas and research by Thomas

Malloy, Michael Allietta, Adam Bergenfield, Nick Kyprianou, Jason

Aron-son, and Rohan Duggal The real estate valuation project example in section

18.6.3 in Chapter 18 is based on ideas by Matt Bujnicki, Matt Enright, and

Alec Kyprianou

We would also like to thank Wendy Gudgeon and Stan Brown from

Palisade Software and Steve Wilcockson, Naomi Fernandes, Meg Vulliez,

Chris Watson, and Srikanth Krishnamurthy of Mathworks for their help

with obtaining most recent versions of the software used in the book and

for additional materials useful for implementing some of the examples

DESSISLAVAA PACHAMANOVA

FRANKJ FABOZZI

xvii

xviii

CHAPTER 1 Introduction

Finance is the application of economic principles to decision making, and

involves the allocation of money under conditions of uncertainty

In-vestors allocate their funds among financial assets in order to accomplish

their objectives Business entities and government at all levels raise funds by

issuing claims in the form of debt (e.g., loans and bonds) or equity (e.g.,

common stock) and, in turn, invest those funds Finance provides the

frame-work for making decisions as to how those funds should be obtained and

then invested

The field of finance has three specialty areas: (1) capital markets and

capital market theory, (2) financial management, and (3) portfolio

man-agement The specialty field of capital markets and capital market theory

focuses on the study of the financial system, the structure of interest rates,

and the pricing of risky assets Financial management, sometimes called

business finance, is the specialty area of finance concerned with financial

de-cision making within a business entity Although we often refer to financial

management as corporate finance, the principles of financial management

also apply to other forms of business and to government entities Moreover,

not all nongovernment business enterprises are corporations Financial

man-agers are primarily concerned with investment decisions and financing

deci-sions within business Making investment decideci-sions that involve long-term

capital expenditures is called capital budgeting Portfolio management deals

with the management of individual or institutional funds This specialty

area of finance—also commonly referred to as investment management,

as-set management, and money management—involves selecting an investment

strategy and then selecting the specific assets to be included in a portfolio

A critical element common to all three specialty areas in finance is the

concept of risk Measuring and quantifying risk is critical for the fair

val-uation of an asset, the selection of capital budgeting projects in financial

management, the selection of individual asset holdings, and portfolio

con-struction in portfolio management The field of risk management includes

1

the identification, measurement, and control of risk in a business entity or

a portfolio

Sophisticated mathematical tools have been employed in order to deal

with the risks associated with individual assets, capital budgeting projects,

and selecting assets in portfolio construction The use of such tools is now

commonplace in the financial industry For example, in portfolio

man-agement, practitioners run statistical routines to identify risk factors that

drive asset returns, scenario analyses to evaluate the risk of their

posi-tions, and algorithms to find the optimal way to allocate assets or execute

a trade

This book focuses on two quantitative tools—optimization and

simula-tion—and discusses their applications in finance In this chapter, we briefly

introduce these two techniques, and provide an overview of the structure of

the book

O P T I M I Z A T I O N

Optimization is an area in applied mathematics that, most generally, deals

with efficient algorithms for finding an optimal solution among a set of

solutions that satisfy given constraints The first application of optimization

in finance was suggested by Harry Markowitz in 1952, in a seminal paper

that outlined his mean-variance optimization framework for optimal asset

allocation Some other classical problems in finance that can be solved by

optimization algorithms include:

Is there a possibility to make riskless profit given market prices of related

securities? (This opportunity is called an arbitrage opportunity and is

discussed in Chapter 13.)

How should trades be executed so as to reach a target allocation with

minimum transaction costs?

Given a limited capital budget, which capital budgeting projects should

be selected?

Given estimates for the costs and benefits of a multistage capital

budget-ing project, at what stage should the project be expanded/abandoned?

Traditional optimization modeling assumes that the inputs to the

algo-rithms are certain, but there is also a branch of optimization that studies the

optimal decision under uncertainty about the parameters of the problem

Fast and reliable algorithms exist for many classes of optimization

prob-lems, and advances in computing power have made optimization techniques

a viable and useful part of the standard toolset of the financial modeler

S I M U L A T I O N

Simulation is a technique for replicating uncertain processes, and evaluating

decisions under uncertain conditions Perhaps the earliest application of

simulation in finance was in financial management Hertz (1964) argued that

traditional valuation methods for investments omitted from consideration an

important component: the fact that many of the inputs were inaccurate He

suggested modeling the uncertainty through probability-weighted scenarios,

which would allow for obtaining a range of outcomes for the value of the

investments and associated probabilities for each outcome These ideas were

forgotten for a while, but have experienced tremendous growth in the last

two decades Simulation is now used not only in financial management,

but also in risk management and pricing of different financial instruments

In portfolio management, for example, the correlated behavior of different

factors over time is simulated in order to estimate measures of portfolio

risk In pricing financial options or complex securities, such as

mortgage-backed securities, paths for the underlying risk factors are simulated; and

the fair price of the securities is estimated as the average of the discounted

payoffs over those paths We will see numerous examples of such simulation

applications in this book

Simulation bears some resemblance to an intuitive tool for modifying

original assumptions in financial models—what-if analysis—which has been

used for a long time in financial applications In what-if analysis, each

un-certain input in a model is assigned a range of possible values—typically,

best, worst, and most likely value—and the modeler analyzes what happens

to the decision under these scenarios The important additional component

in simulation modeling, however, is that there are probabilities associated

with the different outcomes This allows for obtaining an additional piece of

information compared to what-if analysis: the probabilities that specific final

outcomes will happen Probability theory is so fundamental to

understand-ing the nature of simulation analysis, that we include a chapter (Chapter 3)

on the most important aspects of probability theory that are relevant for

simulation modeling

O U T L I N E O F T O P I C S

The book is organized as follows Part One (Chapters 2 through 6)

pro-vides a background on the fundamental concepts used in the rest of the

book Part Two (Chapters 7 through 10) introduces the classical

under-pinnings of modern portfolio theory, and discusses the role of simulation

and optimization in recent developments Part Three (Chapters 11 and 12)

summarizes important models for asset pricing and asset price dynamics.

Understanding how to implement these models is a prerequisite for the

ma-terial in Part Four (Chapters 13 through 16), which deals with the pricing of

financial derivatives, mortgage-backed securities, advanced portfolio

man-agement, and advanced simulation methods Part Five (Chapters 17 and 18)

discusses applications of simulation and optimization in capital budgeting

and real option valuation The four appendices (on the companion web site)

feature introductions to linear algebra concepts, @RISK, MATLAB, and

Visual Basic for Applications in Microsoft Excel

We begin by listing important finance terminology in Chapter 2 This

includes basic theory of interest; terminology associated with equities, fixed

income securities, and trading; calculation of rate of return; and useful

concepts in fixed income, such as spot rates, forward rates, yield, duration,

and convexity

Chapter 3 is an introduction to probability theory, distributions, and

basic statistics We review important probability distributions, such as the

normal distribution and the binomial distribution, measures of central

ten-dency and variability, and measures of strength of codependence between

random variables Understanding these concepts is paramount to

under-standing the simulation models discussed in the book

Chapter 4 introduces simulation as a methodology We discuss

deter-mining inputs for and interpreting output from simulation models, and

explain the methodology behind generating random numbers from

differ-ent probability distributions We also touch upon recdiffer-ent developmdiffer-ents in

efficient random number generation, which provides the foundation for the

advanced simulation methods for financial derivative pricing discussed in

Part Four of the book

In Chapter 5 we provide a practical introduction to optimization We

discuss the most commonly encountered types of optimization problems in

finance, and elaborate on the concept of “difficult” versus “easy”

optimiza-tion problems We introduce optimizaoptimiza-tion duality and describe intuitively

how optimization algorithms work Illustrations of simple finance problems

that can be handled with optimization techniques are provided, including

examples of optimal portfolio allocation and cash flow matching from the

field of portfolio management, and capital budgeting from the field of

fi-nancial management We also discuss dynamic programming—a technique

for solving optimization problems over multiple stages Multistage

opti-mization is used in Chapters 13 and 18 Finally, we review available

soft-ware for different types of optimization problems and portfolio optimization

in particular

Classical optimization methods treat the parameters in optimization

problems as deterministic and accurate In reality, however, these

param-eters are typically estimated through error-prone statistical procedures or

based on subjective evaluation, resulting in estimates with significant

estima-tion errors The output of optimizaestima-tion routines based on poorly estimated

inputs can be at best useless and at worst seriously misleading It is

impor-tant to know how to treat uncertainty in the estimates of input parameters

in optimization problems Chapter 6 provides a taxonomy of methods for

optimization under uncertainty We review the main ideas behind dynamic

programming under uncertainty, stochastic programming, and robust

opti-mization, and illustrate the methods with examples We will encounter these

methods in applications in Chapters 9, 13, 14, and 18

Chapter 7 uses the concept of optimization to introduce the

mean-variance framework that is the foundation of modern portfolio theory

We also present an alternative framework for optimal decision making in

investments—expected utility maximization—and explain its relationship to

mean-variance optimization

Chapter 8 extends the classical mean-variance portfolio optimization

theory to a more general mean-risk setting We cover the most commonly

used alternative risk measures that are generally better suited than

vari-ance for describing investor preferences when asset return distributions are

skewed or fat-tailed We focus on two popular portfolio risk measures—

value-at-risk and conditional value-at-risk—and show how to estimate them

using simulation We also formulate the problems of optimal asset allocation

under these risk measures using optimization

Chapter 9 provides an overview of practical considerations in

imple-menting portfolio optimization We review constraints that are most

com-monly faced by portfolio managers, and show how to formulate them as part

of optimization problems We also show how the classical framework for

portfolio allocation can be extended to include transaction costs, and discuss

index tracking, optimization of trades across multiple client accounts, and

robust portfolio optimization techniques to minimize estimation error

While Chapter 9 focuses mostly on equity portfolio management,

Chapter 10 discusses the specificities of fixed income (bond) portfolio

man-agement Many of the same concepts are used in equity and fixed income

portfolio management (which are defined in Chapter 2); however, fixed

in-come securities have some fundamental differences from equities, so the

concepts cannot always be applied in the same way in which they would be

applied for stock portfolios We review classical measures of bond portfolio

risk, such as duration, key rate duration, and spread duration We discuss

bond portfolio optimization relative to a benchmark index We also give

examples of how optimization can be used in liability-driven bond portfolio

strategies such as immunization and cash flow matching

Chapter 11 transitions from the topic of portfolio management to the

topic of asset pricing, and introduces standard financial models for

explain-ing asset returns—the Capital Asset Pricexplain-ing Model (CAPM), which is based

on the mean-variance framework described in Chapter 7, the Arbitrage

Pricing Theory (APT), and factor models Such models are widely used in

portfolio management—they not only help to model the processes that drive

asset prices, but also substantially reduce the computational burden for

sta-tistical estimation and asset allocation optimization algorithms

Chapter 12 focuses on dynamic asset pricing models, which are based

on random processes We examine the most commonly used types of

ran-dom walks, and illustrate their behavior through simulation The models

discussed include arithmetic, geometric, different types of mean-reverting

random walks, and more advanced hybrid models In our presentation in

the chapter, we assume that changes in asset prices happen at discrete time

intervals At the end of the chapter, we extend the concept of a random walk

to a random process in continuous time

The concepts introduced in Chapter 12 are reused multiple times when

we discuss valuation of complex securities and multistage investments in

Parts Four and Five of the book The first chapter in Part Four, Chapter 13,

is an introduction to the topic of financial derivatives It lists the main classes

of financial derivative contracts (futures and forwards, options, and swaps),

explains the important concepts of arbitrage and hedging, and reviews

clas-sical methods for pricing derivatives, such as the Black-Scholes formula and

binomial trees

Chapter 14 builds on the material in Chapter 13, but focuses mainly on

the use of simulation for pricing complex securities Some of the closed-form

formulas provided in Chapter 12 and the assumptions behind them become

more intuitive when illustrated through simulation of the random processes

followed by the underlying securities A large part of the chapter is dedicated

to variance reduction techniques, such as antithetic variables, stratified

sam-pling, importance samsam-pling, and control variates, as well as quasi–Monte

Carlo methods Such techniques are widely used today for efficient

imple-mentation of simulations for pricing securities and estimating sensitivity to

different market factors We provide specific examples of these techniques,

and detailed VBA and MATLAB code to illustrate their implementation

The numerical pricing methods in Chapter 15 are based on similar

techniques to the ones discussed in Chapter 14, but the context is different

We introduce a complex type of fixed-income securities—mortgage-backed

securities—and discuss in detail a part of the simulation that is specific to

fixed-income securities—generating scenarios for future interest rates and

the entire yield curve

Chapter 16 builds on Chapters 7, 8, 9, 13, and 14, and contains a

discussion of how derivatives can be used for portfolio risk management

and return enhancement strategies Simulation is essential for estimating

the risk of a portfolio that contains complex financial instruments, but the

process can be very slow in the case of large portfolios We highlight some

numerical issues, standard simulation algorithms, and review methods that

have been suggested for reducing the computational burden

Chapters 17 and 18 cover a different area of finance—financial

manage-ment—but they provide useful illustrations for the difference applying

simulation and optimization makes in classical finance decision-making

frameworks Chapter 17 begins with a review of so-called discounted cash

flow (DCF) methodologies for evaluating company investment projects It

then discusses (through a case study) how simulation can be used to estimate

stand-alone risk and enhance the analysis of such projects

Chapter 18 introduces the real options framework, which advocates

for accounting for existing options in project valuation (The DCF analysis

ignores the potential flexibility in projects—it assumes that there will be no

changes once a decision is made.) While determining the inputs for

valu-ation of real options presents significant challenges, the actual techniques

for pricing these real options are based on the techniques for pricing

finan-cial options introduced in Chapters 13 and 14 Simulation and multistage

optimization can again be used as valuable tools in this new context

8

One Fundamental Concepts

9

10

CHAPTER 2 Important Finance Concepts

This chapter reviews important finance concepts that are used throughout

the book We discuss the concepts of the time value of money, different

asset classes, basic trading terminology, calculation of rate of return,

valu-ation, and advanced concepts in fixed income, such as durvalu-ation, convexity,

key rate duration, and total return

One of the most fundamental concepts in finance is the concept of the time

value of money A specific amount of money received today does not have

the same nominal value in the future because of the possibility of investing

the money today and earning interest This section explains the rules for

computing interest, and outlines the basic elements of dealing with cash flows

obtained today and in the future These concepts will reappear many times

throughout the book—they are critical for pricing financial instruments and

making investment decisions

2 1 1 C o m p o u n d I n t e r e s t

Most bank accounts, loans, and investments interest calculations utilize

some form of compounding Simply put, compound interest involves interest

on interest Let us explain the concept with an example If you deposit $100

in a bank deposit that pays 3% per year, at the end of the year you will have

$103 Suppose you keep the money in the bank for a second year, again at

3% interest Compound interest means that the interest during the second

year will be accrued on the entire amount you have in the bank at the end

of the first year—not only on your original deposit of $100, but also on the

interest accrued during the first year Therefore, at the end of the second

year you will have

$103+ 0.03 · $103 = $106.09

11

If there was no compounding, you would have an additional $3 at the

end of the first year, and again at the end of the second year, that is, the total

amount in your account at the end of the second year would be $106.00 In

general, the formula for computing the future value of an initial capital C

invested for n years at interest rate r per year (compounded annually) is

C · (1 + r) n

In our example, computing the interest with and without compounding

made a difference of 9 cents The effect of compounding on the investment,

however, can be substantial, especially over a long time horizon For

ex-ample, you can verify that if you invest $C at an interest rate of 7% per

year with annual compounding, your investment will double in size in

ap-proximately 10 years This increase is significantly larger than if interest

is not compounded, that is, if you simply add the interest on the original

investment over the 10 years (The latter would be 10· 0.07 = 0.70, or 70%

increase in the original investment.)

Interest does not necessarily need to be compounded once per year—it

can be compounded daily, monthly, quarterly, continuously Usually,

how-ever, the interest rate r is still quoted as an annual rate For example, with

quarterly compounding, an interest of r/4 is accrued each quarter on the

amount at the beginning of the quarter At the end of the first quarter, the

original amount C grows to C · (1 + r/4) At the end of the second quarter,

the amount becomes (C · (1 + r/4)·(1 + r/4)) At the end of the first year,

the total amount in the account is C · (1 + r/4)4 After n years, $C of initial

capital grows to

C · (1 + r/4)4· n

In general, if the frequency of compounding is m times per year at an

annual (called nominal) rate r, the amount at the end of n years will be

C · (1 + r/m) m · n The effective annual rate is the actual interest rate that is paid over the

year, that is, the rate reffso that

C · (1 + r/m) m = C · (1 + reff).

So, for example, if there is quarterly compounding and the nominal

annual rate is 3%, the effective interest rate is

r = (1 + 0.03/4)4− 1 = 0.0303 = 3.03%.

Again, the difference between the nominal and the effective annual rate

does not seem that large (only 0.03%); however, the difference increases

with the frequency of compounding

Suppose now that we divide the year into very, very small time intervals

You can think of compounding interest every millisecond So, the number

m in the expression for computing the compound interest rate becomes so

large, it can be considered infinity It turns out that when m tends to infinity,

the expression (1 + r/m) m tends to a very specific number, e r, where the

number e has the value 2.7182 (it has infinitely many digits after the

decimal point).1

Therefore, with continuous compounding, $C of initial capital becomes

C · e r·1at the end of the first year, C · e r·2at the end of the second year, and

C · e r ·n after n years If we are interested in the amount of capital after, say

five months, and we are given the nominal interest rate r as an annual rate,

we first convert five months to years (five months= 5/12 years), and then

compute the future amount of capital as C · e r·(5/12)

Let us provide a concrete example If the nominal interest rate is 3% per

year and we invest $100, then with continuous compounding the amount at

the end of the first year is 100· e0.03 ·1= $103.05 Therefore, the effective

annual rate is 3.05%—higher than the effective annual rate of 3.03% with

quarterly compounding we computed earlier After five months, the amount

in the account will be 100· e0.03 ·(5/12)= $101.26

2 1 2 P r e s e n t V a l u e a n d F u t u r e V a l u e

In the previous section, we explained the concept of interest Suppose you

have $100 today, and you put it in a savings account paying 3% interest per

year At the end of the year, your $100 will become $103 Now suppose that

somebody gives you a choice between receiving $100 today, or $100 one

year from now The two options would not be equivalent to you Given the

opportunity to invest the money at 3% interest, you would demand $103

one year from now to make you indifferent between the two options In

this example, the $103 received at the end of the year can be considered the

future value of $100 received today, whereas $100 is the present value of

$103 received one year from now This is the important concept of the time

value of money—money to be received in the future is less valuable than the

same nominal amount of money received immediately

Formally, the present value (sometimes also called the discounted value)

of a single cash flow CF is the amount of money that must be invested today

to generate the future cash flow The present value of a cash flow depends

on (1) the length of time until the cash flow will be received, and (2) the

interest rate, which is called the discount rate in this context.

The present value (PV) of a cash flow CF received n years from now

when the interest rate r is compounded annually is computed as

PV(CF )= CF

(1+ r) n

The expression

1(1+ r) n

is called the discount factor The discount factor (let us call it d n) is the

number by which we need to multiply the future cash flow to obtain its

present value Note that the discount factor is a number less than 1—the

present value of the cash flow is less than the future value in nominal terms

because it is assumed that the interest accrued between the present and the

future date will be a nonnegative amount

The conversion between present and future value follow the interest

calculation rules we introduced in the previous section For example, if the

annual interest rate r is continuously compounded, the present value of a

cash flow CF received n years from now is

PV(CF )= CF

e r · n = CF · e −r · n

In this case, the discount factor is d n = e −r · n

It is easy to see how the concepts of present and future value extend

when the “present” is not today’s date For example, suppose that we

have invested $100 today for three years in an account paying an annual

rate of 3% compounded continuously At the end of year 1, we will have

$100· e0.03 ·1= $103.05 in the account At the end of year 2, we will have

$100· e0.03 ·2= $106.18 in the account The amounts $103.05 and $106.18

are the future values of $100 on hand today, in year 1 and year 2 dollars

The present values of $103.05 received at the end of year 1 and $106.18

received at the end of year 2 are both $100 ($103.05· e−0.03·1and $106.18·

e−0.03·2, respectively) Note that we can compute the present value of $106.18

received at the end of year 2 in two ways The first is to discount directly

to the present, $106.18· e−0.03·2 The second is to discount $106.18 first to

its present value in year 1 dollars ($106.18· e−0.03·1 = $103.05), and then

discount the year 1 dollars to today dollars ($103.05· e−0.03·1 = $100.00)

The latter technique will be useful when pricing financial derivatives and

real options are discussed in Chapters 13 through 16 and Chapter 18

2 2 A S S E T C L A S S E S

An asset is any possession that has value in an exchange Assets can be

clas-sified as tangible or intangible A tangible asset’s value depends on particular

physical properties of the asset Buildings, land, and machinery are

exam-ples of tangible assets Intangible assets, by contrast, represent legal claims

to some future benefit and their value bears no relation to the form, physical

or otherwise, in which the claims are recorded Financial assets, financial

instruments, or securities are intangible assets For these instruments, the

typical future benefit comes in the form of a claim to future cash

In most developed countries, the four major asset classes are (1) common

stocks, (2) bonds, (3) cash equivalents, and (4) real estate An asset class is

defined in terms of the investment attributes that the members of an asset

class have in common These investment characteristics include (1) the major

economic factors that influence the value of the asset class and, as a result,

correlate highly with the returns of each member included in the asset class;

(2) have a similar risk and return characteristic; and (3) have a common

legal or regulatory structure Based on this way of defining an asset class,

the correlation between the returns of different asset classes would be low

The preceding four major asset classes can be extended to create other

asset classes From the perspective of a U.S investor, for example, the four

major asset classes listed earlier have been expanded as follows by separating

foreign securities from U.S securities: (1) U.S common stocks, (2) non–U.S

(or foreign) common stocks, (3) U.S bonds, (4) non-U.S bonds, (5) cash

equivalents, and (6) real estate

Common stocks and bonds are further partitioned into more asset

classes For example, U.S common stocks (also referred to as U.S equities),

are differentiated based on market capitalization Market capitalization (or

market cap) is computed as the number of shares outstanding times the

market price per share The term is often used as a proxy for the size of a

company Companies are usually classified as large cap, medium cap

(mid-cap), small cap, or micro cap, depending on their market capitalization The

division is somewhat arbitrary, but generally, micro-cap companies have a

market capitalization of less than $250 million, small-cap companies have

a market capitalization between $250 million and $1 billion, mid-cap

com-panies have market capitalization between $1 billion and $5 billion, and

large-cap companies have market capitalization of more than $5 billion

Companies that have market capitalization of more than $250 billion are

sometimes referred to as mega-caps.

With the exception of real estate, all of the asset classes we have

pre-viously identified are referred to as traditional asset classes Real estate and

all other asset classes that are not in the preceding list are referred to as

nontraditional asset classes or alternative asset classes They include hedge

funds, private equity, and commodities

Along with the designation of asset classes comes a barometer to be

able to quantify the performance of the asset class—the risk, return, and

the correlation of the return of the asset class with that of another asset

class The barometer is called a benchmark index, market index, or simply

index An example would be the Standard & Poor’s 500 We describe more

indexes in later chapters The indexes are also used by investors to evaluate

the performance of professional managers whom they hire to manage their

assets

2 2 1 E q u i t i e s

Most generally, equity means ownership in a corporation in the form of

common stock Common stock is securities that entitle the holder to a share

of a company’s success through dividends and/or capital appreciation, and

provide voting rights in a company The terms “equities” and “stocks” are

often used interchangeably

A dividend is a payment (usually, quarterly) disbursed by a company to

its shareholders out of the company’s current or retained earnings Dividends

can be given as cash (cash dividends), additional stock (stock dividends), or

other property Dividends are usually paid out by companies that have

reached their growth potential, so they cannot benefit by reinvesting their

earnings into further expansion

Capital appreciation refers to the growth in a stock price Because of

capital appreciation, investors can make money by investing in a company

that is still in its growth phase, even if the company does not pay dividends

2 2 2 F i x e d I n c o m e S e c u r i t i e s

In its simplest form, a fixed income security is a financial obligation of an

entity that promises to pay a specified sum of money at specified future

dates The entity that promises to make the payment is called the issuer of

the security Some examples of issuers are central governments such as the

U.S government and the French government, government-related agencies

of a central government such as Fannie Mae and Freddie Mac in the United

States, a municipal government such as the state of New York in the United

States and the city of Rio de Janeiro in Brazil, a corporation such as

Coca-Cola in the United States and Yorkshire Water in the United Kingdom, and

supranational governments such as the World Bank

Fixed income securities fall into two general categories: debt obligations

and preferred stock In the case of a debt obligation, the issuer is called the

borrower The investor who purchases such a fixed income security is said to

be the lender or creditor Debt obligations are virtually loans with interest,

where the interest is paid over time in the form of coupons The promised

payments that the issuer agrees to make at the specified dates consist of

two components: interest and principal payments (The principal represents

repayment of the funds borrowed at the end, that is, at the maturity date

for the debt obligation.) Fixed income securities that are debt obligations

include bonds, asset-backed securities (ABSs), and bank loans Bonds are

basically loans taken out by corporations, government entities, or

munici-palities Bank loans are loans by banks to companies or individuals ABSs

are securities backed by pools of loans—mortgages or assets (e.g., cars) The

assets in ABS pools are typically too small or illiquid to be sold individually

Pooling the assets allows them to be sold in pieces to investors, a process

known as securitization The largest number of ABSs by far are backed

by pools of mortgages, and are referred to as mortgage-backed securities

(MBSs) We will discuss MBSs, ABSs, and securitization in more detail in

Chapter 15

In contrast to a fixed income security that represents a debt obligation,

preferred stock represents an ownership interest in a corporation Dividend

payments are made to the preferred stockholder and represent a

distribu-tion of the corporadistribu-tion’s profit Unlike investors who own a corporadistribu-tion’s

common stock, investors who own the preferred stock can only realize a

contractually fixed dividend payment Moreover, the payments that must

be made to preferred stockholders have priority over the payments that a

corporation pays to common stockholders In the case of the bankruptcy

of a corporation, preferred stockholders are given preference over common

stockholders Consequently, preferred stock is a form of equity that has

characteristics similar to bonds

Prior to the 1980s, fixed income securities were simple investment

prod-ucts Holding aside default by the issuer, the investor knew how long

in-terest would be received and when the amount borrowed would be repaid

Moreover, most investors purchased these securities with the intent of

hold-ing them to their maturity date Beginnhold-ing in the 1980s, the fixed income

world changed First, fixed income securities became more complex There

are features in many fixed income securities that make it difficult to

de-termine when the amount borrowed will be repaid and for how long

in-terest will be received For some securities it is difficult to determine the

amount of interest that will be received Second, the hold-to-maturity

in-vestor was replaced by institutional inin-vestors who actively trade fixed income

securities

In this book, we will often use the terms “fixed income securities”

and “bonds” interchangeably Next, we introduce various features of fixed

income securities, and explain how these features affect the risks associated

with investing in fixed income securities This introduction is only cursory

For an in-depth overview of fixed income products, we refer the reader to

Fabozzi (2007)

The term to maturity of a bond is the number of years the debt is

out-standing or the number of years remaining prior to final principal payment

The maturity date of a bond refers to the date that the debt will cease to

ex-ist, at which time the issuer will redeem the bond by paying the outstanding

balance

The par value of a bond is the amount that the issuer agrees to repay

the bondholder at or by the maturity date This amount is also referred to

as the principal value, face value, redemption value, and maturity value.

Because bonds can have a different par value, the practice is to quote

the price of a bond as a percentage of its par value A value of 100 means

“100% of par value.” For example, if a bond has a par value of $1,000 and

the issue is selling for $900, this bond would be said to be selling at 90 If

a bond is quoted at 103 19/32 and has a par value of $1 million, then the

dollar price is (103.59375/100)× $1,000,000 = $1,035,937.50

A bond may trade above or below its par value When a bond trades

below its par value, it is said to be trading at a discount When a bond trades

above its par value, it is said to be trading at a premium.

The coupon rate, which is also called the nominal rate, is the interest

rate the issuer agrees to pay each year The annual amount of the interest

payment made to bondholders during the term of the bond is called the

coupon The coupon is calculated by multiplying the coupon rate by the par

value of the bond In other words,

Coupon= (Coupon rate) · (Par value)For example, a bond with a 5% coupon rate and a par value of $1,000

will pay annual interest of $50 (=0.05 · $1,000)

In the United States, the usual practice is for the issuer to pay the

coupon in two semiannual installments Mortgage-backed securities and

asset-backed securities typically pay interest monthly For bonds issued in

some markets outside the United States, coupon payments are made only

once per year

Not all bonds make periodic coupon payments For example,

zero-coupon bonds do not pay out zero-coupons during the life of the bond The holder

of a zero-coupon bond realizes interest by buying the bond substantially

below its par value (i.e., buying the bond at a discount) Interest is then paid

at the maturity date, where the interest is the difference between the par

value and the price paid for the bond

In addition, the coupon rate on a bond need not be fixed over the bond’s

life Floating-rate securities, sometimes also called floaters or variable-rate

securities, have coupon payments that reset periodically according to some

reference rate The typical formula (called the coupon formula) on certain

determination dates when the coupon rate is reset is as follows:

Coupon rate= (Reference rate) + (Quoted margin)The quoted margin is the additional amount that the issuer agrees to pay

above the reference rate For example, suppose that the reference rate is the

1-month London interbank offer rate (LIBOR).2 Suppose that the quoted

margin is 100 basis points.3Then the coupon formula is

Coupon rate= (1-month LIBOR) + (100 basis points)

An example of a floating rate security is an linked (or

inflation-indexed) bond For example, in 1987, the U.S Department of Treasury

began issuing inflation-adjusted securities referred to as Treasury Inflation

Protection Securities (TIPS) The reference rate for the coupon formula is

the rate of inflation as measured by the Consumer Price Index for All

Ur-ban Consumers (called CPI-U) Corporations and agencies in the United

States also issue inflation-linked bonds For example, in February 1997, J.P

Morgan & Company issued a 15-year bond that pays the CPI plus 400

basis points

A floater may have a restriction on the maximum or minimum coupon

rate that will be paid at any reset date The maximum (respectively, the

minimum) coupon rate is called a cap (respectively, a floor).

Typically, the coupon formula for a floater is such that the coupon rate

increases when the reference rate increases, and decreases when the reference

rate decreases However, there are issues whose coupon rate moves in the

opposite direction from the change in the reference rate Such issues are

called inverse floaters or reverse floaters Such securities give and investor

who believes that interest rates will decline the opportunity to obtain a

higher coupon interest rate.4

A c c r u e d I n t e r e s t Bond issuers do not disburse coupon interest payments

every day Instead, payments are made at prespecified dates (As we

men-tioned earlier, in the United States, for example, coupon interest is typically

paid every six months.) Thus, if an investor sells a bond between coupon

payments and the buyer holds it until the next coupon payment, then the

entire coupon interest earned for the period will be paid to the buyer of the

bond since the buyer will be the holder of record The seller of the bond

gives up the interest from the time of the last coupon payment to the time

until the bond is sold The amount of interest over this period that will

be received by the buyer even though it was earned by the seller is called

accrued interest.

In the United States and in many countries, the bond buyer must pay the

bond seller the accrued interest The amount that the buyer pays the seller

is the agreed upon price for the bond plus accrued interest This amount is

called the full price (Some market participants refer to this as the dirty price.)

The agreed upon bond price without accrued interest is simply referred to

as the price (Some refer to it as the clean price.)

There are exceptions to the rule that the bond buyer must pay the bond

seller accrued interest The most important exception is when the issuer has

not fulfilled their promise to make the periodic interest payments In this

case, the issuer is said to be in default In such instances, the bond is sold

without accrued interest and is said to be traded flat.

P r o v i s i o n s f o r P a y i n g O f f B o n d s The most common structure in the United

States and Europe for paying off the principal for both corporate and

gov-ernment bonds is to pay the entire amount in one lump sum payment at the

maturity date Such bonds are said to have a bullet maturity.

Fixed income securities backed by pools of loans, such as MBSs and

ABSs, often have a schedule of partial principal payments Such fixed income

securities are said to be amortizing securities For many loans, the payments

are structured so that when the last loan payment is made, the entire amount

owed is fully paid

An issue may have a call provision granting the issuer the option to

retire (pay off) all or part of the issue prior to the stated maturity date Some

issues specify that the issuer must retire a predetermined amount of the issue

periodically

C o n v e r s i o n P r i v i l e g e A convertible bond is an issue that grants the

bond-holder the right to convert the bond for a specified number of shares of

common stock Such a feature allows the bondholder to take advantage

of favorable movements in the price of the issuer’s common stock An

ex-changeable bond allows the bondholder to exchange the issue for a specified

number of shares of common stock of a corporation different from the issuer

of the bond

C u r r e n c y D e n o m i n a t i o n The payments that the issuer makes to the

bond-holder can be in any currency For example, an issue in which payments to