Simulation andOptimization in

Finance

The Frank J. Fabozzi SeriesFixed Income Securities, Second Edition by Frank J. FabozziFocus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. AbateHandbook of Global Fixed Income Calculations by Dragomir KrginManaging a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. FabozziReal Options and Option-Embedded Securities by William T. MooreCapital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. FabozziThe Exchange-Traded Funds Manual by Gary L. GastineauProfessional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. FabozziInvesting in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia PilarinuHandbook of Alternative Assets by Mark J. P. AnsonThe Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad ChoudhryThe Handbook of Financial Instruments edited by Frank J. FabozziInterest Rate, Term Structure, and Valuation Modeling edited by Frank J. FabozziInvestment Performance Measurement by Bruce J. FeibelThe Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. FabozziThe Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. MarkowitzFoundations of Economic Value Added, Second Edition by James L. GrantFinancial Management and Analysis, Second Edition by Frank J. Fabozzi and Pamela P. PetersonMeasuring and Controlling Interest Rate and Credit Risk, Second Edition by Frank J. Fabozzi, Steven V. Mann, and

Moorad ChoudhryProfessional Perspectives on Fixed Income Portfolio Management, Volume 4 edited by Frank J. FabozziThe Handbook of European Fixed Income Securities edited by Frank J. Fabozzi and Moorad ChoudhryThe Handbook of European Structured Financial Products edited by Frank J. Fabozzi and Moorad ChoudhryThe Mathematics of Financial Modeling and Investment Management by Sergio M. Focardi and Frank J. FabozziShort Selling: Strategies, Risks, and Rewards edited by Frank J. FabozziThe Real Estate Investment Handbook by G. Timothy Haight and Daniel SingerMarket Neutral Strategies edited by Bruce I. Jacobs and Kenneth N. LevySecurities Finance: Securities Lending and Repurchase Agreements edited by Frank J. Fabozzi and Steven V. MannFat-Tailed and Skewed Asset Return Distributions by Svetlozar T. Rachev, Christian Menn, and Frank J. FabozziFinancial Modeling of the Equity Market: From CAPM to Cointegration by Frank J. Fabozzi, Sergio M. Focardi, and

Petter N. KolmAdvanced Bond Portfolio Management: Best Practices in Modeling and Strategies edited by Frank J. Fabozzi, Lionel

Martellini, and Philippe PriauletAnalysis of Financial Statements, Second Edition by Pamela P. Peterson and Frank J. FabozziCollateralized Debt Obligations: Structures and Analysis, Second Edition by Douglas J. Lucas, Laurie S. Goodman, and

Frank J. FabozziHandbook of Alternative Assets, Second Edition by Mark J. P. AnsonIntroduction to Structured Finance by Frank J. Fabozzi, Henry A. Davis, and Moorad ChoudhryFinancial Econometrics by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo JasicDevelopments in Collateralized Debt Obligations: New Products and Insights by Douglas J. Lucas, Laurie S. Goodman,

Frank J. Fabozzi, and Rebecca J. ManningRobust Portfolio Optimization and Management by Frank J. Fabozzi, Peter N. Kolm, Dessislava A. Pachamanova, and

Sergio M. FocardiAdvanced Stochastic Models, Risk Assessment, and Portfolio Optimizations by Svetlozar T. Rachev, Stogan V. Stoyanov,

and Frank J. FabozziHow to Select Investment Managers and Evaluate Performance by G. Timothy Haight, Stephen O. Morrell, and

Glenn E. RossBayesian Methods in Finance by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. FabozziThe Handbook of Commodity Investing by Frank J. Fabozzi, Roland Fuss, and Dieter G. KaiserThe Handbook of Municipal Bonds edited by Sylvan G. Feldstein and Frank J. FabozziSubprime Mortgage Credit Derivatives by Laurie S. Goodman, Shumin Li, Douglas J. Lucas, Thomas A Zimmerman,

and Frank J. FabozziIntroduction to Securitization by Frank J. Fabozzi and Vinod KothariStructured Products and Related Credit Derivatives edited by Brian P. Lancaster, Glenn M. Schultz, and Frank J. FabozziHandbook of Finance: Volume I: Financial Markets and Instruments edited by Frank J. FabozziHandbook of Finance: Volume II: Financial Management and Asset Management edited by Frank J. FabozziHandbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools edited by Frank J. FabozziFinance: Capital Markets, Financial Management, and Investment Management by Frank J. Fabozzi and Pamela

Peterson-DrakeActive Private Equity Real Estate Strategy edited by David J. LynnFoundations and Applications of the Time Value of Money by Pamela Peterson-Drake and Frank J. FabozziLeveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives by Stephen Antczak,

Douglas Lucas, and Frank J. FabozziModern Financial Systems: Theory and Applications by Edwin NeaveInstitutional Investment Management: Equity and Bond Portfolio Strategies and Applications by Frank J. FabozziQuantitative Equity Investing: Techniques and Strategies by Frank J. Fabozzi, Sergio M. Focardi, Petter N. KolmSimulation and Optimization in Finance: Modeling with MATLAB, @RISK, or VBA by Dessislava A. Pachamanova and

Frank J. Fabozzi

Simulation andOptimization in

FinanceModeling with MATLAB,

@RISK, or VBA

DESSISLAVA A. PACHAMANOVAFRANK J. FABOZZI

John Wiley & Sons, Inc.

Copyright C© 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:

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);978-0-470-88212-2 (ebk)

1. Finance–Mathematical models–Computer programs. I. Fabozzi, Frank J. II. Title.HG106.P33 2010332.0285′53–dc22 2010027038

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Dessislava A. PachamanovaTo my husband, Christian, and my children,

Anna and Coleman

Frank J. FabozziTo my wife, Donna, and my children, Patricia,

Karly, and Francesco

Contents

Preface xi

About the Authors xvi

Acknowledgments xvii

CHAPTER 1Introduction 1

Optimization; Simulation; Outline of Topics

PART ONEFundamental Concepts

CHAPTER 2Important Finance Concepts 11

Basic Theory of Interest; Asset Classes; Basic TradingTerminology; Calculating Rate of Return; Valuation;Important Concepts in Fixed Income; Summary; Notes

CHAPTER 3Random Variables, Probability Distributions, andImportant Statistical Concepts 51

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 Overviewof 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

viii CONTENTS

CHAPTER 4Simulation Modeling 101

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

CHAPTER 5Optimization Modeling 143

Optimization Formulations; Important Types ofOptimization Problems; Optimization ProblemFormulation Examples; Optimization Algorithms;Optimization Duality; Multistage Optimization;Optimization Software; Summary; Software Hints; Notes

CHAPTER 6Optimization under Uncertainty 211

Dynamic Programming; Stochastic Programming;Robust Optimization; Summary; Notes

PART TWOPortfolio Optimization and Risk Measures

CHAPTER 7Asset Diversification and Efficient Frontiers 245

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 8Advances in the Theory of Portfolio Risk Measures 277

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

CHAPTER 9Equity Portfolio Selection in Practice 321

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

Contents ix

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

CHAPTER 10Fixed Income Portfolio Management in Practice 373

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

PART THREEAsset Pricing Models

CHAPTER 11Factor Models 401

The Capital Asset Pricing Model; The Arbitrage PricingTheory; Building Multifactor Models in Practice;Applications of Factor Models in PortfolioManagement; Summary; Software Hints; Notes

CHAPTER 12Modeling Asset Price Dynamics 421

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

PART FOURDerivative Pricing and Use

CHAPTER 13Introduction to Derivatives 477

Basic Types of Derivatives; Important Concepts forDerivative Pricing and Use; Pricing Forwards andFutures; Pricing Options; Pricing Swaps; Summary;Software Hints; Notes

CHAPTER 14Pricing Derivatives by Simulation 531

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

x CONTENTS

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

CHAPTER 15Structuring and Pricing Residential Mortgage-Backed Securities 587

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 16Using Derivatives in Portfolio Management 627

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 FIVECapital Budgeting Decisions

CHAPTER 17Capital Budgeting under Uncertainty 653

Classifying Investment Projects; Investment Decisionsand Wealth Maximization; Evaluating Project Risk;Case Study; Managing Portfolios of Projects; Summary;Software Hints; Notes

CHAPTER 18Real Options 707

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

References 733

Index 743

Preface

S imulation 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 discussingthe practical implications of applying simulation and optimization tech-niques, the book uses simulation and optimization as a means to clarifydifficult concepts in traditional risk models in finance, and explains how tobuild financial models with software. The book covers a wide range of ap-plications and is written in a theoretically rigorous way, which will make itof interest to both practitioners and academics. It can be used as a self-studyaid by finance practitioners and students who have some fundamental back-ground in calculus and statistics, or as a textbook in finance and quantitativemethods courses. In addition, this book is accompanied by a web site wherereaders can go to download an array of supplementary materials. Pleasesee the “Companion Web Site” section toward the end of this Preface formore details.

CENTRAL THEMES

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 uncertaintyconcepts that are necessary to understand the financial applications in laterparts 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-riskand conditional value-at-risk estimation. Part Three, Asset Pricing Models,discusses classical static and dynamic models for asset pricing, such as factormodels and different types of random walks. Part Four, Derivative Pricingand Use, introduces important types of financial derivatives, shows howtheir value can be determined by simulation, reviews advanced simulation

xi

xii PREFACE

methods for efficient implementation of pricing algorithms, and discusseshow derivatives can be employed for portfolio risk management and returnenhancement 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 methodsthat can be used to handle a particular problem in finance. Many of thetopics listed here, especially asset and derivative pricing models, however,have traditionally been out of reach for readers without advanced degrees inmathematics because understanding the theory behind the models and theadvanced methods for modeling requires years of training. Simulation andoptimization formulations provide a framework within which very challeng-ing concepts can be explained through simple visualization and hands-onimplementation, which makes the material accessible to readers with littlebackground in advanced mathematics.

SOFTWARE

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 underdiscussion. The examples themselves are posted on the companion web sitediscussed later in the Preface.

In Simulation and Optimization in Finance, we assume basic familiar-ity with spreadsheets and Microsoft Excel, and use two different platformsto implement concepts and algorithms: the Palisade Decision Tools Suiteand 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 andtheir add-ins versus programming environments such as MATLAB. Specifi-cally, users with finance and social science backgrounds typically prefer anExcel-based implementation, whereas users with engineering and quanti-tative backgrounds prefer MATLAB. Some tasks and implementations areeasier in one environment than in the other, and students who have used thisbook in the form of lecture notes in the past have felt they benefitted fromlearning about both platforms. Basic introductions to the software used inthe book are provided in Appendices B through D, which can be accessed atthe 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

Preface xiii

without necessarily covering every aspect of working in a specific softwareenvironment.

We have, of course, attempted to implement all examples correctly.That said, the code is provided “as is” and is intended only to illustratethe concepts in this book. Readers who use the code for financial decisionmaking are doing so at their own risk. For full information on the termsof use of the code, please see the licensing information in each file on thecompanion web site.

The following web sites provide useful information about Palisade De-cision Tools Suite and MATLAB. Readers can download trial versions orpurchase the software.

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

TEACHING

Simulation and Optimization in Finance: Modeling with MATLAB, @RISK,or VBA covers finance and applied quantitative methods theory, as well asa wide range of applications. It can be used as a textbook for upper-levelundergraduate or lower-level graduate (such as MBA or Master’s) coursesin applied quantitative methods, operations research, decision sciences, orfinancial engineering, finance courses in derivatives, investments or corpo-rate finance with an emphasis on modeling, or as a supplement in a specialtopics course in quantitative methods or finance. In addition, the book canbe used as a self-study aid by students, or serve as a reference for studentprojects.

The book assumes that the reader has no background in finance or ad-vanced quantitative methods except for basic calculus and statistics. Mostquantitative concepts necessary for understanding the notation or applica-tions are introduced and explained in endnotes, software hints, and onlineappendices. This makes the book suitable for readers with a wide range ofbackgrounds and particularly so as a textbook for classes with mixed audi-ences (such as engineering and business students). In fact, the idea for thisbook project matured after years of searching for an appropriate text for acourse with a mixed audience that needed a good reference for both financeand 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. Atthe end of each chapter, there is a summary that contains the most impor-tant discussion points. A Software Hints section provides instructions and

xiv PREFACE

code for implementing the examples in the chapter with both Excel-basedsoftware and MATLAB.

On the companion web site, there are practice sections for selectedchapters. These sections feature examples that complement those foundin their respective chapters. Some practice sections contain cases as well.The cases are more in-depth exercises that focus on a particular practicalapplication not necessarily covered in the chapter, but possible to addresswith 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, namelyAppendix 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 basicmathematical and programming concepts that enable readers to understandthe implementation and the code provided in the Software Hints sections.

The chapters that introduce fundamental concepts all contain code thatcan be found on the companion web site. Some more advanced chapters donot; the idea is that at that point students are sufficiently familiar with theapplications and models to put together examples on their own based on thecode provided in previous chapters. The material in the advanced chapterscan 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 onapplications of optimization for single-period optimal portfolio allocationand risk management. The course then proceeds with Chapters 11 through14, which introduce static and dynamic asset pricing models through sim-ulation as well as derivative pricing by simulation, and ends with Chapters17 and 18, which discuss applications of simulation and optimization incapital budgeting. Chapters 10, 15, and 16 represent good assignments forfinal projects because they use concepts similar to other chapters, but in adifferent context and without as much implementation detail.

Depending on the nature of the course, only some of Chapters 2 through6 will need to be covered explicitly; but the information in these chapters isuseful in case the instructor would like to assign the chapters as reading forstudents who lack some of the necessary background for the course.

COMPANION WEB SITE

Additional material for Simulation and Optimization in Finance can bedownloaded by visiting www.wiley.com/go/pachamanova. Please log in tothe web site using this password: finance123. The files on this companion

Preface xv

web site are organized in the following folders: Appendices, Code, andPractice. The Appendices directory contains Appendix A through D. ThePractice directory contains practice problems and cases indexed by chapter.(Practice problems are present for Chapters 4–16, 18, and Appendix D, as abonus to the content in the book. Please note, however, that only problemsare offered without solutions.) The Code directory has Excel and MATLABsubdirectories that contain files for use with the corresponding software.The latter files are referenced in the main body of the book and the SoftwareHints sections for selected chapters.

The companion web site is a great resource for readers interested inactually implementing the concepts in the book. Such readers should beginby 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 achapter, the chapter’s Software Hints, and, finally, the Excel model files orMATLAB code in the code directory on the companion web site.

NOTES

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. Herresearch interests lie in the areas of portfolio risk management, simulation,high-performance optimization, and financial engineering. She has publisheda number of articles in operations research, finance, and engineering jour-nals, and coauthored the Wiley title Robust Portfolio Optimization andManagement (2007). Dessislava’s academic research is supplemented byconsulting and previous work in the financial industry, including projectswith quantitative strategy groups at WestLB and Goldman Sachs. She holdsan AB in mathematics from Princeton University and a PhD in operationsresearch from the Sloan School of Management at MIT.

Frank J. Fabozzi is Professor in the Practice of Finance in the School ofManagement at Yale University. Prior to joining the Yale faculty, he wasa 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 theAdvisory Council for the Department of Operations Research and FinancialEngineering 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 Yorkin 1972. In 2002 was inducted into the Fixed Income Analysts Society’s Hallof Fame and is the 2007 recipient of the C. Stewart Sheppard Award givenby the CFA Institute. He earned the designation of Chartered Financial Ana-lyst and Certified Public Accountant. He has authored and edited numerousbooks in finance.

xvi

Acknowledgments

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 commentedon chapters or sections of chapters or provided helpful references and intro-ductions:

� Anthony Corr, Brett McElwee, and Max Capetta of Continuum CapitalManagement

� 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 testingsome of the VBA code in the book, such as the VBA implementation of theAmerican option pricing model with least squares in Chapter 14. ProfessorMark Potter of Babson College allowed us to modify his case, “ReebokInternational: Strategic Asset Allocation,” for use as an example in Chapter17, 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 thecases and examples in the book are based on ideas and research by ThomasMalloy, Michael Allietta, Adam Bergenfield, Nick Kyprianou, Jason Aron-son, and Rohan Duggal. The real estate valuation project example in section18.6.3 in Chapter 18 is based on ideas by Matt Bujnicki, Matt Enright, andAlec Kyprianou.

We would also like to thank Wendy Gudgeon and Stan Brown fromPalisade Software and Steve Wilcockson, Naomi Fernandes, Meg Vulliez,Chris Watson, and Srikanth Krishnamurthy of Mathworks for their helpwith obtaining most recent versions of the software used in the book andfor additional materials useful for implementing some of the examples.

DESSISLAVA A. PACHAMANOVA

FRANK J. FABOZZI

xvii

CHAPTER 1Introduction

F inance is the application of economic principles to decision making, andinvolves the allocation of money under conditions of uncertainty. In-

vestors allocate their funds among financial assets in order to accomplishtheir objectives. Business entities and government at all levels raise funds byissuing 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 andthen invested.

The field of finance has three specialty areas: (1) capital markets andcapital market theory, (2) financial management, and (3) portfolio man-agement. The specialty field of capital markets and capital market theoryfocuses on the study of the financial system, the structure of interest rates,and the pricing of risky assets. Financial management, sometimes calledbusiness finance, is the specialty area of finance concerned with financial de-cision making within a business entity. Although we often refer to financialmanagement as corporate finance, the principles of financial managementalso 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 decisions that involve long-termcapital expenditures is called capital budgeting. Portfolio management dealswith the management of individual or institutional funds. This specialtyarea of finance—also commonly referred to as investment management, as-set management, and money management—involves selecting an investmentstrategy and then selecting the specific assets to be included in a portfolio.

A critical element common to all three specialty areas in finance is theconcept of risk. Measuring and quantifying risk is critical for the fair val-uation of an asset, the selection of capital budgeting projects in financialmanagement, the selection of individual asset holdings, and portfolio con-struction in portfolio management. The field of risk management includes

1

2 INTRODUCTION

the identification, measurement, and control of risk in a business entity ora portfolio.

Sophisticated mathematical tools have been employed in order to dealwith the risks associated with individual assets, capital budgeting projects,and selecting assets in portfolio construction. The use of such tools is nowcommonplace in the financial industry. For example, in portfolio man-agement, practitioners run statistical routines to identify risk factors thatdrive asset returns, scenario analyses to evaluate the risk of their posi-tions, and algorithms to find the optimal way to allocate assets or executea trade.

This book focuses on two quantitative tools—optimization and simula-tion—and discusses their applications in finance. In this chapter, we brieflyintroduce these two techniques, and provide an overview of the structure ofthe book.

OPTIMIZATION

Optimization is an area in applied mathematics that, most generally, dealswith efficient algorithms for finding an optimal solution among a set ofsolutions that satisfy given constraints. The first application of optimizationin finance was suggested by Harry Markowitz in 1952, in a seminal paperthat outlined his mean-variance optimization framework for optimal assetallocation. Some other classical problems in finance that can be solved byoptimization algorithms include:

� Is there a possibility to make riskless profit given market prices of relatedsecurities? (This opportunity is called an arbitrage opportunity and isdiscussed in Chapter 13.)

� How should trades be executed so as to reach a target allocation withminimum transaction costs?

� Given a limited capital budget, which capital budgeting projects shouldbe 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 theoptimal 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 techniquesa viable and useful part of the standard toolset of the financial modeler.

Introduction 3

SIMULATION

Simulation is a technique for replicating uncertain processes, and evaluatingdecisions under uncertain conditions. Perhaps the earliest application ofsimulation in finance was in financial management. Hertz (1964) argued thattraditional valuation methods for investments omitted from consideration animportant component: the fact that many of the inputs were inaccurate. Hesuggested modeling the uncertainty through probability-weighted scenarios,which would allow for obtaining a range of outcomes for the value of theinvestments and associated probabilities for each outcome. These ideas wereforgotten for a while, but have experienced tremendous growth in the lasttwo 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 differentfactors over time is simulated in order to estimate measures of portfoliorisk. In pricing financial options or complex securities, such as mortgage-backed securities, paths for the underlying risk factors are simulated; andthe fair price of the securities is estimated as the average of the discountedpayoffs over those paths. We will see numerous examples of such simulationapplications in this book.

Simulation bears some resemblance to an intuitive tool for modifyingoriginal assumptions in financial models—what-if analysis—which has beenused 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 happensto the decision under these scenarios. The important additional componentin simulation modeling, however, is that there are probabilities associatedwith the different outcomes. This allows for obtaining an additional piece ofinformation compared to what-if analysis: the probabilities that specific finaloutcomes 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 forsimulation modeling.

OUTLINE OF TOPICS

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 thebook. Part Two (Chapters 7 through 10) introduces the classical under-pinnings of modern portfolio theory, and discusses the role of simulationand optimization in recent developments. Part Three (Chapters 11 and 12)

4 INTRODUCTION

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 offinancial 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 budgetingand real option valuation. The four appendices (on the companion web site)feature introductions to linear algebra concepts, @RISK, MATLAB, andVisual Basic for Applications in Microsoft Excel.

We begin by listing important finance terminology in Chapter 2. Thisincludes basic theory of interest; terminology associated with equities, fixedincome securities, and trading; calculation of rate of return; and usefulconcepts in fixed income, such as spot rates, forward rates, yield, duration,and convexity.

Chapter 3 is an introduction to probability theory, distributions, andbasic statistics. We review important probability distributions, such as thenormal distribution and the binomial distribution, measures of central ten-dency and variability, and measures of strength of codependence betweenrandom 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, andexplain the methodology behind generating random numbers from differ-ent probability distributions. We also touch upon recent developments inefficient random number generation, which provides the foundation for theadvanced simulation methods for financial derivative pricing discussed inPart Four of the book.

In Chapter 5 we provide a practical introduction to optimization. Wediscuss the most commonly encountered types of optimization problems infinance, and elaborate on the concept of “difficult” versus “easy” optimiza-tion problems. We introduce optimization duality and describe intuitivelyhow optimization algorithms work. Illustrations of simple finance problemsthat can be handled with optimization techniques are provided, includingexamples of optimal portfolio allocation and cash flow matching from thefield of portfolio management, and capital budgeting from the field of fi-nancial management. We also discuss dynamic programming—a techniquefor 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 optimizationin particular.

Classical optimization methods treat the parameters in optimizationproblems as deterministic and accurate. In reality, however, these param-eters are typically estimated through error-prone statistical procedures or

Introduction 5

based on subjective evaluation, resulting in estimates with significant estima-tion errors. The output of optimization routines based on poorly estimatedinputs 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 parametersin optimization problems. Chapter 6 provides a taxonomy of methods foroptimization under uncertainty. We review the main ideas behind dynamicprogramming under uncertainty, stochastic programming, and robust opti-mization, and illustrate the methods with examples. We will encounter thesemethods 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 ininvestments—expected utility maximization—and explain its relationship tomean-variance optimization.

Chapter 8 extends the classical mean-variance portfolio optimizationtheory to a more general mean-risk setting. We cover the most commonlyused alternative risk measures that are generally better suited than vari-ance for describing investor preferences when asset return distributions areskewed or fat-tailed. We focus on two popular portfolio risk measures—value-at-risk and conditional value-at-risk—and show how to estimate themusing simulation. We also formulate the problems of optimal asset allocationunder 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 partof optimization problems. We also show how the classical framework forportfolio allocation can be extended to include transaction costs, and discussindex tracking, optimization of trades across multiple client accounts, androbust 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 incomeportfolio management (which are defined in Chapter 2); however, fixed in-come securities have some fundamental differences from equities, so theconcepts cannot always be applied in the same way in which they would beapplied for stock portfolios. We review classical measures of bond portfoliorisk, such as duration, key rate duration, and spread duration. We discussbond portfolio optimization relative to a benchmark index. We also giveexamples of how optimization can be used in liability-driven bond portfoliostrategies such as immunization and cash flow matching.

Chapter 11 transitions from the topic of portfolio management to thetopic of asset pricing, and introduces standard financial models for explain-ing asset returns—the Capital Asset Pricing Model (CAPM), which is based

6 INTRODUCTION

on the mean-variance framework described in Chapter 7, the ArbitragePricing Theory (APT), and factor models. Such models are widely used inportfolio management—they not only help to model the processes that driveasset 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 basedon random processes. We examine the most commonly used types of ran-dom walks, and illustrate their behavior through simulation. The modelsdiscussed include arithmetic, geometric, different types of mean-revertingrandom walks, and more advanced hybrid models. In our presentation inthe chapter, we assume that changes in asset prices happen at discrete timeintervals. At the end of the chapter, we extend the concept of a random walkto a random process in continuous time.

The concepts introduced in Chapter 12 are reused multiple times whenwe discuss valuation of complex securities and multistage investments inParts 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 classesof 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 andbinomial trees.

Chapter 14 builds on the material in Chapter 13, but focuses mainly onthe use of simulation for pricing complex securities. Some of the closed-formformulas provided in Chapter 12 and the assumptions behind them becomemore intuitive when illustrated through simulation of the random processesfollowed by the underlying securities. A large part of the chapter is dedicatedto variance reduction techniques, such as antithetic variables, stratified sam-pling, importance sampling, and control variates, as well as quasi–MonteCarlo methods. Such techniques are widely used today for efficient imple-mentation of simulations for pricing securities and estimating sensitivity todifferent 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 similartechniques to the ones discussed in Chapter 14, but the context is different.We introduce a complex type of fixed-income securities—mortgage-backedsecurities—and discuss in detail a part of the simulation that is specific tofixed-income securities—generating scenarios for future interest rates andthe entire yield curve.

Chapter 16 builds on Chapters 7, 8, 9, 13, and 14, and contains adiscussion of how derivatives can be used for portfolio risk managementand return enhancement strategies. Simulation is essential for estimatingthe risk of a portfolio that contains complex financial instruments, but the

Introduction 7

process can be very slow in the case of large portfolios. We highlight somenumerical issues, standard simulation algorithms, and review methods thathave 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 applyingsimulation and optimization makes in classical finance decision-makingframeworks. Chapter 17 begins with a review of so-called discounted cashflow (DCF) methodologies for evaluating company investment projects. Itthen discusses (through a case study) how simulation can be used to estimatestand-alone risk and enhance the analysis of such projects.

Chapter 18 introduces the real options framework, which advocatesfor accounting for existing options in project valuation. (The DCF analysisignores the potential flexibility in projects—it assumes that there will be nochanges once a decision is made.) While determining the inputs for valu-ation of real options presents significant challenges, the actual techniquesfor pricing these real options are based on the techniques for pricing finan-cial options introduced in Chapters 13 and 14. Simulation and multistageoptimization can again be used as valuable tools in this new context.

PART

OneFundamental Concepts