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 spreadsheet- based software (@RISK, Solver, and VBA) and mathematical modeling software (MATLAB)
• 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, PHD, 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, PHD, 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.
U L A T IO N A N D O PT IM IZ A T IO N IN F INAN C E + W e b S it e
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Simulation and Optimization in Finance
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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
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Simulation and Optimization in Finance
Modeling with MATLAB,
@RISK, or VBA
DESSISLAVA A. PACHAMANOVA FRANK J. FABOZZI
John Wiley & Sons, Inc.
iii
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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 2010
332.028553–dc22 2010027038
Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
iv
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
Contents
Preface xi
About the Authors xvi
Acknowledgments xvii
CHAPTER 1
Introduction 1
Optimization; Simulation; Outline of Topics
PART ONE
Fundamental Concepts CHAPTER 2
Important Finance Concepts 11
Basic Theory of Interest; Asset Classes; Basic Trading Terminology; Calculating Rate of Return; Valuation;
Important Concepts in Fixed Income; Summary; Notes
CHAPTER 3
Random Variables, Probability Distributions, and
Important Statistical Concepts 51
What is a Probability Distribution?; Bernoulli Probability Distribution and Probability Mass Functions; Binomial Probability Distribution and Discrete Distributions; Normal Distribution and Probability Density Functions; Concept of Cumulative Probability; Describing Distributions; Brief Overview of Some Important Probability Distributions;
Dependence Between Two Random Variables:
Covariance and Correlation; Sums of Random Variables; Joint Probability Distributions and Conditional Probability; From Probability Theory to Statistical Measurement: Probability Distributions and Sampling; Summary; Software Hints; Notes
vii
CHAPTER 4
Simulation Modeling 101
Monte Carlo Simulation: A Simple Example; Why Use Simulation?; Important Questions in Simulation Modeling; Random Number Generation; Summary;
Software Hints; Notes
CHAPTER 5
Optimization Modeling 143
Optimization Formulations; Important Types of Optimization Problems; Optimization Problem Formulation Examples; Optimization Algorithms;
Optimization Duality; Multistage Optimization;
Optimization Software; Summary; Software Hints; Notes
CHAPTER 6
Optimization under Uncertainty 211
Dynamic Programming; Stochastic Programming;
Robust Optimization; Summary; Notes
PART TWO
Portfolio Optimization and Risk Measures CHAPTER 7
Asset Diversification and Efficient Frontiers 245
The Case for Diversification; The Classical
Mean-Variance Optimization Framework; Efficient Frontiers; Alternative Formulations of the Classical Mean-Variance Optimization Problem; The Capital Market Line; Expected Utility Theory; Summary;
Software Hints; Notes
CHAPTER 8
Advances in the Theory of Portfolio Risk Measures 277
Classes of Risk Measures; Value-At-Risk; Conditional Value-At-Risk and the Concept of Coherent Risk Measures; Summary; Software Hints; Notes
CHAPTER 9
Equity Portfolio Selection in Practice 321
The Investment Process; Portfolio Constraints
Commonly Used in Practice; Benchmark Exposure and Tracking Error Minimization; Incorporating
Transaction Costs; Incorporating Taxes; Multiaccount Optimization; Robust Parameter Estimation; Portfolio Resampling; Robust Portfolio Optimization; Summary;
Software Hints; Notes
CHAPTER 10
Fixed Income Portfolio Management in Practice 373
Measuring Bond Portfolio Risk; The Spectrum of Bond Portfolio Management Strategies; Liability-Driven Strategies; Summary; Notes
PART THREE
Asset Pricing Models CHAPTER 11
Factor Models 401
The Capital Asset Pricing Model; The Arbitrage Pricing Theory; Building Multifactor Models in Practice;
Applications of Factor Models in Portfolio Management; Summary; Software Hints; Notes
CHAPTER 12
Modeling Asset Price Dynamics 421
Binomial Trees; Arithmetic Random Walks; Geometric Random Walks; Mean Reversion; Advanced Random Walk Models; Stochastic Processes; Summary;
Software Hints; Notes
PART FOUR
Derivative Pricing and Use CHAPTER 13
Introduction to Derivatives 477
Basic Types of Derivatives; Important Concepts for Derivative Pricing and Use; Pricing Forwards and Futures; Pricing Options; Pricing Swaps; Summary;
Software Hints; Notes
CHAPTER 14
Pricing Derivatives by Simulation 531
Computing Option Prices with Crude Monte Carlo Simulation; Variance Reduction Techniques;
Quasirandom Number Sequences; More Simulation Application Examples; Summary; Software Hints; Notes
CHAPTER 15
Structuring and Pricing Residential Mortgage-Backed Securities 587
Types of Asset-Backed Securities; Mortgage-BackedSecurities: Important Terminology; Types of RMBS Structures; Pricing RMBS by Simulation; Using Simulation to Estimate Sensitivity of RMBS Prices to Different Factors; Structuring RMBS Deals Using Dynamic Programming; Summary; Notes
CHAPTER 16
Using Derivatives in Portfolio Management 627
Using Derivatives in Equity Portfolio Management;
Using Derivatives in Bond Portfolio Management;
Using Futures to Implement an Asset Allocation Decision; Measuring Portfolio Risk When the Portfolio Contains Derivatives; Summary; Notes
PART FIVE
Capital Budgeting Decisions CHAPTER 17
Capital Budgeting under Uncertainty 653
Classifying Investment Projects; Investment Decisions and Wealth Maximization; Evaluating Project Risk;
Case Study; Managing Portfolios of Projects; Summary;
Software Hints; Notes
CHAPTER 18
Real Options 707
Types of Real Options; Real Options and Financial Options; New View of NPV; Option to Expand;
Option to Abandon; More Real Options Examples;
Estimation of Inputs for Real Option Valuation Models; 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 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 simulation, 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
Acknowledgments
I
n 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
F
inance 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 decisions 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 recent developments 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 optimization 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 optimization 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 Pricing 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 sampling, 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 PART
Fundamental Concepts
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CHAPTER 2
Important Finance Concepts
T
his 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 duration, convexity, key rate duration, and total return.2 . 1 B A S I C T H E O R Y O F I N T E R E S T
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
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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
reff=(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, er, 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·er·1at the end of the first year, C·er·2at the end of the second year, and C·er·nafter 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·er·(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 dn) 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
er·n =CF ·e−r·n. In this case, the discount factor is dn=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.
