# Regression

FredaDuan

QUANTITATIVE INVESTMENT

ANALYSIS

Second Edition

Richard A. DeFusco, CFA

Dennis W. McLeavey, CFA

Jerald E. Pinto, CFA

David E. Runkle, CFA

John Wiley & Sons, Inc.

QUANTITATIVE INVESTMENT

ANALYSIS

Second Edition

Richard A. DeFusco, CFA

Dennis W. McLeavey, CFA

Jerald E. Pinto, CFA

David E. Runkle, CFA

John Wiley & Sons, Inc.

Copyright c⃝ 2004, 2007 by CFA Institute. 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:

Quantitative investment analysis / Richard A. DeFusco . . . [et al.].— 2nd ed.

p. cm.—(The CFA Institute investment series) Includes bibliographical references. ISBN-13 978-0-470-05220-4 (cloth) ISBN-10 0-470-05220-1 (cloth)

1. Investment analysis—Mathematical models. I. DeFusco, Richard Armand.

HG4529.Q35 2006 332.601’5195—dc22

2006052578 Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

To Margo, Rachel, and Rebekah R.A.D.

To Jan, Christine, and Andy D.W.M.

In memory of Irwin T. Vanderhoof, CFA J.E.P.

To Patricia, Anne, and Sarah D.E.R.

CONTENTS

Foreword xiii

Acknowledgments xvii

Introduction xix

CHAPTER 1 The Time Value of Money 1

1 Introduction 1 2 Interest Rates: Interpretation 1 3 The Future Value of a Single Cash Flow 3

3.1 The Frequency of Compounding 8 3.2 Continuous Compounding 10 3.3 Stated and Effective Rates 12

4 The Future Value of a Series of Cash Flows 13 4.1 Equal Cash Flows—Ordinary Annuity 13 4.2 Unequal Cash Flows 15

5 The Present Value of a Single Cash Flow 15 5.1 Finding the Present Value of a Single Cash Flow 15 5.2 The Frequency of Compounding 17

6 The Present Value of a Series of Cash Flows 19 6.1 The Present Value of a Series of Equal Cash Flows 19 6.2 The Present Value of an Infinite Series of Equal Cash Flows—Perpetuity 23 6.3 Present Values Indexed at Times Other Than t = 0 24 6.4 The Present Value of a Series of Unequal Cash Flows 26

7 Solving for Rates, Number of Periods, or Size of Annuity Payments 27 7.1 Solving for Interest Rates and Growth Rates 27 7.2 Solving for the Number of Periods 30 7.3 Solving for the Size of Annuity Payments 30 7.4 Review of Present and Future Value Equivalence 35 7.5 The Cash Flow Additivity Principle 36

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CHAPTER 2 Discounted Cash Flow Applications 39

1 Introduction 39 2 Net Present Value and Internal Rate of Return 39

2.1 Net Present Value and the Net Present Value Rule 40 2.2 The Internal Rate of Return and the Internal Rate of

Return Rule 42 2.3 Problems with the IRR Rule 45

3 Portfolio Return Measurement 47 3.1 Money-Weighted Rate of Return 47 3.2 Time-Weighted Rate of Return 49

4 Money Market Yields 54

CHAPTER 3 Statistical Concepts and Market Returns 61

1 Introduction 61 2 Some Fundamental Concepts 61

2.1 The Nature of Statistics 62 2.2 Populations and Samples 62 2.3 Measurement Scales 63

3 Summarizing Data Using Frequency Distributions 65 4 The Graphic Presentation of Data 72

4.1 The Histogram 73 4.2 The Frequency Polygon and the Cumulative Frequency

Distribution 74 5 Measures of Central Tendency 76

5.1 The Arithmetic Mean 77 5.2 The Median 81 5.3 The Mode 84 5.4 Other Concepts of Mean 85

6 Other Measures of Location: Quantiles 94 6.1 Quartiles, Quintiles, Deciles, and Percentiles 94 6.2 Quantiles in Investment Practice 98

7 Measures of Dispersion 100 7.1 The Range 100 7.2 The Mean Absolute Deviation 101 7.3 Population Variance and Population Standard

Deviation 103 7.4 Sample Variance and Sample Standard Deviation 106 7.5 Semivariance, Semideviation, and Related Concepts 110 7.6 Chebyshev’s Inequality 111 7.7 Coefficient of Variation 113 7.8 The Sharpe Ratio 115

8 Symmetry and Skewness in Return Distributions 118 9 Kurtosis in Return Distributions 123 10 Using Geometric and Arithmetic Means 127

Contents ix

CHAPTER 4 Probability Concepts 129

1 Introduction 129 2 Probability, Expected Value, and Variance 129 3 Portfolio Expected Return and Variance of Return 152 4 Topics in Probability 161

4.1 Bayes’ Formula 161 4.2 Principles of Counting 166

CHAPTER 5 Common Probability Distributions 171

1 Introduction 171 2 Discrete Random Variables 171

2.1 The Discrete Uniform Distribution 173 2.2 The Binomial Distribution 175

3 Continuous Random Variables 185 3.1 Continuous Uniform Distribution 186 3.2 The Normal Distribution 189 3.3 Applications of the Normal Distribution 197 3.4 The Lognormal Distribution 200

4 Monte Carlo Simulation 206

CHAPTER 6 Sampling and Estimation 215

1 Introduction 215 2 Sampling 215

2.1 Simple Random Sampling 216 2.2 Stratified Random Sampling 217 2.3 Time-Series and Cross-Sectional Data 219

3 Distribution of the Sample Mean 221 3.1 The Central Limit Theorem 222

4 Point and Interval Estimates of the Population Mean 225 4.1 Point Estimators 225 4.2 Confidence Intervals for the Population Mean 227 4.3 Selection of Sample Size 233

5 More on Sampling 235 5.1 Data-Mining Bias 236 5.2 Sample Selection Bias 238 5.3 Look-Ahead Bias 240 5.4 Time-Period Bias 240

CHAPTER 7 Hypothesis Testing 243

1 Introduction 243 2 Hypothesis Testing 244

x Contents

3 Hypothesis Tests Concerning the Mean 253 3.1 Tests Concerning a Single Mean 254 3.2 Tests Concerning Differences between Means 261 3.3 Tests Concerning Mean Differences 265

4 Hypothesis Tests Concerning Variance 269 4.1 Tests Concerning a Single Variance 269 4.2 Tests Concerning the Equality (Inequality) of Two Variances 271

5 Other Issues: Nonparametric Inference 275 5.1 Tests Concerning Correlation: The Spearman Rank

Correlation Coefficient 276 5.2 Nonparametric Inference: Summary 279

CHAPTER 8 Correlation and Regression 281

1 Introduction 281 2 Correlation Analysis 281

2.1 Scatter Plots 281 2.2 Correlation Analysis 282 2.3 Calculating and Interpreting the Correlation Coefficient 283 2.4 Limitations of Correlation Analysis 287 2.5 Uses of Correlation Analysis 289 2.6 Testing the Significance of the Correlation Coefficient 297

3 Linear Regression 300 3.1 Linear Regression with One Independent Variable 300 3.2 Assumptions of the Linear Regression Model 303 3.3 The Standard Error of Estimate 306 3.4 The Coefficient of Determination 309 3.5 Hypothesis Testing 310 3.6 Analysis of Variance in a Regression with One Independent Variable 318 3.7 Prediction Intervals 321 3.8 Limitations of Regression Analysis 324

CHAPTER 9 Multiple Regression and Issues in Regression Analysis 325

1 Introduction 325 2 Multiple Linear Regression 325

2.1 Assumptions of the Multiple Linear Regression Model 331 2.2 Predicting the Dependent Variable in a Multiple Regression Model 336 2.3 Testing Whether All Population Regression Coefficients Equal Zero 338 2.4 Adjusted R2 340

3 Using Dummy Variables in Regressions 341 4 Violations of Regression Assumptions 345

4.1 Heteroskedasticity 345 4.2 Serial Correlation 351 4.3 Multicollinearity 356

Contents xi

4.4 Heteroskedasticity, Serial Correlation, Multicollinearity: Summarizing the Issues 359

5 Model Specification and Errors in Specification 359 5.1 Principles of Model Specification 359 5.2 Misspecified Functional Form 360 5.3 Time-Series Misspecification (Independent Variables Correlated

with Errors) 368 5.4 Other Types of Time-Series Misspecification 372

6 Models with Qualitative Dependent Variables 372

CHAPTER 10 Time-Series Analysis 375

1 Introduction 375 2 Challenges of Working with Time Series 375 3 Trend Models 377

3.1 Linear Trend Models 377 3.2 Log-Linear Trend Models 380 3.3 Trend Models and Testing for Correlated Errors 385

4 Autoregressive (AR) Time-Series Models 386 4.1 Covariance-Stationary Series 386 4.2 Detecting Serially Correlated Errors in an Autoregressive Model 387 4.3 Mean Reversion 391 4.4 Multiperiod Forecasts and the Chain Rule of Forecasting 391 4.5 Comparing Forecast Model Performance 394 4.6 Instability of Regression Coefficients 397

5 Random Walks and Unit Roots 399 5.1 Random Walks 400 5.2 The Unit Root Test of Nonstationarity 403

6 Moving-Average Time-Series Models 407 6.1 Smoothing Past Values with an n-Period Moving Average 407 6.2 Moving-Average Time-Series Models for Forecasting 409

7 Seasonality in Time-Series Models 412 8 Autoregressive Moving-Average Models 416 9 Autoregressive Conditional Heteroskedasticity Models 417 10 Regressions with More than One Time Series 420 11 Other Issues in Time Series 424 12 Suggested Steps in Time-Series Forecasting 425

CHAPTER 11 Portfolio Concepts 429

1 Introduction 429 2 Mean–Variance Analysis 429

2.1 The Minimum-Variance Frontier and Related Concepts 430 2.2 Extension to the Three-Asset Case 439 2.3 Determining the Minimum-Variance Frontier for Many Assets 442 2.4 Diversification and Portfolio Size 445

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2.5 Portfolio Choice with a Risk-Free Asset 449 2.6 The Capital Asset Pricing Model 458 2.7 Mean–Variance Portfolio Choice Rules: An Introduction 460

3 Practical Issues in Mean–Variance Analysis 464 3.1 Estimating Inputs for Mean–Variance Optimization 464 3.2 Instability in the Minimum-Variance Frontier 470

4 Multifactor Models 473 4.1 Factors and Types of Multifactor Models 474 4.2 The Structure of Macroeconomic Factor Models 475 4.3 Arbitrage Pricing Theory and the Factor Model 478 4.4 The Structure of Fundamental Factor Models 484 4.5 Multifactor Models in Current Practice 485 4.6 Applications 493 4.7 Concluding Remarks 509

Appendices 511

References 521

Glossary 527

About the CFA Program 541

About the Authors 543

Index 545

FOREWORD

HOW QUANTITATIVE INVESTMENT ANALYSIS CAN IMPROVE PORTFOLIO DECISION MAKING

I am a Quant. By my own self-admission, I use quantitative investment techniques in the management of investment portfolios. However, when I tell people that I am a Quant, they often respond: ‘‘But Mark, aren’t you a lawyer?’’ Well, yes, but . . .

The fact is that Quants come from all walks of life. Whether we are called Quants, Quant Jocks, Gear Heads, Computer Monkeys, or any of the other monikers that are attached to investors who like to scribble equations on a piece of paper, we all share a common denominator—the use of quantitative analysis to make better investment decisions. You don’t have to be a rocket scientist with a Ph.D. in an esoteric mathematical field to be a Quant (although there are, I suspect, several former rocket scientists who have found working in the financial markets to be both fun and profitable). Anyone can become a Quant—even a lawyer.

But let’s take a step back. Why should any investor want to use quantitative tools in the management of investment portfolios? There are three reasons why Quants are so popular.

First, the financial markets are very complicated places. There are many interwoven variables that can affect the price of securities in an investment portfolio. For example, the stock price of a public company can be affected by macroeconomic factors such as the level of interest rates, current account deficits, government spending, and economic cycles. These factors may affect the cost of capital at which a corporation finances its new projects, or influence the spending patterns of the company’s customers, or provide economic impetus through government spending programs.

In addition to macro variables, the value of a company’s stock can be affected by factors that are peculiar to the company itself. Factors such as cash flow, working capital, book- to-market value, earnings growth rates, dividend policy, and debt-to-equity ratios affect the individual value of each public company. These are considered to be the fundamental factors that have an impact on the specific company as opposed to the broader stock market.

Then we come to the financial market variables that affect a company’s valuation. Its ‘‘beta’’ or measure of systematic risk will impact the expected return for the company and, in turn, its stock price. The famous Capital Asset Pricing Model that measures a stock’s beta is really just a linear regression equation of the type described in Chapter 8.

Last, there are behavioral variables that can affect security values. Such behavior as herding, overconfidence, overreaction to earnings announcements, and momentum trading can all impact the price of a company’s stock. These behavioral variables can have a lasting impact on a stock price (remember the technology bubble of 1998–2001 when tech stocks were going to take over the world?) as well as generate a significant amount of ‘‘noise’’ around a security’s true value.

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Considering all of these variables together at one time to determine the true value of a security can be an overwhelming task without some framework in which to analyze their impact. It is simply not possible for the human mind alone (at least, not mine) to be able to weigh the impact of individual company specific factors such as price-to-earnings ratios, macroeconomic variables such as government spending programs, investor behavioral patterns such as momentum trading, and other potentially influential variables in a rigorous fashion within the human brain.

This is where Quantitative Investment Analysis can help. Factor modeling techniques such as those described in Chapter 11 can be used to supplement the intuition of the human mind to produce a quantitative framework that digests the large number of plausible variables that can impact the price of a security. Further, given the many variables that can affect a security’s value, it is not possible to consider each variable in isolation. The economic factors that cause a security’s price to go up or down are interwoven in a complex web such that the variables must be considered together to determine their collective impact on the price of a security.

This is where the value of Chapters 8 and 9 are most useful. These two chapters provide the basic knowledge for building regression equations to study the impact of economic factors on security prices. The regression techniques provided in Chapters 8 and 9 can be used to filter out which variables have a significant impact on the price of a security, and which variables just provide ‘‘noise.’’

In addition, Chapter 9 introduces the reader to ‘‘dummy variables.’’ Despite their name, you don’t have to be a dummy like me to use them. Dummy variables are a neat way to study different states of the world and their impact on security prices. They are often referred to as ‘‘binary’’ variables because they divide the world into two states for observation, for example, financial markets up versus financial markets down; Republicans in control of the White House versus Democrats in control of the White House; Chicago Cubs win (almost never) versus Chicago Cubs lose; and so on. This last variable—the record of the Chicago Cubs—I can attest has no impact on security valuations, although, as a long-standing and suffering Cub fan, it does have an impact on my morale.

As another example, consider a recent research paper where I studied the behavior of private equity managers in the way they price their private equity portfolios depending on whether the public stock markets were doing well versus when the public stock markets were doing poorly. To conduct this analysis, I ran a regression equation using dummy variables to divide the world into two states: up public stock markets versus down public stock markets. By using dummy variables in this manner, I was able to observe different behavioral patterns of private equity managers in how they marked up or down their private equity portfolios depending on the performance of the public stock markets.

The second reason Quantitative Investment Analysis will add value to the reader is that it provides the basic tools to consider a breadth of economic factors and securities. It is not only the fact that there are many interwoven economic variables that impact the value of a security, the sheer number of securities in the market place can be daunting. Therefore, most investors only look at a subset of the investable securities in the market.

Consider the U.S. stock market. Generally, this market is divided into three categories based on company size: large-cap, mid-cap, and small-cap stocks. This division is less so because there might be ‘‘size’’ effects in valuation, but rather, because of the pragmatic limitation that asset managers simply cannot analyze stocks beyond a certain number. So traditional fundamental investors select different parts of the U.S. stock market in which to conduct their security analysis. However, the division of the stock market into size categories effectively establishes barriers for investment managers. There is no reason, for example, why a portfolio

Foreword xv

manager with insight into how earnings surprises affect stock prices cannot invest across the whole range of stock market capitalization.

This is where Chapters 6 and 7 can be useful. The quantitative skills of sampling, estimation, and hypothesis testing can be used to analyze large baskets of data. This allows portfolio managers to invest across a broad universe of stocks, breaking down traditional barriers such as cap-size restrictions. When viewed in this light, quantitative analysis does not displace the fundamental stock picking skills of traditional asset managers. Rather, quantitative analysis extends the portfolio manager’s insight with respect to company, macro, and market variables to a broader array of investment opportunities.

This also has implications for the statistical tools and probability concepts provided in Chapters 3 and 4. The larger the data set to be analyzed the greater the reliability of the parameter estimation derived from that data set. Breadth of economic analysis will improve not only the statistical reliability of the quantitative analysis, but will also increase the predictability of the relationships between economic factors and stock price movement. The statistical tools provided in this book allow the portfolio manager to realize the full potential of his or her skill across a larger universe of securities than may have previously been achieved.

Another example might help. Every year the California Public Employees’ Retirement System (CalPERS), my former employer, publishes a list of the most poorly governed companies in the United States. This list has now been published for 16 years and has been very successful. Early on in the process, the selection was conducted on a subset of the U.S. stock market. However, this process has evolved to consider every U.S. stock held in CalPERS’s portfolio regardless of stock market capitalization range. This requires the analysis of up to 1,800 stocks every year based on both economic factors and governance variables. The sheer number of securities in this data sample could not be analyzed without the application of quantitative screening tools to expand the governance universe for CalPERS.

Last, Quantitative Investment Analysis can provide a certain amount of discipline to the investment process. We are all human, and as humans, we are subject to making mistakes. If I were to recount all of the investment mistakes that I have made over my career, this Foreword would exceed the length of the chapters in this book. Just as a brief example, one of my ‘‘better calls’’ was Starbucks Coffee. Early on when Starbucks was just getting started, I visited one of their shops to see what the buzz was all about. At that time a Latte Grande was selling for about $1.50. I recall that I thought this was an outrageous price and I can remember distinctly saying: ‘‘Oh, this is a dumb idea, this will never catch on!’’ Ah yes . . .

So back to quantitative techniques—how can they help? In this instance, they could have helped me remove my human biases and to think more analytically about Starbucks’ prospects. If I had taken the time to conduct an empirical review using the quantitative tools provided in this text, I would have seen the fundamental value underlying that buck-fifty Latte.

The fact is that we are all subject to behavioral biases such as overconfidence, momentum, and overreaction. Not only can these be analyzed as discussed above, they can be revealed and discounted when we make our investment decisions. Perhaps the single biggest behavioral hurdle to overcome for investors is the inability to sell a security when its value declines. All too often we become almost emotionally attached to the securities in our portfolio such that we find it hard to sell a security that begins to decline in price.

Yet, this is precisely, where Quantitative Investment Analysis can help because it is dispassionate. Quantitative tools and modeling techniques can take the emotion and cognitive biases out of the portfolio decision-making process. As portfolio managers, our goal is to be objective, critical, and demanding. Unfortunately, sometimes our embedded habits and opinions can get in the way. However, quantitative models are unemotional and they can root

xvi Foreword

out our cognitive biases in a way that we simply cannot do ourselves by looking in the mirror (in fact, when I look in the mirror I see someone who is six feet and four inches tall and incredibly good looking but then my wife Mary reminds me that I am only six feet and one inch tall and she had better offers).

All in all, the investor will appreciate the methods, models, and techniques provided in this text. This book serves as an excellent introduction to those investors who are just beginning to use quantitative tools in their portfolio management process as well as an excellent reference guide for those already converted. Quantitative investing is not difficult to grasp—even a lawyer can do it.

Mark J. P. Anson CEO, Hermes Pensions Management CEO, British Telecomm Pension Scheme [email protected]

ACKNOWLEDGMENTS

W e would like to thank the many individuals who played important roles in producingthis book. Robert R. Johnson, CFA, Managing Director of the CFA and CIPM Programs Division,

saw the need for specialized curriculum materials and initiated this project. We appreciate his support for the timely revision of this textbook. Senior executives in the CFA Program Division have generously given their advice and time in the writing of both editions of this book. Philip J. Young, CFA, provided continuous assistance in writing the book’s learning outcome statements and participated in final manuscript reviews. Jan R. Squires, CFA, contributed an orientation stressing motivation and testability. Mary K. Erickson, CFA, made contributions to the accuracy of the text. John D. Stowe, CFA, supplied suggestions for revising several chapters.

The Executive Advisory Board of the Candidate Curriculum Committee provided invaluable input: James Bronson, CFA, Chair; Peter Mackey, CFA, Immediate Past Chair; and members, Alan Meder, CFA, Victoria Rati, CFA, and Matt Scanlan, CFA, as well as the Candidate Curriculum Committee Working Body.

The manuscript reviewers for this edition were Philip Fanara, Jr., CFA; Jane Farris, CFA; David M. Jessop; Lisa M. Joublanc, CFA; Asjeet S. Lamba, CFA; Mario Lavallee, CFA; William L. Randolph, CFA; Eric N. Remole; Vijay Singal, CFA; Zoe L. Van Schyndel, CFA; Charlotte Weems, CFA; and Lavone F. Whitmer, CFA. We thank them for their excellent work.

We also appreciate the many comments received from those who used the first edition. Jacques R. Gagne, CFA, Gregory M. Noronha, CFA, and Sanjiv Sabherwal provided

highly detailed proofreading of the individual chapters. We thank each for their dedicated and painstaking work. We are also indebted to Dr. Sabherwal for his expert assistance in running regressions, revising in-chapter examples, and creating some of the end-of-chapter problems/solutions.

Fiona D. Russell provided incisive copyediting that substantially contributed to the book’s accuracy and readability.

Wanda A. Lauziere of the CFA Program Division, the project manager for the revision, expertly guided the manuscript from planning through production and made many other contributions to all aspects of the revision.

Finally, we thank Ibbotson Associates of Chicago for generously providing us with EnCorr AnalyzerTM.

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INTRODUCTION

CFA Institute is pleased to provide you with this Investment Series covering major areas in the field of investments. These texts are thoroughly grounded in the highly regarded CFA Program Candidate Body of Knowledge (CBOK®) that draws upon hundreds of practicing investment professionals and serves as the anchor for the three levels of the CFA Examinations. In the year this series is being launched, more than 120,000 aspiring investment professionals will each devote over 250 hours of study to master this material as well as other elements of the Candidate Body of Knowledge in order to obtain the coveted CFA charter. We provide these materials for the same reason we have been chartering investment professionals for over 40 years: to improve the competency and ethical character of those serving the capital markets.

PARENTAGE

One of the valuable attributes of this series derives from its parentage. In the 1940s, a handful of societies had risen to form communities that revolved around common interests and work in what we now think of as the investment industry.

Understand that the idea of purchasing common stock as an investment—as opposed to casino speculation—was only a couple of decades old at most. We were only 10 years past the creation of the U.S. Securities and Exchange Commission and laws that attempted to level the playing field after robber baron and stock market panic episodes.

In January 1945, in what is today CFA Institute Financial Analysts Journal , a funda- mentally driven professor and practitioner from Columbia University and Graham-Newman Corporation wrote an article making the case that people who research and manage portfolios should have some sort of credential to demonstrate competence and ethical behavior. This person was none other than Benjamin Graham, the father of security analysis and future mentor to a well-known modern investor, Warren Buffett.

The idea of creating a credential took a mere 16 years to drive to execution but by 1963, 284 brave souls, all over the age of 45, took an exam and launched the CFA credential. What many do not fully understand was that this effort had at its root a desire to create a profession where its practitioners were professionals who provided investing services to individuals in need. In so doing, a fairer and more productive capital market would result.

A profession—whether it be medicine, law, or other—has certain hallmark characteristics. These characteristics are part of what attracts serious individuals to devote the energy of their life’s work to the investment endeavor. First, and tightly connected to this Series, there must be a body of knowledge. Second, there needs to be some entry requirements such as those required to achieve the CFA credential. Third, there must be a commitment to continuing education. Fourth, a profession must serve a purpose beyond one’s direct selfish interest. In this case, by properly conducting one’s affairs and putting client interests first, the investment

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professional can work …