Quantitative Investment Analysis
Description
Part of the CFA Institute Investment Series, this authoritative guide is relevant the world over and will facilitate your mastery of quantitative methods and their application in todays investment process. This updated edition provides all the statistical tools and latest information you need to be a confident and knowledgeable investor. This edition expands coverage of Machine Learning algorithms and the role of Big Data in an investment context along with capstone chapters in applying these techniques to factor modeling, risk management and backtesting and simulation in investment strategies.
The authors go to great lengths to ensure an even treatment of subject matter, consistency of mathematical notation, and continuity of topic coverage that is critical to the learning process. Well suited for motivated individuals who learn on their own, as well as a general reference, this complete resource delivers clear, example-driven coverage of a wide range of quantitative methods. Inside you'll find:
* Learning outcome statements (LOS) specifying the objective of each chapter
* A diverse variety of investment-oriented examples both aligned with the LOS and reflecting the realities of todays investment world
* A wealth of practice problems, charts, tables, and graphs to clarify and reinforce the concepts and tools of quantitative investment management
You can choose to sharpen your skills by furthering your hands-on experience in the Quantitative Investment Analysis Workbook, 4th Edition (sold separately)--an essential guide containing learning outcomes and summary overview sections, along with challenging problems and solutions.
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CFA Institute is the global association of investment professionals that sets the standard for professional excellence and credentials. The organization is a champion for ethical behavior in investment markets and a respected source of knowledge in the global financial community. The end goal: to create an environment where investors' interests come first, markets function at their best, and economies grow. CFA Institute has more than 155,000 members in 165 countries and territories, including 150,000 CFA® charterholders, and 148 member societies. For more information, visit www.cfainstitute.org.
Content
Preface xv
Acknowledgments xvii
About the CFA Institute Investment Series xix
Chapter 1 The Time Value of Money 1
Learning Outcomes 1
1. Introduction 1
2. Interest Rates: Interpretation 2
3. The Future Value of a Single Cash Flow 4
3.1. The Frequency of Compounding 9
3.2. Continuous Compounding 11
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 14
4.2. Unequal Cash Flows 15
5. The Present Value of a Single Cash Flow 16
5.1. Finding the Present Value of a Single Cash Flow 16
5.2. The Frequency of Compounding 18
6. The Present Value of a Series of Cash Flows 20
6.1. The Present Value of a Series of Equal Cash Flows 20
6.2. The Present Value of an Infinite Series of Equal Cash Flows-Perpetuity 24
6.3. Present Values Indexed at Times Other than t = 0 25
6.4. The Present Value of a Series of Unequal Cash Flows 27
7. Solving for Rates, Number of Periods, or Size of Annuity Payments 27
7.1. Solving for Interest Rates and Growth Rates 28
7.2. Solving for the Number of Periods 30
7.3. Solving for the Size of Annuity Payments 31
7.4. Review of Present and Future Value Equivalence 35
7.5. The Cash Flow Additivity Principle 37
8. Summary 38
Practice Problems 39
Chapter 2 Organizing, Visualizing, and Describing Data 45
Learning Outcomes 45
1. Introduction 45
2. Data Types 46
2.1. Numerical versus Categorical Data 46
2.2. Cross-Sectional versus Time-Series versus Panel Data 49
2.3. Structured versus Unstructured Data 50
3. Data Summarization 54
3.1. Organizing Data for Quantitative Analysis 54
3.2. Summarizing Data Using Frequency Distributions 57
3.3. Summarizing Data Using a Contingency Table 63
4. Data Visualization 68
4.1. Histogram and Frequency Polygon 68
4.2. Bar Chart 69
4.3. Tree-Map 73
4.4. Word Cloud 73
4.5. Line Chart 75
4.6. Scatter Plot 77
4.7. Heat Map 81
4.8. Guide to Selecting among Visualization Types 82
5. Measures of Central Tendency 85
5.1. The Arithmetic Mean 85
5.2. The Median 90
5.3. The Mode 92
5.4. Other Concepts of Mean 92
6. Other Measures of Location: Quantiles 102
6.1. Quartiles, Quintiles, Deciles, and Percentiles 103
6.2. Quantiles in Investment Practice 108
7. Measures of Dispersion 109
7.1. The Range 109
7.2. The Mean Absolute Deviation 109
7.3. Sample Variance and Sample Standard Deviation 111
7.4. Target Downside Deviation 114
7.5. Coefficient of Variation 117
8. The Shape of the Distributions: Skewness 119
9. The Shape of the Distributions: Kurtosis 121
10. Correlation between Two Variables 125
10.1. Properties of Correlation 126
10.2. Limitations of Correlation Analysis 129
11. Summary 132
Practice Problems 135
Chapter 3 Probability Concepts 147
Learning Outcomes 147
1. Introduction 148
2. Probability, Expected Value, and Variance 148
3. Portfolio Expected Return and Variance of Return 171
4. Topics in Probability 180
4.1. Bayes' Formula 180
4.2. Principles of Counting 184
5. Summary 188
References 190
Practice Problem 190
Chapter 4 Common Probability Distributions 195
Learning Outcomes 195
1. Introduction to Common Probability Distributions 196
2. Discrete Random Variables 196
2.1. The Discrete Uniform Distribution 198
2.2. The Binomial Distribution 200
3. Continuous Random Variables 210
3.1. Continuous Uniform Distribution 210
3.2. The Normal Distribution 214
3.3. Applications of the Normal Distribution 220
3.4. The Lognormal Distribution 222
4. Introduction to Monte Carlo Simulation 228
5. Summary 231
References 233
Practice Problems 234
Chapter 5 Sampling and Estimation 241
Learning Outcomes 241
1. Introduction 242
2. Sampling 242
2.1. Simple Random Sampling 242
2.2. Stratified Random Sampling 244
2.3. Time-Series and Cross-Sectional Data 245
3. Distribution of the Sample Mean 248
3.1. The Central Limit Theorem 248
4. Point and Interval Estimates of the Population Mean 251
4.1. Point Estimators 252
4.2. Confidence Intervals for the Population Mean 253
4.3. Selection of Sample Size 259
5. More on Sampling 261
5.1. Data-Mining Bias 261
5.2. Sample Selection Bias 264
5.3. Look-Ahead Bias 265
5.4. Time-Period Bias 266
6. Summary 267
References 269
Practice Problems 270
Chapter 6 Hypothesis Testing 275
Learning Outcomes 275
1. Introduction 276
2. Hypothesis Testing 277
3. Hypothesis Tests Concerning the Mean 287
3.1. Tests Concerning a Single Mean 287
3.2. Tests Concerning Differences between Means 294
3.3. Tests Concerning Mean Differences 299
4. Hypothesis Tests Concerning Variance and Correlation 303
4.1. Tests Concerning a Single Variance 303
4.2. Tests Concerning the Equality (Inequality) of Two Variances 305
4.3. Tests Concerning Correlation 308
5. Other Issues: Nonparametric Inference 310
5.1. Nonparametric Tests Concerning Correlation: The Spearman Rank Correlation Coefficient 312
5.2. Nonparametric Inference: Summary 313
6. Summary 314
References 317
Practice Problems 317
Chapter 7 Introduction to Linear Regression 327
Learning Outcomes 327
1. Introduction 328
2. Linear Regression 328
2.1. Linear Regression with One Independent Variable 328
3. Assumptions of the Linear Regression Model 332
4. The Standard Error of Estimate 335
5. The Coefficient of Determination 337
6. Hypothesis Testing 339
7. Analysis of Variance in a Regression with One Independent Variable 347
8. Prediction Intervals 350
9. Summary 353
References 354
Practice Problems 354
Chapter 8 Multiple Regression 365
Learning Outcomes 365
1. Introduction 366
2. Multiple Linear Regression 366
2.1. Assumptions of the Multiple Linear Regression Model 372
2.2. Predicting the Dependent Variable in a Multiple Regression Model 376
2.3. Testing Whether All Population Regression Coefficients Equal Zero 378
2.4. Adjusted R2 380
3. Using Dummy Variables in Regressions 381
3.1. Defining a Dummy Variable 381
3.2. Visualizing and Interpreting Dummy Variables 382
3.3. Testing for Statistical Significance 384
4. Violations of Regression Assumptions 387
4.1. Heteroskedasticity 388
4.2. Serial Correlation 394
4.3. Multicollinearity 398
4.4. Heteroskedasticity, Serial Correlation, Multicollinearity: Summarizing the Issues 401
5. Model Specification and Errors in Specification 401
5.1. Principles of Model Specification 402
5.2. Misspecified Functional Form 402
5.3. Time-Series Misspecification (Independent Variables Correlated with Errors) 410
5.4. Other Types of Time-Series Misspecification 414
6. Models with Qualitative Dependent Variables 414
6.1. Models with Qualitative Dependent Variables 414
7. Summary 422
References 425
Practice Problems 426
Chapter 9 Time-Series Analysis 451
Learning Outcomes 451
1. Introduction to Time-Series Analysis 452
2. Challenges of Working with Time Series 454
3. Trend Models 454
3.1. Linear Trend Models 455
3.2. Log-Linear Trend Models 458
3.3. Trend Models and Testing for Correlated Errors 463
4. Autoregressive (AR) Time-Series Models 464
4.1. Covariance-Stationary Series 465
4.2. Detecting Serially Correlated Errors in an Autoregressive Model 466
4.3. Mean Reversion 469
4.4. Multiperiod Forecasts and the Chain Rule of Forecasting 470
4.5. Comparing Forecast Model Performance 473
4.6. Instability of Regression Coefficients 475
5. Random Walks and Unit Roots 478
5.1. Random Walks 478
5.2. The Unit Root Test of Nonstationarity 482
6. Moving-Average Time-Series Models 486
6.1. Smoothing Past Values with an n-Period Moving Average 486
6.2. Moving-Average Time-Series Models for Forecasting 489
7. Seasonality in Time-Series Models 491
8. Autoregressive Moving-Average Models 496
9. Autoregressive Conditional Heteroskedasticity Models 497
10. Regressions with More than One Time Series 500
11. Other Issues in Time Series 504
12. Suggested Steps in Time-Series Forecasting 505
13. Summary 507
References 508
Practice Problems 509
Chapter 10 Machine Learning 527
Learning Outcomes 527
1. Introduction 527
2. Machine Learning and Investment Management 528
3. What is Machine Learning? 529
3.1. Defining Machine Learning 529
3.2. Supervised Learning 529
3.3. Unsupervised Learning 531
3.4. Deep Learning and Reinforcement Learning 531
3.5. Summary of ML Algorithms and How to Choose among Them 532
4. Overview of Evaluating ML Algorithm Performance 533
4.1. Generalization and Overfitting 534
4.2. Errors and Overfitting 534
4.3. Preventing Overfitting in Supervised Machine Learning 537
5. Supervised Machine Learning Algorithms 539
5.1. Penalized Regression 539
5.2. Support Vector Machine 541
5.3. K-Nearest Neighbor 542
5.4. Classification and Regression Tree 544
5.5. Ensemble Learning and Random Forest 547
6. Unsupervised Machine Learning Algorithms 559
6.1. Principal Components Analysis 560
6.2. Clustering 563
7. Neural Networks, Deep Learning Nets, and Reinforcement Learning 575
7.1. Neural Networks 575
7.2. Deep Learning Neural Networks 578
7.3. Reinforcement Learning 579
8. Choosing an Appropriate ML Algorithm 589
9. Summary 590
References 593
Practice Problems 593
Chapter 11 Big Data Projects 597
Learning Outcomes 597
1. Introduction 597
2. Big Data in Investment Management 598
3. Steps in Executing a Data Analysis Project: Financial Forecasting with Big Data 599
4. Data Preparation and Wrangling 603
4.1. Structured Data 604
4.2. Unstructured (Text) Data 610
5. Data Exploration Objectives and Methods 617
5.1. Structured Data 618
5.2. Unstructured Data: Text Exploration 622
6. Model Training 629
6.1. Structured and Unstructured Data 630
7. Financial Forecasting Project: Classifying and Predicting Sentiment for Stocks 639
7.1. Text Curation, Preparation, and Wrangling 640
7.2. Data Exploration 644
7.3. Model Training 654
7.4. Results and Interpretation 658
8. Summary 664
Practice Problems 665
Chapter 12 Using Multifactor Models 675
Learning Outcomes 675
1. Introduction 675
2. Multifactor Models and Modern Portfolio Theory 676
3. Arbitrage Pricing Theory 677
4. Multifactor Models: Types 683
4.1. Factors and Types of Multifactor Models 683
4.2. The Structure of Macroeconomic Factor Models 684
4.3. The Structure of Fundamental Factor Models 687
4.4. Fixed-Income Multifactor Models 691
5. Multifactor Models: Selected Applications 695
5.1. Factor Models in Return Attribution 696
5.2. Factor Models in Risk Attribution 698
5.3. Factor Models in Portfolio Construction 703
5.4. How Factor Considerations Can Be Useful in Strategic Portfolio Decisions 705
6. Summary 706
References 707
Practice Problems 708
Chapter 13 Measuring and Managing Market Risk 713
Learning Outcomes 713
1. Introduction 714
2. Understanding Value at Risk 714
2.1. Value at Risk: Formal Definition 715
2.2. Estimating VaR 718
2.3. Advantages and Limitations of VaR 730
2.4. Extensions of VaR 733
3. Other Key Risk Measures-Sensitivity and Scenario Measures 735
3.1. Sensitivity Risk Measures 736
3.2. Scenario Risk Measures 740
3.3. Sensitivity and Scenario Risk Measures and VaR 746
4. Using Constraints in Market Risk Management 750
4.1. Risk Budgeting 751
4.2. Position Limits 752
4.3. Scenario Limits 752
4.4. Stop-Loss Limits 753
4.5. Risk Measures and Capital Allocation 753
5. Applications of Risk Measures 755
5.1. Market Participants and the Different Risk Measures They Use 755
6. Summary 764
References 766
Practice Problems 766
Chapter 14 Backtesting and Simulation 775
Learning Outcomes 775
1. Introduction 775
2. The Objectives of Backtesting 776
3. The Backtesting Process 776
3.1. Strategy Design 777
3.2. Rolling Window Backtesting 778
3.3. Key Parameters in Backtesting 779
3.4. Long/Short Hedged Portfolio Approach 781
3.5. Pearson and Spearman Rank IC 785
3.6. Univariate Regression 789
3.7. Do Different Backtesting Methodologies Tell the Same Story? 789
4. Metrics and Visuals Used in Backtesting 792
4.1. Coverage 792
4.2. Distribution 794
4.3. Performance Decay, Structural Breaks, and Downside Risk 797
4.4. Factor Turnover and Decay 797
5. Common Problems in Backtesting 801
5.1. Survivorship Bias 801
5.2. Look-Ahead Bias 804
6. Backtesting Factor Allocation Strategies 807
6.1. Setting the Scene 808
6.2. Backtesting the Benchmark and Risk Parity Strategies 808
7. Comparing Methods of Modeling Randomness 813
7.1. Factor Portfolios and BM and RP Allocation Strategies 814
7.2. Factor Return Statistical Properties 815
7.3. Performance Measurement and Downside Risk 819
7.4. Methods to Account for Randomness 821
8. Scenario Analysis 824
9. Historical Simulation versus Monte Carlo Simulation 828
10. Historical Simulation 830
11. Monte Carlo Simulation 835
12. Sensitivity Analysis 840
13. Summary 848
References 849
Practice Problems 849
Appendices 855
Glossary 865
About the Authors 883
About the CFA Program 885
Index 887
CHAPTER 1
THE TIME VALUE OF MONEY
Richard A. DeFusco, PhD, CFA
Dennis W. McLeavey, DBA, CFA
Jerald E. Pinto, PhD, CFA
David E. Runkle, PhD, CFA
LEARNING OUTCOMES
The candidate should be able to:
- interpret interest rates as required rates of return, discount rates, or opportunity costs;
- explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk;
- calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;
- solve time value of money problems for different frequencies of compounding;
- calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;
- demonstrate the use of a time line in modeling and solving time value of money problems.
1. INTRODUCTION
As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing, if we are shopping for a loan. As investment analysts, much of our work also involves evaluating transactions with present and future cash flows. When we place a value on any security, for example, we are attempting to determine the worth of a stream of future cash flows. To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore, a smaller amount of money now may be equivalent in value to a larger amount received at a future date. The time value of money as a topic in investment mathematics deals with equivalence relationships between cash flows with different dates. Mastery of time value of money concepts and techniques is essential for investment analysts.
The chapter1 is organized as follows: Section 2 introduces some terminology used throughout the chapter and supplies some economic intuition for the variables we will discuss. Section 3 tackles the problem of determining the worth at a future point in time of an amount invested today. Section 4 addresses the future worth of a series of cash flows. These two sections provide the tools for calculating the equivalent value at a future date of a single cash flow or series of cash flows. Sections 5 and 6 discuss the equivalent value today of a single future cash flow and a series of future cash flows, respectively. In Section 7, we explore how to determine other quantities of interest in time value of money problems.
2. INTEREST RATES: INTERPRETATION
In this chapter, we will continually refer to interest rates. In some cases, we assume a particular value for the interest rate; in other cases, the interest rate will be the unknown quantity we seek to determine. Before turning to the mechanics of time value of money problems, we must illustrate the underlying economic concepts. In this section, we briefly explain the meaning and interpretation of interest rates.
Time value of money concerns equivalence relationships between cash flows occurring on different dates. The idea of equivalence relationships is relatively simple. Consider the following exchange: You pay $10,000 today and in return receive $9,500 today. Would you accept this arrangement? Not likely. But what if you received the $9,500 today and paid the $10,000 one year from now? Can these amounts be considered equivalent? Possibly, because a payment of $10,000 a year from now would probably be worth less to you than a payment of $10,000 today. It would be fair, therefore, to discount the $10,000 received in one year; that is, to cut its value based on how much time passes before the money is paid. An interest rate, denoted r, is a rate of return that reflects the relationship between differently dated cash flows. If $9,500 today and $10,000 in one year are equivalent in value, then $10,000 - $9,500 = $500 is the required compensation for receiving $10,000 in one year rather than now. The interest rate-the required compensation stated as a rate of return-is $500/$9,500 = 0.0526 or 5.26 percent.
Interest rates can be thought of in three ways. First, they can be considered required rates of return-that is, the minimum rate of return an investor must receive in order to accept the investment. Second, interest rates can be considered discount rates. In the example above, 5.26 percent is that rate at which we discounted the $10,000 future amount to find its value today. Thus, we use the terms "interest rate" and "discount rate" almost interchangeably. Third, interest rates can be considered opportunity costs. An opportunity cost is the value that investors forgo by choosing a particular course of action. In the example, if the party who supplied $9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent on the money. So we can view 5.26 percent as the opportunity cost of current consumption.
Economics tells us that interest rates are set in the marketplace by the forces of supply and demand, where investors are suppliers of funds and borrowers are demanders of funds. Taking the perspective of investors in analyzing market-determined interest rates, we can view an interest rate r as being composed of a real risk-free interest rate plus a set of four premiums that are required returns or compensation for bearing distinct types of risk:
r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium- The real risk-free interest rate is the single-period interest rate for a completely risk-free security if no inflation were expected. In economic theory, the real risk-free rate reflects the time preferences of individuals for current versus future real consumption.
- The inflation premium compensates investors for expected inflation and reflects the average inflation rate expected over the maturity of the debt. Inflation reduces the purchasing power of a unit of currency-the amount of goods and services one can buy with it. The sum of the real risk-free interest rate and the inflation premium is the nominal risk-free interest rate.2 Many countries have governmental short-term debt whose interest rate can be considered to represent the nominal risk-free interest rate in that country. The interest rate on a 90-day US Treasury bill (T-bill), for example, represents the nominal risk-free interest rate over that time horizon.3 US T-bills can be bought and sold in large quantities with minimal transaction costs and are backed by the full faith and credit of the US government.
- The default risk premium compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount.
- The liquidity premium compensates investors for the risk of loss relative to an investment's fair value if the investment needs to be converted to cash quickly. US T-bills, for example, do not bear a liquidity premium because large amounts can be bought and sold without affecting their market price. Many bonds of small issuers, by contrast, trade infrequently after they are issued; the interest rate on such bonds includes a liquidity premium reflecting the relatively high costs (including the impact on price) of selling a position.
- The maturity premium compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended, in general (holding all else equal). The difference between the interest rate on longer-maturity, liquid Treasury debt and that on short-term Treasury debt reflects a positive maturity premium for the longer-term debt (and possibly different inflation premiums as well).
Using this insight into the economic meaning of interest rates, we now turn to a discussion of solving time value of money problems, starting with the future value of a single cash flow.
3. THE FUTURE VALUE OF A SINGLE CASH FLOW
In this section, we introduce time value associated with a single cash flow or lump-sum investment. We describe the relationship between an initial investment or present value (PV), which earns a rate of return (the interest rate per period) denoted as r, and its future value (FV), which will be received N years or periods from today.
The following example illustrates this concept. Suppose you invest $100 (PV = $100) in an interest-bearing bank account paying 5 percent annually. At the end of the first year, you will have the $100 plus the interest earned, 0.05 × $100 = $5, for a total of $105. To formalize this one-period example, we define the following terms:
PV = present value of the investment FVN = future value of the investment N periods from...