Level I - Volume 1 Quantitative Methods (CFA Institute) PDF
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This document is a study guide for the CFA Program's Level I, Volume 1, covering Quantitative Methods. It details various concepts in finance, including rates and returns, time value of money, and statistical measures of asset returns as well as simulation methods and portfolio mathematics.
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© CFA Institute. For candidate use only. Not for distribution. QUANTITATIVE METHODS CFA® Program Curriculum 2025 LEVEL 1 VOLUME 1 © CFA Institute. For candidate use only. Not for distribution. ©2023 by CFA Institute. All rights reserved. This copyright covers material written expressly for t...
© CFA Institute. For candidate use only. Not for distribution. QUANTITATIVE METHODS CFA® Program Curriculum 2025 LEVEL 1 VOLUME 1 © CFA Institute. For candidate use only. Not for distribution. ©2023 by CFA Institute. All rights reserved. This copyright covers material written expressly for this volume by the editor/s as well as the compilation itself. It does not cover the individual selections herein that first appeared elsewhere. Permission to reprint these has been obtained by CFA Institute for this edition only. Further reproductions by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval systems, must be arranged with the individual copyright holders noted. CFA®, Chartered Financial Analyst®, AIMR-PPS®, and GIPS® are just a few of the trademarks owned by CFA Institute. To view a list of CFA Institute trademarks and the Guide for Use of CFA Institute Marks, please visit our website at www.cfainstitute.org. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering legal, accounting, or other professional service. If legal advice or other expert assistance is required, the services of a competent pro- fessional should be sought. All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only. ISBN 9781953337986 (paper) ISBN 9781961409101 (ebook) May 2024 © CFA Institute. For candidate use only. Not for distribution. CONTENTS How to Use the CFA Program Curriculum vii CFA Institute Learning Ecosystem (LES) vii Designing Your Personal Study Program vii Errata viii Other Feedback viii Quantitative Methods Learning Module 1 Rates and Returns 3 Introduction 3 Interest Rates and Time Value of Money 5 Determinants of Interest Rates 6 Rates of Return 8 Holding Period Return 8 Arithmetic or Mean Return 9 Geometric Mean Return 10 The Harmonic Mean 14 Money-Weighted and Time-Weighted Return 19 Calculating the Money Weighted Return 19 Annualized Return 28 Non-annual Compounding 29 Annualizing Returns 30 Continuously Compounded Returns 32 Other Major Return Measures and Their Applications 33 Gross and Net Return 33 Pre-Tax and After-Tax Nominal Return 34 Real Returns 34 Leveraged Return 36 Practice Problems 38 Solutions 42 Learning Module 2 Time Value of Money in Finance 45 Introduction 45 Time Value of Money in Fixed Income and Equity 46 Fixed-Income Instruments and the Time Value of Money 47 Equity Instruments and the Time Value of Money 55 Implied Return and Growth 60 Implied Return for Fixed-Income Instruments 60 Equity Instruments, Implied Return, and Implied Growth 65 Cash Flow Additivity 69 Implied Forward Rates Using Cash Flow Additivity 71 Forward Exchange Rates Using No Arbitrage 74 Option Pricing Using Cash Flow Additivity 76 Practice Problems 81 © CFA Institute. For candidate use only. Not for distribution. iv Contents Solutions 84 Learning Module 3 Statistical Measures of Asset Returns 87 Introduction 87 Measures of Central Tendency and Location 89 Measures of Central Tendency 90 Dealing with Outliers 92 Measures of Location 93 Measures of Dispersion 100 The Range 101 Mean Absolute Deviations 101 Sample Variance and Sample Standard Deviation 101 Downside Deviation and Coefficient of Variation 102 Measures of Shape of a Distribution 110 Correlation between Two Variables 117 Scatter Plot 117 Covariance and Correlation 119 Properties of Correlation 120 Limitations of Correlation Analysis 121 Practice Problems 127 Solutions 130 Learning Module 4 Probability Trees and Conditional Expectations 133 Introduction 133 Expected Value and Variance 134 Probability Trees and Conditional Expectations 136 Total Probability Rule for Expected Value 137 Bayes' Formula and Updating Probability Estimates 141 Bayes’ Formula 142 Practice Problems 151 Solutions 152 Learning Module 5 Portfolio Mathematics 153 Introduction 153 Portfolio Expected Return and Variance of Return 155 Covariance 155 Correlation 158 Forecasting Correlation of Returns: Covariance Given a Joint Probability Function 164 Portfolio Risk Measures: Applications of the Normal Distribution 167 References 173 Practice Problems 174 Solutions 175 Learning Module 6 Simulation Methods 177 Introduction 177 Lognormal Distribution and Continuous Compounding 178 The Lognormal Distribution 178 © CFA Institute. For candidate use only. Not for distribution. Contents v Continuously Compounded Rates of Return 181 Monte Carlo Simulation 184 Bootstrapping 189 Practice Problems 193 Solutions 194 Learning Module 7 Estimation and Inference 195 Introduction 195 Sampling Methods 197 Simple Random Sampling 197 Stratified Random Sampling 198 Cluster Sampling 199 Non-Probability Sampling 200 Sampling from Different Distributions 202 Central Limit Theorem and Inference 205 The Central Limit Theorem 205 Standard Error of the Sample Mean 207 Bootstrapping and Empirical Sampling Distributions 209 Practice Problems 214 Solutions 215 Learning Module 8 Hypothesis Testing 217 Introduction 217 Hypothesis Tests for Finance 219 The Process of Hypothesis Testing 219 Tests of Return and Risk in Finance 224 Test Concerning Differences between Means with Dependent Samples 228 Test Concerning the Equality of Two Variances 229 Parametric versus Nonparametric Tests 236 Uses of Nonparametric Tests 237 Nonparametric Inference: Summary 237 Practice Problems 238 Solutions 242 Learning Module 9 Parametric and Non-Parametric Tests of Independence 245 Introduction 245 Tests Concerning Correlation 246 Parametric Test of a Correlation 247 Non-Parametric Test of Correlation: The Spearman Rank Correlation Coefficient 251 Tests of Independence Using Contingency Table Data 255 Practice Problems 263 Solutions 264 Learning Module 10 Simple Linear Regression 265 Introduction 265 Estimation of the Simple Linear Regression Model 267 © CFA Institute. For candidate use only. Not for distribution. vi Contents Introduction to Linear Regression 267 Estimating the Parameters of a Simple Linear Regression 270 Assumptions of the Simple Linear Regression Model 277 Assumption 1: Linearity 277 Assumption 2: Homoskedasticity 279 Assumption 3: Independence 281 Assumption 4: Normality 282 Hypothesis Tests in the Simple Linear Regression Model 284 Analysis of Variance 284 Measures of Goodness of Fit 285 Hypothesis Testing of Individual Regression Coefficients 287 Prediction in the Simple Linear Regression Model 297 ANOVA and Standard Error of Estimate in Simple Linear Regression 297 Prediction Using Simple Linear Regression and Prediction Intervals 299 Functional Forms for Simple Linear Regression 304 The Log-Lin Model 305 The Lin-Log Model 306 The Log-Log Model 308 Selecting the Correct Functional Form 309 Practice Problems 312 Solutions 325 Learning Module 11 Introduction to Big Data Techniques 329 Introduction 329 How Is Fintech used in Quantitative Investment Analysis? 330 Big Data 331 Advanced Analytical Tools: Artificial Intelligence and Machine Learning 334 Tackling Big Data with Data Science 337 Data Processing Methods 337 Data Visualization 338 Text Analytics and Natural Language Processing 339 Practice Problems 341 Solutions 342 Learning Module 12 Appendices A-E 343 Appendices A-E 343 Glossary G-1 © CFA Institute. For candidate use only. Not for distribution. vii How to Use the CFA Program Curriculum The CFA® Program exams measure your mastery of the core knowledge, skills, and abilities required to succeed as an investment professional. These core competencies are the basis for the Candidate Body of Knowledge (CBOK™). The CBOK consists of four components: A broad outline that lists the major CFA Program topic areas (www.cfainstitute.org/programs/cfa/curriculum/cbok/cbok) Topic area weights that indicate the relative exam weightings of the top-level topic areas (www.cfainstitute.org/en/programs/cfa/curriculum) Learning outcome statements (LOS) that advise candidates about the specific knowledge, skills, and abilities they should acquire from curricu- lum content covering a topic area: LOS are provided at the beginning of each block of related content and the specific lesson that covers them. We encourage you to review the information about the LOS on our website (www.cfainstitute.org/programs/cfa/curriculum/study-sessions), including the descriptions of LOS “command words” on the candidate resources page at www.cfainstitute.org/-/media/documents/support/programs/cfa-and -cipm-los-command-words.ashx. The CFA Program curriculum that candidates receive access to upon exam registration Therefore, the key to your success on the CFA exams is studying and understanding the CBOK. You can learn more about the CBOK on our website: www.cfainstitute.org/programs/cfa/curriculum/cbok. The curriculum, including the practice questions, is the basis for all exam questions. The curriculum is selected or developed specifically to provide candidates with the knowledge, skills, and abilities reflected in the CBOK. CFA INSTITUTE LEARNING ECOSYSTEM (LES) Your exam registration fee includes access to the CFA Institute Learning Ecosystem (LES). This digital learning platform provides access, even offline, to all the curriculum content and practice questions. The LES is organized as a series of learning modules consisting of short online lessons and associated practice questions. This tool is your source for all study materials, including practice questions and mock exams. The LES is the primary method by which CFA Institute delivers your curriculum experience. Here, candidates will find additional practice questions to test their knowledge. Some questions in the LES provide a unique interactive experience. DESIGNING YOUR PERSONAL STUDY PROGRAM An orderly, systematic approach to exam preparation is critical. You should dedicate a consistent block of time every week to reading and studying. Review the LOS both before and after you study curriculum content to ensure you can demonstrate the © CFA Institute. For candidate use only. Not for distribution. viii How to Use the CFA Program Curriculum knowledge, skills, and abilities described by the LOS and the assigned reading. Use the LOS as a self-check to track your progress and highlight areas of weakness for later review. Successful candidates report an average of more than 300 hours preparing for each exam. Your preparation time will vary based on your prior education and experience, and you will likely spend more time on some topics than on others. ERRATA The curriculum development process is rigorous and involves multiple rounds of reviews by content experts. Despite our efforts to produce a curriculum that is free of errors, in some instances, we must make corrections. Curriculum errata are periodically updated and posted by exam level and test date on the Curriculum Errata webpage (www.cfainstitute.org/en/programs/submit-errata). If you believe you have found an error in the curriculum, you can submit your concerns through our curriculum errata reporting process found at the bottom of the Curriculum Errata webpage. OTHER FEEDBACK Please send any comments or suggestions to info@cfainstitute.org, and we will review your feedback thoughtfully. © CFA Institute. For candidate use only. Not for distribution. Quantitative Methods © CFA Institute. For candidate use only. Not for distribution. © CFA Institute. For candidate use only. Not for distribution. LEARNING MODULE 1 Rates and Returns by Richard A. DeFusco, PhD, CFA, Dennis W. McLeavey, DBA, CFA, Jerald E. Pinto, PhD, CFA, David E. Runkle, PhD, CFA, and Vijay Singal, PhD, CFA. Richard A. DeFusco, PhD, CFA, is at the University of Nebraska-Lincoln (USA). Dennis W. McLeavey, DBA, CFA, is at the University of Rhode Island (USA). Jerald E. Pinto, PhD, CFA, is at CFA Institute (USA). David E. Runkle, PhD, CFA, is at Jacobs Levy Equity Management (USA). Vijay Singal, PhD, CFA, is at Virginia Tech (USA). LEARNING OUTCOMES Mastery The candidate should be able to: interpret interest rates as required rates of return, discount rates, or opportunity costs and 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 different approaches to return measurement over time and describe their appropriate uses compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses calculate and interpret major return measures and describe their appropriate uses INTRODUCTION Interest rates are a critical concept in finance. In some cases, we assume a particular 1 interest rate and in others, the interest rate remains the unknown quantity to deter- mine. Although the pre-reads have covered the mechanics of time value of money problems, here we first illustrate the underlying economic concepts by explaining the meaning and interpretation of interest rates and then calculate, interpret, and compare different return measures. © CFA Institute. For candidate use only. Not for distribution. 4 Learning Module 1 Rates and Returns LEARNING MODULE OVERVIEW An interest rate, r, can have three interpretations: (1) a required rate of return, (2) a discount rate, or (3) an opportu- nity cost. An interest rate reflects the relationship between differently dated cash flows. An interest rate can be viewed as the sum of the real risk-free inter- est rate and a set of premiums that compensate lenders for bearing distinct types of risk: an inflation premium, a default risk premium, a liquidity premium, and a maturity premium. The nominal risk-free interest rate is approximated as the sum of the real risk-free interest rate and the inflation premium. A financial asset’s total return consists of two components: an income yield consisting of cash dividends or interest payments, and a return reflecting the capital gain or loss resulting from changes in the price of the financial asset. A holding period return, R, is the return that an investor earns for a single, specified period of time (e.g., one day, one month, five years). Multiperiod returns may be calculated across several holding periods using different return measures (e.g., arithmetic mean, geometric mean, harmonic mean, trimmed mean, winsorized mean). Each return computation has special applications for evaluating investments. The choice of which of the various alternative measurements of mean to use for a given dataset depends on considerations such as the presence of extreme outliers, outliers that we want to include, whether there is a symmetric distribution, and compounding. A money-weighted return reflects the actual return earned on an investment after accounting for the value and timing of cash flows relating to the investment. A time-weighted return measures the compound rate of growth of one unit of currency invested in a portfolio during a stated measurement period. Unlike a money-weighted return, a time-weighted return is not sensitive to the timing and amount of cashflows and is the preferred performance measure for evaluating portfolio managers because cash withdrawals or additions to the portfolio are generally outside of the control of the portfolio manager. Interest may be paid or received more frequently than annually. The periodic interest rate and the corresponding number of compounding periods (e.g., quarterly, monthly, daily) should be adjusted to compute present and future values. Annualizing periodic returns allows investors to compare different investments across different holding periods to better evaluate and compare their relative performance. With the number of compound- ing periods per year approaching infinity, the interest is compound continuously. Gross return, return prior to deduction of managerial and adminis- trative expenses (those expenses not directly related to return gener- ation), is an appropriate measure to evaluate the comparative perfor- mance of an asset manager. © CFA Institute. For candidate use only. Not for distribution. Interest Rates and Time Value of Money 5 Net return, which is equal to the gross return less managerial and administrative expenses, is a better return measure of what an investor actually earned. The after-tax nominal return is computed as the total return minus any allowance for taxes on dividends, interest, and realized gains. Real returns are particularly useful in comparing returns across time periods because inflation rates may vary over time and are particularly useful for comparing investments across time periods and perfor- mance between different asset classes with different taxation. Leveraging a portfolio, via borrowing or futures, can amplify the port- folio’s gains or losses. INTEREST RATES AND TIME VALUE OF MONEY 2 interpret interest rates as required rates of return, discount rates, or opportunity costs and explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk The time value of money establishes the equivalence between cash flows occurring on different dates. As cash received today is preferred to cash promised in the future, we must establish a consistent basis for this trade-off to compare financial instruments in cases in which cash is paid or received at different times. An interest rate (or yield), denoted r, is a rate of return that reflects the relationship between differently dated – timed – cash flows. If USD 9,500 today and USD 10,000 in one year are equivalent in value, then USD 10,000 – USD 9,500 = USD 500 is the required compensation for receiving USD 10,000 in one year rather than now. The interest rate (i.e., the required compensation stated as a rate of return) is USD 500/USD 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 to accept an investment. Second, interest rates can be considered discount rates. In the previous example, 5.26 percent is the discount rate at which USD 10,000 in one year is equivalent to USD 9,500 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 course of action. In the example, if the party who supplied USD 9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent by consuming rather than saving. So, we can view 5.26 percent as the opportunity cost of current consumption. © CFA Institute. For candidate use only. Not for distribution. 6 Learning Module 1 Rates and Returns Determinants of Interest Rates Economics tells us that interest rates are set by the forces of supply and demand, where investors supply funds and borrowers demand their use. 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 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. (1) The real risk-free interest rate is the single-period interest rate for a com- pletely 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 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 Treasury bills (T-bills), for example, do not bear a liquidity premium because large amounts of them 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 typically reflects a positive maturity premium for the longer-term debt (and possibly different inflation premiums as well). The sum of the real risk-free interest rate and the inflation premium is the nominal risk-free interest rate: The nominal risk-free interest rate reflects the combination of a real risk-free rate plus an inflation premium: (1 + nominal risk-free rate) = (1 + real risk-free rate)(1 + inflation premium). In practice, however, the nominal rate is often approximated as the sum of the real risk-free rate plus an inflation premium: Nominal risk-free rate = Real risk-free rate + inflation premium. Many countries have short-term government debt whose interest rate can be considered to represent the nominal risk-free interest rate over that time horizon in that country. The French government issues BTFs, or negotiable fixed-rate discount Treasury bills (Bons du Trésor à taux fixe et à intérêts précomptés), with maturities of up to one year. The Japanese government issues a short-term Treasury bill with maturities of 6 and 12 months. The interest rate on a 90-day US T-bill, for example, represents the nominal risk-free interest rate for the United States over the next three © CFA Institute. For candidate use only. Not for distribution. Interest Rates and Time Value of Money 7 months. Typically, interest rates are quoted in annual terms, so the interest rate on a 90-day government debt security quoted at 3 percent is the annualized rate and not the actual interest rate earned over the 90-day period. Whether the interest rate we use is a required rate of return, or a discount rate, or an opportunity cost, the rate encompasses the real risk-free rate and a set of risk premia that depend on the characteristics of the cash flows. The foundational set of premia consist of inflation, default risk, liquidity risk, and maturity risk. All these premia vary over time and continuously change, as does the real risk-free rate. Consequently, all interest rates fluctuate, but how much they change depends on various economic fundamentals—and on the expectation of how these various economic fundamentals can change in the future. EXAMPLE 1 Determining Interest Rates Exhibit 1 presents selected information for five debt securities. All five invest- ments promise only a single payment at maturity. Assume that premiums relating to inflation, liquidity, and default risk are constant across all time horizons. Exhibit 1: Investments Alternatives and Their Characteristics Maturity Interest Rate Investment (in years) Liquidity Default Risk (%) 1 2 High Low 2.0 2 2 Low Low 2.5 3 7 Low Low r3 4 8 High Low 4.0 5 8 Low High 6.5 Based on the information in Exhibit 1, address the following: 1. Explain the difference between the interest rates offered by Investment 1 and Investment 2. Solution: Investment 2 is identical to Investment 1 except that Investment 2 has low liquidity. The difference between the interest rate on Investment 2 and In- vestment 1 is 0.5 percent. This difference in the two interest rates represents a liquidity premium, which represents compensation for the lower liquidity of Investment 2 (the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly). 2. Estimate the default risk premium affecting all securities. Solution: To estimate the default risk premium, identify two investments that have the same maturity but different levels of default risk. Investments 4 and 5 both have a maturity of eight years but different levels of default risk. Investment 5, however, has low liquidity and thus bears a liquidity premium relative to Investment 4. From Part A, we know the liquidity premium is 0.5 percent. The difference between the interest rates offered by Investments 5 and 4 is 2.5 percent (6.5% − 4.0%), of which 0.5 percent is a liquidity premium. This © CFA Institute. For candidate use only. Not for distribution. 8 Learning Module 1 Rates and Returns implies that 2.0 percent (2.5% − 0.5%) must represent a default risk premium reflecting Investment 5’s relatively higher default risk. 3. Calculate upper and lower limits for the unknown interest rate for Invest- ment 3, r3. Solution: Investment 3 has liquidity risk and default risk comparable to Investment 2, but with its longer time to maturity, Investment 3 should have a higher ma- turity premium and offer a higher interest rate than Investment 2. Therefore, the interest rate on Investment 3, r3, should thus be above 2.5 percent (the interest rate on Investment 2). If the liquidity of Investment 3 was high, Investment 3 would match Invest- ment 4 except for Investment 3’s shorter maturity. We would then conclude that Investment 3’s interest rate should be less than the interest rate offered by Investment 4, which is 4 percent. In contrast to Investment 4, however, Investment 3 has low liquidity. It is possible that the interest rate on Invest- ment 3 exceeds that of Investment 4 despite Investment 3’s shorter maturity, depending on the relative size of the liquidity and maturity premiums. How- ever, we would expect r3 to be less than 4.5 percent, the expected interest rate on Investment 4 if it had low liquidity (4% + 0.5%, the liquidity premi- um). Thus, we should expect in the interest rate offered by Investment 3 to be between 2.5 percent and 4.5 percent. 3 RATES OF RETURN calculate and interpret different approaches to return measurement over time and describe their appropriate uses Financial assets are frequently defined in terms of their return and risk characteristics. Comparison along these two dimensions simplifies the process of building a portfolio from among all available assets. In this lesson, we will compute, evaluate, and compare various measures of return. Financial assets normally generate two types of return for investors. First, they may provide periodic income through cash dividends or interest payments. Second, the price of a financial asset can increase or decrease, leading to a capital gain or loss. Some financial assets provide return through only one of these mechanisms. For example, investors in non-dividend-paying stocks obtain their return from price movement only. Other assets only generate periodic income. For example, defined benefit pension plans and retirement annuities make income payments over the life of a beneficiary. Holding Period Return Returns can be measured over a single period or over multiple periods. Single-period returns are straightforward because there is only one way to calculate them. Multiple-period returns, however, can be calculated in various ways and it is important to be aware of these differences to avoid confusion. © CFA Institute. For candidate use only. Not for distribution. Rates of Return 9 A holding period return, R, is the return earned from holding an asset for a single specified period of time. The period may be one day, one week, one month, five years, or any specified period. If the asset (e.g., bond, stock) is purchased today, time (t = 0), at a price of 100 and sold later, say at time (t = 1), at a price of 105 with no dividends or other income, then the holding period return is 5 percent [(105 − 100)/100)]. If the asset also pays income of two units at time (t = 1), then the total return is 7 percent. This return can be generalized and shown as a mathematical expression in which P is the price and I is the income, as follows: _( P1 − P0 ) + I1 R = P , (1) 0 where the subscript indicates the time of the price or income; (t = 0) is the begin- ning of the period; and (t = 1) is the end of the period. The following two observations are important. We computed a capital gain of 5 percent and an income yield of 2 per- cent in this example. For ease of illustration, we assumed that the income is paid at time t = 1. If the income was received before t = 1, our holding period return may have been higher if we had reinvested the income for the remainder of the period. Return can be expressed in decimals (0.07), fractions (7/100), or as a percent (7 percent). They are all equivalent. A holding period return can be computed for a period longer than one year. For example, an analyst may need to compute a one-year holding period return from three annual returns. In that case, the one-year holding period return is computed by compounding the three annual returns: R = [(1 + R1) × (1 + R2) × (1 + R3)] − 1, where R1, R2, and R3 are the three annual returns. Arithmetic or Mean Return Most holding period returns are reported as daily, monthly, or annual returns. When assets have returns for multiple holding periods, it is necessary to normalize returns to a common period for ease of comparison and understanding. There are different methods for aggregating returns across several holding periods. The remainder of this section presents various ways of computing average returns and discusses their applicability. The simplest way to compute a summary measure for returns across multiple periods is to take a simple arithmetic average of the holding period returns. Thus, three annual returns of −50 percent, 35 percent, and 27 percent will give us an average of 4 percent per year = ( − 50 % + 35 ________________ 3 % + 27% ). The arithmetic average return is easy to compute and has known statistical properties. _ In general, the arithmetic or mean return is denoted by R iand given by the following equation for asset i, where Rit is the return in period t and T is the total number of periods: _ Ri1 + R i2 + … + Ri,T−1 + R iT T R i = ____________________ T 1 = _ T ∑ Rit . (2) t=1 © CFA Institute. For candidate use only. Not for distribution. 10 Learning Module 1 Rates and Returns Geometric Mean Return The arithmetic mean return assumes that the amount invested at the beginning of each period is the same. In an investment portfolio, however, even if there are no cash flows into or out of the portfolio the base amount changes each year. The previous year’s earnings must be added to the beginning value of the subsequent year’s investment— these earnings will be “compounded” by the returns earned in that subsequent year. We can use the geometric mean return to account for the compounding of returns. A geometric mean return provides a more accurate representation of the growth in portfolio value over a given time period than _ the arithmetic mean return. In general, the geometric mean return is denoted by R Giand given by the following equation for asset i: _ _________________________________________ R Gi = √ ( ) × … × (1 + Ri,T−1 ) × (1 + Ri2 ) × (1 + RiT ) − 1 T 1 + Ri1 (3) ____________ √∏ T T = (1 + Rt ) − 1 , t=1 where Rit is the return in period t and T is the total number of periods. In the example in the previous section, we calculated the arithmetic mean to be 4.00 percent. Using Equation 4, we can calculate the geometric mean return from the same three annual returns: _ 3 _____________________________ R Gi = √ ( 1 − 0.50) × (1 + 0.35) × (1 + 0.27) − 1 = − 0.0500. Exhibit 2 shows the actual return for each year and the balance at the end of each year using actual returns. Exhibit 2: Portfolio Value and Performance Actual Return Year-End Amount Year-End Amount for the Year Year-End Using Arithmetic Using Geometric (%) Amount Return of 4% Return of −5% Year 0 EUR1.0000 EUR1.0000 EUR1.0000 Year 1 −50 0.5000 1.0400 0.9500 Year 2 35 0.6750 1.0816 0.9025 Year 3 27 0.8573 1.1249 0.8574 Beginning with an initial investment of EUR1.0000, we will have a balance of EUR0.8573 at the end of the three-year period as shown in the fourth column of Exhibit 2. Note that we compounded the returns because, unless otherwise stated, we earn a return on the balance as of the end of the prior year. That is, we will receive a return of 35 percent in the second year on the balance at the end of the first year, which is only EUR0.5000, not the initial balance of EUR1.0000. Let us compare the balance at the end of the three-year period computed using geometric returns with the balance we would calculate using the 4 percent annual arithmetic mean return from our earlier example. The ending value using the arithmetic mean return is EUR1.1249 (=1.0000 × 1.043). This is much larger than the actual balance at the end of Year 3 of EUR0.8573. In general, the arithmetic return is biased upward unless each of the underlying holding period returns are equal. The bias in arithmetic mean returns is particularly severe if holding period returns are a mix of both positive and negative returns, as in this example. We will now look at three examples that calculate holding period returns over different time horizons. © CFA Institute. For candidate use only. Not for distribution. Rates of Return 11 EXAMPLE 2 Holding Period Return 1. An investor purchased 100 shares of a stock for USD34.50 per share at the beginning of the quarter. If the investor sold all of the shares for USD30.50 per share after receiving a USD51.55 dividend payment at the end of the quarter, the investor’s holding period return is closest to: A. −13.0 percent. B. −11.6 percent. C. −10.1 percent. Solution: C is correct. Applying Equation 2, the holding period return is −10.1 per- cent, calculated as follows: R = (3,050 − 3,450 + 51.55)/3,450 = −10.1%. The holding period return comprised of a dividend yield of 1.49 percent (= 51.55/3,450) and a capital loss of −11.59 percent (= −400/3,450). EXAMPLE 3 Holding Period Return 1. An analyst obtains the following annual rates of return for a mutual fund, which are presented in Exhibit 3. Exhibit 3: Mutual Fund Performance, 20X8–20X0 Year Return (%) 20X8 14 20X9 −10 20X0 −2 The fund’s holding period return over the three-year period is closest to: A. 0.18 percent. B. 0.55 percent. C. 0.67 percent. Solution: B is correct. The fund’s three-year holding period return is 0.55 percent, calculated as follows: R = [(1 + R1) × (1 + R2) × (1 + R3)] − 1, R = [(1 + 0.14)(1 − 0.10)(1 − 0.02)] − 1 = 0.0055 = 0.55%. © CFA Institute. For candidate use only. Not for distribution. 12 Learning Module 1 Rates and Returns EXAMPLE 4 Geometric Mean Return 1. An analyst observes the following annual rates of return for a hedge fund, which are presented in Exhibit 4. Exhibit 4: Hedge Fund Performance, 20X8–20X0 Year Return (%) 20X8 22 20X9 −25 20X0 11 The fund’s geometric mean return over the three-year period is closest to: A. 0.52 percent. B. 1.02 percent. C. 2.67 percent. Solution: A is correct. Applying Equation 4, the fund’s geometric mean return over the three-year period is 0.52 percent, calculated as follows: _ R G= [(1 + 0.22)(1 − 0.25)(1 + 0.11)](1/3) − 1 = 1.0157(1/3) − 1 = 0.0052 = 0.52%. EXAMPLE 5 Geometric and Arithmetic Mean Returns 1. Consider the annual return data for the group of countries in Exhibit 5. Exhibit 5: Annual Returns for Years 1 to 3 for Selected Countries’ Stock Indexes 52-Week Return (%) Average 3-Year Return Index Year 1 Year 2 Year 3 Arithmetic Geometric Country A −15.6 −5.4 6.1 −4.97 −5.38 Country B 7.8 6.3 −1.5 4.20 4.12 Country C 5.3 1.2 3.5 3.33 3.32 Country D −2.4 −3.1 6.2 0.23 0.15 Country E −4.0 −3.0 3.0 −1.33 −1.38 Country F 5.4 5.2 −1.0 3.20 3.16 Country G 12.7 6.7 −1.2 6.07 5.91 Country H 3.5 4.3 3.4 3.73 3.73 Country I 6.2 7.8 3.2 5.73 5.72 © CFA Institute. For candidate use only. Not for distribution. Rates of Return 13 52-Week Return (%) Average 3-Year Return Index Year 1 Year 2 Year 3 Arithmetic Geometric Country J 8.1 4.1 −0.9 3.77 3.70 Country K 11.5 3.4 1.2 5.37 5.28 Calculate the arithmetic and geometric mean returns over the three years for the following three stock indexes: Country D, Country E, and Country F. Solution: The arithmetic mean returns are as follows: Annual Return (%) Arithmetic Sum 3 Mean Return Year 1 Year 2 Year 3 ∑ Ri (%) i=1 Country D −2.4 −3.1 6.2 0.7 0.233 Country E −4.0 −3.0 3.0 −4.0 −1.333 Country F 5.4 5.2 −1.0 9.6 3.200 The geometric mean returns are as follows: 1 + Return in Decimal Form (1 + Rt) Product 3rd root Geometric 1⁄ 3 ( 1 + Rt ) [∏ ] T T mean Year 1 Year 2 Year 3 ∏ t t (1 + Rt ) return (%) Country D 0.976 0.969 1.062 1.00438 1.00146 0.146 Country E 0.960 0.970 1.030 0.95914 0.98619 −1.381 Country F 1.054 1.052 0.990 1.09772 1.03157 3.157 In Example 5, the geometric mean return is less than the arithmetic mean return for each country’s index returns. In fact, the geometric mean is always less than or equal to the arithmetic mean with one exception: the two means will be equal is when there is no variability in the observations—that is, when all the observations in the series are the same. In general, the difference between the arithmetic and geometric means increases with the variability within the sample; the more disperse the observations, the greater the difference between the arithmetic and geometric means. Casual inspection of the returns in Exhibit 5 and the associated graph of means in Exhibit 6 suggests a greater variability for Country A’s index relative to the other indexes, and this is confirmed with the greater deviation of the geometric mean return (−5.38 percent) from the arithmetic mean return (−4.97 percent). How should the analyst interpret these results? © CFA Institute. For candidate use only. Not for distribution. 14 Learning Module 1 Rates and Returns Exhibit 6: Arithmetic and Geometric Mean Returns for Country Stock Indexes, Years 1 to 3 Country A B C D E F G H I J K 6 –4 –2 0 2 4 6 8 Mean Return (%) Geometric Mean Arithmetic Average The geometric mean return represents the growth rate or compound rate of return on an investment. One unit of currency invested in a fund tracking the Country B index at the beginning of Year 1 would have grown to (1.078)(1.063)(0.985) = 1.128725 units of currency, which is equal to 1 plus Country B’s geometric mean return of 4.1189 per- cent compounded over three periods: [1 + 0.041189]3 = 1.128725. This math confirms that the geometric mean is the compound rate of return. With its focus on the actual return of an investment over a multiple-period horizon, the geometric mean is of key interest to investors. The arithmetic mean return, focusing on average single-period performance, is also of interest. Both arithmetic and geometric means have a role to play in investment management, and both are often reported for return series. For reporting historical returns, the geometric mean has considerable appeal because it is the rate of growth or return we would have to earn each year to match the actual, cumulative investment performance. Suppose we purchased a stock for EUR100 and two years later it was worth EUR100, with an intervening year at EUR200. The geometric mean of 0 percent is clearly the compound rate of growth during the two years, which we can confirm by compounding the returns: [(1 + 1.00)(1 − 0.50)]1/2 − 1 = 0%. Specifically, the ending amount is the beginning amount times (1 + RG)2. However, the arithmetic mean, which is [100% + −50%]/2 = 25% in the previous example, can distort our assessment of historical performance. As we noted, the arithmetic mean is always greater than or equal to the geometric mean. If we want to estimate the average return over a one-period horizon, we should use the arithmetic mean because the arithmetic mean is the average of one-period returns. If we want to estimate the average returns over more than one period, however, we should use the geometric mean of returns because the geometric mean captures how the total returns are linked over time. The Harmonic Mean _ The harmonic mean, X H, is another measure of central tendency. The harmonic mean is appropriate in cases in which the variable is a rate or a ratio. The terminology “harmonic” arises from its use of a type of series involving reciprocals known as a harmonic series. © CFA Institute. For candidate use only. Not for distribution. Rates of Return 15 Harmonic Mean Formula. The harmonic mean of a set of observations X1, X2, …, Xn is: _ X H = _ n n , (4) ∑(1 / Xi ) i=1 with Xi > 0 for i = 1, 2, …, n. The harmonic mean is the value obtained by summing the reciprocals of the observations, n ∑ (1 / Xi ), i=1 the terms of the form 1/Xi, and then averaging their sum by dividing it by the number of observations, n, and, then finally, taking the reciprocal of that average, _ n n . ∑(1 / Xi ) i=1 The harmonic mean may be viewed as a special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude. For example, if there is a sample of observations of 1, 2, 3, 4, 5, 6, and 1,000, the harmonic mean is 2.8560. Compared to the arithmetic mean of 145.8571, we see the influence of the outlier (the 1,000) to be much less than in the case of the arithmetic mean. So, the harmonic mean is quite useful as a measure of central tendency in the presence of outliers. The harmonic mean is used most often when the data consist of rates and ratios, such as P/Es. Suppose three peer companies have P/Es of 45, 15, and 15. The arithmetic mean is 25, but the harmonic mean, which gives less weight to the P/E of 45, is 19.3. The harmonic mean is a relatively specialized concept of the mean that is appro- priate for averaging ratios (“amount per unit”) when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units. The concept is best explained through an illustration. A well-known application arises in the investment strategy known as cost averaging, which involves the periodic investment of a fixed amount of money. In this application, the ratios we are averaging are prices per share at different purchase dates, and we are applying those prices to a constant amount of money to yield a variable number of shares. An illustration of the harmonic mean to cost averaging is provided in Example 6. EXAMPLE 6 Cost Averaging and the Harmonic Mean 1. Suppose an investor invests EUR1,000 each month in a particular stock for n = 2 months. The share prices are EUR10 and EUR15 at the two purchase dates. What was the average price paid for the security? Solution: Purchase in the first month = EUR1,000/EUR10 = 100 shares. Purchase in the second month = EUR1,000/EUR15 = 66.67 shares. The investor purchased a total of 166.67 shares for EUR2,000, so the average price paid per share is EUR2,000/166.67 = EUR12. The average price paid is in fact the harmonic mean of the asset’s prices at the purchase dates. Using Equation 5, the harmonic mean price is 2/[(1/10) + (1/15)] = EUR12. The value EUR12 is less than the arithmetic mean pur- chase price (EUR10 + EUR15)/2 = EUR12.5. © CFA Institute. For candidate use only. Not for distribution. 16 Learning Module 1 Rates and Returns Because they use the same data but involve different progressions in their respec- tive calculations, the arithmetic, geometric, and harmonic means are mathematically related to one another. We will not go into the proof of this relationship, but the basic result follows: Arithmetic mean × Harmonic mean = (Geometric mean)2. Unless all the observations in a dataset are the same value, the harmonic mean is always less than the geometric mean, which, in turn, is always less than the arithmetic mean. The harmonic mean only works for non-negative numbers, so when working with returns that are expressed as positive or negative percentages, we first convert the returns into a compounding format, assuming a reinvestment, as (1 + R), as was done in the geometric mean return calculation, and then calculate (1 + harmonic mean), and subtract 1 to arrive at the harmonic mean return. EXAMPLE 7 Calculating the Arithmetic, Geometric, and Harmonic Means for P/Es Each year in December, a securities analyst selects her 10 favorite stocks for the next year. Exhibit 7 presents the P/Es, the ratio of share price to projected earnings per share (EPS), for her top 10 stock picks for the next year. Exhibit 7: Analyst’s 10 Favorite Stocks for Next Year Stock P/E Stock 1 22.29 Stock 2 15.54 Stock 3 9.38 Stock 4 15.12 Stock 5 10.72 Stock 6 14.57 Stock 7 7.20 Stock 8 7.97 Stock 9 10.34 Stock 10 8.35 1. Calculate the arithmetic mean P/E for these 10 stocks. Solution: The arithmetic mean is calculated as: 121.48/10 = 12.1480. 2. Calculate the geometric mean P/E for these 10 stocks. Solution: The geometric mean P/E is calculated as: _ _____________________ _ √ P 10 _ _ _ _ E = P P P P E × E × … × E × E Gi 1 2 9 10 10 __________________________ = √ 22.29 × 15.54…× 10.34 × 8.35 © CFA Institute. For candidate use only. Not for distribution. Rates of Return 17 10 ______________ = √ 38, 016, 128, 040 = 11.4287. The geometric mean is 11.4287. This result can also be obtained as: _ ln(38,016,128,040) ln(22.29×15.54…×10.34×8.35) ______________________ ______________ _ P 24.3613/10 10 = e 10 = e E = e Gi = 11.4287. 3. Calculate the harmonic mean P/E for the 10 stocks. Solution: The harmonic mean is calculated as: _ X H = _ n n , ∑ (1 / Xi ) i=1 _ 10 X H = _______________________________ 1 , ( 22.29 ) + ( 15.54 ) + … + ( 10.34 ) + ( 8.35 ) _ 1 _1 _ 1 _ _ X H = 10 / 0.9247 = 10.8142. In finance, the weighted harmonic mean is used when averaging rates and other multiples, such as the P/E ratio, because the harmonic mean gives equal weight to each data point, and reduces the influence of outliers. These calculations can be performed using Excel: To calculate the arithmetic mean or average return, the =AVERAGE(return1, return2, … ) function can be used. To calculate the geometric mean return, the =GEOMEAN(return1, return2, … ) function can be used. To calculate the harmonic mean return, the =HARMEAN(return1, return2, … ) function can be used. In addition to arithmetic, geometric, and harmonic means, two other types of means can be used. Both the trimmed and the winsorized means seek to minimize the impact of outliers in a dataset. Specifically, the trimmed mean removes a small defined percentage of the largest and smallest values from a dataset containing our observation before calculating the mean by averaging the remaining observations. A winsorized mean replaces the extreme observations in a dataset to limit the effect of the outliers on the calculations. The winsorized mean is calculated after replacing extreme values at both ends with the values of their nearest observations, and then calculating the mean by averaging the remaining observations. However, the key question is: Which mean to use in what circumstances? The choice of which mean to use depends on many factors, as we describe in Exhibit 8: Are there outliers that we want to include? Is the distribution symmetric? Is there compounding? Are there extreme outliers? © CFA Institute. For candidate use only. Not for distribution. 18 Learning Module 1 Rates and Returns Exhibit 8: Deciding Which Measure to Use Collect Sample Include all values, Yes Arithmetic Mean including outliers? Yes Compounding? Geometric Mean Yes Harmonic mean, Extreme Trimmed mean, outliers? Winsorized mean QUESTION SET A fund had the following returns over the past 10 years: Exhibit 9: 10-Year Returns Year Return 1 4.5% 2 6.0% 3 1.5% 4 −2.0% 5 0.0% 6 4.5% 7 3.5% 8 2.5% 9 5.5% 10 4.0% 1. The arithmetic mean return over the 10 years is closest to: A. 2.97 percent. B. 3.00 percent. C. 3.33 percent. Solution: B is correct. The arithmetic mean return is calculated as follows: © CFA Institute. For candidate use only. Not for distribution. Money-Weighted and Time-Weighted Return 19 _ R = 30.0%/10 = 3.0%. 2. The geometric mean return over the 10 years is closest to: A. 2.94 percent. B. 2.97 percent. C. 3.00 percent. Solution: B is correct. The geometric mean return is calculated as follows: _ 10 _____________________________________________ R G = √ ( 1 + 0.045) × (1 + 0.06) × … × (1 + 0.055) × (1 + 0.04) − 1 _ 10 _ R G = √ 1.3402338 − 1= 2.9717%. MONEY-WEIGHTED AND TIME-WEIGHTED RETURN 4 compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures The arithmetic and geometric return computations do not account for the timing of cash flows into and out of a portfolio. For example, suppose an investor experiences the returns shown in Exhibit 2. Instead of only investing EUR1.0 at the start (Year 0) as was the case in Exhibit 2, suppose the investor had invested EUR10,000 at the start, EUR1,000 in Year 1, and EUR1,000 in Year 2. In that case, the return of –50 percent in Year 1 significantly hurts her given the relatively large investment at the start. Conversely, if she had invested only EUR100 at the start, the absolute effect of the –50 percent return on the total return is drastically reduced. Calculating the Money Weighted Return The money-weighted return accounts for the money invested and provides the investor with information on the actual return she earns on her investment. The money-weighted return and its calculation are similar to the internal rate of return and a bond’s yield to maturity. Amounts invested are cash outflows from the investor’s perspective and amounts returned or withdrawn by the investor, or the money that remains at the end of an investment cycle, is a cash inflow for the investor. For example, assume that an investor invests EUR100 in a mutual fund at the beginning of the first year, adds another EUR950 at the beginning of the second year, and withdraws EUR350 at the end of the second year. The cash flows are presented in Exhibit 10. © CFA Institute. For candidate use only. Not for distribution. 20 Learning Module 1 Rates and Returns Exhibit 10: Portfolio Balances across Three Years Year 1 2 3 Balance from previous year EUR0 EUR50 EUR1,000 New investment by the investor (cash inflow for the mutual fund) at the start of the year 100 950 0 Net balance at the beginning of year 100 1,000 1,000 Investment return for the year −50% 35% 27% Investment gain (loss) −50 350 270 Withdrawal by the investor (cash outflow for the mutual fund) at the end of the year 0 −350 0 Balance at the end of year EUR50 EUR1,000 EUR1,270 The internal rate of return is the discount rate at which the sum of present values of cash flows will equal zero. In general, the equation may be expressed as follows: T _ C Ft ∑ t = 0, (5) t=0 ( 1 + IRR) where T is the number of periods, CFt is the cash flow at time t, and IRR is the internal rate of return or the money-weighted rate of return. A cash flow can be positive or negative; a positive cash flow is an inflow where money flows to the investor, whereas a negative cash flow is an outflow where money flows away from the investor. The cash flows are expressed as follows, where each cash inflow or outflow occurs at the end of each year. Thus, CF0 refers to the cash flow at the end of Year 0 or beginning of Year 1, and CF3 refers to the cash flow at end of Year 3 or beginning of Year 4. Because cash flows are being discounted to the present—that is, end of Year 0 or beginning of Year 1—the period of discounting CF0 is zero. CF 0 = − 100 CF 1 = − 950 CF 2 = + 350 CF 3 = + 1, 270 CF C F CF 2 CF 3 . _ 0 _ 1 _ _ ( 0 + 1 + 2 + 3 1 + IRR 1 + IRR 1 + IRR 1 + IRR ) ( ) ( ) ( ) − 100 _ − 950 + 350 + 1270 = _ _ _ 1 + ( 1 + IRR) 1 + (1 + IRR) 2 + (1 + IRR) 3 = 0 IRR = 26.11% The investor’s internal rate of return, or the money-weighted rate of return, is 26.11 percent, which tells the investor what she earned on the actual euros invested for the entire period on an annualized basis. This return is much greater than the arithmetic and geometric mean returns because only a small amount was invested when the mutual fund’s return was −50 percent. All the above calculations can be performed using Excel using the =IRR(values) function, which results in an IRR of 26.11 percent. Money-Weighted Return for a Dividend-Paying Stock Next, we’ll illustrate calculating the money-weighted return for a dividend paying stock. Consider an investment that covers a two-year horizon. At time t = 0, an inves- tor buys one share at a price of USD200. At time t = 1, he purchases an additional share at a price of USD225. At the end of Year 2, t = 2, he sells both shares at a price of USD235. During both years, the stock pays a dividend of USD5 per share. The t =1 dividend is not reinvested. Exhibit 11 outlines the total cash inflows and outflows for the investment. © CFA Institute. For candidate use only. Not for distribution. Money-Weighted and Time-Weighted Return 21 Exhibit 11: Cash Flows for a Dividend-Paying Stock Time Outflows 0 USD200 to purchase the first share 1 USD225 to purchase the second share Time Inflows 1 USD5 dividend received from first share (and not reinvested) 2 USD10 dividend (USD5 per share × 2 shares) received 2 USD470 received from selling two shares at USD235 per share To solve for the money-weighted return, the first step is to group net cash flows by time. For this example, we have −USD200 for the t = 0 net cash flow, −USD220 = − USD225 + USD5 for the t = 1 net cash flow, and USD480 for the t = 2 net cash flow. After entering these cash flows, we use the spreadsheet’s (such as Excel) or calculator’s IRR function to find that the money-weighted rate of return is 9.39 percent. CF 0 = − 200 CF 1 = − 220 CF 2 = + 480 CF 0 + _ _ CF 1 + _ CF 2 . )0 )1 1 + IRR 1 + IRR 1 + IRR ( ( ( )2 − 200 − 220 480 = _ _ _ 1 + ( )1 + ( )2 = 0 1 + IRR 1 + IRR IRR = 9.39% All these calculations can be performed using Excel using the =IRR(values) function, which results in an IRR of 9.39 percent. Now we take a closer look at what has happened to the portfolio during each of the two years. In the first year, the portfolio generated a one-period holding period return of (USD5 + USD225 − USD200)/USD200 = 15%. At the beginning of the second year, the amount invested is USD450, calculated as USD225 (per share price of stock) × 2 shares, because the U
