2023-L1V1.pdf

© CFA Institute. For candidate use only. Not for distribution. QUANTITATIVE METHODS CFA® Program Curriculum 2023 LEVEL 1 VOLUME 1 © CFA Institute. For candidate use only. Not for distribution. ©2022 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 978-1-950157-96-9 (paper) ISBN 978-1-953337-23-8 (ebook) 2022 © CFA Institute. For candidate use only. Not for distribution. CONTENTS How to Use the CFA Program Curriculum ix Errata ix Designing Your Personal Study Program ix CFA Institute Learning Ecosystem (LES) x Feedback x Quantitative Methods Learning Module 1 The Time Value of Money 3 Introduction 3 Interest Rates 4 Future Value of a Single Cash Flow 6 Non-Annual Compounding (Future Value) 10 Continuous Compounding 12 Stated and Effective Rates 14 A Series of Cash Flows 15 Equal Cash Flows—Ordinary Annuity 15 Unequal Cash Flows 16 Present Value of a Single Cash Flow 17 Non-Annual Compounding (Present Value) 19 Present Value of a Series of Equal and Unequal Cash Flows 21 The Present Value of a Series of Equal Cash Flows 21 The Present Value of a Series of Unequal Cash Flows 25 Present Value of a Perpetuity 26 Present Values Indexed at Times Other than t = 0 27 Solving for Interest Rates, Growth Rates, and Number of Periods 28 Solving for Interest Rates and Growth Rates 29 Solving for the Number of Periods 31 Solving for Size of Annuity Payments 32 Present and Future Value Equivalence and the Additivity Principle 36 The Cash Flow Additivity Principle 38 Summary 39 Practice Problems 40 Solutions 45 Learning Module 2 Organizing, Visualizing, and Describing Data 59 Introduction 59 Data Types 60 Numerical versus Categorical Data 61 Cross-Sectional versus Time-Series versus Panel Data 63 Structured versus Unstructured Data 64 Data Summarization 68 Organizing Data for Quantitative Analysis 68 Summarizing Data Using Frequency Distributions 71 Summarizing Data Using a Contingency Table 77 indicates an optional segment © CFA Institute. 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Not for distribution. iv Contents Data Visualization 82 Histogram and Frequency Polygon 82 Bar Chart 84 Tree-Map 87 Word Cloud 88 Line Chart 90 Scatter Plot 92 Heat Map 96 Guide to Selecting among Visualization Types 98 Measures of Central Tendency 100 The Arithmetic Mean 101 The Median 105 The Mode 106 Other Concepts of Mean 107 Quantiles 116 Quartiles, Quintiles, Deciles, and Percentiles 117 Quantiles in Investment Practice 122 Measures of Dispersion 123 The Range 123 The Mean Absolute Deviation 124 Sample Variance and Sample Standard Deviation 125 Downside Deviation and Coefficient of Variation 128 Coefficient of Variation 131 The Shape of the Distributions 133 The Shape of the Distributions: Kurtosis 136 Correlation between Two Variables 139 Properties of Correlation 140 Limitations of Correlation Analysis 143 Summary 146 Practice Problems 151 Solutions 164 Learning Module 3 Probability Concepts 173 Probability Concepts and Odds Ratios 174 Probability, Expected Value, and Variance 174 Conditional and Joint Probability 179 Expected Value and Variance 191 Portfolio Expected Return and Variance of Return 197 Covariance Given a Joint Probability Function 202 Bayes' Formula 206 Bayes’ Formula 206 Principles of Counting 212 Summary 218 References 220 Practice Problems 221 Solutions 228 Learning Module 4 Common Probability Distributions 235 Discrete Random Variables 236 indicates an optional segment © CFA Institute. For candidate use only. Not for distribution. Contents v Discrete Random Variables 237 Discrete and Continuous Uniform Distribution 241 Continuous Uniform Distribution 243 Binomial Distribution 246 Normal Distribution 254 The Normal Distribution 254 Probabilities Using the Normal Distribution 258 Standardizing a Random Variable 260 Probabilities Using the Standard Normal Distribution 260 Applications of the Normal Distribution 262 Lognormal Distribution and Continuous Compounding 266 The Lognormal Distribution 266 Continuously Compounded Rates of Return 269 Student’s t-, Chi-Square, and F-Distributions 272 Student’s t-Distribution 272 Chi-Square and F-Distribution 274 Monte Carlo Simulation 279 Summary 285 Practice Problems 288 Solutions 296 Learning Module 5 Sampling and Estimation 303 Introduction 304 Sampling Methods 304 Simple Random Sampling 305 Stratified Random Sampling 306 Cluster Sampling 308 Non-Probability Sampling 309 Sampling from Different Distributions 313 The Central Limit Theorem and Distribution of the Sample Mean 315 The Central Limit Theorem 315 Standard Error of the Sample Mean 317 Point Estimates of the Population Mean 320 Point Estimators 320 Confidence Intervals for the Population Mean and Sample Size Selection 324 Selection of Sample Size 330 Resampling 332 Sampling Related Biases 335 Data Snooping Bias 336 Sample Selection Bias 337 Look-Ahead Bias 339 Time-Period Bias 340 Summary 341 Practice Problems 344 Solutions 349 Learning Module 6 Hypothesis Testing 353 Introduction 354 Why Hypothesis Testing? 354 indicates an optional segment © CFA Institute. 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Not for distribution. vi Contents Implications from a Sampling Distribution 355 The Process of Hypothesis Testing 356 Stating the Hypotheses 357 Two-Sided vs. One-Sided Hypotheses 357 Selecting the Appropriate Hypotheses 358 Identify the Appropriate Test Statistic 359 Test Statistics 359 Identifying the Distribution of the Test Statistic 360 Specify the Level of Significance 360 State the Decision Rule 362 Determining Critical Values 363 Decision Rules and Confidence Intervals 364 Collect the Data and Calculate the Test Statistic 365 Make a Decision 366 Make a Statistical Decision 366 Make an Economic Decision 366 Statistically Significant but Not Economically Significant? 366 The Role of p-Values 367 Multiple Tests and Significance Interpretation 370 Tests Concerning a Single Mean 373 Test Concerning Differences between Means with Independent Samples 377 Test Concerning Differences between Means with Dependent Samples 379 Testing Concerning Tests of Variances 383 Tests of a Single Variance 383 Test Concerning the Equality of Two Variances (F-Test) 387 Parametric vs. Nonparametric Tests 392 Uses of Nonparametric Tests 393 Nonparametric Inference: Summary 393 Tests Concerning Correlation 394 Parametric Test of a Correlation 395 Tests Concerning Correlation: The Spearman Rank Correlation Coefficient 397 Test of Independence Using Contingency Table Data 399 Summary 404 References 407 Practice Problems 408 Solutions 419 Learning Module 7 Introduction to Linear Regression 429 Simple Linear Regression 429 Estimating the Parameters of a Simple Linear Regression 432 The Basics of Simple Linear Regression 432 Estimating the Regression Line 433 Interpreting the Regression Coefficients 436 Cross-Sectional vs. Time-Series Regressions 437 Assumptions of the Simple Linear Regression Model 440 Assumption 1: Linearity 440 Assumption 2: Homoskedasticity 442 Assumption 3: Independence 444 indicates an optional segment © CFA Institute. For candidate use only. Not for distribution. Contents vii Assumption 4: Normality 445 Analysis of Variance 447 Breaking down the Sum of Squares Total into Its Components 448 Measures of Goodness of Fit 449 ANOVA and Standard Error of Estimate in Simple Linear Regression 450 Hypothesis Testing of Linear Regression Coefficients 453 Hypothesis Tests of the Slope Coefficient 453 Hypothesis Tests of the Intercept 456 Hypothesis Tests of Slope When Independent Variable Is an Indicator Variable 457 Test of Hypotheses: Level of Significance and p-Values 459 Prediction Using Simple Linear Regression and Prediction Intervals 460 Functional Forms for Simple Linear Regression 464 The Log-Lin Model 465 The Lin-Log Model 466 The Log-Log Model 468 Selecting the Correct Functional Form 469 Summary 471 Practice Problems 474 Solutions 488 Appendices 493 indicates an optional segment © CFA Institute. For candidate use only. Not for distribution. © CFA Institute. For candidate use only. Not for distribution. ix 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) Topic area weights that indicate the relative exam weightings of the top-level topic areas (www.cfainstitute.org/programs/cfa/curriculum) Learning outcome statements (LOS) that advise candidates about the spe- cific knowledge, skills, and abilities they should acquire from curriculum content covering a topic area: LOS are provided in candidate study ses- sions and 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. The CFA Program curriculum that candidates receive 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 entire curriculum, including the practice questions, is the basis for all exam questions and is selected or developed specifically to teach the knowledge, skills, and abilities reflected in the CBOK. ERRATA The curriculum development process is rigorous and includes multiple rounds of reviews by content experts. Despite our efforts to produce a curriculum that is free of errors, there are instances where we must make corrections. Curriculum errata are periodically updated and posted by exam level and test date online 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. 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 that you have mastered the © CFA Institute. For candidate use only. Not for distribution. x How to Use the CFA Program Curriculum applicable content and can demonstrate the knowledge, skills, and abilities described by the LOS and the assigned reading. Use the LOS 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 study sessions than on others. CFA INSTITUTE LEARNING ECOSYSTEM (LES) Your exam registration fee includes access to the CFA Program Learning Ecosystem (LES). This digital learning platform provides access, even offline, to all of the curricu- lum content and practice questions and is organized as a series of short online lessons with associated practice questions. This tool is your one-stop location for all study materials, including practice questions and mock exams, and the primary method by which CFA Institute delivers your curriculum experience. The LES offers candidates additional practice questions to test their knowledge, and some questions in the LES provide a unique interactive experience. FEEDBACK Please send any comments or feedback to [email protected], and we will review your suggestions carefully. © 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 The Time Value of Money by Richard A. DeFusco, PhD, CFA, Dennis W. McLeavey, DBA, CFA, Jerald E. Pinto, PhD, CFA, and David E. Runkle, 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). LEARNING OUTCOME Mastery 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 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 calculate the solution for time value of money problems with different frequencies of compounding calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding INTRODUCTION As individuals, we often face decisions that involve saving money for a future use, or 1 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 transac- tions 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 © CFA Institute. For candidate use only. Not for distribution. 4 Learning Module 1 The Time Value of Money 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 reading1 is organized as follows: Section 2 introduces some terminology used throughout the reading 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 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 In this reading, 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. 1 Examples in this reading and other readings in quantitative methods at Level I were updated in 2018 by Professor Sanjiv Sabherwal of the University of Texas, Arlington. © CFA Institute. For candidate use only. Not for distribution. Interest Rates 5 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 sup- ply 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 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 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 reflect- ing 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 2 Technically, 1 plus the nominal rate equals the product of 1 plus the real rate and 1 plus the inflation rate. As a quick approximation, however, the nominal rate is equal to the real rate plus an inflation premium. In this discussion we focus on approximate additive relationships to highlight the underlying concepts. 3 Other developed countries issue securities similar to US Treasury bills. The French government issues BTFs or negotiable fixed-rate discount Treasury bills (Bons du Trésor àtaux fixe et à intérêts précomptés) with maturities of up to one year. The Japanese government issues a short-term Treasury bill with matur- ities of 6 and 12 months. The German government issues at discount both Treasury financing paper (Finanzierungsschätze des Bundes or, for short, Schätze) and Treasury discount paper (Bubills) with maturities up to 24 months. In the United Kingdom, the British government issues gilt-edged Treasury bills with maturities ranging from 1 to 364 days. The Canadian government bond market is closely related to the US market; Canadian Treasury bills have maturities of 3, 6, and 12 months. © CFA Institute. For candidate use only. Not for distribution. 6 Learning Module 1 The Time Value of Money 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 FUTURE VALUE OF A SINGLE CASH FLOW 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 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 today r = rate of interest per period For N = 1, the expression for the future value of amount PV is FV1 = PV(1 + r) (1) For this example, we calculate the future value one year from today as FV1 = $100(1.05) = $105. Now suppose you decide to invest the initial $100 for two years with interest earned and credited to your account annually (annual compounding). At the end of the first year (the beginning of the second year), your account will have $105, which you will leave in the bank for another year. Thus, with a beginning amount of $105 (PV = $105), the amount at the end of the second year will be $105(1.05) = $110.25. Note that the $5.25 interest earned during the second year is 5 percent of the amount invested at the beginning of Year 2. Another way to understand this example is to note that the amount invested at the beginning of Year 2 is composed of the original $100 that you invested plus the $5 interest earned during the first year. During the second year, the original principal again earns interest, as does the interest that was earned during Year 1. You can see how the original investment grows: Original investment $100.00 Interest for the first year ($100 × 0.05) 5.00 Interest for the second year based on original investment ($100 × 0.05) 5.00 © CFA Institute. For candidate use only. Not for distribution. Future Value of a Single Cash Flow 7 Interest for the second year based on interest earned in the first year (0.05 × 0.25 $5.00 interest on interest) Total $110.25 The $5 interest that you earned each period on the $100 original investment is known as simple interest (the interest rate times the principal). Principal is the amount of funds originally invested. During the two-year period, you earn $10 of simple interest. The extra $0.25 that you have at the end of Year 2 is the interest you earned on the Year 1 interest of $5 that you reinvested. The interest earned on interest provides the first glimpse of the phenomenon known as compounding. Although the interest earned on the initial investment is important, for a given interest rate it is fixed in size from period to period. The com- pounded interest earned on reinvested interest is a far more powerful force because, for a given interest rate, it grows in size each period. The importance of compounding increases with the magnitude of the interest rate. For example, $100 invested today would be worth about $13,150 after 100 years if compounded annually at 5 percent, but worth more than $20 million if compounded annually over the same time period at a rate of 13 percent. To verify the $20 million figure, we need a general formula to handle compounding for any number of periods. The following general formula relates the present value of an initial investment to its future value after N periods: FVN = PV(1 + r)N (2) where r is the stated interest rate per period and N is the number of compounding periods. In the bank example, FV2 = $100(1 + 0.05)2 = $110.25. In the 13 percent investment example, FV100 = $100(1.13)100 = $20,316,287.42. The most important point to remember about using the future value equation is that the stated interest rate, r, and the number of compounding periods, N, must be compatible. Both variables must be defined in the same time units. For example, if N is stated in months, then r should be the one-month interest rate, unannualized. A time line helps us to keep track of the compatibility of time units and the interest rate per time period. In the time line, we use the time index t to represent a point in time a stated number of periods from today. Thus the present value is the amount available for investment today, indexed as t = 0. We can now refer to a time N periods from today as t = N. The time line in Exhibit 1 shows this relationship. Exhibit 1: The Relationship between an Initial Investment, PV, and Its Future Value, FV 0 1 2 3... N–1 N PV FVN = PV(1 + r)N In Exhibit 1, we have positioned the initial investment, PV, at t = 0. Using Equation 2, we move the present value, PV, forward to t = N by the factor (1 + r)N. This factor is called a future value factor. We denote the future value on the time line as FV and © CFA Institute. For candidate use only. Not for distribution. 8 Learning Module 1 The Time Value of Money position it at t = N. Suppose the future value is to be received exactly 10 periods from today’s date (N = 10). The present value, PV, and the future value, FV, are separated in time through the factor (1 + r)10. The fact that the present value and the future value are separated in time has important consequences: We can add amounts of money only if they are indexed at the same point in time. For a given interest rate, the future value increases with the number of periods. For a given number of periods, the future value increases with the interest rate. To better understand these concepts, consider three examples that illustrate how to apply the future value formula. EXAMPLE 1 The Future Value of a Lump Sum with Interim Cash Reinvested at the Same Rate 1. You are the lucky winner of your state’s lottery of $5 million after taxes. You invest your winnings in a five-year certificate of deposit (CD) at a local financial institution. The CD promises to pay 7 percent per year compound- ed annually. This institution also lets you reinvest the interest at that rate for the duration of the CD. How much will you have at the end of five years if your money remains invested at 7 percent for five years with no withdraw- als? Solution: To solve this problem, compute the future value of the $5 million investment using the following values in Equation 2: PV = $5, 000, 000 r = 7 % = 0.07 N = 5 N VN = PV (1 + r) F                 = $5,000,000 ( 1.07)5 = $5,000,000( 1.402552) = $7,012,758.65 At the end of five years, you will have $7,012,758.65 if your money remains invested at 7 percent with no withdrawals. In this and most examples in this reading, note that the factors are reported at six decimal places but the calculations may actually reflect greater precision. For exam- ple, the reported 1.402552 has been rounded up from 1.40255173 (the calculation is actually carried out with more than eight decimal places of precision by the calculator or spreadsheet). Our final result reflects the higher number of decimal places carried by the calculator or spreadsheet.4 4 We could also solve time value of money problems using tables of interest rate factors. Solutions using tabled values of interest rate factors are generally less accurate than solutions obtained using calculators or spreadsheets, so practitioners prefer calculators or spreadsheets. © CFA Institute. For candidate use only. Not for distribution. Future Value of a Single Cash Flow 9 EXAMPLE 2 The Future Value of a Lump Sum with No Interim Cash 1. An institution offers you the following terms for a contract: For an invest- ment of ¥2,500,000, the institution promises to pay you a lump sum six years from now at an 8 percent annual interest rate. What future amount can you expect? Solution: Use the following data in Equation 2 to find the future value: PV = ¥2, 500, 000 r = 8 % = 0.08 N = 6 N VN = PV (1 + r) F                 = ¥2, 500, 000 ( 1.08)6 = ¥2, 500, 000( 1.586874) = ¥3, 967, 186 You can expect to receive ¥3,967,186 six years from now. Our third example is a more complicated future value problem that illustrates the importance of keeping track of actual calendar time. EXAMPLE 3 The Future Value of a Lump Sum 1. A pension fund manager estimates that his corporate sponsor will make a $10 million contribution five years from now. The rate of return on plan assets has been estimated at 9 percent per year. The pension fund manager wants to calculate the future value of this contribution 15 years from now, which is the date at which the funds will be distributed to retirees. What is that future value? Solution: By positioning the initial investment, PV, at t = 5, we can calculate the future value of the contribution using the following data in Equation 2: PV = $10 million r = 9 % = 0.09 N = 10 N VN = PV (1 + r) F                 = $10,000,000 ( 1.09)10 = $10,000,000( 2.367364) = $23,673,636.75 This problem looks much like the previous two, but it differs in one im- portant respect: its timing. From the standpoint of today (t = 0), the future amount of $23,673,636.75 is 15 years into the future. Although the future value is 10 years from its present value, the present value of $10 million will not be received for another five years. © CFA Institute. For candidate use only. Not for distribution. 10 Learning Module 1 The Time Value of Money Exhibit 2: The Future Value of a Lump Sum, Initial Investment Not at t=0 As Exhibit 2 shows, we have followed the convention of indexing today as t = 0 and indexing subsequent times by adding 1 for each period. The additional contribution of $10 million is to be received in five years, so it is indexed as t = 5 and appears as such in the figure. The future value of the investment in 10 years is then indexed at t = 15; that is, 10 years following the receipt of the $10 million contribution at t = 5. Time lines like this one can be extremely useful when dealing with more-complicated problems, especially those involving more than one cash flow. In a later section of this reading, we will discuss how to calculate the value today of the $10 million to be received five years from now. For the moment, we can use Equation 2. Suppose the pension fund manager in Example 3 above were to receive $6,499,313.86 today from the corporate sponsor. How much will that sum be worth at the end of five years? How much will it be worth at the end of 15 years? PV = $6,499,313.86 r = 9 % = 0.09 N = 5 N VN = PV (1 + r) F                 = $6,499,313.86 ( 1.09)5 = $6,499,313.86( 1.538624) = $10,000,000 at the five-year mark and PV = $6,499,313.86 r = 9 % = 0.09 N = 15 N VN = PV (1 + r) F                  = $6,499,313.86 ( 1.09)15 = $6,499,313.86( 3.642482) = $23,673,636.74 at the 15-year mark These results show that today’s present value of about $6.5 million becomes $10 million after five years and $23.67 million after 15 years. 4 NON-ANNUAL COMPOUNDING (FUTURE VALUE) calculate the solution for time value of money problems with different frequencies of compounding © CFA Institute. For candidate use only. Not for distribution. Non-Annual Compounding (Future Value) 11 In this section, we examine investments paying interest more than once a year. For instance, many banks offer a monthly interest rate that compounds 12 times a year. In such an arrangement, they pay interest on interest every month. Rather than quote the periodic monthly interest rate, financial institutions often quote an annual interest rate that we refer to as the stated annual interest rate or quoted interest rate. We denote the stated annual interest rate by rs. For instance, your bank might state that a particular CD pays 8 percent compounded monthly. The stated annual interest rate equals the monthly interest rate multiplied by 12. In this example, the monthly interest rate is 0.08/12 = 0.0067 or 0.67 percent.5 This rate is strictly a quoting convention because (1 + 0.0067)12 = 1.083, not 1.08; the term (1 + rs) is not meant to be a future value factor when compounding is more frequent than annual. With more than one compounding period per year, the future value formula can be expressed as rs mN FVN = PV (1 + _ m ) (3) where rs is the stated annual interest rate, m is the number of compounding periods per year, and N now stands for the number of years. Note the compatibility here between the interest rate used, rs/m, and the number of compounding periods, mN. The periodic rate, rs/m, is the stated annual interest rate divided by the number of compounding periods per year. The number of compounding periods, mN, is the number of compounding periods in one year multiplied by the number of years. The periodic rate, rs/m, and the number of compounding periods, mN, must be compatible. EXAMPLE 4 The Future Value of a Lump Sum with Quarterly Compounding 1. Continuing with the CD example, suppose your bank offers you a CD with a two-year maturity, a stated annual interest rate of 8 percent compounded quarterly, and a feature allowing reinvestment of the interest at the same interest rate. You decide to invest $10,000. What will the CD be worth at maturity? Solution: Compute the future value with Equation 3 as follows: PV = $10,000 r s = 8 % = 0.08 m = 4 rs / m = 0.08 / 4 = 0.02 N = 2 mN          = 4(2) = 8 interest              periods rs mN FVN = PV (1 + _ m ) = $10,000 ( 1.02)8 = $10,000( 1.171659) = $11,716.59 At maturity, the CD will be worth $11,716.59. 5 To avoid rounding errors when using a financial calculator, divide 8 by 12 and then press the %i key, rather than simply entering 0.67 for %i, so we have (1 + 0.08/12)12 = 1.083000. © CFA Institute. For candidate use only. Not for distribution. 12 Learning Module 1 The Time Value of Money The future value formula in Equation 3 does not differ from the one in Equation 2. Simply keep in mind that the interest rate to use is the rate per period and the expo- nent is the number of interest, or compounding, periods. EXAMPLE 5 The Future Value of a Lump Sum with Monthly Compounding 1. An Australian bank offers to pay you 6 percent compounded monthly. You decide to invest A$1 million for one year. What is the future value of your investment if interest payments are reinvested at 6 percent? Solution: Use Equation 3 to find the future value of the one-year investment as fol- lows: PV = A$1,000,000 r s = 6 % = 0.06 m = 12 rs / m = 0.06 / 12 = 0.0050 N = 1 mN = 12                          (1) = 12 interest periods rs mN FVN = PV (1 + _ m ) = A$1,000,000 ( 1.005)12 = A$1,000,000( 1.061678) = A$1,061,677.81 If you had been paid 6 percent with annual compounding, the future amount would be only A$1,000,000(1.06) = A$1,060,000 instead of A$1,061,677.81 with monthly compounding. 5 CONTINUOUS COMPOUNDING calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding calculate the solution for time value of money problems with different frequencies of compounding The preceding discussion on compounding periods illustrates discrete compounding, which credits interest after a discrete amount of time has elapsed. If the number of compounding periods per year becomes infinite, then interest is said to compound continuously. If we want to use the future value formula with continuous compound- ing, we need to find the limiting value of the future value factor for m → ∞ (infinitely many compounding periods per year) in Equation 3. The expression for the future value of a sum in N years with continuous compounding is FVN = PV ers N (4) © CFA Institute. For candidate use only. Not for distribution. Continuous Compounding 13 The term e rs N is the transcendental number e ≈ 2.7182818 raised to the power rsN. Most financial calculators have the function ex. EXAMPLE 6 The Future Value of a Lump Sum with Continuous Compounding Suppose a $10,000 investment will earn 8 percent compounded continuously for two years. We can compute the future value with Equation 4 as follows: PV = $10,000 rs = 8 % = 0.08 N = 2 r N F          VN = PV e s       = $10,000 e0.08( 2) = $10,000( 1.173511) = $11,735.11 With the same interest rate but using continuous compounding, the $10,000 investment will grow to $11,735.11 in two years, compared with $11,716.59 using quarterly compounding as shown in Example 4. Exhibit 3 shows how a stated annual interest rate of 8 percent generates different ending dollar amounts with annual, semiannual, quarterly, monthly, daily, and contin- uous compounding for an initial investment of $1 (carried out to six decimal places). As Exhibit 3 shows, all six cases have the same stated annual interest rate of 8 percent; they have different ending dollar amounts, however, because of differences in the frequency of compounding. With annual compounding, the ending amount is $1.08. More frequent compounding results in larger ending amounts. The ending dollar amount with continuous compounding is the maximum amount that can be earned with a stated annual rate of 8 percent. Exhibit 3: The Effect of Compounding Frequency on Future Value Frequency rs/m mN Future Value of $1 Annual 8%/1 = 8% 1×1=1 $1.00(1.08) = $1.08 Semiannual 8%/2 = 4% 2×1=2 $1.00(1.04)2 = $1.081600 Quarterly 8%/4 = 2% 4×1=4 $1.00(1.02)4 = $1.082432 Monthly 8%/12 = 0.6667% 12 × 1 = 12 $1.00(1.006667)12 = $1.083000 Daily 8%/365 = 0.0219% 365 × 1 = 365 $1.00(1.000219)365 = $1.083278 Continuous $1.00e0.08(1) = $1.083287 Exhibit 3 also shows that a $1 investment earning 8.16 percent compounded annually grows to the same future value at the end of one year as a $1 investment earning 8 percent compounded semiannually. This result leads us to a distinction between the stated annual interest rate and the effective annual rate (EAR).6 For an 8 percent stated annual interest rate with semiannual compounding, the EAR is 8.16 percent. 6 Among the terms used for the effective annual return on interest-bearing bank deposits are annual percentage yield (APY) in the United States and equivalent annual rate (EAR) in the United Kingdom. By contrast, the annual percentage rate (APR) measures the cost of borrowing expressed as a yearly © CFA Institute. For candidate use only. Not for distribution. 14 Learning Module 1 The Time Value of Money Stated and Effective Rates The stated annual interest rate does not give a future value directly, so we need a for- mula for the EAR. With an annual interest rate of 8 percent compounded semiannually, we receive a periodic rate of 4 percent. During the course of a year, an investment of $1 would grow to $1(1.04)2 = $1.0816, as illustrated in Exhibit 3. The interest earned on the $1 investment is $0.0816 and represents an effective annual rate of interest of 8.16 percent. The effective annual rate is calculated as follows: EAR = (1 + Periodic interest rate)m – 1 (5) The periodic interest rate is the stated annual interest rate divided by m, where m is the number of compounding periods in one year. Using our previous example, we can solve for EAR as follows: (1.04)2 − 1 = 8.16 percent. The concept of EAR extends to continuous compounding. Suppose we have a rate of 8 percent compounded continuously. We can find the EAR in the same way as above by finding the appropriate future value factor. In this case, a $1 investment would grow to $1e0.08(1.0) = $1.0833. The interest earned for one year represents an effective annual rate of 8.33 percent and is larger than the 8.16 percent EAR with semiannual compounding because interest is compounded more frequently. With continuous compounding, we can solve for the effective annual rate as follows: EAR = e rs − 1 (6) We can reverse the formulas for EAR with discrete and continuous compounding to find a periodic rate that corresponds to a particular effective annual rate. Suppose we want to find the appropriate periodic rate for a given effective annual rate of 8.16 per- cent with semiannual compounding. We can use Equation 5 to find the periodic rate: 0.0816 = ( 1 + Periodic rate)2− 1 1.0816 = ( 1 + Periodic rate)2             (    1.0816)1/2− 1 = Periodic rate ( 1.04)− 1 = Periodic rate 4% = Periodic rate To calculate the continuously compounded rate (the stated annual interest rate with continuous compounding) corresponding to an effective annual rate of 8.33 percent, we find the interest rate that satisfies Equation 6: 0.0833 = ers − 1    1.0833 = e rs To solve this equation, we take the natural logarithm of both sides. (Recall that the natural log of e r s is ln e r s = r s.) Therefore, ln 1.0833 = rs, resulting in rs = 8 percent. We see that a stated annual rate of 8 percent with continuous compounding is equivalent to an EAR of 8.33 percent. rate. In the United States, the APR is calculated as a periodic rate times the number of payment periods per year and, as a result, some writers use APR as a general synonym for the stated annual interest rate. Nevertheless, APR is a term with legal connotations; its calculation follows regulatory standards that vary internationally. Therefore, “stated annual interest rate” is the preferred general term for an annual interest rate that does not account for compounding within the year. © CFA Institute. For candidate use only. Not for distribution. A Series of Cash Flows 15 A SERIES OF CASH FLOWS 6 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 In this section, we consider series of cash flows, both even and uneven. We begin with a list of terms commonly used when valuing cash flows that are distributed over many time periods. An annuity is a finite set of level sequential cash flows. An ordinary annuity has a first cash flow that occurs one period from now (indexed at t = 1). An annuity due has a first cash flow that occurs immediately (indexed at t = 0). A perpetuity is a perpetual annuity, or a set of level never-ending sequen- tial cash flows, with the first cash flow occurring one period from now. Equal Cash Flows—Ordinary Annuity Consider an ordinary annuity paying 5 percent annually. Suppose we have five sepa- rate deposits of $1,000 occurring at equally spaced intervals of one year, with the first payment occurring at t = 1. Our goal is to find the future value of this ordinary annuity after the last deposit at t = 5. The increment in the time counter is one year, so the last payment occurs five years from now. As the time line in Exhibit 4 shows, we find the future value of each $1,000 deposit as of t = 5 with Equation 2, FVN = PV(1 + r)N. The arrows in Exhibit 4 extend from the payment date to t = 5. For instance, the first $1,000 deposit made at t = 1 will compound over four periods. Using Equation 2, we find that the future value of the first deposit at t = 5 is $1,000(1.05)4 = $1,215.51. We calculate the future value of all other payments in a similar fashion. (Note that we are finding the future value at t = 5, so the last payment does not earn any interest.) With all values now at t = 5, we can add the future values to arrive at the future value of the annuity. This amount is $5,525.63. Exhibit 4: The Future Value of a Five-Year Ordinary Annuity | | | | | 0 1 2 3 4 5 $1,000(1.05)4 = $1,215.506250 $1,000 $1,000(1.05)3 = $1,157.625000 $1,000 $1,000(1.05)2 = $1,102.500000 $1,000 $1,000(1.05)1 = $1,050.000000 $1,000 $1,000(1.05)0 = $1,000.000000 Sum at t = 5 $5,525.63 © CFA Institute. For candidate use only. Not for distribution. 16 Learning Module 1 The Time Value of Money We can arrive at a general annuity formula if we define the annuity amount as A, the number of time periods as N, and the interest rate per period as r. We can then define the future value as FVN = A[ (1 + r)N−1+ ( 1 + r)N−2+ ( 1 + r)N−3+ … + (1 + r)1+ ( 1 + r)0] which simplifies to FVN = A[ _ ] (1 + r)N− 1 r (7) The term in brackets is the future value annuity factor. This factor gives the future value of an ordinary annuity of $1 per period. Multiplying the future value annuity factor by the annuity amount gives the future value of an ordinary annuity. For the ordinary annuity in Exhibit 4, we find the future value annuity factor from Equation 7 as [_ 0.05 ] = 5.525631 (1.05)5− 1 With an annuity amount A = $1,000, the future value of the annuity is $1,000(5.525631) = $5,525.63, an amount that agrees with our earlier work. The next example illustrates how to find the future value of an ordinary annuity using the formula in Equation 7. EXAMPLE 7 The Future Value of an Annuity 1. Suppose your company’s defined contribution retirement plan allows you to invest up to €20,000 per year. You plan to invest €20,000 per year in a stock index fund for the next 30 years. Historically, this fund has earned 9 percent per year on average. Assuming that you actually earn 9 percent a year, how much money will you have available for retirement after making the last payment? Solution: Use Equation 7 to find the future amount: A = €20,000 r = 9% = 0.09 N = 30 (1 + r)N− 1 (1.09)30− 1 FV annuity factor = _ r = _ 0.09 = 136.307539 FVN = €20,000(136.307539) = €2,726,150.77 Assuming the fund continues to earn an average of 9 percent per year, you will have €2,726,150.77 available at retirement. Unequal Cash Flows In many cases, cash flow streams are unequal, precluding the simple use of the future value annuity factor. For instance, an individual investor might have a savings plan that involves unequal cash payments depending on the month of the year or lower © CFA Institute. For candidate use only. Not for distribution. Present Value of a Single Cash Flow 17 savings during a planned vacation. One can always find the future value of a series of unequal cash flows by compounding the cash flows one at a time. Suppose you have the five cash flows described in Exhibit 5, indexed relative to the present (t = 0). Exhibit 5: A Series of Unequal Cash Flows and Their Future Values at 5 Percent Time Cash Flow ($) Future Value at Year 5 t=1 1,000 $1,000(1.05)4 = $1,215.51 t=2 2,000 $2,000(1.05)3 = $2,315.25 t=3 4,000 $4,000(1.05)2 = $4,410.00 t=4 5,000 $5,000(1.05)1 = $5,250.00 t=5 6,000 $6,000(1.05)0 = $6,000.00 Sum = $19,190.76 All of the payments shown in Exhibit 5 are different. Therefore, the most direct approach to finding the future value at t = 5 is to compute the future value of each payment as of t = 5 and then sum the individual future values. The total future value at Year 5 equals $19,190.76, as shown in the third column. Later in this reading, you will learn shortcuts to take when the cash flows are close to even; these shortcuts will allow you to combine annuity and single-period calculations. PRESENT VALUE OF A SINGLE CASH FLOW 7 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 Just as the future value factor links today’s present value with tomorrow’s future value, the present value factor allows us to discount future value to present value. For example, with a 5 percent interest rate generating a future payoff of $105 in one year, what current amount invested at 5 percent for one year will grow to $105? The answer is $100; therefore, $100 is the present value of $105 to be received in one year at a discount rate of 5 percent. Given a future cash flow that is to be received in N periods and an interest rate per period of r, we can use the formula for future value to solve directly for the present value as follows: FVN = PV (1 + r)N = FVN[_       PV ( 1 ) N ] (8) 1 + r PV = FVN( 1 + r)−N We see from Equation 8 that the present value factor, (1 + r)−N, is the reciprocal of the future value factor, (1 + r)N. © CFA Institute. For candidate use only. Not for distribution. 18 Learning Module 1 The Time Value of Money EXAMPLE 8 The Present Value of a Lump Sum 1. An insurance company has issued a Guaranteed Investment Contract (GIC) that promises to pay $100,000 in six years with an 8 percent return rate. What amount of money must the insurer invest today at 8 percent for six years to make the promised payment? Solution: We can use Equation 8 to find the present value using the following data: FVN = $100,000 r = 8 % = 0.08 N = 6 PV = FVN(1 + r)−N                = $100,000[ _ ( 1 ) 6 ] 1.08 = $100,000( 0.6301696) = $63,016.96 We can say that $63,016.96 today, with an interest rate of 8 percent, is equivalent to $100,000 to be received in six years. Discounting the $100,000 makes a future $100,000 equivalent to $63,016.96 when allowance is made for the time value of money. As the time line in Exhibit 6 shows, the $100,000 has been discounted six full periods. Exhibit 6: The Present Value of a Lump Sum to Be Received at Time t=6 0 1 2 3 4 5 6 PV = $63,016.96 $100,000 = FV EXAMPLE 9 The Projected Present Value of a More Distant Future Lump Sum 1. Suppose you own a liquid financial asset that will pay you $100,000 in 10 years from today. Your daughter plans to attend college four years from to- day, and you want to know what the asset’s present value will be at that time. © CFA Institute. For candidate use only. Not for distribution. Non-Annual Compounding (Present Value) 19 Given an 8 percent discount rate, what will the asset be worth four years from today? Solution: The value of the asset is the present value of the asset’s promised payment. At t = 4, the cash payment will be received six years later. With this informa- tion, you can solve for the value four years from today using Equation 8: VN = $100,000 F r = 8 % = 0.08 N = 6 PV = FVN( 1 + r)−N                = $100,000_ 1 )6 ( 1.08 = $100,000( 0.6301696) = $63,016.96 Exhibit 7: The Relationship between Present Value and Future Value 0... 4... 10 $46,319.35 $63,016.96 $100,000 The time line in Exhibit 7 shows the future payment of $100,000 that is to be received at t = 10. The time line also shows the values at t = 4 and at t = 0. Relative to the payment at t = 10, the amount at t = 4 is a projected present value, while the amount at t = 0 is the present value (as of today). Present value problems require an evaluation of the present value factor, (1 + r)−N. Present values relate to the discount rate and the number of periods in the following ways: For a given discount rate, the farther in the future the amount to be received, the smaller that amount’s present value. Holding time constant, the larger the discount rate, the smaller the present value of a future amount. NON-ANNUAL COMPOUNDING (PRESENT VALUE) 8 calculate the solution for time value of money problems with different frequencies of compounding Recall that interest may be paid semiannually, quarterly, monthly, or even daily. To handle interest payments made more than once a year, we can modify the present value formula (Equation 8) as follows. Recall that rs is the quoted interest rate and equals the periodic interest rate multiplied by the number of compounding periods in each year. In general, with more than one compounding period in a year, we can express the formula for present value as © CFA Institute. For candidate use only. Not for distribution. 20 Learning Module 1 The Time Value of Money rs −mN PV = FVN(1 + _ m ) (9) where m = number of compounding periods per year rs = quoted annual interest rate N = number of years The formula in Equation 9 is quite similar to that in Equation 8. As we have already noted, present value and future value factors are reciprocals. Changing the frequency of compounding does not alter this result. The only difference is the use of the periodic interest rate and the corresponding number of compounding periods. The following example illustrates Equation 9. EXAMPLE 10 The Present Value of a Lump Sum with Monthly Compounding 1. The manager of a Canadian pension fund knows that the fund must make a lump-sum payment of C$5 million 10 years from now. She wants to invest an amount today in a GIC so that it will grow to the required amount. The current interest rate on GICs is 6 percent a year, compounded monthly. How much should she invest today in the GIC? Solution: Use Equation 9 to find the required present value: FVN = C$5,000,000 r s = 6 % = 0.06 m = 12 rs / m = 0.06 / 12 = 0.005 N = 10                      mN = 12 (10) = 120 rs −mN PV = FVN( m ) 1 + _ = C$5,000,000 ( 1.005)−120 = C$5,000,000( 0.549633) = C$2,748,163.67 In applying Equation 9, we use the periodic rate (in this case, the monthly rate) and the appropriate number of periods with monthly compounding (in this case, 10 years of monthly compounding, or 120 periods). © CFA Institute. For candidate use only. Not for distribution. Present Value of a Series of Equal and Unequal Cash Flows 21 PRESENT VALUE OF A SERIES OF EQUAL AND UNEQUAL CASH FLOWS 9 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 Many applications in investment management involve assets that offer a series of cash flows over time. The cash flows may be highly uneven, relatively even, or equal. They may occur over relatively short periods of time, longer periods of time, or even stretch on indefinitely. In this section, we discuss how to find the present value of a series of cash flows. The Present Value of a Series of Equal Cash Flows We begin with an ordinary annuity. Recall that an ordinary annuity has equal annuity payments, with the first payment starting one period into the future. In total, the annuity makes N payments, with the first payment at t = 1 and the last at t = N. We can express the present value of an ordinary annuity as the sum of the present values of each individual annuity payment, as follows: A A A A A PV = _ _ _ _ _ ( 1 + r)+ (1 + r)2+ (1 + r)3+ … + (1 + r)N−1+ (1 + r)N (10) where A = the annuity amount r = the interest rate per period corresponding to the frequency of annuity payments (for example, annual, quarterly, or monthly) N = the number of annuity payments Because the annuity payment (A) is a constant in this equation, it can be factored out as a common term. Thus the sum of the interest factors has a shortcut expression: ( 1 ) N 1−_ [ ] _ 1 + r V = A r P (11) In much the same way that we computed the future value of an ordinary annuity, we find the present value by multiplying the annuity amount by a present value annuity factor (the term in brackets in Equation 11). © CFA Institute. For candidate use only. Not for distribution. 22 Learning Module 1 The Time Value of Money EXAMPLE 11 The Present Value of an Ordinary Annuity 1. Suppose you are considering purchasing a financial asset that promises to pay €1,000 per year for five years, with the first payment one year from now. The required rate of return is 12 percent per year. How much should you pay for this asset? Solution: To find the value of the financial asset, use the formula for the present value of an ordinary annuity given in Equation 11 with the following data: A = €1,000 r = 12% = 0.12 N=5 ( 1 ) N 1−_ [ ] _ 1 + r PV = A r ( 1 ) 5 1−_ [ ] _ 1.12 = €1,000 0.12 = €1,000(3.604776) = €3,604.78 The series of cash flows of €1,000 per year for five years is currently worth €3,604.78 when discounted at 12 percent. Keeping track of the actual calendar time brings us to a specific type of annuity with level payments: the annuity due. An annuity due has its first payment occurring today (t = 0). In total, the annuity due will make N payments. Exhibit 8 presents the time line for an annuity due that makes four payments of $100. Exhibit 8: An Annuity Due of $100 per Period | | | | 0 1 2 3 $100 $100 $100 $100 As Exhibit 8 shows

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