Quantitative Investment Analysis E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: The CFA Institute Series
- Sprache: Englisch
Whether you are a novice investor or an experienced practitioner, Quantitative Investment Analysis, 4th Edition has something for you. Part of the CFA Institute Investment Series, this authoritative guide is relevant the world over and will facilitate your mastery of quantitative methods and their application in todays investment process. This updated edition provides all the statistical tools and latest information you need to be a confident and knowledgeable investor. This edition expands coverage of Machine Learning algorithms and the role of Big Data in an investment context along with capstone chapters in applying these techniques to factor modeling, risk management and backtesting and simulation in investment strategies. The authors go to great lengths to ensure an even treatment of subject matter, consistency of mathematical notation, and continuity of topic coverage that is critical to the learning process. Well suited for motivated individuals who learn on their own, as well as a general reference, this complete resource delivers clear, example-driven coverage of a wide range of quantitative methods. Inside you'll find: * Learning outcome statements (LOS) specifying the objective of each chapter * A diverse variety of investment-oriented examples both aligned with the LOS and reflecting the realities of todays investment world * A wealth of practice problems, charts, tables, and graphs to clarify and reinforce the concepts and tools of quantitative investment management You can choose to sharpen your skills by furthering your hands-on experience in the Quantitative Investment Analysis Workbook, 4th Edition (sold separately)--an essential guide containing learning outcomes and summary overview sections, along with challenging problems and solutions.
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CFA Institute is the premier association for investment professionals around the world, with over 160,000 members in 165 countries and territories. Since 1963 the organization has developed and administered the renowned Chartered Financial Analyst® Program. With a rich history of leading the investment profession, CFA Institute has set the highest standards in ethics, education, and professional excellence within the global investment community and is the foremost authority on investment profession conduct and practice. Each book in the CFA Institute Investment Series is geared toward industry practitioners along with graduate-level finance students and covers the most important topics in the industry. The authors of these cutting-edge books are themselves industry professionals and academics and bring their wealth of knowledge and expertise to this series.
QUANTITATIVE INVESTMENT ANALYSIS
Fourth Edition
Richard A. DeFusco, CFA
Dennis W. McLeavey, CFA
Jerald E. Pinto, CFA
David E. Runkle, CFA
Cover image: © r.nagy/Shutterstock
Cover design: Wiley
Copyright © 2004, 2007, 2015, 2020 by CFA Institute. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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ISBN 978-1-119-74362-0 (Hardcover)
ISBN 978-1-119-74365-1 (ePDF)
ISBN 978-1-119-74364-4 (ePub)
CONTENTS
Cover
Title Page
Copyright
PREFACE
ACKNOWLEDGMENTS
ABOUT THE CFA INSTITUTE INVESTMENT SERIES
CHAPTER 1: THE TIME VALUE OF MONEY
LEARNING OUTCOMES
1. INTRODUCTION
2. INTEREST RATES: INTERPRETATION
3. THE FUTURE VALUE OF A SINGLE CASH FLOW
4. THE FUTURE VALUE OF A SERIES OF CASH FLOWS
5. THE PRESENT VALUE OF A SINGLE CASH FLOW
6. THE PRESENT VALUE OF A SERIES OF CASH FLOWS
7. SOLVING FOR RATES, NUMBER OF PERIODS, OR SIZE OF ANNUITY PAYMENTS
8. SUMMARY
PRACTICE PROBLEMS
CHAPTER 2: ORGANIZING, VISUALIZING, AND DESCRIBING DATA
LEARNING OUTCOMES
1. INTRODUCTION
2. DATA TYPES
3. DATA SUMMARIZATION
4. DATA VISUALIZATION
5. MEASURES OF CENTRAL TENDENCY
6. OTHER MEASURES OF LOCATION: QUANTILES
7. MEASURES OF DISPERSION
8. THE SHAPE OF THE DISTRIBUTIONS: SKEWNESS
9. THE SHAPE OF THE DISTRIBUTIONS: KURTOSIS
10. CORRELATION BETWEEN TWO VARIABLES
11. SUMMARY
PRACTICE PROBLEMS
CHAPTER 3: PROBABILITY CONCEPTS
LEARNING OUTCOMES
1. INTRODUCTION
2. PROBABILITY, EXPECTED VALUE, AND VARIANCE
3. PORTFOLIO EXPECTED RETURN AND VARIANCE OF RETURN
4. TOPICS IN PROBABILITY
5. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 4: COMMON PROBABILITY DISTRIBUTIONS
LEARNING OUTCOMES
1. INTRODUCTION TO COMMON PROBABILITY DISTRIBUTIONS
2. DISCRETE RANDOM VARIABLES
3. CONTINUOUS RANDOM VARIABLES
4. INTRODUCTION TO MONTE CARLO SIMULATION
5. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 5: SAMPLING AND ESTIMATION
LEARNING OUTCOMES
1. INTRODUCTION
2. SAMPLING
3. DISTRIBUTION OF THE SAMPLE MEAN
4. POINT AND INTERVAL ESTIMATES OF THE POPULATION MEAN
5. MORE ON SAMPLING
6. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 6: HYPOTHESIS TESTING
LEARNING OUTCOMES
1. INTRODUCTION
2. HYPOTHESIS TESTING
3. HYPOTHESIS TESTS CONCERNING THE MEAN
4. HYPOTHESIS TESTS CONCERNING VARIANCE AND CORRELATION
5. OTHER ISSUES: NONPARAMETRIC INFERENCE
6. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 7: INTRODUCTION TO LINEAR REGRESSION
LEARNING OUTCOMES
1. INTRODUCTION
2. LINEAR REGRESSION
3. ASSUMPTIONS OF THE LINEAR REGRESSION MODEL
4. THE STANDARD ERROR OF ESTIMATE
5. THE COEFFICIENT OF DETERMINATION
6. HYPOTHESIS TESTING
7. ANALYSIS OF VARIANCE IN A REGRESSION WITH ONE INDEPENDENT VARIABLE
8. PREDICTION INTERVALS
9. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 8: MULTIPLE REGRESSION
LEARNING OUTCOMES
1. INTRODUCTION
2. MULTIPLE LINEAR REGRESSION
3. USING DUMMY VARIABLES IN REGRESSIONS
4. VIOLATIONS OF REGRESSION ASSUMPTIONS
5. MODEL SPECIFICATION AND ERRORS IN SPECIFICATION
6. MODELS WITH QUALITATIVE DEPENDENT VARIABLES
7. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 9: TIME-SERIES ANALYSIS
LEARNING OUTCOMES
1. INTRODUCTION TO TIME-SERIES ANALYSIS
2. CHALLENGES OF WORKING WITH TIME SERIES
3. TREND MODELS
4. AUTOREGRESSIVE (AR) TIME-SERIES MODELS
5. RANDOM WALKS AND UNIT ROOTS
6. MOVING-AVERAGE TIME-SERIES MODELS
7. SEASONALITY IN TIME-SERIES MODELS
8. AUTOREGRESSIVE MOVING-AVERAGE MODELS
9. AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY MODELS
10. REGRESSIONS WITH MORE THAN ONE TIME SERIES
11. OTHER ISSUES IN TIME SERIES
12. SUGGESTED STEPS IN TIME-SERIES FORECASTING
SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 10: MACHINE LEARNING
LEARNING OUTCOMES
1. INTRODUCTION
2. MACHINE LEARNING AND INVESTMENT MANAGEMENT
3. WHAT IS MACHINE LEARNING?
4. OVERVIEW OF EVALUATING ML ALGORITHM PERFORMANCE
5. SUPERVISED MACHINE LEARNING ALGORITHMS
6. UNSUPERVISED MACHINE LEARNING ALGORITHMS
7. NEURAL NETWORKS, DEEP LEARNING NETS, AND REINFORCEMENT LEARNING
8. CHOOSING AN APPROPRIATE ML ALGORITHM
9. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 11: BIG DATA PROJECTS
LEARNING OUTCOMES
1. INTRODUCTION
2. BIG DATA IN INVESTMENT MANAGEMENT
3. STEPS IN EXECUTING A DATA ANALYSIS PROJECT: FINANCIAL FORECASTING WITH BIG DATA
4. DATA PREPARATION AND WRANGLING
5. DATA EXPLORATION OBJECTIVES AND METHODS
6. MODEL TRAINING
7. FINANCIAL FORECASTING PROJECT: CLASSIFYING AND PREDICTING SENTIMENT FOR STOCKS
8. SUMMARY
PRACTICE PROBLEMS
CHAPTER 12: USING MULTIFACTOR MODELS
LEARNING OUTCOMES
1. INTRODUCTION
2. MULTIFACTOR MODELS AND MODERN PORTFOLIO THEORY
3. ARBITRAGE PRICING THEORY
4. MULTIFACTOR MODELS: TYPES
5. MULTIFACTOR MODELS: SELECTED APPLICATIONS
6. SUMMARY
REFERENCES
PRACTICE PROBLEMS
CHAPTER 13: MEASURING AND MANAGING MARKET RISK
LEARNING OUTCOMES
1. INTRODUCTION
2. UNDERSTANDING VALUE AT RISK
3. OTHER KEY RISK MEASURES—SENSITIVITY AND SCENARIO MEASURES
4. USING CONSTRAINTS IN MARKET RISK MANAGEMENT
5. APPLICATIONS OF RISK MEASURES
6. SUMMARY
REFERENCE
PRACTICE PROBLEMS
CHAPTER 14: BACKTESTING AND SIMULATION
LEARNING OUTCOMES
1. INTRODUCTION
2. THE OBJECTIVES OF BACKTESTING
3. THE BACKTESTING PROCESS
4. METRICS AND VISUALS USED IN BACKTESTING
5. COMMON PROBLEMS IN BACKTESTING
6. BACKTESTING FACTOR ALLOCATION STRATEGIES
7. COMPARING METHODS OF MODELING RANDOMNESS
8. SCENARIO ANALYSIS
9. HISTORICAL SIMULATION VERSUS MONTE CARLO SIMULATION
10. HISTORICAL SIMULATION
11. MONTE CARLO SIMULATION
12. SENSITIVITY ANALYSIS
13. SUMMARY
REFERENCES
PRACTICE PROBLEMS
APPENDICES
GLOSSARY
ABOUT THE AUTHORS
ABOUT THE CFA PROGRAM
INDEX
END USER LICENSE AGREEMENT
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
Preface
Acknowledgments
About the CFA Institute Investment Series
Begin Reading
Appendices
Glossary
About the Authors
About the CFA Program
Index
End User License Agreement
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PREFACE
We are pleased to bring you Quantitative Investment Analysis, Fourth Edition, which focuses on key tools that are needed for today’s professional investor. In addition to classic areas such as the time value of money and probability and statistics, the text covers advanced concepts in regression, time series, machine learning, and big data projects. The text teaches critical skills that challenge many professionals, and shows how these techniques can be applied to areas such as factor modeling, risk management, and backtesting and simulation.
The content was developed in partnership by a team of distinguished academics and practitioners, chosen for their acknowledged expertise in the field, and guided by CFA Institute. It is written specifically with the investment practitioner in mind and is replete with examples and practice problems that reinforce the learning outcomes and demonstrate real-world applicability.
The CFA Program Curriculum, from which the content of this book was drawn, is subjected to a rigorous review process to assure that it is:
Faithful to the findings of our ongoing industry practice analysis
Valuable to members, employers, and investors
Globally relevant
Generalist (as opposed to specialist) in nature
Replete with sufficient examples and practice opportunities
Pedagogically sound
The accompanying workbook is a useful reference that provides Learning Outcome Statements that describe exactly what readers will learn and be able to demonstrate after mastering the accompanying material. Additionally, the workbook has summary overviews and practice problems for each chapter.
We are confident that you will find this and other books in the CFA Institute Investment Series helpful in your efforts to grow your investment knowledge, whether you are a relatively new entrant or an experienced veteran striving to keep up to date in the ever-changing market environment. CFA Institute, as a long-term committed participant in the investment profession and a not-for-profit global membership association, is pleased to provide you with this opportunity.
ACKNOWLEDGMENTS
Special thanks to all the reviewers, advisors, and question writers who helped to ensure high practical relevance, technical correctness, and understandability of the material presented here.
We would like to thank the many others who played a role in the conception and production of this book: the Curriculum and Learning Experience team at CFA Institute, with special thanks to the curriculum directors, past and present, who worked with the authors and reviewers to produce the chapters in this book; the Practice Analysis team at CFA Institute; and the Publishing and Technology team for bringing this book to production.
ABOUT THE CFA INSTITUTE INVESTMENT SERIES
CFA Institute is pleased to provide the CFA Institute Investment Series, which covers major areas in the field of investments. We provide this best-in-class series for the same reason we have been chartering investment professionals for more than 45 years: to lead the investment profession globally by setting the highest standards of ethics, education, and professional excellence.
The books in the CFA Institute Investment Series contain practical, globally relevant material. They are intended both for those contemplating entry into the extremely competitive field of investment management as well as for those seeking a means of keeping their knowledge fresh and up to date. This series was designed to be user friendly and highly relevant.
We hope you find this series helpful in your efforts to grow your investment knowledge, whether you are a relatively new entrant or an experienced veteran ethically bound to keep up to date in the ever-changing market environment. As a long-term, committed participant in the investment profession and a not-for-profit global membership association, CFA Institute is pleased to provide you with this opportunity.
THE TEXTS
Corporate Finance: A Practical Approach is a solid foundation for those looking to achieve lasting business growth. In today’s competitive business environment, companies must find innovative ways to enable rapid and sustainable growth. This text equips readers with the foundational knowledge and tools for making smart business decisions and formulating strategies to maximize company value. It covers everything from managing relationships between stakeholders to evaluating merger and acquisition bids, as well as the companies behind them. Through extensive use of real-world examples, readers will gain critical perspective into interpreting corporate financial data, evaluating projects, and allocating funds in ways that increase corporate value. Readers will gain insights into the tools and strategies used in modern corporate financial management.
Equity Asset Valuation is a particularly cogent and important resource for anyone involved in estimating the value of securities and understanding security pricing. A well-informed professional knows that the common forms of equity valuation—dividend discount modeling, free cash flow modeling, price/earnings modeling, and residual income modeling—can all be reconciled with one another under certain assumptions. With a deep understanding of the underlying assumptions, the professional investor can better understand what other investors assume when calculating their valuation estimates. This text has a global orientation, including emerging markets.
Fixed Income Analysis has been at the forefront of new concepts in recent years, and this particular text offers some of the most recent material for the seasoned professional who is not a fixed-income specialist. The application of option and derivative technology to the once staid province of fixed income has helped contribute to an explosion of thought in this area. Professionals have been challenged to stay up to speed with credit derivatives, swaptions, collateralized mortgage securities, mortgage-backed securities, and other vehicles, and this explosion of products has strained the world’s financial markets and tested central banks to provide sufficient oversight. Armed with a thorough grasp of the new exposures, the professional investor is much better able to anticipate and understand the challenges our central bankers and markets face.
International Financial Statement Analysis is designed to address the ever-increasing need for investment professionals and students to think about financial statement analysis from a global perspective. The text is a practically oriented introduction to financial statement analysis that is distinguished by its combination of a true international orientation, a structured presentation style, and abundant illustrations and tools covering concepts as they are introduced in the text. The authors cover this discipline comprehensively and with an eye to ensuring the reader’s success at all levels in the complex world of financial statement analysis.
Investments: Principles of Portfolio and Equity Analysis provides an accessible yet rigorous introduction to portfolio and equity analysis. Portfolio planning and portfolio management are presented within a context of up-to-date, global coverage of security markets, trading, and market-related concepts and products. The essentials of equity analysis and valuation are explained in detail and profusely illustrated. The book includes coverage of practitioner-important but often neglected topics, such as industry analysis. Throughout, the focus is on the practical application of key concepts with examples drawn from both emerging and developed markets. Each chapter affords the reader many opportunities to self-check his or her understanding of topics.
All books in the CFA Institute Investment Series are available through all major booksellers. And, all titles are available on the Wiley Custom Select platform at http://customselect.wiley.com/ where individual chapters for all the books may be mixed and matched to create custom textbooks for the classroom.
CHAPTER 1THE TIME VALUE OF MONEY
Richard A. DeFusco, PhD, CFA
Dennis W. McLeavey, DBA, CFA
Jerald E. Pinto, PhD, CFA
David E. Runkle, PhD, CFA
LEARNING OUTCOMES
The candidate should be able to:
interpret interest rates as required rates of return, discount rates, or opportunity costs;
explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk;
calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;
solve time value of money problems for different frequencies of compounding;
calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;
demonstrate the use of a time line in modeling and solving time value of money problems.
1. INTRODUCTION
As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing, if we are shopping for a loan. As investment analysts, much of our work also involves evaluating transactions with present and future cash flows. When we place a value on any security, for example, we are attempting to determine the worth of a stream of future cash flows. To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore, a smaller amount of money now may be equivalent in value to a larger amount received at a future date. The time value of money as a topic in investment mathematics deals with equivalence relationships between cash flows with different dates. Mastery of time value of money concepts and techniques is essential for investment analysts.
The chapter1 is organized as follows: Section 2 introduces some terminology used throughout the chapter and supplies some economic intuition for the variables we will discuss. Section 3 tackles the problem of determining the worth at a future point in time of an amount invested today. Section 4 addresses the future worth of a series of cash flows. These two sections provide the tools for calculating the equivalent value at a future date of a single cash flow or series of cash flows. Sections 5 and 6 discuss the equivalent value today of a single future cash flow and a series of future cash flows, respectively. In Section 7, we explore how to determine other quantities of interest in time value of money problems.
2. INTEREST RATES: INTERPRETATION
In this chapter, we will continually refer to interest rates. In some cases, we assume a particular value for the interest rate; in other cases, the interest rate will be the unknown quantity we seek to determine. Before turning to the mechanics of time value of money problems, we must illustrate the underlying economic concepts. In this section, we briefly explain the meaning and interpretation of interest rates.
Time value of money concerns equivalence relationships between cash flows occurring on different dates. The idea of equivalence relationships is relatively simple. Consider the following exchange: You pay $10,000 today and in return receive $9,500 today. Would you accept this arrangement? Not likely. But what if you received the $9,500 today and paid the $10,000 one year from now? Can these amounts be considered equivalent? Possibly, because a payment of $10,000 a year from now would probably be worth less to you than a payment of $10,000 today. It would be fair, therefore, to discount the $10,000 received in one year; that is, to cut its value based on how much time passes before the money is paid. An interest rate, denoted r, is a rate of return that reflects the relationship between differently dated cash flows. If $9,500 today and $10,000 in one year are equivalent in value, then $10,000 − $9,500 = $500 is the required compensation for receiving $10,000 in one year rather than now. The interest rate—the required compensation stated as a rate of return—is $500/$9,500 = 0.0526 or 5.26 percent.
Interest rates can be thought of in three ways. First, they can be considered required rates of return—that is, the minimum rate of return an investor must receive in order to accept the investment. Second, interest rates can be considered discount rates. In the example above, 5.26 percent is that rate at which we discounted the $10,000 future amount to find its value today. Thus, we use the terms “interest rate” and “discount rate” almost interchangeably. Third, interest rates can be considered opportunity costs. An opportunity cost is the value that investors forgo by choosing a particular course of action. In the example, if the party who supplied $9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent on the money. So we can view 5.26 percent as the opportunity cost of current consumption.
Economics tells us that interest rates are set in the marketplace by the forces of supply and demand, where investors are suppliers of funds and borrowers are demanders of funds. Taking the perspective of investors in analyzing market-determined interest rates, we can view an interest rate r as being composed of a real risk-free interest rate plus a set of four premiums that are required returns or compensation for bearing distinct types of risk:
r
=
Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium
The
real risk-free interest rate
is the single-period interest rate for a completely risk-free security if no inflation were expected. In economic theory, the real risk-free rate reflects the time preferences of individuals for current versus future real consumption.
The
inflation premium
compensates investors for expected inflation and reflects the average inflation rate expected over the maturity of the debt. Inflation reduces the purchasing power of a unit of currency—the amount of goods and services one can buy with it. The sum of the real risk-free interest rate and the inflation premium is the
nominal risk-free interest rate
.
2
Many countries have governmental short-term debt whose interest rate can be considered to represent the nominal risk-free interest rate in that country. The interest rate on a 90-day US Treasury bill (T-bill), for example, represents the nominal risk-free interest rate over that time horizon.
3
US T-bills can be bought and sold in large quantities with minimal transaction costs and are backed by the full faith and credit of the US government.
The
default risk premium
compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount.
The
liquidity premium
compensates investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly. US T-bills, for example, do not bear a liquidity premium because large amounts can be bought and sold without affecting their market price. Many bonds of small issuers, by contrast, trade infrequently after they are issued; the interest rate on such bonds includes a liquidity premium reflecting the relatively high costs (including the impact on price) of selling a position.
The
maturity premium
compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended, in general (holding all else equal). The difference between the interest rate on longer-maturity, liquid Treasury debt and that on short-term Treasury debt reflects a positive maturity premium for the longer-term debt (and possibly different inflation premiums as well).
Using this insight into the economic meaning of interest rates, we now turn to a discussion of solving time value of money problems, starting with the future value of a single cash flow.
3. THE FUTURE VALUE OF A SINGLE CASH FLOW
In this section, we introduce time value associated with a single cash flow or lump-sum investment. We describe the relationship between an initial investment or present value (PV), which earns a rate of return (the interest rate per period) denoted as r, and its future value (FV), which will be received N years or periods from today.
The following example illustrates this concept. Suppose you invest $100 (PV = $100) in an interest-bearing bank account paying 5 percent annually. At the end of the first year, you will have the $100 plus the interest earned, 0.05 × $100 = $5, for a total of $105. To formalize this one-period example, we define the following terms:
PV =
present value of the investment
FV
N
=
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
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: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.
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
Interest for the second year based on interest earned in the first year (0.05 × $5.00 interest on interest)
0.25
Total
$110.25
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 compounded 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:
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 Figure 1 shows this relationship. In Figure 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 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.
FIGURE 1 The Relationship between an Initial Investment, PV, and Its Future Value, FV
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
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 compounded 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 withdrawals?
Solution: To solve this problem, compute the future value of the $5 million investment using the following values in Equation 2:
At the end of five years, you will have $7,012,758.65 if your money remains invested at 7 percent with no withdrawals.
Note that in this and most examples in this chapter, the factors are reported at six decimal places but the calculations may actually reflect greater precision. For example, 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
EXAMPLE 2
The Future Value of a Lump Sum with No Interim Cash
An institution offers you the following terms for a contract: For an investment 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:
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
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:
This problem looks much like the previous two, but it differs in one important 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.
FIGURE 2 The Future Value of a Lump Sum, Initial Investment Not at t = 0
As Figure 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 chapter, 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?
and
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.
3.1. The Frequency of Compounding
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
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
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:
At maturity, the CD will be worth $11,716.59.
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 exponent is the number of interest, or compounding, periods.
EXAMPLE 5
The Future Value of a Lump Sum with Monthly Compounding
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 follows:
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.
3.2. Continuous 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 compounding, 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
The term 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:
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.
TABLE 1 The Effect of Compounding Frequency on Future Value
Frequency
r
s
/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
$1.00
e
0.08(1)
= $1.083287
Table 1 shows how a stated annual interest rate of 8 percent generates different ending dollar amounts with annual, semiannual, quarterly, monthly, daily, and continuous compounding for an initial investment of $1 (carried out to six decimal places).
As Table 1 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.
Table 1 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.
3.3. Stated and Effective Rates
The stated annual interest rate does not give a future value directly, so we need a formula 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 Table 1. 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:
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:
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 percent with semiannual compounding. We can use Equation 5 to find the 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:
To solve this equation, we take the natural logarithm of both sides. (Recall that the natural log of is ln .) 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.
4. THE FUTURE VALUE OF A SERIES OF CASH FLOWS
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 sequential cash flows, with the first cash flow occurring one period from now.
4.1. Equal Cash Flows—Ordinary Annuity
Consider an ordinary annuity paying 5 percent annually. Suppose we have five separate 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 Figure 3 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 Figure 3 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.
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
which simplifies to
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 Figure 3, we find the future value annuity factor from Equation 7 as
FIGURE 3 The Future Value of a Five-Year Ordinary Annuity
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
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:
Assuming the fund continues to earn an average of 9 percent per year, you will have €2,726,150.77 available at retirement.
4.2. 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 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 Table 2, indexed relative to the present (t = 0).
All of the payments shown in Table 2 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 $19,190.76, as shown in the third column. Later in this chapter, you will learn shortcuts to take when the cash flows are close to even; these shortcuts will allow you to combine annuity and single-period calculations.
TABLE 2 A Series of Unequal Cash Flows and Their Future Values at 5 Percent
Time
Cash Flow ($)
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
5. THE PRESENT VALUE OF A SINGLE CASH FLOW
5.1. Finding the Present Value of a Single Cash Flow
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:
We see from Equation 8 that the present value factor, (1 + r)−N, is the reciprocal of the future value factor, (1 + r)N.
EXAMPLE 8
The Present Value of a Lump Sum
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:
FIGURE 4 The Present Value of a Lump Sum to Be Received at Time t = 6
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 Figure 4 shows, the $100,000 has been discounted six full periods.
EXAMPLE 9
The Projected Present Value of a More Distant Future Lump Sum
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 today, and you want to know what the asset’s present value will be at that time. 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 information, you can solve for the value four years from today using Equation 8:
FIGURE 5 The Relationship between Present Value and Future Value
The time line in Figure 5 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)
