Qlik sense not equal set analysis

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QUANTITATIVE INVESTMENT ANALYSIS Third Edition

Richard A. DeFusco, CFA

Dennis W. McLeavey, CFA

Jerald E. Pinto, CFA

David E. Runkle, CFA

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CONTENTS 1. Foreword 2. Preface 3. Acknowledgment 4. About the CFA Institute Investment Series 5. CHAPTER 1: The Time Value of Money

1. 1. Introduction 2. 2. Interest Rates: Interpretation 3. 3. The Future Value of a Single Cash Flow 4. 4. The Future Value of a Series of Cash Flows 5. 5. The Present Value of a Single Cash Flow 6. 6. The Present Value of a Series of Cash Flows 7. 7. Solving for Rates, Number of Periods, or Size of Annuity Payments 8. 8. Summary 9. Problems 10. Notes

6. CHAPTER 2: Discounted Cash Flow Applications 1. 1. Introduction 2. 2. Net Present Value and Internal Rate of Return 3. 3. Portfolio Return Measurement 4. 4. Money Market Yields 5. 5. Summary 6. References 7. Problems 8. Notes

7. CHAPTER 3: Statistical Concepts and Market Returns 1. 1. Introduction 2. 2. Some Fundamental Concepts 3. 3. Summarizing Data Using Frequency Distributions 4. 4. The Graphic Presentation of Data

5. 5. Measures of Central Tendency 6. 6. Other Measures of Location: Quantiles 7. 7. Measures of Dispersion 8. 8. Symmetry and Skewness in Return Distributions 9. 9. Kurtosis in Return Distributions 10. 10. Using Geometric and Arithmetic Means 11. 11. Summary 12. References 13. Problems 14. Notes

8. CHAPTER 4: Probability Concepts 1. 1. Introduction 2. 2. Probability, Expected Value, and Variance 3. 3. Portfolio Expected Return and Variance of Return 4. 4. Topics in Probability 5. 5. Summary 6. References 7. Problems 8. Notes

9. CHAPTER 5: Common Probability Distributions 1. 1. Introduction to Common Probability Distributions 2. 2. Discrete Random Variables 3. 3. Continuous Random Variables 4. 4. Monte Carlo Simulation 5. 5. Summary 6. References 7. Problems 8. Notes

10. CHAPTER 6: Sampling and Estimation 1. 1. Introduction 2. 2. Sampling

3. 3. Distribution of the Sample Mean 4. 4. Point and Interval Estimates of the Population Mean 5. 5. More on Sampling 6. 6. Summary 7. References 8. Problems 9. Notes

11. CHAPTER 7: Hypothesis Testing 1. 1. Introduction 2. 2. Hypothesis Testing 3. 3. Hypothesis Tests Concerning the Mean 4. 4. Hypothesis Tests Concerning Variance 5. 5. Other Issues: Nonparametric Inference 6. 6. Summary 7. References 8. Problems 9. Notes

12. CHAPTER 8: Correlation and Regression 1. 1. Introduction 2. 2. Correlation Analysis 3. 3. Linear Regression 4. 4. Summary 5. Problems 6. Notes

13. CHAPTER 9: Multiple Regression and Issues in Regression Analysis 1. 1. Introduction 2. 2. Multiple Linear Regression 3. 3. Using Dummy Variables in Regressions 4. 4. Violations of Regression Assumptions 5. 5. Model Specification and Errors in Specification 6. 6. Models with Qualitative Dependent Variables

7. 7. Summary 8. References 9. Problems 10. Notes

14. CHAPTER 10 : Time-Series Analysis 1. 1. Introduction to Time-Series Analysis 2. 2. Challenges of Working with Time Series 3. 3. Trend Models 4. 4. Autoregressive (AR) Time-Series Models 5. 5. Random Walks and Unit Roots 6. 6. Moving-Average Time-Series Models 7. 7. Seasonality in Time-Series Models 8. 8. Autoregressive Moving-Average Models 9. 9. Autoregressive Conditional Heteroskedasticity Models 10. 10. Regressions with More than One Time Series 11. 11. Other Issues in Time Series 12. 12. Suggested Steps in Time-Series Forecasting 13. 13. Summary 14. Problems 15. Notes

15. CHAPTER 11: An Introduction to Multifactor Models 1. 1. Introduction 2. 2. Multifactor Models and Modern Portfolio Theory 3. 3. Arbitrage Pricing Theory 4. 4. Multifactor Models: Types 5. 5. Multifactor Models: Selected Applications 6. 6. Summary 7. References 8. Problems

16. Appendices 17. Glossary

18. About the Editors and Authors 19. About the CFA Program 20. Index 21. Advert 22. EULA

List of Tables

1. Chapter 1

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

2. Chapter 2

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 7

8. TABLE 8

3. Chapter 3

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 7

8. TABLE 8

9. TABLE 9

10. TABLE 10

11. TABLE 11

12. TABLE 12

13. TABLE 13

14. TABLE 14

15. TABLE 15

16. TABLE 16

17. TABLE 17

18. TABLE 18

19. TABLE 19

20. TABLE 20

21. TABLE 21

22. TABLE 22

23. TABLE 23

24. TABLE 24

25. TABLE 25

26. TABLE 26

27. TABLE 27

28. TABLE 28

29. TABLE 29

30. TABLE 30

4. Chapter 4

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 7

8. TABLE 8

9. TABLE 9

10. TABLE 10

11. TABLE 11

12. TABLE 12

13. TABLE 13

5. Chapter 5

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 7

8. TABLE 8

9. TABLE 9

6. Chapter 6

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

7. Chapter 7

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 7

8. TABLE 8

9. TABLE 9

10. TABLE 10

11. TABLE 11

8. Chapter 8

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 2

8. TABLE 7

9. TABLE 8

10. TABLE 9

11. TABLE 10

12. TABLE 11

13. TABLE 12

9. Chapter 9

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 2

5. TABLE 1

6. TABLE 4

7. TABLE 5

8. TABLE 6

9. TABLE 7

10. TABLE 8

11. TABLE 9

12. TABLE 10

13. TABLE 11

14. TABLE 12

15. TABLE 13

16. TABLE 14

17. TABLE 15

18. TABLE 16

19. TABLE 17

20. TABLE 18

10. Chapter 10

1. TABLE 1

2. TABLE 2

3. TABLE 3

4. TABLE 4

5. TABLE 5

6. TABLE 6

7. TABLE 7

8. TABLE 8

9. TABLE 9

10. TABLE 10

11. TABLE 11

12. TABLE 12

13. TABLE 13

14. TABLE 14

15. TABLE 15

16. TABLE 16

17. TABLE 17

18. TABLE 18

19. TABLE 1

20. TABLE 2

21. TABLE 3

22. TABLE 4

23. TABLE 5

24. TABLE 6

25. TABLE 7

26. TABLE 8

27. TABLE 9

28. TABLE 10

List of Illustrations

1. Chapter 1

1. Figure 1 The Relationship between an Initial Investment, PV, and Its Future Value, FV

2. Figure 2 The Future Value of a Lump Sum, Initial Investment Not at t = 0

3. Figure 3 The Future Value of a Five-Year Ordinary Annuity

4. Figure 4 The Present Value of a Lump Sum to Be Received at Time t = 6

5. Figure 5 The Relationship between Present Value and Future Value

6. Figure 6 An Annuity Due of $100 per Period

7. Figure 7 The Present Value of an Ordinary Annuity with First Payment at Time t = 10 (in Millions)

8. Figure 8 Solving for Missing Annuity Payments (in Thousands)

9. Figure 9 The Additivity of Two Series of Cash Flows

2. Chapter 3

1. Figure 1 Histogram of S&P 500 Annual Total Returns: 1926 to 2012

2. Figure 2 Histogram of S&P 500 Monthly Total Returns: January 1926 to December 2012

3. Figure 3 Frequency Polygon of S&P 500 Monthly Total Returns: January 1926 to December 2012

4. Figure 4 Cumulative Absolute Frequency Distribution of S&P 500 Monthly Total Returns: January 1926 to December 2012

5. Figure 5 Center of Gravity Analogy for the Arithmetic Mean

6. Figure 6 Properties of a Normal Distribution (EV 5 Expected Value)

7. Figure 7 Properties of a Skewed Distribution

8. Figure 8 Leptokurtic: Fat Tailed

3. Chapter 4

1. Figure 1 Addition Rule for Probabilities

2. Figure 2 BankCorp’s Forecasted EPS

3. Figure 3 BankCorp’s Forecasted Operating Costs

4. Chapter 5

1. Figure 1 One-Period Stock Price as a Bernoulli Random Variable

2. Figure 2 A Binomial Model of Stock Price Movement

3. Figure 3 Continuous Uniform Distribution

4. Figure 4 Continuous Uniform Cumulative Distribution

5. Figure 5 Two Normal Distributions

6. Figure 6 Units of Standard Deviation

7. Figure 7 Two Lognormal Distributions

5. Chapter 6

1. Figure 1 Student’s t-Distribution versus the Standard Normal Distribution

6. Chapter 7

1. Figure 1 Rejection Points (Critical Values), 0.05 Significance Level, Two- Sided Test of the Population Mean Using a z-Test

2. Figure 2 Rejection Point (Critical Value), 0.05 Significance Level, One- Sided Test of the Population Mean Using a z-Test

7. Chapter 8

1. Figure 1 Scatter Plot of Annual Money Supply Growth Rate and Inflation Rate by Country, 1980–2012

2. Figure 2 Variables with a Correlation of 1

3. Figure 3 Variables with a Correlation of –1

4. Figure 4 Variables with a Correlation of 0

5. Figure 5 Variables with a Strong Nonlinear Association

6. Figure 6 US Inflation and Stock Returns: 1990–2013

7. Figure 7 Actual Change in Euro Area HICP versus Predicted Change

8. Figure 8 Fitted Regression Line Explaining the Inflation Rate Using Growth in the Money Supply by Country, 1980–2012

9. Figure 9 Actual Change in Euro Area HICP versus Predicted Change

10. Figure 10 Fitted Regression Line Explaining Stock Returns by Inflation

during 1990–2013

11. Figure 11 Fitted Regression Line Explaining Enterprise Value/Invested Capital Using ROIC–WACC Spread for the Food Industry

8. Chapter 9

1. Figure 1 Regression with Homoskedasticity

2. Figure 2 Regression with Heteroskedasticity

3. Figure 3 Value of the Durbin–Watson Statistic

4. Figure 4 Linear Regression When Two Variables Have a Linear Relation

5. Figure 5 Linear Regression When Two Variables Have a Nonlinear Relation

6. Figure 6 Plot of Two Series with Changing Means

9. Chapter 10

1. Figure 1 Swiss Franc/US Dollar Exchange Rate, Monthly Average of Daily Data

2. Figure 2Monthly US Retail Sales

3. Figure 3 Monthly CPI Inflation, Not Seasonally Adjusted

4. Figure 4Monthly CPI Inflation with Trend

5. Figure 5Starbucks Quarterly Sales by Fiscal Year

6. Figure 6 Starbucks Quarterly Sales with Trend

7. Figure 7 Residual from Predicting Starbucks Sales with a Trend

8. Figure 8Natural Log of Starbucks Quarterly Sales

9. Figure 9 Monthly CPI Inflation

10. Figure 10 Log of AstraZeneca’s Quarterly Sales

11. Figure 11 Log Difference, AstraZeneca’s Quarterly Sales

12. Figure 12 Monthly US Real Retail Sales and 12-Month Moving Average of Retail Sales

13. Figure 13 Monthly Europe Brent Crude Oil Price and 12-Month Moving Average of Prices

14. Figure 1 Predicted and Actual Civilian Unemployment Rates

15. Figure 2 Lightweight Vehicle Sales

16. Figure 3 Change in Civilian Unemployment Rate

17. Figure 4 Lightweight Vehicle Sales

18. Figure 5 Change in Natural Log of Lightweight Vehicle Sales

19. Figure 6 Quarterly Sales at Cisco

20. Figure 7 Quarterly Sales at Avon

FOREWORD “Central limits,” “probability distributions,” “hypothesis test”— investors have a bit of trouble generating enthusiasm for such terms. Yet, they should be enthusiastic because every investor needs these tools to analyze, compete, and succeed in today's economic environment. The financial markets and the participants in them become increasingly sophisticated every year. So, at times, it seems like you need a PhD in mathematics just to keep up with the markets.

Fortunately, a PhD is not necessary to succeed. In fact, the financial market battlefield is littered with the credentials of highly educated individuals who have failed spectacularly despite their intense education. Nonetheless, the better equipped you are with the basic tools of financial calculus, the better your chance of success.

Quantitative Investment Analysis provides the necessary utensils for success. In this volume, you will find all the statistical gadgets you need to be a confident and knowledgeable investor. Math need not be a four letter word. It can make your wealth analysis sharper, your investment theme more precise, your portfolio construction more successful.

Furthermore, this book is chock full of examples, practice problems (with answers!), charts, tables, and graphs that bring home in clear detail the concepts and tools of financial calculus. Whether you are a novice investor or an experienced practitioner, this book has something for you. In fact, as I read the book in preparation for writing this foreword, I kept getting unconsciously pulled into the examples; unwittingly, I became engaged in the book before I knew it. But that effect is part of the beauty of this book: It is an easy-to-read and easy-to-use handbook. I wanted to know more with each example I read. I know that you, too, will find that this book stimulates your curiosity while having the same ease of use that I found. Enjoy!

Mark J. P. Anson, PhD, CFA, CAIA, CPA President & Chief Investment Officer

Acadia Capital Bass Family Office

PREFACE We are pleased to bring you Quantitative Investment Analysis, Third Edition, which focuses on key tools that are needed for today's professional investor. In addition to classic time value of money, discounted cash flow applications, and probability material, the text covers advanced concepts such as correlation and regression that ultimately figure into the formation of hypotheses for purposes of testing. The text teaches critical skills that challenge many professionals, including the ability to distinguish useful information from the overwhelming quantity of available data.

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, which 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 hope 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.

ACKNOWLEDGMENT We would like to thank Eugene L. Podkaminer, CFA, for his contribution to revising coverage of multifactor models. We are indebted to Professor Sanjiv Sabherwal for his painstaking work in updating self-test examples in all chapters except the chapter on multifactor models. His contribution was essential to the fresh look of this third edition.

We are indebted to Wendy L. Pirie, CFA, Gregory Siegel, CFA, and Stephen E. Wilcox, CFA, for their help in verifying the accuracy of the text. Margaret Hill, Wanda Lauziere, and Julia MacKesson and the production team at CFA Institute provided essential support through the various stages of production. We thank Robert E. Lamy, CFA, and Christopher B. Wiese, CFA, for their encouragement and oversight of the production of a third edition.

ABOUT THE CFA INSTITUTE INVESTMENT SERIES CFA Institute is pleased to provide you with 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.

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.

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-

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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 practitionerimportant 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.

One of the most prominent texts over the years in the investment management industry has been Maginn and Tuttle's Managing Investment Portfolios: A Dynamic Process. The third edition updates key concepts from the 1990 second edition. Some of the more experienced members of our community own the prior two editions and will add the third edition to their libraries. Not only does this seminal work take the concepts from the other readings and put them in a portfolio context, but it also updates the concepts of alternative investments, performance presentation standards, portfolio execution, and, very importantly, individual investor portfolio management. Focusing attention away from institutional portfolios and toward the individual investor makes this edition an important and timely work.

The New Wealth Management: The Financial Advisor's Guide to Managing and Investing Client Assets is an updated version of Harold Evensky's mainstay reference guide for wealth managers. Harold Evensky, Stephen Horan, and Thomas Robinson have updated the core text of the 1997 first edition and added an abundance of new material to fully reflect today's investment challenges. The text provides authoritative coverage across the full spectrum of wealth management and serves as a comprehensive guide for financial advisors. The book expertly blends investment theory and real-world applications and is written in the same thorough but highly accessible style as the first edition.

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

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CHAPTER 1 THE TIME VALUE OF MONEY Richard A. DeFusco, CFA

Dennis W. McLeavey, CFA

Jerald E. Pinto, PhD, CFA

David E. Runkle, PhD, CFA

LEARNING OUTCOMES

After completing this chapter, you will be able to do the following:

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 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: Interpretation 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. 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:

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).