CH-12 Financial Return and Risk Concepts – Introduction to Finance: Markets, Investments, and Financial Management : Book Guide

Chapter 12 Financial Return and Risk Concepts CHAPTER PREVIEW This chapter on financial return and risk is a fitting capstone to Part 2’s discussion of investments. It discusses basic tenets that lie at the heart of modern financial theory: portfolio risk and return, market efficiency, and the capital asset pricing model. It also provides a tie-in to Part 3’s discussion of financial management, since management uses information and data from the financial markets to make capital budgeting and capital structure decisions. This is a quantitative chapter, so the text and your lectures should present intuitive explanations and illustrative examples so your students won’t get lost in the calculations without seeing the practical usefulness of the information in this chapter. The chapter starts by reviewing some basic concepts of average return, both historical and expected, as well as measures of risk such as variance, standard deviation, and coefficient of variation. Next the concept of market efficiency is introduced with the idea that the only means of earning higher expected returns is to carry higher risk. Portfolio return and risk measures are introduced, with the concepts of diversification and correlation playing important parts. This leads into a basic discussion of the CAPM and its uses. The Learning Extension delves into some of the quantitative aspects of how to estimate beta and the security market line. LEARNING OBJECTIVES Know how to compute arithmetic averages, variances, and standard deviations using return data for a single financial asset. Understand the sources of risk Know how to compute expected return and expected variance using scenario analysis. Know the historical rates of return and risk for different securities. Understand the concept of market efficiency and explain the three types of efficient markets. Explain how to calculate the expected return on a portfolio of securities. Understand how and why the combining of securities into portfolios reduces the overall or portfolio risk. Explain the difference between systematic and unsystematic risk. Understand the importance of ethics in investment-related positions. Learning Extension: Estimating Beta Security Market Line CHAPTER OUTLINE I. Historical Return and Risk for a Single FINANCIAL Asset A. Arithmetic Average Annual Rates of Return B. Variance as a Measure of Risk Standard Deviation as a Measure of Risk II. WHERE DOES RISK COME FROM? III. Expected Measures of Return and Risk Iv. Historical Returns and Risk of Different Assets V. Efficient CAPITAL Markets Vi. Portfolio Returns and Risk A. Expected Return on a Portfolio Variance and Standard Deviation of Return on a Portfolio To Diversify or Not To Diversify? D. Portfolio Risk and the Number of Investments in the Portfolio E. Systematic and Unsystematic Risk Capital Asset Pricing Model summary LEARNING EXTENSION: bETA AND THE SECURITY MARKET LINE ESTIMATING BETA II. SECURITY MARKET LINE LECTURE notes I. HISTORICAL RETURN AND RISK FOR A SINGLE FINANCIAL ASSET Annual holding period returns can be combined to find the arithmetic average annual return. Once an average has been computed, we can use the year-by-year deviations from that average to form the basis of a measure of risk or variability. The variance is computed by summing the squared deviations and dividing by the number of observations less one. The problem with variance as a risk measure is its units; as a sum of squared deviations, units of dollars squared or percent squared have little intuitive meaning for managers or analysts. The square root of the variance—the standard deviation—helps give us a risk measure with understandable units. It can also be given a practical statistical interpretation if the data is normally distributed. The coefficient of variation controls for size. It allows us to better compare a high return/high standard deviation security to a low return/low risk one. The coefficient of variation measures risk per unit of return. As such, it is a pure, unitless relative measure of an asset’s risk. You may want to query students about other measures of risk. Some possibilities include the semideviation (which is similar to the standard deviation except only return observations less than the average are included in the calculation) and variance/standard deviation based on deviations from a trend line. Students may look at the size of the annual returns on Walgreens and Microsoft and question the desirability of diversifying. Noting the large down years (2000 for MSFT, 2001 for WAG) and the behavior of the S&P and NASDAQ over 2000-2002 will help adjust students’ perceptions of risk and the desirability of putting their funds in only one, or in a small set, of stocks. (Use Discussion Questions 1 through 3 here.) WHERE DOES RISK COME FROM? Variability of returns is a financial fact of life. But it just doesn’t happen (although stock market commentator are quick to describe a day’s price reaction as due to “market jitters.” This section starts to lay the ground for Part 3’s financial management discussion in drawing links between financial market behavior (interest rates, exchange rates), business finance (changes in sales, costs, and profits) and the risk behavior of individual securities. (Use Discussion Questions 4 through 6 here.) IiI. EXPECTED MEASURES OF RETURN AND RISK Financial decisions are made by looking into the uncertain future. Historical data can help us make decisions today, but we also must consider current and expected future economic and market conditions. In this section we develop measures of expected return and risk based upon likely (and even some unlikely) future scenarios. Each possible scenario is a state of nature, each with its own set of return possibilities. The text example uses only three scenarios for illustrative purposes. A more formal analysis will have more. Students sometimes get confused interpreting the expected return and risk measures from this analysis. That is, they get caught up in the calculations and gloss over the meaning of the resulting numbers. The expected return is not the most likely outcome; that is reserved for the scenario with the highest probability of occurrence. Neither is the expected return the return that will occur next period—otherwise there would no longer be any risk! Try to help the students learn the proper interpretation: the expected return is the long-run average outcome if the current situation could be replicated many, many times. Implicit in this is that our scenario forecasts and associated probabilities are correct. The risk measures that arise from scenario analysis are based upon the probabilities of the different scenarios and the expected value found previously. Unless the probabilities of the possible scenarios are normally distributed, it is difficult to develop an intuitive interpretation to the variance and standard deviation measures. About all we can say is that large variances imply greater risk (relative to expected return) than smaller variances. Footnote 2 in this chapter provides another perspective on estimating expected returns: use the current T-bill rate and add various risk premiums to it to reflect expected future market conditions; this perspective is based on the CAPM which is discussed later in the chapter. This section also shows how the market acts as if it does perform scenario analysis in the context of its reaction to interest rate cuts by the Fed. (Use Discussion Question 7 here.) Iv. HISTORICAL RETURNS AND RISK OF DIFFERENT ASSETS The base return in the economy is the T-bill rate; it (should) incorporate the real risk-free rate of return and inflationary expectations. Risk premiums are added to the T-bill rate; the more the risk, the larger the risk premium. Risk levels should determine return levels, or similarly, risk drives return. Table 12.4 presents historical data that shows riskier assets enjoy higher average returns over time. This can be a jumping-off point for a discussion of financial market indexes if you chose, especially since the quirks of index construction lead long-term government bonds to have lower returns with higher risk than long-term corporate bonds in Table 12.4. (Use Discussion Question 8 here.) V. EFFICIENT CAPITAL MARKETS The concept of informational market efficiency is a cornerstone of modern finance. Yet puzzling examples of inefficiencies seemingly persist (quarterly earnings announcements, low P/E stocks, generally poor performance of IPO stocks compared to a control group, etc.). For the most part, some of the apparently anomalous results may result from data-mining and be nothing more than spurious results. Others may disappear over time (witness the demise of the “January effect” in recent years). Informing students of anomalies in the context of the EMH can provide an interesting “point-counterpoint” class discussion and show students that we don’t know all the answers yet. Certainly, most of the evidence supports the concept of efficiency, but “pockets” or times of inefficiency may exist, and that is when a superior stock-picker or portfolio manager can add value to their own portfolio or those of their clients. Perhaps the most convincing argument for the existence of wide-spread market efficiency is the poor performance of equity mutual funds (as a group) over time compared to the S&P 500 index. (Use Discussion Questions 9 through 13 here.) Vi. PORTFOLIO RETURNS AND RISK The math in this section may cause students to lose sight of intuition and application of these concepts; try to stress the practical aspects of this part of the chapter and why the calculations are needed and useful. Up until this point we have implicitly focused on single assets or single asset groups. But in the real world, people don’t own just one asset. Investments occur in many assets, from human capital to stocks and bonds to real estate and precious metals. Corporations can be considered to be a portfolio of assets dedicated to providing a product or service. Some key points developed in this section include: Portfolio risk is more complicated than just a weighted average of the components’ variances. Diversification effects determine portfolio risk. Covariances and correlations can best be described intuitively through the use of graphical relationships before formulas are reviewed. Diversification is a key component in finance as it lays the groundwork for practical investment plans. A well-positioned diversified portfolio, with risk levels determined by the investor’s risk preferences, should be the goal of every investor. Table 12.5 can be nicely supplemented by reviewing problem 10 in this chapter. 3. There is a limit to diversification effects as additional assets are added to a portfolio. This occurs because assets have exposures to common influences, such as the ups and downs of the overall market. 4. Some of an asset’s or portfolio’s risk can be diversified away (unsystematic risk) and some cannot (systematic or market risk). A well-diversified portfolio of 20–30 U.S. stocks (15–20 international stocks) has very little unsystematic risk exposure; most of the portfolio risk will be systematic. 5. Beta is a measure of systematic risk. It is a relative measure, not an absolute risk measure, such as variance. (Use Discussion Questions 14 through 22 here.) DISCUSSION QUESTIONS AND ANSWERS 1. Explain how a percentage return is calculated and describe the calculation of an arithmetic average return. There are two sources of return: income and capital gain (or loss). The percentage return is computed by adding the dollar return of these two sources and then dividing the sum by the purchase price. The arithmetic average return is a sum of annual percentage returns divided by the number of observations or years. 2. Describe how the variance and standard deviation are calculated and indicate how they are used as measures of risk. A deviation is the difference between an observed return and the arithmetic average return. The variance is computed by squaring each deviation, adding them together, and dividing the sum by the number of observations less one. The standard deviation is the square root of the variance. Both the variance and standard deviation show the dispersion in returns from the average. The variance’s units, dollars squared or percent squared, are not intuitive to most users. The square root of the variance, the standard deviation, has units of dollars or percent. If the returns are normally distributed, about 68 percent of the observed return should fall within one standard deviation of the average return; 95 percent should fall within two standard deviations; 99 percent should fall within three standard deviations. Thus, the standard deviation gives the user an idea of the range of the data. 3. What is meant by the coefficient of variation? How is it used as a measure of risk? The coefficient of variation is the standard deviation divided by the average. It is a measure of risk per unit of return. By itself, the standard deviation may be difficult to use to determine which asset is more or less risky because of differences in asset average returns. The coefficient of variation controls for the size of the average return. 4. Business risk has three possible sources. What are they? Business risk is caused by changes in quantity sold, changes in the difference between price and variable cost (the price−cost margin, and its level of fixed costs. 5. What are sources of risk facing a firm which are reflected on its income statement? Sources of risk that are reflected in the income statement include business risk, exchange rate risk, purchasing power risk, financial risk, interest rate risk, and tax risk. A firm’s revenues and expenses may vary because of business, exchange rate, and purchasing power risks. Interest expense may vary due to interest rate and financial risks. Taxes paid, of course, are influenced by tax risk—changes in the tax code and regulations. 6. Suppose the U.S. dollar strengthens in the past year against other currencies. Explain its effect on U.S. dollar revenues and expenses for a global firm headquartered in the U.S. If the dollar gets stronger, a dollar will purchase more units of foreign currency; conversely, foreign currencies will buy less dollars. In terms of a firm’s revenues, goods sold overseas will lead to less dollar-based revenue. For a firm which needs to send dollars overseas to pay for supplies or labor expenses, dollar-based costs will fall as less dollars will be needed to pay the overseas bill if the dollar gets stronger. 7. Describe the meaning of a “state of nature” and explain how this concept is used to provide expected measures of return and risk. A state of nature is a possible future outcome. It can be used to help provide estimates of expected return and risk as follows. First, enumerate all possible future states of nature over the planning horizon. Second, develop estimates of asset returns for each state of nature. Third, develop estimates for the probability of each state of nature occurring. Equations 12.5 and 12.6 can then be used to estimate expected return and variance. 8. Explain the historical relationships between return and risk for common stocks versus corporate bonds. As seen in Table 12.4, higher risk assets generally have higher historical returns. Historical risk premiums are larger the greater an asset’s risk. 9. Explain what is meant by “market efficiency.” What are the characteristics of an efficient market? A market is efficient if the economic effects of random news events are quickly reflected in asset prices. A result of this is that no one investor can consistently outperform the market averages on a risk-adjusted basis. Efficient markets have many participants, each seeking to maximize their own wealth. Information flows occur randomly, and the economic impact of information is rapidly incorporated in current asset prices. 10. What are the differences among the weak, semi-strong, and strong forms of the efficient market hypothesis? The basic difference is the amount of information that is absorbed into asset price levels. Under the weak form, only historical information is reflected in asset prices. Under the semi-strong form, both past and currently known information is reflected in prices. Under the strong form, all information, past and present, publicly known and private, is reflected in prices. 11. What type of market efficiency—none, weak, semi-strong, or strong—exists under each of the following statements? a. I know which stocks are going to rise in value by looking at their price changes over the past two weeks. None. Weak-form assumes past information such as price trends are reflected in current prices. A market is inefficient if past price trends can help predict future price trends. b. Returns earned by company officers trading their own firm’s stock are no better than those of other investors. Strong-form as even corporate insiders cannot earn above-average risk-adjusted returns. c. If a firm announces lower-than-expected earnings, you know the price will fall over the next quarter. Weak as past information can be used to predict future price trends over a quarter of a year. d. By the time I heard the news about the dividend increase the stock had already risen by a substantial amount. Semi-strong as the current news resulted in a quick price reaction. e. Whatever the stock market does in January, it will continue to move in that direction for the rest of the year. None. This is an example of an inefficient market. f. As soon as the chairman of the Federal Reserve give his testimony to Congress about future monetary policy, interest rates rose and stock prices dropped. Semi strong. Current news is quickly reflected in current asset prices. 12. Explain if you agree or disagree with this statement: “After the merger announcement the stock price greatly increased. Then it fell for the next 1-2 days before becoming relatively stable. This is proof against the efficient market hypothesis.” Disagree; it is not proof showing the EHM is false. The EMH states that, given current information, we cannot consistently invest and trade securities and do better than the market averages on a risk-adjusted basis. There will be times when the market appears inefficient or an investor appears “skillful” (or “lucky”). But the EMH cannot be proven or disproven on the basis on one example. A truer test of the statement in this question is: could the investor have profitably traded on this price pattern immediately after the merger announcement became public? And can they consistently do so for subsequent firms announcing mergers? Perfect hindsight does not assist us when making current investments. 13. How do mutual fund return data present evidence for or against efficient markets? Explain. Mutual funds, managed by highly-educated, credentialed individuals or teams of individuals, on average earn lower returns than their corresponding benchmark or index. In any year some mutual funds will outperform the index—but in most years general equity mutual funds do not outperform a broad market index. And when returns are compounded over longer time periods—such as 5-years or 10-years—the evidence is clear: the average mutual fund underperforms the broad market indexes. 14. Define what is meant by a portfolio and describe how the expected return on a portfolio is computed. A portfolio is a collection of assets. Expected return on a portfolio is a weighted average of the expected return of the portfolio’s components. 15. Explain the terms diversification and correlation in the context of forming portfolios. Diversification means that we invest in different assets, not only one. When we diversify, portfolio risk may fall as the asset returns may not be highly correlated with each other. Correlation is a measure of the strength of the linear relationship between two sets of numbers. It falls between –1 (perfect negative correlation) and +1 (perfect positive correlation). The less positive or the more negative the correlation, the greater the risk reduction from diversification. 16. Explain the fallacy of this statement: “I’d rather put my money into a single high earning asset than in a portfolio of diversified investments; I’ll earn more money with the single asset.” The fallacy is the hidden assumption that the investor can identify a single asset that will earn high returns in the future. There are no high return “sure things.” And this leads us to a second hidden assumption: there is no risk with the “single high earning asset.” For example, over time, the S&P 500 index has had attractive returns. Certainly, looking backward, there are periods of time, such as the late 1990s, when it performed quite well. But as markets are efficient it is difficult to predict future periods when the S&P 500 will perform well. In addition, the gyrations of the S&P 500 index over time shows its risk. Owning a diversified portfolio with an appropriate risk exposure is an important component of long-term investment success. 17. Describe what happens to portfolio risk as more and more assets are added to a portfolio. Are there advantages to international diversification? Because of diversification effects, portfolio risk initially declines. But as all assets are exposed to a common economic and market environment, after a while, additional increases in the number of assets (estimated to be 20–30 stocks in the U.S. stock market) do not continue to lower portfolio risk. There are advantages to international diversification. As economies are not perfectly linked or correlated, portfolio risk can be further reduced by investing in international securities. 18. How does systematic risk differ from unsystematic risk? Systematic risk is the risk that is inherent in the system; it is also known as market risk, the risk of investing in risky assets. It cannot be diversified away. Unsystematic risk arises from microeconomic (firm- or industry-specific) influences; it can be reduced to virtually zero in a well-diversified portfolio. 19. Classify each of the following as an example of systematic or unsystematic risk. a. The labor unions at Caterpillar, Inc. declared a strike yesterday. Unsystematic as it is company-specific. b. Contrary to what polls stated, the President was re-elected. Systematic as it affects the country as a whole. c. Disagreement about inflation policy leads to a fall in the Euro relative to the dollar. Systematic as inflation and exchange rates affect national economies and all industries. d. The computer industry suffers lower profits because of aggressive pricing strategies on new desktop computers. Unsystematic as the information deals with a single industry and investors in that industry. e. Every Christmas selling season there is a “hot” toy that many parents try to purchase for their child. Unsystematic as the new of a popular toy to company-specific; one firm and its investors reap the benefits. 20. What is meant by the Capital Asset Pricing Model? Describe how it relates to expected return and risk. The CAPM is a theory of the relationship between expected return and risk. It states that expected return is a linear function of systematic risk, beta. Expected return is also influenced by the risk-free rate and the market risk premium (the extra expected return earned from investing in the risky market portfolio). 21. Define the concept of “beta” and describe what it measures. Beta is a measure of an asset’s systematic risk. It measures the volatility of an asset’s returns relative to the market’s returns. 22. What is the market portfolio? Can we invest in such a portfolio? It is the portfolio comprised of all risky assets, the weights being proportionate to their market values. For example, if there are only three risky assets in the world with market values of 10, 20, and 30 (total market value of the risky assets is 60), their weights in the market portfolio will be 10/60 or 1/6; 20/60 or 1/3, and 30/60 or ½. We cannot invest in a true market portfolio. All risky assets comprises all stocks, bonds, and other financial assets; it also includes real estate, farmland, buildings, and hard-to-measure assets such as human capital. PROBLEMS and answers 1. From the information listed in the text, compute the average annual return, the variance, standard deviation, and coefficient of variation for each asset. Asset A: Average return = (Sum of returns)/n = 34/4 = 8.5% Variance =  (Ri – Average)2/ (n – 1) = 77/(4 – 1) = 25.67%2 Standard deviation = 25.67 1= 5.07% Coefficient of variation = Standard deviation/average = 5.07/8.5 = 0.60 Asset B: Average return = 4.2% Variance = 119.2%2 Standard deviation = 10.92% CV = 2.6 Asset C: Average return = 14.67% Variance = 6.33%2 Standard deviation = 2.52% CV = 0.17 Asset D: Average return = 1% Variance = 186.4%2 Standard deviation = 13.65% CV = 13.65 2. Based upon your answers to problem 1, which asset appears riskiest based on standard deviation? Based on coefficient of variation? The highest standard deviation is for asset D, 13.65%. The highest coefficient of variation is also asset D, 13.65. 3. Recalling the definitions of risk premiums from chapter 8 and using the Treasury bill return in Table 12.4 as an approximation to the nominal risk-free rate, what is the risk premium from investing in each of the other asset classes listed in Table 12.4? Stock risk premium = 11.1 – 3.8 = 7.3% Treasury bond risk premium = 5.4– 3.8 = 1.6% 4. What is the real, or after-inflation, return from each of the asset classes listed in Table 12.4? Stocks 11.1% - 3.2% = 7.9% Treasury bonds 5.4% - 3.2% = 2.2% U.S Treasury bills 3.8% - 3.2% = 0.6% 5. RCMP, Inc. shares rose 10 percent in value last year while the inflation rate was 3.5 percent. What was the real return on the stock? If an investor sold the stock after one year and paid taxes on the investment at a 15 percent tax rate what is the real after-tax return on the investment? The nominal return is 10% and the inflation rate is 3.5%. The real return on RCMP’s shares is 10%- 3.5% = 6.5%. Taking taxes into consideration, using a 15% tax rate the nominal after-tax return is 10% (1-0.15) = 8.5%. Subtracting the inflation rate, the real after-tax return is 8.5% - 3.5% = 5.0%. 6. Find the real return on the following investments: Stock Nominal Return Inflation A 10% 3% B 15% 8% C -5% 2% The real return is computed as nominal return – inflation rate as follows: Stock A: 10% - 3% = 7% Stock B: 15% - 8% = 7% Stock C: -5% - 3% = -8% 7. Find the real return, nominal after-tax return, and real after-tax return on the following: Stock Nominal Return Inflation Tax Rate X 13.5% 5% 15% Y 8.7% 4.7% 25% Z 5.2% 2.5% 28% Real return is nominal return minus the inflation rate: Stock X: 13.5% - 5% = 8.5% Stock Y: 8.7% - 4.7% = 4.0% Stock Z: 5.2% - 2.5% = 2.7% Nominal after-tax return is nominal return (1-tax rate): Stock X: 13.5% (1- 0.15) = 11.48% Stock Y: 8.7% (1- 0.25) = 6.53% Stock Z: 5.2% (1-0.28) = 3.74% The real after-tax return is the nominal after-tax return minus the inflation rate: Stock X: 11.48% - 5% = 6.48% Stock Y: 6.53% - 4.7% = 1.83% Stock Z: 3.74% - 2.5% = 1.24% 8. The countries of Stabilato and Variato have the following average returns and standard deviations for their stocks, bond, and short-term government securities. What range of returns should you expect to earn 95% of the time for each asset class if you invested in Stabilato’s securities? From investing in Variato’s securities? Stabilato Asset Average return Standard Deviation 95% range (Average +/- 2 σ) Stocks 8 3 14 2 Bonds 5 2 9 1 Short-term Government debt 3 1 5 1 Variato Average return Standard Deviation 95% range (Average +/- 2 σ) Stocks 15 13 41 -11 Bonds 10 8 26 -6 Short-term Government debt 6 3 12 0 9. Using the information below, compute the percentage returns for the following securities: Price today Price one year ago Dividends received Interest received Dollar Return= change in price + income Percentage Return=Dollar return/initial price a) RoadRunner stock $20.05 $18.67 $0.50 $1.88 10.07% b)Wiley Coyote stock $33.42 45.79 $1.10 -$11.27 -24.61% c)Acme long-term bonds $1,015.38 $991.78 $100.00 $123.60 12.46% d) Acme short-term bonds $996.63 $989.84 $45.75 $52.54 5.31% e) Xlingshot stock $5.43 $3.45 $0.02 $2.00 57.97% 10. Given her evaluation of current economic conditions, Ima Nutt believes there is a 20 percent probability of recession, a 50 percent change of continued steady growth, and a 30 percent probability of inflationary growth. For each possibility, Ima has developed an interest rate forecast for long-term Treasury bond interest rates. This information is listed in the text. a. What is the expected interest rate under Ima’s forecast? E(rate) = .2(6%) + .5(9%) + .3(14%) = 9.9% b. What is the variance and standard deviation of Ima’s interest rate forecast? Variance = .2(6 – 9.9)2 + .5(9 – 9.9)2 + .3(14 – 9.9)2 = 8.49%2 Standard deviation = = 2.91% c. What is the coefficient of variation of Ima’s interest rate forecast? CV = Standard deviation / E(rate) = 0.29 d. If the current long-term Treasury bond interest rate is 8 percent, should Ima consider purchasing a Treasury bond? Why or why not? No. The expected rate of 9.9% is higher than the current rate of 8%. Thus, bond prices are expected to fall. 11. Ima is considering a purchase of Wallnut Company stock. Using the same scenarios and probabilities as in problem 10, she estimates Wallnut’s return is -5% in a recession, 20 percent in constant growth, and 10% in inflation. a. What is Ima’s expected return forecast for Wallnut stock? b. What is the standard deviation of the forecast? c. If Wallnut’s current price is $20 a share and is expected to pay a dividend of $0.80 a share next year, what price does Ima expect Wallnut to sell for in one year? Scenario Probability Wallnut return Recession 20% -5% Constant growth 50% 20% Inflation 30% 10% a. Expected return 12.00% =20% (-5%) + 50%(20%) + 30% (10%) b. Variance 91.00% =20% (-5%-12%)^2 + 50%(20%-12%)^2 + 30% (10%-12%)^2 Standard Deviation 9.54% c. The return is computed as the (change in price + income)/beginning price. If the expected return is 12.00% (or 0.12 in decimal form) we have: (Expected price - $20) + 0.80 = 0.12 $20 or (Expected price - $20) + 0.80 = 0.12 ($20) = $2.40 = Expected price - $19.20 = $2.40 Solving, we see the expected price = $21.60. 12. Ima’s sister, Uma, has completed her own analysis of the economy and Wallnut’s stocks. Uma used recession, constant growth and inflation scenarios but with different probabilities and expected stock returns. Uma believes the probability of recession is quite high, at 60 percent and that in a recession Wallnut’s stock return will -20 percent. Uma believes the scenarios of constant growth and inflation are equally likely and that Wallnut’s returns will be 15 percent in the constant growth scenario and 10 percent under the inflation scenario. a. What is Uma’s expected return forecast for Wallnut stock? b. What is the standard deviation of the forecast? c. If Wallnut’s current price is $20 a share and is expected to pay a dividend of $0.80 a share next year, what price does Uma expect Wallnut to sell for in one year? With the probability of recession set at 60 percent, the probability of not having a recession is 1-0.60 or 0.40. As the probability of the constant growth and inflation scenarios are equally likely, there probabilities are 0.40/2 or 0.20 (20%) each. Using these probabilities we have: Scenario Probability Wallnut return Recession 60% -20% Constant growth 20% 15% Inflation 20% 10% a. Expected return -7.00% = 60% (-20%) + 20%(15%) + 20% (10%) Variance 256.00% = 60% (-20%-(-7%))^2 + 20%(15%-(-7%))^2 + 20% (10%-(-7%))^2 b. Standard Deviation 16.00% c. The return is computed as the (change in price + income)/beginning price. If the expected return is -7.00% (or -0.07 in decimal form) we have: (Expected price - $20) + 0.80 = -0.07 $20 or (Expected price - $20) + 0.80 = -0.07 ($20) = -$1.40 = Expected price - $19.20 = -$1.40 Solving, we see the expected price = $17.80. 13. Scenario analysis has many practical applications in addition to being used to forecast security returns. In this problem, scenario analysis is used to forecast an exchange rate. Jim Danday’s forecast for the Euro/dollar exchange rate depends upon what the U.S. Federal Reserve and European central bankers do to their country’s money supply. Jim is considering the scenarios and exchange rate forecasts that are listed in the text. a. What is Jim’s expected exchange rate forecast? E(ER) = .2(1.15) + .3(1.05) + .35(0.95) + .15(0.85) = 1.005 b. What is the variance of Jim’s exchange rate forecast? Variance = .2(1.15 – 1.005)2 + .3(1.05 – 1.005)2 + .35(0.95 – 1.005)2 + .15(0.85 – 1.005)2 = 0.009475 c. What is the coefficient of variation of Jim’s exchange rate forecast? CV = Standard deviation/E(ER) = 0.09734/1.005 = 0.0969 14. Using the data in Table 12.4, calculate and interpret the coefficient of variation for each asset class. Stocks CV = 20.4/11.1 = 1.84 U.S. Treasury bonds CV = 7.6/5.4 = 1.41 U.S. Treasury bills CV = 3.0/3.8 = 0.79 Interpretation: CV measures the risk per unit of return of each asset class. 15. Below is annual stock return data on Krahamco and M.J. Edit, Inc. Year Krahamco M.J. Edit 2007 10% -3% 2008 15% 0% 2009 -10% 15% 2010 5% 10% a. What is the average return, variance, and standard deviation for each stock? Average return = (Sum of returns)/n Krahamco: 20/4 = 5.0% M. J. Edit: 22/4 = 5.5% Variance =  (Ri – Average)2/ (n – 1) Krahamco = 350/(4 – 1) = 116.67%2 M. J. Edit= 213/(4 – 1) = 71.0%2 Standard deviation Krahamco = 116.67= 10.80% M. J. Edit = 71.0= 8.43% b. What is the expected portfolio return on a portfolio comprised of i. 25% Krahamco and 75% M. J. Edit? ii. 50% Krahamco and 50% M. J. Edit? iii. 75% Krahamco and 25% M. J. Edit? E(portfolio return) = .25(5%) + .75(5.5%) = 5.375% E(portfolio return) = .5(5%) + .5(5.5%) = 5.25% E(portfolio return) = .75(5%) + .25(5.5%) = 5.125% c. Without doing any calculations, would you expect the correlation between the returns on Krahamco and M.J. Edit's stock to be positive, negative, or zero? Why? Probably negative, as the changes in returns from year-to-year moved in opposite directions in two of the three years (2007-2008: return rose for both Krahamco and M. J. Edit; 2008-2009: return fell for Krahamco, rose for M. J. Edit; 2009-2010: return rose for Krahamco, fell for M. J. Edit). 16. Below is annual stock return data on AAB Company and YYZ, Inc. Year AAB YYZ 2006 0% 5% 2007 5% 10% 2008 10% 15% 2009 15% 20% 2010 -10% -20% What is the average return, variance, and standard deviation for each stock? Average return = (Sum of returns)/n AAB: 20/5 = 4.0% YYZt: 30/5 = 6.0% Variance =  (Ri – Average)2/ (n – 1) AAB = 370/(5 – 1) = 92.5%2 YYZ = 970/(5 – 1) = 242.5%2 Standard deviation AAB =  92.5 = 9.62% YYZ = 242.5 = 15.57% b. What is the expected portfolio return on a portfolio comprised of i. 25% AAB and 75% YYZ? ii. 50% AAB and 50% YYZ? iii. 75% AAB and 25% YYZ? E(portfolio return) = .25(4%) + .75(6%) = 5.5% E(portfolio return) = .5(4%) + .5(6%) = 5.0% E(portfolio return) = .75(4%) + .25(6%) = 4.5% Without doing any calculations, would you expect the correlation between the returns on AAB and YYZ's stock to be positive, negative, or zero? Why? Probably positive, as the changes in returns from year-to-year moved in the same direction in all four of the year-to-year changes (2006-2007: return rose for both AAB and YYZ; 2007-2008: return rose for both AAB and YYZ; 2008-2009: return rose for both AAB and YYZ; 2009-2010; return fell for both AAB and YYZ). 17. Estimate the weights (wi) for assets in the three portfolios given the following information about the portfolio holdings: price Number of securities Market value Portfolio weights Percentages to one decimal place a. Stock A $25 200 $5,000 0.22523 22.5% Stock B $53 100 $5,300 0.23874 23.9% Stock C $119 100 $11,900 0.53604 53.6% Total value $22,200 b. Bond A $975 10 $9,750 0.19295 19.3% Bond B $1,020 20 $20,400 0.40372 40.4% Bond C $888 10 $8,880 0.17574 17.6% Bond D $1,150 10 $11,500 0.22759 22.8% Total value $50,530 c. Stock A $25 1000 $25,000 0.23256 23.3% Stock C $119 500 $59,500 0.55349 55.3% Bond D $1,150 20 $23,000 0.21395 21.4% Total value $107,500 d. Stock B $53 1000 $53,000 0.56025 56.0% Stock C $119 100 $11,900 0.12579 12.6% Bond A $975 20 $19,500 0.20613 20.6% Bond B $1,020 10 $10,200 0.10782 10.8% Total value $94,600 18. a. Tim’s portfolio contains two stocks, Lightco and Shineco. Last year his portfolio returned 14 percent. Lightco’s return as 5 percent and Shineco returned 20 percent. What are the weights of each in Tim’s portfolio? Let wL be the weight of Lightco; with only two stocks in the portfolio, the weight of Shineco must be 1- wL . The return on the portfolio is computed as follows using equation 12-7: Portfolio return = Lightco weight x Lightco return + Shineco weight x Shineco return 14% = wL (5%) + (1 - wL ) 20% = (5%) wL + 20% - (20%)wL Solving for wL , -6% = (-15%)wL so wL = 6/15 = 0.40. Lightco is 40% of the portfolio and Shineco is 60%. Checking these numbers we have: 0.40 (5%) + 0.60(20%) = 14% which agrees with the information given in the problem. b. The following year Tim adds a third stock, Brightco, and reallocates his funds among the three stocks. Lightco and Shineco have the same weight in the portfolio and Brightco’s weight is one-half of Lightco. During the year Lightco returns 10 percent, Shineco returns 12 percent and Brightco loses 5 percent. What was the return on his portfolio? In the new portfolio the weights of Lightco and Shineco are the same; call it x. The portfolio weight of Brightco is one-half of Lightco, or .5x. Together the weights of the three stocks must sum to 1.0: x + x + .5x = 1.0  2.5x = 1.0  x = 1 / 2.5 = 0.40. Lightco and Shineco each comprise 40% of the portfolio and Brightco’s weight is 20%. The portfolio’s return is: 0.40(10%) + 0.40 (12%) + 0.20(-5%) = 7.8%. 19. Spreadsheets are useful for computing statistics: averages, standard deviation, variance, and correlation are included as built-in functions. Below is recent monthly stock return data for ExxonMobil (XOM) and Microsoft (MSFT). Using a spreadsheet and its functions, compute the average, variance, standard deviation, and correlation between the returns for these stocks. What does the correlation between the returns imply for a portfolio containing both stocks? Month XOM Return MSFT Return November -4.6% 10.4% October 0.1% 13.6% September -1.9% -10.3% August -3.3% -13.8% July -4.4% -9.3% June -1.6% 5.5% May 0.7% 2.1% April 9.4% 23.9% March -0.1% -7.3% February -3.2% -3.4% Answer: XOM MSFT Average Return -0.89 1.14 Variance 16.5299 148.2071 Standard deviation 4.0657 12.1740 Correlation 0.6704 The two stocks have a moderately high positive correlation, i.e. their returns move together over time. If future returns are expected to behave similarly, the benefits of including both stocks in a portfolio may mean little for risk diversification. 20. If the conditions in the future are expected to be like those in the past, what is the expected portfolio return and standard deviation in a portfolio comprised of a. 25% XOM and 75% MSFT? b. 50% XOM and 50% MSFT? c. 75% XOM and 25% MSFT? The chapter does not introduce the formula for finding variance of a two-asset portfolio using the correlation of the two assets. Thus, students will likely solve this by constructing portfolios and estimating the expected return and standard deviation from the 10 monthly observations. You may want to compare their “long-way” standard deviation estimates with those derived from the two-asset portfolio variance equation: σp2 = w12 σ12 + w22 σ22 + 2 w1 w2 ρ σ1 σ2 Of course, the answers will be the same. Expected Portfolio Return Standard Deviation 25% XOM/75% MSFT 0.6325 9.840873 50% XOM/50% MSFT 0.125 7.600996 75% XOM/25% MSFT -0.3825 5.568139 21. Construct a spreadsheet to replicate the analysis of Table 12.5. That is, assume $10,000 is invested in a single asset which returns 7 percent annually for 25 years and $2,000 is placed in 5 different investments, earning returns of –100%, 0%, 5%, 10%, 12%, respectively, over the 25 year time frame. For each of the questions below, begin with the original scenario presented in Table 12.5. a. Experiment with the return on the fifth asset. How low can the return go and still have the diversified portfolio earn a higher return than the single-asset portfolio? Approximately 10.42 percent b. What happens to the value of the diversified portfolio if the first two investments are both a total loss? Asset number 1 2 3 4 5 Initial Investment $2,000 $2,000 $2,000 $2,000 $2,000 Number of years 25 25 25 25 25 Base return rate -100% -100% 5% 10% 12% Return $0.00 $0.00 $6,772.71 $21,669.41 $34,000.13 Total Return $62,442.25 The value of the portfolio falls by $2,000 c. Suppose the single asset portfolio earns a return of 8 percent annually. How does the return of the single asset portfolio compare to that of the 5-asset portfolio? How does it compare if the single asset portfolio earns a 6 percent annual return? Single Investment $10,000 Number of years 25 Base return 8% Total return $68,484.75 Return is about $4,000 higher than that of the diversified portfolio Single Investment $10,000 Number of years 25 Base return 6% Total return $42,918.71 Return is lower than that of diversified portfolio by over $20,000 d. Assume that Asset 1 of the diversified portfolio remains a total loss (-100% return) and asset two earns no return. Make a table showing how sensitive the portfolio returns are to a 1 percentage point change in the return of each of the other three assets. That is, how is the diversified portfolio’s value affected if the return on asset 3 is 4 percent and 6 percent? If the return on asset 4 is 9 percent or 11 percent? If the return on asset 5 is 11 percent? 13 percent? How does the total portfolio value change if each of the three asset’s returns are 1 percentage point lower than in Table 12.5? If they are one percentage point higher? Portfolio Difference ($) Return ($) with Base Portfolio Asset 3 earns 4 percent $63,001.21 -$1,441.04 Asset 3 earns 6 percent $66,253.28 $1,811.03 Asset 4 earns 9 percent $60,019.00 -$4,423.25 Asset 4 earns 11 percent $69,943.77 $5,501.52 Asset 5 earns 11 percent $57,613.05 -$6,829.20 Asset 5 earns 13 percent $72,903.21 $8,460.96 All three 1 percent lower $51,748.76 -$12,693.49 All three 1 percent higher $80,215.75 $15,773.50 e. Using the sensitivity analysis of parts c and d, explain how the two portfolios differ in their sensitivity to different returns on their assets. What are the implications of this for choosing between a single asset portfolio and a diversified portfolio? A one percentage point change in the value of the single asset caused its value to range from $42,918.71 to $68,484.75—a range in dollar terms of over $25,000 and, in percentage terms, a decline of 20.92 percent to an increase of 26.2 percent. The worst range of the diversified portfolio given a one percentage point change in an asset was for the change in asset 5 when the portfolio’s value ranged from $57,613.05 to $72,903.21, a range of only about $15,000. When assets 3, 4, and 5 earned one percentage point less than their base case, the portfolio’s value, $51,748.76, was about $12,700 (19.7 percent) less than its “base case”—which is only $2,500 less than the $54,274.33 “base case” of the single asset portfolio. The best case scenario—when assets 3, 4, and 5 all earn one percent more than expected—the portfolio’s return is $80,215.75 (24.5 percent higher than the base case). It appears the single asset portfolio is more sensitive to changes in its return assumptions while the diversified portfolio is less sensitive. SUGGESTED QUIZ 1. What does a correlation measure? 2. Why should only systematic risk affect the expected return on an asset? 3. How does beta affect the size of an asset’s reaction to a systematic event? 4. Find the average and standard deviation of the following returns: 5%, 9%, 15%, 12% Solution: The average of these numbers is the sum of the returns divided by 4: (5 + 9 + 15 + 12)/4 = 41/4 = 10.25% To find the standard deviation, we must find the variance and then take its square root: [(5 – 10.25)2 + (9 – 10.25)2 + (15 – 10.25)2 + (12 – 10.25)2]/(4 – 1) = 18.25%2 The standard deviation is the square root of 18.25, which is 4.27 percent. Learning Extension 12a Estimating Beta and The Security Market Line The capital asset pricing model was developed to determine the relationship between expected return and risk (as measured by systematic risk; unsystematic risk can be diversified away). The CAPM assumes investors hold diversified portfolios. Expected return is affected by: a. The risk-free rate of return b. The market risk premium, or the expected reward investors receive from holding the risky market portfolio The asset’s beta, or systematic risk Which is expressed by the security market line? The beta of an asset is estimated by the slope of the regression line when the asset’s return (vertical axis) is graphed against the market return (horizontal axis). The beta of a portfolio of assets is simply a weighted average of the betas of the assets of the portfolio. The practical uses of the CAPM include evaluating asset risk (estimating its beta); forecasting investment returns on assets and portfolios; estimating a company’s cost of capital (discussed in Chapter 18). PROBLEMS AND ANSWERS 1. Stock market forecasters are predicting that the stock market will rise a modest 5 percent next year. Given the beta of each stock listed in the text, what is the expected change in each stock’s value? Beta represents the relative volatility of an asset compared to the market’s volatility. It is estimated from the regression equation: R: = a + (beta) RMKT + e a. BCD (1.25)(5%) = 6.25% b. NOP (0.70)(5%) = 3.50% c. WXY (1.10)(5%) = 5.50% d. ZYX (1.00)(5%) = 5.00% 2. Suppose the estimated security market line is: E(Ri ) = 4.0 + 7( i ) The standard form of the SML is: E(Ri) = RFR + [E(RMKT) – RFR] beta; a. What is the current Treasury bill rate? 4.0% b. What is the current market risk premium? 7.0% c. What is the current expected market return? E(RMKT) – RFR = 7 = E(RMKT) – 4  E(RMKT) = 11% d. Explain what beta ( ) measures. Beta represents the relative volatility of an asset compared to the market’s volatility. It is estimated from the regression equation: Ri = a + (beta) RMKT + e 3. Financial researchers at Smith Sharon, an investment bank, estimate the current security market line as: E(Ri ) = 4.5 + 6.8(i ) a. Explain what happens to expected return as beta increases from 1.0 to 2.0. Expected return rises, from an expected return of 11.3% [E(R) = 4.5 + 6.8(1)] to an expected return of 18.1% [E(R) = 4.5 + 6.8(2)] b. Suppose an asset has a beta of –1.0. What is the expected return on this asset? Would anyone want to invest in it? Why or why not? E(R) = 4.5 + 6.8 (–1) = –2.3%. With a negative expected return, no rational investor will buy this asset. It would be better to invest funds in treasury bills. 4. a. What was the risk-free rate in the economy for the year if Reilly Incorporated’s stock has a 14 percent return, a beta of 0.85, and the market return is 15 percent? The security market line is given by: Expected security return = risk-free rate + [market return – risk-free rate] x beta Using the known information, we have 14% = RFR + [15%-RFR](0.85) = (0.15) RFR + 12.75 so RFR = (14% - 12.75%)/0.15 = 8.33 percent b. The stock of another company had a return of 20 percent. What would you estimate its beta to be? We estimated the risk-free rate to be 8.33 percent in part a). Once again using the security market line: Expected security return = risk-free rate + [market return – risk-free rate] x beta and using the known information, we have 20% = 8.33% + [15%-8.33%](beta) = 8.33% + (6.67%)(beta) so beta = (20% - 8.33%)/6.67% = 1.75 5. You’ve collected data on the betas of various mutual funds. Each fund and its beta is listed in the text. a. Estimate the beta of your fund holdings if you held equal proportions of each of the above funds. Betaportfolio = .25(0.23) + .25(0.77) + .25(1.05) + .25(1.33) = 0.845 b. Estimate the beta of your fund holdings if you had 20 percent of your investments in the Weak fund, 40 percent in Fido, 15 percent in Vanwatch, and the remainder in Temper. Betaportfolio = .20(0.23) + .40(0.77) + .15(1.05) + .25(1.33) = 0.844 6. Using your answers to Parts a and b in Problem 5, estimate your portfolio’s expected return if the security market line is estimated as: E(Ri ) = 5.2 + 8.4( i ) a. Expected return = 5.2 + 8.4(0.845) = 12.30% b. Expected return = 5.2 + 8.4(0.844) = 12.29% 7. As mentioned above, spreadsheets can do the work for us of computing beta. Use Excel’s “slope” function to estimate the beta of Microsoft using the data in Table LE12.1. Use the “intercept” function to estimate the alpha or intercept term of the regression line. S&P 500 Microsoft Return Return –1.70% –2.00% –1.70 4.80 –1.60 –6.00 1.20 –4.10 1.70 1.00 5.10 6.50 Slope 1.618 Intercept -1.750 8. Below is nine month’s return data for Walgreens and the S&P 500. Month Walgreens Return S&P500 Return 1 0.0203 7.5% 2 -0.0595 1.8% 3 0.0023 -8.2% 4 0.0203 -6.4% 5 -0.0221 -1.1% 6 -0.1426 -2.5% 7 -0.0597 0.5% 8 0.0485 7.7% 9 -0.0794 -6.4% a. Estimate the intercept (alpha) and beta for Walgreen’s stock using spreadsheet functions. Alpha -2.754 Beta 0.340 R squared 0.107 Interpret what the slope estimate means to a stock analyst. The slope estimate, or beta, is a measure of the stock’s relative volatility compared to the market. Over this short nine-month time frame, Walgreen’s beta is 0.34, meaning WAG was about one-third as volatile as the S&P 500. c. Compute the R-squared of the regression using Excel’s RSQ function. What does the R-squared tell us about the relationship between Walgreens’ returns and those of the market? The R-squared is 0.107. This means that 10.7 percent of the variation of WAG’s stock returns from their mean is explained by the variation in S&P 500’s returns from its mean over the time period examined. In finance terms, 10.7 percent of WAG’s returns arose from systematic risk and 89.3 percent from unsystematic risk. Solution Manual for Introduction to Finance: Markets, Investments, and Financial Management Ronald W. Melicher, Edgar A. Norton 9780470561072, 9781119560579, 9781119398288