CH-9 Risk and Return – Practical Financial Management : Book Guide

CHAPTER 9 RISK AND RETURN FOCUS Our initial focus is on defining risk in financial terms and understanding how that concept fits into portfolio theory. As we gain a more sophisticated understanding of risk, we're able to focus on the concept of beta and how to apply it through the SML. PEDAGOGY The study of Risk and Return presents the biggest pedagogical challenge in basic finance. Therefore motivating the study and developing ideas patiently is especially important. Students are easily confused early in the discussion by the transition from the everyday notion of risk to its financial representation as variation in return. We therefore take pains to present these ideas carefully through an intuitive illustration. Risk and Return is also the area in which textbook treatments using mathematical statistics get students who aren't good at math into the most trouble. The approach used here presents statistical concepts graphically and in words to overcome this pedagogical road block. It's worth noting that while we minimize the statistics used in the theoretical development of the CAPM, we don't skimp on the algebraic math required to apply the SML. TEACHING OBJECTIVES Instruction should begin motivating the study of risk and return by explaining that higher long-term returns are available on equity than on debt but that there's an associated risk. Point out that the objective of investing is to take advantage of the high returns of equity while avoiding as much risk as possible, and that this is done by investing in diversified portfolios. The issue is how do we choose stocks to go into those portfolios, and what are the pricing implications of those choices. After studying this chapter students should have a good understanding of the relationship between risk and return in both everyday terms and in the more precise terms of the CAPM. They should understand the basic assumptions behind the model and be able to do simple problems using the SML. OUTLINE I. WHY STUDY RISK AND RETURN Equities yield higher returns, but also have higher risk than debt investments. A. The General Relationship Between Risk & Return Higher rewards just about always come with higher risk. B. The Return on an Investment Defining the return on a stock investment precisely. Expected and required returns. C. Risk - A Preliminary Definition The probability that return will be less than expected. A correct but incomplete definition. II. PORTFOLIO THEORY A. Return as a Random Variable A review of random variables and probability distributions. The return on an investment as a continuous random variable. B. Risk Redefined as Variance Risk as statistical variance and intuitively as the variability (up and down movement) of return. C. Risk Aversion An important characteristic of most human behavior, precisely defined and explained. D. Decomposing Risk - Systematic (Market) and Unsystematic (Business-specific) Risk Market and Business-specific risk defined and explained. E. Portfolios A portfolio's own risk and return. The goal of investors to minimize portfolio risk. F. Diversification - How Risk is Affected When Stocks Are Added to a Portfolio How diversification eliminates unsystematic and reduces systematic risk. The importance of market risk in portfolio theory. G. Measuring Market Risk - The Concept of Beta Beta defined and explained. How betas are developed. H. Using Beta - The Capital Asset Pricing Model (CAPM) The CAPM and the SML, an explanation of how required returns are determined. The equation and what each term means. Adjusting to changes in the risk free rate and risk aversion. How the SML helps to determine stock price. Making decisions based on the SML and the portfolio concept of risk. I. The Validity and Acceptance of the CAPM and its SML The controversy over the CAPM's practical value. QUESTIONS 1. What is the fundamental motivation behind portfolio theory? That is, what are people trying to achieve by investing in portfolios of stocks rather than in a few individual stocks or in debt? What observations prompted this view? Answer: It has been observed that over the long term, investments in equities (stocks) outperform investments in debt (bonds, savings accounts, etc.) by a factor of roughly three to one. However, the return on equity investments tends to vary a great deal over shorter periods. Indeed it is sometimes negative. This variation is characterized as risk. Stock portfolios are an attempt to capture the high average return of equity investing while avoiding as much of the associated risk as possible. 2. What is the general (in words) relationship between risk and return? Answer: In general, risk and return increase together. In other words, ventures in which high rewards can be expected usually come with a good chance of doing poorly or incurring a loss. 3. Define and discuss (words only, no equations) the concepts of expected return and required return. Answer: The expected return is the return most people plan on receiving when they make an investment even though they know that the actual return is likely to be somewhat different. Statistically the expected return is the mean (expected value) of the probability distribution of returns. An investor's required return for a particular stock is the expected return it must offer to just keep him or her in the stock. If it is expected to return less than the required return, the investor will sell. 4. Give a verbal definition of risk that's consistent with the way we use the word in everyday life. Discuss the weaknesses of that definition for financial theory. Answer: Risk is the probability that something bad will happen. In finance it's the probability that the return on an investment will turn out to be less than expected. The theoretical problem is that the definition is one sided. In statistical terms it represents the left side of the probability distribution of returns. That makes it difficult to work with mathematically. 5. Define risk aversion in words without reference to probability distributions. If people are risk averse, why are lotteries so popular? Why are trips to Las Vegas popular? (Hint: Think in terms of the size of the amount risked and entertainment value.) Answer: Risk aversion describes a characteristic of most (but not all) people. Most of us prefer a lower risk venture (investment, gamble) to one with a higher risk if the expected returns are close to equal. In other words, most people don't like to play "long shots." People are risk averse when the amount of money placed at risk is substantial to the investor (gambler), and there isn't any entertainment involved. When the amount of money is small and the process is "fun," people are frequently not risk averse. This explains the popularity of the lottery in which you're almost sure to lose your investment. The amount is insignificant and worth the thrill. Las Vegas is the same idea. The likely gambling loss is viewed as the cost of a good time. Of course, compulsive gamblers are not risk averse. In finance we assume risk aversion, because we're dealing with substantial sums in a serious setting, and the vast majority of people exhibit risk averse behavior. 6. The following definition applies to both investing and gambling: Putting money at risk in the hope of earning more money. In spite of this similarity, society has very different moral views of the two activities. a. Develop an argument reconciling the differences and similarities between the two concepts. That is, why do people generally feel good about investing and bad about gambling? (Hint: Think of where the money goes and what part of income is used.) b. Describe the difference between investing and gambling by drawing a probability distribution to represent each. (Hint: Think of the expected return and the probability of a big gain or a big loss.) Answer: a. Money invested through primary market transactions as described in this text is used for economically beneficial purposes by companies. These activities create jobs and products. The money used in gambling isn't put to work to create economic wealth. It is simply redistributed among the gamblers. b. Investing is generally characterized by a probability distribution whose most likely (expected) return is modestly positive (5-15%) with relatively small probabilities of large losses or large gains. In gambling, the most likely return is usually a loss, and the probability of total loss is significant. However, there is a larger probability of a big win. 7. Why does it make sense to think of the return on a stock investment as a random variable? Does it make sense to think of the return on a bond investment that way? How about an investment in a savings account? Answer: The return on a stock investment is the net result of the business's performance and the influence of the stock market. Both of these processes involve thousands of individual pressures pushing the return up and down at the same time. This fits the nature of a random variable. A bond's return on the other hand is largely insulated from such disturbances. Unless the issuer fails, it must pay the contracted interest and principal regardless of its own business performance or market conditions. Hence, the return on a bond is unlikely to vary from its expected value. The return on a savings account is virtually guaranteed, since federal insurance eliminates even the risk of bank failure. Hence returns on these debt investments are not properly viewed as random variables. 8. In everyday language, risk means the probability of something bad happening. Risk in finance, however, is defined as the variance of the probability distribution of returns. a. Why do these definitions seem contradictory? b. Reconcile the two ideas. Answer: a. The ideas seem contradictory because standard deviation (or variance) includes the probability that good things happen as well as bad things. This isn't consistent with the everyday notion of risk. b. Financial probability distributions are generally fairly symmetrical, so a large standard deviation implies a large probability of significant bad news. Understanding that, theorists use the whole variance definition, because it's easier to work with mathematically than a one sided definition. 9. Analyze the shape of the probability distribution for a high risk stock versus that of a low risk stock. (Hint: Think in terms of where the area under the curve lies.) Answer: Consider the probability distributions for high and low risk stocks. Notice from the diagram that both curves have about half of their areas on either side of expected value. So both have about a fifty percent chance of producing a return that's worse than expected. That doesn't tell us much. The thing that really makes the risky stock risky is that it has more area far away from the mean on the left side than the low risk stock. That means there's a lot greater chance of a substantially bad return with the risky stock than with the low risk issue. 10. Describe risk in finance as up and down movement of return. Does this idea make sense in terms of the variance definition? Answer: Risk is variability of return. In probability terms that means a large standard deviation (variance). In terms of actual observations it means that the return will take on a wide variety of values over time. In other words, risk refers to the difference between the high and low values taken on by a stock's return as time passes. A high-risk stock routinely goes through big swings between its highs and lows, while a low risk stock's return is relatively stable. The probabilistic definition just says that in any period the probability of a wide swing in return is larger for a risky stock than a low risk stock. This idea is entirely consistent with the up and down movement idea. 11. Define and discuss the idea of separating risk into two parts. Describe each part carefully. Answer: Risk is the variability, or up and down movement, of return. In general, this up and down movement is caused by stimuli that act on the firm, the market and the economic environment. Hence risk can be thought of as the responsiveness of securities to economic and business events. The overall movement of a firm's return can be decomposed into two parts related to the cause of the variation. Movement caused by things that affect only the firm in question or its industry is called business-specific risk. A change in return due to a labor dispute at a particular company or in a single industry is an example. The remainder of the movement in return is caused by things that affect all stocks, although they aren't all impacted equally. This movement is called market risk, because the stimuli influence all securities in the market. The general state of the economy is an example. Market risk is commonly called systematic risk, because it affects the entire financial system. Business-specific risk is then called unsystematic risk. 12. Describe the goal of a portfolio owner in terms of risk and return. How does he or she evaluate the risk characteristics of stocks being considered for addition to the portfolio? Answer: The owner's goal is to construct a portfolio that captures the high average return associated with equities, but eliminates as much of the risk (variation in return) as possible. In other words, he or she wants to construct a portfolio whose overall return doesn't vary much. This is done by selecting stocks such that the movements in their returns offset each other as much as possible. In selecting stocks, the owner is concerned with the way their returns impact the overall portfolio's return rather than with how they behave as individuals. 13. Discuss lowering portfolio risk through diversification. Consider a. Unsystematic (business-specific) risk. b. Systematic (market) risk. Answer: a. Unsystematic risk can be eliminated by selecting a wide variety of stocks. Viewed over a large number of stocks, business-specific events are more or less random occurrences that tend to offset one another if the securities in the portfolio are from several industries. Hence we say that business-specific risk can be diversified away by choosing the stocks of different companies in different industries. b. Systematic risk generally cannot be entirely eliminated through diversification, because the returns on most stocks tend to move together. However, since returns don't move exactly together, it's possible to choose stocks such that the variations in their returns partially offset one another. This is done by paying attention to the timing of the variations rather than just to their magnitudes. 14. Describe the concept of beta. Include what it measures and how it's developed. Answer: The key measure of risk in a portfolio is the degree to which stocks move together. A stock's beta measures the degree to which its return moves with the return on the market as a whole. Hence, beta measures market risk which is the movement in a stock’s return that’s caused by things that affect all stocks. Beta is developed for a particular stock by regressing its historical return against the market's return for the same periods. Beta is the slope of the regression line. It represents the average relationship between changes in the market's return and changes in the stock's return. 15. Describe the SML in words. What is it saying about how investors form required rates of return? Thoroughly evaluate the implications of the SML's message. Answer: The SML purports to describe the way in which investors form required returns for individual stocks. It says that the required return is the risk free rate plus a risk premium. Further, it claims that the risk premium is just the average market risk premium multiplied by the stock's beta, which is a measure of its market risk. In other words, the SML says that the only thing about the company that influences its required return is its market risk as measured by beta. This is very important because the required return ultimately determines a stock's price. That means that an important way a firm can influence the market price of its stock is to do things that affect its stability and beta. 16. How does the SML determine the price of a security? Answer: In theory, the SML determines a firm's required rate of return. Then, given a dividend pattern, the required rate of return determines price. We can see this in the constant growth model of stock valuation, The dividend pattern is implied by D1 and g. If these are fixed, price depends on k. If k is determined by beta in accordance with the SML, price ultimately depends on beta, which measures market risk. 17. How is risk aversion reflected in the SML? Answer: Risk aversion is reflected in the SML by the market risk premium, (kM - kRF). A larger risk premium means that people are more averse to risk, since they require a greater reward for bearing it. Graphically the market risk premium is equivalent to the slope of the SML. Hence a steeper SML implies a more risk averse investing community. 18. The CAPM purports to explain how management decisions about risk can influence the well-being of stockholders. Describe in words the mechanism through which this works. Answer: If a management action makes a company more risky, its return will fluctuate more in response to various market stimuli. That means its beta will increase. A higher beta implies that investors will increase their required returns for the stock in accordance with the SML. But a higher required return implies a lower stock price all other things equal (e.g. in the Gordon Model). Hence the stock's price will fall making stockholders poorer. 19. Is the CAPM a true and accurate representation of the securities world? Answer: All economic models are abstractions of reality. They simplify the world and leave a lot out. Indeed, no one could ever build a model that included the complexity of the real world. The question is whether or not the model captures enough of the substance of things to make accurate predictions about reality. The CAPM is a simplification of financial markets that clearly leaves a lot out. It's very unlikely that people actually form required returns based solely on betas as the CAPM says. It is debatable, however, whether or not that representation captures enough of the essence of financial markets to make reasonably accurate predictions about prices. Reputable scholars argue both ways, that it does and that it doesn't. BUSINESS ANALYSIS 1. You've just begun work at the brokerage firm of Dewey, Cheatam, and Howe as a stock analyst. This morning you read an article in the paper that said a large-scale reduction in defense spending is eminent. Fred Fastback, a broker at the firm, has several clients who are elderly retirees. You recently learned that he's actively putting those clients into several defense industry stocks he describes as low risk. Fred has told you that he feels the stocks are low risk because they have betas of 1.0 or less. How would you advise Fred? Consider the real meaning of beta and its constancy over time. Answer: Fred is forgetting that betas can change over time in response to changing conditions. Betas for the defense industry stocks were developed in the past, probably during periods when defense spending was high and steady. That made the stocks' returns stable as reflected by low betas. Now if spending cuts are coming the defense industry won’t be as lucrative or steady as it was. That will probably mean the companies in the field will become more risky and their betas will rise. Hence in this case, a historical beta isn't appropriate to assess future risk. Fred may also be confused about what beta means. It measures risk in a portfolio sense. If his retiree clients don't have diversified portfolios, beta isn't the right measure of risk for them. PROBLEMS Expected and Required Returns, Equation 9.2 (page 395) 1. The Duncan Company's stock is currently selling for $15. People generally expect its price to rise to $18 by the end of next year. They also expect that it will pay a dividend of $.50 per share during the year. (Hint: Apply Equation 9.2 page 395.) a. What is the expected return on an investment in Duncan's stock? b. Recalculate the expected return if next year's price is forecast to be only $17 and the dividend $.25. c. Calculate the actual return on Duncan if at the end of the year the price turns out to be $13 and the dividend actually paid was just $.10. Solution: 2. The Rapscallion Company’s stock is selling for $43.75. Dave Jones has done some research on the firm and its industry, and thinks it will pay dividends of $5 next year and $7 the following year. After those two years Dave thinks its market price will peak at $50. His strategy is to buy now, hold for the two years and then sell at the peak price. If Dave is confident about his financial projections, but requires a return of 25% before investing in stocks like Rapscallion. Should he invest in this opportunity? Comment on Rapscallion’s risk level. (Hint: Approximate the annual return by extending Equation 9-2 to two years and taking half of the result.) Solution: The expected return on Dave’s investment over a two year holding period is k = [D1 + D2 +(P2 – P0) / P0 = [$5 + $7 + ($50 - $43.75] / = $18.25 / $43.75 = 41.7% Divide by 2 to approximate an annual rate. k = 41.7% / 2 = 20.9% Since this is less than Dave’s required return of 25% he should not invest. These returns, both expected and required imply that Rapscallion is a rather risky company. Calculating the Mean and Standard Deviation of a Discrete Probability Distribution: Examples 9-1, 9-2, and 9-3 (pages 397, 399, and 400) 3. Wayne Merritt drives from Cleveland to Chicago frequently and has noticed that traffic and weather make a big difference in the time it takes to make the trip. As a result, he has a hard time planning activities around his arrival time. To better plan his business, Wayne wants to calculate his average driving time as well as a measure of how much an actual trip is likely to vary from that average. To do that he clocked ten trips with the following results: a. Calculate the mean, standard deviation and coefficient of variation of Wayne’s driving time to Chicago. (Hint: Treat the ten trips as the ten possible outcomes of a discrete probability distribution each of which has a probability of 0.1.) b. Calculate the average variation in driving time. Compare the standard and average variations. Is the difference significant? Which is more meaningful to Wayne? Solution: Since all trips have the same .10 probability we can multiply the sum of the squared deviations by that probability rather than multiplying each individually so we have Variance = 28,440 ×.10 = 2,844, and Standard Deviation = (2,844)½ = 53.33 So, Mean = 4160/10 = 416 minutes Standard Deviation = 53.33 minutes CV = 53.33/416 = .128 b. The average variation is simply the average of the figures in the difference column computed as follows: Average variation = (56 + 41 + 26 + 26 + 11 + 11 + 11 + 4 + 34 + 144) / 10 = 364/10 = 36.4 It’s important to notice that at 36 minutes the average variation is significantly lower than the standard deviation of 53 minutes. That’s because the squaring process involved in calculating the standard deviation gives larger variations from the mean proportionately more weight than smaller variations. To see this look at the influence of the last trip on the standard deviation calculation. That one observation makes up almost 73% of the sum of the squared deviations. Because of this disproportionate influence of extreme values, presumably due to unusual weather or traffic conditions that are often predictable, an average deviation would probably be more meaningful to Wayne. Discrete Probability Distributions: Example 9-1 and Footnote 1 (page 397) 4. Suppose dice had four sides instead of six, so rolling a single die would produce equally likely numbers from one to four, and rolling two dice would produce numbers from two to eight. a. Compute the probability distribution of outcomes from rolling two dice. Mean and Standard Deviation: Examples 9-2 and 9-3, (pages 399 and 400) b. Calculate the mean, standard deviation and coefficient of variation of the distribution. Solution: Write down all 16 possible combinations of two numbers ranging from 2-4. Each pair has a probability of one-sixteenth or .0625. NOTE: Problems 5-8 assume discrete probability distributions for the returns on stocks to keep the computations simple. Evaluating Stand-Alone Risk: Example 9-4, (page 406) 5. Conestoga Ltd. has the following estimated probability distribution of returns. Calculate Conestoga’s expected return, the variance and standard deviation of its expected return and the return’s coefficient of variation. Solution: 6. The probability distribution of the return on an investment in Omega Inc.'s common stock is as follows: Graph the probability distribution. Calculate the expected return, the standard deviation of the return, and the coefficient of variation. Solution: 7. Calculate the expected return on an investment in Delta Inc.'s stock if the probability distribution of returns is as follows. Plot the distribution on the axis with Omega Inc. of the previous problem. Based on the graph, which company has the lower risk/variance? If offered the choice between making an investment in Delta and in Omega Inc., which would most investors choose? Why? Solution: Most investors would choose Omega, because it has a lower risk/variance coupled with the same expected return as Delta. 8. The Manning Company's stock is currently selling for $23. It has the following prospects for next year: Calculate Manning's expected return for a one-year holding period. Solution: Risk Aversion – Figure 9-6: (page 406) 9. Imagine making choices in the following situation to test your degree of risk aversion. Someone offers you the choice between the following game and a sure thing. The Game: A coin is tossed. If it turns up heads, you get a million dollars. If tails, you get nothing. The Sure Thing: You're given $500,000. a. What is the expected value of each option? b. Which option would you choose? c. Viewing the options as probability distributions, which has the larger variance? What is the variance of the sure thing? d. Suppose the game is changed to offer a payoff of $1.2 million for a head but still offers nothing for a tail. The sure thing remains $500,000. What is the expected value of each option now? Which option would you choose now? e. Most people will have chosen the sure thing in part d. Assuming you did too, how much would the game's payoff have to increase before you would choose it over the sure thing? f. Relate this exercise to Figure 9-6. Solution: Portfolio Return: Example 9-5, (page 415) 10. A portfolio consists of the following four stocks. 11. Laurel Wilson has a portfolio of five stocks. The stocks’ actual investment performance last year is given below along with an estimate of this year’s performance. Projecting Returns with Beta: Example 9-6 (page 417) 12. Threads Inc. manufactures stylish clothing for teenagers. The firm has a beta of 1.4 and earned a return on equity of 20% last year. However, a new financial crisis has just hit the stock market and Wall Street experts think the return on an average stock will be cut in half this year. The market has been producing equity returns of about 18% lately. Estimate this year’s return on an equity investment in Thread’s Solution: Beta represents the past average change in Threads’ return relative to changes in the market’s return. The market’s return is expected to drop from 18% to half of that value or by 9%, over the current year, and bThreads is 1.4. Portfolio Beta: Example 9-9 (page 420) 13. The stocks in the problem 11 have the following betas. Calculate Laurel’s portfolio beta for last year and for this year. Assume that the changes in investment (value) come from changing stock prices rather than buying and selling shares. What has happened to the riskiness of Laurel’s portfolio? Should she be concerned? Solution: Overall, the portfolio has become slightly more risky as a result of the change in weights in the various stocks. However, the change in the risk profile is relatively small, so Laurel probably shouldn’t be concerned. 14. A four-stock portfolio is made up as follows Determination of Beta: Figure 9-8 (page 415) 15. The Charming Co. manufactures decorating products. Treasury bills currently yield 5.4%, and the market is returning 8.1%. a. Calculate the Charming Co.'s beta from its characteristic line as depicted following: b. What expected return would an average investor require to buy shares of Charming? c. Would the answer to part b be a "fair" return? Why? (Hint: Think in terms whether the investor has a diversified portfolio or just a few stocks and the risk she faces in each case.) Solution: c. We can think of a return as "fair" if it compensates an investor appropriately for risk. This return is therefore fair if the investment is viewed in the portfolio sense in which market risk is all that matters. If an investor does not have a broadly diversified portfolio and is therefore subject to total risk, he or she may not view this return as adequate or fair. 16. The return on Holland-Wilson Inc. (HWI) stock over the last three years is shown below along with the market’s return for the same period. Plot HWI’s return against that of the market in each of the three years. Make three estimates of HWI’s beta by drawing characteristic lines between pairs of data points (1 and 2; 1 and 3; 2 and 3). What does this range of betas imply about the stock’s risk relative to an average stock? Solution: Using 20x1 and 20x2 data, the slope would be 4/3 or 1.33 Using 20x1 and 20x3 data, the slope would be 8/7 or 1.14 Using 20x2 and 20x3 data, the slope would be 4/4 or 1.00 The range is 1.00 to 1.33 so the actual beta is probably greater than 1.00. This would make HWI stock slightly more risky than the average stock. 17. You have recently purchased stock in Topical Inc. which has returned between 7% and 9% over the last three years. Your friend, Bob, has criticized your purchase and insists that you should have invested in Combs Inc., as he did, because it’s been returning between 10% and 12% in the last three years. Bob knows nothing about financial theory. Topical’s beta is 0.7 and Combs’ is 1.2. Treasury bills are currently yielding the risk free rate of 4.2%, while the stock market is returning an average rate of 9.4%. a. What return should you expect from Topical? What return should Bob expect from Combs? b. Write a few words explaining to Bob why these expected returns aren’t the whole story. Solution: b. Bob, expected return isn’t everything in an investment, risk also has to be considered. Based on past movements in their returns, Topical is a relatively low risk stock while Combs’ is fairly risky. You might make a lot on Combs, but there’s a good chance you’ll lose money too. Topical tends to be much more stable, and while I may not make as much if conditions are good, I’m far less likely to lose anything if conditions are bad in the coming years. 18. Erin Behlen has a three stock portfolio and is interested in estimating its overall return next year. She has $25,000 invested in Forms Corp., which has a beta of 1.3; $75,000 in Crete Corp. with a beta of .8 and $20,000 in Stalls Corp, which has a beta of 1.45. The stock market is currently returning 10.2% and Treasury securities are yielding the risk free rate of 4.6%. What return should Erin anticipate on her portfolio? Solution: First calculate Erin’s portfolio’s beta which is the investment dollar weighted average of the betas of the stocks it contains. First calculate the total dollar investment. Portfolio Dollar Value = $25,000 + $75,000 + $20,000 = $120,000 Then the weighted average beta is bP = ($25,000/120,000)1.3 + ($75,000/$120,000).8 + $20,000/$120,000)1.45 = = .27 +.5 +.24 = 1.01 Next use the SML to estimate the portfolio’s required return. kP = kRF + (kM – kRF)bP = 4.6 + (10.2 – 4.6)1.01 = 10.26% which is just above the market’s current return. 19. The CFO of Ramekin Pottery Inc. is concerned about holding up the price of the company’s stock. He’s asked you to do an analysis starting with an estimate of the return investors are likely to require before they will invest in the firm. The overall stock market is currently returning 16%, 90 day treasury bills yield 6%, and the return on Ramekin’s stock typically responds to changes in the political and economic environment only about 60% as vigorously as does that of the average stock. a. Prepare an estimate of the firm’s required return using the CAPM. b. Is a higher or lower required return good for the company? Why? c. Suppose the CFO asks you what management can do to improve the required return. How will you respond? d. What will you tell him if he wants it done in within the next three months? Solution: a. Estimate the required return using the SML of the CAPM kR = kRF + (kM – kRF)bR = 6 + (16 – 6).60 = 6 + 6 = 12% b. A lower required return is better because it is more likely to be exceeded by investors expected returns, which implies more people will be interested in the stock which will tend to bid up its price. Since a high stock price achieves management’s goal of maximizing shareholder wealth, a lower required return is desirable. c. In terms of the SML, the only thing management can do is take actions that will lower beta, since that’s the only element of the equation that’s controllable by the firm. This implies stabilizing the business so that returns are less variable. The best way for management to do that is probably by attempting to stabilize earnings from year to year. That might be difficult in this case because the firm’s beta is already quite low. d. Influencing beta in so short a time is probably impossible. Beta is derived by developing a relationship between a stock’s return and the market return over many years. That isn’t likely to be changeable in three months. 20. You are a junior treasury analyst at the Palantine Corporation. The treasurer believes the CAPM gives a good estimate of the company's return to equity investors at any time, and has asked you to prepare an estimate of that return using the SML. Treasury bills currently yield 6%, but may go up or down by 1%. The S&P 500 shows a return of 10% but may vary from that figure up to 12%. Palantine's beta is .8. Construct a table showing all possible values of kPalantine for 1% increments of kRF and kM (nine lines). For this problem treat kM and kRF separately. That is, do not assume a change in kM when kRF changes. Solution: Valuing (Pricing) a Stock with CAPM: Example 9-10 (page 425) 21. The Framingham Company expects to grow at 4% indefinitely. Economists are currently asserting that investment opportunities in short term government securities (treasury bills) are readily available at a risk free rate of 5%. The stock market is returning an average rate of 9%. Framingham’s beta has recently been calculated at 1.4. The firm recently paid an annual dividend of $1.68 per share. At what price should shares of Framingham stock be selling? Solution: First calculate a required return for investment in Framingham stock using the SML. 22. Whole Foods Inc. paid a quarterly dividend of $0.47 recently. Treasury bills are yielding 4%, and the average stock is returning about 11%. Whole Foods is a stable company. The return on its stock responds to changes in the political and economic environment only about 70% as vigorously as that of the average stock. Analysts expect the firm to grow at an annual rate of 3.5% into the indefinite future. Calculate a reasonable price that investors should be willing to pay for Whole Foods stock. Solution: Calculate a required return for Whole Foods using the SML. kWF = kRF + (kM – kRF) bWF = 4 + (11 – 4) .7 = 4 + 4.9 = 8.9% Determine a price using the Gordon Model first noticing that the annual dividend rate is the recent quarterly payment times 4. Hence: 23. Seattle Software Inc. recently paid an annual dividend of $1.95 per share and is expected to grow at a 15% rate indefinitely. Short term federal government securities are paying 4% while an average stock is earning its owner 11%. Seattle is a very volatile stock, responding to the economic climate two and a half times violently as an average stock. This is, however, typical of the software industry. a. How much should a share of Seattle be worth? b. Do you see any problems with this estimate? Change one assumption to something more reasonable and compare the results. Solution: a. First solve the SML for Seattle’s required return. b. The growth assumption is probably too aggressive even for the software industry. Growth of 15% forever is unlikely. Recalculate the second step of the estimating process with an 8% growth assumption. P0 = D0(1 + g) / (k – g) = $1.95(1.08) / (.215 - .08) = $2.11 / .135 = $15.63 24. The Aldridge Co. is expected to grow at 6% into the indefinite future. Its latest annual dividend was $2.50. Treasury bills currently earn 7% and the S&P 500 yields 11%. a. What price should Aldridge shares command in the market if its beta is 1.3? b. Evaluate the sensitivity of Aldridge's price to changes in expected growth and risk by recalculating the price while varying the growth rate between 5% and 7% (increments of 1%) and varying beta between 1.2 and 1.4 (increments of .1). Solution: 25. Bergman Corp. has experienced zero growth over the last seven years paying an annual dividend of $2.00 per share. Investors generally expect this performance to continue. Bergman stock is currently selling for $24.39. The risk-free rate is 3.0% and Bergman’s beta is 1.3. a. Calculate the return investors require on Bergman’s stock. b. Calculate the market return. c. Suppose you think Bergman is about to announce plans to grow at 3.0% into the foreseeable future. You also believe investors will accept that prediction and continue to require the same return on its stock. How much should you be willing to pay for a share of Bergman’s stock. Solution: a. Use the Gordon Model with g=0 to find Bergman’s required rate of return P0 = D0(1 + g) / (k-g) = D0 / k $24.39 = $2.00 / k k = .082 or 8.2% b. Use the SML to find the market return, kM. kX = kRF + (kM-kRF)bX 8.2 = 3.0 + (kM-3.0)1.3 from which kM = 7.0% c. P0 = D0(1+g) / (k-g) = [$2.00 (1.03)]/(.082 - .03) = $39.62 26. Weisman Electronics just paid a $1.00 dividend, the market yield is yielding 10%, the risk-free rate is 4%, and Weisman’s beta is 1.5. How fast do investors expect the company to grow in the future if its stock is selling for $27.25. Solution: First solve for Weisman’s required return using the SML kX = kRF + (kM-kRF)bX = 4 + (10 – 4)1.5 = 13% The solve for Weisman’s growth rate using the Gordon Model; P0 = D0(1+g) / (k-g) $27.25 = [$1.00 (1 + g)]/(.13 – g) from which g = .09 or 9% Strategic Decisions Based on CAPM: Example 9-11 (page 430) 27. Weisman Electronics of the previous problem is considering acquiring an unrelated business. Management thinks the move could change the firm’s stock price by moving its beta up or down and decreasing its growth rate. A consultant has estimated that Weisman’s beta after the acquisition could be anywhere between 1.3 and 1.7 while the growth rate could remain at 9% or decline to as little as 5%. Calculate a range of values for Weisman’s stock based on best and worst-case scenarios. Solution: Worst case: 5% growth rate and 1.7 beta Required return = 4 + 6 (1.7) = 14.2% Stock price = $1.05/(.142 - .05) = $11.41 Best case: 9% growth rate and 1.3 beta Required return = 4 + 6 (1.3) = 11.8% Stock price = $1.09/(.118 - .09) = $38.93 28. Broken Wing Airlines just paid an annual dividend of $2, has a beta of 1.3, and a growth rate of 6% for the foreseeable future. The current return on the market is 10% and Treasury bills earn 4%. If the rate on Treasury bills drops by 0.5% and the market risk premium [(kM - kRF)] increases by 1.0%, what growth rate would keep Broken Wing’s stock price constant? (Hint: Calculate the price before rates change, substitute into the Gordon Model and solve for g.) Solution: First calculate Broken Wing’s required return using the SML: kX = kRF + (kM-kRF)bX = 4 + (10 – 4) (1.3) = 11.8% Then calculate its price using the Gordon Model P0= [$2.00 (1.06)]/(.118 - .06) = $36.55 If the rate on Treasury bills drops by 0.5% and the market risk premium increases by 1.0%, Broken Wing’s new require return would be: kX = kRF + (kM-kRF)bX = 3.5 + 7 (1.3) = 12.6% Then solve the Gordon Model for a growth rate that will maintain the intrinsic value of $36.55 P0 = D0 (1+g) / (k-g) $36.55 = [$2.00 (1 + g)] / (.126 – g) (.126 – g) 36.55 = 2 + 2g 4.6053 – 36.55g = 2 + 2g 2.6053 = 38.55g g = .068 or 6.8% 29. Lipson Ltd. expects a constant growth rate of 5% in the future. Treasury bills yield 8% and the market is returning 13% on an average issue. Lipson's last annual dividend was $1.35. The company's beta has historically been .9. The introduction of a new line of business would increase the expected growth rate to 7% while increasing its risk substantially. Management estimates the firm's beta would increase to 1.2 if the new line is undertaken. Should Lipson undertake the new line of business? Solution: Hence the new line of business appears to have a net positive effect on stock price and should be adopted. The SML and Changing Market Conditions: Example 9.12 (page 429) 30. The Picante Corp's beta is .7. Treasury bills yield 5% and an average stock yields 10%. a. Write and sketch the SML, and locate Picante on it. Calculate Picante's required rate of return and show it on the graph. b. Assume the yield on Treasury bills suddenly increases to 7% with no other changes in the financial environment. Write and sketch the new SML, calculate Picante's new required rate, and show it on the new line. c. Now assume that besides the change in part b investors' risk aversion increases so that the market risk premium is 7%. Write and sketch the resulting SML, calculate Picante's required return and show it on the last line. Solution: COMPUTER PROBLEMS 31. Problem 22 in Chapter 8 concerned the Rollins Metal Company, which is engaged in long-term planning. The firm is trying to choose among several strategic options that imply different future growth rates and risk levels. Reread that problem now. The CAPM gives some additional insight into the relation between risk and required return. We can now define risk as beta, and evaluate its effect on stock price by constructing a chart similar to the one called for in Problem 22 of Chapter 8 replacing k on the left side with beta (b). Rollins' beta calculated from historical data is .8. However, the risky strategies being considered could influence that figure significantly. Management feels beta could rise to as much as 2.0 under certain strategic options. Treasury bills currently yield 3%, while the S&P index is showing a return of 8%. Recall that Rollins' last dividend was $2.35. a. Use the CAPMVAL program to construct the following chart. b. The effect of beta on required return and price is influenced by the general level of risk aversion, which in the CAPM is represented by (kM  kRF), the market risk premium (also the slope of the SML). In part a of this problem the market risk premium is 8% – 3%= 5%. Economists, however, predict a recession, which could sharply increase risk aversion. Reconstruct the chart above assuming the market risk premium increases to 7% (kM rises to 10% with no change in kRF). c. Do your charts give any new insights into the risk-return-growth relationship? That is, how does the reward for bearing more risk in terms of stock price change in recessionary times? Write the implied required return on your charts next to the values of beta. Then compare the charts with the one from problem 23 of chapter 8. d. Does the inclusion of beta and the CAPM really make management's planning job any less intuitive? In other words, is it any easier to associate a strategy's risk level with a beta than directly with a required return? Solution: c. The reward in terms of stock price for bearing risk decreases because investors are more risk averse. I.e., their increased dislike for risky strategies is reflected in the lower prices they're willing to pay for a firm’s undertaking risky ideas. (Notice that the b = .8, g = 8% cell isn't meaningful. It’s just a computational anomaly that arises from ke and g being close together.) d. Probably not. In fact, it may be more difficult as beta is more of a conceptual abstraction than growth. Developing Software 32. Write a spreadsheet program to calculate the expected return and beta for a portfolio of ten stocks given the expected returns and betas of the stocks in the portfolio and their dollar values. The calculation involves taking a weighted average of the individual stock's expected returns and betas where the weights are based on the dollar values invested in the stocks. Set up your spreadsheet like this: The computational procedure is as follows. 1. Input the names of the stocks, their dollar values, their betas and their kes. 2. Sum the value column. 3. Calculate the weight column by dividing each row's value cell by the cell carrying the sum of the values. 4. Calculate the beta and ke factors by multiplying the individual beta and ke cells by the cells in the weight column on the same row. 5. Sum the two factor columns for the results indicated. Is your program general in that it will handle a portfolio of up to ten stocks, or will it only work for exactly ten? If it is general, what do you have to be careful about with respect to inputs? Extra: Assume you have $1M to invest in stocks. Look up several stocks' betas in Value Line and estimate ke for each. Look up the current price of each stock in The Wall Street Journal and form a hypothetical portfolio by allocating your money among the stocks. Find your portfolio's expected return and beta using your program. Solution Manual for Practical Financial Management William R. Lasher 9781305637542