Risk and management by ljismeok

C H AP T ER

8 Risk and Rates of Return

A MANAGERIAL PERSPECTIVE

he performance of the major stock markets from 1995 through 1998 can best be described as remarkable—a period that investors would love to repeat again and again. During that four-year stretch, stocks traded on U.S. stock markets earned an average return greater than 20 percent per year. In 1998, companies such as Microsoft and WorldCom MCI more than doubled in value. The value of some Internet companies, such as America Online, Amazon.com, and Yahoo!, increased by more than 500 percent. Consider the return you would have earned in 1998 if you had purchased Amazon.com at the beginning of the year for $30.13 and then sold it at the end of the year for $321.25: a one-year return of 966 percent. On the other hand, if you had waited until January 2000 to buy Amazon.com and then held it until the end of the year, you would have lost approximately 80 percent of your investment because the company’s stock decreased significantly during the year. In fact, during 2000 the values of most Internet company stocks declined significantly. Indeed, many Internet companies did not survive the ‘‘Internet skepticism’’ that existed during the year. By comparison, if you had purchased the stock of Enron at the beginning of 2000, your investment would have nearly doubled in value by the end of the year. But if you still held Enron in mid-2003, the value of your investment would have declined to $0.05 per share

because the company was in bankruptcy at that time. More recently, the price of Google increased 140 percent in 2005, but then it declined 28 percent in the first two months of 2006 before it recovered to generate a positive 9 percent return for the entire year. If you had bet all your money on the stock of a single company, you would have essentially ‘‘put all your eggs in one basket’’ and faced considerable risk. For example, you would have won big if you chose to invest in Amazon.com for one year either in 1998 or 2002. But you would have lost big if you chose to invest in Amazon.com in 2000. Investors who diversified by spreading their investments among many stocks, perhaps through mutual funds, would have earned a return somewhere between the extraordinary increases posted by Amazon.com in 1998 and 2002 and the extraordinary decreases posted by Amazon.com and other Internet companies in 2000. Large ‘‘baskets’’ of such diversified investments would have earned returns fairly close to the average of the stock markets. Investing is risky! Although the stock markets performed well from 1995 through 1998, they also go through periods characterized by decreasing prices or negative average returns. For instance, in 1990, 1994, and 2000, the average stock listed on the New York Stock Exchange decreased in value by 7.5 percent, 3.1 percent, and 5.9 percent, respectively. More recently, 305

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Chapter 8 Risk and Rates of Return

during the first few months of 2006, the stock market was fairly fickle. At the beginning of the year, the Dow Jones Industrial Average (DJIA) was 10,718. One month later, the DJIA was at about the same level, which means investors earned an average rate of return of approximately 0 percent during the month of January. In mid-May, however, the DJIA was 11,630. Investors who ‘‘got in the market’’ in January and ‘‘got out of the market’’ in May earned an equivalent annual return equal to about 26 percent (noncompounded), and investors who waited to ‘‘get in the market’’ in February and then got out in May earned an equivalent annual return equal to about 34 percent. One month later, however, the DJIA was back to its beginning-ofthe-year value. During 2006 the DJIA at times experienced periods of substantial increases and other times it decreased substantially, but by the end of the year the index had increased 16 percent, which represented a

Chapter Essentials —The Questions

higher-than-average market return. What a roller-coaster ride! What risk! Who knows what the stock market will be doing when you read this book. It could be an up market (referred to as a ‘‘bull’’ market) or it could be a down market (referred to as a ‘‘bear’’ market). Whatever the case, as times change, investment strategies and portfolio mixes need to be changed to meet new conditions. For this reason, you need to understand the basic concepts of risk and return and to recognize how diversification affects investment decisions. As you will discover, investors can create portfolios of securities to reduce risk without reducing the average return on their investments. After reading this chapter, you should have a better understanding of how risk affects investment returns and how to evaluate risk when selecting investments such as those described here.

After reading this chapter, you should be able to answer the following questions:

What does it mean to take risk when investing? How are the risk and return of an investment measured? How are the risk and return of an investment related? For what type of risk is an average investor rewarded? How can investors reduce risk? What actions do investors take when the return they require to purchase an investment is different from the return the investment is expected to produce?

In this chapter, we take an in-depth look at how investment risk should be measured and how it affects assets’ values and rates of return. Recall that in Chapter 5, when we examined the determinants of interest rates, we defined the real risk-free rate, r , to be the rate of interest on a risk-free security in the absence of inflation. The actual interest rate on a particular debt security was shown to be equal to the real risk-free rate plus several premiums that reflect both inflation and the riskiness of the security in question. In this chapter, we define the term risk more precisely in terms of how it relates to investments, we examine procedures used to measure risk, and we discuss the relationship between risk and return. Both investors and financial managers should understand these concepts and use them when considering investment decisions, whether the decisions concern financial assets or real assets. We will demonstrate that each investment—each stock, bond, or physical asset—is associated with two types of risk: diversifiable risk and nondiversifiable risk. The sum of these two components is the investment’s total risk. Diversifiable risk is not important to rational, informed investors because they will eliminate its effects by diversifying it away. The really significant risk is nondiversifiable risk; this risk is bad in the sense that you cannot eliminate it, and if you invest in anything other than risk-free assets, such as short-term Treasury bills, you will be exposed to it. In the balance of the chapter, we will describe these risk concepts and consider how risk enters into the investment decision-making process.

Defining and Measuring Risk

DEFINING

AND

307

MEASURING RISK

Webster’s Dictionary defines risk as ‘‘a hazard; a peril; exposure to loss or injury.’’ As this definition suggests, risk refers to the chance that some unfavorable event will occur. If you engage in skydiving, you are taking a chance with your life: Skydiving is risky. If you bet on the horses, you risk losing your money. If you invest in speculative stocks (or, really, any stock), you are taking a risk in the hope of receiving an appreciable return. Most people view risk in the manner just described—as a chance of loss. In reality, however, risk occurs any time we cannot be certain about the outcome of a particular activity or event, so we are not sure what will happen in the future. Consequently, risk results from the fact that an action such as investing can produce more than one outcome in the future. When multiple outcomes are possible, some of the possible outcomes are considered ‘‘good’’ and some of the possible outcomes are considered ‘‘bad.’’ To illustrate the riskiness of financial assets, suppose you have a large amount of money to invest for one year. You could buy a Treasury security that has an expected return equal to 5 percent. This investment’s anticipated rate of return can be determined quite precisely because the chance of the government defaulting on Treasury securities is negligible; the outcome is essentially guaranteed, which means that the security is a risk-free investment. Alternatively, you could buy the common stock of a newly formed company that has developed technology that can be used to extract petroleum from the mountains in South America without defacing the landscape and without harming the ecology. The technology has yet to be proved economically feasible, so the returns that the common stockholders will receive in the future remain uncertain. Experts who have analyzed the common stock of the company have determined that the expected, or average longrun, return for such an investment is 30 percent. Each year, the investment could yield a positive return as high as 900 percent. Of course, there also is the possibility that the company might not survive, in which case the entire investment will be lost and the return will be 100 percent. The return that investors receive each year cannot be determined precisely because more than one outcome is possible; this stock is a risky investment. Because there is a significant danger of earning considerably less than the expected return, investors probably would consider the stock to be quite risky. There is also a very good chance that the actual return will be greater than expected, which, of course, is an outcome you would gladly accept. This possibility could not exist if the stock did not have risk. Thus, when we think of investment risk, along with the chance of receiving less than expected, we should consider the chance of receiving more than expected. If we consider investment risk from this perspective, then we can define risk as the chance of receiving an actual return other than expected. This definition simply means that there is variability in the returns or outcomes from the investment. Therefore, investment risk can be measured by the variability of all the investment’s returns, both ‘‘good’’ and ‘‘bad.’’ Investment risk, then, is related to the possibility of earning an actual return other than the expected one. The greater the variability of the possible outcomes, the riskier the investment. We can define risk more precisely, however, and it is useful to do so.

Probability Distributions An event’s probability is defined as the chance that the event will occur. For example, a weather forecaster might state: ‘‘There is a 40 percent chance of rain today and a 60 percent chance that it will not rain.’’ If all possible events, or outcomes, are listed, and

risk

The chance that an outcome other than the expected one will occur.

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probability distribution

A listing of all possible outcomes, or events, with a probability (chance of occurrence) assigned to each outcome.

if a probability is assigned to each event, the listing is called a probability distribution. For our weather forecast, we could set up the following simple probability distribution: Outcome

Probability

Rain No rain

0.40 ¼ 40% 0.60 ¼ 60 1.00 100%

Here the possible outcomes are listed in the left column, and the probabilities of these outcomes, expressed both as decimals and as percentages, are given in the right columns. Notice that the probabilities must sum to 1.0, or 100 percent. Probabilities can also be assigned to the possible outcomes (or returns) from an investment. If you buy a bond, you expect to receive interest on it; those interest payments will provide you with a rate of return on your investment. This investment has two possible outcomes: (1) the issuer makes the interest payments, or (2) the issuer fails to make the interest payments. The higher the probability of default on the interest payments, the riskier the bond; the higher the risk, the higher the rate of return you would require to invest in the bond. If you invest in a stock instead of buying a bond, you will again expect to earn a return on your money. As we saw in Chapter 7, a stock’s return includes dividends plus capital gains. Again, the riskier the stock—that is, the greater the variability of the possible payoffs—the higher the stock’s expected return must be to induce you to invest in it. With this idea in mind, consider the possible rates of return (dividend yield plus capital gains yield) that you might earn next year on a $10,000 investment in the stock of either Martin Products Inc. or U.S. Electric. Martin manufactures and distributes equipment for the data transmission industry. Because its sales are cyclical, the firm’s profits rise and fall with the business cycle. Furthermore, its market is extremely competitive, and some new company could develop better products that could force Martin into bankruptcy. U.S. Electric, on the other hand, supplies electricity, which is considered an essential service. Because it has city franchises that protect it from competition, this firm’s sales and profits are relatively stable and predictable. Table 8-1 shows the rate-of-return probability distributions for these two companies. As shown in the table, there is a 20 percent chance of a boom, in which case both companies will have high earnings, pay high dividends, and enjoy capital gains. There is a 50 percent probability that the two companies will operate in a normal economy and offer moderate returns. There is a 30 percent probability of a recession, which will mean low earnings and dividends as well as potential capital losses. Notice, however, that Martin’s rate of return could vary far more dramatically than that of U.S.

TABLE 8-1 Probability Distributions for Martin Products and U.S. Electric State of the Economy Boom Normal Recession

Probability of This State Occurring 0.2 0.5 0.3 1.0

Rate of Return on Stock If Economic State Occurs Martin Products

U.S. Electric

110% 22 60

20% 16 10

Expected Rate of Return

309

TABLE 8-2 Calculation of Expected Rates of Return: Martin Products and U.S. Electric Martin Products State of the Economy (1)

Probability of This State Occurring (2)

Return If This State Occurs (3)

0.2 0.5 0.3 1.0

110% 22 60

Boom Normal Recession

Product: (2) (3) ¼ (4)

^rUS

22% 11 18 ¼ 15%

U.S. Electric Return If This State Occurs (5)

Product: (2) (5) ¼ (6)

20% 16 10 ^rUS

4% 8 3 ¼ 15%

Electric. There is a fairly high probability that the value of Martin’s stock will vary substantially, possibly resulting in a loss of 60 percent or a gain of 110 percent; conversely, there is no chance of a loss for U.S. Electric, and its maximum gain is 20 percent.1

What does investment risk mean? Set up illustrative probability distributions for (1) a bond investment and (2) a stock investment.

EXPECTED RATE

RETURN

Table 8-1 provides the probability distributions showing the possible outcomes for investing in Martin Products and U.S. Electric. We can see that the most likely outcome is for the economy to be normal, in which case Martin will return 22 percent and U.S. Electric will return 16 percent. Other outcomes are also possible, however, so we need to summarize the information contained in the probability distributions into a single measure that considers all these possible outcomes. That measure is called the expected value, or expected rate of return, for the investments. Simply stated, the expected value (return) is the weighted average of the outcomes, with each outcome’s weight being its probability of occurrence. Table 8-2 shows how we compute the expected rates of return for Martin Products and U.S. Electric. We multiply each possible outcome by the probability it will occur and then sum the results. We designate the expected rate of return, ^r , which is termed ‘‘r hat.’’2 We insert the ‘‘hat’’ over the r to indicate that this return is uncertain because we do not know when each of the possible outcomes will occur in the future. For example, Martin products will return its stockholders 110 percent when the economy is booming, but we do not know in which year the economy will be booming.

1It is, of course, completely unrealistic to think that any stock has no chance of a loss. Only in hypothetical examples could this situation occur. 2In Chapter 6, we used r to signify the return on a debt instrument, and in Chapter 7, we used r to signify the return d s

on a stock. In this section, however, we discuss only returns on stocks; thus, the subscript ‘‘s’’ is unnecessary, and we use the term ^r rather than ^rs to represent the expected return on a stock.

expected value (return), ^r

The rate of return expected to be realized from an investment; the mean value of the probability distribution of possible results.

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Chapter 8 Risk and Rates of Return

The expected rate of return can be calculated using the following equation:

8–1

Expected rate of return ¼ rˆ ¼ Pr1 r1 þ Pr2 r2 þ þ Prn rn n X ¼ Pri ri i¼1

Here ri is the ith possible outcome, Pri is the probability that the ith outcome will occur, and n is the number of possible outcomes. Thus, rˆ is a weighted average of the possible outcomes (the ri values), with each outcome’s weight being its probability of occurrence. Using the data for Martin Products, we compute its expected rate of return as follows: rˆ ¼ Pr1 ðr1 Þ þ Pr2 ðr2 Þ þ Pr3 ðr3 Þ ¼ 0:2ð110%Þ þ 0:5ð22%Þ þ 0:3ð 60%Þ ¼ 15:0% Notice that the expected rate of return does not equal any of the possible payoffs for Martin Products given in Table 8-1. Stated simply, the expected rate of return represents the average payoff that investors will receive from Martin Products if the probability distribution given in Table 8-1 does not change over a long period of time. If this probability distribution is correct, then 20 percent of the time the future economic condition will be termed a boom, so investors will earn a 110 percent rate of return; 50 percent of the time the economy should be normal and the investment payoff will be 22 percent; and 30 percent of the time the economy should be in recession and the payoff will be a loss equal to 60 percent. On average, then, Martin Products’ investors should earn 15 percent over some period of time. We can graph the rates of return to obtain a picture of the variability of possible outcomes, as shown in Figure 8-1. The height of each bar in the figure indicates the probability that a given outcome will occur. The probable returns for Martin Products

FIGURE 8-1 Probability Distribution of Martin Products’ and U.S. Electric’s Rate of Return A. Martin Products

B. U.S. Electric

Probability of Occurrence

0.5

0.4

0.3

0.2

0.1

60 40 20

ˆrMartin = 15%

100 120 Rate of Return (%)

Rate of

ˆrUS = 15% Return (%)

Expected Rate of Return

311

range from þ110 percent to –60 percent, with an expected return of 15 percent. The expected return for U.S. Electric is also 15 percent, but its range is much narrower.

Continuous versus Discrete Probability Distributions So far, we have assumed that only three states of the economy can exist: recession, normal, and boom. Under these conditions, the probability distributions given in Table 8-1, are called discrete because the number of outcomes is limited, or finite. In reality, of course, the state of the economy could actually range from a deep depression to a fantastic boom, with an unlimited number of possible states in between. Suppose we had the time and patience to assign a probability to each possible state of the economy (with the sum of the probabilities still equaling 1.0) and to assign a rate of return to each stock for each state of the economy. We would then have a table similar to Table 8-1, except that it would include many more entries in each column. We could use this table to calculate the expected rates of return as described previously, and we could approximate the probabilities and outcomes by constructing continuous curves such as those presented in Figure 8-2. In this figure, we have changed the assumptions so that there is essentially a zero probability that Martin Products’ return will be less than –60 percent or more than 110 percent, or that U.S. Electric’s return will be less than 10 percent or more than 20 percent. Virtually any return within these limits is possible, however. Such probability distributions are called continuous because the number of possible outcomes is unlimited. For example, U.S. Electric’s return could be 10.01 percent, 10.001 percent, and so on.

FIGURE 8-2 Continuous Probability Distributions of Martin Products’ and U.S. Electric’s Rates of Return Probability Density

U.S. Electric

Martin Products −60

Rate of 110 Return (%)

Expected Rate of Return Note: The assumptions regarding the possibilities of various outcomes have been changed from those in Figure 8-1. There the probability of obtaining exactly 16 percent return for U.S. Electric was 50 percent; here it is much smaller because there are many possible outcomes instead of just three. With continuous distributions, it is more appropriate to ask what the probability is of obtaining at least some specified rate of return than to ask what the probability is of obtaining exactly that rate. This topic is covered in detail in statistics courses.

discrete probability distribution

The number of possible outcomes is limited, or finite.

continuous probability distribution

The number of possible outcomes is unlimited, or infinite.

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Chapter 8 Risk and Rates of Return

The tighter the probability distribution, the less variability there is and the more likely it is that the actual outcome will approach the expected value. Consequently, under these conditions, it becomes less likely that the actual return will differ dramatically from the expected return. Thus, the tighter the probability distribution, the lower the risk assigned to a stock. Because U.S. Electric has a relatively tight probability distribution, its actual return is likely to be closer to its 15 percent expected return than is that of Martin Products.

Measuring Total (Stand-Alone) Risk: The Standard Deviation

standard deviation, s

A measure of the tightness, or variability, of a set of outcomes.

Because we have defined risk as the variability of returns, we can measure it by examining the tightness of the probability distribution associated with the possible outcomes. In general, the width of a probability distribution indicates the amount of scatter, or variability, of the possible outcomes. To be most useful, any measure of risk should have a definite value; thus, we need a measure of the tightness of the probability distribution. The measure we use most often is the standard deviation, the symbol for which is s, the Greek letter ‘‘sigma.’’ The smaller the standard deviation, the tighter the probability distribution, and, accordingly, the lower the total risk associated with the investment. To calculate the standard deviation, we take the following steps, as shown in Table 8-3: 1. Calculate the expected rate of return using Equation 8–1. For Martin, we previously found rˆ ¼ 15%. 2. Subtract the expected rate of return, rˆ , from each possible outcome, ri, to obtain a set of deviations from rˆ : Deviationi ¼ ri rˆ

variance, s2

The deviations are shown in column 3 of Table 8-3. 3. Square each deviation (shown in column 4), multiply the result by the probability of occurrence for its related outcome (column 5), and then sum these products to obtain the variance, s2, of the probability distribution, which is shown in column 6:

The standard deviation squared; a measure of the width of a probability distribution.

8–2

Variance ¼ s2 ¼ ðr1 ^rÞ2 Pr1 þ ðr2 ^rÞ2 Pr2 þ þ ðrn ^rÞ2 Prn n X ¼ ðri ^rÞ2 Pri i¼1

TABLE 8-3 Calculating Martin Products’ Standard Deviation Expected Return ^r (2)

Payoff ri (1) 110% 22 60

15% 15 15

Deviation rj ^r (1) (2) ¼ (3) ¼ ¼ ¼

95 7 75

(rj ^r)2 ¼ (4) 9,025 49 5,625

ðrj ^rÞ2 Pri (4) (5)

Probability (5) 0.2 0.5 0.3

9,025 49 5,625 Variance

Standard deviation ¼ s ¼

0.2 0.5 0.3 s2

¼ ¼ ¼ ¼ ¼

s2 (6) 1,805.0 24.5 1,687.5 3,517.0

pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi s2 ¼ 3; 517 ¼ 59:3%

Expected Rate of Return

4. Take the square root of the variance to get the standard deviation shown at the bottom of column 6: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr1 ^rÞ2 Pr1 þ ðr2 ^rÞ2 Pr2 þ þ ðrn ^rÞ2 Prn sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X ¼ ðri ^rÞ2 Pri

Standard deviation ¼ s ¼

8–3

i¼1

As you can see, the standard deviation is a weighted average deviation from the expected value, and it gives an idea of how far above or below the expected value the actual value is likely to be. As shown in Table 8-3 Martin’s standard deviation is s ¼ 59.3%. Using these same procedures, we find U.S. Electric’s standard deviation to be 3.6 percent. The larger standard deviation for Martin indicates a greater variation of returns for this firm, and hence a greater chance that the actual, or realized, return will differ significantly from the expected return. Consequently, Martin Products would be considered a riskier investment than U.S. Electric, according to this measure of risk. To this point, the example we have used to compute the expected return and standard deviation is based on data that take the form of a known probability distribution. That is, we know or have estimated all of the future outcomes and the chances that these outcomes will occur in a particular situation. In many cases, however, the only information we have available consists of data over some past period. For example, suppose we have observed the following returns associated with a common stock: Year

2008 2009 2010 2011

15% 5 20 22

We can use this information to estimate the risk associated with the stock by estimating standard deviation of returns. The estimated standard deviation can be computed using a series of past, or observed, returns to solve the following formula:

Estimated s ¼ s ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP un u ð€rt rÞ2 tt¼1

8–4

n 1

Here €rt represents the past realized rate of return in Period t, and r (‘‘r bar’’) is the arithmetic average of the annual returns earned during the last n years. We compute rt as follows: n P

r¼

€rn

€r1 þ €r2 þ þ €rn t¼1 ¼ n n

8–5

313

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Chapter 8 Risk and Rates of Return

Continuing our current example, we would determine the arithmetic average and estimate the value for s as follows:3 r¼

15 þ ð 5Þ þ 20 þ 22 ¼ 13:0% 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð15 13Þ2 þ ð 5 13Þ2 þ ð20 13Þ2 þ ð22 13Þ2 4 1 rffiffiffiffiffiffiffiffi 458 ¼ 12:4% ¼ 3

Estimated s ¼ s ¼

coefficient of variation (CV)

A standardized measure of the risk per unit of return. It is calculated by dividing the standard deviation by the expected return.

8–6

The historical standard deviation is often used as an estimate of the future standard deviation. Much less often, and generally incorrectly, rt for some past period is used as an estimate of ^rt , the expected future return. Because past variability is likely to be repeated, s might be a good estimate of future risk. It is much less reasonable, however, to expect that the past level of return (which could have been as high as +100 percent or as low as –50 percent) is the best expectation of what investors think will happen in the future.

Coefficient of Variation (Risk/Return Ratio) Another useful measure to evaluate risky investments is the coefficient of variation (CV), which is the standard deviation divided by the expected return: Coefficient of variation ¼ CV ¼

Risk s ¼ Return ^r

The coefficient of variation shows the risk per unit of return. It provides a more meaningful basis for comparison when the expected returns on two alternatives differ. Because both U.S. Electric and Martin Products have the same expected return, it is not necessary to compute the coefficient of variation to compare the two investments. In this case, most people would prefer to invest in U.S. Electric because it offers the same expected return with lower risk. The firm with the larger standard deviation, Martin, must have the larger coefficient of variation because the expected returns for the two stocks are equal, but the numerator in Equation 8–6 is greater for Martin. In fact, the coefficient of variation for Martin is 59.3%/15% ¼ 3.95; for U.S. Electric, CV ¼ 3.6%/15% ¼ 0.24. Thus Martin is more than 16 times riskier than U.S. Electric using this criterion. The coefficient of variation is more useful when we consider investments that have different expected rates of return and different levels of risk. For example, Biobotics Corporation is a biological research and development firm that, according to stock analysts, offers investors an expected rate of return equal to 35 percent with a standard deviation of 7.5 percent. Biobotics offers a higher expected return than U.S. Electric, but it is also riskier. With respect to both risk and return, is Biobotics or U.S. Electric a better investment? If we calculate the coefficient of variation for Biobotics, we find that it equals 7.5%/35% ¼ 0.21, which is slightly less than U.S. Electric’s coefficient of variation of 0.24. Thus, Biobotics actually has less risk per unit of return than U.S. Electric, even though its standard deviation is higher. In this case, the additional return

3You should recognize from statistics courses that a sample of four observations is not sufficient to make a good estimate. We use four observations here only to simplify the illustration.

Expected Rate of Return

315

FIGURE 8-3 Comparison of Probability Distributions and Rates of Return for U.S. Electric and Biobotics Corporation Probability Density U.S. Electric

Biobotics

offered by Biobotics is more than sufficient to compensate investors for taking on the additional risk. Figure 8-3 graphs the probability distributions for U.S. Electric and Biobotics. As you can see in the figure, U.S. Electric has the smaller standard deviation and hence the more peaked probability distribution. As the graph clearly shows, however, the chances of a really high return are much better with Biobotics than with U.S. Electric because Biobotics’ expected return is so high. Because the coefficient of variation captures the effects of both risk and return, it is a better measure for evaluating risk in situations where investments differ with respect to both their amounts of total risk and their expected returns.

Risk Aversion and Required Returns Suppose you have worked hard and saved $1 million, which you now plan to invest. You can buy a 10 percent U.S. Treasury note, and at the end of one year you will have a sure $1.1 million—that is, your original investment plus $100,000 in interest. Alternatively, you can buy stock in R&D Enterprises. If R&D’s research programs are successful, the value of your stock will increase to $2.2 million. Conversely, if the firm’s research is a failure, the value of your stock will go to zero, and you will be penniless. You regard R&D’s chances of success or failure as being 50-50, so the expected value of the stock investment is 0.5($0) þ 0.5($2,200,000) ¼ $1,100,000. Subtracting the $1 million cost of the stock leaves an expected profit of $100,000, or an expected (but risky) 10 percent rate of return: Expected rate Expected ending value Beginning value ¼ Beginning value of return $1;100;000 $1;000;000 $100;000 ¼ ¼ 0:10 ¼ 10:0% ¼ $1;000;000 $1;000;000 In this case, you have a choice between a sure $100,000 profit (representing a 10 percent rate of return) on the Treasury note and a risky expected $100,000 profit

Expected Rate of Return (%)

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Chapter 8 Risk and Rates of Return

risk aversion

Risk-averse investors require higher rates of return to invest in higher-risk securities.

risk premium (RP)

The portion of the expected return that can be attributed to the additional risk of an investment. It is the difference between the expected rate of return on a given risky asset and the expected rate of return on a less risky asset.

(also representing a 10 percent expected rate of return) on the R&D Enterprises stock. Which one would you choose? If you choose the less risky investment, you are risk averse. Most investors are risk averse, and certainly the average investor is risk averse, at least with regard to his or her ‘‘serious money.’’ Because this is a well-documented fact, we shall assume risk aversion throughout the remainder of the book. What are the implications of risk aversion for security prices and rates of return? The answer is that, other things held constant, the higher a security’s risk, the higher the return investors demand, and thus the less they are willing to pay for the investment. To see how risk aversion affects security prices, we can analyze the situation with U.S. Electric and Martin Products stocks. Suppose each stock sold for $100 per share and had an expected rate of return of 15 percent. Investors are averse to risk, so they would show a general preference for U.S. Electric because there is less variability in its payoffs (less uncertainty). People with money to invest would bid for U.S. Electric stock rather than Martin stock, and Martin’s stockholders would start selling their stock and using the money to buy U.S. Electric stock. Buying pressure would drive up the price of U.S. Electric’s stock, and selling pressure would simultaneously cause Martin’s price to decline. These price changes, in turn, would alter the expected rates of return on the two securities. Suppose, for example, that the price of U.S. Electric stock was bid up from $100 to $125, whereas the price of Martin’s stock declined from $100 to $75. This development would cause U.S. Electric’s expected return to fall to 12 percent, whereas Martin’s expected return would rise to 20 percent. The difference in returns, 20% 12% ¼ 8%, is a risk premium (RP). The risk premium represents the compensation that investors require for assuming the additional risk of buying Martin’s stock. This example demonstrates a very important principle: In a market dominated by risk-averse investors, riskier securities must have higher expected returns, as estimated by the average investor, than less risky securities. If this situation does not hold, investors will buy and sell investments and prices will continue to change until the higher-risk investments have higher expected returns than the lower-risk investments. Figure 8-4 illustrates this relationship. We will consider the question of how much higher the returns on risky securities must be later in the chapter, after we examine how diversification affects the way risk should be measured.

FIGURE 8-4 Risk/Return Relationship Return, r

rHigh rAvg Return = r = rRF + RP rLow

Payment for Risk = Risk Premium = RP

rRF Risk-Free Return, rRF = r* + Inflation Premium = r* + IP 0

Below Average Risk

Average Risk

Above Average Risk

Risk

Portfolio Risk—Holding Combinations of Assets

Which of the two stocks graphed in Figure 8-2 is less risky? Why? How do you calculate the standard deviation associated with an investment? Why is the standard deviation used as a measure of total, or stand-alone, risk? Which is a better measure of total risk: the standard deviation or the coefficient of variation? Explain. What is meant by the following statement: ‘‘Most investors are risk averse’’? How does risk aversion affect relative rates of return? Suppose you own a stock that provided returns equal to 5 percent, 8 percent, –4 percent, and 15 percent during the past four years. What is the average annual return and standard deviation of the stock? (Answer: r ¼ 6%; s ¼ 7.9%)

PORTFOLIO RISK—HOLDING COMBINATIONS

ASSETS

In the preceding section, we considered the riskiness of an investment held in isolation—that is, the total risk of an investment if it is held by itself. Now we analyze the riskiness of investments held in portfolios.4 As we shall see, holding an investment—whether a stock, bond, or other asset—as part of a portfolio generally is less risky than holding the same investment all by itself. In fact, most financial assets are not held in isolation but rather as parts of portfolios. Banks, pension funds, insurance companies, mutual funds, and other financial institutions are required by law to hold diversified portfolios. Even individual investors—at least those whose security holdings constitute a significant part of their total wealth—generally hold stock portfolios rather than the stock of only one firm. From an investor’s standpoint, then, the fact that a particular stock goes up or down is not very important. What is important is the return on his or her portfolio and the portfolio’s risk. Logically, the risk and return characteristics of an investment should not be evaluated in isolation; instead, the risk and return of an individual security should be analyzed in terms of how that security affects the risk and return of the portfolio in which it is held. To illustrate, consider an investment in Payco American, a collection agency company that operates several offices nationwide. The company is not well known, its stock is not very liquid, its earnings have fluctuated quite a bit in the past, and it doesn’t even pay a dividend. This suggests that Payco is risky and that its required rate of return, r, should be relatively high. Even so, Payco’s r always has been quite low relative to the rates of return offered by most firms with similar risk. This information indicates that investors regard Payco as being a low-risk company despite its uncertain profits and its nonexistent dividend stream. The reason for this somewhat counterintuitive fact relates to diversification and its effect on risk. Payco’s stock price rises during recessions, whereas the prices of other stocks tend to decline when the economy slumps. Therefore, holding Payco in a portfolio of ‘‘normal’’ stocks tends to stabilize returns on the entire portfolio.

4A portfolio is a collection of investment securities or assets. If you owned some General Motors stock, some ExxonMobil stock, and some IBM stock, you would be holding a three-stock portfolio. For the reasons set forth in this section, the majority of all stocks are held as parts of portfolios.

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expected return on a portfolio, ^rp

The weighted average expected return on stocks held in a portfolio.

Portfolio Returns The expected return on a portfolio, ^rp , is simply the weighted average of the expected returns on the individual stocks in the portfolio, with each weight being the proportion of the total portfolio invested in each stock:

8–7 Portfolio return ¼ ^rp ¼ w1^r1 þ w2^r2 þ þ wN^rN ¼

N X

wj^rj

j¼1

Here the ^rj values are the expected returns on the individual stocks, the wj values are the weights, and the portfolio includes N stocks. Note two points: (1) wj is the proportion of the portfolio’s dollar value invested in Stock j, which is equal to the value of the investment in Stock j divided by the total value of the portfolio, and (2) the wj values must sum to 1.0. Suppose security analysts estimate that the following returns could be expected on four large companies: Company AT&T Citigroup General Electric Microsoft

Expected Return, ^r 8% 13 19 16

If we formed a $100,000 portfolio, investing $25,000 in each of these four stocks, our expected portfolio return would be 14.0 percent: ^rp ¼ wATT^rATT þ wCiti^rCiti þ wGE^rGE þ wMicro^rMicro ¼ 0:25ð8%Þ þ 0:25ð13%Þ þ 0:25ð19%Þ þ 0:25ð16%Þ ¼ 14:0% realized rate of return, €r

The return that is actually earned. The actual return (€r) usually differs from the expected return (^r).

Of course, after the fact and one year later, the actual realized rates of return, €r, on the individual stocks will almost certainly differ from their expected values, so €rp will be somewhat different from ^rp ¼ 14%. For example, Microsoft’s stock might double in price and provide a return of þ100 percent, whereas General Electric’s stock might have a terrible year, see its price fall sharply, and provide a return of –75 percent. Note, however, that those two events would somewhat offset each other, so the portfolio’s return might still approach its expected return, even though the individual stocks’ actual returns were far from their expected returns.

Portfolio Risk As we just saw, the expected return of a portfolio is simply a weighted average of the expected returns of the individual stocks in the portfolio. Unlike returns, the riskiness of a portfolio (sP) generally is not a weighted average of the standard deviations of the individual securities in the portfolio. Instead, the portfolio’s risk usually is smaller than the weighted average of the individual stocks’ standard deviations. In fact, it is theoretically possible to combine two stocks that by themselves are quite risky as measured by their standard deviations and form a completely risk-free portfolio—that is, a portfolio with sP ¼ 0. To illustrate the effect of combining securities, consider the situation depicted in Figure 8-5. The bottom section of the figure gives data on the rates of return for Stock

Portfolio Risk—Holding Combinations of Assets

319

FIGURE 8-5 Rate of Return Distribution for Two Perfectly Negatively Correlated Stocks (r ¼ 1.0) and for Portfolio WM A. Rates of Return r¨W

Stock W

¨rM

Stock M

2008

Portfolio WM

¨rWM

2008

B. Probability Distribution of Returns Probability Density

Probability Density

Stock M

Stock W

0 15

Percent

Probability Density

0 15

Percent

rˆM

ˆrw

Year 2004 2005 2006 2007 2008 Average return, rStandard deviation, s

Portfolio WM

0 15

Percent

rˆWM

Stock W (r¨W)

Stock M (r¨M)

Portfolio WM (r¨WM)

40 % 10 35 5 15 15 % 22.6 %

10 % 40 5 35 15 15 % 22.6 %

15% 15 15 15 15 15% 0%

Note: To construct Portfolio WM, 50 percent of the total amount invested is invested in Stock W and 50 percent is invested in Stock M.

W and Stock M individually as well as rates of return for a portfolio invested 50 percent in each stock. The three top graphs show the actual historical returns for each investment from 2004 to 2008, and the lower graphs show the probability distributions of returns, assuming that the future is expected to be like the past. The two stocks would be quite risky if they were held in isolation. When they are combined to form Portfolio WM, however, they are not risky at all. (Note: These stocks are called W and M because their returns graphs in Figure 8-5 resemble a W and an M.) The reason Stocks W and M can be combined to form a risk-free portfolio is because their returns move in opposite directions. That is, when W’s returns are low,

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correlation coefficient, r

A measure of the degree of relationship between two variables.

M’s returns are high, and vice versa. The relationship between any two variables is called correlation, and the correlation coefficient, r, measures the direction and the strength of the relationship between the variables.5 In statistical terms, we say that the returns on Stock W and Stock M are perfectly negatively correlated, with r ¼ 1.0.6 The opposite of perfect negative correlation—that is, r ¼ 1.0—is perfect positive correlation—that is, r ¼ þ1.0. Returns on two perfectly positively correlated stocks would move up and down together, and a portfolio consisting of two such stocks would be exactly as risky as the individual stocks. This point is illustrated in Figure 8-6, in which we see that the portfolio’s standard deviation equals that of the individual stocks. As you can see, there is no diversification effect in this case—that is, risk is not reduced if the portfolio contains perfectly positively correlated stocks. Figure 8-5 and Figure 8-6 demonstrate that when stocks are perfectly negatively correlated (r ¼ 1.0), all risk can be diversified away; conversely, when stocks are perfectly positively correlated (r ¼ þ1.0), diversification is ineffective. In reality, most stocks are positively correlated, but not perfectly so. On average, the correlation coefficient for the returns on two randomly selected stocks would be about þ0.4. For most pairs of stocks, r would lie in the range of þ0.3 to þ0.6. Under such conditions, combining stocks into portfolios reduces risk but does not eliminate it completely. Figure 8-7 illustrates this point with two stocks for which the correlation coefficient is r ¼ þ0.67. Both Stock W and Stock Y have the same average return and standard deviation— r ¼ 15% and s ¼ 22.6%. A portfolio that consists of 50 percent of both stocks has an average return equal to 15.0 percent, which is exactly the same as the average return for each of the two stocks. The portfolio’s standard deviation, however, is 20.6 percent, which is less than the standard deviation of either stock. Thus the portfolio’s risk is not an average of the risks of its individual stocks—diversification has reduced, but not eliminated, risk. From these two-stock portfolio examples, we have seen that risk can be completely eliminated in one extreme case (r ¼ 1.0), whereas diversification does no good in the other extreme case (r ¼ þ1.0). In between these extremes, combining two stocks into a portfolio reduces, but does not eliminate, the riskiness inherent in the individual stocks. What would happen if the portfolio included more than two stocks? As a rule, the riskiness of a portfolio will be reduced as the number of stocks in the portfolio increases. If we added enough stocks, could we completely eliminate risk? In general, the answer is no, but the extent to which adding stocks to a portfolio reduces its risk depends on the degree of correlation among the stocks: The smaller the positive correlation among stocks included in a portfolio, the lower its total risk. If we could find a set of stocks whose correlations were negative, we could eliminate all risk. In the typical case, in which the correlations among the individual stocks are positive but less than þ1.0, some—but not all—risk can be eliminated.

5The correlation coefficient, r, can range from þ1.0 (denoting that the two variables move in the same direction with exactly the same degree of synchronization every time movement occurs) to 1.0 (denoting that the variables always move with the same degree of synchronization, but in opposite directions). A correlation coefficient of zero suggests that the two variables are not related to each other—that is, changes in one variable occur independently of changes in the other. 6Following is the computation of the correlation coefficient that measures the relationship between Stock W and Stock M shown in Figure 8-5. The average return and standard deviation for both stocks are the same: r ¼ 15% and s ¼ 22.6%.

ð40 15Þð 10 15Þ þ ð 10 15Þð40 15Þ þ ð35 15Þð 5 15Þ þ ð 5 15Þ þ ð35 15Þ þ ð15 15Þð15 15Þ 5 1 ¼ 512:5

Covariance ¼

Correlation ¼ r ¼ Covariance/(sWsM) ¼ 512.5/[(22.6)(22.6)] ¼ 1.0

Portfolio Risk—Holding Combinations of Assets

FIGURE 8-6 Rate of Return Distributions for Two Perfectly Positively Correlated Stocks (r ¼ þ1.0) and for Portfolio MM0 A. Rates of Return r¨M Stock M

r¨M′

¨rMM′

Stock M′

2008

Percent

Year 2004 2005 2006 2007 2008 Average return, rStandard deviation, s

2008

B. Probability Distribution of Returns Probability Density Probability Density

0 15 = rˆM

Portfolio MM′

0 15 =rˆM′

Probability Density

Percent

0 15 =rˆp

Stock M (r¨M)

Stock M′ (r¨M′)

Portfolio MM′ (r¨MM′)

10 % 40 5 35 15 15 % 22.6 %

10% 40 5 35 15 15% 22.6%

Percent

Note: To construct Portfolio MM0 , 50 percent of the total amount invested is invested in Stock M and 50 percent is invested in Stock M0 .

To test your understanding, consider the following question: Would you expect to find higher correlations between the returns on two companies in the same industry or in different industries? For example, would the correlation of returns on Ford’s and General Motors’ stocks be higher, or would the correlation coefficient be higher between either Ford or GM and Procter & Gamble (P&G)? How would those correlations affect the risk of portfolios containing them? Answer: Ford’s and GM’s returns have a correlation coefficient of approximately 0.9 with one another because both are affected by the factors that affect auto sales. They have a correlation coefficient of only 0.4 with the returns of P&G.

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FIGURE 8-7 Rate of Return Distributions for Two Partially Correlated Stocks (r ¼ þ0.67) and for Portfolio WY A. Rates of Return r¨W

Stock W

Stock Y

r¨Y

r¨WY

2008

Portfolio WY

2008

B. Probability Distribution of Returns Probability Density Portfolio WY

Stocks W and Y

0 15 = rˆp

Percent

Year

Stock W r¨W

Stock Y ¨rY

Portfolio WY ¨rWY

2004 2005 2006 2007 2008 Average return, rStandard deviation, s

40 % 10 35 5 15 15 % 22.6 %

28 % 20 41 17 3 15 % 22.6 %

34% 5 38 11 9 15% 20.6%

Note: To construct Portfolio WY, 50 percent of the total amount invested is invested in Stock W and 50 percent is invested in Stock Y.

Implications: A two-stock portfolio consisting of Ford and GM would be riskier than a two-stock portfolio consisting of either Ford or GM plus P&G. Thus, to minimize risk, portfolios should be diversified across industries.

Firm-Specific Risk versus Market Risk As noted earlier, it is very difficult—if not impossible—to find stocks whose expected returns are not positively correlated. Most stocks tend to do well when the national

Portfolio Risk—Holding Combinations of Assets

FIGURE 8-8 Effects of Portfolio Size on Portfolio Risk for Average Stocks Portfolio Risk, ␴p (%) 28

Company-Specific, or Diversifiable, Risk (Unsystematic Risk)

σM = 15.0

Minimum Attainable Risk in a Portfolio of Average Stocks 10

Total Risk

Nondiversifiable Risk Related to Market Fluctuations (Systematic Risk)

All NYSE Stocks

economy is strong and to do poorly when it is weak.7 Thus, even very large portfolios end up with substantial amounts of risk, though the risks generally are less than if all of the money was invested in only one stock. To see more precisely how portfolio size affects portfolio risk, consider Figure 8-8. This figure shows how portfolio risk is affected by forming ever-larger portfolios of randomly selected stocks listed on the New York Stock Exchange (NYSE). Standard deviations are plotted for an average one-stock portfolio, for a two-stock portfolio, and so on, up to a portfolio consisting of all common stocks listed on the NYSE. As the graph illustrates, the riskiness of a portfolio consisting of average NYSE stocks generally tends to decline and to approach some minimum limit as the size of the portfolio increases. According to the data, s1, the standard deviation of a one-stock portfolio (or an average stock), is approximately 28 percent. A portfolio consisting of all of the stocks in the market, which is called the market portfolio, would have a standard deviation, sM, of about 15 percent (shown as the horizontal dashed line in Figure 8-8). Figure 8-8 shows that almost half of the riskiness inherent in an average individual stock can be eliminated if the stock is held as part of a reasonably well-diversified 7It is not too difficult to find a few stocks that happened to rise because of a particular set of circumstances in the past while most other stocks were declining. It is much more difficult to find stocks that could logically be expected to go up in the future when other stocks are falling. Payco American, the collection agency discussed earlier, is one of those rare exceptions.

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Chapter 8 Risk and Rates of Return

firm-specific (diversifiable) risk

That part of a security’s risk associated with random outcomes generated by events, or behaviors, specific to the firm. It can be eliminated by proper diversification. market (nondiversifiable) risk

The part of a security’s risk associated with economic, or market, factors that systematically affect firms. It cannot be eliminated by diversification. relevant risk

The portion of a security’s risk that cannot be diversified away; the security’s market risk. It reflects the security’s contribution to the risk of a portfolio.

portfolio—namely, a portfolio containing 40 or more stocks. Some risk always remains, so it is virtually impossible to diversify away the effects of broad stock market movements that affect almost all stocks. That part of the risk of a stock that can be eliminated is called diversifiable, or firm-specific, or unsystematic, risk; that part that cannot be eliminated is called nondiversifiable, or market, or systematic, risk. Although the name given to the risk is not especially important, the fact that a large part of the riskiness of any individual stock can be eliminated through portfolio diversification is vitally important. Firm-specific, or diversifiable, risk is caused by such things as lawsuits, loss of key personnel, strikes, successful and unsuccessful marketing programs, the winning and losing of major contracts, and other events that are unique to a particular firm. Because the actual outcomes of these events are essentially random (unpredictable), their effects on a portfolio can be eliminated by diversification—that is, bad events in one firm will be offset by good events in another. Market, or nondiversifiable, risk, on the other hand, stems from factors that systematically affect all firms, such as war, inflation, recessions, and high interest rates. Because most stocks tend to be affected similarly (negatively) by these market conditions, systematic risk cannot be eliminated by portfolio diversification. We know that investors demand a premium for bearing risk. That is, the riskier a security, the higher the expected return required to induce investors to buy (or to hold) it. However, if investors really are primarily concerned with portfolio risk rather than the risk of the individual securities in the portfolio, how should we measure the riskiness of an individual stock? The answer is this: The relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified portfolio. In other words, the riskiness of General Electric’s stock to a doctor who has a portfolio of 40 stocks or to a trust officer managing a 150-stock portfolio is the contribution that the GE stock makes to the entire portfolio’s riskiness. The stock might be quite risky if held by itself, but if much of this total risk can be eliminated through diversification, then its relevant risk—that is, its contribution to the portfolio’s risk—is much smaller than its total, or stand-alone, risk. A simple example will help clarify this point. Suppose you are offered the chance to flip a coin once. If a head comes up, you win $20,000; if the coin comes up tails, you lose $16,000. This proposition is a good bet: The expected return is $2,000 ¼ 0.5($20,000) þ 0.5( $16,000). It is a highly risky proposition, however, because you have a 50 percent chance of losing $16,000. For this reason, you might refuse to make the bet. Alternatively, suppose you were offered the chance to flip a coin 100 times; you would win $200 for each head but lose $160 for each tail. It is possible that you would flip all heads and win $20,000. It is also possible that you would flip all tails and lose $16,000. The chances are very high, however, that you would actually flip about 50 heads and about 50 tails, winning a net of about $2,000. Although each individual flip is a risky bet, collectively this scenario is a low-risk proposition because most of the risk has been diversified away. This concept underlies the practice of holding portfolios of stocks rather than just one stock. Note that all of the risk associated with stocks cannot be eliminated by diversification: Those risks related to broad, systematic changes in the economy that affect the stock market will remain. Are all stocks equally risky in the sense that adding them to a well-diversified portfolio would have the same effect on the portfolio’s riskiness? The answer is no. Different stocks will affect the portfolio differently, so different securities have different degrees of relevant (systematic) risk. How can we measure the relevant risk of an individual stock? As we have seen, all risk except that related to broad market movements can, and presumably will, be diversified away. After all, why accept risk that we can easily eliminate? The risk that remains after diversifying is market risk

Portfolio Risk—Holding Combinations of Assets

325

(that is, risk that is inherent in the market), and it can be measured by evaluating the degree to which a given stock tends to move up and down with the market.

The Concept of Beta (b) Recall that the relevant risk associated with an individual stock is based on its systematic risk, which in turn depends on the sensitivity of the firm’s operations to economic events such as interest rate changes and inflationary pressures. Because the general movements in the financial markets reflect movements in the economy, we can measure the market risk of a stock by observing its tendency to move with the market or with an average stock that has the same characteristics as the market. The measure of a stock’s sensitivity to market fluctuations is called its beta coefficient, designated with the Greek letter b. An average-risk stock is defined as one that tends to move up and down in step with the general market as measured by some index, such as the Dow Jones Industrial Average, the S&P 500 Index, or the New York Stock Exchange Composite Index. Such a stock will, by definition, have a beta (b) of 1.0. This value indicates that, in general, if the market moves up by 10 percent, the stock price will also increase by 10 percent; if the market falls by 10 percent, the stock price will decline by 10 percent. A portfolio composed of such b ¼ 1.0 stocks will move up and down with the broad market averages, and it will be just as risky as the averages. If b ¼ 0.5, the stock’s relevant (systematic) risk is only half as volatile as the market, and a portfolio of such stocks will be half as risky as a portfolio that includes only b ¼ 1.0 stocks—it will rise and fall only half as much as the market. If b ¼ 2.0, the stock’s relevant risk is twice as volatile as an average stock, so a portfolio of such stocks will be twice as risky as an average portfolio. The value of such a portfolio could double—or halve—in a short period of time. If you held such a portfolio, you could quickly become a millionaire—or a pauper. Figure 8-9 graphs the relative volatility of three stocks. The data below the graph assume that in 2006 the ‘‘market,’’ defined as a portfolio consisting of all stocks, had a total return (dividend yield plus capital gains yield) of rM ¼ 14%, and Stocks H, A, and L (for high, average, and low risk) also had returns of 14 percent. In 2007 the market rose sharply, and the return on the market portfolio was rM ¼ 28%. Returns on the three stocks also increased: the return on H soared to 42 percent; the return on A reached 28 percent, the same as the market; and the return on L increased to only 21 percent. In 2008 the market dropped, with the market return falling to rM ¼ 14%. The three stocks’ returns also fell, H plunging to 42 percent, A falling to 14 percent, and L declining to 0 percent. As you can see, all three stocks moved in the same direction as the market, but H was by far the most volatile; A was just as volatile as the market; and L was less volatile than the market. The beta coefficient measures a stock’s volatility relative to an average stock (or the market), which has b ¼ 1.0. We can calculate a stock’s beta by plotting a line like those shown in Figure 8-9. The slopes of these lines show how each stock moves in response to a movement in the general market. Indeed, the slope coefficient of such a ‘‘regression line’’ is defined as a beta coefficient. Betas for literally thousands of companies are calculated and published by Merrill Lynch, Value Line, and numerous other organizations. Table 8-4 provides the beta coefficients for some well-known companies. Most stocks have betas in the range of 0.50 to 1.50, and the average for all stocks is 1.0 by definition.8

8In theory, betas can be negative. For example, if a stock’s returns tend to rise when those of other stocks decline, and vice versa, then the regression line in a graph such as Figure 8-9 will have a downward slope, and the beta will be negative. Note, however, that few stocks have negative betas. Payco American, the collection agency company, might have a negative beta.

beta coefficient, b

A measure of the extent to which the returns on a given stock move with the stock market.

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Chapter 8 Risk and Rates of Return

FIGURE 8-9 Relative Volatility of Stocks H, A, and L Return on Stock j, r¨i(%) 50 Stock H, High Risk: β = 2.0

Stock A, Average Risk: β = 1.0 Stock L, Low Risk: β = 0.5

20 14 10

Return on the Market, ¨rM (%) 10

20 30 40

2006 2007 2008

r¨H

¨rA

¨rL

r¨M

14 42 42

14 28 14

14 21 0

14 28 14

If we add a higher-than-average-beta stock (b > 1.0) to an average-beta (b ¼ 1.0) portfolio, then the beta, and consequently the riskiness, of the portfolio will increase. Conversely, if we add a lower-than-average-beta stock (b < 1.0) to an average-risk portfolio, the portfolio’s beta and risk will decline. Thus, because a stock’s beta measures its contribution to the riskiness of a portfolio, theoretically beta is the correct measure of the stock’s riskiness. We can summarize our discussion to this point as follows: 1. A stock’s risk consists of two components: market risk and firm-specific risk. 2. Firm-specific risk can be eliminated through diversification. Most investors do diversify, either by holding large portfolios or by purchasing shares in mutual funds. We are left, then, with market risk, which is caused by general movements in the stock market and which reflects the fact that most stocks

Portfolio Risk—Holding Combinations of Assets

327

TABLE 8-4 Beta Coefficients for Selected Companies Company I. Above Average Market Risk: b > 1.0 Nortel Networks Corporation Yahoo! Inc. E TRADE Group Inc. Sun Microsystems eBay II. Average Market Risk: b 1.0 Dow Jones & Company Ryland Group Scotts Corporation Krispy Kreme Toyota Motor Corporation III. Below Average Market Risk: b < 1.0 Barnes & Noble Kroger Company Walgreen Company Gillette Company Progress Energy

Beta

Industry/Product

4.18 3.40 2.87 2.80 1.76

Communications equipment; telephone equipment Computer services/global Internet communications Investment services/online financial services Computers and peripherals Retail (specialty nonapparel)/web-based auction

1.02 1.01 0.99 0.99 0.99

Publishing and printing (newspapers) Home building Pesticide, fertilizer, and agricultural chemicals Snack and nonalcoholic beverage bars Auto and truck manufacturer

0.75 0.50 0.28 0.28 0.17

Specialty retailing; bookstores Food retailing; supermarkets Retail drugs; pharmacies and drugstores Personal and household products Electric utilities; electric power generation

Source: Standard & Poor’s Research Insight, 2006.

are systematically affected by major economic events such as war, recessions, and inflation. Market risk is the only risk that is relevant to a rational, diversified investor because he or she should already have eliminated firmspecific risk. 3. Investors must be compensated for bearing risk. That is, the greater the riskiness of a stock, the higher its required return. Such compensation is required only for risk that cannot be eliminated by diversification. If risk premiums existed on stocks with high diversifiable risk, well-diversified investors would start buying these securities and bidding up their prices, and their final (equilibrium) expected returns would reflect only nondiversifiable market risk. An example might help clarify this point. Suppose half of Stock A’s risk is market risk (it occurs because Stock A moves up and down with the market). The other half of Stock A’s risk is diversifiable. You hold only Stock A, so you are exposed to all of its risk. As compensation for bearing so much risk, you want a risk premium of 8 percent higher than the 5 percent Treasury bond rate. That is, you demand a return of 13 percent (¼ 5% þ 8%) from this investment. But suppose other investors, including your professor, are well diversified; they also hold Stock A, but they have eliminated its diversifiable risk and thus are exposed to only half as much risk as you are. Consequently, their risk premium will be only half as large as yours, and they will require a return of only 9 percent (¼ 5% þ 4%) to invest in the stock. If the stock actually yielded more than 9 percent in the market, other investors, including your professor, would buy it. If it yielded the 13 percent you demand, you would be willing to buy the stock, but the well-diversified investors would compete with you for its acquisition.

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They would bid its price up and its yield down, which would keep you from getting the stock at the return you need to compensate you for taking on its total risk. In the end, you would have to accept a 9 percent return or else keep your money in the bank. Thus, risk premiums in a market populated with rational investors—that is, those who diversify—will reflect only market risk. 4. The market (systematic) risk of a stock is measured by its beta coefficient, which is an index of the stock’s relative volatility. Some benchmark values for beta follow: b ¼ 0.5: The stock’s relevant risk is only half as volatile, or risky, as the average stock. b ¼ 1.0: The stock’s relevant risk is of average risk. b ¼ 2.0: The stock’s relevant risk is twice as volatile as the average stock. 5. Because a stock’s beta coefficient determines how the stock affects the riskiness of a diversified portfolio, beta ( ) is a better measure of a stock’s relevant risk than is standard deviation ( ), which measures total, or standalone, risk.

Portfolio Beta Coefficients A portfolio consisting of low-beta securities will itself have a low beta because the beta of any set of securities is a weighted average of the individual securities’ betas:

8–8

Portfolio beta ¼ bp ¼ w1 b1 þ w2 b2 þ þ wN bN ¼

N X

w j bj

j¼1

Here bP, the beta of the portfolio, reflects how volatile the portfolio is in relation to the market; wj is the fraction of the portfolio invested in the jth stock; and bj is the beta coefficient of the jth stock. For example, if an investor holds a $105,000 portfolio consisting of $35,000 invested in each of three stocks, and each of the stocks has a beta of 0.7, then the portfolio’s beta will be bP1¼ 0.7: bP1 ¼ ð1=3Þð0:7Þ þ ð1=3Þð0:7Þ þ ð1=3Þð0:7Þ ¼ 0:7 Such a portfolio will be less risky than the market, which means it should experience relatively narrow price swings and demonstrate relatively small rateof-return fluctuations. When graphed in a fashion similar to Figure 8-9 , the slope of its regression line would be 0.7, which is less than that for a portfolio of average stocks. Now suppose one of the existing stocks is sold and replaced by a stock with bj ¼ 2.5. This action will increase the riskiness of the portfolio from bP1¼ 0.7 to bP2 ¼ 1.3: bP2 ¼ ð1=3Þð0:7Þ þ ð1=3Þð0:7Þ þ ð1=3Þð2:5Þ ¼ 1:3 Had a stock with bj ¼ 0.4 been added, the portfolio beta would have declined from 0.7 to 0.6. Adding a low-beta stock, therefore, would reduce the riskiness of the portfolio.

The Relationship Between Risk and Rates of Return (CAPM)

329

Explain the following statement: ‘‘A stock held as part of a portfolio is generally less risky than the same stock held in isolation.’’ What is meant by perfect positive correlation, by perfect negative correlation, and by zero correlation? In general, can we reduce the riskiness of a portfolio to zero by increasing the number of stocks in the portfolio? Explain. What is meant by diversifiable risk and nondiversifiable risk? What is an average-risk stock? Why is beta the theoretically correct measure of a stock’s riskiness? If you plotted the returns on a particular stock versus those on the Dow Jones Industrial Average index over the past five years, what would the slope of the line you obtained indicate about the stock’s risk? Suppose you have a portfolio that includes two stocks. You invested 60 percent of your total funds in a stock that has a beta equal to 3.0 and the remaining 40 percent of your funds in a stock that has a beta equal to 0.5. What is the portfolio’s beta? (Answer: 2.0)

THE RELATIONSHIP BETWEEN RISK OF RETURN (CAPM)

AND

RATES

In the preceding section, we saw that beta is the appropriate measure of a stock’s relevant risk. Now we must specify the relationship between risk and return. For a given level of beta, what rate of return will investors require on a stock to compensate them for assuming the risk? To determine an investment’s required rate of return, we use a theoretical model called the Capital Asset Pricing Model (CAPM). The CAPM shows how the relevant risk of an investment as measured by its beta coefficient is used to determine the investment’s appropriate required rate of return. Let’s begin by defining the following terms: ^rj ¼ Expected rate of return on the jth stock; is based on the probability distribution for the stock’s returns. rj ¼ Required rate of return on the jth stock; rj is the rate that investors demand for investing in Stock j. If ^rj < rj, you would not purchase this stock, or you would sell it if you owned it; if ^rj > rj, you would want to buy the stock; and, you would be indifferent if ^rj ¼ rj. rRF ¼ Risk-free rate of return. In this context, rRF is generally measured by the return on long-term U.S. Treasury securities. bj ¼ Beta coefficient of the jth stock. The beta of an average stock is bA ¼ 1.0. rM ¼ Required rate of return on a portfolio consisting of all stocks, which is the market portfolio. rM is also the required rate of return on an average (bA ¼ 1.0) stock.

Capital Asset Pricing Model (CAPM)

A model used to determine the required return on an asset, which is based on the proposition that any asset’s return should be equal to the risk-free return plus a risk premium that reflects the asset’s nondiversifiable risk.

330

Chapter 8 Risk and Rates of Return

RPM ¼ (rM rRF) ¼ Market risk premium. This is the additional return above the risk-free rate required to compensate an average investor for assuming an average amount of risk (bA ¼ 1.0). RPj ¼ (rM rRF) bj ¼ Risk premium on the jth stock ¼ (RPM) bj. The stock’s risk premium is less than, equal to, or greater than the premium on an average stock, depending on whether its relevant risk as measured by beta is less than, equal to, or greater than an average stock, respectively. If bj ¼ bA ¼ 1.0, then RPj ¼ RPM; if bj > 1.0, then RPj > RPM; and, if bj < 1.0, then RPj < RPM. market risk premium (RPM)

The additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.

8–9

The market risk premium (RPM) depends on the degree of aversion that investors on average have to risk.9 Let’s assume that at the current time, Treasury bonds yield rRF ¼ 5% and an average share of stock has a required return of rM ¼ 11%. In this case, the market risk premium is 6 percent: RPM ¼ rM rRF ¼ 11% 5% ¼ 6% It follows that if one stock is twice as risky as another, its risk premium should be twice as high. Conversely, if a stock’s relevant risk is only half as much as that of another stock, its risk premium should be half as large. Furthermore, we can measure a stock’s relevant risk by finding its beta coefficient. Therefore, if we know the market risk premium, RPM, and the stock’s risk as measured by its beta coefficient, bj, we can find its risk premium as the product RPM bj. For example, if bj ¼ 0.5 and RPM ¼ 6%, then RPj is 3 percent: Risk premium for stock j ¼ RPM bj ¼ 6% 0:5 ¼ 3:0% As Figure 8-4 shows, the required return for any investment j can be expressed in general terms as

8–10

Required return ¼ Risk-free return þ Premium for risk rj ¼ rRF þ RPj Based on our previous discussion, Equation 8–10 can also be written as

8–11

rj ¼ rRF þ ðRPM Þbj ¼ Capital Assest Pricing Model ðCAPMÞ ¼ rRF þ ðrM rRF Þbj ¼ 5% þ ð11% 5%Þð0:5Þ ¼ 5% þ 6%ð0:5Þ ¼ 8% 9This concept, as well as other aspects of CAPM, is discussed in more detail in Chapter 3 of Eugene F. Brigham and Phillip R. Daves, Intermediate Financial Management, 9th ed. (Cincinnati, OH: South-Western College Publishing, 2007). Note that we cannot measure the risk premium of an average stock, RPM ¼ rM rRF, with great precision because we cannot possibly obtain precise values for the expected future return on the market, rM. Empirical studies suggest that where long-term U.S. Treasury bonds are used to measure rRF and where rM is an estimate of the expected return on the S&P 500, the market risk premium varies somewhat from year to year. It has generally ranged from 4 to 8 percent during the past 20 years. Chapter 3 of Intermediate Financial Management also discusses the assumptions embodied in the CAPM framework. Some of the assumptions of the CAPM theory are unrealistic. As a consequence, the theory does not hold exactly.

The Relationship Between Risk and Rates of Return (CAPM)

331

FIGURE 8-10 The Security Market Line (SML) Required Rate of Return (%)

SML: ri = rRF (rM rRF)βi

rHigh = 17

Relatively Risky Stock’s Risk Premium: 12%

rM = rA = 11

Safe Stock Risk Premium: 3%

rLow = 8

Market (Average Stock) Risk Premium: 6%

rRF = 5 Risk-Free Rate: 5% 0

0.5

1.0

1.5

2.0

Equation 8–11, which is the CAPM equation for equilibrium pricing, is called the security market line (SML). If some other stock were riskier than Stock j and had bj2 ¼ 2.0, then its required rate of return would be 17 percent: rj2 ¼ 5% þ ð6%Þ2:0 ¼ 17% An average stock, with b ¼ 1.0, would have a required return of 11 percent, the same as the market return: rA ¼ 5% þ ð6%Þ1:0 ¼ 11% ¼ rM Equation 8–11 (the SML equation) is often expressed in graph form. Figure 8-10, for example, shows the SML when rRF ¼ 5% and rM ¼ 11%. Note the following points: 1. Required rates of return are shown on the vertical axis, and risk (as measured by beta) is shown on the horizontal axis. This graph is quite different from the one shown in Figure 8-9, in which the returns on individual stocks are plotted on the vertical axis and returns on the market index are shown on the horizontal axis. The slopes of the three lines in Figure 8-9 represent the three stocks’ betas. In Figure 8-10, these three betas are plotted as points on the horizontal axis. 2. Risk-free securities have bj ¼ 0; therefore, rRF appears as the vertical axis intercept in Figure 8-10. 3. The slope of the SML reflects the degree of risk aversion in the economy. The greater the average investor’s aversion to risk, (a) the steeper the slope of the line, (b) the greater the risk premium for any stock, and (c) the higher the

Risk, βj

security market line (SML)

The line that shows the relationship between risk as measured by beta and the required rate of return for individual securities.

332

Chapter 8 Risk and Rates of Return

FIGURE 8-11 Shift in the SML Caused by a 2 Percent Increase in Inflation Required Rate of Return (%) SML2 SML1

rM2 = 13 rM1 = 11

rRF2 = 7 rRF1 = 5

Increase in Inflation = 2% Original IP = 2%

r* = 3 Real Risk-Free Rate = 3% 0

0.5

1.0

1.5

2.0

Risk, βj

required rate of return on stocks.10 These points are discussed further in a later section. 4. The values we worked out for stocks with bj ¼ 0.5, bj ¼ 1.0, and bj ¼ 2.0 agree with the values shown on the graph for rLow, rA, and rHigh. Both the SML and a company’s position on it change over time because of changes in interest rates, investors’ risk aversion, and individual companies’ betas. Such changes are discussed in the following sections.

The Impact of Inflation As we learned in Chapter 5, interest amounts to ‘‘rent’’ on borrowed money, or the price of money. In essence, then, rRF is the price of money to a risk-free borrower. We also learned in Chapter 5 that the risk-free rate as measured by the rate on U.S. Treasury securities is called the nominal, or quoted, rate, and it consists of two elements: (1) a real inflation-free rate of return, r , and (2) an inflation premium, IP, equal to the anticipated average rate of inflation.11 Thus, rRF¼ r þ IP. If the expected rate of inflation rose by 2 percent, rRF would also increase by 2 percent. Figure 8-11 illustrates the effects of such a change. Notice that under the CAPM, the increase in rRF also causes an equal increase in the rate of return on all 10Students sometimes confuse beta with the slope of the SML. This is a mistake. The slope of any line is equal to the ‘‘rise’’ divided by the ‘‘run,’’ or (y1 – y0)/(x1 – x0). Consider Figure 8-10. If we let y ¼ r and x ¼ b, and we go from the origin to bM ¼ 1.0, we see that the slope is (rM – rRF)/(bM – bRF) ¼ (11% – 5%)/(1 – 0) ¼ 6%. Thus, the slope of the SML is equal to (rM – rRF), the market risk premium. In Figure 8-10, rj ¼ 5% þ (6%)bj, so a doubling of beta (for example, from 1.0 to 2.0) would produce an 8-percentage-point increase in rj. In this case, the total risk premium on Stock j would double—that is, RPj ¼ (8%)2.0 ¼ 16%.

Treasury bonds also contain a maturity risk premium (MRP). Here we include the MRP in r to simplify the discussion.

11Long-term

The Relationship Between Risk and Rates of Return (CAPM)

FIGURE 8-12 Shift in the SML Caused by Increased Risk Aversion Required Rate of Return (%) 21

SML2

SML1

rM2 = 13 rM1 = 11 New Market Risk Premium, rM2 rRF 8%

9 8

rRF = 5

Original Market Risk Premium, rM1 rRF 6%

0.5

1.0

1.5

2.0

Risk, βj

risky assets because the inflation premium is built into the required rate of return of both risk-free and risky assets.12 For example, the risk-free return increases from 5 percent to 7 percent, and the rate of return on an average stock, rM, increases from 11 percent to 13 percent. Thus, all securities’ returns increase by 2 percentage points.

Changes in Risk Aversion The slope of the security market line reflects the extent to which investors are averse to risk. The steeper the slope of the line, the greater the average investor’s risk aversion. If investors were indifferent to risk, and if rRF was 5 percent, then risky assets would also provide an expected return of 5 percent. If there was no risk aversion, there would be no risk premium, so the SML would be horizontal. As risk aversion increases, so does the risk premium and, therefore, so does the slope of the SML. Figure 8-12 illustrates an increase in risk aversion. In this case, the market risk premium increases from 6 percent to 8 percent, and rM increases from rM1 ¼ 11% to rM2¼ 13%. The returns on other risky assets also rise, with the effect of this shift in risk aversion being more pronounced on riskier securities. For example, the required return on a stock with bj ¼ 0.5 increases by only 1 percentage point, from 8 percent to 9 percent. By comparison, the required return on a stock with bj ¼ 2.0 increases by

12Recall that the inflation premium for any asset is equal to the average expected rate of inflation over the life of the asset. In this analysis, we must therefore assume either that all securities plotted on the SML graph have the same life or that the expected rate of future inflation is constant. Also note that rRF in a CAPM analysis can be proxied by either a long-term rate (the T-bond rate) or a short-term rate (the T-bill rate). Traditionally, the T-bill rate was used, but a movement toward use of the T-bond rate has occurred in recent years because a closer relationship exists between T-bond yields and stocks than between T-bill yields and stocks. See Stocks, Bonds, Bills, and Inflation, 2006 Yearbook (Chicago: Ibbotson & Associates, 2007) for a discussion.

333

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Chapter 8 Risk and Rates of Return

4 percentage points, from 17 percent to 21 percent. Because RPj ¼ RPM(bj) ¼ (13% – 11%)bj ¼ (2%)bj, the changes in these risk premiums are computed as follows: 1. if bj ¼ 0:5; DRPj ¼ ð2%Þ0:5 ¼ 1% 2. if bj ¼ 2:0; DRPj ¼ ð2%Þ2:0 ¼ 4% Thus, when the average investor’s aversion to risk changes, investments with higher beta coefficients experience greater changes in their required rates of return than investments with lower betas.

Changes in a Stock’s Beta Coefficient As we will see later in this book, a firm can affect its beta risk by changing the composition of its assets and by modifying its use of debt financing. External factors, such as increased competition within a firm’s industry or the expiration of basic patents, can also alter a company’s beta. When such changes occur, the required rate of return, r, changes as well, and, as we saw in Chapter 7, this change will affect the price of the firm’s stock. For example, consider Genesco Manufacturing, with a beta equal to 1.0. Suppose some action occurred that caused this firm’s beta to increase from 1.0 to 1.5. If the conditions depicted in Figure 8-10 held, Genesco’s required rate of return would increase from r1 ¼ rRF þ ðrM rRF Þbj ¼ 5% þ ð11% 5%Þ1:0 ¼ 11% to r2 ¼ 5% þ ð11% 5%Þ1:5 ¼ 14% Any change that affects the required rate of return on a security, such as a change in its beta coefficient or in expected inflation, will affect the price of the security.

A Word of Caution A word of caution about betas and the CAPM is in order here. First, the model was developed under very restrictive assumptions. Some of the assumptions include the following: (1) all investors have the same information, which leads to the same expectations about future stock prices; (2) everyone can borrow and lend at the risk-free rate of return; (3) stocks (or any other security) can be purchased in any denomination or fraction of shares; and (4) taxes and transaction costs (commissions) do not exist. Second, the entire theory is based on ex ante, or expected, conditions, yet we have available only ex post, or past, data. The betas we calculate show how volatile a stock has been in the past, but conditions could certainly change. The stock’s future volatility, which is the item of real concern to investors, might therefore differ quite dramatically from its past volatility. Although the CAPM represents a significant step forward in security pricing theory, it does have some potentially serious deficiencies when applied in practice. As a consequence, estimates of rj found through use of the SML might be subject to considerable error. For this reason, many investors and analysts use the CAPM and the concept of b to provide ‘‘ballpark’’ figures for further analysis. The concept that investors should be rewarded only for taking relevant risk makes sense. And the CAPM provides an easy way to get a rough estimate of the relevant risk and the appropriate required rate of return of an investment.

Stock Market Equilibrium

Differentiate between the expected rate of return ( ^r ) and the required rate of return (r) on a stock. Which would have to be larger to persuade you to buy the stock? What are the differences between the relative volatility graph (Figure 8-9 ), in which ‘‘betas are made,’’ and the SML graph (Figure 8-10 ), in which ‘‘betas are used’’? Consider the methods of constructing the graphs and the purposes for which they were developed. What happens to the SML graph (1) when inflation increases or (2) when inflation decreases? What happens to the SML graph (1) when risk aversion increases or (2) when risk aversion decreases? What would the SML look like if investors were indifferent to risk—that is, had zero risk aversion? How can a firm influence its market, or beta, risk? Stock F has a beta coefficient equal to 1.2. If the risk-free rate of return equals 4 percent and the expected market return equals 10 percent, what is Stock F’s required rate of return? (Answer: rF ¼ 11.2%)

STOCK MARKET EQUILIBRIUM Based on our previous discussion, we know that we can use the CAPM to find the required return for an investment (say, Stock Q), which we designate as rQ. Suppose the risk-free return is 5 percent, the market risk premium is 6 percent, and Stock Q has a beta of 1.5 (bQ ¼ 1.5). In this case, the marginal, or average, investor will require a return of 14 percent on Stock Q: rA ¼ 5% þ 6%ð1:5Þ ¼ 14% This 14 percent return is shown as a point Q on the SML in Figure 8-13. The average investor will want to buy Stock Q if the expected rate of return exceeds 14 percent, will want to sell it if the expected rate of return is less than 14 percent, and will be indifferent (and therefore will hold but not buy or sell Stock Q) if the expected rate of return is exactly 14 percent. Now suppose the investor’s portfolio contains Stock Q, and he or she analyzes the stock’s prospects and concludes that its earnings, dividends, and price can be expected to grow at a constant rate of 4 percent per year forever. The last dividend paid was D0 ¼ $3, so the next expected dividend is ^ 1 ¼ $3:00ð1:04Þ ¼ $3:12 D Our ‘‘average’’ (marginal) investor observes that the current price of the stock, P0, is $34.67. Should he or she purchase more of Stock Q, sell the current holdings, or maintain the current position? Recall from Chapter 7 that we can calculate Stock Q’s expected rate of return as follows (see Equation 7–6): ^rQ ¼

^1 $3:12 D þ 0:04 ¼ 0:09 þ 0:04 ¼ 0:13 ¼ 13% þg¼ P0 $34:67

335

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Chapter 8 Risk and Rates of Return

FIGURE 8-13 Expected and Required Returns on Stock Q Required Rate of Return (%)

SML: ri rRF (rM rRF)βi

rQ 14 rˆQ 13

rM 11

rRF 5

equilibrium

The condition under which the expected return on a security is just equal to its required return,^r ¼ r, and the price is stable.

1.0

1.5

Risk, βj

This value is plotted on Figure 8-13 as Point Q, which is below the SML. Because the expected rate of return, ^rQ ¼ 13%, is less than the required return, rQ ¼ 14%, this marginal investor would want to sell the stock, as would other holders. Because few people would want to buy at the $34.67 price, the current owners would be unable to find buyers unless they cut the price of the stock. The price would therefore decline, and this decline would continue until the stock’s price reaches $31.20. At that point, the market for this security would be in equilibrium because the expected rate of return, 14 percent, would be equal to the required rate of return: ^rQ ¼

$3:12 þ 0:04 ¼ 0:10 þ 0:04 ¼ 0:14 ¼ 14% $31:20

Had the stock initially sold for less than $31.20—say, $28.36—events would have been reversed. Investors would have wanted to buy the stock because its expected rate of return ( ^r ¼ 15%) would have exceeded its required rate of return, and buy orders would have driven the stock’s price up to $31.20. To summarize, two conditions must hold in equilibrium: 1. The expected rate of return as seen by the marginal investor must equal the required rate of return: ^rj ¼ rj. 2. The actual market price of the stock must equal its intrinsic value as esti^0 . mated by the marginal investor: P0 ¼ P ^ 0 > P0 hence Of course, some individual investors might believe that ^rj > rj and P they would invest most of their funds in the stock. Other investors might ascribe to the opposite view and sell all of their shares. Nevertheless, it is the marginal investor who ^ 0 : If these establishes the actual market price. For this investor, ^rj ¼ rj and P0 ¼ P conditions do not hold, trading will occur until they do hold.

Different Types of Risk

When a stock is in equilibrium, what two conditions must hold? If a stock is not in equilibrium, explain how financial markets adjust to bring it into equilibrium. Suppose Porter Pottery’s stock currently sells for $32. The company, which is growing at a constant rate, expects its next dividend to equal $3.20. Analysts have determined that the market value of the stock is currently in equilibrium and that investors require a rate of return equal to 14 percent to purchase the stock. If the price of the stock increases to $35.56 tomorrow after Porter’s yearend financial statements are made public, what is the stock’s expected return? Assume that the company’s growth rate remains constant. (Answer: 13%)

DIFFERENT TYPES

RISK

In Chapter 5, we introduced the concept of risk in our discussion of interest rates, or the cost of money. At that point, we stated that the nominal, or quoted, rate of return, r, can be written as follows: Rate of return ðinterestÞ ¼ r ¼ Risk-free rate þ Risk premium þ RP ¼ rRF ¼

½r þ IP

þ ½DRP þ LP þ MRP

Remember that here r ¼ Quoted, or nominal, rate of interest on a given security. There are many different securities, hence many different quoted interest rates. rRF ¼ Nominal risk-free rate of return. r ¼ Real risk-free rate of interest, which is the interest rate that would exist on a security with a guaranteed payoff if inflation is expected to be zero during the investment period. IP ¼ Inflation premium, which equals the average inflation rate expected over the life of the security. DRP ¼ Default risk premium, which reflects the chance that the borrower will not pay the debt’s interest or principal on time. LP ¼ Liquidity, or marketability, premium, which reflects the fact that some investments are more easily converted into cash on a short notice at a ‘‘reasonable price’’ than are other securities. MRP ¼ Maturity risk premium, which accounts for the fact that longer-term bonds experience greater price reactions to interest rate changes than do short-term bonds. The discussion in Chapter 5 presented an overall view of interest rates and general factors that affect these rates. But we did not discuss risk evaluation in detail; rather, we described some of the factors that determine the total risk associated with debt, such as default risk, liquidity risk, and maturity risk. In reality, these risks also affect other types of investments, including equity. Equity does not represent a legal contract that requires the firm to pay defined amounts of dividends at particular times or to ‘‘act’’ in specific ways. There is, however, an expectation that positive returns will be generated

337

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Chapter 8 Risk and Rates of Return

through future distributions of cash because dividends will be paid, capital gains will be generated through growth, or both. Investors also expect the firm to behave ‘‘appropriately.’’ If these expectations are not met, investors generally consider the firm in ‘‘default’’ of their expectations. In such cases, as long as no laws have been broken, stockholders generally do not have legal recourse, as would be the case for a default on debt. As a result, investors penalize the firm by selling their stock holdings, which causes the value of the firm’s stock to decline. In this chapter, we build on the general concept that was introduced in Chapter 5 by showing how the risk premium associated with any investment should be determined (at least in theory). The basis of our discussion is Equation 5–3, which we develop further in this chapter as follows: rj ¼ Risk-free rate þ Risk premium ¼

þ ðrM rRF Þbj ¼ CAPM

rRF

According to the CAPM, investors should not expect to be rewarded for all of the risk associated with an investment—that is, its total, or stand-alone, risk—because some risk can be eliminated through diversification. The relevant risk, and thus the risk for which investors should be compensated, is that portion of the total risk that cannot be ‘‘diversified away.’’ Thus, in this chapter we show the following: Total risk ¼ s ¼ Systematic risk

þ Unsystematic risk

¼ MarketðeconomicÞrisk þ Firm-specific risk ¼ Nondiversifiable risk þ Diversifiable risk ¼ Cannot be eliminated þ Can be eliminated Relevant risk ¼ Nondiversifiable risk þ Diversifiable risk ðeliminatedÞ ¼ Systematic risk Systematic risk is represented by an investment’s beta coefficient, b, in Equation 8–11. The specific types and sources of risk to which a firm or an investor is exposed are numerous and vary considerably depending on the situation. A detailed discussion of all the different types of risks and the techniques used to evaluate risks is beyond the scope of this book. But you should recognize that risk is an important factor in the determination of the required rate of return (r), which, according to the following equation, is one of the two variables we need to determine the value of an asset: Value ¼

^ 1 CF ð1 þ rÞ

^ 2 CF ð1 þ rÞ

þ þ

n X ^ n ^ t CF CF n ¼ t ð1 þ rÞ t¼1 ð1 þ rÞ

This equation was first introduced in Chapter 1, and it was discussed in greater detail in Chapter 6 and Chapter 7. What is important to understand here is that the value of an asset, which could be a stock or a bond, is based on the cash flows that the asset is expected to generate during its life and the rate of return investors require to ‘‘put up’’ their money to purchase the investment. In this chapter, we provide you with an indication as to how the required rate of return, r, should be determined, and we show that investors demand higher rates of return to compensate them for taking greater amounts of ‘‘relevant’’ risks. Because it is an important concept and has a direct effect on value, we will continue to discuss risk in the remainder of the book. Although there are instances in which the discussions focus on the risk to which investors are exposed, most of the discussions

Different Types of Risk

339

TABLE 8-5 Different Types (Sources) of Risk General Type of Risk

Name of Risk

Brief Description

I. Systematic risks Interest rate risk (nondiversifiable risk; market risk; relevant risk) Inflation risk

When interest rates change, (1) the values of investments change (in opposite directions) and (2) the rate at which funds can be reinvested also changes (in the same direction). The primary reason short-term interest rates change is because investors change their expectations about future inflation. Maturity risk Long-term investments experience greater price reactions to interest rate changes than do short-term bonds. Liquidity risk Reflects the fact that some investments are more easily converted into cash on a short notice at a ‘‘reasonable price’’ than are other securities. Exchange rate risk Multinational firms deal with different currencies; the rate at which the currency of one country can be exchanged into the currency of another country—that is, the exchange rate—changes as market conditions change. Political risk Any action by a government that reduces the value of an investment.

II. Unsystematic risks (diversifiable risk; firm-specific risk)

Business risk

Financial risk Default risk III. Combined risks (some systematic risk and some unsystematic risk)

Total risk

Corporate risk

Risk that would be inherent in the firm’s operations if it used no debt—factors such as labor conditions, product safety, quality of management, competitive conditions, and so forth, affect firmspecific risk. Risk associated with how the firm is financed—that is, its credit risk. Part of financial risk—the chance that the firm will not be able to service its existing debt. The combination of systematic risk and unsystematic risk; also referred to as stand-alone risk, because this is the risk an investor takes if he or she purchases only one investment, which is tantamount to ‘‘putting all your eggs into one basket.’’ The riskiness of the firm without considering the effect of stockholder diversification; based on the combination of assets held by the firm (inventory, accounts receivable, plant and equipment, and so forth). Some diversification exists because the firm’s assets represent a portfolio of investments in real assets.

focus on risks that affect corporations. Because we discuss different types of risk throughout the book, we thought it might be a good idea to summarize and describe these risks in brief terms. Table 8-5 shows the risks that are discussed in the book and indicates whether each risk is considered a component of systematic (nondiversifiable) or unsystematic (diversifiable) risk. Note that (1) this table oversimplifies risk analysis because some risks are not easily classified as either systematic or unsystematic, and (2) some of the risks included in the table will be discussed later in the book. Even so, this table should show the relationships among the different risks discussed in the book.

Classify default risk, maturity risk, and liquidity risk as either diversifiable or nondiversifiable risk.

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Chapter 8 Risk and Rates of Return

Chapter Essentials —The Answers

To summarize the key concepts, let’s answer the questions that were posed at the beginning of the chapter:

What does it mean to take risk when investing? In finance, risk is defined as the chance of receiving a return other than the one that is expected. Thus, an investment is considered risky if more than one outcome (payoff) is possible. Every risky investment has both ‘‘bad’’ risk—that is, the chance that it will return less than expected—and ‘‘good risk’’—that is, the chance that it will return more than expected. In simple terms, risk can be defined (described) using one word: variability.

How are the risk and return of an investment measured? How are the risk and return of an investment related? An investment’s risk is measured by the variability of its possible payoffs (returns). Greater variability in returns indicates greater risk. Investors require higher returns to take on greater risks. Thus, generally speaking, investments with greater risks also have higher returns. The expected return of an investment is measured as a weighted average of all of the possible returns the investment can generate in the future, with the weights being the probability that the particular return will occur.

How can investors reduce risk? Risk can be reduced through diversification. Investors achieve diversification by forming portfolios that contain numerous financial securities (perhaps stocks and bonds) that are not strongly related to each other. For example, an investor can form a well-diversified portfolio by purchasing the stocks of 40 or more companies in different industries, such as transportation, utilities, health care, entertainment, food services, and so forth. Total risk, which is equal to market (systematic) risk plus firm-specific (unsystematic) risk, can be reduced through diversification because little or no unsystematic risk should exist in a well-diversified investment portfolio. For what type of risk is an average investor rewarded? Investors should be rewarded only for risk that they must take. Because firm-specific, or unsystematic, risk can be reduced or eliminated through diversification, investors who do not diversify their investment portfolios should not be rewarded for taking such risk. Consequently, an investment’s relevant risk is its systematic, or market, risk, which is the risk for which investors should be rewarded. Systematic risk cannot be reduced through diversification. An investment’s ‘‘irrelevant’’ risk is its firm-specific, or unsystematic, risk because it is this portion of the total risk that can be eliminated (at least theoretically) through diversification.

What actions do investors take when the return they require to purchase an investment is different from the return the investment is expected to produce? Investors will purchase a security only when its expected return, ^r, is greater than its required return, r. When ^r < r, investors will not purchase the security and those who own the security tend to sell it, which causes the security’s price to decrease and its expected return to increase until ^r ¼ r.

ETHICAL DILEMMA RIP—Retire in Peace Retirement Investment Products (RIP) offers a full complement of retirement planning services and a diverse line of retirement investments that have

varying degrees of risk. With the investment products available at RIP, investors could form retirement funds with any level of risk preferred, from risk

Chapter Essentials—Personal Finance

free to extremely risky. RIP’s reputation in the investment community is impeccable because the service agents who advise clients are required to fully inform their clients of the risk possibilities that exist for any investment position, whether it is recommended by an agent or requested by a client. Since 1950, RIP has built its investment portfolio of retirement funds to $60 billion, which makes it one of the largest providers of retirement funds in the United States. You work for RIP as an investment analyst. One of your responsibilities is to help form recommendations for the retirement fund managers to evaluate when making investment decisions. Recently, Howard, a close friend from your college days who now works for SunCoast Investments, a large brokerage firm, called to tell you about a new investment that is expected to earn very high returns during the next few years. The investment is called a ‘‘Piggy-back Asset Investment Device,’’ or PAID for short. Howard told you that he really does not know what this acronym means or how the investment is constructed, but all the reports he has read indicate PAIDs should be a hot investment in the future, so the returns should be very handsome for those who get in now. The one piece of information he did offer was that a PAID is a rather complex investment that consists of a combination of securities whose values are based on numerous debt instruments issued by government agencies, including the Federal National Mortgage Association, the Federal Home Loan Bank, and so on. Howard made it clear that he would like you to consider

341

recommending to RIP that PAIDs be purchased through SunCoast Investments. The commissions from such a deal would bail him and his family out of a financial crisis that resulted because they had bad luck with their investments in the 2001 financial markets. Howard has indicated that somehow he would reward you if RIP invests in PAIDs through SunCoast because, in his words, ‘‘You would literally be saving my life.’’ You told Howard you would think about it and call him back. Further investigation into PAIDs has yielded little additional information beyond what previously was provided by Howard. The new investment is intriguing because its expected return is extremely high compared with similar investments. Earlier this morning, you called Howard to quiz him a little more about the return expectations and to try to get an idea concerning the riskiness of PAIDs. Howard was unable to adequately explain the risk associated with the investment, although he reminded you that the debt of U.S. government agencies is involved. As he says, ‘‘How much risk is there with government agencies?’’ The PAIDs are very enticing because RIP can attract more clients if it can increase the return offered on its investments. If you recommend the new investment and the higher returns pan out, you will earn a very sizable commission. In addition, you will be helping Howard out of his financial situation because his commissions will be substantial if the PAIDs are purchased through SunCoast Investments. Should you recommend the PAIDs as an investment?

The concepts presented in this chapter should help you to better understand the relationship between investment risk and return, which is an important concept in finance. If you understand the basic concepts we discussed, you should be able to construct an investment portfolio that has the level of risk with which you are comfortable. What important principles should I remember from this chapter when investing? First, remember that risk and return are positively related. As a result, in most cases, when you are offered an investment that promises to pay a high return, you should conclude that the investment has high risk. When considering possible investments, never separate ‘‘risk’’ and ‘‘return’’—that is, do not consider the return of an investment without also considering its risk. Second, remember that you can reduce some investment risk through diversification, which can be achieved by purchasing different investments that are not highly positively related to each other. In many instances, you can reduce risk without reducing the expected rate of return associated with your investment position. How can I diversify if I don’t have enough money to purchase 40 different securities? Mutual funds, which we briefly discussed in Chapter 3, provide investors with the opportunity to diversify their investment positions because these

Chapter Essentials —Personal Finance

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investments consist of large portfolios often containing more than 50 to 100 securities that are well diversified. Many types of mutual funds with various investment objectives exist. Shares in most mutual funds can be purchased for as little as $500; thus, you don’t have to be rich to diversify. Individuals are well advised to follow an old adage when investing: ‘‘Don’t put all your eggs in one basket.’’ How can I use the concepts presented in the chapter to construct a portfolio that has a level of risk with which I am comfortable? Remember that (1) a stock’s (investment’s) beta coefficient gives a measure of its ‘‘relevant’’ risk, and (2) a portfolio’s beta equals the weighted average of the betas of all of the investments contained in the portfolio. Thus, if you can determine their beta coefficients, you can choose those investments that provide the risk level you prefer when they are combined to form a portfolio. If you prefer lower risk to higher risk, you should purchase investments with low betas, and vice versa. In addition, you can adjust the riskiness of your portfolio by adding or deleting stocks with particular risks—that is, to reduce a portfolio’s risk, you can either add securities with low betas or delete from the portfolio (sell) securities with high betas. Beta coefficients for most large companies’ stocks are easy to find—they are posted on numerous Internet sites, contained in various financial publications that are available in public libraries, published by investment organizations, and so forth. How can I determine the required and expected rates of return for an investment? Many investors examine the past performance of an investment to determine its expected return. Care must be taken with this approach because past returns often do not reflect future returns. However, you might be able to get a rough idea as to what you expect a stock’s long-term growth will be in the future by examining its past growth, especially if the firm is fairly stable. Investors also rely on information provided by professional analysts to form opinions about expected rates of return. To determine an investment’s required rate of return, investors often evaluate the performances of similar-risk investments. In addition, as we discussed in this chapter, some investors use the CAPM to get a ‘‘ballpark figure’’ for an investment’s required rate of return. The beta coefficients for most large companies can be obtained from many sources, including the Internet; the risk-free rate of return can be estimated using the rates on existing Treasury securities; and the expected market return can be estimated by evaluating market returns in recent years, the current trend in the market, and predictions made by economists and investment analysts. When investing your money, keep these words of wisdom in mind: ‘‘If you lose sleep over your investments or are more concerned with the performance of your portfolio than with your job, then your investment position probably is too risky.’’ If you find yourself in such a position, use the concepts discussed in this chapter to adjust the riskiness of your portfolio.

QUESTIONS 8-1 8-2 8-3 8-4

‘‘The probability distribution of a less risky expected return is more peaked than that of a riskier return.’’ Is this a correct statement? Explain. What shape would the probability distribution have for (a) completely certain returns and (b) completely uncertain returns? Give some events that affect the price of a stock that would result from unsystematic risk. What events would result from systematic risk? Explain. Explain why systematic risk is the ‘‘relevant’’ risk of an investment and why investors should be rewarded only for this type of risk.

Self-Test Problems

8-5

Security A has an expected return of 7 percent, a standard deviation of expected returns of 35 percent, a correlation coefficient with the market of 0.3, and a beta coefficient of 0.5. Security B has an expected return of 12 percent, a standard deviation of returns of 10 percent, a correlation coefficient with the market of 0.7, and a beta coefficient of 1.0. Which security is riskier? Why?

8-6

Suppose you owned a portfolio consisting of $250,000 of long-term U.S. government bonds. a. Would your portfolio be risk-free? b. Now suppose you hold a portfolio consisting of $250,000 of 30-day Treasury bills. Every 30 days your bills mature and you reinvest the principal ($250,000) in a new batch of bills. Assume that you live on the investment income from your portfolio and that you want to maintain a constant standard of living. Is your portfolio truly risk-free?

8-7

c. Can you think of any asset that would be completely risk-free? Could someone develop such an asset? Explain. A life insurance policy is a financial asset. The premiums paid represent the investment’s cost. a. How would you calculate the expected return on a life insurance policy? b. Suppose the owner of a life insurance policy has no other financial assets—the person’s only other asset is ‘‘human capital,’’ or lifetime earnings capacity. What is the correlation coefficient between returns on the insurance policy and returns on the policyholder’s human capital? c. Insurance companies have to pay administrative costs and sales representatives’ commissions; hence, the expected rate of return on insurance premiums is generally low, or even negative. Use the portfolio concept to explain why people buy life insurance despite the negative expected returns.

8-8 8-9

If investors’ aversion to risk increased, would the risk premium on a highbeta stock increase more or less than that on a low-beta stock? Explain. Do you think it is possible to construct a portfolio of stocks that has an expected return that equals the risk-free rate of return?

8-10 Suppose the beta coefficient of a stock doubles from b1 ¼ 1 to b2 ¼ 2. Logic says that the required rate of return on the stock should also double. Is this logic correct? Explain.

SELF-TEST PROBLEMS (Solutions appear in Appendix B at the end of the book.) ST-1 Define the following terms, using graphs or equations to illustrate your answers whenever feasible: a. Risk; probability distribution b. Expected rate of return, ^r ; required rate of return, r c. Continuous probability distribution; discrete probability distribution d. Standard deviation, s; variance, s2; coefficient of variation, CV e. Risk aversion; realized rate of return, €r f. Risk premium for Stock j, RPj; market risk premium, RPM g. Expected return on a portfolio, ^rP

key terms

343

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h. Correlation coefficient, r i. Market risk; company-specific risk; relevant risk j. Beta coefficient, b; average stock’s beta, bM k. Capital Asset Pricing Model (CAPM); security market line (SML); SML equation l. Slope of SML as a measure of risk aversion beta coefficient

ST-2 Of the $10,000 invested in a two-stock portfolio, 30 percent is invested in Stock A and 70 percent is invested in Stock B. If Stock A has a beta equal to 2.0 and the beta of the portfolio is 0.95, what is the beta of Stock B?

required rate of return

ST-3 If the risk-free rate of return, rRF, is 4 percent and the market return, rM, is expected to be 12 percent, what is the required rate of return for a stock with a beta, b, equal to 2.5? ST-4 Stock A and Stock B have the following historical returns:

realized rates of return

Year

Stock A’s Returns, €rA

Stock B’s Returns, €rB

2004 2005 2006 2007 2008

10.00% 18.50 38.67 14.33 33.00

3.00% 21.29 44.25 3.67 28.30

a. Calculate the average rate of return for each stock during the period 2004–2008. Assume that someone held a portfolio consisting of 50 percent Stock A and 50 percent Stock B. What would have been the realized rate of return on the portfolio in each year from 2004 through 2008? What would have been the average return on the portfolio during this period? b. Calculate the standard deviation of returns for each stock and for the portfolio. Use Equation 8–4. c. Looking at the annual returns data on the two stocks, would you guess that the correlation coefficient between returns on the two stocks is closer to 0.9 or to 0.9? ST-5 Stocks R and S have the following probability distributions of returns: Returns Probability 0.5 0.1 0.4

Stock R 2% 10 15

Stock S 20% 12 2

a. Calculate expected return for each stock. b. Calculate the expected return of a portfolio consisting of 50 percent of each stock. c. Calculate the standard deviation of returns for each stock and for the portfolio. Which stock is considered riskier with respect to total risk? d. Compute the coefficient of variation for each stock. According to the coefficient of variation, which stock is considered riskier?

Problems

345

e. Looking at the returns in the probability distributions of the two stocks, would you guess that the correlation coefficient between returns on the two stocks is closer to 0.9 or to 0.9? f. If you added more stocks at random to the portfolio, which of the following is the most accurate statement of what would happen to sP? (1) sP would remain constant, no matter how many stocks are added. (2) sP would approach 15 percent as more stocks are added. (3) sP would decline to zero if enough stocks were included

PROBLEMS 8-1

8-2

8-3

Based on the following probability distribution, what is the security’s expected return? State

Probability

1 2 3

0.2 0.4 0.5

5.0% 10.0 30.0

What is the expected return of the following investment? Probability

Payoff

0.3 0.2 0.5

30.0% 10.0 2.0

expected return

Susan’s investment portfolio currently contains three stocks that have a total value equal to $100,000. The beta of this portfolio is 1.5. Susan is considering investing an additional $50,000 in a stock that has a beta equal to 3. After she adds this stock, what will be the portfolio’s new beta? Suppose rRF ¼ 5%, rM ¼ 12%. What is the appropriate required rate of return for a stock that has a beta coefficient equal to 1.5?

portfolio beta

8-5

The current risk-free rate of return, rRF, is 4 percent and the market risk premium, RPM, is 5 percent. If the beta coefficient associated with a firm’s stock is 2.0, what should be the stock’s required rate of return?

required return

8-6

Following is information for two stocks:

coefficient of variation

8-4

Investment Stock D Stock E

Expected Return, ^r

Standard Deviation, s

10.0% 36.0

8.0% 24.0

required return

Which investment has the greater relative risk? 8-7

ZR Corporation’s stock has a beta coefficient equal to 1.8 and a required rate of return equal to 16 percent. If the expected return on the market is 10 percent, what is the risk-free rate of return, rRF?

risk-free return

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Chapter 8 Risk and Rates of Return portfolio return

expected returns

8-8

8-9

Currently, the risk-free return is 3 percent and the expected market rate of return is 10 percent. What is the expected return of the following three-stock portfolio? Amount Invested

Beta

$400,000 500,000 100,000

1.5 2.0 4.0

The market and Stock S have the following probability distributions: Probability

0.3 0.4 0.3

15% 9 18

20% 5 12

a. Calculate the expected rates of return for the market and Stock S. b. Calculate the standard deviations for the market and Stock S. portfolio return

c. Calculate the coefficients of variation for the market and Stock S. 8-10 Marvin has investments with the following characteristics in his portfolio: Expected Investment

Amount Return, ^r

ABC EFG QRP

Invested

30% 16 20

$10,000 50,000 40,000

What is the expected return of Marvin’s portfolio of investments, ^rp ? expected returns

8-11 Stocks X and Y have the following probability distributions of expected future returns: Probability

0.1 0.2 0.4 0.2 0.1

10% 2 12 20 38

35% 0 20 25 45

a. Calculate the expected rate of return for Stock Y, ^rY , ( ^rX ¼ 12%).

required return

b. Calculate the standard deviation of expected returns for Stock X (sY ¼ 20.35%). Also, calculate the coefficient of variation for Stock Y. Is it possible that most investors might regard Stock Y as being less risky than Stock X? Explain. 8-12 Yesterday Susan determined that the risk-free rate of return, rRF, is 3 percent, the required return on the market portfolio, rM, is 10 percent, and the required rate of return on Stock K, rK, is 17 percent. Today Susan received

Problems

347

new information that indicates investors are more risk averse than she thought, such that the market risk premium, RPM, actually is 1 percent higher than she estimated yesterday. When Susan considers the effect of this change in risk premium, what will she determine the new rK to be? 8-13 Terry recently invested equal amounts in five stocks to form an investment portfolio, which has a beta equal to 1.2—that is, bP ¼ 1.2. Terry is considering selling the riskiest stock in the portfolio, which has a beta coefficient equal to 2.0, and replacing it with another stock. If Terry replaces the stock with b ¼ 2.0 with a stock with b ¼ 1.0, what will be the new beta of his investment portfolio? Assume that the equal amounts are invested in each stock in the portfolio.

portfolio beta

8-14 Thomas has a five-stock portfolio that has a market value equal to $400,000. The portfolio’s beta is 1.5. Thomas is considering selling a particular stock to help pay some university expenses. The stock is valued at $100,000, and if he sells it the portfolio’s beta will increase to 1.8. What is the beta of the stock Thomas is considering selling?

portfolio beta

8-15 Suppose rRF ¼ 8%, rM ¼ 11%, and rB ¼ 14%.

beta computation

a. Calculate Stock B’s beta, Bb. b. If Stock B’s beta were 1.5, what would be its new required rate of return? 8-16 Suppose rRF ¼ 9%, rM ¼ 14%, and bX ¼ 1.3. a. What is rX, the required rate of return on Stock X?

SML and CAPM

b. Now suppose rRF (1) increases to 10 percent or (2) decreases to 8 percent. The slope of the SML remains constant. How would each change affect rM and rX? c. Assume rRF remains at 9 percent, but rM (1) increases to 16 percent or (2) decreases to 13 percent. The slope of the SML does not remain constant. How would these changes affect rX? 8-17 Suppose you hold a diversified portfolio consisting of a $7,500 investment in each of 20 different common stocks. The portfolio beta is equal to 1.12. You have decided to sell one of the stocks in your portfolio with a beta equal to 1.0 for $7,500 and to use the proceeds to buy another stock for your portfolio. Assume that the new stock’s beta is equal to 1.75. Calculate your portfolio’s new beta.

portfolio beta

8-18 Stock R has a beta of 1.5, Stock S has a beta of 0.75, the expected rate of return on an average stock is 15 percent, and the risk-free rate of return is 9 percent. By how much does the required return on the riskier stock exceed the required return on the less risky stock?

required rates of return

8-19 Suppose you are the money manager of a $4 million investment fund. The fund consists of four stocks with the following investments and betas:

portfolio required return

Stock A B C D

Investment

Beta

$ 400.000 600,000 1,000,000 2,000,000

1.50 0.50 1.25 0.75

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Chapter 8 Risk and Rates of Return

expected returns and risk

If the market required rate of return is 14 percent and the risk-free rate is 6 percent, what is the fundâ&#x20AC;&#x2122;s required rate of return? 8-20 Following is information about Investment A, Investment B, and Investment C: Return on Investment Economic Condition Boom Normal Recession ^r s

Probability

0.5 0.4 0.1

25.0% 15.0 5.0 18.0% ?

40.0% 20.0 40.0 24.0% 23.3%

5.0% 10.0 15.0 ? 3.3%

a. Compute the expected return, ^r , for Investment C. b. Compute the standard deviation, s, for Investment A. c. Based on total risk and return, which of the investments should a riskaverse investor prefer? expected returns

8-21 Suppose you win the Florida lottery and are offered a choice of $500,000 in cash or a gamble in which you would get $1 million if a head is flipped but zero if a tail comes up. a. What is the expected value of the gamble? b. Would you take the sure $500,000 or the gamble? c. If you choose the sure $500,000, are you a risk averter or a risk seeker? d. Suppose you take the sure $500,000. You can invest it in either a U.S. Treasury bond that will return $537,500 at the end of one year or a common stock that has a 50-50 chance of being either worthless or worth $1,150,000 at the end of the year. (1) What is the expected dollar profit on the stock investment? (The expected profit on the T-bond investment is $37,500.) (2) What is the expected rate of return on the stock investment? (The expected rate of return on the T-bond investment is 7.5 percent.) (3) Would you invest in the bond or the stock? (4) Exactly how large would the expected profit (or the expected rate of return) have to be on the stock investment to make you invest in the stock, given the 7.5 percent return on the bond? (5) How might your decision be affected if, rather than buying one stock for $500,000, you could construct a portfolio consisting of 100 stocks with $5,000 invested in each? Each of these stocks has the same return characteristics as the one stockâ&#x20AC;&#x201D;that is, a 50-50 chance of being worth either zero or $11,500 at yearend. Would the correlation between returns on these stocks matter?

Problems

8-22 The McAlhany Investment Fund has total capital of $500 million invested in five stocks: Stock A B C D E

Investment $160 120 80 80 60

349

portfolio return

Stock’s Beta Coefficient

million million million million million

0.5 2.0 4.0 1.0 3.0

The current risk-free rate is 8 percent. Market returns have the following estimated probability distribution for the next period: Probability

Market Return

0.1 0.2 0.4 0.2 0.1

10% 12 13 16 17

a. Compute the expected return for the market. b. Compute the beta coefficient for the investment fund. (Remember, this problem involves a portfolio.) c. What is the estimated equation for the security market line? d. Compute the fund’s required rate of return for the next period. e. Suppose John McAlhany, the president, receives a proposal for a new stock. The investment needed to take a position in the stock is $50 million, it will have an expected return of 18 percent, and its estimated beta coefficient is 2.0. Should the firm purchase the new stock? At what expected rate of return should McAlhany be indifferent to purchasing the stock? 8-23 Stock A and Stock B have the following historical returns:

Year

Stock A’s Returns, €rB

2004 2005 2006 2007 2008

18.0% 33.0 15.0 0.5 27.0

Stock B’s Returns, €rB 14.5% 21.8 30.5 7.6 26.3

a. Calculate the average rate of return for each stock during the period 2004 2008. b. Assume that someone held a portfolio consisting of 50 percent Stock A and 50 percent Stock B. What would have been the realized rate of return

realized rates of return

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Chapter 8 Risk and Rates of Return

on the portfolio in each year from 2004 through 2008? What would have been the average return on the portfolio during this period? c. Calculate the standard deviation of returns for each stock and for the portfolio. Use Equation 8–4. d. Calculate the coefficient of variation for each stock and for the portfolio. e. If you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or the portfolio? Why?

Integrative Problem 8-24 Assume you recently graduated with a major in finance, and you just landed a job in the trust department of a large regional bank. Your first assignment is to invest $100,000 from an estate for which the bank is trustee. Because the estate is expected to be distributed to the heirs in approximately one year, you have been instructed to plan for a one-year holding period. Furthermore, your boss has restricted you to the following investment alternatives, shown with their probabilities and associated outcomes. (For now, disregard the blank spaces in the table; you will fill in the blanks later.)

risk and rates of return

Estimated Returns on Alternative Investments State of the Economy Recession Below Average Average Above Average Boom ^r s CV

Probability

T-Bills

High Tech

Collections

U.S. Rubber

Market Portfolio

0.1 0.2 0.4 0.2 0.1

8.0% 8.0 8.0 8.0 8.0

22.0% 2.0 20.0 35.0 50.0

28.0% 14.7 0.0 10.0 20.0

10.0% 10.0 7.0 45.0 30.0

13.0% 1.0 15.0 29.0 43.0

Two-Stock Portfolio

The bank’s economic forecasting staff has developed probability estimates for the state of the economy, and the trust department has a sophisticated computer program that was used to estimate the rate of return on each alternative under each state of the economy. High Tech Inc. is an electronics firm, Collections Inc. collects past-due debts, and U.S. Rubber manufactures tires and various other rubber and plastics products. The bank also maintains an ‘‘index fund’’ that includes a market-weighted fraction of all publicly traded stocks; by investing in that fund, you can obtain average stock market results. Given the situation as described, answer the following questions. a. (1) Why is the risk-free return independent of the state of the economy? Do T-bills promise a completely risk-free return? (2) Why are High Tech’s returns expected to move with the economy whereas Collections’ are expected to move counter to the economy? b. Calculate the expected rate of return on each alternative and fill in the row for rˆ in the table. c. You should recognize that basing a decision solely on expected returns is appropriate only for risk-neutral individuals. Because the beneficiaries of the

Problems

trust, like virtually everyone, are risk averse, the riskiness of each alternative is an important aspect of the decision. One possible measure of risk is the standard deviation of returns. (1) Calculate this value for each alternative, and fill in the row for s in the table. (2) What type of risk does the standard deviation measure? (3) Draw a graph that shows roughly the shape of the probability distributions for High Tech, U.S. Rubber, and T-bills. d. Suppose you suddenly remembered that the coefficient of variation (CV) is generally regarded as being a better measure of total risk than the standard deviation when the alternatives being considered have widely differing expected returns and risks. Calculate the CVs for the different securities, and fill in the row for CV in the table. Does the CV measure produce the same risk rankings as the standard deviation? e. Suppose you created a two-stock portfolio by investing $50,000 in High Tech and $50,000 in Collections. (1) Calculate the expected return ( ^rp ), the standard deviation (sp), and the coefficient of variation (CVp) for this portfolio and fill in the appropriate rows in the table. (2) How does the riskiness of this two-stock portfolio compare to the riskiness of the individual stocks if they were held in isolation? f. Suppose an investor starts with a portfolio consisting of one randomly selected stock. What would happen (1) to the riskiness and (2) to the expected return of the portfolio as more randomly selected stocks are added to the portfolio? What is the implication for investors? Draw two graphs to illustrate your answer. g. (1) Should portfolio effects influence the way that investors think about the riskiness of individual stocks? (2) If you chose to hold a one-stock portfolio and consequently were exposed to more risk than diversified investors, could you expect to be compensated for all of your risk? That is, could you earn a risk premium on the part of your risk that you could have eliminated by diversifying? h. The expected rates of return and the beta coefficients of the alternatives as supplied by the bank’s computer program are as follows: Security High Tech Market U.S. Rubber T-bills Collections

Return ( ^r ) 17.40% 15.00 13.80 8.00 1.74

Risk (b) 1.29 1.00 0.68 0.00 0.86

(1) What is a beta coefficient, and how are betas used in risk analysis? (2) Do the expected returns appear to be related to each alternative’s market risk? (3) Is it possible to choose among the alternatives on the basis of the information developed thus far? (4) Use the data given at the beginning of the problem to construct a graph that shows how the T-bill’s, High Tech’s, and Collections’ beta coefficients are calculated. Discuss what beta measures and explain how it is used in risk analysis. i. (1) Write out the SML equation, use it to calculate the required rate of return on each alternative, and then graph the relationship between the

351

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Chapter 8 Risk and Rates of Return

expected and required rates of return. (2) How do the expected rates of return compare with the required rates of return? (3) Does the fact that Collections has a negative beta coefficient make any sense? What is the implication of the negative beta? (4) What would be the market risk and the required return of a 50-50 portfolio of High Tech and Collections? Of a 50-50 portfolio of High Tech and U.S. Rubber? j. (1) Suppose investors raised their inflation expectations by 3 percentage points over current estimates as reflected in the 8 percent T-bill rate. What effect would higher inflation have on the SML and on the returns required on high- and low-risk securities? (2) Suppose instead that investors’ risk aversion increased enough to cause the market risk premium to increase by 3 percentage points. (Inflation remains constant.) What effect would this change have on the SML and on returns of highand low-risk securities?

Computer-Related Problem Work the problem in this section only if you are using the problem spreadsheet. realized rates of return

8-25 Using File C08, rework Problem 8-23, assuming that a third stock, Stock C, is available for inclusion in the portfolio. Stock C has the following historical returns: Year

Stock C’s Return, rc

2004 2005 2006 2007 2008

32.00% 11.75 10.75 32.25 6.75

a. Calculate (or read from the computer screen) the average return, standard deviation, and coefficient of variation for Stock C. b. Assume that the portfolio now consists of 33.33 percent Stock A, 33.33 percent Stock B, and 33.33 percent Stock C. How does this composition affect the portfolio return, standard deviation, and coefficient of variation versus when 50 percent was invested in A and in B? c. Make some other changes in the portfolio, making sure that the percentages sum to 100 percent. For example, enter 25 percent for Stock A, 25 percent for Stock B, and 50 percent for Stock C. (Note that the program will not allow you to enter a zero for the percentage in Stock C.) Notice that ^rp remains constant and that spchanges. Why do these results occur? d. In Problem 8–23, the standard deviation of the portfolio decreased only slightly because Stocks A and B were highly positively correlated with one another. In this problem, the addition of Stock C causes the standard deviation of the portfolio to decline dramatically, even though sC ¼ sA ¼ sB. What does this change indicate about the correlation between Stock C and Stocks A and B? e. Would you prefer to hold the portfolio described in Problem 8-23 consisting only of Stocks A and B or a portfolio that also included Stock C? If others react similarly, how might this fact affect the stocks’ prices and rates of return?