besley 14e ch08

CHAPTER

8

A MAN A GERIAL PE R S P E C T I V E

T he performance of the major stock marketsfrom 1995 through 1998 can best be describedas remarkable—a period that investors would

love to repeat again and again. During that four-yearstretch, stocks traded on U.S. stock markets earned anaverage return greater than 20 percent per year. In 1998,companies such as Microsoft and WorldCom MCI morethan doubled in value. The value of some Internetcompanies, such as America Online, Amazon.com, andYahoo!, increased by more than 500 percent. Considerthe return you would have earned in 1998 if you hadpurchased 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 otherhand, if you had waited until January 2000 to buyAmazon.com and then held it until the end of the year,you would have lost approximately 80 percent of yourinvestment because the company’s stock decreasedsignificantly during the year. In fact, during 2000 thevalues of most Internet company stocks declined sig-nificantly. Indeed, many Internet companies did notsurvive the ‘‘Internet skepticism’’ that existed during theyear. By comparison, if you had purchased the stock ofEnron at the beginning of 2000, your investment wouldhave nearly doubled in value by the end of the year. Butif you still held Enron in mid-2003, the value of yourinvestment 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 percentin 2005, but then it declined 28 percent in the firsttwo months of 2006 before it recovered to generate apositive 9 percent return for the entire year.

If you had bet all your money on the stock of asingle company, you would have essentially ‘‘put allyour eggs in one basket’’ and faced considerable risk.For example, you would have won big if you chose toinvest in Amazon.com for one year either in 1998 or2002. But you would have lost big if you chose to investin Amazon.com in 2000. Investors who diversified byspreading their investments among many stocks,perhaps through mutual funds, would have earned areturn somewhere between the extraordinary increasesposted by Amazon.com in 1998 and 2002 and theextraordinary decreases posted by Amazon.com andother Internet companies in 2000. Large ‘‘baskets’’ ofsuch diversified investments would have earnedreturns fairly close to the average of the stock markets.

Investing is risky! Although the stock markets per-formed well from 1995 through 1998, they also gothrough periods characterized by decreasing prices ornegative average returns. For instance, in 1990, 1994,and 2000, the average stock listed on the New YorkStock Exchange decreased in value by 7.5 percent, 3.1percent, and 5.9 percent, respectively. More recently,

Risk and Ratesof Return

305

during the first few months of 2006, the stock marketwas fairly fickle. At the beginning of the year, the DowJones Industrial Average (DJIA) was 10,718. Onemonth later, the DJIA was at about the same level,which means investors earned an average rate ofreturn of approximately 0 percent during the month ofJanuary. In mid-May, however, the DJIA was 11,630.Investors who ‘‘got in the market’’ in January and ‘‘gotout of the market’’ in May earned an equivalent annualreturn equal to about 26 percent (noncompounded),and investors who waited to ‘‘get in the market’’ inFebruary and then got out in May earned an equivalentannual return equal to about 34 percent. One monthlater, however, the DJIA was back to its beginning-of-the-year value. During 2006 the DJIA at times experi-enced periods of substantial increases and other times itdecreased substantially, but by the end of the year theindex had increased 16 percent, which represented a

higher-than-average market return. What a roller-coasterride! What risk!

Who knows what the stock market will be doingwhen you read this book. It could be an up market(referred to as a ‘‘bull’’ market) or it could be a downmarket (referred to as a ‘‘bear’’ market). Whatever thecase, as times change, investment strategies andportfolio mixes need to be changed to meet newconditions. For this reason, you need to understandthe basic concepts of risk and return and to recognizehow diversification affects investment decisions. Asyou will discover, investors can create portfolios ofsecurities to reduce risk without reducing the averagereturn on their investments. After reading thischapter, you should have a better understanding ofhow risk affects investment returns and how toevaluate risk when selecting investments such as thosedescribed 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 bemeasured and how it affects assets’ values and rates of return. Recall that inChapter 5, when we examined the determinants of interest rates, we defined thereal risk-free rate, r�, to be the rate of interest on a risk-free security in the absenceof inflation. The actual interest rate on a particular debt security was shown to beequal to the real risk-free rate plus several premiums that reflect both inflation andthe riskiness of the security in question. In this chapter, we define the term riskmore precisely in terms of how it relates to investments, we examine proceduresused to measure risk, and we discuss the relationship between risk and return. Bothinvestors and financial managers should understand these concepts and use themwhen considering investment decisions, whether the decisions concern financialassets or real assets.

We will demonstrate that each investment—each stock, bond, or physical asset—isassociated with two types of risk: diversifiable risk and nondiversifiable risk. The sum ofthese two components is the investment’s total risk. Diversifiable risk is not importantto rational, informed investors because they will eliminate its effects by diversifying itaway. The really significant risk is nondiversifiable risk; this risk is bad in the sense thatyou cannot eliminate it, and if you invest in anything other than risk-free assets, such asshort-term Treasury bills, you will be exposed to it. In the balance of the chapter, wewill describe these risk concepts and consider how risk enters into the investmentdecision-making process.

Chapter Essentials—The Questions

306 Chapter 8 Risk and Rates of Return

DEFINING AND MEASURING RISK

Webster’s Dictionary defines risk as ‘‘a hazard; a peril; exposure to loss or injury.’’ Asthis definition suggests, risk refers to the chance that some unfavorable event willoccur. If you engage in skydiving, you are taking a chance with your life: Skydiving isrisky. If you bet on the horses, you risk losing your money. If you invest in speculativestocks (or, really, any stock), you are taking a risk in the hope of receiving an appreciablereturn.

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 particularactivity or event, so we are not sure what will happen in the future. Consequently, riskresults from the fact that an action such as investing can produce more than one outcomein the future. When multiple outcomes are possible, some of the possible outcomes areconsidered ‘‘good’’ and some of the possible outcomes are considered ‘‘bad.’’

To illustrate the riskiness of financial assets, suppose you have a large amount ofmoney to invest for one year. You could buy a Treasury security that has an expectedreturn equal to 5 percent. This investment’s anticipated rate of return can bedetermined quite precisely because the chance of the government defaulting onTreasury securities is negligible; the outcome is essentially guaranteed, which meansthat the security is a risk-free investment.

Alternatively, you could buy the common stock of a newly formed company thathas developed technology that can be used to extract petroleum from the mountains inSouth America without defacing the landscape and without harming the ecology. Thetechnology has yet to be proved economically feasible, so the returns that the commonstockholders will receive in the future remain uncertain. Experts who have analyzedthe common stock of the company have determined that the expected, or average long-run, return for such an investment is 30 percent. Each year, the investment could yielda positive return as high as 900 percent. Of course, there also is the possibility that thecompany might not survive, in which case the entire investment will be lost and thereturn will be �100 percent. The return that investors receive each year cannot bedetermined precisely because more than one outcome is possible; this stock is a riskyinvestment. Because there is a significant danger of earning considerably less than theexpected return, investors probably would consider the stock to be quite risky. There isalso a very good chance that the actual return will be greater than expected, which, ofcourse, is an outcome you would gladly accept. This possibility could not exist if thestock did not have risk.

Thus, when we think of investment risk, along with the chance of receiving less thanexpected, we should consider the chance of receiving more than expected. If weconsider investment risk from this perspective, then we can define risk as the chance ofreceiving an actual return other than expected. This definition simply means that there isvariability in the returns or outcomes from the investment. Therefore, investment riskcan 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 otherthan the expected one. The greater the variability of the possible outcomes, the riskierthe investment. We can define risk more precisely, however, and it is useful to do so.

Probability DistributionsAn event’s probability is defined as the chance that the event will occur. For example, aweather forecaster might state: ‘‘There is a 40 percent chance of rain today and a60 percent chance that it will not rain.’’ If all possible events, or outcomes, are listed, and

riskThe chance that anoutcome other thanthe expected one willoccur.

Defining and Measuring Risk 307

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:

Here the possible outcomes are listed in the left column, and the probabilities of theseoutcomes, expressed both as decimals and as percentages, are given in the rightcolumns. Notice that the probabilities must sum to 1.0, or 100 percent.

Probabilities can also be assigned to the possible outcomes (or returns) from aninvestment. If you buy a bond, you expect to receive interest on it; those interestpayments will provide you with a rate of return on your investment. This investmenthas two possible outcomes: (1) the issuer makes the interest payments, or (2) the issuerfails to make the interest payments. The higher the probability of default on theinterest payments, the riskier the bond; the higher the risk, the higher the rate of returnyou would require to invest in the bond. If you invest in a stock instead of buying abond, you will again expect to earn a return on your money. As we saw in Chapter 7, astock’s return includes dividends plus capital gains. Again, the riskier the stock—thatis, the greater the variability of the possible payoffs—the higher the stock’s expectedreturn must be to induce you to invest in it.

With this idea in mind, consider the possible rates of return (dividend yield pluscapital gains yield) that you might earn next year on a $10,000 investment in the stockof either Martin Products Inc. or U.S. Electric. Martin manufactures and distributesequipment for the data transmission industry. Because its sales are cyclical, the firm’sprofits rise and fall with the business cycle. Furthermore, its market is extremelycompetitive, and some new company could develop better products that could forceMartin into bankruptcy. U.S. Electric, on the other hand, supplies electricity, which isconsidered an essential service. Because it has city franchises that protect it fromcompetition, this firm’s sales and profits are relatively stable and predictable.

Table 8-1 shows the rate-of-return probability distributions for these two com-panies. As shown in the table, there is a 20 percent chance of a boom, in which caseboth 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 normaleconomy 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.

Outcome Probability

Rain 0.40 ¼ 40%No rain 0.60 ¼ 60

1.00 100%

probability distributionA listing of all possibleoutcomes, or events,with a probability(chance of occurrence)assigned to each out-come.

TABLE 8-1 Probability Distributions for Martin Products and U.S. Electric

State ofthe Economy

Probability ofThis State Occurring

Rate of Return on StockIf Economic State Occurs

Martin Products U.S. Electric

Boom 0.2 110% 20%Normal 0.5 22 16Recession 0.3 �60 10

1.0


Electric. There is a fairly high probability that the value of Martin’s stock will varysubstantially, 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 is20 percent.1

EXPECTED RATE OF RETURN

Table 8-1 provides the probability distributions showing the possible outcomes forinvesting in Martin Products and U.S. Electric. We can see that the most likelyoutcome is for the economy to be normal, in which case Martin will return 22 percentand U.S. Electric will return 16 percent. Other outcomes are also possible, however, sowe need to summarize the information contained in the probability distributions into asingle measure that considers all these possible outcomes. That measure is called theexpected value, or expected rate of return, for the investments.

Simply stated, the expected value (return) is the weighted average of theoutcomes, with each outcome’s weight being its probability of occurrence. Table 8-2shows 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 andthen 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 uncertainbecause 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 theeconomy is booming, but we do not know in which year the economy will bebooming.

What does investment risk mean?

Set up illustrative probability distributions for (1) a bond investment and (2) astock investment.

TABLE 8-2 Calculation of Expected Rates of Return: Martin Products and U.S. Electric

Martin Products U.S. Electric

State of theEconomy

(1)

Probability of ThisState Occurring

(2)

Return If ThisState Occurs

(3)Product:

(2) � (3) ¼ (4)

Return If ThisState Occurs

(5)Product:

(2) � (5) ¼ (6)

Boom 0.2 110% 22% 20% 4%Normal 0.5 22 11 16 8Recession 0.3 �60 �18 10 3

1.0 rUS ¼ 15% rUS ¼ 15%

expected value (return), rThe rate of returnexpected to be realizedfrom an investment; themean value of theprobability distributionof possible results.

1It is, of course, completely unrealistic to think that any stock has no chance of a loss. Only in hypothetical examplescould this situation occur.

2In Chapter 6, we used rd to signify the return on a debt instrument, and in Chapter 7, we used rs to signify the returnon a stock. In this section, however, we discuss only returns on stocks; thus, the subscript ‘‘s’’ is unnecessary, and weuse the term r rather than rs to represent the expected return on a stock.

Expected Rate of Return 309

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

8–1 Expected rate of return ¼ r ¼ Pr1r1 þ Pr2r2 þ � � � þ Prnrn

¼Xn

i¼1

Priri

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, r is a weighted average of the possibleoutcomes (the ri values), with each outcome’s weight being its probability ofoccurrence. Using the data for Martin Products, we compute its expected rate ofreturn as follows:

r ¼ Pr1ðr1Þ þ Pr2ðr2Þ þ Pr3ðr3Þ¼ 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 forMartin Products given in Table 8-1. Stated simply, the expected rate of returnrepresents the average payoff that investors will receive from Martin Products if theprobability 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 futureeconomic condition will be termed a boom, so investors will earn a 110 percent rate ofreturn; 50 percent of the time the economy should be normal and the investmentpayoff will be 22 percent; and 30 percent of the time the economy should be inrecession and the payoff will be a loss equal to 60 percent. On average, then, MartinProducts’ 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 possibleoutcomes, as shown in Figure 8-1. The height of each bar in the figure indicates theprobability 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

120100806040200�20�40�60

0.5

0.4

0.3

0.2

0.1

Probability of Occurrence

Rate of Return (%)rMartin = 15%

B. U.S. Electric

2520151050�5�10

0.5

0.4

0.3

0.2

0.1

Probability of Occurrence

Rate of Return (%)ˆ rUS = 15%ˆ


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

Continuous versus Discrete Probability DistributionsSo far, we have assumed that only three states of the economy can exist: recession,normal, and boom. Under these conditions, the probability distributions given inTable 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 deepdepression 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 ofthe economy (with the sum of the probabilities still equaling 1.0) and to assign a rate ofreturn to each stock for each state of the economy. We would then have a table similarto Table 8-1, except that it would include many more entries in each column. We coulduse this table to calculate the expected rates of return as described previously, andwe could approximate the probabilities and outcomes by constructing continuouscurves such as those presented in Figure 8-2. In this figure, we have changed theassumptions so that there is essentially a zero probability that Martin Products’ returnwill be less than –60 percent or more than 110 percent, or that U.S. Electric’s returnwill be less than 10 percent or more than 20 percent. Virtually any return within theselimits is possible, however. Such probability distributions are called continuousbecause the number of possible outcomes is unlimited. For example, U.S. Electric’sreturn 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

0

Probability Density

15−60 110Rate of

Return (%)

Expected Rateof Return

U.S. Electric

Martin Products

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 smallerbecause there are many possible outcomes instead of just three. With continuous distributions, it is moreappropriate to ask what the probability is of obtaining at least some specified rate of return than to ask what theprobability is of obtaining exactly that rate. This topic is covered in detail in statistics courses.

continuous probabilitydistributionThe number of possibleoutcomes is unlimited,or infinite.

discrete probabilitydistributionThe number of possibleoutcomes is limited, orfinite.


The tighter the probability distribution, the less variability there is and the morelikely 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 dra-matically from the expected return. Thus, the tighter the probability distribution, thelower the risk assigned to a stock. Because U.S. Electric has a relatively tight prob-ability distribution, its actual return is likely to be closer to its 15 percent expectedreturn than is that of Martin Products.

Measuring Total (Stand-Alone) Risk:The Standard DeviationBecause we have defined risk as the variability of returns, we can measure it by examiningthe tightness of the probability distribution associated with the possible outcomes. Ingeneral, the width of a probability distribution indicates the amount of scatter, or vari-ability, of the possible outcomes. To be most useful, any measure of risk should have adefinite 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 probabilitydistribution, 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, wepreviously found r ¼ 15%.

2. Subtract the expected rate of return, r , from each possible outcome, ri, toobtain a set of deviations from r :

Deviationi ¼ ri � r

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 sumthese products to obtain the variance, s2, of the probability distribution,which is shown in column 6:

8–2Variance ¼ s2 ¼ ðr1 � rÞ2Pr1 þ ðr2 � rÞ2Pr2 þ � � � þ ðrn � rÞ2Prn

¼Xn

i¼1

ðri � rÞ2Pri

TABLE 8-3 Calculating Martin Products’ Standard Deviation

Payoff ri

(1)

ExpectedReturn r

(2)

Deviationrj � r

(1)� (2) ¼ (3)(rj � r)2 ¼

(4)Probability

(5)ðrj � rÞ2Pri

(4)� (5) ¼s2

(6)

110% � 15% ¼ 95 9,025 0.2 9,025 � 0.2 ¼ 1,805.022 � 15 ¼ 7 49 0.5 49 � 0.5 ¼ 24.5�60 � 15 ¼ �75 5,625 0.3 5,625 � 0.3 ¼ 1,687.5

Variance ¼ s2 ¼ 3,517.0

Standard deviation ¼ s ¼ffiffiffiffiffis2p

¼ffiffiffiffiffiffiffiffiffiffiffiffi3; 517

p¼ 59:3%

variance, s2

The standard deviationsquared; a measure ofthe width of a prob-ability distribution.

standard deviation, sA measure of thetightness, or variability,of a set of outcomes.


4. Take the square root of the variance to get the standard deviation shown atthe bottom of column 6:

8–3Standard deviation ¼ s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr1 � rÞ2Pr1 þ ðr2 � rÞ2Pr2 þ � � � þ ðrn � rÞ2Prn

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

ðri � rÞ2Pri

s

As you can see, the standard deviation is a weighted average deviation from theexpected value, and it gives an idea of how far above or below the expected value theactual value is likely to be. As shown in Table 8-3 Martin’s standard deviation iss¼ 59.3%. Using these same procedures, we find U.S. Electric’s standard deviationto be 3.6 percent. The larger standard deviation for Martin indicates a greatervariation of returns for this firm, and hence a greater chance that the actual, orrealized, 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 andstandard deviation is based on data that take the form of a known probability dis-tribution. That is, we know or have estimated all of the future outcomes and thechances that these outcomes will occur in a particular situation. In many cases,however, the only information we have available consists of data over some pastperiod. For example, suppose we have observed the following returns associated with acommon stock:

We can use this information to estimate the risk associated with the stock byestimating standard deviation of returns. The estimated standard deviation canbe computed using a series of past, or observed, returns to solve the followingformula:

8–4

Estimated s ¼ s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnt¼1ð€rt � rÞ2

n� 1

vuuut

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

as follows:

8–5

r ¼ €r1 þ €r2 þ � � � þ €rn

n¼

Pnt¼1

€rn

n

Year r

2008 15%2009 �52010 202011 22


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

r ¼ 15þ ð�5Þ þ 20þ 22

4¼ 13:0%

Estimated s ¼ s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið15� 13Þ2 þ ð�5� 13Þ2 þ ð20� 13Þ2 þ ð22� 13Þ2

4� 1

s

¼ffiffiffiffiffiffiffiffi458

3

r¼ 12:4%

The historical standard deviation is often used as an estimate of the future standarddeviation. Much less often, and generally incorrectly, rt for some past period is usedas an estimate of rt , the expected future return. Because past variability is likely to berepeated, 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 percentor as low as –50 percent) is the best expectation of what investors think will happen inthe 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:

8–6Coefficient of variation ¼ CV ¼ Risk

Return¼ s

r

The coefficient of variation shows the risk per unit of return. It provides a moremeaningful 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 isnot 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 thesame expected return with lower risk. The firm with the larger standard deviation,Martin, must have the larger coefficient of variation because the expected returns forthe two stocks are equal, but the numerator in Equation 8–6 is greater for Martin. Infact, the coefficient of variation for Martin is 59.3%/15% ¼ 3.95; for U.S. Electric,CV¼ 3.6%/15%¼ 0.24. Thus Martin is more than 16 times riskier than U.S. Electricusing this criterion.

The coefficient of variation is more useful when we consider investments that havedifferent expected rates of return and different levels of risk. For example, BioboticsCorporation is a biological research and development firm that, according to stockanalysts, offers investors an expected rate of return equal to 35 percent with a standarddeviation 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 abetter investment? If we calculate the coefficient of variation for Biobotics, we findthat it equals 7.5%/35%¼ 0.21, which is slightly less than U.S. Electric’s coefficient ofvariation 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 goodestimate. We use four observations here only to simplify the illustration.

coefficient of variation(CV)A standardized mea-sure of the risk per unitof return. It is calcu-lated by dividing thestandard deviation bythe expected return.


offered by Biobotics is more than sufficient to compensate investors for taking on theadditional risk.

Figure 8-3 graphs the probability distributions for U.S. Electric and Biobotics. Asyou can see in the figure, U.S. Electric has the smaller standard deviation and hencethe more peaked probability distribution. As the graph clearly shows, however, thechances of a really high return are much better with Biobotics than with U.S. Electricbecause Biobotics’ expected return is so high. Because the coefficient of variationcaptures the effects of both risk and return, it is a better measure for evaluating risk insituations where investments differ with respect to both their amounts of total risk andtheir expected returns.

Risk Aversion and Required ReturnsSuppose you have worked hard and saved $1 million, which you now plan to invest. Youcan 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, thevalue of your stock will increase to $2.2 million. Conversely, if the firm’s research is afailure, the value of your stock will go to zero, and you will be penniless. You regardR&D’s chances of success or failure as being 50-50, so the expected value of the stockinvestment is 0.5($0)þ 0.5($2,200,000)¼ $1,100,000. Subtracting the $1 million costof the stock leaves an expected profit of $100,000, or an expected (but risky) 10 percentrate of return:

Expected rate

of return¼ Expected ending value� Beginning value

Beginning value

¼ $1;100;000� $1;000;000

$1;000;000¼ $100;000

$1;000;000¼ 0:10 ¼ 10:0%

In this case, you have a choice between a sure $100,000 profit (representing a10 percent rate of return) on the Treasury note and a risky expected $100,000 profit

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

U.S. Electric

Biobotics

Expected Rateof Return (%)

0 15 35

Probability Density


(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 riskaverse. 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-documentedfact, 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 higherthe return investors demand, and thus the less they are willing to pay for theinvestment. To see how risk aversion affects security prices, we can analyze thesituation 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 torisk, so they would show a general preference for U.S. Electric because there is lessvariability in its payoffs (less uncertainty). People with money to invest would bid forU.S. Electric stock rather than Martin stock, and Martin’s stockholders would startselling their stock and using the money to buy U.S. Electric stock. Buying pressurewould drive up the price of U.S. Electric’s stock, and selling pressure wouldsimultaneously cause Martin’s price to decline. These price changes, in turn, wouldalter the expected rates of return on the two securities. Suppose, for example, that theprice of U.S. Electric stock was bid up from $100 to $125, whereas the price of Martin’sstock declined from $100 to $75. This development would cause U.S. Electric’sexpected return to fall to 12 percent, whereas Martin’s expected return would rise to20 percent. The difference in returns, 20%� 12%¼ 8%, is a risk premium (RP). Therisk premium represents the compensation that investors require for assuming theadditional risk of buying Martin’s stock.

This example demonstrates a very important principle: In a market dominated byrisk-averse investors, riskier securities must have higher expected returns, as estimatedby 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 thehigher-risk investments have higher expected returns than the lower-risk investments.Figure 8-4 illustrates this relationship. We will consider the question of how muchhigher the returns on risky securities must be later in the chapter, after we examinehow diversification affects the way risk should be measured.

risk premium (RP)The portion of theexpected return thatcan be attributed to theadditional risk of aninvestment. It is thedifference between theexpected rate of returnon a given risky assetand the expected rateof return on a less riskyasset.

FIGURE 8-4 Risk/Return Relationship

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

Payment for Risk = Risk Premium = RP

Risk

0

Return, r

Return = r = rRF + RP

rRF

rLow

rAvg

rHigh

Below Average Risk

Average Risk

Above Average Risk

risk aversionRisk-averse investorsrequire higher rates ofreturn to invest inhigher-risk securities.


PORTFOLIO RISK—HOLDING COMBINATIONS OF ASSETS

In the preceding section, we considered the riskiness of an investment held inisolation—that is, the total risk of an investment if it is held by itself. Now we analyzethe riskiness of investments held in portfolios.4 As we shall see, holding aninvestment—whether a stock, bond, or other asset—as part of a portfolio generallyis less risky than holding the same investment all by itself. In fact, most financialassets are not held in isolation but rather as parts of portfolios. Banks, pension funds,insurance companies, mutual funds, and other financial institutions are required bylaw to hold diversified portfolios. Even individual investors—at least those whosesecurity holdings constitute a significant part of their total wealth—generally holdstock portfolios rather than the stock of only one firm. From an investor’s stand-point, 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 beevaluated in isolation; instead, the risk and return of an individual security shouldbe analyzed in terms of how that security affects the risk and return of the portfolio inwhich it is held.

To illustrate, consider an investment in Payco American, a collection agencycompany that operates several offices nationwide. The company is not well known, itsstock is not very liquid, its earnings have fluctuated quite a bit in the past, and it doesn’teven pay a dividend. This suggests that Payco is risky and that its required rate ofreturn, r, should be relatively high. Even so, Payco’s r always has been quite lowrelative to the rates of return offered by most firms with similar risk. This informationindicates that investors regard Payco as being a low-risk company despite its uncertainprofits and its nonexistent dividend stream. The reason for this somewhat counter-intuitive fact relates to diversification and its effect on risk. Payco’s stock price risesduring recessions, whereas the prices of other stocks tend to decline when theeconomy slumps. Therefore, holding Payco in a portfolio of ‘‘normal’’ stocks tends tostabilize returns on the entire portfolio.

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 thecoefficient 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 averageannual return and standard deviation of the stock? (Answer: r¼ 6%; s¼ 7.9%)

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

Portfolio Risk—Holding Combinations of Assets 317

Portfolio ReturnsThe expected return on a portfolio, rp, is simply the weighted average of theexpected returns on the individual stocks in the portfolio, with each weight being theproportion of the total portfolio invested in each stock:

8–7Portfolio return ¼ rp ¼ w1r1 þ w2r2 þ � � � þ wNrN ¼

XN

j¼1

wjrj

Here the rj values are the expected returns on the individual stocks, the wj values arethe weights, and the portfolio includes N stocks. Note two points: (1) wj is the pro-portion of the portfolio’s dollar value invested in Stock j, which is equal to the value ofthe investment in Stock j divided by the total value of the portfolio, and (2) the wj valuesmust sum to 1.0.

Suppose security analysts estimate that the following returns could be expected onfour large companies:

If we formed a $100,000 portfolio, investing $25,000 in each of these four stocks, ourexpected portfolio return would be 14.0 percent:

rp ¼ wATTrATT þ wCitirCiti þ wGErGE þ wMicrorMicro

¼ 0:25ð8%Þ þ 0:25ð13%Þ þ 0:25ð19%Þ þ 0:25ð16%Þ ¼ 14:0%

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 mightdouble in price and provide a return of þ100 percent, whereas General Electric’sstock 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 eachother, so the portfolio’s return might still approach its expected return, even thoughthe individual stocks’ actual returns were far from their expected returns.

Portfolio RiskAs we just saw, the expected return of a portfolio is simply a weighted average of theexpected returns of the individual stocks in the portfolio. Unlike returns, the riskinessof a portfolio (sP) generally is not a weighted average of the standard deviations of theindividual securities in the portfolio. Instead, the portfolio’s risk usually is smaller thanthe weighted average of the individual stocks’ standard deviations. In fact, it is the-oretically possible to combine two stocks that by themselves are quite risky asmeasured by their standard deviations and form a completely risk-free portfolio—thatis, a portfolio with sP ¼ 0.

To illustrate the effect of combining securities, consider the situation depicted inFigure 8-5. The bottom section of the figure gives data on the rates of return for Stock

Company Expected Return, r

AT&T 8%Citigroup 13General Electric 19Microsoft 16

realized rate of return, €rThe return that isactually earned. Theactual return (€r) usuallydiffers from theexpected return (r).

expected return on aportfolio, rp

The weighted averageexpected return onstocks held in a portfolio.


W and Stock M individually as well as rates of return for a portfolio invested 50 percentin each stock. The three top graphs show the actual historical returns for eachinvestment from 2004 to 2008, and the lower graphs show the probability distributionsof returns, assuming that the future is expected to be like the past. The two stockswould be quite risky if they were held in isolation. When they are combined to formPortfolio WM, however, they are not risky at all. (Note: These stocks are called W andM 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 isbecause their returns move in opposite directions. That is, when W’s returns are low,

FIGURE 8-5 Rate of Return Distribution for Two Perfectly Negatively Correlated Stocks (r ¼ �1.0) and forPortfolio WM

25

15

20080

�10

Portfolio WM

25

15

rM

20080

�10

Stock M

25

15

rW

0

�10

Stock W

A. Rates of Return

2008

B. Probability Distribution of Returns

0

Probability Density

15 Percent

Stock W

� rwˆ

0

Probability Density

15 Percent

Stock M

0

Probability Density

15 Percent

Portfolio WM

� rMˆ � rWMˆ

¨ ¨ rWM¨

Year

2004 2005 2006 2007 2008

Portfolio WM(rWM)¨

1515151515150

%

%%

Stock M(rM)¨

�1040

�5351515

22.6%

%

%

Stock W(rW)¨

40 �10

35�515

15 22.6

%

%%

-Average return, rStandard deviation, s

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.


M’s returns are high, and vice versa. The relationship between any two variables iscalled correlation, and the correlation coefficient, r, measures the direction and thestrength of the relationship between the variables.5 In statistical terms, we say that thereturns on Stock W and Stock M are perfectly negatively correlated, with r ¼ �1.0.6

The opposite of perfect negative correlation—that is, r¼�1.0—is perfect positivecorrelation—that is, r ¼ þ1.0. Returns on two perfectly positively correlated stockswould move up and down together, and a portfolio consisting of two such stocks wouldbe exactly as risky as the individual stocks. This point is illustrated in Figure 8-6, inwhich 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 reducedif the portfolio contains perfectly positively correlated stocks.

Figure 8-5 and Figure 8-6 demonstrate that when stocks are perfectly negativelycorrelated (r ¼ �1.0), all risk can be diversified away; conversely, when stocks areperfectly positively correlated (r¼þ1.0), diversification is ineffective. In reality, moststocks are positively correlated, but not perfectly so. On average, the correlationcoefficient for the returns on two randomly selected stocks would be aboutþ0.4. Formost 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 isr ¼ þ0.67. Both Stock W and Stock Y have the same average return and standarddeviation—�r ¼ 15% and s ¼ 22.6%. A portfolio that consists of 50 percent of bothstocks has an average return equal to 15.0 percent, which is exactly the same as theaverage 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 theportfolio’s risk is not an average of the risks of its individual stocks—diversification hasreduced, but not eliminated, risk.

From these two-stock portfolio examples, we have seen that risk can be completelyeliminated in one extreme case (r¼�1.0), whereas diversification does no good in theother extreme case (r¼þ1.0). In between these extremes, combining two stocks into aportfolio reduces, but does not eliminate, the riskiness inherent in the individualstocks.

What would happen if the portfolio included more than two stocks? As a rule, theriskiness 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 answeris no, but the extent to which adding stocks to a portfolio reduces its risk depends on thedegree of correlation among the stocks: The smaller the positive correlation amongstocks included in a portfolio, the lower its total risk. If we could find a set of stockswhose correlations were negative, we could eliminate all risk. In the typical case, inwhich 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 fromþ1.0 (denoting that the two variables move in the same direction withexactly the same degree of synchronization every time movement occurs) to �1.0 (denoting that the variablesalways move with the same degree of synchronization, but in opposite directions). A correlation coefficient of zerosuggests that the two variables are not related to each other—that is, changes in one variable occur independently ofchanges in the other.

6Following is the computation of the correlation coefficient that measures the relationship between Stock W andStock M shown in Figure 8-5. The average return and standard deviation for both stocks are the same: �r ¼ 15% ands ¼ 22.6%.

Covariance ¼ ð40� 15Þð�10� 15Þ þ ð�10� 15Þð40� 15Þ þ ð35� 15Þð�5� 15Þ þ ð�5� 15Þ þ ð35� 15Þ þ ð15� 15Þð15� 15Þ5� 1

¼ �512:5

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

correlation coefficient, rA measure of the degreeof relationship betweentwo variables.


To test your understanding, consider the following question: Would you expect tofind higher correlations between the returns on two companies in the same industry orin different industries? For example, would the correlation of returns on Ford’s andGeneral Motors’ stocks be higher, or would the correlation coefficient be higherbetween either Ford or GM and Procter & Gamble (P&G)? How would thosecorrelations affect the risk of portfolios containing them?

Answer: Ford’s and GM’s returns have a correlation coefficient of approximately0.9 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.

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

Year

2004 2005 2006 2007 2008

Portfolio MM′(rMM′)¨

�10 40

�5351515

22.6

%

%%

Stock M′(rM′)¨

�1040

�5351515

22.6%

%

%

Stock M(rM)¨

�10 40

�53515

15 22.6

%

%%

-Average return, rStandard deviation, s

25

15

20080

�10

25

15

20080

�10

25

15

0

�10

Probability Density Probability DensityProbability Density

2008

Portfolio MM′ Stock M′Stock M¨ ¨rM rM′ rMM′¨

A. Rates of Return

0 15 Percent 0 15 Percent 0 15 Percentˆ= rM ˆ=rM′ ˆ=rp


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.


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

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

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


25

15

20080

�15

Portfolio WY

25

15

20080

�15

Stock Y

25

15

0

�15

Stock W

A. Rates of Return

2008

rW¨ rY¨ rWY¨

0

ProbabilityDensity

15 Percent

Portfolio WY

Stocks W and Y

= rpˆ

Year

2004 2005 2006 2007 2008

Average return, Standard deviation, s

Stock WrW¨

Stock YrY¨

Portfolio WYrWY¨

34 5

38 �11

9 15

20.6

%

%%

282041

�173

1522.6

%

%

%

40 �10

35�515

15 22.6

%

%%

-r

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.


economy is strong and to do poorly when it is weak.7 Thus, even very large portfoliosend up with substantial amounts of risk, though the risks generally are less than if all ofthe 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 ofrandomly selected stocks listed on the New York Stock Exchange (NYSE). Standarddeviations 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. Asthe graph illustrates, the riskiness of a portfolio consisting of average NYSE stocksgenerally tends to decline and to approach some minimum limit as the size of theportfolio increases. According to the data, s1, the standard deviation of a one-stockportfolio (or an average stock), is approximately 28 percent. A portfolio consisting of allof the stocks in the market, which is called the market portfolio, would have a standarddeviation,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 individualstock can be eliminated if the stock is held as part of a reasonably well-diversified

FIGURE 8-8 Effects of Portfolio Size on Portfolio Risk for Average Stocks

10

5

Portfolio Risk, �p (%)

0All NYSEStocks

20

σM = 15.0

25

28

403020101

TotalRisk Nondiversifiable

Risk Related toMarket Fluctuations(Systematic Risk)

Minimum Attainable Risk in aPortfolio of Average Stocks

Company-Specific, orDiversifiable, Risk(Unsystematic Risk)

7It is not too difficult to find a few stocks that happened to rise because of a particular set of circumstances in the pastwhile most other stocks were declining. It is much more difficult to find stocks that could logically be expected to goup in the future when other stocks are falling. Payco American, the collection agency discussed earlier, is one ofthose rare exceptions.


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 marketmovements that affect almost all stocks.

That part of the risk of a stock that can be eliminated is called diversifiable, orfirm-specific, or unsystematic, risk; that part that cannot be eliminated is callednondiversifiable, or market, or systematic, risk. Although the name given to the risk isnot especially important, the fact that a large part of the riskiness of any individual stockcan be eliminated through portfolio diversification is vitally important.

Firm-specific, or diversifiable, risk is caused by such things as lawsuits, loss ofkey personnel, strikes, successful and unsuccessful marketing programs, the winningand 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 inone 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 affectedsimilarly (negatively) by these market conditions, systematic risk cannot be eliminatedby portfolio diversification.

We know that investors demand a premium for bearing risk. That is, the riskier asecurity, 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 thanthe risk of the individual securities in the portfolio, how should we measure theriskiness of an individual stock? The answer is this: The relevant riskiness of anindividual stock is its contribution to the riskiness of a well-diversified portfolio. Inother words, the riskiness of General Electric’s stock to a doctor who has a portfolioof 40 stocks or to a trust officer managing a 150-stock portfolio is the contributionthat the GE stock makes to the entire portfolio’s riskiness. The stock might be quiterisky if held by itself, but if much of this total risk can be eliminated throughdiversification, then its relevant risk—that is, its contribution to the portfolio’srisk—is much smaller than its total, or stand-alone, risk.

A simple example will help clarify this point. Suppose you are offered the chanceto 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 ¼0.5($20,000) þ 0.5(�$16,000). It is a highly risky proposition, however, because youhave a 50 percent chance of losing $16,000. For this reason, you might refuse to makethe bet. Alternatively, suppose you were offered the chance to flip a coin 100 times; youwould win $200 for each head but lose $160 for each tail. It is possible that you wouldflip 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 50heads and about 50 tails, winning a net of about $2,000. Although each individual flip isa risky bet, collectively this scenario is a low-risk proposition because most of the riskhas been diversified away. This concept underlies the practice of holding portfolios ofstocks rather than just one stock. Note that all of the risk associated with stocks cannotbe eliminated by diversification: Those risks related to broad, systematic changes in theeconomy that affect the stock market will remain.

Are all stocks equally risky in the sense that adding them to a well-diversifiedportfolio 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 dif-ferent degrees of relevant (systematic) risk. How can we measure the relevant risk of anindividual stock? As we have seen, all risk except that related to broad marketmovements can, and presumably will, be diversified away. After all, why accept riskthat we can easily eliminate? The risk that remains after diversifying is market risk

relevant riskThe portion of a secur-ity’s risk that cannot bediversified away; thesecurity’s market risk. Itreflects the security’scontribution to the riskof a portfolio.

firm-specific(diversifiable) riskThat part of a security’srisk associated withrandom outcomesgenerated by events, orbehaviors, specific tothe firm. It can beeliminated by properdiversification.

market (nondiversifiable)riskThe part of a security’srisk associated witheconomic, or market,factors that systemati-cally affect firms. Itcannot be eliminated bydiversification.


(that is, risk that is inherent in the market), and it can be measured by evaluating thedegree 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 sys-tematic risk, which in turn depends on the sensitivity of the firm’s operations toeconomic events such as interest rate changes and inflationary pressures. Because thegeneral movements in the financial markets reflect movements in the economy, we canmeasure the market risk of a stock by observing its tendency to move with the marketor with an average stock that has the same characteristics as the market. The measureof a stock’s sensitivity to market fluctuations is called its beta coefficient, designatedwith the Greek letter bb.

An average-risk stock is defined as one that tends to move up and down in step withthe general market as measured by some index, such as the Dow Jones IndustrialAverage, the S&P 500 Index, or the New York Stock Exchange Composite Index. Sucha stock will, by definition, have a beta (b) of 1.0. This value indicates that, in general, ifthe market moves up by 10 percent, the stock price will also increase by 10 percent; ifthe market falls by 10 percent, the stock price will decline by 10 percent. A portfoliocomposed of such b ¼ 1.0 stocks will move up and down with the broad marketaverages, 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 willbe half as risky as a portfolio that includes only b¼ 1.0 stocks—it will rise and fall onlyhalf as much as the market. If b¼ 2.0, the stock’s relevant risk is twice as volatile as anaverage 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 youheld 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 graphassume that in 2006 the ‘‘market,’’ defined as a portfolio consisting of all stocks, had atotal return (dividend yield plus capital gains yield) of rM¼ 14%, and Stocks H, A, andL (for high, average, and low risk) also had returns of 14 percent. In 2007 the marketrose sharply, and the return on the market portfolio was rM ¼ 28%. Returns on thethree stocks also increased: the return on H soared to 42 percent; the return on Areached 28 percent, the same as the market; and the return on L increased to only 21percent. 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 samedirection as the market, but H was by far the most volatile; A was just as volatile as themarket; and L was less volatile than the market.

The beta coefficient measures a stock’s volatility relative to an average stock (or themarket), which has b ¼ 1.0. We can calculate a stock’s beta by plotting a line like thoseshown in Figure 8-9. The slopes of these lines show how each stock moves in response to amovement in the general market. Indeed, the slope coefficient of such a ‘‘regression line’’ isdefined as a beta coefficient. Betas for literally thousands of companies are calculated andpublished by Merrill Lynch, Value Line, and numerous other organizations. Table 8-4provides the beta coefficients for some well-known companies. Most stocks have betas inthe range of 0.50 to 1.50, and the average for all stocks is 1.0 by definition.8

beta coefficient, bbA measure of the extentto which the returns ona given stock move withthe stock market.

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 benegative. Note, however, that few stocks have negative betas. Payco American, the collection agency company,might have a negative beta.


If we add a higher-than-average-beta stock (b> 1.0) to an average-beta (b¼ 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-riskportfolio, the portfolio’s beta and risk will decline. Thus, because a stock’s betameasures its contribution to the riskiness of a portfolio, theoretically beta is the correctmeasure 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 investorsdo diversify, either by holding large portfolios or by purchasing shares inmutual funds. We are left, then, with market risk, which is caused by generalmovements in the stock market and which reflects the fact that most stocks

FIGURE 8-9 Relative Volatility of Stocks H, A, and L

20

10

Return on Stock j, ri(%)

0

�10

30

30

40

50

2010�10�20

�20

�30

�40

�50

Stock H,High Risk: β = 2.0

Stock A,Average Risk: β = 1.0

Stock L,Low Risk: β = 0.5

14

Return on the Market, (%)

14

rM¨

¨

¨ ¨ ¨ rM

1428

�14

1421 0

1428

�14

1442

�42

2006 2007 2008

rH rA rL


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 firm-specific risk.

3. Investors must be compensated for bearing risk. That is, the greater theriskiness of a stock, the higher its required return. Such compensation isrequired only for risk that cannot be eliminated by diversification. If riskpremiums existed on stocks with high diversifiable risk, well-diversifiedinvestors would start buying these securities and bidding up their prices, andtheir final (equilibrium) expected returns would reflect only nondiversifiablemarket risk.

An example might help clarify this point. Suppose half of Stock A’s risk ismarket 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 youare exposed to all of its risk. As compensation for bearing so much risk, youwant a risk premium of 8 percent higher than the 5 percent Treasury bondrate. That is, you demand a return of 13 percent (¼ 5% þ 8%) from thisinvestment. But suppose other investors, including your professor, are welldiversified; they also hold Stock A, but they have eliminated its diversifiablerisk 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 areturn 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, otherinvestors, including your professor, would buy it. If it yielded the13 percent you demand, you would be willing to buy the stock, but thewell-diversified investors would compete with you for its acquisition.

TABLE 8-4 Beta Coefficients for Selected Companies

Company Beta Industry/Product

I. Above Average Market Risk: b >> 1.0Nortel Networks Corporation 4.18 Communications equipment; telephone equipmentYahoo! Inc. 3.40 Computer services/global Internet communicationsE�TRADE Group Inc. 2.87 Investment services/online financial servicesSun Microsystems 2.80 Computers and peripheralseBay 1.76 Retail (specialty nonapparel)/web-based auction

II. Average Market Risk: b �� 1.0Dow Jones & Company 1.02 Publishing and printing (newspapers)Ryland Group 1.01 Home buildingScotts Corporation 0.99 Pesticide, fertilizer, and agricultural chemicalsKrispy Kreme 0.99 Snack and nonalcoholic beverage barsToyota Motor Corporation 0.99 Auto and truck manufacturer

III. Below Average Market Risk: b << 1.0Barnes & Noble 0.75 Specialty retailing; bookstoresKroger Company 0.50 Food retailing; supermarketsWalgreen Company 0.28 Retail drugs; pharmacies and drugstoresGillette Company 0.28 Personal and household productsProgress Energy 0.17 Electric utilities; electric power generation

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


They would bid its price up and its yield down, which would keep you fromgetting the stock at the return you need to compensate you for taking on itstotal risk. In the end, you would have to accept a 9 percent return or elsekeep your money in the bank. Thus, risk premiums in a market populatedwith rational investors—that is, those who diversify—will reflect onlymarket 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 forbeta follow:

b ¼ 0.5: The stock’s relevant risk is only half as volatile, or risky, as theaverage 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 riski-ness of a diversified portfolio, beta (�) is a better measure of a stock’s rele-vant risk than is standard deviation (�), which measures total, or stand-alone, risk.

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

8–8 Portfolio beta ¼ bp ¼ w1b1 þ w2b2 þ � � � þ wNbN

¼XN

j¼1

wjbj

Here bP, the beta of the portfolio, reflects how volatile the portfolio is in relation tothe market; wj is the fraction of the portfolio invested in the jth stock; and bj is thebeta coefficient of the jth stock. For example, if an investor holds a $105,000 port-folio consisting of $35,000 invested in each of three stocks, and each of the stocks hasa 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 shouldexperience relatively narrow price swings and demonstrate relatively small rate-of-return fluctuations. When graphed in a fashion similar to Figure 8-9 , the slopeof its regression line would be 0.7, which is less than that for a portfolio of averagestocks.

Now suppose one of the existing stocks is sold and replaced by a stock withbj ¼ 2.5. This action will increase the riskiness of the portfolio from bP1¼ 0.7 tobP2 ¼ 1.3:

bP2 ¼ ð1=3Þð0:7Þ þ ð1=3Þð0:7Þ þ ð1=3Þð2:5Þ ¼ 1:3

Had a stock with bj¼ 0.4 been added, the portfolio beta would have declined from0.7 to 0.6. Adding a low-beta stock, therefore, would reduce the riskiness of theportfolio.


THE RELATIONSHIP BETWEEN RISK AND RATES

OF RETURN (CAPM)

In the preceding section, we saw that beta is the appropriate measure of a stock’srelevant risk. Now we must specify the relationship between risk and return. For agiven level of beta, what rate of return will investors require on a stock to compensatethem for assuming the risk? To determine an investment’s required rate of return, weuse a theoretical model called the Capital Asset Pricing Model (CAPM). The CAPMshows how the relevant risk of an investment as measured by its beta coefficient is usedto 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 theprobability distribution for the stock’s returns.

rj ¼ Required rate of return on the jth stock; rj is the ratethat investors demand for investing in Stock j. If rj < rj,you would not purchase this stock, or you would sell itif you owned it; if rj > rj, you would want to buy thestock; and, you would be indifferent if rj¼ rj.

rRF ¼ Risk-free rate of return. In this context, rRF is generallymeasured by the return on long-term U.S. Treasurysecurities.

bj ¼ Beta coefficient of the jth stock. The beta of an averagestock is bA ¼ 1.0.

rM ¼ Required rate of return on a portfolio consisting of allstocks, which is the market portfolio. rM is also therequired rate of return on an average (bA ¼ 1.0) stock.

Explain the following statement: ‘‘A stock held as part of a portfolio is generallyless 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 thenumber of stocks in the portfolio? Explain.

What is meant by diversifiable risk and nondiversifiable risk? What is anaverage-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 JonesIndustrial Average index over the past five years, what would the slope of theline you obtained indicate about the stock’s risk?

Suppose you have a portfolio that includes two stocks. You invested 60 percentof your total funds in a stock that has a beta equal to 3.0 and the remaining40 percent of your funds in a stock that has a beta equal to 0.5. What is theportfolio’s beta? (Answer: 2.0)

Capital Asset PricingModel (CAPM)A model used todetermine the requiredreturn on an asset,which is based on theproposition that anyasset’s return should beequal to the risk-freereturn plus a risk pre-mium that reflects theasset’s nondiversifiablerisk.

The Relationship Between Risk and Rates of Return (CAPM) 329

RPM ¼ (rM � rRF) ¼ Market risk premium. This is the additional return abovethe risk-free rate required to compensate an average in-vestor 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’srisk premium is less than, equal to, or greater thanthe premium on an average stock, depending onwhether its relevant risk as measured by beta is lessthan, 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.

The market risk premium (RPM) depends on the degree of aversion thatinvestors on average have to risk.9 Let’s assume that at the current time, Treasurybonds yield rRF¼ 5% and an average share of stock has a required return of rM¼ 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 betwice as high. Conversely, if a stock’s relevant risk is only half as much as that of anotherstock, its risk premium should be half as large. Furthermore, we can measure a stock’srelevant risk by finding its beta coefficient. Therefore, if we know the market riskpremium, RPM, and the stock’s risk as measured by its beta coefficient, bj, we can findits risk premium as the product RPM�bj. For example, ifbj¼ 0.5 and RPM¼ 6%, thenRPj is 3 percent:

8–9 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 returnþ 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 ðRPMÞbj ¼ Capital Assest Pricing Model ðCAPMÞ¼ rRF þ ðrM � rRFÞbj

¼ 5%þ ð11%� 5%Þð0:5Þ¼ 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. Brighamand Phillip R. Daves, Intermediate Financial Management, 9th ed. (Cincinnati, OH: South-Western College Pub-lishing, 2007). Note that we cannot measure the risk premium of an average stock, RPM ¼ rM� rRF, with greatprecision 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 isan estimate of the expected return on the S&P 500, the market risk premium varies somewhat from year to year. Ithas generally ranged from 4 to 8 percent during the past 20 years. Chapter 3 of Intermediate Financial Managementalso discusses the assumptions embodied in the CAPM framework. Some of the assumptions of the CAPM theoryare unrealistic. As a consequence, the theory does not hold exactly.

market risk premium(RPM)The additional returnover the risk-free rateneeded to compensateinvestors for assumingan average amount ofrisk.


Equation 8–11, which is the CAPM equation for equilibrium pricing, is called thesecurity market line (SML).

If some other stock were riskier than Stock j and had bj2 ¼ 2.0, then its requiredrate 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, thesame 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 followingpoints:

1. Required rates of return are shown on the vertical axis, and risk (as measuredby beta) is shown on the horizontal axis. This graph is quite different from theone shown in Figure 8-9, in which the returns on individual stocks are plottedon the vertical axis and returns on the market index are shown on the hori-zontal axis. The slopes of the three lines in Figure 8-9 represent the threestocks’ betas. In Figure 8-10, these three betas are plotted as points on thehorizontal axis.

2. Risk-free securities have bj ¼ 0; therefore, rRF appears as the vertical axisintercept 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 ofthe line, (b) the greater the risk premium for any stock, and (c) the higher the

security market line(SML)The line that shows therelationship betweenrisk as measured bybeta and the requiredrate of return for indi-vidual securities.

FIGURE 8-10 The Security Market Line (SML)

Required Rateof Return (%)

Risk-FreeRate: 5%

Safe StockRiskPremium: 3%

Market (AverageStock) RiskPremium: 6%

Relatively RiskyStock’s RiskPremium: 12%

rRF = 5

0 Risk, βj1.5 2.01.00.5

rLow = 8

rHigh = 17

rM = rA = 11

SML: ri = rRF � (rM � rRF)βi


required rate of return on stocks.10 These points are discussed further in alater section.

4. The values we worked out for stocks with bj ¼ 0.5, bj ¼ 1.0, and bj ¼ 2.0agree 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 changesin interest rates, investors’ risk aversion, and individual companies’ betas. Suchchanges are discussed in the following sections.

The Impact of InflationAs we learned in Chapter 5, interest amounts to ‘‘rent’’ on borrowed money, or theprice 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 twoelements: (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 by2 percent. Figure 8-11 illustrates the effects of such a change. Notice that under theCAPM, the increase in rRF also causes an equal increase in the rate of return on all

FIGURE 8-11 Shift in the SML Caused by a 2 Percent Increase in Inflation


Real Risk-FreeRate = 3%

0 Risk, βj1.5 2.01.00.5

rM2 = 13

SML2

rM1 = 11

rRF2 = 7

rRF1 = 5

r* = 3Original IP = 2%

Increase inInflation = 2%

SML1

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 y¼ r and x¼ b, and we go fromthe origin to bM¼ 1.0, we see that the slope is (rM – rRF)/(bM –bRF)¼ (11% – 5%)/(1 – 0)¼ 6%. Thus, the slope ofthe SML is equal to (rM – rRF), the market risk premium. In Figure 8-10, rj ¼ 5% þ (6%)bj, so a doubling ofbeta (for example, from 1.0 to 2.0) would produce an 8-percentage-point increase in rj. In this case, the total riskpremium on Stock j would double—that is, RPj ¼ (8%)2.0 ¼ 16%.

11Long-term Treasury bonds also contain a maturity risk premium (MRP). Here we include the MRP in r� tosimplify the discussion.


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

Changes in Risk AversionThe slope of the security market line reflects the extent to which investors are averse torisk. 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 wouldalso provide an expected return of 5 percent. If there was no risk aversion, there wouldbe no risk premium, so the SML would be horizontal. As risk aversion increases, sodoes 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 riskpremium increases from 6 percent to 8 percent, and rM increases from rM1 ¼ 11% torM2¼ 13%. The returns on other risky assets also rise, with the effect of this shift in riskaversion being more pronounced on riskier securities. For example, the requiredreturn on a stock with bj¼ 0.5 increases by only 1 percentage point, from 8 percent to9 percent. By comparison, the required return on a stock with bj ¼ 2.0 increases by

FIGURE 8-12 Shift in the SML Caused by Increased Risk Aversion


rRF = 5

0 Risk, βj1.5 2.01.00.5

17

21

rM2 = 13

rM1 = 11

98

SML2

SML1

New Market RiskPremium, rM2 � rRF � 8%

Original Market RiskPremium, rM1 � rRF � 6%

12Recall that the inflation premium for any asset is equal to the average expected rate of inflation over the life of theasset. In this analysis, we must therefore assume either that all securities plotted on the SML graph have the samelife 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-termrate (the T-bill rate). Traditionally, the T-bill rate was used, but a movement toward use of the T-bond rate hasoccurred in recent years because a closer relationship exists between T-bond yields and stocks than between T-billyields and stocks. See Stocks, Bonds, Bills, and Inflation, 2006 Yearbook (Chicago: Ibbotson & Associates, 2007) fora discussion.


4 percentage points, from 17 percent to 21 percent. Because �RPj ¼ �RPM(bj) ¼(13% – 11%)bj¼ (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 higherbeta coefficients experience greater changes in their required rates of return thaninvestments with lower betas.

Changes in a Stock’s Beta CoefficientAs we will see later in this book, a firm can affect its beta risk by changing thecomposition 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 basicpatents, can also alter a company’s beta. When such changes occur, the required rate ofreturn, r, changes as well, and, as we saw in Chapter 7, this change will affect the priceof the firm’s stock. For example, consider Genesco Manufacturing, with a beta equal to1.0. Suppose some action occurred that caused this firm’s beta to increase from 1.0 to1.5. If the conditions depicted in Figure 8-10 held, Genesco’s required rate of returnwould increase from

r1 ¼ rRF þ ðrM � rRFÞbj

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 itsbeta coefficient or in expected inflation, will affect the price of the security.

A Word of CautionA word of caution about betas and the CAPM is in order here. First, the model wasdeveloped under very restrictive assumptions. Some of the assumptions include thefollowing: (1) all investors have the same information, which leads to the same expec-tations about future stock prices; (2) everyone can borrow and lend at the risk-free rateof return; (3) stocks (or any other security) can be purchased in any denomination orfraction 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 haveavailable only ex post, or past, data. The betas we calculate show how volatile a stock hasbeen 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 dramaticallyfrom its past volatility.

Although the CAPM represents a significant step forward in security pricingtheory, it does have some potentially serious deficiencies when applied in practice. As aconsequence, estimates of rj found through use of the SML might be subject toconsiderable error. For this reason, many investors and analysts use the CAPM andthe concept of b to provide ‘‘ballpark’’ figures for further analysis. The concept thatinvestors should be rewarded only for taking relevant risk makes sense. And the CAPMprovides an easy way to get a rough estimate of the relevant risk and the appropriaterequired rate of return of an investment.


STOCK MARKET EQUILIBRIUM

Based on our previous discussion, we know that we can use the CAPM to find therequired return for an investment (say, Stock Q), which we designate as rQ. Supposethe risk-free return is 5 percent, the market risk premium is 6 percent, and Stock Q hasa beta of 1.5 (bQ ¼ 1.5). In this case, the marginal, or average, investor will require areturn 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, andwill be indifferent (and therefore will hold but not buy or sell Stock Q) if the expectedrate of return is exactly 14 percent. Now suppose the investor’s portfolio contains StockQ, 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 yearforever. The last dividend paid was D0 ¼ $3, so the next expected dividend is

D1 ¼ $3:00ð1:04Þ ¼ $3:12

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, ormaintain the current position?

Recall from Chapter 7 that we can calculate Stock Q’s expected rate of return asfollows (see Equation 7–6):

rQ ¼D1

P0þ g ¼ $3:12

$34:67þ 0:04 ¼ 0:09þ 0:04 ¼ 0:13 ¼ 13%

Differentiate between the expected rate of return ( r ) and the required rate ofreturn (r) on a stock. Which would have to be larger to persuade you to buy thestock?

What are the differences between the relative volatility graph (Figure 8-9 ), inwhich ‘‘betas are made,’’ and the SML graph (Figure 8-10 ), in which ‘‘betas areused’’? Consider the methods of constructing the graphs and the purposes forwhich they were developed.

What happens to the SML graph (1) when inflation increases or (2) wheninflation decreases?

What happens to the SML graph (1) when risk aversion increases or (2) whenrisk aversion decreases? What would the SML look like if investors wereindifferent 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 equals4 percent and the expected market return equals 10 percent, what is StockF’s required rate of return? (Answer: rF ¼ 11.2%)

Stock Market Equilibrium 335

This value is plotted on Figure 8-13 as Point Q, which is below the SML. Because theexpected rate of return, rQ ¼ 13%, is less than the required return, rQ ¼ 14%, thismarginal investor would want to sell the stock, as would other holders. Because fewpeople would want to buy at the $34.67 price, the current owners would be unable tofind 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 ofreturn, 14 percent, would be equal to the required rate of return:

rQ ¼$3:12

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

Had the stock initially sold for less than $31.20—say, $28.36—events would havebeen reversed. Investors would have wanted to buy the stock because its expected rateof return ( r ¼ 15%) would have exceeded its required rate of return, and buy orderswould 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 therequired rate of return: rj¼ rj.

2. The actual market price of the stock must equal its intrinsic value as esti-mated by the marginal investor: P0 ¼ P0 .

Of course, some individual investors might believe that rj > rj and P0 > P0 hencethey would invest most of their funds in the stock. Other investors might ascribe to theopposite view and sell all of their shares. Nevertheless, it is the marginal investor whoestablishes the actual market price. For this investor, rj ¼ rj and P0 ¼ P0: If theseconditions do not hold, trading will occur until they do hold.

FIGURE 8-13 Expected and Required Returns on Stock Q


SML: ri � rRF � (rM � rRF)βi

rRF � 5

0 Risk, βj1.5

Q

1.0

rQ � 13

rQ �14

rM � 11

ˆ

equilibriumThe condition underwhich the expectedreturn on a securityis just equal to itsrequired return, r¼ r,and the price is stable.


DIFFERENT TYPES OF RISK

In Chapter 5, we introduced the concept of risk in our discussion of interest rates, orthe 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 rateþ Risk premium

¼ rRF þ RP

¼ ½r� þ IP� þ ½DRPþ LPþMRP�

Remember that here

r¼Quoted, or nominal, rate of interest on a given security. There are manydifferent securities, hence many different quoted interest rates.

rRF¼Nominal risk-free rate of return.

r�¼Real risk-free rate of interest, which is the interest rate that would existon a security with a guaranteed payoff if inflation is expected to be zeroduring the investment period.

IP¼ Inflation premium, which equals the average inflation rate expected overthe life of the security.

DRP¼Default risk premium, which reflects the chance that the borrower willnot pay the debt’s interest or principal on time.

LP¼Liquidity, or marketability, premium, which reflects the fact that someinvestments are more easily converted into cash on a short notice at a‘‘reasonable price’’ than are other securities.

MRP¼Maturity risk premium, which accounts for the fact that longer-termbonds experience greater price reactions to interest rate changes thando short-term bonds.

The discussion in Chapter 5 presented an overall view of interest rates and generalfactors that affect these rates. But we did not discuss risk evaluation in detail; rather, wedescribed some of the factors that determine the total risk associated with debt, such asdefault risk, liquidity risk, and maturity risk. In reality, these risks also affect othertypes of investments, including equity. Equity does not represent a legal contract thatrequires the firm to pay defined amounts of dividends at particular times or to ‘‘act’’ inspecific ways. There is, however, an expectation that positive returns will be generated

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 itinto equilibrium.

Suppose Porter Pottery’s stock currently sells for $32. The company, which isgrowing at a constant rate, expects its next dividend to equal $3.20. Analystshave determined that the market value of the stock is currently in equilibriumand that investors require a rate of return equal to 14 percent to purchase thestock. If the price of the stock increases to $35.56 tomorrow after Porter’s year-end 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 of Risk 337

through future distributions of cash because dividends will be paid, capital gains willbe generated through growth, or both. Investors also expect the firm to behave‘‘appropriately.’’ If these expectations are not met, investors generally consider thefirm in ‘‘default’’ of their expectations. In such cases, as long as no laws have beenbroken, stockholders generally do not have legal recourse, as would be the case for adefault 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 5by showing how the risk premium associated with any investment should be deter-mined (at least in theory). The basis of our discussion is Equation 5–3, which wedevelop further in this chapter as follows:

rj ¼ Risk-free rateþ Risk premium

¼ rRF þ ðrM � rRFÞbj ¼ CAPM

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

Total risk ¼ s ¼ Systematic risk þUnsystematic risk

¼MarketðeconomicÞriskþ Firm-specific risk

¼ Nondiversifiable risk þDiversifiable risk

¼ Cannot be eliminated þ Can be eliminated

Relevant risk ¼ Nondiversifiable risk þDiversifiable risk ðeliminatedÞ¼ Systematic risk

Systematic risk is represented by an investment’s beta coefficient, b, inEquation 8–11.

The specific types and sources of risk to which a firm or an investor is exposed arenumerous and vary considerably depending on the situation. A detailed discussion ofall the different types of risks and the techniques used to evaluate risks is beyond thescope of this book. But you should recognize that risk is an important factor in thedetermination of the required rate of return (r), which, according to the followingequation, is one of the two variables we need to determine the value of an asset:

Value ¼ CF1

ð1þ rÞ1þ CF2

ð1þ rÞ2þ � � � þ CFn

ð1þ rÞn ¼Xn

t¼1

CFt

ð1þ rÞt

This equation was first introduced in Chapter 1, and it was discussed in greater detail inChapter 6 and Chapter 7. What is important to understand here is that the value of anasset, which could be a stock or a bond, is based on the cash flows that the asset isexpected 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 anindication as to how the required rate of return, r, should be determined, and we showthat investors demand higher rates of return to compensate them for taking greateramounts of ‘‘relevant’’ risks.

Because it is an important concept and has a direct effect on value, we will continueto discuss risk in the remainder of the book. Although there are instances in which thediscussions focus on the risk to which investors are exposed, most of the discussions


focus on risks that affect corporations. Because we discuss different types of riskthroughout the book, we thought it might be a good idea to summarize and describethese risks in brief terms. Table 8-5 shows the risks that are discussed in the book andindicates whether each risk is considered a component of systematic (nondiversifiable)or unsystematic (diversifiable) risk. Note that (1) this table oversimplifies risk analysisbecause 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, thistable should show the relationships among the different risks discussed in the book.

Classify default risk, maturity risk, and liquidity risk as either diversifiable ornondiversifiable risk.

TABLE 8-5 Different Types (Sources) of Risk

General Type of Risk Name of Risk Brief Description

I. Systematic risks(nondiversifiable risk;market risk; relevant risk)

Interest rate risk When interest rates change, (1) the values of investments change(in opposite directions) and (2) the rate at which funds can bereinvested also changes (in the same direction).

Inflation risk The primary reason short-term interest rates change is becauseinvestors change their expectations about future inflation.

Maturity risk Long-term investments experience greater price reactions tointerest rate changes than do short-term bonds.

Liquidity risk Reflects the fact that some investments are more easily convertedinto cash on a short notice at a ‘‘reasonable price’’ than are othersecurities.

Exchange rate risk Multinational firms deal with different currencies; the rate atwhich the currency of one country can be exchanged into thecurrency of another country—that is, the exchange rate—changesas market conditions change.

Political risk Any action by a government that reduces the value of aninvestment.

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

Business risk Risk that would be inherent in the firm’s operations if it used nodebt—factors such as labor conditions, product safety, quality ofmanagement, competitive conditions, and so forth, affect firm-specific risk.

Financial risk Risk associated with how the firm is financed—that is, its credit risk.Default risk Part of financial risk—the chance that the firm will not be able to

service its existing debt.

III. Combined risks(some systematic riskand some unsystematicrisk)

Total risk The combination of systematic risk and unsystematic risk; alsoreferred to as stand-alone risk, because this is the risk an investortakes if he or she purchases only one investment, which istantamount to ‘‘putting all your eggs into one basket.’’

Corporate risk The riskiness of the firm without considering the effect of stock-holder diversification; based on the combination of assets held bythe firm (inventory, accounts receivable, plant and equipment,and so forth). Some diversification exists because the firm’s assetsrepresent a portfolio of investments in real assets.

Different Types of Risk 339

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

� What does it mean to take risk when investing? In finance, risk is defined as thechance of receiving a return other than the one that is expected. Thus, aninvestment 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 returnless than expected—and ‘‘good risk’’—that is, the chance that it will return morethan 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 andreturn of an investment related? An investment’s risk is measured by the variabilityof 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 ofan investment is measured as a weighted average of all of the possible returns theinvestment can generate in the future, with the weights being the probability thatthe particular return will occur.

� How can investors reduce risk? Risk can be reduced through diversification.Investors achieve diversification by forming portfolios that contain numerousfinancial securities (perhaps stocks and bonds) that are not strongly related toeach other. For example, an investor can form a well-diversified portfolio bypurchasing the stocks of 40 or more companies in different industries, such astransportation, utilities, health care, entertainment, food services, and so forth.Total risk, which is equal to market (systematic) risk plus firm-specific (unsyste-matic) risk, can be reduced through diversification because little or no unsyste-matic risk should exist in a well-diversified investment portfolio.

� For what type of risk is an average investor rewarded? Investors should berewarded only for risk that they must take. Because firm-specific, or unsystematic,risk can be reduced or eliminated through diversification, investors who do notdiversify their investment portfolios should not be rewarded for taking such risk.Consequently, an investment’s relevant risk is its systematic, or market, risk, which isthe risk for which investors should be rewarded. Systematic risk cannot be reducedthrough diversification. An investment’s ‘‘irrelevant’’ risk is its firm-specific, orunsystematic, 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 aninvestment 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 itsrequired return, r. When r < r, investors will not purchase the security and thosewho own the security tend to sell it, which causes the security’s price to decreaseand its expected return to increase until r ¼ r.

Chapter Essentials—The Answers

ETHICAL DILEMMA

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Retirement Investment Products (RIP) offers a fullcomplement of retirement planning services and adiverse line of retirement investments that have

varying degrees of risk. With the investment prod-ucts available at RIP, investors could form retire-ment funds with any level of risk preferred, from risk


The concepts presented in this chapter should help you to better understand the rela-tionship 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 aninvestment portfolio that has the level of risk with which you are comfortable.

What important principles should I remember from this chapter wheninvesting? First, remember that risk and return are positively related. As a result, inmost 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 possibleinvestments, never separate ‘‘risk’’ and ‘‘return’’—that is, do not consider the return ofan investment without also considering its risk. Second, remember that you can reducesome investment risk through diversification, which can be achieved by purchasingdifferent investments that are not highly positively related to each other. In manyinstances, you can reduce risk without reducing the expected rate of return associatedwith your investment position.

How can I diversify if I don’t have enough money to purchase 40 differentsecurities? Mutual funds, which we briefly discussed in Chapter 3, provide inves-tors with the opportunity to diversify their investment positions because these

Chapter Essentials—Personal Finance

free to extremely risky. RIP’s reputation in theinvestment community is impeccable because theservice agents who advise clients are required to fullyinform their clients of the risk possibilities that existfor any investment position, whether it is recom-mended by an agent or requested by a client. Since1950, RIP has built its investment portfolio ofretirement funds to $60 billion, which makes it oneof the largest providers of retirement funds in theUnited States.

You work for RIP as an investment analyst. One ofyour responsibilities is to help form recommendationsfor the retirement fund managers to evaluate whenmaking investment decisions. Recently, Howard, aclose friend from your college days who now works forSunCoast Investments, a large brokerage firm, calledto tell you about a new investment that is expected toearn very high returns during the next few years. Theinvestment is called a ‘‘Piggy-back Asset InvestmentDevice,’’ or PAID for short. Howard told you that hereally does not know what this acronym means orhow the investment is constructed, but all the reportshe has read indicate PAIDs should be a hot invest-ment in the future, so the returns should be veryhandsome for those who get in now. The one piece ofinformation he did offer was that a PAID is a rathercomplex investment that consists of a combination ofsecurities whose values are based on numerous debtinstruments issued by government agencies, includingthe Federal National Mortgage Association, theFederal Home Loan Bank, and so on. Howardmade it clear that he would like you to consider

recommending to RIP that PAIDs be purchasedthrough SunCoast Investments. The commissionsfrom such a deal would bail him and his family out ofa financial crisis that resulted because they had badluck with their investments in the 2001 financialmarkets. Howard has indicated that somehow hewould reward you if RIP invests in PAIDs throughSunCoast because, in his words, ‘‘You would literallybe saving my life.’’ You told Howard you would thinkabout it and call him back.

Further investigation into PAIDs has yieldedlittle additional information beyond what pre-viously was provided by Howard. The new invest-ment is intriguing because its expected return isextremely high compared with similar investments.Earlier this morning, you called Howard to quiz hima little more about the return expectations and to tryto get an idea concerning the riskiness of PAIDs.Howard was unable to adequately explain the riskassociated with the investment, although hereminded you that the debt of U.S. governmentagencies is involved. As he says, ‘‘How much risk isthere with government agencies?’’

The PAIDs are very enticing because RIP canattract more clients if it can increase the return offeredon its investments. If you recommend the newinvestment and the higher returns pan out, you willearn a very sizable commission. In addition, you willbe helping Howard out of his financial situationbecause his commissions will be substantial if thePAIDs are purchased through SunCoast Investments.Should you recommend the PAIDs as an investment?

Chapter Essentials—Personal Finance 341

investments consist of large portfolios often containing more than 50 to 100 securitiesthat are well diversified. Many types of mutual funds with various investmentobjectives 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 oldadage when investing: ‘‘Don’t put all your eggs in one basket.’’

How can I use the concepts presented in the chapter to construct aportfolio 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) aportfolio’s beta equals the weighted average of the betas of all of the investmentscontained in the portfolio. Thus, if you can determine their beta coefficients, you canchoose those investments that provide the risk level you prefer when they arecombined to form a portfolio. If you prefer lower risk to higher risk, you shouldpurchase investments with low betas, and vice versa. In addition, you can adjust theriskiness of your portfolio by adding or deleting stocks with particular risks—that is, toreduce a portfolio’s risk, you can either add securities with low betas or delete from theportfolio (sell) securities with high betas. Beta coefficients for most large companies’stocks are easy to find—they are posted on numerous Internet sites, contained invarious financial publications that are available in public libraries, published byinvestment organizations, and so forth.

How can I determine the required and expected rates of return for aninvestment? Many investors examine the past performance of an investment todetermine its expected return. Care must be taken with this approach because pastreturns often do not reflect future returns. However, you might be able to get a roughidea as to what you expect a stock’s long-term growth will be in the future by examiningits past growth, especially if the firm is fairly stable. Investors also rely on informationprovided by professional analysts to form opinions about expected rates of return.

To determine an investment’s required rate of return, investors often evaluate theperformances 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 requiredrate of return. The beta coefficients for most large companies can be obtained frommany sources, including the Internet; the risk-free rate of return can be estimatedusing the rates on existing Treasury securities; and the expected market return can beestimated by evaluating market returns in recent years, the current trend in themarket, and predictions made by economists and investment analysts.

When investing your money, keep these words of wisdom in mind: ‘‘If you losesleep over your investments or are more concerned with the performance of yourportfolio than with your job, then your investment position probably is too risky.’’ If youfind yourself in such a position, use the concepts discussed in this chapter to adjustthe riskiness of your portfolio.

QUESTIONS

8-1 ‘‘The probability distribution of a less risky expected return is more peakedthan that of a riskier return.’’ Is this a correct statement? Explain.

8-2 What shape would the probability distribution have for (a) completely certainreturns and (b) completely uncertain returns?

8-3 Give some events that affect the price of a stock that would result fromunsystematic risk. What events would result from systematic risk? Explain.

8-4 Explain why systematic risk is the ‘‘relevant’’ risk of an investment and whyinvestors should be rewarded only for this type of risk.


8-5 Security A has an expected return of 7 percent, a standard deviation ofexpected returns of 35 percent, a correlation coefficient with the marketof �0.3, and a beta coefficient of �0.5. Security B has an expected return of12 percent, a standard deviation of returns of 10 percent, a correlationcoefficient with the market of 0.7, and a beta coefficient of 1.0. Whichsecurity 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-dayTreasury bills. Every 30 days your bills mature and you reinvest theprincipal ($250,000) in a new batch of bills. Assume that you live on theinvestment income from your portfolio and that you want to maintain aconstant standard of living. Is your portfolio truly risk-free?

c. Can you think of any asset that would be completely risk-free? Couldsomeone develop such an asset? Explain.

8-7 A life insurance policy is a financial asset. The premiums paid represent theinvestment’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 financialassets—the person’s only other asset is ‘‘human capital,’’ or lifetimeearnings capacity. What is the correlation coefficient between returns onthe insurance policy and returns on the policyholder’s human capital?

c. Insurance companies have to pay administrative costs and sales repre-sentatives’ commissions; hence, the expected rate of return on insurancepremiums is generally low, or even negative. Use the portfolio concept toexplain why people buy life insurance despite the negative expected returns.

8-8 If investors’ aversion to risk increased, would the risk premium on a high-beta stock increase more or less than that on a low-beta stock? Explain.

8-9 Do you think it is possible to construct a portfolio of stocks that has anexpected 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. Logicsays that the required rate of return on the stock should also double. Is thislogic 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 youranswers 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

Self-Test Problems 343

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); SMLequation

l. Slope of SML as a measure of risk aversion

ST-2 Of the $10,000 invested in a two-stock portfolio, 30 percent is invested inStock A and 70 percent is invested in Stock B. If Stock A has a beta equal to2.0 and the beta of the portfolio is 0.95, what is the beta of Stock B?

ST-3 If the risk-free rate of return, rRF, is 4 percent and the market return, rM, isexpected to be 12 percent, what is the required rate of return for a stock witha beta, b, equal to 2.5?

ST-4 Stock A and Stock B have the following historical returns:

a. Calculate the average rate of return for each stock during the period2004–2008. Assume that someone held a portfolio consisting of 50 per-cent Stock A and 50 percent Stock B. What would have been the realizedrate of return on the portfolio in each year from 2004 through 2008? Whatwould have been the average return on the portfolio during this period?

b. Calculate the standard deviation of returns for each stock and for theportfolio. Use Equation 8–4.

c. Looking at the annual returns data on the two stocks, would you guess thatthe correlation coefficient between returns on the two stocks is closer to0.9 or to �0.9?

ST-5 Stocks R and S have the following probability distributions of returns:

a. Calculate expected return for each stock.

b. Calculate the expected return of a portfolio consisting of 50 percent ofeach stock.

c. Calculate the standard deviation of returns for each stock and for theportfolio. Which stock is considered riskier with respect to total risk?

d. Compute the coefficient of variation for each stock. According to thecoefficient of variation, which stock is considered riskier?

Year Stock A’s Returns, €rA Stock B’s Returns, €rB

2004 �10.00% �3.00%2005 18.50 21.292006 38.67 44.252007 14.33 3.672008 33.00 28.30

Returns

Probability Stock R Stock S

0.5 �2% 20%0.1 10 120.4 15 2

beta coefficient

required rate of return

realized rates of return


e. Looking at the returns in the probability distributions of the two stocks,would you guess that the correlation coefficient between returns on thetwo stocks is closer to 0.9 or to �0.9?

f. If you added more stocks at random to the portfolio, which of the fol-lowing 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 Based on the following probability distribution, what is the security’sexpected return?

8-2 What is the expected return of the following investment?

8-3 Susan’s investment portfolio currently contains three stocks that have a totalvalue equal to $100,000. The beta of this portfolio is 1.5. Susan is consideringinvesting an additional $50,000 in a stock that has a beta equal to 3. After sheadds this stock, what will be the portfolio’s new beta?

8-4 Suppose rRF ¼ 5%, rM ¼ 12%. What is the appropriate required rate ofreturn for a stock that has a beta coefficient equal to 1.5?

8-5 The current risk-free rate of return, rRF, is 4 percent and the market riskpremium, RPM, is 5 percent. If the beta coefficient associated with a firm’sstock is 2.0, what should be the stock’s required rate of return?

8-6 Following is information for two stocks:

Which investment has the greater relative risk?

8-7 ZR Corporation’s stock has a beta coefficient equal to 1.8 and a required rateof return equal to 16 percent. If the expected return on the market is10 percent, what is the risk-free rate of return, rRF?

State Probability r

1 0.2 �5.0%2 0.4 10.03 0.5 30.0

Probability Payoff

0.3 30.0%0.2 10.00.5 �2.0

Investment Expected Return, r Standard Deviation, s

Stock D 10.0% 8.0%Stock E 36.0 24.0

expected return

expected return

portfolio beta

required return

required return

coefficient of variation

risk-free return

Problems 345

8-8 Currently, the risk-free return is 3 percent and the expected market rate ofreturn is 10 percent. What is the expected return of the following three-stockportfolio?

8-9 The market and Stock S have the following probability distributions:

a. Calculate the expected rates of return for the market and Stock S.

b. Calculate the standard deviations for the market and Stock S.

c. Calculate the coefficients of variation for the market and Stock S.

8-10 Marvin has investments with the following characteristics in his portfolio:

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

8-11 Stocks X and Y have the following probability distributions of expected futurereturns:

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

b. Calculate the standard deviation of expected returns for Stock X (sY ¼20.35%). Also, calculate the coefficient of variation for Stock Y. Is itpossible that most investors might regard Stock Y as being less risky thanStock X? Explain.

8-12 Yesterday Susan determined that the risk-free rate of return, rRF, is 3 per-cent, the required return on the market portfolio, rM, is 10 percent, and therequired rate of return on Stock K, rK, is 17 percent. Today Susan received

ExpectedInvestment Amount Return, r Invested

ABC 30% $10,000EFG 16 50,000QRP 20 40,000

Probability rX rY

0.1 �10% �35%0.2 2 00.4 12 200.2 20 250.1 38 45

Probability rM rS

0.3 15% 20%0.4 9 50.3 18 12

Amount Invested Beta

$400,000 1.5500,000 2.0100,000 4.0

portfolio return

expected returns

portfolio return

expected returns

required return


new information that indicates investors are more risk averse than shethought, such that the market risk premium, RPM, actually is 1 percenthigher than she estimated yesterday. When Susan considers the effect of thischange 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 investmentportfolio, which has a beta equal to 1.2—that is, bP ¼ 1.2. Terry is consid-ering selling the riskiest stock in the portfolio, which has a beta coefficientequal to 2.0, and replacing it with another stock. If Terry replaces the stockwith b ¼ 2.0 with a stock with b ¼ 1.0, what will be the new beta of hisinvestment portfolio? Assume that the equal amounts are invested in eachstock in the portfolio.

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 tohelp pay some university expenses. The stock is valued at $100,000, and if hesells it the portfolio’s beta will increase to 1.8. What is the beta of the stockThomas is considering selling?

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

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?

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 affectrM 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 investmentin 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 equalto 1.0 for $7,500 and to use the proceeds to buy another stock for yourportfolio. Assume that the new stock’s beta is equal to 1.75. Calculate yourportfolio’s new beta.

8-18 Stock R has a beta of 1.5, Stock S has a beta of 0.75, the expected rate ofreturn on an average stock is 15 percent, and the risk-free rate of return is9 percent. By how much does the required return on the riskier stock exceedthe required return on the less risky stock?

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

Stock Investment Beta

A $ 400.000 1.50B 600,000 �0.50C 1,000,000 1.25D 2,000,000 0.75

portfolio beta

portfolio beta

beta computation

SML and CAPM

portfolio beta

required rates of return

portfolio required return

Problems 347

If the market required rate of return is 14 percent and the risk-freerate is 6 percent, what is the fund’s required rate of return?

8-20 Following is information about Investment A, Investment B, and Invest-ment C:

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 risk-averse investor prefer?

8-21 Suppose you win the Florida lottery and are offered a choice of $500,000 incash or a gamble in which you would get $1 million if a head is flipped butzero 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 acommon 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? (Theexpected 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 is7.5 percent.)

(3) Would you invest in the bond or the stock?

(4) Exactly how large would the expected profit (or the expectedrate of return) have to be on the stock investment to makeyou invest in the stock, given the 7.5 percent return on thebond?

(5) How might your decision be affected if, rather than buyingone stock for $500,000, you could construct a portfolio consistingof 100 stocks with $5,000 invested in each? Each of thesestocks has the same return characteristics as the one stock—that is,a 50-50 chance of being worth either zero or $11,500 at year-end. Would the correlation between returns on these stocksmatter?

Return on Investment

Economic Condition Probability A B C

Boom 0.5 25.0% 40.0% 5.0%Normal 0.4 15.0 20.0 10.0Recession 0.1 �5.0 �40.0 15.0

r 18.0% 24.0% ?

s ? 23.3% 3.3%

expected returnsand risk

expected returns


8-22 The McAlhany Investment Fund has total capital of $500 million invested infive stocks:

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

a. Compute the expected return for the market.

b. Compute the beta coefficient for the investment fund. (Remember, thisproblem 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 newstock. The investment needed to take a position in the stock is $50million, it will have an expected return of 18 percent, and its estimatedbeta coefficient is 2.0. Should the firm purchase the new stock? At whatexpected rate of return should McAlhany be indifferent to purchasingthe stock?

8-23 Stock A and Stock B have the following historical returns:

a. Calculate the average rate of return for each stock during the period2004�2008.

b. Assume that someone held a portfolio consisting of 50 percent Stock Aand 50 percent Stock B. What would have been the realized rate of return

YearStock A’s

Returns, €rB

Stock B’sReturns, €rB

2004 �18.0% �14.5%2005 33.0 21.82006 15.0 30.52007 �0.5 �7.62008 27.0 26.3

Probability Market Return

0.1 10%0.2 120.4 130.2 160.1 17

Stock Investment Stock’s Beta Coefficient

A $160 million 0.5B 120 million 2.0C 80 million 4.0D 80 million 1.0E 60 million 3.0

portfolio return

realized rates of return

Problems 349

on the portfolio in each year from 2004 through 2008? What would havebeen the average return on the portfolio during this period?

c. Calculate the standard deviation of returns for each stock and for theportfolio. Use Equation 8–4.

d. Calculate the coefficient of variation for each stock and for theportfolio.

e. If you are a risk-averse investor, would you prefer to hold Stock A, StockB, or the portfolio? Why?

Integrative Problem

8-24 Assume you recently graduated with a major in finance, and you just landed ajob in the trust department of a large regional bank. Your first assignment isto invest $100,000 from an estate for which the bank is trustee. Because theestate 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, shownwith their probabilities and associated outcomes. (For now, disregard theblank spaces in the table; you will fill in the blanks later.)

The bank’s economic forecasting staff has developed probability estimatesfor the state of the economy, and the trust department has a sophisticatedcomputer program that was used to estimate the rate of return on eachalternative under each state of the economy. High Tech Inc. is an electronicsfirm, Collections Inc. collects past-due debts, and U.S. Rubber manufacturestires and various other rubber and plastics products. The bank also maintainsan ‘‘index fund’’ that includes a market-weighted fraction of all publicly tradedstocks; 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 HighTech’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 therow for r in the table.

c. You should recognize that basing a decision solely on expected returns isappropriate only for risk-neutral individuals. Because the beneficiaries of the

Estimated Returns on Alternative Investments

State of theEconomy Probability T-Bills

HighTech Collections

U.S.Rubber

MarketPortfolio

Two-StockPortfolio

Recession 0.1 8.0% �22.0% 28.0% 10.0% �13.0%Below Average 0.2 8.0 �2.0 14.7 �10.0 1.0Average 0.4 8.0 20.0 0.0 7.0 15.0Above Average 0.2 8.0 35.0 �10.0 45.0 29.0Boom 0.1 8.0 50.0 �20.0 30.0 43.0

r

s

CV

risk and ratesof return


trust, like virtually everyone, are risk averse, the riskiness of each alternativeis an important aspect of the decision. One possible measure of risk is thestandard 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 standarddeviation measure? (3) Draw a graph that shows roughly the shape of theprobability 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 thestandard deviation when the alternatives being considered have widelydiffering expected returns and risks. Calculate the CVs for the differentsecurities, and fill in the row for CV in the table. Does the CV measureproduce the same risk rankings as the standard deviation?

e. Suppose you created a two-stock portfolio by investing $50,000 in HighTech and $50,000 in Collections. (1) Calculate the expected return ( rp ),the standard deviation (sp), and the coefficient of variation (CVp) for thisportfolio and fill in the appropriate rows in the table. (2) How does theriskiness of this two-stock portfolio compare to the riskiness of the indi-vidual stocks if they were held in isolation?

f. Suppose an investor starts with a portfolio consisting of one randomlyselected stock. What would happen (1) to the riskiness and (2) to theexpected return of the portfolio as more randomly selected stocks areadded to the portfolio? What is the implication for investors? Draw twographs to illustrate your answer.

g. (1) Should portfolio effects influence the way that investors think aboutthe riskiness of individual stocks? (2) If you chose to hold a one-stockportfolio and consequently were exposed to more risk than diversifiedinvestors, 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 haveeliminated by diversifying?

h. The expected rates of return and the beta coefficients of the alternativesas supplied by the bank’s computer program are as follows:

(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 marketrisk? (3) Is it possible to choose among the alternatives on the basis of theinformation developed thus far? (4) Use the data given at the beginning ofthe problem to construct a graph that shows how the T-bill’s, High Tech’s,and Collections’ beta coefficients are calculated. Discuss what betameasures and explain how it is used in risk analysis.

i. (1) Write out the SML equation, use it to calculate the required rate ofreturn on each alternative, and then graph the relationship between the

Security Return ( r ) Risk (b)

High Tech 17.40% 1.29Market 15.00 1.00U.S. Rubber 13.80 0.68T-bills 8.00 0.00Collections 1.74 �0.86

Problems 351

expected and required rates of return. (2) How do the expected rates ofreturn compare with the required rates of return? (3) Does the fact thatCollections has a negative beta coefficient make any sense? What is theimplication of the negative beta? (4) What would be the market risk andthe required return of a 50-50 portfolio of High Tech and Collections? Ofa 50-50 portfolio of High Tech and U.S. Rubber?

j. (1) Suppose investors raised their inflation expectations by 3 percentagepoints over current estimates as reflected in the 8 percent T-bill rate.What effect would higher inflation have on the SML and on the returnsrequired on high- and low-risk securities? (2) Suppose instead thatinvestors’ risk aversion increased enough to cause the market risk pre-mium to increase by 3 percentage points. (Inflation remains constant.)What effect would this change have on the SML and on returns of high-and low-risk securities?

Computer-Related Problem

Work the problem in this section only if you are using the problem spreadsheet.

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

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, 33.33percent Stock B, and 33.33 percent Stock C. How does this compositionaffect the portfolio return, standard deviation, and coefficient of variationversus when 50 percent was invested in A and in B?

c. Make some other changes in the portfolio, making sure that the percen-tages 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 programwill not allow you to enter a zero for the percentage in Stock C.) Noticethat rp remains constant and that spchanges. Why do these results occur?

d. In Problem 8–23, the standard deviation of the portfolio decreased onlyslightly because Stocks A and B were highly positively correlated with oneanother. In this problem, the addition of Stock C causes the standarddeviation of the portfolio to decline dramatically, even though sC ¼ sA ¼sB. What does this change indicate about the correlation between Stock Cand Stocks A and B?

e. Would you prefer to hold the portfolio described in Problem 8-23 con-sisting 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 andrates of return?

YearStock C’sReturn, rc

2004 32.00%2005 �11.752006 10.752007 32.252008 �6.75

realized ratesof return


besley 14e ch08

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