Risk and Rates 6of ReturnCHAPTERSOURCE: Beard, William Holbrook (1823–1900). New York Historical Society/The Bridgeman Art LibraryInternational,
NO PAIN NO GAIN$f someone had invested $1,000 in a portfolio ofaround, you’re not tied to the fickleness of a givenlarge-company stocks in 1925 and then reinvestedmarket, stock, or industry.... Correlation, inIall dividends received, his or her investment wouldportfolio-manager speak, helps you diversify properlyhave grown to $2,845,697 by 1999. Over the samebecause it describes how closely two investments tracktime period, a portfolio of small-company stocks wouldeach other. If they move in tandem, they’re likely tohave grown even more, to $6,641,505. But if instead hesuffer from the same bad news. So, you should combineor she had invested in long-term government bonds, theassets with low correlations.”$1,000 would have grown to only $40,219, and to . investors tend to think of “the stock market” asmeasly $15,642 for short-term . stock market. However, . stocks amount toGiven these numbers, why would anyone invest inonly 35 percent of the value of all stocks. Foreignbonds? The answer is, “Because bonds are less risky.”markets have been quite profitable, and they are notWhile common stocks have over the past 74 yearsperfectly correlated with . markets. Therefore, globalproduced considerably higher returns, (1) we cannot bediversification offers . investors an opportunity tosure that the past is a prologue to the future, and (2)raise returns and at the same time reduce risk. However,stock values are more likely to experience sharp declinesforeign investing brings some risks of its own, mostthan bonds, so one has a greater chance of losingnotably “exchange rate risk,” which is the danger thatmoney on a stock investment. For example, in 1990 theexchange rate shifts will decrease the number of dollarsaverage small-company stock lost percent of itsa foreign currency will , and large-company stocks also suffered the central thrust of the Business WeekBonds, though, provided positive returns that year, asarticle was on ways to measure and then reduce risk, itthey almost always point out that some recently created instrumentsOf course, some stocks are riskier than others, andthat are actually extremely risky have been marketed aseven in years when the overall stock market goes up,low-risk investments to naive investors. For example,many individual stocks go down. Therefore, putting allseveral mutual funds have advertised that theiryour money into one stock is extremely risky. Accordingportfolios “contain only securities backed by the a Business Weekarticle, the single best weapongovernment” but then failed to highlight that the fundsagainst risk is diversification: “By spreading your moneythemselves are using financial leverage, are investing in231
“derivatives,” or are taking some other action thatSOURCES:“Figuring Risk: It’s Not So Scary,” Business Week,November 1, 1993, 154–155; “T-Bill Trauma and the Meaning ofboosts current yields but exposes investors to hugeRisk,” The Wall Street Journal,February 12, 1993, C1; , Bonds, Bills, and Inflation: (Valuation Edition) 2000Yearbook(Chicago: Ibbotson Associates, 2000).When you finish this chapter, you should understandwhat risk is, how it is measured, and what actions canbe taken to minimize it, or at least to ensure that youare adequately compensated for bearing it. In this chapter, we start from the basic premise that investors like returns and dis-like risk. Therefore, people will invest in risky assets only if they expect to receivehigher returns. We define precisely what the term riskmeans as it relates to in-vestments, we examine procedures managers use to measure risk, and we discussthe relationship between risk and return. Then, in Chapters 7, 8, and 9, we extendthese relationships to show how risk and return interact to determine securityprices. Managers must understand these concepts and think about them as theyplan the actions that will shape their firms’ you will see, risk can be measured in different ways, and different conclu-sions about an asset’s riskiness can be reached depending on the measure analysis can be confusing, but it will help if you remember the following: financial assets are expected to produce cash flows,and the riskiness ofan asset is judged in terms of the riskiness of its cash riskiness of an asset can be considered in two ways: (1) on a stand-alonebasis,wheretheasset’scashflowsareanalyzedbythemselves,or(2)ina portfolio context,where the cash flows from a number of assets are com-1bined, and then the consolidated cash flows are is an im-portant difference between stand-alone and portfolio risk, and an asset thathas a great deal of risk if held by itself may be much less risky if it is heldas part of a larger a portfolio context, an asset’s risk can be divided into two components:(a) diversifiable risk,which can be diversified away and thus is of little con-1A portfoliois a collection of investment securities. If you owned some General Motors stock, someExxon Mobil stock, and some IBM stock, you would be holding a three-stock portfolio. Because di-versification lowers risk, most stocks are held in 6 RISK AND RATES OF RETURN
cern to diversified investors, and (b) market risk, which reflects the risk ofa general stock market decline and which cannot be eliminated by diversifi-cation, hence doesconcern investors. Only market risk is relevant—diver-sifiable risk is irrelevantto rational investors because it can be asset with a high degree of relevant (market) risk must provide a rela-tively high expected rate of return to attract investors. Investors in generalare averse to risk,so they will not buy risky assets unless those assets havehigh expected this chapter, we focus on financial assetssuch as stocks and bonds, butthe concepts discussed here also apply to physical assetssuch as computers,trucks, or even whole plants. INVESTMENT RETURNSWith most investments, an individual or business spends money today with theexpectation of earning even more money in the future. The concept of returnprovides investors with a convenient way of expressing the financial perfor-mance of an investment. To illustrate, suppose you buy 10 shares of a stock for$1,000. The stock pays no dividends, but at the end of one year, you sell thestock for $1,100. What is the return on your $1,000 investment?:Dollar return Amount received Amount invested $1,100 $1,000 $ at the end of the year you had sold the stock for only $900, your dollar re-turn would have been $,twoproblemsarise:(1)Tomakeameaningfuljudgmentaboutthereturn,youneedtoknowthescale(size)oftheinvestment;a$100returnona$100investmentisagoodreturn(assumingtheinvestmentisheldforoneyear),buta$100returnona$10,000investmentwouldbeapoorreturn.(2)Youalsoneedtoknowthetimingofthereturn;a$100returnona$100investmentisaverygoodreturnifitoccursafteroneyear, solution to the scale and timing problems is to express investment resultsasratesofreturn,,therateofreturnonthe1-year stock investment, when $1,100 is received after one year, is 10 percent:Amount received Amount invested Rate of return Amount investedDollar return$100 Amount invested$1,000 10%.The rate of return calculation “standardizes” the return by considering the re-turn per unit of investment. In this example, the return of , or 10 percent,indicates that each dollar invested will earn ($) $. If the rate ofINVESTMENT RETURNS233
return had been negative, this would indicate that the original investment wasnot even recovered. For example, selling the stock for only $900 results in a 10 percent rate of return, which means that each dollar invested lost 10 $10returnona$100investmentproducesa10percentrateofreturn,whilea$10returnona$1,, rates of return on an annual basis, which is typically done inpractice, solves the timing problem. A $10 return after one year on a $100 in-vestment results in a 10 percent annual rate of return, while a $10 return afterfive years yields only a percent annual rate of return. We will discuss all thisin detail in Chapter 7, which deals with the time value of we illustrated return concepts with one outflow and one inflow, inlater chapters we demonstrate that rate of return concepts can easily be appliedin situations where multiple cash flows occur over time. For example, whenIntel makes an investment in new chip-making technology, the investment ismade over several years and the resulting inflows occur over even more now, it is sufficient to recognize that the rate of return solves the two majorproblems associated with dollar returns, size and timing. Therefore, the rate ofreturn is the most common measure of investment -TEST QUESTIONSDifferentiate between dollar return and rate of is the rate of return superior to the dollar return in terms of account-ing for the size of investment and the timing of cash flows?STAND-ALONE RISKRiskRiskis defined in Webster’sas “a hazard; a peril; exposure to loss or injury.”The chance that some unfavorableThus, risk refers to the chance that some unfavorable event will occur. If youevent will in skydiving, you are taking a chance with your life—skydiving is you bet on the horses, you are risking your money. If you invest in specula-tive stocks (or, really, anystock), you are taking a risk in the hope of making anappreciable asset’s risk can be analyzed in two ways: (1) on a stand-alone basis, wherethe asset is considered in isolation, and (2) on a portfolio basis, where the assetStand-Alone Riskis held as one of a number of assets in a portfolio. Thus, an asset’s stand-aloneThe risk an investor would face ifriskis the risk an investor would face if he or she held only this one asset. Ob-he or she held only one asset. viously, most assets are held in portfolios, but it is necessary to understandstand-alone risk in order to understand risk in a portfolio illustrate the riskiness of financial assets, suppose an investor buys$100,000 of short-term Treasury bills with an expected return of 5 percent. Inthis case, the rate of return on the investment, 5 percent, can be estimated quiteprecisely, and the investment is defined as being essentially risk free. However,if the $100,000 were invested in the stock of a company just being organized toprospect for oil in the mid-Atlantic, then the investment’s return could not be234CHAPTER 6 RISK AND RATES OF RETURN
estimated precisely. One might analyze the situation and conclude that the ex-pectedrate of return, in a statistical sense, is 20 percent, but the investor shouldalso recognize that the actualrate of return could range from, say, 1,000 per-cent to 100 percent. Because there is a significant danger of actually earningmuch less than the expected return, the stock would be relatively investment will be undertaken unless the expected rate of return is high enoughto compensate the investor for the perceived risk of the our example, itis clear that few if any investors would be willing to buy the oil company’s stockif its expected return were the same as that of the assets rarely produce their expected rates of return—generally, riskyassets earn either more or less than was originally expected. Indeed, if assets al-ways produced their expected returns, they would not be risky. Investment risk,then, is related to the probability of actually earning a low or negative return—the greater the chance of a low or negative return, the riskier the , risk can be defined more precisely, and we do so in the next event’s probabilityis defined as the chance that the event will occur. For ex-ample, a weather forecaster might state, “There is a 40 percent chance of raintoday and a 60 percent chance that it will not rain.” If all possible events, oroutcomes, are listed, and if a probability is assigned to each event, the listing isProbability Distributioncalled a probability our weather forecast, we could set up theA listing of all possible outcomes,following probability distribution:or events, with a probability(chance of occurrence) assigned toOUTCOMEPROBABILITYeach outcome.(1)(2) 40%No 60% 100%The possible outcomes are listed in Column 1, while the probabilities of theseoutcomes,expressedbothasdecimalsandaspercentages,aregiveninCol-umn2. Notice that the probabilities must sum to , or 100 can also be assigned to the possible outcomes (or returns) froman investment. If you buy a bond, you expect to receive interest on the bondplus a return of your original investment, and those payments will provide youwith a rate of return on your investment. The possible outcomes from this in-vestment are (1) that the issuer will make the required payments or (2) that theissuer will default on the payments. The higher the probability of default, theriskier the bond, and the higher the risk, the higher the required rate of you invest in a stock instead of buying a bond, you will again expect to earna return on your money. A stock’s return will come from dividends plus capitalgains. Again, the riskier the stock—which means the higher the probabilitythat the firm will fail to perform as you expected—the higher the expected re-turn must be to induce you to invest in the this in mind, consider the possible rates of return (dividend yield pluscapital gain or loss) that you might earn next year on a $10,000 investment inthe stock of either Martin Products Inc. or . Water Company. Martin man-STAND-ALONE RISK235
Probability Distributions for Martin Products and . WaterTABLE 6-1RATE OF RETURN ON STOCKIF THIS DEMAND OCCURSDEMAND FOR THE PROBABILITY OF THIS COMPANY’S PRODUCTSDEMAND OCCURRINGMARTIN . %20%(70) and distributes computer terminals and equipment for the rapidlygrowing data transmission industry. Because it faces intense competition, itsnew products may or may not be competitive in the marketplace, so its futureearnings cannot be predicted very well. Indeed, some new company could de-velop better products and literally bankrupt Martin. . Water, on the otherhand, supplies an essential service, and because it has city franchises that pro-tect it from competition, its sales and profits are relatively stable and rate-of-return probability distributions for the two companies areshown in Table 6-1. There is a 30 percent chance of strong demand, in whichcase both companies will have high earnings, pay high dividends, and enjoycapital gains. There is a 40 percent probability of normal demand and moder-ate returns, and there is a 30 percent probability of weak demand, which willmean low earnings and dividends as well as capital losses. Notice, however, thatMartin Products’ rate of return could vary far more widely than that of . There is a fairly high probability that the value of Martin’s stock willdrop substantially, resulting in a 70 percent loss, while there is no chance of a2loss for . we multiply each possible outcome by its probability of occurrence and thenˆExpected Rate of Return, ksum these products, as in Table 6-2, we have a weighted averageof rate of return expected to beThe weights are the probabilities, and the weighted average is the expectedrealized from an investment; the3ˆrate of return, k,called “k-hat.”The expected rates of return for both Mar-weighted average of thetin Products and . Water are shown in Table 6-2 to be 15 percent. This typeprobability distribution of table is known as a payoff is, of course, completely unrealistic to think that any stock has no chance of a loss. Only in hy-pothetical examples could this occur. To illustrate, the price of Columbia Gas’s stock dropped from$ to $ in just three hours a few years ago. All investors were reminded that any stock isexposed to some risk of loss, and those investors who bought Columbia Gas learned that lesson thehard Chapters 8 and 9, we will use kand kto signify the returns on bonds and stocks, , this distinction is unnecessary in this chapter, so we just use the general term, k, to sig-nify the expected return on an 6 RISK AND RATES OF RETURN
Calculation of Expected Rates of Return: Payoff MatrixTABLE 6-2MARTIN . WATERDEMAND FOR PROBABILITY RATE OF RETURN RATE OF RETURN THE COMPANY’S OF THIS DEMAND IF THIS DEMAND PRODUCT: IF THIS DEMAND PRODUCT: PRODUCTS OCCURRING OCCURS (2) (3) OCCURS (2) (5) (1)(2)(3) (4)(5) (6)%30%20%6%(70)(21)11013%ˆˆ 15%k 15%The expected rate of return calculation can also be expressed as an equation4that does the same thing as the payoff matrix table:ˆExpected rate of return k Pk Pk Pk1122nnn Pk.(6-1)iiai 1Here kis the ith possible outcome, Pis the probability of the ith outcome, andiiˆn is the number of possible outcomes. Thus, kis a weighted average of the pos-sible outcomes (the kvalues), with each outcome’s weight being its probabilityiof occurrence. Using the data for Martin Products, we obtain its expected rateof return as follows:ˆk P(k) P(k) P(k)112233 (100%) (15%) ( 70%) 15%.. Water’s expected rate of return is also 15 percent:ˆk (20%) (15%) (10%) 15%.We can graph the rates of return to obtain a picture of the variability of pos-sible outcomes; this is shown in the Figure 6-1 bar charts. The height of eachbar signifies the probability that a given outcome will occur. The range ofprobable returns for Martin Products is from 70 to 100 percent, with an ex-pected return of 15 percent. The expected return for . Water is also 15 per-cent, but its range is much far, we have assumed that only three situations can exist: strong, nor-mal,,ofcourse,demandcouldrangefromadeepde-pression to a fantastic boom, and there are an unlimited number of possibilities4Thesecondformoftheequationissimplyashorthandexpressioninwhichsigma( )means“sumup,” 1,thenPk Pk;ifi 2,thenPk Pk;andsoii11ii22non until i n, the last possible outcome. The symbol simply says, “Go through the following ai 1process: First, let i 1 and find the first product; then let i 2 and find the second product; thencontinue until each individual product up to i n has been found, and then add these individualproducts to find the expected rate of return.”STAND-ALONE RISK237
Probability Distributions of Martin Products’ FIGURE 6-1and . Water’s Rates of Returna. Martin Productsb. . WaterProbability ofProbability –70015100Rate of Return0101520Rate of Return(%)(%)Expected RateExpected Rateof Returnof Returnin between. Suppose we had the time and patience to assign a probability toeach possible level of demand (with the sum of the probabilities still ) and to assign a rate of return to each stock for each level of demand. Wewould have a table similar to Table 6-1, except that it would have many moreentries in each column. This table could be used to calculate expected rates ofreturn as shown previously, and the probabilities and outcomes could be ap-proximated by continuous curves such as those presented in Figure 6-2. Herewe have changed the assumptions so that there is essentially a zero probabilitythat Martin Products’ return will be less than 70 percent or more than 100percent, or that . Water’s return will be less than 10 percent or more than20 percent, but virtually any return within these limits is tighter, or more peaked, the probability distribution, the more likely it is thatthe actual outcome will be close to the expected value, and, consequently, the less likelyit is that the actual return will end up far below the expected return. Thus, the tighterthe probability distribution, the lower the risk assigned to a stock. Since . Waterhas a relatively tight probability distribution, its actual returnis likely to becloser to its 15 percent expected returnthan is that of Martin -ALONERISK: THESTANDARDDEVIATIONRiskisadifficultconcepttograsp,,acommondefinition,andonethatissatisfactoryformanypurposes,isstatedintermsofprobabilitydistri-238CHAPTER 6 RISK AND RATES OF RETURN
Continuous Probability Distributions of Martin Products’ FIGURE 6-2and . Water’s Rates of ReturnProbability . WaterMartin Products–70015100Rate of Return(%)ExpectedRate of ReturnNOTE:The assumptions regarding the probabilities of various outcomes have been changed from thosein Figure 6-1. There the probability of obtaining exactly 15 percent was 40 percent; here it is muchsmallerbecause there are many possible outcomes instead of just three. With continuous distributions,it is more appropriate to ask what the probability is of obtaining at least some specified rate of returnthan to ask what the probability is of obtaining exactly that rate. This topic is covered in detail instatistics -2:Thetightertheprobabilitydistribu-tionofexpectedfuturereturns,, be most useful, any measure of risk should have a definite value—weneed a measure of the tightness of the probability distribution. One such mea-Standard Deviation, sure is the standard deviation, the symbol for which is , pronounced “sigma.”A statistical measure of theThe smaller the standard deviation, the tighter the probability distribution,variability of a set of , accordingly, the lower the riskiness of the stock. To calculate the standarddeviation, we proceed as shown in Table 6-3, taking the following steps: the expected rate of return:nˆExpected rate of return k 1ˆFor Martin, we previously found k 15%. ˆ the expected rate of return (k) from each possible outcome (k)iˆto obtain a set of deviations about kas shown in Column 1 of Table 6-3:ˆDeviation k -ALONE RISK239
Calculating Martin Products’ Standard DeviationTABLE 6-322ˆˆˆk k(k k)(k k)Piiii(1)(2)(3)100 15 857,225(7,225)() 2, 15 00(0)() 70 15 857,225(7,225)() 2, 4, deviation 4, 33 5 %. each deviation, then multiply the result by the probability of oc-currence for its related outcome, and then sum these products to obtain2Variance, the varianceof the probability distribution as shown in Columns 2 and 3The square of the standardof the table:ˆVariance (k k)P.(6-2)iiai , find the square root of the variance to obtain the standard devia-tion:n2ˆStandard deviation (k k)P.(6-3)iiaBi 1Wilshire AssociatesThus, the standard deviation is essentially a weighted average of the deviationsprovides a download sitefrom the expected value, and it provides an idea of how far above or below thefor various returns seriesexpected value the actual value is likely to be. Martin’s standard deviation isfor indexes such as theseen in Table 6-3 to be %. Using these same procedures, we findWilshire 5000 and . Water’s standard deviation to be percent. Martin Products has theWilshire 4500 at standard deviation, which indicates a greater variation of returns in Microsoft Excelthus a greater chance that the expected return will not be realized. Therefore, Products is a riskier investment than . Water when held a probability distribution is normal, the actualreturn will be within 1standard deviation of the expectedreturn percent of the time. Figure 6-3illustrates this point, and it also shows the situation for 2 and 3 . ForˆˆMartin Products, k 15% and %, whereas k 15% and %for . Water. Thus, if the two distributions were normal, there would be percent probability that Martin’s actual return would be in the range of15 percent, or from to percent. For . Water, percent range is 15 percent, or from to percent. Withsuch a small , there is only a small probability that . Water’s return wouldbe significantly less than expected, so the stock is not very risky. For the aver-age firm listed on the New York Stock Exchange, has generally been in the5range of 35 to 40 percent in recent the example, we described the procedure for finding the mean and standard deviation when thedata are in the form of a known probability distribution. If only sample returns data over some pastperiod are available, the standard deviation of returns can be estimated using this formula:(footnote continues)240CHAPTER 6 RISK AND RATES OF RETURN
FIGURE 6-3Probability Ranges for a Normal %%%ˆ–3σ–2σ–1σk+1σ+2σ+3σNOTES: area under the normal curve always equals , or 100 percent. Thus, the areas under any pair ofnormal curves drawn on the same scale, whether they are peaked or flat, must be of the area under a normal curve is to the left of the mean, indicating that there is a 50percent probability that the actual outcome will be less than the mean, and half is to the right ofk, indicating a 50 percent probability that it will be greater than the the area under the curve, percent is within 1 of the mean, indicating that theprobability is percent that the actual outcome will be within the range k 1 to k 1 . exist for finding the probability of other ranges. These procedures are covered instatistics a normal distribution, the larger the value of , the greater the probability that the actualoutcome will vary widely from, and hence perhaps be far below, the expected, or most likely,outcome. Since the probability of having the actual result turn out to be far below the expected resultis one definition of risk, and since measures this probability, we can use as a measure of definition may not be a good one, however, if we are dealing with an asset held in a diversifiedportfolio. This point is covered later in the chapter.(Footnote 5 continued)n2(k k)tAvgat 1Estimated S .(6-3a)Rn 1Here k (“k bar t”) denotes the past realized rate of return in Period t, and k is the average an-tAvgnual return earned during the last n years. Here is an example:—YEARkt199915%2000 5200120 (15 5 20) k %.Avg3222(15 10) ( 5 10) (20 10) Estimated (or S) C3 1350 %.B2(footnote continues)STAND-ALONE RISK241
MEASURINGSTAND-ALONERISK: THECOEFFICIENTOFVARIATIONIfachoicehastobemadebetweentwoinvestmentsthathavethesameexpectedreturnsbutdifferentstandarddeviations,mostpeoplewouldchoosetheonewiththelowerstandarddeviationand,therefore,,givenachoicebetweentwoinvestmentswiththesamerisk(standarddeviation)butdifferentexpectedreturns,,thisiscommonsense—returnis“good,”riskis“bad,”and,consequently, of Variation (CV)answerthisquestion,weuseanothermeasureofrisk,thecoefficientofvaria-Standardized measure of the risktion(CV),whichisthestandarddeviationdividedbytheexpectedreturn:per unit of return; calculated as the standard deviation divided byCoefficient of variation CV .(6-4)ˆthe expected ,,,Martin,, , a case where the coefficient of variation is necessary, consider Projects Xand Y in Figure 6-4. These projects have different expected rates of return anddifferent standard deviations. Project X has a 60 percent expected rate of returnand a 15 percent standard deviation, while Project Y has an 8 percent expectedreturn but only a 3 percent standard deviation. Is Project X riskier, on a rela-tive basis, because it has the larger standard deviation? If we calculate the coef-ficients of variation for these two projects, we find that Project X has a coeffi-cient of variation of 15/60 , and Project Y has a coefficient of variationof 3/8 . Thus, we see that Project Y actually has more risk per unit ofreturn than Project X, in spite of the fact that X’s standard deviation is , even though Project Y has the lower standard deviation, accordingto the coefficient of variation it is riskier than Project Y has the smaller standard deviation, hence the more peaked proba-bility distribution, but it is clear from the graph that the chances of a really low(Footnote 5 continued)The historical is often used as an estimate of the future . Much less often, and generally incor-rectly, k for some past period is used as an estimate of k, the expected future return. Because pastAvgvariability is likely to be repeated, may be a good estimate of future risk, but it is much less rea-sonable to expect that the past levelof return (which could have been as high as 100% or as lowas 50%) is the best expectation of what investors think will happen in the 6-3a is built into all financial calculators, and it is very easy to use. We simply enterthe rates of return and press the key marked S (or S) to get the standard deviation. Note, though,xthat calculators have no built-in formula for finding where probabilistic data are involved; thereyou must go through the process outlined in Table 6-3 and Equation 6-3. The same situation holdsfor computer spreadsheet 6 RISK AND RATES OF RETURN
Comparison of Probability Distributions and Rates of Return FIGURE 6-4for Projects X and YProbabilityDensityProject YProject X0860Expected Rateof Return (%)return are higher for Y than for X because X’s expected return is so high. Be-cause the coefficient of variation captures the effects of both risk and return, itis a better measure for evaluating risk in situations where investments have sub-stantially different expected $1million,,andattheendofoneyearyouwillhaveasure$,whichisyouroriginalinvestmentplus$50,,youcanbuystockinR&&D’sresearchprogramsaresuccessful,yourstockwillincreaseinvalueto$-ever,iftheresearchisafailure,thevalueofyourstockwillgotozero,&D’schancesofsuccessorfailureasbeing50-50,($0) ($2,100,000) $1,050,$1millioncostofthestockleavesanexpectedprofitof$50,000,oranexpected(butrisky)5percentrateofreturn:Expected ending value Cost Expected rate of return Cost$1,050,000 $1,000,000 $1,000,000$50,000 5%.$1,000,000Thus, you have a choice between a sure $50,000 profit (representing a 5 per-cent rate of return) on the Treasury note and a risky expected $50,000 profit(also representing a 5 percent expected rate of return) on the R&D Enterprisesstock. Which one would you choose? If you choose the less risky investment, you areSTAND-ALONE RISK243
THE TRADE-OFF BETWEEN RISK AND RETURNhe table accompanying this box summarizesthe historicaldifferent investments. For T-bills, however, the standard devia-Ttrade-off between risk and return for different classes of in-tion needs to be interpreted carefully. Note that the tablevestments from 1926 through 1999. As the table shows, thoseshows that Treasury bills have a positive standard deviation,assets that produced the highest average returns also had thewhich indicates some risk. However, if you invested in a one-highest standard deviations and the widest ranges of Treasury bill and held it for the full year, your realized re-For example, small-company stocks had the highest average an-turn would be the same regardless of what happened to thenual return, percent, but their standard deviation of re-economy that year, and thus the standard deviation of your re-turns, percent, was also the highest. By contrast, would be zero. So, why does the table show a percentTreasury bills had the lowest standard deviation, percent,standard deviation for T-bills, which indicates some risk? Inbut they also had the lowest average return, , a T-bill is riskless if you hold it for one year, but if you in-When deciding among alternative investments, one needs tovest in a rolling portfolio of one-year T-bills and hold the port-be aware of the trade-off between risk and return. While therefolio for a number of years, your investment income will varyis certainly no guarantee that history will repeat itself, returnsdepending on what happens to the level of interest rates inobserved in the past are a good starting point for estimatingeach year. So, while you can be sure of the return you will earninvestments’ returns in the future. Likewise, the standard devi-on a T-bill in a given year, you cannot be sure of the return youations of past returns provide useful insights into the risks ofwill earn on a portfolio of T-bills over a period of Realized Returns, 1926–1999AVERAGE RETURNSTANDARD %%Large-company -term corporate -term goverment . Treasury : Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2000 Yearbook(Chicago:Ibbotson Associates, 2000), averse. Most investors are indeed risk averse, and certainly the average investor isrisk averse with regard to his or her “serious money.” Because this is a well-documentedRisk Aversionfact, we shall assume risk aversion throughout the remainder of the -averse investors dislike riskWhataretheimplicationsofriskaversionforsecuritypricesandratesofre-and require higher rates of returnturn?Theansweristhat,otherthingsheldconstant,thehigherasecurity’srisk,as an inducement to buy , 6 RISK AND RATES OF RETURN
$,,andMartin’’sstock,andsellingpres-surewouldsimultaneouslycauseMartin’,inturn,,forexample,’sstockpricewasbidupfrom$100to$150,whereasMartin’sstockpricedeclinedfrom$100to$’sexpectedreturntofallto10percent,whileMar-tin’,20% Risk Pr10% 10%,isariskpremium,RP,whichrepresentstheadditionalcompensa-emium, RPThe difference between rate of return on a givenThis example demonstrates a very important principle: In a market dominatedrisky asset and that on a less riskyby risk-averse investors, riskier securities must have higher expected returns, as by the marginal investor, than less risky securities. If this situation does not exist,buying and selling in the market will force it to will consider the questionof how much higher the returns on risky securities must be later in the chapter,after we see how diversification affects the way risk should be measured. Then,in Chapters 8 and 9, we will see how risk-adjusted rates of return affect theprices investors are willing to pay for different -TEST QUESTIONSWhat does “investment risk” mean?Set up an illustrative probability distribution for an is a payoff matrix?Which of the two stocks graphed in Figure 6-2 is less risky? Why?How does one calculate the standard deviation?Which is a better measure of risk if assets have different expected returns:(1) the standard deviation or (2) the coefficient of variation? Why?Explain the following statement: “Most investors are risk averse.”How does risk aversion affect rates of return?RISK IN A PORTFOLIO CONTEXTIn the preceding section, we considered the riskiness of assets held in we analyze the riskiness of assets held in portfolios. As we shall see, anasset held as part of a portfolio is less risky than the same asset held in , most financial assets are held as parts of portfolios. Banks, pensionfunds, insurance companies, mutual funds, and other financial institutions areSTAND-ALONE RISK245
required by law to hold diversified portfolios. Even individual investors—atleast those whose security holdings constitute a significant part of their totalwealth—generally hold portfolios, not the stock of only one firm. This beingthe case, from an investor’s standpoint the fact that a particular stock goes upor down is not very important; what is important is the return on his or her port-folio, and the portfolio’s risk. Logically, then, the risk and return of an individual se-curity should be analyzed in terms of how that security affects the risk and return ofthe portfolio in which it is illustrate, Pay Up Inc. is a collection agency company that operates na-tionwide through 37 offices. The company is not well known, its stock is notvery liquid, its earnings have fluctuated quite a bit in the past, and it doesn’t paya dividend. All this suggests that Pay Up is risky and that its required rate of re-turn, k, should be relatively high. However, Pay Up’s required rate of return in2001, and all other years, was quite low in comparison to those of most othercompanies. This indicates that investors regard Pay Up as being a low-riskcompany in spite of its uncertain profits. The reason for this counterintuitivefact has to do with diversification and its effect on risk. Pay Up’s earnings riseduring recessions, whereas most other companies’ earnings tend to declinewhen the economy slumps. It’s like fire insurance—it pays off when otherthings go bad. Therefore, adding Pay Up to a portfolio of “normal” stockstends to stabilize returns on the entire portfolio, thus making the portfolio ˆExpected Return on aThe expected return on a portfolio, k,is simply the weighted average of thepˆPortfolio, kexpected returns on the individual assets in the portfolio, with the weightspThe weighted average of thebeing the fraction of the total portfolio invested in each asset:expected returns on the assets heldˆˆˆˆk wk wk wkp1122nnin the ˆ wk.(6-5)iiai 1ˆHere the k’s are the expected returns on the individual stocks, the w’s are theiiweights, and there are n stocks in the portfolio. Note (1) that wis the fractioniof the portfolio’s dollar value invested in Stock i (that is, the value of the in-vestment in Stock i divided by the total value of the portfolio) and (2) that thew’s must sum to that in August 2001, a security analyst estimated that the followingreturns could be expected on the stocks of four large companies:ˆEXPECTED RETURN, %General we formed a $100,000 portfolio, investing $25,000 in each stock, the ex-pected portfolio return would be %:246CHAPTER 6 RISK AND RATES OF RETURN
ˆˆˆˆˆk wk wk wk wkp11223344 (12%) (%) (10%) (%) %.Realized Rate ofOf course, after the fact and a year later, the actual realized rates of return, k , Return, k The return that was actuallyon the individual stocks—the k , or “k-bar,” values—will almost certainly beiˆearned during some past from their expected values, so k will be different from k %.ppThe actual return (k ) usually turnsFor example, Coca-Cola stock might double in price and provide a return ofout to be different from the 100%, whereas Microsoft stock might have a terrible year, fall sharply, andˆexpected return (k) except forhave a return of 75%. Note, though, that those two events would be some-riskless offsetting, so the portfolio’s return might still be close to its expected re-turn, even though the individual stocks’ actual returns were far from their ex-pected we just saw, the expected return on a portfolio is simply the weighted aver-age of the expected returns on the individual assets in the portfolio. However,unlikereturns,theriskinessofaportfolio, ,isgenerallynottheweightedpaverage of the standard deviations of the individual assets in the portfolio; theportfolio’s risk will be smallerthan the weighted average of the assets’ ’s. Infact, it is theoretically possible to combine stocks that are individually quiterisky as measured by their standard deviations and to form a portfolio that iscompletely riskless, with illustrate the effect of combining assets, consider the situation in Figure6-5. The bottom section gives data on rates of return for Stocks W and M in-dividually, and also for a portfolio invested 50 percent in each stock. The threetop graphs show plots of the data in a time series format, and the lower graphsshow the probability distributions of returns, assuming that the future is ex-pected to be like the past. The two stocks would be quite risky if they were heldin isolation, but when they are combined to form Portfolio WM, they are notrisky at all. (Note: These stocks are called W and M because the graphs of theirreturns in Figure 6-5 resemble a W and an M.)The reason Stocks W and M can be combined to form a riskless portfolio isthat their returns move countercyclically to each other—when W’s returnsfall, those of M rise, and vice versa. The tendency of two variables to move to-Correlationgether is called correlation,and the correlation coefficient, r, measures this6The tendency of two variables statistical terms, we say that the returns on Stocks W and M aremove negatively correlated,with r opposite of perfect negative correlation, with r , is perfect positiveCorrelation Coefficient, rcorrelation,with r . Returns on two perfectly positively correlated stocksA measure of the degree ofrelationship between correlation coefficient, r,can range from , denoting that the two variables move up anddown in perfect synchronization, to , denoting that the variables always move in exactly op-posite directions. A correlation coefficient of zero indicates that the two variables are not related toeach other—that is, changes in one variable are independentof changes in the is easy to calculate correlation coefficients with a financial calculator. Simply enter the returnson the two stocks and then press a key labeled “r.” For W and M, r IN A PORTFOLIO CONTEXT247
Rate of Return Distributions for Two Perfectly Negatively CorrelatedFIGURE 6-5Stocks (r ) and for Portfolio WMa. Rates of Return___Stock WStock MPortfolio WMk (%)k (%)k (%) W Mp252525151515000200120012001–10–10–10b. Probability Distributions of ReturnsProbabilityProbabilityProbabilityDensityDensityDensityStock WStock MPortfolio WM015Percent015Percent015Percentˆˆˆ(= k )(= k )(= k )WMpSTOCK W STOCK M PORTFOLIO WM YEAR(k )( k)( k)%(%)%1998()()()%%%Average %%%Standard %%%248CHAPTER 6 RISK AND RATES OF RETURN
(M and M ) would move up and down together, and a portfolio consisting oftwo such stocks would be exactly as risky as the individual stocks. This point isillustrated in Figure 6-6, where we see that the portfolio’s standard deviation isequal to that of the individual stocks. Thus, diversification does nothing to reducerisk if the portfolio consists of perfectly positively correlated 6-5 and 6-6 demonstrate that when stocks are perfectly negativelycorrelated (r ), all risk can be diversified away, but when stocks are per-fectly positively correlated (r ), diversification does no good reality, most stocks are positively correlated, but not perfectly so. On aver-age, the correlation coefficient for the returns on two randomly selected stockswould be about , and for most pairs of stocks, r would lie in the range of to . Under such conditions, combining stocks into portfolios reduces risk butdoes not eliminate it 6-7 illustrates this point with two stockswhose correlation coefficient is r . The portfolio’s average return is 15percent, which is exactly the same as the average return for each of the twostocks, but its standard deviation is percent, which is less than the standarddeviation of either stock. Thus, the portfolio’s risk is notan average of the risksof its individual stocks—diversification has reduced, but not eliminated, these two-stock portfolio examples, we have seen that in one extremecase (r ), risk can be completely eliminated, while in the other extremecase (r ), diversification does nothing to limit risk. The real world liesbetween these extremes, so in general combining two stocks into a portfolio re-duces, but does not eliminate, the riskiness inherent in the individual would happen if we included more than two stocks in the portfolio?As a rule, the riskiness of a portfolio will decline as the number of stocks in the portfo-lio we added enough partially correlated stocks, could we completelyeliminate risk? In general, the answer is no, but the extent to which addingstocks to a portfolio reduces its risk depends on the degree of correlationamongthe stocks: The smaller the positive correlation coefficients, the lower the riskin a large portfolio. If we could find a set of stocks whose correlations were zeroor negative, all risk could be eliminated. In the real world, where the correlationsamong the individual stocks are generally positive but less than , some, but not all,risk can be test your understanding, would you expect to find higher correlations be-tween the returns on two companies in the same or in different industries? Forexample, would the correlation of returns on Ford’s and General Motors’ stocksbe higher, or would the correlation coefficient be higher between either Fordor GM and AT&T, and how would those correlations affect the risk of portfo-lios containing them?Answer:Ford’s and GM’s returns have a correlation coefficient of about one another because both are affected by auto sales, but their correlationis only about with AT&:Atwo-stockportfolioconsistingofFordandGMwouldbelesswelldiversifiedthanatwo-stockportfolioconsistingofFordorGM,plusAT&,tominimizerisk, leaving this section we should issue a warning—in the real world, itis impossibleto find stocks like W and M, whose returns are expected to be per-fectly negatively correlated. Therefore, it is impossible to form completely risklessstock can reduce risk, but it cannot eliminate it. Thereal world is closer to the situation depicted in Figure IN A PORTFOLIO CONTEXT249
Rate of Return Distributions for Two Perfectly Positively CorrelatedFIGURE 6-6Stocks (r ) and for Portfolio MM a. Rates of Return___Stock MStock M´Portfolio MM´k (%)k (%)k (%)MMp252525151515000200120012001–10–10–10b. Probability Distributions of ReturnsProbabilityProbabilityProbabilityDensityDensityDensity015Percent015Percent015Percentˆˆˆ(= k )(= k )(= k )MMpSTOCK M STOCK M PORTFOLIO MM YEAR(k )(k )( k)MMp1997(%)(%)(%)()()()%%%Average %%%Standard %%%250CHAPTER 6 RISK AND RATES OF RETURN
Rate of Return Distributions for Two Partially Correlated Stocks FIGURE 6-7(r ) and for Portfolio WYa. Rates of Return___Stock Yk (%)Stock Wk (%)k (%)Portfolio WYWYp252525151515000200120012001–15–15–15b. Probability Distributions of ReturnsProbabilityDensityPortfolio WYStocks W and Y015Percentˆ(= k )pSTOCK W STOCK Y PORTFOLIO WY YEAR(k )( k)( k)%%%1998()()()()%%%Average %%%Standard %%%RISK IN A PORTFOLIO CONTEXT251
DIVERSIFIABLERISKVERSUSMARKETRISKAs noted above, it is difficult if not impossible to find stocks whose expected re-turns are negatively correlated—most stocks tend to do well when the national7economy is strong and badly when it is , even very large portfoliosend up with a substantial amount of risk, but not as much risk as if all themoney were invested in only one see more precisely how portfolio size affects portfolio risk, consider Fig-ure 6-8, which shows how portfolio risk is affected by forming larger and largerportfolios of randomly selected New York Stock Exchange (NYSE) deviations are plotted for an average one-stock portfolio, a two-stockportfolio, and so on, up to a portfolio consisting of all 2,000-plus commonstocks that were listed on the NYSE at the time the data were graphed. Thegraph illustrates that, in general, the riskiness of a portfolio consisting of large-company stocks tends to decline and to approach some limit as the size of theportfolio increases. According to data accumulated in recent years, , the stan-1dard deviation of a one-stock portfolio (or an average stock), is approximatelyMarket Portfolio35 percent. A portfolio consisting of all stocks, which is called the marketA portfolio consisting of all , would have a standard deviation, , of about percent, which isMshown as the horizontal dashed line in Figure , almost half of the riskiness inherent in an average individual stock can beeliminated if the stock is held in a reasonably well-diversified portfolio, which is onecontaining 40 or more risk always remains, however, so it is virtuallyimpossible to diversify away the effects of broad stock market movements thataffect almost all ’sriskthatcanbeeliminatediscalleddiversifiablerisk,, RiskDiversifiable riskis caused by such random events as lawsuits, strikes, suc-That part of a security’s riskcessful and unsuccessful marketing programs, winning or losing a major con-associated with random events; ittract, and other events that are unique to a particular firm. Since these eventscanbe eliminated by properare random, their effects on a portfolio can be eliminated by diversification— events in one firm will be offset by good events in another. Market risk,on the other hand, stems from factors that systematically affect most firms: war,Market Riskinflation, recessions, and high interest rates. Since most stocks are negativelyThat part of a security’s risk thataffected by these factors, market risk cannot be eliminated by eliminated byWe know that investors demand a premium for bearing risk; that is, the riskiness of a security, the higher its expected return must be to in-duce investors to buy (or to hold) it. However, if investors are primarily con-cerned with the riskiness of their portfoliosrather than the riskiness of the indi-7It is not too hard to find a few stocks that happened to have risen because of a particular set ofcircumstances in the past while most other stocks were declining, but it is much harder to findstocks that could logically be expectedto go up in the future when other stocks are risk is also known as company-specific,or unsystematic,risk. Market risk is also knownas nondiversifiable,or systematic,or beta,risk; it is the risk that remains after 6 RISK AND RATES OF RETURN
FIGURE 6-8Effects of Portfolio Size on Portfolio Risk for Average StocksPortfolio Risk, σp(%)3530Diversifiable Risk25σ= Attainable Risk in aPortfolio of Average StocksPortfolio's15Stand-AlonePortfolio'sRisk:Market Risk:DeclinesRemains Constantas Stocks10Are Added501102030402,000+Number of Stocksin the Portfoliovidual securities in the portfolio, how should the riskiness of an individual stockCapital Asset Pricing Modelbe measured? One answer is provided by the Capital Asset Pricing Model(CAPM)(CAPM), an important tool used to analyze the relationship between risk and9A model based on the propositionrates of primary conclusion of the CAPM is this: The relevant risk-that any stock’s required rate ofiness of an individual stock is its contribution to the riskiness of a well-diversified port-return is equal to the risk-free other words, the riskiness of General Electric’s stock to a doctor whoof return plus a risk premium thatreflects only the risk remainingafter , the 1990 Nobel Prize was awarded to the developers of the CAPM, Professors HarryMarkowitz and William F. Sharpe. The CAPM is a relatively complex subject, and only its basic el-ements are presented in this text. For a more detailed discussion, see any standard investments basic concepts of the CAPM were developed specifically for common stocks, and, there-fore, the theory is examined first in this context. However, it has become common practice to ex-tend CAPM concepts to capital budgeting and to speak of firms having “portfolios of tangible as-sets and projects.” Capital budgeting is discussed in Chapters 11 and IN A PORTFOLIO CONTEXT253
has a portfolio of 40 stocks or to a trust officer managing a 150-stock portfoliois the contribution the GE stock makes to the portfolio’s riskiness. The stockmight be quite risky if held by itself, but if half of its risk can be eliminated bydiversification, then its relevant risk, which is its contribution to the portfolio’sRelevant RiskThe risk of a security that cannotrisk,is much smaller than its stand-alone diversified away, or its . This reflects a security’,youwin$20,000,butifcontribution to the riskiness of aatailcomesup,youlose$16,—($20,000) ( $16,000) $2,,itisahighlyriskyproposition,becauseyouhavea50percentchanceoflosing$16,,,supposeyouwereof-feredthechancetoflipacoin100times,andyouwouldwin$200foreachheadbutlose$$20,000,anditisalsopossiblethatyouwouldflipalltailsandlose$16,000,butthechancesareveryhighthatyouwouldactuallyflipabout50headsandabout50tails,winninganetofabout$2,-dividualflipisariskybet,,exceptthatwithstocksalloftheriskcannotbeeliminatedbydiversification—thoserisksre-latedtobroad,-diversified portfolio would have the same effect on the portfolio’s riskiness?The answer is no. Different stocks will affect the portfolio differently, so dif-ferent securities have different degrees of relevant risk. How can the relevantrisk of an individual stock be measured? As we have seen, all risk except that re-lated to broad market movements can, and presumably will, be diversified all, why accept risk that can be easily eliminated? The risk that remainsafter diversifying is market risk, or the risk that is inherent in the market, and it canbe measured by the degree to which a given stock tends to move up or down with the next section, we develop a measure of a stock’s market risk, andthen, in a later section, we introduce an equation for determining the requiredrate of return on a stock, given its market tendency of a stock to move up and down with the market is reflected inBeta Coefficient, bits beta coefficient, b. Beta is a key element of the CAPM. An average-risk stockA measure of market risk, which isis definedas one that tends to move up and down in step with the general mar-the extent to which the returns onket as measured by some index such as the Dow Jones Industrials, the S&P 500,a given stock move with the stockor the New York Stock Exchange Index. Such a stock will, by definition, be a beta, b, of , which indicates that, in general, if the market moves upby 10 percent, the stock will also move up by 10 percent, while if the marketfalls by 10 percent, the stock will likewise fall by 10 percent. A portfolio of suchb stocks will move up and down with the broad market averages, and itwill be just as risky as the averages. If b , the stock is only half as volatileas the market—it will rise and fall only half as much—and a portfolio of suchstocks will be half as risky as a portfolio of b stocks. On the other hand,if b , the stock is twice as volatile as an average stock, so a portfolio of254CHAPTER 6 RISK AND RATES OF RETURN
such stocks will be twice as risky as an average portfolio. The value of such aportfolio could double—or halve—in a short time, and if you held such aportfolio, you could quickly go from millionaire to 6-9 graphs the relative volatility of three stocks. The data below thegraph assume that in 1999 the “market,” defined as a portfolio consisting of allstocks, had a total return (dividend yield plus capital gains yield) of k 10%,Mand Stocks H, A, and L (for High, Average, and Low risk) also all had returnsof 10 percent. In 2000, the market went up sharply, and the return on the mar-ket portfolio was k 20%. Returns on the three stocks also went up: HMsoaredto30percent;Awentupto20percent,thesameasthemarket;andLonly went up to 15 percent. Now suppose the market dropped in 2001, andthe market return was k 10%. The three stocks’ returns also fell, HMplunging to 30 percent, A falling to 10 percent, and L going down only to k 0%. Thus, the three stocks all moved in the same direction as the mar-Lket, but H was by far the most volatile; A was just as volatile as the market; andL was less ’svolatilityrelativetoanaveragestock,whichbydefinitionhasb ,andastock’—indeed,theslopecoefficientofsucha“regressionline”isdefinedasabetacoefficient.(Proceduresforactuallycalcu-latingbetasaredescribedinAppendix6A.)BetasforliterallythousandsofcompaniesarecalculatedandpublishedbyMerrillLynch,ValueLine,andnu-merousotherorganizations,,, it is possible for a stock to have a negative beta. In this case,the stock’s returns would tend to rise whenever the returns on other stocks practice, we have never seen a stock with a negative beta. For example, ValueLinefollows more than 1,700 stocks, and not one has a negative beta. Keep inmind, though, that a stock in a given year may move counter to the overallmarket, even though the stock’s beta is positive. If a stock has a positive beta,we would expectits return to increase whenever the overall stock market , company-specific factors may cause the stock’s realized return to de-cline, even though the market’s return is a stock whose beta is greater than is added to a b portfolio, thenthe portfolio’s beta, and consequently its riskiness, will increase. Conversely, ifa stock whose beta is less than is added to a b portfolio, the portfo-lio’s beta and risk will decline. Thus, since a stock’s beta measures its contributionto the riskiness of a portfolio, beta is the theoretically correct measure of the stock’ preceding analysis of risk in a portfolio context is part of the CapitalAsset Pricing Model (CAPM), and we can summarize our discussion to thispoint as follows: stock’s risk consists of two components, market risk and ,andmostinvestorsdoindeeddiversify,eitherbyholdinglargeportfoliosorbypurchasingRISK IN A PORTFOLIO CONTEXT255
THE BENEFITS OF DIVERSIFYING OVERSEAShe size of the global stock market has grown steadily overResearchers and practitioners alike have struggled to under-Tthe last several decades, and it passed the $15 trillion markstand this reluctance to invest overseas. One explanation isduring 1995. . stocks account for approximately 41 percentthat investors prefer domestic stocks because they have lowerof this total, whereas the Japanese and European markets con-transaction costs. However, this explanation is not completelystitute roughly 25 and 26 percent, respectively. The rest of theconvincing, given that recent studies have found that investorsworld makes up the remaining 8 percent. Although the . eq-buy and sell their overseas stocks more frequently than theyuity market has long been the world’s biggest, its share of thetrade their domestic stocks. Other explanations for the domes-world total has decreased over bias focus on the additional risks from investing overseasThe expanding universe of securities available internation-(for example, exchange rate risk) or suggest that the typicalally suggests the possibility of achieving a better . investor is uninformed about international investmentstrade-off than could be obtained by investing solely in . se-and/or views international investments as being extremely riskycurities. So, investing overseas might lower risk and simultane-or uncertain. More recently, other analysts have argued that asously increase expected returns. The potential benefits of diver-world capital markets have become more integrated, the corre-sification are due to the facts that the correlation between thelation of returns between different countries has increased, andreturns on . and international securities is fairly low, and re-hence the benefits from international diversification have de-turns in developing nations are often quite . A third explanation is that . corporations are them-Figure 6-8, presented earlier, demonstrated that an investorselves investing more internationally, hence . investors arecan significantly reduce the risk of his or her portfolio by hold-de facto obtaining international a large number of stocks. The figure accompanying this boxWhatever the reason for the general reluctance to hold in-suggests that investors may be able to reduce risk even furtherternational assets, it is a safe bet that in the future . in-by holding a large portfolio of stocks from all around the world,vestors will shift more and more of their assets to overseas in-given the fact that the returns of domestic and are not perfectly the apparent benefits from investing overseas, thetypical . investor still dedicates less than 10 percent of hisSOURCE:Kenneth Kasa, “Measuring the Gains from International Portfolio Diversi-or her portfolio to foreign stocks—even though foreign stocksfication,” Federal Reserve Bank of San Francisco Weekly Letter,Number 94-14, Aprilrepresent roughly 60 percent of the worldwide equity , ,then,withmarketrisk,whichiscausedbygeneralmovementsinthestockmarketandwhichreflectsthefactthatmoststocksaresystematicallyaffectedbyeventslikewar,recessions,, must be compensated for bearing risk—the greater the riski-ness of a stock, the higher its required return. However, compensation existed on stocks due to diversifiable risk, well-diversifiedinvestors would start buying those securities (which would not be espe-cially risky to such investors) and bidding up their prices, and the stocks’final (equilibrium) expected returns would reflect only nondiversifiablemarket this point is not clear, an example may help clarify it. Suppose halfof Stock A’s risk is market risk (it occurs because Stock A moves up anddown with the market), while the other half of A’s risk is diversifiable. You256CHAPTER 6 RISK AND RATES OF RETURN
Portfolio Risk, σp(%). . and International StocksNumber of Stocksin the Portfoliohold only Stock A, so you are exposed to all of its risk. As compensationforbearingsomuchrisk,youwantariskpremiumof8percentoverthe10 percent T-bond rate. Thus, your required return is k 10% A8% 18%. But suppose other investors, including your professor, arewell diversified; they also hold Stock A, but they have eliminated its di-versifiable risk and thus are exposed to only half as much risk as , their risk premium will be only half as large as yours, andtheir required rate of return will be k 10% 4% 14%.AIf the stock were yielding more than 14 percent in the market, diver-sified investors, including your professor, would buy it. If it were yielding18 percent, you would be willing to buy it, but well-diversified investorswould bid its price up and its yield down, hence you could not buy it at aprice low enough to provide you with an 18 percent return. In the end,you would have to accept a 14 percent return or else keep your money inthe bank. Thus, risk premiums in a market populated by rational, diver-sified investors can reflect only market IN A PORTFOLIO CONTEXT257
FIGURE 6-9Relative Volatility of Stocks H, A, and L_Return on Stock i, k i(%)Stock H,High Risk: b = A,Average Risk: b = L,Low Risk: b = –20–100102030_Return on the Market, k(%)M–10–20–30YEAR k k k kHALM199910%10%10%10%2000302015202001(30)(10)0(10)NOTE:These three stocks plot exactly on their regression lines. This indicates that they are exposedonly to market risk. Mutual funds that concentrate on stocks with betas of 2, 1, and would havepatterns similar to those shown in the market risk of a stock is measured by its beta coefficient, which is anindex of the stock’s relative volatility. Some benchmark betas follow:b : Stock is only half as volatile, or risky, as an average : Stock is of average : Stock is twice as risky as an average 6 RISK AND RATES OF RETURN
Illustrative List of Beta CoefficientsTABLE 6-4STOCKBETAMerrill & District is a gas distribution company. It has a monopoly in much of Alabama, and its prices areadjusted every three months so as to keep its profits relatively : Value Line,September 2000, portfolio consisting of low-beta securities will itself have a low beta, be-cause the beta of a portfolio is a weighted average of its individual secu-rities’ betas: b wb wb wbp1122nnn wb.(6-6)iiai 1Here bis the beta of the portfolio, and it shows how volatile the portfo-plio is in relation to the market; wis the fraction of the portfolio investediin the ith stock; and bis the beta coefficient of theith stock. For exam-iple, if an investor holds a $100,000 portfolio consisting of $33, in-vested in each of three stocks, and if each of the stocks has a beta of ,then the portfolio’s beta will be b :pb () () () a portfolio will be less risky than the market, so it should experiencerelatively narrow price swings and have relatively small rate-of-returnfluctuations. In terms of Figure 6-9, the slope of its regression line wouldbe , which is less than that for a portfolio of average suppose one of the existing stocks is sold and replaced by a stockwithb :p1p2b () () ()p2 IN A PORTFOLIO CONTEXT259
Had a stock with b been added, the portfolio beta would have de-iclined from to . Adding a low-beta stock, therefore, would reducethe riskiness of the portfolio. Consequently, adding new stocks to a port-folio can change the riskiness of that a stock’s beta coefficient determines how the stock affects the riskiness of adiversified portfolio, beta is the most relevant measure of any stock’s -TEST QUESTIONSExplain the following statement: “An asset held as part of a portfolio is gen-erally less risky than the same asset held in isolation.”What is meant by perfect positive correlation, perfect negative correlation,and zero correlation?In general, can the riskiness of a portfolio be reduced to zero by increasingthe number of stocks in the portfolio? is an average-risk stock? What will be its beta?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 DowJones Index over the past five years, what would the slope of the regressionline you obtained indicate about the stock’s market risk?THE RELATIONSHIP BETWEEN RISK AND RATES OF RETURNIn the preceding section, we saw that under the CAPM theory, beta is the ap-propriate measure of a stock’s relevant risk. Now we must specify the relation-ship between risk and return: For a given level of risk as measured by beta, whatrate of return will investors require to compensate them for bearing that risk?To begin, let us define the following terms:ˆk expectedrate of return on the ith ˆthatifkislessthank,youwouldnotpur-iichasethisstock,oryouwouldsellitifyouˆ,youwouldiiwanttobuythestock,becauseitlookslikeaˆ k realized, after-the-fact return. One obvi-ously does not know kat the time he or sheis considering the purchase of a risk-free rate of return. In this context, kisRFRFgenerally measured by the return on long-term . Treasury 6 RISK AND RATES OF RETURN
IS THE DOW JONES HEADING TO 36,000?n the 18-year period since 1982, the Dow Jones IndustrialSiegel acknowledges that stocks are riskier for short-term in-IAverage has risen from 777 to over 10,526, or an increase ofvestors. This point is confirmed when we compare the averageapproximately 1,255 percent! Although millions of investorsannual standard deviation of stock market returns (20 percent)have profited from this increase, many analysts believe thatwith that of bonds (9 percent). The higher volatility of stocksstocks are now overvalued. These analysts point to record P/Eoccurs because stocks get hit harder than bonds in the shortratios as an indication that stock prices are too high. Federalrun when the economy weakens or inflation increases. However,Reserve Chairman Alan Greenspan made the same point, warn-stocks have always eventually recovered, and over longer peri-ing about the dangers of “irrational exuberance.”ods they have outperformed sharp contrast to this bearish perspective, James Glass-Glassman and Hassett contend that more and more investorsman and Kevin Hassett, co-authors of a book, Dow 36,000,are viewing stocks as long-term investments, and they are con-make the following argument:vinced that the long-run risk of stocks is fairly low. This has ledinvestors to put increasing amounts of money in the stock mar-Using sensible assumptions, we are comfortable with stockket, pushing up stock prices and driving stocks’ returns evenprices rising to three or four times their current levels. Ourhigher. These positive results, in turn, lower the perceived risk-calculations show that with earnings growing at the sameiness of stocks, and that leads to still more buying and furtherrate as the gross domestic product and Treasury bond yieldsstock market 6 percent, a perfectly reasonable level for the DowTo put all of this in perspective, we need to address threewould be 36,000—tomorrow, not 10 or 20 years from points. First, the relevant market risk premium is for-ward looking—it is based on investors’ perceptions of the rel-How do Glassman and Hassett reach this conclusion? They claimative riskiness of stocks versus bonds in the future, and it willthat the market risk premium (k k) has declined, and thatMRFchange over time. Most analysts acknowledge that the risk pre-it will continue to decline in the future. Investors require a riskmium has fallen, but few agree with Glassman and Hassett thatpremium for bearing risk, and the size of that premium dependsit is or should be zero. Most believe that investors require aon the average investor’s degree of risk aversion. From 1926premium in the neighborhood of at least 3 to 5 percent as anthrough 1999, large-company stocks have produced average an-inducement for holding stocks. Moreover, the risk premiumnual returns of percent, while the returns on long-termwould probably rise sharply if something led to a sustainedgovernment bonds have averaged percent, suggesting a riskmarket of % % %. However, Glassman andSecond, if the risk premium were to stabilize at a relativelyHassett make the following assertion:low level, then investors would receive low stock returns in theWhat has happened since 1982, and especially during thefuture. For example, if the T-bond yield were 5 percent and thepast four years, is that investors have become calmer andmarket risk premium were 3 percent, then the required returnsmarter. They are requiring a much smaller extra return, oron the market would be 8 percent. In this situation, it would“risk premium,” from stocks to compensate for their unreasonable to expect stock returns of 12 to 13 percent inThat premium, which has averaged about 7 percent in mod-the history, is now around 3 percent. We believe that it isThird, investors should be concerned with real returns,headed for its proper level: zero. That means that stockwhich take inflation into account. For example, suppose theprices should rise -free nominal rate of return were percent and the mar-ket risk premium were 3 percent. Here, the expected nominalAdecliningriskpremiumleadstoalowerrequiredreturnonreturn on an average stock would be percent. If ,inturn,impliesthatstockpricesshouldrisebe-were percent, the real return would be only looking at things in terms of real returns ,GlassmanandHassettcitere-that with low market risk premiums, stocks will have a hardsearchbyJeremySiegeloftheUniversityofPennsylvania’sWhar-time competing with inflation-indexed Treasury securities,-sellingbook,StocksfortheLongRun,which currently provide investors with only a slightly lower realSiegeldocumentsthatoverthelongrunstockshavenotbeenreturn with considerably less ,basedonhisresearch,Siegelcon-cludesthat,“Thesafestlong-terminvestmentforthepreserva-SOURCE:James K. Glassman and Kevin A. Hassett, “Stock Prices Are Still Far Tootionofpurchasingpowerhasclearlybeenstocks,notbonds.”Low,” The Wall Street Journal,March 17, 1999, IN A PORTFOLIO CONTEXT261
b beta coefficient of the ith stock. The beta ofian average stock is b required rate of return on a portfolio con-Msisting of all stocks, which is called the mar-ket portfolio. kis also the required rate ofMreturn on an average (b ) (k k) risk premium on “the market,” and also onMMRFan average (b ) stock. This is the addi-tional return over the risk-free rate requiredto compensate an average investor for as-suming an average amount of risk. Averagerisk means a stock whose b b (k k)b (RP)b risk premium on the ith stock. The stock’siMRFiMirisk premium will be less than, equal to, orgreater than the premium on an averagestock,RP,dependingonwhetheritsbetaMisless than, equal to, or greater than . Ifb b , then RP Risk Premium, RPThe market risk premium, RP,shows the premium investors require forMMThe additional return over thebearing the risk of an average stock. The size of this premium depends on therisk-free rate needed toperceived risk of the stock market and investors’ degree of risk aversion. Let uscompensate investors for assumingassume that at the current time Treasury bonds yield k 6% and an averageRFan average amount of of stock has a required rate of return of k 11%. Therefore, the mar-Mket risk premium is 5 percent calculated as:RP k k 11% 6% 5%.MMRFItshouldbenotedthattheriskpremiumofanaveragestock,k k,isMRFhardtomeasurebecauseitisimpossibletoobtainpreciseestimatesofthe10expectedfuturereturnofthemarket,,, historical estimates might be a good starting point for estimating themarket risk premium, historical estimates may be misleading if investors’ atti-tudes toward risk change considerably over time. (See the Industry Practice Boxentitled “Estimating the Market Risk Premium.”) Indeed, many analysts haveargued that the market risk premium has fallen in recent years because an in-creasing number of investors have been willing to bear the risks of the stockmarket. If this claim is correct, the market risk premium may be considerablylower than what would be implied using historical data. (See the Industry Prac-tice Box entitled “Is the Dow Jones Heading to 36,000?” for a discussion of10This concept, as well as other aspects of the CAPM, is discussed in more detail in Chapter 3 ofEugene F. Brigham and Phillip R. Daves, Intermediate Financial Management, 7th ed.(Fort Worth,TX: Harcourt College Publishers, 2002). Chapter 3 of Intermediate Financial Managementalso dis-cusses the assumptions embodied in the CAPM framework. Some of these are unrealistic, and be-cause of this the theory does not hold 6 RISK AND RATES OF RETURN
how changes in investor risk aversion may have influenced the market risk pre-mium and stock returns in recent years.)Whilethemarketriskpremiumrepresentstheriskpremiumfortheentirestockmarket,,ifonestockweretwiceasriskyasanother,itsriskpremiumwouldbetwiceashigh,whileifitsriskwereonlyhalfasmuch,,wecanmeasureastock’,RP,andthestock’sriskasmea-Msuredbyitsbetacoefficient,b,wecanfindthestock’sriskpremiumastheiproduct(RP),ifb 5%,:Risk premium for Stock i RP (RP)b(6-7)iMi (5%)() %.As the discussion in Chapter 5 implied, the required return for any invest-ment can be expressed in general terms asRequired return Risk-free return Premium for -freereturnincludesapremiumforexpectedinflation,,therequiredreturnforStockicanbewrittenasfollows:Required returnRisk-freeMarket riskStock i'sSML Equation: ababon Stock iratepremiumbetak k (k k)b(6-8)iRFMRFi k (RP)bRFMi 6% (11% 6%)() 6% 5%() %.Equation 6-8 is called the Security Market Line (SML).If some other Stock j were riskier than Stock i and had b , then its re-jquired rate of return would be 16 percent:k 6% (5%) 16%.jAn average stock, with b , would have a required return of 11 percent, thesame as the market return:k 6% (5%) 11% Market Line (SML)As noted above, Equation 6-8 is called the Security Market Line (SML)The line on a graph that showsequation, and it is often expressed in graph form, as in Figure 6-10, whichthe relationship between risk asshows the SML when k 6% and k 11%. Note the following points:RFMmeasured by beta and the requiredrate of return for ,whileriskassecurities. Equation 6-8 is for the -9, RELATIONSHIP BETWEEN RISK AND RATES OF RETURN263
ESTIMATING THE MARKET RISK PREMIUMhe Capital Asset Pricing Model (CAPM) is more than just aical) risk premium. In this situation, an analyst would be se-Ttheory describing the trade-off between risk and return. Theriously missing the boat if he used the historical risk premiumCAPM is also widely used in practice. As we will see in Chapterto approximate the expected risk premium. To further illustrate9, investors use the CAPM to determine the discount rate forthis point, the strong performance in the stock market over thevaluing stocks. Later, in Chapter 10, we will also see that cor-past several years has produced high historical premiums—in-porate managers use the CAPM to estimate the cost of equity fi-deed, Ibbotson Associates estimate that the market risk averaged percent a year during the period betweenThemarketriskpremiumisanimportantcomponentofthe1995 and 1999. Nobody would seriously suggest that future ,whatwewouldideallyliketouseinthevestors require a percent premium to invest in the stockCAPMistheexpectedmarketriskpremium,whichgivesanin-market! Given these concerns, Ibbotson and others suggest thatdicationofinvestors’,wecannothistorical estimates are more reliable if estimated over longerdireclyobserveinvestors’,academicianstime -A second concern is that historical estimates may be because they only include the returns of firms that haveHistorical premiums are found by taking the differences be-survived and do not take into account the performances of fail-tween actual returns of the overall stock market and the risk-ing firms. Stephen Brown, William Goetzmann, and Stephenfree rate. Ibbotson Associates provide perhaps the most com-Ross discussed the implications of this “survivorship bias” in aprehensive estimates of historical risk premiums. Their1995 Journal of Financearticle. Putting these ideas into prac-estimates indicate that the equity risk premium has averagedtice, Tom Copeland, Tim Koller, and Jack Murrin have recentlyabout 8 percent a year over the past 75 that this “survivorship bias” increases historical re-1Analysts have pointed out some of the shortcomings ofturns by 1⁄2to 2 percent a year. For that reason, they suggestusing an historical estimate as a proxy for the expected riskthat practitioners trying to estimate a forward-looking expected1premium. First, historical estimates may be very misleading atrisk premium subtract 1⁄2to 2 percent from their historical risktimes when the market risk premium is changing. As we men-premium in an earlier box entitled “Is the Dow Jones Heading toSOURCES:Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2000 Yearbook36,000?,” many analysts believe that the expected risk premium(Chicago: Ibbotson Associates, 2000); Stephen J. Brown, William N. Goetzmann,has fallen in recent years. It is important to recognize that aand Stephen A. Ross, “Survival,” The Journal of Finance,Vol. 50, No. 3, July 1995,sharp drop in the expected risk premium (perhaps because of853–873; and Tom Copeland, Tim Koller, and Jack Murrin, Valuation: Measuringlower perceived risk and/or declining risk aversion) pushes upand Managing the Value of Companies, 3rd edition,(New York: McKinsey & Com-stock prices, and that ironically increasesthe observed (histor-pany, 2000).threelinesinFigure6-9wereusedtocalculatethethreestocks’betas, 0;therefore,,(5%inFigure6-10)reflectsthedegreeofriskaversionintheeconomy—thegreatertheaverageinvestor’saversiontorisk,then(a)thesteepertheslopeoftheline,(b)thegreaterthe264CHAPTER 6 RISK AND RATES OF RETURN
FIGURE 6-10The Security Market Line (SML)Required RateSML: k = k + (k – k ) biRFMRFiof Return (%)= 6% + (11% – 6%) bi= 6% + (5%) b ik = 16HighRelativelyMarket RiskRisky Stock’sk = k = 11MAPremium: 5%.Risk Premium: 10%Applies Also toan Average Stock,k = Is the SlopeSafe Stock’sCoefficient in theRiskSML EquationPremium: %k = 6RFRisk-FreeRate, , b iriskpremiumforallstocks,and(c) values we worked out for stocks with b , b , and b with the values shown on the graph for k, k, and ’spositiononitchangeovertimeduetochangesininterestrates,investors’aversiontorisk,andindividualcompanies’ sometimes confuse beta with the slope of the SML. This is a mistake. The slope of anystraightlineisequaltothe“rise”dividedbythe“run,”or(Y Y)/(X X).ConsiderFigure10106-10. If we let Y k and X beta, and we go from the origin to b , we see that the slope is(k k)/(b b) (11% 6%)/(1 0) 5%.Thus,theslopeoftheSMLisequaltoMRFMRF(k k), the market risk premium. In Figure 6-10, k 6% 5%b, so a doubling of beta (forMRFiiexample, from to ) would produce a 5 percentage point increase in RELATIONSHIP BETWEEN RISK AND RATES OF RETURN265
THEIMPACTOFINFLATIONAswelearnedinChapter5,interestamountsto“rent”onborrowedmoney,,,orquoted,rate,anditconsistsoftwoelements:(1)arealinfla-tion-freerateofreturn,k*,and(2)aninflationpremium,IP,,k k* -termTreasurybondsRFhashistoricallyrangedfrom2to4percent,-fore,ifnoinflationwereexpected,,astheexpectedrateofinflationincreases,,the6percentkshownRFinFigure6-10mightbethoughtofasconsistingofa3percentrealrisk-freerateofreturnplusa3percentinflationpremium:k k* IP 3% 3% 6%.RFIf the expected inflation rate rose by 2 percent, to 3% 2% 5%, thiswould cause kto rise to 8 percent. Such a change is shown in Figure that under the CAPM, the increase in kleads to an equalincrease inRFthe rate of return on all risky assets, because the same inflation premium is built13into the required rate of return of both riskless and risky example,the rate of return on an average stock, k, increases from 11 to 13 risky securities’ returns also rise by two percentage slope of the Security Market Line reflects the extent to which investors areaverse to risk—the steeper the slope of the line, the greater the average in-vestor’s risk aversion. Suppose investors were indifferent to risk; that is, theywere not risk averse. If kwere 6 percent, then risky assets would also provideRFan expected return of 6 percent, because if there were no risk aversion, therewould be no risk premium, and the SML would graph as a horizontal line. Asrisk aversion increases, so does the risk premium, and this causes the slope ofthe SML to become 6-12 illustrates an increase in risk aversion. The market risk premiumrises from 5 to percent, causing kto rise from k 11% to k %. The returns on other risky assets also rise, and the effect of this shift inrisk aversion is more pronounced on riskier securities. For example, the re-quired return on a stock with b increases by only percentage points,ifrom to percent, whereas that on a stock with b increases percentage points, from to -term Treasury bonds also contain a maturity risk premium, MRP. Here we include theMRP in k* to simplify the that the inflation premium for any asset is equal to the average expected rate of inflationover the asset’s life. Thus, in this analysis we must assume either that all securities plotted on theSML graph have the same life or else that the expected rate of future inflation is -termrate(theRFT-bondrate)orashort-termrate(theT-billrate).Traditionally,theT-billratewasused,,Bonds,Bills,andInflation:(ValuationEdition)2000Yearbook(Chicago:IbbotsonAssociates,2000) 6 RISK AND RATES OF RETURN
FIGURE 6-11Shift in the SML Caused by an Increase in Inflation Required Rateof Return (%)SML= 8% + 5%(b)2iSML= 6% + 5%(b)1ik = 13M2k = 11M1k = 8RF2Increase in Anticipated Inflation, IP = 2%k = 6RF1Original IP = 3%k* = 3Real Risk-Free Rate of Return, k*, b iCHANGESINASTOCK’SBETACOEFFICIENTAs we shall see later in the book, a firm can influence its market risk, hence itsbeta, through changes in the composition of its assets and also through its useof debt. A company’s beta can also change as a result of external factors such asincreased competition in its industry, the expiration of basic patents, and thelike. When such changes occur, the required rate of return also changes, and,as we shall see in Chapter 9, this will affect the firm’s stock price. For example,consider Allied Food Products, with a beta of . Now suppose some actionoccurred that caused Allied’s beta to increase from to . If the condi-tions depicted in Figure 6-10 held, Allied’s required rate of return would in-crease from 13 to 16 percent:k k (k k)b1RFMRFi 6% (11% 6%) 13%to k 6% (11% 6%) 16%.As we shall see in Chapter 9, this change would have a dramatic impact on Al-lied’s stock RELATIONSHIP BETWEEN RISK AND RATES OF RETURN267
FIGURE 6-12Shift in the SML Caused by Increased Risk AversionSML = 6% + %(b)2iRequired Rateof Return (%) = 6% + 5%(b)1ik = = 11M1New Market , k– k = %M2 = 6RFOriginal Market RiskPremium, k– k = 5%M1 , b iSELF-TEST QUESTIONSˆDifferentiate among the expected rate of return (k), the required rate of re-turn (k), and the realized, after-the-fact return ( k) on a stock. Which wouldˆˆhave to be larger to get you to buy the stock, kor k? Would k, k, and ktyp-ically be the same or different? are the differences between the relative volatility graph (Figure 6-9),where “betas are made,” and the SML graph (Figure 6-10), where “betas areused”? Discuss both how the graphs are constructed and the informationthey happens to the SML graph in Figure 6-10 when inflation increases ordecreases?What happens to the SML graph when risk aversion increases or decreases?What would the SML look like if investors were indifferent to risk, that is, ifthey had zero risk aversion?How can a firm influence its market risk as reflected in its beta?268CHAPTER 6 RISK AND RATES OF RETURN
PHYSICAL ASSETS VERSUS SECURITIESInabookonfinancialmanagementforbusinessfirms,whydowespendsomuchtimediscussingtheriskinessofstocks?Whynotbeginbylookingattheriskinessofsuchbusinessassetsasplantandequipment?Thereasonisthat,foramanagementwhoseprimarygoalisstockpricemaximization,theover-ridingconsiderationistheriskinessofthefirm’sstock,andtherelevantriskofanyphysicalassetmustbemeasuredintermsofitseffectonthestock’,supposeGoodyearTireCompanyisconsideringamajorinvestmentinanewproduct,,henceearningsonthenewoperation,arehighlyuncertain,,supposereturnsintherecapbusinessarenegativelycorrelatedwithGoodyear’sregularoperations—whentimesaregoodandpeoplehaveplentyofmoney,theybuynewtires,butwhentimesarebad,,returnswouldbehighonregularoperationsandlowontherecapdivisionduringgoodtimes,, analysis can be extended to the corporation’s stockholders. BecauseGoodyear’s stock is owned by diversified stockholders, the real issue each timemanagement makes an asset investment is this: How will this investment affectthe risk of our stockholders? Again, the stand-alone risk of an individual proj-ect may look quite high, but viewed in the context of the project’s effect onstockholders’ risk, it may not be very large. We will address this issue again inChapter 10, where we examine the effects of capital budgeting on companies’beta coefficients and thus on stockholders’ -TEST QUESTIONSExplain the following statement: “The stand-alone risk of an individual proj-ect may be quite high, but viewed in the context of a project’s effect onstockholders’ risk, the project’s true risk may not be very large.”How would the correlation between returns on a project and returns on thefirm’s other assets affect the project’s risk?SOME CONCERNS ABOUT BETA AND THE CAPMTheCapitalAssetPricingModel(CAPM)ismorethanjustanabstracttheorydescribedintextbooks—itisalsowidelyusedbyanalysts,investors,,despitetheCAPM’sintuitiveappeal,,arecentstudybyEu-geneFamaoftheUniversityofChicagoandKennethFrenchofYalefoundnoPHYSICAL ASSETS VERSUS SECURITIES269
historicalrelationshipbetweenstocks’returnsandtheirmarketbetas, beta does not determine returns, what does? Fama and French found twovariablesthatareconsistentlyrelatedtostockreturns:(1)thefirm’ssizeand(2)itsmarket/book ratio. After adjusting for other factors, they found thatsmaller firms have provided relatively high returns, and that returns are higheron stocks with low market/book ratios. By contrast, they found no relationshipbetween a stock’s beta and its an alternative to the traditional CAPM, researchers and practitionershave begun to look to more general multi-beta models that encompass theCAPM and address its shortcomings. The multi-beta model is an attractivegeneralization of the traditional CAPM model’s insight that market risk—riskthat cannot be diversified away—underlies the pricing of assets. In the multi-beta model, market risk is measured relative to a set of risk factors that deter-mine the behavior of asset returns, whereas the CAPM gauges risk only relativeto the market return. It is important to note that the risk factors in the multi-beta model are all nondiversifiable sources of risk. Empirical research investi-gating the relationship between economic risk factors and security returns isongoing, but it has discovered several systematic empirical risk factors, includ-ing the bond default premium, the bond term structure premium, and ,-sentsasignificantstepforwardinsecuritypricingtheory,itdoeshavesomedeficiencieswhenappliedinpractice,-TEST QUESTIONAre there any reasons to question the validity of the CAPM? VERSUS RISKBefore closing this chapter, we should note that volatility does not necessarilyimply risk. For example, suppose a company’s sales and earnings fluctuatewidely from month to month, from year to year, or in some other this imply that the company is risky in either the stand-alone or portfoliosense? If the fluctuations follow seasonal or cyclical patterns, as for an ice creamdistributor or a steel company, they can be predicted, hence volatility would notsignify much in the way of risk. If the ice cream company’s earnings droppedabout as much as they normally did in the winter, this would not concern ,“TheCross-SectionofExpectedStockReturns,”JournalofFinance,,1992,427–465;,“CommonRiskFactorsintheReturnsonStocksandBonds,”JournalofFinancialEconomics,,1993,3– 6 RISK AND RATES OF RETURN
vestors, so the company’s stock price would not be affected. Similarly, if thesteel company’s earnings fell during a recession, this would not be a surprise, sothe company’s stock price would not fall nearly as much as its earnings. There-fore, earnings volatility does not necessarily imply investment consider some other company, say, Wal-Mart. In 1995 Wal-Mart’searnings declined for the first time in its history. That decline worried investors—they were concerned that Wal-Mart’s era of rapid growth had ended. Theresult was that Wal-Mart’s stock price declined more than its earnings. Again,we conclude that while a downturn in earnings does not necessarily imply risk,it could, depending on let’s consider stock price volatility as opposed to earnings volatility. Isstock price volatility more likely to imply risk than earnings volatility? The an-swer is a loud yes! Stock prices vary because investors are uncertain about thefuture, especially about future earnings. So, if you see a company whose stockprice fluctuates relatively widely (which will result in a high beta), you can betthat its future earnings are relatively unpredictable. Thus, biotech companieshave less predictable earnings than utilities, biotechs’ stock prices are volatile,and they have relatively high conclude, keep two points in mind: (1) Earnings volatility does not nec-essarily signify risk—you have to think about the cause of the volatility beforereaching any conclusion as to whether earnings volatility indicates risk. (2)Stock price volatility does signify -TEST QUESTIONSDoes earnings volatility necessarily imply risk? Explain. Whyisstockpricevolatilitymorelikelytoimplyriskthanearningsvolatility?Inthischapter,,wedifferentiatedbetweenstand-aloneriskandriskinaportfoliocontext,,wedevelopedtheCAPM,,wewillgiveyouthetoolstoestimatetherequiredratesofreturnforbonds,preferredstock,andcommonstock,,thecostofcapitalisanimportantelementinthefirm’ VERSUS RISK271
Riskcan be defined as the chance that some unfavorable event will occur. Theriskinessofanasset’scashflowscanbeconsideredonastand-alonebasis(eachassetbyitself)orinaportfoliocontext,wheretheinvestmentiscombinedwithotherassetsanditsriskisreducedthroughdiversification. Most rational investors hold portfolios of assets,and they are more con-cerned with the riskiness of their portfolios than with the riskiness of in-dividual assets. The expected returnon an investment is the mean value of its probabil-ity distribution of returns. Thegreatertheprobabilitythattheactualreturnwillbefarbelowtheex-pectedreturn, measures of stand-alone risk are the standard deviationand the co-efficient of variation. The average investor is risk averse,which means that he or she must becompensated for holding risky assets. Therefore, riskier assets have higherrequired returns than less risky assets. An asset’s risk consists of (1) diversifiable risk, which can be eliminatedby diversification, plus (2) market risk,which cannot be eliminated by di-versification. The relevant riskof an individual asset is its contribution to the riskinessof a well-diversified portfolio,which is the asset’s market risk cannot be eliminated by diversification, investors must becompensated for bearing it. The Capital Asset Pricing Model is a model based on the propositionthat any stock’s required rate of return is equal to the risk-free rate of re-turn plus a risk premium that reflects only the risk remaining after diver-sification. A stock’s beta coefficient, b,is a measure of its market risk. Beta mea-sures the extent to which the stock’s returns move relative to the market. Ahigh-betastockismorevolatilethananaveragestock, . The beta of a portfoliois a weighted averageof the betas of the indi-vidual securities in the portfolio. The Security Market Line (SML)equation shows the relationship be-tween a security’s market risk and its required rate of return. The returnrequired for any security i is equal to the risk-free rateplus the marketrisk premiumtimes the security’s beta: k k (k k) Even though the expected rate of return on a stock is generally equal toits required return, a number of things can happen to cause the requiredrate of return to change: (1) the risk-free rate can changebecause ofchanges in either real rates or anticipated inflation, (2) a stock’s beta canchange,and (3) investors’ aversion to risk can change. Because returns on assets in different countries are not perfectly corre-lated, global diversificationmay result in lower risk for multinationalcompanies and globally diversified 6 RISK AND RATES OF RETURN
In the next three chapters, we will see how a security’s rate of return affects itsvalue. Then, in the remainder of the book, we will examine the ways in whicha firm’s management can influence a stock’s riskiness and hence its -1The probability distribution of a less risky expected return is more peaked than that ofa riskier return. What shape would the probability distribution have for (a) completelycertain returns and (b) completely uncertain returns?6-2Security A has an expected return of 7 percent, a standard deviation of expected returnsof 35 percent, a correlation coefficient with the market of , and a beta coefficient of . Security B has an expected return of 12 percent, a standard deviation of returns of10 percent, a correlation with the market of , and a beta coefficient of . Which se-curity is riskier? Why?6-3Suppose you owned a portfolio consisting of $250,000 worth of long-term . govern-ment your portfolio be riskless? suppose you hold a portfolio consisting of $250,000 worth of 30-day Treasurybills. Every 30 days your bills mature, and you reinvest the principal ($250,000) in anew batch of bills. Assume that you live on the investment income from your port-folio and that you want to maintain a constant standard of living. Is your portfoliotruly riskless? you think of any asset that would be completely riskless? Could someone de-velop such an asset? -4A life insurance policy is a financial asset. The premiums paid represent the investment’ would you calculate the expected return on a life insurance policy? the owner of a life insurance policy has no other financial assets—the per-son’s only other asset is “human capital,” or lifetime earnings capacity. What is thecorrelation coefficient between returns on the insurance policy and returns on thepolicyholder’s human capital? insurance companies have to pay administrative costs and sales representatives’commissions; hence, the expected rate of return on insurance premiums is generallylow, or even negative. Use the portfolio concept to explain why people buy life in-surance in spite of negative expected -5If investors’ aversion to risk increased, would the risk premium on a high-beta stock in-crease more or less than that on a low-beta stock? -6If a company’s beta were to double, would its expected return double?6-7Is it possible to construct a portfolio of stocks that has an expected return equal to therisk-free rate?6-8A stock had a 12 percent return last year, a year in which the overall stock market de-clined in value. Does this mean that the stock has a negative beta?SELF-TEST PROBLEMS(SOLUTIONS APPEAR IN APPENDIX B)ST-1Define the following terms, using graphs or equations to illustrate your answers wher-Key termsever feasible:-alone risk; risk; probability distributionˆ rate of return, probability deviation, ; variance, ; coefficient of variation, aversion; realized rate of return, k premium for Stock i, RP; market risk premium, Asset Pricing Model (CAPM)SELF-TEST PROBLEMS273
ˆ return on a portfolio, k; market coefficient, r; risk; diversifiable risk; relevant coefficient, b; average stock’s beta, Market Line (SML); SML of SML as a measure of risk aversionST-2Stocks A and B have the following historical returns:Realized rates of returnYEARSTOCK A’S RETURNS, kSTOCK B’S RETURNS, kAB1997(%)(%) the average rate of return for each stock during the period 1997 through2001. Assume that someone held a portfolio consisting of 50 percent of Stock A and50 percent of Stock B. What would have been the realized rate of return on the port-folio in each year from 1997 through 2001? What would have been the average re-turn on the portfolio during this period? calculate the standard deviation of returns for each stock and for the Equation 6-3a in Footnote at the annual returns data on the two stocks, would you guess that the cor-relation coefficient between returns on the two stocks is closer to or to you added more stocks at random to the portfolio, which of the following is themost accurate statement of what would happen to ?p(1) would remain (2) would decline to somewhere in the vicinity of 21 (3) would decline to zero if enough stocks were and required rate of returnofits business coming from each of the subsidiaries, and their respective betas, are asfollows:SUBSIDIARYPERCENTAGE OF BUSINESSBETAElectric utility60% is the holding company’s beta? that the risk-free rate is 6 percent and the market risk premium is 5 is the holding company’s required rate of return? is considering a change in its strategic focus; it will reduce its reliance on theelectric utility subsidiary, so the percentage of its business from this subsidiary will be50 percent. At the same time, ECRI will increase its reliance on the international/special projects division, so the percentage of its business from that subsidiary willrise to 15 percent. What will be the shareholders’ required rate of return if ECRIadopts these changes?274CHAPTER 6 RISK AND RATES OF RETURN
STARTER PROBLEMS6-1A stock’s expected return has the following distribution:Expected returnRATE OF RETURN DEMAND FOR THE PROBABILITY OF THIS IF THIS DEMAND COMPANY’S PRODUCTSDEMAND (50%)Below (5) the stock’s expected return, standard deviation, and coefficient of -2An individual has $35,000 invested in a stock that has a beta of and $40,000 investedPortfolio betain a stock with a beta of . If these are the only two investments in her portfolio, whatis her portfolio’s beta?6-3Assume that the risk-free rate is 5 percent and the market risk premium is 6 and required rates of returnWhat is the expected return for the overall stock market? What is the required rate ofreturn on a stock that has a beta of that the risk-free rate is 6 percent and the expected return on the market is 13Required rate of returnpercent. What is the required rate of return on a stock that has a beta of stock has a required return of 11 percent. The risk-free rate is 7 percent, and the mar-Beta and required rate of returnket risk premium is 4 is the stock’s beta? the market risk premium increases to 6 percent, what will happen to the stock’s re-quired rate of return? Assume the risk-free rate and the stock’s beta remain -TYPE PROBLEMSTheproblemsincludedinthissectionaresetupinsuchawaythattheycouldbeusedasmultiple-choice exam -6The market and Stock J have the following probability distributions:Expected %20% the expected rates of return for the market and Stock the standard deviations for the market and Stock the coefficients of variation for the market and Stock -7Stocks X and Y have the following probability distributions of expected future returns:Expected (10%)(35%)-TYPE PROBLEMS275
ˆˆ the expected rate of return, k, for Stock Y. (k 12%.).(.) k 5%, k 10%, and k 12%.RFMARequired rate of Stock A’s Stock A’s beta were , what would be A’s new required rate of return?6-9Suppose k 9%, k 14%, and b rate of is k, the required rate of return on Stock i? suppose k(1) increases to 10 percent or (2) decreases to 8 percent. The slopeRFof the SML remains constant. How would this affect kand k? assume kremains at 9 percent but k(1) increases to 16 percent or (2) fallsRFMto 13 percent. The slope of the SML does not remain constant. How would thesechanges affect k?i6-10Suppose you hold a diversified portfolio consisting of a $7,500 investment in each of 20Portfolio betadifferent common stocks. The portfolio beta is equal to . Now, suppose you havedecided to sell one of the stocks in your portfolio with a beta equal to for $7,500 andto use these proceeds to buy another stock for your portfolio. Assume the new stock’sbeta is equal to . Calculate your portfolio’s new -11Suppose you are the money manager of a $4 million investment fund. The fund consistsPortfolio required returnof 4 stocks with the following investments and betas:STOCKINVESTMENTBETAA$ 400,,000()C1,000,,000, the market’s required rate of return is 14 percent and the risk-free rate is 6 percent,what is the fund’s required rate of return?6-12You have a $2 million portfolio consisting of a $100,000 investment in each of 20 dif-Portfolio betaferent stocks. The portfolio has a beta equal to . You are considering selling $100,000worth of one stock that has a beta equal to and using the proceeds to purchase an-other stock that has a beta equal to . What will be the new beta of your portfolio fol-lowing this transaction?6-13Stock R has a beta of , Stock S has a beta of , the expected rate of return on anRequired rate of returnaverage stock is 13 percent, and the risk-free rate of return is 7 percent. By how muchdoes the required return on the riskier stock exceed the required return on the less riskystock?6-14Stock X has an expected return of 10 percent, a beta coefficient of , and a standardEvaluating risk and returndeviation of expected returns of 35 percent. Stock Y has an expected return of per-cent, a beta coefficient of , and a standard deviation of expected returns of 25 risk-free rate is 6 percent, and the market risk premium is 5 percent. each stock’s coefficient of stock is riskier for diversified investors? each stock’s required rate of the basis of the two stocks’ expected and required returns, which stock would bemost attractive to a diversified investor? the required return of a portfolio that has $7,500 invested in Stock X and$2,500 invested in Stock the market risk premium increased to 6 percent, which of the two stocks wouldhave the largest increase in their required return?276CHAPTER 6 RISK AND RATES OF RETURN
PROBLEMS6-15Suppose you won the Florida lottery and were offered (1) $ million or (2) a gambleExpected returnsin which you would receive $1 million if a head were flipped but zero if a tail came is the expected value of the gamble? you take the sure $ million or the gamble? you choose the sure $ million, are you a risk averter or a risk seeker? you actually take the sure $ million. You can invest it in either a bond that will return $537,500 at the end of a year or a common stock thathas a 50-50 chance of being either worthless or worth $1,150,000 at the end of theyear.(1)What is the expected dollar profit on the stock investment? (The expected profiton the T-bond investment is $37,500.)(2)What is the expected rate of return on the stock investment? (The expected rateof return on the T-bond investment is percent.)(3)Would you invest in the bond or the stock?(4)Exactly how large would the expected profit (or the expected rate of return) haveto be on the stock investment to makeyouinvest in the stock, given the per-cent return on the bond?(5)How might your decision be affected if, rather than buying one stock for $,youcouldconstructaportfolioconsistingof100stockswith$5,000invested in each? Each of these stocks has the same return characteristics as theone stock—that is, a 50-50 chance of being worth either zero or $11,500 atyear-end. Would the correlation between returns on these stocks matter?6-16The Kish Investment Fund, in which you plan to invest some money, has total capital ofSecurity Market Line$500 million invested in five stocks:STOCKINVESTMENTSTOCK’S BETA COEFFICIENTA$160 beta coefficient for a fund like Kish Investment can be found as a weighted averageof the fund’s investments. The current risk-free rate is 6 percent, whereas market returnshave the following estimated probability distribution for the next period:PROBABILITYMARKET % is the estimated equation for the Security Market Line (SML)? (Hint: First de-termine the expected market return.) the fund’s required rate of return for the next Bridget Nelson, the president, receives a proposal for a new stock. The in-vestment needed to take a position in the stock is $50 million, it will have an expectedPROBLEMS277
return of 15 percent, and its estimated beta coefficient is . Should the new stockbe purchased? At what expected rate of return should the fund be indifferent to pur-chasing the stock?6-17Stocks A and B have the following historical returns:Realized rates of returnYEARSTOCK A’S RETURNS, kSTOCK B’S RETURNS, kAB1997(%)(%)()() the average rate of return for each stock during the period 1997 that someone held a portfolio consisting of 50 percent of Stock A and 50percent of Stock B. What would have been the realized rate of return on the portfo-lio in each year from 1997 through 2001? What would have been the average returnon the portfolio during this period? the standard deviation of returns for each stock and for the the coefficient of variation for each stock and for the you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or theportfolio? Why?6-18You have observed the following returns over time:Financial calculator needed; Expectedand required rates of returnYEARSTOCK XSTOCK YMARKET199714%13%12%1998197101999 16 5 1220003112001201115Assume that the risk-free rate is 6 percent and the market risk premium is 5 percent. are the betas of Stocks X and Y? (Hint: See Appendix 6A.) are the required rates of return for Stocks X and Y? is the required rate of return for a portfolio consisting of 80 percent of StockX and 20 percent of Stock Y? Stock X’s expected return is 22 percent, is Stock X under- or overvalued?SPREADSHEET PROBLEM6-19Bartman Industries’ stock prices and dividends, along with the Wilshire 5000 Index, areEvaluating risk and returnshown below for the period 1995-2000. The Wilshire 5000 data are adjusted to INDUSTRIESREYNOLDS INCORPORATEDWILSHIRE 5000YEARSTOCK PRICEDIVIDENDSTOCK PRICEDIVIDENDINCLUDES $$$$,,,,,, 6 RISK AND RATES OF RETURN
the data given to calculate annual returns for Bartman, Reynolds, and theWilshire 5000 Index, and then calculate average returns over the 5-year period.(Hint: Remember, returns are calculated by subtracting the beginning price from theending price to get the capital gain or loss, adding the dividend to the capital gain orloss, and dividing the result by the beginning price. Assume that dividends are al-ready included in the index. Also, you cannot calculate the rate of return for 1995 be-cause you do not have 1994 data.) the standard deviations of the returns for Bartman, Reynolds, and theWilshire 5000. (Hint: Use the sample standard deviation formula given in Footnote5 to this chapter, which corresponds to the STDEV function in Excel.) Bartman,Reynolds, a scatter diagram graph that shows Bartman’s and Reynolds’ returns on thevertical axis and the market index’s returns on the horizontal Bartman’s and Reynolds’ betas by running regressions of their returnsagainst the Wilshire 5000’s returns. Are these betas consistent with your graph? risk-free rate on long-term Treasury bonds is percent. Assume that the av-erage annual return on the Wilshire 5000 is nota good estimate of the market’s re-quired return—it is too high, so use 11 percent as the expected return on the mar-ket. Now use the SML equation to calculate the two companies’ required you formed a portfolio that consisted of 50 percent of Bartman stock and 50 per-cent of Reynolds stock, what would be the beta and the required return for the port-folio? an investor wants to include Bartman Industries’ stock in his or her portfo-lio. Stocks A, B, and C are currently in the portfolio, and their betas are , ,and , respectively. Calculate the new portfolio’s required return if it consists of25 percent of Bartman, 15 percent of Stock A, 40 percent of Stock B, and 20 percentof Stock information related to the cyberproblems is likely to change over time, due to the releaseof new information and the ever-changing nature of the World Wide Web. With thesechanges in mind, we will periodically update these problems on the textbook’s web site. Toavoid problems, please check for these updates before proceeding with the -20The tendency of a stock’s price to move up and down with the market is reflected inRisk and rates of returnits beta coefficient. Therefore, beta is a measure of an investment’s market risk and isa key element of the this exercise you will find betas using Yahoo!Finance, located at . To find a company’s beta, enter the desired stock symbol andrequest a basic quote. Once you have the basic quote, select the “Profile” option inthe “More Info” section of the basic quote screen. Scroll down this page to find thestock’s to Yahoo!Finance, what is the beta for a company called ELXSI, whosestock symbol is ELXS? Yahoo!Finance obtain a report on MBNA America Bank’s holding company,KRB, whose stock symbol is KRB. What is KRB’s beta?!Finance’slook-upfeaturetoobtainExxonMobil’,clickonsymbollookup,typepartofthecompanyname,sayExxon,andthenclickonLookup.(Hint:Youshouldfindthatthecompany’sstocksymbolisXOM.) and view a report on Ford Motor Company, and identify its beta. UseYahoo!Finance’s look-up feature to obtain Ford’s trading you made an equal dollar investment in each of the four stocks above, ELXSI,KRB, Exxon Mobil, and Ford Motor Company, what would be your portfolio’sbeta?CYBERPROBLEM279
MERRILL FINCH .(1) Why is the T-bill’s return independent of the state of6-21RiskandReturnAssumethatyourecentlygraduatedthe economy? Do T-bills promise a completely risk-freewithamajorinfinance,andyoujustlandedajobasafinancialreturn? (2) Why are High Tech’s returns expected toplannerwithMerrillFinchInc.,alargefinancialservicescor-move with the economy whereas Collections’ are $100, to move counter to the economy? the expected rate of return on each alternativeˆyear,youhavebeeninstructedtoplanfora1-yearholdingpe-and fill in the blanks on the row for kin the table , should recognize that basing a decision solely on ex-vestmentalternativesinthetablebelow,shownwiththeirpected returns is only appropriate for risk-neutral indi-probabilitiesandassociatedoutcomes.(Disregardfornowtheviduals. Since your client, like virtually everyone, is riskitemsatthebottomofthedata;youwillfillintheblankslater.)averse, the riskiness of each alternative is an importantMerrill Finch’s economic forecasting staff has developedaspect of the decision. One possible measure of risk isprobability estimates for the state of the economy, and its se-the standard deviation of returns. (1) Calculate this valuecurity analysts have developed a sophisticated computer pro-for each alternative, and fill in the blank on the row forgram, which was used to estimate the rate of return on each in the table below. (2) What type of risk is measuredalternative under each state of the economy. High Tech the standard deviation? (3) Draw a graph that showsis an electronics firm; Collections Inc. collects past-dueroughlythe shape of the probability distributions fordebts; and . Rubber manufactures tires and various otherHigh Tech, . Rubber, and and plastics products. Merrill Finch also maintains you suddenly remembered that the coefficient“market portfolio” that owns a market-weighted fraction ofof variation (CV) is generally regarded as being a betterall publicly traded stocks; you can invest in that portfolio,measure of stand-alone risk than the standard deviationand thus obtain average stock market results. Given the sit-when the alternatives being considered have widely dif-uation as described, answer the following expected returns. Calculate the missing CVs, and280CHAPTER 6 RISK AND RATES OF RETURN
RETURNS ON ALTERNATIVE INVESTMENTSESTIMATED RATE OF RETURNSTATE OFHIGH -STOCKTHE %(%)%%*(%)%Below ()() ()()ˆ%%% *,whentheeconomyisbelowaverage,,iftheeconomyisinaflat-outrecession,,’ in the blanks on the row for CV in the table above.(1) What is a beta coefficient, and how are betas used inDoes the CV produce the same risk rankings as the stan-risk analysis? (2) Do the expected returns appear to bedard deviation?related to each alternative’s market risk? (3) Is it you created a 2-stock portfolio by investingto choose among the alternatives on the basis of the in-$50,000inHighTechand$50,000inCollections.(1) Cal-formation developed thus far? Use the data given at theˆculatetheexpectedreturn(k),thestandarddeviation( ),start of the problem to construct a graph that shows howppand the coefficient of variation (CV) for this portfoliothe T-bill’s, High Tech’s, and Collections’ beta coeffi-pand fill in the appropriate blanks in the table above. (2)cients are calculated. Then discuss what betas measureHow does the riskiness of this 2-stock portfolio compareand how they are used in risk the riskiness of the individual stocks if they ,thatis,long-termheld in isolation? an investor starts with a portfolio consisting ofquently,MerrillFinchassumesthattherisk-freerateisone randomly selected stock. What would happen (1) to8percent.(1)WriteouttheSecurityMarketLinethe riskiness and (2) to the expected return of the port-(SML)equation,useittocalculatetherequiredrateoffolio as more and more randomly selected stocks werereturnoneachalternative,andthengraphtherelation-added to the portfolio? What is the implication for ? Draw a graph of the two portfolios to illustrate(2)Howdotheexpectedratesofreturncomparewithyour Should portfolio effects impact the way investorslectionshasanexpectedreturnthatislessthantheT-think about the riskiness of individual stocks? (2) If youbillratemakeanysense?(4)Whatwouldbethemarketdecided to hold a 1-stock portfolio, and consequentlyriskandtherequiredreturnofa50-50portfolioofwere exposed to more risk than diversified investors,HighTechandCollections? you expect to be compensated for all of your risk;Rubber?that is, could you earn a risk premium on that part ofj.(1) Suppose investors raised their inflation expectationsyour risk that you could have eliminated by diversifying?by 3 percentage points over current estimates as expected rates of return and the beta coefficients offlected in the 8 percent risk-free rate. What effect wouldthe alternatives as supplied by Merrill Finch’s computerhigher inflation have on the SML and on the returns re-program are as follows:quired on high- and low-risk securities? (2) Suppose in-stead that investors’ risk aversion increased enough toˆSECURITYRETURN (k)RISK (BETA)cause the market risk premium to increase by 3 percent-age points. (Inflation remains constant.) What effectHigh % this have on the SML and on returns of high- -risk securities?. ()INTEGRATED CASE281
6ACALCULATING BETA COEFFICIENTSTheCAPMisanexantemodel,whichmeansthatallofthevariablesrepresentbefore-the-fact,,thebetacoefficientusedintheSMLequationshouldreflecttheexpectedvolatilityofagivenstock’,peoplegen-erallycalculatebetasusingdatafromsomepastperiod,andthenassumethatthestock’ illustrate how betas are calculated, consider Figure 6A-1. The data at thebottom of the figure show the historical realized returns for Stock J and for themarket over the last five years. The data points have been plotted on the scat-ter diagram, and a regression line has been drawn. If all the data points hadfallen on a straight line, as they did in Figure 6-9 in Chapter 6, it would be easyto draw an accurate line. If they do not, as in Figure 6A-1, then you must fit theline either “by eye” as an approximation or with a what the term regression line,or regression equation, means: The equa-tion Y a bX e is the standard form of a simple linear regression. It statesthat the dependent variable, Y, is equal to a constant, a, plus b times X, whereb is the slope coefficient and X is the independent variable, plus an error term,e. Thus, the rate of return on the stock during a given time period (Y) dependsonwhathappenstothegeneralstockmarket,whichismeasuredby X the data have been plotted and the regression line has been drawn ongraph paper, we can estimate its intercept and slope, the a and b values in Y a bX. The intercept, a, is simply the point where the line cuts the verticalaxis. The slope coefficient, b, can be estimated by the “rise-over-run” involves calculating the amount by which k increases for a given increaseJin k. For example, we observe in Figure 6A-1 that k increases from toMJ percent (the rise) when kincreases from 0 to percent (the run).MThus, b, the beta coefficient, can be measured as follows:Rise ( ) Beta that rise over run is a ratio, and it would be the same if measured usingany two arbitrarily selected points on the regression line equation enables us to predict a rate of return for StockJ, given a value of k . For example, if k 15%, we would predict k MMJ % (15%) %. However, the actual return would probably dif-fer from the predicted return. This deviation is the error term, e, for the year,Jand it varies randomly from year to year depending on company-specific fac-tors. Note, though, that the higher the correlation coefficient, the closer thepoints lie to the regression line, and the smaller the errors. In actual practice, monthly, rather than annual, returns are generally usedfor kandk ,andfiveyearsofdataareoftenemployed;thus,therewouldbeJM282APPENDIX 6A CALCULATING BETA COEFFICIENTS
FIGURE 6A-1Calculating Beta CoefficientsHistoric Realized Returns_on Stock J, k(%)JYear 140Year 530__k= a+ bk+ eJJ JM J_= – + + eM J20Year 310Year –100102030Historic Realized Returns_on the Market, k (%)_Ma= Intercept = –%J ∆k = % + % = 16%J–10__Rise∆k16J∆k= 10%_b ==== ∆k10M–20Year 2YEARMARKET ( k)STOCK J ( k)%%2()()%%Average k %% %% k5 12 60 data points on the scatter diagram. Also, in practice one would usethe least squares methodfor finding the regression coefficients a and b. This pro-cedure minimizes the squared values of the error terms, and it is discussed instatistics least squares value of beta can be obtained quite easily with a financialcalculator. The procedures that follow explain how to find the values of betaand the slope using either a Texas Instruments, a Hewlett-Packard, or a Sharpfinancial 6A CALCULATING BETA COEFFICIENTS283
TEXASINSTRUMENTSBA, BA-II, ORMBA 2ndModeuntil “STAT” shows in the the first X value (k in our example), press , andM then enter the first Y value (k ) and press . Step 2 until all values have been 2ndb/ato find the value of Y at X 0, which is the value of the x yY intercept (a), , and then press to display the value of theslope (beta), could also press 2ndCorrto obtain the correlation coefficient, r,which is it all together, you should have this regression line: k kJMr Clear allto clear your memory the first X value (k in our example), press INPUT, andM then enter the first Y value ( k ) and press . Be sureto enterJthe X variable Step 2 until all values have been display the vertical axis intercept, press 0 ˆ,my. Then display the beta coefficient, b, press SWAP. Then shouldappear.ˆ, obtain the correlation coefficient, press xand then SWAPto get r it all together, you should have this regression line: k JMr 2nd FModeuntil “STAT” shows in the lower right corner of 2nd FCAto clear all memory registers.(x,y) the first X value (k in our example) and press . (This Mis the RM key; do not press the second F key at all.) Then enter the firstY value ( k ), and press DATA. (This is the M key; again, do notJpress the second F key.)1The Hewlett-Packard 17B calculator is even easier to use. If you have one, see Chapter 9 of theOwner’s 6A CALCULATING BETA COEFFICIENTS
Step 3 until all values have been 2nd Fato find the value of Y at X 0, which is the value of the Y intercept (a), , and then press 2nd Fbto display the value ofthe slope (beta), can also press 2nd Frto obtain the correlation coefficient, r, whichis it all together, you should have this regression line: k kJMr ’,,,thefilecanberetained,andwhennewdatabecomeavailable,,,,the“truebeta”mightactuallybehigherorlower,,thespreadsheetcanbeusedtocalculatereturnsdatafromhistoricalstockpriceanddividendinformation,-portant,becausestockmarketdataaregenerallyprovidedintheformofstockpricesanddividends,-1You are given the following set of data:Beta coefficients and rates of returnHISTORICAL RATES OF RETURN ( k)YEARSTOCK Y ( k)NYSE ( k)%%()()5()()()%%%% a scatter diagram graph (on graph paper) showing the relationship betweenreturns on Stock Y and the market as in Figure 6A-1; then draw a freehand approx-APPENDIX 6A A 285CALCULATING BETCOEFFICIENTS
imation of the regression line. What is the approximate value of the beta coefficient?(If you have a calculator with statistical functions, use it to calculate beta.) a verbal interpretation of what the regression line and the beta coefficient showabout Stock Y’s volatility and relative riskiness as compared with other (1)thefirm’sriskifthestockwereheldina1-assetportfolioand(2)theactualriskpremiumonthestockiftheCAPMheldexactly?Howwouldthedegreeofscatter(orthecorrelationcoef-ficient)affectyourconfidencethatthecalculatedbetawillholdtrueintheyearsahead? the regression line had been downward sloping and the beta coefficient hadbeen negative. What would this imply about (1) Stock Y’s relative riskiness and (2) itsprobable risk premium? an illustrative probability distribution graph of returns (see Figure 6-7) forportfolios consisting of (1) only Stock Y, (2) 1 percent each of 100 stocks with betacoefficients similar to that of Stock Y, and (3) all stocks (that is, the distribution of re-turns on the market). Use as the expected rate of return the arithmetic mean as givenpreviously for both Stock Y and the market, and assume that the distributions areˆnormal. Are the expected returns “reasonable”—that is, is it reasonable that k Yˆk %?, suppose that in the next year, Year 12, the market return was 27 percent, butFirm Y increased its use of debt, which raised its perceived risk to investors. Do youthink that the return on Stock Y in Year 12 could be approximated by this historicalcharacteristic line?ˆˆk % (k) % (27%) %., suppose k in Year 12, after the debt ratio was increased, had actually been 0Ypercent. What would the new beta be, based on the most recent 11 years of data (thatis, Years 2 through 12)? Does this beta seem reasonable—that is, is the change inbeta consistent with the other facts given in the problem?6A-2You are given the following historical data on market returns, k , and the returns onMSecurity Market LineStocks A and B, k and k :ABYEARk k k %%%()(), the risk-free rate, is 9 percent. Your probability distribution for kfor next year isRFMas follows:(14%) 6A CALCULATING BETA COEFFICIENTS
graphically the beta coefficients for Stocks A and the Security Market Line, and give its the required rates of return on Stocks A and B.ˆ a new stock, C, with k 18 percent and b , becomes available. IsCCthis stock in equilibrium; that is, does the required rate of return on Stock C equalits expected return? Explain. If the stock is not in equilibrium, explain how equilib-rium will be 6A 287CALCULATING BETA COEFFICIENTS
