Chapter2 TheCapitalAssetPricing

2

Chapter 2

The Capital Asset PricingModel: An Application ofBivariate RegressionAnalysis

“In investing mone “Un investing money, the amount of interest you want should depen( 8 ‘) vou want shion whether you want to eat well or sleep well, {depend

¥. KENFIELD MORLEY. “SomeThings I Believe,’ the Rotarian, February 1937

ottahen.his i one ofthe peculiarly dangerous monthsto3 in stacksin. Theothers are July, Januar; mst ‘ 1, ry, September,April, November, May, March, June, December, ‘August and .February,”

MARK TWAIN, Pudo'nhead Wilson's Calendar (1899). p, 108

“Natureis the realization aftheae simplest conceivable mathematical

ALBERT EINSTEIN, ideas ang Opinions (1954). p. 274

“Financialforecaslook respectable.”

Ing Gppears to bea science that makes astrology

BURTON MALKIEL, 4 Random Walk Down Wail Streat(1988), p. 182

eT

24

Oneof the most sought-afier possessions for s typical private investoror securities market analyst is a reliable equation predicting the return onalternative securities, A first step in developing and empirically implementingsuch tin equation involves gaining an understanding of why a particularstockhas a low or high rate of return, In this chapter we focus attention onthecapital asset pricing model (CAPM), a modelthat helps considerably in devel-oping such understanding. As we shall see, a remarkable feature of CAPM isthat its most important parameters can be estimated by using the very sim-plest of econometric techniques, namely, a bivariate linear model in which adependent variableis regressed on 4 constant and a single independent vari-able. The simple CAPM framework therefore provides a useful introductionto empirical econometrics.

The empirical analysis of stock markets has had a very importantrolein the development of econometrics. In 1932, Alfred Cowles [1], a quantita.tively oriented investment analyst, provided leadership io found the Econo-metric Society.' Cowles also initiated funding support to establish the CowlesCommission for Researchin Economics. Some of the most important devel-opments in econometric theory, including the theory underlying simuita-neous equations estimation and identification, were conceived by researchersat the Cowles Commission,first at ihe University of Chicago and now at YaleUniversity. In this contextit is interesting to note that in the first volume ofEconometrica, the official publication of the Econometric Society, Cowlespublished an article purporting to show that the most successful records ofstock marketforecasters “arelittle, ifany, better than what might be expected

from pure chance, There is some evidence, on the other hand,to indicate thatthe least successful records are worse than what could reasonably be attributedto chance.”

It is therefore appropriate that ourfirst application in this hands-oneconometrics text involves an empirical examination of stock markets. Webegin this chapter with a summary discussion ofthe financial theory under-tying the CAPM, considertherole of diversification, derive the principal estimating equations, interpret them, and then consider issues in empiricalimplementation.’ Finally, in the hands-on applications we examine ten yearsof monthly returns data for a variety of companies and for the market as awhole, estimate company-specific betas using bivariate regression procedures,assess why gold is a special asset, interpret the R? measure in terms oftheproportion oftotal risk that is nondiversifiable, evaluate properties ofcertainportfotios, conduct event studies, estimate a generalized version of CAPM,and then test assumptions concerning stochastic specification.

DEFINITIONS AND BASIC FINANCE CONCEPTS

Assumethat when investors act in the securities markets, their behavior isperfectly rational in the sense that their only concern is the myopic ane ofassessing returns from their own investments, Define the rate of return on an

21

22 The Capital Asset Pricing Model

investment as

Pt d= Pyref

t

OoP

y

Po(Qa)

where

P.®™ price ofsecurity ut the end of the time periric us h period,de dividends {if any) paid during the time period,Po price of security at the beginning of the time period,

Al . .Teennear cean be easily calculated ex post (once the investment

i aide), is course uncertain ex ante (before the inves ision has been made), Hereattet inte a wedoverante rteseens er, we interpret ras the expected orex ante rate

sake)ann investors (other than those who enjoy gambling for its own.sae); ‘hen4only interested in the most likely or expected return onan invest-

‘ Iso concerned with the possible distributi iitaken as a random variable. The ri. ing poteinveoteeot i¢ able. The risk accompanying a possiblei isn : ¢ risk ac investment istypically characterized bythedistribution of such possible returns, Returns

1 0 be distributed normally, and in such ; distrition can be completely described b: . spestedvalue aedthen y two measures, the expected va aai . s I + pected value and thefonUnderenee root of the variance, o, called the standard devia-! . assumption, in the empirical finance Hieri e ris

is ‘ypically measured by the standard deviation «‘ ee Heerature risklower Wrile neytaeae unanimous in preferring higher returns to

equal, it is also the case that most investors are river ones, other things equal tsare risk-eersean investors prefer 4 tower standard deviation 10 a higher one,

expected return, This implies thatif the risk an} {or portfolio of investments appea ge, investors arelikely10accept‘ 3 pears to be large, investors likel acesuch high risk only if it is acc i i id returns similarly,t y ‘companied by a high expected return: similaan investment with a low expected ret i savittase; a urn will be acceptable onl: it has asmall risk.* But how muchof i ill i sone ie ordenta. of at premium wil estors require jsea eatee rekd i investors require in order to

omaneons were to purchase an asset having zerorisk, they would stilldemand 3 enanner “ postpone current consumption. Such

tur z risk-free rate of return, and we denote it hit n Is x a Ih here by #3anOST the USsecurities market often employas a measure of% eda 5. Treasury bilt rate held until maturity, appareinvestors believe that the probability defi Fee dealsof default on such a security is vizero, We canuse these concepts to ion forti ‘nest° pts to define compensation fo: he ripremiumonthe jth asset ¢ EXCESS Te t isk we thetpn it set, us the excess return over the risk-free rate 7, that

Risk premium, 5 7, — 7; (2.2)

Withthese definitions in mi_ ; mind we now turn toa considerati iversification and risk management, sonsideration ofiver-

2.2. Diversification and Portfolio Optimitity 23

2.2. DIVERSIFICATION AND PORTFOLIO OPTIMALITY

How do intelligent investors manage risk on their investments? To examine

the risk managementprocess,it is use! 1 10 introduce the notion of diversifi-

cation, Although mathematical discussions of the diversification process can

easily becomevery involved, here we summarize the principal results, using

a combination ofrelatively simple analysis and intuition, based in large part

on the pioneering work of Harry M. Markowitz.”

If aninvestor helds wo securities, the expected return on the total part+

folio r, is simply the weighted average of the expected returns on cachof the

assets, the weights being the relative shares invested in each of the Iwo

securities,fy = Wy + HG (2.3)

where 1, is the proportion of total funds invested in asset j, 7 = 1,2, and wy

© Wy = LO, Further, the total variance of the portfolio, 4}, is

oh = det + dod + 2 wun, & tel + ded + 2 wompame 2)

where

oo variance ofreturn on security jj = 1.2,

2, = standard deviution of return on security jj = 1. 2.

o,, 2 covariance of returns on securities | and 2, and

fa = simple correlation between returns on securities | and 2.

The second equality in Eq. (2.4) holds,since by definition, 0.2 = audir

We now wantto show thatfor a given amount offunds to be invested,

diversification generally reduces risk. To see this. assume first the unlikely

situation in which returns on securities | and 2 are perfectly correlated, that

is, assume that pp, the simple correlation coefficient between returns on assets

1 and 2, equals 1.0, In this case, 7%) = 12 whichis the largest possible value

that ¢,, can take. But as can be seen hy inspecting Eq. (2.4), whenever a17 is

ata maximum, given ¢, and a2, S0 too js the variance on thetotal portfalio

of}, As the covariance and therefore the correlation betweenreturns on assets

tand 2 decreases and becomesless than perfect, the final term in Eq. (2.4)

becomes smaller, and so too doesthe total portfolio variance a, Thisis

itively appealing: By holding twoassets whose returns do not move lo;

in perfect harmony, the lower return on one asset can be partially off

relatively higher return on the other asset, resulting in a reasonable overill

portfolio return yet a reduved total portfalio risk.

It is instructive to demonstrate this with several examples of portfolio

and return underalternative diversification behavior, Consider the simple

case in which the expected returns on securities 1 and 2 both equal 10%,

where the standard deviation of returns ¢ for each is equal to 2.0 and where

it is initially assumed that returns on the two securities are perfectly corre-

lated, that is, p2 = 10, which implies that a; = 4,0.

24 The Capital Asset Pricing Modet

HARRY M. MARKOWITZFather ofModern Portfolio T!heory

he basic financetheory underlyingmodern portfolio

analysis is due in largePart to the pioneeringwork of Harry M. Mar-kowitz, Markowitz is aProduct of Chicago. Hewas born there in 1927, @attended its elementaryand secondary schools,andreceived three degreesfrom the University ofChicago—a bachelor’s, master's, anddoctorate in economics. In his Ph.D.dissertation, Markowitz developedthe basic portfolio mode! summa-Tized in this chapter. His seminalanalysis appeared in a 1952 issue ofthe Journal of Finance, and a morecomplete treatment was published in1959 as a Cowles Foundation reportby John Wiley and Sons.

Markowitz's professional careerhas been in and out of academia andin and out offinance. Leaving Chi-cago in 1952 with all but his disser-tation completed, Markowitz joinedthe Rand Corporation in Sunta Mon-ica, California, In 1960 he took a po-sition at the General Electric Corpo-ration, and then in 1961 he returnedtoRand. At Rand and at GE, Marko-witz wrote computer code for pro-gramsthitt simulated logistic and/ormanufacturing Processes. His inter.ests in simulation Programming

came Chairman of theBoard and Technical Di-rector of ConsolidatedAnalysis Centers, Inc, a

y computer sofiware firm.One year after assumingan academic position atUCLA in 1968, Marko.witz became portfolioManager and tater pres}-dent of an investmentfirm

—

called ArbitrageManagement Company. From 1974to 1983 he served as a Research StaffMemberat IBM's ThomasJ, WatsonResearch Center, focusing on the de-Sign ofdata base computer languages,

Markowitz is a Fellow of the

grew, and in 1963 he te |

Econometric Society and of theAmerican Academyof Arts and Sci-ences and hasserved as President ofthe American Finance Association.In 1989 he was awardedthe John vonNeumann Theory Prize by the Op-erations Research Society of Americaand The Institute of ManagementScience, in part for his research onportfolio theory and the SIMSCRIPTProgramming language, Although heig still very actively involved in real-world financial matters, Harry M.Markowitz currently holds the Mar-vin Speiser Distinguished Professorof Finance and Economics chair atBaruchCollege at the City Universityof New York,

2.2. Diversification and Portfolio Optimality 25

One possible investmentstrategy is to place funds entirely into security1, implying that w, = 1.0 and w, = 0.0; we name this Case A, andits con-sequences are presented in the first row of Table 2.1, In Cuse A the expectedportfolio return, based on Eq. (2.3), is calculated usr, = 1.0 (10%) + 0.0(10%) = 10%, Substituting the Case A values into Eq. (2.4) yields a portfoliovariance of 4.0, or a standard deviation of 2.0,

A secondinvestmentstrategy, called Case B, involves putting all fundsinto security 2 and none into security , implying that w, = 0.0 and ww, =1.0; this case is presented in the second row of entries in Table 2.1. Use ofEqs. (2.3) and (2.4) again implies that, = 10%, while o} = 4.0 and risk «equals 2.0, Since the risk and return consequencesoffollowing Case A or CaseB are identical, investors will be indifferent between these two cases,

A third alternative investmentstrategy, Case C,is to diversify the port-folio by purchasing equa? amountsofsecurities | and 2, implying thats, =w, = 0,5. If one substitutes Case C entries in Table 2.1 into Eqs. (2.3) and(2.4), one again obtains the same portfolio return of10%and standard devi-ation of 2.0, Notice that in each of these three Cases A, B, and C, because ofthe perfect correlation assumption, the portfolio risk and return are the samewhether the investor holds only security |, only security 2, or a combinationofthese assets. If returns on these two securities had not been perfectly cor-related, however, the portfolio variance would have been, smaller.

To see this, first consider Case D in Table 2.1. Here the correlationbetween returns on assets | and 2 is positive but less than perfect, thatis, p..= 0.5. All other features of this case are the same as in Case C. Notice thatowing to diversification (equal amountsofsecurities 1 and 2 are purchased),the investor is able to exploit the less than perfect corretation between assetreturns and obtain the sameportfolio return of 10%, yet at a reduced varianceof 3.0 and standard deviation ofabout 1.7. On the basis of Eqs, (2,3) and (2.4)it is simple to show that, in the presence ofless than pertect correlationbetween asset returns, had the investor not diversified and purclased onlysecurity } (Case E) or only security 2 (Case F), the same 10%return wouldhave beenattained but at a higher variance of 4.0 and standard deviation of

Table 2.4 Examples of Risk and Return UnderAlternative Portfolio

Olversifications

Sphn wr By om ay Pu an ty oh Risk

Case A 10% 10 0.0 20 2.0 Lo 40 10% 40 20

Case B 10% 00 10 20° 20 10 40 10% 4.0 20

Case C 10% 08 85 20 20 Lo 40 10% 40 20

Case D 10% 08 O85 20 20 0.5 2.0 10% 3.0 17

Case E 10% 10 00 20 20 08 20° 10% 40 20

Case F 10% 00 10 20 20 05 20 10% 40 20

Case G 10% 05 05 20 20 -L0 -40 10% 0.0 00

26 ‘The Capital Asset Pricing Model

2.0,as in Cases A, B, and C. Cases D, E, and F therefore clearly demonstratethe benefits of diversification in reducingrisk,

Finally, in the most unlikely case that returns on assets 1 and 2 wereperfectly negatively correlated, diversification could eliminate risk entirely.For example, in Case G, a. = — 1.0, butif'w, = w, = 05,0) = «, = 0,

Nowconsider the case of an investordiversifying by holding #securities,where #1 can belarger than 2. As before, the expected return onthetotal port-folio is a weighted average of security-specific expected returns #, where thesv, weights are shares oftotal funds invested in each asset, thatis,

me wy, (2.5)=

Once again, with #securities the total variance of the portfotio depends notonly on the variances of the #1 individual securities, but alsa on their covari-ances. Specifically, the portfolio varianceis calculated as

no * noonah DY way, = D> who? + 2+ > D> wove, (2.6)

ret Jat it ir aes

whereo, is the covariance between returns on securities (and j ando?is thevariance, Note that the total portfolio variance in Eq. (2.6) has # varianceferms and a(n — 2) covariances, with a(? — 1)/2 of them being different.Therefore thelargeris n, other things equal, the greateris the relative impor-tance ofasset covariancesto total portfolio variance. For example, when #1is5, there are five variances and 20 covariances; when # is doubled to 10, thenumberof variances doubles to 10, but the number of covariance terms inEq. (2.6) increases to 90! As » becomes very large, the portfolio varianceapproaches a (weighted) average of the covariances. Hence covariances areextremely importantin the diversitication process.

The above discussion focused on the average return and variance of adiversified portfolio. For portfolio decision-making purposes, marginalreturns and variancesare also important. Suppose that in the ini portfolioof # assets held by an investor, there were zero holdingsofsecurity &, implyingthat initially 1, = 0. Next, assumethat the investor decided to acquire a verysmall positive amount of security A, but that other holdings remainedunchanged. Define the marginal returnofthe Rih asset on, as the change infy given a small change in 1, From Eq, (2.5) this marginal return is simplyequalto f;:

Marginal return, © dr,/div, =r (2.7)

This smatl change in asset holdings also affects the portfolio variance.Definethe marginal varianceof the kthasset as the changein o? given x smallchange in 1, From Eq. (2.6) und using the fact that a weighted sum ofindi-vidual covarianceswith security k equalsthe covarianceofsecurity & with the

portfolio—itself a weighted sumof other securities

ginal variance is simply

2.3 Derivation of a Lineur Relationship 27

ak

iy

ye odMarginal variance, 2s 22D way

=

oyA

where o,,is the covariance bet

marginal variance—the change in total portfolio variance

chungein the holdings of asse

relurns on asset & and the po:

rot

tween Se

it follows that the mar-

(2.8)

curity K and the portfolio p." Hencetheasa result ofa small

t k—-depends simply on the covariance between

rifolio.

Given these definitions, we can now present an important

portfolio optimality derived in finance theory. If two securi

principle ofes in a portfolio

have the same marginal variance but different expected returns, then that

portfolio cannot be optimalin thesense of providing a maximumreturn for

givenrisk, The reason such a portfolio coulId not be optimalis that it would

be possible to obtain a higher return without increasing risk by holding more

of the asset with the higher return (he

are assumed to be identical}. Therefore if a portfolio is eptimal,marginal variances of the two asseis

all securities

with the same marginal variance must have identical expected returns.

The marginal variance and the variances and covariances in Eqs. (2.6)

and (2.8) all depend on units ‘of measurement. Like the economists’ notion

ofelasticity, fina

measures that are independentol

known relative measure is the beta value

Since a security's beta value depends on its covariant

closely related to its marginal variance, one can combine

jal economists hi

Betas = ai)/a}

to derive a factor of proportionality between beta and marginal

Marginal variance, = 201) = 2a}, > beta,

Given this relationship,

can be expressed equivalently in term t h

variances. Specifically, if a portfolio is optimal, then all securi

samebeta valuerelative to the portfolio must have identical expr

2.3 DERIVATION OF A LINEAR RELATIONSHIP BETWEEN

RISK AND RETURN

To this point we haverelate

beta values, and we have prese!

mality, But how might one

an empirically implementable rel

next few pages we summiari

dV:

ariances, covariances, marginal variance: c

nted an important principle of portfolio opti-

ave foundit convenient to adapt relative

f units of measurement. Perhaps the bes

for security k, computed simply as

(2.9)

ce, which in turn isEqs.(2.8) and (2.9)

variance:

the previous discussion on portfolio optimatity

rms of bela values rather than marginals with therd returns,

move from these insights 1o portfolio choice and

ze the very important contributionlationship between tisk and return? In the

of the CAPM


to facilitating relatively simple empirical analysis, and we show that the rela-tionship between risk and return is a linear one.’

Suppose an investor has a portfolio, called @, consisting of a mix of woassets. The blend ofthese two assets generates an expected portfolio return 1,and has a variance of 2, Nowlet there be arisk-free asset whose returnis MNandlet the investor be able to borrow or lend indefinitely at the risk-free rateofreturn r,. One possibility facing this investoris to combineportfolio a withtherisk-free asset into a new portfolio. In such case the expected return onthe new portfolio would equal

ry (= wey + weary (2.10)

where iv, is the proportion oftotal funds invested in portfolio a. The varianceof this new portfolio would be

oh = wal + (1 — wo? + ie(L — wow (2.91)

where ¢,,is the covariance between the expected return ofportfolio a and thaton therisk-free asset. However, since by definition the risk-free asset has azero variance return, this risk-frec return is also uncorrelated with that on anyothersecurity, implying that a? = a, = 0, Thus Eq.(2.11) reduces to

ea wel or a, = Waxy (2.12)Rearranging the sccond expression in Eq. (2.12) yields w. = a,/e, und

{1 — w,) = | — ¢,/ewhich, after substituting back into Eq. (2.10) andcollecting terms, gives us

Py Bh

tm ty]al| % (2.13)

In Eq, (2.13) we have a simple linear relationship between portfolioreturn #, and portfolio risk ¢,, one that even Albert Einstein would admire(recall the Einstein quote at the beginning of this chapter}. Specitically, thetotal portfolio return 7, is the sum of two terms: the risk-free rate of return 7,and (7, ~ 4)/o, timesthe portfolio risk o,. This linearity is shownin Fig, 2.1in which the expected return is measured onthe vertical axis, risk is on thehorizontalaxis, the intercept term is r,, and the slope term is (¢, — 4,ou

Severalfeatures of Fig. 2.1 are worth noting. Fi stor chose

a st, if the investto invest only in the risk-free asset such that w, = 0, then by Eg. (2.10), 7,would equal r,, and by (2.11), ¢, would equal zero. Second, if the investorinstead chose to invest only in the @ portfolio and entirely avoided the riskefree asset (say, at point @ in Fig. 2,1), then w, = 1,= rand o,'= ¢,. Third,the slopeofthetine in Fig. 2,1 represents the reward to the investor ofaccepteing increased risk, that is, of increasing the proportion of fundsinvested inthe risky portfolio a.

Portfolio «is, of course, but one of manypossible risky portfolios con-structed by ourinvestor; assets 1 and 2 could have been combined in numer-ousalternative combinations, This raises the interesting issue of what the

2.3. Derivation of a Linear Relationship 29

y

Note that theslope ofthe line is equal to W7,- 1/0,

FIGURE 2.1 The linear relationship between risk and

return,

return fronticr would looklike for an investor whois consideringomer alter:

native possibilities of a portfolio blended from the! eWwO ns y ss ie 2

Suppose we place two securities on a risk-relurn dingram asin ae

Let asset 1 be low risk-low return, while asset 2 is high risk-higl +

ther, let the correlation between returns on ussets | and 2 be Jess than perfect.

As was pointed out in Eq. (2.3), a combination of the twostocks will yield an

expected return of 4, iti + Wats a weighted average ofreturns on thew

assets. However, because ofdiversification, the total portfolio risk o, wi

smaller than the weighted average of the standard deviations, since (#19,

+ Wy0,)' is Jess than the right-hand side of Eq. (2.4) wheneve! oaee

i < ¢ risk~ frontier for various Sesult, provided that yy, < 1, the risk-return various co ms

of aseels 1 and 2 will look like the concave curve depicted in Fig. 23

dentally, it is worth noting that as pi: —* 1, the concave curve convergt

aight line. , - 7 .

“ yur investor is now faced with the following problem: Among afte

possiblerisky asset portfolio combinations in Fi g 3, whichblen oan

folio of risky assets with the risk-free asset will yield the maximum re

Asset 2 100%vase Assat 2

100%* Asset 1 asset 1

Q

FIGURE 2.3FIGURE 2.2

Rt] The Capital Asset Pricing Model

Slope equals (7, =ry/c,

FIGURE 2.4

any given portfolio risk? The CAPM soluti is optimal: is iifact ute Gent tion to this optimality problemis in

Onepossible strategy is one th: i i\ g at we considered earlier when we con-aeeae iRaidingassets 1 and 2 in the proportions corresponding, say, @ in Fig. 2.4, Note that at point a on the (4, — a) b ito, s Note t ottom linene 2.4, the total portfoliorisk is o* and the return is raWe oon show he" t such a strategy would be feasible, it would not be optimal in the senseleninanee send attain an even higher return given total risk o® by4 portion of funds, say, 1 — w,, at the riskfree ra ni

I ortic nds, say, ” sks! te Handtheninvestng the remaining Portion of funds, ¥), according to the portfolio proportions5 resent f some other Point on the concave risk-return frontier in Fig.a(as we si ia soon sec, portfolio 4), An investor who followed this mixedegy wou! id be able toreach point ¢, where the return is larger than at point@, yet the o” risk is the sumeas at point «a. ueToseethis, consider another iwa this, Portfolio(valied 5) and thessolities available when this risky porifolio is combined with the risk-freeasst pecitically. ifwe re] ated Our anulysis that began just above Eg. (2.10)andgore a jncar relationship betweenrisk and return for various mixes ofwith the risk-free asset, we would obtain alinear i i

olio. K-tire + eu equation analo-yous toEa. (2.13), having intercept rand slope equalto(7, Mew Suchaae s the middle one drawn in Fig.2.4;its stope is larger than that obtained‘i qarious mixes of the carlier Portfotio a and the risk-free asset, implyingwat ie rewards to riskare higher when portiolio 4is mixed withthe risk-freeasset tnstend ofportfolio a. An important result here is that with this 4 port-as ts shownin Fig, 2.4, one could hold a mixture ofit and the risk-free

risk-return pos-

2.3 Derivation of a Linear Relationship 31

asset to attain any desired return onthe straight line through points r,and 4;except at the intercept point where ¢, = 0, for any given risk the portfolioreturn y, is larger on the portfolio } line than on the portfolio aling, In thissense, portfolio 6 dominates portfolio a.

But whystop with portfolio 4? Using the very sume reasoning as withportfolios a and A, consider yet another portfolio combination ofas land2, say, that at point ¢ on the concave risk-return fromier in Fig. 2.4, Analo-gousto Eq. (2.13), one could now blend portfolio d with a risk-free asset innumerouswaysto obtain a risk-return linear equationrelatingr, to ¢,, havingintercept rand slope now equalto (ry — #,)/oy. Such a straight line is the topone in Fig, 2.4; note thatits slope is larger than that based on porttolio 4.Points on this 7, — line indicate the expected return 7, corresponding withalternative levelsofrisk ¢,.

Note that with « strategy based on a mix ofportfolio d withthe risk-freeasset, the investor could alwaysattain a higher return for the samerisk thanwith the # mixture, since the straight tine connecting r, through point d isalways above the r, — dline exceptat the point where 6, = 0. Furthermore,since this r, — line is tangentto the concaverisk-return frontier, there is nootherportfolio superiorto ¢, for any other line emanating from r, and havinga larger slope would not touch the concaverisk-return frontier and thereforewould not be feasible, Portfolio ¢ is said 10 be efficient.

The implication ofthis analysis is startling: Each investor should holdportfotio d, regardless of his or her risk-return preferences, and then achievethe desired amountofrisk by borrowing or lendingat the risk-free rate 7, Inparticular, if o* is the desired maximum amountofrisk given the investor'spreferences, the optimal procedure for the investor is to mix a | and 2according to portfolio d and then blend by borrowingor{in this cuse) tendingfundsat therisk-free rate until point cis attained. Note that each investoristherefore involved in only two investments—arisky portfolio d and borrow:ing or lending on risk-free tan.”

This sumeline ofreasoning can easily be generatized to more realisticsituations in which the numberofrisky securitics available to investors isgreater than two. In the case of 7 securities the CAPM best struegy for eachinvestor is 10 hold the assets in optimal proportions on the concave risk-return frontier and then adjust for the personespecitic desired risk level byborrowing orlending at the risk-free rate.'!

The thoughtful reader might now ask, "But just whereis this optimportfolio ¢really locuted?" Although in general it might in fact be very hardto plot out the entire concave frontier ofattainable risk-return combinationsand its tungency with the risk-free lending tine, under CAPM assumptionsthis calculation is simple, perhaps even unnecessary. Specifically, if une makesthe assumption thatall investors have the same information and opportuni-ties and that there are na taxes ortransactions costs, then, even if people'sattitudes towardrisk differ, everyone will process informationin the same wayand will view their investment prospects identically, In such a vase, everyone

RY4 The Capital Asset Pricing Model

wil warepaactythe same porvolio d mix ofsecurities | and 2, but cach per-c portfolio

@

and therisk-free asset to suit hi:Tisk-return tastes. In this case th lio forallesos wli sles, total market portfolio for all investors wi* a stors willbeerofpertolio ad. Alternatively, cach investor's portfolio will

¢ clone of the market us a whole. Therefore, accordiCAPM ihe ceiett t a herefore, according to the, egy is to invest in securit! the sume i

as they are in the overall market, si: as those ofthebesth t, Since these are the same as th fattainable portfolio, and then adj ihe rick protoremsesb 0, just for person-specific risk erencesborrowing or lending at therisk-free rate, . isk preferences by

2.4 FURTHER INTERPRETATION OF THE CAPMRISK-RETURN LINEAR RELATIONSHIP

We have shown thatdiversification is ellective iificationis ellective in reducing ri ist i ig tisk because pricesoa ferent stocks are imperfectly correlated. We nowexaminerisk in greaterHorie a oussic experimental study by Wayne Wagner and Sheila Lauveh was8 hown thatdiversification reducesrisk very rapidly at first, but

ie point, additional diversification has little effec isk or variability. Spectheay ang c ica as little effect on tisk or vari-1 8 portfoliosofdifferent size drawn fr an histori

sample of stocks, Wagner and Lau d "giversieationeanWi lemonstrated that diversification caalmost halve the variability of 1 renetitcan be0 ‘eturns but that most of this benefie caobtained by holding relativel: i becomes cathorly few stocks; the improvement become:small when the number of securities is increased beyond,say, ten.'? S rathercan be eenshane course, 4 inate risk entirely. The risk that

climinated by diversification is culled specific, unittent Linate d Specific, unique, orsneerHak:Specific tisk derivesfrom thefact that manyofthe dangers

ities that surround an individual comps iiTOI pany are peculiar to thatcompany and, perhaps, its immediate competi ific ri .pan s. titors; specific risk hereforebe eliminated by holdin, iversified p oe, 2 a well-diversified portfolio. But there i:tisk that cannot be avoided no oe divessies This ekematter how muchonediversifies. This risk is: . This riBencrally Known a8 market or systematic risk, Market risk derives rakefet neteedboata economywide and global dangers and opportunities

2 all businesses,

The

fact that stocks have dency

to

“movetogether”reflects the presence of isk, ci: cannot beeliminatedmarketrisk, risk that ca be elimi.Whroughdiversification. Note that this risk will evenhee anonhdi . this ris ine a ipertnnat tisk wilt remain even when an optimal

Toexplore furtherthis de]c pendence ofstock returns on marketriore fi ¢ isk, notePateanecationofthe CAPM is that the risk of a well-diversified port-

i market risk ofthe securities included ifolio, Suppose, therefor , sled porto Gelio, . re, that you had a well-diversified folmicroscopic cloneof the entire stock ma i moe aarket portfolio) and that you wa:

re rymscnsititat dependence of stock returns on risk further by computing ite

@ particular company’s stocks in your i: inicular portfolio, suy, thosecompany J, to variationsin the overall market return. Obviously, you would

2.4 Further Interpretationof the CAPM 33

not want to compute this by looking at company j's returns in isolation.

Rather, you would want to use the covariance information trom Eqs. (2.8)

and (2.9)relative to the market as a whole

Specifically, recall that we noted earlier that one measure of the relative

murginal variance of a security, say, the Ath asset, is calledits beta value rel-

ative to the portfolio, in Eq. (2.9) this was defined as beta, = o,/a;. One

interpretation of thisrelative variance notionis that if the return on the port-

folio is expected to increuse by, say, 1%, then the return on the Ath assetis

expected to increase by bela, times 1%. The investment beta is therefore a

measure ofthe sensitivity of the return on the Ath asset to variation in returns

on the portfolio; beta, summarizes the dependence on portfoliorisk.

11 will be useful to think ofthe portfolio asthe entire market’s portfolio,

Let us therefore deline an investmentbeta for, say, company j relative to the

entire market portfolio as

Beta, @ oyq/o% (2.14)

where o,, is the covariance between company/’srewurn und that of the market

asa whole and a, is the variance of the market’s returns.

There is one problem, however, in relating beta, to the CAPM frame-

work, The covariance and variance termsfor beta, in Eq, (2.14) refer to the

total returns onassets, whereas by contrast in the CAPM developmenttothis

point we have dealt with variations.in risk premiums, thatis, the excess return

overthe riskfree rate, such as 7,, — ty, where 7, is the return on the entire

market's assets, Can one rewrite Eq. (2.14) in terms ofrisk premiums rather

than total returns? As we now show, we can, since beta, is unaffected by this

change.

To see this, note that since the ratio of the covariance term a,to the

variance term o?, is unaffected by subtracting the risk-free return from the

total return,the investment beta ratio Eq. (2. 14) holds even whenit is defined

in termsofrisk premiums rather than in terms oftotal returns. This has an

important implication in the CAPM context, where we deal with risk premi-

ums rather than with total returns. Specifically, since Eq, (2.14) is invariant

to rescaling in termsofrisk premiumsrather than total returns, the value of

beta for a particular company equals the covarianceofits risk premium with

the market portfolio risk premium, divided by the variance of the murkel’s

risk premium. This suggests that the beta, summary measure of dependence

on market risk has wide applicability.

Securities vary considerably in the value oftheir investment betas; some,

for example, have values as large as 2, in cating that a 1%rise or fall in the

market results in a 2%rise orfall in the value of that security. Such securities

are relatively risky. Othertruly “blue chip”stocks, on the other hand,are not

as sensitive to market movements and have much smaller betus of, say, 0.5,

implying that a [4 rise or fall in the market results in a}:rise or fall in their

value. Traditionally. holding stocks with betas greater than | is called an

34

28

The Capital Asset Pricing Mudel

“aggressive stance, while holding stocks with betas less than unity is calied‘defensive. As we shail see in Exercise 3 ofthis chapter, for someassets, betacan even be negative—these are “superdefensive" assets! ,___,, Investmentbetascan also be defined for portfolios of assets (rather thanindividual assets) relative to the market as a whole. For exumple, consider aPortfolio q consisting of n assets, and define its beta value relative to the mar-ket as a whole asbeta, = ¢,,/o2, Given the def of covariances, otan LO inition mces, one

Beta,, = S~ w, + beta, (2.15)=where w, is the proportionofportfolio g invested in asset f and beta,,, is assetPs beta value relative to the market's portfolio. Hence the beta value ofaportfolio is simply a weighted average of the beta values of the componentsecurities,the weights being the portfolio investment shares, °Obviously, for the stock market as a whole the covariation withitself’ isthe Sameasits variance, implying that the beta ratio for the stock market asa whole is 1.0. Moreover, since by Eq. (2.15) the beta of a well-diversifiedentire market portfolio depends on the weighted averagebetasofthe securitie:included in the portfotio,it follows that, on average, individual st ks hav vabeta of 1.0. Finally, it is warth noting that since the covariance ofa rishlesssetwith the market portfolio is zero, beta,,, equals zero whenever assetjis

ECONOMETRIC ISSUES INVOLVEDiMe Can IN IMPLEMENTING

We now want to move toward econometric i iv toward metric implementation of the CAPMframework, Our first objective is to derive an estimable equation. Consider asmall portfolio B whose sole security is janda large, well-diversified portfolioen whichis the entire market's portfolio, Substituting j for p and mfor a intoG. (2.13), we can rewrite the CAPMlinear relationship (2.13) as

To te = (afan) + (tm — 4) (2.16)where r, and 7, ure returns to security j a risk. i’ 'y Jandto a risk-free asset, respectively; r,ite return on theoverall securities market portfolio: and alesis the aioa the standard deviations of returns on asset j and the market portfolio i.me term y= is the risk premium for security j, while r,, — 5 representsre De market : tisk premium. Bythe way,overthelast 60 of so years inited States 3 i i iyen les the average market risk premium has been about 8.44% per

According to Eq. (2.16), the risk premium on jt is sito Eq,(2.16), th the jth asset is simply afactor of proportionality o,/¢,, times the marketrisk Premium;this factorofProportionality expresses the dependence ofsecurity 7's return on the market

2.5 Econometric Issues 35

return, a dependencehighlighted by the CAPM.Such reasoning suggests thatthe proportionality factoro,/,, must be related in some wayto the investmentbeta, discussed in the previoussection.

Toexplore this relutionship further,let us generalize Eq. (2.16) by add-ing to it an intercept term «, and a stochastic disturbance term ¢, and thendefine a new parameter @, as being equalto the proportionality factor, thatis,8, = a,/2m. This gives us an estimable equationrelating the total risk premiumof security j 10 the market risk premium and to the stochastic disturbanceterm:

no at Bn ty (2.17)

In Eg. (2.17) the stochastic disturbance term ¢, reflects the effects of specific(unsystematic) and diversifiable tisk, We will assume that ¢, has an expectedvalue of zero and a variance of ? and thatit is also independently and iden-tically normally distributed.

Now comes the clincher: The least squares estimate of 2, in Eq. (2.17)is in fact identical to the investment beta defined in Eq. (2.14)! To see this,consider the bivariate regression model y = a + Ax + «. The least squaresestimate of # is COV(x, p)/VAR(x). Now let y= 7, — 7, from Eq,(2.17), andlet. x © 7, — 7. Then the least squares estimate of 4, is simply 8, = COV(y,= tn tn — HYVAR(t, ~ 14)But this is precisely equal to Omf92, the invest-ment beta defined in Eq. (2.14), Intuitively, the least squares estimate of thefactor of proportionality 4, is simply the ratio of the standard deviations ofthe risk premiumsfor asset j and for the market#2.

These results imply that, for any security j, one can estimate 8, usingbivariate least squares procedures on Eq. (2.17). Graphically, the covariatvariance ratio is the least squares estimate of the slope of a regression linerelating the risk premium ofa particular security j on the vertical axis to themarket risk premium onthe horizontalaxis; the linearityof this relationshipderives from Eq. (2.13). Although one can use least squares procedures,in thebivariate regression framework, estimating 8,is in fact trivial: Once one hasvalues ofthe appropriate covariances and variances,all one need dois simplycompute fi,as their ratio.

To estimate the #, parameter based on time series data of individualcompanies, it must of course be assumed that, for a particular company. 8,istelatively stable over time. Quite frequently, monthly data are employed thatare based on returns from the New York Stock Exchange. Econometric stud-ies based on such data have found that in many studies (but with some nota-ble exceptions), 8, has tended to be relatively stable over a five-year (60-month) time span." There are cases, however, in which the conditions in anindustry or a firm abruptly change, implying that the relevant 8, might alsovary. Oil companystocks, for example, had a beta below unity before the1973 OPECoi] embargo,but this soon changed after 1973, and since then oilcompany betus have typically increased. Similarly, when theairline industryin the United States was deregulated in 1978, the betas of most major U.S.


airline companies rose; analogous changesin betas occurred forelectric util-ities, particularly for those with substantin! nuclear-generating capacity, afierthe Chernobyl nuclear power accident in [986. This stability of beta remainsa problematicissue, however, and therefore in Exercises 6 and 7 of this chap-ter we will encourage you to examine the variability of estimated betas overtime, using statistical techniques known as Chow tests and event studymethodologies.

While the parameter 8, is of obvious interest and importance, in ourestimable Eq.(2.17) another parameter appears that was added in an ad hocmanner, namely, a, Specifically, recall that «x, does not appearin Eq. (2.16),the linear equation analogousto Eq.(2. 3). On the basisofthe finance theoryunderlying the CAPM and summarized in Eq. (2.16), one should thereforeexpectestimates of «, to be close to zero on average (more onthis later, how-ever). In fact, a typical empiricalresult obtained when oneestimates param-eters of Eq,(2.17) is that the least squares estimate of ey, is insignificantly dif.ferent from zero, The null hypothesis that a, = O can be tested of coursesimply by determining whetherthe s-statistic correspondingto the estimate ofa, is greater than thecritical value at some predetermined significancelevel.Moreover, if one wants to imposetherestriction that ey = 0 in (2.17), mostcomputerregression programsprovide optionsthat allow the user to omit anintercept term,

It is worth noting that in some investment houses, portfolio managers’performances have occasionally been evaluated by computing the predictedTeturns the manager would have earned given the betas of the companiesinhis or her portfolio and subtracting this Predicted return from realizedteturns, thereby obtaining an implicit estimate of the manager's w; if the real-ized return were larger than that predicted by the overall portfolio 2, then theportfolio manager wassaid to have produced a positive a, and he or she wascompensated accordingly. Such simple compensation schemes are seldom.used today, but they suggest an interesting interpretation of « for portfoliomanagers, ”*

Suppose that a securities market analys{ employed monthly timeseriesdata on returns ofa particular company over the preceding five years andestimated the parameters a and § using conventionallinear regression tech-niques. Perhaps the analyst obtained the empiricalfindingthatfor a particularcompanythe estimateof « was positive andsignificantly different from zero.This would imply that even if the market as a whole were expected to earnnothing(that is, if 7, ~ in Eq. (2.17) equalled zero), investors in this com-pany expected

a

positive rate ofprice appreciation, (Note that, for example,an estimate of 0.67 for w based on monthly datais interpreted as an annualTate of price appreciation of approximately 12 times 0.67, or about 8% peryear.) For some other company the analyst might find that the estimated wwas negative and significantly different from zero. True udherents to theCAPM would therefore argue that, on average, one should expect estimatesof « to be zero,


This raises the importantissue of whether one can test the CAPM ive

remarks are worth makingin this context. First, the finance theory unt eriving

the CAPM explicitly employs expected or ex ante returns, whereasi can

observe is realized or eX posi returns. This makes rigorous testing ofthe

CAPM moredifficult, although hot impossible if more sophisticated ce

tric techniques are employed. . : :

eeTeating Hi the CAPM,the market portiolio shouldsnelude

alf risky investments, whereas most market indexes and estimates 0 co

tain only a sample of stocks, say, those traded an the New 3 ons ok

Exchange (thereby excluding. for example, all risky asseis trad ed in he ms

of the world—even humancapital, private businesses, and private reales ae)

Collectingsufficient data on risky assets to estimate Fn reliably cou! pe a ior

midable and prohibitively expensive task. In this contextit is wort nati i

that early empirical studies based on the CAPM discovered. that estimate o

@ depended critically on the choice of ra’ specifically, Tn measures “don

the Dow Jones 30 Industrials Index provided different results tom re

based on the Standard & Poor $00 Index or the Wilshire 5000.' ' ada,

fundedin large part by Merrill, Lynch,Pierce, Fenner & Smith, Ine.the “en

ter for Research on Securities Prices (CRSP) ut the University i ‘ cae

makes available to researchers estimates of r,, based on the valucsweig ed

transactions of all stocks listed on the New York and Americ 0 peck

Exchanges. Fortunately, such data series ON Fn have been made aval eee

us, andin the hands-on applicationsofthis chapter we will employ the C!

es of P,. ; ca ag,

aehind, the typical measure of a risk-free asset is something ike the3

day U.S. Treasury bill rate, whichreally isrisk-free onlyif it is hel Aa ma u

rity. Further, it is risk-free only in a nominal sense, for even if it is held

maturity, the uncertainty ofinflation makes thereal rate of rewurn une na

Thereforeit is difficult, if not impossible, to obtain a good measure oft he sk

free return. This makes testing of the CAPMmodel even more trou iesom

Fourth,there is a strong tradition within statistics that argues that i i

notpossible totest u specific model unless there is an alternative viable model

against which it can be compared. In the case of the CAPM, one a enn we

that might be used is the ingenious arbitrage pricing model (ari M) ey opel

by Stephen Ross, which in its simplest interpretation essentiallyinval ws “

ing right-hand variables to Eq. (2.17), such as the rate of unexpect ved ree

inflation.'* Provided that the above issueswere properly taken into ace , a

one could in principle test the CAPM against theAPM merely ytestingine

null hypothesis that the parameters on the additional Fight aries

were simultaneously equal 10 zero against re alternative hypothesis

‘ameters were simultaneously not equal to zero. | .

Peathandtnally, amore informal method of testing the CAPM involves

checking whetherits predictions are consistent with what isobserved. me

context, two aspects of the CAPM model appear to be inconsisten

observation,

38 The Capitat Asset Pricing Model

STEPHEN A. ROSS

Generalizing the CAPM

Ithough the capi- ital asset pricingmodel of modern

finance theory is widelytaught and implementedtoday, the validity of the #fCAPM hinges on a num-berofrestrictive assump-tions, One very usefulgeneralization of theCAPMis called the arbi-trage =pricing =model i{APM). The APM allowsforfactors other than the marketriskPremium to affect returns on securi-ties. The APM was developed in1976 by Stephen A. Ross,a financialcconomist then at the Universityof Pennsylvania,

Ross's research on the APM hasattracted a great deal of interest andcontroversy and has raised importantmethodologicalissues of whetherit ispossible to test the CAPM asa specialcase of the APM (see thetext for fur-therdetails and references).

Born in Boston in (944, StephenRoss developed a strong interest inapplied mathematics. Hereceived hisB.S, degree with honorsfrom the Cal-ifornia Institute of Technotogy, ma-joring in physics, His attentionshifted to economics and finance,

however, and in 1969 heearned a doctorate ineconomics from Harvard,

Although Ross iswidely recognized asa rig-oroustheoreticalfinancialresearcher, he is also re-spected as an unusuallyclear expositor. At theUniversity of Pennsylva-

i nia, for example, he was" nominated for distinguished teaching awards

three years in succession. In 1977,Ross moved to Yale University, andtoday he holdsits Sterling Professorof Economies and Finance chair, achair previously held by Nobel Lau-feate James Tobin.

In addition to undertaking aca-demic research in finance theory,Rossregularly travels from Yale toWall Street and New York City,where he engages in financial con-sulting and serves as a principat withRichard Roll in Roll and Ross AssetManagement Corporation, Ross is‘on the editorial board of numerousprofessional journats, is a Fellowofthe Econometric Society, and in1988 was elected President of theAmerican Finance Association,

. !. One important implication of the CAPM framework is that the rela-tionship between risk and return is not only linear butis also positive. As hasbeen noted by Fischer Black, Michael Jensen, and Myron Scholes [1972],however, there have been a numberofinstances in which this relationship hasappeared to be negative rather than positive. Specifically, Black, Jensen, und


Scholes found that over the entire nine-year period from April 1957 throughDecember (965, securities with higher ds produved lower returns than lessrisky (lower-8) securities. Why this happened is not yet entirely clear and con-stitutes a puzzting contradiction of the CAPM framework.

2, Another implication of the CAPM framework is that a security with2 zero @ should give a return exactly equalto therisk-free rate. Black, Jensen,and Scholes exhaustively studied security returns on the New York StockExchange over a 35-year period and found instead that the measured zero-#rate of return exceeded therisk-free rate, implying that some unsystematic (ornon-) risk makes the return higherfor the zero-# portfolio than is predictedby the CAPM. Moreover, the actual risk-return relationship examined byBlack, Jensen, and Scholtes appeared to be flatter than that predicted by theCAPM.It is therefore not clear what factors other than the market risk pre-miumare being vatued in the marketplace. According to the CAPM,it is onlymarket risk that matters. since unsystematic risk can be diversified away.Muchresearch is currently underway thit attempts to examine whether facetors otherthan ,, affect marketrisk,

In this contextit is worth noting that some securities firms such as Mer-rill, Lynch, Pierce, Fenner and Smith regularly publish a “beta book” inwhich they report estimates of « and # based on standard OLS regressionmethods, as well as “adjusted 8”estimates that attemptto deal with the zero-8 portfolio problem mentioned above, using more complex Bayesianstatis-tical procedures, Although these Bayesian procedures are of considerableinterest, they are beyond the scopeofthis chapter.

Several other econometric issues should be briefly noted. The printedoutput of computer regression programstypically includes measures of R?,the standard error ofthe regression, and é-statistics. These and other standardstatisticul measures have a particularly interesting interpretation and appli-cation within the CAPM. Consider, for example, the simple correlationbetweenthe risk premiumon the security j (7, — 7;) and the market risk pre-mium(+, — ¢;)—variables that are on the left- and right-handsides, resitively, of the CAPM regression equation (2.17). The sample correlation coef-ficient between them can be rewritten as follows:

2 in 2 bn ineeeea, BG, (2.18)

where Gj, Gm @? are sample covariances and variancesfor r,, — r, and 7, —7, and 8, is the least squares estimate of , Hence the sample correlationbetween the porifolio and market risk premiumsis simply the product of theleust squares estimate of 8, and the relative sample standard deviations of themarket and j-portfolio risk premiums,

The standard errorofthe residual in the regression equation (2.17) alsohas a useful interpretation. Specifically, while the left-handside of Eq. (2.17)reflects the effects of bothspecific (unsystematic) and market (systematic) riskon the portfolio in company /, the Br. ~ 7) term on the right-hand side


reflects only the impact of marketrisk, It therefore follows that the estimatedresidual in Eq.(2.17) incorporates only the eflects ofspecific (unsystematic)risk, The Standard errorofthe residual (also often called (he standard error ofthe regression), computed as the square root of s?, defined asz

=P dn - 2 (2.19)™where @, is the least Squares residual for ic

quares the 7h observation, therefore mea-suresthe standard deviation ofthespecific {unsystematie) risk—portfolioriskhats Nolresponsive to market fluctuations. A large standard error of theOfenerhen eepe manth, would indicate that a substantial amount¢ Portfolio j risk premium could ni explaiin the marketrisk premium, ot be explained by changesFurther, since the &? value from i. varue Irom regression computer output indicateswhat Proportion ofthe variation in the dependent variable is explained bypifiation in the right-hand or independent variables, in the CAPM context ofa. or). R measures the market (systematic) portionoftotal risk. On theol orhand, 1 ~ Ris the proportion of totalrisk that is specific {unsystem-atic). ' iam F, Sharpe (1985, Pp. 167] notes that for an individual companya typical R? measure from a CAPM equation is about .30 but that as oneAssets intoa larger portfolio, the R? measureiss ¢ ion of specific risk through diversification,3 hin“ notethat, since in the bivariate regression madel R?= then values do not necessurily vorrespond withlarge estimate:Tosee this, note that from Eq. (2.15), " large estimates ore

é imR= pl = B- (2.20)It follows that forsomestocks with very large variance 4}, R? can be low evenwhile the estimate of 8, is high; in such cases the reaction of the particularstock (or portfolio) to marketvariationsis very sharp, yet market variationexplains only a small portion ofthe stock's large variability. The regeessiequation for otherstacks might have a high R? but a low 4, estimate; this canoccur when ation in the stock's (or portfolio's) risk premium issmall inrelation fo variation in the market risk premium,that is, the ratio ofsam pleVariances in Eq.(2.20) is large. Moreover, note that a very low R? docs wotoraageCAPM framework; rather, it simply indicates that the total risknt Cs any's assets is a it ‘a if ,

tatdo tteen ve anaes is almost entirely company-specifiv, unre-

The finaltypical regression outputofparticular interest to us here is thet-statistic, Earlier, it was noted that the t-statistic on the estimate of « can byused to test directly the null hypothesis that a = 0 against the alternativehypothesis that a # 0, Failure to reject this null hypothesis might be viewedas evidenee in support of the CAPM, : me

2.6 Hands On with the CAPM 41

The t-statistic on the estimate of # corresponds tv an analogous nullhypothesis, namely. that 8 = 0, agains! the alternative hypothesis thal 6 40. Quite frequently, it is of interestto test a different hypothesis, namely, thatthe movementof asset prices of a particular company is the same as that ofthe market as a whole; this correspondsto test ofthe null hypothesis that 8= 1 against the alternative hypothesis that 6 # 1. To conduct such anhypothesis test, simply take the estimated standard error of the #2 estimate(usually provided on computer regression outputs), construct a confidenceinterval given some reasonable level of confidence (e.g., 95% or 99%), andthen determine whether # = [ falls within this confidence interval. If it does,the null hypothesisis not rejected; if ¢= 1 does notfull within this confidenceinterval, the null hypothesis is rejected.

Oneother remark is worth making before we go on to hands-on empir-icat implementations. Since within the CAPM framework, 2 enables one tocalculate whit the required return on a particular stock mightbe,it also indi-rectly allows one to computetheall-equity firm's cost of capital. For a welmanagedfirm thatis considering 4 particular investment project, the com!nition of expected return andits # should, of course, place the project aboveits cost ofcapital if the project is to be accepted. However, if the companyiconsidering a new project that is morerisky than the company’s average proj-ects, a larger expected return should be required by the firnt before investing,since such & project would increase the average risk of the companyand,according to the CAPM model, would result in investors’ requiring a higherreturn. This implies that, analogous to Eq. (2.15), projects also have their ownbetas, and if the # of a particular projectis higher than the company’s average8, So too should be the required expected rate of return."

This completes our discussion of the theory underlying the CAPM andour brief review of econometric issues involved in estimating its parameters.We are now ready to becomedirectly involved in implementing the CAPMempirically.

HANDS ON WITH THE CAPM

The purpose of the exercises in this chapter is to help you gain an empiricalhands-on experience and understanding of the CAPM using bivariate leastsquares regression techniques. Exercise | is particularly useful, since in it youbecome acquainted with duta used in subsequent exercises. In Exercise 2you obtain and interpretleast squares estimates of #, while in Exercise 3 youencounterthe 8 ofa very interesting asset—gold, In Exercise 4 you are giventhe opportunity to Jearn how you can recover from mistakes-—runningregression of Y on Yrather than of Y on X for Delia Airlines, In Exercise 5you gain a better understanding ofthe benefitsofdiversification by employingthe CAPM toconstruct a portfolio,

Thelast five exercises ofthis chapter employslightly more sophisticated


econometric techniques; therefore you might wantto return to these exerciseslater after becoming more familiar with the requisite econometric tools. Inparticular, in Exercise 6 you test for the stability of @ using the Chow testprocedure, In Exercise 7 you employ dummyvariables in “event studies” forthe Three Mite [sland nuclear power accident and for the bidding warbetween DuPont and Dow Chemicalin their takeover attempt of Conoco. InExercise 8 you employ Chow tests and dummyvariables to assess whetherJanuary returns are different from those of other months, as has apparentlybeen observed. Then in Exercise 9 you assess an alternative to the CAPMframework,the arbitrage pricing model, by comparing results from multivar-iate and bivariate regressions. In thefinal application, Exercise 10, you con-duct a numberoftests to determine the empirical validity of the stochasticspecification.

.On the data diskette provided, you will find a subdirectory called

CHAP2.DATwith numerousdatafiles, In these data files a numberof dataseries are provided, including data on ten years of monthly returns for 17companies over the time period January 1978 to December 1987 (values forr,). In another data file named MARKET,youwill find a data series takenfrom the Center for Research on Securities Prices (often called CRSP) thatmeasures fH, specifically, MARKET is a value-weighted composite monthlymarket return based on transactions from the New York Stock Exchange andthe American Exchange over the same ten-year time span. Note that theCRSP measure ofr,, includes data on all stocks listed at the New York andAmerican Stock Exchanges, notjust the 30 used for the Dow Jones index orthe Standard & Poor 500. Anotherdata file is called RKFREE,andit providesyou with a measure of r,, namely, the return on 30-day U.S. Treasury bills,The 17 companies whose returns are in the masterfile of CHAP2.DAToper-ate primarily in eight industries:

Industry Company File NameOil Mobil MOBIL

Texaco TEXACOComputers International Business Machines {BM

Digital Equipment Company DECData General DATGEN

Electric utilities Consolidated Edison CONEDPublic Service of New PSNHHampshire

Forest products Weyerhauser WEYERBoise BOISE

Electronic components Motorola MOTORTandy TANDY

Airlines Pan American Airways PANAM

Delta DELTA

Exercises 43

Banks Continental Iinois CONTIL

Citicorp CITCRP

Foods Gerber GERBER

General Mills GENMIL

i ivs dit si : variation in risk and returns.These firms and industries display considerable variation in tis

Several otherdata files are in the CHAP2.DAT subdirectory, suchas GOLD,

EVENTS,and APM. The contents of those data files are discussed in the exer-

cises below. . . as

“ At numeroustimesin the following exercises we will ask you lo test an

hypothesis using a “reasonable level ofsignificance." Since whatis "reuson-

able”is to some extent discretionary, you are asked to follow the convention

of stating precisely whatpercentsignificance tevel is being used to test hypoth-

eses; in that way, others with different preferences can still draw their own

inferences. . ; ante files

, Note: Rememberthat before the data diskette can be used, ihe data files

? et er i i efe: ck 10 Chuptermust be properly formatted. For further information, refer buel ! Nt

1, Section 1.3. MAKE SURE THAT ALL DATA FILES TO BE USED

BELOW ARE FORMATTED.

EXERCISES

EXERCISE 1: Getting Started with the Data

i ise i eC re fumiliar with theThe purpose ofthis exercise is to help you become mor F with

data series in your subdirectory CHAP2.DATand to have you speculate on

what values company-specific betas mighttake.

! sing data in the file named MARKET,print and plot returns for the

“) faaonthe in the dataseries, from January 1985 through December

1987, (The output of most computer regression programs looks rather

messyifthefull time period of 120 monthsis plotted.) sina the

(b) Next, using data from the MARKETand RKFREE files covering the

entire [20-month time span, construct the risk premium measure ( hy

1) for any company of your choice and the overall market risk premium

(ry 7 1). (Note:1 is possible that for some monthsthe tisk Bremaium

of a particular company’s assets, or that for the market asa whole, is

negative. Does this occur in your sample?In particular, check for Octo-

ber 1987. What happened in that month?) Compute andprint out same

ple meansfor Fp fan Msp — A ANG Ty — Fy Note that these returns are

monthly, not annual, One way of converting the mean monthly returns

to annual equivalents is based on the formula

Pama & 1+ FP I

44 The Capital Asset Pricing Mudet

where Pou 5 the annual return and F is the mean monthly return. Aresulting value of .060, for example, corresponds io a 6.04% annualreturn. Compute annual returnsfor r,, fm tn fp — tn and ty — 4. Dothese values appear plausible? Why or why not?

(c) Next plot both the company's and the market's risk premium forthetasi 36 monthsin the data series, from January 1983 through December1987 (makesure monthscoincidefor the two series). Are there any note-worthy aspects of these plots? Do you think that this company’s # forthis time period will be greater than or less than unity? Why? Doesthismakeintuitive sense?

(d) Finally, catcutate the variance and the standard deviation of the com-pany’s and market'srisk premiumsover the same 36-month time periodas in part (c), as well as the simple correlation coefficient between them,Using Eq. (2.18) and the above values, calculate the implied 8 for thiscompany.Is this 8 roughly equal to what you expected?

EXERCISE 2: Least Squares Estimatesof ¢

Fromthelist of industries on pages 42-43, choose one industry chat you thinkis highty risky and anotherindustry that youthinkis relatively “sale,” Divideyour sample into the first halfJanuary 1978-December 1982) and the secondhalf (January 1983-December 1987) and choose the half with which you willwork,

{a) Using your computerregression software, the 60 observations you havechosen, and Eq.(2.17), estimate by ordinary least squares the parame-ters « and fi for one firm in eachofthese two industries. Do the esti-mates of 8 correspond well with your prior intuition orbeliefs? Whyorwhy not?

(b) For one ofthese companies, make a timeplot ofthe historicat companytisk premium, the companyrisk premiumpredicted by the regressionmodel, and the associated residuals. Are there uny episodes or dates thatappear to correspond with unusually large residuals? If so, atiempt 10interpret them, .

(c) For each ofthe companies, test the null hypothesis that « = 0 againstthe alternative hypothesis that a # 0, using a significance level of 95%.Would rejection ofthis null hypothesis imply that the CAPM has beeninvalidated? Why or why not?

(d) For each company, construct a 95%confidence intervalfor 8. Then testthe null hypothesis that the company's risk is the same as the averagetisk over the entire market, thatis, test that 8 = [ against the alternativehypothesis that 8 # 1. Did you find any sueprises?

(ec) For each of the two companies, compute the proportion oftotal risk thatis market risk, also called systematic and nondiversifiable. William F.

Exercises 45

Sharpe (1985, p. 167] states that “Uncertainty about the overall market.,. accounts for only 30% of the uncertainty about the prospects fortypical stock.” Does evidence from the two companies you have chosencorrespondto Sharpe's typical stock? Why or why not? Whatis the pro-Portion oftotalrisk that is specific and diversifiable? Dothese propor-tions surprise you? Why? an ,

(f) In your sample, do large estimates of ¢ correspond with higher 2? val-ues? Would you expect this always to be the case? Why or why not?

EXERCISE 3: Why Gold Is Special

The purpose of this exercise is to acquaint you with features of a ratherremarkable asset whose peculiar covariance of returns with the market as awhole often makesit attractive to investors.

(a) There is one assetin the data directory whose datafile is named GOLD.The GOLDdata file contains series on monthly returns for GOLD, aswell as data series for the market (MARK76) andrisk-free (RKFR76)variables, ail for the January 1976-December 1985 time period. Usingthe January 1976-December 1979 four-year time period and theCAPM,generute variables measuring the GOLD-specitic and marketrisk premiums, and then estimatethe 8 for GOLD, Computea 95%con-fidence interval for 8. Do yourestimates make sense? Why might suchan asset be particularly desirable to an investor who is attempting toreduce risk through diversification? What does this imply concerningthe expected return on such au asset?

(b) Now estimate the 8 for GOLD using data from January 1980-Decem-ber 1985. Construct a 95% confidence interval for f. Has anythingchanged? Comment on supply and demand shift factors possibly alter-ing the 8 of GOLD.

EXERCISE 4: Consequencesof Running the Regression Backward

The purpose of this exercise is to explore the consequences of running aregression backwards, that is, of regressing 1 on Yrather than Yon X, and todiscover how one can recoverthe “correct” estimates fromthe computer out-put of the “incorrect” regression,

(a) Using data from January 1983 through December 1987 for Delta Air-lines in the data file DELTA,as weil as data on r,, in MARKETand onyin RKFREE, construct the risk premium for Delta Airlines and forthe market as a whole. To simplify notation, now define Y, = 7, —tn(the risk premium for Delta Airlines) and X= ty. — %, (the market riskpremium), 7 = 1,..., 120.


Suppose that instead ofspecifying the “correct CAPM regressionequation

Yama + BX, +6 (2.21)

you inadvertently specified the “incorrect” reciprocal regression

equation

Xe ht y¥ ty, (2,22)

Show that in the incorrect equation, one can still find the original

CAPM parameters and disturbances, that is, solve Eq, (2.21) for 4, in

terms of Y, and ¢, and show that 8 = —a/#. y = 1/8, and v, =

{= [/B)e.(b) Nowestimate by OLS the parametersin the incorrect equation (2.22),

and denote these estimates of 8 and + by d and g,respectively. What is

the R? from this incorrect regression? At a reasonable level of signifi-

cance,lest the null hypothesis that y = O against the alternative hypoth-

esis that y # 0, Next, construct implicit estimates of the CAPM and

@ parameters from this incorrect regression as b, = I/g and a, =

—d/g (the subscript x indicates that the estimate is obtained from a

regression of ¥ on Y).(c) Having suddenly discovered your mistake, now run the currect regres-

sion and estimate by OLS the « and # parameters in Eq. (2.21), and

denote these parameterestimates as a, and , (the subscript yindicatesthat the estimate is obtained from aregression of Y on A’). Whatis theR? from this regression? At a reasonable level ofsignificance, test thenull hypothesis that @ = 0 against the alternative hypothesis that @

0(d) Notice that the 2? from the correct equation (2,21) in part (¢)is identical

10 the R? fromthe incorrect equation (2.22) in part (b} and that so too

are the é-statistics computed in parts (b) and (c). Why does this occur?

Using the formula for R? from your econometric theory textbook and

for the OLS estimates of @ in Eq, (2.21) and + in Eq. (2.22), show that

Rea bg = bib, (2.23)

Notice that what Eq. (2.23) implies is that if you run the wrong regres-sion (Eq. 2.22) and obtain g rather than 4,, you can simply use your R?from this incorrect regression, your value of g and Eq. (2.23) to solvefor b,—the estimated value of # you would have obtained had you runthe correct regression! Verify numerically that this relationship amongR}, b,, g, and 6, occurs with your data for Delta Airlines.

{e) Finally, notice that since R? is alwaysless than unity, Eq.(2.23) impliesthat b, < b,, Which estimate of 6 do you prefer—the smaller 4, or thelarger b.2 Why?

Exercises 47

EXERCISE 5: Using the CAPM to Construct Portfolios

The purpose ofthis exercise is 10 provide you with a greater appreciation ofthe benefits of diversification, Imagine that in December 1982 yourrich uncledied and bequeathed to you one million dolars, which he specified that youmust invest immediately in risky assets. Further, accordingto the terms oftheestate, you could not draw on any funds until you graduated, Therefore inlate 1982 you needed to decide how to invest the proceeds of this estate sothat upon graduation you would have ample fundsfor the future, You willnow examine the effects ofdiversification upon risk and return, given threealternative portfolios that you might have chosen in January 1983,

Choose one industry that you think is relatively risky und another indus-trythatis relatively safe, from the list on pages 42-43, Choose two companiesin the safe industry and twoin therelatively riskier industry,

(a) Calculate the means and standard deviationsofthe returns for each ofthe four companies over the January 1983-December 1987 timeperiod.Do the risk-return patterns for these companies correspond with yourprior expectations? Why or why not? In which might your uncle haveinvested? Why?

{b) Construct three alternative (one million dollar total) portfolios as fol-lows, Portfolio I: 50%in a company in the safe industry and $0%in acompany from therisky industry. Portfolio [1: 50% in each of the twocompanies in the safe industry, Portfolio II; 30%in each of the twocompaniesin the risky industry. Calculate the sample correlation coefficient between the two companyreturnsin each ofthe three portfolios.Commentonthe size and interpretationofthese correlations. For eachofthe three portfolios, calculate the means and standard deviations ofreturns over the January 1983-December [987 time period, Are there

any surprises?(c) Whichofthe threealternative portfolio diversifications would have been

most justifiable in terms of reducing the unsystematic risk of investment? Why? (Choose your words carefully.)

(d) Next estimate the CAPM equation (2.17) for each of these three por-folios over the same January 1983~December 1987 time period. For¢ach portfolio, using a reasonable level of significance, test the nullhypothesis that 8 = | against the alternative hypothesis that # 4 t.Whichportfolio had the smallest proportion of unsystematic risk?

(¢) For Portfolio 1, compare the X? fromthe portfolio regression in part (d)

with the 2? from the separate regressionsfor the two companies. Would

you expect the R? from the portfolio equation to be higher than

that from the individual equations? Why or why not? Interpret yourfindings.

4B ‘The Capital Asset Pricing Model

EXERCISE 6: Assessing the Stability of ¢ over Time and AmongCompanies

An important maintained assumption of much empirical work in the CAPMframeworkis that the 8 parameteris stable over time. To the extent that com-panies within the sume industry are relatively similar, one mightalso expectthat the fs are equal across companies in the same industry. In this exercise,such assumptionsare tested, using the Chow test for parameterstability,”

(a) From your datafile, choose two industries. To ullow for possible strucetural change, for each companyin the sample,divide the historical dataseries into 1wo halves, one for January 1978-December 1982 and theother for January 1983-December 1987, Using Eq. (2.7) and a reason-ablelevel of significance, test the null hypothesis that for each companythe parameters « and @ are equal over the two hulf-samples, thatis, thatfor each company these parameters ure constant overtime.

(b) Now, for each industry, using a reasonable levelofsignificance, test thenull hypothesis that the parameters « and g are identicalfor ail firmsinthe industry over the entire January 1978-December 1987 time period.(Be particularly careful in how you compute degreesoffreedom.)

(c) Finally, for each industry, using a reasonable ievel ofsignificance, testthe null hypothesis that the parameters « and # are identicatforall com-panies in the industry and are equal over bur the January 1978-December 1982 and January 1983-December 1987 timeintervals, (Beparticularly careful in calculating degrees of freedom.) What do youconclude concerning the stability of parameters over companies andtime?

EXERCISE 7: Three Mile island and the Conoco Takeover: EventStudies

An important behavioral assumption underlying the CAPM is thatall inves-tors efficiently use whatever information is available to them. It is also typi+cally assumed thatall investors have equal access to information, One impli-cation of thisis that ifan unexpected event occurs that generates informationto all investors concerning changes in the expected future profitability of aparticular company, such a release of information should immediately raiseor lowertheprice of that company’s securities. This sharp changein the valueof a company'sassets will often, but not always, tend to be a “once and forall” shift, and thereafter movements in the company's securities are oftenagain determined by its 8 and the overall market risk premium.

In this exercise we examine and quantify the effects of certain eventsthat generated information relevant to investors. We consider two cases:

Exercises 49

(1) the effect of the Three Mile Island nuclear powerplant accident on March28, 1979,on the returns earned by General Pubic Utilities, the parent holdingcompanyof the Three Mile Island generating plant?'; and (2) the effect ofthebidding war between, among others, DuPont and Dow Chemical in theirattempt to take over Conoco in June-August 1981.7 Incidentally, in bothcases it wil be implicitly assumed that the companies’ subsequent betas wereunaffected by the new information.

In your subdirectory CHAP2.DATis a data file named EVENTS,con-taining 120 months' data on monthly returns for GPU, DUPONT, and DOWas well as the RKFREEreturn and the overat] MARKETreturn; these datacoverthe time period from January 1976 through December 1985. This filealso includes data for CONOCO,but the data cover only the time period ofJanuary 1978 through September 1981, since Conoco wastaken overat thattime,

(a) First, generate values of the company-specific and overall market riskpremiums for GPU, DUPONT, and DOW over the January 1976-December 1985 time period and for CONOCO covering the January1976-September 1981 time interval. Compute sample means andprintthem out.

{b) Then, using the CAPM equation (2,17), estimate the a and # of GeneralPublic Utilities over the January 1976-December 1985 timeperiod, butexctude from your sample the month of April 1979. On the basis of yourTegression model, compute the fitted value for April 1979 and then theresidual for April 1979, the period just after the Three Mile Islandnuclear power accident occurred. (Note: The monthly data are dailyaverages, and since the accident occurredin thelast days of March 1979,it does not substantially affect the monthly March (979 data.) How doyou interpret this residual?

(c) Now form a dummyvariable (denoted TMIDUM)whose value in April(979 is unity and whose value is zero in al! other months. Using leastsquares regression methods, estimate an expanded CAPM model inwhich Eq, (2.17) is extended to include the dummyvariable TMIDUM.Compare the value of the estimated coefficient on the TMIDUM vari-able with the value of the residual in purt (b) for Apri! 1979, Also com-pare the estimated slope coefficients in the two regressions, Why atethese equal? Finally, using a reasonablelevel ofsignificance,test the nullhypothesis that the coefficient on the TMIDUM variable is equalto zeroagainst the alternative hypothesis that it is not equal to zero, Was theThree Mile Island accidenta significant event?

(d) Using the CAPM equation (2.17) and us Jengthy a time period sampleas possible, estimate the « and @ ofthe three companies involved in theConocotakeover event: DuPont(the eventually successful takeover bid-der), Dow (an unsuccessful bidder), and Conoco (the takeover target);note that for Conoco, the data series terminates in September 1981,


since at that time the takeover was completed. For cach company, con-struct the average of the regression residuals for the months of June,July, August, and September 1981.

(e) Next, construct for DuPont, Dow, and Conoco a dummy variablewhose valueis | during the four months of June, July, August, and Sep-tember 1981 and whose value is 0 in all other months, Using leastsquares regression methods, estimate an expanded CAPM model overthe January 1976-December 1985 time period for DuPont and Dow{and for January 1976-September 1981 for Conaco), in which Eq.(2.17) is extended to include the June-September 1981 dummy vari-able. Compare the value of the estimated coefficient on this dummyvariable for each company with the arithmetic average valueofits resid-uals over the June-September 1981 time period from part (d), Thencompare the estimated betas, Why does the equality relutionship ofpart(d) no longer hold here? How do you interpret the signs ofthese esti-mated coefficients; in particular, which shareholders were the winnersand which were the losers during the Conoco takeoverefforts? Is thisplausible? Finally, using a reasonable fevel ofsignificance, for each ofthe three companiestest the null hypothesis that the coefficient on theJune-September 1981 dummyvariable is equalto zero agains! the alter-native hypothesis thatit is not equal to zero,

EXERCISE 8: Is January Different?

There is some tentative evidence supporting the notion that sock returns inthe month of January are, other things being equal, higher than in othermonths, especially for smaller companies.” Why this might be the case is notclear, since even if investors soid losing stocks during December for tax rea-sons, the expectation that January returns would be higher would shift supplyand demand curves and thereby would tend to equilibrate returns over theyearvia the possibility of intertemporal arbitrage.” However, the “January isdifferent” hypothesis does seem worth checking out empirically, In this exer-cise you investigate how this hypothesis might be tested.

(a) First, if the ‘January premium”affected the overall market reiura F,,and the risk-free return r, by the same amount,say, j,,. Show that themarketrisk premium would be unaffected. In this case, could the “Jan-uary is different” hypothesis be tested within the CAPM framework ofEq,(2.17)? Would it not make more sense, however, 1o assume that the“January is different” hypothesis referred only to risky assets? Why orwhy not?

(b) Suppose instead, therefore, that the “January premium" did not affectriskefree assets. In this case, one might hypothesize that in January theoverall market return would change from r,, 10 75, where 74, fin + Jon

(e)

(a)

(c}

Exercises 51

and j, is the January premium, Would the market risk premium beaflected? Why or why not? Further, if the CAPM model were true andthe «and @ parameters were constunt, show that in January the expectedportfolio return should increase to r;, where 7, 1, + 8 + jn RewriteEq. (2.17) using the right-handsides of the above r;, and 7; expressionsin place of ¢,, and ¢,. respectively: then, noting that @ > j,, is unobserva-ble, subtract 8 - j,, from both sides, With what are you left compared toEq, (2.17)? What doesthis indicute concerning the possibility of testingthe “Januaryis different" hypothesis within he CAPM framework?Given the conclusions that emerged from parts (a) and (b), it wouldseem to make sense to abandon the CAPM frameworkandinstead toexaminealternative ways oftesting the “January is differenthypathe-sis, To do that, choose any three industries in your January 1978-December 1987 sample. Form a dummy variable called DUMJ thattakes the value of unity if the month is January andis zeroforall othermonths; also ensure that for each companythe variable +,is accessiblefrom yourdata files (nof the +, — ry, risk premium variable used in theCAPMregression), Now for each company run the regression of 7, onan intercept term and on the dummyvariable DUMJ, Usinga reason-able level of significance, test the null hypothesis that the coefficient onthe DUMJ variable is zero against the alternative hypothesis that it isnot equal to zero, Is Junuary different?Now supposethaiin spite of the reservations developed in parts (a) and(b) concerning the possibility of testing the “January is different”hypothesis within the CAPM framework, sheer perseverance compelsyou still to estimate Eq. (2.17), More specifically, for each company inthe same sample as in part (c), form a subsample that consists only ofthe ten January observations (one from each year). Then formthe com-plementary subsample consisting of the remaining 11 months of eachyear for each company. Next, using the Chowtest procedure.test thenull hypothesis that the parameters of the CAPM in January are equalto thoseofthe remaining monthsofthe year. Interpret yourresults care-fully. Whatis the alternative hypothesis in this casc?Yet one other way of examining this “January is different” hypothesiswithin the CAPM frameworkis to assume that the @ parameter is con-stant overall months but that the January intercept term might diffefrom that for the remaining t{ months ofthe year. Set up and estimaa CAPM model, using the same companies as in parts (c) and (d), inwhich the slope coefficient g is the sameforall months, while the intercept term for January and the commonintercept term for all othermonthsof the year are permitted to difler. Using a reasonable level ofsignificance, test the null hypothesis that “Januaryis different.” Thentest the null hypothesis that “January is better.”Commenton the varioustests and test results in parts (a) through (c).What do you conclude? Is Januury different?


EXERCISE 9: Comparing the Capital Asset and Arbitrage PricingModels

Aswasnoted in thetext, an alternative mode! ofasset pricing has been devel-oped by Stephen A. Ross and has been called the arbitrage pricing model{APM).*In our context the APM is ofinterest in that the CAPM might beviewed as a testable special case of a more general model. More specifically,in the APM,securities are allowed to respond differentially to economywide“surprises,” such as unexpected oil price shocks, or unexpected changes inthe overall rate of inflation, In this exercise, the CAPM is treated as a specialcase of the APM using the multivariate linear regression model. It is worthnoting here that while typical applications of the APM use quite sophisticatedempirical methods, such as factor analysis, for pedagogical purposes here wewill simply treat the CAPM and the bivariate regression model as a specialcase of the APM in the multiple regression context.

A data file named APM in the subdirectory CHAP2.DAT containsmonthly timeseries for three data series: the consumerprice index (CPI), theprice of domestic crude oil (POIL), and the Federal Reserve Board index ofindustrial production (FRBIND), Make sure that all these data series areproperly formatted (see Chapter1, Section 1.3 for further details).

{a) Choose two industries whose companies, in your judgment, might beparticularly sensitive to aggregate economic conditions, For each firm inthis sample, using the entire January 1978-December 1987 time span,first construct a data series called RINFreflecting the rate of inflation(CPI in month ¢ minus CPI in month ¢ — | divided by CPI in month 1— 1); another dataseries called ROIL—the growth rate in the real priceof oil—computed as (POIL/CPI), — (POIL/CPI), ,, all divided by(POIL/CPI),.,; and finally, a data series GIND reflecting growth inindustrial production, computed as FRBIND in month ¢ minusFRBINDin month ¢ — |, divided by FRBIND in month ¢ — 1, (Note:The data series for CPI, POIL, and FRBINDbegin in December 1977,not January 1978; this permits you to compute changes in these vari-ablesfor the full January 1978-December 1987 time period.) Computethe sample means of RINF, ROIL, and GIND and then generate simple“surprise” variables SURINF, SUROIL, and SURIND as RINF minusits sample mean, ROIL minusits sample mean, and GIND minusitssample mean, respectively. Print out the data serics for SURINF,SUROIL,and SURIND.

(b) Next, estimate the standard CAPM model for your chosen companyand then estimate a multipte regression model in which the right-handvariables include a constant, the market risk premium of the CAPMmodel, and the “susprise” variables SURINF, SUROIL, and SURIND.

Exercises 53

Using a reasonablelevel of significance, test the joint null hypothesisthat the coefficients of these three additional variables are simulta-neously equal to zero, Interpret the results.

(c) On the basis of your results in part (b), develop and defend a modelspecification for each company that best reflects the factors affectingcompanyreturns. Do you conclude that the CAPM modelis supported,or do you reject it? Is marketrisk entirely captured by the marketriskpremium, or do other variables affect market risk? Are any oftheseresults surprising? Why?

EXERCISE 10: What about Our Assumptions ConcerningDisturbances? Did We Mass Up?

In orderfor the ordinary least squares estimator to have the optimal proper-ties, it has been assumed that the ¢, disturbancesure independently and iden-tically normally distributed with mean zero and variance 0%, Inthis exercise‘wetest these assumptions. Notethatparts(a), (b), (c), and (d) are independentquestions and,in particular, are not sequential; hence any subset of them canbe performed oncethe requisite econometric procedures have been learned.

(a) Testing for homoskedasticity: A numberoftests have been proposed totest the null hypothesis of homoskedasticity against an alternativehypothesis consisting ofeither a specific or some unspecified form ofheteroskedasticity. Here we employ a very simple test, due to Halbert J.White [1980},

From your masterdatafile in CHAP2.DAT,choose any two firms.Forthe first firm in the sample, on the basis ofthe entire January 1978-December 1987 dataseries, estimate parameters in the CAPM equation(2.17) using ordinary least squares. Retrieve the residuals from this OLSregression and then generate a new variable defined as the squareoftheresidualat cach observation. (Note: Take a took at the residual for Octo-her 1987, Is it unusual? Why mightthis be the case?) Next, run an aux+itary OLS regression equation in which the dependent variableis thesquared residual series from thefirst regression and the right-hand vari-ables include a constant term, the market cisk premium 7, and thesquare of the market risk premium,thatis, 7,. [f one assumesthat theoriginal disturbances are homokurtic (thatis, the expected valueofcl isa constant), then under the null hypothesis, 7 (the sample size, here,120)timesthe R?from this auxiliary regressionis distributed asymptot-ically as a chi-square random variable with two degrees of freedom.Computethis chi-squaretest statistic for homoskedasticity and compareit to the 95%critical value. Are your data from thefirst firm consistentwith the homoskedasticity assumption? Perform the same Whitetest for


the second firm. If the nul! hypothesis of homoskedasticity is rejectedfor any firm, make the appropriate adjustments and reestimate theCAPMequation using an appropriate weighted least squares procedure.In such cases, does adjusting for heteroskedasticity affect the parameterestimatessignificantly? The estimated standard errors? The f-stutistics ofsignificance? Is this what you expected? Why or why not?

(b) Testing for first-order autoregressive disturbances: From your masterdatafile in CHAP2.DAT, choose two industries, For each firm in theJanuary 1978-December 1987 sample of these two industries, use theDurbin-Watsonstatistic and test for the absence ofa first-order autore-gressive stochastic process employing an appropriate level of signifi-cance. If the Durbin-Watsonstatistic for any firm reveals that the nullhypothesis of no autocorrelation eitheris rejected or is in the range of“inconclusive”inference, reestimate the CAPM equationforthosefirmsusing either the Hildreth-Lu or Cochrane-Orcuttestimation procedure.”Are the estimated parameters, standard errors, and/or t-statisticsaffected in any important way? Are any of the ubove results surprising?Why?

(c) Testing for first-order moving average disturbances: From your masterdata file in CHAP2.DAT, choose two industries. For each firm in theJanuary 1978-December 1987 sample of these 1wo industrics, estimatethe CAPM modelallowing fora first-order movingaverage disturbanceprocess. Using either an asymptotic ¢-test or a likelihood ratio test sta-tistic and a reasonable level of significance,test the null hypothesis thatthe moving average parameter is equal to zero against the alternativehypothesisthat it is not equal to zera.*

(d) Testing for normality: From yourdata file, choose two industries. Foreach firm in the January 1978~December 1987 sample ofthese indus-tries, use residuals from CAPM regression equations and the chi-squaregoodness-of-fit test (or the Kolmogorov-Smirnov test procedure) to testfor normality ofthe residuals.”

CHAPTER NOTES

1, For an historical account ofthe founding of the Econometric Society, see Curl F,

Christ [1983].2. Alfred Cowles 11] {1933}, p. 324.

3, For a more complete discussion of the CAPM,see any textbook in modern cor-porate finance, e.g. Richard Brealey and Stewart Myers [1988], Chapters 7-9, orWiltiam Sharpe (1985], Chapters 6-8, as well as the references cited therein.

4. It might be argued thatrisk should be measured only by downside surprises, How-ever,if the distribution of returns is symmetric, as is the case with the normaldistribution,then use of either the varianceor the standard deviation as a meusureofrisk is appropriate. Incidentally, although the issue remains somewhat contro-versial, there is a substantial amountofevidence suggesting that returns are dis-

e

15,16,

17,1B,

. Other useful studies demonstrating the rapid benefits of dive:

Notes $5

tributed reasonably symmetrically and approximately normally; see, for example,Franco Modigliani and Gerald A. Pogue (1974a,b].

. There is a substantial bodyofliterature demonstrating that, on average, investors.in the United States have in fact received higher rates ofreturn for bearinggreaterrisk. See, for example, the empirical study by Roger Ibbutson und Rex Sinqueficld(1986) covering the 60-year time span from 1926 to 1985.

. Nate, however, that the price ofthis asset is guaranteed onlyif it is held to matue

rity. Further, although the xomtinal return is known with certainty if the asset isheid to maturity, the extent of inflation is uncertain, and thusits rea/ rate of returnis not risk-free.

. For a more complete treatmentofdiversification,see the seminal article by HarryM. Markowitz [1952].

. This exposition is admittedly very simple and ignores complicationsofshares add-

ing to unity, as well as the source of funding for the additional purchases of secu-rity k, For a more detailed and rigorous discussion, see Harry M. Markowitz[1952, 1959], An intuitive exposition using discrete changesis found in WilliamF. Sharpe [1985}, pp. 154-156.

. Two seminal articles on the CAPM are those by William F. Sharpe [1964] andJohn Lintner [1965]. A third pioneering article on CAPM byJack L. Treynor{1961} hus neverbeen published,

. This two-step procedure has becn called the separation theorem and was firstderived hy James Tobin [1958].

. Obviously, the concave risk-return frontier in this case represents alternative optic

mal portfolios composed of # assets, To ensure that the portfolio on the frontiertather than underit, each portfolio mix must fulfill the portfoliv optimality con-ditions discussed nearthe end of Section 2.2.

ation are by‘ttional context,

Franco Modigliani and Gerald Pogue (1974a, b] and, in an interBruno Solnik [1974].

. Richard A, Brealey and Stewart C. Myers {1988], p. 136, takenfrom Roger tbbot-son and Rex Sinquefield [1986}.

. Studies examining the historical stability of ¢ include Marshall E. Blume [1971];Robert A. Levy [1971]; William F, Sharpe and Guy M. Cooper (1972); and Fischer Black, MichaelC. Jensen, and Myron Scholes [1972].

Forfurtherdiscussion, see Burton Malkict (1985), Chapter 9,Forone attemptto circumventthe ex ante, ex post problem. see Eugene F. Famaand James D. MacBeth [1973].

g model, see modernfinance text-bookssuchas Richard A. Brealey und Stewart C, Myers [1988], 163-164, orWilliam F. Sharpe [1985}, Chapter 8, pp. 182-201. The classic anticle on the arbi-truge pricing theoryis that by Stephen A. Ross [1976]. Em;

ofthe APM typically involve rather sophisticated econometric techniques, inctud-ing various types offactor models. Some examples ofAPM empirical implemen-tations include Richard Roll and Stephen A. Ross (1980), Edwin Burmeister andKent D, Wall [1986]; Nai-Fu Chen, Richard Roll, and Stephen A. Ross [1986];

and Marjorie B. McElroy and Edwin Burmeister [1988a, b].Most modernfinance texts have discussions on proceduresforestimating project-specific betas and incorporating them properly into cost of capital calculations.

56 ‘The Capital Asset Pricing Modet

See, for example, Richard A, Brealey and Stewart C, Myers [1988], Chapter9, pp.173-203, An analytical framework for envisaging such project-specific betas hasbeen developed by William F, Sharpe [1977],

20, The Chowtest is described in moststandard econometrics textbooks, The originalreference is Gregory C. Chow [1960]: un expository treatment is provided inFranklin M. Fisher [1970], und a further discussionis found in Damodar N, Guja-

rati (1970).Forfurtherdiscussion ontheeffects ofthe Three Mite [sland accidentonsecuritiesprices of GPU and otherelectric utility companies, see Curl R. Chen [1984] andThomasJ. Lastavie [1981]; for an analysis of the effects of the Chernobyl nuclearpoweraccident on U.S. electric utility security prices, see Juanita M. Haydel(1988).

. For a description and analysis of the Conoco takeover, see Richard $. Ruback[1982}.

23, Sec, for example, Richard H. Thaler [1987a); Philip Brown, Allan W. Kteidon,and Terry A. Marsh (1983); Donald B. Keim (1983); Josef Lakonishok und Sey-mour Smidt (1984); and Richard Roll [1983]. incidentally, empirical evidence husalso been reported supporting the notion of Monday, end of weekend, holiday,and even intradayeffects; see, for example, Kenneth R. French {1980] and, foradditional discussion and references, Richard Thaler [1987]. Finally, the May/June 1978 (Vol, 6, No. 2/3) issue of the Journal ofFinancial Econumics is devoted

entirely to articles discussing u variety of anomalous returns.24, However, see Burton G, Malkicl [1985], pp. 179-180,for a discussion ofthe role

of relatively large transactionscosts for securitics of small cumpunies in reducingpossible intertemporal arbitrage.

25, Far references on the Chowtest, see footnote 20,26. Forreferences, see footnote 18.27, The Durbin-Watson test statistic and the Hildreth-i.u and Cochrane-Orcutt

mution procedures are described in considerable detuil in most standard econo-metrics textbooks, Your instructor can provide you with further details,

28, Most graduate econametrics textbooks now include discussionsofestimation andinference in models with moving average disturbances. A useful and readable spe-

cialized reference text on timeseries estimation and inference is Charles Nelson(973).

29, Chi-square goodness-of-fit tests are described in muststatistics textbuoks, is theKalmogoroy-Smirnovtest; in the contest ofeconometrics textbooksa brit ewoftests for normatity is found in Judge et al. [1985], pp, 826-827, Analternativetest procedure is the studentized range test procedure, Por an application in theCAPMcontext andfor tables, see Eugene F, Fama [1976], especially pp. 8 [| andTable 1.9, p. 40.

2h.

we 8

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Weekend, Holiday. Turn of the Month, and Intraday Effects.” Josrad! of Eco:

nontic Perspectives. Pull, 169-177,mes M. (1958), “Liquidity Preference as Behavior toward Risk.” Review of

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ince Matrix Esti-

May, 817-

FURTHER READINGS

Cragg, John Gi. and Burton G, Malkiel [1982], Expectations and theStructureof Share

Prices, Chicago: University of Chicago Press. A rigarous empirical analysis ofmarket returns.

Fama, 1976]. Foundations ofFinance, New York: Basic Bouks. Especially

Chapters 1-4. A readable and usefulclassic text.

Huang, Chi-fu und Robert H. nberger [1988], Mondarions for Finanvial Boo

nomics, Amsterdam: North-Holland. Especially Chapter 10, “Econometric Issuesin Testing the Capita! Asset Pricing Model.” Containsa usefisl summary of recent

researchissuics,

Malkiel, Burton G.[1985], 4 RandamWatk down Wall Street, NewYork: WW. Nor

ton, Especially Chapters § and 9, Enjoyable and stimulating reading.

Markowitz, Harry M. [1959], Poryfolie Selection: Efficient Diversification of Invest-ments, New York: John Wiley & Sons. A classic treatmentofportfolio theory.

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