Samizdat

In Defense of the CAPM

Jack L. Treynor, TreynorCapital Management

Many textbooks repeat two com-mon complaints about the CapitalAsset Pricing Model (CAPM):

1. Evidence that it takes morethan one factor to explainthe shared, or systematic,risk in securities refutes theCAPM.

2. In demonstrating that therisk premium on an assetdepends only on its system-atic factor loadings, Arbi-trage Pricing Theory (APT)provides investors with a re-sult of great practical valuethat the CAPM does not pro-vide.

It may not be coincidental thatsome of the same books thatmake the first complaint don'tactually discuss the CAPM. Somediscuss the market model, andsome discuss (usually without at-tribution) the Vasicek-McQuownpricing model. Both build on Wil-liam Sharpe's Diagonal Model pa-per, which suggested (followingup on a footnote in HarryMarkowitz's book) that systematicrisk could usefully bt" accountedfor by a single factor."

Vasicek and McQuown's argu-ment proceeds as follows. As-sume a single systematic risk fac-tor exists, such that residual riskscan be uncorrelated both withthis factor and with each other. Bytaking appropriately long posi-tions in some securities and ap-propriately short positions in oth-ers, the individual investor canhedge away all systttmatic risk.

* This material will appear as an appendix lothe second edition of the Guide lo PortfolioManagement by James L. Farrell Jr., to bepublished by McGraw-Hill Book Company.

Thus he won't bear systematicrisk unless it competes success-fully with residual risk for inclu-sion in his ponfolio. But if heincludes enough positions (longor short) in his portfolio, he candrive the ponfolio's residual riskto zero, Now, if residual risk ispriced, this offers him an infinitereward-to-risk ratio. In order formarket risk to compete success-fully, it must offer an infinite ex-pected return. Otherwise the in-vestor will choose not to holdany systematic risk, and the mar-ket won't clear.

The market model takes the as-sumptions of the Diagonal Modeland the result of the Vasicek andMcQuown model and concludesthat the actual, ex post systematicreturn—surprise plus risk premi-um—of every asset will be pro-portional to the asset's marketsensitivity. Both Vasicek and Mc-Quown and the market modelthus assume a single systematicfactor; but the CAPM in its variousforms (Sharpe, Treynor, Lintner,Mossin, Black) makes no assump-tion about factor structure. Inparticular, it does not make theone-factor. Diagonal Model as-sumption that Vasicek and Mc-Quown and the market modelmake.

Actually, a CAPM that abandonsthe assumptions of homogeneousexpectations (as Lintner did) andriskless borrowing and lending(as Black did) doesn't make manyother assumptions, All it does as-sume is that asset risk can beadequately expressed in terms ofvariances and covariances (withother assets); that these measuresexist; that investors agree onthese risk measures; and that in-vestors can trade costlessly to in-crease expected portfolio returnor reduce ponfolio variance, to

which they are all averse. Short-selling is permitted.

There is thus nothing about factorstructure in the CAl'M's assump-tions. But there is also nothingabout factor structure in theCAPM's conclusions. Factor struc-ture is about surprise—in partic-ular, the nature of correlationsbetween surprises in different as-sets. In the Diagonal Model, asingle factor accounts for all cor-relation. In richer, more complexfactor structures, correlations inasset surprises can be due to avariety of pervasive, market-wideinfluences. The conclusion oftheCAPM—that an asset's expectedreturn is proportional to its cova-riance with the market portfo-lio—makes no assertions aboutthe surprise in that asset's return.Because expected return and sur-prise are mutually exclusive ele-ments in ex post, actual return,there can be no conflict betweenthe CAPM and factor structure.

This point is the key to the secondcomplaint. In order to demon-strate that APT's principal conclu-sion—that an asset's risk pre-mium depends only on itssystematic factor loadings—alsoholds for the CAPM, we can as-sume a perfectly general factorstructure u'ithout fear of trippingon either the assumptions of theCAPM or its conclusions.

Consider a market in which abso-lute surprise for the ith asset, x,,obeys:

Eq. 1

+where Uj are the systematic fac-tors in the market and ê is aresidual unique to the ith asset.

5

-z.

o

1/1

<

<I

<

11

The absolute surprise for themarket as a whole, x, is simply thesum of absolute surprises for theindividual assets:

X = +The CAPM asserts that any riskpremium of the ith asset dependsonly on its covariance with themarket—i.e., on the expectationof the following product:

Eq. 2

Bearing in mind that the u's ande's are surprise, and assumingthat all covariances between e'sand between u's and e's are zero,we have for this expectation:

Eq. 3

or

Eq. 4

where

Eq. 5

o-jk^ =

and

Eq. 6

We see that an asset's covariancewith the market portfolio (abso-lute surprise with absolute sur-prise) has two terms—one forsystematic factors and one for theasset's unique surprise. A key is-sue is the relative size of thesetwo terms.

Figure A Systematic vs. Unique Risk

Same Order of Magnitude

I \

rTOrders of Magnitude Bigger

atic part zero by choosing a set offactors that are orthogonal. Thenthe covariance expression be-comes:

Eq. 7

+ . . . + Sî .

Among the many possible choicesof orthogonal factors, we canchoose one in which only onefactor correlates with the marketportfolio, Assign the factor indexto the factors so that the first termin the bracketed expression is thevariance term for that factor. Thenwe can ignore for the momentthe other systematic terms andfocus on the first and last terms inthe covariance expression. Equa-tion (7). Factoring the respectiveterms into standard deviationsgives us:

Eq. 8

We can, of course, make all the12 covariance terms in the system-

Consider the first term. One fac-tor reflects the absolute factorweight of the asset, the other theabsolute factor weight of the mar-ket portfolio. The former is of thesame order of magnitude as thestandard deviation of the asset'sunique risk; the latter is orders ofmagnitude larger (see Figure A).This is because all the systematicrisk in the market ponfolio's ab-solute gain or loss is refiected inthe market's factor weight,whereas the unique risk in thesecond term is that for a singleasset. But this means that the sys-tematic product is orders of mag-nitude larger than the uniqueproduct.

To keep our exposition simple,we chose a special set of system-atic factors. Had we chosen differ-ent systematic factors, the marketportfolio could have been corre-lated with most or all of them,rather than merely one. Then thesystematic portion of an asset'scovariance with the market wouldhave non-zero terms for morethan one factor. Spreading itacross several factors, however,won't diminish the magnitude ofthe aggregate systematic portion.Whether one systematic factor orseveral are correlated with themarket portfolio, then, the sys-tematic portion of an asset's cova-riance with the market will beorders of magnitude bigger thanthe unique portion.

If we drop the unique term andrelax the assumption that onlyone systematic factor correlateswith the market, then the expres-sion within brackets in Equation(4) will in general have a differentvalue for each of the systematicfactors. But the value doesn't de-pend on i—the index that distin-guishes between individual as-sets. What is outside the bracketsis the factor loadings ^jj for the ithindividual asset on the factors u,in the factor structure. But this isthe principal conclusion of APT—that an asset's risk premiumshould depend only on its factorloadings. Any corollaries that fiowfrom this APT result also fiowfrom the CAPM.

One of the differences betweenAPT and the CAPM is the specifi-cation by CAPM of how systematicfactors should be priced in rela-tion to each other. The expres-sion in brackets in the first termof Equation (4) is, to some con-stant of proportionality, the priceof risk. The value of the expres-sion, hence that price, is generallygoing to be different for differentfactors. APT can't specify the valueof this expression; therefore itcan't specify how factors will bepriced in relation to each other.

What the APT does do is weakenthe assumptions necessary to

reach this conclusion. APT ap-peals to the older, better estab-lished tradition of arbitrage argu-ments exemplified by Modiglianiand Miller's 1958 paper. APT joinsthe idea of specifying security riskin terms of a factor structure (Far-rars I960 doctoral thesis at Har-vard; Hester and Feeney's con-

temporaneous work at Yale) withModigliani and Miller's arbitrageargument. Some may argue that,in effect, APT substitutes system-atic risk factors for Modiglianiand Miller's risk classes. Othersmay argue that the real prove-nance of APT is not Modiglianiand Miller, but rather Vasicek and

McQuown, with multiple risk fac-tors substituted for their singlerisk factor.

This writer has no desire to take aposition in such controversies.His only purpose is to clear upcertain misunderstandings re-garding the CAPM.

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