4299haim Levi

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HAIM LEVY

Hebrew university and

Center of Law and Business

September 2008

THE CAPM: ALIVE AND WELL?

A REVIEW AND SYNTHSIS

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ABSTRACTMeanMean--Variance analysis and the CAPM are two pillarsVariance analysis and the CAPM are two pillarsof modern finance. Yet, these two models are stronglyof modern finance. Yet, these two models are strongly

criticized on theoretical and empirical grounds.criticized on theoretical and empirical grounds.Theoretically, it is claimed that expected utility isTheoretically, it is claimed that expected utility isinvalid, some of the other assumptions whichinvalid, some of the other assumptions whichunderline these models are invalid, and that the Meanunderline these models are invalid, and that the Mean--Variance criterion may lead to paradoxical choices.Variance criterion may lead to paradoxical choices.These models are criticized empirically because theThese models are criticized empirically because thedistributions of rates of return are far from beingdistributions of rates of return are far from beingNormal and the CAPM has only negligible explanatoryNormal and the CAPM has only negligible explanatorypower. We show in this paper that the Mpower. We show in this paper that the M--V and theV and the

CAPM survive the theoretical criticisms, though theCAPM survive the theoretical criticisms, though theexpected utility model does not. Also, it is shown thatexpected utility model does not. Also, it is shown thatdespite the negative empirical results, with exdespite the negative empirical results, with ex--anteanteparameters the CAPM can not be rejected.parameters the CAPM can not be rejected.Furthermore, experimental studies which use exFurthermore, experimental studies which use ex--anteanteparameters strongly support the CAPM.parameters strongly support the CAPM.

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The Main Theoretical Criticism

1. Allais (1953)

EU2. Roy (1952) EU

3. Risk Aversion? (F&S, 1948),Markowitz (1952), Swalm (1966),K&T (1979), (1992)

4. Baumol (1963), Leshno & Levi(2002) Paradoxes of M-V rule.

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The Main Theoretical Criticisms of EUT,hence of the CAPM

1. Allais (1953) criticizes EUT. He shows

that using EUT in making choicesbetween pairs of alternatives, particularlywhen small probabilities are involved,may lead to some paradoxes within EUTtheory. Hence, it casts doubt on the

validity of EUT which is the foundation ofthe M-V rule and CAPM. This paradoxmotivated the idea of using decision

weights, see below.

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2. Roy (1952) also criticizes EUT. He asserts

that,"A man who seeks advice about his actions willnot be grateful for the suggestion that he

maximizes expected utility" (see Roy, 1952 p.433).

He suggests that people should rely on Safety

First (SF) rule rather than EUT. If one acceptRoy's claim, EUT is generally invalid, hencealso the M-V and the CAPM is not intact

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3. Even if EUT is intact some fundamental papers

question the validity of the risk aversionassumption. Just to mention a few of thesestudies, Friedman and Savage (1948),

Markowitz (1952b), Swalm (1966), Levy (1969)and Kahneman and Tversky (1979) claim thatthe typical preference must include risk-averse

as well as risk-seeking segments. Thus, thevariance can not be an index for risk, whichcasts doubt on the validity of the CAPM.

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4. Prospect Theory (PT) of Kahneman and

Tversky (1979) and its modified version,Cumulative Prospect Theory (CPT) of Tverskyand Kahneman (1992) show that subjects

behave in contradiction to what is predicted byEUT, hence they reject EUT which, onceagain, indirectly casts doubt on the validity of

the M-V analysis and the CAPM. It is worthnoting that the CPT's criticism of EUT is quietgeneral and has various dimensions, beyond

the criticism of the shape of the preferencementioned in point 3) above.

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5. Baumol (1963), Leshno and Levy (2002) and

Levy, Leshno and Leibowitz (2008) claim thatthe M-V rule is a sufficient but not a necessaryinvestment decision rule, hence it is not an

optimal rule, leading to an elimination of aportion ,or portions, of the M-V efficient frontierfrom the efficient set. Therefore, the market

portfolio may be also eliminated from theefficient set, which has an ambiguous effect onthe CAPM.

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The Main Criticisms of the M-V and CAPM

1. Normal Distributions?2. CAPM Test: F&F (1992)

3. Negative Weights in the Efficient

Portfolios

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1. The M-V criterion and the CAPM rely on the Normal

distribution assumption. Numerous studies test thegoodness of fit of actual rates of return distributions to

the Normal distribution. Almost in all cases the null

hypothesis asserting that the distribution of rates ofreturn is Normal is strongly rejected, hence one of the

main justifications of the M-V analysis and the CAPM

loses ground.

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2. Testing the CAPM directly reveals only little support for

the expected linear risk-return relationship, and insome cases it reveals a strong rejection of the CAPM,

when beta reveals almost no explanatory power of the

variation in mean returns. Numerous studies revealthis result, where the most famous and well cited

paper falling in this category is the one by Fama and

French (1992). For an excellent summary of the

empirical results, see Fama and French (2004).

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3. Deriving the M-V efficient set, it is generally found that some of

the investment weights of the tangency portfolio are negative.Moreover, as the number of assets increases, it is shown

empirically that the percentage of assets corresponding to the

tangency portfolio with negative weights approach 50%. These

findings contradict the CAPM, as to guarantee equilibrium theinvestment weights of the tangency portfolio must be all positive.

In addition, if most investors in practice choose mainly a portfolio

with positive weights, it implies that they do not choose by the M-

V rule, as selecting an optimal portfolio by the M-V rule yieldsmany negative investment weights. Therefore, the existence

negative weights imply that, in practice, investments are not

selected by the M-V rule; hence the CAPM is not valid.

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Allais ParadoxAllais paradox may be solved once one employs

decision weights (DW), rather than objective

probabilities (see Kahneman and Tversky (1979)). As

can be seen later on in the paper, when Normal

distributions is assumed and monotonic DW are

employed, as suggested by Quiggin (1982) andTversky and Kahneman (1992), all investors select

their portfolios from those portfolios located on the

Capital Market Line (CML), hence the CAPM is intact.Thus, one can use, on the one hand, DW to solve the

Allais paradox, and on the other hand, with DW as

suggested by CPT, the CAPM is valid.

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Roy's Safety First Rule

)( dxPMin rp

< )1(

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Roy's Safety First RuleUsing Chebyceff's inequality Roy shows that the following holds,

(2)

Of course, this inequality is meaningful only if k is larger than 1.

Choosing k=

This can be rewritten as,

(3)This implies a fortiorithat,

Hence,(4)

Thus the goal according to Roy is to minimize the ratio, , or

alternatively to maximize the ratio,

(4a)

{ } 2r k1kxP >

( ) d

( ){ } ( )2

2r ddxP >

( ) ( ){ } ( )22r ddxP >

( ) ( )22r ddxP < ( )d

( ) d

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d 1

d 2

Standard Deviation

r

m

Mean

m1

m2

pp1

*

p1

Figure 1

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U(x)

d x

How is the severe criticism of Roy on EUT reconciledwith the fact that the CAPM which is derived in the EUTframework remains is intact?

Figure 2

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U(x)

d x

Figure 3

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To summarize Roy's criticism we have thefollowing possible cases:

Case a: Here it is assumed that the riskless assetprevails and that for all investors that di is smaller thanr. Also, it is assumed that risky assets do notdominate by FSD the riskless asset, and that there isno requirement to invest some portion of the wealth inthe risky assets. This case is unacceptable because

by Roy's rule, in equilibrium all risky assets vanishfrom the market in contradiction to the observed facts.We rule out this case, as also Roy does not claim thatin equilibrium there will be no risky assets.

Case b: In this case the riskless asset exists andthe CAPM is intact as long as di< r and there is aconstraint that some portion of the investment must beallocated to the risky assets.

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To summarize Roy's criticism we have thefollowing possible cases:

Case c: The riskless asset is not available,a case where the Zero Beta CAPM holds.

Case d: In our view this is the most

relevant case. The riskless asset may prevailand if it prevails it may be smaller or greater(or equal) than di. There is no constraint on

investing in the risky or riskless asset. Thepreference is given by Figure 3, when both,SF and monotonicity are accounted for

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Overcoming the risk - aversion assumption

Theorem 1: Let F and G denotes twoNormal distributions, then F dominatesG by FSD iff,

a)

and (5)b)

( ) ( )xExE GF >

( ) ( )xx GF =

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Figure 4a

Standard Deviation

r

m

F

G

Meanr '

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Figure 4b

Return

F

G

CumulativeDistributions

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Cumulative Prospect TheoryAs T&K modified their Prospect Theory and suggestCumulative Prospect Theory (CPT), we focus here

only on CPT. By CPT the investor maximizes a valuefunction of the form,

(6)

(7)

( )

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Overcoming Baumol's and Almost M-V criticisms

21Standard Deviation,

505Mean,

Prospect FProspect G

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Overcoming Baumol's and Almost M-V criticisms

While Baumol's rule is based solely on an intuition, in arecent study Leshno and Levy (Leshno and Levy (2002)suggest new rules, called Almost Stochastic Dominance

(ASD) and Almost M-V(AMV) denoted by SD* and M-V*rules, respectively. Also, Baumol published his page beforethe CAPM was published hence he analyzed theimplication to the M-V efficient set while L&L analyzed alsoto implication to the CAPM. L&L show that there are caseswhere neither F nor G dominates the other yet, theysuspect that in practice such dominance exists. To see this

consider the following two Normal distributions:

( ) ( )

( ) ( )2,10N,N~F

1,1N,N~G

=

=

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Figure 5a

Return

F

G

CumulativeDistributions

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Figure 5b

Standard Deviation

r

U2

F

G

Mean

U1

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Baumol suggets the following investmentrule instead of the M-V rule:

F dominates G iff,

a)and (8)

b)where k is larger than 1.

( ) ( )xExE GF

( ) ( ) ( ) ( ) GGGFFF kxExLkxExL ==

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Figure 6a

Standard Deviation

r

a

Mean

m

b

c

r'

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Figure 6b

Standard Deviation

r a

Mean

m

c

r'

b

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(9)

(10)

p

m

mp

rr

+=

constant=

m

m r

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Figure 7Mean

Standard Deviation

r

m

b

a

r' r''

m'

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Overcomingthe empirical

Criticism of the

CAPM

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The Normality assumption

It is well known that Normality (or an Ellipticdistribution) is very crucial to the derivation ofthe CAPM. We also assume Normality in

showing the validity of the CAPM in variousscenarios. Numerous studies examine theNormality hypothesis with a clear cut result:the Normality distribution is statisticallystrongly rejected

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The Normality assumption The M-V rule is justified as an approximation

to expected utility even when distributions arenot Normal on two grounds:

i) Levy and Markowitz (1979) have shownempirically that the M-V rule is an excellent

approximation to expected utility (see alsoKroll, Levy and Markowitz (1984) andMarkowitz (1991)).

The utility loss induced by adopting thisapproach has been found empirically to be

negligible.

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The Normality assumptionii) Other studies estimate directly the financial loss

rather than the utility loss due to the assumption of

Normality, when the empirical rates of return actuallyare not distributed Normaly.

For example, Duchin and Levy who employ themyopic preference find that the loss per 10,000 dollarsinvestment is merely $2-$6, depending on the degreeof the relative risk aversion parameter. To put thingsin perspective, suppose that the planned investmenthorizon is one year. The mean rate of return on riskyassets is about 12% (see, Ibbotson 2007). Then,loosing $2-$6 per $10,000 investment implies that themean rate of return drops, on average, from 12% to11.94%-11.98%, a negligible loss.

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The empirical tests of the CAPMFrom these studies one is tempted to conclude that

the CAPM is, at least empirically, invalid, whichallegedly drastically reduces its value. We claim belowthat this is not the case: though the CAPM is rejectedwith expost parameters it can not be rejected on

empirical grounds with ex-ante parameters, inparticular with ex-ante beta or some components ofthis beta, quite a strong statement. Recall that theCAPM is stated by Sharpe and Lintner with ex-ante

and not ex-post parameters.

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The empirical tests of the CAPMIn the early tests of the CAPM the firstpass

and secondpass regressions where definedas follows:

First Pass: (11)

Second Pass: (12)

itmtiiit eRR ++=

ii10iR ++=

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We turn now to the difference betweenex-post and ex-ante parameters. Thereare few approaches to incorporate these

differences. Generally, it can be shownthat accounting for even small possibledifferences between ex-post and ex-ante

parameters the CAPM can not berejected. Let us elaborate.

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1) Exante betaIn testing the CAPM (see eq. (12)), it is assumed that betaestimated by eq.(11) is the correct ex-ante beta. We claim

that taking into account possible difference between ex-postand ex-ante beta the CAPM can not be rejected. Indeed,Levy (1983), test the CAPM when such differences aretaken into account. He shows that regardless of thepossible measurement errors involved in the measurementof rates of returns, the CAPM can not be rejected on ex-ante basis.

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2) The efficiency of the market portfolio

Taking into account the possible difference

between ex-post and ex-ante parameters, in arecent paper, Levy & Roll (2008) show that whenonly small changes in the sample means andstandard deviations are done, the observed marketportfolio is M-V efficient, which according to Roll(1977) implies that the linear CAPM is intact. Theyemploy a novel "reverse engineering" approach.

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"Surprisingly, slight"Surprisingly, slightvariations of the samplevariations of the sample

parameters, well within theparameters, well within theestimation error bounds,estimation error bounds,suffice to make the proxysuffice to make the proxy

efficient. Thus, manyefficient. Thus, manyconventional market proxiesconventional market proxies

could be perfectlycould be perfectlyconsistent with the CAPM".consistent with the CAPM".

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3) Negative investment weightsUsing historical rates of return to derive the M-V efficient setit is generally found that some of the weights are negative.

Of course with a small number of assets, e.g. 3-5 assets itis possible to get that all weights are positive. However, inthe relevant case for testing the CAPM, when hundreds ifnot thousands of assets should be incorporated, negativeinvestment weights always exists. Moreover, thepercentage of negative weights becomes close to 50% ofthe assets included in the study when the number of assets

increases. And one does not need to have astronomy largenumber of assets to obtain this result: Levy (1983) showsthat even with 15 assets the percentage of negative weightsis about 50%

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"We show that the probability of obtaining a"We show that the probability of obtaining apositive tangency portfolio based on samplepositive tangency portfolio based on sample

parameters converge to zero exponentially withparameters converge to zero exponentially withthe number of assets. However, at the same time,the number of assets. However, at the same time,very small adjustments in the return parameters,very small adjustments in the return parameters,well within the estimation error, yield a positivewell within the estimation error, yield a positivetangency portfolio. Hence looking for positivetangency portfolio. Hence looking for positiveportfolios in parameter space is somewhat likeportfolios in parameter space is somewhat like

looking for rational numbers on the number line: iflooking for rational numbers on the number line: ifa point in the parameter space is chosen ata point in the parameter space is chosen at

random it almost surely corresponds to nonrandom it almost surely corresponds to non--positive portfolio (an irrational number); however,positive portfolio (an irrational number); however,

one can find very close points in parameter spaceone can find very close points in parameter spacecorresponding to positive portfolios (rationalcorresponding to positive portfolios (rationalnumbers)"numbers)"

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4) Experimental studies which use exanteparameters

Using this technique Levy (1997) find a

strong support for the CAPM with more than70% explanatory power. Levy concludes

"..mean return and risk are strongly"..mean return and risk are stronglypositively related when thesepositively related when these

parameters are determined on an exparameters are determined on an ex--

ante basis , as claimed by Sharpeante basis , as claimed by Sharpe--LintnerLintnermodel"model"(see p.145).

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4) Experimental studies which use exante

parameters

Bossaerts and Plott ( 2002 ), also find an

experimental support for the CAPM .In theirwords

"when interpreted as the equilibrium to"when interpreted as the equilibrium towhich a complex financial marketwhich a complex financial market

system has a tendency to move, thesystem has a tendency to move, the

CAPM received support in theCAPM received support in theexperiments reported here"experiments reported here"

(see p. 1110).

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SUMMARYa) Theoretical Criticisms of M-V and CAPM

The Criticism

Allais paradox

Roy SF

Risk-Seeking(F&S, Swalm, K&T)

Prosoect Theory

Baumol and Leshno & Levi

M-V leads to paradoxes in

choices

Solution

DW+FSD

Zero Beta Model

Constraint on Investment

A SF Preference With Monotonicity

No need for Risk Aversion FSD

FSD: All will choose portfolios

from M-V efficient frontier with

CPT (Using FSD)

M-V efficient frontier is modified

but the CAPM is intact

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b) Empirical Criticisms

The Criticism

The Normality is Rejected

Direct CAPM Tests

(F&F 1992,2004)

Negative Weights of theTangency Portfolio (1983)

Solution

Approximation with a small utility

log and small Llinear log (2$ per

10,000$)

With the use Ex-Ante Data on beta

(Levy) or other parameters (Levi &

Roll), one can not reject the CAPM

With A Small change in [Ex-Ante]

a XXX portfolio is found

Experimental Test With ExExperimental Test With Ex--AnteAnteParameters Show That TheParameters Show That The

CAPM is Strongly SupportedCAPM is Strongly Supported

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