the capm is alive and well - a review and synthesis

8/9/2019 The CAPM is Alive and Well - A Review and Synthesis

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European Financial Management, Vol. 16, No. 1, 2010, 4371doi: 10.1111/j.1468-036X.2009.00530.x

The CAPM is Alive and Well: A Review

and Synthesis

Haim LevyThe Hebrew University and The Center of Law and Business

E-mail: [email protected]

Abstract

Mean-Variance (M-V) analysis and the CAPM are derived in the expected utilityframework. Behavioural Economists and Psychologists (BE&P) advocate thatexpected utility is invalid, suggesting Prospect Theory as a substitute paradigm.

Moreover, they show that the M-V rule, which is the foundation of the CAPM,is not always consistent with peoples choices. Thus, BE&P cast doubt on thevalidity of expected utility paradigm and of the M-V rule, hence the CAPM istheoretically questionable. In addition, there is very little empirical support tothe CAPM. We show in this study that the CAPM is theoretically valid evenwhen one accepts the BE&P framework and even when expected utility is invalid.

Moreover, within the BE&P framework there is a strong experimental support for

the CAPM.

Keywords: CAPM,M-V,expected utility,Prospect Theory

JEL classification: D81,C91

1. Introduction

Since the emergence of modern finance, only a few papers, which are highly cited andconsidered to be the pillars of the profession, have been published. There is no doubt that

Markowitzs (1952a) Mean-Variance (M-V) efficiency analysis and Sharpe (1964) andLintners (1965) Capital Asset Pricing Model (CAPM) are two such pillars. The CAPMis the base model on which hundreds of academic studies rely. The impact of the CAPMis not confined to academic research. Practitioners also frequently use the Sharpe Ratio,which relies on the M-V model and use beta, which is derived from the CAPM as therisk index. The list of studies which rely on the M-V analysis and the CAPM is toolong to mention here. However, it is sufficient to open any finance textbook to see thegreat impact of the M-V rule and the CAPM on the profession. Indeed, in recognitionof these important contributions to science, Markowitz and Sharpe won the Nobel Prizein economics in 1990.

The author acknowledges the helpful comments of Moshe Levy Frank Fabozzi and Harry


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44 Haim Levy

Yet, the CAPM is under severe and ongoing theoretical and empirical academicattacks. Most of the early attacks emerge from numerous empirical studies in financeand economics, which reveal that the CAPM does not fit empirical asset pricing well(for an excellent survey of the empirical results, see Fama and French, 2004) However,the more severe attacks come from an unexpected angle: behavioural economists andpsychologists cast doubt on the validity of von Neumann and Morgensterns (1953)Expected Utility Theory (EUT); thereby also casting indirect doubt on the validityof the M-V rule and the CAPM, which are derived within the EUT framework (thiscriticism, however, does not apply to the APT derivation of the CAPM by Ross, (1976),which does not rely on expected utility).

In a breakthrough series of studies, Kahneman and Tversky (K&T) and Tverskyand Kahneman (T&K) (see for example, 1979, 1992) show that the typical investor isnot always a rational and efficient machine, as assumed by EUT economists. Thisimplies that the foundations of EUT are not valid, further implying that all models -including the M-V efficiency analysis and the CAPM, derived in the EUT framework -

are questionable. Kahneman and Tversky first suggested Prospect Theory (PT) and, lateron, a modified version of PT, called Cumulative Prospect Theory (CPT), as substitutesfor EUT. The influence of PT and CPT on economics and finance has been enormous.Indeed, in recognition of these contributions to academic research, Kahneman won theNobel Prize in economics in 2002. Thus, we have two conflicting paradigms - EUT andCPT - and if CPT is correct and EUT is rejected, the M-V efficiency analysis and theCAPM have no theoretical justification. As a result, these models seem to lose ground.

Nevertheless, despite these criticisms, in the current paper we show that M-V analysisand the CAPM are, in fact, alive and doing well. We do not claim that the other alternatetheories are wrong. Rather, we attempt to show that a modified version of M-V analysis

and the classic CAPM can also be justified and safely used in the CPT framework,despite the fact that under CPT, EUT is rejected. In this paper we focus on the theoreticalcriticisms of the M-V and the CAPM, but we also briefly discuss the implications ofthe empirical and experimental results to the validity of the CAPM. We show that theCAPM cannot be rejected as was previously thought, as long as ex ante rather thanex postparameters are employed in the CAPM tests, or when ex ante experimentalparameters are employed.

The structure of this paper is as follows: In Section 2, we discuss the commonassumptions needed to justify the M-V rule and the CAPM within the EUT framework,and provide a brief review of the CAPM and M-V criticisms. In Section 3, we provide

the detailed theoretical criticisms of the CAPM, and show that the CAPM easilywithstands the barrage of theoretical criticisms; it is valid even in the non-expectedutility framework. Namely, EUT is criticised (and rightfully so); however, the CAPMremains intact - quite a surprising result. In Section 4, we briefly discuss the empiricalcriticisms of M-V and the CAPM and show that the CAPM cannot be empiricallyrejected. Moreover, we show that experimental studies which use ex ante parametersstrongly support the CAPM. Section 5 concludes the paper.

2. The Main Criticisms of M-V Analysis and the CAPM

To understand the criticisms of the CAPM and M-V efficiency analysis in a transparentway let us first be more specific regarding the role of Normality and the shape of


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The CAPM is Alive and Well: A Review and Synthesis 45

is generally common to require that investors maximise expected utility with a quadraticpreference, or alternatively, to require that the distributions of rates of return are Normaland risk aversion prevails.1 With some additional assumptions (e.g. homogeneousexpectations, no transaction costs, unrestricted borrowing, etc.), one can derive thelinear risk-return relationship, as implied by the CAPM. As the quadratic preference istoo specific and since it also suffers from other severe drawbacks (see Arrow, 1965), wefocus here on the case of Normality and risk aversion. After defining the assumptionsneeded to justify the CAPM, let us now turn to the criticisms of this model.

2.1 Theoretical criticisms of EUT and CAPM

The main theoretical criticisms are as follows:

Allais (1953) criticises EUT. He shows that using EUT in making choices between

pairs of alternatives, particularly when small probabilities are involved, may lead tosome paradoxes within EUT theory. Thus, it casts doubt on the validity of EUT, whichis the foundation of both the M-V rule and CAPM. This paradox motivated the idea ofusing decision weights instead of objective probabilities (for more details, see below).

Roy (1952) also criticises EUT. He asserts that,

A man who seeks advice about his actions will not be grateful for the suggestionthat he maximises expected utility (see Roy, 1952, p. 433).

He suggests that people should rely on the Safety First (SF) rule, rather than on EUT.

If one accept Roys claim, EUT is generally invalid; hence, the M-V and the CAPMare also invalid.2

Even if EUT is valid, some fundamental papers question the validity of the risk-aversion assumption. Just to mention a few of these studies, Friedman and Savage(1948), Markowitz (1952b), Swalm (1966), Levy (1969) and Kahneman and Tversky(1979) all claim that the typical preference must include risk-averse, as well as risk-seeking, segments. Thus, the variance cannot be an index for risk, which casts doubton the validity of the CAPM.

Kahneman and Tverskys (1979) Prospect Theory (PT) and its modified version -Tversky and Kahnemans (1992) Cumulative Prospect Theory (CPT) - show that

subjects behave in contradiction to what is predicted by EUT; hence, they reject EUTwhich, once again, indirectly casts doubt on the validity of the M-V analysis and theCAPM. It is worth noting that the CPTs criticism of EUT is quiet general and hasvarious dimensions, beyond the criticism of the shape of the preference mentioned inPoint 3, above.

1 One can justify the M-V rule with the more general family of distributions, the Ellipticdistributions, which include - apart from the Normal distribution - other distributions, e.g.,the Logistic distribution, see Chamberlain (1983) and Berk (1997). However, we focus hereon the Normal distribution, and the results can be extended to the other distributions included

in the Elliptic family.2 Though Roy explicitly rejects EUT, there is one specific degenerate utility function whichconforms to the SF rule However this preference has a severe drawback and is thus


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46 Haim Levy

Providing some simple examples, Baumol (1963) and Leshno and Levy (2002) showthat there are cases where there is no M-V dominance of one prospect over the other,yet allegedly all subjects would prefer one of these prospect. Therefore they claimthat the M-V rule is a sufficientbut not anecessaryinvestment decision rule; thus, itis not anoptimalrule, leading to an elimination of a portion, or portions, of the M-Vefficient frontier from the efficient set. Therefore, the market portfolio may also beeliminated from the efficient set, which has an ambiguous effect on the CAPM.

2.2 Empirical criticisms of the M-V and the CAPM

The main empirical criticisms are as follows:

The M-V criterion and the CAPM rely on the Normal distribution assumption.Numerous studies test the goodness of fit of actual rates of return distributions to

the Normal distribution. In almost all cases, the null hypothesis asserting that thedistribution of rates of return is Normal is strongly rejected; hence, one of the mainjustifications of the M-V analysis and the CAPM loses ground.

Testing the CAPM directlyreveals only minimal support for the expected linear risk-return relationship; in some cases it reveals a strong rejection of the CAPM, whenbeta reveals almost no explanatory power of the variation in mean returns.

Deriving the M-V efficient set, it is generally found that some of the investmentweights of the tangency portfolio are negative. Moreover, as the number of assetsincreases, it is empirically shown that the percentage of assets corresponding to thetangency portfolio with negative weights approaches 50%.3 These findings contradict

the CAPM because, in order to guarantee equilibrium, the investment weights of thetangency portfolio must all be positive.4

Thus, the M-V criterion and the CAPM are theoretically and empirically attacked.One may argue that some or all of these criticisms are invalid. We choose a differentroute: we accept all these criticisms and assume that they are valid, and examine theireffect on the CAPM. Taking this approach, one would suspect that the M-V analysis andthe CAPM would not survive the above list of harsh criticisms. Surprisingly, however,this is not the case.

In this study, we show that the M-V analysis and the CAPM survive all these criticisms.

Indeed, the M-V efficiency analysis has to be slightly modified to incorporate some ofthe above theoretical criticisms, but the CAPM is found to be alive and doing well, evenafter these modifications. We conclude that unless we observe in the future new andmore convincing evidence refuting the M-V framework and the CAPM, we can safelycontinue to include these two fundamental models in our teaching curriculum, in futureacademic research, as well as in practice.

3 For studies which investigate the mathematical conditions needed to obtain positive weightsportfolios, and for the empirical results showing that negative weights always exist, see for

example, Roll (1977), Rudd (1977), Levy (1983), Green (1986) and Best and Grauer (1992).4 According to the CAPM, it is assumed that unrestricted short selling is allowed, which isan unrealistic assumption However the end result of the CAPM is that all investors hold


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3. Overcoming the Theoretical Criticisms of the CAPM

Let us examine each criticism separately:

3.1 The Allais paradoxThe Allais paradox may be solved once one employs decision weights (DW), rather thanobjective probabilities (see Kahneman and Tversky, 1979). As can be seen later on inthe paper, when Normal distributions are assumed and monotonic DW are employed,as suggested by Quiggin (1982) and Tversky and Kahneman (1992), all investors selecttheir portfolios from those portfolios located on the Capital Market Line (CML); hence,the CAPM is intact. Thus, on the one hand, one can use DW to solve the Allais paradox,while on the other hand, with DW as suggested by CPT, the CAPM is valid (for moredetails, see Points 3 and 4 discussed below). However, it is important to emphasise thatthe original Allais paradox is presented with discrete distributions, and hence, is not

directly relevant to our study, as we assume Normality. Yet, if a paradox like the onepresented by Allais can also be constructed with Normal distributions, then we suggestthat monotonic DW may solve the paradox without violating the CAPM, even whenEUT is rejected (see discussion at the end of Point 4 below).

3.2 Roys Safety First Rule

Roy advocates that the risk of disaster is the main factor one should consider in makingan investment choice; therefore, expected utility, in its classic form, should not be takenseriously. In his 1952 model, Roy goes to the extreme and sets the goal of the investor

solely as the minimisation of the probability of disaster. Thus, according to this extremeSafety First (SF) rule, one should select the diversification strategy, which minimisesthe probability of disaster. Namely, the goal is:

Minp

Pr(x


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48 Haim Levy

We would now like to discuss the claim that various versions of the CAPM are alsointact in Roys framework, regardless of whether the riskless asset prevails. To showthis, we consider various scenarios. As we shall see, all these interpretations, except forone (which is not accepted by economists, as it implies that risky assets do not exist inthe market), are consistent with the classic M-V analysis and with various forms of theCAPM.

Let us first take Roys rule strictly as a given by Eq. (1). Though Roy does not discussthe allocation of the investment between risky and riskless assets, we first assume (thisassumption will be relaxed later on in the paper) that the investor decides to invest someportion of her wealth in risky assets; the only question is how to diversify the allocatedamount to risky assets according to Roys rule among the available individual riskyassets.

Using Chebysheffs inequality, the following holds,

Pr{|x | >k} 1/k2 (2)

Of course, this inequality is meaningful only if k is larger than 1.Choosing k= ( d)/, Roy shows that Eq. (2) can be rewritten as,

Pr{|x | >( d)} 2/( d)2 (3)

This impliesa fortiorithat,

Pr{( x)> ( d)} 2/( d)2

Hence, the upper bound of the probability of disaster is given by,

Pr(x


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d1

d2

Standard Deviation

r

m

Mean

m1

m2

pp1

*

p1

Fig. 1. The optimum portfolios when the riskless asset does not prevail are denoted by m, m1and m2, corresponding to disaster levels d= r, d1 and d2, respectively. When the riskless

asset prevails, and there is a constraint that some proportion of the wealth must be invested inthe risky assets, the optimal portfolios, according to Roys rule, are given by Portfolio P

corresponding to m, and Portfolio P1 corresponding to m1(the portfolio corresponding to m2is omitted to provide a transparent picture). However, when the riskless asset prevails, having

a disaster level d1, the investor would benefit from shifting from m 1to m; hence, shiftingfrom p1 to, say, p

1 , the probability of disaster corresponding to d1 decreases. Therefore, with

the riskless asset, all investors will choose some combination of m and r, regardless of thedisaster level di.

Suppose now, as before, that some portion of the assets must be invested in riskyassets, that the riskless asset prevails, and that the disaster level, di, is not necessarilyequal to r. Moreover, di may vary across investors, but all di are smaller than r (see thevalues d1and d2in Figure 1).

7 In this case, we claim that the classic Separation Theoremholds true as all investors will select the same portfolio of risky assets, Portfolio m, i.e,the market portfolio, as advocated by the CAPM. To better understand this, first recallthat Portfolio m1 minimises the probability of disaster when d1 is the disaster level andthe riskless asset does not prevail. However, with the riskless asset, for any combinationof the riskless asset and Portfolio m1 located on line rm1, there is a combination of

Portfolio m and the riskless asset located on line rm, which dominates it in accordancewith Roys rule. For example, suppose that by combining m1 with r, Portfolio p1 isselected (see Figure 1). Then, by combining m with r, Portfolio p1 can be achieved andthis portfolio dominates Portfolio p1, as it has the same mean return and a lower risk ofa disaster, when d1is the disaster level. The reason for this is that the slope of lined1p

1

is higher than the slope of lined1p1. To sum up, when the investors goal is to minimisethe risk of disaster, as suggested by Roy, one of the versions of the CAPM holds true; theselected version depends on the relationship between r and d, and whether the risklessasset prevails.

7 The case where d is greater than r is not considered in this paper, as according to Roysrule, it leads to an infinite borrowing, which contradicts equilibrium. To better understandthis claim recall that with d > r moving to the right on the CML the slope ( d)/


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50 Haim Levy

U(x)

d x

Fig. 2. The only preference which is consistent with Roys strict Safety-First rule.

How can Roys severe criticism of EUT be reconciled with the fact that the CAPM,which is derived within the EUT framework, remains intact? To answer this questionone must analyse the preference implied by Roys rule. It can be shown that the onlyutility function which is consistent with Roys strict SF rule is presented in Figure 2.Thus, if r is greater than d, and if the investor is not constrained by having to investsome proportion of the wealth in the risky asset (and in addition, if the risky asset doesnot dominate the safe asset by First degree Stochastic Dominance (FSD), as occurs

with Normal distributions), then it is easy to see that with the preference presented inFigure 2, risky assets vanish from the market, since investing 100% in the riskless assetis optimal, resulting in a zero probability of disaster. This result is intact for any valued, as long as it is smaller than r. It is clear from the above analysis and from Roys paperthat Roy did not mean to employ this extreme strategy, as in such a case risky assetsvanish; and hence, all the mathematical M-V analyses conducted by him are redundant.

The above interpretation of the SF rule is, of course, unacceptable. As explainedabove, in order to receive acceptable results, one can alternatively assume that thereis no riskless asset, or that, as explained previously, the riskless asset exists, but thereis a constraint: some portion of the money must be allocated to the risky assets. In

this realistic scenario, Blacks Zero Beta CAPM is intact. However, with the risklessasset and with the above imposed constraint, the optimal investment policy will not beconsistent with a maximisation of expected utility, as dictated by Figure 2.

Alternatively, one can relax the assumption that some portion of the assets must beinvested in the risky assets and assume another utility function, which is still in linewith Roys approach, but also takes monotonicity into account. To be more specific,the preference presented in Figure 2 is unacceptable because it ignores the magnitudeof the outcomes. Namely, according to the strict SF rule, a utility of $10 or one milliondollars is the same, as long as these two values are greater than d. Indeed, in a laterpaper even Roy (1956), himself, suggests an extension of his SF rule to a more realistic

framework. He suggests that the disaster level d may be uncertain implying that theutility function is continuous.

How can one keep the main component of Roys SF rule in an EU framework without


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U(x)

d x

Fig. 3. The preference which is consistent with the modified SafetyFirst rule: There is apenalty when the outcomes are below the disaster level di, but the preference is monotonic.

or a million dollars? Indeed, in a recent paper, Levy and Levy (2008) suggest a synthesisof Roys rule with EUT; however, this time they suggest a preference which, on the onehand, in the spirit of Roy, penalises the potential investment if the return is below somedisaster level, d, and on the other hand, does not imply indifference to all outcomesabove d, and the larger the outcome the better off the investor is. This preference ispresented in Figure 3.

The suggested preference given in Figure 3 is neither continuous nor completelyconcave. Yet, as long as the distribution of returns is Normal, in order to obtain theCAPM there is no need to assume risk aversion; rather, it is sufficient to assume anon-decreasing preference (see Point 3 below). In addition, continuity of the preferenceis not needed (see Kroll and Levy, 1982). Thus, with this interpretation of the SF rule,this rule as well as the CAPM is valid in an expected utility framework, even when theconstraint that some portion of the wealth must be invested in the risky assets is relaxed(see Point 3 below).

To summarise Roys criticism then, we put forward the following possible cases:

Case A: In this case, it is assumed that the riskless asset prevails and that for all investors

di is smaller than r. Furthermore, it is assumed that no risky asset dominates theriskless asset by First degree Stochastic Dominance (FSD) and that there is norequirement to invest some portion of the wealth in the risky assets. This caseis unacceptable because, according to Roys rule, in equilibrium all risky assetsvanish from the market in contradiction to the observed facts. As such, we ruleout this case, also because Roy does not claim that in equilibrium there will beno risky assets.

Case B: In this case, the riskless asset exists and the CAPM is intact, as long as di


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52 Haim Levy

on investing in either the risky or riskless asset. The suggested preference ispresented in Figure 3, when both SF and monotonicity are accounted for. Thus,in this scenario, Figure 3, rather than Figure 2, is relevant for Roys analysis.

In the next section, we will show that with Normal distribution the CAPM is intact

with the preference presented in Figure 3, despite the fact that this preference is neitherconcave nor continuous. Thus, according to Roys modified version of the SF rule, onthe one hand, EUT is not violated, while on the other hand, the CAPM is intact. If theriskless asset does not exist with this preference, the Black version of the CAPM isintact.

3.3 Overcoming the risk-aversion assumption

We next show that with Normal distributions one does not need to assume risk aversionand that the CAPM also holds true with risk seeking preference. This is crucial, as itallows the CAPM to survive most of the above theoretical criticisms. To be more specific,we advocate that with Normal distributions, all investors - including risk-seekers - willchoose their portfolio from those located on the CML. To prove our claim, we employFirst degree Stochastic Dominance (FSD) asserting that prospect F dominates prospectG, for all investors with non-decreasing utility functions, if for all values x we haveF(x) G(x), when F and G are the relevant cumulative distributions (for a proof, seeHadar and Russell (1969) and Hanoch and Levy (1969)).

With Normal distributions, the FSD condition is represented by Theorem 1 below. Asthis case is heavily employed in this paper, let us write down the precise conditions fordominance.

Theorem 1. Let F and G denote two Normal distributions. Then, F dominates G byFSD if,

a) EF(x)> EG(x) (5)

and

b) F(x) = G(x)

where E and stand for the mean and standard deviation of the return. For a proof and discussion of the equivalence of Theorem 1 in the Normal case to the general FSDcondition, see Hanoch and Levy (1969) and Levy (2006).

Armed with the FSD criterion, corresponding to the Normal case, let us nowshow that the CAPM is also intact for risk-seeking investors. Suppose the investorconsiders investing in either Portfolio F or Portfolio G (see Figure 4a). For any possibleinterior portfolio G, there is a portfolio denoted by F located vertically above it. Asboth portfolios have the same variance, the corresponding two cumulative Normaldistributions do not intersect, and the one with the highest mean dominates the other byFirst degree Stochastic Dominance (FSD). For the relative location of the two cumulativedistributions, see Figure 4b. Therefore, according to Theorem 1, for risk averters, risk-seekers, as well as any other non-decreasing preference, Portfolio F dominates Portfolio

G. Therefore, for any portfolio located below the CML, there is a portfolio on the CMLwhich dominates it by FSD.

As can be seen to justify the CAPM we rely on the FSD criterion which is derived


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Standard Deviation

r

m

F

G

Meanr'

Fig. 4a. The Mean-Variance efficient frontier: Portfolio F dominates Portfolio G by theMean-Variance rule, as well as by the First degree Stochastic Dominance rule.

Return

F

G

Cumulative

Distributions

Fig. 4b. The cumulative distributions of F and G, where F dominates G by First degreeStochastic Dominance rule.

EUT, to refute most of the CAPMs criticisms asserting that the EUT is not intact? Mostcompeting decision-making paradigms, even though they criticise EUT, accept FSD as afoundation that should not be contradicted. For example, Kahneman and Tversky realisethat the original PT (1979) may violate FSD, an unacceptable characteristic; hence,suggesting the modified theory, CPT, which does not contradict FSD. Similarly, othergeneralisations or extensions of EUT do not violate FSD (see for example Quiggins,

1982 and 1993).8

Thus, FSD is a stronger foundation, even more general than EUT, tobuild on.


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54 Haim Levy

Finally, with the above analysis, we see that risk seekers and risk averters alike willchoose their optimal portfolios from portfolios located on the CML (see Figure 4a);hence, we claim that risk aversion is not necessary to justify the CAPM equilibrium.However, as regards risk-seeking, one must impose a restriction on borrowing in order toguarantee equilibrium.9 Otherwise, we may have a risk seeker who wishes to borrow aninfinite amount of money which, of course, contradicts equilibrium. Such a restrictionseems reasonable since in practice such restrictions exist regarding the amount borrowed.It can be shown that if the restrictions on borrowing are ineffective, then the Sharpe-Lintner CAPM is intact. However, if the restriction is effective, it is possible that notall investors will hold the tangency portfolio - a case where the Black version of theCAPM is intact (see Kroll et al., 1988).

3.4 Cumulative prospect theory

As Kahneman and Tversky modified their Prospect Theory and suggested the Cumula-tive Prospect Theory (CPT), we focus here only on CPT. According to CPT, the investormaximises a value function of the form,

V(x) =

x if x 0

(x ) if x< 0(6)

where 0 <


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dominates G(x),where F and G denote two prospects under consideration, w denotes theinitial wealth and x denotes the change of wealth. Thus, the claim of CPT that investorsmake investment decisions based on change of wealth, rather than total wealth, has noimpact on the M-V efficient frontier. It is worth mentioning that the FSD efficient setis also not affected by the initial wealth (for more details see Levy, 2006).

Now, let us turn to the more challenging difference between CPT and the CAPM.Suppose that investors examine various portfolios, e.g. mutual funds or ETFs. Denoteby F and G the cumulative distribution of two portfolios under consideration and byF and G the transformed distributions, as suggested by the DW function representedby Eq. (7). It is well known that if F and G are Normal, F and G are no longerNormal; hence, with DW one cannot employ the M-V rule, as it is no longer an optimalinvestment decision rule. Therefore, it is not clear whether the CAPM is intact.

Surprisingly, the competing paradigm of the M-V rule the stochastic dominanceparadigm comes to the rescue of the M-V rule and the CAPM. To better perceive this,recall that CPTs DW are constructed such that they do not violate FSD. Suppose that F

and G are as presented in Figure 4a. As by assumption F and G are Normal, F dominatesG according to the M-V rule. Moreover, as the variances of F and G are identical, F alsodominates G according to FSD; therefore, such dominance is also intact for the valuefunction suggested by CPT (see Theorem 1 and Figure 4b). However, F also dominatesG according to CPT, since CPT does not violate FSD.12 By the same token, for anyportfolio below the CML, there is a portfolio on the CML which dominates it by EUTas well as CPT. Thus; all CPT investors will choose a portfolio located on the CML,implying that the Separation Theorem and the CAPM are intact. The same results alsohold true as regards to Quiggins Rank Dependent Expected Utility, since according tohis suggested DW system, FSD is not violated.

Finally, note that while the compositions of the efficient M-V set, as well as theSD efficient sets, are not affected by the investors initial wealth, the selection of theoptimalportfolio from the CML is generally affected by the initial wealth. However, thisselection does not affect the CAPM, as all selections are made from the M-V efficientfrontier, i.e., from the CML. Therefore, all investors hold a combination of the tangencyportfolio and the riskless asset.

To conclude, under both the CAPM and CPT (and the RDEU), all investors choosetheir optimal portfolio from the CML. Although the optimal selected portfolio underM-V and CPT are not necessarily identical, the Separation Theorem is intact; therefore,the CAPM is intact - quite a surprising result.

Having noted this reconciliation between the CAPM and CPT, we can now returnto the Allais paradox and to Roys SF rule discussed above. Suppose that one can alsoestablish a paradox - similar to the Allais paradox - regarding Normal distributions (ifsuch a paradox does not hold true with Normal distributions, then Allaiss criticism ofEUT is irrelevant to the CAPM). The paradox is generally resolved once one employsDW, when the weight of a very small probability dramatically increases. Suppose thatCPTs decision weights solve the paradox. Then, one can use these weights which, ofcourse, contradict EUT. However, we have seen above that although EUT is contradictedby CPT, the CAPM is still intact. Thus, using CPTs decision weights may, on the onehand, solve the paradox, while on the other hand it is consistent with the generalised

12 It is interesting to note that although F is located vertically above G, F is not necessarilylocated above G However we proved that for any portfolio like G (corresponding to


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Return

F

G

CumulativeDistribution

Fig. 5a. Neither F nor G dominates the other by First degree Stochastic Dominance rule orby MeanVariance rule; yet, F dominates G by Almost First degree Stochastic Dominance

(FSD) rule and by Almost Mean Variance (M-V) rule.

Standard Deviation

U2

F

G

Mean

U1

Fig. 5b. Portfolios F and G in Figure 5a are shown in the M-V space. Utility U1 is M-Vrelevant, while it is M-V irrelevant.

To understand this more clearly, consider the following two Normal distributions:

G N(, ) = N(1, 1)

F N(, ) = N(10, 2)

Though neither FSD nor M-V dominance exists, Leshno and Levy claim that afterremoving economically irrelevant preferences, FSD and M-V dominance of F over Gexists.

As Figure 5 plainly shows, there is no dominance between the two distributions F

and G; yet, presumably all investors will select Prospect F. Leshno and Levy claim thatthe SD rules are established forallpossiblemathematicalpreferences, while in practicemany of these preferences do not fit any realistic investor; hence they are economically


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58 Haim Levy

rule (e.g. FSD) and investors realistic choices. To overcome this difficulty of the SDrules, as well as the M-V rule, Leshno and Levy developed new rules which eliminatedirrelevant preferences. These new rules reveal a preference for option F, in the aboveexample, over option G, and this preference holds true for 100% of the investors.

Leshno and Levy analyse the FSD rule, while Baumol focuses on the M-V rule.However, if the distributions are Normal, then the M-V and FSD rules coincide;13 hence,Leshno and Levy, like Baumol, eliminate a segment from the M-V efficient frontier,which may affect the validity of the CAPM, especially if the market portfolio is locatedon the relegated segment. In terms of Figure 5b, this implies that while, according to M-V both F and G are efficient, according to M-V, G is inefficient because the preferencethat tangents to Point G, in practice, does not actually exist.

We show below that although Baumol and Leshno and Levy relegate some segment ofthe M-V efficient set to the inefficient set, and despite the fact that the market portfoliomay be eliminated from the efficient set, when the riskless asset prevails the CAPMis, once again, intact. Thus, the M-V efficiency analysis may be affected by Baumol

and Leshno and Levys reduction in the efficient set, while the CAPM is not. Let uselaborate:

Baumol suggests the following investment rule instead of the M-V rule: F dominatesG if,

a) EF(x) EG(x) (8)

and

b) LF(x) = EF(x) kF LG(x) = EG(x) kG

where k is larger than 1.

As can be seen, the risk index is the lower floor L, rather than the standard deviation.Of course, the higher the value L, the lower the risk involved with the investment underconsideration. By taking the first derivative of L with respect to E, and treating it asa function of E, it can easily be shown that the lower portion of the M-V efficient setis relegated to the inefficient set according to Baumols criterion. Namely, L/E


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Standard Deviation

r

a

Mean

m

b

c

r'

Fig. 6a. Segment ab of the M-V frontier is inefficient according to Baumols rule, wherePortfolio m remains on the Baumol efficient set given by segment bc.

Standard Deviation

ra

Mean

m

c

r'

b

Fig. 6b. Segment ab is inefficient according to Baumols rule, where Portfolio m is locatedon the inefficient segment bc.

below that with the riskless asset, Portfolio m is always efficient, regardless of whetherit is relegated to the inefficient asset or not in the case where the riskless asset does notprevail. First, note that the inefficient set, according to Baumol, contains the segmentwhere the derivativeE/ >k. Therefore, the lower part of the efficient frontier maybe inefficient. Suppose that segment ab in Figure 6b is inefficient and segment bc isefficient; hence, when the riskless asset does not prevail Portfolio m is inefficient.

We turn now to the case with a riskless asset. The M-V efficient set becomes line rr

of Figure 6b represented by the CML formula,

p = r+m r

mp (9)


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60 Haim Levy

According to Baumol, the efficient set is still defined by the derivative E/ k(which also implies that E/ >k). Namely, as we move to theright on the CML, both E and L increase; therefore, each point on the CML dominatesall points with a lower mean, a case wherein infinite borrowing is optimal. 2) Thederivative on the CML fulfills the condition E/


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Standard Deviation

r

m b

a

Meanr' r''

m'

Fig. 7. The M-V and M-V efficient frontiers: Left to point b is the M-V inefficientsegment, where m is located on this segment.

Unlike the case of Baumol, in the case of Leshno and Levy not all portfolios locatedon rr are necessarily efficient. It is possible that Portfolio m is inefficient becausePortfolio m dominates it by MV. However, as m is a linear combination of m andthe riskless asset, all investors will end up holding a combination of m and the risklessasset. As a result, the CAPM is also valid when investors employ either the SD orM-V rule. Finally, even if the riskless asset does not exist, according to both Baumol

and Leshno and Levy, the optimal portfolios will be selected from the reduced M-V frontier, and in equilibrium Blacks (1972) Zero Beta CAPM holds true. This alsooccurs because according to Baumols criterion and M-V, all interior portfolios arealways inefficient.

4. Overcoming the Empirical Criticism of the CAPM

Let us examine each criticism separately:

4.1 The normality assumption

It is well known that Normality (or an Elliptic distribution) is very crucial to thederivation of the CAPM. We also assume Normality by showing the validity of theCAPM in various scenarios. Numerous studies examine the Normality hypothesis with aclear cut result: the Normality of the return distributions is statistically strongly rejected.The primary findings of these studies show that the empirical distributions are skewedand that they have fatter tails, relative to the Normal distribution. Prominent studies onthis topic include Mandelbrot (1963), Fama (1965), Officer (1972), Clark (1973), Grayand French (1990), Zhou (1993), Mantegna and Stanley (1995), Focardi and Fabozzi

(2003), and Levy and Duchin (2004).As regards to finding the distribution which best fits best the empirical data, Harvey

et al (2002) found that the skewed Normal provides the best fit while Levy and Duchin


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62 Haim Levy

distribution providing the best fit when monthly data is employed. However, they alsofound that for longer investment horizons, other distributions best fit the data with theLognormal distribution, as expected, providing the best fit for the longest investmenthorizon studied in their paper (four years).15 As the CAPM strongly relies on theNormality assumption, we next analyse the effect of the departure from Normalityon the CAPM validity.

The M-V rule is justified as an approximation to expected utility, even whendistributions are not Normal on two grounds:

Levy and Markowitz (1979) have shown that with empirical return distributions whichare not normal, the M-V rule is an excellent approximation for expected utility (seealso Kroll et al., 1984 and Markowitz, 1991). Kroll et al. and Markowitz show thatvirtually all relevant risk-averse utility functions can be approximated locally by aquadratic function; hence, a direct and precise expected utility maximisation yieldsalmost the same expected utility obtained by selecting the best portfolio only from

the set of efficient portfolios located on the M-V efficient frontier. This conclusion isvalid as long as the range of returns is not too wide. Thus, Levy and Markowitz claimthat in most cases one can safely use the M-V rule, despite the statistical rejection ofNormality. The utility loss induced by adopting this approach has been empiricallyfound to be negligible. Other studies directly estimate thefinancialloss, rather than theutility loss, due to the assumption of Normality, when the empirical rates of return arenot really distributed Normally. The procedure employed is as follows: First, maximisethe expected utility directly and find the certainty equivalent of the selected portfolioinvestment. Then, once again, maximise expected utility, but this time by assumingNormality. Hence, by construction, a lower certainty equivalent is obtained because

in practice distributions are not Normal. As a result, one deviates from the optimalinvestment diversification. The difference between the two certainty equivalent sumsmeasures the financial loss induced by the deviation of the empirical distribution fromNormality. For these types of studies, which make different assumptions regardingthe investors preference, see Simaan (1993), Tewet al. (1991), and Duchin and Levy(2008).

All these studies reveal that the financial loss is negligible, despite the strongstatistical rejection of Normality. For example, Duchin and Levy, who employ the myopicpreference, find that the loss per $10,000 investment is merely $2-$6, depending on the

degree of the relative risk-aversion parameter. To put things in perspective, suppose thatthe planned investment horizon is one year. The mean rate of return on risky assets isabout 12% (see Ibbotson, 2007). Then, a loss of $2-$6 per $10,000 investment impliesthat the mean rate of return drops, on average, from 12% to 11.94%-11.98%, a negligibleloss.16

15 With a very long horizon, the distributions become positively skewed and approaching theLognormal distribution. For the role of skewness in efficiency analysis, see Arditti and Levy(1975) and Harvey and Siddique (2000).16 One may claim that even this small loss will induce the investor to select her portfolio

by expected utility maximisation; hence, an interior portfolios is selected, a case where theCAPM does not hold true. However, for this scenario to hold true, one must assume theexistence of very sophisticated investors who select their portfolio according to a complex


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Finally, recall that the direct optimal diversification - without the Normality assump-tion - is the correct one and any optimisation with Multivariate-Normality assumption (aswas done by Duchin and Levy, 2008) induces a loss, since the distribution of empiricaldistribution is, in practice, not Multivariate-Normal. However, the two calculations yieldidentical results: i.e. a zero loss is involved, when the preference is quadratic, or whenthe empirical distributions are indeed Multivariate-Normal. The surprising result isthat although the Multivariate-Normal distribution is strongly statistically rejected, thefinancial loss induced by making the normality assumption is minimal, and one cansafely assume Normality and employ the M-V criterion for investment selection - a veryencouraging result from the point of view of the CAPM.

4.2 The CAPM empirical and experimental tests

Classical economists employ empirical test to examine the validity of a given theory.Behavioural economists and psychologists rely on laboratory experiment. We showin this section that with ex ante parameters the CAPM can not be rejected. Evenmore important, employing behavioural economics approach the CAPM has a strongexperimental support.

Shortly after the theoretical model of the CAPM was published, testing the modelempirically was in fashion and was a subject researched by top researchers in finance.From these empirical studies, one is tempted to conclude that the CAPM is empirically,questionable, which allegedly drastically reduces its value.

We claim below that the empirical results are inconclusive: Although the CAPMis rejected with expost parameters, it cannot be rejected with ex ante parameters,in particular with ex ante beta. Recall that the CAPM is defined by Sharpe and

Lintner in terms of the ex ante, rather than ex post, parameters. Unfortunately, inthe empirical studies ex post parameters are employed, simply because the ex anteparameters are unknown. Obviously, there are differences between the ex postand theex ante parameters, and it can be shown that with small changes in the parameters -which take into account these possible differences - the CAPM cannot be rejected.

In the early CAPM tests, the first-passandsecond-passregressions were defined asfollows:

First-Pass : Ri t= i + i Rmt + eit (11)

Second Pass : Ri = 0 + 1i + i (12)where: Rit and Rmt stand for the rate of return on the ith asset and the market portfolioduring Period t, respectively, while i is the estimate of the ith asset risk, as estimatedin the first-pass regression.17

The CAPM relates to the expected rate of return of each security i and to its risk asfollows:

i = r+ (m r)i (12a)

17 The above descrbed empirical test of te CAPM is the old static test. Of course,with decades

of empirical tests there are more sophisticated tests, like the conditional CAPM test wherebetas are allowed to change overtime. For extension of the CAPM test see for example,Cochrane (2001) Roll and Ross (1994) Kandel and Stambaugh (1995) Levhari and Levy


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64 Haim Levy

where r is the risk-free interest rate andm is the expected rate of return on the marketportfolio. If the CAPM is intact, we expect that 0will not be significantly different fromthe risk free interest rate and that 1 will not be significantly different from m r.

18

Virtually all empirical tests reveal that this is not the case; therefore, the CAPM isrejected. Moreover, the R2 is generally low - about 4% with monthly data and about20% with annual data - as long as individual assets, rather than portfolios, are considered,as required by the CAPM tests (see Levy, 1978).19

We now turn to the difference between ex post and ex ante parameters. There areseveral approaches for incorporating these differences. Generally, it can be shown thataccounting for even small possible differences between ex postandex ante parametersthe CAPM cannot be rejected. Let us elaborate:

4.2.1. Ex ante beta. In testing the CAPM (see Eq. (12)), it is assumed that beta,estimated by Eq. (11), is the correct ex ante beta. We claim that taking into accountpossible differences between theex postandex antebeta, the CAPM cannot be rejected.

Indeed, Levy (1983) tested the CAPM when such differences were taken into account. Heshowed that regardless of the possible measurement errors involved in the measurementof rates of returns, the CAPM cannot be rejected on an ex ante basis. This is clear ifwe simply recall that the beta used in the second-passregression is an estimate of theunknown trueex antebeta. Using joint confidence intervals for all betas, and taking intoaccount this possible difference between ex ante andex postbetas, he showed that theCAPM cannot be rejected, and that an almost perfectsecond-passregression is obtainedwith a possibleex antevector of betas.20

Finally, even with the ex postdata, the CAPM empirical results are not always negative.Actually, the results depend on the sub-period selected for such testing. For example, for

the ten portfolios ordered by their book to market ratio (B/M), Fama and French (2004)report a negative relationship between average return and beta for the period 19632003(see Figure 8a). However, using the same data, Levy (2008b) reports a strong positiverelationship between these two variables for the sub-period 1927-1962 (see Figure 8b),and even a stronger relationship (with 84% explanation power) for the whole period19272007 (see Figure 8c). Thus, even with ex postdata, the empirical results remainambiguous.

4.2.2. The efficiency of the market portfolio. In a breakthrough article, Roll (1977)showed that with the above procedure of testing the CAPM, the only relevant CAPM

test is to examine whether the employed market portfolio is M-V efficient. If it is,then in the second-pass regression the CAPM would reveal a perfect fit, and this isa technical result. Moreover, the fact that in the empirical tests a less than perfect fitis obtained merely indicates that the market portfolio is not M-V efficient; hence, itis commonly concluded that the CAPM is invalid. Taking into account the possibledifference between ex postandex ante parameters, in a recent paper, Levy and Roll(2008) show that when only small changes in the sample means and standard deviations

18 When excess returns are employed0 is expected not to be significantly different fromzero.19

However, to avoid some statistical errors, some studies employ the rates of return onportfolios rather than individual assets.20 The standard CAPM regressions take into account possible differences between sample


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Fig. 8. The regression results for ten portfolios ordered by book to market ratio (B/M), using

K. French data base. (For more details, see http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html)


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66 Haim Levy

are applied, the observed market portfolio is M-V efficient, which according to Roll(1977) implies that the linear CAPM is intact. They employ a novel reverse engineeringapproach. To be more specific, they take a given market proxy portfolio and investigatethe minimal necessary changes that must be applied to the parameters, such that thisportfolio will be located on the M-V efficient frontier. Surprisingly, they find that onlysmall changes are required, well within the allowed differences between sample meansand true ex ante means. Therefore, they conclude that one cannot reject the marketportfolio efficiency; hence, one cannot reject the CAPM. In their words:

Surprisingly, slight variations of the sample parameters, well within the estimationerror bounds, suffice to make the proxy efficient. Thus, many conventional market

proxies could be perfectly consistent with the CAPM.

4.2.3. Negative investment weigh. Using historical rates of return to derive the M-V

efficient set, it is generally found that some of the weights are negative; of course, with asmall number of assets, e.g. 35 assets, it is possible to find that all weights are positive.However, in the relevant case for testing the CAPM, where hundreds if not thousandsof assets need to be incorporated, negative investment weights always exists. Moreover,the percentage of negative weights becomes close to 50% of the assets included in thestudy, when the number of assets increases. Furthermore, one does not need to have anastronomically large number of assets to obtain this result: Levy (1983) shows that evenwith 15 assets the percentage of negative weights is about 50%. Thus, having negativeweights on the efficient frontier - and in particular about 50% negative weights, whenthe market portfolio by definition is composed of only positive weights - is disturbing

because it indicates that the market portfolio is interior to the M-V efficient frontier.Once again, according to Roll (1977), this is evidence against the validity of the

CAPM.21 This empirical evidence attacks the CAPM in a similar way to the previouspoint (see Point 3 above), but the attack comes from a slightly different angle. Therefore,it is not surprising that the remedy, i.e. saving the CAPM from this attack, is verysimilar to the previous point, as suggested by Levy and Roll (2008). Indeed, in a recentpaper, Levy (2008a) conducts a technique to examine whether with ex ante data it ispossible that all weights will be positive, even though withex postdata, we find negativeweights corresponding to portfolios located on the M-V frontier. He shows that withsmall changes in the parameters to account for the possible difference between ex post

and ex ante parameters, a tangency portfolio with only positive weights is obtained.Levy shows that the fact that empirically negative weights are almost everywhere doesnot constitute any evidence against the CAPM. Moreover, this result is rather expectedin the CAPM framework. To cite from his paper,

We show that the probability of obtaining a positive tangency portfolio basedon sample parameters converge to zero exponentially with the number of assets.

However, at the same time, very small adjustments in the return parameters, well

21 Levy (2008) has shown that when there is a reasonable heterogeneity in the end of period

distributions of the individual risky assets then positive prices imply that the risk premiummust be close to zero, an unacceptable result. He suggests that in such a case the generalisedsegmented CAPM of Levy (1978) and Merton (1987) may serve as a substitute to the CAPM


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within the estimation error, yield a positive tangency portfolio. Hence, looking forpositive portfolios in parameter space is somewhat like looking for rational numberson the number line: if a point in the parameter space is chosen at random it almost

surely corresponds to non-positive portfolio (an irrational number); however, onecan find very close points in parameter space corresponding to positive portfolios(rational numbers).

This finding sheds light on the empirical results, which consistently show negativeweights corresponding to M-V efficient portfolios: an infinite number of portfolios withpositive weights exist; however, there is zero chance to discover them - quite an amazingresult. Thus, the empirically obtained negative weights cannot prove that the CAPM isinvalid. The positive tangency portfolios indeed exist - in large numbers - but we cannotfind them.

4.2.4. Experimental studies of the CAPM. Finally, in the above analysis an effort has

been made to account for the possible difference between ex postandex anteparameters.Can we test the CAPM withex- anteparameters? Empirically, we cannot, as the ex anteparameters are not available. However, one can set up an experiment wherein the subjectssimultaneously determine the mean rate of return, beta and equilibrium prices; hence,the CAPM can be tested experimentally with ex anteparameters. For example, supposethat the end of a period return is given by $100(1+R), where the return, R, is a randomvariable whose distribution is given to the subjects. What price is one willing to payfor this asset today? At what price is one willing to sell such an asset today? Havingmany assets similar to this one - and many subjects - one can look at the aggregatedemand and aggregate supply and find the equilibrium prices which clear the market.

Note that by determining the equilibrium prices one also simultaneously determines themean return and beta of each asset, and these parameters are ex anteparameters. Usingthis technique, Levy (1997) found strong support for the CAPM with more than 70%explanatory power. Levy concludes,

. . . mean return and risk are strongly positively related when these parameters aredetermined on an ex ante basis, as claimed by the Sharpe-Lintner model (see p.145).

Bossaerts and Plott (2002), also found experimental support for the CAPM. In theirwords,

when interpreted as the equilibrium to which a complex financial market systemhas a tendency to move, the CAPM received support in the experiments reportedhere (see p. 1110).

To sum up, empirically we have no evidence that rejects the CAPM, and experimen-tally, withex ante parameters, we strongly support it.

5. Concluding Remarks

The M-V efficiency analysis of Markowitz (1952) and the Sharpe-Lintner CAPM areboth derived from the expected utility framework which was long ago criticised by both


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the capm is alive and well - a review and synthesis

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