Statistical Analysis of the CAPM

II. Black CAPM

Brief Review of the Black CAPM

• The Black CAPM assumes that

(i) all investors act according to the µ− σ rule,(ii) face no short selling constraints, and(iii) exhibit perfect agreement with respect to the

probability distribution of asset returns.

• It is not assumed that they can lend and borrow ata common risk–free rate.

• Under these assumptions, the market portfolio is amean–variance efficient portfolio.

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• Thus, there is a portfolio Z, i.e., the zero–betaportfolio with respect to the market portfolio, suchthat for each risky asset or portfolio of risky assetsi, we have

µi = µz + βi(µm − µz), (1)

where µm is the expected return of the marketportfolio.

2

Framework for Estimation and Testing

• The CAPM relationship (1) is expressed in terms ofexpected values, which are not observable.

• To obtain a model with observable quantities, wedescribe returns using the following market model :

rit = αi + βirm,t + ϵit i = 1, . . . , N (2)

E(ϵit) = 0, i = 1, . . . , N (3)

E(ϵitϵjt′) =

{σij if t = t′

0 if t ̸= t′i, j = 1, . . . , N (4)

E(rm,tϵi,t) = 0, i = 1, . . . , N. (5)

• Here ri,t is the return of asset i in period t, andrm,t is the return of the market portfolio in periodt.

• This is very similar to the framework employed fortesting the Sharpe–Lintner CAPM.

3

• However, in contrast to the market model consideredlast week, the relation (2) is not stated in terms ofexcess returns.

• According to equation (4), the asset–specific errorterms may be correlated.

• Thus, we allow for a non-diagonal covariance matrix,Σ, of the vector ϵt = [ϵ1t, . . . , ϵNt]

′,

COV (ϵt) = Σ =

σ21 σ12 · · · σ1N

σ12 σ22 · · · σ2N

... ... . . . ...σ1N σ2N · · · σ2

N

• Conditional on the excess return of the market, wethen also have

COV (rt) = Σ, (6)

where rt = [r1t, . . . , r2t]′.

4

• We will also assume that the error terms follow amultivariate normal distribution, i.e.,

ϵtiid∼ N(0,Σ). (7)

• The Black CAPM implies a restriction on the inter-cept terms in (2), namely,

αi = (1− βi)µz, i = 1, . . . , N, (8)

or, using vector notation,

α = (1N − β)µz. (9)

• Equation (9) imposes a nonlinear restriction on theparameters, because µz is not known and has to beestimated, along with the further unknown parame-ters of the (restricted) model, i.e., β and Σ.

5

Estimation of the Parameters

• Write the market model as

rt = α+ βrm,t + ϵt, t = 1, . . . , T,

ϵtiid∼ N(0,Σ),

where α = [α1, . . . , αN ]′, and β = [β1, . . . , βN ]′.

• The maximum likelihood estimator (MLE) for theunconstrained model has been derived last week,and is given by

α̂ = r̄ − β̂r̄m, (10)

β̂ =

∑Tt=1(rt − r̄)(rm,t − r̄m)∑T

t=1(rm,t − r̄m)2(11)

=

∑Tt=1(rt − r̄)(rm,t − r̄m)

T σ̂2m

=

∑Tt=1(rm,t − r̄m)rt

T σ̂2m

,

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and

Σ̂ =1

T

T∑t=1

ϵ̂tϵ̂′t (12)

=1

T

T∑t=1

(rt − α̂− β̂rm,t)(rt − α̂− β̂rm,t)′.

where

r̄ =1

T

T∑t=1

rt, r̄m =1

T

T∑t=1

rm,t,

σ̂2m =

1

T

T∑t=1

(rm,t − r̄m)2.

7

Estimation of the Restricted Model

• Recall that the Black CAPM imposes

α = (1N − β)µz. (13)

• The parameters to estimate are µz, β and Σ, andthe log–likelihood function is

logL(µz, β,Σ) = −NT

2log(2π)− T

2log |Σ|

−1

2

T∑t=1

ϵ̂′tΣ−1ϵ̂t,

where

ϵ̂t = rt − (1N − β̂)µ̂z − β̂rm,t. (14)

8

• Note that, by the same arguments as last week,whatever the estimators of β and µz will be, theestimator of Σ is

Σ̂(µ̂z, β̂) =1

T

T∑t=1

ϵ̂tϵ̂′t (15)

=1

T

T∑t=1

(rt − (1N − β̂)µ̂z − β̂rm,t)

×(rt − (1N − β̂)µ̂z − β̂rm,t)′.

• Moreover, for any given µ̂z, β̂ will be theequation–by–equation OLS estimator of the regres-sion through the origin

(rt − 1N µ̂z) = β(rm,t − µ̂z), t = 1, . . . , T.

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It follows that

β̂(µ̂z) =

T∑t=1

(rt − 1N µ̂z)(rm,t − µ̂z)

T∑t=1

(rm,t − µ̂z)2. (16)

• From last week’s analysis, we also know that thelog–likelihood function, evaluated at the MLE, is

logL = −NT

2[log(2π) + 1]− T

2log |Σ̂|.

• Thus, we have to find β̂ and µ̂z such that

log |Σ̂| = log

∣∣∣∣∣ 1TT∑

t=1

(rt − µ̂z(1N − β̂)− β̂rm,t)

× (rt − µ̂z(1N − β̂)− β̂rm,t)′∣∣∣ (17)

is minimized.

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• But, as we have seen in (16), β̂ can be written as afunction of µ̂z, namely

β̂(µ̂z) =

T∑t=1

(rt − 1N µ̂z)(rm,t − µ̂z)

T∑t=1

(rm,t − µ̂z)2. (18)

• Thus, (17) can be written as a function of just asingle variable, µ̂z.

• Hence, we can find the MLE of the restricted modelby first identifying µ̂z.

• This can be done, for example, by conducting asimple grid–search.

• Then compute β̂ via (18) and finally evaluate Σ̂using equation (15).

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Likelihood Ratio (LR) Test

• Having estimated the parameters of both the un-restricted as well as those of the restricted marketmodel, we can conduct a likelihood ratio (LR) test.

• If

– Σ̂1 denotes the estimated error term covariancematrix under the unrestricted model, and

– Σ̂0 is the estimated error term covariance matrixunder the null hypothesis,

then, by the same arguments as last week, the LRtest statistic is

LR = T[log |Σ̂0| − log |Σ̂1|

]asy∼ χ2(N − 1).

• Note that the degrees of freedom of the null dis-tribution is N − 1. Relative to the Sharpe–Lintnermodel, we lose one degree of freedom because µz isa free parameter.

12

0 2 4 6 8 10 12 14 16−1

0

1

2

3

4

5

portfolio standard deviation

po

rtfo

lio m

ea

n

MVS with short salesDAXindividual assets

13

0 2 4 6 8 10 12 14 16−1

0

1

2

3

4

5

portfolio standard deviation

po

rtfo

lio m

ea

n

MVS with short salesMVS without short salesDAXindividual assets

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0 0.5 1 1.5 2 2.5−4634.5

−4634

−4633.5

−4633

−4632.5

−4632

−4631.5

−4631

µz

log

−lik

elih

oo

d a

s f

un

ctio

n o

f µ

z

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5

3

3.5

4

estimated betas

sam

ple

mean r

etu

rn

• We have also seen that the asymptotic likelihoodratio test may exhibit poor performance in finitesamples.

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• To mitigate these effects, the adjusted statistic

LR⋆ =

(T − N

2− 2

)[log |Σ̂0| − log |Σ̂1|

]asy∼ χ2(N − 1)

has been shown to more closely match the χ2 dis-tribution in finite samples.

• There also exists a further device that provides auseful check.1

• This also gives rise to a closed–from estimator forthe zero-beta rate, µZ.

1Cf. Shanken, J. (1986) Testing Portfolio Efficiency when the Zero–BetaRate in Unknown: A Note. Journal of Finance 41, 269-276

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Lower Bound for the Exact Distribution

• Suppose for the moment that µz is known.

• Then, we can proceed as last week when testingthe Sharpe–Linter model, i.e., we can consider the“excess return” market model

rt − µz1N = α+ β(rm,t − µz) + ϵt. (19)

• The zero–beta CAPM is true if α = 0.

• The estimates of the unrestricted model are

α̂1 = r̄ − 1Nµz − β̂(r̄m − µz)

β̂1 =

∑t(rt − r̄)(rm,t − r̄m,t)∑

t(rm,t − r̄m)2

Σ̂1 =1

T

∑t

[rt − r̄ − β̂1(rm,t − r̄m)]

×[rt − r̄ − β̂1(rm,t − r̄m)]′.

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Note that β̂1 and Σ̂1 do not depend on µz, and,thus, the value of the log–likelihood function doesalso not depend on µz, as, at the maximum,

logL1 = −NT

2[log(2π) + 1]− T

2log |Σ̂1|.

• The MLE under the restriction that α = 0 is

β̂0(µz) =

∑t(rt − 1Nµz)(rm,t − µz)∑

t(rm,t − µz)2(20)

Σ̂0(µz) =1

T

∑t

(rt − µz(1N − β̂0)− β̂0rm,t)

×(rt − µz(1N − β̂0)− β̂0rm,t)′.

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• The value of the constrained log–likelihood is

logL0(µz) = −NT

2[log(2π) + 1]

−T

2log |Σ̂0(µz)|,

which can be viewed as a function of only onevariable, i.e., µz.

• Consequently, the likelihood ratio test statistic,

LR(µz) = T[log |Σ̂0(µz)| − log |Σ̂1|

], (21)

can be viwed as a function of only µz.

• Obviously, the value of µz which minimizes thelikelihood ratio statistic will be the MLE of µz.

• (Recall that |Σ̂1| does not depend on µZ.)

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• In the last week, we have developed a relationbetween the LR test and the F test for the Sharpe–Linter CAPM, which used a formula expressing |Σ̂0|in terms of |Σ̂1| and α̂.

• Repeating the same line of arguments shows that(21) can be written as

LR(µz) = T log

[α̂′Σ̂−1

1 α̂σ̂2m

(r̄m − µz)2 + σ̂2m

+ 1

],

(22)where

α̂ = (r̄ − β̂1r̄m)− (1N − β̂1)µz. (23)

is a function of µz

21

• Thus, the MLE of µz is the value which minimizes

g(µz) = α̂′Σ̂−11 α̂

σ̂2m

(r̄m − µz)2 + σ̂2m

=[µ2

za− 2bµz + c]σ̂2m

σ̂2m + (r̄m − µz)2

,

where

a = (1N − β̂1)′Σ̂−1

1 (1N − β̂1),

b = (1N − β̂1)′Σ̂−1

1 (r̄ − β̂1r̄m),

c = (r̄ − β̂1r̄m)′Σ̂−11 (r̄ − β̂1r̄m).

22

• It follows that we can find the MLE of µz by solving

dg(µz)

dµz=

(2aµz − 2b)[(r̄m − µz)2 + σ̂2

m]

[(r̄m − µz)2 + σ̂2m]2

−(µ2za− 2bµz + c)2(µz − r̄m)

[(r̄m − µz)2 + σ̂2m]2

= 0,

that is, µz is a root of the quadratic

Aµ2z +Bµz + C = 0, (24)

where

A = b− ar̄m,

B = a(σ̂2m + r̄2m)− c,

C = −b(σ̂2m + r̄2m) + cr̄m.

• This is a closed–form solution for µ̂z.

23

• It can be shown that the roots of (24) are real.

• The maximum likelihood estimator, then, is theroot which corresponds to the smaller value oflog(det(Σ̂(µz))).

• Once µ̂z is determined, we can apply the closedform expressions (20) to determine the MLE for theother parameters.

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• Now let us return to our objective of finding a testwhich does not rely on asymptotic arguments.

• Recall that, with µz known, we could use the stati-stic

J(µz) =T −N − 1

N

[1 +

(r̄m − µz)2

σ̂2m

]−1

α̂′Σ̂−11 α̂

(25)to conduct an exact finite–sample test.

• This uses the result that, in this case, (25) has anF distribution with N degrees of freedom in thenumerator and T −N −1 degrees of freedom in thedenominator.

• As µz is not known, this cannot be done.

• The MLE of µz, µ̂z, is the value which minimizesthe LR test statistic, LR(µz).

• As we know that J(µz) is a monotonic transforma-tion of LR(µz), µ̂z also minimizes J(µz).

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• Thus,J(µ̂z) ≤ J(µz), (26)

where µz is the true value of the zero–beta portfo-lio’s mean.

• That is, the F test based on µ̂z and using the“exact” F distribution will accept too often.

• But we know that, if it rejects, it will reject for anyvalue of µz, and we need not resort to asymptoticapproximations in this case.

• This is a useful check because we have seen that theasymptotic likelihood ratio test rejects too often.

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