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Statistical Analysis of the CAPM
I. Sharpe–Linter CAPM
Brief Review of the Sharpe–Lintner CAPM
• The Sharpe–Lintner CAPM assumes that all
investors act according to the µ−σ rule, can
lend and borrow any desired amount at a
common risk–free rate rf and exhibit per-
fect agreement with respect to the probabil-
ity distribution of asset returns.
• Under these (key) assumptions, the market
portfolio is given by
xm =Σ−1(µ− rf1N)
1′NΣ−1(µ− rf1N)
. (1)
• The central equation of the Sharpe–Lintner
CAPM is a direct consequence of (1) and is
given by
µi = rf + βi(µm − rf), i = 1, . . . , N, (2)
where rf is the risk–free rate, µm is the ex-
pected return of the market portfolio and
βi = COV (Ri, Rm)/σ2m, where Ri is the re-
turn of asset i and Rm is the return of the
market portfolio.
• Equation (2) states that there is a linear re-
lation between the excess return of asset i
(over the risk–free) rate and the excess re-
turn of the market portfolio, with zero inter-
cept.
Framework for Estimation and Testing
• The CAPM relationship (2) is expressed in
terms of expected values, which are not ob-
servable.
• To obtain a model with observable quan-
tities, we describe excess returns using the
excess return market model:
rit = αi + βirm,t + εit i = 1, . . . , N (3)
E(εit) = 0, i = 1, . . . , N (4)
E(εitεjt′) =
σij if t = t′
0 if t 6= t′i, j = 1, . . . , N
(5)
E(rm,tεi,t) = 0, i = 1, . . . , N. (6)
• Here ri,t is the excess return on asset i in
period t (over risk–free rate), and rm,t is the
excess return on the market portfolio in pe-
riod t (over risk–free rate).
• At first glance, the market model we will
be using looks similar to the Single–Index
Model (SIM), but there are important dif-
ferences:
– All returns involved are excess returns over
the risk–free rate rf .
– According to equation (5), the asset–specific
error terms may be correlated. Thus, we
allow for a non-diagonal covariance ma-
trix, Σ, of the vector εt = [ε1t, . . . , εNt]′,
COV (εt) = Σ =
σ21 σ12 · · · σ1N
σ12 σ22 · · · σ2N
... ... . . . ...
σ1N σ2N · · · σ2N
Conditional on the excess return of the
market, we then also have
COV (rt) = Σ, (7)
where rt = [r1t, . . . , r2t]′.
– Note, however, that we still assume that
there is no correlation over time, i.e. E(εtε′t′)
= 0 for t 6= t′, and that the covariance
matrix Σ is constant over time.
• We will assume that the betas are constant
over time. This is by no means self-evident
and can, in principle, be tested using econo-
metric techniques for detecting structural breaks
(see Greene, Chapter 7).
• We will also assume that the error terms fol-
low a multivariate normal distribution, i.e.,
εtiid∼ N(0,Σ). (8)
• The Sharpe–Lintner CAPM implies that the
intercept in the excess return market model
is zero, i.e., α = 0. That is, a test of this
model corresponds to a test of the hypoth-
esis
H0 : αi = 0, i = 1, . . . , N. (9)
• To perform such a test, it is necessary to
estimate the parameters of the model and
to derive an appropriate test statistic.
Test based on Time Series Regression
• Write our excess return market model as
rt = α + βrm,t + εt, t = 1, . . . , T,
εtiid∼ N(0,Σ),
where α = [α1, . . . , αN ]′, and β = [β1, . . . , βN ]′.
• Given our assumptions about the distribu-tional properties of the error process, εt,the density of excess returns, conditional onthe market return, rm,t, is
f(rt|rm,t)
=exp
−1
2(rt −α− βrm,t)′Σ
−1(rt −α− βrm,t)
(2π)N/2|Σ|1/2,
and the joint density is
f(r1, . . . , rT |rm,1, . . . , rT,1) (10)
=
T∏
t=1
f(rt|rm,t)
=
exp
−1
2
T∑t=1
(rt −α− βrm,t)′Σ−1(rt −α− βrm,t)
(2π)NT/2|Σ|T/2
• To estimate the unknown parameters, α, β,
and Σ, of this density, we use the method
of maximum likelihood. To do so, we define
the log–likelihood function, i.e., the log of
the joint density viewed as a function of the
unknown parameters.
• The maximum likelihood estimator is then
found by maximizing this function with re-
spect to its arguments, i.e., the unknown
parameters.
• From (10), the log–likelihood function is
logL(α, β,Σ) (11)
= −NT
2log(2π)− T
2log |Σ|
−1
2
T∑
t=1
(rt −α− βrm,t)′Σ−1
×(rt −α− βrm,t),
which we want to maximize with respect to
α, β and Σ.
• From (11), it is clear that the estimates of
α and β are determined by minimizing
S =T∑
t=1
(rt −α− βrm,t)′Σ−1(rt −α− βrm,t)
=T∑
t=1
r′tΣ
−1rt − 2r′tΣ−1(α + βrm,t)
+(α + βrm,t)′Σ−1(α + βrm,t)
.
• Note that this is a seemingly unrelated re-
gression with identical regressors,∗ so that
the MLE is identical to OLS equation by
equation. In fact, the first order conditions
are
∂S
∂α= −2Σ
−1T∑
t=1
(rt −α− βrm,t) (12)
= 0
∂S
∂β= −2Σ
−1T∑
t=1
rm,t(rt −α− βrm,t)(13)
= 0,
implying
α = r − βrm (14)
∗ See Greene (Chapter 14.2), and Theil (Chapter 7).
and†
β =
∑Tt=1(rt − r)(rm,t − rm)∑T
t=1(rm,t − rm)2(15)
=
∑Tt=1(rt − r)(rm,t − rm)
T σ2m
=
∑Tt=1(rm,t − rm)rt
T σ2m
where
r =1
T
T∑
t=1
rt, rm =1
T
T∑
t=1
rm,t,
σ2m =
1
T
T∑
t=1
(rm,t − rm)2.
†The third equality makes use of the basic identitysxy = T−1
∑t(xt − x)(yt − y) = T−1
∑t(xt − x)yt, which
is easily seen as both expressions are equal to xy− xy.
• To find the MLE of Σ, we make use of the
following differentiation rules for a symmet-
ric matrix X:
∂tr(XA)
∂X= A + A′ − diag(A) (16)
and
∂ log |X|∂X
= 2X−1 − diag(X−1). (17)
The log–likelihood function can be written
as
logL = −NT
2log(2π)− T
2log |Σ|
−1
2
T∑
t=1
tr(ε′Σ−1ε
)
= −NT
2log(2π) +
T
2log |Σ−1|(18)
−1
2
T∑
t=1
tr(Σ−1εε′
), (19)
where εt = rt − α− βrm,t, (18) uses |A−1| =|A|−1, and (19) uses the permutation rule
tr(ABC) = tr(BCA).
Thus, using (16) and (17), we require
∂ logL
∂Σ−1
=T
2[2Σ− diag(Σ)]
−1
2
2
T∑
t=1
εtε′t − diag
T∑
t=1
εtε′t
= 0,
implying
Σ =1
T
T∑
t=1
εtε′t (20)
=1
T
T∑
t=1
(rt − α− βrm,t)(rt − α− βrm,t)′.
• The OLS estimators of α and β are unbi-
ased, normally distributed and have covari-
ance matrices
COV (β) =1
T σ4m
COV
T∑
t=1
(rm,t − rm)rt
=1
T σ4m
T∑
t=1
(rm,t − rm)2COV (rt)
=1
T σ2m
Σ,
and
COV (α) = COV (r − βrm)
= COV
1
T
T∑
t=1
rt − rm
T∑
t=1
(rm,t − rm)rt
T σ2m
=1
T2σ4m
T∑
t=1
[σ2m − rm(rm,t − rm)]2COV (rt)
=1
T2σ4m
T [σ4m + r2mσ2
m]Σ
=1
T
(1 +
r2mσ2
m
)Σ. (21)
• It can be shown that T Σ has a Wishart dis-
tribution, WN(T − 2,Σ), which is a matrix
generalization of the χ2.‡
Moreover, Σ is independent of both α and
β.
‡ See, for example, Zellner (1971).
Testing for α = 0
• We discuss two tests of the null hypothe-
sis α = 0, in historical order. The first is a
likelihood ratio (LR) test relying on asymp-
totic arguments,§ while the second is an ex-
act finite–sample F-test. Subsequently, the
relation between the tests will be considered.
Likelihood Ratio (LR) Test
• To conduct the likelihood ratio test, we first
compute the Maximum Likelihood Estima-
tor under the null hypothesis that α = 0,
which is a regression through the origin. De-
note the corresponding estimators by β0 and
Σ0. They are given by
β0 =
∑Tt=1 rtrm,t∑Tt=1 r2m,t
, (22)
§This has been proposed by Gibbons, M. (1982). Mul-tivariate Tests of Financial Models: A New Approach.Journal of Financial Economics 10, 3-28.
and
Σ0 =1
T
T∑
t=1
ε0t ε0t (23)
=1
T
T∑
t=1
(rt − β0rm,t)(rt − β0rm,t)′,
where ε0t = rt − β0rm,t.
• The Likelihood Ratio Test is based on the
comparison between the log–likelihood val-
ues of the unconstrained model and the con-
strained model.
• More precisely, the LR test statistic is given
by
LR = −2(logL0 − logL1), (24)
where logL0 is the log–likelihood function
of the constrained model, and logL1 is the
log–likelihood function of the unconstrained
model, each evaluated at the respective MLE.
• The asymptotic distribution of LR defined in
(24) is χ2 with degrees of freedom equal to
the number of parameter restrictions implied
by the null hypothesis. In our situation, this
corresponds to N degrees of freedom (N is
the number of assets), because the CAPM
implies that αi = 0 for i = 1, . . . , N .
• Now
logL1 = −NT
2log(2π)− T
2log |Σ1|
−1
2
T∑
t=1
tr
(Σ−11 εε′
)
= −NT
2log(2π)− T
2log |Σ1|
−1
2
T∑
t=1
tr
1
T
T∑
t=1
εtε′t
−1
εε′
= −NT
2log(2π)− T
2log |Σ1|
−T
2tr
T∑
t=1
εtε′t
−1 T∑
t=1
εε′
= −NT
2log(2π)− T
2log |Σ1| −
T
2tr(IN)
= −NT
2(log(2π) + 1)− T
2log |Σ1|.
• By the same line of arguments,
logL0 = −NT
2(log(2π) + 1)− T
2log |Σ0|.
Consequently,
LR = T[log |Σ0| − log |Σ1|
]. (25)
F Test
• The finite–sample F test is based on the
following result:
Result: If N–dimensional random variable X
is N(0,Ω), the N × N random matrix A is
Wishart(T,Ω), and X and A are indepen-
dent, then
T −N + 1
NX ′A−1X ∼ FN,T−N+1, (26)
i.e., the quantity [(T−N+1)/N ]X ′A−1X has
an F distribtuion with N degrees of freedom
in the numerator and T − N + 1 degrees of
freedom in the denominator.
• Using, in (26), X =√
T [1+r2m/σ2m]−1/2α and
A = T Σ, and recalling the results we have
for α (in particular, normality and (21)), the
statistic
J =T −N − 1
N
(1 +
r2mσ2
m
)−1
α′Σ−1
α (27)
has an F distribution with N degrees of free-
dom in the numerator and T −N −1 degrees
of freedom in the denominator, i.e.,¶
J ∼ FN,T−N−1. (28)
¶This was developed in Gibbons/Ross/Shanken(1989): A Test of the Efficiency of a Given Portfolio.Econometrica 57, 1121-1152.
Economic Interpretation of the CAPM F Test
• Apart from following a known finite–sample
distribution, the test statistic J defined in
(27) also has economic interpretation.
• Recall that the key testable implication of
the CAPM is that the market portfolio is a
µ− σ efficient portfolio.
• In the presence of a risk–free rate, this means
that the market portfolio is the tangency
portfolio.
• It can be shown that‖
J =
(T −N − 1
N
)θ?2 − θ2
m
1 + θ2m
, (29)
where θ? is the Sharpe ratio of the ex post
(i.e., using the sample mean vector and the
sample covariance matrix) efficient portfo-
lio formed from the risky assets under study
(including our market proxy) and θm is the
Sharpe ratio of the portfolio used as a mar-
ket proxy in our analysis.
• Equation (29) is particularly interesting be-
cause it uncovers what we are actually test-
ing: We test whether our market proxy is so
far away from the ex post efficient portfolio
that we are not willing to believe that it is
the population tangency portfolio, where the
distance is measured in terms of the Sharpe
Ratio.
‖Gibbons/Ross/Shanken (1989): A Test of the Effi-ciency of a Given Portfolio. Econometrica 57, 1121-1152.
Proof of (29)
• Comparing (27) and (29), the equality be-tween these quantities follows if we showthat α′
Σ−1
α = θ?2 − θ2m.
• Let r = [rm, r′]′. The (sample) covariancematrix of these variables is
V =
[σ2
m σ2mβ′
σ2mβ Σ + σ2
mββ′
]. (30)
• We know that the efficient portfolio usingthe assets in r is characterized by the weightvector
w =V −1r
1′V −1r, (31)
and, thus, it has squared Sharpe ratio
θ?2 =(w′r)2
w′V w=
(r′V −1r)2
r′V −1r= r′V −1r. (32)
• Next, it is easily checked that the inverse of(30) is
V −1 =
[σ−2
m + β′Σ−1β −β′Σ−1
−Σ−1β Σ−1
](33)
• Using (33), we get by straightforward com-
putation, and using (32),
θ?2 = r′V −1r = [rm, r′]V −1[rm, r′]′
=r2mσ2
m+ (r − βrm)′Σ−1(r − βrm)
= θ2m + α′Σ−1α,
recalling that α = r − βrm.
Relation between F and LR tests
• The finite–sample F test can also be inter-
preted as a likelihood ratio test.
• To see this, first note that for the uncon-
strained MLE of β, denoted by β1,
β1 =
∑t(rm,t − rm)rt
T σ2m
=
∑t rm,trt − rm
∑t rt
T σ2m
=r2mσ2
mβ0 −
rmr
σ2m
=r2mσ2
mβ0 −
rm
σ2m
(r − β1rm)− r2mσ2
mβ1
=r2mσ2
mβ0 −
rm
σ2m
α− r2mσ2
mβ1.
Rearranging and using the basic identity
σ2m = r2m − r2m =
1
T
T∑
t=1
r2m,t − r2m,
shows that
β0 = β1 +rm
σ2m + r2m
α. (34)
Inserting (34) into Σ0 (see equation (23))
and noting that the normal equations (12)
and (13) imply
T∑
t=1
(rt − α− β1rm,t)′(1− rmrm,t
r2m + σ2m
)α = 0,
we arrive at
Σ0 = Σ1 +
(σ2
m
r2m + σ2m
)αα′. (35)
From our discussion of the Common Cor-
relation Model, we are already familiar with
the Sherman–Morrison formula for the de-
terminant, |A+uv′| = |A|(1+v′A−1u), which,
applied to (35), gives
|Σ0| = |Σ1|[1 +
σ2m
r2m + σ2m
α′Σ−11 α
]
Thus, (25) may be written as
LR = T log|Σ0||Σ1|
= T log
[1 +
σ2m
r2m + σ2m
α′Σ−11 α
]
= T log
[N
T −N − 1J + 1
],
where J is the F–statistic given by (27), or,
equivalently,
J =T −N − 1
N
[exp
LRT
− 1
], (36)
which, as (36) is a monotonic transforma-
tion of LR, shows that J may also be inter-
preted as a likelihood ratio test.
• As the F–test based on (27) is exact, it is,
for realistic sample sizes, clearly preferable
compared to the likelihood ratio test relying
on asymptotic arguments.
Cross-Sectional Regressions
• The classical tests (with Fama and Mac-
beth (1973)∗∗ one of the most prominent
examples) of the CAPM concentrated on the
implication of the model that “beta” com-
pletely captures the variation of expected ex-
cess returns.
• To outline the procedure, assume initially
that the betas are known.
• Then, for each point of time, t, we esti-
mate a cross-sectional regression, where we
regress the excess returns of the N assets
on the betas,
ri,t = γ0,t + γ1,tβi + εi, i = 1, . . . , N. (37)
This gives us estimates γ0,t and γ1,t.
∗∗Fama, E. and J. MacBeth (1973). Risk, Return,and Equilibrium: Empirical Tests. Journal of Polit-ical Economy 71, 607-636.
• Running the regression (37) for t = 1, . . . , T ,
we obtain the time series of regression coef-
ficients, γ0,tt=1,...,T and γ1,tt=1,...,T .
• Implications of the CAPM are E(γ0) = 0 and
E(γ1) > 0 (positive risk premium), which
can be tested by standard t–tests, assuming
that the returns are normally distributed.
• Note that, in the regression (37), further
variables can be included to investigate whether
beta completely captures cross-sectional vari-
ation in expected excess returns. For exam-
ple, we can consider the extended version of
(37)
ri,t = γ0,t + γ1,tβi + γ2,tβ2i + γ3,tσi + εi,
i = 1, . . . , N. (38)
Here, with the inclusion of the squared βi,
one can test whether there is a nonlinear
relationship between risk and expected (ex-
cess) return (i.e., the security market line is
nonlinear), while the σi represents the firm–
specific risk, which, according to the CAPM,
does not produce excess return. Thus, in
(38), we test for E(γ0) = E(γ2) = E(γ3) =
0.
• The cross-sectional regression methodology
has the drawback that, in practice, the betas
are not known but have to be estimated.
• This introduces an errors-in-variables prob-
lem which gives rise to biased estimators of
the γi’s in the second-pass regression (37)
or (38).
Roll’s Critique
• Roll (1977)†† emphasizes that tests of the
CAPM really only reject the mean–variance
efficiency of the market proxy we use in the
test (recall equation (29)).
• This implies that the CAPM is essentially
untestable, because “the theory is not testable
unless the exact composition of the true mar-
ket portfolio is known and used in the tests.
This implies that the theory is not testable
unless all individual assets are included in the
sample”.
• Roll argues that using a proxy for the market
portfolio is subject to two difficulties: “First,
the proxy itself might be mean–variance ef-
ficient even when the true market portfolio
††R. Roll (1977). A Critique of the Asset Pricing The-ory’s Tests. Part I: On Past and Potential Testabilityof the Theory. Journal of Financial Economics 4, 129-176.
is not. This is a real danger since every
sample will display efficient portfolios that
satisfy perfectly all of the theory’s implica-
tions. (...) On the other hand, the chosen
proxy may turn out to be inefficient; but ob-
viously, this alone implies nothing about the
true market portfolio’s efficiency”.
• Thus, what we essentially test is a joint hy-
pothesis: The CAPM and the hypothsis that
the portfolio used in the tests as the market
proxy is the true market portfolio.
• When we reject this hypothesis, we can con-
clude that either
(a) the CAPM is false, or
(b) the portfolio used was not the true mar-
ket portfolio, or
(c) both (a) and (b).
• Clearly, it is extremely difficult to measure
the “market portfolio”, because this entity
can, in principle, include not just traded fi-
nancial assets, but also consumer durables,
real estate, and human capital.
Homework:
Read the following paper:
http://gsbwww.uchicago.edu/fac/finance/
papers/capm2004a.pdf.
You may also consider this one:
http://minneapolisfed.org/research/QR/QR1941.pdf
References
• Campbell/Lo/MacKinlay (1997). The Econo-
metrics of Financial Markets. Princeton Uni-
versity Press: Princeton.
• Greene, W. H. (2003). Econometric Analy-
sis, Fifth Edition. Upper Saddle River: Pren-
tice Hall.
• Theil, H. (1971). Principles of Economet-
rics. Amsterdam: John Wiley & Sons.
• Zellner, A. (1971). Introduction to Bayesain
Inference in Econometrics. New York: John
Wiley & Sons.