Validating the CAPM and the Fama-French three-factor model Michael Michaelides Department of Economics, Virginia Tech, USA Aris Spanos Department of Economics, Virginia Tech, USA January 2016 Abstract The primary aim of this paper is to revisit the empirical adequacy of the structural CAPM and the Fama-French three-factor model. By distinguishing between the structural and statistical models, where the latter comprises the probabilistic assumptions imposed on the data, the paper shows that these structural models are statistically misspecified. This raises questions about the Fama-French three-factor model which was justified in terms of the significance of the added factors. If the original CAPM is statistically misspecified, however, the significance of the added factors is questionable because it is likely that the nominal error probabilities used by these tests are very different from the actual ones. The paper respecifies the statistical models underlying the CAPM and Fama-French three-factor structural models. The respecification results in a Student’s t VAR(1) and Dynamic Linear Regression models which provide a sound basis for revisiting the empirical adequacy of the structural models. The statistical inference results, based on the Fama-French data, call into question the empirical adequacy of these models. 1
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Validating the CAPM and the Fama-French
three-factor model
Michael Michaelides
Department of Economics,
Virginia Tech, USA
Aris Spanos
Department of Economics,
Virginia Tech, USA
January 2016
Abstract
The primary aim of this paper is to revisit the empirical adequacy of the
structural CAPM and the Fama-French three-factor model. By distinguishing
between the structural and statistical models, where the latter comprises the
probabilistic assumptions imposed on the data, the paper shows that these
structural models are statistically misspecified. This raises questions about the
Fama-French three-factor model which was justified in terms of the significance
of the added factors. If the original CAPM is statistically misspecified, however,
the significance of the added factors is questionable because it is likely that the
nominal error probabilities used by these tests are very different from the actual
ones. The paper respecifies the statistical models underlying the CAPM and
Fama-French three-factor structural models. The respecification results in a
Student’s t VAR(1) and Dynamic Linear Regression models which provide a
sound basis for revisiting the empirical adequacy of the structural models. The
statistical inference results, based on the Fama-French data, call into question
the empirical adequacy of these models.
1
1 IntroductionAsset pricing theory is concerned with the analysis of prices or values of claims on
financial assets in a world of uncertainty. Evaluation of financial asset riskiness and
associated premium is one of the most fundamental issues faced by individual investors
and financial institutions for both pricing and investing in such assets. How does
the risk of an investment affects, or should affect, its expected return? Why do
some financial assets tend to pay higher average returns than others? How are the
various types of risk measured? Asset pricing theory is concerned with providing
answers to these questions and continues to be one of the most important theoretical
underpinning of portfolio management and investment analysis.
The early work of Markowitz (1952) marked a new era in asset pricing theory. In
his pioneering paper, Markowitz argued that a risk averse investor who focuses on
the mean and variance of the returns of individual assets contained in a portfolio, is
motivated by the highest possible profit given the lowest possible risk. By providing
a mathematical justification, he showed that an investor wants to maximize the mean
and minimize the variance, and therefore, the optimal portfolio is one whose mean-
variance combination is in the efficient frontier. Yet, it was Tobin’s (1958) separation
theorem that extended the logic of efficient frontier using one’s attitude towards risk.
He demonstrated that an investor who already holds the optimal combination of risky
assets and is able to borrow or lend at the risk free rate, can decide whether to borrow
or lend, depending on his attitude towards risk. His main conclusion was that the
single market portfolio is the efficient frontier that dominates any other combination.
Although the groundwork of Markowitz and Tobin was simple and intuitive, it
presented difficulties in the computation of the variance-covariance matrix when the
number of assets was very large. The latter limitation, together with the absence of
a theory for estimating the correct cost of capital of an investment in the influential
work of Modigliani and Miller (1958), provided the primary motivation for Treynor
(1962, published 1999), Sharpe (1964), Lintner (1965a, 1965b) and Mossin (1966), to
independently introduce the most popular model in the asset pricing literature; the
Capital Asset Pricing Model (CAPM).
The theoretical CAPM has the following form:
(−)=(−) =1 (1)
where, is the expected return of asset , is the expected return of the risk free
asset, (−) is the expected excess asset return (risk premium) of asset is thebeta coefficient of asset , is the expected return of the market, and (−) isthe expected excess market return (market premium).
The model relates − and − via , which is a standardized measure ofsystematic risk - not the total risk - based upon an asset’s covariance with the mar-
ket portfolio. The theoretical underpinnings of the CAPM depended on a number
of strong assumptions. First, investors are rational and risk averse, they all share
homogeneous expectations, and behave in a manner as to maximize their economic
2
utility. Second, they are broadly diversified across a range of assets and can bor-
row/lend unlimited amounts under the risk free rate of interest. Third, the stock
market functions under conditions of perfect competition and equilibrium. Investors,
- who all have access to available information - have the right to buy or/and sell any
amount of shares, at any period of time, having a very little impact, if any, on the
market price. Fourth, investors are not burdened of any transaction costs and at
the same time taxation and any other sort of restrictions are similarly absent. Fifth,
returns are Normally distributed, with known means, variances and covariances.
For inference purposes the CAPM is usually embedded into a statistical model of
the form:
(−)= + (−) + =1 =1 (2)
where, is the returns of asset , is the returns of a risk free asset, is
the intercept (Jensen’s alpha), is the beta coefficient and is the returns of
the market. The error term, , is assumed to satisfy the following probabilistic
assumptions:
(i) ()=0 (ii) ()=2−22 (iii) ( −)=0
(iv) ( )6=0 for 6= =1 (v) ( )=0 for 6= =1 (3)
Despite its simplicity, the CAPM is hampered by its very strong set of assump-
tions. The fact that every assumption is likely to be violated in the real world,
motivated academics and professionals to extend the CAPM, in an attempt to im-
prove it by relaxing some of the assumptions. Some of the well-known extensions to
the model include, the Black (1972) CAPM, the Intertemporal CAPM (ICAPM) by
Merton (1973), and the Consumption CAPM (CCAPM) by Breeden (1979). Further,
subsequent work by Ross (1976), Chen et al. (1986), Fama and French (1992, 1993),
Carhart (1997), among others, has introduced several sources of risk - called factors -
that determine returns besides the original market beta. The extended CAPM takes
the generic form:
−=+1(−)+P
=2 + =1 =1 (4)
Did the CAPM developed into a better pricing model since its introduction? From
the theory perspective, the extensions of CAPM provide a clearer, more refined, and
theoretically a more appealing framework which attempts to bridge the gap between
theory and data in more elaborated ways.
The bridging of the gap between theory and data, however, has a weak link. The
validity of the probabilistic assumptions (directly or indirectly) imposed on the data
is not thoroughly tested. As a result, the various improvements of the original CAPM
have not been subjected to adequate empirical scrutiny. The inclusion of additional
factors provides sound empirical improvements to the original structural model only
when they are statistically justified. The latter requires that the estimated model
3
in the context of which the additional factors are tested is statistically adequate: its
probabilistic assumptions are valid for the data in question.
The primary objective of this paper is to revisit the CAPM and the Fama-French
extension with a view to evaluate their empirical validity vis-a-vis the data by thor-
oughly testing the probabilistic assumptions (implicitly) imposed on the data. It is
shown that both the original CAPM and the Fama-French three-factor models are
empirically invalidated using the Fama and French (1993) data.
2 Statistical vs. Substantive modelsThe current situation where a modeler attempts to improve the scope of a theoretical
(structural) model by adding potentially relevant variables has a serious flaw. The
reliability of the significance test for the coefficients of such variables depends crucially
on whether the original model is statistically adequate. When that is not the case,
the tests of significance are likely to be unreliable because of sizeable discrepancies
between their nominal and actual error probabilities. This serious flaw stems from
not distinguishing between a theoretical (structural) model and a statistical model;
see Spanos and McGuirk (2001), Spanos (2009).
A structural model is a mathematical formulation of a theory that aims to ap-
proximate a real-world phenomenon of interest with a view to provide an adequate
explanation. Justifiably, such models could be described as simplifications of the
real world because they typically rely on strong assumptions. Moreover, by replacing
some of the unrealistic assumptions with more realistic ones the theory’s fecundity
can be enhanced. On the other hand, a statistical model represents the probabilistic
assumptions imposed (directly or indirectly) on the data. Its role is to ensure the
error-reliability of all statistical inferences based on the estimated model since the in-
ference procedures invoke these assumptions. In practice, these assumptions are not
clearly brought out because they are imposed on the data indirectly via the structural
model in conjunction with the error term assumptions, such as (3), and not the ob-
servable random variables involved. Hence, the statistical premises are rarely tested
thoroughly, and as a result the endeavors to enhance the substantive adequacy of
the estimated structural model are often of questionable value on empirical grounds.
To avoid this conundrum one needs to establish the adequacy of the implicit statis-
tical model before probing for the appropriateness of substantive refinements of the
original structural model.
In the asset pricing literature, more often than not, statistical adequacy is not
properly secured. As a result, this often undermines the credibility of inferences
relating to the inclusion of additional potentially relevant variables. Indeed, no trust-
worthy evidence for or against the substantive theory can be secured when the implicit
statistical model is misspecified.
In order to avoid unreliable inference results, one needs to test the validity of the
probabilistic assumptions imposed on the particular data, or more precisely, on the
stochastic process underlying the data. In the case of the statistical CAPM and its
4
extensions, the statistical Generating Mechanism (GM) used in the literature is:
= + β>X + =1 2 ∈N:=(1 2 )
where B>:=(>1 >2
>) and X:=(12 ) Hence, the probabilistic as-
sumptions imposed on the underlying observable process are given in table 1.
Table 1 - Multivariate Normal/Linear Regression (LR) Model
In summary, the joint M-S testing using auxiliary regressions indicate clearly that
the Multivariate Normal LR model suffers from serious statistical misspecifications.
The residuals indicate major departures from the all the statistical assumptions and
the only assumption which seems to hold is Linearity. Taken together, the above M-S
testing results call into question the empirical credibility of the Fama-French exten-
sion of the CAPM because their substantive adequacy was based on a statistically
11
misspecified model. Before the Fama-French three-factor model can be substantively
assessed in an empirically reliable way, one needs to respecify the implicit statistical
model associated with the CAPM with a view to account for the chance regularities
in the data that are not accounted for by the original model.
3.2 Respecification: Normal VAR(1)In light of the act that the Normality plays a crucial role in the original Markowitz
(1952) optimal portfolio theory as well as the CAPM and its extensions, an attempt
will be made to account for all the other departures indicated by the above M-S
testing results without changing Normality.
Table 9 - Normal Vector Autoregressive [VAR(1)] model
Statistical GM: Z=α0+
X=1
γ+
−1X=1
δ+A>1 Z−1+u ∈N
[1] Normality (Z|Z0−1;θ) Z0−1:=(Z−1 Z1) is Normal,
[2] Linearity (Z|(Z0−1))=α0+X=1
γ+
−1X=1
δ+A>1 Z−1
[3] Homosk/city (Z|(Z0−1))=Ω is free of Z0−1
[4] Markov Z ∈N is a Markov process,
[5] t-invariance: θ:=
½(i) coefficients (α0A1)
(ii) covariance (Ω)
α0 = μ−A>1 μ A1 = Σ−10 Σ1 Ω = Σ0−Σ>
1Σ−10 Σ1
The respecification takes the form of replacing the original reduction assumptions im-
plicitly imposed on the observable stochastic process Z:=(X) =1 2 ∈Nwith more appropriate ones. Departures from assumptions [4] and [5] suggest that the
IID assumptions are clearly invalid and need to be replaced with more appropriate
assumptions that allow for temporal dependence and certain forms of heterogeneity.
The generic way to do that is to use lags, trends and seasonal dummy variables. For-
mally, assuming that Z:=(X) =1 2 ∈N is Normal, Markov (M) andmean heterogeneous but covariance stationary (S) gives rise to the Normal VAR(1)
heterogeneous model given in table 9. The probabilistic reduction takes the form:
(Z1Z2 Z;φ(t))= (Z1;ψ1)
Q
=2(Z|Z−1;ψ(t)) == (Z1;ψ1)
Q
=2(Z|Z−1;ψ(t))(15)
where ψ(t) includes both the Gram-Schmidt trend polynomials and seasonal dum-
mies. Note that making the further reduction:
Q=2
(Z|Z−1;ψ(t)) =Q=2
(|XZ−1;ψ1())·(X|Z−1;ψ2()) (16)
12
gives rise to the Dynamic Normal/Linear Regression model specified in terms of
(|XZ−1;ψ1) =1 2
Table 10 - Dynamic Normal/Linear Regression model
Statistical GM: =0+β>1x + β
>2Z−1 + ∈N,
[1] Normality: (|xZ−1;θ) is Normal,[2] Linearity: (|X=xZ−1)=0+β
>1x+β
>2Z−1
[3] Homosked.: (|X=xZ−1)= 2 [4] Markov: Z ∈N is a Markov process,[5] t-invariance: θ:=(0β1β2
2 ) are constant ∀∈N.
To appraise the statistical adequacy of the Normal VAR(1) model in table 9, anal-
ogous M-S testing is applied. It is important to emphasize that the choice of the
degree of the trend polynomials and the number of lags need to be decided on statis-
tical adequacy grounds. The M-S results for the selected portfolios are presented in
table 11.Table 11 - M-S testing for Normality
[1.1] 0969[000] 0977[000] 0925[000]
[1.2] 1996[000] 0994[013] 5252[000]
[1.3] 23061[000] 29927[000] 45276[000]
[1.1] 0977[000] 0979[000] 0925[000]
22 [1.2] 0891[023] 0863[027] 5486[000]
[1.3] 30595[000] 29741[000] 45201[000]
[1.1] 0976[000] 0978[000] 0935[000]
33 [1.2] 1298[002] 0903[021] 4846[000]
[1.3] 31363[000] 30033[000] 40581[000]
[1.1] 0987[004] 0977[000] 0950[000]
44 [1.2] 0686[073] 1047[009] 3676[000]
[1.3] 13651[001] 31281[000] 33738[000]
[1.1] 0991[041] 0977[000] 0925[000]
[1.2] 0606[115] 1020[011] 5148[000]
[1.3] 8921[012] 31184[000] 46105[000]
The heterogeneous Normal VAR(1) model appears to account for the dependence
and heterogeneity in the data. The results indicate that mean homogeneity, includ-
ing seasonality, for all the portfolios is accounted for while variance homogeneity is
accounted for with a few exceptions.
The distribution of the residuals violates the assumption of Normality, without
any exceptions. The p-values are very close to zero, for all the equations in the system
and for all the portfolios. Likewise, the assumption of homoskedasticity is seriously
violated for at least two equations in the system of each portfolio. Moreover, the
assumption of linearity is fully satisfied for two individual equations in the system
13
model but often violated for the third equation. A possible explanation for this
violation can be the skewness of the probability distribution of or a symptom of
the existence of dynamic heteroskedasticity in the form of second order dependence.
Given that the trend polynomials and seasonal dummies are statistically significant,
the initial results of Fama and French are called into question.
5 Fama-French model: substantive adequacyIn order to embed the CAPM in the statistical adequate model, the Student’s t VAR
needs to be reparameterized into a system of Student’s t Dynamic Linear Regression
(DLR) model for =1 :
=+1+2+
X=1
+
−1X=1
+3−1+4−1+5−1+ ∈N(18)
17
One can use the estimated Student’s t DLR model as a sound basis for testing the
CAPM and Fama-French model over-identifying restrictions:
0: G(θϕ)=0 vs. 1: G(θϕ)6=0 for θ∈Θ ϕ∈ΦThe statistical adequacy of the Student’s t DLR model ensures the reliability of
inference; the actual error probabilities approximate closely the nominal ones. The
relevant test is based on the likelihood ratio statistic:
(Z)=max∈Φ (;Z)
max∈Θ (;Z)=
(;Z)(;Z) ⇒−2 ln(Z) 0v
2() (19)
For =20 and =01 =3756 the observed test statistic for the CAPM yields:
−2 ln(Z0)=244863[0000000] (20)
and for the Fama-French model:
−2 ln(Z0)=151453[0000000] (21)
where the number in square brackets denotes the p-value. These testing results are
typical of the results pertaining to the validity of the over-identifying restrictions
imposed by the original CAPM and the Fama-French three-factor model. The tiny
p-values suggest that the data provide strong evidence against all these structural
models.
The DLR model in (18), as a reparameterization of a statistically adequate model
[Student’s t VAR(1)], is also statistically adequate. The validity of statistical premises
secures the error-reliability of the resulting inferences. The latter allows one to pose
questions of substantive adequacy. By estimating the relevant statistically adequate
model of CAPM and by viewing the omitted variables problem as a substantive
misspecification problem (Spanos, 2006b), one can draw reliable conclusions.
In the case of the statistically adequate CAPM, the intercepts of the smallest
size and highest BE/ME portfolios are significantly greater than the intercepts of the
biggest size and lowest BE/ME portfolios, respectively. By including the SMB and
HML to the model, the magnitude of some of the intercepts decreases significantly but
there is no sign of convergece to any values close to 0 The latter comes as no surpise
since the magnitude of the intercepts is affected by the trends, dummies and lags in
the model. What is more interesting, is the considerable change in terms of their
statistical significance. More than half of the intercepts are statistically significant
for the CAPM, while the significance diminishes for the three-factor model; only 4
intercepts are significant.
The convergence toward 1 of the estimated coefficients for the market might be
indicative of the strong correlations between the market and SMB or HML, but
evaluating these correlations is misleading in light of the fact that the mean of the
data is not constant. The additional restriction resulting frommodeling excess returns
turns out to be empirical invalid for every portfolio that has been estimated. This
is clearly demonstrated in table 15 where the estimated coefficients of the various
models are shown for one such portfolio. The only reliable comparisons one can make
18
is on the basis of the unrestricted Student’s t DLR models, which are statistically
adequate. In the case of the CAPM, when the two unrestricted estimated coefficients
of and are added up the result is −121 and for the three-factor model −26;both of which are miles away from 1. The approximate equality of the estimated
coefficients between the restricted Normal LR and restricted Student’s t DLR provides
no indication that the Fama-French findings are ball park correct, since both models
are statistically misspecified. The overidentifying restrictions tests (20)-(21), based
on Student’s t DLR model, show that both restricted forms of the Student’s t DLR
model are statistically misspecified, and one learns nothing pertaining to empirical