Empirical likelihood based nonparametric testing for
CAPM
Pietro Coretto and Maria Lucia ParrellaAbstract.The Capital Asset Pricing Model (CAPM) predicts a linear relation between assets’
return and their betas. However, there is empirical evidence that such a relationship does not
necessarily occur, and in some cases it might even be nonlinear. In this paper we explore
a nonparametric approach where the linear specification is tested against a nonparametric
alternative. This methodology is implemented on S&P500 data.Key words:CAPM, goodness-of-fit test, empirical likelihood1 Introduction
An asset pricing model provides a method for assessing the riskiness of cash flows
from a project. The model provides an estimate of the relationship between that risk-
iness and the cost of capital. According to the “capital asset pricing model” (CAPM),
the only relevant measure of a project’s risk is a variable unique to this model, known
as the project’s beta. In the CAPM, the cost of capital, i.e., the return, is a linear func-
tion of the the beta of the project being evaluated. A manager who has an estimate
of the beta of a potential project can use the CAPM to estimate the cost of capital for
the project. If the CAPM captures investors’ behaviour adequately, then the historical
data should reveal a positive linear relation between return on financial assets and
their betas. Also, no other measure of riskshould be able to explain the differences in
average returns across financial assets that are not explained by CAPM betas. The fact
that CAPM theory predicts the existence of a cross-section linear relation between
returns and betas can be empirically tested. To this end we propose a nonparametric
testing methodology (see [10] and [3] among others).
The first test of the CAPM was run by Fama and MacBeth [7] and their study
validated the theory. The authors tested the linearity against some parametricnonlinear
alternatives. However subsequent empirical analysis highlighted that the validity of
the CAPM could depend on the testing period. There is a huge amount of literature
on this topic (for a comprehensive review see [11]), however, final conclusions have
not been made.M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010