104 P. Coretto and M.L. Parrella
The famous Fama-MacBeth contribution (and the following) tests the linear spec-
ification against a number of nonlinear parametric specifications. The main contri-
bution of this paper is that we test the linear specification of the CAPM against
a nonlinear nonparametric specification. And by this we do not confine the test to
a specific (restricting) nonlinear alternative. Our testing method is based on kernel
smoothing to form a nonparametric specification for the null hypothesis that the re-
lation between returns and betas is linear against the alternative hypothesis that there
is a deviation from the linearity predicted by the CAPM. We apply our methodology
to the S&P 500 market.
The paper is organised as follows: we introduce the theoretical model, we intro-
duce the Fama and MacBeth two-stage parametric estimation procedure, we outline
the nonparametric testing methodology and finally we discuss some empirical findings
based on the analysis of the S&P 500 market.
2 The CAPM in a nutshell
CAPM was first developed by Sharpe and Treynor; Mossin, Lintner and Black brought
the analysis further. For a comprehensive review see [5] and [1]. We will refer to SLB
as the Sharpe-Lintner-Black version of the model. The SLB model is based on the
assumption that there is a positive trade-off between any asset’s risk and its expected
return. In this model, the expected return on an asset is determined by three variables:
the risk-free rate of return, the expected return on the market portfolio and the asset’s
beta. The last one is a parameter that measures each asset’s systematic risk, i.e., the
share of the market portfolio’s variance determined by each asset.
2.1 Theoretical model
The CAPM equation is derived by imposing a number of assumptions that we discuss
briefly. An important building-block of the CAPM theory is the so-calledperfect
market hypothesis. This is about assuming away any kind of frictions in trading and
holding assets. Under the perfect market hypothesis, unlimited short-sales of risky
assets and risk-free assets are possible.
The second assumption is that all investors choose their portfolios based on mean
(which they like) and variance (which they do not like). This assumption means that
people’s choices are consistent with Von-Neumann- Morgenstern’s axiomatisation.
All investors make the same assessment of the return distribution. This is referred
to as “homogenous expectations”. The implication of this hypothesis is that we can
draw the same minimum-variance frontier for every investor.
Next is the “market equilibrium” hypothesis (i.e., supply of assets equals demand).
The market portfolio is defined as the portfolio of assets that are in positive net supply,
weighted by their market capitalisations. Usually it is assumed that the risk-free
instrument is in zero net supply. On the demand side, the net holdings of all investors
equal aggregate net demand. The last assumption states that all assets are marketable,
i.e., there is a market for each asset.