Capital Asset Pricing Model: Nonparametric Estimation of Beta Parameter by the Bootstrap Statistical Methodology

Abstract

Investors are interested in receiving the highest return on a portfolio, subject on some constraint in the risk involved. The Capital Asset Pricing Model (CAPM) is a linear model of the form: rs = α βrm ε, where rs, rm, are the returns of a particular security (s) and of the market as a whole correspondingly, and ε is a random error with E(c) = 0 and var(ε) = σ2 . The above model states that the risk for any security can he separated into two parts: the systematic risk from the market as a whole and the specific risk of the particular security. The systematic risk is measured by the coefficient β (beta) in the CAPM model and the specific risk by the standard error of estimation Sc. The parameter beta of a security measures its responsiveness to changes in the market. It is very important for an investor to estimate, as accurately as possibly the parameter beta because it can assess the competitiveness of a company.

An investor holding stocks with beta values greater than I is taking on aggressive stance: holding stocks with beta values less than I represents a defensive stance.

In this paper, a nonparametric method, called Bootstrap (Efron, 1982), is employed to estimate the parameters beta of seven assets of London Stock Exchange. The coefficients α and β, the standard errors of the estimates a and b and the 95% Confidence Intervals for the parameter β are estimated by the Classical least Squares and Weighted Least Squares Methods, as well as, by the Bootstrap Method. The findings are: (I) Bootstrap Method produces 95% Confidence Intervals for β which are shorter in length than the Confidence Intervals produced by the more Classical Methods. (ii) The estimate of the Standard Error for beta is similar for the Classical and Bootstrap Methods. (iii) The computer intensive method of Bootstrap frees the researcher from the need to assume the Normality of the errors in the CAPM model, because of its non-parametric nature.

Keywords: CAPM Model, Bootstrap, Risk

JEL Classification: E22, G31, J24, J41