The Discrete Time Beta

Abstract

Empirical studies of the CAPM commonly utilise returns measured over discrete time intervals. In such a sampled data frame, the correct choice of the risk free rate is far from clear-it is commonly taken as the yield on a discount bill or bond, whose time to maturity coincides with the sampling interval. It is also unclear whether use of discrete data preserves the independence of the CAPM relationship from idiosyncratic variance elements. This paper relates the continuous beta to what one would observe from discrete data, assuming continuous underlying no-arbitrage trading processes. It is shown that the risk free rate can be eliminated, as a nuisance variable, and the CAPM cast in tenns of two benchmarks, namely the market rate of return and the yield on a default free discount bond; the latter more or less corresponds to the 'risk free rate' as it appears in the usual empirical study. In such tenns, the bond beta contaminates the measured stock beta, powering up the latter, so that high beta stocks appear even higher, and low beta stocks even lower. It is shown that the bond beta is equal to one half times the proportion of the overall market portfolio devoted to risky assets. Thus one can expect the powering up - and hence the measured stock betas -to vary over the business cycle. Moreover, idiosyncratic variance elements · appear as an intercept term in the CAPM relationship, though this effect vanishes as the sampling interval becomes small.