The Predictable and Misleading Consequences When Using Periodic Returns in Traditional Tests of the Capital Asset Pricing Model

Abstract

This paper offers a new understanding of the statistical results obtained by Black, Jensen, and Scholes (1972) and Miller and Scholes (1972) in their tests of the Capital Asset Pricing Model (CAPM). Numerical examples and simulations are used to illustrate how the empirical results of both papers are what we should expect to observe under the null hypothesis that the CAPM does not hold and the returns, either periodic or continuous, are variable. This paper employs geometric Brownian motion (GBM) as a reasonable first-order approximation for the return generating process to capture the predictable, asymmetrical effects of compounding variable returns. The empirical results of the cross-section and time-series tests in both papers can be explained by the simple mathematical dependence of periodic returns upon the underlying variance of the continuously compounded returns. The numerical examples and simulations reproduce the empirical results by matching the variance of the continuously compounded returns on the market factor in the model of GBM to the variance of the returns on the market portfolio in each sample period, although the securities are not priced in the model to reflect aversion to either beta or the variance of the continuously compounded returns.

Keywords: Asset pricing tests, CAPM, continuously compounded returns, periodic returns, geometric Brownian motion

JEL Classification: G12