Effects of Measurement Errors on Systematic Risk and Performance Measure of a Portfolio

In this paper, we examine the effects of errors in measurement of the two independent variables, return on market (Rm) and return on risk-free assets (Rf), in the traditional one-factor capital asset pricing model (CAPM). After discussing Sharpe-Lintner's CAPM and both Jensen and Fama's specifications thereof, we review briefly the recent results of Friend and Blume [6], hereafter FB; Black, Jensen and Scholes [1], hereafter BJS; and Miller and Scholes (11], hereafter MS. In Section II, we first explore possible sources of measurement errors for both Rm and Rf; then we specify these errors mathematically and derive analytically their effects on estimates of systematic risk of a security or portfolio, , and the Jensen's measure of performance, . In Section III, we derive an analytical expression for the regression coefficient of estimated b's where we estimate the equation . The result is then examined to find the conditions under which errors in measurement of Rm and Rf can cause b to have a positive or negative value even if the true b is zero. The conditions are then used to examine FB's results and their interpretation. In Section IV, an alternative hypothesis testing procedure for the CAPM is examined. We show that the empirical results so derived are also affected by the measurement errors and the sample variation of the systematic risk. The relative advantage between the two different testing hypothesis procedures is then explored. Finally, we comment on the relevance of the result to the popular zero-beta model and indicate areas for further research.