The Sampling Relationship Between Sharpe's Performance Measure and its Risk Proxy: Sample Size, Investment Horizon and Market Conditions

Abstract

Sharpe's, Treynor's and Jensen's measures have been extensively used for the performance evaluation of mutual funds or portfolios. These three widely used performance measures have been found to be highly correlated with their corresponding risk measures by a number of empirical studies. This paper focuses the investigation on the possible sources of the bias associated with the empirical relationship between the estimated Sharpe's measure and its estimated risk measure. In general, the sample size, the investment horizon and the market conditions are three important factors in determining the strong relationship between the ex-post Sharpe's measure and its estimated risk surrogate. The interesting findings of this study are as follows: (1) the estimated Sharpe's measure is uncorrelated with the estimated risk measure either when the risk free rate of interest equals the expected return on the market portfolio over the sample period or when the sample size is infinite, (2) the estimated Sharpe's measure is positively (or negatively) correlated with the estimated risk measure if the risk-free rate of interest is greater than (or less than) the expected return on the market portfolio, (3) an observation horizon shorter than the true investment horizon can reduce the dependence of the estimated Sharpe's measure on its estimated risk measure, and (4) an observation horizon longer than the true investment horizon will magnify the dependence. The results have indicated that, in conducting empirical research, a shorter observation horizon and a large sample size should be used to reduce the bias associated with the estimated Sharpe's measure.