Alternative Methods to Derive Statistical Distribution of Sharpe Performance Measure: Review, Comparison, and Extension

Abstract

A crucial assumption for the single-period representation of a portfolio’s performance, namely, the Sharpe ratio, to be valid is the stationary in the distributions of a portfolio’s excess returns R1,…,RT. In literature, the most often adopted stationary assumption is the i.i.d. excess returns R1,…,RT. When the excess returns R1,…,RT are i.i.d. normally distributed, Lee and Chen (1979) and Jobson and Korkie (1981) derived the exact non-central t and the asymptotic normal distribution, respectively, for the ex post Sharpe ratio. Lo (2002) derived the same asymptotic normal distribution as that of Jobson and Korkie (1981) under the i.i.d. non-normal assumption. Merten (2002) improves the asymptotic normal distribution under the i.i.d. non-normal assumption, which takes into consideration the skewed and leptokurtic characteristics of non-normal excess returns. Lo (2002), Christie (2005) and Opdyke (2007) considered a less restricted stationary assumption for the excess returns R1,…,RT, i.e., strictly-stationary-ergodicity, and derived the asymptotic normal distribution of the ex post Sharpe ratio. This study reviews the previous distributional results and concludes that the asymptotic normal distributions under the i.i.d. and the strictly-stationary-ergodic assumptions coincide. In addition, this study improves the aforementioned asymptotic normal distributions by adopting the least restricted stationary assumption on the excess returns R1,…,RT, namely, weakly-stationary-ergodicity. Using monthly excess returns of the value-weighted index of all the CRSP firms from Jan. 1947 to Dec. 1968, Jan. 1969 to Dec. 1990, and Jan. 1991 to Dec. 2012, respectively, it is shown the excess returns in the three time periods are neither iid nor strictly stationary. On the other hand, the weakly stationary assumption holds for the first time period, and is slightly violated for the second time period. By comparing the asymptotic variances of the asymptotic normal distributions during the first and the second time periods, the derived asymptotic normal distribution fits better than those derived under the i.i.d. or strictly-stationary-ergodic assumption. It is hoped more accurate statistical inference of the Sharpe performance measure can be obtained.