Connecting Sharpe ratio and Student t-statistic, and beyond

TL;DR

This paper derives the exact distribution of the Sharpe ratio under normal return assumptions, relates it to the Student t-distribution, and explores its properties and estimation errors in various statistical settings.

Contribution

It provides the exact distribution of the Sharpe ratio for independent normal returns and extends analysis to autocorrelated and heteroscedastic data, linking it to the Student t-distribution.

Findings

Sharpe ratio follows a scaled non-centered Student distribution under normality.

Empirical Sharpe ratio asymptotically achieves the Cramer-Rao bound.

Derived formulas for Sharpe ratio distribution under autocorrelation and heteroscedasticity.

Abstract

Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions (independent normally distributed - and identically distributed returns). In this paper, we provide the exact distribution of the Sharpe ratio for independent normally distributed return. In this case, the Sharpe ratio statistic is up to a rescaling factor a non centered Student distribution whose characteristics have been widely studied by statisticians. The asymptotic behavior of our distribution provide the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general formula in this case of heteroscedasticity and autocorrelation.