Asymptotic Likelihood Inference for Sharpe Ratio

The Sharpe ratio is one of the most widely used measures of the performance of an investment with respect to its return and risk. Since William Sharpe (1966) defined the ratio, as the funds excess return per unit of risk measured by standard deviation, investments have been often ranked and evaluated on the basis of Sharpe ratio by both private as well as institutional investors. Our study on Sharpe ratio estimator is focused on its finite sample statistical properties which have being given less attention in practice.

Approximations aimed at improving the accuracy of likelihood method have been proposed over the past three decades. Among them, Lugannani and Rice (1980) and Barndorff-Nielsen (1986) introduced two widely used tail area approximations with third order of convergence. Furthermore, Fraser(1988; 1990), Fraser and Reid (1995), Fraser, Reid and Wu (1999) improved their methods and developed a general tail probability methodology, based on the tangent exponential model.

The objective of this paper is to use the third order asymptotic likelihood-based statistical method to obtain highly accurate inference on Sharpe ratio. Since the methodology is demonstrated to work well generally for any parametric distribution, our study will assume the market log returns are independent identically distributed (IID) normal, or follow an autoregressive process of order one (AR(1)) with Gaussian white noise.

While most literature address large sample properties of the Sharpe ratio statistic (Lo 2002, Mertens 2002, Christie 2005, Bailey and Lopez de Prado 2012); it is important to compare the performance of investments when only small sample observations are available, especially before and after markets change direction. Our research would address this issue. New tests are developed for testing hypothesis on the Sharpe ratio calculated from one sample and on the difference of two Sharpe ratios. Comparison between our method and the currently existing methods in the literature are conducted by simulations. The p-values and confidence intervals for Sharpe ratio are calculated and various applications are illustrated.