Statistical Inference for Sharpe Ratio

Sharpe ratios (Sharpe 1966) are the most popular risk-adjusted performance measure for investment portfolios and investment funds. Given a riskless security as a benchmark, its Sharpe ratio is defined by S R = μ − z σ 2 $$SR = \frac{{\mu - z}}{{\sqrt {{\sigma ^2}} }}$$ where μ and σ2 denote the portfolio’s mean return and return volatility, respectively, and z represents the riskless return of the benchmark security. From an investor’s point of view, a Sharpe ratio describes how well the return of an investment portfolio compensates the investor for the risk he takes. Financial information systems, for example, publish lists where investment funds are ranked by their Sharpe ratios. Investors are then advised to invest into funds with a high Sharpe ratio. The rationale behind this is that, if the historical returns of two funds are compared to the same benchmark, the fund with the higher Sharpe ratio yields a higher return for the same amount of risk. Though (ex post) Sharpe ratios are computed using historical returns, it is assumed that they have a predictive ability (ex ante). We refer to Sharpe (1994) for related discussions and further references.