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Deflated and Probabilistic Sharpe Ratios

A reported Sharpe ratio is an estimate, not a fact: computed from a finite, skewed, fat-tailed sample, after a search. Two related tools turn that sentence into numbers.

The Sharpe ratio is a random variable

An observed Sharpe SR^\widehat{SR} over TT periods estimates the true one with error, and the error bars widen when returns are non-normal. The standard deviation of the estimator is approximately

σSR^≈1−γ3SR^+γ4−14SR^2T−1\sigma_{\widehat{SR}} \approx \sqrt{\frac{1 - \gamma_3 \widehat{SR} + \frac{\gamma_4 - 1}{4}\widehat{SR}^2}{T - 1}}

where γ3\gamma_3 is the skewness of returns and γ4\gamma_4 the kurtosis. Negative skew and fat tails, the signature of strategies that sell insurance, inflate the error: a smooth premium-collecting return stream reports a precise-looking Sharpe with a wide true uncertainty.

The probabilistic Sharpe ratio

The probabilistic Sharpe ratio uses this to answer a clean question: what is the probability that the true Sharpe exceeds some benchmark SR∗SR^*?

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