What is the Probabilistic Sharpe Ratio?
The Probabilistic Sharpe Ratio (PSR) is the probability that a strategy's true Sharpe ratio exceeds a chosen benchmark, given the Sharpe you observed, the length of the track record, and the skewness and kurtosis of the returns. It converts a point estimate into a confidence statement. A Sharpe of 1.5 over 24 months with negatively skewed, fat-tailed returns can easily carry a PSR below 0.90, meaning the evidence does not support the claim at conventional confidence.
The problem it solves
A Sharpe ratio is an estimate from a sample, so it carries estimation error. Two strategies both reporting 1.5 are not equally credible if one has three years of history and the other has three months. Worse, the standard error of the Sharpe estimator depends on the shape of the return distribution: negative skew and excess kurtosis — exactly what option-selling and carry strategies produce — inflate it. Those strategies look best precisely where the metric is least reliable.
PSR, introduced by Bailey and López de Prado, restates the question: instead of “what is the Sharpe?” it asks “what is the probability the true Sharpe beats a threshold?”
Formula
Read the denominator carefully, because that is where the intuition lives.
Negative skew (g1 < 0) makes the term -g1*SR_hat positive,
which enlarges the denominator, shrinks the statistic and lowers PSR. Excess
kurtosis (g2 > 3) does the same. Longer track records raise
sqrt(n-1) and push PSR up.
Worked example
Two years of monthly returns, n = 24, both strategies reporting the same headline Sharpe of 1.50:
| Input | Strategy A | Strategy B |
|---|---|---|
| Observed Sharpe | 1.50 | 1.50 |
| Skewness γ₁ | 0.00 | −1.20 |
| Kurtosis γ₂ | 3.0 | 7.0 |
| Denominator | 1.4577 | 2.4850 |
| z, against SR* = 0 | 4.935 | 2.895 |
| PSR, against SR* = 0 | 1.0000 | 0.9981 |
| z, against SR* = 1.0 | 1.645 | 0.965 |
| PSR, against SR* = 1.0 | 0.9500 | 0.8327 |
Working strategy A's denominator explicitly, because the kurtosis term catches people out — it does not vanish at γ₂ = 3:
Against a zero benchmark both look fine. Raise the benchmark to a Sharpe of 1.0 — the question an allocator with a passive alternative actually asks — and they separate sharply: A lands exactly on the 0.95 bar, while B falls to 0.83 and fails it. Identical headline Sharpe, materially different evidence. That is the entire point of the measure.
Note the annualisation trap in this example: if SR_hat is annualised, n must still be the number of observations (24 months), and SR_hat and SR* must be expressed at the same frequency. Mixing an annualised Sharpe with a monthly observation count is the single most common implementation error.
How to use it
- 0.95 is the conventional bar. Below it, the track record does not support the claim at 95% confidence — which is a statement about evidence, not about the strategy being bad.
- Set SR* to something meaningful. Testing against zero is a weak hurdle. Testing against the Sharpe of a passive alternative is the question an allocator actually cares about.
- Use it to size track-record requirements. Inverting the formula gives the minimum track record length needed for a given observed Sharpe to clear 0.95 — often far longer than practitioners expect.
Limitations
- It does not correct for selection bias. PSR evaluates one track record. If you tested 200 strategies and are reporting the best, PSR is the wrong tool — the Deflated Sharpe Ratio extends it to account for the number of trials.
- It assumes IID returns. Serial correlation, common in illiquid or smoothed portfolios, inflates the apparent Sharpe and is not handled by the standard formulation.
- Higher moments are themselves estimated. Skewness and kurtosis from short samples are noisy, and the correction is only as good as those estimates.
- Backtest Sharpe is not live Sharpe. PSR says nothing about costs, capacity, or whether the strategy was fitted to the sample.
References
- Bailey, D. & López de Prado, M. (2012). “The Sharpe Ratio Efficient Frontier.” Journal of Risk 15(2), 3–44.
- Bailey, D. & López de Prado, M. (2014). “The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality.” Journal of Portfolio Management 40(5), 94–107.
- Lo, A. (2002). “The Statistics of Sharpe Ratios.” Financial Analysts Journal 58(4), 36–52.