Probabilistic Sharpe Ratio: A Worked Calculation
With an observed Sharpe of 1.50, 24 observations, skewness of −1.20 and kurtosis of 7.00, the Probabilistic Sharpe Ratio against a zero benchmark is 99.81%. Here is the arithmetic behind that result, including the denominator that is easy to miscalculate.
Sharpe and the benchmark must be expressed at the observation frequency. For monthly observations, use monthly Sharpe values; do not enter an annualized Sharpe alongside a count of monthly returns. Kurtosis here is ordinary kurtosis, where a normal distribution has value 3, not excess kurtosis.
Inputs
| Observed Sharpe SR | 1.50 per observation period |
|---|---|
| Benchmark SR* | 0.00 per observation period |
| Observations n | 24 |
| Skewness g1 | −1.20 |
| Kurtosis g2 | 7.00 |
Calculate without intermediate rounding
The normal-return case still has a kurtosis term. Setting skewness to zero and kurtosis to 3 gives sqrt(1 + 0.5 × 1.50²) = 1.4577, not 1. Zero excess kurtosis does not remove (g2 − 1)/4 from this formula.
Change the question
Raise the benchmark from 0 to 1 while keeping the other inputs fixed. The z-statistic becomes 0.9650 and PSR falls to 83.27%, below a 95% threshold. The observed record is the same; the claim being tested is stronger.
Open the free calculator with these default inputs. It calculates locally in your browser and displays both the denominator and z.
Reproduce with Python
This check uses Python's standard library. The browser calculator uses an error-function approximation, so compare at the displayed precision rather than requiring identical last digits.
What the percentage does not mean
PSR is not the probability of a profitable next trade. Its interpretation depends on the statistical assumptions, including independent returns and reliable moment estimates. It does not account for costs or the number of strategy variants tried. Selection across many backtests needs a separate correction such as the Deflated Sharpe Ratio.
Arithmetic checked 4 September 2026. The earlier worked example printed z = 2.8955; the correct value rounded to four decimals is 2.8949. This correction leaves the displayed PSR of 99.81% unchanged.