Probabilistic Sharpe Ratio Formula
The probabilistic Sharpe ratio (PSR) is the probability that a strategy's true Sharpe ratio exceeds a chosen benchmark, given the observed Sharpe, the sample length, and the return distribution's skew and kurtosis. It converts a point estimate into a confidence level, correcting for the fact that short, skewed, fat-tailed samples make a high Sharpe far less certain than it looks.
Formula
Copy the exact expression or work through it step by step below.
PSR(SR_0) = Phi( ( (SR_hat - SR_0) x sqrt(T - 1) ) / sqrt(1 - g3 x SR_hat + ((g4 - 1)/4) x SR_hat^2) ) Variables
SR_hat
Observed Sharpe ratio
The per-period Sharpe estimated from the sample, before annualization. All other terms calibrate how much to trust it.
SR_0
Benchmark Sharpe
The threshold the true Sharpe must beat, often zero (is there any edge at all?) or the Sharpe of a competing strategy. PSR reports the probability SR_hat reflects a true Sharpe above this bar.
T
Number of observations
Sample length. The sqrt(T - 1) term means longer track records sharpen confidence: the same Sharpe over more periods yields a higher PSR.
g3
Skewness
Third standardized moment of returns. Negative skew increases the denominator and lowers PSR, because occasional large losses make a high Sharpe less reliable.
g4
Kurtosis
Fourth standardized moment. Fat tails (g4 above 3) widen the Sharpe estimate's standard error, reducing PSR for the same observed Sharpe.
Phi
Standard normal CDF
Maps the standardized statistic to a probability in [0, 1], the PSR itself.
Step By Step
- 1
Estimate the per-period Sharpe, skewness, and kurtosis from the return series, and count the observations T.
Per-period Sharpe 0.10, skew 0, kurtosis 3 (normal), T = 250.
- 2
Choose the benchmark Sharpe SR_0, commonly zero to test for any edge.
Set SR_0 = 0.
- 3
Compute the denominator, the standard error term, using skew and kurtosis.
With skew 0 and kurtosis 3: sqrt(1 - 0 + ((3 - 1)/4) x 0.10^2) = sqrt(1 + 0.5 x 0.01) = sqrt(1.005) = 1.0025.
- 4
Form the standardized statistic and apply the normal CDF.
((0.10 - 0) x sqrt(249)) / 1.0025 = (0.10 x 15.78) / 1.0025 = 1.578 / 1.0025 = 1.574; Phi(1.574) = 0.942.
Worked Example
PSR for a daily strategy, normal returns, one year of data
Per-period Sharpe (SR_hat)
0.10
Observations (T)
250
Benchmark (SR_0)
0
Skew / kurtosis
0 / 3
Denominator = sqrt(1 - 0 x 0.10 + ((3 - 1)/4) x 0.10^2) = sqrt(1 + 0.5 x 0.01) = sqrt(1.005) = 1.0025. Numerator = (0.10 - 0) x sqrt(250 - 1) = 0.10 x sqrt(249) = 0.10 x 15.780 = 1.5780. Statistic = 1.5780 / 1.0025 = 1.574. PSR = Phi(1.574) = 0.942.
PSR of about 0.94, just under the conventional 0.95 confidence bar. Despite an annualized Sharpe near 1.6 (0.10 x sqrt(250)), one year of data leaves a 6% chance the true Sharpe is actually below zero. The lesson: a strong-looking annualized Sharpe is not yet statistically established over a single year, even with perfectly normal returns.
Common Variations
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Sources & References
- The Sharpe Ratio Efficient Frontier — David H. Bailey and Marcos Lopez de Prado, Journal of Risk (2012)
- The Statistics of Sharpe Ratios — Andrew W. Lo, Financial Analysts Journal (2002)
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