Methodology: Deflated Sharpe Ratio Calculator

What problem this solves

An observed Sharpe ratio looks impressive in isolation. After K trials of a backtest grid, the maximum observed Sharpe is biased upward by selection. Deflated Sharpe Ratio (DSR) corrects this bias and reports the probability the underlying skill is positive.

Probabilistic Sharpe Ratio (PSR)

Equation 2 of Bailey-López (2014). For an observed (per-period) Sharpe SR_obs, n observations, sample skew γ₃, and sample kurtosis γ₄:

z = (SR_obs − SR_*) · √(n − 1) / √(1 − γ₃·SR_obs + ((γ₄ − 1) / 4) · SR_obs²)
PSR(SR_*) = Φ(z)

PSR is the probability that the true Sharpe exceeds the benchmark SR_*.

Maximum Sharpe under the null

For K independent trials with true Sharpe = 0, the expected maximum observed Sharpe is approximated by:

E[max] ≈ (1 − γ_E) · Φ⁻¹(1 − 1/K) + γ_E · Φ⁻¹(1 − 1/(K·e))

where γ_E ≈ 0.5772 is the Euler-Mascheroni constant. This is the per-period quantity; we annualize by √(periods/year).

Deflated Sharpe Ratio

DSR = PSR computed against the deflated benchmark E[max]:

DSR = PSR(SR_obs; SR_* = E[max])

Output as a probability between 0 and 1. Above 95% means the observed Sharpe survives the multiple-testing haircut at the 5% level.

References

Bailey, D. H., 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. DOI: 10.3905/jpm.2014.40.5.094.
Bailey, D. H., López de Prado, M. (2012). "The Sharpe Ratio Efficient Frontier." Journal of Risk 15(2): 3–44.

Limitations

K-trial estimate assumes independent trials. Correlated grid searches need a smaller effective K (Bailey-López appendix).
Sample skew and kurtosis are noisy with n < 250; the resulting z-statistic widens accordingly.
Annual conversion uses √(periods/year). Assumes i.i.d. returns; serial correlation overstates the sample size.

How Deflated Sharpe Ratio works