Formula, in Backtesting & Validation

Deflated Sharpe Ratio Formula

The deflated Sharpe ratio (DSR) is the probability that a strategy's true Sharpe ratio exceeds the Sharpe ratio expected from the best of N unskilled trials, after adjusting for sample length, skewness and kurtosis. It was published by Bailey and López de Prado in the Journal of Portfolio Management 40(5), 94-107, in 2014.

8 variables. Published May 26, 2026. Updated Oct 11, 2026. By AI Fin Hub Research.

Formula

DSR = Z( (SR_hat - SR_0) x sqrt(T - 1) / sqrt(1 - g3 x SR_hat + ((g4 - 1) / 4) x SR_hat^2) )

SR_0 = sqrt(V[SR_n]) x ( (1 - gamma) x Z^-1(1 - 1/N) + gamma x Z^-1(1 - 1/(N x e)) )

All Sharpe ratios per period (not annualized); g4 is full kurtosis (3 for normal returns); gamma = 0.5772...

Variables

SR_hat

Selected strategy's Sharpe ratio

The per-period Sharpe ratio of the strategy you picked, estimated from its T returns. Divide an annualized figure by the square root of periods per year first.

SR_0

Expected maximum Sharpe under the null

The Sharpe ratio expected from the best of N independent trials when every true Sharpe ratio is zero. It is the rejection threshold the observed Sharpe must clear, and it grows with N and with the spread of the trials.

V[SR_n]

Variance of the trials' Sharpe ratios

The variance of the estimated Sharpe ratios across all N trials, in per-period units. This is a cross-trial dispersion, not the standard error of one estimate.

Number of independent trials

How many independent configurations were tested before selecting this one. The paper's Appendix 3 discusses estimating N when trials are correlated.

Number of return observations

Sample length of the selected strategy's return series.

Skewness

Third standardized moment of the selected strategy's returns. Negative skew enlarges the denominator and lowers the DSR.

Kurtosis

Fourth standardized moment (full kurtosis, 3 for a normal distribution). Fat tails raise the variance of the Sharpe estimate.

gamma, e, Z

Constants and normal CDF

gamma is the Euler-Mascheroni constant (about 0.5772), e is Euler's number, Z is the standard normal CDF and Z^-1 its inverse.

Step by step

1

Convert the selected strategy's Sharpe ratio to per-period units and estimate its skewness and kurtosis from the T returns.

Annualized 2.5 on daily data with 250 periods a year: SR_hat = 2.5 / sqrt(250) = 0.1581.
2

Count the independent trials N and compute the variance of their Sharpe ratios V[SR_n] in per-period units.

N = 100 trials with an annualized variance of 0.5: V = 0.5 / 250 = 0.002.
3

Compute the expected maximum E[max] = (1 - gamma) Z^-1(1 - 1/N) + gamma Z^-1(1 - 1/(N e)) and multiply by sqrt(V) to get SR_0.

For N = 100, E[max] = 2.5306, so SR_0 = sqrt(0.002) x 2.5306 = 0.1132 per period (1.79 annualized).
4

Plug SR_hat, SR_0, T, skewness and kurtosis into the DSR expression and apply the normal CDF.

With T = 1250, skew -3 and kurtosis 10: z = 1.284 and DSR = Z(1.284) = 0.9004.
5

Compare the DSR with your confidence level. The paper's example uses 95%.

0.9004 is below 0.95, so the 2.5 Sharpe ratio is not a significant discovery after 100 trials.

Worked example

The numerical example in Bailey and López de Prado (2014): a Treasury auction-cycle strategy found after 100 trials

Selected Sharpe (annualized, daily data)

2.5 over T = 1,250 days

Independent trials N

100

Variance of trial Sharpe ratios (annualized)

0.5

Skewness / kurtosis

-3 / 10

SR_hat = 2.5 / sqrt(250) = 0.1581; SR_0 = sqrt(0.5/250) x 2.5306 = 0.1132; denominator = sqrt(1 + 3 x 0.1581 + (9/4) x 0.1581^2) = 1.2372; z = (0.1581 - 0.1132) x sqrt(1249) / 1.2372 = 1.284; DSR = Z(1.284) = 0.9004.

DSR of 0.90, below the 95% bar, which is why the investor in the paper declines. The paper reports that with N = 46 trials the DSR would have been 0.9505, and that with normal returns (skew 0, kurtosis 3) the 95% bar holds up to N = 88. The site's calculator, which assumes V[SR_n] = 1/(T - 1) when the trials' dispersion is unknown, returns a benchmark of 1.13 annualized and a DSR of 0.993 for the same inputs, so supply the real cross-trial variance when you have it.

Common variations

Probabilistic Sharpe ratio: the same expression with a fixed benchmark SR* in place of SR_0 (no multiple-testing adjustment), from Bailey and López de Prado's 2012 Sharpe ratio efficient frontier paper.

Calculator convention: when V[SR_n] is unknown, the site's engine uses the null sampling variance 1/(T - 1), so SR_0 = E[max of N] x sqrt(periods_per_year / (T - 1)) in annualized units.

Minimum track record length: invert the probabilistic Sharpe ratio to solve for the T needed to reach a target confidence.

Probability of backtest overfitting: cross-validates in-sample and out-of-sample ranks instead of deflating one Sharpe ratio.

Run the numbers next

Calculators Backtest Overfitting Score
Upload a backtest trade log and compute Probability of Backtest Overfitting (PBO), Deflated Sharpe Ratio, and the odds your edge survives live trading.
Calculators Risk-Adjusted Returns Calculator
Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.

Sources

The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting, and Non-Normality , Bailey and López de Prado. Journal of Portfolio Management 40(5), 94-107 (2014). DOI 10.3905/jpm.2014.40.5.094. SSRN 2460551
The Deflated Sharpe Ratio (author copy, July 2014) , David H. Bailey
The Sharpe ratio efficient frontier , Bailey and López de Prado. Journal of Risk 15(2), 3-44 (2012)

Keep the topic connected

Formula Sharpe Ratio Formula
The Sharpe ratio formula: excess return over the risk-free rate divided by return volatility, then annualized. Every variable defined, with a worked example.
Learn Sharpe Ratio
Sharpe ratio defined, when it lies (skew, fat tails, autocorrelation), and how to read a Sharpe number you didn't compute yourself.
Learn Overfitting
Overfitting in trading-strategy backtests: how multiple-testing inflates apparent edges and the diagnostics that catch it.
Learn Probability of Backtest Overfitting (PBO): Bailey et al.
Probability of Backtest Overfitting (PBO): Bailey, Borwein, López de Prado and Zhu, Journal of Computational Finance 20(4), 2017, SSRN 2326253, via CSCV.