Deflated Sharpe Ratio calculator

A backtest arrives as a Sharpe ratio, a sample length and little else. Deflating it asks a different question: how often would a search of this size produce a number this good with no skill behind it. Fill in the backtest below and every quantity updates as you type.

dsr deflate --sharpe 2.5000 --annual --periods-per-year 250 --n-obs 1250 --trials 100 --trial-variance 0.5000 --skew -3.0000 --kurtosis 10.0000

from deflated_sharpe import deflated_sharpe_ratio

result = deflated_sharpe_ratio(
    sharpe=0.15811388,   # per period
    n_obs=1250,
    n_trials=100,
    var_trials=0.0020000000,
    skew=-3.0000,
    kurtosis=10.0000,
)
result.dsr   # 0.9004

The mathematics

All quantities are per period: a daily Sharpe ratio is the mean daily return over its standard deviation, and T counts daily observations. The Probabilistic Sharpe Ratio (Bailey and López de Prado 2012) is the probability that the true Sharpe ratio exceeds a benchmark SR*, allowing for the skewness γ₃ and raw kurtosis γ₄ of the returns:

If N strategies with no skill are tried and their estimated Sharpe ratios have variance V, the best of them is expected to show

where γ ≈ 0.5772 is the Euler–Mascheroni constant. The Deflated Sharpe Ratio is the Probabilistic Sharpe Ratio with that expected maximum as the benchmark, $\widehat{DSR}=\widehat{PSR}({\widehat{SR}}_0)$, so it corrects for the search and for non-normal returns at the same time.

Two length bounds follow from the same algebra. The minimum track record length is the number of observations at which PSR reaches a chosen confidence 1 − α, and the minimum backtest length is the number of years of data before N trials stop producing an annualised Sharpe ratio of E[max] by chance (Bailey, Borwein, López de Prado and Zhu 2014):

Those four expressions produce every number in the calculator above. It opens on the worked example in Bailey and López de Prado (2014), an annualised Sharpe ratio of 2.5 over five years of daily returns, chosen from 100 configurations whose Sharpe ratios have variance 0.5, with skewness −3 and kurtosis 10. That backtest reaches 0.9004 and falls short of the 95% bar, while the same returns would have passed as the best of 46 trials.

Why a Sharpe ratio needs deflating

A researcher who tries a hundred versions of a strategy and reports the best one has run a hundred experiments and published the luckiest. Even when none of the versions has any skill, the best of them shows a respectable Sharpe ratio while a longer search produces an even better-looking winner. The Deflated Sharpe Ratio of Bailey and López de Prado (2014) estimates the probability that the true Sharpe ratio of the selected strategy is above zero once that search is taken into account.

The demo below makes the problem concrete. Every strategy in it has a true Sharpe ratio of exactly zero, yet the best of sixty usually shows an annualised Sharpe ratio above 1.

How long a track record has to be

Frequency matters even at the same annualised Sharpe ratio, because fewer observations per year carry less information, and fat tails matter more than frequency. Monthly returns with the moments of the HFR hedge fund index need almost five years to show that a Sharpe ratio of 2 beats 1, against under three for daily returns.

The trial count that matters

N should be the number of distinct configurations tried, including the ones abandoned early. Because strategy variants are correlated, replacing it with an effective count such as the eigenvalue participation ratio looks more careful, but when V is measured from the trials themselves it corrects for correlation twice. A shared factor moves every trial’s Sharpe ratio together and so already shrinks their spread, which puts the correlation inside √V before N is ever touched.

Running the batch in the first demo checks this. With the distinct count, the best of a zero-skill corpus clears SR₀ about half the time at any correlation, which is what an expected maximum should do. With the participation ratio, 73 to 85 percent of correlated null corpora clear it in the package’s reference simulation, so a bar built that way lets most pure noise through.

Validation

The package reproduces every numerical example printed in the three source papers, and each one is a test that runs on Python 3.10 to 3.13 across Linux, macOS and Windows.

Monte Carlo tests check the rest. PSR crosses 0.95 on about 5% of zero-skill samples, and the expected maximum matches simulated maxima, overstating them by 2.5% at ten trials and under 1% from a hundred. The minimum track record length lands exactly where PSR reaches its confidence level.

What a DSR does not tell you

A DSR above 0.95 says the selection is unlikely to be explained by the search and the shape of the returns alone. It says nothing about transaction costs, capacity, look-ahead in the data, regime change or trials that were never recorded. Nor does a DSR below 0.95 prove the absence of skill, particularly on short samples.

Every formula also assumes serially independent returns. Positions held across several bars make returns autocorrelated, so the package provides an AR(1) effective sample size to pass in place of the raw observation count.