The Deflated Sharpe Ratio and Backtest Overfitting

How the Deflated Sharpe Ratio corrects for multiple testing, skew and kurtosis, and how it relates to the probability of backtest overfitting.

Published 2026-10-11 · Educational explainer, not investment advice

Selection bias under multiple testing

If you try enough strategy variants, the best one will show a high Sharpe ratio by chance alone. Reporting only that winner without the number of trials is the most common source of false discoveries in quantitative research.

Deflating the Sharpe ratio

The Deflated Sharpe Ratio (DSR) is the probability that the true Sharpe exceeds the maximum Sharpe expected from N independent trials, adjusting for sample length, skewness and kurtosis of returns.

from scipy.stats import norm
import numpy as np
def expected_max_sr(sr_var, n_trials):
    g = 0.5772156649
    return np.sqrt(sr_var) * ((1 - g) * norm.ppf(1 - 1 / n_trials)
                              + g * norm.ppf(1 - 1 / (n_trials * np.e)))
def deflated_sr(sr, sr_var, n_trials, T, skew, kurt):
    sr0 = expected_max_sr(sr_var, n_trials)
    denom = np.sqrt(1 - skew * sr + (kurt - 1) / 4 * sr**2)
    return norm.cdf((sr - sr0) * np.sqrt(T - 1) / denom)

Probability of backtest overfitting

Combinatorially symmetric cross-validation estimates how often the in-sample best configuration underperforms the median out of sample. Together with DSR it gives reviewers a quantitative view of how much a result depends on search.

Source: López de Prado, M. (2018). Advances in Financial Machine Learning. Wiley. Chapters 11–14; Bailey & López de Prado (2014), The Deflated Sharpe Ratio.

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