The Deflated Sharpe Ratio
A Sharpe of 2.5 looks decisive — until you ask how many variants were tried. Bailey & López de Prado (2014) show how to raise the bar for N independent trials, and how skew, kurtosis, and sample length still matter. Numbers below are from their paper only.
- Why the best of many trials has an expected maximum Sharpe even when true skill is zero
- How DSR is a PSR with that raised rejection threshold (plus non-Normality and T)
- The paper’s worked treasury-seasonality example: annualized SR 2.5, N = 100, and a DSR that fails 95%
- When to stop testing (the 1/e rule of thumb they cite)
The problem DSR is built for
Backtests and track records are usually ranked by the Sharpe ratio (SR). That is fine for a single, pre-specified trial. It is not fine when someone searched a large space of parameters, kept the winner, and reported only that path.
Bailey & López de Prado call out two inflation channels that pile on top of each other:
- Selection bias under multiple testing — the more independent trials
N, the higher the expected maximum estimated SR, even if every strategy has true SR = 0. - Non-Normal returns — short samples, skewness, and kurtosis make a raw SR look more “significant” than it is. Their earlier Probabilistic Sharpe Ratio (PSR) already folded in sample length
Tand the first four moments.
The Deflated Sharpe Ratio (DSR) is PSR with the rejection threshold no longer a free user choice: it is set to the expected maximum SR under the null, given N and the variance of SRs across trials.
Expected maximum SR after N trials
Assume the estimated Sharpes across a strategy class are i.i.d. Normal with mean E[{SR̂}] and variance V[{SR̂}]. Appendix 1 of the paper approximates the expected maximum after N independent trials with an Euler–Mascheroni expansion (γ ≈ 0.5772):
Under a null of no skill, E[{SR̂}] = 0, so the threshold that a “winner” must beat grows with N and with how dispersed the trial Sharpes are. That threshold is what they plug into DSR as SR̂0.
N. For teaching, treat N as the count of independent experiments — not every line in a grid search if those lines are highly correlated.DSR = PSR with a raised bar
Write SR̂ for the selected strategy’s estimated Sharpe, T for its sample length, and γ̂3, γ̂4 for sample skewness and kurtosis. DSR (their Eq. (2)) is the Normal CDF of a studentized gap between SR̂ and SR̂0, with a non-Normal variance correction in the denominator:
with
In words: five extra ingredients beyond a textbook two-moment Sharpe — non-Normality (γ̂3, γ̂4), length T, trial-SR variance, and independent trial count N.
The paper’s numerical example (use these numbers)
A strategist researching U.S. Treasury auction seasonality backtests many configurations (pre-/post-auction windows, tenors, holding periods, stop-losses, …). Many combinations print an annualized SR of 2; one prints 2.5 over a daily sample of five years. The investor asks for the full disclosure set. The strategist answers with the values in the table — all from the paper’s “A Numerical Example” section.
| Input | Paper value |
|---|---|
| Claimed annualized SR̂ | 2.5 |
| Sample length T | 1250 days (5 years) |
| Independent trials N | 100 |
| Variance of trial SRs, V[{SR̂n}] | 1/2 |
| Skewness γ̂3 | −3 |
| Kurtosis γ̂4 | 10 |
| Observations per year (for annualization) | 250 |
Source: Bailey & López de Prado (2014), “A Numerical Example.” Only values printed in that section.
Working non-annualized (divide annualized SR by √250), the paper reports:
SR̂0 ≈ 0.1132under N = 100DSR̂ ≈ 0.9004
That is below a 95% confidence bar. The investor’s reading in the paper: there is only about a 90% chance that the true SR is greater than zero after N = 100 paths — so the offer is declined.
What would have cleared 95%?
Same non-Normal returns, but only N = 46 independent trials → DSR̂ = 0.9505 (above 95%). If returns had been Normal instead (γ̂3 = 0, γ̂4 = 3), the paper says the investor could have accepted up to N = 88 trials at that same 0.9505 cutoff. Non-Normality and multiple testing both matter; DSR handles them jointly.
When should we stop testing?
Every extra independent trial permanently raises the bar. The paper points to the classical secretary problem / 1/e-law of optimal choice (Bruss, 1984): from the set of theoretically justified configurations, sample about 1/e ≈ 37% at random and record their performance; then keep drawing one by one until you find a candidate that beats all previous. That stopping rule is a useful cost on the search, not a license to brute-force the rest of the grid.
How this fits Quant for Free
Module 5’s validation gauntlet already treats DSR as the multiple-testing gate. This page is the paper explainer behind that gate: same chrome, full disclosure of the Bailey–López de Prado numerical example, and a direct link into the calculator.
When a scaled trend book (or any pretty curve) shows up in a teardown, pair the story with trial counts the way TSMOM & volatility scaling pairs scaled vs unscaled tables — then run the numbers through the tool rather than trusting the headline Sharpe.
- Sharpe ratio (SR)
- Return per unit of risk — the usual performance score, fragile under search.
- PSR
- Probabilistic Sharpe Ratio: P(true SR > threshold), adjusting for T and non-Normality.
- DSR
- Deflated Sharpe Ratio: PSR with threshold = expected max SR under N trials.
- SR̂0
- That expected-max threshold (Euler–Mascheroni approximation in Eq. (1)/(2)).
- Selection bias
- Reporting only the winners from many trials, which inflates Type I error.
- 1/e stopping
- Sample ~37% of justified configs, then take the first record-breaker thereafter.
Where to go next
ToolDeflated Sharpe Ratio — plug in your SR, N, skew, kurtosis, and T Learn · Module 5
The validation gauntlet — overfitting, out-of-sample, multiple testing, costs Learn · Deep dive
TSMOM & volatility scaling — another paper explainer in the same Learn chrome