Our own calculation, 7 November 2026, for the article "Tuning backtest parameters without fooling yourself". Two ways: the closed-form approximation of the expected maximum of N independent standard normal draws given by Bailey, Borwein, Lopez de Prado and Zhu (2014, equation 2.4, with the Euler-Mascheroni constant), and the exact expectation by numerical integration of x times N times phi(x) times Phi(x) to the power N minus 1. The standard error of an annual Sharpe ratio with no skill over T years is the square root of 1 over T (Lo, 2002, with SR = 0): over 5 years, 0.447.
What we found
- The expected best in-sample Sharpe ratio of N independent no-skill configurations on 5 years of data is the expected maximum of N standard normals times 0.447: for 45 configurations about 0.99, which is the paper's statement that on five years no more than 45 independent configurations should be tried before a Sharpe ratio of 1 appears by luck alone.
- For 100 configurations the expected best is about 2.5 standard errors, a Sharpe ratio of about 1.1 from luck alone; the article's 1.1.
- For the article's 190-pair grid, treated as 190 independent tries, the best of no-skill pairs averages about 2.7 standard errors, a Sharpe ratio of about 1.22; for the 495-pair grid, about 3.0 standard errors, about 1.36.
- Neighbouring moving-average pairs are strongly correlated, so a grid of 190 pairs is fewer than 190 independent tries; the figures above are the ceiling of what luck alone produces, not the exact inflation.
- The two ways agree to within 0.035 in the expected maximum at every N; the approximation runs slightly high.
Check it yourself
| Configurations N | Expected maximum, Bailey eq. 2.4 | Expected maximum, exact integral | Times 0.447: Sharpe from luck, 5 years |
|---|---|---|---|
| 7 | 1.387 | 1.352 | 0.60 |
| 45 | 2.236 | 2.208 | 0.99 |
| 100 | 2.531 | 2.508 | 1.12 |
| 190 | 2.749 | 2.729 | 1.22 |
| 495 | 3.050 | 3.034 | 1.36 |
What this cannot tell you: the inflation for a particular correlated grid (it needs the trials' correlation, or a holdout); anything about a strategy with real skill, whose in-sample ratio is skill plus the same luck.