AlphaAssayThe Deflated Sharpe Ratio, explained — with a calculator
The Deflated Sharpe Ratio (DSR) answers one question: after accounting for how many strategy variants you tried, what is the probability that your backtest's Sharpe ratio reflects a genuine edge rather than selection luck? It takes your observed Sharpe, the length and shape of your return series (skewness, kurtosis), and — the input everyone omits — the number of trials behind the result, then raises the benchmark your Sharpe must clear. Bailey and López de Prado introduced it in 2014 because trying forty variants and keeping the best one manufactures a „great" backtest from pure noise, mathematically guaranteed. The calculator below runs entirely in your browser; nothing is uploaded. Rule of thumb: a DSR probability below 0.95 means your result is statistically indistinguishable from an accident. This same statistic, with cumulative trial accounting, runs as gate 2 of every AlphaAssay verdict.
The calculator
Annualised Sharpe is de-annualised internally (SR ÷ √periods-per-year); the cross-sectional variance of no-edge trial Sharpes defaults to 1/T (their sampling variance) — the theoretically neutral assumption when you have not measured it. The calculator implements Bailey & López de Prado (2014): expected maximum Sharpe under N trials via extreme-value approximation, then the Probabilistic Sharpe Ratio against that benchmark.
What does the DSR look like as trials pile up? (computed with the calculator below — same formula, same defaults)
| observed Sharpe | history | trials N | DSR probability | reading |
|---|---|---|---|---|
| 1.8 | 3y daily | 1 | ≈ 1.0 | a single pre-registered test — strong |
| 1.8 | 3y daily | 10 | ≈ 0.94 | a modest grid search already erodes it |
| 1.8 | 3y daily | 50 | ≈ 0.8 | typical „I tuned it for a weekend" |
| 1.8 | 3y daily | 200 | ≈ 0.64 | indistinguishable from selection luck |
Same backtest, same Sharpe — the only thing that changed is honesty about the search. Run your own numbers above.
Why does the ordinary Sharpe ratio flatter you?
Because it prices a single experiment, and you did not run a single experiment. If forty variants of an idea are tried and the best is kept, the winner's Sharpe contains selection luck by construction — Bailey, Borwein, López de Prado and Zhu (2014) showed a „great" backtest is mathematically guaranteed given enough trials, from pure noise. The full mechanism, with the published base rates: why backtests flatter everyone.
The formula, step by step
Step 1 — the benchmark your Sharpe must beat. Under N independent trials of no-edge strategies,
the expected maximum observed Sharpe is approximately
E[max SR] ≈ σ_trials · [(1−γ)·z(1−1/N) + γ·z(1−1/(N·e))],
where γ ≈ 0.5772 (Euler–Mascheroni) and z is the standard-normal quantile. More trials → higher hurdle.
Step 2 — the Probabilistic Sharpe Ratio against that hurdle.
DSR = Φ( ((SR − SR*) · √(T−1)) / √(1 − γ₃·SR + ((γ₄−1)/4)·SR²) )
— your observed SR versus the hurdle SR*, scaled by track-record length T and penalised for skewness γ₃
and kurtosis γ₄ (fat tails and negative skew make a Sharpe less trustworthy).
Step 3 — read it as a probability. DSR ≥ 0.95: the edge is unlikely to be a selection artifact. Below: the honest answer is „not proven".
The input nobody tracks: your true trial count
Every parameter grid, every discarded variant, every „just one more tweak" is a trial — including the
ones you deleted. This is why AlphaAssay accounts trials cumulatively per strategy
family: the battery remembers what your family has spent even when you don't, and
family_budget_exhausted is the failure code that says the honest answer to another tweak is
already known.
What the DSR cannot see
The DSR corrects for selection — nothing else. Leaked future information, unrealistic costs and capacity limits all survive a perfect DSR, which is why it is one gate of four, not the whole trial: the four gates.