1.2.4 The Deflated Sharpe Ratio

Basic Prerequisites: 03 · The Multiple-Testing Problem (expected maximum Sharpe) and Probability & Measure Theory.

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1. Intuition & Practical Objective

A raw Sharpe ratio conflates two very different claims: (i) “this strategy made money relative to its risk” and (ii) “this strategy is statistically distinguishable from luck”. The Probabilistic Sharpe Ratio (PSR) answers (ii) for a single trial. The Deflated Sharpe Ratio (DSR) answers it after admitting that the strategy was selected as the best of $N$ trials - and that its returns are not normal.

The practical objective: turn a strategy’s reported Sharpe, sample length, skew/kurtosis, and the number of trials into one number in $[0,1]$ - the probability that the true Sharpe exceeds the selection-bias-adjusted threshold ${\widehat{SR}}_0$. Pass/fail conventions: $\text{DSR}\ge 0.95$ = significant at 95%, i.e. the strategy is not explainable as the best of $N$ null flukes.

The one-line essence. “DSR is a PSR whose rejection threshold is raised to the height expected of the maximum of $N$ zero-skill trials: ${\widehat{SR}}_0=\sqrt{V[\{{\widehat{SR}}_n\}]}\mathbb{E}[{\max }_n{Z}_n]$, and whose standard error is widened by skewness and kurtosis.”

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2. Mathematical Ground Truth & Derivations

2.1 The Probabilistic Sharpe Ratio (single trial)

The Sharpe estimator’s variance under non-normal returns (Lo 2002; Bailey & López de Prado 2012) is

$$\widehat{\mathrm{Var}}[\widehat{SR}]=\frac{1-{\hat{\gamma }}_3\widehat{SR}+\frac{{\hat{\gamma }}_4-1}{4}{\widehat{SR}}^2}{T-1},$$

where ${\hat{\gamma }}_3$ is skewness and ${\hat{\gamma }}_4$ is Pearson kurtosis ($=3$ for normal). Hence, against a user-chosen benchmark ${\widehat{SR}}^{\ast }$,

$$\text{PSR}({\widehat{SR}}^{\ast })=\Phi \left(\frac{(\widehat{SR}-{\widehat{SR}}^{\ast })\sqrt{T-1}}{\sqrt{1-{\hat{\gamma }}_3\widehat{SR}+\frac{{\hat{\gamma }}_4-1}{4}{\widehat{SR}}^2}}\right).$$

Note the non-normality penalty: negative skew inflates the denominator (raises the bar); fat tails (kurtosis $>3$) do the same. Two strategies with identical Sharpe but different higher moments get different PSRs.

2.2 The Deflated Sharpe Ratio

Replace the arbitrary benchmark by the expected maximum under the null of $N$ trials: ${\widehat{SR}}^{\ast }\to {\widehat{SR}}_0$ with

$$\boxed{{\widehat{SR}}_0=\sqrt{V[\{{\widehat{SR}}_n\}]}\cdot [(1-\gamma ){\Phi }^{-1}(1-\frac{1}{N})+\gamma {\Phi }^{-1}(1-\frac{1}{Ne}\left)\right]}$$

and define

$$\boxed{\text{DSR}=\Phi \left(\frac{(\widehat{SR}-{\widehat{SR}}_0)\sqrt{T-1}}{\sqrt{1-{\hat{\gamma }}_3\widehat{SR}+\frac{{\hat{\gamma }}_4-1}{4}{\widehat{SR}}^2}}\right).}$$

Everything is in non-annualized (per-period) units - DSR is scale-invariant only if $T$ matches the return frequency and $\widehat{SR}$, ${\widehat{SR}}_0$ share the period.

What DSR adds over a raw Sharpe - five extra inputs, each of which can only lower significance:

1. $N$ - the number of independent trials (raises ${\widehat{SR}}_0$),
2. $V[\{{\widehat{SR}}_n\}]$ - how dispersed the trials were (scales ${\widehat{SR}}_0$),
3. $T$ - sample length (longer = more certain),
4. ${\hat{\gamma }}_3,{\hat{\gamma }}_4$ - non-normality (widens the denominator).

2.3 Minimum Track Record Length

Inverting DSR $=1-\alpha$ for the required sample length (PSR threshold) gives the MinTRL:

$$\text{MinTRL}=1+[1-{\hat{\gamma }}_3\widehat{SR}+\frac{{\hat{\gamma }}_4-1}{4}{\widehat{SR}}^2]{\left(\frac{{\Phi }^{-1}(1-\alpha )}{\widehat{SR}}\right)}^2.$$

It answers “how long must this track record be before it is significant?” - and it grows with negative skew and fat tails, the empirical signature of option-selling and carry strategies.

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3. Computational Implementation - DSR end-to-end

Stdlib only. Reproduces the canonical Bailey & López de Prado worked example, sweeps $N$, contrasts normal vs non-normal returns, and computes MinTRL.

Read the story: a strategy with an annualized Sharpe of 2.5 over 5 years is only fundable if it survived at most ~46 trials ($\text{DSR}=0.950$). At the 100 trials it actually survived, $\text{DSR}=0.900$ - below the 95% bar. The normal-returns column isolates the second source of inflation: holding $N$ fixed, non-normality (skew $-3$, kurt $10$) costs ~4 percentage points of DSR at $N=100$ (0.942 $\to$ 0.900). MinTRL shows the flip side: a Sharpe-2.5 record needs only ~110 days (~0.44 yr) ignoring non-normality, stretched to ~168 days (0.7 yr) once skew $-3$ / kurt $10$ are priced in - negative skew is expensive.

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4. Failure Modes & First-Principles Breakdowns

1. Unit mismatch. DSR mixes $T$, $\widehat{SR}$, ${\widehat{SR}}_0$, and $V[\{{\widehat{SR}}_n\}]$; all must be in the same (per-period) units. Annualizing some but not others is the classic implementation bug (annualize the final threshold, not the inputs).
2. Kurtosis convention. The formula uses Pearson kurtosis ($3$ for normal). Passing excess kurtosis ($0$ for normal) silently biases DSR; scipy.stats.kurtosis(..., fisher=False) is the safe call.
3. $V[\{{\widehat{SR}}_n\}]$ ignored or guessed. $V$ is the variance of the trial Sharpes, not the strategy’s own variance. Using the strategy’s return variance is a category error; using a made-up $V$ is unverifiable.
4. $N$ understated. Every dependent/derived trial counts (page 03). Understating $N$ makes DSR too generous - the exact failure DSR exists to prevent.
5. Threshold mis-read. $\text{DSR}=0.90$ does not prove fraud; it says the result is not distinguishable from the best of $N$ flukes at 95%. Conversely $\text{DSR}\ge 0.95$ is necessary, not sufficient - it says nothing about costs, capacity, or decay.
6. Short samples + fat tails. Both inflate the standard error; a 6-month record with kurtosis 12 can never reach DSR 0.95 regardless of Sharpe. MinTRL tells you this before you deploy.

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5. References

• Bailey, D. H. & López de Prado, M.: The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality, JPM 40(5), 94–107 (2014)
• Bailey, D. H. & López de Prado, M.: The Sharpe Ratio Efficient Frontier, Journal of Risk 15(2) (2012)
• Lo, A.: The Statistics of Sharpe Ratios, Financial Analysts Journal 58(4) (2002)
• Ingersoll, Spiegel, Goetzmann & Welch: Portfolio Performance Manipulation and Manipulation-Proof Performance Measures, RFS 20(5) (2007)

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6. Connected Graph Bridges

• Back: 03 · Multiple Testing · Index Hub
• Forward: 05 · Failure Modes & Practice → 06 · Purged CV, PBO & Reality Check
• Siblings: 02 · Why Backtests Lie · Feature Engineering & Labeling (triple-barrier labels drive the skew that DSR penalizes)