Draft:Deflated Sharpe Ratio

The Deflated Sharpe Ratio (DSR) is a statistical method used to determine whether the Sharpe Ratio of an investment strategy is statistically significant, after correcting for selection bias, backtest overfitting, and non-normality in return distributions. It provides a more reliable test of financial performance, especially when many strategies are evaluated.

Background

The Sharpe Ratio, developed by William F. Sharpe, is a widely used measure of risk-adjusted return, calculated as the ratio of excess return over the risk-free rate to the standard deviation of returns. While useful, the Sharpe Ratio has limitations, especially when applied to multiple strategy evaluations. Issues such as selection bias, where the best-performing strategy is chosen from a large set, and backtest overfitting, where a strategy is tailored to past data, can inflate the Sharpe Ratio, leading to misleading conclusions about a strategy's effectiveness. Additionally, the Sharpe Ratio assumes normally distributed returns, an assumption often violated in practice, and it does not take into account sample length.

Mathematical Definition

The DSR is defined as:

${\text{DSR}}=\Phi \left({\frac {({\hat {SR}}-SR_{0})\cdot {\sqrt {T-1}}}{\sqrt {1-{\hat {\gamma }}_{3}{\hat {SR}}+{\frac {{\hat {\gamma }}_{4}-1}{4}}{\hat {SR}}^{2}}}}\right)$

Where:

• ${\hat {SR}}$ is the observed Sharpe Ratio (not annualized),
• $SR_{0}$ is the threshold Sharpe Ratio that reflects the highest Sharpe Ratio expected from $N$ unskilled strategies,
• ${\hat {\gamma }}_{3}$ is the skewnewss of the returns,
• ${\hat {\gamma }}_{4}$ is the kurtosis of the returns,
• $T$ is the returns' sample length.
• $\Phi$ is the standard normal cumulative distribution function.

The threshold $SR_{0}$ is approximated by:

$SR_{0}={\sqrt {\mathbf {V} [{\hat {SR}}_{n}]}}\left((1-\gamma )\Phi ^{-1}\left[1-{\frac {1}{N}}\right]+\gamma \Phi ^{-1}\left[1-{\frac {1}{Ne}}\right]\right)$

Where:

• $\mathbf {V} [{\hat {SR}}_{n}]$ is the cross-sectional variance of Sharpe Ratios across trials,
• $\gamma$ is the Euler-Mascheroni constant (approx. 0.5772),
• $e$ is Euler's number,
• $\Phi ^{-1}$ is the inverse standard normal CDF,
• $N$ is the number of independent strategy trials.

False Strategy Theorem: Statement and Proof

The False Strategy Theorem provides the theoretical foundation for the Deflated Sharpe Ratio (DSR) by quantifying how much the best Sharpe Ratio among many unskilled strategies is expected to exceed zero purely due to chance. Even if all tested strategies have true Sharpe Ratios of zero, the highest observed Sharpe Ratio will typically be positive and statistically significant—unless corrected. The DSR corrects for this inflation.

Statement

Let $\{{\hat {SR}}_{1},{\hat {SR}}_{2},\dots ,{\hat {SR}}_{N}\}$ be $N$ Sharpe Ratios independently drawn from a normal distribution with mean zero and variance $\sigma ^{2}$. Then the expected maximum Sharpe Ratio among these $N$ trials is approximately:

$SR_{0}={\sqrt {\sigma ^{2}}}\cdot \left((1-\gamma )\Phi ^{-1}\left(1-{\frac {1}{N}}\right)+\gamma \Phi ^{-1}\left(1-{\frac {1}{Ne}}\right)\right)$

Where:

• $\Phi ^{-1}$ is the quantile function (inverse CDF) of the standard normal distribution,
• $\gamma \approx 0.5772$ is the Euler–Mascheroni constant,
• $e\approx 2.718$ is Euler’s number,
• $N$ is the number of independent trials.

This value $SR_{0}$ is the **expected maximum Sharpe Ratio** under the null hypothesis of no skill. It represents a benchmark that any observed Sharpe Ratio must exceed in order to be considered statistically significant.

Proof Sketch

Let $X_{1},X_{2},\dots ,X_{N}\sim {\mathcal {N}}(0,1)$ be independent standard normal variables. The expected maximum of $N$ such variables is approximated by:

$\mathbb {E} [\max(X_{1},\dots ,X_{N})]\approx (1-\gamma )\Phi ^{-1}\left(1-{\frac {1}{N}}\right)+\gamma \Phi ^{-1}\left(1-{\frac {1}{Ne}}\right)$

Now let ${\hat {SR}}_{i}\sim {\mathcal {N}}(0,\sigma ^{2})$ for each $i$. Then:

$\mathbb {E} \left[\max({\hat {SR}}_{1},\dots ,{\hat {SR}}_{N})\right]=\sigma \cdot \mathbb {E} [\max(X_{1},\dots ,X_{N})]$

Combining the two expressions gives:

$SR_{0}=\sigma \cdot \left((1-\gamma )\Phi ^{-1}\left(1-{\frac {1}{N}}\right)+\gamma \Phi ^{-1}\left(1-{\frac {1}{Ne}}\right)\right)$

If $\sigma ^{2}$ is estimated as the cross-sectional variance of Sharpe Ratios $\mathbf {V} [{\hat {SR}}_{n}]$, then:

$SR_{0}={\sqrt {\mathbf {V} [{\hat {SR}}_{n}]}}\cdot \left((1-\gamma )\Phi ^{-1}\left(1-{\frac {1}{N}}\right)+\gamma \Phi ^{-1}\left(1-{\frac {1}{Ne}}\right)\right)$

This completes the derivation.

Implication for the DSR

The False Strategy Theorem shows that in large-scale testing, even unskilled strategies will produce apparently "significant" Sharpe Ratios. To correct for this, the DSR adjusts the observed Sharpe Ratio by subtracting the expected maximum from noise, $SR_{0}$, and scaling by its standard error:

${\text{DSR}}=\Phi \left({\frac {{\hat {SR}}-SR_{0}}{\sigma _{\hat {SR}}}}\right)$

This yields the probability that the observed Sharpe Ratio reflects true skill, not selection bias or overfitting.

Estimating the Effective Number of Trials (N)

In practice, many strategy trials are not independent due to overlapping features. To estimate the effective number of independent trials $N_{\text{eff}}$, López de Prado (2018) proposes clustering similar strategies using unsupervised learning techniques such as hierarchical clustering on the correlation matrix of their returns.

This approach yields a conservative lower bound for $N$. Alternatively, spectral methods (e.g. eigenvalue distribution of the correlation matrix) can also provide estimates of $N_{\text{eff}}$.

Confidence and Power of the Sharpe Ratio under Multiple Testing

To assess the significance of Sharpe Ratios under multiple testing, López de Prado (2018) derives closed-form expressions for the Type I and Type II errors.

Confidence

The probability that a discovered strategy is not a false positive (i.e., the confidence) is:

${\text{Confidence}}=1-\alpha _{K}=\left(\Phi \left({\frac {{\hat {SR}}\cdot {\sqrt {T-1}}}{\sqrt {1-\gamma _{3}{\hat {SR}}+{\frac {\gamma _{4}-1}{4}}{\hat {SR}}^{2}}}}\right)\right)^{K}$

Where:

• $T$ is the number of return observations,
• $\gamma _{3}$ and $\gamma _{4}$ are the skewness and kurtosis of returns,
• $K$ is the number of effectively independent trials.

Power

The power of the test, i.e., the probability of correctly identifying a strategy with true Sharpe Ratio $SR^{*}$, is:

${\text{Power}}=1-\beta _{K}=1-\left(\Phi \left(\Phi ^{-1}\left[(1-\alpha _{K})^{1/K}\right]-\theta \right)\right)^{K}$

With:

$\theta ={\frac {SR^{*}\cdot {\sqrt {T-1}}}{\sqrt {1-\gamma _{3}{\hat {SR}}+{\frac {\gamma _{4}-1}{4}}{\hat {SR}}^{2}}}}$

These equations quantify the reliability of observed Sharpe Ratios under multiple testing and return non-normality.

See also

• Sharpe Ratio
• Backtesting
• Overfitting
• Selection bias
• Multiple comparisons problem