# Backtesting of outperformance of a benchmark using the Deflated Sharpe Ratio

I want to test whether, let's say, strategy A outperforms strategy B.

In Marcos López de Prado's book Advances in Financial Machine Learning he presents the following statistics:

1. The Probalistic Sharpe Ratio: the probability that a strategy's observed Sharpe Ratio $$\hat{SR}$$ exceeds a benchmark Sharpe Ratio $$SR^*$$ follows a normal distribution when we consider the skew $$\hat{\gamma_3}$$ and kurtosis $$\hat{\gamma_4}$$ of the strategy's return: $$\hat{PSR}[SR^*] = Z[\frac{(\hat{SR} - SR^*)\sqrt{T-1}}{\sqrt{1-\hat{\gamma_3}\hat{SR} + \frac{\hat{\gamma_4} - 1}{4}\hat{SR}}}]$$

However, if we were to run many simulations, just by random chance, at some point we would find some strategy that outperforms our benchmark significantly. So a new statistic is introduced accounting the multiplicity of trials:

1. The Deflated Sharpe Ratio, $$\hat{DSR}=\hat{PSR}[SR^\star]$$ where $$SR^\star=\sqrt{Var(\{\hat{SR_n}\})}((1-\gamma)Z^{-1}[1-\frac{1}{N}]+\gamma Z^{-1}[1-\frac{1}{N}e^{-1}])$$, where $$\gamma$$ is the Euler-Mascheroni constant and $$e$$ is Euler's constant. In fact, $$SR^\star$$ is the maximum expected Sharpe Ratio in the series of Sharpe Ratios (calculated over all backtests). The Deflated Sharpe Ratio tests whether a given strategy's excess returns are significantly higher than zero.

You can refer to this article for more information.

So far so good. In the book, López de Prado summarises saying that the DSR can be computed on both absolute as relative returns.

My question then is fairly simple. I have strategy A and strategy B, and ran $$N$$ simulations. If I want to test whether strategy A is expected to outperform strategy B, I can simply take A's excess returns over B, and calculate the deflated Sharpe Ratio over that return series?

Thank you all!