# Can we use White's reality check to compare two Sharpe ratios?

I read a paper from Ledoit and Wolf that proposes a method to compare two Sharpe ratios and a paper from White that proposes a method to compare $n$ trading rules.

My question is: Can we use White's method to compare two Sharpe ratios? I prefer this method because it's computationally simpler.

• Did you see the source code provided by one of the authors of the Ledoit and Wolf paper? – Bob Jansen Mar 13 '12 at 19:18
• The point of White is to adjust for multiplicity. Why use it to compare only two Sharpe ratios? Or maybe you want to use it for $n >> 2$ Sharpe ratios? – James Sep 12 '14 at 14:28
• @ColinTBowers apparently it was under patent until Jan 2017, as a simple google search would have revealed: patents.google.com/patent/US5893069A/en?oq=5893069 . And yes, Hansen's loglog trick is certainly an improvement on WRC. Can I ask what the "tests proposed in some recent publications" are? – steveo'america Feb 18 at 19:15
• ah, thanks. I only recalled White's patent because I remember feeling subversive when implementing it. I too reflexively doubt patentability of 'obvious' uses of known methods, but I also worked for a company that lost +500M in a patent case for using Kalman Filters (!). A recent survey of several methods for MHT correction on Sharpe is Pav, 2019. Again, none of these are required for $n=2$ strategies. – steveo'america Feb 19 at 1:09
• @ColinTBowers Yes, the CLT/Delta method analysis was proposed by Jobson & Korkie, later turned into F-test by Leung & Wong, and a $\chi^2$ test by Wright et al. – steveo'america Feb 20 at 18:30

As James has pointed out in the comments, White's Reality Check is specifically designed to control the family-wise error rate given $$k > 2$$ statistics. The theory does not depend on $$k$$ asymptotics, so there is nothing invalid about using White's Reality Check for $$2$$ statistics, but in practice there would be little point to doing this. Further, as stevoamerica points out above, the Reality check had a patent on it until two years ago - whether it would be enforced in a court case though is another question entirely...
In particular, for $$k=2$$, it is fairly straightforward to construct a simple statistical test for the difference in two Sharpe ratios. Presumably there is some insight in Ledoit and Wolf's paper that makes their statistic superior to what I am about to suggest. Also, see stevoamericas comments on the question for references to some other sophisticated testing measures. But if what you're after is simplicity, then the following is still perfectly valid:
Let $$R_{1,t}$$, and $$R_{2,t}$$ denote returns on the two assets of interest. In this framework, I define the Sharpe ratio: $$$$S_1 = \frac{\mathbb{E} R_{1,t}}{\sqrt{\mathbb{V} R_{1,t}}}$$$$ For any random variable $$X_t$$ the sample mean is defined: $$$$\bar{X} = \frac{1}{T} \sum_{t=1}^T X_t$$$$ Let: $$$$\bar{\sigma}_1 = \sqrt{\frac{1}{T} \sum_{t=1}^T (R_{1,t} - \bar{R}_1)^2}$$$$ A natural estimator for $$S_1$$ is: $$$$\hat{S}_1 = \frac{\bar{R}_1}{\bar{\sigma}_1}$$$$ I assume suitable regularity conditions on $$R_{1,t}$$ such that $$\bar{R}_1 \overset{\mathbb{P}}{\rightarrow} \mathbb{E} R_{1,t}$$, $$\sqrt{T} \bar{R}_1 \overset{d}{\rightarrow} \mathcal{N}$$, and $$\bar{\sigma}_1 \overset{\mathbb{P}}{\rightarrow} \sqrt{\mathbb{V}(R_{1,t})}$$ (e.g. weak dependence and suitably bounded moments). By Slutsky's theorem, these conditions are sufficient for: $$$$\hat{S}_1 \overset{\mathbb{P}}{\rightarrow} S_1$$$$ and note that by Cramer's theorem: $$$$\sqrt{T} \bar{S}_1 = \frac{\sqrt{T} \bar{R}_1}{\bar{\sigma}_1} \overset{d}{\rightarrow} \mathcal N$$$$ since the numerator is converging in distribution to a Normal, and the denominator is converging in probability to a constant strictly greater than $$0$$.
So we have a CLT for our statistic. For the purposes testing a difference in two statistics, it is easier if our statistic can be phrased as a single sample mean. This is straightforward. Let: $$$$Y_{1,t} = (\bar{\sigma}_1)^{-1} R_{1,t} ,$$$$ where it is worth emphasizing that it immediately follows that: $$$$\hat{S}_1 = \bar{Y}_1 .$$$$ Incorporating the second asset, we now define: $$$$d_t = Y_{1,t} - Y_{2,t} .$$$$ The theory thus far is sufficient to show that under: $$$$H_0 : S_1 = S_2 ,$$$$ we have: $$$$\bar{d} \overset{d}{\rightarrow} \mathcal{N}(0, \alpha) .$$$$ So we've literally transformed the problem into testing whether a sample mean is equal to zero, with a CLT existing for the sample mean. If you think $$d_t$$ exhibits time-series dependence, then you will need to estimate $$\alpha$$ using a HAC estimator, or else you could just bootstrap the statistic. Both are likely to give you similar outcomes. If you aren't worried about time-series dependence then just estimate $$\alpha$$ using the sample standard deviation of $$d_t$$ over $$\sqrt{T}$$.