Can a Sharpe ratio of 1.55 be better than a Sharpe ratio of 1.63 in a 1 year track-record? Not necessarily. Sharpe ratios are not comparable, unless we control the skewness and kurtosis of the returns.

In this post we are going to analyze the advantages of the Probabilistic Sharpe Ratio exposed by Marcos López de Prado in this paper. It will include an example coded in Python.

Context

The Sharpe Ratio (SR) is the most common risk-reward ratio when evaluating different investment strategies (although there are other alternatives). And, as all we know it is calculated as:

\(\begin{equation}S R=\frac{\mu}{\sigma}\end{equation}\) ,

where \({\mu}\) is the mean return and \({\sigma}\) is its standard deviation.

Since the true \({\mu}\) and \({\sigma}\) are usually unknown, we estimated them with the historical returns. Therefore, the Sharpe ratio we are usually calculating is not the true SR, it is an estimation, and as every estimation, it will have its variance and confidence level.

Lo (2002) exposed that, assuming Normal returns (skewness=0, kurtosis=3), the estimated Sharpe ratio (\(\widehat{SR}\)) would follow a Normal distribution where the standard deviation is then given by:

\(\begin{equation}\hat{\sigma}(\widehat{SR})=\sqrt{\frac{1}{n-1}\left(1+\frac{1}{2} \widehat{SR}^{2}\right)}\end{equation}\)

Above all, Martens (2002) concludes that Normality assumption on returns could be dropped, and still the estimated Sharpe ratio would follow a Normal distribution with the next standard deviation:

\(\begin{equation}\hat{\sigma}(\widehat{SR})=\sqrt{\frac{1}{n-1}\left(1+\frac{1}{2} \widehat{SR}^{2}-\gamma_{3} \widehat{SR}+\frac{\gamma_{4}-3}{4} \widehat{SR}^{2}\right)}\end{equation}\)

In the prior formula we can see how a skewness (\(\gamma_{3}\)) of 0 and a kurtosis (\(\gamma_{4}\)) of 3 would reduce the Martens formula to Lo formula.

Although skewness and kurtosis does not affect the point estimate of Sharpe ratio, it greatly impacts its confidence bands, and consequently its statistical significance.

Marcos López de Prado

So, an important conclusion is that, despite of the Non Normality of the returns distributions, the \(\widehat{SR}\) would always follows a Normal distribution with the next parameters:

\(\begin{equation}(\widehat{S R}-S R) \stackrel{a}{\rightarrow} N\left(0, \frac{1+\frac{1}{2} S R^{2}-\gamma_{3} S R+\frac{\gamma_{4}-3}{4} S R^{2}}{n-1}\right)\end{equation}\)

The SR 2.0: Probabilistic Sharpe Ratio

As a solution for the problem exposed in the previous section, Marcos López de Prado developed the Probabilistic Sharpe Ratio (PSR).

Given a predefined benchmark Sharpe Ratio (\(SR^{*}\)), the estimated Sharpe ratio \(\widehat{SR}\) can be expressed in probabilistic terms as:

\(\begin{equation}P S R\left(S R^{*}\right)=\operatorname{Prob}[S R \leq \widehat{S R}]\end{equation}\)

After some algebra the prior formula can be expressed as:

\(\begin{equation}\widehat{P S R}\left(SR^{*}\right)=Z\left[\frac{\left(\widehat{SR}-SR^{*}\right)}{\hat{\sigma}(\widehat{S R})}\right]\end{equation}\)\(\begin{equation}=Z\left[\frac{\left(\widehat{SR}-SR^{*}\right) \sqrt{n-1}}{\sqrt{1+\frac{1}{2} \widehat{SR}^{2}-\gamma_{3} \widehat{SR}+\frac{\gamma_{4}-3}{4} \widehat{SR}^{2}}}\right]\end{equation}\),

where \(Z\) is the cdf of the Standard Normal distribution.

From the above formula it is derived that, for a given \(SR^{*}\), \(PSR\) increases with greater \(\widehat{SR}\) (in the original sampling frequency, i.e. non-annualized), or longer track-record (\({n}\)), or positively skewed returns (\(\gamma_{3}\)), but it decreases with fatter tails (\(\gamma_{4}\)).

A practical example in Python

In this last section, we will briefly review a practical example, coded in Python, in which the concepts we have seen so far can help us for comparing different investment strategies, and select the “really” best one.

Fantasy world

First of all, we will create two random returns distributions with scipy, that will simulate the weekly returns of two different Hedge Funds in which we are interested to invest.

The next table presents the true statistics for both returns distributions. But keep in mind that in the real world we do not know these stats unless we have a very long track record, that is not the case in the hedge fund industry…

As you can see we have made the distributions in a way that in the long term the Strategy 2 will have a better Sharpe ratio than Strategy 1 (SR = mean / std). But remember, in the real world we do not have access to these stats… so let’s move away from Fantasy World, and go to the Real world!

Real world

The harsh reality is that for choosing in which Hedge Fund invest our savings, we only have a one-year track-record of weekly returns. So we have the last 52 weekly returns of both Hedge Funds. Let’s look at their stats:

Mmm… it seems like the Hedge Fund 1 has a bigger Sharpe ratio, let’s invest in it! Wait a moment! What is about Probabilistic Sharpe Ratio, how confident can we be with our SR estimations?

Ohh, now we can see that despite the bigger \(\widehat{SR}\) of the Hedge Fund 1 it seems more reasonable to invest our money in Hedge Fund 2!

This is because we can not have the same confidence in our \(\widehat{SR}\) estimations. With HF1 we have “only” a certainty of 92.99% that in the future its true SR will be greater than 0 (\(SR^{*}\)), but with HF2 we have more statistically certainty that in the future will have positive returns: 95.19% of chances.

Finally, we can check that if we generate more samples of both returns distributions, the annualized Sharpe ratio for Hedge Fund 2 is 3.85, and for Hedge Fund 1 is 2.42. (you can get these results dividing the true mean by the true stDev showed in the first table, too)

Thanks to having calculated the PSR instead of only the SR, we have been able to select an strategy with a Sharpe ratio 59% higher.

In five years this will turn into winning three times more.

Key takeaways

• The estimated Sharpe Ratio always follows a Normal Distribution even if the returns do not.
• The estimation errors on the SR increases with negatively skewed returns and fatter tails (big kurtosis). In those cases, it will be needed a longer track-record for decreasing these errors.
• The Probabilistic Sharpe Ratio is a powerful statistic that gives us the confidence level associated with a particular SR estimation.

To check the code used for this post, and the formulas in Python of the \(\hat{\sigma}(\widehat{SR})\) and \({PSR}\), take a look at my GitHub repository.

And if you have any questions or feedback, let me know in the comments below. See you soon in the next post!

References

• [1] Marcos López de Prado and David Bailey (2012). The Sharpe ratio efficient frontier.
• [2] Geng Deng and Craig McCann (2013). Robust Portfolio Optimization with Value-At-Risk Adjusted Sharpe Ratios.