# Is this a common variation of sharpe ratio?

As an aside on his answer on another question Freddy said:

Sharpe ratio is an often cited metric, though I do not like it too much because you are penalized for out-sized positive returns while I would only define negative returns as risk.

At first glance this seems like a good point, but a quick hunt around the site found no mention of it. (To clarify: "It" means calculating variance of just the strategy losses and using that as the divisor.)

What are the arguments against? Does it have a name and is anyone using it? Is there any R function that already implements it?

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Darren, you could have asked me directly in that related question but here goes :-)

The measure you are looking for is called "Sortino Ratio", here a quick wiki and a rather excellent (as its concise but to the point) treatise of the issue at hand:

http://en.wikipedia.org/wiki/Sortino_ratio

http://www.edge-fund.com/Hard02.pdf

and yes there is an R library: http://braverock.com/brian/R/PerformanceAnalytics/html/SortinoRatio.html

I believe that the more risk metrics one looks at in order to evaluate a strategy or portfolio the better. However, if one had to chose Sharpe over Sortino ratio I would not hesitate a second to recommend focusing on Sortino, basically on downside risk. Mathematically it is possible that portfolio1 with no negative but stable low returns results in a higher Sharpe ratio than portfolio2 with no negative returns either but much higher despite more volatile returns, even if portfolio2 has much higher total returns. Obviously there is no one-size-fits-all answer but risk in a portfolio and performance return context is for me still a measure of potential downside, a.k.a. losses. But you need to decide for yourself.

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Thanks; now I know what I'm looking for I see R also has functions for DownsideDeviation, SemiVariance, etc. I didn't ask in the other question as: a)it was off-topic; b) you'd already expressed your bias, and I wanted to hear arguments against ;-) –  Darren Cook Aug 22 '12 at 8:08
pair points, well, I hope the answer helps a little. –  Matt Wolf Aug 22 '12 at 11:32

There are several arguments against using the Sharpe ratio. First is that the Sharpe ratio can be gamed by managers:

1. Illiquid stocks or infrequent marking-to-market raises the sharpe ratio. An example of this is using the NACREF appraisal index to measure the return & volatility of real-estate assets as opposed to the NAREIT index which is marked-to-market much more frequently.

2. Lengthening the measurement interval (to monthly instead of daily returns, for example). This lowers the estimated volatility. Longer holding periods increase the ratio by approximately the square root of time. Digression: This practice is quite pervasive. Whenever I see a strategy tear-sheet I immediately flip to the definition of sharpe ratio and often find that the manager uses monthly sharpe ratios instead of daily.

3. Several strategies such as buy-write have high sharpe ratios that mask severe downside risk for several years. For example, writing out of the money puts and calls generates premium which has a high sharpe in good times. Similarly, strategies that take on default risk, liquidity risk, have the ability to bias upwards the sharpe ratio in normal times (see Long-Term Capital Management)

4. Engage in a return swap with a broker dealer to eliminate the highest returning and lower returning months in the portfolio will increase Sharpe by eliminating extreme returns

5. Smoothing of returns with derivatives

6. The Sharpe ratio can be gamed by adjusting the universe of analysis. For example, a manager with a Sharpe ratio of 1.5 performing security selection the S&P 500 universe has better active management skill than a manager who achieves the same Sharppe ratio on the Russell 5000.

7. To use Sharpe ratio to compare manager performance across strategies there is an assumption that i) investors care about the 1st two moments of returns, and ii) that when the Sharpe ratio is used to compare across strategies that strategy returns are normally distributed.

There are various non-parametric and monte carlo techniques that can improve upon the limitations identified above. Also there are other measures such as Sterling Ratio, Return over Max Drawdown (RAMOD), that can inform one's perspective when used in concert with the Sharpe ratio.

Also, attached is a paper by Andrew Lo that is a nice critique of the Sharpe ratio. His conclusion:

The results presented in this article provide one way to gauge the accuracy of these estimators, and it should come as no surprise that the statistical properties of Sharpe ratios depend intimately on the statistical properties of the return series on which they are based. This suggests that a more sophisticated approach to interpreting Sharpe ratios is called for, one that incorporates information about the investment style that generates the returns and the market environment in which those returns are generated. For example, hedge funds have very different return characteristics from the characteristics of mutual funds; hence, the comparison of Sharpe ratios between these two investment vehicles cannot be performed naively.

As mentioned above, the R Performance Analytics package has a number of performance measurement tools.

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I would slightly adjust point 7 to say that you implicitly assume that all the investor cares about is the first two moments, rather than assuming normality. You can still calculate Sharpe ratios on non-normal distributions. –  John Aug 22 '12 at 15:48
@John - I like your point as an add'l new point (added). I should clarify my point to say that the comparability of the statistic across strategies assumes the joint normally distributed returns. –  Quant Guy Aug 22 '12 at 15:52
@QuantGuy, while I agree with you that statistics can be gamed most of your points have nothing at all to do with Sharpe ratio or any other PARTICULAR measure. Points 1, 3, 4, 5, 6 are general points how the system is gamed, while point 2 is more related but I would not even follow your rational here, often Sharpe ratios drop when increasing the measurement interval b/c the drop in excess return outweighs the decrease in vol, hardly fitting in with your argument. Point 7: Which investors care about higher moments when 96% or so of total risk (relevent to investors) is captured by the first 2? –  Matt Wolf Aug 22 '12 at 23:55
@Freddy - Most of the critiques against the Sharpe nature are statistical in nature. True they apply to other measures but it seems an answer to the question. On Point 7: Most investors do not have mean-variance utility functions (see Prospect theory and Behavioral finance literature). On point 2, at higher intervals intra-month or intra-period volatility is drowned out. Monthly vol measurements miss entirely episodes such as Aug 2007 quant meltdown, flash crash, etc. There are scenarios where daily Sharpe can be > monthly sharpe but it isn't usually the case. –  Quant Guy Aug 23 '12 at 4:15
@Freddy btw, I don't disagree with your answer in fact I +1'd it -- I think there are some other dimensions to bring out that's all. Cheers –  Quant Guy Aug 23 '12 at 4:21

The statement is accurate.

The "argument against" is that some investors prefer stability of returns over time instead of returns with high variance, even if all returns in the series are positive. For example if a manager has a 5 year track record with no losing years and averages 20%, but one of the years he has a 2% return that's a important for the investor to understand.

In other words, excluding positive returns from the variance calculation can hides information.

Here's a good paper on "down side risk adjusted performance measures". These are a class of information measures that can be setup to incorporate a variety of factors, including separating out upside return variance from the measure...

http://corporate.morningstar.com/us/documents/MethodologyDocuments/ResearchPapers/UnifiedApproach.pdf

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so you rather penalize such manager by assigning him a Sharpe ratio almost the same as a manager who managed to make 7-8% a year? The question here is what the definition of risk is in the context of portfolio performance measurement. So far I have not heard of investors who prefer stable low returns over high but volatile returns, given total returns are maximized under the constraint that there are no negative returns. As I showed, Sortino ratio accounts for all down side risk, so should a loss show up it would immediately penalize the metric. –  Matt Wolf Aug 22 '12 at 6:26

In addition to the Sortino ratio, another option is calculating the excess return to conditional Value at Risk (CVaR or also called ES) ratio. CVaR is a common measure of tail risk that effectively measures the expected loss (or return) below a certain percentile. You may need to take the absolute value of CVaR.

It is possible to use a weighted average of CVaRs at several different percentiles in the ratio. This is called mixed CVaR and it enables you to express your preferences on different amounts of downside risk. In addition, subtracting out the expected return from CVaR is called CVaR deviation (you can also create a mixed CVaR deviation). This metric is perhaps closer to the standard deviation and can be a better alternative since it will not change signs (it is possible for CVaR to change signs).

CVaR can be calculated in the same library as the Sortino ratio using the ES function.

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