Downside Risk Metrics and the Limits of the Sharpe Ratio

# Is this a common variation of sharpe ratio?


# 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?

## Answer by Matt Wolf (score 8, accepted)

https://quant.stackexchange.com/a/3988

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.

## Answer by Ram Ahluwalia (score 5)

https://quant.stackexchange.com/a/3989

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

- 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.

- 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.

- 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)

- 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

- Smoothing of returns with derivatives

- 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.

- 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.

## Answer by glyphard (score 2)

https://quant.stackexchange.com/a/3987

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

## Answer by John (score 2)

https://quant.stackexchange.com/a/3993

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.