Sortino ratio: A better measure of risk

January 31, 2013 06:00 PM

Sortino ratio calculation

In this example, we will calculate the annual Sortino ratio for a hypothetical CTA with positive skew with the following set of annual returns:

Annual Returns:  2%, 1%, –1%, 18%, 8%,–2%, 1%, –1%
Target Return:  0%

Although in this example we use a target return of 0%, any value may be selected, depending on the application, i.e., a futures trading system developer comparing different trading systems vs. a pension fund manager with a mandate to achieve 8% annual returns. Of course using a different target return will result in a different value for the target downside deviation. If you are using the Sortino ratio to compare managers or trading systems, you should be consistent in using the same target return value.

First, we will calculate the numerator of the Sortino ratio, the average period return minus the target return:

Average annual return – Target return  =  3.25% – 0%  =  3.25%

Next, we will calculate the target downside deviation:

1)  For each data point, calculate the difference between that data point and the target level. For data points above the target level, set the difference to 0%. The result of this step is the underperformance data set.

min(0, 2% – 0%)  =  0%
min(0, 1% – 0%)  =  0%
min(0, –1% – 0%)  =  –1%
min(0, 18% – 0%)  =  0%
min(0, 8% – 0%)  =  0%
min(0, –2% – 0%)  =  –2%
min(0, 1% – 0%)  =  0%
min(0, –1% – 0% ) =  –1%

2)  Next, calculate the square of each value in the underperformance data set determined in Step 1.

0%  ^  2  =  0%
0%  ^  2  =  0%
–1%  ^  2  =  0.01%
0%  ^  2  =  0%
0%  ^  2  =  0%
–2%  ^  2  =  0.04%
0%  ^  2  =  0%
–1%  ^  2  =  0.01%

3)  Then, calculate the average of all squared differences determined in Step 2. Notice that we do not “throw away” the 0% values.

Average  =  (0% + 0% + 0.01% + 0% + 0% + 0.04% + 0% + 0.01%) / 8  =  0.0075%

4)  Then, take the square root of the average determined in Step 3. This is the target downside deviation used in the denominator of the Sortino ratio.

Target Downside Deviation  =  Square root of 0.0075%  =  0.866%

Finally, we calculate the Sortino ratio:

Sortino Ratio  =  3.25% / 0.866% = 3.75 

This is a strong score and indicative of the return stream from which we calculated it. Calculating the Sharpe ratio on the same set of returns would have produced a Sharpe ratio (0% RFR) of 0.52, a mediocre one that indicates more volatility by penalizing the outsized positive returns. 

Sortino vs. Sortino

Often in trading literature and trading software packages we have seen the Sortino ratio, and in particular the target downside deviation, calculated incorrectly. Most often, we see the target downside deviation calculated by “throwing away all the positive returns and taking the standard deviation of negative returns.” We hope that by reading this article, you can see how this is incorrect. Specifically:

In Step 1, the difference with respect to the target level is calculated, unlike the standard deviation calculation where the difference is calculated with respect to the mean of all data points. If every data point equals the mean, then the standard deviation is zero, no matter what the mean is. Consider the following return stream: [–10, –10, –10, –10]. The standard deviation is 0; while the target downside deviation is 10 (assuming target return is 0).

In Step 3, all above target returns are included in the averaging calculation. The above target returns set to 0% in Step 1 are not thrown away.

The Sortino ratio takes into account both the frequency of below-target returns as well as the magnitude of them. Throwing away the zero underperformance data points removes the ratio’s sensitivity to frequency of underperformance. Consider the following underperformance return streams: [0, 0, 0, –10] and [–10, –10, –10, –10]. Throwing away the zero underperformance data points results in the same target downside deviation for both return streams, but clearly the first return stream has much less downside risk than the second.

In this article we presented the definition of the Sortino ratio and the correct way to calculate it. While the Sortino ratio addresses and corrects some of the weaknesses of the Sharpe ratio, we feel there is one measure that is even better yet: The Omega Ratio. We look forward to tackling the Omega Ratio in our next article. 

Tom Rollinger is director of new strategies development for Sunrise Capital Partners. Previously he was a portfolio manager for quantitative hedge fund legend Edward O. Thorp. Scott Hoffman is the founder of CTA Red Rock Capital Management.

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