From the February 2013 issue of Futures Magazine • Subscribe!

Sortino ratio: A better measure of risk

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. 

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