Earlier this week we discussed the Red Rock Capital research paper discussing different metrics used to evaluate CTA risk adjusted performance.  Sharpe has long been considered the go to statistic commonly referred to by brokers and CTAs in Managed Futures. Today we intend to cover the differences and make some conclusions on the Sortino vs. Sharpe ratio debate and give a different perspective on analyzing a CTA.

The differences are as follows:

1. In the Target Downside Deviation calcula­tion, the deviations of Xi from the user select­able target return are measured, whereas in the Standard Deviation calculation, the deviations of Xi from the average of all Xi is meaasured.
2. In the Target Downside Deviation calcu­lation, all Xi above the target return are set to zero, but these zeros are still included in the summa­tion. The calculation for Standard Deviation has no Min() function.

Standard deviation is a measure of dispersion of data around its mean, both above and below. Target Downside Deviation is a measure of dis­persion of data below some user selectable target return with all above target returns treated as underperformance of zero. Big difference.

Example Sortino Ratio Calculation

In this example, we will calculate the annual Sor­tino ratio for the following set of annual returns:

Annual Returns: 17%, 15%, 23%, -5%, 12%, 9%, 13%, -4%  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, calculate the numerator of the Sortino ratio, the average period return minus the target return:

Average Annual Return – Target Return: (17% + 15% + 23% – 5% + 12% + 9% + 13% – 4%) ÷ 8 – 0% = 10%

Next, 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 under ­performance data set.
2. Next, calculate the square of each value in the under performance data set determined in the first step. Note that percentages need to be expressed as decimal values before squaring, i.e. 5% = 0.05.
3. Then, calculate the average of all squared differences determined in Step 2. Notice that we do not “throw away” the 0% values.
4. Lastly, take the square root of the average determined in #3.  This is the Target Downside Deviation used in the denominator of the Sortino Ratio.

Conclusion

The main reason we wrote this article is because in both literature and trading software packages, we have seen the Sortino ratio, and in particular the target downside deviation, calculated incorrectly more often than not. Most often, we see the target downside deviation calculated by “throwing away all the positive returns and take the standard deviation of negative returns”. We hope that by reading this article, you can see how this is incorrect. Specifically:

1.  In Step 1 above, 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).
2. In Step 3 above, all above target returns are included in the averaging calculation. The above target returns set to 0% in step 1 are kept.
3. The Sortino ratio takes into account both the frequency of below target returns as well as the magnitude of below target returns. Throwing away the zero underperformance data points removes the ratio’s sensitivity to frequency of under performance.  Consider the following under performance return streams: [0, 0, 0, -10] and [-10, -10, -10, -10]. Throwing away the zero under performance 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.

This paper 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, neither statistic measures ongoing and future risks; they both measure the past “goodness” of a manager’s or investment’s return stream.

Red Rock Capital is a systematic global macro hedge fund (CTA) located on Chicago’s Magnificent Mile.  The firm is helmed by Thomas Rollinger, most notably a former protege and devoted pupil of quantitative hedge fund legend Edward O. Thorp. Red Rock recently celebrated its 10th anniversary and is well-positioned to grow and thrive in the alternative investment arena.