# The Sortino Ratio

## by CI Staff

In the last couple of installments of Fundamental Focus, we have discussed risk-adjusted performance measures, namely the Sharpe and Treynor ratios (these articles appeared in the Fourth Quarter 2012 and First Quarter 2013 issue, respectively, and are available online at ComputerizedInvesting.com).

Both ratios assess a money manager’s ability to generate excess return per unit of total risk. Where the two ratios differ is in their risk measure. Sharpe’s original work focused on total risk, as measured by historical standard deviation of returns, whereas Treynor used systematic or market risk, as measured by the portfolio’s beta. These articles spurred many reader comments, some of which pointed out that these risk measures handle upside and downside risk equally. However, most investors are only concerned with the downside risk (few will complain if their portfolios are rising in value!).

Several financial measures have been developed that attempt to measure the downside risk of an asset without penalizing an asset or strategy with large positive performance deviations. One of them is the Sortino ratio.

## Sortino Ratio vs. Sharpe Ratio

The Sortino ratio, named after Frank A. Sortino, measures the risk-adjusted return of an individual asset or a portfolio, as do the Sharpe and Treynor ratios. However, it only concerns itself with returns that fall below a user-specific minimum or required rate of return (minimum accepted return, or MAR). In other words, it measures the excess return against the risk of not meeting the minimum return. This differs from the other risk-adjusted return measures we’ve discussed lately, which treat upside and downside volatility equally.

The Sortino ratio overcomes the shortcomings of the Sharpe ratio, namely, that it relies on standard deviation, which assumes that returns have a normal or symmetrical distribution, and that it uses the mean return as a target return.

The Sortino ratio avoids these downsides by, first, incorporating a relative target rate of return instead of simply using the average return, and, secondly, by quantifying downside volatility without penalizing upside volatility.

That is not to say that the Sortino ratio is perfect. In researching this article, we ran across multiple versions of the calculation, with differing downside deviation calculations. Some methods consider all periodic returns, changing any returns that exceed the minimum accepted return to zero, and then calculating the standard deviations across all returns. However, there is an argument that using zeros in the calculation underestimates the volatility. The other option is to ignore any positive excess returns and calculate the standard deviation of only the negative returns. The argument here is that since the Sortino ratio is intended to measure downside risk, this seems to capture the spirit of what Sortino was trying to accomplish.

## Calculating the Sortino Ratio

The Sortino ratio is calculated as follows:

### S = (R – T) / DR

Where:

- R is the realized return of the asset or portfolio,
- T is the minimum accepted return (MAR) and
- DR is the downside deviation as measured by the standard deviation of negative asset or portfolio returns.

Figure 1 presents two forms of the Sortino ratio for two different assets (a copy of this spreadsheet is available online). The calculations differ in their handling of excess returns. The first (Sortino 1) simply changes positive excess returns to zeros, while the other (Sortino 2) only uses negative excess returns in the calculation of downside risk.

For both assets, our minimum accepted monthly return is 0.8%, which approximates a 10% annual return (1.00812 – 1). To calculate the downside risk, we use the lower partial second moment of the data with respect to zero. In layman’s terms, we calculate the square root of the average squared distance of the realized return from the MAR. For Sortino 1, we square the lesser of zero and the negative excess return for each month, add the results together, divide the sum by 12 and then take the square root of the average. However, for Sortino 2, we only square the negative excess returns and calculate the square root of the average, using 5 for Asset 1 and 10 for Asset 2. These are the number of months that Asset 1 and Asset 2 had negative excess returns, respectively.

From Figure 1, we can see the impact on the downside risk of including zeros for those months where the excess return is positive versus using only the negative returns. For both assets, using zeros lowers the downside risk. The impact on the downside risk for Asset 1 was much more pronounced than for Asset 2, which had twice the number of months where the excess return was negative. Sortino 1’s downside risk was 35% lower than that of Sortino 2 for Asset 1.

For both assets, the average excess return was simply the difference between the average monthly return and the minimum accepted return. Here, we used all 12 months for both Sortino calculations. Using only negative excess returns impacts only the downside risk (denominator) of the Sortino ratio’s calculation.

Finally, the Sortino ratio is the quotient of dividing the average excess return by the downside risk. In this example, Sortino 2 is higher for both assets, as the higher downside risk in the denominator lowers the overall value. Some argue that using only the number of returns that fall below the MAR in calculating the average of the sum of the squared deviations, as we did for Sortino 2, may significantly underestimate downside risk. Comparing Asset 1 to Asset 2, Asset 1 had much higher Sortino ratio values, due to the higher average excess return, which in turn led to lower downside risk.

## Interpretation

Calculating a Sortino ratio for a single asset or portfolio is meaningless. Like the Sharpe ratio, the Sortino ratio is intended to be used to compare assets or portfolios. Like the Sharpe ratio, the higher the Sortino ratio, the better the risk-adjusted performance. Therefore, in our example in Figure 1, Asset 1 has a better risk-adjusted return. If you were to compare the Sharpe ratio to the Sortino ratio of an asset or portfolio, you would get an idea of what portion of the volatility is related to outperformance versus underperformance.

## Caveats

While the Sortino ratio may improve on some of the shortcomings found with the Sharpe ratio, it is not perfect. As we have shown, there are multiple means of calculating the Sortino ratio, so you have to be sure you are aware of the underlying formula when examining the numbers.

In addition, the Sortino ratio, like the Sharpe ratio, is based on past returns and past performance is not a guarantee of future returns. However, investors can use the ratios to help forecast potential future returns.

## Conclusion

The Sortino ratio is another means of calculating risk-adjusted returns. Since most investors are not concerned with upside volatility, it is an attractive method of comparing portfolios or investment strategies by only taking into consideration downside volatility.

As we have shown, calculating downside risk is a more difficult endeavor, and the calculation you use can have a significant impact on the final results. Furthermore, upside volatility shouldn’t be ignored altogether, and the Sortino ratio isn’t a complete measure of risk. While upside volatility may generate impressive gains over time, we cannot forget that these gains are the result of taking on additional risk. Risk is a two-way street and the risk that leads to upside outperformance can, and often does, lead to downside underperformance.

John Duguidfrom NJ posted about 1 year ago: