Sortino Ratio

What is the Sortino Ratio?

The Sortino ratio is a statistical tool which is used in order to evaluate the return from the investment for the given level of the bad risk and it is calculated by subtracting the risk-free rate of return from the expected return of the portfolio and dividing the resultant from the standard deviation of the negative portfolio (downside deviation).

Formula

The Sortino Ratio Formula is given below:<R>-Rf/σd

Sortino Ratio Formula = (Rp – Rf) / σd

where

• Rp is the expected rate of return of the portfolio
• Rf is a risk-free or minimum acceptable rate of return
• σd is the standard deviation of negative asset return

So it is the extra return over and above the target rate of return or risk-free rate of return per unit downward risk.

Sortino ratio calculation is similar to the Sharpe ratio, which is a common measure of risk-return trade-off, the only difference being that the latter uses both upside and downside volatility while evaluating the performance of a portfolio; however, the former uses only downside volatility. Just like the Sharpe ratio, a higher Sortino ratio is better.

How to Calculate Sortino Ratio?

Let us consider an example to understand the importance of this ratio. Let there be two different investment portfolio schemes A & B, with annualized returns of 10% & 15%, respectively. Assuming that the downward deviation of A is 4%, whereas for B is 12%. Also, considering the fixed deposit risk free rate of 6%.

• Sortino ratio calculation for A is: (10-6)/4  = 1
• Sortino ratio calculation for B is: (15-6)/12 =  0.75

Now even though B has a greater annualized return than A, its Sortino ratio is less than that of A’s. So if investors are more concerned about the downside risks associated with the scheme than the expected returns, then they will go for scheme A as it is earning more return per unit of bad risk it also takes it has a greater probability of avoiding any large loss.

Example

Sortino ratio was named after Frank A Sortino, who developed it in order to differentiate between good volatility and bad volatility, which was not possible with the Sharpe ratio. The evaluation of portfolio performance using the Sharpe ratio is indifferent to the direction of volatility, i.e., the treatment of volatility is the same for upward or downward deviation. The downward deviation is used for Sortino ratio calculation, whereby it considers only those periods when the rate of return was lower than the target or risk-free rate of return.

To illustrate these, let us take another example; assuming an investment portfolio scheme with the below returns in 12 months:

Other parameters:

The risk-free rate of return: 6%

We can derive the standard deviation of the sample from the above table using the formula:

• σ = sqrt(variance/n-1) where n is the size of the sample
• σ = sqrt(6.40%/11) à  σ = 7.63%

and the Sharpe ratio can be calculated using the formula:

• (Rp-Rf)/ σ

Sharpe ratio formula = (7% – 6%)/7.63%

Sharpe ratio = 0.1

It can be clearly observed from the table above that the variance in column (R-R(Avg)² seems to ignore the direction of volatility like if we compare period 5 & period 10, where there are equal but opposite differences between the actual return and the average rate of return still the variance is same for both, irrespective of the upside or downside deviation from the average rate.

So we can say that even if the +13% difference between the return and the average return for period eight would have been -13%, the standard deviation would still be the same, which is definitely not an appropriate evaluation; a substantial negative variance would mean a lot riskier portfolio. It can give a similar evaluation for portfolios with different risks associated as this measure is indifferent to whether the return is above or below the average rate of return.

Now if we look at how we calculate the Sortino ratio below:

Here, for the calculation of a downward deviation, only negative variances are considered i.e., only those periods when the rate of return was less than the target or risk-free rate of return as highlighted in yellow in the table, ignoring all the positive variances and taking them as zero.

We can derive the downward deviation of the sample from the above table using the formula:

• σd = sqrt(2.78%/12) à  σ = 4.81%

and the Sortino ratio can be calculated using the formula:

• Soriano Ratio Formula = (Rp-Rf)/ σd
• Sortino ratio = (7% – 6%)/4.81%
• = 0.2

Observations

• It can be seen that the Sortino ratio is a bit higher than the Sharpe ratio for this investment portfolio reason being there were very few breaches of the target or risk-free rate of return
• Also, Sharpe ratio sort of big generalized deviations like 13%, which was actually not a risky shift and, in fact, good for the investors
• As mentioned earlier, we can see how the Sortino ratio is able to differentiate between good and bad variances through its calculation of a downward deviation.
• Its calculation is especially useful for those retail investors who look to invest with certain defined goals and a target rate of return.
• It is also a better tool for measurement of the performance of a fund manager whose returns are positively skewed as it will ignore all the positive variances while calculating volatility or risk and provide a more appropriate evaluation.

The limitation of the Sortino ratio is that there should be enough bad volatility events for the calculation of a downward deviation to be statistically significant.

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