# Sortino Ratio

Sortino ratio measures excess return per unit of downside risk. It is a modification of Sharpe ratio which is calculated by dividing the difference between portfolio return and risk-free rate by the standard deviation of negative returns. A higher Sortino ratio is better.

Rational investors are inherently risk averse and they should take risk only if it is compensated by additional return. Sharpe ratio is a common measure of risk-return trade-off. It compares excess return with total standard deviation of the portfolio’s investment returns, a measure of both the deviations above the mean return and those below the mean return. But upside deviations are good for an investor, so the real risk which investors should worry about is the risk of returns falling below the mean. Sortino ratio defines risk as the risk of downside variation only and provides a better picture of risk-adjusted performance.

## Formula

The following formula shows calculation of Sortino ratio:

$$Sortino\ Ratio=\frac{R_p-r_f}{\sigma_d}$$

Where Rp is the actual/expected return on the portfolio, rf is the risk-free rate and σd is the downside standard deviation.

The downside standard deviation is calculated :

1. identify the reference point below which the return is considered bad, let’s call it minimum acceptable return (MVR), it might be the mean return, the risk-free rate or 0.
2. find deviation of each return value from the minimum acceptable return, if the value is above MVR, ignore it and if the value is below MVR, square it.
3. Sum up all the squared values in Step 2.
4. Divided values obtained in Step 3 by n, i.e. the total number of observations
5. Take square root of the value in Step 4.

## Example

Let’s calculate Sortino ratio for Apple and Google from 1 January 2017 to 31 December 2017.

The monthly return data is as follows:

Jan 3.32% 12.89%
Feb 0.77% 4.87%
Mar 9.21% -0.01%
Apr 6.50% 6.34%
May -5.82% -5.72%
Jun 2.40% 3.27%
Jul 0.95% 10.27%
Aug 2.11% -6.02%
Sep 6.00% 9.68%
Oct 0.47% 1.66%
Nov 2.45% -1.52%
Dec 11.81% 1.79%

Let's say the risk free rate is 5% and the monthly minimum acceptable return is 2%.

The following table shows calculation of squared deviations where the investment return is lower than MAR:

Return Dev.2 Return Dev.2
Jan 3.32% 0.00% 12.89% 0.00%
Feb 0.77% 0.02% 4.87% 0.00%
Mar 9.21% 0.00% -0.01% 0.04%
Apr 6.50% 0.00% 6.34% 0.00%
May -5.82% 0.61% -5.72% 0.60%
Jun 2.40% 0.00% 3.27% 0.00%
Jul 0.95% 0.01% 10.27% 0.00%
Aug 2.11% 0.00% -6.02% 0.64%
Sep 6.00% 0.00% 9.68% 0.00%
Oct 0.47% 0.02% 1.66% 0.00%
Nov 2.45% 0.00% -1.52% 0.12%
Dec 11.81% 0.00% 1.79% 0.00%

The monthly squared deviations lower than MAR sum up to 0.66% and total observations are 12, so:

$$Monthly\ downside\ deviation\ (Google)\\=\sqrt{\frac{0.66\%}{12-1}}\\=2.45\%$$

The monthly downside deviation is annualized as follows:

$$Annualized\ downside\ deviation\\=2.45\%\times\sqrt{12}\\=8.49\%$$

Similarly, we work out that annualized downside deviation for Apple is 12.39%.

Return over the period for Google and Apple works out to 46.83% and 41.95% respectively.

Now, we calculate Sortino ratios:

$$Sortino\ Ratio\ (Google)\\=\frac{46.8\%-5\%}{8.49\%}\\=4.93$$

$$Sortino\ Ratio\ (Apple)\\=\frac{41.95\%-5\%}{12.39\%}\\=2.98$$

Even there is not very significant difference between returns on both stock, there risk-return tradeoff is quite different. Google has way better return per unit of downside risk.