Sortino ratio

The Sortino ratio is essentially a modification of the Sharpe ratio which compares return on a portfolio to downside risk (i.e. the risk of under-performing the benchmark) rather than volatility. The aim is to ensure that we only adjust performance for risk of loss, not “risk” of out-performance. The formula is the same except that the denominator is the downside risk:

d/θd where d = rp - rb

where rp is the return on a portfolio,
rb is the return on a benchmark,
d is called the differential return, and,
θ is the semi-standard deviation of d. The semi-standard deviation is calculated in the same way as the standard deviation, except that numbers above or below a certain threshold are set equal to the threshold. In this case, we simple replace all positive values of d with zero.

If the distribution of risk is symmetrical, then this will make little difference to how portfolios compare. However, if the risk is asymmetrical, it does make a difference: for example, if a portfolio contained investments with a positive fat tail, the Sharpe ratio will be lower because of this risk, whereas the Sortino ratio will be the same as for a portfolio without the fat tail.

The original formulation of the Sortino ratio used an arbitrary rate called the minimum acceptable return (abbreviated to MAR) in place of r above. It makes more sense to use a benchmark as basis for comparison. In practice the benchmark used is the risk free rate to allow comparison of the widest possible range of portfolios.