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# Sortino ratio

The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target, or required rate of return, while the Sharpe ratio penalizes both upside and downside volatility equally. It is thus a measure of risk-adjusted returns that is demonstrably more accurate than the Sharpe ratio.

The ratio is calculated as:

$S = frac\left\{R-T\right\}\left\{DR\right\}$,

where R is the asset or portfolio realized return; T is the target or required rate of return for the investment strategy under consideration, (T was originally known as the minimum acceptable return, or MAR); DR is the downside risk. The downside risk is the target semideviation = square root of the target semivariance (TSV). TSV is the return distribution's lower-partial moment of degree 2 (LPM2).

$DR = left\left(int_\left\{-infty\right\}^T \left(T - x\right)^2,f\left(x\right),dx right\right)^\left\{1/2\right\},$

where $T$ is often taken to be the risk free interest rate and $f\left(\right)$ is the pdf of the returns. $DR$ can also be thought of, and calculated from a sequence of historical returns $x$ as, the root mean square underperformance $U$, where $U = x - T$ if $x - T < 0$, otherwise $U = 0$.

Thus, the ratio is the actual rate of return in excess of the investor's target rate of return, per unit of downside risk.

The ratio was created by Brian M. Rom in 1986 as an element of Investment Technologies' Post-Modern Portfolio Theory portfolio optimization software.