The upside-potential ratio is a measure of a return of an investment asset relative to the minimal acceptable return. The measurement allows a firm or individual to choose investments which have had relatively good upside performance, per unit of downside risk.
U = ∑ min + ∞ ( R r − R min ) P r ∑ − ∞ min ( R r − R min ) 2 P r = E [ ( R r − R min ) + ] E [ ( R r − R min ) − 2 ] , {\displaystyle U={{\sum _{\min }^{+\infty }{(R_{r}-R_{\min }})P_{r}} \over {\sqrt {\sum _{-\infty }^{\min }{(R_{r}-R_{\min }})^{2}P_{r}}}}={\frac {\mathbb {E} [(R_{r}-R_{\min })_{+}]}{\sqrt {\mathbb {E} [(R_{r}-R_{\min })_{-}^{2}]}}},}
where the returns R r {\displaystyle R_{r}} have been put into increasing order. Here P r {\displaystyle P_{r}} is the probability of the return R r {\displaystyle R_{r}} and R min {\displaystyle R_{\min }} which occurs at r = min {\displaystyle r=\min } is the minimal acceptable return. In the secondary formula ( X ) + = { X if X ≥ 0 0 else {\displaystyle (X)_{+}={\begin{cases}X&{\text{if }}X\geq 0\\0&{\text{else}}\end{cases}}} and ( X ) − = ( − X ) + {\displaystyle (X)_{-}=(-X)_{+}} . The upside-potential ratio may also be expressed as a ratio of partial moments since E [ ( R r − R min ) + ] {\displaystyle \mathbb {E} [(R_{r}-R_{\min })_{+}]} is the first upper moment and E [ ( R r − R min ) − 2 ] {\displaystyle \mathbb {E} [(R_{r}-R_{\min })_{-}^{2}]} is the second lower partial moment. The measure was developed by Frank A. Sortino.
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