Farinelli and Tibiletti ratio and stochastic dominance

Abstract

Farinelli and Tibiletti (F–T) ratio, a general risk-reward performance measurement ratio, is popular due to its simplicity and yet generality that both Omega ratio and upside potential ratio are its special cases. The F–T ratios are ratios of average gains to average losses with respect to a target, each raised by a power index, p and q. In this paper, we establish the consistency of F–T ratios with any nonnegative values p and q with respect to first-order stochastic dominance. Second-order stochastic dominance does not lead to F–T ratios with any nonnegative values p and q, but can lead to F–T dominance with any $p<1$ and $q\ge 1$. Furthermore, higher-order stochastic dominance ($n\ge 3$) leads to F–T dominance with any $p<1$ and $q\ge n-1$. We also find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, we can get the necessary relationship between the stochastic dominance with the F–T ratio after imposing some conditions on the means. There are many advantages of using the F–T ratio over other measures, and academics and practitioners can benefit by using the theory we developed in this paper. For example, the F–T ratio can be used to detect whether there is any arbitrage opportunity in the market, whether there is any anomaly in the market, whether the market is efficient, whether there is any preference of any higher-order moment in the market, and whether there is any higher-order stochastic dominance in the market. Thus, our findings enable academics and practitioners to draw better decision in their analysis.