Farinelli and Tibiletti ratio and Stochastic Dominance

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 ≥ 1. Furthermore, higher-order stochastic dominance (n ≥ 3) leads to F-T dominance with any p < 1 and q ≥ 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. Our findings enable academics and practitioners to draw better decision in their analysis.