Inter-order relations between equivalence for Lp-quantiles of the Student's t distribution

Abstract

In the statistical and actuarial literature, Lp-quantiles, p∈[1,+∞), represent an important class of risk measures defined through an asymmetric p-power loss function that generalize the classical (L1-)quantiles. By exploiting inter-order relations between partial moments, we show that for a Student's t distribution with ν∈[1,+∞) degrees of freedom the Lν−j-quantile and the Lj+1-quantile always coincide for any j∈[0,ν−1]. For instance, for a Student's t distribution with 4 degrees of freedom, the L4-quantile and L1-quantile are equal and the same holds for the L3-quantile and L2-quantile; for this distribution, closed form expressions for the Lp-quantile, p=1,2,3,4 are provided. Explicit formulas for the central moments are also established. The usefulness of exact formulas is illustrated on real-world financial data.