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

Abstract

In the statistical and actuarial literature, $L_p$-quantiles, $p\in [1,+\infty )$, represent an important class of risk measures defined through an asymmetric p-power loss function that generalize the classical ($L_1$-)quantiles. By exploiting inter-order relations between partial moments, we show that for a Student's t distribution with $\nu \in [1,+\infty )$ degrees of freedom the $L_{\nu -j}$-quantile and the $L_{j+1}$-quantile always coincide for any $j\in [0,\nu -1]$. For instance, for a Student's t distribution with 4 degrees of freedom, the $L_4$-quantile and $L_1$-quantile are equal and the same holds for the $L_3$-quantile and $L_2$-quantile; for this distribution, closed form expressions for the $L_p$-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.