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.

Bignozzi, V., Merlo, L., Petrella, L. (2024). Inter-order relations between equivalence for Lp-quantiles of the Student's t distribution. INSURANCE MATHEMATICS & ECONOMICS, 116(May 2024), 44-50 [10.1016/j.insmatheco.2024.02.001].

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

Bignozzi, V
Primo
;
2024

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.
Articolo in rivista - Articolo scientifico
Expectiles; Generalized quantiles; Partial moments; Quantiles; Risk measures;
English
13-feb-2024
2024
116
May 2024
44
50
open
Bignozzi, V., Merlo, L., Petrella, L. (2024). Inter-order relations between equivalence for Lp-quantiles of the Student's t distribution. INSURANCE MATHEMATICS & ECONOMICS, 116(May 2024), 44-50 [10.1016/j.insmatheco.2024.02.001].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/468379
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