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

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  • Bignozzi, Valeria
  • Merlo, Luca
  • Petrella, Lea

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.

Suggested Citation

  • Bignozzi, Valeria & Merlo, Luca & Petrella, Lea, 2024. "Inter-order relations between equivalence for Lp-quantiles of the Student's t distribution," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 44-50.
    • Handle: RePEc:eee:insuma:v:116:y:2024:i:c:p:44-50
    DOI: 10.1016/j.insmatheco.2024.02.001
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    More about this item

    Keywords

    Expectiles; Generalized quantiles; Partial moments; Quantiles; Risk measures;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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