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Probability equivalent level of Value at Risk and higher-order Expected Shortfalls

Author

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  • Barczy, Mátyás
  • K. Nedényi, Fanni
  • Sütő, László

Abstract

We investigate the probability equivalent level of Value at Risk and nth-order Expected Shortfall (called PELVEn), which can be considered as a variant of the notion of the probability equivalent level of Value at Risk and Expected Shortfall (called PELVE) due to Li and Wang (2022). We study the finiteness, uniqueness and several properties of PELVEn, we calculate PELVEn of some notable distributions, PELVE2 of a random variable having generalized Pareto excess distribution, and we describe the asymptotic behaviour of PELVE2 of regularly varying distributions as the level tends to 0. Some properties of nth-order Expected Shortfall are also investigated. Among others, it turns out that the Gini Shortfall at some level p∈[0,1) corresponding to a (loading) parameter λ⩾0 is the linear combination of the Expected Shortfall at level p and the 2nd-order Expected Shortfall at level p with coefficients 1−2λ and 2λ, respectively.

Suggested Citation

  • Barczy, Mátyás & K. Nedényi, Fanni & Sütő, László, 2023. "Probability equivalent level of Value at Risk and higher-order Expected Shortfalls," Insurance: Mathematics and Economics, Elsevier, vol. 108(C), pages 107-128.
    • Handle: RePEc:eee:insuma:v:108:y:2023:i:c:p:107-128
    DOI: 10.1016/j.insmatheco.2022.11.004
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    References listed on IDEAS

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    Cited by:

    1. Georgios I. Papayiannis & Georgios Psarrakos, 2025. "A tail-shape actuarial index based on equal level relationships between Value at Risk and Expected Shortfall," Papers 2507.13562, arXiv.org, revised Jan 2026.
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    3. Ortega-Jiménez, Patricia & Pellerey, Franco & Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2024. "Probability equivalent level for CoVaR and VaR," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 22-35.
    4. Baishuai Zuo & Chuancun Yin, 2025. "Analyzing distortion riskmetrics and weighted entropy for unimodal and symmetric distributions under partial information constraints," Papers 2504.19725, arXiv.org, revised Nov 2025.
    5. Zou, Zhenfeng & Hu, Taizhong, 2024. "Adjusted higher-order expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 1-12.

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    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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