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Asymptotic results on tail moment and tail central moment for dependent risks

Part of: Multivariate analysis Distribution theory - Probability Applications

Published online by Cambridge University Press:  08 May 2023

Jinzhu Li
Jinzhu Li*
Affiliation:
Nankai University
*
*Postal address: School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, P. R. China. Email address: lijinzhu@nankai.edu.cn
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Abstract

In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.

Keywords

Tail momenttail central momentrisk measureasymptotic independenceregular variationGumbel max-domain of attraction

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 60E05: Distributions
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1116 - 1143
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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