Second order regular variation and conditional tail expectation of multiple risks

Lei Hua and Harry Joe

Insurance: Mathematics and Economics, 2011, vol. 49, issue 3, 537-546

Abstract: For the purpose of risk management, the study of tail behavior of multiple risks is more relevant than the study of their overall distributions. Asymptotic study assuming that each marginal risk goes to infinity is more mathematically tractable and has also uncovered some interesting performance of risk measures and relationships between risk measures by their first order approximations. However, the first order approximation is only a crude way to understand tail behavior of multiple risks, and especially for sub-extremal risks. In this paper, we conduct asymptotic analysis on conditional tail expectation (CTE) under the condition of second order regular variation (2RV). First, the closed-form second order approximation of CTE is obtained for the univariate case. Then CTE of the form E[X1∣g(X1,…,Xd)>t], as t→∞, is studied, where g is a loss aggregating function and (X1,…,Xd)≔(RT1,…,RTd) with R independent of (T1,…,Td) and the survivor function of R satisfying the condition of 2RV. Closed-form second order approximations of CTE for this multivariate form have been derived in terms of corresponding value at risk. For both the univariate and multivariate cases, we find that the first order approximation is affected by only the regular variation index −α of marginal survivor functions, while the second order approximation is influenced by both the parameters for first and second order regular variation, and the rate of convergence to the first order approximation is dominated by the second order parameter only. We have also shown that the 2RV condition and the assumptions for the multivariate form are satisfied by many parametric distribution families, and thus the closed-form approximations would be useful for applications. Those closed-form results extend the study of Zhu and Li (submitted for publication).

Keywords: Second order approximation; Value at risk; Conditional tail expectation; Asymptotic analysis; Sub-extremal multiple risk (search for similar items in EconPapers)
Date: 2011
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (42)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0167668711000941
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:insuma:v:49:y:2011:i:3:p:537-546

DOI: 10.1016/j.insmatheco.2011.08.013

Access Statistics for this article

Insurance: Mathematics and Economics is currently edited by R. Kaas, Hansjoerg Albrecher, M. J. Goovaerts and E. S. W. Shiu

More articles in Insurance: Mathematics and Economics from Elsevier
Bibliographic data for series maintained by Catherine Liu ().