@article { ISI:000313478200008,
	title = {Tail comonotonicity and conservative risk measures},
	journal = {ASTIN Bulletin},
	volume = {42},
	number = {2},
	year = {2012},
	month = {NOV},
	pages = {601-629},
	publisher = {Peeters},
	type = {Article},
	abstract = {Tail comonotonicity, or asymptotic full dependence, is proposed as a reasonable conservative dependence structure for modeling dependent risks. Some sufficient conditions have been obtained to justify the conservativity of tail comonotonicity. Simulation studies also suggest that, by using tail comonotonicity, one does not lose too much accuracy but gain reasonable conservative risk measures, especially when considering high scenario risks. A copula model with tail comonotonicity is applied to an auto insurance dataset. Particular models for tail comonotonicity for loss data can be based on the BB2 and BB3 copula families and their multivariate extensions.},
	keywords = {Archimedean copula, asymptotic full dependence, conditional tail expectation, Copula, Dependence modeling, Laplace transform, regular variation},
	issn = {0515-0361},
	doi = {10.2143/AST42.2.2182810},
	author = {Hua, Lei and Joe, Harry}
}
@article { ISI:000297832100025,
	title = {Second order regular variation and conditional tail expectation of multiple risks},
	journal = {Insurance Mathematics \& Economics},
	volume = {49},
	number = {3},
	year = {2011},
	month = {NOV},
	pages = {537-546},
	publisher = {Elsevier Science BV},
	type = {Article},
	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 vertical bar X-1 vertical bar g(X-1, ..., X-d) > t], as t -> infinity. is studied, where g is a loss aggregating function and (X-1, ..., X-d) := (RT1, ..., RTd) with R independent of (T-1, ..., T-d) 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 -alpha 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). (C) 2011 Elsevier B.V. All rights reserved.},
	keywords = {Asymptotic analysis, conditional tail expectation, Second order approximation, Sub-extremal multiple risk, Value at risk},
	issn = {0167-6687},
	doi = {10.1016/j.insmatheco.2011.08.013},
	author = {Hua, Lei and Joe, Harry}
}
