Abstract
Evaluating risk measures, premiums, and capital allocation based on dependent multi-losses is a notoriously difficult task. In this paper, we demonstrate how this can be successfully accomplished when losses follow the multivariate Pareto distribution of the second kind, which is an attractive model for multi-losses whose dependence and tail heaviness are influenced by a heavy-tailed background risk. A particular attention is given to the distortion and weighted risk measures and allocations, as well as their special cases such as the conditional layer expectation, tail value at risk, and the truncated tail value at risk. We derive formulas which are either of closed form or follow well-defined recursive procedures. In either case, their computational use is straightforward.
Highlights
In the insurance literature, loss is usually viewed as a non-negative random variable, say X ≥ 0.Denote its cumulative distribution function by FX
For additional information on the multivariate Pareto distribution of the second kind, we refer to Arnold [28] and Yeh [33,34], where we find statistical inferential results concerning the parameters μ, σ, and α
We have argued that the multivariate Pareto distribution of the second kind is an attractive model for multidimensional risks/losses
Summary
Loss is usually viewed as a non-negative random variable, say X ≥ 0.Denote its cumulative distribution function (cdf) by FX. Loss is usually viewed as a non-negative random variable, say X ≥ 0. For more details on premium calculation principles, their construction and underlying axioms, we refer to, for example, Denuit et al [1], Pflug and Römisch [2], Tsanakas and Desli [3], Wang et al [4], Young [5], and references therein. Many of the premium calculation principles are related to non-expected utility theories. On the latter topic, we refer to the monographs by Puppe [6], Quiggin [7], Wakker [8], as well as to the review articles by Machina [9,10], and references therein
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