Aggregation of Risks and Asymptotic independence

Details Aggregation of Risks and Asymptotic independence Aggregation of Risks and Asymptotic independence

2008-07-25

arxiv.org/abs/0807.4200v2

Mathematics

Probability

29 pages. slight change in title. minor changes in proof of Theorem 1. Added comparisons and remarks. This will appear in Adva

Scientific paper

We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks $(X,Y)$ such that $P(X + Y > x) \sim (const)P (X > x)$. With the further assumption of non-negativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of $X$ and $Y$ are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored leading to an approximate solution of an optimization problem which is applied to portfolio design.

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