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the University of Cantabria, Santander, Spain
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**‘Some Analytical Solutions for the Problem of Aggregation of Dependent Risks’**

Abstract: The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and many branches of applied probability. In this talk, we discuss the problem of aggregation of dependent risks in two relevant cases, for which we obtain analytical expressions for the probability density function of the aggregated distribution. In the first case, we consider the problem of aggregation when the risks have different tails, which are of Pareto and Gamma types, and the dependence structure is a Farlie-Gumbel-Morgenstern copula. We also consider several extensions of this case. In the second case, we study the problem of aggregation when the risks are modeled according to a multivariate mixture of exponential distributions. We begin by studying the properties of the mixture model, including dependence conditions, moments, copula (which is Archimedean) and other relevant features. We continue with the analytical formulation for the pdf and the cdf of the aggregated risks. Then, we study in detail some specific claim families with Pareto, Gamma, Weibull and other parent distributions. For these models, we compute some risk measures including VaR and TVaR. Explicit ruin formulas are discussed for some particular cases. An extension of the basic model is studied based on mixtures of gamma and gamma product-ratio claims.

HEC-DSA

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