A Multivariate Dependence Measure for Aggregating Risks

Abstract

To evaluate the aggregate risk in a financial or insurance portfolio, a risk analyst has to calculate the distribution function of a sum of random variables. As the individual risk factors are often positively dependent, the classical convolution technique will not be sufficient. On the other hand, assuming a comonotonic dependence structure will likely overrate the real aggregate risk. In order to choose between both approximations, or perhaps use a weighted average, we should have an indication on the accuracy. Clearly this accuracy will depend on the copula between the individual risk factors, but it is also influenced by the marginal distributions. In this paper we introduce a multivariate dependence measure that takes both aspects into account. This new measure differs from other multivariate dependence measures, as it focuses on the aggregate risk rather than on the copula or the joint distribution function itself. We prove several interesting properties of this new measure and discuss its relation to other dependencemeasures. We also give some comments on the estimation and conclude with examples and numerical results.

Keywords: comonotonic copula, independence, aggregate distribution, concordance order, positive