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The benefits of diversifying risks are difficult to estimate quantitativelybecause of the uncertainties in the dependence structure between the risks.Also, the modelling of multidimensional dependencies is a non-trivial task.This paper focuses on one such technique for portfolio aggregation, namely theaggregation of risks within trees, where dependencies are set at each step ofthe aggregation with the help of some copulas. We define rigorously thisprocedure and then study extensively the Gaussian Tree of quite arbitrary sizeand shape, where individual risks are normal, and where the Gaussian copula isused. We derive exact analytical results for the diversification benefit of theGaussian tree as a function of its shape and of the dependency parameters.  Such a "toy-model" of an aggregation tree enables one to understand the basicphenomena's at play while aggregating risks in this way. In particular, it isshown that, for a fixed number of individual risks, "thin" trees diversifybetter than "fat" trees. Related to this, it is shown that hierarchical treeshave the natural tendency to lower the overall dependency with respect to thedependency parameter chosen at each step of the aggregation. We also show thatthese results hold in more general cases outside the gaussian world, and applynotably to more realistic portfolios (LogNormal trees). We believe that anyinsurer or reinsurer using such a tool should be aware of these systematiceffects, and that this awareness should strongly call for designing trees thatadequately fit the business.  We finally address the issue of specifying the full joint distributionbetween the risks. We show that the hierarchical mechanism does not require norspecify the joint distribution, but that the latter can be determined exactly(in the Gaussian case) by adding conditional independence hypotheses betweenthe risks and their sums.

Copula-based Hierarchical Aggregation of Correlated Risks. The behaviour of the diversification benefit in Gaussian and Lognormal Trees