Copula based hierarchical risk aggregation - Tree dependent sampling and the space of mild tree dependence

TL;DR

This paper analyzes copula-based hierarchical risk aggregation, proposing a modified sampling algorithm and exploring the space of feasible distributions without the conditional independence assumption, providing insights for better risk model design.

Contribution

It introduces a modified sampling algorithm with proof of convergence and characterizes the space of feasible distributions beyond the conditional independence assumption.

Findings

The modified algorithm approximates the distribution specified by the conditional independence assumption.

The space of feasible distributions is characterized when dropping the independence assumption.

Input parameters and tree structure significantly influence the aggregation model.

Abstract

The ability to adequately model risks is crucial for insurance companies. The method of "Copula-based hierarchical risk aggregation" by Arbenz et al. offers a flexible way in doing so and has attracted much attention recently. We briefly introduce the aggregation tree model as well as the sampling algorithm proposed by they authors. An important characteristic of the model is that the joint distribution of all risk is not fully specified unless an additional assumption (known as "conditional independence assumption") is added. We show that there is numerical evidence that the sampling algorithm yields an approximation of the distribution uniquely specified by the conditional independence assumption. We propose a modified algorithm and provide a proof that under certain conditions the said distribution is indeed approximated by our algorithm. We further determine the space of feasible distributions for a given aggregation tree model in case we drop the conditional independence assumption. We study the impact of the input parameters and the tree structure, which allows conclusions of the way the aggregation tree should be designed.