Risk in a Large Claims Insurance Market with Bipartite Graph Structure

Abstract

We model the influence of sharing large exogeneous losses to the reinsurance market by a bipartite graph. Using Pareto-tailed claims and multivariate regular variation we obtain asymptotic results for the value-at-risk and the conditional tail expectation. We show that the dependence on the network structure plays a fundamental role in their asymptotic behaviour. As is well known in a nonnetwork setting, if the Pareto exponent is larger than 1, then for the individual agent (reinsurance company) diversification is beneficial, whereas when it is less than 1, concentration on a few objects is the better strategy. An additional aspect of this paper is the amount of uninsured losses that are covered by society. In our setting of networks of agents, diversification is never detrimental to the amount of uninsured losses. If the Pareto-tailed claims have finite mean, diversification is never detrimental, to society or individual agents. By contrast, if the Pareto-tailed claims have infinite mean, a conflicting situation may arise between the incentives of individual agents and the interest of some regulator to keep the risk for society small. We explain the influence of the network structure on diversification effects in different network scenarios.