Bounds for randomly shared risk of heavy-tailed loss factors

Abstract

For a risk vector $V$, whose components are shared among agents by some random mechanism, we obtain asymptotic lower and upper bounds for the individual agents' exposure risk and the aggregated risk in the market. Risk is measured by Value-at-Risk or Conditional Tail Expectation. We assume Pareto tails for the components of $V$ and arbitrary dependence structure in a multivariate regular variation setting. Upper and lower bounds are given by asymptotically independent and fully dependent components of $V$ with respect to the tail index $\alpha$ being smaller or larger than 1. Counterexamples, where for non-linear aggregation functions no bounds are available, complete the picture.