Bounds for randomly shared risk of heavy-tailed loss factors

For a risk vector $V$, whose components are shared among agents by some random mechanism, we obtain asymptotic lower and upper bounds for the agents' exposure risk and the systemic 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 asymptotic independent and fully dependent components of $V$ in dependence of the tail index $\al$ being smaller or larger than 1. Counterexamples complete the picture.