Weighted Comonotonic Risk Sharing Under Heterogeneous Beliefs

Abstract

We study a weighted co-monotonic risk sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess-of-loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is Value-at-Risk or Tail-Value-at-Risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

Keywords: Distortion Risk Measure, Heterogeneous Beliefs, Individual Rationality, Value-at-Risk, Tail-Value-at-Risk

JEL Classification: C61, G10