Optimal Reinsurance with Multiple Reinsurers: Competitive Pricing and Coalition Stability

Abstract

This paper studies economic pricing of reinsurance contracts via competition of an insurer with multiple reinsurers. All firms are assumed to be endowed with distortion risk measures or expected exponential utilities. Reinsurance contracts are required to be Pareto optimal, individually rational, and satisfy a competition constraint that we call coalition stability. As shown in the literature, it holds that Pareto optimality is equivalent to a structure on the indemnities. This paper characterizes the corresponding premiums by a competition argument. The competition among reinsurers imposes constraints on the premiums that the reinsurers are able to charge and this may lead to a strictly positive profit for the insurer. When the firms use distortion risk measures, this constraint yields stability for sub-coalitions, which is a condition akin to the core in cooperative game theory. The premiums and the profit of the insurer are derived in closed-form. This paper illustrates this premium function with the Mean Conditional Value-at-Risk and the GlueVaR. If the firms use expected exponential utilities, the premium is represented by an exponential premium.

Keywords: reinsurance, multiple reinsurers, competition, premiums, Mean Conditional Value-at-Risk