Optimal dynamic risk sharing under the time-consistent mean-variance criterion

In this paper, we consider a dynamic Pareto optimal risk-sharing problem under the time-consistent mean-variance criterion. A group of n insurers is assumed to share an exogenous risk whose dynamics is modeled by a Lévy process. By solving the extended Hamilton–Jacobi–Bellman equation using the Lagrange multiplier method, an explicit form of the time-consistent equilibrium risk-bearing strategy for each insurer is obtained. We show that equilibrium risk-bearing strategies are mixtures of two common risk-sharing arrangements, namely, the proportional and stop-loss strategies. Their explicit forms allow us to thoroughly examine the analytic properties of the equilibrium risk-bearing strategies. We later consider two extensions to the original model by introducing a set of financial investment opportunities and allowing for insurers' ambiguity towards the exogenous risk distribution. We again explicitly solve for the equilibrium risk-bearing strategies and further examine the impact of the extension component (investment or ambiguity) on these strategies. Finally, we consider an application of our results in the classical risk-sharing problem of a pure exchange economy.