🤖 AI Summary
This study addresses the issue of suboptimal, non-comonotonic risk sharing that arises when regulatory or contractual constraints undermine incentives for risk-averse agents. The paper introduces “quantile-convex order robustness” as a sufficient condition on the feasible set under which a comonotonic improvement exists for all preferences consistent with the convex order, thereby restoring the comonotonicity between Pareto-optimal allocations and aggregate losses. This condition encompasses common risk management constraints—such as value-at-risk (VaR) caps and individual deductibles—and is validated within the mean-variance framework. The result provides a unified and tractable theoretical foundation for constrained risk-sharing problems.
📝 Abstract
Regulatory and contractual constraints on individual exposures are standard in insurance and reinsurance markets, but a poorly designed constraint can distort the economic incentives of risk-averse agents. In the unconstrained problem, the classical comonotonic improvement theorem guarantees Pareto-optimal allocations that are nondecreasing in the aggregate loss. A constraint that is not stable under risk reduction can destroy this property. We show by example that Value-at-Risk caps lead to optimal allocations that are non-comonotonic in the aggregate loss. We identify componentwise convex-order solidity as a sufficient condition on the feasible set that restores the comonotonic improvement under constraints. If replacing any agent's allocation by a less risky one preserves feasibility, then every feasible allocation admits a feasible comonotonic improvement for all convex-order-consistent preferences. This criterion covers many constraints typical in risk management, but excludes Value-at-Risk caps and idiosyncratic deductibles. We illustrate the implications of our main result in a mean-variance risk-sharing application.