Comonotonic improvement under feasibility constraints

TL;DR

This paper examines how certain feasibility constraints in risk-sharing affect the optimality of allocations, proposing conditions under which comonotonic improvements are preserved.

Contribution

It introduces componentwise convex-order solidity as a condition that maintains comonotonic improvement under specific risk constraints.

Findings

Value-at-Risk caps can disrupt comonotonic optimality.

Componentwise convex-order solidity restores comonotonic improvement.

The criterion applies to many risk management constraints, excluding some like V@R caps.

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.