This paper studies a problem of optimal reinsurance design under asymmetric information. The insurer adopts distortion risk measures to quantify his/her risk position, and the reinsurer does not know the functional form of this distortion risk measure. The risk-neutral reinsurer maximizes his/her net profit subject to individual rationality and incentive compatibility constraints. The optimal reinsurance menu is succinctly derived under the assumption that one type of insurer has a larger willingness to pay than the other type of insurer for every risk. Some comparative analyses are given as illustrations when the insurer adopts the value at risk or the tail value at risk as preferences.

We study a weighted comonotonic 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.

In distinction to the Borch’s model of a reinsurance market, this paper treats the problem of optimal risk exchange in an insurance market where treaties are allowed between the insurer and each insured only, not among insureds themselves. A characterization of the Pareto-optimal contract is found. It is shown that the indemnity function in the contract is of a coinsurance kind. We also present a way of finding Pareto-optimal contracts under individual rationality constraints. The obtained results are compared with those of the known model of risk exchange in a reinsurance market.