Pareto-optimal reinsurance under the mean-variance framework with background risk*

In this paper, we study Pareto-optimal reinsurance under the mean-variance framework with background risk. To avoid moral hazard, both the ceded loss functions and the retained loss functions are asked to be increasing. The reinsurance premium is calculated by a function of the expected indemnity. The implicit characterization of the optimal ceded loss function is given. By utilizing the implicit characterization, the optimal parametric ceded loss function is provided piecewise. For some special dependence structures between the background risk and the insurable risk, the optimal ceded loss functions have a much simpler form. Under the condition that the sum of the background risk and the insurable risk is strongly negatively expectation dependent (SNED) with respect to the insurable risk, full reinsurance with upper limits is shown to be optimal, and no reinsurance is shown to be optimal if the weighting factor of the insurer is not smaller than that of the reinsurer. When the background risk is independent of the insurable risk, the optimal ceded loss function is a combination of the stop-loss reinsurance and quota share reinsurance. Finally, we give two numerical illustrations to analyze the impact of correlation coefficients, weighting factors and risk aversion coefficients on the reinsurance policy.