Non-optimality of a linear combination of proportional and non-proportional reinsurance

For the subclass of reinsurance contracts with maximum deductible contained in the class of all bivariate comonotonic risk-exchange structures associated to a given risk, we consider optimality with respect to a long-term actuarial mean self-financing property and competitiveness of the insurance premium. For arbitrary varying risks, the linear combination of proportional and stop-loss reinsurance is not optimal unless it is a pure stop-loss contract, at least if the variance premium principle is used to set insurance prices. By known distribution of the risk, it is shown how an optimal deductible of a stop-loss contract can be determined. Some applications to insurance and finance are briefly mentioned.