Optimality condition of multiple-risk insurance

The insurance optimal conditions in case of a single loss are generalized to optimal contract which provide coverage against n comonotonic risks in one insurance policy. In order to consider the tails of the insured risks, we determine the insurance price of catastrophic risks by the distorted insurance premium calculation principle instead of the expected value insurance premium calculation principle; and discuss general properties of the insurance products; and prove that the optimal insurance policy will still offer full coverage above a deductible in case of multiple losses without regard to the trade costs. Finally, the expression of optimal indemnity by an example is obtained when the utility function is given.