Some new ordering relations for policy limits and deductibles with homogeneous and non-homogeneous risks

Policy limits and deductibles are two common insurance coverage that frequently used by the insurance company. In the policy limit coverage, the company indemnifies the policyholder’s loss up to pre-specified value such as known as the limit amount and if the loss exceeds the remaining loss will be indemnified by the policyholder. In policy deductible coverage, the policyholder is self insured up to pre-specified value such as known as deductible amount and if the loss exceeds the remaining loss will be covered by the company. Let X_1,...,X_n be n risks faced by the policyholder. In policy limit coverage with limit amount l, the policyholder should divide l in to n non-negative values l_1,...,l_n such that each l_i is a limit value, taking care of the risk X_i and /sum_{i=1}^n l_i=l Parallelly, in the policy deductible coverage with deductible amount d the policyholder should divide d in to n non-negative values d_1,...,d_n such that each d_i is a deductible value, taking care of the risk X_i and /sum_{i=1}^n d_i=d One main concern ,considered in the literature, is how to find an allocation of limits or deductibles that maximizes the expectation of policyholder’s utility or minimizes the remaining risk. Most of the previous works that had been done in this area only focused on the independent or comonotonic risks. In order to extend the previous results, we consider the dependent risks. We assume that the faced risks are exchangeable, arrangement increasing or arrangement decreasing and obtain the best and the worst allocation in each case, for both policy limits and deductibles.