Some New Results on Policy Limit Allocations

Suppose that a policyholder faces $n$ risks X1, ..., Xn which are insured under the policy limit with the total limit of l. Usually, the policyholder is asked to protect each Xi with an arbitrary limit of li such that ∑ni=1li=l. If the risks are independent and identically distributed with log-concave cumulative distribution function, using the notions of majorization and stochastic orderings, we prove that the equal limits provide the maximum of the expected utility of the wealth of policyholder. If the risks with log-concave distribution functions are independent and ordered in the sense of the reversed hazard rate order, we show that the equal limits is the most favourable allocation among the worst allocations. We also prove that if the joint probability density function is arrangement increasing, then the best arranged allocation maximizes the utility expectation of policyholderchr('39')s wealth. We apply the main results to the case when the risks are distributed according to a log-normal distribution.