Optimal allocation of policy deductibles for exchangeable risks

Let X1,…,Xn be a set of n continuous and non-negative random variables, with log-concave joint density function f, faced by a person who seeks for an optimal deductible coverage for these n risks. Let d=(d1,…dn) and d∗=(d1∗,…dn∗) be two vectors of deductibles such that d∗ is majorized by d. It is shown that ∑i=1n(Xi∧di∗) is larger than ∑i=1n(Xi∧di) in stochastic dominance, provided f is exchangeable. As a result, the vector (∑i=1ndi,0,…,0) is an optimal allocation that maximizes the expected utility of the policyholder's wealth. It is proven that the same result remains to hold in some situations if we drop the assumption that f is log-concave.