Optimal allocation of policy deductibles for exchangeable risks

Publication date: Available online 1 September 2016
Source:Insurance: Mathematics and Economics
Author(s): Sirous Fathi Manesh, Baha-Eldin Khaledi, Jan Dhaene
Let $X_1,\dots ,X_n$ 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 this $n$ risks. Let $d=(d_1,\dots d_n)$ and $d^{\ast }=(d_1^{\ast },\dots d_n^{\ast })$ be two vectors of deductibles such that $d^{\ast }$ is majorized by $d$. It is shown that ${\sum }_{i=1}^n(X_i\wedge d_i^{\ast })$ is larger than ${\sum }_{i=1}^n(X_i\wedge d_i)$ in stochastic dominance, provided $f$ is exchangeable. As a result, the vector $({\sum }_{i=1}^n{d}_i,0,\dots ,0)$ 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.