OPTIMAL INSURANCE

S. David Promislow, Department of Mathematics & Statistics, York University Virginia R. Young, Department of Mathematics, University of Michigan

ABSTRACT

For a given loss X, suppose that one can purchase partial insurance I(X) where 0 _≤ I(x) _≤ x for all x, subject to a premium principle H(I). The object is to choose I to optimize

some quantity G I, H(I) . A classical result of this type is a theorem of Arrow which shows that deductible insurance maximizes the expected utility of resulting wealth, when H(I) is some nondecreasing function of E(I). In this paper we present a unifying framework for determining optimal insurance for general G and H. To perform the required analysis, we need only a weak form of derivative of a functional( weaker than the Gateaux derivative). This observation allows us to include previous results within our framework, including Arrow's Theorem, Young's work on Wang's premium principle, and the work of Gajek and Zagrodny on minimizing the variance of retained claims subject to a standard deviation premium principle. We focus on when optimal insurance is piecewise linear because this form of insurance is what one observes in the marketplace.

JEL CLASSIFICATION: D81

KEYWORDS: Deductible insurance, (piecewise) linear insurance, expected utility, premium principles

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