Choosing a Deductible for Insurance Contracts: Best or Worst Insurance Policy?

Abstract

This paper considers the second-order condition for the choice of an optimal deductible in insurance contracts. An equivalent condition is derived in terms of comparing growth rates of the net price of insurance and a relative utility weighting reflecting the degree of risk aversion. Mathematical models of economic choice often examine the properties of the best economic choice without adequately ruling out the possibility that they may actually be examining the worst choice instead. The choice of a best insurance contract by an individual is no exception. In an earlier paper in this journal, Schlesinger [7] showed that the necessary conditions for the choice of an optimal deductible for a given insurance policy can lead to a worst (minimal-expected-utility) outcome if one is not careful. A sufficient condition for a best (expected-utility-maximizing) choice of deductibles was presented, but its interpretation seems rather artificial (Schlesinger [ 198 1; p. 476 eqn. (18)]). This sufficient condition is hardly trivial as is evidenced by a recent paper of Boyle and Mao [1] in which more than half the paper is an appendix, the sole purpose of which is to verify this sufficient condition in a relatively simple model. This note reexamines this sufficient condition, showing it depends upon the growth rate of the net price of insurance coverage and a utility weight that reflects a measure of risk aversion. The paper begins with a streamlined presentation of the necessary firstorder condition for a best deductible choice. This necessary condition also Harris Schlesinger is Assistant Professor of Economics at Vanderbilt University and is currently a Research Fellow at the International Institute of Management in Berlin, West Germany. This content downloaded from 157.55.39.92 on Wed, 22 Jun 2016 07:06:04 UTC All use subject to http://about.jstor.org/terms Deductible for Insurance Contracts 523 receives a more canonical interpretation than in previous literature (cf. Gould [3], Pashigian, et al. [5] and Schlesinger [7]). The second order condition sufficient for a best deductible is then discussed and interpreted. For this purpose, consider a risk averter in a simple one period model in which a loss x occurs. This loss is the realization of a random variable with density function f(x) and corresponding cumulative density F(x). For simplicity, assume F(x) is differentiable for x > 0. A straight deductible insurance policy fully indemnifies the individual for losses in excess of some predetermined deductible level. The model allows for the purchase of an insurance policy with any desired deductible level D at a price of P(D) called the insurance premium. At the individual's choice of a deductible level D, it must follow that -1 < P'(D) < 0.1 The actuarial value of a policy with deductible D is given by A(D) J (x-D)dF(x). (1) D