The Optimal Deductible for an Insurance Policy When Initial Wealth Is Random

Individual wealth typically includes risky assets. Often some of these risks are insurable; but other risks, such as market valuation of stocks, inflation, and other general economic conditions, are usually not insurable. A series of recent papers by Kihlstrom, Romer, and Williams (1981), Ross (1981), Nachman (1982), and Pratt (1982) have shown that the introduction of background risk requires stronger definitions of risk aversion. We will use some of these results to examine the choice of the optimal deductible in an insurance contract when initial wealth is random. An examination of insurance purchases within a broader portfolio is considered by Doherty (1981) and by Mayers and Smith (1983). These papers use a mean-variance framework to examine the separability of the insurance decision. In general, portfolio relationships have nontrivial implications for insurance-buying strategies. In particular, correlation between insurable and noninsurable risk may affect the optimal level of insurance coverage. Deductible insurance is considered mainly because of its widespread use by many insurers. Furthermore, deductibles The literature concerning the choice of optimal deductibles for insurance policies assumes that wealth is nonstochastic except for the value being insured. However, a decision maker's asset portfolio might easily contain other sources of risk, some of which might be uninsurable. This paper examines the choice of a deductible insurance contract in the presence of uninsurable background risk. Existing theorems concerning the optimality of full coverage are shown to hold only under restricted conditions concerning the correlation of insurable risk with other risky assets. More general propositions are established and the effects of risk aversion are considered.