EVERAL types of insurance policies have a deductibility provision that waives the liability of the insurance company for losses less than the deductible. The insured may select a deductible from among several possibilities. The reader confronts this situation whenever he insures his automobile for collision losses. In most states he may insure his auto with a deductible of $50, $100, $150, or a larger amount. This paper is concerned with the following problem: Given that an individual plans to purchase collision insurance, what is the optimum value for the deductible? This is a problem in decision-making under uncertainty since the insured must estimate the probabilities of different size losses. In addition, the insured must somehow balance the greater protection that comes from small deductibles with the corresponding increase in the annual premium. At one extreme the insured could select the smallest deductible and, hence, self-insure for only very small losses. At the other extreme the insured could select a large deductible or not insure at all and, hence, self-insure for all but the very large losses or for all losses. How much of the liability should an individual assume?' MAXIMIZATION OF EXPECTED UTILITY