The problem of selecting a deductible when an asset is exposed to risk is analysed in the framework of consumption theory. The riskbearing budget is defined as the difference between the optimal consumption level under complete certainty and the optimal consumption level in a context of risk. How this risk-bearing budget should be divided between buying insurance and building up a contingency reserve is investigated. The wealth effect on both the risk-bearing budget and the deductible also is analysed. Insurance, perceived as an instrument of risk redistribution, undoubtedly plays a vital role in the theory of risk-bearing. Individuals as well as corporations are concerned about finding some optimal insurance arrangements when confronted with risk situations. At the beginning of the sixties, Borch introduced utility theory into contemporary actuarial thought. He was concerned particularly with the problem of risk redistribution through the reinsurance market. Borch [3] (reprinted in Borch [4]), then found that an insurer seeking to maximize the variance reduction of its claim distribution for a given net premium should seek to conclude a Stop Loss treaty. In his classical analysis of the economics of medical care, Arrow [1] generalized Borch's result by demonstrating the superiority of a deductible form of insurance for a risk averting individual facing the risk of an income reduction. The works of Borch and Arrow generated further research and a theory for the demand of insurance gradually took form. On the theoretical side, the contributions of Mossin [7] and Smith [9] are among the most interesting ones. Along the lines of Arrow, the approach commonly adopted in attempting to build a decision model to determine the optimal insurance coverage is based on the maximization of the expected utility of terminal wealth. The two major deficiencies in this approach are: First, it completely ignores saving as an alternative to insurance buying. Denis Moffet is Assistant Professor of actuarial science, Universite Laval in Qu6bec. This paper is based on a chapter of Essays in the Economics of Insurance which was awarded the 1976 Ernst Meyer Prize by the Geneva Association. Comments by Karl Borch, Jan Mossin, Agnar Sandmo, Baruch Berliner and two referees have been most useful. An earlier version of this paper was presented at the 1975 Risk Theory Seminar. ( 669 ) This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 06:10:57 UTC All use subject to http://about.jstor.org/terms 670 The Journal of Risk and Insurance Second, it neglects the effect upon current consumption of any outlay intended specifically for the purpose of risk-bearing. By considering a simple risk situation, it was illustrated (Moffet [5]) how the insurance buying problem could be formulated in terms of consumption theory (see, for example, Sandmo [8]). This present study, however, concentrates on a more realistic problem, namely the one of choosing an appropriate deductible level when a given asset is subject to a risk of deterioration. Also attention is focused on some actuarial considerations about the premium for a given deductible. The analysis flows from the determination of an optimal consumption level, assuming an initial wealth. The risk-bearing budget is defined as the difference between the optimal consumption level under complete certainty and the optimal consumption level in a context of risk. How this risk-bearing budget should be divided between buying insurance and building up a contingency reserve fund then is investigated. The Model A single-period model is used. At the beginning of the period the consumer's net worth is W. One of the consumer's assets, the value of which is N < W, can suffer a loss amounting to x during the period, subject to the following probability law: p: probability that no loss will be incurred, i.e. x = 0 q: probability that a full loss will be incurred, i.e. x = N f(x): probability density function over the open interval N (o,N) such that p + f(x)dx + q 1 where 0 < p < 1 and 0 < q < 1 This risky situation can be confronted by buying insurance on the market and/or by building a contingency reserve fund. It is assumed that only one claim can occur during the period and that its settlement is made at the end of the period. Insurance can be bought with a deductible D, that is the insurer will pay the amount in excess of D if a loss is incurred. The insurance premium is denoted P(D) and when there is no danger of confusion it is written P instead of P(D). The consumption level for the period and the consumer's net worth at the end of the period are denoted by C and Y. Assume that the consumer's preferences are represented by a utility function V having the form