On the Theory of Rational Insurance Purchasing in a Continuous-Time Model

Abstract

The determination of the optimal insurance coverage has raised a growing academic interest. The seminal works by Arrow [1963], Smith [1968] and Mossin [1968] contributed to establish the cornerstone results of insurance economics. Indeed, it is now a well-known result that a risk averse expected utility maximizer will never purchase full insurance coverage if the insurance premium is actuarially unfair A positive amount deductible is always optimal from the insured's point of view if the premium is based on expected insurance payments and includes a positive proportional loading. Arrow [1974] extended this result to the case of state-dependent utility functions. Some authors have however provided results which stand in sharp contrast with the above mentiond rule. Razin [1976], for instance, argued that the nature of results obtained very much depends on the decision framework. Using a binominal distribution of losses and Savage's minimax regret criterion as an alternative to the expected utility framework, he showed that it is always optimal to purchase insurance with a positive amount deductible even if the insurance premium is fair. In the same vein, Briys and Louberge [1985] obtained that, under the Hurwicz criterion, full insurance is very often optimal even under a non zero insurance loading. These results may seem conflicting. They nevertheless have a common basis: they focus on insurance in isolation. The optimal insurance coverage is determined in a single risk setting where consumption and portfolio decicions are not considered. In other words, previous studies assume either explicitly or implicitly that the insurance decision is perfectly separable from any other decisions. This strong assumption is obviously questionable. As recently pointed out by Doherty and Schlesinger [1983], the result by Smith and Mossin only holds if insurable and non insurable losses are non positively correlated. For a positive correlation between these losses more than full coverage is requested by risk averse individuals operating in incomplete markets.