Academic journal article Journal of Risk and Insurance

Multiple Losses, Ex Ante Moral Hazard, and the Implications for Umbrella Policies

Academic journal article Journal of Risk and Insurance

Multiple Losses, Ex Ante Moral Hazard, and the Implications for Umbrella Policies

Article excerpt


Under certain cost conditions the optimal insurance policy offers full coverage above a deductible, as Arrow and others have shown. However, many insurance policies currently provide coverage against several losses although the possibilities for the insured to affect the loss probabilities by several prevention activities (multiple moral hazard) are substantially different. This article shows that optimal contracts under multiple moral hazard generally call for complex reimbursement schedules. It also examines the conditions under which different types of risks can optimally be covered by a single insurance policy and argues that the case for umbrella policies under multiple moral hazard is limited in practice.


In the real world many insurance contracts cover different kinds of losses although insured's prevention activities affect the probability of each kind of loss in different ways. Take health insurance as an example, where one finds that a single policy covers risks like rectal cancer (where prevention does not have much effect) and lung cancer (where protection is highly effective). Other insurance contracts cover different losses although prevention technologies vary for each kind of loss. Consider a household contents policy, covering the risk of burglary (the prevention of which requires one type of technology) as well as the risk of a lightning strike (which can be reduced by an entirely different type of technology). In both situations, intuition would suggest a different treatment of different types of risk in insurance contracts to give scope for well-directed incentives for prevention to be in the insured's interest, even though each risk might lead to the same monetary amount of loss. On the other hand, indemnity payments that depend on the type of loss suffered contradict the insured's preferences for a safe income in exchange for a single premium. Therefore, as has been stressed in the principal-agent literature (Holmstrom, 1979; Laffont and Martimort, 2002), an optimal insurance contract has to strike a balance between providing appropriate incentives for prevention and the individuals' demand for insurance protection.

Optimal insurance contracts have been discussed widely in the literature since the early contribution by Arrow (1963). Arrow, not taking into account informational asymmetries, states that under certain conditions concerning a risk-neutral insurer's cost function, an optimal insurance policy will offer full coverage above a nonnegative deductible. While full insurance (a deductible of zero) is optimal at actuarially fair premiums that cover exactly the expected value of indemnities, partial insurance turns out to be attractive if premiums contain a constant loading. Compared to any other feasible insurance arrangements a deductible policy concentrates indemnity on higher losses and thereby minimizes risk averse insured's loss of expected utility due to the cost of insurance. We will refer to this result as the standard insurance contract. (1) Authors like Raviv (1979) and others (2) have confirmed and extended Arrow's result. At the end of his article Raviv (1979, p. 261) addresses the question of how to deal with multiple losses and concludes that "the results regarding optimal insurance policies hold unchanged when the insured faces more than one risk when the loss considered is the loss from all those risks." This first-best result for multiple losses is also confirmed by Gollier and Schlesinger (1995). As a consequence, they stress the desirability of an extremely simple form of an umbrella policy that contains a single deductible but provides protection against the full range of risks individuals are exposed to.

In this article, it will be shown that the introduction of multiple ex ante moral hazard generally changes the optimal insurance schedule in a way that is significantly more complex than insurance contracts we usually find in practice. …

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