Abstract
In problems of optimal insurance design, Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub only gave a characterization of these optimal indemnity schedules in the special case of an MLR. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub as corollaries. Finally, we formalize Marshall’s argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any”shape.
Highlights
The problem of optimal insurance design under uncertainty dates back to the seminal work of Arrow: Theorem 5 (Arrow) [1] who showed that when the insured, or Decision Maker (DM), is a risk-averseExpected-Utility (EU)-maximizer, the insurer is a risk-neutral EU-maximizer, the two parties assign the same distribution to the insurable random loss and the premium principle depends on the actuarial value of the indemnity; full insurance above a deductible is optimal for the DM
We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub [9] as corollaries
When P “ Q, we recover the classical framework of Arrow: Theorem 5 (Arrow)
Summary
The problem of optimal insurance design under uncertainty dates back to the seminal work of Arrow [1] who showed that when the insured, or Decision Maker (DM), is a risk-averse. The probability of a zero loss can be seen as a proxy for the DM’s optimism, and Marshall shows that if the insurer is risk-neutral, the optimal insurance indemnity is a deductible contract; and the deductible level increases with the DM’s optimism This is a rather restrictive approach to belief heterogeneity, since this heterogeneity is reduced only to the likelihood that each party attaches to the event of a zero loss. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub [9] as corollaries. Some proofs and related analyses are collected in Appendices A–F
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