Abstract
In the classical expected utility framework, a problem of optimal insurance design with a premium constraint is equivalent to a problem of optimal insurance design with a minimum expected retention constraint. When the insurer has ambiguous beliefs represented by a non-additive probability measure, as in Schmeidler, this equivalence no longer holds. Recently, Amarante, Ghossoub and Phelps examined the problem of optimal insurance design with a premium constraint when the insurer has ambiguous beliefs. In particular, they showed that when the insurer is ambiguity-seeking, with a concave distortion of the insured’s probability measure, then the optimal indemnity schedule is a state-contingent deductible schedule, in which the deductible depends on the state of the world only through the insurer’s distortion function. In this paper, we examine the problem of optimal insurance design with a minimum expected retention constraint, in the case where the insurer is ambiguity-seeking. We obtain the aforementioned result of Amarante, Ghossoub and Phelps and the classical result of Arrow as special cases.
Highlights
In the classical problem of optimal insurance design, it is well known since the work of Arrow [1]that when the insured is a risk-averse expected utility (EU) maximizer and the insurer is a risk-neutralEU maximizer, the indemnity schedule that maximizes the insured’s expected utility ofterminalwealth subject to a premium constraint is a deductible indemnity of the form Y “ max 0, Xd, where X is the loss random variable and d ě 0 is a given constant deductible.For a given indemnity schedule Y, the premium constraint is a constraint of the form: ż Π ě p1 ` ρqYdP where Π ě 0 is the premium paid and ρ ě 0 is a loading factor
We focus in this paper on Choquet integration as an aggregation concept for decision-making under ambiguity, and we consider the Choquet expected utility (CEU) model of Schmeidler [8] as a model of decision-making under ambiguity
Motivated by empirical evidence suggesting that insurers tend to exhibit more ambiguity than the insured individuals (e.g., [13]), AGP [14] study the problem of optimal insurance design in a setting where the insurer has ambiguous beliefs about the realizations of the insurable loss, whereas the insured does not
Summary
In the classical problem of optimal insurance design, it is well known since the work of Arrow [1]. The constraint given in Equation (1) is often referred to as a minimum expected retention constraint It requires that the indemnity schedule Y be such that the associated retention random variable has a minimum pre-specified expectation under the insurer’s beliefs. The seminal work of Knight [5] suggested that there might be situations where the information available to a decision maker is too coarse for him or her to be able to formulate an additive probability measure over the list of contingencies. These occurrences are typically referred to as situations of decision under Knightian uncertainty, or ambiguity. We focus in this paper on Choquet integration as an aggregation concept for decision-making under ambiguity, and we consider the CEU model of Schmeidler [8] as a model of decision-making under ambiguity
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