Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer

Abstract

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.

Budget-constrained optimal insurance without the nonnegativity constraint on indemnities

“…Amarante and Ghossoub [2] study the problem of optimal insurance design with a retention constraint and in the presence of the nonnegativity constraint on indemnities, but with ambiguity only on the side of the insurer, represented by a distortion T 2 of the probability measure P . Ghossoub [24] extends the setting of Amarante and Ghossoub [2] to also account for ambiguity on the side of the DM, represented by a distortion T 1 of the probability measure P . He shows that the optimal indemnity is a variable deductible schedule, with a state-contingent deductible that depends on the sate of the world only through T 1 and T 2 .…”

“…All the results of this section would still hold, with adequate modifications. Moreover, the assumption of absolute continuity of Q with respect to P can be dropped, and one can use the technique developed in Ghossoub [23,26] and Amarante and Ghossoub [2], based on a Lebesgue decomposition of P with respect to Q.…”

“…Bernard et al [5], Xu et al [51], and Ghossoub [24] extend Arrow's setting to the case of a fixed total insurance budget and an expected-value premium principle, but where the DM behaves according to Rank-Dependent Expected-Utility [42,52]. Amarante et al [3] and Amarante and Ghossoub [2] consider the case of a fixed total insurance budget and an EU-maximizing DM, but distortion premium principles and more general Choquet premium principles. Cui et al [16], Zhuang et al [55], Cheung et al [11], Cheung and Lo [13], and Lo [38,39] examine the problem of minimizing a distortion risk measure of the insurer's retained risk, with a fixed reinsurance budget, and under a distortion reinsurance premium principle.…”

“…We first consider a setting where the DM is a Rank-Dependent Expected-Utility (RDEU)-maximizer (as in [42,52]), who distorts the true probability distribution of the random loss, due to some ambiguity on his side, and where the premium principle is an expected-value premium principle. We relax the nonnegativity constraint and we assume that there are no costs associated with handling an insurance claim 2 . This allows us to test whether the existence of a deductible is due to the variable cost of insurance under ambiguity on the side of the DM.…”

Application of Choquet Integral-Fuzzy Measures for Aggregating Customers’ Satisfaction

“…According to Heilpern, [10] the CI-FM can be applied to many real decision-making cases in economics and business. It is also reported that CI-FM has been used in specific areas such as insurance [11], green building [12], airline services [13], business education evaluation [14], loan market matching [15], and financial risk [16]. In customer service-oriented research, Pasrija and Srivastava [17] investigated the software quality using the CI-FM.…”