Optimal Reinsurance with Heterogeneous Reference Probabilities

This article attempts to extend Arrow’s theorem of the deductible to the case of belief heterogeneity, which allows the insured and the insurer to have different beliefs about the distribution of the underlying loss. Like Huberman et al. [(1983) Bell Journal of Economics14(2), 415–426], we preclude ex post moral hazard by asking both parties in the insurance contract to pay more for a larger realization of the loss. It is shown that, ceteris paribus, full insurance above a constant deductible is always optimal for any chosen utility function of a risk-averse insured if and only if the insurer appears more optimistic about the conditional loss given non-zero loss than the insured in the sense of monotone hazard rate order. We derive the optimal deductible level explicitly and then examine how it is affected by the changes of the insured’s risk aversion, the insurance price and the degree of belief heterogeneity.

Section: Discussionmentioning

confidence: 99%

On the Optimality of a Straight Deductible Under Belief Heterogeneity

Chi

2018

“…Remark 3.1. Similar idea of the proof of Theorem 3.3 (i) was also used in and Boonen and Ghossoub (2019), in which the fundamental theorem of Lebesgue integral calculus plays a key role. However, Theorem 3.3(i) studies a general risk-sharing problem while the latter study reinsurance problems.…”

Section: Weighted Risk Sharing With Distortion Risk Measuresmentioning

confidence: 91%

“…Comonotonicity of an optimal allocation is a necessary condition for an Arrow-Debreu equilibrium to exist in a complete market; see the discussions in Boonen et al (2018). We are interested in comonotonic allocations also because in the context of insurance, comonotonicity of insurance contracts rules out moral hazard, so that a policyholder will not increase the ground-up loss purposely in order to receive more coverage from an insurer; see, for example, Boonen and Ghossoub (2019). In economics, comonotonicity is a no-hedge condition; see Yaari (1987).…”

Section: Introductionmentioning

confidence: 99%

Weighted Comonotonic Risk Sharing Under Heterogeneous Beliefs

Liu¹

2020

We study a weighted comonotonic risk-sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess of loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is value at risk or tail value at risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

“…Note that F 1 may be different from F Q , which is assumed by the insurer. Thus, the problem becomes a special case of a more general optimal insurance problem with heterogeneous beliefs, which was studied, for example, by Boonen (2016) and Boonen and Ghossoub (2019). In particular, Boonen and Ghossoub (2019) considered one insurer and multiple reinsurers who have different distributional assumptions.…”

Section: Corollary 31 the Solution To Problem 3 Is Given Bymentioning

confidence: 99%

Optimal Insurance Contracts Under Distortion Risk Measures With Ambiguity Aversion

Jiang¹,

Escobar‐Anel

Ren

2020

This paper presents analytical representations for an optimal insurance contract under distortion risk measure and in the presence of model uncertainty. We incorporate ambiguity aversion and distortion risk measure through the model of Robert and Therond [(2014) ASTIN Bulletin: The Journal of the IAA, 44(2), 277–302.] as per the framework of Klibanoff et al. [(2005) A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.]. Explicit optimal insurance indemnity functions are derived when the decision maker (DM) applies Value-at-Risk as risk measure and is ambiguous about the loss distribution. Our results show that: (1) under model uncertainty, ambiguity aversion results in a distorted probability distribution over the set of possible models with a bias in favor of the model which yields a larger risk; (2) a more ambiguity-averse DM would demand more insurance coverage; (3) for a given budget, uncertainties about the loss distribution result in higher risk level for the DM.