On the Optimality of a Straight Deductible Under Belief Heterogeneity

Abstract

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.

Optimum Insurance Contracts With Background Risk and Higher-Order Risk Attitudes

“…Proof of Proposition 5.1. The proof of this proposition is a slight modification to that of Theorem 4.2 in Chi (2017).…”

On the existence of a representative reinsurer under heterogeneous beliefs

“…Proof: The proof is similar to that of Theorem 4.1 in Chi (2019), but adapted to the present setting. Suppose that HR is non-increasing on its domain.…”

“…The following result provides such an extension of Theorem 4.3, as well as a converse in the general case. The proof is similar to that of Theorem 4.1 in Chi (2019), and it is therefore omitted. (ii) The hazard ratio HR is non-increasing on its domain.…”

“…That is, the insurer does not decide how much premium to spend on reinsurance, but is only interested in the optimal reinsurance indemnity schedule for the given premium level, treated as a budget constraint. On the other hand, Boonen (2016) and Chi (2019) optimize the utility of the insurer's net wealth given a premium principle of the reinsurer. Boonen (2016) studies the case where the insurer minimizes a distortion risk measure as proposed by Wang (1996).…”

“…Boonen (2016) studies the case where the insurer minimizes a distortion risk measure as proposed by Wang (1996). Chi (2019) solves the problem with expected utility maximizing preferences for the insurer, and he determines the optimal reinsurance contract under the condition of a monotone hazard ratio. Here, we extend the results of Boonen (2016) and Chi (2019) to a setting with multiple reinsurers and more general preference for the insurer.…”