Optimal insurance in the presence of reinsurance

Abstract

This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution. ARTICLE HISTORY

How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk

“…We say that g : [0, 1] → [0, 1] is a distortion function if g is a non-decreasing function, such that g(0 + ) := lim x↓0 g(x) = 0 and g(1) = 1. Unlike Zhuang et al [11], no (left-or right-) continuity assumptions are imposed on g, neither are convexity or concavity conditions, because our analysis is valid without these extraneous technical assumptions and therefore holds in high generality, with the agent being at complete liberty to choose a distortion function that best aligns with his/her risk preference. Corresponding to a distortion function g, the DRM of a random variable Y is defined by…”

“…More remarkably, the striking symmetry between Formula (2) and Formula (4), when viewed in the correct light, will substantially facilitate understanding the marginal costs and benefits of insurance and reinsurance. Compared with the reinsurance premium principles in Asimit et al [13] and Zhuang et al [11], Formula (4) possesses two subtle differences. First, unlike Asimit et al [13], no allowance is made in Formula (4) for the possibility of default when the reinsurer charges the reinsurance premium.…”

“…We note in passing that setting δ = 1 or sending β ↑ 1 retrieves the default-free framework of Zhuang et al [11].…”

“…Synthesizing the discussions in Asimit et al [13] and Zhuang et al [11], we begin with a precise description of a DRM-based insurance-reinsurance model in which three agents, namely a policyholder, an insurer and a default-prone reinsurer, coexist. In this model, the risk faced by an agent is evaluated by means of distortion risk measures (DRMs), whose definition calls for the notion of a distortion function (see Denneberg [15] and Wang [16]).…”

“…Moreover, when the prospect of default is incorporated, as in Asimit et al [13], the discrepancy between the beliefs of the insurer and reinsurer about the recovery rate of the loss given default may give rise to more aggressive risk transfer from the insurer and therefore counter-intuitive optimal reinsurance contracts. Second, it was shown in Proposition 3.1 of Zhuang et al [11] that the insurance premium function h I should be set as the distortion function of the policyholder, provided that the policyholder is also a DRM-adopter. Such a specification provides the minimum incentive for the policyholder to "participate" in the insurance contract.…”