Prevention efforts, insurance demand and price incentives under coherent risk measures

Abstract

This paper studies an equilibrium model between an insurance buyer and an insurance seller, where both parties' risk preferences are given by convex risk measures. The interaction is modeled through a Stackelberg type game, where the insurance seller plays first by offering prices, in the form of safety loadings. Then the insurance buyer chooses his optimal proportional insurance share and his optimal prevention effort in order to minimize his risk measure. The loss distribution is given by a family of stochastically ordered probability measures, indexed by the prevention effort. We give special attention to the problems of self-insurance and self-protection. We prove that the formulated game admits a unique equilibrium, that we can explicitly solve by further specifying the agents criteria and the loss distribution. In self-insurance, we consider also an adverse selection setting, where the type of the insurance buyers is given by his loss probability, and study the screening and shutdown contracts. Finally, we provide case studies in which we explicitly apply our theoretical results.

A continuous-time model of self-protection

“…As a function of the price θ, the coverage choice is decreasing: it can decrease continuously from 1 to 0 if (1 + θ 2 ) = εe γ , or it will jump suddenly to no-coverage if not. This differs from some results in the literature, for instance [2], where the optimal proportion of coverage α is either 0 or 1.…”

“…[20] and [36] both determine the form of the optimal insurance contract, under expected utility, respectively in case of self-protection, where a deductible is optimal, and in case of self-insurance, where this time low losses are fully covered while large losses are partially covered. When the criteria of both the protection buyer and seller are given by coherent risk measures, [2] shows under moral hazard, that if the buyer's risk measure decreases faster in effort than his expected loss, optimal effort is non-decreasing in the insurance price with a potential discontinuity when optimal coverage switches from full to zero. On the contrary, if the decrease of the buyer's risk measure is slower than the expected loss, optimal effort may or may not be non-decreasing with the insurance price.…”

Risk mitigation services in cyber insurance: optimal contract design and price structure

“…Our approach is based on the work of Bensalem et al. ( 2020 ), by using the framework of distortion risk measures and stochastic ordering of loss distributions, respectively, to capture risk assessment of all parties and the effects of risk mitigation services, and by modelling the interaction between insurer and insurance buyer(s) as a Stackelberg game. We extend their setting to a bivariate problem for the insurer, allowing her to choose the price for both risk transfer and risk mitigation, and analyse the results of the corresponding buyer’s problem [which is conceptually similar to Bensalem et al.…”

“…We extend their setting to a bivariate problem for the insurer, allowing her to choose the price for both risk transfer and risk mitigation, and analyse the results of the corresponding buyer’s problem [which is conceptually similar to Bensalem et al. ( 2020 )] in the cyber insurance context. Furthermore, we transcend from the study of an interaction with a single buyer to examples of (sequential or simultaneous) interactions with several buyers with dependent losses.…”

The Day After Tomorrow: Mitigation and Adaptation Policies to Deal with Uncertainty

“…As it is commonly assumed in the literature (Bensalem et al, 2020), we model the cost of the eort as a quadratic (convex) function of the eort itself:…”