Optimal prevention strategies in the classical risk model

Abstract

In this paper, we propose and study a first risk model in which the insurer may invest into a prevention plan which decreases claim intensity. We determine the optimal prevention investment for different risk indicators. In particular, we show that the prevention amount minimizing the ruin probability maximizes the adjustment coefficient in the classical ruin model with prevention, as well as the expected dividends until ruin in the model with dividends. We also show that the optimal prevention strategy is different if one aims at maximizing the average surplus at a fixed time horizon. A sensitivity analysis is carried out. We also prove that our results can be extended to the case where prevention starts to work only after a minimum prevention level threshold.

Optimal prevention of large risks with two types of claims

“…and we require p < p lim . Since functions λ 2 , h : p → λ1µ1+λ2(p)µ2 c−p and derivatives of h follow similar properties as the ones in Proposition 1 of Gauchon et al (2020), it is possible to show that the function h is either increasing or first decreasing and then increasing with p. Thus, the intermediate value theorem ensures the existence and the uniqueness of p lim when λ1µ1+λ2(0)µ2 c < 1, which is assumed in the rest of the paper.…”

“…are similar to the ones imposed on λ(.) in Gauchon et al (2020) and we refer the reader to this latter paper for a discussion about these assumptions. The Poisson processes N 1 and N 2 are assumed to be independent.…”

“…The impact of prevention (sometimes combined with self-insurance, that can be assimilated to reinsurance in our framework) has been investigated in economics by Ehrlich and Becker (1972), Dionne and Eeckhoudt (1985) or Courbage (2001) among others. In ruin theory, a first risk model with prevention has been recently proposed by Gauchon et al (2020), with a unique set of claims for 2 which prevention has some effect. The authors consider an insurance company with initial surplus U (0) = u.…”

“…random variables, independent from N , and with cumulative distribution function F X , such that E(X) = µ < ∞. Gauchon et al (2020) assume that λ(.) is a decreasing, strictly convex, positive, and C 2 function defined on [0, c].…”

“…Exceptions include Courbage and Rey (2012) who consider an independent background risk, Hofmann and Peter (2015) who study the interplay of early and late prevention, and Courbage et al (2017) who investigate prevention of correlated risks. When it comes to prevention by insurers, Gauchon et al (2020) focus on a single source of risk. We instead distinguish between large claims and small claims and assume that prevention reduces only the intensity of large claims.…”