Irreversible reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost

TL;DR

This paper models an insurance company's decision-making process for irreversible reinsurance contracts with fixed costs, aiming to minimize capital injections and activation costs through optimal retention and timing strategies.

Contribution

It introduces a novel model combining fixed transaction costs with dynamic retention and timing choices, providing explicit solutions and numerical insights.

Findings

Explicit solution involving nonlinear optimization and optimal stopping problems.

Optimal strategies depend on model parameters and reinsurance type.

Numerical analysis illustrates the impact of parameters on decisions.

Abstract

We propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cram\'er-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any specific functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model's parameters.