Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Salvatore Federico, Giorgio Ferrari, Maria-Laura Torrente

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 Cramér-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.

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

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 Cramér-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.
Paper Structure (12 sections, 7 theorems, 126 equations, 8 figures)

This paper contains 12 sections, 7 theorems, 126 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Solution to the problem
  4. Reformulation of the problem
  5. Optimal reinsurance
  6. Optimal reinsurance timing
  7. Examples and Illustrations
  8. Proportional reinsurance
  9. Excess-of-loss reinsurance
  10. The case $Z \sim \textrm{Exp}(1/\mu)$
  11. The case $Z \sim \textrm{Pareto}(\zeta,\alpha)$
  12. Comparison of the value function in the case of proportional and excess-of-loss reinsurance

Key Result

Theorem 3.2

Recall OP and fb. Let $b^*\in\mathcal{B}^{*}$ and let $\tau^*(b^*)\in\mathcal{T}$ such that with the convention $e^{-\rho\tau}f_{b^*}(X^{x}_{\tau})=0$ on $\{\tau=\infty\}$. Then, the couple $a^{*}:=(\tau^*(b^*),b^{*})\in\mathcal{A}$ is an optimal reinsurance strategy (with $b^{*}$ thought of as a constant random variable).

Figures (8)

  • Figure 1: Function $f_{b^*}$ in the case $b^* \in [0,1)$.
  • Figure 2: Dependency of $b^*$ with respect to $\rho$, $\mu$ and $\sigma^2$.
  • Figure 3: Dependency of $x^*_{b^*}$ with respect to $\rho$, $\mu$, $\sigma^2$ and $K$.
  • Figure 4: Dependency of $b^*$ with respect to $\rho$ and $\mu$.
  • Figure 5: Dependency of $x^*_{b^*}$ with respect to $\rho$, $\mu$, and $K$.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • Theorem 3.6
  • ...and 11 more