On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model

TL;DR

This paper develops a model for fair reinsurance premiums where capital injections occur at ruin times caused by large jumps, providing explicit formulas in a perturbed risk setting with practical numerical illustrations.

Contribution

It introduces an explicit formula for reinsurance premiums based on discounted expected capital injections in a perturbed risk model with jump and Brownian components.

Findings

Explicit premium formulas derived for models with jumps and Brownian perturbations

Reinsurance premiums depend on the nature of ruin events (jumps vs. oscillations)

Numerical examples demonstrate the applicability of the formulas

Abstract

We consider a risk model where deficits after ruin are covered by a new type of reinsurance contract that provides capital injections. To allow the insurance company's survival after ruin, the reinsurer injects capital only at ruin times caused by jumps larger than a chosen retention level. Otherwise capital must be raised from the shareholders for small deficits. The problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014) on successive ruin events, we show that an explicit formula for these reinsurance premiums exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. Here ruin events are due either to Brownian oscillations or jumps and reinsurance capital injections only apply in the latter case. The results are illustrated explicitly for two specific risk models and in some numerical examples.