This paper studies the optimal dividend for a multi-line insurance group, in which each subsidiary runs a product line and is exposed to some external credit default risk. The external default contagion is considered in the sense that one default event can affect the default probabilities of all surviving subsidiaries. The total dividend problem is formulated for the insurance group and we reveal for the first time that the optimal singular dividend strategy is still of the barrier type. Furthermore, we show that the optimal barrier for each subsidiary is modulated by the current default state, namely how many and which subsidiaries have defaulted will determine the dividend threshold for each surviving subsidiary. These interesting conclusions are based on our analysis of the associated recursive system of Hamilton-Jacobi-Bellman variational inequalities (HJBVIs), which is new to the literature. The existence of the classical solution is established and the rigorous proof of the verification theorem is provided. For the case of two subsidiaries, the value function and optimal barriers for each subsidiary are explicitly constructed. Some numerical examples are also presented to illustrate the economic insights.
- Pub Date:
- September 2019
- Quantitative Finance - Risk Management;
- Mathematics - Optimization and Control
- Keywords: Insurance group, external default contagion, optimal dividend, reflection control, default-state-modulated barriers, recursive system of HJBVIs