Optimal Dividend, Reinsurance and Capital Injection Strategies for Collaborating Business Lines: The Case of Excess-of-Loss Reinsurance

TL;DR

This paper derives optimal strategies for dividends, reinsurance, and capital injections for two collaborating insurance lines, using a diffusion model to maximize discounted dividends until ruin, with explicit strategies and numerical illustrations.

Contribution

It provides closed-form solutions for optimal strategies and characterizes their structure, including threshold and barrier types, in a multi-line reinsurance setting.

Findings

Optimal dividend strategy is threshold or barrier type depending on dividend rate bounds.

Pure excess-of-loss reinsurance is optimal among reinsurance options.

Optimal capital injections depend on reserve levels to prevent ruin.

Abstract

This paper considers an insurer with two collaborating business lines that must make three critical decisions: (1) dividend payout, (2) a combination of proportional and excess-of-loss reinsurance coverage, and (3) capital injection between the lines. The reserve level of each line is modeled using a diffusion approximation, with the insurer's objective being to maximize the weighted total discounted dividends paid until the first ruin time. We obtain the value function and the optimal strategies in closed form. We then prove that the optimal dividend payout strategy for bounded dividend rates is of threshold type, while for unbounded dividend rates it is of barrier type. The optimal combination of proportional and excess-of-loss reinsurance is shown to be pure excess-of-loss reinsurance. We also show that the optimal level of risk ceded to the reinsurer decreases as the aggregate reserve level increases. The optimal capital injection strategy involves transferring reserves to prevent the ruin of one line. Finally, numerical examples are presented to illustrate these optimal strategies.