Optimal Dividend, Reinsurance, and Capital Injection Strategies for an Insurer with Two Collaborating Business Lines

TL;DR

This paper derives optimal strategies for dividends, reinsurance, and capital injections for a two-line insurer, providing closed-form solutions and insights into how these strategies depend on model parameters.

Contribution

It offers a complete analytical solution for a complex multi-decision problem in insurance risk management, including explicit strategies and value functions.

Findings

Optimal dividend strategy is a threshold policy with the more important line having a lower threshold.

Reinsurance proportion decreases as the aggregate reserve level increases.

Capital injections are used solely to prevent ruin of individual business lines.

Abstract

This paper considers an insurer with two collaborating business lines, and the risk exposure of each line follows a diffusion risk model. The manager of the insurer makes three decisions for each line: (i) dividend payout, (ii) (proportional) reinsurance coverage, and (iii) capital injection (from one line into the other). The manager seeks an optimal dividend, reinsurance, and capital injection strategy to maximize the expected weighted sum of the total dividend payments until the first ruin. We completely solve this problem and obtain the value function and optimal strategies in closed form. We show that the optimal dividend strategy is a threshold strategy, and the more important line always has a lower threshold to pay dividends. The optimal proportion of risk ceded to the reinsurer is decreasing with respect to the aggregate reserve level for each line, and capital injection is only used to prevent the ruin of a business line. Finally, numerical examples are presented to illustrate the impact of model parameters on the optimal strategies.