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

2025-11-14

AI Summary

This paper investigates the optimal joint strategy of dividend distribution, reinsurance, and capital injection for an insurer operating two interdependent business lines, aiming to maximize the weighted discounted dividends paid prior to ruin. Method: The reserve dynamics are modeled via a diffusion approximation; a Hamilton–Jacobi–Bellman (HJB) equation is formulated and solved using stochastic control theory. Contribution/Results: Theoretical analysis reveals that the optimal reinsurance is pure excess-of-loss, with the retention level decreasing in total surplus; optimal dividend payouts follow a threshold policy under bounded surplus and a barrier policy under unbounded surplus; capital injections are applied solely to prevent ruin in either line. For the first time, closed-form expressions are derived for all optimal strategies and the value function. Numerical experiments confirm the economic plausibility and robustness of the solutions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Optimizing dividend payout strategies for collaborating insurance business lines
• Determining optimal reinsurance mix between proportional and excess-of-loss coverage
• Establishing capital injection strategies to prevent ruin between lines

Innovation

• Combines proportional and excess-of-loss reinsurance strategies
• Uses threshold and barrier type dividend payout methods
• Implements capital injection to prevent business line ruin