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

2025-08-11

AI Summary

This paper investigates the optimal joint strategy of dividend distribution, proportional reinsurance, and cross-line capital injections for a two-line collaborative insurer, aiming to maximize the expected weighted cumulative dividends prior to ruin. A two-dimensional diffusion risk model is adopted to formulate a stochastic control framework that, for the first time, integrates all three decision instruments into a unified optimization problem. By solving the associated Hamilton–Jacobi–Bellman (HJB) equation via dynamic programming, closed-form expressions for both the value function and the optimal feedback controls are derived. Key theoretical findings include: (i) the more critical business line exhibits a lower dividend threshold; (ii) capital injections serve solely as a ruin-protection mechanism with no incentive for proactive dividend enhancement; and (iii) the optimal reinsurance proportion decreases with increasing risk exposure. Numerical experiments corroborate parameter sensitivity and confirm the economic implications of the derived strategies.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Optimize dividend, reinsurance, and capital injection strategies for insurers
• Maximize expected dividend payments until first ruin occurs
• Determine closed-form solutions for value functions and optimal strategies

Innovation

• Optimal dividend strategy with threshold approach
• Dynamic proportional reinsurance based on reserves
• Capital injection to prevent business line ruin