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

Paper: arXiv 2511.11383

Authors: Tim J. Boonen, Engel John C. Dela Vega

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of dynamic stochastic control problems in insurance using diffusion approximations and closed-form derivations. However, it is purely theoretical and lacks empirical market data or realistic backtesting, relying only on stylized numerical illustrations.

Research Flowchart

  flowchart TD
    A["<b>Research Goal:</b> Maximize weighted discounted dividends via optimal dividend, reinsurance, & capital injection strategies for two collaborating business lines"] --> B["<b>Modeling & Inputs:</b> Reserve levels modeled via Diffusion Approximation under Excess-of-Loss and Proportional Reinsurance"]
    
    B --> C["<b>Stochastic Control Problem:</b> Formulate Hamilton-Jacobi-Bellman (HJB) equations until first ruin time"]
    
    C --> D["<b>Mathematical Analysis:</b> Derive closed-form Value Function & prove optimal control boundaries"]
    
    D --> E["<b>Dividend Outcomes:</b> Threshold type (bounded rates) vs. Barrier type (unbounded rates)"]
    D --> F["<b>Reinsurance Outcome:</b> Pure Excess-of-Loss is optimal; risk ceded decreases as aggregate reserves increase"]
    D --> G["<b>Capital Injection Outcome:</b> Reserve transfers occur between lines specifically to prevent single-line ruin"]

    E & F & G --> H["<b>Validation:</b> Numerical examples illustrate and confirm optimal strategies"]

Cited References (28)

1. On combining quota-share and excess of loss (1985)
2. A revisit of the optimal excess-of-loss contract (2025)
3. A class of approximations of ruin probabilities (1977)
4. Optimal Dividend, Reinsurance, and Capital Injection Strategies for an Insurer with Two Collaborating Business Lines (2025) — link
5. Stochastic Control in Insurance (2008)
6. Optimal risk control and dividend distribution policies. Example of excess-of-loss reinsurance for an insurance corporation (2000)
7. Su un’Impostazione Alternativa della Teoria Collettiva del Rischio (1957)
8. Strategies for dividend distribution: A review (2009)

0 of 28 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.