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

Tim J. Boonen, Engel John C. Dela Vega

TL;DR

This paper addresses optimizing dividends, reinsurance, and inter-line capital injections for a two-line insurer under diffusion-reserve dynamics. It proves that the optimal reinsurance contract is pure excess-of-loss and derives closed-form value functions, revealing threshold-type dividends for bounded rates and barrier-type dividends for unbounded rates. The analysis yields a precise, region-based policy structure, including a reinsurance threshold $w_0$ and dividend thresholds $u_1,u_2$, with uniform capital-injection rules to avert ruin. Numerical examples confirm the theoretical insights and illustrate how optimal risk ceded to reinsurers decreases with higher aggregate reserves. The results inform practical risk management and capital-allocation decisions across collaborating lines, highlighting the trade-offs between dividend payouts, reinsurance retention, and systemic solvency.

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.

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

TL;DR

This paper addresses optimizing dividends, reinsurance, and inter-line capital injections for a two-line insurer under diffusion-reserve dynamics. It proves that the optimal reinsurance contract is pure excess-of-loss and derives closed-form value functions, revealing threshold-type dividends for bounded rates and barrier-type dividends for unbounded rates. The analysis yields a precise, region-based policy structure, including a reinsurance threshold and dividend thresholds , with uniform capital-injection rules to avert ruin. Numerical examples confirm the theoretical insights and illustrate how optimal risk ceded to reinsurers decreases with higher aggregate reserves. The results inform practical risk management and capital-allocation decisions across collaborating lines, highlighting the trade-offs between dividend payouts, reinsurance retention, and systemic solvency.

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.

Paper Structure

This paper contains 13 sections, 28 theorems, 147 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Model
  3. The Gain of Pure Excess-of-Loss Reinsurance
  4. Bounded Dividend Rates
  5. Unbounded Dividend Rates
  6. Numerical Examples
  7. Proof of Main Results
  8. Proof of Theorem \ref{['thm:bdd w0<u1<u2']}
  9. Proof of Theorem \ref{['thm:bdd u1<w0<u2']}
  10. Proof of Theorem \ref{['thm: bdd u1<u2<w0 or M infty']}
  11. Proof of Theorem \ref{['thm:ubdd M1 finite']}
  12. Proof of Theorem \ref{['thm: unbdd M1 infty']}
  13. Conclusion

Key Result

Lemma 3.1

Define Then, $h_i(s)$ is an increasing function for $s>0$.

Figures (7)

  • Figure 1: Regions for Capital Injection Decisions
  • Figure 2: Regions for Dividend Payout and Capital Injection Decisions
  • Figure 3: Uniformly distributed claims: $\kappa_1=4$, $\kappa_2=2$, $\delta =0.5$, $a=0.3$, $\overline c_1=3$, $\overline c_2=2$ ($M_1=1$, $M_2=1.5$, $w_0=0.19 < u_1=0.24<u_2=0.87$, corresponding to Theorem \ref{['thm:bdd w0<u1<u2']})
  • Figure 4: Uniformly distributed claims: $\kappa_1=4$, $\kappa_2=2$, $\delta =0.5$, $a=0.3$, $\overline c_1=3$, $\overline c_2=1$ ($M_1=1$, $M_2=1.5$, $u_1=0.17 < w_0=0.22<u_2=0.57$, corresponding to Theorem \ref{['thm:bdd u1<w0<u2']})
  • Figure 5: Uniformly distributed claims: $\kappa_1=4$, $\kappa_2=2$, $\delta =0.5$, $a=0.3$, $\overline c_1=1$, $\overline c_2=0.5$ ($M_0=0.71$, $M_1=1$, $M_2=1.5$, $u_1=0.09 <u_2=0.19$, corresponding to Theorem \ref{['thm: bdd u1<u2<w0 or M infty']})
  • ...and 2 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Theorem 4.2
  • Theorem 4.3
  • ...and 45 more