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Optimal Dividend, Reinsurance, and Capital Injection for Collaborating Business Lines under Model Uncertainty

Tim J. Boonen, Engel John C. Dela Vega, Len Patrick Dominic M. Garces

Abstract

This paper considers an insurer with two collaborating business lines that faces three critical decisions: (1) dividend payout, (2) reinsurance coverage, and (3) capital injection between the lines, in the presence of model uncertainty. The insurer considers the reference model to be an approximation of the true model, and each line has its own robustness preference. The reserve level of each line is modeled using a diffusion process. The objective is to obtain a robust strategy that maximizes the expected weighted sum of discounted dividends until the first ruin time, while incorporating a penalty term for the distortion between the reference and alternative models in the worst-case scenario. We completely solve this problem and obtain the value function and optimal (equilibrium) strategies in closed form. We show that the optimal dividend-capital injection strategy is a barrier strategy. The optimal proportion of risk ceded to the reinsurer and the deviation of the worst-case model from the reference model are decreasing with respect to the aggregate reserve level. Finally, numerical examples are presented to show the impact of the model parameters and ambiguity aversion on the optimal strategies.

Optimal Dividend, Reinsurance, and Capital Injection for Collaborating Business Lines under Model Uncertainty

Abstract

This paper considers an insurer with two collaborating business lines that faces three critical decisions: (1) dividend payout, (2) reinsurance coverage, and (3) capital injection between the lines, in the presence of model uncertainty. The insurer considers the reference model to be an approximation of the true model, and each line has its own robustness preference. The reserve level of each line is modeled using a diffusion process. The objective is to obtain a robust strategy that maximizes the expected weighted sum of discounted dividends until the first ruin time, while incorporating a penalty term for the distortion between the reference and alternative models in the worst-case scenario. We completely solve this problem and obtain the value function and optimal (equilibrium) strategies in closed form. We show that the optimal dividend-capital injection strategy is a barrier strategy. The optimal proportion of risk ceded to the reinsurer and the deviation of the worst-case model from the reference model are decreasing with respect to the aggregate reserve level. Finally, numerical examples are presented to show the impact of the model parameters and ambiguity aversion on the optimal strategies.

Paper Structure

This paper contains 18 sections, 11 theorems, 91 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Problem Formulation
  4. Verification Theorem
  5. Analytical Solutions
  6. The Case of $0<\frac{\rho}{1+\beta_2\frac{\gamma_1}{1-\gamma_1}}\leq \frac{\mu_1 /\sigma_1}{\mu_2/\sigma_2}\leq \frac{1+\beta_1\frac{\gamma_1}{1-\gamma_1}}{\rho}$
  7. Remaining cases
  8. The Case of No Model Uncertainty
  9. Numerical Illustrations
  10. Proof of Main Results
  11. Proof of Lemma \ref{['lemma: gamma1 suff cond']}
  12. Proof of Theorem \ref{['theorem 1']}
  13. Proof of Theorem \ref{['theorem 2']}
  14. Proof of Theorem \ref{['theorem 3']}
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 2.4

Let $X^{u}(t):=X^{u}_1(t)+X^{u}_2(t)$. Suppose there exist: such that Then, Moreover, $u^*$ and $\theta^*$ are the optimal strategies; that is, $(u^*,\theta^*)$ is the Nash equilibrium of the zero-sum stochastic differential game.

Figures (11)

  • Figure 1: Regions for dividend payout and capital injection decisions.
  • Figure 2: Value function $g(x)$, optimal reinsurance strategies $\bar{\pi}_1(x), \bar{\pi}_2(x)$, and optimal distortion strategies $\bar{\theta}_1(x), \bar{\theta}_2(x)$ as a function of the (total) reserve $x$ when $\rho = 0.60$. This parameter configuration, corresponding to Theorem \ref{['theorem 1']}, yields $w_0 = 0.67 < b^* = 1.95$.
  • Figure 3: Value function $g(x)$, optimal reinsurance strategies $\bar{\pi}_1(x), \bar{\pi}_2(x)$, and optimal distortion strategies $\bar{\theta}_1(x), \bar{\theta}_2(x)$ as a function of the (total) reserve $x$ when $\rho = 0$. This parameter configuration, corresponding to Theorem \ref{['theorem 2']}, yields $w_0 = 0.50 < b^* = 1.86$.
  • Figure 4: Reinsurance threshold $w_0$ and values of $w_1$ and $w_2$ as a function of $\beta_i$ when $\beta_{3-i} = 5$ is fixed. In Figure \ref{['fig:beta2 fixed']} (resp. Figure \ref{['fig:beta1 fixed']}), the region A to the left of the dotted graph represents the values of $\beta_1$ (resp. $\beta_2$) for which $w_2 > w_1$ (resp. $w_1 > w_2$). The region B to the right of the dotted graph represents values of $\beta_1$ (resp. $\beta_2$) for which $w_1 > w_2$ (resp. $w_2 > w_1$). When $\beta_1$ is variable, the dot-dashed graph indicates the upper bound on $\beta_1$ up to which $w_0$ exists.
  • Figure 5: The barrier $b^*$ corresponding to the optimal dividend-capital injection strategy and the ratio $w_0 / b^*$ as a function of $\beta_i$ when $\beta_{3-i} = 5$ is fixed, for $i=1,2$. The legend indicates which parameter is variable. When $\beta_1$ is variable, the dot-dashed graph indicates the upper bound on $\beta_1$ up to which either $w_0$ exists or $b^*$ is nonzero.
  • ...and 6 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Verification Theorem
  • proof
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • ...and 18 more