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Optimal Ratcheting of Dividends with Irreversible Reinsurance

Tim J. Boonen, Engel John C. Dela Vega

TL;DR

The paper addresses optimal dividend payouts under a ratcheting constraint and irreversible proportional reinsurance in a diffusion-approximation insurance model. It develops a Hamilton–Jacobi–Bellman framework and proves the value function is the unique viscosity solution, from which threshold-type policies emerge. For finite reinsurance and dividend sets, it provides a backward-recursive construction of optimal thresholds y^* and z^* via scale-function arguments and a detailed verification argument. Numerical illustrations show dividends are typically increased before expanding reinsurance, offering practical guidance for insurers on implementing finite-set ratcheting strategies.

Abstract

This paper considers an insurance company that faces two key constraints: a ratcheting dividend constraint and an irreversible reinsurance constraint. The company allocates part of its reserve to pay dividends to its shareholders while strategically purchasing reinsurance for its claims. The ratcheting dividend constraint ensures that dividend cuts are prohibited at any time. The irreversible reinsurance constraint ensures that reinsurance contracts cannot be prematurely terminated or sold to external entities. The dividend rate and reinsurance level are modeled as nondecreasing processes, thereby satisfying the constraints. Claims are modeled using a Brownian risk model. The main objective is to maximize the cumulative expected discounted dividend payouts until the time of ruin. The reinsurance and dividend levels are restricted to a finite set. The optimal value function is shown to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. A threshold strategy is constructed and shown to be optimal. Finally, numerical examples are presented to illustrate the optimality conditions and optimal strategies.

Optimal Ratcheting of Dividends with Irreversible Reinsurance

TL;DR

The paper addresses optimal dividend payouts under a ratcheting constraint and irreversible proportional reinsurance in a diffusion-approximation insurance model. It develops a Hamilton–Jacobi–Bellman framework and proves the value function is the unique viscosity solution, from which threshold-type policies emerge. For finite reinsurance and dividend sets, it provides a backward-recursive construction of optimal thresholds y^* and z^* via scale-function arguments and a detailed verification argument. Numerical illustrations show dividends are typically increased before expanding reinsurance, offering practical guidance for insurers on implementing finite-set ratcheting strategies.

Abstract

This paper considers an insurance company that faces two key constraints: a ratcheting dividend constraint and an irreversible reinsurance constraint. The company allocates part of its reserve to pay dividends to its shareholders while strategically purchasing reinsurance for its claims. The ratcheting dividend constraint ensures that dividend cuts are prohibited at any time. The irreversible reinsurance constraint ensures that reinsurance contracts cannot be prematurely terminated or sold to external entities. The dividend rate and reinsurance level are modeled as nondecreasing processes, thereby satisfying the constraints. Claims are modeled using a Brownian risk model. The main objective is to maximize the cumulative expected discounted dividend payouts until the time of ruin. The reinsurance and dividend levels are restricted to a finite set. The optimal value function is shown to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. A threshold strategy is constructed and shown to be optimal. Finally, numerical examples are presented to illustrate the optimality conditions and optimal strategies.
Paper Structure (11 sections, 11 theorems, 133 equations, 1 figure, 1 table)

This paper contains 11 sections, 11 theorems, 133 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Main Results
  4. Singleton Case
  5. Finite Set Case
  6. Derivation of the Optimal Strategy
  7. Numerical Illustrations
  8. Conclusion
  9. Proofs of Section \ref{['section:model']}, and Intermediate Lemmas and Propositions
  10. Proofs of the Results in Section \ref{['section:hjb']}
  11. Value Function Derivation Using Scale Functions

Key Result

Proposition 2.1

The optimal value function $V(x,a,c)$ is bounded above by $\frac{\overline{c}}{q}$, nondecreasing in $x$ and $a$, and nonincreasing in $c$.

Figures (1)

  • Figure 1: Examples 1, 2, and 3

Theorems & Definitions (31)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 3.1
  • Definition 3.1
  • Theorem 3.2
  • ...and 21 more