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Optimal Reinsurance under Endogenous Default and Background Risk

Zongxia Liang, Zhaojie Ren, Bin Zou

Abstract

This paper studies an optimal reinsurance problem for a utility-maximizing insurer, subject to the reinsurer's endogenous default and background risk. An endogenous default occurs when the insurer's contractual indemnity exceeds the reinsurer's available reserve, which is random due to the background risk. We obtain an analytical solution to the optimal contract for two types of reinsurance contracts, differentiated by whether their indemnity functions depend on the reinsurer's background risk. The results shed light on the joint effect of the reinsurer's default and background risk on the insurer's reinsurance demand.

Optimal Reinsurance under Endogenous Default and Background Risk

Abstract

This paper studies an optimal reinsurance problem for a utility-maximizing insurer, subject to the reinsurer's endogenous default and background risk. An endogenous default occurs when the insurer's contractual indemnity exceeds the reinsurer's available reserve, which is random due to the background risk. We obtain an analytical solution to the optimal contract for two types of reinsurance contracts, differentiated by whether their indemnity functions depend on the reinsurer's background risk. The results shed light on the joint effect of the reinsurer's default and background risk on the insurer's reinsurance demand.
Paper Structure (26 sections, 16 theorems, 109 equations, 5 figures)

This paper contains 26 sections, 16 theorems, 109 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Summary and Contributions of the Paper
  4. Related Literature
  5. Model
  6. Optimal Loss- and Background-Risk-Dependent Indemnities
  7. Optimal Contract
  8. Comparative Statics
  9. Optimal Loss-Dependent Indemnities
  10. Conclusion
  11. Proofs of Section \ref{['sec:main']}
  12. Proof of Lemma \ref{['lem:a_bar']}
  13. Proof of Proposition \ref{['prop:a_bar']}
  14. Proof of Theorem \ref{['thm:step_one']}
  15. Proof of Corollary \ref{['cor:unique_d']}
  16. ...and 11 more sections

Key Result

Proposition 2.1

If $S \le 0$ almost surely (i.e., $\mathbb{P}(S \le 0) = 1$), then the optimal strategy to Problem prob:main is no reinsurance with $I^* \equiv 0$.

Figures (5)

  • Figure 1: Optimal contract $I^*_a$ in \ref{['eq:I_op']}.
  • Figure 2: Optimal premium $a^*$ (left) and deductible $d^*$ (right) with respect to $\eta$
  • Figure 3: Optimal premium $a^*$ (left) and deductible $d^*$ (right) with respect to $\gamma$
  • Figure 4: Optimal contract $I^*_S$ in \ref{['eq:opI']} when $N = 2$.
  • Figure 5: Indemnity $I_1$ in \ref{['eq:I1_def']} on $x \in (x_{i-1},x_i]$.

Theorems & Definitions (46)

  • Remark 2.1
  • Proposition 2.1
  • proof
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 36 more