Pareto-optimal reinsurance under dependence uncertainty

Abstract: This paper studies Pareto-optimal reinsurance design in a monopolistic market with multiple primary insurers and a single reinsurer, all with heterogeneous risk preferences. The risk preferences are characterized by a family of risk measures, called Range Value-at-Risk (RVaR), which includes both Value-at-Risk (VaR) and Expected Shortfall (ES) as special cases. Recognizing the practical difficulty of accurately estimating the dependence structure among the insurers' losses, we adopt a robust optimization approach that assumes the marginal distributions are known while leaving the dependence structure unspecified. We provide a complete characterization of optimal indemnity schedules under the worst-case scenario, showing that the infinite-dimensional optimization problem can be reduced to a tractable finite-dimensional problem involving only two or three parameters for each indemnity function. Additionally, for independent and identically distributed risks, we exploit the argument of asymptotic normality to derive optimal two-parameter layer contracts. Finally, numerical applications are considered in a two-insurer setting to illustrate the influence of the dependence structures and heterogeneous risk tolerances on optimal strategies and the corresponding risk evaluation.

• The paper establishes that robust Pareto-optimal reinsurance treaties can be characterized via a finite-parameter search over indemnity functions.
• It demonstrates that layered contracts, including stop-loss and hybrid proportional-excess indemnities, achieve optimality under various risk measures.
• Numerical results reveal that dependence uncertainty significantly influences risk allocation, highlighting the critical balance between contract design and reinsurance value.

Pareto-optimal Multilateral Reinsurance under Dependence Uncertainty

Problem Formulation and Motivation

The paper analyzes the design of Pareto-optimal reinsurance treaties in monopolistic markets composed of multiple primary insurers and a single reinsurer, recognizing practical estimation difficulties regarding the dependence structure of insurers' losses. Heterogeneous risk preferences among market participants are modeled using Range Value-at-Risk (RVaR), subsuming both Value-at-Risk (VaR) and Expected Shortfall (ES) as limiting cases. The core problem is to characterize reinsurance agreements such that no participant's evaluated risk can be reduced without increasing that of another, subject to robust modeling constraints: marginal loss distributions are assumed known, while joint dependence is left unconstrained.

The approach leverages a robust optimization paradigm, transforming the inherently infinite-dimensional problem into a tractable finite-dimensional parameterization of indemnity functions. The framework incorporates uncertain risk aggregation—minimizing system-wide risk quantified as a weighted sum of participants’ RVaR exposures—under the worst-case distributional scenarios for dependence.

Characterization of Pareto-optimal Reinsurance Contracts

The main theoretical contribution is proof that the robust Pareto-optimal contract for the entire multilayer system is equivalent to the solution of a single aggregate risk minimization problem. Formally, this optimization considers indemnity schedules $f_i$ forming the ceded loss functions, and premiums $\pi_i$ specified exogenously. The system objective function is

$$V(\mathbf{f}) = \sum_{i=1}^n \mathrm{RVaR}_{\beta_i, \alpha_i}(X_i - f_i(X_i)) + \sup_{\mathbf{X} \in \mathcal{E}_n(\mathbf{F})} \mathrm{RVaR}_{\beta, \alpha}\left(\sum_{i=1}^n f_i(X_i)\right)$$

The paper demonstrates that under dependence uncertainty, layered contracts—in particular, classically constructed layer ($g_{a,b}$) and hybrid proportional-excess ( $r_{a,b,c}$ ) indemnities—are sufficient to attain optimality. For ES criteria ($\beta=0$), the optimal contract for each cedant is a two-parameter stop-loss structure; for RVaR under convex tail assumptions, tractable three-parameter solutions ($r_{a,b,c}$) are shown to be optimal, enabling finite search over parameter spaces. These results are substantially more constructive than previously published works relying only on implicit contract forms.

Explicit characterization is provided (Theorem 1), showing worst-case robust optimization reduces to a finite parameter search, and existence and computability of optimal indemnity function parameters is established.

Special Case: VaR-based Optimization

In the specific case where risk assessment is restricted to VaR, the paper gives parameterizations of optimal schedules for both two-insurer settings and certain classes of marginal distributions (convex/concave in the tail). Through the use of Makarov-type bounds for $n=2$, it is shown that the worst-case VaR can be exactly computed as an infimum over feasible allocations. This framework allows for transparent computation and comparison across i.i.d., comonotonic, and ambiguous dependence regimes.

Asymptotic Results for i.i.d. Risks

For large portfolios it is shown, leveraging asymptotic normality, that aggregate risk under i.i.d. scenarios converges to layered contracts, which coincide with classic reinsurance solutions under mean-standard deviation premium principles. In these regimes, the reduction from infinite- to finite-dimensional optimization persists, and explicit formulas for optimal retention levels and layer limits are derived; as the number of insureds tends to infinity, optimal attachment points vanish, reinforcing the predictability afforded by risk pooling.

Numerical Results and Practical Implications

Numerical simulations demonstrate that worst-case system risk under ambiguous dependence is not necessarily attained under comonotonic structures: in some parameter regimes, the i.i.d. case yields higher VaR than the comonotonic benchmark. Sensitivity analysis with different confidence levels and distributional configurations illustrates the nuanced relationship between reinsurance benefit and both the reinsurer's and cedants' risk assessment thresholds.

Additionally, it is shown that with sufficiently conservative (high-alpha) risk measures on the part of insurers, the value of reinsurance is diminished, and optimal contracts may not outperform a no-reinsurance benchmark. The structural impact of marginal loss distributions is highlighted: convex/concave tail shapes limit the functional form and region of effectiveness for layer contracts.

Theoretical and Practical Implications

The findings provide rigorous foundations for robust multilateral contract design under model uncertainty. Practically, the reduction to finite-dimensional optimization facilitates real-world implementation in settings where joint loss distributions are difficult to estimate and regulatory or contractual requirements enforce risk-tolerance heterogeneity among insurers.

This work advances robust insurance contract theory, showing that computational tractability and economic interpretability can be retained under minimal distributional assumptions, and it supplies practitioners with explicit forms of indemnity functions readily suited for algorithmic optimization.

Future Directions

Natural extensions include relaxing the assumption of fixed premiums to study endogenously priced treaties, inclusion of regulatory constraints or default risk, and further generalization of robust risk measures beyond the RVaR family. Exploring dynamic contracts where indemnity schedules are periodically updated with evolving portfolio data also presents a promising avenue for future research.

Conclusion

The paper provides a thorough treatment of Pareto-optimal reinsurance design under deep model uncertainty, supplying explicit, economically interpretable, and computationally feasible solutions for robust risk sharing. Its results bridge theoretical optimality and practical feasibility, significantly contributing to both the actuarial science literature and the operational practice of insurance risk management.

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Reference:

"Pareto-optimal reinsurance under dependence uncertainty" (2512.11430)