Modern Computational Methods in Reinsurance Optimization: From Simulated Annealing to Quantum Branch & Bound

TL;DR

This paper tackles catastrophe excess-of-loss reinsurance contract optimization under multi-faceted business constraints. It develops a practical, data-driven local-search framework using simulated annealing, supported by preprocessing and caching to rapidly evaluate contract performance, and analyzes a forward-looking quantum-accelerated branch-and-bound approach on a simplified problem with application-specific bounds. Key contributions include an efficient Markov-chain optimization pipeline, an iterative profit-and-risk bound for pruning in branch-and-bound, and a realistic assessment of quantum practicality, data-loading costs, and potential crossovers. The work provides practitioners with a robust numerical framework for today and a concrete roadmap for future quantum enhancements in reinsurance optimization.

Abstract

We propose and implement modern computational methods to enhance catastrophe excess-of-loss reinsurance contracts in practice. The underlying optimization problem involves attachment points, limits, and reinstatement clauses, and the objective is to maximize the expected profit while considering risk measures and regulatory constraints. We study the problem formulation, paving the way for practitioners, for two very different approaches: A local search optimizer using simulated annealing, which handles realistic constraints, and a branch & bound approach exploring the potential of a future speedup via quantum branch & bound. On the one hand, local search effectively generates contract structures within several constraints, proving useful for complex treaties that have multiple local optima. On the other hand, although our branch & bound formulation only confirms that solving the full problem with a future quantum computer would require a stronger, less expensive bound and substantial hardware improvements, we believe that the designed application-specific bound is sufficiently strong to serve as a basis for further works. Concisely, we provide insurance practitioners with a robust numerical framework for contract optimization that handles realistic constraints today, as well as an outlook and initial steps towards an approach which could leverage quantum computers in the future.

Figures (8)

• Figure 1: (a) Layered reinsurance profile: Layer 1 has attachment 3MM and limit 5MM, Layer 2 has attachment 8MM and limit 8MM. (b) Distribution of 5MM from a single loss event (red): The first 3MM in losses are retained by the cedant. The remaining 2MM are ceded to the reinsurer of layer 1, which is now 40% exhausted (all of which could be reinstated, if designated in the contract). (c) Assuming this is the first applicable loss in the treaty period, the distribution of 10MM in losses between insurer and reinsurers: The first 3MM in losses are retained, the next 5MM are ceded to the reinsurer of layer 1, exhausting that layer. The remaining 2MM are ceded to the reinsurer of layer 2. In a typical case, if not reinstated, layer 1 will not cover any future claims and layer 2 still has 8MM - 2MM = 6MM available for future claims that exceed the attachment of layer 2 (still 8MM). I.e., it is now equivalent to a 6̀MM in excess of 8MM" layer.
• Figure 2: (a) Multi-attachment reinsurance profile: Layer 1 has two peril subgroups ($s_{1}$ & $s_{2}$) with respective attachments of 3MM & 1MM and limit 5MM. Layer 2 has respective attachments of 8MM & 6MM for $s_{1}$ & $s_{2}$. Neither layer has a reinstatement clause. (b) Example: Distribution of 8MM loss 1 to perils $s_{1}$ (red) and 7MM loss 2 to perils $s_{2}$ (orange) between insurer and reinsurers: First, loss 1 occurs and the first 3MM of losses are retained by the insurer, then the next 5MM are ceded to the reinsurer of Layer 1. Secondly, loss 2 occurs and the first 1MM in losses are retained by the insurer. Layer 1 has been exhausted by loss 1 so cannot be used -- a further 5 MM which is not recovered. The remaining 1MM in losses between 6MM and 7MM are ceded to the reinsurer of Layer 2.
• Figure 3: Visualization of the different Markov chain moves from an initial state (center: one tower with two layers), and their reversal (to the center). (i) Creation and removal of a subgroup. (ii) Removal and addition of a layer. (iii) Splitting and joining of a layer. (iv) Adjustment of layer boundary: Layer 1 grows while Layer 2 shrinks. (v) Adjustment of layer boundary with shift above: Layer 2 retains the same limit. (vi) Subgroup shift from (i).
• Figure 4: While substantial improvements in the objective function can typically be found early on (depending on the starting point), these increments become smaller as the optimizer approaches the best solution found. While simulated annealing is a heuristic and cannot provide a guarantee about optimality, we can increase the probability of finding a solution that is close to optimal by (i) ensuring sampling in each Markov chain is long enough to reach this region of small increments and (ii) running several Markov chains in parallel (with different random seeds).
• Figure 5: Rapid objective function evaluation is key to a numerical optimization approach. When computed directly from the full event data set, this takes several seconds for each evaluation. Dataset normalization and cumulative sum representation reduce this time to below one second. With caching, a capable database backend and multi-threading, we can evaluate more than 80 contracts per second (Ryzen 9 5950X, 96GB ram).