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Catastrophe Insurance: An Adaptive Robust Optimization Approach

Dimitris Bertsimas, Cynthia Zeng

TL;DR

The paper tackles the challenge of pricing catastrophe insurance under increasing climate-related uncertainty. It proposes an Adaptive Robust Optimization (ARO) framework that blends Central Limit Theorem–based and machine learning–based risk sets to price premiums while maintaining solvency and smoothing premium evolution. By applying the approach to the NFIP using historical NFIP data and ML-derived risk forecasts, the authors demonstrate that RO2 and ARO can reduce insolvencies and deliver favorable surplus–deficit profiles compared with historical and CMA baselines, with ARO offering robust performance and smoother premium adjustments under high uncertainty. The framework is shown to be flexible and generalizable to other catastrophic risks such as wildfires and droughts, supporting policymakers in building resilience through robust, data-informed pricing.

Abstract

The escalating frequency and severity of natural disasters, exacerbated by climate change, underscore the critical role of insurance in facilitating recovery and promoting investments in risk reduction. This work introduces a novel Adaptive Robust Optimization (ARO) framework tailored for the calculation of catastrophe insurance premiums, with a case study applied to the United States National Flood Insurance Program (NFIP). To the best of our knowledge, it is the first time an ARO approach has been applied to for disaster insurance pricing. Our methodology is designed to protect against both historical and emerging risks, the latter predicted by machine learning models, thus directly incorporating amplified risks induced by climate change. Using the US flood insurance data as a case study, optimization models demonstrate effectiveness in covering losses and produce surpluses, with a smooth balance transition through parameter fine-tuning. Among tested optimization models, results show ARO models with conservative parameter values achieving low number of insolvent states with the least insurance premium charged. Overall, optimization frameworks offer versatility and generalizability, making it adaptable to a variety of natural disaster scenarios, such as wildfires, droughts, etc. This work not only advances the field of insurance premium modeling but also serves as a vital tool for policymakers and stakeholders in building resilience to the growing risks of natural catastrophes.

Catastrophe Insurance: An Adaptive Robust Optimization Approach

TL;DR

The paper tackles the challenge of pricing catastrophe insurance under increasing climate-related uncertainty. It proposes an Adaptive Robust Optimization (ARO) framework that blends Central Limit Theorem–based and machine learning–based risk sets to price premiums while maintaining solvency and smoothing premium evolution. By applying the approach to the NFIP using historical NFIP data and ML-derived risk forecasts, the authors demonstrate that RO2 and ARO can reduce insolvencies and deliver favorable surplus–deficit profiles compared with historical and CMA baselines, with ARO offering robust performance and smoother premium adjustments under high uncertainty. The framework is shown to be flexible and generalizable to other catastrophic risks such as wildfires and droughts, supporting policymakers in building resilience through robust, data-informed pricing.

Abstract

The escalating frequency and severity of natural disasters, exacerbated by climate change, underscore the critical role of insurance in facilitating recovery and promoting investments in risk reduction. This work introduces a novel Adaptive Robust Optimization (ARO) framework tailored for the calculation of catastrophe insurance premiums, with a case study applied to the United States National Flood Insurance Program (NFIP). To the best of our knowledge, it is the first time an ARO approach has been applied to for disaster insurance pricing. Our methodology is designed to protect against both historical and emerging risks, the latter predicted by machine learning models, thus directly incorporating amplified risks induced by climate change. Using the US flood insurance data as a case study, optimization models demonstrate effectiveness in covering losses and produce surpluses, with a smooth balance transition through parameter fine-tuning. Among tested optimization models, results show ARO models with conservative parameter values achieving low number of insolvent states with the least insurance premium charged. Overall, optimization frameworks offer versatility and generalizability, making it adaptable to a variety of natural disaster scenarios, such as wildfires, droughts, etc. This work not only advances the field of insurance premium modeling but also serves as a vital tool for policymakers and stakeholders in building resilience to the growing risks of natural catastrophes.
Paper Structure (29 sections, 1 theorem, 57 equations, 8 figures, 6 tables)

This paper contains 29 sections, 1 theorem, 57 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Optimization Framework for Catastrophe Insurance
  4. Robust Optimization
  5. Nominal Formulation
  6. Robust Optimization Formulation
  7. Uncertainty Set from Central Limit Theorem
  8. Uncertainty Set from Machine Learning Risk Models
  9. The Robust Counterpart
  10. Choices of demand damping function f(p)
  11. Adaptive Robust Optimization Formulation
  12. Case Study for US National Flood Insurance
  13. Data
  14. Optimization Model Parameter Estimation
  15. Machine Learning Model
  16. ...and 14 more sections

Key Result

Proposition 1

The overall min-max problem is equivalent to where

Figures (8)

  • Figure 1: Number of major disasters globally since 1900, maintained by the EM-DAT database emdat. A disaster is defined as an event which overwhelms local capacity, necessitating a request to the national or international level for external assistance. Disasters include: flood, storm, earthquake, drought, landslide, extreme temperature, wildfire, volcanic activity, mass movement (dry), glacial lake outburst, fog, etc.
  • Figure 2: Surplus (or loss) computed during 2012 to 2022 across all states when vary different levels of $\gamma_2$. Two dotted lines demonstrate the level of surplus from two baseline models: historical surplus calculated from the actual premiums collected, cma surplus is computed using the cumulative moving average.
  • Figure 3: The scatter plot visualizes the efficient frontier, showing how different values of $\gamma_2$ affect the number of insolvent states (x-axis) and the total surplus (or deficit) (y-axis) computed as the total premium charged minus actual loss over the testing period. Note that CMA and Hist are plotted as static points because their values do not change with varying $\gamma_2$ values.
  • Figure 4: (a) Demand damping estimation for Louisiana state (LA).
  • Figure 5: (b) Demand damping estimation for New York state (NY).
  • ...and 3 more figures

Theorems & Definitions (2)

  • Proposition
  • proof