Monopoly Pricing of Weather Index Insurance

Paper: arXiv 2512.01623

Authors: Tim J. Boonen, Wenyuan Li, Zixiao Quan

Abstract

This study models the monopoly pricing of weather index insurance as a Bowley-type sequential game involving a profit-maximizing insurer (leader) and a farmer (follower). The farmer chooses an insurance payoff to minimize a convex distortion risk measure, while the insurer anticipates this best response and selects a premium principle and its parameters to maximize profit net of administrative costs. For the insurer, we adopt three different premium-principle parameterizations: (i) an expected premium with a single risk-loading factor, (ii) a two-parameter distortion premium based on a power transform, and (iii) a fully flexible pricing kernel drawn from the general Choquet integral representation with nondecreasing distortions. For the farmer, we model index payoffs using neural networks and compare solutions under fully connected architectures with those under convolutional neural networks (CNNs). We solve the game using a penalized bilevel programming algorithm that employs a function-value-gap penalty and delivers convergence guarantees without requiring the lower-level objective to be strongly convex. Based on Iowa’s soybean yields and high-dimensional PRISM weather data, we find that CNN-based designs yield smoother, less noisy payoffs that reduce basis risk and push insurer profits closer to indemnity insurance levels. Moreover, expanding pricing flexibility from a single loading to a two-parameter distortion premium, and ultimately to a flexible pricing kernel, systematically increases equilibrium profits.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous game-theoretic framework applying Bowley solutions to index insurance pricing via neural networks and bilevel optimization. It combines sophisticated mathematical modeling (distortion risk measures, non-strongly convex bilevel algorithms) with solid empirical evaluation using high-dimensional PRISM weather and Iowa crop yield data.

Research Flowchart

  flowchart TD
    A["Research Question: How to optimize monopoly pricing <br>and design of weather index insurance?"] --> B["Data Inputs: Iowa Soybean Yields <br>& High-Dimensional PRISM Weather Data"]
    
    B --> C["Model Formulation: Bowley Sequential Game<br>• Farmer: Minimizes Distortion Risk (NN / CNN)<br>• Insurer: Maximizes Profit (3 Pricing Kernels)"]
    
    C --> D["Computational Optimization:<br>Penalized Bilevel Programming Algorithm<br>with Function-Value-Gap Penalty"]
    
    D --> E["Outcome 1: CNN Payoff Design<br>Yields smoother payoffs, reduces basis risk, <br>& elevates profits near indemnity levels"]
    
    D --> F["Outcome 2: Flexible Pricing Kernels<br>Expanding from single loading to Choquet kernel <br>systematically increases equilibrium profits"]

Cited References (34)

1. Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes (2013)
2. On penalty-based bilevel gradient descent method (2025) — link
3. Managing weather risk with a neural network-based index insurance (2024)
4. Bowley solution under the reinsurer’s default risk (2024)
5. Risk-adjusted B owley reinsurance under distorted probabilities (2019)
6. Investigating bi-level optimization for learning and vision from a unified perspective: A survey and beyond (2021)
7. Is there a viable market for area-based crop insurance? (2007)
8. Relaxing heteroscedasticity assumptions in area-yield crop insurance rating (2011)

0 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.