Monopoly Pricing of Weather Index Insurance

TL;DR

This paper models monopoly pricing of weather index insurance as a sequential game, exploring different premium schemes and neural network-based payoff modeling, demonstrating increased profits with flexible pricing.

Contribution

It introduces a bilevel programming approach for monopoly pricing with neural network payoffs and compares various premium parameterizations for improved profit outcomes.

Findings

CNN-based payoffs reduce basis risk and noise.

Flexible premium parameterizations increase profits.

Neural networks effectively model complex index payoffs.

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.