Effective a Posteriori Ratemaking with Large Insurance Portfolios via Surrogate Modeling

A posteriori ratemaking in insurance uses a Bayesian credibility model to upgrade the current premiums of a contract by taking into account policyholders' attributes and their claim history. Most data-driven models used for this task are mathematically intractable, and premiums must be then obtained through numerical methods such as simulation such MCMC. However, these methods can be computationally expensive and prohibitive for large portfolios when applied at the policyholder level. Additionally, these computations become ``black-box" procedures as there is no expression showing how the claim history of policyholders is used to upgrade their premiums. To address these challenges, this paper proposes the use of a surrogate modeling approach to inexpensively derive a closed-form expression for computing the Bayesian credibility premiums for any given model. As a part of the methodology, the paper introduces the ``credibility index", which is a summary statistic of a policyholder's claim history that serves as the main input of the surrogate model and that is sufficient for several distribution families, including the exponential dispersion family. As a result, the computational burden of a posteriori ratemaking for large portfolios is therefore reduced through the direct evaluation of the closed-form expression, which additionally can provide a transparent and interpretable way of computing Bayesian premiums.