Pricing foreseeable and unforeseeable risks in insurance portfolios

TL;DR

This paper introduces a Bayesian method for pricing insurance that accounts for both foreseeable and unforeseeable risks by modeling risk heterogeneity with a mixed Poisson process and heavy-tailed distributions, improving premium responsiveness.

Contribution

It presents a novel Bayesian approach using mixed priors and heavy-tailed distributions to better capture and price unforeseen risks in insurance portfolios.

Findings

Bayesian premiums are more reactive to claim trends.

Heavy-tailed distributions effectively model unforeseeable risks.

Mixture of posterior means estimates premiums considering risk heterogeneity.

Abstract

In this manuscript we propose a method for pricing insurance products that cover not only traditional risks, but also unforeseen ones. By considering the Poisson process parameter to be a mixed random variable, we capture the heterogeneity of foreseeable and unforeseeable risks. To illustrate, we estimate the weights for the two risk streams for a real dataset from a Portuguese insurer. To calculate the premium, we set the frequency and severity as distributions that belong to the linear exponential family. Under a Bayesian setup , we show that when working with a finite mixture of conjugate priors, the premium can be estimated by a mixture of posterior means, with updated parameters, depending on claim histories. We emphasise the riskiness of the unforeseeable trend, by choosing heavy-tailed distributions. After estimating distribution parameters involved using the Expectation-Maximization algorithm, we found that Bayesian premiums derived are more reactive to claim trends than traditional ones.