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

Pricing foreseeable and unforeseeable risks in insurance portfolios