  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.
