On the risk of credibility premium rules

A discrete-time risk process is considered where the full distribution of the claim size X is not completely known to the insurance company. Rather, it assumes that the distribution of X given $Z=\zeta $Z=ζ is $F_\zeta $Fζ where Z is some structural random variable for which a prior is available. The main emphasis of the paper is the unconditional ruin probability $\psi (u) $ψ(u) in this setting where the premium is either updated according to incoming information about the claim distribution or computed by the expected value principle. This is in turn studied via the conditional ruin probability $\psi _\zeta (u) $ψζ(u), for which large deviations estimates are available. Rigorous proofs are given only for the case of the $F_\zeta $Fζ forming a scale parameter family, including the classical case of gamma claims with a gamma prior. However, the analysis readily suggests what should be the behaviour of $\psi (u) $ψ(u) in different models for the claims.