The Use of Collateral Data in Credibility Theory: A Hierarchical Model

In classical credibility theory, a linearized Bayesian forecast of the fair premium for an individual risk contract is made using prior estimates of the collective fair premium and individual experience data. However, collateral data from other contracts in the same portfolio is not used, in spite of intuitive feelings that this data would contain additional evidence about the quality of the risk collective from which the portfolio was drawn. By using a hierarchical model, one makes the individual risk parameters exchangeable, in the sense of de Finetti, and a modified credibility formula is obtained which uses the collateral data in an intuitively satisfying manner. The homogeneous formula of Buehlmann and Straub is obtained as a limiting case when the hyperprior distribution becomes "diffuse"