Credibility Theory

This survey of actuarial credibility theory traces its origins, describes its evolutionary history, gives an account of its main issues and results, and assesses its merits with a view to related work in statistical science. Key-words: Experience rating, limited fluctuation theory, greatest accuracy theory, Bayes, linear Bayes, empirical Bayes, hierarchical models, Hilbert space methods, recursive computation, history of actuarial science. A. Credility what it was and what it is. In actuarial parlance the term credibility was originally attached to experience rating formulas that were convex combinations (weighted averages) of individual and class estimates of the individual risk premium. Credibility theory, thus, was the branch of insurance mathematics that explored model-based principles for construction of such formulas. The development of the theory brought it far beyond the original scope so that in today’s usage credibility covers more broadly linear estimation and prediction in latent variable models. B. The origins. The advent of credibility dates back to Whitney [46], who in 1918 addressed the problem of assessing the risk premium m, defined as the expected claims expenses per unit of risk exposed, for an individual risk selected from a portfolio (class) of similar risks. Advocating the combined use of individual risk experience and class risk experience, he proposed that the premium rate be a weighted average of the form m̄ = z m̂ + (1− z)μ , (1) where m̂ is the observed mean claim amount per unit of risk exposed for the individual contract and μ is the corresponding overall mean in the insurance portfolio. Whitney viewed the risk premium as a random variable. In the language of modern credibility theory, it is a function m(Θ) of a random element Θ representing the unobservable characteristics of the individual risk. The random nature of Θ expresses the notion of heterogeneity; the individual risk is a random selection from a portfolio of similar but not identical risks, and the distribution of Θ describes the variation of individual risk characteristics across the portfolio.