Experience Rating and Credibility

TL;DR

Bühlmann reviews seven papers on credibility and experience rating, arguing that experience rating gradually adjusts an initial rate during coverage and that Bayesian and credibility methods can handle heterogeneous risks. He also offers an equilibrium justification for the posterior mean and derives a distribution-free credibility formula.

Key findings

• Experience rating is described as the gradual alteration of an initially fixed rate during the risk-coverage period, implemented through devices such as profit participation, no-claims discounts, bonus-malus scales, sliding-scale premiums or commissions, and reinsurance arrangements.
• Among the reviewed papers, Bohman finds that maximum business volume is generally not reached at the lowest possible premiums; Benckert tests tariff-class adequacy using a lognormal distribution for individual claim amounts; Johansen treats risk parameters as random variables to identify accident-prone risks; and Lundberg proves consistency of a Bayes estimate from a Poisson process while showing that estimates based on time of absence are not consistent.
• Franckx derives the credibility formula from least squares without assuming a distribution for individual risks, though still depending on the prior distribution of risk parameters; Derron computes credibility improvement for a Pareto-type distribution when the largest claim is excluded.
• Bühlmann proposes equilibrium across experience-rated subclasses as a justification for using the posterior mean, and derives a credibility relation with a factor of the form \(b = n/(n+k)\), noting that it makes no assumption about the individual-risk distribution or the structural distribution.

Methods

• The text is a discussion/report on seven submitted papers plus the author’s own technical remarks; it does not present an empirical dataset or sample.
• It reviews sequential statistical methods, Bayes procedures, and credibility techniques, then develops mathematical arguments using conditional expectation, least-squares linear approximation, and an equilibrium postulate.
• The main analysis is conceptual and mathematical, including variance-component expressions and a distribution-free credibility derivation.

Limitations

• The paper states that classical sequential methods such as Wald’s Sequential Probability Ratio Test cannot go beyond testing an entire tariff class and fail for the individual-risk case; it also notes that only one submitted paper specifically addresses credibility techniques, and that direct estimation of \(E[\sigma^2(\theta)]\) and \(\mathrm{Var}[\mu(\theta)]\) from a priori observations has not yet been attempted.
• It observes that Bichsel treats random changes in individual risks over time, but suggests that trend-type changes remain untreated.
• Evident from the text: the work is a commentary/report rather than an empirical study, and some formulas are incomplete in the provided text.