Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Credibility Models for Claim Counts 123
insurance. In an empirical Bayesian setting, the prediction is derived from the expectation
of a random effect with respect to a posterior distribution taking into account the history of
the individual.
3.1.5 Linear Credibility
Bayesian statistics offer an intellectually acceptable approach to greatest accuracy credibility
theory. Nevertheless, practical applications involve numerical methods to perform integration
with respect to a posteriori distribution, making more elementary approaches desirable (at
least to get an easy-to-compute approximation of the result). In that respect, linear credibility
formulas are especially useful. Basically, the actuary still resorts to a quadratic loss function
but the shape of the credibility predictor is constrained ex ante to be linear.
3.1.6 Financial Equilibrium
When the insurer uses past claims history to reevaluate the amount of premium to be charged
to the policyholders, the increases and decreases granted to the policyholders in the portfolio
must exactly balance each other. The credibility mechanism indeed has little effect on the
number of claims filed to the company. Therefore, the number of claims with and without
credibility is very much the same.
For this reason, we expect that the a posteriori corrections average to unity. This ensures
that the average number of claims without credibility equals the average number of claims
with credibility, and that the premium income will be enough to compensate the claims.
3.1.7 Combining a Priori and a Posteriori Ratemaking
The amount of premium paid by the policyholder depends on the rating factors of the current
period (think for instance of the type of the car or of the occupation of the policyholder) but
also on the claim history. The insurance premium is the product of a base premium and of a
credibility coefficient. The base premium is a function of the current rating factors whereas
the credibility coefficient usually depends on the history of claims at fault. Clearly, a priori
and a posteriori ratings interact. To the extent that good drivers are rewarded in their base
premiums (through other rating variables) the size of the bonus they require for equity is
reduced.
The claims history of each policyholder consists of a short integer-valued sequence of
yearly claim counts. The basic model used for experience rating is based on the Negative
Binomial distribution. This probability law can be seen as a Poisson mixture distribution with
Gamma mixing. Therefore, it allows for serial dependence of claim counts, by introducing
Gamma-distributed unobserved individual heterogeneity. The serial dependence in claim
counts sequences is generated by integrating the unobserved factor, and by updating its
prediction when individual information increases. Alternative models with LogNormal or
Inverse Gaussian unobserved heterogeneity have also been considered in the actuarial
literature (and reviewed in Chapter 2).