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

124 Actuarial Modelling of Claim Counts
3.1.8 Loss Function
Whatever the model selected for the number of claims, the a posteriori premium correction
is derived from the application of a loss function. The standard choice is a quadratic loss.
In this case, the credibility premium is the function of past claim numbers that minimizes
the expected squared difference with the next year claim number. It is well known that the
solution is given by the a posteriori expectation.
The penalties obtained in a credibility system calling upon a quadratic loss function
are often so severe that it is almost impossible to implement them in practice, mainly for
commercial reasons. In order to avoid this problem, some authors have proposed resorting to
an exponential loss function: the hope is that breaking the symmetry between the overcharges
and the undercharges leads to reasonable penalties. This reduces the maluses and the bonuses,
and results in a financially balanced system.
3.1.9 Agenda
Section 3.2 introduces the basics of credibility models taking into account a priori
characteristics. It starts with a simple introductory example that contains all the ideas of
credibility theory. Then, the probabilistic tools used in this context are briefly recalled.
Section 3.3 is devoted to credibility formulas based on a quadratic loss function. The
optimal predictor is then shown to be the conditional expectation of future claims given
past claims history. In the particular case of Gamma distributed risk parameters, explicit
expressions are available for a posteriori premium corrections. Discrete Poisson mixtures
are considered in detail, providing approximate credibility formulas. Also, linear credibility
predictions are derived.
In Section 3.4, the quadratic loss function is replaced with an exponential one. Again, the
general formulas simplify in the Negative Binomial case. Linear credibility predictions are
considered as approximations to exact premium corrections.
Section 3.5 discusses the type of dependence generated by the credibility construction. It
is shown that the risk parameter, as well as future claim numbers, increases with the number
of claims recorded in the past, and that future and past claim numbers are positively related.
This confirms the intuition behind the actuarial credibility model.
The final Section 3.6 gives the references, and discusses further issues.
3.2 Credibility Models
3.2.1 A Simple Introductory Example: the Good Driver / Bad Driver
Model
Consider an insurance portfolio where 60 % of the policyholders are good drivers. The
probability that a good driver reports k claims during the year is given by the oi G
distribution with G = 005. The remaining 40 % of the policyholders are bad drivers. The
probability that they report k claims is given by the oi B distribution with B = 015.