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

8
Transient Maximum Accuracy
Criterion
8.1 Introduction
8.1.1 From Stationary to Transient Distributions
All developments so far have been based on the stationary distribution of the Markov process
describing the trajectory of the policyholder in the bonus-malus scale. As Borgan, Hoem
& Norberg (1981) objected, an asymptotic criterion is moderately relevant for bonusmalus
systems needing relatively long periods to reach their steady state, since policies
are in force only during a limited number of insurance periods. These authors modified
the criterion in order to take into account the rating error for new and young policies.
As Norberg (1976), Borgan ET AL. (1981) measured the performances of a bonus-malus
system by a weighted average of the expected squared rating errors for selected insurance
periods.
In Chapter 4, the relativities associated with the levels of the bonus-malus scale were
computed on the basis of an asymptotic criterion. The implicit assumption behind the results
in the preceding chapters is thus that the Markov process reaches its steady state after a
relatively short period, as is the case for the −1/top bonus-malus scale for instance. If a
majority of the policies are far from the steady state, it seems desirable to modify the criterion
so as to take into account the rating error for new policies and for policies of a moderate
age as well.
8.1.2 A Practical Example: Creating a Special Scale for New Entrants
Before entering into the mathematical developments, let us describe a concrete situation
where the asymptotic criterion used in the previous chapters is no longer relevant, and
Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems
S. Pitrebois and J.-F. Walhin © 2007 John Wiley & Sons, Ltd
M. Denuit, X. Maréchal,