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

Bonus-Malus Scales 213
We observe that the different metrics we have chosen do not provide very different results.
Because the expected posterior random effect has a financial meaning (i.e. it is the multiplier
of the average cost to get the a posteriori premium), we may be tempted to choose its
corresponding metric to transfer a policyholder from one scale to the other. Note also that
the a priori characteristics of the driver influence the way the policy is transferred from
one scale to the other. This results in a number of rules according to the risk classification
scheme applied by the insurance company.
Remark 4.5 Transferring a policyholder from the bonus-malus scale of a given insurer
to the bonus-malus scale of another insurer remains a more complicated task. Indeed, the
actuary needs to know the a posteriori random effect in both situations. However the a
priori random effect may be different due to another type of a priori tariff or due to adverse
selection. The distance to minimize then extends to d 1 L 1 = l 1 2 L 2 = l 2 .
4.9 Dependence in Bonus-Malus Scales
It is clear that the sequence L 1 L 2 is CIS, but we do not have in general that given
L = l increases with l. The reason is as follows: Because of the finite number of levels,
it does not automatically follow that a policyholder in a higher level filed more claims in
the past. To show this, consider the scale −1/top. A policyholder in level 3 in year 3 may
occupy that level having filed two claims in year 1 and no claim after. On the contrary, a
policyholder in level 4 could have filed one claim in year 2. From (3.18) we conclude that
should be larger (in the ≼ ST -sense) for policyholder 1 than for policyholder 2 despite the
fact that policyholder 1 is in an inferior level.
As a consequence, we cannot be sure that the Bayesian relativities obtained with a quadratic
loss function are increasing with the level occupied in the scale. This is why linear relativities
are so useful in practice.
Remark 4.6 This counterintuitive fact becomes reasonable if we allow for random effects
to vary in time. Provided the autocorrelogram is decreasing, old claims have less predictive
power than recent ones. Then, we have to compare two old claims to one recent claim, and
it becomes less obvious that policyholder 1 is more dangerous than policyholder 2.
4.10 Further Reading and Bibliographic Notes
Chapter 7 in Rolski ET AL. (1999) offers an excellent introduction to Markov chains, with
applications to bonus-malus systems. Several parts of this chapter are directly inspired
from this source. For the most part, this chapter is based on Pitrebois, Denuit &
Walhin (2003b), following on from Taylor (1997) for the extension of Norberg’s (1976)
pioneering work on segmented tariffs; on Pitrebois, Denuit & Walhin (2004) for the
linear relativities; on Pitrebois, Walhin & Denuit (2006c) for Section 4.8; and on
Pitrebois, Denuit & Walhin (2003a) for the Belgian bonus-malus scale and its special
bonus rule.
Norberg (1976), Borgan, Hoem & Norberg (1981) and Gilde & Sundt (1989)
assumed that the bonus-malus system forms a first order Markov chain. Centeno &
Andrade e Silva (2002) considered bonus-malus systems that are not first order Markovian