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

342 Actuarial Modelling of Claim Counts
Table 9.5 Parameters and and financial equilibrium for different values of .
= 15 = 20 = 25
Financial
equilibrium
Financial
equilibrium
Financial
equilibrium
0.0507 0.0133 1.0419 0.0393 0.0133 1.0403 0.0321 0.0133 1.0393
9.4 Further Reading and Bibliographic Notes
This chapter is based on Pitrebois, Denuit & Walhin (2006b). Despite its apparent
difference with bonus-malus scales, the French bonus-malus system can be treated as a scale
with many levels. Kelle (2000) followed this route, and used a Markov chain with 530
states to analyse the French system. The large amount of states needed is due to the malus
reduction in the case of claims with shared responsibility, forcing the author to consider the
pair (number of claims with whole responsibility, number of claims with partial liability) to
make the computation.
In this chapter, we did not consider all the characteristics of the bonus-malus system
in force in France. We have disregarded the special bonus rule (which suppresses all
the penalties after two claim-free years). The French law imposes other specific rules on
insurance companies. For instance, the French bonus-malus system is such that drivers never
pay more than 350 % of the base premium nor less than 50 % of the base premium. Therefore
the minimization process has to be carried with an adapted CRM coefficient of the form
r ∗ = max05 min35r
Several simplifying assumptions can be considered to ease the numerical computations.
For instance, we could work with binary annual claim numbers: either the policyholder
does not report any claim or he reports a single claim to the company. Such an assumption
replacing N t by minN t 1 leads to smaller discounts and higher penalties, which is a prudent
strategy for the insurer.
Even if the vast majority of bonus-malus systems appear as scales in which policyholders
move according to their claims history, there are some exceptions (such as the system
in force in France). We refer the reader to Neuhaus (1988) for another example, where
the malus after a claim is expressed by a fixed monetary amount (instead of a relativity).
This interesting mechanism restores some fairness in case of differentiated a priori
price lists.
The multivariate version of Panjer’s recursive formula has been derived by Sundt
(1999) and Ambagaspitya (1999). Sundt (1999) provided a proof based on conditional
expectations whereas Ambagaspitya (1999) used a proof based on generating functions.
Sundt (1999) also showed that the following recursive formula can be used :
f S s =
1
1 − af X 0
s∑
x≠0
(
a + b s )
1 +···+s k
f
x 1 +···+x X xf S s − x
k
x > 0