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

276 Actuarial Modelling of Claim Counts
Table 6.6 Results for the bonus-malus systems −1/ + 2/ + 3, −1/ + 2, −1/ + 3 and −1/top for
Portfolio B.
Level l −1/ + 2/ + 3 −1/ + 2 −1/ + 3 −1/top
l r l l r l l r l l r l
5 59 % 2159% 54 % 2232% 92 % 1840% 163 % 1469%
4 62 % 1695% 60 % 1713% 78 % 1566% 125 % 1293%
3 64 % 1434% 58 % 1489% 117 % 1174% 99 % 1172%
2 112 % 1118% 115 % 1121% 95 % 1086% 80 % 1080%
1 92 % 1046% 93 % 1050% 78 % 1016% 66 % 1007%
0 611% 745% 620% 747% 540% 720% 466% 706%
Portfolio B Table 6.6 gives for each of the six levels of the bonus-malus scale the proportion
of the portfolio in that level (column 2) and the relativity attached to that level (column 3)
for the system −1/ + 2/ + 3. About 60 % of the portfolio is in level 0 and enjoys a discount
of about 25 %. The rest of the portfolio is spread out among levels 1–5. The r l s range from
74.5 % to 215.9 %.
In order to compare the results with those of traditional bonus-malus scales, we have also
recalled the results given by the three bonus-malus scales −1/ + 2, −1/ + 3 and −1/top
already considered in the previous chapters. The scale −1/ + 2/ + 3 is close to the scale
−1/ + 2 (again, this is because the majority of the claims only induce material damage).
Nevertheless, r 5 is reduced from 223.2 % to 215.9 % when the claims with bodily injuries
are more severely penalized.
6.4 Further Reading and Bibliographic Notes
The second part of this chapter is based on Pitrebois, Denuit & Walhin (2006a).
Lemaire (1995, Chapter 13) applied a model proposed by Picard (1976) to Belgian data,
distinguishing the accidents that caused property damage only, from those that caused bodily
injuries. The credibility model proposed by Lemaire (1995) is based on a Poisson-Gamma
mixture, and assumes that given the expected annual claim frequency of the policyholder,
the frequency of claims with bodily injuries conforms to a Beta distribution. This approach
can be extended to several categories of claims using a Dirichlet distribution (that is, using
a suitable multivariate Beta distribution).
With the aid of multi-equation Poisson models with random effects, Pinquet (1998)
designed an optimal credibility model for different types of claims. As an example, claims
are separated into two groups according to fault with respect to a third party. See also
Pinquet (1997) on allowing for the costs of the claims.
This chapter gives only basic methods to deal with different types of claims in a posteriori
ratemaking. Advanced statistical and econometrics models could certainly improve the
actuarial analysis. For instance, Wedel, Böckenholtb & Kamakurac (2003) developed a
general class of factor-analytic models for the analysis of multivariate (truncated) count data.
These models provide a parsimonious and easy-to-interpret representation of multivariate
dependencies in counts that extend the general linear latent variable model.