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

Bonus-Malus Scales 189
Table 4.7 Numerical characteristics for the system −1/ + 3 and for Portfolio A.
Level l PrL = l r l = EL = l
without a priori
ratemaking
r l = EL = l
with a priori
ratemaking
EL = l
5 73 % 2571 % 2308 % 17.4 %
4 59 % 2194 % 2009 % 16.7 %
3 90 % 1517 % 1452 % 15.5 %
2 73 % 1365 % 1331 % 15.2 %
1 60 % 1240 % 1230 % 15.0 %
0 645% 578% 642 % 14.1 %
Table 4.8 Numerical characteristics for the system −1/ + 3 and for Portfolio B.
Level l PrL = l r l = EL = l
with a priori
ratemaking
EL = l
5 92 % 1840% 200%
4 78 % 1566% 197%
3 117 % 1174% 192%
2 95 % 1086% 190%
1 78 % 1016% 189%
0 540% 720% 184%
are displayed in Table 4.8. The comparison with Portfolio A shows that the relativities are
much less dispersed here.
4.5.3 Interaction between Bonus-Malus Systems and a Priori Ratemaking
Since the relativities attached to the different levels are the same whatever the risk class
to which the policyholders belong, those scales overpenalize a priori bad risks. Let us
explain this phenomenon, put in evidence by Taylor (1997). Over time, policyholders will
be distributed over the levels of the bonus-malus scale. Since their trajectory is a function
of past claims history, policyholders with low a priori expected claim frequencies will
tend to gravitate to the lowest levels of the scale. Conversely for individuals with high a
priori expected claim frequencies. Consider for instance a policyholder with a high a priori
expected claim frequency, a young male driver living in a urban area, say. This driver
is expected to report many claims (this is precisely why he has been penalized a priori)
and so to be transferred to the highest levels of the bonus-malus scale. On the contrary, a
policyholder with a low a priori expected claim frequency, a middle-aged lady living in
a rural area, say, is expected to report few claims and so to gravitate to the lowest levels
of the scale. The level occupied by the policyholders in the bonus-malus scale can thus be
partly explained by their observable characteristics included in the price list. It is thus fair to
isolate that part of the information contained in the level occupied by the policyholder that