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

Efficiency and Bonus Hunger 247
to the insurer if its cost exceeds a random threshold, assumed to be LogNormally distributed
with parameters specific to the level occupied in the scale. Note that we deal here with
moderate claim sizes only. The reason is that large claims are not subject to bonus hunger
and are systematically reported to the insurer. The frequency of large claims will have to be
added to the corrected frequency of moderate claims to get the actual number of accidents
caused each year.
Let l i be the level occupied by policyholder i in the bonus-malus scale at the beginning
of the period, and let RL i l i ∼ N or i 2 be the random optimal retention, with a linear
predictor of the form
i = 0 +
p∑
j x ij + fl i
j=1
specific to policyholder i, where the function f· expresses the effect of occupying level l i
in the bonus-malus scale. Considering the values obtained for the optimal retention in the
literature (reaching a maximum somewhere in the middle of the scale, and decreasing when
approaching uppermost and lowermost levels), we will use here a quadratic effect f of l i .
Note that other approaches are possible (see the references in the closing section for more
details).
This means that policyholder i will report all the accidents with a cost larger than RL i l i ,
and defray himself all those with a cost less than RL i l i . At the portfolio level, the
RL i l i s are assumed to be independent. Now, let CA ik be the cost of the kth accident
caused by policyholder i. We assume that for each i the random variables CA i1 CA i2
are
∑
independent and identically distributed, with CA ik ∼ N or i 2 , where i = 0 +
p
j=1 jx ij . Moreover, the CA ik s and RL i l i are mutually independent. We consider here
the explanatory variables selected in the LogNormal analysis of the censored claim costs,
presented in Table 5.5.
Now, denoting as c i1 c ini the costs of the n i moderate claims filed by policyholder i,
the likelihood is
2 = ∏ n
∏ i
f i c ik
f i c ik =
in i >0 k=1
where f i · denotes the probability density function of CA ik given CA ik >RL i l i . Each
factor involved in the likelihood can be written is
1
√ exp
(− ln c ( )
ik − i 2 ln cik − i
2cik
2 2 )
(
1 − − )
i −
√ i
2 + 2
The estimators of the parameters , , 2 , and are determined by maximizing the likelihood
2 .
This basic model could be refined in different respects. Firstly, the retention limit could
depend on the number of claims previously filed by the policyholder during the same
year. The retention for the second claim depends on the level to which the policyholder is