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

268 Actuarial Modelling of Claim Counts
and the best linear predictor for the expected frequency of claims with bodily injuries in
year T i + 1is
bod
iT i +1
1 + mat
i•
( )
2
mat
− bm + bm Ni•
mat + ( bod 2 + mat i•
2 mat 2 bod − 2 bm ) Ni•
bod
1 + mat
i• mat1 2 + bod
i• bod 2 −
mat i• bod
i•
2 bm
6.2.9 Numerical Illustration for Portfolio A
A Priori Ratemaking
The observed annual frequency for claims with bodily injuries is 16 %. The observed annual
frequency for claims with material damage only is 130%.
In this portfolio, the two types of claims we consider are positively correlated. This can
be seen from Table 6.1, where the conditional expectation of the number of claims of one
type is computed given the number of claims of the other type. The more claims of one type
reported, the higher this conditional expectation, resulting in positive dependence.
A Posteriori Corrections
Let us now update the claim frequencies with the help of the formulas obtained with linear
credibility. To this end, we first estimate . This gives
Let us now consider two types of drivers:
̂ 2 mat = 08458
̂
bod 2 = 11188
̂ bm = 06255
• a good driver with mat
it
= 0083 and bod
it
= 0010, and
• a bad driver with mat
it
= 0246 and bod
it
= 0030.
Table 6.1 Observed annual frequency of claims with bodily injuries, given the number
of claims with material damage only; and observed annual frequency of claims with
material damage only, given the number of claims with bodily injuries; for portfolio A.
Conditional expectation of the
number of claims with bodily
injuries
Conditional expectation of the
number of claims with
material damage only
Given the number of claims
with material damage only
Given the number of claims
with bodily injuries
= 0 is 1.5 % = 0 is 12.9 %
= 1 is 2.7 % = 1 is 22.8 %
= 2 is 3.6 % = 2 is 37.7 %