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

266 Actuarial Modelling of Claim Counts
c mat/bod
i
c bod/bod
i
evaluates the information contained in past material claims on the occurrence of
future claims with bodily injuries;
evaluates the information contained in past claims with bodily injuries on the
occurrence of future claims with bodily injuries.
The values of these coefficients are determined by minimizing simultaneously mat and bod
that may be rewritten as
and as
mat = E [( mat
bod = E [( bod
iT i +1 mat i
iT i +1 bod i
− c mat
i
− c bod
i
− c mat/mat
i
− c mat/bod
i
N mat
i•
N mat
i•
− c bod/mat
i
− c bod/bod
i
N bod
i•
N bod
i•
) 2 ]
) 2 ]
Setting to zero the partial derivatives of mat and bod with respect to the six parameters gives:
c mat
i
c bod
i
= mat
= bod
0 = mat
iT i +1 − cmat/mat i
mat
i•
iT i +1 − cmat/bod i
mat
i•
iT i +1 Emat i
N mat
i•
− c bod/mat
i
0 = mat
EN mat
i•
N bod
iT i +1 Emat i
N bod
i•
− cbod/mat i
bod
i•
(6.2)
− cbod/bod i
bod
i•
(6.3)
− cmat
i
mat
i•
− cmat/mat i
E [ N mat
i• 2]
i• (6.4)
− cmat
i
bod
i•
− cmat/mat i
EN mat
i•
N bod
i•
− c bod/mat
i
E [ N bod
i• 2] (6.5)
0 = bod
iT i +1 Ebod i
N mat
i•
− c bod/bod
i
0 = bod
EN mat
i•
N bod
iT i +1 Ebod i
N bod
i•
− cbod
i
mat
i•
− cmat/bod i
E [ N mat
i• 2]
i• (6.6)
− cbod
i
bod
i•
− cmat/bod i
EN mat
i•
N bod
i•
− c bod/bod
i
E [ N bod
i• 2] (6.7)
The expectancies involved in this system are given by
E mat
i
E bod
i
E mat
i
E bod
i
N mat
i•
N bod
i•
N bod
i•
N mat
i•
= mat
i•
= bod
i•
2 mat + mat i•
2 bod + bod i•
= bod
bm + bod
i•
i•
= mat
bm + mat
i•
i•