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

Multi-Event Systems 265
of the true expected claim frequency mat
iT i +1 mat i
minimizes
⎡(
mat = E ⎣
mat
iT i +1 mat i
Similarly, the best linear predictor
of bod
iT i +1 bod i
minimizes
⎡(
bod = E ⎣
∑
T i
c bod
i0
+
t=1
bod
iT i +1 bod i
− c mat
i0
T
∑ i
−
t=1
c mat/bod
it
N mat
− c bod
i0
c mat/mat
it
∑
T i
it
+
t=1
T
∑ i
−
t=1
c mat/bod
it
N mat
it
T
∑ i
−
t=1
c bod/bod
it
N bod
it
N mat
it
T
∑ i
−
t=1
c bod/mat
it
N bod
it
c bod/bod
it
N bod
it
) ⎤ 2
⎦
) ⎤ 2
⎦
The optima are obtained by setting to zero the derivatives of mat with respect to ci0
mat
to cis
mat/mat , that is,
c mat
i0
c mat/mat
is
T i
= mat
iT i +1 − ∑
= mat
t=1
c mat/mat
it
mat
it
T
∑ i
−
t=1
T
iT i +1 2 mat − ∑ i
2 mat
c mat/mat
it
mat
it
t=1
c bod/mat
it
bod
it
T
∑ i
− bm
t=1
c bod/mat
it
bod
it
The last relation shows that cis
mat/mat does not depend on s. Similarly, one can check that
cis
mat/bod , cis
bod/mat and cis
bod/bod do not depend on s. This justifies the approach based on aggregate
data Ni•
mat and Ni•
bod (that are an exhaustive summary of past claims histories in the credibility
model with static random effects).
Denoting as ci
mat/mat (ci
bod/mat , ci
mat/bod and ci
bod/bod , respectively) the common values of the
cis
mat/mat (cis
bod/mat , cis
mat/bod and cis
bod/bod , respectively), the best linear predictors are thus of the form
for mat
for bod
follows:
iT i +1 mat i
iT i +1 bod i
, and
c mat
i
c bod
i
+ c mat/mat
i
+ c mat/bod
i
N mat
i•
N mat
i•
+ c bod/mat
i
N bod
i•
+ c bod/bod
i
N bod
i•
. The meaning of the coefficients involved in these linear predictors is as
and
c mat/mat
i
c bod/mat
i
evaluates the information contained in past material claims on the occurrence of
future material claims;
evaluates the information contained in past claims with bodily injuries on the
occurrence of future material claims;