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

Multi-Event Systems 261
score bod
it
= bod
p∑
0
+
j=1
bod
j
x itj
We make the following assumptions about the dependence structure of the random variables:
(i) Given bod
i
, the Nit
bod s, t = 1 2T i , are independent.
, the Nit
mat s, t = 1 2T i , are independent.
i
bod , the sequences Nit
bod t = 1 2T i and Nit
mat t = 1 2T i
are independent.
(ii) Given mat
i
(iii) given mat
i
As explained in the previous chapter, overdispersion and serial correlation are induced by
missing explanatory variables, whose effect is modelled with the help of the random effects
and bod
mat
i
i
.
6.2.3 Bayesian Credibility Approach
The Bayesian approach requires numerical integration, which sometimes prevents the
practical implementation of the resulting formulas (even if, nowadays, numerical integration
has become straightforward with modern computers). It consists of deriving the conditional
distribution of the random effects mat
i
and bod
i
conditional distribution then drives a posteriori premium corrections.
Let us denote as
∑
T i
k mat
i•
= k mat
it
t=1
, given the past claims history. This
the total number of claims with material damage only filed by policyholder i during the T i
coverage periods, and as
∑
T i
k bod
i•
= k bod
it
t=1
the total number of claims with bodily injuries filed by this policyholder. The corresponding
expected claim frequencies are
The joint distribution of the N mat
it
PrN mat
it
∫
=
0
∑
T i
mat
i•
= mat
it
t=1
and bod
∑
T i
i•
=
t=1
s and N bod s is given by
= k mat
it
N bod
it
= k bod
it
for t = 1T i
∫
0
exp− mat
i
× f mat
i
mat
i•
− bod i
it
bod
i
d mat
i
d bod
i
bod
i•
mat i
kmat i•
bod
it
bod
i
kbod i•
∏ Ti
t=1 mat it
kmat it
∏ Ti
t=1 kmat it !kit bod !
bod
it
kbod it