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

Multi-Event Systems 263
6.2.5 Variance-Covariance Structure of the Random Effects
For deriving linear credibility formulas, we only need the moment structure of the risk
variables. Let us introduce the variance-covariance matrix of i
bod i
mat that is denoted as
( )
2
= bod
bm
bm mat
2
In words, 2 bod and 2 mat
are the variances of bod
i
covariance between bod
i
and mat
i
and i
mat
. Note that the following inequalities
, respectively, and bm is the
2 bod ≥ 0 2 mat ≥ 0 and bm≤ bod mat
must be fulfilled to ensure that is positive definite. The estimated variances and covariance
have to fulfill the same constraints. If not, this rules out the linear credibility model.
6.2.6 Variance-Covariance Structure of the Annual Claim Numbers
Let us now compute the variance and covariance of Ni• mat and Ni• bod.
Since
Ni•
mat ∼ oi mat
i•
mat i
, we have
Similarly, from N bod
i•
VN mat
i•
∼ oi bod
i•
bod i
VN bod
i•
= mat
i•
we get
= bod
i•
+ ( ) 2
mat 2
i• mat
+
( ) 2
bod 2
i• bod
To have an idea about the dependence existing between the numbers of claims with material
damage only and with bodily injuries, let us now compute the covariance between Ni•
mat
and Ni• bod:
[
]
CN mat
i•
Nbod i• = E N mat
i•
− mat bod
i•
Ni•
− bod
i•
[
= E E [ N mat
i•
− mat
i•
[
= E E [ ∣
N mat
i•
− mat
i•
[
= E
= mat
i•
mat
i•
(
mat
i
bod i•
bm
As expected, the covariance bm between mat
i
bod
Ni•
∣ mat
i
− bod
i•
∣ ∣ mat
] [ ∣
E N
bod
− bod
i•
− 1 ) (
bod
i•
bod
i
− 1 )]
and bod
i
i
i•
bod
i
∣ bod
i
] ]
] ]
drives the covariance of Ni•
mat and Ni• bod.