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

330 Actuarial Modelling of Claim Counts
Some algebra immediately leads to the following system of equations to solve:
∑
a t t
t=1
∫
∑
= 1 + a t t
0
t=1
2 (e − e − ) t−1e f d
∫
0
(
t−1e
e 2+2 + e − 2 2+
− 2) 2 f d
and
∑
a t t
t=1
∫
∑
= 1 − a t t
0
t=1
( ) t−1e
e − e − − f d
∫
0
(
e 2+2 + e − 2 − 2) t−1e − f d
Again, this system does not admit any closed-form solution, but can be solved numerically
(using an appropriate SAS/IML optimization algorithm).
Remark 9.1 Note that here, we have made an averaging with respect to the age structure
of the portfolio. In the case where the portfolio is partitioned into a series of risk classes, an
average with respect to the composition of the portfolio (in terms of classification variables)
could also be performed. If some explanatory variables are correlated with A, care must be
taken in the second averaging.
9.2.6 Multivariate Panjer and De Pril Recursive Formulas
Notations
In Sections 9.2.7 and 9.3.3, we will need the bivariate and trivariate extensions of the
Panjer algorithm that was described in Section 7.2. The present section is devoted to the
presentation of this method as well as a particular case for the sum of independent and
identically distributed random vectors, known as multivariate De Pril’s recursive formula.
Assume independent and identically distributed realizations of possibly dependent losses
X i = X i1 X ik T affected by a common event, denoted by the counting variable N . For
example N may count the number of hurricanes hitting the United States and the X j s may
represent the cost of the hurricane in state numbered j, j = 1k. It is natural to try to
obtain the distribution of the aggregate claim:
( ) T
∑ N N∑
S = S 1 S k T = X i1 X ik (9.3)
Even when the components of X are independent, the components of S will have some
positive dependence due to the common counter N .
When N belongs to the Panjer family of counting random variables, let us show that a
multivariate version of Panjer’s recursive formula emerges. To this end, we will use the
following notations:
i=1
i=1