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

Actuarial Analysis of the French Bonus-Malus System 327
9.2.2 Probability Generating Functions of Random Vectors
In this chapter we will use multivariate models for counting random vectors. Specifically, let
us consider random vectors M = M 1 M n T valued in n . The multivariate probability
mass function of M is
p M k 1 k n = PrM 1 = k 1 M n = k n
Throughout the chapter we will extensively use the multivariate extension of the probability
generating function introduced in Chapter 1, which is defined as
M z = z M 1
1 ···z M n
n
∑ ∑
= z k 1
1 ···zk n
n p Mk 1 k n
k 1 =0
k n =0
Let us now point out several interesting properties of the multivariate probability generating
functions. If any function that is known to be a multivariate probability generating function
for a random vector M is expanded as a power series in z, then the coefficient of z k 1
1 ···zk n
n
must be p M k 1 k n . Furthermore,
• z ↦→ M zzz is the probability generating function of M 1 +···+M n ;
• z ↦→ M z 00 is the probability generating function of M 1 ;
• M z 1 z n = M1
z 1 ··· Mn
z n when the random variables M 1 M n
independent.
are
9.2.3 CRM Coefficients
We will assume that the CRM coefficients only depend on the observed number of reported
claims and not on their severity. Therefore the base premium is simply multiplied by a
constant (essentially the expected cost of a claim).
Let t be the ‘reduction’ coefficient and t be the ‘majoration’ coefficient applying to a
policyholder who has been covered for t years. The CRM coefficient for years 1 to t then
becomes
with
where I j is defined as
r t t
N • I • t= 1 + t N •
1 − t I •
N • =
t∑
N j and I • =
j=1
t∑
I j (9.1)
j=1
{
1ifNj = 0
I j =
0ifN j ≥ 1