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

Actuarial Analysis of the French Bonus-Malus System 333
and the auxilliary random variable N ∼ inn 1 − f0.
The probability generating function of W 11 +···+W 1N W k1 +···+W kN T is given by
(
n (
1 − 1 − f01 − W u)
= X u ) n
from which we conclude that S = W 11 +···+W 1N W k1 +···+W kN T .
Applying Property 9.1 with
a = f0 − 1
f0
and b = 1 − f0 n + 1
f0
we get the desired result.
□
9.2.7 Analysis of the Financial Equilibrium of the French Bonus-Malus
System
An interesting property of the relativities associated with Markovian bonus-malus systems
and obtained through Norberg’s least-squares criterion is that they make the bonus-malus
system financially balanced, i.e. the premium income of the insurer does not increase nor
decrease over time (on average). In this section, we would like to check whether or not the
French-type bonus-malus system enjoys this property.
More precisely, once t and t have been obtained, we would like to verify whether
Er t t
N • I • t is equal to 1, where N • and I • are as defined in (9.1). The computation of
Er t t
N • I • t requires knowledge of the joint distribution of the random couple N • I • .
Let us denote as
fx y = PrN 1 = x I 1 = y =
the joint discrete mass function of the random couple N 1 I 1 , conditional on = , and as
f ⋆t x y = PrN • = x I • = y =
the joint discrete mass function of the random couple N • I • defined in (9.1), conditional
on = . We then have the following result.
Property 9.3
For fixed , the following recursive formulas
g ⋆t x y = f ⋆t x t − y for 0 ≤ y ≤ t and x>0
g ⋆t 0 0 = e −t
−
x
fx 0 = e for x>0
x!
f0 1 = e −
( )
x∑ t + 1
g ⋆t x y = e − 1 g ⋆t x − u y − 1gu 1 for y ≥ 1 and x ≥ 1
y
u=1