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

Bonus-Malus Systems with Varying Deductibles 281
7.2.3 Panjer Algorithm
Direct Convolution Approach
The Panjer algorithm then allows us to compute the distribution function of the discretized
version of S, defined as
S =
N∑
C k
k=1
To realize the merit of the Panjer approach, let us first write the formulas in the direct
approach. Let us denote as f i = PrC1
= i the probability mass function of C 1 . The
probability mass function of C1 +···+C k
is
[ ]
k∑
f ⋆k
i = Pr C j
= i
j=1
In the applications we have in mind, the distribution function for the claim costs satisfies
F0 = 0. Clearly, with Discretization Method (7.3), we then have f 0 = 0 so that f ⋆k
i = 0if
i ≤ k − 1 (since C1 ⋆k
≥ with probability 1). The fi
s satisfy
f ⋆k
i−k+1
∑
i = f ⋆k−1
i−j f j if i ≥ k (7.5)
Then, the probability mass function g i = PrS = i of S satisfies
g i =
i∑
k=0
j=1
PrN = kf ⋆k
i i∈ (7.6)
This direct computation of the probability mass function of S requires a lot of computation
time. Things are even worse in the Discretization (7.4) for which f 0 > 0.
Panjer Family
The Panjer formula holds for a class of probability distributions referred to as the Katz
family in statistical circles, and as the Panjer family in the actuarial literature. This family
contains all the counting distributions such that the relation
(
p k = a + b )
p
k k−1 k= 1 2 (7.7)
is fulfilled for some a and b. The Panjer family contains three elements. Specifically, the
probability distributions satisfying (7.7) are
(i) the Poisson distribution, obtained with a = 0 and b>0;
(ii) the Negative Binomial distribution, obtained with 0