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

282 Actuarial Modelling of Claim Counts
Panjer Algorithm with f 0 = 0
The Panjer algorithm is easily established with probability generating functions. Here, we
use another approach with conditional expectations. The proof of the Panjer formula is based
on the following technical lemma.
Lemma 7.1
The relation
f ⋆n
j
= n j
j∑
i=1
if i f ⋆n−1
j−i
is valid for j ≥ n.
Proof
The proof consists of noting that, on the one hand, for any j ≥ n
[ ∣ ] [ ∣ ]
∣∣∣∣ n∑
E X 1 X k = j = 1 n∑ ∣∣∣∣ n∑
k=1
n E X k X k = j = j
k=1 k=1
n
and on the other hand, if f ⋆k
j > 0,
[ ∣ ] [
]
∣∣∣∣ n∑
j∑
n∑
E X 1 X k = j = i Pr X 1 = i
X
∣ k = j
k=1
i=1
k=1
∑ j
i=1
=
i Pr X 1 = i and ∑ n
k=2 X k = j − i
Pr ∑ n
k=1 X k = j
∑ j
i=1
=
if if ⋆n−1
j−i
f ⋆n
j
Equating these two formulas yields the expected result.
□
We are now ready to derive the Panjer algorithm if f 0 = 0 (as is the case with Discretization
Method (7.3)).
Proposition 7.1 If the probability distribution of N belongs to the Panjer family and if
f 0 = 0, then the g i s are obtained recursively from
starting with g 0 = p 0 .
g j =
j∑
i=1
(
a + i b )
f
j i g j−i (7.8)
Proof
Since C k ≥ almost surely, we clearly have
PrS = 0 = PrN = 0 = p 0