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The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S = ∑ i = 1 N X i {\displaystyle S=\sum _{i=1}^{N}X_{i}\,} where both N {\displaystyle N\,} and X i {\displaystyle X_{i}\,} are random variables and of special types. In more general cases the distribution of S is a compound distribution. The recursion for the special cases considered was introduced in a paper [1] by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo[2]). It is heavily used in actuarial science (see also systemic risk).
We are interested in the compound random variable S = ∑ i = 1 N X i {\displaystyle S=\sum _{i=1}^{N}X_{i}\,} where N {\displaystyle N\,} and X i {\displaystyle X_{i}\,} fulfill the following preconditions.
We assume the X i {\displaystyle X_{i}\,} to be i.i.d. and independent of N {\displaystyle N\,} . Furthermore the X i {\displaystyle X_{i}\,} have to be distributed on a lattice h N 0 {\displaystyle h\mathbb {N} _{0}\,} with latticewidth h > 0 {\displaystyle h>0\,} .
In actuarial practice, X i {\displaystyle X_{i}\,} is obtained by discretisation of the claim density function (upper, lower...).
The number of claims N is a random variable, which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the probability distribution of N has to be a member of the Panjer class, otherwise known as the (a,b,0) class of distributions. This class consists of all counting random variables which fulfill the following relation:
for some a {\displaystyle a} and b {\displaystyle b} which fulfill a + b ≥ 0 {\displaystyle a+b\geq 0\,} . The initial value p 0 {\displaystyle p_{0}\,} is determined such that ∑ k = 0 ∞ p k = 1. {\displaystyle \sum _{k=0}^{\infty }p_{k}=1.\,}
The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following W N ( x ) {\displaystyle W_{N}(x)\,} denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.
In the case of claim number is known, please note the De Pril algorithm.[3] This algorithm is suitable to compute the sum distribution of n {\displaystyle n} discrete random variables.[4]
The algorithm now gives a recursion to compute the g k = P [ S = h k ] {\displaystyle g_{k}=P[S=hk]\,} .
The starting value is g 0 = W N ( f 0 ) {\displaystyle g_{0}=W_{N}(f_{0})\,} with the special cases
and
and proceed with
The following example shows the approximated density of S = ∑ i = 1 N X i {\displaystyle \scriptstyle S\,=\,\sum _{i=1}^{N}X_{i}} where N ∼ NegBin ( 3.5 , 0.3 ) {\displaystyle \scriptstyle N\,\sim \,{\text{NegBin}}(3.5,0.3)\,} and X ∼ Frechet ( 1.7 , 1 ) {\displaystyle \scriptstyle X\,\sim \,{\text{Frechet}}(1.7,1)} with lattice width h = 0.04. (See Fréchet distribution.)
As observed, an issue may arise at the initialization of the recursion. Guégan and Hassani (2009) have proposed a solution to deal with that issue .[5]
