Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution of a compound random variable

S = \sum_{i=1}^N X_i.\,.

where both N\, and 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 of Harry Panjer[1]. It is heavily used in actuarial science.

Contents

Preliminaries

We are interested in the compound random variable S = \sum_{i=1}^N X_i\, where N\, and X_i\, fulfill the following preconditions.

Claim size distribution

We assume the X_i\, to be i.i.d. and independent of N\,. Furthermore the X_i\, have to be distributed on a lattice h \mathbb{N}_0\, with latticewidth h>0\,.

f_k = P[X_i = hk].\,

Claim number distribution

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:

P[N=k] = p_k= (a %2B \frac{b}{k}) \cdot p_{k-1},~~k \ge 1.\,

for some a and b which fulfill a%2Bb \ge 0\,. The initial value p_0\, is determined such that \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)\, denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.

Recursion

The algorithm now gives a recursion to compute the g_k =P[S = hk] \,.

The starting value is g_0 = W_N(f_0)\, with the special cases

g_0=p_0\cdot \exp(f_0 b)\text{ if }a = 0,\,

and

g_0=\frac{p_0}{(1-f_0a)^{1%2Bb/a}}\text{ for }a \ne 0,\,

and proceed with

g_k=\frac{1}{1-f_0a}\sum_{j=1}^k \left( a%2B\frac{b\cdot j}{k} \right) \cdot f_j \cdot g_{k-j}.\,

Example

The following example shows the approximated density of \scriptstyle S \,=\, \sum_{i=1}^N X_i where \scriptstyle N\, \sim\, \text{NegBin}(3.5,0.3)\, and \scriptstyle X \,\sim \,\text{Frechet}(1.7,1) with lattice width h = 0.04. (See Fréchet distribution.)

References

  1. ^ Panjer, Harry H. (1981). "Recursive evaluation of a family of compound distributions." (PDF). ASTIN Bulletin (International Actuarial Association) 12 (1): 22–26. http://www.casact.org/library/astin/vol12no1/22.pdf. 

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