Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S=∑i=1NXi where both N and Xi 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).

## Contents

* 1 Preliminaries
* 1.1 Claim size distribution
* 1.2 Claim number distribution
* 2 Recursion
* 3 Example
* 4 References
* 5 External links

## Preliminaries

We are interested in the compound random variable S=∑i=1NXi where N and Xi fulfill the following preconditions.

### Claim size distribution

We assume the Xi to be i.i.d. and independent of N. Furthermore the Xi have to be distributed on a lattice hℕ0 with latticewidth h>0.

fk=P[Xi=hk].

In actuarial practice, Xi is obtained by discretisation of the claim density function (upper, lower...).

### 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]=pk=(a+bk)⋅pk−1,k≥1.

for some a and b which fulfill a+b≥0. The initial value p0 is determined such that ∑k=0∞pk=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 WN(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 discrete random variables.[4]

## Recursion

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

The starting value is g0=WN(f0) with the special cases

g0=p0⋅exp⁡(f0b) if a=0,

and

g0=p0(1−f0a)1+b/a for a≠0,

and proceed with

gk=11−f0a∑j=1k(a+b⋅jk)⋅fj⋅gk−j.

## Example

The following example shows the approximated density of S=∑i=1NXi where N∼NegBin(3.5,0.3) and X∼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]

## References

1. ↑ Panjer, Harry H. (1981). "Recursive evaluation of a family of compound distributions.". ASTIN Bulletin (International Actuarial Association) 12 (1): 22–26. doi:10.1017/S0515036100006796. http://www.casact.org/library/astin/vol12no1/22.pdf.
2. ↑ CV, actuaries.org; Staff page, math.uwaterloo.ca
3. ↑ Vose Software Risk Wiki: http://www.vosesoftware.com/riskwiki/Aggregatemodeling-DePrilsrecursivemethod.php
4. ↑ De Pril, N. (1988). "Improved approximations for the aggregate claims distribution of a life insurance portfolio". Scandinavian Actuarial Journal 1988 (1–3): 61–68. doi:10.1080/03461238.1988.10413837.
5. ↑ Guégan, D.; Hassani, B.K. (2009). "A modified Panjer algorithm for operational risk capital calculations". Journal of Operational Risk 4 (4): 53–72. doi:10.21314/JOP.2009.068.

## External links

* Panjer recursion and the distributions it can be used with

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Panjer recursion. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Panjer_recursion&oldid=189195"