The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable where both and 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).

Preliminaries

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We are interested in the compound random variable   where   and   fulfill the following preconditions.

Claim size distribution

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We assume the   to be i.i.d. and independent of  . Furthermore the   have to be distributed on a lattice   with latticewidth  .

 

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

Claim number distribution

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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   and   which fulfill  . The initial value   is determined such that  

The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following   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   discrete random variables.[4]

Recursion

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The algorithm now gives a recursion to compute the  .

The starting value is   with the special cases

 

and

 

and proceed with

 

Example

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The following example shows the approximated density of   where   and   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

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  1. ^ Panjer, Harry H. (1981). "Recursive evaluation of a family of compound distributions" (PDF). ASTIN Bulletin. 12 (1). International Actuarial Association: 22–26. doi:10.1017/S0515036100006796. S2CID 15372040.
  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. CiteSeerX 10.1.1.413.5632. doi:10.21314/JOP.2009.068. S2CID 4992848.
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