Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution approximation 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 ^[1] by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo^[2]). It is heavily used in actuarial science (see also systemic risk).

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].\,

In actuarial practice, $X_{i}\,$ 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]=p_{k}=\left(a+{\frac {b}{k}}\right)\cdot p_{k-1},~~k\geq 1.\,

for some $a$ and $b$ which fulfill $a+b\geq 0\,$ . 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.

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 $g_{k}=P[S=hk]\,$ .

The starting value is $g_{0}=W_{N}(f_{0})\,$ with the special cases

g_{0}=p_{0}\cdot \exp(f_{0}b)\quad {\text{ if }}\quad a=0,\,

and

g_{0}={\frac {p_{0}}{(1-f_{0}a)^{1+b/a}}}\quad {\text{ for }}\quad a\neq 0,\,

and proceed with

g_{k}={\frac {1}{1-f_{0}a}}\sum _{j=1}^{k}\left(a+{\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.)

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

^ 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.
^ CV, actuaries.org; Staff page, math.uwaterloo.ca
^ Vose Software Risk Wiki: http://www.vosesoftware.com/riskwiki/Aggregatemodeling-DePrilsrecursivemethod.php
^ 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.
^ 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.

External links

Panjer recursion and the distributions it can be used with

[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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