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 {{cite journal|last=Panjer|first=Harry H.|year=1981|title=Recursive evaluation of a family of compound distributions.| journal=ASTIN Bulletin|volume=12|issue=1|pages=22–26|publisher=International Actuarial Association|url=http://www.casact.org/library/astin/vol12no1/22.pdf|doi=10.1017/S0515036100006796|s2cid=15372040 }} by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo[http://www.actuaries.org/COUNCIL/Documents/CV_Panjer.pdf CV], actuaries.org; [https://math.uwaterloo.ca/statistics-and-actuarial-science/about/people/harry-panjer Staff page], math.uwaterloo.ca). 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:

: $P[N=k] = p_k= \left(a + \frac{b}{k} \right) \cdot p_{k-1},~~k \ge 1.\,$

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

In the case of claim number is known, please note the De Pril algorithm.Vose Software Risk Wiki: http://www.vosesoftware.com/riskwiki/Aggregatemodeling-DePrilsrecursivemethod.php This algorithm is suitable to compute the sum distribution of $n$ discrete random variables.{{Cite journal | doi = 10.1080/03461238.1988.10413837| title = Improved approximations for the aggregate claims distribution of a life insurance portfolio| journal = Scandinavian Actuarial Journal| volume = 1988| issue = 1–3| pages = 61–68| year = 1988| last1 = De Pril | first1 = N. }}

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

: $g_0=\frac{p_0}{(1-f_0a)^{1+b/a}} \quad \text{ for } \quad a \ne 0,\,$

and proceed with

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

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.)

Image:Expba07.jpg

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

.{{cite journal

|last1 = Guégan |first1 = D.

|last2 = Hassani |first2 = B.K.

|title = A modified Panjer algorithm for operational risk capital calculations

|year = 2009

|journal = Journal of Operational Risk

|volume = 4

|issue = 4

|pages = 53–72

|doi = 10.21314/JOP.2009.068