Panjer recursion explained

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

S = \sum ^{N} _{i =1} 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 ^{N} _{i =1} 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 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 + (b)/(k) \right) ⋅ 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 ^{inf ty} _{k =0} 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 casesg _{0=p} _{0 ⋅} \exp(f _{0} b) if a = 0 ,

and

g _{0= (p _{0})/(( 1- f ^{1+ b/a} _{0 a)})} for a \ne 0 ,

and proceed with

g _{k = (1)/(1- f _{0a})} \sum ^{k} _{j =1} \left( a + (b ⋅ j)/(k) \right) ⋅ f _{j} ⋅ g _{k-j} .

Example

The following example shows the approximated density of

\scriptstyle S = \sum ^{N} _{i =1} X _{i} where \scriptstyle N \sim NegBin ( 3 . 5 , 0 . 3 ) and \scriptstyle X \sim 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. 12. 1. 22–26. International Actuarial Association. 10.1017/S0515036100006796. 15372040 .
2. http://www.actuaries.org/COUNCIL/Documents/CV_Panjer.pdf CV
3. Vose Software Risk Wiki: http://www.vosesoftware.com/riskwiki/Aggregatemodeling-DePrilsrecursivemethod.php
4. 10.1080/03461238.1988.10413837. Improved approximations for the aggregate claims distribution of a life insurance portfolio. Scandinavian Actuarial Journal. 1988. 1–3. 61–68. 1988. De Pril . N. .
5. Guégan . D. . Hassani . B.K. . A modified Panjer algorithm for operational risk capital calculations . 2009 . Journal of Operational Risk . 4 . 4 . 53–72 . 10.21314/JOP.2009.068 . 4992848 . 10.1.1.413.5632.

External links

• Panjer recursion and the distributions it can be used with