Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable ${\displaystyle S=\sum _{i=1}^{N}X_{i}\,}$. where both ${\displaystyle N\,}$ and ${\displaystyle X_{i}\,}$ 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 of Harry Panjer.^[1] It is heavily used in actuarial science.

Preliminaries

We are interested in the compound random variable ${\displaystyle S=\sum _{i=1}^{N}X_{i}\,}$  where ${\displaystyle N\,}$  and ${\displaystyle X_{i}\,}$  fulfill the following preconditions.

Claim size distribution

We assume the ${\displaystyle X_{i}\,}$  to be i.i.d. and independent of ${\displaystyle N\,}$ . Furthermore the ${\displaystyle X_{i}\,}$  have to be distributed on a lattice ${\displaystyle h\mathbb {N} _{0}\,}$  with latticewidth ${\displaystyle h>0\,}$ .

${\displaystyle f_{k}=P[X_{i}=hk].\,}$ 

In actuarial practice, ${\displaystyle X_{i}\,}$  is obtain 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:

${\displaystyle P[N=k]=p_{k}=(a+{\frac {b}{k}})\cdot p_{k-1},~~k\geq 1.\,}$ 

for some a and b which fulfill ${\displaystyle a+b\geq 0\,}$ . The initial value ${\displaystyle p_{0}\,}$  is determined such that ${\displaystyle \sum _{k=0}^{\infty }p_{k}=1.\,}$ 

The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following ${\displaystyle W_{N}(x)\,}$  denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.

Recursion

The algorithm now gives a recursion to compute the ${\displaystyle g_{k}=P[S=hk]\,}$ .

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

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

and

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

and proceed with

${\displaystyle g_{k}={\frac {1}{1-f_{0}a}}\sum _{j=1}^{k}\left(a+{\frac {b\cdot j}{k}}\right)\cdot f_{j}\cdot g_{k-j}.\,}$ 

Example

The following example shows the approximated density of ${\displaystyle \scriptstyle S\,=\,\sum _{i=1}^{N}X_{i}}$  where ${\displaystyle \scriptstyle N\,\sim \,{\text{NegBin}}(3.5,0.3)\,}$  and ${\displaystyle \scriptstyle X\,\sim \,{\text{Frechet}}(1.7,1)}$  with lattice width h = 0.04. (See Fréchet distribution.)

 

References

1. Panjer, Harry H. (1981). "Recursive evaluation of a family of compound distributions" (PDF). ASTIN Bulletin. 12 (1). International Actuarial Association: 22–26.

External links

• Panjer recursion and the distributions it can be used with