Panjer recursion

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

${\displaystyle S=\sum _{i=1}^{N}X_{i}.}$.

where both ${\displaystyle N}$ and ${\displaystyle X_{i}}$ are stochastic. It 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 number distribution

${\displaystyle N}$  is the "claim number distribution", i.e. ${\displaystyle N\in \mathbb {N} _{0}}$ . ${\displaystyle N}$  is assumed to be independent of the ${\displaystyle X_{i}}$ .

Furthermore, ${\displaystyle N}$  has to be a member of the Panjer class. The Panjer class consists of all counting random variables which fulfill the following relation: ${\displaystyle p_{k}=(a+{\frac {b}{k}})\cdot p_{k-1},~~k\geq 1.}$  for some ${\displaystyle a}$  and ${\displaystyle b}$  which fulfill ${\displaystyle a+b\geq 0}$ . the value ${\displaystyle p_{0}}$  is determined such that ${\displaystyle \sum _{k=0}^{\infty }p_{k}=1.}$ 

Sundt proved in the paper ^[2] that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to the Panjer class. They have the parameters and values as described in the following table. ${\displaystyle W_{N}(x)}$  denotes the probability generating function.

Distribution	${\displaystyle P[N=k]}$	${\displaystyle a}$	${\displaystyle b}$	${\displaystyle p_{0}}$	${\displaystyle W_{N}(x)}$	${\displaystyle E[N]}$	${\displaystyle Var(N)}$
Binomial	${\displaystyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}$	${\displaystyle {\frac {p}{p-1}}}$	${\displaystyle {\frac {p(n+1)}{1-p}}}$	${\displaystyle (1-p)^{n}}$	${\displaystyle (px+(1-p))^{n}}$	${\displaystyle np}$	${\displaystyle np(1-p)}$
Poisson	${\displaystyle e^{-\lambda }{\frac {\lambda ^{k}}{k!}}}$	${\displaystyle 0}$	${\displaystyle \lambda }$	${\displaystyle e^{-\lambda }}$	${\displaystyle e^{\lambda (s-1)}}$	${\displaystyle \lambda }$	${\displaystyle \lambda }$
negative binomial	${\displaystyle {\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}}$	${\displaystyle 1-p}$	${\displaystyle (1-p)(r-1)}$	${\displaystyle p^{r}}$	${\displaystyle \left({\frac {p}{1-z(1-p)}}\right)^{r}}$	${\displaystyle {\frac {r(1-p)}{p}}}$	${\displaystyle {\frac {r(1-p)}{p^{2}}}}$

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 ${\displaystyle h>0}$ .

${\displaystyle f_{k}=P[X_{i}=hk].}$ 

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.
2. B. Sundt and W. S. Jewell (1981). "Further results on recursive evaluation of compound distributions" (PDF). ASTIN Bulletin. 12 (1). International Actuarial Association: 27–39.