Panjer recursion

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

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

where both $N$ and $X_{i}$ are stochastic. It was introduced in a paper of Harry Panjer . It is heavily used in actuarial science.

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 number distribution

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

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

Sundt proved in the paper 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. $W_{N}(x)$ denotes the probability generating function.

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

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 $h>0$.

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

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

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

and

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

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.)