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 and of a special type. 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 $S=\sum _{i=1}^{N}X_{i}\,$ 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].\,

Claim number distribution

$N\,$ is the "claim number distribution", i.e. $N\in \mathbb {N} _{0}\,$ .

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: $P[N=k]=p_{k}=(a+{\frac {b}{k}})\cdot p_{k-1},~~k\geq 1.\,$ 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 ^[2] that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to the Panjer class, depending on the sign of $a\,$ . 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	${\binom {n}{k}}p^{k}(1-p)^{n-k}\,$	${\frac {-p}{1-p}}$	${\frac {p(n+1)}{1-p}}$	$(1-p)^{n}\,$	$(px+(1-p))^{n}\,$	$np\,$	$np(1-p)\,$
Poisson	$e^{-\lambda }{\frac {\lambda ^{k}}{k!}}\,$	$0\,$	$\lambda \,$	$e^{-\lambda }\,$	$e^{\lambda (s-1)}\,$	$\lambda \,$	$\lambda \,$
negative binomial	${\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}\,$	$1-p\,$	$(1-p)(r-1)\,$	$p^{r}\,$	$\left({\frac {p}{1-x(1-p)}}\right)^{r}\,$	${\frac {r(1-p)}{p}}\,$	${\frac {r(1-p)}{p^{2}}}\,$

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.)