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 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 ${\displaystyle S=\sum _{i=1}^{N}X_{i}\,}$  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 ${\displaystyle h>0\,}$ .

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

Claim number distribution

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

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: ${\displaystyle P[N=k]=p_{k}=(a+{\frac {b}{k}})\cdot p_{k-1},~~k\geq 1.\,}$  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, depending on the sign of ${\displaystyle a\,}$ . 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	${\displaystyle {\binom {n}{k}}p^{k}(1-p)^{n-k}\,}$	${\displaystyle {\frac {-p}{1-p}}}$	${\displaystyle {\frac {p(n+1)}{1-p}}}$	${\displaystyle (1-p)^{n}\,}$	${\displaystyle (px+(1-p))^{n}\,}$	${\displaystyle np\,}$	${\displaystyle np(1-p)\,}$
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	${\displaystyle {\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}\,}$	${\displaystyle 1-p\,}$	${\displaystyle (1-p)(r-1)\,}$	${\displaystyle p^{r}\,}$	${\displaystyle \left({\frac {p}{1-x(1-p)}}\right)^{r}\,}$	${\displaystyle {\frac {r(1-p)}{p}}\,}$	${\displaystyle {\frac {r(1-p)}{p^{2}}}\,}$

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.