Panjer algorithm

The Panjer recursion (also known as Panjer algorithm ) is an algorithm for the probability distribution of a special composite random variable

${\displaystyle S:=\sum _{i=1}^{N}X_{i}:=\sum _{n=0}^{\infty }\chi _{\{\omega \in \Omega |N(\omega )=n\}}\sum _{i=1}^{n}X_{i}}$

to calculate. Here and are random variables which form a collective model and denote the indicator function . ${\displaystyle N}$ ${\displaystyle X_{i}}$ ${\displaystyle \chi }$

The algorithm was first published in a publication by Harry Panjer . It is widely used in insurance .

Preconditions

We are interested in the special composite random variable , where and must meet the following preconditions: ${\displaystyle \textstyle S=\sum _{i=1}^{N}X_{i}}$ ${\displaystyle N}$ ${\displaystyle X_{i}}$

Number of claims distribution

${\displaystyle N}$is a "damage number distribution", i. H. . is independent of . ${\displaystyle N\in \mathbb {N} _{0}}$ ${\displaystyle N\,}$ ${\displaystyle X_{i}\,}$

There must also be an element of the Panjer class . The Panjer class consists of all random variables with values ​​in which satisfy the following relation: with and for and with . The value is determined so that is satisfied. ${\displaystyle N}$ ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle \textstyle p_{k}=\left(a+{\frac {b}{k}}\right)\cdot p_{k-1}}$ ${\displaystyle ~~k\geq 1}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a+b\geq 0}$ ${\displaystyle p_{0}\,}$ ${\displaystyle \textstyle \sum _{k=0}^{\infty }p_{k}=1}$

Sundt proved in the paper that only the binomial distribution , the Poisson distribution and the negative binomial distribution are in the Panjer class. You have the parameters and values ​​as described in the following table, where the probability generating function is designated. ${\displaystyle W_{N}(x)\,}$

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	${\displaystyle e^{-\lambda }{\frac {\lambda ^{k}}{k!}}}$	${\displaystyle 0\,}$	${\displaystyle \lambda \,}$	${\displaystyle e^{-\lambda }\,}$	${\displaystyle e^{\lambda (x-1)}\,}$	${\displaystyle \lambda \,}$	${\displaystyle \lambda \,}$
Negative binomial	${\displaystyle {\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}}$	${\displaystyle 1-p\,}$	${\displaystyle (1-p)(r-1)\,}$	${\displaystyle p^{r}\,}$	${\displaystyle \left({\frac {p}{1-x(1-p)}}\right)^{r}\,}$	${\displaystyle {\frac {r(1-p)}{p}}\,}$	${\displaystyle {\frac {r(1-p)}{p^{2}}}\,}$

Distribution of individual losses

We assume that there are identically distributed independent random variables that are independent of . Furthermore, it must be distributed on a grid with grid length . ${\displaystyle X_{i}\,}$ ${\displaystyle N\,}$ ${\displaystyle X_{i}\,}$ ${\displaystyle h\mathbb {N} _{0}}$ ${\displaystyle h>0\,}$

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

Recursion

The algorithm uses recursion to calculate the probabilities . ${\displaystyle g_{k}=P[S=hk]\,}$

The starting value is: ${\displaystyle g_{0}=W_{N}(f_{0})\,}$

with the special cases

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

and

${\displaystyle g_{0}={\frac {p_{0}}{(1-f_{0}a)^{1+b/a}}}{\mbox{ für }}a\neq 0.}$

The following values ​​can be calculated as follows:

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

example

The example shows the approximated density function of where and . The individual damage distribution was discretized with a grid width (see also Fréchet distribution ). ${\displaystyle \textstyle S\,=\,\sum _{i=1}^{N}X_{i}}$ ${\displaystyle \textstyle N\,\sim \,{\text{NegBin}}(3.5,0.3)}$ ${\displaystyle \textstyle X\,\sim \,{\text{Frechet}}(1.7,1)}$ ${\displaystyle h=0.04}$

See also

• Panjer distribution

literature

• Schmidt, Klaus D .: Actuarial Mathematics , Springer Dordrecht Heidelberg London New York 2009, ISBN 978-3-642-01175-7 .

Individual evidence

1. ^ Harry H. Panjer: Recursive evaluation of a family of compound distributions. . (PDF) In: ASTIN Bulletin . 12, No. 1, 1981, pp. 22-26. doi : 10.1017 / S0515036100006796 .
2. ^ B. Sundt and WS Jewell: Further results on recursive evaluation of compound distributions . (PDF) In: ASTIN Bulletin . 12, No. 1, 1981, pp. 27-39. doi : 10.1017 / S0515036100006802 .