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 [ cite journal|last=Panjer|first=Harry H.|year=1981|title=Recursive evaluation of a family of compound distributions.| journal=ASTIN Bulletin|volume=12|issue=1|pages=22–26|publisher=International Actuarial Association|url=http://www.casact.org/library/astin/vol12no1/22.pdf|format=PDF] . 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:$P\; [N=k]\; =\; p\_k=\; (a\; +\; frac\{b\}\{k\})\; cdot\; p\_\{k-1\},~~k\; ge\; 1.,$for some $a,$ and $b,$ which fulfill $a+b\; ge\; 0,$.the value $p\_0,$ is determined such that $sum\_\{k=0\}^infty\; p\_k\; =\; 1.,$

Sundt proved in the paper [cite journal|author=B. Sundt and W. S. Jewell|title=Further results on recursive evaluation of compound distributions|journal=ASTIN Bulletin|volume=12|issue=1|year=1981|pages=27–39|publisher=International Actuarial Association|url=http://www.casact.org/library/astin/vol12no1/27.pdf|format=PDF ] 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.

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

:$g\_0=p\_0cdot\; exp(f\_0\; b)\; ext\{\; if\; \}a\; =\; 0,,$

and

:$g\_0=frac\{p\_0\}\{(1-f\_0a)^\{1+b/a\; ext\{\; for\; \}a\; e\; 0,,$

and proceed with

:$g\_k=frac\{1\}\{1-f\_0a\}sum\_\{j=1\}^k\; left(\; a+frac\{bcdot\; j\}\{k\}\; ight)\; cdot\; f\_j\; cdot\; g\_\{k-j\}.,$

Example

The following example shows the approximated density of $scriptstyle\; S\; ,=,\; sum\_\{i=1\}^N\; X\_i$ where $scriptstyle\; N,\; sim,\; ext\{NegBin\}(3.5,0.3),$ and $scriptstyle\; X\; ,sim\; ,\; ext\{Frechet\}(1.7,1)$ with lattice width "h" = 0.04. (See Fréchet distribution.)

References

External links

* [http://www.vosesoftware.com/ModelRiskHelp/index.htm#Aggregate_distributions/Aggregate_modeling_-_Panjer_s_recursive_method.htm Panjer recursion and the distributions it can be used with]