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# Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution of a compound random variable
$S = sum_\left\{i=1\right\}^N X_i.,$.
where both $N,$ and $X_i,$ are stochastic and of a special type. It was introduced in a paper of Harry Panjer . It is heavily used in actuarial science.

## Preliminaries

We are interested in the compound random variable $S = sum_\left\{i=1\right\}^N X_i,$ 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 $h mathbb\left\{N\right\}_0,$ with latticewidth $h>0,$.

$f_k = P\left[X_i = hk\right].,$

### Claim number distribution

$N,$ is the "claim number distribution", i.e. $N in mathbb\left\{N\right\}_0,$.

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: $P\left[N=k\right] = p_k= \left(a + frac\left\{b\right\}\left\{k\right\}\right) cdot p_\left\{k-1\right\},~~k ge 1.,$ for some $a,$ and $b,$ which fulfill $a+b ge 0,$. the value $p_0,$ is determined such that $sum_\left\{k=0\right\}^infty p_k = 1.,$

Sundt proved in the paper that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to the Panjer class, depending on the sign of $a,$. They have the parameters and values as described in the following table. $W_N\left(x\right),$ denotes the probability generating function.

Distribution $P\left[N=k\right],$ $a,$ $b ,$ $p_0,$ $W_N\left(x\right),$ $E\left[N\right],$ $Var\left(N\right),$
Binomial $binom\left\{n\right\}\left\{k\right\} p^k \left(1-p\right)^\left\{n-k\right\} ,$ $frac\left\{-p\right\}\left\{1-p\right\}$ $frac\left\{p\left(n+1\right)\right\}\left\{1-p\right\}$ $\left(1-p\right)^n,$ $\left(px+\left(1-p\right)\right)^\left\{n\right\} ,$ $np,$ $np\left(1-p\right) ,$
Poisson $e^\left\{-lambda\right\}frac\left\{ lambda^k\right\}\left\{k!\right\},$ $0,$ $lambda ,$ $e^\left\{- lambda\right\},$ $e^\left\{lambda\left(s-1\right)\right\} ,$ $lambda,$ $lambda ,$
negative binomial $frac\left\{Gamma\left(r+k\right)\right\}\left\{k!,Gamma\left(r\right)\right\},p^r,\left(1-p\right)^k ,$ $1-p,$ $\left(1-p\right)\left(r-1\right),$ $p^r ,$ $left\left(frac\left\{p\right\}\left\{1 - x\left(1-p\right)\right\}right\right) ^r ,$ $frac\left\{r\left(1-p\right)\right\}\left\{p\right\} ,$ $frac\left\{r\left(1-p\right)\right\}\left\{p^2\right\} ,$

## Recursion

The algorithm now gives a recursion to compute the $g_k =P\left[S = hk\right] ,$.

The starting value is $g_0 = W_N\left(f_0\right),$ with the special cases

$g_0=p_0cdot exp\left(f_0 b\right)text\left\{ if \right\}a = 0,,$

and

$g_0=frac\left\{p_0\right\}\left\{\left(1-f_0a\right)^\left\{1+b/a\right\}\right\}text\left\{ for \right\}a ne 0,,$

and proceed with

$g_k=frac\left\{1\right\}\left\{1-f_0a\right\}sum_\left\{j=1\right\}^k left\left(a+frac\left\{bcdot j\right\}\left\{k\right\} right\right) cdot f_j cdot g_\left\{k-j\right\}.,$

## Example

The following example shows the approximated density of $scriptstyle S ,=, sum_\left\{i=1\right\}^N X_i$ where $scriptstyle N, sim, text\left\{NegBin\right\}\left(3.5,0.3\right),$ and $scriptstyle X ,sim ,text\left\{Frechet\right\}\left(1.7,1\right)$ with lattice width h = 0.04. (See Fréchet distribution.)

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