Panjer recursion

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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.

1 Preliminaries
- 1.1 Claim size distribution
- 1.2 Claim number distribution
2 Recursion
3 Example
4 References

[edit] 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.

[edit] 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 $h>0\,$ .

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

[edit] 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 ^[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	$\frac{\Gamma(r+k)}{k!\,\Gamma(r)}\,p^r\,(1-p)^k \,$	$1-p\,$	$(1-p)(r-1)\,$	$p^r \,$	$\left( \frac{p}{1 - x(1-p)}\right) ^r \,$	$\frac{r(1-p)}{p} \,$	$\frac{r(1-p)}{p^2} \,$

[edit] 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

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

and proceed with

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

[edit] 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.)