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 random variables and of special types. In more general cases the distribution of S is a compound distribution. The recursion for the special cases considered was introduced in a paper of Harry Panjer. It is heavily used in actuarial science.

Preliminaries

We are interested in the compound random variable $S = \sum_{i=1}^N X_i\,$ 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

The number of claims N is a random variable, which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the probability distribution of N has to be a member of the Panjer class, otherwise known as the (a,b,0) class of distributions. This 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 initial value $p_0\,$ is determined such that $\sum_{k=0}^\infty p_k = 1.\,$

The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following $W_N(x)\,$ denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.

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

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

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.)

External links

• Panjer recursion and the distributions it can be used with