Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution approximation 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 by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo ). It is heavily used in actuarial science (see also systemic risk).

The Panjer recursion is an algorithm to compute the probability distribution approximation 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 by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo ). It is heavily used in actuarial science (see also systemic risk). We are interested in the compound random variable S = \sum_{i=1}^N X_i\, where N\, and X_i\, fulfill the following preconditions. f_k = P[X_i = hk].\, In actuarial practice, X_i\, is obtained by discretisation of the claim density function (upper, lower...). P[N=k] = p_k= \left(a + \frac{b}{k} \right) \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. In the case of claim number is known, please note the De Pril algorithm. This algorithm is suitable to compute the sum distribution of n discrete random variables. 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) \quad \text{ if } \quad a = 0,\, and g_0=\frac{p_0}{(1-f_0a)^{1+b/a}} \quad \text{ for } \quad 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}.\, 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.) As observed, an issue may arise at the initialization of the recursion. Guégan and Hassani (2009) have proposed a solution to deal with that issue . ==External links== Panjer recursion and the distributions it can be used with

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