Panjers recursive method

See also: Aggregate distributions introduction, Aggregate modeling in ModelRisk, Vose Aggregate Panjer

Panjer's recursive method (Panjer 1981, Panjer and Willmot 1992) applies where the number of variables n being added together follows one of these distributions: Binomial, Geometric, Negative Binomial, Poisson, Polya

The technique begins by taking the claim size distribution and discretising it into a number of values with increment C. Then the probability is redistributed so that the discretised claim distribution has the same mean as the continuous variable. There are a few ways of doing this, but if the discretisation steps are small they give essentially the same answer. A simple method is to assign the value (i*C) the probability si as follows:

In the discretisation process we have to decide on a maximum value of i (called r) so we don't have an infinite number of calculations to perform.  Now comes the clever part. The above discrete distributions lead to a simple one time summation through a recursive formula to calculate the probability p(j) that the aggregate distribution will equal j*C:

 

The formula works for all frequency distributions for n that are of the (a,b,0) class which means that from P(n=0) up we have a recursive relationship between P(n=i) and P(n=i-1) of the form:

a and b are fixed values that depend on which of the discrete distributions is used and their parameter value. The specific formula for each case is given below for the (a,b,0) class of discrete distributions:

  • For the Binomial(n,p): ,   

  • For the Geometric(p):   ,     

  • For the NegBin(s,p):   ,

  • For the Poisson(λ):     ,   a = 0 , b = λ

  • For the Polya(α,β):     ,

The output of the algorithm is two arrays {i}, p(i} that can be constructed into a distribution, for example as VoseDiscrete({i}, p{i})*C. Panjer's method can occasionally numerically 'blow up' with the binomial distribution, but when it does so it generates negative probabilities so is immediately obvious.

A small change to Panjer's algorithm allows the formula to be applied to (a,b,1) distributions, which mean that the recursive formula works from P(n=1) onwards. This allows us to include the Logarithmic distribution using the formulae:

Panjer's method cannot, however, be applied to the Delaporte distribution. Panjer's method requires a bit of hands-on management because one has to experiment with the maximum value r to ensure sufficient coverage and accuracy of the distribution. ModelRisk uses two controls for this: MaxP specifies the upper percentile value of the distribution of X at which the algorithm will stop, and Intervals specifies how many steps will be used in the discretisation of the X distribution. In general the larger one makes Intervals, the more accurate the model will be but at the expense of computation time. The MaxP value should be set high enough to realistically cover the distribution of X but if one sets it too high for a long tailed distribution, there will be an insufficient number of increments in the main body of the distribution.

Panjer's recursive method is implemented in ModelRisk with the Vose Aggregate Panjer window. One can compare the exact moments of the aggregate distribution with those of the Panjer constructed distribution to ensure that the two correspond with sufficient accuracy for the analyst's needs.

Read on: Aggregate modeling - Fast Fourier Transform (FFT) method
 

 

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