Aggregate Panjer | Vose Software

# Aggregate Panjer

### Introduction

The sum of a random number (frequency) of randomly sized (severity) variables is in itself again a distribution, called the aggregate distribution.

Panjer's recursive method is an efficient method for directly constructing an approximation of the aggregate distribution, where the frequency distribution is any of the following: Poisson, Polya, Negative Binomial, Geometric, Logarithmic, Delaporte.

There are a lot of advantages to being able to construct the aggregate distribution directly, among which are:

• We can determine tail probabilities to a high precision.

• It is much faster than Monte Carlo simulation.

• We can manipulate the aggregate distribution as with any other in Monte Carlo simulation, e.g. correlate it with other variables.

A continuous distribution (e.g. a Gamma) can be fitted to the aggregate distribution (by matching moments), and this fitted distribution can in turn be inserted in the spreadsheet (see below).

The Max P parameter specifies the upper percentile value of the claim size distribution (called X from now on) at which the algorithm will stop, and the Intervals parameter 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. In ModelRisk 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.

You can read more about the mathematical details of Panjer's recursive algorithm here.

To see the output functions of this window, click here.

### Window elements

In the Aggregate parameters area, the Frequency and Severity distributions can be chosen: you can insert these manually, link dynamically to a Distribution Object in the spreadsheet, or select a distribution from the Select Distribution window.

In the two other fields listed you can specify the Number of Intervals and MaxP parameters for Panjer's algorithm. You can read the details about these in the topic about Panjer's recursive method.

Preview graphs for respectively the claim frequency distribution, claim size distribution, and aggregate distribution are shown.

Different types of output can be specified by selecting the appropriate option under the preview graph:

• Object - to insert the constructed distribution as a distribution Object in the spreadsheet.

• Simulation - (default) to generate random values from the constructed distribution.

• f(x) and F(x) - to calculate the probability density function and the cumulative distribution function of some x value(s) (an extra parameter x values will appear on the left side of the window).

• F-1(U) - to calculate the inverse cumulative when a U-value is entered.

The preview graph of the aggregate distribution below has the following special buttons in its graphics toolbar:

From left to right, these allow you to:

• Overlay one of several fitted distribution (by matching moments) to the calculated aggregate distribution.

• Insert the aggregate distribution in the spreadsheet.

• Insert the fitted overlay curve in the spreadsheet in different ways.

##### Using aggregate moments to check for accuracy

Whilst the aggregate calculation techniques offered by ModelRisk are generally very accurate, it is wise for the user to ensure that the numerical result is within the level of accuracy required.

The most direct way of testing the required accuracy is to compare the moments of the constructed aggregate distribution to the exact values that can be determined through manipulation of the frequency and claim size distributions.

That is why we have included the exact aggregate moment values for comparison in the ModelRisk aggregate De Pril, Panjer and FFT windows, in the exact column of the summary statistics table:

For explanations about other fields, buttons, graphs and summary statistics tables in this window, see

### Useful tips and tricks

See also: Graphics, workflow and error handling in ModelRisk

##### Using View Function to return to a window

The output of ModelRisk windows always corresponds to VoseFunctions (the functions ModelRisk adds to Excel) being entered into one or more spreadsheet cells.

You can always re-open the window for a ModelRisk function that is in a spreadsheet cell by using View Function. Select the spreadsheet cell and then select View Function from the ModelRisk menu/toolbar/ribbon.