Aggregate FFT

See also: Aggregate modeling in ModelRisk, Aggregate distributions introduction, Aggregate modeling - Fast Fourier Transform (FFT) method

Introduction

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

The Aggregate FFT window from ModelRisk

The Aggregate FFT window directly constructs the aggregate distribution using the Fast Fourier Transform method. 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.

In the FFT algorithm the severity distribution is divided into a number m=2^n of discrete steps. By default n=12 is chosen. Optionally n can be increased with the 'Density level' field: increasing n by one - doubles - the number of discrete steps, yielding a higher accuracy at the cost of a slower calculation. This can be necessary when working with a long-tailed severity distribution.

Compare the FFT moments with the exact moments in the summary statistics table ('FFT' and 'Exact' columns) to check the calculation's accuracy with the chosen density level and increase if necessary.

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 FFT method is explained in more mathematical detail here.

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

Window elements

In the Aggregate parameters region, you can specify the Frequency distribution (a discrete distribution object) and the Severity distribution (a continuous distribution object) in the fields labeled accordingly.

You can also specify the Density level. If omitted, this will have a default value of 12.

Preview graphs of the frequency, severity and resulting 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: