# Aggregate Multivariate FFT

### Introduction

The Aggregate Multivariate FFT window is a lot like the Aggregate FFT window, in that it calculates the sum of a random number of randomly sized distributions.

However, in the Aggregate Multivariate FFT window you can enter multiple pairs of severity/frequency distributions to be added.

So, for example, you could model a portfolio of two related insurance policies (e.g. one for car accidents and one for trucks) aggregating a Poisson number of Lognormal-sized variables together with a Polya number of Normal-sized 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 FFT method is explained in more mathematical detail here.

### Window elements

On the upper left of the window is the list of Frequency/Severity distribution pairs. These should be discrete, respectively continuous Vose Distribution Objects.

To add a new distribution to the list, click anywhere in the white area. To remove a frequency/severity distribution pair from the list, select it (by clicking on it) and then click the x button on the right below the list.

Preview graphs for 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:

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

### Useful tips and tricks

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.