# Aggregate Monte Carlo

The sum of a random number (frequency) of randomly sized (severity) variables is in itself again a distribution, called the aggregate distribution. Use the Aggregate Monte Carlo window to directly generate sums of random variables.

We randomly sample a positive integer N from the (discrete) frequency distribution. Next, we simulate N random values from the severity distributions and add them together. The outcome is a new random value for total or "compound" severity.

Note that the severity distribution is specified as a Distribution Object. To incorporate a modified severity distribution (e.g. to model a deductible): just use the proper Object.

Note that while Aggregate MC is the most straightforward way to calculate the aggregate distribution, there are algorithms for constructing the aggregate distribution directly with any desired accuracy and speed: Aggregate DePril, Aggregate Panjer, Aggregate FFT.

Especially in insurance modeling, where modeling extreme scenarios is often of crucial importance, the power of these algorithms may be preferred for modeling, as they allow for modeling to any desired precision much faster.

Aggregate distributions are most often associated with insurance risk analysis, but have applications in virtually every type of risk analysis. Not properly understanding when to simulate aggregate distributions is one of the most common errors in modeling.

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).

### Window elements

In the Frequency Distribution field you can insert the distribution that governs the number of random variables to be added together. This should be a discrete non-negative Vose Distribution, or even a positive integer, not a Vose Distribution Object, since we are simply taking a random sample from this distribution, not manipulating it in any other way.

In the Severity Distribution field you can insert the distribution that governs the size of the individual summed samples. This should be a continuous Vose Distribution object.

In the Chart options region, you can enter the number of random samples and the approximate number of bars shown for the preview graph of the aggregate distribution.

Preview graphs for the frequency, severity and resulting aggregate distribution are shown.

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 in different ways.

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

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.