VoseAggregateMoments | Vose Software

# VoseAggregateMoments

=VoseAggregateMoments(Frequency distribution,Severity distribution)

This array function directly calculates the four first statistical moments (mean, variance, skewness, kurtosis) of the aggregate distribution that is the result of the aggregation of the Frequency distribution and the Severity distribution.

• Frequency - a discrete distribution Object. Alternatively you can use an integer number, to calculate the moments of the sum of n variables.

• Severity - any distribution object

The output is a 4x1 (4x1) array, in which case the numerical values of the moments will be returned, or a 4x2 (2x4) array in which case the numerical values will be returned with labels, as shown in the image on the right.

The moments of an aggregate distribution can be calculated directly from those of the frequency and severity distributions.

For example, if the frequency distribution has mean, variance and skewness of , , and respectively, and the severity distribution has mean, variance and skewness of , , and  respectively, then these are the formulas for the first three moments:

 Aggregate distribution moments Mean                   Variance             Skewness

##### Using aggregate moments to demonstrate CLT

A nice way to see CLT at work is using the VoseAggregateMoments function with a number as frequency argument provides a nice illustration of Central Limit Theorem. The larger you make n, the closer the skewness and kurtosis will approach the Normal skewness and kurtosis of 0 and 3 respectively. Try inserting

{=VoseAggregateMoments(n,VoseTriangleObject(0,1,4))}

using larger and larger values of n. As you use larger n, the skewness and kurtosis (indicating the shape of the aggregate distribution) will approach more and more the Normal values of 0 and 3.

##### Using aggregate moments to check for accuracy of an aggregate calculation

ModelRisk offers several aggregate functions designed to directly determine the distribution of the sum of a random number of random variables independently drawn from the same distribution. The Panjer and FFT algorithms are based on well-known and commonly implemented numerical techniques. Whilst these techniques 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 aggregate distribution to the precise values that can be determined through manipulation of the frequency and claim size distributions.

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

Also have a look at the example model Aggregate_moments that demonstrates how one can use the direct calculation of aggregate moments to check for the accuracy of a Panjer or FFT calculation.