Calculation of aggregate moments
An example of a Monte Carlo simulation risk analysis model for the insurance industry
Minimum software requirements: ModelRisk Complete edition
Technical difficulty: 3
Techniques used: Monte Carlo simulation in Excel
ModelRisk functions used: VoseAggregateMoments, Panjer, De Pril , FFT
An aggregate distribution is the sum of a random number of random variables. The number of variables to be summed is defined by a frequency distribution, and the random variables being summed are independent samples from the same severity distribution. If one knows the moments of both the frequency and severity distributions it is possible to directly determine the moments of the aggregate distribution.
The ModelRisk function VoseAggregateMoments will return the aggregate distribution for any combination of applicable univariate distributions available with ModelRisk, including when either or both distribution is truncated or shifted.
This is a powerful tool for ensuring that approximations are sufficiently accurate, which is why these values are included for comparison in the Panjer, De Pril and FFT windows, in the exact column of the summary statistics table, for example:
The example model Aggregate_moments demonstrates how one can use the direct calculation of aggregate moments to check for the accuracy of a Panjer or FFT calculation.
One application of directly determining aggregate moments has been to then use Method of Moments to fit some parametric distribution. If there is essentially no probability of the aggregate distribution taking a value of zero one can fit one of the continuous parametric distributions.
For example, the Gamma distribution with a positive shift is quite popular because one can fit to the first three moments (mean, variance, skewness). The AggregateMC, Panjer, FFT, De Pril and MultiFFT windows in ModelRisk allow the user to fit a distribution based on matching aggregate moments and place the fitted distribution in a spreadsheet. We don't recommend this method for critical analysis, and suggest that you at least compare the fitted parametric distribution to a Panjer or FFT first.