# VoseAggregateProduct

=VoseAggregateProduct(Frequency distribution, Exposure distribution, Lossfraction distribution, U)

Directly constructs the distribution of the sum of a random number of random variables, where the variables to be summed are each the product of an Exposure variable multiplied by a LossFraction variable on [0,1]:

This function is useful when modeling credit risk, where we generally have separate distributions for the amount of exposure a debt holder has, and the fraction of that exposure that is realized as a loss.

• Frequency distribution - a claim frequency distribution object.

• Exposure distribution - a claim size distribution object.

• Lossfraction distribution - a distribution object with domain between zero and one.

• U - optional parameter specifying the cumulative percentile of the distribution. If omitted the function generates random values.

In the routine performed by this function, fL(x) for the individual loss distribution is calculated as follows:

where fF( ) is the density function for the loss fraction distribution and fE( ) is the density function for the exposure distribution. The aggregate distribution is then constructed directly using a Fast Fourier Transform meaning we can do probability calculations on it and take advantage of the U-parameter.

Example

Say we want to model the total loss of an insurance policy. We assume a Poisson(1000) number of claims each of LogNormal(100,20) size, but for each claim we can recover part of the payout again. so the eventual cost of an individual claim event is only a fraction of the payout:

Lossfraction*Exposure

where Lossfraction is a Beta(13,15) variable. The total cost is then modeled by:

=VoseAggregateProduct(VosePoissonObject(1000,VoseLogNormalObject(100,20,VoseBetaObject(13,15))

##### VoseFunctions for this distribution

VoseAggregateProduct generates values from this distribution or calculates a percentile.

VoseAggregateProductObject constructs a distribution object for this distribution.

VoseAggregateProductProb returns the probability density or cumulative distribution function for this distribution.

VoseAggregateProductProb10 returns the log10 of the probability density or cumulative distribution function.