Moments of an aggregate distribution

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See also: Aggregate distributions introduction, Aggregate modeling in ModelRisk, Insurance and finance risk analysis modeling introduction

There are general formulas for determining the moments of an aggregate distribution given that one has the moments for the frequency distribution for n and the severity distribution for X.

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 the aggregate distribution has the following moments:

There is also a formula for kurtosis but it is rather ugly. The ModelRisk function VoseAggregateMoments determines the first four moments of an aggregate distribution for any frequency and severity distribution, even if they are bounded and/or shifted.

The equations shown above deserve a little more exploration. First, let's consider the situation where n is a fixed value, so , , and . Then we have moments for the aggregate distribution of:

You can see that this gives support to Central Limit Theorem which says that if n is large enough the aggregate distribution approaches a Normal distribution with mean  and variance . The skewness equation shows that the aggregate skewness is proportional to the skewness of X but decreases rapidly at first with increasing n, then more slowly, and asymptotically towards zero

Another interesting example is to consider the aggregate moments equations when n follows a Poisson() distribution, which is very commonly the most appropriate distribution for n, and also has the convenience of being described by just one parameter. Now we have

, ,  

and the aggregate moments are:

The mean and variance equations are simple formulae. We can see the skewness decreases with  in the same way as it does for a fixed value for n. If X is symmetrically distributed then, for a given , the skewness is at its maximum when the mean and standard deviation of X are the same and at its lowest when the standard deviation is very high. Thus, the aggregate distribution will be more closely Normal when V_c is large.

Being able to determine the aggregate moments is pretty useful. One can directly compare sums of random variables. One can also match these moments to some parametric distribution and use that as an approximation to the aggregate distribution. An aggregate distribution is almost always right skewed, so we can select from a number of right-skewed distributions like the Lognormal and Gamma and match moments. For example, a Gamma(a,b) distribution shifted by a value T has:

Thus, matching skewness gives us a value for a. Then matching variance gives us b, and finally matching means gives us T. Adding a shift gives us three parameters to estimate so we can match three moments. The following model gives an example:

Example model Aggregate_moments for determining aggregate moments

Cells C3:C5 are the parameters for the model. Cells D3 and D4 use ModelRisk functions to create distribution objects. B8:C11 and D8:E11 use the VoseMoments function to calculate the moments of the two distributions. Alternatively, you can use the equations we have listed for every univariate distribution in ModelRisk.  F8:F10 manually calculates the first three aggregate moments, and G8:H11 calculates all four using the VoseAggregateMoments function as a check. In C15:C17 the moment equations are inverted to determine the Gamma distribution parameters. Finally, G14:H17 uses the VoseMoments function again to determine the moments of the Gamma distribution. You can see that they match the mean, variance and skewness of the aggregate distribution - as they should - but also that the kurtosis is very close, so the Gamma distribution would likely be a good substitute for the aggregate distribution.

To be sure we would need to plot the two together. In every aggregate calculation window in ModelRisk we can fit shifted versions of the Gamma, Lognormal, Pearson 5, Pearson 6 and Fatigue distributions to constructed aggregate distributions and overlay the distributions for an extra visual comparison.

Read on: Panjer's recursive method