Aggregate Discrete window

The sum of a random number (frequency) of randomly sized (severity) variables is in itself again a distribution, called the aggregate distribution. The Aggregate Discrete function allows you to determine the distribution of the sum of a number of independent variables that follow a Discrete distribution

Output functions of this window:  VoseAggregateMC

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 the Distribution Object function for one of the allowable distribution types. 

In the Severity Distribution X field you insert a list of values that the individual severity distribution may take. This can be a list contained within { ... } or, more usually, a reference to a range in the spreadsheet.

In the Severity Distribution P field you insert a list of probabilities associated with  the values that the individual severity distribution may take. This must be a list of the same length as the previous field. Again, this can be a list contained within { ... } or, more usually, a reference to a range in the spreadsheet.

Step is an optional parameter. It defines the length of the increments used in the algorithm explained below.

MaxP is an optional parameter. It defines the cumulative probability at which the algorithm will finish evaluating the severity distribution, as explained below. It should be a value very close to 1. By default, it takes the value 0.9999 if omitted.

Charts

The top left chart displays the frequency distribution. If a single value is entered, it will show this value as a vertical line.

The top right chart shows the discrete severity distribution, i.e. the values entered in the Severity Distribution X field on the horizontal axis against the values entered in the Severity Distribution P field, which have been normalized to sum to 1.

The chart below shows the aggregate distribution. Sliders at the left and right allow you to read off cumulative probabilities.

Table

The table to the right compares the theoretical moments of the exact aggregate distribution against the approximation being produced by the VoseAggregateDiscrete function with these settings (Step and MaxP). Comparing the two columns allows the user to determine whether a sufficiently accurate approximation has been reached. If it has not, the MaxP value can be increased and the Step value decreased.

Output type

The user must enter a spreadsheet location for the VoseAggregateDiscrete function in the Output Location field. The user can select between five different types of output:

Object              Insert a distribution object function (VoseAggregateDiscreteObject)
Simulation        Simulate from the aggregate distribution (VoseAggregateDiscrete)
f(x)                   Calculate a probability mass (VoseAggregateDiscreteProb(....,FALSE))
F(x)                  Calculate a cumulative mass (VoseAggregateDiscreteProb(....,TRUE))
F-1(U)               Use the U parameter to simulate from the aggregate distribution (VoseAggregateDiscrete(...,U))

Algorithm

ModelRisk uses an adaptive algorithm that mathematically constructs the aggregate distribution. The approach is based on a well-known recursive relationship, with an adaptive component that allows the algorithm to handle a very large set of possible values.

The algorithm rounds off the discrete severity distribution provided into one whose values are integer numbers of Step apart. Thus, for example, using Severity distribution X = {1,2,3,4.1,5} and Step = 1 the algorithm will round the X values to {1,2,3,4,5}. If Step = 0.1, the X values will be unchanged.

If the Severity distribution is entered as a discrete distribution with a very long right tail, the algorithm has to deal with a huge set of possible value. The MaxP function is used to apply a limit to the amount that the tail is evaluated. Setting a MaxP value of 0.9999, for example, will ignore any tail values in the last 0.01% of the distribution.

The algorithm use works with a category of distributions known as (a,b,1). Thus, the allowed frequency distributions are: Geometric, Logarithmic, Negative Binomial, Polya, and Poisson.

 

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.

Note that there are other functions for constructing the aggregate distribution from continuous severity distribution directly with any desired accuracy and speed: Aggregate DePril, Aggregate Panjer, Aggregate Discrete. The VoseAggregateMC function also provides a generic simulation method for evaluating aggregates.

 

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