Aggregate De Pril | Vose Software

# Aggregate De Pril

### Introduction

De Pril's recursive method is a method often used in insurance risk analysis modeling.

It calculates the aggregate payout distribution of a portfolio of J independent life insurance policies that each have a claim probability of pj. To put it in life insurance terminology, we classify policies by their different mortality rates.

The held policies are categorised according to the payout amount and the probability of claim. The possible payout amounts are discretised into M multiples of a base, i.e. base, 2*base, ..., M*base. The probability of payout is also discretised into J possible values: p1, p2, ... pJ. njm is the number of held policies with payout m*base are deemed to have probability pj of being claimed within the cover period, giving a total of MxJ different types of payout events to be modelled.

The output is the aggregate payout distribution - note that it has a certain probability attached to a zero outcome (by default this is the green vertical line on the window's preview graph).

The algorithm for calculating this aggregate payout is exact, but very computationally intensive. Specifying (optionally) a non-zero integer K gets a faster, but approximated result. K governs the payout size below which payout events are ignored in the calculation: the lower K, the faster the algorithm (at the cost of a cruder approximation).

The method is explained in more mathematical detail here.

### Window elements

In the parameters region, you can fill in the following fields:

• {probabilities} - this should be a 1xJ array, with J being the number of different policy's payout probability possibilities.

• {n} - an JXM array of the elements njm being the number of policies associated with probability pj and claim size m*base.

• base - the base number for the benefit payouts. This is typically a value like \$1000 or \$5000.

• K - optional integer parameter (>0) for using approximate rather than exact formulas in the calculations, for higher speed. If omitted, the exact payout distribution will be calculated.

The upper preview graph window plots njm against (m*base) for each of the J different probabilities of claim.

The lower pane shows a graph of the calculated aggregate distribution.

Different types of output can be specified by selecting the appropriate option under the preview graph:

• Simulation - (default) to generate random values from the constructed distribution.

• f(x) and F(x) - to calculate the probability density function and the cumulative distribution function of some x value(s) (an extra parameter x values will appear on the left side of the window).

• F-1(U) - to calculate the inverse cumulative when a U-value is entered.

##### Using aggregate moments to check for accuracy

Whilst the aggregate calculation techniques offered by ModelRisk 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 constructed aggregate distribution to the exact values that can be determined through manipulation of the frequency and claim size distributions.

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

For explanations about other fields, buttons, graphs and summary statistics tables in this window, see

### 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.