20.4 Examples of simulation in actuarial modeling
20.4.1 Aggregate loss calculations
The recursive method for calculating aggregate loss distribution presented in Chapter 9 has three features. First, the recursive method is exact up to the level of the approximation introduced. The only approximation involves replacing the true severity distribution with an arithmetized approximation. The approximation error can be reduced by increasing the number of points (that is, reducing the span). Second, it assumes that aggregate claims can be written as S = X1 + ··· + XN with N, X1, X2, … independent and the Xjs identically distributed. Third, the recursive method assumes that the frequency distribution is in the (a, b, 0) or (a, b, 1) classes.
There is no need to be concerned about the first feature because the approximation error can be made as small as desired, though at the expense of increased computing time. However, the second restriction may prevent the model from reflecting reality. The third restriction means that if the frequency distribution is not in one of these classes (or can be constructed from them, such as with compound distributions), we will need to find an alternative to the recursive method. Simulation is one such alternative.
In this section, we indicate some common ways in which the independence or identical distribution assumptions may fail to hold and then demonstrate how simulation can be used to obtain numerical values of the distribution of aggregate losses. ...