Fast Fourier Transform (FFT) method | Vose Software

# Fast Fourier Transform (FFT) method

The density function f(x) of a continuous random variable can always be converted into its Fourier transform fx(t) (also called it characteristic function) as follows:

and we can transform back using:

Characteristic functions are really useful for determining the sums of random variables because fX+Y(t) = fX(t)* fY(t), i.e. we just multiply the characteristic functions of variables X and Y to get the characteristic function of (X+Y). For example, the characteristic function for a Normal distribution is

.

Thus, for variables X = Normal(mX, sX) and Y = Normal(mY, sY) we have:

In this particular example the function form of  equates to another Normal distribution with mean  and variance , so we don't have to apply a transformation back - we can already recognise the result.

The fast Fourier transform method of constructing an aggregate distribution where there are a random number n of identically distributed random variables X to be summed is described fully in Robertson (1992). The technique involves discretising the severity distribution X like Panjer's method so that one has two sets of discrete vectors, one each for the frequency and severity distributions.

The mathematics involves complex numbers and is based on the convolution theory of discrete Fourier transforms, which says that to obtain the aggregate distribution one multiplies the two discrete Fourier transforms of these vectors pointwise and then computes the inverse discrete Fourier transform. The fast Fourier transform is used as a very quick method for computing the Discrete Fourier transform for long vectors.

The main advantage of the FFT method is that it is not recursive so when one has a large array of possible values the FFT won't suffer the same error propagation that Panjer's recursion will. The FFT can also take any discrete distribution for its frequency distribution (and, in principle, any other non-negative continuous distribution if one discretises it). The FFT can also be started away from zero, whereas the Panjer method must calculate the  probability of every value staring at zero. Thus, as a rough guide, consider using Panjer's method where the frequency distribution does not take very large values and where it is one of those for which Panjer's method applies, otherwise use the FFT method. ModelRisk offers a version of the FFT method

The FFT method is implemented in ModelRisk with some adjustments to improve efficiency and allow for a continuous aggregate distribution in the Aggregate FFT window. One can compare the exact moments of the aggregate distribution with those of the FFT constructed distribution to ensure that the two correspond with sufficient accuracy for the analyst's needs.

FFT methods can also be extended to a group of {n,X} paired distributions. This is called multivariate FFT, which ModelRisk makes available via the Aggregate Multi FFT window.