Modelling the Reserving Cycle Using the Fourier Transform
The paper will demonstrate an application of the discrete Fourier transform in reserving by presenting a distribution free model of the reserving cycle. A parallelogram of data is organized as a periodic sequence, which is interpolated by a Fourier series. A simple criteria is given for separating the signal from the noise, revealing the loss emergence pattern. Finding the least squares trend takes a slightly different form in Fourier analysis, where Parseval’s theorem relates the total sum of squares to the sum of the squared Fourier coefficients. The reserving cycle is identified as being generated by the lowest frequencies in the remaining noise. This presents the actuary with the interesting choice of finding the trend that minimizes either the systemic risk (reserving cycle) or the total risk.