Compound Distribution Approximations: FFT and Recursion
Compute compound distributions using FFT, Panjer recursion, and approximation methods for Exam MAS-II.
Fast Fourier Transform Method
The FFT efficiently computes the PMF of a compound distribution S = X_1 + ... + X_N. Since the characteristic function of a compound distribution is phi_S(t) = P_N(phi_X(t)), the algorithm is: (1) discretize severity onto a grid, (2) compute the discrete Fourier transform (DFT) of the severity PMF, (3) apply the frequency PGF to each element of the DFT, (4) apply the inverse DFT to obtain the compound distribution PMF. The FFT reduces computation from O(n^2) to O(n*log(n)). Aliasing errors from discretization and truncation must be managed by choosing an adequate grid size and span.
Recursion and Approximations
Panjer recursion computes exact compound PMFs for (a,b,0) class frequencies (Poisson, negative binomial, binomial) in O(n^2) time. For non-Panjer frequencies, the recursive formula does not apply, and FFT or simulation is needed. Analytical approximations include the normal approximation (fast but inaccurate in tails), translated gamma (matches three moments: mean, variance, skewness), and NP approximation (normal power, adjusts the normal quantile for skewness). The Wilson-Hilferty approximation transforms chi-squared quantiles. Exam MAS-II tests FFT implementation, Panjer recursion, and the accuracy trade-offs of various approximation methods for aggregate loss distributions.