Compound Distribution Approximations: FFT and Recursion

The Compound Distribution Problem

Many actuarial applications require computing the distribution of aggregate losses S = X_1 + X_2 + ... + X_N, where N is a random count (claim frequency) and X_i are independent, identically distributed claim amounts (severities). The distribution of S is called a compound distribution. Computing this distribution exactly is challenging because it requires convolving the severity distribution N times and averaging over all possible values of N. Three main computational approaches exist: recursion (Panjer), transform methods (FFT), and analytical approximations.

The choice of method depends on the frequency distribution, the severity distribution, the required accuracy, and the available computational resources. Each approach has strengths and limitations that actuaries must understand to select the right tool for each application.

Panjer Recursion

Panjer recursion provides an exact method for computing the compound PMF when the frequency distribution belongs to the (a,b,0) class. A counting distribution is in this class if the ratio of consecutive probabilities satisfies p_k / p_{k-1} = a + b/k for all k >= 1. The three main members are: Poisson (a = 0, b = lambda), negative binomial (a = beta/(1+beta), b = (r-1)*beta/(1+beta)), and binomial (a = -q/(1-q), b = (m+1)*q/(1-q)).

The recursion formula for the compound PMF is: f_S(x) = (1/(1 minus a*f_X(0))) * sum from y=1 to x of (a + b*y/x) * f_X(y) * f_S(x minus y), with the initial value f_S(0) = P_N(f_X(0)), the probability generating function of N evaluated at the probability of zero severity. For the Poisson case, f_S(0) = exp(-lambda * (1 minus f_X(0))). The recursion computes f_S(x) for x = 1, 2, 3, ... in sequence, each step using previously computed values.

Panjer recursion requires the severity distribution to be discretized onto a lattice with spacing h (for example, h = 100 for claim amounts rounded to the nearest 100). The computation has O(n^2) complexity where n is the number of grid points, because each f_S(x) requires summing over all previous values. For moderate grid sizes (up to several thousand points), this is efficient. For very fine grids or very long tails, the O(n^2) cost becomes prohibitive and the FFT method is preferred.

Fast Fourier Transform Method

The FFT exploits a fundamental property of compound distributions: the characteristic function (or moment generating function or probability generating function) of S factors as phi_S(t) = P_N(phi_X(t)), where P_N is the PGF of the frequency distribution and phi_X is the characteristic function of the (discretized) severity. The algorithm proceeds in four steps.

First, discretize the severity distribution onto a grid of m points (where m is a power of 2 for FFT efficiency). Second, compute the discrete Fourier transform (DFT) of the severity PMF, producing m complex values. Third, apply the frequency PGF to each of the m Fourier-domain values: for Poisson frequency, this means computing exp(lambda * (phi_k minus 1)) at each frequency point phi_k. Fourth, apply the inverse DFT to the resulting values to obtain the compound distribution PMF on the original grid. The FFT reduces the computation from O(m^2) to O(m * log(m)), enabling much finer grids and longer tails than Panjer recursion.

Managing FFT Errors

The FFT method introduces two types of numerical errors that must be controlled. Discretization error arises from approximating a continuous severity distribution on a discrete grid. The grid spacing h should be small enough that the discretized distribution accurately represents the original continuous distribution. Methods for discretizing include the rounding method (placing the probability mass at the nearest grid point), the mean-preserving method (distributing mass between adjacent points to preserve the mean), and the unbiased method (matching both the mean and the cell probability).

Aliasing error arises from the periodic nature of the DFT. The DFT treats the PMF as periodic with period m*h, so probability that falls beyond the grid wraps around and contaminates the lower end of the distribution. To minimize aliasing, choose the grid span m*h large enough to capture essentially all of the probability mass of the compound distribution. Tilting (exponentially shifting the severity distribution before the FFT and reversing the shift afterward) can reduce aliasing by concentrating the distribution within the grid.

Analytical Approximations

When exact computation is unnecessary or infeasible, analytical approximations provide quick estimates of the compound distribution or its quantiles. The normal approximation uses the central limit theorem: S is approximately N(E[S], Var(S)), where E[S] = E[N]*E[X] and Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. This approximation is fast and easy to compute but inaccurate in the tails, especially for skewed distributions. It tends to underestimate the probability of large aggregate losses.

The translated gamma approximation matches the first three moments of S to a shifted gamma distribution. The parameters of the gamma distribution are determined by matching the mean, variance, and skewness of S. Since the gamma distribution is right-skewed (like most aggregate loss distributions), this approximation captures the asymmetry that the normal approximation misses. It provides substantially better tail estimates than the normal approximation, especially at moderate confidence levels (up to about the 95th percentile).

The normal power (NP) approximation adjusts the standard normal quantile to account for skewness. The p-th quantile of S is approximated as E[S] + sigma_S * (z_p + (z_p^2 minus 1) * gamma_1 / 6), where z_p is the p-th standard normal quantile, sigma_S is the standard deviation of S, and gamma_1 is the skewness coefficient of S. This formula is easy to compute and provides a quick estimate of specific quantiles without requiring the full distribution.

Comparing Methods

Panjer recursion provides exact results (subject to discretization) for (a,b,0) class frequencies but has O(n^2) complexity and does not apply to frequencies outside this class. The FFT provides exact results for any frequency distribution (as long as the PGF or characteristic function is known) with O(n*log(n)) complexity but requires careful management of discretization and aliasing errors. Analytical approximations are the fastest but least accurate, with accuracy depending on how well the true distribution matches the approximating family.

For exam purposes, know when each method applies. Panjer recursion is the method of choice for small to moderate problems with Poisson, negative binomial, or binomial frequencies and discretized severities. The FFT is preferred for large-scale computations or non-Panjer frequencies. Approximations are useful for quick estimates, sanity checks, and situations where the full distribution is not needed and only specific quantiles or moments are required.

Common Pitfalls

A frequent error with Panjer recursion is using the wrong initial value f_S(0). This value depends on both the frequency distribution and the probability of zero severity, and getting it wrong produces an incorrect entire distribution. For the Poisson case, f_S(0) = exp(-lambda * (1 minus p_0)), where p_0 is the probability of zero severity. If the severity distribution has no mass at zero (the usual case for loss amounts), then p_0 = 0 and f_S(0) = exp(-lambda).

With the FFT, students sometimes forget that the grid must be a power of 2 for the standard FFT algorithm to apply. Using a non-power-of-2 grid either produces an error or requires a slower DFT implementation. Also remember that the FFT produces the compound PMF on a discrete grid, and interpolation may be needed to estimate probabilities at non-grid points. Finally, when comparing approximation methods, always check against exact results (from Panjer or FFT) on a simple example to calibrate the approximation error before relying on the approximation for a more complex problem.