Recursion that computes the aggregate loss distribution exactly without simulation, when frequency belongs to the (a, b, 0) class and severities are discretized.
Computing an aggregate loss distribution by direct convolution costs time that explodes with claim count. Harry Panjer showed in 1981 that if the frequency law satisfies the recurrence p(n) equals (a plus b over n) times p(n minus 1), which Poisson, binomial and negative binomial do and nothing else does, then the aggregate distribution is built by a simple recursion over a grid of discretized severities. You get a distribution that is exact up to discretization, hence free of simulation noise, which matters when reading an extreme percentile: ten thousand simulations still leave visible uncertainty on the 99.5th percentile, the recursion leaves none. Two precautions govern its use. Severity discretization introduces a bias you control by preserving the mean within each cell, and accumulated rounding error can destabilize the recursion at very high intensities, a case where fast Fourier transform regains the advantage.
Pricing an excess of loss treaty, 2026 underwriting year, Poisson frequency with intensity 12 above the attachment point, severities discretized in 50K EUR steps over 400 cells. Panjer recursion returns the full aggregate distribution in under a second and puts the 99.5th percentile at 23.4M EUR, against a 22.9M to 23.9M EUR interval from 10,000 Monte Carlo simulations of the same model.
récursion de Panjer, classe (a, b, 0), Panjer algorithm, recursive aggregate loss