Panjer recursion
Recursion that computes the aggregate loss distribution exactly without simulation, when frequency belongs to the (a, b, 0) class and severities are discretized.
Panjer, 1981, ASTIN Bulletin: Harry Panjer showed that if the frequency distribution satisfies p(n) equals (a plus b over n) times p(n minus 1), the aggregate loss distribution can be built by a simple recursion over a grid of discretized severities. Poisson, binomial and negative binomial satisfy that relation, and nothing else does. Direct convolution, by comparison, has a run time that blows up with the number of claims. The result is exact up to discretization, with no simulation noise, which is what matters when you read an extreme percentile: ten thousand simulations still leave visible uncertainty on the 99.5th percentile, the recursion leaves none. Two caveats. Severity discretization introduces a bias, kept in check by preserving the mean within each cell. And at very high intensities, accumulated rounding error can destabilize the recursion, at which point the fast Fourier transform takes over.
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.
Panjer recursion, Panjer algorithm, (a, b, 0) class, recursive aggregate loss
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