The distribution of a book's aggregate loss obtained by summing a Poisson number of independent claim amounts, the backbone of the collective risk model and of any annual loss simulation.
The collective risk model separates what needs separating: claim count on one side, claim amount on the other. If the count follows a Poisson law with parameter lambda and amounts are mutually independent and independent of the count, aggregate loss follows a compound Poisson distribution. Two results follow immediately and get used daily: expected loss is lambda times average cost, and its variance is lambda times the second moment of severity, so relative volatility decays as the square root of lambda, which is the exact statement of the law of large numbers in insurance. The distribution itself has no closed form, and you get it through Panjer recursion, fast Fourier transform, or simulation. Its weakness is the Poisson assumption, which forces variance to equal mean: as soon as frequency is overdispersed, whether from contagion, weather cycles or portfolio heterogeneity, you switch to a negative binomial, and the aggregate loss tail thickens noticeably.
Small business cyber book, 2026 underwriting year: 4,000 policies, annual frequency 1.5%, so lambda equals 60 expected claims, average cost 85K EUR, severity standard deviation 220K EUR. Expected loss 5.1M EUR, variance equals 60 times (85,000 squared plus 220,000 squared), giving a standard deviation of 1.83M EUR and a coefficient of variation of 36%. With a negative binomial frequency of the same mean and twice the variance, the standard deviation rises to 2.3M EUR.
modèle collectif, collective risk model, charge agrégée, compound distribution, fréquence-sévérité agrégée