Compound Poisson distribution
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.
Lundberg, 1903, Uppsala doctoral thesis: the collective risk model separates claim count from claim amount. 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 get used daily. Expected loss is lambda times average cost; its variance is lambda times the second moment of severity. Relative volatility therefore decays as one over the square root of lambda, which is the insurer's law of large numbers stated exactly. The distribution itself has no closed form, and you get it through Panjer recursion (ASTIN Bulletin, 1981), fast Fourier transform or simulation. Its weak point is the Poisson assumption, which forces variance to equal mean. As soon as frequency is overdispersed, from contagion, weather cycles or portfolio heterogeneity, you switch to a negative binomial, and the aggregate loss thickens.
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 1.94M EUR.
Follow the research
One letter a week. First, the week's underwriting note, when one is published: a dated, argued decision, signed by Paul Bernardski. Then what the observatory published on cyber risk, political risk and insurance. No account, no tracking.
No spam, one-click unsubscribe, no data ever sold.