compound Poisson model
/ PWAH-sson /
Take the collective risk model and make one specific, friendly choice: let the number of claims follow a Poisson distribution. The result — a Poisson number of claims, each of a random size, summed into a total — is the compound Poisson model. It is the single most-used model for aggregate insurance losses, because the Poisson assumption unlocks a stack of clean mathematical results.
Here S = X1 + ... + XN with N Poisson(lambda) and the Xi an independent severity sample. Because the Poisson has equal mean and variance (both lambda), the aggregate moments simplify beautifully: E[S] = lambda times E[X], and Var(S) = lambda times E[X^2] (the second moment of severity), so the variance depends only on lambda and the size distribution's second moment. The crown-jewel property is additivity: the sum of independent compound Poisson variables is again compound Poisson, with the lambdas added and the severities mixed in proportion. This is why you can merge many compound Poisson sub-portfolios, or split one by claim type, and stay inside the same tractable family.
The compound Poisson is the backbone of aggregate loss modelling, ruin theory, and reinsurance pricing, precisely because of that additivity and the simple moments. It also pairs perfectly with the Panjer recursion, since Poisson sits in the (a,b,0) class, giving an exact computation of the whole aggregate distribution. The honest caveat inherits the Poisson's weakness: it forces the claim count's variance to equal its mean. When counts are over-dispersed (the common real case), the compound Poisson understates aggregate variability, and actuaries switch to a compound negative binomial or a mixed Poisson — which, conveniently, is itself often a compound Poisson with a mixed structure.
Two independent compound Poisson books: book A has lambda = 30 with average claim 1,000, book B has lambda = 20 with average claim 2,500. Their combined claims are again compound Poisson with lambda = 50, and the merged severity is a mixture — 30/50 weight on book A's sizes, 20/50 on book B's. Expected combined loss = 30 times 1,000 + 20 times 2,500 = 80,000, computed directly without re-deriving anything.
Poisson frequency gives E[S] = lambda E[X], Var(S) = lambda E[X^2], and additivity across books.
The compound Poisson inherits the Poisson's mean = variance constraint on counts. With over-dispersed claim counts it understates aggregate variability; a compound negative binomial or mixed Poisson is the standard remedy.