Concept

Compound Poisson Distribution

A compound Poisson model adds independent claim severities over a Poisson claim count. It is the basic aggregate-loss model behind FAM, ASTAM, Panjer recursion, and many ruin-theory examples.

Quick Answer

A compound Poisson aggregate loss has the form S = X_1 + ... + X_N, where N is Poisson with mean lambda and the severities X_i are independent of N. If N = 0, then S = 0.

The July 2026 FAM syllabus places aggregate risk models inside the severity, frequency, and aggregate models block. The Spring 2026 ASTAM syllabus then asks candidates to perform calculations for sums of compound Poisson models.

Mean And Variance

The compound Poisson formulas are simple because the Poisson count has mean and variance both equal to lambda. The aggregate mean is frequency times severity mean. The aggregate variance is lambda times the second raw moment of severity.

That second formula is often the exam trap. It is not lambda times Var(X) unless the severity mean is zero, which is not true for claim amounts.

Compound Poisson aggregate loss

S=\sum_{i=1}^{N}X_i,\qquad N\sim \operatorname{Poisson}(\lambda)

Mean and variance

E[S]=\lambda E[X],\qquad \operatorname{Var}(S)=\lambda E[X^2]

Generating Function View

The moment generating function of S is the Poisson probability generating function evaluated at the severity moment generating function. This is the compact way to remember why the compound Poisson family behaves so neatly under sums.

If two independent compound Poisson aggregate losses have rates lambda_1 and lambda_2, their sum is also compound Poisson with rate lambda_1 + lambda_2. The new severity distribution is a rate-weighted mixture of the original severities.

MGF of a compound Poisson aggregate

M_S(t)=\exp\{\lambda(M_X(t)-1)\}

Worked Example

A policy has claim count N ~ Poisson(3). Individual claim amounts have mean 1,000 and standard deviation 2,000. Then E[X^2] = Var(X) + E[X]^2 = 4,000,000 + 1,000,000 = 5,000,000.

The aggregate expected loss is 3(1,000) = 3,000. The aggregate variance is 3(5,000,000) = 15,000,000, so the aggregate standard deviation is about 3,873.

Where It Shows Up

In FAM, compound Poisson appears as a core aggregate-risk model. In ASTAM, it connects to aggregate models, sums of compound Poisson models, discretization, and Panjer recursion. In ruin theory, it is the standard claim arrival model in the classical risk process.

For GLMs, the compound Poisson idea also explains why Tweedie pure-premium models can have both a positive probability of zero loss and a continuous distribution for positive loss.

Common Traps

Trap 1: using lambda Var(X) for aggregate variance and forgetting the severity mean-squared term.

Trap 2: treating the aggregate distribution as Poisson. The count is Poisson; the aggregate payment is not, unless severities are a special degenerate case.

Trap 3: copying a continuous severity directly into Panjer recursion. Panjer needs a discrete severity distribution, so continuous severities must be discretized first.