Superposition of Compound Poisson Risks¶

Theorem¶

Let independent compound Poisson losses $S_j$ have rates $\lambda_j\ge0$ and severity cumulative distribution functions (CDFs) $F_j$ , for $j=1,\ldots,m$ $1, \dots, m$ . If $\displaystyle \lambda=\sum_j\lambda_j>0$ $0$ , their sum is compound Poisson with rate $\lambda$ and severity CDF $\displaystyle F=\sum_j(\lambda_j/\lambda)F_j$ $j \sum (λ_{j} / λ) F_{j}$ .

Interpretation¶

Pooling portfolios (see Poisson merging) adds their claim-arrival rates. A randomly chosen pooled claim belongs to a portfolio with probability proportional to that portfolio's arrival rate.

Three time lines: portfolio 1 with rate 2 and claims of 10, portfolio 2 with rate 3 and claims of 20, and the pooled stream with rate 5 that contains both kinds of claim.

The figure uses the Example's portfolios with illustrative arrival times; each bar's height is the claim amount. Pooling keeps every claim, so the pooled stream has rate $2+3=5$ $5$ . A pooled claim comes from portfolio 1 and equals 10 with probability $2/5$ , and from portfolio 2 and equals 20 with probability $3/5$ (in the stretch drawn, 4 of the 10 claims are 10s), so the pooled severity is the mixture $F=\tfrac{2}{5}F_1+\tfrac{3}{5}F_2$ $\frac{3}{5} F_{2}$ . The mean total loss is $5(\tfrac{2}{5}\cdot10+\tfrac{3}{5}\cdot20)=80$ $80$ .

Change the two rates and claim sizes: the pooled compound Poisson law with rate $\lambda_1+\lambda_2$ and the mixed severity agrees with the convolution of the two portfolio totals at every point.

The pooled severity law is a mixture distribution: first choose the originating portfolio with probability $\lambda_j/\lambda$ , then draw a claim from that portfolio's severity law. Those probabilities are proportional to claim rates, not to portfolio counts.

Key Results¶

Pooled severity weights are $\lambda_j/\lambda$ , not equal portfolio weights unless rates are equal.
Independent losses have additive means and variances.
Multiplying the component moment generating functions (MGFs) yields $\exp[\lambda(M_X(t)-1)]$ when the MGFs exist.
The result also holds without a finite positive MGF, by viewing the pooled claims as one Poisson arrival stream in which each arrival independently carries a portfolio label $j$ (probability $\lambda_j/\lambda$ ) and a severity from $F_j$ ; compare Poisson splitting.

Examples¶

Two portfolios have rates 2 and 3 and fixed claim amounts 10 and 20. Their pooled rate is 5 and severities are 10 with probability $2/5$ and 20 with probability $3/5$ . Mean total loss is 80.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 4

Aliases: sum of independent compound Poisson losses, pooled severity mixture, compound Poisson superposition, sum of compound Poisson risks Keywords: pooling portfolios, λ = λ_1 + ... + λ_m, severity weights λ_j/λ, Poisson merging, mixture of severity distributions Symbols: lambda, sum

Last updated: 16 October 2026 22:28