Model insurance claims in aggregate.

A reinsurer quoting an excess layer needs the distribution of total annual losses, not just claim counts and severities separately. The aggregate model fuses frequency and severity into one random variable $S$, and almost every pricing, reserving, and capital question on MAS-I traces back to its mean, variance, and tail.

Two model architectures. The individual risk model sums a fixed number $n$ of independent policy losses: $S=Y_1+\cdots +Y_n$, where each $Y_j$ can be zero. The collective risk model sums a random number $N$ of positive claim amounts: $S=X_1+\cdots +X_N$. Collective is preferred for portfolio aggregates because $N$ and $X$ are estimated from separate data.

KEY: Independence has two layers. The $X_i$ are iid, and $N$ is independent of the $X_i$. Both must hold for the compound formulas to apply.

Mean and variance of $S$. Using the law of total expectation and total variance:

$$E[S]=E[N]E[X],\text{Var}(S)=E[N]\text{Var}(X)+\text{Var}(N)E[X{]}^2.$$

The variance has a frequency-uncertainty piece and a severity-uncertainty piece. Pure-frequency uncertainty (the $\text{Var}(N)E[X{]}^2$ term) often dominates when claims are rare and severities concentrated.

Common mistakes

• Using $\text{Var}(X)$ instead of $E[X^2]$ in compound Poisson variance. The shortcut formula $\text{Var}(S)=\lambda E[X^2]$ collapses both terms only because $E[N]=\text{Var}(N)$. Plugging in the wrong second moment understates risk.
• Mixing individual and collective formulas. In the individual model $N$ is fixed at $n$; there is no $\text{Var}(N)E[X{]}^2$ term. Applying the collective variance formula to a 1,000-policy individual model double-counts uncertainty.
• Forgetting that Panjer requires discrete severity. A continuous $X$ must be discretized (method of rounding or local moment matching) before the recursion runs.

Bottom line

• Collective risk model: $S={\sum }_{i=1}^N{X}_i$ where $N$ is claim count and $X_i$ are iid severities independent of $N$.
• Compound mean and variance: $E[S]=E[N]E[X]$; $\text{Var}(S)=E[N]\text{Var}(X)+\text{Var}(N)E[X{]}^2$.
• Compound Poisson shortcut: if $N\sim \text{Poisson}(\lambda )$, then $\text{Var}(S)=\lambda E[X^2]$.
• Sums of independent compound Poissons are again compound Poisson, with severity a mixture weighted by each line's ${\lambda }_i$.

Exam shortcut

If the problem says "Poisson frequency," go straight to $\text{Var}(S)=\lambda E[X^2]$; skip the law-of-total-variance derivation. When asked for a stop-loss premium and given only mean and variance, default to the normal approximation formula $\sigma [\varphi (z)-z(1-\Phi (z))]$ with $z=(d-E[S])/\sigma$. For combined-line aggregates, always sum the Poisson rates first and form the weighted severity mixture before any moment calculation; this avoids running parallel computations per line.

The full lesson (about 1,733 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B6

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