The Compound Poisson Distribution

A trading desk gets hit by margin calls at random times through the year (a Poisson process), and each margin call demands a random amount of cash. You want the total cash demanded over the year. That total isn't just "number of calls times average size," because both the count and each individual size are random and add their uncertainty together. The compound Poisson distribution is exactly the model for this: a random number of events, Poisson-distributed, each contributing an independent random amount, summed up.

Formally, if $N$ is Poisson with rate $\lambda$ over a period, and $X_1, X_2, \dots$ are i.i.d. sizes independent of $N$, the total is $S = X_1 + \dots + X_N$. Its mean and variance follow simple combination rules: $\mathbb{E}[S] = \lambda \, \mathbb{E}[X]$, and $\mathrm{Var}(S) = \lambda \, \mathbb{E}[X^2]$ — note the variance depends on the second moment of the sizes, not just their spread, so a few large, rare sizes can dominate the total's variability even if most events are small.

Suppose margin calls arrive at $\lambda = 12$ per year, and each call averages $50,000 with $\mathbb{E}[X^2] = 3.6 \times 10^9$ (reflecting occasional very large calls). Expected total cash demanded: 12 \times 50{,}000 = \600{,}000$per year. Variance of the total:$12 \times 3.6 \times 10^9 \approx 4.3 \times 10^{10}, giving a standard deviation of roughly \208,000 — a spread that's large relative to the mean precisely because size variability, not just count variability, feeds into it.