compound (aggregate) loss distribution

You have decided that next year your portfolio will produce a random number of claims, each of a random size. The thing you ultimately care about — total dollars paid out across the whole year — is itself a random quantity with its own distribution. That distribution of the grand total, built from a random count of random-sized claims, is the compound (aggregate) loss distribution.

It is the distribution of S = X1 + ... + XN, the output of the collective risk model. 'Compound' signals that it is built by compounding two randomness layers: the random frequency N and the random severity X. Two facts make it manageable. Its mean is tidy: E[S] = E[N] times E[X]. Its variance combines both sources of uncertainty: Var(S) = E[N] times Var(X) + Var(N) times (E[X])^2 — the first term is uncertainty in claim sizes, the second is uncertainty in how many claims occur. But the full shape of S has no simple closed form in general; you obtain it by convolution, by the Panjer recursion, by a normal or other approximation, or by simulation.

The aggregate distribution is what pricing, reserving, reinsurance, and capital ultimately rest on, because the questions that matter — 'what premium covers expected losses plus a margin?', 'how much capital to survive a 1-in-200 year?' — are questions about the tail and quantiles of S. The recurring honest warning is against the normal approximation: it is tempting to treat S as bell-shaped via the central limit theorem, and that is fine near the mean for large, well-behaved portfolios, but it badly underestimates the right tail when severity is heavy or claim counts are low. The expensive answers live in the tail, exactly where the normal approximation is least trustworthy.