collective risk model
Instead of tracking 1,000 specific policies one by one, picture standing at the door of an insurer and watching claims arrive as a stream: first a random number of claims will come in over the year, and then each arriving claim has a random size. Add them all up and you have the total cost. Modelling the portfolio this way — a random count of claims, each of random size — is the collective risk model.
Formally the aggregate loss is S = X1 + X2 + ... + XN, where N is a random number of claims (drawn from a frequency distribution) and each Xi is an independent claim size (drawn from a severity distribution), with the sizes independent of the count. The key difference from the individual risk model is that the number of terms N is itself random, not a fixed list. This is precisely the frequency-severity decomposition assembled into a single quantity: frequency supplies N, severity supplies the Xi, and S is their compound sum. The resulting distribution of S is called the compound or aggregate loss distribution.
The collective risk model is the dominant framework for non-life pricing, reserving, and capital, because it matches how claims really occur — a random number of incidents of random size from a pool, rather than a fixed census of named lives. Its great convenience is that the mean and variance of S have clean formulas in terms of the frequency and severity moments, and when frequency is Poisson the model becomes especially tractable (the compound Poisson). The honest caveat is its core assumption: claim sizes are independent of each other and of the count. Real catastrophes violate both — one storm causes many claims at once, all large — so the plain collective model must be extended for correlated or catastrophe risk.
An insurer models annual claims with frequency N Poisson(lambda = 50) and severity each averaging 4,000 with standard deviation 6,000. Expected aggregate loss E[S] = 50 times 4,000 = 200,000. By the compound-Poisson variance formula, Var(S) = 50 times (6,000^2 + 4,000^2) = 50 times 52,000,000 = 2,600,000,000, giving a standard deviation of about 51,000 around the 200,000 mean.
S = X1 + ... + XN with N random — a random count of random-sized claims.
The model assumes claim sizes are mutually independent and independent of the count. A catastrophe breaks both — one event yields many claims at once, often all large — so the plain collective model can severely understate catastrophe and correlated risk.