Compound Distributions: Frequency and Severity Models

The Compound Model

In the compound (or collective risk) model, the aggregate loss S = X1 + X2 + ... + XN, where N is the random number of claims (frequency) and Xi are iid claim amounts (severity), independent of N. This model separates "how many claims" from "how large each claim is," which is natural in insurance.

Using the double expectation formula: E[S] = E[N] * E[X], where X represents a single claim. For the variance: Var(S) = E[N] * Var(X) + Var(N) * (E[X])^2. These formulas are derived from the total variance formula by conditioning on N.

Common Frequency-Severity Combinations

If N ~ Poisson(lambda) and Xi ~ any distribution with mean mu and variance sigma^2, then E[S] = lambda * mu and Var(S) = lambda * (sigma^2 + mu^2) = lambda * E[X^2]. The simplification Var(S) = lambda * E[X^2] for Poisson frequency is a key Exam P shortcut.

If N ~ Negative Binomial or Binomial, the general formula applies. The MGF of S (when it exists) is M_S(t) = M_N(ln(M_X(t))), where M_N is the MGF of N evaluated at ln(M_X(t)).

Exam P Applications

Typical problems give you the frequency and severity distributions and ask for E[S], Var(S), or P(S > threshold) (using the normal approximation via CLT). You may also be asked to find the stop-loss premium E[(S - d)+] for a retention d, or to compare aggregate losses under different deductible and limit structures. Always start by identifying which distribution governs frequency and which governs severity, then apply the compound model formulas.