Collective Risk Model¶

Definition¶

The collective risk model writes $\displaystyle S=\sum_{i=1}^NX_i$ $i = 1 \sum N X_{i}$ , where $N$ is a nonnegative integer claim count and the claim severities $X_i$ are independent, identically distributed nonnegative variables, independent of $N$ . The sum is zero when $N=0$ $0$ .

Interpretation¶

The number of claims is random, and the summands describe actual claims rather than all policies. A policy can generate more than one claim unless the frequency model rules that out.

Diagram of five policies above three claims: policy 2 produces claims X_1 and X_2, policy 4 produces X_3, the other policies none, so N = 3 and S = X_1 + X_2 + X_3.

The figure shows one year of a small portfolio. The policies form a fixed set, but the model sums over the claims they produce: here policy 2 has two claims, policy 4 has one and policies 1, 3 and 5 have none, so $N=3$ and $S=X_1+X_2+X_3$ $X_{3}$ . In another year $N$ could be 0, giving $S=0$ $0$ , or exceed the number of policies. In the Example $N\sim\operatorname{Pois}(5)$ $Pois (5)$ , so the portfolio produces five claims on average.

Change $\lambda$ and the severity mean and standard deviation: $\mathbb E[S]=\lambda\mu$ and $\operatorname{Var}(S)=\lambda(\sigma^2+\mu^2)$ $μ^{2})$ , so the claim-size mean contributes to the spread of $S$ even when $\sigma$ is small.

The individual risk model sums losses over a fixed set of policies; here the summation is over a random number of actual claims. Assuming i.i.d. random variables gives all severities the same law. The independent random variables assumptions ensure that fixing $N=n$ leaves a sum of $n$ independent draws from that law.

Key Results¶

Conditional on $N=n$ $n$ , total loss has the distribution of a sum of $n$ severities.
Mean and variance follow from the random-sum formulas on the compound distribution page.
If severities are strictly positive, $\mathbb{P}(S=0)=\mathbb{P}(N=0)$ $0)$ ; otherwise zero severities also matter.
Frequency–severity independence is a substantive modelling assumption.

Examples¶

With $N\sim\operatorname{Pois}(5)$ and independent severities of mean 100 and variance 400, mean aggregate loss is 500 and variance is $5(400+100^2)=52000$ $52000$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 3

Aliases: frequency-severity model, random-claim-count aggregate loss, collective model Keywords: S = X_1 + ... + X_N, claim count and claim severity, aggregate loss, frequency–severity independence, P(S = 0) = P(N = 0) Symbols: lambda, mu, sigma, sum

Last updated: 16 October 2026 22:28