Collective and Individual Risk Models | Free Exam FAM Lesson

An auto insurer covers 10,000 policies. It needs the expected total payout and how wildly that total could swing. Getting either wrong by 5% misprices the entire book. Two frameworks dominate: the collective risk model (random claim count, i.i.d. severities) and the individual risk model (fixed policyholders, each with own claim probability).

HIGH-FREQUENCY: The collective risk model mean/variance formulas appear on nearly every FAM exam.

$S = X_{1} + X_{2} + \dots + X_{N}$

$N$ is random, $X_{i}$ are i.i.d., $N$ is independent of the $X_{i}$ .

Mean: $E [S] = E [N] \cdot E [X]$ .

Variance (law of total variance):

$Var (S) = E [N] \cdot Var (X) + Var (N) \cdot (E [X])^{2}$

First term: process variance. Second term: mixing variance.

KEY: Process variance captures severity variability (how big each claim is); mixing variance captures frequency variability (how many claims occur). Both must be computed separately, they are not interchangeable.

HIGH-FREQUENCY: For compound Poisson ( $E [N] = Var (N) = λ$ ):

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Common mistakes

Using wrong variance formula for compound Poisson. General: $E [N] Var (X) + Var (N) (E [X])^{2}$ . Poisson simplifies to $λ E [X^{2}]$ . Applying the Poisson shortcut to NB frequency is wrong.
Confusing $E [X^{2}]$ with $(E [X])^{2}$ . In $λ E [X^{2}]$ , you need the second raw moment. $E [X^{2}] \neq = (E [X])^{2}$ unless $Var (X) = 0$ .
Forgetting to square $E [X]$ in mixing variance. The term is $Var (N) \cdot (E [X])^{2}$ . Omitting the square gives wrong units.

Bottom line

Collective: $S = X_{1} + \dots + X_{N}$ , with $E [S] = E [N] \cdot E [X]$ and $Var (S) = E [N] Var (X) + Var (N) (E [X])^{2}$
Compound Poisson shortcut: $Var (S) = λ E [X^{2}]$ , because $E [N] = Var (N) = λ$
Individual: $S = \sum I_{i} B_{i}$ over a fixed $n$ policyholders, with $E [S] = \sum q_{i} E [B_{i}]$
Individual variance (fixed benefit): $Var (S) = \sum b_{i}^{2} q_{i} (1 - q_{i})$ , using $q_{i} (1 - q_{i})$ rather than $q_{i}^{2}$

Exam shortcut

When given $E [X]$ and $E [X^{2}]$ , jump straight to $λ E [X^{2}]$ . Do not waste time computing $Var (X)$ first. For non-Poisson frequency, you need the full two-term formula, always check which distribution $N$ follows. "ENE-X" (E of S = E[N] times E[X]. "Process + Mix") process variance from severity variability, mixing variance from frequency variability. "Poisson means equal", the two terms collapse.

The full lesson (about 1,941 words, 13 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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