Collective and Individual Risk Models

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.

$$S = X_1 + X_2 + \cdots + X_N$$

$N$ 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):

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

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 ($E[N] = \text{Var}(N) = \lambda$):

Common mistakes

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

Bottom line

• Collective: $S = X_1 + \cdots + X_N$, with $E[S] = E[N] \cdot E[X]$ and $\text{Var}(S) = E[N]\text{Var}(X) + \text{Var}(N)(E[X])^2$
• Compound Poisson shortcut: $\text{Var}(S) = \lambda E[X^2]$, because $E[N] = \text{Var}(N) = \lambda$
• 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): $\text{Var}(S) = \sum b_i^2 q_i(1-q_i)$, using $q_i(1-q_i)$ rather than $q_i^2$

Exam shortcut

When given $E[X]$ and $E[X^2]$, jump straight to $\lambda E[X^2]$. Do not waste time computing $\text{Var}(X)$ 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 1,967 words, 13 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2i

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