Normal and Lognormal Approximations for Aggregate Distributions

You computed $E[S]$ and $\text{Var}(S)$ for 500 Poisson-frequency claims. To estimate the probability that total claims exceed a specified amount, FAM uses normal and lognormal approximations. Their suitability depends on the aggregate model and the requested threshold.

Choose the normal or log-normal approximation according to the aggregate model and the requested tail. Both methods match the aggregate mean and variance, but the log-normal model also preserves positive support.

HIGH-FREQUENCY: The normal approximation for aggregate claims is one of the most heavily tested mechanics on FAM.

A central limit approximation is supported when many independent finite-variance contributions produce a roughly symmetric aggregate:

$$\Pr(S > s) \approx 1 - \Phi\!\left(\frac{s - E[S]}{\sqrt{\text{Var}(S)}}\right)$$

Use when many independent contributions and limited skewness support the approximation, or the problem explicitly requests it. A large expected claim count alone is insufficient.

TRAP: The normal approximation is symmetric and assigns positive probability to $S < 0$, which is impossible for aggregate claims. For small portfolios or highly skewed severities, this can materially distort tail probabilities.

Common mistakes

• Forgetting to take the log when standardizing lognormal. Standardize $\ln s$, not $s$. Using $s$ gives a z-value in the hundreds.
• Confusing lognormal parameters with mean/variance of $S$. The parameters $\mu$ and $\sigma^2$ are the mean and variance of $\ln S$, not of $S$. The mean of $S$ is $e^{\mu + \sigma^2/2}$.
• Using $\text{Var}(X)$ instead of $E[X^2]$ in compound Poisson variance. $\text{Var}(S) = \lambda E[X^2]$, not $\lambda \text{Var}(X)$. Compute $E[X^2] = \text{Var}(X) + (E[X])^2$ first.

Bottom line

• Normal: $\Pr(S > s) \approx 1 - \Phi((s - E[S])/\sigma_S)$. Suitable when many independent finite-variance contributions yield a roughly symmetric aggregate.
• Lognormal: match moments, solve $\sigma^2 = \ln(\text{Var}(S)/(E[S])^2 + 1)$, $\mu = \ln(E[S]) - \sigma^2/2$. Standardize $\ln s$, not $s$.
• Lognormal preserves positive support and right skewness. Compare both tail probabilities at the requested threshold; neither is always larger.
• Both need same inputs: $E[S]$ and $\text{Var}(S)$ from LO 2i.

Exam shortcut

Write $E[S]$ and $\text{Var}(S)$ immediately and box them. For lognormal, compute $\sigma^2$ first, it uses only $\text{Var}(S)/(E[S])^2$. If your right-tail answer exceeds 0.5 when $s > E[S]$, something is wrong. "Log then look" (take log of $s$, then look up $\Phi$. "Sigma-squared first." "1 minus Phi") every tail probability question subtracts from 1.

The full lesson (about 1,825 words, 12 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2j

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