Normal and Lognormal Approximations for Aggregate Distributions

You computed $E[S]$ and $\text{Var}(S)$ for 500 Poisson-frequency claims. Now: "What is the probability total claims exceed $600,000?" The exact compound Poisson CDF has no closed form. You need an approximation you can look up in a normal table. Two dominate FAM: normal (fast, large portfolios) and lognormal (better for skewed, smaller portfolios).

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

By the CLT, when $E[N]$ is large:

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

Use when: $E[N]$ is large ($\lambda >25$ roughly), severity not extremely skewed, or the problem instructs "use normal approximation."

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.

Match $E[S]$ and $\text{Var}(S)$ to a lognormal:

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: $\mathrm{Pr}(S>s)\approx 1-\Phi ((s-E[S])/{\sigma }_S)$. Fast, works when $E[N]$ is large.
• 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 captures right skewness. Normal is symmetric (assigns probability to $S<0$).
• 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 1,700 words, 11 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2j

Browse all free Exam FAM lessons or jump into free Exam FAM practice questions.