Moments and Percentiles of Severity Distributions

An actuary prices excess-of-loss reinsurance attaching at $500,000. She needs the expected payment per claim above that point. If she confuses the excess loss variable with the limited expected value, the premium is mispriced by millions.

HIGH-FREQUENCY: Moment calculations for exponential, Pareto, gamma, lognormal, and Weibull appear repeatedly on FAM.

The $k$-th raw moment:

$$E[X^k] = \int_0^\infty x^k f(x)\,dx$$

The survival function form is often simpler:

$$E[X] = \int_0^\infty S(x)\,dx$$

Variance: $\text{Var}(X) = E[X^2] - (E[X])^2$. Never confuse the second raw moment with the second central moment.

TRAP: The second raw moment $E[X^2]$ is not the variance. Variance subtracts $(E[X])^2$. Mixing these up is one of the most common calculation errors.

Exponential($\theta$): $E[X^k] = k!\,\theta^k$. $\text{Var}(X) = \theta^2$.

Gamma($\alpha, \theta$): $E[X] = \alpha\theta$. $E[X^2] = \alpha(\alpha+1)\theta^2$. $\text{Var}(X) = \alpha\theta^2$.

Pareto($\alpha, \theta$): $E[X] = \theta/(\alpha - 1)$ requires $\alpha > 1$. $E[X^2] = 2\theta^2/[(\alpha-1)(\alpha-2)]$ requires $\alpha > 2$.

Common mistakes

• Confusing per-loss and per-payment. Per-loss is $E[X] - E[X \wedge d]$. Per-payment divides by $S(d)$. On Pareto($3, 2000, d=500$): trap is 640 (per loss) when answer is 1,250 (per payment).
• Using wrong second moment formula for gamma. Gamma second moment is $\alpha(\alpha+1)\theta^2$, not $\alpha^2\theta^2$. The latter gives variance = 0.
• Applying exponential quantile to Pareto. At 90th percentile, Pareto($2, 1000$) gives 2,162. Exponential gives 2,303. Both are plausible, choose the right formula.

Bottom line

• Raw moments: $E[X^k] = \int_0^\infty k x^{k-1} S(x)\,dx$; the density and survival forms give the same answer. Variance $= E[X^2] - (E[X])^2$.
• LEV: $E[X \wedge d] = \int_0^d S(x)\,dx$; excess $= E[X] - E[X \wedge d]$.
• For a finite mean and $S(d)>0$, mean excess loss is $e(d) = (E[X] - E[X \wedge d]) / S(d)$. Exponential gives $\theta$; Pareto gives $(d+\theta)/(\alpha-1)$ when $\alpha>1$.
• Pareto moments exist only when $k < \alpha$; the lognormal mean carries the $\sigma^2/2$ term while the median does not.

Exam shortcut

Set up the payment structure algebraically before computing. Write $E[X \wedge (d+u)] - E[X \wedge d]$ first. Then look up the LEV formula. Most errors happen because candidates jump to computing without clarifying what quantity the question asks: per loss, per payment, or capped payment. "Per loss, no division. Per payment, divide by survival." Recite before every deductible problem. "Pareto moments stop at alpha." Alpha = 2 means the mean exists but variance does not.

The full lesson (about 2,357 words, 16 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2a

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