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 $=E[X]-E[X\wedge d]$.
• Mean excess loss: $e(d)=(E[X]-E[X\wedge d])/S(d)$; exponential is constant $\theta$, Pareto is $(d+\theta )/(\alpha -1)$, increasing.
• 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 2,279 words, 15 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2a

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