Understand and interpret the characteristics of severity distributions.

Severity is the size of a claim given that one occurs. Picking the right distribution and reading its fingerprints (mean, tail, hazard, skew) is half the battle on ASTAM.

What severity distributions describe. Severity $X$ is the dollar size of one claim, conditional on the claim occurring. It is non-negative, continuous (usually), and almost always right-skewed. A few large losses dominate the mean.

The four lenses. Read every severity through:

• CDF $F(x)=\mathrm{Pr}(X\le x)$: the probability a claim is at most $x$.
• Survival $S(x)=1-F(x)$: the right tail. Reinsurance pricing lives here.
• Density $f(x)=F^{\prime }(x)$: the shape.
• Hazard $h(x)=f(x)/S(x)$: the instantaneous failure rate at size $x$. Increasing hazard means large claims get rarer faster; decreasing hazard means once a claim is big, it tends to keep growing.

KEY: Two distributions can share a mean and variance yet price a high deductible reinsurance treaty very differently. The tail $S(x)$, not $E[X]$, drives excess-layer premium.

Common mistakes

• Treating CV near 1 as proof of exponential. Many families hit CV = 1 without being memoryless.
• Quoting a Pareto mean when $\alpha \le 1$. The mean is infinite, not $\theta /(\alpha -1)$.
• Claiming lognormal is light-tailed because all moments exist. It is heavy-tailed (subexponential).

Bottom line

• Severity $X$ is a non-negative, right-skewed continuous random variable; characterize it by $F(x)$, $S(x)$, $f(x)$, and the hazard $h(x)=f(x)/S(x)$.
• Light-tailed families (gamma, Weibull $\tau >1$) have all moments; heavy-tailed families (Pareto, lognormal, Weibull $\tau <1$) have explosive or infinite tails.
• A tail is heavy when the MGF fails to exist or the hazard tends to zero; a decreasing hazard signals a heavy tail.
• Pareto mean is finite only for $\alpha >1$; variance only for $\alpha >2$.

Exam shortcut

For exponential, exploit memorylessness: conditional excess above any $d$ is exponential$(\theta )$ again. For Pareto, memorize $e(d)=(d+\theta )/(\alpha -1)$ and $E[X\wedge u]$ in closed form; both appear constantly. Tail-weight check on the fly: compare $S(x)\cdot e^{tx}$ for small $t$. If it blows up, the tail is heavy.

The full lesson (about 1,325 words, 9 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1c

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