Severity

Study guides: Exam FAM, Exam ASTAM, Exam MAS-I, Exam MAS-II, Exam 5, Exam 8, Exam GI-101

Severity is the average cost per claim — total losses divided by claim count — as distinct from how often claims occur.

\text{Severity} = \frac{\text{Total Losses}}{\text{Number of Claims}}

\text{Pure Premium} = \text{Frequency} \times \text{Severity}

• Severity is the inflation-sensitive half of the pure premium. Medical costs, wage replacement, repair costs, jury awards and legal expense all feed it, which is why severity trend usually exceeds general inflation in liability lines — the phenomenon called social inflation.
• Severity is far less credible than frequency at the same volume, because claim size distributions are heavily skewed: a handful of large losses can move an average by a wide margin while the underlying cost level is unchanged.
• For that reason severity is usually studied on capped or basic limits data, with the excess layer priced separately through increased limits factors. Trending unlimited severity mixes a genuine cost trend with the leveraged growth of the excess layer.
• Severity develops, and it develops in two directions at once: upward through IBNER on known claims, downward through the dilution of late-reported small claims. This is why average severity triangles are often built on a Report Year basis, where the count denominator is fixed.
• The interaction with fixed policy provisions is leveraged: with a fixed deductible or limit, a 10\% increase in ground-up loss produces more than 10\% increase in the insurer's severity — see Inflation.

Projecting Ultimate Severity An accident year has 600 reported claims and \$4{,}200{,}000 of reported losses at 24 months. The severity development factor to ultimate is 1.20 and ultimate claim count is projected at 650.

Estimate ultimate losses.

Answer

\begin{align*}
\text{Severity at 24 mo} &= \frac{\$4{,}200{,}000}{600} = \$7{,}000 \\[4pt]
\text{Ultimate severity} &= \$7{,}000 \times 1.20 = \$8{,}400 \\[4pt]
\text{Ultimate losses} &= 650 \times \$8{,}400 = \$5{,}460{,}000
\end{align*}

The implied loss development factor is \$5{,}460{,}000 / \$4{,}200{,}000 = 1.30 — which decomposes into 1.20 of severity development and 650/600 = 1.083 of count development. Splitting the two is the whole point of the frequency-severity approach: the actuary can see whether the development is coming from claims not yet reported or from claims already known getting worse.

Capped vs. Unlimited Severity Trend A liability book's average severity, on both an unlimited and a \$250{,}000-capped basis:

AY	Unlimited	Capped at \$250K
2020	\$21{,}000	\$18{,}500
2021	\$23{,}500	\$19{,}400
2022	\$21{,}800	\$20{,}300
2023	\$29{,}500	\$21{,}300
2024	\$26{,}000	\$22{,}300

Which series should be used to select a severity trend?

Answer The capped series. Its year-over-year changes are +4.9\%, +4.6\%, +4.9\%, +4.7\% — a clean 4.8\% trend that an exponential fit will reproduce with a high R^2.

The unlimited series swings +11.9\%, -7.2\%, +35.3\%, -11.9\%. Fitting a trend to that measures the accident of when the large claims happened to fall, not the rate at which costs are rising. Two actuaries fitting different windows of the same data would reach materially different answers.

The correct procedure is:

1. Select the trend from capped data — here 4.8\%.
2. Price the excess layer separately with ILFs or an excess loading, recognizing that the excess layer trends faster than the capped layer (leveraged trend).
3. Recombine.

Dropping step 2 and simply using the capped trend on unlimited losses understates the projection, because the excess layer's own growth is never picked up.