Actuarial Notes

Severity Analysis

Study guide: Exam 5

Severity Analysis is the study of average claim cost across accident years, maturities, classes and coverages — used both to select a severity trend for pricing and to diagnose what a triangle is doing.

\text{Average Severity} = \frac{\text{Losses}}{\text{Claim Counts}}

\text{Annual Trend} = \left(\frac{S_n}{S_0}\right)^{1/n} - 1

  • The numerator and denominator must be consistently defined. Reported losses over reported counts, paid over closed, ultimate over ultimate — mixing them (reported losses over closed counts is the classic) produces a series that measures nothing.
  • Severity triangles are read down the column, not across the row: comparing accident years at the same maturity isolates the change in cost level, while movement along a row is development.
  • Three distinct forces move a severity series and the analysis must separate them:
    • Genuine cost inflation — medical, wage, repair, legal.
    • Large-loss noise — a shock claim in one year, which is why severity is studied on capped data.
    • Mix and dilution — late-reported small claims pulling averages down, or a shift in the mix of coverages and limits.
  • As a reserving diagnostic, average case outstanding per open claim is the sharpest single indicator: a jump across all accident years in one calendar year is case strengthening, while a rise concentrated in one accident year is a genuine change in that year's claims.
  • Severity analysis is what the Frequency-Severity Method rests on: project counts and severities separately, then multiply.

Fitting a Severity Trend Auto bodily injury ultimate figures: AY 2021 — \$5{,}000{,}000 on 1{,}000 claims; AY 2023 — \$6{,}600{,}000 on 1{,}100 claims.

Compute the annual severity trend.

Answer

\begin{align*}
S_{2021} &= \frac{\$5{,}000{,}000}{1{,}000} = \$5{,}000 \\[4pt]
S_{2023} &= \frac{\$6{,}600{,}000}{1{,}100} = \$6{,}000 \\[6pt]
\text{Trend} &= \left(\frac{6{,}000}{5{,}000}\right)^{1/2} - 1 \\
&= 1.0954 - 1 \\
&= 9.5\%
\end{align*}

Note what the two-point fit hides: it uses no information from AY 2022, and it cannot distinguish a steady 9.5\% from a flat year followed by a 20\% jump. Two points give a number, not a trend — a defensible selection needs the full series and a fitted regression.

Diagnosing a Triangle with Average Case Outstanding Average case outstanding per open claim, by accident year and maturity:

AY 12 mo 24 mo 36 mo
2021 \$9{,}000 \$14{,}500 \$21{,}000
2022 \$9{,}300 \$15{,}100 \$27{,}500
2023 \$9{,}500 \$19{,}600
2024 \$12{,}400

What is happening, and how does it affect the reserving methods?

Answer Read down each column and the picture is clear:

  • At 12 months: 9{,}000 \to 9{,}300 \to 9{,}500, about +2.7\% a year — then \$12{,}400, a +30.5\% jump.
  • At 24 months: 14{,}500 \to 15{,}100 (+4.1\%) — then \$19{,}600, +29.8\%.
  • At 36 months: 21{,}000 — then \$27{,}500, +31.0\%.

The jump sits on the latest diagonal at every maturity and is the same size at each. That is a case reserve strengthening during the 2024 calendar year, not a change in any accident year's underlying cost.

Consequences:

  • Reported chain ladder is now biased high: historical factors were built on weaker reserves and are being applied to a strengthened diagonal, developing the strengthening a second time.
  • Paid chain ladder is unaffected — payments have not changed — so a divergence between paid and reported ultimates should now appear, confirming the diagnosis.
  • The fix is Berquist-Sherman: restate the historical reported triangle to current case adequacy by trending average case outstanding at the selected severity trend, rebuild the triangle, and re-select factors from the restated history.