Frequency-Severity Method

Study guides: Exam 5, Exam GI-101

Frequency-Severity Method projects claim counts and average severity separately and multiplies them, instead of developing loss dollars directly.

U = N_{\text{ult}} \times S_{\text{ult}}

U = \text{Exposures} \times \text{Frequency}_{\text{ult}} \times S_{\text{ult}}

Friedland gives three variants:

• Development on counts and severities. Develop the count triangle to ultimate counts; develop an average severity triangle (dollars ÷ counts at each age) to ultimate severity; multiply. The straightforward version.
• Counts and trended severity. Develop counts, but estimate severity by trending a mature year's severity forward rather than developing an immature one — useful where severity development is unstable but severity trend is well understood.
• Disposal rate (Berquist-Sherman) approach. Project the number of claims closing in each future period from historical disposal rates, value them at a trended severity for that maturity, and sum the future payments. This is the variant that handles a changing settlement rate head on.

Further points:

• The value of the split is diagnostic. A dollar triangle tells you losses grew; frequency and severity tell you why, and the two causes call for different responses and carry different persistence.
• Counts are more credible and less inflation-sensitive; severity is where the trend lives. Projecting each with the tool it suits — development for counts, trend for severity — is often better than developing either alone.
• The method needs consistent count definitions across the whole triangle, and it is sensitive to large losses in the severity series, so severities are usually capped and the excess layer added back separately.
• A pitfall: average severity triangles mix two opposing forces — IBNER pushing severity up, late-reported small claims pulling it down. Report-year severities avoid the dilution; accident-year severities do not.

Straightforward Frequency-Severity Projection An accident year has 10{,}000 earned car-years. Projected ultimate frequency is 0.08 claims per car-year and projected ultimate severity is \$12{,}500.

Estimate ultimate losses.

Answer

\begin{align*}
N_{\text{ult}} &= 10{,}000 \times 0.08 = 800 \text{ claims} \\[4pt]
U &= 800 \times \$12{,}500 = \$10{,}000{,}000
\end{align*}

Equivalently a pure premium of 0.08 \times \$12{,}500 = \$1{,}000 per car-year. The decomposition matters when the estimate is challenged: an actuary can defend 0.08 against the exposure base and \$12{,}500 against severity trend and settled-claim data, where a bare \$1{,}000 has no supporting evidence of its own.

Diagnosing an Adverse Year An accident year's ultimate estimate has risen 22\% since the previous valuation. The components:

	Prior valuation	Current
Ultimate counts	1{,}200	1{,}215
Ultimate severity	\$9{,}000	\$10{,}830

What happened, and what should change in the analysis?

Answer

\begin{align*}
\text{Count change} &= \frac{1{,}215}{1{,}200} - 1 = +1.3\% \\[4pt]
\text{Severity change} &= \frac{10{,}830}{9{,}000} - 1 = +20.3\%
\end{align*}

1.013 \times 1.203 = 1.219 \;\checkmark

Counts are essentially where they were — reporting is complete and the earlier count projection was good. The entire deterioration is severity.

That narrows the causes sharply. It is not a reporting-pattern change, not a frequency shock, and not a volume issue. The candidates are: claims already known getting worse (IBNER, possibly from a case adequacy change), an inflation or legal shock hitting open claims, or one or two large claims developing.

Two things follow for the analysis. First, check whether this is one accident year or all of them — a severity jump across every open year on the same diagonal is a calendar-year effect requiring a triangle restatement, not a year-specific reserve increase. Second, if severity trend has genuinely stepped up, the pricing severity trend selection is stale too, and the reserving finding is also a rate-adequacy finding.