Frequency

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

Frequency is the number of claims per unit of exposure — how often losses happen, as distinct from how much they cost.

\text{Frequency} = \frac{\text{Number of Claims}}{\text{Earned Exposures}}

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

• Frequency is only meaningful relative to its exposure base and its claim definition. Claims per car-year, per \$100 of payroll and per house-year are different quantities, and counting reported claims, closed claims or claims with payment gives three different frequencies from the same data.
• Frequency is usually the more stable and more credible of the two components: claim counts are less skewed than claim dollars, so a modest volume of exposure supports a credible frequency estimate long before it supports a credible severity estimate.
• Frequency is also less inflation-sensitive. Severity tracks medical, wage and repair-cost inflation; frequency responds to behaviour, exposure quality, weather and coverage terms. This is why the two are trended separately (see Loss Trend).
• Claims-only frequency understates true development. Frequency computed on reported counts at an immature age is biased low, since claims are still being reported — the counts must be developed to ultimate with their own count triangle.
• Frequency is the diagnostic that separates a coverage or exposure problem from a cost problem: rising frequency at flat severity points at underwriting, mix or the claim environment; flat frequency with rising severity points at inflation or the legal environment.

Computing Frequency and the Pure Premium A workers compensation book has 25{,}000 employee-years of exposure in accident year 2023 and 1{,}750 ultimate reported claims. Ultimate average severity is projected at \$18{,}000.

Compute frequency and pure premium.

Answer

\begin{align*}
\text{Frequency} &= \frac{1{,}750}{25{,}000} \\
&= 0.070 \text{ claims per employee-year} \\[6pt]
\text{Pure premium} &= 0.070 \times \$18{,}000 \\
&= \$1{,}260 \text{ per employee-year}
\end{align*}

Both figures must be on an ultimate basis. Using reported counts at 24 months instead of developed counts would understate frequency, and — because severity is losses over counts — would simultaneously overstate severity, partially masking the error in the pure premium. Checking frequency and severity separately against expectation is what catches it.

Decomposing a Deteriorating Loss Ratio A personal auto book's accident year results, all at ultimate:

AY	Exposures	Claims	Losses
2022	50{,}000	3{,}000	\$12{,}000{,}000
2024	52{,}000	3{,}744	\$17{,}160{,}000

The pure premium rose from \$240 to \$330 per exposure. Which component drove it, and what does that imply?

Answer

\begin{align*}
\text{Freq}_{2022} &= \frac{3{,}000}{50{,}000} = 0.0600 \\
\text{Freq}_{2024} &= \frac{3{,}744}{52{,}000} = 0.0720 \\[6pt]
\text{Sev}_{2022} &= \frac{\$12{,}000{,}000}{3{,}000} = \$4{,}000 \\
\text{Sev}_{2024} &= \frac{\$17{,}160{,}000}{3{,}744} = \$4{,}583
\end{align*}

Over two years:

\begin{align*}
\text{Frequency change} &= \frac{0.0720}{0.0600} - 1 = +20.0\% \\
\text{Severity change} &= \frac{4{,}583}{4{,}000} - 1 = +14.6\%
\end{align*}

Check: 1.200 \times 1.146 = 1.375, which reproduces the +37.5\% move in pure premium (\$240 \to \$330) exactly — the decomposition is multiplicative, not additive.

Both components moved, but frequency moved more, and that points somewhere different from inflation: more accidents per car-year suggests a mix shift toward higher-risk drivers or territories, a relaxation in underwriting, or a change in what is being reported as a claim. The corrective actions differ — a severity problem is fixed with rate and limits, a frequency problem with underwriting and classification.