Why the Exposure Denominator Matters
A frequency or pure premium is meaningful only when its denominator has a clear interpretation. “Claims per policy” might describe claims per policy written, claims per policy in force at a particular date, or claims per earned policy-year. Those are different measures. They can all be useful, but they are not interchangeable without information about how much exposure each policy contributed.
For frequency-severity analysis, an exposure basis is the unit used to express the amount of insurance risk observed. Depending on the product and purpose, that unit might be a policy-year, vehicle-year, location-year, payroll unit, or another measurable quantity. The selected unit should match the risk being analyzed and the unit in which a resulting rate will be used.
An exposure unit need not be a whole policy. If a policy is in force for half of a year, it generally contributes one-half of a policy-year of exposure, subject to the applicable exposure measurement rules. A policy count, by contrast, counts policies. A count of policies in force on a particular date is a snapshot, not an amount of exposure accumulated over a period.
Match the Denominator to the Question
Start by asking what one unit of the intended rate represents. If the rate is meant to describe annual risk per insured vehicle, vehicle-years are usually more directly interpretable than policy-years. If a policy covers several vehicles, a policy-year and a vehicle-year measure different amounts of insured risk. Similarly, a written-policy count may be appropriate for a measure per contract issued, but it does not by itself give an annual rate when contract durations vary.
The numerator and denominator must also refer to the same experience population and period. Claims from a year of earned exposure should not be divided by a count of policies in force only at year-end unless the intended measure specifically uses that snapshot and its limitations are understood. A midyear cancellation, a policy written partway through the year, or a changing portfolio can make the snapshot quite different from the exposure actually observed.
| Denominator | What one unit represents | Typical caution |
|---|---|---|
| Earned policy-years | One policy in force for one year, accumulated over the observation period | Do not substitute a policy count without accounting for duration. |
| Policies written | One contract issued during a specified period | Policies may contribute different amounts of time at risk. |
| Vehicle-years | One insured vehicle covered for one year | A policy-year may cover multiple vehicles. |
| Point-in-time policy count | One policy in force at a specified date | A snapshot is not the same as exposure accumulated over a year. |
Worked Examples
Worked Example: Earned Policy-Years Versus Policies Written
A hypothetical insurer writes 1,000 contracts during a year. Those contracts contribute 800 earned policy-years during the year. The experience produces 64 claims and $192,000 of losses. Calculate frequency, severity, and pure premium per earned policy-year. Then calculate the corresponding measures per policy written and identify a denominator error. Treat the supplied totals as deterministic observed data; no interest, mortality, or probability distribution is assumed.
Method. Calculate each measure on a clearly stated basis. Frequency and pure premium per earned policy-year use 800 policy-years:
Severity is based on claims, not exposure:
Thus pure premium per earned policy-year is \(0.08\times \$3{,}000=\$240\). Direct division checks the result: \(\$192{,}000/800=\$240\) per policy-year.
On a per-policy-written basis, the corresponding frequency is \(64/1{,}000=0.064\) claims per policy written, and the loss cost is \(\$192{,}000/1{,}000=\$192\) per policy written. These are valid measures for that basis, but they do not describe the annual rate per earned policy-year. The average earned duration is \(800/1{,}000=0.8\) policy-years per written policy, and \(\$192/0.8=\$240\) per policy-year.
Common mistake avoided: Calling \(0.064\times \$3{,}000=\$192\) an annual pure premium per policy-year. That multiplication uses claims per policy written, so its result is dollars per policy written. For the annual policy-year measure, use \(0.08\times \$3{,}000=\$240\).
Worked Example: Choosing Between Policy-Years and Vehicle-Years
A hypothetical book has 80 policies, each covering an average of 2.5 vehicles for a full year. The experience is 20 claims and $75,000 of losses. Calculate the frequency and pure premium per vehicle-year and per policy-year. Treat these as deterministic observed totals; no interest, mortality, or probability distribution is assumed.
Method. The total vehicle-year exposure is \(80\times 2.5=200\). The policy-year exposure is 80. The two bases describe the same experience but answer different rate questions.
On a vehicle-year basis, frequency is \(20/200=0.10\) claims per vehicle-year. Severity is \(\$75{,}000/20=\$3{,}750\) per claim. Therefore pure premium is \(0.10\times \$3{,}750=\$375\) per vehicle-year. Direct division confirms \(\$75{,}000/200=\$375\).
On a policy-year basis, frequency is \(20/80=0.25\) claims per policy-year. The same claim severity applies, so pure premium is \(0.25\times \$3{,}750=\$937.50\) per policy-year. The direct check is \(\$75{,}000/80=\$937.50\).
The policy-year pure premium is 2.5 times the vehicle-year pure premium because each policy-year represents 2.5 vehicle-years on average: \(2.5\times \$375=\$937.50\). The check on total losses is \(200\times \$375=\$75{,}000\), and also \(80\times \$937.50=\$75{,}000\).
Common mistake avoided: Reporting the vehicle-year frequency of 0.10 as though it were claims per policy-year. On the policy-year basis, the frequency is 0.25. Using the wrong frequency with the per-claim severity would understate the policy-year pure premium.
Worked Example: Unequal Policy Durations
A hypothetical portfolio writes 120 policies. Their total earned exposure is 90 policy-years, equivalent to an average of \(90/120=0.75\) policy-years per written policy. The portfolio has 9 claims and $54,000 of losses. Calculate the frequency and pure premium per policy-year and per policy written. Treat the totals as deterministic observed data; no interest, mortality, or probability distribution is assumed.
Method. Per policy-year frequency is \(9/90=0.10\) claims per policy-year. Severity is \(\$54{,}000/9=\$6{,}000\) per claim. Pure premium per policy-year is \(0.10\times \$6{,}000=\$600\), which agrees with \(\$54{,}000/90=\$600\).
Per policy written, frequency is \(9/120=0.075\) claims per policy written, and pure premium is \(\$54{,}000/120=\$450\) per policy written. The conversion is consistent: \(\$450/0.75=\$600\) per policy-year on average.
Common mistake avoided: Treating the 120 policies as 120 full policy-years. Doing so would use the wrong exposure total and report \(\$54{,}000/120=\$450\) as the annual pure premium. That figure is per policy written; the annual figure is \(\$600\) per policy-year.
Worked Example: Pooling Rates With Exposure Weights
A hypothetical portfolio has two segments measured in the same policy-year basis. Segment A has 100 policy-years, 10 claims, and $10,000 of losses. Segment B has 900 policy-years, 9 claims, and $18,000 of losses. Calculate the pooled frequency, severity, and pure premium. Then compare the pooled frequency with the unweighted average of the segment frequencies. Treat the data as deterministic; no interest, mortality, or probability distribution is assumed.
Method. First add the underlying totals. Total exposure is \(100+900=1{,}000\) policy-years, total claims are \(10+9=19\), and total losses are \(\$10{,}000+\$18{,}000=\$28{,}000\).
Pooled frequency is \(19/1{,}000=0.019\) claims per policy-year. Pooled severity is \(\$28{,}000/19=\$1{,}473.6842\ldots\) per claim. Rounded to cents, that is \(\$1{,}473.68\). Pooled pure premium is \(0.019\times(\$28{,}000/19)=\$28\) per policy-year; direct division gives the same result, \(\$28{,}000/1{,}000=\$28\).
The segment frequencies are \(10/100=0.10\) and \(9/900=0.01\). Their unweighted average is \((0.10+0.01)/2=0.055\), which is not the pooled frequency. Segment A represents one-tenth of total exposure, while Segment B represents nine-tenths. Pooling claim counts and exposures, rather than averaging the segment rates equally, preserves the portfolio’s actual exposure mix.
Common mistake avoided: Averaging segment frequencies without regard to exposure size. The correct pooled rate is \(0.019\), not \(0.055\).
Why the Frequency-Severity Product Works
The identity connecting frequency, severity, and pure premium follows directly from their definitions. For \(N>0\), calculate frequency and severity on the same set of claims and the same exposure basis. Multiplying the two ratios cancels the claim count:
The result is the total loss per unit of exposure. This identity requires the claim count, loss total, and exposure total to refer to compatible experience. If one rate uses vehicle-years while another uses policy-years, the cancellation of \(N\) does not repair the mismatched units. When \(N=0\), severity \(L/N\) is undefined, but pure premium \(L/E_b\) remains defined if exposure is positive.
For multiple segments, use the same principle on totals: divide the sum of claim counts by the sum of exposures for pooled frequency, and divide the sum of losses by the sum of exposures for pooled pure premium. An average of segment rates is appropriate only if the intended weighting matches the question; it is not automatically the portfolio rate.
Practical Checks Before Using a Rate
A denominator choice is not merely a formatting decision. It determines what the rate means and how it can be applied. Before comparing two estimates or using a rate in a projection, check that both use compatible exposure definitions, observation periods, and populations.
- Name the unit. Write “claims per vehicle-year” or “dollars per policy written,” rather than an unqualified “frequency” or “pure premium.”
- Use earned exposure for time-at-risk measures. A policy count does not account for partial-year coverage, cancellations, or varying policy duration.
- Match the risk unit. If a policy covers several separately exposed items, a policy-year may not represent the same quantity of risk across policies.
- Aggregate numerators and denominators before calculating a pooled rate. Do not use an unweighted mean of segment rates unless equal segment weighting is intentional.
- Check the units of any product. Frequency per exposure unit multiplied by dollars per claim gives dollars per that same exposure unit.
The central discipline is to state the exposure basis before calculating. Once the basis is explicit, frequency, severity, and pure premium can be checked against the same underlying claims, losses, and exposure. This supports comparisons that are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus and helps prevent a numerically correct ratio from being interpreted on the wrong unit.
Check Your Understanding
For each problem, state the exposure basis with the calculated rates. Assume deterministic observed totals; no interest, mortality, or probability distribution is needed.
- A hypothetical book has 450 earned policy-years, 36 claims, and $108,000 of losses. What are frequency, severity, and pure premium per policy-year? (Answer: 0.08 claims per policy-year; $3,000 per claim; $240 per policy-year.)
- A fleet has 300 earned vehicle-years, 30 claims, and $90,000 of losses. Calculate frequency, severity, and pure premium per vehicle-year. (Answer: 0.10 claims per vehicle-year; $3,000 per claim; $300 per vehicle-year.)
- A portfolio has 200 policies written, 160 earned policy-years, 12 claims, and $48,000 in losses. Give the frequency and pure premium per earned policy-year and per policy written. (Answer: 0.075 claims per policy-year and $300 per policy-year; 0.06 claims per policy written and $240 per policy written.)
- Segment A has 50 policy-years, 4 claims, and $20,000 in losses. Segment B has 150 policy-years, 6 claims, and $60,000 in losses. Calculate the pooled frequency and pure premium. (Answer: 0.05 claims per policy-year and $400 per policy-year.)
- A hypothetical exposure basis consists of units of $1,000 in payroll. The experience has 300 such exposure units, 15 claims, and $120,000 in losses. Calculate frequency, severity, and pure premium per exposure unit. (Answer: 0.05 claims per $1,000 of payroll; $8,000 per claim; $400 per $1,000 of payroll.)