Why the Claim-Count Definition Matters
A frequency estimate is not determined by division alone. The numerator must specify what is being counted: for example, unique claims arising from a defined accident period, claims reported by a particular valuation date, or claims closed by that date. Those choices can produce different rates from the same underlying experience.
For loss estimation, frequency is useful only when its count definition is explicit and its exposure denominator refers to the corresponding population and period. A reported-claim count divided by accident-period exposure answers a different question from a count of claims reported during a calendar period divided by exposure earned in that period. Closed claims introduce another distinction: closure depends on claim handling and settlement, not just on whether an insured event occurred.
Examples of exposure units include earned policy-years and insured vehicle-years. Use the unit relevant to the experience and keep it consistent when comparing or combining observations. The earlier material on choosing an exposure denominator addresses the choice of exposure base; here, the emphasis is on defining the count that belongs over that base.
Reported Claims and Closed Claims
A reported claim is a claim that has been reported to the insurer, according to the chosen recording rule. A count for an accident period may include claims from that period reported by a specified valuation date. It is therefore important to state both the period to which claims are assigned and the date through which reports are included.
A closed claim is a claim whose file has been recorded as closed by the valuation date under the insurer’s administrative definition. A closed claim is generally a subset of reported claims for the same cohort and valuation date. Some reported claims remain open; claims may also reopen after closure. For a simple comparison, this tutorial treats the reported and closed counts as unique claims in the same cohort, with each claim counted at most once in each count.
A closed-claim frequency is not automatically an estimate of the rate at which insured events generate claims. It reflects both claim occurrence and the process by which files are closed. Differences in settlement speed, claim complexity, or closure practices can change the closed count at a valuation date without an equivalent change in underlying claim incidence.
| Count definition | What the numerator includes | Interpretation to keep in mind |
|---|---|---|
| Reported claims | Unique claims reported for the defined cohort by the valuation date | May omit claims not yet reported by that date |
| Closed claims | Unique claims in the cohort recorded as closed by the valuation date | Depends on the reported claims and their closure status by that date |
| Claims reported during a calendar period | Claims received during that period, potentially from several accident periods | Does not necessarily match exposure earned in the calendar period |
Calculating and Pooling Frequency
For a single defined group, calculate frequency as the selected count divided by the matching exposure. If the question asks for reported-claim frequency, use reported claims in the numerator; if it asks for closed-claim frequency, use closed claims. State the count definition alongside the result, rather than reporting an unqualified number.
When combining groups that use the same count definition and exposure unit, divide the total count by total exposure. This produces an exposure-weighted frequency. Taking an unweighted average of group frequencies gives each group equal weight, regardless of how much exposure each represents, and generally answers a different question.
Worked Example: Reported Versus Closed Frequency
A hypothetical insurer reviews one accident-year cohort with 1,200 earned policy-years. By the valuation date, 348 unique claims from that cohort have been reported, and 261 of those claims are recorded as closed. Calculate the reported-claim frequency and the closed-claim frequency. Interpret the difference.
Method. Calculate each frequency using the same accident-year cohort and exposure, changing only the numerator to match the requested count definition.
Reported-claim frequency. The numerator is 348 reported claims. Therefore,
The reported-claim frequency is 0.29 claims per earned policy-year.
Closed-claim frequency. The numerator is 261 claims recorded as closed. Therefore,
The closed-claim frequency is 0.2175 closed claims per earned policy-year. The difference in these counts is \(348-261=87\) claims, so 87 reported claims in this cohort are not recorded as closed at the valuation date.
Check and common mistake. The closed count, 261, is less than the reported count, 348, as expected under the stated definitions. A common mistake is to describe 0.2175 as the cohort’s overall claim-occurrence frequency. It is specifically a closed-claim frequency at the stated valuation date; the closure status is part of what it measures.
Worked Example: Pooling Two Groups Correctly
A hypothetical insurer has experience for two vehicle groups using the same accident-period definition, valuation date, and earned-vehicle-year exposure unit. Group A has 600 earned vehicle-years, 150 reported claims, and 105 closed claims. Group B has 1,400 earned vehicle-years, 280 reported claims, and 210 closed claims. Find the pooled reported-claim and closed-claim frequencies.
Method. For each count definition, add the group counts and divide by the combined exposure. This weights each group by its exposure.
The total exposure is
The reported-claim total is \(150+280=430\). Thus the pooled reported-claim frequency is
The closed-claim total is \(105+210=315\). Thus the pooled closed-claim frequency is
Check and common mistake. The group reported frequencies are \(150/600=0.25\) and \(280/1{,}400=0.20\). Their unweighted average is \((0.25+0.20)/2=0.225\), not the pooled result of 0.215. The unweighted calculation gives each group half the influence even though Group A has less than half the exposure. The pooled result is verified directly by \(430/2{,}000=0.215\).
Why the Ratio Estimates a Frequency
A simple probability model gives a formal justification for the count-to-exposure ratio. Suppose a homogeneous group has a constant expected claim rate \(\lambda\) per exposure unit, and claim counts follow a Poisson process: for fixed exposure \(E\), the total count \(N\) satisfies \(N\sim\operatorname{Poisson}(\lambda E)\). This model assumes a common rate across the exposure being combined and a count definition and reporting basis that are consistently applied. It is not appropriate without adjustment when the group has materially different underlying rates, or when the observed count is incomplete in a way the model does not represent.
Under this model, the likelihood of observing \(N=n\) claims is
Taking logarithms gives \(\log L(\lambda)=-\lambda E+n\log(\lambda E)-\log(n!)\). Differentiating with respect to \(\lambda\) and setting the derivative equal to zero gives
Solving yields \(\hat{\lambda}=n/E\). The second derivative is \(-n/\lambda^2\), which is negative for \(n>0\), so this stationary point is a maximum. If \(n=0\), the likelihood is maximized at the boundary \(\lambda=0\). Thus, under the stated model, the observed count divided by exposure is the maximum-likelihood estimate of the rate. This model-based justification does not remove the need to define the claims consistently: the model’s \(N\) must still be the count relevant to the analysis.
Worked Example: Matching an Accident-Year Count to Exposure
A hypothetical insurer has 1,600 earned policy-years for accident year 2025. By the valuation date, 352 claims assigned to that accident year have been reported, of which 264 are closed. During calendar year 2025, the insurer received 410 claims in total: 320 arose from accident year 2025 and 90 arose from earlier accident years. Calculate the reported-claim and closed-claim frequencies for the 2025 accident-year cohort, and identify the result of using the calendar-year report count incorrectly.
Method. Match the numerator to the exposure cohort. For accident-year frequency, use claims assigned to accident year 2025, not every claim received during calendar year 2025.
The accident-year reported-claim frequency is
The accident-year closed-claim frequency is
The 410 claims reported during calendar year 2025 include 90 claims from earlier accident years. Dividing that report-period total by the 2025 accident-year exposure gives \(410/1{,}600=0.25625\), but that quotient mixes a calendar-period count with an accident-period denominator. It is not the requested accident-year frequency. The report-period count is also not equal to the 352 accident-year claims reported by the valuation date: the latter is the specified count for that accident-year cohort.
Check and common mistake. The closed count is a subset of the accident-year reported count, and \(264/352=0.75\), so the closed frequency is 75% of the reported frequency. The incorrect 0.25625 results from including the 90 prior-year claims in the numerator; matching the cohort gives the correct reported frequency, 0.22.
Practical Definition Checks
Before calculating or comparing a frequency, write down the counting rule and verify the following points:
- Unit counted: Decide whether one record represents one claim, one coverage feature, or another unit. A claim with several features should not silently become several claims.
- Assignment period: State whether claims are assigned by accident date, report date, or another basis. Use exposure for the corresponding population and period.
- Valuation date and status: For reported and closed counts, state the valuation date. A reported claim may still be open, and closure status can change.
- Consistency across groups: Apply the same claim definition and exposure unit before pooling or comparing frequencies.
- Use in loss estimation: The claim-count population should be compatible with the loss data used in the analysis. If the count includes claims that the associated loss measure excludes, frequency and loss estimates may not describe the same experience.
A frequent exam trap is to treat every available claim count as interchangeable. It is not. A count of claims reported during a calendar year, a count of claims arising from an accident-year cohort, and a count of closed files can all be valid measures, but they answer different questions. Name the measure, show the numerator and denominator, and interpret the result in those terms.
A clear frequency statement therefore includes the count definition, the cohort or period, the valuation date when relevant, the exposure unit, and the resulting rate. That precision makes the estimate interpretable and helps prevent a mismatch when the frequency is used in loss estimation. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each problem, identify the appropriate count definition before calculating.
- A hypothetical book has 750 earned policy-years for one accident-year cohort. By the valuation date, 165 unique claims from that cohort have been reported, and 120 are closed. Find the reported-claim and closed-claim frequencies per earned policy-year. (Answer: 0.22 reported claims and 0.16 closed claims per earned policy-year.)
- Two groups have the same claim definition and exposure unit. Group A has 400 exposure units and 92 reported claims; Group B has 1,600 exposure units and 288 reported claims. Find the pooled reported frequency, and compare it with the unweighted average of the group frequencies. (Answer: pooled frequency \(380/2{,}000=0.19\); unweighted average \((0.23+0.18)/2=0.205\).)
- A hypothetical accident-year cohort has 900 earned vehicle-years, 171 reported claims, and 119 closed claims by the valuation date. Calculate each frequency to four decimal places. (Answer: 0.1900 reported claims and 0.1322 closed claims per earned vehicle-year.)
- During a calendar year, 240 claims are reported; 180 arose in the current accident year and 60 arose in earlier years. The current accident year has 1,000 earned exposure units. What is the current accident-year reported frequency, and why is dividing 240 by 1,000 inappropriate for that measure? (Answer: \(180/1{,}000=0.18\); 240 includes claims from earlier accident years.)
- A hypothetical portfolio has 500 earned policy-years and 90 unique reported claims. Eighteen of those claims each have two separately recorded coverage features; all other claims have one feature. Find the claim frequency and the feature-record rate. (Answer: claim frequency \(90/500=0.18\); feature-record rate \((90+18)/500=0.216\) per earned policy-year.)