What a Frequency Observation Contains
A claim-frequency calculation is only meaningful when its numerator and denominator describe the same insured risk over the same period. A count of claims for one coverage cannot sensibly be divided by exposure for a different coverage; likewise, claims from a full year cannot be paired with only six months of exposure. The essential task is therefore to construct a matched pair before calculating a rate.
For observation \(i\), let \(N_i\) be the number of eligible claims and let \(e_i\) be the earned exposure on the selected basis. The basis might be policy-years, vehicle-years, or another defined unit. A frequency observation is the pair \((N_i,e_i)\). The observed frequency is \(N_i/e_i\) when \(e_i>0\). The count is the numerator; the earned exposure is the denominator.
The eligible count follows the counting rule established for the analysis: for example, whether claims are counted by occurrence or by coverage claim. The exposure follows the selected exposure basis and is earned only for time when the relevant coverage applies. These definitions must be settled before calculating frequency. A claim record that exists in the data is not automatically eligible for every coverage’s frequency count.
Constructing a Matched Count and Denominator
Use the same four questions for each policy or segment: what risk is being measured, what period is included, which claims are eligible, and what exposure unit is used? Coverage periods and earned exposure determine how much denominator is accumulated; the claim-counting rule determines which events enter the numerator. The matching is essential even when the final result is reported as a single portfolio rate.
Specify whether a record represents a policy, a policy-coverage combination, or another defined risk unit, and state the experience period.
Apply the intended claim-counting rule and retain only claims belonging to the selected risk, coverage, and period.
Earn exposure only while the selected risk and coverage are in force, and express it in the stated unit.
Verify that the numerator and denominator share the same scope, then calculate the rate or aggregate the pairs.
For example, a vehicle-year basis counts one vehicle insured for one full year as one unit of exposure. Two covered vehicles insured for half a year each contribute \(2(0.5)=1.0\) vehicle-year. Fractional exposure is normal; it should not be rounded to a whole unit before aggregation. If the required unit is policy-years rather than vehicle-years, the same contract may contribute a different number of exposure units. State the basis rather than treating “exposure” as a unit-free quantity.
For a portfolio of observations, the aggregate observed frequency is total eligible claims divided by total earned exposure:
This is not generally the unweighted average of the individual rates. A policy with more exposure represents more of the portfolio experience and therefore receives more weight in the aggregate calculation.
Why the Aggregate Formula Uses Total Exposure
The exposure weighting follows directly from the definitions. For any observation with positive exposure, write its observed rate as \(r_i=N_i/e_i\), so \(N_i=e_i r_i\). Substituting this identity into the aggregate rate gives:
Thus the aggregate is an exposure-weighted average of the observed rates. An observation with twice as much exposure has twice the weight. Averaging rates equally would answer a different question: the mean rate for a randomly selected record, regardless of how much exposure that record represents.
A common statistical justification uses a simple Poisson model. Suppose observations are independent and \(N_i\sim\operatorname{Poisson}(\lambda e_i)\), where \(\lambda\) is a common claim rate per exposure unit and each \(e_i\) is measured accurately. The sum of independent Poisson counts is Poisson with mean \(\lambda\sum_i e_i\). Therefore, the expected total count is proportional to total exposure, and the likelihood estimate of the common rate is \(\sum_i N_i/\sum_i e_i\). This model assumes a common rate and independent counts; it is not required to compute the descriptive observed frequency. If rates vary across risks or counts are dependent, the ratio remains the portfolio’s observed claims per exposure unit, but the simple common-rate model may not adequately describe the data.
Worked Example: Combining Policy-Level Frequency Observations
A hypothetical insurer is reviewing one liability coverage over a common experience period. Its records show the following eligible claim counts and earned policy-year exposures: Policy A, 2 claims and 0.75 policy-years; Policy B, 0 claims and 0.25 policy-years; Policy C, 3 claims and 1.00 policy-year; Policy D, 1 claim and 0.50 policy-years. Calculate the aggregate frequency and compare it with the unweighted mean of the policy rates.
Method and assumptions. Use the count divided by earned policy-year exposure. All counts are already eligible under the stated coverage and period. The calculations use fixed observed data; no interest or mortality assumption applies, and no probability distribution is needed to compute these observed rates.
Step 1: Total the counts. The eligible count is \(2+0+3+1=6\) claims.
Step 2: Total the matching exposure. Earned exposure is \(0.75+0.25+1.00+0.50=2.50\) policy-years.
Step 3: Divide total claims by total exposure.
As a check, the policy-level rates are \(2/0.75=2.6667\), \(0/0.25=0\), \(3/1.00=3\), and \(1/0.50=2\) claims per policy-year. Their exposure-weighted average is \([0.75(2.6667)+0.25(0)+1.00(3)+0.50(2)]/2.50=(2+0+3+1)/2.50=2.4\), agreeing with the aggregate calculation.
The unweighted mean is \((2.6667+0+3+2)/4=1.9167\) claims per policy-year. It differs because it gives the policy with 0.25 exposure the same weight as the policy with 1.00 exposure. The common mistake is to average the policy rates without considering their exposure. The correction is to weight each rate by its exposure, equivalently to divide total claims by total exposure.
Match Coverage and Time Before Dividing
Coverage-specific matching is particularly important when one policy contains several coverages. A claim can be eligible for one coverage but not another, and exposure can differ by coverage even though both appear on the same policy. Likewise, a claim should enter only the experience period defined for the analysis. A report date alone does not establish which coverage period generated the exposure; the claim’s applicable coverage and the analysis’s timing rules must be checked.
Worked Example: Matching a Coverage Count to Vehicle Exposure
A hypothetical fleet policy provides comprehensive coverage on two vehicles from January 1 through June 30 and on three vehicles from July 1 through December 31. The experience period is the calendar year. During that year, two eligible comprehensive claims occur while the coverage is in force. A collision claim also occurs, and a comprehensive loss from before the policy began is reported during the year. Calculate comprehensive frequency per vehicle-year.
Method and assumptions. Count only the two claims eligible for comprehensive coverage in the specified period. Exposure is earned in vehicle-years using the covered vehicle count in each half-year. The event counts are given as exact observations; no interest, mortality, or probability-distribution assumption is needed.
Step 1: Identify the numerator. The two in-period comprehensive claims are eligible, so \(N=2\). The collision claim is not a comprehensive claim. The earlier loss falls outside the coverage period being measured, so it is also excluded.
Step 2: Calculate vehicle-year exposure. In the first half-year, two vehicles contribute \(2(0.5)=1.0\) vehicle-year. In the second half-year, three vehicles contribute \(3(0.5)=1.5\) vehicle-years. Total exposure is \(1.0+1.5=2.5\) vehicle-years.
Step 3: Calculate frequency.
The exposure check is \(2(0.5)+3(0.5)=2.5\) vehicle-years, and the numerator check is two claims for the same coverage and year. A tempting but wrong calculation would count all three reported events and divide by 2.5, producing 1.2. That mixes a collision claim and a pre-inception loss into the comprehensive count. The correction is to establish claim eligibility before dividing, not to count every claim record attached to the policy.
Unit Conversion and Segment Aggregation
Rates can be compared or combined only after exposure units are made consistent. If one segment’s exposure is recorded in vehicle-months and another’s in vehicle-years, either convert both to the same unit or calculate the rates separately and combine them with the appropriate exposure weights. Since 12 vehicle-months equal one vehicle-year, a count per vehicle-month and a count per vehicle-year differ by a factor of 12.
Suppose segment \(j\) has count \(N_j\) and exposure \(e_j\). The combined frequency is \(\sum_j N_j/\sum_j e_j\), provided the segments use compatible claim eligibility rules, periods, and exposure units. It is not generally appropriate to take the simple mean of the segment rates, because segments may represent different amounts of exposure.
Worked Example: Converting Exposure Before Combining Segments
A hypothetical book has two segments, with counts defined under the same coverage and experience period. Segment A has 24 eligible claims and 12,000 earned vehicle-months. Segment B has 18 eligible claims and 6,000 earned vehicle-months. Calculate each segment’s rate and the combined frequency in claims per vehicle-year.
Method and assumptions. Convert vehicle-months to vehicle-years by dividing by 12, then divide eligible claims by exposure. The observed counts are fixed; no interest, mortality, or probability distribution is required for this calculation.
Step 1: Convert Segment A exposure. \(12{,}000/12=1{,}000\) vehicle-years. Its rate is \(24/1{,}000=0.024\) claims per vehicle-year.
Step 2: Convert Segment B exposure. \(6{,}000/12=500\) vehicle-years. Its rate is \(18/500=0.036\) claims per vehicle-year.
Step 3: Combine counts and exposures. Total claims are \(24+18=42\), and total exposure is \(1{,}000+500=1{,}500\) vehicle-years. Therefore:
The check using segment rates is \([1{,}000(0.024)+500(0.036)]/1{,}500=(24+18)/1{,}500=0.028\). The unweighted mean of the rates is \((0.024+0.036)/2=0.030\), which is not the combined portfolio rate because Segment A has twice Segment B’s exposure. A second common error is to divide claims by vehicle-months and label the result “per vehicle-year”; the unit conversion must be explicit.
Common Data Checks
A well-constructed pair is auditable: another analyst can tell what each claim represents and reproduce the denominator from the coverage and exposure rules. Before using a frequency observation, check the following:
- Scope: Does each counted claim belong to the selected risk unit and coverage?
- Period: Are claims and earned exposure defined for the same experience period?
- Unit: Are all exposure values on the same basis, such as policy-years or vehicle-years?
- Eligibility: Have exclusions and claim-counting rules been applied consistently?
- Validity: Is exposure positive for any record with a positive claim count? Such a pair usually signals a scope, timing, or data error that must be resolved.
The resulting observed rate describes claims per stated exposure unit. It is not automatically a prediction for a future period, and it does not by itself explain differences in risk among policies. Its value is that it puts claim counts on a denominator that reflects how much of the defined risk was actually insured.
Check Your Understanding
For each problem, keep the claim scope and exposure basis aligned before calculating.
- A hypothetical portfolio has 5 eligible claims on 1.5 policy-years of exposure and 2 eligible claims on 2.0 policy-years. What is the combined frequency per policy-year? (Answer: 2 claims per policy-year)
- A policy has 1 eligible claim during a period with 0.4 policy-years of exposure, no claims during a further 0.6 policy-years, and 3 eligible claims during another 1.0 policy-years. Find the total-period frequency. (Answer: 2 claims per policy-year)
- A segment has 9 eligible claims and 9,000 earned vehicle-months. Convert the exposure and calculate frequency per vehicle-year. (Answer: 0.012 claims per vehicle-year)
- Segment A has 12 eligible claims and 6 vehicle-years of exposure; Segment B has 36 eligible claims and 24 vehicle-years. Calculate combined frequency and the unweighted mean of the two segment rates. (Answer: combined frequency 1.6 claims per vehicle-year; unweighted mean 1.75 claims per vehicle-year)
- A hypothetical coverage has 4 eligible claims in one segment with 2 vehicle-years of exposure and 6 eligible claims in a second segment with 6 vehicle-years. A separate claim is recorded for a different coverage, with 3 vehicle-years of exposure on that coverage. What is the frequency for the specified coverage across the first two segments? (Answer: 1.25 claims per vehicle-year)