Why an Aggregate Frequency Can Mislead
A rise or fall in observed claim frequency does not necessarily mean that the claim propensity of comparable risks changed. The portfolio may contain a different mix of risk groups, use a different exposure denominator, or have different coverage terms that affect whether an event becomes a counted claim. If these changes are mistaken for a frequency trend, the selected trend can distort a projection.
The key is to compare like with like. First define the claim count and exposure consistently. Then ask whether the mix of exposures and the conditions for counting claims remained comparable. When they did not, an aggregate frequency trend should not automatically be applied to the next period. This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
If the group exposures cover the same population and claim definition, let \(w_{jt}=e_{jt}/\sum_j e_{jt}\) be group \(j\)'s share of total exposure. Then aggregate frequency is a weighted average of group frequencies. The weights change when the exposure mix changes, even if each group's frequency is unchanged.
What Makes Two Periods Comparable?
A frequency comparison is meaningful only when the numerator and denominator describe the same exposure and claim basis in both periods. A claim count should follow a consistent rule: for example, whether separate reports from one event count as one claim or several, and whether claims are counted when reported or when incurred. The denominator should also represent the risk exposure during the period, not merely a convenient count taken at one date.
A policy-year, vehicle-year, payroll amount, and number of locations are not interchangeable exposure bases. A change in average time insured can make policy counts misleading: 1,000 policies each insured for a full year represent more exposure than 1,000 policies each insured for half a year. Similarly, a change in the number of vehicles per policy can make policy counts a poor substitute for vehicle-years. Use the basis that matches the risk unit being counted, and keep its time unit consistent.
Coverage can affect the numerator even when the underlying event rate is stable. A higher deductible may mean fewer events generate reported claims or payments; a changed limit can also alter which events meet a reporting rule. In that case, the observed claim frequency is frequency under the specified coverage and claim-count definition. It is not automatically a measure of the underlying frequency of accidents or other events.
Fixed-Mix Comparisons
When group frequencies can be calculated on a consistent basis, a fixed-mix comparison helps separate a change in group frequencies from a change in the proportions of exposures across groups. Choose reference weights \(w_j^*\) that sum to one. The standardized frequency for period \(t\) is the frequency that would have resulted if period \(t\)'s group frequencies had been applied to those same reference weights:
The reference weights could be the first period's mix, the latest period's mix, or a specified prospective mix. State the choice: a standardized comparison depends on it. If the standardized frequency changes, group frequencies changed on the chosen basis. If the aggregate frequency changes while the standardized frequency does not, the difference is explained by the altered group proportions, subject to the quality of the grouping and data.
Standardization is useful only when the groups and claim bases are sufficiently comparable. It cannot correct a changed claim definition by itself, nor does it prove why a group's frequency changed. If group definitions are too broad, important differences can remain within a group. For selecting a projection, separately state the frequency trend assumption and the anticipated future exposure mix.
Worked Example: Stable Group Rates, Changing Risk Mix
A hypothetical insurer groups earned vehicle-years into standard and higher-risk classes. In Period 1, it records 900 standard vehicle-years with 18 claims and 100 higher-risk vehicle-years with 6 claims. In Period 2, it records 500 vehicle-years in each class, with 10 standard-class claims and 30 higher-risk claims. Assume claim definitions, coverage, and exposure measurement are unchanged, and that the counts are complete for both periods. Calculate the aggregate frequencies and determine whether the data show a change in either class frequency.
Method. Calculate each class frequency and the total claims per total vehicle-year. Then compare the period totals with a fixed mix.
Period 1 standard frequency is \(18/900=0.02\), or 2 claims per 100 vehicle-years. Higher-risk frequency is \(6/100=0.06\), or 6 per 100. The aggregate is \((18+6)/(900+100)=24/1{,}000=0.024\), or 2.4 per 100.
Period 2 standard frequency is \(10/500=0.02\); higher-risk frequency is \(30/500=0.06\). The aggregate is \((10+30)/(500+500)=40/1{,}000=0.04\), or 4 per 100. The observed aggregate increased from 0.024 to 0.04, a relative increase of \(0.04/0.024-1=0.6667\), or 66.67%.
The class frequencies did not change. To verify the mix explanation, use Period 1's weights: 90% standard and 10% higher-risk. The standardized frequency for Period 2 is \(0.90(0.02)+0.10(0.06)=0.018+0.006=0.024\). It equals Period 1's aggregate frequency. The apparent 66.67% increase is entirely due to the higher share of higher-risk exposures in Period 2 under the stated assumptions.
Check. Period 2 has 50% higher-risk exposure, so its aggregate must be halfway between 0.02 and 0.06: \((0.02+0.06)/2=0.04\), matching the direct calculation. The common mistake is to treat the 66.67% aggregate increase as a trend in comparable risks. A projection should instead use supportable class frequencies and an explicit future class mix.
Coverage Changes and Claim Propensity
A shift in coverage can affect observed claim counts even when exposure is measured correctly. For example, if a claim is counted only when a loss exceeds the deductible, raising the deductible can reduce counted claims without changing how often losses occur. The comparison therefore needs either a consistent coverage basis or a clearly defined frequency measure that accounts for the coverage difference.
A coverage adjustment cannot be inferred merely from the aggregate counts. It requires appropriate information about the events or losses that would be counted under each coverage condition. If that information is unavailable, report the observed frequencies on their respective coverage bases and avoid interpreting the difference as a pure change in claim propensity.
Worked Example: A Deductible Shift Changes Reported Frequency
A hypothetical book has 800 vehicle-years in each of two periods. In Period 1, all vehicles have a low deductible. Assume there are 40 accident events, and 80% produce a counted claim under that deductible. In Period 2, 400 vehicle-years retain the low deductible and 400 have a higher deductible. Each half has 20 accident events; under the low deductible, 80% produce a counted claim, while under the higher deductible, 40% do. Assume the accident-event frequency is unchanged and claim reporting probabilities are as stated. Find the observed counted-claim frequencies and compare them with accident-event frequency.
Method. Count claims separately by deductible before dividing by matching vehicle-years. Keep the event count distinct from the claim count.
In Period 1, counted claims are \(40(0.80)=32\). The observed frequency is \(32/800=0.04\), or 4 counted claims per 100 vehicle-years. Accident-event frequency is \(40/800=0.05\), or 5 events per 100.
In Period 2, the low-deductible half produces \(20(0.80)=16\) counted claims; the higher-deductible half produces \(20(0.40)=8\). Total counted claims are \(16+8=24\), so observed frequency is \(24/800=0.03\), or 3 per 100. Accident events total 40, so event frequency remains \(40/800=0.05\).
The reported-claim frequency falls by \(0.03/0.04-1=-0.25\), or 25%, but the assumed accident-event frequency is unchanged. If the future book is entirely low-deductible and the assumptions continue, the appropriate counted-claim frequency is \(0.05(0.80)=0.04\), not 0.03. The common mistake is to apply the observed 25% decline as a trend in accident propensity, or to use the mixed-coverage rate as though it described an all-low-deductible future book.
Check. The Period 2 claim count is 24, not 40: 40 is the number of events, of which some do not produce a counted claim under the specified deductible conditions. The coverage-specific count calculation reconciles to \(16+8=24\).
Consequences for Projections
A projection combines a frequency assumption with an exposure forecast. If group frequencies are projected separately, expected claims for a future period can be calculated as the sum of projected group frequencies times forecast group exposures. This makes the assumed future mix visible rather than burying it in a single historical aggregate.
This calculation assumes that each projected group frequency applies to the corresponding future exposure, with claim definitions and coverage matched to the frequency basis. If only an aggregate frequency is available, applying it to forecast total exposure assumes that the aggregate rate is appropriate for the future mix. That assumption may be reasonable when the mix is stable, but should not be hidden when the mix is expected to change.
Worked Example: Avoiding a False Aggregate Trend in a Projection
A hypothetical portfolio has low- and high-risk groups. In Period 1, the group frequencies are 0.020 and 0.060, with 80% and 20% of exposure, respectively. In Period 2, both group frequencies are 10% higher, while exposure is split equally between the groups. For the next period, assume each group frequency will again increase by 10% from Period 2, and forecast 10,000 vehicle-years with a 70% low-risk and 30% high-risk mix. Calculate the expected claims. Compare with the result from incorrectly applying Periods 1-to-2 aggregate trend to Period 2's aggregate frequency.
Method. First calculate the two aggregate frequencies and the group-based future projection. Then apply the naive aggregate trend as a check on the size of the distortion.
Period 1 aggregate frequency is \(0.80(0.020)+0.20(0.060)=0.016+0.012=0.028\). Period 2 group frequencies are \(0.020(1.10)=0.022\) and \(0.060(1.10)=0.066\). With equal exposure shares, Period 2 aggregate frequency is \(0.50(0.022)+0.50(0.066)=0.044\).
The observed aggregate factor from Period 1 to Period 2 is \(0.044/0.028=1.5714\), an increase of 57.14%. But the group frequencies each increased only 10%; the rest of the aggregate change reflects the move from an 80% low-risk mix to a 50% low-risk mix.
For the next period, projected group frequencies are \(0.022(1.10)=0.0242\) and \(0.066(1.10)=0.0726\). Applying the specified future mix gives \(0.70(0.0242)+0.30(0.0726)=0.01694+0.02178=0.03872\). Expected claims are \(10{,}000(0.03872)=387.2\).
If the Period 1-to-2 aggregate factor is incorrectly applied to Period 2's aggregate rate, the next aggregate rate would be \(0.044(1.5714)=0.06914\) approximately, implying about \(10{,}000(0.06914)=691.4\) claims. This repeats a historical mix shift as though it were a continuing within-group trend and ignores the stated future mix. The correct projection is 387.2 expected claims under the assumptions. Expected claims need not be an integer.
Check. The future frequency of 0.03872 lies between the projected low-risk and high-risk rates, 0.0242 and 0.0726, and is closer to the low-risk rate because 70% of the future exposure is low-risk. This is consistent with the calculated future mix.
Common Traps and Practical Checks
- Using mismatched denominators. Do not compare claim counts per policy with claims per vehicle-year as if they were the same rate. Convert to a consistent exposure basis when the necessary information is available.
- Ignoring exposure duration. A policy count at a date does not necessarily represent the amount of exposure during a full observation period. Use earned exposure on the defined time basis.
- Calling every aggregate change a trend. Calculate group frequencies and inspect their exposure shares before selecting an aggregate trend.
- Mixing coverage bases. A deductible or reporting-rule change may alter the count of claims while leaving event frequency unchanged. Label which frequency is being measured.
- Standardizing without stating the weights. A fixed-mix frequency is conditional on its reference mix. State that mix and use it consistently in the comparison.
- Projecting a historical mix shift forward automatically. Use the forecast exposure mix or justify an assumption that the aggregate frequency remains suitable for future exposure.
Check Your Understanding
For each problem, identify the relevant frequency basis and distinguish an exposure or coverage change from a change in group frequency.
- A hypothetical insurer records 12 claims in 600 standard-risk vehicle-years and 8 claims in 200 higher-risk vehicle-years in Period A. In Period B it records 18 claims in 900 standard-risk vehicle-years and 12 claims in 300 higher-risk vehicle-years. Find the two aggregate frequencies and say whether either group frequency changed. (Answer: Period A \(20/800=0.025\); Period B \(30/1{,}200=0.025\); neither group rate changed: standard 0.020 and higher-risk 0.040 in both periods.)
- In a hypothetical book, 500 low-deductible vehicle-years have 25 counted claims, while 500 high-deductible vehicle-years have 15 counted claims. Find the aggregate counted-claim frequency and the two coverage-specific frequencies per 100 vehicle-years. (Answer: Aggregate \(40/1{,}000=0.040\); low-deductible frequency 5 per 100; high-deductible frequency 3 per 100.)
- A portfolio has group frequencies of 0.030 and 0.050, with 70% and 30% of exposure in the two groups. Calculate its aggregate frequency. If the next period has 40% exposure in the first group and group frequencies are unchanged, calculate the new aggregate frequency and identify the source of the change. (Answer: Initial \(0.70(0.030)+0.30(0.050)=0.036\); new \(0.40(0.030)+0.60(0.050)=0.042\); the change is due to exposure mix.)
- A hypothetical portfolio has 2,000 one-year policies in a period, each covering an average of 1.5 vehicles for the full year, and 90 claims. Calculate frequency per policy-year and per vehicle-year. Which measure is appropriate for comparing claim counts per insured vehicle-year? (Answer: \(90/2{,}000=0.045\) per policy-year; vehicle exposure is \(2{,}000(1.5)=3{,}000\) vehicle-years, so \(90/3{,}000=0.030\) per vehicle-year; use the latter for that comparison.)
- Two risk groups have current frequencies of 0.018 and 0.048. Each is assumed to increase by 5% next period. Forecast exposure is 4,000 units in the first group and 1,000 in the second. Calculate projected expected claims. (Answer: Projected rates are 0.0189 and 0.0504; expected claims are \(4{,}000(0.0189)+1{,}000(0.0504)=75.6+50.4=126\).)