From Selected Frequency to Expected Count
A selected claim frequency describes the expected number of claims per unit of exposure on a stated basis. Projecting expected counts requires one more input: the exposure expected for the forecast period. The central task is to match those quantities consistently. A frequency expressed per 100 policy-years cannot be multiplied directly by a number of policies unless the exposure has first been converted to policy-years and the rate basis has been handled.
This tutorial treats the selected frequency as applicable to the forecast period and to the exposure definition being used. If a frequency trend is supplied, first determine the rate for the forecast period; then apply that rate to the matching forecast exposure. The result is an expected count, which need not be an integer. It is a mean, not a prediction that a fractional number of claims will occur.
The units must agree. For example, if \(f_s\) is claims per 100 policy-years, \(E_s\) must be measured in policy-years, and the count is \(f_sE_s/100\). If the rate is already expressed as claims per policy-year, no further scaling is needed. Frequency may be estimated or selected from experience, a model, or other appropriate analysis; this calculation does not by itself determine which frequency is appropriate.
The projection also depends on the exposure definition. If the selected rate is per earned vehicle-year, the forecast exposure must represent earned vehicle-years, not simply the number of vehicles at a point in time. A policy that contributes only part of a year contributes the corresponding fraction of an exposure unit when that is how the selected frequency is defined.
Why Frequency Times Exposure Gives the Mean Count
The formula follows from the definition of frequency. For a segment with positive exposure \(E_s\), frequency is the expected count divided by exposure. Multiplying both sides by exposure recovers the expected count. For a more explicit justification, suppose the forecast exposure is divided into \(m\) comparable units and let \(I_j\) be the claim count in unit \(j\). If each unit has expected count \(f_s\), then the total count is \(N_s=\sum_{j=1}^{m} I_j\), and linearity of expectation gives the result below.
Here \(m\) is the number of exposure units. The argument does not require the counts in different units to be independent; linearity of expectation is sufficient. In practice, exposure may be fractional, and the frequency is interpreted as an expected rate per unit, so the same multiplication applies. A Poisson count model is not required for calculating the expected count. If a Poisson model is appropriate, its mean would be this expected count, but the projection formula alone does not establish that assumption.
For multiple segments, the portfolio count is the sum of the segment counts. By the same linearity property, the portfolio expected count is the sum of segment expected counts, whether or not segment counts are independent. The portfolio frequency, when desired, is therefore an exposure-weighted rate, not generally the simple average of the segment rates.
The displayed expression assumes each \(f_s\) is per one exposure unit. For rates stated per \(k\) units, divide each segment product by \(k\). If all segments use the same exposure unit and rate basis, the aggregate portfolio frequency is
This weighted rate is useful as a summary, but calculating expected counts segment by segment is often clearer, particularly when segments have different selected frequencies or different future exposure forecasts.
Projecting One Segment
Worked Example: Applying a Two-Year Frequency Trend
A hypothetical insurer selects a frequency of 0.080 claims per 100 policy-years at the current rate level. Its selected annual frequency trend is 2.5%, compounded for two years. Forecast exposure for the target period is 12,000 policy-years. Project the expected claim count. State the method and avoid mixing the rate basis with the exposure units.
Method. Project the selected frequency to the target period, then multiply by the number of 100-policy-year exposure blocks. This assumes the selected trend applies for both full years and the target exposure is measured on the same policy-year basis.
The two-year trend factor is \(1.025^2=1.050625\). Thus the target-period frequency is \(0.080\times1.050625=0.08405\) claims per 100 policy-years.
The forecast exposure contains \(12{,}000/100=120\) blocks of 100 policy-years. Therefore the expected count is \(0.08405\times120=10.086\) claims, or 10.09 expected claims when rounded to two decimal places. As a check, the count before applying the trend is \(0.080\times120=9.6\); the projected count is larger, consistent with a positive trend.
The common mistake is to multiply 0.08405 by 12,000 as though the frequency were per one policy-year. That calculation gives 1,008.6 and is too large by a factor of 100. The correction is to divide exposure by 100, or equivalently divide the product of the rate and exposure by 100.
Projecting and Aggregating Multiple Segments
Worked Example: Combining Three Exposure Segments
A hypothetical equipment insurer has three segments. Forecast exposures are 4,000, 6,000, and 2,500 equipment-years. Their selected frequencies are respectively 0.120, 0.075, and 0.040 claims per 100 equipment-years. Calculate the expected count in each segment, the total expected count, and the aggregate frequency. Identify why an unweighted average of the three frequencies is not the correct portfolio rate.
Method. Convert each exposure to blocks of 100 equipment-years, apply that segment’s selected frequency, and add the segment counts. The selected rates and exposure units are assumed to be comparable within each segment.
For segment 1, \(4{,}000/100=40\) blocks, so the expected count is \(40\times0.120=4.8\). For segment 2, \(6{,}000/100=60\) blocks, so the expected count is \(60\times0.075=4.5\). For segment 3, \(2{,}500/100=25\) blocks, so the expected count is \(25\times0.040=1.0\).
The total exposure is \(4{,}000+6{,}000+2{,}500=12{,}500\) equipment-years. The total expected count is \(4.8+4.5+1.0=10.3\) claims. The aggregate frequency is \(10.3/(12{,}500/100)=10.3/125=0.0824\) claims per 100 equipment-years. Checking the result, \(0.0824\times125=10.3\).
The unweighted mean of the segment rates is \((0.120+0.075+0.040)/3=0.07833\) claims per 100 equipment-years, approximately. Applying it to all 125 exposure blocks would give about \(0.07833\times125=9.7917\) claims, not 10.3. This is wrong because the segments do not have equal exposure. The correct aggregate rate weights each segment rate by its share of the total exposure.
Worked Example: Forecasting Segment Rates Before Aggregation
A hypothetical specialty insurer forecasts 15,000 exposure units in segment A and 6,000 in segment B. At the current rate level, the selected frequencies are 0.52 and 0.90 claims per 1,000 exposure units, respectively. The selected annual frequency trends are 2% for A and −1% for B, and the target period is two years away. Project the target-period counts by segment and in total. Use the segment-specific trends and state the rounding.
Method. Trend each selected frequency for two years, multiply by the segment exposure expressed in blocks of 1,000, and then sum. This assumes each stated trend is compounded annually for both years and that the forecast exposure uses the same units as the selected rates.
For A, the trend factor is \(1.02^2=1.0404\). The target frequency is \(0.52\times1.0404=0.541008\) claims per 1,000 units. Exposure is \(15{,}000/1{,}000=15\) blocks, so A’s expected count is \(0.541008\times15=8.11512\).
For B, the trend factor is \(0.99^2=0.9801\). The target frequency is \(0.90\times0.9801=0.88209\) claims per 1,000 units. Exposure is \(6{,}000/1{,}000=6\) blocks, so B’s expected count is \(0.88209\times6=5.29254\).
The total expected count is \(8.11512+5.29254=13.40766\), or 13.41 claims to two decimal places. As a check, the total exposure is 21 blocks of 1,000; the exposure-weighted target frequency is \(13.40766/21=0.63846\) claims per 1,000 units, and \(0.63846\times21=13.40766\). The common mistake is to apply one common trend to both segments or to average their trends first. Since the segment frequencies and exposures differ, calculate each segment’s projected count using its own selected rate and trend before aggregating.
Practical Checks and Limitations
A count projection is only as meaningful as its inputs. Before calculating, confirm that the frequency’s claim definition, segment, time basis, and exposure denominator match the forecast. A rate based on one definition of claim count should not be applied to a differently defined count without justification. Likewise, projected exposure should represent the period and population for which the selected frequency is intended.
The calculation does not determine whether the selected frequency is credible, whether a trend is appropriate, or whether future exposure will actually emerge as forecast. Those are input-selection questions. Here, the frequency and exposure are treated as given. If either input is uncertain, the projected count is conditional on the selected values; the multiplication alone does not quantify that uncertainty.
Keep the result in perspective: an expected count such as 10.09 is a mean and can be fractional even though actual claims are whole events. Do not round intermediate segment counts before aggregation unless a reporting convention specifically requires it. Rounding early can create avoidable differences between the sum of segment projections and a portfolio calculation.
Check Your Understanding
For each problem, use the stated frequency basis and show the unit conversion before calculating. Answers are rounded only as stated.
- A selected frequency is 0.065 claims per 100 service-years. Forecast exposure is 18,000 service-years. What is the expected count? (Answer: 11.70 expected claims.)
- A selected rate is 0.30 claims per 1,000 inspection-years, with a 4% annual frequency trend for three years. Forecast exposure is 25,000 inspection-years. What is the projected count, rounded to two decimals? (Answer: 8.44 expected claims.)
- Two hypothetical segments have forecast exposures of 3,200 and 6,800 units, with selected frequencies of 0.11 and 0.07 claims per 100 units. Find the two segment counts, total count, and aggregate frequency per 100 units. (Answer: 3.52 and 4.76 expected claims; 8.28 total; 0.0828 claims per 100 units.)
- Segment P has a selected frequency of 0.52 claims per 1,000 units, a 2% annual trend, and 15,000 forecast units. Segment Q has a selected frequency of 0.90 per 1,000, a −1% annual trend, and 6,000 forecast units. Project each rate for two years, then find the total expected count, rounded to two decimals. (Answer: P = 8.11512, Q = 5.29254; total = 13.41 expected claims.)
- A forecast of 8,400 policy-years is split 60% to segment M and 40% to segment N. Their selected frequencies are 0.075 and 0.120 claims per policy-year. Calculate the segment exposures and expected counts, then the aggregate frequency. (Answer: M exposure = 5,040 policy-years and count = 378; N exposure = 3,360 policy-years and count = 403.2; total = 781.2 expected claims and aggregate frequency = 0.093 claims per policy-year.)