Why an Integrated Indication Requires Consistent Inputs
A pure premium indication combines selected losses with the exposures that produced them. When experience spans multiple periods, the losses may need to be trended to a common prospective period, while the exposures must remain on the same basis and correspond to those losses. A sound calculation therefore depends on more than applying a formula: the selections, time periods, trend factors, and rate assumptions must fit together.
This tutorial develops a complete indication from multi-period experience, adds supplied rate loads, and checks the indicated change by the loss ratio method. The calculations are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The denominator is exposure, not premium: the pure premium measures loss cost per exposure unit. Premium enters the loss ratio method as the amount to which the loss ratio applies. For a valid comparison, both methods must use the same selected losses, exposure base, trend assumptions, current rate level, and target loss ratio. The examples assume that exposures need no separate trend adjustment and that the supplied trend factors apply to losses.
Build the Indication in a Consistent Order
Choose losses and exposures from corresponding periods and on the same exposure basis. Confirm that the selected losses represent the measure required by the problem, such as selected ultimate losses.
Multiply each period’s losses by its supplied trend factor. Do not trend one period to a different prospective date from another.
Add trended losses across periods and divide by total matching exposures. This gives an exposure-weighted pure premium.
Divide the pure premium by the target loss ratio to obtain the indicated rate per exposure.
Compare the indicated rate with the current rate. Where the assumptions permit, reproduce the same change using trended losses and premium on the current rate level.
Why the Rate Formula Holds
Suppose the target loss ratio is \(1-q\), and let \(R\) be the rate per exposure required for the prospective period. With \(E\) exposures, premium at that rate is \(RE\). If \(L^*\) is the projected total loss, the rate is adequate under the target loss ratio when projected losses equal the target proportion of premium:
Dividing by \((1-q)E\) gives the required rate. Since projected loss per exposure is \(L^*/E\), the formula is:
This result assumes the same exposure unit is used throughout, losses are consistently selected and trended, and \(1-q\) is an appropriate positive target loss ratio. It does not, without an adjustment, incorporate a fixed dollar amount per exposure; the formula above treats the load through a percentage target loss ratio. If the problem specifies a different rate structure, use that structure rather than forcing it into this formula.
Worked Example: Two Periods, One Prospective Indication
A hypothetical insurer selects ultimate losses of $420,000 on 1,000 vehicle-years from the first period and $660,000 on 1,500 vehicle-years from the second. The supplied trend factors to the prospective period are 1.10 and 1.05, respectively. The current rate is $550 per vehicle-year, and the target loss ratio is 0.80. Calculate the indicated rate and percentage change, then check the result using the loss ratio method.
Method and matching. The selected losses and vehicle-years refer to the same periods and exposure base. Trend each loss amount to the prospective period:
Total trended losses are $462,000 + $693,000 = $1,155,000. Total exposures are 1,000 + 1,500 = 2,500 vehicle-years. Thus the projected pure premium is $1,155,000 / 2,500 = $462 per vehicle-year.
The target loss ratio is 0.80, so the indicated rate is $462 / 0.80 = $577.50 per vehicle-year. Relative to the current rate, the indicated change is ($577.50 - $550) / $550 = $27.50 / $550 = 0.05, or 5.00%.
Loss ratio check. At the current rate, premium on the matching exposure base is 2,500($550) = $1,375,000. The ratio of trended losses to this premium is $1,155,000 / $1,375,000 = 0.84. The indicated change is 0.84 / 0.80 - 1 = 0.05, or 5.00%. Both methods agree. The calculation avoids the common mistake of dividing losses by premium to obtain a pure premium; premium is used here only for the loss ratio check.
Exposure Weighting and the Loss Ratio Check
The aggregate calculation weights each period’s loss cost by its exposure volume. In general, averaging period pure premiums equally does not give the aggregate pure premium. Algebra makes the distinction clear: the aggregate is \(\sum_j E_j(L_jT_j/E_j) \,/\, \sum_j E_j\), so a period with more exposures receives more weight.
For a loss ratio check, the premium denominator must be at the same rate level used for the comparison. If the experience premiums reflect different historical rates, they must first be placed on a common current rate level using the supplied on-level factors. When a single current rate \(R_0\) applies to all matching exposures and the target loss ratio is \(1-q\), the two methods give the same indicated change:
Dividing this ratio by the target loss ratio and subtracting one gives the percentage change. The equivalence requires matching exposure and premium bases and consistent assumptions. A mismatch in either denominator can make two correct-looking calculations appear to disagree.
Worked Example: Different Exposure Volumes and a Weighted Result
A hypothetical book has selected losses of $360,000 on 800 exposures in 2023 and $520,000 on 1,000 exposures in 2024. Trend factors to the prospective period are 1.12 and 1.06. The current rate is $625 per exposure, and the target loss ratio is 0.82. Find the indicated rate and change, and verify the result.
Method. Apply the period-specific factors: $360,000(1.12) = $403,200 and $520,000(1.06) = $551,200. Total trended losses are $403,200 + $551,200 = $954,400; total exposures are 800 + 1,000 = 1,800. The aggregate pure premium is $954,400 / 1,800 = $530.2222 per exposure before final rounding.
Divide by the target loss ratio: $530.2222 / 0.82 = $646.6125. The indicated rate is $646.61 per exposure. Its change from $625 is ($646.6125 - $625) / $625 = 0.03458, or 3.46%.
Check. Premium at the current rate is 1,800($625) = $1,125,000. The trended loss ratio is $954,400 / $1,125,000 = 0.8483556. Therefore the loss ratio indication is 0.8483556 / 0.82 - 1 = 0.03458, or 3.46%.
A tempting but incorrect calculation is the unweighted average of the two trended period pure premiums: $403,200 / 800 = $504 and $551,200 / 1,000 = $551.20, whose average is $527.60. That gives an indicated rate of $527.60 / 0.82 = $643.41, not $646.61. The 800-exposure period should not receive the same weight as the 1,000-exposure period.
On-Level Premiums When Historical Rates Differ
The loss ratio check must use premium at a common rate level. Historical earned premium is not automatically comparable with a current rate: if rates changed during the experience period, the premium must be adjusted using the provided on-level factors. This adjustment is separate from trending losses. Trending moves losses to a prospective cost level; on-leveling moves premium to a common rate level.
Worked Example: Reconciling Historical Premium and Selected Losses
A hypothetical portfolio has 600 exposures in 2023, when its rate was $600 per exposure, and 900 exposures in 2024, when its rate was $625. Selected losses are $240,000 and $420,000, with trend factors of 1.25 and 1.10 to the prospective period. The current rate is $650 per exposure, and the target loss ratio is 0.75. Calculate the pure premium indication and verify it using on-level premium.
Pure premium method. Trended losses are $240,000(1.25) = $300,000 and $420,000(1.10) = $462,000. Their total is $762,000; total exposures are 600 + 900 = 1,500. The projected pure premium is $762,000 / 1,500 = $508. The indicated rate is $508 / 0.75 = $677.33 per exposure when rounded to cents. Using the unrounded rate, the change from $650 is ($677.3333 - $650) / $650 = 0.04205, or 4.21%.
Loss ratio check. On-level the 2023 premium using $650 / $600 = 1.0833333: 600($600) = $360,000, and $360,000(1.0833333) = $390,000. On-level the 2024 premium using $650 / $625 = 1.04: 900($625) = $562,500, and $562,500(1.04) = $585,000. Total on-level premium is $390,000 + $585,000 = $975,000. The trended loss ratio is $762,000 / $975,000 = 0.7815385. The indication is 0.7815385 / 0.75 - 1 = 0.04205, or 4.21%. The check agrees because the on-level premium is exactly 1,500($650).
The example avoids comparing trended losses with unadjusted historical premium. That would mix prospective loss costs with premium at two different rate levels and would not provide a valid check.
Common Traps and Practical Checks
- Using mismatched data. A loss total and exposure total must describe the same periods, coverage, and exposure unit. Check that the numerator and denominator correspond before calculating.
- Applying a factor to the wrong quantity. Use each supplied trend factor on the specified losses and period. Do not apply it to exposures or historical premium unless the assumptions explicitly require that.
- Averaging period pure premiums equally. Aggregate trended losses and exposures first; this automatically weights periods by exposure volume.
- Comparing with the wrong premium level. On-level historical premium before calculating a loss ratio indication when rates differ by period.
- Confusing a rate with a rate change. The indicated rate is a dollar amount per exposure. The percentage change is the difference from the current rate divided by that current rate.
- Applying a percentage load additively. With target loss ratio \(1-q\), the rate is pure premium divided by \(1-q\), not pure premium multiplied by \(1+q\).
As a final audit, check units, recompute the total trended losses and exposures, confirm the target loss ratio, and verify the percentage change from the unrounded indicated rate. A loss ratio check is especially useful for detecting inconsistent premium levels or exposure denominators, but agreement alone cannot repair selections that do not represent the intended experience.
Check Your Understanding
For each problem, use the stated trend factors and loads. Round dollar rates to cents and percentage changes to two decimal places.
- A hypothetical book has $210,000 of selected losses on 500 exposures and a trend factor of 1.08. The current rate is $430 per exposure and the target loss ratio is 0.80. Find the indicated rate and percentage change. (Answer: $567.00 per exposure; increase of 31.86%)
- Selected losses are $300,000 on 750 exposures in the first period and $525,000 on 1,250 exposures in the second. The trend factors are 1.10 and 1.04. With target loss ratio 0.88 and current rate $480, calculate the indicated rate and percentage change. (Answer: $497.73 per exposure; increase of 3.69%)
- A hypothetical portfolio has $480,000 of selected losses on 1,200 exposures with trend factor 1.15, and $320,000 on 800 exposures with trend factor 1.05. The target loss ratio is 0.75 and current rate is $600. Calculate the indicated rate and change. (Answer: $592.00 per exposure; decrease of 1.33%)
- Trended losses are $410,000 on 1,000 matching exposures. The current rate is $520 per exposure and the target loss ratio is 0.80. Use the loss ratio method to check the pure premium indication and state the percentage change. (Answer: $512.50 per exposure; decrease of 1.44%)
- Two periods have already-trended pure premiums of $300 and $600 per exposure, with 900 and 100 exposures, respectively. The current rate is $350 and the target loss ratio is 0.90. Calculate the exposure-weighted pure premium, indicated rate, and percentage change; do not average the two pure premiums equally. (Answer: $330.00 pure premium; $366.67 indicated rate; increase of 4.76%)