From Experience Losses to an Indicated Rate
A pure premium indication connects selected claim experience to a prospective price per exposure. When the experience period differs from the period for which rates are being set, losses may need to be trended to a common prospective period. The resulting loss cost is then converted to a rate using the loading convention specified in the problem.
The sequence matters: trend the selected losses, divide by the corresponding exposures, and only then apply the stated loading. An exposure is the chosen unit of insurance risk, such as one vehicle-year or one building-year; it is not premium. This method is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Trend the Losses to a Common Period
A trend factor adjusts a loss amount for the expected change in loss costs between its experience period and the prospective period. A problem may supply the cumulative factor directly. If it supplies an annual trend rate \(t\) and a trend interval of \(n\) years, the standard compound-trend model gives
This formula assumes a constant annual trend rate, compounding at the stated annual rate, and a time interval measured in years. The interval should be the one selected for the loss measure in the problem; it is not automatically the number of calendar years between policy effective dates. A stated factor already incorporates the chosen interval and should not be compounded again.
Apply the factor to losses, not to exposures, when the intended calculation is a prospective loss cost per historical exposure. The denominator remains the exposure associated with the selected losses, provided the exposure base and coverage are consistent across the experience being combined. If exposures or coverage are not comparable, dividing the trended losses by their unadjusted total may not produce a meaningful indication.
Worked Example: Combining Two Trended Experience Segments
A hypothetical insurer selects $420,000 of losses from one experience segment and $510,000 from a second. The corresponding exposures are 4,000 and 4,500, respectively. The supplied cumulative trend factors to the prospective period are 1.12 and 1.06. A multiplicative loading of 8% on pure premium is specified. Calculate the indicated rate per exposure.
Method. Trend each segment's losses to the prospective period, add the trended losses, divide by total exposures, and apply the multiplicative loading.
Step 1: Trend the first segment. $$ \$420{,}000(1.12)=\$470{,}400. $$
Step 2: Trend the second segment. $$ \$510{,}000(1.06)=\$540{,}600. $$
Step 3: Combine the losses and exposures. Total trended losses are \(\$470{,}400+\$540{,}600=\$1{,}011{,}000\). Total exposures are \(4{,}000+4{,}500=8{,}500\).
Step 4: Calculate the pure premium. $$ PP=\frac{\$1{,}011{,}000}{8{,}500}=\$118.941176\ldots $$
Step 5: Apply the loading. The loading is a markup on pure premium, so $$ R=\$118.941176\ldots(1.08)=\$128.456470\ldots $$ Rounded to the nearest cent, the indicated rate is $128.46 per exposure.
Check. The markup is \(\$118.941176\ldots(0.08)=\$9.515294\ldots\); adding it to the pure premium gives \(\$128.456470\ldots\), consistent with the multiplication. The common mistake avoided is averaging the two segment pure premiums without weighting for their exposures.
Why the Combined Pure Premium Is Exposure-Weighted
The aggregate formula is not an arbitrary averaging rule. It follows directly from the definition of loss cost per exposure. For each segment, its trended loss cost is \(L_jT_j/E_j\). To combine segments, the total trended losses are added and divided by total exposures:
To verify the equality, distribute the exposure weight \(E_j/\sum_k E_k\) across the segment loss cost \(L_jT_j/E_j\). The \(E_j\) terms cancel, leaving \(L_jT_j/\sum_k E_k\). Summing these terms gives \(\sum_j L_jT_j/\sum_k E_k\), the aggregate formula. Thus the segment loss costs are weighted by their shares of total exposure, not given equal weights unless the exposure amounts happen to be equal.
This result also explains why total premium is not the denominator in a pure premium calculation: the denominator defines the amount of insured risk being priced. Premium is a price produced by rating, whereas exposure is the unit over which the loss cost is measured.
Annual Trend Rates and Fractional Periods
When the annual rate and time interval are supplied instead of a cumulative trend factor, calculate the factor first. Under the compound-trend model, a two-year interval at an annual trend rate of 5% gives \(1.05^2=1.1025\), not \(1+2(0.05)=1.10\). The latter is a simple-trend approximation, not the compound model. Use simple trend only if the problem explicitly instructs you to do so.
Worked Example: Calculating a Trend Factor from an Annual Rate
A hypothetical portfolio has $360,000 of selected losses from an experience midpoint exactly two years before the prospective midpoint. The selected annual loss trend rate is 5%. There are 3,150 corresponding exposures, and the specified multiplicative loading on pure premium is 10%. Calculate the indicated rate.
Method. Use compound annual trending for two years, calculate the pure premium from trended losses per exposure, and apply the markup.
Step 1: Calculate the cumulative trend factor. $$ T=(1.05)^2=1.1025. $$
Step 2: Trend the losses. $$ \$360{,}000(1.1025)=\$396{,}900. $$ The increase is \(\$396{,}900-\$360{,}000=\$36{,}900\).
Step 3: Calculate the pure premium. $$ PP=\frac{\$396{,}900}{3{,}150}=\$126.00. $$
Step 4: Apply the loading. $$ R=\$126.00(1.10)=\$138.60. $$ The indicated rate is $138.60 per exposure.
Check. Ten percent of \(\$126.00\) is \(\$12.60\), and \(\$126.00+\$12.60=\$138.60\). The common mistake avoided is using one year of trend because the rate is stated annually, even though the selected period is two years.
Applying a Multiplicative Loading Correctly
A problem's loading language determines the rate formula. A multiplicative markup of \(m\) on pure premium means that the rate is the pure premium plus \(m\) times the pure premium. Factoring out the pure premium gives \(R=PP(1+m)\). In contrast, if a variable provision is \(v\) of the final rate, then the portion of rate available for losses is \(1-v\), so \(PP=R(1-v)\) and \(R=PP/(1-v)\).
For example, a 20% markup on a pure premium of $100 gives \(\$100(1.20)=\$120\). A 20% provision as a share of final rate gives \(\$100/(1-0.20)=\$125\). The difference arises because the provision in the second case is calculated as a percentage of the larger final rate, not the pure premium. Do not infer one convention from the word “loading” alone; use the definition stated in the problem.
Worked Example: Preserving Precision Before Applying the Loading
A hypothetical program has $1,169,958 of selected losses and 10,000 corresponding exposures. The supplied cumulative trend factor to the prospective period is 1.10. The problem specifies an 8% multiplicative markup on pure premium. Calculate the indicated rate per exposure to the nearest cent.
Method. Apply the cumulative factor once, divide by exposures, retain the resulting precision through the loading calculation, and round the final rate only.
Step 1: Trend the selected losses. $$ \$1{,}169{,}958(1.10)=\$1{,}286{,}953.80. $$
Step 2: Calculate pure premium. $$ PP=\frac{\$1{,}286{,}953.80}{10{,}000}=\$128.69538. $$
Step 3: Apply the 8% markup. $$ R=\$128.69538(1.08)=\$138.9910104. $$ Rounded to the nearest cent, the indicated rate is $138.99 per exposure.
Check. Eight percent of \(\$128.69538\) is \(\$10.2956304\). Adding that amount to the pure premium gives \(\$138.9910104\), which rounds to \(\$138.99\). A common arithmetic trap is to report \(\$139.03\) for this multiplication; that value is incorrect. Another trap is rounding the pure premium to \(\$128.70\) before applying the loading, which gives \(\$138.996\), a different unrounded result. Keep precision until the final rounding step.
Common Errors and a Reliable Calculation Sequence
A sound indication depends on consistent choices as well as correct arithmetic. In particular, the selected losses, trend periods, exposure base, and loading definition must all refer to the same intended calculation. Use the following sequence to make each choice visible:
Identify the loss amounts, corresponding exposures, prospective period, and whether trend factors are supplied or must be calculated.
Apply each appropriate cumulative factor once. If using an annual rate, calculate the factor from the stated time interval and compounding model.
Add the trended losses and the corresponding exposures. Confirm that the exposure units and coverage are comparable.
Divide total trended losses by total exposures, not by premium and not by the number of experience segments.
Use multiplication for a markup on pure premium and division by \(1-v\) for a provision stated as a share of final rate.
Recompute the arithmetic, verify the dollar-per-exposure units, and round only at the requested final precision.
The most common errors are using the wrong trend interval, applying a supplied cumulative factor a second time, changing the exposure denominator without a stated reason, and confusing a markup with a share of final rate. A brief check can catch many of these: trend-adjusted losses should reflect the direction and size of the selected trend, and the final rate should exceed the pure premium when a positive markup is specified.
Check Your Understanding
For each problem, identify the trend calculation, exposure denominator, and loading convention before calculating. Round indicated rates to the nearest cent.
- A hypothetical group has $240,000 of selected losses, 2,000 exposures, a cumulative trend factor of 1.06, and a 5% markup on pure premium. What is the indicated rate per exposure? (Answer: $133.56 per exposure)
- Two hypothetical experience segments have losses of $400,000 and $500,000, exposures of 4,000 and 5,000, and cumulative trend factors of 1.04 and 1.00. Apply a 6% markup to the aggregate pure premium. What rate is indicated? (Answer: $107.88 per exposure)
- A hypothetical account has $250,000 of selected losses from a midpoint two years before the prospective midpoint, 2,200 exposures, and an annual compound trend rate of 4%. Apply a 9% markup. What is the indicated rate? (Answer: $133.97 per exposure)
- A hypothetical portfolio's trend-adjusted pure premium is $150 per exposure. A variable provision is specified as 20% of final rate, not as a markup on pure premium. What is the indicated rate? (Answer: $187.50 per exposure)
- A hypothetical insurer selects $300,000 of losses with a cumulative trend factor of 1.10 and $400,000 with a factor of 1.05. The corresponding exposures are 2,500 and 3,500. If the loading is an 8% markup on pure premium, what rate is indicated? (Answer: $135.00 per exposure)