Why Combine Experience Periods?
A rate indication based on one experience period may be unduly affected by random variation, unusual events, or a small amount of business. Combining several selected periods can provide a broader view. The essential task is not merely to add rows: the amounts must first be made comparable, and each period must contribute in relation to the amount of business it represents.
For example, an annual loss ratio from a period with $500,000 of premium should not automatically have the same influence as one from a period with $2,000,000 of premium. A pooled loss ratio reflects the premium volume underlying each period. Similarly, a pooled pure premium reflects the exposure volume. This tutorial focuses on assembling and combining those inputs; it does not determine which periods should be selected or how to make a separate judgment about their future representativeness.
What Must Be Consistent Across Periods?
The period-level figures should be prepared using the choices established for the indication: the selected experience period, premium base, exposure base, loss measure, and prospective projection basis. Earlier material on on-level premium and loss trending explains those adjustments. Here, the important point is that the adjustments must be applied period by period before the figures are combined.
Let \(P_j^{OL}\) denote the on-level premium for period \(j\), measured at the current rate level. Let \(L_j^{proj}\) denote that period’s selected loss amount projected to the prospective period on the chosen loss basis. Let \(E_j\) denote the corresponding exposure. The symbols are period-specific, but their definitions and units must remain the same across all included periods.
A practical dataset generally has one row per period and columns for the raw records, relevant adjustments, and final selected inputs. Keeping both raw and adjusted amounts makes the calculation auditable: a reviewer can see what was changed and can trace each combined total to its source period.
| Period-level field | Role in the combined dataset | Consistency check |
|---|---|---|
| On-level premium, \(P_j^{OL}\) | Denominator for a pooled loss ratio | All periods expressed at the same rate level |
| Projected losses, \(L_j^{proj}\) | Numerator for a pooled loss ratio or pure premium | Same loss definition, valuation basis, and prospective projection basis |
| Exposure, \(E_j\) | Denominator for a pooled pure premium | Same exposure definition and unit in every period |
| Period label and source detail | Supports review and reconciliation | Records belong to the selected periods and are not inadvertently omitted or counted twice |
Combining Periods by Aggregating Amounts
For a pooled loss ratio, add the projected losses across the included periods and divide by the sum of their on-level premiums. For a pooled pure premium, divide the sum of projected losses by the sum of the corresponding exposures. The aggregation occurs in the numerator and denominator before the ratio is calculated.
These are summaries of the selected experience inputs, not complete rate indications by themselves. A full indication may require other components, such as expense provisions or a profit and contingencies provision. Those components are outside the calculation here.
Why a Pooled Ratio Is Not Usually an Average of Annual Ratios
Suppose each period has a loss ratio \(LR_j=L_j^{proj}/P_j^{OL}\). Since \(L_j^{proj}=P_j^{OL}LR_j\), substitution into the pooled formula gives:
Thus, the pooled loss ratio is an average of annual loss ratios in which each period’s share is determined by its share of total on-level premium. This result follows directly from the definitions: each period’s loss amount equals its premium multiplied by its loss ratio, and summing those losses gives the total numerator. The corresponding pure premium is likewise an exposure-volume-weighted average of the period pure premiums, provided every period uses the same exposure definition.
An unweighted average answers a different question: it gives each period the same numerical influence, regardless of how much premium or exposure it contains. It may be appropriate only if equal period weighting is deliberately intended for a separate analytical reason. It is not the pooled result obtained by combining the underlying amounts.
Worked Example: Pooling Three Period Loss Ratios
A hypothetical insurer has selected three experience periods. The on-level premiums are $1,200,000, $2,000,000, and $800,000. The corresponding projected losses are $780,000, $1,180,000, and $520,000. Calculate the pooled loss ratio and compare it with the unweighted average of the period loss ratios.
Method and assumptions. This is a pooled loss-ratio calculation. The premiums are all on the same current rate-level basis, and the losses are all projected to the same prospective period on the same selected loss basis. No discounting is applied: the stated amounts are used on a common valuation basis. No mortality assumption, probability distribution, or model parameters are used in this deterministic aggregation.
Step 1: Calculate each period’s loss ratio. For period 1, \(780{,}000/1{,}200{,}000=0.65\), or 65%. For period 2, \(1{,}180{,}000/2{,}000{,}000=0.59\), or 59%. For period 3, \(520{,}000/800{,}000=0.65\), or 65%.
Step 2: Add the underlying amounts. Total projected losses are \(780{,}000+1{,}180{,}000+520{,}000=\$2{,}480{,}000\). Total on-level premium is \(1{,}200{,}000+2{,}000{,}000+800{,}000=\$4{,}000{,}000\).
Step 3: Calculate the pooled result. The pooled loss ratio is \(2{,}480{,}000/4{,}000{,}000=0.62\), or 62.00%.
Step 4: Compare the unweighted average. The arithmetic average of the three annual ratios is \((65\%+59\%+65\%)/3=63\%\). The difference arises because the middle period has the most premium and a lower loss ratio than the other two.
Check. The premium shares are 30%, 50%, and 20%. Applying those shares to the period ratios gives \(0.30(65\%)+0.50(59\%)+0.20(65\%)=19.5\%+29.5\%+13.0\%=62.0\%\), matching the pooled result. The common mistake avoided is treating each calendar period as equally influential when pooling actual premium and loss amounts.
Combining Periods by Exposure
The same logic applies when the selected measure is pure premium. The exposure denominator must match the projected losses in scope. A period with more exposure contributes more to the pooled result than a period with less exposure, even if both periods are one year long.
Worked Example: Pooling Pure Premiums Across Different Exposure Volumes
A hypothetical specialty vehicle program has three selected periods. Its exposure is 12,000 vehicle-years, 20,000 vehicle-years, and 8,000 vehicle-years. Projected losses on the same coverage basis are $1,080,000, $1,900,000, and $840,000. Calculate the pooled pure premium per vehicle-year.
Method and assumptions. This is an exposure-based pure-premium calculation. All three exposure counts use vehicle-years, and the losses correspond to those same exposures and coverage. The losses are stated on the same prospective basis. No interest rate or discounting is applied, and no mortality assumption, probability distribution, or distribution parameters are needed.
Step 1: Calculate each period’s pure premium. Period 1: \(1{,}080{,}000/12{,}000=\$90\) per vehicle-year. Period 2: \(1{,}900{,}000/20{,}000=\$95\) per vehicle-year. Period 3: \(840{,}000/8{,}000=\$105\) per vehicle-year.
Step 2: Add the losses and exposures. Total projected losses are \(1{,}080{,}000+1{,}900{,}000+840{,}000=\$3{,}820{,}000\). Total exposure is \(12{,}000+20{,}000+8{,}000=40{,}000\) vehicle-years.
Step 3: Divide the pooled amounts. The pooled pure premium is \(\$3{,}820{,}000/40{,}000=\$95.50\) per vehicle-year.
Step 4: Verify using exposure shares. The exposure shares are \(12{,}000/40{,}000=30\%\), \(20{,}000/40{,}000=50\%\), and \(8{,}000/40{,}000=20\%\). The exposure-weighted period pure premium is \(0.30(\$90)+0.50(\$95)+0.20(\$105)=\$27+\$47.50+\$21=\$95.50\).
Check. The weighted calculation agrees with the pooled calculation. The common mistake avoided is averaging the three period pure premiums equally: that would give \((\$90+\$95+\$105)/3=\$96.67\), which does not represent the combined 40,000 vehicle-years.
Build the Dataset Before Calculating the Overall View
In practice, the calculation is only as reliable as the dataset. A useful process is to establish the period rows, check each period’s inputs and adjustments, and then calculate the combined result. The process below emphasizes reconciliation rather than selecting the periods or making a judgment about how well they represent future conditions.
Use one row for each included experience period and identify the period clearly. Confirm that the records belong to the selected period structure.
Record the on-level premium, projected loss, and exposure where relevant. Apply the established adjustments to each period, rather than applying a single adjustment to an already combined amount when the periods require different factors.
Confirm that premium is at a common rate level, losses share a common loss basis, and exposure has the same meaning and unit throughout.
Sum the displayed period values and compare the result with the stated combined totals. Investigate any difference before calculating a ratio or pure premium.
Divide aggregate projected losses by aggregate on-level premium for a pooled loss ratio, or by aggregate matching exposure for a pooled pure premium.
Worked Example: Combining Periods After Period-Specific Adjustments
A hypothetical property program uses three selected periods. The source premiums are $1,000,000, $1,500,000, and $800,000. The respective factors to the common rate level are 1.20, 1.00, and 0.90. The selected ultimate losses are $700,000, $900,000, and $480,000. The respective factors projecting losses to the prospective period are 1.10, 1.05, and 1.15. Find the pooled loss ratio using the adjusted amounts.
Method and assumptions. This is a period-by-period adjustment followed by a pooled loss-ratio calculation. The supplied premium factors convert each premium to the same current rate level; the supplied loss factors project each selected ultimate loss to the same prospective period. The loss and premium bases are otherwise consistent. No discounting or interest assumption is used, and no mortality assumption, probability distribution, or distribution parameters are required.
Step 1: Adjust each premium. Period 1: \(\$1{,}000{,}000(1.20)=\$1{,}200{,}000\). Period 2: \(\$1{,}500{,}000(1.00)=\$1{,}500{,}000\). Period 3: \(\$800{,}000(0.90)=\$720{,}000\).
Step 2: Project each loss. Period 1: \(\$700{,}000(1.10)=\$770{,}000\). Period 2: \(\$900{,}000(1.05)=\$945{,}000\). Period 3: \(\$480{,}000(1.15)=\$552{,}000\).
Step 3: Sum the adjusted premiums and projected losses. Total on-level premium is \(\$1{,}200{,}000+\$1{,}500{,}000+\$720{,}000=\$3{,}420{,}000\). Total projected losses are \(\$770{,}000+\$945{,}000+\$552{,}000=\$2{,}267{,}000\).
Step 4: Calculate the pooled loss ratio. The ratio is \(\$2{,}267{,}000/\$3{,}420{,}000=0.662865\ldots\), or 66.29% when rounded to two decimal places.
Check. The adjusted period ratios are 64.17%, 63.00%, and 76.67%. Weighting them by adjusted premium shares of \(1.2/3.42\), \(1.5/3.42\), and \(0.72/3.42\) gives approximately 66.29%. The common mistake avoided is combining unadjusted source amounts, which would mix rate levels and projection dates in one ratio.
Common Traps When Combining Periods
- Averaging annual ratios without a reason. The arithmetic average gives each period the same influence. For a pooled ratio, aggregate the numerators and denominators first.
- Using inconsistent denominators. A loss ratio requires premium on a consistent rate-level basis; a pure premium requires matching exposure on a consistent definition and unit.
- Applying period-specific adjustments to the wrong quantity. If factors vary by period, apply them to the corresponding period amounts before summing.
- Combining periods with different loss definitions. For example, amounts that include different expense components do not become comparable just because they are in the same currency.
- Rounding too early. Retain sufficient precision in adjusted period amounts and ratios, then round the final reported result as appropriate.
Check Your Understanding
For each problem, combine the stated amounts on the specified basis. Assume the period inputs are otherwise consistent unless the problem says otherwise.
- A hypothetical liability portfolio has on-level premiums of $600,000 and $900,000 in two periods, with projected losses of $390,000 and $495,000. Calculate the pooled loss ratio. (Answer: 59.00%)
- A hypothetical equipment program has 5,000 and 15,000 equipment-years of exposure, with projected losses of $475,000 and $1,350,000. Calculate the pooled pure premium per equipment-year. (Answer: $91.25 per equipment-year)
- A hypothetical insurer has on-level premiums of $750,000, $1,250,000, and $1,000,000, with projected losses of $450,000, $875,000, and $650,000. Calculate the pooled loss ratio to two decimal places. (Answer: 65.83%)
- A hypothetical service contract portfolio has 8,000, 12,000, and 20,000 contract-years in three periods, with projected losses of $680,000, $1,020,000, and $1,800,000. Calculate the pooled pure premium and explain why an unweighted average of period pure premiums would answer a different question. (Answer: $87.50 per contract-year; the pooled measure weights each period by its contract-year exposure.)
- Two periods have on-level premiums of $400,000 and $1,600,000 and projected losses of $300,000 and $960,000. Calculate the pooled loss ratio and the unweighted average of the two period loss ratios. (Answer: pooled loss ratio 63.00%; unweighted average 67.50%)