Why Maturity Matters When Applying Selected Factors
A selected development pattern is usually written as a sequence of age-to-age factors. But accident years in a valuation triangle are not all observed at the same age. One year may have cumulative losses through 12 months, another through 24 months, and an older year through 48 months. Each year therefore needs a different portion of the selected pattern to reach ultimate.
The essential operation is to start at the year’s latest observed development age and multiply only the factors from that age onward. Applying factors for development that has already occurred overstates the projection. Omitting a factor that remains understates it. This tutorial focuses on matching each accident year to the right portion of a selected pattern; it does not address how the factors themselves are selected.
For example, if factors are selected for 12-to-24 months, 24-to-36 months, and 36 months to ultimate, a year currently at 24 months needs the second and third factors. It does not need the 12-to-24-month factor. A year already at 36 months needs only the age-to-ultimate factor.
Here \(C_{AY,j}\) is the accident year’s cumulative loss at its current development age \(j\), \(M_j\) is the remaining development multiplier, and \(\widehat{U}_{AY}\) is its projected ultimate loss. The final factor in the product may be a selected age-to-ultimate factor. If no such factor or other justified path to ultimate is available, the listed age-to-age factors alone do not establish an ultimate estimate.
Why the Remaining Factors Are Multiplied
The product follows from the definition of each age-to-age factor. Suppose a year has cumulative losses \(C_j\) at age \(j\). The selected factor gives projected losses at the next age as \(C_{j+1}=C_j f_{j,j+1}\). At the following age, \(C_{j+2}=C_{j+1}f_{j+1,j+2}\). Substituting the first expression into the second gives \(C_{j+2}=C_j f_{j,j+1}f_{j+1,j+2}\). Continuing in the same way yields the product of all factors still required. Factors for ages earlier than \(j\) do not appear because they describe transitions already represented in \(C_j\).
These calculations assume the selected factors apply to the loss measure and development ages being projected, the cumulative observations are on a consistent basis, and the factors provide a valid path from each current age to ultimate. The calculation does not correct for changes in development behavior or other departures from the selected pattern.
A Row-by-Row Projection Process
A reliable calculation starts with the accident year’s maturity rather than with the first factor in the pattern. The same selected pattern can therefore produce different multipliers for different rows.
Read the accident year’s current cumulative loss and identify its development age, such as 24 or 36 months.
Begin with the factor that starts at the current age. Include every subsequent selected factor needed to reach ultimate, and exclude earlier factors.
Multiply the remaining factors together, then multiply that result by the current cumulative loss.
For a paid triangle, subtract current cumulative paid losses from projected ultimate to obtain the unpaid estimate.
Sum the year-specific ultimates and unpaid amounts. Keep full precision in intermediate calculations and round only as instructed.
Worked Example: Three Accident Years at Different Ages
A hypothetical insurer has selected the following paid-loss development pattern, with losses measured in thousands of dollars: 12-to-24 months, 1.50; 24-to-36 months, 1.20; 36-to-48 months, 1.10; and 48 months to ultimate, 1.05. At the valuation date, accident year A is at 36 months with cumulative paid losses of 800, year B is at 24 months with 950, and year C is at 12 months with 700. Project each ultimate and the aggregate unpaid amount. State results to the nearest thousand dollars.
Method. Apply only the factors remaining from each year’s current age. Since these are paid losses, ultimate less current cumulative paid is the indicated unpaid amount.
Year A, current age 36 months. The remaining factors are 1.10 and 1.05. The multiplier is \(1.10\times1.05=1.155\). Projected ultimate is \(800\times1.155=924\). The unpaid amount is \(924-800=124\).
Year B, current age 24 months. The remaining factors are 1.20, 1.10, and 1.05. The multiplier is \(1.20\times1.10=1.32\), then \(1.32\times1.05=1.386\). Projected ultimate is \(950\times1.386=1{,}316.7\). The unpaid amount is \(1{,}316.7-950=366.7\).
Year C, current age 12 months. All four factors remain. The multiplier is \(1.50\times1.20=1.80\), \(1.80\times1.10=1.98\), and \(1.98\times1.05=2.079\). Projected ultimate is \(700\times2.079=1{,}455.3\). The unpaid amount is \(1{,}455.3-700=755.3\).
The current cumulative paid total is \(800+950+700=2{,}450\). The projected ultimate total is \(924+1{,}316.7+1{,}455.3=3{,}696\), so aggregate unpaid is \(3{,}696-2{,}450=1{,}246\). In thousands, the answers rounded to the nearest thousand dollars are ultimate losses of $3,696,000 and unpaid losses of $1,246,000.
Check. The year-specific unpaid amounts sum to \(124+366.7+755.3=1{,}246\), matching aggregate ultimate less aggregate current paid losses. The common mistake avoided is applying all four factors to every accident year.
Keep Precision Through the Aggregate Calculation
A displayed ultimate may be rounded for reporting, but calculations that feed a portfolio total should use unrounded values until the final step. A small difference in one row can carry through to the aggregate unpaid estimate. The next example shows the effect explicitly, using inputs in thousands of dollars and reporting final amounts to the nearest thousand dollars.
Worked Example: A Mature Year and the Correctly Rounded Total
A hypothetical paid triangle uses selected factors of 1.50 from 12 to 24 months, 1.25 from 24 to 36 months, 1.08 from 36 to 48 months, and 1.03 from 48 months to ultimate. Year A is at 48 months with cumulative paid losses of 3,300 thousand dollars; year B is at 36 months with 2,400 thousand; and year C is at 24 months with 1,800 thousand. Find the ultimate and unpaid totals, rounding only the final reported amounts to the nearest thousand dollars.
Method. Map each row to its current age, multiply only its remaining factors, and retain full precision in thousands until aggregation.
Year A, current age 48 months. Only the 48-to-ultimate factor remains. Ultimate is \(3{,}300\times1.03=3{,}399\). Unpaid is \(3{,}399-3{,}300=99\).
Year B, current age 36 months. The remaining multiplier is \(1.08\times1.03=1.1124\). Ultimate is \(2{,}400\times1.1124=2{,}669.76\). Unpaid is \(2{,}669.76-2{,}400=269.76\). For this row alone, the ultimate rounds to 2,670 thousand dollars and unpaid rounds to 270 thousand dollars; retain 2,669.76 and 269.76 when totaling.
Year C, current age 24 months. The remaining multiplier is \(1.25\times1.08=1.35\), then \(1.35\times1.03=1.3905\). Ultimate is \(1{,}800\times1.3905=2{,}502.9\). Unpaid is \(2{,}502.9-1{,}800=702.9\).
The current paid total is \(3{,}300+2{,}400+1{,}800=7{,}500\). The ultimate total before rounding is \(3{,}399+2{,}669.76+2{,}502.9=8{,}571.66\). Aggregate unpaid is \(8{,}571.66-7{,}500=1{,}071.66\), also equal to \(99+269.76+702.9\). Rounded to the nearest thousand dollars, the aggregate ultimate is $8,572,000 and unpaid is $1,072,000.
Check. The aggregate unpaid reconciles both as ultimate less current paid and as the sum of the row-level unpaid amounts. The common mistake avoided is replacing the exact year B ultimate of 2,669.76 with 2,670 before calculating the portfolio total.
Do Not Reapply Development Already Reflected in the Current Loss
The latest cumulative observation already incorporates development through its stated age. A year at 36 months must not be projected as though it were still at 12 months. The full pattern is appropriate only for a year whose current age is the first age in that pattern. This distinction is particularly important when a question supplies a common set of factors but lists accident years with different maturities.
Worked Example: Detecting an Overstated Projection
A hypothetical paid-loss pattern has factors of 1.45 from 12 to 24 months, 1.20 from 24 to 36 months, 1.08 from 36 to 48 months, and 1.03 from 48 months to ultimate. Accident year A is at 12 months with 900 thousand dollars paid; year B is at 36 months with 2,000 thousand paid. Calculate both ultimates and unpaid amounts, and show the error from applying the full pattern to year B. Give results in thousands to two decimal places.
Method. Build a separate remaining multiplier for each accident year. The full pattern applies to year A, but year B needs only the 36-to-48-month and 48-to-ultimate factors.
Year A, current age 12 months. Its multiplier is \(1.45\times1.20=1.74\), \(1.74\times1.08=1.8792\), and \(1.8792\times1.03=1.935576\). Ultimate is \(900\times1.935576=1{,}742.0184\). Unpaid is \(1{,}742.0184-900=842.0184\).
Year B, current age 36 months. Its multiplier is \(1.08\times1.03=1.1124\). Ultimate is \(2{,}000\times1.1124=2{,}224.8\). Unpaid is \(2{,}224.8-2{,}000=224.8\).
The correct aggregate ultimate is \(1{,}742.0184+2{,}224.8=3{,}966.8184\), and aggregate unpaid is \(842.0184+224.8=1{,}066.8184\). To two decimal places, these are 3,966.82 and 1,066.82 thousand dollars.
If the full multiplier 1.935576 were wrongly applied to year B, its projected ultimate would be \(2{,}000\times1.935576=3{,}871.152\), rather than 2,224.8. The overstatement would be \(3{,}871.152-2{,}224.8=1{,}646.352\) thousand dollars. The error comes from multiplying year B by factors for 12-to-24 and 24-to-36 months even though its current cumulative paid amount is already at 36 months.
Check. Each correct unpaid amount is nonnegative and equals its ultimate less current paid losses. The common mistake avoided is treating every row as though it had the same valuation age.
Common Exam Traps and Practical Checks
- Starting at the first listed factor automatically. Start at the factor whose starting age matches the accident year’s current age.
- Reading the latest triangle column without its age label. A value’s position in a table matters because it identifies how much development is already reflected in that value.
- Using an ultimate multiplier intended for another age. A cumulative-to-ultimate multiplier from 12 months cannot be applied to a 36-month cumulative loss without adjusting for the development already completed.
- Confusing ultimate with unpaid. Ultimate is the projected total loss; on a paid basis, unpaid is ultimate minus current cumulative paid loss.
- Rounding each row too early. Keep unrounded projected values for totals, then round the reported results at the end.
- Assuming a final age-to-age factor reaches ultimate. It does so only if the selected pattern defines that factor as age-to-ultimate or otherwise supplies a valid endpoint.
Check Your Understanding
For each problem, identify the factors still required from the stated current age. Unless indicated otherwise, losses are cumulative paid losses and all amounts are in thousands of dollars.
- A selected pattern has factors 1.30 from 12 to 24 months, 1.15 from 24 to 36 months, and 1.04 from 36 months to ultimate. An accident year is at 24 months with cumulative paid losses of 500. Find its ultimate and unpaid amount. (Answer: Ultimate = 598; unpaid = 98.)
- A year is at 36 months with cumulative paid losses of 1,250. The selected factors are 1.06 from 36 to 48 months and 1.02 from 48 months to ultimate. Find the remaining multiplier and unpaid amount. (Answer: Multiplier = 1.0812; unpaid = 101.50.)
- A pattern has factors 1.40 from 12 to 24 months, 1.20 from 24 to 36 months, 1.10 from 36 to 48 months, and 1.05 from 48 months to ultimate. One year is at 24 months with 600 paid, and another is at 48 months with 1,500 paid. Find aggregate ultimate and unpaid. (Answer: Aggregate ultimate = 2,406.60; unpaid = 306.60.)
- A selected pattern has factors 1.25 from 12 to 24 months, 1.12 from 24 to 36 months, and 1.04 from 36 months to ultimate. One year is at 12 months with 800 paid, and another is at 24 months with 1,000 paid. Find each year’s unpaid amount and the aggregate unpaid amount. (Answer: Year at 12 months = 364.80; year at 24 months = 164.80; aggregate unpaid = 529.60.)
- A triangle has cumulative paid losses at 48 months, but the only selected factors supplied are 12-to-24 months, 24-to-36 months, and 36-to-48 months. Can these factors alone produce a justified ultimate projection for that row? (Answer: No; a selected 48-month-to-ultimate factor or another justified path from 48 months to ultimate is needed.)