Worked Paid Triangle Factor and Tail Selection

From Paid Triangle to Selected Development

A paid triangle records cumulative amounts paid by accident period as they mature. To use the triangle for a projection, an analyst calculates how paid amounts have changed between development ages, selects factors for future development, and considers whether payments after the last measured age require a tail. The arithmetic matters, but so does keeping each factor attached to the correct age interval.

The calculations below use hypothetical loss data. They focus on paid development factors and a tail selection; they do not imply that any one mechanical selection is appropriate for every portfolio.

Definition: Let $C_{i,j}$ be the cumulative paid loss for accident year $i$ at development age $j$. The observed age-to-age link ratio from age $j$ to age $j+1$ is $C_{i,j+1}/C_{i,j}$, for an accident year with observations at both ages. A selected factor, written $\hat f_j$, is the factor chosen to represent development from age $j$ to age $j+1$. A tail factor $T_J$ is a multiplier for development after the final measured age $J$, through ultimate.

A link ratio describes one interval, not all future development. For example, a factor from 12 to 24 months is applied to a 12-month cumulative amount to estimate its 24-month amount. To move farther, apply the subsequent age-to-age factors in order. A tail-to-ultimate factor begins only after the last age covered by those factors.

Calculating and Selecting Factors

For each development interval, calculate the available accident-year link ratios. One common summary is the volume-weighted factor, also called the ratio of sums:

\hat f_j^{\text{vol}}= \frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}}

The sums include only accident years observed at both ages. Other selections, including a simple average or a judgmental selection, may be appropriate depending on the data and the purpose of the estimate. The selected factor should be coherent with the development pattern: each factor applies to its stated interval, and factors should not imply an implausible pattern without an explicit reason.

Why the volume-weighted formula works: Write each observed link ratio as $r_i=C_{i,j+1}/C_{i,j}$. Then $C_{i,j+1}=C_{i,j}r_i$, so $\sum_i C_{i,j+1}/\sum_i C_{i,j}=\sum_i(C_{i,j}/\sum_k C_{k,j})r_i$. Thus the ratio of sums is a weighted average of the individual ratios, with weights proportional to the cumulative amounts at the beginning of the interval. It gives larger starting volumes more influence.

The ratio-of-sums calculation requires meaningful denominators. If a beginning cumulative amount is zero or negative, the corresponding individual ratio may be undefined or misleading; a volume-weighted result can also be unsuitable if its aggregate denominator is zero, negative, or not representative. In those cases, investigate the data and use a defensible alternative rather than forcing the formula.

Worked Example: Selecting Factors From a Paid Triangle

A hypothetical insurer has the following cumulative paid losses, in thousands of dollars. Calculate the observed factors, select a factor at each available age, and project each accident year's paid losses to ultimate. The selected tail-to-ultimate factor after 48 months is 1.02.

Accident year	12 months	24 months	36 months	48 months
2021	100	150	180	189
2022	120	174	209	—
2023	140	196	—	—
2024	160	—	—	—

Step 1: Calculate the 12-to-24-month ratios. The three observed ratios are $150/100=1.5000$, $174/120=1.4500$, and $196/140=1.4000$. Their volume-weighted value is $(150+174+196)/(100+120+140)=520/360=1.4444$ to four decimals.

Step 2: Calculate the 24-to-36-month ratios. The observed ratios are $180/150=1.2000$ and $209/174=1.2011$. The volume-weighted value is $(180+209)/(150+174)=389/324=1.2006$ to four decimals.

Step 3: Calculate the 36-to-48-month ratio. The only observed ratio is $189/180=1.0500$. With just one observation, the indicated factor is 1.0500, though the selection still requires judgment about whether that observation is representative.

Step 4: Make and state the selections. For this illustration, select 1.45 from 12 to 24 months, 1.20 from 24 to 36 months, and 1.05 from 36 to 48 months. These selections are consistent with the observed ratios; they are choices for this hypothetical example, not automatic rounding rules. Use the given 1.02 tail factor after 48 months.

Step 5: Project each latest observed amount. The selected development multiplier from 12 months to ultimate is $1.45\times1.20\times1.05\times1.02=1.86354$. Thus 2024 projects to $160\times1.86354=298.1664$. For 2023, observed at 24 months, the multiplier is $1.20\times1.05\times1.02=1.2852$, giving $196\times1.2852=251.8992$. For 2022, observed at 36 months, it is $1.05\times1.02=1.071$, giving $209\times1.071=223.839$. For 2021, observed at 48 months, it is $1.02$, giving $189\times1.02=192.78$.

Check. The multipliers shorten as the observed age gets later: 1.86354 from 12 months, 1.2852 from 24 months, 1.071 from 36 months, and 1.02 from 48 months. Rounded only at the final step to two decimals, the projected ultimate paid losses are 298.17, 251.90, 223.84, and 192.78 thousand dollars, respectively.

Method and trap avoided. This is a selected-factor projection by development age. The common mistake is to apply the 48-month tail factor directly to the 12-month amount and stop there: $160\times1.02=163.20$ thousand dollars. That is wrong because 1.02 covers only development after 48 months; the three intervening age-to-age factors are also needed.

Estimating a Tail From Late Development

A tail factor summarizes development beyond a specified final age. If selected annual link ratios describe successive intervals after that age, their product gives the combined multiplier over those intervals. If the modeled intervals stop short of ultimate, the remaining development must also be included or otherwise allowed for. The tail factor must be defined by its starting age so it is not confused with an ordinary age-to-age factor.

T_J=\prod_{k=J}^{K-1} \hat f_k

Here the selected factors cover consecutive intervals from age $J$ through age $K$, with the final interval reaching ultimate or an explicitly defined ultimate endpoint. This multiplication assumes the factors are applicable to the same loss measure and form a consistent sequence of development intervals.

Worked Example: Building a Tail Factor From Late Ratios

For a hypothetical paid portfolio, the selected late-development ratios after 48 months are 1.012 from 48 to 60 months, 1.006 from 60 to 72 months, and 1.002 from 72 months to ultimate. Estimate the 48-month tail factor and apply it to a cumulative paid amount of $2,400,000 at 48 months.

Step 1: Identify the covered intervals. The three factors are consecutive and begin at 48 months; the last factor reaches ultimate. Their product therefore estimates development from 48 months to ultimate.

Step 2: Multiply the factors. First, $1.012\times1.006=1.018072$. Then, $1.018072\times1.002=1.020108144$. The indicated tail factor is therefore 1.020108144, or 1.0201 to four decimals.

Step 3: Apply it to the 48-month paid amount. The projected ultimate is $2{,}400{,}000\times1.020108144=\$2{,}448{,}259.5456$. Rounded only at the final step to the nearest cent, this is $2,448,259.55.

Check. The increase is $\$2{,}448{,}259.55-\$2{,}400{,}000=\$48{,}259.55$, approximately 2.0108% of the 48-month amount. This is consistent with a multiplier slightly above 1.02.

Method and trap avoided. This is a product of successive late-development factors. Adding the percentages, $1.2\%+0.6\%+0.2\%=2.0\%$, gives only an approximation because later percentages apply to amounts already increased. The correct combined factor is the product, 1.020108144.

Applying a Selected Schedule at the Correct Age

For a cumulative amount observed at age $j$, the projection uses only the selected factors from age $j$ onward, followed by the tail factor whose starting age matches the final selected link ratio. It does not use factors for intervals that have already emerged. This is a practical way to check that the factor schedule is being applied consistently.

Worked Example: Projecting a Newer Accident Year

A hypothetical specialty insurer selects a 12-to-24-month factor of 1.38 and a 24-to-36-month factor of 1.15. Its selected tail factor from 36 months to ultimate is 1.03. The newest accident year has cumulative paid losses of $150,000 at 12 months. Project its ultimate paid losses.

Step 1: Write the applicable path. The 12-month amount must develop through 24 months, then through 36 months, and then through the tail. The multiplier is $1.38\times1.15\times1.03$.

Step 2: Multiply in sequence. $1.38\times1.15=1.587$, and $1.587\times1.03=1.63461$. The combined multiplier from 12 months to ultimate is 1.63461.

Step 3: Apply the multiplier. $150{,}000\times1.63461=\$245{,}191.50$. Rounded to the nearest cent, the projected ultimate is $245,191.50.

Check. The implied development is $\$245{,}191.50-\$150{,}000=\$95{,}191.50$, a 63.461% increase over the 12-month amount, as implied by a multiplier of 1.63461.

Method and trap avoided. This applies the selected factors from the accident year's current age forward. Applying only the tail would give $150{,}000\times1.03=\$154{,}500$, which omits development from 12 to 36 months. Conversely, applying factors for ages before 12 months would count development that has already occurred.

Selection Checks and Common Traps

A factor calculation is not complete when the division is finished. Before using a selection, check what data contributed to it, the age interval it represents, whether it is consistent with adjacent factors, and whether the tail starts at the correct age. A factor can be arithmetically correct and still be misapplied.

Check	Question to ask	Why it matters
Age interval	Does each factor correspond to the stated start and end ages?	Prevents omitting or duplicating development.
Triangle eligibility	Were only accident years observed at both ages included?	Prevents using unavailable or mismatched data.
Selection basis	Is the selected factor supported by the ratios and a stated rationale?	Distinguishes an actuarial selection from an unexamined mechanical result.
Tail boundary	Does the tail begin after the last age-to-age factor?	Prevents applying the tail twice or at the wrong age.
Units and rounding	Are amounts in consistent units, with rounding deferred?	Avoids scaling errors and accumulated rounding differences.

For example, a 1.02 tail factor means a 2% increase over the amount at its stated starting age. It is a multiplier, not an amount to add to the loss, and it is not automatically the total multiplier from every earlier age. Also distinguish a selected factor from the data-based indication: reporting both makes the judgment visible and easier to review.

These calculations support loss development analysis aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. Their reliability depends on the relevance of the selected development pattern to the hypothetical portfolio, not simply on multiplying accurately.

Check Your Understanding

For each problem, identify the relevant interval before calculating. Amounts and selections are hypothetical.

For 12-to-24-month cumulative paid amounts of 75 to 90, 100 to 118, and 125 to 142.5, calculate the volume-weighted link ratio. Give the factor to four decimals. (Answer: $350.5/300=1.1683$.)
A tail after 60 months consists of selected ratios 1.010 from 60 to 72 months, 1.005 from 72 to 84 months, and 1.002 from 84 months to ultimate. Calculate the tail factor to four decimals. (Answer: $1.010\times1.005\times1.002=1.0170801$, or 1.0171.)
A cumulative paid amount of $240,000 is observed at 24 months. The selected factors are 1.25 from 24 to 36 months, 1.08 from 36 to 48 months, and a 1.02 tail after 48 months. Project ultimate to the nearest cent. (Answer: $240{,}000\times1.25\times1.08\times1.02=\$330{,}480.00$.)
A portfolio has $500,000 paid at 36 months, a selected 36-to-48-month factor of 1.04, and a tail factor of 1.015 after 48 months. Calculate ultimate paid losses and state the total multiplier from 36 months. (Answer: multiplier $1.04\times1.015=1.0556$; ultimate $=\$527{,}800.00$.)
Observed 12-to-24-month amounts are 80 to 104 and 120 to 150. Use the volume-weighted ratio as the selected 12-to-24-month factor. If the newest year has $200,000 paid at 12 months, with a selected 24-to-36-month factor of 1.10 and a tail factor of 1.02 after 36 months, project ultimate to the nearest cent. (Answer: factor $254/200=1.27$; ultimate $=200{,}000\times1.27\times1.10\times1.02=\$284{,}988.00$.)

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