Why the Starting Age and Endpoint Matter
A tail factor summarizes the remaining development beyond a specified age. When selected annual link ratios are available for that period, the calculation is a product: each ratio carries the cumulative loss from one development age to the next. The arithmetic is simple, but the interpretation depends on choosing exactly the ratios that connect the stated starting age to the stated endpoint.
This tutorial focuses on calculating that product. It does not address how the annual ratios are selected. The selected ratios are treated as inputs, and the task is to verify their age labels, include each required interval once, and report the factor with appropriate precision.
The labels determine what to multiply. To get from age 8 to age 11, the intervals are 8-to-9, 9-to-10, and 10-to-11. A factor labelled 7-to-8 is before the starting age and is excluded. A factor labelled 11-to-12 is beyond the endpoint and is also excluded.
A terminal ratio may be labelled, for example, “15-to-ultimate” rather than “15-to-16.” It is included when the requested endpoint is ultimate. Do not assume that a factor ending at a named development age is automatically a tail-to-ultimate factor: the stated endpoint must match the calculation.
Why the Ratios Multiply
The product follows directly from the definition of a link ratio. Let \(C_k\) denote the cumulative loss measure at development age \(k\), for the same accident-year or other loss group throughout. The selected link ratio for the interval from \(k\) to \(k+1\) represents the ratio of the later cumulative amount to the earlier one. In an idealized exact sequence, \(f_k=C_{k+1}/C_k\). Therefore, the successive ratios telescope:
Every intermediate cumulative amount appears once in a numerator and once in a denominator, so those terms cancel. The remaining ratio compares the endpoint amount with the starting-age amount. Selected link ratios are estimates rather than necessarily exact ratios for a particular future sequence, but their cumulative factor is calculated by the same product rule.
Record the starting age and the endpoint, such as age 8 through age 11 or age 12 through ultimate.
For age 8 through age 11, list 8-to-9, 9-to-10, and 10-to-11 in order.
Include each interval that connects the starting age to the endpoint exactly once; exclude ratios outside that span.
Confirm the first and last included labels, retain precision during multiplication, and round only the final factor as instructed.
Worked Calculations
Worked Example: From Age 8 Through Age 11
A hypothetical long-tail liability schedule has selected annual link ratios of 1.020 from age 7 to 8, 1.014 from age 8 to 9, 1.009 from age 9 to 10, and 1.004 from age 10 to 11. Calculate the cumulative factor from age 8 through age 11. Report the factor to four decimal places.
Method and assumptions. This is a deterministic product of selected development factors. No interest rate, mortality assumption, or probability distribution is used. The requested starting age is 8, so the age 7-to-8 ratio is outside the span. The endpoint is age 11, so the ratios needed are 8-to-9, 9-to-10, and 10-to-11.
First multiply the 8-to-9 and 9-to-10 factors:
Then include the final interval, 10-to-11:
Rounding only the final result to four decimal places gives a factor of 1.0272.
Check and common mistake. The calculation includes three intervals, starting at age 8 and ending at age 11. Multiplying only 1.009 and 1.004 gives 1.013036, but that omits the 8-to-9 interval and is a factor from age 9, not age 8. The 1.020 ratio is not included because it ends at the requested starting age.
Worked Example: A Tail Factor Ending at Ultimate
A hypothetical claims portfolio is observed through development age 12. The selected ratios beyond that age are 1.008 for age 12-to-13, 1.004 for age 13-to-14, 1.002 for age 14-to-15, and 1.001 for age 15-to-ultimate. Calculate the tail-to-ultimate factor from age 12, to six decimal places.
Method and assumptions. This is a deterministic product of the four supplied factors. No interest rate, mortality assumption, or probability distribution is used. Because the requested endpoint is ultimate, the 15-to-ultimate ratio is included along with the three age-to-age ratios.
Multiply the first two ratios:
Include the age 14-to-15 factor:
Finally, include the terminal age 15-to-ultimate factor:
Rounded to six decimal places, the tail-to-ultimate factor from age 12 is 1.015070.
Check and common mistake. The factor is slightly greater than 1, consistent with all four supplied ratios being greater than 1. Stopping after age 15 would omit the 1.001 terminal factor and give 1.014056064 instead. That is the factor through age 15, not the factor to ultimate.
Worked Example: Calculating Factors From Two Starting Ages
For a hypothetical runoff schedule, the selected ratios are 1.011 from age 10 to 11, 1.006 from age 11 to 12, 1.003 from age 12 to 13, and 1.001 from age 13 to ultimate. Calculate the factor from age 10 to ultimate and the factor from age 11 to ultimate. Round each answer to four decimal places.
Method and assumptions. Both calculations are deterministic products of the supplied ratios. No interest rate, mortality assumption, or probability distribution is used. The age 10 factor includes all four ratios; the age 11 factor excludes the 10-to-11 ratio because that interval is already before its starting age.
For the age 10 factor, multiply in sequence:
and then include the terminal ratio:
Thus, the age 10-to-ultimate factor is 1.0211 to four decimal places.
For the age 11 factor, use only the intervals from age 11 onward:
Thus, the age 11-to-ultimate factor is 1.0100 to four decimal places.
Check and common mistake. The age 10 factor must equal the 10-to-11 ratio multiplied by the age 11 factor, using unrounded values: \(1.011 \times 1.010027018 = 1.021137315198\). Using the already rounded 1.0100 in that check introduces a small rounding difference. More importantly, applying the 1.011 ratio to the age 11 factor would be wrong when calculating from age 11, because it would include development before the starting age.
Interpreting and Reporting the Product
A cumulative factor is dimensionless: it is a multiplier, not a dollar amount or a percentage increase. For example, a factor of 1.0272 represents a cumulative level 1.0272 times the starting level, or a relative increase of 0.0272 (2.72%) over that starting level. The factor itself should still be reported as 1.0272 when the requested quantity is a development factor.
The multiplication rule applies to a consecutive sequence of selected ratios. If the selected schedule skips an age, do not silently fill the gap with a ratio of 1.000; first determine whether the supplied factors actually connect the stated starting age and endpoint. Likewise, if a ratio is labelled from age 15 to ultimate, treat its endpoint as ultimate rather than relabelling it as an ordinary 15-to-16 ratio.
Keep the full precision of the supplied factors and intermediate products until the last step. Rounding each partial product can change the final displayed result, particularly when several small factors are multiplied. A useful final check is to write the included age labels beside the product and confirm that they form a continuous chain with no missing or repeated interval.
Check Your Understanding
For each hypothetical schedule, identify the intervals needed before calculating. Round each requested factor to four decimal places unless the question specifies otherwise. These are deterministic factor calculations; no interest, mortality, or probability distribution assumption is needed.
- A schedule gives ratios 1.030 from age 5 to 6, 1.015 from age 6 to 7, 1.007 from age 7 to 8, and 1.003 from age 8 to 9. Find the factor from age 6 through age 9. (Answer: 1.0252)
- From age 14, selected ratios are 1.009 for 14-to-15, 1.004 for 15-to-16, and 1.002 for 16-to-ultimate. Find the tail-to-ultimate factor from age 14. (Answer: 1.0151)
- A hypothetical schedule has ratios 1.020 for 10-to-11, 1.010 for 11-to-12, 1.005 for 12-to-13, 1.002 for 13-to-14, and 1.001 for 14-to-ultimate. Calculate the factor from age 11 to ultimate; explain why the first ratio is excluded. (Answer: 1.0181; exclude 10-to-11 because it precedes age 11)
- The selected factor from age 8 to ultimate is 1.0300. The factors from age 8-to-9 and 9-to-10 are 1.012 and 1.006. What factor from age 10 to ultimate is consistent with these values? (Answer: 1.0117)
- From age 9, the selected ratios are 1.013 for 9-to-10, 1.007 for 10-to-11, 1.002 for 11-to-12, and 1.001 for 12-to-ultimate. Find the age 9-to-ultimate factor, retaining full precision until the final rounding. (Answer: 1.0232)