Select Tail Factors for Paid and Incurred Losses

Why Paid and Incurred Tail Factors Need Separate Selections

At a late development age, much of a claim portfolio may have already been paid or recorded as incurred. The remaining development can still matter: a tail factor projects from the latest observed age to ultimate. But paid and incurred amounts reflect different processes. Paid losses emerge as cash is issued; incurred losses combine paid losses with case reserves, which can change as claim estimates are revised. A factor that is reasonable for one basis is not automatically reasonable for the other.

The selection question is therefore not simply “what is the tail factor?” It is “what factor is supported for this measure of loss, at this valuation age, under this claim-settlement and reserving pattern?” The earlier material on tail factors and competing tail estimates provides relevant context. Here the emphasis is on selecting and interpreting paid and incurred factors separately.

Definition: At development age $d$, let $P_d$ denote cumulative paid loss and $I_d$ denote cumulative incurred loss, where incurred loss is paid loss plus case reserves. A paid tail factor $T_P$ multiplies $P_d$ to estimate ultimate paid loss: $U_P=P_dT_P$. An incurred tail factor $T_I$ multiplies $I_d$ to estimate ultimate incurred loss: $U_I=I_dT_I$. Each factor is measured from the stated development age to ultimate; it does not include development before age $d$.

A tail factor of 1.04 means a 4% increase relative to the amount at the selected age. For example, if cumulative paid loss is $10 million and the selected paid tail is 1.04, projected ultimate paid loss is $10.4 million. The 4% is not a percentage of the already unpaid amount; it is a multiplier applied to the current cumulative paid amount.

What Each Basis Captures

Basis	What emerges over time	Evidence to assess	Potential weakness
Paid	Claim payments, including settlement and payment timing	Late-age paid development, relevant historical paid patterns, and suitable external paid benchmarks	Payment timing can be affected by settlement practices, litigation, or operational delays
Incurred	Paid losses plus case-reserve estimates	Late-age incurred development, case-reserving practices, and relevant incurred benchmarks	Reserve strengthening or release can change incurred development without a matching change in cash payments

The two factors can differ for sound reasons. For example, a portfolio with slow settlements may have substantial future paid emergence. If case reserves already recognize much of the expected claim cost, future incurred emergence can be smaller. Conversely, reserve releases can reduce incurred amounts even while paid losses continue to develop. These are tendencies to investigate, not rules that determine the factor in every portfolio.

Important distinction: $T_P$ projects ultimate from cumulative paid loss, while $T_I$ projects ultimate from cumulative incurred loss. If the selected factors produce different ultimate estimates, that difference is a signal to review the evidence and assumptions. It is not a reason to silently substitute one factor for the other.

Why the Tail Factor Is a Multiplier to Ultimate

Suppose the remaining development after age $d$ consists of age-to-age factors $f_{d,d+1}, f_{d+1,d+2}, \ldots, f_{m-1,m}$, followed by a tail factor $T_m$ from age $m$ to ultimate. The total multiplier from age $d$ is the product of the remaining factors. This reasoning applies separately to paid and incurred development, using factors appropriate to the selected basis.

T_d=f_{d,d+1}f_{d+1,d+2}\cdots f_{m-1,m}T_m

The product follows because each factor applies to the amount produced by the preceding factor. If an amount $L_d$ develops first to $L_df_{d,d+1}$, then to $L_df_{d,d+1}f_{d+1,d+2}$, and so on, the final projected amount is the original amount times the product. A tail factor selected from age $d$ must cover all development after age $d$; multiplying by factors that are already included in that tail would count development twice.

A Practical Selection Process

Fix the basis and valuation age.
Record whether the data are paid or incurred, the cumulative amount at the latest age, and the age from which the tail begins. Keep paid and incurred selections distinct.

Review relevant evidence.
Compare late-age patterns, fitted tail estimates, and external benchmarks only when their basis and claim characteristics are relevant. Consider operational distortions and case-reserve changes when interpreting incurred evidence.

Select a factor for each basis.
State the evidence and judgment supporting the paid selection and the incurred selection. Do not average unlike evidence or assume the factors should be equal.

Calculate and check the implications.
Multiply each current cumulative amount by its selected factor. Compare the resulting ultimates and interpret differences in light of paid emergence and case reserves.

Worked Selections and Checks

Worked Example: Selecting Separate Factors from Late-Age Evidence

A hypothetical specialty liability portfolio is observed at age 10. Cumulative paid loss is $4,800 thousand and cumulative incurred loss is $5,000 thousand. A review of relevant paid evidence supports a tail range of 1.020 to 1.030; the actuary selects 1.025. Relevant incurred evidence supports 1.015 to 1.020; the actuary selects 1.018. Calculate both projected ultimates and assess the difference.

Method. Apply each selected factor to its own cumulative amount. The factors are supported by different evidence because paid and incurred development measure different processes.

Paid ultimate. The paid selection is 1.025, so the amount above current paid loss is 0.025 times current paid loss:

$$ 4{,}800(1.025)=4{,}800+4{,}800(0.025) =4{,}800+120=4{,}920 $$

Projected ultimate paid loss is $4,920 thousand. The projected future paid amount is therefore $4{,}920-4{,}800=120$ thousand.

Incurred ultimate. The incurred selection is 1.018:

$$ 5{,}000(1.018)=5{,}000+5{,}000(0.018) =5{,}000+90=5{,}090 $$

Projected ultimate incurred loss is $5,090 thousand. The indicated incurred development above the current incurred amount is $5{,}090-5{,}000=90$ thousand.

Check and interpretation. Current case reserves are $5{,}000-4{,}800=200$ thousand. The incurred projection implies total unpaid loss of $5{,}090-4{,}800=290$ thousand, or $90 thousand more than current case reserves. In contrast, the paid projection implies only $120 thousand of future paid loss. The projected ultimates differ by $5{,}090-4{,}920=170$ thousand. That gap calls for reviewing the evidence and assumptions; the two projections do not reconcile automatically.

Common mistake avoided: Do not describe the incurred projection as $120 thousand of future paid loss plus a $90 thousand reserve increase. The $120 thousand comes from the paid projection, while the $90 thousand is incurred development from a different projection. Combining them would mix two independently selected estimates.

Worked Example: Reconciling a Coherent Paid and Incurred Projection

A hypothetical commercial property portfolio has $3,600 thousand cumulative paid loss and $4,200 thousand cumulative incurred loss at a late valuation age. The selected paid tail is 1.20, based on evidence that payment emergence remains material. The incurred selection is 1.028571, reflecting evidence that most expected claim cost is already included in case reserves. Calculate the projections and interpret their relationship.

Paid projection. Applying the paid tail:

U_P=3{,}600(1.20)=4{,}320

Future paid loss under this projection is $4{,}320-3{,}600=720$ thousand.

Incurred projection. Applying the incurred tail:

U_I=4{,}200(1.028571)\approx4{,}320

The factor is approximately $4{,}320/4{,}200=1.028571$, so the incurred projection is also $4,320 thousand, to the nearest thousand.

Unpaid and reserve check. Current case reserves are $4{,}200-3{,}600=600$ thousand. The projected unpaid amount relative to current paid loss is $4{,}320-3{,}600=720$ thousand. This is $120 thousand above current case reserves. Equivalently, projected incurred development is $4{,}320-4{,}200=120$ thousand. The relationship can be checked as:

\text{current incurred} +\text{future paid} -\text{current case reserves} =4{,}200+720-600=4{,}320

The $720 thousand is total future payment implied by the paid ultimate. The $120 thousand is the additional incurred development implied by the incurred ultimate; it is not an extra amount to add to the $720 thousand. The calculation is consistent because both selected factors project to the same ultimate.

Common mistake avoided: Do not add future paid loss to the incurred projection as though both were separate additions to current incurred loss. Future payment of amounts already in case reserves reduces those reserves as claims are paid.

Worked Example: Adjusting Judgment for a Case-Reserving Change

A hypothetical professional liability portfolio has current cumulative paid loss of $7,200 thousand and incurred loss of $7,800 thousand at age 12. The paid evidence supports a factor of 1.060, while an incurred extrapolation suggests 1.010. During the period covered by the incurred observations, the company strengthened case reserves. A relevant incurred benchmark from portfolios with stable reserving supports 1.025. Select and calculate factors, then explain how the reserve change affects the decision.

Selection judgment. The paid factor is selected at 1.060 because the paid evidence is not directly affected by case-reserve revisions and is supported by the stated evidence. The raw incurred extrapolation is treated cautiously: reserve strengthening can elevate incurred amounts at observed ages and distort the apparent remaining incurred development. Given the stated benchmark and reserving change, select an incurred tail of 1.025 rather than mechanically adopting 1.010.

Paid ultimate.

U_P=7{,}200(1.060)=7{,}200+432=7{,}632

The paid projection implies $7{,}632-7{,}200=432$ thousand of future paid loss.

Incurred ultimate.

U_I=7{,}800(1.025)=7{,}800+195=7{,}995

The incurred projection implies $7{,}995-7{,}800=195$ thousand of incurred development. Current case reserves are $7{,}800-7{,}200=600$ thousand, and the incurred ultimate implies total unpaid of $7{,}995-7{,}200=795$ thousand. Thus the indicated unpaid amount is $795-600=195$ thousand above current case reserves.

Assessment. The paid and incurred ultimates differ by $7{,}995-7{,}632=363$ thousand. The selected incurred factor is higher than the raw extrapolation because the observed incurred pattern is affected by a known reserve change. This selection is a judgment supported by the stated benchmark, not a universal adjustment formula. The difference between the projections should be investigated further rather than hidden by substituting the paid factor for the incurred factor.

Common mistake avoided: Do not interpret a low incurred extrapolation as proof that little future loss remains when case-reserve strengthening has altered the observations. First assess whether the incurred pattern represents underlying claim emergence or changes in reserving practice.

Common Traps in Selecting and Applying the Factors

Using one factor for both bases. Paid and incurred observations can have different emergence patterns. Select on the appropriate basis, even if the resulting factors happen to be similar.
Forgetting the valuation age. A tail from age 10 is not interchangeable with a tail from age 12. Check exactly which future development the factor includes.
Confusing a factor with a percentage increase. A factor of 1.025 adds 2.5% of the current cumulative amount, not 2.5% of some separately estimated unpaid amount.
Ignoring case-reserve changes. Incurred link ratios and tail evidence can reflect changes in case adequacy or practice as well as claim emergence.
Forcing paid and incurred ultimates to agree. Agreement is a useful check when supported by the evidence, but it is not guaranteed by the definitions. A material difference warrants investigation and transparent judgment.

Key takeaway: Select paid and incurred tail factors separately, using evidence that reflects each basis. Apply each factor to the corresponding cumulative amount, then compare the implied ultimates, unpaid amounts, and case reserves. A difference is information to investigate, not permission to mix the two projections.

Check Your Understanding

For each problem, use the stated basis and valuation age. Treat all figures as hypothetical and in the units specified.

A portfolio has $2,500 thousand cumulative paid loss at age 9. Select a paid tail factor of 1.040. Calculate projected ultimate paid loss and the implied future paid amount. (Answer: Ultimate paid loss = $2,600 thousand; future paid = $100 thousand.)
A portfolio has $3,200 thousand incurred loss and $3,000 thousand paid loss at age 11. The selected incurred tail is 1.025. Calculate incurred ultimate, indicated incurred development, current case reserves, and projected unpaid relative to current paid loss. (Answer: Incurred ultimate = $3,280 thousand; incurred development = $80 thousand; current case reserves = $200 thousand; projected unpaid = $280 thousand.)
At age 10, a hypothetical book has $5,000 thousand paid and $5,100 thousand incurred. The selected factors are 1.030 paid and 1.010 incurred. Calculate both ultimates and their difference. State whether the factors should be made equal solely to remove that difference. (Answer: Paid ultimate = $5,150 thousand; incurred ultimate = $5,151 thousand; difference = $1 thousand; no, not solely to force agreement.)
At a late age, a portfolio has $6,000 thousand paid and $6,500 thousand incurred. Case reserves are therefore $500 thousand. A paid tail of 1.050 and an incurred tail of 1.020 are selected. Calculate both ultimates, future paid implied by the paid projection, and unpaid relative to current paid implied by the incurred projection. Compare the two ultimate estimates. (Answer: Paid ultimate = $6,300 thousand; future paid = $300 thousand; incurred ultimate = $6,630 thousand; incurred-projection unpaid relative to paid = $630 thousand; the incurred estimate is $330 thousand higher.)
An incurred tail extrapolation is below a stable-reserving benchmark, but the observed incurred data include a period of case-reserve strengthening. Name the key diagnostic issue and explain why the raw extrapolation should not automatically control. (Answer: The incurred pattern may be distorted by changes in case-reserving practice; review the impact of strengthening and the relevance of the benchmark before selecting the incurred tail.)

What You'll Learn