Reconciling Divergent Paid and Incurred Estimates

Why Paid and Incurred Estimates Can Diverge

Paid and incurred chain ladder estimates use different views of claim emergence. Cumulative paid losses reflect amounts already paid. Cumulative incurred losses combine paid amounts with case outstanding estimates on reported claims. Because those starting values and their development patterns differ, the two methods can produce different ultimates and different implied unpaid amounts.

A material difference is a reason to investigate, not a reason to average the estimates automatically or choose whichever is larger. The actuary should compare the estimates at the same development age, review the stability of the selected factors, and consider whether payment timing or case reserve practices have changed. This diagnostic approach is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For accident year $i$ at development age $j$, let $P_{i,j}$ be cumulative paid losses and $C_{i,j}$ be cumulative incurred losses. In the usual case, $C_{i,j}=P_{i,j}+K_{i,j}$, where $K_{i,j}$ is case outstanding at that evaluation date. Let $F^P_j$ and $F^I_j$ be the selected age-to-ultimate factors for paid and incurred losses. The chain ladder projections are $U^P_{i,j}=P_{i,j}F^P_j$ and $U^I_{i,j}=C_{i,j}F^I_j$. The corresponding implied unpaid amounts are $R^P_{i,j}=U^P_{i,j}-P_{i,j}$ and $R^I_{i,j}=U^I_{i,j}-C_{i,j}$.

These formulas describe projections from a common evaluation date and a specified set of selected factors. They do not establish that either selection is correct. In particular, a factor selected from historical development can be misleading if the current claims, settlement practices, or case reserve methods do not resemble the experience underlying that factor. Chain ladder projection also relies on development patterns being sufficiently stable for the selected factors to be informative.

A Reconciliation Identity

The difference between the implied unpaid amounts can be separated into two parts: the difference between projected ultimates and the difference between the current incurred and paid values. This identity helps prevent an analyst from confusing a reserve difference with an ultimate loss difference.

R^P_{i,j}-R^I_{i,j} =(U^P_{i,j}-P_{i,j})-(U^I_{i,j}-C_{i,j}) =(U^P_{i,j}-U^I_{i,j})+(C_{i,j}-P_{i,j}).

The justification is direct: expand the two reserve definitions, collect the ultimate terms, and then collect the current cumulative loss terms. When the paid and incurred estimates have the same ultimate $U$, the identity reduces to $R^P-R^I=C-P=K$. Thus, even if the estimated ultimate is identical, the paid-based implied unpaid amount exceeds the incurred-based implied unpaid amount by the existing case outstanding. If the ultimates differ, that ultimate difference must also be considered.

Worked Example: Separating Ultimate and Unpaid Differences

For a hypothetical accident year at the latest evaluation, paid losses are $500 and incurred losses are $650. The selected paid age-to-ultimate factor is 2.00, and the selected incurred age-to-ultimate factor is 1.60. Calculate each ultimate and implied unpaid amount, then reconcile the difference.

Method. Apply each method’s selected factor to its own cumulative loss basis. The case outstanding is $C-P=\$650-\$500=\$150$.

Paid projection. The paid ultimate is $U^P=\$500(2.00)=\$1{,}000$. Its implied unpaid amount is $R^P=\$1{,}000-\$500=\$500$.

Incurred projection. The incurred ultimate is $U^I=\$650(1.60)=\$1{,}040$. Its implied unpaid amount is $R^I=\$1{,}040-\$650=\$390$.

Check. The paid-based implied unpaid amount exceeds the incurred-based amount by $\$500-\$390=\$110$. The identity gives $(\$1{,}000-\$1{,}040)+(\$650-\$500)=-\$40+\$150=\$110$, the same result. The ultimates differ by $40, while the case outstanding is $150.

Common mistake avoided. Multiplying the incurred amount by 1.60 does not give $1,000: $\$650(1.60)=\$1{,}040$. Using $1,000 would incorrectly make the two ultimates equal and change the incurred implied unpaid amount.

Compare Maturity and Factor Stability

A comparison is most informative when the paid and incurred observations refer to the same accident year and evaluation date. If one estimate uses a more mature observation than the other, or if selected factors cover different development horizons, the apparent disagreement may partly be a comparison problem rather than a difference in underlying claims.

Maturity matters because an immature cumulative amount is more heavily multiplied to reach ultimate. A small change in a selected factor can therefore have a larger dollar effect at an early age. Factor stability matters because a factor that varies substantially across accident years may be sensitive to mix, random variation, or a change in claim handling. A stable factor is not automatically appropriate, but an unstable one calls for closer review before it is applied to the current year.

Worked Example: Checking Unequal Factor Stability

A hypothetical insurer is evaluating one accident year at 12 months. Its cumulative paid losses are $300 and cumulative incurred losses are $450. The three historical 12-to-24-month paid link ratios are 1.50, 1.60, and 1.90; the corresponding incurred link ratios are 1.20, 1.25, and 1.22. For this illustration, select the simple arithmetic mean of each set, then use tail factors of 1.10 for paid and 1.05 for incurred. Estimate both ultimates and implied unpaid amounts.

Paid factor. The arithmetic mean of the paid link ratios is $(1.50+1.60+1.90)/3=5.00/3=1.6667$. Multiplying by the paid tail factor gives a paid age-to-ultimate factor of $1.6667(1.10)$, or $1.8333$ using the unrounded mean $5/3$.

Paid ultimate and implied unpaid. Using the unrounded mean, $U^P=\$300(5/3)(1.10)=\$550.00$. Then $R^P=\$550.00-\$300=\$250.00$.

Incurred factor. The arithmetic mean of the incurred link ratios is $(1.20+1.25+1.22)/3=3.67/3=1.223333$. Including the tail factor gives $1.223333(1.05)=1.2845$.

Incurred ultimate and implied unpaid. $U^I=\$450(1.223333)(1.05)=\$578.025$, or $578.03 rounded to the nearest cent. Therefore $R^I=\$578.025-\$450=\$128.025$, or $128.03.

Interpretation and check. The paid link ratios range from 1.50 to 1.90, while the incurred ratios range only from 1.20 to 1.25. The paid selection is less stable in this small illustrative set, so its projection deserves particular scrutiny. The implied unpaid difference is $\$250.00-\$128.025=\$121.975$, or $121.98. The reconciliation identity gives $(\$550-\$578.025)+(\$450-\$300)=-\$28.025+\$150=\$121.975$. Calculations retain unrounded factors until the final amounts are rounded to cents.

Common mistake avoided. Comparing the two selected link ratios alone would ignore the different starting cumulative amounts, tail factors, and case outstanding. The full projections use each factor on its own matched 12-month cumulative value.

Use Claim Patterns to Interpret the Difference

After matching the evaluation age, consider whether claim payment or case reserve patterns have changed. If settlements have slowed, paid losses may emerge more slowly than in the historical data. A paid chain ladder using older, faster-settlement experience may then project an inappropriate future payment pattern. Conversely, accelerated settlements can alter paid development in the other direction. These are diagnostic possibilities, not conclusions established by the factors alone.

Incurred results can be affected by case reserve adequacy and reserving practices. If case reserves have become more or less conservative, the incurred cumulative amounts and their link ratios may shift even if the ultimate cost of claims has not changed. A change in case reserving can therefore cause paid and incurred projections to diverge, or can make one method appear stable while its underlying assumptions have changed.

Worked Example: Recognizing a Possible Settlement-Speed Change

For a hypothetical coverage, three historical 12-to-24-month paid link ratios are 1.35, 1.40, and 1.85. The corresponding incurred link ratios are 1.16, 1.17, and 1.18. A latest accident year, also at 12 months, has $400 paid and $650 incurred. For this illustration, use the simple arithmetic mean of each set of historical ratios, with paid and incurred tail factors of 1.10 and 1.05, respectively. Calculate the projections and identify the main diagnostic signal.

Paid selection. The mean paid link ratio is $(1.35+1.40+1.85)/3=4.60/3=1.533333$. Including the tail factor gives $1.533333(1.10)=1.686667$.

Paid projection. Retaining the unrounded mean, $U^P=\$400(4.60/3)(1.10)=\$674.6667$, or $674.67 to the nearest cent. The implied unpaid amount is $\$674.6667-\$400=\$274.6667$, or $274.67.

Incurred selection and projection. The mean incurred link ratio is $(1.16+1.17+1.18)/3=3.51/3=1.17$. Including its tail factor gives $1.17(1.05)=1.2285$. Thus $U^I=\$650(1.17)(1.05)=\$798.525$, or $798.53. The implied unpaid amount is $\$798.525-\$650=\$148.525$, or $148.53.

Interpretation and check. The most recent paid link ratio, 1.85, is substantially above the other two, whereas the incurred ratios are close together. This pattern could be consistent with a change in settlement speed, but it is not proof of one. The paid estimate is $123.86 below the incurred estimate. The case outstanding is $\$650-\$400=\$250$, and the identity reconciles the unpaid difference: $(\$674.6667-\$798.525)+\$250=\$126.1417$, or $126.14. This equals $\$274.6667-\$148.525$, with both final amounts rounded to cents.

Common mistake avoided. A single high paid link ratio should not be treated as conclusive evidence that the current year will develop like that observation. Review the underlying claim settlement pattern and the factor selection before relying on the arithmetic mean.

A Practical Reconciliation Sequence

A useful review separates numerical reconciliation from actuarial interpretation. First verify that the paid and incurred data use the same accident year, valuation date, coverage, and loss basis. Then compare their ages and factors, quantify the difference in ultimate and implied unpaid amounts, and investigate whether claims handling explains the pattern. The sequence does not mechanically select a best estimate; it makes the reasons for disagreement explicit.

Match the data.
Confirm accident period, evaluation date, development age, coverage, and whether the figures use the same loss and expense basis.

Recalculate both projections.
Apply each selected age-to-ultimate factor to its corresponding cumulative paid or incurred amount. Calculate each implied unpaid amount from its own basis.

Inspect factor behavior.
Review the historical link ratios and selected factors for stability, unusual observations, and differences in development horizon.

Investigate claim patterns.
Consider whether settlement speed, case reserve adequacy, claim mix, or other operational changes could affect the paid or incurred pattern.

State what remains uncertain.
Document which assumptions drive the difference and whether either selected pattern may not represent current experience. Do not resolve unexplained disagreement by averaging automatically.

Common Exam Traps

Comparing unpaid amounts as if they were ultimates. The paid-based implied unpaid amount and incurred-based implied unpaid amount start from different current values. Compare their ultimates separately, then reconcile the unpaid difference.
Using a factor on the wrong cumulative value. The paid factor multiplies cumulative paid losses; the incurred factor multiplies cumulative incurred losses.
Ignoring the evaluation age. A difference is harder to interpret if one projection starts at a different maturity or uses a different remaining development horizon.
Assuming stable incurred ratios prove case reserves are adequate. Stable ratios describe the observed pattern, but they do not by themselves establish that case estimates are appropriate.
Assuming the larger estimate is safer or more accurate. A higher projection may reflect a valid signal, an unstable factor, a handling change, or an issue in the data. The cause must be investigated.

Key takeaway: Reconcile the components before judging the estimates. A paid-incurred ultimate difference, a current case outstanding, and uncertainty in the selected development factors are distinct quantities. Matched maturity and claim-handling diagnostics help explain how they combine.

Check Your Understanding

For each problem, calculate carefully and distinguish projected ultimate from implied unpaid amount.

A hypothetical accident year has $240 paid and $360 incurred. The selected paid and incurred age-to-ultimate factors are 2.50 and 1.75. Find both ultimates and both implied unpaid amounts. (Answer: paid ultimate $600, paid implied unpaid $360; incurred ultimate $630, incurred implied unpaid $270.)
At a common evaluation age, paid losses are $520 and incurred losses are $760. Selected factors are 1.80 for paid and 1.30 for incurred. Find the ultimate difference and the difference between implied unpaid amounts. (Answer: paid ultimate $936, incurred ultimate $988; paid implied unpaid $416, incurred implied unpaid $228; paid-based implied unpaid exceeds incurred-based by $188.)
A hypothetical book has three historical 12-to-24-month paid link ratios of 1.40, 1.60, and 2.00, and incurred ratios of 1.20, 1.25, and 1.15. Use the simple arithmetic mean for each set and tail factors of 1.10 for paid and 1.05 for incurred. At 12 months, paid is $300 and incurred is $500. Estimate both ultimates to the nearest cent. (Answer: paid ultimate $550.00; incurred ultimate $630.00.)
At one valuation date, paid is $450 and incurred is $700. Selected paid and incurred factors are 1.80 and 1.30. Calculate each implied unpaid amount and use the reconciliation identity to check their difference. (Answer: paid implied unpaid $360; incurred implied unpaid $210; difference $150, equal to the $250 case outstanding plus the $810 minus $910 ultimate difference.)
At the same development age, paid ratios have risen across recent years while incurred ratios remain nearly level. State one possible operational explanation and one reason that this pattern alone does not establish which ultimate estimate is correct. (Answer: Slower settlement could raise paid development ratios; the pattern alone does not prove a settlement change or establish that either selected factor set is appropriate.)

What You'll Learn