Why Reconcile Adjusted Paid and Incurred Results?
Berquist-Sherman adjustments put historical paid or incurred data on a basis intended to reduce distortions from changes in settlement speed or case-reserving adequacy. Once adjusted triangles and development factors have been selected, each method produces an estimate of ultimate losses. Comparing those estimates is useful, but the comparison is not a requirement that the two answers match exactly. They use different observed amounts and can retain different development patterns, data limitations, and estimation variability.
A useful reconciliation asks two questions. First, how large is the difference in ultimate estimates, stated on a consistent basis? Second, how much of that difference comes from the adjusted cumulative amounts and how much from the selected development factors? These steps describe the discrepancy; they do not by themselves establish which method is more accurate.
An age-to-ultimate factor is the selected multiplier that projects a cumulative amount at a specified maturity to ultimate. A factor of 1.20 means that the projected ultimate is 1.20 times the adjusted amount at that age. It is not a percentage reserve; the estimated unpaid amount under that method is the ultimate estimate less the relevant cumulative paid amount. When comparing methods, keep the amount basis and valuation date consistent, and state the sign convention for the difference.
The usual chain-ladder interpretation assumes that the selected development factors are appropriate for the adjusted data and that the adjusted development pattern is sufficiently stable for the projection. Factors may nevertheless be selected differently for paid and incurred data because the two triangles have different definitions and development behavior. These projections are not guaranteed to be equal, even if both adjustments are reasonable.
Calculating and Decomposing the Difference
The direct comparison is to calculate each ultimate separately, then subtract in the stated order. A further decomposition can reveal whether the difference is mainly associated with the adjusted amounts or with the selected factors. Use the paid factor as a reference and add and subtract \(I_aF^P_a\):
The first term is the adjusted-amount contribution, measured by applying the paid factor to the difference between adjusted incurred and adjusted paid. The second is the factor contribution, measured by applying the incurred adjusted amount to the difference between the two factors. The sum is exactly the incurred-minus-paid ultimate gap. This identity is an accounting decomposition, not a causal explanation: choosing a different reference factor gives a different allocation between the two components, while leaving the total gap unchanged.
The identity holds by ordinary algebra: expanding the right-hand side gives \(I_aF^P_a-P_aF^P_a+I_aF^I_a-I_aF^P_a\), so the two \(I_aF^P_a\) terms cancel, leaving \(I_aF^I_a-P_aF^P_a\). Thus, the decomposition does not approximate the difference. It exactly reconciles it, provided the amounts and factors are paired as defined.
Worked Example: Reconciling Two Estimates at the Same Maturity
A hypothetical liability portfolio has adjusted cumulative paid losses of 360 and adjusted cumulative incurred losses of 510, both in thousands of dollars. The selected paid age-to-ultimate factor is 1.40, and the selected incurred factor is 1.12. Calculate both ultimate estimates, the incurred-minus-paid gap, and its decomposition using the paid factor as the reference. All amounts in this example are in thousands of dollars.
Method. Multiply each adjusted amount by its own factor, subtract paid ultimate from incurred ultimate, and then apply the decomposition. This avoids comparing an adjusted amount directly with an ultimate estimate.
Paid ultimate. \(U^P=360(1.40)=504\). The paid method estimates ultimate losses of 504 thousand dollars.
Incurred ultimate. \(U^I=510(1.12)=571.2\). The incurred method estimates ultimate losses of 571.2 thousand dollars.
Difference. \(\Delta=571.2-504=67.2\). Incurred ultimate exceeds paid ultimate by 67.2 thousand dollars.
Adjusted-amount contribution. \((510-360)(1.40)=150(1.40)=210\).
Factor contribution. \(510(1.12-1.40)=510(-0.28)=-142.8\).
Check. \(210+(-142.8)=67.2\), which agrees with the directly calculated gap. The larger adjusted incurred amount contributes positively, while the lower incurred factor offsets part of that contribution. The decomposition identifies the arithmetic sources of the gap, not whether either selection is preferable.
Reading Differences Across Maturities
Maturity matters because the amount already observed and the remaining projected development change with age. At a more mature age, the age-to-ultimate factor is often closer to 1, so a factor difference may have a different effect than it would at an immature age. However, maturity alone does not determine the size or direction of the paid-incurred gap: adjusted amounts and selected factors both matter.
To compare different accident years or maturity groups, calculate each method's ultimate for each row before summing. Do not multiply a pooled amount by an unweighted average factor unless that average is specifically weighted to reproduce the sum of the row-level projections. In general, the total is \(\sum_j A_jF_j\), not \((\sum_j A_j)\) times the simple arithmetic average of the \(F_j\).
Worked Example: Comparing a Younger and an Older Maturity
A hypothetical insurer compares adjusted paid and incurred estimates for two separate accident-year cohorts. For the younger cohort, adjusted paid is 360, adjusted incurred is 510, and the selected paid and incurred factors are 1.40 and 1.12. For the older cohort, the corresponding amounts are 720 and 900, with factors 1.15 and 1.06. All amounts are in thousands of dollars. Calculate the estimates by cohort, the gap for each cohort, and the total gap.
Method. Treat the cohorts separately because their factors differ. For each one, calculate paid ultimate and incurred ultimate, then take incurred minus paid.
Younger cohort. Paid ultimate is \(360(1.40)=504\). Incurred ultimate is \(510(1.12)=571.2\). The gap is \(571.2-504=67.2\), with incurred higher.
Older cohort. Paid ultimate is \(720(1.15)=828\). Incurred ultimate is \(900(1.06)=954\). The gap is \(954-828=126\), with incurred higher.
Portfolio total. Paid ultimate is \(504+828=1{,}332\). Incurred ultimate is \(571.2+954=1{,}525.2\). The total incurred-minus-paid gap is \(1{,}525.2-1{,}332=193.2\) thousand dollars. As a check, the cohort gaps sum to \(67.2+126=193.2\).
The older cohort has a larger absolute gap even though its factors are closer to 1. That is possible because the adjusted amounts and the difference between the two methods' factors also affect the result. The example does not imply that a more mature cohort should always have a larger gap.
Interpreting the Adjusted Amount and Factor Contributions
When a gap is material, the decomposition provides a disciplined starting point for discussion. A large adjusted-amount contribution points to a difference between the adjusted cumulative paid and incurred amounts at the comparison age. A large factor contribution points to different selected age-to-ultimate factors. Either can dominate, and they can have opposite signs. The net difference can therefore be small even when both components are large.
The reference-factor choice must be stated. The decomposition above uses \(F^P_a\) for the amount contribution and \(I_a\) for the factor contribution. An equally valid alternative is to use the incurred factor as the reference:
This alternative also sums exactly to the same gap, but it allocates the gap differently between the two components. Therefore, do not treat either component as uniquely determined evidence about the cause of a difference. The direct ultimate estimates and their total gap are invariant to the chosen decomposition.
Worked Example: Separating Amount and Factor Effects
For a hypothetical accident year, adjusted paid losses are 450 and adjusted incurred losses are 630, in thousands of dollars. The selected paid factor is 1.40 and the selected incurred factor is 1.10. Calculate the ultimate estimates, decompose the incurred-minus-paid gap using the paid factor as reference, and verify the result.
Paid ultimate. \(U^P=450(1.40)=630\) thousand dollars.
Incurred ultimate. \(U^I=630(1.10)=693\) thousand dollars.
Direct gap. \(\Delta=693-630=63\) thousand dollars, so incurred ultimate exceeds paid ultimate by 63.
Adjusted-amount contribution. \((630-450)(1.40)=180(1.40)=252\) thousand dollars.
Factor contribution. \(630(1.10-1.40)=630(-0.30)=-189\) thousand dollars.
Check. \(252-189=63\) thousand dollars, equal to the direct gap. Here the adjusted incurred amount contributes \(+252\), while the lower incurred factor contributes \(-189\). Reporting only the positive amount contribution would materially misstate the net reconciliation.
Common Reconciliation Traps
A reconciliation is only as reliable as its comparison basis. Before interpreting a difference, confirm that both amounts refer to the same valuation date, accident-year definition, currency, coverage, and gross-or-net treatment. Confirm that each factor corresponds to the maturity and adjusted triangle from which its base amount comes. If one factor is applied to a raw amount and the other to an adjusted amount, the resulting difference mixes methods and is not a clean comparison.
- Mixing amounts and units. If values are stated in thousands, \(360(1.40)=504\) means 504 thousand dollars, not 504,000 in the same table units. Carry the stated unit through every calculation.
- Reversing the sign. With \(\Delta=U^I-U^P\), a positive result means incurred ultimate is higher. Reversing the subtraction reverses the interpretation.
- Using the wrong factor. Apply the selected factor for that method and maturity to its corresponding adjusted cumulative amount.
- Averaging factors without appropriate weights. Calculate row-level ultimates and add them when factors vary by row; a simple average of factors generally does not preserve the total.
- Assuming adjusted methods must agree. Adjustments address specified distortions; they do not make paid and incurred data identical or eliminate estimation uncertainty.
- Calling a decomposition causal. The components depend on the reference factor. The total gap does not, but the allocation between components does.
Check Your Understanding
For all problems, amounts are in thousands of dollars. Use incurred minus paid as the signed difference. Show each method's ultimate before subtracting.
- Adjusted paid is 240 and its age-to-ultimate factor is 1.25. Adjusted incurred is 280 and its factor is 1.10. Calculate both ultimates and the gap. (Answer: paid 300; incurred 308; gap +8.)
- For one cohort, adjusted paid is 400 with factor 1.30, and adjusted incurred is 500 with factor 1.08. Calculate the two ultimates and say which is higher. (Answer: paid 520; incurred 540; incurred exceeds paid by 20.)
- Adjusted paid is 520, adjusted incurred is 600, the paid factor is 1.25, and the incurred factor is 1.10. Using the paid factor as reference, find both contributions and verify their sum against the direct gap. (Answer: adjusted-amount contribution +100; factor contribution −90; gap +10.)
- For cohort A, adjusted paid is 300 with factor 1.20 and adjusted incurred is 310 with factor 1.15. For cohort B, adjusted paid is 500 with factor 1.10 and adjusted incurred is 540 with factor 1.03. Calculate the portfolio totals by summing cohort ultimates. (Answer: paid total 910; incurred total 912.7; gap +2.7.)
- Two cohorts have adjusted paid amounts and factors of 200 and 1.50, then 600 and 1.10. Their adjusted incurred amounts and factors are 250 and 1.30, then 550 and 1.05. Calculate the paid and incurred portfolio ultimates and the signed gap. (Answer: paid total 960; incurred total 902.5; gap −57.5, so paid exceeds incurred by 57.5.)