Why an Integrated Case Is Useful
Berquist-Sherman adjustments address a particular problem in loss development: observed triangle movement may reflect changes in claim settlement speed or case reserve practices, as well as the underlying emergence of losses. An adjustment puts observations on a more comparable basis before selecting development factors. It does not make the data complete or remove every source of distortion.
This case study follows a hypothetical liability portfolio through closure-rate adjustment of paid losses, case-reserve adjustment of incurred losses, factor selection, and ultimate estimates. The computations are shown in thousands of dollars unless otherwise stated. Each adjustment depends on an assumption about what would have happened under a common claims-handling basis, so the assumptions must be reviewed alongside the arithmetic.
Adjusting Paid Losses for Closure Rates
Let \(P\) be cumulative paid loss at a given development age, \(c\) the observed cumulative closure proportion at that age, and \(c^*\) the selected closure proportion for the common basis. Under the simplifying assumption that the closed claims are representative in payment amount, a basic closure-rate adjustment is:
The adjustment scales the observed paid amount by the ratio of target to observed closure proportions. If claims have closed more slowly than the target pattern, \(c^*/c\) exceeds one and the standardized paid amount increases. This is not a prediction that the insurer will immediately pay the difference; it is an estimate of the paid amount that would be associated with the target closure level under the model.
The assumption matters. The adjustment can mislead if claims closed by a particular age differ materially in severity from claims still open, or if payment amounts per closed claim have changed for reasons other than settlement speed. Claim mix, large losses, and the reliability of closure counts should therefore be considered before using the result.
Worked Example: Standardizing a Paid Amount
A hypothetical accident period has $4,800 thousand in cumulative paid losses at 24 months. By that age, 60% of claims are closed. The selected common-basis closure proportion is 75%. Estimate the standardized paid amount and state what it represents.
Method. Apply the closure-rate adjustment \(P^*=P(c^*/c)\). Here \(P=4{,}800\), \(c^*=0.75\), and \(c=0.60\).
Adjustment ratio. \(0.75/0.60=1.25\).
Standardized paid amount. \(4{,}800(1.25)=6{,}000\) thousand.
The adjusted value is therefore $6,000 thousand. It is a common-basis analytical amount, not a statement that actual payments at 24 months were $6,000 thousand. This calculation assumes that the average payment characteristics of closed claims are suitable for standardizing the difference in closure proportions. The common mistake is to add 15% to paid losses because closure increased by 15 percentage points; the correct adjustment uses the ratio \(75\%/60\%=1.25\), not an additive percentage-point change.
Adjusting Incurred Losses for Case Reserve Adequacy
Incurred loss at an evaluation age is paid loss plus case reserves. If the average case reserve per open claim changes because of a reserving-practice change, incurred development may change even when the underlying claim costs have not. A common-basis adjustment replaces the observed case reserve per open claim with a selected target amount.
Here \(I^*\) is adjusted incurred loss, \(P\) is paid loss, \(N_{\mathrm{open}}\) is the number of open claims, and \(r^*\) is the selected case reserve per open claim on the target basis. If the observed case reserve total is \(R\) and the observed average is \(r=R/N_{\mathrm{open}}\), the same adjustment can be written:
This follows because \(R=N_{\mathrm{open}}r\). Multiplying \(R\) by \(r^*/r\) gives \(N_{\mathrm{open}}r^*\), the case reserve total implied by the target average. The adjustment assumes the open-claim count and the selected target average are appropriate for the claims being standardized; it does not correct inaccurate claim counts or an unrepresentative claim mix.
Worked Example: Restating Case Reserves
A hypothetical accident period has $2,400 thousand paid and 160 open claims at an evaluation date. The case reserve total is $1,120 thousand. The selected common-basis case reserve is $6 thousand per open claim. Calculate raw and adjusted incurred loss.
Observed reserve per open claim. \(1{,}120/160=7\) thousand per claim.
Raw incurred loss. \(2{,}400+1{,}120=3{,}520\) thousand.
Target case reserve total. \(160(6)=960\) thousand.
Adjusted incurred loss. \(2{,}400+960=3{,}360\) thousand.
The adjusted incurred amount is $3,360 thousand, which is $160 thousand below raw incurred loss. Equivalently, \(1{,}120(6/7)=960\), so the reserve adjustment gives the same result. The common trap is to multiply the entire incurred amount by \(6/7\); only the case reserve component is being restated, while paid loss remains unchanged.
From Adjusted Triangles to Ultimate Losses
Once the relevant triangle entries have been adjusted, development factors can be selected using the adjusted observations. A link ratio compares cumulative losses at successive development ages. A selected age-to-ultimate factor is formed from the applicable selected link ratios, with a tail factor used when development beyond the last observed age is material. If \(L_a\) is the adjusted cumulative loss at the latest observed age and \(F_{a:\mathrm{ult}}\) is the selected age-to-ultimate factor, the projection is:
The factor must correspond to the exact age and adjusted basis of the loss amount being projected. Applying a factor selected from unadjusted data to adjusted losses, or applying a tail factor twice, changes the basis of the projection and is not justified.
Select target closure proportions and case reserve levels by development age, and document why they are appropriate.
Standardize paid losses for closure-rate changes and incurred losses for case reserve adequacy. Preserve the distinction between paid amounts and reserve estimates.
Use adjusted triangle movement and judgment to select age-to-age factors and any tail factor. Check that the observations in each ratio share a comparable basis.
Project each accident period to ultimate, aggregate the results, and compare paid- and incurred-based estimates without assuming they must be identical.
Integrated Hypothetical Case
Consider a hypothetical portfolio with three accident periods. The paid method below uses selected factors derived from the closure-adjusted paid data. The incurred method uses adjusted latest incurred values and separately selected age-to-ultimate factors. The selections are illustrative; in practice, they would be supported by the full adjusted triangles, diagnostics, and actuarial judgment.
Worked Example: Projecting and Reconciling Three Accident Periods
For the paid projection, the selected calculations use closure adjustments of \(0.75/0.70\), \(0.50/0.40\), and \(0.50/0.45\), as applicable. The paid selections also include a 1.10 tail factor. For the incurred projection, the adjusted latest incurred amounts are $9,400 thousand, $7,200 thousand, and $4,600 thousand, with selected age-to-ultimate factors of 1.02, 1.20, and 1.55, respectively. Calculate each method’s ultimate by accident period, aggregate each method, and compare the totals.
Paid, accident period 1. The latest adjusted paid amount is $8,000 thousand. Applying the selected tail factor gives \(8{,}000(1.10)=8{,}800.00\) thousand.
Paid, accident period 2. The adjusted benchmark is \(5{,}500(0.75/0.70)=5{,}892.8571\) thousand. The selected multiplier is \(8{,}000/[5{,}500(0.75/0.70)(1.10)]=1.23416\) (rounded here for display). Applying that multiplier to the latest $6,500 thousand and then the 1.10 tail gives:
Paid, accident period 3. The selected adjusted-data ratio is:
The additional selected multiplier is \(8{,}000/[5{,}500(0.75/0.70)]=1.357576\). Thus the accident-period-3 paid ultimate is:
Aggregate paid estimate. Adding the unrounded period estimates gives \(8{,}800.00+8{,}824.2424+7{,}163.8710=24{,}788.1134\) thousand, or $24,788.11 thousand rounded to two decimals. The period estimates are $8,800.00 thousand, $8,824.24 thousand, and $7,163.87 thousand.
Incurred, accident period 1. \(9{,}400(1.02)=9{,}588\) thousand.
Incurred, accident period 2. \(7{,}200(1.20)=8{,}640\) thousand.
Incurred, accident period 3. \(4{,}600(1.55)=7{,}130\) thousand.
Aggregate incurred estimate. \(9{,}588+8{,}640+7{,}130=25{,}358\) thousand. The incurred-based aggregate exceeds the paid-based aggregate by \(25{,}358-24{,}788.1134=569.8866\) thousand, or $569.89 thousand rounded to two decimals.
The difference is a diagnostic, not an automatic error. The two methods use distinct adjusted data and selected factors, so their estimates need not match exactly. Review the accident-period pattern, factor selections, tail treatment, and the assumptions behind both adjustments. The common mistake this calculation avoids is carrying a rounded or incorrectly placed factor into an aggregate: retain full precision in intermediate steps, apply the stated tail exactly once in each formula, and round only the final amounts.
How to Interpret the Reconciliation
A credible integrated analysis should explain both the numerical result and its limitations. Paid and incurred methods can differ because their data respond differently to claims handling: payment timing affects paid development, while case reserve practices affect incurred development. Adjustments aim to reduce those distortions, but they rely on selected target levels and may be sensitive to claim mix, large claims, and sparse observations.
A useful reconciliation checks whether the difference is concentrated in one accident period or spread across the portfolio, whether paid and incurred factors are based on comparable adjusted observations, and whether the selected tail is consistent with the development ages available. If an adjustment produces an implausible result, revisit its assumptions rather than treating the formula as a mechanical correction.
Check Your Understanding
Work each problem through the adjustment and projection before checking the answer.
- A hypothetical portfolio has $3,600 thousand paid at an age when 45% of claims are closed. The target closure proportion is 60%. What is the standardized paid amount? (Answer: $4,800 thousand.)
- A segment has $1,500 thousand paid, 100 open claims, and $800 thousand in case reserves. The target case reserve is $7 thousand per open claim. Find raw and adjusted incurred loss. (Answer: Raw incurred is $2,300 thousand; adjusted incurred is $2,200 thousand.)
- An adjusted latest paid amount is $5,200 thousand. The selected age-to-ultimate factor is 1.18 and the tail factor is 1.05. Calculate ultimate paid loss. (Answer: \(5{,}200(1.18)(1.05)=\$6{,}442.80\) thousand.)
- Two accident periods have adjusted incurred amounts of $6,400 thousand and $5,000 thousand, with selected factors of 1.25 and 1.40. Calculate each ultimate and the aggregate. (Answer: $8,000 thousand and $7,000 thousand; aggregate $15,000 thousand.)
- A paid-based aggregate is $18,250 thousand and an incurred-based aggregate is $18,900 thousand. What is the dollar difference, which estimate is higher, and what is the appropriate interpretation? (Answer: The incurred estimate is higher by $650 thousand; investigate the adjusted data and selections rather than assuming either method is automatically wrong or forcing equality.)