Why Rebase Case Reserves?
In an incurred-loss triangle, reported incurred loss is the sum of paid loss and case outstanding reserves. If case-reserving practices change, the case component may change even when the underlying claims do not. For example, a hypothetical insurer might begin setting more conservative case reserves at the same development age. The resulting increase in incurred loss could then be mistaken for a change in claim emergence or ultimate loss.
The Berquist-Sherman incurred adjustment addresses this issue by retaining paid losses and replacing the observed case outstanding reserves with amounts consistent with a selected case-reserve adequacy pattern. The adjustment is performed at each development age, so different ages can have different selected adequacy targets. Within a given age, the target is applied consistently across the valuation periods being standardized.
This definition makes the adequacy measure explicit: \(c^*\) is a share of adjusted incurred loss, not a multiplier applied directly to the observed case reserve. A selected pattern is a set of targets \(c^*_j\), one for each development age \(j\). Its purpose is to put incurred losses from different valuation periods on a comparable case-reserving basis.
Deriving the Adjustment
Once the paid amount is fixed, the target adequacy condition determines the replacement case reserve. At a given development age, the required relationship is \(C^*/I^*=c^*\), while \(I^*=P+C^*\). Substituting the second equation into the first gives the adjustment formula.
Adding retained paid loss gives adjusted incurred loss:
The derivation shows why the target applies to adjusted incurred loss, not to the original incurred loss. The case reserve must be solved together with the adjusted incurred amount because the replacement case reserve is itself part of that amount.
The method is cell-specific: use the target for that cell's development age, not the target for its valuation year. It does not change the paid amount, estimate a new ultimate loss directly, or determine what adequacy pattern should be selected. The target pattern is an input, selected from the analysis of case-reserving adequacy.
Applying the Formula
Use the cumulative paid loss for the valuation period and the development age of the cell.
Assign the selected adequacy share \(c^*_j\) for that development age.
Compute \(C^*_j=[c^*_j/(1-c^*_j)]P_j\).
Add the replacement case reserve to paid loss, or equivalently compute \(I^*_j=P_j/(1-c^*_j)\).
Confirm that \(I^*_j-P_j=C^*_j\) and \(C^*_j/I^*_j=c^*_j\), subject to rounding.
Worked Example: Replacing Reserves in One Development-Age Cell
A hypothetical liability portfolio has cumulative paid loss of $720,000 at 18 months. Observed case outstanding reserves are $480,000, so observed incurred loss is $1,200,000. The selected case-reserve share for 18 months is 25%. Find adjusted case reserves and adjusted incurred loss. Use the formula method and state the check.
Method. The paid amount remains $720,000. The selected target is \(c^*=0.25\), so the replacement reserve must be 25% of adjusted incurred loss.
Step 1: Calculate the adjusted case reserve.
Step 2: Calculate adjusted incurred loss.
Check. Adjusted case reserves are 25% of adjusted incurred loss because \(240{,}000/960{,}000=0.25\). Also, adjusted incurred less retained paid is \(960{,}000-720{,}000=\$240{,}000\), the adjusted case reserve. The observed case-reserve share was \(480{,}000/1{,}200{,}000=40\%\), so the adjustment lowers incurred loss to reflect the selected 25% target. The common mistake avoided here is multiplying the original $480,000 reserve by 25%; 25% is the target share of adjusted incurred, not a reserve adjustment factor.
Worked Example: Applying Age-Specific Targets Across Two Cells
A hypothetical commercial package has the following cumulative data, in thousands of dollars. The selected adequacy targets are 30% at 12 months and 20% at 24 months. Calculate the adjusted case reserves and incurred losses at both ages, and then sum the adjusted amounts. Assume the two entries represent separate cells in the data being summarized.
| Development age | Paid loss | Observed case reserves | Selected target |
|---|---|---|---|
| 12 months | 500 | 500 | 30% |
| 24 months | 800 | 400 | 20% |
12 months. Use the 30% target for this age and paid loss of 500:
24 months. Use the 20% target for this age and paid loss of 800:
Totals. The adjusted case reserves sum to \(214.2857+200=414.2857\) thousand, and adjusted incurred losses sum to \(714.2857+1{,}000=1{,}714.2857\) thousand. The retained paid total is \(500+800=1{,}300\) thousand, and \(1{,}300+414.2857=1{,}714.2857\) thousand. The age-specific checks are \(214.2857/714.2857=30\%\) and \(200/1{,}000=20\%\). Reported to the nearest thousand dollars, the totals are adjusted case reserves of $414,000 and adjusted incurred loss of $1,714,000. The common mistake avoided is applying one target to both cells or averaging the two targets before calculating.
Interpreting the Result and Its Limits
The adjustment changes the case component, not the paid component. If the selected target is higher than the observed case-reserve share, adjusted case reserves and adjusted incurred losses increase; if it is lower, both decrease. For \(P>0\), this follows because \(I^*=P/(1-c^*)\) increases as \(c^*\) increases over the interval from 0 to 1.
The calculation also has limits. A target of \(c^*=1\) makes the denominator \(1-c^*\) zero and does not yield a finite adjusted incurred amount. A target below zero or above one is not a valid case-reserve share. If paid loss is zero and the selected target is less than one, the formulas give adjusted case reserves and incurred loss of zero. That result follows algebraically, but it cannot create an estimate of positive losses from a cell with no paid loss; separate information would be needed for that purpose.
The target must use the same definition and basis as the data being adjusted. In this tutorial, it is the ratio of case outstanding to incurred loss, \(C/I\). If a different adequacy measure is used, such as case outstanding divided by an estimate of unpaid ultimate loss, the formula must be re-derived for that definition rather than applied unchanged. Likewise, the arithmetic does not establish that the selected pattern is appropriate: the target pattern should be supported by the adequacy analysis and applied consistently.
Worked Example: Distinguishing a Target Share from a Multiplier
A hypothetical specialty insurer reports $1,500,000 paid and $1,000,000 in case outstanding at a particular age. Its selected case-reserve share of adjusted incurred is 40%. A reviewer proposes multiplying the observed case reserve by 40% and adding the result to paid loss. Calculate the correct adjustment, the reviewer's result, and explain the difference.
Correct method. The target share is \(c^*=0.40\), and paid loss is retained at $1,500,000:
Reviewer's calculation. Multiplying the observed reserve by 40% gives \(0.40(1{,}000{,}000)=\$400{,}000\); adding paid produces $1,900,000. But that result has case-reserve share \(400{,}000/1{,}900{,}000=0.2105\), or about 21.05%, not 40%.
Check and lesson. The correct result has \(1{,}000{,}000/2{,}500{,}000=40\%\). The proposed calculation treats the selected adequacy percentage as a multiplier on the old reserve, even though it is defined as a target ratio of the new reserve to adjusted incurred. That is the central exam trap: solve for the replacement reserve using the target relationship rather than scaling the old reserve mechanically.
Check Your Understanding
For each problem, use the case-reserve share of adjusted incurred as the adequacy definition. Retain paid loss, apply the target for the stated age, and check the ratio where practical.
- A hypothetical machinery account has paid loss of $360,000 and a 25% selected target at 12 months. Find adjusted case reserves and adjusted incurred loss. (Answer: Adjusted case reserves $120,000; adjusted incurred loss $480,000.)
- A hypothetical product-liability cohort has paid loss of $1,200,000 and observed case reserves of $800,000. Its selected target at 24 months is 20%. Find the observed case-reserve share and the adjusted amounts. (Answer: Observed share 40%; adjusted case reserves $300,000; adjusted incurred loss $1,500,000.)
- A hypothetical property portfolio has paid loss of $700,000 at 12 months and $900,000 at 24 months. The selected targets are 30% and 10%, respectively. Calculate adjusted case reserves and incurred losses at each age, then the totals. (Answer: At 12 months, adjusted case reserves $300,000 and incurred $1,000,000; at 24 months, adjusted case reserves $100,000 and incurred $1,000,000; totals are $400,000 and $2,000,000, respectively.)
- A hypothetical account has $450,000 paid and $150,000 observed case reserves at a development age where the selected target is 40%. A reviewer reports adjusted incurred loss of $630,000. Is that correct? Give the correct amount and the reviewer’s implied case-reserve share if the paid amount was retained. (Answer: No. Correct adjusted incurred is $750,000 and adjusted case reserves are $300,000; the reviewer’s implied case reserves are $180,000 and implied share is \(180{,}000/630{,}000=28.57\%\).)
- A hypothetical cell has $250,000 paid and a proposed selected target of 100%. Can the standard adjustment produce finite adjusted case reserves and incurred loss? Explain using the formula condition. (Answer: No. At \(c^*=1\), \(1-c^*=0\), so the formula has a zero denominator and no finite solution.)