Why Separate the Sources of Change?
A Berquist-Sherman adjustment can change the cumulative losses used in an estimate, the development pattern applied to those losses, or both. The resulting ultimate estimate may therefore change for more than one reason. To interpret the result, compare raw and adjusted estimates on a consistent basis and identify which part of the difference comes from the adjusted latest loss amount and which part comes from the selected development factor.
This distinction matters in practice because a higher or lower ultimate estimate is not, by itself, evidence that the adjustment increased or decreased the underlying claim cost. The adjustment may change the observed loss values, alter the development factors selected from the triangle, or do both. A clear reconciliation helps explain the estimate and makes it easier to detect inconsistent inputs. This topic is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The superscripts \(R\) and \(A\) distinguish raw and adjusted quantities; they do not indicate different kinds of ultimate loss. For a valid comparison, the two latest cumulative amounts must refer to the same accident period, valuation date, loss basis, and coverage of claims. The selected factors must also be appropriate to their respective raw or adjusted triangles. In particular, a factor selected from an adjusted triangle should not be applied to a raw cumulative amount without an explicit reason.
Reconcile the Change in Two Parts
A useful convention is to change the latest loss amount first, holding the raw factor fixed, and then change the factor to its adjusted selection. This convention creates a transparent two-step bridge from the raw estimate to the adjusted estimate.
Calculate \((C_i^A-C_i^R)F_i^R\). This is the change in estimated ultimate when the cumulative amount changes but the raw factor is held fixed.
Calculate \(C_i^A(F_i^A-F_i^R)\). This is the additional change from replacing the raw selected factor with the adjusted selected factor.
Add the two effects and verify that their sum equals \(C_i^A F_i^A-C_i^R F_i^R\), the direct difference between adjusted and raw ultimates.
The identity follows by adding and subtracting the intermediate estimate \(C_i^A F_i^R\). That intermediate estimate uses the adjusted cumulative amount with the raw factor:
Factoring each parenthesis gives the two terms in the reconciliation formula. The identity is exact; it does not depend on a statistical model. It does, however, depend on the chosen order of the bridge. If the factor is changed first and the cumulative amount second, the allocation between the two effects will differ, although the total change will be the same. State the convention used whenever reporting the components.
Worked Reconciliations
Worked Example: Separating an Amount Adjustment from a Factor Change
Method: Use the two-step bridge, changing the latest cumulative amount first. All loss amounts below are in thousands of dollars. A hypothetical insurer has raw latest cumulative loss of \(600\) for an accident period and an adjusted amount of \(660\). The raw selected age-to-ultimate factor is based on aggregate link-ratio data with numerator \(1{,}500+1{,}040\) and denominator \(1{,}000+800\). The adjusted selection uses numerator \(1{,}350+1{,}188\) and denominator \(900+880\). Calculate both ultimate estimates and reconcile their difference.
1. Calculate the raw selected factor. Its numerator is \(1{,}500+1{,}040=2{,}540\), and its denominator is \(1{,}000+800=1{,}800\). Therefore,
The raw ultimate estimate is \(600(1.411111\ldots)=846.6667\) thousand dollars.
2. Calculate the adjusted selected factor. Its numerator is \(1{,}350+1{,}188=2{,}538\), and its denominator is \(900+880=1{,}780\). Thus,
The adjusted ultimate estimate is \(660(1.4258426966\ldots)=941.0562\) thousand dollars.
3. Isolate the amount-adjustment effect. First apply the adjusted cumulative amount to the raw factor. The intermediate estimate is \(660(1.411111\ldots)=931.3333\) thousand dollars. Relative to the raw estimate, the amount-adjustment effect is \(931.3333-846.6667=84.6667\) thousand dollars. Equivalently, \((660-600)(1.411111\ldots)=84.6667\).
4. Isolate the factor-selection effect. At the adjusted cumulative amount, changing the factor adds \(660(1.4258426966\ldots-1.411111\ldots)=9.7228\) thousand dollars.
5. Check. The two components sum to \(84.6667+9.7228=94.3895\) thousand dollars. The direct difference is \(941.0562-846.6667=94.3895\) thousand dollars, so the bridge reconciles. Reported to the nearest \(0.01\) thousand dollars, the total increase is \(94.39\) thousand dollars.
Common mistake avoided: Using \(660F^R=931.3333\) as the adjusted ultimate would ignore the adjusted factor selection. Conversely, adding an independently calculated factor effect to an already-computed adjusted ultimate would double count it. Use either the direct adjusted ultimate or the intermediate estimate plus the factor effect, not both.
Worked Example: Opposing Effects Across Two Accident Periods
Method: Reconcile each accident period separately before aggregating. A hypothetical specialty insurer compares two accident periods, with amounts in thousands of dollars. For period A, \(C_A^R=300\), \(C_A^A=315\), \(F_A^R=1.50\), and \(F_A^A=1.45\). For period B, \(C_B^R=400\), \(C_B^A=380\), \(F_B^R=1.25\), and \(F_B^A=1.30\). Calculate each raw and adjusted ultimate and the two effects.
Period A. The raw ultimate is \(300(1.50)=450\). The adjusted ultimate is \(315(1.45)=456.75\). The amount effect is \((315-300)(1.50)=22.50\). The factor effect is \(315(1.45-1.50)=-15.75\). Their sum is \(22.50-15.75=6.75\), equal to \(456.75-450\).
Period B. The raw ultimate is \(400(1.25)=500\). The adjusted ultimate is \(380(1.30)=494\). The amount effect is \((380-400)(1.25)=-25\). The factor effect is \(380(1.30-1.25)=19\). Their sum is \(-25+19=-6\), equal to \(494-500\).
Portfolio check. The raw total is \(450+500=950\), and the adjusted total is \(456.75+494=950.75\), an increase of \(0.75\) thousand dollars. The amount effects total \(22.50-25=-2.50\), while the factor effects total \(-15.75+19=3.25\). Thus, \(-2.50+3.25=0.75\), matching the direct aggregate difference.
The small net change conceals larger, offsetting movements. Reporting only the aggregate would obscure both the increase for period A and the decrease for period B. The amount and factor effects should therefore be reviewed by accident period before they are summed.
Common mistake avoided: Do not infer that an almost unchanged portfolio ultimate means the adjustment had little effect in every period. Here, material period-level changes mostly offset.
Worked Example: When an Adjusted Ultimate Falls Despite a Higher Factor
Method: Apply the same bridge even when the adjusted factor is higher, because the adjusted cumulative amount may move in the opposite direction. For a hypothetical accident period, raw cumulative loss is \(720\), adjusted cumulative loss is \(684\), the raw selected factor is \(1.18\), and the adjusted selected factor is \(1.22\). Amounts are in thousands of dollars.
1. Raw estimate. \(U^R=720(1.18)=849.60\) thousand dollars.
2. Adjusted estimate. \(U^A=684(1.22)=834.48\) thousand dollars.
3. Amount effect. The cumulative amount falls by \(684-720=-36\). At the raw factor, the effect is \((-36)(1.18)=-42.48\) thousand dollars.
4. Factor effect. The factor rises by \(1.22-1.18=0.04\). At the adjusted cumulative amount, the effect is \(684(0.04)=27.36\) thousand dollars.
5. Check. The effects sum to \(-42.48+27.36=-15.12\) thousand dollars. The direct difference is \(834.48-849.60=-15.12\) thousand dollars. The adjusted ultimate is lower because the decrease in the cumulative amount more than offsets the increase in the selected factor.
Common mistake avoided: A higher development factor does not necessarily imply a higher ultimate. Compare the products \(C^R F^R\) and \(C^A F^A\), and reconcile the difference rather than interpreting either input alone.
Selection, Comparability, and Common Traps
Adjusted development factors are selected from adjusted development experience, which may differ from raw experience because the underlying triangle has been restated. The factor comparison is meaningful only when the selections correspond to the same development age and ultimate definition. If the raw and adjusted estimates use different valuation dates, loss bases, or treatment of claims, their difference cannot be attributed cleanly to the adjustment or factor selection.
The formula also assumes that the ultimate estimate for the accident period is represented by the latest cumulative amount multiplied by the selected age-to-ultimate factor. It is not the appropriate comparison if a different estimation method or an explicit tail adjustment is applied to only one side of the comparison. In such cases, include every changed component in the bridge, or limit the comparison to the portion that uses this multiplicative form.
Common errors include mixing a raw cumulative amount with an adjusted factor, comparing factors selected at different ages, and rounding each component too early. Keep sufficient precision through the calculations, then round the reported results at the end. When selecting a factor changes several age-to-age factors, use the resulting age-to-ultimate factors consistently in the reconciliation.
| Question | Consistent approach |
|---|---|
| Did the adjusted cumulative amount change? | Measure its effect while holding the raw factor fixed. |
| Did the selected age-to-ultimate factor change? | Measure its effect using the adjusted cumulative amount under the stated bridge convention. |
| Do the components reconcile? | Compare their sum with adjusted ultimate minus raw ultimate. |
| Did the portfolio total change only slightly? | Check accident-period results for offsetting movements. |
| Do the two estimates use comparable inputs? | Confirm matching accident period, valuation age, loss basis, and ultimate definition. |
Check Your Understanding
For each problem, use the amount-first bridge and keep the components separate until the final check.
- A hypothetical accident period has raw cumulative loss \(240\), adjusted cumulative loss \(252\), and selected factor \(1.40\) on both bases. Find both ultimates and the amount and factor effects. (Answer: Raw ultimate \(336\); adjusted ultimate \(352.80\); amount effect \(16.80\); factor effect \(0\), all in the same monetary units.)
- Raw cumulative loss is \(700\), adjusted cumulative loss is \(665\), the raw factor is \(1.20\), and the adjusted factor is \(1.24\). Find the two ultimates and reconcile the change. (Answer: Raw \(840\); adjusted \(824.60\); amount effect \(-42.00\); factor effect \(26.60\); total change \(-15.40\).)
- For a hypothetical period, raw cumulative loss is \(500\), adjusted cumulative loss is \(525\), the raw factor is \(1.32\), and the adjusted factor is \(1.28\). Calculate the estimate change and its two components. (Answer: Raw \(660\); adjusted \(672\); amount effect \(33\); factor effect \(-21\); total change \(12\).)
- For period A, raw and adjusted amounts are \(200\) and \(210\), with factors \(1.50\) and \(1.45\). For period B, the amounts are \(400\) and \(380\), with factors \(1.25\) and \(1.30\). Find the raw and adjusted portfolio totals, and aggregate each effect. (Answer: Raw total \(800\); adjusted total \(798.50\); aggregate amount effect \(-10\); aggregate factor effect \(8.50\); total change \(-1.50\).)
- A hypothetical period has raw cumulative loss \(360\), adjusted cumulative loss \(378\), and raw factor \(1{,}800/1{,}500\). The adjusted selected factor is \(1{,}900/1{,}520\). Calculate the raw and adjusted ultimates and the two bridge effects. (Answer: Raw factor \(1.20\); adjusted factor \(1.25\); raw ultimate \(432\); adjusted ultimate \(472.50\); amount effect \(21.60\); factor effect \(18.90\); total change \(40.50\).)