Purpose of an Incurred-Loss Adjustment
Incurred losses combine cumulative paid losses with case reserves for claims that are not fully settled. If case reserve practices change over time, two incurred-loss observations at the same development age may not be comparable: one may reflect more generous or more conservative case estimates than the other. A Berquist-Sherman incurred-loss adjustment puts observed case reserves onto a selected, consistent adequacy basis before forming adjusted incurred losses.
The calculation is cell-specific. For each accident-period and development-age cell, retain the cumulative paid loss, adjust the observed case reserve using the selected ratio for that development age, and add the two amounts. The selected ratio is not a loss development factor and does not adjust paid losses.
Under this convention, \(q_a>1\) means observed case reserves are higher than the standard-basis reserves, so the adjustment reduces them. Conversely, \(0<q_a<1\) means observed reserves are lower than the standard-basis amount, so the adjustment increases them. Because other sources may use the reciprocal convention, always identify what the ratio means before deciding whether to multiply or divide.
Why the Adjustment Formula Holds
The adjustment rests on a proportional model for case reserves at a given development age. The model says that the observed reserve is the selected adequacy ratio times the reserve on the desired standard basis. It does not say that paid losses are subject to the same practice change.
The ordinary incurred amount is the sum of paid loss and case reserve. Replacing only the case reserve with its standard-basis amount gives the adjusted value:
This follows directly from the definition of incurred loss; the adjustment does not alter what has already been paid. The model assumes that the selected \(q_a\) applies to the relevant claims in that development-age cell, is positive, and is expressed using the observed-to-standard direction defined above. It also assumes the adjustment is intended to address case reserve adequacy rather than a separate change in claim mix, loss inflation, or payment timing. If those effects are present, this calculation alone does not remove them.
Cell-by-Cell Calculation
Use the selected ratio associated with each cell's development age, not its accident period. The following sequence makes both the calculation and its audit trail clear.
Record cumulative paid loss \(P_{i,a}\) and observed case reserves \(C_{i,a}\) for the accident period and age.
Use the selected \(q_a\) for that development age, and confirm it is observed reserves divided by standard-basis reserves.
Calculate \(C^*_{i,a}=C_{i,a}/q_a\).
Calculate \(I^*_{i,a}=P_{i,a}+C^*_{i,a}\), then compare it with \(I_{i,a}=P_{i,a}+C_{i,a}\).
Worked Example: Adjusting a Small Incurred Triangle
A hypothetical insurer has cumulative paid losses and observed case reserves, in thousands of dollars, for three accident periods. The selected observed-to-standard adequacy ratios are \(q_{12}=1.25\) at 12 months and \(q_{24}=0.80\) at 24 months. Calculate observed and adjusted incurred loss for every available cell.
| Accident period | Age (months) | Paid \(P\) | Observed case reserve \(C\) |
|---|---|---|---|
| 2023 | 12 | 300 | 240 |
| 2023 | 24 | 520 | 180 |
| 2024 | 12 | 360 | 300 |
| 2025 | 12 | 280 | 200 |
Method. For each cell, calculate observed incurred as paid plus observed case reserves, then divide the case reserve by the selected ratio for that age and add the result to paid loss.
2023 at 12 months. Observed incurred is \(300+240=540\). The adjusted case reserve is \(240/1.25=192\), so adjusted incurred is \(300+192=492\). The adjustment reduces incurred by \(540-492=48\).
2023 at 24 months. Observed incurred is \(520+180=700\). The adjusted case reserve is \(180/0.80=225\), so adjusted incurred is \(520+225=745\). The adjustment increases incurred by \(745-700=45\).
2024 at 12 months. Observed incurred is \(360+300=660\). The adjusted case reserve is \(300/1.25=240\), so adjusted incurred is \(360+240=600\). The adjustment reduces incurred by \(660-600=60\).
2025 at 12 months. Observed incurred is \(280+200=480\). The adjusted case reserve is \(200/1.25=160\), so adjusted incurred is \(280+160=440\). The adjustment reduces incurred by \(480-440=40\).
Check. Paid losses are unchanged in all four cells. The 12-month adjusted reserves are each 80% of their observed amounts, because \(1/1.25=0.80\). The 24-month adjusted reserve is higher than observed because \(1/0.80=1.25\). The calculations are consistent with the ratio directions.
Common mistake avoided. Multiplying the 12-month reserve of 240 by 1.25 would produce 300 and adjusted incurred of 600. That reverses the stated ratio convention: 1.25 means the observed reserve is 125% of the standard amount, so the correct standard reserve is \(240/1.25=192\), not \(240(1.25)\).
Interpreting Each Accident Period
An adjustment should be reported by accident period and age, not only as a grand total. The amount of change in a cell is
This identity shows that the adjustment is driven by the observed case reserve and the selected ratio. If \(q_a=1\), the cell is unchanged. If \(q_a>1\), the difference is negative; if \(0<q_a<1\), it is positive. The paid amount cancels because it appears in both observed and adjusted incurred.
For an accident period with more than one observed age, calculate each age separately. Do not apply a single average adequacy ratio to its total case reserve unless that average has been specifically selected and justified for that purpose. Different ages can have different ratios, and a combined ratio can conceal increases in some cells and decreases in others.
Worked Example: Reading the Direction and Size of Changes
For a hypothetical marine equipment portfolio, the selected ratios are \(q_{12}=1.20\) and \(q_{24}=0.75\). Accident period X has paid loss of 450 and case reserves of 360 at 12 months, and paid loss of 700 and case reserves of 210 at 24 months. Amounts are in thousands of dollars. Calculate both observed and adjusted incurred values and explain the direction of each change.
At 12 months. Observed incurred is \(450+360=810\). The adjusted case reserve is \(360/1.20=300\), so adjusted incurred is \(450+300=750\). The change is \(750-810=-60\), a reduction of 60.
At 24 months. Observed incurred is \(700+210=910\). The adjusted case reserve is \(210/0.75=280\), so adjusted incurred is \(700+280=980\). The change is \(980-910=70\), an increase of 70.
Compare the cells. The 12-month ratio exceeds one, so observed reserves are above standard and the adjustment lowers incurred. The 24-month ratio is below one, so observed reserves are below standard and the adjustment raises incurred.
Check. The 12-month difference formula gives \(360(1/1.20-1)=360(0.833333\ldots-1)=-60\). The 24-month formula gives \(210(1/0.75-1)=210(1.333333\ldots-1)=70\). Both agree with the direct calculations.
Common mistake avoided. Applying the 12-month ratio to both ages would ignore the selected age-specific pattern. At 24 months, the selected ratio is 0.75, so the correct adjusted reserve is 280, not \(210/1.20=175\).
Period Comparisons and Totals
A useful comparison table includes the accident period, development age, observed incurred, adjusted incurred, and the change. This keeps the adjustment transparent and makes it easier to see whether a period's change is caused by a high or low selected ratio. In an incomplete triangle, compare only cells that are available; do not fill missing cells by treating them as zero.
Summing adjusted and unadjusted cells can be a useful arithmetic check, but a sum across different development ages is not itself a like-for-like accident-period estimate. For example, one accident period may contribute a 24-month cell while another contributes only a 12-month cell. Those amounts represent different points in development. The adjustment itself does not complete the triangle or select a development factor.
Worked Example: Reconciling Cell Changes to a Total
A hypothetical property insurer reviews four observed cells, all in thousands of dollars. The selected ratios are \(q_{12}=1.25\) and \(q_{24}=0.80\). Calculate the unadjusted and adjusted incurred amounts, the change in each cell, and the total change across the listed cells.
| Accident period | Age | Paid | Observed case reserve |
|---|---|---|---|
| A | 12 | 400 | 300 |
| A | 24 | 650 | 200 |
| B | 12 | 500 | 240 |
| C | 12 | 350 | 180 |
Accident period A at 12 months. Observed incurred is \(400+300=700\). Adjusted reserve is \(300/1.25=240\), so adjusted incurred is \(400+240=640\). The change is \(-60\).
Accident period A at 24 months. Observed incurred is \(650+200=850\). Adjusted reserve is \(200/0.80=250\), so adjusted incurred is \(650+250=900\). The change is \(+50\).
Accident period B at 12 months. Observed incurred is \(500+240=740\). Adjusted reserve is \(240/1.25=192\), so adjusted incurred is \(500+192=692\). The change is \(-48\).
Accident period C at 12 months. Observed incurred is \(350+180=530\). Adjusted reserve is \(180/1.25=144\), so adjusted incurred is \(350+144=494\). The change is \(-36\).
Total check. Unadjusted incurred across the four listed cells is \(700+850+740+530=2{,}820\). Adjusted incurred is \(640+900+692+494=2{,}726\). The total change is \(2{,}726-2{,}820=-94\). The cell changes reconcile: \(-60+50-48-36=-94\).
Common mistake avoided. The total change is not found by applying one ratio to all four reserves. Three cells use the 12-month ratio and one uses the 24-month ratio. Also, this cross-cell total is only a reconciliation of the listed observations; it does not compare accident periods at a common maturity.
Assumptions, Checks, and Exam Traps
The adjustment is most informative when its basis is explicit. Before calculating, confirm that the selected ratios have the same direction as the convention used here and correspond to the cell ages. Confirm also that the input case reserves and paid losses are cumulative values measured at the relevant valuation date. A ratio of zero cannot be used because division by zero is undefined; an implausibly small positive ratio can create an unusually large adjusted reserve and warrants review.
- Do not adjust paid losses. The formula changes case reserves only; cumulative paid loss is carried through unchanged.
- Do not reverse the ratio. With \(q=\text{observed}/\text{standard}\), divide observed case reserves by \(q\).
- Do not confuse age with accident period. The selected ratio is keyed to development age.
- Do not mistake an adjustment for an ultimate estimate. It standardizes the incurred observation; further reserving methods are separate.
- Check the sign. A ratio above one must lower the case reserve, while a positive ratio below one must raise it.
The calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus because it requires candidates to connect cumulative paid, case reserves, and age-specific assumptions without reversing the selected basis. A sound answer shows the ratio definition, each adjusted reserve, the resulting incurred value, and a reconciliation of the change.
Check Your Understanding
For each problem, use the convention that the selected ratio is observed case reserves divided by standard-basis case reserves. Amounts are in thousands of dollars unless stated otherwise.
- At 12 months, paid loss is 420, observed case reserves are 250, and \(q_{12}=1.25\). Find observed and adjusted incurred loss. (Answer: observed incurred \(=670\); adjusted incurred \(=620\).)
- At 24 months, paid loss is 600, observed case reserves are 160, and \(q_{24}=0.80\). Find the adjusted case reserve and adjusted incurred. (Answer: adjusted case reserve \(=200\); adjusted incurred \(=800\).)
- For one accident period, the 12-month cell has paid loss 300, case reserves 220, and \(q_{12}=1.10\); the 24-month cell has paid loss 500, case reserves 180, and \(q_{24}=0.90\). Find observed and adjusted incurred in each cell. (Answer: 12 months, observed \(=520\), adjusted \(=500\); 24 months, observed \(=680\), adjusted \(=700\).)
- Two cells have paid/case reserve pairs of \(400/250\) at 12 months and \(700/150\) at 24 months. The ratios are \(q_{12}=1.25\) and \(q_{24}=0.75\). Compare the sum of observed incurred with the sum of adjusted incurred. (Answer: both sums equal \(1{,}500\); the first cell decreases by 50 and the second increases by 50.)
- A 36-month cell has paid loss 480, observed case reserves 315, and a selected ratio \(q_{36}=0.90\). The ratio is defined as observed reserves divided by standard-basis reserves. Find the implied standard-basis reserve and adjusted incurred. (Answer: standard-basis reserve \(=350\); adjusted incurred \(=830\).)