Interpreting the Analysis, Not Just the Arithmetic
A Berquist-Sherman analysis is most useful when it does more than produce adjusted loss figures. The analyst must connect the adjustments to the operational change that caused them, select benchmarks that make sense for the affected claims, and interpret what the adjusted development says about unpaid liabilities. A correct calculation can still support a poor conclusion if, for example, a closure-rate benchmark is not comparable or an indicated liability is treated as a point estimate without regard to the data limitations.
This tutorial uses paid loss, reported incurred loss, case outstanding, closure proportions, development factors, and tail factors. The earlier material on Berquist-Sherman adjustments establishes the adjustment methods; here the focus is on applying and explaining them under integrated, time-constrained conditions. The approach is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The two adjustments address different mechanisms. A change in case reserve adequacy affects the unpaid case reserves included in incurred loss; it does not directly change paid loss. A change in settlement speed affects how much loss has been paid by a given development age; it does not, by itself, imply that claim severity or ultimate loss has changed.
For the paid adjustment, if observed closure is faster than the benchmark, then \(p_o>p_b\) and \(p_b/p_o<1\), so adjusted paid loss decreases. If observed closure is slower, then \(p_o<p_b\) and the multiplier is greater than 1, so adjusted paid loss increases. For incurred loss, when case reserves are less adequate than the benchmark, the selected multiplier \(a\) is greater than 1 and adjusted incurred loss increases. The direction of each result should be checked against the operational diagnosis before interpreting any development factors.
A Timed Sequence for an Integrated Problem
Under exam conditions, first identify the operational change and the data it affects. Then state the benchmark and adjustment formula, calculate adjusted observations, select development factors from those adjusted observations, and finally convert an ultimate estimate to an unpaid indication on the correct basis. Keeping those steps separate helps prevent a plausible-looking result from concealing a reversed adjustment or a mismatched benchmark.
Decide whether the evidence points to reserve adequacy, settlement speed, or both. State which loss measure is directly affected.
Use a closure or adequacy basis that is comparable by development age and, where material, claim mix and handling conditions.
Normalize the affected observations, then calculate development from the adjusted data rather than mixing adjusted and raw values.
For an ultimate loss indication, unpaid loss is indicated ultimate less actual paid loss to date. Explain the method, selections, and limitations.
Worked Integrated Case
The following hypothetical insurer has experienced faster settlement at some development ages and a change in case reserve adequacy. Amounts are in thousands of dollars. The closure proportions and adequacy multipliers shown are assumed to be selected from comparable benchmark analyses; they are not inferred from the loss values alone.
| Accident year and age | Paid | Incurred | Observed closure | Benchmark closure | Adequacy multiplier |
|---|---|---|---|---|---|
| 2022, 12 months | 400 | 700 | 40% | 45% | 1.20 |
| 2022, 24 months | 640 | 800 | 64% | 60% | 1.25 |
| 2023, 12 months | 500 | 850 | 50% | 45% | 1.10 |
| 2023, 24 months | 700 | 900 | 70% | 60% | 1.00 |
| 2024, 12 months | 420 | 780 | 35% | 40% | 1.25 |
Worked Example: Adjusting the Triangle and Interpreting Its Indications
For accident year 2024 at 12 months, use the supplied closure benchmark and adequacy multiplier. Then calculate 12-to-24-month development factors from the two earlier accident years, select factors of 1.333 for adjusted paid and 1.06 for adjusted incurred, and apply tail factors of 1.05 and 1.02, respectively. Estimate ultimate and unpaid loss by each method. Current actual paid loss is 420.
Method. Normalize paid loss to the benchmark closure proportion and normalize case outstanding to the benchmark adequacy. Calculate development using adjusted historical observations. The selected development and tail factors are hypothetical assumptions for this exercise.
Step 1: Adjust paid loss for settlement speed. At 12 months in 2024, observed closure is 35%, below the 40% benchmark; settlement is slower than the benchmark. The paid adjustment factor is \(0.40/0.35=1.142857\). Therefore:
Step 2: Adjust incurred loss for reserve adequacy. Case outstanding is \(780-420=360\). Applying the adequacy multiplier of 1.25 gives adjusted case outstanding of \(360(1.25)=450\). Thus:
Step 3: Calculate adjusted historical development. For 2022, adjusted paid at 12 months is \(400(0.45/0.40)=450\), and at 24 months it is \(640(0.60/0.64)=600\). Its paid link ratio is \(600/450=1.3333\). For 2023, adjusted paid is \(500(0.45/0.50)=450\) at 12 months and \(700(0.60/0.70)=600\) at 24 months, also giving \(600/450=1.3333\). The selected paid factor of 1.333 is consistent with both adjusted observations.
For incurred loss in 2022, case outstanding is \(700-400=300\) at 12 months and \(800-640=160\) at 24 months. Adjusted incurred is \(400+300(1.20)=760\) and \(640+160(1.25)=840\), giving \(840/760=1.1053\). For 2023, adjusted incurred is \(500+350(1.10)=885\) at 12 months and \(700+200(1.00)=900\) at 24 months, giving \(900/885=1.0169\). The arithmetic mean is \((1.1053+1.0169)/2=1.0611\), which supports the supplied selected factor of 1.06.
Step 4: Project ultimate and unpaid loss. Apply each selected age-to-ultimate factor, including its tail factor, to the corresponding adjusted 12-month value:
The paid method indicates unpaid loss of \(672-420=252\). The incurred method indicates unpaid loss of \(940.644-420=520.644\). Amounts are in thousands; calculations retain unrounded values until the final presentation. The paid and incurred indications differ substantially. That difference is a result to investigate, not a reason to average automatically: consider the stability and comparability of the adjusted data, the selected factors and tail, and whether the two methods capture the operational changes consistently.
Check. The 2024 closure rate is below its benchmark, so the paid adjustment must increase 420 to 480. The adequacy multiplier exceeds 1, so adjusted incurred must exceed reported incurred: 870 is greater than 780. Both unpaid amounts subtract actual paid of 420, not adjusted paid of 480.
Why the Paid Adjustment Has This Form
The paid adjustment follows from expressing paid loss per claim exposed to closure, then applying that amount to the benchmark closure proportion. Consider a fixed population of claims at one development age. If \(P\) is cumulative paid loss and \(p_o\) is the observed proportion closed, then \(P/p_o\) is the paid amount per claim in the population implied by the observed aggregate, under the normalization model. Applying the benchmark closure proportion \(p_b\) gives the normalized paid amount:
This justification relies on more than algebra. It assumes the closure proportion is a suitable measure of settlement progress, and that the claims represented by the observed and benchmark proportions are sufficiently comparable for the normalization. It does not establish that claims close randomly, that all closed claims have identical payments, or that payment amounts are unaffected by claim mix. If those assumptions are doubtful, the adjusted result needs qualification or another method.
The adequacy adjustment has a direct accounting interpretation. Reported incurred is paid loss plus case outstanding, \(I=P+C\). If the case outstanding is judged to require multiplier \(a\) to reach the selected benchmark basis, leave paid loss unchanged and replace \(C\) with \(aC\). Therefore \(I_{\text{adj}}=P+aC\), not \(aI\). Multiplying the entire incurred amount would also alter the paid component, which is not the purpose of a case-reserve adequacy adjustment.
Worked Example: Normalizing Paid Loss for Faster Settlements
A hypothetical portfolio has cumulative paid loss of 900 at a development age when 50% of claims are closed. A comparable benchmark closure proportion is 40%. Calculate normalized paid loss and explain the direction.
Step 1: Calculate the multiplier. The observed closure proportion exceeds the benchmark, so settlement is faster than the benchmark. The multiplier is \(0.40/0.50=0.80\).
Step 2: Adjust paid loss.
The normalized paid amount is 720, which is lower than the observed 900 because it represents the lower benchmark closure proportion. This normalization does not mean that 180 has been recovered or that ultimate loss has fallen; it changes the development basis used to compare observations.
Common Exam Trap: Reversing the Multiplier
A frequent error is to multiply paid loss by observed closure divided by benchmark closure. That reverses the normalization. In the faster-settlement example, the incorrect calculation would be \(900(0.50/0.40)=1{,}125\), increasing paid loss even though the target benchmark has fewer claims closed. The correct calculation is \(900(0.40/0.50)=720\). A quick direction check catches the error: if observed closure is higher than benchmark closure, the paid adjustment factor must be below 1; if observed closure is lower, it must be above 1.
A related error is to adjust the full reported incurred amount for reserve adequacy. Suppose paid loss is 1,800, reported incurred is 2,100, and the benchmark adequacy multiplier is 1.20. Current case outstanding is \(2{,}100-1{,}800=300\); adjusted case outstanding is \(300(1.20)=360\). Adjusted incurred is \(1{,}800+360=2{,}160\). The incorrect calculation \(2{,}100(1.20)=2{,}520\) wrongly inflates the paid component as well.
Worked Example: Separating the Reserve Adjustment from the Paid Amount
For a hypothetical line, paid loss is 1,200 and reported incurred loss is 1,650. A comparable benchmark analysis indicates that current case outstanding should be multiplied by 1.08. Calculate adjusted incurred loss and identify the component that changes.
Step 1: Find case outstanding. \(C=1{,}650-1{,}200=450\).
Step 2: Adjust case outstanding. \(C_{\text{adj}}=450(1.08)=486\).
Step 3: Add unchanged paid loss.
Adjusted incurred is 1,686, an increase of 36 over reported incurred. Only case outstanding changes; paid loss remains 1,200. As a check, the increase \(486-450=36\) equals \(450(1.08-1)\).
Interpreting the Indicated Liability and Its Limits
For either method in the integrated example, the unpaid indication is calculated as indicated ultimate loss less actual cumulative paid loss. This is because the unpaid amount is the portion of ultimate loss not yet paid. The adjusted paid amount is an intermediate normalization used to project development; it is not automatically the actual paid amount to subtract from ultimate. Likewise, adjusted incurred is an intermediate estimate of incurred on a normalized basis, not a cash payment.
The adjustments should be understood as conditional estimates, not as proof that an operational change is the sole cause of observed development. Benchmark selection matters: closure proportions should reflect comparable development ages, and adequacy benchmarks should reflect comparable claim populations and reserving conditions. Changes in claim mix, large losses, coverage, policy limits, inflation, or data practices can affect the comparisons. The earlier material on data limitations and when not to apply Berquist-Sherman adjustments is relevant when these conditions are material.
A disciplined interpretation reports both the numerical indication and what it depends on. For example: “On the selected adjusted paid basis, the indicated unpaid loss is 252 thousand; the incurred-based indication is 520.644 thousand. The difference warrants review of factor stability, benchmark comparability, and tail assumptions.” This is more informative than presenting one result as certain or averaging incompatible indications without justification.
Check Your Understanding
For each problem, use the stated hypothetical assumptions and retain full precision until the final answer.
- At a selected development age, cumulative paid loss is 560, observed closure is 35%, and benchmark closure is 42%. Calculate normalized paid loss and state whether the adjustment increases or decreases the observed amount. (Answer: 672; it increases.)
- Paid loss is 1,200 and incurred loss is 1,650. The selected case-reserve adequacy multiplier is 1.08. Calculate adjusted incurred loss. (Answer: 1,686.)
- Adjusted paid loss at 12 and 24 months is 360 and 540 for one accident year, and 400 and 600 for another. Calculate each 12-to-24-month link ratio and their arithmetic average. (Answer: 1.50 for each year; average 1.50.)
- Adjusted paid loss at the latest age is 720. The selected age-to-age factor is 1.25 and the tail factor is 1.04. Actual paid loss is 650. Calculate indicated ultimate and unpaid loss. (Answer: ultimate 936; unpaid loss 286.)
- A hypothetical accident year has paid loss 480, incurred loss 780, observed closure 40%, and benchmark closure 50%. Its case-reserve adequacy multiplier is 1.20. Selected incurred development and tail factors are 1.10 and 1.02. Calculate adjusted paid and incurred, then calculate incurred-based ultimate and unpaid loss. (Answer: adjusted paid 600; adjusted incurred 840; ultimate 942.48; unpaid loss 462.48.)