The Berquist-Sherman Adjustment Problem

Why Historical Development Can Mislead

Loss development methods use observed claim emergence to estimate what losses will ultimately cost. Their interpretation depends on historical periods being reasonably comparable. If claim handling changes, however, a triangle can show a different development pattern even when the underlying claim costs have not changed. Two important sources of distortion are case reserve adequacy and claim settlement rates.

Case reserves are estimates of unpaid amounts on reported, open claims. If adjusters become more or less adequate in setting those reserves, incurred losses change even though the amounts already paid do not. Settlement rates describe how quickly claims are closed. If claims close faster or slower, paid losses can emerge at a different pace even when ultimate claim costs are unchanged. The Berquist-Sherman adjustments are designed to address these separate comparability problems: standardize incurred data for changes in case reserve adequacy and paid data for changes in settlement rates.

The central task here is to recognize the distortion and the information needed to address it. A development pattern is not automatically evidence of a change in ultimate loss cost: it may partly reflect how claims have been reserved or settled.

Definition: For reported claims at a valuation date, paid losses are cumulative amounts paid to date. Case reserves are the outstanding estimates carried for reported open claims. Incurred losses are paid losses plus case reserves. Case reserve adequacy describes how case reserves compare with the unpaid amounts ultimately expected on those claims. A settlement rate describes the proportion of claims that have been closed by a stated maturity, using a clearly specified claim cohort and denominator.

$$ I = P + C $$

In this identity, $I$ denotes incurred losses, $P$ cumulative paid losses, and $C$ case reserves at the same valuation date. It applies to a consistently defined incurred measure and does not mean that case reserves are the same as ultimate unpaid losses. In particular, reported incurred losses may exclude amounts for claims not yet reported.

Two Different Sources of Distortion

A case-reserving change directly affects $C$, and therefore $I$, through the identity $I=P+C$. If adjusters set higher case reserves for otherwise comparable open claims, incurred losses appear higher at that maturity. Lower reserves make incurred losses appear lower. Paid losses are not changed merely by changing a case estimate.

A settlement-rate change directly affects how much of the claim cost has been paid by a given maturity. Faster closure can increase paid emergence at that age; slower closure can reduce it. Paid development can therefore change even if the eventual total cost is stable. Settlement patterns can also influence which claims remain open and what case reserves are carried, so an incurred pattern may reflect more than one handling change at once.

The Berquist-Sherman framework treats these as distinct problems rather than assuming that a single adjustment to a triangle will resolve both. Incurred data require information that helps assess case reserve practices. Paid data require information that reveals settlement behavior. Comparing loss dollars alone does not identify which process changed.

Why Case Reserve Adequacy Changes Incurred Development

Suppose ultimate loss $U$ for reported claims is the amount already paid $P$ plus the amount that will eventually be paid on those claims. Let that ultimate unpaid amount be $Q$. Then $U=P+Q$, while reported incurred is $I=P+C$. Subtracting gives:

$$ U-I = Q-C $$

This is the key justification for the incurred adjustment problem. If case reserves $C$ are lower than the ultimate unpaid amount $Q$, incurred losses are below the eventual total for those claims by the difference $Q-C$. If case reserves increase toward $Q$, incurred losses rise even if paid losses and ultimate loss do not change. Thus, an observed change in incurred development can result from a change in the estimate of unpaid loss rather than a change in underlying claim cost.

The equation does not imply that the actuary knows $Q$ exactly at an immature valuation. It explains the mechanism. To compare incurred development across periods, the analyst needs a way to distinguish changes in actual claim emergence from changes in case reserve adequacy, while allowing for differences in claim mix, maturity, and other relevant conditions.

Worked Example: A Case Reserve Practice Change

Two hypothetical accident-year groups have the same exposure, claim mix, and ultimate loss of $12 million. At an earlier comparable maturity, each group had $8 million of incurred losses. At the next comparable maturity, the first group has $6 million paid and $4 million in case reserves; the second has the same $6 million paid but $6 million in case reserves. Assume the difference is solely a change in case reserve adequacy. Compare the observed incurred development and explain what it suggests.

Method. Calculate incurred losses as paid plus case reserves, then compare each later incurred amount with the common earlier amount. This isolates the effect of the stated reserve change; it does not estimate a full adjusted triangle.

For the first group, later incurred losses are $6+4=10$ million dollars. Its observed development ratio from the earlier maturity is $10/8=1.25$.

For the second group, later incurred losses are $6+6=12$ million dollars. Its observed development ratio is $12/8=1.50$.

The second ratio is higher by $1.50-1.25=0.25$, even though paid losses are identical and ultimate losses are assumed to be the same. The reason is that the second group carries $2 million more in case reserves at the later maturity. In this hypothetical, the different incurred signal reflects case reserve adequacy, not a different ultimate loss.

Check. The later incurred amounts are $10 million and $12 million, both no greater than the assumed $12 million ultimate. The first group has $2 million between incurred and ultimate; the second has none. Common mistake avoided: treating the larger incurred development ratio as proof that the second group has higher ultimate loss cost, without checking whether case reserve practices changed.

Why Settlement Rates Change Paid Development

Paid losses are actual payments made by the valuation date, but the timing of those payments depends on claim handling. A useful diagnostic is the proportion of a comparable reported claim cohort that has closed by a given maturity. That proportion is not itself a dollar development factor. It helps explain whether paid losses may have emerged at a different speed.

The link between closure and paid dollars depends on the claim circumstances. Claims can receive partial payments while open; settled claims can differ in size from claims still open; and closure definitions can vary. Consequently, claim closure counts should be interpreted alongside paid losses and claim mix, not substituted for them. A simple example can make the settlement effect clear when its assumptions are deliberately restrictive.

Worked Example: Faster Closure and Paid Emergence

Two hypothetical cohorts each contain 100 identical claims. Each claim ultimately costs $120,000. In each cohort, a claim is paid in full when it closes, receives no partial payments beforehand, and never reopens. At the same maturity, 40 claims in cohort A are closed and 55 claims in cohort B are closed. Calculate paid losses, the closed-claim proportions, and the simple ultimate-to-paid ratios. Explain why those ratios differ.

Method. Under the stated full-payment-at-closure assumption, multiply the number of closed claims by $120,000 to obtain paid losses. Divide the closed count by 100 for the closure proportion, and divide the known ultimate by paid losses for the simple ratio.

Cohort A has paid losses of $40 \times \$120{,}000=\$4{,}800{,}000$. Its closed-claim proportion is $40/100=40\%$. Its total ultimate is $100 \times \$120{,}000=\$12{,}000{,}000$, so its ultimate-to-paid ratio is $\$12{,}000{,}000/\$4{,}800{,}000=2.50$.

Cohort B has paid losses of $55 \times \$120{,}000=\$6{,}600{,}000$. Its closed-claim proportion is $55/100=55\%$. Its ultimate-to-paid ratio is $\$12{,}000{,}000/\$6{,}600{,}000=1.81818\ldots$, or 1.818 when rounded to three decimals.

Both cohorts have the same $12 million ultimate, but cohort B has more paid by this maturity because more claims have closed. Its lower simple ultimate-to-paid ratio reflects faster paid emergence under the assumptions, not a lower ultimate. Check. The remaining unpaid totals are $7.2 million in A and $5.4 million in B; adding each to paid losses gives $12 million. Common mistake avoided: interpreting a lower paid development ratio as a lower cost estimate without considering a change in settlement speed. The example’s ratio is illustrative, not a complete adjustment procedure.

Read Paid and Incurred Signals Together

Paid and incurred data can respond differently to the same handling environment. Paid losses reflect payments made; incurred losses combine those payments with case reserves. A shift toward faster settlement may increase paid emergence. At the same time, changes in reserve adequacy can raise or lower incurred losses. The effects can reinforce each other or partially offset.

The next example uses a controlled hypothetical population to show why neither triangle should be interpreted in isolation. The known ultimate is included to check the arithmetic and explain the mechanism; in practice, ultimate loss is what the analysis seeks to estimate.

Worked Example: Settlement and Reserve Changes Together

Two hypothetical cohorts each contain 100 identical claims, each with a $120,000 ultimate cost, for a total ultimate of $12 million per cohort. At a common maturity, cohort A has 40 closed claims, $4.8 million paid, and $4.2 million in case reserves. Cohort B has 55 closed claims, $6.6 million paid, and $2.7 million in case reserves. Assume every closed claim has been paid in full and the remaining claims are open. Calculate incurred losses, average case reserves per open claim, and the ultimate-to-incurred ratios. Identify the two handling differences.

Method. First calculate incurred as paid plus case reserves. Then calculate open counts as total claims less closed claims, and divide aggregate case reserves by open counts. Finally, compare the known ultimate with incurred.

For cohort A, open claims are $100-40=60$. Incurred losses are $\$4.8+\$4.2=\$9.0$ million. Average case reserves per open claim are $\$4{,}200{,}000/60=\$70{,}000$. The ultimate-to-incurred ratio is $\$12{,}000{,}000/\$9{,}000{,}000=1.333\ldots$, or 1.333 to three decimals.

For cohort B, open claims are $100-55=45$. Incurred losses are $\$6.6+\$2.7=\$9.3$ million. Average case reserves per open claim are $\$2{,}700{,}000/45=\$60{,}000$. The ultimate-to-incurred ratio is $\$12{,}000{,}000/\$9{,}300{,}000=1.29032\ldots$, or 1.290 to three decimals.

Cohort B has a higher closure proportion, $55\%$ rather than $40\%$, and higher paid losses. It also has lower average case reserves per open claim. Under the hypothetical assumptions, that combination reflects faster settlement and less case reserve carried per open claim. The incurred totals are close, but their composition differs: B has $1.8 million more paid and $1.5 million less in case reserves.

Check. Remaining ultimate unpaid is $7.2 million for A and $5.4 million for B. Case reserves fall short of those amounts by $3.0 million and $2.7 million, respectively; adding those differences to incurred gives $12 million in each cohort. Common mistake avoided: concluding that similar incurred totals mean claim handling is comparable. The paid amounts, closure proportions, and case reserves per open claim tell a different story.

Data Features Needed for Comparability

A meaningful diagnosis requires more than two aggregate loss totals. The observations must be aligned by maturity and defined consistently. Useful information includes:

Accident or claim cohort and valuation date. The analyst needs to know which claims are grouped together and the age at which each loss observation was recorded. Comparing different maturities can confuse development with handling changes.
Cumulative paid losses and case reserves. These must refer to the same cohort and valuation date so incurred losses can be reconciled as paid plus case reserves.
Claim counts and status. Reported, open, closed, and, where relevant, reopened claim counts help interpret settlement behavior and the population for which case reserves are held.
Consistent claim definitions and coverage. Changes in reporting rules, coverage, limits, or inclusion criteria can change the population being compared.
Relevant mix information. Claim severity, type, and other material differences can affect both average case reserves and closure patterns. A change in an aggregate measure is not automatically a change in practice.
Handling context. Information about changes in reserving guidelines, settlement authority, or claim operations can support or challenge a suspected explanation.

These features support the central distinction: incurred data are sensitive to case reserve adequacy, while paid data are sensitive to settlement timing. Neither sensitivity makes one basis universally preferable. It means that the actuary should understand how the observed data were produced before relying on their development.

Common Exam Traps and Practical Interpretation

Calling case reserves paid losses. Case reserves are estimates of outstanding amounts; they enter incurred losses but not cumulative paid losses.
Equating incurred with ultimate. Incurred equals paid plus case reserves, not necessarily the final cost. Its difference from ultimate depends in part on case reserve adequacy and on losses not yet included in the reported measure.
Equating a closure proportion with a paid proportion. The two may move together under restrictive assumptions, but partial payments, severity differences, and closure practices can break that relationship.
Interpreting a changed link ratio as a changed loss level. Before attributing a different development pattern to ultimate costs, check for changes in maturity, reserve practices, settlement speed, and claim mix.
Using an average per open claim without context. Aggregate case reserves divided by open claims can be a diagnostic, but it is influenced by the types and severities of claims that remain open. It is not, by itself, a definitive measure of adequacy.

The practical value of the Berquist-Sherman problem is the discipline of asking what has changed in the claim process before projecting historical development. Comparable data should reflect a consistent basis for case reserves and a consistent settlement environment, or the differences should be recognized and addressed. The adjustment goal is comparability, not changing the underlying claim experience or forcing paid and incurred views to agree.

Key takeaway: Changes in case reserve adequacy can distort incurred development; changes in settlement rates can distort paid development. Diagnose the two mechanisms separately, using aligned loss amounts, valuation ages, claim counts, and claim-status information before interpreting historical patterns.

Check Your Understanding

For each hypothetical situation, identify the relevant distortion and calculate the requested quantities.

At a comparable maturity, a cohort has $3.6 million paid and $2.4 million in case reserves. A second comparable cohort has the same paid amount and $3.0 million in case reserves. Calculate both incurred amounts and their difference. What change could explain the difference without a change in paid emergence? (Answer: $6.0 million and $6.6 million of incurred, a $0.6 million difference; a change in case reserve adequacy could explain it.)
A group of 80 identical claims, each ultimately costing $50,000, has 32 claims closed at a stated maturity. Assume full payment at closure and no partial payments. Calculate the closure proportion, paid losses, and ultimate-to-paid ratio. (Answer: 40%; $1,600,000; 2.50.)
Two cohorts each have $9 million ultimate losses. At the same maturity, cohort X has $3 million paid and $3 million in case reserves; cohort Y has $4 million paid and $2 million in case reserves. Calculate incurred for each and explain why equal incurred amounts do not imply identical handling. (Answer: $6 million each; Y has $1 million more paid and $1 million less in case reserves, so the composition differs.)
A cohort has 120 reported claims, 72 closed claims, $5.4 million paid, and $2.4 million in case reserves at a common maturity. Calculate its closure proportion, open count, incurred losses, and average case reserves per open claim. (Answer: 60%; 48 open claims; $7.8 million incurred; $50,000 per open claim.)
Two periods show the same paid losses at a given maturity, but incurred losses are higher in the later period. Claim mix and valuation age are comparable. Name the direct component that must differ in the incurred identity, and give one additional data feature needed to judge whether a reserving-practice change is plausible. (Answer: Case reserves must be higher in the later period; useful evidence includes open-claim counts or information about reserving guidelines and claim mix.)

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