Putting Adjusted Paid and Incurred Triangles on a Common Basis

Why a Common Basis Matters

Berquist-Sherman adjustments change the basis on which historical loss experience is viewed. An adjusted paid triangle may standardize the effect of settlement rates, while an adjusted incurred triangle may standardize case reserve adequacy. Those adjustments address different distortions. Before comparing the resulting triangles, the actuary must also confirm that each selected cell describes the same accident period, valuation date, development age, and claim population, with consistent definitions of the amounts included.

A cell labeled “12 months” is not fully identified by its age alone. It also belongs to a particular accident period, is measured as of a particular valuation date, and may include a particular set of coverages or claim types. If any of those differ, a difference between two cell values may reflect the mismatch rather than a change in paid or incurred development.

Definition: Let $P^{adj}_{i,j}$ denote adjusted cumulative paid loss for accident period $i$ at development age $j$, and let $I^{adj}_{i,j}$ denote adjusted cumulative incurred loss for the same accident period and age. Each cell must also be understood to have a valuation date and a defined claim and loss population. A common-basis comparison matches those dimensions and uses consistent amount conventions, such as currency, gross or net of recoveries, and loss-only or loss-plus-expense scope. Matching the basis does not make paid and incurred values equal: paid measures amounts paid to date, while incurred generally combines paid amounts with case reserves.

Identify the Coordinates Before Comparing Values

An accident period identifies the cohort to which claims are assigned, such as an accident year or accident quarter. Development age measures elapsed time from the stated origin of that cohort to the valuation date. The convention for the origin matters: a quarterly triangle might measure age from the end of the accident quarter, while an annual triangle might measure it from the end of the accident year.

For a fixed accident period and a fixed age convention, the valuation date follows from the cohort origin and elapsed age. In a simple date representation,

\text{valuation date}=\text{accident-period origin}+\text{development age}.

The equation is a dating rule, not a development model. It explains why a cell for the same cohort at 12 months cannot also be a 15-month cell as of a later valuation date. Comparisons between different valuation snapshots must use cells representing the intended common age, or explicitly account for the different ages. A common age on different accident periods still refers to different cohorts; whether that is an appropriate comparison depends on the question being asked.

Claim population is another coordinate, not a footnote. The population may be defined by coverage, claim type, policy form, geographic scope, or treatment of catastrophe claims. A paid cell excluding one category cannot be fairly paired with an incurred cell that includes it unless both are restricted to a common population or both are brought to the same intended scope using supportable data.

Identify the accident period.
Confirm that the paid and incurred cells represent the same cohort definition, not merely rows with similar labels.

Establish the valuation date and age convention.
Check the actual valuation dates and how age is measured from the accident-period origin.

Match claim and amount scope.
Reconcile coverage, claim inclusion, loss components, currency, and gross or net treatment.

Compare the adjusted measures only after matching.
Interpret the resulting paid-versus-incurred difference as a difference between measures, not as a failure of the cells to reconcile.

Worked Comparisons

Worked Example: Matching Valuation Date and Development Age

A hypothetical quarterly liability triangle measures development age from the end of each accident quarter. For accident quarter Q1 2024, adjusted paid amounts are $1.10 million at 6 months and $1.24 million at 9 months. Adjusted incurred amounts are $1.22 million at 6 months and $1.48 million at 9 months. The 6-month cells are valued at September 30, 2024, and the 9-month cells at December 31, 2024. An analyst proposes comparing the 9-month paid value with the 6-month incurred value because both are adjusted. Identify a valid same-age comparison and explain the error.

Method. Match the accident period first, then select the same development age. The accident-period origin is March 31, 2024. Six elapsed months leads to September 30, 2024; nine elapsed months leads to December 31, 2024.

At 6 months, the matched comparison is adjusted paid of $1.10 million against adjusted incurred of $1.22 million. The difference is $1.22 million − $1.10 million = $0.12 million, or $120,000.

At 9 months, the other matched comparison is adjusted paid of $1.24 million against adjusted incurred of $1.48 million. The difference is $1.48 million − $1.24 million = $0.24 million, or $240,000.

The proposed comparison, $1.24 million paid at 9 months against $1.22 million incurred at 6 months, mixes development ages and valuation dates. Its arithmetic difference is only $20,000, but that number is not a like-for-like paid-versus-incurred comparison. The correct comparison depends on the target age: pair the two 6-month cells or the two 9-month cells. Common mistake avoided: treating the word “adjusted” as a substitute for matching the cell coordinates. The values are checked against the stated cells and the subtractions above.

Worked Example: Restricting to a Common Claim Population

For a hypothetical accident year at 12 months and one valuation date, an adjusted paid triangle includes ordinary claims only, with $480,000 paid. The adjusted incurred triangle includes ordinary claims with $620,000 incurred and catastrophe claims with $180,000 incurred. The actuary wants to compare ordinary claims only. Calculate the valid comparison and show why comparing the displayed totals is misleading.

Method. Select the same claim population from both triangles. The paid cell already represents ordinary claims, so its common-population value is $480,000. The incurred cell includes an additional catastrophe component; restrict it by removing that component:

I^{adj}_{\text{ordinary}}=\$620{,}000,\qquad I^{adj}_{\text{all}}=\$620{,}000+\$180{,}000=\$800{,}000.

The ordinary-claims comparison is therefore $480,000 paid against $620,000 incurred, a difference of $620,000 − $480,000 = $140,000. If the intended population were all claims, the paid triangle would instead need its catastrophe paid component. Suppose claim-level data show that amount to be $120,000. Then all-claims adjusted paid is $480,000 + $120,000 = $600,000, and the all-claims incurred value is $800,000; the difference is $200,000.

Comparing $480,000 paid with $800,000 incurred gives a $320,000 difference, but that comparison mixes an ordinary-only paid population with an all-claims incurred population. It is not the difference for either defined population. Common mistake avoided: assuming a missing population component can be repaired by scaling the included amount without evidence. Here the valid values come from the stated component data, not an assumed proportional adjustment. The ordinary-only and all-claims totals each add back to their stated components.

Worked Example: Converting Gross Amounts to a Net Basis

At the same accident period, 18-month age, and valuation date, a hypothetical adjusted paid cell is $735,000 gross of recoveries. Recoveries associated with paid amounts are $35,000. The adjusted incurred cell is $910,000 gross, and its corresponding recognized recoverables are $60,000. Calculate the comparison on a net basis and identify the result of mixing bases.

Method. Subtract the corresponding recoveries from each gross cell. Net adjusted paid is $735,000 − $35,000 = $700,000. Net adjusted incurred is $910,000 − $60,000 = $850,000. The like-for-like net comparison is therefore $700,000 paid against $850,000 incurred, a difference of $150,000.

If the actuary instead compares gross paid of $735,000 with net incurred of $850,000, the difference is $115,000. That calculation mixes gross and net measures and understates the net-basis difference by $35,000, exactly the paid recoveries omitted from the conversion. Alternatively, a gross-basis comparison would require bringing the incurred cell to its stated gross amount: $735,000 against $910,000, a difference of $175,000.

Common mistake avoided: adjusting one triangle for recoveries while leaving the other gross. The net calculation is checked by adding the recoveries back: $700,000 + $35,000 = $735,000 and $850,000 + $60,000 = $910,000.

Why the Alignment Rule Holds

A triangle cell is an observation about a cohort at a particular stage of development. Write its coordinate as $(i,j)$, where $i$ is the accident period and $j$ is development age. Given a fixed age convention, that coordinate determines the valuation date for that cohort. Consequently, comparing a paid cell $(i,j)$ with an incurred cell $(i,j)$ removes differences caused simply by using another cohort or another elapsed time. This is the reason for matching coordinates before interpreting differences.

\begin{aligned} \text{same }i &\Rightarrow \text{same accident cohort},\\ \text{same }j\text{ and age convention} &\Rightarrow \text{same elapsed development time},\\ \text{same }i,\ j\text{ and convention} &\Rightarrow \text{same valuation date}. \end{aligned}

The conclusion assumes the two triangles use the same accident-period definitions and the same age convention. If, for example, one triangle measures annual age from the beginning of the accident year and the other from year-end, identical age labels do not establish identical elapsed time. The dates must be reconciled directly. Likewise, matching $(i,j)$ does not establish a common population or amount basis; those must be checked separately.

Finally, matching the basis does not imply that adjusted paid and adjusted incurred should agree. At a given age, incurred can exceed paid because it includes outstanding case reserves. The purpose of alignment is to make the comparison interpretable, not to force the measures to coincide. When the earlier material on adjusted incurred losses or standardized paid losses is used to construct the triangles, retain the definitions and assumptions underlying each adjusted cell so the comparison is auditable.

Common Exam Trap

A frequent error is to compare cells because their age labels match, while overlooking that one triangle includes a different accident-period convention, valuation date, or claim set. Another is to subtract gross paid from net incurred and describe the result as a net difference. In either case, the arithmetic can be correct while the comparison is invalid.

Key takeaway: Before interpreting a paid-versus-incurred difference, verify four dimensions: accident period, valuation date, development age, and claim population. Also match the amount basis, including currency, recoveries, and included loss components. A common basis makes the comparison meaningful; it does not make paid and incurred amounts interchangeable or equal.

Check Your Understanding

For each problem, first identify the intended basis before comparing the adjusted values.

A hypothetical Q4 2024 annual cohort uses year-end as its age origin. At September 30, 2025, adjusted paid is $410,000 and adjusted incurred is $530,000. At December 31, 2025, the corresponding values are $455,000 and $590,000. Which pair is the valid 9-month comparison, and what is its incurred-minus-paid difference? (Answer: September 30, 2025 values, $410,000 versus $530,000; difference $120,000.)
For one cohort and valuation date, an adjusted paid cell at 18 months is $2.40 million. An incurred cell at 24 months is $3.10 million, while the incurred cell at 18 months is $2.80 million. Which incurred value should be paired with paid, and what is the difference? (Answer: $2.80 million; incurred exceeds paid by $0.40 million.)
At a common cohort and age, paid is $360,000 for ordinary claims and $90,000 for catastrophe claims. Incurred is $470,000 for ordinary claims and $140,000 for catastrophe claims. Give the ordinary-only comparison and the all-claims comparison, including each difference. (Answer: Ordinary only, $360,000 versus $470,000, difference $110,000; all claims, $450,000 versus $610,000, difference $160,000.)
At matched coordinates, paid is $1.08 million gross with $80,000 recoveries, and incurred is $1.32 million gross with $120,000 recoverables. What are the net paid and net incurred amounts, and the net difference? (Answer: $1.00 million paid, $1.20 million incurred, difference $0.20 million.)
A triangle pair has the same accident year and claim population, but the paid cell is valued at 15 months and the incurred cell at 12 months. The incurred triangle also has a 15-month cell of $760,000; the paid 15-month cell is $640,000. If the purpose is a 15-month comparison, which incurred cell applies and what is the difference? (Answer: Use the $760,000 incurred cell at 15 months; incurred exceeds paid by $120,000.)

What You'll Learn