Reconciling Raw and Adjusted Triangle Movement

Why Reconcile Raw and Adjusted Triangle Movement?

A Berquist-Sherman adjustment is easier to review when it can be traced from the observed triangle value to the adjusted value. A single adjusted incurred amount can hide two different changes: replacing case reserves that may be inadequate or excessive, and standardizing for a different proportion of claims closed by a given development age. Separating these changes makes the calculation auditable and helps explain why adjusted age-to-age movement differs from raw movement.

The bridge below uses a specified benchmark closure rate and a specified adequacy-standardized case reserve per open claim. Those selections must be supported and held consistently across the ages being compared. This is a reconciliation framework, not a rule that any particular benchmark is appropriate for every portfolio.

Definition: At development age $a$, let $N_a$ be the number of claims in the consistently defined claim population, $q_a$ the proportion closed, $P_a$ cumulative paid losses, and $C_a$ case outstanding losses. Raw incurred losses are $I_a=P_a+C_a$. Let $\bar{c}_a=C_a/[N_a(1-q_a)]$ be the average case outstanding per open claim when $q_a<1$. Let $q^*$ be the selected benchmark closure rate and $r^*$ the selected adequacy-standardized case reserve per open claim.

For the paid adjustment, define $\bar{p}_a=P_a/(N_aq_a)$ as the paid-per-closed-claim proxy used in this method. It is a ratio of aggregate paid losses to closed claim count; it need not be the literal payment on claims that are closed. The standardization assumes that this proxy is suitable for estimating paid losses at the benchmark closure rate. The case reserve adjustment assumes that $r^*$ is appropriate for open claims after the relevant mix and data considerations have been addressed.

The Reconciliation Bridge

Use the same sequence at each development age. First replace raw case reserves while holding the observed open-claim count fixed. Then standardize both paid losses and the number of open claims to the benchmark closure rate, using the same selected reserve amount per open claim. The first change isolates reserve adequacy; the second isolates the settlement-rate standardization.

Replace case reserves at the observed closure rate.
Set $C_a^A=N_a(1-q_a)r^*$. The reserve-replacement effect is $C_a^A-C_a$.

Standardize paid losses.
Set $P_a^S=N_aq^*\bar{p}_a=P_a(q^*/q_a)$, provided $q_a>0$. This applies the observed paid-per-closure proxy to the benchmark number of closed claims.

Standardize case outstanding.
Set $C_a^S=N_a(1-q^*)r^*$, the benchmark number of open claims times the standardized reserve per open claim.

Reconcile the total.
Adjusted incurred is $I_a^S=P_a^S+C_a^S$. The settlement-rate effect is $(P_a^S-P_a)+(C_a^S-C_a^A)$.

The settlement-rate effect includes two components: the paid change caused by using the benchmark closed-claim count, and the case change caused by using the benchmark open-claim count. Reporting only the paid component would not reconcile incurred losses. Keep the effect signs: a negative amount reduces the adjusted value, and a positive amount increases it.

Why the Bridge Reconciles

The raw value is $I_a=P_a+C_a$. Add and subtract the intermediate reserve-replaced value $C_a^A$, then add and subtract the raw paid amount. This separates the total difference into the two defined components:

\begin{aligned} I_a^S-I_a &=(P_a^S+C_a^S)-(P_a+C_a)\\ &=(C_a^A-C_a)+\big[(P_a^S-P_a)+(C_a^S-C_a^A)\big]. \end{aligned}

The first term is exactly the reserve-replacement effect. The bracketed terms are exactly the settlement-rate effect. Therefore, raw incurred plus both effects equals adjusted incurred; there is no unexplained residual unless data definitions or rounding differ.

For two development ages $a$ and $b$, define the adjustment balance $A_a=I_a^S-I_a$ at each age. Subtracting the raw age-to-age movement from the adjusted movement gives:

$$ I_b^S-I_a^S=(I_b-I_a)+(A_b-A_a). $$

This identity is the key movement check. The adjusted movement is not the raw movement plus the adjustment at the later age alone; it is the raw movement plus the change in the adjustment balance between ages.

Worked Reconciliations

Worked Example: Reconciling Two Ages for One Claim Cohort

A hypothetical liability portfolio has 100 claims in a consistently tracked cohort. All amounts are in thousands of dollars. At age 12, 60 claims are closed, cumulative paid losses are $600, and case outstanding is $480. At age 24, 75 claims are closed, cumulative paid losses are $825, and case outstanding is $200. Use $q^*=0.50$ and $r^*=\$15$ thousand per open claim. Reconcile incurred at each age and compare the raw and adjusted movement.

Method and assumptions. Use the age-specific paid-per-closure proxy and replace case reserves before standardizing settlement rates. The claim population is held at 100 at both ages; the selected benchmark and reserve amount are held fixed.

Age 12. The closure rate is $q_{12}=60/100=0.60$, leaving $100-60=40$ open claims. Raw incurred is $600+480=1{,}080$. The raw average case reserve is $480/40=12$, so reserve replacement gives $C_{12}^A=40(15)=600$, an increase of $600-480=120$. The paid-per-closure proxy is $600/60=10$. Benchmark paid is $P_{12}^S=100(0.50)(10)=500$. Benchmark case outstanding is $C_{12}^S=100(1-0.50)(15)=750$. Thus the settlement effect is $(500-600)+(750-600)=-100+150=50$. Adjusted incurred is $500+750=1{,}250$, also $1{,}080+120+50=1{,}250$.

Age 24. The closure rate is $75/100=0.75$, leaving 25 open claims. Raw incurred is $825+200=1{,}025$. Reserve replacement gives $C_{24}^A=25(15)=375$, an increase of $375-200=175$. The paid-per-closure proxy is $825/75=11$, so $P_{24}^S=100(0.50)(11)=550$. Benchmark case outstanding remains $750$. The settlement effect is $(550-825)+(750-375)=-275+375=100$. Adjusted incurred is $550+750=1{,}300$, also $1{,}025+175+100=1{,}300$.

Movement check. Raw movement from age 12 to 24 is $1{,}025-1{,}080=-55$. The adjustment balances are $A_{12}=1{,}250-1{,}080=170$ and $A_{24}=1{,}300-1{,}025=275$; their change is $275-170=105$. Therefore adjusted movement is $-55+105=50$, which agrees with $1{,}300-1{,}250=50$. The common mistake avoided here is adding the age-24 adjustment balance, 275, to raw movement; the correct amount to add is the change in balances, 105.

Worked Example: Paid Losses Rise While Adjusted Incurred Falls

At a development age, a hypothetical book has 100 claims, a closure rate of 0.40, paid losses of $360 thousand, and case outstanding of $840 thousand. Use a benchmark closure rate of 0.60 and a standardized reserve of $12 thousand per open claim. Find the reserve-replacement effect, settlement effect, and adjusted incurred.

Method. First hold the observed 60 open claims fixed for the reserve replacement. Then recalculate paid and case amounts at the benchmark closure rate.

Raw incurred is $360+840=1{,}200$. There are $100(1-0.40)=60$ open claims, so $C^A=60(12)=720$. The reserve-replacement effect is $720-840=-120$. The paid-per-closure proxy is $360/[100(0.40)]=360/40=9$. Benchmark paid is $P^S=100(0.60)(9)=540$. Benchmark case outstanding is $C^S=100(1-0.60)(12)=480$. The settlement effect is $(540-360)+(480-720)=180-240=-60$. Adjusted incurred is $540+480=1{,}020$, and the bridge checks: $1{,}200-120-60=1{,}020$.

Paid losses increase by 180, but adjusted incurred decreases by 180 overall because the reserve replacement and the change in benchmark open-claim count reduce case outstanding by more. A common mistake is to assume that an increase in standardized paid must increase adjusted incurred. The calculation shows why the paid and case components must both be included.

Worked Example: Reconciling Paid and Case Movement Separately

A hypothetical cohort contains 80 claims. At age 12, $q_{12}=0.50$, paid losses are $320 thousand, and case outstanding is $400 thousand. At age 24, $q_{24}=0.75$, paid losses are $540 thousand, and case outstanding is $140 thousand. Use $q^*=0.625$ and $r^*=\$9$ thousand per open claim. Reconcile each age and explain the movement.

Age 12. There are $80(0.50)=40$ closed and 40 open claims. Raw incurred is $320+400=720$. Reserve replacement is $40(9)=360$, a change of $360-400=-40$. The paid-per-closure proxy is $320/40=8$, giving standardized paid $80(0.625)(8)=400$. Standardized case outstanding is $80(1-0.625)(9)=30(9)=270$. The settlement effect is $(400-320)+(270-360)=80-90=-10$. Adjusted incurred is $400+270=670$; the bridge gives $720-40-10=670$.

Age 24. There are $80(0.75)=60$ closed and 20 open claims. Raw incurred is $540+140=680$. Reserve replacement is $20(9)=180$, a change of $180-140=40$. The paid-per-closure proxy is $540/60=9$, giving standardized paid $80(0.625)(9)=450$. Standardized case outstanding is again 270. The settlement effect is $(450-540)+(270-180)=-90+90=0$. Adjusted incurred is $450+270=720$; the bridge gives $680+40+0=720$.

Movement check. Raw incurred movement is $680-720=-40$, while adjusted movement is $720-670=50$. The adjustment balances are $670-720=-50$ at age 12 and $720-680=40$ at age 24, a change of $40-(-50)=90$. Thus $-40+90=50$, as required. The adjusted paid movement is $450-400=50$, and adjusted case movement is $270-270=0$; together they also give adjusted incurred movement of 50.

Review Checks and Limitations

A reviewer should be able to trace each age from observed amounts to the adjusted total without relying on an unexplained net factor. Present the closure rate, paid-per-closure proxy, observed open-claim count, selected $q^*$ and $r^*$, and both effect amounts. Then verify the total bridge and the age-to-age movement identity.

Keep claim definitions aligned. The count used for closure rates must match the population underlying the paid and case amounts. Changes in reporting, claim eligibility, or claim count definitions can undermine the comparison.
Do not reverse the paid ratio. If $q_a=0.40$ and $q^*=0.60$, the paid-per-closure approach multiplies raw paid by $0.60/0.40=1.5$, not $0.40/0.60$. The benchmark has more closed claims, so the standardized paid amount is higher under the stated proxy assumption.
Do not treat the paid proxy as a causal estimate. If claims closing earlier differ materially in severity or payment pattern, the paid-per-closure proxy may not represent the claims that would close under the benchmark rate. Mix changes and large losses can also distort it.
Do not force the formula when closure information is absent. If $q_a=0$, the ratio $P_a(q^*/q_a)$ is undefined. The available data do not support this paid standardization; another supported method or additional information is needed.
Preserve signs and units. A reserve replacement can be negative, and a settlement effect can offset it. Perform the reconciliation in consistent currency units and round only after calculating the component amounts.

Key takeaway: At each age, adjusted incurred equals raw incurred plus the case reserve replacement effect plus the settlement-rate effect. Across ages, adjusted movement equals raw movement plus the change in the adjustment balance—not the full adjustment at either endpoint.

Check Your Understanding

Use the bridge definitions above. Unless stated otherwise, all dollar amounts are in thousands.

A hypothetical cohort has 120 claims, $q_a=0.50$, $P_a=480$, and $C_a=360$. Use $q^*=0.75$ and $r^*=5$. Find adjusted incurred and the total adjustment from raw incurred. (Answer: adjusted incurred $=870$; total adjustment $=+30$.)
A book has 200 claims, $q_a=0.80$, $P_a=1{,}600$, and $C_a=400$. Use $q^*=0.60$ and $r^*=8$. Find the reserve-replacement effect, settlement effect, and adjusted incurred. (Answer: reserve replacement $=-80$; settlement effect $=-80$; adjusted incurred $=1{,}840$.)
Raw incurred rises from 950 at age $a$ to 1,020 at age $b$. The adjustment balances are $A_a=-40$ and $A_b=25$. Find adjusted age-to-age movement and verify it from adjusted endpoint values. (Answer: adjusted movement $=135$, from adjusted values 910 and 1,045.)
A hypothetical cohort has 50 claims, $q_a=0.40$, $P_a=200$, and $C_a=120$. Use $q^*=0.50$ and $r^*=3.5$. Calculate both bridge effects and adjusted incurred. (Answer: reserve replacement $=-15$; settlement effect $=+32.5$; adjusted incurred $=337.5$.)
At a development age, the reported closure rate is zero and cumulative paid losses are positive. Can the paid-per-closure standardization $P_a(q^*/q_a)$ be applied as written? (Answer: no; division by zero makes it undefined, so additional information or another supported method is required.)

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