A Worked Paid-Loss Settlement Adjustment

Why Adjust Cumulative Paid Losses?

Cumulative paid losses can change because claim payments develop over time, but they can also be affected by how quickly claims close. If one accident year has settled a smaller share of its reported claims by a given development age, its cumulative paid amount may be lower even when the underlying claim costs are not lower. A settlement-rate adjustment puts paid losses on a selected common settlement basis so that comparisons are less affected by differences in closure speed.

The calculation uses the reported claim count for each accident year, the cumulative number of claims closed by each development age, and the corresponding cumulative paid loss. It is important to keep those quantities aligned: the settlement rate at an age is cumulative, so it must be compared with the cumulative paid loss at that same age.

Definition: For accident year $i$ and development age $t$, let $N_i$ be the number of reported claims, $C_{i,t}$ the cumulative number of those claims closed by age $t$, and $P_{i,t}$ the cumulative paid loss by age $t$. The observed cumulative settlement rate is $r_{i,t}=C_{i,t}/N_i$. Let $q_t$ be the selected standard cumulative settlement rate at age $t$. The settlement-rate-adjusted cumulative paid loss is $P^*_{i,t}=P_{i,t}(q_t/r_{i,t})$, provided $r_{i,t}>0$.

The standard pattern $q_t$ is the benchmark settlement-rate pattern selected for the adjustment. It should be defined at the same development ages as the paid triangle. The adjustment factor $q_t/r_{i,t}$ is greater than one when the observed settlement rate is below standard, equal to one when it matches standard, and less than one when it is above standard.

This procedure is an approximation. It treats cumulative paid loss as moving proportionally with the cumulative settlement rate, as though average paid amounts per settled claim and other relevant features were held constant. It does not separately model payments on open claims, differences in closed-claim severity, or changes in claim mix. If no claims have closed at an age, $r_{i,t}=0$ and the formula is undefined; a finite adjustment cannot be calculated from that cell without additional assumptions or another method.

Why the Adjustment Factor Has This Form

The factor follows from the proportionality assumption. At a fixed age, suppose cumulative paid loss is approximately proportional to the share of reported claims settled. Write that relationship as $P_{i,t}\approx k_{i,t}r_{i,t}$, where $k_{i,t}$ represents the paid-loss level associated with a unit of settlement rate. Holding that level constant while replacing the observed settlement rate with the standard rate gives the adjusted amount.

P^*_{i,t}\approx k_{i,t}q_t \quad\text{and}\quad P_{i,t}\approx k_{i,t}r_{i,t} \quad\Longrightarrow\quad P^*_{i,t}=P_{i,t}\frac{q_t}{r_{i,t}}.

Thus, the ratio is standard over observed, not observed over standard: it changes the observed paid amount by the factor needed to move from the cohort’s settlement rate to the benchmark rate. This derivation justifies the calculation under its stated assumption; it does not establish that paid loss is truly proportional to settlement rate in every portfolio.

A Consistent Calculation Process

For every available accident-year and development-age cell, use the same sequence. Counts and losses must refer to the same cohort and valuation age, and $P_{i,t}$ must be cumulative when $C_{i,t}$ is cumulative.

Calculate the observed rate.
Divide cumulative claims closed by the accident year's reported claim count: $r_{i,t}=C_{i,t}/N_i$.

Compare with the selected standard.
At the same development age, calculate $q_t/r_{i,t}$. Confirm that the two rates use the same cumulative basis.

Adjust the paid amount.
Multiply cumulative paid loss by the factor: $P^*_{i,t}=P_{i,t}(q_t/r_{i,t})$.

Rebuild and check the triangle.
Place each adjusted value in its original accident-year and development-age cell. Compare adjusted and original values and check that each calculation used the intended standard rate.

Worked Applications

Worked Example: Adjusting Two Ages for One Accident Year

A hypothetical accident year has 150 reported claims. By 12 months, 45 claims have closed and cumulative paid loss is $270,000. By 24 months, 90 claims have closed and cumulative paid loss is $600,000. The selected standard cumulative settlement rates are 40% at 12 months and 75% at 24 months. Calculate the adjusted paid losses at both ages. The method is the cumulative settlement-rate ratio adjustment.

At 12 months, observed settlement rate. The observed rate is $45/150=0.30$, or 30%. The standard rate is 0.40, so the adjustment factor is $0.40/0.30=4/3$.

At 12 months, adjusted paid loss. Multiply the original cumulative paid loss by the factor: $$270{,}000\times 4/3=$360{,}000$.

At 24 months, observed settlement rate. The observed rate is $90/150=0.60$, or 60%. The adjustment factor is $0.75/0.60=1.25$.

At 24 months, adjusted paid loss. The result is $$600{,}000\times 1.25=$750{,}000$.

The adjusted values are therefore $360,000 at 12 months and $750,000 at 24 months. Both exceed their original values because the observed settlement rates are below the selected standards at both ages. The common mistake this method avoids is multiplying by observed over standard, which would reduce the amounts despite the cohort being less settled than the benchmark.

Worked Example: Rebuilding a Small Paid Triangle

A hypothetical portfolio has the following reported and cumulative closed claim counts. Cumulative paid losses are in thousands of dollars. The selected standard cumulative settlement rates are 50% at 12 months, 80% at 24 months, and 90% at 36 months. Calculate every available adjusted paid cell.

Accident year	Reported claims	Closed at 12 months	Closed at 24 months	Closed at 36 months	Paid at 12 months	Paid at 24 months	Paid at 36 months
A	100	50	80	90	400	720	810
B	120	48	84	—	360	700	—
C	80	32	—	—	240	—	—

Accident year A. Its observed rates are $50/100=0.50$, $80/100=0.80$, and $90/100=0.90$. Each equals the selected standard at that age, so the factors are $0.50/0.50=1$, $0.80/0.80=1$, and $0.90/0.90=1$. The adjusted paid values are $400(1)=400$, $720(1)=720$, and $810(1)=810$, all in thousands.

Accident year B at 12 months. The observed rate is $48/120=0.40$. The factor is $0.50/0.40=1.25$, giving adjusted paid loss $360(1.25)=450$ thousand.

Accident year B at 24 months. The observed rate is $84/120=0.70$. The factor is $0.80/0.70=8/7$, giving $700(8/7)=800$ thousand.

Accident year C at 12 months. The observed rate is $32/80=0.40$. The factor is $0.50/0.40=1.25$, giving $240(1.25)=300$ thousand.

The adjusted triangle, in thousands, is:

\begin{array}{c|ccc} &12\text{ months}&24\text{ months}&36\text{ months}\\ \hline A&400&720&810\\ B&450&800&\text{--}\\ C&300&\text{--}&\text{--} \end{array}

The unavailable cells remain unavailable; an adjustment does not create additional observations. The calculation avoids applying a later-age standard rate to an earlier-age cell. Each adjustment uses the standard rate for that cell's development age.

Worked Example: The Incremental-Rate Trap

A hypothetical accident year has 200 reported claims. By 12 months, 60 have closed; by 24 months, 120 have closed. Cumulative paid losses are $300,000 and $540,000, respectively. Standard cumulative settlement rates are 40% and 70%. Calculate adjusted paid losses and identify the error in using the 24-month incremental closure rate.

At 12 months, the observed cumulative rate is $60/200=0.30$. The factor is $0.40/0.30=4/3$, and the adjusted paid loss is $$300{,}000(4/3)=$400{,}000$.

At 24 months, the observed cumulative rate is $120/200=0.60$. The factor is $0.70/0.60=7/6$, and the adjusted cumulative paid loss is $$540{,}000(7/6)=$630{,}000$.

A tempting but incorrect calculation uses only the closures between 12 and 24 months: $ (120-60)/200=0.30$. It then divides the 24-month standard cumulative rate by this incremental rate, obtaining $0.70/0.30$, and calculates $$540{,}000(0.70/0.30)=$1{,}260{,}000$. This is wrong because $540,000 is cumulative paid loss through 24 months, while 0.30 describes only the additional claims closed during the interval. The matching cumulative observed rate is $120/200=0.60$, so the correct result is $630,000.

The check is the basis match: cumulative paid at 24 months is adjusted using the proportion of all reported claims closed by 24 months, not the proportion newly closed since 12 months.

Interpreting the Adjusted Triangle

An adjusted paid triangle is a standardized comparison, not a record of actual cash payments. In the rebuilt example, accident year B's adjusted 12-month amount is 450 thousand rather than 360 thousand because its 40% observed settlement rate is below the 50% standard. The adjustment says what the paid amount would be under the proportionality assumption at the standard rate; it does not mean the insurer actually paid an additional 90 thousand.

When the observed rate exceeds the standard, the adjustment reduces the paid amount. When the rates agree, the cell is unchanged. The adjustment can therefore make accident-year paid amounts more comparable with respect to closure speed, while remaining sensitive to the chosen standard pattern and the assumption that paid loss scales with settlement rate.

A frequent exam trap is to treat the adjusted triangle as if it had removed every influence of claim handling. It has only standardized paid amounts using settlement rates. Differences in average paid amount per closed claim, payments on open claims, claim mix, and other features are not separately resolved by this calculation. Interpret the result as a targeted adjustment, not a complete explanation of why paid losses differ.

Key takeaway: At each accident year and development age, divide cumulative claims closed by reported claims, compare that observed cumulative settlement rate with the selected standard rate, and multiply cumulative paid loss by standard over observed. Keep the age and cumulative basis matched, and interpret the result in light of the proportionality assumption.

Check Your Understanding

For each problem, use cumulative claim counts and cumulative paid losses at the stated age.

A hypothetical accident year has 250 reported claims. At 18 months, 100 claims have closed and cumulative paid loss is $800,000. If the standard cumulative settlement rate is 50%, what is the adjusted paid loss? (Answer: $1,000,000)
A cohort has 160 reported claims. At 12 and 24 months, cumulative closures are 48 and 112, and cumulative paid losses are $240,000 and $560,000. Standard rates are 40% and 80%. Calculate both adjusted paid losses. (Answer: $320,000 at 12 months and $640,000 at 24 months)
A cohort has 90 reported claims. By 12 months, 27 have closed and cumulative paid loss is $135,000; by 24 months, 63 have closed and cumulative paid loss is $378,000. Standards are 40% at 12 months and 70% at 24 months. What are the adjusted paid amounts? (Answer: $180,000 at 12 months and $378,000 at 24 months)
A cohort has 400 reported claims, of which 240 have closed by the valuation age. Cumulative paid loss is $1,200,000, and the standard cumulative settlement rate is 70%. Calculate the adjusted amount and explain the direction of change. (Answer: $1,400,000; it increases because the observed rate is 60%, below the 70% standard)
A cohort has 300 reported claims, 180 cumulative closures, and $900,000 cumulative paid loss at a particular age. The selected standard rate is 75%. By what percentage does the adjusted amount exceed the original, and what is the adjusted amount? (Answer: 25%; $1,125,000)

What You'll Learn