Diagnosing Settlement-Speed Distortions

Why Paid Development Can Be Misleading

Paid losses often develop more slowly when claims take longer to settle, even if the underlying cost of claims has not changed. Conversely, a change in the amount paid on closed claims can raise paid losses without any change in settlement speed. A paid triangle alone may not distinguish these explanations: its movement combines claim closure, payment amounts, and payments on claims that remain open.

A useful diagnostic compares two measures at the same development age: the proportion of claims closed and the average amount paid on closed claims. This comparison helps determine whether a change in paid development is consistent with faster or slower settlement, higher or lower paid severity, or both. It is a diagnostic, not proof of causation. Comparisons should use consistently defined claim populations and should account for payments on open claims when reconciling to total paid losses.

Definition: For accident year $a$ at development age $d$, let $N_{a,d}$ be the consistently defined number of reported claims in the cohort, $C_{a,d}$ the number closed by that age, and $P^c_{a,d}$ the cumulative paid amount attributable to claims closed by that age. Define the closure rate $q_{a,d}=C_{a,d}/N_{a,d}$, and the paid severity among closed claims $s_{a,d}=P^c_{a,d}/C_{a,d}$, when $C_{a,d}>0$. The closed-claim paid amount per reported claim is $m_{a,d}=P^c_{a,d}/N_{a,d}=q_{a,d}s_{a,d}$.

The measures require careful definitions. “Closed” should mean the same thing in every cohort, and reopened claims should be treated consistently. The reported-claim denominator must also be comparable: if one cohort has substantial late reporting still to come, its measured closure rate may be low partly because the denominator is incomplete. Comparisons are most informative when cohorts have similar claim mix and are evaluated at the same maturity.

The product $q_{a,d}s_{a,d}$ is an exact identity for the closed-claim paid component as defined above. Total cumulative paid losses may also include payments on open claims, so $m_{a,d}$ is not automatically equal to total paid losses divided by reported claims. That distinction matters when using the diagnostic to explain a paid triangle.

How the Decomposition Works

Suppose two cohorts, labelled 1 and 2, are compared at the same development age. Their closed-claim paid amounts per reported claim satisfy:

m_1=q_1s_1,\qquad m_2=q_2s_2,\qquad \frac{m_2}{m_1}=\frac{q_2}{q_1}\frac{s_2}{s_1},

provided $m_1>0$. The ratio form separates the total change into a closure-rate ratio and a paid-severity ratio. If the closure-rate ratio is above 1 while the severity ratio is about 1, the pattern is consistent with more claims being settled by that age. If the closure-rate ratio is about 1 while the severity ratio is above 1, the pattern is consistent with higher paid amounts among closed claims. If both ratios differ materially from 1, both components contribute.

The identity also gives an exact additive decomposition, but the allocation of the interaction between the two components depends on the order chosen:

$$ m_2-m_1=(q_2-q_1)s_1+q_2(s_2-s_1). $$

This version first changes the closure rate while holding severity at cohort 1’s level, then changes severity at cohort 2’s closure rate. Reversing the order gives $(q_2-q_1)s_2+q_1(s_2-s_1)$, which has the same total but allocates the interaction differently. For diagnosis, reporting the component ratios is often clearer than implying that there is one uniquely correct dollar allocation.

Why the formula holds: Among $N$ reported claims, $C$ are closed. If their average paid amount is $s=P^c/C$, then the paid amount on those closed claims is $P^c=Cs$. Dividing by the full reported count gives $P^c/N=(C/N)s=qs$. Thus a rise in the closed-claim paid component per reported claim can arise from a higher fraction closed, a higher average paid amount per closed claim, or both.

Worked Comparisons

Worked Example: A Change Consistent With Faster Settlement

A hypothetical insurer compares two otherwise comparable cohorts at 12 months. Each cohort has 200 reported claims. In the reference cohort, 100 claims are closed and their closed-claim paid amount totals $1,200,000. In the newer cohort, 130 are closed and their closed-claim paid amount totals $1,560,000. Calculate the closure rate, paid severity, and closed-claim paid amount per reported claim for each cohort. Diagnose the pattern.

Method. Calculate $q=C/N$, $s=P^c/C$, and $m=qs=P^c/N$ separately for each cohort. The data specify paid amounts attributable to closed claims, so the decomposition applies directly.

Reference cohort. The closure rate is $100/200=0.50$, or 50%. Paid severity is $\$1{,}200{,}000/100=\$12{,}000$. Therefore $m=0.50(\$12{,}000)=\$6{,}000$ per reported claim. Directly, $\$1{,}200{,}000/200=\$6{,}000$.

Newer cohort. The closure rate is $130/200=0.65$, or 65%. Paid severity is $\$1{,}560{,}000/130=\$12{,}000$. Thus $m=0.65(\$12{,}000)=\$7{,}800$ per reported claim; directly, $\$1{,}560{,}000/200=\$7{,}800$.

The paid severity is unchanged, while the closure rate rises by 15 percentage points. The closed-claim paid amount per reported claim rises by $\$7{,}800-\$6{,}000=\$1{,}800$, or 30%. The ratio check is $0.65/0.50=1.30$ for closure and $12{,}000/12{,}000=1.00$ for severity, giving a total ratio of $1.30(1.00)=1.30$. This pattern is consistent with faster settlement, assuming the cohorts are otherwise comparable. The check $qs=P^c/N$ matches the direct calculation for both cohorts.

Common mistake avoided: Treating the 30% increase in this paid measure as a 30% increase in claim severity. The average paid amount per closed claim did not change; the observed increase comes from a larger proportion closed by 12 months.

Worked Example: Standardizing the Closure Rate

At a fixed development age, a reference closure rate is 55%. A newer hypothetical cohort has 200 reported claims, 140 closed claims, and $2,100,000 paid on those closed claims. Calculate its observed measures, then estimate the closed-claim paid amount per reported claim if its closure rate had remained at the reference level while its observed paid severity stayed unchanged.

Method. First calculate the newer cohort’s actual closure rate and paid severity. Then standardize by replacing its closure rate with the reference rate while holding the newer cohort’s severity fixed. This is a counterfactual diagnostic, not a claim that the reference rate would necessarily have occurred.

The observed closure rate is $140/200=0.70$, or 70%. Paid severity is $\$2{,}100{,}000/140=\$15{,}000$. The observed closed-claim paid amount per reported claim is $0.70(\$15{,}000)=\$10{,}500$, also $\$2{,}100{,}000/200$.

At the reference closure rate and the same $15,000 severity, the standardized amount per reported claim is $0.55(\$15{,}000)=\$8{,}250$. For 200 claims, this corresponds to $200(\$8{,}250)=\$1{,}650{,}000$. The difference from the observed closed-claim amount is $\$2{,}100{,}000-\$1{,}650{,}000=\$450{,}000$, or $\$10{,}500-\$8{,}250=\$2{,}250$ per reported claim.

The result isolates the effect of substituting the reference closure rate under the stated fixed-severity assumption. It does not remove a severity difference from another cohort, nor does it estimate payments on open claims. The direct check is $200(0.55)(\$15{,}000)=\$1{,}650{,}000$.

Common mistake avoided: Applying the 55% closure rate to the old cohort’s severity or dividing the standardized amount by 140. Standardization holds the newer severity fixed and uses the full 200-claim denominator.

Worked Example: Identifying Combined Effects at Two Ages

Two hypothetical cohorts each have 100 reported claims. At 12 months, the older cohort has 40 closed claims with $400,000 paid on those claims; the newer cohort has 55 closed claims with $660,000 paid. At 24 months, the corresponding figures are 70 closed and $770,000 paid for the older cohort, and 75 closed and $990,000 paid for the newer cohort. Calculate closure rate, paid severity, and $m$ at each age, then interpret the differences.

At 12 months, older cohort. $q=40/100=0.40$, $s=\$400{,}000/40=\$10{,}000$, and $m=0.40(\$10{,}000)=\$4{,}000$ per reported claim.

At 12 months, newer cohort. $q=55/100=0.55$, $s=\$660{,}000/55=\$12{,}000$, and $m=0.55(\$12{,}000)=\$6{,}600$. The closure ratio is $0.55/0.40=1.375$; the severity ratio is $12{,}000/10{,}000=1.20$. Their product is $1.375(1.20)=1.65$, matching $6{,}600/4{,}000=1.65$. Both components contribute to the 65% increase.

At 24 months, older cohort. $q=70/100=0.70$, $s=\$770{,}000/70=\$11{,}000$, and $m=0.70(\$11{,}000)=\$7{,}700$.

At 24 months, newer cohort. $q=75/100=0.75$, $s=\$990{,}000/75=\$13{,}200$, and $m=0.75(\$13{,}200)=\$9{,}900$. The closure ratio is $0.75/0.70\approx1.0714$, while the severity ratio is $13{,}200/11{,}000=1.20$. The product is approximately $1.2857$, matching $9{,}900/7{,}700\approx1.2857$.

The newer cohort’s closure advantage is larger at 12 months than at 24 months, which is consistent with a settlement-speed difference that narrows with maturity. Paid severity is higher at both ages, so the data also indicate a severity component. These observations do not establish why either measure changed. Each $m$ value has been checked against $P^c/N$.

Common mistake avoided: Attributing the entire increase in paid amounts to settlement speed because more newer-cohort claims are closed. The severity ratio is 1.20 at both ages and materially contributes to the difference.

Reading Patterns Carefully

The following interpretations are useful when the compared cohorts have the same definitions, similar maturity, and sufficiently comparable claim populations:

Observed pattern	What it is consistent with	What to check
Closure rate changes; paid severity is stable	A settlement-speed effect on the closed-claim paid component	Whether reporting completeness and closure definitions are comparable
Closure rate is stable; paid severity changes	A change in paid amounts among closed claims	Whether claim populations and payment practices are comparable
Both closure rate and paid severity change	A combination of settlement and severity effects	How much each ratio contributes to the total change
Neither measure explains the total paid-triangle movement	Other components may be important	Payments on open claims, data consistency, and the relationship to total paid losses

A higher closure rate at a given age is evidence about the proportion closed by that age, not a complete measure of claim handling or ultimate claim cost. Similarly, paid severity among closed claims is affected by which claims have closed: claims closed early need not be representative of claims still open. If claim mix differs between cohorts, the measured severity difference may reflect that difference rather than a change in the cost level for comparable claims.

For this reason, the strongest conclusion is usually conditional: the observed pattern is consistent with a settlement-speed effect, a severity effect, or both, given comparable populations and definitions. The diagnostic helps assess whether raw paid development may be distorted by settlement timing; it does not, by itself, prove that a change in settlement practice caused the pattern or determine the correct ultimate loss estimate.

Common Exam Traps

Using inconsistent denominators. A closure rate requires a clearly defined reported-claim count. Do not compare a denominator that includes late-reported claims with one that does not.
Confusing a total paid measure with closed-claim paid amounts. Payments on open claims can make total paid losses differ from $P^c$. Do not assert the identity $m=qs$ for a different numerator.
Comparing different maturities. A cohort at 12 months should not be compared directly with one at 24 months as though age were held constant.
Calling an association a cause. Higher closure rates and higher paid severity identify a pattern; they do not establish the operational or economic cause.
Ignoring the severity ratio. A change in paid losses is not automatically a settlement-speed effect. Calculate both $q_2/q_1$ and $s_2/s_1$ before interpreting the total change.

Key takeaway: At a common development age, compare closure rates and paid severity among closed claims separately. Their product explains the closed-claim paid amount per reported claim; differences in either component can affect paid development. Use the result as a diagnostic, and reconcile it carefully to total paid losses.

Check Your Understanding

For each problem, assume the stated closed-claim paid amounts use a consistent definition. Interpret the measures cautiously when the problem asks for a diagnosis.

A hypothetical cohort has 160 reported claims, 72 closed claims, and $1,080,000 paid on closed claims at 12 months. Calculate its closure rate, paid severity, and closed-claim paid amount per reported claim. (Answer: 45%; $15,000; $6,750 per reported claim.)
At a common age, a reference cohort has closure rate 50% and paid severity $8,000. A newer cohort has closure rate 60% and paid severity $9,200. Calculate each cohort’s amount per reported claim, the percentage change, and the sequential decomposition that changes closure first and severity second. (Answer: $4,000 and $5,520; a 38% increase; $800 from closure and $720 from severity, totaling $1,520.)
A new cohort has 250 reported claims, 180 closed claims, and $2,520,000 paid on closed claims. A reference closure rate is 60%. Calculate the observed amount per reported claim and the amount standardized to the reference closure rate while holding the new cohort’s paid severity fixed. (Answer: observed closure 72%, severity $14,000, and amount $10,080 per reported claim; standardized amount $8,400 per reported claim, or $2,100,000 total.)
At 18 months, an older cohort has closure rate 60% and paid severity $11,000; a newer cohort has closure rate 60% and paid severity $12,650. Calculate the paid amount per reported claim for each and identify the component that accounts for the change. (Answer: $6,600 and $7,590 per reported claim; a 15% increase attributable to the paid-severity component in this decomposition.)
At 12 months, an older cohort’s closure rate and severity are 35% and $16,000; a newer cohort’s are 50% and $16,000. At 24 months, the corresponding values are 68% and $17,000 for the older cohort, and 70% and $18,700 for the newer cohort. Calculate the newer-to-older ratios of closure rate, severity, and amount per reported claim at each age, then interpret the pattern. (Answer: at 12 months, ratios are 1.4286, 1.0000, and 1.4286; at 24 months, they are approximately 1.0294, 1.1000, and 1.1324. The early difference is closure-driven; at 24 months both components contribute, with the severity difference larger.)

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