Why Settlement-Rate Shifts Matter
Paid losses accumulate as claims are settled and payments are made. If claims close more slowly in one accident period than another, paid-loss development can differ even when the underlying claim costs have not changed. A change in the observed closure pattern is therefore important to identify before interpreting paid development. It is a diagnostic signal, however, not proof by itself that a claims-handling process changed: reporting delays, claim mix, and data conventions can also change the observed pattern.
The central comparison is between claims at the same development age. Development age is the elapsed time from the start of an accident period to a valuation date, commonly measured in months or years. Comparing one period at 12 months with another at 24 months would confound settlement speed with the amount of time available for claims to close.
In practice, an actuary may also examine a conditional closure rate by development interval. If \(N_j\) claims are open at the start of interval \(j\), and \(D_j\) of those claims close during that interval, the rate is \(h_j=D_j/N_j\). For a fixed cohort without reopened claims, the cumulative closure proportion through interval \(k\) is \(C_k\), and it is related to the conditional rates by:
The cumulative rate answers “what proportion has closed by this age?” The conditional rate answers “of those still open, what proportion closed in this interval?” These are different questions and should not be compared as if they were the same measure.
Reading Comparable Closure Patterns
A useful comparison holds the measurement basis steady. Use the same definition of a claim, the same treatment of reopened claims, the same cohort basis, and the same development ages. Confirm whether the denominator is claims reported by each valuation date or a fixed cohort count. If claims can be reported late, a denominator that grows with valuation age can make the cumulative closure rate change for two reasons: more claims have entered the observed population, and more claims have closed.
A practical diagnostic starts with a table of reported, open, and closed claim counts by accident period and valuation age. From those counts, calculate cumulative closure rates and, where the data support it, conditional rates by interval. Look for differences in level, shape, and timing. A lower rate at early ages followed by convergence at later ages may indicate slower early closure rather than a permanent difference in ultimate closure. A persistent gap across ages may suggest a more sustained difference, but still requires investigation.
The comparison should also account for calendar time. A claims-handling change may affect several accident periods at once as they pass through the same calendar period. Conversely, an accident-period comparison can mix development-age effects with different calendar environments. Plotting or tabulating several cohorts at common ages helps reveal a pattern; reviewing the calendar dates at which those ages occurred helps assess plausible explanations.
Worked Example: Comparing Closure Speed at Common Ages
A hypothetical insurer compares two accident-period cohorts, each containing 200 reported claims. Assume reporting is complete for both cohorts, claim definitions are unchanged, and no closed claims reopen. The older cohort has 80 claims closed by 12 months, 140 by 24 months, and 170 by 36 months. The newer cohort has 110, 160, and 180 closed at those respective ages. Calculate cumulative and conditional closure rates, then describe the pattern.
Method. First divide cumulative closed counts by 200. Then calculate each interval’s closures and divide by the claims open at the interval’s start.
For the older cohort, cumulative closure rates are \(80/200=40\%\) at 12 months, \(140/200=70\%\) at 24 months, and \(170/200=85\%\) at 36 months. For the newer cohort, they are \(110/200=55\%\), \(160/200=80\%\), and \(180/200=90\%\).
For the older cohort, 120 claims remain open at 12 months. During months 12–24, 60 close, so the conditional rate is \(60/120=50\%\). At 24 months, 60 remain open; 30 close during months 24–36, giving \(30/60=50\%\).
For the newer cohort, 90 claims remain open at 12 months. During months 12–24, 50 close, giving \(50/90=55.56\%\). At 24 months, 40 remain open; 20 close during months 24–36, giving \(20/40=50\%\).
The newer cohort has higher cumulative closure rates at all three ages, with the largest difference at 12 months: 55% versus 40%, a 15 percentage-point gap. The interval rates show that the cohorts’ later closure patterns are much closer: 55.56% versus 50% in months 12–24 and 50% versus 50% in months 24–36. This is evidence of faster early closure in the newer cohort, subject to the stated comparability assumptions. It does not, by itself, identify the cause.
Check. The older cohort’s conditional rates imply \(1-(1-0.50)(1-0.50)=75\%\) closed by 36 months only if the first two intervals were the only intervals. Here there are three intervals, and the first rate is \(80/200=40\%\); the full calculation is \(1-(1-0.40)(1-0.50)(1-0.50)=85\%\), matching \(170/200\). The calculation confirms the rates are consistent.
Common mistake avoided: comparing the newer cohort at 24 months with the older cohort at 36 months would make the newer cohort appear slower (80% versus 85%), but the ages are unequal. Compare like with like.
Why Conditional Rates Help
The relationship between cumulative and conditional closure rates follows from tracking the probability that a claim remains open. Let \(h_j\) be the fraction of claims open at the start of interval \(j\) that close during it. Then the fraction that stays open through that interval is \(1-h_j\). A claim remains open through all \(k\) intervals only if it stays open in each one, so the fraction still open is the product of those interval-specific fractions:
Subtracting both sides from 1 gives \(C_k=1-\prod_{j=1}^{k}(1-h_j)\). This derivation assumes a consistent cohort and status definition, and that the closure count in each interval is drawn from claims open at its start. If claims reopen and are counted as closed again, or if the cohort denominator changes because of late reporting, the simple relationship may not describe the observed table without adjustments to the definitions.
Conditional rates are useful because they separate the amount of closure in an interval from the number of claims already closed. But they can become volatile when few claims remain open. Always show the underlying counts alongside a percentage; a rate based on 10 open claims is less stable than one based on 1,000.
Separate Settlement Speed from Claim Mix
A cohort’s aggregate closure rate can change even when closure behavior within each claim segment is unchanged. For example, a cohort with a larger share of complex claims may close more slowly than one dominated by straightforward claims. Compare patterns within relevant, consistently defined segments—such as injury type, claim size band, or coverage—before attributing an aggregate shift to handling speed. Segmentation is a diagnostic; small segment counts and changes in classification practices can themselves make comparisons unreliable.
Worked Example: An Aggregate Shift Caused by Claim Mix
Two hypothetical cohorts each contain 100 claims. Each has simple and complex claims. At 12 months, 80% of simple claims and 20% of complex claims have closed in both cohorts. The older cohort contains 80 simple and 20 complex claims; the newer cohort contains 20 simple and 80 complex claims. Calculate each aggregate closure rate and assess what the comparison does and does not show.
Older cohort. Simple claims closed: \(80\times0.80=64\). Complex claims closed: \(20\times0.20=4\). Total closed: \(64+4=68\). The aggregate closure rate is \(68/100=68\%\).
Newer cohort. Simple claims closed: \(20\times0.80=16\). Complex claims closed: \(80\times0.20=16\). Total closed: \(16+16=32\). The aggregate closure rate is \(32/100=32\%\).
The aggregate rate falls by 36 percentage points, from 68% to 32%. Yet the rate within each stated claim segment is identical across cohorts: 80% for simple claims and 20% for complex claims. The arithmetic shows that the different segment proportions alone are sufficient to produce the aggregate gap in this example. It would be misleading to describe the gap as evidence of slower handling without investigating the changed claim mix.
Check. The segment counts reconcile to the cohort totals: \(64+4=68\) claims closed in the older cohort, and \(16+16=32\) in the newer cohort. The calculated percentages therefore use the stated denominator of 100 in each case.
Common mistake avoided: treating a different aggregate rate as direct evidence of a process change. First check whether the cohorts contain comparable types of claims and whether segment definitions have remained stable.
Reporting Practices Can Distort the Comparison
A closure rate based on claims reported by the valuation date is affected by when claims enter the observed cohort. If one period has more late-reported claims, many of its reported claims may have had little time to close by a given accident age. This can create an apparent settlement-speed shift even if the time from report to closure has not changed. Review claim counts reported by age and, where available, the elapsed time between report and closure.
Other reporting changes can matter as well. A new system may record a claim as closed at a different point in the administrative process; a change in claim identification may combine or split claims; or a change in the treatment of reopened claims may alter the recorded counts. These are changes in measurement or data practice, not necessarily changes in the underlying pace of settlement.
Worked Example: Detecting a Reporting-Lag Signal
At 12 months after accident, two hypothetical cohorts have the following counts. Cohort A has 80 claims reported: 70 were reported at least six months earlier, and 10 were reported within the last six months. Of the earlier-reported claims, 56 are closed; of the recent reports, 2 are closed. Cohort B has 60 claims reported: 20 were reported at least six months earlier, and 40 were reported within the last six months. Of the earlier-reported claims, 16 are closed; of the recent reports, 8 are closed. Calculate aggregate and report-age-segment closure rates.
For cohort A, \(56+2=58\) claims are closed, so the aggregate rate is \(58/80=72.5\%\). Among claims reported at least six months earlier, the rate is \(56/70=80\%\); among recent reports, it is \(2/10=20\%\).
For cohort B, \(16+8=24\) claims are closed, so the aggregate rate is \(24/60=40\%\). Among claims reported at least six months earlier, the rate is \(16/20=80\%\); among recent reports, it is \(8/40=20\%\).
The aggregate rate is 32.5 percentage points lower in cohort B, but the closure rates within the two report-age segments are identical. Cohort B has a larger share of recently reported claims: \(40/60=66.67\%\), compared with \(10/80=12.5\%\) in cohort A. The pattern is therefore consistent with reporting lag affecting the aggregate comparison. It does not prove that reporting lag is the only cause; the counts and other relevant data still need review.
Check. The segment counts reconcile: cohort A has \(70+10=80\) reported and \(56+2=58\) closed; cohort B has \(20+40=60\) reported and \(16+8=24\) closed.
Common mistake avoided: reading the 72.5% versus 40% aggregate comparison as a handling change without checking how long claims have been in the system. The equal segment rates point to a plausible composition effect from reporting timing.
A Practical Diagnostic Sequence
Use a disciplined sequence before drawing a conclusion about a settlement-rate shift:
Compare the same development ages, cohort definitions, valuation basis, claim units, and closure-status rules.
Tabulate reported, open, and closed claims; calculate cumulative rates and, when useful, interval conditional rates. Note small denominators.
Check reporting emergence, time since report, claim mix, reopening practices, and data-system or classification changes.
State the observed difference and the plausible explanations. A closure-rate pattern can flag a change; it cannot alone establish its cause.
The point of this analysis is to decide whether observed closure patterns are comparable and what questions the data raise. An apparent shift may reflect real changes in settlement speed, but it may also reflect the composition of the claims entering the measure or the way claims are recorded. Establishing which explanation is credible requires consistent data and supporting operational context.
Check Your Understanding
For each problem, use the stated denominator and keep the comparison at the specified development age.
- A cohort has 150 reported claims, of which 90 are closed by 18 months. Calculate the cumulative closure rate. (Answer: 60%)
- At the start of months 12–24, 80 claims are open; 36 close during that interval. Calculate the conditional closure rate for the interval. (Answer: 45%)
- A cohort has conditional closure rates of 40% in the first interval and 50% in the second. Assuming a fixed cohort and no reopenings, calculate the cumulative closure rate by the end of the second interval. (Answer: 70%)
- At 12 months, cohort X has 70 closed claims out of 100 reported, while cohort Y has 84 closed out of 120 reported. Compare their cumulative closure rates. (Answer: Both are 70%; the closed counts differ, but the rates do not.)
- At 18 months, a cohort has 40 claims reported at least six months earlier, with 30 closed, and 60 recently reported claims, with 12 closed. Calculate the aggregate closure rate and the rates in both report-age segments. What diagnostic issue should be considered? (Answer: Aggregate \(42/100=42\%\); earlier-reported \(30/40=75\%\); recent \(12/60=20\%\). The larger share of recent reports may depress the aggregate rate and should be investigated.)