Why Measure Claim Settlement Rates?
In Berquist-Sherman work, the pace at which claims close matters because a paid-loss pattern can change when claim handling or settlement speed changes. Before assessing such changes, an actuary needs a clear and reproducible measure of closure by development age. The basic measure is a proportion: among claims reported for a defined cohort by a specified age, what share has closed by that age?
The key is consistency. A rate is meaningful only when its numerator and denominator refer to the same claim population, the same cohort definition, the same development age, and a stated meaning of “closed.” A ratio built from mismatched counts can look plausible while measuring no coherent quantity.
Counts, Cohorts, and Development Age
A claim cohort is a group of claims selected by a consistent origin period, such as accident year. Development age measures elapsed time from that origin to the valuation point. If the cohort is accident year 2023 and the valuation is 12 months after the end of that accident year, its development age is 12 months. Other cohort definitions can be used, but they must be applied consistently to both counts.
Let \(R_a\) be the cumulative number of claims reported for a cohort by development age \(a\), and let \(C_a\) be the number of those claims that have experienced first closure by age \(a\). Under this counting convention, \(0 \leq C_a \leq R_a\). The closure rate is:
The complement, \(1-q_a\), is the proportion of claims reported by age \(a\) that have not yet experienced first closure by that age. It is not necessarily the proportion currently open if some previously closed claims have reopened. Rates are undefined when \(R_a=0\), because there are no reported claims in the denominator.
When counts are collected consistently, both reported counts and first-closure counts are cumulative within the cohort and cannot decrease as development age increases. The rate itself should not be assumed to increase at every age: additional reports can enter the denominator before they close. If a data extract instead records the number currently closed at each valuation, that is a status snapshot, not a cumulative first-closure count. Such a ratio may fall when claims reopen and should be labelled and interpreted accordingly.
Why the Ratio Represents a Closure Proportion
The formula follows directly from the claim-level classification. At age \(a\), consider the \(R_a\) reported claims in the cohort. Under the first-closure convention, each is either in the set of claims closed at least once by age \(a\), counted by \(C_a\), or in the set not yet closed by age \(a\), counted by \(R_a-C_a\). These sets are disjoint and together contain all \(R_a\) claims.
Thus \(C_a/R_a\) is the observed fraction of the reported claims that have closed by that age, and \((R_a-C_a)/R_a\) is its complement. This is an empirical proportion for the specified cohort and age; it does not by itself predict the closure experience of another cohort.
Worked Example: Calculating Rates Across Development Ages
A hypothetical insurer groups claims by accident year. For one accident-year cohort, its cumulative count records show 80 claims reported and 32 first closures by 12 months; 100 reported and 65 first closures by 24 months; 110 reported and 88 first closures by 36 months; and 115 reported and 103 first closures by 48 months. Calculate the closure rate at each age and the not-yet-closed proportion at 36 months.
Method. At each age, divide the cumulative first-closure count by the cumulative reported count for the same cohort and age. The data satisfy \(C_a \leq R_a\) at every age.
At 12 months: \(q_{12}=32/80=0.4000\), or 40.00%.
At 24 months: \(q_{24}=65/100=0.6500\), or 65.00%.
At 36 months: \(q_{36}=88/110=0.8000\), or 80.00%. The not-yet-closed proportion is \(1-0.8000=0.2000\), or 20.00%. Equivalently, \(110-88=22\) claims are not yet closed, and \(22/110=0.2000\).
At 48 months: \(q_{48}=103/115=0.895652\ldots\), or 89.57% when rounded to two decimal places.
Check and common mistake. The counts are internally feasible: reported and first-closure counts are nondecreasing with age, and closures never exceed reports. A common mistake is to divide the 36-month closures by the 48-month reported count, obtaining \(88/115=76.52\%\). That uses a later-age denominator and no longer measures the fraction of claims reported by 36 months that had closed by 36 months. The correct 36-month rate is \(88/110=80.00\%\).
Comparing or Combining Cohorts
To compare settlement rates across cohorts, calculate each rate at the same development age and use the same claim-status convention. A 12-month rate for one cohort should not be compared directly with a 24-month rate for another as though they represented the same maturity. When a single pooled rate is required at a given age, add the relevant claim counts first, then divide. This weights each cohort according to its reported claim count.
Here \(j\) indexes the included cohorts. This pooled rate is generally not the unweighted arithmetic mean of their individual rates. The two quantities coincide only in special cases, such as when the cohorts have equal reported counts.
Worked Example: Pooling Two Cohorts at the Same Age
At 24 months, two hypothetical accident-year cohorts have the following counts. Cohort A has 80 claims reported and 56 first closures. Cohort B has 120 claims reported and 72 first closures. Calculate each cohort’s rate, the pooled rate, and the unweighted average of the two rates.
Method. First calculate each rate using its own 24-month counts. Then calculate the pooled rate from combined counts.
For cohort A, \(q_{A,24}=56/80=0.7000\), or 70.00%. For cohort B, \(q_{B,24}=72/120=0.6000\), or 60.00%.
The pooled reported count is \(80+120=200\). The pooled first-closure count is \(56+72=128\). Therefore, \(q_{24}^{\text{pooled}}=128/200=0.6400\), or 64.00%.
The unweighted average of the cohort rates is \((70.00\%+60.00\%)/2=65.00\%\). It differs from the pooled rate because the cohorts have different reported counts. Cohort B, with 120 reported claims, receives more weight in the pooled result than cohort A, with 80.
Check and common mistake. The pooled result lies between 60.00% and 70.00%, as expected, and can also be checked as \((80/200)(70.00\%)+(120/200)(60.00\%)=64.00\%\). Treating the unweighted average, 65.00%, as the pooled closure rate would give the smaller cohort the same influence as the larger one.
Claim Status and Data Checks
A closure-rate calculation depends on how claim status is recorded. A first-closure count is cumulative: once a claim has closed, it remains in that count at later ages even if it reopens. In contrast, a current-status count classifies each claim as open or closed at the valuation date. If a closed claim reopens, it moves out of the current-closed count. Neither convention is automatically wrong, but they answer different questions. The counts must not be mixed within one ratio.
If source data provide only a current-status snapshot, describe the result as a current closed-to-reported proportion at that valuation age. Do not call it a cumulative first-closure rate unless first closures can actually be identified. Conversely, if first-closure counts are available, do not subtract reopened claims from them while retaining a cumulative interpretation.
Worked Example: Handling Reopened Claims Consistently
At 30 months, a hypothetical cohort has 100 reported claims. Claim records show that 70 distinct claims have experienced a first closure by that age. Of those 70, eight later reopened and are currently open. Calculate the cumulative first-closure rate and, separately, the current closed-to-reported proportion.
Method. Keep the two status definitions separate. The cumulative first-closure numerator includes every distinct claim that has closed at least once. The current closed numerator excludes claims currently open after reopening.
The cumulative first-closure rate is \(70/100=0.7000\), or 70.00%. The number currently closed is \(70-8=62\), so the current closed-to-reported proportion is \(62/100=0.6200\), or 62.00%.
Check and common mistake. The difference is \(8/100=8.00\) percentage points, exactly the share of reported claims that closed and subsequently reopened. Using 62 as the numerator for the cumulative first-closure measure would incorrectly remove claims that did have a closure by 30 months. Using 70 as the current closed count would incorrectly classify the eight reopened claims as currently closed.
Practical Checks Before Reporting a Rate
Before using a closure rate, verify that it is calculated from compatible counts. These checks are especially useful when counts have been extracted from separate claim-system reports or assembled into a development-age table.
State the origin period and confirm that both reported and closed counts include the same claims.
Use counts measured as of the same age and valuation basis; do not combine counts from different maturities.
Decide whether closures mean first closure by age or current closed status, and ensure the numerator follows that definition.
For a first-closure measure, check \(0 \leq C_a \leq R_a\), and check that cumulative counts do not decline as age increases.
Compute \(C_a/R_a\), state the age, cohort, and closure convention, and round only after the ratio is calculated.
These checks prevent common interpretation errors: using an ultimate or later-age reported count as the denominator, averaging cohort rates without considering their sizes, or treating “ever closed” as synonymous with “currently closed.” A well-labelled rate is not just a percentage; it is a percentage for a defined population, at a defined age, under a defined status rule.
Check Your Understanding
For each problem, calculate using the cohort, age, and status convention stated.
- A hypothetical cohort has 75 claims reported and 30 first closures at 18 months, then 100 reported and 70 first closures at 30 months. Calculate both cumulative closure rates and the not-yet-closed proportion at 30 months. (Answer: 40.00%, 70.00%, and 30.00%, respectively.)
- At 12 months, cohort A has 50 reported and 20 first closures; cohort B has 90 reported and 45 first closures. Calculate the pooled closure rate. (Answer: \(65/140=46.43\%\).)
- At 24 months, cohort A has 80 reported and 56 first closures; cohort B has 120 reported and 72 first closures. Find the pooled rate and compare it with the unweighted mean of the two cohort rates. (Answer: pooled rate \(=128/200=64.00\%\); unweighted mean \(=65.00\%\).)
- A cohort has 120 reported claims and 84 distinct claims that have experienced first closure by 24 months. Nine of those claims subsequently reopened and are currently open. Find the cumulative first-closure rate and the current closed-to-reported proportion. (Answer: 70.00% and 62.50%, respectively.)
- A cohort’s cumulative counts are: at 12 months, 40 reported and 10 first closures; at 24 months, 55 reported and 33 first closures; at 36 months, 65 reported and 52 first closures. Calculate the three age-specific closure rates. Explain which denominator belongs in the 24-month calculation. (Answer: 25.00%, 60.00%, and 80.00%; use the 24-month reported count of 55.)