Following One Cohort Through Time
A reserve balance is a valuation-date estimate of unpaid obligations, not a direct measure of how much a cohort’s estimated total cost has changed. As claims are paid, the balance will often run off. At the same time, information about the remaining claims can revise the estimate of the cohort’s total cost upward or downward. A useful run-off analysis follows the same defined cohort at successive valuation dates and records both developments separately.
For this tutorial, a cohort is a fixed group of claims, such as claims arising from one accident year. Use a consistent loss basis and scope at each valuation date. The calculations below use undiscounted amounts and assume no changes in currency basis, coverage scope, or treatment of recoveries between valuations. They concern losses only; if an analysis instead includes other liability components, those components must be included consistently throughout.
The estimated unpaid liability \(R_t\) is a snapshot: it is the estimated amount still unpaid at date \(t\). The estimated ultimate \(U_t\) is the cohort’s estimated total losses, including losses already paid and those still unpaid. Thus the two snapshots answer different questions. Comparing \(R_t\) and \(R_{t+1}\) shows how the unpaid estimate changed; comparing \(U_t\) and \(U_{t+1}\) shows how the estimated total cost changed.
A run-off report should make the distinction visible. For each valuation date, show cumulative paid losses, the unpaid estimate, and their sum. For each interval, show the payments made and the change in estimated ultimate losses. This layout makes it possible to reconcile the movement rather than infer its cause from a reserve balance alone.
The Run-Off Reconciliation
The change in the unpaid estimate is linked to two movements: payments reduce the amount still unpaid, while a revision to the estimated ultimate changes the estimated total cost. Define \(\Delta U=U_{t+1}-U_t\) and \(\Delta R=R_{t+1}-R_t\). Starting from \(R=U-C\), the bridge is:
This derivation uses only the definition \(R_t=U_t-C_t\) and the cohort payment identity \(C_{t+1}-C_t=P_{t,t+1}\). It is a bookkeeping reconciliation, not a claim-development method: it does not explain why the estimated ultimate changed, nor does it establish whether an estimate is adequate.
A negative \(\Delta R\) is a decrease in the estimated unpaid liability; a positive \(\Delta R\) is an increase. Neither sign alone identifies whether the estimated ultimate strengthened or weakened, because payments also affect the liability balance. Similarly, a decrease in the estimated ultimate is not the same thing as a payment: payments transfer amounts from unpaid to paid, while a revision changes the current estimate of total cohort losses.
Reading a Cohort Run-Off Schedule
A practical schedule can be built from one row per valuation date and one bridge for each interval. Use the same cohort definition and valuation basis in every row. The interval payment amount should reconcile the cumulative-paid figures; if it does not, the data need investigation before interpreting the reserve movement.
| Item | What it records | How to use it |
|---|---|---|
| Cumulative paid, \(C_t\) | Total cohort losses paid through valuation date \(t\) | Check that the increase equals interval payments |
| Unpaid estimate, \(R_t\) | Estimated losses still unpaid at date \(t\) | Compare snapshots, but do not interpret the change alone |
| Estimated ultimate, \(U_t\) | Cumulative paid plus estimated unpaid losses | Track revisions to the cohort’s estimated total losses |
| Interval payments, \(P_{t,t+1}\) | Losses paid between the two valuation dates | Separate cash emergence from estimate revisions |
| Ultimate revision, \(\Delta U\) | Change in estimated ultimate losses | Positive means an upward revision; negative means a downward revision |
| Reserve movement, \(\Delta R\) | Change in the unpaid estimate | Reconcile as \(\Delta U-P_{t,t+1}\) |
Worked Example: A Reserve Decrease Despite an Ultimate Strengthening
A hypothetical accident-year cohort has cumulative paid losses of $620,000 and an unpaid estimate of $480,000 at the first valuation date. During the next year, $250,000 is paid. At the second valuation date, the unpaid estimate is $310,000. Calculate the estimated ultimate at each date, the change in ultimate, and the reserve movement. State what the figures say about the cohort.
Method. Calculate each estimated ultimate as cumulative paid plus unpaid estimate. Then reconcile the change in unpaid estimate against the interval payments and ultimate revision.
At the first date:
Cumulative paid at the second date is:
The second-date estimated ultimate is:
The ultimate estimate increased by:
The reserve movement is:
Check the bridge: \(\Delta U-P_{t,t+1}=\$80{,}000-\$250{,}000=-\$170{,}000\), which agrees with the direct reserve comparison. The unpaid estimate fell by $170,000 even though the estimated ultimate increased by $80,000, because $250,000 was paid during the interval. The common mistake this avoids is calling the reserve decrease a favorable ultimate revision.
Several Valuation Dates
With more than two valuation dates, apply the same bridge to each interval, then reconcile from the first date to the last. The intermediate rows help reveal when changes occurred; the endpoint comparison checks the full-period movement. A multi-date schedule is especially useful when a cohort’s liability is reported over several financial periods.
Worked Example: Reconciling Two Successive Run-Off Intervals
A hypothetical accident-year cohort has the following valuation estimates, in thousands of dollars. The interval payments are stated separately. Calculate each estimated ultimate, each interval’s ultimate revision and reserve movement, and the total bridge from the first to the final valuation date.
| Valuation date | Cumulative paid | Unpaid estimate | Payments since prior date |
|---|---|---|---|
| December 31, 2023 | $400 | $600 | — |
| December 31, 2024 | $700 | $380 | $300 |
| December 31, 2025 | $950 | $220 | $250 |
Method. Add cumulative paid and unpaid estimates at each date; for each interval, subtract its payments from the change in estimated ultimate to obtain the reserve movement.
At December 31, 2023, the estimated ultimate is \( \$400+\$600=\$1{,}000\) thousand. At December 31, 2024, it is \( \$700+\$380=\$1{,}080\) thousand. Therefore, the first interval has an ultimate revision of \( \$1{,}080-\$1{,}000=\$80\) thousand, and a reserve movement of \( \$380-\$600=-\$220\) thousand. The bridge gives \( \$80-\$300=-\$220\) thousand.
At December 31, 2025, the estimated ultimate is \( \$950+\$220=\$1{,}170\) thousand. The second interval’s ultimate revision is \( \$1{,}170-\$1{,}080=\$90\) thousand. Its reserve movement is \( \$220-\$380=-\$160\) thousand, and the bridge gives \( \$90-\$250=-\$160\) thousand.
Across the full period, payments total \( \$300+\$250=\$550\) thousand. The estimated ultimate increased by \( \$1{,}170-\$1{,}000=\$170\) thousand, and the unpaid estimate changed by \( \$220-\$600=-\$380\) thousand. The full-period check is \( \$170-\$550=-\$380\) thousand. Each cumulative-paid figure also reconciles: \( \$400+\$300=\$700\), then \( \$700+\$250=\$950\). The cohort’s unpaid estimate ran off at both intervals, while its estimated ultimate strengthened at both. The interval-by-interval bridge prevents those two observations from being conflated.
When the Unpaid Estimate Rises
A liability estimate can increase even while claims are being paid. This occurs when the upward revision to estimated ultimate losses is greater than interval payments. Conversely, a downward ultimate revision may coexist with a smaller reserve decrease than the payments alone would suggest. The bridge quantifies these effects without implying a cause for the revisions.
Worked Example: Payments Do Not Prevent a Reserve Increase
For a hypothetical fixed cohort, the first valuation shows $1,250,000 cumulative paid and a $750,000 unpaid estimate. During the following period, $300,000 is paid. At the next valuation, cumulative paid is therefore $1,550,000 and the unpaid estimate is $950,000. Calculate the change in estimated ultimate and the reserve movement, then verify their relationship.
Method. Compare the total estimated losses at the two dates and use the run-off bridge to separate the $300,000 payment from the estimate revision.
The first estimated ultimate is \( \$1{,}250{,}000+\$750{,}000=\$2{,}000{,}000\). The second is \( \$1{,}550{,}000+\$950{,}000=\$2{,}500{,}000\). Thus \(\Delta U=\$2{,}500{,}000-\$2{,}000{,}000=\$500{,}000\).
The reserve movement is \(\Delta R=\$950{,}000-\$750{,}000=\$200{,}000\). The bridge confirms it: \(\$500{,}000-\$300{,}000=\$200{,}000\). The unpaid estimate increased by $200,000 because the $500,000 upward revision to estimated ultimate exceeded the $300,000 paid. The common mistake this avoids is assuming that any interval with payments must show a lower unpaid estimate.
Common Run-Off Errors and Uses
Run-off analysis is useful for explaining how a cohort’s reported liability changes from one valuation to another. It also gives a direct consistency check on the reported figures. But a reconciliation is not, by itself, a diagnosis of reserve adequacy or a causal explanation of changing estimates. The information shown supports statements about amounts and directions of movement; additional analysis is needed to explain why the estimates changed.
- Comparing reserve balances as if they were ultimate estimates. A lower unpaid estimate may reflect payments, a lower estimated ultimate, or both. Calculate \(U_t=C_t+R_t\) at each date.
- Subtracting payments twice. The estimated ultimate already includes cumulative paid losses. To calculate the new ultimate, add the second-date cumulative paid amount to the second-date unpaid estimate; do not subtract interval payments again.
- Mixing cohorts or bases. New accident years, different coverage scopes, or inconsistent treatment of recoveries can make a comparison cease to be a fixed-cohort bridge. Confirm that the figures refer to the same cohort and basis.
- Ignoring the cumulative-paid check. If \(C_{t+1}-C_t\) does not equal reported interval payments, investigate timing, scope, or data consistency before interpreting the reserve change.
The displayed identity assumes undiscounted amounts on a consistent basis. If liability estimates are discounted, a change in the unpaid estimate may also reflect the valuation of payment timing and the discount basis. In that situation, use a reconciliation appropriate to the discounted measure rather than applying the undiscounted bridge without adjustment.
Check Your Understanding
For each problem, follow one fixed hypothetical cohort and keep amounts on the stated basis.
- At the first valuation, a cohort has $320,000 cumulative paid losses and a $680,000 unpaid estimate. During the interval, $240,000 is paid, and the ending unpaid estimate is $510,000. Calculate the ending cumulative paid, both estimated ultimates, the ultimate revision, and the reserve movement. (Answer: Ending cumulative paid $560,000; estimated ultimates $1,000,000 and $1,070,000; ultimate strengthens $70,000; reserve decreases $170,000.)
- A cohort begins with $700,000 cumulative paid and a $900,000 unpaid estimate. Another $350,000 is paid before the next valuation, when the unpaid estimate is $620,000. Find the ending estimated ultimate and reconcile the reserve movement. (Answer: Ending estimated ultimate $1,670,000; ultimate strengthens $70,000; reserve decreases $280,000, since $70,000 − $350,000 = −$280,000.)
- At the first date, cumulative paid is $2.4 million and the unpaid estimate is $1.6 million. In the next two intervals, $0.9 million and then $0.6 million are paid. At the final date, the unpaid estimate is $0.45 million. Calculate the final cumulative paid and ultimate, then reconcile the full-period reserve movement. (Answer: Final cumulative paid $3.9 million; final estimated ultimate $4.35 million; ultimate strengthens $0.35 million; reserve decreases $1.15 million, since $0.35 million − $1.50 million = −$1.15 million.)
- A cohort has $800,000 cumulative paid and a $500,000 unpaid estimate at the opening valuation. The interval payment is $180,000; the closing cumulative-paid figure is $980,000 and the closing unpaid estimate is $440,000. Determine whether the paid figures reconcile, and calculate the ultimate revision and reserve movement. (Answer: Paid figures reconcile; estimated ultimate increases from $1,300,000 to $1,420,000, a $120,000 strengthening; reserve decreases $60,000, since $120,000 − $180,000 = −$60,000.)
- A hypothetical cohort starts with $500,000 cumulative paid and a $900,000 unpaid estimate. For the next valuation, assume $260,000 will be paid and the estimated ultimate will be $1,450,000. Calculate the projected closing cumulative paid and unpaid estimate, then verify the reserve movement. (Answer: Closing cumulative paid $760,000; closing unpaid estimate $690,000; ultimate strengthens $50,000; reserve decreases $210,000, since $50,000 − $260,000 = −$210,000.)