Why an Unusual Link Ratio Needs Investigation
A link ratio summarizes the change in cumulative losses between two development ages. One unusually high or low ratio can materially affect a selected factor, but its size alone does not tell you whether it is wrong. It may reflect a data defect, a genuine large claim, a change in claim handling, or a real shift in the business being measured. The actuarial task is to trace the ratio to its underlying losses and decide whether those losses are both valid and relevant to the projection.
This tutorial focuses on investigating unusual observations, not on choosing a universal statistical cutoff. A screen can help identify ratios to review, but it cannot establish that a value is an error or determine whether it belongs in a selected factor.
The identity \(f_i=1+\Delta_i/C_{i,1}\) gives the first diagnostic. A high ratio can result from unusually large incremental emergence, a comparatively small starting cumulative amount, or both. Once the denominator is known to be sound, tracing \(\Delta_i\) is often more informative than looking at the ratio alone. For this topic, examples use deterministic cumulative loss amounts in dollars, with no discounting, interest, mortality, or stochastic loss distribution assumed.
A Practical Investigation
Begin with a signal, not a verdict. Compare an observation with the other ratios at the same development age and, where relevant, with the same accident-year, calendar-year, or operational context. A simple plot or robust summary can help prioritize review. But rows in a development triangle are not automatically independent observations, and a small set of ratios does not support treating a screening statistic as proof.
Then reconcile the triangle values to their source. Verify that the cumulative amounts are drawn from the intended valuation dates, loss basis, and accident years. Check that claims, recoveries, and adjustments were not duplicated, omitted, or assigned to the wrong period. If the data reconcile, identify which claims or transactions produced the incremental emergence. Ask whether the development came from an isolated large claim, a group of claims, a procedural change, a coverage decision, or a broader shift that could recur.
Calculate the ratio and incremental emergence. Compare them with observations at the same development age, without assuming a fixed cutoff makes a value wrong.
Trace each cumulative amount to its valuation and source records. Confirm that the loss basis and triangle placement are consistent.
Identify the claims, transactions, or operational changes that explain the increment. Separate verified events from unexplained discrepancies.
Decide whether the observation represents development expected in the future, a valid but unusual event, or an error requiring correction.
Retain, correct, or handle the observation separately only after its cause and effect on the selected factor are understood.
A useful way to quantify leverage is to compare the volume-weighted factor with and without the row under review. For positive denominators, write the factor including all rows as \(F\), and the factor excluding row \(j\) as \(F_{-j}\). If \(w_j=C_{j,1}/\sum_i C_{i,1}\), then
Since \(F_{-j}\) is the weighted average of the remaining ratios, the change caused by including row \(j\) is
This follows by writing \(F=(W_{-j}F_{-j}+C_{j,1}f_j)/(W_{-j}+C_{j,1})\), where \(W_{-j}\) is the sum of the other denominators, and subtracting \(F_{-j}\). The identity explains why an extreme ratio with little starting volume may have limited influence, while a moderately unusual ratio on a large volume can matter more. It measures influence, not whether an observation is valid.
Worked Investigations
Worked Example: A High Ratio From a Verified Large Claim
A hypothetical liability portfolio has the following cumulative incurred losses, in thousands of dollars, from 12 to 24 months. Investigate the unusual ratio for Accident Year 2022 and quantify its effect on the volume-weighted factor. Assume the triangle values are correctly recorded; no interest, mortality, or probability distribution is used.
| Accident year | 12-month cumulative | 24-month cumulative | Link ratio |
|---|---|---|---|
| 2019 | 1,000 | 1,100 | 1.10 |
| 2020 | 1,250 | 1,400 | 1.12 |
| 2021 | 900 | 999 | 1.11 |
| 2022 | 800 | 1,240 | 1.55 |
| 2023 | 1,100 | 1,199 | 1.09 |
Step 1: Calculate the unusual emergence. For 2022, the increment is \(1{,}240-800=440\) thousand dollars. The ratio is \(1{,}240/800=1.55\). The other ratios range from 1.09 to 1.12, so 1.55 is a clear investigation priority.
Step 2: Trace the increment. A hypothetical claim-level review confirms that 300 thousand dollars of the increase is a documented reserve increase on one large claim, and the remaining 140 thousand dollars is ordinary emergence. The claim is valid, not duplicated, and belongs to this accident year. The ratio is therefore genuine rather than a data error.
Step 3: Measure its impact. The total 12-month amount is \(1{,}000+1{,}250+900+800+1{,}100=5{,}050\). The total 24-month amount is \(1{,}100+1{,}400+999+1{,}240+1{,}199=5{,}938\). Thus the factor including 2022 is \(5{,}938/5{,}050=1.17584\), or 1.176 to three decimals. Excluding 2022 for comparison gives \(4{,}698/4{,}250=1.10541\), or 1.105 to three decimals.
Conclusion and mistake avoided. The verified claim explains the high ratio, but that does not automatically make 1.55 representative of future development. The actuary should consider the claim’s relevance to the projection and document any separate treatment. The common mistake is to delete a genuine observation merely because it is extreme. A deletion would remove valid experience without explaining how the associated loss is handled.
Worked Example: Correcting a Data Error Before Recalculating
A hypothetical property portfolio has 12-to-24-month cumulative paid losses, in thousands of dollars. One ratio is unusually high. Reconcile the observation and recalculate the factor after the source record is checked. Treat the listed values as deterministic; no interest, mortality, or probability distribution is assumed.
| Accident year | 12-month cumulative | 24-month cumulative as recorded |
|---|---|---|
| 2020 | 1,000 | 1,100 |
| 2021 | 1,200 | 1,320 |
| 2022 | 900 | 990 |
| 2023 | 1,100 | 1,760 |
Step 1: Flag the row. The first three ratios are \(1{,}100/1{,}000=1.10\), \(1{,}320/1{,}200=1.10\), and \(990/900=1.10\). The 2023 ratio as recorded is \(1{,}760/1{,}100=1.60\), with an increment of \(1{,}760-1{,}100=660\) thousand dollars.
Step 2: Reconcile to the source. A hypothetical transaction-level audit finds that a 550-thousand-dollar payment batch was duplicated in the 24-month cumulative amount. The corrected amount is \(1{,}760-550=1{,}210\) thousand dollars. Its corrected increment is \(1{,}210-1{,}100=110\), and its corrected ratio is \(1{,}210/1{,}100=1.10\).
Step 3: Recalculate the factor. The total 12-month amount is \(1{,}000+1{,}200+900+1{,}100=4{,}200\). The recorded 24-month total was \(1{,}100+1{,}320+990+1{,}760=5{,}170\), giving an erroneous factor of \(5{,}170/4{,}200=1.23095\), or 1.231 to three decimals. After correction, the 24-month total is \(1{,}100+1{,}320+990+1{,}210=4{,}620\), so the corrected factor is \(4{,}620/4{,}200=1.100\).
Conclusion and mistake avoided. Correct the affected cumulative amount and retain the row; do not discard the entire accident year. The common mistake is to remove the outlier row without reconciling the source. Correction preserves the valid exposure and loss experience while removing the duplicate transaction.
Worked Example: Using a Robust Screen Without Treating It as a Verdict
For a hypothetical professional-liability portfolio, five 12-to-24-month incurred link ratios are 1.08, 1.10, 1.09, 1.35, and 1.11. Assess the high value using a median-based screen, then interpret a documented cause. Assume the ratios are calculated from valid, positive cumulative amounts; no probability distribution, interest, or mortality assumption is imposed.
Step 1: Calculate the screen. The ordered ratios are 1.08, 1.09, 1.10, 1.11, and 1.35, so the median is 1.10. The absolute deviations from the median are 0.02, 0.01, 0, 0.01, and 0.25. Ordered, these are 0, 0.01, 0.01, 0.02, and 0.25, so the median absolute deviation (MAD) is 0.01. The 1.35 ratio is far from the center and merits investigation. This calculation is a descriptive screen, not a formal significance test or a claim that observations follow a particular distribution.
Step 2: Trace the high ratio. Suppose the corresponding accident year has cumulative incurred losses of 800 thousand dollars at 12 months and 1,080 thousand dollars at 24 months. The increment is \(1{,}080-800=280\) thousand dollars. Claim-level review confirms that 180 thousand dollars arose from a documented coverage decision affecting several claims in that accident year; the remaining 100 thousand dollars is other emergence.
Step 3: Assess the decision’s scope. The records confirm a valid development event, not a posting error. The 1.35 observation should not be called an error merely because it is distant from the median. If the decision is specific to this cohort and is not expected to recur, the actuary may consider a factor representing other development while separately accounting for the valid claim amounts. If similar decisions are expected in future cohorts, excluding the observation without further adjustment could understate development.
Conclusion and mistake avoided. A robust screen helps locate the unusual observation; claim and context review determine what it means. The common mistake is to turn a statistical flag into an automatic exclusion rule. There is no universal MAD threshold that can replace source validation and a reasoned assessment of future relevance.
Common Traps and Documentation
- “Far from the average” means “wrong.” An extreme ratio is a prompt to investigate, not evidence of a data defect.
- Checking only the ratio. Also calculate the incremental emergence and inspect the underlying claims. The same ratio can arise from very different dollar amounts and causes.
- Removing a row before reconciling it. Verify the source values first. If there is an error, correct the affected value and recalculate rather than discarding valid data.
- Keeping every valid event in an ordinary factor without judgment. Valid losses still need to be assessed for relevance to future development. A one-time event may warrant separate consideration, but its losses do not cease to exist.
- Using a screen as a probability model. A median or other descriptive diagnostic can prioritize review. It does not establish independence, a loss distribution, or a formal rejection rule for triangle observations.
A useful workpaper records the original ratio, the comparison used to flag it, the source records reviewed, the cause of the emergence, any correction made, and the rationale for the factor treatment. This makes the decision reproducible and distinguishes “unusual but valid” from “unexplained” or “incorrect.”
Check Your Understanding
For each problem, identify the calculation or diagnostic that matters and decide what the evidence supports.
- A hypothetical accident year has cumulative losses of 720 thousand dollars at 12 months and 864 thousand dollars at 24 months. Calculate its link ratio and incremental emergence. (Answer: Link ratio \(864/720=1.20\); incremental emergence is 144 thousand dollars.)
- Three hypothetical rows have 12-month amounts of 1,000, 1,200, and 800, and 24-month amounts of 1,100, 1,320, and 1,120, all in thousands. Calculate the all-row volume-weighted factor and the factor excluding the third row. What does the comparison show? (Answer: All-row factor \(3{,}540/3{,}000=1.180\); excluding the third row, \(2{,}420/2{,}200=1.100\). The third row materially raises the aggregate factor and warrants investigation, but the calculation alone does not justify excluding it.)
- A hypothetical row has 900 thousand dollars at 12 months and 1,530 thousand dollars at 24 months. An audit confirms that 540 thousand dollars of the latter amount was duplicated; the corrected amount is valid otherwise. Calculate the recorded and corrected ratios and state the treatment. (Answer: Recorded ratio \(1{,}530/900=1.70\); corrected amount \(1{,}530-540=990\), corrected ratio \(990/900=1.10\). Correct the cumulative value and retain the row.)
- Five hypothetical ratios are 1.04, 1.05, 1.06, 1.07, and 1.30. Find the median and MAD, and state what the screen tells you. (Answer: Median \(=1.06\); absolute deviations are 0.02, 0.01, 0, 0.01, and 0.24, so MAD \(=0.01\). The 1.30 value is a clear review priority, not proof of an error.)
- A high ratio is traced to a documented, valid coverage decision affecting a particular accident-year cohort. The decision is not expected to recur, but the resulting claims remain in the data. What should the analysis distinguish? (Answer: Preserve the valid claim amounts, but assess whether the cohort’s development represents future cohorts; if not, consider handling that development separately rather than treating the high ratio as a data error or automatically using it as an ordinary factor.)