Excluding Anomalous Link Ratios

Why an Unusual Link Ratio Needs Investigation

A link ratio that is much higher or lower than the others at the same development age can have a substantial effect on a selected development factor. But an unusual value is a signal to investigate, not automatic grounds for exclusion. It may reflect a recording error, a one-time event that is not expected to recur, or genuine claims development that should remain part of the experience.

The investigation starts with the underlying triangle cells. A ratio is calculated from a pair of cumulative amounts, so the reason for an unusual ratio may be in its numerator, its denominator, or both. Reviewing only the ratio can hide that distinction. The decision to exclude should rest on the facts about the observation and the intended use of the selected factor, not on whether the ratio makes the result inconvenient.

Definition: For accident period \(i\) and development age \(j\), the age-to-age link ratio is \(C_{i,j+1}/C_{i,j}\), where \(C_{i,j}\) is the cumulative loss at age \(j\). A selected factor may be estimated by the volume-weighted ratio \(\sum_i C_{i,j+1}/\sum_i C_{i,j}\), using the included observations. An anomalous link ratio is one that appears unusually high or low relative to other observations for the same age; the term describes a pattern, not a conclusion that the observation is wrong.

In a volume-weighted calculation, the complete observation consists of the two cumulative cells that form the ratio. If an observation is excluded, its numerator and denominator must both be removed from their respective totals. Keeping one cell while removing the other no longer calculates the ratio from a consistent set of observations.

A Disciplined Investigation

A practical review proceeds from the triangle to the supporting records and then to the selection decision. First confirm that the two cells are from the correct accident period and development ages, use the same loss basis, and have the intended units. Then compare the pair with the other rows and inspect whether the numerator, denominator, or change between them is exceptional. If the ratio remains unusual, seek an explanation in the underlying data or known claims-handling circumstances.

Confirm the calculation.
Recalculate the ratio from the two cumulative cells. Check the row, development ages, units, and loss measure before treating the value as unusual.

Inspect the underlying amounts.
Compare the base-age and next-age cumulative values with other rows. Ask whether an unusually small denominator, unusually large increase, or unusual decrease explains the ratio.

Seek supporting evidence.
Check source records and relevant information about the observation. A verified duplicate, coding mistake, or clearly nonrecurring event provides a stronger basis for exclusion than the ratio's size alone.

Make and document the selection decision.
Include a valid observation even if it is extreme. If evidence supports exclusion, record which observation was excluded, why, and how the selected factor changes.

Why Volume-Weighted Recalculation Requires Both Cells

The volume-weighted factor is a ratio of totals, not an average of the included ratios with equal weights. This can be seen by writing each row's next-age amount as its base-age amount multiplied by its link ratio. For included rows, let \(r_i=C_{i,j+1}/C_{i,j}\). Then the selected factor is a weighted average of those link ratios, with weights proportional to the base-age amounts:

\frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}} = \frac{\sum_i C_{i,j}r_i}{\sum_i C_{i,j}} = \sum_i \left(\frac{C_{i,j}}{\sum_k C_{k,j}}\right)r_i.

The weights sum to one because \(\sum_i C_{i,j}/\sum_k C_{k,j}=1\). Excluding one row therefore removes that row's weight and its contribution to the next-age total. This is why an exclusion must remove both cells. It also explains why a high ratio from a small base amount can affect a volume-weighted factor less than the same ratio from a large base amount.

Worked Examples

Worked Example: A High Ratio Caused by a Verified Data Error

A hypothetical insurer reviews four cumulative paid-loss observations, in thousands of dollars, from age 12 to age 24. The age-12 amounts are \(1{,}000\), \(1{,}500\), \(2{,}000\), and \(2{,}500\). The corresponding age-24 amounts are \(1{,}200\), \(1{,}800\), \(2{,}600\), and \(5{,}000\). An audit finds that the last age-24 amount includes a duplicated \(2{,}000\) transaction; the corrected amount is \(3{,}000\). Calculate the link ratios and the volume-weighted factor before and after correction. State the appropriate treatment.

Method. Calculate each link ratio from its pair of cells, then compare the original and corrected volume-weighted ratios. Because the audit establishes the correct cell value, correcting the data is preferable to treating the erroneous value as valid experience.

The four original link ratios are:

\frac{1{,}200}{1{,}000}=1.20,\qquad \frac{1{,}800}{1{,}500}=1.20,\qquad \frac{2{,}600}{2{,}000}=1.30,\qquad \frac{5{,}000}{2{,}500}=2.00.

The last ratio is noticeably higher than the other three. The original volume-weighted factor is:

\frac{1{,}200+1{,}800+2{,}600+5{,}000} {1{,}000+1{,}500+2{,}000+2{,}500} = \frac{10{,}600}{7{,}000} =1.5143.

The audit establishes that the last numerator should be \(3{,}000\), so its corrected link ratio is \(3{,}000/2{,}500=1.20\). The corrected volume-weighted factor is:

\frac{1{,}200+1{,}800+2{,}600+3{,}000} {1{,}000+1{,}500+2{,}000+2{,}500} = \frac{8{,}600}{7{,}000} =1.2286.

The large change from \(1.5143\) to \(1.2286\) confirms that the audited error materially affected the factor. The correct action is to use the verified correction, with the audit trail documented. If the correct value could not be established, the observation could be excluded from selection with the data problem recorded; it should not be silently altered.

Common mistake. Removing the questionable \(5{,}000\) numerator while leaving its \(2{,}500\) denominator gives \(8{,}600/7{,}000\) only if the numerator has already been replaced by \(3{,}000\); simply dropping the \(5{,}000\) from the original total but retaining all denominators gives \(5{,}600/7{,}000=0.8000\). That calculation mixes different sets of observations. Use the audited corrected numerator, or remove both cells if excluding the observation.

Worked Example: A Low Ratio From a Nonrecurring Event

A hypothetical insurer examines four age-12-to-age-24 incurred-loss observations, in thousands of dollars. The age-12 amounts are \(1{,}000\), \(1{,}500\), \(2{,}000\), and \(2{,}500\); the age-24 amounts are \(1{,}100\), \(1{,}680\), \(1{,}500\), and \(2{,}775\). Supporting records show that the third observation reflects a one-time settlement program that will not be repeated and is not representative of the periods for which the factor is being selected. Calculate the ratios and factors with and without that observation.

Method. Inspect the low ratio in context, use the documented information to assess representativeness, and calculate the volume-weighted factor both ways to show the effect of the selection decision.

The four link ratios are:

\frac{1{,}100}{1{,}000}=1.10,\qquad \frac{1{,}680}{1{,}500}=1.12,\qquad \frac{1{,}500}{2{,}000}=0.75,\qquad \frac{2{,}775}{2{,}500}=1.11.

The factor using all four observations is:

\frac{1{,}100+1{,}680+1{,}500+2{,}775} {1{,}000+1{,}500+2{,}000+2{,}500} = \frac{7{,}055}{7{,}000} =1.0079.

The settlement-program observation contributes both \(1{,}500\) to the age-24 total and \(2{,}000\) to the age-12 total. Excluding the complete observation gives:

\frac{1{,}100+1{,}680+2{,}775} {1{,}000+1{,}500+2{,}500} = \frac{5{,}555}{5{,}000} =1.1110.

The records provide a reason for exclusion beyond the fact that \(0.75\) is low: the observation reflects a known, nonrecurring program and is not considered representative of the intended selection. The factor excluding it is \(1.1110\). This decision and its rationale should be documented; another purpose or target population could require a different judgment.

Common mistake. Excluding the observation because it produces a more convenient factor, without confirming the settlement program or its relevance, is not a defensible selection rationale. An extreme ratio by itself does not establish that the experience should be omitted.

Worked Example: A High Ratio That Should Remain

A hypothetical insurer has four age-12-to-age-24 paid-loss observations, in thousands of dollars. The age-12 amounts are \(500\), \(1{,}000\), \(1{,}500\), and \(2{,}000\); the age-24 amounts are \(900\), \(1{,}100\), \(1{,}650\), and \(2{,}200\). The first period's \(900\) is verified against source records and reflects a large claim reported during the interval. The claim type and reporting process are expected to remain relevant. Calculate the factor and assess whether the first observation should be excluded solely because its ratio is high.

Method. Calculate each ratio and the volume-weighted factor, then weigh the verified explanation against the intended use of the factor.

The link ratios are:

\frac{900}{500}=1.80,\qquad \frac{1{,}100}{1{,}000}=1.10,\qquad \frac{1{,}650}{1{,}500}=1.10,\qquad \frac{2{,}200}{2{,}000}=1.10.

The first ratio is much higher than the other three, but its base amount is also relatively small. The volume-weighted factor using all observations is:

\frac{900+1{,}100+1{,}650+2{,}200} {500+1{,}000+1{,}500+2{,}000} = \frac{5{,}850}{5{,}000} =1.1700.

For comparison, excluding the first observation would give:

\frac{1{,}100+1{,}650+2{,}200} {1{,}000+1{,}500+2{,}000} = \frac{4{,}950}{4{,}500} =1.1000.

The source data are verified, and the claim development is considered relevant to the periods being modeled. There is no stated basis for treating it as an error or nonrepresentative event. Therefore, the high ratio should not be excluded solely because it differs from the others. The sensitivity calculation shows why the judgment matters: excluding it would reduce the factor from \(1.1700\) to \(1.1000\).

Common mistake. Calling the highest ratio an outlier and deleting it automatically confuses an unusual observation with an invalid one. Investigate the source and relevance first; retain valid experience when no sound exclusion rationale exists.

Common Traps and Documentation

The examples illustrate that exclusion is a judgment about data quality and relevance, not a mechanical outlier rule. A defensible review separates three questions: whether the ratio is unusual, what underlying circumstance explains it, and whether that circumstance makes the observation inappropriate for the factor being selected. Answering only the first question is not enough.

Do not remove only one cell. An observation in a volume-weighted ratio contributes a numerator and a denominator; exclude the pair, or correct a cell when the correct value is verified.
Do not assume every extreme value is erroneous. Genuine development can be unusual and still relevant to the expected future pattern.
Do not treat a plausible story as proof. Seek records or other support for a proposed data error or nonrecurring event.
Do not hide the effect of the decision. State the selected factor with and without the observation when the impact is material.
Do not change the selection reason after seeing the answer. Apply a consistent standard based on the intended use of the factor.

A useful selection record identifies the development age, accident period, two cumulative cells, calculated ratio, source of the investigation, reason for inclusion or exclusion, and the factor's sensitivity to the decision. This makes the judgment reviewable and helps distinguish an evidence-based adjustment from a change made merely to obtain a preferred estimate. The method is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Key takeaway: An unusual link ratio is a prompt to inspect its underlying cumulative cells and supporting evidence. Correct verified errors, exclude complete observations only for a documented reason, and retain valid experience even when it is extreme.

Check Your Understanding

For each problem, use the stated evidence rather than treating unusual size alone as grounds for exclusion.

A hypothetical triangle has age-12 and age-24 amounts of \(800\) and \(960\), \(1{,}200\) and \(1{,}440\), and \(1{,}600\) and \(2{,}000\), all in thousands of dollars. Calculate the ratios and the volume-weighted factor. No observation has a known data problem. (Answer: Ratios \(1.20, 1.20, 1.25\); factor \(4{,}400/3{,}600=1.2222\); no exclusion is justified by the information given.)
Three incurred-loss observations have age-12 amounts \(1{,}000\), \(2{,}000\), and \(3{,}000\), and age-24 amounts \(1{,}100\), \(2{,}200\), and \(1{,}800\), in thousands. Records confirm that the third reflects a one-time adjustment not expected to recur. Calculate the factor excluding it. (Answer: \( (1{,}100+2{,}200)/(1{,}000+2{,}000)=1.1000\).)
Three observations have age-12 amounts \(500\), \(1{,}000\), and \(1{,}500\), and age-24 amounts \(900\), \(1{,}100\), and \(1{,}650\), in thousands. The first ratio is verified as genuine and relevant. Calculate the factor using all three; state whether the first observation should be removed just because its ratio is higher. (Answer: \(3{,}650/3{,}000=1.2167\); no, the verified relevant observation should not be excluded solely for being high.)
An analyst excludes a row with age-12 amount \(2{,}000\) and age-24 amount \(1{,}500\) from a volume-weighted factor, but removes only the \(1{,}500\) from the age-24 total. What is the conceptual error? (Answer: The calculation retains the excluded row's denominator while removing its numerator; a valid exclusion removes both cells from their respective totals.)
A triangle produces a very high ratio from a small age-12 amount. The source records confirm the amount, but the underlying claim development is not yet understood. What should the analyst do before selecting or excluding the observation? (Answer: Investigate the underlying cells and supporting claim information, assess whether the experience is relevant to the factor's intended use, and document the decision; the ratio alone does not justify exclusion.)

What You'll Learn