Economic Modelling · Run-off triangles

Run-off Triangle Assumptions, Checks and Practical Issues

Updated 11 October 2026 · Fact-checked

Run-off triangle methods project past claims patterns into the future, so they rely on assumptions such as stable development, consistent data and steady business mix. You check results with ratios, diagnostics and comparison to other methods, add a tail factor for development beyond the data, and state the uncertainty clearly.

Understand Assumptions, Checks and Practical Issues

Run-off triangle reserving is general insurance reserving. It is not a topic in the 2026 CM2 Economic Modelling syllabus, which covers Rational economic theory, Measures of investment risk, Asset valuations, Liability valuations and Option theory. It does not appear in the 2026 topic lists for the other Core Principles subjects either. Treat this page as background or supporting study, and check the current IAI syllabus for the subject you are sitting.

A run-off triangle shows claims by accident (or underwriting) year down the rows and by development year across the columns. Reserving methods such as the chain ladder use the past pattern to fill the empty lower-right part of the triangle. The result is only as good as the assumption that the future will look like the past.

Every method has assumptions. The basic chain ladder assumes that development factors are stable across accident years, that past patterns will continue, and that the data are consistent. It also assumes no change in claim settlement speed, case estimate strength, mix of business, inflation, reinsurance or legal environment. If any of these has changed, the projection can be badly wrong. Methods such as Bornhuetter-Ferguson and loss ratio rely instead on an external prior estimate of ultimate claims, so they lean on a different assumption.

A tail factor covers development beyond the last column of the triangle. Long-tailed classes such as liability can still have claims emerging after the data end. If you ignore the tail, you understate the reserve. The tail is usually found by extrapolating the fitted pattern of development factors, using a curve, or using industry or reinsurer benchmarks. It is judgement-heavy, so say so.

Data problems are common. Examples are late booking of claims, claims reclassified between years, large one-off claims, changes in the definition of a claim, missing years, and mixing gross and net of reinsurance data. Changes in business mix, for example a shift from motor own damage to liability, make older rows unrepresentative. Faster settlement makes payments appear earlier. If you apply all-year average paid factors that include slower-settling older years to the recent years, the chain ladder on paid data tends to overstate the ultimate. This is a directional tendency, not a certainty, so check it against the data. Stronger case estimates raise early incurred amounts. Past incurred factors applied to recent years would then tend to overstate ultimates, provided the older years had weaker case estimates.

Checking means testing reasonableness. Look at the development factors by year for trends or outliers. Compare projected ultimate loss ratios across years. Compare the paid and incurred results. Compare with other methods. Review the implied tail and the emerging experience against what you projected. Finally, remember that a reserve is a best estimate with uncertainty from process risk, parameter risk and model risk.

Key rules to remember

Age-to-age (development) factor: f(j) = Σ C(i, j+1) ÷ Σ C(i, j), summed over accident years i with both values observed; C(i, j) is cumulative claims for accident year i at development year j. This is the volume-weighted chain ladder factor. A simple average of individual ratios is an alternative.
Projected ultimate with tail: Ultimate(i) = C(i, latest) × f(latest) × ... × f(last) × tail factor; The tail factor is 1 only if you believe claims are fully developed at the end of the triangle.
Reserve: Reserve(i) = Ultimate(i) − Paid to date(i); Ultimate − Paid to date gives the total outstanding reserve, which is case reserves plus IBNR. With incurred data, Ultimate − Incurred to date gives IBNR in the wide sense, covering both IBNR and IBNER.
Implied ultimate loss ratio: Ultimate loss ratio(i) = Ultimate(i) ÷ Premium(i); Use as a reasonableness check across accident years. Premiums should be on a consistent basis (earned, same gross or net treatment).
Paid-to-incurred ratio: Paid ÷ Incurred at each development year; A rising ratio at the same age can indicate faster settlement or weaker case estimates, because incurred is then lower. A falling ratio can indicate slower settlement or stronger case estimates. Check other data, such as claim closure rates and average case estimates, to tell the causes apart.

How to solve Assumptions, Checks and Practical Issues questions

Use this method for any question that asks you to comment on, check or improve a run-off triangle reserve.

1Identify the data type (paid, incurred, number of claims), whether it is gross or net, and the periods used.
2List the assumptions of the method in use, such as stable development factors, consistent data and no change in mix or settlement.
3Compute or read the development factors and ultimates. Look for trends, outliers and unusual years.
4Test each assumption against the facts in the question, for example faster settlement, new products, large claims or inflation.
5State the direction of the bias. Say whether the reserve is likely to be too high or too low and why.
6Deal with the tail. Say whether a tail factor is needed and how you would estimate it.
7Suggest adjustments or alternatives, such as a second method, separating large claims, splitting by class or adjusting for inflation.
8Conclude with the reserve as a best estimate and describe the sources of uncertainty.

Quickest way: Assumption-change-effect-action

When to use it: Use this for short written questions asking how a change affects reserves or what to check.

Name the change in the question (settlement speed, mix, case estimate strength, data, inflation).
State which chain ladder assumption it breaks.
State the effect on the factors and on the reserve, giving direction.
Name one check and one adjustment (alternative method, data split, adjusted factors).
Add one line on tail and uncertainty.

Common mistakes in Assumptions, Checks and Practical Issues

Saying the chain ladder is always reliable if the arithmetic is right.
Students focus on the calculation and forget that the method only projects past patterns.
Fix: Always state the assumptions and test them against the information given in the question.
Getting the direction of bias wrong when settlement speeds up.
Students think faster settlement means lower claims.
Fix: Faster settlement raises early paid amounts. If you apply all-year average factors from slower-settling years to recent years on paid data, the ultimates tend to be overstated. This is a tendency, not a certainty, so confirm it with the data. The ultimate cost is not lower; the timing changes.
Ignoring or setting the tail factor to 1 without comment.
The tail is not visible in the triangle, so it seems unnecessary.
Fix: Check whether the last development factors are still above 1 and whether the class is long-tailed. Justify the tail you choose.
Treating a large one-off claim as part of the normal pattern.
Students apply the method mechanically to all cells.
Fix: Identify large claims, remove or cap them, project the rest and add the large claims back separately.
Mixing gross and net data, or changing premium bases, when comparing years.
The triangle looks consistent on the page.
Fix: Confirm the basis of each row and column before analysing. Say in your answer that you checked it.
Giving a single reserve with no mention of uncertainty.
Students treat the projection as exact.
Fix: Describe process, parameter and model uncertainty, and say you would give a range or sensitivity.

Worked examples

Example 1

A cumulative paid triangle gives development factors of 2.00 from year 0 to 1, 1.25 from year 1 to 2 and 1.10 from year 2 to 3. The latest accident year has paid ₹40,00,000 at development year 0. Past data show claims still emerge after year 3, and you estimate a tail factor of 1.05. Calculate the projected ultimate and the reserve, and comment.

Show the solution

Cumulative factor to year 3 = 2.00 × 1.25 × 1.10 = 2.75.
Include the tail: 2.75 × 1.05 = 2.8875.
Ultimate = ₹40,00,000 × 2.8875 = ₹1,15,50,000.
Reserve = ultimate − paid to date = ₹1,15,50,000 − ₹40,00,000 = ₹75,50,000.
Without the tail, the ultimate would be ₹40,00,000 × 2.75 = ₹1,10,00,000, so ignoring the tail would understate the ultimate by ₹5,50,000.
Comment: the tail is a judgement and the result is sensitive to it, so you should test other tail values and say the figure is a best estimate.

Answer: Ultimate ₹1,15,50,000 and reserve ₹75,50,000. The tail adds ₹5,50,000 to the ultimate and is a key source of uncertainty.

Example 2

An insurer's paid claims triangle for motor liability shows that in the last two accident years, a higher proportion of claims was paid in development year 0 than in earlier years. Management says this is because the claims team now settles claims faster. The reserving actuary has used chain ladder on paid data with all-year average factors. Explain the effect on the reserve and what you would do.

Show the solution

Identify the change: faster settlement, not higher ultimate claims.
The chain ladder assumes the pattern of payments is stable. Here recent years pay out earlier than older years.
Effect on factors: all-year averages include older years with slower payment, so the factors from year 0 to year 1 are likely to be too high for recent years. Applied to the recent years' higher paid amounts at year 0, they tend to overstate the ultimate and the reserve. This is a directional tendency, not a certainty.
Check: compare paid to incurred ratios, case estimate averages and claim counts closed at each age to confirm that settlement speed has changed and case estimates have not.
Action: use only recent diagonals or adjusted factors for the recent years, use an incurred or claim-count based method as a cross-check, or reflect the speed-up explicitly in the pattern.
Mention uncertainty: the true size of the speed-up is estimated, so give a range and monitor actual against expected claims.

Answer: The reserve from paid chain ladder with all-year factors is likely to be overstated, though this is a tendency and not certain. Confirm the speed-up with the paid and incurred data, then adjust the factors or use an alternative method.

Exam tips

Check the current IAI syllabus for your subject before spending time on this topic. Run-off triangle reserving is not in the 2026 CM2 syllabus.
Link every comment to a named assumption. Markers of reserving questions reward 'this breaks the stable-pattern assumption' more than a general remark.
Always give the direction of the effect (reserve too high or too low) and the reason.
Distinguish paid from incurred data when discussing settlement speed and case estimate strength, because the effects differ.
Give at least one check, one alternative method and one source of uncertainty in a discussion question.
In computations, show the factors, the tail and the subtraction of paid to date, so you earn method marks even if an arithmetic slip occurs.

Practice questions from Run-off triangles

Assumptions, Checks and Practical Issues: frequently asked questions

Is run-off triangle reserving part of the CM2 Economic Modelling syllabus?

No. The 2026 CM2 syllabus covers Rational economic theory, Measures of investment risk, Asset valuations, Liability valuations and Option theory. Run-off triangles are not listed. Use this page as background and confirm the current syllabus with IAI.

What is a tail factor in claims reserving?

A tail factor is a multiplier applied to the cumulative claims at the last observed development year to allow for claims that develop beyond the triangle. It is 1 only if development is complete. It is usually estimated by extrapolating the factors or using benchmarks.

How do I check the reasonableness of chain ladder results?

Look at development factors by year for trends and outliers, and compare ultimate loss ratios across accident years. Compare paid and incurred projections, and compare with another method such as Bornhuetter-Ferguson. Test the result against known changes in the business.

How do changes in claims settlement affect run-off triangles?

Faster settlement makes paid amounts appear earlier. If all-year average factors from slower-settling years are applied to recent years on paid data, the ultimates tend to be overstated. This is a tendency, not a certainty. Slower settlement tends to have the opposite effect. Check closure rates and paid-to-incurred ratios and adjust the method.

Why does a change in business mix matter for reserving?

Different classes develop at different speeds and have different tails. If the mix changes, the old pattern no longer fits the new business. Splitting the data by class or adjusting the factors is the usual response.