Economic Modelling · Run-off triangles

Average Cost per Claim Method in Run-off Triangles

Updated 11 October 2026 · Fact-checked

The average cost per claim method projects two things separately from run-off triangles: the number of claims and the average cost of each claim. You develop each triangle to ultimate, usually with chain ladder factors, then multiply ultimate number by ultimate average cost for each accident year. Subtract payments to date to get the reserve.

Understand Average Cost per Claim Method

The basic chain ladder method projects total claim amounts in one step. The average cost per claim method splits the problem in two. Total claims = number of claims × average cost per claim. You project each part on its own, then multiply.

Why split? Claim numbers and claim sizes behave differently. Numbers are usually reported and settled fairly quickly. Average sizes depend on inflation, claim mix and how long large claims take to settle. Projecting them separately lets you see which part is driving the result, and lets you apply different assumptions to each.

You need two triangles on a consistent basis. The first is a cumulative number of claims triangle, for example claims reported by accident year and development year. The second is an average cost per claim triangle, found by dividing cumulative claim amounts by the matching cumulative number of claims. If the amounts are incurred, divide by reported claims. If the amounts are paid, divide by the number of claims that the payments relate to. Keep the basis the same in numerator and denominator.

You then project each triangle to ultimate. In the simplest version you apply chain ladder development factors to both. For the average cost, you might instead assume claim inflation, or a stable pattern of development ratios. For each accident year, ultimate claims = ultimate number × ultimate average cost. The reserve is the ultimate minus the amount already paid (or already incurred, if you want the IBNR-style figure). State your basis clearly.

The method assumes that the development pattern of numbers and of average costs is stable across accident years, and that the two are independent enough to be projected separately. If claim numbers and claim sizes are linked, for example a faster settlement of small claims, these assumptions weaken and you should say so.

Key rules to remember

Ultimate claims for an accident year: Ultimate claims = Ultimate number of claims × Ultimate average cost per claim; Apply this to each accident year, then sum for the total.
Average cost per claim (triangle cell): Average cost = Cumulative claim amount ÷ Cumulative number of claims; Use the same basis for both: incurred amounts with reported numbers, or paid amounts with the matching number of claims.
Development factor for claim numbers: f(j) = Σ N(i, j+1) ÷ Σ N(i, j), summed over accident years i with both values observed; N is the cumulative number of claims. Use only years that have both columns.
Development factor for average cost: g(j) = Σ A(i, j+1) ÷ Σ A(i, j), over years with both values observed; A is the cumulative average cost per claim. This ratio of column sums weights each year by the size of its average cost in the earlier column, so years with higher average costs count for more. If you want equal weights, use the simple average of the individual year-to-year ratios A(i, j+1) ÷ A(i, j) instead. State which one you use.
Projection to ultimate: Ultimate N(i) = latest N(i) × product of f(j) for the remaining development years; similarly for A(i) using g(j); Multiply the latest diagonal value by all later factors.
Inflation-based average cost: Average cost for year i+1 = Average cost for year i × (1 + claim inflation rate); Use when the question gives a claim inflation rate and asks you to adjust average costs between accident years.
Reserve: Reserve = Ultimate claims − Claims paid to date; If you use incurred amounts, the difference from incurred to date is the IBNR-type provision. Say which one you are giving.

How to solve Average Cost per Claim Method questions

Use this order for any question on the average cost per claim method. Write each assumption you make, because markers give credit for it.

1Identify the bases. Note whether the numbers are reported or settled, and whether the amounts are paid or incurred. Check that they match.
2Check whether the data is cumulative. If it is incremental, build the cumulative triangles first. If you are given only totals and numbers, form the average cost triangle by dividing cell by cell.
3Calculate development factors for the claim numbers triangle. Sum each column and the matching earlier column over the years with both observed, then take the ratio.
4Calculate development factors for the average cost triangle in the same way, or use the inflation or pattern the question gives.
5Project both triangles to ultimate. Multiply the latest diagonal value in each row by the remaining factors.
6Multiply ultimate number by ultimate average cost for each accident year. Sum the years for the total.
7Subtract payments to date (or incurred to date) to get the reserve, if asked. Label which one you used.
8State assumptions and comment: stable development patterns, no change in claim mix, consistent treatment of inflation. Compare with chain ladder if asked.

Quickest way: Fast route for a small triangle

When to use it: Use this when the question gives both triangles directly and asks for ultimate claims or a reserve, with limited time.

Work row by row for the latest diagonal only. You do not need to fill in the whole projected triangle.
Find each development factor once and write it beside the column. Reuse it for every row.
For each row, compute the cumulative factor to ultimate for numbers, and for average cost. For the youngest year this is the product of all factors.
Multiply latest number × its cumulative factor, and latest average cost × its cumulative factor, then multiply the two results.
Do a quick check: ultimate for each accident year should be larger than the latest known value. Totals should look sensible compared with earlier years.

Common mistakes in Average Cost per Claim Method

Mixing bases, for example dividing paid amounts by reported claim numbers.
Students take the numbers triangle and the amounts triangle as given and divide without checking what each represents.
Fix: Write the basis beside each triangle before dividing. Match incurred with reported and paid with the matching settled or paid number, or state the assumption you are making.
Projecting total claims with chain ladder and then multiplying by something, or multiplying two cumulative triangles together by mistake.
The idea of splitting the product gets confused with the single-triangle chain ladder.
Fix: Keep two separate projections: numbers and average cost. Multiply only at the end, using the ultimate values.
Including the wrong rows when calculating a development factor.
Students sum the full columns, including years that have no value in the next column.
Fix: For each factor, use only the accident years that have both columns observed. The numerator and denominator must cover the same rows.
Using the average of cell-by-cell ratios when the question expects the ratio of column sums, or the reverse, without saying so.
Both are called a development factor, and the method is not always stated.
Fix: Follow the question's instruction. If none is given, use the ratio of column sums and state that this is your choice.
Ignoring inflation when the average cost pattern clearly rises by accident year.
Students treat the development of average cost as purely a function of development year.
Fix: Check the first column across accident years. If it rises steadily, comment on claim inflation and use an inflation adjusted method if the question allows.
Giving the ultimate as the reserve.
The final multiplication feels like the end of the question.
Fix: Re-read what is asked. A reserve is ultimate minus amounts already paid, or minus incurred to date for an IBNR-type figure.

Worked examples

Example 1

An insurer has the following cumulative number of claims reported, by accident year and development year (0, 1, 2), and the cumulative average incurred cost per reported claim (in ₹). Numbers: 2021: 100, 150, 165; 2022: 120, 180; 2023: 140. Average cost (₹): 2021: 20,000, 24,000, 27,600; 2022: 21,000, 25,200; 2023: 22,000. Using the average cost per claim method with development factors from the ratio of column sums for both triangles, estimate the ultimate incurred claims for each accident year and in total.

Show the solution

Numbers, factor from development year 0 to 1: (150 + 180) ÷ (100 + 120) = 330 ÷ 220 = 1.5.
Numbers, factor from year 1 to 2: only 2021 has both columns, so 165 ÷ 150 = 1.1.
Ultimate numbers: 2021 = 165. 2022 = 180 × 1.1 = 198. 2023 = 140 × 1.5 × 1.1 = 140 × 1.65 = 231.
Average cost, factor 0 to 1: (24,000 + 25,200) ÷ (20,000 + 21,000) = 49,200 ÷ 41,000 = 1.2.
Average cost, factor 1 to 2: 27,600 ÷ 24,000 = 1.15.
Ultimate average cost: 2021 = ₹27,600. 2022 = 25,200 × 1.15 = ₹28,980. 2023 = 22,000 × 1.2 × 1.15 = 22,000 × 1.38 = ₹30,360.
Ultimate claims: 2021 = 165 × 27,600 = ₹45,54,000. 2022 = 198 × 28,980 = ₹57,38,040. 2023 = 231 × 30,360 = ₹70,13,160.
Total = 45,54,000 + 57,38,040 + 70,13,160 = ₹1,73,05,200.

Answer: Ultimate incurred claims: 2021 ₹45,54,000; 2022 ₹57,38,040; 2023 ₹70,13,160. Total ₹1,73,05,200. This assumes stable development patterns for numbers and average cost.

Example 2

For accident year 2023, the ultimate number of claims is projected as 400. The ultimate average cost per claim for accident year 2022 was ₹30,000. Claim inflation is assumed to be 5% a year and the same pattern applies in both years. Claims paid to date for 2023 are ₹80,00,000. Estimate the ultimate claims and the outstanding reserve for 2023.

Show the solution

Ultimate average cost for 2023 = 30,000 × 1.05 = ₹31,500.
Ultimate claims = 400 × 31,500 = ₹1,26,00,000.
Reserve = ultimate − paid to date = 1,26,00,000 − 80,00,000 = ₹46,00,000.
State the assumptions: 5% inflation applies fully to the average cost, and the claim mix is unchanged between the two years.

Answer: Ultimate claims ₹1,26,00,000; outstanding reserve ₹46,00,000.

Exam tips

Show both factor calculations clearly, with the rows used in each sum. Method marks are given even if arithmetic slips.
Write the basis (reported or settled, paid or incurred) at the start. Many written answers lose marks by leaving it out.
If asked to compare with the chain ladder method, say the split method shows what drives the result and handles changes in claim numbers and size separately. Its drawbacks are more data, and the assumption that numbers and sizes are independent.
In computer-based questions, build the average cost triangle with a formula, then fill factors and ultimates by formulas, so you can check for errors.
If the question gives an inflation rate, check whether it should be applied to average costs only. Do not apply it to claim numbers.

Practice questions from Run-off triangles

Average Cost per Claim Method: frequently asked questions

What is the difference between the average cost per claim method and the chain ladder method?

Chain ladder projects the total claim amounts in one triangle. The average cost per claim method projects numbers and average sizes in two triangles and multiplies the results. The split shows where changes come from and lets you adjust each part separately.

How do I project claim numbers and average claim size in a run-off triangle?

Calculate development factors for each triangle from the ratio of successive column sums, using only years with both columns observed. Apply the remaining factors to the latest value in each row. Then multiply the two ultimates for each accident year.

When is the average cost per claim method better than chain ladder?

It works better when claim numbers and claim sizes develop in different ways, or when the mix of claims or the rate of settlement has changed. It also helps when you want to apply inflation to average costs but not to numbers.

Can I use inflation with this method?

Yes. If average costs rise with claim inflation, you can adjust the average cost triangle to a common price level, project, and then re-inflate. If the question gives a simple rate, you can scale the average cost from one accident year to the next.