Economic Modelling · Run-off triangles

Chain Ladder Method: Step-by-Step Reserve Calculation

Updated 11 October 2026 · Fact-checked

The chain ladder method projects a cumulative claims triangle to ultimate. You calculate development factors as the ratio of column totals for adjacent development years, multiply the latest cumulative figure by the remaining factors to get ultimate claims, then subtract claims paid to date to get the outstanding reserve.

Understand Chain Ladder Method

A run-off triangle shows claims by origin year (when the claim event happened) and development year (how many years later the figure is measured). Only the upper-left part is known. The lower-right part is what you must estimate.

The chain ladder method assumes each origin year develops in the same proportional way. If claims grew by 50% in the first year for older origin years, the newest origin year will also grow by about 50%. The growth from one development year to the next is the development factor.

You work on cumulative data. Each factor is the sum of the later column divided by the sum of the earlier column, using only origin years where both values are known. This is the volume-weighted (all-years) average. Examiners may also ask for a simple average of the individual year-to-year ratios, so read the question.

To project, take the latest known cumulative figure for each origin year and multiply it by the factors still to come. This gives the ultimate claims. The outstanding reserve is ultimate minus the amount already paid (or, for incurred data, the ultimate minus incurred to date gives the IBNR-type amount, so check what the triangle measures).

The method is simple and uses only the data. Its weakness is that it needs stable patterns. It does not allow for inflation, changes in claim handling, changes in business mix or large one-off claims. The most recent origin years rest on very little data, so their projections are the least reliable.

Key rules to remember

Development factor (volume-weighted): f(j) = Σ C(i, j+1) ÷ Σ C(i, j), summed over origin years i for which C(i, j+1) is known; C(i, j) is the cumulative claims for origin year i at development year j. Use the same origin years in the numerator and denominator.
Individual development ratio: d(i, j) = C(i, j+1) ÷ C(i, j); A simple-average factor is the mean of these ratios. Use it only if the question asks for it.
Ultimate claims: Ultimate(i) = C(i, latest) × f(latest) × f(latest+1) × … × f(last) × tail factor; If no tail factor is given, assume the last factor completes development (tail factor = 1).
Outstanding reserve: Reserve(i) = Ultimate(i) − C(i, latest); Valid when the triangle is of cumulative paid claims. Total reserve is the sum over all origin years.
Cumulative-to-ultimate factor: F(j) = f(j) × f(j+1) × … × f(last); Calculate once for each development year, then apply to every origin year. It saves time.

How to solve Chain Ladder Method questions

Use this method for any chain ladder question. Check first whether the triangle is cumulative or incremental, and whether it is paid or incurred.

1Check the data. If the triangle is incremental, convert it to cumulative by adding across each row.
2Mark the latest known diagonal. These are the values you will project from.
3Calculate each development factor. For each column pair, add the later column over the origin years where it exists, and divide by the sum of the earlier column over the same origin years.
4State your assumptions: factors stay the same in future, no inflation effect, and a tail factor of 1 unless told otherwise.
5Work out the cumulative-to-ultimate factors by multiplying factors from the right.
6Multiply each latest diagonal value by its cumulative-to-ultimate factor to get ultimate claims.
7Subtract claims paid to date from each ultimate to get the reserve for each origin year, then add to get the total.
8Do a sense check. Older years should have small reserves, and each ultimate should be at least as large as its latest value.

Quickest way: Column totals and a single multiplier row

When to use it: Use this for any triangle of four or more years when you need ultimates and reserves under time pressure.

Write the column sums for the earlier and later columns in the margin before dividing.
Divide once per column pair and keep at least three decimal places.
Build the cumulative-to-ultimate factors from right to left in one row beneath the triangle.
Multiply each diagonal value by the matching factor in the row.
Check the oldest year: its reserve should be zero when no tail factor applies.
Add up reserves last and compare the total with the sum of the ultimates minus the sum of the diagonal.

Common mistakes in Chain Ladder Method

Using incremental figures to calculate factors
The triangle given in the question is incremental and students do not notice the heading.
Fix: Always read the heading. Convert to cumulative before calculating any ratio.
Including the wrong origin years in a factor
Students sum the whole earlier column, even for origin years that have no later value.
Fix: Only sum the earlier column for the origin years where the later column is known. The numerator and denominator must cover the same years.
Confusing the ultimate with the reserve
Students stop after projecting the final column.
Fix: Subtract the latest cumulative paid figure from each ultimate. The reserve is the outstanding amount only.
Applying the factors to the wrong development year
Students multiply by factors that have already happened for that origin year.
Fix: For each origin year, apply only the factors from its latest development year onward.
Ignoring the tail factor
Students assume the triangle is fully developed at the last column.
Fix: If a tail factor is given, multiply every ultimate by it. If none is given, say you have assumed development is complete.
Presenting the answer with no assumptions
Students treat it as pure arithmetic.
Fix: State that you assume past development patterns continue, there is no change in inflation or claim settlement, and the data is cumulative paid claims.

Worked examples

Example 1

The table shows cumulative paid claims (₹ lakh) by origin year and development year. Origin year 1: 1,000; 1,500; 1,800; 1,890. Origin year 2: 1,100; 1,650; 1,980. Origin year 3: 1,200; 1,800. Origin year 4: 1,300. Assuming no further development after development year 3, calculate the total outstanding reserve using the chain ladder method.

Show the solution

The data is cumulative, so no conversion is needed. The latest values are 1,890; 1,980; 1,800 and 1,300.
f(0→1) = (1,500 + 1,650 + 1,800) ÷ (1,000 + 1,100 + 1,200) = 4,950 ÷ 3,300 = 1.5.
f(1→2) = (1,800 + 1,980) ÷ (1,500 + 1,650) = 3,780 ÷ 3,150 = 1.2.
f(2→3) = 1,890 ÷ 1,800 = 1.05.
Cumulative-to-ultimate factors: from development year 2 it is 1.05; from year 1 it is 1.2 × 1.05 = 1.26; from year 0 it is 1.5 × 1.26 = 1.89.
Ultimates: origin year 1 = 1,890. Origin year 2 = 1,980 × 1.05 = 2,079. Origin year 3 = 1,800 × 1.26 = 2,268. Origin year 4 = 1,300 × 1.89 = 2,457.
Reserves: year 1 = 0; year 2 = 2,079 − 1,980 = 99; year 3 = 2,268 − 1,800 = 468; year 4 = 2,457 − 1,300 = 1,157.
Total = 99 + 468 + 1,157 = 1,724.

Answer: The total outstanding reserve is ₹1,724 lakh, assuming past development patterns continue and there is no tail beyond development year 3.

Example 2

Cumulative paid claims (₹ lakh): origin year 1: 800; 1,000; 1,100. Origin year 2: 900; 1,125. Origin year 3: 1,000. Calculate the development factors, then the ultimate claims and total reserve, using a tail factor of 1.02 beyond the last development year.

Show the solution

f(0→1) = (1,000 + 1,125) ÷ (800 + 900) = 2,125 ÷ 1,700 = 1.25.
f(1→2) = 1,100 ÷ 1,000 = 1.1.
Tail factor = 1.02, so the factor from development year 2 to ultimate is 1.02.
Cumulative-to-ultimate factors: from year 2 = 1.02; from year 1 = 1.1 × 1.02 = 1.122; from year 0 = 1.25 × 1.122 = 1.4025.
Ultimates: origin year 1 = 1,100 × 1.02 = 1,122. Origin year 2 = 1,125 × 1.122 = 1,262.25. Origin year 3 = 1,000 × 1.4025 = 1,402.5.
Reserves: year 1 = 1,122 − 1,100 = 22. Year 2 = 1,262.25 − 1,125 = 137.25. Year 3 = 1,402.5 − 1,000 = 402.5.
Total reserve = 22 + 137.25 + 402.5 = 561.75.

Answer: The factors are 1.25 and 1.1. The ultimates are ₹1,122 lakh, ₹1,262.25 lakh and ₹1,402.5 lakh, and the total reserve is ₹561.75 lakh.

Exam tips

Write down each factor as a fraction first, for example 4,950 ÷ 3,300, so you can earn method marks even if the arithmetic slips.
Always list the assumptions of the method. Written questions often award marks for them.
Be ready to comment on limitations: the latest origin years rely on little data, and inflation, large claims and changes in settlement speed distort factors.
In multiple-choice questions, check whether the question wants a simple-average or volume-weighted factor, and whether the answer is the ultimate or the reserve.
In computer-based papers, show the formula for the factor in your code or spreadsheet and label rows and columns so the result can be followed.

Practice questions from Run-off triangles

Chain Ladder Method: frequently asked questions

What is the difference between volume-weighted and simple-average development factors?

The volume-weighted factor divides the sum of a later column by the sum of the earlier column over the same origin years. The simple average takes each year's ratio and averages them. Unless the question says otherwise, the volume-weighted version is the standard chain ladder factor.

Why does the chain ladder method use cumulative data?

The method models how total claims grow as a multiple of what is already known. Cumulative figures make that proportional growth meaningful. Incremental figures must be converted to cumulative before you calculate factors.

What are the main assumptions of the chain ladder method?

It assumes the pattern of claims development is the same for every origin year, and that the factors stay stable in future. It also assumes no change in inflation, claim handling or business mix. If these fail, the projection is unreliable.

Do I need a tail factor?

A tail factor is needed when claims are still developing after the last column of the triangle. If the question gives one, apply it to every ultimate. If it does not, state that you assume development is complete.