The chain ladder method, worked step by step

CM2 · Financial Engineering & Loss Reserving · 12 min read

The problem a reserve solves

A general insurer sells a year of motor cover. Some of the accidents that happen in that year are reported and settled within weeks. Others are reported late. Others are reported promptly and then take years to settle, because liability is disputed or an injury has not yet stabilised. So at the end of the year the insurer has paid out some money and knows it will pay out more, but cannot know how much.

It must nevertheless put a number in its accounts, because that number determines whether the company is solvent and whether last year's business made a profit. That number is the outstanding claims reserve, and the chain ladder is the oldest and most widely used way of estimating it.

The idea behind it is simple enough to state in one sentence: look at how claims from earlier years grew as they developed, and assume the recent years will grow the same way.

How the data is arranged

Claims are set out in a run-off triangle. The rows are accident years — when the loss happened. The columns are development years — how long after the accident year the payment was made. Development year 0 is the accident year itself.

The reason it is a triangle and not a rectangle is worth stating because it is the whole of the problem. Today is a diagonal cut across the rectangle. The oldest accident year has had four years to develop and so has four figures; the newest has had one. Everything below and to the right of the diagonal has not happened yet, and estimating it is the job.

Two conventions matter and a question will always make clear which is in use:

Cumulative figures — each entry is everything paid so far on that accident year. These increase along a row.
Incremental figures — each entry is what was paid in that development year alone.

The chain ladder works on cumulative figures. If a question gives incremental ones, the first line of your answer is the cumulative triangle, and it costs almost nothing to produce.

Development factors

A development factor is the ratio by which cumulative claims grew from one development year to the next. The chain ladder uses a single factor per column, obtained by adding up and dividing — not by averaging the individual rows:

f_j = ( ∑ C_i,j+1 ) / ( ∑ C_i,j )

In words: add the column to the right, add the column to the left over the same accident years, and divide. Only accident years that have both figures are included.

This is the volume-weighted average, and it is the standard chain ladder. It is not the same as the simple average of the individual year-on-year ratios, and the difference is not an accident: the volume-weighted factor gives more influence to accident years with more claims, which are the years with more information. A simple average lets a small, volatile year pull the factor about. The worked example below gives both so the gap can be seen.

Projecting to ultimate

Once each column has a factor, the triangle is filled in by multiplying along the rows, and the figure at the end of the row is the ultimate claims for that accident year. Two equivalent ways to get there, and a question may reward either:

ultimate = C_latest × f_j × f_j+1 × … × f_n−1
ultimate = C_latest × F_j, where F_j is the product of all the remaining factors

F_j is the cumulative development factor from development year j to ultimate, and it is worth computing once per column because it does three jobs at a stroke. It gives the ultimate in one multiplication. Its reciprocal, 1/F_j, is the proportion of the claims already developed, which is the figure the Bornhuetter-Ferguson method needs. And F_j − 1 multiplied by the latest cumulative figure gives the reserve directly, without computing the ultimate at all:

reserve for accident year i = C_latest × (F_j − 1)
total reserve = (total ultimate) − (total paid to date)

The second line is the check to run every single time. Compute the reserve row by row, compute it again as total ultimate minus total paid, and the two must agree exactly.

What the method assumes

The chain ladder is almost assumption-free in appearance and heavily loaded with assumptions in fact. A question asking when it is unreliable is asking which of these has broken.

Each accident year develops in the same pattern. This is the central assumption and everything rests on it.
The future development of a year depends only on claims paid to date, not on which accident year it is or how large it is.
Nothing about the business has changed. No shift in the mix of policies, in the deductible, in the geography, in the reinsurance.
Nothing about the claims process has changed. No change in how quickly claims are settled, in reserving philosophy, in court practice.
Past claims inflation will continue at the same rate. Note what this means: the basic method does not ignore inflation, it bakes the past rate into the factors and projects it forward. If future inflation will differ, the factors are wrong, and the standard repair is the inflation-adjusted chain ladder, which restates the triangle in current prices, develops it, and then inflates forward at an explicit assumed rate.

The weakness that the next method exists to fix

Look at where the estimate for the newest accident year comes from. It is one number — the claims paid in the first year — multiplied by the largest cumulative development factor in the table. On the example below that factor is 1.8844, so the entire reserve for the newest year rests on a single figure being representative.

That is a thin foundation, and it is thinnest exactly where the reserve is largest. If the newest year happened to have a slow start for a reason that has nothing to do with its eventual cost — a systems change, a quiet quarter, a delay in the claims department — the chain ladder will project that slow start all the way to ultimate and understate the reserve badly. It will do so with equal confidence in the other direction if one large claim was settled early.

This is the single most examinable criticism of the chain ladder, and it is the reason the Bornhuetter-Ferguson method exists: that method brings in outside information, in the form of premiums and an expected loss ratio, so that the recent years do not have to carry their own weight.

Where marks are lost

Using incremental figures where cumulative ones belong. Development factors must come from cumulative claims. Build the cumulative triangle first and label it, so a marker can see it was done.

Including an accident year in a column where it has no later figure. The ratio for a column uses only the accident years that have figures in both columns, so the denominator for an early column excludes the newest years. Using the full column sum underneath puts too much in the denominator and biases every factor downwards.

Averaging the individual ratios when the volume-weighted factor was wanted. Both are defensible estimators and the standard method is volume-weighted. State which you have used.

Forgetting that the oldest accident year needs no projection. If the question says claims are fully run off after development year 3, then the oldest accident year's reserve is exactly zero. It is a free mark, and quietly multiplying it by a factor anyway is a visible error.

Reporting the ultimate when the question asked for the reserve. The reserve is what is still outstanding: ultimate minus paid to date. Reporting the ultimate overstates the answer by the whole of the claims already settled, which is usually most of it.

Mixing units. Triangles are given in thousands, lakhs or crores. Write the unit at the top of the working and keep it there.

Questions students ask

Why is it called a chain ladder?

Because the factors are chained together — each one links one development year to the next, and climbing from a known figure to the ultimate is a matter of going up the rungs. The name describes the arithmetic rather than any statistical idea.

Should the earliest accident years be used in the factors at all?

That is a judgement a question may invite you to make. The usual argument for dropping them is that the oldest years describe a business that no longer exists, so a factor based on them is answering the wrong question. The argument against is that dropping data to get a result you prefer is not estimation. In practice a reserving actuary looks at the individual ratios in a column and, if one row is plainly out of line, says so, excludes it, and records why. What is not acceptable is excluding it silently.

Does the chain ladder give an indication of uncertainty?

Not on its own. It produces a single number with no standard error attached, which is one of its real limitations — a reserve is an estimate and the accounts carry it as though it were a measurement. Stochastic reserving methods exist to put a distribution around it, and the simplest practical check available within the basic method is to look at the spread of the individual development ratios in each column: a column whose ratios run from 1.19 to 1.21 is telling you something very different from one running from 1.05 to 1.60.

What if a development factor comes out below 1?

Then cumulative paid claims fell, which cannot happen with payments alone. It is common and legitimate with incurred claims, where case estimates are revised downwards as claims settle for less than was reserved. A factor below 1 on a paid triangle, by contrast, means a recovery, a salvage, or an error in the data, and the right response in an exam is to say which you think it is.

A worked example

The table below shows cumulative claims paid, in ₹ lakh, for a portfolio of general insurance policies. Claims are fully run off by the end of development year 3.

The triangle is invented for this example. It is not a real insurer's experience.

Accident year	Dev 0	Dev 1	Dev 2	Dev 3
2022	400	612	728	764
2023	440	654	789	—
2024	500	742	—	—
2025	560	—	—	—

Claims paid to date are 764 + 789 + 742 + 560 = ₹2,855 lakh.

Step 1: the development factors

Each factor uses only the accident years with a figure in both columns.

f_0→1 = (612 + 654 + 742) / (400 + 440 + 500) = 2,008 / 1,340 = 1.4985075
f_1→2 = (728 + 789) / (612 + 654) = 1,517 / 1,266 = 1.1982622
f_2→3 = 764 / 728 = 1.0494505

Note the denominators. The first factor excludes 2025, which has no dev 1 figure; the second excludes 2024 and 2025.

Look at the individual ratios before going on

Accident year	Dev 0 → 1	Dev 1 → 2	Dev 2 → 3
2022	1.5300	1.1895	1.0495
2023	1.4864	1.2064	—
2024	1.4840	—	—
Volume-weighted	1.4985	1.1983	1.0495

The first column's ratios run from 1.4840 to 1.5300, a spread of under five points — tight enough that the chain ladder's central assumption looks reasonable on this data. The simple average of those three ratios is 1.5001, against the volume-weighted 1.4985. The volume-weighted figure is slightly lower because 2022, the year with the highest ratio, is also the smallest year and so carries less weight. On this triangle the choice moves the final reserve by well under one per cent; on a triangle with one small, erratic year it can move it a great deal, and that is why the method specifies which average to use.

Step 2: cumulative development factors to ultimate

F₃ = 1 (fully run off)
F₂ = 1.0494505 = 1.0494505
F₁ = 1.1982622 × 1.0494505 = 1.2575170
F₀ = 1.4985075 × 1.2575170 = 1.8843986

And the proportions already developed, which are simply the reciprocals:

Development year	Cumulative factor to ultimate	Proportion developed
0	1.8843986	53.07%
1	1.2575170	79.52%
2	1.0494505	95.29%
3	1.0000000	100.00%

Step 3: ultimates and reserves

Accident year	Paid to date	Factor	Ultimate	Reserve
2022	764	1.0000000	764.00	0.00
2023	789	1.0494505	828.02	39.02
2024	742	1.2575170	933.08	191.08
2025	560	1.8843986	1,055.26	495.26
Total	2,855	—	3,580.36	725.36

The outstanding claims reserve is ₹725.36 lakh.

The second route, as a check

Those ultimates came from one multiplication each. Step the factors along the row instead, which uses the figures in a different order:

2024: 742 × 1.1982622 = 889.1106, then × 1.0494505 = 933.0775
2025: 560 × 1.4985075 = 839.1642
then × 1.1982622 = 1,005.5387
then × 1.0494505 = 1,055.2631

Against 933.0776 and 1,055.2632 from the single cumulative factor. The two routes agree to three decimal places, and the fourth moves because the factors printed above were rounded to seven places before being multiplied together. Carry them unrounded and the routes are identical. That is rounding, not a mistake, and knowing the difference is most of what checking an answer is.

The third route: the reserve without the ultimate

Multiply the paid figure by F − 1 and the reserve appears directly:

2023: 789 × 0.0494505 = 39.0165
2024: 742 × 0.2575170 = 191.0776
2025: 560 × 0.8843986 = 495.2632
total = 725.3573

And the arithmetic check that closes the question: total ultimate minus total paid is 3,580.3572705 − 2,855 = 725.3572705, which is the same ₹725.3573 lakh. Three routes, one answer.

Reading the answer

The reserve is ₹725.36 lakh against ₹2,855 lakh already paid, so the insurer is reporting that it still owes about a quarter as much again as it has paid out. The bulk of it — ₹495.26 lakh, over two-thirds of the total — belongs to the single newest accident year.

That concentration is the thing to say out loud in an answer. The ₹495.26 lakh figure rests entirely on one number, the ₹560 lakh paid in 2025, multiplied by 1.8844. A 10% misreading of that single figure moves the total reserve by about ₹50 lakh, or 7%, with nothing in the method to flag it.

One more reading is available for free. Dividing each ultimate by that year's earned premium gives the ultimate loss ratio the chain ladder is implying. With premiums of ₹1,000, ₹1,100, ₹1,250 and ₹1,400 lakh, those ratios are 76.4%, 75.3%, 74.6% and 75.4% — remarkably stable, which is a point in favour of the projection. If one year had come out at 95% while the others sat near 75%, that would be the finding of the question, and the Bornhuetter-Ferguson method would be the tool for doing something about it.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

The chain ladder method: estimating outstanding claims from a run-off triangle