CM2 · explained

The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in

The basic chain ladder does not ignore claims inflation. It bakes the past rate into its development factors and projects it silently forward. This page restates a triangle into current money, develops it, and inflates the result at an assumption you can argue about.

CM2 · Financial Engineering & Loss Reserving · 11 min read

After this page you can

Restate a run-off triangle into current money, develop it, and inflate the projected payments forward at an explicit future assumption

The hidden assumption in the basic method

A development factor is a ratio of money paid at one date to money paid at an earlier date. If prices rose between those dates, the factor contains that rise. It cannot do otherwise: there is nothing in the arithmetic that separates "more claims settled" from "the same claims cost more".

So the basic chain ladder does not ignore claims inflation. It measures the past rate and projects it forward, without saying so. That is fine as long as future inflation will match past inflation. When it will not — and whenever there is a reason to think it will not, the question will say so — the projection is wrong, and wrong in a direction nobody has stated.

The inflation-adjusted chain ladder makes the assumption explicit. Take the past inflation out of the data, develop a triangle that is free of it, then put a chosen future rate back in. The development factors then measure only the speed at which claims emerge, which is what they were supposed to measure all along.

Why the payment date matters, not the accident year

This is the one structural idea on the page and everything depends on getting it right.

Inflation acts on money when it is paid. In a run-off triangle, payments made in the same calendar year lie on the same diagonal, because the payment year is the accident year plus the development year:

payment year = accident year + development year

So the restatement multiplies diagonals, not rows and not columns. The 2022 accident year's development-year-3 payment and the 2025 accident year's development-year-0 payment were both made in 2025 and both need the same factor, which is 1. The 2022 accident year's first payment was made three years earlier and needs three years of inflation.

That is also why the method needs incremental figures. A cumulative entry is a mixture of payments made in different years, and there is no single factor that restates a mixture. Convert to incrementals first, restate, then build cumulatives back up. A question that gives a cumulative triangle is asking for that conversion as the first line of the answer.

The method, in four steps

1. Restate every incremental payment into money of one chosen date, usually the latest. Multiply each by (1 + ipast)n, where n is the number of years from the payment year to the base year.

2. Build cumulatives from the restated incrementals and run an ordinary chain ladder on them. The factors that come out now describe emergence alone, with inflation removed.

3. Project to ultimate and difference back to incrementals, giving the future payments expressed in base-year money.

4. Inflate each future payment forward to the year it will actually be paid, at the assumed future rate: multiply by (1 + ifuture)m where m is the number of years from the base year to that payment year.

Step 4 is where the method differs from everything else, and it is where the marks are. The reserve is a sum of payments made in different future years, each inflated by a different amount, and getting the exponents right requires writing the payment year next to every projected figure.

What the adjustment is worth

The direction is always the same and worth stating before any arithmetic, because it is a free check on the answer:

  • if future inflation is assumed lower than past inflation, the adjusted reserve is lower than the basic chain ladder's;
  • if it is assumed higher, the adjusted reserve is higher;
  • if it is assumed equal, the two land in nearly the same place.

That last point deserves a word, because "nearly" is the honest version and an exam may probe it. Setting the future rate equal to the past rate does not reproduce the basic chain ladder exactly. The restatement multiplies different diagonals by different factors, so the restated triangle is not a scalar multiple of the original one, and the volume-weighted development factors therefore come out slightly different. On the worked example below the two agree to within 0.07% — close enough to confirm that the basic method really is projecting past inflation forward, and not identical. Claiming exact equality is wrong; claiming they are unrelated is also wrong.

Where this leaves the other two run-off methods

Four methods appear in the syllabus and it is worth knowing which problem each one solves, because a question will ask you to choose.

  • Basic chain ladder. Emergence and inflation together, projected forward unchanged. The default, and the right choice when nothing has changed.
  • Inflation-adjusted chain ladder. As above, but with the inflation assumption pulled out and made explicit. The right choice when future inflation will differ from past, which is the usual reason a question introduces it.
  • Average cost per claim. Splits the reserve into how many claims and how much each, and develops the two separately. The right choice when claim frequency and claim severity are moving differently.
  • Bornhuetter-Ferguson. Keeps the emergence pattern from the triangle and takes the overall level from premiums and a loss ratio. The right choice when the recent accident years have too little data to carry their own level.

They are not rivals so much as different repairs to the same method, and a reserving exercise runs several and investigates where they disagree.

Questions students ask

Which inflation index should be used?

Whichever one drives the claims, which is rarely general consumer prices. Motor repair costs track vehicle parts and garage wages; liability claims track earnings and court awards; property claims track building materials and labour. Using a general index when a specific one exists is a judgement a question may well ask you to criticise, and the direction of the error depends on which index has been rising faster.

What if past inflation was not constant?

Then use the actual year-by-year rates: the factor for a payment made n years before the base year is the product of the observed inflation factors over those n years, rather than a single rate raised to a power. The method is unchanged and the arithmetic is only a little longer. Using an average rate when the actual figures are given is a loss of accuracy with no compensating gain.

Does this fix the chain ladder's other problems?

No, and saying so is part of a good answer. It fixes one assumption. The method still assumes every accident year emerges in the same pattern, still rests the newest accident year's reserve on very little data, and still produces a single figure with no indication of its uncertainty. Bornhuetter-Ferguson addresses the second of those; nothing on this page addresses the first or the third.

Why not just discount the future payments as well?

Many reserving bases do, and in that case the inflating-forward and the discounting are applied to the same future payments in the same step, which is why both need the payment dates. The two partly offset, so a reserve inflated at 5% and discounted at 7% is smaller than an undiscounted one in today's money. Whether to discount at all is a reporting-basis question rather than a methods question, and a run-off triangle exercise will tell you which basis to use.

A worked example

The same portfolio as the chain ladder page, reprinted so this page stands alone, but now as incremental claims paid in ₹ lakh. Claims are fully run off by the end of development year 3.

Past claims inflation has run at 8% a year. The reserving basis assumes 5% a year in future. The triangle and both rates are invented for this example.

Accident yearDev 0Dev 1Dev 2Dev 3
202240021211636
2023440214135—
2024500242——
2025560———

Cumulating these gives 400, 612, 728, 764 for 2022 and so on — the same triangle the basic chain ladder page used, whose reserve was ₹725.36 lakh. Paid to date is ₹2,855 lakh.

  1. Restate into 2025 money

    Payment year is accident year plus development year, so the factors go on the diagonals:

    Payment yearYears to 2025Factor
    202231.083 = 1.2597120
    202321.082 = 1.1664000
    202411.0800000
    202501.0000000

    Applying them payment by payment:

    Accident yearDev 0Dev 1Dev 2Dev 3
    2022503.8848247.2768125.280036.0000
    2023513.2160231.1200135.0000—
    2024540.0000242.0000——
    2025560.0000———

    Check one entry by hand: the 2022 accident year's dev 1 payment of 212 was made in 2023, two years before the base, so 212 × 1.1664 = 247.2768. And note that 2022 dev 3, 2023 dev 2, 2024 dev 1 and 2025 dev 0 were all paid in 2025 and are all unchanged — that diagonal is the easiest place to spot a misplaced factor.

  2. Cumulate and develop

    Accident yearDev 0Dev 1Dev 2Dev 3
    2022503.8848751.1616876.4416912.4416
    2023513.2160744.3360879.3360—
    2024540.0000782.0000——
    2025560.0000———
    f0→1 = 2,277.4976 / 1,557.1008 = 1.4626526
    f1→2 = 1,755.7776 / 1,495.4976 = 1.1740424
    f2→3 = 912.4416 / 876.4416 = 1.0410752

    Compare these with the basic method's 1.4985075, 1.1982622 and 1.0494505. Every one is lower, and that is the inflation coming out: the basic factors were measuring emergence plus 8% a year, and these measure emergence alone.

    cumulative factors to ultimate: F0 = 1.7877512, F1 = 1.2222664, F2 = 1.0410752

  3. Project, and difference back to payments

    Each row is projected to ultimate in 2025 money and the cumulatives differenced, giving the future payments and the year each will be made.

    Accident yearDevPaid inIn 2025 moneyInflated at 5%
    20233202636.118937.9248
    202422026136.1012142.9062
    20243202737.711241.5766
    202512026259.0855272.0398
    202522027142.5556157.1676
    20253202839.499645.7257
    Total651.0719697.3406

    The accident year 2022 is fully run off, so its reserve is zero and it appears nowhere in this table.

    Check one line in full. The 2025 accident year's dev 3 payment will be made in 2028, three years after the base, so the factor is 1.053 = 1.1576250 and 39.4996 × 1.1576250 = 45.7257. And the first line's factor is 1.051, not 1.052, because 2026 is one year after 2025.

  4. The answer

    reserve in 2025 money = ₹651.07 lakh
    reserve inflated to the payment dates = ₹697.34 lakh

    Those are two different things and a question will specify which it wants. The second is the money that will actually leave the company; the first is what it is worth in today's prices. The ₹46.27 lakh between them is three years of 5% inflation on money that has not been paid yet.

  5. Against the basic chain ladder

    MethodReserve
    Basic chain ladder on the nominal triangle₹725.36 lakh
    Inflation-adjusted, 5% future inflation₹697.34 lakh
    Difference−₹28.02 lakh, or 3.9%

    Lower, as predicted, because the basic method is implicitly projecting the past 8% and the basis assumes 5%. The direction was known before the arithmetic started, which is why it is worth stating first: an adjusted reserve that came out higher on these assumptions would mean a mistake somewhere.

  6. The check: set the future rate back to 8%

    Run the whole method again with future inflation at the past 8% rather than 5%. If the basic chain ladder really is projecting past inflation forward, the two should land in nearly the same place.

    inflation-adjusted at 8% future = ₹725.83 lakh
    basic chain ladder = ₹725.36 lakh
    gap = ₹0.47 lakh, or 0.07%

    Nearly, and not exactly. The 0.07% is real and has a cause worth understanding: restating the triangle multiplies each diagonal by a different factor, so the restated triangle is not a scaled copy of the original, and the volume-weighted development factors therefore place slightly different weights on the accident years. The two methods are doing the same thing by different routes and arriving 0.07% apart.

    That gap is the honest answer to "does the adjustment collapse back to the basic method?" It collapses back to within a rounding error, which is enough to confirm that the basic method's hidden assumption is exactly past inflation projected forward — and it does not collapse back identically, which is why the claim has to be made carefully.

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.

Other CM2 topics explained

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.