Reserve risk · RiskAtoms Academy

By the end you will be able to

1Read a claims triangle and project it to ultimate with the chain ladder, splitting the reserve into case reserves and IBNR.
2Compare paid and incurred projections, and explain prior-year development using the book's motor liability strengthening.
3Measure reserve uncertainty with Mack's standard error and turn it into a 1-in-200 reserve.
4Explain why long-tail lines carry more reserve risk than short-tail lines.

Why it matters

RiskAtoms Insurance SE's largest non-life liability is not next year's claims. It is €7,497m owed for accidents that have already happened, €2,896m of it for claims nobody has reported yet. Each year the estimate moves. In 2025 it moved by €116m the wrong way on motor liability alone. Reserve risk is the chance that the past costs more than the balance sheet says.

Core explanation

Estimating what the past will cost

Illustrative dataRiskAtoms Insurance SE's insurance book

The same composite European insurer as Modules 1.8 and 2.2, in € millions at 31 December 2025: four non-life lines with ten years of claims triangles, a 1.2 million vehicle-year motor pricing data set, a 17,000-event catastrophe set, its reinsurance, its life books, its assets and an outline of its Solvency II standard formula. Every policy, claim, event and figure is made up for teaching and seeded so it never changes; none is any insurer's data. Every lesson in Module 3.8 uses this book, so the figures agree.

This lesson uses the book's paid and incurred triangles, accident years 2016 to 2025, for four lines. They are generated from stated payment patterns with seeded noise, so the chain ladder needs no tail factor (every line is fully paid within ten years). The book selects the incurred chain ladder for every line, and treats the four lines' errors as independent when it adds them up.

The triangle

Claims are filed by accident year, then followed as they develop. Put each accident year in a row and each year of development in a column, and the known figures form a triangle: 2016 has ten years of history, 2025 only one. This is general liability's incurred triangle (paid plus case reserves):

General liability, cumulative incurred claims, €m. Bold: the latest diagonal, the position at 31 December 2025.
AY	1	2	3	4	5	6	7	8	9	10
2016	238	352	448	534	543	587	607	616	619	623
2017	314	407	528	584	633	650	669	683	687
2018	258	449	549	622	644	685	700	704
2019	296	426	522	599	644	664	670
2020	288	434	551	598	651	671
2021	287	460	556	647	651
2022	363	584	734	777
2023	365	574	692
2024	315	533
2025	463

Read down a column and the amounts grow with the business. Read along a row and each accident year keeps growing for a decade, as late claims arrive and court awards come in. The 2025 row has one number, €463m; the question is what it will grow to.

The chain ladder

The chain ladder assumes each accident year will grow the way earlier ones did. For each pair of columns, the development factor is one column sum over another (M-A08: Summation notation). Multiplying the factors still to come takes each row's latest value to its ultimate:

The same triangle completed by the chain ladder (italics). f under column j takes development year j to j + 1; CDF is the product of the factors from there to the end.
AY	1	2	3	4	5	6	7	8	9	10	Ultimate
2016	238	352	448	534	543	587	607	616	619	623	623
2017	314	407	528	584	633	650	669	683	687	691	691
2018	258	449	549	622	644	685	700	704	708	712	712
2019	296	426	522	599	644	664	670	679	683	687	687
2020	288	434	551	598	651	671	687	696	700	704	704
2021	287	460	556	647	651	681	696	706	710	714	714
2022	363	584	734	777	816	854	873	885	890	896	896
2023	365	574	692	776	816	853	873	884	889	895	895
2024	315	533	662	743	781	816	835	846	851	857	857
2025	463	717	891	999	1,050	1,098	1,124	1,139	1,145	1,152	1,152
f	1.549	1.243	1.122	1.051	1.046	1.023	1.014	1.005	1.006
CDF	2.489	1.607	1.293	1.153	1.097	1.050	1.026	1.012	1.006	1.000

The ultimate less what is already incurred is IBNR; the ultimate less what is already paid is the reserve. For the whole book:

End-2025, €m, gross and undiscounted, incurred chain ladder. Paid covers accident years 2016–2025 to date.
Line	Paid	Case reserves	IBNR	Reserve	IBNR share	Mack SE	CV
Motor liability	10,152	2,072	947.1	3,019.1	31%	170.6	5.6%
Motor own damage	7,895	165	79.5	244.5	33%	10.1	4.1%
Property (home and commercial)	13,708	797	407.9	1,204.9	34%	50.0	4.1%
General liability	4,904	1,567	1,461.6	3,028.6	48%	163.0	5.4%
Total	36,659	4,601	2,896.1	7,497.1	39%	241.4	3.2%

Paid against incurred

The chain ladder can run on either triangle. Paid figures are facts but slow; incurred figures carry the claims handlers' knowledge and move sooner, but change when reserving practice changes. When the two disagree, something is happening. On motor own damage and property they agree within €6m. On general liability the paid projection is €181m higher than the incurred one: either case reserves are too low, or payments are running faster than they used to. A reserving actuary would investigate before choosing.

How wrong could it be? Mack's standard error

The chain ladder gives one number. In 1993 Thomas Mack showed how to attach a standard error to it using only the triangle: estimate how much the individual development ratios scatter around each factor, then carry that variance (M-C03: Variance and standard deviation) through to the ultimate. It has two parts: process error (the future is random even if the factors are right) and estimation error (the factors themselves come from a few noisy rows). On the book the total is €241.4m, 3.2% of the reserve, if the four lines move independently; if their errors simply added, it would be €393.7m.

Prior-year development: motor liability

Run the chain ladder on the triangle as it stood a year ago (drop the latest diagonal), then on today's, and compare the ultimates of the same accident years. The difference is prior-year development. In 2025 the book released €55.4m overall, but the lines went different ways:

2025, €m; + is a strengthening (a cost in the year), − a release.
Line	Prior-year development	As % of a year ago
Motor liability	+€116.3m	+1.02%
Motor own damage	−€7.4m	−0.10%
Property (home and commercial)	−€17.2m	−0.13%
General liability	−€147.1m	−2.12%
Total	−€55.4m

Motor liability's +€116.3m sits in its two most recent prior years: +€57.4m on 2023 and +€95.6m on 2024, as the first development factor rose from 1.2434 to 1.2535. The years before them went the other way: 2022 released €34.2m and 2021 €19.7m. 2023 and 2024 were reserved on the assumption that claims inflation would ease, but costs kept running ahead of it (L3.8.04: Premium risk and the underwriting cycle). Those years also still have most of their bodily injury claims open or not yet reported. The strengthening alone takes motor liability's combined ratio above 100% for the year (L3.8.01: How insurers take risk).

Long tail, short tail

Line	Tail	Paid in year 1	Reserve ÷ ultimate	Mean term (years)	IBNR share
Motor liability	medium	30%	22.9%	2.6	31%
Motor own damage	short	80%	3.0%	1.2	33%
Property (home and commercial)	short	55%	8.1%	1.7	34%
General liability	long	10%	38.2%	3.1	48%

In a short-tail line, most of the cost is known within a year; the reserve is small and an error shows up fast. In a long-tail line the reserve is large, it is paid over years, and much of it is IBNR, so it is exposed for longer to inflation, court decisions and new kinds of claim. Note that general liability's CV (5.4%) is no higher than motor liability's (5.6%): Mack measures the noise the triangle has shown, not the shocks it has not. Asbestos never appeared in a triangle until it did.

Sort itLong tail or short tail?

Sort each cover by how long its claims take to be known and paid.

Windscreen and accident damage to a policyholder's own car.
Employers' liability for industrial disease.
Storm damage to homes.
Bodily injury claims under motor liability.
Professional indemnity for architects.

The 1-in-200 reserve

Capital asks a quantile question (L2.5.06: Value at risk: the idea): what could the reserve turn out to be in a 1-in-200 case? A common shortcut treats the reserve as lognormal with the chain ladder's mean and Mack's standard error. For general liability that gives €3,473.5m, €444.9m above the best estimate. The Solvency II standard formula, using its market-wide standard deviation of 11% for the segment and 3σ, would put the same reserve's stand-alone charge at €999.4m, more than twice as much.

Why the two measures differ

Three reasons, each pulling a different way. Mack measures the run-off to ultimate on this triangle's own history; Solvency II asks how much the estimate can move in one year, which is usually smaller (Merz and Wüthrich, 2008). The standard formula's σ is calibrated on many insurers' data, including shocks this book has never seen. And the standard formula's charge is on the discounted best estimate net of reinsurance, here compared gross and undiscounted for simplicity.

The formal version

Mack's model. Mack (1993) makes three assumptions: the expected next value is the current value times a factor; accident years are independent; and the variance of the next value is proportional to the current one.

\begin{aligned} E[C_{i,j+1} \mid C_{i,1},\dots,C_{i,j}] &= f_j\, C_{i,j} \\ \operatorname{Var}[C_{i,j+1} \mid C_{i,1},\dots,C_{i,j}] &= \sigma_j^2\, C_{i,j} \\ \hat\sigma_j^2 &= \frac{1}{n-j-1}\sum_{i=1}^{n-j} C_{i,j}\Big(\frac{C_{i,j+1}}{C_{i,j}} - \hat f_j\Big)^2 \end{aligned}

The last $\sigma^2$ cannot be estimated from one ratio and is extrapolated as $\min(\hat\sigma_{n-2}^4/\hat\sigma_{n-3}^2,\ \hat\sigma_{n-3}^2,\ \hat\sigma_{n-2}^2)$ . The mean squared error of accident year $i$ 's reserve is then

\widehat{\text{mse}}(\hat R_i) = \hat C_{i,n}^{\,2} \sum_{j = n+1-i}^{n-1} \frac{\hat\sigma_j^2}{\hat f_j^{\,2}}\Big(\underbrace{\frac{1}{\hat C_{i,j}}}_{\text{process}} + \underbrace{\frac{1}{\sum_{k=1}^{n-j} C_{k,j}}}_{\text{estimation}}\Big)

The total adds cross terms, because every accident year uses the same estimated factors. The standard error is the square root. For the 1-in-200 reserve with mean $R$ and standard error $s$ :

\begin{aligned} \sigma^2 &= \ln\!\big(1 + (s/R)^2\big), \qquad \mu = \ln R - \sigma^2/2 \\ R_{99.5\%} &= e^{\mu + z_{0.995}\,\sigma}, \qquad z_{0.995} = 2.5758 \end{aligned}

Try it

Run the chain ladder on any line, paid or incurred, now or a year ago. Compare long-tail liability with short-tail own damage, then switch motor liability to end-2024 to see where the €116m came from.

Try itRun the chain ladder

Line

Triangle

As at

Cumulative incurred claims, €m. Bold: the latest diagonal. Italics: the chain ladder's projection. f under column j takes year j to j + 1.
AY	1	2	3	4	5	6	7	8	9	10	Ultimate
2016	238	352	448	534	543	587	607	616	619	623	623
2017	314	407	528	584	633	650	669	683	687	691	691
2018	258	449	549	622	644	685	700	704	708	712	712
2019	296	426	522	599	644	664	670	679	683	687	687
2020	288	434	551	598	651	671	687	696	700	704	704
2021	287	460	556	647	651	681	696	706	710	714	714
2022	363	584	734	777	816	854	873	885	890	896	896
2023	365	574	692	776	816	853	873	884	889	895	895
2024	315	533	662	743	781	816	835	846	851	857	857
2025	463	717	891	999	1,050	1,098	1,124	1,139	1,145	1,152	1,152
f	1.549	1.243	1.122	1.051	1.046	1.023	1.014	1.005	1.006
CDF	2.489	1.607	1.293	1.153	1.097	1.050	1.026	1.012	1.006	1.000

Reserve (ultimate − paid)€3,028.6m

1-in-200 reserve (lognormal)€3,473.5m

Ultimate, all accident years€7,932.6m

IBNR (ultimate − incurred)€1,461.6m

Mack standard error€163.0m

Coefficient of variation5.4%

Gross of reinsurance, undiscounted, no tail factor. The 1-in-200 assumes a lognormal reserve with the chain ladder's mean and Mack's standard error; z = 2.5758. Illustrative data.

Worked example

General liability, from triangle to 1-in-200

The book's general liability incurred triangle, € millions.

The reserve and its uncertainty

Step 1 of 5

1
The first factor
Accident years 2016–2024 have both development years 1 and 2. Add each column over those rows:
$\hat f_1 = \frac{\sum C_{i,2}}{\sum C_{i,1}} = \frac{352 + 407 + 449 + 426 + 434 + 460 + 584 + 574 + 533}{238 + 314 + 258 + 296 + 288 + 287 + 363 + 365 + 315} = \frac{4{,}219}{2{,}724} = 1.5488$

Regulatory lens

Under Solvency II the claims reserve is a best estimate: the probability-weighted, discounted value of the future payments, held with a risk margin on top. Reserve risk sits in the non-life premium and reserve risk sub-module (Delegated Regulation (EU) 2015/35, Articles 115 to 117): its volume is the best estimate for claims outstanding, net of reinsurance, per segment, and Annex II sets its standard deviation, for example 9% for motor vehicle liability, 10% for fire and other damage to property and 11% for general liability. EIOPA's calibration targets a 99.5% level over one year. L3.8.04: Premium risk and the underwriting cycle combines it with premium risk; M4.6: Solvency II and insurance regulation covers technical provisions and the risk margin in detail, and M4.8: Insurance accounting: IFRS 17 (overview) the IFRS 17 liability for incurred claims.

Spark Atoms

Spark AtomThe triangle that broke Lloyd's

Asbestos and pollution claims on policies written decades earlier helped drive the “unprecedented run of losses” that hit Lloyd's from the late 1980s. Under its 1996 Reconstruction and Renewal plan, a new company, Equitas, reinsured virtually all of the market's 1992 and prior non-life liabilities, so that new members would not inherit the old ones. In 2006 Berkshire Hathaway's National Indemnity agreed to reinsure Equitas in turn. No chain ladder on the 1970s triangles would have seen it coming.

Spark AtomOne cell, nine factors

General liability's 2025 accident year has one known number, €463m, and the chain ladder multiplies it by 2.49. Its IBNR of €689m has a standard error of €110m, a CV of 16%. Actuaries often prefer methods that lean on the expected loss ratio for the youngest years, such as Bornhuetter–Ferguson, for exactly this reason.

Key takeaways

A claims triangle shows each accident year's development; the chain ladder projects it with volume-weighted factors, each a column sum over a column sum.
On the book the reserve is €7,497.1m: €4,601m of case reserves and €2,896.1m of IBNR. Paid and incurred projections should agree; general liability's differ by €181m.
Mack's standard error measures the chain ladder's own uncertainty: €241.4m (3.2%) on the book, treating lines as independent.
Prior-year development is news about old years: €55.4m released overall in 2025, but motor liability strengthened by €116.3m on accident years 2023 and 2024, where claims costs ran ahead of the reserving assumptions.
Long-tail lines carry the most reserve risk: general liability is 48% IBNR with a mean term of 3.1 years. Its 1-in-200 excess is €445m by Mack against €999m from the standard formula's 3σ.

Reaction

Check what stuck

Question 1 · objective 1

In a small incurred triangle, the accident years that have both columns sum to 1,200 at development year 1 and 1,500 at development year 2. What is the development factor from year 1 to 2?

Question 2 · objective 1

The latest accident year has incurred claims of €400m at development year 1, and the cumulative factor to ultimate from year 1 is 2.5. What is its IBNR, in €m?

€m

Question 3 · objective 2

In 2025 RiskAtoms Insurance SE strengthened motor liability reserves by €116.3m, mostly on accident years 2023 and 2024. What is the most likely cause?

AClaims inflation: claims costs ran ahead of what was assumed when those years were reservedBA new windstorm in 2025CLower premium rates in 2025DThe chain ladder always strengthens the most recent years

Question 4 · objective 3

A reserve has a best estimate of €1,000m and a standard error of €100m. Treating it as lognormal, what is its 1-in-200 value, in €m (to the nearest million)?

€m

Question 5 · objective 4

General liability's reserve is €3,029m, motor own damage's €245m, on similar premiums. Which statement is right?

ALiability is long-tail: only 10% is paid in the first year, 48% of its reserve is IBNR, and its mean term is 3.1 yearsBOwn damage is under-reservedCLiability has more claims than own damageDThe two reserves carry the same risk per euro

Glossary

Accident year: Accident year: the year in which the insured event happened, whenever it is reported or paid.
Claims triangle: A table of claims by accident year (rows) and development year (columns). Only cells up to the latest diagonal are known.
Paid claims: Claims actually paid to date.
Case reserves: Case reserves: the claims handlers' estimates of what is still to pay on claims already reported.
Incurred claims: Paid plus case reserves: the cost of the claims the insurer knows about.
Development factor: Development factor: how much a column grows to the next, measured on the accident years that have both.
Chain ladder: A reserving method that projects each accident year to ultimate by multiplying its latest value by the development factors still to come.
Ultimate: The total that an accident year's claims will finally cost, once every claim is closed.
IBNR: Incurred but not reported: ultimate less incurred. Claims not yet reported, plus too-low case reserves on those that are.
Mack's standard error: Thomas Mack's 1993 formula for the standard error of a chain-ladder reserve, using only the triangle itself.
Prior-year development: Prior-year development: the change in the estimated cost of earlier accident years during the year. A rise is a strengthening, a fall a release.
Long-tail and short-tail: How long claims take to be reported and paid. Long-tail lines have reserves that are large, long and uncertain.

Deep Dives & Origins

Deep Dives

T. Mack, “Distribution-free calculation of the standard error of chain ladder reserve estimates”, ASTIN Bulletin 23(2), 1993, pp. 213–225. Advanced.
M. Merz and M. V. Wüthrich, “Modelling the claims development result for solvency purposes”, CAS E-Forum, Fall 2008, pp. 542–568: the one-year view. Advanced.

Origins

T. Mack, “Distribution-free calculation of the standard error of chain ladder reserve estimates”, ASTIN Bulletin 23(2), 1993 (assumptions, σ² estimator and extrapolation, Theorem 3 and its corollary).
M. Merz and M. V. Wüthrich, “Modelling the claims development result for solvency purposes”, CAS E-Forum, Fall 2008, pp. 542–568 (one-year against full run-off uncertainty).
Commission Delegated Regulation (EU) 2015/35, Articles 115–117 and Annex II, version at 31 December 2020 (legislation.gov.uk; EIOPA Single Rulebook).
EIOPA, Calibration of the Premium and Reserve Risk Factors in the Standard Formula of Solvency II, Report of the Joint Working Group, EIOPA-11/163, 12 December 2011.
HM Revenue & Customs, Lloyd's Manual, LLM1020: market developments 1992 to 2002.
A. Wilkinson and P. Hertz (Cadwalader, Wickersham & Taft), “The Lloyd's Reconstruction and Renewal: a success story or is the jury still out?”, 1998, via FindLaw (losses of 1988 to 1992 from asbestosis and pollution claims and the LMX spiral).

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