Claims triangulation is the arrangement of an insurer's claims data in a table that shows how each year's claims have developed at successive valuation dates. Because recent years have fewer valuations than older ones, the filled cells form a triangle, called a loss triangle, a claims development triangle or a run-off triangle.
This entry explains how a triangle is laid out and grouped, what data it can hold, a worked example from triangle to projected ultimate claims, how triangles are used as diagnostics, what distorts them and how insurers publish claims development tables in their accounts.
What a claims triangle shows
The Casualty Actuarial Society's reserving text by Jacqueline Friedland describes three dimensions in a development triangle: rows, columns and diagonals.1 Each row is an origin period, such as an accident year, so each row follows a fixed group of claims. Each column is an age, measured in months from the start of the origin period. Each diagonal is a valuation date, so a triangle with four diagonals holds the claims position at four successive year ends.1
Three ways to read a development triangle
Friedland writes that development triangles are used both as a diagnostic tool and as the primary input to numerous techniques for estimating unpaid claims.1 The Faculty and Institute of Actuaries' Claims Reserving Manual (1997) defines the basic chain ladder link ratio for each delay as the sum of one column divided by the sum of the column before it, excluding the last entry, and among the weaknesses it lists is that trends and anomalies in the data are not allowed for if the method is operated blindly.2
Accident year, underwriting year and report year triangles
The rows of a triangle can be defined in several ways, and the choice changes what the triangle measures. Friedland lists report year, underwriting year, treaty year, policy year and fiscal year as alternatives to accident year, and states that accident year is by far the most common basis for triangles in the US and Canada, that reinsurers often organise claims by underwriting year, and that in some areas, such as Lloyd's of London, financial reporting by underwriting year is more common than accident year.1
| Basis | Claims grouped by | Advantage described | Disadvantage described |
|---|---|---|---|
| Accident year | Date of occurrence | Claims occur over a shorter period, so ultimate claims should become reliably estimable sooner; useful for tracking inflation, law changes and major events | Possible mismatch between claims and exposures, and claims from policies priced at more varied times |
| Policy or underwriting year | Year the policy (or reinsurance contract) incepted | A true match between claims and premiums, useful when underwriting or pricing changes | An annual policy year spans 24 months of claims, so ultimate claims take longer to estimate; harder to isolate a single large event |
| Report year | Date the claim is reported to the insurer | The number of claims is fixed at the close of the year, giving more stable data; used for claims-made covers | Measures development on known claims only, so pure IBNR needs other methods |
| Calendar year | Date of the transaction | No future development, and readily available from financial reporting | Cannot address development, so rarely used for projection; used for exposures and diagnostic testing |
Underwriting year triangles behave differently in their first year. Lloyd's notes in its claims development table that, because the table is on an underwriting year basis, there is an apparent jump from the end of the underwriting year to one year later, as a large proportion of premiums are earned in the year of account's second year of development.3 The PRA noted a related distortion in 2019: on its chart of incurred movements, years 2004 and 2010 were distorted for firms reporting on an underwriting year basis, because Hurricane Katrina in 2005 and the 2011 natural catastrophes also affected the previous underwriting year for some firms.4
Paid, incurred and claim count triangles
The cells can hold different measures. Friedland's list of common types includes reported claims, case outstanding, cumulative paid claims, incremental paid claims, reported claim counts, closed claim counts and outstanding claim counts.1 Reported claims are cumulative paid claims plus case outstanding at the valuation date, so a reported triangle is the paid triangle plus the case outstanding triangle.1 Friedland uses "reported" rather than "incurred" because the term incurred losses, used by many throughout the industry, can be misunderstood as to whether it includes IBNR, and many actuaries use "case incurred" to show that it does not.1 Figures can be gross or net of reinsurance and before or after recoveries such as subrogation.1 A cumulative triangle shows the total to date in each cell; an incremental triangle shows only the movement in each period, and Friedland labels incremental triangles specifically because her examples use cumulative values unless stated.1
Paid and reported triangles develop at different speeds. In Friedland's example based on industry-aggregated US private passenger automobile data for accident years 1998 to 2007, net of reinsurance, the cumulative factors for paid claims are greater than those for reported claims at every age, and the text describes this as a typical relationship for many lines of property and casualty insurance.1
| Age (months) | Reported: factor to next age | Paid: factor to next age | Reported: cumulative factor to ultimate | Paid: cumulative factor to ultimate |
|---|---|---|---|---|
| 12 | 1.164 | 1.702 | 1.292 | 2.390 |
| 24 | 1.056 | 1.186 | 1.110 | 1.404 |
| 36 | 1.027 | 1.091 | 1.051 | 1.184 |
| 48 | 1.012 | 1.044 | 1.023 | 1.085 |
| 60 | 1.005 | 1.019 | 1.011 | 1.040 |
| 120 (tail to ultimate) | 1.000 | 1.002 | 1.000 | 1.002 |
A cumulative factor of 2.390 at 12 months means the paid amount is projected to grow to 2.390 times its 12 month value by ultimate, against 1.292 for reported claims. The difference arises because reported claims already include the case outstanding on open claims, which paid claims do not yet include.
A worked triangle, step by step
Friedland sets out the development technique, also known as the chain ladder technique, in seven steps.1 Its underlying assumption is that claims recorded to date will continue to develop in a similar manner in the future.1
The seven steps of the development technique
The example below uses invented figures for a small portfolio of paid claims, in thousands of pounds, to show the arithmetic. It follows the same steps but is not taken from any source.
| Accident year | 12 months | 24 months | 36 months | 48 months | 60 months |
|---|---|---|---|---|---|
| 2021 | 1,000 | 1,800 | 2,160 | 2,376 | 2,400 |
| 2022 | 1,100 | 2,090 | 2,420 | 2,640 | |
| 2023 | 1,200 | 2,160 | 2,640 | ||
| 2024 | 1,300 | 2,350 | |||
| 2025 | 1,400 |
Each age-to-age factor divides one cell by the cell to its left: accident year 2021 grew from 1,000 at 12 months to 1,800 at 24 months, a factor of 1.800. Doing this for every pair of cells gives the factor triangle below, with the averages beneath it. The volume-weighted average divides column totals, using only the years that have reached both ages: for 12 to 24 months, 1,800 plus 2,090 plus 2,160 plus 2,350 is 8,400, against 1,000 plus 1,100 plus 1,200 plus 1,300, or 4,600, which gives 1.826. The simple average of the four individual factors is 1.827. The medial average leaves out the highest and lowest factors, here 1.900 and one of the two factors of 1.800, and averages the rest, 1.800 and 1.808, to give 1.804.
| 12 to 24 | 24 to 36 | 36 to 48 | 48 to 60 | 60 to ultimate | |
|---|---|---|---|---|---|
| 2021 | 1.800 | 1.200 | 1.100 | 1.010 | |
| 2022 | 1.900 | 1.158 | 1.091 | ||
| 2023 | 1.800 | 1.222 | |||
| 2024 | 1.808 | ||||
| Simple average | 1.827 | 1.193 | 1.095 | 1.010 | |
| Volume-weighted average | 1.826 (8,400 ÷ 4,600) | 1.193 (7,220 ÷ 6,050) | 1.095 (5,016 ÷ 4,580) | 1.010 (2,400 ÷ 2,376) | |
| Medial average | 1.804 | 1.200 | 1.095 | 1.010 | |
| Selected factor | 1.826 | 1.193 | 1.095 | 1.010 | 1.005 (assumed tail) |
| Cumulative factor to ultimate | 2.419 | 1.325 | 1.111 | 1.015 | 1.005 |
The selections here are the volume-weighted averages, and the tail factor of 1.005 is an assumption that paid claims grow by a further 0.5% after 60 months. The cumulative factors are built from the oldest age backwards, rounding to three decimal places at each step: 1.010 times 1.005 is 1.015 at 48 months; 1.095 times 1.015 is 1.111 at 36 months; 1.193 times 1.111 is 1.325 at 24 months; and 1.826 times 1.325 is 2.419 at 12 months.
| Accident year | Age at valuation (months) | Paid to date | Cumulative factor | Projected ultimate | Ultimate less paid |
|---|---|---|---|---|---|
| 2021 | 60 | 2,400 | 1.005 | 2,412 | 12 |
| 2022 | 48 | 2,640 | 1.015 | 2,680 | 40 |
| 2023 | 36 | 2,640 | 1.111 | 2,933 | 293 |
| 2024 | 24 | 2,350 | 1.325 | 3,114 | 764 |
| 2025 | 12 | 1,400 | 2.419 | 3,387 | 1,987 |
| Total | 11,430 | 14,526 | 3,096 |
The final column, 3,096 in total, is the projected amount still to be paid. Because this is a paid triangle, it covers both the case reserves already set on known claims and claims incurred but not reported; the IBNR entry works through how a reported triangle separates the two. The newest year carries the most leverage: accident year 2025 accounts for 1,987 of the 3,096, because its cumulative factor applies to only one year of payments.
The choice of average matters. Had the medial average of 1.804 been selected for 12 to 24 months, the cumulative factor for 2025 would be 1.804 times 1.325, or 2.390, and its projected ultimate 1,400 times 2.390, or 3,346, which is 41 lower, while every other year would be unchanged. In the same way, the factor from 48 to 60 months rests on accident year 2021 alone, so one year's experience sets the pattern applied to every later year.
Friedland describes the trade-off between stability, from including more years in an average, and responsiveness, from using only the most recent years.1
Using triangles as diagnostics
Friedland describes triangles of ratios and averages built from the basic data, and explains what each one tests.1
| Diagnostic triangle | What it tests | Caution in the text |
|---|---|---|
| Paid-to-reported claims ratio | Changes in case outstanding adequacy or in settlement patterns | A change may come from paid claims or from case outstanding, and an unchanged ratio can hide offsetting changes in both |
| Closed-to-reported claim counts ratio | Changes in the speed of settlement | Can move for operational reasons, such as a catastrophe that disrupts claims handling, a new call centre or new claim set-up guidelines |
| Average case outstanding (case outstanding divided by open claims) | Changes in case outstanding adequacy: a fall down a column suggests weakening, a rise suggests strengthening | Can also change with the mix of business; in a stable environment it would rise down a column at the rate of inflation |
| Average paid and average reported claim values | How average claim values evolve at each age | Large claims, present or absent, can distort averages |
The PRA has used the same diagnostic. Its November 2019 letter to chief actuaries reported continued evidence of weakening case reserves, pointing to higher paid-to-incurred claims ratios on more recent years in Casualty Financial and Professional Liability, Medical Malpractice and US General Liability classes, which it said indicates a possible weakening in the case reserving basis, and its appendix chart came from PRA analysis of Solvency II triangles as at year end 2018.4
What distorts a triangle
Because the technique projects past patterns forward, anything that changes those patterns changes the answer. Friedland lists other important assumptions of the development method as consistent claim processing, a stable mix of types of claims, stable policy limits and stable reinsurance retention limits.1
| Cause | Effect on the triangle |
|---|---|
| Stronger case reserves1 | Reported claims and the latest age-to-age factors both rise, so without adjustment the reported projection overstates ultimate claims |
| Faster or slower settlement1 | Paid development assumes no significant change in the speed of claims closure and payment; a change moves paid factors along the latest diagonals |
| Large claims1 | With too little data, large claims can greatly distort age-to-age factors and projected ultimates; actuaries may exclude them and project them separately |
| Claims inflation4 | Reserving assumptions can lag the inflation seen in claims: the PRA found in 2019 that some claims teams' views of annual claims inflation were 5% to 10% higher than allowed for in reserving, pricing and planning assumptions |
| Law and tort reform14 | Historical development may no longer predict the future; the PRA's 2019 letter named the Ogden rate, MedCo, Claims Portal and whiplash reforms as motor examples |
| Latent claims2 | Claims emerge at once across many origin years, so large movements occur across each affected diagonal and conventional triangulation methods will not work |
| Delays in recording case reserves4 | The PRA observed excessive delays between claim notification and case reserves being recorded, and inconsistencies where third parties such as lawyers and delegated authorities set case reserves |
Friedland's text adds that the development technique is particularly suitable for high-frequency, low-severity lines with stable and relatively timely reporting.1 The 1997 Claims Reserving Manual gives asbestosis as an example of latent claims, where claims start to be reported at roughly the same time irrespective of the underwriting year to which they relate.2 In delegated authority business, claims data can reach the risk carrier through bordereaux, whose claims fields Lloyd's sets out in its Coverholder Reporting Standards.5 With fields like these, bordereaux can supply the data for each new diagonal of paid and reported triangles.
Claims development tables in published accounts
Insurers reporting under IFRS 17 publish a form of triangle in their accounts. Paragraph 130 of the standard, reproduced in the Australian Accounting Standards Board's AASB 17, which incorporates IFRS 17, requires an entity to disclose actual claims compared with previous estimates of the undiscounted amount of the claims.6 The disclosure starts with the period of the earliest material claims still uncertain at the reporting date, but need not start more than 10 years before the end of the reporting period, and is not required for claims whose uncertainty is typically resolved within one year; the entity must reconcile it with the carrying amount of its insurance contracts.6 On first application, an entity need not disclose previously unpublished claims development earlier than five years before the end of the year in which it first applies the standard, but must say so if it does not.6
Beazley's 2025 annual report shows how this looks in practice. Its gross table starts with the 2019 underwriting year, being five years before the end of the reporting period in which IFRS 17 was first applied by the group, revalues historic periods at current exchange rates, and includes expected claims, reinsurance recovery cash flows and claims handling expenses but not the risk adjustment.7 Beazley's estimate for its 2021 underwriting year was $2,732.5m at the end of the underwriting year, $3,004.2m one year later, $2,815.0m two years later, $2,639.2m three years later and $2,815.9m four years later.7 Beazley reconciles the table, with other items including its 2018 and prior underwriting years, to a gross discounted liability for incurred claims of $7,979.1m; the reconciling items include an impact of discounting of $(639.3)m and a risk adjustment for non-financial risk of $705.1m.7
The Lloyd's market publishes a gross and a net claims development table in its Aggregate Accounts, which are prepared where practicable in accordance with UK GAAP, including FRS 102 and FRS 103.3 The tables show estimates of earned ultimate cumulative claims for syndicates in aggregate by Lloyd's underwriting year, converted at 2025 year end exchange rates.3
| Underwriting year | At end of underwriting year | One year later | Latest estimate | Gross claims paid | Gross claims reserve |
|---|---|---|---|---|---|
| 2016 | 8,608 | 18,863 | 22,117 | 20,301 | 1,816 |
| 2017 | 16,445 | 26,211 | 29,465 | 27,118 | 2,347 |
| 2018 | 13,303 | 24,097 | 28,200 | 24,811 | 3,389 |
| 2019 | 10,129 | 22,329 | 22,990 | 19,215 | 3,775 |
| 2020 | 11,635 | 21,109 | 21,297 | 16,511 | 4,786 |
| 2021 | 13,322 | 24,954 | 28,080 | 19,823 | 8,257 |
| 2022 | 14,422 | 24,244 | 24,374 | 14,746 | 9,628 |
| 2023 | 11,910 | 22,063 | 22,240 | 9,556 | 12,684 |
| 2024 | 15,114 | 27,533 | 27,533 | 7,506 | 20,027 |
| 2025 | 13,479 | Not yet available | 13,479 | 1,322 | 12,157 |
Compare the second and third columns: the jump appears in every year with two valuations. Read along a row and the re-estimates show the direction of development: in the full table, the 2016 estimate rose at every valuation from 18,863 one year later to 22,117 nine years later, while the 2020 estimate peaked at 21,862 two years later and fell to 21,297 five years later.3
Lloyd's market gross claims development table totals, 31 December 2025
Lloyd's states that the two most critical assumptions for claims provisions are that the past is a reasonable predictor of the likely level of future claims development and that the rating and other models used for current business are fair reflections of likely ultimate claims, and that adjustments to provisions set in prior years are reflected in the period in which they are made.3 It reported that 2025 marked the 20th consecutive year of prior year releases for the market, with a prior year development of (3.3)% against (2.5)% in 2024, releases being shown in brackets.3
Show References Hide References
- Friedland, J. (Casualty Actuarial Society). Estimating Unpaid Claims Using Basic Techniques. Version 3, July 2010.
- Faculty and Institute of Actuaries. Claims Reserving Manual, volume 1, Section 2: Introduction to Reserving. September 1997.
- Lloyd's. 2025 Aggregate Accounts. 2026.
- Prudential Regulation Authority. Feedback from recent PRA reserving reviews. Letter to chief actuaries, 5 November 2019.
- Lloyd's. Coverholder Reporting Standards User Guide, Version 5.2. 20 August 2019.
- Australian Accounting Standards Board. AASB 17 Insurance Contracts, compiled standard incorporating IFRS 17. Compilation no. 5, Federal Register of Legislation, registered 9 September 2026.
- Beazley plc. Annual Report and Accounts 2025. 2026.
Frequently Asked Questions
What is the difference between a loss development factor and a cumulative development factor?
A loss development factor, also called an age-to-age, report-to-report or link ratio, measures growth over a single interval, such as from 12 to 24 months. A cumulative development factor, also called an age-to-ultimate factor, is the product of all the remaining interval factors and measures total growth from one age to ultimate. The terminology varies by market. The Casualty Actuarial Society text notes that many actuaries use LDF to mean the loss development factor to ultimate, while in South Africa actuaries often use LDF for the incremental factor and UDF for the cumulative factor, so the meaning of an abbreviation needs checking in any analysis.
Selecting the interval factors is a matter of judgement. The same text lists what actuaries look for: a smooth progression of factors across development periods, with development decreasing steadily as claims mature; a small range of factors within each column; enough volume and homogeneity for the experience to be credible; systematic patterns that suggest operational or external change; and whether the history still applies to the current book. It also describes comparing new selections with the prior year's, which shows how much of a change in projected ultimate claims comes from the factors and how much from actual experience.
Why do paid and incurred loss triangles give different results?
The two triangles rest on different assumptions, about the adequacy of case reserves and the speed of payment, and the Casualty Actuarial Society text tests what happens when each changes, using constructed scenarios.
In a steady state, both methods produce an estimated IBNR equal to the actual figure. When claim ratios rise with no change in reserving, both remain correct: projected ultimate claims for all years were $10,249,350 against $8,804,525 in the steady state, and the reported and paid results matched. When case reserve adequacy was raised by 6% in 2007 and 25% in 2008 (for the latest four accident years only), case outstanding for accident year 2008 rose from $380,075 to $475,094, and the actual IBNR fell to $253,336 from $438,638 because true ultimate claims had not changed. The reported projection gave total unpaid claims of $1,478,573, which the text calculates as 20% greater than the actual $1,230,997, while the paid triangle, and so the paid projection, was unchanged from the steady state.
What is a tail factor in a loss triangle?
A tail factor carries claims from the oldest age in the triangle to their final value. It is needed when the factors at the most mature ages are still significantly greater than 1.000, so development has not stopped by the last column. The Casualty Actuarial Society text says the point beyond which no tail factor is required varies tremendously by line: short-tail claims are generally settled within months or a few years, while for long-tail lines such as US medical professional liability and workers compensation, some claims can take more than fifteen years to settle. For lines with significant subrogation after payment, factors at the end of the triangle may approach 1.000 from below.
Because the tail factor multiplies every origin year's projection, the text warns that it can create disproportionate leverage on total estimated unpaid claims. It describes three approaches: relying on industry benchmark factors; fitting a curve to the observed or selected factors and extrapolating it, with exponential decay a common assumption; and, for paid development where reported claims are already considered to be at ultimate, using reported-to-paid ratios at the latest age. It quotes Joseph Thorne in 1978 that care must be taken in projecting the tail from older years to recent ones, because trends in benefits, claim types and settlement practices can lengthen or shorten it.
How many years of data does a loss triangle need?
The Casualty Actuarial Society text says that, if possible, development should be analysed out to the point at which it ceases, when the selected factors equal 1.000. The number of periods needed varies by line, jurisdiction and data type: paid claims typically need more development periods than reported claims, and reported claims often need more than reported claim counts. Automobile physical damage claims settle much more quickly than general liability claims, so they need fewer periods.
The interval does not have to be annual. The same text describes monthly, quarterly and semi-annual triangles, with the credibility of the experience and the stability of development as the main considerations in choosing. Volume matters as much as length. An insurer entering a new line or territory, or a smaller insurer with a limited portfolio, may lack credible development history; the text notes that the development technique is still often used in these cases, with factors selected from benchmark patterns, such as comparable lines or industry data, and applied to the insurer's own latest claims.
Can a claims triangle be built from bordereaux data?
Yes, provided each claim's transactions carry the dates needed to place them in the right row and column. The Lloyd's Coverholder Reporting Standards (version 5.2, 2019) set out claims bordereau fields that include date of loss, date claim first advised or made, paid this month and previously paid amounts for indemnity and fees, reserves for indemnity and fees, and date first reserve established. The same guide requires risk and claims data under coverholder appointment agreements with Lloyd's Brussels to be provided monthly to support financial and actuarial processing and oversight. With fields like these, bordereaux from a coverholder can supply the data for each new diagonal of paid and reported triangles.
The quality of the triangle depends on how the data is recorded upstream. In its 2019 letter to chief actuaries, the PRA reported that some information in claims watch-lists had not been incorporated in case reserves, and that some claims teams monitored development less closely for follow business than for lead business, without separating the two when tracking trends. The Casualty Actuarial Society text makes a related point about claims-made covers: extended reporting endorsements that are not coded as a new policy can allow development beyond 12 months even on annual policies, so the actuary must understand how data enters and is tracked in the insurer's systems before choosing how to aggregate it.
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