IAI Actuarial Core Principles · Economic Modelling

Run-off Triangles: Reserving Methods for IAI Exams

A run-off triangle arranges claims data by accident year and development year so you can project unpaid claims. You calculate development factors from past years, apply them to the latest diagonal to get ultimate claims, then subtract payments to date to get the reserve. Methods differ in how they use data and prior views.

What this chapter covers

This chapter covers how to estimate the outstanding cost of claims from past data. You start with the triangle itself: claims grouped by accident (or origin) year along one axis and development year along the other, in incurred or paid, cumulative or incremental form. Every reserving method then builds on that layout.

You then learn the main projection methods. The chain ladder method uses development factors from the data. The inflation-adjusted chain ladder removes past claim inflation first and puts it back for future payments. The average cost per claim method projects claim numbers and average sizes separately. The Bornhuetter-Ferguson method blends an a priori estimate with actual experience. Loss ratio methods rely mainly on premium and an expected ratio. The chapter ends with the assumptions behind each method, the checks you run, and the practical issues that make a result unreliable.

The chapter ties into the rest of your modelling work through its focus on valuing uncertain liabilities, stating assumptions clearly and judging whether a model suits the data. Check the current IAI syllabus and past papers for your session to see exactly where this material is examined and whether it comes up in the written paper, the computer-based paper, or both.

Run-off triangles are calculation-heavy, which makes them a reliable place to earn marks if your method is tidy. The work is mechanical once you know the steps, so careful students can score well. The written parts also reward you for comparing methods and stating assumptions, which separates a full answer from a numbers-only one. If a computer-based paper includes this topic, practising the same calculations in a spreadsheet or in R saves time and cuts arithmetic slips.

Run-off triangles: topics in the order to study them

1Run-off Triangles and Claims Data BasicsEvery method reads the triangle, so you must be fluent with origin years, development years, cumulative versus incremental and the latest diagonal first.
2Chain Ladder MethodThis is the core method and the base for the others, so learn factors, projection and reserve calculation here.
3Inflation-Adjusted Chain LadderIt is the chain ladder with an added deflate-then-reinflate step, so it is easy once the basic method is solid.
4Average Cost per Claim MethodIt reuses the chain ladder idea on claim numbers and average cost, so it comes after you can project a triangle.
5Bornhuetter-Ferguson MethodIt needs the chain ladder development pattern plus an a priori ultimate, so you learn it after the data-driven methods.
6Loss Ratio and Other Reserving MethodsThese methods lean on premium and prior views, which makes them easier to compare once you know BF.
7Assumptions, Checks and Practical IssuesYou can only judge assumptions and choose between methods after you have worked through all of them.

How to prepare Run-off triangles

Build skill in layers: layout first, then one method at a time, then comparison. Always work from a small triangle on paper before you use a spreadsheet.

Draw a 4 by 4 triangle and practise reading it: identify the latest diagonal, convert between cumulative and incremental, and say what each cell means.
Do the chain ladder by hand: development factor = Σ C(i, j+1) ÷ Σ C(i, j) over the origin years where both are observed. Then multiply the latest cumulative figure by the remaining factors and subtract payments to date.
Repeat the same triangle with each other method, so you see how the answers differ and why.
For inflation adjustment, write the three steps every time: deflate to a common price base, project, then reinflate using the stated future inflation rate for each payment year.
For BF, write the formula in words and symbols: ultimate = claims to date + a priori ultimate × (1 − proportion developed). Practise getting the proportion developed from the chain ladder factors.
Write short answers on assumptions and when each method suits the data, and practise stating them in your own words.
Redo your hand calculations in a spreadsheet or in R, if your paper has a computer-based component, and check they match.

Common mistakes in Run-off triangles

Using the wrong cells to compute development factors
Fix: Write the sum ranges next to each factor and use only the origin years where both columns are observed.
Confusing cumulative and incremental figures
Fix: Check the triangle label first. Convert to cumulative before calculating factors, and convert back only if you need incremental payments.
Reporting ultimate claims as the reserve
Fix: Finish every question with the line: reserve = ultimate − paid to date. Label which one you are giving.
Applying inflation in the wrong direction or at the wrong date
Fix: Fix one price base at the start. Deflate past payments to it, project, then inflate each future payment from that base to its payment year.
Mixing up the BF proportion developed
Fix: The proportion developed is 1 ÷ the cumulative factor to ultimate. The BF formula applies the undeveloped share, (1 − that proportion), to the a priori ultimate.
Giving numbers with no discussion of assumptions
Fix: Add a line on which assumption drives the result, such as a stable pattern, steady inflation or a reasonable a priori loss ratio, and say when it could fail.

Last-day revision: Run-off triangles

A triangle shows claims by origin year and development year, and only the top-left part is known.
Chain ladder factors are ratios of column totals, using only the years where both columns are observed.
Ultimate = latest cumulative claims × product of the remaining development factors.
Reserve = ultimate − claims paid to date (for a paid triangle).
Chain ladder assumes future development follows the same pattern as past development.
For inflation adjustment, deflate, project, then reinflate at the assumed future rates.
Average cost per claim projects the number of claims and the average cost separately, then multiplies them.
BF ultimate = claims to date + a priori ultimate × (1 − proportion developed).
BF is more stable than chain ladder on immature years because it leans on the a priori estimate.
The loss ratio method sets ultimate = premium × expected loss ratio and ignores actual experience.
Check that the triangle is consistent across years: look for changes in claim handling, mix of business or reserving practice.
State your assumptions clearly in every answer, including inflation and the treatment of the tail.

Run-off triangles practice questions

Run-off triangles: frequently asked questions

What is a run-off triangle?

It is a table of claims data grouped by the year the claim arose and the number of years since then. Only the top-left part is known, and the missing lower-right part is what you project to estimate outstanding claims.

When should I use the Bornhuetter-Ferguson method instead of chain ladder?

Use BF when recent origin years have little data, so chain ladder factors would multiply a small, volatile figure. BF adds an a priori estimate of ultimate claims, which makes the result steadier. Its quality depends on how good that a priori estimate is.

Do I need to know how to do these calculations in a spreadsheet?

Check your paper's syllabus and past papers. If a computer-based paper covers this chapter, spreadsheet or R skill helps a lot. Whatever the format, practise the hand method first so you understand each step.

How do I handle inflation in a run-off triangle?

Convert the past claims to a common price base using past inflation, then apply the chain ladder to those adjusted figures. Next, reinflate the projected payments using the assumed future inflation for each payment year.