Chain Ladder vs Bornhuetter-Ferguson Loss Reserve Calculator

The method

Every number on this page is built from cumulative reported losses (paid plus case reserves) by accident year and age, in a 10 × 10 triangle valued at 12 to 120 months.

Development factors. The age-to-age factor from 12 to 24 months is the sum of the 24-month losses divided by the sum of the 12-month losses, over every accident year that has both. That is the volume-weighted average. Enter a selected factor at any age to override it. A tail factor takes losses from 120 months to ultimate. Multiplying the factors from an age onward gives the cumulative factor to ultimate (CDF), and its reciprocal is the percent reported.

Chain ladder ultimate = reported losses × CDF

Expected loss ratio ultimate = earned premium × a priori expected loss ratio

Bornhuetter-Ferguson ultimate = reported losses + earned premium × a priori loss ratio × (1 − percent reported)

BF keeps what has been reported and adds only the losses expected to be reported later. Algebraically it is a blend of the other two methods, weighted by percent reported:

BF ultimate = percent reported × chain ladder ultimate + (1 − percent reported) × expected loss ratio ultimate

The workbook checks that identity for every accident year. IBNR is ultimate less reported, so it includes development on known claims as well as claims not yet reported.

The worked book is a liability line with $490,000,000 of earned premium over 2016 to 2025, $262,422,000 reported, and a 65% a priori loss ratio from pricing. The 2025 year includes one $3,500,000 claim already reported at 12 months. The method in use is the common maturity rule: BF for any year less than 70% reported, chain ladder for the rest. Here that means BF for 2024 and 2025 and chain ladder for 2016 to 2023, for $69,779,916 of IBNR.

As a check on the arithmetic, the workbook reproduces the worked chain ladder example in Wikipedia's article. The article multiplies reported losses by cumulative factors rounded to three decimals (1.292 at 12 months, 1.110 at 24 months, and so on). Enter its triangle, a tail factor of 1.000, and each selected factor as the ratio of those printed cumulative factors (=1.292/1.110 for 12 to 24 months, and so on), and every accident year's ultimate matches the published figure within 40 cents. The total reserve is exactly $25,690,869.

This spreadsheet is the form above: paste your own triangle in and every figure on this page recalculates. Your own version could become the reserve review your team reruns each quarter with the new diagonal, or an API that feeds selected ultimates straight into your close.

Chain ladder vs Bornhuetter-Ferguson: which is more accurate?

Neither, in general. They agree on mature years and diverge on immature ones, and the gap is mostly about how much each method trusts losses reported early. On the same triangle with the same factors, chain ladder reserves $77,231,734 and BF reserves $67,872,288. That's a $9,359,447 difference.

From 2016 to 2021 the two methods agree within $80,000 a year, because there is little left to develop. By 2025, only 37.53% is reported, and chain ladder multiplies those reported losses by 2.664. Any noise in that 12-month figure is multiplied too.

ASOP No. 43 asks the actuary to consider multiple methods suited to the data, and to explain a reliance on a single method. Showing all three on one triangle is that comparison, done the same way every quarter.

How does one large loss affect chain ladder IBNR?

Chain ladder develops a known claim as if it were the first sign of many more. One $3,500,000 claim in the 2025 year adds $5,825,291 of IBNR, and it adds nothing under BF. That one claim accounts for nearly two-thirds of the gap between the methods.

The arithmetic is exact. A claim reported at 12 months is in the latest diagonal, so chain ladder takes it to ultimate at the 12-month CDF of 2.664. The $3,500,000 becomes $9,325,291, and the $5,825,291 increase is pure IBNR. The leverage is linear: a $7,000,000 claim would add $11,650,582. BF builds its unreported losses from premium, not from what was reported, so the claim stays at its reported value.

The decision this changes: under the maturity rule, 2025 is on BF, so the large claim doesn't move the $69,779,916 reserve at all. Switch the whole book to chain ladder and the same claim inflates the 2025 loss ratio to 82.18%, which is 17.18 points above the a priori 65%. That 82.18% is how a single claim ends up in next year's rate filing.

When does Bornhuetter-Ferguson give the wrong answer?

When the a priori loss ratio is wrong. BF trusts it for everything not yet reported, so on a deteriorating book it lags. In 2023, chain ladder points to a 72.41% loss ratio against the 65% a priori.

2023 is 70.14% reported, just over the 70% threshold, so the maturity rule uses chain ladder. Raise the threshold to 75% and 2023 moves to BF. The reserve then falls by $1,194,527, because BF assumes the unreported 29.86% will come in at 65% when the reported losses suggest worse.

The a priori ratio drives BF directly. In the workbook's tests, with every year on BF, the reserve is $57,430,397 at a 55% a priori loss ratio and $78,314,178 at 75%. Chain ladder doesn't move at all. Every point of a priori loss ratio is worth $1,044,189 of BF reserve here. Write down where the a priori came from (pricing, the plan, prior years' indications) and update it when the experience says it is stale.

How do you select loss development factors?

Start from the volume-weighted averages, then select. The selection is where actuarial judgment goes, and it can move the reserve as much as the choice of method. On Wikipedia's published triangle, the volume-weighted averages give a reserve of $25,819,851. The article's selected factors give $25,654,736, or $25,690,869 as printed, because the article rounds the cumulative factors to three decimals.

In the calculator, each of the nine ages has a selected-factor input. Leave it at 0 and the volume-weighted average is used. Enter a value and it overrides the average at that age only, and the CDFs and every method downstream follow. The tail factor, 1.007 at the defaults, carries the triangle from 120 months to ultimate. On long-tail lines it can be the single largest selection in the review.

Common reasons to override an average:

• A diagonal distorted by a change in case reserving practice.
• One accident year with a large loss sitting inside a factor.
• Too few years at older ages for the average to be stable.

Record the reason next to the selection. The next person to run the review should be able to see why a factor isn't the average.

What this calculator does not do

Stated plainly, so you know when to reach for a reserving system:

• Reported losses only. It doesn't separately project paid losses or reconcile the paid and reported indications, which most reviews do.
• One averaging method. All-year volume-weighted averages, with no simple, medial or latest-three-year averages to compare. Use the selected factors for those.
• No large-loss adjustment. The large loss input shows the leverage; it doesn't remove the claim from the triangle or develop it separately. Excess-layer and capped-triangle techniques are outside its scope.
• One a priori loss ratio for every year. In practice it often varies by year with rate changes and trend. Cape Cod, which derives it from the data, isn't included.
• Point estimates only. No Mack standard error, no bootstrap, no range of reasonable estimates.
• Annual 10 × 10 triangles, gross of reinsurance, undiscounted, no LAE.

Sources

• Chain-ladder method, Wikipedia (worked example)
• Bornhuetter–Ferguson method, Wikipedia
• Actuarial Standard of Practice No. 43, Property/Casualty Unpaid Claim Estimates
• Mack, "Parameter Estimation for Bornhuetter/Ferguson", CAS Forum, Fall 2006
• Sigma Actuarial Consulting, "The Bornhuetter-Ferguson Method"
• Loss Data Analytics, Chapter 11: Loss Reserving (Open Actuarial Textbooks Project)