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Mini-course 3 of 6

Bornhuetter-Ferguson, Benktander and Cape Cod

When an origin year has little information, projecting reported losses amplifies noise. The methods in this mini-course blend what has been observed with a prior expectation and let you decide how much to trust each part.

30 min
Estimated time
4
Lessons
5
Questions
ObjectivesLessonsWorked exampleSelf-assessmentReadings
01

Objectives

By the end of this mini-course you will be able to:

  • Compute Bornhuetter-Ferguson IBNR and interpret it as a credibility blend.
  • Choose and defend a prior expected loss ratio (ELR).
  • Apply the Benktander method and explain why it is usually more accurate.
  • Estimate the ELR with the Cape Cod method and use it to check the prior.
02

Lessons

Lesson 1

Bornhuetter-Ferguson: an expectation for what is missing

Bornhuetter and Ferguson (1972) proposed estimating only the unreported part from an external expectation, instead of projecting it from reported losses:

\[IBNR_i^{BF}=P_i\cdot ELR_i\cdot\left(1-\frac{1}{CDF_i}\right)\]

If the latest year had a large loss reported early, the chain ladder multiplies it by the CDF; BF does not, because reported losses are added only once. The result can be written as a blend: BF ultimate = Z × chain ladder ultimate + (1 − Z) × expected ultimate, with Z = 1/CDF, the percentage reported.

Lesson 2

Choosing the expected loss ratio

BF inherits the bias of its ELR. Typical sources are the loss ratio in the pricing basis, the business plan, or the experience of mature years brought to current rate and cost trend levels. Each must be adjusted for rate and coverage changes between years.

An ELR taken from the sales budget rather than from experience is the most common mistake: if the rate turned out to be inadequate, BF hides it for years. That is why the ELR is checked against Cape Cod and against the loss ratios implied by mature years.

Lesson 3

Benktander: iterate once

Benktander (1976) proposed using the BF ultimate as a new prior and applying the formula again:

\[\hat U_i^{GB}=C_i+q_i\,\hat U_i^{BF},\qquad q_i=1-\frac{1}{CDF_i}\]

This is equivalent to giving the chain ladder a credibility of \(1-q_i^2\), higher than in BF. Mack (2000) showed that, under a simple model, the Benktander mean squared error is almost always smaller than that of BF and of the chain ladder. If you iterate indefinitely, the result converges to the chain ladder.

Lesson 4

Cape Cod: the ELR the data imply

The Cape Cod method, also known as Stanard-Bühlmann, estimates a common ELR from the data: it divides reported losses by used-up premium, which is each year's premium multiplied by its percentage reported.

\[ELR^{CC}=\frac{\sum_i C_i}{\sum_i P_i/CDF_i}\]

That ELR is then used in the BF formula. It assumes the loss ratio is the same in every year once adjusted for rate and trend, so premium must be brought to current level first. It is an excellent check on the prior: if Cape Cod comes out well above the chosen ELR, the prior is probably out of date.

03

Worked example

Using the series triangle, the pricing team proposes a 62% expected loss ratio for every year, taken from the original pricing basis. We compare the four methods. Amounts in millions of pesos.

Step 1. Cape Cod ELR

YearPremium% reportedUsed-up premiumIncurred
20213,60099.0%3,564.42,380
20223,80096.1%3,651.72,570
20234,10090.2%3,697.82,780
20244,30075.4%3,240.62,450
20254,60040.7%1,874.41,450
Total20,40016,028.911,630
Cape Cod ELR = 11,630 / 16,028.9 = 72.56%.

Step 2. IBNR by method

Year% unreportedChain-ladderBF (ELR 62%)BenktanderCape Cod (ELR 72.6%)
20211.0%23.822.123.825.9
20223.9%104.491.9103.9107.6
20239.8%302.4249.4297.2291.8
202424.6%800.9656.8765.4768.7
202559.3%2,108.41,689.91,860.41,977.6
Total3,339.92,710.13,050.73,171.5
IBNR by method, millions of pesos.
  1. For 2025, the percentage unreported is 59.3%. BF gives 4,600 × 62% × 0.5925 = 1,689.9; the chain ladder gives 2,108.4.
  2. Benktander takes the BF ultimate (3,139.9) as the prior: 1,450 + 0.5925 × 3,139.9 = 3,310.4, that is, IBNR of 1,860.4.
  3. Cape Cod estimates an ELR of 72.6%, ten points above the proposed 62%. The chain ladder implied loss ratios for mature years (66.8% and 70.4%) are also above 62%.
  4. Conclusion: the 62% prior is stale and BF understates. A reasonable selection for 2025 lies between Cape Cod (1,977.6) and the chain ladder (2,108.4), depending on what the claims department explains about reporting speed.
Key point: The methods do not compete: they cross-examine each other. Total IBNR ranges from 2,710.1 (BF) to 3,339.9 (chain ladder), and almost all of the gap comes from 2025 and from a prior ELR the data do not support.
04

Self-assessment

Choose one answer per question: you will see right away whether it is correct and why. At the end, check your score. Nothing is stored or sent.

1. Which part of the reserve does BF estimate with the expected loss ratio?
2. A recent year had a large loss reported in its first month. Which method reacts most to that loss?
3. If a year's CDF is 2.0, what credibility does Benktander give the chain ladder?
4. What does the Cape Cod method assume?
5. In the example, Cape Cod gives 72.6% against a 62% prior. What would a prudent actuary do?
05

Readings

Primary sources and reference texts, all publicly accessible at the time of publication.

  1. Bornhuetter, R. and Ferguson, R. (1972). The Actuary and IBNR. Proceedings of the CAS, vol. LIX.
  2. Mack, T. (2000). Credible Claims Reserves: The Benktander Method. ASTIN Bulletin 30(2).
  3. Friedland, J. (2010). Estimating Unpaid Claims Using Basic Techniques, version 3. Casualty Actuarial Society (free full text).

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