Friedland09.BornFerg

Reading: Friedland, J.F., Estimating Unpaid Claims Using Basic Techniques, Casualty Actuarial Society, Third Version, July 2010. The Appendices are excluded.

Chapter 9: Bornhuetter-Ferguson Method

BattleTable

Based on past exams, the main things you need to know (in rough order of importance) are:

• fact A...
• fact B...

reference	part (a)	part (b)	part (c)	part (d)
E (2019.Spring #17)	ultimate: - reported B-F	ultimate: - reported B-F	combining data: - argument against
E (2019.Spring #19)	Friedland07.Development	ultimate: - reported B-F (tort reform)
E (2018.Spring #17)
E (2018.Spring #19)
E (2017.Spring #18)	Friedland08.ExpectedClms	unpaid & IBNR: - reported B-F
E (2016.Fall #21)	IBNR: - B-F method	ECR assumption: - assess reasonableness	alternate method: - recommend & justify
E (2016.Spring #16)	Friedland08.ExpectedClms	Friedland07.Development	ultimate: - paid B-F
E (2015.Fall #18)	ultimate: - reported B-F	CDFs < 1.0: - does B-F still work?
E (2015.Spring #19)	ultimate: - reported B-F	selected ultimates: - discuss appropriateness	ultimate: - Benktander	unpaid: - explain Benktander iterations
E (2014.Fall #18)	IBNR: - Benktander	credibility weighting: - explain Benktander
E (2014.Spring #14)	ultimate: - reported B-F	rptd B-F vs rptd dvlpt: - situations where equal	rptd B-F vs rptd dvlpt: - situations B-F is better
E (2013.Fall #17)	unpaid: - Benktander	unpaid: - Benktander 1000 iterations
E (2013.Spring #4)	ultimate: - reported B-F
E (2013.Spring #18)	key assumption: - B-F method	credibility-weighting: - explain for B-F	credibility-weighting: - violation of assumption	(d) paid vs rptd B-F for rising LRs (e) B-F vs Cape Cod

In Plain English!

Example A: Intro to BF Method

The following information is as of Dec 31, 2023. Calculate the ultimate for AY 2023 using the reported Bornhuetter-Ferguson method.

EP	1,000
ECR	65%
reported claims12	375
reported CDF12→ult	1.4

First, note how easy it is to calculate the ultimate using the reported development method:

• Ult_r-Dev   =   375 x 1.4   =   525

And the ultimate for the ECR method is equally easy:

• Ult_ECR   =   65% x 1,000   =   650

The reported BF method is just a weighted average of these where the weight, z, given to the reported development method is:

z   =   %reported   =   1/CDF

So,

• z   =   1 – 1/CDF_12→ult   =   1/1.4   =   0.714

and therefore,

• Ult_r-BF   =   [ 0.714 x 525 ] + [ (1 – 0.714) x 650 ]   =   561

Actually, there's a simpler but equivalent formula for the BF ultimate:

Ultr-BF   =   (reported claims) + %unreported x UltECR   =   (reported claims) + (1 – 1/CDF) x UltECR

The proof that these formulations of the BF method are equivalent is not likely to be asked on the exam but you can click the link if you'd like to see it. It's easy. And it's helpful to understand the interpretation of the BF method as a weighted average of the development and ECR methods. Anyway, let's verify that this simpler formula gives the same result.

• Ult_r-BF   =   (reported claims) + %unreported x Ult_ECR   =   375 + (1 – 1/CDF) x 650   =   375 + (1 – 1/1.4) x 650   =   561

Here's a straightforward exam problem on the ECR and reported BF methods. Note that you have to choose LDFs so you might not get exactly the same answer as in the examiner's report. You just have to make sure you make reasonable selections and write a short phrase to justify your choices. In this problem, the triangle is so small that unless there is an obvious outlier in the LDF triangle is should be ok to select an all period weighted average.

E (2017.Spring #18)

There's also a paid version of the BF method and it works exactly like the reported version. Just substitute paid amounts in place of reported amounts and the paid CDF in place of the reported CDF. (The ultimate for the ECR method would be the same for both the paid and reported BF methods.)

Ultp-BF   =   (paid claims) + %unpaid x UltECR   =   (paid claims) + (1 – 1/CDF) x UltECR