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

Pop Quiz

What is one way of coming up with the ECR used in the ECR method? Click for Answer 

Study Tips

VIDEO: F-09 (001) Bornhuetter-Ferguson Method → 5:00 Forum

Study the examples. Do the BattleActs practice problems. Memorize the concepts. Work the old exam problems.

Estimated study time: 3 days (not including subsequent review time)

BattleTable

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

• calculating ultimate / unpaid / IBNR
• understanding that BF is a weighted average of the development and ECR methods

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)	ultimate: - rptd devlpt (tort reform)	ultimate: - reported B-F (tort reform)
(2018.Spring #17)	Excel Practice Problems
(2018.Spring #19)	Excel Practice Problems
E (2017.Fall #5)	ultimate: - reported B-F 1
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: - where is B-F better
E (2013.Fall #17)	unpaid: 2 - 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

¹ Requires calculating earned premium at current rate level which is discussed in Pricing - Chapter 5. Note however that sample answer 1 did not do the on-level calculation correctly. There they assumed the rate change of 5% applied to all in-force policies (like a law change that produces a vertical line, not a diagonal line, in the CRL diagram.) Sample answer 2 correctly calculated the average rate level and current rate level just like in an EP@CRL problem where the diagonal splits the CY square in half.

² There is an error in the statement of this problem. The problem asks you to calculate the unpaid amount, but this is not possible from the given information because you are not given paid losses, only reported losses. The question should have asked for the IBNR amount.

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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/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

Observation 1: The second term on the right side of the formula: %unreported x ECR is the same thing as unreported claims which is actually just the IBNR.

Observation 2: If you're using the paid BF method then the second term on the right side of the formula: %unpaid x ECR is the same thing as total unpaid which is just the case + IBNR.

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 it 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

A few practice problems...

Practice: 4 problems on the basic BF method   (thx jj96!)

And the quiz...

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Example B: Tort Reform with the BF Method

We're going to take a quick look at a BF method problem that requires an extra adjustment for tort reform. The key is precisely understanding the effect of the tort reform and how to properly adjust for it in your calculation. You'll see it's different (easier actually) from how you adjust for tort reform with the ECR method.

Recall that the ultimate loss is the sum of the paid and unpaid loss. Suppose you're given:

• unpaid = 1,000
• paid = 400

Then the ultimate loss is 1,000 + 400 = 1,400. Now, if tort reform is introduced and expected to reduce all future paid amounts by 15% then:

• adjusted unpaid = 1,000 x (1 - 15%) = 850 (remember that unpaid is the same thing as future paid)

So taking into account tort reform, the adjusted ultimate loss is 850 + 400 = 1,250. Here are a couple of practice problems based on this idea:

Practice: 2 problems on BF method with tort reform   ← typo in question: you need to find the ultimate for AY 2024 not 2025

Here's a exam problem just like the tort reform practice problems:

E (2019.Spring #19)

And the quiz...

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Example C: Benktander Variation on BF

The Benktander method is an iterative method and is a very simple extension of the BF method. Recall the formula for the reported BF method:

• Ult_r-BF   =   (reported claims) + (1 – 1/CDF) x Ult_ECR

The Benktander variation looks almost the same except you substitute Ult_r-BF for Ult_ECR on the right side.

UltBenk   =   (reported claims) + (1 – 1/CDF) x Ultr-BF

Let's calculate the Benktander estimate of ultimate using our example from Intro to BF Method. The first step is to calculate the "normal" reported BF ultimate:

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

Then we take the result, 561, and substitute it into the Benktander formula:

• Ult_Benk   =   375 + (1 – 1/CDF) x 561   =   535

The Benktander method is called an iterative method because you can continue indefinitely. Just take the the result on the left side of the equation:

• next iteration Ult_Benk   =   375 + (1 – 1/1.4) x 535   =   528

• next iteration Ult_Benk   =   375 + (1 – 1/1.4) x 528   =   526

Pop Quiz A!    :-o

• What answer do you get if you apply the iterative Benktander approach an infinite number of times? Click for Answer 

The quiz has old exam problems using the Benktander method.

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BF Method Concepts

You've learned how to apply the BF method but you also need to learn when to apply it. Like any reserving method, it works well in certain situations and not so well in others.

BF Method Key Assumption: unreported (or unpaid) claims will emerge in accordance with expected claims

Let's examine what this means. We know the BF method can be thought of as a weighted average of the development method and ECR method, but the formula we generally use when doing a calculation is this:

• Ult_r-BF   =   (reported claims) + %unreported x Ult_ECR      ← formula for the reported BF method (discussion is similar for paid BF method)

The second term on the right side of the equation is just the unreported claims (or unpaid claims in the paid BF formula) and you can see it's completely determined by the ECR. (thx ES!) We are implicitly assuming that the reported claims provide no information about the emergence of the unreported claims. That always seemed a little strange to me. Let's look at a simple example:

• suppose your reported-claims-to-date = 500
• suppose your Ult_ECR = 1,000 and the reported CDF = 1.5 so that the %unreported = (1 - 1/CDF) = 0.333
• based on this, you expect 0.333 x 1,000 = 333 to be reported in the future (this is the emergence)

So Ult_r-BF = 500 + 333 = 833. But notice that the 333 has nothing to do with the reported-claims-to-date of 500. You could have had 1,000,000 of reported claims but the expected emergence of 333 would be the same because the 333 is based only on the ECR, not any claims activity to date. That's an extreme example but my point is that there are situations where the BF assumption makes sense and some where it doesn't.

Recall our discussions from prior chapters, Chapter 7 - Development Method - Summary & Observations and Chapter 8 - ECR Method Concepts. There we looked at situations where each of those methods works well and situations where they don't. Since the BF method is a weighted average of those methods, the situations where BF works well is going to be a mixture. In other words, it's a bit of a mishmash where BF works and where it doesn't, but here are a few general comments.

Question: in what situations is the key assumption for the BF method more likely to be satisfied [Hint: NxDx]

• New line or territory (where data is thin or volatile)
• Development method may not work well for less mature periods (LDFs could be highly leveraged, especially for lines with longer emergence and settlement patterns)

(the HINT is similar to the one for situations where the ECR method works but with the 2^nd and 4^th items deleted and marked with an x)

Another idea to keep in mind when selecting an appropriate reserving method is responsiveness versus stability. We discussed this in Chapter 8 - ECR Method Concepts in the context of the development and ECR methods, so let's see where BF fits.

Responsiveness versus Stability: how does the BF method compare to the development & ECR methods regarding responsiveness & stability

• since BF is a weighted average of the development & ECR methods, the responsiveness & stability of BF should lie between those methods

→ responsiveness: ECR < BF < development method

(BF is more responsive than the ECR method but less responsive than the development method.)

→ stability: development method < BF < ECR

(BF is more stable than the development method but less stable than the ECR method.)

Occasionally, we run into a little bit of a problem with the BF method:

CDF < 1.0: describe the problem with the BF method when the reported (or paid) CDF is less than 1.0

• BF is a weighted average between the development and ECR methods
• the weight given to the development method is 1/CDF

→ problem: if CDF < 1.0 then the development method weight is > 1.0 which doesn't make sense (violates a principle of credibility-weighting)

→ soultion: set a lower bound of 1.0 for the CDF when using BF

Here are a couple of good exam problems with short answer components about BF concepts. Remember, it isn't enough to simply know the mechanics of a reserving method – you also have to understand when it works and where it doesn't.

E (2014.Spring #14)

E (2013.Spring #18)

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A Few Extra Exam Problems

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POP QUIZ ANSWERS

• Use the development method to calculate a preliminary ultimate then divide by EP to come up with an ECR. The final ECR selection for the ECR method is judgmental.

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Pop Quiz A - ANSWER

→ Continued application of the Benktander method converges to the development method estimate of ultimate. In this case, 525.

• You can prove this algebraically but it's easier to understand conceptually. Recall the reported BF ultimate is a weighted average between the development method and the ECR method. For our example we had:

Ult_r-Dev   <   Ult_r-BF <   Ult_ECR

• By the same reasoning, the first iteration of Benktander is a weighted average of the development method and the BF method so we have:

Ult_r-Dev   <   Ult_Benk   <   Ult_r-BF   <   Ult_ECR

• And again, the next iteration of Benktander will be a weighted average of the development method and the first iteration of Benktander:

Ult_r-Dev   <   next iteration Ult_Benk   <   Ult_Benk   <   Ult_r-BF   <   Ult_ECR

→ You can see what's happening: Each successive iteration of Benktander is squeezed between the previous iteration and the development method. The result is that is must converge to the development method estimate. (If the development method had originally been higher than the ECR method, then the inequalities would be reversed but the reasoning is the same.)

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