Friedland13.BerqSherm

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

Chapter 13: Berquist-Sherman Methods

Pop Quiz

Given:

• Suppose you have paid & reported claim (loss) triangles, and paid & reported count triangles.

Questions:

• What are the best 3 diagnostic triangles to look at for evidence of changes in case reserve adequacy. (Hint: See Chapter 6 (Diagnostics) - Changes in Case Reserve Adequacy)
• What are the best 3 diagnostic triangles to look at for evidence of claim settlement rate. [Hint: See Chapter 6 (Diagnostics) - Changes in Settlement Rate)

BONUS:

• For an increase in case reserve adequacy, do the diagnostics show increases or decreases down columns?
• For an increase in claim settlement rate, do the diagnostics show increases or decreases down columns?

Click for Answer 

Study Tips

There are 2 versions of the Berquist-Sherman method:

reported method - adjusts for changes in case reserve adequacy (full method is frequently asked on exam)

paid method - adjusts for changes in settlement rate of claims (full method is not frequently asked on exam)

Both methods have time-consuming calculations but the reported method is easier and is asked more often. The paid method has one particular step (the interpolation step) that is very confusing and you are not often asked to do the calculations form start to finish. For the paid method you're more likely to be asked concept questions or to perform only a portion of the calculations.

There's a short section at the beginning of the chapter on Data Selection & Rearrangement. The idea is you should first just try to select "good" data before going to the trouble of making complicated adjustments as in the Berquist-Sherman methods.

Estimated study time: 1 week (not including subsequent review time)

BattleTable

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

• reported Berquist-Sherman method - calculate ultimate / unpaid / IBNR
• paid Berquist-Sherman method - calculate ultimate / unpaid / IBNR

reference	part (a)	part (b)	part (c)	part (d)
E (2019.Fall #21)	reported BS: - calc unpaid
E (2019.Spring #14)	accuracy of paid devlpt: - evaluate
E (2019.Spring #22)	reported BS: - calc ultimate
E (2018.Spring #21)
E (2017.Fall #24)	reported BS: - calc IBNR	alternate method: - propose & justify
E (2017.Spring #22)	paid BS - claim count adjustment
E (2016.Fall #23)	reported BS, BF - calc unpaid
E (2016.Spring #20)	paid BS - calc ultimate
E (2016.Spring #21)	Friedland06.Diagnostics	ultimate: - suggest method
E (2015.Fall #21)	paid BS - disposal rate	paid BS - distortion
E (2015.Spring #22)	impact: - settlement rate change	response: - to settlement rate change	related methods: - rptd B-S, rptd devlpt
E (2015.Spring #23)	evaluate methods: - paid & rptd devlpt	adjustment to method: - propose & justify
E (2014.Fall #20)	reported BS, paid BS - calc ultimate
E (2014.Spring #18)	ultimate: - select & apply 2 methods
E (2013.Fall #6)	reported BS: - calc ultimate 1
E (2013.Fall #21)	paid BS - calc ultimate
E (2013.Fall #23)	operational change: - explain	investigating the change: - questions for clms mgmt 2	diagnostics: - to check results 3
E (2013.Spring #23)	reported BS: - adjust rptd triangle	compare IBNR: adjusted vs unadjusted

¹ This question requires knowledge of pricing. You may need to come back to it later if you haven't covered the pricing material yet.

² See Chapter 4 - Meetings for potential questions.

³ Review Chapter 6 - Diagnostics but note that valid answers can include diagnostics other than those specifically discussed in chapter 6.

In Plain English!

Data Selection and Rearrangement

In Chapter 6, we discussed the effects of changes in case reserve strength and claim settlement rate on diagnostic triangles. In Chapter 7, we investigated how changes in case reserve strength affect the development method. Here are 2 facts, you absolutely must know:

• a change in case reserve strength affects reported data (paid data is not affected)
• a change in settlement rate affects paid data (reported data is not affected)

So, if I gave you a paid loss triangle and a reported loss triangle and told you there had been a change in case reserve strength, which triangle would you rather use to develop estimates of ultimate? It's easy: use the paid triangle because it is not affected by changes in case strength. The reported loss triangle could have distortions that cause the reported development method to either over or under-estimate the true ultimate.

But what if I told you there had been a change in the settlement rate of claims, then which triangle would you rather use? Again, it's easy: use the reported triangle because it is not affected by changes in settlement rate. In this situation, it's the paid triangle that's distorted so the paid development method is less likely to be accurate.

What we did above is called data selection. In other words, select data that's more likely to be free of distortions and try not to use data that you know may lead to inaccurate results.

Here are a few other examples...

situation	data selection
• change in definition of counts (claim versus claimant) • see Chapter 11 - FS Method Key Assumptions for a simple example	• use exposures in place of claim counts
• change in policy limits between policy years	• use policy year data in place of accident year data
• court decisions that correlate with report date • see Chapter 11 - Disposal Rate Method for an exam problem involving a court decision	• use report year data in place of accident year data
• significant growth or shrinkage causing a shift in average accident date	• use accident quarters in place of accident years
• change in mix of business • see Chapter 7 - Scenario 6 (Change in Product Mix) for an example of a change in mix	• subdivide data into more homogeneous groups

mini BattleQuiz 1 You must be logged in or this will not work.

Berquist-Sherman Paid Method

Description of Method

Here's the first fact you have to memorize:

Purpose of Berquist-Sherman Paid Method: Adjust paid loss triangle for changes in claim settlement rate.

In practice, the first thing you should do in a reserve analysis is check for things like changes in the claim settlement rate. If there have not been changes, then you don't need the Berquist-Sherman paid adjustment. In the Pop Quiz at the top of this wiki article, we reviewed diagnostics for increases in settlement rate:

→ paid/reported loss, paid/reported counts or average case O/S loss

But there's another very useful diagnostic if you also know UC (Ultimate counts), It's called the CDR (Claims Disposal Rate).

CDR   =   CPC / UC

Recall CPC is Cumulative Paid Counts. And very often in exam problems on the BS Paid Method, you will be given ultimate counts. That's the case in the first BS Paid exam problem we're going to look at below. So if the ultimate counts for AY 2020 is 100, and at 12 months development you have 50 paid counts, then CDR₁₂ = 50/100 = 0.5. If at 24 months the paid counts is 80 then CDR₂₄ = 80/100 = 0.8, and so on.

This problem is an example of the full Berquist-Sherman paid method. Don't bother trying to figure out the solution in the examiner's report because Alice's solution is much clearer. (See further down.)

E (2016.Spring #20)

Before looking at Alice's solution however, here's how the method works:

• Normally you're given CPC, UC, CPL

(For a review of these abbreviations, see Chapter 11 - Disposal Rate Method.)

Step 1

Step 1a: calculate CDR

→ This is to check for changes in claim settlement rate. Increases down columns generally indicate an increase in settlement rate.

→ Make CDR selections for each column. Do not select the average. Do select the most recent diagonal.

Step 1b: restate CPC using the CDR from step 1a

→ This is done by multiplying the UC for each AY by the vector of selected CDRs

Step 2

• restate CPL using either linear or exponential interpolation

(Many students find this step confusing. We'll discuss it in detail further down. Don't worry if you don't follow it the first time through.)

Step 3

• apply the development method to the restated CPL triangle

(Just apply the standard development to the restated triangle of cumulative paid losses. This should be easy for you! Theoretically, this restated CPL is now corrected for any changes in the claim settlement rate. All AYs should be at the same level and the development method applied to this triangle shouldn't be distorted.)

Here is Alice's solution to the exam problem referenced above. It looks complicated but don't be alarmed. It's all quite easy except for the interpolation step, but don't worry if you don't completely understand the interpolation the first time through. We'll cover that in much more detail in the next section.

Solution: 2016.Spring #20

Interpolation Step in BS Paid Method

If you've read the Chapter 13 in the Friedland source text, you may have noticed that linear interpolation is not discussed. The source text only explains exponential interpolation, but here's an exam problem that specifically asks you to use linear interpolation. Don't do it now. I just wanted to point that out.

E (2013.Fall #21)

We're going to isolate and study the interpolation step of the Berquist-Sherman paid method in detail. Then when you attempt a full BS paid problem, it should go smoothly.

Example A: Linear Interpolation

Suppose you're given 2 points on the real plane:

• (x₀, y₀) = (10,500)
• (x₁, y₁) = (20,600)

Question 1: Use linear interpolation to find the y-value corresponding to x = 16.

Answer:

• y  =   [(16 - 10) / (20 - 10)] * (600 - 500) + 500   =   560

Actually, the formula makes it harder to see what's going on. Because the numbers are so simple, you can easily see that x = 16 is 60% of the way between 10 and 20. (That's the quantity inside the square brackets.) Then the corresponding y-value must also be 60% of the way between 500 and 600 which is obviously 560.

Of course, if the numbers aren't so simple you might not be able to do the calculation in your head. Suppose you're now given:

• (x₀, y₀) = (21.2, 378.6)
• (x₁, y₁) = (64.7, 982.0)

Question 2: Use linear interpolation to find the y-value corresponding to x = 31.2.

Answer:

• y = [(31.2 - 21.2) / (64.7 - 21.2)] * (982.0 - 378.6) + 378.6 = 517.3

You can write out the formula as shown below, but I don't think this is the most helpful way to think about it:

• y   =   [ (x - x₀) / (x₁ - x₀) ]   *   (y₁ - y₀) + y₀

Think about it like this instead:

linear interpolation:   y  =   [ proportional distance of x from x0 ]   *   (y1 - y0)   +   y0

In terms of high school algebra, you're basically just fitting a linear equation y = mx + b to the given points and then using that equation to get the new y-value corresponding to the given x-value. But you don't actually have to find m and b. That would take too long. If you understand how interpolation works intuitively, you can do it much quicker.

Example B: Linear Interpolation Applied to Paid Data

Instead of x-values and y-values we're now going to use paid counts and paid losses which is what you have in the Berquist-Sherman paid method. Suppose you're given the following data for AY 2020.

data type	12-months	24-months	36-months	48-months
CPC	10	20	25	28
CPL	500	600	640	670

Remember the first step of the BS paid method is to calculate CDR (Claims Disposal Rate) then use it to restate CPC (Cumulative Paid Counts). Let's suppose you did that and found the restated CPC values to be:

data type	12-months	age 24-months	age 36-months	age 48-months
restated CPC	15	18	24	28
restated CPL	?	?	?	?

The interpolation step is finding the corresponding restated CPL-values in the above table. Let's do them one by one:

12-months:

→ since restated CPC₁₂ = 15 is within the 12-24 interval of the original CPC-values, we do the interpolation using original CPC and CPL values as follows:

• (x₀, y₀)   =   (CPC₁₂, CPL₁₂)   =   (10, 500)
• (x₁, y₁)   =   (CPC₂₄, CPL₂₄)   =   (20, 600)

→ restated CPL₁₂   =   [ (15 - 10) / (20 - 10) ] * (600 - 500) + 500   =   550

24-months:

→ since restated CPC₂₄ = 18 is still within the 12-24 interval of the original CPC-values, we do the interpolation using the same original CPC and CPL values as before:

• (CPC₁₂, CPL₁₂)   =   (10, 500)
• (CPC₂₄, CPL₂₄)   =   (20, 600)

→ restated CPL₂₄   =   [ (18 - 10) / (20 - 10) ] * (600 - 500) + 500   =   580

36-months:

→ since restated CPC₃₆ = 24 is within the 24-36 interval of the original CPC-values, we do the interpolation using original CPC and CPL values as follows:

• (CPC₂₄, CPL₂₄)   =   (20, 600)
• (CPC₃₆, CPL₃₆)   =   (25, 640)

→ restated CPL₃₆   =   [ (24 - 20) / (25 - 20) ] * (640 - 600) + 600   =   632

48-months:

→ if we assume the 48-month value is the lastest diagonal then the restated CPC and CPL values are the same as the orignal values and no calculation is necessary

→ restated CPL₄₈   =   original CPL₄₈   =   670

Bonus Question 1: what if restated CPC₃₆ = 26 (instead of 24)

→ if restated CPC₃₆ = 26 then you have to use a different interval and this is the part that confuses people. You have to notice that 26 is in the original 36-48 CPC-interval and then use the original CPC and CPL values corresponding to that interval:

• (CPC₃₆, CPL₃₆)   =   (25, 640)
• (CPC₄₈, CPL₄₈)   =   (28, 670)

→ restaed CPL₃₆   =   [ (26 - 25) / (28 - 25) ] * (670 - 640) + 640   =   650

Bonus Question 2: what if restated CPC₁₂ = 9 (instead of 15)

→ if restated CPC₁₂ = 9 then you have a choice. You could use the same interval as we did for interpolating the paid value of 15 and you would get:

• CPL₁₂ = [ (9 - 10) / (20 - 10) ] * (600-500) + 500 = 490

→ Or, use the origin (0,0) as your other interpolation point as follows, which I think makes more sense because if CPC = 0 then CPL = 0 also. (Note you may get a different answer but the source text doesn't specify a method so either should be acceptable.)

• (CPC₀, CPL₀)   =   (0, 0)
• (CPC₁₂, CPL₁₂)   =   (10, 500)

→ restaed CPL₁₂   =   [ (9 - 0) / (10 - 0) ] * (500 - 0) + 0   =   450

Now you can attempt (2013.Fall #21). The examiner's report performs their linear interpolation using selected disposal rates without explicitly calculating the restated paid counts. This is potentially quicker than my method but it's easier to make a mistake. You can decided for yourself how you'd rather do it. There's a thread on that topic in Actuarial Outpost.

Alice's solution below shows both linear and exponential interpolation even though the official problems asked only for linear interpolation. (If you're asked to perform exponential interpolation on the exam, you should always be given the parameters for the exponential regression.)

E (2013.Fall #21)

Solution: 2013.Fall #21

And here's the solution to (2016.Spring #16) using both linear and exponential interpolation. (The question asked only for exponential interpolation.)

E (2016.Spring #20)

Solution: 2016.Spring #16

Example C: Exponential Interpolation

Let's revisit the first problem from Example A: Linear Interpolation but this time we're going to use exponential intepolation. Suppose you're given 2 points on the real plane:

• (x₀, y₀) = (10,500)
• (x₁, y₁) = (20,600)

Suppose you're also given the regression parameters a = 417 and b = 0.01840 for the exponential regression y = ae^bx.

Question 1: Use exponential interpolation to find the y-value corresponding to x = 16.

Answer: All you have to do is substitute x = 16.

• y   =   417 * e^{(0.01840 * 16)}   =   559.7

This is very close to the answer of 560 you get with linear interpolation. Exponential interpolation in a Berquist-Sherman problem is generally easier than linear interpolation because you will be given the exponential regression parameters.

Let's now do a problem that's closer to what you'll have to do on the exam using paid counts and paid losses. We did this exact problem earlier using linear interpolation but here you are also given the exponential regression parameters for each successive pair of (count, loss) values.

data type	12-months	24-months	36-months	48-months
CPC	10	20	25	28
CPL	500	600	640	670

AY 2020	0-12		12-24		24-36		36-48
	a	b	a	b	a	b	a	b
	use 12-24 values	use 12-24 values	417	0.01840	463	0.01299	437	0.1539

The regression parameters for each interval are based on these pairs of given values:

12-24: uses (10, 500) & (20, 600)

24-36: uses (20, 600) & (25, 640)

36-48: uses (25, 640) & (28, 670)

Remember the first step of the BS paid method is to calculate CDR (Claims Disposal Rate) then use it to restate CPC (Cumulative Paid Counts). Let's suppose you did that and found the restated CPC values to be:

data type	12-months	age 24-months	age 36-months	age 48-months
restated CPC	15	18	24	28
restated CPL	?	?	?	?

The interpolation step is finding the corresponding restated CPL-values in the above table. Let's do them one by one:

12-months:

→ since restated CPC₁₂ = 15 is within the 12-24 interval of the original CPC-values, we use the a and b for the 12-24 range:

→ restated CPL₁₂   =   417 * e^{(0.01840 * 15)}   =   549.5

24-months:

→ since restated CPC₂₄ = 18 is still within the 12-24 interval of the original CPC-values, we use the a and b for the 12-24 range as before:

→ restated CPL₂₄   =   417 * e^{(0.01840 * 18)}   =   580.7

36-months:

→ since restated CPC₃₆ = 24 is within the 24-36 interval of the original CPC-values, we use the a and b for the 24-36 range:

→ restated CPL₃₆   =   463 * e^{(0.01299 * 24)}   =   632.4

48-months:

→ if we assume the 48-month value is the lastest diagonal then the restated CPC and CPL values are the same as the orignal values and no calculation is necessary

→ restated CPL₄₈   =   original CPL₄₈   =   670

Bonus Question 1: what if restated CPC₃₆ = 26 (instead of 24)

→ if restated CPC₃₆ = 26 then you have to use a different interval and this is the part that confuses people. You have to notice that 26 is in the original 36-48 CPC-interval and then use the a and b for the 36-48 range:

→ restaed CPL₃₆   =   437 * e^{(0.01539 * 26)}   =   652.0

Bonus Question 2: what if restated CPC₁₂ = 9 (instead of 15)

→ if restated CPC₁₂ = 9 then you have a choice. You could use the same interval as we did for interpolating the paid value of 15 and you would get:

• CPL₁₂ = [ (9 - 10) / (20 - 10) ] * (600-500) + 500 = 490

→ Or, use the origin (0,0) as your other interpolation point as follows, which I think makes more sense because if CPC = 0 then CPL = 0 also. (Note you may get a different answer but the source text doesn't specify a method so either should be acceptable.)

• (CPC₀, CPL₀)   =   (0, 0)
• (CPC₁₂, CPL₁₂)   =   (10, 500)

→ restaed CPL₁₂   =   [ (9 - 0) / (10 - 0) ] * (500 - 0) + 0   =   450

POP QUIZ ANSWERS

• To detect changes in case reserve adequacy the best 3 diagnostics to look at are:

→ paid/reported loss, average reported loss, average case O/S loss

• To detect changes in claim settlement rate the best 3 diagnostics to look at are:

→ paid/reported loss, paid/reported counts or average case O/S loss

• BONUS: For an increase in case reserve adequacy and an increase in claim settlement rate, green font indicates an increase down columns in the diagnostic triangles and red font indicates a decrease.

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