Friedland10.CapeCod

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

Chapter 10: Cape Cod Method

Pop Quiz

The BF method is a weighted average of 2 other methods. What are those 2 methods? Click for Answer 

Study Tips

VIDEO: F-10 (001) Cape Cod Method → 4:30Forum

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

• When you finish this chapter, you should probably pause and review chapters 7-10 before moving on to the Frequency & Severity method in chapter 11.

Breaking News!!

• AvR emailed us with a link to a research paper related to the: Cape Cod Method. It explains the origin of the name "Cape Cod":

→ "In response, Hans Bühlmann and James Stanard developed a method to estimate this expected loss ratio in the early 1980s. (See Stanard 1985.) The method was so named as it sprang out of an actuarial conference held on Cape Cod."

• Awesome shout-out to AvR!!

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 incorporating trending and tort reform
• understanding that CC method is a weighted average of the development and ECR methods
• understanding how BF and CC are similar and different

reference	part (a)	part (b)	part (c)	part (d)
E (2019.Fall #19)	ultimate: - rptd CC (adjustments)
E (2019.Spring #18)	ultimate claims: - rptd CC (trend)	identify scenario: - paid CC works better
(2018.Spring #18)	BA PowerPack Step 4
E (2017.Fall #21)	ultimate: - rptd CC (tort reform)
E (2017.Spring #23)	ultimate: - paid devlpt (tort reform)	ultimate: - paid CC (tort reform)
E (2016.Spring #18)	Cape Cod vs B-F: - compare	Cape Cod vs B-F: - adjustments to rptd loss	Cape Cod vs B-F: - adjustments to EP	court decision: - identify best method
E (2015.Spring #17)	IBNR: - rptd CC (trend, rate chg)
E (2014.Spring #15)	IBNR: - rptd BF	IBNR: - rptd CC (trend)	B-F vs Cape Cod: - rising claims, thin data
E (2013.Fall #20)	IBNR: - rptd CC (adjustments)

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In Plain English!

Example A: Intro to the CC Method

Great news! You can cover the CC method pretty quickly because it's very similar to the BF method. The formulas for Ult_r-BF and Ult_r-CC look exactly the same as you can see below:

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

The difference is in how Ult_ECR is calculated.

BF: You calculate Ult_ECR using the ECR method. The ECR method does adjust for trend and tort reform but the final ECR selection is judgmental. For a quick refresher, see: ECR Method Trend Adjustment and ECR Method Tort Reform Adjustment.

CC: You calculate Ult_ECR using a formula. There is no judgment involved. The formula adjusts the losses for trends and tort reform but also adjusts to the EP to current rate level.

Before we look at an easy example of the CC method, you need to understand the concept of used-up premium or UUP for short.

• Let's take a step back a look at how earned premium works: If an insurer writes an auto policy for $1,200 then we have $1,200 in written premium and this written premium is earned uniformly over the policy term. That means that in each month, 1/12 x $1,200 = $100 of premium is earned.
• Now let's get back to used-up premium. If we start with $1,200 of written premium, this written premium gets "used-up" according to the loss development pattern of the claim corresponding to the policy, which is not uniform over the policy term. Of course, used-up premium doesn't really make sense (Thx AvR!!) for an individual policy because an individual policy may not even have a claim; rather, it must be done in aggregate. (And it's based on CY earned premium, not written premium.)

Suppose you're given the data set below. Normally if you wanted an overall loss ratio, you'd first multiply the reported claims by the CDF, take the sum, then divide by the sum of the EP. If you check you'll see that works out to 3,312.5 / 3,000 = 110.4%. But now we're going to do it kinda backwards: Instead of multiplying the reported claims by their CDF, we divide the EP by the CDF as an alternate way of putting the claims and EP on the same level. This is shown in the table where UUP = Used Up Premium.

CY/AY	reported claims	CDF to ult	EP	UUP = EP / CDF
2023	750	1.5	1,000	667 = 1,000 / 1.5
2024	475	2.5	1,000	400 = 1,000 / 2.5
2025	250	4.0	1,000	250 = 1,000 / 4.0
total	1,475	--		1,317

The CC method then uses 1,475 / 1,317 = 112.0% as the ECR. No judgment, it's just a formula. Note that that it doesn't match the 110.4% calculated above. It's close, but algebraically they are not equal. Friedland describes the concept of used-up premium as follows:

The used-up premium represents the premium corresponding to the claims that are expected to be reported (paid) through the valuation date.

Note: Do you remember what 1/CDF represents conceptually? For a reported CDF, the quantity 1/CDF is the %reported. Using that interpretation, dividing EP by the CDF to get UUP makes perfect sense.

Further down, there are a couple of relatively easy practice problems. Note there is 1 extra step in these problems because you have to apply the pure premium trend to the losses before calculating the ECR. Be careful: If you're given a pure premium trend, remember that it applies to losses not premiums. Without digressing too much, we learned (or will learn) from pricing that:

• frequency = counts / exposures
• severity = dollars / counts

Then if

• pure premium = dollars / exposures

by simple algebra we have

• pure premium = frequency x severity

(It's a common mistake to apply the pure premium trend to the premiums instead of the losses.)

There is a step in the solution to the practice problems where the ECR is adjusted by backing out the pure premium trend. You can ignore that here because the adjustment is 1.00 anyway. We'll cover that in the next section.

Practice: 2 problems on the basic CC method

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Example B: Apply CC to Specific Years

We're going to revisit the practice problems from the previous section but instead of calculating the ultimate for the most recent AY, we're going to do it for earlier AYs and that requires an extra adjustment. We had 3 AYs: 2023, 2024, 2025. In both, you were asked to calculate the ultimate for AY 2025 which was the most recent AY. The first two practice problems below are the same except in the first one you have to instead calculate the ultimate for AY 2024; and in the second, for AY 2023.

CC method for earlier AYs: To calculate the ultimate for a particular AY, you have to restate the ECR to be at the level of that AY

The reason for this extra adjustment is that the way we applied the CC method, the calculated ECR was at the level of the most recent AY. But if we want the ultimate for a different AY, we have to put the ECR at the level appropriate for that AY. For this problem, that means detrending the ECR by an appropriate number of years. (There are potentially other adjustments as well but we'll get to those in the next section.)

Practice: 2 problems on CC method with detrending

Ok, now you're ready to give this old exam problem a try. Note that part (b) asks you to compare the reported CC method with the paid CC method. The paid CC method works exactly the same way but you use paid claims instead of reported claims. If you need a hint for part (b), take a look back at Chapter 7 - Influences of a Changing Environment. That section demonstrates how the reported development method compares to the paid development method. Pay particular attention to Scenario 3. You can re-watch the video or just check Alice's notes.

E (2019.Spring #18)

Example C: The Full CC Method

In the previous two sections, we covered special cases of the CC method. Let's now put it all together. For any CC problem you must be given the following at a minimum:

• claims (either reported or paid), CDFs, EP

If that's all you're given, you'd use the CDFs to get the UUP, then ECR = claims/UUP, then you can apply the formula for Ult_r-CC. But it might not be very accurate without trends, rate changes, and tort reform information.

To properly adjust the given claims, we also need:

• trend factors
• tort reform factors

Sometimes you're not given the factors directly. If that's the case the problem is harder. You may instead be given frequency & severity trends so that you'd have to calculate the trend factors yourself. Not terribly difficult but it does represent an extra step. For tort reform, you may only be told that a certain reform is expected to reduce all claims by 10% after a certain date so you'd have to calculate the appropriate factors yourself. We covered how to do that in tort reform with the ECR method. But however you get to these factors, you would calculate the adjusted claims for each AY as follows: (adjusted to the level of the most recent AY)

adjusted claims   =   (unadjusted claims) x (trend factor) x (tort reform factor)

To properly adjust the given EP, we need:

• premium trend factors
• CDFs (we would always be given this in a Cape Cod problem)
• rate level adjustment factors

Again, you might not be given the rate level adjustment factors directly. Sometimes you're only given the rate change history so you would have to calculate the factors yourself. This is a major topic in pricing which you can find by clicking here. In any case, you would then calculate the used-up premium or UUP for each year as follows:

UUP   =   EP x (premium trend) x (rate level adjustment factors) / CDF

The next step is calculating the ECR (at the level of the most recent AY)

selected ECR   =   Σ (adjusted claims) / Σ UUP

If you're calculating the ultimate for a year that is not the most recent AY, you have to make further adjustments to the calculated ECR to put the ECR on the level appropriate for that AY. (We covered a simple version of this in a previous section.) That means you have to back out:

• premium trends (from the UUP)
• rate level changes (from the UUP)
• trends (from the adjusted claims)
• tort reform adjustments (from the adjusted claims)

Common mistake: Rate level changes are backed out by multiplying the selected ECR by the on-level changes. That's because EP is in the denominator of the selected ECR formula. (Trends and tort reform adjustments are backed out in the "normal" way by dividing the selected ECR by those factors because claims are in the numerator of the selected ECR formula.)

These practice problems do not use a premium trend (although they do use a pure premium trend, which is the same thing as a claim trend). See the next section for a difficult exam problem that does incorporate a premium trend.

Practice: 4 problems on CC method with multiple adjustments (shout-out to KL for finding an error in the solution! → has been corrected - was using 3 years instead of 4 to find ECR)

And here's a very similar exam problem but note the examiner's report presents the solution slightly differently. I would not recommend using their method.

E (2013.Fall #20)

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Example D: Harder CC Problems

Once you understand the basic version of the CC method, here are some harder problems for you to try.

Problem 1: E (2019.Fall #19)

Why it's harder:

• They don't give you any rate level adjustment factors (on-level factors). You have to use your knowledge of pricing to figure them out. Click here to jump to that section.
• They also don't give you the pure premium (claim) trend factors or the premium trend factors but they're easy to calculate because they give you the percentage trends. You would use this formula:

(1 + trend)^{appropriate integer exponent}.

Problem 2: E (2017.Spring #23)

Why it's harder:

• They don't give the CDFs for calculating the used-up premium. (Cumulative Development Factors). You have to calculate them yourself from the paid data triangle but it's very tricky because you first have to adjust the triangle in part (a) to take into account the tort reform. If you don't make that adjustment correctly, you'll get the wrong CDFs.
• In general, in an exam situation, if you get stuck don't linger for too long. Just do the best you can and keep moving. Every exam has 1 or 2 really difficult problems that everyone will have trouble with. It's absolutely imperative you keep to your time management plan and leave enough time to answer all the questions you do know how to do.

Problem 3: E (2015.Spring #17)

Why it's harder:

• They don't give you pure premium trend factors. You have to come up with these yourself by first calculating the pure premium, then looking at year-over-year changes, then judgmentally selecting your own trend. The trend factor for each year is then:

(1 + trend)^{appropriate integer exponent}.

• They don't give you any rate level adjustment factors (on-level factors). All you're told is that there have been significant rate changes but no change in mix of business and that the policies are written uniformly throughout the year. If you get stuck and can't for the life of you figure out what the on-level premium is then just make up some reasonable-looking premium numbers. That way you can still proceed and demonstrate you know how to apply the Cape Cod method, even if some of your intermediate calculations are wrong. And obviously, make sure you write down all the relevant Cape Cod formulas. You will get at least some partial credit for that.

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CC Method Concepts

We already discussed how the CC and BF methods are both similar and different but let's tie everything together.

Question: how are the CC and BF methods similar

• formula for ultimate is the same (for both paid & reported versions of each method)

paid version:        Ult_r-CC = Ult_r-BF    =   (paid claims)        + %unpaid x Ult_ECR         =   (paid claims)        + (1 – 1/CDF) x Ult_ECR

reported version: Ult_r-CC = Ult_r-BF    =   (reported claims) + %unreported x Ult_ECR   =   (reported claims) + (1 – 1/CDF) x Ult_ECR

• Recall that for the paid version, the second term on the right represents total unpaid and does not depend on the amount paid to date. Same comment for the reported version except that second term on the right represents IBNR and does not depend on the amount reported to date. This is the key assumption of the BF and CC methods. See further down.

Question: how are the CC and BF methods different

• The difference is in how ECR and expected claims are calculated

→ CC uses a formula (uses internal company data, no judgment involved)

→ BF uses results of ECR method (may use internal company data or external industry data, incorporates judgment)

Ok, here's a biggie. We've discussed this already and it's the same as the key assumption of the BF method.

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

Given a set of data, the actuary must decide which method(s) to use and/or which to weight more heavily in coming up with a final estimate for the reserve. To do this, the actuary must understand the pros and cons of each method and where each is likely to provide accurate estimates.

Question: identify advantages of the CC method

• uses reported (paid) claims in the calculation of the ECR, therefore it will respond (at least partially) to changes in claims ratios

(note that significant trends in the claims ratio then may not be adequately reflected in the CC formula for ECR)

Question: identify disadvantages of the CC method

• dependent on the availability and accuracy of the rate level adjustment factor (can use CC without adjusting for CRL but accuracy may be lower)
• thin data increases volatility (consider using BF in this situation because we can incorporate judgment into the ECR selection)

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Review of Methods Through Chapter 10

The quiz has an old exam problem asking you to compare the BF and CC methods. I also created a web-based problem that asks you assess the accuracy of the reserving methods from previous chapters in different situations.

Quick-Vid: F-10 (050) Assess Accuracy of Method ~ 1:00

Here's a review and summary of what you need for the web-based problem:

The following table summarizes whether or not each method is accurate in the given situation. Explanations are provided below the table. The mix change situation can have many different variations and we'll look at 3 of them. (Recall that's where one segment or LOB is growing at a different rate from the other segment it's being combined with. We covered that here in chapter 7)

A - development patterns for both segments are stable and equal, LRs are equal

B - development patterns for both segments are stable and equal, LRs are NOT equal

C - development patterns for both segments are stable but NOT equal

In the table:

→ yes means accurate

→ no means not accurate but that you can't tell whether it will be HIGH or low

Method	Stable	Loss Ratio inc / dec	Case Strength inc / dec	Settlement Rate inc / dec	Mix Change A / B / C
development method: PAID	yes	yes	yes	HIGH / low	yes / yes / no
development method: REPORTED	yes	yes	HIGH / low	yes	yes / yes / no
ECR Method	yes	low / HIGH	yes	yes	yes / no / no
BF method: PAID	yes	low / HIGH	yes	HIGH / low	yes / no / no
BF method: REPORTED	yes	low / HIGH	HIGH / low	yes	yes / no / no

• The CC method isn't included in this table because qualitatively it performs roughly the same way as the BF method. So if you get a question about the accuracy of the CC method, you can generally answer the same way you would for the BF method. That isn't to say there's no difference between CC and BF – you can create scenarios where either is more accurate than the other – but the error in each will generally be in the same direction, either HIGH or low. Note that both CC and BF permit multiple adjustments for things like trends, tort reform, and rate changes, so their accuracy depends greatly on how these adjustments are made. In other words, the more complex a method is, the more difficult it is to make general statements about accuracy in given situations because estimates could be HIGH, low, or accurate depending on how appropriately the data is adjusted.

Brief Explanations:

Situation: Stable

• This is the base case which was first discussed here in chapter 6 in the context of diagnostics for changes in case reserve adequacy and also here in chapter 7. All methods are accurate when operations are stable.

Situation: Loss Ratio is changing

• The effect on the development method is discussed here in chapter 7. The development method is accurate under a changing loss ratio because the underlying development pattern is not affected.
• If the loss ratio increases (decreases) then the ECR method will produce estimates of ultimate that are too low (high) if not adjusted. (If the actuary recognizes the decreasing loss ratio and makes appropriate adjustments to the ECR then this method can be accurate.)
• Both BF methods are not accurate for the same reason as for ECR but by a smaller amount. Recall that BF a weighted average of the development and ECR methods and the development method is accurate under a changing loss ratio.

Situation: Case Strength is changing

• The effect on the development method is discussed here in chapter 7.

→ The paid development method is accurate because a change in case strength affects only the reported loss triangle. (Paid loss and case O/S loss are separate components of total loss.)

→ If case strength increases (decreases) the reported development method will overstate (understate). Recall that the ultimate does not change but changing the case strength alters the development pattern so historical LDFs are no longer appropriate for claims at the new level of case strength.

• The ECR method will remain accurate because as noted above, changing the case strength does not change the ultimate.
• The BF paid method will be accurate. It's a weighted average of paid development and ECR both of which are accurate in this situation.
• The BF reported method will not be accurate for the same reason as for the reported development method but by a smaller amount.

Situation: Settlement Rate is changing

• We briefly discussed changes in settlement rate here in chapter 6 in the context of diagnostic triangles but Friedland did not cover this scenario in chapter 7. Fortunately, the effects of this change are similar to changes in case strength – the main difference being that it's the paid data that's affected versus the reported data.

→ If the settlement rate increases (decreases) the paid development method will overstate (understate). Recall that the ultimate does not change but changing the settlement rate shifts losses between development periods and therefore alters the development pattern. The result is that historical LDFs are no longer appropriate for claims shifted to different development periods. (It might be a good idea to rewatch the video for that section from chapter 6.)

→ The reported development method is accurate because a change in settlement rate affects only paid data triangles. (One way to understand why reported losses are not affected is that the change in settlement rate just shifts losses from the case category to the paid category at a different rate but leaves to the total reported unchanged. Recall that reported = paid + case.)

• The ECR method will remain accurate because changing the settlement rate does not change the ultimate.
• The BF paid method will not be accurate for the same reason as for the paid development method but by a smaller amount.
• The BF reported method will be accurate. It's a weighted average of reported development and ECR both of which are accurate in this situation.

Situation: Mix of Business is changing

• If a method is not accurate in this situation, you can't tell whether it will overstate or understate without seeing the actual data. In the quiz then, you must answer i for inaccurate rather than + or - which would indicate HIGH or low.
• Both development methods will be accurate when each segment has the same, stable development pattern. It doesn't matter whether the loss ratios for each are different. The stable development pattern of the resulting combined data triangles will capture any shifts in loss ratio. This means both development methods will be accurate for A and B. The development methods will not be accurate for C however (unless the LRs are the same for each segment.) This is exactly the case that was discussed here in chapter 7, scenario 6.
• The ECR method will only be accurate if both segments have the same LRs. That means it will be accurate for A. If the loss ratios are different and the segments are growing at different rates then the ultimate loss ratios of the combined triangle will gradually move towards that of the larger segment and the ECR would have to be continuously adjusted. So in general, the ECR method will not be accurate for B and C.
• The BF methods will be accurate for A because the development methods and ECR method are both accurate. The BF methods will not be accurate for B or C because the ECR method isn't accurate for B and neither the development nor ECR method is accurate for C.

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POP QUIZ ANSWERS

The BF method is weighted average of these 2 methods:

• development method (either paid or reported)
• ECR method

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