Friedland07.Development

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

Chapter 7: Development Technique(or Chain Ladder Technique)

Study Tips

This is a long chapter, almost 60 pages, but the development method, also called the chain-ladder method, is not hard. Once you practice it a few times, you'll get the hang of it. It might already be familiar to you anyway. The source text is organized like this:

first 15 pages: (page 84-98) mechanics of the development method

• this the single most important topic on the Exam 5 syllabus
• source text explains the method in words but it's simpler just to look at examples (provided in this wiki article)

next 8 pages: (page 98-105) influences of a changing environment

• You need to understand how changes in the environment affect the accuracy of the development method (this is important)
• I have used my own policy & claims simulation software called SimPolicy to create illustrative examples
• video explanations are provided for selected examples

last 25 pages: (page 106-130) PDFs of Excel spreadsheet to illustrate the material

• these examples in the source text are very detailed (maybe a little too detailed)
• you can take a look at these if you'd like but make sure you look at my examples first

Note: If you're reading this chapter as part of your first pass through the pricing material, you only need to know the first 15 pages on the mechanics of the method. If you're studying the reserving material, then you need to study the whole chapter.

Estimated study time: 2-5 days depending on whether you're covering this material for pricing or reserving (not including subsequent review)

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 #16)	ultimate claims: - rptd devlpt	industry data: - considerations	impact: - of operational change
E (2019.Spring #19)	ultimate claims: - rptd devlpt (tort reform)	Friedland09.BornFerg
E (2018.Spring #20)
E (2017.Fall #17)	reserving data: - advantages of subdividing	reserving data: - disads. of subdividing
E (2017.Fall #19)	ultimate claims: - rptd devlpt	diagnostic: - operational changes	accuracy issues: - unpaid claim data
E (2017.Spring #20)	CY paid claim amounts: - uses development method
E (2016.Fall #25)	ultimate claims: - rptd development	claims & ALAE: - combining data	ultimate ALAE: - development method
E (2016.Spring #14)	ultimate claims: - paid development	ultimate claims: - using industry data	ultimate claims: - final estimate
E (2016.Spring #16)	Friedland08.ExpectedClms	ultimate: - paid devlpt	Friedland09.BornFerg
E (2014.Spring #13)	ultimate claims: - rptd development	assumptions: - rptd development method	alternative: 1 - if assumptions violated
E (2013.Spring #16)	ultimate claims: - development method	accident half-years: - advantage over full years

¹ To answer part (c) you need knowledge of reserving techniques from later material.

In Plain English!

Before getting to the first example Alice-the-Actuary wanted me to mention the different terms for a very important concept regarding the development method. These terms all mean the same thing:

• age-to-age factor
• development factor
• Loss Development Factor
• link ratio

Friedland uses both age-to-age factor and Loss Development Factor. But the latter has a very nice abbreviation: LDF.

Here is a closely related concept that also has more are than one label:

• age-to-ultimate factor
• Cumulative Development Factor

The nice thing about Cumulative Development Factor is that it has the simple abbreviation CDF. Be careful if you read the source text however. They sometimes also refer to a "Claim Development Factor" which is the same thing as "Loss Development Factor" and is not abbreviated by CDF.

Development Method - A Simple Example

You will never have a problem this simple on the exam or in real life. It's purely to teach you the mechanics of the method. Enjoy it while you can.

Example: calculate the ultimate loss for each AY using the given data

Cumulative paid loss triangle:

AY	12	24	36	48
2020	48	140	201	240
2021	48	140	201
2022	48	140
2023	48

Method: paid loss development

→ Use historical patterns to predict future losses.

→ The text breaks this method into 7 steps but Alice likes to condense it into just 4 steps as follows:

Step A: calculate age-to-age factors or LDFs

• This is incredibly simple:
  • The 1^st age-to-age factor (or LDF) for AY 2020 = 140/48 = 2.92. (You just divide the 24-month value by the 12-month value.)
  • The 2^nd age-to-age factor for AY 2020 = 201/140 = 1.43 (Divide the 36-month value by the 24-month value.)
  • The 3^rd age-to-age factor for AY 2020 = 240/201 = 1.19 (Divide the 36-month value by the 24-month value.)

AY	12-24	24-36	36-48
2020	2.92	1.44	1.19
2021	2.92	1.44
2022	2.92

• Now do the same thing for AY 2021 and AY 2022. For this very simple example the age-to-age factors are the same for each AY. We'll build to more complicated examples once we've covered the basic method. (There is no row for AY 2023 in the LDF triangle because AY 2023 has only 1 data point at age = 12. You would have to wait another year to get the next data point at age = 24 to be able to calculate the 12-24 LDF for AY 2023.)
• Note how the column labels changed. Instead of 12 for the first column, we now use 12-24. That's because the value of 2.92 represents how the value at 12 months ($48) develops to get the value at 24 months ($140).

Step B: select an age-to-age factor (or LDF) for each column

• This is potentially the most complex step because it requires actuarial judgment. The idea is to scan the column from top to bottom and select a representative value that according to Friedland:

...represents the growth anticipated in the subsequent development interval

• It's like an IQ-test! (Pick the next number in the pattern.) And actuaries tend to be very sensitive about selecting LDFs (age-to-age factors). But for certain high-volume line, short-tail lines of business like auto insurance, the computer can make pretty good selections for most development periods. That means the actuary can spend more time on the parts of the analysis that really do require human insight. Click for a Funny Story About Selecting LDFs.

• Anyway, selecting the LDFs is very simple for this example because every value in each column is the same. (Remember, we will build to more complex examples once you understand the basic method.) Our selected LDFs are as follows: (Even Ian-the-Intern could have figured this out!)

	12-24	24-36	36-48	48-ultimate*
selected	2.92	1.44	1.19	1.00   (tail factor)

^* Note the additional column for the tail factor LDF. We'll return to this later but for now just assume there is no development on claims past 48 months. In other words, every development period past 48 months (60, 72,...) will have paid loss equal to the 48-month value of 240. If you were then to calculate the LDFs for 48-60, 60-72,...etc, they would all equal 1.00.

Step C: calculate age-to-ultimate LDFs

• This step does not require judgment. It is just arithmetic.
• In Step B, we calculated age-to-age development factors. Now we multiply them together to get age-to-ultimate development factors, also called Cumulative Development Factors or 'CDFs. Here's the result. (Explanation below.)

	12-ult	24-ult	36-ult	48-ultimate
selected	5.00	1.71	1.19	1.00   (tail factor)

• To get these values, you have to work backwards, from right to left:

48-ult: No calculation required in this example. Just copy the 1.00 tail factor from Step B.

36-ult: (selected age-to-age LDF) x (prior [age-ult])

= 1.19 x 1.00

= 1.19

24-ult: (selected age-to-age LDF) x (prior [age-ult])

= 1.19 x 1.19

= 1.71

12-ult: (selected age-to-age LDF) x (prior [age-ult])

= 2.92 x 1.71

= 5.00

• When you reproduce these calculations, you may see minor difference due to rounding because I'm only showing 2 decimal places. I didn't want to clutter the presentation with a bunch of extra decimals.

Step D: calculate ultimate losses based on the latest diagonal of paid losses (the latest diagonal comprises the values in brown font from the original triangle)

• Here's the result. I've reproduced the latest diagonal from the original paid loss triangle just for convenience but you don't technically have to do that.

	AY 2023	AY 2020	AY 2021	AY 2020
diagonal	48	140	201	240
ultimate	240	240	240	240

AY 2020: diagonal x (age-ult)

= 240 x 1.00

= 240

AY 2021: diagonal x (age-ult)

= 201 x 1.19

= 240

AY 2021: diagonal x (age-ult)

= 140 x 1.71

= 240

AY 2021: diagonal x (age-ult)

= 48 x 5.0

= 240

• Again, please ignore the minor rounding differences. Take a quick look at the following link, which lays out the solution to the above problem concisely. There are also a few extra comments included within the solution that might be worth looking at.

Demo F-07 (002): Development Method - Simple Example

Development Method - The Key Assumption

Notice that Steps A, C, D in the development method were formula-based calculations: No actuarial judgment required. But Step B was different. The actuary had to select LDFs for each column that "fit the pattern" from previous years. And your final estimates of AY ultimates could vary greatly depending on what you selected. In the simple example from above, even Ian-the-Intern could select good LDFs. That's because the development pattern was consistent from year to year. In other words, the way losses (or claims) developed from one period to the next was the same for every row or AY. Mathematically, that means the calculated age-to-age development factors were the same within each column.

Key Assumption: The development method assumes future loss development is similar to development in prior years

• Mathematically, this means the LDFs within each column are roughly the same.
• Sometimes this is matter of judgment and sometimes you can use diagnostics to assist in determining whether this stability/consistency assumption holds.

Pop Quiz A!    :-o

• Identify which of these triangles satisfy the key assumption of the development method. Click for Answer 

(You have to ask yourself whether the development from year to year is similar enough. It's never going to be exact.)

Triangle 1	12	24	36	48
2020	48.1	141.2	200.7	240.0
2021	47.4	140.5	201.0
2022	48.2	139.6
2023	48.0

Triangle 2	12	24	36	48
2020	48.0	140.4	200.7	240.0
2021	43.6	136.0	198.6
2022	40.0	132.1
2023	39.6

Triangle 3	12	24	36	48
2020	42.4	145.7	202.3	240.0
2021	56.1	144.0	205.3
2022	52.1	137.2
2023	42.2

Triangle 4	12	24	36	48
2020	46.8	144.3	196.9	240.0
2021	53.3	145.4	203.1
2022	60.0	151.2
2023	68.6

Triangle 5	12	24	36	48
2020	48.0	140.0	201.0	240.0
2021	52.0	152.0	217.0
2022	56.0	164.0
2023	60.0

• After you've looked at the quick answer under Pop Quiz Answers, see if you can apply the development method to each of these triangles. Even if the key assumption doesn't hold, go ahead and do the calculations. Since these triangles were all created using my simulation software SimPolicy you can check your estimate of ultimate losses against the "real" ultimate losses. Use the link below to see the full solutions.
• There are also extra comments within the solutions that will help prepare you for upcoming material. The development method is a good basic method but it does have shortcomings. Some of those can be addressed by being a little more sophisticated in how it's applied, but others must be addressed by different methods entirely. Friedland covers several of those other reserving methods in subsequent chapters.

Demo F-07 (004): Development Method - Full Solutions to Pop Quiz A

Okay, that was a long pop quiz so I won't torture you too much more in this section. One last thing though: the source text mentions a second main assumption of the development method as follows:

• claims observed for an immature period provide information about claims yet to be observed

This is just another way of saying that history repeats itself, except there's specific reference to immature periods. A set of claims at 12-months development would be considered immature but those claims become progressively more mature at each successive period. In general, estimates of ultimates based on immature claims will be less accurate than estimates based on mature claims. And you don't know for certain the ultimate value of a claim until it has been closed. (Even then it could be reopened, but that doesn't happen very often.)

Friedland lists other "assumptions" but they are all more or less implied by the Key Assumption. Rather than "assumptions", I would describe these as situations where the Key Assumption is satisfied and where the development method is likely to work well.

Question identify situations where the development method is likely to work well

• consistent claim processing
• stable mix of types of claims
• stable policy limits
• stable reinsurance & retention limits
• high frequency & low severity lines of business

You should give a little thought as to why these situations satisfy the Key Assumption although this might be easier once you've worked through more examples and understand better the mechanics of the method. It always comes back to the idea of stability & consistency. The reason the development method works well high frequency & low severity lines of business is that:

• high frequency means there will be more claims and therefore more credibility in the data (reduces noise in the triangle)'
• low severity reduces chance of large losses creating outlier data points in development triangle that could distort the development pattern

Consistent claims processing means (among other things) that claims are settled at roughly the same rate from one year to the next, and that the level of case reserves is consistent for similar types of claims. If either of those don't hold, the the development method could produce distorted results. The Berquist-Sherman method, discussed in a later chapter, is specifically designed to correct for this.

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

Development Method - LDF Selections

The examples we've look at so far only had 4 AYs and 4 development periods. That made Step B, where you had to select LDFs, pretty simple because you could just eyeball it. There are essentially only 3 possibilities in such a small triangle:

• if the LDFs in a column appear stable, your selected LDF could be be the average
• If there is a trend in a column, your selected LDF could reflect that trend
• if the LDFs do not appear stable and there is no trend, then the development method may not work (but you might just go ahead and select the average anyway then hope for the best!)

We're going to look at a loss triangle with 12 AYs and 12 development periods, but the application of the development method to this triangle has an extra "sub-step" as part of Step B.

Step B (sub-step): calculate candidate LDF selections then select your LDF from among these candidates

These candidate LDF selections often include:

• average of all values in column
• average excluding high and low values (source text calls this the "medial average")
• weighted average (where the loss dollars are the weights)
• median
• average of last n values (where n can be anything from 1 to the number of values in the column)
• geometric average (nth root of the product of n LDFs from the column, although this is not common)

Of course in the end you don't have to select one of your candidates:

• you can use your own judgment
• or you may conclude the development method isn't appropriate and use a different method entirely

Anyway, here's a short video explaining the Excel file given below. The Excel file has a lot going on so it's a good idea to watch the video first. (There will be a quiz so pay attention!)

VIDEO: F-07 (010) Candidate LDFs → 6:30

And here is the Excel file from the video with candidate LDFs for you to play with.

• See how changing the selected LDFs changes the estimates of ultimate loss. The great thing about creating triangles with a simulation is that you know the true ultimate loss. Experimenting with LDF selections teaches you how to make good selections in real-life situations where you don't know the answer in advance.

Excel Demo F-07 (010): Development Method - Candidate LDFs

Assignment: The Excel file below has 5 examples of paid loss triangles with either 4 AYs or 6 AYs. I'd suggest solving the small triangles with pencil, paper, and calculator. Once the method is firmly embedded in your brain, you can solve the larger triangles by typing the appropriate formulas into Excel.

Excel Assignment F-07 (011): Development Method - Calculate Ultimates

Something to keep in mind: One of the reasons I like math is that there is virtually always one correct answer. No arguments. But actuarial work is more than just math and when applying reserving methods, there is no such thing as "the" single correct answer. On the exam, as long as your method is correct and you choose your LDFs in a semi-intelligent way, you should get full credit. About the triangles in the assignment: Since these triangles were created by SimPolicy, we do in fact know the real ultimate losses, unlike in the real world, but you'll see there's a limit to how accurate your estimates can be if you're only given the triangle of losses. It's partly due to normal random variation but also because there are just too many moving parts in such a complex system for the human brain to handle. I always find it humbling when I think I've made really good LDF selections, but my estimates turn out to be less accurate than if I had just chosen the all-period average. It a good reminder to myself that I'm not as smart as I think I am. Remember that next time you're in a meeting and people are arguing about their LDF selections or trends or whatever. Don't take yourself too seriously. :-)

Development Method - Tail Factors

In all the examples so far, the tail factor for the development triangle had been 1.000. That meant at least 1 accident year was complete: all its claims were paid and settled so there would be no more development past that point. In the very simple example at the top of this article, the oldest AY had all its claims settled by 48 months and we assumed the other AYs would also be complete by 48 months.

In the earlier video and spreadsheet example, we assumed development was complete after 12 years or t=12. The next example uses the same data but the simulation at stops at t=6. That means we do not have any complete AYs. And recall the 6-ultimate CDF was something like 1.07 (depending on how you selected your LDFs) so if we didn't include this tail factor, we would be under-estimating by roughly 7%.

Step B (sub-step): identify methods for estimating the tail factor in a development triangle

• industry benchmarks
• curve-fitting using existing LDFs to extrapolate the tail factors
• use reported-to-paid ratios at the latest observed paid development period

• works only for paid development triangles
• requires the reported development triangle have at least 1 complete AY

→ Friedland states this method is beyond the scope of the text so it should not appear on the exam

One simple way of using curve-fitting is to use a "square-root" pattern. If your latest LDF is 1.10, you can extrapolate to the next period with an LDF of 1.10^1/2 = 1.048, and so taking square roots on until you get values very close or essentially equal to 1.000. Something more sophisticated would be to use all your selected LDFs to fit an exponential curve that decreases to 1.00. That seems like it would work better but in practice you often gain very little from that extra work. It's largely a crap-shoot. You can try out for yourself though in the spreadsheet to see if you can get a better SSE. (Remember, this is simulated data so we know the answer in advance. That means you can try to methods for coming up with a tail factor and see right away which is best.)

Note that sometimes tail factors can be less than 1.00. This may happen in physical damage coverages where salvage reduces the insurer's losses after the claim is settled. But the square root trick still works because successive square-roots of a number between 0 and 1 results in a bigger number and approaches 1.00 in the limit. Take a look at the video then download the spreadsheet so you can play around with creating the tail factor.

VIDEO: F-07 (020) Tail Factors → 3:00

And here is the Excel file from the video with tails factor selections you can play with.

• Try setting the tail factor equal to 1.00, which is the same as not having a tail factor because you're assuming no further development. Then try fitting an exponential curve and using the regression to see if you can improve on the square root method that I inserted.

Excel Demo F-07 (020): Development Method - Tail Factors

Here is Alice's solution to an exam problem involving tail factors and 4 similar practice problems:

Solution: 2017.Fall #19

Practice: 4 problems like 2017.Fall #19

Development Method - A Few Comments

All the above examples of the development method used paid loss data, but the development method works whenever you can organize data into a triangle. Most commonly, the development method is applied to:

• paid losses
• reported losses
• paid counts
• reported counts

Sometimes it's applied to ALAE or Salvage & Subrogation triangles but the data in those triangles is often sparse and there are other more reliable methods. We cover those in Friedland14.Recoveries and Friedland16.ALAE.

Did you also notice that all the development method examples we looked used Accident Years. The method applies equally well however to any of the following:

• Accident Year
• Policy Year
• Underwriting Year
• Report Year
• Fiscal Year

And then there's the level of aggregation. We used year but you can just as easily group by quarter or even month. Do you think the development method would still work? Well, yes. Yes it does! As long as the Key Assumption holds:

• future loss development is similar to development in prior years.

If you need a quick review, click Development Method - Key Assumption. Of course if you aggregate data by accident quarter and development quarter you'll have less data in each cell which reduces credibility, but theoretically it can be done.

Side story: When I worked at National General in North Carolina, we had accident month data by development month. For the larger states it worked very well, plus we could update our estimates every month so that any inaccuracies in initial estimates improved very quickly. The software, called the Triangle System, was 100% automated so the results were available in a just a few days after month-end. The system included some basic AI that then flagged specific areas for human analysis. The idea was to let the computer do the grunt work so humans like Alice-the Actuary could focus on the higher-order thinking. (Our trends and expected loss ratios were input in advance based on both industry and company data.)

Development Method - Estimating Unpaid Amounts

So far we looked at estimating ultimate losses or claims. The real goal however is to estimate the unpaid losses, but that's a easy once you have the ultimates. This next example is similar to Exhibit 1 in the source text, but it's smaller, only 6 AYs instead of 10. It shows 3 things:

1. estimating ultimate loss (or ultimate claims) using both paid loss triangles and reported loss triangles
   • (the method is exactly the same for each but you often get different results – more on this below)
2. calculating Case O/S (case outstanding)
3. estimating unpaid loss, both IBNR and total amounts

Item 1 above is just the plain old development method but we apply it to both a paid loss triangle and a reported loss triangle. Item 2 is a simple formula based on the given data. For item 3 you need to know the ultimate loss but once you do, it' simple. Here's the video:

VIDEO: F-07 (030) Paid & Reported Development → 5:30

Here's the corresponding Excel file:

Excel Demo F-07 (030): Development Method - Paid & Reported Development

And the formulas used to create the summary table shown:

Case	=   (reported loss) – (paid loss)   at the same evaluation date
IBNR	=   (ultimate loss) – (reported loss)
Total unpaid loss	=   (ultimate loss) – (paid loss)

These formulas are valid for separate AYs but also in total. An obvious implication of these formulas is:

Total unpaid loss

=   Case + IBNR

Let's return to the comment I made under Item 1. You can estimate ultimate losses using either paid or reported data but you often get different answers.

paid vs reported development: why do you get different answers

• Recall the formula for the ultimate loss for each AY from Step D in Development Method - A Simple Example:

→ ultimate   =   diagonal x CDF

• Difference 1: The diagonal entry, which is just the raw data, is going to be different in the paid loss triangle versus the reported loss triangle

(paid loss should be ≤ reported loss)

• Difference 2: paid losses develop differently (more slowly) than reported losses, which means the paid CDF will generally be different from the reported CDF

(paid CDFs are generally ≥ reported CDFs)

We often refer to paid and reported development as 2 different methods, but that isn't strictly accurate. It's the same method applied to different sets of data. The distinction is terribly important however.

paid vs reported development: why is it good to use both methods

• each method is a check on the other

(since these methods only provide estimates, not exact answers, they are subject to distortion)

Then the final estimate of ultimate is often a judgment-weighted-average paid & reported development estimates.

A very important topic in reserving is why different methods produce distorted results in specific situations. We'll begin discussing this later in this wiki article and also in subsequent chapters from Friedland. A simple example is that the paid development method tends to over-estimate the true ultimate loss if claim settlement rates increase. The increase is sometimes an effect of hiring more claims staff. Many, many different things can distort the accuracy of reserving methods and we gotta learn how to cope with all that!

Development Method - Summary & Observations

Here's a summary of the various steps in the development method. This isn't something you have to memorize. Once you've practiced the calculations several times, you won't have any trouble with it. Remember that Step B is the only step where you have to use judgment. The other steps are just formulas.

Step A: calculate age-to-age factors or LDFs

Step B: select an age-to-age factor (or LDF) for each column Step B (sub-step): calculate candidate LDF selections then select your LDF from among these candidates Step B (sub-step): identify methods for estimating the tail factor in a development triangle

Step C: calculate age-to-ultimate LDFs

Step D: calculate ultimate losses based on the latest diagonal of paid losses

Something Friedland mentions in chapter 7 is the cumulative % paid and cumulative % reported. It's easy but we won't spend much time on it now because it isn't really needed until later chapters. It's a very simple formula. If you have the CDFs from the development method then each age i:

cumulative % paidi	= 1 / (paid CDFi)
cumulative % reportedi	= 1 / (reported CDFi)

In my mind, I just think: 1/CDF. Let's use the same paid loss data as in our original Simple Example. The CDFs we came up are as listed in the table below.

paid CDF	cumulative % paid	incremental % paid
5.00	1/5.00 = 0.200 or 20.0%	(current row) – (prior row) = 80.0%
1.71	1/1.71 = 0.585 or 58.5%	38.5%
1.19	1/1.19 = 0.840 or 84.0%	25.5%
1.00	1/1.00 = 1.000 or 100%	16.0%

It's useful also to keep in mind the corresponding formula for the % unpaid at age i: 1 – 1/CDF_i. If you think about it for just a moment, you can see why the formulas make sense. Anyway, let's move on. (I'll remind you when you need this later on.)

• Include Friedland's observations (these will make good flash cards)
  • If all selected LDFs are ≥ 1.00 then CDFs for less mature years are greater than CDFs for more mature years. Mathematically, this means:
    • CDF_i ≤ CDF_i+1 ∀

• when it works and when it doesn't (this leads into influence of changing environments)

Influence of a Changing Environment - Intro

Start with questions. Give internal & external changes. Ask students to take a guess. Then we will investigate using simulated data.

→ bad start: There are many influences, both internal and external, that can cause the development to totally mess up. But, there are changes that do not mess it up.

• BQ: provide triangles then ask simple questions

(next section, provides triangles but ask which shock is most likely and give choice of several. Then ask how to account for and give choice of several)

Friedland's examples of changing environment

• Changes in Claim Ratios and Case Outstanding Adequacy

• Scenario 1 is a steady-state environment where claim ratios are stable and there are no changes from historical levels of case outstanding strength (U.S. PP Auto Steady-State)
• Scenario 2 is an environment of increasing claim ratios and no change in case outstanding strength (U.S. PP Auto Increasing Claim Ratios)
• Scenario 3 is an environment of stable claim ratios with an increase in case outstanding strength (U.S. PP Auto Increasing Case Outstanding Strength)
• Scenario 4 is an environment where there are increases in both claim ratios and case outstanding strength (U.S. PP Auto Increasing Claim Ratios and Case Outstanding Strength)

• Changes in Product Mix

• Scenario 5: no change in product mix
• Scenario 6: product mix is changing

Pop Quiz A - Answer

To answer this question you have to do Step A of the development method where you calculate the LDFs (or link ratios or age-to-age factors or whatever you like to call them.)

• Triangle 1: satisfies stability/consistency assumption for loss development to work reasonably well
• Triangle 2: doesn't satisfy stability/consistency assumption (LDFs are increasing significantly within columns)
• Triangle 3: hard to tell - the underlying development pattern is the same for each AY but the simulation introduced moderate random variation which confounds the data
• Triangle 4: doesn't satisfy stability/consistency assumption (LDFs are decreasing in the 12-24 column)
• Triangle 5: satisfies stability/consistency assumption for loss development to work reasonably well (losses increased year-over-year, but the development pattern didn't change)

Triangle 1 LDFs	12-24	24-36	36-48
2020	2.936	1.421	1.196
2021	2.963	1.430
2022	2.897

Triangle 2 LDFs	12-24	24-36	36-48
2020	2.924	1.430	1.196
2021	3.116	1.461
2022	3.302

Triangle 3 LDFs	12-24	24-36	36-48
2020	3.436	1.388	1.186
2021	2.568	1.426
2022	2.636

Triangle 4 LDFs	12-24	24-36	36-48
2020	3.085	1.364	1.219
2021	2.726	1.397
2022	2.520

Triangle 5 LDFs	12-24	24-36	36-48
2020	2.924	1.430	1.196
2021	2.924	1.430
2022	2.924

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