Friedland05.Triangles

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

Chapter 5: The Development Triangle

Pop Quiz

Give an example of a change in mix of business that might impact how you do the reserve analysis? (Many answers are possible.) Click for Answer 

Study Tips

VIDEO: F-05 (001) Triangles → 4:30 Forum

A loss development triangle (or claim development triangle, or just triangle for short) is a fundamental concept in both pricing and reserving. Here are 4 question you must know how to answer:

1. what does a loss development triangle look like
2. what information does it give you
3. how do you construct a triangle from raw claims data (equivalently raw loss data)
4. how do you use a triangle to estimate ultimate claims (equivalently ultimate losses)

We cover #1,2,3 in this chapter. Most of the exam questions are about point #3, constructing various types of triangles from raw claims data. Point #4, using triangles, is what most of the rest of the Friedland text is about. There are many methods for estimating ultimate losses (or claims) that have triangles as their starting point.

Estimated study time: 2 days (not including subsequent review time)

BattleTable

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

• how to construct AY, RY, PY claim triangles from the data provided
• pros & cons of various triangle aggregation method
• how to calculate CY, AY, RY, PY claim totals from the data provided

reference	part (a)	part (b)	part (c)	part (d)
E (2016.Fall #16)	construct: - AY paid triangle	construct: - AY reported triangle	Friedland06.Diagnostics	Friedland15.Evaluation
E (2016.Spring #15)	calculate: - CY rptd clms	Friedland03.Data	Friedland03.Data	Friedland07.Development
E (2015.Fall #16)	construct: - RY rptd, AY paid triangles	Friedland03.Data
E (2015.Spring #15)	construct: - AY rptd triangle	construct: - RY rptd triangle
E (2014.Fall #14)	construct: - AY paid, rptd triangles	Friedland06.Diagnostics
E (2013.Fall #14)	construct: - AY paid triangle	construct: - RY paid triangle	data aggregation: - RY versus AY	data aggregation: - annual versus quarterly

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In Plain English!

The Development Triangle: Layout

Below is what a development triangle looks like. The data in this example is shown in blue font and is paid claims or equivalently paid losses. But it can be any kind of data that develops over time (O/S loss, paid loss, LAE, counts,...)

	Development Period (Age)
AY	12	24	36	48
2020	400	600	700	750
2021	300	400	500
2022	400	600
2023	350

Each cell in this accident year triangle is identified by 3 dimensions:

1. The row labels are Accident Years and you'll often see the abbreviation AY.
2. The column labels are Development Period or Age. Those terms are interchangeable. In this triangle the periods are months and that's very common.
3. The skew diagonals represent Calendar Year, abbreviated CY

Let's take a moment to explain these terms by way of a couple of simple examples:

• Suppose someone has a minor accident. (and let's also assume they report the accident to their insurer on the same day.)

→ If the accident occurred on July 1, 2020 then the accident year would be 2020.

→ If the accident occurred on September 25, 2021 then the accident year would be 2021.

accident year: the year in which the accident occurred (easy!)

• At some later date, suppose the person's claim is paid by their insurer

→ If the accident occurred in 2020 and was paid in 2020, then we say the accident was paid by age 12 months

(it doesn't matter when in 2020 the accident occurred and was paid)

→ If the accident occurred in 2020 and was paid in 2021, then we say the accident was paid by age 24 months

(it doesn't matter when in 2020 the accident occurred or when in 2021 the accident was paid)

development age: the lag time between when the accident occurred and when it was paid (measured from the beginning of the year of occurrence to the end of the year it paid)

• The calendar year diagonals will be explained in the video in the next section.

Pop Quiz A!    :-o

Find the value of n and m. Click for Answer 

• If an accident occurred in 2022 and was paid in 2024, then the accident was paid by age n months.
• If an accident occurred in 2022 and was paid in 2022, then the accident was paid by age m months.

Side note:

• The rows in a development triangle are often AYs but a triangle can also be constructed on different bases such as RY or PY (Report Year, Policy Year)
• The rows don't have to be years. It isn't uncommon for the rows to represent quarters or half-years. Half-years are normally abbreviated AHY-1 for January to June and AHY-2 for July to December. Certain lines of business exhibit seasonality and looking at half-years can make the seasonality effect more apparent. Examples are snowmobile insurance which would have the majority of its claims in AHY-1 whereas boat insurance would have more claims in AHY-2. There are some great examples of seasonality later on in Chapter 11 - Frequency & Severity Methods
• The columns don't have to represent yearly increments. It's possible for the column labels to represent quarters 3, 6, 9, 12,... (This is often the case if the rows are accident quarters.)

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The Development Triangle: Information

We know the data in the above (cumulative) triangle represents paid losses but how are the numbers in each cell interpreted? Let's look at just the first row, the row corresponding to AY 2020. There are 4 entries, each corresponding to one of the development periods 12, 24, 36, 48:

(2020, 12) = 400 : This $400 represents claims with an accident date in 2020 that were paid in the 1^st year, between 1/1/2020 and 12/31/2020

(2020, 24) = 600 : The difference (2020, 24) - (2020, 12) = $600 - $400 = $200 represents claims with an accident date in 2020 that were paid in the 2^nd year, between 1/1/2021 and 12/31/2021

(2020, 36) = 700 : The difference (2020, 36) - (2020, 24) = $700 - $600 = $100 represents claims with an accident date in 2020 that were paid in the 3^rd year, between 1/1/2022 and 12/31/2022

(2020, 48) = 750 : The difference (2020, 48) - (2020, 36) = $750 - $700 = $50 represents claims with an accident date in 2020 that were paid in the 4^th year, between 1/1/2023 and 12/31/2023

Pop Quiz B!    :-o

Find and interpret the following values using the small triangle provided in the previous section. Click for Answer 

(a) (2021, 12)

(b) (2021, 24)

(c) (2021, 36)

Here's a video that further explains the concept of a triangle and the information it gives you.

VIDEO: F-05 (010) Information in Triangles → 7:00

Alice's pro tips on the alternate triangle format from the video

• First, you might want to print the following: (or open in separate tab for reference)

Reference: Alternate Triangle Format

• Notice that in both formats, the rows are AYs (Accident Years) but the column labels are different.

• in the standard triangle format → columns show claim activity at the same ages: 12, 24,...
• in the alternate triangle format → columns show claim activity for the same CYs: 2020, 2021,...

• Another way of saying this is:

• in the standard triangle format → columns show claim activity for different CYs
• in the alternate triangle format → columns show claim activity at different ages

• For example,

• in the standard triangle format → the 12-month column tells us that every AY had a cumulative paid loss of 1,800 by age 24 (but the CY corresponding to age 24 is different for each AY)
• in the alternate triangle format → the 2021 column tells us the cumulative paid loss for each AY at Dec 31, 2021 (but the age corresponding to these values is different for each AY)

(This is something that's probably easier to grasp just by getting practice working with triangles.)

Here's a set of supplementary problems based on the video to test basic knowledge of how to read information from triangles.

Practice: 6 easy problems based on video

Here's a short quiz to test your understanding.

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The Development Triangle: Construction

The previous section discussed triangles in a more conceptual way. Here we look at a detailed example of how to actually construct a triangle using raw claims data. There's also an example in the source text that's worth looking at after you've watched the video and done the practice problems. It looks a lot more complicated than it really is. The trick is keeping a ruler handy so you can draw horizontal lines to break the data table into manageable sections. This is explained in the video.

VIDEO: F-05 (020) Contructing Triangles → 5:00

Here is Alice's solution to an exam problem on building a triangle and 4 similar practice problems. Pay attention to whether the questions ask for AY (Accident Year) or RY (Report Year) triangles.

Solution: 2014.Fall #14

Practice: 4 problems like 2014.Fall #14   

Once you're confident with those practice problems, parts (a) and (b) of this exam problem should be a breeze: (parts (c) and (d) will be discussed in later chapters)

E (2016.Fall #16)

The other exam problems are somewhat similar but they give you the claim information in a different format. That shouldn't be a huge problem though. If you understand the basic version of the problem discussed in the video and the practice problems, you'll be able to figure them out.

Another pro tip from Alice: Sometimes it's helpful to rewrite the given information so that it matches a type of problem you already know how to do.

Give Alice's tip a try for this problem:

E (2015.Spring #15)

The exam questions above are included in this quiz so you can check them off and have a record that you've done them.

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The Development Triangle: Use

The discussion of how triangles are used to calculate ultimate claims (losses) and total unpaid claims (losses) begins in Chapter 7 - The Development Method. Next up however, in Chapter 6 - Diagnostics we'll delve more deeply into the information you can extract from triangles, or combinations of triangles. Often, changes in company operations will be evident in certain triangles and these are things you have to be able to recognize to do a proper analysis. You can deduce a surprising amount from triangles if you know what to look for. Alice will gently guide you on this fascinating journey. :-)

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POP QUIZ ANSWERS

An example of a change in mix of business from accident year 2020 to 2022 could be the following:

• accident year 2020 has 25% of policies with deductibles of $200 and 75% of policies with deductibles of $500
• accident year 2022 has 40% of policies with deductibles of $200 and 60% of policies with deductibles of $500

If you were combining the data for all deductible levels when doing your analysis, this change may impact the accuracy of your results. It might be necessary to perform a separate analysis on policies with different deductibles. This is discussed in detail in Friedland07.Development: Influence of a Changing Environment - Scenarios 5 & 6.

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Pop Quiz A - Answer

• n = 36, m = 12

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Pop Quiz B - Answer

(2021, 12) = 300 : This $300 represents claims with an accident date in 2021 that were paid in the 1^st year, between 1/1/2021 and 12/31/2021

(2021, 24) = 400 : The difference (2021, 24) - (2021, 12) = $400 - $300 = $100 represents claims with an accident date in 2021 that were paid in the 2^nd year, between 1/1/2022 and 12/31/2022

(2021, 36) = 500 : The difference (2021, 36) - (2021, 24) = $500 - $400 = $100 represents claims with an accident date in 2021 that were paid in the 3^rd year, between 1/1/2023 and 12/31/2023

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