CM2-21: Run-off triangles page 6

[yuli2513]

In the example question on page 6 of chapter 21, the subquestion (ii) says if the table shows the amounts of claims paid, tabulated by reporting year, the IBNR won't be included. I am struggling to understand this.

In my understanding, the IBNR is the incurred but not reported part, and if we know that there was an accident 2 years ago, we can still include it this year (since we are making the table based on the reporting year).

The difference between (i) where the table is tabulated based on accident year and (ii) where the table is tabulated based on reporting year should only be a formatting difference and the total number we are looking at from the left triangle (known part) of the table should be the same.

Can anybody help to point out where my logic goes wrong here? Thanks in advance!

 

[Steve Hales]

Hi. I think the issue might be with the statement "and if we know that there was an accident 2 years ago". How would we know unless it was reported? In which case it won't appear in the IBNR figure, because it's been reported.
Does this match with what you're asking?
Thanks
Steve

 

[yuli2513]

Hi. I think the issue might be with the statement "and if we know that there was an accident 2 years ago". How would we know unless it was reported? In which case it won't appear in the IBNR figure, because it's been reported.
Does this match with what you're asking?
Thanks
Steve

Hi Steve,

Thanks a lot for the answer. I still don't quite get it. Let me refine my description: if there was one claim from an accident 2 years ago and we are informed of it today, we would naturally know from this claim report (from the policyholder) when exactly was the accident and also we know the reporting year. In this case, we could easily split out the IBNR numbers from the reports.

I have the feeling I am referring too much to my personal experience here, could you please kindly help me to see where I assumed too much and where my logic went wrong?

 

[Steve Hales]

Hi
I'm afraid I'm still not entirely sure what the question is
If a claim is reported today (in 2021) for an accident which occurred two years ago (in 2019) then depending on how the run-off triangle is defined that claim amount will either sit in the 2021 or the 2019 row. It will appear in the 2021 row (Development Year 0) if we're using "reporting year" or in the 2019 row (Development Year 2) if we're using "accident year".
Remember that the numbers in the top-left triangle of the table represent the known claim amounts that have been paid. The question on page 6 has to do with reserves - which is the estimate of future claim payments.
Please let me know if this still doesn't help and I'll ask someone else to chip in to the conversation.
Thanks
Steve

 

[yuli2513]

Hi
I'm afraid I'm still not entirely sure what the question is
If a claim is reported today (in 2021) for an accident which occurred two years ago (in 2019) then depending on how the run-off triangle is defined that claim amount will either sit in the 2021 or the 2019 row. It will appear in the 2021 row (Development Year 0) if we're using "reporting year" or in the 2019 row (Development Year 2) if we're using "accident year".
Remember that the numbers in the top-left triangle of the table represent the known claim amounts that have been paid. The question on page 6 has to do with reserves - which is the estimate of future claim payments.
Please let me know if this still doesn't help and I'll ask someone else to chip in to the conversation.
Thanks
Steve

Hi Steve,

Again, thanks a lot for the very fast reply. I think I am getting it, but just to make sure I am getting it right:

What the question means is if we only have one of the triangles, so in this case, if we only have the triangle with the claim information based on reporting years, we don't know within each year which ones are reported in time and which ones are IBNR (unlike normally in real life). Since only the reserves are asked about, and since we have no idea of the IBNR in the future, there would be no reserve for IBNR, as a reserve is needed only for future IBNR, not the past ones.

Is this right?

 

[Mark Willder]

Hi Steve,

Again, thanks a lot for the very fast reply. I think I am getting it, but just to make sure I am getting it right:

What the question means is if we only have one of the triangles, so in this case, if we only have the triangle with the claim information based on reporting years, we don't know within each year which ones are reported in time and which ones are IBNR (unlike normally in real life). Since only the reserves are asked about, and since we have no idea of the IBNR in the future, there would be no reserve for IBNR, as a reserve is needed only for future IBNR, not the past ones.

Is this right?

Hi

A numerical example might help. Consider an insurance contract where 90% of claims are reported in the first year and 10% of claims are reported in the second year. All claims are for an amount of 100, but 50% of this is paid in the year of notification and 50% in the year after.

Let's say that there are 20 claims in 2018, 30 in 2019 and 40 in 2020.

Let's first look at accident year. The run off triangle for the incremental claims then looks like this (with commas between the various entries)

2018 20 x 0.9 x 0.5 x 100, 20 x 0.9 x 0.5 x 100 + 20 x 0.1 x 0.5 x 100, 20 x 0.1 x 0.5 x 100
2019 30 x 0.9 x 0.5 x 100, 30 x 0.9 x 0.5 x 100 + 30 x 0.1 x 0.5 x 100
2020 40 x 0.9 x 0.5 x 100

this gives:

2018 900, 1,000, 100
2019 1,350, 1,500
2020 1,800

We'll then make these numbers cumulative and work out the ratios to project the missing values. This will give us the reserve for the outstanding claim payments and the IBNR for the accident years 2019 and 2020. The IBNR will be included for 2020 as we can see in the table that the second and third delay years for 2018 have also included the claims from 2018 that were notified late.

Let's now look at reporting year. We now put the 10% of claims incurred in 2018 that are reported late into the 2019 reporting year. So the run off triangle for the incremental claims now looks like this

2018 20 x 0.9 x 0.5 x 100, 20 x 0.9 x 0.5 x 100, 0
2019 30 x 0.9 x 0.5 x 100 + 20 x 0.1 x 0.5 x 100, 30 x 0.9 x 0.5 x 100 + 20 x 0.1 x 0.5 x 100
2020 40 x 0.9 x 0.5 x 100 + 30 x 0.1 x 0.5 x 100

this gives:

2018 900, 900, 0
2019 1,450, 1,450
2020 1,950

We'll then make these numbers cumulative and work out the ratios to project the missing values. This will give us the reserve for the outstanding claim payments but not the IBNR for the accident years 2019 and 2020. We can see that the 2020 projected numbers will be only 1950 and 0 for the second and third years, so will only include the claims reported so far and will miss the 4 claims reported late.

I'll leave it as an exercise for you to complete the triangles and calculate the reserves. You'll see that the reserve calculated using the reporting year triangle is smaller than the reserve for the accident year triangle, the difference being the IBNR.

I hope this helps.

Best wishes

Mark

 

[yuli2513]

Hi

A numerical example might help. Consider an insurance contract where 90% of claims are reported in the first year and 10% of claims are reported in the second year. All claims are for an amount of 100, but 50% of this is paid in the year of notification and 50% in the year after.

Let's say that there are 20 claims in 2018, 30 in 2019 and 40 in 2020.

Let's first look at accident year. The run off triangle for the incremental claims then looks like this (with commas between the various entries)

2018 20 x 0.9 x 0.5 x 100, 20 x 0.9 x 0.5 x 100 + 20 x 0.1 x 0.5 x 100, 20 x 0.1 x 0.5 x 100
2019 30 x 0.9 x 0.5 x 100, 30 x 0.9 x 0.5 x 100 + 30 x 0.1 x 0.5 x 100
2020 40 x 0.9 x 0.5 x 100

this gives:

2018 900, 1,000, 100
2019 1,350, 1,500
2020 1,800

We'll then make these numbers cumulative and work out the ratios to project the missing values. This will give us the reserve for the outstanding claim payments and the IBNR for the accident years 2019 and 2020. The IBNR will be included for 2020 as we can see in the table that the second and third delay years for 2018 have also included the claims from 2018 that were notified late.

Let's now look at reporting year. We now put the 10% of claims incurred in 2018 that are reported late into the 2019 reporting year. So the run off triangle for the incremental claims now looks like this

2018 20 x 0.9 x 0.5 x 100, 20 x 0.9 x 0.5 x 100, 0
2019 30 x 0.9 x 0.5 x 100 + 20 x 0.1 x 0.5 x 100, 30 x 0.9 x 0.5 x 100 + 20 x 0.1 x 0.5 x 100
2020 40 x 0.9 x 0.5 x 100 + 30 x 0.1 x 0.5 x 100

this gives:

2018 900, 900, 0
2019 1,450, 1,450
2020 1,950

We'll then make these numbers cumulative and work out the ratios to project the missing values. This will give us the reserve for the outstanding claim payments but not the IBNR for the accident years 2019 and 2020. We can see that the 2020 projected numbers will be only 1950 and 0 for the second and third years, so will only include the claims reported so far and will miss the 4 claims reported late.

I'll leave it as an exercise for you to complete the triangles and calculate the reserves. You'll see that the reserve calculated using the reporting year triangle is smaller than the reserve for the accident year triangle, the difference being the IBNR.

I hope this helps.

Best wishes

Mark

Hi Mark,

Thank you very much for this very detailed reply. I can tell I was terribly wrong earlier. Thank you very much for explaining this so well.