Actuarial Outpost > CAS Challenge: 2017 Q10(a) -- Bootstrapping and CY Trend
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#1
01-02-2018, 03:01 PM
 |B|rad Member CAS Join Date: Jul 2009 Posts: 1,026
Challenge: 2017 Q10(a) -- Bootstrapping and CY Trend

Here is the challenge:

Create a 6x6 triangle of incremental paid activity with a clear calendar-year trend such that, if you apply the bootstrap model process, you get residuals that resemble the solution in the examiners report.

Quote:
 Originally Posted by Problem 10. (2 points) An actuary is reviewing the diagnostic results of a bootstrapping model. a. (1 point) Construct and label a plot of residuals that would indicate a calendar year trend due to inflation which is not reflected in the model, and describe how the residual pattern shows inflation.
Quote:
 Originally Posted by Solution
#2
01-02-2018, 03:02 PM
 |B|rad Member CAS Join Date: Jul 2009 Posts: 1,026

I attempted this myself, but maybe I don't understand bootstrapping as well as I should.

I started with an oversimplified scenario where, for any accident year, 2000 is paid in year 1, then 1500, 1200, 1000, 700, and 400 for years 2, 3, 4, 5, 6.

Then, I added a +10% calendar year trend by multiplying the second diagonal by 1.1, the third diagonal by 1.21, etc.

This doesn't work at all, because the model just increases the age-to-age factors by 10% and all residuals are zero. Alternatively, if you use a GLM approach where you don't allow gammas (or force all gammas to zero), your alphas and betas absorb the constant inflation and you get a perfect fit again.

I introduced randomness, but that just gave me random residuals.

I tried using increasing inflation (5%, 8%, 10%, 12%, 15% for CYs 2 through 6), and I got a U-shaped residuals graph. Decreasing inflation (15%, 12%, 10%, 8%, 5%) gave me a rainbow-shaped graph.
#3
01-02-2018, 03:19 PM
 Abelian Grape Meme-ber                         Meme-ber CAS Join Date: Jul 2014 Favorite beer: Allagash Curieux Posts: 42,017

I think the key words in the problem are: "due to
inflation which is not reflected in the model"

What do you get if you recalculate the residuals where the fitted triangle is based on the triangle WITHOUT inflation? I tested this myself and got this:

Last edited by Abelian Grape; 01-02-2018 at 03:24 PM..
#4
01-02-2018, 03:42 PM
 |B|rad Member CAS Join Date: Jul 2009 Posts: 1,026

Thanks for the response, Grape.

I agree that I must be misinterpreting the part of the question that you highlighted. I basically took it to mean "use a GLM approach but you're not allowed to use gammas".

A few things though:

1. The "actuals with 10% inflation applied" should just be labeled "actuals", and your fit should be based on these. I don't think it makes sense to fit on something that didn't happen.

2. The sample solution starts with negative residuals at early CYs and goes to positive residuals for later CYs.

3. I don't think the bootstrap process will ever give materially non-zero residuals for the (1, 6) or (6, 1) coordinates.

I follow your process, but I think it's too far removed from the syllabus bootstrap.
#5
01-02-2018, 03:47 PM
 Abelian Grape Meme-ber                         Meme-ber CAS Join Date: Jul 2014 Favorite beer: Allagash Curieux Posts: 42,017

Quote:
 Originally Posted by |B|rad Thanks for the response, Grape. I agree that I must be misinterpreting the part of the question that you highlighted. I basically took it to mean "use a GLM approach but you're not allowed to use gammas". A few things though: 1. The "actuals with 10% inflation applied" should just be labeled "actuals", and your fit should be based on these. I don't think it makes sense to fit on something that didn't happen. 2. The sample solution starts with negative residuals at early CYs and goes to positive residuals for later CYs. 3. I don't think the bootstrap process will ever give materially non-zero residuals for the (1, 6) or (6, 1) coordinates. I follow your process, but I think it's too far removed from the syllabus bootstrap.
1. That's why the problem had the disclaimer that I highlighted. Because if you fit based on actuals and capture inflation in your fit, then you wouldn't be getting the residual plots in the problem.

2. I'll let you figure this one out. Hint: The top-left of the actuals triangle doesn't necessarily have to equal that of the fitted.

3. It will in the event that the fit doesn't capture something like inflation.
#6
01-02-2018, 04:10 PM
 |B|rad Member CAS Join Date: Jul 2009 Posts: 1,026

Quote:
 Originally Posted by Abelian Grape Hint: The top-left of the actuals triangle doesn't necessarily have to equal that of the fitted.
Ok, I think my disconnect is that your "fitted" triangle is not created using any process described in Shapland. I feel like the question constrains you to using Shapland, but you're just not allowed to allow CY parameters.

Shapland has example files where he fits a GLM but does not allow CY parameters, but he doesn't have anything exactly like this problem.