Question on Additional 132-Ult Column in CDF Output

[Eddie-Zhu-IRL]

Hi Team,

I am using ChainLadder in R and Python to validate the results from an Excel tool. When extracting development factors, I found a difference in the length of cdf_ & ldf_ given by Python MackChainLadder and ChainLadder, compared with those listed in the summary table.

Summary table has AccidentYear ranging from 1998 to 2007, which is a 10-year window and the cdfs are calculated as Ultimate/Latest correspond to these 10 years. My understanding is that, for this 10-year triangle, the development periods to ultimate would range from 12-Ult to 120-Ult. However, the cdf_ output contains an additional 132-Ult column for both outputs from ChainLadder and MackChainLadder. In the screenshot below, cdf_ for 132-Ult are equal to 1.0000.

Could you please clarify why the additional 132-Ult development period is included in cdf_ , and how this value should be interpreted when reconciling the ChainLadder results with those the summary table?

Regards,
Eddie Zhu

[by-ae]

The extra column is the tail. Chainladder and MackChainladder both need an ldf_ that runs past the last development age. If you don't fit a tail yourself, they apply TailConstant() with its defaults (tail=1.0, projection_period=12). See MethodBase.validate_X in chainladder/methods/base.py.

TailConstant pads ldf_ with factors of 1.0 for 120-132 and 132-144 (tails/base.py). That's where cdf_ gets its 120-Ult and 132-Ult entries. Reproduced with chainladder 0.10.1 on the 10x10 genins sample:

import chainladder as cl
t = cl.load_sample("genins")
cl.Development().fit(t).cdf_     # 12-Ult ... 108-Ult      (9 columns, no tail)
cl.Chainladder().fit(t).cdf_     # 12-Ult ... 132-Ult      (11 columns)
#                                  ... 108-Ult 1.017725, 120-Ult 1.0, 132-Ult 1.0

For reconciling with your summary table:

• 120-Ult is the CDF for your oldest year (latest age 120). With no tail it's 1.0, which matches Ultimate/Latest = 1.0 for 1998.
• 132-Ult is past the end of the triangle, so no accident year uses it. You can ignore it. With the default tail it's always 1.0.

If you do fit a tail, the column becomes meaningful. For example, cl.TailConstant(1.05) gives 120-Ult = 1.05, and 132-Ult holds the part of the tail still left after the 12-month projection_period. To compare per accident year, use model.ultimate_ / model.latest_diagonal, which already lines each year up with the right CDF.

[Eddie-Zhu-IRL]

Thanks AE.