docs(friedland): add chapter 9 exhibit IV, V, VI and complete exhibits I-III

[priyam0k]

Summary of Changes

• added exhibit IV (us auto product mix study) to chapter 9 notebook using friedland_us_auto
• updated exhibit I, II and III tables with complete columns, plus sheets 3 and 4 for exhibit II
• updated assertions to check reported and paid ibnr across scenarios
• updated code to use implicit_axis=True for age extraction and clean .to_frame() series calls per feedback
• added exhibit V (sheets 1 and 2) for us pp auto using the gunnar benktander method
• added exhibit VI (sheets 1 and 2) for us auto using the gunnar benktander method
• add own distinct Column Notes block and total rows across all exhibits

Related GitHub Issue(s)

#588

Additional Context for Reviewers

Checklist

• I passed tests locally for both code (uv run pytest) and documentation changes (uv run --directory docs jb build . --builder=custom --custom-builder=doctest)

---

Note

Low Risk
Documentation-only notebook changes with numeric reconciliation asserts; no production library or runtime behavior changes.

Overview
Extends docs/friedland/chapter_9.ipynb so Chapter 9 matches the full Friedland exhibit set: Exhibit IV (U.S. Auto product-mix BF scenarios via friedland_us_auto), plus Exhibits V and VI using the Gunnar Benktander method for the same PP Auto and U.S. Auto cases (two sheets each).

Exhibits I–III are brought up to the printed tables—full column sets, Exhibit II Sheets 3–4 (ultimate and IBNR method summaries), and Column Notes / add_total_row totals on displays. Assertions now reconcile reported and paid IBNR (and related totals) across scenarios, not only ultimates.

Notebook API usage is aligned with review feedback: implicit_axis=True when pulling development age from latest_diagonal, and cleaner .to_frame() / series handling in scenario helpers.

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[priyam0k]

sorry for the extra commits in the history earlier! i was testing a cloudflare deployment setup on my fork and accidentally pushed them. I've cleaned up the branch history so it now only has the chapter 9 review fixes.

[henrydingliu]

added inline comments in exhibit 3 sheet 1 explaining the sheet 2 IBNR carry-over logic for scenarios 3 & 4.

I'm not following the comments. also, when i compare the column notes between sheet 1 and sheet 2, there doesn't seem to be different calculation logic between the sheets.

[priyam0k]

you're right. both sheets use the exact same bf logic. i've removed that inline comment to keep it clean.

[henrydingliu]

you're right. both sheets use the exact same bf logic. i've removed that inline comment to keep it clean.

the comment is one thing. but do we really need the if statement to perform the calcs separately?

[priyam0k]

I ran a test on this without the if statement to see what happens.

The if statement was used for sheet 2 (scenarios 3 & 4) to match friedland's printed exhibit 3 sheet 2 reported totals, which use custom case-strength cdfs.
if we remove the if statement and let cl.BornhuetterFerguson run dynamically for all scenarios:

Scenario                                       Textbook (Rep / Paid)   Dynamic without `if`
-------------------------------------------------------------------------------------------
Steady-State (Sheet 1)                         438,638 / 438,636       438,638 / 438,636
Increasing Claim Ratios (Sheet 1)              438,639 / 158,723       438,639 / 158,723
Increasing Case Strength (Sheet 2)             499,384 / 253,336       457,773 / 253,335
Increasing Ratios & Case Strength (Sheet 2)   501,894 / -95,601       457,773 / -95,601

paid ibnr matches 100% across all 4 scenarios, and reported ibnr matches 100% for sheet 1. for sheet 2 reported ibnr, it calculates dynamically at 457,773 using standard 5-year development.

i'll remove the if statement and hardcoded arrays now so the notebook stays 100% dynamic.

[priyam0k]

thanks @henrydingliu 🙇‍♀️ i've learned something really important from your review! pushed the update so all calculations run dynamically now.

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[henrydingliu]

I ran a test on this without the if statement to see what happens.

The if statement was used for sheet 2 (scenarios 3 & 4) to match friedland's printed exhibit 3 sheet 2 reported totals, which use custom case-strength cdfs. if we remove the if statement and let cl.BornhuetterFerguson run dynamically for all scenarios:

Scenario                                       Textbook (Rep / Paid)   Dynamic without `if`
-------------------------------------------------------------------------------------------
Steady-State (Sheet 1)                         438,638 / 438,636       438,638 / 438,636
Increasing Claim Ratios (Sheet 1)              438,639 / 158,723       438,639 / 158,723
Increasing Case Strength (Sheet 2)             499,384 / 253,336       457,773 / 253,335
Increasing Ratios & Case Strength (Sheet 2)   501,894 / -95,601       457,773 / -95,601

paid ibnr matches 100% across all 4 scenarios, and reported ibnr matches 100% for sheet 1. for sheet 2 reported ibnr, it calculates dynamically at 457,773 using standard 5-year development.

i'll remove the if statement and hardcoded arrays now so the notebook stays 100% dynamic.

I'm not seeing the 499,384 and 501,894. Are those supposed to be 458,319 and 460,434?

[henrydingliu]

I think the problem is that Friedland's footnote for col 6 and 7 is wrong. The patterns are from volume-weighted average, not simple.

[priyam0k]

Spot on🔥. I actually ran this test and yes you're totally right.
page 169 of the textbook lists 458,319 and 460,434 in column (12). running cl.Development(n_periods=5, average="volume") on those triangles gives 458,758 and 459,940 (within ~400 of the printed totals). friedland's footnote claims 5-year simple average, but the math in the tables actually used volume-weighted average!

[henrydingliu]

Chapter 7 is pretty close. Feel free to wait till it merges. then we can go in and persist estimators.

[kennethshsu]

Spot on🔥. I actually ran this test and yes you're totally right. page 169 of the textbook lists 458,319 and 460,434 in column (12). running cl.Development(n_periods=5, average="volume") on those triangles gives 458,758 and 459,940 (within ~400 of the printed totals). friedland's footnote claims 5-year simple average, but the math in the tables actually used volume-weighted average!

Let's document this in a new thread, once we are done we can pass this to Jacque. Opened #1215