[FEAT] One-year claims development result (Merz–Wüthrich) for MackChainladder #1543

Description

MackChainladder measures reserve uncertainty to ultimate: mack_std_err_ is the prediction error of the reserve for all future payments. Solvency II sets reserve capital over a one-year horizon, so it needs a different quantity: how far the estimate of ultimate could move over the next year, once one more diagonal is observed and the chain ladder is re-run. That change is the claims development result (CDR). chainladder-python has no way to estimate its uncertainty today. R's ChainLadder added CDR() for MackChainLadder and BootChainLadder in 0.2.0 (2015). In #30, Peter England wrote that, as far as he could see, chainladder-python computes Mack's standard errors but not the standard deviation of the CDR for a sequence of one-year views to ultimate, and that he dropped his chainladder-python example for that reason.

Merz & Wüthrich, "Modelling the claims development result for solvency purposes" (CAS E-Forum, Fall 2008), give closed-form estimators under Mack's distribution-free model. They need only the triangle and the factors and sigmas that MackChainladder already holds. The technique table in #680 lists "A Practical Way to Estimate One-year Reserve Risk" (Robbin, CAS E-Forum, Summer 2012). Section 6.2.1 of that paper reviews the Merz–Wüthrich estimators, and Robbin's own method builds one-year risk from an ultimate reserve-risk estimate plus payment and reporting patterns. MackChainladder already produces the ultimate estimate.

Is your feature request aligned with the scope of the package?

• Yes, absolutely!
• No, but it's still worth discussing.
• N/A (this request is not a codebase enhancement).

Describe the solution you'd like, or your current workaround.

Two fitted attributes on MackChainladder, alongside mack_std_err_ and total_mack_std_err_, rather than a new estimator:

• cdr_std_err_: the one-year CDR standard error by origin
• total_cdr_std_err_: the same for all origins together, including the covariance between origins

A first PR would keep to the paper's setting:

• volume-weighted development factors over all periods, with no exclusions
• no tail beyond the default constant 1.0 (R's documentation says tail factors are not supported)
• a square triangle, one diagonal ahead, which is one year when the development periods are annual

Other configurations would raise a ValueError.

I have a prototype that computes both quantities from a fitted MackChainladder's own ldf_, sigma_, std_err_, latest diagonal and ultimate_, vectorized over the index and column axes. Against R's CDR() on the two Merz–Wüthrich samples that already ship with the package:

Per origin, the prototype agrees with R to the last digit R prints (4 decimals for mw2008, 7 for mw2014). On the paper's own example it reproduces Table 2's factors exactly and its sigmas to within 0.015, the observable CDR of −40,075 in Table 3 (computed from the next diagonal the paper prints), and every entry of Table 4 to within 0.2%. The largest gap is one Mack figure, 1,563.8 against the paper's 1,566 (accident year 2 in the paper's numbering), and R gives 1,563.8 there too. On clrd grouped by LOB (paid and incurred, 12 triangles), the vectorized version matches the same formulas applied one triangle at a time to within 2e-12 relative.

These figures use Development(sigma_interpolation="mack"), which matches R's est.sigma="Mack". With the default "log-linear" the last sigma changes, and both results move with it: on mw2008 the one-year figure becomes 81,336.66 and total_mack_std_err_ 108,732.16.

Questions before a PR:

1. Are attributes on MackChainladder the right home, and are these names acceptable? "CDR" is the paper's term and R's, though cdr_ sits close to cdf_; one_year_std_err_ is an alternative.
2. Should unsupported configurations raise, or return NaN with a warning?
3. Should the first PR stop at one year? The sequence Peter England was looking for is what R's CDR(dev="all") returns: one standard error per future calendar year, from Merz & Wüthrich, "Claims Run-Off Uncertainty: The Full Picture" (Swiss Finance Institute Research Paper No. 14-69, 2014). I'd start with the first year, which the 2008 paper's example checks, and add the later years in a follow-up checked against R's dev="all" output.

Do you have any additional supporting notes?

• Merz, M. and Wüthrich, M. V., "Modelling the claims development result for solvency purposes", CAS E-Forum, Fall 2008, pp. 542–568. The paper's Results 3.1–3.3 and 3.5 give the estimators, and section 4 works the example.
• R ChainLadder's CDR reference page, with the outputs quoted above: https://mages.github.io/ChainLadder/reference/CDR.html
• Robbin, I., "A Practical Way to Estimate One-year Reserve Risk", CAS E-Forum, Summer 2012: https://www.casact.org/abstract/practical-way-estimate-one-year-reserve-risk

Would you be willing to contribute this ticket?

• Yes, absolutely!
• Yes, but I would like some help.
• No.

(This issue was written with Árbol, an AI assistant, under my direction.)