actuarialmath Uniform shortcuts use the wrong conditioning age and reverse limited-life weights

GitHub report · Evidence archive · Developer issue

Archive SHA-256: ec9f55b79b3a9f35e16ed984ec923536b0b128ba2adbde0eedd3b37c36b4b36f. Original English report and executable evidence.

Status: confirmed locally; reported to the maintainer. The bounded novelty review does not establish absolute priority. Developer report: https://github.com/terence-lim/actuarialmath/issues/9.

Affected source

• Upstream: terence-lim/actuarialmath
• Current remote main: 7d18f11ad304898f177b7922b3c53f70e4c2b4f4
• Official release copy: PyPI 1.1.0, taken from the independently hash-verified wheel retained by the earlier actuarial audit
• File: src/actuarialmath/mortalitylaws.py
• The current and release target files are byte-identical, SHA-256 ae0a9757a1bb580b4f3a62b666afc28a387f02d8f3eec0b6475b86e35d885833.

Mathematical discrepancy

For De Moivre's law at selection age x, duration s and limiting age omega, the remaining lifetime is uniform on [0, L], where

L = omega - (x + s).

Therefore:

E[T]              = L / 2
Var[T]            = L^2 / 12
E[min(T, t)]      = t - t^2/(2L),                 0 <= t <= L
Pr[T > t]         = (L - t)/L.

The current Uniform.e_x complete shortcuts use omega - x, discarding s. Its limited expectation computes

t_p_x = t / (omega - x)
return t_p_x * t + (1 - t_p_x) * (t / 2)

but t/(omega-x) represents death within the term. The death branch should receive the conditional mean t/2, while the survival branch should receive the full term t. The two weights are reversed. Uniform.E_x also uses omega-x instead of omega-(x+s).

Minimal public examples

At omega=100, x=40, s=0, t=10, zero interest and a continuous unit payment, the exact value is

integral_0^10 (1 - u/60) du = 10 - 100/120 = 55/6
                              = 9.166666666666...

Observed results:

Public call	Current / 1.1.0	Exact	Candidate
e_x(40, t=10, curtate=False)	5.833333333333334	9.166666666666667	9.166666666666668
temporary_annuity(40, t=10, discrete=False)	5.0	9.166666666666667	9.166666666666664
E_x(40, s=15, t=10)	0.8333333333333334	0.7777777777777778	0.7777777777777778

The first two failures occur even at s=0; they are therefore distinct from the previously reported continuous Annuity.a_x selection-duration defect.

Candidate correction

The isolated candidate:

1. computes remaining = omega - (x+s) in Uniform.e_x and Uniform.E_x;
2. uses remaining/2 and remaining^2/12 for complete moments;
3. assigns the t/2 value to the death-within-term probability and t to survival in the limited expectation;
4. computes the pure-endowment survival probability as (remaining-t)/remaining.

The exact patch is candidate.patch.

Executed verification

Each variant was imported in a separate process through the actual public classes. The grid covers:

• x = {10, 40, 60};
• s = {0, 5, 15} with x+s < 100;
• t = {0, 1, 5, 10, 20} inside the lifetime support;
• force of interest delta = {0, 0.01, 0.05};
• pure-endowment raw moments 1 and 2;
• continuous temporary annuities with unit payments.

Independent 80-decimal closed forms produced 468 comparisons per variant:

Check family	Checks	Current failures	Candidate failures	Restored	Release 1.1.0
Complete expectation	9	6	0	6	6
Complete variance	9	6	0	6	6
Limited expectation	45	35	0	35	35
Pure-endowment raw moments	270	144	0	144	144
Continuous temporary annuity	135	84	0	84	84
Total	468	275	0	275	275

The full grid was repeated at 120 decimal digits with the same counts. All 468 restored outputs and all 468 release outputs are bit-identical to current. Of 193 originally passing controls, 174 remain bit-identical under the candidate; the remaining 19 change only through algebraic evaluation order, by at most 3.552713678800501e-15, and all still match the oracle tolerance. The saved patch also applies to a fresh copy and reverses to the original SHA-256 exactly; see evidence/PATCH_REPLAY.json.

Duplicate review

The review covered all eight current upstream issue/PR records, their five comments, the relevant source history and focused GitHub issue/code searches. No issue or correction for these expressions was found. Exact code search for the limited-expectation denominator returned only the upstream source file. The expression dates to commit a5d2797ac722b6abcb2d1459311b85c20130a018 from 4 June 2023, so this is not described as a newly introduced regression.

Issue #7 reports a different Annuity.a_x call-site defect and explicitly left Uniform.temporary_annuity untriaged. The earlier published report likewise excluded this shortcut. Absolute novelty is not claimed.

Limits

• This verifies Uniform shortcut formulas for the stated finite domain; it is not a proof for every mortality law or contract method.
• Unit annuity benefits were used to isolate this defect. A separately observed zero-interest benefit-scaling discrepancy is excluded.
• The discrete zero-interest path and pure-endowment variance selector expose separate exceptions and are excluded pending independent analysis.
• Optional plotting and notebook-display modules were stubbed at import time; the tested numerical methods do not call them.
• The complete upstream test suite was not run.
• No insurer deployment, policy record, reserve, customer loss, regulatory breach or bounty eligibility was tested.

Developer disclosure

The reproducer, formulas, candidate diff and validation totals were submitted as actuarialmath issue #9 before broad distribution. No maintainer response, accepted patch or released correction is claimed. The investigation and report were AI-assisted; executable evidence and independent closed-form checks are retained.