actuarialmath continuous annuity: dropping elapsed years changes a value and a decision

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Archive SHA-256: 8784b39fc462e56b810752f7117da5ce8a385c4e37edebe106b7bae753073de9. Original English report and executable evidence. Verified distribution links appear above.

20 September 2026 · Independent GERO research by Xamit Kadirbekov · AI-assisted investigation and preparation.

A continuous annuity factor that should be 8.75 becomes 9.166666666666666 in the public actuarialmath implementation. The function accepts years since selection, s, but substitutes zero in its continuous survival calculation. An equivalent attained-age input returns 8.75. This report documents one implementation defect, a narrow local candidate and an executed synthetic document/decision consequence.

Minimal public-API example

from actuarialmath import Uniform
life = Uniform(omega=100).set_interest(i=0)
print(life.a_x(40, s=20, t=10, discrete=False))  # 9.166666666666666
print(life.a_x(60, s=0, t=10, discrete=False))   # 8.75

The annuity API defines x as selection age and s as elapsed years. The survival API carries both quantities into the survival function.

For this ultimate uniform-lifetime model, both inputs describe a person aged 60. Remaining lifetime is uniform from 0 to 40. At zero interest, a continuous payment rate of one unit per year, for up to ten years while alive, has expected present value

integral from 0 to 10 of (1 - y/40) dy
= 10 - 100/80
= 8.75.

Using selection age 40 without its elapsed 20 years instead gives remaining lifetime 60 and 10 - 100/120 = 55/6. Same-attained-age equivalence is asserted for this ultimate model, not for arbitrary select mortality models.

Implementation and narrow candidate

The pinned source passes s in the discrete branch but calls self.S(x, 0, t=t+u) in the continuous integrand. The candidate changes only that survival argument:

- self.S(x, 0, t=t+u)
+ self.S(x, s, t=t+u)

Executed source: main 7d18f11ad304898f177b7922b3c53f70e4c2b4f4. Separately executed release: official PyPI 1.1.0, wheel SHA-256 b19990e4378aaa19fe6bc1182b4269faec6617cb62b0677fea1e624fbbb3ff6f. All 22 copied release Python modules were compared with that wheel. The affected file is identical in current and release snapshots.

Independent oracle and checks

The main grid calls the real Annuity.a_x with a supplied uniform survival function. With L = 100 - x - s, payment rate B, deferral u, term t and force of interest delta, the continuous reference is

B * integral from u to u+t of exp(-delta*y) * (1 - y/L) dy.

At zero force it is B * [t - (u*t + t*t/2)/L]. At positive force an antiderivative is exp(-delta*y) * [(y-L)/delta + 1/delta²] / L, multiplied by B. Discrete controls use an independent discounted finite sum. The reference uses 80-digit Decimal arithmetic, with 59 controls per variant repeated at 120 digits. Nominal decimal rates are compared with the library's float64 counterparts at fixed absolute tolerance 2e-12 * max(1, B*t); the reference does not represent exact binary64 input arithmetic.

The 2,160 cases per variant cover selection ages 20/40/60; elapsed years 0/1/5/10/20; terms 0/1/5/10; deferrals 0/2/7; forces 0/.01/.05; constant rates 1/1000; continuous and discrete branches. All tested intervals stay inside lifetime support.

Executed variant	Failures / 2,160
Pinned current source	648
Official release 1.1.0	648
Local one-line candidate	0
Original argument restored	648

All 1,512 previously passing outputs are exactly unchanged in the tested environment. Separate executions of the package's Uniform class verify the minimal example. A fresh portable packet replay applies and reverses the patch, verifies the source/wheel hashes, and reproduces all four numerical grids, Uniform controls, written memos and read-back decisions.

Environment: Python 3.12.14, NumPy 2.5.3, SciPy 1.18.1, pandas 3.0.6, matplotlib 3.10.8 and IPython 9.17.1; one configured numerical worker, CPU only. No full upstream test-suite or performance claim.

Executed formula → document → decision chain

For the same example with a continuous payment rate of 1,000 units per year, a GERO adapter writes a JSON expected-value memo. A separate consumer reads the stored rounded amount and compares it with a synthetic ceiling of 9,000.

Stage	Original / release / restored	Local candidate
Calculated expected present value	9,166.666666666668	8,750.0
Amount written in JSON	9,166.67	8,750.00
Read-back ceiling decision	OVER_LIMIT	WITHIN_LIMIT

The displayed amount changes by 416.67 units. The pricing formula is real library code. Memo creation, rounding, the ceiling and the consumer are explicitly GERO demonstrations. This is not an insurer's quote, a required reserve, an actual underwriting decision or evidence of customer loss.

Prior art, disclosure and limits

The bounded duplicate review inspected all pre-existing six issue/PR bodies and four comments, the complete PR #2 diff, eight annuity.py history records, the guide repository's issues, focused searches and GERO catalogues. A fresh check before distribution found unchanged main, seven issue/PR records including our own #7, and no additional PR or response. No exact earlier report or correction was found in that scope; no global priority claim is made. The source pattern dates to 2023, so this is not presented as a recently introduced regression. The internal static lead preceded its executed confirmation.

Developer issue #7 was submitted and its exact body verified before publication. No upstream acknowledgment, accepted patch or released fix is claimed.

The correction is limited to this argument in Annuity.a_x. Other annuity convenience methods, arbitrary survival laws, variable benefits, whole-life limits and negative forces are not validated here. In particular, Uniform.temporary_annuity gives a separate unexpected zero-rate result that remains unchanged; it is not used as an oracle or described as repaired. Other initial candidates are excluded. No real insurer system, policyholder record, deployment, loss or regulatory breach was tested.

Original report: CC BY 4.0. GERO scripts and candidate: MIT. Upstream source retains Terence Lim's MIT license. See the complete evidence archive, source ledger, executable runner, raw outputs and explicit environment requirements.