Implementation Notes

Status: Draft, 2026-08-29; citations re-verified against the primary documents 2026-08-30. Built from products/fondsgebundene_rentenversicherung/technical-notes.md; the product it implements is specified in product-spec.md.

This is a mechanics demonstration, not a pricing, reserving or disclosure result. The mechanics are common ground in German practice and several are cited: the Beitragsverrechnung order and the purchase of Anteileinheiten at the Anteilspreis [S1], the Höchstzillmersatz of 25 ‰ of the Beitragssumme R12 R13 REG-R16 and the even spreading of the acquisition charge over the first five contract years R1 REG-R28, the Beitragsrückgewähr death benefit [S2], the Zeitwert Rückkaufswert and the conditions on a Stornoabzug R1 REG-R36, the survival of the fund-based charges into a beitragsfrei contract R3, the max(guaranteed, current) Rentenfaktor rule [S4] R22, and the unisex tariff REG-R34. delib was drafted with HTTP egress blocked and nothing retrieved; the citations have since been re-verified against the primary documents, and eighteen of the forty-four entries in sources.md now record a document that was opened and read, twenty-four record none. That verifies mechanics of the kind cited above and none of the levels below. The levels are almost entirely std: no lapse rate and no expense or commission scale was established anywhere, and outside the single carrier whose rates could be read [S2] [S15] no charge rate and no Rentenfaktor was established at any carrier — that one tariff is a comparator the Standardizations table below measures the shipped levels against, not their source. The DAV tables behind both mortality bases — DAV 2008 T for the Risikobeitrag R17 REG-R48, DAV 2004 R behind the Rentenfaktor R16 REG-R49 — are Deutsche Aktuarvereinigung property and are cited by name, never shipped. Replace the charge scale, the decrement tables and the fund path with a real tariff and company data before drawing any conclusion from a number. Nothing here is an Effektivkostenquote and nothing here may be compared with a PRIIPs performance scenario R8 R9 REG-R32.

Run it

python products/fondsgebundene_rentenversicherung/run.py
python products/fondsgebundene_rentenversicherung/run.py 7    # the beitragsfrei cell

Three lines to the same thing:

import modelx as mx
model = mx.read_model("products/fondsgebundene_rentenversicherung/FRV_DE_S")
model.Projection[1].result_cf()

Projection takes a point_id and Projection[1] is the worked-example anchor cell. result_cf() returns a tidy DataFrame indexed by policy month t with one column per cash flow line, and result_fund() the per-policy unit side — the Anteilspreis, the unit count, the four within-month Fondsguthaben balances, the Beitragsrückgewähr base, the net amount at risk and the three decrement rates, which is roughly what a German Standmitteilung reports [S17]. model.doc describes the product and the basis, model.Projection.doc maps the notes’ symbols onto the cells names, and model.Data.doc says what each input file is and what a replacement must preserve.

The time index and the frame

t is the 0-based policy month counted from the contract’s own inception: t = 0 is the inception month, period t runs from time t to time t + 1, and t = 60 means the same thing on every model point — the first month after the acquisition-charge instalment ends. proj_len() = 12 × (annuity_age − entry_age) is the number of policy months and so the frame’s exclusive end, so result_cf() covers range(proj_start(), proj_len()) — the last row is t = proj_len() − 1, which is 359 on the anchor cell, and len(result_cf()) is proj_len() − proj_start(). That is lifelib’s own convention (basiclife/BasicTerm_S, savings/CashValue_SE: for t in range(proj_len())), and it is what the conventions suite asserts for every model point.

proj_start() = duration_init_m — 0 for new business and 96 for the in-force cell. duration_init_m is an elapsed count and is therefore already 0-based, so it is the first projected index itself and needs no + 1; an in-force model point simply opens partway through a frame whose origin is still inception. pols_if(proj_start()) == pols_if_init() exactly, and pols_if(proj_len()) = 0, one past the last row.

The contractual policy year is the 1-based label policy_year(t) = t // 12 + 1, derived from t and never indexed by, and age(t) = entry_age + policy_year(t) − 1 = entry_age + t // 12. Because a 0-based frame has no t = −1, the opening Anteilspreis is its own cells, unit_price_open(t) — unit_price_init at t = proj_start() and unit_price(t − 1) after it — which is what av_pp and units_bought_pp read; the ledgers cum_prem_pp and cum_charge_acq_pp likewise seed inside the frame’s first month rather than one before it.

What the CSVs did and did not do

File	Column	Decision
model_point_table.csv	duration_init_m	Unchanged. An elapsed count, 0-based by nature; 96 stays 96 and is now proj_start() itself
model_point_table.csv	pup_month	Shifted −1: a point on the frame’s time axis. Model point 7 goes beitragsfrei at t = 120 (was 121). 0 stays the “never” sentinel, not the inception month
model_point_table.csv	topup_month	Shifted −1: model point 9 tops up at t = 120 (was 121). 0 stays the “none” sentinel
model_point_table.csv	wd_month	Shifted −1: model point 9 withdraws at t = 240 (was 241). 0 stays the “none” sentinel
model_point_table.csv	entry_age, annuity_age, prem_term_y, prem_mode_months	Unchanged. Ages, a term in years and a frequency in months — none is a point on the frame
lapse_table.csv	policy_year	Unchanged. The contractual 1-based label; lapse_rate_base(t) maps through policy_year(t) = t // 12 + 1
fund_scenario_table.csv	policy_year	Unchanged, for the same reason
charge_table.csv	alpha_spread_months	Unchanged. A count of months, not an index
mort_table.csv	age	Unchanged. An attained age
rentenfaktor_table.csv	annuity_age	Unchanged. An age at Rentenbeginn

No provenance column was touched, and no row order changed.

The insurer guarantees the number of units, not their value

Everything else follows from that sentence, and it is what makes this a unit-linked model rather than a translation of the general-account ones beside it. There is no Rechnungszins in the accumulation phase, no Deckungskapital, no Zinsüberschuss — and, because § 125 VAG makes the covering assets a segregated Anlagestock held in the very units the liability is denominated in R15 REG-R7, no investment-mismatch term anywhere in the model. The state variable is therefore the unit count, and euro are derived from it:

units_pp(t + 1) = units_pp(t) + units_bought_pp(t) − units_cancelled_pp(t)
av_pp(t)        = units_pp(t) × unit_price_open(t)

check_units_roll_fwd() asserts the first line and check_av_roll_fwd() the account identity that carries the price. They look redundant and are not, which is why both ship: the unit identity has no price term in it at all, so it fails when a charge is taken in euro without the matching units being cancelled while every euro total still looks plausible; the account identity fails when the return is applied at the wrong point in the order. An implementation can pass either one alone. The fund’s TER is a third category and is deliberately given no charge_* cells: borne inside the Anteilspreis, it never appears in a policy ledger, so fund_return_net_ann(t) = gross − ter nets it off the assumed return instead. Charging it explicitly double-counts; ignoring it overstates the policyholder’s return.

net_cf is the non-unit stream

Every benefit paid before Rentenbeginn — the death benefit up to the fund, the Rückkaufswert, the Teilentnahme, the capital released at Rentenbeginn — is funded by cancelling the policyholder’s own units, so a gross presentation counts the same money twice. delib’s first ruling requires every model to publish check_net_cf(); the identity here is one line:

net_cf = charge_acq + charge_admin_prem + charge_admin_fund + charge_policy_fee
         + charge_risk + stornoabzug − expenses − commissions − death_strain

Charges in, the insurer’s own expenses, its commission and the death strain out, and nothing else. expenses excludes commission and commissions is its own cells and its own result_cf() column — the delib convention, stated the same way on KLV_DE_S, Basis_DE_S, Riester_DE_S and RLV_DE_S, and the opposite of the frlib chassis, where commission sits inside the expense total. Taking both conventions at once double-counts the commission. check_net_cf_resid(t) does not restate that formula: it rebuilds the first two terms by a different route, as premiums − prem_to_av, which is what the Beitragsverrechnung leaves behind — so the check crosses the unit / non-unit boundary rather than asserting the code against itself. The gross flows are published beside it rather than dropped. premiums, prem_to_av, claims_death, claims_lapse, claims_maturity, withdrawals and av_releases are result_cf() columns, all excluded from net_cf, and check_benefit_funding() asserts that they net exactly:

claims_death + claims_lapse + claims_maturity + withdrawals + stornoabzug
    = av_releases + death_strain

Booking the whole Fondsguthaben as an insurer outgo is this product’s first-order failure mode — every column still looks reasonable and the liability is overstated by the entire fund — and publishing the excluded columns is what lets a reader see what was excluded. The scale of it on the anchor: 64 869,36 € of benefits against 40 586,28 € of premiums, of which 4,39 € is an insurer cost. Seven check_*() cells travel with the model, each a bool over all t with a check_*_resid(t) companion — check_net_cf, check_prem_split, check_units_roll_fwd, check_av_roll_fwd, check_benefit_funding, check_pols_roll_fwd, check_acq_charge — and all seven are True on all thirteen model points.

Withheld from the premium, or cancelled out of the fund

This is the distinction the product turns on and the model’s most easily-hidden error. The Beitragsverrechnung withholds the acquisition instalment and the premium-based administration charge before any unit exists:

prem_to_av_pp(t) = B(t) + Z(t) − charge_acq_pp(t) − beta_rate × B(t)
units_bought_pp(t) = prem_to_av_pp(t) / unit_price_open(t)

while the kapitalbezogene Verwaltungskosten, the Stückkosten and the Risikobeitrag are levied after the month’s return, by cancelling units that already exist. check_prem_split() asserts that the premium splits exactly three ways, so a model that also netted the Stückkosten or the fund-based charge out of the Beitrag fails there. What the model avoids is the shortcut that looks identical while premiums are paid: netting gamma out of the Beitrag is right until the premium stops and wrong from then on. Model point 7 goes beitragsfrei at t = 120 on a zero-return fund, and from there premiums(t) is zero while charge_admin_fund(t), charge_policy_fee(t) and charge_risk(t) continue and the fund decays — the product fact § 165 VVG makes possible R3 REG-R28, not a modelling artefact. A Zuzahlung pays its own Zuzahlungskosten and no beitragsbezogene charge, being no regular Beitrag; it is booked in charge_acq, which is why charge_acq(120) is non-zero on model point 9.

The acquisition charge, its window, and the in-force cell

charge_acq_total() = alpha_rate × beitragssumme() — 2,50 % of the sum of premiums payable, the Höchstzillmersatz itself R12 REG-R16 — spread in equal instalments over acq_window_months() = min(alpha_spread_months, 12 × prem_term_y) months at the policy’s own premium frequency R1 REG-R28. The composite takes the cap rather than a guessed interior point, so the reference implementation demonstrates the binding constraint. On the anchor cell that is 1 800,00 € over 60 instalments of 30,00 € — 15 % of each of the first sixty premiums, t = 0 … 59, and nothing from t = 60, where the Anlagebeitrag steps from 162,00 € to 192,00 €. That cliff is the characteristic shape of a German unit-linked contract’s early values and it is why this model runs monthly: an annual grid cannot place the sixtieth month.

The instalment count is the window divided by the frequency — 60 monthly, 20 quarterly, 10 half-yearly, 5 annual, and 24 on model point 12, whose premium term is two years; a shortened term still spread over sixty months would understate every instalment. check_acq_charge() closes the ledger against an expectation counted rather than accumulated — instalment dates elapsed times the instalment, plus the Zuzahlungskosten on any Zuzahlung received — so a window running one month too long fails there.

beitragssumme() is the sum of premiums payable at the initial level and is invariant: it does not shrink on lapse or Beitragsfreistellung and does not grow with a Beitragsdynamik increment or a Zuzahlung. A real tariff re-zillmers each accepted increment over its own sixty months, and an increment cannot be assumed at inception; the bias that leaves is stated rather than hidden. Letting S follow the premiums actually paid would make the acquisition charge a function of the lapse assumption — wrong, and circular. An in-force model point opens after the window has closed: model point 6 starts at t = 96, so charge_acq(t) is zero at every projected month and commissions(96) carries no Abschlussprovision, comm_acq_pp and expense_acq_pp both falling at t = 0 and only there. That is the whole of the difference between an in-force cell and a new-business one on this chassis.

The Beitragsrückgewähr, and two mortality bases at once

The composite death benefit is max(Fondsguthaben, Summe der gezahlten Beiträge) [S2], so the net amount at risk is max(cum_prem_pp(t) − F, 0) — positive early, vanishing once the fund overtakes the premiums paid (t = 94 on the anchor cell), returning after a market fall. That makes cum_prem_pp a genuine state variable rather than a reporting convenience, and the risk charge a quantity recomputed every month. It is the premiums paid, gross: cum_prem_pp(59) = 12 000,00 € against 9 720,00 € actually invested, so reading the floor off the invested amount would understate the death benefit by 19 %.

The floor at zero in nar_pp() is load-bearing. Without it the contract would pay the insurer a negative charge in every month the fund is above the floor and death_strain would turn negative, silently booking the fund’s growth as insurance profit. And db_pp(t) = av_pp_at(t, "BEF_DECR") + nar_pp(t) rather than max(floor, fund): writing it as a sum keeps the two sides apart, the first term being the policyholder’s money and the second the insurer’s.

The Risikobeitrag is priced on a death table and the conversion guarantee on an annuity table. mort_rate_tariff_at_age reads mort_table.csv, a std first-order DAV 2008 T proxy R17 REG-R48; rentenfaktor_guar reads rentenfaktor_table.csv, standing in for DAV 2004 R R16 REG-R49. No cells reads both files — the arithmetic form of the statement that a German fondsgebundene contract carries two mortality bases at once; a model pricing the death charge on an annuitant table understates it. The projection then decrements on a third rate, the second-order best estimate mort_be_factor = 0.75 times the tariff rate REG-R47. The wedge is the Risikoergebnis, and because the factor is flat it is exactly 25 % of the Risikobeitrag collected — 5,849973 less 4,387480 = 1,462493 € on the anchor — a closed form a reader can check with a calculator. One basis for both makes the risk result identically zero.

The two monthly conversions are deliberately different. Mortality is split linearly, mort_rate_mth = mort_rate / 12, because the tariff’s Risikobeitrag is q(x)/12 times the riskiertes Kapital and charge and decrement must share the split, or the model manufactures a risk result out of a rounding convention. Lapse is split geometrically, 1 − (1 − lapse_rate)^(1/12), because nothing is priced off it and the annual rate is the observable to reproduce; the fund return compounds geometrically for the same reason, while gamma_rate_mth() is gamma_rate_ann() / 12 because a German tariff quotes charge rates nominally.

The Rückkaufswert is the Fondsguthaben

§ 169 VVG sends a fondsgebundene contract to the Zeitwert, and on a pure unit-linked contract with no insurer-given guarantee the Zeitwert is the fund R1 REG-R28. There is no discounting, no Rechnungszins, no mortality basis, no Zillmerung residue and no second-basis Mindestrückkaufswert anywhere in this model — the cleanest surrender rule of the ten delib products. The protection sits earlier, in the sixty-month spreading, which is why the surrender value is positive from the first month. A Stornoabzug is permissible only if vereinbart, beziffert and angemessen, and never for unamortised acquisition costs R1 REG-R36. stornoabzug_pp(t) is therefore a flat rate on the Fondsguthaben and deliberately not a function of charge_acq_total() − cum_charge_acq_pp(t): that prohibition is what stops an insurer recovering through the deduction what the five-year spreading denies it. Only std_high carries a rate and only model point 5 uses it.

Beitragsfreistellung is a model point election, pup_month, not a cohort decrement — the one place the model reproduces a mechanic exactly on one cell rather than approximately on all of them. A paid-up policy’s fund and its Beitragsrückgewähr base both depend on the month it went paid-up, so a cohort rate would need one sub-cohort per month: a two-dimensional recursion over 360 months for a second-order effect. The std 1 % p.a. rate is recorded and not implemented, and the omission biases charge income upward. Storno and Beitragsfreistellung stay two different things: one an exit paying the Rückkaufswert, the other a change of state paying nothing R2 R3.

The last month, the age at Rentenbeginn, and the reduction in yield

lapse_rate_mth(proj_len() − 1) = 0 std. The end of the last projected month is Rentenbeginn, so a surrender and an annuitisation are the same event releasing the same Fondsguthaben, and the whole surviving cohort is booked as pols_maturity. No cash flow moves either way; the convention decides only the split between the lapse total and the maturity count, and it is what the closure identity reproduces — deaths 0,04377181 plus lapses 0,65322937 plus maturity 0,30299882 = 1,00000000. It is frlib’s convention on TD_FR_S and delib adopts it.

age(proj_len() − 1) = annuity_age − 1, because the annuity begins at the end of that month, and the Rentenfaktor is read at annuity_age: 25,00 at 67 on the anchor, not the 24,45 an off-by-one fetches at 66, a 2,2 % understatement of the pension. The rule applied is max(rentenfaktor_guar(), rentenfaktor_curr()) [S4] R22 — a guarantee with upside, so a model applying only the guaranteed factor understates the benefit whenever the current tariff is richer. On std_2026 the two are equal, so the max() is exercised without an unsourced uplift; model point 13 carries rich_current, 12 % higher, where it visibly bites. Only the conversion terms are guaranteed; the capital they multiply is the market’s, so a guaranteed Rentenfaktor is not a guaranteed pension.

reduction_in_yield() is the product’s defining metric, because on a contract with no Rechnungszins the charge stack is the economics. It is gross_return_ref() − irr_ann(), computed on a single persisting contract — no survivorship, no lapse — because a reduction in yield is a statement about one policy. On the anchor 5,0000 % less 3,6593 % = 1,3407 % p.a., and across the four shipped charge scales it moves by a factor of five. It is a delib-defined measure and not the statutory Effektivkostenquote, which is aligned to the total-cost-indicator method of the PRIIPs RTS over a specified recommended holding period R7 R9 REG-R31 REG-R32; this model implements neither.

Inputs are external files

The six input CSVs live in this directory, beside run.py — not inside the model folder. FRV_DE_S/ holds nothing but formulas:

products/fondsgebundene_rentenversicherung/
  model_point_table.csv  mort_table.csv  lapse_table.csv     <- inputs live here
  charge_table.csv  fund_scenario_table.csv  rentenfaktor_table.csv
  run.py
  product-spec.md  technical-notes.md  model.md  sources.md  <- the documents
  FRV_DE_S/                                                  <- formulas only
    __init__.py  _system.json                                   (model docstring)
    Data/__init__.py                (reads the CSVs, once per model)
    Projection/__init__.py          (the by-policy projection)

This follows lifelib’s annuallife/TradLife_A, which keeps its inputs beside the model and reads them at run time. It is the opposite of basiclife/BasicTerm_S, which stores its inputs inside the model through modelx’s IOSpec machinery — hence no _data/ directory and no embedded values here at all.

Read once, in Data

Projection is parameterized by point_id, so every Projection[N] is a separate ItemSpace with its own cells cache; readers placed there would re-read every file for every policy. They live instead in an unparameterized Data Space, which Projection references as data — so each file is read once per model however many policies are projected. The conventions suite counts the reads over a full sweep and asserts the file set, not merely the count. Data.input_dir() resolves the location from _model.path.parent at read time, so it works wherever the repository is checked out.

Reference	Cells	File
model_point_file	model_point_table()	model_point_table.csv
mort_file	mort_table()	mort_table.csv
lapse_file	lapse_table()	lapse_table.csv
charge_file	charge_table()	charge_table.csv
fund_scenario_file	fund_scenario_table()	fund_scenario_table.csv
rentenfaktor_file	rentenfaktor_table()	rentenfaktor_table.csv

The trade-off: the model is not portable on its own — copy FRV_DE_S/ without the CSVs and it reads fine, then fails on first evaluation — but a diff of the model shows logic changes only, and an input can be swapped in place. Every file but model_point_table.csv carries a provenance column, one tag per row, asserted by the conventions suite: delib’s second ruling, the citation discipline reaching the data files rather than stopping at the prose. The model point table is the one exemption, a model point being a configuration and not an assumption.

File	Contents	Provenance
model_point_table.csv	Thirteen model points. Point 1 is the anchor cell (M37 / monthly 200,00 € / 30 years / Rentenbeginn at 67 / Beitragsrückgewähr / std_gross / base). Points 2–13 exercise the Einmalbeitrag, all four frequencies, a pct_fund and a sum_assured death benefit, an in-force cell opening at duration 96, a beitragsfrei cell on a zero-return fund, a Zuzahlung and a Teilentnahme, a Beitragsdynamik, a Nettotarif on an ETF, a two-year premium term on a stress path, a non-zero Stornoabzug, and a Rentenbeginn at 70 where the max() bites	anchor cell std, the notes’ worked example
mort_table.csv	First-order annual death rates, ages 18–100	std Gompertz proxy 0.00080 × 1.10^(age − 37), anchored at q(37) = 0.00080 — the value the worked example rests on. Not a DAV table; those are cited and never shipped R17 REG-R48. It stands for whichever first-order death basis a tariff uses, which at the one carrier that could be read is 65 % of DAV 1994 T rather than DAV 2008 T [S2]. A replacement must preserve the anchor, an insured-lives gradient and a first-order margin above best estimate
lapse_table.csv	Annual lapse: 6 % in years 1–5, 3 % in 6–10, 2 % in 11–12, 3 % from 13	std — no German unit-linked Stornoquote was established anywhere. The front-loading is inferred from the exit terms R1 R2 REG-R45; the ×2.5 tax step lives in Projection, not here, depending on age as well as duration
charge_table.csv	Four tariffs: std_gross, std_netto, std_high, std_low	std but for alpha_rate on std_gross, the 25 ‰ cap R12 R13 REG-R16, and the 60-month spread R1 REG-R28. One carrier’s levels are now known — 2,50 % acquisition, 6,90 % of premium, 0,42 % p.a. of fund, 18 €/yr [S15] — and std_gross is lighter than that on every line but the policy fee; nine other named carriers still supply none. The std_gross-to-std_netto gap is the acquisition load [S18]
fund_scenario_table.csv	Gross return and TER by (scenario_id, policy_year): base, etf, zero, stress	std deterministic paths. Not forecasts and not PRIIPs scenarios R8 R9 REG-R32 — and note that on the profession’s own standard a German Schicht-3 unit-linked annuity is a PRIIP Kategorie 4 product whose scenarios come from a stochastic capital-market model, not from an underlying’s own return history R18
rentenfaktor_table.csv	Guaranteed and current factors by (factor_id, annuity_age), ages 60–75	std and derived, not observed: 10 000 / (12 · T_eff(x)) with T_eff(x) = 100/3 − 0.75 (x − 67) at a 0 % Rechnungszins — confirmed for a fondsgebundene tariff on DAV 2004 R [S2] [S10] R16 R22 REG-R49 — giving exactly 25,00 at 67. Observed guaranteed factors at Rentenbeginn 67 are 25,22 / 24,12 / 22,91 / 21,83 € at deferments of 12 / 20 / 30 / 40 years [S15], so the shipped table is about 9 % generous at the anchor cell and, being flat in the deferment, cannot reproduce the generational gradient. std_2026 sets the current factor equal to the guaranteed one; rich_current 12 % higher

Modules that are off in the base run

Module	Switch	Off value	What it does
Dynamic lapse	lapse_dyn_beta	0.0	lapse_dyn_add(t) = β · max(0, 1 − av_pp(t)/cum_prem_pp(t)) raises the lapse rate while the contract is under water against the premiums paid — unit-linked lapse is market-sensitive precisely because the exit is at fund value on short notice R1 R2. 0.15 is the reference value; no German calibration for a coefficient of any size exists in this corpus. Switched on it bites hardest on model point 12, whose stress path leaves the fund far below premiums paid for years
Ablaufmanagement	ablauf_flag (model point)	False on twelve of thirteen	A linear ramp of the gross return from the scenario’s rate to mmkt_return_ann = 1.50 % over the last glide_months = 60 months. With one fund and a deterministic return a reallocation and a change of assumed return are the same thing, so this is the honest representation of what is known — and nothing about a real Ablaufmanagement was established: not whether it is opt-in, not the ramp length, not the destination. Model point 8 switches it on
Überschussbeteiligung	—	not implemented	A unit-linked contract’s surplus arises from the risk and cost results only R5 R14 REG-R9 REG-R18; the model computes the risk result and credits none of it back. The omission biases the projected Fondsguthaben downward, the honest direction for a charge demonstration

Hybrid and guarantee designs — statisches and dynamisches Hybrid, Zwei- and Drei-Topf-Hybride, i-CPPI, Wertsicherungsfonds — are described in the product specification and deliberately not implemented: each is a rule for reallocating between a guaranteed pot and a risky pot along a path, and a deterministic projection has one smooth path, so the rule either never triggers or triggers on a hand-chosen shock. What would have to be added is named instead — a multi-scenario asset model, a monthly reallocation rule, a guaranteed pot accreting at a Rechnungszins, a Wertsicherungsfonds return model — and that is a different model. kapitalwahl is a fourth switch and changes no cash flow by design: both routes release the same Fondsguthaben, the annuity being published rather than projected. It is carried because the two tax regimes genuinely differ R19 R20 REG-R41 REG-R45 and because take-up is the largest behavioural unknown here; no take-up rate was established, so the base run annuitises.

Sign convention

net_cf is income positive — charges in, expenses, commission and the death strain out — the notes’ own orientation and the library-wide sign. liability_cf publishes the same stream outgo-positive, liability_cf(t) = −net_cf(t) exactly, and both are result_cf() columns so the identity is verifiable in the frame rather than only in prose. A Solvency II best estimate of the non-unit liability is Σ v(t) × liability_cf(t) over the relevant risk-free term structure, with the unit liability — the Fondsguthaben itself, backed one-for-one by the Anlagestock — added at market value R15 REG-R6 REG-R7. Nothing here discounts.

expenses excludes commission: it is the issue expense at t = 0, the inflating monthly maintenance expense, and the per-event expenses of a death, a surrender and an annuitisation. The Abschluss- and Bestandsprovision are commissions, their own column, and net_cf subtracts each once — the delib convention, and the opposite of the frlib chassis, where commission sits inside the expense total; taking both at once charges the commission twice. The worked example fixes the reading: expenses(0) = 200,00 + 4,00 + 0,0075 + 0,2571 = 204,26 € and commissions(0) = 1 800,00 + 3,00 = 1 803,00 €, together the 2 007,26 € of acquisition and first-month cash. Expect on a new-business cell a large negative net_cf in the inception month t = 0 — −1 966,22 € on the anchor, commission and issue expense falling there while the charge that funds them arrives over sixty months — then a thin positive margin growing with the fund. On the Einmalbeitrag cell the sign reverses: t = 0 is +1 060,45 €, the 3 250,00 € withheld at inception more than covering the acquisition cost with no recovery to wait for.

Naming

Cells follow lifelib’s basiclife/BasicTerm_S where it has an analogue and savings/CashValue_SE for the account-value vocabulary: pols_* for policy counts, av_* for the account value, *_pp for per-policy amounts, *_rate for rates, claims(t, kind) with an uppercase kind string, and av_pp_at(t, timing) / pols_if_at(t, timing) for the within-month reads. The technical notes use compact actuarial symbols; the full mapping is in the Projection Space docstring. The chassis is shared with frlib/products/assurance_vie_uc/UC_FR_S, the French unités de compte contract, and the On UC_FR_S column says where a shared name means the same thing there and where it does not:

Notes	Cells	On UC_FR_S	Why
F(t), F_τ(t)	av_pp / av_pp_at / av_at	names shared, readings differ: there av_pp(t) is the end-of-month fund (av_pp_at(t, "BEF_DECR")), av_pp_at names five timings, and av_at weights by the after-decrement count	The fund at the start of the month, at four named points inside it, and weighted by pols_if. The timings are the processing order made addressable
u(t), Δu(t)	units_pp / units_bought_pp / units_cancelled_pp	units / fee_units / wd_units	The state variable and its two movements; delib puts every cancellation in one cells because the identity checking them has no price term
K(t), —	nar_pp / death_strain	nar / plancher_strain	The riskiertes Kapital / the French garantie plancher amount at risk, floored at zero in both, and the only part of a death benefit the insurer funds
W(t), A(t)	av_releases / withdrawals / prem_to_av_pp	same	The unit-side total the benefit-funding identity reconciles against; an owner election, never claims_wd, a Teilentnahme being no claim; and the Anlagebeitrag that buys units

Three German terms of art keep their German form in the cells names, each naming a quantity with a statutory definition and no English equivalent that would not mislead: beitragssumme(), the base of the Höchstzillmersatz and not “total premiums”; stornoabzug(t), a deduction whose validity conditions are statutory and not a “surrender charge” — the euro amount retained, with the fraction it is struck at as stornoabzug_rate(), which is the spelling RV_DE_S and Riester_DE_S use for that rate too; and the three rentenfaktor_*(), euro per 10 000 € and not an annuity factor. Five further cases needed care:

Notes	Cells	Why
qᴵ(t), q(t)	mort_rate_tariff / mort_rate, and their _mth forms	Two rates: the first-order table prices the Risikobeitrag, the second-order one produces the claims, and their difference is the Risikoergebnis. mort_rate is the projection’s own decrement, per the library’s shared vocabulary
w(t)	lapse_rate_base / lapse_tax_step / lapse_dyn_add / lapse_rate / lapse_rate_mth	The library requires an annual lapse_rate beside the monthly one; the table rate, the tax multiplier and the dynamic addition are separate cells so each is testable alone
α(t)	charge_acq_pp / cum_charge_acq_pp / charge_acq_total	The instalment, the ledger and the total the ledger must reach — check_acq_charge needs all three
D(t)	db_floor_pp / db_pp	The guaranteed floor and what a death actually pays; keeping them apart makes death_strain exactly the net amount at risk
(expense / commission)	expense_acq_pp / comm_acq_pp, expenses / commissions	Two lines, not one. expenses is the insurer’s own outgo excluding commission and commissions is the Abschluss- and Bestandsprovision, which is what those two names mean on every delib model that has a commission to publish

sex and kapitalwahl drive no formula — the tariff is unisex from 21 December 2012 REG-R34 and the capital option is a reporting split — but both are exposed as documented cells rather than dropped, because the notes’ model point attribute table lists them.

Standardizations used

Every entry is std, and the rationale is what makes it honest.

Standardization	Value	Rationale
Acquisition rate, composite, and the Zuzahlungskosten	2.50 % of the Beitragssumme; 2.50 % of a Zuzahlung	The rate is the Höchstzillmersatz — “Der Zillmersatz darf 25 Promille der Summe aller Prämien nicht überschreiten”, § 4 Abs. 1 Satz 2 DeckRV R12 REG-R16 — and taking the cap demonstrates the binding constraint instead of inventing a level. It turns out to be the level one real tariff charges: 2,50 % der kumulierten Anlage [S15]. The Zuzahlungskosten is set equal to it, and on a single premium it is the whole acquisition charge, which is what a real wording does — “Bei Verträgen gegen Einmalbeitrag und bei Zuzahlungen entnehmen wir alle Abschluss- und Vertriebskosten sofort” [S2]
Premium-based admin beta_rate	4.00 %	Middle of an argued 2 %–10 % range. 6,90 % of each premium observed at one carrier [S15]; no other carrier level was established
Fund-based admin gamma_rate_ann	0.30 % p.a., taken as /12	Middle of an argued 0.10 %–1.20 % range. 0,42 % p.a. of the fund observed at one carrier [S15]. Divided, not compounded, because a German tariff quotes a nominal monthly charge
Stückkosten policy_fee_mth	3.00 € per month	Middle of an argued 0–5 € range; 18 € per year — 1,50 €/month — observed at one carrier [S15], so the composite is the dearer of the two. A euro amount, which is why it consumes a small paid-up fund — and why BaFin warns that a large absolute Stückkosten charge makes the Effektivkosten vary sharply with premium size R10
Stornoabzug, composite	0.00 %	§ 169 Abs. 5 VVG permits one only where it is vereinbart, beziffert and angemessen and voids any deduction for unamortised acquisition cost R1 REG-R36; zero avoids an unsourced number on a contested clause. A real one exists and is a flat 150 €, not a percentage [S2], so both the level and the shape are standardizations. 2.00 % on std_high exercises the machinery
The three charge variants	std_netto, std_high, std_low	The ends of the argued range plus the commission-free tariff [S18]; the gap to std_gross is the acquisition load
Mortality proxy	0.00080 × 1.10^(x − 37), ages 18–100	DAV tables are not redistributed R17 REG-R48. Anchored at q(37) = 0.00080 so the worked example reproduces; the 10 % gradient is insured-lives. It stands for whichever first-order death table the tariff uses — DAV 2008 T is the modern one, and the one fondsgebundene tariff whose bases could be read prices its Risikobeiträge on a unisex order at 65 % of DAV 1994 T instead [S2]
mort_be_factor	0.75, flat	Crude, and said to be. It buys a Risikoergebnis that is exactly 25 % of the Risikobeitrag REG-R47
Monthly conversions	mortality /12; lapse and return geometric	Charge and decrement must share a split; the lapse and return annual rates are the observables to reproduce
Lapse table, the 40 % cap and the zero final month	6 / 3 / 2 / 3 % by duration band; lapse_cap = 40 %; lapse_rate_mth(n − 1) = 0	No German unit-linked Stornoquote exists in this corpus; the front-loading is inferred from the exit terms R1 R2. The cap stops a dynamic module producing an absurd rate; the zero final month is because a surrender and an annuitisation are then one event
Tax-threshold lapse step	×2.5 for twelve months from max(13, 62 − entry_age + 1)	The 12/62 rule is statutory — EStG § 20 Abs. 1 Nr. 6 Satz 2, which names the 60th year of life, raised to the 62nd by § 52 Abs. 28 “für Vertragsabschlüsse nach dem 31. Dezember 2011”, and applying on surrender as well as at maturity R20 REG-R45. It is the strongest single driver of German surrender behaviour; keying it on duration alone fires fourteen years early on the anchor. The ×2.5 magnitude is std
Fund paths and TER	5.00 % gross, 0.45 % TER on base; etf, zero, stress	Round, clearly-labelled assumptions. Not forecasts, not PRIIPs scenarios R8 R9 REG-R32
Kickback credited back, and the Ausgabeaufschlag	0.00 % p.a.; fully waived	A passive fund pays no trail. The two questions this once sidestepped are answered: insurers do receive rebates out of the fund’s Verwaltungsvergütung and must test them for Fehlanreize and consider passing them back R10 R15, and the market pays a weighted mean just over 0,30 % p.a. of the fund, up to over 1,20 %, on about a third of new business R11 — so 0 % models the cheap end of a real flow. Only the PRIIPs treatment of a credited rebate is still open R7 R8. The Ausgabeaufschlag waiver is confirmed: “Ausgabeaufschläge und Depotkosten fallen nicht an”, and units are bought at the Rücknahmepreis [S2]
Guaranteed Rentenfaktor	25,00 € per 10 000 € at 67, from 10 000/(12 T_eff)	Derived arithmetic, not a market observation — and now measurable against one. The 0 % Rechnungszins on DAV 2004 R is confirmed for a fondsgebundene tariff [S2] [S10] R16. The level is not: at this model’s own anchor cell the observed guaranteed factor is 22,91 € against the shipped 25,00 €, and the observed factor falls with the deferment — 25,22 / 24,12 / 22,91 / 21,83 € at 12 / 20 / 30 / 40 years to age 67 — where the shipped table is flat in it [S15]. Not changed in this pass: the table, the worked example and the golden tests move together
Current Rentenfaktor	equal to guaranteed (std_2026); +12 % (rich_current)	Exercises the max() without an unsourced uplift, and makes it visibly bite on one cell. Consumer sources describe real guaranteed factors at 50–70 % of the current one [unverified] R22, so both settings are conservative
Expenses and commission (expenses excludes commission; commissions is its own column)	acquisition commission 2.50 % of S + 200,00 € issue; 4,00 €/month at 2 % inflation; renewal 1.5 %; 150 / 50 / 100 € per event	No German commission scale was established. The acquisition commission equals the acquisition charge, so the model shows the financing problem the Höchstzillmersatz and the five-year spread exist to regulate. comm_acq_rate is a flat scalar, so on std_netto and std_low the assumed commission exceeds the tariff’s own charge and those cells carry a projected loss — the flat assumption showing, not a product fact
Timing, processing order and the negative-fund safeguards	premium in advance, return, fund charges, Teilentnahme, Risikobeitrag, deaths before lapses; min(.., remaining) on the Stückkosten and the Risikobeitrag	The Bewertungsstichtag lag disappears on a monthly grid; observing the amount at risk before the charge that prices it makes death_strain exactly the riskiertes Kapital. The floors are safeguards, not tariff terms, and no shipped model point triggers one
Beitragsfreistellung as an election	pup_month, no cohort paid-up rate	A cohort rate needs one sub-cohort per paid-up month for a second-order effect; the std 1 % p.a. is recorded, not implemented, and the omission biases charge income upward
Modules off in the base run	lapse_dyn_beta = 0, ablauf_flag = False, no Überschuss credit	Base-run values, so the worked example reproduces with the machinery still there
The thirteen model points	see the input table above	Configurations, not assumptions — which is why they are the one provenance-exempt file

The quantities that are not standardizations are the acquisition-charge cap, § 4 Abs. 1 Satz 2 DeckRV R12 REG-R16, and the five-year spreading, § 169 Abs. 3 VVG — which reaches this contract through Abs. 4’s “im Übrigen gilt Absatz 3” and through tariff practice rather than directly R1 R13 REG-R28; the Beitragsverrechnung order, read verbatim at [S2] § 14 Abs. 1 as well as [S1]; the Beitragsrückgewähr shape [S2] § 2 Abs. 7; the max(guaranteed, current) factor rule [S2] § 2 Abs. 2, [S4] R22; the Zeitwert Rückkaufswert, § 169 Abs. 4, and the conditions on a Stornoabzug, Abs. 5 R1 REG-R36; the survival of the fund-based charges into a beitragsfrei contract R3 [S2] § 14 Abs. 2; the unisex tariff REG-R34, now visible in a real Rechnungsgrundlage [S2]; and the rule that the insurer guarantees units and not their value, which VAG § 124 Abs. 2 Satz 2 Nr. 1 and § 125 Abs. 5 make structural [S1] R15 REG-R7.

Three shapes this model implements differently from the one tariff that could be read, none of them changed in this pass and all of them recorded so a calibration pass knows where to look: the acquisition charge falls to zero at t = 60, where a real tariff continues a slice of it for the whole premium term as a percentage [S2] § 18 Abs. 2; the Stornoabzug is a percentage of the fund, where a real one is a flat euro amount [S2] § 17 Abs. 4; and a Teilentnahme reduces the fund but not the Beitragsrückgewähr floor, where a real clause reduces both [S2] § 2 Abs. 7.

Tests

tests/test_fondsgebundene_rentenversicherung_de.py asserts the notes’ worked example — all seventeen printed rows of Panel A to the cent and pols_if to six decimals, its expenses and commissions columns among them, Panel B’s benefit columns, Panel C’s per-policy unit side, and every column total at full precision — the notes’ three independent rebuilds (t = 0 from the tariff alone, t = 60 at the cliff, the reduction in yield as a savings account), the four closure identities, the Einmalbeitrag variant’s printed table, the four-tariff reduction-in-yield comparison, the seven check_*() identities with their residuals, and one test per listed modeling pitfall — eighteen of them, named for the pitfall they guard. The library-wide house style — layout, docstrings, naming, the retired-name register, the check_net_cf ruling, the provenance ruling, the model point sweep and the round trip — is asserted separately, once, in tests/test_model_conventions_de.py, which owns the only whole-table sweep in the library.

python -m pytest lifelib/libraries/delib/tests -q