Technical Notes

Status: Draft, 2026-08-29 (sources accessed 2026-08-29); citations re-verified against the primary documents 2026-08-30.

Scope note. These notes specify a reference liability cash-flow projection model — model name RV_DE_S, monthly grid — for the standardized composite German klassische aufgeschobene private Rentenversicherung defined in product-spec.md (same directory). This is not any single insurer’s contract. [S#]/[R#] tags refer to the source list in sources.md (numbering carried from _research/klassische_rentenversicherung.md; frozen); [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md (its own frozen numbering). std marks standardizations introduced for the reference implementation; unverified marks claims no retrieved document confirms. Parameter values are identical to those in product-spec.md. Cells names, model-point columns and CSV headers are English lower_snake_case; German terms of art keep their German form in prose.

Retrieval conditions. These notes were drafted with direct HTTP egress from the build environment blocked, on WebSearch result summaries alone and a budget exhausted after eighteen queries on this product; that policy has since been lifted and the citations were re-verified against the primary documents on 2026-08-30. Thirty-six of the forty-three entries in sources.md now read Retrieved: yes — including every VVG, EStG and DeckRV section these notes rest on, read as canonical XML with its amendment Stand recorded, and the carrier wordings whose § numbers appear below. Four entries were reached only in part and three not at all, and each says which it is: treat a citation here as sound where its entry says Retrieved: yes, and as a pointer to the instrument rather than a certificate where it does not. The consequence for these notes is unchanged, and still specific and large: the corpus establishes this product’s mechanics thoroughly and its levels barely at all. No charge parameter and no behavioural rate was established at any German carrier for any year, and the Rentenfaktor range R19 R24 and the declared rate [S15] the pass did establish are not the levels this model ships, so every number in assumption classes (b) and (c) below, and most of class (a)’s levels, is std. What the citations do establish is the shape of each recursion, and that is what this file is for.

---

Model scope and conventions

• Purpose. Project gross best-estimate liability cash flows, undiscounted — premiums in; death benefits, surrender values, the Kapitalabfindung, annuity instalments and insurer expenses out — for a single-policy model point on an expected (probability-weighted) basis, together with the two account balances that make the product what it is: the Deckungskapital and the Ansammlungsguthaben.

• Out of scope, and said so rather than left to be discovered. Discounting; the statutory Deckungsrückstellung REG-R54; the Zinszusatzreserve REG-R17; the RfB and the MindZV arithmetic REG-R18 REG-R19; the Sicherungsbedarf test REG-R20; the Solvency II best estimate, risk margin, SCR and MCR REG-R6; IFRS 17 REG-R55; and all taxation — every cash flow is gross of Kapitalertragsteuer, Solidaritätszuschlag and Kirchensteuer REG-R38. Also not modeled, each for a stated reason: the Bonusrente surplus system R24; Zuzahlung, which no source names (gap 15); the survivor’s-annuity and BU riders [S10] [S4]; § 163 VVG adjustment of the guaranteed Rentenfaktor R3 R17 REG-R27; and § 169 Abs. 6 R1.

• Projection frequency. Monthly grid over an annual product, so the model runs on two clocks. The contract’s own natural period is the Versicherungsjahr: the Rechnungszins is credited annually, the Überschussbeteiligung is declared annually and the Ansammlungsguthaben’s interest is credited at each policy year end and on termination [S4] § 3 Abs. 6, [S15], the § 165 paid-up value is “stated in the contract for each insurance year” R2, and Kündigung and Beitragsfreistellung both take effect “for the end of the current insurance period” R1 R2. Every one of those constructions therefore keeps the policy year as its argument. What the monthly step is for is the other half of the product: the monthly annuity [S13] R24, which the annual grid could only compress and which these notes listed as a pitfall for that reason; the Rentengarantiezeit, which guarantees 12m instalments rather than m lumps; and the decrements, which now fall in the month they happen.

• What t counts. t indexes the policy month, counted from inception, and is 0-based: t = 0 is the first policy month. duration(t) = t // 12 is the 0-based policy year k, and every annual construction in this file takes that k; policy_year(t) = duration(t) + 1 is the contractual 1-based label the input files are keyed on. The attained age and the calendar year both step on the anniversary — age(t) = issue_age + duration(t), calendar_year(t) = issue_year + duration(t) — so one generational mortality surface and one declared-rate path serve a book of mixed vintages, every policy year carrying its own attained age and calendar year REG-R14 REG-R49. A new-business model point opens at t = 0. An in-force model point that has already run duration_init complete policy years opens at t_start() = 12 × duration_init, carrying its opening balances on the model point; k_start() = duration_init is the same instant on the annual clock. is_anniv(t) is t % 12 == 11, the month the annual machinery acts in.

• proj_len() is the exclusive end of the frame in months, per the library-wide ruling (tests/test_model_conventions_de.py), so the frame is t = t_start() … proj_len() − 1 and result_cf().index[-1] == proj_len() − 1:

  proj_len_y() = omega_age − issue_age          the number of policy years
  proj_len()   = 12 × proj_len_y()              the frame, in months
  

  with omega_age = 121 std the terminal age of the shipped mortality proxy, at which mort_rate = 1. The projection therefore ends when the annuitant cannot survive further, not at a fixed horizon: a life annuity has no term, and truncating one at, say, 40 years silently drops the tail the Rentenfaktor was priced for. For the anchor cell, proj_len_y() = 121 − 50 = 71, so the frame is t = 0 … 851 and the last policy year ends at attained age 121.

• The Rentenbeginn is the end of the deferment period of n = aufschub_y years, which is the end of month 12n − 1 and the same instant as the start of month 12n. Accumulation-phase months are t < 12n; payout-phase months are t ≥ 12n. The Kapitalabfindung is paid in month 12n − 1; the first annuity instalment falls in month 12n.

• Two speeds on each decrement. mort_rate(t) and lapse_rate(t) return the annual rate of the policy year t falls in — the vectors tabulated below, flat across the year’s twelve months — and mort_rate_mth(t) and lapse_rate_mth(t), each 1 − (1 − r)^(1/12), are what the recursion applies. The twelfth is geometric and never r / 12, so twelve compound back to the year’s rate exactly and pols_if at every anniversary is what the annual grid gave. Two rates of 1 are certainties rather than rates — the terminal q = 1 of the proxy, and the § 165 cash-out — and both are placed in the anniversary month instead of being twelfth-rooted, since spreading a certainty geometrically empties the cohort eleven months early.

• Timing conventions std. The premium at the start of the policy year, which is what § 12 Abs. 1 VVG makes the Versicherungsperiode for this tariff, so prem_due(t) is true in the first month of each policy year and nowhere else; charges deducted immediately after; the Rechnungszins credited on the post-premium, post-charge balance at the end of the year; the declared surplus credited to the Ansammlungsguthaben at the end of the year; death and surrender at the end of each month, deaths before surrenders; the Rentenbeginn events after the decrements of month 12n − 1; and annuity instalments at the start of each payout month, monthly in advance, which is what a Rente is. The ordering of premium credit, charge deduction and interest accrual is not established by any source in this corpus [S11] and is the single most consequential std in this file (processing order, and pitfall 2).

• What a mid-year exit is paid. The value struck at the end of the current Versicherungsperiode — § 169 Abs. 3 VVG’s zum Schluss der laufenden Versicherungsperiode, and not the cancellation date R1 — so a policy leaving in any month of policy year k is paid cv_pp(k) or db_pp(k), the same amount the annual grid paid. It is the consistent reading: the Beitrag was payable in advance for the whole period, so the period’s value is what the contract earned. What the monthly grid changes is when the claim falls and how many policies are exposed to it, not what it is worth.

• Currency, sign and rounding. EUR throughout. net_cf(t) is income-positive (premiums +, benefits and expenses −), with the outgo-positive orientation published as liability_cf(t) = −net_cf(t). Intermediate values at full precision; displayed cash flows to euro cents, pols_if to six decimals std.

• Unisex pricing is a hard constraint, not a convention. A model point carries sex for the decrement — the underlying DAV tables are sex-specific raw material REG-R47 REG-R49 — and sex must not enter the premium, the charge scale or the Rentenfaktor, sex-based differences being prohibited for contracts concluded from 21 December 2012 REG-R34.

---

Model point attributes

Thirty columns. model_point_table.csv is indexed by point_id and is the one input file exempt from the provenance rule, because a model point is a configuration rather than an assumption. The right-hand column names the points that exercise each attribute away from its base value.

Attribute	Type	Meaning	Exercised by
point_id	int	Key; Projection is parameterized by it	all
policy_id	str	Label, DE-RV-nnnn	all
sex	enum {M, F}	Decrement only; never priced REG-R34	2, 4, 7, 9, 11, 13 are F
issue_age	int	Age last birthday at inception	all
issue_year	int	Calendar year of inception; drives the generational mortality index and fixes the guarantee vintage REG-R14 REG-R49	6 (2005), 14 (2017)
duration_init	int	Complete policy years already elapsed; 0 = new business. Already 0-based, so the frame opens at t_start() = 12 × duration_init	6 (20), 14 (8)
pols_if_init	float	Policies represented; pols_if(t_start())	all (1.0)
premium_form	enum {laufend, einmal}	Recurring or single premium [S11] REG-R53	2 (einmal)
prem_gross_pp	EUR p.a.	Annual Bruttobeitrag before the frequency loading	all but 2
premium_single_pp	EUR	Einmalbeitrag	2
prem_freq	enum {annual, half_yearly, quarterly, monthly}	Zahlweise; keys freq_load_table.csv for the Ratenzahlungszuschlag. The collection is annual in advance whatever the elected frequency, § 12 Abs. 1 VVG making the Versicherungsperiode the year for this tariff	3 (monthly), 4 (quarterly), 5 (half-yearly)
prem_term_y	int	Premium-paying years from inception, a count: premiums fall in policy years k < prem_term_y	4 (20 of 22), 5 (25 of 27)
aufschub_y	int	Deferment in years, a count: Rentenbeginn at the end of month t = 12 × aufschub_y − 1	all
int_rate_guar	rate	The contract’s Rechnungszins — a model-point attribute, not a global assumption R7 REG-R14 REG-R15	6 (2,75 %), 14 (0,90 %)
charge_id	str	Key into charge_table.csv; zillmer_25 or the pre-2015 zillmer_40 REG-R16	6 (zillmer_40)
annuity_rate_guar	EUR/month per 10 000 €	garantierter Rentenfaktor, fixed at inception [S8] R24	all (levels std)
rf_scenario_id	str	Key into rentenfaktor_table.csv for the aktueller Rentenfaktor	5 (high), 13 (low)
decl_scenario_id	str	Key into decl_rate_table.csv for the declared laufende Verzinsung	14 (low)
guar_capital_pp	EUR	Minimum guaranteed contract value at Rentenbeginn [S9]; 0 = not stated	13 (60 000)
death_benefit_form	enum {prem_refund, deckungskapital, max}	The three documented designs [S1] R24; max std [S19]	2, 12 (deckungskapital); 5, 13 (max)
db_incl_surplus	int 0/1	Whether the Ansammlungsguthaben is added to the death benefit R24	4, 12
rgz_years	int	Rentengarantiezeit in years R24 [S9] [S13]; the model pays 12 × rgz_years guaranteed instalments	9 (0), 11 (5), 3/14 (15), 5/10 (20)
kapitalwahl_rate	float	Fraction electing the Kapitalwahlrecht at Rentenbeginn [S12] R6 R21	9 (1.00), 2/10 (0.00), 5 (0.20)
pup_year	int	Contractual policy year of Beitragsfreistellung, 1-based; 0 = never R2. The contract is paid up from policy year k = pup_year − 1	7 (10), 8 (3, trips the minimum)
dynamik_rate	rate	Dynamik annual premium increase [S4]; 0 = option off	12 (5 %)
payout_system	enum {konstant, teildynamisch, volldynamisch}	Überschussverwendung in payment R19 R20 R24	4, 11 (teildynamisch); 5, 10 (volldynamisch)
av_pp_init	EUR	Opening Deckungskapital	6, 14
av_sur_pp_init	EUR	Opening Ansammlungsguthaben	6, 14
prem_cum_pp_init	EUR	Premiums paid before the valuation date — the Beitragsrückgewähr base	6, 14
alpha_amort_pp_init	EUR	Acquisition charge already amortised before the valuation date	6 (2 592), 14 (900)

The fourteen model points. Point 1 is the worked example’s anchor cell. Between them the points carry both premium forms, all four payment frequencies, two in-force cells on two legacy guarantee vintages, both charge sets, all three death-benefit forms with and without surplus, all three payout systems, five Rentengarantiezeit durations including zero, Kapitalwahlrecht take-ups of 0 %, 20 %, 30 % and 100 %, the Dynamik, both Beitragsfreistellung branches, and four boundary cases: the paid-up conversion that fails the Mindestversicherungsleistung and is cashed out (8), full commutation at Rentenbeginn (9), the guaranteed Rentenfaktor binding over a lower current one together with a binding guar_capital_pp (13), and a Rentengarantiezeit of zero (9).

One constraint the table must satisfy and the model asserts. av_spread_pp (below) is seeded equal to av_pp_init on an in-force point, which is exact only once the acquisition charge is amortised under both treatments, so every in-force point has duration_init ≥ alpha_spread_years = 5 std. Points 6 and 14 open at durations 20 and 8.

---

State variables

Variable	Description	Updated
proj_len_y, proj_len	omega_age − issue_age policy years, and 12 × that as the exclusive end of the frame in months	once per model point
duration(t), policy_year(t), is_anniv(t)	The bridge between the two clocks: the 0-based policy year of month t, its 1-based label, and whether t is the year’s last month	derived
age(t)	Attained age in month t = issue_age + duration(t); steps on the anniversary	annual step
calendar_year(t)	issue_year + duration(t); the second index of the generational mortality surface, also stepping on the anniversary	annual step
pols_if(t)	Policies in force at the start of month t; pols_if(t_start()) = pols_if_init()	monthly recursion
pols_annuity(t)	The count the annuity instalment is paid on — the annuitised count inside the Rentengarantiezeit, survivors after it, zero before Rentenbeginn	monthly
av_pp(k)	Deckungskapital per policy at the start of policy year k	annual recursion
av_pp_at(k, timing)	"BEF_PREM", "AFT_PREM", "AFT_INT"	within policy year k
av_sur_pp(k), av_sur_pp_at(k, timing)	Ansammlungsguthaben per policy, and its within-year points	annual recursion
av_spread_pp(k), av_spread_pp_at(k, timing)	The parallel Deckungskapital in which the acquisition charge is spread evenly over the first five contract years — the § 169 Abs. 3 VVG floor REG-R28	annual recursion
spread_diff_pp(k)	av_spread_pp(k) − av_pp(k); the only quantity by which the two accounts differ	annual recursion
capital_conv_pp	Conversion capital struck at Rentenbeginn	once
annuity_rate_appl	max(annuity_rate_guar, annuity_rate_curr) [S4]	once
annuity_guar_mth_pp	garantierte Rente, monthly, struck at Rentenbeginn	once
annuity_sur_mth_pp(k)	Überschussrente, monthly, by payout system; steps on the anniversary	annual
annuity_pp(t)	The instalment actually paid in month t: G + U(duration(t))	monthly
mort_rate_guar(t), mort_rate(t)	First-order (tariff) and second-order (best-estimate) annual mortality, flat across a policy year	lookup
mort_rate_mth(t), lapse_rate_mth(t)	The monthly rates the recursion applies, each 1 − (1 − r)^(1/12)	derived
lapse_rate(t)	Annual surrender rate; 0 from Rentenbeginn	lookup

There is no paid-up sub-population state: Beitragsfreistellung is a deterministic election at a stated policy year rather than a continuous decrement, for the reason given under Policyholder behaviour modelling. There is no Bonusrente ledger in the accumulation phase, and no survivor’s-annuity state.

---

Assumption inputs

The eight input files. All inputs are external CSVs beside run.py, read once per model by the unparameterized Data Space (the annuallife/TradLife_A layout). Every file but the model point table carries a per-row provenance column, this library’s second ruling.

File	Index columns	Value columns
model_point_table.csv	point_id	the thirty model-point attributes above (provenance-exempt)
mort_table.csv	sex, age	q_base, improve, provenance
decl_rate_table.csv	scenario_id, calendar_year	decl_rate, provenance
rentenfaktor_table.csv	rf_scenario_id, age	annuity_rate_curr, provenance
charge_table.csv	charge_id, item	value, provenance
lapse_table.csv	duration	lapse_rate, provenance
freq_load_table.csv	prem_freq	freq_load, n_instalments (recorded, not applied), provenance
param_table.csv	item	value, provenance

param_table.csv holds every scalar std assumption that is neither a charge nor a rate table: expense_acq_pp, expense_maint_pp, expense_annuity_pp, expense_claim_pp, expense_infl, mort_be_factor, mort_base_year, omega_age, val_reserve_rate, sur_ann_rate, sur_ann_growth, sur_ann_theta and roll_fwd_tol. Keeping them in a file rather than in Projection References is what gives each of them its own provenance tag.

(a) Contractual and guaranteed elements (cited)

Input	Value / rule	Basis
Deckungskapital recursion	The Sparbeitrag — the premium net of the tariff costs and the Risikobeitrag — accumulated at the Rechnungszins: “indem wir die eingezahlten Beiträge abzüglich der tariflichen Kosten und Risikobeiträge mit dem tariflichen Garantiesatz von 0,90 Prozent p. a. verzinsen”	[S8] § 1 Abs. 2; [S11] § 27 Abs. 1; ordering std
Rechnungszins	A model-point attribute; 1,00 % for 2026 issues, 2,75 % and 0,90 % on the two legacy points	R7 R11 REG-R14 REG-R15
Höchstzillmersatz	25 ‰ of the Beitragssumme from 1 January 2015 — § 4 Abs. 1 Satz 2 DeckRV, “Der Zillmersatz darf 25 Promille der Summe aller Prämien nicht überschreiten” — 40 ‰ before, the rate at conclusion applying for the whole term (§ 4 Abs. 4)	R7; REG-R16 REG-R20; restated by four carriers as “2,5 % der Beiträge” [S1] § 14, [S4] § 11, [S8] § 8, [S9] § 16
§ 169 Abs. 3 surrender floor	At least the Deckungskapital that results from spreading the charged acquisition and distribution costs evenly over the first five contract years, “die aufsichtsrechtlichen Regelungen über Höchstzillmersätze bleiben unberührt”	R1 at article level; REG-R28; [S1] § 12 Abs. 3, [S4] § 10 Abs. 3, [S8] § 7 Abs. 3, [S9] § 16 Abs. 4, [S11] § 34 Abs. 2
§ 169 Abs. 5 Stornoabzug	Permitted only if agreed, quantified and appropriate; a deduction for unamortised acquisition costs is void	R1 REG-R28
§ 165 paid-up value	Computed on the premium calculation basis, on the basis of the § 169 Abs. 3–5 surrender value, and tabulated per insurance year	R2 REG-R28
§ 165 minimum benefit	Below the Mindestversicherungsleistung the contract is cashed out at the surrender value including profit shares, not made paid-up	R2
Death benefit before Rentenbeginn	Beitragsrückgewähr (premiums paid, ohne Zinsen), the accumulated Deckungskapital, or the larger of the two; optionally plus the attributable surplus; or no benefit at all where no extension is bought	[S8] § 1 Abs. 1; [S9] § 1 Abs. 3 for the max form; [S4] § 1 Abs. 2–3; R24
Conversion capital	Includes Überschussbeteiligung and Bewertungsreserven, subject to a minimum guaranteed contract value	[S9]
Conversion rule	monthly annuity = capital / 10 000 × Rentenfaktor — “wie viel Rente wir Ihnen monatlich je 10.000 Euro … zahlen”	[S11] § 52 Abs. 1; [S14] § 2 Abs. 5; [S18]; R24
Rentenfaktor applied	max(garantierter, aktueller) — the annuity on the bases current at Rentenbeginn against the garantierte Mindestrente, the higher paid, tested at NÜRNBERGER at each monthly instalment	[S9] § 1 Abs. 1; [S14] § 2 Abs. 3 and 6; [S18]; R24
Bewertungsreserven	hälftige participation under § 153 Abs. 3 Satz 2 VVG, crystallised at the transition to annuity payment — § 153 Abs. 4, “Bei Rentenversicherungen ist die Beendigung der Ansparphase der nach Absatz 3 Satz 2 maßgebliche Zeitpunkt” — and continuing in payment by contract	R4 at article level; [S4] § 3 Abs. 2; [S15]; REG-R24
Annuity	Monthly in advance, for life, from Rentenbeginn — “monatlich, jeweils zum Monatsersten”; garantierte Rente plus Überschussrente, only the first guaranteed, the RfB-financed part promised “jeweils nur für ein Versicherungsjahr”	[S9] § 1 Abs. 1; [S4] § 3 Abs. 7; R20 R24
Rentengarantiezeit	Payment continues until the agreed years expire, “unabhängig davon, ob die versicherte Person diesen Termin erlebt”; commutable to a present value at the payout-phase Rechnungszins	[S1] § 1 Abs. 4; [S4] § 1 Abs. 4 and § 10 Abs. 14; [S9] § 1 Abs. 5; R17 R24
Surrender in payment	None	REG-R28; reading std
Mortality basis	DAV 2004 R (Aggregattafel), a Generationentafel, first order carrying safety margins over second order; carriers derive their own tables from it (NÜRNBERGER Tafel 2013R, Debeka 01/21 R)	[S4] § 1 Abs. 6, [S5] [S6] [S7] [S16]; [S9] [S11] for the derived tables; R12 R13 REG-R47 REG-R49
Unisex tariff	Sex may not enter premium or benefit for contracts from 21 December 2012	REG-R34

(b) Insurer-discretionary current elements (snapshot; revisable annually)

Everything in this class is a declaration, not a promise: the GDV model wording says so in terms — “Die Höhe der Überschussbeteiligung hängt von vielen Einflüssen ab, die nicht vorhersehbar und von uns nur begrenzt beeinflussbar sind. … Die Höhe der künftigen Überschussbeteiligung kann also nicht garantiert werden. Sie kann auch Null Euro betragen.” [S1] § 2 Abs. 7. One carrier’s rates are now established (gap 4 closed): Bayern-Versicherung’s Überschussverteilung 2026 declares the annuity Zinsüberschussanteil for tariff generations 2015–2025 as 3 % less the Rechnungszins before the Rentenbeginn and 3,35 % less it during the Rentenbezug, against 2,25 % and 2,5 % for 2025 [S15] — so the total interest credited is 3,00 % for 2026. The values below are unchanged and remain std; they are a scenario path, not that declaration, and model.md records the divergence.

Input	Value	Basis
Declared laufende Verzinsung decl_rate	2,55 % p.a. level on the base path; 1,50 % on the low path	std (i)
Surplus rate bonus_rate(t)	max(0, decl_rate(t) − int_rate_guar) — 1,55 % on the 1,00 % vintage, 0 % on the 2,75 % vintage	mechanic R24 REG-R53; std level
Interest credited on the Ansammlungsguthaben	at decl_rate(t), the full declared rate	mechanic R24; std (ii)
Bewertungsreserven rate at Rentenbeginn val_reserve_rate	1,5 % of the accumulated value	mechanic [S4] R4; level std (iii)
aktueller Rentenfaktor at Rentenbeginn	32,00 € per month per 10 000 € at age 67 on the base path; 25,50 € low; 35,00 € high	mechanic [S13] R24; level std (iv)
Überschussrente rate sur_ann_rate (konstant)	12 % of the garantierte Rente, level	mechanic R20; level std (v)
Überschussrente growth sur_ann_growth (volldynamisch)	1,5 % p.a. compound on the garantierte Rente	mechanic R20 R24; level std (v)
teildynamisch split sur_ann_theta	0,5 — half the constant increment plus half the growth rate	mechanic R20 R24; std (v)

(i) Anchored on the only public market averages the library has: the average laufende Verzinsung for 2025 was 2,53 % Klassik / 2,58 % Neue Klassik, and for 2026 the sources give 2,6–2,7 %, 2,87 % and 2,54 % — three incompatible averages REG-R53. 2,55 % sits inside the 2025 pair and is a market average, not a carrier’s declaration. A carrier’s declaration is now available and is not this number: Bayern-Versicherung’s Überschussverteilung 2026 credits a total of 3,00 % before the Rentenbeginn (2025: 2,25 %) and 3,35 % during the payout phase (2025: 2,5 %) on tariff generations 2015–2025 [S15]. The shipped path stays at 2,55 % because it is a market-average scenario rather than one carrier’s book, and because changing it moves the worked example and the golden tests; the divergence is recorded rather than absorbed. The declared rate is the Garantieverzinsung plus the laufende Zinsüberschussbeteiligung, never a surplus on top of the guarantee REG-R53; that is what the max(0, ·) in bonus_rate implements, and it is why the 2,75 % legacy point receives no interest surplus at all — a real and important German result, not a modelling artefact. (ii) The Ansammlungsguthaben mechanic is established in a carrier’s own definition — the surpluses are credited to an Überschussguthaben and accumulated with interest, the interest credited at each policy year end, on termination, and for an annuity also at the start of annuity payment or on commutation [S15]; [S4] § 3 Abs. 6 says the same and adds that the balance is paid out on death, surrender or commutation. Neither states the rate. Crediting the full declared rate rather than the Rechnungszins is std; the alternative is a documented variant and moves the anchor cell’s final Ansammlungsguthaben materially. (iii) The mechanic is now statutory rather than restated: § 153 Abs. 3 Satz 2 VVG allots the amount determined at termination “zur Hälfte”, and § 153 Abs. 4 makes the Beendigung der Ansparphase the relevant moment for an annuity R4; [S4] § 3 Abs. 2 and [S15] apply it. No amount, ratio or reserve level is established anywhere. 1,5 % of the accumulated value is a placeholder sized to be visible without dominating. (iv) See product spec footnote 9. Chosen so that the anchor cell’s max() resolves upward (the current factor wins) and point 13’s resolves downward (the guarantee binds), because a rule with one branch never exercised is a rule no test covers. (v) The three systems are established and their directions are established — the constant form is set from a whole-period projection and falls if the insurer earns less, the fully dynamic form adjusts annually to actual surplus development, and the partial form is a stated combination of the two R20 R24 — and two carriers now name them in their own AVB: Garantie-PLUS-Rente / Bonus-PLUS-Rente / Bonusrente [S4] § 3 Abs. 7, and dynamische Überschussrente / teildynamische Bonusrente [S9] § 2 Abs. 5 c). The fully dynamic form is confirmed to ratchet — “Die jeweils erreichte Rentenhöhe kann nicht mehr sinken” [S9] — while the level form is confirmed not to be guaranteed, the RfB-financed part being promised “jeweils nur für ein Versicherungsjahr” [S4]. No level, rate or split is established for any of them.

(c) Behavioural and experience assumptions (the modeller’s view)

Every input in this class is std and none has any evidence behind it (gap 20): no German lapse rate, no Beitragsfreistellung rate, no Kapitalwahlrecht take-up rate and no market Stornoquote was returned by any search for this product.

Mortality. The first-order basis is DAV 2004 R, a Generationentafel which is the property of the Deutsche Aktuarvereinigung, is not public and is not shipped [S8] R12 R13 REG-R49. mort_table.csv is a std proxy with the structure the real table has and none of its values: a sex-distinct base table q_base(sex, x) for base year 2005 — the year DAV 2004 R was intended for new business R13 — and an age-dependent annual improvement rate improve(x), combined as

mort_rate_guar(t) = q_base(sex, x(t)) × (1 − improve(x(t)))^( calendar_year(t) − 2005 )

which is the generational form the reference library requires of any annuity proxy, built the way it recommends: a base table times a cumulative improvement factor, anchored to Destatis’s own Generationensterbetafeln as the free redistributable analogue REG-R49 REG-R52. The proxy is anchored so that q_base(M, 50) = 0.002000 exactly, and that anchor is stated in the model’s Data docstring; a substitute table must preserve it if the worked example is still to close. improve(x) is 1,5 % p.a. below age 60, grading linearly to 0,5 % at age 100 and to zero at 110 std — a deliberate simplification of the Starttrend / Zieltrend structure the German construction actually uses, and documented as one rather than presented as a replication REG-R49.

The second-order (best-estimate) basis is the first-order one loaded:

mort_rate(t) = mort_rate_guar(t) × mort_be_factor,    mort_be_factor = 1.15  [std]

The factor is above one and that is the whole point. For an annuity, prudence means assuming mortality lower than expected, so the first-order table sits below best estimate, and the safety margin runs in two dimensions — level and trend REG-R47 REG-R49. The same table is used for the accumulation-phase death benefit, which is the German peculiarity worth naming: an annuity table prices a death benefit, so that benefit is systematically under-charged relative to a death-business basis such as DAV 2008 T REG-R48, and the Beitragsrückgewähr design exists partly because it makes the mismatch immaterial — the benefit is the premiums, not a sum insured.

Lapse (Storno). lapse_table.csv, keyed by the contractual policy year (duration, 1-based, so the lookup goes through policy_year(t) = duration(t) + 1 and the twelve months of a policy year all read the same row), and zero from Rentenbeginn:

Duration	1	2	3	4–7	8–11	12	13+	payout
lapse_rate std	6,0 %	5,0 %	4,5 %	4,0 %	3,5 %	6,0 %	3,0 %	0 %

The duration-12 step is the only shaped feature and it is the one with a reason: § 20 Abs. 1 Nr. 6 EStG makes half the gain taxable only where the contract has run at least twelve years and payment falls after completion of the 62nd year of life, so German Schicht-3 surrenders are suppressed approaching duration 12 and spike at it R6 REG-R45. That is the German analogue of the eight-year threshold that drives French assurance vie behaviour, and delib models it the same way frlib does — as a duration-dependent shape with the threshold named and the level std. The level of every cell is unsourced.

Election rates. kapitalwahl_rate is a model-point attribute, base 30 % std. It is not a behavioural formula, and the notes say why: the annuitise-or-commute decision is a tax comparison — 18 % of each instalment at the marginal rate against half the Unterschiedsbetrag taxed once R5 R6 REG-R41 REG-R45 — and this model computes no tax, so the rate stands in for a calculation it does not perform. pup_year is likewise a deterministic election rather than a rate.

Expenses (all levels std; no German carrier publishes any of them, gap 14).

Input	Value	Note
Acquisition expense_acq_pp	400,00 € per policy at issue, new business only	std
Maintenance, accumulation expense_maint_pp	45,00 € per policy p.a., inflating; a twelfth a month	std
Administration, payout expense_annuity_pp	30,00 € per policy p.a., inflating; a twelfth a month	std
Settlement expense_claim_pp	120,00 € per death, surrender or commutation event	std
Inflation expense_infl	2,0 % p.a., compounding in policy years and stepping on the anniversary	std

Charges — and the distinction from expenses is load-bearing. A charge is a deduction the tariff makes from the premium or the Deckungskapital; it moves money inside the contract and produces no cash flow. An expense is the insurer’s actual outgo and is a cash flow. Confusing them is pitfall 6. charge_table.csv is keyed by (charge_id, item) so each number carries its own provenance tag:

Item	zillmer_25	zillmer_40	Basis
alpha_rate	0,025 of the Beitragssumme	0,040	cap REG-R16; use of the cap std
alpha_spread_years	5	5	§ 169 Abs. 3 REG-R28
beta_rate	0,040 of each gross premium	0,040	std
gamma_rate	0,0020 p.a. of the Deckungskapital	0,0020	std
gamma_pup_rate	0,0030 p.a. while premium-free	0,0030	std
stornoabzug_rate	0,020 of the pre-deduction value	0,000	conditions R1 REG-R28; level and flat, duration-free shape std — observed forms are none [S8] [S9], a flat 250 EUR [S4], or percentages tapering to nil over the last ten years [S11]
min_annuity_mth	30,00 € a month	30,00 €	§ 165 threshold R2; level std, against 25,00 € a month at two carriers [S4] [S9]
annuity_admin_rate	0,015 of each instalment	0,015	std; recorded, not applied — see pitfall 12

freq_load_table.csv carries the Ratenzahlungszuschlag std (gap 14): annual 1,000 (1 instalment), half-yearly 1,020 (2), quarterly 1,030 (4), monthly 1,050 (12). No retrieved wording publishes one; every carrier refers the amounts to the Kostenausweis nach § 2 VVG-InfoV or the Persönlicher Vorschlag [S1] § 14 Abs. 1, [S4] § 11 Abs. 1, [S9] § 16 Abs. 1, which is why this gap is structural.

The loading is the whole of what the Zahlweise does here, and the n_instalments column stays documentation even on a monthly grid. § 12 Abs. 1 VVG makes the Versicherungsperiode the year where premiums are not measured in shorter periods, the Deckungskapital this model turns on is defined at anniversaries, and § 169 Abs. 3 pays a mid-period exit the value struck at the end of that period — so the consistent reading is a premium payable in advance for the whole year and a value earned for the whole year, and prem_due(t) is true in the first month of each policy year only. Modelling the instalments would require an unearned-premium convention on mid-year exits that no source in this corpus establishes. KLV_DE_S is the contrast and the reason this is a decision rather than an omission: there the echt / unecht distinction turns on whether the Versicherungsperiode is genuinely monthly, two model points differ in nothing else, and the instalment stream is therefore in the frame.

---

Cash flow components and recursions

Notation, defined once and used throughout

Symbol	Cells	Meaning
t, t0, N	—	the 0-based policy-month index, t = t0 … N − 1; t0 = t_start() = 12 × duration_init; N = proj_len() = 12 × proj_len_y()
k, k0, N_y	duration, k_start, proj_len_y	the 0-based policy-year index, k = t // 12; k0 = duration_init; N_y = omega_age − issue_age. Every annual construction below takes k
x(t), τ(t)	age, calendar_year	attained age issue_age + k(t); calendar year issue_year + k(t); both step on the anniversary. The contractual policy year is k + 1
n, m, κ	aufschub_y, rgz_years, kapitalwahl_rate	deferment years; guarantee period; commutation take-up. n and m are counts of years, so the last accumulation month is t = 12n − 1 and the guarantee window is 12n ≤ t < 12n + 12m
l(t)	pols_if	policies in force at the start of month t; l(t0) = pols_if_init()
a(t)	pols_annuity	the count the annuity instalment is paid on, monthly
V(k)	av_pp	Deckungskapital per policy at the start of policy year k
A(k)	av_sur_pp	Ansammlungsguthaben per policy at the start of policy year k
Ṽ(k), Δ(k)	av_spread_pp, spread_diff_pp	the five-year-spread parallel account, and Ṽ(k) − V(k)
P(k)	prem_pp	gross premium per policy for policy year k, after the frequency loading, payable in advance
α(k), α̃(k)	charge_acq_pp, charge_acq_spread_pp	zillmered and evenly-spread acquisition charge
β(k), γ(k), ρ(k)	charge_prem_pp, charge_admin_pp, charge_risk_pp	premium, reserve-based and risk charges
S(k)	prem_to_av_pp	Sparbeitrag — the premium net of what the charges take from it
C(k)	charge_from_av_pp	the part of the charges the premium could not meet, taken from V
i, d(k), b(k)	int_rate_guar, decl_rate, bonus_rate	Rechnungszins; declared rate; max(0, d(k) − i)
D(k), Ď(k)	db_pp, db_base_pp	death benefit paid on a death in policy year k; its start-of-year measure, used only for ρ
R(k), R̄(k), R̲(k)	cv_pp, cv_tariff_pp, cv_floor_pp	surrender value; the tariff value net of the Stornoabzug; the § 169 Abs. 3 floor
q*(t), q(t), w(t)	mort_rate_guar, mort_rate, lapse_rate	first-order and best-estimate mortality, and the surrender rate — all three annual, flat across a policy year
q^m(t), w^m(t)	mort_rate_mth, lapse_rate_mth	the monthly rates the recursion applies, each 1 − (1 − r)^(1/12)
K, f_g, f_c, f	capital_conv_pp, annuity_rate_guar, annuity_rate_curr, annuity_rate_appl	conversion capital; the two Rentenfaktoren and the applied one
G, U(k)	annuity_guar_mth_pp, annuity_sur_mth_pp	garantierte Rente and Überschussrente, monthly amounts stepping once a year

Rates are per annum and dimensionless; V, A, P, D, R, K are EUR per policy; every cash flow in result_cf() is EUR for the model point as a whole.

The premium and the Beitragssumme

P(k) = 0                                                 if k ≥ n, or k ≥ prem_term_y,
                                                         or (pup_year > 0 and k ≥ pup_year − 1)
     = premium_single_pp                                 if premium_form = einmal and k = 0
     = prem_gross_pp × freq_load × (1 + dynamik_rate)^k   if premium_form = laufend

prem_due(t) = ( t mod 12 == 0 )        the first month of each policy year, and no other
premiums(t) = P(duration(t)) × l(t)    where prem_due(t), else 0

beitragssumme_pp = Σ_{u=0..min(prem_term_y, n)−1} P_sched(u)
alpha_total_pp   = alpha_rate × beitragssumme_pp

P_sched is the premium schedule as written at inception, ignoring any later Beitragsfreistellung: § 4 DeckRV takes the Zillmersatz on the sum of all premiums payable under the contract REG-R16, and a later election does not retrospectively shrink that base. The frequency loading is inside P, so it is inside the Beitragssumme too.

The premium decomposition

α(k)  = min( P(k), max(0, alpha_total_pp − alpha_cum_pp(k)) )        zillmered
α̃(k)  = alpha_total_pp / alpha_spread_years   for k < 5, else 0      § 169 Abs. 3 [REG-R28]
β(k)  = beta_rate × P(k)
γ(k)  = (paid_up(k) ? gamma_pup_rate : gamma_rate) × V(k)
Ď(k)  = prem_cum_pp(k) + P(k)                     if death_benefit_form = prem_refund
      = V(k)                                      if death_benefit_form = deckungskapital
      = max(of the two)                           if death_benefit_form = max
ρ(k)  = q*(12k) × max(0, Ď(k) − V(k))             for k < n, else 0
charge_due_pp(k)      = α(k) + β(k) + γ(k) + ρ(k)
charge_from_prem_pp(k)= min( P(k), charge_due_pp(k) )
C(k)                  = charge_due_pp(k) − charge_from_prem_pp(k)
S(k)                  = P(k) − charge_from_prem_pp(k)

Two conventions are doing work here and both are std. First, the risk charge is struck on start-of-year quantities, Ď(k) and V(k), rather than on the post-premium balance — otherwise ρ depends on V after ρ, and the recursion is circular. It reads the annual first-order rate q*(12k), the rate of the first month of policy year k, which is the year’s rate: the Risikobeitrag the tariff strikes is an annual charge and is not twelfth-rooted. Second, charges are met from the premium where there is one and from the Deckungskapital where there is not: [S11] describes premium-based deductions, but a premium-free contract still bears administration and mortality cost, and C(k) is what makes Beitragsfreistellung cost something instead of being free. Note that for death_benefit_form = deckungskapital the net amount at risk is identically zero, so ρ ≡ 0 — which is correct, and is a good invariance test.

The Deckungskapital and the Ansammlungsguthaben

av_pp_at(k,"BEF_PREM") = V(k)
av_pp_at(k,"AFT_PREM") = V(k) + S(k) − C(k)
int_credited_pp(k)     = i × av_pp_at(k,"AFT_PREM")
av_pp_at(k,"AFT_INT")  = av_pp_at(k,"AFT_PREM") + int_credited_pp(k)

bonus_credited_pp(k)      = b(k) × av_pp_at(k,"AFT_PREM") + d(k) × A(k)
av_sur_pp_at(k,"AFT_INT") = A(k) + bonus_credited_pp(k)

The Ansammlungsguthaben is a second, parallel account with its own credited rate, settling at year end and at exit R24. Its interest-surplus credit is b(k) = max(0, d(k) − i) applied to the same base the guarantee is applied to, so the two together deliver the declared laufende Verzinsung d(k) and never more — the arithmetic the reference library names as the commonest error in describing a German contract REG-R53.

The § 169 Abs. 3 floor, as a difference recursion

The two accounts differ only in the acquisition charge, so the model carries the difference rather than a second full recursion:

Δ(k0) = 0
spread_diff_pp_at(k,"AFT_INT") = ( Δ(k) + α(k) − α̃(k) ) × (1 + i)
Δ(k+1) = spread_diff_pp_at(k,"AFT_INT")
av_spread_pp_at(k,"AFT_INT") = av_pp_at(k,"AFT_INT") + spread_diff_pp_at(k,"AFT_INT")

with γ and ρ deliberately taken at the same euro amount in both accounts std, which is what makes the difference exact. Two consequences are worth stating because they are not obvious. The difference is large in the first five years — on the anchor cell the whole 25 ‰ is taken in the first year against one fifth of it — and it never returns to zero, because the spread account earns the Rechnungszins on the amounts not yet deducted. So on a zillmered tariff with a positive Rechnungszins the § 169 Abs. 3 floor is above the tariff Deckungskapital at every duration, not only in the first five years.

Surrender, and the Beitragsfreistellung election

surr_charge_pp(k) = stornoabzug_rate × ( av_pp_at(k,"AFT_INT") + av_sur_pp_at(k,"AFT_INT") )
R̄(k) = av_pp_at(k,"AFT_INT") + av_sur_pp_at(k,"AFT_INT") − surr_charge_pp(k)
R̲(k) = av_spread_pp_at(k,"AFT_INT")
R(k) = max( R̄(k), R̲(k) )

A surrender in any month of policy year k is paid R(k): § 169 Abs. 3 VVG strikes the value zum Schluss der laufenden Versicherungsperiode and not at the cancellation date R1, and the Beitrag was payable in advance for that same period.

The floor is the § 169 Abs. 3 Deckungskapital alone, because Abs. 3 speaks of the Deckungskapital; profit shares sit on top of the statutory minimum rather than inside it, which is the reading § 165 Abs. 2’s “surrender value … including profit shares” R2 supports. The alternative reading, in which the floor also carries the Ansammlungsguthaben, is not implemented and would make the floor bind at every duration; the chosen reading lets the floor bind early and stop binding once the surplus account has outgrown the interest residual and the Stornoabzug, so both branches of the max() are exercised on the anchor cell alone.

Beitragsfreistellung is a deterministic election in the contractual policy year pup_year, which is policy year k = pup_year − 1, tested at the end of the preceding year:

pup_value_pp = max( av_pp_at(pup_year−2,"AFT_INT"), av_spread_pp_at(pup_year−2,"AFT_INT") )
pup_cashout  = ( pup_value_pp / 10 000 × f_g ) < min_annuity_mth

If pup_cashout is false the contract continues premium-free: P(k) = 0 from k = pup_year − 1, the Deckungskapital is reset to pup_value_pp — the § 165 rule that the paid-up benefit is computed on the § 169 Abs. 3–5 value, on the premium basis R2 REG-R28 — the Ansammlungsguthaben is untouched, Δ is set to zero because the two accounts have merged, and the reserve-based charge switches to gamma_pup_rate. The uplift is real money and it is published, as pup_uplift(pup_year − 2) = ( pup_value_pp − av_pp_at(pup_year−2,"AFT_INT") ) × l(12(pup_year − 1)), so that the fund-level roll-forward still closes. It is booked in the transition year k = pup_year − 2, because that is the year whose roll-forward needs it: the uplift is the step between that year’s zillmered end-of-year balance and the reset opening balance of policy year k = pup_year − 1, and check_av_roll_fwd() closes in every policy year only if it is credited on the earlier of the two. On the anchor cell it is identically zero; on point 7 it is 28,39 € at k = 8. If pup_cashout is true the contract is cashed out instead, the whole surviving cohort leaving in the last month of policy year k = pup_year − 2 through the surrender decrement at R(k) R2. That is one of the two places w(k) is 1 rather than a rate, and w^m places the certainty in the anniversary month rather than spreading it geometrically: § 165 makes the cash-out fall at the end of the Versicherungsperiode, not a twelfth of the way into it. No Stornoabzug is applied on the paid-up route std: Abs. 5 is drafted for a payout on Kündigung, and here the contract continues. The alternative is a documented variant.

Decrements and the in-force recursion

Monthly, with deaths before surrenders std inside each month, which is the annual grid’s own order taken a twelfth at a time. Accumulation phase, t < 12n:

q^m(t) = 1 − (1 − q(t))^(1/12)            the year's annual rate, geometrically twelfthed
w^m(t) = 1 − (1 − w(t))^(1/12)
pols_death(t) = l(t) × q^m(t)
pols_lapse(t) = ( l(t) − pols_death(t) ) × w^m(t)
D(k)          = ( prem_refund ? prem_cum_pp(k) + P(k)
                : deckungskapital ? av_pp_at(k,"AFT_INT")
                : max(of the two) )  + ( db_incl_surplus ? av_sur_pp_at(k,"AFT_INT") : 0 )
claims(t,"DEATH") = D(duration(t)) × pols_death(t)
claims(t,"LAPSE") = R(duration(t)) × pols_lapse(t)
l(t+1)        = l(t) − pols_death(t) − pols_lapse(t) − pols_commutation(t)

Payout phase, t ≥ 12n: mortality only, no surrender, no premium, no death benefit.

pols_death(t) = l(t) × q^m(t)
pols_lapse(t) = 0
l(t+1)        = l(t) − pols_death(t)

Because (1 − q^m)^12 (1 − w^m)^12 = (1 − q)(1 − w) exactly, l(12k) at every anniversary is the annual grid’s l(k) to the last bit. What the finer grid changes is the split of a year’s exits between the two decrements — competing monthly rather than in a fixed annual order — not their total, and the shift is small: 16,87 € of claims from death to lapse over the anchor cell’s whole run.

Note that D(k) uses end-of-year balances while Ď(k) in the risk charge uses start-of-year ones; they are different quantities with deliberately similar names, and the model publishes both. Both are annual: a death in any month of policy year k is paid D(k).

The Rentenbeginn

All of it happens at the end of the last accumulation month t = 12n − 1 — which is the end of policy year k = n − 1 — on the survivors of that month’s decrements:

capital_gross_pp = av_pp_at(n−1,"AFT_INT") + av_sur_pp_at(n−1,"AFT_INT")
val_reserve_pp   = val_reserve_rate × capital_gross_pp
K                = max( guar_capital_pp, capital_gross_pp + val_reserve_pp )
f                = max( f_g, f_c )
G                = K / 10 000 × f

pols_surv_rb              = l(12n−1) − pols_death(12n−1) − pols_lapse(12n−1)
pols_commutation(12n−1)   = κ × pols_surv_rb
pols_annuitization(12n−1) = (1 − κ) × pols_surv_rb
claims(12n−1,"COMMUTATION") = K × pols_commutation(12n−1)
l(12n)                    = pols_annuitization(12n−1)

f_c is read from rentenfaktor_table.csv at (rf_scenario_id, issue_age + n) — the annuitant’s attained age at Rentenbeginn. The commuting policyholders receive K, the same capital the annuitants convert, Bewertungsreserven included [S9]: the corpus gives no basis for paying them less, and inventing one would be a charge no source supports.

The annuity in payment

U(k) = sur_ann_rate × G                                                  konstant
     = G × ( (1 + sur_ann_growth)^(k − n) − 1 )                          volldynamisch
     = θ·sur_ann_rate·G + G × ( (1 + θ·sur_ann_growth)^(k−n) − 1 )       teildynamisch, θ = 0.5

annuity_pp(t)   = G + U(duration(t))         one instalment, paid in advance
a(t)            = 0                            for t < 12n
                = pols_annuitization(12n−1)    for 12n ≤ t < 12n + 12m   the Rentengarantiezeit
                = l(t)                         for t ≥ 12n + 12m
annuity_payments(t) = annuity_pp(t) × a(t)

The instalment is the product’s own unit and the model now pays it as one. The timing is established — “Wir zahlen die Rente monatlich, jeweils zum Monatsersten” [S9] § 1 Abs. 1 — and the annual grid these notes were first written for could only compress twelve of them into one start-of-year payment, a std that is now gone. U steps once a year, on the anniversary, because an Überschussrente is redeclared annually.

a(t) is the mechanic the Rentengarantiezeit consists of: inside the guarantee window the instalment is due whether or not the annuitant is alive, so it is weighted by the annuitised count and not by survivors R17 R24, and the window is 12m instalments. Because l(t) ≤ pols_annuitization(12n−1) throughout the payout phase, a(t) = max(l(t), 1{12n ≤ t < 12n+12m} × pols_annuitization(12n−1)), which is how check_annuity_guarantee() states it. Inside the window the annual and monthly grids give the same money to the cent, the count being fixed; they part company after it, where twelve monthly measurements of a falling count are not one annual one, and that difference is the whole −369,14 € the conversion moved on the anchor cell.

Expenses and the cash flow statement

base(t)      = l(t)  for t < 12n,   a(t)  for t ≥ 12n
expenses(t)  = expense_acq_pp × 1{t = t_start() and duration_init = 0}
             + ( t < 12n ? expense_maint_pp : expense_annuity_pp ) / 12
               × (1+expense_infl)^duration(t) × base(t)
             + expense_claim_pp × ( pols_death(t) + pols_lapse(t) + pols_commutation(t) )

net_cf(t)      = premiums(t) − claims_death(t) − claims_lapse(t) − claims_commutation(t)
                 − annuity_payments(t) − expenses(t)
liability_cf(t) = − net_cf(t)

The administration expense is a twelfth of the year’s amount each month rather than a monthly level of its own, so a year of it on a closed cohort is exactly the annual grid’s charge; the inflation factor steps on the anniversary, because it compounds in policy years and a twelfth-rooted inflation would be a different assumption wearing the same number. A policy leaving mid-year now bears administration only for the months it was there, which is worth 23,00 € over the anchor cell’s run.

result_cf() returns a DataFrame indexed by the 0-based month t (index.name == "t"), contiguous from t_start() to proj_len() − 1, with these columns in this order:

pols_if, pols_annuity, premiums, claims_death, claims_lapse, claims_commutation,
annuity_payments, expenses, liability_cf, net_cf

result_cf_annual() sums it into policy years, indexed by the 1-based policy_year, which is the view the worked example above is stated on. The account movements av, av_sur, prem_to_av, int_credited and bonus_credited are state, not cash flows, they move once a policy year, and they live in result_pols() with the rest of the annual state: the Sparbeitrag and the two credits move money inside the contract and never cross the boundary, and a cash flow statement whose columns do not all sum to its bottom line is one a reader has to know which columns to skip. The six that do cross the boundary are premiums, the three claims_*, annuity_payments and expenses, and those six are exactly what check_net_cf() reconciles.

The published identities

Cells	Clock	Identity
check_net_cf()	month	net_cf(t) = premiums − claims_death − claims_lapse − claims_commutation − annuity_payments − expenses, and liability_cf(t) = −net_cf(t)
check_pols_roll_fwd()	month	pols_if(t+1) = pols_if(t) − pols_death(t) − pols_lapse(t) − pols_commutation(t), and pols_if(t) ≥ 0
check_decrement_closure()	month	Σ_t (pols_death + pols_lapse + pols_commutation) + pols_if(N) = pols_if_init(), with pols_if(N) = 0 — one past the last month — because mort_rate is 1 in the policy year at omega_age − 1
check_av_roll_fwd()	year	av(k) + prem_to_av(k) − charge_from_av(k) + int_credited(k) + pup_uplift(k) − av_release(k) = av(k+1)
check_av_sur_roll_fwd()	year	av_sur(k) + bonus_credited(k) − av_sur_release(k) = av_sur(k+1)
check_prem_split()	year	prem_pp(k) = prem_to_av_pp(k) + charge_from_prem_pp(k) and charge_due_pp(k) = charge_from_prem_pp(k) + charge_from_av_pp(k)
check_cv_floor()	year	cv_pp(k) = max(cv_tariff_pp(k), cv_floor_pp(k)) and cv_pp(k) ≥ cv_floor_pp(k) REG-R28
check_annuity_conv()	scalar	annuity_guar_mth_pp × 10 000 = capital_conv_pp × annuity_rate_appl(), annuity_rate_appl() = max(f_g, f_c) ≥ f_g, and capital_conv_pp ≥ guar_capital_pp [S4] [S9]
check_annuity_guarantee()	month	pols_annuity(t) = max(pols_if(t), 1{12n ≤ t < 12n+12m} × pols_annuitization(12n−1)) for t ≥ 12n, and = 0 for t < 12n

Each has a residual companion beside it, and the residual’s argument follows its cells’ clock: check_*_resid(t) over months for the six monthly identities, check_*_resid(k) over policy years for the four annual ones. A monthly residual for the Deckungskapital roll-forward would first have had to invent a monthly reserve, which is the error the two-clock split exists to make impossible. Each check_*() takes no argument and returns a bool. check_net_cf() is required of every delib model by the library’s first ruling and is asserted in tests/test_model_conventions_de.py.

---

Processing order, on two clocks

The annual block runs once per policy year k, at its start, and is the whole of steps A1 to A6. The monthly block runs in every month t of that year, and is steps M1 to M5. Steps A1 to A6 and M3 run only in the accumulation phase; step A7 only at the boundary; step M4 only in the payout phase.

The annual block, for k = k0 … N_y − 1:

A1. Open the year. age_y(k) = issue_age + k, calendar_year_y(k) = issue_year + k. The opening state is av_pp(k), av_sur_pp(k), spread_diff_pp(k), alpha_cum_pp(k) and prem_cum_pp(k); the population is pols_if(12k). A2. Test the phase and the elections. Accumulation if k < n, payout if k ≥ n; paid_up(k) is true if pup_year > 0 and k ≥ pup_year − 1. If k = pup_year − 1 and the contract is converting, av_pp(k) has already been reset to pup_value_pp by A1’s recursion and spread_diff_pp(k) to zero; pup_uplift(pup_year − 2) — booked one year earlier, on the transition year — records the difference. A3. Premium in advance, decomposed. prem_pp(k), then charge_acq_pp, charge_prem_pp, charge_admin_pp and charge_risk_pp — the last two on start-of-year balances — then charge_from_prem_pp, charge_from_av_pp and the Sparbeitrag prem_to_av_pp. A4. Credit the Deckungskapital and accrue the guarantee. av_pp_at(k,"AFT_PREM") = av_pp(k) + prem_to_av_pp(k) − charge_from_av_pp(k), then int_credited_pp(k) = int_rate_guar × av_pp_at(k,"AFT_PREM") and av_pp_at(k,"AFT_INT"). This is the std ordering — premium, then charges, then interest on what is left. The retrieved wordings fix the decomposition ([S8] § 1 Abs. 2, [S11] § 27 Abs. 1) but none of them fixes the within-year sequence, which stays a standardization. A5. Credit the surplus. decl_rate(k) from the declared-rate path, bonus_rate(k) = max(0, decl_rate(k) − int_rate_guar), then bonus_credited_pp(k) to the Ansammlungsguthaben, at year end R24. A6. Roll the spread account and strike the year’s values. spread_diff_pp_at(k,"AFT_INT"), hence av_spread_pp_at(k,"AFT_INT"), the surrender value cv_pp(k) and the death benefit db_pp(k). These are what a claim in any month of the year is paid. A7. End of month 12n − 1 — the Rentenbeginn. Strike capital_gross_pp, add val_reserve_pp, apply max(guar_capital_pp, ·) to get K; determine f = max(annuity_rate_guar, annuity_rate_curr); split the survivors between pols_commutation(12n−1) and pols_annuitization(12n−1); pay claims(12n−1,"COMMUTATION"); strike G. Both account balances go to zero.

The monthly block, for t = t_start() … N − 1:

M1. Open the month. duration(t) = t // 12 selects the year’s state; the population is pols_if(t). M2. Look up the year’s rates and twelfth them. mort_rate_guar(t) from the generational surface at (sex, age(t), calendar_year(t)), mort_rate(t) = mort_rate_guar(t) × mort_be_factor, lapse_rate(t) from the duration table at policy_year(t) (zero if duration(t) ≥ n), then mort_rate_mth(t) and lapse_rate_mth(t), each 1 − (1 − r)^(1/12). M3. Premium, where one is due. prem_due(t) in the first month of the policy year; premiums(t) = prem_pp(duration(t)) × pols_if(t). M4. Payout phase, start of the month. annuity_pp(t) = G + U(duration(t)) paid on pols_annuity(t), which is the annuitised count inside the Rentengarantiezeit and survivors after it. M5. End of the month — decrements, expenses, publish. Death at mort_rate_mth(t) paying db_pp(duration(t)); then surrender on the survivors at lapse_rate_mth(t) paying cv_pp(duration(t)). Where duration(t) = pup_year − 2, pup_cashout is true and is_anniv(t), the whole surviving cohort leaves through the surrender decrement. Expenses: acquisition at t = t_start() for new business; a twelfth of the year’s maintenance or annuity administration per policy on base(t), inflated by policy year; settlement expense on the month’s deaths, surrenders and commutations. Then pols_if(t+1), and net_cf(t) = premiums − claims_death − claims_lapse − claims_commutation − annuity_payments −     expenses, with liability_cf(t) = −net_cf(t).

---

Known modeling pitfalls

These are the specific ways an implementation of this product looks right and is wrong. Each one becomes a test in tests/test_klassische_rentenversicherung_de.py.

1. Adding the declared rate on top of the guarantee. The laufende Verzinsung is the Garantieverzinsung plus the laufende Zinsüberschussbeteiligung REG-R53. Assert int_credited_pp(k) + bonus_rate(k)·av_pp_at(k,"AFT_PREM") = decl_rate(k)·av_pp_at(k,"AFT_PREM") whenever decl_rate(k) ≥ int_rate_guar, and that on model point 6 — a 2,75 % vintage against a 2,55 % declaration — bonus_rate(k) = 0 in every policy year while int_credited_pp(k) > 0. A model that credits 1,00 % and a further 2,55 % overstates the anchor cell’s year-one crediting by 39 % (56,82 € against 40,82 €) and its accumulated value at Rentenbeginn by 8,5 % (63 768,69 € against 58 788,98 €), the whole of the error sitting in the Ansammlungsguthaben (12 698,26 € against 7 718,55 €).

2. Getting the within-year order wrong. Premium, then charges, then interest on the balance std. Crediting interest before the charges, or on the opening balance only, changes year-one interest by the whole of i × (S(0) − C(0)). Assert int_credited_pp(k) equals int_rate_guar × av_pp_at(k,"AFT_PREM") exactly, and that av_pp_at(k,"AFT_INT") ≠ (av_pp(k))·(1+i) + S(k) on the anchor cell.

3. Applying only the guaranteed Rentenfaktor. The rule is max(guaranteed, current) [S4] R24. Assert annuity_rate_appl() = 32.00 on the anchor (the current factor wins) and = annuity_rate_guar on point 13 (the guarantee binds), and that annuity_rate_appl() ≥ annuity_rate_guar on every point. A model taking the guaranteed factor alone understates the anchor’s annuity by 12,5 %.

4. Weighting the guaranteed annuity by survivors. Inside the Rentengarantiezeit the instalment is due whether the annuitant lives or not R17 R24. Assert pols_annuity(t) = pols_annuitization(12n−1) for 12n ≤ t < 12n + 12m and = pols_if(t) after, and that the two differ at t = 12n + 12m − 1 on the anchor. Assert also that point 9, with rgz_years = 0, has pols_annuity(t) = pols_if(t) at every payout month. The window is 12m instalments, which is what a Rentengarantiezeit guarantees.

5. Treating Beitragsfreistellung as a lapse. They are separate decrements with different consequences: the paid-up contract keeps its guarantee vintage and its guaranteed Rentenfaktor and pays a reduced benefit; the surrendered one is gone for cash R1 R2. Assert that on point 7 pols_if is unbroken through the paid-up year k = pup_year − 1, prem_pp(k) = 0 from it, int_rate_guar is unchanged, and that the conversion itself moves no policy: lapse_rate in policy year pup_year − 2 is the ordinary duration-9 table rate of 3,5 % and not 1, and av_pp(pup_year − 1) = pup_value_pp > av_pp_at(pup_year − 2,"AFT_INT"). What is not true is that surrender ceases: a beitragsfrei contract keeps its § 168 VVG Kündigung right, so claims_lapse stays positive from the paid-up year on — 764,60 € in policy year 10 on point 7 — and only the § 165 cash-out branch of point 8 empties the cohort, in the anniversary month of policy year pup_year − 1.

6. Booking the Kostenbeitrag as an expense. The charges are internal deductions that move money inside the contract; expenses is the insurer’s best-estimate outgo. Assert expenses(t) ≠ charge_due_pp(duration(t)) × pols_if(t) and that expenses(t) is invariant to beta_rate and gamma_rate, while av_pp(k+1) is not. Double-counting them inflates outgo by the whole charge load and is the commonest way to make a German model look conservative.

7. Computing the surrender value off the zillmered reserve. § 169 Abs. 3 floors it at the reserve with acquisition costs spread evenly over the first five contract years REG-R28. Assert cv_pp(k) = max(cv_tariff_pp(k), cv_floor_pp(k)), that the floor binds in the anchor’s early years and stops binding later, and that omitting it changes claims_lapse in the second policy year by the whole of spread_diff_pp_at(1,"AFT_INT").

8. Letting the Stornoabzug recover acquisition costs. § 169 Abs. 5 permits a deduction only if agreed, quantified and appropriate, and voids one for unamortised acquisition costs R1 REG-R28. Assert that surr_charge_pp(k) is a flat percentage of the pre-deduction value and carries no duration term, and that cv_pp(k) never falls below cv_floor_pp(k) however large stornoabzug_rate is set.

9. Using one mortality basis where the product uses two. The first-order basis fixes the risk charge and the guaranteed benefits; the second-order basis drives the projection REG-R47. Assert mort_rate(t) = mort_rate_guar(t) × mort_be_factor with mort_be_factor > 1, that charge_risk_pp uses the annual mort_rate_guar while pols_death uses the monthly mort_rate_mth derived from mort_rate, and that swapping them changes net_cf. Assert also that 1 − (1 − mort_rate_mth(t))^12 = mort_rate(t): the twelfth is geometric, and dividing the annual rate by twelve instead would leave pols_if above the annual grid’s at every anniversary.

10. Using a period mortality table. DAV 2004 R is a Generationentafel; a period-table proxy priced at an annuitisation decades ahead understates the liability by a margin that dwarfs every other assumption REG-R49. Assert mort_rate_guar depends on calendar_year(t) as well as age(t): on the anchor, mort_rate_guar at attained age 67 in 2043 must be strictly below the same age’s rate for a life reaching 67 in 2026.

11. Charging the risk premium on a zero net amount at risk. With death_benefit_form = deckungskapital the death benefit is the reserve, so there is nothing at risk. Assert charge_risk_pp(k) = 0 in every policy year on points 2 and 12, and that charge_risk_pp(k) > 0 in every accumulation year on the anchor, where the benefit is the premiums paid. On that cell the net amount at risk rises to a peak of 4 587,95 € at k = 5 and then declines, ending the deferment at 3 204,24 € rather than at zero: with the whole 25 ‰ zillmered out of year 1, a 4 % premium charge and a 1,00 % Rechnungszins, the Deckungskapital never overtakes the premiums paid inside seventeen years, which is precisely why Beitragsrückgewähr is real risk cover on this design and not a formality. The risk charge itself rises monotonically, from 4,37 € to 15,39 €, because first-order mortality more than triples over the deferment while the amount at risk moves by a third.

12. Deducting the payout-phase administration charge from the annuity. The Rentenfaktor is exogenous here and already carries the tariff’s payout loading, so annuity_admin_rate is recorded in charge_table.csv and not applied. Assert annuity_payments(t) = (G + U(duration(t))) × pols_annuity(t) exactly, with no charge term, and that the model’s payout-phase expenses are a twelfth of the inflated per-policy expense_annuity_pp on the exposed count plus the expense_claim_pp settlement cost of that month’s deaths, and nothing else. The settlement line survives into the payout phase although the death pays no benefit, because stopping an annuity and running the Rentengarantiezeit succession is administration the insurer still performs.

13. Paying a death benefit after Rentenbeginn. The reference model pays none: claims_death(t) = 0 for every t ≥ 12n on every model point, and the test asserts it. This is now a modelling choice, not an absence of evidence. [S4] § 1 Abs. 5 offers Beitragsrückgewähr während der Rentenzahlungszeit as an alternative to the Rentengarantiezeit — premiums paid less rider premiums less annuities already received at their inception-guaranteed level, the claim extinguishing once instalments exceed premiums — and § 3 Abs. 7 accumulates the surplus attributable to it with interest until it is paid or the claim lapses. Modelling it would add a second post-Rentenbeginn benefit path and move the worked example, so it is recorded here and deliberately left out of this pass. The other two documented mechanics are the Rentengarantiezeit [S1] [S4] [S9] and the survivor’s-annuity rider, which begins only after any guarantee period expires [S10] § 1 Abs. 3.

14. Letting the Kapitalwahlrecht leave the account behind. Commuting policyholders receive capital_conv_pp — the same capital annuitants convert, Bewertungsreserven included [S9]. Assert claims_commutation(12n−1) = capital_conv_pp × kapitalwahl_rate × pols_surv_rb, that both account balances are zero from k = n, and that on point 9 (kapitalwahl_rate = 1.00) pols_if(t) = 0 and every cash flow is zero for t ≥ 12n.

15. Forgetting that the guarantee vintage is a model-point attribute. Existing contracts keep the Rechnungszins they were written on R7 REG-R14. Assert that points 1, 6 and 14 credit 1,00 %, 2,75 % and 0,90 % respectively, and that re-running point 6 on a single global 1,00 % rate moves its Deckungskapital at Rentenbeginn from 82 833,38 € to 76 439,87 €, −7,7 %, while its Ansammlungsguthaben goes the other way, 3 629,35 € to 9 292,76 €, +156 %. The conversion capital barely moves — 87 759,66 € against 87 018,62 €, −0,8 % — which is the point: a single global rate leaves the total looking almost right and puts the money in the wrong account, where it carries no guarantee.

16. Letting sex reach the tariff. Unisex has been compulsory since 21 December 2012 REG-R34. Assert that two model points identical but for sex produce identical prem_pp, identical charge_* except through mort_rate_guar, and identical annuity_rate_appl().

17. Amortising the acquisition charge against a shrunken Beitragssumme. The § 4 DeckRV base is the sum of all premiums payable under the contract as written REG-R16, not the premiums a later Beitragsfreistellung leaves behind. Assert alpha_total_pp on point 7 is unchanged by pup_year, and that alpha_cum_pp never exceeds alpha_total_pp.

18. Truncating the payout phase. A life annuity has no term; proj_len() = omega_age − issue_age. Assert result_cf().index[-1] == proj_len() − 1, that pols_if(proj_len()) = 0 — one past the last row — and that the decrement closure sums to pols_if_init() exactly. A 40-year horizon on the anchor cell would drop a real, if small, tail of annuity payments beyond attained age 90.

---

Policyholder behaviour modelling

Every formula here is std; nothing in the corpus calibrates any of it (gap 20).

• Base surrender. The duration table above, with the duration-12 step at the § 20 Abs. 1 Nr. 6 EStG twelve-year threshold R6 REG-R45. The shape is the assumption; the levels are placeholders. A German Schicht-3 projection with a lapse rate flat in duration has ignored the strongest single driver of German surrender behaviour.

• No dynamic surrender. The obvious German dynamic term would key the surrender rate on the gap between a market rate and the declared laufende Verzinsung, as frlib’s euro-fund model keys it on the Livret A gap. It is not implemented here, for a reason worth stating: on this product the policyholder who surrenders forfeits a guaranteed Rentenfaktor struck on bases decades old, and the value of that forfeited option is exactly what a rate-gap formula does not capture. A model that adds a naive rate-gap term to a book of 4,00 % vintages will lapse precisely the contracts a real policyholder would never surrender.

• Beitragsfreistellung as an election, not a rate. A scalar per-policy account cannot carry two sub-populations with different Deckungskapital, and splitting the account would double the accumulation-phase state for a mechanic whose rate no source establishes. The model therefore carries pup_year as a deterministic election on the model point, exercises both of its statutory branches (conversion, and cash-out below the Mindestversicherungsleistung) on points 7 and 8, and says here that a portfolio model needs the sub-population split this one does not have.

• The Kapitalwahlrecht as a take-up rate. kapitalwahl_rate is a model-point attribute, base 30 % std. The decision it stands for is a tax comparison — the Ertragsanteil at 18 % of each instalment against half the Unterschiedsbetrag once R5 R6 REG-R41 REG-R45 — and this model computes no tax, so the rate is a stand-in for a calculation it does not perform, not an estimate of one.

• What is deliberately absent. No Widerruf decrement (it sits inside the year-1 lapse rate, REG-R23); no premium-default path, although § 166 VVG makes German lapse a three-way decrement in reality REG-R28 REG-R30; no Wiederinkraftsetzung [S11]; no selective-lapsation mortality loading; and no take-up modelling for the Dynamik, whose parameters are unestablished (gap 15).

---

Worked example

Configuration. Model point 1, point_id = 1, policy_id = DE-RV-0001: sex = M; issue_age = 50; issue_year = 2026; duration_init = 0, so the frame opens at t = 0; pols_if_init = 1.0; premium_form = laufend; prem_gross_pp = 3 000,00 €; premium_single_pp = 0,00 €; prem_freq = annual, hence freq_load = 1,000; prem_term_y = 17; aufschub_y = 17, so the Rentenbeginn falls at the end of month t = 203 — the end of the seventeenth policy year — at attained age 67; int_rate_guar = 1,00 %, the 2026 vintage REG-R15; charge_id = zillmer_25; annuity_rate_guar = 28,00 € per month per 10 000 €; rf_scenario_id = base; decl_scenario_id = base; guar_capital_pp = 0,00 €, so the guaranteed-contract-value floor is inoperative on this cell; death_benefit_form = prem_refund; db_incl_surplus = 0; rgz_years = 10; kapitalwahl_rate = 0,30; pup_year = 0; dynamik_rate = 0,0000; payout_system = konstant; av_pp_init = 0,00 €; av_sur_pp_init = 0,00 €; prem_cum_pp_init = 0,00 €; alpha_amort_pp_init = 0,00 €. Hence proj_len_y() = 121 − 50 = 71 and proj_len() = 852, so the frame is t = 0 … 851: the accumulation phase is t = 0 … 203, the Rentengarantiezeit covers t = 204 … 323 — a hundred and twenty guaranteed instalments — and the survivor-weighted annuity runs from t = 324 to t = 851.

Assumptions, each tagged. Contractual and cited: the Rechnungszins i = 1,00 % p.a. REG-R15; the Höchstzillmersatz alpha_rate = 25 ‰ of the Beitragssumme REG-R16, giving beitragssumme_pp = 17 × 3 000,00 = 51 000,00 € and alpha_total_pp = 1 275,00 €, zillmered — taken in full from the first year’s premium and nil thereafter; the § 169 Abs. 3 five-year spread, alpha_spread_years = 5, giving α̃(k) = 255,00 € for k = 0 … 4 REG-R28; the § 165 Mindestversicherungsleistung test, not triggered on this cell R2; the death benefit Beitragsrückgewähr, the premiums paid to date, premiums only [S1] R24; the conversion rule monthly annuity = capital / 10 000 × Rentenfaktor R24; and the applied factor max(garantiert, aktuell) [S4] R24. Insurer-discretionary current: the declared laufende Verzinsung decl_rate = 2,55 % p.a. level on the base path std, hence bonus_rate = max(0; 2,55 % − 1,00 %) = 1,55 % p.a. std, with the Ansammlungsguthaben itself credited at the full 2,55 % std; the Bewertungsreserven crystallisation val_reserve_rate = 1,5 % of the accumulated value at Rentenbeginn std [S4] R4; the aktueller Rentenfaktor at age 67 on the base path, 32,00 € per month per 10 000 € std, which exceeds the guaranteed 28,00 € and therefore wins the max(); and the Überschussrente under the konstant system, sur_ann_rate = 12 % of the garantierte Rente, level std. Charges (all levels std): beta_rate = 4,0 % of each gross premium; gamma_rate = 0,20 % p.a. of the Deckungskapital; the Risikobeitrag ρ(k) = mort_rate_guar(12k) × max(0, prem_cum_pp(k) + prem_pp(k) − av_pp(k)); stornoabzug_rate = 2,0 % of the pre-deduction value, subject to the § 169 Abs. 3 floor R1 REG-R28; annuity_admin_rate = 1,5 %, recorded and not applied (pitfall 12). Behavioural and experience (all std): first-order mortality from the shipped generational proxy, mort_rate_guar(t) = q_base(M, x(t)) × (1 − improve(x(t)))^(τ(t) − 2005), annual and flat across a policy year, with the anchor q_base(M, 50) = 0,002000 and improve(x) = 1,5 % below age 60 grading to 0,5 % at 100 and to zero at 110; best-estimate mortality mort_rate(t) = mort_rate_guar(t) × 1,15; surrender 6,0 % / 5,0 % / 4,5 % / 4,0 % (durations 4–7) / 3,5 % (8–11) / 6,0 % at duration 12 / 3,0 % thereafter, and zero from Rentenbeginn; the Kapitalwahlrecht take-up 30 %; and expenses of 400,00 € acquisition at issue, 45,00 € per policy p.a. in the accumulation phase and 30,00 € p.a. in the payout phase, both inflating at 2,0 % p.a., plus 120,00 € per death, surrender or commutation event. omega_age = 121 std. No Dynamik, no Beitragsfreistellung, no guar_capital_pp floor, no behavioural modules.

The frame. The model projects months; the table below is RV_DE_S.Projection[1].result_cf_annual(), the monthly frame summed into policy years, which is the only view on which a year’s worth of arithmetic can be read at once. Money to the cent and pols_if to six decimals. The index is written as the 0-based policy year k = policy_year − 1, so row k covers months 12k … 12k + 11. Rows 0–16 are the whole accumulation phase; rows 17, 26, 27, 39, 54 and 70 sample the payout at the first annuity year, the last guaranteed year, the first survivor-weighted year and three points down the tail. av and av_sur are the two balances at the start of the row’s year, for the model point as a whole, taken from result_pols(); they are state rather than cash flow and they move once a year, which is why they are not columns of result_cf().

k	pols_if	av	av_sur	premiums	claims_death	claims_lapse	claims_commutation	annuity_payments	expenses	net_cf
0	1.000000	0.00	0.00	3,000.00	4.88	158.67	0.00	0.00	451.10	2,385.34
1	0.938426	1,517.10	23.28	2,815.28	9.89	249.12	0.00	0.00	47.87	2,508.41
2	0.889902	4,032.52	84.53	2,669.71	15.14	320.11	0.00	0.00	45.76	2,288.69
3	0.848216	6,335.16	179.84	2,544.65	20.70	362.97	0.00	0.00	43.99	2,116.98
4	0.812600	8,474.57	306.74	2,437.80	26.62	445.58	0.00	0.00	42.92	1,922.68
5	0.778360	10,439.40	461.52	2,335.08	32.85	526.51	0.00	0.00	41.87	1,733.85
6	0.745440	12,238.83	641.09	2,236.32	39.41	601.95	0.00	0.00	40.83	1,554.13
7	0.713787	13,881.56	842.56	2,141.36	46.41	588.10	0.00	0.00	39.48	1,467.37
8	0.686908	15,455.93	1,068.70	2,060.72	53.95	648.43	0.00	0.00	38.70	1,319.64
9	0.660906	16,904.27	1,313.89	1,982.72	61.92	705.09	0.00	0.00	37.94	1,177.77
10	0.635750	18,232.03	1,575.91	1,907.25	70.35	758.18	0.00	0.00	37.18	1,041.55
11	0.611408	19,444.42	1,852.62	1,834.22	78.94	1,384.79	0.00	0.00	37.86	332.64
12	0.572603	20,013.45	2,086.42	1,717.81	87.98	713.03	0.00	0.00	34.50	882.30
13	0.553206	21,091.10	2,390.82	1,659.62	99.18	752.73	0.00	0.00	33.97	773.74
14	0.534287	22,078.63	2,706.76	1,602.86	111.26	790.09	0.00	0.00	33.43	668.08
15	0.515827	22,978.50	3,032.52	1,547.48	124.27	825.12	0.00	0.00	32.90	565.19
16	0.497806	23,793.00	3,366.34	1,493.42	138.27	857.84	8,596.26	0.00	49.65	−8,148.61
17	0.336143	0.00	0.00	0.00	0.00	0.00	0.00	862.65	14.36	−877.01
26	0.311032	0.00	0.00	0.00	0.00	0.00	0.00	862.65	17.36	−880.01
27	0.307034	0.00	0.00	0.00	0.00	0.00	0.00	782.87	16.14	−799.01
39	0.229120	0.00	0.00	0.00	0.00	0.00	0.00	576.71	15.74	−592.45
54	0.055062	0.00	0.00	0.00	0.00	0.00	0.00	128.34	5.64	−133.99
70	0.000000	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	−0.00
Total	—	—	—	35,986.30	1,022.04	10,688.30	8,596.26	23,115.89	1,646.77	−9,082.96

At the start of policy year 71 the surviving fraction is 2,15 × 10⁻⁷ and every cash flow rounds to nil; the year is kept because result_cf().index[-1] == proj_len() − 1 is a machine-checked property of this library, and because pols_if(852) — one past the last month — is the exact zero the closure identity below needs.

The Total row is summed over all 852 months at full precision and then rounded, not summed from the rounded cells, and the two differ: summing the 71 rounded annual rows gives net_cf = −9 082,97 € against −9 082,96 €. The largest single-column gap is annuity_payments, 23 115,83 € against 23 115,89 €; claims_death and expenses each lose two cents likewise. No rounded quantity is re-used anywhere: every row is computed at full precision and rounded once, at the point of display.

The shape is a long positive accumulation phase, a −8 148,61 € spike in policy year 17 where the Kapitalabfindung falls on 30 % of the survivors, and a negative annuity tail running another fifty-four years. Undiscounted, the cell collects 35 986,30 € and pays out 45 069,26 €.

What the monthly grid moved, against the annual-step model this replaced. Premiums are identical — 35 986,30 € — and so is every account balance, every surrender value and the conversion capital: the whole annual layer is bit-identical, on all fourteen model points. Three columns moved, each for a reason:

Column	Annual grid	Monthly grid	Why
annuity_payments	23 485,03 €	23 115,89 €	Twelve instalments paid through the year on a cohort that loses lives, instead of twelve paid at the start of it. Inside the Rentengarantiezeit the two agree to the cent — 862,65 € a year — because the count is fixed there; the whole −369,14 € is in the survivor-weighted tail
claims_death	1 038,91 €	1 022,04 €	Deaths and surrenders now compete month by month rather than once a year
claims_lapse	10 670,70 €	10 688,30 €	The other side of the same shift: total exits are unchanged at every anniversary, the split is not
expenses	1 669,77 €	1 646,77 €	A policy leaving mid-year bears administration for the months it was there

Independent checks

These rebuild table cells a different way — from the tariff parameters rather than from the recursion — in arithmetic a reader can follow with a calculator.

Check 1 — the first policy year, k = 0, from the tariff parameters alone. The account arithmetic is annual and nothing here reads the model:

Beitragssumme = 17 x 3,000.00                         = 51,000.00
alpha_total   = 0.025 x 51,000.00                     =  1,275.00  (P(0) covers it: all in year 1)
beta(0)       = 0.040 x 3,000.00                      =    120.00
gamma(0)      = 0.0020 x 0.00                         =      0.00  (the account is empty)
q*(0)         = 0.002000 x (1 - 0.015)^(2026-2005)
              = 0.002000 x 0.72804940                 = 0.00145610
rho(0)        = 0.00145610 x (0.00 + 3,000.00 - 0.00) =      4.3683
charges due   = 1,275.00 + 120.00 + 0.00 + 4.3683     =  1,399.3683  (all met from the premium)
S(0)          = 3,000.00 - 1,399.3683                 =  1,600.6317
int_credited  = 0.0100 x 1,600.6317                   =     16.0063
V at year end = 1,600.6317 + 16.0063                  =  1,616.6380
Delta after 0 = (0.00 + 1,275.00 - 255.00) x 1.01     =  1,030.2000
cv_floor(0)   = 1,616.6380 + 1,030.2000               =  2,646.8380
cv_tariff(0)  = (1,616.6380 + 24.8098) x 0.98         =  1,608.6189  (the floor wins)

Every line of that is unchanged from the annual-step model, which is the point of the two-clock split. The decrements are where the month enters, and the year’s totals are a sum of twelve:

q(0)          = 0.00145610 x 1.15                     = 0.00167451   annual
q^m(0)        = 1 - (1 - 0.00167451)^(1/12)           = 0.00013965   monthly
w(0)          = table duration 1                      = 0.06         annual
w^m(0)        = 1 - (1 - 0.06)^(1/12)                 = 0.00514301   monthly
deaths, mth 0 = 1.000000 x 0.00013965                 = 0.00013965
lapses, mth 0 = (1.000000 - 0.00013965) x 0.00514301  = 0.00514229
deaths, yr 0  = sum over the twelve months            = 0.00162796
lapses, yr 0  = sum over the twelve months            = 0.05994608
claims_death  = 3,000.00 x 0.00162796                 =      4.8839
claims_lapse  = 2,646.8380 x 0.05994608               =    158.6676
expenses(0)   = 400.00 + sum of twelve x 45.00/12 x l
                + 120.00 x 0.06157405                 =    451.1043
net_cf, yr 0  = 3,000.00 - 4.8839 - 158.6676 - 451.1043 = 2,385.3442

which is the table’s first row to the cent. Two things to read off it. The twelve monthly deaths sum to 0,001628 against the annual grid’s 0,001675 and the twelve monthly lapses to 0,059946 against 0,059900 — the competing-decrement shift, and the survivors at the anniversary are the same 0,938426 either way, because (1 − q^m)^12 (1 − w^m)^12 = (1 − q)(1 − w) exactly. And the surrender value the leavers are paid is cv_pp(0) in every month of the year, § 169 Abs. 3 VVG striking it at the end of the Versicherungsperiode.

The load-bearing line of the annual block is the seventh: the acquisition charge takes 42,5 % of the year-one premium, leaving a Sparbeitrag of 1 600,63 €, and the § 169 Abs. 3 floor then stands 1 030,20 € above the tariff Deckungskapital at the end of the first year. That gap is the whole of Zillmerung in one line, and it is why the floor binds through policy year 4 and stops binding in policy year 5.

Check 2 — the declared rate contains the guarantee. The two credits are struck on the same base and must together be the declared rate, never the declared rate on top of the guarantee:

guarantee  0.0100 x 1,600.6317 = 16.0063170368
surplus    0.0155 x 1,600.6317 = 24.8097914070
sum                             = 40.8161084438
declared   0.0255 x 1,600.6317 = 40.8161084438     identical to ten decimals

A model crediting 1,00 % and a further 2,55 % puts 56,82 € into the first year instead of 40,82 €, 39 % too much; rolled forward on the same premium and charge schedule that reaches 63 768,69 € of accumulated value at Rentenbeginn against the correct 58 788,98 €, an 8,5 % overstatement sitting entirely in the Ansammlungsguthaben (12 698,26 € against 7 718,55 €). The mirror image is model point 6 below.

Check 3 — the Rentenbeginn, rebuilt from the two balances. At the end of month t = 203 — the end of the seventeenth and last accumulation year — the Deckungskapital stands at 51 070,4278 € per policy and the Ansammlungsguthaben at 7 718,5532 €:

capital_gross  = 51,070.4278 + 7,718.5532           = 58,788.9809
val_reserve    = 0.015 x 58,788.9809                =    881.8347
K              = max(0.00, 58,788.9809 + 881.8347)  = 59,670.8156
f              = max(28.00, 32.00)                  =     32.00    (the current factor wins)
G              = 59,670.8156 / 10,000 x 32.00       =    190.9466  EUR a month
U              = 0.12 x 190.9466                    =     22.9136  EUR a month
annuity_pp     = 190.9466 + 22.9136                 =    213.8602  EUR a month, in advance
pols_surv_rb   = 0.481648 - 0.000222 - 0.001220     =   0.480205
commutations   = 0.30 x 0.480205                    =   0.144061
claims_commut. = 59,670.8156 x 0.144061             =  8,596.2645
annuitisations = 0.70 x 0.480205                    =   0.336143   = pols_if(204)
annuity_pay    = 213.8602 x 0.336143                =     71.8877  EUR a month
                 x 12 months of the guarantee       =    862.6523  EUR in policy year 18

matching the table at policy years 17 and 18. The instalment is the product’s own unit, and the annual figure is now a sum of twelve of them rather than a year’s annuity paid at once; inside the Rentengarantiezeit the count is fixed, so the year’s total is the same 862,65 € the annual grid gave and the two models agree to the cent. They part company from policy year 28, where the count is survivors and twelve monthly measurements of it are not one annual one. Applying the guaranteed 28,00 € instead would give G = 167,0783 €, 87,5 % of the right answer and 12,5 % of the whole payout phase lost, the annuity scaling linearly in f.

Closure, twice. The decrements account for the whole policy — summed over all 852 months at full precision, deaths 0,371014 + surrenders 0,484924 + commutations 0,144061 + survivors at t = 852 — one past the last month — 0,000000 = 1,000000 exactly, against pols_if_init() = 1,000000. The survivor term is exactly zero because mort_rate is 1 in the policy year at attained age 120 = omega_age() − 1, which is what makes proj_len_y() = omega_age() − issue_age the right horizon: a 40-year run would strand 0,219599 of a policy at attained age 90 and silently drop most of the tail of the payout phase. And the account rolls forward — in policy year 0, at fund level, 0.00 + 1,600.6317 − 0.00 + 16.0063 − 99.5429 = 1,517.0951 = av(1), where 99,5429 € is the end-of-year balance of 1 616,6380 € carried out by the 0,061574 of a policy that died or surrendered at any point in the year. The same identity closes in every policy year, including across the Rentenbeginn, where av_release(16) is the whole balance. That is check_av_roll_fwd(), and it is stated per policy year: the Deckungskapital is defined at anniversaries, so a monthly residual for it would first have had to invent a monthly reserve.

One thing the table does not show, and it is the point of the Rentengarantiezeit. In policy year 27, the last guaranteed year, pols_annuity is 0,336143 — the count that annuitised — in every one of its twelve months, while pols_if has fallen to 0,307366 by the last of them. The instalment is due either way, so the year’s outgo is 862,65 € and not the roughly 800 € a survivor weighting would give. From policy year 28 the two coincide. check_annuity_guarantee() asserts it in every month of every model point.

Variant A — the Einmalbeitrag form (model point 2)

A 55-year-old woman paying a single 50 000,00 € premium for a twelve-year deferment to the same attained age 67, with death_benefit_form = deckungskapital and kapitalwahl_rate = 0.00.

k	pols_if	av	av_sur	premiums	claims_death	claims_lapse	annuity_payments	expenses	net_cf
0	1.000000	0.00	0.00	50,000.00	76.88	2,891.05	0.00	451.10	46,580.97
1	0.938426	44,310.12	680.01	0.00	78.45	2,267.01	0.00	47.87	−2,393.33
11	0.611187	31,246.12	5,757.88	0.00	117.65	2,222.96	0.00	37.85	−2,378.46
12	0.572308	0.00	0.00	0.00	0.00	0.00	1,548.54	22.06	−1,570.60
23	0.532497	0.00	0.00	0.00	0.00	0.00	1,433.87	25.74	−1,459.61
39	0.370668	0.00	0.00	0.00	0.00	0.00	982.17	25.57	−1,007.74
65	0.000285	0.00	0.00	0.00	0.00	0.00	0.77	0.07	−0.84
Total	—	—	—	50,000.00	1,128.88	21,374.96	46,844.86	1,908.71	−21,257.42

Totals again at full precision then rounded; the rounded-cell net_cf is −21 257,45 €. Three things read off it. The Beitragssumme is the single premium, so alpha_total_pp is 1 250,00 € and the first year’s Sparbeitrag is 46 750,00 € — 93,5 % of the premium against the anchor’s 53,4 %, which is the whole economic difference between the two forms. The net amount at risk is identically zero in every policy year, the benefit being the Deckungskapital itself, so the Risikobeitrag disappears from the decomposition. And the conversion capital is 62 913,28 € against the anchor’s 59 670,82 €, a garantierte Rente of 201,32 € a month, on a contract that paid 50 000,00 € once rather than 51 000,00 € over seventeen years.

Variant B — the 2,75 % legacy vintage (model point 6)

An in-force contract written in 2005 on a 2,75 % Rechnungszins and the 40 ‰ Höchstzillmersatz, twenty policy years elapsed, so the frame opens at t_start() = 240 — policy year 21 — and the Rentenbeginn falls at the end of month 299, the end of policy year 25. Its opening balances are the balances this model’s own recursion produces for the same contract run from inception, so the cell is the continuation of a projectable contract rather than a guess.

k	pols_if	av	av_sur	premiums	int_credited	bonus_credited	claims_death	claims_lapse	claims_commutation	annuity_payments	net_cf
20	1.000000	61,190.90	3,200.00	2,592.00	1,747.81	81.60	259.29	2,052.23	0.00	0.00	210.52
21	0.965315	63,039.53	3,167.78	2,502.10	1,796.18	80.78	283.18	2,104.86	0.00	0.00	45.22
22	0.931471	64,758.71	3,134.66	2,414.37	1,841.04	79.93	308.66	2,153.51	0.00	0.00	−115.50
23	0.898434	66,348.31	3,100.58	2,328.74	1,882.41	79.06	335.83	2,198.15	0.00	0.00	−271.80
24	0.866171	67,807.93	3,065.46	2,245.12	1,920.26	78.17	364.77	2,238.75	21,974.56	0.00	−22,428.44
25	0.584254	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	2,343.02	−2,372.27
49	0.342195	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	1,337.26	−1,365.93
78	0.000000	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	−0.00
Total	—	—	—	12,082.32	9,187.70	399.55	1,551.74	10,747.50	21,974.56	60,594.20	−84,230.73

Two columns appear here that the anchor table omits, int_credited and bonus_credited, because their ratio is the point. Over the five remaining accumulation years the contract is credited 9 187,70 € of guaranteed interest and 399,55 € of surplus — and every euro of that surplus is the declared 2,55 % paid on the Ansammlungsguthaben’s own balance. Not one cent is interest surplus on the Deckungskapital: bonus_rate = max(0; 2,55 % − 2,75 %) = 0 in every policy year, because a contract already guaranteed more than the insurer is declaring receives no interest surplus at all. A model with one global Rechnungszins cannot produce that row; a model that adds the declared rate to the guarantee produces its opposite. The same cell shows the other legacy asymmetry: a garantierter Rentenfaktor of 34,00 € struck on 2005 bases beats the current 32,00 €, so the max(garantiert, aktuell) resolves to the guarantee; and the 40 ‰ charge set carries no Stornoabzug, so cv_pp(20) is the undeducted 68 586,25 € against a § 169 Abs. 3 floor of 65 304,65 €, inoperative twenty years in.

What changed in these notes, and why

Five statements written before the model existed did not survive contact with it, and were corrected here rather than worked around in the model. (i) pup_uplift moved one row earlier, to the transition year k = pup_year − 2: the fund-level roll-forward needs it there, or check_av_roll_fwd() fails by the whole uplift. The amount is unchanged. (ii) Pitfall 5 no longer asserts that the surrender claim in the paid-up row is zero: a beitragsfrei contract keeps its § 168 VVG Kündigung right, so ordinary surrender continues after a Beitragsfreistellung — 764,60 € in policy year 10 on point 7 — and what distinguishes the election from a lapse is that the conversion itself moves no policy. (iii) Pitfall 1’s and pitfall 11’s magnitudes were wrong for this parameterisation and now carry the model’s own figures: the double-credit error is 8,5 % of the accumulated value rather than “more than half” of the Deckungskapital, and the anchor’s net amount at risk rises to 4 587,95 € at k = 5 and ends the deferment at 3 204,24 € rather than falling towards zero. (iv) Pitfall 15’s “more than a fifth” was wrong, and wrong in an interesting direction: re-running point 6 on a global 1,00 % rate moves its Deckungskapital at Rentenbeginn by −7,7 % and its Ansammlungsguthaben by +156 %, while the conversion capital moves by −0,8 %. The vintage error is a misallocation between the two accounts, not a hole in the total, which is why it survives a reasonableness check on the headline figure. (v) Pitfall 12 overstated the payout-phase expense: expenses(t) there is the inflated expense_annuity_pp on the exposed count plus the expense_claim_pp settlement cost of that month’s deaths, which is a small percentage of the line in the first payout year and grows with mortality down the tail.

---

Valuation and reserve pointers

This library publishes gross best-estimate-style liability cash flows, undiscounted, on a declared grid. The valuation layers consume them and are cited, never reproduced.

• The German statutory Deckungsrückstellung. The HGB reserve of § 341f HGB, computed prospectively on the Rechnungsgrundlagen of the premium calculation REG-R54 and discounted at no more than the § 2 DeckRV rate applicable when the contract was concluded REG-R14. It is not the Solvency II best estimate: an insurer carries two liability measures, and the Überschussbeteiligung, the Zinszusatzreserve and the § 139 VAG Bewertungsreserven test all run on the HGB side REG-R14 REG-R54. av_pp(t) × pols_if(t) is this model’s contribution to that line, not the line itself — a statutory Deckungskapital is a prospective reserve on first-order bases while av_pp is a retrospective account roll-forward, and the two coincide only under assumptions this model does not impose.

• The Zinszusatzreserve. Arises where the § 5 Abs. 3 DeckRV Referenzzins falls below a contract’s tariff rate, and the § 12 MindZV Sicherungsbedarf test compares a Bundesbank month-end swap rate with the highest Rechnungszins applicable to the contract over the next fifteen years — a window that bites hardest on annuity business REG-R17 REG-R18. Model points 6 and 14, on 2,75 % and 0,90 % vintages, are exactly the cells that would carry one. Not computed.

• The surplus layer. The MindZV’s 90 / 90 / 50 minima are a minimum transfer to the RfB, not a minimum payout REG-R18 REG-R10 REG-R19. This model represents the credited outcome — decl_rate and the Ansammlungsguthaben — not the three result sources that fund it; a model of the surplus chassis itself belongs in delib’s kapitallebensversicherung.

• Solvency II. Best estimate plus risk margin REG-R6, BEL = Σ_t v(t) × liability_cf(t) over the stream this model publishes, with the future discretionary benefits — the surplus credit and the Bewertungsreserven crystallisation — the substance of the calculation. No risk-free curve, cost-of-capital rate or contract-boundary rule in this library was read from a retrieved instrument REG-R2 REG-R4, so every such figure would be std. And the guarantees are options: the max(guaranteed, current) Rentenfaktor [S4] is a written option on the insurer’s own future annuity tariff and the Rechnungszins floor a written interest guarantee, neither of which the deterministic path prices. A stochastic-on-deterministic run — this recursion, the crediting rule and the conversion rule re-evaluated per scenario — is what a time-value-of-options-and-guarantees calculation consumes.

• IFRS 17 and professional standards. A profit-participating deferred annuity would be measured under the variable fee approach REG-R55, on this same fulfilment-cash-flow engine; actuarial work sits under the DAV Fachgrundsätze and the § 141 VAG Verantwortlicher Aktuar, distinct from the MaGo’s versicherungsmathematische Funktion REG-R56 REG-R11 REG-R21.

---

Key sensitivities and model risks

In rough order of leverage on a German deferred-annuity block.

1. The Rentenfaktor, and the fact that it is not calibrated to the shipped mortality table. The annuity amount is K / 10 000 × f, so the whole payout phase scales linearly with f, and f is std. Market anchors now exist and the shipped values sit above them: 2025 averages of 24,33–27,18 guaranteed and 27,27–30,40 current by deferment term R24, and a current-factor average of 25,97 for 2022 R19, against a shipped base current factor of 32,00 at age 67. A user recalibrating to market would lower it, and the payout phase scales linearly with the change. The reference library warns that a model publishing a std Rentenfaktor and a std annuity table must say whether the two are consistent and which is authoritative REG-R49. They are not calibrated to each other, and the Rentenfaktor is authoritative: it fixes the benefit amount, while the mortality proxy fixes only how long that amount is paid. The model publishes annuity_due_factor() — the annuity-due present value on the shipped proxy at the guarantee interest basis — purely as a diagnostic, so the gap is visible rather than hidden. Anyone substituting a real DAV 2004 R must re-strike the Rentenfaktoren with it or accept an inconsistency the model will not flag.

2. The declared rate and the guarantee vintage together. bonus_rate = max(0, decl_rate − int_rate_guar) is a difference of two numbers of similar size, so a 25 bp move in decl_rate moves the Ansammlungsguthaben’s accrual by about 16 % on the 1,00 % vintage and by all of it on the 2,75 % vintage, where the rate is already clipped at zero. The declared path is a level std scenario, not a forecast; the market’s own 2026 averages disagree with each other by 33 bp REG-R53, and one carrier’s actual 2026 declaration is 3,00 % [S15] — 45 bp above the shipped path, which on the 1,00 % vintage is a 29 % change in bonus_rate.

3. Mortality, in two dimensions. For a deferred annuity the improvement trend matters more than the level, because the conversion happens decades out: on the anchor cell the annuitant reaches 67 in 2043, 38 improvement years after the proxy’s 2005 base. Both q_base and improve are std, and the trend is the more dangerous of the two — the German construction uses a Starttrend converging to a weaker Zieltrend and the proxy uses a single age-graded rate, which is a documented simplification and not a replication REG-R49.

4. The surrender assumption, and the option it ignores. Cumulative surrender over the anchor’s seventeen accumulation years is material, and every surrendering policy forfeits a guaranteed Rentenfaktor struck on bases that will look generous by 2043. The model’s lapse rates are unconditional; a policyholder who valued the forfeited option would surrender less. The direction of the error is therefore known and one-sided.

5. The Kapitalwahlrecht take-up rate. At 30 % it removes nearly a third of the annuity block at t = 16 and replaces it with a single payment. It is a pure std with no evidence (gap 20), and it substitutes for a tax comparison the model does not perform R6 REG-R45.

6. The charge set, and the § 169 Abs. 3 floor it collides with. alpha_rate at the statutory ceiling makes the year-one Sparbeitrag small and the early Deckungskapital correspondingly thin, but the floor then reverses most of that for surrender purposes REG-R16 REG-R28, so the two parameters must be moved together.

7. The annuity’s timing, now that the grid can carry it. The model pays the instalment monthly in advance, which is what the Rentenfaktor quotes and what the wordings describe — “Wir zahlen die Rente monatlich, jeweils im Voraus”. Neither the exact payment date within the month nor the in-advance/in-arrears basis was established [S13] R24 (gap 19), so the in-advance reading is still std; what is no longer a modelling artefact is the frequency. The annual grid this model ran on compressed the twelve instalments into one start-of-year payment, which was generous to the payout phase by roughly half a year’s interest on one year’s annuity and by a full year of survivorship on instalments a decedent did not live to collect: on the anchor cell that compression was worth 369,14 €, 1,6 % of the payout phase, and the whole of it sat outside the Rentengarantiezeit, where the count is fixed and the two grids agree to the cent.

8. Everything the model does not do. No Bonusrente, no Zuzahlung, no survivor’s annuity, no § 163 VVG adjustment, no Zinszusatzreserve, no MindZV allocation, no Sicherungsbedarf test, no tax. Each is named where it belongs above; together they are the reason this is a mechanics demonstration and not a valuation.