The RV_DE_S Model

Reference liability cash flow model for the klassische aufgeschobene private Rentenversicherung.

RV_DE_S is the executable counterpart of products/klassische_rentenversicherung/technical-notes.md in the lifelib-products library. It projects gross best-estimate liability cash flows, undiscounted, for a single-policy model point of the German Schicht-3 deferred annuity written on the general account — premiums accumulate in the Deckungskapital at the contract’s own Rechnungszins with a declared Überschussbeteiligung beside it, and at the Rentenbeginn the accumulated capital converts at a Rentenfaktor into a lifelong monthly Leibrente or is taken as a lump sum under the Kapitalwahlrecht. The grid is monthly and the contract is mostly not: the Rechnungszins is credited per Versicherungsjahr, the surplus is declared per calendar year, and both Kündigung and Beitragsfreistellung take effect “for the end of the current insurance period”, so those constructions take a policy year k while the in force, the decrements, the claims and the Rente itself take a month t. The annuity is the reason the grid is monthly: the Rentenfaktor is quoted in euro a month and the Rentengarantiezeit guarantees monthly instalments, and an annual grid could only pay a year of them at once.

Four things make this the German deferred-annuity model rather than a translated endowment.

The declared rate contains the guarantee; it does not sit on top of it. The laufende Verzinsung is the Garantieverzinsung plus the laufende Zinsüberschussbeteiligung, so bonus_rate(k) = max(0, decl_rate(k) - int_rate_guar()) and the two credits together deliver the declared rate and never more. On the anchor cell that is 1,00 % guaranteed plus 1,55 % surplus against a 2,55 % declaration. On model point 6, a 2,75 % legacy vintage against the same declaration, bonus_rate is zero in every policy year while interest is still credited at 2,75 % — a real German result and the first listed modeling pitfall, not an artefact. check_av_roll_fwd() and check_av_sur_roll_fwd() keep the two accounts honest about which credit went where.

The *Rechnungszins* is a model-point attribute, not a global assumption. A German life book is a layered stack of guarantee vintages: the rate a contract was written on stays with it for its whole life, so points 1, 6 and 14 credit 1,00 %, 2,75 % and 0,90 % in the same run, from the same tables. Anything that reads a single interest rate off the model has misunderstood the product.

The conversion is an option the insurer wrote. At the Rentenbeginn the applied Rentenfaktor is max(garantierter, aktueller) — the factor fixed at inception against the one the carrier is applying to immediate annuities at that date — and the higher of the two is then guaranteed for the whole payment period. On the anchor cell the current factor wins at 32,00 € against a guaranteed 28,00 €; on point 13 the guarantee binds over a low scenario. A model applying the guaranteed factor alone understates the anchor’s annuity by 12,5 %. check_annuity_conv() asserts the rule and the guaranteed-contract- value floor beside it.

The *Rentengarantiezeit* is paid to the dead. Inside the guarantee window the instalment is due whether or not the annuitant is alive, so the annuity is weighted by the annuitised count and not by survivors: pols_annuity(t) = max(pols_if(t), 1{12n <= t < 12n + 12m} pols_annuitization(12n - 1)), asserted by check_annuity_guarantee() on every model point. On a monthly grid the window is what it says it is — 12m guaranteed instalments.

Spaces. The model contains two:

Data

Reads the eight input CSVs and holds their filename References. It takes no parameters, so each file is read once per model.

Projection

The by-policy projection, parameterized by point_id: Projection[1] is an ItemSpace projecting model point 1. It reaches the input tables through its data Reference, which resolves to the single Data Space.

The split matters for more than tidiness. Because Projection is parameterized, every Projection[N] is a separate ItemSpace with its own cells cache; readers placed there would re-read every file for every policy. In Data they are evaluated once, however many policies are projected.

Input data is external: plain CSVs in the model folder’s parent directory, read at run time rather than stored inside the model. The model folder itself holds no data — no _data/, no IOSpec, no embedded values — so the model and its inputs must travel together. This follows annuallife.TradLife_A; contrast basiclife.BasicTerm_S, which keeps its inputs inside the model.

Projection basis. Monthly steps over an annual product, so the model runs on two clocks and the argument of a cells says which. t counts policy months from inception and is 0-based; duration(t) = t // 12 is the 0-based policy year k, policy_year(t) = duration(t) + 1 the contractual 1-based label, and both the attained age and the calendar year step on the anniversary — age(t) = issue_age + duration(t), calendar_year(t) = issue_year + duration(t) — which is what lets one generational mortality surface and one declared-rate path serve a book of mixed vintages. A new-business point opens at t = 0; an in-force point that has run duration_init complete policy years opens at t = t_start() = 12 duration_init carrying its balances on the model point. proj_len() = 12 x proj_len_y() with proj_len_y() = omega_age() - issue_age is the exclusive end of the frame, so a life annuity is projected to exhaustion rather than truncated at a fixed horizon — on the anchor cell, t = 0 ... 851. The Rentenbeginn falls at the end of the deferment period of n = aufschub_y years: accumulation months are t < 12n, payout months t >= 12n, the Kapitalabfindung is paid in month 12n - 1 and the first annuity instalment in month 12n.

The decrements carry the library’s two speeds — mort_rate(t) and lapse_rate(t) are the annual rates of the policy year, mort_rate_mth and lapse_rate_mth the geometric twelfths the recursion applies — so the whole annual layer is bit-identical to the annual-step model this replaced, on all fourteen model points: every account balance, the § 169 floor, the surrender value, the conversion capital and the premium income are unchanged. What the finer grid changed is the annuity, which is now paid monthly in advance; the split of exits between death and surrender, which now compete month by month; and the expenses, which a mid-year leaver now bears only for the months it was there.

What is sourced and what is not. The contractual mechanics are sourced: the Deckungskapital as the premium net of risk and expense cover accumulated at the Rechnungszins; the Höchstzillmersatz of 25 ‰ from 2015 and 40 ‰ before; the § 169 Abs. 3 surrender floor that spreads acquisition costs over five years; the § 169 Abs. 5 Stornoabzug conditions; the § 165 paid-up rule and its minimum-benefit branch; the three death-benefit designs; the conversion rule and the max(guaranteed, current) Rentenfaktor; the Bewertungsreserven crystallisation at the transition to annuity payment; and the Rentengarantiezeit. Every level is a standardization. No Rentenfaktor, no declared surplus rate, no charge parameter, no expense and no behavioural rate was established for this product at any German carrier for any year, and the DAV tables (DAV 2004 R here) are the property of the Deutsche Aktuarvereinigung, are not public and are cited by name rather than redistributed. This model is a mechanics demonstration, not a pricing or reserving result. Replace the decrement, charge and rate tables with company data before drawing any conclusion from the output.

Model points. Fourteen, covering 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 the surplus account, all three payout systems, five Rentengarantiezeit durations including zero, Kapitalwahlrecht take-ups of 0 %, 20 %, 30 % and 100 %, the Dynamik, both statutory Beitragsfreistellung branches, and the boundary cases: the paid-up conversion that fails the Mindestversicherungsleistung and is cashed out (8), full commutation at Rentenbeginn (9), and the guaranteed Rentenfaktor binding over a lower current one together with a binding guar_capital_pp (13). Model point 1 is the anchor cell of the worked example in the technical notes.

Verification. tests/test_klassische_rentenversicherung_de.py asserts the notes’ worked example to the cent off result_cf_annual() and pols_if to six decimals, the months of the first policy year on the monthly frame beside it, and one test per listed modeling pitfall. Nine check_* identities travel with the model itself and are called on every model point by tests/test_model_conventions_de.py; six of them are monthly and three — the two account roll-forwards and the premium split — are stated per policy year, because the accounts they check are.

Example

>>> import modelx as mx
>>> model = mx.read_model("products/klassische_rentenversicherung/RV_DE_S")
>>> model.Projection[1].result_cf()