The KLV_DE_S Model

Reference liability cash flow model for the German kapitalbildende Lebensversicherung.

KLV_DE_S is the executable counterpart of products/kapitallebensversicherung/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 classic German endowment — the gemischte Versicherung auf den Todes- und Erlebensfall, which pays a guaranteed Erlebensfallleistung at the Ablauf if the insured is then alive and a guaranteed Todesfallleistung on earlier death, both increased by the Überschussbeteiligung — on a monthly grid, over a product whose every contractual mechanic stays on the annual clock it is written on.

This is the Überschussbeteiligung chassis the other nine delib products reuse in modified form, and three things make it the German model rather than a translated one.

The surplus rate multiplies a reserve, not a sum insured and not a premium. The declared laufende Verzinsung is the Garantieverzinsung plus the laufende Zinsüberschussbeteiligung, so a declared 2,70 % on a 1,00 % guarantee is a 1,70 pp credit and never 2,70 pp on top of 1,00 pp: zins_ueberschuss_rate(t) = max(0, decl_rate(t) - rechnungszins()), applied to max(res_pp_at(t, "AFT_INT"), 0). Both max are load-bearing. A gezillmerte Deckungskapital is negative at issue, so a positive rate on an un-floored base would credit a negative surplus; and on the nil scenario the declared rate falls below the guarantee, which the reserve still meets in full, so the credit is zero rather than negative. How long the base stays negative is a parameter question rather than a structural one: at the post-2015 25 ‰ ceiling over a long Beitragszahlungsdauer the first Zillmer premium more than repays the zillmered cost, and the floor is inert on every shipped model point; at the pre-2015 40 ‰ ceiling it is not.

There are three reserves and the product needs all three. res_zill_pp is the gezillmerte Deckungskapital the insurer holds, negative at issue and equal to -alpha_cost there; res_min_pp is the § 169 Abs. 3 VVG floor obtained by spreading the acquisition cost evenly over the first five contract years, and on a long gezillmert contract it normally binds; res_guar_pp is their maximum and is what the customer actually gets. Publishing only the first understates the surrender value at essentially every duration; publishing only the second loses the quantity the Deckungsrückstellung and the beitragsfreie Versicherungssumme are built on.

Making the contract paid-up is not a lapse. § 165 VVG converts the contract to a reduced beitragsfreie Versicherungssumme bought with the § 169 value — the policy stays in pols_if — unless the resulting sum falls below the agreed Mindestversicherungsleistung, in which case the statute obliges the insurer to pay the § 169 value instead and the election becomes a surrender. Model points 11 and 12 exercise the two branches.

Spaces. The model contains two:

Data

Reads the seven 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, the worked example’s anchor cell. 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: 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, so the model and its inputs must travel together.

Projection basis. Monthly steps, over an annual product. t is the policy month, 0-based and counted from issue, and the frame runs t = t_start() ... proj_len() - 1 contiguously with t_start() = 12 x duration_init() and proj_len() = 12 x policy_term() — the frame’s exclusive end, with the Ablauf at the end of the last month. There is no t = proj_len() row.

The model carries two clocks and the argument of a cells says which. Cells that state an annual account take k, the 0-based policy year: the whole pricing block, all three reserves, the § 169 value, the paid-up purchase and every part of the Überschussbeteiligung. Cells that state a month take t: the in force, the claims, the premium instalments and every result_cf() column. duration(t) = t // 12 is the bridge, policy_year(t) = duration(t) + 1 the contractual 1-based label, and is_anniv(t) = (t % 12 == 11) the month the annual machinery acts in. The decrement rates take t and return the year’s annual rate; mort_rate_mth and lapse_rate_mth are what the recursion applies, at 1 - (1 - r)^(1/12), so twelve of each compound back to the year’s rate and pols_if at every anniversary is what an annual-step model carries. That is what leaves the whole annual layer — every reserve, every declared credit, every ledger balance and the Bruttobeitrag itself — bit-identical to the annual-step model this replaced.

A Beitrag instalment on the Zahlweise’s own cycle and a twelfth of the maintenance expense fall at the beginning of the month; the guaranteed Deckungskapital rolls forward over the policy year at the Rechnungszins; the surplus is declared and credited at the anniversary on that year’s closing reserve; death, maturity and surrender fall at the end of the month, surrender after the mortality decrement, and each is paid the balances standing at the end of that month — the year’s own closing figures in an anniversary month, and the ones struck at the last anniversary in the other eleven.

What is sourced and what is not. The contractual mechanics are sourced: the surplus rate as a percentage of the Deckungskapital at the allocation date and the allocation at the Bilanzstichtag; the Rückkaufswert as the Deckungskapital on the Rechnungsgrundlagen der Prämienkalkulation struck at the end of the current Versicherungsperiode and floored by the five-year spreading; the Stornoabzug biting on the guaranteed value alone; the Beitragsfreistellung test of § 165 VVG; the § 161 VVG substitution of the Rückkaufswert for the sum insured on a suicide inside three years; the cessation of premiums on death; and both DeckRV cohort ceilings. Every behavioural and experience assumption is a standardization: no German insurer publishes a mortality basis, an expense loading, a commission scale, a terminal-bonus rate or a lapse rate for this product, and the DAV tables — DAV 2008 T 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, surplus and expense tables with company data before drawing any conclusion from the output.

Model points. Fourteen, covering both premium forms, all four payment frequencies with the echte and unechte readings of a sub-annual one, all three Überschussverwendung systems, an in-force 2012 cohort on a 1,75 % guarantee opening at t = 168, a successful and a failing Beitragsfreistellung, a non-gezillmert tariff, and a short unequal-sums contract at a Risikozuschlag on the nil surplus scenario. Model point 1 is the anchor cell of the worked example in the technical notes.

Verification. tests/test_kapitallebensversicherung_de.py asserts every row of the notes’ twenty-five-year worked example to the cent — on result_cf_annual() — and the twelve months of policy year 1 on the monthly frame beside it, pols_if to six decimals, and one test per listed modeling pitfall. Ten check_*() cells close on every model point, among them check_net_cf() — this library’s first ruling — and check_res_roll_fwd(), the Fackler recursion that proves the premium, the first-order mortality, the interest and the prospective reserve formula are mutually consistent.

Example

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