Implementation Notes

Status: Draft, 2026-08-29. Built from technical-notes.md; the product it implements is specified in product-spec.md.

This is a mechanics demonstration, not a pricing or reserving result. The mechanics are the established German ones and each carries the instrument it must be checked against — the benefit trigger being the statutory Pflegegrad of §§ 14, 15 SGB XI rather than a definition the insurer writes R2 R6, the five-grade Leistungsstaffel scaling one vereinbarte Pflegerente and the Beitragsbefreiung im Leistungsfall [S4], the level Beitrag a Lebensversicherer may adjust only on the narrow § 163 VVG route and never under § 203 VVG R11 REG-R27, the 1,00 % Höchstrechnungszins and 25 ‰ Höchstzillmersatz of the DeckRV R13 REG-R14 REG-R16, the § 169 VVG Rückkaufswert with its five-year cost spread and its Stornoabzug conditions R11 REG-R28, and the unisex pricing rule REG-R34. Every level is a standardization. This model was drafted with nothing retrieved — egress from the build environment was blocked and the session’s search budget was exhausted before the product was researched — and its sources have since been re-verified against the primary documents: one carrier’s Bedingungswerk and Produktbeschreibung were read in full [S4], as were the statutes and the DAV’s own reports, and 27 of the 36 entries in sources.md now read Retrieved: yes. What was not retrieved is the one thing that would fix a level: no Tarifblatt and no premium quotation for any German Pflegerentenversicherung, because none is published [S9] [S13]. DAV 2008 P, the German market’s standard multi-state Pflegetafel, is published by the Deutsche Aktuarvereinigung — the earlier statement here that it is not public was wrong — but it is the DAV’s property and is not redistributed here R15 REG-R51, and neither is DAV 2008 T or DAV 2004 R R16 REG-R48 REG-R49. So every biometric rate, every charge, every lapse rate and the premium itself is std, and the premium is an output of a stated first-order basis rather than a table lookup. Replace the decrement, expense and surrender tables with company data before drawing any conclusion from the numbers.

Run it

python products/pflegerentenversicherung/run.py        # the anchor cell
python products/pflegerentenversicherung/run.py 5      # the statutory bahr Leistungsstaffel
python products/pflegerentenversicherung/run.py 12     # an in-force policy already in claim

import modelx as mx
mx.read_model("products/pflegerentenversicherung/Pflege_DE_S").Projection[1].result_cf()

Projection takes a point_id; Projection[1] is the worked-example anchor cell. result_cf() returns a tidy DataFrame indexed by policy month t carrying pols_if first, the three-way split of it a reader follows the projection with, and one column per cash flow line; result_states() publishes the five Pflegegrad ledgers, the Karenz ledger, the flows between them and the four annual rates beside it. model.Projection.doc holds the full symbol mapping and model.Data.doc says what a replacement for each shipped table must preserve.

The grid is monthly and t is 0-based: t = 0 is the month of issue and age(t) = age_at_entry + t // 12, so the attained age steps at the policy anniversary. The frame starts at duration_mth_init() — 0 for new business, the elapsed duration for an in-force point — and runs to proj_len() - 1, where proj_len() is the number of projected months, the frame’s exclusive end counted from t = 0: 12 × (110 − 45) = 780 on the anchor cell, so range(0, 780) — 780 rows, t = 0 … 779, attained ages 45 to 109, in about seven seconds. It depends on the entry age and the terminal age alone, so an in-force point opening at duration_mth_init() = d0 publishes a shorter frame of proj_len() - d0 rows and still ends at its own proj_len() - 1: duration_mth_init shortens the frame at the front, never at the back — and since proj_len() depends on the entry age, that last index is the point’s own, not the anchor’s 779. Reading proj_len() as the last index, or as a horizon duration_mth_init shifts, is a listed pitfall.

The input CSVs carry no column keyed by t. lapse_table.csv and surrender_table.csv are keyed by policy_year, the 1-based contractual Versicherungsjahr that the model reaches through policy_year(t) = t // 12 + 1; mort_table.csv and incidence_table.csv are keyed by attained age; and model_point_table.csv’s duration_mth_init is an elapsed count of complete months, 0-based by nature. wartezeit_months and karenz_months are contractual lengths in months, not frame indices, so they do not shift either: the gate stays t < wartezeit_months, and month wartezeit_months is the first covered month. None of them moved with this change.

Nine states, and only two of them absorbing

This is what a reader arriving from RLV_DE_S, BasicTerm_S or any single-decrement protection model will get wrong, and it is why the product is worth modelling. A Pflegerente is a multi-state contract whose benefit is a step function of the state, not a cover paying on an event:

   aktiv ──►  PG1  ⇄  PG2  ⇄  PG3  ⇄  PG4  ⇄  PG5
     │  ▲      │        │        │        │       │
     │  └──────┘  Reaktivierung / Herabstufung     │
     ▼           ▼        ▼        ▼        ▼      ▼
   storno                    tot   (absorbing)

The model carries the Pflegegrad explicitly, in pols_pg(t, g), with a second ledger pols_karenz(t, g, z) for lives inside a deferred period and a third, pols_act(t), for active lives. Every transition above is internal to pols_if — lives leave the in-force population only by death or surrender — which check_pols_roll_fwd() and, independently, check_states() assert. Two consequences drive the implementation.

The paying state has three exits and only death is absorbing. A life in Pflegegrad g can die, deteriorate to g + 1 or be downgraded to g − 1; out of grade 1 the downgrade is a Reaktivierung back to the active state, where the life resumes paying its Beitrag and becomes exposed to lapse again. A model that lets the paying state be exited only by death overstates the liability; one that treats every downgrade as a termination understates it, and both errors produce a plausible-looking frame. On the anchor cell the Reaktivierung flow is 0,013768 of a policy over the whole projection and the Herabstufung flow out of grade 2 — the flow that ends an annuity and revives a premium on the delib_std grid — is 0,020835: both small, both non-zero, which is why the tests assert the flow and never a non-monotone stock.

The transitions are allocated, not added. Every shipped rate is annual; every month is stepped with forces held constant over the month, the competing transitions sharing one survival probability in proportion to their forces — p_stay = exp(−Σμ/12) and p_j = (μ_j / Σμ) × (1 − p_stay) — so p_stay + Σ p_j = 1 exactly, by construction. That is what makes check_states() an identity rather than an approximation, and why p_pg_stay, p_pg_death, p_pg_worse and p_pg_better are four published cells rather than folded into the recursion. Adding monthly rates, or applying q/12, differs wherever the forces are large — which on this product is exactly where the money is.

Grade and mortality are correlated, and the highest-paying state is the shortest-lived

In-care mortality is not tabulated. mort_force_care(t, g) is mort_mult(g) times the force of active mortality at the same age — 1,5 at Pflegegrad 1 rising to 9,0 at grade 5 — and mort_rate_care(t, g) is 1 − exp(−mort_force_care(t, g)). The multiple is on the force and not on the rate, and that is load-bearing rather than cosmetic: on forces the ratio is exactly mort_mult(g) at every age below the limiting one, while on rates it compresses towards 1 as the rate saturates — at grade 5 the rate ratio falls from 8,972 at age 45 to 6,854 at 85, 4,308 at 95 and exactly 1,000 at the limiting age. That compression is the scale, not the basis, and a model stating its LTC mortality loading as a rate multiple has a different basis at every age.

What it buys is the product’s central fact: the annuity in payment is short, three to five years rather than the fifteen to twenty a healthy-life annuity would run at the same age, so pricing it on an annuity table — DAV 2004 R is prudent about people living longer R16 REG-R49 — would be prudent in exactly the wrong direction. And because grade and mortality are correlated, claims(t, "ANNUITY") is a grade-by-grade sum, R × Σ_g π_g × esc_pg(t, g), never an average percentage on an average survival curve: applying the entry-mix mean of 0.3815 to Σ_t pols_care(t) = 24.241784 gives 9 248,24 € against the model’s 13 200,11 €, a 30 % understatement of the whole benefit no total would reveal. The substitution is exact at t = 1, where the stock still is the entry mix; the error opens as deterioration moves the stock to a stock-weighted mean of 0.544519.

The Wartezeit and the Karenzzeit are different devices

They are routinely conflated in consumer material and they are implemented in two different places.

The Wartezeit runs from inception and denies cover: inc_force(t) is exactly zero while t < wartezeit_months(), and inc_rate(t) is left alone so it stays the tariff-comparable table rate at every age — a gate on the force, one line, no ledger.

The Karenzzeit runs from onset and defers an admitted claim, so it is a clock per onset rather than a gate on the aggregate — which is why it needs its own ledger dimension, pols_karenz(t, g, z) with 1 ≤ z ≤ K. Lives in it are subject to the same transitions as a served life: they die, deteriorate and recover exactly as if the annuity were running, they simply are not paid, and the clock is discarded on reactivation, because a recovered life who relapses starts a new onset. Where karenz_months() == 0 the ledger is empty and pols_grad degenerates to pols_entry, which is the base run. The cost of the device is the gap between the two: over model point 7’s projection graduations are 0.228177 against entries of 0.253723, 89,9 %, the shortfall being deaths and recoveries recorded inside the six-month deferral — larger than six months of a four-year spell suggests, because mortality is highest immediately after onset. The model uses an aggregate in-care mortality with no select period after onset, so it understates how much a Karenzzeit removes, and point 7’s reduction is a floor rather than an estimate.

The waiver runs with the Leistungsstaffel, not with the diagnosis

pols_waived(t) is the Beitragsbefreiung population and it is not everyone in care: it is Σ_{g : waiver_flag(g)} pols_pg(t, g), restricted to the premium term, with waiver_flag(g) being benefit_pct(g) > 0. Three consequences fall out of the benefit schedule, each a distinct way to get the premium stream wrong:

• a life inside its Karenzzeit is not waived, because no annuity is yet payable;

• a Pflegegrad 1 life is not waived on delib_std, where π_1 = 0, and is waived on bahr, where it is 10 % — the same life, two schedules, opposite answers;

• a life downgraded out of the insured grades leaves the waived population and starts paying again, so pols_prem is structurally non-monotone.

Wiring the waiver to membership of the care ledger instead gets all three wrong at once and still closes every count. check_waiver() asserts the split — pols_prem + pols_waived = pols_in_term — and is arithmetically trivial while pols_prem is a difference; it is published because the failure it guards is not a slip in the subtraction but a disagreement about who belongs on which side, so it is read with the tests that assert the membership itself. The waiver is also in the price, through tar_pols_prem(t): on a contract issued at 45 and claiming at 82 it removes four years or so of remaining Beitrag, and that cost sits inside the level premium.

Two ledgers for one population. esc_pg(t, g) is the escalation-weighted counterpart of pols_pg(t, g): the identical recursion with one extra factor of (1 + d)^(1/12) on the surviving weights, entrants joining at weight 1. Carrying the Leistungsdynamik as a value ledger rather than as a duration-since-onset cohort dimension keeps the model O(n) instead of O(n²); the price is that it reports only the aggregate escalation, which is all the cash flow needs. The annuity is weighted on esc_pg and never on pols_pg — using the head count would silently drop the escalation on every model point that carries one, and no total in the frame would look wrong. With leistungsdynamik = 0 the two ledgers are identical at every t and g, which check_esc_ledger() asserts as an equality on the base run and as a domination (esc_pg ≥ pols_pg) wherever the dynamic is positive.

The dynamic costs more than a four-year spell suggests. On model point 8, d = 2 % raises the projected annuity total from 13 200,11 € to 15 101,44 € — +14,4 % — and the equivalence premium from 64,198409 € to 72,038378 €, +12,2 %: ln(1.144)/ln(1.02) = 6,8 years of payment-weighted elapsed duration against a mean spell in an insured grade of 5,45 years, because the escalation compounds over elapsed time in care, Pflegegrad 1 months included where nothing is paid, and because deterioration puts the largest benefit percentages at the end of a spell.

The Beitrag is a priced quantity, and the pricing engine is a separate ledger

The library publishes undiscounted cash flows. The Beitrag is nevertheless a priced quantity, so the model carries a second, self-contained actuarial-value engine — the tar_* cells — whose only output is premium_mth_pp(). That engine discounts; the projection does not, and rechnungszins() and disc_factor(t) are read by nothing else. Where premium_mth is positive on the model point that is the premium and the engine is never consulted, which is how model points 10, 11 and 12 carry the premium they were actually sold at. Where it is 0.0 — a sentinel, not a free contract — P is struck by equivalence on the first-order (erster Ordnung) bases: every rate times its prudence margin, the sexes blended at unisex_mix_male = 0.50 because sex may not enter a German premium REG-R34, and no lapse at all, which is both German first-order practice and what keeps the model acyclic. Everything that scales with P is linear in it, so

P·U = A + P·D1 + P·a1 + β·P·U + G + C   →   P = (A + G + C) / [U(1 − β) − D1 − a1]
    = (17,789.761930 + 892.884210 + 69.389246) / (313.500018 x 0.970 − 12.000000)
    = 18,752.035386 / 292.095018 = 64.198409 EUR a month

prem_net_level_pp() = A / U = 56.745649, so the whole expense loading is 13,13 %, of which the Zillmerung allowance a1 = 0.025 × 12 × (85 − 45) = 12.000000 units of P alone is 2,53 € a month: strike a1 out of the denominator and the premium falls to 61,665 €. U is 26,13 years’ worth of discounted premium.

check_prem_equiv() closes the same equivalence from the tariff ledgers month by month rather than from the closed form, which is what makes it a real identity: substituting a best-estimate rate into one leg, dropping the Zillmerung term, forgetting the waiver in tar_pols_prem or valuing the annuity on tar_pols_pg instead of tar_esc_pg all make the sum miss zero. Only the sum is the identity — individual months are large and of both signs — and on the anchor cell it is −9,3e−12. Where the model point supplies its own Beitrag the residual is zero by construction. The Risikozuschlag multiplies the gross premium and never the benefit, so claims is invariant to it: point 13 prices at 283,130286 € against an unrated 188,753524 €, exactly 1,50 ×.

There is no published German rate card for this product to reproduce — the single largest difference between this model and frlib’s TD_FR_S, which reproduces a real attained-age grid. The premium here is computed, and the notes sanity-check its level against an argued 50,00–100,00 € band rather than against a citation.

The Zillmerung is charged on the Beitragssumme, not on the annual premium. acq_expense_pp() is acq_permille / 1000 × beitragssumme(), with the per-mille set exactly at the § 4 DeckRV Höchstzillmersatz of 25 ‰ so the ceiling binds visibly REG-R16 REG-R20. A lifelong-premium contract has no finite Beitragssumme without a convention, and beitragssumme_cap_age = 85 std is that convention — P × 12 × (min(prem_end_age, 85) − age_at_entry), and the Einmalbeitrag itself where there is one. On the anchor cell that is 30 815,24 € and a charge of 770,38 €, all of it at t = 0. Charging the per-mille on an annual premium instead understates it by a factor of the paying term — here forty-fold — and is a listed pitfall. Because the charge falls at t = 0 only, an in-force model point never incurs it: its frame opens at duration_mth_init() > 0 and the cost was incurred before the valuation date, which is worth knowing before comparing an in-force point’s first row with a new-business point’s.

Inputs are external files

The nine input CSVs live in this directory, beside run.py, and Pflege_DE_S/ holds nothing but formulas — __init__.py, _system.json, Data/__init__.py and Projection/__init__.py, no _data/, no IOSpec, no embedded values. This follows lifelib’s annuallife/TradLife_A, which keeps its inputs beside the model; it is the opposite of basiclife/BasicTerm_S, which stores its inputs inside the model through modelx’s IOSpec machinery.

Read once, in Data

Projection is parameterized by point_id, so every Projection[N] is a separate ItemSpace with its own cells cache; readers placed there would re-read every file for every policy. They live instead in an unparameterized Data Space that Projection reaches through a data Reference, so each file is read once per model however many policies are projected, and a test counts the reads against a registered file set. Data.input_dir() resolves the location from _model.path.parent, so it works from any checkout. The trade-off: the model is not portable on its own — copy Pflege_DE_S/ without the CSVs and it reads fine, then fails on first evaluation. What you gain is that a diff shows logic changes only, and an input can be swapped in place: point Data.mort_table_file at another same-schema file and the projection follows.

Reference	Cells	File	Contents and provenance
model_point_file	model_point_table()	model_point_table.csv	Fourteen model points. Point 1 is the worked-example anchor cell (F / entry 45 / aktiv / 1 000 € a month at PG5 / delib_std / lifelong monthly Beitrag derived by equivalence / every option off). Points 2–14 exercise the unisex twin, quarterly, half-yearly, annual and single instalments, the bahr grid, an abgekürzte Beitragszahlungsdauer, a Wartezeit with a Karenzzeit, a Leistungsdynamik, a Beitragsrückgewähr, a supplied premium with a Stornoabzug, two in-force points — one of them already in claim — and both ends of the entry-age band. The one file with no provenance column: a model point is a configuration, not an assumption, and it is the only exemption from delib’s second ruling
benefit_scale_file	benefit_scale_table()	benefit_scale_table.csv	The Leistungsstaffel by schedule and Pflegegrad. delib_std is 0 / 30 / 50 / 75 / 100 % std, the flatter, higher shape a Pflegerente aimed at the residential funding gap tends to use; bahr is the 10 / 20 / 30 / 40 / 100 % grid the Pflege-Bahr market writes std — § 127 SGB XI fixes no percentage schedule, only a Geldleistung at every Pflegegrad with a 600 € floor at grade 5 and a ceiling at the SGB XI benefit level R8 — carried for comparison because a Pflegerente cannot be a geförderter Tarif
mort_table_file	mort_table()	mort_table.csv	Annual active-life mortality by sex, ages 18–109. std Gompertz proxy 1 − exp(−B c^age); not DAV 2008 T and not the DAV 2008 P active-life table R15 R16. The anchors a replacement must preserve are q(65) = 1.35 % and q(85) = 10.5 % male, q(65) = 0.75 % and q(85) = 7.0 % female, with mort_rate = 1.0 at age 109
incidence_file	incidence_table()	incidence_table.csv	Annual incidence into any Pflegegrad by sex and age. std min(I0 exp(g(age − 65)), 0.50) with I0_F = 0.0110, g_F = 0.1400, I0_M = 0.0085, g_M = 0.1380. The slope is anchored on prevalence roughly doubling every five years of age above 75, ln 2 / 5 = 0.1386; the observed Pflegequoten put the ratio nearer 1,8 over 70–90, i.e. about 0.117 R18, which is reported in sources.md and not implemented
care_file	care_table()	care_table.csv	The whole in-care basis in five rows: entry_share 0.20 / 0.38 / 0.24 / 0.13 / 0.05, det_rate 0.28 / 0.24 / 0.20 / 0.16 / 0.00, rec_rate 0.10 / 0.06 / 0.04 / 0.02 / 0.01 and mort_mult 1.5 / 2.5 / 3.5 / 6.0 / 9.0 on the force. All std. entry_share is deliberately not the stock distribution of about 9 / 44 / 27 / 14 / 6 % R18: entrants skew lower than the stock because deterioration moves people up over a spell
lapse_file	lapse_table()	lapse_table.csv	Annual lapse from the active state by policy year 1–40, year 40’s rate applying thereafter: 6,0 / 5,0 / 4,0 / 3,5 / 3,0 % then 2,5 / 2,0 / 1,5 %. std, and no lapse rate for a German Pflegerente at any duration was established; the shape is argued from the Zillmerung, and the 14-day Widerruf sits inside year 1 REG-R23
surrender_file	surrender_table()	surrender_table.csv	The guaranteed Rückkaufswert as a fraction of premiums paid to date, by Versicherungsjahr y(t) = t // 12 + 1, 1–40. std shape encoding two cited facts and no cited level — the 25 ‰ Zillmerung allowance, which is why years 1 and 2 are zero REG-R16, and the § 169 Abs. 3 VVG five-year cost spread, which is why it turns positive in year 3 REG-R28
expense_file	expense_table()	expense_table.csv	acq_permille 25,000 ‰ of Beitragssumme, admin_prem_pct 3,0 %, admin_mth_pp 2,00 € a month, claim_expense_pp 1,50 € per annuity payment, expense_infl 1,5 % a year. All levels std; only the 25 ‰ ceiling the first sits exactly at is cited
basis_file	basis_table()	basis_table.csv	rechnungszins 1,00 %, omega_age 110, unisex_mix_male 0.50, rec_age_ref 75, rec_age_decay 0.10, inc_cap 0.50, beitragssumme_cap_age 85, roll_fwd_tol 1e−10 and the five first-order margins inc_margin 1.25, det_margin 1.15, rec_margin 0.80, care_mort_margin 0.85, act_mort_margin 0.90. Only the Rechnungszins is cited REG-R14 REG-R15; everything else is std

Every file but the model point table carries a provenance column, one tag per row — delib’s second ruling, and it is machine-checked.

What a replacement biometric basis must preserve, whether it is DAV 2008 P under licence or a company table: (a) incidence by attained age, sex and grade of entry, because a stroke or a fracture enters directly at grade 3 or 4; (b) deterioration dominating recovery above age 75; (c) mortality in care as a grade-increasing multiple of active mortality; and (d) transition probabilities out of each state summing, with the stay probability, to one.

The published identities

Six check_*() cells, each a no-argument bool over all t with a per-t residual check_*_resid(t), all scaled by roll_fwd_tol from basis_table.csv.

check_net_cf — delib ruling 1, in one line: net_cf(t) = premiums(t) − claims(t,"ANNUITY") − claims(t,"LAPSE") − claims(t,"DEATH") − expenses(t) − claim_expenses(t).

Every term of that is a column of result_cf(), and the residual re-derives the headline number from the three claims kinds separately rather than from their subtotal, so a benefit that stops being included in claims(t), or a column added to the frame without being subtracted, fails here instead of silently changing the answer. The largest residual in the anchor frame is 1,4e−14.

Check	Identity
check_pols_roll_fwd	pols_if(t+1) = pols_if(t) − pols_death(t) − pols_lapse(t): lives leave the in-force population only by death or surrender, and every Pflegegrad transition is internal to it
check_states	pols_act(t) + Σ_{g,z} pols_karenz(t,g,z) + Σ_g pols_pg(t,g) + pols_dead_cum(t) + pols_lapse_cum(t) = pols_if_init(): the three live ledgers and the two absorbing counts partition the initial cohort at every t
check_waiver	pols_prem(t) + pols_waived(t) = pols_in_term(t): the Beitragsbefreiung splits the in-term population and neither loses nor creates a policy
check_esc_ledger	esc_pg(t,g) ≥ pols_pg(t,g) for every t, g, with equality when leistungsdynamik = 0
check_prem_equiv	Σ_t check_prem_equiv_resid(t) ≈ 0: the gross premium closes the first-order equivalence, re-assembled from the tariff ledgers rather than from the closed form

check_pols_roll_fwd and check_states are not one statement made twice: the first telescopes the three ledgers’ own recursions, the second is assembled by direct summation with no reference to them, so it catches a wrong seeding of an in-force point, a life counted in two grades at once, an entrant into care who never leaves the active ledger and a Karenz cohort that graduates twice. Because mort_rate is forced to 1,0 at the limiting age it also closes at the far end: read one month past the last projected row, at t = proj_len() = 780, pols_dead_cum(780) = 0.493968 and pols_lapse_cum(780) = 0.506032 sum to 1,000000000000.

Modules that are off in the base run

Five constructions are implemented and switched off through the model point, so the base run reproduces the worked example while the machinery stays visible and testable.

Module	Switch	Off value	On at	What it does
Wartezeit	wartezeit_months	0	point 7 (36)	Zeroes inc_force(t) while t < wartezeit_months(), so care beginning inside it is not covered at all. Near-universal in the subsidised Pflege-Bahr product, near-absent in the underwritten one, because the Gesundheitsprüfung does the same screening R8
Karenzzeit	karenz_months	0	point 7 (6)	Populates pols_karenz(t, g, z). With K = 0 the ledger is empty and pols_grad degenerates to pols_entry
Leistungsdynamik	leistungsdynamik	0.00	point 8 (0.02)	Escalates the annuity in payment at (1 + d)^(1/12) a month through esc_pg. Not a Beitragsdynamik, which is not modelled at all
Beitragsrückgewähr	beitragsrueckgewaehr	False	point 9 (True)	Makes claims(t,"DEATH") = cum_prem_max_pp(t) × pols_death(t), structurally zero otherwise, and adds the D1 leg to the equivalence
Stornoabzug	stornoabzug_rate	0.00	point 10 (0.05)	Reduces the Rückkaufswert by a contractual fraction. Zero in the base run because a deduction is admissible only if agreed, appropriate and quantified in the contract R11 REG-R28, and no level for any German Pflegerenten tariff was established

Model point 9 is the option worth reading twice. At a Rechnungszins of 1,00 % a gross return of nominal premiums on a death forty years away is close to the whole premium, and with a lifelong Beitragszahlungsdauer the equivalence’s denominator collapses to 3,6 of 313,5 units. The point therefore pays to age 65, which is how the German market writes such a tariff, and its premium is 622,92 € a month — 9,7 times the anchor’s. The implemented form is the gross one, with no offset for annuity already paid: the market’s commoner form nets the annuity off, but that netting is floored at zero per life and these ledgers are aggregates, so netting in aggregate would let a life that received a large annuity subsidise one that received none. The option therefore overstates the death benefit, and that is stated rather than hidden.

Four constructions the notes describe are not implemented, each for a stated reason. No Überschussbeteiligung in any application form — the surplus chassis belongs to products/kapitallebensversicherung/, and a Beitragsverrechnung here would need a declared-rate assumption this corpus supplies nothing for R11 REG-R24. No Beitragsdynamik, whose acceptance rate on each offer is a behavioural assumption with nothing behind it. No Beitragsfreistellung R11 REG-R28, so every voluntary exit is a surrender. And no § 163 VVG re-rating REG-R27, a management action conditional on emerging experience rather than a projected assumption.

Sign convention

net_cf is income positive — Beitrag in, Pflegerente, Rückkaufswert, any Beitragsrückgewähr and both expense lines out — the notes’ own orientation and the library-wide sign. liability_cf publishes the same stream outgo-positive, liability_cf(t) = −net_cf(t) exactly, and both are columns of result_cf() so the identity is verifiable in the frame rather than only in prose. A Solvency II best estimate is Σ v(t) × liability_cf(t) over the relevant risk-free term structure, plus a risk margin REG-R1 REG-R2 REG-R4; nothing in this library discounts, and rechnungszins appears only inside the equivalence.

The shape to expect on the anchor cell is the product’s economic story in three phases. Month 0 is −710,11 €, almost all of it the 25 ‰ Zillmerung allowance charged in one go — expenses(0) = 770,380907 + 2,000000 + 1,925952 = 774,306859 € against a 64,20 € instalment. From t = 1 the contract runs positive, the level Beitrag far above the risk premium, the monthly margin decaying from 59,93 € to 3,45 € by age 65. net_cf crosses zero between t = 251 and t = 252, attained age 66, and the last three decades are run-off: annuity outgo peaks at 49,82 € in month 407 (age 78) and the population in care at 0.092120 in month 417 (age 79). Undiscounted the contract collects 15 857,95 € and pays 17 385,60 €, for −1 527,65 € — not a loss but the consequence of publishing an undiscounted stream whose income falls thirty years before its outgo. That crossing is where the Deckungskapital this model does not compute peaks, and it is the whole economic content of an ageing reserve on a life chassis.

expenses is acquisition and administration only. The Leistungsbearbeitungskosten are claim_expenses, a separate column because they scale with annuity payments made rather than with policies: a Pflegegrad 1 life on delib_std generates none, and neither does a life inside its Karenzzeit. Its level is low, and that is a product fact rather than optimism — the Pflegegrad is determined by the Medizinischer Dienst or MEDICPROOF and not by the insurer R6, so the Nachprüfung is documentation rather than the adversarial re-assessment that drives a Berufsunfähigkeitsrente’s claims cost REG-R29.

Naming

Cells follow lifelib’s basiclife/BasicTerm_S and savings/CashValue_SE wherever those models have an analogue: pols_* for policy counts, plural nouns for cash flows, *_rate for annual rates with *_rate_mth for their monthly equivalents, *_pp for per-policy amounts, claims(t, kind) with an uppercase kind string, pols_if_at(t, timing) for the end-of-month read, and check_*() / check_*_resid(t) for the identities. The technical notes use compact actuarial symbols; the full mapping lives in the Projection Space docstring. The monthly multi-state biometric chassis is shared with frlib’s Dep_FR_S (assurance dépendance) and, inside this library, with BU_DE_S — three models that are not interchangeable, and whose differences are worth naming rather than glossing:

This model	Dep_FR_S	BU_DE_S	Note
pols_pg(t, g), g = 1 … 5	pols_part / pols_tot	pols_dis_dur(t, z)	The ledger dimension differs: a Pflegegrad here, a two-level French severity there, a claim-duration cohort in BU. Only this model’s is a benefit schedule
wartezeit_months	carence_months(cause)	—	Both run from inception; the French one forks by cause of onset. karenz_months / franchise_months / karenz_months all run from onset instead
leistungsdynamik	in-claim revalorisation	leistungsdyn_rate	The escalation of the annuity in payment
mort_force_care(t, g)	mort_rate_partial / mort_rate_total	mort_rate_dis(t, z)	Impaired-life mortality by state; only this model states it as a multiple of the active force. pols_prem — in force less waived — means the same on all three
check_net_cf, check_pols_roll_fwd, check_states	same names	same names	The three identities mean the same thing on all three models

Six names needed care:

Notes	Cells	Why
l_g(t) vs E_g(t)	pols_pg / esc_pg	The same population counted two ways: a head count and an escalation-weighted value. The annuity is weighted on the second and never on the first
l(t) vs the waived split	pols_if / pols_in_term / pols_waived / pols_prem	In force, in force inside the premium term, waived, and paying. pols_care is a fifth thing again and is none of them: it includes the Karenz ledger and a grade-1 life on delib_std
q_A(x) vs q_g(x)	mort_rate / mort_rate_care(t, g)	The active-life table rate and the in-care rate derived from its force. Publishing one rate for both states is the error the pair exists to prevent
i(x) vs ι(t)	inc_rate / inc_force	The table rate, which stays tariff-comparable at every age, and the force the Wartezeit gates to zero
pols_entry vs pols_grad	pols_entry(t, g) / pols_grad(t, g)	Onsets and graduations out of the Karenz ledger. Equal when K = 0; their gap is the cost of a Karenzzeit
σ	stornoabzug_rate()	The Stornoabzug is a fraction here, so it takes the *_rate name the library gives every rate — the spelling RV_DE_S, FRV_DE_S and Riester_DE_S already use for it. Bare stornoabzug is FRV_DE_S’s euro cash flow and its result_cf() column, a different quantity, and the two must not share a name

policy_id, duration_mth(t) and pols_if_init() drive little or nothing here and are exposed as documented cells rather than dropped: a silently missing column is worse than an inert one.

Standardizations used

Everything in this table is std. The product is unusually std-heavy, which is correct rather than a defect: the mechanics are established and cited above, and it is only the levels that no retrievable document supplies. After the provenance pass of 2026-08-30, four of these rows now have a published comparator — the Leistungsstaffel, the first-order margins, the incidence slope and the Stornoabzug implied by surrender_table.csv’s companion parameter. Every one of them is recorded in the Basis column and none was changed: these are pricing inputs, and moving one moves the worked example and the golden tests with it.

Standardization	Value	Rationale
Leistungsstaffel delib_std	0 / 30 / 50 / 75 / 100 %	Inside the observed 0–10 / 10–30 / 30–50 / 60–75 / 100 % market range [unverified]. Grade 1 pays nothing because §§ 36 and 37 SGB XI both begin at grade 2, so grade 1 is not a funding event in the statutory scheme either R3 R4; the middle steps are mid-to-upper range because grades 2 and 3 carry most of the time in care, and the residential grade mix of 0,5 / 16,6 / 37,3 / 31,1 / 14,3 % R18 supports weighting the upper steps. Note the alternative the market also writes: the one carrier’s wording retrieved sells a threshold design — 100 % from grade 2, 3 or 4 — matching the Stufenmodell the DAV’s own bases use [S4] R15. Not implemented
Vereinbarte Pflegerente	1 000,00 € a month	The round number at the lower end of the 1 000–1 500 € the market sells [unverified], sized against the residual a Pflegeheim resident funds after the statutory contribution and an average pension R4 R20. On the retrieved 1 January 2026 Eigenanteil of 3 245 € it covers a little under a third of the first-year gap; the retrieved carrier’s permitted band is 250 € to 4 000 € [S5]. A scaling constant, left where it is
Active-life mortality	Gompertz 1 − exp(−B c^age), anchored q_M(65) = 1.35 %, q_M(85) = 10.5 %, q_F(65) = 0.75 %, q_F(85) = 7.0 %	DAV 2008 T and the DAV 2008 P active-life table are DAV property and are not shipped R15 R16. The anchors are what a replacement must preserve, not the functional form
Incidence	min(I0 exp(g(age − 65)), 0.50), g_F = 0.1400, g_M = 0.1380	DAV 2008 P is not shipped R15. The slope is anchored on prevalence doubling every five years above 75, ln 2 / 5 = 0.1386, and the level on the sex-specific lifetime-risk order of magnitude. The anchor has since been measured and is softer: observed Pflegequoten rise by 1,80 · 1,88 · 1,74 · 1,49 · 1,18 across the five-year bands from 70–75 upward, about ln 1.8 / 5 = 0.117 R18. Reported, not changed
entry_share	0.20 / 0.38 / 0.24 / 0.13 / 0.05	Entrants skew lower than the stock, which Destatis puts at 13,8 / 40,4 / 29,6 / 11,8 / 4,3 % for end-2023 R18 — this table previously quoted 9 / 44 / 27 / 14 / 6 % — because deterioration moves people up over a spell. Using the stock as the entry mix is a listed pitfall, and the model’s own stock share at grades 4 and 5 (21,0 % and 17,6 %) exceeding the entry share is the arithmetic statement of it. No source supplies an entry mix or a per-grade sojourn time [S14]
det_rate / rec_rate	0.28 / 0.24 / 0.20 / 0.16 / 0.00 and 0.10 / 0.06 / 0.04 / 0.02 / 0.01 a year	Deterioration dominating recovery is property (b) a replacement must preserve. Levels are construction; there is no age-at-onset dimension, which a real Pflegetafel has
Recovery damping	exp(−0.10 × max(0, age − 75))	Encodes the one thing about Reaktivierung not in doubt: real after acute events at younger ages, small at the ages where most claims arise R6
mort_mult	1.5 / 2.5 / 3.5 / 6.0 / 9.0 on the force	Carries the research file’s order of magnitude — two to three times an active life at grade 2, five to ten at grade 5 [unverified]. On the force, so the multiple means the same at every age; on rates it would compress to 1 at the oldest ages. The implied spell of three to five years sits at the low end of the sourced mean of about five years for onset after 60 (4,0 male, 5,7 female) [S14]
Terminal age	omega_age = 110, with mort_rate = 1.0 forced at 109 and the force capped at −ln(1e−12)	A modelling choice, not a table fact — the DAV tables run higher. It buys a closed system: check_states() closes exactly instead of leaving a truncation residue
Monthly step	constant forces over the month, exits allocated in proportion to them	One convention applied uniformly to mortality, incidence, deterioration, recovery and lapse. p_stay + Σ p_j = 1 exactly
Processing order	classify → collect Beitrag → pay Rente → start-of-month expenses → transitions → advance the Karenz clock → lapse last, on the survivors of the insured decrements and the reactivation inflow	Both orderings close check_pols_roll_fwd(), which is exactly why the order has to be declared. Applying lapse to the opening cohort instead moves pols_lapse(0) by 6,2e−7 of a policy and materially more once the decrements are large
Lapse table	6,0 % falling to 1,5 % by year 21, active state only, zero after the premium term	No lapse rate for a German Pflegerente at any duration was established. The shape is argued from the Zillmerung; nothing in care lapses, because a claimant with a waived premium has no premium to default on and a live annuity to forfeit
Rückkaufswert table	0 / 0 / 0.05 / 0.12 / 0.20 of premiums paid, rising to 0.70 by year 40	The shape encodes the 25 ‰ Zillmerung allowance REG-R16 and the § 169 Abs. 3 five-year spread REG-R28; no level was established. Whether a pure-risk Pflegerente falls inside § 169 at all is an open question the library states rather than assumes away — § 169 Abs. 1 owes the value where “der Eintritt der Verpflichtung des Versicherers gewiss ist”, and the one carrier’s wording retrieved grants it anyway R11 [S4]. The companion stornoabzug_rate of 0 %, with a 5 % model point, is contradicted by that wording, which agrees 25 % rising to 50 % after a withdrawal. Reported, not changed
beitragssumme_cap_age	85	A lifelong-premium contract has no finite Beitragssumme without a convention. The ceiling the per-mille sits at is cited REG-R16; the base it is struck on is not
Expense levels	25 ‰ once, 3,0 % of premium, 2,00 € a month inflating at 1,5 %, 1,50 € per annuity payment	No charge level of any kind was established for any German Pflegerenten tariff. The acquisition rate sits exactly at the § 4 DeckRV ceiling so the ceiling binds visibly; the rest are placeholders, and the instalment loading is folded into admin_prem_pct rather than shipped as a Ratenzahlungszuschlag no source supports
First-order margins	incidence × 1.25, deterioration × 1.15, recovery × 0.80, in-care mortality × 0.85, active mortality × 0.90, no lapse	The direction is cited REG-R8 REG-R47, and prudence forks by risk: more claims, faster progression, fewer recoveries, longer annuities, and more active lives surviving to claim. Published levels do exist and this table used to say they did not: the DAV’s Gesamtzuschlag on incidence runs 24,5 / 21,4 / 20,5 / 24,0 / 31,2 % by minimum Pflegegrad, its Gesamtabschlag on Invalidensterblichkeit 28,5 / 24,2 / 24,2 / 24,3 / 25,7 %, and on Aktivensterblichkeit 13,6 % R15 REG-R8. inc_margin 1.25 sits inside that range and act_mort_margin 0.90 is close to 0.864; care_mort_margin 0.85 is materially less prudent than the published 0.715–0.758. det_margin and rec_margin have no published counterpart, the Stufenmodell having no per-grade transitions. Reported, not changed
unisex_mix_male	0.50	Sex may not enter a premium concluded from 21 December 2012 REG-R34. Pricing a 50 / 50 mix while writing 60 / 40 is a named model risk — the mismatch is the cross-subsidy, and the mix is endogenous to the price
rechnungszins	1,00 % a year	§ 2 Abs. 1 DeckRV, “wird der Höchstzinssatz … auf 1 Prozent festgesetzt”, read from the consolidated regulation whose Stand is the amending V of 19.7.2024 that set it for new business from 1 January 2025; Abs. 2 attaches the rate used at conclusion to the contract for its whole term R13 REG-R14 REG-R15. Used only in the equivalence — the one genuinely cited pricing assumption in the model
No Ratenzahlungszuschlag	—	The consequence runs the wrong way and is stated rather than hidden: annual mode prices very slightly below monthly here, through the discounting alone, which is the opposite sign to a real German tariff
Age basis	age last birthday advancing at the policy anniversary	The model carries no dates; a date-based implementation carries a fractional offset of at most one year
Timing	Beitrag and Pflegerente both in advance; surrender and death benefits at the end of the month	German Renten are monatlich vorschüssig, and paying in advance puts the annuity on the same weight as the premium it replaces, which is what lets check_waiver() reconcile the two streams against one ledger
The fourteen model points	—	Configuration rather than observation: no rate card, no commercial envelope and no carrier wording was obtained for this product

The only quantities that are not standardizations are the structural rules and two cited numbers: the Pflegegrad trigger and its five-grade scale, thresholds 12,5 / 27 / 47,5 / 70 / 90 and module weights 10 / 15 / 40 / 20 / 15 % read from § 15 SGB XI R2 REG-R51; the Beitragsbefreiung running with the annuity [S4]; the level Beitrag adjustable only under § 163 VVG R11 REG-R27; the unisex rule REG-R34; the § 169 VVG surrender frame R11 REG-R28; the 25 ‰ Höchstzillmersatz of § 4 Abs. 1 DeckRV R13 REG-R16; and the 1,00 % Höchstrechnungszins of § 2 Abs. 1 DeckRV R13 REG-R14 REG-R15. The bahr grid’s 10 / 20 / 30 / 40 / 100 % has been struck off that list: § 127 SGB XI fixes no percentage schedule, only a Geldleistung at every grade with a 600 € floor at grade 5 and a ceiling at the SGB XI benefit level, so the shipped bahr schedule is a market convention and is std like the rest R8.

Tests

tests/test_pflegerentenversicherung_de.py asserts the fourteen printed rows of the notes’ worked example to the cent and the policy counts to six decimals, the full-precision totals against the sum-of-rounded-cells the notes also print, the equivalence premium of 64,198409 € reached two independent ways from A, U, G and C, month 0 rebuilt term by term, the first month’s decrements from the annual rates through the forces, the first annuity payment grade by grade, the closure identity, the shape of the frame — proj_len() == 780 on the anchor cell, index range(0, proj_len()), last index proj_len() - 1, 780 rows — the male twin’s ten printed rows and totals, the four-cell variant table, the six check_* identities with their residuals, and one test per numbered modeling pitfall — seventeen of them. The whole-model-point-table sweep is not here: tests/test_model_conventions_de.py owns the library’s single sweep.

python -m pytest lifelib/libraries/delib/tests/test_pflegerentenversicherung_de.py -q
python -m pytest lifelib/libraries/delib/tests/test_model_conventions_de.py -q -k Pflege_DE_S