Technical Notes#

Status: Draft, 2026-08-29 (research access date 2026-08-29).

Scope note. These notes specify a reference liability cash-flow projection model — model name Pflege_DE_S, monthly grid — for the standardized composite German Pflegerentenversicherung defined in product-spec.md (same directory). This is not any single insurer’s product, and it was drafted with no product document of any kind retrieved for it: direct HTTP egress from the build environment was blocked and the session’s WebSearch budget was exhausted before work on this product began. The citations have since been re-verified against the primary documents, so that 27 of the 36 source entries in sources.md read Retrieved: yes, five read no and four are partial — a tag below is a record of a document read where its entry says yes, and a pointer to be checked where it does not. [S#] / [R#] tags refer to the source list in sources.md (numbering carried from _research/pflegerentenversicherung.md; frozen); [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md (its own R-numbering). std marks standardizations introduced for the reference implementation; unverified marks claims no source could corroborate. On this product every biometric rate, every charge, every lapse rate and the premium itself is std — a fact stated here once and repeated at each assumption table rather than left to be inferred. 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.

Model scope and conventions#

Purpose. Project gross best-estimate liability cash flows, undiscounted — Beiträge, Pflegerente payments, surrender payments, any Todesfallleistung, and expenses — for a single-policy model point, on an expected (probability-weighted) basis. The model publishes what a valuation layer consumes; it does not discount the projected stream, does not compute a Deckungsrückstellung, does not compute a Zinszusatzreserve and does not compute capital. Those layers are referenced under Valuation and reserve pointers, never reproduced.
The one place a discount rate appears. The Beitrag is a priced quantity, struck by equivalence at the Rechnungszins on the first-order (erster Ordnung) bases REG-R8 REG-R47, 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.
Projection frequency. Monthly grid. The Pflegerente is a monthly annuity, the Beitrag a monthly instalment, and the Pflegegrad can change in any month, so the monthly grid is the contract’s own rather than a refinement of an annual one. The _S suffix follows lifelib.
What t counts, and the frame. t is the policy month index, 0-based: t = 0 is the month of issue, and t counts complete months elapsed since issue. age(t) = age_at_entry + t // 12, so the attained age steps at the policy anniversary. The frame starts at t = duration_mth_init(), which is 0 for new business and the elapsed duration for an in-force model point, and ends at proj_len() - 1, so the frame is range(duration_mth_init(), proj_len()) and the policy year is the 1-based label y(t) = t // 12 + 1. Where the frame starts is a product fact and is not asserted by the conventions suite; contiguity is.
proj_len() is the number of projected periods, the exclusive end of the frame counted from t = 0 — the library-wide reading, and lifelib’s own range(proj_len()). Here
```
proj_len() = 12 * (omega_age() - age_at_entry())
```
which depends only on the entry age and the terminal age, not on duration_mth_init(). The anchor cell (entry age 45, omega_age = 110) therefore has proj_len() = 780 and runs to t = 779, 780 monthly rows, attained ages 45 to 109. A point opening at duration_mth_init() = d0 publishes 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.
Terminal age. omega_age = 110 std, with mort_rate(t) = 1.0 forced in the final year of age so the model is a closed system rather than a truncated one and the decrement closure holds exactly. It is a modelling choice, not a table fact — the DAV tables run higher R15 REG-R51 — and it costs nothing material: an active life aged 45 survives to 110 with probability of the order of 1e-4 on the shipped basis.
Timing conventions std. Within month t: the Beitrag is collected at the start of the month, in advance, from the lives then in the premium-paying states; the Pflegerente is paid at the start of the month, in advance, to the lives then in a paying Pflegegrad; per-policy and premium-related expenses are charged at the start of the month; transitions act over the month; and death and surrender benefits fall at the end of the month. pols_if(t) is the count at the start of month t and is the weight on that same result_cf() row’s cash flows, as the house style requires; end-of-period state is reached through pols_if_at(t, "END").
Why the annuity is in advance. German Renten are conventionally 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.
Currency, sign and rounding. EUR throughout. net_cf(t) is income-positive (premiums +, annuity, surrender, death benefit 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 and pols_if to six decimals std.
Out of scope, deliberately, each in one line. No Überschussbeteiligung — the surplus chassis belongs to products/kapitallebensversicherung/. No Beitragsdynamik: the acceptance rate on each offer is a behavioural assumption this corpus cannot support. No Beitragsfreistellung and no § 38 VVG premium-default conversion: every voluntary exit is a surrender, and the direction of the bias is stated in the product specification. No taxation of premium or benefit, the benefit’s treatment being an open question R23 REG-R41. No select mortality after onset of care — the shipped in-care mortality is an aggregate, and the consequence is pitfall 9 and model risk 9. No Pflege-Bahr Zulage, statutorily unavailable R8. And no three-month Nachprüfung notice tail: whether § 174 VVG reaches a Pflegerente through § 177 was not established REG-R29 unverified; § 176 VVG extends §§ 150–170 entsprechend to the Berufsunfähigkeitsversicherung only, which is a point against the extension but not a decision on it.

Model point attributes#

model_point_table.csv is indexed by point_id and is the only input file without a provenance column, a model point being a configuration rather than an assumption (delib ruling 2). “Exercised by” names the points that make the attribute do work.

Attribute	Type	Meaning	Exercised by
`point_id`	int	Index; `Projection` is parameterized by it	all
`policy_id`	str	Human-readable identifier, `PFL-0000NN`	all
`sex`	enum {M, F}	The projection basis. Pricing is unisex REG-R34, so this must not reach `premium_mth_pp()`	1 (F) vs 2 (M), identical otherwise
`age_at_entry`	int	Age last birthday at issue; drives `proj_len()` and every rate lookup	14 (18, the youngest permitted), 13 (65, the oldest)
`duration_mth_init`	int	Complete months elapsed at the projection start; `0` for new business. The frame opens at this `t`	11 (240), 12 (336)
`status`	enum {aktiv, pg1…pg5}	State at the projection start	12 (`pg3`, in claim)
`rente_mth`	float	The vereinbarte Pflegerente at Pflegegrad 5, EUR per month — the scaling constant of the whole benefit	13 (1 500), 14 (500)
`staffel_id`	str	Key into `benefit_scale_table.csv`: `delib_std` or `bahr`	5 (`bahr`)
`prem_end_age`	int	Attained age at which the Beitrag ceases; `110` (= `omega_age`) means lifelong	4 and 9 (65, abgekürzte Beitragszahlungsdauer)
`prem_mode`	enum {monthly, quarterly, half_yearly, annual, single}	Instalment frequency; `single` is the Einmalbeitrag	1, 3, 4, 5, 6
`premium_mth`	float	The contractual monthly Beitrag. `0.0` means “derive by equivalence”	10 (75.00 supplied), 11 and 12 (in-force premiums)
`rating_factor`	float	Risikozuschlag multiplier on the gross premium; `1.00` at standard rates	13 (1.50)
`wartezeit_months`	int	Wartezeit from inception, in months	7 (36)
`karenz_months`	int	Karenzzeit from onset, in months	7 (6)
`leistungsdynamik`	float	Annual escalation of the annuity in payment; `0.0` = off	8 (0.02)
`beitragsrueckgewaehr`	bool	Beitragsrückgewähr death benefit on/off	9 (True)
`stornoabzug_rate`	float	Deduction from the Rückkaufswert, as a fraction; `0.0` = off	10 (0.05)
`pols_if_init`	float	Policy count at the frame’s first `t`; `1.0` everywhere here	all

The three attributes most easily got wrong, each a numbered pitfall: sex is a projection input and must not touch the price REG-R34; premium_mth = 0.0 is a sentinel, not a free contract; and duration_mth_init shifts where the frame starts without changing proj_len().

The fourteen model points#

#	Configuration	What it exercises
1	Anchor. F, entry 45, aktiv, 1 000 €/mth, `delib_std`, lifelong, monthly, premium derived	The worked example
2	M, entry 45, otherwise identical to 1	Unisex pricing against a sex-specific projection: same `premium_mth_pp()`, different claims
3	F, entry 55, quarterly, lifelong, `delib_std`	The upper half of the purchase cluster; quarterly instalments
4	M, entry 40, premiums to age 65, half-yearly	Abgekürzte Beitragszahlungsdauer; half-yearly instalments; the paid-up tail
5	F, entry 50, annual mode, `staffel_id = bahr`	The statutory 10/20/30/40/100 grid, where Pflegegrad 1 is insured and waived
6	F, entry 60, `prem_mode = single`	The Einmalbeitrag: one payment at `t = 0`, no premium-paying ledger thereafter
7	M, entry 50, `wartezeit_months = 36`, `karenz_months = 6`, monthly	Both waiting devices, and the Karenz ledger
8	F, entry 45, `leistungsdynamik = 0.02`, monthly	The escalation ledger; `esc_pg` diverges from `pols_pg`
9	M, entry 45, `beitragsrueckgewaehr = True`, premiums to age 65, monthly	The death-benefit stream, `claims_death` non-zero
10	F, entry 48, `premium_mth = 75.00`, `stornoabzug_rate = 0.05`	A supplied premium instead of a derived one; a non-zero Stornoabzug
11	M, entry 42, `duration_mth_init = 240`, aktiv, monthly	An in-force point opening at `t = 240` with 20 years run
12	F, entry 55, `duration_mth_init = 336`, `status = pg3`	An in-force point in claim: waived premium and a paying state at the frame’s first row
13	Boundary. M, entry 65 (top of the observed band), 1 500 €/mth, `rating_factor = 1.50`	Shortest pre-claim period, highest premium, a Risikozuschlag
14	Boundary. F, entry 18 (bottom of the band), 500 €/mth, monthly	Longest projection (`proj_len() = 1104`, `t = 0 … 1103`), smallest benefit, expense-dominated

Between them the fourteen exercise both premium forms, all four instalment frequencies, both Leistungsstaffeln, every switchable option, two in-force points and both ends of the age band.

Why model point 9 does not pay for life. The Beitragsrückgewähr is the one option whose cost is not a modest loading, and at a Rechnungszins of 1,00 % it is close to the whole premium: returning nominal premiums on a death that is on average forty years away is worth about 0.67 of what was paid, against a premium stream whose own present value is discounted and thinned by survival. Written with a lifelong Beitragszahlungsdauer the equivalence’s denominator, U (1 − β) − D1 − a1, falls to 3.6 of 313.5 units and the model point becomes a knife-edge: the premium is 2,659.56 € a month and a small change in any basis would send it negative. Point 9 therefore pays to 65, which is how the German market writes a Beitragsrückgewähr tariff, and its premium is 622.92 € a month — still 9,7 times the anchor’s, which is the honest measure of what a gross return of premiums costs on this basis rather than the “roughly doubles” the research file reasons to at a higher technical rate.

Input files#

Every file is a plain UTF-8 CSV in the model folder’s parent, read once per model by a reader cells in the Data Space (the annuallife/TradLife_A layout). Every file except model_point_table.csv carries a final provenance column, one tag per row — delib ruling 2.

File	Index columns	Value columns	Content
`model_point_table.csv`	`point_id`	the eighteen attributes above	The fourteen model points. Exempt from `provenance`
`benefit_scale_table.csv`	`staffel_id`, `pflegegrad`	`benefit_pct`, `provenance`	The Leistungsstaffel: 0/30/50/75/100 % for `delib_std`, 10/20/30/40/100 % for `bahr` R8
`mort_table.csv`	`sex`, `age`	`mort_rate`, `provenance`	Annual active-life mortality, ages 18–109, both sexes
`incidence_table.csv`	`sex`, `age`	`inc_rate`, `provenance`	Annual rate of entering any Pflegegrad from the active state, ages 18–109, both sexes
`care_table.csv`	`pflegegrad`	`entry_share`, `det_rate`, `rec_rate`, `mort_mult`, `provenance`	The whole in-care basis in five rows: the distribution of the grade first entered, the annual deterioration rate to the next grade, the annual recovery rate to the previous grade or to active, and the force-of-mortality multiple over an active life of the same age
`lapse_table.csv`	`policy_year`	`lapse_rate`, `provenance`	Annual lapse from the active state, policy years 1–40; year 40’s rate applies to every later year
`surrender_table.csv`	`policy_year`	`rkw_prem_ratio`, `provenance`	The guaranteed Rückkaufswert as a fraction of premiums paid to date, policy years 1–40, clamped as for lapse
`expense_table.csv`	`item`	`value`, `unit`, `provenance`	`acq_permille`, `admin_prem_pct`, `admin_mth_pp`, `claim_expense_pp`, `expense_infl`
`basis_table.csv`	`param`	`value`, `provenance`	`rechnungszins`, `omega_age`, `unisex_mix_male`, `rec_age_ref`, `rec_age_decay`, `inc_cap`, `beitragssumme_cap_age`, `roll_fwd_tol`, and the five first-order prudence margins `inc_margin`, `det_margin`, `rec_margin`, `care_mort_margin`, `act_mort_margin`

Nine files, each read by a reader cells in Data; the suite asserts there are no orphans.

State variables#

The contract is a multi-state risk. The state space is the one § 15 SGB XI defines R2 REG-R51, plus the two absorbing exits a life-assurance contract adds:

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

Forward moves are deterioration, backward moves are Herabstufung and, out of PG1, Reaktivierung; death is reachable from every state. Lapse is reachable only from aktiv std: a claimant whose premium is waived has nothing to lapse from and everything to lose. Pitfall 12 tests it.

Variable	Description	Updated
`proj_len`	Number of projected months, `12 * (omega_age - age_at_entry)`; the frame’s exclusive end	once per point
`duration_mth(t)`, `policy_year(t)`, `age(t)`	Months since issue (equal to `t`), the Versicherungsjahr `t // 12 + 1`, and the attained age `age_at_entry + t // 12`	monthly
`pols_act(t)`, `pols_karenz(t, g, z)`	Lives active at the start of month `t`; lives in Pflegegrad `g` whose Karenzzeit clock stands at `z`, `1 ≤ z ≤ karenz_months`, the ledger being empty when `karenz_months = 0`	monthly recursion
`pols_pg(t, g)`	Lives in Pflegegrad `g` at the start of month `t` whose Karenzzeit has been served — the ledger the annuity is paid on	monthly recursion
`esc_pg(t, g)`	The escalation-weighted counterpart of `pols_pg(t, g)`: the sum over those lives of `(1 + leistungsdynamik)^(months since the annuity began / 12)`. Equal to `pols_pg` exactly when the dynamic is off	monthly recursion
`pols_care(t)`, `pols_if(t)`	Everyone in care, waiting or paid, `Σ_z Σ_g pols_karenz + Σ_g pols_pg`; and `pols_act + pols_care`, the in-force count at the start of month `t`	monthly
`pols_if_at(t, timing)`	`"BEG"` = `pols_if(t)`; `"END"` = `pols_if(t + 1)`. End-of-period state goes through here and never through `pols_if`	monthly
`pols_in_term(t)`, `pols_waived(t)`, `pols_prem(t)`	In-force units inside the premium term (`age(t) < prem_end_age`); of those, the units in a Pflegegrad whose `benefit_pct` is positive — the Beitragsbefreiung population; and the difference, the units that actually pay	monthly
`pols_entry(t, g)`, `pols_grad(t, g)`	Lives entering Pflegegrad `g` from the active state during month `t`, and lives graduating out of the Karenz ledger into `pols_pg(·, g)`	monthly
`pols_death(t)`, `pols_lapse(t)`	Deaths during month `t` from every state; lapses from the active state only, after the insured decrements	monthly
`pols_dead_cum(t)`, `pols_lapse_cum(t)`	Cumulative absorbing counts at the start of month `t`; `cum_prem_max_pp(t)` is premiums payable to date on an uninterrupted path, the base of both the Rückkaufswert and the Beitragsrückgewähr	monthly
`tar_pols_act(t)`, `tar_pols_pg(t, g)`, `tar_pols_prem(t)`	The same ledgers on the first-order basis, without lapse — the pricing engine’s state	monthly

There is no account value and no paid-up state, and the Deckungskapital — a real object of this contract — is deliberately not a state variable: the library publishes undiscounted cash flows and leaves the reserve to the layer that consumes them, with the guaranteed Rückkaufswert entering as data (product spec, footnote 21).

Assumption inputs#

(a) Contractual / guaranteed elements (cited)#

Input	Value	Basis
Benefit	A monthly Pflegerente, paid in advance, equal to `benefit_pct(g) × rente_mth` for the insured’s current Pflegegrad `g`	[S4]
Leistungsstaffel `delib_std`	0 / 30 / 50 / 75 / 100 % across grades 1 to 5	std, product spec footnote 12
Leistungsstaffel `bahr`	10 / 20 / 30 / 40 / 100 % — the Pflege-Bahr grid as the market writes it, not as the statute fixes it. § 127 Abs. 2 Nr. 4 SGB XI requires only a Geldleistung at every Pflegegrad, at least 600 € at grade 5, capped at the SGB XI benefit level; the percentage schedule belongs to the PKV-Verband’s brancheneinheitliche Vertragsmuster under Abs. 2 Satz 2, which this library has not retrieved	shape std; R8 for what the statute fixes
Trigger	The statutory Pflegegrad of §§ 14, 15 SGB XI, determined by the Medizinischer Dienst or by an independent assessor under § 18 Abs. 1, not by the insurer. Wordings commonly pin the statutory text to an edition date or copy it into the conditions rather than tracking the live statute	R2 R6; [S1] [S2] [S4]; REG-R51
Care setting	Irrelevant to the benefit; the same annuity is payable at home and in a Pflegeheim	std, product spec footnote 1
Beitragsbefreiung	Full, from the first month in which any annuity is payable; revived on exit from the paying grades. The one retrieved wording waives on the same trigger but makes the waiver permanent after twelve months’ continuous annuity at grade 4 or 5	[S4]; detail std
Premium form	Level monthly Beitrag, guaranteed for the life of the contract, adjustable only on the § 163 VVG route	R11; REG-R27
Premium cessation	On death; on the start of an insured annuity; at `prem_end_age`	[S4]
Equivalence principle	The gross premium is struck so that, on the first-order bases at the Rechnungszins, the expected present value of premiums equals that of benefits plus expenses	REG-R8 REG-R47
Rechnungszins	1,00 % p.a. — § 2 Abs. 1 DeckRV for new business from 1 January 2025; Abs. 2 keeps the rate used at conclusion for the contract’s whole term	R13; REG-R14 REG-R15
Höchstzillmersatz	25 ‰ of the Summe aller Prämien, § 4 Abs. 1 DeckRV, cut from 40 ‰ by the LVRG from 1 January 2015; the retrieved wording applies exactly this, at “2,5 % der … zu zahlenden Beiträge”	R13; [S4]; REG-R16 REG-R20
Rückkaufswert	The Deckungskapital on the premium bases, floored by the value that results from spreading acquisition and distribution costs evenly over the first five contract years	REG-R28
Stornoabzug	Admissible only if agreed, quantified and appropriate (§ 169 Abs. 5 VVG); a deduction for unamortised acquisition costs is expressly ineffective. *The one retrieved Pflegerenten* wording agrees 25 %**, rising to 50 % after a partial withdrawal — see the model-risk note at the end of this file	R11 REG-R28; [S4]
Unisex	Sex may not enter the premium for contracts concluded from 21 December 2012	REG-R34
Wartezeit / Karenzzeit	Contractual, both 0 in the base run, and the one retrieved underwritten tariff also has Wartezeit keine; § 127 Abs. 2 Nr. 6 SGB XI caps a Pflege-Bahr Wartezeit at “höchstens fünf Jahre”	R8; [S5]; base std

What is not in this table. There is still no cited premium, no cited charge, no cited transition rate and no cited Rückkaufswert level for this product, so this class is unusually thin beside frlib/products/temporaire_deces, which could tabulate a published rate card. Everything numeric that is missing here appears in class (c) as std, and that displacement is the single most important thing to know about this model. What changed in the provenance pass of 2026-08-30 is that three of those gaps now have published comparators rather than nothing at all — the DAV’s first-order bases and prudence loadings for Pflegegrade, the GDV’s in-force average premium, and one carrier’s Stornoabzug — and every one of them is recorded in sources.md and not implemented. See “What the retrieved documents say about the shipped model” there.

(b) Insurer-discretionary current elements#

Thinner than on any savings product in delib, and for a structural reason: a Pflegerente is a life contract, so the insurer’s discretion over price is confined to § 163 VVG and to the surplus rebate REG-R27. There is no bonus rate, no crediting rate and no charge scale to declare.

Input	Snapshot value	Basis
Überschussbeteiligung in any application form	None — no Beitragsverrechnung, no verzinsliche Ansammlung, no Bonus uplift of the vereinbarte Rente	mechanic R11 R12 REG-R24; omission std (1)
The Bruttobeitrag / Zahlbeitrag spread	Zero. The model projects the Bruttobeitrag	REG-R27 REG-R53; std (1)
§ 163 VVG re-rating / tariff drift	Never invoked; the premium is struck once, at issue, on the bases shipped	REG-R27; std (2)
Nachprüfung intensity	Not modelled as a separate decrement: recovery and downgrade are biometric transitions, not claims-management outcomes	R6 [S4]; std (3)

delib publishes gross undiscounted cash flows and demonstrates the Überschussbeteiligung chassis in products/kapitallebensversicherung/; a discretionary Beitragsverrechnung would need a declared-rate assumption this corpus cannot supply (gap 18), and would make the projected premium a discretionary quantity — precisely what the product’s proposition says it is not. The Zahlbeitrag is nevertheless what a customer pays, so a reader comparing this model’s premium with a market quotation is comparing a Bruttobeitrag with a Zahlbeitrag and should expect the model to sit above the quotation.
§ 163 is a management action conditional on emerging experience, not a projected assumption; a model that built one in would be asserting that the guarantee it demonstrates does not hold.
On a Berufsunfähigkeitsrente the insurer runs its own Nachprüfung and claims-management intensity is a real assumption REG-R29. Here the evidence is the statutory determination R6, so re-verification is a documentation exercise and every exit from a paying grade is biometric. That is the sharpest modelling difference between Pflege_DE_S and BU_DE_S.

(c) Behavioural / experience assumptions (the modeller’s view)#

Every input in this class is std. No German source publishes an expense loading, a commission scale, a lapse rate or a surrender-value table for this product (research gaps 2, 3, 19, 20). The proxies below reproduce the mechanics the research file establishes; they are not calibrations, and none is a proxy for DAV 2008 P.

Two statements this section used to make about DAV 2008 P are wrong and are corrected here. The table is not unpublished: the DAV issues the derivation as a free Ergebnisbericht with the bases in Anhänge 1 to 3, and a companion report that re-derives them for the Pflegegrade R15. And a Pflegegrad-shaped basis therefore does exist — the profession built one rather than mapping the scales, which is what the BGH has since held cannot be done REG-R36 REG-R51. delib still redistributes none of it and reproduces no value from it; that is this library’s own choice. Three things about the published bases matter to anyone replacing the tables below: they are a Stufenmodell (“mindestens Pflegegrad g ist erreicht”), not a five-state per-grade chain; they are transitions from Pflegestufen experience rather than Pflegegrad observations, the DAV saying so in terms; and they come with published prudence loadings, which the margins at the end of this section do not match. A replacement must preserve four properties: (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.

Active-life mortality std — a Gompertz proxy, sex-specific, German-population-shaped, and not DAV 2008 T or the DAV 2008 P active-life table R16 REG-R48 REG-R52:

mort_rate(sex, x) = 1 - exp(-B_sex * c_sex ** x)      ages 18 to omega_age - 2
mort_rate(sex, omega_age - 1) = 1.0                   the limiting-age convention

B_M = 1.47884e-05, c_M = 1.110680, B_F = 4.76290e-06, c_F = 1.119962, fixed by two anchors per sex — q(65) = 1.35 % and q(85) = 10.5 % male, 0.75 % and 7.0 % female std. Those anchors are what a substitute table must reproduce for the worked example to close, and the Data docstring states them.

Incidence into care std — the rate at which an active life enters any Pflegegrad:

inc_rate(sex, x) = min(I0_sex * exp(g_sex * (x - 65)), inc_cap)

I0_F = 0.0110, g_F = 0.1400, I0_M = 0.0085, g_M = 0.1380, inc_cap = 0.50. The slope is anchored to prevalence roughly doubling every five years of age above 75, a growth rate of ln 2 / 5 = 0.1386, and the level so that the model’s own probability of reaching an insured grade before death, from the anchor cell’s entry age, is of the order of the 45 % the research file argues, with a mean age at first insured grade in the low eighties unverified. The anchor is now measurable and it is softer than 0.1386. The observed Pflegequoten for 2023 rise by factors of 1,80 · 1,88 · 1,74 · 1,49 · 1,18 across the five-year bands from 70–75 to 95+ R18 — about ln 1.8 / 5 = 0.117 over the range where this model earns its claims, and flattening sharply above 90 where the inc_cap of 0.50 does not yet bind. The shipped slope is left where it is and the divergence is reported: changing it moves the worked example and the golden tests. The female parameters are the higher pair, which is the sex differential the whole unisex tension turns on. The cap is a shape device: without it the exponential exceeds one before age 100.

The in-care basis std, five rows of care_table.csv, and the most consequential table here:

Pflegegrad	`entry_share`	`det_rate` (to `g+1`, annual)	`rec_rate` (to `g-1`/aktiv, annual)	`mort_mult`
1	0.20	0.28	0.10	1.5
2	0.38	0.24	0.06	2.5
3	0.24	0.20	0.04	3.5
4	0.13	0.16	0.02	6.0
5	0.05	0.00	0.01	9.0

entry_share sums to 1.00 and is deliberately not the stock distribution, which the Destatis table for end-2023 puts at 13,8 / 40,4 / 29,6 / 11,8 / 4,3 % R18 — not the 9 / 44 / 27 / 14 / 6 % this file previously quoted. Entrants skew lower than the stock because deterioration moves people up over a spell, and using the stock as the entry mix is pitfall 17; the corrected stock, with 54,2 % rather than 53 % in the two lowest grades, does not disturb that argument. What no source supplies is the entry mix itself, or the time spent at each grade — Assekurata states in terms that no information exists on how long people remain in each Pflegegrad, the grades dating only from 2017 [S14] — so entry_share stays std. mort_mult is a multiple on the force of active mortality at the same age, carrying the research file’s most load-bearing biometric statement — the mortality of a Pflegebedürftiger is of the order of two to three times an active life’s at grade 2 and five to ten times at grade 5 unverified. Recovery is further damped with age, rec_rate(g, x) = rec_rate_g * exp(-rec_age_decay * max(0, x - rec_age_ref)) with rec_age_ref = 75 and rec_age_decay = 0.10 std, which is what makes deterioration dominate recovery above 75 — property (b) of a replacement table.

Three consequences follow, all three pitfalls. The annuity in payment is short, of the order of three to five years, not the fifteen to twenty of a healthy-life pension at the same age — and this is now checkable: the BARMER-Pflegereport 2024, via [S14], puts mean duration of Pflegebedürftigkeit at about five years where care begins after 60, 4,0 for men and 5,7 for women, against about 25 months mean stay in a Pflegeheim. The model’s implied spell sits at the low end of that for women — so pricing it on DAV 2004 R would be prudent in exactly the wrong direction R16 REG-R49. Grade and mortality are correlated, so the highest-paying state is the shortest-lived. And a deferred period bites harder than its length suggests, because a material share of new claimants die inside it.

Lapse std, from the active state only, and zero once the premium term has ended:

Policy year	1	2	3	4	5	6–10	11–20	21–40
`lapse_rate`	6.0 %	5.0 %	4.0 %	3.5 %	3.0 %	2.5 %	2.0 %	1.5 %

Year 40’s rate applies to every later year. No lapse rate for this product at any duration was established (gap 20); the shape is a modeller’s construction, argued from Zillmerung — the Rückkaufswert is near zero for the first years, so an early lapse is expensive to the policyholder and the profile is flatter than a savings product’s, and a paid-up contract has no premium-driven exit at all. The monthly rate is 1 - (1 - lapse_rate(t)) ** (1/12).

The guaranteed Rückkaufswert std, as a fraction of premiums paid to date by Versicherungsjahr y(t) = t // 12 + 1 — the scale-free form, and the form a German contract states REG-R28:

Policy year	1	2	3	4	5	10	15	20	25	30	40
`rkw_prem_ratio`	0.00	0.00	0.05	0.12	0.20	0.42	0.50	0.56	0.60	0.64	0.70

with intermediate years interpolated in the shipped file and year 40’s ratio applying thereafter. Year k’s ratio applies throughout policy year k, t = 12(k − 1) … 12k − 1: it is the current Versicherungsjahr, not the completed one, so a surrender at t = 24 — two completed years — already takes year 3’s 0.05. The shape encodes two cited facts: the 25 ‰ Zillmerung allowance REG-R16, which is why the first two years are zero, and the § 169 Abs. 3 five-year spread floor REG-R28, which is why it turns positive in year three. It never approaches 1.00, the contract having consumed risk premium throughout.

Expenses std, all five levels placeholders, the structure cited:

Item	Value	Unit	Rationale
`acq_permille`	25.0	‰ of Beitragssumme, at `t = 0`	Set exactly at the § 4 DeckRV Höchstzillmersatz REG-R16, so the ceiling binds visibly
`admin_prem_pct`	0.030	fraction of each premium collected	Placeholder; absorbs the Ratenzahlungszuschlag the model does not charge separately
`admin_mth_pp`	2.00	EUR per policy in force per month, `t = 0` prices	Placeholder, a plausible fraction of a mass-market monthly premium
`claim_expense_pp`	1.50	EUR per annuity payment made	Set low — the trigger is determined by a third party R6, so claims cost is well below a Berufsunfähigkeitsrente’s REG-R29
`expense_infl`	0.015	annual	Placeholder; over a 65-year projection it is a factor of about 2.6 on the per-policy line

The Beitragssumme the acquisition charge is struck on is premium_mth_pp() * 12 * (min(prem_end_age, beitragssumme_cap_age) - age_at_entry) with beitragssumme_cap_age = 85 std, and for the single-premium form the Einmalbeitrag itself. A lifelong-premium contract has no finite Beitragssumme without a convention; the cap is that convention, and it is a parameter, not a citation.

The first-order margins std. German practice runs two parallel bases over one contract: erster Ordnung — prudent, statutorily required REG-R8, fixing the Bruttobeitrag — and zweiter Ordnung, the best estimate the projection runs on, with the Sicherheitszuschlag the wedge whose direction forks by risk REG-R47. For care, prudence means higher incidence, faster deterioration, slower recovery, longer duration in care and lower active mortality, because a life that survives is a life that can claim:

Margin	Value	Applied to	Direction
`inc_margin`	1.25	`inc_rate`	more claims
`det_margin`	1.15	`det_rate`	faster progression to the higher-paying grades
`rec_margin`	0.80	`rec_rate`	fewer recoveries, so longer spells
`care_mort_margin`	0.85	in-care mortality	longer annuities
`act_mort_margin`	0.90	active mortality	more lives survive to claim

All five are std and the responsible actuary’s judgment sets them in practice REG-R11 REG-R56. This file used to add that no German source gives a Sicherheitszuschlag level for a Pflegetafel. The DAV does R15 REG-R8, by minimum Pflegegrad: a Gesamtzuschlag on incidence of 24,5 / 21,4 / 20,5 / 24,0 / 31,2 %, a Gesamtabschlag on Invalidensterblichkeit of 28,5 / 24,2 / 24,2 / 24,3 / 25,7 %, and 13,6 % on Aktivensterblichkeit. Against those, inc_margin 1.25 sits inside the published range and act_mort_margin 0.90 is close to 0.864; but care_mort_margin 0.85 is materially less prudent than the published 0.715–0.758, and det_margin and rec_margin have no published counterpart because the Stufenmodell has no per-grade transitions. Nothing here was changed: these are pricing bases, and moving one moves the worked example and the golden tests. The first-order basis carries no lapse at all, which is both German practice and what keeps the model acyclic. Pricing blends the sexes at unisex_mix_male = 0.50 std while the projection uses the point’s own sex; “pricing unisex on a 50 / 50 mix while writing 60 / 40” is model risk 7 REG-R34.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
`t`, `x(t)`, `y(t)`, `g`	policy month, 0-based, `t = d0 … n − 1` with `d0 = duration_mth_init` and `n = proj_len` the number of projected months; attained age `age_at_entry + t // 12`; policy year `t // 12 + 1`; Pflegegrad `1 … 5`
`π_g`	`benefit_pct(g)`, the Leistungsstaffel percentage
`R`	`rente_mth`, the vereinbarte Pflegerente at Pflegegrad 5
`P`	`premium_mth_pp()`, the level monthly gross Beitrag per policy
`m`	`prem_mode_months()` — 1, 3, 6, 12, or 0 for the Einmalbeitrag
`μ_A(t)`, `μ_g(t)`, `ι(t)`	force of active-life mortality at `x(t)`; force of mortality in Pflegegrad `g` = `mort_mult(g) × μ_A(t)`; force of incidence into care, 0 while `t < wartezeit_months`
`s_g`	`entry_share(g)`, the distribution of the grade first entered
`δ_g(t)`, `ρ_g(t)`, `w(t)`	forces of deterioration `g → g+1` and recovery `g → g-1` (from PG1, to active); and the monthly lapse probability from the active state, `lapse_rate_mth`, 0 outside the premium term
`ℓ_A(t)`, `W_{g,z}(t)`, `ℓ_g(t)`	`pols_act(t)`, `pols_karenz(t, g, z)`, `pols_pg(t, g)`
`E_g(t)`	`esc_pg(t, g)`, the escalation-weighted counterpart of `ℓ_g`
`d`, `K`	`leistungsdynamik`, the annual escalation of the annuity in payment; `karenz_months`
`i`	`rechnungszins`; `v = (1 + i) (-1/12)`, used only** in the pricing engine
`α`, `β`, `γ`, `c`, `f`	`acq_permille/1000`, `admin_prem_pct`, `admin_mth_pp`, `claim_expense_pp`, `expense_infl`

Forces are per annum; R, P and every cash-flow component are EUR.

Rates, forces and the monthly step#

Every shipped rate is annual, and every transition inside a month is computed from forces held constant over the month, the competing transitions sharing one survival probability in proportion to their forces. Writing q for an annual rate, the force is μ = -ln(1 - q); with forces μ_1 … μ_k out of a state,

p_stay = exp(-(μ_1 + … + μ_k) / 12)        p_j = (μ_j / Σ_k μ_k) * (1 - p_stay)

so p_stay + Σ_j p_j = 1 exactly, by construction. This is the standard constant-force, proportional-allocation convention, and declaring it is not optional: adding monthly rates instead, or applying q/12, gives different answers wherever the forces are large — which on this product is exactly where the money is. Pitfalls 7 and 8 test it. From the active state, with μ_A and ι, and from Pflegegrad g, with μ_g, δ_g, ρ_g (δ_5 = 0; ρ_1 leads to the active state):

p_act_stay(t)  = exp(-(μ_A + ι) / 12)      p_pg_stay(t, g)   = exp(-(μ_g + δ_g + ρ_g) / 12)
p_act_death(t) = (μ_A / (μ_A + ι)) * (1 - p_act_stay(t))
p_act_care(t)  = (ι   / (μ_A + ι)) * (1 - p_act_stay(t))
p_pg_death(t, g)  = (μ_g / S) * (1 - p_pg_stay(t, g))
p_pg_worse(t, g)  = (δ_g / S) * (1 - p_pg_stay(t, g))
p_pg_better(t, g) = (ρ_g / S) * (1 - p_pg_stay(t, g)),        S = μ_g + δ_g + ρ_g

Both sets are published as cells, and check_states() rests on their summation to one.

The in-force recursions#

Active state. Entrants leave, deaths leave, reactivations arrive, and the survivors of all three are exposed to lapse — lapse acting after the insured decrements, the same ordering frlib’s term model uses and stated because the alternatives give different answers:

pols_entry(t, g) = ℓ_A(t) * p_act_care(t) * s_g
pols_reactiv(t)  = [ ℓ_1(t) + Σ_z W_{1,z}(t) ] * p_pg_better(t, 1)
pols_lapse(t)    = [ ℓ_A(t) * p_act_stay(t) + pols_reactiv(t) ] * w(t)
ℓ_A(t+1)         = [ ℓ_A(t) * p_act_stay(t) + pols_reactiv(t) ] * (1 - w(t))

The Karenz ledger, present only when K > 0. Entrants join at z = 1 and advance one month at a time, subject to the same transitions as a served life; the clock is discarded on reactivation, because the Karenzzeit runs from onset and a recovered life who later relapses starts a new onset:

W_{g,1}(t+1)    = pols_entry(t, g)
W_{g,z+1}(t+1)  = Σ_h W_{h,z}(t) * p_karenz(t, h → g),      1 ≤ z < K
pols_grad(t, g) = Σ_h W_{h,K}(t) * p_karenz(t, h → g)

where p_karenz(t, h → g) is p_pg_stay for h = g, p_pg_worse for g = h + 1, p_pg_better for g = h - 1, and zero otherwise. When K = 0 the ledger is empty and pols_grad(t, g) = pols_entry(t, g) — the degenerate case the base run runs in.

The paying ledger, with the g = 1 recovery term flowing to the active state instead and the g = 5 deterioration term absent:

ℓ_g(t+1) = ℓ_g(t) * p_pg_stay(t, g) + ℓ_{g-1}(t) * p_pg_worse(t, g-1)
         + ℓ_{g+1}(t) * p_pg_better(t, g+1) + pols_grad(t, g)

The escalation ledger is the same population weighted by each life’s own escalation factor since its annuity began: the identical recursion with one extra factor, entrants joining at weight 1.

E_g(t+1) = (1 + d) ** (1/12) * [ E_g(t) * p_pg_stay(t, g) + E_{g-1}(t) * p_pg_worse(t, g-1)
                               + E_{g+1}(t) * p_pg_better(t, g+1) ] + pols_grad(t, g)

When d = 0 this is the ℓ_g recursion exactly, so E_g ≡ ℓ_g in the base run — an invariant check_esc_ledger() asserts. Carrying escalation as a value ledger rather than a duration-since-onset cohort dimension is what 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.

Deaths, and the closure.

pols_death(t)       = ℓ_A(t) * p_act_death(t)
                    + Σ_g [ ℓ_g(t) + Σ_z W_{g,z}(t) ] * p_pg_death(t, g)
pols_dead_cum(t+1)  = pols_dead_cum(t) + pols_death(t)
pols_lapse_cum(t+1) = pols_lapse_cum(t) + pols_lapse(t)

pols_if(t) + pols_dead_cum(t) + pols_lapse_cum(t) = pols_if_init()     for every t

The last line is check_states(). Because mort_rate is forced to 1 in the final year of age, the identity closes at t = n — one past the last projected month — with pols_if = 0, so the decrements sum to pols_if_init() exactly — a closure a reader can check with a calculator on the worked example.

Premium, waiver and the premium-paying population#

pols_in_term(t) = pols_if(t)                      if x(t) < prem_end_age else 0
pols_waived(t)  = Σ_{g : π_g > 0} ℓ_g(t)          restricted to the in-term population
pols_prem(t)    = pols_in_term(t) - pols_waived(t)

Three consequences follow directly from the Leistungsstaffel, and each is a test. A life in a Karenz ledger pays, because no annuity is yet payable and the waiver runs with the annuity. A life at Pflegegrad 1 on the delib_std grid pays, because π_1 = 0; on the bahr grid, where π_1 = 0.10, the same life is waived. And a life that is downgraded out of the paying grades starts paying again, so pols_prem is not monotone.

The instalment is charged only on due months:

premium_due(t)  = (m > 0) and (t % m == 0) and (x(t) < prem_end_age)
premium_pp(t)   = P * m           if premium_due(t) else 0
premiums(t)     = premium_pp(t) * pols_prem(t)

For the Einmalbeitrag (m = 0), premium_pp(0) = P_single and zero thereafter. A waiver that begins between two due dates therefore takes effect at the next due date, which is the German convention for a Beitragsbefreiung on a fractionated contract and is stated because the alternative — refunding the unearned instalment — is a different and equally arguable rule the model does not implement.

cum_prem_max_pp(t) is the premium payable to date on an uninterrupted path, P times the number of premium-months elapsed inside the term (for the single-premium form, P_single from t = 0). It is a deterministic quantity, not a ledger, and it is what both the Rückkaufswert and the Beitragsrückgewähr are struck on.

Benefits#

claims(t, "ANNUITY") = R * Σ_g π_g * E_g(t)
claims(t, "LAPSE")   = rkw_pp(t) * pols_lapse(t)
claims(t, "DEATH")   = brg_pp(t) * pols_death(t)
claims(t)            = claims(t, "ANNUITY") + claims(t, "LAPSE") + claims(t, "DEATH")

with

rkw_pp(t) = rkw_prem_ratio(min(y(t), 40)) * cum_prem_max_pp(t) * (1 - stornoabzug_rate)
brg_pp(t) = cum_prem_max_pp(t)   if beitragsrueckgewaehr else 0.0

Note what claims(t, "ANNUITY") is weighted on: E_g(t), the escalation ledger, not ℓ_g(t) — identical in the base run, not identical with the dynamic on, and using ℓ_g would silently drop the escalation. Note also that the Karenz ledger contributes nothing: a life inside its deferred period is in care, counted in pols_if, pays its premium and receives no annuity.

The Beitragsrückgewähr implemented here is the gross form — return of premiums payable to date, with no offset for annuity already paid. The market’s more common form nets the annuity off [S4]; the model does not, because the netting is floored at zero per life and the ledgers are aggregates, so netting at the aggregate level would let a life that received a large annuity subsidise one that received none. The consequence — the option overstates the death benefit relative to the market-standard form — is stated rather than hidden, and pitfall 11 asserts it.

Expenses and net cash flow#

expense_infl_factor(t) = (1 + f) ** (t / 12)
acq_expense_pp()       = α * beitragssumme()
expenses(t)            = acq_expense_pp() * 1{t = 0}
                       + γ * expense_infl_factor(t) * pols_if(t)
                       + β * premiums(t)
claim_expenses(t)      = c * expense_infl_factor(t) * Σ_{g : π_g > 0} ℓ_g(t)
net_cf(t)              = premiums(t) - claims(t) - expenses(t) - claim_expenses(t)
liability_cf(t)        = -net_cf(t)

The acquisition charge falls at t = 0 only, so an in-force model point never incurs it — its frame opens at t = duration_mth_init > 0. That is correct (the cost was incurred before the valuation date) and it is worth knowing before comparing an in-force point’s first row with a new-business point’s.

claim_expenses is per annuity payment made, so it is weighted on the paying grades only and a Pflegegrad 1 life on the delib_std grid generates none. It is published as its own result_cf() column because it is a per-event cost rather than a per-policy one.

The pricing engine — `premium_mth_pp()`#

Where premium_mth > 0 on the model point, that is the premium and the engine is not consulted. Where it is 0.0, P is struck by equivalence on the first-order bases: every rate multiplied by its margin, blended 50 / 50 across the sexes, no lapse, discounted at v = (1+i)^(-1/12). The tar_* ledgers obey exactly the recursions above with w ≡ 0 and the margined forces. Define, over t = 0 … n − 1 on the tariff ledgers:

A  = Σ_t v**t * R * Σ_g π_g * tar_esc_pg(t, g)                  EPV of the annuity
U  = Σ_{t : premium_due(t)} v**t * m * tar_pols_prem(t)         EPV of premium in units of P
G  = Σ_t v**t * γ * expense_infl_factor(t) * tar_pols_if(t)     EPV of per-policy admin
C  = Σ_t v**t * c * expense_infl_factor(t) * Σ_{g:π_g>0} tar_pols_pg(t, g)
D1 = Σ_t v**t * (premium-months elapsed at t) * tar_pols_death(t)    (0 unless BRG)
a1 = α * 12 * (min(prem_end_age, beitragssumme_cap_age) - age_at_entry)

Everything on the benefit side that scales with P — the Beitragsrückgewähr and the Zillmerung allowance — is linear in P, so P * U = A + P * D1 + P * a1 + β * P * U + G + C solves in closed form:

P = (A + G + C) / [ U * (1 - β) - D1 - a1 ]
premium_mth_pp()    = rating_factor * P
prem_net_level_pp() = A / U            the net level premium, benefits only

For the Einmalbeitrag, U = 1 and the same expression gives the Einmalbeitrag directly. The Risikozuschlag multiplies the gross premium, never the benefit.

check_prem_equiv_resid(t) publishes the per-month discounted imbalance v**t * [premium_pp_tar(t) * tar_pols_prem(t) - benefit_tar(t) - expense_tar(t)], whose sum over t is zero when the equivalence holds; check_prem_equiv() is that sum against a tolerance scaled by P * U. It is not a tautology: both sides are assembled from the ledgers rather than from the closed form, so substituting a best-estimate rate into one leg, or dropping the Zillmerung term, makes it fail.

`result_cf()`#

result_cf() returns a DataFrame indexed by t (df.index.name == "t"), contiguous from duration_mth_init() to proj_len() - 1, in this column order — eleven cash-flow and exposure columns and then liability_cf:

#	Column	Meaning
1	`pols_if`	In force at the start of month `t`; first value equals `pols_if_init()` exactly
2	`pols_act`	Of which active
3	`pols_care`	Of which in care — the Karenz and paying ledgers together
4	`pols_prem`	Of which actually paying a Beitrag
5	`premiums`	Beitrag income, in advance
6	`claims_annuity`	Pflegerente paid, in advance
7	`claims_lapse`	Rückkaufswert paid on surrender, at month end
8	`claims_death`	Beitragsrückgewähr paid on death, at month end; structurally 0 in the base run
9	`expenses`	Acquisition, per-policy administration and premium-related administration
10	`claim_expenses`	Per-annuity-payment claims cost
11	`net_cf`	`premiums - claims_annuity - claims_lapse - claims_death - expenses - claim_expenses`
12	`liability_cf`	The same stream outgo-positive, `-net_cf(t)` exactly. Published as a column because the conventions suite verifies the sign convention in the frame rather than in prose

A second frame, result_states(), publishes the ledgers and rates a reader needs to follow the projection: pols_pg1 … pols_pg5, pols_karenz, pols_entry, pols_grad, pols_reactiv, pols_death, pols_lapse, mort_rate, mort_rate_care_pg5, inc_rate, lapse_rate and premium_pp. It is not part of the house contract and carries no check_*.

The published identities#

check_*() takes no argument and returns a bool over all t; the per-t residual is check_*_resid(t). Six identities are published, and the conventions suite calls every one of them on every model point.

Check	Identity
`check_net_cf` (delib ruling 1)	`net_cf(t) = premiums(t) - claims(t, "ANNUITY") - claims(t, "LAPSE") - claims(t, "DEATH") - expenses(t) - claim_expenses(t)` — `net_cf` rebuilt from the statement’s own published parts, so the headline number is reconciled in code and not only in prose
`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
`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 ledgers plus the absorbed 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` when `leistungsdynamik ≥ 0`, with equality when it is zero
`check_prem_equiv`	`Σ_t check_prem_equiv_resid(t) ≈ 0`: the gross premium closes the first-order equivalence, assembled from the tariff ledgers rather than from the closed form

check_net_cf is mandatory across the library; the other five are this product’s own. All six use roll_fwd_tol from basis_table.csv.

Processing order#

Inside month t, in this order. Nothing here is optional: several of the pitfalls below are simply this list executed in a different sequence.

Set the clocks. x(t) = age_at_entry + t // 12; y(t) = t // 12 + 1. At t = duration_mth_init() seed the ledgers from status and pols_if_init() instead of rolling them in — an active point seeds pols_act, a point in claim seeds pols_pg(·, g) at its grade with the Karenz already served.
Classify the in-force. pols_in_term(t), then pols_waived(t) from the paying grades, then pols_prem(t) as the difference. The Karenz ledger is in-term and unwaived.
Collect the Beitrag, in advance. If premium_due(t), premiums(t) = P × m × pols_prem(t); otherwise zero. Accumulate cum_prem_max_pp(t).
Pay the Pflegerente, in advance, on the escalation ledger: claims(t, "ANNUITY") = R × Σ_g π_g × esc_pg(t, g). The Karenz ledger receives nothing.
Charge start-of-month expenses. acq_expense_pp() at t = 0 only; per-policy administration on pols_if(t), inflated; premium-related administration on the premium just collected; and claim_expenses(t) on the annuity payments just made.
Look up the month’s forces at x(t). μ_A, and ι — zero while t < wartezeit_months; μ_g = mort_mult(g) × μ_A; δ_g; ρ_g, damped above rec_age_ref. Then w(t), zero outside the premium term.
Apply the transitions over the month, constant forces, proportional allocation. From the active state: death, entry into care split by entry_share, or stay. From each grade, in both the Karenz and the paying ledger: death, deterioration, recovery, or stay.
Advance the Karenz clock by one month and graduate the z = K cohort into pols_pg.
Apply lapse, to the survivors of the active state after the insured decrements and after the reactivation inflow. pols_lapse(t) is the result; nothing in care lapses.
Pay the end-of-month benefits. claims(t, "LAPSE") = rkw_pp(t) × pols_lapse(t), and claims(t, "DEATH") = brg_pp(t) × pols_death(t) where the option is on.
Roll the escalation ledger: escalate the surviving weights by (1 + d)^(1/12), then add the graduating entrants at weight 1.
Post the ledgers to t + 1 and form net_cf(t) = premiums - claims_annuity - claims_lapse - claims_death - expenses - claim_expenses.

The projection ends at t = proj_len() - 1. There is no maturity, no survival benefit and no tail state: the contract runs for life, and the closure identity is carried by the decrements.

Known modeling pitfalls#

The specific ways an implementation of this product looks right and is wrong. Each is a test in tests/test_pflegerentenversicherung_de.py.

Applying an average benefit percentage to an average survival curve. Grade and mortality are correlated: Pflegegrad 5 pays most and is lived in shortest. Assert claims(t, "ANNUITY") equals R × Σ_g π_g × esc_pg(t, g), and that replacing it by π̄ × R × pols_care(t), with π̄ the entry-mix mean percentage 0.3815, understates the projected annuity total by 30 % on the anchor cell — 9,248.24 € against 13,200.11 €. The trap is that the substitution is exact at t = 1, where the stock is still the entry mix; the error opens as deterioration moves the stock up the schedule to a stock-weighted mean of 0.544519. Substituting the stock-weighted mean instead reproduces the total by construction and tests nothing.
Pricing the annuity in payment on an annuity table. DAV 2004 R is prudent about people living longer REG-R49; this annuity is paid to a heavily impaired population. Assert mort_rate_care(t, g) ≥ mort_rate(t) for every g and t, strictly increasing in g, and that the force multiple mort_force_care(t, g) / mort_force(t) is exactly mort_mult(g) — 9.0 at grade 5 — at every age below the limiting one. Assert it on forces and not on rates: the rate ratio at grade 5 falls from 8.97 at age 45 to 6.85 at 85, 4.31 at 95 and 1.00 at the limiting age, because a rate saturates and a force does not. That compression is the scale, not the basis.
Treating “in claim” as one state exited only by death. There are three exits. Assert Σ_t pols_reactiv(t) > 0, that recovery moves lives down the grades, and that suppressing recovery and downgrade raises the projected annuity total.
Insuring Pflegegrad 1 by accident. On delib_std, π_1 = 0. Assert claims(t, "ANNUITY") is invariant to pols_pg(t, 1) and that a grade-1 life is counted in pols_prem, not pols_waived.
Waiving the premium at the wrong grade. Assert pols_waived excludes grade 1 on delib_std (point 1) and includes it on bahr (point 5), with check_waiver() closing on both.
Forgetting that the premium revives on a Herabstufung. Assert the flow rather than the stock: Σ_t pols_pg(t,2) × p_pg_better(t,2) > 0 on delib_std, lives leaving the insured grades and starting to pay again; pols_prem(t) ≥ pols_act(t) at every t, strictly wherever pols_pg(t,1) > 0, because an uninsured-grade life and a Karenz life both pay; and check_waiver() closing throughout. The stock pols_prem(t) is nevertheless monotone decreasing on every shipped model point — attrition dominates the revival flow by three orders of magnitude — so a test asserting non-monotonicity would be asserting a coincidence, not the mechanic.
Adding monthly transition probabilities instead of allocating one survival. Assert p_pg_stay + p_pg_death + p_pg_worse + p_pg_better == 1 to 1e-12 for every t and g, and the same for the three active-state probabilities.
Dividing an annual rate by twelve. Assert mort_rate_mth(t) == 1 - (1 - mort_rate(t))**(1/12) and the same form for lapse; that the monthly rate is strictly below the annual one wherever that is positive, which is the house convention; and that q/12 is strictly below 1 - (1 - q)^(1/12), so dividing by twelve understates the monthly decrement. The direction is worth stating because it is the opposite of the intuition: twelve monthly rates added overshoot the annual rate (12 x 0.00006499 = 0.00077984 against q_A(45) = 0.00077956), while twelve of them compounded reproduce it exactly. An earlier draft of these notes asserted the overshoot the wrong way round; the model contradicted it and the model is right.
Treating the Karenzzeit as a benefit gate on the aggregate. It is a deferral clock per onset. On point 7 assert Σ_t Σ_g pols_grad(t,g) < Σ_t Σ_g pols_entry(t,g) strictly, the shortfall equalling deaths and recoveries recorded inside the Karenz ledger.
Escalating the annuity at general-population duration. With d = 0.02 on point 8 the projected annuity total rises from 13,200.11 € to 15,101.44 €, +14,4 %, and the equivalence premium by +12,2 %. Assert both, and esc_pg == pols_pg exactly on point 1. An earlier draft of these notes predicted “less than 5 %” from a four-year spell; the model contradicts it and the model is right on the shipped basis, 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, where the escalation factor is largest. ln(1.144) / ln(1.02) = 6,8 years of payment-weighted elapsed duration against a mean spell in an insured grade of 5,5 years. The comparison that survives is the weaker one: 14,4 % is below the 18 % the same escalation buys on a healthy-life annuity of seventeen-year duration, but not far below it, and a Leistungsdynamik on this basis is not cheap.
Netting the annuity off a Beitragsrückgewähr at the aggregate level. The floor at zero is per life; the model pays the gross form. Assert claims(t, "DEATH") == cum_prem_max_pp(t) × pols_death(t) on point 9 and == 0.0 on point 1.
Paying a surrender value out of the paying state. Assert pols_lapse(t) ≤ pols_act(t), claims(t, "LAPSE") == 0 wherever pols_act(t) == 0, and lapse_rate(t) == 0 once x(t) ≥ prem_end_age (point 4).
Charging the Zillmerung on the wrong base. The 25 ‰ ceiling is a per-mille of the Beitragssumme, not of the annual premium REG-R16. Assert acq_expense_pp() == 0.025 × premium_mth_pp() × 12 × (min(prem_end_age, 85) - age_at_entry), charged at t = 0 only, so an in-force point never incurs it.
Striking the equivalence premium on the projection basis. Assert check_prem_equiv() is True, that premium_mth_pp() is invariant to every value in lapse_table.csv, and that tar_inc_rate(t) exceeds the unisex-blended best-estimate incidence 0.5 i_M(x) + 0.5 i_F(x) by exactly inc_margin at every age below the inc_cap. Against the anchor cell’s own female incidence the ratio is 1.128 at age 45 and 1.089 at 85, not 1.25: the blend and the margin are two separate operations and only the second is prudence.
Pricing on the model point’s own sex. Assert points 1 and 2 — female and male, identical otherwise — have equal premium_mth_pp() and unequal projected annuity totals, the female point’s being larger REG-R34.
Collecting a premium in a month that is not a due date. Assert premiums(t) == 0 at every non-due t on points 3, 4 and 5, and premium_pp(t) == premium_mth_pp() × prem_mode_months() at every due t.
Using the Pflegegrad stock distribution as the entry mix. The stock is 13,8 / 40,4 / 29,6 / 11,8 / 4,3 % at end-2023 R18; entrants are not. Assert Σ_g entry_share(g) == 1.0 and that the model’s own stock share at grades 4 and 5 over the whole projection exceeds entry_share.

Two further errors are worth naming because they are the ones a user will make. Reading a payment-frequency difference as a price difference: the model folds the Ratenzahlungszuschlag into the administration assumption, so annual mode prices slightly below monthly, the opposite sign to a real tariff. And treating proj_len() as the last projected index: it is the number of projected months, the frame’s exclusive end, so the last index is proj_len() - 1; and because the frame starts at duration_mth_init(), an in-force point publishes proj_len() - duration_mth_init() rows, not proj_len().

Policyholder behaviour modelling#

Every dynamic formula here is a std reference construction; no German calibration evidence for any of them exists in this corpus (research gap 20).

Base lapse std. The duration table in class (c), from the active state only, zero after the premium term ends. The argument for the shape, not its level, is Zillmerung: the Rückkaufswert is near zero for the first years REG-R16 REG-R28, so an early lapse is expensive to the policyholder and the profile is flatter than a savings product’s.
No lapse from a paying grade std. A claimant with a waived premium has no premium to default on and a live annuity to forfeit. A Pflegegrad 1 life on delib_std does still pay and could in principle lapse; the population is small and the model does not model it.
Premium-shock lapse std, optional, off. The Beitrag is level and guaranteed, so the affordability shock frlib’s temporaire_deces models has no counterpart here. What could replace it is a Zahlbeitrag shock — a withdrawal of the surplus rebate raises the amount actually called without invoking § 163 REG-R27 REG-R53 — and the model carries no rebate, so the module is empty and is named only so that a user who adds a rebate knows what to add with it.
Selective lapsation std, optional, off. Lapsers are healthier, so persisters’ incidence should be loaded: inc_rate_eff(t) = inc_rate(t) × [1 + λ × max(0, w_cum(t) - w_ref)] with w_ref = 0.30, λ = 0.25 and base run λ = 0. The effect is smaller here than on a term cover, because cumulative lapse is largely complete decades before the risk period.
Not modelled at all, each for a stated reason. Beitragsfreistellung take-up — no split between the three German exits exists in the corpus REG-R28. Beitragsdynamik acceptance — a behaviourally driven second premium path with no evidence behind it. Höherstufung application behaviour — the insured applies and the state re-assesses R6, so grade change is biometric here rather than elective. And anti-selection at purchase, which underwriting removes [S4] and which decays long before the claims arrive.

Worked example#

Configuration. Model point 1, the anchor cell, in full: point_id = 1; policy_id = "PFL-000001"; sex = F; age_at_entry = 45; duration_mth_init = 0; status = aktiv; rente_mth = 1,000.00 EUR per month, the vereinbarte Pflegerente at Pflegegrad 5; staffel_id = delib_std, so the Leistungsstaffel is 0 / 30 / 50 / 75 / 100 % across grades 1 to 5; prem_end_age = 110, equal to omega_age, so the Beitrag is payable for life; prem_mode = monthly, so prem_mode_months = 1 and a Beitrag is due in every month of the term; premium_mth = 0.00, the sentinel that makes the model strike the Beitrag by equivalence; rating_factor = 1.00, standard rates; wartezeit_months = 0; karenz_months = 0; leistungsdynamik = 0.00; beitragsrueckgewaehr = False; stornoabzug_rate = 0.00; pols_if_init = 1.0. Hence proj_len() = 12 × (110 − 45) = 780, the frame runs from t = 0 to t = proj_len() − 1 = 779, and the projection covers attained ages 45 to 109 — 780 monthly rows, of which the table below shows a representative selection together with the full-precision totals.

Assumptions, each tagged. Rechnungszins i = 1.00 % p.a., used only in the pricing engine REG-R14 REG-R15. Active-life mortality mort_rate(F, x) = 1 − exp(−4.76290e−06 × 1.119962^x) std, anchored at q(65) = 0.75 % and q(85) = 7.0 %, with mort_rate(F, 109) = 1.0 by the limiting-age convention std. Incidence inc_rate(F, x) = min(0.0110 × exp(0.1400 × (x − 65)), 0.50) std, anchored on prevalence doubling every five years of age above 75 — an anchor the retrieved Pflegequoten put nearer a factor of 1,8 over 70–90 R18, reported and not changed. Entry mix 0.20 / 0.38 / 0.24 / 0.13 / 0.05 across grades 1 to 5 std. Deterioration 0.28 / 0.24 / 0.20 / 0.16 / 0.00 and recovery 0.10 / 0.06 / 0.04 / 0.02 / 0.01 a year std, the recovery rates damped by exp(−0.10 × max(0, x − 75)) std. In-care mortality multiples 1.5 / 2.5 / 3.5 / 6.0 / 9.0 on the active force std. Lapse from the active state 6.0 / 5.0 / 4.0 / 3.5 / 3.0 % in policy years 1 to 5, 2.5 % in years 6–10, 2.0 % in 11–20 and 1.5 % from year 21 std, converted by 1 − (1 − q)^(1/12). Rückkaufswert 0.00 / 0.00 / 0.05 / 0.12 / 0.20 of premiums paid in policy years 1 to 5, rising to 0.42 by year 10 and 0.70 by year 40 std, with stornoabzug_rate = 0. Expenses std: acquisition 25 ‰ × Beitragssumme at t = 0, the Beitragssumme being P × 12 × (85 − 45) = 480 P; administration 3.0 % of each Beitrag collected plus 2.00 € per policy in force per month; claim expense 1.50 € per annuity payment; expense inflation 1.5 % a year. First-order margins std: incidence × 1.25, deterioration × 1.15, recovery × 0.80, in-care mortality × 0.85, active mortality × 0.90, and no lapse in the tariff basis. Pricing blended 50 / 50 male / female std REG-R34; the projection runs on the female basis. No Wartezeit, no Karenzzeit, no Leistungsdynamik, no Beitragsrückgewähr, no Überschussbeteiligung and no behaviour modules.

The band the research file argues for this configuration is about 50,00 € to 100,00 € a month std (product spec, footnote 9). It is derived arithmetic, not a market observation: a model premium well outside it indicates an error in the bases, and one inside it is not thereby validated. The table below prints the model’s own equivalence premium; the text under it says which end of the band it lands at and why.

The model’s own output#

Every figure is transcribed from Projection[1].result_cf(), money to the cent and policy counts to six decimals. The frame has 780 rows, t = 0 … 779; fourteen are shown. claims_death is a column of the frame, is structurally zero at every t here (beitragsrueckgewaehr = False), and is omitted rather than printed as 780 zeros. The equivalence premium is premium_mth_pp() = 64.198409 € a month.

`t`	`x(t)`	`pols_if`	`pols_care`	`pols_prem`	`premiums`	`claims_annuity`	`claims_lapse`	`expenses`	`claim_expenses`	`net_cf`
0	45	1.000000	0.000000	1.000000	64.20	0.00	0.00	774.31	0.00	-710.11
1	45	0.994793	0.000056	0.994748	63.86	0.02	0.00	3.91	0.00	59.93
2	45	0.989613	0.000111	0.989523	63.53	0.04	0.00	3.89	0.00	59.59
12	46	0.939288	0.000644	0.938758	60.27	0.26	0.00	3.71	0.00	56.29
60	50	0.798781	0.003771	0.795452	51.07	1.86	1.60	3.25	0.01	44.35
120	55	0.698284	0.009864	0.689306	44.25	5.52	3.92	2.95	0.02	31.85
240	65	0.547048	0.035456	0.514343	33.02	21.39	5.65	2.46	0.07	3.45
360	75	0.391509	0.080375	0.318167	20.43	45.97	5.83	1.84	0.17	-33.38
420	80	0.276904	0.092056	0.193720	12.44	49.35	4.20	1.31	0.21	-42.64
480	85	0.143383	0.073044	0.078218	5.02	36.02	1.87	0.67	0.18	-33.71
540	90	0.039295	0.028934	0.013981	0.90	12.90	0.30	0.18	0.07	-12.56
600	95	0.003283	0.003060	0.000696	0.04	1.20	0.01	0.02	0.01	-1.19
660	100	0.000055	0.000053	0.000015	0.00	0.02	0.00	0.00	0.00	-0.02
779	109	0.000000	0.000000	0.000000	0.00	0.00	0.00	0.00	0.00	0.00
Total		268.956131	24.241784	247.014740	15,857.95	13,200.11	2,191.72	1,941.10	52.67	-1,527.65

The Total row is summed over all 780 rows at full precision and then rounded, not summed from the rounded cells, and here the two visibly differ: summing the rounded cells gives premiums 15,857.92, claims_annuity 13,200.02, claims_lapse 2,191.67, expenses 1,941.02, claim_expenses 52.58 and net_cf -1,527.51 — the last of these 0,14 € away. The gap is not noise that cancels. 107 of the 780 claims_annuity cells are positive amounts below half a cent, and every one of them rounds down to 0.00, so the error accumulates in one direction; claim_expenses does the same, its 1,50 € per payment falling on a paying population of the order of 1e-5 for the first two decades.

Three independent checks#

Check 1 — the equivalence premium, from the four actuarial values. The premium is not read off a rate card; it is the solution of the first-order equivalence, and the values that determine it are published as cells: epv_benefits() = A = 17,789.761930, epv_prem_units() = U = 313.500018, epv_admin() = G = 892.884210, epv_claim_expense() = C = 69.389246, with β = admin_prem_pct = 0.030 and the Zillmerung allowance a1 = 0.025 × 12 × (85 − 45) = 12.000000 in units of P:

P = (17,789.761930 + 892.884210 + 69.389246) / (313.500018 x 0.970 - 12.000000)
  = 18,752.035386 / 292.095018  =  64.198409

which is premium_mth_pp() to the sixth decimal. Two by-products are worth reading. U is 26,13 years’ worth of discounted premium, against the “of the order of 27 years’ worth” the research file argues from the other direction. And prem_net_level_pp() = A / U = 56.745649, so the whole expense loading is 64.198409 / 56.745649 = 1.131336 — 13,13 % over the net level premium, of which the Zillmerung allowance alone is 2,53 € a month: strike a1 out of the denominator and the premium falls to 18,752.035386 / 304.095018 = 61.665.

Check 2 — the first month’s decrements, from the annual rates. Nothing here needs the model. Read two annual rates out of the CSVs at age 45 female, convert them to forces, allocate one month’s exits between them in proportion, and apply the lapse rate afterwards:

q_A(45) = 0.00077956    mu_A = -ln(1 - q_A) = 0.00077986
i(45)   = 0.00066891    iota = -ln(1 - i)   = 0.00066913      mu_A + iota = 0.00144900

p_act_stay(0)  = exp(-0.00144900 / 12)                        = 0.99987926
p_act_death(0) = (0.00077986 / 0.00144900) x (1 - 0.99987926) = 0.00006498
p_act_care(0)  = (0.00066913 / 0.00144900) x (1 - 0.99987926) = 0.00005576
                                                        sum   = 1.00000000 exactly

w(0) = 1 - (1 - 0.06)^(1/12) = 0.00514301
pols_act(1)   = 0.99987926 x (1 - 0.00514301) = 0.99473687
pols_lapse(0) = 0.99987926 x 0.00514301       = 0.00514239
pols_death(0) = 0.00006498     pols_care(1) = 0.00005576   (entrants, no Karenzzeit to serve)
pols_if(1)    = 0.99473687 + 0.00005576       = 0.99479262

against the table’s pols_if(1) = 0.994793, and the roll-forward closes on the same three numbers: 1.00000000 - 0.00006498 - 0.00514239 = 0.99479262. Note where the lapse falls — on the survivors of both insured decrements, not on the opening cohort. Applying it to the opening cohort instead would put pols_lapse(0) at 0.00514301, 6,2e-7 of a policy adrift in the first month and materially more once the decrements are large. Both orderings are internally consistent and both close check_pols_roll_fwd(), which is exactly why the processing order has to be declared rather than inferred from a check.

Check 3 — the first annuity payment, grade by grade. The month-0 entrants split across the grades by inc_share, and with karenz_months = 0 they graduate in the same month, so pols_pg(1, g) = esc_pg(1, g) = pols_entry(0, g):

`g`	`inc_share(g)`	`pols_pg(1, g)`	`benefit_pct(g)`	contribution
1	0.20	0.0000111516	0.00	0.00000000
2	0.38	0.0000211880	0.30	0.00635640
3	0.24	0.0000133819	0.50	0.00669095
4	0.13	0.0000072485	0.75	0.00543638
5	0.05	0.0000027879	1.00	0.00278789
	1.00	0.0000557578		0.02127162

1,000.00 × 0.00002127162 = 0.0212716 €, the table’s claims_annuity(1) = 0.02. This is also the arithmetic of pitfall 1. The entry-mix mean percentage is 0.38 × 0.30 + 0.24 × 0.50 + 0.13 × 0.75 + 0.05 × 1.00 = 0.3815, and applying it to pols_care(1) reproduces the annuity exactly, because at t = 1 the stock is the entry mix. It does not stay that way: over the projection the stock share by grade is 9.49 / 24.25 / 27.64 / 21.04 / 17.57 % against an entry share of 20 / 38 / 24 / 13 / 5 %, and the stock-weighted mean percentage is 0.544519. Applying the entry-mix mean to Σ_t pols_care(t) = 24.241784 gives 9,248.24 € against 13,200.11 € — a 30 % understatement of the whole benefit, produced by an error no total in the frame would reveal.

The closure identity. Read one month past the last projected row, at t = proj_len() = 780: pols_dead_cum(780) = 0.493968059928 and pols_lapse_cum(780) = 0.506031940072 sum to 1.000000000000 with pols_if(780) = 1.5e-23: the decrements account for the entire policy, exactly, because mort_rate is forced to 1.0 at age 109. The cash flow statement closes the same way at every t — at t = 0, 64.198409 − 774.306859 = −710.108450, with expenses(0) = 770.380907 + 2.000000 + 1.925952, the allowance 0.025 × 30,815.236279, the month’s per-policy administration and 3,0 % of the Beitrag just collected. The largest check_net_cf_resid(t) anywhere in the frame is 1.4e-14, and check_prem_equiv() closes to 9.3e-12.

Reading the result#

The premium lands at 64,20 € a month, in the lower half of the argued 50,00–100,00 € band std. Three things put it there. The entry mix is weighted to the two lowest grades, which pay 0 % and 30 %, so the stock-weighted average benefit is 54,5 % of the vereinbarte Rente. The model’s own mean spell in an insured grade is 5,5 years, at the top of the three-to-five the research file argues. And the expense loading is 13,13 %, modest because the claims cost is per payment and low.

The sign pattern is the product’s whole economic story. Month 0 is -710,11 €, almost all of it the 25 ‰ Zillmerung allowance charged in one go. From t = 1 the contract runs positive — the level Beitrag is far above the risk premium — and the monthly margin decays from 59,93 € to 3,45 € by age 65. net_cf turns negative for good at t = 252, attained age 66, where the incidence curve overtakes the level premium and, in the layer this model does not compute, where the Deckungskapital peaks. Annuity outgo peaks at 49,82 € in month 407 (age 78) and the population in care at 0.092120 in month 417 (age 79); the last three decades are run-off. Undiscounted the contract collects 15,857.95 € and pays 17,385.60 € of benefit and expense, for -1,527.65 € — not a loss but the consequence of publishing an undiscounted stream whose income falls thirty years before its outgo.

A second scenario: the male twin, model point 2#

Model point 2 is model point 1 with sex = M and nothing else changed — the model’s own statement of the unisex tension: the two cells are priced identically and project differently.

`t`	`x(t)`	`pols_if`	`pols_care`	`pols_prem`	`premiums`	`claims_annuity`	`claims_lapse`	`expenses`	`claim_expenses`	`net_cf`
0	45	1.000000	0.000000	1.000000	64.20	0.00	0.00	774.31	0.00	-710.11
1	45	0.994719	0.000045	0.994683	63.86	0.02	0.00	3.91	0.00	59.93
12	46	0.938452	0.000517	0.938027	60.22	0.21	0.00	3.71	0.00	56.30
120	55	0.688060	0.007570	0.681181	43.73	4.20	3.87	2.91	0.01	32.74
240	65	0.517213	0.024372	0.494872	31.77	14.23	5.44	2.35	0.05	9.71
360	75	0.332372	0.045899	0.291187	18.69	24.19	5.37	1.60	0.10	-12.56
420	80	0.214158	0.047250	0.172470	11.07	22.80	3.80	1.05	0.11	-16.68
480	85	0.098388	0.033682	0.069319	4.45	14.73	1.72	0.49	0.08	-12.58
540	90	0.023361	0.012108	0.013274	0.85	4.73	0.33	0.12	0.03	-4.36
600	95	0.001538	0.001188	0.000612	0.04	0.40	0.01	0.01	0.00	-0.38
Total		251.397804	13.561865	239.326567	15,364.38	6,936.34	2,086.29	1,871.13	28.29	4,442.33

premium_mth_pp() is 64.198409 on both cells, to every decimal, because the pricing engine reads the unisex blend and never sex(). The projections are not close: the male cell’s claims_annuity total is 6,936.34 €, 52,5 % of the female cell’s, and its net_cf total is +4,442.33 € against -1,527.65 €. That 5,969.98 € gap is the cross-subsidy the unisex rule requires, priced on a 50 / 50 mix; a book written 60 / 40 female against this price carries a tenth of it — about 597 € of undiscounted net cash flow per policy — unfunded. Two drivers pull in opposite directions and incidence wins: male active mortality is higher at every age (q(M, 45) = 0.0016637 against q(F, 45) = 0.0007796; q(M, 85) = 0.1049833 against 0.0699052), so fewer men survive to claim, and male incidence is lower (i(M, 85) = 0.1342987 against i(F, 85) = 0.1808911), so fewer of the survivors claim.

The switchable variants beside the anchor#

All four cells run on the same tables and differ only in the model point. Totals over the whole frame, summed at full precision and then rounded:

	1 — anchor	2 — male twin	5 — `bahr` grid	8 — Leistungsdynamik
Configuration	F 45, `delib_std`	M 45, `delib_std`	F 50, `bahr`, annual	F 45, `d = 0.02`
`premium_mth_pp()`	64.198409	64.198409	55.444644	72.038378
`proj_len()`	780	780	720	780
`premiums`	15,857.95	15,364.38	12,289.30	17,794.54
`claims_annuity`	13,200.11	6,936.34	10,110.36	15,101.44
`claims_lapse`	2,191.72	2,086.29	1,482.02	2,459.37
`expenses`	1,941.10	1,871.13	1,556.50	2,093.27
`claim_expenses`	52.67	28.29	57.74	52.67
`net_cf`	-1,527.65	4,442.33	-917.32	-1,912.22

The bahr grid is not simply a cheaper contract. Point 5 enters five years later, so its premium is not comparable with the anchor’s; the shape is. On 10 / 20 / 30 / 40 / 100 % the middle steps are roughly two thirds of delib_std’s while grade 1 is insured for the first time, and the second effect surfaces where a reader would not look for it: claim_expenses is higher on point 5 (57.74 €) than on the anchor (52.67 €) despite a frame sixty months shorter, because a grade-1 life now generates an annuity payment and so a per-payment cost — about a tenth of the care-months that delib_std charges nothing for — and because point 5 sits five years closer to the claim ages. The same life is also waived on bahr and pays on delib_std.

The Leistungsdynamik costs more than the draft of these notes predicted, and pitfall 10 has been rewritten to the model’s number: +14,4 % on the annuity total and +12,2 % on the premium, not the “less than 5 %” a four-year spell suggests. esc_pg == pols_pg exactly on the anchor at all 780 × 5 combinations — check_esc_ledger() asserts it — and diverges on point 8: at t = 480, esc_pg(480, 3) = 0.024913 against pols_pg(480, 3) = 0.022576, a factor of 1.1035 which is exactly the average escalation the lives then in grade 3 have accrued.

What the model and test stages changed in these notes. Six things, listed so the diff is not silent. The worked example above replaced a placeholder. Model point 9 acquired a premium-paying term to age 65, for the reason given under The fourteen model points. Pitfalls 1, 2, 6, 10 and 14 carried numeric predictions the model contradicts and were rewritten to what the model actually produces — respectively the entry-mix rather than the stock-weighted mean percentage, the force rather than the rate multiple, the revival flow rather than a non-monotone stock, +14,4 % rather than “less than 5 %”, and the unisex-blended rather than the sex-specific incidence. Pitfall 8 asserted an inequality in the wrong direction — that twelve times the monthly rate is below the annual rate — and was rewritten when the test module put it to the model: twelve monthly rates added overshoot, and it is q/12 that falls short. The count of sub-cent claims_annuity cells above was 108 in the draft and is 107 in the frame. liability_cf was added to the result_cf() column list as column 12, because the conventions suite verifies the sign convention in the frame. And two paragraphs duplicated in the draft — the Currency, sign and rounding bullet and steps 11 and 12 of the processing order — were reduced to one each. Nothing in the shipped tables was changed to make a prediction come true.

Valuation and reserve pointers#

This library projects gross best-estimate-style liability cash flows, undiscounted, on a declared grid. The valuation layers consume them and are cited, not reproduced.

The Deckungsrückstellung. § 341f HGB requires it prospectively on the tariff bases R12 REG-R54; the DeckRV fixes the Höchstrechnungszins those bases may use — 1,00 % for new business from 1 January 2025, and whatever rate the cohort was written on, for the whole term REG-R14 REG-R15 — and caps the Zillmersatz at 25 ‰ of the Beitragssumme REG-R16. Here the reserve rises for roughly thirty-five years, peaks where the incidence curve crosses the level premium, and runs off: an ageing reserve in economic function and a Deckungsrückstellung in law. The model does not compute it, and the Rückkaufswert enters as contractual data rather than as a derived reserve — which is what keeps check_prem_equiv_resid(t) a closure residual rather than a reserve in disguise.
The Zinszusatzreserve. An HGB reserve with no counterpart elsewhere in this repository, driven by the ten-year Referenzzins of § 5 Abs. 3 DeckRV under the Korridormethode REG-R17. A long-dated, interest-sensitive contract is exactly what attracts it. Not computed here.
The § 169 VVG floor. A model carrying a zillmerised reserve applies two rules separately — what the DeckRV lets the insurer reserve, and what § 169 makes it pay with acquisition costs spread over at least five years — the tighter binding REG-R16 REG-R28. surrender_table.csv encodes the result of both. Whether a pure-risk Pflegerente is inside § 169 at all is open (gap 9).
Solvency II best estimate. Probability-weighted future cash flows discounted at the relevant risk-free term structure, plus a risk margin REG-R1 REG-R2 REG-R6, reaching German life business through the VAG rather than directly: BEL = Σ_t v(t) × liability_cf(t). No cost-of-capital rate, contract-boundary rule or standard-formula shock in this library was read from a retrieved instrument.
The contract boundary, easier here than on a Pflegetagegeld. The premium is level and guaranteed, adjustable only on the narrow § 163 route REG-R27, so the insurer has no unilateral right to reprice the individual contract — the usual trigger for a boundary at the next repricing date. The whole projected stream is therefore inside the boundary on the natural reading; a Pflegetagegeld under § 203 VVG is the opposite case R14 — MB/EPV § 8b recalculates every premium in an observation unit on a breach [S2]. The boundary point itself is still unverified; no Solvency II instrument was retrieved for this product.
Surplus, IFRS 17 and professional standards. The Risikoergebnis — the release of the first-order margins as experience emerges — is the dominant surplus source here REG-R47, distributed under the MindZV and the RfBV REG-R10 REG-R18 REG-R19; the model produces the best-estimate leg and the tar_* engine the first-order leg, and neither becomes a declared rate. IFRS 17 fulfilment cash flows plus a CSM REG-R55 are fed by the same engine, with grouping, CSM and risk adjustment out of scope. The DAV’s Fachgrundsätze bind the responsible actuary who signs the bases REG-R11 REG-R56.

Key sensitivities and model risks#

In rough order of leverage for a German Pflegerente block.

Duration in care — the in-care mortality multiples. The direct multiplier on the liability and still the weakest-evidenced quantity in the model (gap 19); the research file’s own arithmetic shows a mean spell moving from four years to five changing the premium by about a quarter. The shipped multiples are std order-of-magnitude reasoning, and they carry the product. The overall spell is now sourced — about five years where care begins after 60, 4,0 for men and 5,7 for women [S14] — which puts the model’s implied three-to-five years at the low end for women and makes this sensitivity, if anything, sharper. The per-grade sojourn times are not sourced and apparently cannot be, the Pflegegrade dating only from 2017.
Incidence level and slope. The slope is anchored to prevalence doubling every five years above 75 and compounds over sixty-five years, so it is the more dangerous of the two: ±0.01 on g moves incidence at 90 by about a third. The anchor is now measured and is softer than assumed: the observed five-year prevalence ratios are 1,80 · 1,88 · 1,74 · 1,49 · 1,18 from 70–75 upward, about ln 1.8 / 5 = 0.117 against the shipped 0.1386 R18. Shipped unchanged, reported here.
The Pflegegrad definitional break. DAV 2008 P was built on the Pflegestufen, the BGH has held that no inference runs from a Pflegegrad back to a Pflegestufe REG-R36 REG-R51, and the 2017 reform widened the insured population R9. This is still the largest basis risk in the product, it is not a parameter, and no sensitivity here captures it — but it is now a risk with a published response: the DAV’s re-derivation of the bases for Pflegegrade, on a Stufenmodell rather than a per-grade chain, and from Pflegestufen data rather than observations R15.
The Rechnungszins. Benefits fall on average some thirty-five years after issue, so this is the most interest-sensitive product in delib. It is also the one pricing assumption that is genuinely cited REG-R14 REG-R15.
The middle steps of the Leistungsstaffel. The time-weighted average benefit over a spell is about half the top step, so two tariffs with the same 100 % top step differ by more than the headline suggests. Compare model point 1 with model point 5.
Lapse and the Stornoabzug. Nothing supports any lapse level (gap 20). Here lapse is profitable — an early lapser paid for years and never reached the risk period — so the usual protection intuition is inverted, and the pricing basis deliberately excludes it. The Stornoabzug is a separate and newly quantified risk: the model ships 0 % with a 5 % model point, while the one Pflegerenten wording retrieved agrees 25 %, rising to 50 % after a partial withdrawal, justified as compensating the change in the residual book’s risk profile [S4] REG-R28. On a lapse-heavy block the difference between 0 % and 25 % is a first-order effect on surrender outgo, and it is not implemented here.
The unisex mix. Pricing blends 50 / 50 while the projection runs on the point’s own sex; a book written 60 / 40 female against a 50 / 50 price is under-priced by the whole mismatch, and the mix is endogenous to the price REG-R34.
Expense levels and the Zillmerung base. Every level is std (gap 2). At 500 € a month on point 14 the per-policy expense line, not the biometrics, decides whether the cell is viable.
No select mortality after onset. Mortality is highest immediately after onset, especially at grades 4 and 5 unverified; the model uses an aggregate. The model therefore understates how much a Karenzzeit removes, so point 7’s reduction is a floor rather than an estimate.
The terminal age and the payment convention. omega_age = 110 with q = 1 forced in the last year of age, and annuity and premium both in advance — declared conventions rather than facts. Moving the annuity to arrears would shift the benefit stream a month and break the waiver’s alignment with it.