Technical Notes#

Status: Draft, 2026-08-26 (all cited sources accessed 2026-08-26; see sources.md).

Scope note. These notes specify a reference liability cash flow projection model for the standardized composite product defined in product-spec.md (same directory) — the French contrat obsèques in its capital form. This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in _research/obseques.md via sources.md; [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 not confirmed against a retrieved document. Parameter values are identical to those in product-spec.md. Euro amounts use a decimal point and no thousands separator so that every figure is machine-checkable; the sources use the French decimal comma and the figures are transcribed unchanged.

The model is Obseques_FR_S, on a monthly grid. One engine serves three cells, which differ only in the premium_form column of the model point table:

  • RefOBS-VIA — primes viagères: entry 50, capital 5000 €, 336.03 €/year for life, revalorisation 1.00 % p.a. [S14]. This is the worked-example cell.

  • RefOBS-TMP — primes temporaires: entry 50, capital 5000 €, 651.26 €/year for 10 years, revalorisation 1.00 % p.a. [S14].

  • RefOBS-UNI — prime unique: entry 50, capital 5000 €, 4274.04 € once [S5], revalorisation 0.00 % std (the source presents its values sans participation aux bénéfices, so a non-zero rate would be inconsistent with its own surrender scale [S5]).

Two structural differences from the UK guaranteed-acceptance sibling in uklib/products/whole_of_life drive most of the extra machinery here. The sum assured is a state variable that grows out of the participation aux bénéfices, not a constant; and lapse pays money — the surrender value is the provision mathématique [S1] [S8] [S9] — so the “every lapse is a free profit release” arithmetic of the UK design does not carry over.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows (premium income, death outgo, surrender outgo, expenses) for single-policy model points. Reserves, discounting, risk margin and capital are pointed to, not computed.

  • Projection frequency. Monthly std. The contractual premium is annual and payable in advance with instalment options [S1] [S8] [S9]; the monthly grid is required by the twelve-month délai de carence, whose boundary must not be smoothed.

  • Time index. t is the 0-based policy month: t = 0 is the first policy month, month t runs from time t to time t + 1, and the frame is t = 0, 1, …, proj_len − 1, so proj_len is the number of projected months. The policy year is the contractual, 1-based label derived from it, y = policy_year(t) = floor(t/12) + 1, and it is what the premium, lapse and select schedules are keyed by. Where these notes say “policy year 1” they mean the contractual year y = 1, which is t = 0 … 11.

  • Timing conventions std. Premiums at the beginning of the policy month (BOM); deaths resolved at end of month (EOM) against the BOM in-force; surrenders and réductions at EOM after deaths. Revalorisation and premium uprating step at policy anniversaries.

  • Age basis. Différence de millésime — calendar year of subscription minus calendar year of birth [S1] [S8] [S9]; not age last birthday and not age nearest birthday. The true basis increments on 1 January; the model increments at the policy anniversary instead, age(t) = entry_age + floor(t/12) std, exact for January issues.

  • Projection horizon. proj_len = 12 × (omega − entry_age + 1) months with omega = 112 std — the number of projected months, so the last one is t = proj_len − 1 — the tabulation limit of TH 00-02 in the annexe to art. A. 335-1 CA REG-R23; mort_rate is forced to 1 at attained age omega. One insurer’s tables run to attained age 115 [S15], so the horizon is a modelling convention, not a contractual one.

  • Currency / units. EUR. mort_rate and lapse_rate are annual and dimensionless, converted as q_m = 1 − (1 − q)^(1/12) std.

  • Model points. Single-policy expected-value projection. Aggregate capital caps across contracts on one insured (10000 € [S1] [S8], 17580 € [S12]) are per-insured underwriting limits and are not modeled.

  • Claims settlement. Immediate at EOM of the death month std. The post-mortem revalorisation between death and payment — the lower of the twelve-month average TME and the last TME at 1 November of the preceding year [S1] [S8] R8 — and the statutory payment clock of art. L. 132-23-1 CA REG-R31 are settlement-lag refinements, excluded.

  • Sign convention. The notes print the stream outgo-positive as liability_cf; the model publishes net_cf(t) = −liability_cf(t), income-positive, per the library convention.

  • Rounding. Full precision carried; the worked example displays cash flows to the cent and survivorship to five decimals, and the model must reproduce it at that precision.

Deltas against the temporaire décès chassis#

../temporaire_deces/technical-notes.md specifies TD_FR_S, this library’s protection chassis — an individual French death cover on a decrement-and-premium engine that this product reuses rather than reinvents. Those notes are the source of truth for the chassis; this section states only what Obseques_FR_S takes unchanged and where it departs, and does not restate the machinery. Source ids differ between the two products’ research files, so facts belonging to the chassis are pointed at by section rather than re-tagged here.

Inherited unchanged. The age basis: différence de millésime, calendar year of subscription minus calendar year of birth, stepped at the policy anniversary as a std proxy for the true 1 January step [S1] [S8] [S9]. The decrement order: the insured decrement resolves first, the policyholder decrement takes the survivors of it. The shared vocabulary and its sign convention — pols_if, mort_rate, lapse_rate, premiums, claims, expenses, net_cf, result_cf, with net_cf income-positive and the notes’ outgo-positive stream published as liability_cf. The mortality basis posture: TH 00-02 / TF 00-02 are the homologated regulatory tables for a death cover on both products, are cited by name and never shipped REG-R22 REG-R23, and both models run a std Gompertz-form proxy rising 9 % per year of age, anchored at 0.0040 at the reference cell’s own entry age — here carrying a sex dimension as well, because the premium is read from a published rate card and sex therefore enters only the best-estimate decrement. And the absence of an account value: on both products the benefit is a stated capital, not a fund.

Five departures, and each is first-order.

TD_FR_S — temporaire décès

Obseques_FR_S — obsèques

Underwriting

Underwritten issue: a two-tier déclaration de santé escalating to a full questionnaire médical and thence to examinations, with a surprime multiplier (rating_factor) on the tariff rate

Guaranteed acceptance: no questionnaire, no examination, at every retrieved contract [S1] [S11] [S12] [S13]. There is no rating factor and no rated-lives dimension

Anti-selection device

Underwriting itself; a délai d’attente only where the adhesion carried no medical formality, and off (waiting_period_y = 0) in the base run

A 12-month délai de carence on every contract that states a duration [S1] [S8] [S9] [S11] [S13] — the rest reference carences in their tables without giving one [S5] [S14] [S15] [S16] — because underwriting is always waived. Not a variant: it is the chassis of the product

Year-1 exclusion

Suicide voids the death cover in year 1: suicide_factor(t), a multiplicative withholding on claims_death in year 1 only, paying nothing

Suicide is excluded for 12 months [S1] [S8] [S12] [S13] but the insurer pays the valeur de rachat / provision mathématique, not zero [S1] [S8] [S12]. There is no suicide_factor cells here

Sum assured

sum_assured, level or on a fixed benefit_schedule_id; indexation off in the base run

capital_pp(t), a state variable that grows out of the participation aux bénéfices, 1.00 % p.a. guaranteed in the reference cell [S14]

Premium-stops

pols_lapse moves pols_if and nothing else; claims_lapse(t) is structurally zero, by statute

pols_lapse is paid surr_value_pp(t); claims_lapse is non-zero from t = 0 and worth 1005.89 € over the anchor cell’s horizon

Two second-order differences follow from the first five and are worth stating so an implementer does not carry a chassis habit across. Both models run a monthly grid, but they carry different things on it and end in different ways. TD_FR_S is contractually annual end to end — a one-year cover renewed by tacite reconduction and repriced at every renewal, so its tariff, its capital and its decrement vectors all step on the anniversary and its finer grid buys only the timing of claims, expenses and the modal instalment — and its horizon ends at a stated cover_end_age where nothing is payable. Here the monthly grid is forced by the product rather than chosen for resolution, because the carence boundary at twelve months must not be smoothed, and the model has no term at all — proj_len = 12 × (omega − entry_age + 1) with omega = 112 std, and the contract ends only on death, on rachat or on lapse [S1] [S8] [S9] [S11]. And the premium moves in opposite ways: on TD_FR_S the tariff is re-read at the new attained age at every renewal, so the premium rises with age by construction; here it is fixed at inception and the form is final [S1] [S5] [S14], which is exactly what produces the overrun — cumulative premiums grow without bound while the capital grows at most at reval_rate.

The statutory hinge under the last row of the table is one article. Art. L. 132-23 CA withholds réduction and rachat from assurances temporaires en cas de décès and from immediate or in-payment life annuities, and withholds rachat from survivorship capitals, pure endowments and deferred annuities without return of premium. TD_FR_S is squarely the first of those, so it has no surrender value, no paid-up value and no cash-value machinery at any duration. A whole-life funeral contract is none of them: it falls in the residual autres assurances sur la vie class, where “l’assureur ne peut refuser la réduction ou le rachat” R10 — which is why this model needs a surrender scale, a réduction strand and a second population at all, and why the chassis’s “a lapse costs nothing” arithmetic must not be carried over.

Model point attributes#

Attribute

Type

Example (worked configuration, RefOBS-VIA)

point_id

int

1

sex

enum {M, F}

M

entry_age

int (différence de millésime)

50 [S14]

capital_0

currency (€)

5000.00 [S14]

premium_form

enum {single, temporary, lifetime}

lifetime [S1] [S5] [S14]

prem_term_y

int (years of premium payment; 1 = single, 0 = lifetime)

0, the value premium_form = lifetime implies [S1] [S5] [S14]; 5 / 10 / 15 / 20 / 25 are the documented temporary terms [S1] [S5] [S8]

prem_cease_age

int (attained age premiums stop; 0 = never)

0 std (a)

annual_premium

currency (€/year)

336.03 [S14]

prem_freq

int in {1, 2, 4, 12}

1 std (b)

carence_months

int

12 [S1] [S8] [S9]

carence_refund_basis

enum {gross, net_assistance, net_instalment}

gross [S1]

carence_refund_rate

float (interest on the refund, p.a.)

0.00 [S1] [S8] [S9]

accident_mult

float (post-carence accidental multiplier)

1.0 [S1] [S9] [S14]; 2.0 from year 2, subject to accident_cap, variant [S8]

reval_rate

float (annual capital revalorisation)

0.0100 [S14]

reval_prem_linked

bool (remaining premiums uprated too)

false [S14]; true variant [S9] [S10] [S11]

surr_penalty_years

int

0 [S1] [S11]; 10 variant [S8]

surr_penalty_rate

float

0.00 [S1] [S11]; 0.05 variant [S8]

reduction_share

float (share of premium-stops becoming paid-up)

0.00 std (c)

issue_month

int (calendar month of subscription)

1 std (d)

Footnotes:

  • (a) std prem_cease_age = 0: whether lifetime premiums ever stop is not settled by the retrieved tables — one runs them to attained age 115 [S15], one to 95 [S5], one implies cessation near 90 from equal cumulative figures at ages 90 and 95 [S6], and one sells an explicit “to age 80” form [S9] [S10]. Never-ceasing is chosen because it is the documented design that produces the overrun this product is criticised for.

  • (b) std prem_freq = 1: annual in advance, matching the contractual default [S1] and the published rate cards [S5] [S14]. Setting 12 requires the documented 2.2 % instalment loading [S11] to be applied to annual_premium, not a re-tariffing.

  • (c) std reduction_share = 0 in the base cell: réduction is contractually the normal consequence of non-payment R7 [S1] [S8] [S9], but no public source gives any split between voluntary surrender and paid-up conversion — none gives any decrement rate at all. Zero keeps the worked example checkable; the recursion is specified below and the parameter is the sensitivity dial.

  • (d) std issue_month = 1: no retrieved document gives an issue month and nothing in _research/obseques.md bears on one. It exists to make the age-basis approximation of the conventions section auditable rather than invisible. Différence de millésime steps on 1 January [S1] [S8] [S9]; the model steps age at the policy anniversary; the two coincide exactly for a January issue and are out by up to one policy year of mortality otherwise. Every shipped model point sets 1, so the approximation is visible in the data rather than buried in a formula, and a user projecting real dates changes a column instead of a formula.

State variables#

Variable

Description

Updated

pols_if(t) = l(t)

Premium-paying policies in force; l(k) is measured at time k months from issue, l(0) = 1

monthly (deaths, surrenders, réductions)

pols_paid_up(t) = l_r(t)

Paid-up (réduit) policies in force; l_r(k) at time k, l_r(0) = 0

monthly (entries on réduction, exits on death)

capital_pp(t)

Guaranteed capital per policy, uprated by reval_rate

policy anniversaries

cum_prem_pp(t)

Premiums collected per policy to the BOM of month t — the carence refund base

monthly (premium months only)

prem_ann(t)

Current annual premium; rises with the capital when reval_prem_linked

policy anniversaries

surr_value_pp(t)

Surrender value = provision mathématique per policy

monthly

reduced_capital_pp(t)

Paid-up capital fixed at the date of réduction

on conversion

in_carence(t)

Indicator t < carence_months

monthly

age(t)

Attained age = entry_age + floor(t/12)

policy anniversaries

l is a time-point function, and on a 0-based frame it lines up exactly. These notes carry l(k) and l_r(k) at time k months from issue, l(0) = 1 at issue; the model indexes its counts at the start of the month, and the start of month t is time t, so pols_if(t) is l(t) and pols_paid_up(t) is l_r(t) with no offset at all. That is the quantity the worked example prints — its column is headed pols_if(t) — and it is the weight the model applies to every cash flow of month t. The end-of-month count is l(t+1), which the model publishes as pols_if_at(t, "AFT_DECR"). The convention is restated in the Projection docstring’s symbol map, which is the authority for which cells name carries which symbol.

Assumption inputs#

(a) Contractual / guaranteed elements (cited)#

Input

Value

Basis

Cover

Whole life, no term, no maturity, no survival benefit

[S1] [S8] [S9] [S11]

Underwriting

Guaranteed acceptance; no medical questionnaire or examination

[S1] [S11] [S12] [S13]

Délai de carence

12 months; accidental death pays the full capital from day 1; non-accidental death inside it pays the premiums collected

[S1] [S8] [S9] [S11] [S13]

Interest on the refund

None

[S1] [S8] [S9]

Refund netting

Gross premiums collected [S1]; net of the assistance premium of 12 €/year at one insurer [S8]; net of instalment charges at another [S9], quantified only as the 2.2 % annual-to-monthly loading [S11]

[S1] [S8] [S9] [S11]

Accidental-death cap

20000 € on the doubled accidental benefit at the one insurer that doubles it; that insurer’s aggregate cap is 10000 €, or 20000 € where death follows an accident from year 2

[S8]

Accident definition

Sudden, unforeseeable, non-intentional external cause — a near-identical core wording at both contracts that give one [S1] [S8]. The exclusions differ: cerebral and cardio-vascular events are never accidents, whatever their origin, at [S1] only, echoed by the market description, which adds myocardial infarction, coronary conditions and emotional shock R21; [S8] instead excludes acute and chronic illness and harm from medical or surgical treatment

[S1] [S8] R21

Suicide

Excluded in the first 12 months, and for a year after a capital increase

[S1] [S8] [S12] [S13]

Excluded-cause benefit

The valeur de rachat / provision mathématique — not zero

[S1] [S8] [S12]

Premium level

Fixed at inception; the premium form is final

[S1] [S5] [S6] [S7] [S14] [S16]

Revalorisation

1.00 % p.a. of the capital, guaranteed, premiums unchanged

[S14]

First uprating

At the first anniversary — contracts must be in force at least a year

[S1] [S9]

Surrender

Total only, at the provision mathématique. At any time [S1] [S9] [S12]; at one insurer only once one annual premium has been paid [S8]. Settlement within 30 days [S1] [S8], 2 months at another [S9] [S11]

[S1] [S8] [S9] [S11] [S12]; statutory basis R10 REG-R31

Non-payment

10 days, then 40 days’ formal notice with cover suspended, then termination or réduction

R7 [S1] [S8] [S9]

Renonciation

30 calendar days, full refund of all premiums

[S1] [S8] [S11] R7 REG-R29

Capital earmarking

Capital earmarked to the funeral up to its cost; funeral firm is first-rank beneficiary

R2 REG-R38 [S1] [S8] [S9] [S12]

(b) Insurer-discretionary current elements#

Unlike the UK sibling, where this class is nearly empty, it carries real weight here — a French funeral contract is a participating contract and the capital moves.

  • The revalorisation rate. Discretionary at five of the seven insurers — PB credited annually to the capital [S1], by board decision for contracts in force at least a year [S8], through a fonds de revalorisation [S9], “peut être majoré” [S15], and on a published formula [S16] — set annually out of the participation aux bénéfices [S1] [S8] [S9] [S15] [S16]; one publishes its formula — PB equals 90 % of technical and financial profits, after a 1 % management charge on funds under management and after the technical interest guaranteed at inception (art. A 335-1 CA) [S16] REG-R23; one makes it a contractual guarantee at 1 % p.a. [S14], the only guaranteed rate retrieved anywhere. The statutory machinery (art. L. 331-3 CA REG-R14, the compte de participation aux résultats at arts. A. 132-10 to A. 132-15 REG-R15, the eight-year release horizon of the provision pour participation aux bénéfices at art. A. 132-16 REG-R16) is documented once for the library in ../assurance_vie_euro/technical-notes.md and is not restated here; this model consumes a declared rate and does not project a PB account. No insurer’s actually declared PB rate for a funeral contract in any year was found in any public source; the only anchor besides the guarantee is 1.2854 % p.a., derived from a KID scenario [S11].

  • Whether the premiums are uprated with the capital. No at five insurers [S5] [S6] [S7] [S14] [S16]; yes, in the same proportion on the remaining premiums, at one [S9] [S10] [S11]; not stated at a sixth [S1] — unverified there. A first-order fork, carried as reval_prem_linked.

  • The tariff. A model point input. The standardised tables [S5] [S6] [S7] [S10] [S14] [S15] [S16] are the only public rate card, they state that they have no contractual value, and no insurer publishes the mortality table, technical rate, expense loading or margin behind them.

  • The technical rate at inception. Two fragments exist in the whole retrieved set: 0.75 % with table TH 00-02 [S8] and 0 % in a worked example [S1]. The statutory ceiling for a periodic-premium contract is the lower of 3.5 % and 60 % of the reference TME REG-R17.

  • Charge levels within the disclosed maxima. Art. A. 132-8 CA requires charge maxima to be disclosed in the encadré, not limited REG-R30; the observed maxima are in product-spec.md Table 7.

(c) Behavioral / experience assumptions (modeler’s view)#

Nothing in this class is sourced. The research file’s finding is blunt: no public source gives any lapse, surrender or paid-up rate for this product, no mortality experience for guaranteed-issue funeral lives, and no split of deaths between accidental and other causes. Every figure below is a std drafting construction, to be replaced by experience.

Input

Reference basis

Tag

Base mortality q_base(x, sex)

French population mortality by sex and single year of age, from INSEE REG-R24, as a proxy for TH 00-02 / TF 00-02

std (e)

Anti-selection loading f_as

1.25, level across durations and ages

std (f)

Select uplift s(y)

1.60 / 1.30 / 1.15 / 1.00 for policy years 1 / 2 / 3 / 4+

std (f)

Mortality improvement

None in base; a flat 0.8 % p.a. reduction is the sensitivity proxy

std (i)

Accidental share of deaths d_acc

0.05, level

std (g)

Surrender / lapse lapse_rate

6 % / 5 % / 3.5 % / 2.5 % for policy years 1 / 2 / 3–5 / 6+

std (h)

Paid-up share reduction_share

0 in base; 0.5 as the variation

std (c)

Acquisition expense

150 € per policy at t = 0, commission included

std (j)

Maintenance expense

24 € per policy per year, 2.00 €/month, inflating 1.8 % p.a.

std (j)

Claim handling

Folded into maintenance

std (i)

Footnotes:

  • (e) std base mortality: the decrement CSVs are proxies built from the INSEE population series REG-R24 and anchored so that the model reproduces the placeholder rate stated in the worked example exactly. TH 00-02 / TF 00-02 are the regulatory tables here — named by two insurers [S8] [S11], homologated by the arrêté du 20 décembre 2005 REG-R22 and annexed to art. A. 335-1 CA with their décalage d’âge schedules REG-R23 — and are cited by name and never redistributed, per the house rule for restricted tables.

  • (f) std anti-selection, and the direction of it. Acceptance is guaranteed — no medical questionnaire, no examination, entry to 84 or 85 [S1] [S8] [S11] [S12] [S13] — so the pool cannot be better than population and self-selects worse: an applicant who knows their prognosis has every reason to buy, and the only device standing against them is the twelve-month waiting period R21. The loading is therefore upward relative to population mortality, and it is not flat in duration: the excess sits at short durations and decays as the anti-selected cohort dies out. The first-year factor is the largest (1.60) even though a first-year illness death costs only a refund — the deaths still happen, they merely cost less, and moving the excess to year 2 would double-count the protection the carence already gives. The magnitude has no public calibration of any kind; the direction and the shape are the defensible part.

  • (g) std accidental share 5 %: the contractual definition is narrower than external-cause mortality. Cerebral and cardio-vascular events are excluded whatever their origin at the one contract that says so [S1], and the market description adds myocardial infarction, coronary conditions and emotional shock R21; the other contract that defines an accident does not carry that carve-out but excludes acute and chronic illness and harm from medical or surgical treatment instead [S8]; the burden of proof is on the claimant [S1]; and a medical certificate stating the cause is required for any claim inside the waiting period [S1] [S8] [S9]. 5 % is set below any plausible external-cause share for those reasons, and level because no age split was found. It matters only inside the waiting period — where it is the whole of the difference between a refund and a capital.

  • (h) std lapse shape: declining with duration, on the reasoning that a small-premium prévoyance contract bought for one purpose is lapsed early or not at all, reinforced by a surrender value worth far less than the premiums paid for decades (784.01 € after 5 years against 1680.15 € of premiums in the worked cell [S14]). The 30-day renonciation with full refund [S1] [S8] REG-R29 is treated as never-issued business, outside the projection.

  • (i) std two items with no source and no material effect on the base run. Mortality improvement is off in the base projection; France has no publicly available insured-lives projection model comparable to the CMI’s, so the 0.8 % p.a. flat reduction is a sensitivity dial rather than a basis. Claim handling costs are folded into the maintenance expense rather than charged per death, because no retrieved document separates them — the contract documents disclose charges, not the insurer’s own expenses.

  • (j) std expense levels: no French source publishes a currency expense assumption for this product. The anchors are the disclosed charges, which bound expenses from above and leave the margin — acquisition charges of 2.5 % to 5.38 % of the guaranteed capital, i.e. 125 € to 269 € on a 5000 € capital [S8] [S9], and ongoing charges of 0.40 % p.a. of the capital plus 0.57 % p.a. while lifetime premiums are paid, i.e. 48.50 €/year on 5000 € [S1]. The std acquisition expense sits inside that range and the std maintenance expense at about half the ongoing charge. The only aggregate figure published anywhere is a PRIIPs reduction in yield of 1.77 % p.a. over 30 years [S11]. Expense inflation has no source at all.

Cash flow components and recursions#

Notation (defined once, used throughout; product-spec.md states the same mechanics with a 1-based contractual month label, t_spec = t + 1)#

Symbol

Meaning

Cells name

t

policy month, 0-based: t = 0, 1, …, proj_len − 1

—

y

policy year = floor(t/12) + 1 — the contractual, 1-based label

—

x(t)

attained age = entry_age + y − 1 (différence de millésime proxy std)

age

C(y)

guaranteed capital in policy year y

capital_pp

C_0

capital at issue

capital_0

r

annual revalorisation rate

reval_rate

P(t)

premium due at BOM of month t

prem_due_pp

P_a(y)

annual premium in policy year y

prem_ann

K(t)

premiums collected to the BOM of month t

cum_prem_pp

n_car

waiting period in months

carence_months

V(t)

surrender value = provision mathématique

surr_value_pp

C_red(t)

paid-up capital on réduction

reduced_capital_pp

u(x)

single premium per 1 € of whole-life capital at attained age x

single_prem_rate

q(y)

annual mortality rate for policy year y

mort_rate

q_m(y)

monthly mortality = 1 − (1 − q(y))^(1/12) std

mort_rate_mth

w(y), w_m(y)

annual / monthly total premium-stop rate

lapse_rate, lapse_rate_mth

rho

share of premium-stops that become paid-up rather than surrendered

reduction_share

d_acc

accidental share of deaths

acc_share

k_adb

accidental multiplier after the waiting period (1 or 2 [S8])

accident_mult

M_acc

cap on the accidental benefit, 20000 € [S8]

accident_cap

a_ass

assistance premium netted from the refund, 12 €/year [S8]

assistance_prem_pp

phi

annual-to-monthly instalment loading, 2.2 % [S11]

instalment_load

l(t)

premium-paying in force at time t months from issue; l(0) = 1

pols_if(t) — the model indexes at the start of month t, which is time t

l_r(t)

paid-up in force at time t; l_r(0) = 0

pols_paid_up(t), same alignment

surr_scale(t)

the published scale for a rachat in month t; the table is keyed by elapsed months and read at t + 1, the months elapsed at the end of month t

surr_scale_pp

Dimensional check: capital, premiums and benefits are €; q, w, d, rho are dimensionless; every expected cash flow below is € per month per policy issued.

Decrements and survivorship#

  q(y)   = q_base(x(t), sex) x f_as x s(y),   capped at 1, and = 1 at x = omega
  q_m(y) = 1 - (1 - q(y))^(1/12)
  w_m(y) = 1 - (1 - w(y))^(1/12)

  l(t+1)   = l(t) x (1 - q_m(y)) x (1 - w_m(y)),                  l(0)   = 1
  l_r(t+1) = l_r(t) x (1 - q_m(y)) + l(t) x (1 - q_m(y)) x w_m(y) x rho,
                                                                  l_r(0) = 0

with y = policy_year(t), the policy year of the month whose decrements are being applied: l(t) is the count at the start of month t and l(t+1) the count at the end of it, after month t’s deaths and premium-stops.

Deaths resolve before premium-stops. w(y) = 0 for a prime unique cell and for any month after prem_cease_age: there are no premiums left to stop paying, so no premium-stop decrement applies std. Voluntary surrender by a paid-up policyholder is not modeled std.

Premiums#

  P(t)   = P_a(y)   if t is a premium month and t is inside the paying period
         = 0        otherwise
  K(t)   = K(t-1) + P(t),                             K(0) = P(0)
  P_a(y) = annual_premium x (1 + r)^(y-1)             if reval_prem_linked [S9] [S10] [S11]
         = annual_premium                             otherwise [S5] [S14] [S16]

A premium month is t ≡ 0 (mod 12/prem_freq); the paying period is month 0 only (prime unique), months 0 to 12·prem_term_y − 1 (primes temporaires), or every month until prem_cease_age (primes viagères, unbounded when that is 0). The premium is level and fixed at inception unless reval_prem_linked [S1] [S5] [S14].

Death benefit and the délai de carence#

  DB_acc(t)  = C(y)                             if t <  n_car    [S1] [S8] [S9]
             = min( k_adb x C(y), M_acc )        if t >= n_car    [S8]
  R(t)       = K(t)                              gross            [S1]
             = max( 0, K(t) - a_ass x y )        net_assistance   [S8]
             = K(t) / (1 + phi) if prem_freq > 1 else K(t)
                                                 net_instalment   [S9] [S11]
  DB_ill(t)  = R(t) x (1 + i_ref)^((t+1)/12)     if t <  n_car    i_ref = carence_refund_rate = 0
             = C(y)                              if t >= n_car

  claims_death(t) = l(t) x q_m(y) x [ (1 - d_acc) x DB_ill(t) + d_acc x DB_acc(t) ]
                    + l_r(t) x q_m(y) x C_red(t)

The refund is of premiums collected, so with an annual premium in advance it is a step function, flat across the first twelve months, not a monthly accrual. R(t) carries the three carence_refund_basis variants: gross [S1], net of the assistance premium of 12 €/year for each year begun [S8], or net of instalment charges [S9] — the only published quantification of which is the 2.2 % annual-to-monthly loading [S11], so that basis does nothing on an annual-premium cell. The cap M_acc = 20000 € is applied past the waiting period only; inside it the accidental benefit is already the full capital, so doubling it there would double-count the day-one cover. The cap is inert on every shipped model point: the largest accidental benefit any of them reaches is 18532.12 € — the doubled-benefit variant at the far end of the horizon, where the 5000 € capital has revalorised to 9266.06 € — so no figure in the worked example or the shipped run depends on it. It is specified and coded anyway, because that is where the contract puts it and a larger capital_0 reaches it [S8]. A paid-up policy is past the waiting period by construction and pays C_red for any cause.

Revalorisation of the capital#

  C(1)   = C_0
  C(y)   = C_0 x (1 + r)^(y-1)      for y >= 2                    [S14]

The uprating starts at the first anniversary, not at issue: PB is allocated to contracts in force at least one year [S1] [S9]. reval_simple = true replaces the geometric form with C(y) = C_0 x (1 + r x (y-1)) — see product-spec.md footnote (g), where the reading of the contractual wording is unverified. Nothing is uprated inside the waiting period on the illness leg: that benefit is a refund of premiums, not a capital.

Rachat and réduction#

  V(t)              = surr_scale(t) x capital_0 / 5000 x (1 - pen(t))
  pen(t)            = surr_penalty_rate  if t < 12 x surr_penalty_years, else 0
  claims_lapse(t)   = l(t) x (1 - q_m(y)) x w_m(y) x (1 - rho) x V(t)
  C_red(t)          = V(t) / u(x(t))                                on conversion [S1] [S8]

surr_scale(t) is an external input table: the surrender value in € for a 5000 € capital by elapsed months from issue, for the cell’s entry age and premium form, linearly interpolated in elapsed months between the published quinquennial anchors std and held flat beyond the last one. That key is a duration, not the projection’s 0-based month index, and it does not move with the frame. A rachat resolves at the end of month t, by which time t + 1 months have elapsed, so surr_scale(t) reads the table at t + 1: the first projected month, t = 0, is one month into the contract at 13.07 € on the worked cell, and the published five-year anchor of 784.01 € is the value of month t = 59. Each grid carries all nine published anchors, at 60, 120, 180, 240, 300, 360, 420, 480 and 540 months, plus a month-0 anchor that is std; dropping an intermediate one and letting the interpolation stand in for it moves the scale by as much as 11 % and can erase the shape the table exists to show — one temporaire grid peaks at 5074 € at 25 years and then declines [S2], and that peak is an anchor, not an interpolant. The anchors are transcribed from the standardised tables [S2] [S5] [S14] [S15] and already embed that insurer’s own revalorisation, which is why V(t) is not additionally scaled by C(y). surr_scale_table.csv is the source of truth for which anchor came from which document: its provenance column names the insurer, the entry age, the premium form and the source id for every row, and these notes, model.md and product-spec.md restate it rather than define it. The production alternative is a prospective provision mathématique on the tariff basis; it is not the reference implementation because no insurer publishes its tariff basis — the whole retrieved set contains one technical rate with a table (0.75 %, TH 00-02 [S8]) and one rate alone (0 % [S1]).

u(x) is the single premium per 1 € of whole-life capital at attained age x, a second external input table anchored on the published prime unique rate card — 0.854808 at 50, 0.909720 at 60 and 0.963912 at 70, from 4274.04 / 4548.60 / 4819.56 € per 5000 € of capital [S5] — interpolated and extrapolated std. It serves twice: it prices the single premium form, and it turns a mathematical provision into a valeur de réduction.

Cash flow outputs (per policy issued, month t)#

Cash flow

Formula

Column

Premium income

l(t) × P(t)

premiums

Death outgo

per the formula above, both populations

claims_death

Surrender outgo

l(t) × (1 − q_m) × w_m × (1 − rho) × V(t)

claims_lapse

Acquisition expense

150 € at t = 0 std (j)

expenses

Maintenance expense

(l(t) + l_r(t)) × 2.00 × 1.018^(y−1) std (j)

expenses

Paid-up conversion

no cash flow — a state change only

—

Post-mortem revalorisation

excluded std (conventions, above) [S1] [S8] R8 REG-R31

—

Maturity outgo

identically zero — there is no maturity [S1] [S8] [S9]

—

 
  liability_cf(t) = claims_death(t) + claims_lapse(t) + expenses(t) - premiums(t)
  net_cf(t)       = -liability_cf(t)

Monthly processing order std#

At month t — t = 0 for the first — for the l(t) policies in force at its start:

  1. BOM — premium P(t) received if t is a premium month inside the paying period; add it to cum_prem_pp. Paid-up policies skip this step.

  2. BOM — acquisition expense at t = 0; maintenance expense for the month, on both the premium-paying and the paid-up populations.

  3. Anniversary (t ≡ 0 mod 12, t > 0) — uprate capital_pp by reval_rate; uprate prem_ann by the same rate if reval_prem_linked; step age. The order matters: the capital in force during policy year y is the one set at the start of year y, and the premium collected at step 1 of that same month is the uprated one.

  4. EOM — deaths at mort_rate_mth(y) applied to pols_if(t) and pols_paid_up(t); benefit per the carence rules for the first population, reduced_capital_pp for the second.

  5. EOM — premium-stops at lapse_rate_mth(y) on the survivors of step 4: a fraction rho converts to paid-up (state change, no cash flow, reduced_capital_pp fixed at that month’s surr_value_pp / single_prem_rate), and 1 − rho surrenders and is paid surr_value_pp(t). No premium-stop decrement applies where no premium is due.

  6. The survivors are pols_if(t+1) and pols_paid_up(t+1), the counts at the start of the next month.

Known modeling pitfalls#

These are the specific ways an implementation of this product looks right and is wrong. Each is stated so that it can be turned into an assertion, and the figures quoted are from the worked example below.

  1. Paying the full capital inside the carence. The single worst error available here. In the worked cell it turns the first month’s (t = 0) expected death outgo from 0.380884 into 3.345618 (×8.78) and policy-year-1 death outgo from 4.4274 into 38.8893 — an 8.8× overstatement of the front end of the liability. Assert: for t < carence_months, the illness leg of claims_death uses cum_prem_pp, not capital_pp.

  2. Dropping the accident leg inside the carence. The mirror-image error. Accidental death pays the full capital from day 1 [S1] [S8] [S9]; treating the whole waiting period as a refund takes the first month’s death outgo from 0.380884 to 0.224846, understating it by 41 %, and policy-year 1 from 4.4274 to 2.6136. Assert claims_death(0) > q_m(0) × cum_prem_pp(0).

  3. Accruing the refund base monthly when the premium is annual. With prem_freq = 1 the refund base is a step, constant at 336.03 through months t = 0–11. Accruing it as annual_premium × (t + 1) / 12 gives 28.00 at t = 0 and understates policy-year-1 death outgo by 26 % (3.2750 against 4.4274). Assert cum_prem_pp(0) == cum_prem_pp(11) == annual_premium for the annual cell.

  4. Revalorising too early, or revalorising the wrong thing. PB accrues only to contracts in force at least a year [S1] [S9], so uprating at issue makes capital_pp(0) = 5050.00 and overstates the year-1 accidental leg — assert capital_pp(t) == capital_0 for t < 12. And the illness benefit inside the waiting period is a refund of premiums, not a capital: it must not carry reval_rate. carence_refund_rate (zero in every retrieved contract [S1] [S8] [S9]) is a different parameter and must not be confused with it.

  5. Measuring the overrun against the wrong capital, and off by a year. Cumulative premiums first exceed the original 5000 € capital at t = 168 (policy year 15) and the revalorised capital at t = 204 (policy year 18) — three years apart. Separately, the standardised tables date their columns by the age at the end of the year, so their “age 65” column is this model’s attained age 64 during policy year 15. Both conventions are defensible; silently mixing them moves the published crossover by up to four years.

  6. Letting lifetime premiums stop by accident. If prem_cease_age is defaulted to anything non-zero, the viagère overrun disappears and with it the product’s characteristic feature. Assert that a lifetime point with prem_cease_age = 0 still has prem_due_pp > 0 at attained age 100.

  7. Getting the revalorisation coupling backwards. Applying capital uprating without the matching premium uprating on a point configured after the insurer that couples them [S9] [S10] [S11] understates premium income; applying the premium uprating on any of the five that do not [S5] [S6] [S7] [S14] [S16] overstates it. One flag, reval_prem_linked, read from the model point and never hard-coded.

  8. Treating lapse as free. This is where the UK sibling’s model is actively misleading. Rachat pays the provision mathématique [S1] [S8] [S9] [S12], so claims_lapse is non-zero from t = 0. Setting it to zero moves the undiscounted net stream of the worked cell from 2236.92 to 3242.81 — a 45 % overstatement. Assert claims_lapse(t) > 0 for every t with lapse_rate > 0.

  9. Treating réduction as termination. Non-payment produces a paid-up contract, not an exit, wherever the surrender value is sufficient R7 [S1] [S8] [S9]; routing every premium-stop to the exit removes a death liability the contract still owes. With reduction_share = 0.5 the paid-up population must appear in claims_death and must pay reduced_capital_pp, not capital_pp.

  10. Applying a premium-stop decrement where no premium is due. After prem_cease_age, in a prime unique cell and in the paid-up state there is nothing to stop paying, and a lapse decrement there silently destroys liability. Assert lapse_rate_mth(t) == 0 in all three.

  11. Paying zero on an excluded death. Suicide in year 1, war, nuclear and murder by a beneficiary do not extinguish the contract: the insurer pays the valeur de rachat or the provision mathématique [S1] [S8] [S12], so an exclusion modelled as a zero benefit understates outgo by exactly V(t) per excluded death.

  12. Using the wrong age basis, or flattening the loading. The basis is différence de millésime [S1] [S8] [S9]; age last birthday shifts the whole mortality lookup by up to a year at entry, and the décalage d’âge schedules annexed to art. A. 335-1 CA REG-R23 apply on top of it where a homologated table is used. Separately, the anti-selection excess belongs at durations 1–3 and is largest in year 1 even though a year-1 illness death costs only a refund; a flat loading understates the year-2 spike, the largest single step in the worked cell’s death-outgo series.

Policyholder behavior modeling#

All dynamic formulas below are std reference constructions. No public French source gives any lapse, surrender or paid-up rate for this product; the shapes are drafting assumptions with the qualitative anchors cited.

  • Base premium-stop rate std. The duration-declining table above, converted monthly, zero wherever no premium is due. The declining shape follows from the surrender value: for the first two decades it is worth a fraction of the premiums paid (784.01 € against 1680.15 € at five years, 1574.90 € against 3360.30 € at ten [S14]), so an early lapser loses most of their money and a late one has nearly reached a full payout.

  • No carence-completion spike std. Nothing changes for the policyholder at t = 12 except that the cover becomes worth having, so there is no incentive to lapse there; the year-1 rate is set highest instead, for affordability and buyer’s-remorse attrition.

  • Overrun-aware lapse std. Sensitivity module, off in base: lapse_rate(y) = lapse_rate_base(y) × (1 + beta × 1{cum_prem_pp(t) > capital_pp(t)}), beta = 0.5. The rationale is the disclosure the CCSF asked for R13 R15 and the KID warning that total premiums may exceed the capital [S11]; beta is a pure stress dial.

  • Réduction versus rachat std. Where a surrender value exists, stopping payment and taking nothing is strictly dominated by reducing, so the economically rational reduction_share is high, and the contract makes réduction the default outcome of non-payment R7 [S1] [S8] [S9]. Zero in base, 0.5 as the variation, 1.0 as the upper stress. It dominates the late-duration liability and must never be approximated by perturbing the lapse rate instead.

  • The 40-day suspension std. Cover is suspended during the formal-notice window [S1], so a death there pays nothing. Ignoring it is conservative and is what the base model does; modelling it as an immediate exit is not, because the policyholder may still pay and continue.

  • Capital increases and renonciation std. Increases are not modeled: anti-selective on a guaranteed-issue book, mitigated but not removed by the fresh waiting period on the increment [S1] [S8]. The 30-day cooling-off with a full refund [S1] [S8] REG-R29 is modeled as never-issued business, outside the projection.

Worked example#

Cell RefOBS-VIA. Entry age 50 male (différence de millésime), guaranteed capital capital_0 = 5000.00 €, primes viagères of annual_premium = 336.03 € payable annually in advance for life with no cessation age, revalorisation reval_rate = 1.00 % p.a. compound on the capital with the premium unchanged, carence_months = 12 with the illness leg paying the premiums collected and the accident leg the full capital, accident_mult = 1, reduction_share = 0, no surrender penalty. The premium, the revalorisation rate and the surrender-value scale all come from one document [S14]; the waiting-period design comes from the three contracts that state one [S1] [S8] [S9], because that document’s tables reference carences without giving a duration.

Assumptions, each tagged. Mortality std, an illustrative placeholder attributable to no table: q_base(x) = 0.0040 × 1.09^(x−50), anti-selection f_as = 1.25, select uplift s(y) = 1.60 / 1.30 / 1.15 / 1.00 for y = 1 / 2 / 3 / 4+, so mort_rate is 0.008000 in policy year 1, 0.0070850 in year 2 and 0.0167086 in year 15. Lapse std 6 % / 5 % / 3.5 % / 2.5 % for years 1 / 2 / 3–5 / 6+, all of it surrender since reduction_share = 0. Accidental share std d_acc = 0.05. Monthly conversion q_m = 1 − (1 − q)^(1/12) std, so mort_rate_mth = 0.00066912 and lapse_rate_mth = 0.00514301 in policy year 1. Surrender scale [S14], linearly interpolated in elapsed months std between the published anchors 0 / 784.01 / 1574.90 / 2346.97 / 3151.33 / 3980.74 / 4828.57 / 5659.93 / 6429.96 / 7135.11 € at 0 / 60 / 120 / 180 / 240 / 300 / 360 / 420 / 480 / 540 elapsed months, read at t + 1. Expenses are omitted from the table for clarity; age(t) = 49 + y. All amounts in €, full precision carried, displayed rounded. t is the 0-based policy month and y = floor(t/12) + 1 beside it.

t

y

capital_pp

cum_prem_pp

db_illness

surr_value_pp

pols_if(t)

premiums

claims_death

claims_lapse

0

1

5000.00

336.03

336.03

13.07

1.00000

336.03

0.38

0.07

5

1

5000.00

336.03

336.03

78.40

0.97129

0.00

0.37

0.39

11

1

5000.00

336.03

336.03

156.80

0.93793

0.00

0.36

0.76

12

2

5050.00

672.06

5050.00

169.87

0.93248

313.34

2.79

0.68

23

2

5050.00

672.06

5050.00

313.60

0.88387

0.00

2.64

1.18

59

5

5203.02

1680.15

5203.02

784.01

0.77719

0.00

2.39

1.81

119

10

5468.43

3360.30

5468.43

1574.90

0.65347

0.00

3.25

2.17

168

15

5747.37

5040.45

5747.37

2205.42

0.55752

187.34

4.50

2.59

179

15

5747.37

5040.45

5747.37

2346.97

0.53639

0.00

4.33

2.65

204

18

5921.52

6048.54

5921.52

2682.12

0.48896

164.30

5.27

2.76

239

20

6040.54

6720.60

6040.54

3151.33

0.42361

0.00

5.55

2.81

299

25

6348.67

8400.75

6348.67

3980.74

0.31507

0.00

6.72

2.63

359

30

6672.52

10080.90

6672.52

4828.57

0.21337

0.00

7.43

2.16

479

40

7370.61

13441.20

7370.61

6429.96

0.05748

0.00

5.46

0.77

539

45

7746.59

15121.35

7746.59

7135.11

0.01799

0.00

2.88

0.26

db_accident equals capital_pp in every row, at every duration, and is therefore not printed. Over the full horizon (t = 0 to 755, 756 months, ending at attained age 112) the undiscounted totals are: premiums 6184.01, claims_death 2941.20, claims_lapse 1005.89, and net_cf summing to +2236.92 before expenses.

Checks. Survivorship, a different way. The monthly decrements must compound back to the annual ones exactly, so the in-force at the end of policy year 1 is available in closed form: l(12) = (1 − q(1)) × (1 − w(1)) = 0.992 × 0.94 = 0.93248 — the year-1 annual rates — which is the pols_if(t) printed against t = 12. Carrying it one year further, l(24) = 0.93248 × (1 − 0.0070850) × 0.95 = 0.87957971; the row at t = 23 prints l(23), one month earlier, at 0.88387. The capital and the surrender value at t = 204, two ways. Policy year 18 has had 17 upratings, so capital_pp = 5000 × 1.01^17 = 5921.52; equivalently, take the year-15 figure already in the table and carry it three years, 5747.37 × 1.01³ = 5747.37 × 1.030301 = 5921.52. The surrender value at the end of month t = 204 — 205 elapsed months — interpolates between the published 180- and 240-month anchors: 2346.97 + (3151.33 − 2346.97) × 25/60 = 2346.97 + 335.15 = 2682.12. The t = 11 / t = 12 step. Expected death outgo rises from 0.357242 to 2.789356, a factor of 7.8080, and it decomposes exactly into three independent moves: in-force 0.93248/0.93793 = 0.994191, monthly mortality 0.00059234/0.00066912 = 0.885251 (the select uplift drops from 1.60 to 1.30 while the base rate rises 9 %), and benefit 5050.00/569.2285 = 8.871657 — because the blended benefit steps from 0.95 × 336.03 + 0.05 × 5000 = 569.2285 to the full uprated capital. The product 0.994191 × 0.885251 × 8.871657 = 7.8080 reproduces the ratio. That step is the signature discontinuity of the product, and it is the reason the grid must be monthly.

Subsidiary table — the premium-form fork. The same 5000 € capital and entry age 50, priced across three forms on a single published rate card [S5], with cumulative premiums by attained age (age 65 = 15 annual premiums, 75 = 25, 85 = 35, 95 = 45):

Premium form

Annual premium

Cum. to 65

to 75

to 85

to 95

First premium exceeding 5000

prime unique

4274.04 once

4274.04

4274.04

4274.04

4274.04

never

temporaire 10 ans

455.64

4556.40

4556.40

4556.40

4556.40

never

viagère

164.52

2467.80

4113.00

5758.20

7403.40

the 31st, at attained age 80

The viagère row is the whole argument about this product in six numbers: 164.52 × 31 = 5100.12 first exceeds the capital in policy year 31, and by policy year 45 the insured has paid 7403.40 € for 5000 € of cover — while the two other forms stop, permanently, below the capital. The crossover in policy year 31 falls inside the age-80-to-84 band the same rate card produces for entry ages 50 / 60 / 70. The worked cell above, priced by a different insurer at more than twice the lifetime premium (336.03 against 164.52), crosses at policy year 15 instead.

Valuation and reserve pointers#

This library projects gross best-estimate liability cash flows; the valuation layers consume them and are cited, not reproduced.

  • Solvabilité II best estimate. The probability-weighted average of future cash flows discounted at the EIOPA risk-free term structure REG-R1 REG-R4 REG-R5, plus a risk margin. No cost-of-capital rate, contract boundary or standard-formula shock in this library was read from a retrieved instrument REG-R2, so every such figure would be std; the SCR and MCR layers are cited-not-specified.

  • The statutory provision mathématique. The French GAAP balance sheet persists alongside Solvabilité II. The provision mathématique is the difference between the present values of the two parties’ commitments including future management costs, the first of eleven named technical provisions REG-R6 — and, for this product, the quantity the contract makes the surrender value equal to [S1] [S8] [S9] [S12]. A production model computes it prospectively on the tariff basis; this reference model reads a scale, for the reason given above.

  • Participation aux bénéfices. The minimum PB is computed globally on the statutory accounts REG-R14 REG-R15, and amounts parked in the provision pour participation aux bénéfices must be allocated or paid within eight years REG-R16. A funeral capital’s revalorisation is the visible end of that machinery; the machinery itself is modeled once in ../assurance_vie_euro/technical-notes.md. Note the open question recorded in product-spec.md: the 85 % PB floor of art. L. 2223-34-1 CGCT is drafted for the prestations form and may not reach a pure capital contract R3.

  • Technical rate and tables. Any guaranteed rate inside the tariff is capped by art. A. 132-1 CA at the lower of 3.5 % and 60 % of the reference TME for a periodic-premium contract REG-R17. TH 00-02 / TF 00-02 are the homologated non-annuity tables REG-R22, reproduced with their décalage d’âge schedules in the annexe to art. A. 335-1 CA, which permits only homologated or actuary-certified tables REG-R23. They are cited and never shipped; the decrement CSVs are INSEE-derived std proxies REG-R24.

  • Unclaimed contracts. Under the loi Eckert a death benefit unclaimed for ten years transfers to the Caisse des dépôts and becomes State property after twenty years there, with revalorisation continuing until the deposit REG-R39. Small capitals and elderly beneficiaries make this a live item for this product specifically.

  • IFRS 17 and professional standards. Fulfilment cash flows plus a contractual service margin, effective from 1 January 2023 with no French carve-out REG-R45; the same projection feeds it with different discounting and aggregation. NPA 2 Modèles actuariels, a recommended practice in force from 1 January 2016, is the standard this documentation and its test suite sit under REG-R44.

Key sensitivities and model risks#

In order of influence on a guaranteed-acceptance funeral block:

  1. Anti-selection mortality, and its duration shape. The f_as = 1.25 loading and the s(1..3) = 1.60 / 1.30 / 1.15 select uplift are pure std placeholders: no experience study of guaranteed-issue French funeral lives exists in any public source. The interaction with the waiting period is the point — the refund design exists precisely because year-1 non-accidental mortality is anti-selected — so the loading and the carence must be stressed together, never one at a time.

  2. The revalorisation rate. It compounds on the benefit for the whole of a whole-life contract. At 1.00 % p.a. the capital is 7746.59 € at 45 years against 5000 € at issue; the only other numerical anchor available is 1.2854 % p.a. derived from a KID scenario [S11], and five of seven insurers make the rate discretionary [S1] [S8] [S9] [S15] [S16]. Run 0 % / 1 % / 2 %. Where reval_prem_linked is set [S9] [S10] [S11] the sensitivity partly self-hedges, which is exactly why the flag must not be averaged across a book.

  3. Premium form mix. A portfolio fact rather than a sensitivity, and unobserved: the proportion of contracts sold in each form is published nowhere. A viagère book and a prime unique book of the same capital have opposite cash flow shapes — a long premium stream against a slowly rising benefit, versus a single receipt followed by four decades of pure outgo.

  4. Premium-stop behaviour and the surrender / réduction split. With a real surrender value the liability is not lapse-supported in the way the UK design is: in the worked cell zero lapse raises the undiscounted net stream from 2236.92 to 3165.11, because the premiums a lapser stops paying are worth more than the reserve handed back. Run 0.5× / 1× / 2× base lapse, zero lapse, and reduction_share at 0 / 0.5 / 1.0. The discounted, expense-inclusive answer can differ in sign from the undiscounted one and should be checked separately.

  5. Longevity at the top of the table, against a fixed premium. A viagère policy issued at 50 is still paying premiums at 90 and the model is still projecting at 112; improvement assumptions lengthen the premium stream and defer the claim, and the horizon convention (omega = 112 std, against tables running to 115 [S15]) is itself an assumption. Over that horizon maintenance expenses inflate against a premium that by construction cannot move — unless reval_prem_linked is set, the one design in the set that indexes it [S9] [S10] [S11].

  6. Basis mixing. The surrender scale, the premium and the revalorisation rate of the worked cell come from one insurer [S14] precisely so that they are consistent. Feeding one insurer’s premium into another’s surrender scale produces plausible-looking and wrong margins: the lifetime premium for the same capital spans 2.0:1 across the retrieved set at entry age 50, narrowing to 1.7:1 at 60 and 1.5:1 at 70.