Technical Notes#

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

Scope note. These notes turn the standardized composite of product-spec.md (same directory) into a reference liability cash-flow projection model on paper. They describe no single insurer’s product. [S#] and [R#] tags resolve in sources.md, whose numbering is carried verbatim from _research/dependance.md and is frozen; [REG-R#] tags resolve in the cross-product reference library references/regulatory-and-actuarial-references.md, whose R-numbering is separate. std marks a standardization introduced for the reference implementation, always with a rationale and, where one exists, the observed range; unverified marks a claim not confirmed against a retrieved document. Every contractual parameter value here is identical to product-spec.md’s. The model is Dep_FR_S on a monthly grid.

Seven quantities appear here that product-spec.md does not carry, because they are modeling constructs rather than contractual terms and are introduced below as such: the APA prevalence curve, the severity shares that turn public GIR prevalence into insured-state prevalence, the two state-mortality multiples, the aggravation rate, the prevalence-to-incidence identity, the cause mix that weights the three carences, and the lapse table.

The single fact that shapes this model. Every public French number about dependence measures receipt of the allocation personnalisée d’autonomie — an application to a département, granted on the AGGIR grid to GIR 1–4 R2 arts. R. 232-1, R. 232-4 R3. It is a prevalence, not an incidence; it is a public classification, not the insurer’s; and insurer definitions are deliberately stricter, the notice saying in terms that “L’Assureur n’est pas lié par les éventuelles décisions des services publics” [S5 art. 13] [S6 art. 21.1]. DREES notes that APA life expectancy has fallen from 30 to 29.2 months between 2010 and 2022 “traduisant un recours à cette prestation en baisse à âge donné” R7 — a behavioural drift in take-up, not a health improvement. Turning that series into an insured incidence basis is the whole modeling problem of this product, and section (c) does it explicitly rather than by assertion. No public French LTC incidence or continuance table exists: R12 specifies the structure of the laws a model needs but its numerical bases are the insurer’s own experience tables and are not disclosed, and no BCAC-style published reference table for dépendance was located R12 §3.1.3 REG-R28.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows for a single-policy model point of individual assurance dépendance: premiums, rente outgo, the capital d’équipement, the premiums refunded when dependence arises inside the carence, and expenses. Reserves are not computed (see Valuation and reserve pointers).

  • Model structure. A four-state chain — autonomous → partial / total → dead — with a fifth in-force but paid-up state, reduced, reached only by lapse from eight years. The partielle → totale transition is modeled, which is a departure from the only actuarial reference retrieved: R12 §3.1.2 sets that transition to zero for want of a transition law and prices two separate guarantees instead. The contracts themselves do provide for deterioration [S1 §4.3.1.2] [S5 art. 13], so the reference implementation carries it and states the cost of the missing law below.

  • Recovery is not modeled. Contractually the rente stops on improvement out of a covered state [S1 §4.3.1.2] [S6 art. 26] [S7 §4.2.1] and CNP allows the level to move in either direction [S5 art. 13]; R12 §3.1.1 nonetheless sets the probability of return to autonomy to zero, and so does this model. It is a named input held at zero, not an omission, and its direction of error is stated under Known modeling pitfalls.

  • Projection frequency. Monthly, matching the rente mensuelle à terme échu [S1 §4.3.1.2] [S5 art. 16] [S6 art. 26] [S7 §4.2.1] and the monthly premium [S1 §1.2.2] [S2]. t is the policy month and it is 0-based: t = 0 is the first projected month, the frame is t = 0, 1, …, proj_len − 1, proj_len is the number of projected months, and the policy year is the derived label y(t) = floor(t/12) + 1, so policy year 1 is t = 0…11.

  • Timing conventions std. Premium received at the start of month t, and only from lives in pols_auto; maintenance and assistance expense at the start of month t; all benefits, refunds and claim expenses at the end of month t; state transitions at end of month. The revalorisation of the guarantees, of the premium and of the rentes en service, and any tariff revision, all fall at the start of month t = 12, 24, …. Contracts revalue on a calendar date (1 January, or 1 April at the latest) [S1 §4.3.1.3] [S5 art. 15] [S7 §4.2.3]; replacing that with the policy anniversary is a std simplification worth at most six months of index.

  • Age basis. Age at entry by différence de millésimes [S1 §1.1.2.1], advancing at each policy anniversary: age(t) = entry_age + floor(t / 12) std.

  • Claim duration. z = months since the first recognition of a covered state, not since entry into the current state. The franchise clock therefore does not restart on deterioration from partielle to totale std — no retrieved document states that it does, and [S1 §4.3.1.2] makes the higher amount effective from the first day of the month following the opening of the right without mentioning a new franchise.

  • Currency and horizon. EUR; rente, capital and premium in € per month or per event. Cover is viagère with no age limit [S1 §1.1.5] [S5 art. 8], so the projection runs to a terminal age of 110 std: it is proj_len = 12 × (110 − entry_age) months long, 480 for the base cell, so the last projected month is t = proj_len − 1 = 479.

  • Contract boundary. The premium is viagère and the tariff is revisable for the portfolio [S1 §1.2.3] [S5 art. 22] [S7 §4.4]. The model projects all future premiums and benefits inside the boundary; whether a revisable-tariff contract has a Solvabilité II contract boundary shorter than that is a valuation question, is not settled by any retrieved text, and is flagged rather than answered.

  • Discounting. None. The model publishes undiscounted cash flows; EIOPA’s monthly risk-free term structures are the input a market-consistent valuation would apply REG-R5.

  • Rounding. Intermediates at full double precision; displayed state probabilities to six decimals and cash flows to four std. Rounded monthly rows do not re-add to displayed totals; totals are sums of unrounded values.

Model point attributes#

Attribute

Type

Example (worked configuration)

policy_id

str

1

entry_age

int, 40–75 [S1 §1.1.2.1]

70 std

sex

enum {M, F}

F std — decrements are sex-split; the premium is unisex R12 §3.2.1

cover_type

enum {total_only, total_and_partial}

total_and_partial [S1] [S7]

trigger_grid

enum {avq5, avq6, aggir}

avq5 [S1 §2.2]

rente_total_monthly

currency, €/month, 500–3,000

1,000 R8 §2.2

partial_ratio

fraction of the total rente

0.50 [S1] [S2] [S7] [S8] R12 §1.2.1

capital_option

bool

true [S1 §1.1.2.2c]

capital_amount

currency, €

3,500 [S1 §1.1.2.2c]

premium_monthly

currency, €/month at issue

75 R8 §2.2

premium_mode

enum {monthly, quarterly, half_yearly, annual}

monthly [S1 §1.2.2]

carence_accident_months

int

0 [S1 §1.1.5]

carence_illness_months

int

12 [S1 §1.1.5]

carence_neuro_months

int

36 [S1 §1.1.5]

franchise_months

int

3 [S1 §4.3.1.2] [S7 §4.2.1]

reduction_qualifying_years

int

8 [S1 §1.3] [S2] [S7 §4.6] R12 §1.2.1

couple_discount

bool

false std (spec footnote 11)

status

enum {autonomous, reduced, partial, total}

autonomous

claim_duration_months

int (in-claim cells only)

0

years_paid

int (reduced cells only)

0

The std marks in the Example column are base-cell picks, each explained in product-spec.md footnote 2 (entry age, sex, cover, rente and capital levels); every contractual value in the table carries its own citation there. Everything not marked std is contractual and cited in place.

premium_monthly is an input, not a computed quantity. No French insurer publishes a general individual LTC rate table; the only published scale found is CNP Banque de France annexe 1, for a group product on a four-rung severity ladder (2, 3, 4 and 5-or-6 AVQ of 6) sold across five subscribed coverage levels [S5 arts. 13, 16, 17, annexe 1], and no retrieved document discloses a technical rate, a loading or a profit-sharing rule. The base cell’s 75 €/month is the CCSF’s 2013 indicative price for exactly this cover at entry age 70 R8 §2.2. Its shape is corroborated at two other points on the same list — 35 €/month at 50 and 50 €/month at 60 R8 §2.2 — and by the CNP scale’s age gradient, which rises about 3.4× between age 50 and age 74, an average of about 5.2% per year of entry age [S5 annexe 1]. An in-force portfolio also needs claims-in-payment cells, with claim_duration_months and the rente amount in payment as model-point attributes, and paid-up cells with years_paid.

State variables#

Variable

Description

Updated

pols_auto(t)

Probability autonomous, in force, premium-paying, full guarantee, at the start of month t; pols_auto(0) = 1

monthly

pols_red(t)

Probability autonomous, in force, paid-up after mise en réduction: no premium, reduced rente totale only, no capital, no assistance, no further revalorisation of the guarantee

monthly

pols_part(t, z)

Probability in dépendance partielle at start of month t, z months since first recognition

monthly, two-dimensional

pols_tot(t, z)

Probability in dépendance totale, same duration index; a separate ledger pols_totr(t, z) carries lives that entered from pols_red and so hold a reduced rente

monthly, two-dimensional

pols_if(t)

pols_auto + pols_red + Σ_z pols_part + Σ_z pols_tot + Σ_z pols_totr

derived

G(y)

Guaranteed rente totale in policy year y, before claim

at anniversaries

CAP(y)

Guaranteed capital d’équipement in policy year y

at anniversaries

P(y)

Monthly premium in policy year y

at anniversaries

R_T(t, z), R_P(t, z)

Rente in payment for the cohort at duration z

at anniversaries

cum_prem(t)

Premiums actually paid per policy up to and including the start of month t — the contre-assurance refund base

monthly

n_P(t), n_T(t), n_A(t)

Entrants into partielle, into totale direct from autonomy, and aggravations partielle → totale

monthly

carence_exit(t)

Memberships terminated because dependence arose inside the carence for a cause not yet covered

monthly

net_cf(t)

Net cash flow of month t, insurer perspective, income-positive

monthly

Three absences are product facts, not gaps. There is no account value and no surrender value [S1 §7.3] [S11], so no cv_pp exists and lapse before eight years carries no cash flow at all. There is no death benefit on the composite — the optional Capital décès is out of scope [S1 §1.1.4.1] — so claims_death does not exist. And there is no maturity: the cover is viagère [S1 §1.1.5] [S5 art. 8].

pols_red is the one state a naive model omits, and omitting it is a first-order error: lapse from year 8 does not release the liability, it converts it into a smaller one that keeps running for life.

Assumption inputs#

(a) Contractual / guaranteed elements (cited; from the spec)#

Input

Value

Basis

Rente totale

G(1) = 1,000 €/month, monthly in arrears while the state persists

[S1 §4.3.1.2] [S5 art. 16]; amount std, price-paired R8 §2.2

Rente partielle

50% of the rente totale; the two are mutually exclusive

[S1] [S2] [S7] [S8] R12 §1.2.1

Capital d’équipement

3,500 €, once per membership, on first entry into either state, no franchise; extinguished on payment

[S1 §1.1.2.2c, §4.3.2.1] [S2] [S5 art. 17]; no-franchise pick std [S10]

Carence

0 / 12 / 36 months by cause; a claim inside it terminates the membership and refunds all premiums paid

[S1 §1.1.5, §1.1.4.2c] [S2] [S3] [S5 art. 7] [S7 §3.2]; contre-assurance R12 §3.2.1

Franchise

3 months absolute from recognition, so the cohort recognised at end of month s is first paid at end of month s + 4

[S1 §4.3.1.2] [S7 §4.2.1] [S8]; monthly reading std, corroborated by [S2]

Premium

P(1) = 75 €/month, in advance, payable for life until recognition

R8 §2.2; viagère form [S1 §1.2.1] [S5 art. 21] [S7 §4.4]

Premium exonération

From the premium due date following recognition — not from the start of rente payment

[S1 §1.2.4] [S4] [S5 art. 21] [S6 art. 18]

Reduction

From 8 full consecutive years of premiums; reduced rente totale only, no capital, no further revalorisation, assistance ends

[S1 §1.3] [S2] [S5 art. 24.2] [S7 §4.6] R12 §1.2.1; composite std (spec footnote 12)

Reduction scale c(n)

The CNP Banque de France barème, 25% at 8 years rising to 70% at 30

[S5 annexe 2]; re-based to 8 years std (spec footnote 13)

Surrender value

None; lapse before 8 years pays nothing

[S1 §7.3] [S11]

Footnotes to the std entries in this table, none of which is a free choice:

  1. G(1) = 1,000 €/month is a base-cell pick inside the sourced 500–3,000 € band [S1 §1.1.2.2a], chosen because it is the cover for which the only age-graded French price point exists R8 §2.2 (spec footnote 2). The observed market range across insurers is 200–4,000 €/month (spec footnote 6).

  2. The capital d’équipement is paid with no franchise, which only Generali states in terms [S10]; no other retrieved document separates the capital’s franchise from the rente’s (spec footnote 9). The choice moves one one-off payment by three months.

  3. The monthly reading of the three-month franchise — three instalments dropped, the cohort recognised at end of month s first paid at end of month s + 4 — is a standardization of “le 91e jour” onto a monthly grid [S1 §4.3.1.2] [S7 §4.2.1]. It is corroborated rather than assumed: Antarius restores exactly three instalments at the first payment [S2]. The alternative reading, paying at s + 3, would recover one instalment per claim and raise lifetime rente cost by roughly a third of the 7.09% the whole franchise is worth (see Key sensitivities).

  4. The reduction composite — totale only, no capital, no further revalorisation — and the re-basing of the CNP barème from a 5-year to an 8-year qualifying period are spec footnotes 12 and 13; the observed range of qualifying periods is 5 to 8 years across the retrieved contracts, and the CNP scale is the only one published.

(b) Insurer-discretionary current elements#

This class is not thin on this product — it is where the economics live, and every item in it is undisclosed.

Input

Snapshot value

Basis

Revalorisation des garanties g_G

1.0% per policy year, applied to G, to CAP and to the premium in the same proportion

mechanics [S1 §1.2.3] [S5 art. 21] [S7 §3.4]; rate std (1)

Revalorisation des rentes en service g_S

1.5% per policy year, applied to every rente in payment regardless of how long it has been in payment

mechanics [S1 §4.3.1.3] [S5 art. 15] [S7 §4.2.3]; rate std (1)

Tariff revision r(y)

0% in policy years 1–5, 1.5% per year from year 6; hard cap 10% per year excluding revalorisation

cap [S7 §4.4]; path std (2)

Revalorisation of a reduced guarantee

None

[S7 §4.6]

Technical rate, loadings, profit-sharing

Not disclosed in any retrieved document; not modeled

R12 §3.2.1 parameterises r, g, θ symbolically without values

  1. Two rates, deliberately different. Setting g_G = g_S makes the amount in payment depend only on the current policy year and collapses two ledgers into one — which hides a capability the contract requires, because the two indexations are governed by different clauses and, at CNP and Suravenir, by different external references (civil and military pension rates, or the AGIRC point) [S5 arts. 15, 21] [S7 §3.4, §4.2.3]. There is no observed range: no retrieved document states a rate actually served. The CCSF warns that a rente promised fifteen or twenty years ahead can be materially eroded at 2% average inflation R8 §3.3, which is the direction of the risk when g_S is below inflation, as it is here.

  2. A real tariff revision is a management action, not a projected assumption. The column exists so the capability is testable; the base path is arbitrary inside the 0–10% band [S7 §4.4]. See spec footnote 10.

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

Healthy-life mortality std. No French mortality rate was quoted in the research file, and the homologated tables — TH 00-02 / TF 00-02 for non-annuity business REG-R22, TGH05 / TGF05 for annuities REG-R21 — are cited by name and arrêté but not reproduced by this library REG-R23. What is shipped instead is a std proxy into which no retrieved datum enters. INSEE publishes the only freely redistributable French mortality series, and it is what a production implementation would graduate here REG-R24; this table is not read off it. It is a two-parameter Gompertz force,

mu_H(x) = B x c^x,    B = 5.2321459244e-06,  c = 1.11704543
mort_rate(x) = 1 - exp(-mu_H(x))

fitted to the two std anchors mort_rate(60) = 0.00400 and mort_rate(90) = 0.10500 — shaped like a French female population table, with no sourced value behind either anchor. The shipped mort_table.csv records that construction in a provenance column on every row — the formula, both anchors, and that the table is not a copy of any homologated table. Resulting rates: 0.01205 at 70, 0.02087 at 75, 0.03601 at 80, 0.06179 at 85, 0.10500 at 90, 0.17546 at 95, 0.28506 at 100. Above age 109 the rate is forced to 1 std.

State mortality, and why it is not flat. A dependent life’s mortality is far heavier than a healthy life’s at the same age, and this is the largest single lever on the liability. No impaired-life table for either French dependence state exists in any retrieved source. The model applies proportional hazards on the force:

mu_P(x) = k_P x mu_H(x),   k_P = 1.75  [std]
mu_T(x) = k_T x mu_H(x),   k_T = 4.27  [std, calibrated]

k_T is calibrated, not guessed: the CCSF reports a mean duration of receipt of the allocation for heavy dependents (GIR 1–2) of about three years, with mean age at onset of total dependence about 78 for men and 84 for women R9 §2. Setting k_T = 4.27 makes the model’s own expected sojourn in dépendance totale, entered at exact age 84, equal 2.9989 years. k_P = 1.75 has no such anchor and no observed range: it must exceed 1, because GIR 3–4 lives carry excess mortality, and sit well below k_T; at 1.75 the expected sojourn in dépendance partielle entered at age 82, ending in death or aggravation, is 3.14 years, the same order of magnitude as the 29.2-month mean duration of APA receipt DREES reports across all GIRs R7 and the 2.3–3.2-year expected APA durations among beneficiaries in REG-R25. Resulting annual probabilities at 85: healthy 0.06179, partielle 0.10562, totale 0.23841.

Aggravation partielle → totale std. i_A = 0.20 per year, flat in age. There is no public transition law: R12 §3.1.2 models no such transition at all. The value is set so that the sojourn in partielle is about three years (above). Its coupling with incidence is set out under the identity below and is the least obvious property of this model.

Prevalence — the public curve std. What is published is APA prevalence by age and sex. From DREES at end 2023 R7: 7.2% of people aged 60 or over receive APA, 9.1% of women against 4.8% of men, 70% of beneficiaries being women; the rate is 2.3% up to age 79, 17% between 80 and 89 (20% of women, 13% of men), 35% at 85 or over, and about half the population from 90 (54% of women, 40% of men); departmental dispersion of the 60+ rate runs 3.3% to 11.3%. CNSA confirms the same order at December 2022 — 1.3 million beneficiaries, 7.2% of an estimated 18.4 million people aged 60 and over REG-R26. The model fits a logistic in attained age to the two female rates, at representative ages 84.5 (the midpoint of the 80–89 band) and 93 (an approximate mean age of the 90-and-over group), both std picks:

prev(x) = prev_ceil / (1 + exp(-beta x (x - x_mid)))
prev_ceil = 0.90 [std],  beta = 0.195086,  x_mid = 90.921605

so prev(84.5) = 0.20 and prev(93) = 0.54 by construction, and prev(70) = 0.014942, prev(80) = 0.095538, prev(90) = 0.409655, prev(100) = 0.769131 by extrapolation. As in every logistic fit of this kind the curve has three parameters and two anchors, and the unidentified one is prev_ceil, which governs the tail — the region where the claims are. Two checks against rates the fit did not use, weighting each age by survivorship on the std mortality above: the mean of prev over ages 60–79 comes out at 2.15% against the sourced 2.3% R7, and over 85 and above at 41.7% against the sourced 35% R7 (which is an all-sex rate, so a female-anchored curve should sit above it). Over the whole 60+ range the curve gives 11.0% against the sourced 9.1% for women R7 — an overstate of about a fifth, because survivorship weighting from age 60 is not the real age structure of the French 60+ population, which is younger. APA is not available below age 60 R2 art. R. 232-1, so the curve has no anchor at all under 60 and every issue age below 60 in the 40–75 band runs on pure extrapolation.

From public prevalence to insured prevalence std. Public prevalence is APA take-up on GIR 1–4. Insurer definitions are stricter, and the notice says so [S5 art. 13] [S6 art. 21.1]. Two sourced anchors bound the haircut. First, the GIR composition of APA beneficiaries at end 2023 — at home 2% / 18% / 22% / 58% and in establishments 13% / 44% / 19% / 24%, on 815,800 and 549,000 beneficiaries R7 — gives a weighted GIR 1–2 share of 34.9% and a GIR 3–4 share of 65.1%. Second, the market’s own count: 44,200 rentes in payment on sole-and-principal-guarantee contracts against about 1.39 million people covered under such contracts (58% of the 2.4 million covered by insurance undertakings) gives an insured “in rente” prevalence of about 3.2% R10 §2.3 R13 p6 REG-R28, against an APA prevalence of 7.2% of the 60-and-over population R7 REG-R26 — a ratio of about 0.44, on populations whose age structures are not published and are certainly not the same. The model therefore sets

prev_T(x) = s_T x prev(x),   s_T = 0.30  [std]
prev_P(x) = s_P x prev(x),   s_P = 0.15  [std]

with s_T + s_P = 0.45 against the 0.44 the market count implies. s_T = 0.30 sits just below the sourced GIR 1–2 share of 34.9%, which is the direction “the insurer is not bound by the public decision” points. s_P = 0.15 sits far below the GIR 3–4 share of 65.1%, because a 3-of-5-AVQ partielle trigger [S1 §2.2] is far stricter than GIR 3–4 — and the one contract that requires both grids at once equates its 3-of-5 tier with GIR 1–3, not GIR 3–4 [S1 §2.2]. Holding the shares constant across ages is a standardization with a known direction of error: severity mix worsens with age (57% of establishment beneficiaries are GIR 1–2 against 20% at home R7), so the model understates totale prevalence at old ages and overstates it at young ones. For trigger_grid = avq6 the shares should be scaled down and for aggir up, by amounts no retrieved document supports; the shipped severity_share_table.csv carries all three rows and the two non-base rows are std with no anchor whatever.

The prevalence-to-incidence identity. With pi_P, pi_T, pi_H = 1 − pi_P − pi_T the proportions of the living population in each state, i_P, i_T the forces of entry from autonomy, i_A the force of aggravation and mu_H, mu_P, mu_T the forces of mortality, differentiating along the age axis gives, with mubar = mu_H·pi_H + mu_P·pi_P + mu_T·pi_T:

i_P(x) = [ pi_P'(x) + (i_A + mu_P) x pi_P - pi_P x mubar ] / pi_H
i_T(x) = [ pi_T'(x) - i_A x pi_P + mu_T x pi_T - pi_T x mubar ] / pi_H

with pi_G' = s_G · beta · prev · (1 − prev/prev_ceil). This is an identity, not an approximation, and it has three properties an implementation must respect. The mortality terms are not refinements: a rising prevalence understates incidence because the dependent population is simultaneously being drained by its own excess mortality, so dropping mu_T · pi_T understates i_T. i_A and i_T are not independent inputs: raising the aggravation force lowers the direct-to-totale incidence, because the stock of totale lives is pinned by the assumed prevalence. And i_P can go negative at extreme ages, where the prevalence slope flattens while excess mortality does not; the model floors both rates at zero std, which does not bind on the female basis inside the projection and binds at attained age 109 on the male one.

Resulting annual forces on the std basis, and the monthly probabilities i_m = 1 − exp(−i/12):

Attained age

70

75

80

85

90

95

100

i_P

0.000904

0.002365

0.005961

0.013650

0.025599

0.035422

0.034902

i_T

0.000593

0.001823

0.005704

0.017325

0.048116

0.118892

0.262330

The gradient from 70 to 90 is a factor of 28 for i_P and 81 for i_T, and i_T overtakes i_P between ages 80 and 85 — the severity mix worsening with age, arriving here through the mortality terms of the identity rather than through the constant severity shares, which cannot produce it.

The stationary-population assumption std. The cross-sectional APA rate by age is read as the prevalence path a cohort will follow. It is not: the CCSF projects 4 million seniors in loss of autonomy in 2050, 16.4% of the 60+ against 15.3% in 2015, with severe loss of autonomy at 4.3% against 3.7% R9, and take-up at a given age has been falling R7. Two trends in opposite directions, neither modeled.

Cause mix for the carence std. accident 10% / other illness 55% / neurological or psychiatric 35% (spec footnote 8), giving a carence factor S(t) of 0.10 in policy year 1, 0.65 in years 2 and 3, and 1.00 thereafter — the S1 ≤ S2 ≤ S3 ≤ S4 = 100% shape R12 §3.2.1 asks for. No observed range.

Lapse std. No French LTC persistency study is public. The table is anchored on one market fact: individual memberships fell 9.9% in 2024 on 28,400 new subscribers of which 82% individual R10 §2.3 REG-R28, so gross exits from the individual book — deaths, claim entries and lapses together — ran at roughly 11% of the opening portfolio. A lapse table of 3%–8% leaves the balance for mortality and incidence.

Policy year

1

2

3–5

6–10

11+

lapse_rate std

8%

6%

5%

4%

3%

lapse_rate_mth(t) = 1 − (1 − lapse_rate(y))^(1/12). Lapse applies to pols_auto only: lives in a recognised state pay no premium [S1 §1.2.4] and lives in pols_red pay none either, so neither can lapse for non-payment, and with no surrender value there is nothing to surrender for [S1 §7.3].

Expenses (all levels std).

Input

Value

Note

Acquisition

150 € per policy at t = 0

std

Maintenance

3.00 € per policy per month on pols_if, inflating 1.5% p.a. at each anniversary

std

Assistance

1.20 € per policy per month on pols_if − pols_red, inflating 1.5% p.a.

std; the base excludes reduced lives because mise en réduction ends the assistance benefits [S1 §1.3] [S5 art. 24.2]

Claim adjudication

250 € per entrant into either state

std; the AMED file, the médecin-conseil ruling within 45 working days, and the medical arbitration procedure [S5 arts. 19–20] [S6 arts. 23–24]

Rente handling

10 € per instalment paid

std; annual proof of life and of the persisting state [S1 §4.3.1.2] [S6 art. 23]

Expense inflation

1.5% p.a.

std

There is no observed range for any expense level: no retrieved document — notice, IPID, product page or dissertation — discloses an expense assumption, a loading or a commission rate for this product, and R12 §3.2.1 parameterises the loadings r, g and θ symbolically without values. The Note column above is each row’s whole rationale. Only two structural facts are sourced and they are respected: assistance ends on mise en réduction [S1 §1.3] [S5 art. 24.2], so its base excludes pols_red; and claim adjudication is a real, medically supervised process with a 45-working-day deadline and an arbitration route [S5 arts. 19–20] [S6 arts. 23–24], so it carries a per-claim cost an order of magnitude above the per-instalment one.

Cash flow components and recursions#

Notation#

Symbol

Meaning

t

policy month, 0-based: t = 0, 1, …, proj_len − 1; y(t) = floor(t/12) + 1; age(t) = entry_age + floor(t/12)

G(y), CAP(y), P(y)

guaranteed rente totale, capital, monthly premium in policy year y; G(1) = 1,000, CAP(1) = 3,500, P(1) = 75

g_G, g_S, r(y)

revalorisation of guarantees, of rentes en service, tariff revision: 0.010 / 0.015 / 0 then 0.015 std

rho

partial/total rente ratio, 0.50

S(t)

carence factor: 0.10, 0.65, 1.00 std

fr

franchise in months, 3; a cohort is paid when z ≥ fr + 1 = 4

q_H(t), q_P(t), q_T(t)

monthly mortality of autonomous, partielle, totale lives: 1 − (1 − mort_rate(age))^(k/12) with k = 1, k_P, k_T

i_Pm(t), i_Tm(t), i_Am

monthly entry and aggravation probabilities, 1 − exp(−i/12)

w(t)

monthly lapse, applied to pols_auto only

c(n)

reduction coefficient at n completed years of premiums; 0 below 8

e(y), a(y), ec_adj, ec_ren

maintenance, assistance, adjudication and rente-handling expense std

Dimensional check. prev, pi_P, pi_T are dimensionless proportions of a living population; i_P, i_T, i_A, mu are rates per year, and beta carries units of 1/year, which is why pi_G' = s_G · beta · prev · (1 − prev/prev_ceil) is a rate per year and can be added to pi_G · mu, also a rate per year. G, P are € per month, CAP € per event, so G × Σ_z pols_tot and CAP × (n_P + n_T) are both € per policy-month. The error this check catches is the one that dominates this product: multiplying a published APA prevalence — 7.2% of the 60-and-over population R7 — by a rente amount as though it were an annual claim frequency.

The four-state chain#

Write auto = pols_auto, red = pols_red. At end of month t, from the autonomous state, in the order mortality, then lapse, then incidence among the survivors std:

surv(t)  = auto(t) x (1 - q_H(t))
lapse(t) = surv(t) x w(t)
base(t)  = surv(t) - lapse(t)
n_P(t)   = base(t) x i_Pm(t) x S(t)
n_T(t)   = base(t) x i_Tm(t) x S(t)
carence_exit(t) = base(t) x (i_Pm(t) + i_Tm(t)) x (1 - S(t))
auto(t+1) = base(t) x (1 - i_Pm(t) - i_Tm(t))

so that the carence removes exactly the blocked fraction of incidence from the in-force ledger and nothing else — auto(t+1) does not depend on S(t), which is the arithmetic statement of the fact that a carence claim ends the membership rather than being deferred.

From the reduced state, which carries no partial cover and no carence (eight years of premiums have been paid):

surv_r(t) = red(t) x (1 - q_H(t))
n_Tr(t)   = surv_r(t) x i_Tm(t)
red(t+1)  = surv_r(t) - n_Tr(t) + lapse(t) x 1{(t + 1) >= 12 x reduction_qualifying_years}

with the entering rente frozen at G(y) × c(n) at the reduction date and never revalued before claim [S7 §4.6]. Implementations that cannot carry a per-reduction-cohort amount may track red and the probability-weighted mean frozen rente instead; that is exact in expectation because incidence does not depend on the amount.

The indicator counts the instalments already paid: with t 0-based, the lapse at the end of month t becomes a mise en réduction once t + 1 premiums are behind it, so the first such month is t = 12 × 8 − 1 = 95 on the base cell.

From the two dependent states, per duration cohort z, with the aggravated lives paid the partial rente for the month in which they aggravate — the higher amount takes effect from the first day of the following month [S1 §4.3.1.2]:

part_s(t, z) = pols_part(t, z) x (1 - q_P(t))
n_A(t, z)    = part_s(t, z) x i_Am
tot_s(t, z)  = pols_tot(t, z) x (1 - q_T(t))
pols_part(t+1, z+1) = part_s(t, z) - n_A(t, z)
pols_tot(t+1, z+1)  = tot_s(t, z) + n_A(t, z)
pols_part(t+1, 1) = n_P(t)     pols_tot(t+1, 1) = n_T(t)     pols_totr(t+1, 1) = n_Tr(t)

Benefits#

claims_rente(t)   = rho x G_pay(t) x SUM over z >= 4 of part_s(t, z)
                  +       G_pay(t) x SUM over z >= 4 of tot_s(t, z)
                  +                  SUM over z >= 4 of Rred(z) x totr_s(t, z)
claims_capital(t) = CAP(y) x ( n_P(t) + n_T(t) )
refunds_carence(t) = carence_exit(t) x cum_prem(t)

G_pay(t) is the rente in payment for a cohort recognised in policy year y_e, namely G(y_e) × (1 + g_S)^(y − y_e); the reduced ledger carries its own frozen amounts Rred. The capital is paid on entry from pols_auto only — reduced memberships lose the option R12 §1.2.1 — and never twice, so aggravation n_A produces no capital.

Monthly processing order std#

For t = 0, 1, …, proj_len − 1:

  1. Anniversary (start of month, t = 12, 24, …). G(y) = G(y−1) × (1 + g_G); CAP(y) = CAP(y−1) × (1 + g_G); P(y) = P(y−1) × (1 + g_G) × (1 + r(y)); every rente in payment × (1 + g_S). Reduced guarantees are not touched [S7 §4.6].

  2. Premium (start of month). premiums(t) = P(y) × pols_auto(t) — not × pols_if(t). Accumulate cum_prem(t) += P(y).

  3. Expenses (start of month). e(y) × pols_if(t) + a(y) × (pols_if(t) − pols_red(t)), plus acquisition at t = 0.

  4. Look up the age basis. age(t), hence mort_rate, q_H, q_P, q_T, prev, i_P, i_T, and hence i_Pm, i_Tm; w(t) from the policy year; S(t) from t.

  5. End of month — claims. claims_rente(t) on the surviving cohorts with z ≥ 4; claims_capital(t) on n_P(t) + n_T(t); refunds_carence(t); claim expenses ec_adj × (n_P + n_T + n_Tr) + ec_ren × (number of instalments paid).

  6. End of month — decrements and ledger roll, per the recursions above.

Net cash flow#

net_cf(t) = premiums(t) - claims_rente(t) - claims_capital(t)
          - refunds_carence(t) - expenses(t) - claim_expenses(t)

net_cf is income-positive. claims_lapse(t) is identically zero — there is no surrender value [S1 §7.3] — and that zero is a product fact worth publishing. refunds_carence is not a claim: it is a return of premium, and it belongs on its own line because it is the only cash flow that runs backwards through the carence.

Known modeling pitfalls#

  • Flat mortality across states is the biggest single error available here. Applying healthy-life mortality to dependent lives, while leaving the incidence basis unchanged, raises lifetime claims on the worked configuration by +159.7%. A GIR 1–2 life at 84 dies at an annual rate of 0.216 on this basis against 0.055 for a healthy life of the same age. No impaired-life table exists in any retrieved source R12 §3.1.3, which is exactly why the multiple is easy to leave at 1 and catastrophic to leave at 1.

  • Ignoring the mise en réduction turns every lapse into a full release of liability. A paid-up membership keeps a reduced rente totale for life [S1 §1.3] [S2] [S7 §4.6] R12 §1.2.1. Treating lapse from year 8 as an exit understates lifetime claims on the worked configuration by 4.57%, and the ledger it drops peaks at 8.27% of the original policy at month 194 (attained age 86) — the single largest state in the model after pols_auto at that duration. It is the second decrement, not the absence of one.

  • Carence and franchise are different things and a model must implement both. The carence runs from inception, is cause-specific, blocks the benefit and terminates the membership with a full refund of premiums [S1 §1.1.4.2c] [S3] [S5 art. 7] [S7 §3.2]. The franchise runs from recognition, is three months, and only delays payment [S1 §4.3.1.2] [S7 §4.2.1]. Removing the carence raises lifetime claims by +3.99%, removing the franchise by +7.09% — different sizes and different signs of error if either is applied in the other’s place. In policy year 1 of the worked configuration refunds_carence is 0.6141 €, three quarters of the year’s rente and capital claims combined (0.8071 €): during the carence the largest benefit-side cash flow is a premium refund.

  • The franchise is not a premium holiday. Exonération runs from recognition [S1 §1.2.4] [S4] [S5 art. 21] [S6 art. 18], so a life in the three-month franchise pays no premium and receives no rente. Carrying the franchise the way an income-protection deferred period is carried — premium-paying, benefit-free — overstates premium income.

  • i_A and i_T are not independent. The prevalence identity ties them: consistently varying i_A from 0 to 0.20 to 0.40 moves lifetime claims by only +0.54% / 0 / −0.52%, because the stock of totale lives is pinned by prev_T. An implementation that adds an aggravation rate without re-deriving i_T double-counts entries into totale and raises claims by +0.84% while putting the lives in the wrong state — which matters more than the total, because partielle pays half.

  • Premium income rides on pols_auto, never on pols_if. Lives in a recognised state are exonerated [S1 §1.2.4] and reduced lives are paid up [S1 §1.3], so both bands pay nothing. Charging premium to the whole in-force block overstates premium income by the whole of the reduced ledger plus the whole of the claim ledger.

  • A carence claim is a decrement with a cash flow, not a suppressed claim. Modelling the carence as a multiplier on incidence alone leaves the terminated membership in force and omits the refund R12 §3.2.1. Both errors run the same way: they overstate the liability at the front end and the premium income behind it.

  • The capital d’équipement is paid once per membership, not once per state. A life that takes it on entering partielle takes nothing further on aggravating [S1 §4.3.2.1] [S2] [S4] [S5 art. 17]. Paying it again on n_A inflates capital claims by the whole aggravation flow.

  • Two indexations, two ledgers. g_G moves the guarantee and the premium; g_S moves the rente in payment; the reduced guarantee moves with neither [S1 §1.2.3] [S5 art. 21] [S7 §3.4, §4.6]. Collapsing them into one rate happens to work only when g_G = g_S, and the base configuration sets them different so that a test can tell.

  • The duration index runs from first recognition. A cohort that aggravates keeps its z, so it does not serve a second franchise std. Restarting z on aggravation drops three instalments per aggravated life.

  • APA prevalence is a prevalence, and it is public. It is not an incidence, and it is not the insurer’s definition [S5 art. 13] [S6 art. 21.1]. Both conversions — the two-term identity and the severity shares — are explicit std steps, and quoting a model incidence rate as though it carried the R7 provenance of the two prevalence anchors misrepresents where the evidence stops.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions; no French LTC policyholder-behaviour study was retrieved.

  • Lapse stops at recognition, and again at reduction. Once the state is recognised the premium is exonerated [S1 §1.2.4] [S5 art. 21], so there is no premium to miss; once the membership is reduced there is no premium either [S1 §1.3]. w(t) therefore applies to pols_auto alone. On the worked configuration the reduced and dependent bands together are 44.6% of the in-force block at attained age 90 — pols_auto 0.133256, pols_red 0.070326, pols_part 0.010324, pols_tot 0.020960 and pols_totr 0.005782 out of pols_if 0.240648 — so at that age fewer than three in five surviving memberships are still paying anything. The two totale ledgers are separate columns in result_cf() and must be added by hand: 0.020960 + 0.005782 = 0.026742 is the whole totale band, and it is the reduced-entry ledger pols_totr that a naive three-state model loses entirely.

  • Lapse is genuinely a decision to walk away from everything. With no surrender value [S1 §7.3] and a fonds perdu design [S11], a lapse before eight years destroys the whole accumulated value. That is the CCSF’s consumer complaint R8 §4.2 and it is also the reason the std lapse table is set below what one would use on a savings contract.

  • Premium-shock lapse std (optional module, off in the base run). The member may refuse a tariff revision by resiliating within two months of notification, with a possible mise en réduction at the same date [S1 §1.2.3]. The module multiplies lapse in a revision year by M_rev(y) = 1 + 3 × max(0, r(y) − 0.02), so a revision at the 10% cap [S7 §4.4] gives M_rev = 1.24. It is off in the base run because r(y) ≤ 0.015 there, and it is the only place a projected repricing feeds back into the block.

  • Anti-selection sits at the front door. Underwriting is two-stage and the médecin-conseil sets the terms [S5 art. 3] [S7 §2.2]; increases in cover are re-underwritten and restart the carence [S1 §1.1.3] [S7 §3.3]; and the carence itself is a selection device with the sharpest possible teeth — a claim inside it voids the membership entirely [S1 §1.1.4.2c]. No selection loading is applied at issue.

  • Cover changes are held at zero. Increases and decreases are contractually available [S1 §1.1.3] [S7 §3.3] R12 §1.2.1 but they change the guarantee, the premium and the carence together, which is a new model point rather than a decrement.

  • Claim behaviour is not policyholder behaviour here. The insured is often no longer able to claim, which is why every contract and the CCSF urge that relatives be told the contract exists [S7 “Quelques conseils”] R8 §4.4. Late notification is real — the recognition date cannot precede the date the insurer received the claim [S6 arts. 23–24] — and the model does not carry it, so its claim dates are the earliest defensible ones.

Worked example#

Configuration. Female, entry age 70 (différence de millésimes), formule Dépendance Totale et Partielle on the 5-act AVQ grid [S1 §2.2]; rente_total_monthly = 1,000 €, partial_ratio = 0.50, capital_amount = 3,500 €, premium_monthly = 75 € R8 §2.2; carence 0 / 12 / 36 months by cause; franchise 3 months; reduction from 8 years; proj_len = 12 × (110 − 70) = 480 months, so the last projected month is t = 479. Undiscounted. All sixteen rows below sit in policy years 1 and 2, so two sets of rates drive them.

Assumption values used, every one of them:

  • Mortality std. mort_rate(70) = 0.0120506, mort_rate(71) = 0.0134515 from the Gompertz proxy. Monthly: q_H = 0.0010098056 at 70 and 0.0011279321 at 71; q_P = 1 − (1 − q)^(1.75/12) = 0.0017664906 and 0.0019730461; q_T = 1 − (1 − q)^(4.27/12) = 0.0043047564 and 0.0048073955.

  • Lapse std. Policy year 1, 8%: w = 1 − 0.92^(1/12) = 0.0069243826. Policy year 2, 6%: w = 0.0051430128.

  • Prevalence std. prev(70) = 0.01494159, prev(71) = 0.01809552, from the logistic pinned to the sourced female APA rates at 84.5 and 93 R7.

  • Severity shares std. s_P = 0.15, s_T = 0.30, so at age 70 pi_P = 0.002241239, pi_T = 0.004482477, pi_H = 0.993276284.

  • Incidence, from the two-term identity with k_P = 1.75, k_T = 4.27, i_A = 0.20 std: at 70, mubar = 0.012321875 against mu_H = 0.012123789, giving i_P = 0.000904237 and i_T = 0.000592504, hence i_Pm = 0.0000753503 and i_Tm = 0.0000493741; at 71, i_Pm = 0.0000914562 and i_Tm = 0.0000616541. i_Am = 1 − exp(−0.20/12) = 0.0165285462.

  • Carence std. S = 0.10 for t = 0…11, S = 0.65 for t = 12…35.

  • Revalorisation std. At t = 12: G = 1,010.00, CAP = 3,535.00, P = 75 × 1.01 × 1.00 = 75.75; rentes in payment × 1.015, so a year-1 cohort is paid 1,015.00 (total) or 507.50 (partial) from t = 12.

  • Expenses std. Acquisition 150 € at t = 0; maintenance 3.00 €/month and assistance 1.20 €/month in policy year 1, both × 1.015 in year 2; adjudication 250 € per entrant; rente handling 10 € per instalment.

pols_red(t) = 0 throughout the window — the first reduction is at t = 95 — and the expenses column below is the combined expense of the month: maintenance, assistance, claim adjudication, rente handling, and acquisition at t = 0. The model publishes expenses and claim_expenses as two columns of result_cf().

t

pols_auto

pols_part

pols_tot

premiums

claims_rente

claims_capital

refunds_carence

expenses

net_cf

0

1.000000

0.000000

0.000000

75.0000

0.0000

0.0433

0.0084

154.2031

−79.2548

1

0.991949

0.000007

0.000005

74.3962

0.0000

0.0430

0.0166

4.1693

70.1673

2

0.983963

0.000015

0.000010

73.7972

0.0000

0.0426

0.0247

4.1358

69.5942

3

0.976041

0.000022

0.000015

73.2031

0.0000

0.0423

0.0326

4.1025

69.0257

4

0.968183

0.000029

0.000020

72.6137

0.0087

0.0419

0.0404

4.0697

68.4530

5

0.960388

0.000035

0.000025

72.0291

0.0174

0.0416

0.0481

4.0371

67.8849

6

0.952656

0.000042

0.000030

71.4492

0.0260

0.0413

0.0557

4.0048

67.3215

7

0.944987

0.000048

0.000035

70.8740

0.0346

0.0409

0.0631

3.9727

66.7627

8

0.937379

0.000055

0.000041

70.3034

0.0431

0.0406

0.0705

3.9409

66.2083

9

0.929832

0.000061

0.000046

69.7374

0.0516

0.0403

0.0777

3.9093

65.6585

10

0.922346

0.000066

0.000051

69.1759

0.0600

0.0399

0.0847

3.8780

65.1132

11

0.914920

0.000072

0.000057

68.6190

0.0684

0.0396

0.0917

3.8470

64.5723

12

0.907554

0.000078

0.000062

68.7472

0.0779

0.3173

0.0472

3.8930

64.4119

13

0.901730

0.000130

0.000099

68.3060

0.0863

0.3152

0.0505

3.8685

63.9855

14

0.895943

0.000181

0.000137

67.8677

0.0947

0.3132

0.0538

3.8442

63.5619

15

0.890194

0.000230

0.000175

67.4322

0.1029

0.3112

0.0570

3.8200

63.1410

Policy year 1 in aggregate (t = 0…11, all at age 70, all in policy year 1 — the strongest single test target in this file, because it exercises the whole annual cycle on one set of rates), with pols_auto(12) = 0.907554, pols_part(12) = 0.000078, pols_tot(12) = 0.000062 and pols_if(12) = 0.907694:

Line

Policy year 1 total

premiums

861.1983

claims_rente

0.3098

claims_capital

0.4973

refunds_carence

0.6141

expenses (acquisition + maintenance + assistance)

198.2304

claim_expenses

0.0398

net_cf

661.5068

(The totals are sums of unrounded monthly values; the twelve displayed rows do not re-add to them, and the year-1 net_cf differs by €0.0001 from the difference of the six rounded lines above it.)

Checks. Three of these numbers, re-derived a different way.

Month 0, end to end. pols_auto(0) = 1, so premiums(0) = 75 × 1 = 75.0000. Survivors of mortality: 1 × (1 − 0.0010098056) = 0.9989901944. Lapses: 0.9989901944 × 0.0069243826 = 0.0069173903, leaving base(0) = 0.9920728040. Entrants: n_P = 0.9920728040 × 0.0000753503 × 0.10 = 0.0000074753; n_T = 0.9920728040 × 0.0000493741 × 0.10 = 0.0000048983; blocked by the carence, carence_exit = 0.9920728040 × (0.0000753503 + 0.0000493741) × 0.90 = 0.0001113621. So claims_capital(0) = 3,500 × (0.0000074753 + 0.0000048983) = 0.0433, refunds_carence(0) = 0.0001113621 × 75 = 0.0084, claim expense 250 × 0.0000123736 = 0.0031, and expenses(0) = 150 (acquisition) + 3.00 (maintenance) + 1.20 (assistance) + 0.0031 (claim) = 154.2031. Hence net_cf(0) = 75.0000 − 0.0433 − 0.0084 − 154.2031 = −79.2548. Roll forward: pols_auto(1) = 0.9920728040 × (1 − 0.0000753503 − 0.0000493741) = 0.9919490683, printed 0.991949. Note the carence does not appear in that last line — the blocked lives leave the in-force ledger exactly as the covered ones do.

Month 4, the first rente instalment. The cohorts recognised at the end of month 0 are the only ones old enough to be paid: seeded at z = 1 at the start of month 1, they reach z = 4 at the start of month 4 and are paid at its end. The partial cohort survives three months of mortality and aggravation and a fourth month of mortality: 0.0000074753 × [(1 − 0.0017664906)(1 − 0.0165285462)]^3 × (1 − 0.0017664906) = 0.000007060611. The total ledger at z = 4 holds the direct entrants plus the three months of aggravated lives, and comes to 0.000005174632. So claims_rente(4) = 500 × 0.000007060611 + 1,000 × 0.000005174632 = 0.0087, against 0.0000 at t = 3 — the franchise is exactly three dropped instalments, and the model pays the fourth. Nothing else in the month-4 row moves: the rente-handling expense adds 10 × 0.000012235 = 0.0001 to the expense column.

Month 12, the carence step. base(12) = 0.90755402 × (1 − 0.0011279321) × (1 − 0.0051430128) = 0.90186807. Entrants = 0.90186807 × (0.0000914562 + 0.0000616541) × 0.65 = 0.000089755, so claims_capital(12) = 3,535.00 × 0.000089755 = 0.3173 — the jump of 8.0076× on month 11 decomposes exactly as 6.5 × 1.227589 × 1.01 × 0.993611: the carence widening (0.65 / 0.10), the age step 70 → 71 in the incidence identity, the revalorisation of the capital, and the in-force run-off. In the same month the refund falls the other way: refunds_carence(12) = 0.90186807 × 0.0001531103 × 0.35 × 975.75 = 0.0472, where 975.75 = 12 × 75 + 75.75 is the premium actually paid to that point — down from 0.0917 a month earlier even though incidence has risen, because only 35% of causes are still blocked.

The k_T calibration. Running the totale ledger alone from exact age 84 with q_T = 1 − (1 − mort_rate(x))^(4.27/12) and no other decrement gives an expected sojourn of 2.9989 years, against the “about three years” the CCSF reports for heavy dependents R9 §2. At k_T = 2.75 the same calculation gives 4.19 years and at k_T = 3.50, 3.50 years — the sojourn is far more sensitive to k_T than a first look suggests, which is why this is a calibration rather than a pick.

Lifetime totals for the same configuration (undiscounted, per policy issued, over all 480 months): premiums 10,867.00; claims_rente 5,885.08; claims_capital 632.92; refunds_carence 2.86; expenses 828.71; claim_expenses 113.18; net_cf 3,404.24. Claims are 60.0% of premiums and 65.0% of them fall at attained age 85 or over. The reduced ledger peaks at 0.082685 of the original policy at t = 194 (attained age 86), and total expected entrants into a covered state over the lifetime are 0.198 per policy issued, receiving 6.368 rente instalments between them.

Valuation and reserve pointers#

This library projects gross best-estimate liability cash flows. Every valuation layer below consumes them and is cited, never reproduced.

  • Solvabilité II. Technical provisions are a best estimate — the probability-weighted average of future cash flows discounted at the relevant risk-free term structure — plus a risk margin REG-R4 REG-R1 REG-R2. The cash flows above are exactly the input; the curve comes from EIOPA’s monthly publication REG-R5. LTC sits in the Health-SLT underwriting module, life techniques applying because of the long-term commitment R12 §1.1.2.1, and R12 ch. 4 projects the SCR through incidence and longevity shocks combined with the contractual right to revise premiums — which is the point at which the tariff-revision column of section (b) stops being decoration.

  • French statutory provisions. Art. R. 343-3 enumerates the eleven technical provisions, each engagement provisionable under exactly one of them, item 1 being the provision mathématique computed including future management costs REG-R6. This product generates a provision pour risques croissants on autonomous insureds — present value of future commitments less present value of future premiums, allowing for the waiting-period incidence reduction and for the counter-insurance of refunded premiums, computed separately for the rente and the capital — and a provision mathématique des rentes on rentes in payment R12 §3.2.2. A provision d’aggravation for partial dependents who may become total exists in principle and is not used under R12’s two-guarantee model; this model produces the flow it would provide for. The Code des assurances article governing the PRC was not retrieved R18, so none of this is cited to the code.

  • Mortality tables. A tariff must use homologated tables by sex or the undertaking’s own tables certified by an independent actuary REG-R23; TGH05 / TGF05 for annuities REG-R21, TH 00-02 / TF 00-02 otherwise REG-R22. The technical-rate ceiling for a periodic-premium contract is the lower of 3.5% and 60% of the semi-annual average rate of French State borrowings REG-R17. None of these is used in the projection, which is undiscounted.

  • Reinsurance. Quota-share treaties are usual on this risk; the Sogecap product cedes 70% R12 §1.2.1. The model is gross of reinsurance.

  • Professional standards. Institut des actuaires NPA 2, Modèles actuariels, applies to any actuarial model under a principle of proportionality and covers pricing and the technical studies attached to new products REG-R44. NPA 4, on best-estimate provisions in life, was not retrieved and is unverified in this library.

  • IFRS 17. Fulfilment cash flows plus a contractual service margin, effective for reporting periods beginning on or after 1 January 2023 REG-R45; the same projection feeds it with its own discounting and risk adjustment.

Key sensitivities and model risks#

In rough order of leverage on this block. Percentages are changes in undiscounted lifetime claims (claims_rente + claims_capital) on the worked configuration.

  1. State mortality. Flattening it — healthy mortality applied to dependent lives, the incidence basis unchanged — moves claims by +159.7%. Lightening k_T from 4.27 to 2.75 moves them by +9.05% and lifts premiums 2.0%, because the identity also lowers i_T. k_P from 1.75 to 1.0 moves claims +2.08%. There is no impaired-life table for either state in any retrieved source, and the only anchor is a mean duration of about three years for heavy dependents R9 §2.

  2. Lapse. Turning lapse off raises claims +78.1% and premiums +56.8%. On a product whose claims are concentrated fifteen years out, persistency is a first-order assumption, and the only public anchor is a portfolio that shrank 9.9% in 2024 on 28,400 new subscribers R10 §2.3 REG-R28.

  3. The severity shares. Raising s_T and s_P by a tenth relative moves claims +8.06%; raising s_T alone from 0.30 to 0.35 moves them +10.39% and cuts premiums 1.2%. They are the whole of the public-to-insured translation and they are std against two indirect anchors — the GIR 1–2 share of APA beneficiaries R7 and the market’s ratio of rentes in payment to lives covered R10 §2.3.

  4. The prevalence tail. The logistic is pinned at ages 84.5 and 93 and unpinned above 93, and 65.0% of lifetime claims fall at attained age 85 or over. prev_ceil = 0.90 is a std choice with no sourced value behind it and it is the parameter that sets the tail. What would fix it is not a better fit but finer data: DREES publishes the 60+ rate by department and by broad age band R7 but no five-year-band series was retrieved.

  5. The franchise and the carence. Removing the franchise moves claims +7.09%, removing the carence +3.99%. Both are contractual and both are cheap to get wrong in the other’s place.

  6. Revalorisation. g_S compounds over a rente that runs for life and is the inflation exposure the CCSF warns about R8 §3.3; g_G moves the guarantee and the premium together and is close to neutral on the margin. Neither rate is contractual and neither is published.

  7. The tariff-revision path. Zero in the base run for the first five years by construction. It is the insurer’s one real management lever on this product and it is capped at 10% a year at exactly one insurer [S7 §4.4]; treating a projected revision as an assumption rather than an action is the modelling error to avoid, and the premium-shock lapse module is off precisely because turning it on is a joint statement about insurer and policyholder behaviour.

  8. The aggravation rate, which is nearly neutral and easy to misread. Consistently varying i_A over 0 → 0.20 → 0.40 moves lifetime claims by +0.54% / 0 / −0.52%; applying it without re-deriving i_T moves them +0.84% and misallocates lives between a 1,000 €/month benefit and a 500 €/month one. R12 §3.1.2 avoids the question by not modelling the transition at all; this model faces it and the price is that the number has no external anchor.

  9. Recovery, held at zero. The contracts provide for improvement out of a covered state [S1 §4.3.1.2] [S5 art. 13] [S6 art. 26] [S7 §4.2.1] and this model, like R12 §3.1.1, ignores it. The error is one-directional: claims are overstated, and by an amount no retrieved source quantifies.

  10. The supervisor’s view is missing. No ACPR material on this product could be retrieved (see product-spec.md, Regulatory context), so nothing here is calibrated against, or checked by, a supervisory finding on pricing adequacy or provisioning practice.