Technical Notes#

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

Scope note. These notes specify a reference liability cash-flow projection model for the standardized composite rente viagère immédiate defined in product-spec.md (same directory). This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in sources.md, numbering carried verbatim from _research/rente-viagere.md; [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md (its own frozen R1–R49 numbering). std marks standardizations introduced for the reference implementation. Parameter values are identical to those in product-spec.md. The model these notes are implemented as is Rente_FR_S, on a monthly grid. It shares its payout chassis with the UK sibling PA_UK_S and the US sibling SPIA_US_S, and reuses their cells names wherever the machinery is the same — lives_if, lives_death, certain_floor, payment_factor, payment_surv_mth, cum_annuity_pp, annuity_pp, annuity_payments, pols_if, liability_cf. Amounts are in euros with the English decimal point; quoted French text keeps its original comma.

Model scope and conventions#

Purpose. Project gross best-estimate liability cash flows (arrérages to the annuitant, the prorata d’arrérages on death, the reversion stream, the frais d’arrérages retained, maintenance expenses) for one rente viagère in payment. Discounting and reserves are not computed (see Valuation and reserve pointers).
Mortality is the model. After conversion the contract has no premiums, no surrender value R8, no account value and no policyholder option of any kind [S1] [S2] [S3] [S4] [S5] [S6] [S8]. The only decrements are deaths. There is no lapse machinery anywhere in this model and that is a cited product feature, not an omission.
Projection frequency and origin. Monthly grid, 0-based: t = 0, 1, …, proj_len − 1 months from the effective date, which is always the 1st day of a civil month [S2] [S3] [S6]; month t is the whole civil month beginning at the t-th month-start after it, so t = 0 is the civil month of the effective date itself and month t runs from time t to time t + 1. proj_len is the number of months projected, the policy year containing month t is t ÷ 12 + 1 (integer division), and the attained ages step at each 12-month multiple of t. Survival is carried at the time points t = 0, 1, …: l(t) is the probability of being alive at time t, l(0) = 1, so month t opens at l(t) and closes at l(t + 1), and no quantity is ever indexed before time 0.
The model carries the calendar, not just the duration. Revalorisation is credited at 31 December [S2 pt 10.f], so the model point carries the effective date’s calendar year and civil month. Nothing in this product happens on a policy anniversary.
Age basis and generation. Age last birthday at the effective date, incrementing on each 12-month multiple of it std. The generation (millésime, year of birth) is a separate model point attribute and is never derived from the projection year: the tables are generational and the birth year is the table key R1 R19.
No improvement scale. TGH05/TGF05 are prospective generation tables: q(sex, generation, age) already gives the rate the life will experience at that age in calendar year generation + age R1 R19 R25, secondary. A separate improvement projection — which the UK sibling needs, because ONS national life tables are period tables — would double-count the trend. Rente_FR_S has no improve_factor cells and must not acquire one.
Limiting age. ω = 120, the published top age of the tables R25, secondary; the std proxy CSV caps q at 1 there. The construction document states the tables give rates where age + generation > 1995 R19, which the 50–85 issue band respects.
Model points and rounding. EUR throughout; single-policy model points on an expected (probability-weighted) basis, with a scenario mortality basis std carried as in PA_UK_S so the worked example is reproducible row by row. Annuitant and reversionary mortality are independent std — a documented model risk. No intermediate rounding; displayed figures are rounded to the cent independently, so a rounded gross minus a rounded charge can differ by one cent from the rounded net.

Model point attributes#

Attribute	Type	Example (worked configuration)
`purchase_price` C	currency	200,000 std
`effective_year` Y0	int	2026 std
`effective_month` M0	int 1–12 (civil month of the effective date)	4 (April) std
`annuitant_age` x_a0	int, age last birthday at the effective date, 50–85 [S1]	65 std
`annuitant_birth_year` g_a	int (millésime; the generational table key)	1961 std
`annuitant_sex`	enum {`M`, `F`, `mix`}	`M` std
`annuity_rate` ρ	float, taux de rente per unit of capital p.a.	0.0330 std
`reversion_pct` δ	float 0–1 [S2] [S3]; 0 if no réversion	0.60 std
`reversion_coeff` κ	float, definitive reduction of the annuitant’s own annuity	0.76 [S6 Art. 5.4.3]
`reversion_age` x_r0	int, age last birthday at the effective date	61 std
`reversion_birth_year` g_r	int	1965 std
`reversion_sex`	enum {`M`, `F`, `mix`}	`F` std
`guarantee_years`	int, 0 or 5–min(25, e − 5) in 5-year steps [S2] [S3] [S4] [S9]; XOR `reversion_pct` [S2] [S3]	0
`palier_scheme`	enum {`none`, `inc1`, `inc2`, `dec1`, `dec2`} [S2 pt 10.e]	`none`
`palier_step_years` S	int {5, 10} [S2]	0
`payment_freq` m	enum {12, 4, 2, 1} [S2] [S4] [S5] [S6]	12
`payment_timing`	enum {`arrears`, `advance`} — French contracts are all `arrears` [S1]–[S9]	`arrears`
`arrerage_charge_rate` f	float, frais d’arrérages per quittance	0.03 [S5] [S7 Art. 17.3]
`technical_rate` i	float, taux technique priced at conversion	0.0000 [S2 pt 10.d] [S3]
`mort_basis`	enum {`table`, `scenario`}; the scenario switch is std (see Worked example)	`scenario`
`death_mth`	int, month of death on the `scenario` basis, on the 0-based clock; blank or negative = survives	25 (annuitant), −1 (reversionary)

Every std in the Example column is a choice of the worked configuration, not a product feature; each is restated and tagged in the Worked example section below, and the rates among them (ρ, κ, f) carry their own footnotes in product-spec.md and in assumption class (b).

C is the amount actually applied to the annuity: for a wrapper exit the valeur atteinte net of social and tax levies [S4 §7.3.2.3], with any entry charge taken before the model starts [S1] [S6] [S7 Art. 17.1]. The initial gross annual annuity is derived, not carried: A₀ = C ρ κ, because ρ and κ are the two quantities a real barème computes and the model must show the arithmetic rather than take its answer as an input. technical_rate is carried although it appears in no recursion — it reaches the projection only through ρ, and is recorded so a reader can see which rate ρ was struck on. Neither the revalorisation rate nor the charge on the annuity fund is a model point attribute: both are insurer-discretionary, set portfolio-wide, and live in class (b).

State variables#

Variable	Description	Updated
`cal_year_index(t)` k(t)	Number of 31 Decembers strictly before the start of month t	monthly
`revalo_factor(t)` R(t)	Cumulative revalorisation index; R = 1 until the first 1 January	at each 1 January
`palier_factor(t)` Π(t)	Step multiplier of the rente par paliers schedule; 1 when none	at palier boundaries
`annual_income(t)` A(t)	Gross annualised rente in force = A₀ R(t) Π(t)	monthly
`lives_if(t, life)` l(t, ·)	Survival probability to time t, t = 0 at the effective date; l(0) = 1	monthly
`lives_death(t, life)` d(t, ·)	l(t, ·) − l(t + 1, ·), the deaths of month t	monthly
`certain_floor(t)` γ(t)	1 while the annuités garanties run, else 0	monthly
`payment_factor(t)`	max(γ(t), l_a at the payment point) — the annuitant stream’s factor	payment months
`reversion_factor(t)`	δ (1 − l_a(t)) l_r at the payment point	payment months
`cum_annuity_pp(t, kind)` G(t)	Cumulative gross arrérages, as-if-alive (`"ANNUITANT"`) or expected across both streams (`"ALL"`)	payment months
`pols_if(t)`	Probability any payment obligation remains	monthly
`inflation_factor(t)`	Expense inflation index, stepping at each 1 January	at each 1 January

R(t), Π(t) and A(t) are deterministic given the assumption set: the annuity level does not depend on survival, only the payment factors do. G(t) on the "ANNUITANT" kind is therefore a pure schedule and needs no path simulation.

Assumption inputs#

(a) Contractual / guaranteed elements (cited)#

Input	Value	Basis
Instalment	A(t)/m at each payment date, terme échu	[S2] [S3] [S6]
Instalment on the month of death	Due in full: instalments “cessent d’être dus à compter du premier jour du mois qui suit le décès”	[S6]; accrued arrears to the heirs [S1 Art. C17.2] [S7 Art. 7.3]
Mortality table family	TGH05 (male) / TGF05 (female), generational, mandatory for rentes viagères from 1 January 2007	R1 art. 2, verbatim REG-R21
Single-table rule	One table for all lives must be the most prudent — TGF05	R3, verbatim REG-R23
Experience-table floor	An experience-table tariff may never be lower than the homologated-table tariff	R3, verbatim
Taux technique ceiling	min(3.50%, 60% × TME) beyond eight years, on a 0.25-point ladder floored at zero	R4 R5 REG-R17
Reversion	δ × the annuity reached at death, for life, from the 1st day of the month or quarter following death [S6] — the 1st day following death at [S1 Art. C12]; the reduction κ is definitive even if the reversionary predeceases	[S2] [S3] [S1 Art. C12] [S6 Art. 5.4.3]
Reversion coefficient table	Published coefficients by age difference and reversion rate (table below)	[S6 Art. 5.4.3]
Annuités garanties	n months of instalments certain at the same amount, to designated beneficiaries; no lump-sum commutation offered	[S2] [S3] [S4]
Paliers schemes	inc1 100→200%; inc2 100→125→150%; dec1 100→50%; dec2 100→75→50%; first step 5 or 10 years, second step equal	[S2 pt 10.e] [S3 pt 11.d]
Revalorisation date and floor	Credited at 31 December, pro rata temporis in the first partial calendar year, never negative	[S2 pt 10.f] [S3]
Surrender value	None, at any duration	R8, verbatim [S1 Art. C3]
Commutation threshold	€110 per month including majorations légales, × months in the payment period	R10 art. A. 160-2

Reversion coefficient table [S6 Art. 5.4.3], applied to the annuitant’s own annuity. It is the only published option-cost table in the sources; it was built for a points régime and its adoption here as a euro-annuity coefficient is std (spec footnote 15). The key is the difference in millésime between the reversionary and the annuitant.

Age difference (reversionary vs annuitant)	60%	80%	100%
Older by 8 years or more	0.93	0.91	0.89
Older by 4–7 years	0.89	0.86	0.83
Within 3 years either way	0.81	0.76	0.72
Younger by 4–7 years	0.76	0.70	0.65
Younger by 8–15 years	0.66	0.59	0.54
Younger by 16–23 years	0.58	0.51	0.45
Younger by 24–29 years	0.53	0.46	0.40
Younger by 30–34 years	0.49	0.42	0.37
Younger by 35–39 years	0.47	0.40	0.35
Younger by 40–44 years	0.42	0.35	0.30
Younger by 45 years or more	0.35	0.29	0.24

(b) Insurer-discretionary current elements#

Unlike the UK sibling, this class is not empty — it is where the French product’s uprating lives.

Input	Value	Basis
Revalorisation rate ν	1.50% a year, floored at zero	std (i)
Frais sur encours de rentes φ	0.80% a year, entering ν rather than the cash flow	[S2]; level std (ii)
Frais d’arrérages f	3.00% of each quittance	[S5] [S7 Art. 17.3]; composite std (spec footnote 10)
Taux de rente ρ at conversion	3.30% at age 65, unisex TGF05 basis	std (iii)
Guarantee coefficient (when annuités garanties are elected)	0.9820 for a 15-year term at 65	std (iv)

(i) No retrieved document publishes a revalorisation rate, a formula or a history for annuities in payment. The uplift is fed by the annuities’ own profit-sharing account, built under point III of art. A. 132-11 “en incluant le résultat technique généré par ces mêmes rentes”, with “100 % du solde créditeur” attributed to the annuities [S3, verbatim] R7 REG-R15; sums parked in the provision pour participation aux bénéfices must reach policyholders within eight financial years R7 art. A. 132-16, verbatim REG-R16. 1.50% is a round placeholder between the 0.00% taux technique [S2] [S3] and the 2.00% technical-rate ceiling R21, secondary, with the contractual floor of zero as the only cited bound. It is a scenario input, not a contractual parameter. The machinery it stands for is specified in the assurance vie euro technical notes and not restated here; Where ν comes from, under the revalorisation recursion, says what this product inherits from it and where it departs. (ii) The charge bites on the provision mathématique backing the annuity and reduces the profit-sharing base, never the guaranteed annuity, in every retrieved contract [S1 Art. C9] [S2] [S5] [S6] [S7]. It therefore appears in these notes only as a reason ν is lower than the gross return on the annuity fund, and must not be netted from any instalment. Observed range 0.55%–2.3% (spec footnote 11). (iii) Spec footnotes 6 and 7: derived from the TGF05 residual life expectancies published in R19 with a std loading of about 2%. Struck on a 0.00% taux technique [S2] [S3]. The same construction on the male table gives 3.73%. (iv) No retrieved document publishes the cost of annuités garanties. 0.9820 is a std derived figure: the certain-period annuity factor exceeds the life factor by the sum over the guaranteed months of (1 − survival), which on the R19-implied basis is about 0.54 years against a factor of about 29.63. The coefficient therefore varies with the term — 0.9986 at 5 years, 0.9267 at 25 — and is recomputed from the mortality basis rather than carried as a model point attribute.

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

Input	Recommended basis	Basis tags
Base mortality	The mandatory tables are TGH05/TGF05 R1 REG-R21, annexed to the Code des assurances R12; this library does not redistribute them. The reference basis is a std generational proxy keyed on (sex, generation, age), built from INSEE population data REG-R24 and anchored so the tariff annuity factor — the female-table factor the unisex rule selects — reproduces the placeholder ρ of assumption (iii). The best-estimate factor is not anchored: on the male table it gives 3.73%, not ρ	R1 R12 R19 REG-R21 REG-R24; proxy std (v)
Mortality improvements	None applied. The table is generational and carries its own projection R1 R19	R1 R19
Tariff table	TGF05 for every life, regardless of sex, per the most-prudent-single-table rule	R3, verbatim; adoption std (spec footnote 4)
Best-estimate table	The sex-appropriate table where the model point carries a sex; the blend at `portfolio_male_share` where it carries `mix`	R3 R17 R18; blend std (vi)
`portfolio_male_share` θ	0.45	std (vi)
Lapse / surrender	None — no surrender value exists at any duration	R8, verbatim [S1 Art. C3]
Maintenance expense	€30 per contract per annum, payable monthly while any obligation remains, inflating at π	std (vii)
Expense inflation π	1.50% a year, stepping at each 1 January	std (vii)
Proof-of-life suspension	Not modeled	[S1] [S2] [S3]; std (viii)

(v) TGH05/TGF05 are annexed to the Code des assurances and the annexe article carrying TGF05 is itself marked abrogated as of 1 January 2016, with the current location of the tables unidentified R12 unverified. The proxy CSV has the same shape as a generational table — a rate is keyed on (sex, generation, age) and on nothing else — so the model’s indexing is exercised honestly even though the rates are not the regulatory ones. This is the same posture the UK sibling takes on CMI tables. Substituting a licensed basis means replacing the CSV with a same-schema file; no formula changes. (vi) The tariff is unisex by law while the tables are sex-distinct R3; the ministry states the resulting surplus on male lives must in substantial part be returned to policyholders within eight years R17 REG-R16. A portfolio mix is the assumption that reconciles the two: the insurer prices one rate for a cohort whose expected mortality is a blend of the two tables. Art. A. 132-18 does not forbid that blend. Its second table family expressly permits tables built by the undertaking, “with or without sex”, certified by an actuary independent of it on its own or demographically equivalent experience — a blended, non-sex-distinct table is precisely what that permits — subject to a one-sided floor rather than a prohibition: for rentes viagères the tariff from such a table “ne peut être inférieur” to the tariff the appropriate homologated table would give R3, verbatim. The most-prudent-table rule is the other limb; it bites only where a single homologated table is applied to all lives, and then selects TGF05. This model takes that homologated route — prices on TGF05 and projects on the sex-appropriate best estimate — which is one of two lawful constructions, not the only one; a certified blend would remain floored at the TGF05 tariff, so the direction of the prudence margin is the same either way. θ = 0.45 is std — no retrieved document publishes the sex mix of a French annuitant portfolio — and sits below one half because the unisex tariff itself deters male annuitants (see Policyholder behavior modeling). (vii) No insurer publishes expense assumptions. €30 a year is a round placeholder for in-payment administration; note that at the composite’s 3% frais d’arrérages the charge on the worked configuration is about €150 a year, so the French charging structure recovers far more than in-payment administration and the balance funds distribution and margin. Acquisition cost is out of scope (single premium, priced in). (viii) An unreturned attestation valant certificat de vie suspends payment from the following month until it arrives [S2] [S3] [S1 Art. C13]. It shifts timing, not amount, and no source publishes a suspension frequency.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
t	month index from the effective date, 0-based: t = 0, 1, …, proj_len − 1
Y0, M0	calendar year and civil month of the effective date (M0 = 1 for January)
k(t)	completed 31 Decembers strictly before the start of month t
g_a, g_r	millésimes (birth years) of annuitant and reversionary
x_a(t), x_r(t)	attained ages, x(t) = x(0) + floor(t/12) std (Model scope, “Age basis and generation”)
m	payments per year; payment months T = {t : (t + 1) mod (12/m) = 0} (arrears)
C, ρ, κ	capital constitutif, taux de rente, option coefficient
A₀, A(t)	gross annual rente at conversion and in force in month t
ν, R(t)	annual revalorisation rate and its cumulative index
Π(t)	palier step multiplier
δ, n	taux de réversion; annuités garanties in months
f, φ	frais d’arrérages rate; frais sur encours de rentes rate
q(s, g, x)	annual mortality from the generational table, sex s, generation g, age x
l_a, l_r, d_a, d_r	survival probabilities of the two lives, at time points, and their death densities over month t
γ(t)	certain-period indicator, 1{t < n}; C alone is always the capital
h(t)	complete months elapsed since the last payment date, measured at the start of month t
θ	portfolio male share std (assumption (vi))
c_e, π	maintenance expense p.a. and expense inflation

Dimensional check: A, C and every cash flow are currency; ρ, ν, f, φ, δ, κ, θ, q, l and the factors are dimensionless; A/m is currency per payment. Every flow below is currency per month.

Calendar index and the revalorisation recursion#

The annuity is in service for 13 − M0 months of its first calendar year — months t = 0 … 12 − M0 — so

k(t) = 0                                 for t < 13 − M0
k(t) = 1 + floor((t − (13 − M0)) / 12)   otherwise

R(t) = 1                                                     for k(t) = 0
R(t) = (1 + ν · (13 − M0)/12) · (1 + ν)^(k(t) − 1)           for k(t) ≥ 1

The pro-rating factor (13 − M0)/12 implements “les rentes en service depuis moins d’un an au 1er janvier sont revalorisées prorata temporis de la date d’effet au 31 décembre” [S3]; it is 1 for a 1 January effective date, so the general form degenerates correctly. The uplift is credited at 31 December [S2 pt 10.f] and reaches instalments payable from the following 1 January std (spec footnote 12), which is why k(t) counts 31 Decembers strictly before month t. ν ≥ 0 always [S1] [S2] [S3] [S4] [S7].

Where ν comes from. The participation aux bénéfices machinery ν is drawn out of is specified once for this library, in the assurance vie euro technical notes, and is not redeveloped here. Inherited from there: the compte de participation aux résultats built under art. A. 132-11 on a financial account and a technical account, with 85% of the financial balance and the technical balance less the insurer’s share (the greater of 10% of the credit balance and 4.5% of premiums — a clause returned as paraphrase and unverified as to exact formulation) R7 REG-R15; and the provision pour participation aux bénéfices with its eight-year clock R7 art. A. 132-16, verbatim REG-R16, which is where a year’s excess is parked and from which an older vintage is forced out. Where this product deviates, in four places:

A different account. The annuities in payment have their own compte de participation aux bénéfices, built under point III of art. A. 132-11 “en incluant le résultat technique généré par ces mêmes rentes”, with 100% du solde créditeur attributed to them [S3, verbatim] R7. It is not the euro fund’s account and it is not fed by the euro fund’s épargne acquise.
The technical result is not a loading result. On the euro support the death benefit is the account value, so the underwriting result is nil and the technical account is the charges less the expenses. Here the technical result is dominated by the mortality result on the annuities, including the TGF05 prudence margin every male life carries under the unisex rule R3 R17 — which is what makes the revalorisation of an annuity book structurally different from that of a savings book.
ν is an input, not an output. Euro_FR_S derives the credited rate from the constrained allocation and uses the PPB as a lever; this model carries no provision mathématique ledger, no average-provision base and no PPB vintage ledger, so ν is an exogenous std scalar (assumption (b), note (i)). Substituting the euro model’s credited-rate path for the flat ν is the intended extension, and nothing in the recursion above assumes ν is constant.
The uplift lands on the annuity and is irreversible. A euro-fund credit lands in the épargne acquise, which a surrender can take away; R(t) multiplies the rente for the remainder of its life and there is no surrender at any duration R8. There is also no TMG here: i is a pricing rate, not a floor on ν, whose only cited bound is zero.

Palier factor#

palier_scheme = none                Π(t) = 1
two-step (inc1, dec1)               Π(t) = π₁ for t < 12S,  π₂ for t ≥ 12S
three-step (inc2, dec2)             Π(t) = π₁ for t < 12S,  π₂ for 12S ≤ t < 24S,  π₃ after

with (π₁, π₂, π₃) from the scheme table of assumption (a) and S ∈ {5, 10} years; the second step is “d’une durée égale” to the first [S2] [S3]. Π is a step function of duration and nothing compounds — a rente par paliers is not escalation.

Conversion at the effective date#

A₀ = C · ρ · κ

κ = 1 with no option; the reversion coefficient of the [S6] table where réversion is elected; the std guarantee coefficient where annuités garanties are elected. The options are not cumulative [S2] [S3], so exactly one of δ > 0 and n > 0 may hold.

Admission test, not a cash flow. A model point is projectable only if its gross quittance d’arrérages exceeds the statutory threshold:

A₀ · Π(0) / m  >  110 · (12/m)                                [R10 art. A. 160-2]

Below it the insurer may, with the annuitant’s agreement, pay a capital instead R9 R10 [S2] [S3] [S5], so there is no annuity to project. check_commutation_floor() must fail such a point rather than project it.

Generational mortality construction#

q(t, life)     = qtab(basis(life), g(life), x(life, t))       — pure table lookup
q_mth(t, life) = 1 − (1 − q(t, life))^(1/12)                  **[std]**
l(t, life)     = l(t − 1, life) · (1 − q_mth(t − 1, life)),   l(0, ·) = 1
d(t, life)     = l(t, life) − l(t + 1, life)

l is indexed by time and q by the month it applies over: survival to time t takes the rates of months 0 … t − 1, and the deaths of month t are the difference between its opening and closing survivals.

with

basis(life) = "M" or "F"                    where the model point carries a sex
q           = θ q(M, g, x) + (1 − θ) q(F, g, x)   where it carries `mix`   **[std]**

There is no improvement factor and no calendar-year argument. q depends on t only through the attained age x(life, t); the generation g is fixed at the model point. That single line is the largest structural difference from PA_UK_S, whose period base table requires a separate improvement projection to become a cohort view.

On the scenario basis std the survival path is the step function l(t, life) = 1{t ≤ death_mth(life)} — survival to time t, so a life dying in month d is alive at time d, the start of that month, and gone from time d + 1 — with a blank or negative death_mth meaning the life survives the projection; the device PA_UK_S uses, and the basis the worked example runs on.

The tariff table and the best-estimate table are different objects. ρ is struck on TGF05 for every life R3, verbatim while the projection decrements on basis(life); for a male annuitant the two differ by construction, and the gap is the systematic technical surplus that must flow back to policyholders within eight years R17 REG-R16 — which here it does, through ν. Collapsing them destroys both halves of the mechanic (pitfall 3).

Payment factors#

payment_surv_mth(t) = t + 1                 arrears (*terme échu*): the end of month t
                    = t                     advance (*terme à échoir*, unobserved in France):
                                            its start

certain_floor(t)    = 1 if t < n else 0
payment_factor(t)   = max(certain_floor(t), l_a(payment_surv_mth(t)))
reversion_factor(t) = δ · (1 − l_a(t)) · l_r(payment_surv_mth(t))

The max makes the annuités garanties an annuity-certain floor rather than a second stream: while the guarantee runs the full instalment is payable regardless of survival [S2] [S3] [S4], and an additive form would pay 1 + l_a.

The reversion gate is (1 − l_a(t)), the annuitant’s survival to the start of month t, not (1 − l_a(t + 1)), its survival to the end: the survivor’s first instalment falls in the month after the month of death, immediately after the prorata d’arrérages has settled it. What [S6] states for the réversion is that it is payable “from the 1st day of the month or quarter following death”; the annuitant’s own instalments are the ones that “cessent d’être dus à compter du premier jour du mois qui suit le décès” [S6], and that verbatim sentence is the cessation rule quoted in assumption (a), not the reversion-start rule. The model applies the one-month gate at every payment frequency std, which reads the disjunction on its monthly limb: at m = 12 — Préfon’s own periodicity, and the periodicity of every model point that carries a réversion — the two limbs coincide, but on a quarterly contract the gate opens the survivor’s stream up to a quarter earlier than the second limb would. No source states which limb governs which contract, and no shipped model point combines a réversion with m < 12.

Scheduled instalment and cumulative arrérages#

A(t)            = A₀ · R(t) · Π(t)
annuity_pp(t)   = A(t)/m   for t ∈ T, else 0
cum_annuity_pp(t, "ANNUITANT") = cum_annuity_pp(t − 1, ·) + annuity_pp(t)
cum_annuity_pp(t, "ALL")       = cum_annuity_pp(t − 1, ·)
                                 + annuity_pp(t) · (payment_factor(t) + reversion_factor(t))
                                 + prorata_pp(t) · prorata_factor(t)

for t ≥ 1; month 0 is the first projected month, so the accumulation opens there — G(0) is what month 0 itself pays, and there is no month-zero row to carry.

"ANNUITANT" is the deterministic as-if-alive schedule; "ALL" is the expected total paid across both streams. On a probability-weighted run "ALL" is an expectation rather than a path; in a scenario run the two coincide for a surviving annuitant.

Expected cash flows (month t)#

Arrérages.

E[ANN(t)] = pols_if_init · annuity_pp(t) · (payment_factor(t) + reversion_factor(t))

Prorata d’arrérages — the accrued instalment settled on death [S1 Art. C17.2] [S6] [S7 Art. 7.3]. With h(t) = t mod (12/m) complete months since the last payment date,

prorata_pp(t)     = ((h(t) + 1)/(12/m)) · A(t)/m     *terme échu*
                  = 0                                *terme à échoir* **[std]**
prorata_factor(t) = d_a(t) · (1 − γ(t))  +  δ · (1 − l_a(t)) · d_r(t)
E[PRO(t)]         = pols_if_init · prorata_pp(t) · prorata_factor(t)

The second branch is the unobserved terme à échoir switch (spec footnote 9): the instalment covering the month of death was paid at the start of it, so nothing has accrued unpaid and there is nothing to settle. Only shipped model point 11 takes it.

At m = 12, h(t) = 0 for every t, so prorata_pp(t) is exactly one full instalment — the month of death is due in full [S6]; at m = 4 a death in the first month of a quarter settles one third of the quarterly instalment. The (1 − γ(t)) gate suppresses the term while the annuités garanties run, the full instalment being payable there already, and the second term is the symmetric settlement on the reversionary’s own death. There is no “with/without proportion” election in France: the prorata is the rule.

Frais d’arrérages — retained by the insurer out of each quittance [S5] [S7 Art. 17.3]:

E[FRA(t)] = f · (E[ANN(t)] + E[PRO(t)])

Maintenance expense:

pols_if(t) = min(1, max(γ(t), l_a(t + 1)) + 1{δ > 0} · (1 − l_a(t)) · l_r(t + 1))
E[EXP(t)]  = (c_e/12) · (1 + π)^k(t) · pols_if(t)                       **[std]**

Total gross liability cash flow:

liability_cf(t) = E[ANN(t)] + E[PRO(t)] − E[FRA(t)] + E[EXP(t)]
net_cf(t)       = − liability_cf(t)

There is no premium income (C is a pricing input at the effective date), no surrender outgo R8 and no death capital — the representative design is capital aliéné. The frais sur encours de rentes appear nowhere in this recursion by design: they reduce the profit-sharing base, hence ν, and never an instalment.

Monthly processing order#

Advance the calendar: compute k(t). If k(t) > k(t − 1), step the revalorisation index R (pro-rated when k = 1) and the expense inflation index [S2 pt 10.f] [S3]. At t = 0, k = 0 and R = 1 by construction, so there is no earlier month to read.
Update the palier factor Π(t) if t reaches 12S or 24S [S2] [S3].
Set A(t) = A₀ R(t) Π(t). If t ∈ T, record annuity_pp(t).
Decrement mortality: attained ages from the effective-date ages, rates from the generational table at each life’s own millésime; update l_a, l_r, d_a, d_r.
Compute certain_floor(t), payment_factor(t) at payment_surv_mth(t), and reversion_factor(t) using l_a(t), the annuitant’s survival to the start of month t.
Compute E[ANN(t)], then E[PRO(t)] (gated off while the guarantee runs), then E[FRA(t)] on the sum of the two. Update cum_annuity_pp.
Accrue E[EXP(t)] on pols_if(t).
Stop once both lives have passed ω = 120 and the guarantee has run: the projected months are t = 0 … max(n, 12(ω − min x_i)) − 1, so the last of them is the last month of age ω − 1 of the youngest covered life — stopping on the annuitant’s age alone truncates a younger reversionary’s tail.

Known modeling pitfalls#

These are the specific ways an implementation of this product looks right and is wrong. Each is a test.

Applying an improvement scale on top of the generational table. TGH05/TGF05 are prospective; the trend is inside them R1 R19. Any improve_factor double-counts it. Test: q must be unchanged when the effective year moves and the millésime does not.
Indexing the table by projection calendar year instead of birth year. A period-table implementation reads the rate for age 66 in calendar year 2027 and walks diagonally across generations; a generational table reads (g = 1961, x = 66) whatever the projection year. Test: two model points with the same entry age and different millésimes must give different rates at the same attained age.
Using the tariff table as the best estimate. Pricing every life on TGF05 R3 and also projecting a male life on TGF05 makes the unisex prudence margin invisible — no surplus, so no source for the revalorisation the contract shares [S3] R17. Projecting him on TGH05 without crediting the surplus back through ν shows a permanent retained profit the eight-year rule does not allow REG-R16. The two tables must be separate objects.
Revalorising on the policy anniversary instead of 31 December. The uplift is a calendar event [S2 pt 10.f]: on the worked configuration the anniversary convention gives twelve months at the initial level instead of nine and shifts every later step by three months.
Dropping the first-year pro-rata, or uplifting the December instalment. The first uplift is ν · (13 − M0)/12 [S3]; applying the full ν overstates the annuity for the whole of its remaining life, because R(t) is a running product. The credit is at 31 December and reaches instalments payable from 1 January std; applying it to the December arrears instalment adds one instalment a year at the new level.
Losing the arrérage of the month of death. The UK sibling’s default pays nothing for the final partial period; the French rule settles the accrued arrears to the heirs [S1 Art. C17.2] [S6] [S7 Art. 7.3], which at m = 12 is a whole instalment. Test: on the scenario basis the number of instalments paid equals death_mth + 1, the months of service through the month of death, the index being 0-based.
Starting the reversion in the month of death. The gate is (1 − l_a(t)), the annuitant’s survival to the start of month t [S6]; using (1 − l_a(t + 1)), its survival to the end, pays the reversion and the prorata d’arrérages in the same month, so the month of death is paid 1 + δ times.
Paying a prorata during the guarantee period, or adding the certain floor instead of taking a max. While the annuités garanties run the full instalment is already payable regardless of survival [S2] [S3], so the prorata on top double-pays the month of death and an additive certain_floor + l_a pays 1 + l_a for the whole term.
Mishandling the reversion coefficient. κ reduces the annuitant’s own annuity once, permanently, at conversion [S6 Art. 5.4.3]. It must not also scale the reversion stream — the survivor receives δ × the already reduced annuity reached at death [S2] — and it is not released if the reversionary predeceases: “une réduction définitive, même si le bénéficiaire de la réversion vient à décéder antérieurement” [S6 Art. 5.4.3, verbatim].
Modeling a surrender. There is none R8, verbatim. Any lapse rate, surrender value or paid-up value here is a defect. The single exception is not a cash flow but an admission test: check_commutation_floor() must reject a model point whose gross monthly quittance does not exceed €110 R10.
Netting the frais sur encours de rentes off the instalment. They bite on the provision mathématique and reduce the profit-sharing base, never the guaranteed annuity [S1 Art. C9] [S2] [S5] [S6] [S7]; subtracting them from an instalment cuts the annuitant’s income, which no retrieved contract does.
Charging the frais d’arrérages on the annualised rente. The deduction is per quittance d’arrérages [S5] [S7 Art. 17.3]. At a flat percentage the two coincide; at a per-instalment cap [S4 §7.3.2.3] or a flat per-instalment fee R23, secondary they do not, and the payment frequency then changes the total.
Discounting the projected flows at the taux technique. It reaches the projection only through ρ; the best estimate discounts at the risk-free term structure REG-R4 REG-R5. Reusing i as a discount rate produces neither a price nor a reserve.
Applying a palier step to the reversion stream. The survivor receives δ × the annuity reached at death [S2]; a step falling later belongs to the annuitant’s schedule. The representative options are mutually exclusive so it cannot arise here, but a Spirica-style combinable engine [S4] must gate Π(t) on the annuitant being alive.

Policyholder behavior modeling#

There is none to model after conversion, and this is a cited product feature. There is no surrender or reduction of any kind R8, verbatim [S1 Art. C3], no transfer, no premium flexibility, and every option is irrevocable once elected [S1 Art. C15, C16] [S2 pt 10.e] [S3 pt 11.d] [S4 §7.3] [S5] [S6 Art. 5.4.3] [S8]. The model therefore carries no lapse decrement and no dynamic behavior formulas.

Three behaviors sit at the boundary and are handled outside the projection. The commutation election: below the art. A. 160-2 threshold the insurer may pay a capital instead, but only “avec l’accord de l’assuré” [S2] [S3] [S5] R9 R10 — a one-off election at liquidation, implemented as an admission test rather than an in-force option (pitfall 10). Proof of life: failure to return the annual certificate suspends payment from the following month until it arrives [S1 Art. C13] [S2] [S3], a timing effect on an otherwise unchanged obligation, not modeled std. Change of spouse: where the survivor at death is not the person named at liquidation the reversion annuity is recalculated on the survivor’s age at death [S2] [S3] — a live option a single-policy model point cannot carry (spec footnote 16).

Behavior enters the basis, not the projection, as selection effects std.

Annuitization anti-selection. Annuitizing is voluntary outside the versements obligatoires compartment REG-R34 [S2], so voluntary annuitants self-select for longevity. TGH05/TGF05 are annuitant-experience tables built on exactly that population R1 R19, so the effect is already inside the mandatory basis — which is why this library applies no annuitant adjustment factor of the kind the UK sibling needs.
Sex anti-selection, and the direction of θ. The unisex tariff is struck on the female table R3, so a male annuitant receives about 13% less than his own table would give (spec footnote 7). Men therefore annuitize less readily than women, and the realised male share of a French annuitant portfolio sits below the population share. That is the direction, not the magnitude, of the std θ = 0.45; no retrieved document publishes a portfolio mix.
Option selection. Reversion is chosen disproportionately by annuitants with a younger spouse, annuités garanties by annuitants who expect to die early — both push realised experience away from the tariff basis, and no source quantifies either unverified.

Worked example#

Configuration (the worked model point; parameters as in product-spec.md). Capital C = €200,000 std; effective date 1 April 2026 std, so Y0 = 2026 and M0 = 4. Annuitant male, born 1961, age 65 at the effective date std — inside the 50–85 band [S1 Art. C5, C12]. Reversionary female, born 1965, age 61 std — inside the 50–85 reversionary band [S1 Art. C16], and younger by 4 years in millésime. Taux de rente ρ = 3.30% std (assumption (iii)), struck on a taux technique of 0.00% [S2 pt 10.d] [S3]. Réversion at δ = 60% [S2] [S3] [S6] [S9], snapshot std, with coefficient κ = 0.76 from the “younger by 4–7 years / 60%” cell of the published table [S6 Art. 5.4.3]. Monthly, terme échu [S2] [S3] [S6]. Frais d’arrérages f = 3.00% [S5] [S7 Art. 17.3]. Revalorisation ν = 1.50% a year std, credited at 31 December, pro-rated 9/12 in 2026 [S3]. Maintenance expense €30 a year inflating at 1.50% std. No annuités garanties and no paliers — the options are not cumulative [S2] [S3]. Mortality basis scenario std: the annuitant dies in month 25 (May 2028) — the 26th month of service, the index being 0-based — and the reversionary survives throughout. All amounts in euros, unrounded in the model and displayed to the cent.

Conversion: A₀ = C ρ κ = 200,000 × 0.0330 × 0.76 = €5,016.00 a year, so the gross monthly quittance is 5,016.00 / 12 = €418.00. Admission test: 418.00 > 110 R10 art. A. 160-2, and also above the €40 monthly issue floor of [S6] and [S8], so the point projects.

t	Civil month	Event	R(t)	Gross arrérage	Frais (3%)	Net to payee
0	Apr 2026	first arrérage, terme échu	1.000000	418.00	12.54	405.46
8	Dec 2026	last instalment at the initial level	1.000000	418.00	12.54	405.46
9	Jan 2027	31 Dec 2026 uplift, pro-rated 9/12 → 1.125%	1.011250	422.70	12.68	410.02
11	Mar 2027	—	1.011250	422.70	12.68	410.02
20	Dec 2027	—	1.011250	422.70	12.68	410.02
21	Jan 2028	31 Dec 2027 uplift, full 1.50%	1.026419	429.04	12.87	416.17
24	Apr 2028	last instalment to the annuitant	1.026419	429.04	12.87	416.17
25	May 2028	annuitant dies; prorata d’arrérages to the heirs — one whole month	1.026419	429.04	12.87	416.17
26	Jun 2028	réversion begins at 60% of the rente atteinte	1.026419	257.43	7.72	249.70
29	Sep 2028	—	1.026419	257.43	7.72	249.70
32	Dec 2028	—	1.026419	257.43	7.72	249.70
33	Jan 2029	31 Dec 2028 uplift reaches the reversion stream	1.041815	261.29	7.84	253.45
35	Mar 2029	—	1.041815	261.29	7.84	253.45

Checks. Conversion, a different way. The unisex tariff is struck on TGF05 for every life R3; 1/ρ = 1/0.0330 = 30.30 years, against the annuity factor of about 29.63 implied at age 65 for the 1961 generation by the TGF05 life expectancies published in R19 — the gap being the std loading of about 2% (spec footnote 6). Had the same male annuitant been priced on his own table, ρ = 3.73% (spec footnote 7) would give A₀ = 200,000 × 0.0373 × 0.76 = €5,669.60 and a monthly quittance of €472.47 — a 13.0% higher income than the €418.00 he actually receives. That difference is the price of the unisex rule, and it is the surplus that must flow back to policyholders within eight years R17 REG-R16, which in this model it does through ν.

The month-9 instalment, a different way. k(9) = 1, and the first uplift is ν(13 − M0)/12 = 0.015 × 9/12 = 1.125%, so the instalment is 418.00 × 1.01125 = 422.7025, displayed 422.70 — identical to 5,016.00 × 1.011250 / 12. On a policy-anniversary convention the uplift would not arrive until t = 12 and the March 2027 row would still read 418.00 (pitfall 4).

The reversion instalment, a different way. The survivor receives 60% of the rente atteinte at death, i.e. 0.60 × 429.043038 = 257.4258, displayed 257.43 — the same as 0.60 × A(25)/12 with A(25) = 5,016.00 × 1.026419 = 5,148.5165. Note that a lower reversion rate would engage the commutation rule: at δ = 20% the survivor’s quittance would be 0.20 × 429.043038 = €85.81, below the €110 threshold, and CNP applies the rule to the reversion annuity with the réversataire’s agreement [S5] R10.

Cumulative arrérages and total charge. 26 monthly amounts are paid over 26 months of service, t = 0 … 25 — nine at 418.00, twelve at 422.7025 and five at 429.043038, the last of those five being the prorata settled at t = 25 — so cum_annuity_pp(25, "ALL") = 9 × 418.00 + 12 × 422.7025 + 5 × 429.043038 = €10,979.65, of which the insurer retains 3% = €329.39 and the annuitant and his heirs receive €10,650.26. Losing the month-of-death instalment (pitfall 6) would leave 25 amounts and understate the outgo by €429.04.

Full liability cash flow, two months. liability_cf(9) = 422.7025 × 0.97 + (30/12) × 1.015 = 410.021425 + 2.5375 = €412.56; liability_cf(26) = 257.425822 × 0.97 + (30/12) × 1.015² = 249.703048 + 2.575563 = €252.28. net_cf is the negative of each.

The annuités garanties variant. Replace the réversion with annuités garanties of 15 years (n = 180 months, inside the 5-to-min(25, e − 5) range in 5-year steps [S2] [S3] [S4] [S9]) and the coefficient κ = 0.9820 std (assumption (iv)); the options are not cumulative, so δ = 0 [S2] [S3]. Then A₀ = 200,000 × 0.0330 × 0.982002 = €6,481.21 a year, an instalment of €540.10 at the initial level, €546.18 from t = 9 and €554.37 from t = 21. The death in month 25 now changes nothing: certain_floor(t) is 1 through t = 179 — the guarantee covers the 180 months t = 0 … 179 — so the full instalment continues to the designated beneficiaries at the same amount and rises with the same revalorisation index [S2] [S3]; no prorata is due, because the full instalment is already payable (pitfall 8); and from t = 180 the stream stops, there being no reversion in that configuration [S2] [S3]. In expectation the same flows come out of payment_factor(t) = max(certain_floor(t), l_a(payment_surv_mth(t))) with n = 180.

Valuation and reserve pointers#

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

French statutory provisions. An annuity in payment is held in the provision mathématique, the difference between the actuarial present values of the two sides’ commitments including future management costs, one of the eleven technical provisions at art. R. 343-3 REG-R6. The provision pour participation aux bénéfices holds profit shares attributed but not yet payable and must be released within eight financial years R7 art. A. 132-16, verbatim REG-R16; the provision pour risque d’exigibilité attaches to the assets backing it REG-R7. The only statutory reserving basis retrieved for annuities is the commutation barème, which values the annuity on the provision mathématique computed with the tables and interest rates of the règlement ANC n° 2015-11 du 26 novembre 2015 R10 art. A. 160-3.
Solvabilité II. The best estimate is the probability-weighted average of future cash flows discounted at the relevant risk-free term structure, with EIOPA publishing the curves monthly REG-R5. Both statements are carried on EIOPA’s authority REG-R4: neither the directive REG-R1 nor the delegated regulation REG-R2 could be retrieved here — both return an AWS WAF challenge — so no Solvency II article number in this library was read from the instrument itself and every such number is unverified. The liability_cf(t) vector above is exactly that input. Nothing product-specific to rentes viagères was retrieved for the SCR or risk-margin layer, and no cost-of-capital rate in this library was read from a retrieved instrument. The technical rate is not a discount rate: i prices the annuity at conversion [S2] [S4] [S7] and thereafter functions as a lifetime minimum guaranteed return R20, abstract only, never as a valuation rate (pitfall 13).
The liability is structurally prudent and profit-shared. A prudently priced tariff (TGF05 for every life R3), a zero or near-zero technical rate [S2] [S3] R21, and 100% of the annuity profit-sharing account’s credit balance attributed back to the annuities [S3, verbatim] make the French annuity a prudently priced, profit-shared liability rather than a hard-guaranteed one. A best estimate that ignores ν values only the guaranteed floor.
IFRS 17, tax, professional standards. French listed insurers report on IFRS 17 from 2023 with no French carve-out; the fulfilment cash flows are the same vector with the risk adjustment and contractual service margin layered on REG-R45. Policyholder taxation — RVTO fractions by age at entrée en jouissance R13 R14, the RVTG pension regime R13 R15, social levies [S2] [S6] [S7] — does not enter the insurer’s liability cash flows. NPA 1 and NPA 2 (Modèles actuariels), category-3 recommended practices adopted 15 June 2015, frame the assumption-setting and documentation here REG-R43 REG-R44; NPA 4, on best-estimate life provisions, was not retrieved unverified.

Key sensitivities and model risks#

Dominant assumptions, in order.

Longevity level and the generational surface. The liability is a life-contingent payment stream with no offsetting decrement, so the mortality level is the single largest lever on it. The std INSEE-shaped generational proxy REG-R24 stands in for TGH05/TGF05, which cannot be redistributed R12; production work substitutes a same-schema licensed file. Because the trend is inside the table, a basis error is not a level error but a surface error: the wrong generation column mis-states every future year at once.
The unisex reconciliation. The tariff is TGF05 for every life R3 while the best estimate is sex-dependent; θ and the sex mix therefore move the projected surplus directly, and the surplus is the source of ν [S3] R17 REG-R16. A model that sets θ = 1 or collapses the two tables reports a materially different profit signature from one that does not.
The revalorisation path. ν is discretionary, non-negative, unpublished and fed by the annuities’ own technical and financial result net of the frais sur encours [S1] [S2] [S3] [S5]. A deterministic ν values the zero floor at intrinsic only: the floor never binds on a flat positive path, so the option the annuitant holds — an uplift that can rise but never fall — is worth nothing in this projection. The eight- year release rule REG-R16 additionally makes ν path-dependent on the PPB stock, which this model does not carry.
Reversion assumptions. δ, the age difference (which selects κ [S6]) and the survivor’s own generational rates drive the reversion tail. Independence std ignores broken-heart dependence and shared lifestyle, modestly overstating that stream; the change-of-spouse recalculation [S2] [S3] is an unmodeled option that can only increase the liability.
The taux de rente itself, and the charge levels. ρ is std because no French insurer publishes a rate card and the annuity is not guaranteed before liquidation [S4 §7.3.2.3]; every euro figure here scales linearly with it. f is disclosed as a maximum, not a cap REG-R30, and ranges from 0.00% to 3% across the retrieved carriers (spec footnote 10). Expense inflation is second-order against the instalments, but the in-payment term is 30 years and more, so π compounds.
Data risk in the sources themselves. The CJEU judgment was never read R16; the ACPR study of the taux technique returned HTTP 403 and only its abstract is used R20; the “life expectancy minus five years” cap on annuités garanties has no located legal source [S2] [S3] [S4] [S9] against R1 R2 R3; the technical-rate ceiling of 2.00% is a commercial tracker’s reading at 31 July 2026 and moves R21; and the press-reported charge levels date from 2015 R23 R24. Each is flagged where used and none is load-bearing on the recursions.