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 euro support defined in product-spec.md (same directory). This is not any single insurer’s fund. [S#]/[R#] tags refer to the source list in sources.md (numbering carried from _research/assurance-vie-euro.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; unverified marks claims not confirmed against a retrieved document. Parameter values are identical to those in product-spec.md. The mechanics anchors are the insurers’ own booklets [S1] [S2] [S3] [S4] [S9]; the statutory arithmetic is arts. A132-10 to A132-17 of the Code des assurances R5 REG-R15 REG-R16; the quantitative anchor is the ACPR’s 2025 revaluation study R14. The model is Euro_FR_S, a monthly model: t counts policy months from the valuation date and is 0-based. The financial-year layer it carries underneath — the compte de participation aux résultats, the PPB and its eight-year clock, the declared taux servi — is indexed by the projection year y = t // 12.

Model scope and conventions#

Purpose. Project gross best-estimate liability cash flows — versements in; rachats partiels, death and total-surrender claims out; insurer expenses — for single model points on the euro support, together with the two state variables that make the product what it is: the épargne acquise and the provision pour participation aux bénéfices (PPB). Reserves are not computed here (see Valuation and reserve pointers).
The euro support only. The UC compartment of a multisupport contract is a separate liability with a separate levy regime R9, outside the A132-11 machinery R5, art. A132-10; it is the sibling product assurance_vie_uc. Arbitrages between supports are out of scope.
Time index std. t is 0-based and counts policy months: t = 0 is the first projected month, whatever the model point’s completed duration, and month t runs from time t/12 to time (t + 1)/12. The frame is t = 0 … proj_len − 1, so proj_len is the number of projected months, 12 × proj_years = 480, and result_cf() has proj_len rows. Two derived clocks sit on it and nothing is indexed by either. The projection year y = proj_year(t) = t // 12 is the index of the annual layer and the key of fin_rate_table.csv and wd_start_year; where these notes write “year y” as a financial-year statement, the months are t = 12y … 12y + 11. The contract’s own 1-based policy year is policy_year(t) = duration_init + t // 12 + 1, equally its completed policy years at the 31 December that closes it, and it is what the lapse table is read at. is_anniv(t) — t % 12 == 11 — is that 31 December, which on this model’s convention is also the policy anniversary and where every annual contractual event lands.
Projection frequency std: monthly, with the financial year left whole. A monthly grid is not a monthly product. Every contractual mechanic of the crediting machinery is annual and the model keeps it there: the PB is fixed for the closing year and credited at 31 December value date [S1] [S2] [S6] [S7] [S9], the frais de gestion is levied at that same date [S1] [S2] [S9], the levy is withheld as the interest is inscribed R9, art. L136-7 II, and the eight-year PPB clock counts financial years R5, art. A132-16. What the finer grid adds is everything that is not contractually annual: the versements libres programmés and rachats partiels programmés the contract bills monthly [S1], the insurer’s expenses falling where they are incurred, the decrements falling in the month they happen, and — the one answer this changes rather than resolves — the pro rata temporis in-year floor rate a mid-year dénouement actually receives [S1] [S2] [S11], which is now implemented rather than compressed away. BoursoVie’s daily compounding [S1] remains an approximation the monthly grid narrows rather than removes. An annual step remains a special case of the recursions below and reproduces every anniversary value exactly — see Annual equivalence — but it is not what the reference model runs.
Timing conventions std. Versements and rachats partiels are programmés and carry a monthly contractual minimum — EUR 50 and EUR 150 respectively [S1] — so the year’s amounts are collected in twelve equal instalments at the beginning of each month. They enter the crediting base at the mid-month weight (11.5 − k)/12, the fraction of the year still to run from the middle of month k; the twelve weighted twelfths sum to exactly 72/144 = 0.5, so the notes’ historic weight of 0.5 is now derived from “spread evenly through the year” rather than asserted. The year’s revalorisation and the frais de gestion sur encours land whole at 31 December, and the prélèvements sociaux with them. Decrements act at the end of every month, deaths before total surrenders, on monthly rates derived from the annual ones. An exit in the anniversary month therefore takes the full year’s taux servi, exactly as it did on an annual grid; an exit in any of the other eleven takes the contractual rule instead — the announced floor rate pro rata temporis, which with a zero TMG is no in-year interest at all [S1] [S2] [S3]. BoursoVie’s credit of the annual PB to sums surrendered during the year is expressly conditional on the adhesion being in force on the following 1 January [S1], which a rachat total is not; the Afer reading [S11], which tops the in-year floor up to the definitive rate, is a documented switch below, not the base.
The taux servi is a net rate. ts_net(t) is the credited rate in the ACPR’s sense, “net de prélèvements sur encours et avant prélèvements sociaux” R14. The frais de gestion sur encours is inside it and is reported only as a decomposition; deducting it again is the likeliest implementation error and the first pitfall below.
Age basis age last birthday std — no retrieved French document fixes one, and mortality here drives the timing of dénouement, not the benefit amount. Currency EUR; single-policy model points projected on an expected basis, pols_if(t) multiplying per-policy amounts. Rounding full precision internally, reported cash flows to the cent std.
Out of scope, and said so. The HCSF surrender-suspension power under art. L631-2-1 5° ter R8 REG-R13 is not modeled, nor is the exceptional PPB reprise of art. A132-16-1 REG-R16. Both are solvency-stress management actions, both would materially change a mass-surrender projection, and neither has a published trigger a deterministic model could key off. A mass-lapse stress run here is a pre-management-action result.

Model point attributes#

Attribute	Type	Example (worked configuration)
`point_id`	int	1
`policy_id`	str	FR-AVE-0001
`sex`	enum {M, F}	M
`issue_age`	int, age at `adhésion`	55
`duration_init`	int, completed policy years at the valuation date	5
`pols_if_init`	float, policies represented	1.0
`av_pp_init`	currency, `épargne acquise` at the valuation date	100 000.00
`ppb_pp_init`	currency, PPB attributed to the model point	4 000.00
`ppb_vintages_init`	int, equal open vintages the opening PPB is split across	8
`prem_gross_pp`	currency p.a., `versements libres programmés`	2 400.00
`prem_charge_rate`	rate, `frais sur versement`	0.0000
`wd_pp`	currency p.a., `rachats partiels programmés`	3 000.00
`wd_start_year`	int, first projection year the programmed surrender runs, 0-based like `y = t // 12`	5
`fee_rate`	rate p.a., `frais de gestion sur encours`	0.0060
`tmg_rate`	rate p.a., `taux minimum garanti`	0.0000
`ts_target`	rate p.a., the insurer’s target `taux servi`, net	0.0230
`soc_levy_rate`	rate, `prélèvements sociaux`	0.1720
`guarantee_form`	enum {gross, net}	net
`avance_on`	bool, `avance` outstanding	0
`scenario_id`	str, names the `r_fin` path in `fin_rate_table.csv`	base

Every attribute name, every column of model_point_table.csv and every cells name is English lower_snake_case, per the shared vocabulary; the French names stay in the prose, where they are the name of the thing. model_point() selects the row, age(t) is issue_age + duration_init + t // 12 — age last birthday, stepping at the policy anniversary and not inside the year — and proj_len() is 12 × 40 = 480 months std, the number of projected months, so the frame is t = 0 … 479. The horizon stays forty years because the euro support has no term, so it is a modeling choice rather than a contract fact, and fin_rate_table.csv carries forty rows per scenario.

State variables#

Variable	Description	Updated
`av_pp(t)`	`épargne acquise` per policy at the start of month `t`	monthly recursion
`av_pp_at(t, timing)`	the within-month points of the same balance: `"BEF_PREM"`, `"AFT_PREM"`, `"AFT_WD"`, `"AFT_INT"`	within month `t`
`av_at(t, timing)`	`av_pp_at(t, timing) × pols_if(t)`, the fund-level balance	within month `t`
`ppb_pp(y)`	PPB attributed to the model point at the start of financial year `y`	annual recursion
`ppb_vintage_pp(y, v)`	remaining balance of the dotation carried in financial year `v`, at the start of year `y`; released FIFO	annual, both indices
`pb_cum_pp(t)`	cumulative PB credited since the valuation date — the `effet cliquet` ledger	monthly index, steps once a year
`soc_levy_cum_pp(t)`	cumulative `prélèvements sociaux` deducted from the account	monthly index, steps once a year
`guar_floor_pp(t)`	contractual capital floor: `versements` net of `frais sur versement`, less `rachats`, less cumulative `frais de gestion`	monthly; the charge term steps at the anniversary
`ts_net(y)`	`taux servi` credited for financial year `y` (net of charges on encours, before social levies)	annual, crediting rule
`pols_if(t)`	policies in force at the start of month `t`; `pols_if_at(t, timing)` gives `"BEF_DECR"` / `"BEF_LAPSE"` / `"AFT_DECR"`	monthly decrements
`mort_rate(t)`, `lapse_rate(t)`	the annual decrement rates of the policy year containing month `t`	read once a year
`mort_rate_mth(t)`, `lapse_rate_mth(t)`	the monthly rates actually applied, `1 − (1 − r)^(1/12)`	monthly

Assumption inputs#

Three classes are distinguished. Class (a) is contractual or statutory and is cited; class (b) is the insurer’s current discretionary scale, revisable annually within the statutory floor R5 and the eight-year clock REG-R16; class (c) is the modeler’s view of experience.

(a) Contractual / guaranteed elements (cited)#

Input	Value	Basis
`frais sur versement`	0.00%	[S1] [S3] [S13]
`frais de gestion sur encours`	0.60% p.a., levied 31 December, pro rata temporis on in-year movements	[S3]; timing [S1] [S2] [S9]; level choice std, product-spec (4)
Capital guarantee	`versements` net of entry charges, less the annual management charges, measured before social and tax levies	[S3] [S5] [S6] [S7]; measurement [S1] [S2] [S3]
`Effet cliquet`	credited PB is definitively acquired and cannot be called back	[S1] [S9]
`Taux minimum garanti`	0.00% p.a.	std, product-spec (7)
Statutory PB floor	85% of the `compte financier` balance, plus the `compte technique` balance less the greater of 10% of its credit balance and 4.5% of annual premiums	R5, art. A132-11 R6 R14, fn 12 REG-R15
Statutory minimum benefit	that credit balance less interest already credited to mathematical provisions	R5, art. A132-12 REG-R15
PPB release horizon	eight financial years following the year of the dotation	R5, art. A132-16 R6 [S2] REG-R16
Death benefit	the `épargne acquise`, less outstanding `avances` and interest; no additional guarantee	[S3]
Surrender charge	0.00%; settlement two months by statute, 30 days by contract	[S2] [S10] [S13]; R7 REG-R31; [S3] [S6]
`Prélèvements sociaux`	17.2%, on the products as credited to the contract each year	rate [S3]; timing R9, art. L136-7 II

(b) Insurer-discretionary current elements (snapshot; revisable annually)#

Input	Value	Basis
Target `taux servi` `ts_target`	2.30% p.a. net, level	std, product-spec (9); market context R14
Crediting rule	credit `ts_target` where the PPB allows, never below `tmg_rate`, and always allocate the year’s statutory minimum PB in full — credited or carried to the PPB; forced releases override the target upward	std (i)
PPB dotation policy	the excess of the statutory minimum PB over the target, carried to a new vintage	std (i)
PPB release order	FIFO, oldest vintage first	std (ii)
PPB earns no separate return	the return on PPB assets enters the `compte financier` instead, per the A132-14 basis of average technical provisions	REG-R15; convention std (iii)
Opening PPB	4.0% of `av_pp_init`, in eight equal vintages	R14 REG-R47; split std, product-spec (11)
UC-holding bonus	none	R14; exclusion std, product-spec (10)
Year-on-year cap on `ts_net`	none	std (iv)

Footnotes: (i) No insurer publishes its dotation or release policy; only the outer bounds are public — at least 85% of the compte financier and the A132-11 technical share must reach policyholders R5, and the PPB must be released within eight years REG-R16. Aggregate levels are (4.0% of provisions at end-2025 R14; EUR 53.6 bn, −11.1% year on year, at end-2024 REG-R47), but no insurer publishes its own rule. (ii) The statute prescribes no release order; FIFO is the only order that satisfies the eight-year constraint without slack, and it makes ppb_vintage_pp testable. (iii) A132-14 computes the financial result as average technical provisions times a taux de rendement des placements REG-R15 and the PPB is one of those provisions REG-R6, so PPB assets earn inside the compte financier and the vintage balances stay nominal; accreting the vintages and including the PPB in the financial base double-counts. (iv) French insurers do smooth the announced rate, but the PPB is the smoothing device here, and a second cap on the year-on-year change would let the model credit rates the fund cannot fund.

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

No French euro-fund lapse experience is public: the ACPR publishes only aggregate flows — EUR 71.0 bn of surrenders against EUR 1 361 bn of guaranteed-capital encours in 2025 — with no split by duration, age or vintage R15. Every shape below is therefore std, rationalized from the product’s incentive structure.

Input	Recommended basis	Basis tags
Base mortality `mort_rate`	80% of a sex-distinct redistributable proxy table shaped like French population mortality, not the INSEE quotients themselves — the shipped `mort_table.csv` is an analytic Makeham-type curve anchored so that the 80% factor gives the worked example’s q = 0.0060 at age 60 exactly	table std (vii); shape and the only redistributable French series REG-R24; permitted-table framework REG-R23
Base surrender `lapse_rate_base`	4.0% p.a. at policy durations 1–7; 8.0% at duration 8; 5.0% at durations 9+	std; the duration-8 step is the tax threshold R10 R11 REG-R40
Dynamic surrender	additive in the gap between a market reference rate and `ts_net`: see Policyholder behavior modeling	std
Market reference rate `ref_rate`	2.20% p.a. — the 2025 average Livret A rate	R14; use as the dynamic reference std
Partial-surrender utilisation	the programmed annual amount, from projection year `y = wd_start_year`, paid in twelve monthly instalments	std
Insurer expenses	EUR 24 per policy p.a. inflating at 1.5% p.a., plus 0.35% p.a. of the average balance	std (v)
Fund financial rate `r_fin`	scenario path in `fin_rate_table.csv`; base path 3.30% falling to 2.30% over twelve years	std (vi)
`Avance` take-up	0	terms unpublished [S1] [S2] [S3]; exclusion std

(v) Actual unit expenses are not public. The proportional 0.35% is sized so that the loading margin leaves the statutory compte technique small relative to the compte financier, which is what the market outturn implies: a 0.63% average charge rate against a 2.8% asset return and a 2.63% credited rate R14 leaves little technical margin once distribution costs on encours are paid. The fixed/proportional split is a modeling choice. (vi) The path is anchored to the ACPR’s taux de rendement de l'actif — 2.8% in 2025, 2.5% in 2024, near 2.1–2.2% from 2020 to 2023, half of undertakings between 2.4% and 3.3% R14 — and to the reinvestment picture behind it: the 10-year OAT averaged 3.4% in 2025 while about 60% of fixed-coupon bonds maturing within four years still carry a coupon below 3% R14. It is a scenario, not a forecast. (vii) The shipped table is a synthetic curve, not a data extract. INSEE’s T69QMORT quotients are the only freely redistributable French mortality series and the only public shape available REG-R24 — TH 00-02 / TF 00-02 and TGH05 / TGF05 are cited by name and article REG-R23 and never shipped — but what mort_table.csv contains is a Makeham-type curve whose first differences of q by age grow by a constant factor from 18 to 100, with q(M, 60) set to exactly 0.0075 so that the 0.8 best-estimate factor reproduces this table’s 0.0060 placeholder to the digit. Read it as a smooth stand-in of roughly the right level and slope, not as population quotients scaled by 0.8, and do not lift it for any purpose that needs actual French population mortality.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Cells	Meaning
`t`	—	policy month index from the valuation date, 0-based: `t = 0, 1, …, proj_len − 1`, `proj_len = 480`
`y`	`proj_year(t)`	the projection year containing month `t`, `t // 12`; the index of every financial-year statement below
`k`	—	the month’s position inside its year, `t − 12y`, `k = 0 … 11`; `k = 11` is the 31 December, `is_anniv(t)`
`AV(t)`	`av_pp(t)`	`épargne acquise` per policy at the start of month `t`; `AV(0) = av_pp_init`
`P(y)`, `P_g(y)`	`prem_to_av_pp(y)`, `prem_gross_pp(y)`	the year’s `versements` credited net of `frais sur versement`, and before them
`P_m(t)`, `W_m(t)`	`prem_to_av_mth_pp(t)`, `withdrawals_mth_pp(t)`	the month’s instalment of each, `P(y)/12` and `W(y)/12`
`W(y)`	`withdrawals_pp(y)`	the year’s `rachats partiels`, capped once on the year-open balance
`ω(k)`	`prem_wt_mth(k)`	the mid-month crediting weight `(11.5 − k)/12`; `Σ_k ω(k)/12 = 0.5` exactly
`B(y)`	`pm_avg_pp(y)`	crediting base: `AV(12y) + Σ_k ω(k)·(P_m − W_m)`, which is `AV + 0.5·P − 0.5·W` on a level schedule
`c`, `F(y)`	`fee_rate`, `fee_pp(y)`	`frais de gestion sur encours`, 0.60% p.a.; the amount charged at 31 December, `c · B(y)`
`E(y)`, `r(y)`	`expenses_pp(y)`, `r_fin(y)`	the year’s insurer expenses, of which `E(y)/12` is incurred each month; the fund’s financial return rate for year `y`
`Φ(y)`, `T(y)`	`fin_acct_pp(y)`, `tech_acct_pp(y)`	`compte financier` balance `r(y)·(B(y) + Q(y))`; `compte technique` balance `F(y) − E(y)`
`s(y)`	`insurer_tech_share_pp(y)`	insurer’s technical share, `max(0.10·max(T(y),0), 0.045·P_g(y))`
`A(y)`, `A⁺(y)`	`pb_acct_pp(y)`, `pb_min_pp(y)`	`compte de participation aux résultats` balance; the statutory minimum PB
`Q(y)`, `Q_v(y)`	`ppb_pp(y)`, `ppb_vintage_pp(y, v)`	PPB at the start of financial year `y`, `Q(0) = ppb_pp_init`; the remaining balance of the vintage carried in year `v`, `v` on the same 0-based year clock as `y`
`D(y)`, `R(y)`	`ppb_dotation_pp(y)`, `ppb_release_pp(y)`	PPB dotation and release in year `y`
`X(y)`, `I(y)`	`pb_credited_pp(y)`, `int_credited_pp(y)`	PB credited gross of `F(y)`; the net revalorisation added, `X(y) − F(y)`
`I_m(t)`, `L_m(t)`	`int_credited_mth_pp(t)`, `soc_levy_mth_pp(t)`	what actually lands in month `t`: `I(y)` and `L(y)` at 31 December, nil in the other eleven months
`g`, `s*`	`tmg_rate`, `ts_target`	`taux minimum garanti` 0.00%; the insurer’s target `taux servi` 2.30%
`ŝ(y)`, `σ(y)`	`ts_stat(y)`, `ts_net(y)`	statutory floor rate and credited `taux servi`, both net of the charge
`L(y)`, `G(t)`	`soc_levy_pp(y)`, `guar_floor_pp(t)`	`prélèvements sociaux` `0.172·max(I(y),0)`; the contractual capital floor, a monthly balance
`q(y)`, `w(y)`	`mort_rate(t)`, `lapse_rate(t)`	the annual decrement rates of the policy year containing month `t` — the rates these notes tabulate
`q_m`, `w_m`	`mort_rate_mth(t)`, `lapse_rate_mth(t)`	the monthly rates actually applied, `1 − (1 − r)^(1/12)`
`l(t)`	`pols_if(t)`	policies in force at the start of month `t`, `l(0) = pols_if_init`

Monthly rates from annual assumptions std #

Every decrement assumption in this product is published, calibrated and tabulated annually: the mortality table is by attained age, the base surrender vector is by completed policy duration, and the dynamic surrender term keys on two annual rates. Those annual rates keep their names and their meaning — mort_rate(t) and lapse_rate(t) are the rates of the policy year containing month t, constant across its twelve months — and the monthly rates the recursion applies are derived from them at the constant-force conversion:

q_m(t) = 1 − (1 − q(y))^(1/12)
w_m(t) = 1 − (1 − w(y))^(1/12)

so that twelve months compound back to exactly the annual rate, [(1 − q_m)(1 − w_m)]¹² = (1 − q)(1 − w). No retrieved French source states a conversion convention for any decrement, so the choice is std; it is the one that makes the monthly grid reproduce an annual step at every anniversary, and check_decrements_compound() asserts it directly rather than only through its consequence. There is no shock and no contractual surrender date inside a policy year on this product — the duration-8 tax threshold is itself a policy-year boundary — so nothing about either decrement clusters on a month and the whole of each year’s rate is spread.

The taux servi gap that drives lapse_dyn_add is read once, at the year, on ref_rate(y) and ts_net(y), so the annual rate is one rate for the whole policy year. Letting it drift month by month would be a second modeling change smuggled in beside the grid change.

Monthly processing order std #

Once per financial year y, struck at the year open (t = 12y). Nothing in this block is affected by the grid; every formula is the one an annual step used.

The crediting base is accumulated from the year’s twelve monthly movements at the mid-month weight [S1]: B(y) = AV(12y) + Σ_{k=0..11} ω(k)·(P_m(12y+k) − W_m(12y+k)), which on a level programmed schedule is exactly AV(12y) + 0.5·P(y) − 0.5·W(y).
The compte de participation aux résultats is built and the statutory minimum A⁺(y) determined (below).
The crediting rule fixes R(y), D(y) and hence X(y), σ(y), I(y) and L(y) (below). The frais de gestion F(y) = c·B(y) is inside σ(y), not a further deduction.

Each of the twelve months t = 12y + k.

BOM — versement. The month’s instalment is credited net of the entry charge: P_m(t) = P(y)/12; av_pp_at(t, "AFT_PREM") = AV(t) + P_m(t).
BOM — rachat partiel programmé. W_m(t) = W(y)/12; av_pp_at(t, "AFT_WD") = av_pp_at(t, "AFT_PREM") − W_m(t).
Through the month — insurer expense. E(y)/12 is incurred, weighted by the in force of this month. It is a cash flow of the insurer and not a movement on the contract; the amount entering the compte technique stays the year’s total E(y).
31 December only — revalorisation. I_m(t) = I(y) if is_anniv(t), else 0; av_pp_at(t, "AFT_INT") = av_pp_at(t, "AFT_WD") + I_m(t).
31 December only — prélèvements sociaux. L_m(t) = L(y) if is_anniv(t), else 0; av_pp(t+1) = av_pp_at(t, "AFT_INT") − L_m(t).
EOM — decrements, deaths then total surrenders. q_m(t) then w_m(t) on the survivors; each releases av_pp(t+1), the balance the month actually closes on, as a claim. In the anniversary month that carries the whole year’s taux servi; in the other eleven it carries none, which is the contractual floor rate pro rata temporis at a nil TMG.
l(t+1) = l(t)·(1 − q_m(t))·(1 − w_m(t)); guar_floor_pp, soc_levy_cum_pp and pb_cum_pp roll forward, the last two moving only after an anniversary. At the year’s end the PPB vintage ledger rolls forward on its own annual clock.

Annual equivalence. Because the monthly decrement rates compound back to their annual values, the recursion collapses over any twelve months of one policy year to l(12(y+1)) = l(12y)·(1 − q)·(1 − w) — the annual-step recursion, term for term. Because the twelve mid-month weighted twelfths sum to exactly one half, B(y) is the notes’ AV + 0.5·P − 0.5·W to the last bit. And because the participation aux bénéfices is a financial-year account landing whole at 31 December, everything built on it is a function of B(y) and Q(y) alone. Every anniversary value is therefore identical on the two grids, to floating point — l at each t = 12y, AV, B, Φ, T, A, A⁺, Q and every vintage, ŝ, σ, I, L, G and the two cumulative ledgers. Measured against a pre-conversion snapshot of the annual model, over all eleven shipped model points and all forty years, the largest relative difference is 1.1e-14 and most cells are bit-identical. Nothing else agrees, and nothing else should: the cash flows are where the finer grid does its work.

The `épargne acquise` recursion#

av_pp(t+1) = av_pp(t) + prem_to_av_mth_pp(t) − withdrawals_mth_pp(t)
             + int_credited_mth_pp(t) − soc_levy_mth_pp(t)

av(t+1)    = av(t) + premiums(t) − withdrawals(t) + int_credited(t) − soc_levy(t)
             − claims_death(t) − claims_lapse(t)

Both are monthly identities now, and the last two terms of the first line are nil in eleven months of twelve. The second line is the fund-level form, where the releases appear; check_av_roll_fwd() asserts it in every one of the 480 months, with av(t) = av_pp(t)·pols_if(t) and every aggregate the per-policy amount times pols_if(t). The identity is exact because claims are struck on av_pp(t+1), the same balance the survivors carry forward. Measured over all eleven model points, the largest residual is EUR 7.5e-09 on the grouped 250-policy cell.

The `compte de participation aux résultats`#

Built per financial year, per policy, on the two accounts art. A132-11 names R5 REG-R15. Nothing in this block is touched by the monthly grid — it is a financial-year statement, indexed by y, and it reproduces an annual step exactly:

fin_acct_pp(y)          = r_fin(y) · ( pm_avg_pp(y) + ppb_pp(y) )
tech_acct_pp(y)         = fee_pp(y) − expenses_pp(y)
insurer_tech_share_pp(y)= max( 0.10 · max(tech_acct_pp(y), 0), 0.045 · prem_gross_pp(y) )
pb_acct_pp(y)           = 0.85 · fin_acct_pp(y)
                          + tech_acct_pp(y) − insurer_tech_share_pp(y)
pb_min_pp(y)            = max( 0, pb_acct_pp(y) − tmg_rate · pm_avg_pp(y) )
ts_stat(y)              = ( pb_min_pp(y) − fee_pp(y) ) / pm_avg_pp(y)

Four points of substance. The 85% attaches to the financial account and the 90% to the technical account, not the other way round R5, art. A132-11 R14, fn 12 REG-R15. The insurer’s technical share has two limbs and the 4.5%-of-premiums limb often binds, so the policyholder share of a positive technical balance is at most 90% and can be much less. The PPB sits inside the financial base, because A132-14 computes the financial result on average technical provisions REG-R15 and the PPB is one of them REG-R6. And ts_stat(y) is net of the charge: pb_min_pp is gross of fee_pp because the charge is a credit to the technical account, so it is subtracted once to reach the rate the account actually grows by. For the euro support the underwriting result is nil — the death benefit is the account value [S3] — so tech_acct_pp is the loading result alone. A contract with a contractual PB percentage (90% at Suravenir Rendement [S4], 100% at Afer [S9]) replaces the first line with that percentage of the ring-fenced fund’s net financial profits.

The crediting rule, the TMG and the PPB lever#

pb_target_pp(y)  = ts_target · pm_avg_pp(y) + fee_pp(y)
ppb_dotation_pp(y)  = max( 0, pb_min_pp(y) − pb_target_pp(y) )
ppb_discr_rel_pp(y) = min( max(0, pb_target_pp(y) − pb_min_pp(y)), ppb_pp(y) )
ppb_forced_pp(y)    = Σ_v { ppb_vintage_pp(y, v) : v + 8 ≤ y }
ppb_release_pp(y)   = max( ppb_discr_rel_pp(y), ppb_forced_pp(y) )
pb_credited_pp(y)   = pb_min_pp(y) − ppb_dotation_pp(y) + ppb_release_pp(y)
ts_net(y)           = max( tmg_rate,
                           ( pb_credited_pp(y) − fee_pp(y) ) / pm_avg_pp(y) )
int_credited_pp(y)  = ts_net(y) · pm_avg_pp(y)
int_credited_mth_pp(t) = int_credited_pp(y) if is_anniv(t) else 0

The last line is the whole of what the monthly grid does to this block: the year’s revalorisation is fixed by the board for the closing year and credited at 31 December value date [S1] [S2] [S6] [S7] [S9], so it arrives in one month and eleven months of twelve carry none of it.

Read it as three levers on one rate. The statutory floor ts_stat(y) is what the year’s result alone obliges the insurer to credit. The PPB moves the credited rate above or below that floor: a dotation parks this year’s excess, a release spends an earlier year’s. The TMG is a hard floor under the result, and because it guarantees technical interest plus PB together R3 REG-R18 it is a floor on ts_net, not a separate credit stacked on top — with tmg_rate = 0 it never binds here, but a positive TMG binds through the PPB, forcing a release the insurer did not choose. A dotation and a forced release can coexist in one year — this year’s excess goes in while an eight-year-old vintage comes out — and both appear in the worked example’s first three rows.

The PPB and its eight-year clock#

ppb_pp(y+1) = ppb_pp(y) + ppb_dotation_pp(y) − ppb_release_pp(y)

with the vintage ledger ppb_vintage_pp(y, v) carrying the detail: ppb_dotation_pp(y) opens vintage y, ppb_release_pp(y) is drawn FIFO from the oldest open vintage forward, and a vintage carried in year v must be exhausted by the end of year v + 8 R5, art. A132-16 REG-R16. check_ppb_roll_fwd() asserts the balance identity and check_ppb_clock() that ppb_vintage_pp(y, v) = 0 for every v ≤ y − 9 — one year past the deadline, because ppb_vintage_pp(y, v) is a start-of-year balance and the vintage with v = y − 8 is still standing at the start of the year that forces it out.

The clock is not made finer by the monthly grid, and that is the point. Art. A132-16 counts financial years, so both the balance index y and the vintage index v stay on the financial-year clock and v + 8 remains a year deadline on a year balance. Restating the per-vintage ledger at 480 months would carry two clocks in one signature and multiply its cost by twelve for no new information. The opening balance ppb_pp_init is split into ppb_vintages_init equal vintages carried in years −1, −2, …, −8, falling due at y = 7, 6, …, 0 std. The PPB is bounded below by zero — a negative PPB is not a permitted state, and the exceptional reprise of art. A132-16-1 is a supervised recovery measure, not a projection lever REG-R16. When ppb_pp(y) = 0 and ts_stat(y) < ts_target, the model credits ts_stat(y).

`Prélèvements sociaux`#

soc_levy_pp(y)     = soc_levy_rate · max( int_credited_pp(y), 0 )
soc_levy_mth_pp(t) = soc_levy_pp(y) if is_anniv(t) else 0
soc_levy_cum_pp(t) = Σ_{u<t} soc_levy_mth_pp(u)

The levy is taken as the interest is credited, every year, whether or not anything is withdrawn, because the rights are expressed in euros R9, art. L136-7 II; the rate is 17.2% [S3]. Art. L136-7 II charges the products “lors de leur inscription au bon ou contrat”, and the inscription is the 31 December crediting — so on the monthly grid the annual timing is now visible rather than implicit: soc_levy_mth_pp(t) is nil in eleven months of twelve and carries the whole of L(y) in the twelfth, moving with the interest it is struck on. Spreading the levy without spreading the interest would tax interest not yet credited. It sits inside the account roll-forward and outside net_cf. Inside, because it is money that genuinely leaves the contract each year, and a model that defers it to surrender overstates the account and every benefit measured on it. Outside net_cf, because net_cf is the insurer’s liability stream while the levy is a policyholder tax the insurer withholds and remits to the State — neither a benefit nor an insurer expense. It is reported in its own soc_levy column of result_cf(), so a fund-level asset projection adds it back as an outflow in one step. The base is the interest actually inscribed on the contract, i.e. net of the management charge, which is std: art. L136-7 fixes the timing but not the base R9 (product-spec footnote 13).

The capital guarantee floor and the `effet cliquet`#

guar_floor_pp(t+1) = guar_floor_pp(t) + prem_to_av_mth_pp(t) − withdrawals_mth_pp(t)
                     − ( fee_pp(y) if is_anniv(t) else 0 )

check_guar_floor():  av_pp(t) + soc_levy_cum_pp(t) ≥ guar_floor_pp(t)   for all months t
check_cliquet():     pb_cum_pp(12(y+1)) = pb_cum_pp(12y) + max(pb_credited_pp(y), 0)
                     and pb_credited_pp(y) ≥ 0 and ts_net(y) ≥ tmg_rate  for all years y

The floor recursion is the guarantee_form = "net" form [S3] [S5] [S6] [S7]; the "gross" variant drops the charge term [S4] [S8] [S9]. The instalments move the floor every month, but the charge term steps once a year, at the anniversary, which is what “less the annual management charges” says [S6]. check_guar_floor() now runs at every one of the 480 months rather than at forty year-ends, which is a stronger statement: the floor steps down at the anniversary while the account only catches up at the same date, so the tightest month of each year is now looked at. For an in-force cell the premium history before the valuation date is not carried in the model point, so the floor is seeded at guar_floor_pp(0) = av_pp_init std — deliberately conservative, since the true floor on a five-year-old contract sits below its account value by the interest already credited. It is tested on the account value before cumulative social levies, because the published minimum surrender-value tables are stated before social and tax levies [S1] [S2] [S3]. The effet cliquet is a separate and weaker invariant, and conflating the two is a pitfall: what is ratcheted is credited PB, not the balance. On a garantie nette contract the balance can fall in a year that would need ts_net(t) < 0 to cover the charge — the charge keeps biting, the ratchet does not undo it, and both statements are true at once [S6] [S9].

Decrements, claims and cash flow outputs#

pols_death(t)  = pols_if(t) · mort_rate_mth(t)
pols_lapse(t)  = pols_if(t) · (1 − mort_rate_mth(t)) · lapse_rate_mth(t)
pols_if(t+1)   = pols_if(t) − pols_death(t) − pols_lapse(t)
db_pp(t)       = av_pp(t+1)          death benefit: the épargne acquise, no uplift  [S3]
cv_pp(t)       = av_pp(t+1)          surrender value: no penalty  [S2] [S10] [S13]
claims(t, "DEATH") = pols_death(t) · db_pp(t)
claims(t, "LAPSE") = pols_lapse(t) · cv_pp(t)

The benefit formulas are unchanged and their meaning is not: av_pp(t+1) is the balance closing the month of exit, so in the anniversary month it carries the whole year’s taux servi — exactly as on an annual grid — and in the other eleven it carries no in-year revalorisation at all, which is the contractual floor rate pro rata temporis at a nil TMG [S1] [S2] [S3]. There is no maturity decrement: the euro support has no term, and the contract’s stated maturity, where one exists, is renewable annually without limit [S6]. result_cf() is a DataFrame of 480 monthly rows indexed by t, first column pols_if, with

Column	Formula
`premiums`	`prem_to_av_mth_pp(t) · pols_if(t)`
`withdrawals`	`withdrawals_mth_pp(t) · pols_if(t)` — an owner election, not a claim
`claims_death`	`claims(t, "DEATH")`
`claims_lapse`	`claims(t, "LAPSE")`
`expenses`	`expenses_mth_pp(t) · pols_if(t)`
`int_credited`	`int_credited_mth_pp(t) · pols_if(t)` — state movement, reported not summed
`soc_levy`	`soc_levy_mth_pp(t) · pols_if(t)` — excluded from `net_cf`
`liability_cf`	`claims_death + claims_lapse + withdrawals + expenses − premiums`
`net_cf`	`− liability_cf(t)`, income-positive

result_cf_annual() publishes the same frame summed into projection years, indexed by y: every flow column is the total of its twelve months and pols_if is the count entering the year, pols_if(12y), which is the number the annual-step model carried on the same row. It is the frame regrouped, never a second projection, and it is what a reader lays beside the annual-step model this replaced. result_pb() stays a financial-year statement of forty rows indexed by y, and every figure in it is unchanged.

Known modeling pitfalls#

Each of these produces a plausible-looking projection that is wrong, and each becomes a test.

Deducting the management charge twice. ts_net(y) is already net of the frais de gestion sur encours R14; applying av × (1 + ts_net) × (1 − c) costs the policyholder 0.60% a year that was already taken. Test: with ts_net = 0 one year’s movement equals exactly −c · pm_avg_pp(y). The same error in another dress is deducting the fund’s own 0.24% + 0.03% internal costs [S5], which a rate quoted net of contract charges already covers.
Crediting on the closing balance instead of the pro rata temporis base. The PB is allocated “weighted by the time the sums were present on the fund during the year” [S1]; crediting on av_pp(12y) + P(y) − W(y) gives a full year’s interest on a December payment. Test: with P(y) = W(y) = 0 the two agree; with a payment they differ by exactly 0.5 · ts_net(y) · P(y). On the monthly grid there is a second version of the same error: weighting the instalments at the beginning of their month, (12 − k)/12, or at its end, (11 − k)/12, instead of at its middle. Those give 0.541667 and 0.458333 where the mid-month weight gives exactly 0.5, and on the anchor cell they move the year-0 base by EUR 100.00 and the year’s interest by EUR 2.79 — small enough to look like rounding, large enough to break the anniversary equivalence. Test: Σ_k ω(k)/12 == 0.5 exactly, and B(y) == AV(12y) + 0.5·(P(y) − W(y)).
Getting the statutory split backwards. “90% of the financial account and 85% of the technical result” is the popular form and it is wrong: the article says 85% of the compte financier, and the technical balance less the greater of 10% of it and 4.5% of premiums R5, art. A132-11 R14, fn 12 REG-R15. Test: with tech_acct_pp = 0, pb_acct_pp(y) equals 0.85 · fin_acct_pp(y) exactly.
Dropping the 4.5%-of-premiums limb. With a small technical result and a live premium stream, 0.045 · prem_gross_pp(y) exceeds 0.10 · tech_acct_pp(y) and takes the larger bite. Test: in the worked example at y = 5, insurer_tech_share_pp = EUR 108.00, against EUR 28.43 for the 10% limb.
Leaving the PPB out of the financial base. fin_acct_pp is struck on pm_avg_pp + ppb_pp REG-R15 REG-R6; omitting it understates the distributable amount by 0.85 · r_fin · ppb_pp — EUR 41.81 at worked-example y = 5. The mirror error is accreting the vintages as well, which distributes the PPB’s return twice. Test: ppb_vintage_pp(y, v) changes only by releases.
Releasing the PPB LIFO, or letting a vintage age past eight years. Test: ppb_vintage_pp(y, v) = 0 for every v ≤ y − 9 R5, art. A132-16 REG-R16 — the vintage due during year y is the one with v = y − 8, and it is still standing at the start of that year.
Letting the PPB go negative, or losing part of the year’s statutory minimum. A dotation year credits less than ts_stat(y) and that is legal — the balance goes to the PPB, not to the insurer — so the invariant is an allocation identity, not a rate inequality. Test: ppb_pp(y) ≥ 0, ts_net(y) ≥ tmg_rate, and pb_credited_pp(y) + ppb_dotation_pp(y) − ppb_release_pp(y) == pb_min_pp(y) for all y. ts_net(y) ≥ ts_stat(y) happens to hold on every row of the worked example, because the forced release always exceeds the dotation; it is not the invariant, and a model point with no vintage falling due would break it legitimately.
Levying prélèvements sociaux only at surrender. This is the euro fund’s signature mechanic and the commonest foreign-model error: the levy is annual on euro-denominated rights and deferred only on the UC part R9. Test: soc_levy_pp(y) = 0.172 · max(int_credited_pp(y), 0) every year, and the twelve-year total is exactly 17.2% of the twelve-year credited interest.
Levying it on the account rather than on the year’s interest. 17.2% of EUR 100 000 is EUR 17 200; 17.2% of the first year’s (y = 0) EUR 2 827.60 is EUR 486.35.
Testing the effet cliquet as “the account never falls”. Under the garantie nette the balance falls by the management charge in a zero-PB year, and the tables published for exactly that case [S2] [S3] prove it. Ratchet pb_cum_pp, not av_pp(t). Relatedly, compare the guarantee floor to av_pp(t) + soc_levy_cum_pp(t) at every month, because the published minimum surrender values are stated before social and tax levies [S1] [S2] [S3].
Adding a death-benefit uplift. The death capital is the épargne acquise and nothing more [S3]; the optional riders price the UC capital at risk [S3] [S4]. Test: db_pp(t) == cv_pp(t) == av_pp(t+1) in every month, and no surrender penalty anywhere [S2] [S10] [S13].
Giving mid-year exits a full year’s interest. This is the error an annual grid was forced into and the monthly grid retires: the contractual rule is the announced floor rate pro rata temporis, which with a zero TMG is no in-year interest at all [S1] [S2] [S3], and the model now implements it. BoursoVie’s credit of the annual PB to sums surrendered during the year is conditional on the adhesion being in force on the following 1 January [S1], which reaches rachats partiels and not exits. Test: in a non-anniversary month claim_pp(t, "LAPSE") == av_pp_at(t, "AFT_WD") exactly, and in the anniversary month the two differ by I(y) − L(y). The Afer reading [S11] — the declared rate accrued pro rata temporis, I(y)/12 a month — is a documented switch, not the base; it preserves the anniversary equivalence just as exactly, because the year’s interest still sums to I(y), so only the sources decide between them. A positive TMG would have to be accrued month by month inside the year and squared up at 31 December against σ(y); no positive-TMG model point is shipped, so no such cells is either.
Reading an annual table at t rather than at t // 12. The lapse table is keyed by the contract’s 1-based policy year and the scenario path by the projection year, so on a monthly grid both are read through policy_year(t) and proj_year(t). A model that forgot the // 12 would put the anchor cell’s duration-8 surrender step in the third month instead of the third year. Test: lapse_rate_base(t) == 8% for every t = 24 … 35 and for no t outside that block — the step is twelve months wide and lands on the year.

Policyholder behavior modeling#

All formulas are std; the shapes are rationalized from the incentive structure and the aggregate market evidence, and dynamic option-exercise assumptions are the norm this model is built to feed.

Base surrender. 4.0% p.a. at durations 1–7, 8.0% at duration 8, 5.0% at durations 9+, read at policy_year(t) and spread over that policy year’s twelve months at w_m like any other ordinary surrender. The duration-8 step is the tax threshold: the reduced 7.5% rate and the EUR 4 600 / EUR 9 200 annual allowance both switch on at eight years R10 R11 REG-R40, and a French savings projection with no surrender step at duration 8 has ignored the single strongest driver of French partial-surrender timing REG-R40.
The dynamic component — the French mechanic. French surrender behaviour keys on the gap between the taux servi and the rate available elsewhere, most visibly the Livret A:
```
lapse_dyn_add(t)  = a · max( 0, ref_rate(y) − ts_net(y) − tol )
lapse_rate(t)     = min( lapse_cap, lapse_rate_base(t) + lapse_dyn_add(t) )
lapse_rate_mth(t) = 1 − (1 − lapse_rate(t))^(1/12)
```
with a = 4.0, tol = 0.25 point, lapse_cap = 30% std, and ref_rate(y) = 2.20%, the 2025 Livret A average R14. lapse_rate(t) is the annual rate these notes tabulate, read once at the policy year’s gap — both ref_rate and ts_net are annual quantities taken at y = t // 12, so the rate is one number for the whole policy year — and lapse_rate_mth(t) is what the recursion applies. The duration-8 step is therefore twelve months wide. The sign of the relationship is observed rather than assumed: in 2025 the euro rate was 2.63% while the Livret A averaged 2.20% and fell from 2.4% to 1.7% in August and 1.5% in February 2026 R14 R15, and euro supports turned to a +EUR 6.4 bn net inflow after five consecutive years of net outflow R15. The magnitude — a, tol and the cap — has no public calibration and is the most consequential std in this file.
Asymmetry. The dynamic term is one-sided: a taux servi above the reference rate does not push surrenders below the base, because the base already reflects needs-driven withdrawals. A two-sided variant is a scenario switch.
Partial before total. Absent instruction an unspecified withdrawal drains the euro fund before the UC supports [S1] — in a euro-only model, a reminder that withdrawals(t) on a multisupport contract lands here first. The 30-day renunciation unwind REG-R29 [S2] [S6] [S9] is a first-duration effect the anchor cell, at duration 5, is past; a new-business cell needs it.
What the model deliberately does not do. No avance take-up (terms unpublished [S1] [S2] [S3]); no beneficiary-acceptance block on liquidity, which is a real and absolute constraint [S1] [S3]; no mass-surrender scenario with the HCSF response R8 REG-R13. The last is the important one: a mass-lapse stress here is a pre-management-action number, because the supervisor’s power to freeze surrenders for up to six consecutive months is precisely what would change the answer.

Worked example#

Anchor cell, product-spec “Anchor model cell”: av_pp_init = EUR 100 000.00 at duration 5, male age 60; prem_gross_pp = EUR 2 400.00 p.a. and prem_charge_rate = 0, so prem_to_av_pp = EUR 2 400.00 a year, collected as EUR 200.00 at the beginning of each month; withdrawals_pp = EUR 3 000.00 p.a. from year y = 5, likewise EUR 250.00 a month; fee_rate = 0.60%; tmg_rate = 0.00%; ts_target = 2.30%; soc_levy_rate = 17.2%; ppb_pp_init = EUR 4 000.00 in eight equal vintages of EUR 500.00 falling due in years y = 0 to y = 7; expenses EUR 24.00 p.a. inflating at 1.5% plus 0.35% of pm_avg_pp, incurred a twelfth a month; r_fin on the base path below.

The frame is 480 months, t = 0 … 479. Tables 1 and 2 below are financial-year statements and are shown for the first twelve of the forty projection years, y = 0 … 11; every number in them is the number the annual-step model this replaced produced, because the monthly grid leaves anniversary quantities exactly where the annual grid put them. What changed is the index each is read at: av_pp(5) is now av_pp(60) and av_pp(6) is av_pp(72). Table 3 shows the twelve months of year 5 — the one place a reader sees the two layers meet — and the cash-flow extract, which the finer grid genuinely moves, is printed from result_cf_annual(). Currency cells are full-precision model values rounded to the cent, so a printed row reproduces the next row’s opening balance to within EUR 0.01; assertions are to EUR 0.01 and to the displayed precision on rates.

Table 1 — the taux servi and the PPB, by projection year y.

`y`	`r_fin(y)`	`pm_avg_pp(y)`	0.85 × `fin_acct_pp(y)`	policyholder technical share	`pb_min_pp(y)`	`ts_stat(y)`	PPB release (+) / dotation (−)	`ppb_pp(y+1)`	`ts_net(y)`
0	3.30%	101 200.00	2 950.86	121.00	3 071.86	2.4354%	362.94	3 637.06	2.7941%
1	3.25%	105 941.25	3 027.10	132.49	3 159.59	2.3824%	412.70	3 224.36	2.7720%
2	3.20%	110 772.80	3 100.72	144.21	3 244.93	2.3294%	467.48	2 756.88	2.7514%
3	3.10%	115 696.36	3 121.24	156.14	3 277.39	2.2327%	500.00	2 256.88	2.6649%
4	2.95%	120 649.25	3 081.87	168.15	3 250.02	2.0938%	500.00	1 756.88	2.5082%
5	2.80%	124 054.88	2 994.32	176.28	3 170.60	1.9558%	500.00	1 256.88	2.3589%
6	2.65%	125 877.84	2 863.71	180.45	3 044.16	1.8183%	606.30	650.58	2.3000%
7	2.55%	127 675.06	2 781.46	184.55	2 966.01	1.7231%	650.58	0.00	2.2327%
8	2.45%	129 435.30	2 695.49	188.55	2 884.04	1.6282%	0.00	0.00	1.6282%
9	2.40%	130 580.26	2 663.84	191.01	2 854.85	1.5863%	0.00	0.00	1.5863%
10	2.35%	131 695.35	2 630.61	193.39	2 824.00	1.5443%	0.00	0.00	1.5443%
11	2.30%	132 779.35	2 595.84	195.68	2 791.51	1.5024%	0.00	0.00	1.5024%

Table 2 — the épargne acquise roll-forward, by projection year y. av_pp is read at the year’s opening month, av_pp(12y), and the closing column at av_pp(12(y+1)).

`y`	`av_pp(12y)`	`prem_to_av_pp(y)`	`withdrawals_pp(y)`	`int_credited_pp(y)`	`soc_levy_pp(y)`	`av_pp(12(y+1))`
0	100 000.00	2 400.00	0.00	2 827.60	486.35	104 741.25
1	104 741.25	2 400.00	0.00	2 936.65	505.10	109 572.80
2	109 572.80	2 400.00	0.00	3 047.77	524.22	114 496.36
3	114 496.36	2 400.00	0.00	3 083.21	530.31	119 449.25
4	119 449.25	2 400.00	0.00	3 026.13	520.49	124 354.88
5	124 354.88	2 400.00	3 000.00	2 926.27	503.32	126 177.84
6	126 177.84	2 400.00	3 000.00	2 895.19	497.97	127 975.06
7	127 975.06	2 400.00	3 000.00	2 850.54	490.29	129 735.30
8	129 735.30	2 400.00	3 000.00	2 107.43	362.48	130 880.26
9	130 880.26	2 400.00	3 000.00	2 071.36	356.27	131 995.35
10	131 995.35	2 400.00	3 000.00	2 033.83	349.82	133 079.35
11	133 079.35	2 400.00	3 000.00	1 994.84	343.11	134 131.08

The year-5 trace, at full precision, because it is the year in which every lever is active at once. Year 5 is the twelve months t = 60 … 71, and the whole of its int_credited_pp = 2 926.272987 and soc_levy_pp = 503.318954 lands in month t = 71: pm_avg_pp = 124 054.884701 (= 124 354.884701 + 1 200 − 1 500); fee_pp = 744.329308; expenses_pp = 460.046913 (= 0.0035 × 124 054.884701 + 24 × 1.015⁵); fin_acct_pp = 0.028 × (124 054.884701 + 1 756.875780) = 3 522.729293, of which 85% is 2 994.319899; tech_acct_pp = 284.282396; insurer_tech_share_pp = max(28.428, 108.000) = 108.000000 — the 4.5%-of-premiums limb binds; policyholder technical share 176.282396; pb_acct_pp = pb_min_pp = 3 170.602295; ts_stat = 1.955806%; pb_target_pp = 0.023 × 124 054.884701 + 744.329308 = 3 597.591656, so the discretionary release wanted is 426.989361, while the vintage falling due is 500.000000 — the forced release wins; pb_credited_pp = 3 670.602295; ts_net = 2.358853%; int_credited_pp = 2 926.272987; soc_levy_pp = 503.318954; av_pp(72) = 126 177.838734.

Table 3 — the twelve months of year 5 (t = 60 … 71). Eleven months move the account by the level instalments alone, EUR 200.00 in and EUR 250.00 out, netting to −EUR 50.00 a month; the twelfth carries the whole of the year’s revalorisation and levy. Note the arithmetic coincidence worth having: on a level schedule the mid-month-weighted base pm_avg_pp(5) = 124 054.8847 is literally the balance at the middle of the year, the opening of month 66.

`t`	`av_pp(t)`	`prem_to_av_mth_pp(t)`	`withdrawals_mth_pp(t)`	`int_credited_mth_pp(t)`	`soc_levy_mth_pp(t)`	`av_pp(t+1)`
60	124 354.8847	200.00	250.00	0.0000	0.0000	124 304.8847
61	124 304.8847	200.00	250.00	0.0000	0.0000	124 254.8847
62	124 254.8847	200.00	250.00	0.0000	0.0000	124 204.8847
63	124 204.8847	200.00	250.00	0.0000	0.0000	124 154.8847
64	124 154.8847	200.00	250.00	0.0000	0.0000	124 104.8847
65	124 104.8847	200.00	250.00	0.0000	0.0000	124 054.8847
66	124 054.8847	200.00	250.00	0.0000	0.0000	124 004.8847
67	124 004.8847	200.00	250.00	0.0000	0.0000	123 954.8847
68	123 954.8847	200.00	250.00	0.0000	0.0000	123 904.8847
69	123 904.8847	200.00	250.00	0.0000	0.0000	123 854.8847
70	123 854.8847	200.00	250.00	0.0000	0.0000	123 804.8847
71	123 804.8847	200.00	250.00	2 926.2730	503.3190	126 177.8387

Table 4 — decrement and cash-flow extract, pols_if_init = 1, ref_rate = 2.20%, and mort_rate(t) read from the shipped std proxy — 0.0060 at age 60, the placeholder the table is anchored to, and graded upward from there, so 0.007130 at age 62, 0.009262 at age 65 and 0.012060 at age 68. This is the table the finer grid moves, and it is printed from result_cf_annual(): the twelve months of each projection year, summed. The first two columns are read at the year’s opening month and are unchanged from the annual grid, because the monthly decrement rates compound back to the annual ones; the flow columns are not, and the annual grid’s figures are given beneath for comparison.

`y`	`lapse_rate(12y)`	`pols_if(12y)`	`premiums`	`withdrawals`	`claims_death`	`claims_lapse`	`expenses`	`liability_cf`
0	4.0000%	1.000000	2 349.24	0.00	597.66	4 046.14	370.20	2 664.76
2	8.0000%	0.910080	2 096.12	0.00	693.89	8 054.76	360.21	7 012.73
5	5.0000%	0.738099	1 723.18	2 153.97	829.63	4 561.82	330.31	6 152.55
8	6.2873%	0.613775	1 422.28	1 777.85	931.07	4 967.10	284.49	6 538.23

The same four rows on the annual grid this replaced, for comparison: premiums 2 400.00 / 2 184.19 / 1 771.44 / 1 473.06; withdrawals 0.00 / 0.00 / 2 214.30 / 1 841.32; claims_death 628.45 / 743.00 / 862.56 / 968.78; claims_lapse 4 164.51 / 8 276.62 / 4 613.46 / 4 989.75; expenses 378.20 / 375.34 / 339.56 / 294.65; liability_cf 2 771.16 / 7 210.78 / 6 258.43 / 6 621.44. Every one of those moved down, and each for a stated reason: the instalments are collected from a block that decrements every month, the expense is borne by the in force of each month rather than of the year’s first day, and a claim falls at the end of the month of exit carrying no in-year revalorisation.

Checks.

The taux servi from a different direction. ts_net(y) decomposes as 0.85·fin_acct_pp/pm_avg + (policyholder technical share)/pm_avg − fee_rate + (PPB flow)/pm_avg. At y = 5: 2.413706% + 0.142100% − 0.600000% + 0.403047% = 2.358853%, which is the table’s 2.3589%. At y = 8, with the PPB exhausted: 2.082500% + 0.145673% − 0.600000% + 0.000000% = 1.628173%, the table’s 1.6282%. Both are unchanged by the grid.

The twelve-year account identity. Summing Table 2, credited interest is EUR 31 800.82 and social levies EUR 5 469.74, and 5 469.74 / 31 800.82 = 0.172000 exactly — the levy is 17.2% of credited interest and of nothing else. Then 100 000.00 + 28 800.00 − 21 000.00 + 31 800.82 − 5 469.74 = 134 131.08, the closing balance of year 11, av_pp(144). The same total reached the other way: PB credited gross of the charge is EUR 40 538.97 and frais de gestion EUR 8 738.15, and 40 538.97 − 8 738.15 = 31 800.82. Every figure in this paragraph is the annual grid’s, unchanged: these are financial-year quantities and the twelve-year closing balance is read at month 144.

The PPB clock closes. Releases over the twelve years total EUR 4 256.88, against an opening PPB of EUR 4 000.00 plus three dotations (137.06, 87.30, 32.52) of EUR 256.88. Every opening vintage is exhausted by its due year: the release of 606.30 in year 6 clears the last EUR 500.00 vintage, the one carried in year −2, and takes EUR 106.30 from the year −1 vintage, leaving EUR 393.70 to be forced out in year 7 — which year 7’s discretionary need of EUR 650.58 more than covers, so the PPB reaches zero exactly at the clock’s last date. The clock is annual and the monthly grid leaves every figure here untouched.

The guarantee floor. With no PB at all the account falls at exactly fee_rate a year, reproducing the published minimum surrender values: 1 000 × (1 − 0.006)ⁿ gives 994.0000, 988.0360, 982.1078, 976.2151, 970.3578, 964.5357, 958.7485, 952.9960, matching Suravenir’s 994.00 … 952.99 truncated to the cent [S3]; 970 × 0.995ⁿ gives 965.1500, 960.3243, 955.5226, matching MACSF’s 965.15, 960.32, 955.52 [S2]. Here guar_floor_pp(144) = 100 000.00 + 28 800.00 − 21 000.00 − 8 738.15 = 99 061.85 against av_pp(144) + soc_levy_cum_pp(144) = 134 131.08 + 5 469.74 = 139 600.82: the floor never binds on a path with a positive taux servi throughout, and the check now looks at all 144 of those months rather than at twelve year-ends.

The aggregate roll-forward, now a monthly identity. At month t = 0, which collects one instalment and credits nothing: 100 000.00 + 200.00 − 0.00 + 0.00 − 0.00 − 50.24 − 340.11 = 99 809.65, and pols_if(1) × av_pp(1) = 0.996104 × 100 200.00 = 99 809.65. liability_cf(0) = 50.24 + 340.11 + 0.00 + 31.52 − 200.00 = 221.87, so net_cf(0) = −221.87. check_av_roll_fwd() asserts the same identity in each of the 480 months; the annual grid’s year-0 statement, 100 000.00 + 2 400.00 + 2 827.60 − 486.35 − 628.45 − 4 164.51 = 99 948.29, no longer reconstructs anything, because the flows it sums are monthly now — the year total of those monthly flows is the y = 0 row of result_cf_annual().

What year 8 is telling you. At r_fin = 2.45% and a 0.60% charge, the most the account could grow by — if the insurer distributed the whole financial account and kept only its loading margin — is 2.45% − 0.60% = 1.85%. The model credits 1.6282%, and the 0.2218-point wedge is exactly 0.15 × 2.45% = 0.3675% retained from the compte financier less the 0.1457% of the technical account that flows back to policyholders R5, art. A132-11. A 2.30% target is simply not payable on a 2.45% asset return without the PPB, and the model steps down rather than pretending otherwise; the two management actions that would soften it — realising capital gains into the year’s financial account, and the réserve de capitalisation REG-R6 — are outside this model. Years y = 0 to y = 7 credit 2.79% down to 2.23%, inside or just below the band covering 50% of encours in 2025 (2.3%–2.9% R14); y = 8 onward does not, and that step is a model result, not a market forecast.

Valuation and reserve pointers#

Gross best-estimate liability cash flows are what this library produces; valuation layers are cited, not reproduced.

French statutory provisions. The euro support’s liability is the provision mathématique — commitments valued including future management costs, which is why a French PM is not a net-premium reserve — and the PPB is a technical provision in its own right REG-R6. av_pp(t) × pols_if(t) and ppb_pp(t) × pols_if(t) are the model’s contributions to those two lines. The provision pour risque d'exigibilité REG-R7 and the provision pour aléas financiers REG-R8 REG-R9 belong to the general account behind the fund and are not computed here.
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, with EIOPA publishing the curves, the volatility adjustment and the ultimate forward rate monthly REG-R5. The euro fund’s future discretionary benefits — the PPB stock and the discretionary part of the credited rate — are the substance of its best estimate, and the crediting rule above is exactly the management action a market-consistent valuation must model. None of the Solvency II treatment of future discretionary benefits, management actions or the time value of the capital guarantee could be read from a retrieved instrument R18 REG-R2, so it is unverified here; no cost-of-capital rate or lapse shock in this library rests on a retrieved text REG-R2.
The guarantee is an option. The capital floor plus a TMG is a written put on the fund and the deterministic path above prices none of it; a stochastic-on-deterministic run — the crediting rule, the PPB lever and the dynamic surrender formula re-evaluated per scenario — is what a time-value-of-options-and-guarantees calculation consumes.
IFRS 17 and professional standards. The fonds en euros is the archetypal direct-participating contract and would be measured under the variable fee approach; the standard’s landing page confirms the fulfilment-cash-flow plus contractual-service-margin structure but the VFA mechanics were not read and are unverified REG-R45. The fulfilment-cash-flow engine is this same projection. NPA 2 applies “à tout modèle actuariel” under a proportionality principle REG-R44, and the worked example above and the pitfall tests are the documentation it asks for.

Key sensitivities and model risks#

The r_fin path dominates everything. It sets the compte financier, hence the statutory floor, hence how fast the PPB drains. A 50 bp shift in the path moves ts_stat by about 42.5 bp (0.85 × 50) and changes the year in which the PPB is exhausted by several years.
The PPB opening level and its vintage profile. 4.0% of provisions is the market ratio R14 REG-R47, but the vintage split is pure std and it decides when the eight-year clock forces a release. A fund carrying its PPB in young vintages can defer; one carrying it in old vintages cannot.
The crediting rule itself. ts_target level, the absence of a year-on-year cap, and the decision to credit the forced release rather than smooth it are all std choices with no public calibration, and they change the payout path more than any experience assumption.
The dynamic surrender parameters. a = 4.0 and tol = 0.25 point are the largest unanchored numbers in the file. Because the credited rate and the surrender rate move together — a falling ts_net raises surrenders, which shrinks the base the fund earns on — the model has a feedback loop the deterministic run only samples once.
The expense split and the 4.5%-of-premiums limb. The proportional 0.35% and the fixed EUR 24 are std and feed the compte technique, so they move the statutory floor directly; a small-balance model point is dominated by the fixed part and credits materially less. The premiums limb vanishes on a paid-up contract, leaving the insurer only 10% of the technical result, and can exceed the whole technical result on a heavily premium-paying one — two model points identical but for their premium stream credit different rates, and that is the article working as written R5.
Mortality is a timing assumption, not an amount assumption. The death benefit is the account value [S3], so the proxy basis REG-R23 REG-R24 affects only when the account is released — far less than in any protection product.
What the monthly grid corrected, and what timing uncertainty is left. The model now credits an in-year exit the contractual floor rate pro rata temporis [S1] [S2] [S11], which at a zero TMG is nil, instead of the full year’s ts_net an annual step was forced to give it. Measured on the anchor cell over forty years against the annual model it replaced, that correction and the monthly collection of instalments together move premiums −3.56%, withdrawals −4.01%, expenses −3.68%, claims_death −4.14%, claims_lapse −0.55%, int_credited −6.81% and liability_cf −1.57%; pols_if and every financial-year quantity are unchanged. What remains uncertain is which in-year rule the contract means: the Afer variant [S11], which accrues the declared rate pro rata temporis and is a documented switch rather than the base, gives claims_death −3.52%, claims_lapse +0.18% and liability_cf −0.88% instead. That bracket, not one year’s ts_net, is the size of the remaining question.
The HCSF power is unmodeled by construction. A mass-surrender scenario here produces the surrender values the contract owes, not the ones that would be paid if the freeze under art. L631-2-1 5° ter were in force R8 REG-R13.
Data provenance. The TMG is std because no contract publishes one [S1] [S2] [S11]; the avance terms are unverified because all three insurers push them into a separate document [S1] [S2] [S3]; the composition of the 17.2% levy is unverified [S3]; the capital/gain split of a partial surrender is unverified. A calibration pass against an insurer’s own PB policy and its published taux servi history REG-R31 is required before any quantitative use.