Technical Notes#

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

Scope note. These notes specify a reference liability cash-flow projection model, UC_FR_S, for the standardized composite contrat d'assurance vie multisupport defined in product-spec.md (same directory), on a monthly grid. This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in _research/assurance-vie-uc.md (carried into sources.md here); [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md (its own frozen R-numbering). std marks standardizations introduced for the reference implementation; unverified marks claims not confirmed against a retrieved document. Parameter values are identical to those in product-spec.md.

The euro leg is a pointer. UC_FR_S models the unités de compte leg. The fonds en euros enters as a single allocation share carrying an annual credited rate net of its own management charge, because that is all the UC model needs it for: the euro balance is part of the account value that sizes the capital sous risque, and it is the first source from which the garantie plancher premium is levied [S1] [S3] [S4]. Taux minimum garanti, participation aux bénéfices, the provision pour participation aux bénéfices, the effet cliquet and the euro leg’s own margin are specified and implemented in products/assurance_vie_euro/technical-notes.md (model Euro_FR_S) and are not restated, not re-derived and not re-implemented here. net_cf from UC_FR_S is therefore the UC-leg and rider result, not the contract’s total margin.

Model scope and conventions#

Purpose. Project gross liability cash flows for a single-policy model point, decomposed into the unit leg (the unités de compte — a unit count valued at an exogenous liquidation value, matched by the linked assets) and the non-unit cash flows accruing to the insurer: charges collected, less expenses and the garantie plancher death strain. The decomposition is the same one products/unit_linked_bond/ uses for a UK unit-linked bond; what is French about it is that art. A. 132-5 makes the unit count the thing guaranteed R2, so the unit leg is a deterministic sequence and every charge is a unit cancellation. Reserves are not computed (see Valuation and reserve pointers).
Projection frequency. Monthly. Sourced, not chosen: two of the seven retrieved contracts levy the UC management charge monthly [S7] [S13 art. 32.4], and the garantie plancher premium is levied monthly in arrears in three of them [S1] [S3] [S4].
Time index std. t is the policy month and is 0-based: t = 0 is the issue month, month t runs from time t to time t + 1, and the frame is t = 0, 1, …, proj_len − 1, so result_cf() has proj_len rows. The contractual policy year is the 1-based label y = t // 12 + 1; it is derived and never indexed by. The balances at issue — time 0, before month 0 opens — are not a row of the frame: they are written with an init subscript (p_init, n_init, V_init, S_init, B_init), and X_init is the opening balance of month 0 exactly as X(t−1) is the opening balance of month t for t ≥ 1.
Timing conventions std. Within month t, in order: liquidation value moves and the euro leg accrues; the UC management charge is taken on the units held at the start of the month; arbitrages and withdrawals settle; the capital sous risque is observed; the plancher premium is levied; decrements act at end of month. Settlement frictions (J+3 value dating [S10 ART 12.B], next-working-day arbitrage dating [S7], six-month deferral powers [S7]) are ignored.
Age basis. Age last birthday std; policy year y = t // 12 + 1, attained age age(t) = issue_age + t // 12, so the tariff steps at each policy anniversary. The published tariffs are quoted by the insured’s attained age at the calculation date [S4 Annexe I] and are read at age(t).
Currency and precision. EUR; full precision carried, unit counts and liquidation values reported to four decimals (au dix millième [S13 art. 32.2]), money to cents std.
Model points. Single-policy, expected (probability-weighted) basis: survivorship factors multiply per-policy cash flows. One composite UC support std (spec footnote 6).
Unit-price scenario. The liquidation value path is exogenous. The base run uses a deterministic annual UC return, the worked example a stress path. No stochastic generator is specified here and the plancher is not valued as an option — see Key sensitivities.

Model point attributes#

Attribute	Type	Example (worked configuration)
`point_id`	int	1
`issue_age`	int (age last birthday)	65 std
`sex`	enum {M, F}	M std
`premium`	currency, single premium	100,000 std
`prem_charge_rate`	rate — `frais sur versement`	0.0100 std
`uc_alloc`	share of the net premium to the UC leg	0.70 std
`euro_alloc`	share to the euro support, = 1 − `uc_alloc`	0.30 std
`unit_price_init`	liquidation value of the UC support at issue	100.00 std
`mgmt_fee_rate_uc`	annual UC `frais de gestion sur encours`	0.0088 std R13
`euro_credit_rate`	annual rate credited to the euro leg, net of the euro charge	0.0250 std — pointer to `Euro_FR_S`
`arbitrage_fee_rate`	rate on the amount switched	0.0050 [S13]
`plancher_flag`	bool — rider elected	True std (base cell); the rider itself is [S1] [S3] [S4] [S7]
`plancher_basis`	enum {`simple`, `indexee`, `cliquet`}	`simple` std (spec footnotes 14–15)
`plancher_index_rate`	annual indexation used by `indexee`	0.0350 [S1] [S3]
`plancher_ratchet_months`	ratchet period used by `cliquet`	12 std
`plancher_gross_basis`	bool — floor on gross rather than net premiums	False [S4]; True is [S1] [S3] [S13]
`plancher_end_age`	attained age at which the cover ceases	75 [S1] [S3] [S4]
`plancher_cap`	cap on the `capital sous risque`	300,000 [S1] [S3] [S4]
`plancher_levy_source`	enum {`euro_first`, `uc_units`}	`euro_first` [S1] [S3] [S4]
`wd_pattern`	enum {`none`, `one_off`, `programmed`}	`one_off`: 5,000 at t = 5 std
`arb_pattern`	enum {`none`, `one_off`, `progressive`}	`one_off`: 10,000 euro → UC at t = 2 std
`uc_return_scenario`	id of a monthly UC return path in `uc_scenario_table.csv`	`stress_yr1` std
`proj_len`	number of months projected — the exclusive end of the frame `t = 0 … proj_len − 1`	12 in the worked example; 360 in the base run std

State variables#

Variable	Description	Updated
`unit_price(t)`	Liquidation value of the composite UC support at end of month t	scenario input
`units(t)`	Number of UC units held per policy	monthly recursion
`av_uc_pp(t)`	UC account value per policy = `units(t) × unit_price(t)`	derived
`av_euro_pp(t)`	Euro-support account value per policy	monthly recursion
`av_pp_at(t, timing)`	Total account value per policy at `"OPENING"`, `"BEF_FEE"`, `"BEF_WD"`, `"BEF_LEVY"` or `"BEF_DECR"`; `"OPENING"` is the balance the month opens on, `prem_to_av_pp` when `t = 0`	derived
`av_at(t, timing)`	`av_pp_at(t, timing) × pols_if_at(t, "AFT_DECR")` — the in-force account value, a stock weighted by the end-of-month count `l(t+1)`	derived
`cum_prem_net(t)`	Floor base: cumulative premiums net of `frais sur versement`, less partial surrenders	on premium / surrender
`plancher_ratchet(t)`	Highest account value observed at a ratchet date, adjusted proportionally for surrenders (`cliquet` only)	at ratchet dates and on surrender
`plancher_amount(t)`	The floor `F(t)` under the elected basis	monthly
`nar(t)`	`Capital sous risque` — the net amount at risk	monthly
`uc_cost_basis(t)`	Cumulative net amounts invested in the UC leg, less the pro-rata cost of amounts taken out — the `prélèvements sociaux` base	on premium / arbitrage / outflow
`pols_if(t)`	In-force probability at the start of month t — the notes’ `l(t)`, and the weight on every flow of month t; `pols_if(0) = 1`	monthly decrements
`pols_if_at(t, timing)`	The same count at `"BEF_DECR"`, `"BEF_LAPSE"` or `"AFT_DECR"`; `"AFT_DECR"` is the end-of-month count, the notes’ `l(t+1)`	monthly decrements
`age(t)`	Attained age = `issue_age + t // 12`	monthly

Assumption inputs#

Three classes are distinguished explicitly, because on this product they behave very differently: (a) is thin and hard, (b) is where the insurer’s discretion lives, and (c) is where every number is the modeler’s.

(a) Contractual / guaranteed elements (cited)#

Input	Value	Basis
What is guaranteed on the UC leg	The number of units, never their value	R2; reproduced [S1] [S3] [S4 art. 17.1.1] [S7] [S10 ART 9.A] [S13 art. 32.5]
Death benefit	`max(plancher_amount, account value)` = account value + `capital sous risque`, before the cessation age	[S1] [S3] [S4]; basis choice std (spec footnotes 14–15)
Cap on the cover	`capital sous risque` ≤ 300,000 €, the excess reducing the floor	[S1] [S3] [S4]
Cessation of the cover	Attained age 75; also on total surrender or payment of the benefit	[S1] [S3] [S4] [S7] [S11]
Charge base for the rider	The `capital sous risque`, by attained age; nil when the account value is at or above the floor	[S4 Annexe I] [S3 art. 21] [S4 art. 17.1.2]
Levy order for the rider	Euro support first, then the largest UC support by cancelling units	[S1] [S3] [S4]
Surrender value	Account value across all supports; no exit charge	[S1] [S3] [S4] [S7] [S10] [S11] [S13]
Charge mechanism on UC	Percentage rates applied by cancelling units	[S1] [S3] [S4] [S7] [S13 art. 32.4] [S10 ART 12.A]
Partial surrender allocation	Pro rata across supports unless elected	[S10 ART 13.A]; default std
Effect of a surrender on the floor	Reduces the floor base by the amount surrendered	[S1] [S3] [S4 Annexe I]
Effect of an arbitrage on the floor	None — an arbitrage is neither a premium nor a surrender	[S1] [S4] [S7] [S10] [S13]
`Prélèvements sociaux` on UC	17.2%, levied only at `dénouement`	[S4 Annexe II] R8 II, 3°, c)
Unit precision	Four decimals	[S13 art. 32.2]

(b) Insurer-discretionary current elements (snapshot)#

All revisable — art. A. 132-8 requires charge maxima to be disclosed, not levels to be capped REG-R30, and MACSF may renegotiate the plancher tariff with the souscripteur if the group’s demographics or the guarantee’s technical results change [S10 ART 8.D]. The model holds the snapshot.

Input	Snapshot value	Basis
UC management charge `c`	0.88% p.a.	std anchored on the market average R13 R14 REG-R48; range 0.475%–1.50% across [S1]–[S13]
`Frais sur versement` `e`	1.00%	std; range nil–4.50% [S1] [S3] [S4] [S6]–[S8] [S10] [S11] [S13]
`Frais d'arbitrage` `φ`	0.50% of the amount switched	[S13]; level std; range nil [S11] to 2% [S10]
Plancher tariff `PA(a)`	The Spirica published table, annual premium per 10,000 € of `capital sous risque`, ages 12–74	[S4 Annexe I]; shipped as `plancher_rate_table.csv`
`euro_credit_rate`	2.50% p.a., net of the euro management charge	std — the euro leg’s `participation aux bénéfices` machinery and its citations live in `Euro_FR_S`
Indexation of an `indexee` floor	3.50% p.a.	[S1] [S3]; the PRO BTP form sets it annually at the insurer’s discretion [S12] [S13 art. 8.2]
`Gestion pilotée` surcharge	Off (0 bp); +29 bp when enabled	R13; std
Fund-level recurring costs	1.60% p.a., inside `unit_price` — not insurer income	R13; std

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

Everything in this table is std. The research file is explicit that the retrieved documents give a modeler no mortality basis (the tariffs are rate cards, and the table, age definition, loading and margin behind them are undisclosed), no surrender or arbitrage behavior, no unit-return assumption beyond the ±10% p.a. and ±50%-over-8-years disclosure conventions, and no expense basis.

Input	Recommended basis	Basis tags
Best-estimate mortality `mort_rate`	INSEE-derived std proxy, sex-distinct, single year of age, anchored so the model’s base factor reproduces the placeholder below exactly	REG-R24; anchoring std; the permitted-table rule is REG-R23
Placeholder `q` at the worked-example age	1.20% p.a. at male 65	std (1)
Mortality improvement	None in the base	std
Base surrender `lapse_rate`	Table below, with a duration-8 spike	std (2) REG-R40
Dynamic surrender multiplier	Formulas under Policyholder behavior modeling	std
Partial-surrender pattern	`one_off` 5,000 € at t = 5 in the worked cell; `programmed` = 5% of the account value a year in the base run	std
Arbitrage pattern	`one_off` 10,000 € euro → UC at t = 2 in the worked cell	std
Base UC return	4.90% p.a., the five-year average performance of UC supports net of fund charges	R13 REG-R48; use as a projection assumption std
Worked-example UC return	`stress_yr1`: +1.00% a month for months `t = 0`–`5`, −5.00% a month for months `t = 6`–`11`	std (3)
Acquisition expense	400 € per policy at issue	std
Maintenance expense	40 € per policy p.a., level	std

Order-of-magnitude placeholder consistent with the class-(c) proxy, chosen so the worked example’s decrement arithmetic is checkable by hand. It is not the mortality implied by the plancher tariff: 196 € per 10,000 € of capital sous risque at age 65 [S4] is 1.96% of the net amount at risk a year, which would be a very heavy q if it were pure risk premium — but no insurer publishes the split between mortality, expense loading and margin, so the tariff cannot be decomposed [S1] [S3] [S4] [S7]. The model therefore carries the tariff as a price and the mortality as an assumption, and the difference between them is the rider’s expected margin.
Shape rationale: the recommended holding period is 8 years [S5] [S12], and art. 125-0 A CGI makes the eighth anniversary the point at which the withholding rate falls to 7.5% and the 4,600 € / 9,200 € annual abattement becomes available REG-R40 [S4 Annexe II]. A model with no duration-8 spike has ignored the single strongest driver of French surrender timing. Level anchor: UC benefits of 32.6 bn € against UC provisions mathématiques of 666.4 bn € imply an aggregate outflow of roughly 4.9% of provisions in 2025 — a figure derived from R13, not published, and one that mixes surrenders, deaths and maturities.
A deliberate stress, not a best estimate. It is chosen so the worked example crosses the floor: the capital sous risque is zero for the first seven months and positive thereafter, which is the branch an implementation most often gets wrong.

Reference base surrender table std (annual rates on the whole contract):

Policy year	1	2–4	5–7	8	9+
`lapse_rate`	2%	4%	6%	12%	6%

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
`t`	policy month, 0-based: t = 0, 1, …, `proj_len − 1`; `y = t // 12 + 1`; `a = age(t) = issue_age + t // 12`
`X_init`	the value of a state variable at issue — time 0, before month 0 opens; the opening balance of month 0, as `X(t−1)` is the opening balance of month `t ≥ 1`
`P`, `e`	single premium (100,000) and `frais sur versement` rate (0.0100)
`α`	`uc_alloc` (0.70); the euro share is `1 − α`
`p(t)`	`unit_price(t)`, end of month t; `p_init = 100.00`
`n(t)`	`units(t)`, end of month t; `n_init = P(1−e)α / p_init`
`c`, `c_m`	UC management charge 0.0088 p.a.; `c_m = c/12 = 0.000733333` [std 1/12 convention]
`i_e`	`euro_credit_rate` 0.0250 p.a.; monthly factor `(1+i_e)^(1/12) = 1.002059836` std
`V(t)`	`av_euro_pp(t)`, the euro-support balance
`U(t)`	`av_uc_pp(t) = n(t) × p(t)`
`S(t)`	`cum_prem_net(t)`, the floor base
`F(t)`	`plancher_amount(t)`
`R(t)`	`plancher_ratchet(t)` (`cliquet` only)
`K(t)`	`nar(t)`, the `capital sous risque`
`π(a)`	`plancher_rate(a) = PA(a)/10 000`; `π(65) = 0.0196`; monthly `π(a)/12`
`W(t)`, `W_uc(t)`, `W_eur(t)`	`withdrawals(t)` and its pro-rata split
`A(t)`, `φ`	gross amount arbitraged euro → UC; `arbitrage_fee_rate` 0.0050
`B(t)`	`uc_cost_basis(t)`
`τ`	`prélèvements sociaux` rate, 0.172 [S4 Annexe II]
`q_m(t)`, `w_m(t)`	monthly mortality and surrender rates, `1 − (1 − rate_ann)^(1/12)`
`l(t)`	in force at the start of month t, `l(0) = 1`, and the weight on that month’s flows; the cells is `pols_if(t)`
`l(t+1)`	in force at the end of month t, once its decrements have gone; the cells is `pols_if_at(t, "AFT_DECR")`
`E(t)`	maintenance expense = 40/12 per month std

Dimension check: c_m, π(a)/12, q_m, w_m, τ, φ and e are dimensionless per-period rates; n(t) is a pure count; p(t) is EUR per unit; U, V, S, F, R, K, W, A, B, E are EUR; π(a)/12 × K(t) and q_m(t) × K(t) are EUR per policy-month.

Issue (time 0, before month `t = 0`)#

These are the balances the first projected month opens on. They are not a row of the cash-flow frame: nothing happens at time 0 that is not part of month 0.

prem_to_av_pp = P × (1 − e)
p_init = 100.00
n_init = prem_to_av_pp × α / p_init
V_init = prem_to_av_pp × (1 − α)
S_init = prem_to_av_pp                  [net-premium floor basis, S4 Annexe I]
R_init = av_pp_at(0, "OPENING") = prem_to_av_pp
B_init = prem_to_av_pp × α
l(0) = 1,   F_init = S_init,   K_init = 0

K_init = 0 exactly, because the net-premium floor equals the account value at issue. That is an assertable invariant, not a coincidence — see spec footnote 14.

The unit leg#

The UC management charge is taken on the units held at the start of the month and cancels units [S7] [S13 art. 32.4]. Throughout these recursions n(t−1), p(t−1) and V(t−1) mean the opening balance of month t, which in month t = 0 is n_init, p_init and V_init; the model reads them through units_open(t), unit_price_open(t) and av_euro_open_pp(t):

fee_units(t) = n(t−1) × c_m
mgmt_fee_uc(t) = fee_units(t) × p(t)            ← insurer income, in EUR
n'(t) = n(t−1) × (1 − c_m)

then the month’s events settle on the unit count:

n(t) = n'(t) + A(t)(1 − φ)/p(t) − W_uc(t)/p(t)
             − 1{plancher_levy_source = uc_units} × plancher_charge(t)/p(t)

With plancher_levy_source = euro_first the last term vanishes and the unit count is a deterministic function of the event schedule alone — market-independent, exactly as art. A. 132-5 implies R2. With no events at all it collapses to n(t) = n_init × (1 − c_m)^(t+1) — t + 1 monthly levies have fallen by the end of month t — which is the sequence the insurers publish: at 0.1875% a quarter Bourso Vie prints 100 → 99.2521 → 98.5098 → 97.7731 → 97.0418 → 96.3161 → 95.5957 → 94.8808 → 94.1711 over eight years [S3 art. 21], and at 0.25% a quarter Himalia prints 99.0037 → 98.0174 [S2].

U(t) = n(t) × p(t)

The euro leg (pointer)#

V(t) = V(t−1) × (1 + i_e)^(1/12) − A(t) − W_eur(t)
             − 1{plancher_levy_source = euro_first} × plancher_charge(t)

i_e is credited net of the euro management charge, so the euro leg produces no margin line in UC_FR_S. The euro fund’s real machinery — the participation aux bénéfices, the PPB and its eight-year vintage ledger, the effet cliquet, all of it fixed for the closing financial year and credited at 31 December — is Euro_FR_S’s. The 1/12 accrual here is a std smoothing of that annual credit across the months, and it is a simplification rather than a grid artefact: Euro_FR_S runs on the same monthly grid as this model and does not smooth, landing the whole year’s taux servi in the anniversary month. Because twelve monthly factors compound to exactly 1 + i_e, the two readings agree at every anniversary and differ only for a mid-year exit — which Euro_FR_S pays the contractual floor rate pro rata temporis and this model pays a pro-rated share of the year’s credit.

Withdrawals and arbitrages#

A partial surrender is split pro rata across the supports [S10 ART 13.A]:

av_pp_at(t, "BEF_WD") = U_before(t) + V_before(t)
W_uc(t)  = W(t) × U_before(t) / av_pp_at(t, "BEF_WD")
W_eur(t) = W(t) − W_uc(t)
S(t) = S(t−1) − W(t)                    [floor base falls by the nominal amount]

An arbitrage moves A(t) out of the euro leg, pays A(t) × φ to the insurer and invests A(t)(1 − φ) in units at p(t). It does not touch S(t).

The garantie plancher#

Floor, by plancher_basis:

simple    F(t) = S(t)
indexee   F(t) = [F(t−1) × (1 + plancher_index_rate)^(1/12)] − W(t)
cliquet   R(t) = R(t−1) × (1 − W(t)/av_pp_at(t, "BEF_WD"))          on a surrender
          R(t) = max(R(t), av_pp_at(t, "BEF_LEVY"))                 at a ratchet date
          F(t) = max(S(t), R(t))

A ratchet date falls at the end of month t, which is t + 1 months from issue, so an n-month ratchet fires where (t + 1) mod n = 0: the first annual ratchet is the end of t = 11.

The indexee recursion indexes the running floor and then deducts the nominal withdrawal, which is arithmetically identical to indexing the withdrawal forward from its own date and deducting it later — the sources’ rule that surrenders are indexed on the same basis as the floor [S1] [S3]. The cliquet adjustment is proportional, because a ratchet is a value level, not a premium tally; that is the only reason cliquet differs from simple in a year with no ratchet event, and it is asserted in the worked example.

Net amount at risk and charge:

av_pp_at(t, "BEF_LEVY") = U(t) + V(t)          (after fee, arbitrage and withdrawal)
K(t) = 0                                        if not plancher_flag or a ≥ plancher_end_age
K(t) = min(plancher_cap, max(0, F(t) − av_pp_at(t, "BEF_LEVY")))    otherwise
plancher_charge(t) = K(t) × π(a) / 12

K(t) is observed once a month and used for both the charge and the benefit; the published design observes weekly and levies monthly in arrears [S1] [S3] [S4], and the half-month timing difference is the std discretization (spec footnote 16).

Benefits and decrements#

av_pp_at(t, "BEF_DECR") = av_pp_at(t, "BEF_LEVY") − plancher_charge(t)
death benefit per death        = av_pp_at(t, "BEF_DECR") + K(t)
surrender benefit per lapse    = av_pp_at(t, "BEF_DECR")
claims_death(t)  = l(t) × q_m(t) × [av_pp_at(t, "BEF_DECR") + K(t)]
claims_lapse(t)  = l(t) × (1 − q_m(t)) × w_m(t) × av_pp_at(t, "BEF_DECR")
withdrawals(t)   = l(t) × W(t)
l(t+1) = l(t) × (1 − q_m(t)) × (1 − w_m(t))          [deaths before surrenders, **[std]**]

The whole of the account value is funded by cancelling units and by the euro balance, so the insurer’s non-unit cost per death is exactly K(t) — the capital sous risque, and nothing else.

Prélèvements sociaux#

The UC leg is taxed only at dénouement R8 II, 3°, c); the euro leg is taxed annually as interest is credited R8 II, 3°, a) and that flow belongs to Euro_FR_S. On an outflow of X from the UC leg (partial surrender, surrender or death):

B(t) = B_open(t) + A(t)(1 − φ)                        on investments, with
       B_open(t) = B_init for t = 0 and B(t−1) for t ≥ 1
gain(X) = X × (1 − B / U_before)
social_levy_uc = τ × max(0, gain(X))
B := B − B × X / U_before                              pro-rata cost removal

The levy is withheld and remitted — a pass-through, not insurer income or expense. It is reported in its own column and is excluded from net_cf. On a UC loss it is zero, and art. L. 136-7 III bis provides for restitution of an excess already levied on the euro leg where the contract’s final liquidation produces a negative base R8. Whether the plancher top-up above the account value is inside the levy base is not stated in any retrieved document; the model puts it outside and flags the treatment unverified (spec footnote 21).

Non-unit (insurer) cash flow extraction#

Cash flow	Formula	Sign	In-force weight
`Frais sur versement`	`P × e` at t = 0	+	1
UC management charge	`mgmt_fee_uc(t)`	+	`l(t)`
`Frais d'arbitrage`	`A(t) × φ`	+	`l(t)`
Plancher charge	`plancher_charge(t)`	+	`l(t)`
Plancher death strain	`K(t)`	−	`l(t) × q_m(t)`
Maintenance expense	`E(t)`	−	`l(t)`
Acquisition expense	400 std at t = 0	−	1
Account-value benefits (death, surrender, withdrawal)	funded by unit cancellation and the euro balance — no non-unit flow	0	—
Euro-leg margin	out of scope; produced by `Euro_FR_S`	0	—
Fund-level recurring costs	inside `unit_price`, accrue to the fund manager	0	—
`Prélèvements sociaux`	withheld and remitted — pass-through	0	—

net_cf(t) = l(t) × [ mgmt_fee_uc(t) + A(t)φ + plancher_charge(t)
                     − E(t) − q_m(t) × K(t) ]
            + (P·e − 400) × 1{t = 0}

The premium charge and the acquisition expense fall in the first projected month, t = 0. The issue instant is not a row of its own: the issue balances are what month 0 opens on, not a period. Because l(0) = 1 they carry their full per-policy amount. UC_FR_S is the source of truth for the placement: result_cf() has rows t = 0 … proj_len − 1 and prem_charge(0) = 1,000.00, expenses(0) = 403.33, net_cf(0) = 647.99.

net_cf is income-positive; an outgo-positive presentation is simply −net_cf(t), and no liability_cf cells is shipped.

The in-force weight, and the column that publishes it. l(t) above is the count at the start of month t, and it is what result_cf() publishes in its own pols_if column on that same row: pols_if(t) = l(t). Divide any flow on row t by that row’s pols_if and the per-policy amount comes back. The end-of-month count l(t+1) is reached through pols_if_at(t, "AFT_DECR") — the CashValue_SE timing form the library’s shared vocabulary prescribes — and it is what the account-value stock av_at(t, timing) is weighted by.

Monthly processing order std#

For month t, per policy in force at its start:

Advance y, a, E(t); read p(t) from the scenario.
Accrue the euro leg: V ← V × (1 + i_e)^(1/12).
Take the UC management charge on the opening unit count: cancel n(t−1) × c_m units, book mgmt_fee_uc(t) = n(t−1) × c_m × p(t).
Settle any arbitrage: V ← V − A(t); book A(t) × φ; buy A(t)(1 − φ)/p(t) units; add A(t)(1 − φ) to B.
Settle any withdrawal: split W(t) pro rata, cancel W_uc(t)/p(t) units, reduce V by W_eur(t), reduce S by W(t), reduce R proportionally, compute the UC gain component and the prélèvements sociaux, and remove the pro-rata cost from B.
Set av_pp_at(t, "BEF_LEVY") = U(t) + V(t); update F(t) on the elected basis; observe K(t).
Levy plancher_charge(t) = K(t) × π(a)/12 from the euro leg (or by cancelling units if plancher_levy_source = uc_units); set av_pp_at(t, "BEF_DECR").
Decrements at end of month, deaths before surrenders: book claims_death(t) at av_pp_at(t, "BEF_DECR") + K(t) and claims_lapse(t) at av_pp_at(t, "BEF_DECR"); roll l(t) forward to l(t+1).
Extract the non-unit row and accumulate net_cf(t). In month t = 0 that row also carries the frais sur versement and the acquisition expense.

Known modeling pitfalls#

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

Charging the plancher on the account value instead of on the net amount at risk. The charge base is K(t), not av_pp_at(t, ·) [S4 Annexe I]. On the worked cell at t = 11 the correct charge is 16,642.74 × 0.0196/12 = 27.18; on the account value it would be 77,357.26 × 0.0196/12 = 126.35, a factor of 4.6. Test: with the plancher out of the money the charge must be exactly zero, and sum(plancher_charge) == 0 for any path on which av_pp_at(t, "BEF_LEVY") ≥ plancher_amount(t) for all t.
Forgetting that the net amount at risk is floored at zero. max(0, F − av), not F − av. Without the floor the rider pays the insurer a negative charge (a rebate) in every rising month, and the death strain becomes negative — the model silently books the gain on the units as insurance profit.
Applying the cap to the benefit rather than to the risk. The cap is on the capital sous risque, and any excess reduces the floor [S1] [S3] [S4]; capping the death benefit at 300,000 € instead is a different, much cruder contract.
Letting an arbitrage move the floor. S(t) changes on premiums and surrenders only. An arbitrage moves value between the legs, pays a fee and leaves the guarantee untouched; in the worked example the 10,000 € switch at t = 2 leaves plancher_amount = 99,000.00.
Adjusting the cliquet floor by the nominal withdrawal. A ratchet is a value level, so it is reduced proportionally; the simple floor base is reduced nominally. In the worked example the two rules give 94,216.29 and 94,000.00 at t = 11 on the same path.
Charging the management fee on the closing rather than the opening unit count. In a month with an arbitrage the two differ by the arbitrage’s units: at t = 2 the opening-count fee is 52.28 and the closing-count fee would be 59.54. Immaterial monthly, systematic over decades, and a common source of a persistent reconciliation break against an admin system.
Using 1 − (1 − c)^(1/12) instead of c/12. The insurers compound the periodic rate: 0.25% a quarter gives an annual factor of (1 − 0.0025)^4 = 0.99003744, not 1 − 1.00% [S1] [S2]. The model uses c/12 for the same reason. Note that Suravenir’s own published table prints 100 × (1 − 0.60%) = 99.4000 after a year while a monthly 1/12 levy gives 99.4016 [S7] — the two conventions differ in the fourth decimal of the unit count, which is exactly the precision the contract guarantees [S13 art. 32.2].
Levying the plancher premium from the wrong place. With euro_first the UC unit count must be unchanged by the rider. Test: units(11) is 745.036125 under euro_first and 744.044774 under uc_units on the same path — if the two agree, the levy is not being applied at all.
Applying prélèvements sociaux to the UC leg year by year. That is the euro rule R8 II, 3°, a); the UC leg is taxed at dénouement only R8 II, 3°, c). A model that accrues the UC levy annually understates the account value throughout and overstates the charge base the management fee is levied on.
Booking the social levy, the fund-level costs or the euro credited interest as insurer cash flow. All three are pass-throughs or out of scope. On the worked cell, adding the 1.60% fund-level cost to net_cf would inflate the year’s result by 1,136.76 € against a true net_cf of 1,262.66 — both survivorship-weighted at l(t), which is the only way the two are comparable. The unweighted per-policy sum Σ av_uc_pp(t) × 1.60%/12 is 1,152.86 €; putting that figure against a weighted net_cf overstates the distortion by about 16 €, and is the same weighted/unweighted trap as the 630.20 / 621.33 split.
Reading net_cf as the contract’s total margin. It is the UC leg plus the rider. The euro leg’s margin is Euro_FR_S’s output and must be added outside this model.
Letting the guarantee run past the cessation age. K(t) is zero from attained age 75 [S1] [S3] [S4], and the tariff table stops at 74 — an implementation that extrapolates the tariff instead of switching the cover off will silently invent a price.
Treating the plancher charge as a premium for contract-boundary purposes. It is a deduction from an existing account, not a new premium; the rider is elected once at subscription and cannot be restarted [S1] [S3] [S4].

Policyholder behavior modeling#

All dynamic formulas are std reference constructions. No public French persistency or arbitrage study was retrieved, and no insurer document gives a lapse table, an arbitrage frequency or a plancher claims ratio.

Base surrender. lapse_rate(y) per the class-(c) table, converted monthly by w_m = 1 − (1 − lapse_rate)^(1/12).
Duration-8 spike std. The 12% rate at y = 8 is the tax threshold of art. 125-0 A CGI made behavioral: at eight years the withholding falls to 7.5% and the 4,600 € / 9,200 € annual abattement opens REG-R40 [S4 Annexe II], and the recommended holding period in both retrieved DICs is eight years [S5] [S12].
Performance multiplier std. M_perf(t) = min(2.0, 1 + 2.0 × max(0, g_ref − R_12m(t))) where R_12m(t) is the UC return over the twelve completed months ending at the start of month t — so it is undefined, and the multiplier exactly 1, for t < 12 — and g_ref = 4.90% R13. Poor performance raises surrenders; on the deterministic base run M_perf = 1.
Plancher moneyness multiplier std. M_pl(t) = 0.5 while K(t) > 0 and plancher_flag, else 1.0. A policyholder holding an in-the-money floor has a reason not to surrender that a UK bondholder does not — surrendering forfeits the guarantee [S1] [S3] [S4] [S11]. This is the one behavioral assumption specific to this product, it is a pure standardization, and it should be the first thing a user replaces.
Total surrender. lapse_rate(t) = min(0.35, base(y) × M_perf(t) × M_pl(t)) [std cap], where y = t // 12 + 1 is the contractual policy year.
Partial surrender. programmed: 5% of the account value a year, taken monthly and split pro rata std. Rationale: it is the pattern the eight-year tax design encourages, and it keeps the floor base falling in step with the account.
Arbitrage. progressive: a fixed monthly amount from the euro leg into UC, the investissement progressif design [S4 art. 11.2.1] [S7] [S13]. Trigger-based options (sécurisation des plus-values, limitation des moins-values) are specified in product-spec.md and are not implemented in the base recursion; they matter because they systematically move value out of UC after a rise, shrinking the management-charge base and the plancher exposure at the same time.
Renonciation. A 30-day unwind REG-R29 is a real first-month lapse effect and is carried inside the year-1 surrender rate std, not as a separate decrement.
No paid-up state. A single-premium contract carries no premium obligation.

Worked example#

Anchor cell, all parameters std per the tables above: male, issue_age 65, single premium P = 100,000 €, prem_charge_rate 1.00%, uc_alloc 0.70, unit_price_init 100.00 €, mgmt_fee_rate_uc 0.88% p.a., euro_credit_rate 2.50% p.a. net, arbitrage_fee_rate 0.50%, plancher_flag True, plancher_basis simple, plancher_end_age 75, plancher_cap 300,000 €, plancher_levy_source euro_first, plancher_rate = 196 € per 10,000 € of capital sous risque at attained age 65 [S4 Annexe I], i.e. π(65) = 0.0196 and π/12 = 0.001633333.

Events: an arbitrage of 10,000 € from the euro leg to UC at t = 2, the third month; a partial surrender of 5,000 € at t = 5, the sixth month, split pro rata.

Scenario stress_yr1 std: unit_price rises 1.00% a month for months t = 0–5 and falls 5.00% a month for months t = 6–11. Decrements: mort_rate 1.20% p.a. and lapse_rate 2.00% p.a. std, so q_m = 0.001005543 and w_m = 0.001682143. Derived monthly factors: c_m = 0.000733333, (1 + i_e)^(1/12) = 1.002059836.

Per policy in force, EUR; unit prices and counts to four decimals, money to cents. Balances are end-of-month, after that month’s levy, so each row’s av_euro_pp is the next row’s opening euro balance. The init row is the position at issue, before month t = 0 opens; it is not a row of result_av(), whose twelve rows are t = 0 … 11.

t	`unit_price`	`units`	`av_uc_pp`	`av_euro_pp`	`av_pp_at(t,"BEF_DECR")`	`plancher_amount`	`nar`	`mgmt_fee_uc`	`plancher_charge`
init	100.0000	693.0000	69,300.00	29,700.00	99,000.00	99,000.00	0.00	—	—
0	101.0000	692.4918	69,941.67	29,761.18	99,702.85	99,000.00	0.00	51.33	0.00
1	102.0100	691.9840	70,589.29	29,822.48	100,411.77	99,000.00	0.00	51.80	0.00
2	103.0301	788.0502	81,192.89	19,883.91	101,076.80	99,000.00	0.00	52.28	0.00
3	104.0604	787.4723	81,944.69	19,924.87	101,869.55	99,000.00	0.00	60.14	0.00
4	105.1010	786.8949	82,703.44	19,965.91	102,669.35	99,000.00	0.00	60.69	0.00
5	106.1520	748.3227	79,435.96	19,040.29	98,476.25	94,000.00	0.00	61.26	0.00
6	100.8444	747.7739	75,408.83	19,079.51	94,488.34	94,000.00	0.00	55.34	0.00
7	95.8022	747.2256	71,585.85	19,113.43	90,699.28	94,000.00	3,295.34	52.53	5.38
8	91.0121	746.6776	67,956.69	19,141.54	87,098.23	94,000.00	6,890.52	49.87	11.25
9	86.4615	746.1300	64,511.51	19,164.14	83,675.65	94,000.00	10,307.52	47.34	16.84
10	82.1384	745.5829	61,240.99	19,181.47	80,422.46	94,000.00	13,555.40	44.94	22.14
11	78.0315	745.0361	58,136.28	19,193.80	77,330.08	94,000.00	16,642.74	42.66	27.18
Yr 1	—	—	—	—	—	—	—	630.20	82.80

Terminal quantities at t = 11: uc_cost_basis 75,420.62, l(12) 0.968240 — the count once twelve months of decrements have gone, which is pols_if_at(11,"AFT_DECR") and not the start-of-month l(11) = pols_if(11) = 0.970848 the twelfth result_cf row is weighted at — and av_at(11,"BEF_DECR") = 77,330.08 × 0.968240 = 74,874.07.

Plancher basis variants, each run end to end on the same scenario with only plancher_basis (and, for cliquet, plancher_ratchet_months) changed:

Basis	`plancher_amount(11)`	`nar(11)`	`av_pp_at(11,"BEF_LEVY")`	year-1 `plancher_charge`
`simple`	94,000.00	16,642.74	77,357.26	82.80
`indexee`, 3.50% p.a.	97,378.25	20,041.15	77,337.10	108.39
`cliquet`, 12-month ratchet	94,216.29	16,860.46	77,355.83	84.57
`cliquet`, 1-month ratchet	98,476.25	21,155.09	77,321.17	126.04

av_uc_pp(11) is 58,136.28 in all four, because with plancher_levy_source = euro_first the rider never touches the unit count.

Insurer-side extraction, year 1 (per policy, survivorship-weighted at l(t), the start-of-month count, which is the pols_if(t) column of result_cf()):

Frais sur versement at t = 0: +1,000.00
UC management charge: +621.33
Frais d'arbitrage (10,000 × 0.50% at t = 2): +49.73
Plancher charge: +80.67
Plancher death strain (Σ l(t) q_m K(t)): −49.67
Maintenance expense (40 € p.a.): −39.41
Acquisition expense at t = 0: −400.00
net_cf year 1 = +1,262.66

The first and last of those fall in the first projected month, t = 0; the issue instant is not a row of its own, and l(0) = 1 weights them in full. That row reads prem_charge 1,000.00, mgmt_fee_uc 51.33, expenses 403.33 (400 acquisition plus 40/12 maintenance), net_cf 647.99.

Expected benefit and withdrawal flows, year 1: claims_death 1,158.20, claims_lapse 1,852.58, withdrawals 4,933.21. None of the three is a non-unit cash flow.

Settlement arithmetic. Partial surrender at t = 5: av_pp_at(5,"BEF_WD") = 83,469.22 + 20,007.04 = 103,476.25; the UC share is 0.80665095, so W_uc = 4,033.25 and W_eur = 966.75; 4,033.25 / 106.1520 = 37.9951 units are cancelled; the UC gain component is 4,033.25 × (1 − 79,250.00/83,469.22) = 203.87, and the prélèvements sociaux withheld are 17.2% × 203.87 = 35.07 [S4 Annexe II] R8 II, 3°, c); uc_cost_basis falls from 79,250.00 to 75,420.62; cum_prem_net falls from 99,000.00 to 94,000.00. Death at t = 11: the benefit is 77,330.08 + 16,642.74 = 93,972.82, of which 16,642.74 is the insurer’s strain. The UC gain is 58,136.28 − 75,420.62 = −17,284.34, so the UC social levy is zero — and any excess levied year by year on the euro leg is restituted at final liquidation under art. L. 136-7 III bis R8.

Checks.

Unit count. With no events, n(t) = n_init × (1 − c_m)^(t+1): 693.0000 × (1 − 0.000733333)^2 = 691.9840 matches row t = 1 to four decimals. Across the arbitrage, 693 × (1 − c_m)^3 = 691.4765 units survive the third month’s fee and 9,950.00 / 103.0301 = 96.5737 are bought, giving 788.0502 — row t = 2. Across the surrender, 786.3178 − 37.9951 = 748.3227 — row t = 5. From there 748.3227 × (1 − c_m)^6 = 745.0361 — row t = 11, reached without the rider touching a single unit.

Independent reproduction of published tables. The same recursion at 0.1875% a quarter gives 99.2521, 98.5098, 97.7731, 97.0418, 96.3161, 95.5957, 94.8808, 94.1711 — Bourso Vie’s printed eight-year table, digit for digit [S3 art. 21]; at 0.25% a quarter it gives 99.0037 and 98.0174 [S2]; and at an annual 0.60% on 99 units it gives 98.41, 97.82, 97.23, 96.65, 96.07, 95.49, 94.92, 94.35 — MACSF’s pre-70 table [S10 ART 12.A].

Net amount at risk, row t = 8. av_euro_pp in the table is post-levy, so the observation base is 87,098.23 + 11.25 = 87,109.48, and 94,000.00 − 87,109.48 = 6,890.52. The charge is 6,890.52 × 0.0196/12 = 11.25 — the same figure that was added back, which is the arrears convention closing on itself.

Decrements. l(12) = [(1 − q_m)(1 − w_m)]^12 = (1 − 0.012)(1 − 0.020) = 0.968240 exactly — twelve months of decrements, pols_if_at(11,"AFT_DECR") — which is the only sensible test that the monthly rates were derived geometrically rather than by dividing by twelve.

Total row. Yr 1 is the full-precision column sum rounded once (630.1985 → 630.20; 82.7961 → 82.80); adding the printed cells gives 630.18 and 82.79. The survivorship-weighted totals in the extraction above (621.33 and 80.67) are smaller because they are multiplied by l(t) < 1 from the second month on.

Euro leg. 29,700.00 × 1.025^(2/12) = 29,822.48 — row t = 1, before any event.

Valuation and reserve pointers#

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

French statutory. Art. R. 343-3 enumerates eleven technical provisions and defines the provision mathématique as the difference between the actuarial present values of the insurer’s and the insured’s respective commitments, including future management costs REG-R6. It says nothing about unités de compte, nothing about a unit count and nothing about a liquidation-value measurement, and it does not say which of the eleven provisions carries a UC engagement; no retrieved statutory or ACPR text does. The conventional reading — that the UC engagement sits in the provision mathématique and that for unités de compte that provision is the unit count at the liquidation value, which is arithmetic and is reproduced exactly by av_uc_at(t) — is therefore unverified as a statutory proposition. The one retrieved primary document that writes a provision mathématique recursion in units is MACSF’s notice, whose arts. 11–12 set out the provision and the surrender values in units with an eight-year table [S10 ART 11–12]. The garantie plancher is a separate engagement, and no retrieved ACPR or insurer document states how it is provisioned — closed-form option valuation, stochastic projection or unearned premium. This library asserts nothing about it, and a user who needs a plancher reserve must supply the method.
Solvabilité II. Technical provisions are a best estimate plus a risk margin, the best estimate being the probability-weighted average of future cash flows discounted at the relevant risk-free term structure REG-R1 REG-R4. That is stated on EIOPA’s authority: EUR-Lex could not be fetched, so no Solvency II or Delegated Regulation article number in this library was read from the instrument, and no cost-of-capital rate, no lapse shock and no expense-inflation rule here rests on a retrieved text REG-R1 REG-R2. The natural presentation is a unit reserve equal to av_uc_at(t) + av_euro_at(t) plus the non-unit best estimate of the net_cf stream — commonly negative, because future charges exceed future costs.
Mortality basis. Art. A. 335-1 permits only homologated tables (by sex, on INSEE data for non-annuity contracts) or an undertaking’s own experience table certified by an independent actuary REG-R23. TH 00-02 / TF 00-02 are cited by name and article and are not shipped; the decrement CSVs are std proxies built from INSEE’s freely redistributable series and anchored to reproduce the placeholder above REG-R24.
IFRS 17 and professional standards. IFRS 17, effective for periods from 1 January 2023, measures a group of contracts as risk-adjusted fulfilment cash flows plus a contractual service margin REG-R45; a multisupport contract is a candidate for the variable fee approach, but the VFA mechanics were not read from the standard and are unverified. NPA 2 Modèles actuariels — a category 3 pratique recommandée effective 1 January 2016, applying to “tout modèle actuariel” under a principle of proportionality — is the standard this documentation, worked example and test suite are written against REG-R44, with NPA 1 as the general assumption-setting frame REG-R43. NPA 4 (best-estimate provisions in life) was not retrieved and is the standard most directly relevant to the plancher liability.

Key sensitivities and model risks#

In order of influence on this product’s result:

The unit-return path — twice over. Every charge line scales with the account value, and the plancher cost scales with the shortfall of the account value below the floor. Those two exposures point in opposite directions and neither is symmetric: a fall cuts the management charge roughly proportionally and turns the rider on non-linearly. On the worked cell the rider costs nothing for seven months and 27.18 € at t = 11 alone. The base run is deterministic and therefore understates the plancher cost, because E[max(0, F − AV)] exceeds max(0, F − E[AV]). A stochastic or scenario-set run is not an enhancement here; it is the only way to price the rider.
Surrender behavior, and its interaction with the guarantee. A surrender extinguishes the whole future charge stream at no exit cost [S1] [S3] [S4] [S7] [S10] [S11] [S13], and it also extinguishes an in-the-money guarantee. The M_pl multiplier that ties the two together is a pure std invention with no evidence behind it, and it moves the rider’s result in both directions at once — hold the in-the-money policies and the strain rises, but so does the charge income.
The plancher tariff versus the mortality assumption. The tariff is a price [S4]; the mortality is an assumption REG-R24. Their difference is the rider’s margin, and neither the insurers’ mortality basis nor their loading is published, so the sign of that margin at any age is genuinely unknown. Sensitivity-test the tariff and mort_rate independently, never as a single “plancher basis”.
The cessation age, the cap and the charge level. Cover ceases at 75 [S1] [S3] [S4] and the tariff table stops at 74, so moving the cessation age to 80 [S12] [S13] requires a tariff the sources do not contain. The 300,000 € cap never binds on the anchor cell but binds precisely in the deep drawdowns where the guarantee is worth something. And 0.88% p.a. is a market average R13 REG-R48, not a contractual rate: there is no statutory ceiling on any French life charge REG-R30, and retrieved contract rates span 0.475% to 1.50% — a factor of three on the dominant income line.
The euro leg’s credited rate. It enters only through the account value and the levy source, but it does both: a lower credited rate makes the floor bite sooner and shrinks the balance the plancher premium is taken from, which under euro_first eventually forces the levy onto the units and makes the unit count path-dependent.
Macroprudential and liquidity tail. The HCSF may limit surrender payments for up to six consecutive months and defer or restrict arbitrages and advances REG-R13; arts. R. 131-8 to R. 131-12 govern a UC whose underlying fund gates redemptions R7. Neither is modeled, and both are why a French mass-surrender stress is a scenario, not a multiplier.
What the sources do not give, and the model therefore invents. No mortality basis, no lapse or arbitrage experience, no unit-return assumption, no expense basis, no reserving method for the plancher, and no French plancher cliquet design at all. Every one of those is std here, and the honest reading of this model is as a mechanics demonstration whose parameters must be replaced before any of its numbers mean anything.