Technical Notes#

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

Scope note. These notes specify the reference liability cash-flow projection model ADE_FR_S for the standardized composite product defined in product-spec.md (same directory). This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in sources.md, whose numbering is carried verbatim from _research/assurance-emprunteur.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. Every name that becomes a cells or a CSV column is English lower_snake_case; French terms of art are kept in French.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows — premiums, benefit outgo and expenses — for a single-policy model point of French assurance emprunteur on a monthly grid. Reserves are not computed (see Valuation and reserve pointers).

  • What this model inherits, and where it deviates. The death leg is the temporaire_deces chassis (TD_FR_S, products/temporaire_deces/technical-notes.md): the same non-annuity mortality basis — TH 00-02 / TF 00-02 with the annexed décalage d’âge REG-R22 REG-R23, never shipped, replaced by an INSEE-derived std proxy REG-R24 — the same annual-to-monthly conversion, the same “no surrender value, so lapse generates no cash flow” rule, and the same income-positive net_cf sign convention. Four deviations define ADE: (i) the sum insured is not a level capital garanti but the capital restant dû of an amortising loan, recomputed every month from the loan’s own parameters; (ii) the monthly grid the chassis also runs on is here forced by the product rather than chosen for resolution, because the benefit dominating the incapacity side is a monthly échéance and the franchise is counted in days; (iii) the state space is healthy / ITT / IPT / dead, the income_protection three-state chassis (IP_UK_S) with a fourth state and a duration-triggered forced transition; (iv) every guarantee carries its own cover-end age, so the decrements switch off at different times and the premium does not.

  • One head, one loan, deterministic amortisation. A model point is one insured life with a quotité, on one fixed-rate amortising loan whose schedule the model computes. The states are healthy, itt (indexed by claim duration z), ipt and dead, with résiliation and PTIA as additional exits from healthy. Deliberately excluded, with reasons: multi-head aggregation, because the anti-duplication rule and the quotité interaction are a portfolio-level constraint, not a per-life cash flow [S1] [S9] [S12]; perte d’emploi, a separate module with its own carence, franchise, eligibility test, duration cap and decrement that no retrieved source quantifies [S1] [S8]; and IPP and every partial benefit below the 66 % IPT threshold, because the benefit shape is not agreed across the market — a linear ramp (N − 33)/33 at two insurers [S1] [S11] against a flat 50 % at three others [S5] [S9] [S10] — so one reference number would misrepresent half of it. Mi-temps thérapeutique, garantie aide à la famille, invalidité AERAS and the exclusion buy-backs are excluded on the same grounds.

  • Time index std. t is the 0-based policy month: t = 0 is the contract’s first month, the frame is t = 0..proj_len − 1 with proj_len = loan_term_months (240 in the base cell, so the last projected month is t = 239), and the contractual policy year is the 1-based label y = floor(t/12) + 1, derived from t and never indexed by. Month t runs from time t to time t + 1. The quantities carried at a time point keep their own 0-based index k, k = 0 at adhesion: the loan balance crd(k), with crd(0) = capital_initial and crd(T) = 0, and the state probabilities l_h(k), l_itt(k, z) and l_ipt(k), with l_h(0) = 1. Month t therefore opens on crd(t) and l_h(t) and closes on crd(t + 1) and l_h(t + 1).

  • Timing and horizon std. All cover ends at the loan’s contractual expiry [S1] [S9]. Premiums arrive at the beginning of the policy month (BOM) from lives in healthy; the échéance falls at end of month (EOM), so crd(t + 1) is the principal outstanding immediately after the month-t instalment; transitions occur at EOM. Claim benefit for month t is paid at EOM to lives in a paying state at BOM t that have neither recovered nor died during the month — monthly in arrears, so a claim incepting at EOM t is first paid at EOM t+1. Death and PTIA benefits are paid at EOM t against crd(t + 1), the instalment falling on the day of death being deemed due [S9].

  • Age and duration std. age(t) = entry_age + floor(t/12); one insurer computes age by difference of calendar years [S3] and two set the rate by age at adhesion [S9] [S11], so the annual step is a pure convention. z is months since claim payment inception, i.e. since the end of the franchise, running z = 1..itt_max_months on a clock of its own that the policy-month index never touches; the 1 095-day cap [S1] [S11] [S12] gives itt_max_months = 36.

  • Units. EUR. capital_initial, crd and death benefits are amounts; echeance, prem and monthly benefits are EUR per month; rates are probabilities per period unless labelled “per mille”. Model points are projected on an expected basis; an in-force portfolio needs claims-in-payment cells carrying status and claim_duration_months.

Model point attributes#

Attribute

Type

Example (worked configuration)

point_id

int

1

entry_age

int

52 std (spec footnote 1)

sex

enum {M, F}

M std (spec footnote 1); tariffs are sex-rated except where unisex [S3]

capital_initial

currency

200,000 std (spec footnote 1); definition [S9]

loan_rate_annual

float (taux nominal; monthly rate = /12)

0.0300 std (spec footnote 2)

loan_term_months

int

240 std (spec footnote 1); band 1–35 years [S1] [S9]

quotite

float, 0 < q ≤ 1, 1 % steps

1.00 std (spec footnote 1); 1 % steps [S9], ≤100 % per head [S1] [S5] [S9] [S11]

premium_basis

enum {capital_initial, capital_restant_du}

capital_initial [S9] [S11] [S13]; alternative [S2] [S7] [S8] [S10]

premium_rate_annual

float (used when premium_basis = capital_initial)

0.0084 std (spec footnote 7)

indemnity_basis

enum {forfaitaire, indemnitaire}

forfaitaire [S1] [S3] [S6] [S11]; alternative [S10]

income_loss_ratio

float ≤ 1 (used when indemnity_basis = indemnitaire)

1.00 std (9)

franchise_days

enum {30, 60, 90, 120, 180}

90 [S9]; pick std (spec footnote 4)

itt_max_days

int

1095 [S1] [S11] [S12]

ipt_benefit_basis

enum {echeance, crd}

echeance [S5] [S9] [S11]; alternative [S1] [S2] [S7]

deces_end_age

int

85 [S9] [S11]

ptia_end_age

int

70 [S9] [S11]

itt_ipt_end_age

int

70 [S9]

status

enum {healthy, itt, ipt}

healthy

claim_duration_months

int (in-claim cells only)

0

Occupation class and smoker status are not attributes. They are real tariff drivers [S1] [S11], but no public French table is graded by them and no rate card was retrieved; adding a column the shipped rate tables cannot serve would produce model points that do not project.

State variables#

Variable

Description

Updated

crd(k)

Capital restant dû at time k, i.e. after the k-th instalment

at each instalment, deterministic

echeance

Level monthly loan instalment, capital and interest

once, at issue

prem_pp(t)

Monthly premium per policy in force

at each policy anniversary

l_h(k)

Probability in healthy at time k (alive, no claim in payment); l_h(0) = 1

monthly

l_itt(k, z)

Probability in ITT payment at time k at claim duration z

monthly, two-dimensional

l_itt(k)

Total ITT probability = Σ_z l_itt(k, z)

derived

l_ipt(k)

Probability in IPT payment at time k

monthly

n_itt(t)

New ITT claim-payment inceptions in month t (seeds z = 1)

monthly

rec_itt(t), trn_ipt(t), dth_itt(t)

Exits from ITT: recoveries, transitions to IPT, deaths in claim

monthly

cap_itt(t)

ITT mass reaching the 1 095-day assessment at EOM t

monthly

dth_h(t), ptia_h(t), lapses(t)

Exits from healthy: deaths, PTIA claims, résiliations

monthly

dth_ipt(t)

Deaths in IPT

monthly

There is no account value, no surrender value and no unit fund: the state is the insured population plus the deterministic loan schedule. l_h(t) + l_itt(t) + l_ipt(t) + Σ_{s < t} (dth_h + ptia_h + lapses + dth_itt + dth_ipt)(s) = 1 for every t — the states at time t plus everything that has left in months 0 .. t − 1. That identity is check_states().

Assumption inputs#

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

Input

Value

Basis

Loan spine

echeance and crd(k) computed from capital_initial, loan_rate_annual, loan_term_months

read from the échéancier contractually [S1] [S5] [S9]; computed here std (spec footnote 3)

Décès / PTIA benefit

crd(t + 1) × quotite, the balance at the end of month t

[S1] [S5] [S9] [S10] [S11]

ITT / IPT benefit

echeance × quotite, monthly

[S1] [S9] [S11]

Franchise

90 days, embedded in the inception basis

[S9]; pick std (spec footnote 4)

ITT duration cap

1 095 days = 36 months, then a forced consolidation assessment

[S1] [S11] [S12]; consolidation ≤3 years [S10]

IPT threshold

Combined invalidity ≥ 66 % on the barème croisé

[S1] [S5] [S9] [S10] [S11] [S12]

Cover-end ages

Décès 85, PTIA 70, ITT/IPT 70

Décès 85 and PTIA 70 [S9] [S11]; ITT/IPT 70 [S9] alone — MAIF ends ITT/IPT/IPP at 67 [S11]; ranges in the spec

Premium waiver in claim

No premium from lives in itt or ipt

[S5] [S11]; advance-and-refund elsewhere [S9]

Premium levelling

The premium does not fall when the PTIA/ITT/IPT guarantees cease

[S13]

Résiliation

Cancellation at any time from signature of the loan offer; no surrender value

R1 R3 REG-R35

Expiry

All cover and any claim in payment cease at the loan’s contractual expiry

[S1] [S9]

(b) Insurer-discretionary current elements#

Thin by construction: no participation aux bénéfices is credited to the individual contract, there is no bonus and no account value, and the net-of-tax premium is guaranteed for the whole term [S1] [S11]. What discretion exists is recorded, not projected. Premium revision — one insurer revises downward on a risk-reducing change of life habits and may revise the perte d’emploi rate only, at renewal, with three months’ notice [S1]; two others pass on tax changes [S5] [S11]; base model: no revision std. Underwriting outcome — standard terms, a surprime and/or guarantee restrictions, or refusal [S1]; the model projects a standard-terms life only std, the surprime being a rate multiplier rather than a mechanic. One published scale does exist — the AERAS grille de référence states a capped surcharge per guarantee for its list II pathologies R17 — but the grid itself was not retrieved here (spec, Underwriting) and its rates must not be invented; no insurer’s standard-risk rate card is public at all, so there is no base rate to multiply. Claim adjudication — declines are material, 2.5 %–4.4 % on death/PTIA and 7.7 %–16.3 % on incapacity/invalidity for external alternative contracts against 2.5 %–3.8 % and 10.2 %–12.8 % for bank group contracts R12; the base admits every claim std, and a portfolio calibration should scale ben_itt and ben_ipt by an admission ratio. The 2025 garantie aide à la famille is a market undertaking, not a priced element, and is out of scope [S9] R12.

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

No decrement, incidence or termination table for this product was retrieved. Nothing in the corpus gives a mortality basis, an ITT inception rate, a recovery rate or an ITT → IPT transition rate: insurer rate cards are proprietary, the CCSF publishes tariff levels only as chart series R12, and the homologated mortality tables are cited by name but are not redistributable REG-R22 REG-R23. Every rate below is therefore std. The tables are shaped like the quantities a real basis would carry, so licensed tables drop in without changing the recursions.

Input

Reference basis

Basis tags

Healthy-life mortality mort_rate(a)

std proxy table below (male); female = 0.60 × male

values std (1); table names REG-R22 REG-R23; data REG-R24

PTIA incidence ptia_rate(a)

0.10 × mort_rate(a)

ratio std (2)

ITT inception itt_inception_rate(a)

std proxy table below × franchise_factor × sex_factor

values std (3)

ITT termination by duration

std proxy table below: recovery / IPT transition / death in claim

values std (4)

1 095-day assessment split ipt_share_at_cap

0.35 to IPT, 0.65 back to healthy

value std (4)

Mortality in IPT ipt_mort_factor

3.0 × mort_rate(a)

value std (5)

Résiliation lapse_rate(y)

std table below, by policy year

values std (6)

CRD-basis premium scale

std table below, annual rate on the CRD by attained age

values std (7)

Maintenance expense

EUR 30 per policy per year, inflating 1.8 %/yr

std (8)

Claim management expense

EUR 250 per year per claim in payment, inflating 1.8 %/yr

std (8)

income_loss_ratio (indemnitaire only)

1.00 in base

std (9)

Claim admission ratio

1.00 in base

std (10); observed declines R12

Discount

EIOPA risk-free term structure for valuation REG-R5; flat 2.5 %/yr in the worked example

rate std (11)

  1. INSEE population mortality is the only freely redistributable French series and is the data source behind every decrement CSV this library ships REG-R24. It is heavier than medically-selected insured experience, so the proxy overstates death cost for a standard-risk book and understates it for a Lemoine-waiver book written with no medical selection at all. The 0.60 female factor is a pick; one insurer is unisex instead [S3].

  2. No public French PTIA incidence rate exists. PTIA pays the same benefit as Décès and is a subset of severe morbidity, so it is a fixed fraction of the death rate. The ratio matters mainly through the different cover-end ages: above ptia_end_age the PTIA decrement is off while Décès continues.

  3. Shaped as a claim-payment inception rate specific to the franchise, as a real basis would be published per deferred period. franchise_factor = 1.60 / 1.25 / 1.00 / 0.85 / 0.65 for 30 / 60 / 90 / 120 / 180 days and sex_factor = 1.00 male / 1.30 female, both std placeholders; the menu itself is sourced [S9].

  4. Falling recovery and rising IPT transition with duration is the qualitative structure of any disability termination basis: short claims mostly recover, long claims mostly consolidate. The 0.35 split at the cap stands in for the medical assessment against the 66 % barème croisé threshold [S1] [S9]; nothing public quantifies what fraction of three-year ITT claims clears 66 %.

  5. Claimant mortality above healthy-life mortality is universal in disability experience; the ×3.0 factor has no French anchor.

  6. The behavioural heart of the product — see Policyholder behavior modeling.

  7. A rate on the outstanding balance re-read annually with the attained age [S2] [S5] [S7] [S8], calibrated so its present value over the base cell matches the level 0.84 % scale to 0.11 % (Checks) and the margin over the std benefit basis is about 10 %.

  8. No French ADE expense study was retrieved. EUR 30/policy/year is about 1.8 % of the base cell’s annual premium; the claim expense reflects that an incapacity claim is medically managed, unlike a death claim. Both are placeholders.

  9. Indemnitaire contracts cap the benefit at the actual income loss, revenu de référence less revenu de remplacement [S10]. Modeling that properly needs a distribution of employer sick pay and prévoyance cover across the book, which nothing retrieved supplies. At 1.00 the indemnitaire cell equals the forfaitaire cell, which is the honest base — the model exposes the lever rather than inventing its value.

  10. Claims are admitted in full because the model has no way to distinguish an admitted claim from a declined one: the only public French figures are portfolio decline rates by guarantee and contract type R12, with no split between late notice, cover-age breach and medical dispute. A portfolio calibration should set the ratio from its own claims register.

  11. No numeric EIOPA curve value was extracted anywhere in this library, so the worked example’s discount rate is a flat modeling convention REG-R5; it affects only the present values quoted in Checks, never the projected cash flows.

std proxy healthy-life mortality (annual, per mille, male; linear interpolation between pivot ages; female = 0.60 × male):

Age a

30

35

40

45

50

55

60

65

70

75

80

85

mort_rate ‰

0.6

0.8

1.2

2.0

3.2

5.0

7.6

11.5

17.5

27.0

45.0

78.0

std proxy ITT claim-payment inception rates (annual, per mille of lives in healthy; male, franchise 90 days; linear interpolation between pivot ages):

Age a

30

35

40

45

50

55

60

65

69

itt_inception_rate ‰

2.0

2.8

4.0

6.0

9.0

13.5

20.0

28.0

36.0

std proxy ITT termination rates (annual, by claim duration year since payment inception; the three exits compete in the stated order):

Claim duration year (months z)

1 (1–12)

2 (13–24)

3 (25–36)

Recovery itt_recovery_rate

0.55

0.30

0.15

Transition to IPT itt_to_ipt_rate

0.02

0.06

0.12

Death in claim itt_mort_rate

0.02

0.03

0.04

std résiliation table (annual rates from healthy; lives in itt or ipt do not lapse — premiums are waived and the benefit is in payment std):

Policy year y

1

2

3

4–5

6+

lapse_rate

4 %

12 %

12 %

10 %

7 %

std CRD-basis premium scale (annual rate applied to the CRD at the policy anniversary, by attained age; linear interpolation; used only when premium_basis = capital_restant_du):

Age a

30

35

40

45

50

55

60

65

70

Rate

0.14 %

0.18 %

0.26 %

0.40 %

0.62 %

0.95 %

1.45 %

2.10 %

2.90 %

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol

Meaning

t

policy month, 0-based: t = 0..T − 1, T = loan_term_months (240 in the base cell)

k

a time point, 0-based: k = 0 at adhesion, k = t opens month t and k = t + 1 closes it

y, a

policy year floor(t/12) + 1; attained age entry_age + y − 1

i

monthly loan rate = loan_rate_annual / 12 = 0.0025

ech

échéance = capital_initial × i / (1 − (1 + i)^(−T)) = 1 109.1952

crd(k)

ech × (1 − (1 + i)^(−(T − k))) / i, k = 0..T; crd(0) = capital_initial, crd(T) = 0

Q

quotite (1.00 base)

IR

indemnity ratio: 1.00 if indemnity_basis = forfaitaire, else income_loss_ratio

mth(r)

annual-to-monthly conversion = 1 − (1 − r)^(1/12) std (12)

q_h, q_ptia

monthly mth(mort_rate(a)), mth(ptia_rate(a))

w

monthly mth(lapse_rate(y))

ι

monthly mth(itt_inception_rate(a) × franchise_factor × sex_factor)

ρ(z), τ(z), q_s(z)

monthly recovery, IPT-transition and death-in-claim rates at duration z

s_itt(z)

monthly ITT persistency = (1 − ρ(z)) × (1 − τ(z)) × (1 − q_s(z))

q_ipt

monthly mth(min(ipt_mort_factor × mort_rate(a), 1))

D(t), P(t), I(t)

guarantee-in-force indicators for Décès, PTIA, ITT/IPT

e_m(y), ec_m(y)

monthly maintenance and claim expense = 30/12 and 250/12, × 1.018^(y−1)

v(t)

discount factor for month t, whose flows fall at time t + 1 (valuation: EIOPA curve REG-R5; worked example 1.025^(−(t+1)/12), footnote (11))

Dimensional check: q_h, q_ptia, w, ι, ρ, τ, q_s, q_ipt are monthly probabilities; crd and death benefits are EUR; ech, prem and monthly benefits are EUR per month; every cash flow below is EUR per month per policy issued.

  1. Every annual rate in the tables above is converted with the same uniform-force approximation 1 − (1 − r)^(1/12), which makes each monthly rate strictly below its annual rate and keeps the twelve monthly survival factors multiplying back to the annual one. No retrieved source states a conversion convention for any French decrement.

The loan spine and the guarantee indicators#

crd is computed, never read from a table. Two equivalent forms, both of which the model must satisfy to floating-point tolerance — that is check_crd():

crd(k) = ech x (1 - (1 + i)^(-(T - k))) / i
crd(k) = crd(k - 1) x (1 + i) - ech        with crd(T) = 0 exactly

crd is the only thing linking the loan to the insurance: the death benefit in month t is crd(t + 1) × Q, the disability benefit is ech × Q. The guarantee indicators are

D(t) = 1 if a < deces_end_age else 0       (85 base cell -> in force for all t)
P(t) = 1 if a < ptia_end_age  else 0       (70 base cell -> t <= 215)
I(t) = 1 if a < itt_ipt_end_age else 0     (70 base cell -> t <= 215)

At the first month where I(t) = 0, at BOM and before any transition, all l_itt and l_ipt mass moves into l_h: cover has ended, benefit stops, and those lives remain alive, death-covered and premium-paying [S13]. ι is zero from that month.

Premium#

prem(t) = prem_pp(y) x l_h(t)                                           (BOM)

re-read at each policy anniversary (t ≡ 0 mod 12), lives in itt/ipt paying nothing (waiver [S5] [S11]) and prem_pp never falling when P(t) or I(t) does [S13]:

premium_basis = capital_initial    : prem_pp(y) = capital_initial x Q x premium_rate_annual / 12
premium_basis = capital_restant_du : prem_pp(y) = crd(12 x (y - 1)) x Q x crd_rate(a) / 12

Transitions (EOM)#

Out of healthy, in the order death, PTIA, résiliation, ITT inception std:

dth_h(t)  = l_h(t) x q_h
ptia_h(t) = l_h(t) x (1 - q_h) x q_ptia x P(t)
lapses(t) = l_h(t) x (1 - q_h) x (1 - q_ptia x P(t)) x w
n_itt(t)  = l_h(t) x (1 - q_h) x (1 - q_ptia x P(t)) x (1 - w) x i_rate,  i_rate = ι x I(t)
h_stay(t) = l_h(t) x (1 - q_h) x (1 - q_ptia x P(t)) x (1 - w) x (1 - i_rate)

Out of ITT, in the order recovery, transition to IPT, death in claim std, for each duration cohort z:

rec_itt(t, z) = l_itt(t, z) x rho(z)
trn_ipt(t, z) = l_itt(t, z) x (1 - rho(z)) x tau(z)
dth_itt(t, z) = l_itt(t, z) x (1 - rho(z)) x (1 - tau(z)) x q_s(z)
stay(t, z)    = l_itt(t, z) x s_itt(z)

For z < itt_max_months the survivors advance, l_itt(t+1, z+1) = stay(t, z). For z = itt_max_months (36, the 1 095-day cap) they are assessed instead of advanced: cap_itt(t) = stay(t, itt_max_months), of which ipt_share_at_cap passes to IPT and the remainder returns to healthy. Out of IPT there is no recovery — the only exits are death and the age limit: dth_ipt(t) = l_ipt(t) × q_ipt, ipt_stay(t) = l_ipt(t) − dth_ipt(t). State update, from the states opening month t to those closing it:

l_h(t+1)     = h_stay(t) + SUM_z rec_itt(t, z) + (1 - ipt_share_at_cap) x cap_itt(t)
l_itt(t+1,1) = n_itt(t);  l_itt(t+1, z+1) = stay(t, z)  for z < itt_max_months
l_ipt(t+1)   = ipt_stay(t) + SUM_z trn_ipt(t, z) + ipt_share_at_cap x cap_itt(t)

Recovered lives return to healthy and are again exposed to inception std. When ipt_benefit_basis = crd, IPT is not a state at all: the mass that would enter IPT instead triggers a single payment crd(t + 1) × Q and leaves the model, exactly as a death does [S1] [S2] [S7].

Benefit outgo, expenses and net cash flow (EOM)#

ben_deces(t) = crd(t+1) x Q x (dth_h(t) + SUM_z dth_itt(t,z) + dth_ipt(t)) x D(t)
ben_ptia(t)  = crd(t+1) x Q x ptia_h(t)
ben_itt(t)   = ech x Q x IR x SUM_z stay(t, z)
ben_ipt(t)   = ech x Q x IR x (ipt_stay(t) + SUM_z trn_ipt(t, z))
expenses(t)  = e_m(y) x (l_h + l_itt + l_ipt)(t) + ec_m(y) x (l_itt + l_ipt)(t)

liability_cf(t) = ben_deces + ben_ptia + ben_itt + ben_ipt + expenses - prem
net_cf(t)       = -liability_cf(t)

ben_itt includes the capped cohort — a life in ITT throughout month t is paid for that month whether it then stays, passes to IPT at the cap, or returns to healthy — and ben_ipt includes lives that transitioned at EOM t, so the ITT → IPT move creates no unpaid month. New inceptions n_itt(t) are not paid for month t. Death, PTIA, résiliation and expiry generate no other payment: no surrender value, no maturity benefit.

Monthly processing order std#

At month t = 0..T − 1 (nothing survives the frame; at t = T − 1, the last month, all cover and any claim in payment terminate without value [S1] [S9]):

  1. Anniversary (BOM, t = 0, 12, 24, …): advance y and a; set D(t), P(t), I(t); re-read prem_pp(y) on the CRD basis, leave it unchanged on the capital initial basis.

  2. Guarantee-cessation transfer (BOM): if I(t) = 0 and any l_itt/l_ipt mass remains, move all of it into l_h and zero those states.

  3. Premium income (BOM): prem(t) = prem_pp(y) × l_h(t).

  4. Loan instalment (EOM): the schedule carries crd(t) to crd(t + 1) — deterministic, unaffected by any decrement.

  5. Transitions out of healthy (EOM): death, PTIA, résiliation, ITT inception.

  6. Transitions out of ITT (EOM): recovery, IPT transition, death in claim, per duration cohort; then the 1 095-day assessment on cohort z = itt_max_months.

  7. Transitions out of IPT (EOM): death.

  8. State update to time t + 1 for l_h, l_itt(·, z), l_ipt.

  9. Benefit outgo (EOM): ben_deces, ben_ptia, ben_itt, ben_ipt.

  10. Expenses (EOM), then discount at v(t) and accumulate.

Known modeling pitfalls#

Each of these produces a model that looks right and is wrong. They are the test list.

  • Reading the CRD from a table instead of computing it. The whole product hangs off crd; a pasted schedule will not satisfy crd(k) = crd(k−1) × (1 + i) − ech at every k and crd(T) will not be zero. Assert both.

  • Using the wrong CRD, or the wrong rate conversion. crd(t) (opening month t, before its instalment) and crd(t + 1) (closing it, after the instalment) differ by the month’s capital repayment — EUR 609.20 over month t = 0 in the base cell; the convention here is the closing balance crd(t + 1) and whichever is chosen must be used everywhere. A model that indexes crd on the month rather than on the time point pays a whole month’s capital too much or too little. Separately, French loans quote a taux nominal annuel whose monthly rate is nominal ÷ 12, not (1 + nominal)^(1/12) − 1; the effective conversion changes ech, and therefore every benefit and the TAEA.

  • Collapsing Décès and PTIA into one decrement. They pay the identical benefit, so the temptation is strong — and it is wrong, because deces_end_age (85) and ptia_end_age (70) differ. A collapsed decrement either pays PTIA after 70 or stops paying death before 85.

  • Letting the premium fall when the ITT/IPT guarantees cease. The rate is nivelé: the cover shrinks at 70 and the premium does not [S13]. In the base cell that is 24 months × EUR 140.00 = EUR 3 360.00 of premium per surviving policy against death cover alone; a model that switches the premium off with the guarantee understates premium income by exactly that. The mirror error is letting ben_itt or ben_ipt run past the age limit: both must be exactly zero wherever I(t) = 0, and the in-claim mass must be moved, not deleted — deleting it breaks the state identity and destroys the death cover those lives still hold.

  • Collapsing the ITT duration dimension, or dropping the cap. A single ITT bucket with duration-independent terminations misstates runoff badly — the proxy recovery rate falls 0.55 → 0.15 while the IPT transition rate rises 0.02 → 0.12 across the three duration years — and the 1 095-day assessment cannot be expressed at all without z. If cohort z = 36 simply advances to z = 37, ITT claims run for ever and IPT is never fed from the cap: in the base cell that is 35 % of the 0.198077 of each inception still in ITT at three years.

  • Paying the ITT → IPT movers twice, or not at all. A life moving from ITT to IPT at EOM t must be paid exactly once for month t. Assert ben_itt(t) + ben_ipt(t) = ech × Q × IR × (l_itt(t+1) − n_itt(t) + l_ipt(t+1) + (1 − ipt_share_at_cap) × cap_itt(t)) — the paying mass equals the closing disabled mass, less the month’s new inceptions, plus the share of the capped cohort sent back to healthy: those lives were in ITT throughout month t and are paid for it, but they end the month in neither disabled state, so an identity written without that term is short by ech × Q × IR × (1 − ipt_share_at_cap) × cap_itt(t) — up to EUR 0.13 a month in the base cell. Relatedly, benefit is monthly in arrears: including n_itt(t) in ben_itt(t) pays a full month at the instant of inception.

  • Charging premium to lives in claim, or lapsing them. Premiums come from l_h only [S5] [S11]; prem_pp × (l_h + l_itt + l_ipt) overstates premium income and is easy to write by accident when the model also tracks total lives in force. Symmetrically, applying the résiliation decrement to l_itt/l_ipt silently cancels claims in payment.

  • Quotité applied twice, or to the wrong leg. quotite scales the benefit and the premium, once each. Applying it to the CRD and again to the benefit is invisible at quotite = 1.00 — the base cell will not catch it. Test at quotite = 0.60. For the same reason indemnitaire must be the same formula with IR < 1, not a second benefit expression that can drift from the forfaitaire leg.

  • Assuming the “decreasing” premium decreases. On the CRD basis at entry age 52 the premium rises from EUR 125.33 in policy year 1 to a peak of EUR 164.03 in year 10 before falling to EUR 31.65 in year 20, because the attained-age rate climbs faster than the CRD falls. A monotonicity assertion on the CRD-basis premium will fail — correctly.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions. No public French ADE policyholder-behaviour study was retrieved, and the CCSF’s published series are counts of substitution requests, not portfolio lapse rates R12.

  • Base résiliation std. lapse_rate(y) per the table above; monthly w = 1 − (1 − lapse_rate)^(1/12), applied to l_h only. The shape — low in year 1, tripling in year 2, then decaying to a 7 % ultimate — is a reading of the statutory mechanics rather than of data. Cancellation is available à tout moment from signature of the loan offer R1 R3 REG-R35, so nothing legal holds year 1 down; what holds it down is that the borrower has just signed, the lender’s ten-business-day answer and the substitute’s effective-date rule put real friction in the path R1 R3 R8, and the fiche standardisée has only just been read R4 R5. From year 2 the insurer’s own annual reminder of the cancellation right arrives R1, brokers solicit, and the substitution machine engages. This is materially higher than a classic protection lapse: substitution requests to banking networks rose from 99 265 in H1 2021 to 181 600 in H1 2023, the alternative share of insured portfolios rose from 15.3 % to 16.0 % between 2021 and May 2023, and about 215 000 external alternative contracts were added in 17 months R12.

  • Substitution as a response to the premium gap std. Résiliation here is not a lapse in the ordinary sense — the cover does not stop, it moves — so the rate is made to respond to the gap between the premium in force and the price of an equivalent contract in the market:

    gap(y)   = max(0, prem_pp(y) / market_prem_pp(y) - 1)
w_dyn(y) = min(w_max, lapse_rate(y) x (1 + beta x gap(y)))

    with beta = 3.0 and w_max = 0.35 std and market_prem_pp a scenario input (base: equal to prem_pp, so gap = 0 and w_dyn = lapse_rate). The sensitivity is deliberately steep: the CCSF measured bank group tariffs falling 14 %–30 % across the age range between 2019 and 2023 while medically-selected external alternative contracts moved between −40 % and +16 % by age R12, so a book written at an older tariff faces a two-digit gap without doing anything. Acceptance is not automatic — lenders accept 88 %–90 % of requests through banking networks and 70 %–87 % through intermediaries R12 — so an acceptance ratio of 0.88 std multiplies the substitution component, refused requests remaining in force R3.

  • Selective withdrawal [std note]. Substitution requires an equivalent-guarantee offer from another insurer R7 REG-R36, which on a medically-selected contract means passing underwriting again: healthy lives leave, impaired lives stay, and the residual book’s morbidity is worse than the table. The socio-professional skew is consistent — CSP1 are 58 % of substitutions and 69 % of external alternative contracts taken at origination but only 27 % of the banks’ mortgage portfolios R12. The base applies no anti-selection loading std; the scenario lever is itt_inception_rate × (1 + selection_load).

  • Lapse in claim is zero std — premiums are waived and the benefit is in payment [S5] [S11]. Early repayment ends the cover if total [S1] [S9] and rebases the premium on the guaranteed CRD less the amount repaid if partial [S9] [S10]; the base holds the amortisation schedule fixed std, an early-repayment decrement being a documented extension. The questionnaire waiver and the droit à l’oubli are underwriting effects, not projection decrements: they change who is in the book and at what rate, not how a policy runs. The waiver applies only where the insured share of the cumulative encours is ≤ EUR 200 000 and repayment falls before the 60th birthday R1 R2, which is narrow — 58.5 % of borrowers were under the amount threshold but only 23 % of those contracts were eligible, and contracts without medical selection are only 31 % of substitutions R12. Where it does apply, external alternative contracts without medical selection were repriced up by about 10 % on average against 2021 tariffs R12 — an anti-selection loading with a public number. The base carries no such loading and no underwriting_basis column std.

Worked example#

The base cell, run from issue for 15 months. Configuration: male, entry_age 52, capital_initial EUR 200 000, loan_rate_annual 3.00 % (i = 0.0025), loan_term_months 240, quotite 1.00, premium_basis = capital_initial, premium_rate_annual 0.84 %, indemnity_basis = forfaitaire (IR = 1.00), franchise_days 90, itt_max_days 1 095, ipt_benefit_basis = echeance, deces_end_age 85, ptia_end_age 70, itt_ipt_end_age 70. All decrements are the std proxy tables above and the discount is 2.5 %/yr flat std (used only for present values in Checks); every assumption is std except the guarantee structure, the benefit bases and the age limits, which are [S9] [S11] [S13] as tabulated in the spec.

Derived constants: ech = 1 109.1952; prem_pp = 200 000 × 1.00 × 0.0084 / 12 = 140.00. Monthly rates at a = 52: q_h = 0.000327255, q_ptia = 0.000032673, ι = 0.000904486, w = 0.003396053 in year 1 and 0.010596241 in year 2, each from 1 − (1 − r)^(1/12) on the annual rates 0.00392, 0.000392, 0.01080, 0.04 and 0.12. Duration-year-1 terminations: ρ = 0.064376669, τ = q_s = 0.001682143, so s_itt = 0.932478274. At a = 53, q_h = 0.000357368 and ι = 0.000980268.

Months are 0-based, so the first row is t = 0 (policy year y = 1 throughout the first twelve). The CRD and state columns are the closing quantities of month t, i.e. the values at time t + 1; the cash flow columns are the flows of month t.

t

crd(t+1)

l_h(t+1)

l_itt(t+1)

l_ipt(t+1)

prem(t)

ben_deces(t)

ben_ptia(t)

ben_itt(t)

ben_ipt(t)

0

199,390.80

0.995344

0.000901

0.000000

140.00

65.25

6.51

0.00

0.00

1

198,780.09

0.990768

0.001737

0.000001

139.35

65.03

6.46

0.93

0.00

2

198,167.84

0.986267

0.002513

0.000004

138.71

64.79

6.41

1.80

0.00

3

197,554.07

0.981837

0.003232

0.000008

138.08

64.54

6.36

2.60

0.01

4

196,938.76

0.977474

0.003898

0.000013

137.46

64.28

6.32

3.34

0.01

5

196,321.91

0.973174

0.004516

0.000019

136.85

64.01

6.27

4.03

0.02

6

195,703.52

0.968933

0.005088

0.000026

136.24

63.72

6.22

4.67

0.03

7

195,083.58

0.964750

0.005617

0.000034

135.65

63.42

6.17

5.26

0.04

8

194,462.09

0.960620

0.006107

0.000043

135.06

63.12

6.13

5.81

0.05

9

193,839.05

0.956540

0.006561

0.000053

134.49

62.81

6.08

6.32

0.06

10

193,214.46

0.952509

0.006980

0.000063

133.92

62.48

6.04

6.79

0.07

11

192,588.30

0.948524

0.007367

0.000074

133.35

62.16

5.99

7.22

0.08

12

191,960.57

0.937659

0.007789

0.000085

132.79

67.31

6.49

7.62

0.09

13

191,331.28

0.926937

0.008184

0.000099

131.27

66.54

6.40

8.07

0.11

14

190,700.41

0.916356

0.008553

0.000114

129.77

65.77

6.30

8.49

0.13

Column sums over t = 0..14: prem 2 032.99, ben_deces 965.23, ben_ptia 94.16, ben_itt 72.95, ben_ipt 0.71, expenses 38.10.

Supplementary — one ITT cohort through the 1 095-day cap. S(z) is the probability that a claim incepting at z = 0 is still in ITT at the end of duration month z:

z

ρ(z)

τ(z)

q_s(z)

s_itt(z)

S(z)

1

0.064377

0.001682

0.001682

0.932478

0.932478

6

0.064377

0.001682

0.001682

0.932478

0.657404

12

0.064377

0.001682

0.001682

0.932478

0.432180

13

0.029286

0.005143

0.002535

0.963274

0.416308

24

0.029286

0.005143

0.002535

0.963274

0.275843

25

0.013452

0.010596

0.003396

0.972779

0.268335

35

0.013452

0.010596

0.003396

0.972779

0.203620

36

0.013452

0.010596

0.003396

0.972779

0.198077

At z = 36 the surviving 0.198077 is assessed: 0.069327 passes to IPT (35 %) and 0.128750 returns to healthy (65 %). Expected months of ITT payment per inception = Σ_{z=1..36} S(z) = 14.721231, i.e. EUR 16 328.72 of ITT benefit per inception at ech × Q = 1 109.1952.

Checks.

The loan spine, two ways. From the annuity formula, crd(1) = 1 109.1951957 × (1 − 1.0025^(−239)) / 0.0025 = 199 390.8048. By roll-forward, crd(0) × (1 + i) − ech = 200 000 × 1.0025 − 1 109.1951957 = 199 390.8048 — identical. At the far end crd(239) = 1 106.4291 and crd(239) × 1.0025 = 1 109.1952 = ech, so crd(240) = 0 exactly. Total instalments 240 × 1 109.1952 = 266 206.85, of which 66 206.85 is interest.

The state recursion, from the four decrements. l_h(1), the state at the end of month t = 0, should be (1 − 0.000327255)(1 − 0.000032673)(1 − 0.003396053)(1 − 0.000904486) = 0.995344, which is the table’s first row. The residual mass is the four exits of that month — dth_h(0) = 0.000327255, ptia_h(0) = 0.000032662, lapses(0) = 0.003394831, n_itt(0) = 0.000901090 — and 0.995344 plus those four is 1.000000. Over the whole 15 months l_h + l_itt + l_ipt = 0.925024 and cumulative exits = 0.074976, summing to 1.000000000000: check_states(). l_h then falls 0.4598 % per month through policy year 1 (0.995344 → 0.990768) and 1.1455 % per month in year 2 (0.948524 → 0.937659) — the loi Lemoine substitution assumption arriving, monthly w going from 0.003396053 to 0.010596241, a factor of 3.12 diluted by the unchanged mortality and inception decrements. The mortality step at t = 12, the first month of policy year 2, shows separately in ben_deces, which rises 62.16 → 67.31 as a goes 52 → 53 (mort_rate 0.00392 → 0.00428) on a CRD that is still falling.

The in-arrears benefit rule. ben_itt(0) = 0.00 because the only ITT mass at the end of month 0 is that month’s own inception. ben_itt(1) = ech × s_itt(1) × n_itt(0) = 1 109.1952 × 0.932478274 × 0.000901090 = 0.93 — the month-0 inceptions, one month later, net of one month’s terminations (s_itt is on the claim-duration clock z, where the first month in payment is z = 1). ben_deces(0) = crd(1) × q_h = 199 390.8048 × 0.000327255 = 65.25, and ben_ptia(0) / ben_deces(0) = 6.512472 / 65.251648 = 0.0998, the ptia_rate ratio of 0.10 less the month of death exposure that precedes PTIA in the decrement order — the ordering is visible in the arithmetic.

Aggregates over the full 240 months, at 2.5 % flat. PV of premium income is EUR 12 602.19 on the level 0.84 % capital initial basis and EUR 12 588.82 on the CRD basis (ratio 1.001062; equivalent level rate 0.8391 %, rounded to 0.84 % for the spec), the CRD basis being non-monotonic — EUR 125.33 in year 1, EUR 164.03 at its year-10 peak, EUR 31.65 in year 20. PV of outgo is ben_deces 7 170.56 + ben_ptia 635.87 + ben_itt 1 932.71 + ben_ipt 1 293.18 + expenses 334.17 = EUR 11 366.49, a margin of 9.81 % on premium, with death and PTIA 70.8 % of the benefit PV and ITT plus IPT 29.2 % against the market’s published premium split of 69 % / 30 % REG-R37 — a coincidence of calibration rather than evidence, but the only external check available on the shape of the basis. Finally I(t) = 0 from t = 216 (a = 70): crd(216) = EUR 25 806.51 is still owed as that month opens, 0.009266 + 0.013982 of mass in ITT and IPT moves to l_h, ben_itt and ben_ipt are exactly zero for t = 216..239, and premium income continues — EUR 3 360.00 nominal per surviving policy (24 × EUR 140.00), EUR 638.67 survivorship-weighted.

Valuation and reserve pointers#

This library projects gross best-estimate liability cash flows; valuation layers consume them and are NOT reproduced here.

  • Solvabilité II. Technical provisions = best estimate + risk margin, the best estimate being the probability-weighted average of future cash flows discounted at the relevant risk-free term structure REG-R4, with curves published monthly by EIOPA REG-R5 — exactly the projections above, on both the premium and the claim side. No cost-of-capital rate in this library rests on a retrieved instrument REG-R2, so any risk-margin figure would be std; none is specified.

  • French statutory provisions. Art. R. 343-3 of the Code des assurances enumerates eleven technical provisions REG-R6; the two an ADE book touches are the provision mathématique — which for a French contract includes future management costs, so it is not a pure net-premium reserve — and the provision pour égalisation for mortality fluctuations on group death business. How a French insurer actually reserves the increasing-risk pattern created by a nivelé premium was not established by any retrieved source and is unverified (spec, Regulatory context). The model supplies the cash flows such a provision consumes; it does not compute one.

  • Claims in payment are a disabled-life annuity: the expected present value of ech × Q × IR until recovery, transition, death, the age limit or loan expiry, run from (a, z) with the recursions above and l_h(0) = 0; the supplementary table is that annuity’s survival column for z = 0. IFRS 17 REG-R45 consumes the same projections with its own discounting, risk adjustment and CSM layers (cited-not-specified), and the professional frame is NPA 1 REG-R43 and NPA 2 REG-R44, the Institut des actuaires’ pratiques recommandées on general actuarial practice and on actuarial models.

Key sensitivities and model risks#

  1. Résiliation. First-order and product-defining: it sets premium income, the effective duration of the book and, through selective withdrawal, the morbidity of what remains. The whole table is std with no French anchor and the dynamic response to the premium gap is a construction, not a calibration; the observable facts are counts of substitution requests and a 15.3 % → 16.0 % market-share shift R12, neither of which is a lapse rate. Test lapse_rate, beta and the acceptance ratio independently.

  2. ITT inception and termination. Inception drives claim frequency; the recovery / IPT-transition split drives claim length and therefore the whole disabled-life annuity — a small change in itt_recovery_rate compounds over 36 months and again through the 0.35 split at the cap. Both proxy tables are std placeholders.

  3. Mortality level and the CRD profile together. Death is 70.8 % of the benefit PV and its cost is crd(t + 1) × q_h(a), a falling schedule times a rising rate; the peak of that product, not either factor alone, is where the death cost sits, and the loan term moves it.

  4. The premium basis, and the age limits. Level on capital initial against annually re-read on the CRD is a different cash flow shape for the same cover, and the difference is largest exactly where lapse is highest; the two are PV-equivalent here by construction and would not be in a real tariff. The cover-end ages are cited rather than std, but their interaction with loan term is under-appreciated — 24 uncovered months in the base cell, worse for a longer loan or an older entrant, and the CCSF records “maximum cover age exceeded” among the commonest causes of claim decline R12.

  5. The 1 095-day assessment split, indemnitaire exposure, expenses and claim admission. ipt_share_at_cap converts a bounded three-year claim into an annuity that can run to the end of the loan; nothing public quantifies it and the liability is roughly linear in it. At income_loss_ratio = 1.00 the indemnitaire cell equals the forfaitaire cell, but real indemnitaire business pays materially less to employees whose salary is maintained [S6] [S10]. Expenses (EUR 334.17 of PV) and the assumed 100 % claim admission — against observed declines of 7.7 %–16.3 % on incapacity claims R12 — are second-order for the total and first-order for the margin.