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 — model name TD_FR_S, monthly grid — for the standardized composite French assurance temporaire décès defined in product-spec.md (same directory). This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in sources.md (numbering carried from _research/temporaire-deces.md; frozen); [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md (its own 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. Cells names, model-point columns and CSV headers are English lower_snake_case; French terms of art keep their French form in prose.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows — cotisations, death claims, PTIA claims, expenses and commission — for a single-policy model point, on an expected (probability-weighted) basis. Discounting, the provision mathématique recursion, the Solvabilité II risk margin and the capital layers are out of scope (see Valuation and reserve pointers).

  • Projection frequency std: monthly. Monthly steps (monthiversary processing) are the model. A monthly grid is not a monthly product. Every contractual element here is on an annual cycle — the one-year cover renewed by tacite reconduction and repriced at each renewal [S1] [S2] [S3] [S6] [S8] [S9], the benefit schedule, the art. L. 132-7 suicide year, the first-year commission rate, the constante equivalence — and the model keeps every one of them on the anniversary. What the finer grid adds is the timing of everything that is not contractually annual: the modal cotisation collected on the cycle the disclosed frais de fractionnement and frais d’échéance exist to price, claims settled in the month of claim, and maintenance expense accruing where it is incurred. An annual step remains a well-defined special case of the recursion below and reproduces its in-force and every annual contractual quantity at every anniversary exactly — see Annual equivalence — but it is not what the reference model runs.

  • Time index [library-wide]. t is 0-based and counts policy months, as in lifelib’s basiclife/BasicTerm_S and the other monthly models of this library: t = 0 is the first policy month, month t runs from time t to time t + 1, and proj_len is the number of projected months — the exclusive end of the frame — so the projection runs t = 0 … proj_len − 1 and result_cf() has proj_len rows. Because every contractual schedule is annual, the policy year is derived and used as a lookup key: dur(t) = t // 12 is the completed policy years at the start of month t, the contractual policy year is the 1-based label policy_year(t) = dur(t) + 1, and the attained age is x(t) = issue_age + dur(t). The anniversary that opens policy year k is month t = 12(k − 1), and where these notes say “policy year k” as contract language the months are t = 12(k − 1) … 12k − 1. Nothing is indexed by the policy year.

  • Projection horizon. n = proj_len_y = cover_end_age − issue_age is the horizon in policy years, which is how the contract states it, and proj_len = 12n is the frame in months. Month t falls in the policy year at attained age issue_age + dur(t), so the last covered policy year is the one at attained age cover_end_age − 1. For the worked configuration, n = 75 − 58 = 17 and proj_len = 12 × 17 = 204, i.e. t = 0 … 203, the last covered policy year being months 192–203 at attained age 74.

  • Timing conventions std. Monthiversary (BOM) processing. The cotisation instalment at the beginning of each month the elected fractionnement makes it due — annual in advance is the contracts’ base mode [S1] [S2] [S6] [S7] [S8] [S9] and collects in the first month of each policy year; maintenance expense and commission at the beginning of the month; death and PTIA claims at the end of the month of claim, not of the policy year; lapses at the end of the month, after both insured decrements; acquisition expense and the initial commission rate at issue. Decrement rates are quoted annual effective throughout and subscripted m where the monthly rate actually applied is meant.

  • Age basis. Différence de millésime — calendar year minus year of birth, irrespective of birth month [S1] [S2] [S6] [S7]. The age steps at the policy anniversary, x(t) = issue_age + dur(t), and not monthly: the finer grid does not make the age basis finer, and every tariff and cover-limit lookup is therefore flat across a policy year. The real millésime age steps on 1 January, so an implementation on real dates carries a fractional offset of at most one year std.

  • Termination. All states terminate at the end of month t = proj_len − 1: cover ceases at the age limit, nothing is payable, there is no maturity value and no conversion [S3] [S5] [S11] [S15] [S16]. pols_if(proj_len) is never a weight on a cash flow, but it is not unused: it is the survivor term of the closure identity below, and it is what the lapse rate in the final projected policy year — zero, by the convention set out under Lapse — decides.

  • No cash value anywhere. Art. L. 132-23 forbids both rachat and réduction on a temporaire décès R3. The model has no account value, no surrender cells, no paid-up state, and claims_lapse(t) is structurally zero at every t.

  • Currency, sign and rounding. EUR throughout. net_cf(t) is income-positive (cotisations +, claims and expenses −), with the outgo-positive orientation published as liability_cf(t) = −net_cf(t). Intermediate values at full precision; displayed cash flows to euro cents std.

Model point attributes#

Attribute

Type

Example (worked configuration)

point_id

int

1

premium_form

enum {revisable, constante}

revisable

benefit_shape

enum {constant, decreasing}

constant

benefit_schedule_id

str (key into benefit_schedule.csv)

constant

sex

enum {M, F} — reporting only; pricing is unisex R10

M

smoker

enum {N, S} — feeds rating_factor, no published level

N

issue_age

int (différence de millésime)

58

sum_assured

EUR

150,000

cover_end_age

int (death cover ceases at this attained age)

75

ptia_end_age

int (PTIA cover ceases at this attained age)

65

premium_rate_id

str (key into premium_rate_table.csv)

maif_2019

rating_factor

float (surprime multiplier on the tariff rate)

1.00

prem_freq

enum {annual, half_yearly, quarterly, monthly}

annual

level_premium

EUR; 0 = derive by equivalence (constante form only)

0

waiting_period_y

int (délai d’attente, years)

0

accident_multiplier

float (additional accidental capital, 1.00 = option off)

1.00

issue_date

date

—

premium_form is the column a UK or US reader is most likely to get wrong, and it is the first entry in Known modeling pitfalls. sex is carried but must not enter pricing: art. L. 111-7 forbids sex-based premium and benefit differences for contracts written from 21 December 2012 R10, while the homologated valuation tables remain sex-specific R6 REG-R22 — the same tension products/rente_viagere/ faces from the other side.

State variables#

Variable

Description

Updated

proj_len_y

Number of projected policy years = cover_end_age − issue_age

once per model point

proj_len

Number of projected months = 12 × proj_len_y; the frame is t = 0 … proj_len − 1

once per model point

duration_mth(t)

Completed policy months at the start of month t = t

derived

duration(t)

Completed policy years at the start of month t = t // 12

derived

policy_year(t)

Contractual 1-based policy-year label = duration(t) + 1

derived

age(t)

Attained age of month t’s policy year = issue_age + duration(t)

annual, on the anniversary

pols_if(t)

In-force probability at the start of month t; pols_if(0) = 1

monthly recursion

prem_rate(t)

Tariff rate at age(t), read from premium_rate_table.csv

annual, on the anniversary

prem_freq_load(t)

Fractionation multiplier for prem_freq, from freq_loading_table.csv

lookup

prem_instalments()

Instalments per policy year for prem_freq, from the same file

lookup

prem_cycle()

Months between instalments = 12 / prem_instalments()

once per model point

prem_pp(t)

Annual cotisation per in-force policy for month t’s policy year

annual, on the anniversary

prem_inst_pp(t)

Instalment collected at BOM = prem_pp(t) / prem_instalments() in a due month, else 0

monthly

benefit_pp(t)

Capital payable on a month-t death or PTIA claim

annual, on the anniversary (schedule)

mort_rate(t)

Annual dependent rate of the death decrement at age(t)

annual, on the anniversary

ptia_rate(t)

Annual dependent rate of the PTIA decrement; 0 once age(t) ≥ ptia_end_age

annual, on the anniversary

decr_rate(t)

The combined annual insured decrement mort_rate(t) + ptia_rate(t)

annual, on the anniversary

decr_rate_mth(t)

1 − (1 − decr_rate(t))^(1/12), the monthly insured decrement applied

monthly

mort_rate_mth(t)

decr_rate_mth(t) × mort_rate(t) / decr_rate(t), the monthly death rate applied

monthly

ptia_rate_mth(t)

decr_rate_mth(t) × ptia_rate(t) / decr_rate(t), the monthly PTIA rate applied

monthly

lapse_rate(t)

Annual lapse rate of month t’s policy year; 0 through the final policy year

annual, on the anniversary

lapse_rate_mth(t)

1 − (1 − lapse_rate(t))^(1/12), applied at EOM after the insured decrements

monthly

suicide_factor(t)

Death-benefit exclusion factor; < 1 through policy year 1 only, never applied to PTIA

annual, on the anniversary

pols_death(t)

Expected deaths in month t = pols_if(t) × mort_rate_mth(t)

monthly

pols_ptia(t)

Expected PTIA claims in month t = pols_if(t) × ptia_rate_mth(t)

monthly

pols_lapse(t)

Expected lapses in month t, on survivors of both decrements

monthly

premiums(t)

prem_inst_pp(t) × pols_if(t)

monthly

claims_death(t)

benefit_pp(t) × pols_death(t) × suicide_factor(t)

monthly

claims_ptia(t)

benefit_pp(t) × pols_ptia(t)

monthly

claims_lapse(t)

Structurally 0 — a lapse pays nothing R3

monthly

expenses(t)

Acquisition + one twelfth of maintenance + claim expense + commission

monthly

net_cf(t)

Net liability cash flow, income-positive

monthly

There is no account-value state variable, no surrender-value state variable and no paid-up state. That is a statutory fact about the product, not a modeling simplification R3 [S7] [S9].

Assumption inputs#

(a) Contractual / guaranteed elements (cited)#

Input

Value

Basis

Cotisation rule

sum_assured × prem_rate(age(t)) × rating_factor × prem_freq_load, plus the fixed frais d’échéance once a year where the mode is fractionated

[S1] [S2] [S3] [S4] [S6] [S7] [S9]; fee [S1]

Tariff rate table

The published attained-age grid below, ages 18–74

[S3]

Repricing

At the effective date and again at every annual renewal, on attained age

[S1] [S2] [S3] [S4] [S6] [S7] [S9] [S10]

Death benefit

sum_assured, constant, from any cause

[S1] [S2] [S3] [S6] [S7] [S9]

PTIA benefit

The same capital, paid early to the insured; payment ends the contract

[S1] [S2] [S3] [S6] [S8]

Death/PTIA cumulation

Prohibited; the PTIA capital is due only if the insured is alive at payment

[S1] [S2]

PTIA cessation

At ptia_end_age, earlier than cover_end_age at five of the eight carriers but equal at one [S1] and absent at one [S9], so the two ages are separate model-point columns

[S3]; pattern [S2] [S6] [S7] [S8]; exceptions [S1] [S9]

Premium cessation

On death and on recognition of PTIA

[S3] [S7]

Suicide

Death cover “de nul effet” in the first year; covered from the second; clock restarts on an increase, for the increment only. The alinéa 4 immediate cover with its 120 000 € floor is confined to principal-residence loan cover and does not apply here

R1 R2

Surrender / paid-up value

None, by statute; expiry at cover_end_age pays nothing

R3 [S3] [S5] [S7] [S9] [S11]

Fractionation loading prem_freq_load

Annual 1.000 (with a 1 % direct-debit discount at one carrier); half-yearly 1.0250; quarterly 1.0400; monthly 1.0400, plus a fixed frais d’échéance of 3 € / 6 € / 15 € (10 instalments) or 18 € (12)

[S1]

Age basis

Différence de millésime

[S1] [S2] [S6] [S7]

The attained-age tariff table [S3] — tarif de base annuel, in per cent of the guaranteed capital, by attained age. The only complete French standalone temporaire décès rate card in the corpus; shipped as premium_rate_table.csv under rate_id = maif_2019:

Age

Rate

Age

Rate

Age

Rate

Age

Rate

Age

Rate

18–34

0,15 %

42

0,32 %

50

0,64 %

58

1,05 %

66*

2,55 %

35

0,17 %

43

0,36 %

51

0,69 %

59

1,13 %

67*

2,78 %

36

0,17 %

44

0,40 %

52

0,74 %

60

1,56 %

68*

2,88 %

37

0,19 %

45

0,44 %

53

0,79 %

61

1,68 %

69*

3,14 %

38

0,20 %

46

0,48 %

54

0,85 %

62

1,81 %

70*

3,43 %

39

0,22 %

47

0,52 %

55

0,91 %

63

1,97 %

71*

3,74 %

40

0,24 %

48

0,56 %

56

0,93 %

64

2,14 %

72*

4,09 %

41

0,29 %

49

0,60 %

57

0,99 %

65

2,33 %

73*

4,46 %

74*

4,86 %

* Entry is capped at 65, so ages 66–74 are in-force rates only — the carrier’s own footnote reads “la dernière colonne vous indique donc le tarif de base, en cours de contrat, pour couvrir le risque de décès entre 65 et 75 ans” [S3]. The carrier’s own two worked examples fix the rule: 20 000 € at age 34 → 20 000 × 0,15/100 = 30 € for one year; 150 000 € at age 49 → 150 000 × 0,60/100 = 900 € for one year [S3]. Vintage caveat: the grid is a 2019–2021 edition; use it for shape, not level (product spec, footnote 7) [S3] [S4] [S10].

(b) Insurer-discretionary current elements#

Thin, but not empty — and thinner than it looks, because the discretion on this product bites through the rate card, not through a bonus or a charge scale.

Input

Snapshot value

Basis

Tariff drift (experience repricing of the class)

0 % p.a. — the rate card is frozen at its retrieved vintage in the base run std (1)

discretion cited at two carriers: “l’accroissement de la fréquence et/ou du coût moyen des sinistres” [S1] and “les résultats des garanties Assurance Décès” [S6]. A third reserves repricing for legislative or regulatory change only, with 15 days to terminate on a tariff change [S7]

Revalorisation / indexation of capital and cotisation

Off in the base run std (1)

PASS-linked [S1] [S7]; insurer-set rate [S2] [S6]

Surprime level (rating_factor)

1.00 (standard rates) std (2)

mechanics [S1] [S2] [S3] [S6]; no published scale

Participation aux bénéfices

None at policy level. Computed globally across the insurer’s life book where it exists at all

[S1] [S2]; none at all [S9]

Post-death revalorisation

Not projected — a sub-annual window between death and settlement

[S1] [S2] [S3] [S6]; REG-R39

  1. Both levers reprice the contract in force, and both are exogenous to the liability model: an experience re-rating multiplies prem_rate, an indexation multiplies sum_assured and prem_pp together. Setting both to zero keeps the base run reproducible from cited data alone. Note the asymmetry the contracts record: an increase decided by the insurer gives the member 30 days (15 at one carrier) to terminate, while an increase arising from age, index or law “n’ouvre droit ni à contestation ni à résiliation” [S1] [S7].

  2. No insurer publishes a surprime scale [S1] [S2] [S3] [S6] [S7]. The only public French price evidence on rated lives is on borrower cover — average 1,01 % of initial capital before écrêtement and 0,65 % after REG-R37 — which bounds a standard rate from above.

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

Every input in this class is std. No French insurer publishes a mortality table, an A/E factor, a PTIA incidence rate, an expense loading, a commission scale or a lapse rate for this product [S1] [S2] [S3] [S6] [S7] [S8] [S9] [S12].

Mortality. The regulatory non-annuity tables are TH 00-02 (male) and TF 00-02 (female), homologated by the arrêté du 20 décembre 2005 with effect from 1 January 2006 and built by INSEE on French mortality observed over 2000–2002 R6 R9 REG-R22. They are annexed to an arrêté and are cited by name, never shipped in this library REG-R22 REG-R23. Where a single homologated table is used for all insureds it must be the one giving the most prudent tariff — the male table for a death cover R4; the alternative in market practice is a blend, and the Institut des actuaires’ working group uses 60 % TH 00-02 / 40 % TF 00-02 as its unisex death basis R13. Neither choice is prescribed by any retrieved text, so adopting one is std. The shipped mort_table.csv is therefore a std Gompertz-form proxy, not a fitted table:

mort_rate(x) = 0.00400 × 1.09^(x − 58),   ages 18–74

The 9 % per year of age is measured against the one observable French artefact, the published tariff grid, and it sits at the top of what that grid shows rather than inside a tight band. The grid rises at roughly 7–9 % per year of age from age 35 [S3]; over ages 42–58 the step-by-step ratio r(x+1)/r(x) runs from 1,022 (the flat step 55 → 56, 0,91 % → 0,93 %) to 1,125 (42 → 43) with a median near 1,076 [S3]. Compounded, the same grid gives (1,05/0,32)^(1/16) − 1 = 7,7 % a year over 42–58, (4,86/2,55)^(1/8) − 1 = 8,4 % over 66–74, and (4,86/0,17)^(1/39) − 1 = 8,98 % over the whole rated span 35 → 74 — the last of which is what 9 % is anchored to. It is a tariff gradient, not a mortality gradient, so the choice remains std, and of the two unsourced numbers in this basis the slope is the more exposed on a 17-year run (see the sensitivities section). INSEE’s national series REG-R24 is the intended base for a user-supplied replacement; it is population, not insured, mortality, and the reference library records that the INSEE page states no licence or reuse conditions — standard open-data terms are assumed there and that assumption is unverified, so confirm before redistributing derived CSVs REG-R24. The anchor a substitute table must preserve is mort_rate at age 58 = 0.00400, so the worked example still closes.

Décalages d’âge — and why they do not reach this product. The annexed age shifts are imposed by a clause whose scope excludes a death cover: “pour les contrats en cas de vie autres que les contrats de rente viagère, les tables mentionnées au a sont utilisées en corrigeant l’âge de l’assuré conformément aux décalages d’âge ci-annexés” R4 R6 REG-R23. A temporaire décès is a contract en cas de décès, so on the retrieved texts no shift applies to it, and a user who replaces mort_table.csv with TH 00-02 should load that table unshifted. Where the shifts do bite, the profession recommends applying them to the q(x), not to the l(x), because shifting l(x) produces erratic q(x) growth and hence erratic provisions R9. The numeric annexe to the current art. A. 132-18 was not retrieved R4; the abrogated A. 335-1 annexe carried shifts from −11 years at ages 16–32 to 0 at 94+ for TF 00-02 and from −13 years at ages 16–38 to −3 at 75+ for TH 00-02 REG-R23 — a −13-year shift is worth a factor of about 3 on this proxy’s slope, which is why applying one where it is not required is not a rounding error. The std proxy therefore carries no shift, for both reasons: the rule does not reach a death cover, and a shift applied to a proxy that was never a homologated table would be theatre.

PTIA incidence. No retrieved French source gives a PTIA incidence rate at any age. The model uses ptia_rate(x) = ptia_ratio × mort_rate(x) with ptia_ratio = 0.20 std for age(t) < ptia_end_age, and 0 thereafter. The only public French figure that touches PTIA at all is an underwriting-outcome statistic — 87 % of aggravated-risk applications received a PTIA offer with no surprime and no exclusion, against 65 % for death REG-R37 — which says nothing about incidence. 0.20 is a placeholder chosen so PTIA is a visible but clearly secondary decrement; it is the assumption in this file most in need of a real source.

Lapse. No insurer publishes a lapse rate and nothing in the corpus supports one [S1] [S2] [S3] [S6] [S7] [S8] [S9]. The reference table is std, shaped by the one thing the contracts do tell us — that voluntary exit is easy and cheap, because there is nothing to forfeit R3 and notice periods run from “at any time” to one month before the échéance [S1] [S2] [S3] [S8] [S9]:

Policy year (contractual, = dur(t) + 1)

1

2

3

4+

lapse_rate(t) std (3)

12 %

10 %

8 %

6 %

These are annual rates and they stay annual: the table ships as lapse_table.csv keyed by the contractual policy year, so the model reads it through policy_year(t) = dur(t) + 1 — months t = 0 … 11 take the 12 %, and policy years from the fourth take the last row — and they reach the month at w_m(t) = 1 − (1 − w(t))^(1/12) like any ordinary lapse.

In the final projected policy year the lapse rate is zero: w(n − 1) = 0 std. Lapses fall at the end of the month (processing order, step 9), and the end of the last projected policy year is the moment the cover expires — a lapse and an expiry are then the same event paying the same nothing, so the whole surviving cohort is booked as an expiry, l(12n). The zero covers the whole of that policy year, months 12(n − 1) … 12n − 1, and not merely its last month: the convention is stated of a policy year, and zeroing only month 12n − 1 would leave eleven months of 6 % lapse inside the final year. No cash flow moves either way, but the convention is load-bearing for the closure identity: it decides the split between Σ pols_lapse and l(12n), and it is what the worked example’s 0,64859269 / 0,27886852 reproduces — the survivor term being exactly the one an annual step produces. Taking the table rate literally through the last year instead gives roughly 0,666 lapses and 0,266 survivors, the same total and a different split.

  1. No observed range exists — not one of the eight retrieved carriers, and neither secondary guide, publishes a lapse rate [S1] [S2] [S3] [S6] [S7] [S8] [S9] [S13] [S16]. The shape is a modeler’s construction: elevated in the first three years to absorb the 30-day renonciation window REG-R29 [S1] [S2] [S3] and early-duration attrition, then flat. Nothing about the levels is sourced, and a user with experience data should replace the whole table.

Suicide-exclusion factor. Art. L. 132-7 makes the death cover void for suicide in the first year R1. The model applies

suicide_factor(t) = 0.98 for dur(t) = 0,   1.000 for dur(t) ≥ 1   **[std]**

— the first year is contractual policy year 1, months t = 0 … 11 — to death claims only. The statute voids the cover “au cours de la première année du contrat” R1, so the exclusion covers the whole of that policy year; a monthly grid does not turn a statutory year into a month. No retrieved source gives a suicide share of deaths at any age — INSEE cause-of-death data was not fetched for this research — so 0.98 is a placeholder standing for “about 2 % of first-year deaths are excluded suicides”. Setting it to 1.000 is a defensible variant; what is not defensible is applying it to PTIA, or applying it beyond the first year, both of which are pitfalls below.

Expenses and commission (all levels std; the structures are cited where they exist).

Input

Value

Basis

Acquisition expense acq_expense

250 € per policy at issue

std (4)

Initial commission rate comm_rate_init

40 % of the first-year cotisation

std (4)

Renewal commission rate comm_rate_renew

5 % of the cotisation from policy year 2, i.e. dur(t) ≥ 1

std (4)

Maintenance expense maint_expense

25 € per policy p.a., accruing at one twelfth a month and inflating at expense_infl

std (4)

Expense inflation expense_infl

2 % p.a. flat

std (4)

Claim expense claim_expense

150 € per death or PTIA claim

std (4)

Association subscription

0 € (individual wrapper); 1,30 € per member per year in a group wrapper

[S1]; wrapper choice std (4)

Technical rate tech_rate

0,5 % p.a., used only for the constante equivalence, never to discount the published cash flows

std (5)

  1. No observed range exists. No French insurer publishes an expense loading, an acquisition cost or a commission scale for this product [S1] [S2] [S3] [S6] [S7] [S8] [S9] [S12]; the chargements de gestion are built into the tariff and are not separately disclosed anywhere, which is precisely why art. R. 343-3 has to require the provision mathématique to carry them R11. The only disclosed charge figures in the whole corpus are the fractionation loadings and frais d’échéance [S1], the 1,30 € association subscription [S1] and the 3 % annuity conversion charge [S3], and none of them is an expense assumption. The levels above are round-number placeholders sized so that year-one acquisition cost (250 € + 40 % of the cotisation) is of the same order as the year-one cotisation at the anchor age.

  2. The Institut des actuaires’ own illustrations for a death cover use technical rates of 0,5 % and 0 % against a 1 % interest assumption R13; art. A. 132-1 caps a French tariff rate at min(3,5 %, 60 % TME) for contracts à primes périodiques of any duration R5 REG-R17, so 0,5 % is well inside the cap. Adopting 0,5 % rather than 0 % is std.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol

Meaning

t

the 0-based month index, t = 0 … 12n − 1, 12n = proj_len, n = proj_len_y = cover_end_age − issue_age

dur(t)

completed policy years at the start of month t, t // 12

policy_year(t)

the contractual 1-based policy year label, dur(t) + 1

x(t)

attained age of month t’s policy year = issue_age + dur(t)

SA

sum_assured

r(x)

tariff rate at attained age x, from premium_rate_table.csv

f

rating_factor; φ = prem_freq_load

F

prem_freq_fee, the fixed annual frais d’échéance — a euro amount, not a rate, and nil on the annual mode [S1]

k

prem_instalments, the instalments per policy year: 1 / 2 / 4 / 12 [S1]; prem_cycle = 12 / k months

P_tar(t)

prem_tariff_pp(t), the annual tariff cotisation of month t’s policy year, before F

P(t)

prem_pp(t), the annual cotisation charged per in-force policy for month t’s policy year, = P_tar(t) + F

P_inst(t)

prem_inst_pp(t), the instalment collected at the beginning of month t: P(t) / k when t mod prem_cycle = 0, else 0

P_lev

the level cotisation of the constante form, also before F

B(t)

benefit_pp(t), the capital payable on a month-t claim

q_d(t)

mort_rate(t), dependent annual rate of the death decrement of month t’s policy year

q_p(t)

ptia_rate(t), dependent annual rate of the PTIA decrement of the same year

q(t)

decr_rate(t) = q_d(t) + q_p(t), the combined annual insured decrement; q_m(t) is the monthly rate applied

q_dm(t), q_pm(t)

mort_rate_mth(t), ptia_rate_mth(t): q_m(t) split in the ratio q_d : q_p — see Monthly rates from annual assumptions

w(t)

lapse_rate(t), the annual lapse rate of month t’s policy year; w = 0 through the final projected policy year std

w_m(t)

lapse_rate_mth(t) = 1 − (1 − w(t))^(1/12), applied at EOM after both insured decrements [std order]

σ(t)

suicide_factor(t), below 1 through policy year 1

l(t)

pols_if(t), in force at the start of month t (at time t); l(0) = 1

p_τ(y)

tariff survivorship entering policy year y, decrements only, no lapse: p_τ(0) = 1, p_τ(y+1) = p_τ(y)(1 − q_d − q_p) at the year’s rates

v

1 / (1 + tech_rate); v^y is a policy-year discount factor

E0, e(t)

acquisition expense; maintenance expense = (25/12) × 1.02^dur(t) per month

c0, c_r

initial commission rate (0.40); renewal commission rate (0.05)

ec

claim expense (150)

q_d, q_p and w are per-annum probabilities and q_dm, q_pm, w_m per-month ones (all dimensionless); SA, B, P are EUR; P_inst and every cash-flow component is EUR per month.

Monthly rates from annual assumptions std#

Every assumption in this product is published, calibrated and tabulated annually: the mortality proxy is an annual table by attained age, the PTIA rate is a ratio on it, and the lapse vector is by policy year. The monthly rates the recursion applies are derived from them at the constant-force conversion, so that twelve months compound back to exactly the annual rate:

q_m(t)  = 1 − (1 − q_d(t) − q_p(t))^(1/12)
q_dm(t) = q_m(t) × q_d(t) / (q_d(t) + q_p(t))          zero-safe: 0 when q_d + q_p = 0
q_pm(t) = q_m(t) × q_p(t) / (q_d(t) + q_p(t))          ( = ptia_ratio × q_dm(t) )
w_m(t)  = 1 − (1 − w(t))^(1/12)

The two insured rates are converted together and split afterwards, not converted apart. q_d and q_p are dependent rates of one two-decrement table and are therefore additive (pitfall 4), so the rate the annual recursion applies — and therefore the rate the monthly one has to reproduce — is their sum. Converting each separately and adding the results, (1 − (1 − q_d)^(1/12)) + (1 − (1 − q_p)^(1/12)), gives a twelve-month insured survival of 0,9952029339 against the annual 0,9952000000 at the anchor cell: an error of 2,6 × 10⁻⁶ in l at the very first anniversary and 1,0 × 10⁻⁵ by month 204, which would break both the anniversary equivalence and the closure split. The proportional split also keeps q_pm / q_dm = ptia_ratio exactly, so the acceleration ratio the product is built on survives the conversion untouched; and where the PTIA gate has closed, q_p = 0 and the formula degenerates to the plain 1 − (1 − q_d)^(1/12).

No exception is made for any rate. Unlike uslib’s Term_US_S, this product carries no shock lapse tied to a named date inside a policy year, so there is nothing to leave unspread; the one rate these notes do put on a boundary — the zero of the final projected policy year — is a rate for the whole of that year rather than for a month of it. No retrieved French source states a conversion convention for any decrement, so the conversion itself is std; what is not optional is that it reproduce the annual factor.

Cotisation by premium form#

revisable — the French default, and the product’s signature. P_tar(t) = SA × r(x(t)) × f × φ, which changes at every anniversary because r is read at the new attained age [S1] [S2] [S3] [S4] [S6] [S7] [S9] [S10]. The contracts reprice “à chaque échéance annuelle”, so P_tar is flat across the twelve months of a policy year and there are n distinct cotisations over 12n months.

constante — a std construction (product spec, footnote 2). If level_premium > 0 it is used directly; otherwise P_lev is derived by actuarial equivalence with the revisable stream over the whole cover period, on tariff survivorship (insured decrements only, no lapse) and the technical rate:

P_lev = [ Σ_{y=0..n−1} v^y · p_τ(y) · SA · r(x(12y)) · f · φ ] / [ Σ_{y=0..n−1} v^y · p_τ(y) ]

i.e. a survivorship-and-discount-weighted average of the same grid rates; P_tar(t) = P_lev for all t. The sum runs over policy years, not months: the equivalence is an annual construction and stays one on the monthly grid, so v and p_τ take a policy-year argument and P_lev, the annuity-due factor 15,449728 and the present value 60 476,25 € are all unchanged by the conversion. Re-striking the equivalence monthly would move P_lev and break the identity these notes assert.

Then the fee, once a year, under either form:

P(t)      = P_tar(t) + F
P_inst(t) = P(t) / k   if  t mod prem_cycle = 0,   else 0

F is the fixed annual frais d’échéance attached to the payment frequency — 0 € annual, 3 € half-yearly, 6 € quarterly, 18 € monthly [S1]. It is a euro amount, not a second percentage: φ is already a multiplier inside P_tar, and billing F as a further percentage load, or loading the already-loaded cotisation with it, overstates premium income (pitfall 13). P(t) is what the policyholder is charged, so it is what enters premiums(t) and the commission base, while the constante equivalence above is struck on P_tar alone — F is the same amount under either form, so it neither belongs in the equivalence nor changes it. The worked example runs on the annual mode, where F = 0, P(t) = P_tar(t) and k = 1, so the whole cotisation is collected in the first month of each policy year and nothing in the other eleven; the three fractionated model points are where the two differ, e.g. 933,20 € against 915,20 € at t = 0 on model point 4, collected as twelve instalments of 77,77 €.

The instalment is where the finer grid changes an answer rather than its resolution. φ and F exist because the cotisation is paid in instalments, and an annual grid could only charge the loaded amount whole at the start of the year — pricing a service it never modelled. On the monthly grid the instalments after the first are collected on a block that has already lost lives, so the premium-cessation rule of the contracts [S3] [S7] finally bites: the three fractionated model points collect 1,6 % to 3,1 % less than the annual grid did. The fee is divided among the instalments with the rest, which is literally what freq_loading_table.csv records of the monthly mode (“18 EUR frais d’echeance over 12 instalments”), so k instalments still sum to exactly P(t) and pitfall 13 is untouched: P(t) − P_tar(t) = 18,00 € at every t.

Decrements and the in-force recursion#

q_d and q_p are dependent rates — rates of decrement in a two-decrement table, not independent single-decrement rates. Therefore they are additive:

pols_death(t) = l(t) × q_dm(t)
pols_ptia(t)  = l(t) × q_pm(t)
pols_lapse(t) = l(t) × (1 − q_dm(t) − q_pm(t)) × w_m(t)
l(t+1)        = l(t) × (1 − q_dm(t) − q_pm(t)) × (1 − w_m(t)),    l(0) = 1

with q_d(t) + q_p(t) < 1 required at every t, and the PTIA switch-off a hard gate on the attained age rather than a taper: q_p(t) = ptia_ratio × q_d(t) if x(t) < ptia_end_age, else 0. Because x(t) steps on the anniversary, the gate closes on an anniversary too. This is what “the PTIA capital is an anticipated payment of the death capital, and its payment ends the contract” means arithmetically [S1] [S2] [S3] [S6]: a life that leaves through the PTIA decrement is gone from l and can never generate a death claim. Closure identity, which a test should assert:

Σ_{t=0..12n−1} [ pols_death(t) + pols_ptia(t) + pols_lapse(t) ] + l(12n) = 1

Annual equivalence. Because q_dm + q_pm and w_m each compound back to their annual values and both are constant within a policy year, the recursion collapses over any twelve months of one policy year to

l(t + 12) = l(t) × (1 − q_d − q_p) × (1 − w)

— the annual-step recursion, term for term. The in-force at every anniversary is therefore identical on the two grids, to floating point: l(12k) here equals the annual model’s l(k), measured at 2,3 × 10⁻¹⁵ or better across all twelve shipped model points. So is every anniversary-dated contractual quantity — the tariff rate r(x), the cotisation P, the capital B, the annual decrement rates themselves, the level cotisation P_lev and the annuity-due factor behind it — and so is the expiring cohort l(12n). Nothing else agrees, and nothing else should: the cash flows are where the finer grid does its work. result_cf_annual() sums the monthly frame into policy years so the two can be laid side by side.

Benefit amounts and claims#

B(t) = SA × benefit_factor(benefit_schedule_id, policy_year(t)) — benefit_schedule.csv is keyed by the contractual policy year, so the lookup maps through t + 1 — with benefit_factor ≡ 1.0 for benefit_schedule_id = constant, the only schedule shipped [S1] [S2] [S3] [S6] [S7] [S8] [S9]:

claims_death(t) = B(t) × pols_death(t) × σ(t)
claims_ptia(t)  = B(t) × pols_ptia(t)
claims_lapse(t) = 0                                   [R3]

σ never touches claims_ptia: art. L. 132-7 voids the death cover for suicide in the first year, t = 0 R1, and PTIA is not death. With accident_multiplier > 1 an additional capital (accident_multiplier − 1) × B(t) × acc_share is payable on the accidental share of claims [S1] [S2] [S6] [S7] [S9] [S12]; acc_share has no source in the corpus and the base run sets it to 0 std.

Expenses, commission and net cash flow#

premiums(t)    = P_inst(t) × l(t)                        with P(t) = P_tar(t) + F
commissions(t) = c0 × premiums(t)                        for dur(t) = 0
               = c_r × premiums(t)                       for dur(t) ≥ 1
expenses(t)    = E0 · 1{t = 0} + (25/12) · 1.02^dur(t) × l(t)
                 + ec × (pols_death(t) + pols_ptia(t)) + commissions(t)
net_cf(t)      = premiums(t) − claims_death(t) − claims_ptia(t) − expenses(t)
liability_cf(t) = −net_cf(t)

The commission rate is a policy-year rate and steps on the anniversary; the base is the instalment actually collected, so a fractionated payer earns it in instalments too and an annual-mode payer’s commission is unchanged from the annual grid.

Maintenance expense accrues at a twelfth a month and inflates by policy year, 1.02^dur(t), stepping at anniversaries rather than compounding continuously — the frlib house form, shared with Obseques_FR_S, ADE_FR_S and Dep_FR_S, and the way these notes tabulate the charge. (uslib’s Term_US_S uses the continuous 1.02^(t/12); this library does not follow it there, and the difference is measurable: 254,60 € of maintenance over the worked configuration stepped against 256,89 € continuous.) Twelve twelfths of 25 € is the annual charge, so a policy that runs a full year carries what it did on the annual grid; what changes materially is that a decrementing block carries less of it, because the charge is borne by the in-force of each month rather than of the anniversary — 254,60 € against the annual grid’s 263,96 €. The acquisition expense stays a single amount in month 0: it is a flat per-policy cost, not a fraction of any instalment.

result_cf() publishes, indexed by the month t: pols_if, premiums, claims_death, claims_ptia, claims_lapse, expenses, commissions, net_cf, liability_cf. result_cf_annual() publishes the same columns summed into policy years and indexed by policy_year, with pols_if taken at the start of the year, l(12(k − 1)) — the number the annual-step model carried on the same row.

Processing order (monthiversary)#

For t = 0 … 12n − 1, in this order. Each step is tagged by where it lands: [A-anniv] an annual contract term landing whole on the anniversary, [A-spread] an annual assumption spread at 1 − (1 − r)^(1/12), [M] a genuinely monthly quantity.

  1. [A-anniv] Set x(t) = issue_age + dur(t). If x(t) ≥ cover_end_age, stop — the projection is over.

  2. [A-anniv] Anniversary only (t mod 12 = 0) — reprice. Look up r(x(t)), compute P_tar(t) per the premium form and add the fee once, P(t) = P_tar(t) + F. Both are then constant across the twelve months of the policy year: the contracts reprice “à chaque échéance annuelle” [S1] [S2] [S3] [S7], and spreading the repricing over the months would be a different product.

  3. [M] BOM — collect the instalment if the elected mode makes one due, P_inst(t) = P(t)/k when t mod prem_cycle = 0 and 0 otherwise: premiums(t) = P_inst(t) × l(t).

  4. [M] BOM — commission on the cotisation collected: c0 × premiums(t) through policy year 1 (dur(t) = 0) and c_r × premiums(t) thereafter. The rate is [A-anniv]; the base is the instalment.

  5. [M] BOM — expenses on the in-force: one twelfth of the maintenance charge, (25/12) × 1.02^dur(t) × l(t), plus E0 in month 0 only. The inflation factor is [A-anniv], stepping at the anniversary.

  6. [A-anniv] Compute B(t) from the benefit schedule, read at policy_year(t) = dur(t) + 1, and σ(t), which is 0,98 through the whole of policy year 1 and 1 thereafter.

  7. [A-spread] Decrements. Look up the annual q_d(t); set q_p(t) = 0 if x(t) ≥ ptia_end_age, else ptia_ratio × q_d(t). Convert the sum to the month and split it: q_m = 1 − (1 − q_d − q_p)^(1/12), q_dm = q_m q_d/(q_d + q_p), q_pm = q_m q_p/(q_d + q_p).

  8. [M] EOM — claims: claims_death(t) (with σ(t)) and claims_ptia(t), plus the claim expense on both, settled at the end of the month of claim. Claimants have already paid the month’s instalment in step 3; this is the model’s reading of “premium payment ceases at death and at PTIA” [S3] [S7] std.

  9. [A-spread] EOM — lapses: apply w_m(t) = 1 − (1 − w(t))^(1/12) to the survivors of both insured decrements. A lapse pays nothing R3. Through the whole of the final projected policy year, dur(t) = n − 1, w(t) = 0 std: the end of that year is also the moment the cover expires, so the survivors leave as an expiry rather than as a lapse. The two events pay the same nothing, and no cash flow moves — but they land on different sides of the closure identity.

  10. [M] Update l(t+1) = l(t) × (1 − q_dm(t) − q_pm(t)) × (1 − w_m(t)).

  11. [A-anniv] w_cum is read by the selective-lapsation module at the anniversary, so the loaded q_d stays one annual rate for its policy year (behaviour section below).

After t = 12n − 1 the projection ends with no maturity payment and no tail state; the only value defined beyond the frame is the expiring cohort l(12n).

Known modeling pitfalls#

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

  1. Assuming a level cotisation. The French default is revisable, not constante [S1] [S2] [S3] [S6] [S7] [S9] [S10]. Assert that prem_pp(t) varies with t on the revisable form, and specifically that prem_pp(24) / prem_pp(12) = 1.56 / 1.13 = 1.380531 in the worked configuration — policy year 3 against policy year 2. Assert too that it moves only at the anniversary: 17 distinct cotisations over 204 months, not 204.

  2. Paying the capital twice. PTIA is an acceleration, not an addition [S1] [S2] [S3] [S6]. Assert Σ(pols_death + pols_ptia) ≤ 1 and Σ(claims_death + claims_ptia) = SA × Σ(pols_death + pols_ptia) − (1 − σ(0)) × SA × Σ_{t=0..11} pols_death(t) exactly — the suicide withholding covers the twelve months of policy year 1. A life removed by the PTIA decrement must not appear in l(t+1).

  3. Forgetting that PTIA cover stops first. ptia_end_age < cover_end_age in five of the eight retrieved carriers [S2] [S3] [S6] [S7] [S8]. Assert claims_ptia(t) = 0 for every t with x(t) ≥ ptia_end_age — in the worked configuration, exactly zero for t = 84 … 203, the whole of the policy year at attained age 65 onwards.

  4. Mixing the competing-risk conventions. These notes use additive dependent rates (q_d + q_p). An implementation using independent rates, 1 − (1 − q_d)(1 − q_p), gets 0.00479680 against 0.00480000 at t = 0 — a 3.2 × 10⁻⁶ difference in the rate and 0.48 € of first-year expected claims per 150 000 € of capital. Immaterial here, material at older ages and higher rates. Declare the convention and test it. On the monthly grid the same convention decides how the two rates reach the month; see pitfall 15.

  5. Inventing a surrender value. There is none, by statute R3 [S7] [S9]. Assert claims_lapse(t) == 0.0 at every t, and that no av_pp_at / cash-value cells exist.

  6. Getting the age basis wrong. Différence de millésime, not age nearest birthday [S1] [S2] [S6] [S7]. A one-year shift moves prem_pp(0) from 1 575,00 € (age 58) to 1 695,00 € (age 59) — a 7,6 % error in year one that compounds through the whole projection. The age steps at the anniversary, so assert x(11) = 58 and x(12) = 59, not x(1) = 59.

  7. Smoothing the tariff away. The grid steps +38 % from age 59 to 60 against a trend of about +8 % [S3]. Assert prem_rate is a table lookup and that r(60)/r(59) = 1.380531 survives; a fitted curve will not reproduce it.

  8. Misapplying the suicide factor. σ applies to claims_death through the whole of policy year 1, months t = 0 … 11, and never to claims_ptia R1. The statute names a year, so a monthly grid does not shrink it to month 0. Assert claims_death(11) = 0.98 × B(11) × pols_death(11), claims_death(12) = B(12) × pols_death(12) with no factor, and claims_ptia(0) = B(0) × pols_ptia(0) with no factor. Also assert the model does not carry the art. R. 132-5 immediate-cover ceiling of 120 000 €, which belongs to principal-residence loan cover only R1 R2.

  9. Double-counting the premium-cessation rule. Instalments are in advance and claims are at month end, so a claimant has already paid the month’s instalment. Do not additionally multiply premiums(t) by (1 − q_dm − q_pm) — that applies the rule twice. On the monthly grid the item it would double-count is one month’s decrement on one instalment, about 0,04 % of the month’s income at the anchor age, because the finer grid has already put the real effect where it belongs: the instalments after the first are collected on a block that has genuinely lost lives.

  10. Running past the age limit. proj_len = 12 × (cover_end_age − issue_age) is the number of projected months — 204 on the worked configuration — so the last row is t = 203. There is no benefit, no cotisation and no maturity value at t = proj_len, and l(proj_len) is never used in a cash flow [S3] [S5] [S11] — but it is used in the closure identity, and it is only well defined once w = 0 in the final policy year is stated. Assert both: lapse_rate(t) = 0 at every t from 192 to 203 while lapse_rate_base(t) is still the table’s 6 %, lapse_rate(191) = 6 %, and that the four closure terms sum to 1.

  11. Expecting the two premium forms to collect the same total. The constante equivalence is struck on tariff survivorship (no lapse). Once lapses truncate the expensive late years, the projected premium total under constante exceeds the revisable one — 36 367,46 € against 31 999,13 € in the worked configuration. That is correct, not a bug; a test that asserts equality of projected premium totals is testing the wrong identity. The identity that does hold is Σ v^t p_τ(t) P(t) equal across the two forms.

  12. Applying rating_factor to the benefit. A surprime scales the cotisation only, never the capital [S1] [S2] [S3] [S6]. Assert claims_death is invariant to rating_factor.

  13. Double-charging the fractionation loading. prem_freq_load (φ) is a multiplier embedded in the cotisation TTC; the frais d’échéance (F) are a separate fixed fee, in euros [S1]. Applying both as percentage loads, or applying the loading and then also billing the fee as a percentage, overstates premium income. The fee is charged once a year and it is part of what the policyholder pays, so it enters premiums(t) and the commission base but stays out of the constante equivalence. Assert P(0) = 915,20 + 18,00 = 933,20 € on model point 4 (monthly, 200 000 € at attained age 45) and P(t) − P_tar(t) = 18,00 € at every t — and, on the monthly grid, P_inst(0) = 933,20 / 12 = 77,77 € with an instalment due in every month, against prem_cycle = 6 and instalments in months 0 and 6 only on model point 10.

  14. Treating the accidental option as a benefit multiplier. It pays an additional capital on the accidental share of claims [S1] [S2] [S6] [S7], not a uniform uplift on every claim. With acc_share = 0 in the base run, accident_multiplier must have no effect on any cash flow — a good invariance test.

  15. Splitting the two dependent decrements independently on the monthly grid. q_d and q_p are additive (pitfall 4), so it is their sum the annual recursion applies and their sum that must compound back. Converting each apart, (1 − (1 − q_d)^(1/12)) + (1 − (1 − q_p)^(1/12)), misses the annual insured-decrement factor by 2,6 × 10⁻⁶ at the first anniversary and leaves 1,0 × 10⁻⁵ of error in l by month 204 — enough to break the anniversary equivalence and the closure split, and enough to lose the exact q_pm / q_dm = ptia_ratio. Convert the sum and split it in the rates’ own proportion.

  16. Spreading a contract term because the grid is finer. The repricing, the benefit schedule, the suicide year, the first-year commission rate, the constante equivalence and the zero lapse of the final year are all contractual and land on the policy year, whole. A monthly grid is not a monthly product. Assert that each is constant across the twelve months of a policy year.

  17. Collecting a fractionated cotisation whole at the anniversary. The reverse error: it preserves the annual grid’s premium totals and makes φ and the frais d’échéance meaningless, since both exist to price the instalment cycle. Collect P(t)/k on the mode’s own cycle and let the totals move; the three fractionated model points then collect 1,6 % to 3,1 % less than an annual grid reported, which is the premium-cessation rule acting where it could not before.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions; there is no French calibration evidence for any of them.

  • Base lapse std. The duration table above. Channel and wrapper matter and are not modeled: a bancassurance contract terminates when the bank account closes [S8], which is a lapse driver with no actuarial counterpart in the mutual contracts.

  • Premium-shock lapse std (optional module, off in the base run). The revisable form hands the policyholder a rising bill, and the grid’s own +38 % step at age 60 [S3] is exactly where an affordability response would show. Reference multiplier on w(t):

    M_shock(t) = 1 + β × max(0, P(t)/P(t−12) − 1 − g0)

    with g0 = 0.10 and β = 1.5 std. The ratio is between consecutive renewals, so the previous cotisation is read twelve months back: on a monthly grid P(t)/P(t−1) would be 1 in eleven months of twelve and the module would never fire at all. Base run β = 0, so M_shock ≡ 1. M_shock ≡ 1 through the whole of policy year 1, which has no previous renewal to compare with. Switched on in the worked configuration it takes the value 1.420796 through the whole of policy year 3 — months 24 to 35, ratio 1.380531 — and 1 everywhere else; the elevated annual rate is then spread over those twelve months at w_m like any other ordinary lapse.

  • Selective lapsation std (optional module, off in the base run). Lapsers are healthier on average, so persisters’ mortality is loaded:

    q_d_eff(t) = q_d(t) × [ 1 + λ × max(0, w_cum(12·dur(t)) − w_ref) ]

    with w_ref = 0.30 and λ = 0.25 std. Base run λ = 0. w_cum moves every month, but it is read at the anniversary, so the loading is one number for the whole policy year and q_d stays the annual rate of that year — which is what the library’s *_rate / *_rate_mth split requires, and the only basis the module was ever calibrated on. The deterioration is a multiplier on the annual rate, applied before the monthly conversion, so it grades once a year and not once a month. On this product the effect is larger than on a UK level-premium term policy, because cumulative lapse reaches 64,9 % of the original cohort over the worked configuration’s 17 years.

  • No dynamic surrender behavior, no renonciation decrement, no indexation take-up. There is nothing to surrender R3, so the whole of the exit machinery is lapse and a lapse pays nothing; the 30-day renonciation window REG-R29 [S1] [S2] [S3] sits inside the year-1 lapse rate std; and indexation is not modeled because it reprices capital and cotisation together on an exogenous index [S1] [S2] [S6] [S7], with refusal definitive at three carriers [S2] [S6] [S7] — a one-way absorbing state if it were modeled.

Worked example#

Configuration. premium_form = revisable, benefit_shape = constant (benefit_schedule_id = constant, factor 1.0 at every t), issue_age = 58, sum_assured = 150 000 €, cover_end_age = 75, ptia_end_age = 65, premium_rate_id = maif_2019, rating_factor = 1.00, prem_freq = annual (prem_freq_load = 1.000, prem_instalments = 1, so the whole cotisation falls in the first month of each policy year), waiting_period_y = 0, accident_multiplier = 1.00. Hence n = 75 − 58 = 17 policy years, proj_len = 204 months, the frame is t = 0 … 203, and the two tables below are the twelve months of policy year 1 and the entire projection summed into policy years.

Assumptions, each tagged. Tariff rates r(x) for ages 58–74 read from the published grid [S3] — 1,05 / 1,13 / 1,56 / 1,68 / 1,81 / 1,97 / 2,14 / 2,33 / 2,55 / 2,78 / 2,88 / 3,14 / 3,43 / 3,74 / 4,09 / 4,46 / 4,86 % — each read once, at the anniversary opening its policy year, so P(t) is flat across that year’s twelve months. Mortality q_d(t) = 0.00400 × 1.09^dur(t) std, an annual rate. PTIA q_p(t) = 0.20 × q_d(t) for dur(t) ≤ 6 (attained ages 58–64) and 0 from dur(t) = 7, i.e. from month 84 (attained age 65 = ptia_end_age) std. Lapse 12 % / 10 % / 8 % / 6 % from the fourth year std, annual rates read from the policy-year-keyed table at policy_year(t) = dur(t) + 1, with w = 0 through the whole of policy year 17 (months 192–203) because the last projected year ends at expiry (processing order, step 9) — the assumption that fixes the lapse/survivor split in the closure check below, though it moves no cash flow. All four rates reach the month at 1 − (1 − r)^(1/12), the insured pair converted on its sum and split in the ratio q_d : q_p. Suicide factor σ = 0.98 through policy year 1 (months 0–11), 1.000 thereafter, applied to death claims only std R1. Expenses std: E0 = 250 € at issue, (25/12) × 1.02^dur(t) per in-force policy per month, initial commission 40 % of the cotisation collected through policy year 1, renewal commission 5 % of it from policy year 2, claim expense 150 € per death or PTIA claim. No accident option, no indexation, no tariff drift, no behavior modules.

expenses below is the total of acquisition, maintenance, claim expense and commission. All amounts in euros; pols_if to six decimals; cash flows to the cent.

The months of policy year 1 (t = 0 … 11)#

These twelve rows are result_cf() row for row, and they are what an annual-mode policy looks like on a monthly grid: the whole cotisation and the whole acquisition cost fall in month 0 and nothing else does, so month 0 is strongly positive and the eleven months after it carry only claims and a twelfth of the maintenance charge. Model point 4 is the contrast — twelve instalments of 77,77 €.

t

pols_if

premiums

claims_death

claims_ptia

expenses

commissions

net_cf

0

1.000000

1,575.00

49.11

10.02

882.14

630.00

633.73

1

0.989007

0.00

48.57

9.91

2.12

0.00

−60.60

2

0.978135

0.00

48.03

9.80

2.10

0.00

−59.93

3

0.967383

0.00

47.51

9.70

2.07

0.00

−59.28

4

0.956748

0.00

46.98

9.59

2.05

0.00

−58.62

5

0.946231

0.00

46.47

9.48

2.03

0.00

−57.98

6

0.935829

0.00

45.96

9.38

2.01

0.00

−57.34

7

0.925542

0.00

45.45

9.28

1.98

0.00

−56.71

8

0.915367

0.00

44.95

9.17

1.96

0.00

−56.09

9

0.905305

0.00

44.46

9.07

1.94

0.00

−55.47

10

0.895353

0.00

43.97

8.97

1.92

0.00

−54.86

11

0.885510

0.00

43.49

8.87

1.90

0.00

−54.26

expenses(0) = 250 + 2,08 + 0,06 + 630 = 882,14 €: the acquisition cost, one twelfth of the 25 € annual maintenance charge, the claim expense on the month’s 0,00040088 of claim events, and the 40 % initial commission on the whole annual cotisation collected in this month.

The month-0 decrement is the check on the rate conversion. q_d(0) = 0,00400 and q_p(0) = 0,00080, so the annual insured decrement is q(0) = 0,00480 and q_m(0) = 1 − (1 − 0,0048)^(1/12) = 0,00040088, split 0,00033407 / 0,00006681 in the exact ratio 0,20; w(0) = 12 % gives w_m(0) = 1 − (1 − 0,12)^(1/12) = 0,01059624. So l(1) = (1 − 0,00040088)(1 − 0,01059624) = 0,989007 ✓, and twelve such months land on l(12) = 0,875776 — the annual model’s l(1), exactly.

The same frame summed into policy years (years 1–17)#

Each row is the total of its twelve months, produced by result_cf_annual(); pols_if is l(12(k − 1)), the count entering the policy year, which is the number the annual-step model this replaced carried on the same row. The Total row covers all seventeen years.

policy year (months t)

age

r(x)

pols_if

premiums

claims_death

claims_ptia

expenses

net_cf

1 (0–11)

58

1,05 %

1.000000

1,575.00

554.94

113.25

904.22

2.58

2 (12–23)

59

1,13 %

0.875776

1,484.44

546.03

109.21

96.12

733.09

3 (24–35)

60

1,56 %

0.784075

1,834.73

538.15

107.63

111.97

1,076.98

4 (36–47)

61

1,68 %

0.717235

1,807.43

541.82

108.36

109.47

1,047.77

5 (48–59)

62

1,81 %

0.670010

1,819.08

551.70

110.34

109.19

1,047.84

6 (60–71)

63

1,97 %

0.625542

1,848.48

561.45

112.29

109.83

1,064.91

7 (72–83)

64

2,14 %

0.583667

1,873.57

571.01

114.20

110.28

1,078.07

8 (84–95)

65

2,33 %

0.544230

1,902.08

580.35

0.00

110.83

1,210.91

9 (96–107)

66

2,55 %

0.507836

1,942.47

590.28

0.00

112.12

1,240.07

10 (108–119)

67

2,78 %

0.473561

1,974.75

599.98

0.00

113.04

1,261.73

11 (120–131)

68

2,88 %

0.441280

1,906.33

609.40

0.00

108.94

1,187.98

12 (132–143)

69

3,14 %

0.410875

1,935.22

618.48

0.00

109.74

1,207.00

13 (144–155)

70

3,43 %

0.382236

1,966.60

627.16

0.00

110.68

1,228.76

14 (156–167)

71

3,74 %

0.355260

1,993.01

635.36

0.00

111.39

1,246.25

15 (168–179)

72

4,09 %

0.329849

2,023.62

643.01

0.00

112.34

1,268.27

16 (180–191)

73

4,46 %

0.305913

2,046.56

650.03

0.00

112.92

1,283.61

17 (192–203)

74

4,86 %

0.283369

2,065.76

675.04

0.00

113.62

1,277.10

Total

31,999.13

10,094.20

775.29

2,666.69

18,462.95

claims_lapse = 0.00 in every month and is omitted from the table for space; it is a required column of result_cf(). The Total row is the sum at full precision, then rounded — for claims_death that is 10 094,20 € against 10 094,19 € if the seventeen already-rounded cells are added, a one-cent accumulation. Assert the full-precision total.

Cross-checks against the annual grid this replaced. The pols_if column is the same number, row for row, because it is a pure anniversary quantity and the monthly rates compound back to the annual ones; so are age, r(x) and prem_pp. premiums is unchanged to the cent, row for row and in total (31 999,13 €), because an annual-mode cotisation is collected on the anniversary and weighted by the anniversary in-force under either grid — and so is commissions, a policy-year rate on it.

Claims and expenses are not, and the direction is the informative part. Death claims fall from 10 396,90 € to 10 094,20 € (−2,9 %) and PTIA claims from 804,25 € to 775,29 € (−3,6 %), because a block that is losing lives every month is exposed for less of the year than an anniversary weighting assumes; the PTIA gap is the larger of the two because the PTIA years are the early high-lapse ones. Expenses fall from 2 676,38 € to 2 666,69 € (−0,36 %), the net of maintenance falling from 263,96 € to 254,60 € (−3,5 %, the same decrementing-block effect on a charge now borne monthly) against the claim expense following its own smaller base. net_cf is the residual and rises from 18 121,59 € to 18 462,95 € (+1,9 %). One sign changes with it: policy year 1 moves from −38,72 € to +2,58 €, so the shape to describe is no longer “almost no new-business strain” but “the first year is marginally positive” — the year’s cotisation now slightly more than paying the year’s acquisition cost.

Level-premium variant. The same cell with premium_form = constante and level_premium = 0, so P_lev is derived by equivalence at tech_rate = 0,5 %:

P_lev = 60,476.2476 / 15.449728 = 3,914.3891 €   (displayed 3,914.39)

The variant table below carries P_lev unrounded. Every displayed row is stable at two decimals under either treatment; only the premium total moves, to 36 367,47 € if P_lev is rounded to the cent before projecting.

Selected policy years of the resulting projection, from its own result_cf_annual() — decrements, benefits and pols_if are identical to the table above, only the premium and the commission change, and the Total row covers all seventeen years, not only the five displayed:

policy year

age

prem_pp

premiums

claims_death

claims_ptia

expenses

net_cf

1

58

3,914.39

3,914.39

554.94

113.25

1,839.98

1,406.22

2

59

3,914.39

3,428.13

546.03

109.21

193.30

2,579.59

3

60

3,914.39

3,069.17

538.15

107.63

173.69

2,249.70

8

65

3,914.39

2,130.33

580.35

0.00

122.24

1,427.74

17

74

3,914.39

1,109.22

675.04

0.00

65.79

368.39

Total

36,367.46

10,094.20

775.29

3,703.89

21,794.07

P_lev = 3 914,3891 € and the premium total 36 367,46 € are unchanged by the conversion: the equivalence is annual and this point pays annually, so both the striking and the collection live on the anniversary.

The two forms are the whole point of this product. The revisable premium runs from 1 575,00 € to 7 290,00 € — a factor of 4,6286, exactly r(74)/r(58) = 4,86/1,05 and independent of the capital — while the level premium is flat at 3 914,39 €, above the tariff until policy year 9 and below it from policy year 10. The revisable form has almost no new-business strain (policy year 1 is +2,58 €); the level form is strongly positive in policy year 1 (+1 406,22 €) and would carry a real provision mathématique against the later years R11 R13.

Checks.

The cotisation rule, from the source’s own example. The carrier publishes “150 000 € × (0,60 : 100) = 900 € pour un an” at attained age 49 [S3]. The same rule at attained age 58 gives P(0) = 150 000 × 1,05/100 = 1 575,00 €, and the last year’s rate reproduces 150 000 × 4,86/100 = 7 290,00 €. The ratio 7 290,00 / 1 575,00 = 4,6286 equals 4,86 / 1,05 = 4,6286 — the premium multiple over the contract depends only on the grid, not on the capital, which is a one-line test of the whole premium engine.

Policy year 3 rebuilt from scratch, a different way. The in-force rebuild is exact on either grid, because it runs on the annual factors: l(12) = (1 − 0,00400 − 0,00080)(1 − 0,12) = 0,99520 × 0,88 = 0,875776; q_d = 0,00400 × 1,09 = 0,004360 and q_p = 0,000872 in policy year 2, so l(24) = 0,875776 × (1 − 0,005232) × 0,90 = 0,875776 × 0,8952912 = 0,78407455, matching the table’s 0.784075.

The cash-flow rebuild is now a single month’s product, because a policy year’s claims are the sum of twelve of them. Take month 24, the first of policy year 3. The annual rates are q_d = 0,00400 × 1,09² = 0,0047524 and q_p = 0,00095048, so q = 0,00570288 and q_m = 1 − (1 − 0,00570288)^(1/12) = 0,00047649, split q_dm = 0,00039707 and q_pm = 0,00007941. Then claims_death(24) = 150 000 × 0,78407455 × 0,00039707 = 46,70 and claims_ptia(24) = 150 000 × 0,78407455 × 0,00007941 = 9,34. Expenses: (25/12) × 1,02² × 0,78407455 = 1,6995 maintenance, 0,05 × 2 340,00 × 0,78407455 = 91,7367 commission on the whole annual cotisation collected this month, 150 × 0,78407455 × 0,00047649 = 0,0560 claim expense — total 93,49. And 1 834,73 − 46,70 − 9,34 − 93,49 = 1 685,20, the frame’s net_cf(24). The policy-year row above is the sum of that month and the eleven after it: 1 076,98 €.

The decrements close, and nothing is paid twice. Summing the three exits over the 204 months: deaths 0,06737020, PTIA claims 0,00516859, lapses 0,64859269, plus l(204) = 0,27886852 — total 1,00000000 exactly. The survivor term is identical to the annual grid’s, being a pure anniversary quantity; the three exit terms are not, and the reallocation is the expected one — a decrementing block reaches the claim decrements later in the year, so fewer lives leave through them and correspondingly more leave as a lapse (the annual grid read 0,06939268 / 0,00536169 / 0,64637711). The split is what w = 0 in the final policy year decides.

Multiplying total claim events by the capital, 150 000 × (0,06737020 + 0,00516859) = 10 880,82 €, against claims actually paid of 10 094,20 + 775,29 = 10 869,49 €. The difference is 11,33 €, which is precisely the policy-year-1 suicide withholding 0,02 × 150 000 × 0,00377512 — the deaths of the twelve months of policy year 1 — R1, so the exclusion factor is the only thing standing between expected claim events and expected claim amounts, which is what “PTIA is an acceleration, not an addition” means arithmetically.

The level premium is a weighted average of the grid. Independently of the equivalence formula, P_lev / SA should be the v^y p_τ(y)-weighted mean of the seventeen grid rates. That mean is 2,60959276 %, and 150 000 × 0,0260959276 = 3 914,3891 € — the same figure, reached without ever forming the premium stream. The weights sum to 15,449728, the annuity-due factor, and P_lev × 15,449728 = 60 476,25 € equals the present value of the revisable stream on the same basis. Note what this identity does not say: the projected premium totals differ (36 367,46 € against 31 999,13 €), because lapses remove policies before the expensive late years that the level premium has already been charging for. Pitfall 11.

Valuation and reserve pointers#

This library projects gross best-estimate-style liability cash flows, undiscounted, on a declared grid. The valuation layers consume them and are cited, not reproduced.

  • The French statutory provision mathématique. Art. R. 343-3 defines it as the difference between the present values of the two parties’ commitments and requires it to include an estimate of future management costs equal to the chargements de gestion built into the tariff R11 REG-R6. On the revisable form the PM is close to nil at each anniversary — the year’s cotisation buys the year’s risk, so what remains is an unearned-premium and outstanding-claims position R11 R13. On the constante form it builds and releases in the classic way: where the premium rate is flat while the death rate rises, “un montant de PRC est toujours constitué pendant la durée” R13. That contrast is the reason the constante form is carried at all. The provision pour risques croissants of art. R. 343-7 is defined for maladie and invalidité, not death R12; the death-cover analogue is the R. 343-3 PM — “la même provision de prime s’appelle PM en vie et PRC en non-vie” R13. Art. A. 343-1-1 requires acquisition loadings to enter the premium-payer’s commitment and floors the result at zero, at the surrender value and at the reduced-capital provision; the last two are zero here R3 R13, so the operative floor is non-negativity. Art. 142-3 of ANC 2015-11 (as amended by ANC 2016-12) fixes the rate at no more than the tariff rate and the table at the one in force when the tariff was applied, with the option to migrate in-force contracts at each annual inventory and to spread a change of basis over at most eight years R13.

  • Medical selection. The Institut’s illustrative claims abatement for a selected book is 70 % in year 1, 50 % in year 2, 20 % in year 3 R13. It is not applied in the base run — stacking a selection abatement on an already-std mortality proxy would compound two unsourced choices — but it is the first refinement a user with real experience should make, and it changes the sign of the early-duration provision R13.

  • Solvabilité II best estimate. Probability-weighted future cash flows discounted at the relevant risk-free term structure, plus a risk margin REG-R1 REG-R2 REG-R4, with EIOPA publishing the curves monthly REG-R5. BEL = Σ_t v(t) × liability_cf(t) over the recursion above. No cost-of-capital rate, contract-boundary rule or standard-formula shock in this library was read from a retrieved instrument, so every such figure is std REG-R2.

  • Contract boundary — the open question on this product. The contract is a one-year cover renewed by tacite reconduction whose tariff the insurer recomputes at every renewal [S1] [S2] [S3] [S6] [S7] [S9] and, at two of them, may also reprice for class experience [S1] [S6]. Whether the Solvabilité II contract boundary therefore ends at the next renewal — as it would for a reviewable-premium contract — could not be determined: the Delegated Regulation’s boundary rules were not retrievable REG-R2 and the point is unverified. The model’s posture: project to the age limit and publish the full stream; a boundary-truncated view is obtained by truncating result_cf() after the first twelve rows, t = 0 … 11, or equivalently result_cf_annual() after its first row. Do not bake the truncation into the projection.

  • IFRS 17 and professional standards. Fulfilment cash flows plus a contractual service margin, effective from 1 January 2023 with no French carve-out REG-R45; the same expected-cash-flow engine feeds it, and grouping, CSM and risk adjustment are out of scope. Norme de Pratique Actuarielle 2 — Modèles actuariels, adopted 15 June 2015 with effect from 1 January 2016, expressly covers pricing models and the technical studies attached to new products REG-R44; NPA 4, on best-estimate life provisions, was not retrieved and is unverified REG-R44.

Key sensitivities and model risks#

In rough order of leverage for a French protection block:

  1. The premium form. Switching revisable → constante moves projected premium income by +13,7 % (31 999,13 € → 36 367,46 €) and net_cf by +18,04 % on the worked configuration, with no change to a single claim. It is the largest single structural lever in the model, and the constante side of it is std — no French standalone contract in the corpus uses it [S1] [S2] [S3] [S6] [S7] [S9] [S10].

  2. Mortality basis. The reference basis is a std Gompertz proxy because TH 00-02 / TF 00-02 are annexed to an arrêté and not redistributed here R6 REG-R22 REG-R23, and no French insurer publishes a basis [S1]–[S9]. Both the level (q_d at 58) and the slope (9 % per year of age) are unsourced; the slope is the more dangerous of the two on a 17-year run, since it compounds. It is calibrated to the published tariff grid’s own gradient and sits at the top of it — the grid compounds at 7,7 % a year over ages 42–58 and 8,98 % over the whole rated span 35 → 74 [S3] — and a tariff gradient is not a mortality gradient.

  3. Lapse. Nothing in the corpus supports any lapse rate. Cumulative lapse reaches 64,9 % of the original cohort over the worked configuration, so the assumption governs how much of the rising-premium tail is ever collected — and on a revisable contract the late years are the profitable ones, which inverts the usual protection intuition that early lapse is what hurts.

  4. PTIA incidence ratio. ptia_ratio = 0.20 is a pure placeholder with no source at all. It moves 775,29 € of claims in the worked configuration — 7,1 % of total claims — and it interacts with ptia_end_age, since the whole of that exposure sits in the first seven policy years.

  5. Contract boundary. If the boundary is one year rather than the full cover period, the entire projection beyond month 11 leaves the technical provision. Nothing in this library resolves it REG-R2.

  6. Tariff drift. The base run freezes the rate card at its retrieved vintage, but the same carrier’s current page implies about 0,189 % at age 35 against the grid’s 0,17 % [S3] [S4], and two of the eight carriers reserve an explicit right to reprice on class experience [S1] [S6]. A drift assumption is a premium-income assumption, not a mortality one.

  7. Expense levels on small capitals, and the suicide factor. Minimum capitals run from 6 097,96 € [S9] to 100 000 € [S8]; at the representative carrier’s 20 000 € minimum [S3] [S4] the year-one cotisation at age 58 is 210 € against 250 € of acquisition expense std, so the per-policy expense assumption, not mortality, decides whether the cell is viable. The suicide factor is worth only 11,33 € here — immaterial to the result, material to correctness, because an implementation that applies it to PTIA, or to every year, or that imports the 120 000 € immediate-cover ceiling from loan business R1 R2, is wrong in a way the totals will not reveal.