The Projection Space#

The by-policy projection of the Term_JP_A model.

The Space is parameterized by point_id, so Projection[1] is an ItemSpace projecting model point 1:

>>> Projection[1].result_cf()          # the worked example's anchor cell
>>> Projection.point_id = 3            # or switch the default

t counts policy years, 1-based: t = 1 is the first policy year and t = proj_len() the last. There is nothing after it — no 満期保険金 (maturity benefit), no 解約返戻金 (kaiyaku-henreikin, surrender value), no run-off and no tail state of any kind [S1][S8][S10][S14].

Where the horizon ends is a product question rather than a convention. A 歳満了 (sai manryō) contract ends at its stated age and never renews [S1], so proj_len() = policy_term(). A 年満了 (nen manryō) 更新型 contract renews automatically at the end of every 保険期間 until it reaches the renewal ceiling of attained age 80 [S1][S2][S8], so proj_len() = renew_ceiling() - age_at_entry() and a ten-year term issued at 30 is projected for fifty years across five priced terms. contract_boundary = current_term truncates instead at the end of the term in force at the valuation date.

Input data

Inputs are external files: plain CSVs living in the model folder’s parent directory, products/term_life/, read at run time rather than stored inside the model. The model folder therefore holds nothing but formulas — no _data/, no IOSpec, no embedded values — so a diff of the model shows logic changes only, and an input can be edited or swapped without rewriting the model. This follows annuallife.TradLife_A; contrast basiclife.BasicTerm_S, which keeps its inputs inside the model through modelx’s IOSpec machinery.

The consequence worth knowing: the model is not portable on its own. Copying the Term_JP_A folder without its parent’s CSVs produces a model that reads and then fails on first evaluation.

Each table has a filename Reference and a reader Cells, both on Data, reached here through the data Reference:

Reference	Cells	File
model_point_file	data.model_point_table()	model_point_table.csv
mort_table_file	data.mort_table()	mort_table.csv
lapse_table_file	data.lapse_table()	lapse_table.csv
prem_rate_file	data.prem_rate_table()	prem_rate_table.csv

Naming

Cells names follow lifelib’s basiclife.BasicTerm_S wherever that model has an analogue — pols_* for policy counts, plural nouns for cash flows, *_rate for rates, *_pp for per-policy amounts, claims(t, kind) with an uppercase kind string, pols_if_at(t, timing) for the within-year in-force reads. The technical notes use compact actuarial symbols instead. The mapping is:

Notes symbol	Cells	Meaning
(row label)	model_point()	The selected model point row
t	(the index of result_cf)	Policy year, 1-based
x	age_at_entry()	契約年齢, 満年齢
x + t - 1	age(t)	Attained age in year t
(none)	sex()	Rating factor, M or F
(none)	term_type()	nen (renewable) or sai
n	policy_term()	保険期間, in years
w_r	renew_ceiling()	Attained age renewal stops
N	horizon_ceiling()	Years from entry to w_r
t = 1..N	proj_len()	Last projected policy year
(none)	contract_boundary()	ceiling or current_term
k	term_index(t)	Term index, 1 in the first
x_k	term_start_age(k)	Attained age at term k start
m_k	term_len(k)	Length of term k, truncated
SA	sum_assured()	保険金額, level
f	policy_fee_m()	Flat monthly element, 248
r(sex, x, m)	prem_rate_m(t)	Rate per 5,000,000 of cover
qbar(x, m)	mort_table_mean(x, m)	Mean table rate over m years
P_m(k)	premium_mth_pp(t)	Monthly premium, whole yen
P_a(k)	prem_pp(t)	Annualized premium
(table)	mort_rate_at_age(x)	Table rate at an age
(table)	mort_rate_base(t)	Table rate in year t
(margin removal)	mort_be_factor	Best-estimate factor, 0.80
lambda	sel_lapse_lambda	Selective-lapsation loading
l_ref	sel_lapse_ref	Selective-lapsation reference
(none)	sel_lapse_factor(t)	Mortality loading on stayers
q(t)	mort_rate(t)	Death and 高度障害 decrement
w(t)	lapse_rate(t)	Ordinary lapse rate
d(t)	decline_rate(t)	Renewal-decline rate
d_0	decline_base	Flat decline rate, 15%
beta, d_max	decline_beta, decline_max	Decline elasticity module
l(t)	pols_if(t)	In force at start of year t
(within-year)	pols_if_at(t, timing)	BEF_DECR / BEF_LAPSE / …
D(t)	pols_death(t)	Expected claims in year t
(none)	pols_lapse(t)	Ordinary lapses
(none)	pols_decline(t)	Renewal declines
lap(t)	pols_lapse_pool(t)	Reinstatable population
rho	reinstate_rate	Reinstatement rate
(window)	reinstate_window	3 years [S1]
(none)	pols_reinstate(t)	Reinstatements into l(t+1)
(none)	pols_lapse_expire(t)	Pool leavers, window expired
(none)	wop_waived_frac(t)	Fraction with premiums waived
(none)	pols_payer(t)	Policies paying premium
A	ln_amount()	Accelerated amount
i_ln	ln_interest_rate	Six-month discount rate
a(t)	ln_share(t)	Acceleration take-up
(payout formula)	ln_payout_pp(t)	A - interest - premiums
P_a x l	premiums(t)	Premium income
SA x D(t)	claims(t, kind)	Benefit outgo by kind
ec	expense_claim	Claim expense per claim
ec x D(t)	claim_expenses(t)	Claim expense outgo
E0	expense_acq	Acquisition expense, 15,000
e(t)	expenses(t)	Acquisition + maintenance
c0	comm_init_pp()	Initial commission per policy
c_r	comm_renewal_rate	Renewal commission rate, 5%
(footnote 4)	comm_new_term(t)	Commission at a 更新
(none)	commissions(t)	Commission outgo
CF(t)	net_cf(t)	Net cash flow, income positive

Four names needed care.

The notes write q(t) for the decrement and qbar(x, m) for a mean of table rates used by the premium scale. Those are different quantities on different bases — one is best-estimate and one is not — so they get different names: mort_rate() is the decrement, and mort_table_mean() averages mort_rate_at_age(), which reads the table unadjusted. Feeding the best-estimate rate into the premium extension would move a premium scale by an assumption that has nothing to do with pricing.

P_m(k) and P_a(k) are indexed by the term in the notes and by the policy year here. premium_mth_pp() and prem_pp() take t and resolve the term through term_index(), which is what keeps every cash flow line indexed the same way. The premium is nonetheless level within a 保険期間, and check_prem_level() asserts it.

d(t) is spelled decline_rate(), not any variant of lapse. It is a different event at a different time from a different population — see below — and the name is the first line of defence against the two being merged.

lap(t) is spelled pols_lapse_pool() because it is a stock, the lapsed lives still inside the three-year 復活 window, whereas pols_lapse() is the year’s flow into it.

更新 reprices; it does not re-issue

This is the structural difference from this repository’s UK term model and the notes’ second-listed pitfall. At a renewal boundary the premium is recomputed on attained age and on the scale then in force [S1][S4][S8][S12], and nothing else resets:

pols_if is continuous across the boundary. There is no reset to 1.
No acquisition expense and, in the base run, no commission is paid at a renewal (comm_new_term_rate is 0). A 更新 is not new business: no new 保険証券 is issued and no 告知 is taken [S1][S4].
The suicide and contestability clocks run from the original 責任開始日 and do not restart on 更新 [S1][S4][S7][S8]. Only 復活 restarts them [S1]. Neither clock is monetized in the base run, so neither is a cells; both are stated here because treating each renewed term as a fresh policy gets persistency, the strain pattern and both clocks wrong at once.

Truncation at the ceiling shortens the term, not the horizon: a renewal that would carry the policy past attained age 80 renews as an 80歳満了 term instead [S1][S2][S8], so an issue age of 35 has a final term of five years and the projection still ends exactly at 80. That is term_len(), and model point 4 exercises it. Three other market rules exist — shorten to expiry age 90, shorten or lengthen to a 指定年齢, auto-convert to another product [S4][S7][S12] — and importing one changes the horizon.

Renewal decline is not lapse

decline_rate() is non-zero only in a boundary year, and the exits it produces are taken after mortality and after ordinary lapse — the notes’ processing order, steps 3, 4 and 5. It is also the larger decrement where it applies: in year 10 of the anchor cell it removes 0.08235591 of the 0.11175249 lives that leave that year, 74% of all exits. Folding it into lapse_rate() makes the boundary invisible and mis-times most of the cohort’s departure.

Two behaviours roll into the one rate, and a production model should separate them: the policyholder who gives notice to decline, and the policyholder whose first renewed premium goes unpaid through grace, in which case the renewal is treated as never having happened and the contract terminates at the original expiry [S1][S7]. Only the first is a decision. Both leave at the boundary, which is why one rate can carry them — and why neither may appear in force in year t + 1 collecting the renewed premium.

One decrement, one benefit

生保標準生命表2018（死亡保険用）**includes 高度障害** (kōdo shōgai, severe disability) inside its death rate [REG-R20], and the contract pays one sum assured and terminates on whichever of the two events comes first [S1][S8]. mort_rate() is therefore the combined death-and-高度障害 decrement, and there is no disability incidence anywhere in this model. Adding one on top of the table double-counts the benefit — the notes’ first-listed pitfall.

The same reasoning governs the リビング・ニーズ特約 module below: an acceleration is a re-timing and re-pricing of the death benefit, not a second claim, so ln_share() splits the existing decrement rather than adding to it.

Lapse pays nothing, and there is no 自動振替貸付

There is no 解約返戻金 at any duration on this composite [S1][S4][S6][S8][S9][S10][S13][S14], so an ordinary lapse is a pure decrement: it moves pols_if() and pays nothing. claims(t, "LAPSE") exists and returns zero, and result_cf() carries the zero column, because the notes list a non-zero lapse row as a pitfall imported from models with cash surrender values — and because one carrier in eight does write this design with a surrender value [S12], so the zero is asserted from the composite’s sources rather than assumed from the product class.

There is no 自動振替貸付 (jidō furikae kashitsuke, automatic premium loan), stated in terms by one carrier [S7], and no collateral for a 契約者貸付 either — the second an inference from the missing surrender value rather than a citation, since that carrier points its policyholders at the 契約貸付制度 instead [S7] and the document appearing to rule the policy loan out could not be extracted [S11]. Importing the APL mechanic that WholeLife_JP_A carries would create a no-lapse cushion this contract does not have. Grace, then 失効, then 復活-or-not is the whole persistency machinery here.

The premium chassis, and where it stops being sourced

Japanese carriers publish rate cards, so the structure decomposes exactly [S2]:

P_m(k) = f + r(sex, x_k, m_k) * SA / 5,000,000
x_k    = x + (k - 1) * n
m_k    = min(n, w_r - x_k)
P_a(k) = 12 * P_m(k)

with f = 248 per month and P_m rounded to the whole yen before annualization, as rate cards are published [S2][S9][S10]. Four cells are sourced — male ages 30, 40 and 50 and female age 30, all at a ten-year term. Ages 60 and 70 are published by no carrier, and the anchor cell reaches both, so prem_rate_m() extends the scale off the is_anchor row of the matching sex [std]:

r(sex, x, m) = r_anchor * mort_table_mean(x, m) / mort_table_mean(x_a, m_a)

The extension back-casts to ¥958.9 at age 30 against the published ¥974 (-1.5%) and to ¥1,806.4 at age 40 against ¥1,823 (-0.9%), and gives ¥8,976 at 60 and ¥23,881 at 70. Published cells are always used where they exist; the extension fills the gaps. That the back-cast is close is reassuring about the form and says nothing about the level an insurer will charge in 2056 — the notes rate it the third-largest lever on this cell.

The ¥248 is a premium component, not an expense recovery [S2]. It enters the model only through prem_pp(); crediting it against maintenance expense counts it twice.

Modules that are off in the base run

Eight of the notes’ optional constructions are implemented and switched off, so that the base run reproduces the worked example while the machinery stays visible and testable. Three of them are model point columns, five are References:

リビング・ニーズ特約 (living_needs), a discounted acceleration of the death benefit. Off on the anchor cell; on for model points 6 and 9.
保険料の払込の免除 (wop), the premium waiver on an accident-caused 別表4 state. Off on the anchor cell; on for model point 7. 別表4 is a materially lower bar than the 別表3 test for 高度障害 — loss of one eye, deafness in both ears, loss of one limb at the wrist or ankle [S1] — so the waiver incidence is not the 高度障害 incidence and wop_waived_frac() does not reuse mort_rate().
復活 (reinstatement), the lapsed-but-reinstatable population and its three-year window [S1]. Off on the anchor cell; on for model point 8.
Contract boundary (contract_boundary), current_term truncating at the end of the 保険期間 in force at the valuation date. ceiling on the anchor cell; current_term on model point 5. The two differ by more than a rounding — +¥50,400.25 against -¥15,878.74 on the same cell — and the ESR standard-model treatment of a no-underwriting auto-renewal that would settle which is right is [unverified] here [REG-R16], so the model does not rule.
Selective lapsation, q_eff = q (1 + lambda max(0, 1 - l(t)/l_ref)), with sel_lapse_lambda = 0. Stronger here than on a UK term policy: renewal takes no 告知 [S1][S4][S8][S12], so a life that has become uninsurable elsewhere renews while a healthy life re-shops, and the decision recurs four times on the anchor cell.
Renewal-decline elasticity, d = min(d_max, d_0 (P_a(k+1)/P_a(k))^beta), with decline_beta = 0 giving the flat 15%. The premium jump the elasticity would respond to accelerates: 1.87, then 2.16, 2.28 and 2.66.
Age-basis shift, q_x -> sqrt(q_x q_(x+1)), with mort_age_shift = False. 契約年齢 is 満年齢 (man-nenrei, age last birthday) [S1] while 標準生命表2018 is built for 保険年齢 (hoken-nenrei, age nearest birthday) [REG-R20], so reading the table at 満年齢 reads it half a year early and understates mortality. The base run accepts and states that bias; the shift module must move q up, not down — about 0.7% at age 30 and 4.2% at age 40.
Commission at 更新, comm_new_term_rate = 0. Set it to reproduce a scale paying first-year rates on each renewed term, which would change the sign of the cash flow in years 11, 21, 31 and 41. No document in the source set discloses a commission scale at all, so the zero is a choice and not a fact.

Sign convention and the annual-grid bias

The notes’ CF(t) is already income positive — they write + = inflow — which is the library-wide sign of net_cf(), so there is no outgo-positive liability_cf companion to publish: one stream, one sign, one name.

Premiums are annual in advance with no allowance for premiums ceasing at a mid-year death or lapse, which slightly overstates premium income; the offsetting understatement is the end-of-year claim timing. The notes are explicit that these are a matched pair, and that applying a further half-year premium adjustment on top of the end-of-year claim timing would double-count the correction.

What is not modelled, and why

減額 (a reduction in sum assured) changes SA and P_a together, which is a model point re-parameterization rather than a decrement [std scope] [S1][S8]. クーリング・オフ is out of scope: the projection begins with cover in force and the eight-day statutory population already out [REG-R36][S1]. The 復活 arrears, payable at 年6% compound [S1], are not monetized: they settle premiums for years in which this projection collected none, so recognizing them would need a missed-premium ledger the notes do not specify [std scope]. And a partial acceleration under リビング・ニーズ特約 leaves a reduced contract in force at a reduced premium [S1][S7] — a second transition, not one benefit with two amounts — so ln_amount() raises rather than approximating it. It cannot arise inside the composite’s ¥1,000,000-¥30,000,000 envelope, where the ¥30,000,000 per-insured cap is exactly reached at the ceiling and never reduces a single-contract payment: ln_cap_binds() tests that with a strict inequality.

Cells Descriptions#

model_point()[source]#: The selected model point as a Series.

sex()[source]#: The sex (M / F) of the insured, a rating factor of the premium scale [S2].

age_at_entry()[source]#

x: the 契約年齢 (issue age), 満年齢 with fractions discarded [S1].

Age last birthday, not age nearest birthday. 生保標準生命表2018（死亡保険用）is built for a 保険年齢 basis [REG-R20], so reading it here reads it half a year early and understates mortality; mort_age_shift is the optional correction and the Space docstring states the direction. The composite’s envelope is 20-65 [S1][S2].

term_type()[source]#

nen for a 年満了更新型 contract, sai for a 歳満了 one [S1][S4][S7].

The single most consequential model point attribute. A 年満了 contract renews automatically at attained-age rates to the ceiling; a 歳満了 contract has one term, one premium and no repricing [S1], and applying the renewal machinery to it invents cover the contract does not have.

policy_term()[source]#

n: the 保険期間 in years — the priced term, not the projection horizon.

Read from term_y on a 年満了 point and implied by expiry_age on a 歳満了 one [S1][S4]. On a 更新型 contract the horizon is horizon_ceiling(), which is longer, because the contract renews.

renew_ceiling()[source]#

w_r: the attained age at which renewal stops, 80 on the composite [S1][S2][S8].

Observed ceilings run 75 to 99 and 80 is the mode; behaviour at the ceiling varies more than the ceiling does, and the composite truncates into an 80歳満了 term rather than shortening to another expiry age or converting to another product [S4][S7][S12].

sum_assured()[source]#

SA: the 保険金額, level for the whole term and unchanged through 更新.

Paid on death or on a 別表3 高度障害 state, whichever becomes payable first; either terminates the contract and the other is then not paid [S1][S4][S8][S9][S12]. The composite’s envelope is ¥1,000,000-¥30,000,000 in ¥1,000,000 units [S2][S9][S13].

premium_mode()[source]#

Monthly, semiannual or annual premium payment [S1].

Inert on the annual grid, which annualizes either way: P_a = 12 P_m. It is carried because the published rate cards are quoted monthly and because the notes’ monthly-grid sibling model distinguishes them.

contract_boundary()[source]#

ceiling or current_term: how far the liability is projected.

A Japanese 年満了 contract guarantees its premium only within the current 保険期間; at each 更新 the insurer recomputes it on attained age and the scale then in force [S1][S4][S8][S12]. That is a unilateral repricing right exercisable every ten years — but a scale-level right, not an individual one, because renewal takes no 告知 and no fresh underwriting, so the insurer cannot reprice a life for its own deterioration. The ESR coefficients that would settle where the boundary falls were not retrieved [REG-R16], so the model does not rule: it projects to the ceiling in the base run [std] and carries the truncation as a switch. Naming the convention is part of reporting the number.

living_needs()[source]#: Whether the リビング・ニーズ特約 acceleration module is on; false in the base run.

wop()[source]#: Whether the 保険料の払込の免除 module is on; false in the base run [S1][S8].

reinstatement()[source]#

Whether the 復活 module is on; false in the base run [S1].

Off by default because reinstate_rate = 0.10 is an arbitrary placeholder with a material persistency effect: no carrier in the source set publishes a reinstatement rate, no industry statistic in the set gives one, and no observed range can be quoted. A round tenth was chosen for the same reason as ln_take_up — so that no reader reads it as an estimate — and the only defence of it is that the module is off in the base run, leaving the worked example and every published figure independent of it. What is published, and is therefore not [std], is the window — three years, against arrears at 年6% compound [S1], on evidence of health [S8] — which is why reinstate_window is sourced while the rate beside it is not.

horizon_ceiling()[source]#

N: policy years from entry to the renewal ceiling.

renew_ceiling() - age_at_entry() on a 年満了 point, because the contract renews until it gets there [S1][S2][S8]; the term itself on a 歳満了 point, which never renews [S1].

proj_len()[source]#

The last projected policy year: horizon_ceiling(), or the current term.

contract_boundary = current_term truncates at the end of the 保険期間 in force at the valuation date, which for a policy projected from issue is policy_term(). On a 歳満了 point the two coincide.

age(t)[source]#: x + t - 1: the attained age (満年齢) at the start of policy year t.

term_index(t)[source]#

k: the term index — 1 in the original 保険期間, 2 after the first 更新, and so on.

1 + floor((t - 1) / n) on a 年満了 point; always 1 on a 歳満了 one, which never renews [S1]. The premium is a function of k and not of t, which is the state variable a Japanese term model needs and a UK one does not.

term_start_age(k)[source]#: x_k: the attained age at which term k starts, x + (k - 1) n.

term_len(k)[source]#

m_k: the length of term k in years, min(n, w_r - x_k).

Truncation at the ceiling shortens the term, not the horizon: a renewal that would carry the policy past attained age 80 renews as an 80歳満了 term instead [S1][S2][S8]. An issue age of 35 on a ten-year term therefore has a final term of five years and still ends exactly at 80. The truncated term is priced over its own shorter length, which is why this feeds prem_rate_m().

omega_age()[source]#

The highest attained age the shipped mortality table carries for this sex.

Used only to keep the optional age-basis shift from reading past the end of the table; no shipped model point comes near it.

mort_rate_at_age(x)[source]#

The table mortality rate at attained age x, unadjusted.

A [std] proxy for 生保標準生命表2018（死亡保険用）, anchored on the rates the technical notes quote and attribute [REG-R18][R4]; see Data. This is the valuation-table rate with its own margin still in it, and it includes 高度障害 [REG-R20]. It is read directly by the premium scale and only through mort_rate_base() by the decrement.

mort_table_mean(x, m)[source]#

qbar(x, m): the mean table rate over ages x .. x + m - 1.

The shape parameter of the [std] premium extension. Deliberately built on the table rate rather than on mort_rate(): a premium scale is not a best-estimate quantity, and feeding the best-estimate factor in would move a published rate card by an assumption that has nothing to do with pricing.

mort_rate_base(t)[source]#

The table rate at the attained age of policy year t, before the margin removal.

Optionally shifted to sqrt(q_x q_(x+1)) [std] when mort_age_shift is set, which is the annual grid’s correction for reading a 保険年齢 table [REG-R20] at 満年齢 [S1]. The shift raises the rate — about 0.7% at age 30 and 4.2% at age 40 — because the unshifted read is half a year early and understates. Off in the base run.

sel_lapse_factor(t)[source]#

The selective-lapsation loading on mortality in policy year t [std].

1 + lambda max(0, 1 - l(t)/l_ref). Lapsers and decliners are healthier than stayers, so a block that has shed a large proportion of its lives carries impaired mortality on the remainder. The mechanism is stronger here than on a UK term policy and one-directional: renewal takes no 告知 [S1][S4][S8][S12], so a life that has become uninsurable elsewhere renews while a healthy life re-shops — and the decision recurs at every boundary. Off in the base run (sel_lapse_lambda = 0), where it returns 1 in every year. sel_lapse_ref = 1.0 [std] is the cohort at issue, so the loading is driven by the proportion of the original block that has left.

mort_rate(t)[source]#

q(t): the best-estimate death-and-高度障害 decrement in policy year t.

mort_be_factor times the table rate, then the selective-lapsation loading, capped at 1. One decrement carrying one sum assured: 生保標準生命表2018（死亡保険用） includes 高度障害 inside its death rate [REG-R20] and the contract pays once and terminates on whichever event comes first [S1][S8], so projecting the table rate for death and adding a 高度障害 incidence on top double-counts the benefit.

mort_be_factor = 0.80 [std] is this model’s largest single lever and its least evidenced number. The table’s 作成概要 sizes its margin to hold the exceedance probability near 2σ, capped at 130% of the unadjusted rate [REG-R20], and removing a margin at its cap implies 1/1.3 = 0.769; against that the table already carries a forward improvement allowance and its base experience is 2008, 2009 and 2011. 0.80 is a round central choice between the two. No observed range can be given: no Japanese insurer publishes protection experience by duration.

lapse_rate(t)[source]#

w(t): the ordinary lapse rate applied at the end of policy year t [std].

9 / 7 / 6 / 5.5 / 5 percent, from lapse_table.csv; policy years beyond the table take its last row. The level is anchored to the LIAJ’s FY2024 whole-market 解約・失効率 of 5.6% of opening in-force sum insured [REG-R31] — the simple mean over the first ten years is 5.75% and the in-force-weighted mean 5.94%, both a little above it, which is the expected direction for an early-duration protection curve against a figure dominated by long-duration in-force. The shape is a convention with no Japanese published evidence behind it.

A lapse pays nothing: there is no 解約返戻金 at any duration [S1][S4][S8].

decline_rate(t)[source]#

d(t): the renewal-decline rate — non-zero only in a boundary year [std].

A proportion of survivors leave at each 更新 rather than accept the repriced contract. This decrement has no analogue in this repository’s UK or U.S. term models and it is large: on the anchor cell the monthly premium moves from ¥974 to ¥1,823 at the first renewal, a factor of 1.87 [S2]. Against that, renewal is the default — it happens unless notice is given, and the notice period is two weeks [S1][S8], the shortest of the three observed and the design that maximizes renewal by inertia. No carrier publishes a take-up or decline rate, so decline_base = 15% at every boundary.

Zero on a 歳満了 point, which never renews [S1], and zero in the final projected year, where the cover ends at the ceiling rather than renewing. The optional elasticity d = min(d_max, d_0 (P_a(k+1)/P_a(k))^beta) responds to the premium jump, which itself accelerates — 1.87, then 2.16, 2.28 and 2.66 across the anchor cell’s four renewals; decline_beta = 0 in the base run gives the flat rate. decline_max = 0.50 [std] caps it: at beta = 1 the largest jump reaches only 40%, so the cap binds at no boundary, while at beta = 2 it binds at every one.

policy_fee_m()[source]#

f: the flat monthly element inside the premium, ¥248 [S2].

A premium component, not an expense recovery. It enters the model only through prem_pp(); crediting it against expenses() counts it twice. Read from the is_anchor row of the matching sex, which is where the decomposition of the published rate card is recorded.

prem_rate_m(t)[source]#

r(sex, x_k, m_k): the marginal monthly rate per ¥5,000,000 of cover in year t.

The published cell where one exists [S2] — male ages 30, 40 and 50 and female age 30, all at a ten-year term — and otherwise the [std] extension off the is_anchor row of the matching sex:

r(sex, x, m) = r_anchor * qbar(x, m) / qbar(x_a, m_a)

Ages 60 and 70 are published by no carrier and the anchor cell reaches both, so the extension is unavoidable rather than optional. It back-casts to ¥958.9 at age 30 against the published ¥974 (-1.5%) and to ¥1,806.4 at age 40 against ¥1,823 (-0.9%), which is reassuring about the form of the scale and says nothing about the level an insurer will charge decades out.

premium_mth_pp(t)[source]#

P_m: the monthly premium per policy in policy year t, in whole yen.

f + r(sex, x_k, m_k) SA / 5,000,000, rounded half up to the yen before annualization, as published rate cards are quoted [S2][S9][S10]. Level within the 保険期間 and recomputed at each 更新 on attained age [S1][S4][S8][S12]; it is not guaranteed beyond the current term, which is what makes contract_boundary a question rather than a detail.

On the anchor cell this reproduces the published ¥974 exactly: 248 + 2 x 363.

prem_pp(t)[source]#

P_a: the annualized gross premium per policy in policy year t, 12 P_m.

The annualization is [std]: mode discounts and 前納 discounts are insurer-set and unpublished [S1][S7][S9][S14], so premium_mode() does not change the amount. On the anchor cell P_a = 12 x 974 = 11,688.

pols_if_init()[source]#

l(1) = 1: the model point is one policy, projected on an expected basis.

Survivorship multiplies the per-policy cash flows; no aggregation logic is specified in the technical notes and none is implemented.

pols_if(t)[source]#

l(t): the in-force probability at the start of policy year t.

pols_if_init() in year 1, then the notes’ roll-forward l(t+1) = l(t)(1 - q)(1 - w)(1 - d) plus any 復活 reinstatements. This is the weight on every cash flow of the same result_cf() row.

It is continuous across a renewal boundary: a 更新 reprices the contract, it does not re-issue it [S1][S4], so there is no reset to 1 and no new cohort. Defined one year beyond proj_len(), where it is the survivors whose cover expires at the ceiling — on the anchor cell l(51) = 0.026042, 2.6% of the cohort still in force after fifty years and four repricings. That is not a tail state: nothing is paid and nothing runs off, the cover simply ends [S1][S8][S10][S14].

pols_if_at(t, timing)[source]#

The number of policies in force at a point inside policy year t.

The notes’ processing order is death, then ordinary lapse, then the renewal decline — steps 3, 4 and 5 — and each timing reads the population the next decrement is taken from:

"BEF_DECR": l(t), the start of the year, before any decrement; the same number as pols_if() and the weight on that year’s cash flows.
"BEF_LAPSE": after deaths, before ordinary lapse.
"BEF_DECLINE": after ordinary lapse, before the renewal decline. Equal to "AFT_DECR" in every year that is not a renewal boundary.
"AFT_DECR": after all three decrements — the end-of-year state, before any 復活 reinstatement is added back by pols_if().

Zero outside 1 .. proj_len(): the cover has not started or has ended.

pols_death(t)[source]#

D(t) = l(t) q(t): expected death and 高度障害 claims in policy year t.

One decrement covering both, because the table includes 高度障害 [REG-R20] and the contract pays once and terminates on either event [S1][S8].

pols_lapse(t)[source]#

Ordinary lapses at the end of policy year t, from the survivors of mortality.

Pays nothing — there is no 解約返戻金 at any duration [S1][S4][S8] — so this moves pols_if() and nothing else. With the 復活 module on, these lives flow into pols_lapse_pool() rather than leaving for good.

pols_decline(t)[source]#

Renewal declines at the end of a boundary year, after ordinary lapse.

Zero in every year that is not a 更新 boundary, and zero on a 歳満了 point. Where it applies it is the larger exit: 0.08235591 of the 0.11175249 lives leaving in year 10 of the anchor cell. Nothing is paid.

reinstate_rate_eff()[source]#

rho: the 復活 reinstatement rate actually applied — zero unless the module is on.

Keeping the effective rate in one cells is what lets pols_lapse_pool() and check_lapse_pool() carry the same ledger in both positions of the switch: with the module off the pool still tracks the lapsed-but-reinstatable population, and nothing is reinstated out of it.

pols_lapse_pool(t)[source]#

lap(t): lapsed lives still inside the three-year 復活 window at the start of year t.

A stock, where pols_lapse() is the flow into it. Tracked by vintage rather than as one blanket balance, because the window runs from each life’s own 失効 and a single indicator would drop a whole cohort a year early or late:

lap(t) = sum over s in [t - W, t - 1] of pols_lapse(s) (1 - rho)^(t - 1 - s)

with W = reinstate_window = 3 years [S1]. Renewal declines never enter it: a declined renewal is an expiry, not a 失効, and there is nothing to reinstate.

pols_reinstate(t)[source]#

Reinstatements out of the pool into pols_if(t + 1): lap(t) rho.

復活 is available for three years against arrears at 年6% compound [S1], on evidence of health [S8]. The arrears are not monetized — see the Space docstring — so the module is a persistency effect only. Reinstatement is the only event that restarts the suicide and contestability clocks [S1]; 更新 does not. Zero in the base run.

pols_lapse_expire(t)[source]#

Lives leaving the pool in year t because their three-year window has run out.

The vintage that lapsed in year t - W, net of the reinstatements taken out of it along the way. They are gone for good: after the window there is no 復活 [S1].

wop_waived_frac(t)[source]#

The fraction of in-force policies with premiums waived at the start of year t.

A two-state incidence chain [std], u(t+1) = u(t)(1 - rec) + (1 - u(t)) inc, starting from u(1) = 0. The trigger is an accident on or after the 責任開始時 producing a 別表4 state within 180 days [S1][S8][S12][S14]. 別表4 is a materially lower bar than the 別表3 test for 高度障害 — loss of one eye, deafness in both ears, loss of one limb at the wrist or ankle [S1] — so this incidence is not the 高度障害 incidence and must not reuse mort_rate(). It is also largely permanent, which is why wop_rec_rate is zero [std].

wop_inc_rate = 0.0008 is an arbitrary placeholder: no retrieved document gives a 別表4 accident-disability incidence, the FSA and industry statistics in the source set do not publish waiver experience, and no observed range exists to quote. It is not derived from mort_rate() and must not be — 別表4 is a different and much lower bar than the 別表3 test the mortality table’s 高度障害 loading covers, so a number scaled off q would be a false derivation dressed as one. The module is off in the base run; it is live only on model point 7, where the two-state chain and its effect on pols_payer() are what is being shown.

While the waiver runs, premium income stops and cover continues [S1]; mortality and lapse are assumed independent of the waiver state [std], which is what lets the waived population be carried as a fraction rather than as its own decrement. Zero unless the module is on.

pols_waived(t)[source]#: In-force policies whose premiums are waived in policy year t; zero in the base run.

pols_payer(t)[source]#

In-force policies actually paying premium in policy year t.

l(t) less the waived fraction. Equal to pols_if() unless the 保険料の払込の免除 module is on.

ln_cap_binds()[source]#

Whether the リビング・ニーズ特約 cap would reduce a single-contract payment.

sum_assured() > ln_cap, a strict inequality, and False on every shipped model point. The ¥30,000,000 cap is per insured, aggregated across all of that insurer’s contracts [S1][S7][S8][S12] — not per contract — so inside the composite’s ¥1,000,000-¥30,000,000 envelope it is exactly reached at the ceiling and never reduces anything. A model reporting the cap biting at SA = 30,000,000 has a strict-versus-weak inequality error; a model applying it per contract has misread the clause. Model point 9 sits exactly on the boundary and must come back False.

ln_amount()[source]#

A: the accelerated amount under リビング・ニーズ特約, min(SA, cap).

Zero unless the module is on. A full acceleration extinguishes the contract retroactively to the claim date, which is what this model implements; a partial one leaves a reduced contract in force at a reduced premium [S1][S7], which is a second transition and a model point re-parameterization rather than one benefit with two amounts. It cannot arise inside the composite’s envelope, so rather than approximate it ln_amount() rejects it by name.

ln_available(t)[source]#

Whether an acceleration can be claimed in policy year t.

The rider is barred within one year of a non-renewable expiry [S1][S7][S8], which on the annual grid is the final projected policy year — and only that year, since every earlier expiry on a 更新型 point is followed by a renewal. On a 更新型 cell the bar therefore bites only in the ceiling term.

ln_share(t)[source]#

a(t): the share of the year’s decrement arriving as an acceleration [std].

Modelled as a split of the existing death-and-高度障害 decrement rather than as an additional incidence, which is the same ruling the notes make for 高度障害: an acceleration is a re-timing and re-pricing of the death benefit, not a second claim, and a separate incidence on top would double-count it. The trigger is a six-month prognosis, not the twelve months of UK terminal illness cover [S1][S7].

ln_take_up = 0.10 is an arbitrary placeholder, and the honest defence of it is not a rationale but the switch: no retrieved document gives an acceleration take-up for any Japanese carrier, no observed range can be quoted, and nothing in the sources bounds it. A round tenth was chosen because it is visibly round — a number no reader can mistake for an estimate. The module is off in the base run, so the worked example and every figure this model publishes are independent of it; it is live only on model points 6 and 9, where what is being demonstrated is the mechanics of splitting the decrement and never the level of the split.

ln_payout_pp(t)[source]#

The リビング・ニーズ特約 payment per accelerated claim in policy year t.

A - A i_ln / 2 - six months' premiums on A [S1][S7][S8][S12]: the amount is discounted, unlike a UK terminal illness payment, which is the economic reason the rider can be offered without a separate premium. The premium element is pro-rated by A / SA [std], which is exact on the full acceleration this model implements.

ln_interest_rate = 0.02 is an arbitrary placeholder in the same sense. The contract fixes the rate only by reference — the insurer’s rate current at the claim [S1][S7] — and no retrieved document states a level or a range for it. What can be said positively is the size of its effect and nothing more: the rate enters halved, so 2% removes exactly 1% of A and the whole parameter moves the payment by 0.5% for every percentage point. The module is off in the base run, which is the only defence the number has.

premiums(t)[source]#

Premium income at the start of policy year t, an inflow.

P_a(k(t)) on the policies actually paying — pols_payer(), which is pols_if() unless the waiver module is on. Annual in advance with no allowance for premiums ceasing at a mid-year exit, which slightly overstates income; the offsetting understatement is the end-of-year claim timing, and the notes are explicit that the two are a matched pair [std]. Do not apply a further half-year adjustment on top.

claims(t, kind=None)[source]#

Benefit outgo in policy year t, by kind; the total when kind is omitted.

"DEATH": the sum assured on the year’s death and 高度障害 claims, SA (1 - a(t)) D(t). One decrement, one benefit: either event terminates the contract and the other is then not paid [S1][S8].
"LIVING_NEEDS": the discounted リビング・ニーズ特約 acceleration on the share a(t) of the same decrement. Zero in the base run. It is a split of the death benefit, never an addition to it.
"LAPSE": zero, always. There is no 解約返戻金 and no paid-up value at any duration [S1][S4][S6][S8][S9][S10][S13][S14]; the kind exists so that the zero is stated rather than left to inference. One carrier in eight writes this design with a surrender value [S12], so the zero is a fact about the composite and not about the product class.

claim_expenses(t)[source]#

ec D(t): the claim handling expense on the year’s claims [std].

¥30,000 per claim, uninflated, on the whole decrement whether the benefit is paid as a death claim or accelerated. Kept out of expenses() because the notes’ worked example prints the two as separate columns.

inflation_factor(t)[source]#: The expense inflation factor in policy year t: (1 + pi)^(t-1) [std], pi = 1%.

expenses(t)[source]#

E0 and e(t): acquisition and inflating maintenance expense in year t [std].

¥15,000 per policy at issue, then ¥4,000 per policy per year inflating at 1.0%, both at the start of the year. No Japanese public source supplies either level.

No acquisition expense is charged at a 更新. A renewal is not new business — no new 保険証券 is issued and no 告知 is taken [S1][S4] — so the year-1 charge is the only one, however many times the contract renews. ¥4,000 a year against ¥11,688 of premium is a third of the first term’s load, which is why the notes rate the [std] 1.0% inflation rate a poor assumption to leave unexamined.

comm_init_pp()[source]#

c0: initial commission per policy issued [std], 50% of the first year’s P_a.

Paid upfront at issue. With the acquisition expense this is ¥20,844 of year-1 outgo against ¥11,688 of year-1 premium on the anchor cell, which is the deep new business strain the protection shape starts from. No document in the source set discloses a Japanese commission scale, so both this and comm_renewal_rate are levels chosen for the reference implementation.

comm_new_term(t)[source]#

Commission paid at a 更新 [std]; zero in the base run.

A renewal is not new business [S1][S4], so the base run pays no acquisition commission on a renewed term. That is a choice and not a fact: no document in the set discloses a commission scale at all, and a scale paying first-year rates on each renewed term would change the sign of the cash flow in years 11, 21, 31 and 41. Set comm_new_term_rate to switch it on; it then falls in the first year of each term after the first.

commissions(t)[source]#

Commission outgo in policy year t [std].

The initial commission in policy year 1, then 5% of premium income from policy year 2, plus any commission at a 更新 (off in the base run). No clawback on early lapse is modelled: the notes record that no Japanese clawback evidence exists in the source set, so a clawback rule would be an invention rather than a standardization of something observed.

net_cf(t)[source]#

CF(t): the net cash flow of policy year t, income positive.

Premiums less claims, claim expense, maintenance and acquisition expense and commission. The notes’ own sign — they write + = inflow — which is also the library-wide convention, so there is no outgo-positive liability_cf companion to publish.

Lapse and the renewal decline contribute no term: they act only through pols_if(). The shape to expect is a deep first-year strain, thin positive margins through the middle of each term, a negative year immediately before each renewal as the level premium falls behind the rising mortality cost, and a jump back into surplus the year after — on the anchor cell, -¥1,120.47 in year 10 and +¥3,217.97 in year 11.

check_pols_roll_fwd_resid(t)[source]#

The in-force roll-forward residual in policy year t; zero everywhere.

l(t) - l(t+1) - D(t) - lapses - declines + reinstatements, the notes’ identity with the 復活 module’s inflow carried as its own term so that the same residual closes in both positions of the switch. Non-zero would mean the decrements and the roll-forward have drifted apart — most easily by applying the renewal decline to the wrong population, since it is taken after mortality and after ordinary lapse.

check_pols_roll_fwd()[source]#

True when the in-force roll-forward closes in every projected policy year.

The library-wide form of a roll-forward check: no argument, one bool over all t, so one test can call it across every model. check_pols_roll_fwd_resid() gives the signed residual of the year that failed.

check_lapse_pool_resid(t)[source]#

The 復活 pool roll-forward residual in policy year t; zero everywhere.

lap(t) - lap(t+1) - reinstatements - window expiries + lapses. The pool is a stock with one inflow (pols_lapse()) and two outflows (pols_reinstate() and pols_lapse_expire()), and the vintage bookkeeping of the three-year window is exactly where an implementation drops or double-counts a cohort. Closes with the module off as well as on, since the pool is tracked either way.

check_lapse_pool()[source]#: True when the 復活 pool ledger closes in every projected policy year.

check_pols_payer_resid(t)[source]#

The premium-paying population residual in policy year t; zero everywhere.

l(t) - payers - waived. The waiver module carries the waived lives as a fraction of the in-force rather than as a separate decrement, which is only legitimate while the two partition l(t) exactly.

check_pols_payer()[source]#: True when payers and waived lives partition the in-force in every projected year.

check_prem_level_resid(t)[source]#

The premium-level residual in policy year t; zero everywhere.

P_a(t) - P_a(t-1) inside a 保険期間, and zero by definition in the first year of a term. The premium is level within the term and changes only at a 更新 [S1][S4][S8][S12]; a residual here means the premium is drifting with the policy year, which is what happens if the rate lookup is keyed on attained age rather than on the term’s entry age.

check_prem_level()[source]#: True when the premium is level within every 保険期間 of the projection.

check_net_cf_resid(t)[source]#

The published cash flow statement’s ledger residual in policy year t; zero.

net_cf() less the sum of the columns of result_cf(), so a reader adding up the printed statement gets the printed total. It is the check that catches a benefit kind that exists in claims() but was never given a column — which would leave the statement silently short of outgo it is charging.

check_net_cf()[source]#

True when the published cash flow statement adds up in every projected year.

Tested against cash_tol, not the roll_fwd_tol the four decrement checks use. The wider tolerance is a property of what is compared: the other checks close an identity between cells evaluated in one expression, where the residual is exact to a unit or two in the last place of a count near 1.0, while this one re-reads yen amounts of order 1e5 back out of the result_cf() DataFrame, so the round trip through column construction leaves float64 rounding of order 1e-11 in absolute yen. cash_tol = 1e-8 is well above that noise and far below one yen, which is the smallest error a reader adding up the printed statement could observe.

result_cf()[source]#

Result table of cashflows, indexed by policy year t.

pols_if is the start-of-year count, which is the weight applied to every cash flow on the same row. net_cf carries the notes’ own income-positive sign. claims_lapse is a column of zeros by product design — there is no 解約返戻金 — and is published rather than dropped; see the Space docstring.

result_pols()[source]#

Result table of policy counts, decrement rates and the premium, indexed by t.

The renewal machinery is only legible next to the decrements it drives, so term_index and prem_pp are printed here with decline_rate: a boundary year is the row where the decline rate is non-zero and the premium changes on the next row.