Technical Notes#

Status: Draft, 2026-08-20 (all cited sources accessed 2026-08-20).

Scope note. These notes turn the standardized composite of product-spec.md (same directory) into a reference liability cash-flow projection on paper. They describe no single insurer’s contract. [S#] and [R#] resolve against sources.md in this directory, numbering carried verbatim from _research/term-life.md and frozen; [REG-R#] resolves against references/regulatory-and-actuarial-references.md, whose own R-numbering is distinct and must never be read across. std marks a standardization introduced for the reference implementation; unverified marks a claim not confirmed against a retrieved document. Every contractual parameter here is identical to product-spec.md’s. What is new is the assumption basis — a best-estimate mortality adjustment, a lapse table, a renewal-decline rate, an expense and commission structure, and a premium scale beyond the published ages. No retrieved document supplies any of them, so every one is std with its rationale given where it is introduced.

This is the library’s protection chassis. The survivor income term technical notes (収入保障保険) (survivor income term) state only their deltas against this file: the decrement recursion, the premium chassis, the expense and commission structure and the processing order are specified here once and are not restated there.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows — premiums, death and severe disability (kōdo shōgai, 高度障害) claims, claim expenses, maintenance expenses and commission — for a single-policy model point of level term life (teiki hoken, 定期保険), in the sense the ESR current estimate (genzai suikei, 現在推計) requires: probability-weighted future cash flows on assumptions re-set at the 基準日 rather than locked in at issue REG-R15. The same projection is the shape of the item-1 income-and-outgo analysis (ichi-gō shūshi bunseki, 1号収支分析) — premiums, claims, expenses and surrenders by segment over at least ten future years REG-R22.

  • Discounting, MOCE, required capital and reserving are out of scope, cited and not reproduced (see Valuation and reserve pointers). jplib computes no ratio and builds no policy reserve (sekinin-junbikin, 責任準備金).

  • Projection frequency. Annual — the model is Term_JP_A. Nothing in the composite has intra-year contractual structure: the sum assured is level and the premium is level within each policy term (hoken kikan, 保険期間). The one intra-year mechanic that matters, the grace period (yūyo kikan, 猶予期間) of about one month [S1] [S8], sits inside a decrement the annual grid represents as a rate. The survivor income term technical notes (収入保障保険) run monthly because that benefit is a monthly income stream; this product does not need to.

  • Timing conventions std. Premiums at the start of each policy year (annualized, in advance); maintenance expense at the start of the year; acquisition expense and initial commission at issue; death and 高度障害 claims and their claim expense at the end of the policy year in which they arise; ordinary lapse at the end of the year, after deaths; the renewal (kōshin, 更新) decline at the end of a boundary year, after lapse.

  • Age basis std. Issue age (keiyaku nenrei, 契約年齢) is age last birthday (man-nenrei, 満年齢) with fractions discarded [S1]; attained age in year t is x + t − 1. 生保標準生命表2018（死亡保険用）is built for an age nearest birthday (hoken-nenrei, 保険年齢) basis REG-R20, so reading it at 満年齢 reads it half a year early and understates mortality. The base run accepts and states that bias; an optional shift q_x → sqrt(q_x · q_{x+1}) raises the rate by about 0.7% at age 30 and 4.2% at age 40 on the std table below. This resolves the mismatch product-spec.md footnote 3 flags.

  • Currency. JPY throughout [S1]. No FX layer on this product.

  • Model points. Single-policy, on an expected (probability-weighted) basis: survivorship multiplies per-policy cash flows. No aggregation logic is specified here. point_id = 1 is the worked example’s anchor cell.

  • Termination. A to-a-stated-age (sai manryō, 歳満了) contract ends at the end of its term with nothing payable [S1] [S8] [S10] [S14]. A fixed-year (nen manryō, 年満了) 更新型 contract ends only at the renewal ceiling of attained age 80 [S1] [S2] [S8], because until then it renews. No 満期保険金, no surrender value (kaiyaku-henreikin, 解約返戻金), no tail state of any kind [S1] [S4] [S8] [S9] [S10] [S13] [S14].

  • Contract boundary — the paragraph this product forces. A UK term assurance guarantees its premium for the whole term, so the boundary is the term. A Japanese 年満了 contract guarantees it 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 [S1] [S4] [S8] [S12], so the insurer cannot reprice a life for its own deterioration. The ESR standard-model coefficients that would settle where the boundary falls are unverified here — the 柱告示 were not retrieved REG-R16 — so the model does not rule. It projects to the ceiling in the base run std and carries a contract_boundary switch truncating at the end of the current 保険期間. The two differ by more than a rounding: on the anchor cell undiscounted net cash flow is +¥50,400.25 to the ceiling against −¥15,878.74 over the first ten years. Reporting either without naming the convention says nothing.

  • Rounding. Intermediates at full precision; displayed cash flows to two decimals of a yen, in-force to six decimals std. Premium rates round to the whole yen per month before annualization, as published rate cards do [S2] [S9] [S10].

Model point attributes#

Attribute

Type

Anchor cell (point_id = 1)

sex

enum {M, F}

M

issue_age (x)

int, 満年齢, 20–65

30

term_type

enum {nen, sai} — 年満了 / 歳満了

nen

term_y (n)

int years (年満了); implied by expiry_age (歳満了)

10

expiry_age

int (歳満了 only: 60/65/70/80)

—

renew_ceiling (w_r)

int, attained age at which renewal stops

80

sum_assured (SA)

JPY, ¥1,000,000–¥30,000,000 in ¥1,000,000 units

10,000,000

premium_mode

enum {monthly, semiannual, annual}

monthly

living_needs

bool (rider; base run false)

false

wop

bool (保険料の払込の免除; base run false)

false

reinstatement

bool (復活 module; base run false)

false

contract_boundary

enum {ceiling, current_term}

ceiling

The premium is not one of them. P_m (premium_mth_pp) and the flat element f (policy_fee_m, ¥248) are derived from the premium scale, keyed on sex, the entry age of the term in force and that term’s length — which is what makes the repricing at 更新 fall out of the same lookup as the issue premium instead of needing a second column. Carrying the premium on the model point would freeze it at the issue value and hide the repricing.

P_m is nonetheless a published figure at the anchor cell rather than a modeling construction — the sharpest documentary contrast with uklib, where no premium basis is observable and the anchor premium had to be invented. Three carriers price this exact cell at ¥974, ¥980 and ¥1,068 [S2] [S9] [S5]; ¥974 is taken because that carrier publishes enough grid to decompose it. std here covers only the choice of cell and the annualization P_a = 12 × P_m = 11,688.

State variables#

Variable

Description

Updated

l(t)

In-force probability at the start of year t; l(1) = 1

annual recursion

k(t)

Term index: 1 in the original 保険期間, 2 after the first 更新, …

boundary years

P_a(t)

Annualized premium in force in year t; constant within a term

boundary years

q(t)

Best-estimate death-and-高度障害 rate (one decrement)

assumption lookup

w(t)

Ordinary lapse rate (end of year, after deaths)

assumption lookup

d(t)

Renewal-decline rate; non-zero only in a boundary year

boundary years

D(t)

Expected claims in year t = l(t) × q(t)

annual

lap(t)

Lapsed-but-reinstatable population (復活 module; 0 in base run)

annual

CF(t)

Net cash flow of year t, insurer perspective (+ = inflow)

annual

Two of these a Japanese term model needs and a UK one does not: k(t), because the premium is a function of the term index rather than of t, and d(t), because leaving at a renewal boundary is a different event from lapsing mid-term. Two clocks — the three-year suicide window and the two-year contestability window, both running from the 責任開始日 and neither restarting on 更新 [S1] [S4] [S7] [S8] — are contract state the base run tracks but does not monetize; they bind only if the 復活 module is on, since 復活 is the one event that restarts them [S1].

There is deliberately no cv_pp and no account value. The composite has no 解約返戻金 for the whole term [S1] [S4] [S6] [S8] [S9] [S10] [S13] [S14], and with no surrender value there is no automatic premium loan (jidō furikae kashitsuke, 自動振替貸付) to carry the policy through non-payment: one carrier states that absence in terms [S7]. A policy loan (keiyakusha kashitsuke, 契約者貸付) has no collateral either, but that is an inference from the missing surrender value and not a sourced fact — the carrier stating the APL absence points its policyholders at its 契約貸付制度 instead [S7], and the one document that appears to rule the policy loan out could not be text-extracted [S11] unverified. The absence is a product fact: it is why this chassis carries a plain lapse model, and why the APL mechanic is specified in the whole life technical notes (終身保険) and not here.

Assumption inputs#

Three classes, kept separate. The first is cited and the insurer cannot change it; the second is discretionary and, on this product, nearly empty; the third is the modeler’s view and is std throughout, because no Japanese public source supplies it.

(a) Contractual / guaranteed elements (cited)#

Input

Value

Basis

Sum assured

SA level for the term, unchanged through 更新

[S1] [S4] [S8] [S9] [S12]

Death benefit

SA on death in the 保険期間; terminates the contract

[S1] [S4] [S8] [S12]

高度障害保険金

SA, the same amount, on a 別表3 state; terminates the contract

[S1] [S4] [S8] [S9] [S12]

Ordering

Whichever becomes payable first is paid; the other then is not

[S1] [S8]

Premium

Level within the 保険期間; not guaranteed beyond it

[S1] [S4] [S8] [S12]

Premium structure

P_m = f + r(sex, x, n) × SA / 5,000,000, f = 248 per month

[S2]

更新

Automatic unless declined 2 weeks before expiry; same term and SA; repriced on attained age; no 告知

[S1] [S4] [S8] [S12]

Renewal ceiling

Attained age 80; a renewal passing it truncates to an 80歳満了 term

[S1] [S2] [S8]

歳満了

Never renews

[S1]

Clocks

Suicide 3 years, contestability 2 years, from 責任開始日; restart on 復活 only

[S1] [S4] [S7] [S8] REG-R34 REG-R35

猶予期間

To the last day of the month after the 払込期月 (~1 month); then 失効

[S1] [S8]

復活

3 years, against arrears at 年6% compound [S1]; on evidence of health [S8]

[S1] [S8]

解約返戻金

None, whole term

[S1] [S4] [S6] [S8] [S9] [S10] [S13] [S14]

自動振替貸付

Absent — one carrier states it in terms

[S7]

契約者貸付

Not modeled. No 解約返戻金 exists to lend against, but the one document appearing to rule the policy loan out could not be extracted

inference; [S11] unverified

リビング・ニーズ特約 (a tokuyaku, rider)

指定保険金額 − 6 months' interest − 6 months' premiums; cap ¥30,000,000 per insured aggregated across contracts; barred within 1 year of a non-renewable expiry

[S1] [S7] [S8] [S12]

保険料の払込の免除

Accident on or after 責任開始時, 別表4 state within 180 days

[S1] [S8] [S12] [S14]

満期保険金

None

[S1] [S8] [S10] [S14]

(b) Insurer-discretionary current elements#

On 無配当 protection business this class is almost empty, and the emptiness is why the classes are separated at all: on the whole life technical notes (終身保険) and the FX whole life technical notes (外貨建終身保険), declared 契約者配当 and 積立利率 live here. Three residual items:

Input

Snapshot value

Basis

契約者配当

Nil — 「この保険契約については、契約者配当はありません。」

[S1] [S8] [S9] [S10] [S14]; one carrier writes a 有配当 design [S7]

Renewal rate scale

The scale in force at each future renewal is the insurer’s. Modeled as the current scale extended by mortality

mechanic [S1] [S4] [S8] [S12]; scale std

前納 discount, 高額割引

Insurer-set and unpublished; not modeled

[S1] [S7] [S9] [S14]; scope std

The renewal scale is the one genuinely discretionary lever with a large cash-flow effect, and it is invisible. Treating the current scale as persistent is a std assumption, not a neutral one.

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

Mortality — one decrement, and why. 生保標準生命表2018（死亡保険用）includes 高度障害 inside its death rate REG-R20. Death and 高度障害 are therefore one decrement carrying one sum assured, which is what the contract does: either benefit terminates the policy and the other is then not paid [S1] [S8]. Projecting the table qx for death and adding a 高度障害 incidence on top double-counts the benefit.

The table is freely readable at a stable public URL REG-R18 R3 R4 — the sharpest contrast in this library with uklib, whose current CMI tables are subscriber-restricted — but its publisher prohibits reproduction and transmission to third parties without written consent REG-R21. The library therefore cites the table, quotes the rates the worked example needs, and ships mort_table.csv as a std construction whose provenance column points at the IAJ entries.

The shipped file is the canonical jplib proxy, one construction shared by every product in this library rather than a per-product reconstruction, so that a cell carries the same rate and the same provenance wherever it is shipped. Its anchors are the union of the rates read from the table across the library’s research passes, which is why more ages are anchored here than this product’s own pass read; the rows this product ships run from attained age 20 to attained age 80, the range its model points can reach:

Step

Rule

Basis

Anchors

Male 死亡保険用 q20 = 0.00059, q22 = 0.00066, q25 = 0.00067, q30 = 0.00068, q31 = 0.00069, q32 = 0.00070, q33 = 0.00072, q34 = 0.00074, q35 = 0.00077, q40 = 0.00118, q45 = 0.00177, q50 = 0.00285, q55 = 0.00422, q60 = 0.00653, q65 = 0.01015, q70 = 0.01544, q75 = 0.02637, q80 = 0.05006; female 死亡保険用 at the same ages except 31–34, q30 = 0.00037, q60 = 0.00363 among them

read from the table REG-R18, the subset this product’s own pass read also R4

Interpolation

Log-linear in ln q between the two neighbouring anchors, rounded to 5 decimals — the published table’s own granularity. No extrapolation: every non-anchor age lies strictly between two anchors

std

Best estimate

q(t) = 0.80 × q_x^tab

std (1)

Age read

At 満年齢, unshifted; optional sqrt(q_x · q_{x+1}) shift

std

Improvement

None in the base run

std (2)

Suicide-exclusion offset

None: years 1–3 claims are not reduced for excluded suicides

clause [S1]; offset std (3)

  1. The 作成概要 states the margin in the publisher’s words: a risk-theory adjustment sized to hold the exceedance probability near 2σ, capped at 130% of the unadjusted rate REG-R20. Removing a margin at its cap implies 1/1.3 = 0.769. Against that, the table already carries a forward improvement allowance — 2.5% p.a. for five years then 1.0% for three REG-R20 — 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 and the research pass found no equivalent of the FCA’s published data. This factor is the model’s largest lever.

  2. Eight years of population improvement since the table’s effective date are not projected std. A production basis applying an improvement scale must re-derive footnote 1, because part of the 0.80 stands in for it.

  3. Immaterial at these claim levels and unsupported by any incidence split in the sources. Note the direction: the suicide 免責 pays the 責任準備金 to the policyholder rather than nothing [S1], so even a modeled exclusion is not a clean claim saving.

Lapse. Japan’s only published industry-wide persistency figure is the LIAJ’s FY2024 解約・失効率 for 個人保険: 5.6% of opening in-force sum assured, down 0.3 points REG-R31. It is a whole-market number across all shapes and durations, not a duration curve, so the reference table is std and is reconciled to it explicitly:

Policy year

1

2

3

4

5+

Annual lapse w(t) std

9%

7%

6%

5.5%

5%

The simple mean over the first ten years is 5.75% and the in-force-weighted mean 5.94%, both a little above 5.6% — the expected direction, since the industry figure is dominated by long-duration in-force sum assured while this is an early-duration protection curve. The level is anchored; the shape is a convention with no Japanese published evidence behind it. Lapse pays nothing: there is no 解約返戻金 [S1].

Renewal decline. At each 更新 boundary a proportion d of survivors leave rather than accept the repriced contract. This decrement has no uklib analogue 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 2 weeks [S1] [S8], the shortest of the three observed (2 weeks, 1 month, 2 months [S1] [S8] [S7] [S4]), which is the design that maximizes renewal by inertia. No carrier publishes a take-up or decline rate, so d = 15% std at every boundary. Two things roll into it that a production model should separate: the policyholder who gives notice, 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]. Both leave at the boundary; only the first is a decision.

Expenses and commission (all levels std; no Japanese public source exists).

Input

Value

Note

Acquisition expense E0

¥15,000 per policy at issue

std

Initial commission c0

50% of the first-year annualized premium, at issue

std

Renewal commission c_r

5% of premiums from year 2

std

Commission at 更新

None in the base run

std (4)

Maintenance expense e(t)

¥4,000 p.a., inflating 1.0% p.a.

std

Claim expense ec

¥30,000 per death / 高度障害 claim

std

  1. A 更新 is not new business — no new 保険証券 is issued and no 告知 taken [S1] [S4] — so the base run pays no acquisition commission at a renewal. That is a choice, 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. The model exposes it as a switch and the pitfall list tests it.

The ¥248 monthly policy fee is a premium component, not an expense recovery [S2]. It enters the model only through P_a; crediting it against e(t) counts it twice.

Cash flow components and recursions#

Notation#

Symbol

Meaning

cells

t

policy year, t = 1..N; N = w_r − x for 年満了, n for 歳満了

—

x

契約年齢 (満年齢); attained age in year t is x + t − 1

age

k

term index; k = 1 + floor((t − 1) / n) for 年満了

term_index

SA

sum assured, JPY, level

sum_assured

f

flat monthly policy element, ¥248

policy_fee_m

r(sex, x, m)

marginal monthly rate per ¥5,000,000 of cover, entry age x, term m

prem_rate_m

P_m(k), P_a(k)

monthly and annualized premium in term k; P_a = 12 × P_m

premium_mth_pp, prem_pp

q(t)

best-estimate death-and-高度障害 rate

mort_rate

w(t)

ordinary lapse rate

lapse_rate

d(t)

renewal-decline rate; 0 unless t is a boundary year

decline_rate

l(t)

in-force probability at the start of year t; l(1) = 1

pols_if

D(t)

expected claims in year t = l(t) × q(t)

pols_death

E0, e(t)

acquisition expense; maintenance 4,000 × 1.01^(t−1) per policy

expense_acq, expenses

c0, c_r

initial commission 0.50 × P_a(1); renewal rate 0.05

commissions

ec

claim expense per claim, ¥30,000

expense_claim

CF(t)

net cash flow of year t (+ inflow)

net_cf

Dimensional check: q, w, d, l, D are dimensionless probabilities; SA, E0, e, ec, P_a are JPY per policy per year; f and P_m are JPY per month, so every expression using them carries an explicit × 12; r is JPY per month per ¥5,000,000 of cover, so r × SA / 5,000,000 is JPY per month. Every term of CF(t) is JPY per year per policy issued.

Premium chassis#

The published rate structure decomposes exactly [S2]:

P_m(k) = f + r(sex, x_k, m_k) * SA / 5,000,000
x_k    = x + (k - 1) * n              (attained age at the k-th term's start)
m_k    = min(n, w_r - x_k)            (truncation at the ceiling)
P_a(k) = 12 * P_m(k)

At the anchor cell r(M, 30, 10) = 363 and P_m(1) = 248 + 2 × 363 = 974 [S2], reproducing the published figure to the yen; at the first renewal r(M, 40, 10) = 787.5 and P_m(2) = 1,823 [S2]; at the second, r(M, 50, 10) = 1,842.5 and P_m(3) = 3,933 [S2]. Ages 60 and 70 are published by no carrier, and the anchor cell reaches both. The std extension:

r(sex, x, m) = r(sex, 50, 10) * qbar(x, m) / qbar(50, 10)
qbar(x, m)   = mean of q_a^tab over a = x .. x+m-1   (table rates, not best estimate)

anchored on the published age-50 cell and rounded to the whole yen per month. 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%) — close enough to use, and stated so the size of the approximation is visible rather than trusted. It gives P_m = ¥8,976 at age 60 and ¥23,881 at age 70 on ¥10,000,000 of cover.

Decrement recursion and processing order#

For t = 1..N, in this order std:

  1. Start of year. Premium income P_a(k(t)) × l(t); maintenance e(t) × l(t); renewal commission c_r × P_a(k(t)) × l(t) for t ≥ 2. At t = 1 additionally E0 and c0 per policy issued (l(1) = 1).

  2. Decrement lookup. q(t) at attained age x + t − 1; w(t) from the lapse table; d(t) = d if t mod n = 0 and t < N, else 0.

  3. End of year — claims. D(t) = l(t) × q(t); claim outgo SA × D(t); claim expense ec × D(t). One decrement, one benefit: death and 高度障害 are not added.

  4. End of year — ordinary lapse. l(t) × (1 − q(t)) × w(t) leave, applied to survivors of mortality. Nothing is paid — there is no 解約返戻金 [S1].

  5. End of a boundary year — renewal decline. l(t) × (1 − q(t)) × (1 − w(t)) × d(t) leave, applied after lapse. Nothing is paid.

  6. Roll forward, plus any 復活 reinstatements (zero in the base run).

    l(t+1) = l(t) * (1 - q(t)) * (1 - w(t)) * (1 - d(t)) + lap(t) * rho

  7. Repricing at a boundary. k(t+1) = k(t) + 1 and P_a is recomputed at attained age x + t over the term m = min(n, w_r − (x + t)). The projection horizon does not change; the term shortens instead.

At t = N the projection ends: no maturity payment, no run-off, no tail states [S1] [S8] [S10] [S14]. The identity the model must satisfy at every t is

l(t) - l(t+1) = D(t) + lapses(t) + declines(t) - reinstatements(t)

which check_pols_roll_fwd() asserts over all t and returns as a single bool. The reinstatement term is zero in the base run and is carried in the identity anyway, so the same residual closes in both positions of the 復活 switch rather than one form of the check being right for each.

Net cash flow#

CF(t) = P_a(k(t)) * l(t)                          (premiums)
      - SA * D(t)                                 (death and 高度障害 claims)
      - ec * D(t)                                 (claim expense)
      - e(t) * l(t)                               (maintenance)
      - c_r * P_a(k(t)) * l(t) * 1{t >= 2}        (renewal commission)
      - (E0 + c0) * 1{t = 1}                      (acquisition)

net_cf is income-positive, per the library convention. Lapse and decline contribute no term: they act only through l(t). A claims_lapse column exists and is identically zero — the zero is the product fact worth publishing, as it is in Term_UK_A.

Known bias of the annual-in-advance convention std: a full year’s premium is collected at the start of each year with no allowance for premiums ceasing at mid-year exits, so premium income is slightly overstated; the offsetting understatement is the end-of-year claim timing. Do not apply both this convention and a separate half-year premium adjustment.

Optional modules (all off in the base run)#

  • リビング・ニーズ特約. Acceleration incidence a(t) std (no retrieved document gives incidence). On an accelerated amount A ≤ min(SA, 30,000,000):

    payout = A - A * i_ln * 0.5 - 6 months' premiums on A

    with i_ln a std snapshot rate. A full acceleration extinguishes the contract retroactively to the claim date; a partial one reduces SA from that date and the reduced premium continues [S1] [S7] — two transitions, not one benefit with two amounts. Barred within one year of a non-renewable expiry [S1] [S7] [S8]; on a 更新型 cell that bar bites only in the ceiling term.

  • 保険料の払込の免除. A waiver state on the accident-plus-180-days-plus-別表4 test [S1] [S8] [S12] [S14] with std incidence. 別表4 is a materially lower bar than 別表3 — loss of one eye, deafness in both ears, loss of one limb at wrist or ankle [S1] — so the waiver incidence is not the 高度障害 incidence and must not reuse q(t). While the waiver runs, premium income stops, cover continues and alteration rights switch off [S1].

  • 復活. A lapsed-but-reinstatable population with a three-year window W = 3 [S1]. The window runs from each life’s own 失効, so the pool is carried by vintage and not as one balance with an indicator on it — a single indicator drops a whole cohort a year early or late:

    lap(t)              = sum over s in [t - W, t - 1] of lapses(s) * (1 - rho)^(t - 1 - s)
lapses(s)           = l(s) * (1 - q(s)) * w(s)
reinstatements into l(t+1) = lap(t) * rho
window expiries(t)  = lapses(t - W) * (1 - rho)^W

    with rho std. One inflow and two outflows, and the ledger lap(t) − lap(t+1) = reinstatements(t) + expiries(t) − lapses(t) is what check_lapse_pool() asserts — it closes with the module off as well as on, because the pool is tracked either way. Renewal declines never enter it: a declined renewal is an expiry, not a 失効, and there is nothing to reinstate. Off in the base run: any value of rho is an invention with a material persistency effect and no carrier publishes one. Reinstatement restarts both clocks from the new 責任開始 [S1] — the only event that does.

  • Contract boundary. contract_boundary = current_term truncates at the end of the term in force at the valuation date.

What the survivor income term technical notes (収入保障保険) inherits#

Unchanged from this file: the decrement recursion and processing order (steps 1–7), the premium chassis P_m = f + r × SA / 5,000,000 with its renewal repricing, the std mortality construction and its 0.80 factor, the lapse table, the renewal-decline treatment, the expense and commission structure, the age basis and the timing conventions. What changes there: a monthly grid, and a benefit replaced by an annuity-certain income stream with 最低支払保証期間 as its floor — so the in-payment ledger is not decremented, while premium income still carries l(t).

Policyholder behavior modeling#

All dynamic formulas are std reference constructions. Japanese public evidence on behaviour is thinner than the UK’s: one published lapse rate for the whole market REG-R31, and nothing on duration, channel or renewal take-up.

  • Base lapse std. The duration table above, reconciled to 5.6% as shown. Channel is not represented; career agents sold 56.7% of the most recently bought policies R9 — a share of purchases, not of cover — and agent-sold and direct-sold persistency are not the same, but no split is published.

  • Renewal decline std. d = 15%, flat. The refinement the flat rate defers: decline should rise with the premium jump, which itself accelerates — the anchor cell’s premium multiplies by 1.87, then 2.16, then 2.28, then 2.66 across four renewals. A reference elasticity form is

    d(t) = min(d_max, d_0 * (P_a(k+1) / P_a(k))^beta)

    with d_0 = 15%, base run beta = 0 giving the flat rate, and d_max = 50%, all three std. The cap is a guard on the elasticity form rather than a behavioural view: at beta = 1 the largest jump on the anchor cell (×2.66) reaches only 40%, so the cap binds at no boundary, while at beta = 2 even the smallest (×1.87) reaches 53% and it binds at all four. Half the survivors is the round level at which one boundary removes 0.27 of the original cohort — more than three times the 0.08 the base 15% removes, and over half the 0.45 that the ten preceding years of ordinary lapse remove. No document narrows any of the three.

  • Selective lapsation std (optional). Lapsers and decliners are healthier on average, and the effect is stronger here than on a UK term policy because the 更新 decision recurs at ages where the premium is large and health is known:

    q_eff(t) = q(t) * [1 + lambda * max(0, 1 - l(t) / l_ref)]

    with lambda std, base run lambda = 0, and l_ref = 1 std — the reference is the cohort at issue, so the loading is driven by the proportion of the original block that has left, which is the quantity anti-selection is a function of. The mechanism is 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.

  • Rebroking. The Japanese analogue of the UK rebroking driver is the 更新 decision itself rather than a mid-term switch, so it is modeled through d(t) and not as a separate lapse multiplier [std scope].

  • 減額. Permitted above an insurer-set floor, premium reset, no 払戻金 arising [S1] [S8]. Not modeled: it changes SA and P_a together, which is a model-point re-parameterization rather than a decrement [std scope].

  • クーリング・オフ. Out of scope: the model begins with cover in force and the eight-day statutory population REG-R36 already out [S1].

Worked example#

Anchor cell (point_id = 1). Male, 契約年齢 30 (満年齢), 年満了 10年 更新型, renewal ceiling attained age 80, 保険金額 ¥10,000,000, 月払保険料 ¥974 [S2], P_a = ¥11,688 [std annualization]. Horizon N = 80 − 30 = 50 years; boundary years t = 10, 20, 30, 40. Base run: no rider, no waiver, no 復活, no selective lapsation, boundary = ceiling.

Every assumption value the cell uses. The table rates below come from the canonical jplib proxy of 生保標準生命表2018（死亡保険用）男. Attained ages 30–35 and 40 are sourced anchors, read from the published table REG-R18 R4; ages 36–39 are the std log-linear interpolation between the age-35 and age-40 anchors, rounded to 5 decimals. The best-estimate rate is q(t) = 0.80 × q_x^tab std:

attained age

30

31

32

33

34

35

36

37

38

39

40

q_x^tab

0.00068

0.00069

0.00070

0.00072

0.00074

0.00077

0.00084

0.00091

0.00099

0.00108

0.00118

q(t)

0.000544

0.000552

0.000560

0.000576

0.000592

0.000616

0.000672

0.000728

0.000792

0.000864

0.000944

anchor?

yes

yes

yes

yes

yes

yes

no

no

no

no

yes

Lapse w(t) = 9% / 7% / 6% / 5.5% / 5% from year 5 std; renewal decline d = 15% at boundary years std; E0 = ¥15,000, c0 = 0.50 × 11,688 = ¥5,844, c_r = 5% from year 2, e(t) = 4,000 × 1.01^(t−1), ec = ¥30,000, all std.

t

age

l(t)

P_a

Premiums

Claims

Claim exp

Maint. + acq.

Commission

CF(t)

1

30

1.000000

11,688

11,688.00

5,440.00

16.32

19,000.00

5,844.00

−18,612.32

2

31

0.909505

11,688

10,630.29

5,020.47

15.06

3,674.40

531.51

+1,388.85

3

32

0.845373

11,688

9,880.72

4,734.09

14.20

3,449.46

494.04

+1,188.93

10

39

0.578436

11,688

6,760.76

4,997.69

14.99

2,530.51

338.04

−1,120.47

11

40

0.466683

21,876

10,209.17

4,405.49

13.22

2,062.04

510.46

+3,217.97

Trace, year 1. l(1) = 1; premiums = 11,688 × 1 = 11,688.00. q(1) = 0.80 × 0.00068 = 0.000544, so D(1) = 0.000544. Claims = 10,000,000 × 0.000544 = 5,440.00; claim expense = 30,000 × 0.000544 = 16.32. Expenses = E0 + e(1) = 15,000.00 + 4,000.00 = 19,000.00; commission = c0 = 0.50 × 11,688 = 5,844.00. CF(1) = 11,688.00 − 5,440.00 − 16.32 − 19,000.00 − 5,844.00 = −18,612.32. Roll forward: l(2) = 1 × (1 − 0.000544) × (1 − 0.09) = 0.999456 × 0.91 = 0.909505.

Trace, year 2. Premiums = 11,688 × 0.90950496 = 10,630.29. q(2) = 0.80 × 0.00069 = 0.000552; D(2) = 0.90950496 × 0.000552 = 0.00050205; claims = 5,020.47; claim expense = 15.06. Maintenance = 4,000 × 1.01 × 0.90950496 = 4,040 × 0.90950496 = 3,674.40. Renewal commission = 0.05 × 10,630.29 = 531.51. CF(2) = 10,630.29 − 5,020.47 − 15.06 − 3,674.40 − 531.51 = +1,388.85. Roll forward: l(3) = 0.90950496 × (1 − 0.000552) × (1 − 0.07) = 0.845373.

Trace, year 3. Premiums = 11,688 × 0.84537271 = 9,880.72. q(3) = 0.80 × 0.00070 = 0.000560; D(3) = 0.84537271 × 0.000560 = 0.00047341; claims = 4,734.09; claim expense = 14.20. Maintenance = 4,000 × 1.01^2 × 0.84537271 = 4,080.40 × 0.84537271 = 3,449.46. Renewal commission = 0.05 × 9,880.72 = 494.04. CF(3) = 9,880.72 − 4,734.09 − 14.20 − 3,449.46 − 494.04 = +1,188.93.

Trace, year 10 — the boundary year. l(10) = 0.57843599; premiums = 11,688 × 0.57843599 = 6,760.76 on the old premium, because the repricing takes effect at the start of year 11 and not before. q(10) = 0.80 × 0.00108 = 0.000864; D(10) = 0.00049977; claims = 4,997.69; claim expense = 14.99. Maintenance = 4,000 × 1.01^9 × 0.57843599 = 2,530.51; renewal commission = 338.04. CF(10) = 6,760.76 − 4,997.69 − 14.99 − 2,530.51 − 338.04 = −1,120.47. Roll forward, in the processing order and no other: after mortality 0.57843599 × (1 − 0.000864) = 0.57793622; after ordinary lapse × (1 − 0.05) = 0.54903941; after the renewal decline × (1 − 0.15) = 0.46668350. So l(11) = 0.466683, and the three exits in year 10 are 0.00049977 deaths, 0.02889681 lapses and 0.08235591 renewal declines — the decline is 74% of all exits that year, and a model folding it into the lapse rate cannot see it.

Trace, year 11 — the repriced term. Attained age at renewal is 30 + 10 = 40, so P_m(2) = 248 + 2 × 787.5 = 1,823 [S2] and P_a(2) = 21,876. Premiums = 21,876 × 0.46668350 = 10,209.17 — higher than year 10’s despite 19% fewer policies, because the premium multiplied by 1.87. q(11) = 0.80 × 0.00118 = 0.000944; D(11) = 0.00044055; claims = 4,405.49; claim expense = 13.22. Maintenance = 4,000 × 1.01^10 × 0.46668350 = 2,062.04; renewal commission = 0.05 × 10,209.17 = 510.46. CF(11) = 10,209.17 − 4,405.49 − 13.22 − 2,062.04 − 510.46 = +3,217.97.

The renewal ladder, five numbers per renewal:

Renewal

at t

attained age

P_m

P_a

jump

l(t+1)

— (issue)

—

30

974 [S2]

11,688

—

1.000000

1st

10

40

1,823 [S2]

21,876

×1.87

0.466683

2nd

20

50

3,933 [S2]

47,196

×2.16

0.234147

3rd

30

60

8,976 std

107,712

×2.28

0.115211

4th

40

70

23,881 std

286,572

×2.66

0.054121

Cover ends at attained age 80 with l(51) = 0.026042 — 2.6% of the cohort still in force after fifty years and four repricings, paying ¥286,572 a year for ¥10,000,000 of cover. That row is the economically important one and is not a modeling artefact: a renewable term at attained-age rates converges on term-cost pricing, and the reason carriers cap renewal at 80 is that beyond it the product stops being purchasable.

Totals over the fifty years, undiscounted: premiums ¥470,348.54, claims ¥309,768.95, net cash flow +¥50,400.25. Over the first ten years alone — the contract_boundary = current_term answer — net cash flow is −¥15,878.74. The shape is the protection shape: a deep first-year strain (¥20,844 of acquisition cost against ¥11,688 of premium), 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.

Valuation and reserve pointers#

This library projects gross cash flows. Every valuation layer consumes them and is cited, never reproduced:

  • Standard policy reserve (hyōjun sekinin-junbikin, 標準責任準備金). 保険業法第116条第2項 delegates the accumulation method and coefficient levels for the contracts the ordinance specifies REG-R4; 施行規則第68条 fixes the set REG-R7 and 第69条第1項 the taxonomy 保険料積立金 / 未経過保険料 / 払戻積立金 / 危険準備金 REG-R8. 平成8年大蔵省告示第48号 supplies the method — net level premium method (heijun jun-hokenryō-shiki, 平準純保険料式), no Zillmer adjustment — and the table vintage: contracts from 2018-04-01 value on 生保標準生命表2018（死亡保険用）REG-R10 REG-R11. The standard valuation interest rate (hyōjun riritsu, 標準利率) resets annually for regular-premium yen business off three- and ten-year means of 10-year JGB yields with safety coefficients by band, on a 0.5% trigger and 0.25% rounding, effective the following 1 April REG-R10; its current numeric value could not be established from any retrieved document and is unverified.

  • Contingency reserve (kiken junbikin, 危険準備金) is one of the four 第69条第1項 components REG-R8. 価格変動準備金 is asset-driven and outside a liability projection.

  • The valuation table is not this model’s basis. 標準生命表2018 carries an explicit ~2σ margin capped at 130% of the unadjusted rate plus a forward improvement allowance REG-R20, while q(t) here is a std adjustment of it. The same projection cannot serve the statutory reserve and the current estimate without swapping the basis; say which is in use at every point.

  • ESR. From 2026-03-31 insurers report on the 経済価値ベースのソルベンシー規制: assets at fair value, liabilities as 現在推計 + MOCE, required capital at 99.5% over one year, early corrective action below ESR 100% REG-R15. That regime change is why an assumption-parameterized, re-runnable projection is the operative artefact — the old ソルベンシー・マージン比率 basis locked mortality, lapse and interest at issue REG-R15. The old 200% trigger and the new 100% trigger are not comparable quantities REG-R17 REG-R15. jplib computes neither ratio.

  • The professional use of this projection. 保険業法第121条第1項第1号 requires the 保険計理人’s 意見書 REG-R6; the IAJ practice standard turns it into the 1号収支分析 — a forward projection of premiums, claims, expenses and surrenders by 区分経理 segment over at least ten future years, open and closed, sufficiency tested over the first five REG-R22. The recursion above is that projection, per policy.

  • Accounting. IFRS 17 is voluntary in Japan — IFRS applies as 指定国際会計基準 to insurers that elect it REG-R47. One projection feeds three separate measurement bases (J-GAAP REG-R10, ESR REG-R15, IFRS where adopted); conflating them gets wrong which assumptions are locked in.

  • On insurer failure, 生命保険契約者保護機構 covers up to 90% of the 責任準備金 under a rate delegated by 保険業法第270条の3 REG-R40 REG-R41. Cited, never modeled.

Key sensitivities and model risks#

In rough order of leverage:

  1. The best-estimate mortality factor. 0.80 std is the largest single lever and the least evidenced. Its justification is arithmetic on the published margin — a 130% cap implies 1/1.3 = 0.769 — offset by improvement already inside the table REG-R20. A user with own experience should replace it before anything else in this file.

  2. The renewal-decline rate. d = 15% std is the largest structural lever and has no uklib analogue. Undiscounted net cash flow over the fifty years runs +¥92,123.94 at d = 0%, +¥50,400.25 at 15%, +¥22,587.91 at 30% — a factor of four across a range no document narrows.

  3. The renewal rate scale beyond age 50. Nothing is published above the age-50 cell [S2], and the anchor cell spends its last twenty years there. The std extension back-casts to within 1.5% at age 30 and 0.9% at age 40, which is reassuring about the form and says nothing about the level an insurer will charge in 2056.

  4. Contract boundary. Whether the liability runs to the ceiling or to the end of the current 保険期間 changes the sign of the undiscounted answer on this cell. The model does not rule; the ESR coefficient 告示 that would settle it were not retrieved REG-R16.

  5. Selective lapsation across renewals. Renewal takes no 告知 [S1] [S4] [S8] [S12], so the anti-selection is structural and repeats four times on this cell. Base run lambda = 0 understates late-duration claims by construction.

  6. Early-duration lapse against front-loaded acquisition cost. ¥20,844 of year-1 outgo against ¥11,688 of year-1 premium makes the first three lapse rates decide how long the strain takes to recover. No Japanese clawback evidence exists in the sources.

  7. Expense inflation on small premiums, and the age basis. ¥4,000 p.a. of maintenance against ¥11,688 of premium is a third of the first-term load, so the 1.0% std inflation rate is a poor one to leave unexamined; and the 満年齢 / 保険年齢 mismatch [S1] REG-R20 understates q by about 0.7% at age 30 rising to 4.2% at age 40 — small beside item 1, but systematic and in one direction.

Known modeling pitfalls:

  • 高度障害 is not a second decrement. 生保標準生命表2018（死亡保険用）includes 高度障害 in its death rate REG-R20, and the contract pays one benefit and terminates on either event [S1] [S8]. Adding a 高度障害 incidence on top of the table double-counts claims.

  • 更新 reprices; it does not re-issue. l(t) is continuous across the boundary — no reset to 1, no acquisition expense in the base run, no new 保険証券 [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]. Treating each renewed term as a fresh policy gets persistency, 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 [S1] [S2] [S8], so an issue age of 35 has a final term of 5 years and the projection still ends exactly at 80. 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.

  • 歳満了 never renews. Such a model point has one term, one premium and no repricing [S1]; applying the renewal machinery to it invents cover the contract does not have.

  • Lapse pays nothing. There is no 解約返戻金 for the whole term [S1] [S4] [S6] [S8] [S9] [S10] [S13] [S14], so claims_lapse is identically zero and lapse acts only through l(t). One of the eight carriers whose position is documented does have a surrender value [S12] — a Japan term chassis cannot assume the absence the way a UK one can, so the zero is asserted from the composite’s sources, not from the product class.

  • There is no 自動振替貸付 on this chassis. With no 解約返戻金 there is no collateral, and one carrier states the absence in terms [S7]. Importing the APL mechanic from the whole life technical notes (終身保険) creates a no-lapse cushion the contract does not have; the supervisory guideline in any case requires an APL, where one exists, to run at the policyholder’s election rather than automatically REG-R14. Grace → 失効 → 復活-or-not is the whole persistency machinery here.

  • The ¥248 policy fee is premium, not expense. It sits inside P_m [S2] and enters the model only through P_a; crediting it against maintenance expense counts it twice, and P_a must reconstruct as 12 × (248 + 2 × 363) = 11,688 on the anchor cell.

  • Renewal decline is not lapse. It applies only in boundary years, only after mortality and ordinary lapse, and it dominates them: in year 10 of the anchor cell it is 0.08235591 of 0.11175249 total exits. Folding it into w(t) makes the boundary invisible and mis-times most of the cohort’s departure.

  • A failed first renewal premium is an expiry, not a lapse. Where the first premium of the renewed contract goes unpaid through grace, the renewal is treated as never having happened and the contract terminates at the original expiry rather than being 解除 [S1] [S7]. Those lives must not appear in force in year t + 1 collecting the renewed premium, and must not be counted as a mid-term lapse of a term that never began.

  • The living-needs cap is per insured, aggregated across contracts — not per contract [S1] [S7] [S8] [S12]. Inside the composite’s ¥1,000,000–¥30,000,000 envelope it is therefore exactly reached at the ceiling and never reduces a single-contract payment. 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.

  • Read the table at the right age. 契約年齢 is 満年齢 [S1] and 標準生命表2018 is built for 保険年齢 REG-R20. The base run reads at 満年齢 and understates, so the shift module must move q up, not down.

  • Naming the boundary is part of reporting the number. The same cell gives +¥50,400.25 to the ceiling and −¥15,878.74 over the current term. Neither is an ESR current estimate on its own, because the ESR standard-model treatment of a no-underwriting auto-renewal is unverified here REG-R16.