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 model on paper. They describe no single insurer’s product. [S#] and [R#] tags resolve in sources.md, whose numbering is carried verbatim from _research/nursing-care.md and is frozen; [REG-R#] tags resolve in the cross-product reference library references/regulatory-and-actuarial-references.md, whose R-numbering is separate. std marks a standardization introduced for the reference implementation, always with a rationale and, where one exists, the observed range; unverified marks a claim not confirmed against a retrieved document. Every contractual parameter value here is identical to product-spec.md’s. Five quantities appear here that product-spec.md does not carry, because they are modelling constructs rather than contractual terms and are introduced below as such: the certification prevalence curve, the prevalence-to-incidence conversion, the care-state mortality multiple, the recovery rate, and the sub-65 specified-disease (tokutei shippei, 特定疾病) gate.

This document states its deltas against the medical technical notes, the jplib third-sector chassis, whose model is Medical_JP_S. It inherits that file’s monthly grid, its timing conventions, its age basis, its mortality construction from the third-sector (dai-san-bun’ya, 第三分野) standard table 第三分野標準生命表2018, its lapse table, its expense structure and its whole-of-life horizon. It replaces the chassis’s benefit machinery outright. medical is frequency × severity × limit: a daily amount (nichigaku, 日額) multiplied by paid days, capped per hospitalization and again in aggregate, with two day ledgers that on the expectation never bind. Nursing care is incidence into an absorbing state: a lump sum on first entry, an annuity while the insured survives after entry, a premium waiver from a lower entry threshold than either, and a payment counter that does bind. There is no d_pay, no d_ben, no L1, no LA and no agg_days_* ledger anywhere in this model. What replaces them is a three-state chain — healthy, in care, dead — whose entry is certified by a municipality and whose exit, in the base run, is only death.

And the incidence basis is public, which is unique in this library. medical had to construct incidence from 患者調査 prevalence and a mean length of stay; uklib has no public long-term-care morbidity series at all. Japan publishes a national census of certified persons every year, split by sex, age band and all seven certification grades R4 R5 REG-R30. That census is a prevalence, not an incidence, and converting one to the other is the whole modelling problem of this product. Section (c) does it explicitly.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows for a single-policy model point of private nursing-care insurance (kaigo hoken, 介護保険) on the public-scheme-linked (kōteki kaigo hoken rendō-gata, 公的介護保険連動型) design: office premiums, care lump sum (kaigo ichijikin, 介護一時金), care annuity (kaigo nenkin, 介護年金), maintenance and claim expenses, and commission. The intended sense is the current estimate (genzai suikei, 現在推計) the economic-value solvency regime requires — probability-weighted future cash flows on assumptions re-set at a stated reporting date (kijunbi, 基準日) rather than locked in at issue REG-R15 — and it is also the shape of the 1号収支分析, a forward income-and-outgo projection over at least ten future years by 区分経理 segment REG-R22.

  • Out of scope, cited not reproduced. Discounting, MOCE, required capital and every reserving basis. Standard policy reserve (hyōjun sekinin-junbikin, 標準責任準備金), contingency reserve (kiken junbikin, 危険準備金) including its separately identified third-sector limb, the ESR balance sheet and IFRS 17 all consume these cash flows and are pointed at in Valuation and reserve pointers, not computed here.

  • Projection frequency. Monthly grid, inherited from the chassis. The unit of account here is not a day but an annual annuity instalment, and the composite’s premium mode is monthly (月払) — the mode of every published rate table retrieved [S6] [S8], though 半年払 and 年払 are also offered [S1]. t is the policy month, t = 0, 1, …, proj_len − 1, and month t runs from t to t + 1 months after the contract date (keiyakubi, 契約日).

  • Timing conventions std. Office premium received at the start of month t, and only by lives not yet on premium waiver (hokenryō haraikomi menjo, 保険料払込免除); maintenance expense at the start of month t; the lump sum, the annuity instalment and the claim-handling expense at the end of month t; then care incidence, then mortality, then lapse, then benefit-driven termination. Acquisition expense and initial commission at t = 0. The annuity is paid in advance: the first instalment falls on the entry month itself, not a year later [S1] [S7].

  • Claim-date convention std. A certification of need for nursing care (yō-kaigo nintei, 要介護認定) takes effect retroactively to the application date, with the decision due within 30 days R1. The model dates every claim at the month the trigger is met, not at notification, which is the contractually correct date and is up to a month earlier than a notification-dated projection.

  • Age basis. Attained age at 契約日 with the fraction discarded (man-nenrei, 満年齢), incremented at each 年単位の契約応当日 [S1]. age(t) = x + floor(t / 12), x the 契約年齢. 第三分野標準生命表2018 is built for an insurance age (hoken-nenrei, 保険年齢) 方式 — nearest birthday — basis REG-R20, so reading it at 満年齢 understates the valuation age by about half a year; jplib accepts the offset in the base run and marks it std, exactly as medical does.

  • Currency. JPY throughout. Expected values are fractional and displayed to ¥0.01.

  • Model points. One policy at a time on an expected (probability-weighted) basis; Projection is parameterized by point_id. No aggregation logic is specified here.

  • Termination. Whole-of-life cover with whole-of-life premiums [S1] [S4] [S7] [S8] [S10] [S11]. The projection runs to the terminal age of 第三分野標準生命表2018, 116 for males and 118 for females REG-R18 REG-R20, so proj_len = 12 × (terminal_age − x + 1) — 684 months for the anchor cell. There is no maturity benefit, no 死亡保険金 and no surrender value (kaiyaku-henreikin, 解約返戻金), so death and lapse are pure liability-releasing decrements. There is a benefit-driven termination: the contract is extinguished on the tenth annuity instalment, effective retroactively to the date that instalment’s trigger was met [S1].

  • Contract boundary. Level 平準払 premiums, non-participating (mu-haitō, 無配当), no insurer repricing right on the 終身 chassis [S1] [S4] [S7] [S8], so all future premiums and benefits are inside the boundary and the horizon is the whole of life.

  • Rounding. Intermediates at full double precision. Displayed cash flows to ¥0.01, all state probabilities to six decimals std. Monthly rows rounded for display do not re-add to displayed annual totals; totals are sums of unrounded values.

Model point attributes#

Attribute

Type

Anchor cell (point_id = 1)

policy_id

str

—

issue_age (x)

int, 満年齢, 40–79

60

sex

enum {M, F}

M

lump_amount (A_L)

JPY, ¥500,000–¥3,000,000 in ¥100,000 steps

3,000,000

annuity_amount (A_N)

JPY p.a., ¥200,000–¥1,200,000 in ¥100,000 steps

600,000

grade_lump (G_L)

enum 要支援1 … 要介護3

要介護2

grade_annuity (G_N)

enum 要支援1 … 要介護3

要介護3

grade_waiver (G_W)

enum 要支援1 … 要介護3

要介護1

annuity_max (n_A)

int, instalments

10

annuity_test

enum {survival, state}

survival

company_limb

bool — the 180-day / 90-day 満65歳未満 alternative trigger

true

dementia_rider

bool — the 認知症一時金特約 (ninchishō ichijikin tokuyaku), a dementia lump-sum rider (tokuyaku, 特約)

false

dementia_amount

JPY (rider off on the anchor)

1,000,000

mci_fraction

fraction of dementia_amount on 軽度認知障害

0.10

rec_rate

annual recovery out of the care state; read only under annuity_test = state

0.0 std

sel_lapse_lambda

anti-selective lapse loading lam on incidence

0.0 std

waiting_1y

bool — the 1-year 不担保期間 of the simplified-underwriting design

false

premium (P)

JPY per month, office premium, model-point input

11,500 std

prem_mode

enum {monthly, semiannual, annual}

monthly

prem_period

enum {whole_life}

whole_life (終身払)

issue_date

date

—

premium is an input, not a computed quantity. No carrier publishes assumed incidence rate (yotei hasseiritsu, 予定発生率), assumed interest rate (yotei riritsu, 予定利率) or 予定死亡率 for this product, and the regulator confirms there is nothing standard to publish for 第三分野 business R10; the statement of the method of calculating premiums and reserves (sanshutsu hōhō-sho, 算出方法書) is a 基礎書類 filed with the 金融庁 and is not public REG-R2. The ¥11,500 anchor is the sum of two published specimen rates at male 60, 月払, 終身/終身払 — ¥11,550, rounded to the nearest ¥500: ¥5,820 for a ¥3,000,000 要介護2以上 lump sum [S6] and ¥5,730 for a ¥600,000 要介護2以上 annuity on the 5年確定 basis [S6], with a second carrier’s ¥5,817 for a ¥3,000,000 要介護3以上 lump sum corroborating the first to three yen [S8].

State variables#

Variable

Description

Updated

pols_if(t)

In-force probability at the start of month t, alive, not lapsed, not terminated; pols_if(0) = 1. Identically pols_act(t) + care_w(t)

monthly

pols_act(t)

Active: in force and not yet certified at G_W. The premium-paying and lapse-exposed population

monthly

care_w(t)

In force and has entered 要介護1以上 — waiver running, premium zero, lapse suspended

monthly

care_l(t)

In force and has entered 要介護2以上 — the lump sum has been paid to these lives

monthly

care_n(t)

In force and has entered 要介護3以上 — the annuity is in payment

monthly

ann_coh(s, t)

Survivors at t of the cohort that entered G_N in month s; carries the instalment counter

monthly

age(t)

Attained 満年齢 = x + floor(t / 12)

annually

mort_rate(t), mort_rate_care(t)

Annual healthy and care-state mortality at age(t)

lookup

inc_rate_w/l/n(t)

Annual entry rate into 要介護1以上 / 要介護2以上 / 要介護3以上

derived

lapse_rate_mth(t)

Monthly lapse, applied to pols_act only

lookup

net_cf(t)

Net cash flow of month t, insurer perspective, income-positive

monthly

Four absences are product facts, not gaps. There is no cash-surrender-value state. Four of the seven carriers publish an outright nil 解約返戻金 and one carries the fact in the formal product name [S1] [S2] [S7] [S8], so cv_pp does not exist and lapse carries no cash flow. There is no automatic premium loan (jidō furikae kashitsuke, 自動振替貸付) state: with no surrender value there is nothing to lend against [S2] REG-R14. There is no death benefit: the contract terminates on death with nothing payable [S1] [S11], so claims_death does not exist. And there is no day ledger of any kind — the chassis’s agg_days_dis, agg_days_acc and their 支払限度日数 machinery have no counterpart here.

The three care ledgers are nested by construction: care_n ≤ care_l ≤ care_w ≤ pols_if at every t. They are three marginal first-entry distributions riding on one survival ledger, not three disjoint compartments, so they must never be added together — the implementation carries them as cumulative-entry counters and the roll-forward check asserts the ordering, not a sum.

Assumption inputs#

(a) Contractual / guaranteed elements (cited; the insurer cannot change them)#

Input

Value

Basis

介護一時金

A_L = ¥3,000,000, once per contract, on first satisfaction of 要介護2以上; does not terminate the contract

[S1] [S4] [S7] [S12]; threshold std

介護年金

A_N = ¥600,000 p.a. in advance from first satisfaction of 要介護3以上, on the annual anniversaries of the 介護年金支払基準日, at most 10 instalments

[S1] [S7] [S10]; threshold and cap std

Annuity metering

Survival-tested — each instalment needs only that the insured be alive on the payment date; recovery does not stop it

[S4] [S7] [S10] [S12]; std against [S1]

Unpaid lump sum at annuity start

Paid together with the first annuity instalment

[S1]

保険料払込免除

From 要介護1以上, plus 高度障害状態 and a listed 身体障害状態 within 180 days of an accident; waived premiums treated as paid

[S1] [S2] [S8]; threshold std

Company-basis limb

約款-defined dependency state persisting 180 days (90 where dementia-defined), insured 満65歳未満 at diagnosis, with the post-65 continuation relief

[S1] [S2] [S4] [S12]; std

Waiting period

None on the care benefits; 180-day 認知症診断責任開始期 on the dementia rider only

[S1] [S2] against [S4] [S5] [S7]

責任開始期前 rule

Nothing paid where the state results from an illness contracted or an accident occurring before the 責任開始期

[S1] [S2]

Contract termination

On the 10th annuity instalment, retroactive to that instalment’s trigger date; otherwise death, lapse or rescission

[S1]

解約返戻金

None at any duration

[S1] [S2] [S7] [S8]

Grace, 月払

To the last day of the month following the 払込期月; lapse (shikkō, 失効) the day after

[S2]

Reinstatement (fukkatsu, 復活)

Within 1 year of lapse, on fresh declaration (kokuchi, 告知); the 責任開始期 resets to the 復活日

[S1] [S2]

配当金

None — the chassis is 無配当

[S1] [S2] [S4]; REG-R9

(b) Insurer-discretionary current elements#

This class is nearly empty, and its emptiness is the product fact — the same position as medical. There is no 契約者配当, so the 三利源 (死差 / 利差 / 費差) framing and the surplus-distribution methods of 施行規則 第30条の2 do not attach REG-R9; no premium review on the 終身 chassis; no MVA; no non-guaranteed charge scale. What remains:

Input

Snapshot value

Basis

Incidence basis (危険発生率)

The insurer’s own, unpublished, in the 算出方法書

REG-R2; the regulator requires a test, not a table R10 REG-R13

Right to change base rates (基礎率変更権)

Exists for 第三分野, with a numeric exercise standard that must be disclosed at point of sale; not modelled

R10; REG-R39; [std scope]

Adjudication of the company-basis limb

Physician diagnosis against the 約款 definition, which the carrier states differs from the public 要介護認定 standard

[S1] [S2]; not modelled [std scope]

Catastrophe proportionality

War exclusion may be waived where the affected lives would not materially change the liability

[S2]; not modelled [std scope]

The 基礎率変更権 is the one discretionary item with real economic content on this product, and it is deliberately outside the model: what jplib implements is the capability the regime asks for — an incidence basis parameterized so it can be re-set and re-run R10 REG-R15 — not the exercise of the right.

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

Mortality of healthy lives. Inherited from the chassis without change. 第三分野標準生命表2018 is public, free and machine-readable REG-R18 REG-R19, but it is a valuation table whose margin runs the wrong way for a best estimate on a living-benefit product: it is graduated from the national 第21回生命表（2010年）rather than insured experience, it excludes 高度障害, and its risk-theory adjustment is bounded 70% below and 85% above the unadjusted rate R9 REG-R20. Male q60 on it is 0.00548 against 0.00653 on 生保標準生命表2018（死亡保険用）R8 REG-R18. So:

mort_rate(age) = mort_be_factor × q_third_sector(age, sex),   mort_be_factor = 1.25 [std]

mort_be_factor is the reciprocal of 0.80, a round value inside the sourced 70%–85% band REG-R20 — it unwinds the table’s stated margin and nothing more — and it is identical to medical’s, because it is the same table read for the same reason. mort_rate_mth(t) = 1 − (1 − mort_rate(age(t)))^(1/12) std. The 日本アクチュアリー会 site terms restrict reproduction REG-R21, so jplib ships mort_table.csv as a std construction — quoted anchor rates joined by a log-linear graduation — whose provenance column points at REG-R18 and REG-R20 on every row, never as a copy.

Care-state mortality — the differential, and where it comes from. No impaired-life table for the 要介護 state exists in any retrieved source; the 日本アクチュアリー会 publishes a mortality table for this class and no morbidity or impaired-life table at all R8 REG-R23. The model therefore carries a flat multiple:

mort_rate_care(age) = k × mort_rate(age),   k = 2.75 [std]

k is anchored on two sourced numbers. 平均余命 at age 75 on 第23回完全生命表 is 12.54 years for a male R13 REG-R24; the only quantified duration of care in any retrieved source is the household survey’s average of 55.0 months (4 years 7 months) R14. On a constant-force approximation life expectancy is the reciprocal of the force, so the ratio 12.54 / 4.583 = 2.74 is the implied mortality multiple, rounded to 2.75. Both inputs are biased and the biases are named rather than netted: the 55.0-month figure surveys people who provided care, not certified persons, it is truncated (respondents still caring are counted at elapsed duration) and it ends on recovery as well as on death — all of which make it too short, so k = 2.75 is if anything too high; and 12.54 years is population mortality, not healthy-life mortality, which pushes the other way. There is no observed range. Since more than 90% of certified 第1号被保険者 are 75 or over R4, age 75 is the right anchor age for the comparison.

k is not only a post-onset assumption. It feeds back into the derived incidence below, and dropping it (setting k = 1) cuts lifetime lump-sum claims on the anchor cell by 31.0%. That coupling is the least obvious property of this model.

Recovery. The care state is absorbing in the base run — rec_rate = 0 std. This is forced by the sources and it is a real simplification, not a harmless one: the statute provides for a 有効期間 and for the grade to move up or down at 要介護更新認定 R1, a carrier’s own FAQ addresses 「要介護状態が改善された場合」 [S3], and the state-tested annuity design [S1] cannot be modelled at all without it. It is carried as a named input set to zero, with a suggested placeholder of 5% p.a. std when the annuity_test = state switch is on. It is defensible for the composite because the composite’s annuity is survival-tested and one carrier says in terms that recovery does not stop payment [S7], and because the lump sum, having been paid once, cannot be unpaid [S1].

What rec_rate means, and the known limit of the state switch. rec_rate is the rate of falling below the annuity grade G_N, not the rate of returning to health, and it is applied to care_n alone. A life that recovers therefore leaves the annuity ledger, becomes eligible to re-qualify on a new 介護年金支払基準日, and keeps its 保険料払込免除 for the rest of the contract: it stays in care_w, never re-enters pols_act, never pays premium again and is never again exposed to lapse.

For the modelled step that is contractually right. G_N = 要介護3 and G_W = 要介護1, so a downgrade 要介護3 → 要介護2 stops a state-tested annuity while leaving the waiver running, which is what the ladder says [S1] [S2] [S8]. What is not modelled is recovery below G_W — the downgrade that would end the waiver, restore the premium and put the life back in the lapse-exposed population. There is no published transition matrix between grades in any retrieved source, and one rate cannot carry two thresholds, so the model implements the one it can evidence and states the other as a gap. The consequence, and its direction:

  • under annuity_test = state with rec_rate > 0 the waiver, once started, runs for life, so premium income is a lower bound and the waived band is an upper bound. It is a lower bound in the exact sense that act(t), and therefore every premium, is identical to the run with rec_rate = 0 even though the annuity ledger is materially smaller;

  • the one place recovery does reach care_w is through the ten-payment cap, and it moves it upward: a recovered life takes no tenth instalment, so term(t) extinguishes fewer contracts and more of them stay in force on waiver. Recovery never releases a life back into act(t);

  • the base run is untouched — rec_rate = 0 there, and the whole question is a property of a switch that is off;

  • reading rec_rate as a recovery-to-health rate is the error to avoid. It is a recovery-below-G_N rate, and this is a [std scope] limit of the switch, not a bug in it.

Morbidity — turning a prevalence into an incidence. This is the centre of the product.

What is published is 要介護（要支援）認定者数 and certification rate (nintei-ritsu, 認定率) — a point-in-time count of certified persons, not a flow of new certifications. At 31 March 2024, 認定者数 was 7.08 million against 35,890 thousand 第1号被保険者, an 認定率 of 19.4%, split by age as 4.3% at 65–74 and 31.1% at 75 and over R4 R5 REG-R30. The grade composition is 要支援1 14.4% · 要支援2 14.1% · 要介護1 20.7% · 要介護2 16.8% · 要介護3 13.1% · 要介護4 12.6% · 要介護5 8.3% R4 REG-R30, from which 要介護1以上 = 71.5%, 要介護2以上 = 50.8% and 要介護3以上 = 34.0% of all certified persons, derived from that composition R4.

Step 1 — a prevalence curve by age std. Only two age-banded rates were retrieved; the five-year-band rates that a finer basis wants sit in the e-Stat release of the same statistic, which was not fetched R4 REG-R33. The curve is a logistic in attained age:

prev(x) = prev_ceil / (1 + exp(−beta × (x − x_mid)))

pinned to the two sourced rates at representative ages 70 (the midpoint of the 65–74 band) and 82 (the approximate population mean age of the 後期高齢者 group), with a ceiling prev_ceil = 0.95 std. That gives beta = 0.194069 and x_mid = 85.710591, and prev(70) = 0.043, prev(82) = 0.311 by construction, to the six decimals the fitted parameters carry.

The curve is pinned at 70 and 82 and is unpinned above 82, and above 82 is where the claims are. This is the model’s largest single piece of unsourced structure and it is stated here rather than left to be discovered:

  • The logistic has three parameters and two sourced anchors. One degree of freedom is therefore not identified by the data at all, and the free parameter is prev_ceil — the one that governs the tail. prev_ceil = 0.95 is a std choice with no sourced value behind it, not a fitted quantity.

  • Everything the curve says above age 82 is extrapolation. prev(90) = 0.662 and prev(100) = 0.894 are outputs of that unpinned degree of freedom; neither is sourced, and no retrieved document reports a 認定率 at any age above the 75+ band.

  • On the anchor cell 40.2% of lifetime benefit outgo falls at attained age 83 or over — that is, in the extrapolated region. Across the eight shipped model points the share runs from 40.2% (issue age 60) to 74.7% (issue age 79). An extrapolation carrying between two-fifths and three-quarters of the claims is a first-order model risk, not a detail of the fit.

  • Read the fitted tail as an upper bound on the gradient: because prev is convex over the 75+ band, pinning at the population mean age assigns the band average to too young an age and therefore overstates beta.

  • What the free parameter is worth, on the anchor cell: refitting the same two anchors under prev_ceil = 0.60, 0.50 and 0.40 gives prev(90) = 0.517, 0.459 and 0.388 and moves lifetime benefit outgo by −4.2%, −5.8% and −6.7%. The lifetime total is less sensitive than the tail rates are, because a lower ceiling refits to a steeper beta through the anchors and buys back at 70–82 what it gives up at 90+ — which is exactly why the ceiling is easy to overlook and why the timing of the claims moves more than the total does.

  • What would fix it is not a better fit but more data: the five-year-band 認定率 in the unfetched e-Stat release of the same statistic REG-R33 would pin the curve where it currently extrapolates.

Step 2 — grade composition std. prev_G(x) = s_G × prev(x) with s_W = 0.715, s_L = 0.508, s_N = 0.340 R4 REG-R30. Holding the shares constant across ages is a standardization with a known direction of error: severity composition worsens with age, so the model understates 要介護3以上 prevalence at old ages and overstates it at young ones. The published composition is a single all-ages figure, so no observed range exists.

Step 3 — the conversion, which is an identity, not an approximation. In an illness-death model with no recovery, write mu_H and mu_C for the forces of mortality outside and inside the state and i_G for the entry hazard. Differentiating the prevalence prev_G = C / (H + C) along the age axis gives

d(prev_G)/dx = (1 − prev_G) × [ i_G − prev_G × (mu_C − mu_H) ]

and therefore

i_G(x) = prev_G'(x) / (1 − prev_G(x))  +  prev_G(x) × (mu_C(x) − mu_H(x))

Two terms, and the second is not a refinement. A rising prevalence understates incidence because the certified population is simultaneously being drained by its own excess mortality; on the anchor basis the mortality term is 5.8% of i_L at age 60 and a much larger share at the ages where claims actually happen. Substituting the logistic derivative prev_G'(x) = s_G × beta × prev(x) × (1 − prev(x)/prev_ceil) and mu_C − mu_H = (k − 1) × mort_rate(x) gives the form the model implements. inc_rate_G_mth(t) = inc_rate_G(age(t)) / 12 std, uniform within the policy year, exactly as medical treats its incidence.

The conversion rests on a stationary-population assumption std: the cross-sectional 認定率 by age is read as the prevalence path a cohort will follow. Certified persons have grown roughly 2.8-fold in the 23 years since the scheme began R16 and the 認定率 rose 19.0% → 19.4% in one year R5, so the cross-section is not a cohort path; the assumption is the same class of standardization medical makes on 患者調査 REG-R26 REG-R27, and it is stated rather than hidden.

Step 4 — the sub-65 gate std. Below 65 the public limb fires only where the care state arises from one of the 16 特定疾病 listed in 介護保険法施行令 第2条 R1 R3, and the company-basis limb, which partly fills the hole, is restricted to lives 満65歳未満 [S1] [S4] [S12]. So:

inc_rate_G(x) = f_age(x) × [ the two-term identity above ]
f_age(x) = 0.20  for x < 65,   1.00  for x ≥ 65        [std]

f_sub65 = 0.20 is a standardization with a weak anchor and it is named as such: 第2号被保険者 are 131 thousand of the 7,083 thousand certified persons, 1.85% of the total R4, but the 第2号被保険者 denominator was not retrieved so no rate can be computed. The factor is set well above 1.85% because the company-basis limb backfills part of the restriction; a company_limb = false run should use 0.05 std instead. There is no observed range. The gate produces a 6.1× step in incidence between age 64 and age 65 on the anchor basis, which is a real feature of the product and not an artefact to smooth away.

Resulting annual entry rates into 要介護2以上, from the std basis: 0.000134 at 60 · 0.000295 at 64 · 0.001795 at 65 · 0.004795 at 70 · 0.012413 at 75 · 0.030456 at 80 · 0.065003 at 85 · 0.115568 at 90. The 65-to-90 gradient is a factor of 64.

Every rate in that list is an output of the std proxy mortality, not of 第三分野標準生命表2018 itself. The identity’s second term is prev_G × (k − 1) × mort_rate(x), so i_G reads the mortality basis at every age — but the 日本アクチュアリー会 site terms permit these documents to quote only the rates the library actually uses REG-R21. mort_table.csv quotes and attributes a set of anchor rates and interpolates between them (see model.md), so six of the eight ages above — 60, 65, 75, 80, 85 and 90 for a male — sit on a quoted rate and the other two are graduated. The eight entry rates are reproducible from the shipped model; a list computed on the published table would not be, and printing one would publish more of that table than the quoting rule allows.

Lapse std. Inherited from the medical technical notes unchanged, and anchored the same way: the only published industry-wide persistency figure in Japan is 解約・失効率 5.6% p.a. on 個人保険, measured on opening in-force sum assured REG-R31, and no Japanese durational curve is public.

Policy year

1

2

3

4

5

6–20

21+

lapse_rate std

9.0%

7.0%

6.0%

5.5%

5.0%

4.5%

3.0%

lapse_rate_mth(t) = 1 − (1 − lapse_rate(year(t)))^(1/12) std. Lapse applies only to pols_act — see Policyholder behavior modeling.

Expenses and commission (all levels std). Inherited from the chassis with one change.

Input

Value

Basis

Acquisition expense

¥20,000 per policy at t = 0

std, as medical

Initial commission

1.5 × annualized premium at t = 0 (¥207,000 on the anchor)

std, as medical

Renewal commission

3.0% of premiums from policy year 2

std, as medical

Maintenance expense

¥250 per policy per month, inflating 1.0% p.a. at each anniversary

std, as medical

Claim expense

¥5,000 per claim event — the lump sum, and each annuity instalment

std, raised from medical’s ¥3,000

Expense inflation

1.0% p.a. flat

std, as medical

The claim expense is the one deliberate divergence: a care claim requires verification of a municipal certification the insurer does not control, or adjudication of a 180-day persistence test against a 約款 definition the carrier itself says differs from the public standard [S1] [S2], and every annuity instalment carries an annual survival check [S1] [S7]. Maintenance expense is charged on pols_if, including lives on waiver.

Cash flow components and recursions#

Notation#

Symbol

Meaning

t

policy month, t = 0, 1, …, proj_len − 1

x, age(t)

契約年齢 (満年齢); attained age x + floor(t/12)

y(t)

policy year, floor(t/12) + 1

A_L, A_N

介護一時金額 (JPY); 基準介護年金額 (JPY per instalment)

n_A

maximum annuity instalments (10)

s_W, s_L, s_N

grade shares of certified persons: 0.715 / 0.508 / 0.340

prev(x), prev_G(x)

all-grade and grade-G certification prevalence at age x

beta, x_mid, prev_ceil

logistic prevalence parameters: 0.194069 / 85.710591 / 0.95

k

care-state mortality multiple (2.75)

f_age(x)

sub-65 特定疾病 gate: 0.20 below 65, 1.00 at 65 and over

i_W, i_L, i_N

annual entry rates into 要介護1以上 / 要介護2以上 / 要介護3以上; _m = monthly

q_H(t), q_C(t)

monthly healthy and care-state mortality at age(t)

w(t)

monthly lapse rate, applied to pols_act only

n_W, n_L, n_N

expected entrants in month t into each of the three states

S_C(s, t)

care-state survival from month s to month t

P, e(t), ec

monthly office premium; monthly maintenance expense; claim expense per event

E0, c0, c_r

acquisition expense; initial commission; renewal commission rate

Dimensional check. prev and prev_G are dimensionless proportions of a population; i_G is a rate per year and i_G_mth a probability per month; q_H, q_C, w are probabilities per month. beta carries units of 1 / year, which is why prev_G' = s_G · beta · prev · (1 − prev/prev_ceil) comes out as a rate per year and can be added to prev_G · (k − 1) · mort_rate, also a rate per year — the two terms of the identity are dimensionally the same object, and a version of the formula that adds a prevalence to a rate is the commonest way to get this wrong. A_L is JPY per event and A_N JPY per instalment, so A_L × n_L(t) and A_N × ann_count(t) are JPY per policy-month. P, e(t), ec, E0 are JPY. Every net_cf term is JPY per month. The error this check catches is the one that dominates this product: multiplying the published 認定率 — a prevalence, 19.4% — by a benefit amount as if it were an annual claim frequency.

The three-state chain#

Healthy → in care → dead, with the care state absorbing in the base run. Write act(t) = pols_act(t). Entrants in month t:

n_W(t) = act(t)                 × i_W_m(t)
n_L(t) = (pols_if(t) − care_l(t)) × i_L_m(t)
n_N(t) = (pols_if(t) − care_n(t)) × i_N_m(t)

n_W is drawn from the active population alone, because a life already certified at 要介護1以上 cannot enter it again. n_L and n_N are drawn from everyone in force who has not yet reached that grade, which includes lives already in a lower care grade — progression up the ladder is the dominant route into 要介護3以上, not direct entry from health. Because i_N_m ≤ i_L_m ≤ i_W_m at every age (the entry rate is monotone in prev_G, and s_N < s_L < s_W), the nesting care_n ≤ care_l ≤ care_w ≤ pols_if is preserved by the recursion and does not need to be imposed.

Roll-forward, with term(t) the lives extinguished by their tenth annuity instalment:

act(t+1)    = ( act(t) − n_W(t) ) × (1 − q_H(t)) × (1 − w(t))
care_w(t+1) = ( care_w(t) + n_W(t) ) × (1 − q_C(t)) − term(t)
care_l(t+1) = ( care_l(t) + n_L(t) ) × (1 − q_C(t)) − term(t)
care_n(t+1) = ( care_n(t) + n_N(t) ) × (1 − q_C(t)) − term(t)
pols_if(t+1) = act(t+1) + care_w(t+1)

Care-state lives carry q_C and no lapse; active lives carry q_H and lapse. The check_pols() identity is pols_if(t) == act(t) + care_w(t) at every t, and check_nesting() asserts the three inequalities.

The annuity ledger and the ten-payment cap#

Instalments fall on the annual anniversaries of the 介護年金支払基準日, so the cohort entering in month s is paid in months s, s+12, …, s+12(n_A − 1) while it survives [S1] [S7]:

S_C(s, t)     = product over u = s … t−1 of ( 1 − q_C(u) )
ann_count(t)  = sum over j = 0 … n_A−1 of  n_N(t − 12j) × S_C(t − 12j, t)
claims_annuity(t) = A_N × ann_count(t)
term(t)       = n_N(t − 108) × S_C(t − 108, t)          (the tenth instalment)

The j = 0 term is n_N(t) itself: payment in advance, on the entry date. S_C must be computed as a partial product, not as a ratio SC(t)/SC(s) of cumulative products — q_C reaches 1 at the terminal age, so the cumulative product underflows to zero and the ratio form fails exactly where the tail of the liability lives.

term(t) removes the extinguished lives from care_n, care_l, care_w and hence from pols_if, which is the retroactive extinction of [S1] expressed on a monthly grid. On the anchor cell the cap binds: entrants receive 4.62 instalments on average and 14.9% of them reach the tenth, so removing the cap raises lifetime annuity cost by 11.4%. This is the sharpest single contrast with medical, where the 通算 day limit never binds on the expectation and the ledger reads zero forever.

Under the annuity_test = state switch the instalment additionally requires the care state to persist, and a lapse of the state resets the schedule: a life that re-qualifies gets a new 介護年金支払基準日 and starts again from instalment 1 [S1] [S2]. That switch is the only consumer of rec_rate, and with rec_rate = 0 it is a no-op — which is why the switch and the recovery rate must be tested together. Recovery moves care_n only: a recovered life keeps its 保険料払込免除, for the reason and with the limit set out under Recovery above.

Processing order#

For t = 0, 1, …, proj_len − 1:

  1. Start of month. premiums(t) = P × act(t) — not P × pols_if(t); maintenance e(t) × pols_if(t) with e(t) = 250 × 1.01^floor(t/12); renewal commission c_r × premiums(t) for t ≥ 12. At t = 0 additionally E0 and c0 = 1.5 × 12P.

  2. Look up the age basis. age(t), hence mort_rate, mort_rate_care, prev(age(t)) and the three entry rates; hence q_H(t), q_C(t), i_W_m, i_L_m, i_N_m, w(t).

  3. Incidence. n_W(t), n_L(t), n_N(t) per the formulas above; append n_N(t) to the annuity cohort ledger.

  4. End of month — claims.

    claims_lump(t)    = A_L × n_L(t)
claims_annuity(t) = A_N × ann_count(t)
claims(t)         = claims_lump(t) + claims_annuity(t)

    and the claim-handling expense ec × ( n_L(t) + ann_count(t) ).

  5. End of month — decrements, mortality then lapse [std order], on the state each life occupies after step 3; then term(t). Lapse pays nothing — there is no surrender value [S1] [S2] [S7] [S8] — so claims_lapse(t) is identically zero, and that zero is a product fact worth publishing.

  6. Ledger update. Decrement every surviving annuity cohort by 1 − q_C(t); drop cohorts that have taken their tenth instalment.

Net cash flow#

net_cf(t) = premiums(t) − claims_lump(t) − claims_annuity(t)
          − expenses(t) − claim_expenses(t) − commissions(t)

with expenses(t) = e(t)·pols_if(t) + E0·1{t = 0} — acquisition and maintenance only — claim_expenses(t) = ec·(n_L(t) + ann_count(t)) deducted on its own line, and commissions(t) = c0·1{t = 0} + c_r·premiums(t)·1{t ≥ 12}. net_cf is income-positive in the shipped model, per the library convention. The worked example’s table below prints expenses and claim_expenses added together in one column, and says so; the model publishes them as two columns of result_cf().

Policyholder behavior modeling#

  • Lapse stops at the waiver trigger, and that is a structural fact, not a refinement. Once 要介護1以上 is certified the premium is waived and treated as paid on each 払込期月の契約応当日 [S1] [S2], so there is no premium to miss; and with no 解約返戻金 there is nothing to surrender for. Care-state lives therefore carry mortality only. Over the anchor cell’s whole projection the waiver removes 5.53% of the premium income the block would otherwise pay, and at age 85 30.1% of the surviving in-force block is on waiver and paying nothing.

  • The waiver fires strictly before the benefit. G_W = 要介護1 sits one grade below G_L = 要介護2 and two below G_N = 要介護3, which is the market pattern [S1] [S8] and the reverse of medical, where the base waiver is disability-triggered and independent of the benefit. There is a real band of lives — 要介護1 lives, 20.7% of all certified persons R4 — for whom the contract has stopped collecting premium and has not yet paid anything.

  • Lapse is real and immediate for everyone else. No 解約返戻金 means no 自動振替貸付, which REG-R14 treats as a policyholder election in any case, and one carrier’s 約款 says the contract simply lapses the day after grace expires [S2]. No lapse-suppression term belongs in the recursion; the whole_life APL machinery must not be inherited here.

  • 復活 [std scope]. Available within one year on fresh 告知 [S1] [S2], but a reinstated policy is not the policy that lapsed: the 責任開始期 resets to the 復活日, and the entire benefit definition is anchored to 責任開始期以後の傷害または疾病 [S1] [S2]. It belongs in the model as a new model point, not as a negative lapse; the base run treats lapse as absorbing.

  • Anti-selection is unusually direct here, and it is at the front door rather than in the lapse. What is being selected against is an application to a public body that leaves a record, which is why underwriting declines anyone who has ever been certified for, or has ever applied for, 要支援 or 要介護 [S1], or who lives in a 高齢者向け施設 [S11]. The composite is fully underwritten, so no selection loading is applied at issue; the simplified-underwriting design prices its leniency with a 1-year 不担保期間 instead [S7] and is carried as the waiting_1y model-point flag, off in the base run.

  • Anti-selective lapse std (optional module, off in the base run). Healthy lives lapse first, so the persisting block is progressively impaired on the incidence basis: inc_eff(t) = inc_rate_G(age(t)) × [1 + lam × max(0, w_cum(t) − w_ref)] with w_ref = 0.20 and lam = 0.30 std, identical to medical’s module. Base run lam = 0. No Japanese selective-lapse evidence was retrieved.

  • Thresholds and amounts are elected at issue and cannot move. G_L, G_N, G_W, A_L and A_N are model-point attributes; at one carrier the five 保険契約の型 are mutually exclusive and cannot be changed mid-term [S4]. A code path that varies them over t models a contract term that does not exist.

  • クーリング・オフ. Out of scope: an eight-day pre-inception right under 保険業法 第309条 [S1] REG-R36, and modelling it would need a new-business funnel this library does not have.

Worked example#

Anchor cell (point_id = 1). Male, 契約年齢 60 (満年齢), 終身 / 終身払, 介護一時金 A_L = ¥3,000,000 on 要介護2以上, 介護年金 A_N = ¥600,000 per year from 要介護3以上 capped at n_A = 10 instalments and survival-tested, 保険料払込免除 from 要介護1以上, company-basis limb on, 認知症一時金特約 off, 1-year 不担保期間 off, office premium P = ¥11,500 per month. `proj_len = 12 × (116 − 60 +

  1. = 684months. All four rows below sit atage = 60` and in policy year 1, so one set of rates drives them.

Assumption values used, every one of them:

  • Mortality. 第三分野標準生命表2018 男 q60 = 0.00548 R8 REG-R18, quoted here because a worked example needs it and jplib quotes only the rates it uses REG-R21. Best estimate mort_rate(60) = 1.25 × 0.00548 = 0.00685 std; care-state mort_rate_care(60) = 2.75 × 0.00685 = 0.0188375 std. Monthly: q_H = 1 − (1 − 0.00685)^(1/12) = 0.0005726334; q_C = 1 − (1 − 0.0188375)^(1/12) = 0.0015835104.

  • Lapse. Policy year 1, lapse_rate = 9.0% std; w = 1 − (1 − 0.09)^(1/12) = 0.0078284203.

  • Prevalence. prev(60) = 0.95 / (1 + exp(−0.194069 × (60 − 85.710591))) = 0.0064240463, from the logistic std pinned to the sourced 4.3% at age 70 and 31.1% at age 82 R4 R5 REG-R30. Slope prev'(60) = 0.194069 × 0.0064240463 × (1 − 0.0064240463/0.95) = 0.0012382778 per year.

  • Grade shares. s_W = 0.715, s_L = 0.508, s_N = 0.340 R4 REG-R30, so prev_W(60) = 0.0045931931, prev_L(60) = 0.0032634155, prev_N(60) = 0.0021841757.

  • Incidence, from the two-term identity with f_age(60) = 0.20 std. For 要介護2以上: slope term 0.508 × 0.0012382778 / (1 − 0.0032634155) = 0.0006311047; mortality term 0.0032634155 × (2.75 − 1) × 0.00685 = 0.0000391202; annual i_L = 0.20 × (0.0006311047 + 0.0000391202) = 0.0001340450, monthly i_L_m = 0.0000111704. Likewise i_W = 0.0001889030 (i_W_m = 0.0000157419) and i_N = 0.0000896238 (i_N_m = 0.0000074686).

  • Expenses. e(t) = ¥250 (policy year 1); ec = ¥5,000; acquisition ¥20,000; initial commission 1.5 × 12 × 11,500 = ¥207,000; renewal commission 3% from t = 12 — so zero in every row below std.

Every decrement rate above is either quoted from 第三分野標準生命表2018 REG-R18 or derived from 介護保険事業状況報告 R4 R5 REG-R30 with its citation, or is marked std as an illustrative value in the shape of such a table. None of them is an insurer’s basis, and none could be: the 算出方法書 is not published REG-R2 and there is no standard third-sector incidence table to publish R10.

t

pols_if

care_w

premiums

claims_lump

claims_annuity

expenses

commissions

net_cf

0

1.000000

0.000000

11,500.00

33.51

4.48

20,250.09

207,000.00

−215,788.09

1

0.991604

0.000016

11,403.26

33.23

4.44

247.99

0.00

+11,117.59

2

0.983278

0.000031

11,307.33

32.95

4.41

245.91

0.00

+11,024.07

3

0.975022

0.000047

11,212.21

32.67

4.37

243.85

0.00

+10,931.33

The expenses column above is the combined expense of the month — maintenance, claim handling, and acquisition at t = 0. The model keeps them apart, publishing expenses (acquisition and maintenance) and claim_expenses as two columns of result_cf(); the traces below give both parts of every figure in the column.

Trace, month 0. pols_if(0) = 1, care_w(0) = 0, so act(0) = 1. Premium = 11,500 × 1 = 11,500.00. Entrants: n_W = 1 × 0.0000157419 = 0.0000157419; n_L = (1 − 0) × 0.0000111704 = 0.0000111704; n_N = (1 − 0) × 0.0000074686 = 0.0000074686. claims_lump = 3,000,000 × 0.0000111704 = 33.5112. The annuity cohort ledger holds one entry, paid immediately in advance, so ann_count(0) = 0.0000074686 and claims_annuity = 600,000 × 0.0000074686 = 4.4812. Claim expense = 5,000 × (0.0000111704 + 0.0000074686) = 0.0932. expenses = 250.00 (maintenance) + 0.0932 (claim) + 20,000.00 (acquisition) = 20,250.0932. commissions = 1.5 × 138,000 = 207,000.00. net_cf(0) = 11,500.00 − 33.5112 − 4.4812 − 20,250.0932 − 207,000.00 = −215,788.09. Roll forward: act(1) = (1 − 0.0000157419) × (1 − 0.0005726334) × (1 − 0.0078284203) = 0.99158782; care_w(1) = (0 + 0.0000157419) × (1 − 0.0015835104) = 0.00001572; care_l(1) = 0.00001115; care_n(1) = 0.00000746; pols_if(1) = 0.99158782 + 0.00001572 = 0.991604. term(0) = 0 — no cohort is 108 months old.

Trace, month 1. pols_if(1) = 0.991604, care_w(1) = 0.000016, so act(1) = 0.99158782 and premiums = 11,500 × 0.99158782 = 11,403.2599 — note the premium rides on act, not on pols_if, and the two have already parted company in the fifth decimal. Every per-policy rate is unchanged (age(1) = 60, policy year still 1). Entrants: n_L = (0.991604 − 0.00001115) × 0.0000111704 = 0.0000110765, so claims_lump = 33.2295; n_N = (0.991604 − 0.00000746) × 0.0000074686 = 0.0000074059. ann_count(1) sums the cohorts entering at t = 1, −11, −23, …; only t = 1 exists, so ann_count(1) = n_N(1) = 0.0000074059 and claims_annuity = 4.4435 — the month-0 cohort is one month old, not twelve, and is not paid again until t = 12. expenses = 250 × 0.991604 (maintenance, 247.9009) + 5,000 × (0.0000110765 + 0.0000074059) (claim, 0.0924) = 247.9933; no acquisition expense after t = 0, no renewal commission before t = 12. net_cf(1) = 11,403.2599 − 33.2295 − 4.4435 − 247.9933 = +11,117.59. pols_if(2) = 0.983278.

Trace, month 2. Identical structure: act(2) = 0.98324640, premiums = 11,307.3336; n_L = 0.0000109834, claims_lump = 32.9501; n_N = 0.0000073436, claims_annuity = 4.4062; expenses = 245.8194 + 0.0916 = 245.9111; net_cf(2) = +11,024.07. care_w(3) = 0.000047, pols_if(3) = 0.975022.

Policy year 1 in aggregate (t = 0…11, all at age 60, all in policy year 1 — the strongest single test target in this file, because it exercises the whole annual cycle on one set of rates). Σ pols_if(t) = 11.461077 and Σ act(t) = 11.460074; the gap of 0.001003 is the waiver already biting in year 1.

Line

Policy year 1 total

premiums

131,790.85

claims_lump

384.05

claims_annuity

51.36

expenses (acquisition + maintenance)

22,865.27

claim_expenses

1.07

expenses + claim_expenses

22,866.34

commissions

207,000.00

net_cf

−98,510.90

with pols_if(12) = 0.903774, care_w(12) = 0.000179, care_l(12) = 0.000127 and care_n(12) = 0.000085. (The totals are sums of unrounded monthly values; the four displayed rows do not re-add to them, and the year-1 net_cf differs by ¥0.01 from the sum of the rounded monthly values.)

What the numbers say. Year-1 claims of ¥435.41 are 0.33% of year-1 premium — an order of magnitude thinner than medical’s 21.5%, because on the std basis the entry rate into 要介護2以上 at age 60 is 0.000134 a year and at age 90 it is 0.115568, a factor of 862 once the sub-65 gate is included and 64 from 65 to 90. This product prefunds a cost that essentially does not arise for twenty-five years, which makes the lapse assumption, not the incidence basis, the dominant lever: over the whole projection the std table leaves 0.126 of the block in force at age 85, and lifetime claims come to 34.7% of lifetime premium against 53.5% with lapse switched off. Against the two published premium scales the std basis reproduces 41.8% of the lump-sum limb’s premium and 27.3% of the annuity limb’s as undiscounted expected claims [S6] [S8] — low for a retail loss ratio, which is the honest signal that the std prevalence tail, the constant grade composition and the inherited lapse table are all calibration targets rather than results.

Valuation and reserve pointers#

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

  • 標準責任準備金. 保険業法 第116条 requires a 責任準備金 at each 決算期 and delegates the accumulation method REG-R4; 施行規則 第68条 fixes scope REG-R7; 平成8年大蔵省告示第48号 sets the method as net level premium (heijun jun-hokenryō-shiki, 平準純保険料式) on the standard valuation interest rate (hyōjun riritsu, 標準利率) and the standard table REG-R10. For contracts concluded from 1 April 2018 the third-sector valuation mortality is 第三分野標準生命表2018 REG-R11 R8 REG-R18. The 標準利率 applicable to this class could not be established from a retrieved document and is unverified; no value is asserted anywhere in these documents.

  • 危険準備金, and the third-sector limb specifically. 施行規則 第69条 divides the reserve into 保険料積立金, 未経過保険料, 払戻積立金 and 危険準備金 and requires a separately identified 第三分野保険の保険リスクに備える危険準備金 REG-R8. On top of 標準責任準備金 the 金融庁 requires an annual ストレステスト checking that the 予定事故発生率 covers the 99th percentile of incidence risk over a ten-year horizon, a 負債十分性テスト by future cash flow analysis where the 予定発生率 fails to cover risk defined at 97.7%, disclosure of the incidence model used, and a transparent numeric 基礎率変更権 exercise standard disclosed at point of sale R10, computed under 平成10年6月8日大蔵省告示第231号 with the calculating unit separated from internal audit REG-R13 REG-R14. That notification’s own text was not retrieved and its stress magnitudes are unverified. jplib implements the capability the regime demands — an incidence basis parameterized so a shock can be applied per grouping — and not the statutory stress. That distinction must not be blurred downstream.

  • The regulator’s own words are why this model’s morbidity basis is std. For 第三分野 business 「標準死亡率、参考純率といったスタンダードな指標が存在しておらず、公的なデータや各社の実績等から給付事由ごとその発生率を見込まざるを得ない」 R10: each insurer estimates incidence per benefit trigger from public data and its own experience. The reference implementation does exactly that, in public, from R4 and R5.

  • ESR. From 31 March 2026 insurers are supervised on the economic-value 経済価値ベースのソルベンシー規制, liabilities measured as 現在推計 + MOCE at each 基準日 on assumptions re-set then, required capital at 99.5%, early corrective action below an ESR of 100% REG-R15; it supersedes the ソルベンシー・マージン比率 200% trigger REG-R17. jplib computes neither. What it owes the regime is a projection re-runnable on a re-set assumption basis at a stated 基準日 — which, for a third-sector product, is precisely the capability the ストレステスト already demands R10.

  • 1号収支分析. The 保険計理人 appointed under 第120条 REG-R5 gives an 意見書 under 第121条 REG-R6; the 実務基準 turns that into a forward income-and-outgo analysis over 「少なくとも将来10年間」 by 区分経理 segment REG-R22. That ten-year horizon is the same horizon the third-sector ストレステスト uses R10.

  • Three bases, one projection. J-GAAP statutory reserving REG-R10, the ESR economic balance sheet REG-R15 and IFRS 17 — voluntary in Japan REG-R47 — are three measurement bases fed by one set of projected cash flows, which is why these notes keep the cash flows basis-agnostic and undiscounted.

  • Not applicable to this chassis. 契約者配当 and the surplus-distribution methods of 施行規則 第30条の2 REG-R9 do not attach: the contract is 無配当 [S1] [S2] [S4]. 価格変動準備金 under 第115条 is asset-driven and outside a liability projection entirely REG-R3. On insurer failure, contracts are compensated up to 90% of the 責任準備金 through the 生命保険契約者保護機構 REG-R40 REG-R41.

  • Policyholder tax, not modelled. Premiums fall in the 介護医療保険料 basket of the post-2012 生命保険料控除 R11 R12 REG-R43; the anchor’s ¥138,000 annual premium is past the ¥80,000 point at which the basket deduction flattens to ¥40,000, so the marginal premium yen carries no relief. Benefits are not projected net of policyholder tax.

Key sensitivities and model risks#

In rough order of leverage on this block:

  1. Lapse, not incidence. The claims are twenty-five years out and the std lapse table removes most of the block before they arrive: lifetime claims are 34.7% of lifetime premium on the base table and 53.5% with lapse off, and halving the table lifts the ratio to 43.5%. On a product whose liability is concentrated past age 85, persistency is the first-order assumption — and the only sourced anchor is a 5.6% sum-assured-weighted industry figure on a book of death cover REG-R31.

  2. The prevalence tail — an extrapolation carrying the claims. The logistic is pinned at ages 70 and 82 and is unpinned above 82, and above 82 is where the claims are: 40.2% of lifetime benefit outgo on the anchor cell falls at attained age 83 or over, rising to 74.7% at issue age 79. prev(90) = 0.662 and prev(100) = 0.894 are therefore unsourced extrapolations, not fitted values — the curve has three parameters and two anchors, and the unidentified one, prev_ceil = 0.95 std, is precisely the one that sets the tail. Refitting the same anchors at prev_ceil = 0.60 / 0.50 / 0.40 gives prev(90) = 0.517 / 0.459 / 0.388 and moves lifetime outgo by −4.2% / −5.8% / −6.7%; the total moves less than the tail rates do only because a lower ceiling refits to a steeper beta and re-times the claims earlier, so the shape moves more than the sum. Separately, the Jensen bias in the 82 pin overstates beta. This is the model’s largest unsourced structure and its single most improvable input: the five-year-band 認定率 that would pin the tail sits in the unfetched e-Stat release of the same statistic REG-R33.

  3. The care-state mortality multiple k, twice over. It sets how long the annuity runs and it is the second term of the incidence identity. Setting k = 1 cuts lifetime lump-sum claims by 31.0%; k = 4 raises them by 12.4% while cutting annuity claims by 8.3%, because heavier post-onset mortality shortens the annuity it lengthens the incidence into. No impaired-life table exists in any retrieved source.

  4. The constant grade composition. Raising all three shares by 10% relative lifts lifetime claims by 7.5%. Since severity mix worsens with age and the model holds it flat, the direction of the error is known and unquantified R4 REG-R30.

  5. The ten-payment cap. Removing it raises lifetime annuity cost by 11.4%; cutting it to five payments cuts annuity cost by 25.9%. The cap is a contractual parameter at one carrier [S1] and a payout-shape election at another [S7], and it is genuinely live on the expectation — unlike every limit on the medical chassis.

  6. The sub-65 gate. f_sub65 = 0.20 changes almost nothing in the lifetime totals (the loss ratio moves from 34.7% to 36.1% at f_sub65 = 1) but it changes the anchor cell’s first five years by a factor of five, so it dominates every early-duration test and every issue age below 65 — which is most of the 40–79 issue range.

  7. Survival-tested against state-tested annuity. The two designs differ by whether a recovery decrement exists at all [S1] against [S4] [S7] [S10] [S12], not by a parameter. The composite takes the majority survival-tested form; the switch is not a refinement of it.

  8. Longevity is the tail risk, not mortality. On a living-benefit product longer survival means more instalments and more entrants, and the valuation table is set deliberately below national mortality for exactly that reason R9 REG-R20. Using it unadjusted as a best estimate is conservative in the reserving direction and material over a 57-year projection.

Known modeling pitfalls#

  • 認定率 is a prevalence, not an incidence. 19.4% of 第1号被保険者 are certified at a point in time R4 R5 REG-R30; multiplying that (or 50.8% × 19.4%) by a benefit amount, or treating it as an annual claim frequency, is the single commonest error in a Japanese nursing-care model. The conversion is the two-term identity above and is an explicit std step.

  • The excess-mortality term of the identity is not a refinement. Dropping prev_G × (mu_C − mu_H) — which is what happens if the care-state mortality multiple is set to 1 “because there is no impaired-life table” — cuts lifetime lump-sum claims on the anchor cell by 31.0%. A rising prevalence in a population being drained by its own excess mortality implies a higher incidence than the prevalence slope alone.

  • The care state is absorbing and the benefit does not stop. rec_rate = 0 is a named input, not an omission [S3] R1, and one carrier says in terms that recovery does not stop the annuity [S7]. The annuity_test = state switch is a no-op unless rec_rate is moved off zero, so the two must be tested together.

  • Lapse stops at the waiver trigger. Once 要介護1以上 is certified the premium is waived and there is no 解約返戻金 to surrender for [S1] [S2], so lapse must apply to pols_act only. Applying it to pols_if destroys the annuity liability it took thirty years to build.

  • Premium income rides on pols_act, never on pols_if. The waiver fires two grades below the annuity and one below the lump sum [S1] [S8], so a band of lives pays nothing and receives nothing. Charging premium to the whole in-force block overstates lifetime premium income by 5.85% on the anchor cell — the same ¥92,738, read as 5.53% of the in-force-weighted total the waiver takes it out of. At age 85 the band is 30.1% of the surviving block.

  • The three care ledgers are nested, and the nesting must hold at every t. care_n ≤ care_l ≤ care_w ≤ pols_if. Independent entry hazards without the ordering let a life start the annuity before its lump sum has been paid, which the contract forbids — the unpaid lump sum is paid with the first instalment [S1]. They are marginal distributions on one survival ledger and must never be summed.

  • The ten-payment cap binds — do not carry medical’s intuition across. Entrants take 4.62 instalments on average and 14.9% reach the tenth; removing the cap raises annuity cost by 11.4%. The 通算 day ledger that reads zero forever on medical has no counterpart here.

  • The annuity is in advance and starts on the entry date. The j = 0 term of ann_count(t) is n_N(t) itself [S1] [S7]. Deferring the first instalment by a year removes roughly a tenth of the annuity liability and misdates all of it.

  • Compute care-state survival as a partial product. q_C reaches 1 at the terminal age, so a cumulative-product-ratio form of S_C(s, t) divides by zero exactly where the tail of this liability lives.

  • There is a step in incidence at exactly age 65, and it belongs there. Below 65 the public limb fires only on one of the 16 特定疾病 R1 R3 and the company-basis limb is restricted to 満65歳未満 [S1] [S4] [S12]; entry into 要介護2以上 jumps 6.1× between age 64 and 65 on the std basis. A smooth curve through 65 misprices every issue age in the lower half of the 40–79 range.

  • “180日” names two different mechanisms and a model must not implement one of them twice. A 不担保期間 or 認知症診断責任開始期 means cover has not started; the 180-day (90-day for dementia) test inside the company-basis trigger means the care state must have persisted [S1] [S2] [S4] [S5] [S7]. The composite has the second and not the first on the care benefits. Related timing trap: a certification takes effect retroactively to the application date R1, so a claim dated at notification is up to a month late.

  • The prevalence tail is extrapolated, and a reader must not treat it as sourced. The curve is pinned only at ages 70 and 82; prev(90) = 0.662 and prev(100) = 0.894 come from prev_ceil = 0.95 std, a parameter no retrieved document supports, and 40.2% of the anchor cell’s benefit outgo (74.7% at issue age 79) falls in that extrapolated region. Quoting a tail rate as though it carried the R4 REG-R30 provenance of the two anchors misrepresents where this model’s evidence stops.

  • A recovery under annuity_test = state does not restore the premium. rec_rate models the fall below the annuity grade G_N, not a return to health. A life that so falls leaves care_n but stays in care_w, so its 保険料払込免除 runs for the rest of the contract and it is never again exposed to lapse. Under 要介護3 → 要介護2 that is contractually right — the waiver fires at 要介護1, two grades lower — but recovery all the way below 要介護1 is not modelled at all, so premium income under that switch is a lower bound. Reading rec_rate as a recovery-to-health rate overstates the waiver and understates premium.

  • No surrender value, no APL, no death benefit. claims_lapse(t) is identically zero [S1] [S2] [S7] [S8], nothing carries a policy through a missed premium [S2] REG-R14, and claims_death does not exist [S1] [S11]. Importing the whole_life 自動振替貸付 logic suppresses lapses that really happen; adding a death benefit invents one that does not.