The Projection Space

The by-policy projection of the LTC_KR_S model.

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

>>> Projection[1].result_cf()          # the worked example's anchor cell
>>> Projection.point_id = 5            # or switch the default

t counts policy months, 0-based. Month t runs from t to t + 1 months after the 계약일, so t = 0 is the first projected month. proj_len() is the number of projected months, the frame’s exclusive end — 12 x (term_age - issue_age) + 1, so 601 on the anchor cell and 601 rows in result_cf(), indexed 0 to 600, and the frame is range(proj_len()). The last row, t = proj_len() - 1, is the 계약해당일 at 만나이 term_age on which the contract matures with nothing payable: it holds the surviving in-force count and no cash flow.

The age basis

The projection runs on 만나이 (man nai, age last birthday), incremented at each 계약해당일: age(t) = issue_age + t // 12. The contract itself ages on 보험나이 (boheom nai, insurance age — 계약일 현재 실제 만 나이 with a fraction under six months discarded and six months or more rounded up, 표준약관 제21조 [REG-R25]), and the two differ for roughly half of all issue dates. The 만나이 basis is not a concession here, as it is on the cancer chassis; it is the right basis twice over. Every public series the decrements are built from — the 노인장기요양보험 통계연보 연령별 인정률, the 치매역학조사 prevalence and the 생명표 behind the mortality — is published on 만나이; and the benefit definition itself contains a 만나이 test, 「만 65세 이상 노인」 or 「노인성 질병을 가진 만 65세 미만의 자」. The 보험나이 offset therefore survives only in the premium, which enters this model as a model point input, so no conversion is applied anywhere.

Notes symbol map

The technical notes use compact actuarial symbols; the cells use lifelib names.

Notes symbol	Cells	Meaning
—	model_point	the selected model point as a Series
n	proj_len	number of projected policy months
x	issue_age	만나이 at the 계약일
x + floor(t/12)	age	attained 만나이 in month t
P	premium_mth_pp	level monthly office premium
A_B	lump_amount	장기요양진단급여금 sum insured
A_1 / A_2	annuity_high / annuity_low	간병연금 monthly amount by entry grade
G_B	benefit_grade	the contractual 등급 threshold
q(x)	mort_rate	healthy annual mortality
q_C(x)	mort_rate_care	care-state annual mortality
P(x)	prev_rate_at	all-grade certification prevalence
P_C(x)	prev_care_at	prevalence at or above G_B
P_L(x)	prev_light_at	prevalence below G_B
i_D(x)	inc_rate_direct_at	healthy -> care, direct entry
i_L(x)	inc_rate_light_at	healthy -> light grade
rho(x)	prog_rate_at	light grade -> care, progression
l(t)	pols_if	in force at the start of month t
AV(t)	av_pp	계약자적립액 per policy at the start of month t
CV(t)	cv_pp	해약환급금 per policy on a lapse in month t
CF(t)	net_cf	net cash flow, income positive

The four compartments, and why there are four

The contract is a three-state model — healthy, in long-term care, dead — but the care state is not entered in one step, and pretending it is puts the cash flow years too early. Only 13.3% of current 1등급 certifications arose from a first application, against 69.5% from a renewal, whereas at 인지지원등급 the first-application share is 69.8%: severe-grade lives are, in the main, people who entered the scheme years earlier at a light grade and deteriorated. So the in-force block is carried in three compartments and a fourth absorbing one:

pols_healthy()

never certified. Pays premium, is exposed to lapse, carries healthy mortality.

pols_light()

certified at a grade below benefit_grade(). The contract does nothing for these lives — no benefit, no waiver — so they still pay premium and are still exposed to lapse, but they carry an impaired mortality and they are the pool progression draws from.

pols_care()

certified at or above benefit_grade(). The lump sum has been paid, the annuity is running, the premium is waived, surrender is barred by the 약관 and the mortality is the care-state one. Absorbing: the amount and the 감액 are both frozen at first certification and the annuity is metered on survival, not on continued certification, so the contract itself makes the state absorbing for cash-flow purposes.

pols_act() is the first two together — the premium-paying, lapse-exposed population — and pols_if() is all three.

Prevalence to incidence, which is the modelling work

The public statistics are a prevalence. Writing P for the all-grade certification prevalence at 만나이 x, s_G(x) for the share of certified lives at grade G or above, P_C = s_G P and P_L = P - P_C, and mu_H, mu_L, mu_C for the three forces of mortality with mu_bar the population average, the compartment identities in a stationary population are

inflow_C(x) = P_C’(x) + P_C(x) (mu_C(x) - mu_bar(x)) inflow_L(x) = P_L’(x) + P_L(x) (rho(x) + mu_L(x) - mu_bar(x))

The excess-mortality term is not a refinement: a rising prevalence understates entry because the certified population is being drained by its own excess mortality, and that drain is what the 간병연금 is exposed to as well. Two equations carry three unknowns — direct entry, progression and light-grade entry — and the closing assumption is the one the sources leave open: direct_entry_share [std], the share of gross inflow into the care state arriving straight from health rather than by progression, set at 0.20 from the 13.3% / 69.8% first-application split above. Then

i_D(x) = direct_entry_share x inflow_C(x) / (1 - P(x)) rho(x) = (1 - direct_entry_share) x inflow_C(x) / P_L(x) i_L(x) = inflow_L(x) / (1 - P(x))

Three properties of this construction belong in front of a reader rather than in a footnote. It rests on a stationary-population assumption [std]: the cross-sectional 인정률 by age is read as the prevalence path a cohort will follow, and the Korean certified stock grew 71.8% in six years, so the cross-section is not a cohort path and the identity understates entry. It treats the care compartment as leaving only by death, whereas 9.2% of current certifications arose from a 등급변경신청 and grades move both ways, which understates entry again. And below 65 there is no prevalence data at all: the statute admits an under-65 applicant only through the closed 노인성 질병 list, so below 65 the two entry rates are carried down from their age-65 values on the log-gradient of the one disclosed 예정위험률 — 13.0% a year for men, 17.9% for women — which is the only Korean long-term-care incidence anyone publishes.

The level that construction produces is about a fifth of that disclosed 예정위험률 at the same age. disclosed_inc_ratio_at() publishes the ratio rather than hiding it: a 예정위험률 is a loaded rate for a select, underwritten, 180-day-waited population on 보험나이, and the three biases named above all run the same way, so the gap is expected — but it is the largest single uncertainty in this model and the technical notes carry it as a sensitivity.

Input data

Inputs are external files: plain CSVs living in the model folder’s parent directory, products/long_term_care/, read at run time rather than stored inside the model. Each table has a filename Reference and a reader Cells, both on Data, reached here through the data Reference. The consequence worth knowing: the model is not portable on its own. Copying the LTC_KR_S folder without its parent’s CSVs produces a model that reads and then fails on first evaluation.

Cells Descriptions

model_point()

The selected model point as a Series.

policy_id()

The policy identifier of the selected model point.

issue_age()

x: the 만나이 (man nai, age last birthday) at the 계약일.

The composite issues from 30 to 70, the modal envelope of the long-term-care benefit itself rather than of the chassis carrying it [std]. The contract ages on 보험나이; see the Space docstring for why this model does not convert.

sex()

The sex (M / F) of the insured: a rating factor, and a key into every basis table.

It is the only rating factor besides age. No smoker class, occupation class or amount band was disclosed on any long-term-care cover in any retrieved document.

term_age()

The 만나이 at which the 보험기간 ends: 90 on the composite [std].

Observed 85 / 90 / 95 / 100세만기 and 종신; 90세만기 is the modal Korean maturity and the term of both published rate cards. It truncates the exposure at exactly the band carrying the highest certification rate of all — 41.7% at 85 and over, and still rising — so the choice is materially conservative on claim cost, and 95 and 100 are shipped as model points.

prem_period_years()

The 납입기간 in years: 20 on the composite [std].

20년납 is the basis of both published rate cards, it is the 해약공제계수 cap of 별표 14 [REG-R20], and it is structurally necessary to the 미지급형 form — the surrender value is a step at 납입완료, and a 전기납 contract has no step because it has no 납입완료 before maturity.

prem_period_mths()

n_P: the 납입기간 in policy months, 12 x prem_period_years().

prem_mode()

The 납입주기: 월납 on every model point, which is why the grid is monthly.

Monthly is the dominant retail mode and the mode of every published rate card in the file. 감독규정 제7-65조제2항 expressly permits the 계약자적립액 to be computed 「연납보험료를 기준으로」 [REG-R18], which is how a Korean monthly-premium product reconciles a monthly grid with an annually-recursed account.

benefit_grade()

G_B: the contractual 장기요양등급 threshold, cumulative from the top of the scale.

g1 is 1등급 only, g2 is 1~2등급 (the composite), through to g6 for 1~인지지원등급. Every retrieved contract draws its threshold downward from 1등급 and no retrieved document sells a 3등급-only or 5등급-only benefit. Widening the gate is not a re-scaling: at 1~5등급 the benefit is exposed to a population about 7.5 times larger and arriving materially earlier, which the market prices at about 4.5 : 1.

lump_amount()

A_B: the 장기요양진단급여금, ₩10,000,000 on the composite.

Paid 최초 1회한 on the first award at or above benefit_grade(), extinguishing that benefit line but not the contract: the 간병연금 keeps running and the dementia rider stays in force. The level is the unit of both published rate cards, and it is a third of the cancer chassis’s ₩30,000,000 because a long-term-care benefit sits on top of a public scheme that already meets most of the direct service cost.

annuity_on()

True where the 간병연금 rider is attached. On in the base run.

annuity_high()

A_1: the monthly 간병연금 where the entry grade is 1등급, ₩500,000 [S1].

Frozen at first certification — 「그 이후에 장기요양등급이 변경되더라도 지급액은 변경되지 않습니다」 — so a life entering at 2등급 and deteriorating to 1등급 keeps the lower rate for all ten years.

annuity_low()

A_2: the monthly 간병연금 at any other grade inside the gate, ₩300,000 [S1].

The one carrier that grades its annuity by grade grades it in two steps only, so every grade below 1등급 inside the threshold is carried at the lower amount [std].

annuity_max_mths()

n_A: the 간병연금 ceiling in months — 「10년(120개월)을 최고한도로 지급」 [S1].

The cap is the composite’s protection against a post-onset mortality basis nobody publishes, and it is one reason this shape was preferred over the uncapped 종신 form on the shelf. It binds jointly with maturity: the annuity is truncated at the earlier of the cap and the 보험기간, which is the conservative reading of a question no retrieved document resolves [std].

annuity_guar_mths()

The guaranteed months of 간병연금, 「최초 1년(12개월) 보증지급」 [S1].

Instalments inside the guarantee are paid whether or not the insured survives; after it each block of twelve is released only by the annual survival test on the anniversary of the 진단확정일, evidenced by a 주민등록등본.

dementia_rider()

True where the 치매진단급여금 rider is attached. Off on the anchor cell.

A different trigger — the CDR 척도 assessed by a 치매 전문의, not the 등급판정위원회 — with a different sex basis and its own one-year waiting period, which is why it is off in the base run. It is not an independent process: dementia is simultaneously a route into the public grade and a private trigger, so the rider’s incidence is driven off the same underlying certification model.

dementia_amount()

The 치매진단급여금 at CDR 1 이상 (경도치매상태), ₩10,000,000 where attached.

Paid once only across the tier set, behind a one-year 보장개시일 and, inside the definition of the state itself, a 90-day persistence test — together deferring a mild-dementia claim by at least fifteen months from inception.

wait_mths()

The 장기요양상태 보장개시일 in whole months: 3 on the composite [std].

「계약일부터 그 날을 포함하여 90일이 지난 날의 다음 날」, with a carve-back to the 계약일 where the cause is 재해. The observed market runs from no waiting period at all through 90 days to 180 days, and 90 days is the median and aligns this product with the cancer chassis’s 암보장개시일, so the two krlib third-sector products share one waiting-period mechanic. A certification inside the window does not defer the claim: it makes the benefit 무효 and the premiums paid for it come back — see pols_void().

red_mths()

The 감액기간 in months: 12 on the composite [std].

Where the certification is 질병-caused and the 판정일 falls inside the window the benefit is halved; where the cause is 상해/재해 it is paid in full. The observed market runs from no reduction through one year at 50% to two years at 50%. The 감액 decision is frozen at first certification for the whole life of the annuity — a claim starting inside the window stays halved for all ten years — which is the single most easily mis-modelled rule in the product.

cv_form()

The surrender-value form: mijigeup, half_during or pyojun.

Four forms are on the shelf and they are different products, not variants, though the Korean names look almost identical. mijigeup (해약환급금 미지급형) pays nothing during the premium-paying period and 50% of a notional 기본형 value afterwards, and is the composite: 63.8% of Korean 보장성 초회보험료 in 2024 H1 was written in a 무·저해지 form [REG-R27], so a library modelling only 표준형 would be modelling a minority of the market. half_during (납입중50%해약환급금지급형) pays 50% during and 100% after. pyojun (표준형) pays the full 계약자적립액 less the 해약공제액 from year 1.

lapse_form()

The lapse vector: mujihae (the regulated log-linear model) or pyojun.

The vector on a 무해지 form is not free. Among models converging to zero lapse at 완납 the 로그-선형 모형 is the 원칙모형 with a practical convergence point of 0.1% and a post-완납 ultimate of 0.8%; anything else is permitted only against disclosure in the audit report and the 경영공시, external actuarial verification, quarterly reporting of the difference to the FSS in CSM, best-estimate liability, K-ICS ratio and net income, and submission to an on-site inspection [REG-R27]. pyojun is the level comparison vector, carried so the two can be compared — which is the comparison the guidance requires an insurer to disclose.

uw_loading()

The 간편심사 premium multiplier: 1.0 fully underwritten, 1.40 simplified [std].

The cleanest published measure of the price of relaxed underwriting in Korean long-term care is 1.36-1.43x on the main contract at every age and sex. It is carried as a premium multiplier only: no retrieved source gives the simplified pool’s incidence separately, so on a loaded model point the extra premium is pure margin in this model and the true claim cost of that pool is understated. Note that one carrier will not attach its 장기요양 riders to a simplified chassis at all, which is itself a statement about anti-selection on this trigger.

premium_mth_pp()

P: the level monthly office premium per policy, a model point input.

Not a computed quantity. No Korean carrier publishes a long-term-care rate card at the composite’s specification, so the anchor cell’s ₩5,600 is constructed from two rows of the one published card — ₩3,300 for the 주계약 장기요양(1~2등급)급여금 at male 40 plus ₩2,320 for a 재가급여 support rider scaled to the composite’s grade-weighted ₩400,000 a month — and the female cell’s ₩8,400 the same way [std]. On this model’s own basis the anchor premium buys a benefit outgo whose present value at the 예정이율 is 46.0% of the present value of premium income, and the female cell’s 51.5%; the seven other model points carry premiums set at approximately that ratio, landing between 22% and 44% once the threshold, the annuity cap, the term and the 간편심사 loading are applied — the 22% being model point 8, whose 1.40 loading is pure margin here because no retrieved source gives the simplified pool’s own incidence. uw_loading() multiplies it.

pols_if_init()

l(0): the in-force count the projection opens on, one policy.

Every model point is a single new-business policy at duration zero, so the opening exposure is 1 and result_cf() opens at t = 0.

proj_len()

n: the number of projected policy months, 12 (term_age() - issue_age()) + 1.

601 on the anchor cell — a 만나이-40 life to the 90세 계약해당일. proj_len() is the exclusive end of the frame and not the last index: result_cf() carries proj_len() rows indexed 0 to proj_len() - 1 and the frame is range(proj_len()). The + 1 is the terminal row — the 600 months of cover are t = 0 ... proj_len() - 2 and t = proj_len() - 1 is the 90세 계약해당일 itself, the maturity instant: the contract ends there with nothing payable, there being no 만기환급금 on a 순수보장성 contract.

age(t)

age(t): the attained 만나이 in policy month t, x + floor(t / 12).

policy_year(t)

y(t): the policy year containing month t, t // 12 + 1.

The contractual policy-year label, which is 1-based by convention: month t = 0 falls in policy year 1. It is derived from the 0-based index and is never the index itself; only lapse_rate() reads it, the 로그-선형 lapse model being written in policy years.

mort_rate_at_age(x)

The healthy-life annual mortality of this life’s sex at 만나이 x.

A [std] Makeham-Gompertz construction calibrated on the published 제10회 경험생명표 65세 기대여명 [REG-R33]; the table itself is not published [REG-R34]. Read at 만나이 without adjustment, and with no best-estimate factor applied: the calibration anchor is an experience statistic, not a valuation margin, so there is nothing to unwind. Keyed by age rather than by t because the morbidity construction has to evaluate the whole basis at 만나이 65 when the life is younger than that.

mort_rate(t)

q(x): the healthy-life annual mortality at age(t).

mort_rate_mth(t)

q_H(t): monthly healthy mortality, 1 - (1 - q)^(1/12) [std].

mort_rate_light(t)

The annual mortality of a life certified below the benefit threshold.

light_mort_mult x q(x), capped at 1, with light_mort_mult = 1.8 [std]. No retrieved source gives a post-certification mortality table by grade. The multiple sits between healthy and care because the one retrieved study of certified decedents finds a mean 인정점수 at death of 82.1 — squarely inside 2등급 — so the deaths are concentrated in the severe grades and a light-grade life is materially healthier than the cohort that study observed. There is no observed range.

mort_rate_light_mth(t)

q_L(t): monthly light-grade mortality [std].

mort_rate_care(t)

q_C(x): the annual mortality of a life in the certified care state at age(t).

care_mort_mult x q(x), capped at 1, with care_mort_mult = 3.0 [std]. Anchored on two things and neither is a table. The yearbook roll-forward and the application-route estimator agree that the mean duration of a certification is near 4 to 5.5 years, and the mean 만나이 of a certified decedent is over 75; at 만나이 82 on the shipped mortality table a mean duration of 4.5 years implies a force of 0.222 against a healthy force of 0.075, a multiple of 2.96. The one retrieved study measuring time from certification to death — 516.2 days, 8.7% inside a month, 45.6% inside a year — is a right-censored decedent cohort and is a lower bound rather than an estimate of the duration, which is why it fixes the early shape here and not the level.

care_mort_mult is not only a post-onset assumption: it is also the excess-mortality term of the incidence identity in inflow_care_at(), so it moves the entry rate and the annuity’s run-off in opposite directions at once. That coupling is the least obvious property of this model and the technical notes vary it in a sensitivity.

mort_rate_care_mth(t)

q_C(t): monthly care-state mortality [std].

mort_rate_dem_mth(t)

The monthly mortality carried by the dementia rider’s own ledger [std].

dem_mort_mult x q(x) on an annual basis with dem_mort_mult = 2.5, between the light-grade and the care-state multiples: a CDR 1 diagnosis is a lighter state than a 1·2등급 certification and a heavier one than a 3~5등급. No source gives it.

lapse_param(name)

One parameter of the lapse vector, from lapse_table.csv.

lapse_rate(t)

The annual lapse rate in the policy year of month t.

On the mujihae form the regulated log-linear principle model [REG-R27]: the first-year rate decays geometrically in policy year to the 0.1% convergence point in the year 납입완료 falls in, and the post-완납 ultimate rate is 0.8%. On the pyojun form a level rate at every duration, carried for comparison. The first-year level is [std] — no Korean durational persistency series was retrieved — while the shape and the two convergence values are the guidance’s own.

Note what the vector says on a contract that has no soft landing: with no surrender value there is no policy loan and therefore no 보험료 자동대출납입, so a missed premium lapses the contract outright, and the assumption nonetheless has lapse falling toward 납입완료.

lapse_rate_mth(t)

w(t): the monthly lapse rate, 1 - (1 - lapse_rate(t))^(1/12) [std].

Applied to pols_healthy() and pols_light() only. A life in the care state is not exposed: the premium is waived, and the 약관 bars surrender outright once the annuity has started — 「최초 지급사유가 발생한 후에는 이 특약을 해지할 수 없습니다」 — so zero lapse in that state is a constraint and not an assumption.

prev_param(name)

One parameter of the certification-prevalence logistic for this life’s sex.

prev_ceil, prev_beta and prev_x_mid are the [std] parameters of the three-parameter logistic fitted to the five sourced 연령별 인정률 of the 2024 노인장기요양보험 통계연보; the file also carries the five anchors themselves.

prev_rate_at(x)

P(x): the all-grade certification prevalence at 만나이 x.

A logistic in age [std], prev_ceil / (1 + exp(-beta (x - x_mid))), fitted to the five sourced band rates — 1.98% at 65-69 rising to 28.6% at 85 and over for men, 1.63% to 47.3% for women. Two features of the sourced curve survive the fit and matter: the gradient is about 17% per year of age, a factor of 23 over twenty years; and the sex crossover is at about 70 — male certification exceeds female below it and female exceeds male above — which is the reverse of a death-benefit table and is independently confirmed by the disclosed 예정위험률, whose sex ratio crosses one between about 62 and 68.

Nothing above 88.5 is sourced, and that is where the claims are. The fitted ceiling is the parameter the tail is most sensitive to and the fit has one degree of freedom more than the data identifies; the technical notes carry the sensitivity.

prev_slope_at(x)

P’(x): the derivative of prev_rate_at() in age, a rate per year.

The analytic derivative of the logistic, beta P (1 - P / ceil), not a difference quotient: it is the leading term of the incidence identity and a numerical derivative would put noise straight into the claim rate.

share_ge_at(grade, x)

s_G(x): the share of certified lives at grade or above at 만나이 x.

Linear in age between the six sourced band representative ages — 60 for the under-65 band, then 67, 72, 77, 82 and 88.5 — and flat outside them. A proportion of a population, never a rate. Carrying it by age rather than as one all-ages vector is load-bearing: the 1~2등급 share is U-shaped, 22.2% below 65, 11.1% at 80-84 and 14.8% at 85 and over, and a model applying a single vector at all ages mis-prices this benefit by up to a factor of two.

share_slope_at(grade, x)

s_G’(x): the slope of share_ge_at() in age, per year.

Exact for the piecewise-linear share: the constant slope of the bracketing segment, zero outside the sourced range. It enters prev_care_slope_at() as the first half of the product rule, and it is negative over most of the range for a severe threshold, which is the U-shape doing its work.

prev_care_at(x)

P_C(x): the prevalence of certification at or above benefit_grade().

s_G(x) P(x). On the composite’s 1~2등급 gate this is only about a seventh of the all-grade prevalence — 1·2등급 are 13.28% of all certified lives, against 50.8% for the comparable Japanese quantity, so a Korean 「1~2등급」 promise is a far narrower one than a Japanese 「要介護2以上」 promise.

prev_care_slope_at(x)

P_C’(x): the full derivative of prev_care_at(), by the product rule.

s_G'(x) P(x) + s_G(x) P'(x). The first term is usually negative and the second always positive; dropping the first would overstate the entry rate wherever the severe share is falling with age, which is most of the range.

prev_light_at(x)

P_L(x): the prevalence of certification below benefit_grade().

P(x) - P_C(x), the pool progression draws from. Zero by construction where the gate is g6, since 1~인지지원등급 admits every certified life and there is no light state left outside it.

prev_light_slope_at(x)

P_L’(x): the derivative of prev_light_at(), P'(x) - P_C'(x).

mort_force_at(x)

mu_H(x): the force of healthy mortality at 만나이 x, -ln(1 - q(x)).

The compartment identities are written in forces rather than in annual rates, because a rate difference is not a hazard difference and the excess-mortality term of the identity is a difference of hazards.

mort_force_light_at(x)

mu_L(x): the force of mortality of a light-grade life [std].

mort_force_care_at(x)

mu_C(x): the force of mortality of a life in the care state [std].

mort_force_avg_at(x)

mu_bar(x): the population-average force of mortality at 만나이 x.

(1 - P) mu_H + P_L mu_L + P_C mu_C. It is the term that turns a count identity into a proportion identity: prevalence is measured against a living population, and a population that is itself dying faster than a healthy life raises every prevalence it measures. Using mu_C alone in place of mu_C - mu_bar overstates the entry rate, materially so at the ages where the certified share is large.

inflow_care_at(x)

The gross rate of entry into the care compartment per unit of population at x.

P_C'(x) + P_C(x) (mu_C(x) - mu_bar(x)), floored at zero. The second term is where the care state’s own mortality enters the incidence basis, and it is not a refinement: a prevalence that is rising understates entry, because the compartment it measures is being drained by an excess mortality the population around it does not carry. The same care_mort_mult that produces this term also runs the annuity off, so the two move together under a sensitivity and not independently.

prog_rate_at(x)

rho(x): the annual rate at which a light-grade life progresses to the benefit grade.

(1 - direct_entry_share) x inflow_C(x) / P_L(x), capped at prog_rate_cap, and zero where the gate admits every certified life so that there is no light state.

direct_entry_share = 0.20 is the [std] closing assumption of the conversion and the one number the sources leave genuinely open. Its anchor is the yearbook’s own application-route table: only 13.3% of current 1등급 certifications arose from a first application (7,371 of 55,340) against 69.5% from a renewal, whereas at 인지지원등급 — a grade nobody can progress down into — the first-application share is 69.8%. The ratio of the two is where 0.20 comes from. Getting this wrong does not change the lifetime claim count much; it changes when the claim arrives, which on a contract priced at 2.0% over fifty years is most of the answer.

inc_rate_direct_at(x)

i_D(x): the annual rate of direct entry into the care state from health.

direct_entry_share x inflow_C(x) / (1 - P(x)) where there is a light state, and the whole inflow where the gate leaves none. Below 65 the rate is carried down from its age-65 value on sub65_factor_at(), because the statute admits an under-65 applicant only through the closed 노인성 질병 list and no prevalence data exists there at all.

inc_rate_light_at(x)

i_L(x): the annual rate of entry from health into a grade below the threshold.

[P_L'(x) + P_L(x) (rho(x) + mu_L(x) - mu_bar(x))] / (1 - P(x)), floored at zero and carried below 65 on sub65_factor_at() like the direct rate. The floor binds only where the light-grade prevalence is falling faster than progression and mortality can account for, which the fitted curves do not do inside any shipped model point’s range; it is there so that a replacement basis cannot produce a negative entry rate silently.

disclosed_inc_at(x)

The disclosed 예정위험률 for a combined 1·2등급 benefit at 만나이 x.

The sum of the 요양(1등급) and 요양(2등급) 발생률 of the one carrier that publishes them, log-linearly graduated between the three quoted ages and extrapolated on the end gradients outside them. Summing the two rows is an upper bound, the two events being mutually exclusive at first certification. This is a pricing rate for a select, underwritten, 180-day-waited population quoted on 보험나이 and it is not this model’s level; it is read here for its gradient and its sex ratio, and published against the model’s own rate by disclosed_inc_ratio_at().

sub65_gradient()

The log-gradient of the disclosed 예정위험률 per year of age, for this life’s sex.

ln(i(60) / i(40)) / 20 on the combined 1·2등급 rate: 0.1222 for men (13.0% a year) and 0.1648 for women (17.9%). It is the only Korean evidence anywhere on how long-term-care incidence behaves below 65, and it carries the sex ratio with it — female over male on the combined rate runs 0.37 at 40, 0.58 at 50 and 0.87 at 60, so a curve built on this gradient crosses one in the late sixties, exactly where the population data finds the crossover.

sub65_factor_at(x)

The factor carrying the entry rates below 65, exp(-g (65 - x)); 1 at 65 and over.

[std], and it is a scope decision rather than a fitted quantity. The 인정률 series is published for the 65-and-over population only; below 65 the scheme admits an applicant only with one of the 25 노인성 질병 the 시행령 lists — four dementia codes, one Alzheimer code, fourteen cerebrovascular codes, four Parkinson-family codes and four others — with no cancer, no musculoskeletal condition and no frailty category on the list. So the under-65 exposure is both small and concentrated, and the only Korean evidence on its shape is the disclosed 예정위험률’s own gradient, which is what this applies. The progression rate is not scaled: it is a property of a life already certified, not of the gate.

disclosed_inc_ratio_at(x)

The model’s own first-entry rate at x over the disclosed 예정위험률 at x.

Published rather than hidden, because it is the largest single uncertainty in this model. The model’s first-entry rate is direct entry plus progression by lives already certified at a light grade, both read on the same sub-65 convention the projection uses — the whole basis is evaluated at xe = max(x, 65) and carried down on sub65_factor_at():

i_D(x) + P_L(xe) rho(xe) sub65_factor_at(x) / (1 - P_C(xe))

Reading P_L and rho at x itself below 65 instead — the shorter form the identity suggests — mixes an unscaled light-grade prevalence with a scaled direct rate and returns 0.267 at 만나이 40 rather than the 0.240 published here. On the shipped basis the ratio runs at roughly one fifth of the disclosed rate at ages 40 to 60. Four things all point the same way and none of them is quantified by any retrieved source: a 예정위험률 is a loaded pricing rate and not a best estimate; the conversion reads a cross-section as a cohort path in a scheme whose certified stock grew 71.8% in six years; the care compartment is treated as leaving only by death when 9.2% of certifications arose from a 등급변경신청; and the disclosed card is quoted on 보험나이, about half a year older than this model’s 만나이. The technical notes carry the gap as a stated sensitivity rather than closing it with an invented factor.

inc_rate_direct(t)

i_D: the annual direct-entry rate into the care state in month t.

inc_rate_light(t)

i_L: the annual entry rate into a light grade in month t.

prog_rate(t)

rho: the annual progression rate from a light grade to the benefit grade in month t.

inc_rate_direct_mth(t)

The monthly direct-entry rate, i_D / 12 [std], uniform within the policy year.

inc_rate_light_mth(t)

The monthly light-grade entry rate, i_L / 12 [std].

prog_rate_mth(t)

The monthly progression rate, rho / 12 [std].

dem_param(name)

One parameter of the dementia-prevalence logistic, from dementia_table.csv.

dem_prev_at(x)

The prevalence of dementia at CDR 1 or above at 만나이 x, for this life’s sex.

A logistic [std] fitted to the five sourced band prevalences of the 2023 치매역학조사 — 4.99% at 65-69, 5.03% at 70-74, 10.70% at 75-79, 15.57% at 80-84 and 21.18% at 85 and over — times the sourced 65+ sex factor, 0.957 for men and 1.035 for women. Every dementia case is CDR 1 or above by definition, CDR 0.5 being 경도인지장애 and not dementia, so this is the prevalence the composite’s 경도이상 tier is exposed to; a benefit written at CDR 3 이상 would reach about a third of it.

Two weaknesses are named rather than smoothed. The 65-69 and 70-74 anchors are almost equal, which no logistic can reproduce, and the fit is out by up to 31% at 70-74. And the sex factor is flat in age while the sourced series has the male rate above the female at 65-79 and below it at 80 and over — which is why this model does not reproduce the market fact that 치매 covers are priced cheaper for women while 장기요양 covers are priced dearer.

dem_prev_slope_at(x)

The analytic derivative of dem_prev_at() in age.

dem_inc_rate_at(x)

The annual rate of a first CDR 1 이상 diagnosis at 만나이 x.

The same prevalence-to-incidence identity the certification basis uses, with the dementia state’s own excess mortality in it, and carried below 65 on sub65_factor_at(). Building the rider off a sourced prevalence rather than off a share of the certification rate matters, because the two triggers are correlated but not proportional: dementia is the sole qualifying condition for 5등급 and 인지지원등급 and is present in 42.3% of certified decedents, yet most CDR 1 lives are nowhere near a 1·2등급 certification.

dem_inc_rate_mth(t)

The monthly rate of a first CDR 1 이상 diagnosis in month t [std].

Zero unless the rider is attached, and zero for the first dementia_wait_mths months. Fifteen months is the one-year 보장개시일 plus the 90-day persistence test written into the definition of the state itself — 「진단일부터 90일 이상 계속되어 장래에 더 이상의 호전을 기대할 수 없는」 — and the two together are why a naive prevalence-based pricing of a CDR 1 benefit is badly wrong at short durations. The one-year waiting period is not a carrier choice: it is the settled market answer to the 2019 supervisory intervention that followed the 경증치매 boom.

pols_healthy(t)

h(t): in force and never certified, at the start of policy month t.

h(0) = pols_if_init(), then

h(t+1) = ( h(t) - n_L(t) - n_D(t) ) (1 - q_H(t)) (1 - w(t))

Premium-paying and lapse-exposed, on healthy mortality. The lives leaving by pols_entry_care_direct() leave whether the certification triggers the benefit or voids it inside the 보장개시일 window, which is why the same term serves both.

pols_light(t)

l_L(t): in force and certified at a grade below benefit_grade().

l_L(0) = 0, then

l_L(t+1) = ( l_L(t) + n_L(t) - n_P(t) ) (1 - q_L(t)) (1 - w(t))

The contract does nothing for these lives — the benefit, the waiver and the annuity all fire at benefit_grade() and no retrieved Korean contract pays anything at a lower grade — so they keep paying premium and keep lapsing, on an impaired mortality. They are the pool pols_entry_care_prog() draws from, and on the composite’s 1~2등급 gate they are about six sevenths of all certified lives.

pols_care(t)

l_C(t): in force and certified at or above benefit_grade().

l_C(0) = 0, then l_C(t+1) = ( l_C(t) + n_C(t) ) (1 - q_C(t)).

Absorbing, and the contract is drafted so that it is: the 진단급여금 cannot be re-triggered, the 간병연금’s amount is fixed by the grade at first certification and 「그 이후에 장기요양등급이 변경되더라도 지급액은 변경되지 않습니다」, and the instalments are metered on survival rather than on continued certification. No lapse — the premium is waived and the 약관 bars surrender — and no recovery, which is a real simplification the contract makes rather than one this model imposes.

pols_healthy_mid(t)

The never-certified count after the month’s certifications, before mortality.

pols_light_mid(t)

The light-grade count after the month’s certifications, before mortality.

pols_care_mid(t)

The care-state count after the month’s certifications, before mortality.

The month’s voided certifications are not here: inside the 보장개시일 window the benefit is 무효 and the life leaves the model, so they appear in pols_void().

pols_act(t)

The premium-paying, lapse-exposed population: healthy plus light-grade.

Premium income is charged on this, not on pols_if(). Because the waiver fires at the same grade as the benefit rather than below it, there is no band of lives paying nothing and claiming nothing — the Japanese product’s characteristic mis-modelled item does not exist here — but the waiver is still not free: it stops the premium for as long as the insured survives in the care state, which is exactly the quantity the prevalence-to-incidence conversion cannot pin down.

pols_if(t)

l(t): the number of policies in force at the start of policy month t.

pols_act(t) + pols_care(t): alive, not lapsed, not voided and not matured, whether or not the premium is being waived. pols_if(0) = pols_if_init() = 1. This is the weight on every cash flow of the same result_cf() row. On the frame’s last row, t = proj_len() - 1, it is the count that reaches the 90세 계약해당일, and it is paid nothing: there is no 만기환급금 on a 순수보장성 contract.

pols_if_at(t, timing)

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

"BEF_DECR"

l(t), the start of the month, before any decrement; the same number as pols_if() and the weight on that month’s cash flows.

"BEF_LAPSE"

after certification and mortality, before lapse — the processing order is certification, then mortality, then lapse [std order] — so this is the population lapses are taken from, plus the care lives that are not exposed to them.

"AFT_DECR"

l(t+1), the end-of-month state.

pols_entry_light(t)

n_L(t): expected entrants into a light grade in month t, h(t) i_L_m(t).

Drawn from the never-certified population alone.

pols_entry_care_direct(t)

n_D(t): expected direct entrants into the care state, h(t) i_D_m(t).

The catastrophic route — a major stroke, an early dementia — which is the only way in below 65 and is the minority route above it. One fifth of gross inflow into the care state on the shipped direct_entry_share [std].

pols_entry_care_prog(t)

n_P(t): expected entrants into the care state by progression, l_L(t) rho_m(t).

The dominant route above 65, and the reason this model has a light compartment at all: a single-decrement model that treats the 1·2등급 rate as a healthy-life incidence overstates direct entry, understates the delay, and puts the cash flow years too early.

pols_entry_care(t)

n_C(t): expected first certifications at or above the threshold that pay.

n_D(t) + n_P(t), and zero inside the 보장개시일 window: a certification there does not defer the claim, it makes the benefit 무효 — 「특약을 무효로 하며, 이미 납입한 보험료를 돌려드립니다」 — and unlike the cancer chassis there is no cancellation option and no revival, so the benefit is simply gone. Those lives are pols_void().

pols_void(t)

Lives certified inside the 보장개시일 window, whose cover is void.

The benefit is 무효 and the premiums paid for it are returned; the contract, whose main cover is that benefit, ends. A distinct decrement rather than a claim refusal, because the two have different cash flows and the 약관 keep them apart: before the 보장개시일 the contract is void and premiums come back, inside the 감액기간 cover has started and the benefit is merely halved. On the composite the window is three months at an issue age of 40, so the decrement is of order 1e-7 of the cohort — carried because it is a product fact, not because it is material.

pols_death_act(t)

Deaths in month t among lives that have not claimed: healthy and light-grade.

These are the deaths that pay the 계약자적립액: 감독규정 제7-63조제1항제1호 requires a 제3보험 contract to pay it, plus the 미경과보험료, on death from a cause the policy does not cover, and terminate [REG-R17] [REG-R25 제22조]. Death is therefore not a pure decrement on this product, which is a direct consequence of the Korean third-sector design rule and has no counterpart in the Japanese contract.

pols_death_care(t)

Deaths in month t among lives in the certified care state.

These pay nothing: 「지급사유가 발생한 후 사망한 경우에는 별도로 책임준비금을 지급하지 않습니다」. On this product most deaths of claimants happen here, which is what makes the split from pols_death_act() a cash-flow distinction rather than a bookkeeping one.

pols_death(t)

Expected deaths in month t across all three compartments.

pols_lapse(t)

Lapses at the end of month t, from the mortality survivors of the paying population.

Care lives are not exposed — the premium is waived and the 약관 bars surrender. What a lapse pays is cv_pp(), which on the composite’s 미지급형 form is nil for the whole premium-paying period and 50% of a notional 기본형 value afterwards: a cliff, not a curve.

pols_maturity(t)

Policies reaching the 보험기간’s end at t = proj_len() - 1, nil at every other t.

They are paid nothing: 「이 상품은 순수보장성보험으로 보험계약 만기시 지급받는 금액(만기환급금)이 없습니다」. The count is published because the roll-forward has to close on the last row, and because a maturity that pays nothing is still a maturity.

pols_dem(t)

In force and already paid the 치매진단급여금; a first-event counter, not a compartment.

Once only across the tier set. Carried on the dementia state’s own mortality and held at or below pols_if(), so that pols_entry_dem() always draws from a non-negative pool. Nested inside the in-force block and never added to the three compartments.

pols_entry_dem(t)

Expected first CDR 1 이상 diagnoses under the rider in month t.

(l(t) - pols_dem(t)) x dem_inc_rate_mth(t). Zero unless the rider is attached. The two triggers are correlated, not independent: a contract carrying both a 장기요양 and a 치매 benefit pays both on the same underlying event, at different times and on different evidence, and treating them as independent decrements understates the tail. Driving both off the same life with a shared underlying state is what this model does about that.

care_surv(s, t)

S_C(s, t): survival in the care state from month s to month t.

The partial product of (1 - q_C(u)) over u = s ... t-1, computed as a partial product and never as a ratio of cumulative products: q_C reaches 1 at the terminal age of the table, so a cumulative product underflows to zero and the ratio form divides by zero exactly where the tail of this liability lives.

red_factor(t)

r(t): the 감액 factor applying to a certification dated in month t.

1 from red_mths() on; before that 1 - (1 - red_fraction) x disease_share. The 약관 test is on the cause, not on the grade: a 질병-caused certification inside the window is paid at red_fraction = 50%, an 상해/재해-caused one in full. The relative frequency of the two is given by no retrieved source, so disease_share = 0.95 is [std] and the accident carve-out is named rather than dropped; the blended factor is 0.525. Frozen at first certification for the whole life of the annuity — see ann_amount_at().

ann_amount_at(s)

A(s): the monthly 간병연금 of the cohort first certified in month s, frozen at entry.

The blend of annuity_high() at 1등급 and annuity_low() at every other grade inside the gate, weighted by the age-specific grade shares at age(s), times the 감액 factor at s. Both the amount and the reduction are frozen: 「최초 진단 확정일을 기준으로 … 지급액이 결정되며 … 그 이후에 도래하는 매년 진단 확정일이 계약일부터 2년 이상에 해당하더라도 … 지급액은 변경되지 않습니다」. A model that re-tests the 감액 at each instalment date overstates every claim arising in the first policy year, which is the single most easily mis-modelled rule in the product.

Below 만나이 65 the grade blend is read at 65, for the same reason the entry rates are: the whole sub-65 basis is the age-65 basis carried down on sub65_factor_at(), and mixing an age-65 incidence with the under-65 severity mix — which is genuinely more severe, 22.2% of certified lives at 1·2등급 against 16.8% at 65 — would be internally inconsistent [std]. It moves nothing: under-65 claims are of order 1e-5 of the anchor cell’s.

ann_count(t)

The expected number of 간병연금 instalments falling due in month t.

Instalments are monthly; the cohort certified in month s is paid in months s ... s + n_A - 1, of which the first annuity_guar_mths() are guaranteed against death and each later block of twelve is released only by the annual survival test on the anniversary of the 진단확정일:

ann_count(t) = sum over u = 0 ... n_A-1 of  n_C(t - u) x weight(u)
weight(u)    = 1                        for u < annuity_guar_mths()
             = S_C(t - u, t - u + 12 floor(u / 12))   otherwise

The first instalment falls in the month of certification, so the u = 0 term is pols_entry_care() itself. The cap and the maturity truncation bind jointly: nothing is paid on the maturity row t = proj_len() - 1 or after it, which understates the benefit for a late entrant whose ten years of annuity outrun his five years of remaining term.

ann_pay(t)

The 간병연금 outgo of month t, the instalment ledger valued at each cohort’s own amount.

The same sum as ann_count() with ann_amount_at(s) inside it, which is what carries the freeze: a cohort entering at 만나이 68 keeps its own grade-blended amount and its own 감액 decision for all ten years, whatever later cohorts are paid.

ann_tests(t)

The expected number of annual 간병연금 survival tests falling due in month t.

One per surviving claimant on each anniversary of the 진단확정일 after the first, up to the cap. It is an administrative event, not an actuarial abstraction — 「매년 진단 확정일에 피보험자의 주민등록등본을 제출하여야 합니다」 — which is why the claim-handling expense is charged on it rather than on every monthly instalment.

cum_prem_pp(t)

The cumulative office premium paid per policy by the start of month t.

P x min(t, n_P) — the premiums received at 0 ... t-1, that is, everything paid before month t’s own premium. It is the denominator of the published 환급률 progression, which is quoted at policy anniversaries where no further premium has yet fallen due.

The refund on a voided cover is not this quantity but cum_prem_pp(t + 1): the premium of month t is received at the start of the month and the void is recognised at the end of it, so a life voided in month t has paid t + 1 premiums and 「이미 납입한 보험료를 돌려드립니다」 returns all of them. See claims().

prem_accum_factor(t)

The accumulated value at month t of t monthly payments of 1 in advance, at 예정이율.

(1 + j) ((1 + j)^t - 1) / j with j = (1 + prem_int_rate)^(1/12) - 1. prem_int_rate is 2.0% 연단위 복리 and is the one place in krlib where the pricing interest rate is a retrieved figure rather than a [std] one: one carrier states it in terms in a 기초서류 extract — 「주계약 및 특약에 적용한 예정이율은 연단위 복리 2.0%입니다」 — where no Korean carrier publishes one for the cancer chassis. Three cautions travel with it: it is a 2023-vintage rate, it is a 우정사업본부 rate written outside 보험업법, and it prices a different benefit mix. It is preferred to an invented figure, and at 2.0% against the chassis’s [std] 2.50% it is the more conservative of the two on a benefit payable forty years out.

av_ratio_at(fraction)

The 계약자적립액 as a ratio to cumulative premiums, at fraction of the way from 납입완료 to maturity.

Linear between the four sourced anchors of av_table.csv: 0.974 at 납입완료, 1.088 one third of the way on, 1.010 two thirds, and 0.000 at maturity. They are the published 해약환급금 미지급형 환급률 progression — 48.7% / 54.4% / 50.5% / 0.0% at 20, 30, 40 and 50 years on the anchor cell’s own specification — doubled, because that form pays 50% of the notional 기본형 value once the premiums are paid and the 해약공제 has expired. Indexing on the fraction rather than on the policy year is the [std] step that lets one published progression serve every term and every paying period.

net_prem_ratio()

The net premium as a fraction of the office premium, derived not assumed.

av_ratio_at(0) x n_P / prem_accum_factor(n_P): the fraction that, accumulated at the 예정이율 over the premium-paying period, reproduces the sourced 계약자적립액 at 납입완료. On the anchor cell it is 0.7932, implying a 예정사업비 loading of 20.7% — which no Korean 산출방법서 or 사업방법서 was retrieved to check, but which is what the published 환급률 progression and the published 예정이율 imply between them. Deriving it rather than assuming it is what makes the surrender-value cliff reproduce the carrier’s own figures instead of merely resembling them.

av_pp(t)

AV(t): the 계약자적립액 per policy at the start of month t.

Two branches meeting at 납입완료. Up to it, the accumulation of the net premium at the 예정이율; after it, the sourced run-off carried on av_ratio_at. The quantity matters because 감독규정 제7-63조제1항제1호 makes it payable on death from a cause the contract does not cover [REG-R17], and on this product a third of the cohort dies that way before maturity — so death is a decrement with a large cash flow attached, which the Japanese counterpart does not have at all.

Two reconstruction assumptions are named rather than buried. The published progression is the 미지급형’s 환급률 against its own premiums, and the model reads the doubled figure as that contract’s own 계약자적립액; the 기본형 comparator is a product that cannot be bought — 「’기본형’은 … 가입이 불가능하며 … 해지율을 적용하지 않고 계산합니다」 — and its premium is higher, so the two accounts are not in fact the same quantity [std]. And the run-off between the sourced anchors is linear [std], where the real curve bends with the risk cost.

surr_chg_pp(t)

The 해약공제액 per policy in month t, on the 표준형 form only.

surr_chg_ratio x P running off straight-line over the 해약공제기간, which is the premium-paying period capped at seven years [REG-R19 제7-66조제1항]. The level is the supervisor’s own rule of thumb for a 보장성보험 — a 표준해약공제액 of 13 times the monthly premium — which is how 별표 14’s formula in 연납순보험료 and 보험가입금액 is carried in Korean practice [REG-R29] [REG-R20]. The rule of thumb is used here rather than the formula because 별표 15 제9호 computes the notional 보험가입금액 of a contract with no death benefit as a ratio of risk premiums that excludes 「치매 또는 일상생활장해 등 타인의 간병을 필요로 하는 상태」 — read literally, it excludes long-term-care risk premium from the very ratio that gives a care-only contract its 보험가입금액 [REG-R21].

cv_pp(t)

CV(t): the 해약환급금 per policy paid on a lapse in month t.

mijigeup

nil for the whole premium-paying period, then 50% of the 계약자적립액 — a cliff, not a curve, and the strongest of the four forms on the Korean shelf. The legal basis is 감독규정 제7-66조제4항, which lets a 순수보장성보험 priced on a 최적해지율 pay less than the ordinary 계약자적립액 − 해약공제액 floor [REG-R19].

half_during

50% during the paying period, restored to the full account after it.

pyojun

the statutory form, max(AV(t) - 해약공제액, 0) from year 1 [REG-R19 제7-66조제1항].

Reading 「50%」 without reading which side of 납입완료 it attaches to puts the cliff upside down, which is why the form is a model point field and not a switch on a ratio.

premiums(t)

Premium income at the start of month t: P x pols_act(t).

Not P x pols_if(t): the 납입면제 waives the premium of the main contract and of every attached rider from the award of 1·2등급, and waived premiums are treated as paid. Nil from prem_period_mths() on, the contract being 비갱신형 with a fixed 납입기간 — which is the opposite of the Korean medical market’s annual renewal, and is the right answer for a benefit whose claim arrives thirty years after issue.

claims(t, kind=None)

Benefit outgo at the end of month t, by kind.

"LUMP"

A_B r(t) n_C(t): the 장기요양진단급여금, 최초 1회한 on the first award at or above the threshold. Paying it extinguishes that benefit line and leaves the contract, the annuity and the dementia rider running.

"ANNUITY"

ann_pay() — a living benefit, paid monthly while the insured survives in the care state, guaranteed for the first twelve months and capped at 120.

"DEMENTIA"

the 치매진단급여금 rider, once per contract. Zero unless attached.

"DEATH"

AV(t) x pols_death_act(t): the 계약자적립액 on death from a cause the contract does not cover, which 감독규정 제7-63조제1항제1호 requires of every 제3보험 product [REG-R17]. Deaths in the care state pay nothing — 「지급사유가 발생한 후 사망한 경우에는 별도로 책임준비금을 지급하지 않습니다」 — so the split of pols_death() is a cash-flow distinction and not bookkeeping.

"LAPSE"

CV(t) x pols_lapse(t): the 해약환급금, identically zero throughout the paying period on the composite’s 미지급형 form.

"VOID"

cum_prem_pp(t + 1) x pols_void(t): the premiums returned where a certification inside the 보장개시일 window makes the cover 무효. The cover is void ab initio and 「이미 납입한 보험료를 돌려드립니다」, so every premium the life has paid comes back — and a life voided in month t has paid t + 1 of them, month t’s included, because premium falls at the start of the month and the void is recognised at the end of it. cum_prem_pp(t) would leave the insurer holding one month’s premium on a contract it has just declared never to have existed. Tiny, and published because it is a different mechanism from a refused claim.

"MATURITY"

identically zero, and published rather than dropped: 「이 상품은 순수보장성보험으로 보험계약 만기시 지급받는 금액(만기환급금)이 없습니다」.

With no kind, the total of all seven.

inflation_factor(t)

The expense inflation factor in month t: (1 + pi)^floor(t/12) [std].

2.0% a year flat, stepping at each 계약해당일 rather than monthly. No Korean expense inflation assumption was retrieved; the level is the Bank of Korea’s inflation target and is a standardization.

expenses(t)

Acquisition and maintenance expense in month t [std].

expense_acq_mths x P at t = 0, then expense_maint per policy per month inflating at inflation_factor(). Maintenance is charged on pols_if(), including lives on waiver: the policy is still administered when nobody is paying for it, and on this product that is a fortieth of the projection.

No 사업방법서 and no 산출방법서 was retrieved for any Korean long-term-care product, so every expense assumption here is [std] — but neither of the two levels is free. The acquisition expense and the initial commission together are 13 times the monthly premium, exactly the 표준해약공제액 of a 보장성보험 in the supervisor’s own rule of thumb [REG-R29], split 5.2 : 7.8 so that the commission sits at the 60% cap the same release sets, and bounded from above by 별표 14 [REG-R20]. expense_maint is then set so that the present value of the whole expense and commission basis at the 예정이율 lands on the 20.7% loading that net_prem_ratio() implies — the loading the published 환급률 progression and the published 예정이율 imply between them. It lands at 20.3% on the anchor cell. So the expense basis is calibrated to the same two sourced facts as the account, rather than picked.

The claim-handling expense is not here. It is claim_expenses(), deducted on its own line and published in its own column, which is what expenses means everywhere in this library.

claim_expenses(t)

The claim-handling expense of month t [std].

expense_claim per claim event: the first certification, each annual 간병연금 survival test, and a dementia diagnosis. Not per monthly instalment — the proof of life the 약관 requires is annual — and not inflated. The unit is higher than a cancer chassis’s would be because the evidence is a 장기요양인정서 produced by a public body the insurer neither funds nor influences, and because the refusal grounds are administrative: 「허위 또는 부당 판정사실이 확인되는 경우」 nothing is paid.

comm_init_pp()

c0: the initial commission per policy issued, comm_init_mths x P [std].

7.8 months’ premium, which is 60% of the 13-month 표준해약공제액 — the cap the 2019 사업비·모집수수료 reform sets on the annual commission of a 보장성보험, and the same 60% that now sits in 감독규정 제4-32조제8항 [REG-R29] [REG-R22]. Paid up front at t = 0; with the acquisition expense it is what produces the deep new-business strain of the worked example’s first row.

commissions(t)

Commission outgo in month t [std].

The initial commission at t = 0, then 3.0% of premium income from policy year 2. Renewal commission rides on premiums(), so it stops when the waiver stops the premium and when the 납입기간 ends.

net_cf(t)

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

Premiums less the lump sum, the annuity, the dementia rider, the 계약자적립액 on death, the 해약환급금 on lapse, the premiums returned on a voided cover, acquisition and maintenance expense, the claim-handling expense and commission. Income-positive is this library’s sign and the technical notes’ own, so there is no outgo-positive liability_cf companion to publish.

The shape to expect is a deep month-0 strain, thin positive margins for twenty years while the premium runs, and then a long negative tail: the certification rate at 만나이 40 is of the order of one in forty thousand a year and at 85 it is one in twenty, so this contract prefunds a cost that essentially does not arise until the block is old. That is what makes the lapse assumption and the interest rate, not the incidence basis, the dominant levers — and it is why 2024’s supervisory attention on 무·저해지 lapse was aimed at exactly this kind of product.

check_pols_roll_fwd_resid(t)

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

l(t) - l(t+1) - deaths - lapses - voids - maturities. Those four are the only ways a life leaves this contract. A lapse applied to the care population, a voided certification left inside the block, or a maturity not removed on the last row all show up here.

check_pols_roll_fwd()

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

No argument, one bool over all t, the library-wide shape of a check_* cells; check_pols_roll_fwd_resid() gives the signed residual of the month that failed.

check_nesting_resid(t)

The smallest slack in the compartment structure at month t; non-negative everywhere.

The three compartments must be non-negative and must add to pols_if(), and the dementia counter must stay inside the in-force block. A negative value here would mean progression had drained more lives out of the light compartment than were in it, or that the rider’s first-event ledger had outgrown the block it rides on.

check_nesting()

True when the compartments stay non-negative and add to the in-force count.

check_ann_ledger_resid(t)

The 간병연금 ledger residual in month t; zero everywhere.

ann_count() less an independent rebuild that scans every month s in the window t - n_A < s <= t and re-derives each cohort’s weight from its own age, rather than stepping back through the same loop. A ledger that paid the first instalment a year late, that ran past the cap, that lost the twelve-month guarantee or that used a ratio form of care_surv() would show up here.

check_ann_ledger()

True when the 간병연금 ledger closes in every projected month.

check_av_continuity_resid(t)

The residual between the two branches of av_pp() at 납입완료; zero, and zero elsewhere by construction.

The account is an accumulation of net premiums up to 납입완료 and a sourced run-off after it, and the two have to agree at the join. They do because net_prem_ratio() is derived from the run-off’s first anchor rather than assumed — which is the point: the check fails the moment someone replaces the derivation with a round number.

check_av_continuity()

True when the 계약자적립액’s two branches meet at 납입완료.

check_cv_form_resid(t)

The smallest slack in the surrender-value form at month t; non-negative everywhere.

The 해약환급금 is never negative and never exceeds the 계약자적립액, and on the 미지급형 form it is exactly nil for the whole premium-paying period. That last clause is the cliff, and it is asserted with its sign because reading 「50%」 without reading which side of 납입완료 it attaches to puts the cliff upside down — which is the easiest available mistake in this product, four forms being on the shelf under three nearly identical names.

check_cv_form()

True when the surrender-value form behaves in every projected month.

check_net_cf_resid(t)

The residual between net_cf() and the published components; zero everywhere.

net_cf less premiums - claims_lump - claims_annuity - claims_dementia - claims_death - claims_lapse - claims_void - claims_maturity - expenses - claim_expenses - commissions, which are exactly the columns of result_cf(). A cash flow that exists in net_cf but not in the statement, or the reverse, shows up here.

check_net_cf()

True when the cash flow statement re-adds to net_cf() in every month.

The house contract: no model’s headline number is reconciled only in prose.

result_cf()

Result table of cash flows, indexed by policy month t.

pols_if is the start-of-month count, which is the weight applied to every cash flow on the same row; pols_act is the part of it still paying premium, and pols_care the part on waiver and drawing the annuity. claims_annuity is a living benefit — the 간병연금 is paid while the insured survives in the certified state and death stops it — so the name describes the benefit’s form and not its contingency. claims_death is the 계약자적립액, not a death benefit: this contract has none, and what is paid is the account balance 감독규정 제7-63조제1항제1호 requires on a non-covered death. claims_maturity is a column of zeros by product design and is published rather than dropped, as is claims_lapse for every duration inside the premium-paying period. net_cf carries the library’s income-positive sign. The frame is range(proj_len()) and its last row, t = proj_len() - 1, is the 계약해당일 on which the contract matures: an in-force count and no cash flow.

result_pols()

Result table of policy counts, decrement rates and per-policy values, indexed by t.