Technical Notes#

Status: Draft, 2026-09-03 (all cited sources accessed 2026-09-03).

Scope note. These notes turn the standardized composite tax-qualified pension savings contract (yeongeum jeochuk boheom, 연금저축보험) of product-spec.md (same directory) into a reference liability cash-flow projection on paper, and then into Pension_KR_S beside it. They describe no single insurer’s contract. [S#] and [R#] tags resolve against sources.md, whose numbering is carried verbatim from _research/pension-savings.md and is frozen; [REG-R#] tags resolve against the cross-product reference library references/regulatory-and-actuarial-references.md, whose own R1–R62 numbering is separate and also frozen. std marks a standardization introduced for the reference implementation; unverified marks a claim that could not be confirmed against a retrieved document. Every parameter that appears in both documents carries the same value here as in product-spec.md, and every number in the worked example is read off the shipped model rather than recomputed by hand.

This document inherits the surrender-value machinery of the whole life chassis (종신보험) — the 해약공제액 and its 표준해약공제액 (pyojun haeyak gongjeaek) cap under 보험업감독규정 별표 14 REG-R20, and the 해약환급금 (haeyak hwangeupgeum, surrender value) floor of 제7-66조제1항제1호 REG-R19 — and does not restate that machinery. What it does not inherit is that chassis’s 계약자적립액 (gyeyakja jeongnibaek, policyholder account value) recursion. The two products share the name and not the recursion: the chassis’s account is the classical net-level one, dividing by (1 − q) and subtracting a death benefit, and this one is a plain balance with no mortality in it at all. That is the first thing to understand about this product and it is set out under The 계약자적립액 recursion below. What it adds, and specifies here for the first time in the library, is five things:

  1. crediting at the 공시이율 (gongsi iyul, the declared rate) over a stepped 최저보증이율 (choejeo bojeung iyul, guaranteed floor), rather than at a fixed 예정이율;

  2. the 세액공제 (seaek gongje, tax credit) and the 16.5% 기타소득세 (gita sodeukse) on a non-pension withdrawal as policyholder-behaviour drivers that are not insurer cash flows — the reason the lapse assumption on this product is not the savings lapse assumption;

  3. the statutory 연금수령 conditions — five years of contributions, drawing from 만 55세, and the 연금수령한도 — as constraints on the projection R6 제40조의2제3항·제4항;

  4. the annuitisation step: a 100.1%-of-premiums minimum fund, and conversion into a 종신연금형 (life annuity with a guarantee period) or a 확정기간연금형 (annuity-certain) at a factor struck on the 경험생명표 proxy; and

  5. the question of which vintage of that table the factor is struck on — at 가입 or at 연금개시 — which the retrieved contracts settle only by inference and on which two carriers could legitimately differ.

One correction was made in the course of writing these notes and it changed a number. An earlier draft of product-spec.md wrote the payout formula on an annual annuity-due, 연금연액 = 계약자적립액 ÷ ä_n(공시이율) × (1 − 0.005), and called the reconstruction of the published illustration exact. It is not: on an annual factor the reconstruction misses all eight published implied factors by about half a per cent, and it misses them in the wrong direction. The annuity is paid 매월 on every retrieved contract [S1] [S2] [S5] [S6] [S7], and the factor is therefore the annuity-due payable twelve times a year. With that one change the same formula and the same 0.5% charge recover all eight published figures on both interest bases. The evidence is in The annuitisation transition below; product-spec.md now states the monthly form and resolves the residual in its footnote 10, Pension_KR_S implements it, and the worked example is struck on it.

Parameters introduced here that the specification does not carry — each because the specification defers it, or because it is a modelling construct with no contractual counterpart — are flagged new here at the point of introduction. They are: the uniform-force monthly conversions q^m, w^m and j_c and the (f − 1)/(2f) annuity correction; the whole cash-expense and commission set; both lapse vectors; the best-estimate mortality factor and the Makeham construction that generates mort_table.csv; the terminal age ω = 120; the policy-loan rate, draw year and draw fraction; the payment-holiday start year; the definitive 표준해약공제액 computation and the reading of 별표 14 주3 it rests on; the treatment of the 연금수령한도 평가액; and the decision not to round the 연금연액.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows per policy — premiums, deferral-phase death benefits, surrender payments, annuity instalments, expenses and commission — for a single model point, undiscounted and gross of reinsurance. Korea runs three measurement bases over one such stream and all three are live: IFRS 17 (K-IFRS 제1117호, mandatory since 2023-01-01) REG-R60, K-ICS in the same quarter REG-R13, and the 해약환급금준비금 (surrender-value reserve) that has no counterpart anywhere else in this repository REG-R11. This model stops deliberately before all three. Discounting, the risk adjustment, the CSM, 요구자본 and every reserve are out of scope and are cited, not reproduced — see Valuation and reserve pointers.

  • Projection frequency. Monthly (Pension_KR_S), on policy months running 계약해당일 to 계약해당일. On an account product the change of grid is not cosmetic. 감독규정 제7-66조제1항제4호 provides that the 계약자적립액 accrues monthly before 납입완료 and daily afterwards REG-R19, and the contract credits interest 「납입일부터 일자계산을 하여」 from the date each of the twelve monthly instalments is received [S1] [S2]. An annual grid could carry this product only through 제7-65조제2항’s separate permission to compute the account on an annualised premium basis — 「계약자적립액은 … 연납보험료를 기준으로 하여 산출할 수 있다」 REG-R18 — a permission the model used to take, and whose sub-annual timing it had to carry in one explicit factor u(t). The monthly grid needs neither: the 기본보험료 is allocated as the ₩500,000 the contract actually collects, and u(t) is gone.

  • What the monthly grid moved, and what it did not. The assumptions are untouched and stay annual. The 공시이율 is declared as an annual rate REG-R18 REG-R24, the 연금사망률 is an annual table by age, and the lapse vectors are annual by 경과기간, so credit_rate(t), mort_rate(t) and lapse_rate(t) carry exactly the figures tabulated below and the monthly companions are their uniform-force conversions std: j(t) = (1 + i)^(1/12) − 1 on the credited rate and 1 − (1 − q)^(1/12) on each probability, level inside a policy year and stepping at each 계약해당일. Twelve of each compound back to the year’s figure exactly, so the fund and the in-force at every 계약해당일 reproduce the annual-step model this replaced — av_pp to a relative 1.4e-14 and pols_if identically — which is the arithmetic confirmation that u(t) was exactly equivalent to the monthly recursion that now replaces it. What does move is everything inside the policy year, which an annual grid could not state at all: the 환급률 after one month is 95.67% where the first anniversary’s is 96.61%, the sixty months between 납입완료 and 연금개시 each carry their own ₩3,350 charge instead of five annual ₩39,810.71 ones (₩201,000 against ₩199,053.55, the difference being the timing factor that is no longer needed), and a surrender is dated to the month. Two smaller consequences: the 14-day 납입최고(독촉)기간 still collapses, now into the month rather than the year REG-R25 제26조; and 미경과보험료, required on termination by 제7-66조제5항 REG-R19, is still nil on a grid with premiums in advance and surrenders at the month-end std — but the unearned amount a real contract would owe is now at most one month’s premium rather than one year’s.

  • What stays annual because the statute states it annually. 별표 14’s 표준해약공제액 is struck on the 연납순보험료 REG-R20, the 소득세법 세액공제 ceiling is ₩6,000,000 a year and the 연금수령한도 is a per-year figure REG-R41 REG-R42. Each is computed as the annual quantity the statute names and then apportioned across the twelve months of the year — the 세액공제 as ₩82,500 a month, the 한도 as a bound on the year’s twelve instalments — rather than restated on a monthly base the law does not use. The 연금연액 B is likewise the annual amount the contract quotes; what a row pays is B / 12.

  • Timing conventions std. Premium at the start of each policy month, in advance, on the in-force cohort; annuity instalments at the start of the month, in advance; acquisition cash expense at t = 0; maintenance cash expense and commission at the start of each month; any policy-loan advance at the start of the draw month; death benefits and surrender payments at the end of the policy month, deaths before lapses.

  • Time index. t counts completed policy months since issue, 0-based: the first projected policy month is t = 0, premiums fall at t = 0 … 12m − 1, the fund accumulates over t = 0 … n where n = 12(m + d), and the annuity is paid from t = n. The contractual policy year is the derived 1-based label policy_year(t) = t // 12 + 1, so 「제1보험년도」 is t = 0 … 11. pols_if(t) is the in-force count at the start of month t and is the weight on that same result_cf() row. proj_len() is the number of projected months — the exclusive end of the frame, not the last index: the frame is t = 0 … proj_len() − 1, and at the anchor cell proj_len() is 972 = 12 × 81 and result_cf() has 972 rows, t = 0 … 971.

  • Age basis: 보험나이. Every age in the contract is 보험나이 (boheom nai, insurance age): the exact age at the 계약일 with a remainder under six months discarded and six months or more rounded up, incrementing on each 계약해당일 [S6 제20조] REG-R25 제21조. Because 보험나이 increments on the 계약해당일, a monthly grid ages the policy correctly by construction too: the attained age in month t is x + t // 12 exactly, and no fractional-age interpolation is needed anywhere. Korea’s other convention, 만나이 (age last birthday), governs the two statutory tests this product depends on — one retrieved 약관 states the split in terms, 「이 약관에서의 피보험자의 나이는 보험나이를 기준으로 합니다. 다만, 연금개시나이가 만 55세 이상에 해당되는지 여부의 판단은 실제 만 나이를 적용합니다」 [S6 제20조] — and the model reads both the 만 55세 test and the withholding age bands off age(t), which is 보험나이. That is a std simplification worth at most one year. It cannot bite at the anchor cell, whose annuity date is 65 and clears 만 55세 by a decade; it is exactly the case a model point annuitising on 55 would test, and model point 6 is that case.

  • Currency. KRW throughout. Amounts are written ₩ with thousands separators and the Korean 만원 / 억원 convention is given alongside where a Korean reader would expect it: the anchor cell’s fund at annuitisation, ₩160,294,806, is about 1.6억원. run.py prints KRW and pure ASCII.

  • Model points. Single-policy model points projected on an expected (probability-weighted) basis: survivorship and persistency multiply per-policy amounts. point_id parameterizes Projection; point_id = 1 is the worked-example anchor cell. Nine points ship. No aggregation logic is specified here.

  • Termination and horizon. The contract does not mature: there is no 만기보험금 and no maturity date, because the deferral phase ends by conversion rather than by payment. On the 확정기간연금형 form the last projected month is n + 12k − 1, the last instalment, so proj_len = n + 12k and there are no tail states. On the 종신연금형 form there is no natural end, so the horizon is the terminal age of the annuitant table less the issue age — proj_len = 12(ω − x + 1) = 12 × (120 − 40 + 1) = 972 at the anchor cell, ω = 120 being a [std, new here] choice, since no Korean industry table publishes a terminal age any more than it publishes its rates REG-R33 REG-R34. At x + t // 12 = ω the table’s q is 1, and the monthly conversion spreads that certain death uniformly over the twelve months of the final policy year — 1 / (12 − t mod 12) — so the last of the in-force leaves in month 971 rather than all at once on the 계약해당일.

  • Contract boundary. The 기본보험료 is level and guaranteed for the whole 납입기간 with no review right [S1] [S2] [S4] [S6] [S7], so all m premiums sit inside any defensible boundary and the model projects them. The harder question on this chassis is the other one: the 공시이율 is reset monthly at the insurer’s discretion inside a regulated construction REG-R18 REG-R24, which is the classic fact pattern for a direct participating contract under IFRS 17. Whether a 금리연동형 연금저축보험 is measured under the variable fee approach is a live question for a Korean reporter and is unverified here: no retrieved document settles it, and this model computes no IFRS 17 measurement of any kind. What it owes the regime is only that the projection be re-runnable on a basis re-set at a stated valuation date.

  • Rounding. Intermediate values at full double precision. Displayed cash flows and per-policy amounts to two decimal places std, policy counts and rates to ten. That is the precision tests/test_pension_savings_kr.py asserts. The 연금연액 is not rounded [std, new here]: no retrieved document gives a contractual rounding step for it, unlike the ¥100 step Japanese specimens publish, so B is carried unrounded and a model that rounds it is making a contractual assertion the sources do not support.

  • Sign convention. net_cf is income-positive — premiums less benefits, expenses, claim expenses, commission and any loan advanced. That is both these notes’ orientation and the library-wide one, so there is no outgo-positive liability_cf companion: one stream, one sign, one name. A reader comparing the payout rows with an outgo-positive presentation must flip the sign; they are large negatives here.

Model point attributes#

Attribute

Type

Anchor cell (point_id = 1)

policy_id

str

KR-PEN-0001

sex

enum {M, F}

M

issue_age (x)

int, 보험나이, 0 – (Y − m)

40

premium_term_y (m)

int, years, ≥ 5

20

defer_gap_y (d)

int, years, 납입완료 → 연금개시

5

annuity_start_age (Y)

int, 보험나이, = x + m + d, 55–80

65

premium_pp (P)

KRW p.a., level 기본보험료

6,000,000

addl_prem_pp (P_a)

KRW p.a., 추가납입, ≤ 200% of P

0

payout_form

enum {life_guar, certain}

life_guar

payout_term_y (k)

int, 확정기간연금형 term

10 (unused on the life form)

guar_term_y (g)

int, 보증지급기간

10

mort_vintage

enum {issue, commencement, ratchet}

issue

min_fund_on

bool, the 100.1% floor applies

true

surr_chg_rate

float, first-year 해약공제액 ÷ P

0.0

holiday_years (h)

int, 납입유예 spells of one year

0

loan_on

bool, 보험계약대출 module

false

par

bool, 배당 (participating)

false

div_rate

float, declared 계약자배당 rate on the fund

0.000

lapse_basis

enum {pension, savings}

pension

rate_scenario

enum {base, floor, hybrid}

base

Six of these are switches on modules that are off in the base run, which is deliberate: the composite is one contract and the variations are model point values, not code branches. annuity_start_age is derived rather than free — the model rejects a model point where Y ≠ x + m + d, because two spellings of one date is how a projection silently annuitises in the wrong year — and n, prem_end_t and proj_len are derived from it. The contractual terms stay in years, which is how the contract states them; the model converts them to month counts where the grid needs one. At the anchor cell prem_end_t() = 240, annuitisation_t() = 300, annuity_age_eff() = 65 and proj_len() = 972, so the last projected month is t = 971.

The anchor premium is not a modelling invention. It is the annualisation of a published illustration at an identical model point — 남자 40세, 기본보험료 월 500,000원, 20년납, 60세 완납, 65세 개시 [S2] — and it is exactly the 세액공제 ceiling of ₩6,000,000 a year R1 제59조의3제1항 R8 R10, so the anchor saver sits on the corner of the tax schedule. Both facts are what make the calibration below checkable.

State variables#

Variable

Description

Updated

pols_if(t)

Contracts with an obligation open at the start of month t; pols_if(0) = 1

monthly recursion

lives_if(t)

Probability the annuitant is alive at the start of month t; lives_if(0) = 1

monthly recursion

av_pp(t)

계약자적립액 per policy at the start of month t, before that month’s premium

monthly recursion

cv_pp(t)

해약환급금 per policy at time t

derived from av_pp

db_pp(t)

Death benefit for a death in month t − 1, paid at t — the fund itself

derived from av_pp

cum_prem_pp(t)

Cumulative 기본보험료 and 추가납입보험료 paid to time t

monthly recursion

loan_pp(t)

보험계약대출 balance per policy; zero in the base run

monthly recursion

div_acc_pp(t)

Accumulated 계약자배당; zero in the base run

monthly recursion

annuity_pp(t)

Instalment payable at the start of month t; B / 12 from t = n

fixed at t = n

mort_rate(t)

Best-estimate annual mortality for the policy year containing t

assumption lookup

mort_rate_mth(t)

1 − (1 − q)^(1/12), the monthly decrement actually applied

derived

lapse_rate(t)

Best-estimate annual 해지 rate for the policy year containing t

assumption lookup

lapse_rate_mth(t)

1 − (1 − w)^(1/12), applied at the end of month t

derived

credit_rate(t)

Annual rate the fund is credited with in the policy year containing t

assumption lookup

credit_rate_mth(t)

(1 + i)^(1/12) − 1, the rate the recursion applies each month

derived

Two in-force measures are carried and they are not interchangeable. pols_if counts contracts with an obligation open; lives_if counts annuitants alive. In the deferral phase the two separate because a surrender removes a contract without removing a life: at the anchor cell’s annuitisation date pols_if(300) = 0.6096911403 against lives_if(300) = 0.9657433263. In the payout phase they separate for the opposite reason — inside the 보증지급기간 the instalments are unconditional, so pols_if is flat while lives_if runs down on the annuitant basis, and on a 확정기간연금형 that holds for the whole term. Collapsing the two is the most likely way to build this product wrongly, and it is the first pitfall below.

av_pp is an account, not a reserve, and the model is built that way. The 계약자적립액 is a contractual balance defined identically across the retrieved 2026-vintage documents — 「「계약자적립액」이란 순보험료(기본보험료에서 계약체결비용 및 계약관리비용을 뺀 금액)를 「공시이율」로 납입일부터 일자계산을 하여 적립한 금액」 [S1] [S2] — so charges come off the premium, the remainder is credited at the declared rate, and nothing else moves. There is no net-level-premium structure, no survivorship release and no mortality in the recursion at all. cv_pp, not av_pp, is the surrender quantity, as the library’s naming ruling requires; on this composite the two are the same number at every duration, because the adopted 해약공제액 schedule is zero everywhere, and that is a property of the adopted schedule rather than of the product.

Assumption inputs#

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

Input

Value

Basis

기본보험료 P

Level, guaranteed for the whole 납입기간; no review right

[S1] [S2] [S4] [S6] [S7]

Premium frequency

월납, 12 instalments a year, in advance — the grid’s own frequency

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

Interest accrual on premiums

「납입일부터 일자계산을 하여」 — by day from receipt

[S1] [S2]

계약체결비용 α

1.50% of the monthly 기본보험료, policy years 1–7

[S1]

계약관리비용 β

3.00% a month while premiums are due; 0.67% a month after 납입완료, taken from the 적립액

[S1] [S5]

추가납입 charge

2.00%, 계약관리비용 only — the additional premium bears no 계약체결비용

[S1] [S8]

연금수령기간 관리비용 θ

0.5% of the 연금연액

[S1] [S7]

모집수수료

0.00% in every year — a direct-channel product

[S1]

최저보증이율 ladder

1.25% to 5 years, 1.00% to 10 years, 0.50% after; compound annual

[S1] [S2] [S13]

Why a floor exists at all

감독규정 제7-60조제10호 requires every 금리연동형보험 to set one

REG-R16

What the floor guarantees

The credited rate, not the return: charges are still deducted beneath it

[S4] [S8]

해약환급금

계약자적립액 less 해약공제액, floored at zero

[S1] [S8] REG-R19 제7-66조제1항제1호

해약공제액

Zero at every duration on the adopted schedule

[S1]

해약공제기간 cap

The premium term or the acquisition-loading period, capped at 7 years

REG-R19 제7-66조제1항제2호

표준해약공제액

연납순보험료 × 3% (무배당) × min(m, 12), less 주6’s discounted loading

REG-R20 주2·주3·주5·주6 R14

Death benefit before annuitisation

The 계약자적립액 at the date of death; the contract then ends

[S1] [S2] [S4] [S6]

Death cover above the fund

None

[S1] [S2] [S4] [S6]

Why none is required

감독규정 제7-60조제9호 exempts a contract whose premium term ends at 80 or below

REG-R16 제9호

연금개시일

The 계약해당일 at which 보험나이 reaches the elected 연금개시나이

[S1] [S2] [S6]

Minimum fund at 연금개시

100.1% of premiums paid

[S2] [S4 별표1 주10] [S7]

Why 100.1%

감독규정 제7-60조제2호 requires a 저축성보험’s survival benefits to exceed premiums paid

REG-R16 제2호

Floor disapplication

Withdrawn, and the annuity date deferred, where a 납입유예 or a one-instalment reinstatement caused the shortfall

[S4] [S6] [S7]

종신연금형 basis

연금사망률 and 공시이율, per the 산출방법서

[S1] [S2] [S6]

확정기간연금형 basis

공시이율 alone — no mortality, and no survival condition

[S1] [S2] [S6]

Annuity frequency

매월 / 매3개월 / 매6개월, deferred instalments credited at the 공시이율

[S1] [S2] [S5] [S6] [S7]

Death inside the 보증지급기간

The unpaid guaranteed instalments are paid; commutable at the 공시이율

[S1] [S2] [S6]

Death after the 보증지급기간

Nothing further; the contract ends

[S1] [S2] [S6]

Death during a 확정기간 term

The remaining instalments are paid to the count

[S1] [S2] [S4] [S6]

Guaranteed total vs the fund

The guaranteed instalments may total less than the fund at annuitisation

[S1] [S2] [S6]

Surrender after the first instalment

Not available on a 종신연금형

[S2] [S4 제5조③] [S5] [S9]

Contributions after annuitisation

Barred

R6 제40조의2제2항제2호

연금사망률 ratchet

Where a revision increases the annuity, the 연금개시시점 table is substituted

[S1 주6] [S2] [S4 별표1 주11] [S6 별표1 주10] [S9 별표2 주9]

Annual contribution ceiling

₩18,000,000 across every 연금계좌 the saver holds

R6 제40조의2제2항제1호 R11 REG-R56 [S1]

계약이전 (transfer out)

Permitted; not a withdrawal and not taxed

[S1]

Premium waiver (납입면제)

None — 「보험료 납입면제 사유 : 없음」

[S1]

Underwriting

None; 전건 무진단

[S7]

Two absences in that table are product facts and not omissions. There is no death cover above the fund, so the insurer carries no deferral-phase mortality risk at all and the death payment equals the surrender payment at every duration; and there is no premium waiver, so there is no disabled-life state to project. Both are asserted by the sources rather than assumed, and both are testable properties of the model.

(b) Insurer-discretionary current elements#

Input

Base-run value

Basis

공시이율 i(t)

2.15% p.a., level at every duration

[S2]; adoption std (1)

— how it is set

공시기준이율 (외부지표금리 × α + 운용자산이익률 × (1−α), α ≤ 60%) ± 조정률, reset monthly, fixed for the calendar month

[S1] [S2] [S4 제6조①②] REG-R18 REG-R24

— the 조정률

Not modelled: a discretionary margin whose range lives in an unpublished 사업방법서

[S1] [S8]; scope std

Credited rate i_c(t)

max(i(t), i_min(t)) = 2.15% at every duration; the floor never binds in the base run

derived

예정이율 i'

2.50% 연복리 — prices the charge and benefit structure; not a guarantee, not a crediting rate, and it appears nowhere in the fund recursion

[S1] [S5] [S7]

평균공시이율

2.50% for 2026 — a supervisory average; enters only as a constraint, inside the 표준해약공제액

REG-R48 [S14] REG-R9 제1-2조제13호

Annuity-phase charge θ

0.5% of the 연금연액 — a carrier choice, not a market convention

[S1] [S7]; adoption std (2)

Annuitant table vintage

Held at the 가입시점 table; the ratchet is carried and is out of the money

reading [derived] [S1] [S2]; std (3)

해약공제액 schedule

The direct-channel product’s, zero at every duration

[S1]; adoption std (4)

— the alternative

The postal insurer’s front-end 해지공제액, ₩104,000 on a ₩1,200,000 annual premium (8.67%), running off to zero at year 5

[S7]; model point 8

계약자배당

Zero declared. Machinery retained: credit on the fund, accumulate at the 공시이율, apply at t = n as an 증액연금, never in cash

[S2] [S7]; zero base run std (5)

배당 accumulation rate

2.15%, the 공시이율, where a dividend is run

[S2]; std

보험계약대출이율

4.00% where the module is on

std unverified, new here (6)

납입유예

Module off; up to three spells of one year, charges still taken from the fund

[S5] [S7] [S8]; scope std

  1. The observed range of declared rates on 연금저축 books is roughly 2.1% to 3.0% — 3.01% and 2.82% at one carrier on 2026-09-01 [S12], 2.40% at 2026-01 [S1], 2.15% at 2025-12 [S2], 2.3% at 2024-10 [S11] — with a 19-basis-point spread between two vintages inside one carrier on one date [S12] [derived]. The composite takes 2.15%, the conservative arm and the rate on which the anchor illustration and every derived annuity factor are struck [S2]: it is the only choice under which the published fund, the published annuities and the reconstructed factors form one consistent set. The rate is modelled as a step function of policy year, level in the base scenario, because a Korean declared rate is majority-weighted to the insurer’s own realised investment return and moves slowly: one carrier’s published thirteen-month history falls 57 basis points in steps of two to seven, never once reversing [S5] [derived]. It is not a market rate and must not be modelled as one.

  2. Two carriers disclose 「연금수령기간 중의 관리비용: 연금연액의 0.5%」 [S1] [S7]; a third discloses none and its implied factors run about 0.6% the other way [S5] [derived]. The composite takes 0.5% because it is the value that makes the payout formula reconstruct eight published figures on two interest bases (below).

  3. Six independently retrieved contracts carry the same one-way ratchet clause and none states the base vintage, so the reading that the base factor is the 가입시점 one is [derived], corroborated by two carriers publishing the 연금사망률 in the 상품요약서 handed over at inception [S1] [S7] and by trade reporting that a 경험생명표 revision applies to new business only R18 R20. Because revisions have lightened mortality — the 제10회 raised 평균수명 by 2.8 years for men and cut the monthly annuity on a fixed fund by roughly 15% R18 R19 REG-R33 — a revision normally reduces the annuity and the ratchet does not bite. Model point 9 confirms it numerically: mort_table_name() returns annuitant_issue even though the vintage is ratchet.

  4. The composite pairs the only complete published expense schedule [S1] with the only complete published annuitisation illustration [S2], and those are two different products of the same carrier. The consequence is stated openly: the composite’s early-duration surrender values look like the direct-channel product’s (96.6% of premiums after one year) and not like the tied-channel product’s (82.7%), even though the anchor model point is taken from the latter [S1] [S2].

  5. No retrieved carrier publishes a dividend rate on a 연금저축보험, which is one of the two reasons the composite is 무배당; the other is that 별표 14 주5 gives a 무배당 연금저축보험 the tighter surrender-charge coefficient, 3% against 4% REG-R20 R14, so the composite states the tighter constraint.

  6. No retrieved document gives a numeric 보험계약대출이율 for a 연금저축보험 [S1] [S2] [S4] [S5]. The module is off in the base run for exactly that reason; switching it on switches on an invented rate, and the model says so on the cells.

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

Mortality — one table, on the annuitant basis, and it is a construction. 경험생명표 (gyeongheom saengmyeongpyo, the industry experience table), currently the 제10회 applied to new business from 2024-04, is produced by 보험개발원 under the statutory office of 보험요율산출기관, 보험업법 제176조, which carries no publication obligation REG-R4 R16. What is public is the summary — 평균수명 남 86.3세 / 여 90.7세 and 65세 기대여명 남 23.7년 / 여 27.1년 REG-R33 R18 — and not the rates; the KIDI press page is JavaScript-driven and the release could not be opened R17 REG-R34, and the big-data portal refused connections R24. mort_table.csv is therefore a std construction with a provenance column on every row, and it is never presented as the 경험생명표. It ships with its recipe, in three parts, all recorded in mort_anchor_table.csv and re-derived by check_mort_law():

mu(x) = A + B c**y,   q(x) = round(1 - exp(-mu(x)), 8),   y = max(0, x - setback)

Part

Value

Status

Makeham A

5.5583e−04

[std, new here]

Makeham B

2.3281e−06

[std, new here]

Makeham c

1.108956

[std, new here]

Female table

The male law set back 4 years

[std, new here]

annuitant_revised vintage

The issue vintage × 0.85

[std, new here]

Terminal age ω

120, both sexes and both vintages

[std, new here]

The law is fitted jointly to the six annuitant rates two carriers publish in their statutory product summaries — 「연금사망률」 at ages 50/60/70 [S1] and 「개인연금사망률」 at 40/60/80 [S7] — and to the annuity factors the one published annuitisation illustration implies at two interest bases [S2]. A three-parameter law cannot honour both, and the residuals are recorded rather than smoothed: on the male table the fitted rate runs about 1.30× the published rate at 70 and 0.72× at 80, and the female table, being a setback rather than an independent fit, runs 1.9 to 2.8 times the published female rates over ages 50 to 70. An independent female fit was tried and rejected, because it produces a female annuity larger than the male’s at the same age, which no carrier’s rate card does: the sourced sex differential beat the sourced female rates, and both facts sit on the file’s provenance rows.

What the construction says about longevity, stated rather than hidden. On the shipped male table the curtate expectation at 65 is 33.31 years, against the 제10회’s published 23.7 REG-R33; the female table gives 36.97, a gap of 3.66 years against a published 3.4. The first number is not a mistake and it is not this library’s invention: solving ä_n = 23.70 at 2.15% on the same monthly annuity-due this document adopts gives n ≈ 32.9, so the published life annuity is priced as if a 65-year-old male lived to about 98 [S2] [derived]. (On the annual annuity-due the same solve gives 32.5 and about 97; the monthly reading is the consistent one, for the reason set out under The annuitisation transition.) That is what a 연금사망률 loaded on the survival side looks like when it is read back off a published annuity, and the table reproduces the annuity because the annuity is what it was fitted to.

The best-estimate factor, and why its sign is the opposite of a death product. mort_rate(t) = min(1, 1.15 × table rate) [std, new here]. Greater than one, and the direction is the point: the published 연금사망률 is a pricing basis for a longevity liability, so it is loaded on the survival side and a best-estimate death decrement runs heavier than the table, not lighter. The size is a standardization — the only direct evidence of the margin is that the two carriers publishing annuitant rates differ by about 9% at age 60 (0.00150 against 0.00164 for men) [S1] [S7] [derived], and 1.15 sits a little above that. A model that carries a death product’s downward adjustment here has the sign wrong.

The table is used in exactly two places — the annuity factor at the 연금개시일, and the in-force decrement — and it must not be shared with WholeLife_KR_S: one table is loaded for survival and the other for death, and using either for both is wrong in a known direction.

Lapse — argued, not fitted [std, new here]. There is no public Korean lapse statistic for 연금저축보험 by policy year: one carrier’s regulatory disclosure carries a 경과기간별 중도해지율 column in which every row reads 「적용안함」 [S13], the supervisor’s comparison table could not be opened [S19], and the behavioural tables in the 2025 whitepaper sit in attachments that did not convert R13. The reference vector is therefore std and is argued from the contract:

Policy year index t (0-based)

0

1

2

3–4

5–9

10 – (m−1)

m … n−1

≥ n

lapse_rate(t), pension std

4.0%

3.5%

3.0%

2.5%

2.0%

1.5%

1.0%

0%

lapse_rate(t), savings std

8.0%

7.0%

6.0%

5.0%

4.0%

3.0%

2.0%

0%

Four features are load-bearing. The vector is materially flatter than a non-qualified savings contract’s, which the second row carries so the two can be run side by side: a surrender costs 16.5% 기타소득세 on essentially the whole payout once the contributions have been credited [S5] [S8] [derived], a frictional cost nothing else in this repository has. It steps down at 납입완료 rather than up, because the commonest lapse trigger — a premium falling due — is absent between 납입완료 and 연금개시. It is zero from t = n and not from t = n − 1: surrender is available right up to the day before the 연금개시일 [S2] [S4], so the decrement runs through year n − 1 and the contracts leaving there are paid CV(n), a real payment of the full fund. And it is net of 부활 by construction, because reinstatement is not implemented on an annual grid.

It is deliberately not the supervisory 무·저해지 lapse guidance — the log-linear decay to 0.1% at 납입완료 of the 제4차 보험개혁회의 REG-R27 — which is calibrated to 순수보장성 and 무해지 protection business. This product is neither: it has a full surrender value from the first month and no cliff at 납입완료. That guidance is the shape a Korean supervisor expects a lapse curve to have; it is not this product’s numbers.

Expenses and commission. The four cash-expense levels are [std, new here] and the structure is conventional; the two commission rows are not standardizations — they are the published 모집수수료율 of the source product, which is 0.00% in every year [S1].

Input

Value

Basis

Acquisition cash expense E0

₩200,000 per policy at t = 0

[std, new here]

Maintenance cash expense e(t)

₩30,000 p.a. in deferral, ₩20,000 p.a. in payment, taken as a twelfth each month

[std, new here]

Expense inflation π

2.0% p.a. flat, applied as (1 + π)^(t // 12) — a step on the 계약해당일

[std, new here]

Claim expense ec

₩30,000 per death claim, deferral only

[std, new here]

Initial commission c0

0.00% of premium

[S1]

Renewal commission c_r

0.00% of premium

[S1]

These are best-estimate cash expenses and are entirely separate from the 계약체결비용 and 계약관리비용 of class (a), which are contractual loadings living inside av_pp. Charging the loadings against the cash flow, or projecting these into the fund, double-counts expense in one direction and destroys the fund calibration in the other. The commission rows are zero because the source product’s published 모집수수료율 is 0.00% in every year [S1]; commissions() is retained and returns zero, because a zero states the fact where a missing column would hide it.

Option take-up. Annuitisation form: 100% 종신연금형 with a ten-year guarantee std, the composite’s base election; the 확정기간연금형 alternative runs at model points 4, 5 and 6. Commutation of unpaid guaranteed instalments: 0% std — the base run assumes continuation at 100%, under which death inside the guarantee leaves the instalment stream unchanged. 부활 / 간편부활, 계약이전, 의료비인출, the six 부득이한 사유 withdrawals and 배우자 승계 are out of scope [std scope]: each is a real contract term with a real tax effect, and no public frequency exists for any of them.

Cash flow components and recursions#

Notation#

Symbol

Meaning

t

policy month index, 0-based: t = 0, 1, …, proj_len − 1; the contractual policy year is t // 12 + 1; attained 보험나이 in month t is x + t // 12

x, m, d, h

가입나이; 납입기간 in years; the 납입완료 → 연금개시 gap in years; 납입유예 length in years

n

= 12(m + h + d), the policy month of the 연금개시일

k, g

확정기간연금형 term; 보증지급기간 of the 종신연금형

P, P_a

annual 기본보험료; annual 추가납입보험료. The instalments actually collected are P/12 and P_a/12 (prem_mth_pp, addl_prem_mth_pp)

α(t), β(t)

계약체결비용 rate; 계약관리비용 rate, both on the 기본보험료

θ

연금수령기간 관리비용, 0.5% of the 연금연액

i(t), i_min(t)

공시이율; 최저보증이율, both annual

i_c(t)

credited rate, max(i(t), i_min(t)), annual (credit_rate)

j_c(t)

(1 + i_c(t))^(1/12) − 1, the monthly rate the fund rolls on (credit_rate_mth)

f

instalment frequency, 12 for both premiums and annuity

NP(t)

premium credited to the fund (prem_to_av_pp)

C(t)

charge taken from the fund where no premium bears it (charge_from_av_pp)

AV(t)

계약자적립액 at the start of month t, before that month’s premium (av_pp)

SC(t), SC_max

해약공제액; the 표준해약공제액 cap (surr_chg_pp, surr_chg_cap_pp)

CV(t), DB(t)

해약환급금; death benefit (cv_pp, db_pp)

G

100.1% minimum fund at the 연금개시일 (min_fund_pp)

F

연금개시시점 계약자적립액 after the floor (annuity_fund_pp)

ä

the annuity-due factor the annuity is bought at (annuity_due_factor)

B

연금연액, struck once at t = n (annuity_amount_pp)

q(t), w(t)

best-estimate annual mortality; best-estimate annual 해지 rate

q^m(t), w^m(t)

1 − (1 − q)^(1/12), 1 − (1 − w)^(1/12) (mort_rate_mth, lapse_rate_mth)

l(t), L(t)

pols_if(t); lives_if(t)

D(t), W(t)

expected deaths; expected surrenders in month t

E0, e(t), ec

acquisition, maintenance and claim cash expense

CF(t)

net cash flow of month t, insurer perspective, income-positive (net_cf)

Dimensional check: q, w, q^m, w^m, α, β, θ, i, j_c are dimensionless; P, AV, CV, DB, F, B, E0, e, ec are KRW; ä is dimensionless (years of income per unit of annual income); l and L are probabilities. B = F(1 − θ)/ä is therefore KRW per year, and the grid pays B/12 a row — the twelve-per-year structure that the annual grid carried inside u(t) on the way in is now simply the grid itself, and on the way out it is still inside ä, which is and always was the monthly annuity-due.

Processing order#

Within policy month t, in this order. The order is part of the specification, not an implementation detail: three of the pitfalls below are order errors.

  1. Premium, (P/12 + P_a/12) · l(t) at the start of the month, if a premium is due — that is, t < 12(m + h) and not inside a payment holiday.

  2. Annuity instalment, (B/12) · l(t) at the start of the month, if t ≥ n. Paid in advance, so the row t = n carries an instalment.

  3. Acquisition cash expense at t = 0; maintenance cash expense at the start of every month, a twelfth of the annual figure, inflated by (1 + π)^(t // 12) and weighted by l(t); commission at the start (nil on this composite).

  4. Policy-loan advance at the start of the draw month (nil in the base run).

  5. Fund roll-forward, AV(t+1) = (AV(t) + NP(t) − C(t))(1 + j_c(t)). No mortality and no lapse appear here.

  6. Death, at the end of the month, from the whole opening in-force: D(t) = l(t) q^m(t), paid DB(t+1) = AV(t+1) net of any loan, plus a claim expense ec · D(t).

  7. Surrender, at the end of the month, from the survivors of mortality — death before lapse [std order]: W(t) = (l(t) − D(t)) w^m(t), paid CV(t+1) net of any loan.

  8. Roll to t + 1: l(t+1) = (l(t) − D(t))(1 − w^m(t)) in deferral, and the payout-phase rules below from t = n; L(t+1) = L(t)(1 − q^m(t)) throughout.

At t = n exactly, five things happen in one step and in this order: the fund is fixed and floored; B is struck, once, and never recomputed; lapse_rate and the deferral death decrement go to zero; the first instalment is paid at the start of that same month; and AV, CV and DB become zero from t = n + 1. So the row t = n carries an annuity payment and no premium, no death benefit and no surrender.

The premium and its two charges#

NP(t) = (P/12) (1 - alpha(t) - beta(t)) + (P_a/12) (1 - 0.02)      for t < 12(m + h)
      = 0                                                          otherwise

The monthly grid allocates the instalment the contract actually collects, so NP(t) is an exact won amount: ₩500,000 × (1 − 0.015 − 0.030) = ₩477,500 while the 계약체결비용 runs, and ₩500,000 × (1 − 0.030) = ₩485,000 after it stops. The annual grid this replaced had to value the twelve instalments as one start-of-year payment through a factor u(t) = (1/f) Σ (1 + i_c)^(−j/f) = 0.990316187680581 at 2.15% std; that factor is gone, because the timing it stood for is now the grid.

The charges come off the premium, not off the fund, while premiums are being paid — 「동 이율은 납입한 주계약(또는 적립) 보험료에서 계약체결·유지관리에 필요한 경비 … 를 차감한 금액에 대해서만 적용됩니다」 [S4] [S6] — and there is no risk premium to deduct, because there is no cover above the fund. That is what gives an insurance-wrapper pension its negative early-duration return, and the supervisor says so in terms R12.

The 계약체결비용 stops after seven years and the 계약관리비용 does not stop at 납입완료. Both are visible in the model as steps: NP(t) rises from ₩477,500 to ₩485,000 at t = 84 — the seventh 계약해당일, which on this grid is month 84 and not row 7 — and from t = 12(m + h) the fund starts paying

C(t) = (P/12) * beta_paid_up                                for 12(m+h) <= t < n
     = (P/12) * ( alpha(t) + beta_premium_paying )          during a 납입유예
     = 0                                                    for t >= n

which is ₩3,350 a month at the anchor cell, taken from the 적립액 with no premium arriving — 「보험료 납입 완료 후에는 월계약해당일에 계약관리비용 중 유지관련비용(납입후)을 적립액에서 차감합니다」 [S5] [S1], which is a monthly instruction the monthly grid now follows literally. Zero once the annuity is in payment, where the charge is the 0.5% already inside the annuity factor.

The 계약자적립액 recursion#

AV(0)   = 0
AV(t+1) = ( AV(t) + NP(t) - C(t) ) * ( 1 + j_c(t) )        for t = 0 .. n-1
AV(t)   = 0                                                for t > n

There is no mortality in this recursion and there is no lapse in it. The fund is a contractual balance and the death benefit is the balance, so there is no survivorship release to credit and no death strain to subtract; and the surrender release that a deduction would produce is nil, because the 해약공제액 is nil. That is the single largest structural difference from both of the neighbouring accumulation models, and the nearer of the two is the dangerous one. WholeLife_KR_S, the chassis whose surrender-value machinery this product does inherit, runs the classical net-level recursion V(d) = ((V(d−1) + P·1{d <= m})(1 + i) − q·SA)/(1 − q) under the same Korean name, 계약자적립액 — on that model’s anniversary index d (d = 0 at issue), not on this model’s period index t; the deferred annuity on the Japanese page divides by (1 − q') in the same way and pays a larger annuity out of the same premium as a result. A model that ports either shape here does not fail loudly: it silently overstates the 연금개시 fund. check_av_roll_fwd() asserts the recursion above over the whole deferral phase, and it is the check that would catch it.

The accumulation cross-check against the published illustration closes. Rolling that carrier’s published year-20 surrender value forward the five years of the gap at the stated rate gives ₩140,811,363 × 1.0215⁵ = ₩156,613,630 against a published ₩156,420,000, and ₩120,595,257 × 1.005⁵ = ₩123,640,438 against a published ₩123,460,000 [S2] [derived] — residuals of ₩193,630 and ₩180,438, both of the order of five years of the post-payment maintenance charge (0.67% × ₩500,000 × 60 = ₩201,000) [S1]. The two residuals are close but they are not equal: an earlier draft rounded each roll-forward to the nearest ₩100,000 and reported a common ₩180,000, which is a rounding artefact and not a finding. The accumulation is a plain roll-forward at the declared rate net of a small level charge, with no other moving parts.

Deferral-phase benefit amounts, and the statutory cap#

SC(t) = min( surr_chg_rate * P * max(0, (5 - max(t/12, 1)) / 4), SC_max )
CV(t) = max( 0, AV(t) - SC(t) )        for t <= n,   and 0 for t > n
DB(t) = AV(t)                          for t <= n,   and 0 for t > n

CV’s zero floor is 감독규정 제7-66조제1항제1호 written out — 「계약자적립액에서 해약공제액을 공제한 금액이 음(陰)의 값인 경우에는 이를 영(零)으로 처리한다」 REG-R19 — and check_cv_floor() asserts it at every duration. On the composite SC(t) = 0 at every duration [S1], so CV(t) = AV(t) = DB(t): death and surrender pay the same amount and the two decrements differ in their rate, not in their payment. The run-off shape in SC is the postal insurer’s, first-year amount falling linearly to zero at the fifth policy year [S7], and it is exercised at model point 8. The schedule is published by policy year, so the monthly grid interpolates it linearly in the elapsed duration t/12: the quoted figures are reproduced exactly at each 계약해당일 and the first policy year carries the year-1 amount throughout, since the run-off starts from the first anniversary and not from issue std.

The 표준해약공제액, computed in full. 별표 14 gives 표준해약공제액 = 연납순보험료의 5% × 해약공제계수 + 보장성보험의 보험가입금액의 10/1000, with 주2 setting the 저축성보험 coefficient at the premium term capped at 12, 주3 defining the 연납순보험료 as the annual premium less the average loading spread evenly over the payment term capped at 10 years, 주5 replacing the 5% with 4% for a 연금저축보험 and 3% if 무배당, and 주6 subtracting the acquisition amount loaded into the premium discounted at the 평균공시이율 REG-R20 R14. 주4’s 6% concession for a whole-of-life survival annuity is expressly denied to this product, and the second term of the formula is nil because the contract has no 보장성 element. At the anchor cell:

Step

Arithmetic

Value

Whole-term loading

₩6,000,000 × (0.045 × 7 + 0.030 × 13)

₩4,230,000.00

Levelled over min(m, 10)

₩4,230,000 ÷ 10

₩423,000.00

연납순보험료 (주3)

₩6,000,000 − ₩423,000

₩5,577,000.00

Gross cap (주2, 주5)

3% × ₩5,577,000 × min(20, 12)

₩2,007,720.00

주6 deduction

Σ ₩90,000 ÷ 1.025^s, s = 0…6

₩585,731.28

표준해약공제액

₩1,421,988.72

That is about 2.8 months of 기본보험료, the order of magnitude 금융위원회 stated for a 저축성보험 in 2019 REG-R29. The composite uses none of it, and that is the finding: the regulator has singled this product out for the tightest coefficient in the schedule and the reference implementation sits far inside it. The composite’s whole acquisition cost — 1.50% × ₩500,000 × 84 = ₩630,000 — is itself inside the cap, which is a coherent explanation of why the source product’s published 해약공제 table is all zeros [S1]. check_surr_chg_cap() asserts SC(t) ≤ SC_max at every duration on every model point, including the one carrying a real front-end charge.

The reading of 주3 is a choice and it moves the number [std, new here]. The computation above levels the whole term’s loading over the ten-year cap. The alternative reading — levelling only the first ten years’ loading, ₩2,430,000 ÷ 10 = ₩243,000 — gives a 연납순보험료 of ₩5,757,000, a gross cap of ₩2,072,520 and a 표준해약공제액 of ₩1,486,788.72 [derived], 4.56% higher. Neither is excluded by the text. The model implements the first and the alternative is a one-line change; the cap binds nothing on this composite either way, which is why the choice can be stated openly rather than defended.

The annuitisation transition#

At t = n the fund is fixed, floored, and divided by a factor.

The floor. G = 1.001 × cum_prem_pp(n), and F = max(AV(n), G). Three retrieved contracts guarantee it in those words — 「연금개시시의 계약자적립액은 이미 납입한 보험료의 100.1%를 최저보증 합니다」 [S4 별표1 주10] [S2] [S7] — and two more write the functionally identical 「이미 납입한 보험료 + 1,000원」 [S5] [S6]. The base is premiums paid, basic and additional together, because 「이미 납입한 보험료」 is the whole contribution. It is a survival guarantee: a death claim in deferral is not floored at premiums paid on this composite, so the floor is payable only to a policy that reaches the 연금개시일 in force. It is not decorative — on the published guaranteed-rate illustration the fund reaches only 100.5% of premiums at the end of the twenty-year payment term [S2], so on a persistently low-rate path the floor is close to binding, and it is the only element of this contract that behaves like an option rather than an account. Model point 6 is the demonstration: on the guaranteed-rate scenario its fund at annuitisation is exactly ₩30,030,000 = 100.1% × ₩30,000,000, the floor and not the interest. check_min_fund() asserts F ≥ G where the guarantee applies, and model point 9 carries it withdrawn, which is what the contracts do where a 납입유예 or a one-instalment reinstatement caused the shortfall — they defer the annuity date instead [S4] [S6] [S7].

The factor, and the correction to product-spec.md. The 확정기간연금형 is priced on the declared rate alone — 「연금개시시점의 계약자적립액을 기준으로 공시이율을 적용하여 … 계약자가 선택한 확정된 연금지급기간 동안 나누어 계산」 [S1] [S2] [S6] — and the 종신연금형 on the 연금사망률 and the declared rate [S1] [S2] [S6]. Both instalments are paid 매월. The factors are therefore annuities-due payable f = 12 times a year:

adue_certain = ( 1 - v**k ) / d_f,       d_f = f * ( 1 - (1 - d)**(1/f) ),  d = i/(1+i)

adue_life    = sum_{j>=0} max( 1{j < g}, jp_(x+n) ) * v**j  -  (f - 1) / (2 f)

B            = ( F_net / adue ) * ( 1 - theta )

with v = 1/(1 + i_c(n)), survival on the annuitant table at 100% — a pricing basis, not the best-estimate one — and (f − 1)/(2f) = 11/24 the standard correction for instalments payable f times a year [std, new here]. F_net is F plus any accumulated 계약자배당 and less any outstanding 보험계약대출; both are zero in the base run. The factor is unchanged by the move to a monthly grid, because it was always the monthly annuity-due: what changes is that the projection now pays B/12 a row, where the annual grid put the whole of B on one row a year and carried the frequency only inside ä.

Written on an annual annuity-due this is wrong, and an earlier draft of product-spec.md wrote it that way. The evidence is that the monthly form recovers every published figure and the annual form recovers none of them. [S2] publishes one fund at annuitisation and five annuities on each of two interest bases; dividing the fund by the annuity gives the implied factor the carrier actually used:

Form

Published implied factor

Model

Interest basis

확정기간 10년

9.06

9.061

공시이율 2.15%

확정기간 15년

12.92

12.918

2.15%

확정기간 20년

16.39

16.386

2.15%

확정기간 10년

9.81

9.806

최저보증 0.50%

확정기간 15년

14.53

14.528

0.50%

확정기간 20년

19.13

19.134

0.50%

종신 10년보증, 남 65

23.70

23.700

2.15%

종신 10년보증, 남 65

31.18

31.180

0.50%

[derived] from [S2]. Eight published figures, one formula, two interest bases and both annuity forms. On the annual reading the certain factors come out at 9.104 / 12.978 / 16.464 at 2.15% — larger than the published ones, so the reconstructed annuity is smaller than the published annuity, and the 0.5% charge makes the gap worse rather than better. An earlier draft recorded the discrepancy as a uniform 0.995 at 2.15% and an unexplained 1.003 at the guaranteed rate “running the other way”, and told the reader not to over-read it. There is nothing to over-read: it is the monthly instalment, and once the factor is ä^(12) the 0.5% charge falls out with the same sign on both bases.

Which vintage the life factor is struck on is mort_vintage. issue is the composite and the base run; commencement strikes it on the 연금개시시점 table; ratchet implements the contractual clause itself, evaluating both and taking whichever gives the larger annuity — the smaller factor — because the clause bites only where a revision increases the annuity. Since revisions have lightened mortality, the ratchet is out of the money and returns the issue vintage; model points 7 and 9 exercise the other two readings.

The payout forms#

종신연금형 with a 보증지급기간 (base form). Instalments are unconditional for g years — 12g months — and life-contingent after:

pols_if(t) = pols_if(n) * max( 1{t - n < 12g}, L(t) / L(n) )      for t > n

on the best-estimate annuitant basis, so pols_if is flat through the guarantee and then runs off. Death inside the guarantee pays the unpaid guaranteed instalments and the base run assumes continuation at 100% std, under which the stream is unchanged; the contract warns that the guaranteed total may come to less than the fund at annuitisation [S1] [S2] [S6], and at the anchor cell it does — 10B = ₩67,633,745.89 against F = ₩160,294,805.59, 42.19% of it. pols_death(t) in this phase is the run-off of pols_if itself and carries no cash flow: db_pp(t) = 0 for t > n, so claims_death is zero from t = n even where pols_death is not.

확정기간연금형 (module). 12k unconditional instalments at t = n … n + 12k − 1, so

pols_if(t) = pols_if(n)   for n <= t < n + 12k,   and 0 at t = n + 12k

while lives_if continues to run down. pols_death(t) = 0 throughout the term and pols_maturity(t) = pols_if(t) in the final month — the count reaching the scheduled end of the contract, which the in-force roll-forward needs because those survivors neither die nor surrender. There is no claims(t, "MATURITY") on this product, and the absence is a product fact: 연금저축보험 has no maturity benefit and no maturity date.

check_annuity_total() asserts that the guaranteed instalments are level and total gB on the life form and kB on the certain form. A model that had decremented the guaranteed period by mortality, or recomputed B after the 연금개시일, fails it.

In-force recursion#

l(0)   = 1
l(t+1) = ( l(t) - D(t) ) * ( 1 - w_m(t) )       for t < n   (death before lapse)
L(t+1) = L(t) * ( 1 - q_m(t) )                            throughout
D(t)   = l(t) * q_m(t)        for t < n;  the payout-phase rules above from t = n
W(t)   = ( l(t) - D(t) ) * w_m(t)  for t < n;  0 from t = n

q_m(t) = 1 - ( 1 - q(t) )**(1/12)            and 1/(12 - t mod 12) where q(t) = 1
w_m(t) = 1 - ( 1 - w(t) )**(1/12)

Twelve monthly exits compound to the year’s annual rate exactly, so the in-force at every 계약해당일 is the annual-step model’s to the last printed digit.

Net cash flow#

CF(t) = (P + P_a)/12 * l(t) * 1{premium due}                  (premiums)
      - B/12 * l(t) * 1{t >= n, in payment}                   (claims_annuity)
      - DB(t+1) * D(t)                                        (claims_death)
      - CV(t+1) * W(t)                                        (claims_lapse)
      - ec * D(t)                                             (claim_expenses)
      - ( E0 * 1{t = 0} + e(t)/12 * (1+pi)**(t//12) ) * l(t)  (expenses)
      - commission                                            (commissions, nil)
      - loan advance                                          (policy_loans, nil)

Nothing in the tax layer appears here. The 세액공제 is a payment from the state to the saver and the 기타소득세 is a withholding from the saver’s proceeds; neither passes through the insurer’s account, and folding either in would misstate the liability in a way no reconciliation would catch. The tax quantities are published in result_tax() instead, and check_net_cf() — which asserts that the published columns sum to net_cf in every month — is the check that would fail if anyone added one.

The nine checks, each taking no argument and returning a bool, with the signed per-t residual at <name>_resid(t):

Cells

Identity asserted

check_pols_roll_fwd

l(t) − l(t+1) − D(t) − W(t) − pols_maturity(t) = 0

check_av_roll_fwd

(AV(t) + NP(t) − C(t))(1 + j_c(t)) − AV(t+1) = 0 over the deferral

check_cv_floor

CV(t) = max(0, AV(t) − SC(t)) — 감독규정 제7-66조제1항제1호 REG-R19

check_surr_chg_cap

SC(t) ≤ 표준해약공제액 — 별표 14 REG-R20

check_min_fund

F ≥ 100.1% × premiums paid where the guarantee applies

check_annuity_total

the guaranteed instalments are level and total gB (or kB)

check_annuity_limit

no 연금수령연차’s twelve instalments exceed the 연금수령한도 R6 제40조의2제4항

check_mort_law

every shipped rate equals the stated std Makeham construction

check_net_cf

the published columns sum to net_cf in every month

All nine are True on all nine model points.

Optional modules (all off in the base run)#

Module

Switch

What it changes

Model point

확정기간연금형

payout_form

Factor on the declared rate alone; pols_if flat for 12k months; pols_maturity in the last month

4, 5, 6

Mortality vintage

mort_vintage

The table the life factor is struck on; ratchet takes the larger annuity

7, 9

100.1% floor withdrawn

min_fund_on

G = 0; the annuity date is deferred instead

9

연금저축추가납입특약

addl_prem_pp

An additional premium bearing 계약관리비용 only, inside the 200% and ₩18,000,000 caps

8

Front-end 해지공제액

surr_chg_rate

CV separates from AV for five years

8

납입유예

holiday_years

Premiums suspended, charges still taken from the fund, n deferred by 12h months

9

보험계약대출

loan_on

Half the 해약환급금 drawn on the fifteenth 계약해당일 at a std 4.00%, deducted from DB, CV and F

9

계약자배당

par, div_rate

Credits and accumulates a dividend, applied at t = n as an 증액연금; moves 별표 14’s coefficient from 3% to 4%

9

Policyholder behavior modeling#

All dynamic constructions are std reference forms; calibration evidence is cited where any exists, and on this product very little does.

  • Base lapse std. The pension vector of class (c). On the anchor cell its count-weighted mean over the deferral phase — Σ l(t) w(t) ÷ Σ l(t), t = 0 … n − 1, on the annual rates the vector is stated in — is 1.9196%, and the same curve weighted by av_pp instead averages 1.4183% [derived] from the model, because lapse is front-loaded and the fund is back-loaded. On the monthly conversions actually applied the same two weightings are 0.16165% and 0.11903% a month. The weightings are not interchangeable and any future calibration must say which it means. Over the whole deferral phase 39.03% of policies leave before annuitisation, of which 36.53 points are surrenders and 2.50 points are deaths.

  • The tax layer is the behavioural model on this product, and it has a sign change. The saver took a credit of 16.5% of contributions on the way in and pays 16.5% of the surrender value on the way out, so the net tax cost of surrendering is 16.5% × (해약환급금 − cumulative contributions) [derived]: negative while the 환급률 is under 100% and positive after. On the model’s own numbers that is −₩33,575.23 at the first 계약해당일 and +₩42,223.98 at the fifth — the tax turns against the surrendering saver at almost exactly the duration at which the expense loading stops hurting. The two frictions do not overlap; they hand off. That is the whole argument for a lapse vector that is flat rather than steeply front-loaded, and it is why the savings vector is carried as a comparison rather than as this product’s basis.

  • The 16.5% charge falls on essentially the whole payout, not on the gain. The base is the credited money and its return [S8], but on a contract whose contributions were all inside the ₩6,000,000 credit cap that is nearly everything: one carrier’s surrender illustration carries a 세후지급 예상액 column uniformly 83.5% of the surrender value at every duration and on both interest bases [S5] [derived]. surr_tax_pp(t) publishes the charge at every duration and is not deducted from claims: the insurer pays the whole surrender value and the withholding is taken from the policyholder’s proceeds.

  • Part of what looks like lapse is 계좌이체, and it is not a withdrawal. A transfer to another 연금저축 or to an IRP attracts no income tax [S1], and the market moved that way hard in 2025 — 연금저축펀드 reserves +50.7% against 연금저축보험 −1.2% R13 R22. An insurer’s termination count and the wrapper’s persistency are different numbers, and this model’s lapse_rate is the former.

  • 부활 and the 납입최고 state, and why neither survives the monthly grid either std. A premium unpaid at t terminates the contract at t: the 14-day 납입최고(독촉) state REG-R25 제26조 collapses into the month rather than the year, but it still collapses, and there is no reinstatement re-entry — so lapse_rate here is a net-of-부활 rate by construction and a user substituting a gross experience rate will over-decrement. The reinstatement interest ceiling of 평균공시이율 + 1% REG-R25 제27조 is therefore not modelled either.

  • 납입유예 is the contractual alternative to lapsing, and it is not lapse. Up to three spells of one year, charges still deducted from the fund, both the premium dates and the annuity date deferred by h, and the holiday ending prematurely if the fund cannot bear the deduction [S5] [S7] [S8]. Where it runs it also withdraws the 100.1% guarantee [S4] [S6] [S7]. Model point 9 carries a two-year holiday with the guarantee withdrawn: n moves from 300 to 324 and the annuity date with it.

  • No dynamic lapse function is implemented std, and the reason is structural. On a fixed-rate savings contract the economic lapse driver is a new-business rate rising above the rate at issue. Here the declared rate resets monthly on the in-force contract [S1] [S2] [S4 제6조①], so an in-force policy does not go stale in that way and the competitive pressure expresses itself as 계좌이체 to a better-yielding wrapper rather than as surrender. A production model that wants a rate-driven decrement on this product should build it on the spread between this carrier’s declared rate and the market’s, and should route it to transfer rather than to surrender.

  • Annuitisation-election take-up std. 100% 종신연금형 with a ten-year guarantee in the base run. The tax code prices the choice and prices it in a dated, quantified way: a 종신계약 draws a flat 3.3% withholding at every age from 55, against 5.5% until 70 on a fixed-term annuity, the 종신 rate having fallen from 4% to 3% for pensions received on or after 2026-01-01 R5 R9 R21. That is a standing 2.2-percentage-point advantage where before 2026 it was 1.1 points. Against it runs the raw arithmetic of the two factors — at the anchor cell the life form pays 38.23% of what a ten-year certain annuity pays out of the same fund — so the election is a real decision and not a coin flip. One caution travels with the 3.3%: 종신계약 is defined by 소득세법 시행령 제187조의2, whose operative text could not be retrieved R7, so whether a guarantee period of any length is compatible with the status is unverified. If a ten-year guarantee disqualified the contract the anchor cell’s withholding would be 5.5% until 70.

  • The 연금수령한도 constrains the election where it binds, and the contract enforces it. Every retrieved contract makes the default election the tax-recognised maximum — 「연금액은 관련 세법에서 정한 바에 따라 연금소득으로 인정받을 수 있는 범위 이내로 합니다」 [S3 제21조②] [S5] [S7] [S8]. At the anchor cell it does not bind at all: the 연금수령연차 reaches 11 by the 연금개시일 and the formula is disapplied R6 제40조의2제4항. At model point 6, annuitising at 55, the counter is 1 and the limit in the first payment year is 12% of the 평가액 — ₩3,603,600 against an instalment of ₩2,143,455, so the fifteen-year term chosen there clears it. A payout term shorter than about ten years on a contract annuitised as early as it can be would not.

  • Commutation, 자유설계연금형 (fund split), 계약이전, 의료비인출 and the six 부득이한 사유 withdrawals are out of scope [std scope]. Each is a real contract term; none has a public frequency; and each is recorded in product-spec.md rather than guessed at here.

Worked example#

Anchor cell (point_id = 1). Male, 보험나이 40 at issue; level 기본보험료 of ₩500,000 a month (₩6,000,000 a year) payable at t = 0 … 239 (m = 20 years, ₩120,000,000 cumulative); 추가납입 nil; gap d = 5; 연금개시일 at t = n = 300, 보험나이 65; 종신연금형 with a ten-year 보증지급기간; 무배당; annuitant vintage issue; the 100.1% floor on; 해약공제액 nil; 납입유예, 보험계약대출, the dividend and the 추가납입특약 all off; pension lapse basis, base rate scenario.

Assumption values it uses, all listed above, and all of them annual: i(t) = 2.15% at every t and i_min(t) = 1.25 / 1.00 / 0.50% by duration, so i_c(t) = 2.15% at every t and the floor never binds — the monthly rate the fund actually rolls on is j_c(t) = 1.0215^(1/12) − 1 = 0.001774250062; α(t) = 1.50% for t = 0…83 and nil after; β(t) = 3.00% for t < 240 and 0.67% after; θ = 0.5%; f = 12 both ways; E0 = ₩200,000, e(t) = ₩30,000 a year in deferral and ₩20,000 a year in payment (a twelfth a month), π = 2% stepping on each 계약해당일, ec = ₩30,000, commission nil; mortality 1.15 × the std annuitant_issue male table, q(0) = 0.0008065180 (table 0.00070132) and q(300) = 0.0028596130 (table 0.00248662), with monthly companions q^m(0) = 0.0000672347 and q^m(300) = 0.0002386140; lapse 4.0 / 3.5 / 3.0 / 2.5 / 2.0 / 1.5 / 1.0 / 0% by policy year, w^m(0) = 0.0033960532; ω = 120, so proj_len() = 972 = 12 × 81 and result_cf() has 972 rows, t = 0 … 971.

The premium net of both charges is therefore an exact won amount each month,

t = 0..83    NP(t) = 500,000 * (1 - 0.015 - 0.030) = 477,500
t = 84..239  NP(t) = 500,000 * (1 - 0.030)         = 485,000
t >= 240     NP(t) = 0

and the charge taken from the fund after 납입완료 is

t = 240..299  C(t) = 500,000 * 0.0067 = 3,350

Annuitisation quantities, at full precision, all read off the model:

Quantity

Cells

Value

Cumulative premiums to t = n

cum_prem_pp(300)

120,000,000.0000000000

계약자적립액 at 납입완료

av_pp(240)

144,311,957.5668497980

계약자적립액 at 연금개시

av_pp(300)

160,294,805.5909678042

100.1% minimum fund

min_fund_pp()

120,119,999.9999999851 (not binding)

Fund after the floor

annuity_fund_pp()

160,294,805.5909678042

Fund converted

annuity_fund_net_pp()

160,294,805.5909678042

Annuity-due factor, monthly

annuity_due_factor()

23.58191601796395

연금연액

annuity_amount_pp()

6,763,374.5893046195

— the instalment actually paid

annuity_pp(300)

563,614.5491087183

Implied factor F ÷ B

[derived]

23.7004181085 (published 23.70)

Guaranteed total 10B

[derived]

67,633,745.8930462003 (42.19% of F)

Ten-year certain factor on the same fund

annuity_due_certain_factor()

9.01595104056377

표준해약공제액

surr_chg_cap_pp()

1,421,988.7174578153

해약공제액

surr_chg_pp(t)

0.0000000000 at every t

세액공제 a month

tax_credit_pp(0)

82,500.0000000000 — ₩990,000 a year (not an insurer cash flow)

기타소득세 on a surrender at d = 120

surr_tax_pp(120)

10,590,733.1198662706 (not a cash flow)

연금소득세 rate in payment

pension_tax_rate(300)

0.0330000000 (종신계약)

연금수령연차 at t = n

annuity_year_no(300)

11 — the 연금수령한도 is disapplied

Every one of those quantities is the annual-step model’s, to the precision a float64 carries: av_pp(240) and av_pp(300) agree with the annual grid’s av_pp(20) and av_pp(25) to a relative 1.4e-14 and the factor, the annuity and the cap are unchanged. That is the arithmetic confirmation that the annual grid’s u(t) was exactly equivalent to the monthly recursion that has replaced it — and the reason the move to a monthly grid changes the shape of the projection everywhere while leaving the calibration alone.

The implied factor of 23.7004181085 against the published 23.70 is the calibration that matters, and it is the only one the public record offers [S2]. The fund itself is ₩160,294,805.59 against the published ₩156,420,000, and that difference is intended and stated in product-spec.md: the composite takes the tied-channel product’s model point [S2] and the direct-channel product’s lighter expense schedule [S1]. The ratio B ÷ F = 0.0421933485 reproduces the published one (0.042194) to five decimal places.

Policy year 1, month by month, and the turn into year 2. expenses is acquisition plus maintenance; claim_expenses is its own column, as in result_cf(); commissions and policy_loans are zero in every row and are omitted from the printed tables.

t

pols_if(t)

premiums

claims_death

claims_lapse

expenses

claim_expenses

net_cf(t)

0

1.0000000000

500,000.00

32.16

1,624.38

202,500.00

2.02

295,841.44

1

0.9965369404

498,268.47

64.16

3,240.39

2,491.34

2.01

492,470.57

2

0.9930858737

496,542.94

95.99

4,848.05

2,482.71

2.00

489,114.18

3

0.9896467581

494,823.38

127.65

6,447.40

2,474.12

2.00

485,772.22

4

0.9862195525

493,109.78

159.16

8,038.47

2,465.55

1.99

482,444.62

5

0.9828042154

491,402.11

190.49

9,621.29

2,457.01

1.98

479,131.33

6

0.9794007059

489,700.35

221.67

11,195.91

2,448.50

1.98

475,832.30

7

0.9760089829

488,004.49

252.68

12,762.34

2,440.02

1.97

472,547.47

8

0.9726290057

486,314.50

283.54

14,320.63

2,431.57

1.96

469,276.80

9

0.9692607335

484,630.37

314.23

15,870.81

2,423.15

1.96

466,020.22

10

0.9659041259

482,952.06

344.76

17,412.91

2,414.76

1.95

462,777.68

11

0.9625591424

481,279.57

375.14

18,946.96

2,406.40

1.94

459,549.13

12

0.9592257427

479,612.87

414.53

17,871.54

2,446.03

1.98

458,878.80

Three features of that table are the product rather than the arithmetic. claims_lapse runs far ahead of claims_death throughout the deferral phase — 50.5 times at t = 0, 13.6 at t = 120 and 3.8 at t = 299 — because the two decrements pay the same amount per policy, so the ratio is exactly (1 − q^m(t)) w^m(t) ÷ q^m(t), surrenders being taken from the survivors of mortality, and it narrows as mortality rises against a lapse rate that steps down. The rows inside a policy year differ only through the in-force and the fund, because the annual assumptions are flat across the twelve months; what steps at t = 12 is the 계약해당일 itself — mortality rises with 보험나이 41, the lapse rate falls from 4.0% to 3.5% (which is why claims_lapse drops between months 11 and 12 while claims_death jumps), and the expense inflation factor takes its first step, which is why expenses rises there after eleven months of falling with the in-force. And pols_if(12) = 0.9592257427 is the annual model’s pols_if(1) to the last printed digit, which is what twelve compounded monthly exits are for.

The months where the product does something.

t

pols_if(t)

premiums

claims_annuity

claims_death

claims_lapse

expenses

net_cf(t)

83

0.8168532766

408,426.64

0.00

2,802.12

59,476.50

2,299.77

343,846.30

84

0.8154145906

407,707.30

0.00

2,934.75

60,143.29

2,341.64

342,285.60

239

0.6494043424

324,702.17

0.00

14,331.21

117,941.16

2,365.15

190,061.67

240

0.6484877699

0.00

0.00

15,353.93

78,471.19

2,409.05

−96,237.35

241

0.6478387515

0.00

0.00

15,365.43

78,529.92

2,406.64

−96,305.17

299

0.6103365075

0.00

0.00

21,562.84

81,886.17

2,454.22

−105,907.26

300

0.6096911403

0.00

343,630.80

0.00

0.00

1,667.10

−345,297.90

301

0.6096911403

0.00

343,630.80

0.00

0.00

1,667.10

−345,297.90

419

0.6096911403

0.00

343,630.80

0.00

0.00

1,992.34

−345,623.14

420

0.5837918602

0.00

329,033.59

0.00

0.00

1,945.87

−330,979.45

600

0.4669676347

0.00

263,189.75

0.00

0.00

2,094.81

−265,284.56

970

0.0002086081

0.00

117.57

0.00

0.00

1.70

−119.27

971

0.0001043040

0.00

58.79

0.00

0.00

0.85

−59.63

t = 83 and 84 bracket the end of the 계약체결비용 period — the seventh 계약해당일, which on this grid is month 84 — where NP steps up by ₩7,500 a month. t = 239 is the last premium month. t = 240 is 납입완료: the first row with no premium, and the first negative net_cf of the projection. t = 300 is the 연금개시일, where the first instalment is paid, both deferral decrements go to zero and the fund disappears into the annuity. t = 419 is the last guaranteed instalment — the last row on which pols_if is flat — and from t = 420 the in-force runs off on survivorship, which is why claims_annuity starts falling there while B itself never changes. t = 971 is the terminal row: q = 1 in the final policy year and the monthly conversion pays the last of the in-force out over its twelve months.

The fund, the surrender value and the 환급률. On this composite av_pp, cv_pp and db_pp are the same number at every duration, so one column carries all three; the 환급률 is cv_pp(d) ÷ cum_prem_pp(d), the ratio a Korean illustration quotes. d is the month-end, so d = 12y is the 계약해당일 a published grid is quoted at — and d = 1 and 6 are values inside the first policy year that an annual grid could not state at all.

d

cum_prem_pp(d)

av_pp(d) = cv_pp(d) = db_pp(d)

환급률

0

0.00

0.00

—

1

500,000.00

478,347.20

0.956694

6

3,000,000.00

2,882,844.00

0.960948

12

6,000,000.00

5,796,513.76

0.966086

24

12,000,000.00

11,717,652.56

0.976471

36

18,000,000.00

17,766,095.85

0.987005

48

24,000,000.00

23,944,580.67

0.997691

60

30,000,000.00

30,255,902.91

1.008530

84

42,000,000.00

43,288,545.09

1.030680

120

60,000,000.00

64,186,261.33

1.069771

180

90,000,000.00

102,120,559.88

1.134673

228

114,000,000.00

135,510,914.43

1.188692

240

120,000,000.00

144,311,957.57

1.202600

252

120,000,000.00

147,373,998.01

1.228117

288

120,000,000.00

156,960,814.72

1.308007

300

120,000,000.00

160,294,805.59

1.335790

301

120,000,000.00

0.00

—

The 환급률 crosses 100% on the fifth 계약해당일, at 100.85%, having been 96.61% after one year — and 95.67% after the very first month, which is the number a monthly grid adds to this table. That is the shape the adopted expense schedule produces and it is close to the one that schedule’s own carrier publishes: 96.7% at one year and break-even in the fourth year on a 2.40% declared rate [S1], against 96.61% and the fifth year here at 2.15%. A quarter of a point of interest is worth about a year of break-even on this design. The ratio rises every month, not only every anniversary, because the account is credited monthly; on the annual grid it had no value at all between 계약해당일.

The annual assumptions and the monthly conversions applied. Read at the first month of the policy year, t = 12(y − 1), where 보험나이 has just stepped.

policy year

age

q(t)

q^m(t)

w(t)

w^m(t)

1

40

0.0008065180

0.0000672347

0.0400000000

0.0033960532

2

41

0.0008247685

0.0000687567

0.0350000000

0.0029645286

6

45

0.0009199195

0.0000766923

0.0200000000

0.0016821426

11

50

0.0011100950

0.0000925550

0.0150000000

0.0012586770

20

59

0.0018335025

0.0001529204

0.0150000000

0.0012586770

21

60

0.0019635675

0.0001637781

0.0100000000

0.0008371774

25

64

0.0026416305

0.0002204029

0.0100000000

0.0008371774

26

65

0.0028596130

0.0002386140

0.0000000000

0.0000000000

35

74

0.0062631185

0.0005234308

0.0000000000

0.0000000000

36

75

0.0068742400

0.0005746662

0.0000000000

0.0000000000

81

120

1.0000000000

0.0833333333

0.0000000000

0.0000000000

Twelve of each monthly figure compound back to the year’s exactly, and the monthly force is always larger than a twelfth of the annual probability — 0.0000672347 against 0.0000672098 in the first year — which is the convexity a naive q/12 would give away. The last row is the terminal age, where q = 1 and the conversion instead spreads the certain death uniformly: 1/12 in the first month of that policy year, rising to 1 in the twelfth.

Decrements at selected months.

t

pols_death(t)

pols_lapse(t)

lives_if(t)

0

0.0000672347

0.0033958249

1.0000000000

11

0.0000647174

0.0032686823

0.9992606670

12

0.0000659532

0.0028434566

0.9991934820

120

0.0000708146

0.0009629344

0.9908608859

239

0.0000993072

0.0008172653

0.9770032583

240

0.0001062081

0.0005428104

0.9768538545

299

0.0001345199

0.0005108473

0.9659562258

300

0.0000000000

0.0000000000

0.9657433263

419

0.0258992801

0.0000000000

0.9252034626

420

0.0003354854

0.0000000000

0.9247191826

971

0.0001043040

0.0000000000

0.0001652163

Note t = 419 and 420: pols_death is 0.0258992801 at the end of the guarantee and 0.0003354854 the month after, and neither moves a single won of cash flow, because db_pp(t) = 0 once the annuity is in payment. The first of those is the whole of the guarantee period’s accumulated mortality releasing in one row — the obligation was unconditional until then — and it is the run-off of the in-force, not a claim.

Traces#

Six months, term by term, at the precision the model produces. Intermediates are full double precision; the displayed cash flows are those intermediates rounded to two places.

Month 0 — issue. q(0) = 1.15 × 0.00070132 = 0.0008065180, so q^m(0) = 1 − (1 − 0.0008065180)^(1/12) = 0.0000672347; w(0) = 0.04, so w^m(0) = 0.0033960532.

  • Premium: 500,000 × 1.0000000000 = 500,000.00.

  • Allocated: NP(0) = 500,000 × (1 − 0.015 − 0.030) = 477,500. C(0) = 0.

  • Fund: AV(1) = (0 + 477,500 − 0) × 1.001774250062 = 478,347.2044045981, and SC(1) = 0, so CV(1) = DB(1) = the same number.

  • Deaths: D(0) = 1 × 0.0000672347 = 0.0000672347; death outgo = 478,347.2044045981 × 0.0000672347 = 32.16; claim expense = 30,000 × 0.0000672347 = 2.02.

  • Surrenders, from the survivors of mortality: W(0) = (1 − 0.0000672347) × 0.0033960532 = 0.0033958249; surrender outgo = 478,347.2044045981 × 0.0033958249 = 1,624.38.

  • Expenses: E0 + e(0)/12 = 200,000 + 30,000/12 × 1.02⁰ = 202,500.00. Commission nil.

  • CF(0) = 500,000.00 − 32.1615 − 1,624.3833 − 202,500.00 − 2.0170 = +295,841.44.

  • Update: l(1) = (1 − 0.0000672347)(1 − 0.0033960532) = 0.9965369404; L(1) = 1 − 0.0000672347 = 0.9999327653.

Month 0 is +₩295,841.44, and the sign is a real product fact the annual grid could only assert: one month’s ₩500,000 premium covers the whole ₩200,000 계약체결비용 cash expense and the month’s decrements, with ₩295,841 left over. On a grid that collected the year’s premium in one row the same statement was made against ₩6,000,000 and proved much less.

Month 1. Still policy year 1, so q(1) = 0.0008065180 and w(1) = 0.04 are the same rows as month 0; only the in-force and the fund have moved.

  • Premium: 500,000 × 0.9965369404 = 498,268.47.

  • Fund: AV(2) = (478,347.2044045981 + 477,500) × 1.001774250062 = 955,847.2044045981 × 1.001774250062 = 957,543.1163662616.

  • Deaths: D(1) = 0.9965369404 × 0.0000672347 = 0.0000670019; death outgo = 957,543.1163662616 × 0.0000670019 = 64.16; claim expense = 2.01.

  • Surrenders: W(1) = (0.9965369404 − 0.0000670019) × 0.0033960532 = 0.0033840649; surrender outgo = 957,543.1163662616 × 0.0033840649 = 3,240.39.

  • Expenses: 30,000/12 × 1.02⁰ × 0.9965369404 = 2,500 × 0.9965369404 = 2,491.34.

  • CF(1) = 498,268.4702 − 64.1572 − 3,240.3881 − 2,491.3424 − 2.0101 = +492,470.57.

  • Update: l(2) = (0.9965369404 − 0.0000670019)(1 − 0.0033960532) = 0.9930858737.

Month 84 — the 계약체결비용 has stopped. α(84) = 0 while β(84) is still 3.00%, so

NP(84) = 500,000 * (1 - 0.030) = 485,000

against ₩477,500 in each of the eighty-four preceding months — ₩7,500 more into the fund every month from here to 납입완료, which is the whole of the monthly acquisition charge, 1.50% × ₩500,000. Everything else in the row is unchanged in form: premium 500,000 × 0.8154145906 = 407,707.30, death outgo ₩2,934.75 on D(84) = 0.8154145906 × 0.0000820752, surrender outgo ₩60,143.29, maintenance 30,000/12 × 1.02⁷ × 0.8154145906 = 2,341.64, and CF(84) = +342,285.60. The step is small in the month it happens and compounds for 156 months.

Month 240 — 납입완료. No premium is due; the maintenance charge is taken from the fund instead. q(240) = 1.15 × 0.00170745 = 0.0019635675 (q^m = 0.0001637781); w(240) steps down to 0.01 (w^m = 0.0008371774).

  • Premium: 0.00. NP(240) = 0.

  • Charge from the fund: C(240) = 500,000 × 0.0067 = 3,350.00.

  • Fund: AV(241) = (144,311,957.5668497980 − 3,350) × 1.001774250062 = 144,308,607.5668497980 × 1.001774250062 = 144,564,647.1227703094. The fund still grows — ₩3,062,040.45 over the twelve months to d = 252 — because 2.15% on ₩144m is many times a ₩3,350 monthly charge.

  • Deaths: D(240) = 0.6484877699 × 0.0001637781 = 0.0001062081; death outgo = 144,564,647.1227703094 × 0.0001062081 = 15,353.93; claim expense = 30,000 × 0.0001062081 = 3.19.

  • Surrenders: W(240) = (0.6484877699 − 0.0001062081) × 0.0008371774 = 0.0005428104; surrender outgo = 144,564,647.1227703094 × 0.0005428104 = 78,471.19.

  • Expenses: 30,000/12 × 1.02²⁰ × 0.6484877699 = 3,714.8684899459 × 0.6484877699 = 2,409.05.

  • CF(240) = 0 − 15,353.9332 − 78,471.1887 − 2,409.0468 − 3.1862 = −96,237.35.

This is the row that decides the shape of the whole projection. 납입완료 and 연금개시 are different dates, five years apart, and in between the contract is a fund that pays a charge, pays out on death and surrender, and receives nothing. A model that annuitises at 납입완료 loses those sixty months, and with them ₩6,054,921.77 of undiscounted outgo and ₩15,982,848.02 of fund growth.

Month 300 — 연금개시일. The transition, in the order it happens.

  1. Fund fixed: AV(300) = (160,010,906.2306696773) × 1.001774250062 = 160,294,805.5909678042, the last of 300 monthly roll-forwards.

  2. Floor tested: G = 1.001 × 120,000,000 = 120,119,999.9999999851, and F = max(160,294,805.5909678042, 120,119,999.9999999851) = 160,294,805.5909678042. The floor does not bind, with 33.4% to spare.

  3. Factor struck on the annuitant_issue male table at 2.15% with g = 10 and f = 12: ä = 23.58191601796395.

  4. Annuity struck, once: B = 160,294,805.5909678042 ÷ 23.58191601796395 × (1 − 0.005) = 6,797,361.3962860510 × 0.995 = 6,763,374.5893046195 a year, and the instalment the projection actually pays is B/12 = 563,614.5491087183.

  5. First instalment paid, in advance, to every contract with an obligation open: claims_annuity(300) = 563,614.5491087183 × 0.6096911403 = 343,630.80.

  6. Decrements: lapse_rate(300) = 0 and pols_death(300) = 0, so no death benefit and no surrender; AV, CV and DB are zero from t = 301.

  7. Expenses: 20,000/12 × 1.02²⁵ × 0.6096911403 = 2,734.3433241079 × 0.6096911403 = 1,667.10 — the maintenance level drops from ₩30,000 to ₩20,000 a year at annuitisation.

  8. CF(300) = 0 − 343,630.7971 − 1,667.1049 = −345,297.90.

Month 420 — the guarantee ends. The 120th and last guaranteed instalment was paid at t = 419. From here pols_if stops being flat and becomes survivorship:

l(420) = l(300) * L(420) / L(300) = 0.6096911403 * 0.9247191826 / 0.9657433263
       = 0.6096911403 * 0.9575206553 = 0.5837918602

  • Annuity: 563,614.5491087183 × 0.5837918602 = 329,033.59 — the instalment has not changed and never will; the count has.

  • Expenses: 20,000/12 × 1.02³⁵ × 0.5837918602 = 3,333.1492545184 × 0.5837918602 = 1,945.87.

  • CF(420) = −329,033.5861 − 1,945.8654 = −330,979.45, against −345,623.14 the month before: the first fall in payout outgo in ten years, and it is mortality, not arithmetic.

Undiscounted totals, t = 0 … 971#

Column

Total

premiums

94,113,902.7012

claims_annuity

134,873,106.9599

claims_death

2,399,485.4337

claims_lapse

21,424,943.7762

expenses

1,863,197.1962

claim_expenses

750.8770

commissions

0.0000

policy_loans

0.0000

net_cf

−66,447,581.5418

premiums is 78.43% of the ₩120,000,000 nominal, the difference being the decrements; claims_annuity is B/12 × 239.3003, the sum of pols_if over the payout phase; and claims_lapse is 8.93 times claims_death on payments that are identical at every deferral duration, so that ratio is a weighted average of (1 − q^m(t)) w^m(t) ÷ q^m(t). It is far below the 50.5 of the first month because the rate ratio itself collapses — w steps down from 4.0% to 1.0% while q more than triples over the twenty-five years — and not because of anything survivorship does to the weighting.

Every total is a little smaller than the annual grid’s, by between 1.0% and 1.4%, and the reason is the same in each column: a monthly grid loses business a month at a time rather than a year at a time, so the in-force weighting every flow carries is slightly lower from the first month onwards. Premium income falls from ₩95,084,920.76 to ₩94,113,902.70 and the annuity outgo from ₩136,717,952.04 to ₩134,873,106.96, leaving net_cf at −₩66,447,581.54 against −₩68,516,344.57. Nothing in the contract changed; the projection simply stopped pretending that a policy that surrenders in month 3 pays a full year’s premium first.

Reading the shape#

The projection has four distinct regimes and each one is a contractual fact rather than an artefact. Month 0 is strongly positive, +₩295,841.44, because Korean direct-channel acquisition cost is almost nothing even against a single ₩500,000 instalment: no commission at all, a ₩200,000 cash expense, and an acquisition charge that lives inside the fund rather than in the cash flow. Contrast the UK term composite in this repository, which pays 150% of an annualized premium of upfront commission. Then 240 months of declining positive margin, from ₩295,841 to ₩190,062, as surrender outgo grows against a level premium base: by t = 239 the surrender payment of ₩117,941.16 is more than a third of the premium income. Then sixty thin negative months, t = 240…299, totalling −₩6,054,921.77, in which the contract has stopped paying premiums but has not started paying an annuity. Then 672 months of pure outgo, −₩135,826,244.04, flat in B/12 and declining in count.

Undiscounted the projection is −₩66,447,581.54. Discounted at the rate the fund itself credits, 2.15% a year — which on this grid means 1.0215^(t/12), the monthly equivalent — it is +₩3,580,561.01 [derived]. That pair is the product in one line: an account contract with a 4.5%-of-premium loading and a longevity promise is close to a wash at the rate it credits, and the whole of the insurer’s result therefore sits in the spread between what it credits and what it earns, and in whether the annuitants live as long as the factor assumes. Neither of those is in this model, which is the reason it is a mechanics demonstration and not a pricing result.

The nine model points#

m and d are in years, as the contract states them; n and proj_len are in months.

#

sex

x

m

d

n

Y

P

form

vintage

proj_len

F

ä

B

Σ net_cf

1

M

40

20

5

300

65

6,000,000

종신 g=10

issue

972

160,294,805.59

23.58192

6,763,374.59

−66,447,581.54

2

F

40

20

5

300

65

6,000,000

종신 g=10

issue

972

160,294,805.59

25.26673

6,312,383.96

−71,826,694.41

3

M

45

20

0

240

65

3,600,000

종신 g=20

issue

912

86,587,174.54

24.08957

3,576,412.79

−33,616,128.95

4

F

25

20

15

420

60

1,200,000

확정 k=20

issue

660

39,568,543.38

16.30428

2,414,746.68

−16,529,681.76

5

M

30

10

20

360

60

12,000,000

확정 k=10

issue

480

194,438,355.78

9.01595

21,458,209.25

−45,254,564.01

6

F

50

5

0

60

55

6,000,000

확정 k=15

issue

240

30,030,000.00

13.94004

2,143,455.00

−2,445,274.40

7

M

40

20

5

300

65

6,000,000

종신 g=10

commencement

972

160,294,805.59

24.30417

6,562,384.22

−68,821,479.83

8

M

40

20

5

300

65

6,000,000 + 6,000,000

종신 g=10

issue

972

326,721,162.46

23.58192

13,785,459.86

−137,316,414.94

9

M

40

20

5

324

65

6,000,000

종신 g=10

ratchet

972

162,892,867.42

22.70113

4,273,340.12

−36,971,445.78

Four of them are worth a sentence. Point 2 is the anchor cell’s twin with the sex switched: the same fund buys a factor of 25.26673 instead of 23.58192 and an annuity 6.67% smaller, and that is the only thing sex changes on this product. Point 6 is the statutory-minimum contract — five years of premiums, 만 55세 start — run on the floor rate scenario, and it is the one shipped point where the 100.1% guarantee binds exactly: the fund at annuitisation is ₩30,030,000 = 100.1% × ₩30,000,000, against an av_pp(60) of ₩29,573,776.73 that the guarantee tops up. Point 7 is the anchor cell struck on the 연금개시시점 vintage instead: the annuity falls 2.97%, which is the size of the vintage question and the reason it is a switch rather than an assumption. Point 9 carries the ratchet, a two-year 납입유예 (so n = 324), the 100.1% floor withdrawn, a policy loan and a declared dividend all at once, and it confirms the ratchet is out of the money: mort_table_name() returns annuitant_issue because the issue-vintage factor, 22.70113, is smaller than the revised-vintage factor, 23.43728, and the smaller factor is the larger annuity. Its B is 0.15% above the annual grid’s, because the loan it draws and the dividend it accumulates both now accrue month by month.

Valuation and reserve pointers#

This library projects gross cash flows and builds no reserve. Each layer below consumes them and is cited, not reproduced.

  • 책임준비금 (policy reserve). 보험업법 제120조 requires it REG-R3; 감독규정 제6-11조 gives the post-2023 taxonomy — 보험계약부채, 재보험계약부채, 투자계약부채, each of the first two split into 잔여보장요소 and 발생사고요소 — and delegates the calculation to the FSS Governor, which is why the 고시 no longer carries accumulation rules REG-R10. Paragraphs ⑤–⑩ of the old article were deleted on 2022-12-21, and that deletion is the visible trace of the switch from a locked-in statutory reserve to a current-estimate one.

  • 해약환급금준비금, and this product is one of the two it was built for. 감독규정 제6-11조의6 makes the insurer compare, at every balance-sheet date including quarterly interim closes, its IFRS 17 liability for remaining coverage against the aggregate 해약환급금 computed under 제7-66조제1항 plus 미경과보험료, and appropriate any shortfall to a surrender-value reserve inside 이익잉여금 — at company level, not contract level REG-R11. Since 2025-06-11 an insurer whose pre-transitional K-ICS ratio was 130% or more at the previous quarter-end appropriates only 80% of the shortfall. On this product the exposure is maximal by construction: CV(t) = AV(t) at every duration, so the aggregate contractual surrender value is the whole of the fund, and the whole of the gap between it and an IFRS 17 measurement is the earnings profile. The reserve stood at ₩23.7tn at end-2022 and ₩32.2tn at end-2023 REG-R11 REG-R36. Pension_KR_S computes none of it, and cv_pp(t) is the input it would need.

  • 보증준비금 (guarantee reserve). 감독규정 제6-11조의5 requires one inside retained earnings for expected losses on benefit guarantees, junior to the 해약환급금준비금 REG-R10. The 100.1% minimum fund is a benefit guarantee, so this product sits inside that perimeter in principle even though the guarantee is shallow. The basis lives in 시행세칙 별표 24, which was not retrieved REG-R26, so anything further is unverified here.

  • K-ICS. In force since 2023-01-01 alongside IFRS 17 REG-R13 REG-R14. Two of its modules bear directly on this contract and neither is computed here: the 대량해지위험 shock, reported at second hand as a 35% immediate surrender for 저축성보험 contracts REG-R36 and resting on 별표 22, which was not retrieved — so that figure is unverified REG-R26 — and longevity risk on the annuity in payment. The 35% is the one to watch on this product, because a mass surrender pays the full account value.

  • IFRS 17 (K-IFRS 제1117호). Mandatory since 2023-01-01 REG-R60. Whether a 금리연동형 연금저축보험 qualifies as a direct participating contract measured under the variable fee approach is a real question with a large earnings consequence and is unverified here; no retrieved document addresses it. The cash flows above are basis-agnostic input to whichever answer a reporter reaches.

  • 계약자배당. 감독규정 제6-11조의7 and 제6-13조 govern the dividend reserve and its distribution REG-R12. The composite is 무배당 and declares nothing; where a dividend is run it may not be paid in cash before annuitisation and is applied as an 증액연금 [S2].

  • The actuarial opinion. 보험업법 제181조 and 제184조 put the reserve and the basis under the 선임계리사 REG-R5. The shape of result_cf() — a long-dated, undiscounted, assumption-explicit projection — is what that office consumes.

  • Policyholder protection. 예금자보호법 시행령 제18조제7항 sets the limit at ₩100,000,000 from 2025-09-01 and applies it to four separate buckets, the second of which is the combined total of 연금저축계좌 claims REG-R52 REG-R32. A Pension_KR_S policyholder’s protection is therefore separate from the ₩100,000,000 covering their other insurance claims against the same insurer. Not a cash flow in this model.

Key sensitivities and model risks#

In rough order of leverage on this block.

  1. The credited rate, over twenty-five years of accumulation. i_c(t) is the only thing that turns ₩120,000,000 of premium into ₩160,294,805.59 of fund — 300 monthly creditings at (1.0215)^(1/12) − 1 — and it is discretionary within a regulated construction whose external-index weight is capped at 60% REG-R18 REG-R24. The composite’s 2.15% is the conservative arm of an observed 2.1%–3.0% range [S12] [S1] [S2] [S11]. The floor scenario is not a stress but a published illustration column [S2], and at model point 6 it is what makes the 100.1% guarantee bind exactly. Nothing else in the model has comparable leverage.

  2. The annuity factor, and its four separate inputs. B moves inversely with ä, and ä is a function of the table, the vintage, the interest basis and the payment frequency. On the shipped model points: the vintage is worth 2.97% (points 1 vs 7), the sex 6.67% (1 vs 2), the frequency correction 1.91% (the annual factor 24.04025 would give B = ₩6,634,429.17), and the 0.5% charge 0.5% exactly. A production model must re-derive all four; this one exposes each as an input.

  3. Longevity, which is the only insurance risk in the contract. The fitted table’s curtate e(65) of 33.31 years is the illustration’s implied longevity, not a Korean population estimate — 국가데이터처’s 「2024년 생명표 작성 결과」 gives 65세 기대여명 남 19.5 and the 제10회 경험생명표 gives 23.7 REG-R38 REG-R33. Re-striking the factor on a table anywhere near either of those would raise B by tens of per cent, and the model would then no longer reproduce the published annuity. That tension is the product’s, not the library’s, and it is the single most important thing to understand before using this model for anything but mechanics.

  4. The expense schedule, which is a choice between two published products. The composite takes the direct-channel schedule, so its first-year 환급률 is 96.61% rather than the 82.7% of the tied-channel product from which the anchor model point comes [S1] [S2]. That difference is the largest single standardization in the composite and it changes early-duration surrender cost, break-even duration and, through them, any lapse calibration built on 환급률.

  5. Lapse, which nothing public constrains. The vector is std and argued, its count-weighted deferral mean 1.9196% a year and its av_pp-weighted mean 1.4183%. On the savings comparison vector the same chassis loses far more business before annuitisation — model point 5 reaches its annuity date with 0.3831301998 in force. The direction of the error is knowable even where the level is not: this vector is net-of-부활 and net-of-계좌이체, so an experience rate lifted from an insurer’s termination statistics will over-decrement.

  6. The 100.1% floor is an option and it is priced at zero here. It binds at one date on one path, and the model tests it by comparison rather than valuing it. On a persistently low-rate scenario it is the whole of the guarantee, and a stochastic valuation of it belongs in the 보증준비금 REG-R10 REG-R26, not in this projection.

  7. Mass surrender. CV = AV at every duration, so a K-ICS 대량해지 shock on this product pays the entire account value with no deduction REG-R36 unverified, and the 해약환급금준비금 is measured off the same number REG-R11. There is no 무해지/저해지환급형 dispensation to soften it: 제7-66조제4항 does not apply to a contract with a full surrender value from the first month REG-R19.

  8. The 종신계약 withholding rate, which is unverified. The flat 3.3% rests on 소득세법 시행령 제187조의2, whose operative text was not retrieved R7. If a ten-year guarantee period disqualified the contract from 종신계약 status the anchor cell’s withholding would be 5.5% until 70, and the annuitisation-election assumption — 100% life form — would need re-arguing.

  9. The 표준해약공제액 reading. ₩1,421,988.72 on the reading implemented, ₩1,486,788.72 on the alternative, 4.56% apart REG-R20 주3. It binds nothing on this composite; it would matter to a design that used its headroom.

  10. The 세액공제 band. The model takes 16.5%, the lower-income band, because a contract does not know its owner’s income; the alternative is 13.2% R1 R8 R10. The grossing-up of the statutory 15% / 12% is itself unverified arithmetic, because the 지방세법 imposing the surtax was not retrieved REG-R56. Nothing in net_cf depends on it; the lapse argument does.

Known modeling pitfalls#

Each of these is a mistake a competent modeller would actually make on this product, and each is checkable against the shipped model.

  1. Putting a survivorship release into the fund. The 계약자적립액 is an account: AV(t+1) = (AV(t) + NP(t) − C(t))(1 + j_c(t)), full stop. WholeLife_KR_S’s 계약자적립액 is not — it divides by (1 − q) and subtracts a death benefit — and neither is the deferred annuity on the Japanese page. Two neighbours carry the same name and a different recursion, and porting either shape here overstates the 연금개시 fund silently. There is nothing to release: the death benefit is the fund. check_av_roll_fwd() is the check that catches it.

  2. Projecting a deferral-phase mortality strain. db_pp_net(t+1) and cv_pp_net(t+1) are the same number at every duration on this composite, so the strain is exactly zero and no deferral-phase death cover basis is needed. Mortality still matters — it decides how many policies reach the 연금개시일, and hence the charge income and the expense — but a model that prices a death benefit here is pricing a risk the contract does not run.

  3. Applying a death product’s best-estimate adjustment. mort_be_factor is 1.15, greater than one, because the published 연금사망률 is loaded on the survival side. A 0.85 lifted from a death-cover table has the sign wrong and produces both too many survivors and, if it is fed into the factor, too large an annuity.

  4. Getting the decrement order wrong. Deaths are taken from the whole opening in-force and surrenders from the survivors: W(t) = (l(t) − D(t))w^m(t), not l(t)w^m(t). At t = 0 that is 0.0033958249 against 0.0033960532 — small, and wrong in the same direction in every one of the 300 deferral months. check_pols_roll_fwd() closes only on the stated order. A second order error is available on a monthly grid and is not: dividing the annual rate by twelve instead of converting it. q(0)/12 = 0.0000672098 against q^m(0) = 0.0000672347 looks like nothing in the first year and compounds the wrong way for twenty-five.

  5. Stopping the lapse decrement a month early. Surrender is available right up to the day before the 연금개시일 [S2] [S4], so lapse_rate(t) is non-zero through t = n − 1 and zero from t = n. At the anchor cell month 299 pays ₩81,886.17 of surrender benefit on the full fund; zeroing the decrement there deletes it and leaves the in-force too high going into the annuity. On the annual grid the same slip deleted a whole year of it, ₩987,174.98.

  6. Using an annual annuity-due factor. The annuity is paid 매월 and the factor is ä^(12). On the annual factor the anchor cell’s B would be ₩6,634,429.17 instead of ₩6,763,374.59, 1.91% low, and the model would reproduce none of the eight published implied factors. It is the error an earlier draft of product-spec.md contained and the one this document corrected; both documents now carry the monthly form.

  7. Putting mortality into the 확정기간연금형 factor, or leaving it out of the 종신연금형 one. The certain form is priced on the declared rate alone and its instalments are paid to the count whether or not the annuitant lives [S1] [S2] [S6]; the life form is priced on the 연금사망률. At the anchor cell the two factors are 9.01595 and 23.58192 on the same fund — the life annuity pays 38.23% of the certain one — so sharing one code path between them is not a tidy-up, it is a factor of two and a half.

  8. Decrementing pols_if by mortality inside a guaranteed or certain period. Inside the 보증지급기간 the obligation is unconditional, so pols_if is flat from t = 300 to 419 while lives_if falls from 0.9657433263 to 0.9252034626, and not one of those deaths changes a won. Conversely pols_death(419) = 0.0258992801 is a run-off of the in-force, not a claim — the whole guarantee period’s mortality releasing in one row: db_pp(t) = 0 once the annuity is in payment, so claims_death is zero from t = 300.

  9. Recomputing B, or keeping the fund alive after annuitisation. B is struck once at t = n from F_net and never recomputed, and av_pp, cv_pp and db_pp are zero from t = n + 1. A model that keeps rolling the fund forward and also pays the annuity is double-counting the same money. check_annuity_total() asserts the instalments are level and total gB.

  10. Forgetting that the maintenance charge outlives the premium. C(t) = ₩3,350 a month at t = 240…299, ₩201,000 in total, taken from the fund with no premium bearing it [S1] [S5] — and 「월계약해당일에 … 적립액에서 차감합니다」 is exactly what the contract says. A model whose charges are all premium-based charges nothing in the gap and overstates the fund at annuitisation.

  11. Forgetting that the acquisition charge stops. α(t) runs for seven policy years only, so NP steps up by ₩7,500 at t = 84 — the seventh 계약해당일, not row 7. A level-loading model gets the fund wrong in both directions — too high early, too low late — and cannot reproduce the published 환급률 curve; a model that reads the seven-year period as seven rows stops the charge six years early.

  12. Treating the 최저보증이율 as a guarantee on the return. It guarantees the credited rate; the charges are still deducted beneath it [S4] [S8]. So charge_from_av_pp must not consult the floor, and a contract crediting the floor still loses the loading.

  13. Mistaking the 예정이율 for a crediting rate. 2.50% appears in the sources as the rate the charge and benefit structure was priced on, and every document that discloses it says it is not a guarantee [S1] [S5] [S7]. It appears nowhere in the fund recursion. The 평균공시이율, also 2.50% in 2026, is a third rate again and enters only inside the 표준해약공제액 REG-R48.

  14. Putting the tax layer into the cash flow. The 세액공제 (₩82,500 a month, ₩990,000 a year at the anchor cell, ₩19,800,000 over the premium term) is paid by the state to the saver, and the 16.5% 기타소득세 is withheld from the saver’s proceeds. Neither passes through the insurer’s account. Both are published in result_tax(), and check_net_cf() fails the moment either is added to result_cf().

  15. Applying the 연금수령한도 formula where it does not apply. Where the 연금수령연차 reaches 11 the formula is disapplied entirely R6 제40조의2제4항 [S3], and at the anchor cell it is 11 in the first payment year. A model that evaluates 평가액 ÷ (11 − 연금수령연차) × 120/100 regardless divides by zero, or caps an annuity that no rule caps. The 한도 is a per-year figure, so on a monthly grid the test is against the twelve instalments of the 연금수령연차 and not against one of them.

  16. Computing the 표준해약공제액 on the gross premium. 별표 14 주3 works on the 연납순보험료 — the annual premium less the levelled loading, ₩5,577,000 here, not ₩6,000,000 — and 주6 then subtracts the discounted acquisition loading. Using the gross premium gives ₩2,160,000 before 주6 against ₩2,007,720, a 7.6% overstatement of a cap that is meant to bind.

  17. Reading proj_len() as the last index. It is a count of months — the exclusive end of the frame: 972 at the anchor cell, with 972 rows in result_cf() running t = 0 … 971, so the last index is proj_len() − 1. Reading it as the last index appends an empty row at t = proj_len(); the mirror slip, range(proj_len() − 1), silently drops the terminal row, which on the life form is where the last survivors die.

  18. Assuming the 100.1% floor protects a death claim. It is a survival guarantee applied once, at t = n, to a policy in force; a death in deferral is paid the fund, which may be less than premiums paid — and is, for the first four policy years at the anchor cell. Its base is premiums paid including 추가납입, which is why cum_prem_pp adds both.