Technical Notes#

Status: Draft, 2026-08-03. Companion to product-spec.md in this directory — all contractual parameters used here (premiums, fee, modal factors, windows) are the same representative values specified there. This is a standardized composite for reference modeling, not any single insurer’s product. [S#]/[R#] tags cite the product research notes (_research/term-life.md); [REG-R#] tags cite the cross-product reference library (references/regulatory-and-actuarial-references.md; research provenance in _research/regulatory-actuarial.md for R1–R34 and in _research/appp-a820-a821-a822.md and _research/appp-a830.md for the AP&P Manual appendix items cited here, same R-numbering); std marks standardizations introduced for the reference implementation; unverified flags carry over from the research notes.

Model scope and conventions#

  • Scope. Single-life, fully underwritten level premium term per product-spec.md: 10/20/30-year level periods (base cell 20-year), Jump-to-ART post-level term (PLT) with unchanged face to expiry at attained age 95, convertible before min(end of level period, attained age 70), no cash value, non-participating [S2] [S3] [S6]. Gross liability cash flows only; reserves are pointers (see Valuation section).

  • Projection frequency std: monthly. Monthly steps (monthiversary processing) are the model. Every contractual element of the product is on an annual cycle — level premiums, ART renewals at anniversaries, the shock lapse at the level-period end — and the model keeps them there; what the monthly grid adds is the timing of everything that is not: modal premium collection, mid-year claim settlement, monthly expense accrual, and the ability to carry mode-specific behavior (monthly-mode policies show materially lower shock lapse and PLT mortality deterioration R4). An annual step remains a well-defined special case of the recursion below and reproduces its in-force at every anniversary exactly — see Annual equivalence — but it is not what the reference model runs.

  • Time index std. t is 0-based and counts policy months, as in lifelib’s basiclife/BasicTerm_S: the first period is t = 0 (the issue month), period t runs from time t to time t + 1, and the frame is t = 0, 1, …, proj_len − 1 with proj_len = 12(95 − x) the number of policy months projected. Because every contractual schedule is annual, the policy year is derived and used as a lookup key: dur(t) = t // 12 is the completed policy years at the start of month t and the contractual policy year is the 1-based label dur(t) + 1; where these notes say “policy year k” as contract language, the months are t = 12(k − 1) … 12k − 1. Nothing is indexed by the policy year.

  • Timing std. Monthiversary (BOM) processing: premiums and premium-linked expenses at the beginning of the month, in the months the mode makes them due; deaths during the month with claims paid at month end; lapses, shock lapses, and conversions at month end after deaths. The shock lapse is processed at the END of the final level-period month, t = 12n − 1 (equivalently, immediately before the first ART premium falls due at t = 12n) — consistent with VM-20’s “shock lapse in the final year of a level premium period” R2 and the SOA study’s measurement of lapse at the end of the level term R4.

  • Age basis. Age nearest birthday (ANB) [std choice, sourced pattern]: all four carriers with verifiable age rules use ANB [S2] [S3] [S5] [S6], and 2017 CSO / 2015 VBT are published in ANB variants R3 REG-R18. Attained age = issue age + completed policy years, dur(t) = t // 12 completed at the start of month t, so age changes on the anniversary and not on the birthday [S3] [S5] [S6].

  • Model points. Single-policy model points (seriatim); one policy per model point with a count/weight field for grouping. VM-20 NPR is a seriatim quantity R2, so seriatim granularity keeps the projection reusable for valuation feeds.

  • Units. Currency in USD; face in dollars; rates per $1,000 where contractual [S2] [S3] [S5]; decrement rates are quoted annual effective throughout and subscripted m where the monthly rate actually applied is meant.

Model point attributes#

Attribute

Type

Example (specimen anchor cell)

policy_id

str

“TL-000001”

issue_date

date

2026-01-01

issue_age

int (ANB)

35

sex

enum {M, F}

M

rate_class

enum {PPlusNT, PNT, StdNT, StdTob}

StdNT

plan

enum {T10, T20, T30}

T10

face_amount

float ≥ 100,000

100,000

band

int 1–4 (derived from face)

1

premium_mode

enum {A, SA, Q, M}

A (the anchor cell; the mode drives which months collect an instalment)

policy_count

float (weight)

1.0

duration_inforce

int (for in-force runs; 0 at issue)

0

The example column is the specimen anchor cell M35/StdNT/$100k/10-yr [S6], which the worked example below projects. Attribute menu per product-spec.md (issue-age grid std, 4 classes std, 4 bands [S5]/std).

State variables#

Variable

Definition

l(t)

In-force policies at start of month t, i.e. at time t (l(0) = policy_count at issue)

d(t)

Deaths in month t

x(t)

Lapses (incl. shock lapse) at end of month t

c(t)

Conversions at end of month t

AP(t)

Annualized guaranteed gross premium for the policy year containing month t (rate-table row for that policy year, + fee)

P(t)

The modal instalment of AP(t) actually collected at BOM t; zero in a month no instalment is due

dur(t)

Completed policy years at the start of month t, t // 12; the contractual policy year is the 1-based label dur(t) + 1

phase(t)

LEVEL (dur(t) < n), PLT (dur(t) ≥ n, attained age < 95), EXPIRED (t ≥ 12(95 − x))

conv_elig(t)

Boolean: dur(t) < n and attained age < 70

No account value, cash surrender value, loan, or shadow-account state exists for this product [S3] [S6].

Assumption inputs#

Three classes are distinguished; keeping them in separate input structures is deliberate architecture (the same split VM-20 makes between prescribed/guaranteed and prudent-estimate elements R2 REG-R23).

(a) Contractual / guaranteed elements (from the spec — cited)#

Item

Value

Basis

Guaranteed premium scale

Level AP for n years, then guaranteed ART scale to age 95; full schedule printed at issue

[S3] [S6]

Anchor schedule (M35/StdNT/$100k/10-yr)

$140 (yrs 1–10); $764, $830, $992 (yr 15), $1,526 (yr 20), $4,250 (yr 30), $10,946 (yr 40), $30,965 (yr 50), $74,780 (yr 60 — the months t = 708 … 719, attained age 94, the final policy year to expiry at 95)

[S6]

Policy fee

$65/yr, level, inside AP

[S6]

Modal factors

SA 0.52 / Q 0.27 / M 0.08333

[S6]

Death benefit

Level face; proceeds = face + pro-rata unearned premium − due unpaid premium

[S6]

Grace

31 days

[S3] [S6] [S7]

Conversion window / credit

min(n, age 70); credit = one annual premium after year 1

[S2] [S3] [S6]

Expiry

Attained age 95

[S2] [S3] [S5] [S6]

(b) Current non-guaranteed scales#

For this product there are none: premiums and death benefit are fully guaranteed [S3] [S6], and the representative product sets the current PLT scale equal to the guaranteed Jump-to-ART scale std (product-spec fn 10; graded current PLT scales observed in the market R4 are a documented variation, not modeled). This block is intentionally empty so the input schema matches sibling products (UL etc.).

(c) Behavioral / experience assumptions (best estimate)#

Assumption

Recommended public basis

Reference-model standardization

Best-estimate mortality

2015 VBT primary tables (ANB, sex/smoker-distinct) with relative-risk (RR) tables for preferred fit REG-R18, A/E-adjusted to ILEC 2012–2019 inter-company experience R8 REG-R19 (ILEC expected basis 2015 VBT RR100 unverified)

Class factors on 2015 VBT-style base: PPlusNT 0.80, PNT 0.90, StdNT 1.00, StdTob 1.75 std (fn A)

Guaranteed-basis mortality (for reserve feeds)

2017 CSO, ANB, smoker-distinct, loaded R3 REG-R17

Direct table lookup, no adjustment

Level-period lapse

SOA/LIMRA 2015–2022 Term & WL lapse study R6; older full-factor study REG-R20

Duration vector, fn B std

Shock lapse & PLT lapse

SOA U.S. Post-Level Term study (2021) R4 REG-R22

Jump-ratio-keyed table, see Policyholder behavior std

PLT mortality deterioration

Same study R4 REG-R22

Multiplier grading 3.50 → 2.00, see Policyholder behavior std

Conversion rate

SOA 2016 conversion experience study R7 (2009–2023 SOA/LIMRA update in progress R7, partly unverified)

1%/yr while eligible; 2% in final eligible year std (fn C)

Maintenance expense

— (no public basis in research set)

$30/policy/yr inflating 2%/yr std (fn D)

Acquisition expense

—

$300/policy at issue std (fn D)

Commission

—

80% of premium year 1; 5% years 2–n; 2% PLT std (fn D)

Premium tax

—

2.0% of collected premium std (fn D)

Premium mode

Modal factors from the specimen [S6]

A / SA / Q / M as a model point attribute, collected on the mode’s own cycle; mode affects PLT behavior only via the optional R4 factors std

Footnotes

  • (A) Class factors std. The 2015 VBT provides 10 nonsmoker and 4 smoker RR tables for preferred-class fit REG-R18; the four factors {0.80, 0.90, 1.00, 1.75} are a compressed stand-in chosen so that StdNT reproduces the specimen anchor pricing cell [S6] and the NT spread stays inside the RR-table range. Calibration to actual RR tables is an implementation refinement.

  • (B) Level-period lapse std. Annual rates by policy year: 6%, 5%, then 4% for years 3 through n−2, year n−1: 6% (anticipatory rise — lapse rates begin increasing one to two policy years before the end of the level period R6), year n: shock lapse (below). Each is converted to the monthly rate applied within its policy year, except the shock — see Monthly rates from annual assumptions. Detailed study rates by sex/age/band/mode sit behind SOA paid data packages (research notes, Gaps); the vector is an order-of-magnitude standardization consistent with the public highlights: 30-year term lapse rates at attained ages 60+ run 1.0%–1.5% R6, so for T30 the 4% mid-band grades to 1.5% from attained age 60 std.

  • (C) Conversion std. The public 2016 study landing page documents incidence analysis by age/sex/class/size but no headline rate was recorded in the research notes R7; 1%/yr (2% final year) is a placeholder magnitude. Treatment of the conversion cash flow: see Cash flow components.

  • (D) Expenses/commission std. No insurer expense or commission data appear in the retrieved public documents; these are round reference values for a complete gross cash flow statement. Replace with company-specific unit costs in any real application. The policy fee ($65 [S6]) is intended as the contractual funding of per-policy maintenance.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol

Meaning

x

Issue age (ANB); n = level term period in years; F = face amount

t

Month index, 0-based: t = 0, 1, …, 12(95 − x) − 1 (proj_len = 12(95 − x) months). Month t sits in policy year dur(t) + 1

dur(t)

Completed policy years at the start of month t, t // 12

l(t)

In-force count at start of month t (time t); l(0) = 1 per unit model point

q(t)

Best-estimate annual mortality at the attained age of month t, incl. class factor and PLT multiplier; q_m(t) = 1 − (1 − q(t))^(1/12) is the rate applied in the month

w(t)

Annual lapse rate of the policy year containing month t; in policy year n it is the shock lapse. w_m(t) is the rate applied in the month, and the shock is not spread — see below

cv(t)

Annual conversion rate (0 outside eligibility window); cv_m(t) = 1 − (1 − cv(t))^(1/12)

AP(t)

Annualized guaranteed gross premium for the policy year containing month t

P(t)

Modal instalment collected at BOM t: modal factor × AP(t) in a due month, 0 otherwise [S6]

G(t)

Premium income in month t; K(t) commission; E(t) expenses; X(t) premium tax

DC(t)

Death claims incurred in month t; CV(t) conversion credit outflow

M(d)

PLT mortality multiplier at PLT duration d = dur(t) + 1 − n in years (d = 1 in the first PLT policy year, months t = 12n … 12n + 11)

J

Initial premium jump ratio = AP(12n)/AP(12n − 1), the first ART annualized premium (policy year n+1) over the last level one (policy year n), fee included R4 R2 convention

Monthly rates from annual assumptions std#

Every assumption in this product is published, calibrated and tabulated annually: the mortality table is annual, the lapse vector is by policy year, the conversion rate is per year. The monthly rates the recursion applies are derived from them at the constant-force conversion, so that twelve months compound back to exactly the annual rate:

q_m(t)  = 1 − (1 − q(t))^(1/12)
cv_m(t) = 1 − (1 − cv(t))^(1/12)
w_m(t)  = 1 − (1 − w(t))^(1/12)               ordinary lapses only — see the shock below

The shock lapse is the one exception and is NOT spread. w(n−1), the annual rate of the final level-period policy year, is the shock; it is applied in full at the end of the final level-period month, t = 12n − 1, immediately before the first ART premium falls due at t = 12n R2 R4. The other eleven months of that policy year therefore carry w_m = 0. Spreading it instead would lapse policies before the renewal notice that causes them to lapse, and would leave a larger block to pay the first ART premium.

Decrement order and recursion#

Deaths first, then end-of-month voluntary decrements (lapse and conversion) applied to survivors, with conversion and lapse treated as competing rates on the same survivor pool std. Base case l(0) = 1 per unit model point; for t = 0, 1, …:

d(t)  = l(t) · q_m(t)
s(t)  = l(t) · (1 − q_m(t))                   survivors to end of month t
c(t)  = s(t) · cv_m(t)
x(t)  = s(t) · (1 − cv_m(t)) · w_m(t)
l(t+1)= s(t) · (1 − cv_m(t)) · (1 − w_m(t))
      = l(t) · (1 − q_m(t)) · (1 − cv_m(t)) · (1 − w_m(t))

Termination at expiry: l(t) = 0 for x + dur(t) ≥ 95, i.e. for t ≥ proj_len = 12(95 − x) [S2] [S3] [S5] [S6]; the survivors of the last month t = 12(95 − x) − 1 expire rather than decrement (the implementation books them as pols_maturity).

Annual equivalence. Because the ordinary rates compound back to their annual values and the shock sits on a year boundary, the recursion collapses over any twelve months of one policy year to l(t+12) = l(t)(1 − q)(1 − cv)(1 − w) — the annual-step recursion, term for term. The in-force at every anniversary is therefore identical on the two grids, to floating point, and a monthly run can be checked against an annual one on that column alone. Nothing else agrees, and nothing else should: the cash flows are where the finer grid does its work.

Cash flows (per unit in force at issue)#

G(t)  = P(t) · l(t)                           premium, BOM  [S6 schedule and modal scale]
K(t)  = k(t) · G(t)                           commission, BOM  [std]
X(t)  = 0.02 · G(t)                           premium tax, BOM  [std]
E(t)  = 300 · 1{t=0} + (30/12) · 1.02^(t/12) · l(t)
                                              maintenance/acquisition, BOM  [std]
DC(t) = F · d(t)                              death claims, EOM  [S6]
CV(t) = AP(t) · c(t) · 1{dur(t)≥1}            conversion credit, EOM  [S6]
NetCF(t) = G(t) − K(t) − X(t) − E(t) − DC(t) − CV(t)

with k(t) = 80% in policy year 1, 5% in policy years 2…n, 2% after std, applied to the premium collected — so a modal payer earns it in instalments too. Three quantities stay on the annualized premium AP and so do not move with the mode: the jump ratio J and the conversion credit CV(t). The acquisition expense is mode-independent too, for a different reason — it is a flat per-policy amount, not a fraction of any premium.

Maintenance expense accrues at a twelfth a month and inflates continuously, 1.02^(t/12), rather than stepping at anniversaries. Twelve twelfths of $30 is the annual charge, and the continuous factor puts a full policy year of it about 0.9% above the anniversary-stepped figure; what changes materially against the annual grid is that a decrementing block carries less of it, because it is borne by the in-force of each month rather than of the anniversary.

Simplifications std: (i) the pro-rata unearned-premium refund on death and the due-unpaid-premium deduction [S6] are not modeled — on the monthly grid the item is bounded by one modal instalment, so it is immaterial by construction for a monthly payer and at most one annual premium on the deceased cohort for an annual one; (ii) grace-period mechanics [S3] [S6] are not separately modeled — lapse is treated as effective at the monthiversary; (iii) reinstatement [S3] [S6] is not modeled as a decrement reversal.

Conversion treatment [std choice — explained]#

Two defensible treatments exist:

  1. Decrement with cost load (adopted). Conversion removes the policy from the term block (c(t) above); the direct cash flow charged to the term product is the contractual conversion credit of one annual premium [S6]. The post-conversion mortality anti-selection documented by the SOA conversion studies R7 is borne by the permanent product’s model, not double-counted here. Adopted because it keeps the term model self-contained, uses only contractual cash flows, and matches how the conversion credit is actually paid (against the new policy’s initial premium [S6]).

  2. Transfer-out (alternative). Model conversion as a zero-cash-flow transfer to a companion permanent model point (lifelib-style linked runs). Preferable when the library is run as a linked term+permanent projection; the switch is an output-routing choice, not a different liability.

Processing order (monthiversary)#

Numbered order each month:

  1. Check expiry (attained age 95) and terminate [S2] [S3] [S5] [S6].

  2. Collect the modal premium if one is due this month — modal factor × AP, the factors being A 1.0 / SA 0.52 / Q 0.27 / M 0.08333 [S6], with the modal load inside the factor. Instalments start at issue and repeat on the mode’s cycle, so annual mode collects in the first month of each policy year and monthly mode in every month.

  3. Pay commission and premium tax on the premium collected std.

  4. Incur 1/12 of the annual maintenance expense; acquisition expense in month 0 only std.

  5. Apply deaths at q_m; pay claims at end of month: F (simplification (i) above).

  6. Apply conversions at cv_m if within the eligibility window; pay the conversion credit of one annualized premium [S2] [S3] [S6] (before any lapse, so that conversion and lapse compete on the same survivor pool).

  7. At the level-period-end monthiversary only — month 12n − 1 — apply the shock lapse to survivors in full std (per R2 R4 timing).

  8. Apply ordinary lapses at w_m to the remaining survivors std.

  9. Roll forward l.

Policyholder behavior modeling#

All dynamic formulas in this section are std standardizations calibrated to the ranges published in the SOA 2021 PLT study R4 REG-R22; none is itself a published industry formula.

Shock lapse at end of level period#

Keyed to the initial premium jump ratio J = AP(12n)/AP(12n−1) — the first ART annualized premium (policy year n+1) over the last level one (policy year n), the two months either side of the level-period boundary — with the policy fee included in both premiums — the jump definition used by both the SOA 2021 study R4 and VM-20’s prescribed-shock table (premium increase per $1,000 including the policy fee) R2. The shock is w(n−1), the lapse rate of the last level-period year:

J (jump ratio)

Shock lapse w(n−1) std

≤ 2.0

35%

2.0 – 4.0

55%

4.0 – 6.0

80%

6.0 – 8.0

85%

> 8.0

90%

Rationale: the study’s observed Jump-to-ART shock lapses span 27%–96% and increase with the jump ratio and attained age R4; the bucket values sit inside that envelope. The anchor cell (J ≈ 5.46 [S6]-derived) takes 80% — which coincidentally equals the VM-20 prescribed NPR shock for its 10-year level period jumping to ART with a ≥400% increase R2, but note the two are conceptually distinct (best estimate vs prescribed). Optional refinements supported by the study: +5 pts at attained ages 60+ and −15 pts for monthly-mode policies (monthly mode shows materially lower shock lapse R4) std.

PLT lapse after the shock#

Elevated but declining by PLT duration in policy years, d = dur(t) + 1 − n R4: 30% in the first post-level-term year (d = 1), 15% in the second (d = 2), 10% from the third (d ≥ 3) std, until expiry. These are annual rates and are spread over their twelve months at w_m like any other ordinary lapse.

PLT mortality deterioration (anti-selection)#

Multiplicative on the best-estimate base table:

q(t)   = q_base(x+dur(t)) · class_factor · M(d),   d = dur(t) + 1 − n ≥ 1  (i.e. dur(t) ≥ n)
M(1)   = min(8.0, 1 + 0.55 · (J − 1))          [std]
M(d)   = max(2.0, M(1) − 0.15 · (d − 1))       [std]  (grade to 200%, then level)

(M(d) = 1 during the level period, dur(t) < n. The deterioration is a multiplier on the annual rate, applied before the monthly conversion, so it grades once a year as the study measures it and not once a month.)

For the anchor cell J ≈ 5.46 gives M(1) = 3.45 ≈ 3.50 (the worked example uses 3.50). Rationale: first-year Jump-to-ART deterioration observed at 154%–1,066% of level-period mortality, increasing with the jump; deterioration declines over PLT durations, falling below 200% after roughly 10 years R4 — M(d) reaches 2.00 at d = 11 and stays level. Monthly-mode policies show lower deterioration R4; an optional 0.75 multiplier on (M(d) − 1) for monthly mode is supported std.

Anticipatory lapse#

The annual rate of policy year n−1 is set 2 points above the mid-duration level (6% vs 4% in the base vector), because lapse rates begin to rise one to two policy years before the end of the level period R6 std.

Conversion#

cv(t) = 1% per year while conv_elig, 2% in the final eligible policy year (option value is highest just before the window closes) std; zero otherwise, and spread over the months at cv_m. Anti-selective conversion interacts with PLT deterioration — converters are disproportionately impaired lives R7 scope; magnitude not recorded — so implementations linking term and permanent blocks should not apply both a conversion cost load and full PLT deterioration to the same lives (see Conversion treatment above).

Worked example#

Specimen anchor-cell model point M35 / Standard NT / $100,000 / 10-year plan / annual mode, unit in-force. Contractual premiums from the specimen guaranteed schedule: AP = $140 for the months of policy years 1–10 (t = 0 … 119), AP = $764 in policy year 11 (t = 120 … 131) and $830 in policy year 12 [S6]; J = 764/140 ≈ 5.46. Assumptions: illustrative best-estimate annual q_base rising from 0.00080 (age 35) to 0.00160 (age 44) — vector 0.00080, 0.00085, 0.00090, 0.00095, 0.00100, 0.00110, 0.00120, 0.00130, 0.00145, 0.00160 — then 0.00180/0.00200 (ages 45/46) with M(1) = 3.50, M(2) = 3.35 std; annual lapse vector 6%, 5%, 4%×6, 6% (anticipatory), 80% (shock, at month 119 in full), 30%, 15% std; commission 80%/5%/2%, premium tax 2%, maintenance $30/yr accruing monthly and inflating 1.02^(t/12), acquisition $300 std. All flows per the recursion above (premium/commission/tax/expense BOM, claims EOM, no discounting).

The months of policy year 1 (t = 0 … 11)#

The annual premium falls in month 0 and nothing else does, which is what an annual-mode policy on a monthly grid looks like: one premium, one acquisition charge, then eleven months of claims and a twelfth of the maintenance charge each.

t

l(t)

Premium G

Claims DC

Comm K

Maint+Acq E

Tax X

Net CF

l(t+1)

0

1.000000

140.00

6.67

112.00

302.50

2.80

−283.97

0.994791

1

0.994791

0.00

6.63

0.00

2.49

0.00

−9.13

0.989608

2

0.989608

0.00

6.60

0.00

2.48

0.00

−9.08

0.984453

3

0.984453

0.00

6.57

0.00

2.47

0.00

−9.04

0.979325

4

0.979325

0.00

6.53

0.00

2.46

0.00

−9.00

0.974223

5

0.974223

0.00

6.50

0.00

2.46

0.00

−8.95

0.969148

6

0.969148

0.00

6.46

0.00

2.45

0.00

−8.91

0.964099

7

0.964099

0.00

6.43

0.00

2.44

0.00

−8.87

0.959077

8

0.959077

0.00

6.40

0.00

2.43

0.00

−8.83

0.954081

9

0.954081

0.00

6.36

0.00

2.42

0.00

−8.78

0.949111

10

0.949111

0.00

6.33

0.00

2.41

0.00

−8.74

0.944167

11

0.944167

0.00

6.30

0.00

2.40

0.00

−8.70

0.939248

The month-0 decrement is the check on the rate conversion: q(0) = 0.00080 and w(0) = 0.06 give q_m = 1 − (1 − 0.0008)^(1/12) = 0.00006669 and w_m = 1 − (1 − 0.06)^(1/12) = 0.00514301, so l(1) = (1 − 0.00006669)(1 − 0.00514301) = 0.994791 ✓, and twelve such months land on l(12) = 0.939248 — the annual model’s l(1), exactly.

The same frame summed into policy years (years 1–12)#

Each row is the total of its twelve months; l at BOY is l(12(k−1)), the count entering the policy year, which is the number the annual grid carried on the same row.

Policy year

months t

l at BOY

Premium G

Claims DC

Comm K

Maint+Acq E

Tax X

Net CF

1

0–11

1.000000

140.00

77.78

112.00

329.42

2.80

−381.99

2

12–23

0.939248

131.49

77.99

6.57

28.32

2.63

15.98

3

24–35

0.891527

124.81

78.76

6.24

27.55

2.50

9.77

4

36–47

0.855096

119.71

79.73

5.99

26.95

2.39

4.65

5

48–59

0.820112

114.82

80.50

5.74

26.36

2.30

−0.08

6

60–71

0.786520

110.11

84.92

5.51

25.79

2.20

−8.30

7

72–83

0.754229

105.59

88.84

5.28

25.22

2.11

−15.86

8

84–95

0.723191

101.25

92.28

5.06

24.67

2.02

−22.79

9

96–107

0.693361

97.07

97.74

4.85

23.89

1.94

−31.36

10

108–119

0.650814

91.11

104.13

4.56

23.53

1.82

−42.92

11

120–131

0.129955

99.29

69.90

1.99

4.08

1.99

21.33

12

132–143

0.090395

75.03

56.28

1.50

3.15

1.50

12.59

Reading the table: the 80% shock lapse at the end of month 119 (the last month of policy year 10, the last level-period year) collapses in-force from 0.649859 to 0.129955; in policy year 11 the premium per survivor jumps 5.46× while expected claims per survivor reflect q = 0.00180 × 3.50 = 0.0063 — the anti-selected PLT block barely clears its own claims [pattern per R4]. Conversion is switched off (cv = 0) in this table to keep it to one decrement narrative; enabling cv(t) per the behavior section removes a further ~1%/yr of s(t) during policy years 1–10 and adds the CV(t) outflow. (This worked example uses guaranteed premiums that are contractual [S6]; every decrement/expense number is illustrative std — it is a mechanics check, not a pricing result.)

Cross-checks. The premium, commission and tax columns are the same numbers the annual grid produced, because an annual-mode premium is collected on the anniversary and weighted by the anniversary in-force under either grid: 91.11 in policy year 10 either way. Claims and expenses are not, and the direction is the informative part — policy year 11 pays 69.90 of claims against the annual grid’s 81.87, because a block losing 30% of its lives over the year is exposed for less of it than an anniversary weighting assumes, and the maintenance charge falls from 4.75 to 4.08 for the same reason. The shock arithmetic: l(120) = l(119)(1 − q_m(119)) × (1 − 0.80) = 0.649859 × (1 − 0.00013343) × 0.20 = 0.129955 ✓, which is also l(9)(1 − 0.0016)(1 − 0.80) on the annual grid, since the eleven ordinary-lapse-free months before it carry only mortality.

Valuation and reserve pointers#

This library projects gross liability cash flows. Reserve layers consume those flows but are not reproduced here:

  • VM-20 minimum reserve = seriatim NPR + max(0, DR − NPR-aggregate) etc., with term NPR on 2017 CSO, prescribed interest, prescribed lapses (6%/10% by level-period length, prescribed shock 25%–80%, 0% after final premium) and an NPR floor at the cost of insurance to the next paid-to-date; the deterministic exclusion test no longer applies to term R2 REG-R3. The DR for post-2017 issues must assume 100% lapse at the end of the level term where PLT would otherwise be profitable — PLT profits cannot be capitalized; PLT losses must be reflected R2. A projection feeding VM-20 must therefore be able to run with (a) prudent-estimate behavior per these notes and (b) the prescribed NPR/PLT-override assumption sets, from the same cash flow engine.

  • Pre-PBR in-force (A-830, the model regulation known outside the manual as “Regulation XXX”): basic reserves = max(segmented, unitary) under the contract segmentation method REG-R154 ¶21; deficiency reserves as quantity A less the basic reserve REG-R154 ¶17, with X-factor select mortality confined to the first segment REG-R154 ¶18. The valuation table is date-split, not 1980 CSO flat: 1980 CSO with elective select factors before 1 January 2004, and the 2001 CSO Mortality Table from 1 January 2004 for basic reserves, deficiency reserves and the tabular cost of insurance REG-R154 ¶¶16, 17, 23. The quantitative substrate A-830 does not restate — what a basic reserve is (¶¶11–13), the minimum reserve behind the deficiency definition (¶¶19–20) and the maximum valuation interest rates (¶¶7–10) — is A-820 REG-R153. Both appendices are now read at first hand and this pointer no longer rests on Model #830 alone R1 REG-R6.

  • Asset adequacy / cash flow testing sits under VM-30/ASOP 22 REG-R29 with ASOP 7 governing the cash flow analysis itself REG-R27 and ASOP 56 governing the model REG-R32; VM-20 practice detail in the Academy practice note REG-R23 and assumption governance in the Academy resource manual REG-R25.

  • Tax reserves: 92.81% of the NAIC-method reserve, floored at net surrender value (zero for term), capped at statutory REG-R16. GAAP/LDTI: the same projected cash flows feed the LFPB with annually updated assumptions and single-A discounting through OCI REG-R34 [unverified — source not fetched; corroborated summaries only]. Reinsurance reserve financing of XXX term: AG 48 / Model #787 REG-R11 REG-R12.

Key sensitivities and model risks#

Dominant assumptions, in rough order of economic impact for a level-term block:

  1. PLT shock lapse × mortality deterioration. These two are jointly calibrated to the premium jump R4; moving one without the other misstates the PLT tail badly. Because VM-20 forces PLT profits to zero in the DR R2, PLT optimism cannot help statutory results but PLT pessimism (deterioration above premium loadings) flows straight through.

  2. Best-estimate mortality level and slope. The level-period margin is thin (see worked example — premiums ≈ expected claims at Standard NT); a few basis points of A/E R8 REG-R19 move the block’s lifetime result materially.

  3. Level-period lapse. Term with no cash value is lapse-supported in early durations (acquisition strain recovery) and lapse-sensitive before the shock (each anticipatory lapse in policy year n − 1 — policy year 9 of a T10 — R6 forfeits a year of level premium against no benefit).

  4. Conversion incidence. Converts remove healthy-ish premium payers and (in linked models) deliver anti-selected lives to the permanent block R7; sensitivity grows with the conversion window length.

  5. Expenses/commission std matter mainly through the acquisition strain and the tiny PLT in-force tail (fixed per-policy costs on a shrinking block).

Known modeling pitfalls:

  • Shock timing double-count. Applying the shock lapse both at the end of month 12n − 1 and the start of month 12n, or spreading it across the twelve months of policy year n at w_m, changes the PLT premium base materially. The second of those is the live hazard on a monthly grid, and it is not a rounding difference: spreading an 80% annual rate lapses roughly half the block before the ART renewal notice that is the cause of the lapse, and hands a smaller block the first ART premium. The shock belongs at the single point immediately before that premium — month 12n − 1, in full R2 R4 std.

  • Jump ratio definition. Include the policy fee in both numerator and denominator — the 2021 SOA study defines the jump including the fee (the 2014 study did not) R4, and VM-20’s shock table keys on premium increase per $1,000 including the fee R2. Fee-in/fee-out inconsistency silently shifts a policy across shock buckets. The formulaic engine uses the opposite convention, so the two must not be conflated: A-830 ¶5’s segmentation ratio is on guaranteed gross premium per thousand of face amount, “ignoring policy fees only if level for the premium paying period” — and the $65 fee is level for the whole period [S6], so the fee comes out there REG-R154 ¶5. One product, two premium-ratio conventions: fee-in for behaviour and the VM-20 NPR shock R2 R4, fee-out for A-830 segmentation. At the anchor cell they differ by nearly a factor of two (≈5.46 against ≈9.32) [S6]-derived.

  • Deterioration base. M(d) multiplies the best-estimate base mortality, not the guaranteed/valuation table; applying it to 2017 CSO (already loaded R3) double-counts margin.

  • Converting the rate at the wrong point. q_m and w_m are taken from the fully loaded annual rate — base × class factor × M(d) for mortality — not from the base table before the multipliers. Converting first and multiplying after gives a different number wherever a multiplier is not 1, which is the whole post-level term: at M(1) = 3.50 the two differ by 0.21% of the monthly rate, compounding over the PLT tail.

  • Modal factor double-count. The modal load lives in the modal factor [S6], so the premium collected is factor × AP and nothing else scales it. Three quantities stay on the annualized AP and must not be modalized: the jump ratio (the shock buckets are calibrated on annualized premiums R4) and the conversion credit (contractually one annual premium [S6]). The acquisition expense must not be modalized either, being a flat per-policy charge rather than a fraction of premium.

  • ANB/ALB mismatch. Model ages, rate table lookups, and mortality tables must share the ANB basis [S2] [S3] [S5] [S6] R3; a silent ALB table import shifts mortality by half a year of age.

  • Expiry handling. The guaranteed schedule ends at attained age 95 [S6]; projecting ART premiums past 95, or terminating a year early, corrupts the tail. The correct rule is l(t) = 0 for x + dur(t) ≥ 95: the last projected month is t = 12(95 − x) − 1, the last month of the policy year running from attained age 94 to expiry at 95; testing x + dur(t) ≥ 94, or ending the frame twelve months early, drops that whole year.

  • Banding on face decrease. A requested face decrease re-scales premium excluding the fee (((a − b) × c) + b [S6]) and can cross a band boundary [S3]; implementations that re-derive band from face_amount each period handle this automatically.

Companion documents: product-spec.md (contract terms), sources.md (citations).