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. Annual steps are the default; a monthly mode is provided as an option. Annual is adequate because all decrements are contractually annual-cycle (level premiums, ART renewals at anniversaries) and there is no account value requiring monthiversary processing. Monthly mode matters when modal premium cash flow timing, mid-year claim timing, or mode-specific behavior (monthly-mode policies show materially lower shock lapse and PLT mortality deterioration R4) is in scope.

  • Timing std. Anniversary (BOY/BOM) processing: premiums and premium-linked expenses at the beginning of the period; deaths during the period with claims paid at period end; lapses, shock lapses, and conversions at period end after deaths. The shock lapse is processed at the END of the final level-period year (equivalently, immediately before the first ART premium falls due) — 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 x+t = issue age + completed policy years 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 annual effective unless subscripted m for monthly.

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

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 period t (l(1) = policy_count at issue)

d(t)

Deaths in period t

x(t)

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

c(t)

Conversions at end of period t

AP(t)

Annualized guaranteed gross premium for policy year t (from rate table + fee)

dur(t)

Policy year (curtate duration + 1)

phase(t)

LEVEL (dur ≤ n), PLT (n < dur, attained age < 95), EXPIRED

conv_elig(t)

Boolean: dur ≤ 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, age 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 persistency (modal)

Annual-mode base cell; mode mix optional

Mode affects PLT behavior only via R4 factors, optional 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, 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). 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

Policy year, t = 1, 2, …, 95 − x (annual model)

l(t)

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

q(t)

Best-estimate annual mortality at attained age x+t−1, incl. class factor and PLT multiplier

w(t)

Annual lapse rate for year t (w(n) = shock lapse)

cv(t)

Annual conversion rate (0 outside eligibility window)

AP(t)

Annualized guaranteed gross premium for year t

G(t)

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

DC(t)

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

M(d)

PLT mortality multiplier at PLT duration d = t − n

J

Initial premium jump ratio = AP(n+1)/AP(n), fee included R4 R2 convention

Decrement order and recursion (annual model)#

Deaths first, then end-of-year voluntary decrements (lapse and conversion) applied to survivors, with conversion and lapse treated as competing rates on the same survivor pool std:

d(t)  = l(t) · q(t)
s(t)  = l(t) · (1 − q(t))                     survivors to end of year t
c(t)  = s(t) · cv(t)
x(t)  = s(t) · (1 − cv(t)) · w(t)
l(t+1)= s(t) · (1 − cv(t)) · (1 − w(t))
      = l(t) · (1 − q(t)) · (1 − cv(t)) · (1 − w(t))

Termination at expiry: l(t) = 0 for x + t − 1 ≥ 95 S2 S3 S5 S6.

Cash flows (annual model, per unit in force at issue)#

G(t)  = AP(t) · l(t)                          premium, BOY  [S6 schedule]
K(t)  = k(t) · G(t)                           commission, BOY  [std]
X(t)  = 0.02 · G(t)                           premium tax, BOY  [std]
E(t)  = 300 · 1{t=1} + 30 · 1.02^(t−1) · l(t) maintenance/acquisition, BOY  [std]
DC(t) = F · d(t)                              death claims, EOY  [S6]
CV(t) = AP(t) · c(t) · 1{t>1}                 conversion credit, EOY  [S6]
NetCF(t) = G(t) − K(t) − X(t) − E(t) − DC(t) − CV(t)

Simplifications std: (i) the pro-rata unearned-premium refund on death S6 is ignored in the annual model (it is a half-premium-sized timing item on the deceased cohort; in monthly mode it becomes immaterial by construction); (ii) grace-period mechanics S3 S6 are not separately modeled — lapse is treated as effective at the anniversary; (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.

Monthly option — processing order (monthiversary)#

Monthly decrements std: q_m = 1 − (1 − q)^(1/12), w_m = 1 − (1 − w)^(1/12) for ordinary lapses; the shock lapse w(n) is NOT spread — it is applied in full at the final level-period monthiversary (month 12n). Numbered order each month:

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

  2. Collect modal premium if due this month (monthly mode: 0.08333 × AP S6); annualized modal load is implicit in the modal factor.

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

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

  5. Apply deaths at q_m; pay claims at end of month: F + pro-rata unearned premium − due unpaid premium S6.

  6. Apply conversions at cv_m if within the eligibility window; pay conversion credit S2 S3 S6 (before any lapse, matching the annual recursion’s conversion-before-lapse order).

  7. At the level-period-end monthiversary only: apply shock lapse to survivors std (per R2 R4 timing).

  8. Apply ordinary lapses at w_m to 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(n+1)/AP(n) 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:

J (jump ratio)

Shock lapse w(n) 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 R4: w(n+1) = 30%, w(n+2) = 15%, w(n+d) = 10% for d ≥ 3 std, until expiry.

PLT mortality deterioration (anti-selection)#

Multiplicative on the best-estimate base table:

q(n+d) = q_base(x+n+d−1) · class_factor · M(d)
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)

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#

w(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% while conv_elig, 2% in the final eligible year (option value is highest just before the window closes) std; zero otherwise. 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(1..10) = $140, AP(11) = $764, AP(12) = $830 S6; J = 764/140 ≈ 5.46. Assumptions: illustrative best-estimate 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; lapse vector 6%, 5%, 4%×6, 6% (anticipatory), 80% (shock), 30%, 15% std; commission 80%/5%/2%, premium tax 2%, maintenance $30 × 1.02^(t−1), acquisition $300 std. All flows per the recursion above (premium/commission/tax/expense BOY, claims EOY, no discounting).

t

l(t)

Premium G

Claims DC

Comm K

Maint+Acq E

Tax X

Net CF

l(t+1)

1

1.000000

140.00

80.00

112.00

330.00

2.80

−384.80

0.939248

2

0.939248

131.49

79.84

6.57

28.74

2.63

13.71

0.891527

3

0.891527

124.81

80.24

6.24

27.83

2.50

8.01

0.855096

4

0.855096

119.71

81.23

5.99

27.22

2.39

2.88

0.820112

5

0.820112

114.82

82.01

5.74

26.63

2.30

−1.86

0.786520

6

0.786520

110.11

86.52

5.51

26.05

2.20

−10.16

0.754229

7

0.754229

105.59

90.51

5.28

25.48

2.11

−17.79

0.723191

8

0.723191

101.25

94.01

5.06

24.92

2.02

−24.78

0.693361

9

0.693361

97.07

100.54

4.85

24.37

1.94

−34.63

0.650814

10

0.650814

91.11

104.13

4.56

23.33

1.82

−42.73

0.129955

11

0.129955

99.29

81.87

1.99

4.75

1.99

8.69

0.090395

12

0.090395

75.03

60.56

1.50

3.37

1.50

8.09

0.076321

Reading the table: the 80% shock lapse at the end of year 10 collapses in-force from 0.651 to 0.130; year-11 premium per survivor jumps 5.46× while year-11 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 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 table was computed mechanically from the recursion exactly as specified above; l(11) = 0.650814 × (1 − 0.0016) × (1 − 0.80) = 0.129955 ✓; monthly q from annual 0.0016 would be 1 − (1 − 0.0016)^(1/12) = 0.00013343 ✓.

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 year-9 anticipatory lapse 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 end of year n and start of year n+1, or spreading it across months, changes the PLT premium base materially; it belongs at the single point immediately before the first ART premium 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.

  • 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 at 94 (off-by-one on x + t − 1 ≥ 95), corrupts the tail.

  • 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).