Technical Notes

Status: Draft, 2026-08-03; cross-product [REG-R#] citations extended 2026-08-06 with the AP&P Manual appendix items read at first hand. Product sources [S#]/[R#] were accessed 2026-08-03; the [REG-R#] entries carry their own access dates per entry in sources.md. Companion to product-spec.md in this directory; both use identical parameter values. This is a standardized composite for reference modeling, not any single insurer’s product. [S#]/[R#] cite _research/variable-ul.md; [REG-R#] cites the cross-product reference library references/regulatory-and-actuarial-references.md (research provenance: _research/regulatory-actuarial.md, same R-numbering); std marks standardizations introduced for the reference implementation. Facts flagged unverified in the research notes stay flagged here.

Model scope and conventions

• Scope. Single-policy seriatim projection of gross liability cash flows for the representative VUL contract of product-spec.md, baseline without riders or no-lapse guarantee std. Reserve calculations are out of scope (see “Valuation and reserve pointers”).

• Base chassis. The monthiversary processing order (premium → withdrawal/loan → DB and NAAR → monthly deduction → growth → EOM decrements, deaths before lapses) follows the universal-life reference notes (products/universal_life/technical-notes.md). One sourced deviation from that base chassis: the VUL prospectuses define the NAAR as death benefit − account value with no one-month discount S2, so this model omits the one-month guaranteed-rate discount used in the fixed-UL base recursion.

• Projection frequency. Monthly, on policy monthiversaries; contractual daily accruals (M&E, fund expenses, fixed-option interest S1) are approximated by monthly factors std.

• Timing. Beginning-of-month (BOM) monthiversary processing: premium receipt, withdrawals, loan activity, and the monthly deduction occur at the monthiversary; investment growth accrues over the month; decrements (death, lapse/surrender) and claim/surrender payments occur at end of month (EOM), deaths before lapses std.

• Age basis. Age nearest birthday (ANB) std, consistent with the 2017 CSO ultimate ANB tables cited for guaranteed COI maxima S2 S4. Attained age advances on policy anniversaries.

• Model points. One policy per model point; decrements applied as probabilities (in-force measure l_t), not stochastic lives std. Cash flows are probability-weighted per unit policy in force at issue (or at the projection start for in-force points).

• Compliance mechanics. The corridor is enforced in the DB formula S2 R3. Guideline premium limits and 7-pay/MEC testing R3 R4 are not enforced in the baseline; premiums are assumed within limits std.

Model point attributes

Attribute	Type	Example
policy_id	str	“VUL-000001”
issue_age (ANB)	int	45
sex	enum {M, F}	M
risk_class	enum	standard nonsmoker
face_amount F_0	currency	500,000
db_option	enum {A, B}	A
s7702_test	enum {GPT, CVAT}	GPT (baseline std)
planned_premium (annualized)	currency	6,000
premium_mode	enum	monthly
premium_allocation α = (α_1, α_2, α_F)	vector, Σ = 1	(0.60, 0.40, 0.00)
duration_inforce (months, for in-force points)	int	24
initial_subaccount_values	vector	(30,000, 20,000)
initial_fixed_value	currency	0
initial_loan_balance	currency	0
nlg_rider	bool	false (baseline std)

The two-subaccount lineup (1 = equity, fund expense e_1 = 0.75% p.a.; 2 = bond, e_2 = 0.55% p.a.) is a std collapse of observed menus; observed fund-expense ranges 0.29%–1.18% S1, 0.55%–2.88% gross S2, 0.46%–2.54% S3, 0.08%–1.93% S4.

State variables

Variable	Meaning
t	policy month index (t = 0 at issue or projection start)
x_t	attained age (ANB), advancing on anniversaries
SA_{i,t}	value of subaccount i (separate account)
FA_t	fixed-option value (general account)
LA_t	loan-account (collateral) value (general account) S3
D_t	outstanding policy debt (principal + capitalized interest)
AV_t	total account value = Σ_i SA_{i,t} + FA_t + LA_t S1 S2 S3 S4
F_t	face amount (reduced by Option A withdrawals S1 S2)
DB_t	death benefit per option and corridor
NAAR_t	net amount at risk = max(0, DB_t − AV_t) S2 (floor std)
SC_t	surrender charge (per schedule, std scale)
CSV_t	cash surrender value = AV_t − SC_t − D_t S1
l_t	probability policy is in force at start of month t
status	in force / grace / lapsed / matured (age 121)

Assumption inputs

Three classes are distinguished explicitly; a model implementation should keep them in separate input structures.

(a) Contractual / guaranteed elements (from the spec; cited)

Item	Value	Basis
Premium load ceiling	6.0% of each premium	S2
COI guaranteed maxima	2017 CSO sex-distinct S/NS ultimate ANB, monthly per $1,000 NAAR; cap $83.34 (observed $83.33–$83.34)	S2 S4 R12; cap S1 S2 S3 S4
Per-policy charge	$10.00/month	S2 S4
Per-$1,000 face charge	$0.20 per $1,000 of F_0 per month (current = guaranteed std)	S2
M&E guaranteed max	0.60% p.a.	std (spec footnote 8)
Fixed-option floor	1.0% effective annual	S1
Loan charged/credited rates	2.0%/1.0% years 1–9; 1.05%/1.0% from year 10	S1
Surrender charge	$18.00 per $1,000 initial, linear to 0 over 14 years	std (spec footnote 10)
Corridor factors κ	250% (≤40), 215% (45), 185% (50), 150% (55), 130% (60), to 100% at 90–95; linear interpolation	S2 R3; interpolation std
Grace / default	default when AV − SC − D ≤ 0; 61-day grace	S1 R8
Age-121 rule	no premiums or monthly deductions after attained age 121; asset charges continue	S1 S2 S4

(b) Current non-guaranteed scales (insurer-declared; snapshot)

Governed as NGEs under ASOP No. 2 (by class; no recouping of past losses) R11.

Item	Value	Basis
Premium load — current	4.0% flat	std (spec footnote 3)
COI — current scale	input vector c_t; default placeholder 50% of guaranteed 2017 CSO; disclosed anchor male 45 std NS year 1: current $0.04 (gtd $0.22)	std (spec footnote 5); anchor S4
M&E — current	0.45% p.a.	S1
Declared fixed rate	1.0% (= floor; declared rates not public)	std (spec footnote 13)
Credits (persistency credit, expense reductions)	none in baseline	std; variations S1 S2

(c) Behavioral / experience assumptions

Item	Recommended public basis	Basis
Best-estimate mortality	2015 VBT (sex/smoker-distinct, RR tables for preferred fit), calibrated with ILEC 2012–2019 A/E experience	REG-R18 REG-R19
Base lapse/surrender	LIMRA/SOA U.S. Individual Life Persistency (2009–2013, includes VUL plans); 2015–2021 UL lapse/surrender study for modern levels (VUL not broken out separately — applied to VUL by analogy, flagged)	REG-R20 REG-R21
Premium persistency	2015–2021 UL premium persistency study (flexible-premium payment behavior); VUL by analogy	REG-R21
Dynamic behavior	fund-performance-sensitive multipliers, see “Policyholder behavior modeling”	std
Insurer expenses	$75/policy/year maintenance + 2% of premium collection expense (placeholders; internal expense assumptions are not public)	std
Decrement mortality vs COI	Note: the COI charge uses the current COI scale (class (b), revenue); the death decrement uses best-estimate mortality (this class). They must never be conflated.	std convention

VUL-specific policyholder-behavior studies were not retrieved; premium persistency and dynamic lapse for VUL remain unsourced unverified — hence the std placeholders below.

Cash flow components and recursions

Notation (defined once; used in both documents)

• t: policy month; x_t: attained age; l_t: in-force probability at BOM.

• P_t: premium paid at monthiversary t; γ: premium load rate (current 0.04).

• α_i: allocation share to account i (subaccounts i = 1,2; F = fixed).

• SA_{i,t}, FA_t, LA_t, D_t, AV_t, F_t, DB_t, NAAR_t, SC_t, CSV_t: state above.

• r_{i,t}: gross fund return of subaccount i in month t (scenario input).

• e_i: fund expense ratio (annual); m: M&E rate (annual; current 0.45% = 0.0045); i_fix: declared fixed rate; i_L, i_C: loan charged/credited rates.

• c_t: current monthly COI rate per $1,000 NAAR; e_pol = 10; e_face = 0.20.

• κ_t: corridor factor at x_t.

• q^d_t: best-estimate monthly mortality; q^w_t: monthly lapse; ρ_t: premium persistency factor.

• Monthly conversions: q^d_t = 1 − (1 − q^d,annual)^{1/12}; likewise lapse std.

Monthly processing order (monthiversary t → t+1)

1. Advance to monthiversary t; on an anniversary, advance x_t and the policy-year dependent parameters (loan tier, SC_t, corridor κ_t). If x_t ≥ 121: skip steps 2–4 and 6 (no premiums, no monthly deduction) S1 S2 S4.

2. Premium. P_t = ρ_t × planned modal premium. Load: γ·P_t to insurer. Net premium allocation: SA_{i,t} += α_i·(1−γ)·P_t; FA_t += α_F·(1−γ)·P_t.

3. Withdrawal (if modeled): reduce accounts by withdrawal + $25 fee; Option A reduces F_t proportionately S1 S2. Baseline: none std.

4. Loan activity (if modeled): new loans/repayments move value between investment options and LA_t S3; D_t accrues at i_L, LA_t at i_C, monthly compounding (1+i)^{1/12} std (contractually interest is due/capitalized annually S1).

5. Death benefit and NAAR (post-premium values):

   • Option A: DB_t = max(F_t, κ_t·AV_t)

   • Option B: DB_t = max(F_t + AV_t, κ_t·AV_t)

   • NAAR_t = max(0, DB_t − AV_t)

6. Monthly deduction.

   • COI_t = c_t · NAAR_t / 1000, with c_t ≤ min(2017 CSO max, 83.34) S2 S4 R12

   • MD_t = COI_t + e_pol + e_face·F_0/1000

   • Allocated pro rata across unloaned accounts std: each unloaned account j pays MD_t · V_j / Σ_unloaned V (loan account LA is excluded).

7. Investment growth over the month:

   • Subaccounts (unit-value dynamics): SA_{i,t+1} = SA’{i,t} · (1 + r{i,t}) · (1 − e_i/12) · (1 − m/12), where SA’ is the post-deduction value. In the contract, fund expenses and (for S1) M&E accrue daily in the unit value S1; the monthly product form is a std approximation. Insurers deducting M&E monthly S2 S3 S4 are captured by the same factor.

   • Fixed option: FA_{t+1} = FA’_t · (1 + i_fix)^{1/12}, i_fix ≥ 1.0% S1.

   • Loan account: LA_{t+1} = LA_t · (1 + i_C)^{1/12}; debt D_{t+1} = D_t · (1 + i_L)^{1/12} std monthly accrual.

8. EOM decrements and payments (deaths before lapses std; balances here are EOM values after step 7, so outstanding debt is D_{t+1}):

   • Death: probability l_t·q^d_t; claim outflow = DB_t^{EOM} − D_{t+1} (debt repaid internally) S1 S3, where DB_t^{EOM} recomputes the option/corridor formula on EOM account values std.

   • Surrender/lapse: probability l_t·(1 − q^d_t)·q^w_t; outflow = CSV_t^{EOM} = AV_{t+1} − SC_t − D_{t+1} S1.

   • Maintenance expense outflow: l_t · (75/12) std; premium expense 2%·P_t at step 2 std.

   • Survivorship: l_{t+1} = l_t · (1 − q^d_t) · (1 − q^w_t).

9. Status checks. If CSV_t ≤ 0 (and no NLG): default → grace; the baseline model lapses the policy at the next monthiversary if not cured, collapsing the 61-day grace and notice mechanics S1 R8 into a one-month lag std. At x_t = 121, switch to the age-121 regime S1 S2 S4.

Scenario requirement

Subaccount gross returns r_{i,t} are exogenous scenario inputs. The reference model runs either (a) deterministic scenarios (level or path-specified gross returns — e.g., illustration-style level returns net of specified charges), or (b) stochastic sets of gross-return paths. For statutory use, VM-20 defines a Deterministic Reserve (Section 4) and a Stochastic Reserve (Section 5), with economic scenarios addressed in its Appendix 1 R7; GAAP long-duration (LDTI) measurement consumes the same projected cash flows with different assumption-update and discounting overlays REG-R34 — source not fetched; summary-based, flagged. Declared fixed-option rates would in practice vary with general-account yields; the baseline holds i_fix at the 1.0% floor std.

Separate-account vs general-account cash flow split

Account location: SA_{i,t} are separate-account assets; FA_t and LA_t are general-account liabilities/assets S1 S3. The model reports two views:

• Gross (policyholder) view — the reference model’s primary projection std.

  • Inflow: premiums l_t·P_t (full premium; the net premium is a pass-through into the accounts, the load is insurer revenue).

  • Outflows: death claims l_t·q^d_t·(DB_t^{EOM} − D_{t+1}); surrenders l_t(1−q^d_t)q^w_t·CSV_t^{EOM}; withdrawals; insurer expenses.

• Net-of-account (general-account strain) view — derived report.

  • Insurer margins collected: premium loads γP_t, monthly deductions MD_t, M&E collected via unit values, loan spread (i_L − i_C on D_t), surrender charges SC_t on surrender.

  • Net mortality cost: l_t·q^d_t·NAAR_t^{EOM} — the general-account cost of a death after seizing the account.

  • Account transfers (memo): on death, Σ_i SA_{i,t} moves separate account → general account; FA/LA release internally; on surrender the separate account liquidates to fund CSV.

Projected output columns (per month t, per scenario; probability-weighted by l_t) std naming:

Column	Definition	View
prem_gross	l_t · P_t	gross inflow
load_income	l_t · γ · P_t	net (margin)
md_income	l_t · MD_t (COI + per-policy + per-$1,000)	net (margin)
me_income	l_t · M&E collected via unit values	net (margin)
loan_spread	l_t · (i_L − i_C) accrual on D_t	net (margin)
claim_gross	l_t · q^d_t · (DB_t^{EOM} − D_{t+1})	gross outflow
claim_net	l_t · q^d_t · NAAR_t^{EOM}	net (GA strain)
surr_outgo	l_t (1−q^d_t) q^w_t · CSV_t^{EOM}	gross outflow
sc_income	l_t (1−q^d_t) q^w_t · SC_t	net (margin)
expense	l_t · (maintenance + premium expense)	both
sa_transfer	account transfers separate ↔ general (memo)	memo
av_eop, naar, l_t	state snapshots for reconciliation	memo

Reconciliation identity (per month): net GA cash flow = load_income + md_income + me_income + loan_spread + sc_income − claim_net − expense; the gross view must reproduce it after adding back the account pass-throughs (net premiums in, account releases out) std.

Warning — a common specification error: “death benefit paid = DB − AV from the separate account” is NOT the insurer’s claim cash flow. The insurer’s liability outflow is the full death benefit (less policy debt); seizing the account value is the funding of part of that outflow, and DB − AV (= NAAR) is the net general-account strain. Projecting only DB − AV as the claim understates gross benefit outgo and breaks reconciliation with statutory exhibits; projecting full DB and separately expensing NAAR double counts. The reference model projects the gross view and derives the net view arithmetically from the same run std.

Policyholder behavior modeling

All dynamic formulas are std: no public VUL-specific dynamic-behavior study was retrieved [unverified gap], so forms are standardized with rationale, calibrated to the base tables in assumption class (c).

• Funding ratio. φ_t = AV_t / AV*_t, where AV*_t is the account value projected at issue under the pricing path (level 6% gross subaccount return, current charges, planned premiums) std. φ_t < 1 means performance/funding shortfall.

• Dynamic lapse. q^w_t = q^w,base_t · λ_t, λ_t = min(2.0, max(0.5, 1 + β·(1 − φ_t))), β = 0.5 std. Rationale: in a protection-oriented VUL a performance shortfall raises the premium required to sustain coverage, pushing marginal policyholders to lapse (and underfunded policies drift toward the default test of step 9); overfunded policies are stickier. Bounds prevent extreme extrapolation.

• Premium persistency. ρ_t = ρ^base_t · min(1.3, max(0.7, φ_t^{−δ})), δ = 0.25 std; ρ^base_t from the UL premium persistency study levels REG-R21 (placeholder grading: 1.00 year 1 → 0.85 year 5 → 0.80 thereafter std). Rationale: shortfalls induce catch-up funding by retained policyholders (φ < 1 ⇒ ρ up); strong performance induces premium holidays (φ > 1 ⇒ ρ down) — the signature flexible-premium behavior the UL studies measure REG-R21.

• Surrender at surrender-charge cliff. Optional spike multiplier on q^w in the month after SC_t reaches zero (end of year 14) std; magnitude an input.

• No dynamic mortality. Anti-selective lapse interaction (lapse-supported effects) is not modeled in the baseline std.

Worked example — one month, two subaccounts

Model point: male 45 standard nonsmoker, F_0 = 500,000, Option A, GPT; policy year 3 (SC factor 12/14); planned premium $500/month paid; allocation 60/40; no fixed balance, no debt; current scales as above (γ = 4%; c = $0.04 per $1,000 S4 — illustrative current rate at the disclosed representative point; e_1 = 0.75%, e_2 = 0.55%, m = 0.45% S1); scenario month: r_1 = +1.00%, r_2 = −0.50% gross. Premium level is illustrative only std. Corridor κ(45) = 215% S2.

Step	Item	SA_1 (equity)	SA_2 (bond)	Total AV
0	BOM balances	30,000.00	20,000.00	50,000.00
2	Premium 500.00; load 4% = 20.00; net 480.00 split 60/40	+288.00	+192.00	50,480.00
5	DB = max(500,000; 2.15 × 50,480 = 108,532.00) = 500,000.00; NAAR = 449,520.00	—	—	—
6	COI = 0.04 × 449.520 = 17.98; expense = 10.00 + 0.20 × 500 = 110.00; MD = 127.98, pro rata 60/40	−76.79	−51.19	50,352.02
7	Growth factor: (1+r)(1−e/12)(1−m/12) → SA_1: 1.0100 × 0.999375 × 0.999625 = 1.008990; SA_2: 0.9950 × 0.999542 × 0.999625 = 0.994171	×1.008990 → 30,482.82	×0.994171 → 20,023.41	50,506.23
—	Memo: M&E collected via unit values ≈ 11.44 + 7.51 = 18.95; insurer margin this month = 20.00 + 127.98 + 18.95 = 166.93	—	—	—
—	Memo: SC = 18.00 × (12/14) × 500 = 7,714.29; CSV = 50,506.23 − 7,714.29 = 42,791.94	—	—	—
—	Memo: EOM DB = 500,000.00; EOM NAAR = 449,493.77 (net GA strain if death this month; gross claim outflow = 500,000.00)	—	—	—

EOM decrements (step 8) then weight the claim, surrender, and survivorship flows by l_t·q^d_t and l_t(1−q^d_t)q^w_t; they are omitted from the table, which tracks the account recursion per policy in force.

Valuation and reserve pointers

This library projects gross liability cash flows; reserve layers are cited, not reproduced. Statutory: VM-20 minimum reserve = NPR floor plus excess of max(DR, SR) over aggregate NPR (less due/deferred premium asset); VUL without secondary guarantees is in the “All Other” reserving category (product code 080), with secondary guarantees in the ULSG category (code 090); variable life may not use the SET certification method R7. GMDB reserves per AG XXXVII; separate-account investment rules per AG XXIII (both texts still unretrieved, cited through R7); Model 270 requires reserves for variable benefits held in the separate account on a basis consistent with the Standard Valuation Law R7 R8 REG-R1 — its AP&P print, A-270, has been read but carries no reference id, so nothing is stated from it. The formulaic appendix items sitting under the NPR are A-820 REG-R153, the A-585 UL adaptation whose reach to a variable contract is unresolved REG-R155, and A-830, which excludes VUL by its own terms REG-R154. Current Valuation Manual edition: Jan. 1, 2026 (VM-01/02/20/31, VM-C/M/G/V) REG-R3. Practice guidance: ASOP 52 (VM governs in conflict) R9; AAA VM-20 practice note R10; ASOP 7 (cash flow analysis) REG-R27; ASOP 56 (model governance for this implementation itself) REG-R32. Tax reserves: greater of net surrender value and 92.81% of the NAIC-method reserve, capped at statutory REG-R16. GAAP: LDTI (ASU 2018-12) overlays measurement on the same projected cash flows REG-R34 — not fetched; summary-based, flagged.

Key sensitivities and model risks

Dominant assumptions (roughly in order):

1. Separate-account return scenario (level and volatility): drives AV, hence NAAR, COI revenue, M&E revenue, corridor DB, and the default/lapse dynamics — the defining VUL sensitivity. Results are scenario-distributions, not points.

2. Current COI scale (the 50%-of-CSO placeholder std): COI is the largest charge; disclosed year-1 current/guaranteed ratios (e.g., 0.04/0.22 S4) show the placeholder is conservative early and the select-to-ultimate shape matters.

3. Premium persistency ρ_t: flexible premiums are the UL-family assumption with the widest behavioral range REG-R21; funding level feeds back into lapse and default.

4. Lapse and dynamic lapse (λ_t): level from dated/analogous studies REG-R20 REG-R21 with the dynamic form unsourced std unverified.

5. Best-estimate mortality vs 2015 VBT/ILEC REG-R18 REG-R19; NAAR-weighted, so it interacts with the return scenario.

Known modeling pitfalls:

• Conflating COI-scale mortality (charge) with decrement mortality (experience).

• Projecting DB − AV as the death outflow (see the warning above).

• Forgetting the NAAR floor at zero, or letting corridor factors create discontinuous DB jumps at quinquennial ages instead of interpolating std.

• Applying M&E both in the unit-value factor and as a monthly deduction (double counting across insurer conventions — pick one; this model uses the unit-value factor std).

• Pro-rata deduction allocation breaking on zero unloaned balances (guard the denominator; deduction shortfall triggers the default test).

• Ignoring the loan account: loaned value earns i_C, not fund returns; debt compounds at i_L; DB and CSV are debt-reduced S1 S3.

• Missing the age-121 regime switch (charges stop; asset drags continue) S1 S2 S4.

• Grace-period collapse std accelerates lapses by up to two months versus the contractual 61-day mechanics S1 R8 — immaterial for most uses, material for short-horizon liquidity studies.

• The NLG variation changes the risk profile qualitatively (lapse floor under poor performance → higher NAAR persistence); see products/guaranteed_ul/technical-notes.md for shadow-account mechanics and S4 for the rider’s notional-load design.