Technical Notes#

Status: Draft, 2026-08-03 (underlying research accessed 2026-08-03).

Scope note: these notes specify a reference liability cash flow projection model (lifelib/modelx style) for the standardized composite products defined in product-spec.md (“RefWL-Par” participating whole life; “RefWL-FE” non-par final-expense whole life). They do not describe any single insurer’s model. [S#]/[R#] tags cite the product research file (_research/whole-life.md); [REG-R#] tags cite the cross-product reference library (references/regulatory-and-actuarial-references.md; research provenance in _research/regulatory-actuarial.md, same R-numbering). std marks standardizations introduced for the reference implementation. Parameter values are identical to those in product-spec.md.

Model scope and conventions#

  • Projection frequency: annual, on policy years (anniversary to anniversary) std. Rationale: the contract’s cash flow drivers — level annual premium, annual dividend declaration, anniversary loan-interest capitalization S1 — are all annual. No monthiversary processing is performed; monthly modal premiums would enter only as a premium-income refinement via modal factors S1 and are excluded by the annual-mode standardization (product-spec Table 2 note (f)).

  • Timing conventions std: premiums and premium-linked expenses at the beginning of the policy year (BOY); death claims, dividends, surrenders, and maturity at the end of the policy year (EOY), in the processing order given below. State variables are stored at EOY (= policy anniversary t).

  • Age basis: age nearest birthday (ANB) std (product-spec Table 1 note (a)); the 2017 CSO set provides ANB tables R8. Attained age at anniversary t is x + t.

  • Projection horizon: to the anniversary at attained age 100, where the model pays a maturity benefit and terminates std. The contract itself matures at 121 S1, but the guaranteed CV equals face at 100 and PUA CV equals PUA face at 100 S1 S3, so from age 100 the policy is economically an endowment at face; truncating at 100 changes only the timing of the terminal payment between ages 100–121 (mortality vs. maturity), not its amount per survivor.

  • Model points: single-policy model points, projected seriatim; results scale linearly in face within a band-free specification std. Amounts are U.S. dollars per policy; probabilities are per policy year.

  • Decrement model: annual rates; deaths before surrenders at EOY; dividends credited to policies in force at EOY before surrender processing std (order list below).

  • Sex-distinct rates throughout (unisex only as a variant) S1 S3.

Model point attributes#

Attribute

Type

Example

policy_id

str

“WLPAR-000001”

product

enum {WL_PAR, WL_FE_LEVEL, WL_FE_GRADED}

WL_PAR

premium_period

enum {TO_100, PAY_10, PAY_20, TO_65}

TO_100 std (product-spec Table 1 note (b); menu S1 S3)

issue_age (x)

int

45

sex

enum {M, F}

M

risk_class

enum {PREF_NT, STD_NT, TOB} std

STD_NT

face_amount (F)

float

100,000 std

annual_premium (G)

float

1,800.00 [std illustrative] (product-spec Table 2 note (c))

dividend_option

enum {CASH, REDUCE_PREM, ACCUM, PUA}

PUA (default S1 S2)

pua_rider_premium (A_t)

float per year

0.00

term_blend_target

float (0 = off)

0.00 (variant: 2 × F std)

loan_utilization

float in [0,1]

0.00 (variant: 0.20 std)

duration_inforce (t0)

int (0 for new business)

0

puaf_inforce

float (PUA face at t0)

0.00

loan_inforce

float

0.00

State variables#

Variable

Meaning

Initialization

l_t

Probability in force at anniversary t (per issued policy)

l_0 = 1

CV_t

Guaranteed cash value per policy (base), EOY t

table input; CV_{100−x} = F S1 S3

PUAF_t

Paid-up additions face in force, EOY t

PUAF_0 = puaf_inforce

PUACV_t

PUA cash value, EOY t

PUAF_t · NSP_{x+t} std

DA_t

Dividend accumulation balance (ACCUM option only)

0

L_t

Loan balance incl. capitalized interest, EOY t

L_0 = loan_inforce

DB_t

Death benefit payable on death in year t

formula below

D_t

Dividend credited at EOY t

recursion below

Assumption inputs#

The model distinguishes three assumption classes. Keeping them in separate input structures is deliberate: (a) is locked by contract, (b) is an insurer-declared snapshot that re-rates annually, (c) is the modeler’s experience basis.

(a) Contractual / guaranteed elements (from the product spec)#

Input

Value

Basis

Guarantee interest i_g

4.00%

S1; Model 808 floor R1

Guarantee mortality q^g_{x+t}

2017 CSO composite, sex-distinct, ANB

S1 R3 R8; ANB std

Guaranteed CV schedule CV_t

Table input per model point (generated on the above basis)

S1 R1; see below

Gross premium G

Model point input (level, guaranteed)

S1 S3

Loan rate i_L

6.00% fixed, in arrears

S1

Endowment/maturity

CV = F at age 100; model maturity at 100

S1 S3; truncation std

FE premium rates

Per $1,000 rate table + $36 fee

S7

FE graded DB

110% of premiums paid, natural death in years 1–2

S6 S7

(b) Current non-guaranteed scale (insurer-declared; snapshot)#

Input

Value

Basis

Dividend interest rate i_d

6.00% (2026-scale snapshot)

std, within observed 5.75%–6.60% S4 S14

Experience mortality in scale q^{sc}_{x+t}

AE^{sc} · q^{2015VBT}_{x+t} with AE^{sc} = 0.70 of 2017 CSO in the worked example

[std illustrative]; structure per S4 R6, tables REG-R18

Expense margin in scale e^{m}_t

$25 per policy per year

std

Dividend floor

D_t ≥ 0

std (dividends are non-negative distributions of surplus R6)

PUA purchase basis

NSP_{x+t} on 2017 CSO / 4%, unloaded (dividend purchases); 10% load on rider payments

std / S3 (product-spec Table 3 note (k), Riders)

Accumulation option credit rate

i_d

S2 rate declared annually; reuse of DIR std

Non-guaranteed scales are constrained in illustration use by the disciplined-current-scale and self-support / lapse-support machinery of Model 582 R2 and ASOP 24 REG-R30; the model’s “current scale” should be interpreted as a currently-payable-scale snapshot, not a projection of future scale changes.

Cash flow components and recursions#

Notation (defined once, used throughout)#

x           issue age (ANB)                     t   policy year, t = 1 … 100 − x
F           base face amount                    G   gross annual premium
i_g         guaranteed interest (4.00%)         i_d dividend interest rate (6.00%)
i_L         policy loan rate (6.00%)            v_g = 1 / (1 + i_g)
q^g_{y}     2017 CSO rate at attained age y     q^e_{y}  best-estimate rate at age y
w_t         lapse rate in policy year t         l_t  in-force probability at EOY t
CV_t        guaranteed cash value (base), EOY t
NSP_y       net single premium per 1 of paid-up (endow-at-100) WL face at age y,
            on 2017 CSO / 4%:  NSP_y = A_{y:(100−y)|}  (endowment insurance to 100)
ä_{y:n|}    annuity-due, n years, on 2017 CSO / 4%
D_t         dividend credited at EOY t          PUAF_t, PUACV_t  PUA face / cash value
DA_t        dividend accumulation balance       L_t  loan balance at EOY t
DB_t        death benefit for deaths in year t  E_t  expense outgo in year t

Guaranteed cash value: conceptual formula and practical treatment#

Conceptual (Standard Nonforfeiture Law minimum, adjusted-premium / nonforfeiture-net-level- premium method) R1:

NNLP      = F · NSP_x / ä_{x:(100−x)|}                       (net level premium, NF basis)
EA        = 0.01 · F + 1.25 · min(NNLP, 0.04 · F)            (expense allowance)  [R1]
P_adj     such that  P_adj · ä_{x:m|} = F · NSP_x + EA       (m = premium period)  [R1]
CV_t^min  = F · NSP_{x+t} − P_adj · ä_{x+t:(m−t)|}           (t < m; second term 0 for t ≥ m)

on 2017 CSO / 4% S1 R1 R3. Properties to verify: CV_{100−x}^min = F (since NSP_100 = 1), and smooth progression by duration R1.

Practical treatment std: the reference implementation reads CV_t (per $1,000 of face) from a table input, because contractual CV tables are policy-form documents not publicly available for the surveyed carriers (research gap noted in _research/whole-life.md). The shipped table is generated from the formula above; an implementer replacing it with a carrier table changes no other logic. Contractual CV_t ≥ CV_t^min always R1.

Dividend recursion (three-factor contribution formula)#

Anchor (published mechanics, Northwestern Mutual) S4:

D_t = ( CV_{t−1} + G − MEC_t ) · (1 + i_d) − CV_t

where MEC_t is the mortality-and-expense charge based on actual company results — i.e., the dividend is the excess of an experience-basis accumulated value over the guaranteed value S4.

Reference parametrization std (exact carrier factor formulas are proprietary; this is the classic three-factor contribution decomposition consistent with S4 and the contribution principle R6):

D_t = D^int_t + D^mort_t + D^exp_t ,   floored at 0
D^int_t  = (i_d − i_g) · (CV_{t−1} + NP_g)                       (interest margin)
D^mort_t = (q^g_{x+t−1} − q^{sc}_{x+t−1}) · (F − CV_t)           (mortality margin)
D^exp_t  = e^m_t                                                  (expense margin)

with NP_g = NNLP (the nonforfeiture net level premium, so the interest margin applies to the guaranteed fund including the year’s net premium) std, q^{sc} the scale’s experience mortality (class (b)), and e^m_t the per-policy expense margin (class (b)). Dimensions: every term is dollars per policy per year. Refinements observed in practice — interest on the mortality margin, premium-timing adjustments, banded factors S1 S3 — are absorbed into the calibration of q^{sc} and e^m_t std.

Dividends on the PUA block (PUAs are dividend-eligible S14) std:

D^PUA_t = (i_d − i_g) · PUACV_{t−1} + (q^g_{x+t−1} − q^{sc}_{x+t−1}) · (PUAF_{t−1} − PUACV_{t−1})

No dividend is credited for policy year 1 (D_1 = D^PUA_1 = 0) std (product-spec Table 3 note (j); Guardian pays none S1, MassMutual pays a first-year dividend S3).

Direct recognition (loaned values) std parametrization of S1 S3: replace i_d with i_L on the loaned portion:

D^int_t (adjusted) = (i_d − i_g) · (CV_{t−1} + NP_g − L_{t−1}) + (i_L − i_g) · L_{t−1}

With i_L = 6.00% S1 and the snapshot i_d = 6.00% std the adjustment is zero — a coincidence of the snapshot, not a model property.

Dividend application (by option)#

  • PUA (default S1 S2): ΔPUAF_t = (D_t + D^PUA_t) / NSP_{x+t}; PUAF_t = PUAF_{t−1} + ΔPUAF_t; PUACV_t = PUAF_t · NSP_{x+t} std (valuing all PUA face at the attained-age NSP on the guarantee basis; exact at issue of each layer and at age 100, approximate between std). At age 100, NSP_100 = 1 so PUACV = PUAF S1.

  • CASH: dividend paid out; policyholder cash flow at EOY.

  • REDUCE_PREM: offsets next year’s BOY premium: G^{net}_{t+1} = max(G − D_t, 0), excess to PUAs std (excess-to-PUA per MassMutual RPD S3).

  • ACCUM: DA_t = DA_{t−1} · (1 + i_d) + D_t; balance adds to death and surrender proceeds S1 S2.

PUA rider (in-scope rider)#

Rider payment A_t (BOY, within limits set at issue S3 S11): ΔPUAF^rider_t = A_t · (1 − 0.10) / NSP_{x+t−1} — 10% load std from the observed 7.5%–10% range S3. Rider PUAs merge into PUAF_t.

Term-blend rider (in-scope rider, simplified std)#

Target face TF = 2 F std (within observed caps: ≤ 9× base S2, ≤ 300% of base S3). Each year, OYT face = max(TF − F − PUAF_t, 0); the dividend first pays the OYT cost q^{sc}_{x+t} · OYT_t · v_g std, remainder buys PUAs; crossover when PUAF_t ≥ TF − F, after which the rider is pure PUA S2 S3 S11. Death benefit while blended: TF + excess PUAs − L_t.

Benefit amounts#

DB_t   = F + PUAF_{t−1} + DA_{t−1} − L_{t−1}                 (PUA/ACCUM components as elected)
CSV_t  = CV_t + PUACV_t + DA_t − L_t                          (surrender value, EOY t)
MAT    = F + PUAF_T + DA_T − L_T   at T = 100 − x             (model maturity [std])

DB per the contractual formula S1, reduced to modeled components std. Deaths in year t are assumed to occur at EOY before the year-t dividend is credited, so DB_t carries the prior year’s PUA face std (terminal-dividend and premium-refund items not modeled, product-spec Table 3 note (m)).

Annual processing order (policy year t, per unit in force l_{t−1})#

  1. BOY: collect gross premium G (if t ≤ premium period) and PUA rider premium A_t; pay premium tax and acquisition/maintenance expense E_t.

  2. BOY: apply REDUCE_PREM offset from D_{t−1} if elected.

  3. During year: interest accrues implicitly (CV table on i_g S1; loan at i_L S1).

  4. EOY — deaths: probability q^e_{x+t−1}; outgo q^e_{x+t−1} · l_{t−1} · DB_t.

  5. EOY — loan interest capitalization: L_t = L_{t−1} · (1 + i_L) less repayments S1.

  6. EOY — dividend: credit D_t + D^PUA_t to survivors (from t = 2 std); apply per dividend option; update PUAF_t, PUACV_t, DA_t.

  7. EOY — surrenders: probability w_t applied to survivors l_{t−1} · (1 − q^e_{x+t−1}); outgo = CSV_t per surrendering policy.

  8. Update in force: l_t = l_{t−1} · (1 − q^e_{x+t−1}) · (1 − w_t).

  9. At T = 100 − x: pay MAT · l_T; terminate std.

Ordering (deaths → dividend → surrenders at EOY) is std; it makes surrender values include the just-credited dividend, consistent with anniversary processing.

Net liability cash flow (per issued policy, year t)#

NetCF_t = − G^{net}_t · l_{t−1} − A_t · l_{t−1} + E_t · l_{t−1}          (BOY items, sign: outgo +)
          + q^e · l_{t−1} · DB_t + w_t · l_{t−1}(1 − q^e) · CSV_t        (EOY benefits)
          + D^{cash}_t · l_{t−1}(1 − q^e) + MAT · l_T · 1{t=T}           (cash dividends, maturity)

Internal dividend applications (PUA, ACCUM, REDUCE_PREM) are not cash flows when credited; they emerge later through DB, CSV, and MAT std. Loans are modeled on the offset view: see next.

Loans (offset treatment — brief)#

Base run: loan_utilization = 0. Variant std: L_t = 0.20 · CV_t maintained by borrowing/repaying at EOY; borrowed amounts are policyholder cash outflows from the insurer, loan interest received is an inflow, and DB/CSV/MAT are net of L_t S1 S3 S9. Under direct recognition the dividend adjustment above applies S1 S3. Economically the loan is an offsetting asset; the reference model reports gross liability flows plus a separate loan account rather than netting into a “net amount at risk” presentation std.

RefWL-FE variant deltas#

  • Premium: G = (F/1000) · rate(x, sex, tobacco) + 36 S7; no dividends (non-par unverified; modeled non-par).

  • Graded plan: for natural-cause deaths in years 1–2, DB_t = 1.10 · (cumulative premiums paid); accidental deaths pay F from day 1 S6 S7. Accidental split requires an accidental-death fraction of q^e std (reference value 3% of deaths std).

  • Maturity at age 100 (120 in FL — not modeled std) pays F − L_T S8.

  • CV schedule: reuse of the par nonforfeiture machinery std (product-spec Table 5 note (r)).

  • Lapse: FE simplified-issue business lapses higher than par WL; reference schedule 12% year 1, 10% year 2, grading to 6% level by year 5 std (no FE-specific study in the research base; flagged as an open issue).

Policyholder behavior modeling#

Base behavior is static (schedules in class (c)). Dynamic overlays, all std:

  • Interest-sensitive lapse multiplier (for scenario runs): w_t^dyn = w_t · min(1 + 2.0 · max(0, r^{cmp}_t − i_d − 0.01), 3.0) where r^{cmp}_t is the competitor/market rate in the scenario. Rationale: par WL cash values are liquid at book value, so sustained rate spreads induce excess surrender; the low base level reflects the strong persistency of dividend-paying WL. Calibration is judgmental std — the research base records no dynamic-lapse study for WL.

  • Premium offset behavior: once D_t ≥ G (dividend covers the premium), a fraction 0.50 std of policyholders switch to REDUCE_PREM/premium-offset behavior (offset is a real product feature: Guardian option S S2; MassMutual APO S3). This shifts premium income to internal dividend application in later durations.

  • Loan utilization: static 0%/20% variants only std; no dynamic loan take-up (the 6%-fixed direct-recognition design largely neutralizes loan arbitrage S1 S3).

  • No dynamic mortality (anti-selection) on lapse for the base par product std; selective-lapse mortality loading is documented mainly for term post-level-period designs (see the SOA persistency/PLT study family around REG-R20), not level-premium par WL.

Worked example#

Single-year walk-through of the core recursion: RefWL-Par, male Standard NT, x = 45, F = 100,000 std, G = 1,800 [std illustrative], PUA dividend option, no rider, no loan. Policy year t = 10 (attained age 55 at EOY). All table values are illustrative std (the shipped CV/NSP tables are generated on 2017 CSO / 4% as specified above); i_g = 4.00% S1, i_d = 6.00% std.

Step

Item

Formula

Value

1

Guaranteed CV, BOY (EOY 9)

CV_9 (table)

9,500.00 std

2

Guaranteed CV, EOY

CV_10 (table)

11,200.00 std

3

Net level premium (NF basis)

NP_g

1,300.00 std

4

Guarantee mortality, age 54

q^g_54

0.00320 std

5

Scale mortality, age 54

q^{sc}_54 = 0.70 · q^g_54

0.00224 std

6

Interest margin

(0.06 − 0.04) · (9,500 + 1,300)

216.00

7

Mortality margin

(0.00320 − 0.00224) · (100,000 − 11,200)

85.25

8

Expense margin

e^m_10

25.00 std

9

Dividend

D_10 = 216.00 + 85.25 + 25.00

326.25

10

NSP at age 55

NSP_55 (table)

0.42 std

11

PUA face purchased

ΔPUAF = 326.25 / 0.42

776.79

12

PUA face, EOY (prior 4,100.00 std)

PUAF_10 = 4,100.00 + 776.79

4,876.79

13

PUA cash value, EOY

PUACV_10 = 4,876.79 × 0.42

2,048.25

14

Death benefit for year 11 deaths

F + PUAF_10

104,876.79

15

Surrender value, EOY 10

CV_10 + PUACV_10

13,248.25

(For clarity the PUA-block dividend D^PUA_10 is omitted from this table; in the model it adds (0.02 · PUACV_9) + (0.00096 · (PUAF_9 − PUACV_9)) to the amount in step 9 std.)

Valuation and reserve pointers (brief)#

This library projects gross liability cash flows; statutory, tax, and GAAP measurement are separate layers, cited not reproduced:

  • Statutory: Standard Valuation Law root REG-R1, codified in the AP&P Manual as Appendix A-820 and now read in full — ¶11 CRVM, ¶¶7–10 the valuation interest rate, ¶16 the aggregate nonforfeiture floor, ¶¶19–20 deficiency reserves, ¶¶24 and 27 the formulaic/PBR boundary REG-R153; A-830 likewise REG-R154, though ¶3.b routes no calculation paragraph to a level-premium level-benefit whole life. Both were “not retrieved” behind the VM-A index entry REG-R110 and no longer are. For issues on/after 2020-01-01 — a date that is the PBR accreditation year, the statutory-law trigger A-820 ¶¶3–4 prints being 1 January 2017 — VM-20 minimum reserve = f(net premium reserve, deterministic reserve, stochastic reserve) with exclusion tests; seriatim NPR on 2017 CSO; traditional par WL typically passes the deterministic exclusion test (valuation net premiums ≤ guaranteed gross premiums) and many WL blocks hold NPR only R3. Small companies under the Life PBR Exemption (< $300M) value under VM-A/VM-C (pre-PBR CRVM) R3. ASOP 52 governs the actuary’s PBR work REG-R31.

  • Tax: IRC §807 — greater of net surrender value and 92.81% of the CRVM/VM reserve, capped at statutory REG-R16; the statutory engine plus a haircut/cap wrapper.

  • GAAP: LDTI (ASU 2018-12) rewrites long-duration GAAP (annually updated cash flow assumptions, single-A discounting through OCI) REG-R34 — not fetched; characterization corroborated only by secondary summaries. Same projected cash flows, different measurement overlay — the reason projection and measurement are separated in this library.

  • Model governance: ASOP 56 (modeling) REG-R32 and, for cash-flow analysis engagements, ASOP 7 REG-R27 — listed in the regulatory bibliography frame validation/documentation expectations for the implementation itself.

Key sensitivities and model risks#

Dominant assumptions (in typical order of impact on par WL liability value):

  1. Dividend scale vs. guarantee spread (i_d − i_g, mortality margin, expense margin): drives dividends, hence PUA growth, hence death benefit and surrender value trajectories — compounding because PUAs themselves earn dividends S14. The DIR snapshot is a declared, changeable rate (observed 5.75%–6.60% for 2026 alone S4 S14); scale-change dynamics are a scenario input, not a model constant.

  2. Best-estimate mortality (level and improvement vs. 2015 VBT REG-R18, A/E per ILEC R9): sets both claim outgo and the mortality margin of the dividend; note the same table family feeds two places with opposite signs — a consistency trap.

  3. Lapse: low and level for par WL, but long-duration liabilities are convex in lapse; illustration regulation exists precisely because lapse-supported scales misstate value R2. Verify the model is not inadvertently lapse-supported when testing dividend scales.

  4. Expense inflation on per-policy maintenance for a product with 55+-year horizons.

  5. Loan utilization under direct recognition S1 S3: shifts dividend composition and net cash flow timing; the fixed-6%/DIR-6% snapshot coincidence (zero adjustment) will not survive a scale change.

Known modeling pitfalls:

  • CV-table vs. first-principles mismatch: if the CV table input and the NSP/annuity functions come from different bases, PUACV ≠ PUAF at age 100 and the dividend recursion leaks. Regenerate all guarantee-basis quantities from one 2017 CSO / 4% source S1 R1 R8.

  • Dividend floor and negative margins: with D_t floored at 0 std, adverse experience does not claw back — asymmetry matters in stochastic runs.

  • First-dividend timing (year 1 vs 2) shifts early-duration PUA compounding; it is a real cross-carrier difference S1 S3, keep it a parameter.

  • MEC administration on limited-pay variants: 10-pay premiums approach 7-pay limits; face decreases can retroactively create MECs and PUA-rider payments consume 7-pay room R5 S3 S1. The reference model does not police §7702/§7702A limits R4 R5 — flag model points that would fail rather than silently projecting them std.

  • Truncation at age 100 std is exact for surrender/maturity amounts but reallocates age-100–121 payments from death to maturity; do not use the truncated model for mortality-timing-sensitive measures beyond age 100 S1.

  • State variations (FL maturity 120, WA face minimums, ND suicide, MT unisex) S6 S7 S8 S1 are not modeled; the reference is a generic-state contract std.