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: monthly, on policy months std. The contract’s annual drivers stay annual and are processed at the anniversary — the dividend declaration and the capitalization of loan interest [S1] — and the guaranteed cash value schedule is an anniversary table. What the monthly grid adds is everything that is not contractually annual: modal premium collection, which retires the annual-mode standardization of product-spec Table 2 note (f) now that the grid can express the sourced modal factors [S1] [S7]; death claims settled in the month of death rather than at the anniversary; monthly expense accrual; and a surrender value that means something between anniversaries. An annual step remains a 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: t is 0-based and counts policy months. t = 0 is the issue month, period t runs from time t to time t + 1, and the frame is t = 0 … T − 1 with T = 12(100 − 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, never indexed by — “policy year 10” below is the months t = 108 … 119. The anniversary is the end of the last month of a policy year, t ≡ 11 (mod 12), and that is where every annual event lands. An in-force model point opens at t = 12·t0, t0 = duration_inforce being the policy years already elapsed, so the frame always opens on an anniversary. A state variable subscripted t is its value at the end of month t; the value entering month t is the closing value of month t − 1, or the model point’s opening state at the first projected month, and an annual quantity reads the balance entering its policy year, twelve months back. The one exception is l_t, the in-force probability, which is read at the start of month t so that it weights that month’s cash flows.

  • Timing conventions std: premiums and premium-linked expenses at the beginning of the month (BOM), in the months the elected mode makes them due; death claims and surrenders at the end of the month (EOM); dividends, loan-interest capitalization and maturity at the anniversary, in the processing order given below. State variables are stored at EOM.

  • Decrement conversion std: every rate in this product is published and tabulated annually, and the monthly rates the recursion applies are derived from them at the constant-force conversion, q_m = 1 − (1 − q)^(1/12) and w_m = 1 − (1 − w)^(1/12), so that twelve months compound back to exactly the annual rate.

  • Age basis: age nearest birthday (ANB) std (product-spec Table 1 note (a)); the 2017 CSO set provides ANB tables R8. The attained age entering the policy year of month t is x + dur(t); at the anniversary that ends it, x + dur(t) + 1. Age changes on the anniversary, not on the birthday and not monthly.

  • 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. The last projected month is t = 12(100 − x) − 1.

  • 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, applied monthly at the converted rates above; deaths before surrenders at EOM; the dividend is credited to the policies in force at the anniversary, after that month’s deaths and before its surrenders 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])

premium_mode

enum {A, SA, Q, M}

A (the anchor cell; the mode drives which months collect an instalment, at the sourced modal factors [S1] [S7])

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, policy years elapsed (0 for new business); the frame opens at month 12·t0

0

puaf_inforce

float (PUA face entering t = 12·t0)

0.00

loan_inforce

float

0.00

State variables#

Variable

Meaning

Initialization

l_t

Probability in force at the start of month t (per issued policy)

l_0 = 1

CV_t

Guaranteed cash value per policy (base), EOM t

anniversary table input, straight-line between anniversaries std; CV_{T−1} = F at T = 12(100 − x) [S1] [S3]

PUAF_t

Paid-up additions face in force, EOM t

opening balance at t = 12·t0 is puaf_inforce

PUACV_t

PUA cash value, EOM t

PUAF_t · NSP_m(t) std, NSP_m the same straight line

DA_t

Dividend accumulation balance (ACCUM option only), EOM t

opening balance 0; accrues at (1 + i_d)^{1/12} monthly

L_t

Loan balance incl. capitalized interest, EOM t

opening balance at t = 12·t0 is loan_inforce

DB_t

Death benefit payable on death in month t

formula below

D_t

Dividend credited at the anniversary ending month t; 0 in every other month

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+dur(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+dur(t)}

AE^{sc} · q^{2015VBT}_{x+dur(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+dur(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   month index, t = 0 … 12(100 − x) − 1
dur(t)      completed policy years, t // 12         (policy year dur(t) + 1)
F           base face amount                    G   gross annual premium
P(t)        modal instalment collected at BOM t: modal factor x G^net in a due month [S1][S7]
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
            (both annual; q_m = 1 − (1 − q)^(1/12) is what the month applies)
w_t         annual lapse rate of the policy year of month t;  w_m likewise
l_t         in-force probability, BOM of t
CV_t        guaranteed cash value (base), EOM t: the anniversary table, straight-lined
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)
NSP_m(t)    the same, straight-lined from NSP_{x+dur(t)} to NSP_{x+dur(t)+1}  [std]
ä_{y:n|}    annuity-due, n years, on 2017 CSO / 4%
D_t         dividend credited at the anniversary ending month t (0 otherwise)
PUAF_t, PUACV_t  PUA face / cash value          DA_t  dividend accumulation balance
L_t         loan balance at EOM t
DB_t        death benefit, deaths in month t    E_t  expense outgo in month t
            (every state subscript t is an end-of-month t value)

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+1} − P_adj · ä_{x+t+1:(m−t−1)|}     (second term 0 once t ≥ m − 1)

on 2017 CSO / 4% [S1] R1 R3, written at the anniversary that ends policy year k, the formula’s t being k − 1. Properties to verify: the schedule reaches F at attained age 100 (since NSP_100 = 1), and smooth progression by duration R1.

Practical treatment std: the reference implementation reads the schedule (per $1,000 of face) from a table input keyed by policy year, 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 ≥ CV^min always R1.

Between anniversaries std. The schedule is printed at anniversaries only, so on a monthly grid CV_t is straight-lined:

CV_t = CV^anniv_{dur(t)} + (CV^anniv_{dur(t)+1} − CV^anniv_{dur(t)}) · (t mod 12 + 1)/12

with CV^anniv_k the schedule’s policy-year-k row and CV^anniv_0 = 0. It is exact at the anniversary, where t mod 12 = 11, so no anniversary quantity moves against the annual grid — not the dividend’s interest margin, not the net amount at risk, not the anniversary surrender value. Pro-rating for elapsed time is the ordinary policy-form convention and is what makes a mid-year surrender value mean anything; the Standard Nonforfeiture Law requires an adjustment for lapse of time in any case R1. NSP is straight-lined the same way, as NSP_m(t), so the paid-up-additions block is valued on the same clock.

Dividend recursion (three-factor contribution formula)#

Anchor (published mechanics of one surveyed carrier) [S4]:

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

with CV_{t−12} the guaranteed cash value entering the policy year (zero at issue), and 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        (t an anniversary month)
D^int_t  = (i_d − i_g) · (CV_{t−12} + NP_g)                      (interest margin)
D^mort_t = (q^g_{x+dur(t)} − q^{sc}_{x+dur(t)}) · (F − CV_t)     (mortality margin)
D^exp_t  = e^m_t                                                  (expense margin)

evaluated once a year, at the anniversary, and zero in every other month: the declaration is annual [S1] and the monthly grid does not make it monthly. CV_{t−12} is the guaranteed cash value entering the policy year the anniversary closes (zero at issue), which is the annual grid’s CV_{t−1} under the month index. 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} is the scale’s experience mortality (class (b)) at the annual rate, 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−12} + (q^g − q^{sc}) · (PUAF_{t−12} − PUACV_{t−12})

on the block entering the policy year the anniversary closes, valued at NSP_{x+dur(t)}. No dividend is credited for policy year 1 std (product-spec Table 3 note (j); one carrier pays none [S1], another 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−12} + NP_g − L_{t−12}) + (i_L − i_g) · L_{t−12}

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]): at the anniversary, ΔPUAF_t = (D_t + D^PUA_t) / NSP_{x+dur(t)+1}; PUAF_t = PUAF_{t−1} + ΔPUAF_t; PUACV_t = PUAF_t · NSP_m(t) std (the purchase falls at the anniversary, so it is priced at the attained age reached there; valuing all PUA face at the attained-age NSP on the guarantee basis is 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 the anniversary.

  • REDUCE_PREM: offsets the next policy year’s premium: G^{net} = max(G − D_{prev anniv}, 0), and the modal instalments then split what is left; excess to PUAs std (excess-to-PUA per one carrier’s reduce-premium option [S3]).

  • ACCUM: DA_t = DA_{t−1} · (1 + i_d)^{1/12} + D_t; the balance accrues monthly and the dividend lands on it at the anniversary, so twelve months compound to exactly the annual credit; the balance adds to death and surrender proceeds [S1] [S2].

PUA rider (in-scope rider)#

Rider payment A per year (within limits set at issue [S3] [S11]), billed with the base premium and so collected on the same modal cycle std: ΔPUAF^rider_t = (modal instalment of A) · (1 − 0.10) / NSP — priced at NSP_{x+dur(t)} for the instalment that opens a policy year and at NSP_m(t−1) for the ones inside it — with a 10% load std from the observed 7.5%–10% range [S3]. Rider PUAs merge into PUAF_t, so a monthly-mode rider buys twelve small layers through the year rather than one at the anniversary.

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

Target face TF = 2 F std (within observed caps: ≤ 9× base [S2], ≤ 300% of base [S3]). It is a one-year term layer: it is bought at the anniversary that closes a policy year and is level through that year, so OYT is constant within the policy year and the cost is charged once, at the anniversary. OYT face = max(TF − F − PUAF_{prev anniv}, 0) on the block entering the policy year (writing it on the block leaving the year is circular — see model.md); the dividend first pays the OYT cost q^{sc}_{x+dur+1} · OYT · v_g std, remainder buys PUAs; crossover when PUAF ≥ TF − F, after which the rider is pure PUA [S2] [S3] [S11]. Death benefit while blended: TF + excess PUAs − L.

Benefit amounts#

DB_t   = F + PUAF_{t−1} + OYT_t + DA_{t−1} − L_{t−1}          (PUA/ACCUM/OYT as elected)
CSV_t  = CV_t + PUACV_t + DA_t − L_t                          (surrender value, EOM t)
MAT    = F + PUAF_{T−1} + DA_{T−1} − L_{T−1}                  (at t = T − 1,
                                                               T = 12(100 − x);
                                                               model maturity [std])

DB per the contractual formula [S1], reduced to modeled components std. Deaths in month t are assumed to occur at EOM and, at an anniversary, before that month’s dividend is credited, so DB_t carries the PUA face, accumulation balance and loan entering the month std (terminal-dividend and premium-refund items not modeled, product-spec Table 3 note (m)).

Monthly processing order (month t, per unit in force l_t)#

  1. BOM: collect the modal instalment of the gross premium G^net (if dur(t) < premium period m) and of the PUA rider premium A, in the months the mode makes them due; pay premium tax on what was collected and the month’s twelfth of the maintenance expense, with the acquisition expense in month 0 only.

  2. BOM: the REDUCE_PREM offset is applied to the policy year’s annual premium, from the dividend credited at the anniversary that opened it; the instalments split what is left.

  3. During the month: interest accrues implicitly (CV schedule on i_g [S1], straight-lined between anniversaries; loan at i_L [S1], capitalized annually).

  4. EOM — deaths: probability q^e_m; outgo q^e_m · l_t · DB_t.

  5. Anniversary only — loan interest capitalization: L = L_{prev anniv} · (1 + i_L) less repayments [S1].

  6. Anniversary only — dividend: credit D_t + D^PUA_t to survivors (from policy year 2 std); apply per dividend option; update PUAF_t, PUACV_t, DA_t.

  7. EOM — surrenders: probability w_m applied to survivors l_t · (1 − q^e_m); outgo = CSV_t per surrendering policy.

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

  9. At t = T − 1, T = 12(100 − x): pay MAT to the survivors l_{T−1} · (1 − q^e_m) — w = 0 through the final policy year, so nobody surrenders out of it — and terminate std.

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

Annual equivalence. Because the monthly rates compound back to their annual values and every annual event stays on the anniversary, twelve months of step 8 collapse to l_{t+12} = l_t · (1 − q^e) · (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 so is every anniversary-dated quantity: the guaranteed cash value, the dividend and all three of its margins, the paid-up additions purchased and in force, and the anniversary surrender and death benefit amounts. What differs is the cash flows, which is the point of the finer grid.

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

NetCF_t = − P(t) · l_t − A(t) · l_t + E_t · l_t                    (BOM items, sign: outgo +)
          + q^e_m · l_t · DB_t + w_m · l_t(1 − q^e_m) · CSV_t      (EOM benefits)
          + D^{cash}_t · l_t(1 − q^e_m)                            (cash dividends)
          + MAT · l_t(1 − q^e_m) · 1{t = T−1}                      (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 as the cash value moves, so on the monthly grid the advances spread through the year while the interest still capitalizes once, at the anniversary. 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 policy years 1–2 — the months t = 0 … 23 — DB_t = 1.10 · (cumulative premiums paid to BOM t), which on the monthly grid now grows with each instalment rather than once a year; 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−1} [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% in policy year 1, 10% in year 2, grading to 6% level by year 5 std, annual rates spread over their months at w_m (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 the prior anniversary’s dividend covers the annual premium, a fraction 0.50 std of policyholders switch to REDUCE_PREM/premium-offset behavior (offset is a real product feature: a lettered dividend option at one carrier [S2]; a named automatic offset option at another [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], annual mode, PUA dividend option, no rider, no loan. Policy year 10 — the months t = 108 … 119, with the anniversary at the end of month 119 (attained age 55 there). 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.

Every step below is an anniversary quantity, and the monthly grid leaves anniversary quantities exactly where the annual grid put them — which is why the walk-through is unchanged by the change of grid. What the grid changed is the index each one is read at: CV_8 is now CV_107, the close of the month before the policy year opens, and CV_9 is CV_119.

Step

Item

Formula

Value

1

Guaranteed CV entering the policy year

CV_107 (table, policy year 9)

9,500.00 std

2

Guaranteed CV at the anniversary

CV_119 (table, policy year 10)

11,200.00 std

3

Net level premium (NF basis)

NP_g

1,300.00 std

4

Guarantee mortality, age 54

q^g_54 (annual)

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

25.00 std

9

Dividend

D_119 = 216.00 + 85.25 + 25.00

326.25

10

NSP at age 55

NSP_55 (table)

0.42 std

11

PUA face purchased

ΔPUAF_119 = 326.25 / 0.42

776.79

12

PUA face at the anniversary (prior 4,100.00 std)

PUAF_119 = 4,100.00 + 776.79

4,876.79

13

PUA cash value at the anniversary

PUACV_119 = 4,876.79 × 0.42

2,048.25

14

Death benefit for deaths in month t = 120

F + PUAF_119

104,876.79

15

Surrender value at the anniversary, EOM t = 119

CV_119 + PUACV_119

13,248.25

(For clarity the PUA-block dividend D^PUA is omitted from this table; in the model it adds (0.02 · PUACV_107) + (0.00096 · (PUAF_107 − PUACV_107)) to the amount in step 9 std.)

The steps the monthly grid adds sit between those anniversaries. The mortality rate applied in each of the twelve months is q^e_m = 1 − (1 − 0.70 × 0.00320)^{1/12} = 0.00018686, and twelve of them compound back to 1 − (1 − 0.00018686)^{12} = 0.00224 exactly — the annual rate of step 5’s basis. The guaranteed cash value climbs from 9,500.00 to 11,200.00 in twelve straight-line steps of 141.67, so a surrender in month t = 113 (the sixth month of the policy year) is valued at CV_113 = 9,500 + 6 × 141.67 = 10,350.00 plus the paid-up additions at NSP_m(113), rather than at either anniversary’s figure. And the annual premium of 1,800.00 is still collected once, in month 108, because the anchor cell is annual mode; model point 15 is the same policy on monthly mode, which collects 0.085833 × 1,800 = 154.50 in each of the twelve.

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. On the monthly grid it compounds continuously, 1.02^{t/12}, rather than stepping at anniversaries.

  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 (policy year 1 vs 2) shifts early-duration PUA compounding; it is a real cross-carrier difference [S1] [S3], keep it a parameter.

  • Making an annual event monthly. The dividend declaration, the capitalization of loan interest and the purchase of paid-up additions are annual by contract [S1]. Crediting a twelfth of the dividend each month, or capitalizing loan interest monthly, changes the answer and is not a refinement — it is a different product. On the monthly grid these events belong on the anniversary month and nowhere else.

  • Converting the rate at the wrong point. q_m and w_m are taken from the fully loaded annual rate, and the annual rate is what the dividend’s mortality margin uses: q^g − q^{sc} in D^mort is an annual margin on an annual net amount at risk, and substituting monthly rates there understates the dividend twelvefold.

  • Interpolating the wrong quantity. CV and NSP are straight-lined between anniversaries std, which is exact at the anniversary. Interpolating PUAF as well would double-count — the block is a step function that only changes when a purchase is made — and interpolating the dividend would make it monthly (see above).

  • 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.