Technical Notes#

Status: Draft, 2026-08-04 (all cited sources accessed 2026-08-04).

Scope note. These notes specify a reference liability cash-flow projection model for the standardized composite product defined in product-spec.md (same directory). This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in _research/fixed-deferred-annuity.md; [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md, whose shared R-numbering now runs R1–R157 with most of the R73–R149 block unused (provenance: _research/regulatory-actuarial.md for R1–R34, _research/regulatory-actuarial-annuities.md for R35–R72, and the AP&P Manual appendix extractions _research/appp-ag33.md, appp-ag35.md, appp-a820-a821-a822.md and appp-a585-a250-a255-a270.md for R151–R157, all accessed 2026-08-06). std marks standardizations introduced for the reference implementation. Parameter values are identical to those in product-spec.md. This is the deferred annuity base chassis: the fixed-indexed annuity notes reference the surrender-benefit composition order and the Model #805 floor construction below rather than restating them, but restate — with FIA-specific parameters — the account-value roll-forward (index-credit driven), the MVA family, the death benefit and the lapse architecture; do not carry this file’s recursions or rates into an FIA model unexamined. The variable annuity notes do not, and must not — a VA’s separate account is outside Model #805, which reaches only a VA fixed account via Model #250 §7.B REG-R42 REG-R43.

Model scope and conventions#

  • Purpose. Project gross liability cash flows (single premium in; free withdrawals, excess withdrawals, full surrenders, death benefits, annuitization transfers, and expenses out) for a single-contract model point of a 5-year MYGA with a market value adjustment. Reserves are not computed (see Valuation and reserve pointers).

  • Projection frequency: monthly std. The contract credits interest daily, quoted as an annual effective rate S4 S5 S16, and surrender charges/MVA step on contract-year boundaries S8 S10. Monthly is the coarsest grid resolving both: it hits every contract anniversary exactly, and it resolves the 30-day guarantee-period-end window and the shock-lapse boundary to within one step. Finer grids buy nothing on a book-value chassis with no daily-valued index.

  • Crediting discretization. Monthly compounding at (1 + i_cr)^(1/12). Because the declared rate is an effective annual rate under both conventions, twelve monthly factors reproduce the annual accretion exactly — the discretization affects only the placement of interest within a month. Do not additionally compound daily; document the convention when reconciling to an admin system.

  • Timing. Elective transactions (withdrawals, owner-elected surrenders, annuitization elections) at the beginning of the policy month (BOM); interest credited at end of month (EOM); decrements applied at EOM std, with decrement benefits valued on the post-crediting AV(t).

  • Age basis: age nearest birthday (ANB) std. The VM-22 prescribed mortality basis (2012 IAM Basic with Scale G2 and the Table 6.7 factors) is stated ANB, and the Valuation Manual supplies the ALB conversion q(x)_ALB = [q(x)_ANB + (1 − q(x)_ANB) × q(x+1)_ANB] / (2 − q(x)_ANB) rather than a native ALB table R2 §6.B.8 R9; the SOA/LIMRA fixed-rate deferred surrender study is also ANB, with a Balducci exposure adjustment R8.

  • Model points. Single-contract model points on an expected (probability-weighted) basis: an in-force factor l(t) multiplies per-contract cash flows. Grouping is a caller concern.

  • Contract-year indexing. y(t) = ceil(t / 12); anniversaries at t = 12, 24, …. Guarantee period n = 5, so the initial guarantee and surrender charge periods both end at t = 60.

  • Rounding. Full precision internally; cash flows reported to cents std.

Model point attributes#

Attribute

Type

Example (anchor cell)

issue_age

int (ANB)

60

sex

enum {M, F}

M

tax_status

enum {NQ, IRA, Roth, inherited}

NQ std

premium

currency

100,000 S11 rate band ≥$100,000

issue_date

date

contract month 0

guarantee_period_years

int

5 S10 S11

declared_rate_initial

rate p.a.

0.0445 S11

gmir

rate p.a.

0.0025 S11

gmsv_rate (i_nf)

rate p.a.

0.0280 S11

sc_schedule_initial

vector by contract year

(0.09, 0.08, 0.07, 0.06, 0.05) S10

sc_schedule_renewal

vector by contract year

(0.05, 0.04, 0.03, 0.02, 0.01) S2

renewal_architecture

enum {rollover, annual_redeclare}

rollover (Camp A) std

free_wd_rule

enum {pct_av, interest_only, greatest_of}

pct_av at 10% S10

free_wd_mva_exempt

bool

True std (False = Voya/Nationwide convention S3 S4)

mva_family

enum {geometric, linear_duration, declared_differential}

linear_duration S8 S9

mva_cap_rule

enum {sym_sc, min_sc_interest, asym_sc_snfl, gmir_floor, none}

sym_sc S2

mva_ref_yield_at_issue (i0)

rate p.a.

0.0500 std

mva_period_years

int

5 (= surrender charge period) S8 S11

mgsv_withdrawal_convention

enum {gross, net_of_charges}

gross S11

mgsv_annual_charge

currency p.a.

0.00 S11; statutory max 50.00 R1

premium_tax_rate

rate

0.00 std

av_initial, mgsv_initial, tax_basis_initial

currency

100,000 / 87,500 / 100,000

State variables#

Variable

Description

Updated

AV(t)

Account value at end of policy month t

monthly recursion

MGSV(t)

Model #805 minimum guaranteed surrender value at end of month t

monthly recursion

FWB(y)

Free-withdrawal base fixed at the start of contract year y

each anniversary

FW(t)

Unused free-withdrawal allowance remaining in contract year y(t)

on withdrawal / anniversary

i_cr(t)

Declared credited rate in force

at each guarantee-period boundary

gp_end(t)

Months remaining in the current guarantee period

monthly

sc_clock(t)

Months elapsed in the current surrender-charge schedule

monthly (resets on renewal under rollover)

i0_locked

MVA reference yield locked at the start of the current guarantee period

each renewal

basis(t)

Investment in the contract (IRC §72 tax basis)

on withdrawal R6

l(t)

In-force probability at end of month t; l(0) = 1

monthly decrements

Assumption inputs#

Three classes are distinguished explicitly and must never be blended in a parameter file.

(a) Contractual / guaranteed elements (cited; the insurer cannot change them)#

Input

Value

Basis

Surrender charge schedule, initial term

9%, 8%, 7%, 6%, 5%; 0% from year 6

S10

Surrender charge base

amount in excess of the free allowance

S8 S9

Free-withdrawal allowance

10% of premium (year 1); 10% of AV at the last anniversary (years 2+)

S10

Free amount exempt from charge and MVA, incl. at full surrender

yes

S8 S11

MVA formula

μ = (i0 − it) × T

S8 S9

MVA cap

symmetric at the surrender charge amount

S2

MVA excluded from

death benefit, 30-day window, annuitization, RMDs, waiver withdrawals

S2 S4 S5 S8 S13 S16

GMIR (floor on any declared rate)

0.25%

S11

GMSV rate i_nf

2.80%

S11

Model #805 net consideration ratio

87.5% of gross considerations

R1 §4.A(2) REG-R42

Model #805 indexed-rate corridor

min(3.00%, round(5-yr CMT, 1/20%) − 1.25%), floor 0.15%

R1 §4.B REG-R42

Model #805 annual contract charge (max)

$50 p.a., accumulated at i_nf

R1 §4.A REG-R42

Death benefit

full account value; no charge, no MVA; never below the cash surrender benefit

S1 S2 S13 R1 §6

30-day guarantee-period-end window

full account value, no charge, no MVA

S1 S2 S5 S6

Contract fees

none

S5 S10 S13 S16

(b) Insurer-declared current elements (snapshot; non-guaranteed under ASOP 2 REG-R26)#

Input

Value

Basis

Initial declared rate i_cr, months 1–60

4.45% effective annual

S11 (eff. 09/22/25, payments ≥$100,000; 4.10% under $100,000)

Renewal declared rate

i_cr^ren(t) = max(GMIR, MR(t) − s_ren)

rule std; discretion + GMIR floor S1 S2 S11 S16

Renewal spread s_ren

0.00% base run; 1.00% scenario

std (a)

Renewal surrender charge schedule

5%, 4%, 3%, 2%, 1%

S2; adoption std

Attained-age cap on renewal charge

4% at 94, 3% at 95, 2% at 96, 1% at 97, 0% at 98–100

S1 S2

MVA reference yield path it

exogenous scalar input series

std; index choice is state-filed S8 S12

The renewal surrender charge schedule and its attained-age cap are printed as contract terms in S1 S2, not as declared elements; they are listed in (b) only because they attach at a renewal the insurer also re-rates. Load them from the guaranteed-element file and treat only the renewal rate as non-guaranteed.

(a) There is no public evidence on renewal-rate setting: Voya “observes no specific formula” S3 and Nationwide “observes no specific method”, both citing fixed-income yields, competitive considerations, administrative costs and general economic trends S3 S4. The base run sets s_ren = 0 so the credited rate equals the competitor rate and the dynamic-lapse term is exactly zero — the same discipline the UL notes use. The 1.00% scenario exercises the dynamic term. Renewal declarations are non-guaranteed elements: ASOP 2’s scope expressly covers fixed deferred annuities REG-R26.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol

Meaning

t

policy month index, t = 1, 2, …; y = y(t) = ceil(t/12) contract year

n

guarantee period in years (5); the surrender charge and MVA periods equal it

P

single purchase payment (100,000)

i_cr(t)

declared credited rate in force, effective annual

f(t)

monthly crediting factor = (1 + i_cr(t))^(1/12)

i_nf

GMSV / minimum-nonforfeiture accumulation rate (0.0280); g = (1 + i_nf)^(1/12)

sc(y)

surrender charge rate in contract year y of the current schedule

FWB(y)

free-withdrawal base: P for y = 1, else AV(12(y−1)) S10

FW(t)

unused free allowance in contract year y(t); reset to 0.10 × FWB(y) at each anniversary

W(t)

gross amount removed from the account value at BOM of month t

E(t)

amount exposed to charge and adjustment = max(0, W(t) − FW(t))

μ(t)

MVA rate (signed, dimensionless)

M(t)

MVA amount (signed currency)

C(t)

surrender charge amount (currency, ≥ 0)

AV'(t)

account value after the BOM transaction, before crediting

SV(t)

gross surrender value before the nonforfeiture floor

SB(t)

surrender benefit actually paid

MGSV(t)

minimum guaranteed surrender value (Model #805 floor); the specimen S11 calls it the “GMSV”, products/fixed_indexed_annuity/ the “guaranteed minimum value (MGV)” — one concept

q(t), w(t)

monthly mortality and monthly total surrender rates

a(t)

monthly annuitization election rate

l(t)

in-force probability at end of month t

Dimensional check: μ(t) and sc(y) are both pure rates multiplying the same currency base E(t); M(t), C(t), AV(t), MGSV(t) and every ledger line are currency. T(t) in the MVA is in years, so (i0 − it) × T is rate × years — dimensionless only because it is the first-order duration approximation of the geometric factor [(1+i0)/(1+it)]^T − 1 (see MVA families).

Monthly processing order#

At month t (BOM steps 1–5, EOM steps 6–8):

  1. Roll counters. Set y = y(t). If t ≡ 1 (mod 12) (a contract anniversary has just passed), reset FWB(y) and FW = 0.10 × FWB(y) S10.

  2. Guarantee-period boundary. If the previous month ended a guarantee period (t − 1 ≡ 0 mod 12n): apply the 30-day window (full account value available, no charge, no MVA S1 S2); redeclare i_cr; under rollover, reset sc_clock and start the renewal surrender charge and MVA schedule, and re-lock i0 at the current reference yield S2 S11; under annual_redeclare, set sc(·) ≡ 0 and μ ≡ 0 permanently and redeclare the rate each anniversary thereafter S13.

  3. Elective withdrawal. Compute E(t), C(t), M(t) (below); reduce FW by min(W(t), FW); set AV'(t) = AV(t−1) − W(t); emit the cash flow W(t) + M(t) − C(t).

  4. Annuitization election (only in a 30-day window, t > 12): a fraction a(t) of in-force transfers AV'(t) to the payout model (full account value in the window S1 S2).

  5. Update the tax basis for IRC §72 reporting: withdrawals are income-first, taxable to the extent AV (gross of surrender charge) exceeds basis; basis is reduced only by the non-taxable remainder R6 §72(e)(3)(A) REG-R55. This is a reported quantity, not a liability cash flow.

  6. Credit interest. AV(t) = AV'(t) × f(t).

  7. Roll the nonforfeiture floor. MGSV(t) = [ MGSV(t−1) − d(t) − c(t) ] × g with d(t) = W(t) under the gross convention S11 or W(t) + M(t) − C(t) under net_of_charges S9, and c(t) the monthly slice of the annual contract charge ($0 representative, $50 p.a. statutory maximum R1 S11).

  8. Decrements. Deaths at q(t), then surrenders at w(t) on survivors [std order]: l(t) = l(t−1) × (1 − a(t)) × (1 − q(t)) × (1 − w(t)).

With no withdrawals, steps 3–8 collapse to the core recursion:

AV(t) = AV(t−1) × (1 + i_cr(t))^(1/12)                                [S4] [S5] [S16]
MGSV(t) = MGSV(t−1) × (1 + i_nf)^(1/12)                               [R1] [S11]

with AV(0) = P S5 S10 S16 and MGSV(0) = 0.875 × P R1 §4.A(2) S11.

Surrender benefit — the exact composition order#

The order is account value → MVA → surrender charge → nonforfeiture floor, and it is not interchangeable: both M and C are computed on E(t) before either is deducted S8. For a full surrender at end of month t:

E(t)  = AV(t) − FW(t)                                                 [S8] [S11]
C(t)  = sc(y) × E(t)                                                  [S8] [S10]
M(t)  = cap( μ(t) × E(t) )                                            [S8] [S2]
SV(t) = AV(t) + M(t) − C(t)                                           [S8]
SB(t) = max( SV(t), MGSV(t) )                                         [S8] [S9] [S12]

For a partial withdrawal of gross W(t), replace AV(t) by W(t) in the first and fourth lines; the amount paid is W(t) + M(t) − C(t) and the account value falls by W(t).

MVA inside the free amount — the std convention. The representative model sets free_wd_mva_exempt = True: the free allowance is exempt from both the surrender charge and the MVA, including at full surrender S8 S11. The market is genuinely split. Voya S3 and Nationwide S4 both state that the MVA applies to free-amount withdrawals taken before maturity; the retail MYGAs do not S2 S9 S10 S15 S16. Setting the flag to False gives E(t) = AV(t) and the composition collapses to the multiplicative form:

SB(t) = max( AV(t) × (1 + μ(t) − sc(y)),  MGSV(t) )

which is the identity to use when checking dimensional consistency and when comparing against contracts that quote an MVA factor rather than an MVA rate.

MVA — three formula families, five cap variants#

mva_family selects the rate; mva_cap_rule selects the limit. Both are first-class model parameters, not hard-coded rules, because the cap is the largest single cross-carrier divergence in the source set.

(i) geometric — Treasury/swap discount factor, uncapped in the sources.

Φ(t) = [ (1 + a) / (1 + b + s_adm) ] ^ τ ;   μ(t) = Φ(t) − 1

a = reference yield at deposit; b = reference yield at distribution for a term equal to the remaining period, partial years rounded up to a full year (capped at the guarantee period) S4; s_adm = administrative-expense adder, 25 bp in the Nationwide contract, explicitly covering the cost of liquidating fixed-income investments and structurally biasing the adjustment against the owner S4; τ = days to maturity ÷ 365.25 S4. Voya’s variant has s_adm = 0, uses Treasury notes maturing in the last three months of the term, and τ = x/365 measured from the Wednesday of the week of withdrawal S3. Neither states any cap or collar S3 S4.

(ii) linear_duration — the representative form.

μ(t) = (i0 − it) × T(t)                                               [S8] [S9]
T(t) = (days from the surrender date to the end of the current contract year ÷ 365)
       + whole years remaining in the MVA period                       [S8]

i0 = reference index value at issue (re-locked at each renewal); it = value at surrender; source index = Barclay’s US Credit Index, formula varying by state S8. This is the first-order approximation of (i): at i0 = 5%, it = 6.5%, T = 2.5 the linear form gives −3.750% against −3.484% for (1.05/1.065)^2.5 − 1 — a 27 bp gap, widening with |i0 − it| × T, and always in the contract holder’s disfavour when rates rise.

(iii) declared_differential — the insurer’s own new-money rate. M(t) = W × (Ic − In) × F_s S14, with Ic the rate credited on the money withdrawn, In the rate that would be credited on new money for a guarantee period of the same duration, and F_s a contractual adjustment-factor table by whole years remaining s with partial years interpolated. Specimen table (Ic < 6% / Ic ≥ 6%): s=0 0.00/0.00; 1 0.90/0.90; 2 1.80/1.75; 3 2.60/2.50; 4 3.40/3.15; 5 4.10/3.80; 6 4.80/4.35; 7 5.40/4.85; 8 6.00/5.35; 9 6.50/5.75; 10 7.00/6.15 S14. These are modified-duration factors, which is why the higher-rate column is uniformly lower. Model #245 §4.I recognizes this branch alongside the external-index branch R4 REG-R45.

Cap variants (mva_cap_rule).

Value

Rule

Source design

sym_sc std

M = clamp(M_raw, −C, +C)

Athene NY S2

min_sc_interest

M = clamp(M_raw, −K, +K) with K = min(C, interest credited to date)

Midland S8 S9

asym_sc_snfl

M ≤ +C; on the downside no cap — only SB ≥ MGSV binds

MassMutual Ascend S12

gmir_floor

AV + M ≥ P_accum@GMIR (premiums less prior withdrawals accumulated at the GMIR); the surrender charge may still breach that level

New York Life S13

none

uncapped, fully two-sided

Voya S3, Nationwide S4

μ(t) = 0, unconditionally, when: the surrender is in the 30-day guarantee-period-end window S2; the MVA period has expired S8 S13 S16; the benefit is a death benefit S2 S4 S8 S13 S16; the withdrawal is an RMD or a waiver-rider withdrawal S2 S5 S13; or the contract is being annuitized S16 std.

Minimum guaranteed surrender value (Model #805)#

MGSV(0) = 0.875 × P
MGSV(t) = [ MGSV(t−1) − d(t) − c(t) ] × (1 + i_nf)^(1/12)

with i_nf the contract GMSV rate (2.80% S11). The statute defines the indexed nonforfeiture rate — it is not a band the contract rate sits inside:

i_stat = max( 0.0015,  min( 0.03,  round_{1/20 of 1%}(CMT5) − 0.0125 ) )  [R1 §4.B](#uslib-fixed_deferred_annuity-r1) [REG-R42]

and the contract rate must satisfy i_nf ≥ i_stat; crediting the floor at more than the statutory rate is permitted and simply produces a higher floor, which is exactly what S11 does (“the GMSV rate will not be less than the minimum rate required by each state”). Do not implement the reverse inequality — capping i_nf at round(CMT5) − 1.25% would make the representative 2.80% illegal at any CMT5 below 4.05% and is not what §4.B says. CMT5 is the five-year Constant Maturity Treasury rate reported by the Federal Reserve as of a date, or averaged over a period, specified in the contract and no longer than 15 months before issue or redetermination R1 §4.B.

Do not implement a 1% floor. The retrieved Model #805 print floors the indexed nonforfeiture rate at 15 basis points R1 §4.B REG-R42; the widely repeated 1% figure is unverified against any retrieved document. d(t) is the withdrawal deduction (gross S11 or net_of_charges S9); c(t) is the monthly slice of the annual contract charge, $0 representative and $50 statutory maximum R1 §4.A S11. Premium tax actually paid and indebtedness are additional permitted deductions, both accumulated at i_nf, and are zero here R1 §4.A. The equity-index carve-out of §4.C (an additional reduction of up to 100 bp) does not apply to a book-value MYGA R1 §4.C.

Death benefit and annuitization#

  • Death benefit = AV(t), with no surrender charge and no MVA S1 S2 S13, floored at the cash surrender benefit and hence at MGSV(t) R1 §6. On the base run AV(t) > MGSV(t) at every duration because the 4.45% credited rate exceeds the 2.80% GMSV rate — not unconditionally: the floor accretes at 2.80% while a renewal rate may fall to the 0.25% GMIR, and at the GMIR the floor overtakes the account value after roughly 8.5 further years (≈ contract year 13–14 on the anchor cell). Test the floor on death at every duration std; the alternative “greater of accumulation value and minimum surrender value” design S5 S6 makes it live in the base run too.

  • Annuitization transfers AV'(t) (in the window) or SV(t) (during the surrender charge period) out of the accumulation block S1 S2 S5. Payout factors are not specified here — no retrieved product document contains an annuity rate table S4. The accumulation model emits the transfer as an outgo and hands the amount to the payout model; the statutory maximum valuation rate for the resulting income stream is VM-V §1 REG-R37 and the mortality basis the 2012 IAM/IAR family R9 REG-R59 REG-R60.

Cash flow ledger#

Cash flow

Formula (per contract, month t)

In-force weight

Sign

Single premium

P at t = 0

1

+

Free withdrawal payment

W(t) where W(t) ≤ FW(t)

l(t−1)

−

Excess withdrawal payment

W(t) + M(t) − C(t)

l(t−1)

−

Full surrender payment

SB(t) = max(AV(t) + M(t) − C(t), MGSV(t))

l(t−1)(1 − a(t))(1 − q(t)) w(t)

−

Death benefit

AV(t)

l(t−1)(1 − a(t)) q(t)

−

Annuitization transfer

AV'(t) (window) or SV(t)

l(t−1) a(t)

−

Acquisition commission

0.02 × P at t = 0 std

1

−

Maintenance expense

(50/12) × 1.025^(y−1) std

l(t−1)

−

Premium tax

premium_tax_rate × P std = 0

1

−

Internal transfers are not cash flows. Interest credited to the account value, the surrender charge, the market value adjustment, and the movement of the Model #805 floor are internal accounting entries: they drive AV, MGSV and the benefit amount, but they are never separate ledger lines. Only amounts actually paid to or received from the contract holder, and the insurer’s own expenses, are cash flows. This is the gross-liability convention of the library std. Two corollaries worth stating because they are common implementation errors: (1) a binding nonforfeiture floor is not a separate “top-up” cash flow — it raises SB(t), and the difference MGSV(t) − SV(t) is a reconciliation quantity only; (2) the IRC §72 taxable-income split is a reported quantity and generates no insurer cash flow R6 REG-R55.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions built on the VM-22 prescribed functional form R2 §6.B.5, which is the only publicly specified dynamic-lapse formula for this product and is therefore the natural skeleton even for a best-estimate run.

Base lapse by renewal architecture#

VM-22 Table 6.5 (fixed annuities with no guaranteed living benefit) is keyed on years before/after surrender-charge expiry and on whether the contract year contains an interest-guarantee-period (IGP) expiry R2:

Years before/after SC expiry

IGP ≤ 1 yr

IGP > 1 yr, not an IGP-expiry year

IGP-expiry year (IGP > 1 yr)

3+ after

3.0%

2.0%

55.0%

2 after

7.5%

2.0%

65.0%

1 after

10.0%

2.0%

75.0%

Upon expiry

25.0%

6.0%

75.0%

1 to expiry

2.5%

1.0%

70.0%

2 to expiry

2.5%

1.0%

70.0%

3+ to expiry

2.5%

1.0%

70.0%

Applying the mapping evidenced by the guideline’s own 3-year worked examples R2 to the representative 5-year IGP / 5-year surrender charge period gives, as a std extension:

renewal_architecture

Contract-year base lapse

rollover (Camp A: new 5-year IGP + new SC schedule) S1 S2 S5 S11

1%, 1%, 1%, 1%, 1%, 75%, 1%, 1%, 1%, 1%, 75%, … (period 5)

annual_redeclare (Camp B: annual rates, no SC) S13

1%, 1%, 1%, 1%, 1%, 75%, 10%, 7.5%, 3%, 3%, 3%, …

These are the two patterns the guideline’s Examples 1 and 2 generate for a 3-year contract (1,1,1,75,10,7.5,3 and 1,1,1,75,1,1,75 respectively) R2, scaled to a 5-year term. The architecture switch, not the level, is the first-order modeling decision: Camp A creates a repeating shock every five years, Camp B a single shock followed by ordinary interest-sensitive lapse. Monthly conversion: w_base_m = 1 − (1 − w_base_annual)^(1/12).

Experience corroboration, not calibration: the SOA/LIMRA 2023–2024 fixed-rate deferred study reports surrender rates peaking in the year the surrender charge expired and remaining elevated afterwards, decreasing as the GMIR band rose, decreasing as the credited rate rose, and increasing with the excess of market over credited rate — that relationship “well defined in the years after surrender charge expiry” but muted during the charge period — and that “in the year the surrender charge expired, high ‘shock’ surrender rates were observed that were not necessarily impacted or driven by market interest rate sensitivity” R8 REG-R63. Detailed tables sit behind the paid package and were not retrieved.

Dynamic (interest-sensitive) lapse std#

w_annual(t) = clamp( Base(y) × G + Rate(t) × Φ_MVA(t),  0.005,  0.90 )
Rate(t)     = Market(t) × max( 0, 1 − 5 × (1 − CSV(t)/AV(t)) )
Market(t)   = −1.25 × (CR − MR)^X          if CR ≥ MR
            = 0                             if MR > CR ≥ MR − BF
            = +1.25 × (MR − BF − CR)^X      if CR < MR − BF

with, per the prescribed parameterization R2 §6.B.5:

  • G = GMIR Factor. Fixed annuities: 1.25 if GMIR ≤ 1.0%; 1.00 if 1.0% < GMIR ≤ 2.5%; 0.70 if GMIR > 2.5%. Representative GMIR 0.25% S11 → G = 1.25.

  • BF = buffer factor = 50 bp — the band inside which no dynamic response occurs.

  • X = 2.0 during the surrender charge period, 2.5 at the shock and thereafter.

  • CR = current crediting rate; MR = market competitor rate. For fixed annuities with an interest guarantee period of 5 ≤ IGP < 7 years, MR = the 7-year Treasury rate plus a 50% A / 50% AA spread minus the Pricing Spread, with Pricing Spread = 0% R2.

  • CSV(t)/AV(t) = SB(t)/AV(t), so 1 − CSV/AV is exactly the combined surrender-charge and negative-MVA haircut; the dynamic term switches off entirely once that haircut reaches 20%.

  • ITM Factor = 1 for an Accumulation-category contract with no guaranteed living or death benefit — the guideline says so explicitly R2.

  • Φ_MVA = MVA Factor. The prescribed value is 0 while an MVA is in effect and 1 otherwise R2, i.e., the regulator’s view is that an in-force MVA completely neutralizes interest-rate-driven disintermediation, leaving only Base × G.

std departure for best estimate. The reference model exposes Φ_MVA as a parameter with the prescribed values {0 in force, 1 expired} as the statutory default and 0.35 while in force as the best-estimate default. Rationale: the prescribed 0 is a regulatory simplification, and the industry study found a “notable increase” in surrender rates between a 0% and a 3% market-minus-credited spread during the surrender charge period R8. That study does not separate MVA from non-MVA contracts, so 0.35 is a judgement, not a calibration — sensitivity-test it (see model risks).

Base deterministic run. MR(t) = CR(t) = 4.45% and s_ren = 0, so Market(t) = 0, Rate(t) = 0, and w_annual(t) = Base(y) × 1.25, floored at 0.5% and capped at 90%. Contract years 1–5 therefore run at 1.25% and contract year 6 at 90% (75% × 1.25 = 93.75%, capped) R2.

Partial withdrawal and annuitization behavior#

  • Partial withdrawals. 0% in the base run std; the VM-22 Table 6.2 age-banded rates are the variant (see assumption note (b)). Where an RMD module is switched on, RMD amounts are free of charge and MVA even above the free allowance S15 std, and are modeled as a withdrawal with E(t) = 0.

  • Annuitization. a(t) = 1.0% in each 30-day guarantee-period-end window after contract year 1, 0% elsewhere std (assumption note (c)). Statutory alternative: a(t) ≡ 0 R2 §6.B.6.

  • Free-withdrawal utilization. Base run 0; a utilization variant takes u × FW(y) each contract year with u a std input. Note the interaction: taking the free amount each year both lowers AV and lowers the future free base, and — under the gross Model #805 convention S11 — reduces the nonforfeiture floor by the same amount.

Worked example#

Anchor cell: Male 60 ANB, non-qualified, P = $100,000, 5-year guarantee period, i_cr = 4.45% S11, i_nf = 2.80% S11, surrender charge 9/8/7/6/5 S10, free withdrawal 10% S10. Monthly factors: f = 1.0445^(1/12) = 1.0036348, g = 1.028^(1/12) = 1.0023039 (both derived). A free withdrawal of $4,000 is taken at BOM of month 13 (contract year 2 allowance = 10% × AV(12) = $10,445.00, so the whole amount is free of charge and MVA S10 S11). Full surrender at end of month 30. All figures in dollars; full precision carried, displayed to cents.

t

Event

AV(t−1)

W(t)

AV'(t)

AV(t)

MGSV(t)

1

—

100,000.00

0.00

100,000.00

100,363.48

87,701.59

2

—

100,363.48

0.00

100,363.48

100,728.28

87,903.65

3

—

100,728.28

0.00

100,728.28

101,094.40

88,106.17

12

1st anniversary

104,071.72

0.00

104,071.72

104,450.00

89,950.00

13

free withdrawal

104,450.00

4,000.00

100,450.00

100,815.11

86,148.02

24

2nd anniversary

104,540.04

0.00

104,540.04

104,920.03

88,356.60

30

full surrender

106,840.74

0.00

106,840.74

107,229.09

89,585.05

Checks on the table: AV(12) = 100,000 × 1.0445 = 104,450.00 exactly, and AV(24) = 100,450 × 1.0445 = 104,920.025 (displayed 104,920.03) — twelve monthly factors reproduce the annual effective rate exactly. MGSV(0) = 0.875 × 100,000 = 87,500.00; MGSV(12) = 87,500 × 1.028 = 89,950.00; the month-13 withdrawal is deducted gross (not reduced by charges or MVA) under the S11 convention, giving MGSV(24) = (89,950 − 4,000) × 1.028 = 88,356.60. The surrender traces below are computed from the cent-rounded values shown, so they reproduce by hand.

Surrender trace, end of month 30 (contract year 3, sc = 7% S10; MVA reference yield i0 = 5.00% at issue, it = 6.50% at surrender, both std):

  • Free allowance for contract year 3: FW = 0.10 × AV(24) = 10,492.00 S10, unused.

  • E = 107,229.09 − 10,492.00 = 96,737.09 S8 S11.

  • C = 0.07 × 96,737.09 = 6,771.60 S8 S10.

  • T = 0.5 + 2 = 2.5 years (six months to the end of contract year 3, plus contract years 4 and 5 remaining in the 5-year MVA period) S8.

  • μ = (0.0500 − 0.0650) × 2.5 = −0.037500 S8 S9; M_raw = −3,627.64.

  • Symmetric cap: |M| ≤ C = 6,771.60 S2 — not binding, so M = −3,627.64.

  • SV = 107,229.09 − 3,627.64 − 6,771.60 = 96,829.85.

  • MGSV(30) = 88,356.60 × 1.028^(1/2) = 89,585.05 — floor not binding.

  • SB(30) = 96,829.85, of which $10,492.00 is the untouched free amount and $86,337.85 the adjusted, charged excess.

A case where the floor binds. Same contract, full surrender at end of month 6 (contract year 1, sc = 9% S10) with the reference yield at 10.00% (a stress level, std, chosen to force both the cap and the floor to bind): AV(6) = 102,200.78; FW = 0.10 × 100,000 = 10,000.00 (year-1 base is purchase payments S10); E = 92,200.78; C = 8,298.07; T = 0.5 + 4 = 4.5; μ = −0.225; M_raw = −20,745.18, capped to −8,298.07 by the symmetric rule S2; SV = 102,200.78 − 8,298.07 − 8,298.07 = 85,604.64; MGSV(6) = 87,500 × 1.028^(1/2) = 88,716.54. SB(6) = 88,716.54 — the Model #805 floor binds and adds $3,111.90. Note the ordering lesson: with a symmetric cap the worst case is AV − 2·sc·E (= AV × (1 − 2·sc) only when the free amount is zero — here it is $102,200.78 − 2 × 0.09 × $92,200.78 = $85,604.64, not $102,200.78 × 0.82 = $83,804.64), and it is only at short durations with a high surrender charge that this falls below 0.875 × P × (1 + i_nf)^t.

Geometric-branch unit test S4. For mva_family = geometric the Nationwide contract supplies fully worked arithmetic that a regression test should reproduce exactly: a 5-year GPO, $10,000 allocation, Specified Interest Rate 8.5%, 5-year swap at deposit a = 8%, surrender 985 days from maturity, Specified Value $12,067.96, Φ = [(1 + a)/(1 + b + 0.0025)]^(985/365.25). At b = 7%: Φ = 1.01897, surrender value $12,296.89. At b = 9%: Φ = 0.96944, surrender value $11,699.17 S4. Recomputing from the printed five-decimal factors reproduces both to within three cents; assert the factors, not the dollar figures. Two implementation details: the contract selects b’s maturity by rounding 985/365.25 = 2.69 up to 3 years while the exponent uses the exact day count S4; and the Appendix A sensitivity table for a 10-year GPO with a = 8% shows −2.06% at b = 8% with 9 years remaining — the pure effect of the 25 bp expense adder, and a second regression target S4.

Valuation and reserve pointers#

This library projects gross liability cash flows. Reserve layers consume them and are cited, not reproduced:

  • VM-22 principle-based reserves for non-variable annuities R2 REG-R36. Constitutes CARVM for in-scope contracts R2 §1.A; applies for valuation dates on or after January 1, 2026 R2 §2.B; three-year elective transition on VM-A/VM-C/VM-M/VM-V for business issued in the first three years, mandatory prospectively thereafter R2 §2.B (2029 is arithmetic, not quotation unverified). A MYGA is in the Accumulation Reserving Category R2. Aggregate reserve = SR (CTE70) + DR for contracts passing the Single Scenario Test + formulaic reserves for excluded contracts; the additional standard projection amount is disclosure-only under VM-31 R2 §3.

  • Formulaic CARVM — A-820 ¶¶14–15 as interpreted by AG 33 REG-R153 REG-R151, with the guideline family indexed at VM-C REG-R41. AG 33’s printed title is “Determining CARVM Reserves for Annuity Contracts With Elective Benefits” and its printed Effective Date block reads “This guideline shall be effective on December 31, 1998, affecting all contracts issued on or after January 1, 1981” REG-R151 Effective Date. The December 31, 1995 date and the alternative title this file previously carried come from IRS Rev. Rul. 2002-6, describing a differently-titled instrument R7 REG-R39. Both are recorded and the reconciliation is unresolved — the extracted pages carry no amendment history, so “a later revision” is an inference, not a fact from either source; the 1 January 1981 issue-date reach is common to both, and the 33⅓ / 66⅔ / 100% grade-in ran off by December 31, 2000. The mechanics are no longer unverified: they are in the primary-text extractions _research/appp-a820-a821-a822.md and _research/appp-ag33.md. VM-V §1 carries the statutory maximum valuation interest rate on the post-annuitization payout stream REG-R37.

  • Tax and GAAP. IRC §807: greater of net surrender value and, post-TCJA, 92.81% of the NAIC-prescribed method reserve (CARVM), capped at statutory R7 REG-R16. LDTI (ASU 2018-12) with ASOP No. 10 on the U.S. GAAP basis REG-R34 REG-R71.

  • Standards for the modeling work itself. ASOP 7 (life/health cash flow analysis — the standard for exactly this disintermediation/reinvestment/MVA work) REG-R27; ASOP 22 (asset adequacy) REG-R29; ASOP 56 (modeling) REG-R32; ASOP 2 (non-guaranteed elements — the declared renewal rate) REG-R26; ASOP 54 (pricing) REG-R70.

Key sensitivities and model risks#

Dominant assumptions, in rough order of impact on a MYGA block:

  1. The shock lapse at surrender-charge / guarantee-period expiry, and the renewal architecture switch that positions it. Base lapse moves from 1% to 75% in a single contract year under the prescribed table R2, and Camp A repeats that every five years while Camp B does it once S11 S13. Nothing else in the model moves the liability duration as much. Run both architectures before quoting a duration.

  2. The renewal declared rate s_ren, jointly with dynamic lapse. The credited-minus- competitor spread drives both the interest margin and the surrender rate, in opposite directions; the 50 bp buffer and the quadratic/2.5-power response make the sensitivity strongly convex around CR = MR R2.

  3. Φ_MVA — whether the MVA suppresses dynamic lapse. The prescribed value of 0 R2 and the best-estimate std 0.35 bracket a large range; at 0 the entire dynamic term vanishes during the surrender charge period.

  4. The MVA cap rule. Symmetric-at-charge, min(charge, interest credited), asymmetric-with-nonforfeiture-floor, GMIR-floored, and uncapped produce materially different tail surrender values on the same rate path S2 S3 S4 S8 S9 S12 S13. The cap, not the formula family, is where the money is.

  5. The Model #805 floor at short durations. As the worked example shows, the floor binds early (high charge, large negative MVA) and not later; a model that tests the floor only at full surrender in later durations will miss it entirely.

Known modeling pitfalls:

  • Composition order. MVA → surrender charge → floor, with both computed on the pre-deduction excess E S8. Applying the charge first and the MVA to the net figure understates the adjustment by sc × |M|; applying the floor before the MVA silently removes the downside protection.

  • The free-amount / MVA interaction. free_wd_mva_exempt is a real product difference S2 S3 S4 S9 S10 and changes both the surrender value and the 1 − CSV/AV haircut that gates dynamic lapse. Do not hard-code it.

  • Gross vs net withdrawals. W(t) is the gross amount removed from the account value; contracts promising a stated net check need a gross-up solve — Voya’s prospectus works the case, $2,099.08 withdrawn to deliver a $2,000 check at a 0.9528 factor S3.

  • The Model #805 withdrawal convention. gross S11 versus net_of_charges S9 are both live and give different floors; the difference then compounds at i_nf for the rest of the contract.

  • The 15 bp floor. Implementing the folklore 1% floor overstates the Model #805 minimum in low-rate environments — the retrieved statute says 15 bp R1 §4.B REG-R42.

  • Surrender-charge clock on renewal. Under rollover the clock resets S1 S2 S11; Voya and Nationwide run it from the original purchase payment date so it never restarts S3 S4. Getting this wrong relocates the shock lapse by years.

  • Mortality table plumbing. The prescribed formula uses the 2012 IAM Basic table (VM-M §2.C) with Scale G2 and the VM-22 F_x factors R2 §6.B.8, not the 2012 IAM Period/IAR valuation table. Where the IAR generational table is used, the Valuation Manual’s rounding trap applies: round from the 2012 period rate each time, never compound an already-rounded prior-year rate REG-R59. Do not substitute life bases — annuitant mortality is a different and lighter basis than the 2017 CSO / 2015 VBT / ILEC families REG-R59 REG-R60 REG-R61, and annuity surrender behavior is structurally unlike life lapse REG-R63. Deferred-period annuitant mortality is under-evidenced: only a 2011–2015 study and a 2006 analysis were identified, neither retrieved REG-R65. Fortunately mortality is second-order here — death pays full account value with no charge and no MVA S1 S2 S13.

  • Era and snapshot caveats. The 4.45% declared rate, 2.80% GMSV rate and 0.25% GMIR S11 are a September 2025 snapshot of one product; the same insurer’s 2023 brochure stated the GMIR “will be 1% or higher” S10. Levels are era-representative; mechanics are stable.