Technical Notes#

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

Scope note. A reference liability cash-flow projection model for the standardized composite product defined in product-spec.md (same directory); not any single insurer’s product. [S#] / [R#] tags refer to _research/fixed-indexed-annuity.md; [REG-R#] tags refer to references/regulatory-and-actuarial-references.md, whose shared numbering now runs R1–R157 with most of the R73–R149 block unused: R1–R34 originate in _research/regulatory-actuarial.md, R35–R72 in _research/regulatory-actuarial-annuities.md, and R151–R157 are the seven AP&P Manual appendix items read at first hand on 2026-08-06 — R151 AG 33, R152 AG 35, R153 A-820 with A-821 and A-822, R154 A-830, R155 A-585, R156 A-250, R157 A-255. std marks standardizations introduced for the reference implementation; unverified marks claims the research file could not confirm. An unverified flag leaves that state only when the primary text is read — which is what closed the AG 33 and AG 35 mechanics below REG-R151 REG-R152; nothing is upgraded on recollection, and every flag not closed that way still stands. Parameter values are identical to product-spec.md.

Inherited versus new. The base contract is the fixed deferred annuity chassis in _research/fixed-deferred-annuity.md and products/fixed_deferred_annuity/ — the surrender-benefit composition order and the Model #805 floor construction — whose structure is referenced, not restated; the schedules, rates and recursions below are this composite’s own and are not that file’s. In particular the account-value roll-forward here is index-credit driven, and the lapse architecture below is not that file’s renewal/shock-lapse architecture (see “Policyholder behavior modeling”: an in-force GLWB suppresses the shock lapse). MGV below is the same Model #805 floor that file calls MGSV — one quantity, two source labels. Two base-contract items are restated rather than inherited, because the FIA composite chooses differently from the fixed-deferred composite: the MVA (ratio form [(1+i₀)/(1+iₜ)]^(n/12) − 1 with the Nassau limit S10, against that file’s linear (i₀ − iₜ) × T with a symmetric surrender-charge cap) and the death benefit (max(AV, MGV) S1 S2 S5 S10, against that file’s full account value floored at the cash surrender benefit). New here: the index crediting engine, the premium bonus with vesting and clawback, and the GLWB rider (benefit base, rider charge, lifetime withdrawal, excess-withdrawal adjustment, post-depletion phase).

Difference from the indexed UL segment engine (products/indexed_ul/technical-notes.md): the FIA shares the vocabulary of segments, caps, participation rates, floors and credit bases, but has no cost of insurance, no net amount at risk and no death benefit corridor, hence no COI/NAAR/corridor circularity. There is no premium load, no per-unit charge and no face amount. The ladder is a single annual segment per indexed account, not a monthly sweep ladder of up to twelve concurrent segments. The rider is a guaranteed lifetime withdrawal benefit that survives account-value exhaustion and pays for life, not a no-lapse guarantee on a death benefit.

Model scope and conventions#

Purpose. Project gross liability cash flows (premium, guaranteed and excess withdrawals, surrender payments, death benefits, expenses, and the post-depletion guaranteed income stream) for a single-contract model point. Reserves are not computed.
Projection frequency: annual std, with the contract anniversary as the single event date. Every mechanic in the composite is annual — annual point-to-point crediting S2 S4 S10, the rider charge at the end of each contract year S9, the annual benefit base update S9, and the annual lifetime withdrawal. A monthly grid is needed only for excluded variants: monthly-sum crediting S4 R1, Athene’s monthly charge deduction S1 S2, daily interim values S10 S11, mid-year withdrawal crediting S3 S10 S11.
Timing. All transactions occur at the anniversary and are processed as the last events of the contract year ending there std. The contract-year surrender charge percentage, vesting percentage and free withdrawal amount therefore all apply to a withdrawal at anniversary t, and the index credit for year t is computed on the balance carried from anniversary t−1 — reproducing Athene’s rule that “withdrawals are not credited with index interest in the year they are taken” S1.
Age basis: age nearest birthday (ANB) std — the statutory annuity tables are published on that basis (VM-M / Model #821 print the 2012 IAM Period Table for female and male, age nearest birthday) REG-R59, and the AP&P print of the same table at A-821 Appendices I–IV independently carries the “Age Nearest Birthday” heading for both sexes REG-R153. The std stands: it marks the model’s choice of a single age basis, not the tables’ basis. Attained age at anniversary t = issue_age + t.
Model points. Single-contract, projected on an expected (probability-weighted) basis; survivorship and persistency factors multiply per-contract cash flows. No aggregation logic specified.
Decrement order: death before surrender std. Rounding: full precision internally, cents on reported cash flows std. State basis: one composite state basis std.

Model point attributes#

Attribute	Type	Example (anchor cell)
`issue_age`	int (ANB)	62 std
`sex`	enum {M, F}	M
`tax_status`	enum {NQ, Q}	NQ std
`single_premium`	currency	100,000 std
`bonus_rate`	rate	0.07 S5
`alloc_fixed` / `alloc_indexed`	fraction (sums to 1)	0.00 / 1.00 std
`glwb_elected` / `glwb_basis`	bool / enum {single, joint}	true / single std
`joint_age`	int (ANB), joint only	n/a
`income_start_age`	int (ANB), ≥ 50 S2 S3 S9	70 std
`utilization_intensity`	fraction of `LW` actually withdrawn	1.00 std
`sc_schedule`	vector, contract years 1–11+	9.1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 % S5
`vesting_vector`	vector, contract years 1–11+	0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 % S5
`mva_index_at_issue` `i₀`	rate	0.0300 std
`av_initial`, `bb_initial`, `mgv_initial`	currency (in-force cells)	107,000 / 100,000 / 87,500

State variables#

Variable	Description	Updated
`F(t)`, `A(t)`	Fixed / indexed account balance at anniversary `t` after all processing	annually
`AV(t)`	Account value = `F(t) + A(t)`	annually
`MGV(t)`	Guaranteed minimum (nonforfeiture) value	annually
`BB(t)`	GLWB benefit base — notional, no cash value S1 S9	annually
`RB(t)`	Rollup base for the guaranteed simple rollup S2 S9	annually / on withdrawal
`LW(t)`	Locked annual lifetime withdrawal amount (0 before exercise)	at exercise; ratchet; excess withdrawals
`phase(t)`	`ACCUM` / `INCOME` / `DEPLETED` / `TERMINATED`	annually
`v(t)`, `sc(t)`	Vested bonus percentage; surrender charge percentage for year `t` S5	schedule lookup
`FW(t)`	Free withdrawal amount = `0.10 × AV(t−1)`	annually
`Wcum(t)`	Cumulative gross withdrawals	on withdrawal
`l(t)`	In-force probability at end of year `t`; `l(0) = 1`	annual decrements
`rider_in_force(t)`	Boolean; false once the rider terminates S9	on events
`depletion_cause(t)`	Flag set when an excess withdrawal, surrender charge or MVA touches the account value	in step 5

Assumption inputs#

Class (a) is contractual and cannot be changed by the insurer; class (b) is the insurer-declared current scale, a non-guaranteed element under ASOP No. 2 R6 REG-R26; class (c) is the modeler’s view of experience. They must not be mixed in the code.

(a) Contractual / guaranteed elements#

Input	Value	Basis
Index credit floor `f`	0%	S1 S4 S10 R1
Guaranteed minimum annual cap `c_min`	0.25%	S4; selection std
Guaranteed minimum fixed rate `i_F,min`	1.00%	S10; selection std
Surrender charge schedule `sc(t)`	9.1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0%	S5
Surrender charge base	gross withdrawal − free withdrawal amount	S10
Free withdrawal	10% of the prior-anniversary account value, no carry-forward	S1 S3 S5 S6 S9 S10; combination std
Bonus vesting vector `v(t)`	0, 10, …, 100%	S5
Bonus clawback	`(1 − A) × [B/(1+B)] × C`	S10
MGV base / rate `i_nf`	87.5% of premium excluding the bonus / 1.00% inside the 0.15%–3% corridor	S10 R2; level std
Guaranteed simple rollup `g(t)`	5.00% (yrs 1–10), 2.00% (yrs 11–20), 0% after	S2
Rollup base convention	flat dollar increment on premium adjusted for withdrawals	S2 S9
Stacking factor `m`	150% of dollar credits, floored at 0	S8 S9
Growth period `T_g`	min(first lifetime withdrawal, contract year 20)	S1 S2
Lifetime withdrawal percentages `π`	age-band table (`product-spec.md`); joint = single − 0.50%	S3 S1; 80+ band std
Minimum age for lifetime withdrawals	50	S2 S3 S9
Rider charge maximum	1.50%, changeable only after contract year 15	S9
Post-depletion guarantee	income continues if depletion is caused by guaranteed withdrawals or rider charges; terminates if caused by excess withdrawals, surrender charges or MVA	S1 S9
Death benefit ≥ cash surrender benefit	statutory constraint	R2 §6

(b) Insurer-declared current elements (snapshot; revisable NGEs R6 REG-R26)#

Input	Value	Basis
Declared annual cap `c`	5.25%	S2 (rates effective 07/01/2022); selection std
Declared fixed account rate `i_F`	2.30%	S2
Rider charge rate `φ`	0.95% of the benefit base	S9
Allocation / strategy charge	0%	current 0%, max 2.50% S3 S4; choice std
MVA reference index level `iₜ`	scenario input	generic std; Barclay’s US Credit Index in the linear-form products S6 S7

Caveat. Every value here is a non-guaranteed element captured only as of the date on its source document, and Athene’s current rate sheets could not be fetched [S-f1]. Insurers review caps, participation rates, spreads and triggers frequently (e.g. monthly), targeting the priced product option budget and considering investment yields, option costs, volatility, premium volumes, competition and profit objectives R1. Reference re-declaration rule std: set c(t) so the one-year call-spread cost equals the option budget b_opt(t) = earned_rate(t) − required_spread, subject to c(t) ≥ c_min R1 R6. The base projection holds the snapshot scale level.

(c) Behavioral and experience assumptions#

Input	Recommended public basis	Tags
Annuitant mortality	2012 IAM Basic / 2012 IAR generational with Projection Scale G2: `q_x^{2012+n} = q_x^{2012} × (1 − G2_x)^n`, rounding applied from the 2012 period rate each time, never by compounding an already-rounded rate	REG-R59 REG-R60
Mortality A/E deviation	2020–2024 Individual Payout Annuity Mortality Experience Study (23 parent groups, >80% of industry sales, 3.1m contract-years, 143,190 deaths), presented against the 2012 IAM Table	REG-R61; A/E factor 100% std
Deferred-period mortality	materially under-evidenced — only a 2011–2015 deferred annuity mortality study and a 2006 analysis are public; qualified contracts show lower A/E than non-qualified, and FIAs without GLWBs showed an anomalous increasing A/E by account-value band	R1 REG-R65
Base surrender	shape: low early, rising through the surrender charge period, spiking in the shock year, then falling back but staying above pre-shock levels	R1; table std
Shock lapse	in the year the surrender charge expires, 10% with a GLWB rider versus 33% without	R8; corroborated unverified at REG-R62
GLWB withdrawal incidence	37% of GLWB contracts took withdrawals in 2019–2020 versus fewer than 30% without	R8
GLWB withdrawal efficiency	the majority of users withdraw 95%–105% of the maximum; contracts in that band have the lowest surrender rates; activated GLWBs lapse least	R1
GLWB activation timing	clusters at the required-minimum-distribution age; withdrawal rates rise with attained age, highest for qualified contracts at 70+	R1; REG-R64 unverified; assumption framework REG-R67
Additional premium	rare: 2.5% of contracts in years 2–10, 1.9% with a GLWB, 3.3% without	R8
Maintenance expense	$80 per contract per year, inflating 2.5% p.a.	std
Acquisition expense	6.0% of single premium at issue	std
Premium tax	0.0% (composite state basis)	std

The Academy’s guaranteed-living-benefit resource guide is the checklist to test a utilization assumption against — moneyness, age and RMD timing, qualified versus non-qualified, distribution channel, systematic withdrawal plan enrolment, rider type — and is explicitly non-binding, “a list of considerations and resources” REG-R67. The detailed FIA surrender and utilization tables are behind paid subscriptions R9 REG-R62, so every class (c) number is an order-of-magnitude anchor, not a calibration target.

Cash flow components and recursions#

Notation (defined once)#

Symbol	Meaning
`t`	contract year index, `t = 1, 2, …`, and the anniversary ending it; `x` = issue age (ANB); attained age at `t` is `x + t`
`P`, `b`, `v(t)`	single premium; bonus rate (0.07 S5); vested percentage in year `t` S5
`F(t)`, `A(t)`, `AV(t)`	fixed / indexed / total account value after all processing at `t`
`I(t)`, `R(t)`	index level; `R(t) = I(t)/I(t−1) − 1` (price index, dividends excluded S6 R1)
`c`, `c_min`, `f`	declared cap (0.0525 S2); guaranteed minimum cap (0.0025 S4); floor (0)
`p`, `s`, `d`, `c_m`	participation rate; spread / index margin; trigger rate; monthly cap (variants)
`cr(t)`, `IC(t)`, `FI(t)`	credit rate; index credit amount; fixed account interest
`i_F`, `i_F,min`	declared / guaranteed minimum fixed rate (0.0230 S2 / 0.0100 S10)
`φ`, `Φ(t)`	rider charge rate (0.0095 S9); rider charge amount
`BB(t)`, `RB(t)`, `g(t)`, `m`	benefit base; rollup base; rollup rate S2; stacking factor (1.50 S8 S9)
`LW(t)`, `π(a, basis)`	annual lifetime withdrawal amount; payout percentage at attained age `a` S3
`G(t)`, `E(t)`, `ρ(t)`	gross withdrawal; excess above `LW`; proportional reduction factor
`FW(t)`, `X(t)`	free withdrawal amount `= 0.10 × AV(t−1)`; chargeable amount
`SC(t)`, `CB(t)`, `MVA(t)`	surrender charge; non-vested bonus clawback; MVA (signed)
`MGV(t)`, `i_nf`	guaranteed minimum value; nonforfeiture accumulation rate (0.0100 std)
`i₀`, `iₜ`, `n`	MVA index at issue, at withdrawal; months remaining in the MVA period
`q(t)`, `w(t)`, `l(t)`	annual mortality rate; surrender rate; in-force probability

Dimensional check. cr is dimensionless, so IC = base × cr is currency; Φ = φ × BB is currency — a rate applied to a notional amount (the benefit base has no cash value S1 S9) producing a real deduction from the account value; LW = π × BB is currency per year; ρ is dimensionless. All account-value terms are currency.

Initialisation (`t = 0`)#

A(0) = alloc_indexed × P × (1 + b)     F(0) = alloc_fixed × P × (1 + b)      [S5]
AV(0) = P × (1 + b) = 107,000
BB(0) = P = 100,000        RB(0) = P = 100,000                               [S9]
MGV(0) = 0.875 × P = 87,500            (bonus excluded)                      [S10] [R2]
LW(0) = 0     phase(0) = ACCUM     rider_in_force(0) = true

Processing order at anniversary `t` std#

Fixed for the reference model, following Nassau’s stated sequence — the rider charge is deducted after index credits are added S9 — with the benefit-base update after the charge, so the charge is always assessed on the opening base as Φ(t) = φ × BB(t−1).

Index credit and fixed interest. 2. Rider charge. 3. Benefit base: rollup, stack, step-up.
Lifetime / excess withdrawal. 5. Charges on the excess and proportional reduction of the guarantee. 6. Guaranteed minimum value roll. 7. Phase transition (incl. depletion test).
Decrements. Steps 1–3 are skipped in DEPLETED; steps 1–7 in TERMINATED.

Step 1 — index credit.

cr(t) = max( f , min( c , R(t) ) )                              [S2] [S4] [S10] [R1]
IC(t) = A(t−1) × cr(t)      FI(t) = F(t−1) × i_F
AV⁽¹⁾(t) = AV(t−1) + IC(t) + FI(t)

Variants, all floored at f: max(f, p × R) S4 S10 R1; max(f, min(c, p × R)) — worked at R1 as min(80% × 10%, 6%) = 6%; max(f, p × R − s) S8 R1; d × 1{R ≥ 0} R1; max(f, Σ_{k=1..12} min(R_k, c_m)) S4 R1.

Segment bookkeeping. One annual segment per indexed account, created at anniversary t−1 with balance A(t−1), maturing at t; the credit locks at maturity and cannot be lost to later declines S1. The credit base is the segment’s opening balance less withdrawals from that account during the segment — Midland’s “Interest Credit Basis” S6 — which collapses to A(t−1) here because all transactions occur at anniversaries std. Reallocation is permitted at each anniversary S1 S5; dividends are excluded from R(t) S6 R1. The floor applies to the credit, not to the account value — charges do reduce the account value below its prior balance S7. On a monthly grid, mid-segment conventions are: no credit in the year of withdrawal S1; prorated for the portion of the year the money stayed in the allocation S3; G × PAR/(1 + PAR), PAR being Nassau’s Protected Account Return and not a participation rate S10; full earnings-to-date on the free amount and pro rata above S11.

Step 2 — rider charge.

Φ(t) = φ × BB(t−1)          AV⁽²⁾(t) = AV⁽¹⁾(t) − Φ(t)                        [S9]

Deducted from the fixed account first, then proportionately across indexed accounts S9. φ is fixed for 15 contract years, then resettable but never above 1.50% S9. Variants: charge on the contract value S5; no explicit charge, the guarantee funded through lower caps, participation rates or higher index margins S3 S8. The composite does not deduct the charge from MGV; Athene deducts from both the accumulated value and the minimum guaranteed contract value except in certain states S1 S2, and Allianz deducts its allocation charge from the guaranteed minimum value in most states S3 S4 — implement as a switch std.

Step 3 — benefit base.

rollup(t) = g(t) × RB(t−1)                    if t ≤ T_g, else 0        [S2] [S9]
stack(t)  = m × max( 0 , IC(t) + FI(t) )      if t ≤ T_g, else 0        [S8] [S9]
BB⁽³⁾(t)  = BB(t−1) + rollup(t) + stack(t)
BB⁽⁴⁾(t)  = max( BB⁽³⁾(t) , AV⁽²⁾(t) )              annual step-up **[std]**

T_g = the earlier of the first lifetime withdrawal and contract year 20 S1 S2. The rollup is a flat dollar increment, not simple interest on the grown base — Athene computes it on premium less withdrawals S2, Nassau on the adjusted initial base S9, and Nassau’s 15-year table confirms a constant $3,000 per year on a $100,000 adjusted initial base S9. The stack is on realised dollar credits net of any strategy fee, floored at zero S9.

The step-up is a std generalisation. No retrieved document describes an automatic annual ratchet during deferral; documented instead are an at-exercise step-up to the contract value S5, an annual benefit amount computed on the greater of base and account value at exercise S9, and a never-decreasing income amount once withdrawals begin S3. Testing once at exercise reproduces S5/S9; testing annually is the superset. Under the blended baseline the step-up rarely binds — an extra dollar of credit adds $1 to the account value and $1.50 to the base — so it binds mainly in the pure-rollup variant (see the worked example). Rarely, not never: the account-value bonus starts AV(0) = 107,000 above BB(0) = 100,000, so a first contract year with a zero index credit gives AV⁽²⁾(1) = 106,050 against BB⁽³⁾(1) = 105,000 and the step-up binds. Test it at every anniversary rather than assuming the stack dominates.

Three growth mechanisms must be expressible; the baseline is (c):

	Mechanism	Configuration	Observed at
(a)	Guaranteed deferral rollup only	`m = 0`; `g` simple on `RB`, or compound on `BB`	S1 S2 simple, S5 compound (Options 1, 3, 5), S9 simple
(b)	Index-credit stacking only, no guaranteed rollup	`g = 0`; `m` = 1.50 or 2.50	S3 S4 (150% Balanced / 250% Accelerated)
(c)	Blended — baseline std	`g` = 5.00%/2.00%, `m` = 1.50	S2 S8 S9

In design (b) the account value is deliberately starved: Allianz’s Accelerated option credits 250% of index interest to the benefit base but only 50% to the account value, and the contract defaults to Balanced (150%/100%) once lifetime withdrawals begin S3. Model this as an account-value interest factor κ ∈ {0.50, 1.00} applied to IC(t) in step 1, with the benefit-base factor m S3 S4. In the benefit-base-only bonus designs the bonus is added to BB(0) and never touches the account value or the surrender benefit S3 S4 S8.

Step 4 — lifetime and excess withdrawal. Exercise is permitted from attained age 50 S2 S3 S9:

LW(t) = π( x + t , basis ) × BB⁽⁴⁾(t)                                    [S3] [S9]

π is locked at first exercise std; joint = single − 0.50% on the younger life S1 S3. After exercise the ratchet still applies — LW(t) = max( LW(t−1) , π × BB⁽⁴⁾(t) ) — so income never decreases S3. For a gross withdrawal G(t):

guaranteed portion = min( G(t) , LW(t) )      E(t) = max( 0 , G(t) − LW(t) )
AV⁽⁵⁾(t) = AV⁽²⁾(t) − G(t)

Withdrawals up to LW carry no surrender charge, no MVA and no bonus clawback even if LW exceeds the free withdrawal amount S9; unused LW does not carry forward S9 (Allianz accumulates it without interest as a “cumulative withdrawal amount” S3).

Step 5 — charges on the excess and proportional reduction.

X(t)   = max( 0 , G(t) − FW(t) )                      pre-exercise
X(t)   = max( 0 , E(t) − remaining free withdrawal )  post-exercise **[std]**
SC(t)  = X(t) × sc(t)                                                    [S5] [S10]
CB(t)  = ( 1 − v(t) ) × [ b / (1 + b) ] × X(t)                           [S10]
MVA(t) = X(t) × { [ (1 + i₀) / (1 + iₜ) ]^(n/12) − 1 }                   [S10]

Whether the guaranteed withdrawal consumes the free withdrawal amount is a std choice. S9 says only that withdrawals up to the annual benefit amount carry no charge “even if greater than the Free Withdrawal Amount”; it does not say whether they exhaust it. The convention above — remaining free withdrawal = max(0, FW(t) − LW(t)) — is the insurer-favourable reading and is what the worked example uses; the alternative leaves the full FW(t) available against the excess.

MVA is signed — negative when the reference yield has risen S10 — and is limited to |MVA| ≤ max(0, G(t) − SC(t) − CB(t) − MGV(t)), so a negative MVA combined with charges never reduces the surrender value below the guaranteed minimum value and the maximum positive MVA cannot exceed the maximum negative MVA S10. MVA = 0 outside the MVA period and on the death benefit S5 S6 S7 S10.

pre-exercise    ρ(t) = G(t) / AV⁽²⁾(t)                          [S1] [S3] [S5] [S9]
post-exercise   ρ(t) = E(t) / ( AV⁽²⁾(t) − LW(t) )                       [S9]
BB(t) = BB⁽⁴⁾(t) × (1 − ρ)     LW(t) ← LW(t) × (1 − ρ)     RB(t) = RB(t−1) × (1 − ρ)

with ρ = 0 when E(t) = 0, and ρ = 1 (base to zero, rider terminates S9) if the post-exercise denominator is non-positive. The post-exercise denominator is the account value after the guaranteed payment has notionally been taken — verbatim at S9: account value $100,000, base $200,000, annual benefit amount $10,000, withdrawal $28,000 → denominator $90,000, excess $18,000, reduction 20%, base → $160,000, benefit amount → $8,000. An RMD above LW is not an excess withdrawal after exercise; before exercise it reduces the base pro rata S1 S9. The alternative rollup-base convention — Athene’s dollar subtraction, “Premium minus Withdrawals” S2 — is RB(t) = max(0, RB(t−1) − G(t)).

Step 6 — guaranteed minimum value.

MGV(t) = max( 0 , MGV(t−1) × (1 + i_nf) − G(t) )                         [R2] [S10]

with MGV(0) = 0.875 × P excluding the bonus S10. Model #805 §4A also permits an accumulated $50 annual contract charge and accumulated premium tax to be deducted R2; both are set to zero std because no retrieved product declares an actual annual policy fee, making the modeled floor slightly conservative. Contractually i_nf = min(3%, max(0.15%, CMT₅ − 125 bp − Δ)) where Δ ≤ 100 bp is the FIA additional reduction available while the contract provides substantive participation in an equity indexed benefit R2; Δ requires an annualized option cost of the guaranteed index features ≥ 25 bp and then equals min(100 bp, annualized option cost), certified annually R3. Whether the 15 bp floor of §4B(3) survives the §4C reduction is not stated in the retrieved text unverified; Nassau’s “the interest rates will range between 0.15% and 3%” suggests it does S10. Correction: the §4B floor is 15 basis points, not 1% R2; the composite’s 1.00% is a std pick inside the corridor, not the statutory floor.

Step 7 — phase transitions and the post-depletion liability.

ACCUM    → INCOME      first lifetime withdrawal, attained age ≥ 50      [S2] [S3] [S9]
INCOME   → DEPLETED    AV ≤ 0 attributable only to guaranteed withdrawals
                       and rider charges                                 [S1] [S9]
INCOME   → TERMINATED  AV ≤ 0 attributable to an excess withdrawal, a
                       surrender charge or an MVA                        [S1] [S5] [S9]
any      → TERMINATED  death, full surrender, or BB reaching zero        [S9]
DEPLETED → TERMINATED  death of the covered person only                  [S1] [S9]

DEPLETED is where the economic value of the guarantee sits. In it the insurer pays LW annually from its own funds for the rest of the covered life S1 S3 S9 R1; there is no account value, so no rider charge is deducted S9 and no index credit is computed; the surrender value and death benefit are zero; lapse is impossible, so every surrender and dynamic-lapse formula must be switched off and l(t) = l(t−1) × (1 − q(t)); under the joint option the payment continues to the survivor S1 S9. The attribution test is not cosmetic — an account value run to zero by an excess withdrawal loses the guarantee entirely S1 S5 S9. Implement it as depletion_cause, set in step 5 whenever E(t) > 0, SC(t) > 0 or MVA(t) < 0, and evaluate it before the depletion test. Athene’s confinement and terminal illness waivers are themselves excess withdrawals that terminate the income rider S1 — a trap if waivers are added.

Step 8 — decrements and cash flow outputs. l(t) = l(t−1) × (1 − q(t)) × (1 − w(t)), with w(t) = 0 in DEPLETED.

Cash flow	Formula	Weight
Premium income (+)	`P` at `t = 0`	1
Guaranteed withdrawal (−)	`min(G(t), LW(t))`	`l(t−1)`
Excess withdrawal (−)	`E(t) − SC(t) − CB(t) + MVA(t)`	`l(t−1)`
Surrender (−)	`CSV(t) = max( AV(t) − SC(t) − CB(t) + MVA(t) , MGV(t) )`	`l(t−1) × (1 − q(t)) × w(t)`
Death benefit (−)	`max( AV(t) , MGV(t) )`, full bonus vesting, no charges	`l(t−1) × q(t)`
Post-depletion income (−)	`LW` while `phase = DEPLETED`	`l(t−1)`
Acquisition expense (−)	6.0% of `P` at `t = 0` std	1
Maintenance expense (−)	`80 × 1.025^(t−1)` std	`l(t−1)`

SC, CB and Φ(t) are internal transfers within the account value, not separate cash flows — they reduce what is ultimately payable. Reporting them as fee income while also projecting the account value net of them double-counts.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions; the qualitative and order-of-magnitude evidence is cited, and the tables that would calibrate them are behind paid subscriptions R9 REG-R62.

Base surrender table std — shape from R1: low early, rising through the surrender charge period, spiking at expiry, then falling back but staying above pre-shock levels.

Contract year	1–3	4–6	7–9	10	11 (shock)	12+
`w_base`	2%	3%	4%	5%	see below	6%

Shock lapse with rider suppression std — the single most important behavioral fact in the product. In the year the surrender charge expires the surrender rate was 10% with a GLWB rider versus 33% without R8:

w_shock = 0.33   no GLWB rider
w_shock = 0.10   GLWB in force but not activated
w_shock = 0.05   GLWB activated (phase = INCOME)

The third row extrapolates std from the qualitative finding that “contracts with GLWBs lapse less than those without” and “activated GLWBs lapse least,” with the lowest rates where the withdrawal is 95%–105% of the maximum R1. Applying a plain fixed-deferred shock lapse — reported at roughly 52%–56% for fixed-rate deferred annuities [REG-R63, unverified](#uslib-reg-r63) — to an FIA with an in-force rider will materially understate the tail this product is sold for.

Rider moneyness multiplier std. Surrender is further suppressed when the guarantee is in the money:

M_money(t) = clamp( 1 − 0.6 × max(0, BB(t)/AV(t) − 1) , 0.2 , 1.0 )
w(t) = min( 0.35 , w_base(t) × M_shock(t) × M_money(t) ),   w(t) = 0 in DEPLETED

Rational surrender destroys a guarantee worth BB − AV in benefit-base terms; the observed direction is documented R1 R8, the functional form is not.

GLWB activation (utilization timing) std. Activation clusters at the required-minimum-distribution age R1 [REG-R64 unverified](#uslib-reg-r64), which makes the RMD regulations a behavioral input, not merely a tax one REG-R57 REG-R58:

h(a) = 0.00  for a < 60      h(a) = 0.05  for 60 ≤ a < rmd_age
h(a) = 0.40  at a = rmd_age  h(a) = 0.15  for a > rmd_age

with rmd_age = 73 std as a configurable model parameter — the statutory age is set by IRC §401(a)(9) as amended by SECURE 2.0 and finalized in T.D. 10001 REG-R57 REG-R58, is not printed in the retrieved research material, and must not be hard-coded. The deterministic base run activates at the model point’s income_start_age instead.

Withdrawal intensity given activation std. The majority of users withdraw 95%–105% of the maximum R1, and 37% of GLWB contracts took withdrawals in 2019–2020 versus fewer than 30% without a rider R8. Base assumption: withdraw exactly LW, with sensitivities at 0.95 and 1.05. The 1.05 case is an excess withdrawal and triggers the pro-rata reduction in step 5 — a 5% overdraw permanently reduces the guarantee, which is why efficiency and excess-withdrawal assumptions cannot be set independently.

Excess withdrawal incidence std: zero in the base run; any non-zero assumption must route through step 5 and the INCOME → TERMINATED attribution test S1 S5 S9. Additional premium std: none — deposits occur on 2.5% of contracts in years 2–10 and only 1.9% with a GLWB R8. Annuitization std: not modeled; where offered, payments are based on the greater of account value and cash surrender value — not the benefit base S3 S10, so annuitization is generally dominated by the GLWB S3.

Worked example#

Anchor cell: Male 62 ANB, single life, P = $100,000, b = 7%, GLWB elected at issue, first lifetime withdrawal at anniversary 8 (attained age 70). Parameters as specified: c = 5.25% S2, f = 0% S1, φ = 0.95% S9, g = 5.00% for years 1–10 S2, m = 1.50 S8 S9, π(70, single) = 5.20% S3, sc(8) = 3% S5, v(8) = 70% S5, i_nf = 1.00% std. Opening state at anniversary 7 (illustrative balances, broadly consistent with a seven-year deferral at these parameters) std: AV(7) = 128,000.00 (100% indexed), BB(7) = 180,000.00, RB(7) = 100,000.00, MGV(7) = 93,811.84 (= 87,500 × 1.01⁷), Wcum(7) = 0.

#	Item	Formula	Value
1	Index return, year 8	`5,450 / 5,000 − 1`	9.0000%
2	Credit rate	`max(0, min(5.25%, 9.00%))`	5.2500%
3	Index credit `IC(8)`	`128,000.00 × 0.0525`	6,720.00
4	Account value after credit	`128,000.00 + 6,720.00`	134,720.00
5	Rider charge `Φ(8)`	`0.0095 × 180,000.00`	1,710.00
6	Account value after charge `AV⁽²⁾`	`134,720.00 − 1,710.00`	133,010.00
7	Guaranteed rollup	`0.0500 × 100,000.00`	5,000.00
8	Stacking credit	`1.50 × 6,720.00`	10,080.00
9	Benefit base before step-up	`180,000.00 + 5,000.00 + 10,080.00`	195,080.00
10	Step-up test	`max(195,080.00, 133,010.00)`	195,080.00 (does not bind)
11	Lifetime withdrawal `LW(8)`	`0.0520 × 195,080.00`	10,144.16
12	Free withdrawal amount `FW(8)`	`0.10 × 128,000.00`	12,800.00
13	Excess `E(8)`	`max(0, 10,144.16 − 10,144.16)`	0.00 → no SC, MVA or clawback S9
14	Account value `AV(8)`	`133,010.00 − 10,144.16`	122,865.84
15	Guaranteed minimum value `MGV(8)`	`93,811.84 × 1.01 − 10,144.16`	84,605.80
16	Closing benefit base `BB(8)`	unchanged by a guaranteed withdrawal S9	195,080.00

Surrender test at the same anniversary. A full surrender of G = AV(8) = 122,865.84 with 12,800.00 − 10,144.16 = 2,655.84 of free amount remaining gives X = 120,210.00; SC = 3% × 120,210.00 = 3,606.30 S5 S10; clawback = 0.30 × (0.07/1.07) × 120,210.00 = 2,359.26 S10; with i₀ = 3.00%, iₜ = 3.50% and n = 24 months remaining, MVA = 120,210.00 × [(1.03/1.035)² − 1] = 120,210.00 × (−0.00963850) = −1,158.64 S10, inside the limit max(0, 122,865.84 − 5,965.56 − 84,605.80) = 32,294.48 S10. Net proceeds = 122,865.84 − 3,606.30 − 2,359.26 − 1,158.64 = 115,741.64, and CSV = max(115,741.64, 84,605.80) = 115,741.64.

Where the step-up binds. Under variant (a) — 3% simple rollup on RB, no stacking S9 — the same cell carries BB(7) = 121,000.00, so Φ(8) = 1,149.50, AV⁽²⁾ = 133,570.50 and BB⁽³⁾ = 124,000.00. The step-up then binds: BB(8) = 133,570.50 and LW(8) = 0.0520 × 133,570.50 = 6,945.67. General result: the step-up matters when realised index credits outrun the guaranteed rollup, and is dominated whenever a stacking factor above 1.0 is present.

Where the liability lands. Holding index credits at zero from anniversary 8, the account value drains by LW + Φ = 10,144.16 + 0.0095 × 195,080.00 = 11,997.42 a year and is exhausted during contract year 19, at attained age about 81. From that point the insurer pays $10,144.16 a year for the rest of the contract holder’s life, with no account value, no surrender value, no death benefit and no possibility of lapse S1 S3 S9 R1. That stream is the guarantee.

Valuation and reserve pointers#

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

Formulaic statutory (CARVM). AG 33 — printed title “Determining CARVM Reserves for Annuity Contracts With Elective Benefits” — constructs and values the integrated benefit streams for annuity contracts with elective benefits; AG 35 specifies how the index feature enters that greatest-present-value calculation, through four computational methods with quarterly certification and change-notification requirements, “Type 1” and “Type 2” being the guideline’s own printed section headings, not industry shorthand REG-R151 REG-R152. Both texts have now been read in full from the AP&P Manual Appendix C print — a free download, not the paid publication recorded earlier REG-R33 — so their mechanics are no longer unverified; titles and continued incorporation remain independently confirmed by the VM-C index REG-R41. AG 33’s effective date, recorded on both sides because the two do not reconcile: its own Effective Date block reads “This guideline shall be effective on December 31, 1998, affecting all contracts issued on or after January 1, 1981”, with a grade-in of 33⅓ / 66⅔ / 100% that completed on December 31, 2000 and therefore has no live effect on any current valuation REG-R151; the library elsewhere carries December 31, 1995 for a differently-titled instrument from IRS Rev. Rul. 2002-6. The 1981 issue-date reach is common to both, the extracted pages carry no amendment history, and the reconciliation is unresolved — that the guideline was later revised is an inference, not something either source states. AG 35 prints no date at all; its only temporal language is “regardless of the date of issue” REG-R152. Two documents in this chain remain unavailable and are named rather than glossed: AG IX-B, which AG 35 points at three times as an alternative source of the valuation interest rate for an indexed contract and which this library holds only as a VM-C index entry REG-R41 REG-R152, and the NAIC Interest-Indexed Annuity Contracts Model Regulation, Sections 5 and 6 of which AG 35 supersedes and which is not in this library at all REG-R152.
Principle-based statutory. VM-22, effective for valuation dates on or after January 1, 2026, with a three-year elective transition and mandatory prospective application three years after the effective date REG-R36; the Academy paper states elective 1/1/2026 and required 1/1/2029 R1; the VM-22 (A) Subgroup handles post-launch monitoring R7. An FIA sits in the Accumulation reserving category, which expressly includes fixed income streams from guaranteed living benefits after account exhaustion — the DEPLETED phase of this model — and GLB utilization risk is named among the risks to be reflected; the stochastic reserve is CTE70 REG-R36. Maximum valuation interest rates for formulaic income-annuity reserves are in VM-V Section 1, not VM-22 REG-R36 REG-R37. Enabling statute: Model #820 REG-R1; parent document REG-R3.
Asset adequacy. ASOP No. 22, meaning the same projection must serve CARVM and cash flow testing REG-R29 REG-R27. AG 35 does not itself impose the requirement — it directs that reserves be tested “to the extent required by law, regulation, or regulatory requirements” REG-R152. The operative NAIC requirement is VM-30, with SVL §6.B, codified as A-822 ¶3, behind it, and any shortfall becomes an additional reserve REG-R100 REG-R1 REG-R153.
Nonforfeiture floor. Model #805 §4 and Model #806 §7 as implemented in step 6 R2 R3 REG-R42.
Tax reserve. IRC §807: the greater of net surrender value and 92.81% of the NAIC-prescribed method — CARVM for annuities — capped at the statutory reserve REG-R16.
GAAP (LDTI). The index feature is an embedded derivative, fair-valued on expected current and future index credits (current index period closed-form Black-Scholes); the GLWB is a market risk benefit at fair value with an adjustment for explicit fees; remaining cash flows form the host contract discounted at a host accrual rate set so the total liability at issue equals the premium; DAC, DSI and URL are the intangibles R1 REG-R34 REG-R71.
Standards for the modeling work. ASOP No. 56 (modeling) REG-R32; ASOP No. 54 (pricing, if a profit-metric mode is added) REG-R70; ASOP No. 2 for any NGE re-declaration logic R6 REG-R26.

Key sensitivities and model risks#

Dominant assumptions, in rough order of impact on the value of the guarantee:

GLWB activation timing and intensity. The liability is a function of when the holder starts and whether they take 95%, 100% or 105% of the maximum R1 R8 REG-R64 REG-R67; activation five years earlier compounds directly into the depletion date.
Surrender, specifically the rider-suppressed shock lapse. 10% versus 33% in the shock year R8 governs how much of the block survives to reach DEPLETED at all. Applying a fixed-deferred shock lapse to a rider-in-force FIA is the most consequential error available here.
Longevity in the DEPLETED phase. The payment is a life annuity written at a payout percentage set decades earlier: use the 2012 IAM/IAR family with Scale G2 REG-R59 REG-R60 and test against the 2020–2024 payout experience REG-R61; deferred-period annuitant mortality, which governs who reaches the income phase, is served publicly by only two dated studies REG-R65.
Cap re-declaration and the option budget. The declared cap drives the account value, the stacking credit and hence the benefit base; it is an NGE reset against the option budget R1 R6 REG-R26 REG-R68. Holding the snapshot cap constant for 40 years is a strong implicit assumption.
Benefit base growth form. Blended versus pure rollup versus pure stacking changes both the level of the guarantee and its correlation with index performance; pure stacking shifts the deferral guarantee from insurer to market and is materially cheaper to hedge S3 S4.

Known pitfalls:

The behavioral assumptions above must not be reused in a CARVM valuation. Every dynamic lapse, activation, utilization and excess-withdrawal formula in “Policyholder behavior modeling” is an experience assumption. AG 33 states that for elective benefits “incidence rates should not be based on tables reflecting past company experience, industry experience or other expectations” — the elective assumption is not an assumption at all but a decision variable maximised over, with all rates between 0% and 100% theoretically in scope and the greatest present value typically occurring at 0% or 100% REG-R151. Wiring the projection’s behavior module into the reserve run is a silent compliance error. The non-elective side is the opposite case: incidence comes from the SVL-prescribed tables where one exists, otherwise from company or industry experience with margins for conservatism (unquantified), and the SVL-prescribed annuity mortality table discounts every payment in every integrated benefit stream for survivorship — the elective surrender and withdrawal streams included, so a cash-value stream is not valued on a mortality-free basis REG-R151.
“Efficient policyholder selection” is not AG 33’s language and should not be attributed to it. The phrase appears nowhere in the guideline; the actual construction is the prohibition, the trial sets, and the direction to “consider, not necessarily test” all potential integrated benefit streams REG-R151.
The 0% floor is on the index credit, not the account value. Rider charges and strategy fees can exceed interest credited, “which would result in loss of premium” S7. Flooring the account value silently removes the charge drag that produces depletion.
The clawback factor is b/(1+b), not b S10 — the account value already contains the bonus; using b over-recovers by (1+b).
The “simple rollup” is a flat dollar increment, on premium less withdrawals S2 or on the adjusted initial base S9 — never simple interest on the current grown base. Compounding it inflates the base and every downstream charge and payment.
Attribution at depletion. Survival of the income stream depends on the cause S1 S5 S9; a model testing only AV ≤ 0 will either give the guarantee away after an excess withdrawal or destroy it after a legitimate one.
No lapse in DEPLETED. Leaving the surrender decrement on silently truncates the most expensive part of the liability.
Rider charge base and ordering. The charge is on the benefit base, not the account value S1 S2 S9, and is taken after index credits S9. In the worked example the benefit base closes at 1.59× the account value at anniversary 8 (195,080.00 against 122,865.84), so charging on the account value understates the deduction by a growing margin.
Excess-withdrawal denominator. Post-exercise it is the account value net of the guaranteed amount S9; pre-exercise it is the gross account value S1 S3 S5 S9 — the two differ by exactly LW.
MVA sign and collar. Negative when yields rise S10; applies only above the free amount, only inside the MVA period, never to the death benefit, never below the nonforfeiture minimum S1 S5 S6 S7 S10. The linear form (i₀ − iₜ) × T S6 S7 is unbounded and must be collared separately; the ratio form used here is not.
Monthly-sum floor convention is ambiguous. The Academy’s worked example applies a 0% monthly floor as well as a 1% monthly cap R1, which is unusual — most monthly-sum designs cap the upside monthly but let negative months subtract in full, and the Allianz rate sheet declares a 1.70% monthly cap with a 0.50% guaranteed minimum without stating the floor S4. Verify against a contract before implementing that variant.
Interim values and index costs. The baseline has none; adding one requires a different structure — Nassau’s Daily Account Value / Protected Account Value with a 90% protection level S10 or Nationwide’s Balanced Allocation Value S11, both daily marks of the embedded option rather than interpolations. Proprietary volatility-controlled indices separately deduct embedded servicing, transaction and financing costs (0.50% p.a. at BNPP MAD 5 and AiPEX S2; 2 bp change-in-notional plus 12 bp annualized replication at S&P 500 Dynamic Intraday TCA S10), which reduce R(t) before the cap or participation rate.
Stale and state-varying parameters. Declared rates are dated 07/01/2022 S2 and the access date S4; Athene’s rate sheets [S-f1], American Equity’s official host [S-f3] and the current Nationwide brochure [S-f4] could not be fetched. State variation in surrender charges, vesting, MVA availability and waiver terms is extensive and deliberately not modeled S2 S3 S6 S7.