Technical Notes#

Status: Draft, 2026-08-04 (cited sources accessed 2026-08-04, except the AP&P Manual appendix items R151–R157, accessed 2026-08-06 — see sources.md).

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/registered-index-linked-annuity.md; [REG-R#] tags refer to the shared cross-product numbering space, which now runs R1–R157 with most of the R73–R149 block unused, curated at references/regulatory-and-actuarial-references.md (R1–R34 from _research/regulatory-actuarial.md, R35–R72 from _research/regulatory-actuarial-annuities.md). std marks standardizations introduced for the reference implementation. Parameter values are identical to those in product-spec.md; the mechanics anchor is the Brighthouse Shield Level II prospectus, Appendix F S2.

Read this first. A conforming RILA model requires an option-pricing routine and a market-data interface (discount curve, implied volatility surface, dividend yield) — not as a refinement but as a precondition. Actuarial Guideline LIV makes the Interim Value — the value at which every mid-term withdrawal, surrender, death benefit, annuitization, transfer and fee deduction settles — the market value of a hypothetical replicating portfolio of European options plus a fixed income proxy R2, and the source prospectuses implement exactly that with Black-Scholes S2 S6 or an equivalent market-standard European model S4. No other product in this library has a contractual value that cannot be computed without a derivatives pricer. Architecturally the crediting engine, the interim-value engine and the market-data provider are three separate components, and the interim-value engine is called at every projection step for every open option.

Model scope and conventions#

Purpose. Project gross liability cash flows (premium, surrender and withdrawal payments, death claims, annuitization outgo, expenses) for a single-contract model point. Reserves are not computed (see Valuation and reserve pointers).
Projection frequency. Monthly std. The contractual interim value is a daily quantity S2 S4 S5 S6; the model evaluates it at each month end. Terms are integer years, the withdrawal-charge schedule runs by complete contract years and free-withdrawal limits reset annually S1 S2, so a monthly grid captures every contractual boundary. A daily sub-grid is needed only for a path-dependent Performance Lock election module S2.
Timing convention. Month index t = 0, 1, 2, … denotes month ends, t = 0 being the Issue Date. Within a month std: market state, then term-end crediting (if the month end is a Term End Date), then interim values, then contract-holder transactions, then decrements. Full ordering below.
Age basis. Age nearest birthday (ANB) std — the 2012 IAM Period Table printed in Model #821 and VM-M is stated age nearest birthday REG-R59, and VM-21 prescribes percentages of the 2012 IAM Basic Table with Scale G2 for prudent-estimate mortality on contracts with VAGLBs and roll-up GMDBs REG-R35 REG-R59 — a class that does not include the return-of-premium GMDB modeled here, so the prudent-estimate basis is a scope reference, not a prescription for this design.
Model points. Single-contract model points on an expected (probability-weighted) basis: in-force factors multiply per-contract cash flows. A model point is one contract holding one index-linked option; multi-option contracts are a vector of options sharing one contract-level decrement and one contract-level guarantee base.
Index basis. All representative indices are price return S1 S2, so the dividend yield is a live pricing input — omitting it overprices every call in the portfolio.
Rounding. Full precision internally; cash flows reported to cents std.

Model point attributes#

Attribute	Type	Example (anchor cell std)
`issue_age`	int (ANB)	60
`sex`	enum {M, F}	M
`issue_date`	date	2026-01-01
`single_premium`	currency	100,000
`option_id`	int (index-linked bucket)	1
`index_code`	enum {SPX, RTY, MXEA, NDX} — price return	SPX
`term_years` `T`	int {1, 3, 6}	6
`buffer` `b`	rate	0.10
`crediting_type`	enum {CAP, STEP, EDGE} (FLOOR module optional)	CAP
`declared_cap` `c`	rate (NGE, reset each term)	1.00
`declared_step` `s`	rate (NGE)	0.08
`declared_edge` `e`	rate (NGE)	0.06
`participation` `PR`	rate (NGE)	1.00
`guar_min_cap(T)`	rate	0.02 / 0.06 / 0.08 for T = 1 / 3 / 6 S2
`term_start_index` `I_s`	index level	100.00 (normalized)
`term_start_mvr` `r_0`	annual effective rate	0.0400
`investment_amount` `IA`	currency	100,000
`fixed_account_value`	currency	0
`holding_account_value`	currency	0
`rop_base`	currency (GMDB return-of-premium base)	100,000
`wc_schedule`	vector by complete contract year	(0.07, 0.07, 0.06, 0.05, 0.04, 0.03, 0.00) S1 S2
`lock_flag`	bool (Performance Lock exercised this term)	false
`lock_value`	currency (valid when `lock_flag`)	—

State variables#

Variable	Description	Updated
`IA_k(t)`	Investment Amount (AG 54 “Index Strategy Base”) of option k	term-end crediting; proportional withdrawal reduction
`V_k(t)`	Interim Value of option k at month end t	every month (option repricing)
`beta_k`	Initial option budget per unit of notional for the current term	at each Term Start Date
`I(t)`	Index level	every month (scenario input)
`r(t)`	Market Value Rate (CMT at the term’s maturity)	every month (scenario input)
`FA(t)`, `HA(t)`	Fixed Account / Holding Account values	monthly accrual
`AV(t)`	Account Value = sum of V_k(t) + FA(t) + HA(t)	monthly
`ROP(t)`	Return-of-premium GMDB base	proportional reduction on withdrawal
`FW_used(y)`	Free withdrawal amount consumed in contract year y	on withdrawal; resets annually
`AV_anniv(y)`	Account Value at the prior Contract Anniversary (free-withdrawal base)	annually S1 S2
`cy(t)`	Complete contract years since issue = floor(t/12)	monthly
`tau_k(t)`	Years remaining in option k’s term = (days remaining)/365 S2	monthly
`l(t)`	In-force probability at end of month t; `l(0) = 1`	monthly decrements
`lock_flag_k`, `LV_k(t)`	Performance Lock state and locked value	on election

Assumption inputs#

Three assumption classes are distinguished, plus a fourth block that is peculiar to this product: the contractual formula itself consumes market data.

(a) Contractual / guaranteed elements#

Input	Value	Basis
Buffer `b` (guaranteed for the term)	0.10	S1 S2
Term length `T`	6 years (menu 1/3/6)	S1 S2; menu std
Minimum guaranteed Cap Rate	2% / 6% / 8% for T = 1 / 3 / 6	S1 S2
Minimum guaranteed Step Rate, Edge Rate	2%, 2%	S2
Minimum guaranteed interest rate (Fixed / Holding Account)	1%	S1 S2
Withdrawal charge schedule `wc(cy)`	7, 7, 6, 5, 4, 3, 0 %	S1 S2
Free Withdrawal Amount	0 in contract year 1; thereafter 10% of `AV_anniv`, non-cumulative	S1 S2
Death benefit	max(Account Value, ROP base) for issue ages ≤ 80	S2
Value used for every mid-term transaction	Interim Value	S1 S2 S4 S6
Interim value formula	family (a), straight-line budget amortization, CMT discount	S2; selection std
Transfer Period	5 calendar days after the Contract Anniversary coinciding with a Term End Date; `V = IA` during it	S1 S2
Maturity Date	later of (anniversary after oldest owner’s age 90) and 10 years	S2

(b) Insurer-declared current elements (NGEs, revisable under ASOP No. 2 R5 REG-R26)#

Input	Snapshot value	Basis
Declared Cap Rate, T = 6 / 3 / 1	100% / 55% / 12%	std — no current rate sheet was retrievable (both insurer rate pages returned HTTP 403 / WAF rejection; research gap 1). Observed illustrative prospectus values: 10% (1yr) S1; 75% (3yr, 10% buffer) S3; 12% (1yr) and 100% (6yr) S6; 60% with 110% participation (6yr, 20% buffer) S4
Declared Step Rate (1 yr)	8%	std; observed illustrative 8% S1, 5% with 90% participation S3, 12.5% trigger S6
Declared Edge Rate (1 yr)	6%	std — no Edge value appears in any retrieved document; set below the Step Rate because the Edge design pays down to −b
Participation Rate	100%	S4
Declared Fixed / Holding Account rate	3.00%	std — only the 1% contractual minimum is public S1 S2
Trading cost factor `kappa`	0.10% of the sum of absolute option market values	std — AG 54 requires a Trading Cost provision R2 but no prospectus quantifies it (research gap 8); the Academy example implies costs of order 0.1% of option value R6

NGE redetermination rule std. At each Term Start Date the declared cap is the solution c of

Pi( c ; b, T, sigma, r, q )  =  beta_target(T)
beta_target(T)               =  1 - ( 1 + y_e - spread ) ^ (-T)

floored at the contractual minimum guaranteed cap S2, where Pi(.) is the per-unit replicating-portfolio value defined below, y_e is the projected earned rate on supporting assets and spread is the target margin (std 2.22%, the level implied by the snapshot 6-year cap — see Worked example). This is the machinery ASOP No. 2 governs: a determination policy, an NGE framework, NGE scales, policy classes and periodic review of in-force NGEs R5. Base projection holds the snapshot scale level.

(c) Behavioral and experience assumptions#

Input	Recommended public basis	Basis tags
Base mortality (deferral period)	2012 IAM Basic Table (ANB) — the unloaded table underlying the Period Table, VM-M §2.C — with generational Projection Scale G2, `q_x^{2012+n} = q_x^{2012,Basic} x (1 − G2_x)^n`	REG-R59
Valuation / guaranteed-purchase-rate basis	2012 IAM Period Table with Scale G2 (i.e. the 2012 IAR), `q` rounded to three decimals per 1,000 from the 2012 period rate each time — never by compounding an already-rounded prior-year rate	REG-R59
Mortality A/E	2020–2024 Individual Payout Annuity Mortality Experience Study (23 parent groups, >80% of sales, 3.1m contract-years, 143,190 deaths), reported against the 2012 IAM Basic basis	REG-R61; A/E factor std 100%
Deferred-period annuitant mortality	Materially under-evidenced publicly — only a 2011–2015 deferred annuity mortality study and a 2006 analysis are indexed, neither fetched	REG-R65 unverified
Base surrender	VA / RILA contract-holder behavior study 2022–2024 (17 companies, ~48% of new premium for VAs and RILAs, 11.5m contracts, $1.5tn, >625,000 surrender events) — detailed tables are behind a paid data package	REG-R64; reference table std
Charge-expiry shock lapse	Order-of-magnitude anchors only: ~10% (with GLWB) vs ~33% (without) for FIA, ~52%/~56% for fixed-rate deferred	REG-R62 REG-R63 unverified
Withdrawal / partial-surrender utilization	Same VA/RILA study (4m withdrawal transactions, $56.7bn withdrawn)	REG-R64; reference rule std
Maintenance expense	$60 per contract per year, inflating 2.5% p.a.	std
Acquisition expense	6% of premium plus $200 per contract	std
Premium tax	0%	S2; value std

Mortality base — the trap. Do not run best-estimate mortality off the 2012 IAM Period Table. The Period Table (and hence the 2012 IAR) carries the valuation margin built in at construction — 10% at all ages up to and including 100, grading down 1% per year above it REG-R60 — so a 100% A/E applied to the Period Table sets deferral-phase mortality roughly 10% below the unloaded basis before any experience adjustment. The experience study reports against the Basic table, which is why the base row above is the Basic table. The payout chassis in products/immediate_annuity/technical-notes.md calibrates the same study to 108.4% of 2012 IAM Basic projected with G2; that factor is payout-annuitant-select and is deliberately not imported here, because the RILA deferral-phase population is not annuitant-select and public deferred-period annuitant mortality is thin REG-R65 unverified. Hence the std 100% A/E, which is a placeholder, not a measurement.

Reference base annual surrender table std (shape only; calibration is the user’s). w_base is the un-shocked annual rate; the charge-expiry shock enters separately as M_sc(y) in the Total surrender formula below, so the two are applied multiplicatively and neither is baked into the other:

Contract year	1–6	7	8+
`w_base(y)`	2.0%	2.0%	6.0%
`M_sc(y)`	1.0	3.0	1.0

giving a reference year-7 rate of 6.0% before the moneyness multiplier. Note that this std shape places the shock year and the ultimate at the same level, so the reference table exhibits no post-shock reversion; it is a shape placeholder and a calibrated run should separate the two.

Contract year 7 is the first year in which the withdrawal charge is zero S1 S2 and the first year following a 6-year Term End Date — the two events coincide on this chassis, which is why the shock is applied there.

(d) Market data required by the contractual formula (a RILA-specific input class)#

Input	Snapshot value	Basis
Market Value Rate curve (CMT, term maturity, linearly interpolated)	4.00% annual effective, flat	mechanic S2; level std
Risk-free rate for option pricing `r`	4.00% annual effective (`r_cc = ln 1.04 = 3.9221%`)	std
Dividend yield `q` (price-return indices)	2.00% annual effective (`q_cc = ln 1.02 = 1.9803%`)	std
Implied volatility `sigma`	20.00%, flat surface	std

AG 54 requires these to be “consistent with the observable market prices of derivative assets over the Index Strategy Term, whenever possible” R2. A production implementation must supply a surface, not a scalar: Equitable documents the interpolation procedure explicitly — quotes are taken for the closest maturities above and below the remaining time and, for each, the closest moneyness above and below the actual moneyness; then interpolate to the target moneyness at the shorter maturity, repeat at the longer maturity, and linearly interpolate between the two for the remaining time S4. The Academy’s demonstration scenario set uses volatilities of 20% and 25% and index term performance from −30% to +30% in 5% steps R6 — a usable regression grid.

Cash flow components and recursions#

Notation (defined once)#

Symbol	Meaning
`t`	month index (month ends), `t = 0` at issue; `cy(t) = floor(t/12)` = complete contract years
`k`	index-linked option (bucket) index
`I(t)`, `I_s`	index level at t; index level at the current Term Start Date
`R_k(t)`	index performance to date = `I(t)/I_s − 1`; `R_k` at term end is the crediting input
`b`, `c`, `s`, `e`, `PR`, `f`	buffer, cap, step, edge, participation rate, floor (all positive fractions)
`T`	term length in years; `tau(t)` = days remaining / 365 S2
`IA_k(t)`	Investment Amount (Index Strategy Base)
`Pi(I, tau)`	per-unit-of-notional market value of the replicating option portfolio
`beta`	initial option budget = `Pi(I_s, T)`
`B_k(t)`	amortized initial budget in currency
`F_k(t)`, `D_k(t)`, `TC_k(t)`	fixed income asset proxy, derivative asset proxy, trading cost
`V_k(t)`	Interim Value
`r_0`, `r(t)`	Market Value Rate at term start / at t (annual effective)
`r_cc`, `q_cc`, `sigma`	continuously-compounded risk-free and dividend rates, implied volatility
`C(I,K,tau)`, `P(I,K,tau)`, `BC(I,K,tau)`	Black-Scholes European call, put, and cash-or-nothing binary call paying 1
`AV(t)`	Account Value = `sum_k V_k(t) + FA(t) + HA(t)`
`G`	gross withdrawal removed from the contract; `FW` free withdrawal amount; `wc(cy)` charge rate
`ROP(t)`	return-of-premium GMDB base
`q_m(t)`, `w_m(t)`	monthly mortality and surrender rates; `l(t)` in-force probability

Black-Scholes, continuous parameters:

d1 = [ ln(I/K) + (r_cc − q_cc + sigma^2/2) tau ] / (sigma sqrt(tau)),   d2 = d1 − sigma sqrt(tau)
C  = I e^(−q_cc tau) N(d1) − K e^(−r_cc tau) N(d2)
P  = K e^(−r_cc tau) N(−d2) − I e^(−q_cc tau) N(−d1)
BC = e^(−r_cc tau) N(d2)

Dimensional check: Pi is dimensionless (a fraction of notional), beta dimensionless, IA, B, F, D, TC, V, AV, G are currency, tau and T are years, all rates are per annum.

Term-end crediting#

Let R be the index performance over the completed term. Piecewise crediting rate g:

BUFFER + CAP      g = min(R, c)                if R >= 0
                  g = min(0, R + b)            if R <  0                    [S1] [S2]

BUFFER + CAP + PR g = min(PR * R, c)           if R >= 0
                  g = min(0, R + b)            if R <  0                    [S4]

BUFFER + STEP     g = s                        if R >= 0
                  g = min(0, R + b)            if R <  0                    [S1] [S2]

BUFFER + EDGE     g = e                        if R >= -b
                  g = R + b                    if R <  -b                   [S2]

FLOOR + CAP       g = min( max(R, -f), c )                                  [S5]

The min(0, ·) in the buffer branch is load-bearing: “The Performance Rate can never be greater than zero if the Index Performance is negative” S1. Note the deliberate discontinuities the source documents flag — a Step design pays the full step at R = 0.00% and zero at R = −0.01% S4, and a dual/absolute-return design flips sign at the buffer edge S4. They are contractual, not artifacts; do not smooth them.

Roll-forward S1 S2:

IA_k(term end) = IA_k(term start, adjusted for withdrawals) * (1 + g)

Replicating portfolios (per unit of notional; each option has notional equal to `IA` S4)#

CAP:        Pi = [ C(I, I_s, tau) − C(I, I_s(1+c), tau) − P(I, I_s(1−b), tau) ] / I_s     [S2] [S3] [S5] [S6]
CAP + PR:   Pi = PR * [ C(I, I_s, tau) − C(I, I_s(1+c/PR), tau) ] / I_s
                 − P(I, I_s(1−b), tau) / I_s                                              [S4]
STEP:       Pi = s * BC(I, I_s, tau) − P(I, I_s(1−b), tau) / I_s                          [S2] [S5]
EDGE:       Pi = e * BC(I, I_s(1−b), tau) − P(I, I_s(1−b), tau) / I_s                     [S2] [S5]
FLOOR:      Pi = [ C(I, I_s, tau) − C(I, I_s(1+c), tau)
                   − P(I, I_s, tau) + P(I, I_s(1−f), tau) ] / I_s                         [S5]
TIERED PR:  Pi = [ C(I, I_s, tau) + (PR2 − PR1) * C(I, I_s(1+L), tau)
                   − P(I, I_s(1−b), tau) ] / I_s                                          [S3]

L is the tier level. The floor design needs four options, not three — buy the ATM call spread, sell an ATM put, and buy back an OTM put struck at the floor so the short put exposure stops there; Allianz states that “the out-of-the-money put will almost always reduce, and never exceed, the negative impact of the at-the-money put for the Index Guard Strategy” S5. For the Edge/dual-precision design the binary call is in the money because it pays when the index ratio is at or above 1 − b S2 S5. If a Cap option is uncapped, the out-of-the-money call is valued at zero S2.

Verification identity (implement as a unit test). At tau = 0, every Pi above collapses to the corresponding g. For CAP with I/I_s = 1 + R: max(R,0) − max(R − c, 0) − max(−b − R, 0) = g. If a model’s interim value at term end does not reproduce IA * (1 + g) exactly, the strike set or the notional convention is wrong.

Interim value — the std baseline (family (a), the AG 54-literal form S2 R2)#

B_k(t)  = beta * IA_k(t) * tau(t) / T                                (straight-line amortization [S2])
F_k(t)  = [ IA_k(t) − B_k(t) ] * [ (1 + r_0) / (1 + r(t)) ] ^ tau(t)
D_k(t)  = IA_k(t) * Pi( I(t), tau(t) )
TC_k(t) = kappa * IA_k(t) * sum of | per-unit option component values |   for tau(t) > 0
        = 0                                                              at tau(t) = 0
V_k(t)  = F_k(t) + D_k(t) − TC_k(t)

B is the market value of the Derivative Asset Proxy under initial market conditions, amortized straight-line to term end S2; r_0 and r(t) are the Market Value Rates (CMT at the term’s maturity, linearly interpolated between adjacent CMT maturities) on the Term Start Date and the calculation date S2; tau is days remaining / 365 S2. Because B_k(t) is defined as beta times the current IA_k(t), V_k(t) is homogeneous of degree one in IA_k(t) std — a convention the model must impose so that a withdrawal reduces the interim value by exactly the cash removed (below).

Boundary conditions (both are AG 54 requirements and both are exact under this form):

tau = T: B = beta * IA, F = IA(1 − beta), D = beta * IA, so F + D = IA. AG 54 requires the Index Strategy Base to equal the Strategy Value at term start R2.
tau = 0: B = 0, F = IA, D = IA * g, and TC = 0 — there are no options left to exit, and the Academy example likewise shows no trading cost at term end R6 — so V = F + D = IA(1 + g). Implement TC_k as strictly interior to the term (kappa_effective = kappa * 1{tau > 0} std); leaving the trading-cost provision on at tau = 0 breaks the term-end verification identity stated above by kappa * IA * |g|.

The Academy’s worked example confirms both numerically and notes that the interim value is undefined at term start and term end — those points are Strategy Values, not Interim Values R6.

Interim value — the two alternative families (implement as switchable strategies)#

(b) Full notional discounted at a current rate, plus an always-positive expense rebate S4 S6:

V_k(t) = IA_k(t) / (1 + rate(t)) ^ tau(t)  +  D_k(t)  +  CCF_k(t)
CCF_k(t) = E_0 * tau(t) / T          (Cap Calculation Factor; always positive, declines)

Equitable discounts the full Segment Investment with no subtraction of an initial option budget and adds the Cap Calculation Factor, “a return of estimated expenses for the portion of the Segment Duration that has not elapsed” (worked: $10 of estimated expenses on a one-year segment gives $6 with 219 days remaining, since 10 x 219/365 = 6); the rate is an investment-grade rate (risk-free plus a spread) that Equitable notes is above swap rates and therefore “will result in a lower value for that component” S4. Lincoln’s C x [1/(1+E)^D x (1+E)^D/(1+F)^D] is presented as accretion-times-MVA but collapses algebraically to C / (1+F)^D S6.

(c) A delta applied to the notional rather than a value, with no interest-rate adjustment term S5:

DailyAdjustment(t) = [ ( Pi(I(t), tau(t)) − Pi(I_s, T) ) + Pi(I_s, T) * (1 − tau(t)/T) ] * IA_k(t)
V_k(t)             = IA_k(t) + DailyAdjustment(t)

Allianz describes the second bracketed term (“proxy interest”) as “approximated by the value of amortizing the cost of the Proxy Investment over the Term to zero” S5 — the same option-budget amortization that appears as term B in families (a) and (b), added rather than subtracted. There is no interest-rate adjustment factor in this form at all S5.

Non-obvious equivalence worth knowing. Expanding (c) and using beta = Pi(I_s, T) gives V = IA[1 + Pi(I(t), tau) − beta * tau/T], which is exactly family (a) with the MVA factor set to 1. Families (a) and (c) differ only in whether the fixed leg is marked for interest-rate movement. That is the direct consequence of AG 54’s project history: specific MVA requirements were deliberately removed from the guideline because consensus was unreachable on whether MVAs should be permitted at all, leaving the “equity” principle to state review R2.

Option-budget amortization switch std. B_k(t) may be computed either as beta * IA * tau/T (straight-line, the std baseline S2, and the mandated form in Pennsylvania for S3) or as Pi_initial-conditions(I_s, tau(t)) * IA (initial market conditions with updated time to expiry, the national form for S3). One insurer needs both, so this is a configuration flag, not a modeling opinion.

Legacy contrast module (pre-AG 54) S1. The older design uses no option pricing: accrue each rate linearly and apply the term-end rules to the accrued rates — AccruedCapRate = c x (days elapsed)/(days in term), likewise for the Shield and Step Rates, 365 days assumed per calendar year of a term S1. Worked: $50,000, Shield 10, 10% Cap, 1-year term, index 500 → 600 at day 183 gives an accrued cap of 5%, a 5% Performance Rate and an interim value of $52,500 S1. Useful as a tractable first implementation target and as a regression contrast; it predates AG 54’s July 1, 2024 effective date R2 and would not satisfy the Hypothetical Portfolio requirement without a material-consistency demonstration.

The universal proportional rule for withdrawals#

Every insurer in the sample reduces the index-linked notional in proportion to the reduction in interim value, not dollar-for-dollar S2 S3 S4 S6. For a gross withdrawal G_k taken from option k at time t:

IA_k(t+) = IA_k(t−) * ( 1 − G_k / V_k(t−) )
V_k(t+)  = V_k(t−) − G_k                      (follows from homogeneity of V in IA)

The reduction in notional is Delta_IA = G_k * IA_k(t−) / V_k(t−), so

Delta_IA − G_k = G_k * ( IA_k(t−) / V_k(t−) − 1 )   >  0   whenever  V_k(t−) < IA_k(t−)

i.e. the notional lost can exceed the cash received. Numeric illustration at the worked-example parameters: IA = $100,000, V = $84,803.11, G = $8,000 → Delta_IA = $9,433.62, an excess of $1,433.62 over the cash withdrawn, and the remaining notional is $90,566.38. Brighthouse works the same rule at $50,000 x (1 − $20,000 / $53,514.77) = $31,313.57 S2; Prudential works it at a 71.429% ratio and gives a second case in which a $14,000 withdrawal against a $14,000 interim value zeroes a $14,285.71 base S3. The prospectus states the asymmetry directly: a withdrawal when the interim value is below the investment amount “will cause a greater percentage reduction in the Investment Amount that remains” S2.

Exception: after a Performance Lock, the locked value is reduced dollar-for-dollar S2 — the option leg is gone and the bucket is a fixed accrual to term end.

Withdrawal charge, free amount, and allocation#

FW(t)        = 0                                            if cy(t) = 0
             = 0.10 * AV_anniv(y) − FW_used(y)              otherwise            [S1] [S2]
chargeable   = max( 0, G_total − FW(t) )
WC(t)        = wc( cy(t) ) * chargeable                                          [S1] [S2]
net proceeds = G_total − WC(t)

The charge is deducted from the amount withdrawn and is not grossed up on this chassis S1 S2 (contrast S4, where “any amount deducted to pay withdrawal charges is also subject to that same withdrawal charge percentage”). Verification against the prospectus example S2: $100,000 payment, $80,000 Account Value at the start of contract year 6, full withdrawal → FW = $8,000, chargeable $72,000, wc(5) = 3%, charge $2,160, cash value $77,840. G_total is allocated across open options pro rata to interim value std (the prospectuses do not prescribe an allocation for an unspecified withdrawal).

Contract-level values and benefits#

AV(t)   = sum_k V_k(t) + FA(t) + HA(t)
CSV(t)  = AV(t) − wc(cy(t)) * max(0, AV(t) − FW(t))                 (full surrender)  [S1] [S2]
ROP(t+) = ROP(t−) * ( 1 − G_total / AV(t−) )                        (proportional)    [S1] [S2]
DB(t)   = max( AV(t), ROP(t) )        for issue ages <= 80;  = AV(t) for 81+          [S2]
Annuitization value = AV(t), with each open option contributing V_k(t)                [S1]

Note the GMDB’s interaction with the interim value: AV(t) is depressed exactly when the option leg is deep out of the money, so the return-of-premium guarantee bites in equity stress S1 S2. The reduction ratio applied to ROP uses the gross amount removed from the contract, i.e. including any withdrawal charge S1.

Fixed and Holding Accounts accrue monthly at the declared rate, floored at the 1% guaranteed minimum S1 S2: FA(t) = FA(t−1) * (1 + max(i_declared, 0.01))^(1/12).

Monthly processing order std #

At each month end t:

Market state. Refresh I(t), the CMT curve, sigma, q from the scenario.
Term-end crediting. For each option whose Term End Date is t: compute R_k = I(t)/I_s − 1, apply g, set IA_k <- IA_k (1 + g). Set the interim value equal to IA_k for the Transfer Period S2.
Renewal / transfer. Apply the term-end roll rule (below): renew into the same option at the new declared rate (floored at the guaranteed minimum cap S2), transfer to the Fixed Account, or route to the Holding Account if the option and the Fixed Account are both unavailable S2. Reset I_s, r_0, T, tau, recompute beta = Pi(I_s, T).
Interim values. For every open option, compute F_k, D_k, TC_k, V_k.
Accounts. Accrue FA, HA. Set AV(t). On a Contract Anniversary, snapshot AV_anniv and reset FW_used.
Contract-holder transactions. Scheduled and dynamic partial withdrawals: allocate, compute the withdrawal charge, reduce IA_k proportionally, reduce ROP proportionally, recompute V_k and AV(t).
Decrements (end of month). Death at q_m(t), then surrender at w_m(t) on survivors [std order]; plus the discrete term-end surrender fraction if t is a Term End Date. Update l(t) = l(t−1) (1 − q_m(t)) (1 − w_m(t)).
Maturity. At the Maturity Date, force annuitization of AV(t) S2.

Cash flow outputs (per contract, month t, before in-force weighting)#

Cash flow	Formula	Sign
Single premium	`single_premium` at t = 0	+
Death claims	`DB(t)`	−
Full surrender	`CSV(t)`	−
Partial withdrawal	`G_total − WC(t)`	−
Withdrawal charge income	`WC(t)` (retained; not a separate cash flow if `CSV`/net proceeds are used)	+
Annuitization outgo	`AV(t)` converted to a payout stream at the Maturity Date	−
Acquisition expense	`0.06 x premium + 200` at t = 0 std	−
Maintenance expense	`60/12 x 1.025^(y−1)` std	−
Option budget / hedge cost	not a liability cash flow — it is an asset-side flow; the liability model sees it only through the declared cap (see note)	n/a

Note on annuitization: this file specifies the deferral phase only. The payout stream bought at the Maturity Date is not re-derived here — survivorship weighting, period-certain floors and joint-life continuance are the payout chassis in products/immediate_annuity/technical-notes.md, applied to the two forms this contract offers (Life with 10 Years of Annuity Payments Guaranteed; Joint and Last Survivor with 10 Years Guaranteed) S2. Two deltas against that chassis: this contract offers no refund forms (neither cash refund nor installment refund) S2, so that branch is unused; and the survivor continuance percentage on the joint form is documented only by name in the retrieved prospectuses, so 100% last-survivor continuance is assumed std unverified and the reduced-percentage branch is likewise unused. Purchase rates themselves were not located in any retrieved document (research gap 2), so the reference model computes factors on the std basis in product-spec.md.

Note on the option budget: on this chassis the cap is the fee — “While no fees or charges are deducted from the amounts held in the Index Strategies, the available Cap Rates, Participation Rates, Tier Levels, and Step Rates reflect the expenses related to the Index Strategies” S3; Equitable and Lincoln call the cap an “implicit ongoing fee” S4 S6. A gross-liability projection must therefore not deduct a charge from the index-linked value; the margin appears as the spread between the earned rate and the option budget implied by the declared cap.

Aggregate expected cash flows weight each row: expenses by l(t−1); death claims by l(t−1) q_m(t); surrenders by l(t−1) (1 − q_m(t)) w_m(t) [std timing].

Policyholder behavior modeling#

All dynamic formulas below are std reference constructions; calibration sources are cited where they exist, and the public RILA-specific data are thin (REG-R64 reports aggregate counts only; the detailed tables sit behind a paid data package).

Base surrender std. Annual w_base(y) per the table above, converted monthly: w_m = 1 − (1 − w_annual)^(1/12).
Charge-expiry shock std. Multiplier M_sc(7) = 3.0, unity elsewhere (table above) — contract year 7 being the first year with a zero withdrawal charge S1 S2, which on the 6-year chassis coincides with the first Term End Date. Order-of-magnitude anchors from adjacent products: ~33% shock without a living-benefit rider and ~10% with one for FIA, ~52%/~56% for fixed-rate deferred REG-R62 REG-R63 unverified. The representative RILA carries no living-benefit rider, which argues for the un-suppressed end; the offsetting force is the term structure (below), which is why the std shock is set well below the FIA “without rider” anchor.
Interim-value moneyness suppression std — the RILA-specific effect. Surrendering mid-term crystallizes the interim value, which is punitive when the option leg is out of the money. Reference form:
```
M_iv(t) = min( 1.0, max( 0.25, V_k(t) / IA_k(t) ) ) ^ 2
```
so a bucket at V/IA = 0.85 carries a 0.72 multiplier and one at or above par carries 1.0. Rationale: the prospectuses warn repeatedly that the interim value “may be less than” the value at term end S1 and can be negative even with the index up S2; a rational holder defers.
Term-end concentration std. During the Transfer Period the interim value equals the Investment Amount — no option adjustment S2 — so the economic penalty for exiting vanishes for five days each term. Model this as a discrete surrender fraction at each Term End Date: phi = 10% when the withdrawal charge is zero, phi = 3% otherwise std, applied in addition to the background monthly rate.
Total surrender. w_annual(y,t) = min( 0.50, w_base(y) x M_sc(y) x M_iv(t) ) [std cap], plus the discrete phi at term ends.
Term-end roll behavior std. The contractual default is automatic renewal into the same option at the new declared rate S1. Reference split at each Term End Date, after the phi surrender: 80% renew into the same option, 15% transfer to the Fixed Account, 5% transfer to a different index-linked option (modeled as renewal at the same parameters). Renewals are re-struck at the prevailing index level and the new declared cap; if the declared cap would fall below the guaranteed minimum, it is floored there S2, which mechanically raises the option budget and compresses the spread — the reason the guaranteed-minimum table is a genuine tail exposure, not decoration.
Partial withdrawals std. Base rule: 0% in contract year 1 (the free amount is zero S1 S2); thereafter 2% of Account Value per year, taken at contract anniversaries and capped at the Free Withdrawal Amount so no withdrawal charge is incurred in the base run. RMD-driven withdrawals for qualified cells begin at the applicable age; RMD timing is a behavioral input, not merely a tax one REG-R58 REG-R64.
Performance Lock [std, optional module]. Election rule: lock when V_k(t) / IA_k(t) >= 1 + theta with theta = 0.15, once per term S2. After a lock the option leg is removed, the bucket accrues to term end, and withdrawals reduce the locked value dollar-for-dollar S2. Switched off in the base run.

Worked example#

Anchor cell: male 60, single premium $100,000, 100% to one 6-year option, S&P 500 price return, buffer b = 10%, Cap crediting with a declared cap c = 100% std. Market inputs std: r = 4.00% annual effective, q = 2.00%, sigma = 20.00%, index normalized to I_s = 100. Strikes: ATM call at 100, OTM call at I_s(1+c) = 200, OTM put at I_s(1−b) = 90; each option’s notional is the Investment Amount S4. Trading cost kappa = 0.10% of the sum of absolute option values std.

Derived at term start: Pi(100, 6) = 0.215679 − 0.033445 − 0.081602 = 0.100632, so the option budget is beta = 10.0632% = $10,063.19 and the fixed leg opens at $89,936.81. The equivalent geometric accretion yield on the fixed leg is 1.7834%, i.e. a 2.22% spread against the 4.00% market rate — that spread is the std input to the NGE cap-solve rule above. The Market Value Rate is 4.00% at term start and 5.00% at t = 3 (a 100 bp rise, to exercise the MVA factor).

#	Point	t (yrs)	Index	R	Fixed proxy	ATM call	− OTM call	− OTM put	Deriv. proxy	Trading cost	Interim value	Investment Amount
0	Term start (Strategy Value, not an IV R6)	0	100	0.00%	89,936.81	21,567.95	−3,344.55	−8,160.20	10,063.19	33.07	— (base = 100,000.00)	100,000.00
A1	Scenario A, mid-term	3	120	+20.00%	92,280.78	29,145.94	−2,182.94	−2,726.33	24,236.66	34.06	116,483.39	100,000.00
B1	Scenario B, mid-term	3	80	−20.00%	92,280.78	5,778.47	−85.24	−13,151.88	−7,458.66	19.02	84,803.11	100,000.00
B2	Scenario B, immediately after an $8,000 withdrawal	3	80	−20.00%	83,575.37	5,233.35	−77.20	−11,911.19	−6,755.04	17.22	76,803.11	90,566.38
A2	Scenario A, term end — credit `min(40%, 100%) = +40%`	6	140	+40.00%	100,000.00	40,000.00	0.00	0.00	40,000.00	0.00	— (Strategy Value 140,000.00)	140,000.00
B3	Scenario B, term end — credit `min(0, −25% + 10%) = −15%`	6	75	−25.00%	90,566.38	0.00	0.00	−13,584.96	−13,584.96	0.00	— (Strategy Value 76,981.42)	76,981.42

Trace and checks:

Row 0. F = 100,000 (1 − 0.100632 x 6/6) x (1.04/1.04)^6 = 89,936.81; F + D = 100,000.00 exactly — AG 54’s requirement that the Index Strategy Base equal the Strategy Value at term start R2. The interim value is undefined here R6.
Rows A1/B1. tau = 3, so B = 0.100632 x 100,000 x 3/6 = 5,031.60 and the unadjusted fixed leg is 94,968.40. The MVA factor (1.04/1.05)^3 = 0.971690 reduces it to 92,280.78 — a $2,687.62 cost of the 100 bp rate rise. Without the rate move the interim values would be $119,171.01 (A) and $87,490.73 (B).
Row B1 is the case the prospectuses warn about: the index is down 20%, well beyond the buffer’s mid-term protection, and the short out-of-the-money put alone is worth −$13,151.88. Even in Row A1 — index up 20% — the short put still subtracts $2,726.33 S2.
Row B2, the proportional rule. G = $8,000 (assumed within the Free Withdrawal Amount, so no withdrawal charge). Ratio 8,000 / 84,803.11 = 9.4336%; IA <- 100,000 x (1 − 0.094336) = 90,566.38. The notional falls $9,433.62 for $8,000 of cash — an excess of $1,433.62, exactly G (IA/V − 1). Every component of the interim value scales by the same 0.905664 factor, so the interim value falls by exactly the $8,000 withdrawn: 84,803.11 − 8,000 = 76,803.11. The ROP GMDB base falls in the same proportion as the Account Value S1 S2.
Rows A2/B3. At tau = 0 the replicating portfolio reproduces the crediting formula exactly: +40% capped at 100% gives Pi = 0.40; −25% with a 10% buffer gives Pi = −0.15 = min(0, R + b). Scenario B’s term-end Investment Amount is 90,566.38 x 0.85 = 76,981.42 — the withdrawal’s proportional bite persists to term end.

Valuation and reserve pointers#

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

Nonforfeiture. AG 54 is the governing standard and is unusual in that the nonforfeiture value is the model’s interim value: contracts issued on or after July 1, 2024 must produce Interim Values materially consistent with the Hypothetical Portfolio less Trading Costs, with an actuarial memorandum and certifications filed with each product R2 REG-R44. Nonforfeiture benefits follow Model #250 Section 7, excluding §7.B R2 REG-R43. Model #805 does not apply if and only if AG 54 is satisfied REG-R42 REG-R44; if it did apply, note the indexed nonforfeiture rate is floored at 15 basis points, not 1% REG-R42.
Statutory reserve. VM-21 constitutes CARVM for in-scope contracts; aggregate reserve = stochastic reserve (CTE70) + additional standard projection amount, with contract-holder behavior in §10 and prudent-estimate mortality in §11 REG-R35. Scope test: VM-21 §2.A.3 excludes separate-account contracts that guarantee an index and offer no GMDB/VAGLB, so a bare accumulation RILA is out of scope, while the representative return-of-premium GMDB design is in R3 S2. AG 43 remains the scoping shell that pulls pre-2017 business onto the VM-21 calculation REG-R35 REG-R38; implementation guidance in the Academy’s VM-21 practice note supplement REG-R66. An out-of-scope contract falls back to formulaic CARVM, and that fallback is no longer unsourced — AG 33 has been read and reaches any annuity contract subject to CARVM with elective benefits available REG-R151, while AG 35 has been read and does not address this design REG-R152.
Capital. C-3 Phase II: TAR at CTE 90, RBC = TAR − statutory reserves, subject to the Standard Scenario floor REG-R47; VM-21 §§4.A–4.E and the RBC requirements are identical apart from the elective tax treatment, so one projection serves both REG-R35. Reform background in REG-R48.
Tax. IRC §807: greater of net surrender value and 92.81% of the NAIC-prescribed method (CARVM), capped at statutory REG-R16; distribution taxation under §72 REG-R55; §817(h) diversification REG-R15.
U.S. GAAP. Index credits and annuity guarantees are the paradigm market risk benefits at fair value through earnings under LDTI REG-R34, with ASOP No. 10 the professional-standards counterpart REG-R71.
Standards for the modeling work itself. ASOP No. 7 (life cash flow analysis) REG-R27; ASOP No. 22 (asset adequacy) REG-R29; ASOP No. 54 (pricing) REG-R70; ASOP No. 56 (modeling, validation, model risk) REG-R32; ASOP No. 2 for the NGE determination process R5 REG-R26. There is no ASOP for principle-based reserves for annuities — ASOP 52 is scoped to VM-20 life products REG-R31, and the nearest guidance is the non-binding Academy practice note REG-R66.

Key sensitivities and model risks#

Dominant assumptions, in rough order:

Implied volatility. The interim value is a derivative price; sigma moves it directly, and the sign differs by strategy: increases in expected volatility hurt dual-precision, precision and 1-year performance strategies, while decreases hurt the floor (Index Guard) strategy S5. A flat-surface approximation is the largest single simplification in this model.
Interest rates through the MVA factor. At the worked-example parameters a 100 bp rise costs $2,687.62 of interim value at the term midpoint — larger than the entire trading-cost provision by two orders of magnitude. Family (c) has no such term at all S5, so the choice of interim-value family is itself a rate-sensitivity assumption.
The NGE renewal rule for caps. It sets every future option budget and hence future interim values, surrender behavior and margin. Floored at the guaranteed minimum S1 S2, which converts a low-rate environment into a direct margin compression.
Surrender timing relative to term boundaries. Because the interim value equals the Investment Amount during the Transfer Period S2, surrenders concentrate there; a model that spreads surrenders uniformly across the term systematically over-collects the negative interim value adjustment.
Proportional withdrawal accounting. Modeling withdrawals as dollar-for-dollar reductions of the notional overstates remaining notional in down markets by G (IA/V − 1) per withdrawal and compounds through the rest of the term.
GMDB moneyness correlation. The return-of-premium guarantee is most in the money exactly when interim values are depressed S1 S2 — the guarantee and the account are not independent, so a deterministic run understates its cost.

Known modeling pitfalls:

Price return vs total return. All representative indices are price return S1 S2; omitting the dividend yield overprices every call and inflates interim values throughout.
Applying the cap annually on a multi-year term. “We do not apply the Cap and any Participation Rate annually on a 3-year or 6-year Term Index Option” S5 — the cap applies to the whole-term return.
Losing homogeneity. If B_k(t) is frozen at the term-start notional rather than scaled with the current IA_k(t), a withdrawal no longer reduces the interim value by exactly the cash withdrawn and the contract silently gains or loses value on every one.
Term-end mismatch. If V_k at tau = 0 does not equal IA_k (1 + g) to the cent, the strike set, the notional convention, the buffer sign — or a trading-cost provision left switched on at tau = 0 — is wrong; unit-test it for every crediting type.
Discount-rate reference mismatch. CMT S2, CMT plus corporate spread S6, a credit index at a duration that need not match the term S3, or an investment-grade rate above swap rates S4. Reconciling to one insurer requires that insurer’s reference.
Amortization convention. Straight-line S2 vs updated time to expiry S3 — one insurer uses both, split by state S3.
Negative interim value with the index up. “You could have negative Interim Value, even if the Index Value has increased at the time of the calculation” S2; flooring the interim value at zero, or at the notional, is not implementing the contract.
Smoothing the crediting discontinuities. Step and Edge designs are genuinely discontinuous at R = 0 and R = −b S2 S4; the binary options are what make the interim value track that, and smooth approximations break the term-end identity.
Era mixing. The pre-AG 54 pro-rata design S1 and the hypothetical-portfolio design S2 are both live in in-force blocks (AG 54 applies to issues on or after July 1, 2024 R2); an in-force model must carry both engines and key them off issue date.
Trading costs are a free parameter. No retrieved prospectus quantifies them S2 S4 S6; the std 0.10% matches only the order of magnitude implied by R6.
Regression vectors exist — use them. Lincoln publishes interim-value grids across index moves of −30%/−10%/+20%/+40% for 1- and 6-year terms and for cap, trigger and dual-trigger accounts S6; Prudential a three-strategy grid at ±20% S3; Brighthouse a single fully decomposed case S2; and the Academy a six-year path with all Black-Scholes inputs disclosed plus an Excel Lambda library reproducing the calculation R6. These are the only public conformance tests available.