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#] refer to _research/deferred-income-annuity.md; [REG-R#] refers to references/regulatory-and-actuarial-references.md, whose shared numbering now runs R1–R157 as one space, with most of the R73–R149 block unused (R1–R34 of life origin, provenance _research/regulatory-actuarial.md; R35–R72 annuity-specific, provenance _research/regulatory-actuarial-annuities.md; R150–R157 the AP&P Manual appendix and actuarial-guideline prints read on 2026-08-06). std marks standardizations introduced for the reference implementation; unverified marks claims not confirmed against a retrieved document. Parameter values are identical to those in product-spec.md.

Two structural facts govern the whole design.

There is no account value. No credited rate, no index crediting, no M&E or rider charge, no surrender charge, no free-withdrawal corridor, no market value adjustment, no benefit base, no interim value S1 fn.1 S2 S4 R13. Consequently there is no lapse decrement — VM-22’s standard-projection lapse section is expressly “not applicable” to contracts with no account value or surrender benefit R9 — and no annuitization decrement (prescribed at 0% R9). The full list is in product-spec.md, “Parameters that do not exist for this product”. Do not synthesize any of them.
The income phase is the immediate-annuity payout chassis. Payout forms, refund and certain-period guarantees, survivor reduction and its interaction with a guarantee period, COLA escalation and payment-survivorship weighting are specified in products/immediate_annuity/technical-notes.md and are not restated here. These notes cover the deferral phase, the transition, the in-force options and the QLAC overlay, and define only the payout-phase quantities the DIA changes — chiefly the refund base, which is cumulative premiums rather than a single premium.

Model scope and conventions#

Purpose. Project gross expected liability cash flows — premiums in; deferral death benefits, income payments, refund benefits, acceleration and commutation payments, maintenance expenses out — for a single-contract model point. Reserves are not computed (see “Valuation and reserve pointers”).
Projection frequency. Monthly, indexed t = 0, 1, 2, … from issue std. Monthly is natural because the modal payment frequency is monthly S1 S2 S3 S4 and because the 13-month minimum deferral and the 13-month premium cut-off are expressed in months S2 S4 R9.
Timing conventions std. Premiums are received at the start of month t. Income is paid in arrears at the end of each payment period, matching products/immediate_annuity/product-spec.md; the model exposes pay_timing ∈ {advance, arrears} because no retrieved DIA document states the convention. Deferral death benefits are paid at the end of the month of death. T is a month index and the income start date is the start of month T (exactly T/12 years after issue); under arrears the first payment falls one payment period later, 12/m months after T — at m = 12, at the end of month index T, i.e. T + 1 months from issue. Consequently deaths in months t < T are deferral-phase deaths and deaths in months t ≥ T are payout-phase deaths.
Age basis. Age nearest birthday (ANB) std: MassMutual states contract issue age on an ANB basis S2; prescribed VM-22 payout mortality is ANB with a conversion formula supplied for age-last-birthday companies R9; the 2012 IAM Basic and Period tables were developed ANB R15 REG-R59.
Model points. Single-contract model points on an expected (probability-weighted) basis: the in-force factor l(t) multiplies every per-contract cash flow. Joint cells carry two lives and a joint status. No aggregation logic is specified here.
Decrement set. Mortality only; l(t) never decrements for lapse or surrender R9.
Rounding. Full precision internally, cash flows to cents std. Generational mortality rates follow the Valuation Manual rule — three decimal places per 1,000, computed from the 2012 period rate each time, never by compounding an already-rounded prior-year rate R9 REG-R59. A-821 ¶14 prints the same rule and the same worked counter-example, so this is now sourced twice over rather than once REG-R153.

Model point attributes#

Attribute	Type	Example (anchor cell std)
`issue_age`, `sex`	int (ANB); enum {M, F}	60; F
`joint`, `joint_age`, `joint_sex`	bool; int; enum	false
`survivor_pct`, `reduction_trigger`	enum {0.50, 0.6667, 0.75, 1.00}; enum {either, primary}	n/a
`convertible_joint`	bool	false
`market_type`	enum {NQ, TradIRA, RothIRA, QLAC}	NQ
`income_form` `f`	enum {LO, LO_ROP, CR, IR, PC(n)} + joint variants	CR
`certain_period` `n`	years, 10–30 (PC forms only)	n/a
`income_start_month` `T`	int months from issue; `13 ≤ T ≤ 360` and start age ≤ 85	240
`premium_schedule`	list of (month, amount)	[(0, 100000), (60, 50000)]
`db_form`	enum {ROP, NONE}	ROP
`cola_rate` `c`	0.00, or 0.01–0.04	0.00
`pay_frequency` `m`, `pay_timing`	enum {12, 4, 2, 1}; enum {arrears, advance}	12; arrears std
`adjust_right`, `adjust_uses`	bool; int	true; 1
`accel_uses`, `accel_months`	int 0–5; int {3, 6}	2; 6
`commutation_allowed`	bool (extended case only)	false
`guaranteed_future_rates`	bool — whether purchase rates for future premiums are guaranteed R13 §1.B(1)(h)	false std
`qlac_premium_room`	currency (QLAC only)	n/a

Anchor cell std: Female 60 ANB, nonqualified, $100,000 at issue plus $50,000 at the start of policy year 6, income start at attained age 80, Life with Cash Refund, monthly in arrears, return-of-premium death benefit in deferral, no COLA. Used identically in product-spec.md and in the worked example.

State variables#

Variable	Description	Updated
`B(t)`	Guaranteed annual income purchased to date, before COLA	on each premium; on adjustment exercise
`CP(t)`	Cumulative premiums paid — the deferral death benefit base and the refund base	on each premium
`l(t)`	In-force (survival) probability at the start of month `t`; `l(0) = 1`	monthly
`phase(t)`	{deferral, payout, terminated}	at `t = T`; on death; on exhaustion of a non-life-contingent form
`T(t)`	Current income start month (mutable once)	on adjustment exercise
`RG(t)`	Remaining refund balance `= max(0, CP(T) − income scheduled through the end of month t−1)`; the `t−1` is the arrears convention (an instalment due at the end of the death month has not been paid)	monthly in payout, refund forms
`n_g`	Derived guarantee period in years	fixed at `T`; re-derived if `B` changes
`adjust_used`, `accel_used`	bool; int	on exercise
`accel_blackout(t)`	months of suspended payments remaining	monthly
`commuted(t)`, `resume_month`	extended case: guaranteed payments commuted; month the life-contingent tail resumes	on exercise
`sel_mult(t)`	mortality selection multiplier applied after an adjustment exercise	on exercise
`qlac_room(t)`	Indexed QLAC limit less premiums paid to this and any other intended QLAC	on each premium

Assumption inputs#

Class (a) is contractual and fixed at issue; class (b) is the only insurer-declared element in the product; class (c) is the modeler’s view of experience.

(a) Contractual / guaranteed elements#

Input	Value	Basis
Guaranteed income per slice	Fully guaranteed at the time each premium is paid	R13 §3.H(1)
Deferral death benefit	100% of cumulative premiums, no interest, lump sum	S1 S2 S3 S4 R13 §3.I(1)(a)
Permitted DB calculation methods	% of premiums; % of premiums plus interest; flat dollar; combination	R13 §3.I(1)
Forms with no deferral DB	Life Only, Joint Life Only; and Single Life — No Death Benefit (deferral ≥ 10 yrs, start date locked)	S1 S2 S3 S4
Minimum / maximum deferral; max start age	13 months / 30 years; attained age 85	S2 S4 R9; choices std
Premium cut-off	No premium within 13 months of the income start date	S2 S4
Income start date adjustment	One-time ±5 years; new date ≥ 13 months after the last premium; option, day of month and frequency locked	S1 S2 S3 S4
Adjustment repricing inputs	Originally scheduled payment; new date; Moody’s Seasoned Baa Corporate Bond Yield at the request date; Annuity 2012 Mortality Table; contractual interest-rate-change adjustment	S2 (A2000 in the NYL formulation S1)
COLA	Fixed compound 1%–4% on each income-start anniversary; elected at issue, irrevocable	S2 S3 S4; menu std
Payment acceleration	6 monthly payments in one sum then 5 months without; 2 uses; age 59½; nonqualified	S1 S4; count std
Commutation (extended)	≤100% of the PV of remaining guaranteed payments; interest-rate adjustment applies; life-contingent tail resumes	S4 S5 R13 §3.F
Loans, surrender, withdrawals in deferral	None; prohibited	S1 S2 R13 §3.P
Explicit charges	None disclosed in any source	S1 S2 S3 S4
Minimum monthly income	$100	S2
Small-benefit termination right	Company may terminate for present value after 2 years without considerations if the paid-up benefit is under $20 monthly	R10 §3.B; not modeled std

(b) Insurer-declared current elements#

There is exactly one: the annuity purchase rate applied to each premium, set “at the time each purchase payment is made” S3 on “the attained age of the annuitant, the specified income commencement date and specified income option, and the company’s then current annuity purchase rates” R13 §3.B(1)(b), floored at the income a new contract of the same class would buy R13 §3.B(1)(c).

No purchase-rate table was obtained, and none is published. The Compact expressly relieves the insurer of disclosing the deferral-period basis: “Since the premium and income benefit are fully defined in the contract, the mortality table and interest rate used in the deferral period and for determining the contractually specified income payable do not need to be disclosed in the contract or the Actuarial Memorandum” R13 §1.B(1)(a). The purchase-rate function below is therefore an explicit std construction, not a sourced parameter. An in-force model that reads B from an administration extract does not need it; it is required only for new business, subsequent premiums and the start-date adjustment.

Input	Value	Basis
Pricing interest rate `i_p`	4.75% annual effective	std (a)
Expense and profit load `L`	6.0% of gross premium	std (b)
Pricing mortality	2012 IAM Basic × Scale G2 generational, ANB, × 100%	std (c)
Interest-rate-change adjustment spread `s_adj`	100 bp deducted from the Baa yield	std (d)
Commutation interest-rate adjustment margin `m_c`	50 bp	std/unverified (e)

(a) A pure modeling assumption; no DIA source discloses a pricing rate. Read it as a long-duration general-account portfolio yield net of default costs; the prescribed VM-V portfolio credit-quality distribution (5% Treasuries / 15% Aa / 40% A / 40% Baa R9) is a reasonable calibration frame. (b) Reconciliation so the load is not arbitrary: on the anchor cell L × 100,000 = $6,000 against roughly $900 of present-valued maintenance expense (item in class (c), survivorship-weighted at i_p, computed on the worked example’s illustrative survival anchors extended past age 85 — the figure is sensitive to that extension), a few thousand of first-year distribution cost, and the balance as profit margin. Sensitivity-test it before relying on any absolute income level. (c) VM-22 prescribes q_x^(2012+n) = q_x^(2012) · (1 − G2_x)^n · F_x on the 2012 IAM Basic table with F_x from Table 6.8 for the standard projection R9; the pricing basis uses the same table and scale without the prescribed F_x loading, at a std A/E of 100%. (d) The contract’s “interest rate change adjustment” is named but never quantified in any retrieved source S1 S2; 100 bp is a placeholder that keeps the repriced income directionally correct S4 S5. (e) The Pacific Life interest-rate adjustment charge formula was not found in the fact sheet S4 or the client guide S5; it would appear only in the contract or the actuarial memorandum. Equation (13) implements only the Compact’s stated principle R13 §3.F(7) and is unverified.

(c) Behavioral and experience assumptions#

Input	Recommended public basis	Tags
Payout-phase mortality	2012 IAR / 2012 IAM Basic with Scale G2, generational, ANB, per Model #821 and VM-M §1.J; the appendix print says the same — A-821 ¶11 prescribes the 2012 IAR table for any individual annuity or pure endowment contract issued on or after January 1, 2015, and A-820 ¶6 makes A-821 the mortality leg of the CARVM triple by direct cross-reference	R9 R14 REG-R59 REG-R60 REG-R153 ¶6
Payout-phase A/E	2020–2024 Individual Payout Annuity Mortality Experience Study — 23 parent groups / 26 companies, >80% of industry sales, 3.1m contract-years, 143,190 deaths, shown against the 2012 IAM table; the study explicitly includes deferred income annuities	R15 REG-R61
Deferral-phase mortality	The weakest link. The only public sources are a 2011–2015 deferred annuity mortality study and a 2006 analysis of mortality during the deferred period, both identified via the SOA index, neither fetched	REG-R65; A/E std = 1.00
Mortality improvement	Scale G2 only (generational); none additional in the base run	std
Lapse / surrender	None. “For contracts in which there is no account value or surrender benefit, such as some contracts within the Payout Annuity Reserving Category …, this section is not applicable”	R9
Annuitization	Not applicable; prescribed at 0% for the standard projection	R9
Maintenance expense	$50 per contract per year escalated 2.5% (VM-22 prescribes $50 for individual Payout Annuity Reserving Category contracts, escalated by `[1.025]^(valuation year − 2015)` in year one and 2.5% thereafter, plus 7 bp on a present-value base for contracts without an account value)	R9; adoption as best estimate std
Premium persistency	Deterministic schedule, factor 1.00	std (f)
Start-date adjustment take-up	1.5% p.a. of exposure, one exercise, 60% defer / 40% advance, with rate and selection overlays	std
Payment acceleration take-up	2% p.a. of exposure among eligible contracts	std
Commutation take-up (extended)	1.5% p.a. of exposure, rate-insensitive in the base	std

(f) DIA subsequent premiums are wholly discretionary and no source publishes a distribution of them. The base model projects the model point’s stated schedule with no attrition and exposes pp(y). Dump-in risk differs structurally from a fixed deferred annuity’s: because a DIA prices each premium at then-current rates there is no rate guarantee to select against — unless the contract guarantees paid-up annuity rates for future premiums, which the Compact requires to be described where offered R13 §1.B(1)(h). VM-22 requires “additional premium dump-ins under high guarantees in low-rate environments” to be reflected REG-R36; that risk is switched on by guaranteed_future_rates and is off in the base std.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
`t`, `y`	policy month index `t = 0, 1, 2, …`; elapsed years `t/12`; policy year `y = floor(t/12) + 1`
`m`	payments per year (12 in the base)
`x`, `x(t)`	issue age ANB; attained age `x + floor(t/12)`
`T`	income start month (`13 ≤ T`; `x + T/12 ≤ 85`)
`k`, `P_k`, `t_k`, `d_k`	premium index, amount, payment month; `d_k = (T − t_k)/12` = deferral of slice `k` in years
`f`	income form elected at issue (immutable S1 S2 S4)
`B(t)`, `CP(t)`, `c`	guaranteed annual income before COLA; cumulative premiums; COLA rate
`l(t)`, `q(t)`	in-force probability at the start of month `t`, `l(0) = 1`; probability of death during month `t`
`i_p`, `v`	pricing interest rate; `v = (1 + i_p)^(−1)`
`L`	expense and profit load, fraction of gross premium
`_s p_x`	probability a life aged `x` survives `s` years on the pricing basis
`a^{(m)}_y(f; i)`	APV at age `y` of $1 per annum of income in form `f`, payable `m`-thly in arrears
`a_def(x, d, f; i)`	APV at the premium date of $1 per annum of income in form `f` beginning `d` years hence
`A_rop(x, d; i)`	APV at the premium date of $1 payable at the end of the month of death within a `d`-year deferral
`pr(x, d, f)`	purchase rate: annual income per $1 of premium
`n_g`, `e`, `g`	derived guarantee period in years; annual maintenance expense per contract; its escalation rate
`i_e(t)`, `i_c(t)`	repricing rate for a start-date change; discount rate for a commutation

Dimensional check. a_def and a^{(m)} are APVs per unit of annual income, so they carry units of years; A_rop and L are dimensionless (per $1 of premium). Hence pr = (dimensionless)/(years) = 1/year, P_k × pr is currency per year, and dividing by m gives the currency amount of one payment.

Pricing kernel: the purchase rate#

a_def(x, d, f; i)  =  v^d  ·  _d p_x  ·  a^{(m)}_{x+d}(f; i)                                       (1)

a^{(m)}_y(f; i) is the APV of the payout form, built from the payment factor Φ(t) = max(C(t), L(t)) defined in products/immediate_annuity/technical-notes.md (certain floor C, life-contingent factor L, survivor percentage and reduction trigger included); for a certain-and-life form with guarantee period n_g,

a^{(m)}_y(f; i)  =  Σ_{j=1..m·n_g} (1/m)·v^{j/m}  +  Σ_{j>m·n_g} (1/m)·v^{j/m}·_{j/m} p_y          (2)

APV of the return-of-premium death benefit over the deferral, benefit paid at the end of the month of death std, with q^[j]_x the probability of death in the j-th month:

A_rop(x, d; i)  =  Σ_{j=1..m·d}  v^{j/m} · _{(j−1)/m} p_x · q^[j]_x                                (3)

The purchase rate follows from the equivalence principle applied per premium:

P·(1 − L)  =  B_slice · a_def(x, d, f; i_p)  +  1{db_form = ROP} · P · A_rop(x, d; i_p)

⇒  pr(x, d, f)  =  [ (1 − L) − 1{ROP} · A_rop(x, d; i_p) ] / a_def(x, d, f; i_p)                   (4)

std, for the reasons in assumption class (b). Equation (4) makes the central design fork quantitative: setting 1{ROP} = 0 raises the numerator from (1 − L) − A_rop to (1 − L), so the no-death-benefit form buys strictly more income for the same premium, by the factor (1 − L)/((1 − L) − A_rop). That is the mortality-gain economics behind MassMutual’s separately-conditioned Single Life — No Death Benefit option S2 and behind the silent removal of the return of premium when Life Only is elected S1 S3 S4.

Refund forms and the circularity they create. The Compact treats income payments made before a return-of-premium death benefit as period certain income R13 definitions, which licenses the closed form. For installment refund, payments “continue in the same amount and frequency until they equal the purchase payments” S2, so the form is exactly a certain-and-life annuity with

n_g  =  CP(T) / B                                                                                  (5)

For cash refund the benefit is instead a lump-sum shortfall paid at death S1 S2, so (2) is an approximation; the exact factor adds a decreasing term benefit

a^{(m)}_y(CR)  =  a^{(m)}_y(life only)
                + (1/B) · Σ_j v^{j/m} · _{(j−1)/m} p_y · q^[j]_y · max(0, CP(T) − (j−1)·B/m)        (6)

Both (5) and (6) put B on both sides. Resolve by fixed-point iteration on B (three iterations from a life-only start are ample), or use the (5) approximation for both refund forms std with (6) as a check. Under arrears, a death between T and T + 1/m yields a cash refund of the full CP(T), no payment having been made.

Deferral-phase recursions#

CP(t)  =  CP(t−1)  +  Σ_{k : t_k = t} P_k,                                        CP(−1) = 0        (7)
B(t)   =  B(t−1)   +  Σ_{k : t_k = t} P_k · pr( x(t_k), (T − t_k)/12, f ),        B(−1)  = 0        (8)
l(t+1) =  l(t) · (1 − q(t)),                                                      l(0)   = 1        (9)

Equation (8) is the core of the product: income is additive across slices, each priced at the annuitant’s attained age and the remaining deferral at its own payment date R13 §3.B(1)(b) S3. Nothing accumulates; there is no balance to roll forward. Admissibility tested at each premium: P_k ≥ 10,000 (first) or ≥ 500 (subsequent) S1 S2 S3 S6; CP(t) ≤ 1,500,000 without approval std; t_k ≤ T − 13 S2 S4; and, on a QLAC, CP(t) ≤ qlac_room.

Monthly processing order#

At month t, with l(t) the in-force probability at the start of the month:

Roll attained age and policy year; look up q(t), applying sel_mult(t) if a start-date adjustment has been exercised.
Premium (deferral only): if a scheduled premium falls at t, test admissibility, compute pr by (4), update CP and B by (7)–(8). Cash flow in: P_k · l(t).
Option exercises (start of month): start-date adjustment (deferral, once); payment acceleration or commutation (payout only). Any of these may reset B, T, accel_blackout or commuted.
Income payment (t ≥ T, arrears — at the end of the month): pay B(t)/m · (1+c)^{floor((t − T)/12)} weighted by the form’s payment survivorship, unless suppressed by an acceleration blackout or a commutation.
Death benefits: deaths during month t at rate q(t); if t < T pay the ROP benefit (or nothing under a no-death-benefit form); if t ≥ T pay the form’s refund benefit.
Expenses: (e/12)·(1+g)^{y−1} · l(t).
Decrement: apply (9).

Order note std: contractual transactions precede the decrement, and the end-of-month income payment is contingent on survival to that point — which is what the arrears convention means.

Expected cash flows (per contract, month `t`)#

Cash flow	Formula	Sign
Premium income	`l(t) · Σ_{k: t_k = t} P_k`	+
Deferral death benefit	`1{t < T} · 1{ROP} · CP(t) · l(t) · q(t)`	−
Income payments	`1{t ≥ T} · (B(t)/m) · (1+c)^{floor((t−T)/12)} · L_pay(t)`	−
Refund / certain-period benefits	per form; cash refund `= l(t)·q(t)·RG(t)` — under arrears the instalment due at the end of the death month has not been paid, which is why `RG` is measured through month `t−1` (same convention as the immediate-annuity notes)	−
Payment acceleration	`accel_months · B(t)/m · l(t) · h_acc(t)` at exercise, offset by suppressed payments in the blackout	−
Commutation (extended)	`CV(t) · l(t) · h_com(t)` at exercise, offset by suppressed guaranteed payments	−
Maintenance expense	`(e/12)·(1+g)^{y−1} · l(t)`	−

L_pay(t) is the form-specific payment-survivorship weight defined in the immediate-annuity notes: l(t) for a life-only payment, certain inside a guarantee period, and the survivor-percentage-weighted joint status for joint forms. The DIA-specific change is only the base of the guarantee: CP(T), the sum of all premiums, not a single premium S2 S4. There is no surrender cash flow row, and none should be added.

Income start date adjustment#

Exercised once at month t_e in deferral, moving T → T′ with |T′ − T| ≤ 60 months, T′ ≥ (last premium month) + 13, and T′ inside the maximum deferral and maximum income-start age S1 S2 S3 S4. The disclosed recalculation keys on the originally scheduled payment, the new date, the Moody’s Seasoned Baa Corporate Bond Yield at the request date, a published annuity mortality table, and a contractual interest-rate-change adjustment S1 S2. Implemented as actuarial equivalence at t_e std:

i_e(t_e)  =  Baa(t_e) − s_adj                                                                     (10)
B′        =  B(t_e) · a_def( x(t_e), (T − t_e)/12, f; i_e ) / a_def( x(t_e), (T′ − t_e)/12, f; i_e )  (11)

Direction check: a_def decreases in d, so T′ > T ⇒ B′ > B and T′ < T ⇒ B′ < B, matching Pacific Life’s statement that advancing reduces and deferring increases the payment S4 S5. Two refinements the disclosed recipe does not mention and the reference model therefore does not apply: the ROP exposure changes with deferral length (the A_rop term in (4)), and CP is unchanged so the derived guarantee period (5) shifts. Both are flagged rather than modeled std. Not available on Life Only or Joint Life Only S1 S3 S4, nor on the Single Life — No Death Benefit option, whose annuity date cannot be changed S2.

Payment acceleration#

At exercise month t_a (payout phase, attained age ≥ 59½, nonqualified, monthly frequency, accel_used < 2) S1 S2 S3 S4 the contract pays n_a = 6 monthly payments in one sum and suspends payments for the following n_a − 1 months S1. Accelerated payments are made unconditionally, so the insurer forgoes the survivorship and interest discount on payments 2 through n_a:

Cost(t_a)  =  (B/m) · Σ_{j=1..n_a−1} [ 1 − v^{j/m} · _{j/m} p_{x(t_a)} ]                          (12)

Small but real, and the reason the feature is capped in uses and gated at 59½ S1 S2 S4; the 59½ gate itself is driven by the IRC §72(q) 10% additional tax R8 REG-R55. Not available on a QLAC S4.

Commutation (extended case only)#

i_c(t)   =  i_p  +  max(0, r_ref(t) − r_ref(0))  +  m_c                                           (13)
CV(t_c)  =  Σ_{j ∈ J_g(t_c)} (B/m) · (1 + i_c(t_c))^{−(j − t_c)/12}                               (14)

J_g(t_c) is the set of remaining guaranteed (non-life-contingent) payment months and r_ref a reference market rate. Equation (13) is one-sided — it rises with rates and does not fall — implementing the Compact’s stated intent that the adjustment “reduce interest risk in the event of rising interest rate after issue” and its required disclosure that “the higher the interest rate the lower the commuted value” R13 §3.F(7). The actual contractual formula is not published anywhere S4 S5, so (13) is std/unverified. After a 100% commutation the payments in J_g are suppressed; if the annuitant is alive at resume_month (the end of the would-be guaranteed period) income resumes until death — the life-contingent tail is not commuted S4, and only Period Certain is fully extinguished. Unavailable on the Life Only family S4; prohibited on a QLAC after the required beginning date other than a rescission period not exceeding 90 days R1 (q)(1)(iv) R2 §202(a)(4); interlocked with acceleration and the start-date adjustment by six-month waiting periods in both directions S4.

Convertible versus non-convertible joint life#

The most model-relevant pricing subtlety in the family, and the only place a joint DIA differs structurally from a joint SPIA. With lives aged x₁, x₂ and survivor percentage s:

Non-convertible — priced on “a single payout assumption that both annuitants will be alive on the annuity date” S2, and if one annuitant dies in deferral the contract continues on the option chosen at issue S2:

P(1−L) − P·A_rop  =  B_J · [ a_def^{(both)} + s · Σ_{i=1,2} a_def^{(only i)} ]                    (15)

Convertible — priced on “two different payout assumptions”: the joint payout if both are alive at the annuity date, and the corresponding single life payout for each annuitant if only one is alive and the contract converts S2:

P(1−L) − P·A_rop  =  B_J^c · a_def^{(both)}  +  Σ_{i=1,2} B_S^{(i)} · a_def^{(only i)}            (16)

a_def^{(both)}    =  v^d · _d p_{x₁x₂}(both alive) · a^{(m)}_joint(f)
a_def^{(only i)}  =  v^d · ( _d p_{x_i} − _d p_{x₁x₂} ) · a^{(m)}_{x_i + d}(single f)

Because B_S^{(i)} (a full single-life payout) exceeds s·B_J (the reduced survivor amount), the right-hand side of (16) carries more value per unit of B_J, so B_J^c < B_J — reproducing MassMutual’s statement exactly: “In general, if both annuitants are alive on the annuity date, the joint life payout will be lower with a convertible joint life annuity option” S2. The convertible variant also caps the period certain at 10 years and is unavailable with the inflation protector S2. Base model: non-convertible std; convertible_joint switches to (16).

QLAC overlay#

A validation-and-restriction layer over the same recursions. It generates no cash flows of its own — it caps premiums, restricts forms, constrains T, disables features and raises compliance flags. Loss of QLAC status changes the owner’s RMD position, not the insurer’s liability cash flows.

Rule	Implementation	Basis
Premium limit	`qlac_room(t) = Limit(year) − CP(t) − (premiums paid to any other contract intended to be a QLAC under any 401(a), 403(a), 403(b), 408 or governmental 457(b) arrangement)`; a breaching premium is rejected	R1 (q)(2)(ii) REG-R57
Limit level and indexing	$200,000 as enacted; indexed like §415(d) limits with base period the calendar quarter beginning July 1, 2022, increments rounded to the next lowest multiple of $10,000; $210,000 for 2026	R1 (q)(4)(ii)(A) R2 §202(a)(2) R3 S4
Percentage-of-account-balance limit	None. SECURE 2.0 § 202(a)(1) directed its elimination and the codified text has no percentage test. Do not implement a 25% test	R2 R1 REG-R58
Latest income start	`T` — the annuity starting date, not the date the first payment lands — no later than the first day of the month next following the 85th anniversary of birth	R1 (q)(1)(ii) S1 S2 S3 S4
Earliest income start	A product rule, not a QLAC rule: after April 1 of the year following the year the owner attains the applicable RMD age	S1 S2
Permitted death benefits	Exhaustive: (i) life annuity to a sole-beneficiary surviving spouse ≤100% of the employee’s payment, commencing no later than the employee’s annuity would have; (ii) life annuity to another beneficiary ≤ the applicable percentage, commencing by the last day of the year following the year of death; or (v) return of premiums up to premiums paid less payments already made, payable by the end of the year following the year of death	R1 (q)(3)
Applicable percentage	MDIB table where there is no pre-annuity-starting-date non-spousal death benefit; Table 6 (≤2 yrs → 100% … 25+ yrs → 20%) where the non-spousal beneficiary is irrevocably set; 0 where the contract provides a return of premium	R1 (q)(3)(iii)
Model consequence	A QLAC carries either the ROP death benefit or a beneficiary life annuity — never both, since the applicable percentage is 0 when ROP is present	R1 (q)(3)(iii)(C)
Liquidity	`commutation_allowed = false`, `accel_uses = 0`, `cola_rate = 0`	R1 (q)(1)(iv) S4; the COLA restriction is a market choice, not a regulatory one (a)
Permitted forms	Restrict `income_form` to {Single Life No Refund, Single Life Cash Refund, Single Life No Death Benefit, Joint & Survivor Cash Refund}; Installment Refund and Period Certain unavailable	S2 S4
RMD exclusion	The contract’s value is excluded from the RMD account balance; not for a Roth IRA	R4 R5 (h)(4)
Failure flags	Excess premium ends QLAC status on the date paid unless returned by the end of the following calendar year; any other failure voids status retroactively to purchase	R1 (q)(4)(i)(B), (q)(4)(iii)(A)
Rescission	A rescission right not exceeding 90 days from purchase does not violate the no-commutation rule	R1 (q)(1)(iv) R2 §202(a)(4)
Divorce	A joint-and-survivor QLAC survives a post-purchase, pre-commencement divorce under QDRO conditions, retroactive to contracts purchased on or after July 2, 2014	R1 (q)(3)(vii) R2 §202(a)(3), §202(c)(1)(B)

(a) Research finding worth carrying into the model as a comment: the regulation expressly permits a QLAC to provide a cost-of-living adjustment described in paragraph (o)(2) R1 (q)(4)(iv), yet NYL, Guardian and Pacific Life all exclude COLA from their QLAC offering S1 S3 S4 and MassMutual limits it on qualified contracts S2. The market is more restrictive than the law; follow the market by default and expose the switch.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions. No DIA behavioral experience study exists in any retrieved source, and VM-22’s standard projection prescribes 0% annuitization and no lapse assumption for this reserving category R9, so these are genuinely modeler’s choices rather than calibrations.

Lapse and surrender: exactly zero, at all durations. Not conservatism — there is no surrender benefit to elect R9 R13. A nonzero lapse assumption in a DIA model is a defect, not a margin.

Income start date adjustment. One-time, deferral only, excluded on Life Only and Joint Life Only S1 S3 S4:

h_adj(t) = h_0 · M_dir(t),   h_0 = 1.5% p.a. of exposure; window policy year 3 to (T/12 − 2)   **[std]**
direction split: 60% defer / 40% advance                                                       **[std]**
M_def(t) = min( 3.0, 1 + 3 · max(0, i_e(t) − i_p − 0.01) )                                     **[std]**
M_adv(t) = min( 3.0, 1 + 3 · max(0, i_p − i_e(t) − 0.01) )                                     **[std]**

Rationale: the option’s value is driven by the spread between the pricing rate locked at issue and the Baa yield at exercise, which is why the right is one-time and why the forms with the strongest health-driven selection are excluded S1 S2 S3 S4. In a higher-rate environment deferring buys proportionally more income (equation (11) with a larger i_e), so M_def rises with rates and M_adv rises when rates fall.

Selection on health. sel_mult = 0.90 for contracts exercising to defer and 1.10 for those advancing, applied to q(t) from the exercise month std. Rationale: insurers exclude the Life Only forms from the adjustment right precisely because that is where anti-selection on health bites hardest S1 S3 S4 — the exclusion is evidence the effect is believed real — and 10% is a deliberately modest reference magnitude with no experience behind it.

Payment acceleration. h_acc(t) = 2% p.a. of exposure among contracts that are nonqualified, on monthly frequency, at attained age ≥ 59½, in the payout phase, with uses remaining, and outside both a blackout and the six-month interlocks std S1 S2 S3 S4. Modeled as pure timing plus the cost in (12).

Commutation (extended case). h_com(t) = 1.5% p.a. of exposure among eligible contracts, rate-insensitive in the base run std: the interest-rate adjustment in (13) is designed to neutralize rate-driven anti-selection R13 §3.F(7), so a rate multiplier would double-count the protection. Expose M_com(t) = min(3.0, 1 + 4·max(0, r_ref(t) − r_ref(0))) for sensitivity testing only.

Spousal continuation on death in deferral. Where the surviving spouse is joint annuitant and sole primary beneficiary the contract may continue instead of paying the death benefit S1 S2 S4. Base std: 100% election of the death benefit. The continuation branch is a switch; under non-convertible continuation no further premiums are allowed S2.

Premium behavior. Per assumption note (f): deterministic schedule, no dump-in dynamics unless guaranteed_future_rates is set std.

Worked example#

Anchor cell std: Female 60 ANB, nonqualified, Life with Cash Refund, monthly in arrears, ROP death benefit in deferral, no COLA. Premiums $100,000 at issue (age 60, 20-year deferral) and $50,000 at the start of policy year 6 (age 65, 15-year deferral). Income start at attained age 80 (T = 240). Pricing basis i_p = 4.75%, L = 6.0% std.

Illustrative factors std. The 2012 IAM Basic / Scale G2 numerical tables were not retrieved — they live at mort.soa.org R16 — so the factors below are mutually consistent illustrative values, not table lookups; a production implementation must substitute table values R9 REG-R59. Anchors: _5p_60 = 0.975000, _10p_60 = 0.920000, _15p_60 = 0.838000, _20p_60 = 0.715000, _5p_80 = 0.780000, with geometric interpolation inside each five-year band, hence _15p_65 = 0.715000/0.975000 = 0.733333. Payout factor a^{(12)}_80(cash refund, female; 4.75%) = 8.60; discount factors v^20 = 0.395293, v^15 = 0.498528. A_rop is computed from the same survival anchors with a mid-band death-timing approximation: A_rop(60, 20) = 0.157900, A_rop(65, 15) = 0.180200.

Slice pricing, equation (4). Slice 1 (age 60, d = 20): a_def = 0.395293 × 0.715000 × 8.60 = 2.430657; pr₁ = (0.940000 − 0.157900)/2.430657 = 0.321765; B₁ = 100,000 × 0.321765 = $32,176.50 per year. Slice 2 (age 65, d = 15): a_def = 0.498528 × 0.733333 × 8.60 = 3.144050; pr₂ = (0.940000 − 0.180200)/3.144050 = 0.241663; B₂ = 50,000 × 0.241663 = $12,083.15 per year. Total from T: B = $44,259.65 per year = $3,688.30 per month. Derived guarantee period (5): n_g = 150,000/44,259.65 = 3.3891 years, so the cash-refund guarantee is exhausted during the fourth payment year, at attained age ≈ 83.4.

The death-benefit fork, same premiums. Setting 1{ROP} = 0 in (4): pr₁ = 0.940000/2.430657 = 0.386727 and pr₂ = 0.940000/3.144050 = 0.298977, so B = 38,672.70 + 14,948.85 = $53,621.55 per year — 21.2% more income for the same $150,000. That difference is the price of the return-of-premium guarantee, and it is mortality gain in the insurer’s hands if the annuitant dies in deferral: at the year-10 death below, the ROP form pays $150,000 and the no-death-benefit form pays nothing while releasing the entire reserve.

Projection (annual display grid; the model runs monthly). E[DB] is the expected deferral death benefit paid in policy year t, = (l(t−1) − l(t)) × CP(t); E[income] is the expected income paid at the end of policy year t on an annual-payment display approximation.

Policy year `t`	Age at start of year (ANB)	Premium at start	`CP(t)`	`B(t)` (annual)	`l(t)`	`E[DB]` in year `t`	`E[income]` at `t`
1	60	100,000	100,000	32,176.50	0.994949	505.08	—
5	64	—	100,000	32,176.50	0.975000	494.95	—
6	65	50,000	150,000	44,259.65	0.963743	1,688.54	—
10	69	—	150,000	44,259.65	0.920000	1,611.90	—
15	74	—	150,000	44,259.65	0.838000	2,369.01	—
19	78	—	150,000	44,259.65	0.738063	3,571.09	—
20	79	—	150,000	44,259.65	0.715000	3,459.50	—
21	80	—	150,000	44,259.65	0.680338	—	30,111.54
25	84	—	150,000	44,259.65	0.557700	—	24,683.61

Trace, policy year 6: the $50,000 premium arrives at the start of the year while the annuitant is alive, so expected premium income is 50,000 × l(5) = $48,750.00; it buys slice 2 at the age-65 / 15-year purchase rate, taking B from $32,176.50 to $44,259.65 by (8) and CP to $150,000 by (7). Deaths during year 6 (l(5) − l(6) = 0.0112569) now attract the larger benefit CP(6) = 150,000, giving E[DB] = $1,688.54. Nothing accumulates and nothing is credited: between premiums the only state change is the survivorship in (9). The income start date is the start of month index T = 240 (20 years from issue); under arrears the first payment falls one month later, 241 months from issue — which is why policy year 20 (months 228–239) still shows a deferral death benefit and policy year 21 (months 240–251) carries the first twelve payments.

Valuation and reserve pointers#

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

Statutory — principle-based. VM-22, “PBR for Non-Variable Annuities”, constitutes CARVM for contracts in scope and names Deferred Income Annuity contracts explicitly in the Payout Annuity Reserving Category R9 §3.F.1.a REG-R36. Effective for valuation dates on or after January 1, 2026, with an elective three-year transition on VM-A/VM-C/VM-M/VM-V for newly issued business and mandatory prospective application three years after the effective date R9 REG-R36. Aggregate reserve = SR (stochastic, CTE70) + DR for contracts passing the Single Scenario Test + formulaic reserves for excluded contracts; the Additional Standard Projection Amount is a VM-31 disclosure item R9 REG-R36. Payout and Accumulation categories may be aggregated only under an integrated risk management process and a single portfolio or portfolios with the same ALM strategy R9.
Statutory — formulaic fallback. For contracts not passing the Stochastic Exclusion Test, VM-V Section 1 “Income Annuities” — not VM-22 — sets the statutory maximum valuation interest rate, its scope expressly including “deferred income annuity contracts issued after Dec. 31, 2017” R9 REG-R37. The rate is a function of the Valuation Rate Bucket (A–D by reference period and initial age; a DIA issued below age 70 with a long reference period lands in Bucket D), the premium determination date — for a DIA the “date consideration is determined and committed to by contract holder”, with an immateriality tolerance of a change under 10% in present value and under $1 million — and jumbo versus non-jumbo status; rates are published daily (jumbo) and quarterly (non-jumbo) by the NAIC R9. VM-V §1 supersedes the interest-rate guidance in AG IX-B and the interest references in AG IX-C REG-R37; the incorporated guideline family is indexed at REG-R41. VM-21 does not apply — it is the variable-annuity standard REG-R35 REG-R36.
Statutory — the formulaic chain itself. CARVM is A-820 ¶¶14–15 (method), ¶¶7–10 (interest) and Appendix A-821 (mortality) REG-R153 ¶6, as interpreted for contracts with elective benefits by AG 33 REG-R151. Both were read from the AP&P Manual print on 2026-08-06. The DIA-specific consequences — which mandatory stream families are empty, why the ±5-year adjustment re-rates the annuitization portion, why a commutation right bars Text 4(B), and why the 7% expense-allowance floor has no base here — follow from those two prints REG-R151 REG-R153.
Prescribed standard-projection assumptions reusable directly: mortality q_x^(2012+n) = q_x^(2012) · (1 − G2_x)^n · F_x on the 2012 IAM Basic table with F_x from Table 6.8 (ANB) R9; lapse: not applicable; annuitization: 0% at all projection intervals; maintenance expense $50 per individual Payout Annuity Reserving Category contract per year escalated at 2.5%, plus 7 basis points applied, for contracts without an account value, to a present-value base R9 (the exact base was truncated at a page break in the research extract R9).
Nonforfeiture. Model #805 applies during deferral but the cash-surrender requirement is conditional and untriggered; the paid-up annuity requirement is satisfied by construction R10 R13 §3.H(1). For Compact filings a comparative-adequacy certification replaces the demonstration R13 §1.B(1)(g). Corrected parameter: the Model #805 indexed nonforfeiture rate is floored at 15 basis points, not 1% REG-R42 — see product-spec.md, Regulatory context.
Tax. IRC §807: the tax reserve is the greater of net surrender value (zero here) and 92.81% of the NAIC-prescribed method — CARVM — capped at the statutory reserve REG-R16. Contract-holder taxation runs through IRC §72’s exclusion ratio and the §72(q) penalty R8 REG-R55.
U.S. GAAP. Under LDTI a payout annuity carries a liability for future policy benefits with annually reviewed assumptions and no market risk benefit — there is no account value for an MRB to attach to REG-R34 REG-R71.
Standards for the modeling work. ASOP 7 (cash flow analysis) REG-R27; ASOP 22 (asset adequacy) REG-R29; ASOP 54 (pricing) REG-R70; ASOP 56 (modeling) REG-R32. There is no ASOP for principle-based reserves for annuities — ASOP 52 is scoped to VM-20 life products REG-R31 REG-R70 context.

Key sensitivities and model risks#

Longevity, levels and improvement. With no lapse decrement nothing offsets a mortality miss; the liability runs to the last survivor. The valuation table is the 2012 IAR/IAM family with Scale G2 R9 REG-R59 REG-R60 while the experience under it has moved — the 2020–2024 payout study measures directly against the 2012 IAM basis and is the right place to look for drift R15 REG-R61. Test the A/E factor and an improvement scale stronger than G2 first.
Deferral-phase mortality, especially on no-death-benefit forms. Equation (4) puts the whole ROP cost in the numerator: on the anchor cell, removing the ROP moves income by 21.2%. Deferred-period annuitant mortality is served publicly by only two dated sources REG-R65 and is the least-evidenced assumption in the model. It is also asymmetric — a higher assumed deferral mortality lets the insurer offer more income — so an error here is not conservative in either direction by default.
The pricing rate i_p and the load L. They set the income level directly through (4). If the model reads B from an administration extract these drop out of the in-force projection entirely, which is the recommended configuration for valuation work and removes the largest std exposure.
Start-date adjustment take-up. A one-time, ±5-year, rate-sensitive option granted without an explicit charge in every retrieved product S1 S2 S3 S4. Take-up, direction split and selection multiplier are all std with no experience behind them.
Commutation in the extended case. The interest-rate adjustment formula is unpublished S4 S5; equation (13) is a construction. Because commutation removes guaranteed payments and leaves the life-contingent tail S4, a wrong adjustment changes the shape of the residual liability, not just its timing.
Timing conventions. Arrears versus advance moves the whole payout stream by one payment period; end-of-month versus mid-month death benefit timing moves deferral claims by half a month. Both are std and must be documented when reconciling to an administration system.

Known pitfalls specific to this product:

Building a lapse module. There is nothing to lapse to R9. A “prudent” 2% lapse assumption is wrong here and understates the liability.
Building an account-value roll-forward. The liability is a schedule of income slices keyed on (premium, purchase date, income start date, income option), not a balance [derived from S1–S4, R13]. Any credited rate, charge or interim value invented to fill a template is fictitious.
Using life-insurance mortality tables. 2017 CSO and 2015 VBT are for insured lives; annuitant longevity must use the 2012 IAM/IAR family and the payout experience studies REG-R59 REG-R61.
Chaining rounded generational rates. q_x^(2012+n) must be computed from the 2012 period rate each time; the Valuation Manual explicitly flags q^2014 = q^2013(rounded) × 0.99 as incorrect R9 REG-R59, and A-821 ¶14 prints the identical prohibition and the identical counter-example REG-R153.
Feeding the behavioral take-up rates into a CARVM run. h_adj, h_acc and h_com are projection assumptions. AG 33 prohibits experience-based elective incidence and makes the elective path a decision variable maximised over, so importing them into a formulaic reserve is a defect, not a refinement REG-R151.
Single-premium refund bases. For a flexible-premium DIA the cash-refund and installment-refund bases are cumulative premiums CP(T), not the initial premium S2 S4.
Circularity in refund forms. (5) and (6) put B on both sides; failing to iterate leaves a systematic bias in the derived guarantee period and hence in the income.
Double-counting the deferral death benefit. Its cost belongs in the purchase rate (4) and in the projected cash flows; it must not additionally be loaded into a_def.
Modeling payment acceleration as a withdrawal. It is a timing shift expressly labelled “not a liquidity feature” S2; the only economic cost is (12).
QLAC arithmetic from stale documents. A 25%-of-account-balance test, a $125,000 or $130,000 cap, or an RMD age of 70½ are all superseded R1 R2 R3, though pre-2023 insurer guides still print them S2 S3. Cite § 1.401(a)(9)-6(q), not “A-17” R1 R6.
Treating the COLA as index-linked. Every observed option is a fixed compound escalator elected at issue, not a CPI adjustment S1 S2 S3 S4.