Technical Notes#

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

Scope note. These notes specify a reference liability cash-flow projection model for the standardized composite product defined in product-spec.md (same directory). This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in sources.md, numbering carried verbatim from _research/immediate-annuity.md; [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md, one shared numbering space now running R1–R157 with most of the R73–R149 block unused (R1–R34 from _research/regulatory-actuarial.md, R35–R72 from _research/regulatory-actuarial-annuities.md, and R150–R157 from the AP&P Manual appendix reading of 2026-08-06 — of which R151 (AG 33) and R153 (A-820 with A-821 and A-822) are cited here). std marks standardizations introduced for the reference implementation. Parameter values are identical to those in product-spec.md; the mechanics anchors are MassMutual RetireEase S1 and Pacific Income Provider S2 S3.

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows (income instalments to the annuitant, joint annuitant and beneficiary; certain-period and refund payments; cash-refund lump sums; commutation payments; maintenance expenses) for a single SPIA in payment. Discounting and reserves are not computed (see Valuation and reserve pointers).

  • Mortality is the model. No premium after outset, no account value, no cash surrender value, no policyholder option other than certain-portion commutation S1 S4 S5. The only decrement is death; the only stochastic driver is longevity. There is no lapse decrement — a point VM-22 makes prescriptively for this reserving category: the prescribed lapse table “is not applicable” for contracts with no account value or surrender benefit, and the prescribed annuitization rate is 0% R2 REG-R36.

  • Projection frequency. Monthly grid, t = 1, 2, … months from the annuity date std. Payment dates fall on the grid per the frequency m; day-count and business-day conventions are not modeled std.

  • Timing conventions std. COLA increases apply at the start of the month containing the anniversary of the annuity date (first at t = 13) S1. Arrears is the default: an instalment due at the end of month t requires survival to the end of month t; on advance timing it is paid at the period start and requires survival to the start. The arrears default follows VM-V’s own prescribed weight-table cash flow model, which assumes “annuity payments are made at the end of each year” R1. Deaths are decremented at end of month; a death during month t means the life does not receive an arrears instalment due at the end of month t, and a survivor reduction likewise takes effect from the instalment due at the end of the month of death — not from the following payment date std. Both follow from evaluating L(t) at end-of-month survival, lᵢ(t), and both must be applied the same way or the two decrements disagree by one payment period.

  • Age basis. Age nearest birthday (ANB) std: MassMutual defines contract age as age nearest birthday S1; the 2012 IAM/IAR family is tabulated ANB R2 R9; IRS Publication 939 uses “the age at the birthday nearest to the annuity starting date” R7. One place prescribes a different basis: VM-V’s “initial age” for valuation-rate bucket selection is the annuitant’s age last birthday at the premium determination date (the younger annuitant on a joint contract, or the rated age if valued as impaired) R1. VM-22 supplies the conversion R2: q(x)_ALB = [q(x)_ANB + (1 − q(x)_ANB)·q(x+1)_ANB] / (2 − q(x)_ANB).

  • Limiting age. ω = 120. No longer std on the valuation side: A-821’s printed 2012 IAM Period Table runs to age 120 for both sexes and prints 1000·q₁₂₀ = 1000.000 there, so ω = 120 is the table’s own terminal age and no extrapolation is required for the valuation basis REG-R153. The printed rates also confirm the ultimate cap the 2012 IAR development report describes — 400.000 per 1,000 = 0.40000 — and expose a terminal-age asymmetry a shared array will hide: female rates reach 400.000 at age 108, male rates are 380.000 at 105 and 400.000 from 106 REG-R153 REG-R60. (The development report’s account of how that cap arises — the 10% margin holds to age 100, then grades down 1% a year until the cap is invoked, where the margin is zero REG-R60 — is a construction property, not a terminal age.) The std extrapolation rule stays for the best-estimate basis, which runs on the 2012 IAM Basic table: A-821 prints only the loaded Period Table, so the Basic table’s own tabulation limit is not sourced here.

  • Model points. USD; single-contract model points projected on an expected (probability-weighted) basis: survival probabilities multiply scheduled per-contract cash flows. No aggregation logic is specified here. Joint-life independence std — the SOA/LIMRA payout study is explicit that its data cannot inform this: “no recognition is given to the secondary annuitant if alive while the primary annuitant is alive”, because of under-reporting of secondary-annuitant deaths R9.

Relation to uk/products/pension-annuity/technical-notes.md. The survival-indexed payment engine is the same object — a scheduled instalment multiplied by a payment factor blending a certain floor with life-contingent survival. The U.S. differences are all structural:

Dimension

UK pension annuity

U.S. SPIA

Best-estimate mortality

CMI-restricted SAPS/PMA-PFA tables, proxied by an ONS population table × α std

2012 IAM Basic × Projection Scale G2, generational, × an A/E factor from a public experience study R3 R9 REG-R59 REG-R61

Valuation mortality

Solvency UK best estimate + risk margin (no prescribed table)

2012 IAR generational table with an explicit no-compound-rounding rule R3 R4 REG-R59

Escalation

RPI 0-floor with catch-up ratchet, LPI-5, fixed

Fixed compound only (1–4%); no RPI/LPI analogue and no CPI-linked option in any retrieved U.S. document S1 S2 S4 S5 S6 S8

Death benefit on a refund basis

Value protection (v × P − payments)

Cash refund (P − payments) and installment refund (payments continue until P is recovered) S1 S3 S5

Guaranteed term

Guarantee period

Period certain (5–30 yrs), plus a derived certain period on refund forms = premium ÷ annualized income S5

Survivor reduction

Dependant percentage δ on the annuitant’s death; the dependant’s pension is an additive second stream that may run alongside the guarantee

δ on either the primary’s death or any annuitant’s death — a switch, not a parameter S1 S2 S3 S7; the reduction rescales the single stream and is suspended by the certain floor S5

Taxation

Income taxed in full as pension income

Exclusion ratio under IRC §72 / IRS Pub. 939 R6 R7 REG-R55

Liquidity

None after cancellation window

Commutation of the certain portion only, net of a declining surrender charge S1

Model point attributes#

Attribute

Type

Example (anchor cell)

premium P

currency

100,000 std

premium_tax_rate τ

float

0.00 std (mechanism S6 S7 S11)

annual_income B(1)

currency p.a.

6,000 std (spec footnote 8)

form

enum {life_only, life_certain, cash_refund, installment_refund, certain_only} — the five forms of S1 S2 S4 S5

life_only (joint) std

joint

bool

true std

primary_age x₁ / primary_sex

int (ANB S1) / enum {M, F}

65 / M std (spec footnote 2)

joint_age x₂ / joint_sex

int (ANB S1) / enum {M, F}

62 / F std (spec footnote 2)

survivor_pct δ

float ∈ {0.50, 2/3, 0.75, 1.00} S1 S2 S5 S6 S7

2/3 std (spec footnote 10)

reduction_trigger

enum {either, primary} S1 S2 S3 S7

switch — both runs shown

certain_months n

int, 0 or 60–360 S1 S2 S4 S5

0 (120 when life_certain) std

frequency m

enum {12, 4, 2, 1} S1 S2

12 std

timing

enum {advance, arrears}

arrears std

cola_rate g

float ∈ {0, .01, .02, .03, .04} S1 S5

0.03 std

commutation_enabled

bool (certain-bearing forms only S1)

false in base std

issue_state_excludes_withdrawal

bool (Oregon S1 S2; NY for S4 riders)

false

qualified

bool

false std (spec footnote 1)

B(1) is a pricing input, not a modeled output: no insurer publishes payout factors or the pricing basis, so it must be taken from a quote or calibrated (spec footnote 8). The model does not derive B(1) from P.

State variables#

Variable

Description

Updated

B(y)

Annualized income in policy year y, unreduced (“as if all annuitants alive”)

anniversaries

l₁(t), l₂(t)

Survival probabilities of primary / joint annuitant to end of month t; l(0) = 1

monthly

d₁(t), d₂(t)

Death densities, dᵢ(t) = lᵢ(t−1) − lᵢ(t)

monthly

l_last(t)

Probability at least one annuitant alive = l₁ + l₂ − l₁·l₂ [std independence]

monthly

d_last(t)

Last-death density = l_last(t−1) − l_last(t)

monthly

G(t)

Cumulative gross instalments scheduled through month t

payment dates

n_R

Derived installment-refund certain period (months)

once, at t = 0

θ_cum(t)

Cumulative commutation fraction applied to certain-period instalments

on withdrawal

CV(t)

Commuted value of the remaining certain instalments

on request

G(t) is a deterministic schedule: instalments payable while any covered life is alive follow the deterministic escalation path, so the refund balance needs no path simulation.

Assumption inputs#

(a) Contractual / guaranteed elements (cited)#

Input

Value

Basis

Instalment

B(y)/m at each payment date

S1 S2

COLA rule

B(y) = B(y−1)(1+g) on each anniversary of the annuity date; compound; irrevocable

S1 S4 S6

Survivor benefit

δ × the current income payment

S2

Reduction trigger

death of either annuitant, or of the primary annuitant only

S1 S2 S3 S7

Certain period

n months of instalments payable regardless of survival

S1 S2 S4 S5

Cash refund

lump sum at death = max(0, P − G(death))

S1 S3 S5

Installment refund

instalments continue until cumulative payments equal P

S1 S4 S5 S6

Refund-implied guaranteed period

premium ÷ annualized income benefit amount

S5

Withdrawal cap / minimum / residual floor

PV of remaining certain payments less surrender charges / $5,000 / $100 per remaining payment

S1

Surrender charge sc(y)

yr 2: 8%; 3: 7%; 4: 6%; 5: 5%; 6: 4%; 7: 3%; 8: 2%; 9: 1%; 10+: 0% (no withdrawal in yr 1)

S1

Effect of a withdrawal on post-certain lifetime payments

none

S1 S2 S5

Cash surrender value

none, at any time

S1 S4 S5

Nonforfeiture floor

none — immediate annuities excluded from Model #805 §2.A

R5 REG-R42

Charges to the policyholder

none (“zero fees”)

S1

(b) Insurer-declared current elements (snapshot)#

For a fixed SPIA this class is nearly empty: no credited rate, no cap, no participation rate, no declared rider terms. Two quantities remain, and neither is published.

Input

Value

Basis

Initial annual income per unit premium (the payout factor)

6.00% of premium p.a. (= $6,000 on $100,000)

std, spec footnote 8 (i)

Commutation discount rate j(t)

4.00% + (10-yr CMT(t) − 10-yr CMT(0)), compound

std unverified, spec footnote 14 (ii)

Commutation discount convention

compound (default std) or simple (per S7)

(ii)

State premium tax τ

0.00%

mechanism S6 S7 S11; rate std (iii)

(i) No insurer publishes payout factors, guaranteed annuity purchase rates or the pricing basis for a fixed SPIA (research gap). The only insurer-sourced anchors are illustrations labelled “for illustrative purposes only”: Joint Life Only, both 65, $230,856 → $1,200/month ⇒ 6.24% annualized; Life with 10-Year Period Certain, age 69, ⇒ ≈7.11%; single life male 65 with 3% Inflation Protection ⇒ ≈5.28% initial S3. A low-reliability broker survey gives male 65 life-only 7.97% with a 5–6% carrier spread S9; NYL’s weekly rate table could not be captured S10. Consequence: no pricing or annuity-rate test against public data is possible. B(1) is exogenous. Note that the 6.00% std level sits above what the COLA-adjusted anchors imply for this cell (≈4.5%; spec footnote 8): it is a round arithmetic anchor that makes the worked example exact, not a price, and must be re-set from a quote before any output is read as one.

(ii) No fixed SPIA issuer publishes a commutation discount formula. MassMutual gives only the cap S1, Pacific Life only “an interest-rate adjustment will apply” S2, NYL only the 10-Year CMT as driver S5. The one explicit formula located is TIAA-CREF Life’s 2008 variable contract: fixed-account commuted value = “the sum of payments less the interest that would have been earned from the effective date of the commuted value calculation to the date each payment would have been made” (simple interest), with 4% on variable accounts S7. The reference implementation therefore assumes a basis and flags it std and unverified; any implementation must carry the same flag.

(iii) Premium tax is deducted before income is determined S6 S7 S11, but no source quantifies a rate and state rates were not researched (research gap).

(c) Behavioral / experience assumptions (modeler’s view)#

Input

Recommended public basis

Basis tags

Base annuitant mortality

2012 IAM Basic Table (unloaded table underlying the 2012 IAM Period Table, developed from the 2002 experience table projected to 2012) with Projection Scale G2, applied generationally

R3 R2 REG-R59

A/E adjustment to best estimate

× 1.084 std, from the 2020–2024 SOA/LIMRA payout study: amount-basis A/E versus 2012 IAM Basic projected with G2 is 108.4% overall (107.5% F, 109.4% M)

R9 REG-R61; adoption std (iv)

Mortality improvement

Scale G2 only, applied generationally; no additional improvement in the base run std

R3 R4; see (iv)

Substandard / rated lives

q_rated = min(1, θ·q_be), θ ≥ 1 (equivalently a rated-age offset); θ = 1 in base

existence S8; VM-V “rated age” R1; overlay std

Lapse / surrender

None — no cash value, no surrender right

S1 S4 S5 R5; VM-22 declares the lapse table inapplicable here R2

Annuitization

Not applicable (already in payout); VM-22 prescribes 0%

R2

Commutation utilization

0% in the base run std

(v)

Maintenance expense

$60 per contract p.a., paid monthly while any payment obligation remains, inflating 2.5% p.a.

std (vi)

(iv) The study measured 99.6% A/E against 2012 IAM Basic unprojected and 108.4% against 2012 IAM Basic projected with Scale G2 over 2020–2024, on 3,109,309 contract-years and 143,190 deaths from 23 parent company groups representing just over 80% of industry sales R9 REG-R61. The reading: G2 has over-projected improvement — actual mortality is running about 8% heavier than the fully projected basis. A flat 1.084 on the projected basis reproduces the study average and is the least-assumption starting point, but it is std, because the study documents gradients the flat factor ignores: by attained age, all groups with ≥65% credibility except 65–74 had A/E above 100%, with 65–69 at 80% and 70–74 at 92%; by annual income band, A/E generally decreases as income rises, from 126% below $2,500 of annual income to 91.5% at $50,000 or more — the classic amount-based socio-economic gradient, directly relevant to any block segmented by policy size R9. A production basis must reflect both. VM-22 additionally requires, for a “longevity segment” (which a SPIA block is), that the industry and credibility-adjusted tables be brought forward for improvement to the valuation date and that future improvement be reflected if it increases the reserve R2 REG-R36.

(v) No public data on SPIA commutation take-up was located; the base run holds utilization at zero so the payment engine is exercised in isolation. (vi) No insurer publishes expense assumptions; MassMutual’s “zero fees” S1 refers to charges to the policyholder, not the insurer’s cost. $60 p.a. is a round placeholder for in-payment administration; acquisition cost is out of scope (single premium, priced in).

Mortality tables are not embedded — and the two that matter here are now in different states. The 2012 IAM Period Table and Projection Scale G2 have been retrieved: the AP&P Manual is a free download, not the paid publication recorded at REG-R33, and A-821 prints both tables in full, both sexes, age nearest birthday, at its Appendices I–IV REG-R153. They are transcribed in _research/appp-a820-a821-a822.md, so the valuation basis (2012 IAR = 2012 IAM Period × Scale G2, generational) is sourceable end to end, and the rounding worked example is now cross-checked against the printed tables rather than standing alone: male age 30, 1000·q^2012 = 0.741, G2₃₀ = 0.010 R3 R4 REG-R153. The 2012 IAM Basic table — the unloaded table this model’s best estimate runs on — is still not retrieved: A-821 prints the loaded Period Table only, and the Basic table is defined in VM-M §2.C R3. Machine-readable versions of either are conventionally obtained from the SOA’s mortality table repository [unverified — not a source in the research file]. The model must load them; it cannot hard-code them, and the two tables must not be interchanged — the loaded Period Table is a valuation object, the Basic table a best-estimate one.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol

Meaning

t

month index from the annuity date, t = 1, 2, …; policy year y(t) = ⌈t/12⌉

m

payments per year (12/4/2/1); payment months T = {12k/m : k = 1, 2, …} (arrears); on advance the k-th instalment falls one full payment period earlier, at the start of month 12(k−1)/m + 1

P, τ

single premium; premium tax rate. P_net = P(1 − τ)

B(y)

unreduced annualized income in policy year y; inst(t) = B(y(t))/m for t ∈ T

g

fixed compound COLA rate (0.03 std, ∈ {1%…4%} S1 S5)

δ

survivor percentage (2/3 std, ∈ {50%, 66⅔%, 75%, 100%})

trig

reduction trigger ∈ {either, primary} S1 S2 S3

n, n_R, n_eff

elected certain months; derived installment-refund months; effective certain months by form

x₁, x₂

issue ages (ANB) of primary and joint annuitant

l₁(t), l₂(t)

survival probabilities; dᵢ(t) = lᵢ(t−1) − lᵢ(t)

l_last(t), d_last(t)

at-least-one-alive probability and last-death density

L(t), C(t), Φ(t)

life-contingent payment factor; certain-floor indicator 1{t ≤ n_eff}; payment factor max(C, L)

G(t)

cumulative gross instalments scheduled through month t

CV(t), j(t), sc(y), θ

commuted value; commutation discount rate; surrender charge rate; withdrawal fraction

c_e, π

maintenance expense p.a. (60 std) and expense inflation (0.025 std)

Dimensional check: B is currency per annum; inst = B/m currency per payment; P, G, CV and refund lump sums currency; Φ, L, C, δ, θ, l, d dimensionless; n, n_eff, n_R months; g, π, j, sc rates. Every cash flow below is currency/month.

COLA update (start of month 12(y−1)+1, y ≥ 2) S1 S4#

B(y) = B(y−1) × (1 + g)

Escalation applies to the unreduced income level and continues after a survivor reduction, because the contract reduces payments to δ “of the current income payment” S2. NYL instead starts the first increase one year after the first income payment S5 — one payment period later than the anniversary-of-annuity-date rule adopted here; the difference is one instalment’s escalation and is a std convention choice.

The payment factor — one formula, five forms, two triggers#

Life-contingent factor L(t) (survival measured at the payment point: end of month t on arrears, end of month t − 12/m on advance — one full payment period earlier, i.e. t − 1 when m = 12 std):

single life:            L(t) = l₁(t)
joint, trig = either:   L(t) = l₁(t)·l₂(t) + δ · [ l₁(t) + l₂(t) − 2·l₁(t)·l₂(t) ]
joint, trig = primary:  L(t) = l₁(t) + δ · [ 1 − l₁(t) ] · l₂(t)
period certain only:    L(t) ≡ 0

The either form pays the full instalment while both are alive and δ × instalment while exactly one is alive S2 S3 — the second bracket is exactly P(at least one alive) − P(both alive). The primary form pays the full instalment while the primary is alive irrespective of the joint annuitant’s status, and δ × instalment only when the primary is dead and the joint annuitant alive S2 S3; this is also the mandatory structure for qualified contracts with a non-spouse joint annuitant, where “if the secondary annuitant dies first, 100% of payments continue while the primary lives” S5.

Certain floor C(t) = 1{t ≤ n_eff}, with

n_eff = n     for life_certain and certain_only, single or joint  [S1] [S2] [S4] [S5]
n_eff = n_R   for installment_refund                              [S1] [S5]
n_eff = 0     for life_only and cash_refund                       [S1] [S2]

Master payment factor and annuity outgo (t ∈ T):

Φ(t) = max( C(t), L(t) )
E[ANN(t)] = inst(t) × Φ(t) × ( 1 − θ_cum(t)·C(t) )

The max makes the certain period an annuity-certain floor rather than an additional stream: during the certain period the full instalment is paid regardless of survival, and max prevents paying 1 + L S1 S2 S5. Two consequences:

  • Because the floor pays the full, unreduced instalment, the construction automatically reproduces NYL’s rule that a survivor reduction “will not be reduced until the end of that period” when the first death falls inside a certain period S5 — no separate flag is needed (spec footnote 11).

  • The (1 − θ_cum·C) term applies a prior commutation to certain-period instalments only; life-contingent payments after the certain period are untouched S1 S2 S5.

Derived installment-refund period. Payments continue until cumulative payments equal the premium S1 S4 S5 S6:

n_R = min{ t ∈ T : G(t) ≥ P }
final instalment at n_R is trimmed to  P − G(n_R − 12/m)          **[std]**

Under a level path (the relevant case — MassMutual does not offer the COLA with Life with Installment Refund S1) this closes to n_R = (12/m)·⌈ m·P / B(1) ⌉ months, which is NYL’s published rule “guaranteed payment period = premium paid ÷ annualized income benefit amount” S5 rounded up to a payment date. Anchor check: P/B(1) = 100,000/6,000 = 16.667 years = 200 months.

Cash refund lump sum (at the death that terminates the income stream):

E[CR(t)] = d_term(t) × max( 0, P − G(t−1) )                        [S1] [S3] [S5]
d_term = d₁       single-life contract
d_term = d_last   joint contract (offered only with δ = 100% [S5])

Measuring the balance at t−1 implements “instalments already paid” for a mid-month death under arrears std; on advance timing an instalment paid at the start of the death month has been paid, so use G(t) in advance payment months or the lump sum is overstated by one instalment.

Maintenance expense (l_alive = l₁ single-life, l_last joint):

IF(t)     = max( C(t), l_alive(t) )                                **[std]**
E[EXP(t)] = (c_e / 12) × (1 + π)^(y(t)−1) × IF(t)                  **[std]**

Total gross liability cash flow:

CF(t) = E[ANN(t)] + E[CR(t)] + E[COMM(t)] + E[EXP(t)]

There is no premium income in the projection (the single premium at t = 0 is a pricing input) and no surrender outgo S1 S4 S5.

Mortality construction#

q_base(x, 2012+k) = q_x^{2012 IAM Basic} × (1 − G2_x)^k               [R3] [R2]
q_be(x, cal)      = min( 1, AE × q_base ),  AE = 1.084                **[std]** from [R9]
q_rated(x, cal)   = min( 1, θ × q_be ),     θ = 1 in base             **[std]**
q_m(t)            = 1 − (1 − q_rated)^(1/12)                          **[std]**
lᵢ(t)             = lᵢ(t−1) × (1 − q_m^{(i)}(t)),  i = 1, 2

k is the number of calendar years from 2012 to the projection year, so the basis is generational, not period: each attained age in each future calendar year uses its own improved rate. The A/E factor is applied to the projected basis, matching the study’s measurement convention R9. The valuation basis is a different table with a different rounding rule and must not be conflated with the best estimate (see Valuation pointers).

Commutation module (optional; certain-bearing forms only S1)#

Eligibility per the composite: commutation_enabled, n_eff > 0, policy year ≥ 2, not Oregon, one withdrawal per contract year S1.

CV(t) = Σ_{ s ∈ T,  t < s ≤ n_eff }  inst(s) · (1 − θ_cum(t)) · v(t, s)

compound (default **[std]**):  v(t, s) = (1 + j(t))^(−(s − t)/12)
simple   (per [S7]):           v(t, s) = max( 0, 1 − j(t)·(s − t)/12 )
j(t) = j₀ + [ CMT10(t) − CMT10(0) ],  j₀ = 4.00%    **[std]** [unverified]

Requested gross withdrawal W, with 5,000 ≤ W ≤ CV(t) and each remaining guaranteed payment staying at or above $100 S1:

surrender charge = sc(y) × W                                        [S1]
E[COMM(t)]       = W × (1 − sc(y))            (paid to the owner)
θ_cum(t⁺)        = θ_cum(t) + (1 − θ_cum(t)) × W / CV(t)            [S5 pro-rata rule](#uslib-immediate_annuity-s5)

The pro-rata reduction implements NYL’s rule that future income payments through the end of the guaranteed period are reduced “by the withdrawal percentage elected”, with full payments resuming for life at the end of that period if the annuitant is alive S5; Pacific Life states the same resumption rule for every form except pure Period Certain S2. On a certain_only contract there is nothing to resume, so a 100% withdrawal ends the contract.

Monthly processing order#

  1. If t = 12(y−1)+1, y ≥ 2: apply B(y) = B(y−1)(1+g) S1.

  2. Decrement mortality: update l₁, l₂, d₁, d₂, l_last, d_last.

  3. If t ∈ T: set inst(t) = B(y(t))/m; compute C(t), L(t), Φ(t); record E[ANN(t)]; update G(t) = G(t−) + inst(t) (deterministic as-if-alive schedule).

  4. Refund: if the form carries a cash refund, accrue E[CR(t)] = d_term(t) × max(0, P − G(t−1)).

  5. Commutation (if enabled and eligible this contract year): evaluate CV(t), apply W, record E[COMM(t)], update θ_cum.

  6. Accrue E[EXP(t)].

  7. Stop when IF(t) < 10⁻⁶, or when every covered life has passed the limiting age (t/12 + x₁ > ω and, if joint, t/12 + x₂ > ω), ω = 120 std — stopping on the primary’s age alone would truncate a younger joint annuitant’s tail.

Policyholder behavior modeling#

There is almost none, and that is a cited product property rather than an omission. The contract is irrevocable, the income option and frequency cannot be changed after issue, there is no account value and no surrender right S1 S2 S3 S4 S5. The model therefore carries no lapse decrement and no dynamic lapse formula — the position VM-22 prescribes for this reserving category R2 REG-R36.

The one live option is commutation of the certain portion, and no public utilization data exists. Reference constructions, both std: a deterministic per-contract-year utilization vector u(y), zero in the base (shape anchors only — the feature requires the owner to be 59½ or older S2 S5, is capped at the PV of remaining certain payments S1 and is barred in Oregon S1 S2); or a rate-driven dynamic take-up, since commutation is worth more when rates have fallen since issue (the interest-rate adjustment raises the payout):

u(y, t) = min( u_max, u_base(y) × max(0, 1 + κ·[ CMT10(0) − CMT10(t) ]) )

with u_max = 0.10, κ = 20 std — pure shape assumptions calibrated to nothing, whose only justification is directional: NYL names the 10-year CMT change as the driver of the withdrawal amount S5 and Pacific Life confirms an interest-rate adjustment applies S2.

Both constructions are best-estimate objects and are barred from a CARVM run. Commutation is an elective benefit under AG 33, and for elective benefits “incidence rates should not be based on tables reflecting past company experience, industry experience or other expectations” — the guideline substitutes trial sets maximised over, theoretically all rates 0% to 100%, with 0% or 100% the typical optimum REG-R151 Definitions 2. A u(y) or u(y, t) vector fed into a reserve calculation is therefore not a conservative approximation of CARVM; it is a different quantity. (Note also that AG 33 reaches this product only because the commutation right exists — see “Valuation and reserve pointers” below.)

Excluded by scope: payment acceleration (borrowing forward with no PV discount) S2 S3 S5; NYL’s 30% Cash Withdrawal, which commutes against life expectancy on a life-only contract and permanently cuts all future income by 30% S5 — the only retrieved feature that commutes a life-contingent stream. Anti-selection enters at outset, not through in-force behavior: voluntary annuitants self-select for longevity, and impaired lives are diverted to age-rated contracts S8 whose valuation is governed by AG 9-C and VM-V’s “rated age” definition R1 REG-R41. The reference model carries this through θ, not through behavior dynamics.

Worked example#

Configuration (anchor cell; parameters identical to product-spec.md). P = $100,000; τ = 0; B(1) = $6,000 p.a. std ⇒ inst = $500.00/month; joint form, primary male ANB 65, joint annuitant female ANB 62 std; monthly (m = 12) in arrears std; fixed compound COLA g = 3% std; survivor percentage δ = 66⅔% std; no certain period (n = 0). Scenario: the joint (secondary) annuitant dies during month 14; the primary survives throughout. The two trigger conventions run side by side — this is the death that distinguishes them.

Income levels: year 1 (t = 1–12) B = 6,000.00 ⇒ 500.00/month; year 2 (from t = 13) B = 6,000 × 1.03 = 6,180.00 ⇒ 515.00/month; year 3 (from t = 25) B = 6,180 × 1.03 = 6,365.40 ⇒ 530.45/month. Reduced amounts: 2/3 × 515.00 = 343.33 and 2/3 × 530.45 = 353.63.

t

Event

Unreduced inst(t)

CF, trig = either

CF, trig = primary

1

first monthly instalment (arrears)

500.00

500.00

500.00

12

12th instalment

500.00

500.00

500.00

13

anniversary: B ← 6,180.00; 13th instalment

515.00

515.00

515.00

14

joint annuitant dies during the month; the instalment due at month end is the first scheduled payment date after the death

515.00

343.33

515.00

15

15th instalment

515.00

343.33

515.00

24

24th instalment

515.00

343.33

515.00

25

anniversary: B ← 6,365.40

530.45

353.63

530.45

Trace and checks.

  • t = 14, trig = either. The death is decremented at the end of month 14, so the scenario values at the payment point are l₁ = 1, l₂ = 0: L = l₁l₂ + δ(l₁ + l₂ − 2l₁l₂) = 0 + (2/3)(1 + 0 − 0) = 0.6667; C = 0, so Φ = 0.6667 and CF = 515.00 × 2/3 = 343.33, and the same at every later payment date. ✔ (Note the timing convention bites here, not at t = 15: the instalment due at the end of the month of death is already the first scheduled payment date after the death.)

  • t = 14, trig = primary, same values: L = l₁ + δ(1 − l₁)l₂ = 1 + 0 = 1, so CF = 515.00. ✔ The joint annuitant’s death is invisible to the payment stream while the primary lives S2 S3 S5.

  • Reverse the death (primary dies in month 14, joint annuitant survives; l₁ = 0, l₂ = 1): L_either = 0 + (2/3)(0 + 1 − 0) = 2/3 and L_primary = 0 + (2/3)(1)(1) = 2/3. Both conventions pay 343.33 from t = 14. The two triggers coincide on the primary’s death and differ only on the secondary’s — which is precisely why the trigger must be a model switch, not a footnote S1 S2 S3 S7.

  • COLA continues after the reduction, because δ applies to the current income payment S2: 343.33 becomes 353.63 at t = 25, not a frozen 343.33.

  • With a 10-year certain period (n = 120): C(t) = 1 for t ≤ 120, so Φ(t) = max(1, L(t)) = 1 and every instalment from t = 14 to t = 120 is the full 515.00 / 530.45 / …, with the reduction to δ beginning only at t = 121 — reproducing NYL’s deferral rule with no extra logic S5.

  • Single-life with cash refund, death in month 14: lump sum = max(0, 100,000 − G(13)) = 100,000 − (12 × 500.00 + 515.00) = 93,485.00 S1 S3 S5. On installment refund (level path, no COLA S1) the derived certain period is 12 × 100,000/6,000 = 200 months S5.

Valuation and reserve pointers#

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

  • Reserve method. CARVM — for SPIAs, deferred annuities in payout and supplementary contracts, “the path of future guaranteed benefits with the highest present value is used to set policy reserves” S7; enabling statute Model #820 REG-R1, codified at AP&P Appendix A-820 ¶15, with the scope gate at ¶14 (qualified-plan group annuity business excepted and routed to a CRVM-consistent method by ¶13.b) and the method/interest/mortality triple at ¶6 REG-R153 ¶¶6, 13.b, 14, 15. AG 33 does not reach the base composite — its applicability requires that elective benefits be available, and its non-elective definition expressly covers immediate annuity benefits “where no benefit options are available”; a commutation right puts the contract inside it REG-R151.

  • VM-22 (PBR), valuation dates on or after 1/1/2026. SPIAs sit in the Payout Annuity Reserving Category; aggregate reserve = SR + DR for contracts passing the Single Scenario Test + reserves for contracts valued under VM-A/VM-C/VM-M/VM-V; SR = CTE70; the additional standard projection amount is disclosure-only under VM-31; a three-year transition election and a $1.0bn/$2.0bn Annuity PBR Exemption apply R2 REG-R36.

  • Prescribed mortality for the VM-22 Standard Projection Amount (also the “little or no data” floor): q_x^(2012+k) = q_x^{2012 IAM Basic}·(1 − G2_x)^k·F_x, with F_x from Table 6.8 (payout-annuity factors, age nearest birthday, reproduced in full at R2) R2 REG-R36. Note this is the same base table and projection scale as the best-estimate construction above, with a prescribed F_x overlay in place of the experience A/E factor.

  • Maximum valuation interest rate. VM-V §1, not VM-22, for immediate annuities issued after 12/31/2017: Iq = R + S − D − E with E = 0.25%, bucketed A–D by reference period and initial age (age last birthday, younger annuitant on a joint contract), rounded to the nearest ¼% quarterly for non-jumbo contracts and to 1/100 of 1% daily for jumbo contracts (initial consideration ≥ $250 million) R1 REG-R37. VM-V §1 supersedes AG 9-B and the interest references in AG 9-C R1 REG-R37 REG-R41. For the older in-force layer VM-V §1 does not reach, the A-820 formulaic rate is I = .03 + W(R − .03) with a flat W = .80 for single premium immediate annuities — no Plan Type and no guarantee-duration lookup — and R the 12-month average of the Moody’s composite yield on seasoned corporate bonds ending June 30 of the calendar year of issue or purchase, rounded “to the nearer one-quarter of one percent (1/4 of 1%)” REG-R153 ¶¶7.a.i(b), 8.b, 9.b. A-820 prints no tie-break for that rounding (the “ties down” convention is VM-20 §3.C.2.a’s, not A-820’s), and its ¶7 trigger — “the effective date of the Codification” — is a date A-820 never prints; both stay unresolved in the primary text REG-R153 ¶7 REG-R3.

  • Valuation mortality. 2012 IAR generational table: q_x^(2012+k) = q_x^{2012 IAM Period}·(1 − G2_x)^k, rounded to three decimal places per 1,000, with the rounding applied to the value computed from the 2012 period rate each time — never by compounding an already-rounded prior-year rate R3 R4 REG-R59. Verified example: male 30, q^2012 = 0.741 ⇒ q^2014 = 0.741 × 0.99² = 0.7262541 → 0.726, not 0.734 × 0.99 = 0.727. Chaining rounded rates is wrong in a way a single-year unit test will not catch. The table-by-issue-date rules are now sourced from the codified appendix that A-820 ¶6 cross-references: Annuity 2000 for individual issues 1/1/2001 through 12/31/2014, 2012 IAR for issues on or after 1/1/2015, 1983 Table “a” without projection for the structured-settlement carve-out (tort and out-of-court settlements, workers’-compensation-type claims, LTD claims where an annuity replaces continuing payments), and 1994 GAR for group-purchased annuities, with no effective date printed for the group rule REG-R153 ¶6, A-821 ¶¶10–12, 15. Still open: A-821 prints no standard for an individual annuity issued before 1/1/2001, and the 1994 GAR, Annuity 2000 and 1983 Table “a” are named and not printed, so A-821’s 1994 GAR formula q_x^(1994+n) = q_x^1994·(1 − AA_x)^n is not computable from library sources REG-R153.

  • Tax and GAAP. IRC §807: tax reserve = greater of net surrender value (zero here) and 92.81% of the NAIC-prescribed method — CARVM for annuities — capped at the statutory reserve REG-R16. Under LDTI, payout annuities carry a liability for future policy benefits with annually reviewed assumptions REG-R34; ASOP 10 governs that work REG-R71.

  • Standards for the modeling work. ASOP 7 (life cash flow analysis) REG-R27; ASOP 22 (asset adequacy — a SPIA block is a classic cash-flow-testing exposure) REG-R29; ASOP 56 (modeling: validation, documentation, model risk) REG-R32; ASOP 54 in pricing mode REG-R70. ASOP 52 is scoped to VM-20 life products REG-R31; the reference library catalogues no VM-21/VM-22 analogue, i.e. no ASOP for principle-based reserves for annuities [unverified as an absolute negative — inferred from the bibliography, not from an ASB statement].

Policyholder taxation is not an insurer cash flow. Under IRC §72 each payment splits at exclusion ratio = investment in the contract ÷ expected return, capped at the unrecovered investment R6 REG-R55, with expected return from the IRS actuarial tables by payout form and a refund feature adjustment reducing the investment in the contract by Table III/VII percentage × min(net cost, total guaranteed return) R7. Worked IRS example: at age 65, $21,053 buying $100/month for life with a full refund feature gives years guaranteed = 21,053/1,200 = 17.54 → 18, a Table VII percentage of 15%, a refund value of $3,158 and adjusted investment of $17,895 R7. Critically for a COLA contract, “the tax-free part remains the same even if the total payment increases due to variation in the annuity amount such as cost of living increases” R7 — the excluded dollar amount is fixed at the first payment, so the taxable proportion of a 3%-escalating SPIA rises every year. This is a policyholder-side computation: it changes no insurer liability cash flow and belongs in an illustration or in-force tax module, not in CF(t).

Key sensitivities and model risks#

Dominant assumptions, in order:

  1. Longevity level. The liability is a life-contingent payment stream with no offsetting decrement — no lapse, no cash value. Lower mortality lengthens every stream with nothing to offset it. The std A/E factor of 1.084 is the weakest calibrated link: the study reports 99.6% against unprojected 2012 IAM Basic but 108.4% against the G2-projected basis R9, so applying the factor to the wrong base misstates the mortality level by about 8%.

  2. Longevity trend. Scale G2 is a fixed, dated improvement scale and the 2020–2024 experience says it has over-projected R9. Sensitivity-test a scaled G2 (e.g. 50% and 150% of tabulated improvement) before anything else. VM-22 requires future improvement to be reflected for longevity segments where it increases the reserve R2.

  3. The survivor-reduction trigger. A structural sensitivity, not a parametric one: on the anchor cell the primary trigger pays 100% for the primary’s whole lifetime whenever the joint annuitant dies first, while either drops to δ. Over a joint 65/62 cell this is a first-order liability movement, and it is invisible if the trigger is buried as a footnote rather than modeled as a switch S1 S2 S3 S7.

  4. Initial income level B(1). The largest source of model error is not an assumption at all: no insurer publishes payout factors or the pricing basis, so B(1) is exogenous and unverifiable (spec footnote 8; assumption note (i)). Every result is conditional on it, and no pricing test against public data is possible.

  5. COLA rate. A 3% compound escalation roughly doubles the instalment over 24 years; liability duration and longevity sensitivity both rise with g. The menu is bounded at 4% for qualified-eligible designs by the sub-5% constant-percentage rule R8.

  6. Commutation basis. Both the discount rate and its functional form are invented std unverified (assumption note (ii)). Latent while utilization is zero, but any run with commutation_enabled inherits an unsupported assumption — flag it in output.

Known modeling pitfalls:

  • Certain-period double-counting. During the certain period the instalment is certain — do not also weight it by survival. max(C, L) prevents paying 1 + L; an additive construction silently doubles the guarantee.

  • Rounding the valuation table by compounding. q^(2012+k) must be rounded from the 2012 period rate every time; chaining rounded rates gives 0.727 where the manual requires 0.726 R3 R4 REG-R59 REG-R153 A-821 ¶14.

  • Applying the A/E factor to the wrong base. 1.084 belongs on 2012 IAM Basic projected with G2; on the unprojected table the study’s own answer is 99.6% R9.

  • Period versus generational. The 2012 IAM Period Table is one calendar year’s rates; the 2012 IAR is that table plus Scale G2 applied generationally R3 R4. Using the Period Table without projection understates longevity throughout.

  • Age-basis mismatch. The tables are ANB; VM-V’s initial age for rate bucketing is ALB R1 R2. Mixing them shifts every lookup by up to a year (conversion in Model scope).

  • Survival-measurement timing. Arrears instalments require survival at the payment date, advance instalments at the period start. Using end-of-period survival for advance payments understates the liability by about one period’s mortality per payment — material at high ages. Symmetrically, refund balance timing: G must net instalments paid before death on arrears (G(t−1)), but an advance instalment paid at the start of the death month has been paid — use G(t) there or the cash refund is overstated by one instalment.

  • Refund-form certain period. n_R is derived from P/B(1) and moves with the pricing input; hard-coding it (e.g. at 200 months) breaks every sensitivity run on B(1).

  • Joint-life independence. The l₁·l₂ products assume independence std; broken-heart and shared-lifestyle dependence overstate the expected survivor stream modestly, and the payout experience study cannot inform the assumption because it gives no recognition to a living secondary annuitant R9.

  • Commutation applied to the wrong slice. A withdrawal reduces certain-period instalments only; applying θ_cum to the life-contingent tail contradicts every retrieved contract S1 S2 S5.

  • Treating exclusion-ratio tax as a cash flow. It is a policyholder computation and generates no insurer flow; modeling it as an outgo distorts the liability.

  • Vintage and scope drift. VM-22 is in its first year of effectiveness with a three-year transition and a pending LATF directive targeted at the 1/1/2027 manual R2; VM-V weight tables, Table X spreads and the published quarterly rates live on the NAIC Industry tab and were not retrieved R1. Re-check both each January.