The DIA_US_S Model

Reference liability cash flow model for U.S. deferred income annuities and QLACs.

DIA_US_S is the executable counterpart of products/deferred_income_annuity/technical-notes.md in the lifelib-products library. It projects gross liability cash flows for a single flexible-premium deferred income annuity: premiums in, each one buying a fully guaranteed paid-up income slice at the then-current purchase rate; a return-of-premium death benefit during the deferral; then, from the income start date, the immediate-annuity payout chassis — life-contingent instalments with a certain floor, refund guarantees, survivor reduction and COLA.

Two structural facts govern the whole design. There is no account value — no credited rate, no charge, no surrender value, no interim value — so there is no lapse decrement and no annuitization decrement; VM-22 declares its standard-projection lapse section “not applicable” to contracts with no account value or surrender benefit and prescribes annuitization at 0% [R9]. Mortality is the only decrement. And the income phase is the immediate-annuity payout chassis: the payment factor, the certain floor, the survivor-reduction triggers and the COLA rule are the same objects as in SPIA_US_S, carrying the same cells names. The DIA-specific change is the base of the guarantee — cumulative premiums CP(T), not a single premium.

Spaces. The model contains two:

Data

Reads the six input CSVs and holds their filename References. It takes no parameters, so each file is read once per model.

Projection

The by-contract projection, parameterized by point_id: Projection[1] is an ItemSpace projecting model point 1. It reaches the input tables through its data Reference, which resolves to the single Data Space.

The split matters for more than tidiness. Because Projection is parameterized, every Projection[N] is a separate ItemSpace with its own cells cache; readers placed there would re-read every file for every contract. In Data they are evaluated once, however many contracts are projected.

Input data is external: CSVs in the model folder’s parent directory, read at run time rather than stored inside the model. The model folder itself holds no data, so the model and its inputs must travel together.

Projection basis. Monthly steps. t counts policy months from issue and is 0-based — t = 0, 1, 2, …, proj_len() — exactly as the technical notes index it (“Projection frequency. Monthly, indexed t = 0, 1, 2, … from issue”). This is a deliberate departure from the 1-based t of Term_US_A and SPIA_US_S: the notes’ income start month T = 240 and premium months 0 and 60 are month indices in the 0-based scheme, and renumbering would silently make T = 240 mean the 241st month. l(t) is the survival probability at the start of month t with l(0) = 1, so lives_if(t) still means “has survived t elapsed months” and carries the same meaning as in SPIA_US_S; what does shift is the death density, lives_death(t) = lives_if(t) - lives_if(t + 1), because month t spans elapsed [t, t+1) here and [t-1, t) there.

The monthly processing order follows the notes: roll the attained age and look up q(t); take any premium at the start of the month, price it by equation (4) and add its income slice to B and its amount to CP; apply any option exercise; pay the income instalment at the end of the month under the default arrears convention, weighted by the form’s payment survivorship; pay the death benefit — the return of CP(t) in deferral, the form’s refund benefit in payout — at the end of the month of death; accrue the maintenance expense; then decrement. Contractual transactions precede the decrement, and the end-of-month instalment is contingent on survival to that point, which is what arrears means. All cash flows are undiscounted; the only discounting in the model is contractual — inside the purchase rate, the repricing of a start-date adjustment and a commuted value.

What is sourced and what is not. The contractual elements come from the composite’s product documents and the Insurance Compact’s uniform standard: the paid-up income slice bought by each premium at then-current rates [R13 §3.B(1)(b)][S3], the 100% return-of-premium deferral death benefit [S1][S2][S3][S4][R13 §3.I(1)(a)], the one-time ±5-year income start date adjustment and its disclosed repricing inputs [S1][S2], six months of payment acceleration [S1][S4], commutation of the guaranteed payments with the life-contingent tail preserved [S4][S5], the QLAC restriction set and its $210,000 2026 premium limit [R1][R2][R3][S4], the $50 maintenance expense escalated at 2.5% [R9], and the absence of lapse and annuitization decrements [R9]. No purchase-rate table exists: the Compact expressly relieves the insurer of disclosing the deferral-period mortality and interest basis [R13 §1.B(1)(a)], so the whole pricing kernel — the 4.75% pricing rate, the 6.0% expense and profit load, the mortality table, the illustrative payout and return-of-premium factors, the 100 bp repricing spread and the 50 bp commutation margin — is a [std] construction. This model is a mechanics demonstration, not a pricing or reserving result.

Not implemented, and named here so the omission is visible: the incidence and selection layer of the in-force options — the 1.5% p.a. start-date-adjustment take-up h_adj with its 60/40 direction split and rate multipliers M_def/M_adv, the 0.90/1.10 health-selection multiplier sel_mult, the 2% p.a. acceleration take-up h_acc and the 1.5% p.a. commutation take-up h_com, all of which split the model point into exercised and unexercised cohorts (each option is instead exercised deterministically at a month named on the model point, and is off by default); the exact cash-refund pricing factor of equation (6), the notes blessing the equation (5) certain-and-life approximation as the [std] base for both refund forms; the convertible-joint pricing of equation (16), the base being non-convertible [std]; spousal continuation on a death in deferral, the base being 100% election of the death benefit [std]; premium admissibility limits and the 13-month premium cut-off, which are validation rather than cash flow; the small-benefit termination right [R10 §3.B]; the exclusion-ratio tax split, which is a policyholder computation and generates no insurer cash flow; and every valuation layer — CARVM, AG 33, VM-22 CTE70 and the 2012 IAR valuation table with its no-compound-rounding rule.

Model points. model_point_table.csv carries seventeen contracts. Point 1 is the notes’ anchor cell — Female 60 ANB, nonqualified, Life with Cash Refund, monthly in arrears, return-of-premium death benefit in deferral, no COLA, $100,000 at issue plus $50,000 at the start of policy year 6, income start at attained age 80 (T = 240). Points 2 and 3 are the same cell on the two readings the notes leave open (see Verification); point 4 is the death-benefit fork; points 5 to 10 walk the payout forms, the joint triggers and the COLA; points 11 and 12 are a compliant and a breaching QLAC; points 13 to 15 exercise the start-date adjustment, payment acceleration and commutation; point 16 runs the generational mortality construction; point 17 pays in advance. The premium schedule is a separate table, premium_schedule.csv, because a flexible-premium contract takes an unbounded number of slices. A test asserts every model point projects.

Verification. tests/test_deferred_income_annuity_us.py asserts every row and column of the notes’ worked example: the two slice purchase rates and the income they buy, the derived guarantee period, the 21.2% death-benefit fork, and all nine rows of the projection table — in-force to six decimals, money to the cent. It also pins the two places the notes are internally inconsistent, on model points 2 and 3, rather than choosing between them.

Example

>>> import modelx as mx
>>> model = mx.read_model("products/deferred_income_annuity/DIA_US_S")
>>> model.Projection[1].result_annual()