Technical Notes#

Status: Draft, 2026-08-03 (all cited sources accessed 2026-08-03; see sources.md).

Scope note. These notes specify a reference liability cash-flow projection model for the standardized composite product defined in product-spec.md (same directory). This is not any single insurer’s product. [S#]/[R#] tags refer to the source list in _research/unit-linked-bond.md (carried into sources.md here); [REG-R#] tags refer to the cross-product reference library references/regulatory-and-actuarial-references.md (its own R-numbering; research provenance in _research/regulatory-actuarial.md). 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; the implementation anchor for mechanics is a single carrier’s KFD + Policy Provisions pair [S1] [S2].

Model scope and conventions#

Purpose. Project gross liability cash flows for a single-policy model point of a clean-charge onshore unit-linked bond, decomposed in the classic UK way into the unit fund (the bid value of units, matched by the linked assets) and the non-unit (“sterling”) cash flows accruing to the insurer: charges collected and fund-based margins, less expenses and death strain. This decomposition is standard UK actuarial practice but is tagged unverified as terminology — the IFoA archive papers evidencing the “sterling reserve” usage could not be text-extracted R9; the rule-level anchor is the Solvency UK requirement that the best estimate reflect all cash in- and out-flows R5 TP 3.2 applied to the product cash flows in [S1]–[S5]. Reserves are not computed (see Valuation and reserve pointers).
Projection frequency. Monthly std. The contract accrues the AMC daily through the unit price [S2 §5.1.1] and prices funds daily/at least monthly [S2 §3.2] [S3 Part E]; the model discretizes to monthly steps with all intra-month flows at the conventions below.
Timing conventions std. Fund growth, tax provision and fund-based charges accrue over the month; withdrawals, adviser charges and rider charges are unit cancellations at end of month (EOM); decrements (death, surrender) are EOM events after cancellations. Settlement frictions (12:00 cut-offs, 2-working-day large deals, 28-day PruFund waits, deferral powers [S2 §4, §8] [S5 Q9]) are ignored.
Age basis. Age last birthday (ALB) std, chosen to index directly into single-year-of-age qx vectors of the ONS national life tables used as the std mortality proxy REG-R32. (Contractual age limits are quoted “next birthday” in the anchor documents [S1]; the difference is immaterial to a product with a 0.1% death strain.)
Currency. GBP. Intermediate values carried at full precision; cash flows reported to pence std.
Model points. Single-policy model points projected on an expected (probability-weighted) basis: survivorship factors multiply per-policy cash flows. A “policy” here is the whole bond of 100 identical segments std (spec footnote 3); per-segment values are the bond values ÷ 100 [S1] [S2 §2.4]. No aggregation logic is specified.
Top-ups. Excluded from the base projection; a top-up is a new model point with its own premium, allowance clock and segments std (spec footnote 4) R2 per-premium allowance arithmetic.

Model point attributes#

Attribute	Type	Example (anchor cell)
`issue_age`	int (ALB)	65 std
`sex`	enum {M, F}	M std
`lives`	enum {single} (joint last-death out of scope, spec footnote 1)	single
`premium`	currency (single premium, net of set-up adviser charge [S2 §1, §12.2])	100,000 std
`n_segments`	int	100 std
`db_uplift`	factor `u`	1.001 [S1] [S2]; choice std
`amc_rate`	annual rate `c`	0.0100 std
`further_costs_rate`	annual rate `f`	0.0010 std
`tax_provision_rate`	rate `t_pf`	0.20 std proxy R6
`wd_pattern`	enum {none, allowance_5pct, custom}	allowance_5pct std
`oac_rate`	annual rate on unit value (ongoing adviser charge)	0 (module value 0.005 [S1] [S2 §7.1 example])
`gmdb_flag`	bool (return-of-premium rider [S1] [S2 §5.2, §10] [S5])	false std
`uf_initial`	currency (premium at issue; >0 for in-force cells)	100,000
`issue_date` / `policy_month_offset`	date / int	month 1

State variables#

Variable	Description	Updated
`UF(t)`	Unit fund = bid value of units at end of month t	monthly recursion
`l(t)`	In-force probability at end of month t; l(0) = 1	monthly decrements
`y`	Policy year = ceil(t/12); insurance year for allowance tracking R2	monthly
`CumWD(n)`	Cumulative withdrawals + ongoing/ad hoc adviser charges to end of insurance year n (allowance-relevant [S2 §12.1.1] [S4] [S5 Q15])	on withdrawal/charge
`CumAllow(n)`	Cumulative allowable element = premium × min(n, 20) × 5% R2	yearly
`ExcessGain(n)`	Excess-event gain at insurance-year end (policyholder-side flag, no insurer cash flow) R1 s498/s507 R2	yearly
`G(t)`	GMDB guaranteed amount = premium − withdrawals − ongoing/ad hoc adviser charges (if `gmdb_flag`) [S2 §10]	on events
`E(t)`	Maintenance expense in month t	monthly

Assumption inputs#

Three classes are distinguished explicitly.

(a) Contractual / guaranteed elements (cited)#

In a clean-charge unit-linked bond the guaranteed layer is thin — that is the point of the design:

Input	Value	Basis
Death benefit	`u × UF`, u = 1.001 (sum assured = 100.1% of bid value of units)	[S1] [S2]; u choice std (spec footnote 6)
Surrender value	`UF` (bid value of units; no penalty)	[S4]; composite scope std (spec footnote 13)
Withdrawal machinery	Regular/partial/segment surrender; 12-month regular cap = max(7.5% of plan value, 7.5% of total paid in) incl. ongoing adviser charges	[S1] [S2 §7.1, §7]
Charge basis	AMC accrues daily through the unit price; adviser/rider charges by unit cancellation	[S2 §5.1.1, §12]
Segmentation	100 identical policies; premium and units divided equally	count std; mechanics [S1] [S2 §2.4]
Liability cap	Benefits derived from fund assets only; no make-whole on external default	[S2 §3.1.9] [S4]

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

All revisable by the insurer (AMC increase provisions are documented on the legacy booklet [S3 Part D]); the model holds the snapshot level:

Input	Snapshot value	Basis
AMC `c`	1.00% p.a.	std — per-fund AMC rate cards not fetched (research gap 5); only the discount tier table is public [S1]
Further costs `f`	0.10% p.a. (fund-borne, not insurer income)	existence [S1] [S2 §3.1.7]; level std
Fund-size discount	Off (level net AMC assumed)	tiers [S1] [S2 §5.1.4]; scope std
Life-fund tax pass-through `t_pf`	20% of gross fund return, in-price, neutral to insurer	mechanism [S2 §3.2.1] [S4] [S5 Q15]; rate proxy std of the policyholder rate R6
GMDB mortality-factor scale	= monthly mortality rate from the class-(c) basis at attained age (cost-of-insurance style), applied to max(0, G − u×UF)	design [S2 §5.2, §10]; scale std — factors not published
MVR / bonus rates	Not applicable — with-profits and PruFund funds out of scope; see `products/with_profits/`	[S2 §3.3] [S3]

The t_pf proxy deliberately ignores I-E timing detail: actual pass-through distinguishes income (as received), realised gains (next charge date), an annual deemed-disposal charge, and full-surrender settlement [S5 Q15] [S4], and the company’s I-E position includes an expense offset and minimum profits test R6. The base model treats collected tax as exactly offsetting tax payable (zero insurer margin impact) std.

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

Input	Recommended public basis	Basis tags
Best-estimate mortality	80% × ONS national life tables qx (single year of age, sex-distinct) std proxy	REG-R32; factor std
Mortality improvement	None in base std; production overlay “CMI_20xx with long-term rate p% std”	REG-R30
Base surrender (full)	std table below	std; design holding period [S1] [S4] [S5]
Withdrawal take-up	anchor cell: 5% of premium p.a., monthly	std (spec footnote 14) R2 [S1] [S4] [S5]
Acquisition expense	£300 per policy at issue	std
Maintenance expense	£60 per policy p.a., inflating 2.5% p.a.	std
Gross fund return scenario `g`	5.0% p.a. (deterministic base)	std

Honesty note on the mortality basis. The CMI’s current assured-lives tables and Projections Model are restricted to Authorised Users (subscribers); older publications are free but current qx cannot be redistributed R8 REG-R30. The canonical teaching tables (AM92/AF92) show the shape an assured-lives basis takes REG-R24, and the ONS national life tables are the only fully redistributable UK mortality source (Open Government Licence; qx by single year of age) REG-R32 — hence the std proxy above, with the caveat that population mortality is heavier than insured-lives experience REG-R32 (the 80% factor is a crude allowance, std). Specific CMI assured-lives table names for this product could not be confirmed from the fetched CMI page and remain unverified (research gap 8) R8. Mortality is nearly irrelevant to this product — the net amount at risk is 0.1% of the unit fund in the composite (0.1%–1% across insurers [S1] [S2] [S3] [S4] [S5]) — unless the GMDB rider is enabled.

Reference base surrender table std (annual rates; to be replaced by portfolio experience; shape rationale: the product is designed to be held 5–10 years or more [S1] [S4] [S5], so surrenders are low early, rise as the advised holding period completes, and settle at a high ultimate level):

Policy year	1	2	3–5	6–10	11+
Annual full-surrender rate `w_base`	2%	3%	5%	8%	10%

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
t	policy month, t = 1, 2, …; y = ceil(t/12); a = attained age (ALB) = issue_age + y − 1
`P`	single premium (100,000)
`UF(t)`	unit fund at end of month t; UF(0) = P
`g`	annual gross fund return (0.05); `g_m` = (1+g)^(1/12) − 1 = 0.0040741 (derived)
`t_pf`	tax-provision rate (0.20) std
`c`, `f`	AMC (0.0100) and further costs (0.0010), annual; `c_m` = c/12 = 0.0008333, `f_m` = f/12 = 0.0000833 [std 1/12 accrual convention]
`u`	death-benefit uplift factor (1.001)
`W(t)`	regular + one-off withdrawals cancelled at EOM of month t (anchor: 5%×P/12 = 416.67)
`AC(t)`	ongoing/ad hoc adviser charges cancelled at EOM (0 in anchor cell)
`GC(t)`	GMDB rider charge (0 unless `gmdb_flag`)
`TX(t)`	tax provision deducted in month t; `AMC$(t)`, `FC$(t)` monetary AMC/further costs
`DS(t)`	death strain per death in month t
`q_m(t)`	monthly mortality rate = 1 − (1 − q_a)^(1/12) from the class-(c) basis; `w_m(t)` monthly surrender rate = 1 − (1 − w_ann)^(1/12)
`l(t)`	in-force probability at end of month t; l(0) = 1
`E(t)`	maintenance expense = 60/12 × 1.025^(y−1) std

Dimension check: g_m, c_m, f_m, t_pf are dimensionless per-month rates or fractions; every product with UF is in GBP; q_m × DS is GBP per policy-month. W, AC, GC, TX, AMC$, FC$, E are GBP per month.

Monthly processing order std#

For month t, per policy in force at t−1:

Update y, a, E(t).
Fund growth and tax provision (within unit price [S2 §3.2.1] [S4] [S5 Q15]): G$(t) = g_m × UF(t−1); TX(t) = t_pf × G$(t); UF_g(t) = UF(t−1) + G$(t) − TX(t) = UF(t−1) × (1 + g_m(1 − t_pf)).
Fund-based charges (AMC accrues via price [S2 §5.1.1]; further costs fund-borne [S2 §3.1.7]): AMC$(t) = c_m × UF_g(t); FC$(t) = f_m × UF_g(t); UF'(t) = UF_g(t) × (1 − c_m − f_m).
Unit cancellations (EOM): withdrawals, adviser charges, rider charge: GC(t) = q_m(t) × max(0, G(t) − u × UF'(t)) if gmdb_flag else 0 (design [S2 §5.2, §10]; scale std); UF(t) = UF'(t) − W(t) − AC(t) − GC(t). Enforce the product cap: rolling-12-month W + AC ≤ max(0.075 × UF, 0.075 × P) [S1] [S2 §7.1].
Death strain per death: DS(t) = (u − 1) × UF(t) + max(0, G(t) − u × UF(t)) × 1{gmdb_flag} — the sum assured is u × UF funded by cancelling the whole unit fund, so the non-unit cost is the 0.1% uplift [S1] [S2] plus any GMDB in-the-money amount [S2 §10] [S5].
Decrements (EOM), deaths before surrenders std: l(t) = l(t−1) × (1 − q_m(t)) × (1 − w_m(t)). Surrender pays UF(t) by cancelling all units — no non-unit cash flow (clean design [S4]; spec footnote 13) — but extinguishes all future margins.
Allowance tracker (insurance-year end, policyholder side only): CumAllow(n) = P × min(n, 20) × 0.05 R2; ExcessGain(n) = max(0, CumWD(n) − CumAllow(n) − Σ prior excess gains) R1 s498/s507 R2. Generates no insurer cash flow; feeds behavior only. Chargeable events on death/full surrender follow s484/s491 R1 and are likewise policyholder-side (the insurer issues certificates [S5 Q15]).

The core unit-fund recursion (anchor cell: AC = GC = 0):

UF(t) = UF(t−1) × (1 + g_m(1 − t_pf)) × (1 − c_m − f_m) − W(t)

Non-unit (insurer) cash flow extraction#

Per policy in force at t−1, before survivorship weighting:

Cash flow	Formula	Sign
AMC margin	AMC$(t) = c_m × UF_g(t)	+
GMDB rider charge	GC(t) (0 in base)	+
Set-up adviser charge / commission	0 — post-RDR adviser charges are pass-throughs facilitated by unit cancellation [S1] [S2 §12] [S4]	0
Maintenance expense	E(t)	−
Acquisition expense (t = 0)	300 std	−
Death strain (per death)	DS(t)	−
Further costs FC$(t)	pass-through to fund costs — excluded from insurer margin std	0
Tax provision TX(t)	pass-through to corporation tax — neutral std (class (b) note) R6	0
Surrender / withdrawal payments	funded by unit cancellation — no non-unit flow (clean design) [S4]	0

Aggregate expected cash flows multiply each row by the in-force factor: AMC, GC and expenses by l(t−1); death strain by l(t−1) × q_m(t); nothing by surrenders (their non-unit flow is zero) [std timing]. The expected net non-unit cash flow:

NUCF(t) = l(t−1) × [ AMC$(t) + GC(t) − E(t) − q_m(t) × DS(t) ]  −  300 × 1{t=0}

Because AMC$(t) ≈ c_m × UF and DS(t) ≈ 0.001 × UF, the insurer’s result is a fund-based margin stream: proportional to the unit fund and to persistency, with mortality contributing only ~0.001 × q of the fund per year. Lapse/withdrawal behavior, not mortality, dominates value. Future margins typically exceed future costs, so the non-unit best estimate is commonly negative (an asset-like offset to the unit reserve) [unverified as standard-practice terminology — see scope note; the rule anchor is R5 TP 3.2].

Policyholder behavior modeling#

All dynamic formulas are std reference constructions; no public UK bond persistency study was fetched (calibration is portfolio-specific).

Withdrawal take-up std. wd_pattern = allowance_5pct: W(t) = 0.05 × P / 12 every month. Rationale: the 5%/20-year tax-deferred allowance R2 is the pattern every fetched KFD leads with [S1] [S4] [S5], it sits inside the 7.5% product cap [S2 §7.1], and adviser charges consume the same allowance [S2 §12.1.1] [S5 Q15] — so rational take-up gravitates to 5% inclusive of charges. Sensitivity: none (accumulation cell) and custom.
Base surrender std. w_base(y) per the class-(c) table, converted monthly.
Dynamic surrender multiplier — market performance std. M_perf(t) = min(2.0, 1 + 2.0 × max(0, g_ref − R_12m(t))), where R_12m is the trailing 12-month gross fund return and g_ref = g (5%). Poor recent performance raises surrender; base deterministic run has R_12m = g_ref so M_perf = 1.
Allowance-exhaustion step std. M_allow(y) = 1.5 for y ≥ 21, else 1.0. After 20 insurance years the cumulative allowance is fully drawn under the anchor withdrawal pattern R2; continued withdrawals then generate immediate excess-event gains R1 s507, pushing policyholders toward full surrender (or advice-driven restructuring).
Total surrender. w_ann(y,t) = min(0.35, w_base(y) × M_perf(t) × M_allow(y)) [std cap].
Segment vs part-surrender election. Whether a policyholder cashes whole segments or part-surrenders across all segments changes their tax [S1] [S4] [S5 Q12] R1 s484/s498, not the insurer’s cash flow (both cancel the same unit value) — carried as a model note only std.
No paid-up state. Single-premium product; no premium obligation exists [unverified as an explicit statement; consistent with S1–S5].

Worked example#

Anchor cell: male 65, P = £100,000, 100 segments (£1,000 each), u = 1.001, c = 1.00%, f = 0.10%, t_pf = 20%, g = 5.0% p.a., W = £416.67/month (5% of premium p.a. R2 allowance), AC = GC = 0; all parameters std per the tables above. Derived monthly rates: g_m = 0.0040741; g_m(1−t_pf) = 0.0032593; c_m = 0.0008333; f_m = 0.0000833. Placeholder mortality for the year: q_a = 1.0% [std order-of- magnitude placeholder consistent with the class-(c) proxy], q_m = 0.000837. Figures in GBP, displayed to pence, full precision carried.

t	UF(t−1)	Gross return G$	Tax TX	AMC$	FC$	W	UF(t)
1	100,000.00	407.41	81.48	83.60	8.36	416.67	99,817.30
2	99,817.30	406.67	81.33	83.45	8.35	416.67	99,634.17
3	99,634.17	405.92	81.18	83.30	8.33	416.67	99,450.61
…	…	…	…	…	…	…	…
12	97,966.60	399.13	79.83	81.90	8.19	416.67	97,779.14
Yr 1	—	4,839.44	967.89	993.10	99.31	5,000.00	97,779.14

Trace, month 1: G$ = 0.0040741 × 100,000 = 407.41; TX = 0.20 × 407.41 = 81.48; UF_g = 100,325.93; AMC$ = 0.0008333 × 100,325.93 = 83.60; FC$ = 8.36; UF’ = 100,233.96; UF(1) = 100,233.96 − 416.67 = 99,817.30. Reconciliation, year 1: 100,000 + 4,839.44 − 967.89 − 993.10 − 99.31 − 5,000.00 = 97,779.14. ✓ Per segment: 977.79.

Insurer-side extraction, year 1 (per policy, survivorship factors ≈ 1 at this q/w):

AMC margin collected: +993.10
Maintenance expense (£60, year 1): −60.00
Expected death strain: Σ q_m × 0.001 × UF(t) = −0.99 (per actual death at month 12 the sum assured would be 1.001 × 97,779.14 = 97,876.92, of which 97,779.14 is funded by cancelling units — strain 97.78)
Tax provision (967.89) and further costs (99.31): pass-throughs, nil margin std
Net non-unit cash flow ≈ +932.11 (acquisition expense −300 falls at issue)

Policyholder-side check (no insurer cash flow): year-1 withdrawals 5,000 = allowable element 100,000 × 1/20 = 5,000 R2 — no excess event; unused allowance carried forward is nil, and the 7.5% product cap (7,500 on paid-in) is not breached [S2 §7.1].

Valuation and reserve pointers#

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

Solvency UK technical provisions. TP = best estimate + risk margin R5 TP 2.4; BE = probability-weighted average of future cash flows discounted at the risk-free term structure, gross of reinsurance, covering all cash in- and out-flows R5 TP 3.1, 3.2. For this product the natural presentation is unit reserve = UF(t) (replicated by the linked assets; cf. the TP 2.5 replication rule R5) plus the non-unit BE of the NUCF stream above — commonly negative [unverified as terminology; R9 archive papers not extractable].
Risk margin. Reformed Solvency UK cost-of-capital formula: CoC = 4%, risk taper λ = 0.9 (floor 0.25) for long-term business, on the notional SCR runoff R5 TP 1.2, 4A.1 REG-R4. SCR aggregation is cited-not-specified in this library.
Matching adjustment / TMTP. Not relevant: unit-linked bond cash flows are neither MA-eligible annuity-style liabilities nor pre-2016 back-book quantities in this composite (new-business model) — no [REG] layer is applied.
IFRS 17. UK-adopted IFRS 17 (effective 1 January 2023) is the accounting frame REG-R38; a unit-linked bond is a candidate for the variable fee approach as direct-participation business [mechanics unverified — general knowledge; flagged as such in the reference library narrative]. The fulfilment-cash-flow engine is the same projection.
Standards for the modeling work. TAS 100 v2.0 (effective 1 July 2023, all technical actuarial work; Principle 5 covers models) R7 REG-R33 same standard; TAS 200 v2.0 (insurance work, effective 1 January 2025) REG-R34.

Key sensitivities and model risks#

Dominant assumptions, in order, for a fund-margin product:

Surrender and withdrawal behavior. Every margin line is proportional to the unit fund and persistency; surrender costs nothing at the point of exit (SV = UF) but truncates the entire future AMC stream. The std base table, the performance multiplier and the year-21 allowance step are the first assumptions to sensitivity-test — no public UK bond persistency study backs them.
Fund return level and path. AMC income scales with UF, so the liability model inherits full market beta on the margin stream; a −20% market move cuts the margin base ~20% and (via M_perf) raises surrenders simultaneously.
AMC snapshot vs expense inflation. The 1.00% std AMC is a snapshot of a discretionary element (per-fund rate cards not public — research gap 5 [S1]); maintenance expenses inflate at 2.5% std while the margin is proportional-to-fund — small-fund cells go margin-negative late in life.
Tax pass-through neutrality. The 20%-of-gross-return in-price proxy std assumes collected tax exactly equals tax payable; the true I-E position has timing (deemed disposals, realised-gain charge dates [S5 Q15] [S4]) and base differences (expense relief, minimum profits test R6) that create insurer-side tax strain or float not captured here.
Mortality — only if GMDB is enabled. Base death strain is 0.001 × UF (≈ £1 p.a. expected per £100k at q = 1%): negligible. With the return-of-premium rider the strain becomes market-contingent (max(0, G − u×UF)) and the unpublished charge scale [S2 §5.2] is a std guess — enable only with its own sensitivity set [S1] [S2 §10] [S5].

Known modeling pitfalls:

Charge-base ordering. AMC accrues on the post-growth, pre-cancellation fund (in-price accrual [S2 §5.1.1]). Charging c_m on UF(t−1) or after withdrawals changes the margin by ~½ month’s growth/withdrawal — small monthly, systematic over decades.
Counting pass-throughs as margin. Further costs [S1] [S2 §3.1.7] and the tax provision [S4] [S5 Q15] reduce the unit fund but are not insurer income; booking them as margin overstates NUCF by ~107% of the AMC in the anchor cell (year-1 tax provision 967.89 ≈ 97% and further costs 99.31 ≈ 10% of the 993.10 AMC).
Treating the 5% allowance as a product feature. It is policyholder tax machinery R1 R2: it never caps what can be withdrawn (the product cap is 7.5% [S2 §7.1]) and generates no insurer cash flow. Model it in behavior only.
Adviser charges are not insurer income. Post-RDR set-up/ongoing/ad hoc adviser charges are facilitated pass-throughs by unit cancellation [S2 §12] [S4]; they reduce UF and consume allowance but add nothing to NUCF.
Segment-level granularity. Modeling at bond level is exact only while all 100 segments stay identical; segment surrenders break symmetry. The composite keeps bond-level modeling and notes the approximation std.
Smoothed funds must not be bolted on. PruFund EGR/smoothing-limit mechanics [S2 §3.3.7–3.3.10] and MVR-bearing with-profits funds [S3] change the unit-price dynamics and add guarantee costs; they belong to the with-profits reference product (products/with_profits/), not this recursion.
Uplift factor slip. 100.1% vs 101% [spec footnote 6] is a ×10 difference in death strain; keep u a parameter, never a hard-coded 1.001.