Technical Notes#

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

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 from _research/term-assurance.md; frozen); [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.

Model scope and conventions#

Purpose. Project gross best-estimate liability cash flows (premiums, death and terminal illness claims by benefit shape, expenses, commission) for a single-policy model point of guaranteed-premium UK term assurance, in the sense required for a Solvency UK best-estimate projection: probability-weighted future cash-flows, gross of reinsurance R1, covering the cash-flow categories of the PRA cash-flows rule (benefits, expenses, premiums, intermediary payments) R2. Discounting, risk margin and capital layers are out of scope (see Valuation and reserve pointers).
Projection frequency. Annual grid std, with a monthly option. The only intra-year contractual structure is the monthly step-down of the decreasing-shape benefit and the monthly FIB instalments [S6] [S8]; the annual grid handles both with mid-year approximations (below), and the monthly grid removes the approximation.
Timing conventions std. Premiums received at the start of each policy year (annualized, in advance); maintenance expenses at the start of each year; death/TI claims paid at the end of the policy year of death; lapses occur at the end of the policy year, after deaths. Acquisition expenses and initial commission at issue (start of year 1).
Age basis. Age nearest birthday at entry, plus curtate policy year — attained age in year t is x + t − 1 std. (The fetched product documents do not state an age basis; UK assured-lives tables are select tables — AM92 has a 2-year select period, TMNL16/TFNL16 a 5-year select period R12 — so the mortality interface must accept select-by-duration rates.)
Currency. GBP throughout. Benefits are paid in sterling to UK bank accounts [S1].
Model points. Single-policy model points projected on an expected (probability-weighted) basis: survivorship factors multiply per-policy cash flows. No aggregation logic is specified here.
Termination. All states terminate at the end of the term: cover expires with no maturity value, no renewal, and no conversion — there is no US-style post-level-term ART tail [S1] [S2] [S6] [S8] R8. The projection horizon is exactly n years.
Contract boundary. Premiums are guaranteed, so the insurer has no unilateral repricing right and the Solvency UK contract boundary is the full term R3: all n years of premiums and benefits are inside the boundary. (Reviewable-premium variants — CI riders, out of scope — would require the rules 3.3/3.7 test R3.)
Rounding. Intermediate values at full precision; displayed cash flows to pence std.

Model point attributes#

Attribute	Type	Example (anchor cell std)
`shape`	enum {level, decreasing, fib}	level
`issue_age`	int (age nearest birthday)	35
`sex`	enum {M, F}	M
`smoker`	enum {N, S}	N
`term_y` (`n`)	int, years (1–50; decreasing 5–50; FIB 5–40)	25
`sum_assured` (`SA0`)	GBP (level/decreasing shapes)	150,000
`fib_income` (`I`)	GBP/month (fib shape)	1,000
`sched_rate` (`j`)	annual effective (decreasing shape)	0.06 std
`joint_first_death`	bool (base model: false)	false
`indexation`	bool (RPI option elected)	false
`wop`	bool (waiver rider; base model: false)	false
`premium_monthly` (`P_m`)	GBP/month	12.00 std
`premium_mode`	enum {monthly, annual}	monthly
`issue_date`	date	—

The anchor premium is a pure modeling value: no UK insurer publishes premium rate tables (quote-engine pricing; only the £5/month minimum is public [S5]), so any reference premium basis is constructed, not observed std.

State variables#

Variable	Description	Updated
`l(t)`	In-force probability at the start of policy year t; `l(1) = 1`	annual recursion
`q(t)`	Mortality rate (incl. TI acceleration) applied in year t	assumption lookup
`w(t)`	Lapse rate applied in year t	assumption lookup
`D(t)`	Expected deaths/TI claims in year t = `l(t) × q(t)`	annual
`DB(t)`	Death benefit payable for deaths in year t (shape-dependent)	annual (schedule)
`idx(t)`	Cumulative indexation factor (1 if option not elected/declined)	on anniversaries
`FIBcum(t)`	Cumulative expected FIB income streams in payment at start of year t = `Σ_{s<t} D(s)` (fib shape)	annual
`CF(t)`	Net liability cash flow of year t (insurer perspective, + = inflow)	annual

The FIB in-payment ledger is not decremented by mortality after the claim: the instalments are an annuity-certain to the end of the term regardless of any life [S6] [S8].

Assumption inputs#

Three classes are distinguished explicitly.

(a) Contractual / guaranteed elements (cited; the insurer cannot change them)#

Input	Value	Basis
Premium `P_m`	Level, guaranteed for the full term	guarantee [S2] [S6] [S9]; level std
Level benefit	`SA0` constant	[S1] [S6]
Decreasing benefit schedule	`B(k)` amortization formula at rate `j` (below)	mechanics [S1] [S6] [S8]; `j` = 6% std
FIB benefit	`I`/month, in arrears, death to end of term; annuity-certain	[S2] [S6] [S8]
Terminal illness	100% acceleration, two-limb 12-month definition, terms ≥ 2 years	[S1] [S6] [S8] R8 [S2] [S4]
Suicide exclusion	12 months, year-one only	[S1] [S6] [S8]
Grace	60 days from due date; then lapse without value	[S1] [S6]
Surrender/paid-up value	None	[S1] [S6] [S8] R8
Indexation option terms	cover +min(max(RPI,0),10%); premium ×(1 + 1.5×increase), cap 15%; removed after 3 declines	[S1] [S2] [S6] [S7]; composite std
Expiry	Cover ceases at end of term; no renewal/conversion	[S1] [S2] [S6] [S8] R8

(b) Insurer-discretionary current elements#

For guaranteed-premium life-only term assurance this class is nearly empty — a deliberate contrast with cash-value products: there are no bonus rates, no MVRs, no reviewable premiums, and no non-guaranteed charge scales on the composite. The two residual discretionary items:

Input	Snapshot value	Basis
FIB commutation basis	Commuted value = PV of remaining instalments at `r_c` = 3.0% p.a. std snapshot; base model take-up 0%	discretion (“fairly and reasonably”) [S6] [S8]; rate std (no insurer publishes the basis)
Underwriting exclusions / rated terms	None on the composite cell (standard rates)	case-by-case schedule exclusions exist [S1] [S3]; scope std

Reviewable-premium mechanics (5-yearly reviews on claims experience, reinsurance cost, lapses, expenses, etc.) exist on CI-type covers at two of the three carriers [S6] [S8] and are documented there as a modeling template, but are out of scope here.

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

Mortality. The current UK protection experience tables are the CMI “16” Series — term assurance mortality including terminal illness and accelerated CI, graduated on 2015–2018 experience R10 — with public confirmation of the table names TMNL16/TFNL16 (male/female non-smoker, 5-year select) via their adoption in the IFoA Formulae and Tables 2025 edition R12. However, CMI tables issued after 1 March 2013 are subscriber-only R11, so the full 16-Series set (including smoker/duration variants) cannot be redistributed in an open reference implementation. The reference basis is therefore a std proxy, stated honestly:

Input	Recommended public basis	Basis tags
Best-estimate mortality (incl. TI)	Shape of the public “00” Series temporary assurance tables — TMN00/TMS00 (male non-smoker/smoker), TFN00/TFS00 (female), 1999–2002 experience — scaled by a std adjustment factor (suggested 75%) to proxy improvement to the 16-Series era; AM92 (2-year select, prior Formulae and Tables basis) is the teaching-table alternative	tables R13 R11; AM92 role R12; factor std
Mortality improvement	None in base std. The CMI Mortality Projections Model is the market-standard overlay — CMI_2024 (June 2025, WP201) R14, superseded by CMI_2025 (March 2026, WP211) REG-R30 — but the model is subscriber-restricted; a production basis would be “x% of TMNL16/TFNL16 with CMI_2025 improvements at a chosen long-term rate”, all subscriber inputs	R14 REG-R30 R11
Population fallback	ONS national life tables (single-age qx, freely redistributable under OGL) — heavier than insured experience; use only as a last-resort open base	REG-R32
TI acceleration timing	None modeled: death and TI are one decrement, one benefit; acceleration shifts payment earlier by less than 12 months, immaterial on an annual grid std	definition [S1] [S6] [S8]; 16-Series mortality includes TI R10
Suicide-exclusion offset	Year-one claims not reduced for excluded suicides std (immaterial; no incidence data in fetched sources)	clause [S1] [S6] [S8]

Lapse. FCA evidence (2024, pure protection in force): average lapse rate 5% p.a.; highest observed early lapse 23% in policy year 1 (non-advised intermediated sales with 4-year clawback); modest lapse spikes just after the 2-year and 4-year commission clawback periods end R9. A full duration curve is not public, so the reference table is std, anchored to the 5% average and the clawback-spike pattern:

Policy year	1	2	3	4	5	6+
Annual lapse `w(t)` std	10%	8%	7%	5%	6%	4%

(Year 3 staying elevated after the 2-year clawback period ends, and the year-5 uptick after the 4-year clawback period ends, echo the post-clawback spike pattern R9; levels are standardized calibrations to be replaced with the user’s experience.)

Expenses and commission (all levels std; structure evidence as cited).

Input	Value	Basis
Initial (acquisition) expense	£150 per policy at issue	std
Initial commission	150% of annualized premium, paid upfront at issue	upfront pattern ~96% of commission R9; level std
Commission clawback	On lapse in years 1–4: clawback of `(48 − months in force)/48` of initial commission (linear, 4-year) — optional module, base model off	clawback periods 2–4 years R9; formula std
Renewal commission	2.5% of premiums from year 2	std
Maintenance expense	£30 per policy p.a., inflating 3% p.a.	std
Claim expense	£250 per death/TI claim	std
Expense inflation	3% p.a. flat	std

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
`t`	policy year, t = 1..n; attained age in year t = x + t − 1 (x = issue age)
`k`	policy month, k = 0..N, N = 12n (monthly grid / benefit schedules)
`P_a`	annualized premium = 12 × P_m = 144.00 (anchor cell) std
`q(t)`	mortality (incl. TI) rate for year t, select-adjusted
`w(t)`	lapse rate for year t (end-of-year, after deaths) [std order]
`l(t)`	in-force probability at start of year t; l(1) = 1
`D(t)`	expected claims in year t = l(t) × q(t)
`SA0`	initial sum assured (level/decreasing)
`I`	FIB monthly income
`j`, `j_m`	decreasing schedule annual rate; j_m = (1+j)^(1/12) − 1
`B(k)`	decreasing-shape benefit after k months (formula below)
`idx(t)`	cumulative indexation factor at start of year t (1 if not indexed)
`E0`, `e(t)`	initial expense (150); maintenance expense = 30 × 1.03^(t−1)
`c0`, `c_r`	initial commission (1.5 × P_a); renewal commission rate (0.025, from year 2)
`ec`	claim expense (250)
`CF(t)`	net cash flow of year t, insurer perspective (+ inflow, − outflow)

Dimensional check: q, w are per-annum probabilities (dimensionless); B, SA0 are GBP; I is GBP/month so FIB outgo terms carry explicit month counts; all CF components are GBP per year.

Benefit amount by shape#

Level: DB(t) = SA0 × idx(t).

Decreasing [S1] [S6] [S8]:

B(k) = SA0 × [(1+j_m)^N − (1+j_m)^k] / [(1+j_m)^N − 1],   B(0) = SA0, B(N) = 0

Annual-grid death benefit uses the mid-year balance std:

DB(t) = B(12(t−1) + 6)

(whole-year identity (1+j_m)^12 = 1+j allows B(12t) = SA0 × [(1+j)^n − (1+j)^t] / [(1+j)^n − 1]; the monthly grid uses B(k) at the exact month of death). Numeric anchor (SA0 = 150,000, j = 6%, n = 25): B(60) = 150,000 × (1.06^25 − 1.06^5) / (1.06^25 − 1) = 150,000 × (4.291871 − 1.338226) / 3.291871 = £134,588 — the benefit after 5 years. Indexation and the decreasing shape are not combined [std scope] (no fetched insurer offers indexed decreasing cover).

Family income benefit [S2] [S6] [S8]: a death in month k triggers N − k monthly instalments of I, in arrears, ending at month N — an annuity-certain independent of survival. On the annual grid, with deaths at mid-year std, a death in year s generates expected instalment outgo:

FIB outgo in year s (year of death):        6 × I × D(s)
FIB outgo in later year u, s < u ≤ n:      12 × I × D(s)

so total FIB claim outgo in year t is

Claims_fib(t) = I × [ 6 × D(t) + 12 × FIBcum(t) ],   FIBcum(t) = Σ_{s<t} D(s)

Optional commutation module std: replace the instalment stream at death with a lump sum CV(k) = I × a(N−k) where a(m) = [1 − (1+r_c)^(−m/12)] / [(1+r_c)^(1/12) − 1] is the m-month annuity-certain factor at the snapshot commutation rate r_c = 3% std (contractually the insurer reduces the sum of remaining instalments “fairly and reasonably” [S6] [S8]). Base model: no commutation.

In-force recursion and processing order#

Annual processing for year t = 1..n std:

Start of year: premium income P_a × idx_p(t) × l(t) (where idx_p(t) is the cumulative premium indexation factor — equal to 1 in the base run); maintenance expense e(t) × l(t); renewal commission c_r × P_a × idx_p(t) × l(t) (from t ≥ 2). At t = 1 additionally E0 and c0 (per policy issued, l(1) = 1).
Benefit schedule: compute DB(t) per shape (mid-year balance for decreasing).
End of year — claims: expected death/TI outgo DB(t) × D(t) (level/ decreasing) or the FIB formula above; claim expense ec × D(t).
End of year — lapses: applied to survivors of mortality [std order: death before lapse]; lapse pays nothing (no surrender value [S6] R8).

Update:

l(t+1) = l(t) × (1 − q(t)) × (1 − w(t))

Anniversary (if indexation elected): with acceptance (behavior section), idx(t+1) = idx(t) × (1 + min(max(RPI, 0), 0.10)) and idx_p(t+1) = idx_p(t) × (1 + min(1.5 × increase, 0.15)) [S1] [S2] [S6].

At t = n the projection ends: no maturity payment, no tail states [S1] [S6] [S8] R8.

Net cash flow#

Level/decreasing shapes:

CF(t) = P_a × idx_p(t) × l(t)                                   (premiums)
      − DB(t) × D(t)                                            (death/TI claims)
      − ec × D(t)                                               (claim expense)
      − e(t) × l(t)                                             (maintenance)
      − c_r × P_a × idx_p(t) × l(t) × 1{t ≥ 2}                  (renewal commission)
      − (E0 + c0) × 1{t = 1}                                    (acquisition)

FIB shape: replace the claims term with Claims_fib(t) and add − ec × D(t) only in the year of death. Premiums stop at death, but FIB instalments continue — premium income always carries l(t), never the FIB ledger.

Monthly-grid variant: the same components at monthly frequency with P_m, monthly decrements q_m = 1 − (1 − q)^(1/12), w_m = 1 − (1 − w)^(1/12) std, exact B(k), and exact FIB instalments; the annualization and mid-year approximations disappear. The annual grid slightly overstates premium income (no allowance for premiums ceasing at mid-year deaths/lapses) — a known bias of the annual-in-advance convention std, quantified in the pitfalls list.

Waiver of premium (optional module, base off)#

With wop = true, an incapacity state is added: incidence inc(t) std (no public UK incidence basis for the WOP work-tasks definitions is in the fetched sources), 26-week deferred period [S1], premiums waived while incapacitated (premium income multiplied by the active-payer probability), mortality unchanged. The WOP extra premium and the incidence/recovery basis are both std placeholders.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions; calibration evidence is cited where it exists.

Base lapse std. Duration table above, anchored to the FCA 5% in-force average and clawback-spike pattern R9. Channel matters: the 23% year-1 observation is specific to non-advised intermediated business with 4-year clawback R9; the composite table is channel-blended.
Selective lapsation std (optional module). Lapsers are healthier on average; persisters’ mortality is loaded:
```
q_eff(t) = q(t) × [1 + λ × max(0, w_cum(t) − w_ref)]
```
with w_cum(t) = cumulative lapse proportion to date, w_ref = 0.20 and λ = 0.25 std. Base run: off (λ = 0).
Rebroking/dynamic lapse std. Guaranteed premiums mean no premium-shock lapse; the economic driver is rebroking when quoted market premiums for the attained age fall below the in-force premium (younger select lives, falling mortality). A reference multiplier:
```
M_reb(t) = min(2.0, max(1.0, P_inforce / P_market(t)))
```
applied to w(t), with P_market(t) an external input; base run P_market = P_inforce, so M_reb = 1.
Indexation take-up std. If indexation = true: each anniversary the increase is accepted with probability 80% std; after 3 consecutive declines the option is removed [S1] [S6] (two at one insurer [S8]). Deterministic base run: always accept, RPI scenario input flat 3% std, giving idx(t+1) = idx(t) × 1.03 and idx_p(t+1) = idx_p(t) × 1.045 (premium factor 1.5 [S1] [S2] [S6]).
GIO exercise. Not modeled: exercises create new policies at then-current rates [S1] [S6] [S8], so they add model points rather than changing this one [std scope].

Worked example#

Anchor cell: male 35 non-smoker, single life, level shape, n = 25, SA0 = £150,000, P_m = £12.00 (P_a = £144.00) std; no indexation, no WOP, no commutation; base lapse table; no selective-lapse or rebroking modules. Mortality placeholders q(1) = 0.00055, q(2) = 0.00060, q(3) = 0.00065 are std illustrative values in the shape of a non-smoker temporary assurance table — they are NOT taken from any CMI table (the current tables are subscriber-only R11; see assumption class (c)). Expenses per the std table: E0 = 150, c0 = 1.5 × 144 = 216.00, e(t) = 30 × 1.03^(t−1), c_r = 2.5% from year 2, ec = 250.

t	l(t)	Premiums `P_a·l(t)`	Claims `SA0·D(t)`	Claim exp `ec·D(t)`	Maint. + initial exp	Commission	Net CF(t)
1	1.000000	144.00	82.50	0.14	180.00	216.00	−334.64
2	0.899505	129.53	80.96	0.13	27.79	3.24	+17.41
3	0.827048	119.09	80.64	0.13	26.32	2.98	+9.02

Trace, year 1: D(1) = 1.0 × 0.00055 = 0.00055; claims = 150,000 × 0.00055 = 82.50; claim expense = 250 × 0.00055 = 0.14; expenses = E0 + e(1) = 150.00 + 30.00 = 180.00; commission = c0 = 216.00; CF(1) = 144.00 − 82.50 − 0.14 − 180.00 − 216.00 = −334.64. Update: l(2) = 1.0 × (1 − 0.00055) × (1 − 0.10) = 0.899505.

Trace, year 2: premiums = 144 × 0.899505 = 129.53; D(2) = 0.899505 × 0.00060 = 0.000540; claims = 150,000 × 0.000540 = 80.96; claim expense = 0.13; maintenance = 30 × 1.03 × 0.899505 = 27.79; renewal commission = 0.025 × 129.53 = 3.24; CF(2) = 129.53 − 80.96 − 0.13 − 27.79 − 3.24 = +17.41. Update: l(3) = 0.899505 × (1 − 0.00060) × (1 − 0.08) = 0.827048.

The pattern is characteristic of guaranteed term: a deep new-business strain in year 1 (upfront commission and acquisition expense against one year’s premium R9) and thin positive margins thereafter — the level premium prefunds the rising mortality cost, so early-duration lapses forfeit margin to the insurer while late-duration lapses relieve it.

Valuation and reserve pointers#

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

Solvency UK best estimate. BEL = probability-weighted average of future cash-flows, discounted at the relevant risk-free term structure, realistic assumptions, gross of reinsurance (recoverables separate) R1; required cash-flow categories per the PRA cash-flows rule — benefits, expenses, premiums, intermediary payments, policyholder-charged taxation R2; contract boundary = full term for guaranteed premiums R3. BEL = Σ_t v(t) × [outgo(t) − income(t)] over the recursion above. Note: for profitable guaranteed term assurance the BEL is commonly negative at issue (PV premiums > PV claims + expenses) — an asset on the regulatory balance sheet; models must not floor it at zero (derivation, no source).
Risk margin. Technical provisions = best estimate + risk margin REG-R1; cost-of-capital method at 4% with life risk-tapering factor λ = 0.9, floor 0.25 REG-R4 — requires an SCR run-off projection, cited-not-specified here.
Regime. PS15/24 completed the restatement of Solvency II assimilated law into PRA rules from end-2024 (“Solvency UK”) R5.
IFRS 17. UK-adopted IFRS 17 (adopted 16 May 2022, effective 1 January 2023) applies to IFRS reporters REG-R38; the fulfilment-cash-flow engine is the same expected-cash-flow projection; grouping, CSM and risk-adjustment layers are out of scope [mechanics beyond the adoption facts: unverified].
Professional standards. Technical actuarial work using this model in scope of UK regulation falls under TAS 100 v2.0 R15 and TAS 200 v2.0 R16.

Key sensitivities and model risks#

Dominant assumptions, in rough order for a protection block:

Mortality basis risk. The reference basis is a std proxy (scaled “00” Series) because the current 16-Series tables are subscriber-only R11 R13; the proxy scaling factor (75% std) is the single largest lever on claims. Production users should substitute subscriber tables (TMNL16/TFNL16 R12 R10) and a CMI projections overlay R14 REG-R30.
Early-duration lapse. With ~96% of commission upfront and 2–4 year clawback R9, year-1–4 lapse rates drive new-business strain recovery; the clawback module changes the sign of the sensitivity inside the clawback window.
Selective lapsation. Guaranteed premiums plus healthy-life rebroking imply persisting lives are progressively impaired; the λ loading materially moves late-duration claims on long terms.
Expense inflation on small premiums. Premiums as low as £5/month [S5] against £30/year maintenance make per-policy expense inflation a solvency-relevant assumption for small-sum-assured blocks.
Shape-specific risks. Decreasing: the schedule rate j is contractual, so the risk is specification error, not experience (mis-implementing the amortization or the monthly convention); FIB: the annuity-certain run-off means claim outgo persists up to n − 1 years after death — omitting the in-payment ledger understates liabilities.
Indexation take-up. The ×1.5 premium factor [S1] [S2] [S6] makes accepted increases premium-margin-accretive if mortality is proportional to cover; selective acceptance (impaired lives accept, healthy decline) reverses the sign std concern; no public take-up data exists in the fetched sources.

Known modeling pitfalls:

TI is not an extra benefit. Death and terminal illness are one decrement and one payment [S1] [S6] [S8]; adding a separate TI decrement double-counts claims. The 16-Series mortality tables already include terminal illness R10.
FIB instalments are certain, not contingent. Do not decrement the in-payment income by mortality or lapse; only new claims depend on l(t) [S6] [S8].
Decreasing-schedule conventions. j_m = (1+j)^(1/12) − 1 std vs a nominal j/12 convention changes B(k) slightly; state the convention and use the B(60) = £134,588 anchor to validate implementations.
Annual-grid biases. Mid-year benefit for the decreasing shape and annual-in-advance premiums are offsetting small biases; the monthly grid is the arbiter. Do not apply both the mid-year claim timing and a separate half-year premium adjustment — pick one convention.
Lapse pays nothing. There is no surrender value [S6] R8; a lapse row in the cash-flow output must be zero-valued (it affects only l(t)), unlike US models with CSV outflows.
No tail states. Terminate everything at month N: no renewal, no conversion, no extended coverage [S1] [S2] [S6] [S8] R8. Importing a US-style post-level-term tail materially misstates UK term liabilities.
Joint life first death. Model as a single joint decrement q_joint = 1 − (1−q_1)(1−q_2) on one policy std; the policy pays once and ends [S1] [S6]. Separation/replacement options create new policies and are out of scope.
Boundary discipline. All guaranteed premiums are inside the contract boundary R3; truncating premium income at an assumed “repricing” point (a Solvency II habit from reviewable business) is wrong for this product.