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/critical-illness.md); [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 model mirrors the term assurance reference model in products/term_assurance/ (base chassis); only CI-specific mechanics are new here.

Model scope and conventions#

Purpose. Project gross best-estimate liability cash flows (premiums, main claims, additional-payment claims, children’s-cover claims, expenses) for a single-policy model point of accelerated (and, as a variant, standalone) Critical Illness Cover. Discounting, reserves and capital are not computed (see Valuation and reserve pointers).
Projection frequency. Monthly grid over the policy term (12 x term months) std. The contract itself has no accumulation account; monthly is chosen for parity with the other reference models in this library.
Timing conventions std. Premiums and maintenance expenses at the beginning of the policy month (BOM); claims and decrements at the end of the policy month (EOM). Annual decrement rates are converted to monthly via q_m = 1 − (1 − q_annual)^(1/12); small frequency loadings may use the rate/12 approximation, stated where used.
Age basis. Age nearest birthday (ANB) std; attained age advances on policy anniversaries. Chosen for consistency with the CMI assured-lives table conventions [unverified — the convention of the restricted tables was not confirmed from a fetched document]; any consistent basis works if used for all lookups.
Currency. GBP. All amounts per single policy.
Model points. Single-policy model points projected on an expected (probability-weighted) basis: survivorship factors multiply per-policy cash flows. Joint life first event is a variant (two-life survivorship product) [std scope: not in base].
Survival period. 14 days [S1] [std pick, see product-spec footnote 9]. In the accelerated base model it is cash-flow-neutral (death within 14 days of diagnosis pays the same SA as a death claim [S1]) and is ignored as a timing refinement std. In the standalone variant it reduces payable claims (below).
Rounding. Intermediate values at full precision; displayed to pence std.

Model point attributes#

Attribute	Type	Example (anchor cell)
`contract_type`	enum {accelerated, standalone}	accelerated
`issue_age`	int (ANB)	40
`sex`	enum {M, F}	M
`smoker`	enum {NS, S}	NS
`sum_assured`	currency (SA)	100,000 std
`term_years`	int (5–50 [S2] [S5])	25 std
`cover_basis`	enum {level} (decreasing/FIB out of scope)	level
`life_basis`	enum {single, joint_first_event}	single std
`premium_guarantee`	enum {guaranteed, reviewable}	guaranteed [S1]
`premium_monthly`	currency	55.00 std (no public rate cards — placeholder)
`premium_mode`	enum {monthly, annual}	monthly std
`children_cover`	bool (automatic on the composite [S1])	true
`indexation`	bool (increasing-cover option; base: false)	false std
`issue_date`	date	month 1

State variables#

Variable	Description	Updated
`l(t)`	In-force probability at end of month t; l(0) = 1	monthly decrements
`t` / `y` / `a`	Policy month; policy year = ceil(t/12); attained age = issue_age + y − 1 (ANB)	monthly
`P(t)`	Premium rate in force (constant under guaranteed premiums; reset at reviews in the reviewable module)	at reviews only
`SA(t)`	Sum assured (constant at SA for level cover; indexation module updates annually)	on events
`grace_flag(t)`	In-grace indicator (60-day grace [S1] [S4]) — deterministic base model does not enter grace	monthly
`n_AP_used`	Additional-payment claims used per condition (contract cap: 1 per condition [S11]) — not tracked in the frequency-loading approximation std	—
`n_child_used`	Children’s claims used (cap 2 [S1]) — not tracked in the frequency-loading approximation std	—

There is no account value, asset share, surrender value, bonus, or MVR state in this product: lapse pays nothing [S1] [S4] [S5] [unverified as explicit statement].

Assumption inputs#

Three classes are distinguished explicitly.

(a) Contractual / guaranteed elements (cited)#

Input	Value	Basis
Main benefit	SA on first of death / TI / CI diagnosis + survival (accelerated); CI only (standalone)	[S1] [S4] [S8] [S11]
Additional-payment benefit `B_AP`	min(0.25 x SA, 25,000) = 25,000 at the anchor cell; non-depleting	[S1] [S4] [S11]
Children’s benefit `B_ch`	min(0.50 x SA, 25,000) = 25,000 at the anchor cell; non-depleting; 2-claim policy cap	[S1]
Child funeral benefit	4,000 — excluded from the base model (de minimis)	[S1]; exclusion std
Survival period	14 days	[S1]; pick std
Premium	Level, guaranteed for the term; 60-day grace, no surrender value	[S1] [S4]
Term / expiry	5–50 years; policy ends by 75th birthday	[S2] [S5]

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

Guaranteed-premium CIC has almost no discretionary machinery — there are no bonus rates, no asset shares, no MVRs. Two snapshot elements exist:

Input	Snapshot value	Basis
Reviewable-premium reviews (variant module only)	Reviews every 5 years from the 5th anniversary; changes driven by claims/industry experience, medical advances, law; one carrier: “no limits” on changes, <2% or 50p ignored; another’s intermediary variant: ±5% tolerance, individual health not a factor. Snapshot: premiums unchanged at each review std	[S3] [S4] [S5]
Indexation basis (if `indexation = true`)	RPI snapshot 3.0% p.a. std → cover +3.0%, premium +4.5% (x1.5 factor), within caps 10%/15%	mechanics [S1] [S4]; RPI level std

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

The CMI’s critical illness investigation covers standalone and full accelerated (death + CI) business, on a diagnosis-rate approach: AC04 insured-lives accelerated-CI diagnosis-rate tables (WP50, 2003–2006 experience), cause-specific rates (WP52, updated WP151), and CIBT93 as the population-based comparison table R8 R9. The current protection base-table generation is the “16” Series (term assurance mortality and accelerated CI, 2015–2018 experience, finalized with WP154) REG-R26; the latest public experience output is WP167 (accelerated CI by cause, 2017–2020) R9. Honest flagging: CMI working papers are public, but current CMI tables and datasets are restricted to Authorised Users (subscribers) REG-R22 R9 — access limits unverified; AC04/16-Series rate values were not obtained. The reference basis below is therefore a std proxy shaped like the named tables, to be replaced by a licensed basis in any real application.

Input	Reference basis	Basis tags
CI diagnosis rates `i_ci(x)`	std proxy table below, shaped like an insured-lives accelerated-CI diagnosis-rate table (AC04/16-Series structure: sex/smoker-distinct, age-increasing)	structure R8 REG-R26; values std
Mortality `q_d(x)`	std proxy table below, shaped like ~0.70 x ONS National Life Tables qx (population mortality is heavier than insured experience; scalar and pivot values are rounded placeholders, not derived ONS data)	ONS tables redistributable REG-R32; values std
Overlap factor `k`	0.10 flat (see combined decrement below)	std
Standalone survival-period slippage `δ`	0.03 (fraction of diagnoses dying within 14 days)	std
Additional-payment frequency	`a(x) = 0.15 x i_ci(x)`, non-terminating	std
Children’s-cover claim frequency	`λ_ch = 0.0004` p.a. per policy, non-terminating, while children_cover active	std
Lapse `w(y)`	std table below; no dynamic lapse in base	std
Mortality/morbidity improvement and CI trend `τ`	0% p.a. in base; if an improvement overlay is required, express as “CMI_20xx with long-term rate p% std”	REG-R30; base std
Expenses	Initial 200 per policy; maintenance 30 p.a. inflating 3% p.a.; claim expense 250 per main claim	std

std proxy diagnosis and mortality rates (annual, male non-smoker; pure placeholders — NOT CMI or ONS values; interpolate log-linearly between pivot ages std):

Age x	40	45	50	55	60	65
`i_ci(x)`	0.0015	0.0025	0.0040	0.0070	0.0110	0.0170
`q_d(x)`	0.0009	0.0014	0.0022	0.0036	0.0060	0.0100

std lapse table (annual rates; protection-book shape, calibration to be replaced by the user’s experience — UK CI lapse studies are proprietary):

Policy year	1	2	3–5	6+
`w(y)`	10%	8%	6%	4%

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
t	policy month, t = 1..12n (n = term_years); y = ceil(t/12); a = attained age (ANB)
`SA`	sum assured (100,000 at the anchor cell)
`P`	monthly premium (55.00 std at the anchor cell)
`i_ci(a)`	annual CI diagnosis rate (first diagnosis of a listed condition, incl. TPD)
`q_d(a)`	annual best-estimate mortality rate
`k`	overlap: proportion of deaths that follow a CI diagnosis that already gave rise to (or would give rise to) a claim in the same year (0.10 std)
`q_claim(a)`	annual combined claim decrement (accelerated), defined below
`q_m(t)`, `w_m(t)`	monthly claim and lapse rates: `1 − (1 − annual)^(1/12)`
`a_m(t)`	monthly additional-payment frequency ≈ `0.15 x i_ci(a) / 12` std
`λ_m`	monthly children’s claim frequency ≈ `λ_ch / 12` = 0.0000333 std
`B_AP`, `B_ch`	25,000 and 25,000 (anchor cell; see contractual inputs)
`E0`, `E_m(y)`, `E_cl`	initial expense 200; maintenance `30/12 x 1.03^(y−1)` per month; claim expense 250 std
`l(t)`	in-force probability at end of month t; l(0) = 1
`δ`	standalone survival-period slippage (0.03 std)
`τ`	CI trend rate (0 in base std)

Dimensional check: all benefit amounts are GBP; q_m, w_m, a_m, λ_m are dimensionless monthly probabilities/frequencies; every cash-flow line below is GBP/month per policy in force at the relevant weighting.

Combined decrement for accelerated CI#

The insured event is death or first CI diagnosis, whichever first — the CMI’s accelerated investigation measures exactly this combined claim incidence with cause-of-claim splits R8 R9. Adding q_d and i_ci naively double-counts lives that are both diagnosed and die in the same period: once the CI claim has been paid (diagnosis + 14-day survival), the subsequent death of that life is not a second claim; and a death within the survival period converts the CI claim into a death claim of the same amount rather than adding one. The classical independent-rates formulation is diagnosis rates plus mortality net of the overlap [unverified as a market-practice statement — recorded as such in the research file]:

q_claim(a) = i_ci(a) x (1 + τ)^(y−1) + q_d(a) x (1 − k)          [std]

where k is the proportion of deaths preceded by a claimable CI diagnosis (deaths “already counted” in i_ci). std simplification: k = 0.10, flat across ages, in the absence of public cause-of-death-linked CI data (the cause-specific splits in WP52/WP151/WP167 R8 R9 are the right calibration source for subscribers). Sensitivity range 0–0.25 (see Key sensitivities). The 14-day survival period needs no further adjustment in the accelerated design: whichever way the overlap resolves, SA is paid once [S1].

Standalone variant deltas#

Death pays nothing; the policy simply terminates. Decrement splits into paying and non-paying parts std:

q_pay(a)  = i_ci(a) x (1 + τ)^(y−1) x (1 − δ)        — CI claims paid (survive 14 days)
q_exit(a) = q_d(a) x (1 − k) + i_ci(a) x (1 + τ)^(y−1) x δ
                                                      — deaths without payment, incl.
                                                        deaths within the survival period

Total decrement q_claim = q_pay + q_exit (same in-force runoff as the accelerated model at these parameters); only the paid part generates claim outgo. Death within the survival period pays nothing on the composite standalone variant [S4] [S11]; a premium-refund-on-death feature exists in some designs [S4] [S11 — recorded jointly in the research file] and is excluded std.

Monthly processing order std #

At BOM of month t:

Premium income: P x l(t−1) (survivors at the start of the month pay).
Maintenance expense: E_m(y) x l(t−1). (Initial expense E0 at t = 1 only, weight 1.)

At EOM of month t:

Main claim decrement: expected claim outgo SA x q_m(t) x l(t−1) (accelerated; standalone uses q_pay_m), plus claim expense E_cl x q_m(t) x l(t−1).
Additional-payment claims (non-terminating — do NOT decrement l): B_AP x a_m(t) x l(t−1).
Children’s-cover claims (non-terminating — do NOT decrement l): B_ch x λ_m x l(t−1).
Lapse applied to non-claiming survivors; update in-force: l(t) = l(t−1) x (1 − q_m(t)) x (1 − w_m(t)) [std order: claim before lapse].
At t = 12n (term end): policy expires; no maturity or surrender value [S1] [S4] [S5].

The frequency-loading treatment of steps 4–5 deliberately ignores the contractual claim-count caps (1 per additional-payment condition [S11]; 2 children’s claims [S1]) and the per-child cross-policy cap (£50,000 [S1]): at the std frequencies the probability of hitting a cap is second-order. Exact treatment would need claim-count state variables (n_AP_used, n_child_used).

Cash flow outputs (per policy, month t)#

Cash flow	Formula	Sign	Timing
Premium income	`P x l(t−1)`	+	BOM
Initial expense	`E0` at t = 1	−	BOM
Maintenance expense	`E_m(y) x l(t−1)`	−	BOM
Main claims	`SA x q_m(t) x l(t−1)` (standalone: `q_pay_m`)	−	EOM
Claim expenses	`E_cl x q_m(t) x l(t−1)`	−	EOM
Additional-payment claims	`B_AP x a_m(t) x l(t−1)`	−	EOM
Children’s-cover claims	`B_ch x λ_m x l(t−1)`	−	EOM
Surrender outgo	0 (no surrender value [S1] [S4] [S5])	—	—

Grace (60 days [S1] [S4]) is not separately modeled in the deterministic base: lapse rates are assumed to already reflect grace-period cures std. Death during grace pays the death benefit less unpaid premiums [term chassis]; immaterial at monthly resolution std.

Reviewable-premium module (variant)#

For premium_guarantee = reviewable: P(t) is constant between reviews; at each 5-yearly review from the 5th anniversary [S3] [S4], P ← P x (1 + ρ_review) where ρ_review is a scenario input (snapshot 0 std). Contractual constraints: one carrier’s form — no limits, changes under 2% or 50p ignored, policyholder may instead reduce cover [S4] [S5]; another’s intermediary form — ±5% tolerance per review, individual health not a factor [S3]. A review-driven lapse response belongs in behavior modeling (below). Premium rates for in-force reviewable business are insurer-discretionary current elements — class (b) snapshots, not guarantees.

Policyholder behavior modeling#

All dynamic formulas are std reference constructions; UK CI lapse experience studies are proprietary, so shapes are stated with rationale and no source is cited for calibration.

Base lapse std. w(y) per the table above, converted monthly. Rationale: protection lapse is duration-skewed (early years highest — buyer’s remorse, remortgaging, distribution churn) and levels off in later durations.
No interest-sensitive lapse. There is no cash value or credited rate to arbitrage; the interest-sensitive dynamic-lapse machinery of the accumulation products in this library is deliberately absent std.
Premium-review shock (reviewable module only) std. w_shock = min(0.30, w(y) + 2.0 x max(0, ρ_review − 0.05)) applied in the 12 months following a review that raises premiums by more than 5%. Rationale: one carrier’s unlimited review changes [S4] make review-driven shocks the dominant behavioral risk on reviewable business; slope and cap are placeholders.
Selective lapsation std. Optional morbidity-anti-selection overlay: after a lapse-shock event, remaining lives carry i_ci x (1 + η) with η = 0.10. Rationale: healthier lives lapse first when premiums rise; magnitude is a placeholder.
Indexation take-up (if indexed) std. Declining an increase 3 years in a row removes the option [S1] [S4]; base model assumes full take-up while active.
GIO / life-change option exercises. Excluded from the base model point std; exercise creates a new policy/increase at current rates without underwriting [S1] [S4] [S11] — an anti-selection cost that a production model should load for.

Worked example#

Anchor cell: male 40 non-smoker, accelerated, SA = £100,000, term 25 years, level guaranteed premium P = £55.00/month std. Age-40 assumptions: i_ci = 0.0015 std, q_d = 0.0009 std, k = 0.10 std, τ = 0 → q_claim = 0.0015 + 0.0009 x 0.90 = 0.00231 annual; q_m = 1 − (1 − 0.00231)^(1/12) = 0.00019270. Year-1 lapse 10% → w_m = 1 − 0.90^(1/12) = 0.0087416. a_m = 0.15 x 0.0015 / 12 = 0.00001875; λ_m = 0.0004 / 12 = 0.0000333. B_AP = B_ch = 25,000. Maintenance E_m = 30/12 = 2.50 (year 1); claim expense 250; initial expense £200 at t = 1 (not shown in the table). Survivor factor per month: s = (1 − q_m)(1 − w_m) = 0.9998073 x 0.9912584 = 0.9910674.

Month t	l(t−1)	Premium `P·l`	Main claim `SA·q_m·l`	Claim exp `250·q_m·l`	Add-pay `B_AP·a_m·l`	Child `B_ch·λ_m·l`	Maint `E_m·l`	Net CF	l(t)
1	1.000000	55.00	19.27	0.05	0.47	0.83	2.50	31.88	0.991067
2	0.991067	54.51	19.10	0.05	0.46	0.83	2.48	31.59	0.982215
3	0.982215	54.02	18.93	0.05	0.46	0.82	2.46	31.31	0.973441

Trace, month 1: premium 55.00 x 1; expected main claim 100,000 x 0.00019270 = 19.27; claim expense 250 x 0.00019270 = 0.05; additional payment 25,000 x 0.00001875 = 0.47; children’s 25,000 x 0.0000333 = 0.83; maintenance 2.50. Net = 55.00 − 23.12 = 31.88 (31.88 − 200 initial expense = −168.12 in total month-1 cash flow). l(1) = 1 x (1 − 0.00019270) x (1 − 0.0087416) = 0.991067. Note the additional-payment and children’s rows do not enter l(t): they are non-terminating loadings [S1] [S3] [S4] [S8] [S11].

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 = best estimate + risk margin (Technical Provisions 2.4); best estimate = probability-weighted cash flows discounted on the risk-free term structure, gross of reinsurance, realistic assumptions, homogeneous risk groups (3.1–3.2, 9.1–9.2, 10.1) R7 REG-R1. Risk margin: cost-of-capital method, CoC 4%, λ = 0.9 taper with floor 0.25, effective 31/12/2024 R7 REG-R4. The matching adjustment is in its own Rulebook Part R7 and is in practice irrelevant to CI term business unverified.
IFRS 17. UK-adopted IFRS 17 (effective 1 January 2023) REG-R38 measures these contracts as fulfilment cash flows plus CSM [mechanics summary: unverified — the standard text was not fetched]; the expected-cash-flow engine is the same projection, with regime-specific discounting, risk adjustment and aggregation.
Professional standards. TAS 100 v2.0 applies to all technical actuarial work from 1 July 2023 R10 REG-R33; TAS 200 v2.0 (insurance) applies from 1 January 2025 REG-R34.

Key sensitivities and model risks#

Dominant assumptions, in order:

CI trend and condition-definition drift. The dominant assumption risk for CI business: diagnosis rates trend with medical practice (earlier and wider diagnosis), and the covered event itself moves when the ABI revises model definitions — the 2021/22 review broadened Alzheimer’s to all dementia, tightened cancer staging exclusions, and excluded myocardial injury from heart attack, with compliance by 31 January 2024 R2 R3; prior reviews 2011, 2014, 2018 R3. Definition changes produce step changes in i_ci that no trend parameter anticipates; sensitivity-test τ at ±2% p.a. std and re-map the incidence basis at each definition-review generation.
Level and shape of the diagnosis-rate proxy. i_ci here is a std placeholder because AC04/16-Series values are subscriber-restricted REG-R22 REG-R26; miscalibration scales claims one-for-one. WP167 also flags COVID-affected 2020 experience R9.
Overlap factor k. Bounds: assuming k = 0 maximally double-counts (overstates combined incidence by the true overlap x q_d per year); k = 0.25 may understate. Calibrate from cause-of-claim data (WP52/WP151/WP167 lineage) where licensed R8 R9.
Lapse. With level guaranteed premiums against steeply age-increasing i_ci, early durations pre-fund later ones: higher-than-assumed late-duration lapses release liability, lower ones extend exposure to the steep part of the incidence curve; the BEL is not monotone in a single lapse scalar. Lapse assumptions must be realistic and condition-dependent under the Rulebook (9.1–9.2) R7.
Expenses and expense inflation. Second-order next to (1)–(4) on this mono-benefit product std placeholders throughout.
Guaranteed vs reviewable premiums. The base model’s premiums are guaranteed — morbidity deterioration cannot be repriced, so items (1)–(3) fall entirely on the insurer. The reviewable module transfers trend risk to policyholders at the cost of review-shock lapse and selective lapsation (anti-selection multiplier η) [S3] [S4] [S5].

Known modeling pitfalls:

Double counting death and CI. Summing q_d + i_ci without the overlap term overstates accelerated claim incidence; conversely, applying k to the standalone paid decrement (instead of to the non-paying death exit) understates claims.
Survival-period misapplication. Applying the 14-day survival reduction δ to the accelerated main benefit is wrong — death within the survival period still pays SA as a death claim [S1]; δ bites only in the standalone variant [S4] [S11].
Depleting the sum assured for partial claims. Additional-payment and children’s claims must not reduce SA or decrement l(t) [S1] [S3] [S4] [S8] [S11]; modeling them as accelerations (the severity-graded plan-account depletion design [S10]) is a different product.
Terminating on additional-payment claims. Same error, opposite sign: only the main benefit ends the policy [S1] [S4] [S11].
Age-basis mismatch. i_ci, q_d and attained-age indexing must share the ANB std basis.
Proxy-basis leakage. The std proxy rates in these notes are placeholders and must not be presented as CMI or ONS values; production work replaces them with a licensed basis and documents the substitution (TAS 100 data/assumption requirements R10 REG-R33).
Premium placeholder. £55/month is not a market rate (no insurer publishes CI rate cards — research-file gap); profitability conclusions from the worked example are meaningless. One carrier’s reviewable reviews have “no limits” [S4] — do not model reviewable business with the guaranteed-premium constraint.