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 fund. [S#]/[R#] tags refer to the source list in sources.md (numbering carried from _research/with-profits.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. Mechanics anchors: the PPFMs of three proprietary insurers [S1] [S4] [S5]; regulatory codification of the asset-share item list: PRA Surplus Funds Part R8; canonical methodology literature: Needleman & Roff (1995) on asset shares and Hibbert & Turnbull (2003) on guarantee costs, as listed on the IFoA SA2 resources page R13.

Model scope and conventions#

Purpose. Project gross best-estimate liability cash flows (premiums in; death, maturity and surrender claims out; expenses; shareholder transfers) for single-policy with-profits model points on the two composite chassis (unitised bond, conventional endowment), with the smoothed-fund (PruFund-style) variation as an alternative crediting module. Reserves are not computed here (see Valuation and reserve pointers).
The asset share is a state variable, not a cash flow. Policy cash flows are premiums, claims (paid at smoothed payouts), expenses and shareholder transfers; the asset share [S1] R8 drives claim amounts through the bonus, smoothing and MVR machinery. The estate absorbs payout-vs-asset-share differences [S1] [S5].
Projection frequency. Monthly std, with the bonus declaration left on its annual cycle. The declaration is the governing act of discretion, it happens once a policy year, and it permanently hardens the guarantee [S1] [S4] [S7] — so it fires in the twelfth month of each policy year and nowhere else, while everything continuous (fund return, charges, mortality charge, decrements, the smoothed payout, the final bonus and the MVR) runs monthly around it. Annual rates are converted with the effective forms (1 + r)^(1/12) and 1 − (1 − r)^(1/12) std, so twelve months compound back to the annual figure exactly and the assumption basis does not move with the grid. The PruFund daily/quarterly smoothing [S9] [S11] remains out of scope: a monthly grid still cannot carry a 5% daily limit or a 2.5% gap trigger that fires and unwinds between two monthly points.
Time index std. t is the 0-based policy month index: t = 0 is the issue month, month t runs from time t to time t + 1, and the projection covers t = 0, 1, …, proj_len − 1, so proj_len is the number of policy months from issue. An in-force cell opens its frame at t = 12 × duration_ifo, its elapsed months, and carries its state in as the opening balances of that month. The contractual policy year containing month t is the 1-based label t // 12 + 1; anniversary k ends month 12k − 1; and the attained age in month t is x + t // 12, advancing on the anniversary.
Timing conventions std. Premiums and partial withdrawals at the start of the month (BOM); fund return accrues over it; proportional charges, the shareholder transfer and the mortality charge at end of month (EOM), in the processing order below; claims and decrements at EOM. The bonus declaration falls at EOM in a declaration month only — (t + 1) mod 12 = 0 — ahead of that month’s mortality charge and payout calculation, so the hardened guarantee is what the month’s claims are measured against.
Age basis. Age nearest birthday std — no retrieved UK document fixes a model age basis; ANB is chosen for symmetry with the library’s US convention (its traditional use in UK assured-lives tables is unverified; the currently marketed bond quotes its issue-age limit on an age-next-birthday basis [S10]).
Currency. GBP. Single-policy model points, projected on an expected (probability-weighted) basis: survivorship factors multiply per-policy cash flows.
Specimen-policy convention. Firms compute asset shares for specimen policies or groups, not necessarily per policy [S1] [S4] [S5] R1 COBS 20.2.5R(2); the reference model computes a per-model-point asset share and treats it as the specimen.
Rounding. Intermediate values at full precision; cash flows reported to pence std.

Model point attributes#

Attribute	Type	Example (anchor cells, product-spec)
`chassis`	enum {UWP_bond, CWP_endowment, SF_prufund}	UWP_bond
`issue_age`	int (ANB)	55 (UWP) / 35 (CWP)
`sex`	enum {M, F}	M
`duration_ifo`	int, completed policy years at valuation	5
`premium_single`	currency (UWP bond)	25,000
`premium_regular`	currency p.a. (CWP: £60/month → 720 p.a.)	720
`sum_assured`	currency (CWP basic SA)	20,000
`term_years`	int (CWP; UWP bond whole-of-life → none)	25
`units`	float (UWP)	25,000
`unit_price`	currency (UWP `Q`; £1.0000 at seed)	1.104081
`attaching_bonus`	currency (CWP `G − SA`)	—
`asset_share_init`	currency (in-force cells); the `AS` the first projected month opens with	30,000
`smoothed_payout_init`	currency; the opening `S`, benchmark for the smoothing cap	29,500
`guarantee_dates`	list of anniversaries (MVR-free); anniversary `k` ends month `12k − 1`	{10}
`mvr_free_wd_rate`	% of original premium p.a.	5%
`tax_basis`	enum {life_net, pension_gross} [S1] REG-R17	life_net
`gao_flag` / `gao_rate`	bool / annuity per £1 cash	false / —
`mutual_dist_flag`	bool (mutual profit distribution variation [S6])	false

State variables#

Variable	Description	Updated
`AS(t)`	Asset share at the end of month t [S1] R8	monthly recursion
`Q(t)`	With-profits unit price (UWP) at the end of month t; never decreases	EOM, declaration months only
`FV(t)`	Unit face value `U(t)·Q(t)` (UWP)	EOM
`G(t)`	Guaranteed benefit `SA` + attaching reversionary bonuses (CWP)	EOM, declaration months only
`b`	Declared annual regular bonus rate for the policy year holding t	once a policy year, setting rule
`S(t)`	Smoothed target payout (after the cap and the corridor)	EOM
`FB(t)`	Final (terminal) bonus payable on a claim in month t	EOM
`MVR(t)`	Market value reduction on non-guaranteed exits	EOM
`CB(t)`	Cost of bonus recognized in month t; nil outside a declaration month	EOM
`ST(t)`	Shareholder transfer = `CB(t)/9` (90:10); nil with it	EOM
`SM(t)`	Smoothing account balance (within estate)	on exits
`CumGC(t)`	Cumulative guarantee-charge deductions (for the 2% lifetime cap [S1])	monthly
`l(t)`	In-force probability at the start of month t (at time t)	BOM, after the previous month’s decrements

Q, FV and G are step functions of the policy year: a declaration moves them in the twelfth month and they are flat through the other eleven. That is the single most important thing to get right when implementing this on a monthly grid — compounding the annual rate b twelve times a year produces a projection that runs, whose roll-forwards close, and whose guarantee is an order of magnitude too large a decade later.

Each state variable except l is a closing balance: AS(t), Q(t), FV(t), G(t), S(t) are the values at the end of month t, so the value a month opens with is the previous month’s close, and for the first projected month it is the carried-in state on the model point. l is the count at a time point: l(t) is the in force at the start of month t, l(0) = 1 at issue, and it is the weight on that month’s cash flows.

Assumption inputs#

Three classes are distinguished explicitly. Class (a) is contractual/guaranteed; class (b) is the insurer’s current discretionary scale (PPFM-governed discretion R2, advised by the With-Profits Actuary R5); class (c) is the modeler’s view of experience.

(a) Contractual / guaranteed elements (cited)#

Input	Value	Basis
Basic sum assured / premium / term (CWP)	£20,000 / £720 p.a. / 25 years	anchor std, product-spec (15)
Bonus hardening	declared regular bonus increases the guaranteed benefit; contractual once added; guaranteed at death/maturity only	[S1] [S8]
Unit-price floor (UWP)	`Q(t) ≥ Q(t−1)`, i.e. `b(t) ≥ 0`	[S1] [S4]
Guarantee events (UWP)	death; contractual guarantee dates (10th anniversary); face value + FB payable without MVR	[S4] [S5]; date choice std, product-spec (12)
Death benefit factor (UWP)	`g_db = 101%` of (FV + FB); MVR never on death	101% std, product-spec (11); no-MVR [S5]
MVR-free withdrawals	≤ 5% p.a. of original premium	std, product-spec (13)
MVR contractual bound	MVR ≤ excess of unit value over underlying asset value	R1 COBS 20.2.16R
PruFund smoothing limits (variation)	daily 5.0% / quarterly 10.0% / gap 2.5% (growth funds); contractual defined terms	[S9] [S11]

(b) Insurer-discretionary current elements (snapshot; revisable under PPFM discipline R2 R5)#

Input	Value	Basis
Regular bonus rate `b` — UWP	2.00% p.a.	std, product-spec (8) — declarations not public in PPFMs
Reversionary bonus rate `b_rev` — CWP	1.50% p.a. compound	std, product-spec (16)
Bonus change cap	±1.00% p.a. in normal circumstances; floor 0	[S1] [S7]; adoption std, product-spec (20)
Guarantee-fill target `θ`	80% of projected maturity asset share	std, product-spec (21); philosophy [S1]
Smoothing y/y cap `σ`	±10%	[S1]; adoption std, product-spec (23)
Target corridor	80%–120% of asset share	[S1] R1; adoption std, product-spec (22)
AMC `c_amc` (UWP)	1.00% p.a.	std, product-spec (9)
Guarantee/smoothing charge `c_g`	0.10% p.a. of asset share; lifetime cap: deductions cease once `CumGC ≥ 2% ×` current asset share	cap [S1]; rate and cap mechanics std, product-spec (10)
Interim bonus rate	= last declared regular bonus rate	practice [S1] [S7]; equality std, product-spec (17)
MVR scale	derived each year from the formulas below (no tabulated scale)	[S5] [S6]; derivation std
EGR (smoothed-fund variation)	5.0% p.a.	std, product-spec (25)
Mutual profit distribution (variation)	0 in base	[S6]; base choice std

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

CMI tables issued after 1 March 2013 are subscriber-restricted R10 REG-R22, so no current CMI rates can be reproduced here: the reference basis is a std proxy on the freely redistributable ONS national life tables REG-R32 (population mortality is heavier than insured experience REG-R32). AM92/AF92 (published 1999) remain the canonical assured-lives shape reference REG-R24; their use in historical with-profits work is unverified convention R10.

Input	Recommended basis	Basis tags
Base mortality	60% × ONS National Life Tables (UK, 2021–2023) qx, sex-distinct	proxy std; source REG-R32; shape cross-check AM92 REG-R24
Mortality improvement	CMI_2025 projections model, long-term rate 1.25% p.a. — named, not reproduced (subscriber-restricted)	model existence REG-R30; LTR choice std
Base surrender rate — UWP bond	5% p.a. flat	std
Base lapse rate — CWP endowment	5% yr 1, 4% yr 2, 3% yr 3, 2% yrs 4+	std
Dynamic surrender multipliers	see Policyholder behavior modeling	std
Paid-up conversion (CWP)	excluded from base model; flag for extension	option exists [S4]; exclusion std
Maintenance expense	£30 per policy p.a., inflating 3.0% p.a.	std
Fund return `r(t)`	5.0% p.a. deterministic base scenario, net of dealing costs [S5]; net of life-fund tax for `tax_basis = life_net` cells [S1] REG-R17	scenario level std
GAO take-up (legacy flag)	90% when in-the-money by >10%, else 30%	std [unverified — no public experience retrieved]

Deterministic single-scenario projection is the base; the cost of guarantees requires stochastic valuation (see Cash flow components, cost-of-guarantees note).

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol	Meaning
`t`	policy month index, 0-based: `t = 0, 1, …, proj_len − 1`; policy year `y(t) = t // 12 + 1`; `x` = age at entry (ANB), attained age `x + t // 12`
`P(t)`	premium received at BOM t (a twelfth of the annual regular premium)
`W(t)`	partial withdrawals paid at BOM t (a twelfth of the annual election)
`E(t)`	insurer maintenance expense in month t ((£30/12) × 1.03^(y(t)−1) std)
`r`, `r_m`	earned fund return, annual (net basis per `tax_basis`) and monthly
`c_amc`, `c_g`	AMC 1.00% p.a.; guarantee/smoothing charge 0.10% p.a. std; `c_amc,m`, `c_g,m` their monthly equivalents
`q(x+t)`, `q_m`	annual mortality rate (class (c) basis) and its monthly equivalent
`w(t)`, `w_m`	annual surrender/lapse rate (incl. dynamic multipliers) and its monthly equivalent, the guarantee-date encashment included
`ε`	guarantee-date encashment rate, 7.5% of the survivors of that month std
`MC(t)`	mortality charge to the asset share in month t [S1]
`b`, `b_rev`	declared annual regular / reversionary bonus rate for the policy year
`Q(t)`, `U(t)`, `FV(t)`	unit price, units, face value (UWP); `FV = U·Q`
`G(t)`	guaranteed benefit (CWP): SA + attaching bonuses
`S(t)`	smoothed target payout after cap and corridor
`FB(t)`, `MVR(t)`, `TB(t)`	final bonus, market value reduction, terminal bonus
`CB(t)`, `ST(t)`	cost of bonus; shareholder transfer = CB/9; both nil outside a declaration month
`θ, κ, σ`	guarantee-fill target 0.80; bonus-smoothing speed 0.5; year-on-year cap 10% std, applied monthly as `(1∓σ)^(1/12)`
`g_db`	UWP death benefit factor 1.01 std
`i_sv`	CWP surrender-basis discount rate 4.0% std; `v_sv = 1/(1+i_sv)`
`n`	CWP term in years (25), so maturity falls at the end of month `12n − 1`; `h` = UWP bonus-setting horizon (10 years std)
`l(t)`	in-force probability at the start of month t (at time t); `l(0) = 1`

Monthly processing order std #

Monthly rates, all std effective conversions of the annual assumptions: r_m = (1+r)^(1/12) − 1, c_amc,m = 1 − (1−c_amc)^(1/12), c_g,m = 1 − (1−c_g)^(1/12), q_m = 1 − (1−q)^(1/12), w_m = 1 − (1−w)^(1/12).

For month t = 0..proj_len−1:

BOM: premium P(t) received — a twelfth of the annual regular premium std, plus the single premium at t = 0; UWP units purchased: U(t) = U(t−1) + α·P(t)/Q(t−1) with allocation α = 100% (product-spec (7)).
BOM: partial withdrawals W(t) paid — a twelfth of the annual election std (MVR applies if outside the MVR-free allowance); asset share reduced pro rata to the pre-MVR policy value [S1].
Fund return r_m accrues on the asset share balance.
EOM: proportional charges: multiply by (1 − c_amc,m − c_g,m); accumulate CumGC; set c_g = 0 for the month once CumGC(t−1) ≥ 2% × AS(t−1) — the cumulative and the asset share the month opens with, so that the charge does not depend on the balance it is deducted from [S1 cap; mechanics std]. The cap is tested every month rather than once a year, so the charge stops the month the cumulative overtakes the threshold.
EOM, declaration months only ((t+1) mod 12 = 0): the annual regular bonus b declared for the policy year per the setting rule below; Q(t) = Q(t−1)(1+b) (UWP) or G(t) = G(t−1)(1+b_rev) (CWP); cost of bonus CB(t) computed on pre-declaration values; shareholder transfer ST(t) = CB(t)/9 deducted from the asset share [S5] R8; product-spec (2). In the other eleven months Q(t) = Q(t−1), G(t) = G(t−1) and CB = ST = 0.
EOM: mortality charge MC(t) = q_m · max(0, DB_g(t) − AS_pre(t)) deducted, where DB_g is the guaranteed death benefit (g_db·FV(t) UWP; G(t) CWP) and AS_pre the balance after step 5 [S1 formula: mortality rate × (death benefit − policy value); guaranteed-only DB in the sum at risk std]. In the eleven months before a declaration the sum at risk is measured against the guarantee as it then stands; the declaration month’s charge is the first to carry the hardened one.
EOM: smoothed payout S(t) computed (cap, then corridor); FB/TB/MVR derived. All three are monthly quantities: a claim in any month is paid on the payout of that month.
EOM: claims paid — deaths at q_m, surrenders at w_m, maturity at the last projected month t = 12n − 1; smoothing account posts (payout − AS(t)) per exiting unit of probability.
Survivorship: l(t+1) = l(t) · (1 − q_m) · (1 − w_m) (maturity month: survivors mature).

Because q_m and w_m compound back to q and w exactly over twelve months, and both are constant within a policy year, l(12y) is the in-force an annual-step projection of the same tables would report at the y-th anniversary. That identity is the cheapest check on a monthly implementation of the decrements.

Regular bonus setting rule std #

The PPFM principles are: rates set from projections; gradual changes (±1% p.a. normal); keep a substantial proportion of the payout in final-bonus form; full discretion to declare zero [S1] [S7]. The reference parametrization:

Project the asset share to the horizon at the expected net return r_e = r_base − c_amc − c_g std — annual rates throughout, because the rate being set is annual — from the balance the declaration month opens with: AS_proj = AS(t−1) · (1+r_e)^(m) + future premiums accumulated to the horizon at r_e, with m = n − y(t) in years, the term less the policy year the declaration closes (CWP), or m = h = 10 (UWP whole-of-life bond).
Supportable rate: the level bonus rate that grows the guarantee to the guarantee-fill target θ = 80% of the projected asset share, measured on the guarantee the declaration month opens with:
- UWP: b_supp = [ θ·AS_proj / FV(t−1) ]^(1/m) − 1
- CWP: b_supp = [ θ·AS_proj / G(t−1) ]^(1/m) − 1
Smoothed declaration with the ±1% discipline [S1] [S7]: b(y) = max( 0, b(y−1) + clamp( κ·(b_supp − b(y−1)), −0.01, +0.01 ) ), κ = 0.5 std. The discipline is per declaration, so the rule is applied once a policy year, at that year’s declaration month, against the rate declared a year earlier — not once a month, which would be a different and far looser rule.

The base projection holds the snapshot rates (2.00% UWP / 1.50% CWP) level; the rule above is the revision module for scenario work.

Smoothed payout, final bonus, terminal bonus#

Raw target = the unsmoothed asset share (payout target 100% of asset share [S5] [S7] [S8] R1). Apply the smoothing cap, then the corridor, every month:

S_raw(t)  = AS(t)
S_cap(t)  = clamp( S_raw(t), (1−σ)^(1/12)·S(t−1), (1+σ)^(1/12)·S(t−1) )   σ = 10%  [S1]
S(t)      = clamp( S_cap(t), 0.80·AS(t), 1.20·AS(t) )                     [S1][R1]

The cap is the year-on-year ±σ discipline [S1] taken to its twelfth root std, so that twelve capped months move the payout by exactly ±σ over the policy year. That conversion is what preserves the rule’s meaning on a monthly grid: a flat ±σ per month would be twelve times as loose, and applying ±σ only at anniversaries would leave the eleven intervening payouts — on which real claims are paid — unsmoothed.

The corridor implements the 80–120% target range deterministically at model-point level; the ≥90%-of-policies test [S1] R1 is a portfolio property, out of scope for a single-policy model std.

UWP final bonus: FB(t) = max(0, S(t) − FV(t)); guarantee-event payout FV(t) + FB(t); death payout g_db · (FV(t) + FB(t)) [S5: no MVR on death].
CWP terminal bonus: TB(t) = max(0, S(t) − G(t)); maturity payout G + TB at the end of the last projected month, t = 12n − 1; death payout G(t) + interim accrual + FB per the same scale [S1] [S4] [S8].
When the guarantee bites (S(t) < FV(t) or S(t) < G(t)), the excess of the guaranteed payout over the asset share is charged to the smoothing/guarantee account within the estate [S1] [S4].

MVR (unitised, non-guaranteed exits)#

MVR(t) = min( max(0, FV(t) − S(t)),  max(0, FV(t) − AS(t)) )
Surrender payout = FV(t) + FB(t) − MVR(t)

The first argument recovers the smoothed-payout shortfall below face value (post-MVR payouts target 100% of asset share, here its smoothed image [S5]); the second is the COBS 20.2.16R bound — the MVR may not exceed the excess of unit value over the underlying asset value R1. Because FB > 0 requires S > FV and MVR > 0 requires S < FV, final bonus and MVR are never simultaneous (the rule observed in one consolidated with-profits fund [S4]; adoption product-spec (24)). MVR-free events: death [S5], guarantee dates [S4] [S5], withdrawals within the 5% allowance std (product-spec (13)).

A guarantee date is a date, and the monthly grid says so: an exit in month 12k − 1 for a guarantee anniversary k is MVR-free, and an exit in any of the other eleven months of that policy year bears the reduction like any other. An annual grid has to treat the whole year as the date.

Cost of bonus and shareholder transfer (90:10 mechanics)#

ST(t) = CB(t) / 9 — one-ninth of the cost of bonus, so that shareholders receive 10% of each 90:10 distribution (product-spec (2); components [S1] [S5] [S8] R1). Measurement of CB std:

Both arise in the declaration month and are nil in the other eleven.

UWP regular bonus: CB_reg(t) = b · FV(t−1) — the face-value uplift delivered by the declaration, on the face value that month opens with.
CWP reversionary bonus: CB_reg(t) = ΔG(t) · v_sv^(n−y(t)) with ΔG(t) = G(t) − G(t−1) — the declared addition discounted to the declaration date over the n − y(t) years still to run (survivorship discount omitted std simplification).
Final/terminal bonus: CB_fb(t) = (FB or TB paid on claims in month t), recognized at payment, so unlike the regular-bonus cost it arises in any month a claim does.

ST is a cash outflow from the fund (distribution to shareholders), reported separately in the model output; per COBS 20.2.17AR, adjustments reducing policyholder distributions below the required percentage require proportionate shareholder-transfer reductions R1 — modeled implicitly by tying ST to actually-declared/paid bonus.

Smoothing account#

On each exit, post the smoothing cost (payout − AS(t)) weighted by the exiting probability to SM(t) (within the estate). Intended broadly neutral over time [S1] [S2] [S5] [S6]; the base model tracks the balance without recycling. Optional module: year-end recycling into credited returns as one insurer operates it (maximum deduction currently 2.5% of asset shares p.a.) [S5].

Cost of guarantees — cited, not specified#

The deterministic charge c_g is a charging proxy, not a valuation. The economic cost of the guarantees (unit-price floor, guarantee-date face value, CWP sum assured plus hardened bonuses, GAO) requires stochastic market-consistent valuation: PRA Technical Provisions 9.2 requires guarantees and options to be valued with realistic dynamic assumptions R7, and the canonical methodology is market-consistent stochastic simulation of the bonus/smoothing/MVR rules (Hibbert & Turnbull 2003; Hare et al. 2000 R13). This model produces the per-scenario cash flows such a valuation consumes; the stochastic layer itself is out of scope.

GAO module (legacy flag)#

Where gao_flag is set (CWP pension cells), the retirement benefit is max( CashFund(T) · OMR(T), CashFund(T) · gao_rate ) — the guaranteed annuity rate floors the open-market conversion. GAOs are present in several closed funds, backed by fixed-interest assets, with interest-rate risk identified as a fund business risk [S4]; the 2000 GAO litigation history is unverified context. gao_rate = £0.09 p.a. per £1 of cash fund std [unverified as typical]; take-up per class (c). The GAO is a valuation-critical option (stochastic interest-rate exposure) — cited, not fully specified.

Cash flow outputs (per month t, probability-weighted by `l`)#

l(t) is the in force at the start of month t, so it is the weight on every flow of that month — the same row of the result table.

Output	Formula
Premium income	`P(t) · l(t)`
Death claims	`q_m · l(t) · DeathPayout(t)`
Surrender claims	`w_m · l(t) · (1 − q_m) · SurrenderPayout(t)`
Maturity claims	`l · (1 − q_m)(1 − w_m) · (G + TB)` — the survivors of the last projected month `12n − 1` (CWP)
Partial withdrawals	`W(t) · l(t)`
Maintenance expenses	`E(t) · l(t)`, a twelfth of the annual expense
Shareholder transfers	`ST(t) · l(t)` — nil outside a declaration month — plus `CB_fb/9` on claims, which arises whenever a claim does

Policyholder behavior modeling#

All dynamic formulas are std — no public UK with-profits lapse experience was retrieved; the shapes are rationalized from the product’s incentive structure, and dynamic option-exercise modeling is a regulatory expectation for the BEL R7.

Base surrender: UWP bond 5% p.a. flat; CWP 5%/4%/3%/2%+ (class (c) table). The table is keyed by the contractual policy year, the 1-based label, so month t reads row t // 12 + 1 and policy years past the table take its last row. The rate is annual and the projection decrements by w_m = 1 − (1 − w)^(1/12) std.
MVR deterrent: w(t) = w_base(t) · 0.6 while MVR(t) > 0 std — an active MVR penalizes exit, and firms may consider exit volumes in setting MVRs within the COBS bound R1 COBS 20.2.16AR. A diffuse tilt, so it multiplies the annual rate.
Guarantee-imminent suppression: w(t) = w_base(t) · 0.8 in the twelve months before a guarantee date std (waiting for the MVR-free window). Also a tilt on the annual rate.
Guarantee-date encashment: an additional ε = 7.5% std of the survivors of the guarantee-date month 12k − 1, and only when FV(t) > AS(t) (the guarantee is in the money), so that w_m(t) = 1 − (1 − w_m,ordinary(t))(1 − ε). MVR-free encashment is rationally exercised precisely in that state and worth nothing otherwise, so the gate is not optional: applying it unconditionally invents anti-selection where there is none.

This is the one place where the monthly grid changes an assumption’s shape rather than its frequency, and deliberately. On an annual grid the exercise could only be a × 2.5 multiplier on the whole guarantee-date year’s surrender rate std, which spreads MVR-free exits across eleven months in which the window is shut. Here it falls in the month the option is actually open, at a rate set so that a guarantee-date year still sheds about the proportion the annual multiplier shed. Neither figure is measured — no public UK with-profits experience was retrieved — and both are rationalized from the incentive structure alone.
Withdrawal utilisation: withdrawing bond cells take the full 5% MVR-free/tax-deferred allowance, a twelfth of it each month std; utilisation 30% of policies std (allowance context [S10] REG-R15).
GAO take-up: 90% when in-the-money by >10%, else 30% std unverified.
Paid-up conversion (CWP): excluded from base std; where modeled, benefits reduce per policy terms and future bonuses may or may not accrue [S4], and asset shares may need separate treatment for altered policies [S6].

Worked example#

Anchor UWP bond cell (product-spec (14)): £25,000 single premium; U = 25,000 units at a seed price of £1.0000; five declarations, one at the end of each of the first five policy years, give an opening price of 1.02^5 = 1.104081 and an opening face value of £27,602.02. The cell is in force at duration 5, so it is projected from t = 60 — the first month of its sixth policy year — and the asset share, the smoothed payout and the unit price are the balances that month opens with. Worked-example state std: AS = £30,000.00, S = £29,500.00. Parameters: c_amc = 1.00% p.a., c_g = 0.10% p.a., q(60) = 0.005 p.a. (illustrative of the class (c) proxy std), g_db = 1.01, σ = 10% year on year. No premium and no withdrawals in the year. Two return scenarios std: A: r = +7.0% p.a.; B: r = −15.0% p.a. (declared bonus cut to 1.00%, the maximum normal reduction [S1] [S7]).

The example projects the sixth policy year, months t = 60 … 71, and the declaration falls at the end of t = 71. Monthly rates:

Rate	Scenario A	Scenario B
`r_m = (1+r)^(1/12) − 1`	+0.565415%	−1.345195%
`c_amc,m = 1 − (1−c_amc)^(1/12)`	0.083718%	0.083718%
`c_g,m = 1 − (1−c_g)^(1/12)`	0.008337%	0.008337%
`q_m = 1 − (1−q)^(1/12)`	0.041762%	0.041762%
cap bounds `(1∓σ)^(1/12)`	0.9912584 / 1.0079741	0.9912584 / 1.0079741

Over the twelve months, and then the closing month t = 71:

Step	Quantity	Scenario A (r = +7.0%)	Scenario B (r = −15.0%)
0	Opening `AS` / `FV` at `t = 60`	30,000.00 / 27,602.02	30,000.00 / 27,602.02
3–4	AMC taken over the twelve months	311.10	274.93
4	Guarantee charge taken over the twelve months	30.98	27.38
6	Mortality charge taken over the twelve months	0.00	4.60
3–4	`AS` after return and charges in month 71	31,747.19	25,216.50
5	Declared bonus `b` for the policy year	2.00%	1.00%
5	`Q(71)`; `FV(71) = 25,000 · Q(71)`	1.126163; 28,154.07	1.115122; 27,878.05
5	Cost of bonus `CB = b · FV(70)`	552.04	276.02
5	Shareholder transfer `ST = CB/9`	61.34	30.67
5	Asset share after `ST`	31,685.86	25,185.83
6	`MC = q_m · max(0, 1.01·FV(71) − AS)` in month 71	0.00	1.24
6	`AS(71)`	31,685.86	25,184.59
7	`S_cap`: clamp against `S(70)` at the monthly bounds	31,685.86 (within)	26,846.96 (floor binds)
7	`S(71)`: corridor clamp to [0.8, 1.2]·AS	31,685.86	26,846.96 (within corridor)
7	Final bonus `FB = max(0, S − FV)`	3,531.79	0.00
7	`MVR = min(max(0, FV−S), max(0, FV−AS))`	0.00	min(1,031.08, 2,693.46) = 1,031.08
8	Guarantee-date payout `FV + FB` (no MVR)	31,685.86	27,878.05 (guarantee bites)
8	Surrender payout `FV + FB − MVR`	31,685.86	26,846.96
8	Death payout `1.01 · (FV + FB)`	32,002.72	28,156.83
8	Smoothing/guarantee cost on exit (payout − AS): guarantee-date / surrender	0.00 / 0.00	2,693.46 / 1,662.37

The Step column is the processing-order step, not the time index: the closing rows are month t = 71 and the three aggregate rows are sums over t = 60 … 71. The closing quantities are the row result_payout() publishes at t = 71; the intermediate steps are asset_share_at(71, …) and the per-claim exit costs are smoothing_cost_pp(71, kind). Note that the payout, the final bonus and the MVR exist in every month of the year, not only its last: a claim in month 65 is paid on month 65’s smoothed payout, which is what a monthly grid is for.

Checks: the asset share ends the year at 105.6% of its opening value in A and 83.9% in B; the smoothed payout ends at 107.4% and 91.0%, both inside the ±10% year-on-year band the monthly cap compounds to. Scenario B’s cap binds in eleven of the twelve months — not the twelfth, because the first month’s asset share was still above the floor — which is why the payout lands a little above the 90.0% an annual step would produce. The MVR (1,031.08) is below the COBS bound FV − AS = 2,693.46 R1; the guarantee-date exit pays full face value with the 2,693.46 excess over asset share borne by the estate’s guarantee/smoothing account [S1] [S4]. On the scenario A guarantee-date claim an additional shareholder transfer of FB/9 = 392.42 accrues at payment (90:10 on the final bonus, ST section). Scenario A pays 100.0% of AS(71); scenario B’s surrender pays 106.6% — both within the 80–120% corridor [S1] R1.

CWP maturity illustration (one line): the twenty-five declarations of a 25-year endowment close at the end of month 12n − 1 = 299, where G(299) = 20,000 · 1.015^25 = £29,018.91 — one declaration per policy year and twenty-five of them, which is the arithmetic a monthly implementation gets wrong if it compounds b monthly. With smoothed maturity target S = £34,000.00 std, TB = 34,000.00 − 29,018.91 = £4,981.09 — 14.7% of the payout in non-guaranteed form, consistent with the substantial-final-bonus philosophy [S1]; the associated shareholder transfer at payment is TB/9 = £553.45 std measurement. (The shipped endowment cell’s own projection reaches a smoothed payout below the guarantee, so the guarantee bites and the terminal bonus is nil; the £34,000 above is the notes’ illustration of the mechanic, not that cell’s output.)

Valuation and reserve pointers#

This library projects gross best-estimate liability cash flows; valuation layers are cited, not reproduced.

Solvency UK BEL. Technical provisions = best estimate + risk margin; the best estimate is the probability-weighted, discounted value of all cash flows R7 REG-R1. For with-profits, the BEL includes future discretionary benefits — future regular and final bonuses expected under PPFM-consistent discretion — because expected payments count “whether or not … contractually guaranteed”, with the surplus-funds carve-out for the unallocated estate R7 R8. The With-Profits Actuary must advise whether the FDB assumptions are consistent with the PPFM R5. Guarantees and options (unit-price floors, guarantee dates, GAOs) must be valued market-consistently with dynamic policyholder behavior R7 — stochastic-on-deterministic use of this model.
Risk margin. Post-reform cost-of-capital method: CoC 4%, risk taper λ = 0.9 (floor 0.25) for long-term business R7 REG-R4. Cited-not-specified.
Ring-fencing and estate. With-profits fund assets must cover the fund’s liabilities R6; surplus funds (the estate) are own funds, excluded from technical provisions R8. TMTP may apply to pre-2016 back-books R7 REG-R3.
Matching adjustment. The guaranteed element of a with-profits immediate or deferred annuity can qualify as an MA “eligible element” REG-R2 — relevant only to the annuity variations, not the composite cells.
IFRS 17. UK-adopted IFRS 17 (effective 1 January 2023) applies to IFRS-reporting insurers REG-R38; with-profits contracts are direct-participation business measured under the variable fee approach [unverified — standard text not fetched]. The fulfilment-cash-flow engine is this same projection.
Conduct overlay. Payout machinery in any valuation must respect the COBS target-range, MVR-bound and required-percentage rules R1 — they are constraints on the FDB discretion, not just conduct background.

Key sensitivities and model risks#

Fund return / equity backing. Asset shares, final bonuses and MVR incidence all key off r(t); the observed strategy ceiling is a benchmark equity backing ratio of 75% (one insurer’s EBR upper limit [S5]). Deterministic base runs materially understate guarantee costs (convexity) — the central model risk here R7 R13.
Bonus discretion path. The split of payout between hardened regular bonus and final bonus changes guarantee costs without changing the target payout: a higher θ or faster κ hardens guarantees. The std parametrization is a genuine modeling choice with no public calibration.
Smoothing parameters. The ±10% cap and 80–120% corridor determine how much of a market shock passes to payouts immediately; firms’ actual limits vary (5%–15% observed [S1] [S5] [S7]) and can be suspended under solvency stress [S5].
MVR application. Whether the discretion is exercised promptly (and the review buffer — one consolidator tolerates up to 10% return variation before an extra MVR review [S4]) drives surrender strain in down markets.
Surrender behavior at guarantee dates. The guarantee-date spike multiplier and MVR deterrent are unverified std shapes; anti-selective exit when guarantees are in the money is the dominant behavioral risk (dynamic assumptions required R7).
Mortality proxy. The 60%-of-ONS basis is a placeholder; insured with-profits experience differs by class and era, and current CMI tables are subscriber-restricted R10 REG-R22 REG-R32.
Expense and charge caps. Where actual expenses exceed capped charges (1% caps [S1] [S5]) the excess falls to the estate — a fund-level, not policy-level, cash flow this single-policy model does not capture.
GAO interest-rate exposure. Legacy GAO cells are long interest-rate optionality [S4]; omitting the stochastic layer understates their cost materially.
Estate interactions. Reattributions, special bonuses and mutual profit distributions [S5] [S6] are fund-level discretions outside the base model; scenario overlays should treat them as management actions.
Data-provenance limits. Snapshot bonus rates, EGRs and MVR scales are std placeholders by design (declarations are not in PPFMs — research gap); a calibration pass against current bonus declarations is required before any quantitative use.