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/pension-annuity.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. Parameter values are identical to those in product-spec.md; the mechanics anchor is one carrier’s pension annuity [S1] [S2].

Model scope and conventions#

  • Purpose. Project gross best-estimate liability cash flows (annuity instalments to annuitant and dependant, guarantee-period payments, value-protection lump sums, maintenance expenses) for a single pension annuity in payment. Discounting, the matching adjustment and reserves are not computed (see Valuation and reserve pointers).

  • Mortality is the model. The contract has no premiums after outset [S2 §1.1], no surrender value [S2 §12] [S5 cl.14.7], no account value and no policyholder options after the cancellation window [S1 p4]. The only decrements are deaths; the only stochastic drivers are longevity and (for indexed options) inflation. This is the design property that makes the liability MA-eligible R1.

  • Projection frequency. Monthly grid, t = 1, 2, … months from the start date std. Payment dates fall on the grid per the frequency m; exact-day mechanics (one carrier’s first-of-month payments and stub proportioning [S5 §§5.2–5.3], another’s working-day adjustment [S2 §2.4]) are not modeled std.

  • Timing conventions std. Escalation is applied at the start of the month containing the policy anniversary (first at t = 13) [S2 §3.3]. Advance instalments are paid at the start of a payment period and require survival at the start; arrears instalments at the end, requiring survival at the payment date. Deaths are decremented at end of month; a death in month t means the life does not receive an arrears payment due at the end of month t [std convention].

  • Age basis. Age last birthday (ALB) std, chosen to index the std ONS life-table proxy by single year of age R13; the ONS convention itself is unverified. Annual rates convert monthly as q_m = 1 − (1 − q_x)^(1/12) std.

  • Limiting age. ω = 115 std: the std base table is extended beyond its maximum tabulated age by log-linear extrapolation of qx, capped at 1 at ω.

  • Currency and model points. GBP throughout [S2 §1.3]. Single-policy model points, projected on an expected (probability-weighted) basis: survival probabilities multiply scheduled per-policy cash flows. No aggregation logic is specified here.

  • Joint-life independence. Annuitant and dependant mortality are independent std (common-shock/”broken-heart” dependence is a documented model risk).

Model point attributes#

Attribute

Type

Example (worked configuration)

purchase_price P

currency

100,000 [S1 p11]

annuitant_age x_a

int (ALB)

65 [S1 p11]

annuitant_sex

enum {M, F}

M std

rating_multiplier θ_a

float ≥ 1 (1 = standard; enhanced overlay)

1.0 std

dependant_present

bool

true

dependant_age x_d

int (ALB)

62 std

dependant_sex

enum {M, F}

F std

dependant_pct δ

float ≤ 1 [S1 p9]

0.50 std

overlap

bool (with/without overlap [S2 §§5.9–5.11])

false std

annual_income A(1)

currency p.a.

5,400 std (see Worked example)

frequency m

enum {12, 4, 2, 1} [S2 §2.2]

4

timing

enum {advance, arrears} [S2 §2.3]

arrears

proportion

bool (arrears only [S2 §4])

false

escalation_type

enum {level, fixed, rpi_catchup, lpi5} (spec menu)

fixed

escalation_rate g

float ≤ 0.10 [S2 §3.2]

0.03 std

guarantee_months n

int, 12–360, 0 if none [S1 p10]; XOR with VP [S2 §§6.7, 7.6]

0

vp_pct v

float ≤ 1 [S1 p11]; v + δ ≤ 1 on first-death basis [S2 §7.3]

0.50 std

vp_basis

enum {first_death, last_survivor} [S2 §7.3]

first_death

The premium P is the amount applied to the annuity after PCLS and adviser charges [S1 p4]; PCLS itself is pre-purchase and outside the model. A(1) is a pricing input: no insurer publishes a rate card, so A(1) is taken from a quote or calibrated to the anchor (£100,000 at 65 buying £6,657 p.a. with 50% VP, January 2026 [S1 p11]; the illustration’s frequency/timing/escalation basis is not recorded).

State variables#

Variable

Description

Updated

A(y)

Annualized income in policy year y (annuitant scale)

anniversaries

peak

Running peak of the RPI reference index (catch-up state)

anniversaries (rpi_catchup only)

G(t)

Cumulative gross instalments scheduled through month t

payment dates

l_a(t)

Annuitant survival probability to end of month t; l_a(0) = 1

monthly

l_d(t)

Dependant survival probability to end of month t; l_d(0) = 1

monthly

d_a(t)

Probability annuitant dies in month t = l_a(t−1) − l_a(t)

monthly

n_rem(t)

Remaining guarantee months = max(0, n − t)

monthly

VPbal(t)

Value-protection balance = max(0, v × P − G(t))

payment dates

Because instalments while the annuitant is alive are deterministic given the escalation path, G(t) and VPbal(t) are deterministic schedules in a deterministic projection — the expected VP outgo needs no path simulation (see recursions).

Assumption inputs#

Three classes are distinguished explicitly.

(a) Contractual / guaranteed elements (cited)#

Input

Value

Basis

Instalment amount

A(y)/m at each payment date

[S1 p8] [S2 §2.2]

Escalation rule

per escalation_type: fixed g ≤ 10%; RPI 0-floor with catch-up (12 months ending six months before the anniversary); LPI = RPI capped 5%, floor 0, September year

[S2 §3.2, §3.3, defs]; LPI floor harmonization std (spec footnote 5)

Dependant’s income

δ × income, same escalation basis; % of the higher of income at death and at guarantee end

[S2 §§5.12–5.13]

Overlap rule

with: dependant stream runs during remaining guarantee; without: starts at guarantee end

[S2 §§5.9–5.11]

Guarantee period

n months of instalments certain, escalation continuing as if alive

[S2 §§6.5–6.6] [S7 §4.2]

Value protection

max(0, v × P − G(death)) on the chosen basis; v + δ ≤ 1 (first-death)

[S1 p11] [S2 §7, §7.3]

Surrender value

none, at any time

[S1 p4] [S2 §12] [S5 cl.14.7]

Charges to policyholder

none (priced into the rate)

[S1 p6]

(b) Insurer-discretionary current elements#

None post-purchase. The contract is non-participating [S7 §7.9] with all options fixed at outset [S1 p4]: there are no bonus rates, no reviewable premiums, no market value reductions, and no discretionary charges — class (b) is empty for this product. The only insurer-discretionary quantity is the annuity rate at purchase (pricing, not an in-force element); its snapshot is the January 2026 anchor quote [S1 p11], and day-to-day rate setting is not publicly documented unverified.

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

Input

Recommended basis

Basis tags

Base annuitant mortality

Proper bases: SAPS S3/S4 pensioner tables (S4 released February 2024, graduated on 2014–2019 data) R10 R11 or the insured-annuitant PMA16/PFA16 family REG-R27. Both are restricted to CMI Authorised Users R11 REG-R22, so the reference basis is a std proxy: latest ONS UK national life table qx by age/sex R13 × annuitant adjustment α = 0.80

R10 R11 R13 REG-R22 REG-R27; α std (i)

Mortality improvements

CMI Mortality Projections Model, cited by name/version: CMI_2024 (WP201, June 2025, calibrated to E&W data to 31 Dec 2024) R12; current version CMI_2025 (WP211, March 2026) REG-R30. Model software restricted; reference fallback is a std deterministic scale: 1.25% p.a. reduction in qx for ages ≤ 90, tapering linearly to 0% at age 110, applied from the base table’s data mid-year

R12 REG-R30; scale std (ii)

Enhanced/impaired rating

Overlay on qx: q_rated = min(1, θ_a × q_base), θ_a ≥ 1 (equivalently a rated-age offset); standard life θ = 1.0

existence [S1 p5] [S4] [S6] [S9]; overlay std (iii)

Lapse / surrender

None — no surrender value exists

[S1 p4] [S2 §12] [S5 cl.14.7] R1

Maintenance expense

£30 per policy per annum, payable monthly while any payment obligation remains, inflating at the RPI assumption

std (iv)

RPI inflation (for indexed options)

3.0% p.a. deterministic

std (v)

(i) The SAPS table naming convention (e.g. S3PMA/S3PFA) is [unverified — not stated on the fetched page] R10. ONS national life tables are period tables of population mortality, freely downloadable and updated annually (latest release dated 10 December 2025 per the fetched dataset page) R13; population mortality is heavier than annuitant experience, hence the α < 1 adjustment. α = 0.80 is a shape-level placeholder, not calibrated to any published annuitant-vs-population comparison — a production basis must license CMI tables R11 REG-R22. (ii) CMI_2025 projects improvements converging to a user-chosen long-term rate with no default recommendation REG-R30 detail marked unverified in the reference library; the std flat-then-taper scale exists only so the reference implementation is runnable without CMI access, and materially understates the age–period–cohort structure of the real model R12. (iii) Insurers’ rating structures (postcode, condition-specific factors [S1 p5] [S9]) are not public; the multiplier form is the simplest overlay that reprices longevity without touching contract mechanics. (iv) No insurer publishes expense assumptions (charges are priced into the rate [S1 p6]); £30 p.a. is a round placeholder for in-payment administration. Acquisition cost is out of scope (single-premium, priced-in). (v) Deterministic RPI cannot value the RPI floor, the catch-up ratchet, or the LPI cap — all inflation options. See Key sensitivities.

Cash flow components and recursions#

Notation (defined once, used throughout)#

Symbol

Meaning

t

month index from start date, t = 1, 2, …; policy year y = ceil(t/12)

m

payments per year (12/4/2/1); payment months T = {12k/m : k = 1, 2, …} (arrears) or {12k/m : k = 0, 1, …} mapped to the start of month 12k/m + 1 (advance)

A(y)

annualized income in policy year y; inst(t) = A(y(t))/m for t ∈ T

g

fixed escalation rate (0.03 std, ≤ 0.10 [S2 §3.2])

I(k), peak

RPI reference index at anniversary k and its running maximum (catch-up state) [S2 defs]

δ

dependant’s percentage (0.50 std, ≤ 1 [S1 p9])

n

guarantee period in months (0 or 12–360 [S1 p10])

v

value-protection percentage (0.50 std, ≤ 1 [S1 p11]); v + δ ≤ 1 on first-death basis [S2 §7.3]

P

purchase price (100,000 [S1 p11])

G(t)

cumulative gross instalments scheduled through month t

q_a(t), q_d(t)

monthly mortality of annuitant/dependant (rated, improved)

l_a(t), l_d(t)

survival probabilities from outset; d_a(t) = l_a(t−1) − l_a(t)

w(t)

dependant-stream availability: 1 if overlap or t > n, else 0 [S2 §§5.9–5.11]

c_e, π

maintenance expense p.a. (30 std) and expense/RPI inflation (0.03 std)

Dimensional check: A(y) is currency per annum; inst = A/m is currency per payment; G, P, VP lump sums are currency; q, l, δ, v, w are dimensionless. Every cash flow below is currency per month.

Escalation update (start of month 12(y−1)+1, y ≥ 2) [S2 §3.3]#

level:        A(y) = A(y−1)
fixed:        A(y) = A(y−1) × (1 + g)
lpi5:         A(y) = A(y−1) × (1 + min(0.05, max(0, rpi_Sep(y−1))))      [S2 §3.2, defs; floor [S5 §7.1.4][S9 §4.3]]
rpi_catchup:  see pseudocode                                              [S2 defs]

RPI catch-up pseudocode (path-dependent ratchet [S2 defs]; a second carrier operates the same rule [S9]):

# I[k] = RPI reference level for anniversary k
# (index for the 12 months ending six months before the anniversary [S2 defs])
peak = I[0]                      # reference level at outset
for k = 1, 2, ...:               # k-th anniversary
    if I[k] > peak:
        A = A * (I[k] / peak)    # increase by the excess over the prior peak
        peak = I[k]
    # else: A unchanged (income frozen until the index exceeds its peak)

Equivalently A(y) = A(1) × max(I(0..y−1)) / I(0): income is indexed to the running peak of the reference index. Under the deterministic RPI assumption (3.0% std) the index is monotone and the ratchet never binds, so rpi_catchup degenerates to fixed-3%; the ratchet has value only under stochastic inflation (see sensitivities).

Scheduled payment schedule (per policy, before survival weighting)#

At each payment month t ∈ T: scheduled annuitant instalment inst(t) = A(y(t))/m; scheduled dependant instalment δ × inst(t). Update G(t) = G(t−) + (instalments scheduled at t). The dependant’s amount uses δ × the income “as if alive” A(y(t)): this implements the contractual “% of the higher of income at death and income at guarantee end” [S2 §5.12] exactly, because under the (non-decreasing std menu) escalation options the as-if-alive income path is monotone, so the higher-of base plus same-basis escalation [S2 §5.13] reproduces δ × A(y(t)) at every later date.

Expected cash flows (month t)#

Annuity outgo (annuitant stream with its guarantee floor, plus dependant stream), for t ∈ T (arrears; for advance replace l(t) with l(t−1) std):

E[ANN(t)] = inst(t) × max(1{t ≤ n}, l_a(t))                — certain during guarantee [S2 §6]
          + inst(t) × δ × (1 − l_a(t)) × l_d(t) × w(t)     — dependant stream [S2 §5]

The first term pays the full instalment regardless of survival while the guarantee runs (annuity-certain floor [S2 §§6.5–6.6] [S7 §4.2]) and l_a(t) × inst(t) thereafter. The second term pays the dependant when the annuitant is dead and the dependant alive, gated by w(t): with overlap both streams run during the remaining guarantee; without overlap the dependant stream starts at guarantee end [S2 §§5.9–5.11]. (Guarantee and VP never coexist in the representative design: n > 0 ⇒ v = 0 [S2 §§6.7, 7.6].)

Proportionate final payment (arrears with proportion only [S2 §4]): for a death in month t, the accrued stub to the next scheduled instalment is approximated as

E[PROP(t)] = d_a(t) × (h(t) + 0.5) / (12/m) × inst(next(t))   **[std half-month accrual]**

where h(t) is the number of complete months since the last payment date. Without proportion (representative default) this term is zero and nothing is paid for the final partial period [S2 §4].

Value protection (first-death basis; n = 0):

E[VP(t)] = d_a(t) × VPbal(t−1),   VPbal(t) = max(0, v × P − G(t))   [S1 p11][S2 §7]

G accumulates gross instalments scheduled while the annuitant is alive; measuring the balance at t−1 implements “instalments already paid” for a mid-month death [std discretization]. On the last-survivor basis, replace d_a(t) with the density of the last death, d_last(t) = d(l_a + l_d − l_a l_d)(t), and let G accumulate the dependant’s instalments too [S2 §7.3] [S5 §8.4]. (One carrier’s variant additionally nets guarantee payments due, excluding future RPI/LPI increases [S7 §4.3] — implementable by extending G with guarantee outflows.)

Maintenance expense:

E[EXP(t)] = (c_e / 12) × (1 + π)^(y−1) × IF(t)                       **[std]**
IF(t) = min(1, max(1{t ≤ n}, l_a(t)) + 1{δ>0} × (1 − l_a(t)) × l_d(t))

IF(t) is the probability any payment obligation remains (guarantee certain, annuitant alive, or dependant stream in payment) std.

Total gross liability cash flow: CF(t) = E[ANN(t)] + E[PROP(t)] + E[VP(t)] + E[EXP(t)]. There is no premium income (single premium at t = 0 is a pricing input, not projected [S2 §1.1]) and no surrender outgo [S2 §12].

Mortality construction#

q_base(x, s)   = ONS qx by age/sex [R13] × α,  α = 0.80              **[std]** (proxy for SAPS S4 [R10][R11] / PMA16-PFA16 [REG-R27])
q_imp(x, c)    = q_base(x) × (1 − f(x))^(c − c_0)                    **[std]** improvement fallback (f = 1.25% p.a. ages ≤ 90, linear taper to 0 at 110; c_0 = base-table data mid-year; production: CMI_2025 with a chosen long-term rate [R12][REG-R30])
q_rated(x, c)  = min(1, θ × q_imp(x, c))                             **[std]** enhancement overlay
q_m            = 1 − (1 − q_rated)^(1/12)                            **[std]**
l(t)           = l(t−1) × (1 − q_m(t)),  separately for annuitant (θ_a) and dependant (θ_d)

Monthly processing order#

  1. If t starts a policy year (t = 12(y−1)+1, y ≥ 2): apply the escalation update (including catch-up state) [S2 §3.3].

  2. If t ∈ T: record scheduled instalments; update G(t).

  3. Decrement mortality: update l_a(t), l_d(t), d_a(t).

  4. Compute expected payment flows E[ANN(t)], E[PROP(t)] using survival to the payment point (arrears: end of month t, i.e. l(t); advance: end of month t−1, i.e. l(t−1)) std.

  5. Compute E[VP(t)] from d_a(t) and VPbal(t−1); update VPbal(t).

  6. Accrue E[EXP(t)].

  7. Stop when IF(t) < 10^-6, or when every in-scope life has passed the limiting age (t/12 + x_a > ω and, if a dependant is present, t/12 + x_d > ω), ω = 115 std — stopping on the annuitant’s age alone would truncate a younger dependant’s tail.

Policyholder behavior modeling#

There is none to model, and this is a cited product feature, not an omission: after the 30-day cancellation window the policyholder holds no options — no surrender or transfer [S1 p4] [S2 §12] [S5 cl.14.7] [S7 §7.5] [S9 §3.9], no alteration of options [S1 p4] [S4] [S6] [S9], and no premium flexibility [S2 §1.1]. Consequently the model has no lapse decrement and no dynamic behavior formulas; the MA eligibility conditions effectively require this shape (no policyholder options beyond a bounded surrender option) R1.

Behavior enters only at outset, outside the projection, as basis-selection effects std to consider when calibrating mortality:

  • Annuitization anti-selection. Since the 2015 pension freedoms annuitization is optional R6, so voluntary annuitants self-select for longevity — a reason annuitant bases sit below population mortality (the direction of α < 1 std).

  • Enhanced-annuity selection. Whole-market enhanced quoting is mandated at the point of sale R5; lives remaining on standard terms are healthier on average. The reference model carries this through θ, not through behavior dynamics.

  • Cancellation window. The 30-day cooling-off [S1 p7] [S2 §13] is ignored (projection starts from a completed purchase) std.

Worked example#

Configuration (the worked model point; parameters as in product-spec.md): P = £100,000 [S1 p11]; annuitant male 65, dependant female 62 std; quarterly (m = 4) in arrears, without proportion [S2 §§2.2–2.3, 4]; fixed escalation g = 3% std; dependant δ = 50% std; value protection v = 50% on the annuitant’s (first) death std — v + δ = 100%, exactly at the contractual bound [S2 §7.3]; no guarantee period (XOR rule [S2 §§6.7, 7.6]). Starting income A(1) = £5,400 p.a. std — an illustrative quote level (no public rate card exists; the cited anchor, £6,657 p.a., is for a 50%-VP basis whose escalation/frequency basis is not recorded [S1 p11], and an escalating joint-life basis starts lower than a level one for the same premium [S1 p8] [S4] [S6]). Scenario: the annuitant dies in month 17; the dependant survives throughout. All amounts in GBP.

Instalments: year 1: 5,400/4 = 1,350.00 per quarter; year 2 (from t = 13): A(2) = 5,400 × 1.03 = 5,562.00, so 1,390.50 per quarter. Dependant income after death: δ × A(2) = 2,781.00 p.a. = 695.25 per quarter, first paid at the next scheduled payment date after death (t = 18) [std convention].

t (month)

Event

Annuitant CF

Dependant CF

VP lump sum

G(t)

3

Q1 instalment (arrears)

1,350.00

—

—

1,350.00

6

Q2 instalment

1,350.00

—

—

2,700.00

9

Q3 instalment

1,350.00

—

—

4,050.00

12

Q4 instalment

1,350.00

—

—

5,400.00

13

Anniversary: A ← 5,400 × 1.03 = 5,562.00

—

—

—

5,400.00

15

Q5 instalment

1,390.50

—

—

6,790.50

17

Annuitant dies. VP = max(0, 0.50 × 100,000 − 6,790.50)

—

—

43,209.50

6,790.50

18

Q6 date: no annuitant payment (arrears, without proportion [S2 §4]); dependant stream starts

0.00

695.25

—

7,485.75

21

Q7 instalment (dependant)

—

695.25

—

8,181.00

24

Q8 instalment (dependant)

—

695.25

—

8,876.25

Checks. VP balance at death uses instalments paid before death: G(16) = 6,790.50, so the lump sum is 50,000 − 6,790.50 = 43,209.50 [S1 p11] [S2 §7]. Had “with proportion” been chosen, a stub of ≈ (1 + 0.5)/3 × 1,390.50 = 695.25 would be paid for the accrued month-and-a-half since t = 15 (std half-month accrual; one carrier would net this stub off the VP fund-value formula [S5 §8.3]). The dependant’s 695.25 continues for her life, escalating 3% at each anniversary on the same basis [S2 §§5.12–5.13].

Guarantee/VP interaction. Had the model point instead carried a 10-year guarantee [std default] and no VP (the XOR rule forbids both [S2 §§6.7, 7.6]), the death in month 17 would change nothing until month 120: instalments of 1,390.50, escalating 3% each anniversary as if the annuitant were alive [S7 §4.2], continue to beneficiaries through t = 120 (annuity-certain floor), and — without overlap — the dependant’s 695.25-style stream would begin only from the first payment date after t = 120, at δ × the income at the end of the guarantee period [S2 §§5.9–5.12]. With overlap, the dependant’s stream would run from t = 18 alongside the guarantee payments [S2 §§5.9–5.11]. In expectation these scenario flows are reproduced by the E[ANN(t)] formula with n = 120 and w(t) as defined.

Valuation and reserve pointers#

This library projects gross best-estimate liability cash flows; valuation layers consume them and are NOT reproduced here:

  • Solvency UK best estimate. Technical provisions = best estimate + risk margin; the best estimate is the probability-weighted average of future cash flows discounted at the relevant risk-free term structure, market-consistently REG-R1. The CF(t) vector above is exactly that input.

  • Matching adjustment. These cash flows feed MA discounting (risk-free + MA) for eligible portfolios: MA permission required; eligibility conditions include no future premiums, restricted underwriting risks, the ≤ 5% BEL mortality-stress test, and no policyholder options R1 REG-R2. Reform context: CP19/23 → PS10/24, effective 30 June 2024 R2 REG-R5; supervisory expectations and matching tests in SS7/18 (October 2025 version) REG-R8. The MA calculation itself is cited-not-specified.

  • Risk margin. Cost-of-capital method at 4% with life-business tapering λ = 0.9 (floor 0.25) per SI 2023/1346 REG-R4; requires an SCR runoff — cited-not-specified.

  • Transitionals. TMTP (simplified regime from 31 December 2024) affects pre-2016 back-books only; it adjusts technical provisions, not projected cash flows REG-R3.

  • IFRS 17. UK-adopted IFRS 17 (adopted 16 May 2022, effective 1 January 2023) REG-R38 measures the same contracts as fulfilment cash flows plus risk adjustment plus CSM (measurement mechanics summarized from general knowledge — unverified in the reference library, which verifies the adoption facts only); the expected-cash-flow engine is identical, with regime-specific discounting and margins layered on.

  • Tax. Pension annuities are pension business — non-BLAGAB, trade-profit basis REG-R17 [S5 §14.11]; no policyholder fund tax enters the projection.

  • Professional standards. Technical actuarial work using this model in the UK falls under FRC TAS 100 v2.0 REG-R33 and TAS 200 v2.0 (effective 1 January 2025) R14. Proxy models fitted on top of heavy annuity cash-flow models — and the outputs the heavy model must expose for them — are treated in the IFoA proxy-model working party paper REG-R36.

Key sensitivities and model risks#

Dominant assumptions, in order:

  1. Longevity level (base table × α × θ). The liability is a life-contingent payment stream with no offsetting decrements; a lower mortality level lengthens every annuity stream. The std α = 0.80 population-proxy adjustment is the weakest link in the reference basis — production work must substitute licensed SAPS S4 / PMA16-era tables R10 R11 REG-R27 REG-R22.

  2. Longevity trend (improvements). The std deterministic scale stands in for CMI_2025 REG-R30; the choice of long-term improvement rate is the single most sensitive judgment in UK annuity valuation, and the CMI model’s user-set long-term rate has no default recommendation REG-R30, detail unverified. The prescribed MA mortality stress (worse of +15% level / +0.15pp additive, ≤ 5% BEL movement) R1 gives a regulatory yardstick for level-risk materiality.

  3. Inflation exposure (RPI/LPI options). RPI-linked instalments make the liability an inflation swap; the 0-floor, catch-up ratchet and LPI 5% cap are inflation option positions [S2 §3.2, defs]. A deterministic 3% path std values them at intrinsic only: the floor and ratchet never bind and the cap never pays off — stochastic inflation (or option-adjusted margins) is required for a market-consistent value. RPI reform risk (index definition) is additional and not modeled.

  4. Dependant assumptions. δ, the age gap, and dependant mortality drive the joint-life tail; the independence assumption std ignores broken-heart dependence and common lifestyle factors, overstating the expected dependant stream modestly.

  5. Expense inflation. Second-order (expenses are small against instalments), but the in-payment term is 30+ years, so the π assumption compounds.

Known modeling pitfalls:

  • Guarantee double-counting. During the guarantee, the annuitant stream is certain — do not also weight it by l_a(t) (the max(1{t≤n}, l_a) form prevents paying 1 + l_a). Symmetrically, VP and guarantee never coexist in the representative design [S2 §§6.7, 7.6]; engines supporting the combinable variant offered by one carrier must net guarantee payments off VPbal [S7 §4.3] or the death benefit is double-paid.

  • Overlap gating. Without overlap the dependant stream is gated on t > n even when the annuitant died mid-guarantee; applying δ from the death date silently converts every without-overlap policy into the more expensive with-overlap form [S2 §§5.9–5.11].

  • Higher-of dependant base. The δ × A(y(t)) simplification relies on non-decreasing escalation; if a decreasing option is configured (one carrier’s pure RPI [S5 §7.1.2]), the contractual “higher of income at death and at guarantee end” [S2 §5.12] must be implemented explicitly.

  • Survival-measurement timing. Arrears payments require survival at the payment date; advance payments at the period start. Using end-of-period survival for advance payments understates the liability by roughly one period’s mortality per payment — material at high ages.

  • Catch-up state. The RPI ratchet is path-dependent: peak must persist across anniversaries. Resetting it each year turns the catch-up into a plain 0-floor and overstates indexed income after deflation-recovery paths [S2 defs].

  • Escalation timing. Increases apply on the anniversary [S2 §3.3], not on payment dates; applying the year-2 rate to the t = 12 arrears instalment (accrued in year 1) overstates income. GMP-bearing policies use different escalation dates (1 April / 1 May at one carrier [S5 §7.2]) — out of scope with GMP generally std.

  • VP balance timing. VPbal must net instalments paid before death; netting the instalment due at the death-month payment date that was never paid (arrears, without proportion) understates the lump sum [S2 §§4, 7]. Symmetrically, on advance timing an instalment paid at the start of the death month has been paid: in advance payment months net it (use VPbal after the month-t advance payment, not VPbal(t−1)) or the lump sum is overstated by one instalment.

  • Population-proxy basis risk. The std ONS × α basis has the wrong shape as well as level versus annuitant tables (socio-economic mix, amounts weighting R10 R11 detail unverified); treat all reference-basis results as mechanics demonstrations, not valuations — and note the CMI restriction honestly rather than shipping approximated “SAPS-like” rates REG-R22.