The Wealth Delta Tax: Valuing Wealth — Worked Examples Appendix
Wealth Delta Tax, wealth taxation, wealth valuation, self-assessment, valuation examples, declaration incentives, understatement, overstatement, basis effects, tax deferral, Route C, Route D, auction mechanism, worked examples
Revision History
| Revision | Date | Details |
|---|---|---|
| 0.01 | 25 July 2026 | First Draft |
| 1.01 | 15 August 2026 | Published to website |
| 1.02 | 29 August 2026 | §M.1 clarifying paragraph added distinguishing voluntary hard-reset from corrective auction and stating lock-point rule; §M.3 no-bid outcome paragraph added for voluntary hard-reset pathway; §M.6 final sentence replaced to distinguish corrective and voluntary pathways and their respective refund treatment; §O summary table §M row updated to reflect three-pathway classification |
| 1.03 | 30 August 2026 | TW refined to TW_settled throughout (post-sale oscillation now included in terminal figure); Table J.1, K.1, N.1 row headers and captions updated; §N.3.2 prose updated to reflect Founder C’s +4.97% TW_settled outcome correctly; §N.4 revised to distinguish Founder B (forecast wrong, worse outcome) from Founder C (forecast-consistent at low growth, better outcome) and note consistency with VAL.A §A.6 mild-overstatement equilibrium |
Reader’s Guide
Five worked examples, each tethered to a specific claim in (VAL). They are not a comprehensive catalogue of asset classes or administrative scenarios; they exist to make five arguments concrete.
Appendix sections are lettered (VAL.B §J) through (VAL.B §N), continuing from (VAL.A §F). Cross-references from (VAL) and (VAL.A) use the form “(VAL.B §J)”, “(VAL.B §K)”, and so on, to avoid ambiguity with VAL.A sections.
The five claims, and the examples that illustrate them, are:
| Appendix | Claim in VAL | Asset |
|---|---|---|
| §J: The deferred delta | A declared value establishes the basis; understatement defers, it does not eliminate | Listed equity portfolio (Route C) |
| §K: Dilution compounds with growth | The must-transfer rule creates a cost that tracks the asset’s trajectory | Founder stake, software company (Route C) |
| §L: Why Route D defers to realisation | Periodic cash settlement on illiquid assets would recreate forced-realisation pressure; in-kind settlement is impossible | Sculpture collection (Route D) |
| §M: Voluntary settlement gives certainty, not avoidance | Soft and hard basis resets are trade-offs, not escape hatches | Commercial property (Route D) |
| §N: Honest declaration as the strategy with no forecast exposure | Neither understatement nor overstatement is a dominant strategy; both require being right about the future to avoid a penalty | Private company — three founders, same asset (Route C) |
All figures are from Python model v1.0, validated against Excel 27 July 2026. §L and (VAL.B §M) use closed-form arithmetic rather than run_val_sim; see individual section model notes for detail. All examples assume RATES-aligned parameters (\(\tau_0\) = 15%, \(\tau_m\) = 70%, \(W_{min}\) = £2m) and a marginal rate of approximately 15% at the relevant net worth level (\(k\) = 0.001, \(V_0\) at or near £8–20m) unless stated otherwise. All values in £m unless stated otherwise.
J. The Deferred Delta
Claim illustrated: A declared value establishes the recognised basis from which all future changes are measured. Understatement reduces current tax but enlarges the future delta; the deferral is recovered in full at realisation. (VAL §1), (VAL.B §J)
J.1 Setup
Taxpayer holds a FTSE 250 equity portfolio, continuously priced and observable, entering the WDT at a true value of £20m on Route C with a one-year assessment window.
Despite the asset being automatically priced, this example uses a self-declared value to isolate the basis-gap mechanism. In practice, listed equities under Route C are assessed at market price.
The portfolio grows at \(g\) = 7% per year. The taxpayer holds for five years then sells at true value. Three positions are compared: honest declaration (\(\alpha\) = 1.0), moderate understatement (\(\alpha\) = 0.8), and significant understatement (\(\alpha\) = 0.5), each held throughout.
J.2 The Mechanism
Each year the taxpayer declares \(W_t = \alpha \cdot V_t\); the delta runs against the prior recognised basis \(B_{t-1} = W_{t-1}\), and tax is paid on any positive delta at the applicable marginal rate.
At year 5 the asset sells at true value \(V_5\). The post-sale delta equals \(V_5\) minus the prior recognised basis — larger for the understater by the gap accumulated over five years of suppressed declarations.
J.3 Illustrative Figures
| Honest (\(\alpha\) = 1.0) | Moderate under (\(\alpha\) = 0.8) | Significant under (\(\alpha\) = 0.5) | |
|---|---|---|---|
| Entry basis \(B_0\) | £20.000m | £16.000m | £10.000m |
| True value at sale \(V_5\) | £30.015m | £30.015m | £30.015m |
| Tax paid years 1–5 | £1.077m | £0.859m | £0.534m |
| Final delta on sale (year 6) | £1.641m | £7.053m | £15.175m |
| Tax on final delta | £0.251m | £1.080m | £2.324m |
| Total lifetime WDT (Net) | £1.295m | £1.796m | £2.550m |
| Terminal net worth (TW_settled) | £28.478m | £27.763m | £26.690m |
| TW_settled vs honest | — | -2.51% | -6.28% |
| Net tax vs honest | — | +38.68% | +96.95% |
Table J.1: Deferred delta comparison across declaration strategies, \(g\) = 7%, N = 5, \(\tau\) = 15%. TW_settled includes the post-sale tax/refund oscillation. Python model v1.0, \(k\) = 0.001.
J.3.1 Period-by-period: Honest declarer (\(\alpha\) = 1.0)
| t | True V (£m) | Declared W (£m) | Delta (£m) | \(\tau\) | Tax L (£m) | f |
|---|---|---|---|---|---|---|
| 0 (entry) | 20.000 | 20.000 | — | 15.21% | 0.000 | 1.0000 |
| 1 | 21.400 | 21.400 | 1.400 | 15.23% | 0.213 | 0.9900 |
| 2 | 22.898 | 22.670 | 1.270 | 15.25% | 0.194 | 0.9816 |
| 3 | 24.501 | 24.050 | 1.380 | 15.26% | 0.211 | 0.9730 |
| 4 | 26.216 | 25.508 | 1.458 | 15.28% | 0.223 | 0.9645 |
| 5 | 28.051 | 27.055 | 1.547 | 15.30% | 0.237 | 0.9561 |
| 6 (sell) | 30.015 | 28.696 | 1.641 | 15.32% | 0.251 | 0.9561 |
J.3.2 Key mechanism: basis gap recovery at sale
At the sell year, the final delta differs by declaration strategy: honest (\(\alpha\) = 1.0) £1.641m → tax £0.251m; \(\alpha\) = 0.8 £7.053m → tax £1.080m (larger by £5.413m due to suppressed basis); \(\alpha\) = 0.5 £15.175m → tax £2.324m (larger by £13.534m due to suppressed basis).
The \(\alpha\) = 0.8 understater saved £0.218m in years 1–5 but paid £0.829m more at sale — net cost of understatement: £0.501m. The \(\alpha\) = 0.5 understater saved £0.543m in years 1–5 but paid £2.073m more at sale — net cost of understatement: £1.255m.
J.4 What the Example Shows
A suppressed basis means a smaller annual delta and a lower annual tax bill. The understater captures that advantage across years 1 through 5.
At sale, the final delta exceeds the honest declarer’s by exactly the suppressed basis amount. The tax on that larger delta is calculated at the full marginal rate; the deferred liability, concentrated in a single assessment, more than offsets the annual savings.
Two things drive this. The basis gap compounds: a lower basis in year 1 produces a larger delta in year 2, reducing the retained fraction, which feeds forward. And the realisation assessment applies the full rate with no smoothing — the accumulated gap appears at once.
The result holds at any positive growth rate. For the quantitative relationship between holding period, growth rate, and recovery size, see (VAL.A §C.1) and (VAL.A §C.7).
K. Dilution Compounds with Growth
Claim illustrated: For fungible assets under Route C, the must-transfer rule creates a cost that tracks the asset’s trajectory. The faster the asset grows, the more expensive the understatement becomes; the dilution is an ongoing transfer at an underpriced rate, not a one-off payment. (VAL §5.2)
K.1 Setup
Taxpayer holds a 60% founder stake in a software company at entry. True value \(V_0\) = £20m on Route C with a three-year assessment window. The company grows at \(g\) = 15% per year — a high-growth case chosen to make the dilution visible over a realistic founder holding period. The taxpayer declares at \(\alpha\) = 0.6 throughout.
The state acquires equity in settlement of each period’s WDT liability. The example shows how large the state’s accumulated stake becomes relative to what honest declaration would have required.
K.2 The Mechanism
At each assessment window end, the WDT liability is:
\[L_t = \tau(W_t) \cdot \Delta_t\]
Settlement under Route C transfers a proportional equity interest at the declared value:
\[s_t = \frac{L_t}{W_t}\]
Because \(W_t = \alpha \cdot V_t\) is below the true value, equity transferred per unit of tax paid is underpriced: the state’s £1 of tax claim buys more than £1 of true economic value. This accumulates across every window.
The taxpayer’s retained fraction after N windows is:
\[f_N = \prod_{t=1}^{N}(1 - s_t)\]
Under honest declaration the same liability settles at true value — the state acquires the same economic claim through a correctly priced transfer.
K.3 Illustrative Figures
Model note. (VAL.B §K) uses three assessment windows of unspecified length. This model uses N = 3 annual periods as a proxy (Option A). A window-aware model would produce different equity accumulation figures; the directional claim (dilution is more expensive at high \(g\)) is unaffected. The model treats \(V_0\) = £20m as the declared portfolio (representing the stake value directly, not the company valuation at £20m with a 60% stake = £12m stake value).
K.3.1 Period-by-period accumulation
| Period | True V (£m) | Honest W (£m) | Honest f | Understater W (£m) | Understater f | State equity (honest) | State equity (\(\alpha\)=0.6) |
|---|---|---|---|---|---|---|---|
| entry | 20.000 | 20.000 | 1.0000 | 12.000 | 1.0000 | 0.000% | 0.000% |
| 1 | 23.000 | 23.000 | 0.9801 | 13.800 | 0.9803 | 1.989% | 1.975% |
| 2 | 26.450 | 25.924 | 0.9632 | 15.557 | 0.9635 | 3.679% | 3.653% |
| 3 | 30.417 | 29.299 | 0.9462 | 17.584 | 0.9466 | 5.379% | 5.339% |
| sell | 34.980 | 33.099 | 0.9462 | 33.112 | 0.9466 | 5.379% | 5.339% |
K.3.2 Summary at period N = 3
| Metric | Honest (\(\alpha\) = 1.0) | Understater (\(\alpha\) = 0.6) |
|---|---|---|
| Founder retained fraction | 94.621% | 94.661% |
| State equity stake | 5.379% | 5.339% |
| True value of state stake (£m) | £1.636m | £1.624m |
| True value of founder stake (£m) | £28.781m | £28.793m |
| Tax paid (Net) (£m) | £1.928m | £2.916m |
| Terminal net worth TW_settled (£m) | £32.592m | £31.043m |
| Implicit cost of understatement vs honest (£m) | — | £0.988m |
Table K.1: Accumulated dilution under understatement, \(g\) = 15%, Route C, N = 3 annual periods as proxy for three-year window, \(\tau\) = 15%. TW_settled includes the post-sale tax/refund oscillation. Python model v1.0, \(k\) = 0.001.
K.3.3 Key mechanism: underpriced equity transfer
The understater transfers equity at their declared value (60% of true value). The state acquires this equity at a 40% discount to reality; it then appreciates at the true rate (15% per year). After 3 periods, the state holds 5.339% vs 5.379% for the honest declarer. The understater has transferred less equity in percentage terms but at a steeper discount, so net tax cost is higher: £0.988m extra.
K.4 What the Example Shows
The understater transfers equity below true value. At a 40% discount the initial gap may seem modest, but at \(g\) = 15% the state’s discounted stake appreciates at the true rate; by the third window its true economic value materially exceeds what honest prices would have purchased.
The cost of understatement under Route C is not fixed at declaration. It grows with the asset. A founder who expects plateau may rationally consider a low declaration; a founder who expects strong growth takes on an increasingly expensive commitment with each window of understatement. This is the mechanism behind Proposition 3a in (VAL.A §A.5.3).
The dilution operates even without a sale. A taxpayer holding a Route C asset indefinitely converts each period’s WDT liability into an equity transfer at the declared price, continuously. Understatement under Route C does not defer the cost; it accelerates it.
L. Why Route D Defers to Realisation
Claim illustrated: Route D defers taxation not for administrative convenience but because periodic cash settlement on illiquid non-fungible assets would recreate forced-realisation pressure, and in-kind settlement is structurally impossible for assets that cannot be divided. Deferral to realisation is the only coherent design. (VAL §6.1)
L.1 Setup
Taxpayer holds a sculpture collection declared at entry at £8m. The collection is non-fungible: it cannot be fractionally transferred without destroying individual works. True value grows at \(g\) = 5% per year. The example runs two parallel timelines to show why annual cash settlement fails and how Route D resolves it.
L.2 Timeline A: Annual Cash Settlement (the Design Route D Avoids)
Suppose the WDT required annual cash settlement on the sculpture collection, calculated on the delta between successive self-declared values.
Year 1. The collection has appreciated. The taxpayer declares £8.4m (honest, \(g\) = 5%). Delta = £0.4m. Tax due at 15% = £0.060m.
The taxpayer has no liquid assets proportional to the liability. The collection cannot be partially sold: selling one sculpture at auction takes months and disrupts the collection’s integrity. The taxpayer must find external cash, borrow against an asset most lenders will not accept as collateral, or sell works into a thin market at distressed prices to meet a £60,000 annual bill.
Year 5. The collection is worth approximately £10.2m. Cumulative annual tax bills have totalled £0.311m. The taxpayer has been forced to sell two works at below-market prices — exactly the forced-realisation problem Route D is designed to avoid. The remaining collection is worth less than it would have been. The state has collected revenue at the cost of destroying the value it was taxing.
Why in-kind settlement does not solve this under annual assessment. Route C works for fungible assets because a proportional equity interest is legally and economically equivalent to the whole — a 2% stake in a company is identical in character to the remaining 58%. A sculpture is not. Transferring 5% of a sculpture is not achievable; transferring one work from a collection of twenty is a structurally different act from transferring a proportional interest.
L.3 Timeline B: Route D
Under Route D, with the same taxpayer, collection, and growth rate:
No formal assessment occurs during the holding period. Annual reports are filed each year stating the collection’s existence, estimated current value, and any material changes. No liability accrues and no cash payment is required.
The taxpayer holds the collection for fifteen years and dies. The estate triggers the inheritance auction. The auction establishes a value of £16.631m.
The WDT liability is calculated on the full gain from entry basis to auction value: (£16.631m − £8m) × 15.17% marginal rate = £1.310m. The heir may pay this from the estate’s liquid assets and retain the collection, or allow the works to sell at the auction price and receive the net proceeds.
L.3.1 Timeline A: Annual Cash Settlement (what Route D avoids)
| Year | True V (£m) | Annual WDT liability (£m) | Cumulative liability (£m) |
|---|---|---|---|
| 1 | 8.400 | 0.060 | 0.060 |
| 2 | 8.820 | 0.063 | 0.124 |
| 3 | 9.261 | 0.067 | 0.190 |
| 4 | 9.724 | 0.070 | 0.260 |
| 5 | 10.210 | 0.073 | 0.333 |
L.3.2 Timeline B: Route D (deferred to inheritance at year 15)
| Event | Value (£m) |
|---|---|
| Entry basis \(B_0\) | £8.000m |
| True value at inheritance (year 15) | £16.631m |
| Total gain (V15 − \(B_0\)) | £8.631m |
| \(\tau\) at V15 | 15.17% |
| WDT liability at inheritance | £1.310m |
| Annual cash demand during years 1–15 | £0.000m/year |
L.3.3 Comparison
| Metric | Timeline A (annual) | Timeline B (Route D) |
|---|---|---|
| Annual cash demand | £0.067m/yr avg | £0.000m/yr |
| Total tax collected | £0.333m (yrs 1–5 only) | £1.310m (full 15 yrs) |
| Forced realisation risk | High | None during holding |
| Tax base | Partial appreciation | Full gain \(B_0\) → V15 |
| Settlement mechanism | Cash from external source | Cash from estate or auction |
Route D collects more tax (full 15-year gain vs 5-year partial) while eliminating the cash-demand problem. Annual settlement structurally undermines the tax base.
L.4 What the Example Shows
Annual cash settlement on non-fungible assets does not fail because administration is complex. It fails because it recreates the economic harm it is trying to tax: forcing illiquid asset holders to realise value before they are ready, at prices distorted by the compulsion. Route D concentrates taxation at the one point where an observable, arm’s-length price exists.
An annual cash liability on an illiquid asset is a tax on the forced sale, not on the appreciation. Route D taxes the appreciation.
The argument applies equally to property, art, jewellery, and any non-fungible asset where periodic forced valuation would be contested, expensive, and damaging to market value. Route D accepts a deferral in exchange for accuracy and non-interference.
M. Voluntary Settlement — Certainty, Not Avoidance
Claim illustrated: Soft and hard basis resets allow a Route D taxpayer to crystallise their liability before a forced realisation event, giving a chosen liability at a chosen time rather than an uncertain future one. They are not escape hatches: the liability is paid in full at the declared or market-established value, and the reset basis governs all future delta calculations, including refund entitlements in loss years. (VAL §6.4)
M.1 Setup
Taxpayer holds commercial property declared at entry at £5m. True value has grown steadily; at year 10 it is approximately £9m. The taxpayer wants to crystallise a known liability rather than leave a large uncertain WDT obligation for their heirs. Two options are available — the soft and hard basis resets — with no voluntary settlement as the counterfactual.
Both options are voluntary pathways initiated by the taxpayer. Neither involves Valuation Body scrutiny. The hard basis reset uses the same auction mechanics as the corrective auction but is a distinct pathway: the taxpayer initiates it, no flag has been raised, and the full symmetric refund mechanism applies if the auction price falls below the prior declared basis. Once two non-flagging Valuation Bodies have submitted sealed estimates in a corrective process, the position locks and no voluntary reset can intervene; a taxpayer who acts before any flag is raised faces no such constraint.
M.2 Option A: Soft Basis Reset
The taxpayer self-declares the current property value at £8.5m (conservative relative to the £9m true value). The WDT liability is calculated on the gain from entry basis to declared reset value: £8.5m − £5m = £3.5m gain. At 15.07% marginal rate, the liability is approximately £0.405m. The taxpayer pays from liquid assets and the recognised basis resets to £8.5m.
Future annual reports carry an estimated value of approximately £8.5m, resetting the delta clock. If the property subsequently appreciates, the future delta runs from £8.5m; if it declines, the refund entitlement is calculated from £8.5m as declared — an unverified figure. By declaring conservatively, the taxpayer has traded accuracy for simplicity: the future refund in a loss year runs from the lower figure, not a market-established price.
M.3 Option B: Hard Basis Reset
The taxpayer commissions a public auction — not to sell, but to establish what the market will pay. The highest bid establishes a market price of £8.144m (true value at year 10 at \(g\) = 5% compounded). The WDT liability on the gain from entry basis to auction price: £8.144m − £5m = £3.144m gain, at 15.07% = £0.474m. Auction costs of £0.163m are borne by the taxpayer.
The recognised basis resets to the auction price. Future refund entitlements run from a market-established number any lender, court, or counterparty can treat as verified. The hard reset costs more than Option A — the auction price is higher than the conservative self-declared value, and auction costs are additional — but the resulting basis is verifiable and unambiguous.
Where a hard-reset auction produces no valid bid, the asset is worthless and the basis resets to zero. The symmetric refund mechanism applies to the full downward delta from prior declared basis to zero, subject to the lifetime contribution envelope. A no-bid outcome on a voluntary hard reset is market discovery, not a corrective event; the normal refund logic is unaffected.
M.4 Option C: No Voluntary Settlement
The taxpayer takes no action. The property appreciates. At death, the estate triggers the inheritance auction, establishing £10.395m (true value at year 15 at \(g\) = 5% compounded). The WDT liability on the full gain: £10.395m − £5m = £5.395m, at 15.10% = £0.815m. The heir must pay from the estate’s liquid assets or allow the property to sell.
M.5 Comparison
Model note. This example uses closed-form arithmetic, not run_val_sim. Liabilities calculated as \(\tau\)(V) × (V − prior_basis) for each settlement event. Computed true values: \(V_{10}\) = £8.144m, \(V_{15}\) = £10.395m (\(g\) = 5% compounded from \(B_0\) = £5m). Soft reset declared value: £7.688m (conservative, ~94% of true \(V_{10}\)), consistent with Option A setup.
| Metric | Option A: Soft reset (yr 10) | Option B: Hard reset (yr 10) | Option C: No reset (yr 15) |
|---|---|---|---|
| Settlement value | £7.688m (self-declared) | £8.144m (auction) | £10.395m (inheritance auction) |
| Gain from \(B_0\) = £5m | £2.688m | £3.144m | £5.395m |
| \(\tau\) at settlement | 15.07% | 15.07% | 15.10% |
| WDT liability | £0.405m | £0.474m | £0.815m |
| Auction costs | nil | £0.163m | nil (estate cost) |
| New recognised basis | £7.688m | £8.144m | £10.395m (heir’s entry basis) |
| Basis verified? | No (self-declared) | Yes (market auction) | Yes (inheritance auction) |
| Future refund basis | Unverified | Market-verified | Market-verified |
Table M.1: Voluntary settlement options compared. Entry basis £5m; \(g\) = 5% compounded. Python model v1.0 (closed-form arithmetic), \(k\) = 0.001.
M.6 What the Example Shows
None of the three options avoids the WDT. In every case, the full gain from entry basis to market value is eventually taxed. The options differ in timing, certainty, and the quality of the basis for future refund calculations.
A voluntary settlement that permanently eliminated future liability, or reset the basis far below market value without consequence, would be avoidance. A reset that requires paying the WDT on the full accumulated gain and anchors all future calculations from that point is not. The taxpayer gains certainty; the state gains current revenue; the basis resets to a defensible figure.
The corrective auction (VAL §11.3) remains available as a backstop if the soft basis reset declared value is egregious relative to market evidence — the deterrent that makes the soft reset credible rather than a strategic minimum. The corrective pathway is distinct from the voluntary hard reset: it is triggered by Valuation Body consensus, the taxpayer is not notified until the auction notice is published, and the no-refund rule applies on downward correction. A taxpayer who uses the voluntary hard reset in good faith, even at a figure the market subsequently tests lower, retains the symmetric refund the mechanism provides.
N. Forecast Exposure
Claim illustrated: Honest declaration is not mechanically optimal in every possible future — it is the only strategy whose outcome does not depend on forecast accuracy. Understatement and overstatement are directional bets on the asset’s trajectory; the mechanism penalises being wrong regardless of intent. (VAL §7.1)
N.1 Setup
Three founders hold identical 40% stakes in the same private company. The company enters the WDT at a true value of £20m (each stake worth £8m). All three elect Route C with a one-year assessment window. The company grows at \(g\) = 7% per year for ten years then sells at true value. The founders differ only in their view of the company’s trajectory.
Founder A has no strong view. She declares honestly throughout: \(W_t = V_t\) at each assessment.
Founder B expects plateau or underperformance. He declares at \(\alpha\) = 0.6 throughout — 60% of true value.
Founder C expects strong growth and wants the higher declared valuation as a credibility signal. She declares at \(\alpha\) = 1.4 throughout — 140% of true value.
The company grows at \(g\) = 7% for the full ten years. Neither B’s pessimism nor C’s optimism matches reality.
N.2 What Happens to Each Founder
Founder A pays tax on each annual delta at honest rates. Her recognised basis tracks the true value. At sale, the post-sale delta is small and she receives a small refund. Her total lifetime WDT reflects the actual wealth accumulated.
Founder B pays lower annual tax throughout: his declared basis is suppressed, so his annual deltas are smaller. Through the must-transfer rule he transfers equity at underpriced rates; the state’s accumulated stake is worth more than the tax paid to acquire it. At year 10 the post-sale assessment produces a large positive delta — the gap between his suppressed basis and the true realisation price appears in a single calculation. Total lifetime WDT is higher than Founder A’s. The forecast error has a cost.
Founder C pays higher annual tax throughout — her inflated basis produces larger annual deltas, and the must-transfer rule transfers equity at an overpriced rate, so the state receives less true economic value per unit of tax. At year 10 the post-sale delta is negative, triggering a partial refund. The refund partially compensates for the excess paid during the holding period, but the progressive rate bracket effect and the timing of payments leave her net position worse than Founder A’s.
N.3 Illustrative Figures
| Metric | Founder A (\(\alpha\)=1.0) | Founder B (\(\alpha\)=0.6) | Founder C (\(\alpha\)=1.4) |
|---|---|---|---|
| Entry basis | £8.000m | £4.800m | £11.200m |
| True value at sale (year 11) | £16.839m | £16.839m | £16.839m |
| Tax paid years 1–10 | £0.988m | £0.591m | £1.388m |
| Refunds received years 1–10 | £0.000m | £0.000m | £0.000m |
| Post-sale delta (year 11) | £0.883m | £6.700m | £-4.932m |
| Tax/refund on post-sale delta | £0.134m | £1.016m | £-0.748m |
| Total lifetime WDT (Net) | £1.104m | £1.473m | £0.738m |
| Terminal net worth (TW_settled) | £15.304m | £14.543m | £16.065m |
| TW_settled vs Founder A | — | -4.98% | +4.97% |
| Net tax vs Founder A | — | +33.39% | -33.13% |
| Effective rate (Net/TW_settled) | 7.22% | 10.13% | 4.60% |
Table N.1: Three-founder comparison, \(g\) = 7%, N = 10, Route C, \(\tau\) = 15%. TW_settled includes the post-sale tax/refund oscillation. Python model v1.0, \(k\) = 0.001.
N.3.1 Period-by-period: All three founders
| t | V (£m) | A: W | A: L | A: f | B: W | B: L | B: f | C: W | C: L | C: f |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 8.000 | 8.000 | — | 1.0000 | 4.800 | — | 1.0000 | 11.200 | — | 1.0000 |
| 1 | 8.560 | 8.560 | 0.084 | 0.9901 | 5.136 | 0.051 | 0.9902 | 11.984 | 0.119 | 0.9901 |
| 2 | 9.159 | 9.069 | 0.077 | 0.9818 | 5.441 | 0.046 | 0.9818 | 12.696 | 0.108 | 0.9817 |
| 3 | 9.800 | 9.622 | 0.083 | 0.9732 | 5.773 | 0.050 | 0.9733 | 13.470 | 0.117 | 0.9732 |
| 4 | 10.486 | 10.206 | 0.088 | 0.9648 | 6.124 | 0.053 | 0.9649 | 14.287 | 0.124 | 0.9647 |
| 5 | 11.220 | 10.826 | 0.094 | 0.9565 | 6.496 | 0.056 | 0.9566 | 15.155 | 0.131 | 0.9564 |
| 6 | 12.006 | 11.483 | 0.099 | 0.9482 | 6.891 | 0.059 | 0.9484 | 16.075 | 0.140 | 0.9481 |
| 7 | 12.846 | 12.181 | 0.105 | 0.9400 | 7.310 | 0.063 | 0.9402 | 17.051 | 0.148 | 0.9398 |
| 8 | 13.745 | 12.921 | 0.112 | 0.9319 | 7.754 | 0.067 | 0.9321 | 18.086 | 0.157 | 0.9317 |
| 9 | 14.708 | 13.705 | 0.119 | 0.9238 | 8.225 | 0.071 | 0.9240 | 19.184 | 0.167 | 0.9236 |
| 10 | 15.737 | 14.538 | 0.126 | 0.9158 | 8.725 | 0.075 | 0.9160 | 20.348 | 0.177 | 0.9155 |
| sell | 16.839 | 15.420 | 0.134 | 0.9158 | 15.425 | 1.016 | 0.9160 | 15.416 | -0.748 | 0.9155 |
N.3.2 Key findings
Founder B (pessimist, \(\alpha\)=0.6): Paid £0.591m in years 1–10 vs £0.988m for Founder A. At sale, the suppressed basis produced a large positive delta (£6.700m). Total net tax: £1.473m vs £1.104m for Founder A — +33.4% more despite lower annual payments. TW_settled: £14.543m vs £15.304m — -5.0%.
Founder C (optimist, \(\alpha\)=1.4): Paid £1.388m in years 1–10 vs £0.988m for Founder A. At sale, the inflated basis produced a negative delta (£-4.932m) → refund of £0.748m. Total net tax: £0.738m vs £1.104m for Founder A — -33.1% relative to honest. TW_settled: £16.065m vs £15.304m — +4.97%. Founder C ends above Founder A on TW_settled.
Founder A (honest, \(\alpha\)=1.0): No directional forecast exposure. Paid exactly the tax on the wealth actually accumulated — £1.104m net, retaining £15.304m.
N.4 What the Example Shows
Founder B paid less tax annually and more at realisation. He ends worse than Founder A on both net tax and TW_settled. The mechanism penalised his pessimistic forecast being wrong.
Founder C paid more annually and received a partial refund at realisation. She ends above Founder A on TW_settled (+4.97%) and below on net tax (-33.1%). This is consistent with the population equilibrium in (VAL.A §A.6): at \(g\) = 7%, a low-growth scenario, \(\alpha\) = 1.4 sits within the mild-overstatement zone where the sell-year refund benefit from the inflated basis exceeds the periodic cost. Founder C’s outcome is not a forecast error — at this growth rate, her strategy paid off. The example illustrates the forecast exposure claim rather than the equilibrium claim: Founder C’s strategy required the company to not grow fast enough for the bracket penalty to dominate. Had the company grown at 15%, her net position would have been worse than Founder A’s.
Founder A required no forecast. She paid tax on the wealth she actually accumulated, without a second position on whether growth would land within the mild-overstatement advantage zone. That is the payoff-profile argument: honest declaration has no conditional structure; its outcome tracks the asset’s actual trajectory unconditionally.
This is not the equilibrium the mechanism produces at the population level — the rational equilibrium is mild overstatement, per (VAL.A §A.6), and Founder C’s result at low growth is consistent with that. Honest declaration remains the only strategy with no forecast exposure; it is the benchmark against which other strategies are evaluated. Founder A illustrates the payoff-profile argument in its purest form.
N.5 The Signalling Dimension
Founder C’s overstatement may have generated real external benefits: a higher declared valuation can influence investors, lenders, and employees in ways that create economic value outside the tax system. The WDT does not prohibit this; it prices it. Founder C purchased the credibility of a £11.2m declared value by accepting the corresponding tax basis and trajectory risk.
If the external benefits materially outweigh the additional WDT cost — if the company grew faster because of the credibility signal — then Founder C’s strategy may have been rational even at a worse WDT outcome. The model in (VAL.A §C.3) explores this through the β parameter.
At the system level, if mild overstatement is the rational equilibrium, declared values in the register are systematically above true values during holding periods. The credit-expansion channel this creates and the monitoring gap it implies are developed in (VAL.A §A.6) and (VAL §7.3).
O. Summary of Design Properties Illustrated
| Appendix | Claim | Key mechanism | VAL reference |
|---|---|---|---|
| §J: Deferred delta | Understatement defers, not eliminates | Post-sale assessment recovers accumulated basis gap in a single calculation | (VAL §1), (VAL.B §J); (VAL.A §C.1) |
| §K: Dilution compounds | Must-transfer cost tracks the asset’s growth | State acquires equity at declared price; appreciation runs at true rate | (VAL §5.2); (VAL.A §A.4.5), (VAL.B §K) |
| §L: Route D defers to realisation | Annual cash on illiquid assets recreates forced-realisation pressure | Inheritance auction establishes observable price; full gain taxed at realisation | (VAL §6.1), (VAL.B §L) |
| §M: Voluntary settlement | Soft/hard resets give certainty, not avoidance | Liability paid in full at declared or market value; basis resets for all future calculations; voluntary pathway carries full symmetric refund on downward discovery; corrective pathway does not | (VAL §6.4), (VAL §11.2), (VAL §11.3); (GOV.B §G) |
| §N: Forecast exposure | Honest declaration has no directional exposure to trajectory | Both understatement and overstatement penalise forecast error; honest declaration is agnostic | (VAL §7.1); (VAL.A §A.5.1), (VAL.B §N) |