WARA–WACC EQUIVALENCE METHOD

Widget Factory: A Practitioner’s Guide

Separating Taxable Tangible Property from Exempt Intangible Value

1. Purpose and Setting

What This Document Is and Why It Matters

This guide shows a property tax assessor, taxpayer representative, or reviewing judge how to apply the WARA–WACC Equivalence Method to determine what proportion of a going concern’s enterprise value represents taxable tangible property. The method was developed to address a problem that has long frustrated ad valorem proceedings: the absence of a market-grounded, internally consistent way to separate the value of physical assets from the value of intangible assets embedded in a single going-concern enterprise.

The subject entity is the Widget Factory, a hypothetical mid-size U.S. manufacturer of industrial components with an enterprise value of $100 million. The calculation is mechanical once the inputs are in hand, but the logic behind it matters—because in an adversarial proceeding, a method that cannot explain its own economic rationale will not survive cross-examination.

2. The Economic Foundation

Why a Firm’s Assets Must Earn Its Cost of Capital

Begin with a fact about how capital markets price companies. When investors buy a firm’s securities—its equity and debt—they pay a price that reflects the risk of the cash flows they expect to receive. The rate of return investors require, given that risk, is the firm’s weighted average cost of capital (WACC). This is not a management choice; it is set by the market.

Now look at the same firm from the asset side. The firm’s assets—its machines, customer relationships, and working capital—generate the cash flows that investors are paying for. It follows that the weighted average return those assets must produce, given their individual risk levels, must equal the WACC. If the assets collectively earned more than WACC, the firm would be creating value out of thin air; if they earned less, investors would not supply capital at observed prices. This is the WARA–WACC identity:

WARA = WACC

wₜ × rₜ + wᵂᶜ × rᵂᶜ + wᵢ × rᵢ = WACC

wₜ + wᵂᶜ + wᵢ = 1.0

Here wₜ, wᵂᶜ, and wᵢ are the fractions of enterprise value attributable to tangible assets, working capital, and intangible assets respectively; rₜ, rᵂᶜ, and rᵢ are the required rates of return on each class. The identity holds in every period, for every firm, in every jurisdiction. It is not a modeling preference. It is the no-arbitrage equilibrium condition for a portfolio of risky assets.

The allocation problem—what share of enterprise value is taxable tangible property? —reduces to solving this system for wₜ and wᵢ, given observable values of WACC and wᵂᶜ, plus empirically estimated values of rₜ, rᵂᶜ, and rᵢ.

Why the Three Asset Classes Carry Different Required Returns

Tangible assets (plant, equipment, inventory) earn a lower required return than intangible assets for a straightforward reason: they can be pledged as collateral, repossessed in default, and sold to alternative users. A creditor who holds a lien on a CNC machine can recover value even if the business fails. The collateral floor reduces the systematic risk investors bear. The required return rₜ is therefore below the firm’s WACC.

Intangible assets (customer relationships, trade names, workforce know-how, order backlog) have no such floor. Their value depends entirely on the going concern’s ability to sustain revenue. A customer relationship is worthless in liquidation. Because investors cannot recover intangible value in distress, they demand a higher return rᵢ to hold it. The risk hierarchy that follows—

rₜ < WACC < rᵢ

Tangible return below WACC • Intangible return above WACC

—is confirmed in the cross-sectional regression across 2,610 firm-year observations reported below. Working capital (cash, receivables, inventory net of payables) earns approximately the risk-free rate rᵂᶜ, because its components are measured at near-realizable values under GAAP and carry no material long-run systematic risk. Working capital is carried at par, not capitalized through a rate multiple.

3. The Parameters

Empirical Estimates from 706 Public Firms

The required returns rₜ and rᵢ are not assumed. They are extracted from the cross-sectional relationship between WACC and balance-sheet asset composition across publicly traded U.S. firms. The identifying insight is that firms differ in the proportions of tangible and intangible capital they employ. If tangibles and intangibles carry different required returns, then a firm with more tangibles should have a systematically lower WACC than an otherwise identical firm with more intangibles. The regression coefficients on the two asset-weight variables are direct estimates of rₜ and rᵢ.

The estimation uses 2,610 firm-year observations from 706 Russell 1000 constituents over 2021–2024, classified by the Fama-French 12-industry system. Financial firms and utilities are excluded. The baseline specification constrains the working capital return to the risk-free rate—a theoretically motivated restriction confirmed by the economic impossibility of the unconstrained result (which implies working capital earns more than intangibles). The manufacturing-sector regression yields the parameters used throughout this example.

3. The Parameters

Empirical Estimates from 706 Public Firms

The tangible-intangible spread—rᵢ − rₜ = 2.77 percentage points—is tighter than the generic examples common in practitioner literature. That matters: a narrower spread makes the allocation more sensitive to WACC and working-capital assumptions, which is why the sensitivity analysis in Section 7 is an indispensable part of any filing.

4. Step 1 — Free Cash Flow Decomposition

Solving for Each Asset Class’s Share of Cash Flows

The derivation begins with the income approach. Enterprise value equals total free cash flows discounted at WACC. If we decompose those free cash flows by asset class and discount each at its own required return, we get the same enterprise value—but now allocated across asset classes.

The portion of free cash flow attributable to working capital is simply rᵂᶜ × WC = 3.36% × $15 M = $0.504 M per year. Working capital itself is worth par—$15 M. The remaining free cash flow, which must be split between tangibles and intangibles, is:

K = WACC × EV − rᵂᶜ × WC

= 6.80% × $100M − 3.36% × $15M

= $6.800M − $0.504M = $6.296M

The operating asset base—enterprise value net of working capital—is $85 M. Together, K and A = $85 M form a two-equation system in the two unknown free cash flows (FCFₜ and FCFᵢ):

Present value constraint: FCFₜ / rₜ + FCFᵢ / rᵢ = A = $85M (i)

WACC constraint: FCFₜ + FCFᵢ = K = $6.296M (ii)

Solving: substitute FCFᵢ = K − FCFₜ from (ii) into (i). With capitalization multiples Mₜ = 1/rₜ = 16.69× and Mᵢ = 1/rᵢ = 11.42×:

FCFₜ = [A − K × Mᵢ] / [Mₜ − Mᵢ]

= [$85M − $6.296M × 11.42] / [16.69 − 11.42]

= [$85M − $71.90M] / 5.27 = $13.10M / 5.27 = $2.487M

FCFᵢ = $6.296M − $2.487M = $3.809M

Discounting at the asset-class rates yields the property values:

PV(FCFₜ) = $2.487M / 5.99% = $41.5M → Taxable tangible property

PV(FCFᵢ) = $3.809M / 8.76% = $43.5M → Exempt intangible property

PV(WC) = WC at par = $15.0M → Exempt (net working capital)

Check: $41.5M + $43.5M + $15.0M = $100.0M = EV ✓

This is the DCF derivation. The same result can be obtained algebraically using weights, as shown in Section 5.

5. Step 2 — The Algebraic Weight Solution

Two Equations, Two Unknowns

For practitioners who prefer to work in weights rather than cash flows, the same result is obtained algebraically. Substitute wᵢ = 1 − wᵂᶜ − wₜ from the adding-up constraint into the WARA = WACC identity and solve for wₜ:

wₜ = (WACC − wᵂᶜ × rᵂᶜ − (1 − wᵂᶜ) × rᵢ) / (rₜ − rᵢ)

= (6.80% − 0.15 × 3.36% − 0.85 × 8.76%) / (5.99% − 8.76%)

= (6.800% − 0.504% − 7.446%) / (−2.770%)

= −1.150% / −2.770% = 0.4152 → wₜ = 41.52%

wᵢ = 1.0 − 0.15 − 0.4152 = 0.4348 → wᵢ = 43.48%

The formula for wₜ has a straightforward structure.

The numerator is the portion of WACC that is not explained by working capital and the assumed intangible return. The denominator is the spread between intangible and tangible required returns. The solution is unique as long as rₜ ≠ rᵢ — which the risk hierarchy guarantees.

6. WARA = WACC Verification

The Internal Consistency Check

Every allocation produced by this method must satisfy the WARA = WACC identity exactly. This is not a hoped-for outcome—it is guaranteed by the algebra. The check is nevertheless important in practice because it makes the internal consistency of the allocation visible and auditable.

The three weighted contributions sum to exactly 6.800% — confirming that the allocation is internally consistent and that the asset-class returns reconcile to the firm’s cost of capital.

7. The Base-Case Allocation

Reading the Numbers

The base-case allocation assigns $41.5 M to taxable tangible property and $43.5 M to exempt intangible property, with $15.0 M of working capital carried at par. The near-symmetry of the tangible and intangible shares deserves explanation, because practitioners accustomed to seeing residual methods often expect intangibles to dominate.

The reason the shares are nearly equal is the 2.77-percentage-point spread between rᵢ and rₜ. Because intangible assets carry a higher required return, each dollar of intangible capital contributes more toward satisfying the WACC constraint per dollar of enterprise value. Fewer intangible dollars are therefore needed to achieve the same WACC contribution. The near-equality of the two weights reflects the fact that the manufacturing-sector WACC of 6.80% sits close to the midpoint of the tangible–intangible return interval (5.99%–8.76%).

The capitalization multiples make the same point more concretely. The market implicitly pays 16.69× normalized earnings for tangible assets (because 1/rₜ = 1/5.99% = 16.69) and 11.42× for intangible assets (because 1/rᵢ = 1/8.76% = 11.42). The tangible multiple is higher because the underlying cash flows are less risky. A firm whose revenues fall cannot repossess a customer relationship, but it can sell a CNC machine.

The multiples are not inputs. They emerge from the regression-estimated discount rates as their reciprocals. They can be compared with observed market transaction multiples for manufacturing-sector asset sales as a reasonableness check.

8. Sensitivity Analysis

How the Allocation Responds to the Two Key Assumption Drivers

Two inputs are legitimately subject to dispute in an assessment proceeding: the working capital weight wᵂᶜ, which depends on how the balance sheet is measured and what counts as operating current assets, and the subject entity’s WACC, which depends on cost-of-capital methodology. The discount rates rₜ and rᵢ are treated as fixed at their regression-derived values; their robustness is addressed by the PP&E sensitivity and temporal stability tests reported in the underlying paper.

The table below varies wᵂᶜ across 10%, 15% (base), and 20% of EV, and WACC across three levels: base minus 10% (6.3%), base (6.8%), and base plus 10% (7.7%). Nine scenarios result.

Three patterns are worth noting.

Higher WACC drives tangible weight down and intangible weight up. Because rᵢ > rₜ, achieving a higher blended return requires loading more of enterprise value into the higher-returning intangible class. Moving WACC from 6.3% to 7.7% at the base working-capital weight shifts $50.6 M from tangible to intangible—a large number from a 140-basis-point WACC variation.

Higher working-capital weight compresses operating assets. As working capital claims a larger share of EV, less remains for tangibles and intangibles. Because working capital earns only 3.36%, a larger WC share forces the remaining operating assets to generate a higher return per dollar, which the algebra resolves by reallocating toward the higher-returning intangible class.

The infeasible cell is an economically informative diagnostic, not a failure.

When wᵂᶜ = 20% and WACC = 7.7%, the equations yield wₜ < 0—no non-negative allocation exists that satisfies both constraints simultaneously. This is not a computational problem. It means the claimed inputs are internally inconsistent: that combination of working capital and overall cost of capital is incompatible with the observed market risk structure of manufacturing assets. In an assessment proceeding, this diagnostic provides a formal basis for requiring the other party to reconcile its inputs before any allocation can proceed.

9. Three Properties That Matter in Adversarial Proceedings

What the Arithmetic Forces You to Confront

9.1 The Feasibility Diagnostic

Any set of inputs that jointly implies a negative asset weight violates the no-arbitrage condition. The formal statement is that the parameter combination must satisfy:

WACC × EV − rᵂᶜ × WC ≤ rᵢ × (EV − WC)

If this inequality fails, no valid allocation exists. The assessor and taxpayer cannot both be right—at least one claimed input must change. This built-in consistency test has no counterpart in residual or excess-earnings methods, which can produce negative intangible values without any automatic signal that the underlying assumptions are incompatible.

9.2 Rate-Environment Invariance

The allocation formula depends on the risk spread S = rᵢ − rₜ, not on the absolute level of discount rates. To see why, write wᵢ = A/S, where A = WACC − wᵂᶜ ⋅ rᵂᶜ − (1 − wᵂᶜ) ⋅ rₜ. If inflation or monetary policy causes all required returns—WACC, rₜ, rᵢ, and rᵂᶜ—to shift upward by the same amount δ, then A and S are both unchanged. The allocation weights wᵢ and wₜ are unaffected.

An expert who adjusts intangible allocations because the risk-free rate rose—without any corresponding change in the relative risk premium between intangible and tangible assets—has made a methodological error.

Reallocation is warranted only when the spread S = rᵢ − rₜ changes. Temporal stability tests on the 2021–2024 sample confirm that the spread was stable across the Federal Reserve’s 2022–2023 tightening cycle (ANOVA F = 0.37, p = 0.77). The allocation is robust to the rate environment of the valuation date.

9.3 A Counterintuitive Direction That Practitioners Must Understand

The WARA identity has an implication that runs directly counter to intuition. When the required return on intangible assets rises—because higher operating risk, higher leverage, or any other factor makes intangible cash flows more volatile—the equilibrium intangible allocation falls, not rises. The formal statement is:

∂wᵢ / ∂rᵢ = −A / S² < 0

A portfolio analogy makes this intuitive. Consider a bond portfolio that must achieve a blended yield of 6.8%. If high-yield bonds pay 10%, you need fewer of them to hit your target than if they pay 7%. In exactly the same way, when intangible assets carry a higher required return, each dollar of intangible capital contributes more toward satisfying the WACC constraint. Fewer intangible dollars are therefore needed in equilibrium. The elasticity of wᵢ with respect to the spread S is exactly −1: a 10% widening of the spread produces a 10% reduction in the intangible allocation.

Practical implication: an assessor who argues for a higher intangible allocation by asserting that intangible assets are especially valuable or especially risky has made an error of direction.

Higher intangible risk produces higher rᵢ, which produces lower wᵢ and therefore lower intangible value. The method does not reward riskier intangibles with higher value. It assigns them a smaller fraction of a given enterprise value precisely because investors demand more return per dollar to hold them.

10. Subject-Specific Adjustment — The FCI Index Method

Adjusting for the Widget Factory’s Fixed-Cost Structure

The base-case rates rₜ = 5.99% and rᵢ = 8.76% are manufacturing-sector means estimated from the full cross-sectional sample. They are the appropriate inputs when the subject entity’s cost structure is typical of the sector. When it differs materially—specifically, when its ratio of depreciation to total production costs is above or below the sector mean—the intangible required return should be adjusted. The tangible return is not adjusted, because the cross-sectional evidence shows no statistically significant relationship between fixed-cost intensity and the required return on tangible assets. Collateral value is independent of the firm’s operating cost structure.

10.1 What Fixed-Cost Intensity Measures

Fixed-Cost Intensity (FCI) measures the structural share of fixed production costs in total production costs. It is computed from two line items available in any audited financial statement:

FCI = D&A / (D&A + COGS)

D&A = depreciation and amortization (from cash flow statement)

COGS = cost of goods sold (from income statement)

A purely variable-cost business has FCI near zero; a capital-intensive automated manufacturer approaches 0.40 or higher. The cross-sectional evidence confirms that a higher FCI widens the intangible–tangible risk spread: the intangible platform (customer relationships, workforce know-how, order backlog) must support a heavier fixed-cost base, making the intangible cash-flow stream more sensitive to revenue volatility. The coefficient on the FCI interaction term is positive and statistically significant in three of five standard-error specifications (Table 5 of the underlying paper).

10.2 The Widget Factory’s Income Statement

The Widget Factory has automated a portion of its production, replacing variable labor with fixed depreciation. The relevant line items are shown below alongside the sector mean for context.

The Widget Factory’s D&A is $0.9 M higher than the sector mean because its automation investment has increased its fixed-asset base. Its COGS is $0.9 M lower for the same reason—variable labor has been replaced by fixed depreciation. The two effects happen to offset in this example, leaving the D&A + COGS denominator unchanged at $30 M. The FCI is 0.180 versus 0.150 for the sector mean, a ratio of 1.20.

10.3 Applying the FCI Index to Adjust rᵢ

The FCI Index is the ratio of the subject’s FCI to the sector mean FCI. The adjusted intangible return is the sector rᵢ multiplied by the FCI Index:

FCI Index = FCIₛᵁᵇᵢᵉᶜᵗ / FCIₛᵉᶜᵗᵒᴿ = 0.180 / 0.150 = 1.20

rᵢ (adjusted) = rᵢ (sector) × FCI Index = 8.76% × 1.20 = 10.51%

Note: rₜ is not adjusted. Tangible required return is FCI-invariant (Section 4.9 of paper).

Two properties of this adjustment.

First, a subject with FCI equal to the sector mean receives no adjustment (FCI Index = 1.0, rᵢ unchanged). The adjustment is proportional to the deviation from the sector mean, not from zero.

Second, use a multi-year average FCI—five years where available—to reduce the effect of one-time asset disposals or D&A step-ups. Adjust for operating leases capitalized on the balance sheet to ensure comparability with the predominantly ownership-based public company sample.

10.4 Revised Allocation: Widget Factory vs. Sector Mean

With rᵢ adjusted to 10.51% and WACC held constant at 6.80%, the standard two-equation system gives:

wₜ = (6.80% − 0.15 × 3.36% − 0.85 × 10.51%) / (5.99% − 10.51%)

= (6.800% − 0.504% − 8.934%) / (−4.520%)

= −2.638% / −4.520% = 0.5836 → 58.4%

wᵢ = 0.85 − 0.5836 = 0.2664 → 26.6%

WARA verification for the adjusted allocation: 58.4% × 5.99% + 15.0% × 3.36% + 26.6% × 10.51% = 3.497% + 0.504% + 2.796% = 6.797% ≈ 6.80% ✓ (rounding in the last digit of wₜ).

The Widget Factory’s above-average depreciation intensity produces $16.9 M more in taxable tangible value compared to a sector-mean firm with identical EV and WACC. The mechanism is the counterintuitive direction established in Section 9.3: higher rᵢ reduces wᵢ and therefore shifts enterprise value toward tangibles. The FCI adjustment is symmetrical—firms with FCI below the sector mean receive a downward rᵢ adjustment and a smaller tangible allocation.

11. Adjusting for the Current Rate Environment

When the Risk-Free Rate Differs from the Sample Period

The regression-derived rates are estimated from a sample spanning 2021–2024, when the FRED 10-year Treasury averaged 3.36%. When the current risk-free rate differs materially—as it does in 2025–2026, with the DGS10 near 4.15%—the question arises whether to re-estimate the regression or adjust the existing estimates analytically.

Because the asset-class required returns were estimated in a CAPM framework (rᵗₜ₞₞ᵗ = Rⁱ + β × MRP), each return embeds an asset-level beta that is a structural characteristic of the asset class, not a function of the current interest rate. The betas can be extracted from the Table 6 estimates:

Implied βₜ = (rₜ − Rᶠₛₚ₞ᵖᵃᵉ) / MRP = (5.99% − 3.36%) / 6.5% = 0.40

Implied βᵢ = (rᵢ − Rᶠₛₚ₞ᵖᵃᵉ) / MRP = (8.76% − 3.36%) / 6.5% = 0.83

With the current 10-year Treasury at 4.15%:

rₜ (current) = 4.15% + 0.40 × 6.5% = 4.15% + 2.60% = 6.75%

rᵢ (current) = 4.15% + 0.83 × 6.5% = 4.15% + 5.40% = 9.55%

Spread = 9.55% − 6.75% = 2.80% (versus 2.77% at sample-period rates—essentially unchanged)

Consistent with rate-environment invariance (Section 9.2), the spread is nearly identical to the sample-period estimate. A practitioner applying a CAPM-derived WACC anchored to the current Treasury yield should use these rate-adjusted inputs rather than the original sample-period estimates, to maintain internal consistency between the WACC input and the asset-class return inputs. The allocation weights will be effectively unchanged, confirming that the base-case result is robust to the current rate environment.

12. Comparison to Alternative Methods

Why Competing Methods Are Less Defensible

None of the first three alternatives provides the feasibility diagnostic of Section 9.1. Residual and excess-earnings methods can produce internally inconsistent results—negative intangible values, or implied asset-class returns that violate the risk hierarchy—without flagging the inconsistency. The WARA–WACC equilibrium condition makes such inconsistencies explicit and formally testable.

13. Practical Checklist

Step-by-Step Application

Confirm enterprise value. Use the transaction price, income-capitalization value, or court-accepted appraisal. EV is the starting denominator.
Identify the Fama-French industry group. Match the subject to a sector and retrieve the empirical rₜ, rᵢ, and sector-mean WACC from Table 6 of the underlying paper. For manufacturing: rₜ = 5.99%, rᵢ = 8.76%, WACC = 6.80%.
Estimate the working capital weight. Use current assets minus current liabilities from the most recent balance sheet, divided by EV. Negative WC is common in retail, technology, and subscription businesses and is not an error—the formula handles it correctly.
1. Compute FCI (optional but recommended). FCI = D&A / (D&A + COGS) from the subject’s income statement and cash flow statement. If FCI differs from the sector mean, compute the FCI Index and adjust rᵢ = rᵢ(sector) × FCI Index. Use a five-year average FCI where available. Do not adjust rₜ.
Solve the two-equation system. Apply the closed-form formula: wₜ = (WACC − wᵂᶜ ⋅ rᵂᶜ − (1 − wᵂᶜ) ⋅ rᵢ) / (rₜ − rᵢ), then wᵢ = 1 − wᵂᶜ − wₜ. If wₜ < 0 or wᵢ < 0, the inputs are internally inconsistent. Revisit WACC or WC assumptions.
Verify WARA = WACC. Multiply each weight by its rate and sum. The result must equal WACC within rounding. Any discrepancy signals a computational error.
Run sensitivity. Vary WACC ±10% and WC weight ±2 percentage points. Document the range of tangible values and identify any infeasible cells. This range is the defensible bracket for the allocation.
Check rate-environment consistency. If the valuation-date risk-free rate differs materially from 3.36%, adjust rₜ and rᵢ using the implied-beta method of Section 11. Verify that the allocation weights are unchanged, consistent with rate-environment invariance.

14. Limitations

What This Method Does and Does Not Do

This method allocates going-concern economic value, not replacement cost. The tangible property value derived here represents the contribution of tangibles to enterprise value under the income approach. It should be reconciled with, but is not identical to, a standalone cost-approach or sales-comparison appraisal. When the income-approach and cost-approach tangible values diverge materially, the difference is a signal worth investigating rather than resolving by assumption.

Sector mean rates are a starting point. The base rates reflect the average manufacturing firm in the sample (mean intangible weight 75%). The FCI Index method adjusts for fixed-cost-intensity deviations. Further firm-specific adjustments may be warranted for very capital-intensive or very asset-light firms that fall outside the sample range.

The sample is dominated by large-cap public firms. The Russell 1000 universe means results may not generalize directly to small private manufacturers. Where private-company data suggests materially different risk characteristics, document the basis for any departure from the sector-mean rates.

Legal definitions of exempt intangibles vary by jurisdiction. The intangible value computed here is an economic concept: organizational capital, defined as EV minus the market-consistent value of tangible assets and working capital. Whether any specific intangible asset is legally exempt must be determined under the applicable state statute. Legal counsel must be consulted before the output is used in a formal assessment proceeding.

Working capital is carried at face value throughout. The method does not capitalize working capital through a rate multiple. This is the correct treatment for short-duration, near-realizable-value assets. For firms with unusual balance-sheet structures—large deferred revenue, significant restricted cash, or non-operating asset holdings—additional adjustments to the WC weight may be necessary.

15. Additional Help

If you’d like help applying the WARA–WACC method in an assessment, audit, or litigation setting, contact me directly.

Email: Info@SampsonValuation.com

WARA–WACC Equivalence Method · Submitted for Peer Review · Journal of Property Tax Assessment & Administration ·

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