DCF valuation under alternative debt policies — a teaching note
A companion to Valuing a company with an AI assistant (The Workbench, Episode 2). This is the teaching note the AI assistant wrote at the end of the episode, when the harness came off and I asked it to explain the model rather than build it — the theory, the propositions, the papers, grounded in this model’s own numbers. I have edited it only lightly: turned its live dataset views into static tables and resolved its interpolated figures to the values they carried when the note was generated. Everything below was sourced from the finished valuation model, and the reconciliations at the end are its own to make. The peer, cost-of-capital and reconciliation figures all match the main post; the glossary there defines the terms.
1. The question
We have one company, one operating forecast, and therefore one stream of unlevered free cash flow. FCFF is what the assets throw off before any financing decision is made; nothing in it knows how the firm is funded.
And yet the equity is worth different amounts depending on what we assume about debt policy. Not because the cash flows change, but because the tax shield on interest changes — how large it is, how risky it is, and therefore what it is worth today.
This note builds three standard valuation methods on the same forecast, shows that under one debt policy they agree to eleven decimal places, and shows that under a different debt policy the answer moves. The disagreement is not error. It is the economics.
2. Step one — the peers and the unlevered beta
We cannot observe our company’s asset risk directly: it is private, and its own equity beta would in any case be contaminated by its own leverage. So we borrow risk from comparable listed firms and strip their leverage out.
The unlevering identity is Modigliani–Miller Proposition I applied to betas: the firm’s assets are a portfolio of the claims on them, so the asset beta is the value-weighted average of the debt and equity betas.
\beta_U = \beta_E \cdot \frac{E}{V} + \beta_D \cdot \frac{D}{V}
This is the risky-debt form. The familiar textbook shortcut \beta_U = \beta_E / (1 + (1-T)D/E) assumes debt is riskless (\beta_D = 0) and a specific tax-shield policy; our peers carry \beta_D > 0, so we use the general form and let the debt bear its share of the risk.
| peer | \beta_E | \beta_D | D/V | \beta_U | r_U |
|---|---|---|---|---|---|
| Tusk | 1.20 | 0.12 | 0.30 | 0.876 | 8.82% |
| Sangyo | 1.40 | 0.20 | 0.40 | 0.920 | 9.06% |
| industry average | — | — | — | 0.898 | 8.94% |
Tusk: 1.2 \times 0.7 + 0.12 \times 0.3 = 0.876. Sangyo: 1.4 \times 0.6 + 0.2 \times 0.4 = 0.92. The two firms differ visibly in equity beta (1.2 against 1.4) but barely at all in asset beta — almost the whole gap was leverage, not business risk. That convergence is the entire justification for the exercise, and it is worth pausing on when it appears.
We take the industry average, \beta_U = 0.898, as our company’s asset beta.
Hamada (1972), The Effect of the Firm’s Capital Structure on the Systematic Risk of Common Stocks, Journal of Finance 27(2), is the origin of the lever/unlever manoeuvre. Note that the specific unlevering formula one should use is not policy-free — it depends on the same debt-policy assumption discussed in §7. Using Hamada’s tax-adjusted formula and then discounting tax shields at r_U is a common and quietly inconsistent mix.
3. Step two — the cost of capital
With an asset beta, CAPM gives the unlevered cost of capital — the return demanded on the firm’s assets if it carried no debt at all:
r_U = r_F + \beta_U \cdot \text{MRP} = 8.939\%
The same CAPM line prices the debt: r_D = r_F + \beta_D \cdot \text{MRP} = 4.44\%. Debt is cheaper than assets because it is safer — not because it is debt.
Modigliani–Miller Proposition II with risky debt then relevers the equity. Leverage does not create value in the MM world; it redistributes risk. The asset return is a weighted average of the claims’ returns, and solving for r_E:
r_E = r_U + (r_U - r_D)\cdot\frac{D}{E} = 11.938\%
At D/V = 0.4 we have D/E = 0.6667, so the equity holders demand 11.938% against the assets’ 8.939%. The spread is pure leverage.
Note what is absent: there is no (1-T) term in the relevering. That omission is a deliberate policy choice, and §7 explains it.
Finally the WACC, in its definitional form — the after-tax weighted average of what the two claims cost:
r_{WACC} = r_E\cdot\frac{E}{V} + r_D(1-T)\cdot\frac{D}{V} = 8.3174\%
| parameter | value | parameter | value | |
|---|---|---|---|---|
| r_F | 4.00% | r_U | 8.939% | |
| market risk premium | 5.50% | r_D | 4.440% | |
| \beta_U | 0.898 | D/E | 0.6667 | |
| \beta_D | 0.080 | r_E | 11.938% | |
| constant debt ratio D/V | 0.40 | r_{WACC} | 8.3174% | |
| tax rate T | 35% | r_{WACC} shortcut | 8.3174% | |
| EV/EBITDA multiple | 6.8 |
The shortcut, and why it is exact
The model also carries rWacc shortcut = r_U - r_D \cdot (D/V) \cdot
T. It reads 8.3174% — the same number. The difference between the two cells is exactly 0.
This is not a coincidence or an approximation. Substitute Proposition II into the definition, with w = D/V:
r_E(1-w) = \left[r_U + (r_U-r_D)\frac{w}{1-w}\right](1-w) = r_U - r_D w
r_{WACC} = (r_U - r_D w) + r_D(1-T)w = r_U - r_D\,w\,T
The WACC is the unlevered rate, reduced by the tax subsidy on the debt the firm is assumed to carry. Read that way, the WACC stops being a formula to memorise and becomes a statement: discounting at the WACC is discounting at r_U while quietly adding the tax shield back. Everything in §7 follows from this.
4. Method one — the WACC benchmark
Discount FCFF at r_{WACC} by backward induction, seeding the last year with the continuation value:
V_L(t) = \frac{FCFF(t+1) + V_L(t+1)}{1 + r_{WACC}}
| €000, base | 2022 | 2023 | 2024 | 2025 | 2026 |
|---|---|---|---|---|---|
| FCFF | 0.00 | 2,649.49 | 3,444.60 | 4,547.53 | 5,964.45 |
| Continuation Value | — | — | — | — | 317,368.01 |
| V_L | 243,845.88 | 261,478.03 | 279,781.60 | 298,504.63 | 317,368.01 |
| Debt | 97,538.35 | 104,591.21 | 111,912.64 | 119,401.85 | 126,947.20 |
| Equity | 146,307.53 | 156,886.82 | 167,868.96 | 179,102.78 | 190,420.80 |
| V_U | 238,422.34 | 257,085.43 | 276,621.70 | 296,801.38 | 317,368.01 |
The continuation value is the exit multiple, EBITDA(2026) \times 6.8 = 317,368.01. Rolling back gives
- V_L(2022) = 243,845.88
- Debt at the policy ratio = V_L \times 0.4 = 97,538.35
- Equity = 146,307.53
The same dataset runs the recursion a second time at r_U to get V_U = 238,422.34 — what these identical cash flows would be worth to an all-equity firm. The gap, 5,423.54, is the entire value of the tax shield. Hold that number.
Notice the self-reference: debt is a fraction of V_L, but V_L is what we are solving for. The WACC method dissolves this circularity by assuming it away — a constant ratio means the ratio is an input, and the debt level is an output. This is exactly why the WACC method cannot represent a fixed debt schedule.
5. Method two — Adjusted Present Value, under two debt policies
Myers (1974), Interactions of Corporate Financing and Investment Decisions, Journal of Finance 29(1), proposed valuing the pieces separately rather than blending them into a discount rate:
V_L = V_U + PV(\text{tax shields})
Value the business as if unlevered, then add the financing side-effects. The virtue is transparency: the tax shield becomes a visible line item with its own cash flows and its own discount rate, instead of being buried inside a WACC.
That last phrase is where the debt policy enters. The tax shield in year t is r_D \cdot D(t-1) \cdot T. How risky is it? It depends entirely on how certain we are about D.
predetD — a predetermined debt schedule. Debt follows the LBO’s actual amortisation plan (90,000 in 2022, drawn straight from FinDebt). The euro amounts are contractual, so the shields are about as risky as the debt itself: discount at r_T = r_D = 4.44\%. This is the Modigliani–Miller (1963) and Myers assumption.
constD/V — a constant debt ratio. Debt is rebalanced to 0.4 of firm value every year. Future debt is therefore unknown today — it rises and falls with the firm — so the shields inherit the risk of the assets, not of the debt: discount at r_T = r_U = 8.939\%. This is Harris and Pringle (1985).
| €000, base — constD/V (r_T = r_U) | 2022 | 2023 | 2024 | 2025 | 2026 |
|---|---|---|---|---|---|
| Debt | 97,538.35 | 104,591.21 | 111,912.64 | 119,401.85 | 126,947.20 |
| Interest | — | 4,330.70 | 4,643.85 | 4,968.92 | 5,301.44 |
| Tax Shield | — | 1,515.75 | 1,625.35 | 1,739.12 | 1,855.50 |
| Tax Shield Value | 5,423.54 | 4,392.60 | 3,159.91 | 1,703.25 | 0.00 |
| V_L | 243,845.88 | 261,478.03 | 279,781.60 | 298,504.63 | 317,368.01 |
| Equity Value | 146,307.53 | 156,886.82 | 167,868.96 | 179,102.78 | 190,420.80 |
| €000, base — predetD (r_T = r_D) | 2022 | 2023 | 2024 | 2025 | 2026 |
|---|---|---|---|---|---|
| Debt | 90,000 | 95,000 | 100,000 | 98,000 | 99,000 |
| Interest | — | 3,996.00 | 4,218.00 | 4,440.00 | 4,351.20 |
| Tax Shield | — | 1,398.60 | 1,476.30 | 1,554.00 | 1,522.92 |
| Tax Shield Value | 5,336.70 | 4,175.05 | 2,884.12 | 1,458.18 | 0.00 |
| V_L | 243,759.04 | 261,260.48 | 279,505.82 | 298,259.56 | 317,368.01 |
| Equity Value | 153,759.04 | 166,260.48 | 179,505.82 | 200,259.56 | 218,368.01 |
Only two lines of the dataset differ between the columns — Debt and rT. Interest, tax shield, the backward induction, V_U, V_L = V_U + tax shield value, and Equity = V_L - Debt are one shared formula each, fanning out across both policies. The model makes the claim structurally: these are not two methods, they are one method under two assumptions.
| predetD | constD/V | |
|---|---|---|
| Debt (2022) | 90,000.00 | 97,538.35 |
| r_T | 4.44% | 8.939% |
| Tax shield value (2022) | 5,336.70 | 5,423.54 |
| V_L (2022) | 243,759.04 | 243,845.88 |
| Equity (2022) | 153,759.04 | 146,307.53 |
6. Method three — Flow to Equity
The most direct method, and the one closest to what a shareholder actually receives. Build the cash flow to equity — FCFF, less after-tax interest, plus net new borrowing — and discount it at the cost of equity:
E(t) = \frac{FCFE(t+1) + E(t+1)}{1 + r_E}
| €000, base | 2022 | 2023 | 2024 | 2025 | 2026 |
|---|---|---|---|---|---|
| FCFF | 0.00 | 2,649.49 | 3,444.60 | 4,547.53 | 5,964.45 |
| less after-tax Interest | — | −2,814.96 | −3,018.50 | −3,229.80 | −3,445.94 |
| change in Debt | — | 7,052.86 | 7,321.43 | 7,489.21 | 7,545.35 |
| FCFE | — | 6,887.39 | 7,747.53 | 8,806.94 | 10,063.86 |
| Equity Continuation Value | — | — | — | — | 190,420.80 |
| Equity Value | 146,307.53 | 156,886.82 | 167,868.96 | 179,102.78 | 190,420.80 |
No tax shield appears anywhere. It does not need to: interest is deducted at its after-tax cost inside the cash flow itself, so the subsidy is already in the numerator rather than in the discount rate. FTE gives Equity = 146,307.53 directly, never computing an enterprise value at all.
The terminal equity value is the exit enterprise value less the debt outstanding then: 317,368.01 - 126,947.20 = 190,420.80.
One caution worth stating aloud: this FCFE is not the FCFE on the cash flow statement. SCF runs the actual LBO debt plan; FTE runs the constant-ratio policy, which borrows an extra 7,053 in 2023 to keep the ratio at 0.4. Different policies, different equity flows. If those two ever agreed, something would be wrong.
7. Why the constant-ratio answers coincide
| Route | Equity (2022) |
|---|---|
| WACC benchmark | 146,307.53 |
| APV, constant ratio | 146,307.53 |
| Flow to Equity | 146,307.53 |
npv() restatement |
146,307.53 |
They agree because they are one proposition written three ways, and the algebra of §3 is the proof.
WACC ≡ APV. We showed r_{WACC} = r_U - r_D w T. Discounting FCFF at r_{WACC} therefore is discounting at r_U and adding back the shield — which is what APV does explicitly. The equivalence needs the shield discounted at r_U; that is precisely the constant-ratio assumption. Confirm it in the cells: the APV tax-shield value of 5,423.54 is exactly the V_L - V_U gap from §4. APV does not merely agree with the benchmark, it decomposes it.
WACC ≡ FTE. Substituting the definitional WACC into the FCFE recursion makes it telescope into V_L - D. The identity that does the work is the definition of the WACC itself:
r_{WACC} = (1-w)\,r_E + w\,r_D(1-T)
So the agreement confirms the arithmetic, not the economics. It is a real check — a sign error or a timing slip breaks it instantly — but three methods agreeing on the same wrong input would agree just as beautifully. Internal consistency is necessary, never sufficient.
8. Why predetermined debt differs — and what it teaches
Equity under predetD is 153,759.04 against 146,307.53 under the constant ratio. Same cash flows, same r_U, same V_U. The difference is the debt assumption alone.
Two effects run in opposite directions, and the naive story gets it backwards:
- The shield is worth less: 5,336.70 against 5,423.54. One might expect the safer, contractual shield discounted at the lower rate r_D to be worth more. It is not — because the constant-ratio policy has debt growing with the firm (126,947 by 2026 against 99,000), generating larger shields. Here the size effect beats the discounting effect.
- Less debt is subtracted: 90,000 of actual debt against 97,538 of policy debt. This effect is far larger, and it decides the sign.
So predetD equity is higher not because the tax shield is worth more, but because the company owes less. The enterprise values are nearly identical (243,759.04 against 243,845.88); almost the whole equity gap is the debt deduction. A reader who reasons only about discount rates gets this exactly wrong.
There is no WACC column for predetD — and that absence is the deepest lesson in this note. A predetermined schedule means D/V drifts every year as the firm grows and the debt amortises. A single constant WACC cannot represent it. One would need a different WACC each year, each requiring the leverage ratio, which requires the value, which is what we are solving for. APV has no such problem: it never needs a debt ratio, only a debt level. This is why APV is the natural language of the leveraged buyout — and why forcing an LBO into a constant-WACC model is not conservative, it is wrong.
9. Which method should a practitioner use?
The methods are equivalent only when their assumptions are made consistent. Choose by which assumption you can actually defend.
WACC — when the firm holds a stable target capital structure. Mature, listed, treasury-managed-to-a-ratio. Its assumption (constant D/V) is then descriptively true, and it is the cheapest to compute and the easiest for an audience to follow. Koller, Goedhart and Wessels’ Valuation makes it the default for going-concern corporate work. It fails whenever leverage is changing — the case it is most often stretched to cover.
APV — when capital structure is changing, and especially when the debt schedule is known: LBOs, project finance, distress, acquisitions with a defined paydown. Myers (1974) introduced it for exactly this; Luehrman, Using APV: A Better Tool for Valuing Operations (HBR, 1997), argued for it as the general default; Inselbag and Kaufold, Two DCF Approaches for Valuing Companies under Alternative Financing Strategies (Journal of Applied Corporate Finance, 1997), is the direct treatment of the choice this note dramatises. Its transparency is also a governance property — the tax shield is a line item a committee can argue about, not a digit hidden in a discount rate.
FTE — when equity cash flow is the natural object and the capital structure is inseparable from operations. The standard case is banks and insurers, where debt is raw material rather than financing and “enterprise value” is barely meaningful. Also natural when a sponsor simply wants the equity cheque discounted at the equity return. Its weakness is fragility: r_E must track leverage year by year, so an unchanging r_E on a deleveraging LBO is a common and serious error.
Capital Cash Flow — worth knowing as the fourth sibling. Ruback, Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows (Financial Management, 2002), folds the shield into the cash flow and discounts everything at r_U; Kaplan and Ruback (1995) found it performed well on highly leveraged transactions. It buys APV’s flexibility with a single discount rate.
The professional failure is rarely picking the “wrong” method. It is mixing assumptions — Hamada’s tax-adjusted relevering with a Harris–Pringle shield, or a constant WACC on a debt paydown schedule. Taggart (1991) is the standard map of which expressions belong together.
10. What this note does not show
Four honest caveats.
The rebalancing convention is Harris–Pringle, not Miles–Ezzell. Our constant-ratio branch discounts every tax shield at r_U and relevers with no tax term, giving r_{WACC} = r_U - r_D w T. Miles and Ezzell (1980) assume debt is set at the start of each period, so the first year’s shield is known and discounted at r_D, later ones at r_U; their WACC carries an extra factor (1+r_U)/(1+r_D). Both are defensible; they are not the same model, and the difference is small here but real.
The exit multiple is treated as an unlevered value. We seed both recursions with the same continuation value, so V_U(2026) = V_L(2026) and the tax-shield value terminates at exactly zero. That says the shield stops at the horizon. A real EV/EBITDA multiple, observed from levered comparables, embeds the market’s view of post-horizon capital structure — so a share of the terminal value is arguably shield we have set to nil. The methods reconcile because of this choice; it is a modelling convention, not a fact.
The tax shield is the only financing effect. No distress costs, no agency costs, no personal taxes, no issuance costs, no debt capacity limit. In this world more leverage is monotonically better, which is why the model cannot be asked what the optimal structure is. Fernández (2004) argues the “present value of tax shields” framing is itself contestable.
One scenario. The figures in this note are the base case. Note also that this equity value of 146,307.53 sits well below the 196,000 equity purchase price in Assumpt — on these assumptions the sponsor is paying more than the constant-ratio DCF supports. That gap, not the reconciliation, is the interesting question. (The main post shows the same model re-run across best- and worst-case scenarios, changing no formulas.)
References
- Modigliani, F. and Miller, M. H. (1958), The Cost of Capital, Corporation Finance and the Theory of Investment, American Economic Review 48(3) — Propositions I and II.
- Modigliani, F. and Miller, M. H. (1963), Corporate Income Taxes and the Cost of Capital: A Correction, American Economic Review 53(3) — the tax shield.
- Hamada, R. S. (1972), The Effect of the Firm’s Capital Structure on the Systematic Risk of Common Stocks, Journal of Finance 27(2).
- Myers, S. C. (1974), Interactions of Corporate Financing and Investment Decisions — Implications for Capital Budgeting, Journal of Finance 29(1) — APV.
- Miles, J. A. and Ezzell, J. R. (1980), The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification, Journal of Financial and Quantitative Analysis 15(3).
- Harris, R. S. and Pringle, J. J. (1985), Risk-Adjusted Discount Rates — Extensions from the Average-Risk Case, Journal of Financial Research 8(3) — this model’s convention.
- Taggart, R. A. (1991), Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes, Financial Management 20(3).
- Kaplan, S. N. and Ruback, R. S. (1995), The Valuation of Cash Flow Forecasts: An Empirical Analysis, Journal of Finance 50(4).
- Inselbag, I. and Kaufold, H. (1997), Two DCF Approaches for Valuing Companies under Alternative Financing Strategies, Journal of Applied Corporate Finance 10(1).
- Luehrman, T. A. (1997), Using APV: A Better Tool for Valuing Operations, Harvard Business Review 75(3).
- Ruback, R. S. (2002), Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows, Financial Management 31(2).
- Fernández, P. (2004), The value of tax shields is not equal to the present value of tax shields, Journal of Financial Economics 73(1).
- Berk, J. and DeMarzo, P., Corporate Finance (Pearson) — textbook treatment of all three methods.
- Koller, T., Goedhart, M. and Wessels, D., Valuation: Measuring and Managing the Value of Companies (McKinsey) — the practitioner standard.