WACC and APV: Project Discount Rates, Debt Policy, and Financing Tax Shields

WACC and APV ask the same valuation question: what are the operations worth after financing effects are accounted for? WACC incorporates those effects through a discount-rate framework. APV separates operating value from financing components. When their assumptions match, their answers should match too.

Learning goals and notation

You should finish able to estimate project-specific costs of capital, unlever and relever comparable betas under a declared model, distinguish fixed debt from a target debt ratio, reconcile WACC with APV, and value a scheduled debt paydown.

Use annual effective rates, year-end cash flows, market-value capital weights, and expected unlevered free cash flow to the firm. Assume immediate usable corporate tax deductions and no other financing frictions in the basic examples. Relax those assumptions explicitly when adding costs.

Symbol	Meaning
$FCF_t$	Unlevered after-tax operating free cash flow
$k_A$	Required return on the unlevered operating business
$k_D$, $k_E$	Required returns on debt and equity
$\tau$	Relevant marginal corporate tax rate
$D$, $E$	Market values of debt and equity
$V_L=D+E$	Levered operating-business value in the simple model
$L=D/V_L$	Debt-to-value ratio
$TS_t$	Usable interest tax savings at date $t$

1. Start with the same operating cash flows

Both methods use

$$FCF_t=EBIT_t(1-\tau)+D\&A_t-CapEx_t-\Delta NWC_t^{op},$$

under the simplified cash-tax assumptions. Here $D\&A$ means depreciation and amortization, not debt. Operating NWC excludes financing debt and excess nonoperating cash.

Do not subtract interest, debt principal repayment, or dividends in unlevered FCF. Do not include an interest tax shield in those operating tax lines. WACC or a separate APV adjustment will account for the financing effect.

This does not remove operating deductions such as depreciation. Those affect the operating tax calculation and remain part of the forecast.

Value versus project NPV

For a business already in place, APV can describe its levered value:

$$V_L=V_U+PV(\text{financing effects}).$$

For a new project with initial operating investment $I_0$:

$$NPV_L=-I_0+PV_{k_A}(FCF)+PV(\text{financing effects}).$$

Subtract initial investment once. Debt proceeds provide financing for it; they are not an additional operating benefit or a second reduction in the investment cost.

2. The WACC formula and its inputs

The standard after-tax expression is

$$\boxed{WACC=k_E\frac{E}{D+E}+k_D(1-\tau)\frac{D}{D+E}.}$$

Use market weights and a marginal rate appropriate to the tax savings actually usable. A historical average tax rate or an old book debt/equity ratio can give misleading weights or benefits.

The debt cost is the current required return for the relevant debt risk and policy, not automatically the coupon on an old bond. For highly risky debt, promised yield can materially exceed expected return, and the simple use of yield in WACC requires care. The tax shield itself follows deductible interest and its usability, not an abstract substitution of any quoted yield.

Project risk, not the source of the check

Imagine the project as a stand-alone business. Estimate its operating risk and sustainable financing policy. The parent’s WACC is suitable only if it is a reasonable comparable for those exposures and tax/financing assumptions.

Borrowing all of a project’s initial cost does not make its valuation debt weight 100%. The incremental capacity it supports within the intended business financing policy can differ from the particular account funding its first payment.

3. Why cheap debt is not a valuation shortcut

Without taxes and other frictions:

$$k_E=k_A+\frac DE(k_A-k_D).$$

The lower-return debt claim leaves a riskier residual equity claim. Their weighted return remains $k_A$.

For $k_A=10\%$, $k_D=4\%$, and $D/E=0.5$, equity cost is 13%. WACC is

$$\frac23(13\%)+\frac13(4\%)=10\%.$$

Holding equity cost fixed while increasing debt produces an artificial decline. With taxes, debt can create a usable shield; distress, incentives, and funding costs can offset it. The after-tax formula does not imply unlimited debt maximizes value.

4. From comparable companies to a project cost of equity

A comparable firm’s equity beta combines business risk with its financing policy. To transfer business risk, unlever each comparable first, form a justified estimate of asset beta, then relever at the project’s target policy.

Under the course’s simplified stable-$D/V$ model, where tax shields are treated as having operating-asset risk, and with relatively safe debt:

$$\beta_A\approx\beta_E\frac{E}{D+E},\qquad \beta_E\approx\beta_A\left(1+\frac DE\right).$$

These are the outline’s formulas with their debt-policy and tax-shield-risk assumptions attached. Do not mix them with a fixed-dollar-debt tax model without checking consistency.

In a no-tax model with risky debt, the more general relationship is

$$\beta_A=\frac{E}{D+E}\beta_E+\frac{D}{D+E}\beta_D.$$

Taxable settings can require additional adjustments for the shield’s risk and value. Cash-rich companies also require separating low-risk excess cash from operating risk before applying the result to a new business.

A common alternative under fixed perpetual debt, usable corporate taxes, and negligible debt beta is

$$\beta_E\approx\beta_A\left[1+(1-\tau)\frac DE\right].$$

The presence or absence of $(1-\tau)$ is not a free choice. It reflects different modeling assumptions. Use matching unlevering and relevering rules for the comparable and the project, and state the model.

Worked comparable example

Use the stable-ratio, asset-risk-shield model for this example:

Comparable	Equity beta	Market $D/V$	Estimated operating asset beta
A	1.2	25%	$1.2(0.75)=0.9$
B	1.5	40%	$1.5(0.60)=0.9$

Their matching estimates give project $\beta_A=0.9$. With target $D/V=40\%$, $D/E=0.4/0.6=2/3$, so

$$\beta_E=0.9\left(1+\frac23\right)=1.5.$$

For risk-free rate 3% and market premium 5%:

$$k_A=3\%+0.9(5\%)=7.5\%,\qquad k_E=3\%+1.5(5\%)=10.5\%.$$

With safe debt cost 3% and tax rate 25%:

$$WACC=0.6(10.5\%)+0.4(3\%)(0.75)=7.2\%.$$

Do not average equity betas first when leverage differs. An average or median asset beta also needs economic judgment about business mix, operating leverage, data quality, and comparability.

5. APV exposes the sources of value

APV starts with unlevered operations and values financing effects separately:

$$\boxed{NPV_L=NPV_U+PV(TS)+PV(OtherBenefits)-PV(OtherCosts).}$$

Potential components include usable interest deductions, issuance fees, a subsidized loan, expected distress losses, and financing-induced investment effects. Specify each component’s dates, cash flows, and risk. Do not subtract an operating loss again if it was already included in base FCF.

The interest shield is

$$TS_t=\tau\times DeductibleInterest_t$$

only when the deduction produces the assumed cash saving at that time. Tax losses, interest limits, changing tax rates, and default can make realized savings differ.

A financing model can be transparent and still wrong: separating a speculative shield forecast from operations does not make its estimated PV reliable.

6. Fixed dollar debt versus a stable debt ratio

Fixed perpetual debt

Suppose a fixed debt amount $D$ is maintained forever, interest is $k_DD$, all deductions are usable, and shield risk supports discounting at $k_D$:

$$PV(TS)=\frac{\tau k_DD}{k_D}=\tau D.$$

Perpetuity and discounting assumptions are essential. Fixed debt lasting three years does not create the same PV as debt maintained forever. A fixed contractual balance does not guarantee tax-shield safety if taxable income is uncertain.

Stable debt-to-value policy

If debt adjusts with business value, future interest and shields change with that value. In the course’s simplified model, the shields have operating-asset risk. For a level expected perpetual shield stream:

$$PV(TS)=\frac{\tau k_DD_0}{k_A}.$$

This is not a universal formula for every constant-ratio policy. Growth changes the stream; periodic versus continuous rebalancing changes which future debt balances and shields are known at each date. A discrete policy can give the next shield different risk from later shields. State the policy and timing rather than inferring one universal discount rate from the label “constant leverage.”

The outline’s suggestion to pick a rate between $k_D$ and $k_A$ for a changing debt schedule is a classroom heuristic, not a general pricing theorem. Model the schedule, deduction usability, and state dependence; there is no guarantee the correct adjustment can be captured by one rate in that interval.

7. Reconcile WACC and APV under one policy

Continue the stable-ratio example: sustainable unlevered FCF is 90 forever, $k_A=7.5\%$, $k_D=3\%$, tax rate 25%, and target $L=D/V=40\%$. Use the stated asset-risk-shield model with level expected FCF and no other side effects.

WACC valuation

$$V_L=\frac{90}{0.072}=1{,}250.$$

Debt is $0.4(1{,}250)=500$ and equity 750.

APV valuation

$$V_U=\frac{90}{0.075}=1{,}200,$$ $$PV(TS)=\frac{0.25(0.03)(500)}{0.075}=50,$$

so $V_L=1{,}200+50=1{,}250$.

Solve the apparent debt/value circularity

Because $D=LV_L$, APV can be written

$$V_L=\frac{FCF}{k_A}+\frac{\tau k_DLV_L}{k_A}.$$

Rearranging:

$$V_L=\frac{FCF}{k_A-\tau k_DL}.$$

The implied WACC is $k_A-\tau k_DL=7.5\%-0.25(3\%)(0.4)=7.2\%$. The two routes agree because they use the same financing policy and shield-risk assumptions.

If initial investment is 1,100, project NPV is 150 by either route. Do not add the shield of 50 once more to the WACC value of 1,250.

8. Same initial debt, different policy, different value

Now change the policy: permanently fix dollar debt at 500 instead of rebalancing it to 40% of value. Keep FCF 90, $k_A=7.5\%$, debt cost 3%, tax rate 25%, and assume debt-risk shield discounting with full usability.

$$V_U=1{,}200,\qquad PV(TS)=0.25(500)=125,$$ $$V_L=1{,}325,\qquad E=825.$$

The matching fixed-debt equity cost is

$$k_E=k_A+(1-\tau)\frac DE(k_A-k_D) \approx9.5455\%.$$

The resulting initial after-tax WACC is approximately 6.7925%. In this particular level-perpetuity example, $90/WACC=1{,}325$ as well.

Policy model	Initial debt	Shield PV	Levered value
Stable 40% ratio; asset-risk shield model	500	50	1,250
Fixed 500 forever; debt-risk shield model	500	125	1,325

The initial debt amount is identical, but future policy and shield risk differ. The valuation difference is not WACC disagreeing with APV. It is the consequence of changing the economic model. A single constant WACC is not generally valid through all states and dates for fixed-debt, finite-life, or growing projects.

9. Worked amortizing-debt APV

A project costs 250 today and produces unlevered FCF of 100, 110, and 120 at the next three year-ends. Its unlevered rate is 9%.

Debt begins at 200, with principal repayments of 60, 60, and 80 at the three year-ends. Assume interest is 4% on each year’s beginning balance, tax rate is 25%, deductions are fully usable, and the scheduled shield has debt-like risk supporting 4% discounting. Financing is assessed separately from the operating outlay; borrowing proceeds are not counted as extra operating NPV.

Year	Beginning debt	Interest	Principal repaid	Tax shield
1	200	8	60	2
2	140	5.6	60	1.4
3	80	3.2	80	0.8

Unlevered operating value is

$$V_U=\frac{100}{1.09}+\frac{110}{1.09^2}+\frac{120}{1.09^3} \approx276.99.$$

Shield value is

$$PV(TS)=\frac2{1.04}+\frac{1.4}{1.04^2}+\frac{0.8}{1.04^3}\approx3.93.$$

If date-0 issuance fees are 2, assumed to have no further tax effect, project APV net of initial investment is

$$NPV_L=-250+276.9899\ldots+3.9287\ldots-2\approx28.92.$$

Principal repayments are required in the financing and liquidity model, but are not deducted from unlevered operating FCF. The initial 200 borrowing funds part of the 250 investment and the 2 fees; the remaining funding must come from equity or another identified source. At each date, check that operating cash plus available liquidity covers interest and principal under downside scenarios.

A debt plug is not proof of lending capacity. For an LBO or project-finance transaction, default, collateral, covenants, and tax capacity can invalidate the assumed debt rate or shield risk even when the arithmetic balances.

10. Other side effects and changing WACC

A subsidized loan can be valued by comparing the contractual funding payments with market-consistent terms for the same risk. Issuance costs, guarantees, contingent obligations, and expected distress effects need separate treatment if they are economically relevant and not already counted elsewhere.

For changing leverage, a theoretically consistent WACC approach can use period-specific rates and recursively consistent values:

$$V_0=\sum_{t=1}^{T}\frac{FCF_t}{\prod_{s=1}^{t}(1+WACC_s)} +\frac{TV_T}{\prod_{s=1}^{T}(1+WACC_s)}.$$

The schedule must follow the modeled risks, policy, and timing; it is not enough to insert arbitrary rates by year. Circular debt/value weights and changing equity costs can make this cumbersome. APV often makes those assumptions easier to inspect, although it still requires careful pricing of financing effects.

11. Choosing a method and auditing the result

Situation	Useful starting approach	Main check
Stable business risk and target market leverage	Constant WACC may be convenient	Are the target and costs appropriate throughout forecast and terminal periods?
Fixed or explicit amortization schedule	APV often makes debt effects clearer	Are shields usable and appropriately priced?
Subsidies, issuance costs, unusual financing	APV can separate components	Are any effects counted twice or omitted from operations?
Material default, hybrids, or changing risks	Richer cash-flow and claim modeling	Are promised payments being confused with expected cash flows?

Do not infer present-day usage percentages from historical lecture comments about which method practitioners prefer. Method choice should follow the economics and the ability to implement assumptions consistently.

Audit business risk, market leverage weights, tax usability, beta adjustments, debt schedule, terminal policy, and value-to-equity reconciliation. Test sensitivity to the market premium, operating forecasts, tax assumptions, and financing policy. A precise WACC estimate can disguise large uncertainty in all of them.

12. Practice questions

1. A project has market debt weight 30%, debt cost 5%, equity cost 11%, and usable tax rate 25%. Calculate WACC.
2. A comparable has equity beta 1.4 and $D/V=25\%$. Under the stated stable-ratio, safe-debt model, find asset beta and relever at target $D/V=40\%$. With risk-free rate 3% and market premium 5%, find equity cost.
3. A no-tax project has $k_A=9\%$, $k_D=3\%$, and $D/E=0.5$. Find equity cost and WACC. Why is holding equity cost fixed incorrect?
4. Permanent fixed debt is 200 at 4%, tax rate is 30%, and shield risk supports 4% discounting. Find annual shield and perpetual shield PV.
5. For a separate asset-risk-shield model with the same initial debt and rates, but $k_A=8\%$, find the PV of a level expected perpetual shield. Why is it different from question 4?
6. Unlevered project value is 500, shield PV 30, and financing-cost PV 5. Initial investment is 450. Find levered value and project NPV.
7. Beginning debt is 100 in year 1 and 50 in year 2, interest is 6%, and the usable tax rate is 25%. Find shields and their PV at an assumed appropriate 6% rate.
8. Explain why a changing repayment schedule does not justify choosing an arbitrary discount rate between debt and asset costs. What should be checked if forecast borrowing exceeds project value?

13. Answers and reasoning

1. $WACC=0.3(5\%)(0.75)+0.7(11\%)=8.825\%$.
2. Asset beta is $1.4(0.75)=1.05$. Relevered beta is $1.05/0.60=1.75$. Equity cost is $3\%+1.75(5\%)=11.75\%$. Use these formulas only under their declared assumptions.
3. Equity cost is $9\%+0.5(9\%-3\%)=12\%$. WACC is $(2/3)(12\%)+(1/3)(3\%)=9\%$. Leverage concentrates risk in equity, changing its required return.
4. Annual shield is $0.30(0.04)(200)=2.40$ and perpetual PV is $2.40/0.04=60$.
5. PV is $2.40/0.08=30$. Different future debt policy and shield risk, not a choice of preferred answer, explain the difference.
6. Levered value net of financing costs is $500+30-5=525$. Project NPV is $525-450=75$.
7. Shields are 1.50 and 0.75. PV is $1.50/1.06+0.75/1.06^2\approx2.08$, conditional on full usability and the specified risk adjustment.
8. Required risk adjustment follows the cash flow’s timing, uncertainty, tax capacity, and financing policy rather than an interval rule. Borrowing feasibility, collateral, outside support, default risk, and resulting tax-shield assumptions must all be examined when debt exceeds stand-alone value.

Further reading

• Dirk Jenter, MIT OpenCourseWare, WACC and APV, Lecture 14 Section A: the course approach to project-specific inputs and financing adjustments.
• Katharina Lewellen, MIT OpenCourseWare, Cost of Capital, Lecture 14 Section B: business risk, capital structure, and valuation assumptions.
• MIT OpenCourseWare, Valuation of Free Cash Flows: the operating cash flows used by both methods.

Feynman study lab

Explain it simply

WACC includes financing effects in a blended discount rate; APV values the operating business and financing effects separately.

Concrete example. Unlevered value 100 plus a financing tax shield worth 5 gives APV 105 if there are no other financing effects.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why should you not add the same tax shield again to a WACC valuation that already includes it?

Check your explanation after you try

That would double count the financing benefit. Each approach must use assumptions consistent with the debt policy and risk of the cash flows.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.