In Week 9, you learned to estimate the cost of equity — the return shareholders require for bearing the risk of owning the firm's stock. You used two approaches: the CAPM (which links expected return to systematic risk via beta) and the dividend discount model (which backs out the required return from the stock price, dividend, and growth rate). Both methods give you rE, the cost of equity capital.
Now we turn to the other major source of financing: debt. The cost of debt is the return that lenders require to provide financing to the firm. Conceptually, it's the interest rate the firm pays on new borrowing. In practice, you estimate it as the yield to maturity (YTM) on the firm's existing publicly traded bonds — the market's current required return on the firm's debt, reflecting both the general level of interest rates and the firm's specific credit risk. Recall from Week 3 that YTM is the discount rate that sets a bond's market price equal to the present value of its remaining coupon and principal cash flows.
But there's a critical twist that doesn't apply to equity: interest payments on debt are tax-deductible. When a firm pays $1 in interest, its taxable income falls by $1, reducing its tax bill by $1 × TC, where TC is the corporate tax rate. The government, in effect, subsidizes part of the firm's borrowing cost. This means the true cost of debt to the firm — the after-tax cost of debt — is lower than the YTM.
The intuition is straightforward: if a firm borrows at 6% and faces a 21% tax rate, the after-tax cost is only 6% × (1 − 0.21) = 4.74%. The IRS absorbs 21% of the interest expense through the tax deduction. This tax shield is one of the fundamental reasons firms use debt in their capital structure — it lowers the effective cost of financing compared to equity, which receives no such tax subsidy. (Dividends are paid from after-tax income.)
A few practical notes. First, the relevant tax rate is the firm's marginal corporate tax rate — the rate applied to the next dollar of income. For most U.S. C-corporations, this is the federal statutory rate of 21%, though state taxes can push the effective marginal rate modestly higher. Second, the tax shield only has value if the firm has enough taxable income to use it. A firm with large tax losses carried forward may not benefit from the deduction immediately. For this course, we'll assume firms are profitable enough to fully use the interest tax shield unless stated otherwise.
Notice how much the tax shield matters. Red Acre's lenders require 6.80%, but the firm's true cost is only 5.37% — a reduction of 1.43 percentage points courtesy of the tax deduction. The bigger the tax rate, the bigger the subsidy.
In practice, you often need to work a bit harder to find rD. If the firm has publicly traded bonds, you estimate YTM from the bond's current market price, coupon rate, and maturity — exactly the bond pricing exercise from Week 3, solved in reverse. If the firm doesn't have traded bonds, you might use the yield on bonds with a similar credit rating and maturity, or look at the interest rate on the firm's most recent bank borrowing.
This example reinforces the Week 3 skill of extracting YTM from bond pricing data and then applies the after-tax adjustment. In practice, financial data providers (Bloomberg, FINRA TRACE) report bond yields directly, so you don't always need to solve for YTM by hand — but you do need to understand what the number means and how to apply the tax adjustment.
A typical firm uses a mix of equity and debt. The firm's overall cost of capital must reflect both sources, weighted by how much of each the firm uses. This blended rate is the weighted average cost of capital (WACC).
The logic is simple: if a firm is financed 70% by equity costing 12% and 30% by debt costing 5% after tax, then the firm's overall cost of capital is a weighted blend of those two rates. Each dollar the firm invests must earn enough to satisfy both its shareholders and its lenders in proportion to their claims.
Several important details deserve emphasis:
Use market values, not book values. The weights E/V and D/V must be based on current market values, not the balance sheet. Why? Because WACC represents the return the firm must earn going forward on new investment. Market values reflect what investors would actually pay for the firm's securities today — they capture current expectations about risk and return. Book values, on the other hand, are historical artifacts that may bear little resemblance to what the securities are actually worth. For equity, the market value is the current stock price times the number of shares outstanding. For debt, it's the market price of the bonds (or a reasonable approximation if bonds don't trade frequently).
The tax adjustment applies only to debt. The (1 − TC) term appears in the debt component because interest is tax-deductible. Equity returns (dividends and capital gains) come from after-tax income — there's no additional tax shield.
WACC is a marginal cost. It reflects the cost of raising the next dollar of capital, assuming the firm maintains its current capital structure proportions. If the firm dramatically changes its financing mix, WACC will change too.
Let's pause and interpret this result. Blue Acre needs to earn at least 9.57% on any new investment of average risk for the firm. Earning less than 9.57% means the project doesn't generate enough return to satisfy both the shareholders (who demand 11.50%) and the bondholders (who demand 6.00% pre-tax) in proportion to their financing contributions. The project would destroy value.
In textbook problems, you're often handed the cost of equity and cost of debt directly. In practice, you need to assemble these from market data — share prices, shares outstanding, bond prices, yields, betas, and so on. This next example walks through the full computation from raw inputs.
Note the market value of debt. The bonds trade at 102% of par — a premium — so the market value ($30.6M) exceeds the face value ($30M). Always use the market price, not the face value. This distinction matters more when bonds trade at a significant premium or discount to par.
How does the mix of debt and equity affect WACC? Since the after-tax cost of debt is almost always lower than the cost of equity (debt is senior, less risky for investors, and tax-deductible), adding more debt to the capital structure generally lowers WACC — up to a point. Beyond some level of leverage, the increased risk of financial distress pushes both the cost of debt and the cost of equity higher, eventually outweighing the tax benefit.
The following example holds the component costs constant and varies only the weights to illustrate the mechanical relationship. In reality, changing the capital structure would also change the component costs — more leverage means more risk for both equity and debt holders — but this simplified comparison builds intuition.
The WACC drops by nearly 1.7 percentage points when the firm shifts from 70/30 to 50/50, because a larger share of the cheaper financing source (after-tax debt at 4.58%) pulls the weighted average down. But remember — this is mechanical. In practice, pushing leverage from 30% to 50% would likely raise both rE and rD, because the firm's financial risk increases. The net effect depends on where the firm sits relative to its optimal capital structure, a question we leave for more advanced courses.
Recall from Week 6 that the NPV rule tells you to accept a project if its present value exceeds its cost — that is, if discounting the project's expected cash flows at the appropriate rate yields a positive NPV. But what is the "appropriate rate"?
The answer hinges on opportunity cost. When a firm invests $1 in a project, that dollar could instead have been returned to the firm's investors — the shareholders and bondholders who supplied the capital. Those investors have alternatives: they could invest in other assets of comparable risk. The return they'd earn on those alternatives is the opportunity cost of the firm using their capital.
WACC captures exactly this opportunity cost. It's the blended return the firm's investors could earn elsewhere on investments of similar risk, weighted by their proportional claims on the firm. If a project earns more than the WACC, it generates value above and beyond what investors require — it makes shareholders richer. If it earns less, investors would have been better off if the firm had simply returned their money.
Think of it this way: the firm is a financial intermediary standing between its investors and real investment opportunities. It raises capital from shareholders and bondholders (at a blended cost of WACC) and deploys that capital into projects. For the intermediation to create value, the projects must earn more than the capital costs. WACC is the hurdle rate — the minimum acceptable return on a new project.
This is the same logic as the NPV rule expressed in terms of rates rather than dollars. A project with a return above WACC will have a positive NPV when you discount its cash flows at WACC. A project with a return below WACC will have a negative NPV. They always agree.
The critical assumption: WACC is the correct discount rate only if the project being evaluated has the same risk as the firm's existing assets and the firm maintains its current capital structure. If either condition fails — the project is riskier (or safer) than average, or the firm is changing its financing mix — then WACC must be adjusted. We'll address this limitation next.
A firm-wide WACC works perfectly when every project the firm considers has the same systematic risk as the firm overall. In practice, many firms operate in multiple lines of business — a conglomerate might have a stable consumer products division and a volatile technology venture. Using a single WACC for both leads to systematic errors.
The problem is straightforward. If the firm's overall WACC is 10%, but Division A (low risk) should face an 8% hurdle and Division B (high risk) should face a 14% hurdle, then applying 10% across the board will:
(a) Reject good projects in the low-risk division. A project earning 9% in Division A creates value (it exceeds the 8% rate appropriate for its risk), but gets rejected because 9% < 10% (the firm-wide WACC).
(b) Accept bad projects in the high-risk division. A project earning 12% in Division B destroys value (it falls short of the 14% rate appropriate for its risk), but gets accepted because 12% > 10%.
Over time, this bias pushes the firm toward riskier and riskier assets — it systematically overinvests in high-risk ventures and underinvests in stable ones. The firm's overall risk profile drifts upward without anyone intending it to.
The solution, conceptually, is to use a divisional cost of capital (or project-specific discount rate) that reflects the risk of the specific investment, not the firm average. In practice, firms estimate divisional betas by looking at comparable "pure-play" companies — publicly traded firms that operate primarily in one line of business similar to the division in question. The pure-play beta serves as a proxy for the division's systematic risk, which then feeds into the CAPM to produce a division-appropriate hurdle rate.
For this course, the key takeaway is to recognize when firm-wide WACC is appropriate and when it isn't. If the firm is reasonably homogeneous — all its projects look roughly like its existing operations — then WACC is a fine hurdle rate. If the firm operates across very different risk profiles, you need to think more carefully.
This section brings the full toolkit together. Given a firm's capital structure, you'll compute the cost of equity (from CAPM or DDM, as in Week 9), determine the after-tax cost of debt, calculate the WACC, and then use that WACC as the discount rate in an NPV analysis. This is the complete chain from firm financing to project evaluation — the capstone of the cost of capital framework.
This example is the payoff for everything you've built across the second half of the course: beta and CAPM (Week 9) give you the cost of equity; bond pricing (Week 3) and the tax adjustment (this week) give you the after-tax cost of debt; capital structure weights combine them into WACC; and NPV analysis (Week 6) tells you whether the project creates value. Each piece depends on the ones before it.
Recall from Week 9 that the dividend discount model and CAPM often produce different estimates of the cost of equity. This isn't a failure — they're different models with different inputs and assumptions. In practice, a common approach is to average the two estimates or use professional judgment to weight the one you consider more reliable. The following example illustrates.
The 1.65 percentage point gap between the two estimates isn't unusual. The DDM is sensitive to the assumed growth rate — a small change in g can shift the result significantly. CAPM depends on beta, the risk-free rate, and the market risk premium, each of which is estimated with uncertainty. Neither model is "right" in a definitive sense; they're complementary lenses on the same question.
The final exam covers Chapters 9, 12, 13, and 14 — corresponding to Weeks 6 through 10. This section works through examples that integrate multiple concepts from across those weeks, helping you see how the pieces connect. Think of these as the kind of multi-step problems you might encounter on the final.
From Week 6, remember that NPV discounts all of a project's cash flows at the required return and sums them. A positive NPV means the project creates value. The IRR is the discount rate that makes NPV exactly zero — it's the project's implied rate of return. When you compare IRR to the required return (or WACC), the decision should agree with the NPV rule for conventional projects.
From Weeks 8 and 9, recall that a portfolio's expected return is the weighted average of its components' expected returns, and its beta is the weighted average of its components' betas. The Security Market Line (SML) from CAPM plots the relationship between beta and required return — any asset sitting above the SML is undervalued (its expected return exceeds what CAPM says it should earn), and any asset below the SML is overvalued.
Notice how cost of capital ties together everything from the second half of the course. The risk-return framework (Weeks 7–8) tells you that investors require higher returns for bearing more systematic risk. CAPM (Week 9) quantifies that relationship and produces the cost of equity. Bond valuation (Week 3) and the tax adjustment (this week) produce the cost of debt. WACC blends them. NPV (Week 6) uses the result to evaluate investments. Each tool depends on the ones that came before.
| Concept | Formula | Variables |
|---|---|---|
| After-Tax Cost of Debt | rD(1 − TC) | rD = YTM, TC = tax rate |
| WACC | (E/V) × rE + (D/V) × rD × (1 − TC) | E, D = market values; V = E + D |
| Cost of Equity (CAPM) | rE = Rf + β × (RM − Rf) | Rf = risk-free rate, β = equity beta, (RM − Rf) = market risk premium |
| Cost of Equity (DDM) | rE = D1/P0 + g | D1 = next dividend, P0 = current price, g = growth rate |
| NPV | NPV = −C0 + Σ [CFt / (1 + r)t] | C0 = initial cost, CFt = cash flow in year t, r = discount rate |
| PV of Annuity | PV = PMT × [1 − (1 + r)−t] / r | PMT = payment, r = rate per period, t = number of periods |
| Portfolio Expected Return | E(RP) = Σ wi × E(Ri) | wi = weight, E(Ri) = expected return |
| Portfolio Beta | βP = Σ wi × βi | wi = weight, βi = asset beta |
| Security Market Line | E(Ri) = Rf + βi × (RM − Rf) | Same as CAPM — plots required return vs. beta |
Red Acre Industries has bonds with a yield to maturity of 7.20%. The corporate tax rate is 21%. Calculate Red Acre's after-tax cost of debt.
Blue Acre Financial has a market value of equity of $300 million and a market value of debt of $120 million. The cost of equity is 12.40%, the pre-tax cost of debt is 5.50%, and the corporate tax rate is 21%. Calculate Blue Acre's WACC.
Green Acre Agriculture has 3.5 million shares outstanding, currently trading at $28 per share. The company has $40 million face value of bonds outstanding, currently trading at 97% of par. The cost of equity is 14.00%, the bonds' yield to maturity is 6.80%, and the corporate tax rate is 21%. Compute Green Acre's WACC.
White Acre Manufacturing has 4 million shares outstanding at $38 per share and $60 million in bonds outstanding at par. The equity beta is 1.20, the risk-free rate is 3.50%, the market risk premium is 7.00%, the bonds' yield to maturity is 5.50%, and the corporate tax rate is 21%. White Acre is evaluating a project that costs $25 million and will generate $7.5 million per year for 5 years. The project has risk similar to the firm's existing operations. Should White Acre accept this project?
Explain in your own words why a firm uses WACC — rather than just the cost of equity or just the cost of debt — as the discount rate for evaluating new projects. Then describe a situation where using the firm-wide WACC would lead to incorrect investment decisions, and explain what should be done instead.
Amber Acre Corporation's stock trades at $62 per share. The firm just paid a dividend of $3.10 (D0 = $3.10), and dividends are expected to grow at 3.50% per year indefinitely. Amber Acre's equity beta is 0.95, the risk-free rate is 4.00%, and the market risk premium is 6.50%. Estimate the cost of equity using both the DDM and CAPM approaches. Comment on why the two estimates differ and how you might reconcile them.
A firm with a WACC of 9.50% is evaluating a project that costs $50,000 and will generate cash flows of $14,000 per year for 5 years. Calculate the NPV. Should the firm accept the project?
The risk-free rate is 3.50% and the market risk premium is 7.00%. Stock X has a beta of 1.10 and an expected return of 14.00%. Stock Y has a beta of 0.70 and an expected return of 9.00%. For each stock, calculate the required return using the Security Market Line. Is each stock undervalued, overvalued, or fairly valued? Explain your reasoning.