Does it matter how a firm is financed? Whether it funds itself with debt, equity, or some mix — does that change what the firm is worth? The Modigliani–Miller theorem gives the surprising benchmark answer, and the weighted average cost of capital is the number that carries the whole argument. A corporate finance tutor returns to this pairing again and again, because capital structure is where a first course stops being about arithmetic and starts being about reasoning. This page builds WACC, the two MM propositions, and the tax shield that breaks the tie.

1 · What WACC is

A firm raises money from two kinds of investor: shareholders, who require a return R_e, and lenders, who require R_d. The weighted average cost of capital blends the two, weighting each by its share of firm value V = E + D:

WACC = (E/V) R_e + (D/V) R_d (1 − T_c)

The (1 − T_c) term is there because interest is tax-deductible — more on that below. WACC is the rate at which the firm discounts its future cash flows, so it is also the hurdle rate for new projects. Lower the WACC and, holding cash flows fixed, the firm is worth more.

That is why capital structure matters, if it matters at all. The central question of this topic is simple to state: does changing the debt-to-equity mix change the WACC?

2 · Proposition I — capital structure is irrelevant

Modigliani and Miller’s first proposition gives a stark answer. In a world with no taxes, no bankruptcy costs, and efficient markets, the value of a firm is independent of how it is financed. WACC is constant, whatever the mix of debt and equity.

The intuition is a pie. The firm’s assets generate a stream of cash flows — that is the size of the pie. Splitting the pie into a debt slice and an equity slice does not change how much pie there is. Financing decides who gets which slice, not how big the whole is.

The formal argument is homemade leverage. If two firms were identical except for their debt, and the levered one sold at a premium, an investor could replicate the levered firm’s payoff by borrowing on their own account and buying the unlevered firm’s shares. Arbitrage would erase any price gap. So the two must have the same value, and the same WACC.

3 · Proposition II — the cost of equity rises with leverage

Here is the part students find genuinely counter-intuitive. If WACC stays flat, and debt is cheaper than equity, why doesn’t loading up on cheap debt pull the average down?

Because equity does not stay still. As a firm borrows more, its equity becomes riskier: debt holders are paid first, so shareholders bear a more variable, more leveraged residual. To compensate, the required return on equity rises. Proposition II makes this exact:

R_e = R_a + (R_a − R_d) (D/E)

where R_a is the cost of capital of the unlevered firm. The cost of equity climbs linearly in the debt-to-equity ratio. And it climbs by precisely the amount needed to offset the greater weight on cheap debt — so the weighted average stays put at R_a. The two effects cancel exactly. That cancellation is Proposition I, seen from the WACC side.

4 · Corporate taxes — the interest tax shield

Now relax the no-taxes assumption, because this is where the real result lives. Interest payments are deductible from taxable profit; dividends are not. So every pound of interest a firm pays saves it T_c pounds in tax. That saving is the interest tax shield, and it is worth having.

Two things change. The cost of debt in the WACC formula becomes an after-tax R_d (1 − T_c), and Proposition II picks up the same factor:

R_e = R_a + (R_a − R_d) (D/E) (1 − T_c)

Put these together and the WACC no longer stays flat. It falls as leverage rises, and it does so along a clean line:

WACC = R_a [1 − T_c (D/V)]

More debt now genuinely lowers the cost of capital, because the government is subsidising it. Firm value rises by the present value of the shield: V_L = V_U + T_c D. Taken at face value, the model says: borrow as much as you possibly can.

5 · The trade-off theory — where MM meets reality

That conclusion — 100% debt — is obviously wrong, and knowing why is the mark of understanding the model rather than reciting it.

MM with taxes leaves out the cost of borrowing too much. As leverage climbs, the probability of financial distress rises: bankruptcy is expensive, and even the threat of it distorts decisions (customers leave, suppliers demand cash, managers under-invest). These costs grow with debt and eventually outweigh the tax shield.

The trade-off theory balances the two. The optimal capital structure is the leverage at which the marginal tax benefit of one more pound of debt just equals the marginal expected cost of distress. That gives an interior optimum — real firms use some debt, not all debt. The pecking-order theory adds a second real-world wrinkle: firms prefer internal funds, then debt, then equity as a last resort, because of the signals each choice sends. Both are the corrections MM’s frictionless benchmark invites.

Worked example — a firm levering up, with and without taxes

Take a firm whose unlevered cost of equity is R_a = 10%, with a cost of debt R_d = 5%. Work it twice: first without taxes, then with a corporate tax rate of T_c = 30%.

Step 1 — Start all-equity. With no debt, WACC is just the cost of equity, which is the unlevered cost: WACC = R_e = R_a = 10%.

Step 2 — No taxes: lever to a 50/50 mix. Move to D/E = 1 (equivalently D/V = 0.5). Proposition II gives the new cost of equity: R_e = 10% + (10% − 5%) × 1 = 15%. Now blend: WACC = 0.5 × 15% + 0.5 × 5% = 7.5% + 2.5% = 10%.

Step 3 — Read the result. WACC did not move. The cost of equity jumped from 10% to 15% — exactly enough to cancel the benefit of funding half the firm with 5% debt. This is Proposition I in numbers: the pie is the same size.

Step 4 — Add a 30% corporate tax, same 50/50 mix. The tax shield now favours debt. Proposition II with taxes gives R_e = 10% + (10% − 5%) × 1 × (1 − 0.30) = 10% + 3.5% = 13.5% — a smaller jump than before. Blend with after-tax debt: WACC = 0.5 × 13.5% + 0.5 × 5% × 0.70 = 6.75% + 1.75% = 8.5%.

Step 5 — Cross-check with the shortcut. The compact formula must agree: WACC = R_a [1 − T_c (D/V)] = 10% × [1 − 0.30 × 0.5] = 10% × 0.85 = 8.5%. It does. Push leverage to D/V = 1 and WACC would fall to 10% × 0.70 = 7% — the floor the tax shield sets.

Step 6 — Interpret. With taxes, leverage lowered the cost of capital from 10% to 8.5%, and raised firm value by the present value of the shield. Taken literally, the model still says borrow more. What stops a real firm short of that is the rising cost of financial distress — the trade-off of Section 5.

Not sure why the cost of equity rises by exactly enough to keep WACC flat? Proposition II is the hinge of the whole Modigliani–Miller argument, and it is exactly the kind of step a one-on-one corporate finance tutor walks you through until it is obvious. Book a trial session.

Practice

Q1. A firm has an unlevered cost of equity of 12% and a cost of debt of 6%, in a world with no corporate taxes. It moves from all-equity to a 50/50 debt–equity mix (D/E = 1). Find the new cost of equity and the WACC.

Q2. The same firm (R_a = 12%, R_d = 6%) now faces a 25% corporate tax rate and adopts a debt-to-value ratio of 0.4. Use the shortcut formula to find its WACC, and state the reduction from the all-equity level.

Q3. A firm has R_a = 8%, R_d = 4%, a 35% tax rate and a debt-to-value ratio of 0.6. Find its levered cost of equity and its WACC, and confirm the component method matches the shortcut.

Answers: Q1: R_e = 12% + (12% − 6%) × 1 = 18%. WACC = 0.5 × 18% + 0.5 × 6% = 12% — unchanged, because there are no taxes (Proposition I). Q2: WACC = 12% × [1 − 0.25 × 0.4] = 12% × 0.9 = 10.8%, a reduction of 1.2 percentage points from 12%. Q3: D/E = 0.6/0.4 = 1.5, so R_e = 8% + (8% − 4%) × 1.5 × (1 − 0.35) = 8% + 3.9% = 11.9%. Component WACC = 0.4 × 11.9% + 0.6 × 4% × 0.65 = 4.76% + 1.56% = 6.32%. Shortcut = 8% × [1 − 0.35 × 0.6] = 8% × 0.79 = 6.32%. They agree.

Key takeaways

• WACC blends the cost of equity and the after-tax cost of debt by their weights in firm value; it is the firm’s discount rate, so a lower WACC means a higher value.
• Modigliani–Miller Proposition I: with no taxes or frictions, firm value and WACC are independent of capital structure. Slicing the pie differently does not change its size.
• Proposition II: the cost of equity rises linearly with the debt-to-equity ratio, by exactly enough to keep WACC constant. Cheaper debt buys you nothing on its own.
• Corporate taxes break the irrelevance. The interest tax shield makes WACC fall with leverage, along WACC = R_a [1 − T_c (D/V)], and raises firm value by T_c D.
• The trade-off theory stops the model short of 100% debt: the tax shield is balanced against the rising costs of financial distress, giving an interior optimal capital structure.

Why New York and London students choose our corporate finance tutoring

• Degree and CFA coverage: our tutors teach WACC and capital structure as both the university syllabus and the CFA curriculum frame them, so the notation and the emphasis match whatever you are sitting.
• Modigliani–Miller specialists: sessions are led by tutors who teach the propositions as a connected argument — irrelevance, then Proposition II, then the tax shield and the trade-off — rather than as disconnected formulas to memorise.
• Exam-first preparation: we work past papers with the mark scheme in view, because the marks in corporate finance are won on the reasoning — why WACC stays flat without taxes, why it falls with them, and why firms still don’t borrow to the hilt.

FAQ

Q: What is the weighted average cost of capital?
A: WACC is the blended required return on a firm’s capital, weighting the cost of equity and the after-tax cost of debt by their shares of firm value. It is the rate used to discount the firm’s cash flows, so it doubles as the hurdle rate for new investment.

Q: What does the Modigliani–Miller theorem say?
A: Its first proposition says that, without taxes, bankruptcy costs or other frictions, a firm’s value is independent of its capital structure — the debt-equity mix does not change what the firm is worth. It is the benchmark against which every real-world capital-structure argument is measured.

Q: Why does the cost of equity rise as a firm takes on more debt?
A: Debt holders are paid before shareholders, so more leverage makes the equity claim riskier and more variable. Shareholders demand a higher return to bear that extra risk, and Proposition II gives the exact linear relationship between the cost of equity and the debt-to-equity ratio.

Q: How does the interest tax shield lower the WACC?
A: Interest is tax-deductible, so debt reduces the firm’s tax bill. That makes the effective cost of debt lower and pulls the weighted average down as leverage rises, along WACC = R_a[1 − T_c(D/V)]. Firm value rises by the present value of the shield, T_c times debt.

Q: If debt lowers the WACC, why don’t firms use 100% debt?
A: Because MM with taxes ignores the cost of borrowing too much. As leverage rises, the risk and cost of financial distress grow. The trade-off theory balances the tax shield against these distress costs, giving a sensible interior optimum rather than all-debt.

Q: Is this examined with the CFA formulas or the academic derivation?
A: Both appear, depending on your course. University exams often want the propositions derived and interpreted; the CFA level wants the formulas applied quickly and correctly. Tell us which you are preparing for and we will pitch the session accordingly.

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Capital structure is two propositions and one tax shield, but the reasoning behind them — irrelevance, the offsetting rise in the cost of equity, the trade-off against distress — is where exam marks and interview questions live. A one-on-one session turns the MM theorem from a set of formulas into an argument you can make in your own words. Tell us your university, course or CFA level, and we will match you with the right tutor this week.