Week 8: Capital Structure Section A & B

Master MM Propositions, tax shields, leverage effects, and optimal capital structure for both exam sections.

      🎯 Exam Relevance: Capital structure is one of four core topics for Section A (news analysis). For Section B, expect WACC calculations, cost of equity derivations, and homemade leverage problems. Always show: Formula → Substitution → Calculation → Interpretation.
    

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MM Proposition I (No Taxes)

The value of the firm is independent of its capital structure. Levered and unlevered firms have the same value.

MM Proposition I

V_L = V_U = E + D

          🎯 Exam Insight: The key intuition: investors can create "homemade leverage" by borrowing personally. If VL ≠ VU, arbitrage opportunities exist.
        

Homemade Leverage Strategy

If a firm levers up but you want unlevered exposure:

Sell a portion of your shares (proportional to new debt ratio)
Lend the proceeds at the risk-free rate
Your total cash flow = original unlevered cash flow

MM Proposition II (No Taxes)

The cost of equity increases linearly with the debt-to-equity ratio.

Cost of Equity (Levered)

r_E = r_A + (D/E) × (r_A − r_D)

WACC (No Taxes)

WACC = (D/(D+E)) × r_D + (E/(D+E)) × r_E = r_A

⚠️ Common Exam Trap: Under MM (no taxes), WACC stays constant regardless of leverage. The increase in r_E exactly offsets the benefit of cheaper debt!

MM with Corporate Taxes

Interest is tax-deductible, creating a tax shield that adds value.

Value with Tax Shield (Perpetual Debt)

V_L = V_U + τ_c × D

WACC with Taxes

WACC = (D/(D+E)) × r_D × (1 − τ_c) + (E/(D+E)) × r_E

✓ Key Implication: With taxes, more debt = lower WACC = higher firm value. This drives the "optimal capital structure" problem.

Beta & Leverage Relationship

Levered Beta (No Tax)

β_L = β_U × [1 + D/E]

Unlevered Beta

β_U = β_L / [1 + D/E]

Use CAPM to find cost of equity:

CAPM

r_E = r_f + β_L × (r_m − r_f)

EBIT Coverage & Credit Rating

For optimal capital structure problems, you'll iterate to find the stable rating:

EBIT Coverage Ratio

EBIT Coverage = EBIT / Interest Expense

⚠️ Iterative Process:
1. Assume current interest rate → Calculate coverage
2. Find new rating from coverage → Get new interest rate
3. Recalculate coverage with new rate
4. Repeat until rating stabilizes

Test Your Understanding

These questions mirror the types of conceptual questions you might see in Section A.

Q1: Under MM Proposition I (no taxes), if a firm increases its debt ratio from 0% to 40%, what happens to firm value?

Firm value increases due to the tax shield Firm value remains unchanged Firm value decreases due to increased bankruptcy risk Firm value increases due to cheaper debt financing

Explanation: Under MM I (no taxes), capital structure is irrelevant. Investors can replicate any leverage through personal borrowing ("homemade leverage"), so firm value stays constant regardless of debt ratio.

Q2: According to MM Proposition II, when a firm increases leverage, what happens to its cost of equity and WACC (no taxes)?

Cost of equity stays constant; WACC decreases Cost of equity increases; WACC increases Cost of equity increases; WACC stays constant Cost of equity decreases; WACC decreases

Explanation: Under MM II, r_E = r_A + (D/E)(r_A − r_D). As D/E rises, equity becomes riskier, so r_E increases. However, this exactly offsets the benefit of using cheaper debt, keeping WACC = r_A.

Q3: A firm has r_A = 12%, r_D = 6%, and D/E = 0.5. What is the cost of equity under MM II (no taxes)?

15% 12% 18% 9%

Explanation: r_E = r_A + (D/E)(r_A − r_D) = 12% + 0.5 × (12% − 6%) = 12% + 3% = 15%

Q4: When a firm announces a recapitalization (increasing debt to pay special dividend) with no taxes, what happens to stock price?

Stock price increases because shareholders receive dividend Stock price decreases because the firm is now riskier Stock price increases then decreases after dividend payment Stock price remains unchanged at announcement

Explanation: Under MM I (no taxes), no value is created or destroyed by changing capital structure. The stock price doesn't move at announcement. After recapitalization, equity value is lower but shareholders received offsetting dividends.

Q5: Under MM with taxes, a firm has V_U = $10M and issues $4M of perpetual debt at τ = 35%. What is V_L?

$10.0 million $11.4 million $14.0 million $12.6 million

Explanation: V_L = V_U + τD = $10M + (0.35 × $4M) = $10M + $1.4M = $11.4M. The tax shield from debt adds $1.4M of value.

Q6: A firm has levered beta of 1.6 at D/E = 0.6. What is the unlevered (asset) beta?

0.96 2.56 1.0 1.25

Explanation: β_U = β_L / (1 + D/E) = 1.6 / (1 + 0.6) = 1.6 / 1.6 = 1.0

Q7: In the homemade leverage example: if a firm adds 40% debt and you own 20 shares, how do you replicate your original all-equity position?

Sell 40% of shares and lend the proceeds at the risk-free rate Borrow 40% of your portfolio value and buy more shares Keep all shares and ignore the leverage change Sell all shares and buy bonds

Explanation: To "unlever" your position, sell shares in proportion to the firm's debt ratio (40%) and lend those proceeds. Your interest income + dividends from remaining shares = original all-equity cash flow.

Q8: Which statement about WACC with taxes is correct?

WACC increases with leverage because equity becomes riskier WACC is always equal to the return on assets The tax shield has no effect on WACC WACC decreases with leverage due to the tax-deductibility of interest

Explanation: With taxes, WACC = (D/V) × r_D × (1−τ) + (E/V) × r_E. The (1−τ) term on debt means the after-tax cost of debt is lower, so WACC falls as leverage increases (ignoring distress costs).

Exam-Style Practice Problems

These problems mirror Section B style. Remember: show formula → substitution → calculation → interpretation for full marks.

MM Propositions Tutorial Problem 2

Problem 1: Leverage Changes and Cost of Capital

Consider a firm with D/V = 0.1. The market value of equity is £9,000. The expected return on debt is 5%, the expected return on equity is 10%, and there are 100 shares outstanding. No taxes or financial distress costs.

D/V: 0.1 (so E/V = 0.9)
Equity value: £9,000
r_D: 5%
r_E: 10%
Shares: 100

Tasks:

Calculate WACC, return on assets (r_A), and enterprise value
If leverage increases to D/V = 0.3 with r_D = 6%, find new r_E and WACC
What happens to stock price at announcement?
What happens to equity value after recapitalization? Are shareholders better off?

Step 1: Find V and WACC

E = 0.9V = £9,000 → V = £10,000

D = 0.1 × £10,000 = £1,000

WACC = 0.1 × 5% + 0.9 × 10% = 0.5% + 9% = 9.5%

By MM II (no taxes): r_A = WACC = 9.5%

Step 2: New cost of equity at D/V = 0.3

Using MM II: r_E = r_A + (D/E)(r_A − r_D)

D/E = 0.3/0.7 = 0.4286

r_E = 9.5% + 0.4286 × (9.5% − 6%) = 9.5% + 1.5% = 11%

New WACC = 0.3 × 6% + 0.7 × 11% = 1.8% + 7.7% = 9.5% (unchanged!)

Step 3: Stock price at announcement

By MM I: no value created or destroyed. Stock price unchanged.

Step 4: After recapitalization

New equity value = 0.7 × £10,000 = £7,000

But shareholders received special dividend = increase in debt = £3,000 − £1,000 = £2,000

Total wealth = £7,000 + £2,000 = £9,000 = original equity

Shareholders are equally well off (just with different composition)

✓ Interpretation for Exam: This demonstrates MM's core insight: in perfect markets, capital structure doesn't create value. Shareholders can replicate any leverage personally.

Tax Shield Tutorial Problem 3

Problem 2: NPV with Tax Shield

Your firm has:

Shares: 20,000 at $25 each
Debt: $500,000 (riskless)
Equity beta: 2
Risk-free rate: 5%
Market risk premium: 8%
Tax rate: 40%

New opportunity: Buy Weeblesoft for $800,000 (100% equity financed currently). FCF next year = $150,000, growing at 3% forever. You'll use your existing D/E ratio for the acquisition.

Tasks:

Calculate NPV including tax benefits of debt
How much does the tax shield contribute to NPV?

Step 1: Calculate cost of equity and current structure

r_E = r_f + β × MRP = 5% + 2 × 8% = 21%

Market cap = 20,000 × $25 = $500,000

D = $500,000, E = $500,000 → V = $1,000,000

D/V = 0.5, E/V = 0.5

Step 2: Calculate WACC (with taxes)

WACC = (D/V) × r_D × (1−τ) + (E/V) × r_E

WACC = 0.5 × 5% × (1−0.4) + 0.5 × 21%

WACC = 0.5 × 3% + 0.5 × 21% = 1.5% + 10.5% = 12%

Step 3: Calculate NPV with tax shield

PV of after-tax FCF = FCF × (1−τ) / (WACC − g)

= $150,000 × 0.6 / (0.12 − 0.03) = $90,000 / 0.09 = $1,000,000

NPV = −$800,000 + $1,000,000 = $200,000

Should invest? YES (NPV > 0)

Step 4: Isolate tax shield contribution

Pre-tax WACC = 0.5 × 5% + 0.5 × 21% = 2.5% + 10.5% = 13%

V_U (NPV without tax shield) = −$800,000 + $90,000/(0.13−0.03)

= −$800,000 + $900,000 = $100,000

PV of tax shield = $200,000 − $100,000 = $100,000

✓ Key Takeaway: Half the NPV ($100K) comes from the tax shield! This shows why firms prefer debt when taxes exist.

Optimal Capital Structure Tutorial Problem 4

Problem 3: Finding Optimal Debt Ratio

Your company has $200M debt and $800M equity (total V = $1B). EBIT = $80M perpetual. Current rating AAA (r_D = 4.3%). Levered beta = 1.5, r_f = 4%, MRP = 5%, tax rate = 30%.

Compare debt ratios: 20%, 30%, 40%

Which maximizes firm value?

Step 1: Find unlevered beta

Current D/E = 200/800 = 0.25

β_U = β_L / (1 + D/E) = 1.5 / 1.25 = 1.2

Step 2: Calculate for each debt ratio

D/V	D/E	β_L	r_E	Interest Rate	WACC
20%	0.25	1.5	11.5%	4.30%	9.80%
30%	0.429	1.714	12.57%	4.63%	9.58%
40%	0.667	2.0	14.0%	4.84%	9.37%

Step 3: WACC calculations shown

At 20%: WACC = 0.2 × 4.3% × 0.7 + 0.8 × 11.5% = 0.60% + 9.20% = 9.80%

At 30%: WACC = 0.3 × 4.63% × 0.7 + 0.7 × 12.57% = 0.97% + 8.80% = 9.58%

At 40%: WACC = 0.4 × 4.84% × 0.7 + 0.6 × 14.0% = 1.36% + 8.40% = 9.37%

✓ Conclusion: 40% debt ratio gives the lowest WACC (9.37%), maximizing firm value. Lower WACC → higher PV of future cash flows.

⚠️ Exam Note: Iterative Interest Rate
The solution required iterating to find stable credit ratings:
• At 30%: Started with 4.3% → Coverage = 6.2 → A+ rating → 4.63% → Coverage = 5.76 (still A+) ✓
• At 40%: Needed 3 iterations to settle on A- rating at 4.84%

Homemade Leverage Tutorial Problem 1

Problem 4: Unlevering Your Position

ABC is converting from all-equity to 40% debt. Currently: 200 shares at £10, EBIT = £500 forever, r_D = 10%, no taxes. You own 20 shares. All earnings paid as dividends.

Show how to replicate your original all-equity cash flow after the firm levers up.

Step 1: Original all-equity cash flow

EPS = £500 / 200 = £2.50

Your cash flow = 20 × £2.50 = £50

Step 2: After 40% debt restructuring

V = £10 × 200 = £2,000

New debt = 0.4 × £2,000 = £800

Shares repurchased = £800 / £10 = 80 shares

Remaining shares = 200 − 80 = 120

Interest = 0.10 × £800 = £80

Net income = £500 − £80 = £420

New EPS = £420 / 120 = £3.50

If you kept 20 shares: 20 × £3.50 = £70 (higher but riskier!)

Step 3: Unlever your position

Sell 40% of your shares: 0.4 × 20 = 8 shares

Proceeds = 8 × £10 = £80

Lend £80 at 10%: Interest income = £8

Keep 12 shares: Dividends = 12 × £3.50 = £42

Total = £8 + £42 = £50 ✓ (same as original!)

✓ Interpretation: This proves MM I: you can "undo" any leverage change through personal lending/borrowing. Capital structure doesn't affect shareholder wealth because investors can adjust on their own.

Interactive Visualizations

See how leverage affects cost of equity and WACC under different scenarios.

MM Proposition II: Cost of Capital vs Leverage

Debt-to-Equity Ratio (D/E) 0.5

Return on Assets (r_A) 12%

Cost of Debt (r_D) 6%

Tax Rate (τ) 0%

15.0%

Cost of Equity (r_E)

12.0%

WACC

33.3%

D/V

66.7%

E/V

Live Calculation

r_E = 12% + 0.5 × (12% − 6%) = 15%

Beta & Leverage Relationship

Unlevered Beta (β_U) 1.0

Debt-to-Equity Ratio (D/E) 0.5

1.50

Levered Beta (β_L)

50%

Risk Increase

Live Calculation

β_L = 1.0 × (1 + 0.5) = 1.50

Tax Shield Value Calculator

Debt Amount ($M) $5M

Tax Rate 35%

$1.75M

PV of Tax Shield (τ × D)

          💡 Insight: With perpetual debt, the tax shield equals τ × D. This is "free" value from using debt — it's why firms prefer debt financing when taxes exist!
        

Exam Readiness Checklist

Check off each item as you master it. Focus on both calculation AND interpretation.

Section A: News Analysis (Capital Structure)

Be ready to apply these concepts to a real article in ~4 sentences:

MM Proposition I: Can explain why capital structure might be irrelevant and when it matters
Trade-off Theory: Can discuss tax shield benefits vs. financial distress costs
Pecking Order: Know the hierarchy (internal funds → debt → equity) and why
Market Timing: Can explain why firms issue equity when stock is "overvalued"
Management Entrenchment: Understand how managers may prefer low debt for job security

          📝 Section A Answer Template:
          Identify the finance issue (e.g., "Company X is increasing leverage...")
Name the theory (e.g., "According to trade-off theory...")
Apply to facts (e.g., "Given their stable cash flows, more debt increases tax shield...")
State implication (e.g., "This should increase firm value, though bankruptcy risk rises")

        

Section B: Calculation Skills

WACC calculation — both with and without taxes
Cost of equity via MM II: r_E = r_A + (D/E)(r_A − r_D)
Cost of equity via CAPM: r_E = r_f + β_L × MRP
Lever/unlever beta: β_L = β_U(1 + D/E)
Tax shield value: V_L = V_U + τD
Homemade leverage: Can show how to replicate/undo firm leverage
EBIT coverage iteration: Can iterate to find stable credit rating

Section B: Interpretation Skills

⚠️ Critical: Interpretation is worth 40-50% of Section B marks! Don't just calculate — explain what it means.

Can explain why WACC stays constant under MM (no taxes)
Can explain why shareholders are "equally well off" after recapitalization
Can interpret what lower WACC implies for firm value
Can explain why equity becomes riskier with more leverage
Can discuss trade-offs in optimal capital structure (tax shield vs. distress)

Common Exam Traps to Avoid

Trap: Thinking lower WACC always means firm is doing better — context matters!
Trap: Forgetting (1 − τ) in WACC with taxes
Trap: Confusing D/V with D/E — double-check which ratio is given
Trap: Not iterating for credit rating when interest rate changes affect coverage
Trap: Skipping interpretation — always state what your number means!

Quick Reference: Key Results

Scenario	Key Result
MM I (no taxes)	V_L = V_U (capital structure irrelevant)
MM II (no taxes)	WACC = r_A (constant regardless of leverage)
MM with taxes	V_L = V_U + τD (debt adds value)
Stock price at recap announcement	Unchanged (no value created)
Shareholder wealth after recap	Same (equity ↓ but dividend ↑)