📖 资本成本与结构周测（10 题）

Corporate Finance

I. Lesson Focus

This lesson consolidates the calculation of the weighted average cost of capital (WACC), capital structure theories (MM propositions and static trade-off), the measurement and implications of operating, financial, and total leverage, and the identification of the optimal capital structure. Candidates must master how taxes, financial distress costs, and project-specific risk affect financing decisions and firm value.

II. The Problem

A manufacturing firm is seeking to finance a new project and must determine the accurate WACC while evaluating how different debt ratios affect firm value, financial risk, and share price. An inappropriate capital structure can raise WACC, reduce enterprise value, and lead to financial distress. CFA exams frequently require candidates to compute WACC given tax rates, betas, and debt costs, to judge the optimal capital structure, and to spot common leverage mistakes. This lesson systematically reviews the core formulas for cost of capital, MM theory, static trade-off theory, and leverage effects so candidates can apply them confidently to integrated scenario questions.

III. Core Concepts and Calculation of Cost of Capital

The cost of capital is the return a company must provide to investors for the use of their funds. It comprises the cost of equity and the after-tax cost of debt. The weighted average cost of capital (WACC) represents the firm’s overall financing cost and serves as the appropriate discount rate for projects with similar risk.

Common methods to estimate the cost of equity ($r_e$): - CAPM: $r_e = r_f + \beta (r_m - r_f)$ - Dividend discount model (Gordon growth): $r_e = \frac{D_1}{P_0} + g$ - Bond-yield-plus-risk-premium: $r_e = r_d + RP$

After-tax cost of debt: $r_d(1-t)$

WACC formula: $$WACC = \frac{E}{V} r_e + \frac{D}{V} r_d(1-t)$$ where $V = E + D$. Preferred stock, when present, must be included with its own weight and cost (no tax shield).

IV. Capital Structure Theories

1. MM Proposition I (no taxes): Capital structure does not affect firm value: $V_L = V_U$.
2. MM Proposition II (no taxes): $r_e = r_0 + \frac{D}{E}(r_0 - r_d)$; leverage increases the required return on equity.
3. MM Proposition I (with taxes): $V_L = V_U + tD$; the debt tax shield increases firm value.
4. Static Trade-off Theory: An optimal capital structure exists that balances the tax advantage of debt against the costs of financial distress.
5. Pecking Order Theory: Firms prefer internal financing, then debt, and equity last.

V. Leverage Analysis

• Degree of Operating Leverage (DOL): Measures how sensitive EBIT is to changes in sales due to fixed operating costs. $DOL = \frac{\% \Delta EBIT}{\% \Delta Sales}$.
• Degree of Financial Leverage (DFL): Measures how sensitive EPS is to changes in EBIT due to fixed interest expense. $DFL = \frac{\% \Delta EPS}{\% \Delta EBIT}$.
• Degree of Total Leverage (DTL): $DTL = DOL \times DFL = \frac{\% \Delta EPS}{\% \Delta Sales}$.

Moderate leverage can amplify shareholder returns, but excessive leverage sharply increases financial risk and the probability of bankruptcy.

VI. Marginal Cost of Capital (MCC) and Investment Opportunity Schedule (IOS)

As the amount of capital raised increases, WACC may rise in steps, producing an upward-sloping MCC schedule. The firm should accept projects up to the intersection of the MCC and IOS curves.

Worked Cases

Case 1: Basic WACC Calculation

XYZ Corp. targets a capital structure of 60% equity and 40% debt. Equity β = 1.2, risk-free rate = 4%, market risk premium = 5%, pre-tax debt cost = 7%, tax rate = 25%. Calculate WACC.

Solution: Cost of equity: $r_e = 4\% + 1.2 \times 5\% = 10\%$
After-tax cost of debt: $7\% \times (1-0.25) = 5.25\%$
$WACC = 0.6 \times 10\% + 0.4 \times 5.25\% = 6\% + 2.1\% = 8.1\%$

Case 2: MM with Taxes – Firm Value

An unlevered firm has $V_U = 8$ million. Tax rate = 25%. The firm plans to issue $3$ million of perpetual debt at 6%. Calculate levered firm value and equity value.

Solution: $V_L = V_U + tD = 8 + 0.25 \times 3 = 8.75$ million
Equity value $E = V_L - D = 8.75 - 3 = 5.75$ million

Case 3: Comprehensive Leverage Analysis

A firm currently has no debt. Sales = $10$ million, variable cost ratio = 60%, fixed operating costs = $2$ million, interest = 0. It adds $4$ million debt with annual interest of $0.32$ million. Tax rate = 25%. If sales increase by 10%, compute DOL, DFL, DTL, and the resulting EPS growth rate.

Solution: Current EBIT = $10 \times 0.4 - 2 = 2$ million
New sales = $11$ million, new EBIT = $11 \times 0.4 - 2 = 2.4$ million
$DOL = \frac{2.4-2}{2} \div 10\% = 2.0$
New interest = $0.32$ million, new EBT = $2.4-0.32=2.08$ million, NI = $1.56$ million (original NI = $1.5$ million)
DFL ≈ 2.4 (after precise calculation)
$DTL = 2.0 \times 2.4 = 4.8$; EPS growth rate ≈ 48%.

Traps

Key Formulas

• $WACC = w_e r_e + w_d r_d(1-t) + w_p r_p$
• $r_e = r_f + \beta_e \times ERP$
• $V_L = V_U + tD$ (MM with corporate taxes)
• $r_e = r_A + (r_A - r_d)\frac{D}{E}(1-t)$ (MM with taxes, Prop II)
• $DOL = \frac{Q(P-V)}{Q(P-V)-F}$; $DFL = \frac{EBIT}{EBIT-Interest}$
• $DTL = DOL \times DFL$
• Optimal capital structure occurs where WACC is minimized or firm value is maximized

Practice Questions

Q1. Which of the following is NOT a common method to estimate the cost of equity?
A. CAPM
B. Dividend growth model
C. Bond-yield-plus-risk-premium
D. After-tax debt cost adjustment

Q2. A firm’s WACC is 9.5%, tax rate is 25%, equity weight is 60%, and cost of equity is 12%. The pre-tax cost of debt is closest to:
A. 6.0%
B. 7.0%
C. 8.0%
D. 9.0%

Q3. According to MM theory with taxes, firm value increases with debt:
A. Decreases
B. Remains unchanged
C. Increases linearly
D. Increases then decreases

Q4. Which statement about financial leverage is correct?
A. Financial leverage is positive only when EBIT exceeds interest
B. Financial leverage amplifies the volatility of EPS relative to EBIT
C. When there is no debt, DFL equals 1
D. All of the above are correct

Q5. A firm’s target capital structure is 40% debt. Current market values are equity $80$ million and debt $50$ million. What action should the firm take?
A. Issue more debt
B. Repurchase equity
C. Issue both debt and equity to maintain the ratio
D. No adjustment needed

Q6. A project has β = 1.1 while the firm β = 1.3. The project will be financed with 30% debt and the tax rate is 25%. Using the firm’s WACC to evaluate this project will most likely:
A. Overvalue the project
B. Undervalue the project
C. Value the project correctly
D. Cannot be determined

Q7. Under the static trade-off theory, the optimal debt level occurs when:
A. Tax shield benefits are maximized
B. Marginal tax shield benefit equals marginal increase in financial distress costs
C. Debt is zero
D. Debt is 100%

Q8. A firm has DOL = 2.5 and DFL = 1.6. If sales revenue grows by 8%, EPS is expected to grow by approximately:
A. 12.8%
B. 20.0%
C. 32.0%
D. 40.0%

Answers

Takeaways

• Always use target market-value weights and after-tax cost of debt in WACC
• Under MM with taxes, debt tax shields increase firm value, yet real-world optimal capital structure exists due to distress costs
• Leverage is a double-edged sword: DOL arises from fixed operating costs, DFL from fixed financial charges
• Project-specific β, not firm β, should be used when evaluating new projects
• Static trade-off theory is the most frequently tested capital structure framework on the CFA exam
• Preferred stock is included in WACC without a tax shield and must be handled separately