📖 资本结构综合练习

Corporate Finance

I. Lesson Focus

This lesson integrates all major capital structure concepts tested at CFA Level I: MM propositions (with and without taxes), the trade-off theory, pecking order theory, EBIT-EPS indifference analysis, WACC minimization, and the impact of taxes versus financial distress costs. Candidates must master both the theoretical formulas and their practical application in corporate decision-making.

II. The Problem

A manufacturing firm currently financed entirely with equity is debating whether to issue bonds to increase leverage. How much value will the interest tax shield create? How much will expected financial distress costs offset that benefit? At different levels of EBIT, will leverage increase earnings per share (EPS)? Within the trade-off theory framework, what target debt ratio maximizes firm value? These are the core real-world and exam questions in corporate finance capital structure decisions. This lesson systematically reviews and integrates the theories, formulas, and scenarios to build mastery.

III. Core Capital Structure Theories Review

Capital structure refers to the mix of long-term debt and equity used to finance the firm. The primary theories are:

1. Modigliani-Miller (MM) Proposition I without Taxes
   In a perfect capital market (no taxes, no bankruptcy costs, no agency costs), firm value is independent of capital structure.
   $V_L = V_U$
   Weighted average cost of capital (WACC) remains constant regardless of leverage.

2. MM Proposition I with Corporate Taxes
   When corporate taxes are introduced, the tax deductibility of interest creates a tax shield that increases firm value.
   $V_L = V_U + t_c \times D$
   where $t_c$ is the corporate tax rate and $D$ is the market value of debt.
   WACC declines as leverage increases.

3. Trade-off Theory
   Firm value = Unlevered value + Present value of tax shields − Present value of financial distress costs.
   An optimal debt level exists where the marginal benefit of the tax shield equals the marginal cost of financial distress.
   Financial distress costs include direct costs (legal and administrative fees) and indirect costs (lost customers, employee turnover, and diverted management attention).

4. Pecking Order Theory
   Due to information asymmetry, firms prefer to finance first with internal funds, then debt, and finally equity.
   There is no well-defined optimal capital structure; observed leverage ratios are the cumulative result of financing deficits.

5. Agency Costs and Signaling
   Debt can reduce equity agency costs (free cash flow hypothesis) but may create conflicts between shareholders and debtholders (asset substitution problem).

IV. EBIT-EPS Analysis and the Indifference Point

EBIT-EPS analysis determines at what level of operating profit leverage improves earnings per share.
The indifference point (where EPS is the same under two financing plans) is found by solving:
$$ \frac{(EBIT - I_1)(1-t) - PD_1}{N_1} = \frac{(EBIT - I_2)(1-t) - PD_2}{N_2} $$ where $I$ = interest expense, $PD$ = preferred dividends, and $N$ = number of shares outstanding.
Above the indifference point, the leveraged plan produces higher EPS; below it, the all-equity plan produces higher EPS.

V. Weighted Average Cost of Capital (WACC) and Optimal Capital Structure

$$ WACC = w_d \times r_d \times (1-t_c) + w_e \times r_e $$ The optimal capital structure is the debt-equity mix that minimizes WACC (and therefore maximizes firm value).
As debt increases, both $r_d$ and $r_e$ rise due to higher risk, causing WACC to follow a U-shaped pattern.

Worked Cases

Case 1: Tax Shield Value under MM with Taxes

ABC Company has an unlevered value $V_U = 80$ million. It plans to issue 30 million of perpetual debt at 7% interest with a corporate tax rate of 25%.
Tax shield value = $t_c \times D = 0.25 \times 30 = 7.5$ million
Levered firm value $V_L = 80 + 7.5 = 87.5$ million
Equity value after leverage = $V_L - D = 87.5 - 30 = 57.5$ million

Conclusion: Debt increases total firm value by 7.5 million, and this entire gain accrues to shareholders.

Case 2: Calculating the EBIT-EPS Indifference Point

XYZ Company has 1 million shares outstanding. It is considering issuing 30 million in bonds (8% coupon) to repurchase shares at the current price of $20 per share; tax rate = 25%.
- All-equity plan: Issue 1.5 million new shares to raise 30 million; no additional interest.
- Debt plan: Additional interest = 30 × 0.08 = 2.4 million; shares outstanding fall to 0.85 million.

Set EPS equal and solve for indifference EBIT ($X$):
$$ \frac{(X - 2.4)(0.75)}{0.85} = \frac{X \times 0.75}{2.5} $$ Solving yields $X = 12$ million.
Thus, when EBIT > 12 million, the debt plan produces higher EPS.

Case 3: Target Capital Structure under Trade-off Theory

A firm has an unlevered value $V_U = 1,000$ million. The present value of tax shields rises with debt, but the present value of financial distress costs rises at an accelerating rate. Data are as follows:

Conclusion: Optimal debt is approximately 400 million (D/V ≈ 37.4%), producing the highest firm value (1,070 million) and lowest WACC (9.35%).

Traps

Key Formulas

• $V_L = V_U + t_c D$ (MM with taxes)
• $V_L = V_U + PV(\text{Tax Shield}) - PV(\text{Financial Distress Costs})$ (Trade-off theory)
• $WACC = w_d r_d (1-t_c) + w_e r_e$
• EBIT-EPS indifference: $\frac{(EBIT-I_1)(1-t)-PD_1}{N_1}=\frac{(EBIT-I_2)(1-t)-PD_2}{N_2}$
• Levered cost of equity: $r_e = r_0 + (r_0 - r_d)(D/E)(1-t_c)$ (MM with taxes)
• Firm value maximization ⇔ WACC minimization

Practice Questions

Q1. According to the MM proposition with taxes, if the corporate tax rate is 25% and the firm issues 50 million of perpetual debt, the present value of the tax shield is closest to:
A. 0 B. 10 million C. 12.5 million D. 50 million

Q2. Under the trade-off theory, the optimal capital structure occurs when:
A. The present value of the tax shield is maximized
B. Financial distress costs are zero
C. The marginal benefit of the tax shield equals the marginal cost of financial distress
D. WACC is maximized

Q3. Which of the following best supports the pecking order theory?
A. More profitable firms have higher leverage ratios
B. More profitable firms have lower leverage ratios
C. Firms maintain a strict target debt ratio
D. The cost of external equity is lower than debt

Q4. A firm has EBIT of 8 million and is currently unlevered. If it issues 20 million of debt at 6% interest (tax rate 25%) and financial distress costs are ignored, firm value will increase by:
A. 5 million B. 3 million C. 20 million D. 0

Q5. In EBIT-EPS analysis, when actual EBIT is above the indifference point:
A. Debt financing produces lower EPS than equity financing
B. Debt financing produces higher EPS than equity financing
C. EPS is the same under both plans
D. Cannot be determined

Q6. As financial leverage increases, which of the following typically rises first?
A. After-tax cost of debt
B. Cost of equity
C. Value of the tax shield
D. Present value of direct bankruptcy costs

Q7. A firm seeks to minimize its WACC. According to trade-off theory, its optimal debt ratio should be lower when:
A. The tax rate is higher
B. Business risk is higher
C. Financial distress costs are higher
D. The present value of the tax shield equals distress costs

Q8. In a perfect capital market with no taxes, according to MM Proposition I without taxes, increasing debt will cause:
A. Total firm value to rise
B. WACC to decline
C. Cost of equity to rise exactly offsetting the lower cost of debt
D. Earnings per share to necessarily increase

Answers

Takeaways

• MM with taxes core formula: $V_L = V_U + t_c D$; the tax shield directly increases firm value
• Under trade-off theory, optimal capital structure balances tax shield benefits against financial distress costs
• The EBIT-EPS indifference point is the critical threshold for determining whether leverage improves EPS
• Pecking order theory emphasizes financing hierarchy due to information asymmetry rather than a fixed target ratio
• WACC follows a U-shape; its minimum corresponds to maximum firm value
• Real decisions must simultaneously consider tax rates, business risk, asset tangibility, and growth opportunities