📖 资本成本与结构复习

Corporate Finance

I. Lesson Focus

II. The Problem

A Chinese listed manufacturing company is considering a new production line expected to generate stable annual free cash flows. The firm currently has a 40% debt ratio, cost of equity of 12%, pre-tax cost of debt of 7%, and a corporate tax rate of 25%. Management needs to know: What is the appropriate discount rate for this project? How will WACC change if leverage is increased to 60%? Will higher leverage necessarily increase firm value? These questions directly determine whether the project should be accepted, whether the financing decision is optimal, and whether shareholder value can be maximized. This lesson systematically reviews the calculation of cost of capital, capital structure theories, and their interactions to equip candidates to handle integrated scenario questions accurately in the exam.

III. Fundamental Concepts and Calculation of Cost of Capital

The cost of capital is the minimum rate of return required by investors and the “price” a company must pay to obtain funds. The central concept in corporate finance is the weighted average cost of capital (WACC), which is the weighted average of the cost of equity and after-tax cost of debt using market-value weights. WACC is used to discount free cash flow to the firm (FCFF) when calculating enterprise value.

WACC formula: $$ WACC = w_e \times r_e + w_d \times r_d \times (1 - t) $$ where: - $w_e$ = market value weight of equity = $E / (E + D)$ - $w_d$ = market value weight of debt = $D / (E + D)$ - $r_e$ = cost of equity - $r_d$ = pre-tax cost of debt - $t$ = corporate tax rate

Three primary methods to estimate cost of equity ($r_e$): 1. CAPM: $r_e = r_f + \beta \times (r_m - r_f)$ 2. Dividend discount model (DDM): $r_e = \frac{D_1}{P_0} + g$ 3. Bond-yield-plus-risk-premium: $r_e = r_d + \text{Equity Risk Premium}$

Cost of debt ($r_d$) is typically estimated using the current yield to maturity (YTM) on comparable bonds. When unavailable, use the market rate corresponding to the firm’s credit rating.

Choice of weights: CFA requires target capital structure market-value weights, not book values or current weights. When the target is unknown, use current market weights or the average of comparable firms.

IV. Capital Structure Theories

Capital structure refers to the mix of debt and equity. The Modigliani-Miller (MM) propositions provide the foundational framework.

MM Proposition I (no taxes): In a perfect market (no taxes, no bankruptcy costs, no agency costs), firm value is independent of capital structure: $V_L = V_U$.

MM Proposition II (no taxes): Increasing leverage raises the cost of equity, exactly offsetting the cheaper debt: $$ r_e = r_0 + (r_0 - r_d) \times \frac{D}{E} $$ where $r_0$ is the cost of capital for an all-equity firm.

With corporate taxes (MM with taxes): The interest tax shield increases firm value: $$ V_L = V_U + t \times D $$ WACC declines with leverage: $$ WACC = r_0 \times (1 - t \times w_d) $$

Trade-off Theory (real world): Firms balance the benefit of interest tax shields against costs of financial distress (bankruptcy and agency costs), leading to an optimal capital structure where WACC is minimized and firm value is maximized.

Pecking Order Theory: Due to information asymmetry, firms prefer internal funds, then debt, and finally equity.

V. Factors Affecting Cost of Capital and Adjustments

• Project risk: If a project’s risk differs from the firm’s average, adjust WACC (pure-play method or β adjustment).
• Country risk: Add a country risk premium (CRP) for cross-border projects.
• Flotation costs: Issuance costs raise the effective cost of capital; adjust the issue price rather than adding directly to WACC.
• Effect of leverage on β: Use the Hamada formula: $$ \beta_L = \beta_U \times [1 + (1 - t) \times \frac{D}{E}] $$

Worked Cases

Case 1: Basic WACC Calculation

XYZ Company has market equity of CNY 800 million and debt of CNY 400 million. Equity β = 1.2, risk-free rate = 4%, market risk premium = 6%, pre-tax cost of debt = 7%, tax rate = 25%. Calculate WACC.

Solution: - $w_e = 8/(8+4) = 2/3 \approx 0.6667$ - $w_d = 4/12 = 0.3333$ - $r_e = 4\% + 1.2 \times 6\% = 11.2\%$ - $WACC = 0.6667 \times 11.2\% + 0.3333 \times 7\% \times (1-0.25) = 7.467\% + 1.75\% = 9.217\% \approx 9.22\%$

Case 2: Effect of Leverage Change on WACC and Value (with taxes)

An unlevered firm has $V_U$ = CNY 100 million, $r_0$ = 12%, tax rate = 25%. It adds permanent debt of CNY 30 million at a pre-tax cost of 8%. Calculate the new WACC and firm value.

Solution: - Tax shield = $0.25 \times 30 = 7.5$ million - $V_L = 100 + 7.5 = 107.5$ million - New weights: $E = 107.5 - 30 = 77.5$ million, $w_d = 30/107.5 \approx 0.2791$, $w_e \approx 0.7209$ - $WACC = 0.7209 \times 12\% \times (1 - 0.25 \times 0.2791) \approx 8.05\%$ - Or directly: $WACC = 12\% \times (1 - 0.25 \times 0.2791) \approx 8.05\%$

Case 3: Hamada Formula to Adjust β and Calculate New Cost of Equity

An unlevered firm has $\beta_U$ = 0.9, tax rate = 25%, plans to move from D/E = 0 to D/E = 1.0. Pre-tax debt cost = 6.5%, risk-free rate = 5%, market risk premium = 5.5%. Calculate new $r_e$ and WACC.

Solution: - $\beta_L = 0.9 \times [1 + (1-0.25)\times1.0] = 0.9 \times 1.75 = 1.575$ - $r_e = 5\% + 1.575 \times 5.5\% = 13.6625\%$ - $w_d = 0.5$, $w_e = 0.5$ - $WACC = 0.5 \times 13.6625\% + 0.5 \times 6.5\% \times (1-0.25) = 6.831\% + 2.4375\% = 9.26875\% \approx 9.27\%$

Traps

Key Formulas

• $WACC = w_e r_e + w_d r_d (1-t)$
• $r_e(\text{CAPM}) = r_f + \beta(r_m - r_f)$
• $V_L = V_U + tD$ (MM with taxes)
• $\beta_L = \beta_U [1 + (1-t)(D/E)]$ (Hamada)
• $r_e = r_0 + (r_0 - r_d)(D/E)$ (MM II no taxes)
• $WACC = r_0 (1 - t \times w_d)$ (tax-adjusted simplification)
• $r_e(\text{DDM}) = D_1/P_0 + g$

Practice Questions

Q1. Which of the following is most likely the correct choice of weights when calculating WACC?
A. Book-value weights
B. Market-value weights based on the target capital structure
C. Current book debt-to-equity ratio
D. Average book weights of comparable firms

Q2. According to MM Proposition II (no taxes), when a firm increases leverage:
A. WACC remains constant
B. Cost of equity declines
C. Cost of equity rises to exactly offset the advantage of cheaper debt
D. Firm value increases

Q3. A firm has β = 1.1, $r_f$ = 4%, market risk premium = 6%, pre-tax cost of debt = 8%, equal 50% weights, and a 25% tax rate. Its WACC is closest to:
A. 7.15%
B. 7.95%
C. 8.35%
D. 9.05%

Q4. After introducing corporate income tax, according to MM theory, firm value is positively related to:
A. Equity proportion
B. Amount of debt
C. Unlevered firm value only
D. Bankruptcy costs

Q5. When using the Hamada formula to adjust levered β, which factor must be included?
A. Market risk premium
B. (1-t)
C. Risk-free rate
D. Dividend growth rate

Q6. If a new project has significantly higher systematic risk than the firm’s existing operations, the most appropriate action is to:
A. Use the company WACC directly
B. Increase the WACC
C. Use only the cost of equity
D. Ignore the risk difference

Q7. Which statement about the trade-off theory is correct?
A. Optimal capital structure occurs when WACC is highest
B. An optimal structure exists when marginal tax-shield benefit equals marginal financial-distress cost
C. Debt should never be used
D. Information asymmetry leads to using only equity

Q8. An all-equity firm has $r_0$ = 10%, tax rate = 30%, and plans to borrow debt equal to 40% of total value at a pre-tax cost of 6%. Using the MM with-taxes framework, WACC is closest to:
A. 7.0%
B. 7.6%
C. 8.2%
D. 9.0%

Answers

Takeaways

• WACC must use market-value target weights; the debt component is adjusted for the tax shield
• Under MM, capital structure is irrelevant without taxes but valuable with taxes; real-world optimal structure balances tax shields against distress costs
• Increasing leverage raises equity β and $r_e$; the Hamada formula is the essential adjustment tool
• Adjust WACC when project risk differs from the firm average
• CAPM is the preferred method for cost of equity; YTM is preferred for cost of debt
• Common exam traps include mixing book weights, omitting tax shields, and discounting the wrong cash flow stream