公司金融（Corporate Finance）
一、本课定位
| 课次 | 主题 | 能力 |
|---|---|---|
| L286 | MM 定理：无税情况 | 解释无税环境下资本结构不影响公司价值，计算无杠杆与有杠杆公司的价值、权益成本及 WACC |
二、我们要解决什么问题？
假设一家公司计划通过发行债券来替换部分股权融资，从而改变资本结构。管理者想知道：在没有公司税和个人税的世界里，这种改变是否会增加公司整体价值？股东是否会因为更高的财务杠杆而要求更高的回报率？如果公司价值不受资本结构影响，那么最优资本结构是否存在？这就是 Modigliani-Miller（MM）定理在无税情况下的核心问题。考试中经常要求考生证明“在完美市场假设下，杠杆不会创造价值”，并计算无杠杆公司（Unlevered）和有杠杆公司（Levered）的价值、权益成本及加权平均资本成本（WACC）。
三、MM 定理的核心假设（完美资本市场）
MM 定理建立在以下严格假设之上，这些假设共同构成了“完美资本市场”（Perfect Capital Market）： - 无交易成本（No transaction costs） - 无税负（No taxes） - 投资者与公司借贷利率相同（Homogeneous expectations, investors and firms can borrow at the same risk-free rate） - 信息对称（Symmetric information） - 无破产成本（No bankruptcy costs） - 个人可进行“ homemade leverage”（投资者可自行借贷复制公司杠杆效果）
在这些假设下，资本结构（Debt/Equity ratio）不会影响公司总价值。
四、MM Proposition I（无税）：公司价值与资本结构无关
核心结论：
有杠杆公司（V_L）的价值等于无杠杆公司（V_U）的价值。
$$ V_L = V_U $$
证明逻辑（套利证明）：
假设存在两家完全相同（同等经营风险、现金流）的公司，一家有杠杆（L），一家无杠杆（U）。若 V_L > V_U，投资者可卖出杠杆公司股票，同时买入无杠杆公司股票并自行借债（homemade leverage），获得无风险套利收益。市场力量会使价格回归均衡，最终 V_L = V_U。
实际意义：公司无法通过改变债务与股权的比例来增加自身价值。价值完全由经营性现金流（Operating Cash Flows）和经营风险决定。
五、MM Proposition II（无税）：权益成本随杠杆线性上升
核心公式：
有杠杆公司权益要求的回报率（r_E）等于无杠杆公司权益成本（r_U）加上风险溢价，该溢价与债务权益比（D/E）成正比。
$$ r_E = r_U + (r_U - r_D) \times \frac{D}{E} $$
其中： - r_U = 无杠杆公司权益成本 = 公司资产的必要回报率（也等于无杠杆 WACC） - r_D = 债务成本（在无税、无破产风险下为无风险利率） - D/E = 债务与权益的市场价值之比
经济直觉：杠杆增加了权益的财务风险（Financial Risk），股东因此要求更高回报。权益成本上升的幅度恰好抵消了更便宜债务带来的好处，使得 WACC 保持不变。
六、加权平均资本成本（WACC）在无税情况下的不变性
在无税环境下，WACC 等于无杠杆公司的权益成本，且与资本结构无关。
$$ \text{WACC} = \frac{E}{V} r_E + \frac{D}{V} r_D = r_U $$
无论公司如何调整 D/E，WACC 始终等于 r_U。这再次印证了 MM I：因为公司价值 V = EBIT / WACC，当 WACC 不变时，V 也不变。
完整案例演算
案例 1：基本价值不变性
ABC 公司预计每年永续 EBIT = 1,000,000 元，无杠杆权益成本 r_U = 12%，债务成本 r_D = 6%。
无杠杆情况：
V_U = EBIT / r_U = 1,000,000 / 0.12 = 8,333,333 元
有杠杆情况（假设发行 4,000,000 元债务替换股权）：
V_L = V_U = 8,333,333 元
E = V_L - D = 8,333,333 - 4,000,000 = 4,333,333 元
D/E = 4,000,000 / 4,333,333 ≈ 0.923
结论：无论是否加杠杆，公司总价值均为 8,333,333 元，证明 MM I 成立。
案例 2：权益成本与 WACC 计算
沿用案例 1 数据，计算有杠杆情况下的 r_E 和 WACC。
r_E = r_U + (r_U - r_D) × (D/E)
= 0.12 + (0.12 - 0.06) × 0.923
= 0.12 + 0.0554 = 0.1754（17.54%）
WACC = (E/V) × r_E + (D/V) × r_D
= (4,333,333/8,333,333) × 0.1754 + (4,000,000/8,333,333) × 0.06
≈ 0.52 × 0.1754 + 0.48 × 0.06 ≈ 0.0912 + 0.0288 = 0.12（12%）
结论：WACC 仍等于 r_U = 12%，验证 MM II 和 WACC 不变性。
案例 3：自制杠杆（Homemade Leverage）套利
假设市场错误定价：V_L = 9,000,000 元（高于 V_U = 8,333,333 元）。投资者持有杠杆公司 1% 股权，价值 90,000 元，年现金流 = 1% × (EBIT - 利息) = 1% × (1,000,000 - 240,000) = 7,600 元。
套利策略：
卖出杠杆公司 1% 股权（获 90,000 元），买入无杠杆公司 1% 股权（需 83,333 元），并自行借款 40,000 元（模仿公司杠杆）。
净现金流出 = 83,333 - 90,000 + 40,000 = 33,333 元（实际为正现金流入）。
每年现金流：1%×EBIT - 利息 = 10,000 - 2,400 = 7,600 元，与原来完全相同，但初始投资更少 → 无风险套利。
市场套利行为将推高 V_U 或压低 V_L，直至二者相等。
易错陷阱对照
| 易错点 | 错误做法 | 正确做法 |
|---|---|---|
| 混淆有税与无税 | 在无税题中仍使用 VL = VU + tD | 严格记住无税时 VL = VU |
| 误以为杠杆能降低 WACC | 认为更多债务会一直降低 WACC | 无税下 WACC 恒等于 r_U，不受 D/E 影响 |
| 错误计算 r_E | 只加 (r_U - r_D) 而不乘 D/E | 必须使用公式 r_E = r_U + (r_U - r_D)(D/E) |
| 忘记 homemade leverage | 认为只有公司能创造杠杆价值 | 个人可通过自制杠杆完全复制公司杠杆效果 |
| 用账面价值而非市场价值 | 用账面 D/E 计算 | 所有 MM 公式必须使用市场价值 |
| 认为 MM 适用于现实 | 直接套用结论到有税、有破产成本的世界 | MM 是理想化基准，现实中税盾和破产成本会改变结论 |
关键公式 / 关系速记
- $V_L = V_U$
- $r_E = r_U + (r_U - r_D) \times \frac{D}{E}$
- $\text{WACC} = r_U$（无税情况下恒定）
- $V_U = \frac{\text{EBIT}}{r_U}$
- $r_A = r_U = \frac{E}{V} r_E + \frac{D}{V} r_D$（资产回报率等于无杠杆权益成本）
- Homemade Leverage：个人借贷可完美替代公司杠杆
练习题（含计算与情景）
Q1. 在 MM 无税世界中，公司价值由以下哪项决定？
A. 资本结构
B. 经营现金流和经营风险
C. 债务利息税盾
D. 破产成本
Q2. 根据 MM Proposition II（无税），当债务权益比上升时：
A. 权益成本下降
B. 权益成本线性上升
C. WACC 下降
D. 公司总价值增加
Q3. 某无杠杆公司 r_U = 15%，若发行债务使 D/E = 1，r_D = 8%，则杠杆后权益成本 r_E 为：
A. 15%
B. 22%
C. 19%
D. 8%
Q4. 在无税完美市场下，公司 WACC 与资本结构的关系是：
A. 随杠杆上升而下降
B. 保持不变
C. 随杠杆上升而上升
D. 先降后升
Q5. 以下哪项不是 MM 定理的无税假设？
A. 无交易成本
B. 存在公司所得税
C. 信息对称
D. 投资者可按相同利率借贷
Q6. 某公司 EBIT = 500,000 元，r_U = 10%，当前无杠杆。若发行 2,000,000 元债务（r_D=6%），公司总价值应为：
A. 5,000,000 元
B. 3,000,000 元
C. 5,000,000 元（保持不变）
D. 无法确定
Q7. Homemade leverage 的主要作用是：
A. 证明个人无法复制公司杠杆
B. 证明投资者可自行创造与公司相同的风险-收益状况
C. 增加公司税盾
D. 降低破产概率
Q8. 若两家公司经营风险完全相同，一家有杠杆一家无杠杆，在无税环境下，它们的：
A. WACC 不同
B. 总价值相同
C. 权益成本相同
D. 债务成本不同
答案与详解
| 题号 | 答案 | 详解 |
|---|---|---|
| Q1 | B | MM I 指出价值仅由经营现金流和资产风险决定，与融资结构无关 |
| Q2 | B | MM II 明确 r_E 随 D/E 线性增加，精确抵消债务低成本的影响 |
| Q3 | B | r_E = 0.15 + (0.15-0.08)×1 = 0.22（22%） |
| Q4 | B | 无税环境下 WACC = r_U，恒定不变 |
| Q5 | B | MM 无税假设明确“无税负”，B 选项错误 |
| Q6 | C | V_L = V_U = 500,000 / 0.10 = 5,000,000 元，资本结构不改变总价值 |
| Q7 | B | Homemade leverage 是 MM 套利证明的核心，投资者可自行复制杠杆效果 |
| Q8 | B | 根据 MM I，V_L = V_U，总价值相同 |
本节要点速记
- 无税完美市场下，资本结构与公司价值无关（V_L = V_U）
- 杠杆增加权益的财务风险，导致 r_E 线性上升
- WACC 在无税情况下恒等于无杠杆权益成本 r_U
- 套利机制（包括 homemade leverage）确保 MM 命题成立
- 所有计算必须使用市场价值而非账面价值
- MM 定理是理想基准，现实中税负和破产成本会打破结论
Corporate Finance
I. Lesson Focus
This lesson explains the Modigliani-Miller (MM) propositions under the assumption of no taxes. Candidates must master the proof that capital structure is irrelevant to firm value, calculate levered cost of equity, demonstrate that WACC remains constant, and understand the arbitrage mechanism of homemade leverage. The material provides the theoretical benchmark against which all later capital-structure topics (tax shields, bankruptcy costs, agency costs) are compared.
II. The Problem
Suppose a firm is considering replacing equity with debt to change its capital structure. In a world without corporate or personal taxes, will this increase the overall value of the company? Will shareholders demand higher returns due to increased financial risk? If firm value is unaffected by leverage, does an optimal capital structure exist? These questions lie at the heart of the MM propositions in a no-tax environment. CFA exams frequently require candidates to prove that “in perfect markets, leverage does not create value” and to compute the values, equity costs, and WACC for both unlevered and levered firms.
III. Core Assumptions of the MM Theorems (Perfect Capital Markets)
The MM theorems rest on strict assumptions that together define a perfect capital market: - No transaction costs - No taxes - Investors and corporations can borrow at the same risk-free rate - Symmetric information - No bankruptcy costs - Investors can create “homemade leverage”
Under these assumptions, capital structure (debt/equity ratio) has no effect on the total value of the firm.
IV. MM Proposition I (No Taxes): Firm Value Is Independent of Capital Structure
Core Conclusion:
The value of a levered firm (V_L) equals the value of an unlevered firm (V_U).
$$ V_L = V_U $$
Arbitrage Proof:
Consider two otherwise identical firms (same operating risk and cash flows), one levered (L) and one unlevered (U). If V_L > V_U, an investor can sell shares in the levered firm, buy shares in the unlevered firm, and borrow personally (homemade leverage) to replicate the same cash-flow risk. This riskless arbitrage profit forces prices back to equilibrium, ensuring V_L = V_U.
Practical Implication: A firm cannot increase its value merely by altering the mix of debt and equity. Value is determined solely by operating cash flows and business risk.
V. MM Proposition II (No Taxes): Cost of Equity Rises Linearly with Leverage
Core Formula:
The required return on levered equity (r_E) equals the unlevered cost of equity (r_U) plus a risk premium proportional to the debt-to-equity ratio.
$$ r_E = r_U + (r_U - r_D) \times \frac{D}{E} $$
where: - r_U = unlevered cost of equity = required return on assets (also equals unlevered WACC) - r_D = cost of debt (risk-free rate in a no-tax, no-bankruptcy world) - D/E = market-value ratio of debt to equity
Economic Intuition: Leverage increases the financial risk borne by equity holders, who therefore demand higher returns. The rise in equity cost exactly offsets the benefit of cheaper debt, leaving WACC unchanged.
VI. Invariance of WACC in a No-Tax World
In the absence of taxes, WACC equals the unlevered cost of equity and is independent of capital structure.
$$ \text{WACC} = \frac{E}{V} r_E + \frac{D}{V} r_D = r_U $$
No matter how the firm adjusts its D/E ratio, WACC remains equal to r_U. This confirms MM Proposition I: because firm value V = EBIT / WACC, constant WACC implies constant V.
Worked Cases
Case 1: Basic Value Invariance
ABC Company expects perpetual EBIT of 1,000,000. The unlevered cost of equity r_U is 12 % and the cost of debt r_D is 6 %.
Unlevered firm:
V_U = EBIT / r_U = 1,000,000 / 0.12 = 8,333,333
Levered firm (issues 4,000,000 of debt to repurchase equity):
V_L = V_U = 8,333,333
Equity value E = V_L – D = 4,333,333
D/E ≈ 0.923
Conclusion: Total firm value remains 8,333,333 regardless of leverage, confirming MM Proposition I.
Case 2: Cost of Equity and WACC Calculation
Using the data from Case 1, compute levered r_E and WACC.
r_E = r_U + (r_U – r_D) × (D/E)
= 0.12 + (0.12 – 0.06) × 0.923
= 0.12 + 0.0554 = 0.1754 or 17.54 %
WACC = (E/V) × r_E + (D/V) × r_D
≈ (0.52)(0.1754) + (0.48)(0.06) = 0.0912 + 0.0288 = 0.12 or 12 %
Conclusion: WACC remains exactly equal to r_U = 12 %, verifying both MM Proposition II and WACC invariance.
Case 3: Homemade Leverage Arbitrage
Suppose the market misprices the levered firm at V_L = 9,000,000 (while true V_U = 8,333,333). An investor owns 1 % of the levered firm (value 90,000), receiving annual cash flow of 7,600 after interest.
Arbitrage strategy: Sell the 1 % levered stake for 90,000, buy 1 % of the unlevered firm for 83,333, and borrow 40,000 personally to replicate the firm’s leverage. Net cash inflow at t = 0 is 6,667 while future cash flows remain identical (7,600). The riskless profit forces prices to converge until V_L = V_U.
Traps
| Common Mistake | Incorrect Approach | Correct Approach |
|---|---|---|
| Confusing no-tax and with-tax cases | Using V_L = V_U + tD in a no-tax question | Strictly apply V_L = V_U when taxes are absent |
| Believing leverage always lowers WACC | Thinking more debt continuously reduces WACC | Recognize WACC equals r_U and is invariant |
| Incorrect r_E formula | Adding only (r_U – r_D) without multiplying by D/E | Must use r_E = r_U + (r_U – r_D)(D/E) |
| Ignoring homemade leverage | Assuming only the firm can create leverage value | Investors can perfectly replicate corporate leverage personally |
| Using book instead of market values | Calculating D/E from accounting figures | All MM formulas require market values |
| Applying MM directly to reality | Stating MM conclusions hold in the real world | MM is an idealized benchmark; taxes and bankruptcy costs change outcomes |
Key Formulas
- $V_L = V_U$
- $r_E = r_U + (r_U - r_D) \times \frac{D}{E}$
- $\text{WACC} = r_U$ (constant in no-tax world)
- $V_U = \frac{\text{EBIT}}{r_U}$
- $r_A = r_U = \frac{E}{V} r_E + \frac{D}{V} r_D$
- Homemade leverage allows investors to replicate corporate leverage perfectly
Practice Questions
Q1. In an MM no-tax world, firm value is determined by:
A. Capital structure
B. Operating cash flows and business risk
C. Debt interest tax shields
D. Bankruptcy costs
Q2. According to MM Proposition II (no taxes), as the debt-to-equity ratio increases:
A. Cost of equity falls
B. Cost of equity rises linearly
C. WACC falls
D. Total firm value increases
Q3. An unlevered firm has r_U = 15 %. If it issues debt so that D/E = 1 and r_D = 8 %, the levered cost of equity r_E is:
A. 15 %
B. 22 %
C. 19 %
D. 8 %
Q4. In a perfect no-tax market, a firm’s WACC:
A. Declines as leverage rises
B. Remains constant
C. Rises as leverage rises
D. First falls then rises
Q5. Which of the following is NOT an assumption of the MM no-tax propositions?
A. No transaction costs
B. Corporate income taxes exist
C. Information is symmetric
D. Investors and firms borrow at the same rate
Q6. A firm has EBIT of 500,000, r_U = 10 %, and is currently unlevered. If it issues 2,000,000 of debt at r_D = 6 %, its total value should be:
A. 5,000,000
B. 3,000,000
C. 5,000,000 (unchanged)
D. Cannot be determined
Q7. The primary role of homemade leverage is to:
A. Prove investors cannot replicate corporate leverage
B. Show investors can create the same risk-return profile themselves
C. Increase corporate tax shields
D. Reduce bankruptcy probability
Q8. Two firms have identical operating risk; one is levered and one is unlevered. In a no-tax world their:
A. WACCs differ
B. Total values are the same
C. Equity costs are the same
D. Debt costs differ
Answers
| Question | Answer | Explanation |
|---|---|---|
| Q1 | B | MM I states value depends only on operating cash flows and asset risk, independent of financing mix |
| Q2 | B | MM II shows r_E increases linearly with D/E, exactly offsetting the lower cost of debt |
| Q3 | B | r_E = 0.15 + (0.15 – 0.08) × 1 = 0.22 or 22 % |
| Q4 | B | Without taxes WACC equals r_U and is invariant to capital structure |
| Q5 | B | The MM no-tax world explicitly assumes no taxes; option B violates the assumption |
| Q6 | C | V_L = V_U = 500,000 / 0.10 = 5,000,000; capital structure does not change total value |
| Q7 | B | Homemade leverage is central to the MM arbitrage proof; investors can perfectly replicate leverage |
| Q8 | B | Per MM I, V_L = V_U so total firm values are identical |
Takeaways
- In a no-tax perfect market, capital structure is irrelevant (V_L = V_U)
- Leverage increases equity’s financial risk, causing r_E to rise linearly
- WACC remains constant and equal to the unlevered cost of equity r_U
- Arbitrage, including homemade leverage, enforces the MM propositions
- All calculations must use market values, never book values
- MM propositions serve as the idealized theoretical benchmark; real-world taxes and bankruptcy costs alter the conclusions