1. The Benchmark: Modigliani and Miller (1958)

Jun He May 31, 2026

公司金融Corporate Finance 资本结构Capital Structure MM定理MM Theorem 无套利No Arbitrage 股利政策Payout Policy 学习笔记Study Note

Note

Modigliani-Miller (1958, MM) 是资本结构模型的基准 (benchmark)：在无摩擦的理想世界里，公司价值只取决于其经营与实体活动，与如何融资（资本结构）无关。Proposition 1.1（资本结构无关）：$V=D+E$ 与 $D/E$ 组合无关——证明靠无套利：买下负债公司 B 的全部股权 + 债权，与买下同等现金流的全股权公司 A，在 $t=1$ 给出完全相同的现金流 $\tilde X$，故 $t=0$ 成本必相等，$V_B=V_A$。Proposition 1.2（股权回报内生于杠杆）：资产回报 $R_A$ 由现金流风险决定，$R_E=R_A+\frac{D}{E}(R_A-R_D)$——杠杆越高股权越险、$R_E$ 越高。Proposition 1.3（股利政策无关）：在无套利下，公司价值 $V_t=\mathbb E_t[\sum_\tau\beta^\tau(\tilde X_{t+\tau}-I_{t+\tau})]$ 只取决于现金流与投资，与分红/回购/股份数无关。影响：MM 首次引入无套利范式；它纠正了「借债拉低加权资本成本→提升公司价值」的朴素错误（错在把 $r_D,r_E$ 当外生——实则 $r_A$ 由现金流性质钉定，$r_D,r_E$ 随杠杆内生调整使加权平均恒等于 $r_A$）。MM 是一面镜子：任何声称找到「最优资本结构」的研究，本质上必依赖某种摩擦 (friction)。

Note

Modigliani-Miller (1958, MM) is the benchmark of capital-structure models: in a frictionless ideal world, a firm's value depends only on its operations and real activities, and is independent of how it raises money (capital structure). Proposition 1.1 (capital structure irrelevance): $V=D+E$ is independent of the $D/E$ mix — proven by no arbitrage: buying all the equity + debt of a levered firm B, versus buying the all-equity firm A with the same cash flow, gives exactly the same $t=1$ payoff $\tilde X$, so the $t=0$ costs must be equal, $V_B=V_A$. Proposition 1.2 (return on equity is endogenous to leverage): the asset return $R_A$ is pinned down by cash-flow risk, and $R_E=R_A+\frac{D}{E}(R_A-R_D)$ — higher leverage means riskier, higher-return equity. Proposition 1.3 (payout policy irrelevance): under no arbitrage, firm value $V_t=\mathbb E_t[\sum_\tau\beta^\tau(\tilde X_{t+\tau}-I_{t+\tau})]$ depends only on cash flows and investment, not on dividends/repurchases/share count. Influence: MM is the first to introduce the no-arbitrage paradigm; it corrects the naive error that "borrowing drags down the weighted cost of capital → raises firm value" (the error is treating $r_D,r_E$ as exogenous — in fact $r_A$ is pinned by the nature of the cash flow, and $r_D,r_E$ adjust endogenously with leverage so that their weighted average always equals $r_A$). MM is a mirror: any study claiming to find an "optimal capital structure" must, in essence, rely on some friction.

1.1 Set-up

1.1.1 Assumptions

MM 模型假设没有摩擦 (no frictions)：

资本市场完全且完美 (complete and perfect)；
任何形式的无税收；
无交易成本；
无破产成本；
完美对称信息，故无委托-代理问题；
所有主体风险中性 (risk neutral)。

Tip

Remark 1.1 「无破产成本」并不意味着公司不能破产。它假设公司可以宣告破产，只是不产生额外成本——既无直接成本（如支付律师费），也无间接成本（如交易对手在破产前不愿签订长期合约）。

The MM model assumes no frictions:

capital markets are complete and perfect;
no taxes in any form;
no transaction costs;
no bankruptcy costs;
perfect symmetric information, so there is no principal-agent problem;
all agents are risk neutral.

Tip

Remark 1.1 "No bankruptcy cost" does not mean firms cannot go bankrupt. It assumes firms can declare bankruptcy but incur no additional cost — neither a direct cost (e.g. paying for the lawyer) nor an indirect cost (e.g. counterparties' unwillingness to sign a long-term contract before bankruptcy).

在简单的两期设定（$t=0,1$）中，MM 把公司/项目视为产生风险现金流的金融证券，用贴现现金流 (DCF) 法估值 (1.1)：

In a simple two-period setting ($t=0,1$), MM views firms/projects as financial securities that generate risky cash flows, and values them by the discounted cash flow (DCF) method (1.1):

$$V_i=\frac{\mathbb E\!\left[\tilde X_i\right]}{R_{A,i}}\tag{1.1}$$

其中 $V_i$ 为公司 $i$ 在 $t=0$ 的全部资产价值，$\tilde X_i$ 为 $t=1$ 产生的总随机现金流，$R_{A,i}$ 为公司 $i$ 资产的特定毛贴现因子 (gross discount factor)。

Tip

Remark 1.2 在此两期模型中，(1.1) 隐含假设所有资产都被用尽以产生现金流，在 $t=1$ 之后无残值。

1.1.2 Discount Rate

公司特定的贴现率 $R_{A,i}$ 可以是：

市场评估的公司 $i$ 资产的毛回报率；或
公司 $i$ 同等价类 (equivalence class) 证券的市场利率；或
公司 $i$ 资产类别的公认机会成本。

无论采用哪个定义，关键是贴现率对公司而言是特定的、由市场钉定，而非私人信念。例如在 CAPM 世界中，$R_{A,i}$ 由下式钉定：

where $V_i$ is firm $i$'s value of all assets at $t=0$, $\tilde X_i$ is the total random cash flow generated at $t=1$, and $R_{A,i}$ is firm $i$'s specific gross discount factor.

Tip

Remark 1.2 In this two-period model, (1.1) implicitly assumes that all assets are used up to generate the cash flow and have no remaining value after $t=1$.

1.1.2 Discount Rate

The firm-specific discount rate $R_{A,i}$ could be:

the market-evaluated gross return rate of firm $i$'s assets; or
the market rate for firm $i$'s equivalence-class securities; or
the commonly agreed opportunity cost of firm $i$'s asset class.

Whichever definition we choose, the key is that the discount rate is specific to the firm and pinned down by the market, not by private beliefs. For example, in a CAPM world, $R_{A,i}$ is pinned down by:

$$\tilde R_{A,i}=R_f+\beta_{A,i}\left(\mathbb E\!\left[\tilde R_m\right]-R_f\right),\qquad \beta_{A,i}=\frac{\operatorname{Cov}\!\left(\tilde R_{A,i},\tilde R_m\right)}{\operatorname{Var}\!\left(\tilde R_m\right)}$$

其中 $\tilde R_{A,i}$ 为公司 $i$ 资产的毛回报，$\tilde R_m$ 为毛市场回报，$R_f$ 为毛无风险利率。

1.1.3 Equity and Debt Arrangements

债务 (Debt)：公司对债权人的预先约定支付；除非公司破产，否则该支付是强制且固定的；破产时，债权人攫取公司清算的全部价值。
股权 (Equity)：在公司偿付完所有债务后，对剩余价值的索取者；破产时，股权持有人一无所获。

1.1.4 Notation

两期 $t=0,1$。
公司价值 $V=D+E$，$D$ 为债务市值、$E$ 为 $t=0$ 股权市值。（注：「市值」不必意味公司公开上市，只是当前公允价值的概念。）
$t=0$ 不产生现金流；$t=1$ 产生风险现金流 $\tilde X$。
债务的面值 (face value) 为 $F$（无破产时 $t=1$ 的偿还额）。
孪生公司 (twin firms) A 与 B：现金流完全相同，$\tilde X_A=\tilde X_B$。

1.2 MM Propositions and Proofs

1.2.1 MM Proposition 1

Important

Proposition 1.1 在 MM 假设下，资本结构与公司价值无关，即 $V=D+E$ 独立于 $D$ 与 $E$ 的组合。

MM 命题 1 的直白含义：蛋糕的大小只由实体活动决定；怎么切分这块蛋糕不影响它的大小。

where $\tilde R_{A,i}$ is the gross return on firm $i$'s asset, $\tilde R_m$ is the gross market return, and $R_f$ is the gross risk-free rate.

1.1.3 Equity and Debt Arrangements

Debt: pre-specified payments from the firm to its creditors; the payment is compulsory and fixed unless the firm goes bankrupt; in bankruptcy, creditors grab all the value of the firm's liquidation.
Equity: the claimant of all the remaining value of a firm after it pays all the debt obligations; in bankruptcy, equity holders get nothing.

1.1.4 Notation

Two periods $t=0,1$.
Value of the firm $V=D+E$, where $D$ is the market value of debt and $E$ the market value of equity at $t=0$. (Note: "market value" does not necessarily mean the firm is publicly listed; it is simply a notion of current fair value.)
No cash flow at $t=0$; a risky cash flow $\tilde X$ is generated at $t=1$.
The debt has a face value $F$ (the repayment amount at $t=1$ if no bankruptcy happens).
Twin firms A and B: exactly the same cash flow, $\tilde X_A=\tilde X_B$.

1.2 MM Propositions and Proofs

1.2.1 MM Proposition 1

Important

Proposition 1.1 Under the MM assumptions, capital structure is irrelevant to the firm's value, i.e. $V=D+E$ is independent of the mix of $D$ and $E$.

The plain meaning of MM Proposition 1: the size of the pie is determined only by real activities; how you slice the pie won't affect its size.

Note

证明 / Proof（Proposition 1.1，无套利）设孪生公司 A、B 有相同现金流 $\tilde X_A=\tilde X_B=\tilde X$。公司 A 纯股权融资（$V_A=E_A$），公司 B 同时用股权与债务融资，$V_B=E_B+D_B$，债务面值为 $F_B$。若投资者在 $t=0$ 买下公司 B 的全部股权与债权，成本为 $E_B+D_B$，在 $t=1$ 获得 $$\underbrace{\max\left\{\tilde X-F_B,\,0\right\}}_{\text{from equity}}+\underbrace{\min\left\{\tilde X,\,F_B\right\}}_{\text{from debt}}=\tilde X.$$ 另一方面，若他在 $t=0$ 买下公司 A 的全部股权，成本 $E_A$，$t=1$ 得 $\tilde X$。两种方案产生完全相同且确定的现金流，故无套利下成本必相等： $$E_B+D_B=E_A\ \Rightarrow\ V_B=V_A.$$ 即两公司 $t=0$ 价值必相等，否则投资者可在 $t=0$ 买入较便宜的公司、卖空较贵的公司套利。$\blacksquare$

1.2.2 MM Proposition 2

Important

Proposition 1.2 股权毛回报 $R_E$ 内生地取决于杠杆。

在命题 1.1 中已论证：资产回报 $R_A$ 纯由现金流的风险特征决定。既然如此，命题 1.2 是说：$R_E$ 与 $R_D$ 必自动调整，使其加权平均对任意杠杆比都等于 $R_A$。因此 $R_E$ 随杠杆内生确定 (1.2)：

Note

证明 / Proof (Proposition 1.1, no arbitrage) Let twin firms A, B have the same cash flow $\tilde X_A=\tilde X_B=\tilde X$. Firm A is purely equity-financed ($V_A=E_A$); firm B is financed with both equity and debt, $V_B=E_B+D_B$, with debt face value $F_B$. If an investor buys all the equity and debt of firm B at $t=0$, it costs $E_B+D_B$, and at $t=1$ he gets $$\underbrace{\max\left\{\tilde X-F_B,\,0\right\}}_{\text{from equity}}+\underbrace{\min\left\{\tilde X,\,F_B\right\}}_{\text{from debt}}=\tilde X.$$ On the other hand, if he buys all the equity of firm A at $t=0$, he spends $E_A$ and gets $\tilde X$ at $t=1$. The two plans generate exactly the same cash flow with certainty, so under no arbitrage their costs must be equal: $$E_B+D_B=E_A\ \Rightarrow\ V_B=V_A.$$ The two firms must have the same value at $t=0$; otherwise the investor could arbitrage by buying the cheaper firm and short-selling the more expensive one at $t=0$. $\blacksquare$

1.2.2 MM Proposition 2

Important

Proposition 1.2 The gross return on equity $R_E$ is endogenously related to leverage.

Proposition 1.1 already argued that the asset return $R_A$ is purely determined by the risk characteristics of the cash flow. Given this, Proposition 1.2 says: $R_E$ and $R_D$ must automatically adjust so that their weighted average equals $R_A$ for any leverage ratio. So $R_E$ is endogenous in leverage (1.2):

$$R_A=\frac{D}{D+E}R_D+\frac{E}{D+E}R_E\ \Longrightarrow\ R_E=R_A+\frac{D}{E}\left(R_A-R_D\right)\tag{1.2}$$

Note

证明 / Proof（Proposition 1.2）由于债权人与股权人在 $t=1$ 瓜分全部现金流，按毛回报定义： $$R_E E+R_D D=\mathbb E\!\left[\tilde X\right]\ \Longrightarrow\ R_E=\frac{\mathbb E\!\left[\tilde X\right]-R_D D}{E}.\tag{1.3}$$ 对风险中性主体，资本的毛回报定义为 $$R_A=\frac{\mathbb E\!\left[\tilde X\right]}{V}.\tag{1.4}$$ 由命题 1.1 代入 $V=D+E$ 到 (1.3)、(1.4)： $$R_A=\frac{\mathbb E\!\left[\tilde X\right]}{D+E}=\frac{R_E E+R_D D}{D+E}\ \Longrightarrow\ R_E E=R_A(D+E)-R_D D,$$ $$\Rightarrow\ R_E=R_A\frac{D+E}{E}-R_D\frac{D}{E}=R_A+\frac{D}{E}\left(R_A-R_D\right),$$ 恰为 (1.2)。$\blacksquare$

Tip

Remark 1.3 命题 1.2 表明：股权的风险随资本结构变化。债务/资产比越高，股权越风险，从而内生地抬高股权回报以补偿增加的风险。

1.2.3 Irrelevant Payout Policy

Important

Proposition 1.3 给定相同现金流，股利政策（如股份回购/增发、分红）对公司价值无影响。

Note

证明 / Proof (Proposition 1.2) Since debt holders and equity holders split up the entire cash flow at $t=1$, by the definition of gross returns: $$R_E E+R_D D=\mathbb E\!\left[\tilde X\right]\ \Longrightarrow\ R_E=\frac{\mathbb E\!\left[\tilde X\right]-R_D D}{E}.\tag{1.3}$$ For risk-neutral agents, the gross return on capital is defined as $$R_A=\frac{\mathbb E\!\left[\tilde X\right]}{V}.\tag{1.4}$$ Plugging $V=D+E$ from Proposition 1.1 into (1.3) and (1.4): $$R_A=\frac{\mathbb E\!\left[\tilde X\right]}{D+E}=\frac{R_E E+R_D D}{D+E}\ \Longrightarrow\ R_E E=R_A(D+E)-R_D D,$$ $$\Rightarrow\ R_E=R_A\frac{D+E}{E}-R_D\frac{D}{E}=R_A+\frac{D}{E}\left(R_A-R_D\right),$$ which is exactly (1.2). $\blacksquare$

Tip

Remark 1.3 Proposition 1.2 implies that the risk of equity changes as the capital structure changes. A higher debt-to-asset ratio means riskier equity, which endogenously raises the return on equity in compensation for that increased risk.

1.2.3 Irrelevant Payout Policy

Important

Proposition 1.3 Given the same cash flows, payout policies such as share repurchase/issuance and dividend payment have no effect on firm value.

Note

证明 / Proof（Proposition 1.3）稍改记号。考虑离散时间 $t=0,1,\dots$，各期贴现因子 $\beta\in(0,1)$ 相同。第 $t$ 期：公司产生净收入 $\tilde X_t$、投资 $I_t$；每股分红 $d_t$、每股价格 $P_t$、股份数 $n_t$。（注：$n_t>n_{t-1}$ 为增发，\(n_t

1.3 Influence

MM 模型是首个引入无套利论证的模型，成为偏均衡金融研究的范式。
经验证据显示：公司借债、增加债务时其价值上升。MM 之前，研究者认为资本成本与债务成本皆外生，于是用如下朴素论证解释该事实：
经验上债务成本 $r_D$ 总低于股权成本 $r_E$；
故公司借债时，作为 $r_D$ 与 $r_E$ 加权平均的资产成本 $r_A$ 因债务比例升高而被拉低；
结果更低的贴现率带来更高的公司价值。
上述论证错在把 $r_D,r_E$ 当外生。有了 MM，$r_A$ 纯由现金流性质钉定，与资本结构无关；故 $r_D,r_E$ 必随资本结构自动调整，使其加权平均恒等于 $r_A$。
尽管不现实，MM 改变了人们对「公司金融中什么才是有趣问题」的看法；其理想世界始终是一面基准镜子，后续研究在其上叠加更多摩擦以贴近现实。
MM 给我们的思维方式：只要一项研究声称找到了最优资本结构，它一定是在讨论某类摩擦。

Tip

注（毛/净回报记号）本节及全书中，小写 $r_D,r_E,r_A$ 为净回报，大写 $R_D,R_E,R_A$ 为毛回报；讨论得出的结论相同。

Note

证明 / Proof (Proposition 1.3) Slightly change the notation. Consider a discrete-time model $t=0,1,\dots$ with the same discount factor $\beta\in(0,1)$ across periods. In period $t$: the firm generates net revenue $\tilde X_t$ and invests $I_t$; the dividend per share is $d_t$, price per share $P_t$, number of shares $n_t$. (Note: $n_t>n_{t-1}$ is share issuance, \(n_t

1.3 Influence

The MM model is the first to introduce the no-arbitrage argument, a paradigm in partial-equilibrium finance research.
Empirical evidence shows a firm's value goes up when it borrows money and increases debt. Before MM, researchers believed both the cost of capital and the cost of debt are exogenous, and explained this fact with a naive argument:
empirically, the cost of debt $r_D$ is always lower than the cost of equity $r_E$;
so when the firm borrows, its cost of assets $r_A$, as a weighted average of $r_D$ and $r_E$, is dragged down simply by the higher proportion of debt;
as a result, a lower discount rate leads to a higher firm value.
The argument above is false in taking $r_D,r_E$ as exogenous. With MM, $r_A$ is purely pinned down by the nature of the cash flow, having nothing to do with capital structure; so $r_D,r_E$ must automatically adjust along with capital structure such that their weighted average coincides with $r_A$.
Unrealistic as it is, MM changed people's view of what the interesting questions in corporate finance are; its ideal world always serves as a benchmark mirror on which following studies add more frictions to mirror reality.
MM lets us think this way: whenever a study claims to have found an optimal capital structure, it must be talking about certain types of frictions.

Tip

Note (gross/net return notation) In this section and throughout the notes, lower-case $r_D,r_E,r_A$ are net returns while upper-case $R_D,R_E,R_A$ are gross returns; the conclusions of the discussion are the same.

References

Modigliani, F. and M. H. Miller (1958). The cost of capital, corporation finance and the theory of investment. The American Economic Review 48(3).