Modigliani-Miller

Model

Let us consider an economy with 2 firms, who stay in the market for 2 periods $t \in \{0,1\}$

• At $t = 1$, they both yield the same cash flow Y

• At $t=0$, 2 firms have different capital structure

  • Firm 1 has equity and risk-free debt
  • Firm 2 has equity but no debt

• At $t = 0$

  • Risk-free rate is r > 0
  • Market value of firm i's quity and debt are: $E_i$ and $D_i$
  • Market value of each firm is $V_t = D_t + E_t$

• At $t = 1$

  • Firm 1's debt holders receive $rD_1$, it's equity holders receive $Y-rD_1$
  • Firm 2's equity holders receive Y

Proposition 1, MM1: $V_1=V_2$

Proof. First, it cannot be that $V_1<V_2$. Suppose $V_1<V_2$, then, at $t=0$, an inverstor could do as follows.

1. Short sell $\alpha\in(0,1)$ of firm 2's share for $\alpha V_2$;
2. Use above income to buy $\frac{\alpha V_2}{V_1}$ of firm 1's debt and equity as $\alpha V_2=\frac{\alpha V_2}{V_1}D_1+\frac{\alpha V_2}{V_1}E_1$

Then at $t=1$, the investor could receive:

For any Y, i.e, an arbitrage opportunity exists. Violate principle of no-arbitrage, then we must have $V_2\leq{V_1}$.

Second, it cannot be true that $V_2<V_1$ . Suppse $V_2<V_1$, then at $t=0$, an investor could do as follows.

1. Short sell $\alpha\in(0,1)$ of firm 1's share for $\alpha E_1$;
2. Sell/bollow $\alpha D_1$
3. Use above income to buy $\frac{\alpha V_1}{V_2}$ of firm 2's equity as $\alpha E_1=\frac{\alpha V_1}{V_2}V_2$

Then at $t=1$, the investor could receive:

For any Y, i.e.,an arbitrage opportunity exists. Violate principle of no-arbitrage, then we must have $V_1\leq{V_2}$

Then we have $V1=V2$.

Market Equilibrium in Capital Market

Let us consider an economy (security market) defined as follows:

• Agent:

  • A signle perishable good is consumed in a 2 periods would: $t=0,1$, price is normalized to 1.

  • Uncertainly in $t=1$ is defined by a finite state space:

    $$\omega \in \{\omega_1,\omega_2,...\omega_{|\Omega|}\}, p_\omega\in(0,1)\ and\ \sum p_\omega =1$$

  • K agents: each $k=1,2,...K$ has

    • Non-negative endowment $e^k_t=0$ and
    $$e^k_{t=1}=[e^k_{t=1,\omega_1},e^k_{t=1,\omega_2},e^k_{t=1,\omega_3}...e^k_{t=1,\omega_{|\Omega|}}]^T\ in\ t=1, and$$
    • Non-negative consumption $e^k_t=0$ and
  $$c^k_{t=1}=[c^k_{t=1,\omega_1},c^k_{t=1,\omega_2},c^k_{t=1,\omega_3}...c^k_{t=1,\omega_{|\Omega|}}]^T\ in\ t=1, and$$
  • All agents have the same information regarding to the sate of the economy and there is no asymmetric information, i.e., no adverse selection or moral hazard
  • All agents have no production technology
  • All agents' preference satisfies the requirement of rationality

• Security market

  We assume that all allocations are determined in a security market. A security market is composed of a set of securities. A security is claim that yields a future payoff: $\{x_\omega\}_{\omega\in}$. We define as a payoff space, i.e., the possible payoff from future. Then a security is a payoff:

  $$\vec{X}(\omega) = [X_{\omega_1}, X_{\omega_2},...,X_{\omega_{|\Omega|}}]^T$$

  We assume there are N securities traded in the market. Let a (column) vetor

  $$\vec{X}(\omega,n)=[X_{\omega_1}(n),X_{\omega_2}(n),...,X_{\omega_{\|\Omega|}}(n)]^T$$

  be the payoff of security n. in the example, we have $n=2$ and for the equity, we have $\vec{X}(\omega, equity)=[1400,900]$. The we have a payoff matrix:

  $$X\equiv[\vec{X}(\omega,1),\vec{X}(\omega,2),...\vec{X}(\omega,n)]=\left[\matrix{ X_{\omega_1}(1)&X_{\omega_1}(2)&...&X_{\omega_1}(n)\\ X_{\omega_2}(1)&X_{\omega_2}(2)&...&X_{\omega_2}(n)\\ .&.&.&.\\ .&.&.&.\\ .&.&.&.\\ X_{\omega_{|\Omega|}}(1)&X_{\omega_{|\Omega|}}(2)&...&X_{\omega_{|\Omega|}}(n)\\ }\right]$$

  which defines the market structure.

• Portfolio

  Portfolio describes a collection of securities. We let $\theta^k=[\theta^k(1),\theta^k(2),...,\theta^k(N)]$ be the portfolio, where $\theta^k(n)$ is the volumne of security n hold by the agent k. We use $\theta^k_{t=0}(n)$ to denote the intial security n hold by agent k. Then we call $\theta_{M(t)}=\sum_k\theta_t^k$ as market portfolio for $t\in\{0,1\}$, which is the total supply of security in the market.

• Transaction process

  We let $S = [S(1),S(2),...,S(N)]$ be the vector of prices for the traded securities. Price is normalized on the price of the only physical good in the economy. Let

  $$\theta^k(S)\equiv[\theta^k(S,1),\theta^k(S,2),...,\theta^k(S,N)]^T$$

  be agent k's demand on security given price S. If the market is clean, then we have

  $$\sum_K\theta_t^k=\sum_K\theta^k_t(S)$$

  i.e., demand = supply. The market clean price can be determined through above condition.

• Perfect/Frictionless Market

  1. All agents participate the market transaction without any cost;
  2. No taxes, no transaction cost;
  3. No limitation on the transaction [sell short or buy long without limitation];
  4. Each agent's transaction will not influence the security price;
  5. Trade at competitive market which implied no arbitirage in equilibrium.

• Market Equilibrium in Security Market

  An equilibrium in a security market consists of a vector of security price $S$, a portfolio allocation $\{\theta^{*k}\}$, and a consumption plan $\{(c_{t=0}^{*k},c_{t=1}^{*k})\}$, such that:

  1. Agent optimization [on the volume choice of securities, i.e., portfolio]

     [Every agent choose the volume of security which can give him the highest payoff]

     We assume agents always solve the following problem:

  $$\max_{\theta^k}\ U^k(c_{t=0}^{k},c_{t=1}^{k})\\ s.t.c_{t=0}^{k}\leq e_{t=0}^{k}-S'\theta^k\\ c_{t=1}^{k}\leq e_{t=1}^{k}+X\theta^k\\ (c_{t=0}^{k},c_{t=1}^{k})\geq(0,0)$$

  The solution to above optimization problem gives agent k's demand for the security as a function of price and endowment, i.e., $\theta^k(\vec{e}, S)$, where $\vec{e}=[\vec{e}_{t=0},\vec{e}_{t=1}]$.

  2. Market cleaning

     [Given a security, demand = supply, i.e., no more transaction would take place]

• Existence and uniqueness of equilibrium

  Theorem 1 If each agent's

  1. consumption set is restrictes to be positive;
  2. Utility functions stricted increasing and concave;
  3. Initial endowment is strictly positive; and
  4. There exists a portoflio with positive payoff

  then there exists a equilibrium in security market.

• Asset pricing

  Let the price vector $S=[S(1),S(2),...S(N)]$, and payoff matrix be $X$. Asset pricing is a mapping from $X$ to $S$, i.e., $S=V(X)$.

• Arbitrage

  An arbitrage opportunity exits if there is a portfolio $\theta=[\theta(1),\theta(2),...,\theta(N)]$ , such that

  1. $S'\theta\leq0$
  2. $X\theta\geq0$
  3. At least on of above inequality is strict.

  i.e., we have the following 3 arbitrage:

  1. $S'(\theta)<0\ and\ X\theta=0$
  2. $S'(\theta)=0\ and\ X\theta>0$
  3. $S'(\theta)<0\ and\ X\theta>0$