The Modigliani-Miller Propositions

A firm’s capital structure is the mix of debt and equity the company uses to finance its investments. The goal of a capital structure decision is to determine the financial leverage that will maximize the value of the company by minimizing the weighted average cost of capital (WACC).

$$\text{r}_{\text{WACC}}=\bigg(\frac{\text{D}}{\text{V}}\bigg)\text{r}_{\text{d}}(1-\text{t})+\bigg(\frac{\text{E}}{\text{V}}\bigg)\text{r}_{\text{e}}$$

Where:

\(\text{r}_{\text{d}}=\) Before-tax marginal cost of debt.

\(\text{r}_{\text{e}}=\) Marginal cost of equity.

\(\text{t}=\) Marginal tax rate.

\(\text{D}=\) Outstanding debt.

\(\text{E}=\) Outstanding equity.

\(\text{V}=\text{D}+\text{E}=\) Value of the company.

Proposition I without Taxes: Capital Structure Irrelevance

The market value of a company is unaffected by the capital structure of the company.

$$\text{Value of a firm levered} (\text{V}_{\text{L}})=\text{Value unlevered} (\text{V}_{\text{U}})=\frac{\text{EBIT}}{\text{r}_{\text{WACC}}}$$

The above relationship implies that the value of a company is determined solely by its cash flows, not by its capital structure. Additionally, the WACC for the company—assuming no taxes—is not affected by its capital structure.

Franco Modigliani and Merton Miller suggested the following assumptions for Proposition I:

1. Investors have similar expectations regarding future cash flows.
2. Bonds and stocks are traded in a perfect capital market.
3. Investors can lend and borrow at a risk-free rate.
4. There are no agency costs.
5. Financing and investing decisions are independent of each other.

Although these assumptions are unrealistic, Modigliani and Miller’s school of thought is that investors can create capital structures they prefer. The capital structure that management chooses does not matter because investors can change it at no cost.

Suppose that management has set the capital structure of a company to consist of 50% debt and 50% equity, and the investor prefers the company’s capital structure to be 60% debt and 40% equity. The investor will use borrowed money to finance his/her share purchase so that the ownership of the company’s assets reflects 60% debt financing. The importance of the Modigliani and Miller theory is that managers cannot use capital structure to change the value of the firm.

Proposition II without Taxes: Higher Financial Leverage Raises the Cost of Equity

Here, Franco Modigliani and Merton Miller remove a few assumptions from proposition Proposition I and state that the cost of equity is a linear function of the company’s debt/equity ratio.

According to this proposition, the cost of equity increases as a company increases its use of debt financing to maintain a constant WACC. The risk of equity is contigent on business risk and financial risk. Business risk determines the cost of capital, while capital structure determines financial risk.

The WACC—still ignoring taxes—is given by:

$$\text{r}_{\text{WACC}}=\bigg(\frac{\text{D}}{\text{V}}\bigg)\text{r}_{\text{d}}+\bigg(\frac{\text{E}}{\text{V}}\bigg)\text{r}_{\text{e}}$$

Where:

\(\text{r}_{\text{wacc}}=\) The weighted average cost of capital.

\(\text{r}_{\text{d}}=\) Before-tax marginal cost of debt.

\(\text{r}_{\text{e}}=\) Marginal cost of equity.

\(\text{t}=\) Marginal tax rate.

\(\text{D}=\) Outstanding debt.

\(\text{E}=\) Outstanding equity.

\(\text{V}=\text{D}+\text{E}=\) Value of the company.

For a company financed by 100% equity, the \(\text{r}_{\text{wacc}}=\text{r}_{0}.\)

To get the cost of equity, we rearrange the formula:

$$\text{r}_{\text{e}}=\text{r}_{0}+(\text{r}_{0}-\text{r}_{\text{d}})\bigg(\frac{\text{D}}{\text{E}}\bigg)$$

Example: MM Proposition II without Taxes

Genghis Investment has an all-equity capital structure. Its characteristics are as follows:

• The expected operating income is $6,000.
• The cost of equity, which also the WACC, is 12%.
• EBIT is perpetual.
• Genghis is planning to issue $18,000 in debt at the cost of 6% to buy back $18,000 worth of its equity.

Calculate the value of Genghis and its cost of equity.

Solution

$$\begin{align*}\text{V}&=\frac{\text{EBIT}}{\text{r}_\text{WACC}}\\&=\frac{$6,000}{0.12}\\&=$50,000\end{align*}$$

When Genghis issues the debt, it pays interest of 6% on the debt.

$$\text{Interest payment}=0.06($18,000)=$1,080$$

Using the MM proposition II, the cost of Genghis’ equity is given by:

$$\text{r}_{\text{e}}=\text{r}_{0}+(\text{r}_{0}-\text{r}_{\text{d}})(1-\text{t})\frac{\text{D}}{\text{E}}$$

Where:

$$\begin{align} \text{E}&=\text{V}-\text{D} \\ &= $50,000 – $18,000 \\ &= $32,000 \end{align}$$

$$\text{r}_{\text{e}}=0.12+(0.12-0.06)\bigg(\frac{$18,000}{$32,000}\bigg)=0.15375=15.38\%$$

Genghis makes \($1,080\) to debtholders and \($6,000-1,080= $4,920\) to equity holders. The value of debt is calculated as follows.

$$\text{V}=\text{D}+\text{E}=\frac{$1,080}{0.06}+\frac{$4,920}{0.15375}=$50,000$$

We can also represent the systematic risk of the assets of the entire company as a weighted average of the systematic risk of the company’s equity and debt:

$$\beta_{a}=\bigg(\frac{\text{D}}{\text{V}}\bigg)\beta_{\text{d}}+\bigg(\frac{\text{E}}{\text{v}}\bigg)\beta_{e}$$

Where:

\(\beta_{\text{a}}=\) Asset beta − Amount of asset non-diversifiable risk

\(\beta_{\text{d}}=\) Beta of debt

\(\beta_{\text{e}}=\) Beta of equity

As the level of debt rises, the risk of default increases, and the costs are borne by equity holders. The equity’s Beta rises as the proportionate use of debt rises.

Question

Krystal Solutions has a capital structure that is fully financed using equity. It has the following characteristics:

• The expected operating income is $ 8,000.
• The WACC is 11%.
• EBIT is perpetual.
• The company plans to issue debt worth $20,000 at the cost of 7% to buy back $20,000 worth of its equity.

The cost of equity of Krystal solution is most likely to be:

1. 12.52%
2. 11%
3. 7%

Solution

The correct answer is A.

We start by first calculating the Value of Krystal solution, which is:

$$V_u={EBIT\over WACC}$$

$$= {$8000\over 0.11} = $72,727.27$$

When Krystal issues the debt, they will pay 7% of the debt.

Interest payment = 0.07($20,000) = $1,400

The cost of Krystal’s equity is calculated as follows:

$$\text{r}_{\text{e}}=\text{r}_{0}+(\text{r}_{0}-\text{r}_{\text{d}})\frac{\text{D}}{\text{E}}$$

Where:

$$ E = V – D$$

$$= $72,727.27 – $ 20,000 = $52,727.27$$

$$\text{r}_{\text{e}}=\text{0.11}+(\text{0.11}-\text{0.07})\frac{\text{20,000}}{\text{ 52,727.27}} = 12.52%$$

Reading 15: Capital Structure

LOS 15 (a) Explain the Modigliani-Miller propositions regarding capital structure.