## Definition

The Modigliani-Miller theory of capital structure proposes that the market value of a firm is irrelevant to its capital structure, i.e., the market value of a levered firm equals the market value of an unlevered firm if they are within the same class of business risk. This approach believes there is no optimal capital structure, and that the valuation of the firm depends on its operating income.

## Modigliani-Miller theory without taxes

### Assumptions

The Modigliani-Miller theory of capital structure developed in 1958 is based on the following assumptions:

- Perfect capital markets
- There are no transactions costs
- There are no flotation costs
- None of the investors can affect the stock price
- Both public and private information are available for any investor
- There are no limitations on buying or selling stock

- All companies within the same class have the same business risk
- The cost of borrowing is fixed (k
_{d}) and always lower than the required rate of return on equity (k_{e}), i.e., k_{d}< k_{e} - All investors have the same estimate of the expected return for each stock
- Companies distribute all net income to dividends
- There are no bankruptcy costs
- There are no taxes

### Formula

The Modigliani-Miller theory believes that valuation of a firm is irrelevant to its capital structure. The equation describing this relationship is as follows:

where V_{U} is the market value of an unlevered firm (capital is represented by equity only), and V_{L} is the market value of a levered firm (capital is represented by a mix of debt and equity).

Thus, the market value of a firm depends on the operating income and business risk rather than its capital structure. Therefore, the market value of an unlevered firm can be calculated using the following formula:

where EBIT is earnings before interest and taxes, and k_{e0} is the required rate of return on equity of an unlevered firm.

The Modigliani-Miller theory of capital structure also believes that the weighted average cost of capital (WACC) is fixed at any level of financial leverage and equals the required rate of return on equity of an unlevered firm (k_{e0}).

### Graph

Assumptions of the Modigliani-Miller theory without taxes are presented in the figure below.

**Debt ratio is used to measure financial leverage*

Modigliani and Miller suggest that the weighted average cost of capital remains fixed because the risk is growing by an increase in financial leverage, and investors will claim a higher return to compensate for it. In other words, the required rate of return on equity will increase as financial leverage increases. Therefore, increasing cheaper debt will be offset by a higher required rate of return on equity. This relationship is described by the following equation:

where k_{eL} is the required rate of return on equity of a levered firm, k_{eU} is the required rate of return on equity of an unlevered firm, D is the market value of debt, E is the market value of equity, and k_{D} is the required rate of return on debt.

Another proof of the Modigliani-Miller theory of capital structure is arbitrage, i.e., simultaneous buying and selling of shares with the same business risk but with different prices. In this case, investors will sell overvalued stock and buy undervalued stock; therefore, the price of overvalued stock will decline, and the price of undervalued stock will increase until they are equal, i.e., until the moment when market equilibrium will occur. When the market reaches equilibrium, arbitrage becomes impossible. Therefore, the market value of firms within the same class of business risk will be the same regardless of their capital structure.

### Example

Let’s assume that earnings before interest and taxes of Total S.E. Inc. is $1,200,000 and the required rate of return on equity for unlevered firms within the same class of business risk is 24%.

The market value of Total S.E. Inc. can be calculated as follows:

The formula used to calculate the required rate of return on equity is as follows:

The weighted average cost of capital can be calculated as shown below

where w_{d} is the proportion of equity (debt ratio), and w_{e} is the proportion of equity.

Let’s calculate the market value of Total S.E. Inc. for each capital structure given in the table above.

__At w _{d} = 0.00__

The market value of an unlevered firm is equal to the market value of its equity.

V = E = $1,200,000 / 0.24 = $5,000,000

WACC = 12% × 0.00 + 24% × (1.00 – 0.00) = 24%

__At w _{d} = 0.20__

D = $5,000,000 × 0.20 = $1,000,000

E = $5,000,000 – $1,000,000 = $4,000,000

I = $1,000,000 × 12% = $120,000

k_{e} = ($1,200,000 – $120,000) / $4,000,000 = 27%

V = $4,000,000 + $1,000,000 = $5,000,000

WACC = 12% × 0.20 + 27% × (1.00 – 0.20) = 24%

__At w _{d} = 0.40__

D = $5,000,000 × 0.40 = $2,000,000

E = $5,000,000 – $2,000,000 = $3,000,000

I = $2,000,000 × 12% = $240,000

k_{e} = ($1,200,000 – $240,000) / $3,000,000 = 32%

V = $3,000,000 + $2,000,000 = $5,000,000

WACC = 12% × 0.40 + 32% × (1.00 – 0.40) = 24%

__At w _{d} = 0.60__

D = $5,000,000 × 0.60 = $3,000,000

E = $5,000,000 – $3,000,000 = $2,000,000

I = $3,000,000 × 12% = $360,000

k_{e} = ($1,200,000 – $360,000) / $2,000,000 = 42%

V = $2,000,000 + $3,000,000 = $5,000,000

WACC = 12% × 0.60 + 42% × (1.00 – 0.60) = 24%

__At w _{d} = 0.80__

D = $5,000,000 × 0.80 = $4,000,000

E = $5,000,000 – $4,000,000 = $1,000,000

I = $4,000,000 × 12% = $480,000

k_{e} = ($1,200,000 – $480,000) / $1,000,000 = 72%

V = $1,000,000 + $4,000,000 = $5,000,000

WACC = 12% × 0.80 + 72% × (1.00 – 0.80) = 24%

The example above illustrates the Modigliani-Miller theorem. As we can see, the required rate of return on equity increases as the proportion of debt increases. Therefore, the weighted average cost of capital and market value of a firm is irrelevant to its capital structure. We can also see that the market value of an unlevered firm equals the market value of a levered firm.

## Modigliani-Miller theory of capital structure with taxes

The theory was further developed by its authors in 1963 by excluding the no taxation assumption.

The main point of the improved theory of capital structure is the hypothesis that valuation of a levered firm will be higher than the valuation of an unlevered firm within the same class of business risk. The reason is that interest expense is an allowable deduction from taxable income; thus, levered firms have a tax shield.

### Formula

The equation describing the relationship between the market value of a levered and an unlevered firm is as follows:

where T is the corporate tax rate, and D is the market value of debt.

In other words, the market value of a levered firm exceeds the market value of an unlevered firm by the amount of tax shield (T×D), assuming the debt is perpetual.

### Graph

The propositions of the Modigliani-Miller theory of capital structure without taxes are illustrated in the figure below.

**Debt ratio is used to measure financial leverage*

As far as the cost of debt is actually lower by the amount of tax shield, an increase in its proportion results in a decrease in the weighted average cost of capital. This relationship can be described by the following equation:

Thus, optimal capital structure exists when the capital of a firm is represented by debt only! Therefore, firms should replace equity by cheaper debt to reduce their WACC and maximize market value.

### Income tax on capital owners

In 1978, the Modigliani-Miller theory of capital structure was further developed by taking into account the income tax on capital owners. The new hypothesis proposed that investor behavior depends on tax preference. The idea behind it is that dividends, capital gains, and interest income are usually taxed at different rates, which suggests that investor behavior is relevant to tax preference.

The equation describing the relationship between the market value of a levered and an unlevered firm is as follows:

where t_{e} is the income tax rate of shareholders, and t_{d} is the income tax rate of debtholders.

## Criticism

The Modigliani-Miller theory of capital structure was criticized because the assumption that capital markets are perfect is completely unrealistic. The arbitrage, as proof of the Modigliani-Miller theory, was also strongly criticized. If there are no perfect capital markets, the arbitrage will be useless because a levered and an unlevered firm within the same class of business risk will have different market values.

The reasons why arbitrage does not allow market equilibrium in real life are as follows:

__Transaction costs__. If there are transactions costs, buying stock will require bigger initial investments, but the return remains the same. Therefore, the market value of a levered firm will be higher than an unlevered one, assuming that both of them are within the same class of business risk.__The cost of borrowing is not the same for individuals and firms__. The cost of borrowing depends on the individual credit rating of the borrower.__Institutional constraints__. Institutional investors slow down arbitrage because they limit the use of financial leverage by their clients.__Bankruptcy cost__. The higher the financial leverage, the higher is the probability of bankruptcy. Therefore, bankruptcy costs have a strong influence on firms.

Many critics of the Modigliani-Miller theory of capital structure believe that assumptions are unrealistic and that the market value of a firm as well as WACC depends on financial leverage.