The Modigliani–Miller theorem is a set of two propositions on corporate capital structure. It was first proposed by Franco Modigliani and Merton Miller in 1958.
Two Propositions (No Taxes)
Modigliani and Miller made some very serious assumptions. The most important two are that there are no taxes and no costs of financial distress. Additional assumptions will be discussed in the next section. The two propositions are still true when the no taxes assumption is relaxed.
Proposition I : the market value of any firm is independent of its capital structure.
This means a firm cannot change its total value just by splitting its cash flows into different streams: The firm’s value is determined by its real assets, not by how it is financed. Thus capital structure is irrelevant as long as the firm’s investment decisions are taken as given.
Firms can not create value simply by changing the company’s capital structure.
Proposition II: the cost of equity is a linear function of the company’s debt/equity ratio.
More specifically, expected return on equity = expected return on assets + (expected return on assets – expected return on debt) * debtequity ratio.
The mathematical representation (which can be derived from the WACC formula) is:
According to this proposition, as the company increases its use of debt financing, the cost of equity rises. We know from MM Proposition I that the value of the company is unchanged and the weighted average cost of capital remains constant if the company changes its capital structure. What Proposition II then means is that the cost of equity increases in such a manner as to exactly offset the increased use of cheaper debt in order to maintain a constant WACC.
The risk of the equity depends on two factors: the risk of the company’s operations (business risk) and the degree of financial leverage (financial risk). Business risk determines the cost of capital, whereas the capital structure determines financial risk.
The expected rate of return on the common stock of a levered firm increases in proportion to the debt–equity ratio (D/E), expressed in market values; the rate of increase depends on the spread between , the expected rate of return on a portfolio of all the firm’s securities, and , the expected return on the debt.
Note that if the firm has no debt.
Assumptions

Expectations are homogeneous. This means investors agree on the expected cash flows from a given investment. This means that all investors have the same expectations with respect to the cash flows from an investment in bonds or stocks.

Bonds and shares of stock are traded in perfect capital markets. This means that there are no transactions costs, no taxes, no bankruptcy costs, and everyone has the same information. In a perfect capital market, any two investments with identical cash flow streams and risk must trade for the same price.

Investors can borrow and lend at the riskfree rate.

There are no agency costs. This means that managers always act to maximize shareholder wealth.

The financing decision and the investment decision are independent of each other. This means that operating income is unaffected by changes in the capital structure.

No costs of asymmetric information

debtholders have prior claim to assets and income relative to equityholders, the cost of debt is less than the cost of equity