Editing Modigliani–Miller theorem (section)

==Without taxes==
===Proposition I===
<math>V_U = V_L \,</math>

where

<math>V_U</math> ''is the value of an unlevered firm'' = price of buying a firm composed only of equity, and <math>V_L</math> ''is the value of a levered firm'' = price of buying a firm that is composed of some mix of debt and equity. Another word for levered is ''geared'', which has the same meaning.<ref>Arnold G. (2007)</ref>

To see why this should be true, suppose an investor is considering buying one of the two firms, U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does.  The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.

This discussion also clarifies the role of some of the theorem's assumptions.  We have implicitly assumed that the [[investor]]'s cost of borrowing money is the same as that of the firm, which need not be true in the presence of asymmetric information, in the absence of efficient markets, or if the investor has a different risk profile than the firm.

===Proposition II===
[[Image:MM2.png|frame|right|Proposition II with risky debt. As [[leverage (finance)|leverage]] ([[Debt to equity ratio|D/E]]) increases, the [[weighted average cost of capital|WACC]] (k0) stays constant.]]

:<math>r_E = r_0 + \frac{D}{E}(r_0 - r_D)</math>

where

* <math>r_E</math> ''is the expected rate of return on equity of a leveraged firm, or [[cost of equity]].''
* <math>r_0</math> ''is the company cost of equity capital with no leverage (unlevered cost of equity, or return on assets with D/E = 0).''
* <math>r_D</math> ''is the expected rate of return on borrowings, or [[cost of debt]].''
* <math>\frac{D}{E}</math> ''is the [[debt-to-equity ratio]].''

A higher debt-to-equity ratio  leads to a higher required return on equity, because of the higher risk involved for equity-holders in a company with debt. The formula is derived from the theory of [[weighted average cost of capital]] (WACC).

These propositions are true under the following assumptions:
* no transaction costs exist, and
* individuals and corporations borrow at the same rates.

These results might seem irrelevant (after all, none of the conditions are met in the real world), but the theorem is still taught and studied because it tells something very important. That is, [[capital structure]] matters precisely because one or more of these assumptions is violated.  It tells where to look for determinants of optimal capital structure and how those factors might affect optimal capital structure.