Editing Modigliani–Miller theorem (section)

==With taxes==
===Proposition I===

:<math>V_L =V_U + T_C D\,</math>

where

* <math>V_L</math> ''is the value of a levered   firm.''
* <math>V_U</math> ''is the value of an unlevered firm.''
* <math>T_C D</math> ''is the tax rate (<math>T_C</math>) x the value of debt (D)"
 Derivation of <math>T_C D</math>- 
 Amount of Annual Interest= Debt x Interest Rate
 Annual Tax Shield= Debt x Interest Rate x Tax Rate
 Capitalisation Value (Perpetual Firm) = (Debt × Interest Rate x Tax Rate) ÷ Cost of Debt
* the term <math>T_C D</math> assumes debt is perpetual

This means that there are advantages for firms to be levered, since corporations can deduct interest payments. Therefore leverage lowers [[tax]] payments. [[Dividend]] payments are non-deductible.

===Proposition II===
:<math>r_E = r_0 + \frac{D}{E}(r_0 - r_D)(1-T_C)</math>

where:

* <math>r_E</math> ''is the required rate of return on equity, or cost of levered equity = unlevered equity + financing premium.''
* <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 required rate of return on borrowings, or [[cost of debt]].''
* <math>{D}/{E}</math> ''is the debt-to-equity ratio.''
* <math>T_c</math> ''is the tax rate.''

The same relationship as earlier described stating that the cost of equity rises with leverage, because the risk to equity rises, still holds. The formula, however, has implications for the difference with the [[Weighted average cost of capital|WACC]]. Their second attempt on capital structure included taxes has identified that as the level of gearing increases by replacing equity with cheap debt the level of the WACC drops and an optimal capital structure does indeed exist at a point where debt is 100%.

The following assumptions are made in the propositions with taxes:
* corporations are taxed at the rate <math>T_C</math> on earnings after interest,
* no transaction costs exist, and
* individuals and corporations borrow at the same rate.