 Modigliani and Miller showed that the leverage does not affect the cost of capital of any firm, but how this phenomena can be reconciled because the cost of capital differ for every security. So then what is the relationship between leverage and the cost of capital or particularly the cost of levered equity. MM1 proposition says that the market value of securities that is the market value of debt and equity is equal to the market values of the underlying assets. This means that the market value of equity and market value of debt and for an unlevered firm is equal to the market value of its assets. This means that the market value of levered firm is equal to the market value of its assets. Whether the firm is levered or it is unlevered. Now we see that the portfolio return is equal to the weighted average returns of the securities involved in it. This means that the relationship between returns of levered equity debt and unlevered equity is basically a model that can be said that the return of unlevered equity is the sum of proportional return of its debt and the proportional return of its equity and the proportional return means the proportion of debt and equity. Now if we solve for the return on unlevered equity using the earlier model we have a simply a little change in the model that says that return on unlevered equity or RE is equal to return on levered equity plus the proportion between debt and equity multiplied by the difference between return on unlevered equity and the return on debt. So this means that the return of unlevered equity is the sum of return on unlevered equity and the additional risk the firm takes due to the leverage. Now this additional risk that the firm takes due to the leverage is termed as the kick. Now this extra effect of this particular kick pushes the return on equity of the levered firm upward if the firm performs well which means that the return on unlevered equity is greater than the return of debt and if the return on unlevered equity is lesser than the return on debt the firm is performing bad this means that the kick is not pushing the return on levered equity upward rather it is pulling down. Now a particular note is that the additional kick or the additional risk that the firm is taking by taking the leverage is subject to the leverage ratio that is measured using the market values of the debt and equity of the firm. So now what the proposition 2 says it says that the cost of levered equity increases with the firm's debt equity ratio or it is the leverage that increases the firm's cost of equity. It means that it's according to this proposition the return on return on levered equity is equal to the sum of unlevered return on equity and the additional risk that the firm takes due to the leverage multiplied by the debt equity ratio. Now how this proposition works let's see an example we see that in our example the money is borrowed for 200 dollars return on unlevered equity is 15 percent as the cash flows are used as collateral so that that is risk free this risk free interest rate is 5 percent the assets market value have 1000 dollars this means the market value of equity will be equal to 800 dollars so in that particular case what is the return on levered equity. So the return on levered equity using the model can be computed at 17.5 percent so this proposition is proving true that the leverage increases the cost of equity for us in this example the unlevered cost of equity was 15 percent in our earlier examples whereas with the increase of leverage into the firm's capital structure the cost of levered equity is 17.5 percent now now what is capital budgeting and wake what is the relationship between these two. M.A. model also allows to check the effect on of leverage on a firm's cost of capital for its new investments or projects. These investments or projects can be financed using the both sources of capital that is the equity and the debt so this means that the riskiness of the assets will be in line or it should be in line with the riskiness of the portfolio of the securities that are used to finance the purchase of these assets or finance this a particular project. Now the overall cost of capital for the firm's assets is the weighted average of its equity and debt cost of capital this means the return on unlevered equity which is also equal to the pre-tax VAC and it is equal to the proportional returns of equity and the proportional return of debt. Now in the perfect capital market as we assume there is no taxes this means that the firm's weighted average cost of capital and its unlevered cost of capital should be equal to each other and similarly the weighted average cost of capital of an unlevered firm should be equal to the return earned by the firm's underlying assets. Now we have a graphics to see the relationship between return on equity return on debt and VAC in the presence of the leverage we see that where we have no debt the firm's VAC is equal to its unlevered cost of equity and that is here 15 percent. Now with the addition of the low cost debt the firm's return on equity is starting increasing due to the fact that return on equity of this levered firm is now equal to unlevered return on equity plus the additional risk in proportion to the firm's leverage. So we see that the net effect on the firm's VAC is also nil. Now we see that as the debt is getting higher it exposes a firm to the default and this raises its debt cost that we see it is going from 5 percent to 15 percent. Now if the firm is using 100 percent debt this means that firm's assets will be equally risky. So the higher leverage raises return on debt and return of unlevered equity both because the more weight is put on the low cost debt but the VAC remains the same. To understand this we have a table here where the debt equity from 1000 to 100 and correspondingly debt from 0 to 900. The return on equity is also increasing from 15 percent to 75 percent because with the addition of the debt the firm is getting riskier so the cost of equity is getting accordingly higher but the return on debt is getting increasing accordingly but at a slower rate. The effect of this combination is that the low cost debt although a low cost but it has no effect on the overall cost of capital of the firm and the VAC in this case remains the same.