Modigliani and Miller Propositions
Capital Structure
A firm's capital structure is the proportion of debt and equity used to finance the firm's assets. We can write the value of the firm as `V = D + E` where `V` is the firm's value, and `D` and `E` are the market values of the firm's debt and equity respectively.
 The idea of an optimal capital structure, is that there is some proportion of debt versus equity financing which maximizes `V`.
 Whether such an capital structure exists, and the method of finding it, has long been of interest in finance.
Modigliani and Miller (MM)
In a series of papers that would lead to a Nobel Prize, M&M made important contributions to understanding the relationship between a firm's capital structure, value, and cost of capital. Their main conclusions can be summarized as:

In the absence of taxes, firm capital structure is irrelevant.

With taxes, a firm's cost of capital can be lowered through issuing debt. This highlights the importance of debt as a tax shield.
MM Proposition I (No Taxes)
First assume that:
 There are no taxes.
 There are no transaction costs.
 Both individials and corporations can borrow at the same rate.
In this case MM show that: The value of the levered and unlevered firms are the same. In notation, `V_L = V_U` where `V_L` and `V_U` are the values of the levered and unlevered firms respectively.
MM Proposition I (No Taxes)
This result rests on the assumption that individials and corporations can borrow at the same rate.

If they do, and leveraged firms are priced higher than unleveraged firms, then investors can buy the unleveraged firms on margin.

By doing so investors create leveraged firms, which increases the values of these firms, earning the investors profits.

The key here is that it does not matter to the investor if the debt is held by the company or by the investor herself. The latter case is termed homemade leverage.
MM Proposition II (No Taxes)
The cost of equity rises linearly with leverage according to the equation: `R_E = R_0 + \frac{D}{E}(R_0  R_D)`, where:
 `R_E` is the return on equity.
 `R_0` is the return on the firm's unlevered equity.
 `R_D` is the return on the firm's debt.
 `D` and `E` are the market values of the firm's debt and equity respectively.
MM Proposition II (No Taxes)
The cost of debt is generally less than the cost of equity. That is if a firm pays a 5% yield on its debt, it will have to earn, say, 9% on its equity. So people often assume firms should borrow to take advantage of the cheaper rate.

What MM II says is that if you increase borrowing to get the cheaper rate, you will also increase the amount you will have to pay on your equity. In fact, the two will exactly offset leaving your (overall) weightedaverage cost of capital the same.

So in effect, you cannot lower your cost of capital by exchanging debt for equity.
MM Proposition I (With Taxes)
If we now allow for taxes, the firm can increase its value by financing with debt. This is because debt allows the firm to pay less in taxes.

Specifically, the tax shield from debt in a given period is `t_cI` where `t_c` is the corporate tax rate and II is the interest paid during the period.

If we assume the firm pays the same amount `I` every period, in perpetuity, then the present value of this tax shield is `t_cD` where `D` is the total amount of outstanding debt. This can be seen by noting the present value of a perpetuity which pays `I` every period is `\frac{I}{r_D}`, and `I` is simply `r_DB` (where `r_D` is the perperiod cost of debt.
MM Proposition I (With Taxes)
So, in the case of perpetual debt the value of the leverd firm is `V_L = V_U + t_cD`.
From previous discussion, it should be clear that debt is valuable to the firm because of its ability to shield the firm from taxes.
 Without taxes the firm does not benefit from financing with debt.
MM Proposition II (With Taxes)
With corporate taxes there is still a positive relationship between leverage and the cost of equity, however the cost of equity is lower than it would be without taxes. The exact relationship is:
`R_E = R_0 + \frac{D}{E}(1  t_c)(R_0  R_D)`
Note, by setting `t_c = 0` the equation reduces to MM Proposition II without taxes.
MM and the WACC
A levered firm's market value can be written as `V_L = E + D` and its weightedaverage cost of capital (WACC) is thus:
`R_{WACC} = \frac{E}{V_L}R_E + \frac{D}{V_L}R_D(1  t_c)`

In the following interactive app you can change the tax rate, and costs of unlevered equity and debt, and see the cost of levered equity, debt, and WACC as a function of the debttoequity ratio.

Note that the benefit of debt on the WACC is increasing in the tax rate. If the tax rate is set to 0%, then there is no benefit of debt on the WACC.
Beyond MM
A common extension of MM says that too much debt increases default probabilities, which raises the cost of capital. Balancing the tax shield with bankruptcy risk may lead to an optimal capital structure.
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