# Modigliani and Miller Theories

Modigliani and Miller theories of capital structure (also called MM or M&M theories) say that (a) when there are no taxes, (i) a company’s value is not affected by its capital structure and (ii) its cost of equity increases linearly as a function of its debt to equity ratio but when (b) there are taxes, (i) the value of a levered company is always higher than an unlevered company and (ii) cost of equity increases as a function of debt to equity ratio and tax rate.

Modigliani and Miller’s (M&M) theories about capital structure offer a good starting point in a company’s quest for optimal capital structure. This is even though they require certain unrealistic assumptions such as: (a) existence of a totally efficient market with no transaction costs, (b) no financial distress and agency costs, (b) ability to borrow and lend at the risk-free rate, etc.

M&M theories offer two propositions in two environments: (a) without tax and (b) with tax.

## M&M Theory: No-Tax Environment

Let’s first discuss the implications of M&M approach in a no-tax environment.

### Proposition 1

The first proposition states that the value of a company is independent of its capital structure. It implies that the value of an all-equity firm is equal to an all-debt firm. Using the theory’s assumptions, Modigliani & Miller demonstrate that an arbitrage opportunity forces the values to converge.

### Proposition 2

The second proposition states the company’s weighted average cost of capital is a function of the company’s business risk and will remain constant regardless of the capital structure. It implies that component cost of capital (i.e. cost of debt and cost of equity) will adjust with any change in debt to equity ratio resulting in a constant weighted-average cost of capital.

This can be expressed as follows:

$$ wacc=\frac{D}{V}\times k_d+\frac{E}{V}\times k_e $$

Where **wacc** is the weighted-average cost of capital, **k _{d}** is the cost of debt,

**k**is the cost of equity,

_{e}**D**is the absolute value of debt,

**E**is the absolute value of equity and

**V**is the value of total assets of the company.

After some mathematical manipulation we arrive at the following equation of cost of equity (k_{e}):

$$ k_e=wacc+(wacc\ -\ k_d)\times\frac{D}{E} $$

The above equation means that with an increase in debt-to-equity ratio (D/E), cost of equity will increase resulting in a constant weighted-average cost of capital (wacc) at any capital structure.

## M&M Theory: Positive Tax Environment

M&M Theory 1’s assumption that there are no taxes is unrealistic. Taxes exist, and interest expense is tax deductible i.e. the ultimate tax burden of a company with debt in its capital structure is lower than a company with zero or lower debt. This brings us to M&M Theory 2 which relaxes the zero-tax assumption.

### Proposition 1

In a tax environment, the value of a levered company is higher than the value of an unlevered company by an amount equal to the product of absolute amount of debt and tax rate.

This can be expressed mathematically as follows:

$$ V_L=V_{UL}+t\times D $$

Where **V _{L}** is the value of levered company i.e. company with some debt in its capital structure,

**V**is the value of an unlevered company i.e. with no or lower debt,

_{UL}**t**is the tax rate and

**D**is the absolute amount of debt.

### Proposition 2

Since interest expense is tax-deductible, our equation for the weighted average cost of capital modifies as follows:

$$ wacc=\frac{D}{V}\times k_d\times(1-t)+\frac{E}{V}\times k_e $$

All other variables are the same as in Proposition 2 of Theory 1 except for the factor of (1-t). **(1 – t)** represents tax shield i.e. the decrease in effective cost of debt due to existence of tax benefit of debt.

After some mathematical adjustment, we get the following function for cost of equity in a positive-tax environment:

$$ k_e=wacc+(wacc\ -\ k_d)\times(1-t)\times\frac{D}{E} $$

The above equation is the same as in Proposition 2 of Theory 1 except for the factor of (1-t). The consequence of debt shield is that cost of equity increases with an increase in D/E but the increase in less pronounced than in a no-tax environment.

The implication of M&M theory with tax is that the capital structure is no longer irrelevant. The value of a company with debt is higher than the value of a company with no or lower debt.

## Example

A company is considering a business in which the expected weighted average cost of capital is 10% keeping in view the associated business risk. It has option to incorporate in Country A which has no taxes or in Country B which as 20% corporate taxes.

If the company’s cost of debt is 6% in both countries, find out its cost of equity in both countries at the following debt-to-equity ratio levels: (a) zero, (b) 1, and (c) 2.

**Country A**

Country A has no taxes, so we can use the cost of equity function as in Proposition 2 of the Theory 1:

$$ k_e\ at\ \ D/E\ of\ 0=10\%+(10\%\ -\ 6\%)\times0=10\% $$

$$ k_e\ at\ \ D/E\ of\ 1=10\%+(10\%\ -\ 6\%)\times1=14\% $$

$$ k_e\ at\ \ D/E\ of\ 1=10\%+(10\%\ -\ 6\%)\times2=18\% $$

We can demonstrate that the weighted average cost of capital at all level of debt-to-equity ratio is the same i.e. 10%. Let’s see what happens at D/E of 1 or D/V of 50%:

$$ wacc=50\%\times6\%+50\%\times14\%=10\% $$

**Country B**

Existence of taxes creates a preference for debt resulting in a lower increase in equity with addition of debt as demonstrated below:

$$ k_e\ at\ \ D/E\ of\ 0=10\%+(10\%\ -\ 6\%)\times(1-20\%)\times0=10\% $$

$$ k_e\ at\ \ D/E\ of\ 1=10\%+(10\%\ -\ 6\%)\times(1-20\%)\times1=13.2\% $$

$$ k_e\ at\ \ D/E\ of\ 1=10\%+(10\%\ -\ 6\%)\times(1-20\%)\times2=16.2\% $$

The consequence of this less pronounced increase in cost of equity is that the weighted average cost of capital decrease with increase in debt-to-equity ratio. Theoretically, the value is maximized for an all-debt company. However, the existence of some other factors such as probability of bankruptcy, etc. causes the cost of debt to increase such that the value of a company is maximized at some intermediate point (i.e. between an all-debt and an all-equity capital structure).