Modigliani and Miller’s (1958) irrelevance theorem is only valid in perfect markets. In real markets, imperfections can make capital structure relevant. In a later publication, Modigliani and Miller (1963) themselves analyzed the effect of corporate income taxes on capital structure. The introduction of corporate taxes means that there is a third party, the tax collector, that claims a part of corporate value. This will reduce the value of the firm. However, interest payments are deductible from the firm’s taxable income, while dividends are not. This means that the tax collector’s part will decrease with debt, so that the sum of levered equity and debt will be larger than unlevered equity. The effect can be shown with our previous example; we will use the normal return scenario and a corporate income tax rate of 20 per cent. The numbers are in Table 5.2.
Table 5.2 (Un)levered cash flows
unlevered levered
profits (EBIT) 15 15
interest (10% of 50) – 5
EBT 15 10
taxes (20%) 3 2
after-tax profits 12 8
total income investors 12 8 + 5 = 13
The total income for investors (debt and equity holders) is higher in the levered firm (13) than in the unlevered (12), because the tax collector’s part is lower. If we assume that all cash flows are perpetuities, so that we can use (2.7), the value of the unlevered firm’s assets and equity can be calculated as 12/0.15 = 80. The value of the levered firm includes not only the assets but also the value of the tax reduction (or the value of its tax shield, as it is usually called). To calculate the latter value we have to know how risky the tax shield is, so that we can use the appropriate discount rate. Following Modigliani and Miller we make the assumption that the amount of debt is predetermined, which means that it is not rebalanced to a fixed proportion of firm value if the latter changes over time. The difference between predetermined and rebalanced debt is analyzed in the next chapter. If the amount of debt is predetermined, then the tax advantage of debt is just as risky as debt itself and, hence, it can be discounted with the interest rate on debt, rd. Under these assumptions the value of the tax advantage is 1/0.1 = 10. This gives a total value of the levered firm of 80 + 10 = 90. Since the value of debt is 5/0.1 = 50, the value of levered equity is 90 − 50 = 40. The calculations are summarized in Table 5.3. We see that, in the presence of corporate income taxes, Vu< Vl: tax is a value flow out of the firm and that flow is larger for the unlevered firm than for the levered.
Table 5.3 (Un)levered returns and values
unlevered levered
ra 0.15 0.15
Value assets (12/0.15) 80 80
Value tax shield (1/0.1) – 10
Value of the firm V (Vu, Vl) 80 90
value debt D (5/0.1) – 50
value equity (Eu, El) 80 90 − 50 = 40
re(reu, rel) 0.15 8/40 = 0.2
These calculations can be formulated in more general terms. If we use x for the firm’s cash flow (earnings before interest and taxes, EBIT), and τ for the corporate income tax rate, then value of the unlevered firm, Vu,is:
Vu= (1 − τ )x ra
The cash flows to the investors in the levered firm consist of two parts:
1. To shareholders: (1 − τ )(x − rdD).
2. To debtholders: rdD.
The first part should be discounted with rel, second part with rd. We sum the two cash flows and work out terms:
(1 − τ )(x − rdD)+ rdD (1 − τ )x − rdD+ τ rdD+ rdD (1 − τ )x + τ rdD
147 5.3 Models of optimal capital structure
The first part, (1 − τ )x, is the cash flow to unlevered equity and the second part, τ rdD, is the tax advantage of debt. The first part should be discounted with ra and the second part with rd,as we just saw. This gives:
Vl = (1 − τ )x
ra +τ rdD rd
Vl = Vu+ τ D (5.3)
This is Modigliani–Miller proposition 1 with taxes: the value of the levered firm is the value of the unlevered firm plus the value of the tax advantage of debt. The latter equals the present value of the tax shield under the assumptions we made (i.e. that debt is fixed and perpetual).
Using the same set of assumptions, we can also derive Modigliani–Miller’s proposition 2 with taxes. As we did in the case without taxes, we can write the balance sheet identity assets = debt + equity in terms of returns. However, in the presence of taxes the levered firm has an additional asset: the value of its tax shields. The balance sheet then becomes:
Assets Al Debt D
Tax shields τ D Equity El
total V total V
Writing this balance sheet in terms of returns, the weighted average cost of capital is (the subscript l is superfluous both for assets and equity and Va is the value of assets):
raVa
This formula for the after-tax WACC can be re-written to gives expressions for raand re: ra= rd(1 − τ ) D
V − τ D + re
E V − τ D
Rewriting for rewe get Modigliani–Miller proposition 2 with taxes:
re = ra+ (1 − τ )(ra− rd)D
E (5.4)
If corporate taxes are the only market imperfection, the value of the levered firm will increase and the WACC will decrease with the debt–equity ratio. The ‘optimal’ solution is then 100 per cent debt financing. Figure 5.2 depicts the decreasing WACC for the example we used.
5.3 ... Models of optimal capital structure
Modigliani and Miller’s conclusions of capital structure irrelevance and optimal capital structure at 100 per cent debt are clearly incompatible with observed capital structures.
For an optimal capital structure with less than 100 per cent debt, other market imperfec-tions have to be included that counterbalance the tax advantage of debt. We will look at some of these imperfections and two models that include them.
0 1 2 3 4 0.05
0.10 0.15 0.20 0.25 0.30 0.35
D/E Returns
WACC equity
debt
Figure 5.2 Modigliani–Miller proposition 2 with taxes