14.1The Modigliani-MillerTheorem

The first step in understanding the firm’s capital structure choice is the Modigliani-

Miller Theorem. Much of the rest of this chapter—indeed, much of the rest of this

text—will build results based on this theorem.

Slicing the Cash Flows of the Firm

Exhibit 14.1 illustrates the total cash flows generated by a firm as a pie chart and the

various claims on those cash flows as slices of the pie. Pie 1 illustrates the case in

which all of the cash flows accrue to debt holders and equity holders; the way that the

pie is sliced does not affect the total cash flows available. This is the assumption of

the Modigliani-Miller Theorem.

Pies 2 and 3 illustrate how a firm’s capital structure affects the total cash flows to

its debt and equity holders. Pie 2 has a piece removed for tax payments to the gov-

ernment. Again, how the pie is sliced does not affect its the total size. However, the

owners of the firm are not interested in the total size of the pie. They are interested

only in those cash flows they can sell to security holders—that is, the debt and equity

portions. Hence, the owners would like to minimize the size of the slice going to the

government, which they cannot sell. When more of the pie is allocated to the debt hold-

ers, less is allocated to the government, implying that the combined debt and equity

slices increase.

Finally, pie 3 includes an additional piece that reflects the possibility that part of

the pie is wasted. In other words, a slice of the pie is lost because of inefficiency, lost

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502Part IVCapital Structure

EXHIBIT14.1Slicing the Cash Flows of the Firm

Pie 1

Pie 2

Pie 3

	Debt
Equity
		Equity	Debt
Debt	Equity
	Govt.		Govt.
		Wasted

Slicing the pie doesn’t	The size of the debt	The size of the debt
affect the total amount	slice can affect the	slice can affect the
available to debt holders	size of the slice	size of the wasted
and equity holders.	going to government.	slice.

opportunities, or avoidable costs. Chapters 16 through 19 describe various reasons why

the size of this wasted slice may be determined partly by how a firm is financed.

This section, and the two sections that follow it, consider the case illustrated by

pie 1, where the total futurecash flows the firm generates for its debt and equity hold-

ers are unaffected by the debt-equity mix. As we will show, if the total cash flows are

unaffected by the debt-equity mix, the total valueof a firm’s debt and equity also is

unaffected by their mix.

The remainder of this chapter focuses on Pie 2. Here, again, the total cash flows

generated by the firm’s assets are unaffected by the debt-equity mix. However, in con-

trast to Pie 1, the corporate profits tax, studied in Section 14.4, and the personal income

and capital gains taxes, studied in Section 14.5, claim a portion of these cash flows and

the size of these government claims depends on the debt-equity mix.

Proof of the Modigliani-Miller Theorem

Modigliani and Miller (1958) proved the irrelevance theorem by showing that if two

firms are identical except for their capital structures, an opportunity to earn arbitrage

profits exists if the total values of the two firms are not the same. To illustrate this,

˜

assume that two firms exist for one year, produce identical pretax cash flows (X)at the

end of that year, and then liquidate. However, the firms are financed differently: com-

pany U is unleveraged (that is, it has no debt) and company Lis leveraged (that is, it

has some debt in its capital structure).

Exhibit 14.2 presents the cash flows of companies U and L, the split of the cash

flows between debt and equity, and the present values of the cash flows. Since com-

˜

pany U has no debt, its uncertain cash flow Xis split only among the firm’s equity

holders. Thus, the current value of company U,or V,is the same as the value of its

U

equity. Company L, in contrast, has a debt obligation in one year of (1 r)Ddol-

D

lars. If, for simplicity, we assume that the debt is riskless, and ris equal to the risk-

D

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Chapter 14

How Taxes Affect Financing Choices

503

EXHIBIT14.2

Liability Cash Flows and TheirMarket Values forTwo Firms with Different Capital Structures

Company U

Company L

Future	Current	Future	Current
Cash Flow	Value	Cash Flow	Value

Debt	0	0		(1 r)DD
				D
Equity	X˜		˜
		V	X	(1 r)DE
		U		DL
Total	˜	V		X˜VD E
	X
		U		LL

less rate, company L’s debt holders will receive (1 r)Dat the end of the year and

D

its equity holders will receive the remaining X˜(1 r)Ddollars. The current value

D

of company L, or V,is the current value of its outstanding debt Dplus its equity E.

LL

According to the Modigliani-Miller Theorem, Vmust equal D E,which can

UL

be seen by applying the tracking approach to valuation introduced in Part II. Since com-

pany U’s equity is perfectly tracked in the future by company L’s debt plus equity, the

value of company U’s equity must equal the combined value of company L’s debt plus

equity. If these values are unequal, an opportunity to earn arbitrage profits exists.

Earning Arbitrage Profits When the Modigliani-MillerTheorem Fails to Hold.

To illustrate this arbitrage opportunity, first consider the case where the value of com-

pany U is greater than the value of company L(V D E). Suppose, for exam-

U L

ple, that company U has an equity value of $100 million and company Lhas an equity

value of $60 million and a debt value of $30 million. If this were the case, a shrewd

investor could profit by buying, for example, 10 percent of the outstanding shares

(equity) of company L(costing $6 million) and 10 percent ($3 million) of its out-

standing debt, and selling short 10 percent (or $10 million) of the shares (equity) of

company U.

Assuming that the investor receives the proceeds of the short sale, this transaction

yields a net cash inflow of $1 million [$10 million ($6 million $3 million)]. How-

ever, when cash flows are realized at year-end, the investor will receive

˜)D] .1(1 r)D

.1[X(1 r

DD

in cash from company L’s stock and bonds and must pay out .1˜to cover the short

X

sale of company U’s stock. Thus, the net combined future cash flow from these trans-

actions is

.1[˜(1 r)D] .1(1 r)D.1˜

XX

DD

which is zero, regardless of the future value that ˜realizes. In other words, the trans-

X

action generates cash for the investor at the beginning of the period, but requires noth-

ing from the investor at the end of the period. Similar opportunities for arbitrage exist

if company U has a lower value than company L, that is, V D E.Since we

U L

assume that such arbitrage opportunities cannot exist, the total value of the two firms

must be the same regardless of how they are financed.

Example 14.1 illustrates this type of arbitrage opportunity in a simple case where

the firm’s cash flows can take on one of only two possible values.

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504Part IVCapital Structure

Example 14.1:An Arbitrage Opportunity If the Modigliani-MillerTheorem