. 27
( 39)


Firm Value,
Time 1
¬le=corpclaims.tex: RP
Section 20·3. More About Stock.

A number of companies have sold such put options on their own stock. For example, in 1995, Some speci¬c corporate
examples [via Ted
Microsoft generated cash by selling puts that speculated that the share price of Microsoft would
drop. By writing these put options on its own shares, Microsoft was able to generate an esti-
mated $2 billion dollars in put premia over a period of ¬ve years. Moreover, because Microsoft™s
share price did not end up dropping, it did not have to pay anything to put purchasers. How-
ever, put options can also back¬re. In 2000, Microsoft had to pay large premiums to purchase
and eliminate put options after its stock price fell below the strike price. The most extreme
put loss may have been that of Maytag. Maytag had written around seven million puts on its
shares by the third quarter of the year 2000, while its shares traded in the thirty dollar range
in 2000. The puts allowed their owners to sell shares at an average price of $51, expiring in
2002. Maytag™s losses from its put speculation were then estimated to be above $100 million.
In general, put options remain rare methods for companies to raise funds. (There are also some
tax disadvantages to writing puts on one™s own stock.)

The Time Dimension
Again, the payo¬ diagram may fail to capture the fact that time can play an interesting role
when the option or warrant contract does not just allow conversion at a particular point in
time, but up until a particular point in time. For example, the value of the call or warrant
can then depend also on the underlying dividend yield that the stock is paying (and changes
therein!)”and indeed the question of when to convert into stock can become quite complex.
And, again, although not entirely accurate, the paradigm of thinking of calls and warrants in
terms of contingent claims usually remains a very useful one.
Solve Now!
Q 20.8 We want to compare a plain bond plus a warrant to a convertible bond.

(a) Firm A has a plain bond that promises to pay $200,000. It has 100,000 equity shares
outstanding. The bond buyers also receive a warrant to purchase 25,000 equity shares at
a price of $8/share. In other words, the bond and warrant form a unit.

(b) Firm B has a convertible bond that promises to pay $200,000 but that can be converted
into one-¬fth of the ¬rm™s equity.

Write down the payo¬ tables (and draw the payo¬ diagrams). Re¬‚ect on how they di¬er.

Q 20.9 Write down the equity payo¬ diagram if the ¬rm has the following capital structure:

• 1,000 senior bonds with promised payo¬s of $100 million, convertible into 50 million new
equity shares.

• 500 junior bonds with promised payo¬s of $50 million, convertible (at the bond holder™s
discretion) into 15 million new shares.

• 100 million equity shares for the rest of the ¬rm.

It is easiest to work with aggregate ¬gures, i.e., consider the ¬rm value in increments of $50
million. Hint: At what equity value and at what ¬rm value (the two are not the same) would the
senior convert?
¬le=corpclaims.tex: LP
522 Chapter 20. Corporate Financial Claims.

20·4. Summary

In the real world, ¬rms are ¬nanced by a whole set of di¬erent ¬nancial claims. The same ¬rm
Large ¬rms usually have
multiple claims may have senior debt, junior debt (perhaps with a conversion feature), equity, and a warrant.
The right way to think about all these claims still often involves the “magic” payo¬ table (and
payo¬ diagram): if the ¬rm ends up worth very little, only the senior debt is paid. If the ¬rm
is worth a little more, both the senior and the junior debt are paid. If the ¬rm is worth even
more, the equity will become worth something; and ¬nally, the warrant and/or the conversion
feature will be valuable.
The chapter covered the following major points:

• The two most basic building blocks of capital structure are debt and equity. These di¬er
in their cash ¬‚ow rights and in their control rights:

“ Debt has ¬rst rights to the distribution of cash ¬‚ows. It is “senior.” It can force the
¬rm into bankruptcy if payments are not made.
“ Equity gets only what is left over after debt has been satis¬ed. It is “junior.” It is in
control of the ¬rm, unless the ¬rm ¬nds itself in ¬nancial distress.

• Payo¬ tables and payo¬ diagrams are often a good way to describe debt and equity, be-
cause they are primarily state-contingent claims, where the ¬rm value is the state. The
paradigm is not always perfect, because the value of claims can also depend on factors
other than ¬rm value.

• Call options and warrants are more junior even than equity. These claims allow their
owners to obtain shares at a given price in the future.

• Preferred equity cannot force bankruptcy, but receives its dividends before common eq-
uity does. The main advantage of preferred equity is that corporate holders pay lower
taxes on preferred equity dividend receipts.

• Convertible bonds allow their owners to convert their bonds into shares. They can there-
fore be considered as part-debt, part-equity.

• Corporate borrowing comes in thousands of di¬erent varieties. It can be plain, convertible,
callable, or convertible and callable. It can be ¬xed-rate or ¬‚oating-rate. It can be short-
term or long-term. It can have detailed covenants of many kinds. And so on.

• The lines between di¬erent ¬nancial instruments are blurry. Issuers regularly introduce
new hybrid securities that carry features that were traditionally associated only with either
debt or equity. Nothing is written in stone. Debt vs. equity (or bank debt vs. public debt)
are nowadays better understood as concepts rather than as clear categories.
¬le=corpclaims.tex: RP
Section 20·4. Summary.

Solutions and Exercises

1. A control right is the right to in¬‚uence decisions, speci¬cally by changing management and/or the board.
2. The fact that all that the owner can lose is his investment. He cannot forfeit his house or other possessions
because the corporation does bad stu¬.

Medical Costs Insurance Payout
$0 $0
$250 $0
$500 $0
$750 $225
$1,000 $450
$2,000 $1,350
¦ ¦
$11,500 $9,900
$11,600 $9,990
$11,611 $10,000
$11,700 $10,000
$12,000 $10,000
$13,000 $10,000
¦ ¦

The “slope” is zero until $500 is reached, then 90% until $11,611.11 is reached, then zero again.

4. See text.

Firm Senior Junior Equity
$0 $0 $0 $0
$50 $50 $0 $0
$100 $100 $0 $0
$150 $100 $50 $0
$200 $100 $100 $0
$250 $100 $150 $0
$300 $100 $200 $0
$350 $100 $200 $50
$400 $100 $200 $100
$450 $100 $200 $150
¦ ¦ ¦ ¦

6. If the ¬rm is worth less than $2 million, the bondholders own the entire ¬rm and shareholders receive nothing.
If the bonds convert, they will be equivalent to 1/4 of all shares. Therefore, at $8 million, bondholders are
indi¬erent between converting and not converting. The payo¬ diagram for the debt is therefore a diagonal
line until $2 million, then a horizontal line until $8 million, and a line with a slope of 0.25 beyond $8 million.
For equity, the line is horizontal until $2 million, then a diagonal until $8 million, and a line with a slope of
75% beyond $8 million.

7. Preferred equity is like a bond in that it does not participate in the upside, and in that common shareholders
do not get their dividends until preferred shareholders have received their dividends. Preferred equity is like
a stock in that its payments are not tax deductible, and in that preferred shareholders have no ability to force
the ¬rm into bankruptcy if their dividends are not paid.
8. All numbers are quoted in thousands.
¬le=corpclaims.tex: LP
524 Chapter 20. Corporate Financial Claims.

Firm A Firm B
Plain Convertible
Plain Bond
Firm Value Bond Equity Warrant + Warrant Bond Equity
$0 $0 $0 $0 $0
$100 $100 $0 $0 $100 $100 $0
$200 $200 $0 $0 $200 $200 $0
$300 $200 $100 $0 $200 $200 $100
$400 $200 $200 $0 $200 $200 $200
$500 $200 $300 $0 $200 $200 $300
$600 $200 $400 $0 $200 $200 $400
$700 $200 $500 $0 $200 $200 $500
$800 $200 $600 $0 $200 $200 $600
$900 $200 $700 $0 $200 $200 $700
a c
$1,000 $200 $800 $0 $200 $200 $800
$20b $220d
$1,100 $200 $880 $220 $880
$1,200 $200 $960 $40 $240 $240 $960
$1,300 $200 $1,040 $60 $260 $260 $1,040
$1,400 $200 $1,120 $80 $280 $280 $1,120
$1,500 $200 $1,200 $100 $300 $300 $1,200

Explanations: a At $1,000 ¬rm value, Firm A™s equity shares are worth $800, which translates into $8/share. At
$8 share, warrant holders are indi¬erent between exercising and not exercising. b At $1,100 ¬rm value, Firm
A™s equity shares are worth $900, which translates into $9/share. At $9/share, warrant holders exercise. They
pay in $8 and receive 25,000/(100,000+25,000)=20% of the ¬rm™s equity value. c Convertible bond holders are
indi¬erent between converting their bond into equity if the ¬rm value is $1,000. After all, they would receive
20% of the ¬rm™s total value, which would come to $200. d Convertible bond holders prefer to convert their
bond into equity, because 20% of $1,100 is more than $200.
In sum, a plain bond plus warrant can be just like a convertible bond.
9. The question seems di¬cult, but it does become easy once you realize the following:
• If the junior does not convert, then the senior™s 50 million in new equity shares would represent 50/150
or one-third of the equity (not the company!). Thus, the senior would convert if the value of the equity
reaches $300 million. This occurs when the ¬rm value reaches $350 million, because the junior creditors
still would have their “$50 million ¬rst” claim.
• If the senior has converted, then the junior™s 15 million in new equity shares would represent 15/165
of the ¬rm. This is about 9.1% of the ¬rm value. Therefore, at a ¬rm value of $550 million (solve
x·15/165 = $50), the junior would be indi¬erent between exercising and not exercising.
These two insights make it easy to write down the payo¬ table (note my odd stepping when convenient for
explanation), all in million dollars:
Firm Senior
Value Bond Bond Equity Remarks
$0 $0 $0 $0
$50 $50 $0 $0
$100 $100 $0 $0
$150 $100 $50 $0
$200 $100 $50 $50
$250 $100 $50 $100
$300 $100 $50 $150
(at V = $350, senior is indi¬erent to converting)
$350 $100 $50 $200
(at $353, senior has converted ” there are now
$353 $101 $50 $202
150 million shares worth $303 in Equity)
$400 $117 $50 $233
$450 $133 $50 $267
$500 $150 $50 $300
(at V = $550, junior is indi¬erent to converting)
$550 $167 $50 $333
(at $561, junior has converted ” there are now
$561 $170 $51 $340
165 million shares in Equity)
$600 $182 $55 $364
V 30.30% 9.09% 60.61%

(All answers should be treated as suspect. They have only been sketched, and not been checked.)
Capital Structure and Capital Budgeting in a
Perfect Market

Should A Company Issue Stocks or Bonds?
last ¬le change: Feb 23, 2006 (15:02h)

last major edit: Feb 2005

This chapter explains how to look at the basics of ¬nancing: in a “perfect” world (no transaction
costs, no taxes, etc.) the value of the ¬rm is determined by the value of the projects (the NPV),
not by whether the ¬rm is ¬nanced with equity or leverage. This chapter also explains the
basics of the weighted average cost of capital formula (WACC). Subsequent chapters will explain
¬nancing in the real world, not in our perfect world.

¬le=capideal.tex: LP
526 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

21·1. Conceptual Basics

21·1.A. The Firm, The Charter, and The Capital Structure

The best way to think about an optimal ¬rm structure is as follows: the current owners of a
The ¬rm™s charter
de¬nes the ¬rm ¬rm want to sell it today for the highest possible price. Their goal is to design a corporate
structure. The
charter that maximizes the total market value of their ¬rm today”that is, the price that new
agreement of who
investors are willing to pay to acquire the ¬rm from them. The corporate charter must not only
receives what part of
future corporate income
specify the voting rules (the procedures to replace incumbent managers or amend the charter
is the Capital Structure
in the future), but also how future earnings are to be split among possibly di¬erent owners and
stakeholders (such as customers, workers, and suppliers). This agreement on how to split up
future earnings is the ¬rm™s ¬nancial structure or capital structure: it is the rules that specify
who receives the proceeds of (usually uncertain) future cash ¬‚ows. The capital structure is
rarely fully explicit or set in stone. Instead, it also encompasses the rules on how the ¬rm may
be governed in the future, which in turn may in¬‚uence the capital structure in the future, which
in turn may in¬‚uence the ¬nancing arrangement today.

21·1.B. Maximization of Equity Value or Firm Value?

Managers are at least in principle appointed by the shareholders. An important question is
Should management
maximize shareholder whether managers should be directed to see themselves as representative of the shareholders
or as representatives of the ¬rm. The common view is that it is the task of management to
maximize shareholder wealth. But equity is only one part of the ¬rm™s capital structure. There
are also liabilities that have claims on the ¬rm™s assets. There are pension obligations, accounts
payable, and ¬nancial debt”and many ¬nancial instruments with both debt and equity features.
(For illustration, we will use ¬nancial debt as a standin for all liabilities.) Does it even make a
di¬erence as to whom management is representing?
The legal situation in the United States is that management has a legal ¬duciary duty to
When is there a
potential problem? shareholders”except if decisions made by the ¬rm can threaten its insolvency. In this case,
management™s legal ¬duciary responsibility extends to creditors, too. In general, managers see
themselves more as representatives of shareholders than as representatives of creditors. When
both bondholders and shareholders bene¬t from a manager™s actions, there is no problem. But
what if there are situations in which optimizing shareholder value (i.e., the value of the equity)
is not the same as optimizing the overall ¬rm™s value? For example, assume it were possible
for managers to increase the value of equity by $1, but at a cost to the value of ¬nancial debt
by $3. (You will later learn how easy it is to do exactly this.) This “expropriative” transaction
would destroy $2 in the net value of the ¬rm. But even in our perfect world, this is the type of
situation that creates a dilemma for management: should management maximize ¬rm value
or shareholder value? Recall that it is shareholders who vote managers into o¬ce and allow
them to stay there. Whether this transaction destroys ¬rm value or not, when the time comes,
managers will ¬nd it in their interest to execute this transaction because doing so raises equity
value and with it management™s chances of being reappointed.
However, there is a ¬‚y in the ointment. Put yourself into the shoes of the old ¬rm owners
Bond buyers understand
future con¬‚icts of today, who are trying to set up the corporate charter that maximizes the value of their ¬rm,
interest. If managers
i.e., for sale to new investors at the highest price. You want to ¬nd the best capital structure
ignored bondholder
today, i.e., before you have found new investors. Clearly, anyone contemplating purchasing
concerns, bondholders
demand a higher
your bonds will take into consideration what managers may do to them in the future, and they
interest rate upfront.
will rationally demand compensation. If you cannot commit the ¬rm today not to undertake
the $3-for-$1 transaction in the future, prospective bond buyers will realize today (before the

Anecdote: German Stakeholders
In Germany, stakeholders have an explicit role in corporations. In the case of joint stock companies, limited lia-
bility companies and cooperatives with more than 500 employees, one third of the members of the Supervisory
Board must be employees. In the case of companies in the iron, coal and steel industry, provision is made for
equal representation on the supervisory board by shareholders and representatives of the work force. A director
representing the employees with responsibility for social a¬airs is also appointed (quali¬ed co-determination).
¬le=capideal.tex: RP
Section 21·1. Conceptual Basics.

fact, or ex-ante) that you (management) will have the incentive to execute it later (after the fact,
or ex-post), no matter what you tell them today.
If the ¬rm were believed to undertake this transaction in the future, what would it be worth Therefore, managers
need to pay attention to
today? It would be worth less than a ¬rm that will not destroy $2 of value in the future. There-
the needs of
fore, management today has a choice: it can either ¬nd a way to commit today not to exploit bondholders”even
bondholders in the future; or it can sell the ¬rm today for a lower net present value, which though they are voted
into power only by
takes into account value destruction tomorrow”because everyone realizes that managers will
equity holders and are
be trapped tomorrow into destroying $2 of ¬rm value. To avoid this, managers should want to always tempted to focus
only on improving
do everything in their power to constrain themselves from expropriating bondholders in the
equity value.
future. Constraining themselves will maximize the value of the ¬rm in the future, which in turn
maximizes the value of the ¬rm today.


• In deciding on an appropriate price to pay, the buyers of ¬nancial securities
take into account what the ¬rm is likely to do in the future.

• The theoretical notion of the optimal capital structure is the structure that
maximizes the value of the ¬rm, not the value of the equity.

In this theoretical world, management should maximize ¬rm owner value, not shareholder Con¬‚icts of interest
arise between
value. Practically, the two objectives di¬er only rarely (and usually only when ¬rms are close to
shareholders and
¬nancial distress), so the popular mantra of “shareholder value maximization” is fortunately bondholders, but they
usually synonymous with “total value maximization.” But strictly speaking, the best capital are dwarfed by the
con¬‚ict between
structure does not have as its goal maximizing shareholder value but ¬rm value. In any case, in
managers and owners.
the real world, managers are far more con¬‚icted with respect to their own welfare (the agency
con¬‚icts we ¬rst discussed in Section 7·6) than they are with respect to favoring shareholders
at the expense of bondholders.
The advantage of a ¬rm that is committed to maximizing ¬rm value in the future is that it Committing to
maximizing ¬rm value
can obtain a better price for its ¬nancial securities (a lower interest rate for its bonds) today.
gives a better interest
Therefore, the ¬rm has the incentive to commit itself today (ex-ante) to treating bondholders rate and higher value
well in the future (ex-post). The ex-ante capital structure that results in the highest ¬rm value today.
today is the optimal capital structure.
This is the most important insight with respect to capital structure, and worth repeating: the The basics of capital
structure theory is to
cost of ex-post actions against bondholders is not only borne by bondholders tomorrow, but it
realize that future
is also borne by the owners today. Another consequence is that caveat emptor (“buyer beware”) events have impact on
applies: bond and stock purchasers can only be hurt to the extent that future opportunistic corporate value today.
The goal is to use capital
actions by management are unforeseen surprises. Thus, it is in management™s (owners™) own
structure to maximize
interest today to commit not to exploit future owners and bondholders tomorrow”especially ¬rm value today.
if everyone knows that when the time comes, management would like to change its mind.

The web chapter on corporate governance will return to this idea.

What would happen if the current management team cannot commit to avoid such bad future Competition among
management teams
$3-for-$1 exchanges? In this case, in our perfect world, another management team that has the
pressures ¬rms to
ability to commit to restrain itself would value the ¬rm more highly than the current manage- improve Capital
ment team. It could purchase the ¬rm and make an immediate pro¬t. Therefore, competition Structures.
among management teams can push ¬rms towards the best capital structure. Again, the gen-
eral thread emphasized throughout this chapter is that ¬rms that can commit to do “the right
thing” tomorrow (ex-post ) are worth more today (ex-ante). It is a direct consequence that ¬rms
that maximize ¬rm value are worth more than ¬rms that maximize just shareholder value.
¬le=capideal.tex: LP
528 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

Solve Now!
Q 21.1 Explain the di¬erence between ex-ante and ex-post, especially in the capital structure
context. Give an example where the two di¬er.

Q 21.2 Can an ex-post maximizing choice be ex-ante bad?

21·2. Modigliani and Miller (M&M), The Informal Way

The famous Modigliani-Miller (M&M) propositions (honored with two Nobel Prizes) are a good
start to understanding ¬rms™ capital structure decisions. Although the M&M theory involves
some complex algebraic calculations, it is actually based on some surprisingly simple ideas”
which the following anecdote explains not only in a funnier but also better way than any complex
calculations. It is an excerpt from an acceptance speech by Merton Miller for an honorary
doctorate at Louvain, Belgium, in 1986. (His coauthor, Franco Modigliani, had just won the ¬rst
Nobel Prize; Merton Miller would receive his own Nobel Prize a few years later.)

Anecdote: M&M: Milk, Cream, and Pizza
How di¬cult it is to summarize brie¬‚y the contribution of these papers was brought home to me very clearly last
October after Franco Modigliani was awarded the Nobel Prize in Economics in part”but, of course, only in part”
for the work in ¬nance. The television camera crews from our local stations in Chicago immediately descended
upon me. “We understand,” they said, “that you worked with Modigliani some years back in developing these
M&M theorems and we wonder if you could explain them brie¬‚y to our television viewers.” “How brie¬‚y?”, I
asked. “Oh, take 10 seconds,” was the reply.
Ten seconds to explain the work of a lifetime! Ten seconds to describe two carefully reasoned articles each
running to more than 30 printed pages and each with 60 or so long footnotes! When they saw the look of
dismay on my face, they said: “You don™t have to go into details. Just give us the main points in simple,
common sense terms.”
The main point of the ¬rst or cost-of-capital article was, in principle at least, simple enough to make. It said that
in an economist™s ideal world of complete and perfect capital markets, and with full and symmetric information
among all market participants, the total market value of all the securities issued by a ¬rm would be governed by
the earning power and risk of its underlying real assets and would be independent of how the mix of securities
issued to ¬nance it was divided between debt and equity capital. Some corporate treasurers might well think
that they could enhance total value by increasing the proportion of debt instruments because yields on debt
instruments, given their lower risk, are, by and large, substantially below those on equity capital. But, under the
ideal conditions assumed, the added risk to the shareholders from issuing more debt will raise required yields
on the equity by just enough to o¬set the seeming gain from use of low cost debt.
Such a summary would not only have been too long, but it relied on shorthand terms and concepts, like perfect
capital markets, that are rich in connotations to economists, but hardly so to the general public. I thought,
instead, of an analogy that we ourselves had invoked in the original paper. “Think of the ¬rm,” I said, “as a
gigantic tub of whole milk. The farmer can sell the whole milk as is. Or he can separate out the cream and sell
it at a considerably higher price than the whole milk would bring. (Selling cream is the analog of a ¬rm selling
low yield and hence high-priced debt securities.) But, of course, what the farmer would have left would be skim
milk, with low butter-fat content and that would sell for much less than whole milk. Skim milk corresponds
to the levered equity. The M&M proposition says that if there were no costs of separation (and, of course, no
government dairy support programs), the cream plus the skim milk would bring the same price as the whole
¬le=capideal.tex: RP
Section 21·2. Modigliani and Miller (M&M), The Informal Way.

Anecdote: (continued.)

The television people conferred among themselves for a while. They informed me that it was still
too long, too complicated and too academic. “Have you anything simpler?”, they asked. I thought
of another way that the M&M proposition is presented which emphasizes the notion of market
completeness and stresses the role of securities as devices for “partitioning” a ¬rm™s payo¬s in each
possible state of the world among the group of its capital suppliers. “Think of the ¬rm,” I said, “as
a gigantic pizza, divided into quarters. If now, you cut each quarter in half into eights, the M&M
proposition says that you will have more pieces, but not more pizza.”
Again there was a whispered conference among the camera crew and the director came back and said:
“Professor, we understand from the press release that there were two M&M propositions. Maybe we
should try the other one.”
He was referring, of course, to the dividend invariance proposition and I know from long experi-
ence that attempts at brief statements of that one always cause problems. The term “dividend” has
acquired too great a halo of pleasant connotations for people to accept the notion that the more
dividends the better might not always be true. Dividends, however, as we pointed out in our article,
do not fall like manna from heaven. The funds to pay them have to come from somewhere”either
from cutting back on real investment or from further sales (or reduced purchases) of ¬nancial in-
struments. The M&M dividend proposition o¬ered no advice as to which source or how much to
tap. It claimed, rather, that once the ¬rm had made its real operating and investment decisions, its
dividend policy would have no e¬ect on shareholder value. Any seeming gain in wealth from raising
the dividend and giving the shareholders more cash would be o¬set by the subtraction of that part
of their interest in the ¬rm sold o¬ to provide the necessary funds. To convey that notion within
my allotted 10 seconds I said: “The M&M dividend proposition amounts to saying that if you take
money from your left-hand pocket and put it in your right-hand pocket, you are no better o¬.”
Once again whispered conversation. This time, they shut the lights o¬. They folded up their equip-
ment. They thanked me for my cooperation. They said they would get back to me. But I knew that I
had somehow lost my chance to start a new career as a packager of economic wisdom for TV viewers
in convenient 10-second sound bites. Some have the talent for it; and some just don™t.
These simple, common sense analogies certainly do less than full justice to the M&M propositions;
crude caricatures or cartoons they may be but they do have some resemblance. So much, in fact, that
looking back now after more than 25 years it is hard to understand why they were so strongly re-
sisted at ¬rst. One writer”David Durand, the same critic who had so strongly attacked the Markowitz
model”even checked out the prices for whole milk, skim milk and cream in his neighborhood super-
market. He found, of course, that the M&M propositions didn™t hold exactly; but, of course, empirical
relations never do.
¬le=capideal.tex: LP
530 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

21·3. Modigliani and Miller (M&M), The Formal Way In Perfect

To begin with, Modigliani and Miller argued that under perfect conditions, the total value of
Consider a perfect
market world. all ¬nancial securities is the same, regardless of whether the ¬rm is ¬nanced by equity or debt
or anything in between. They proved their argument by showing that there would be arbitrage
opportunities if the value of the ¬rm depended on how it is ¬nanced. Because there should
be no arbitrage in real life, it follows that ¬rms should be able to choose any mix of securities
without impact on the ¬rms™ values. This perfect world that M&M describe relies on the familiar
perfect world assumptions (e.g., in Section 6·1).

• There are no transaction costs. In this context, it also means that there are no such
frictions as deadweight losses in bankruptcy. (This includes the assumption that there
are no costs to ¬nancial distress before bankruptcy occurs, either.)

• Capital markets are perfectly competitive, with a large number of investors competing for
many securities.

• There are no taxes.

• There are no di¬erences in opinion and information.

We already know that these assumptions imply that borrowing and lending interest rates are
equal. Of course, these assumptions do not hold in reality. However, once you understand
how the M&M argument works, it becomes easier to understand what happens when these
assumptions are violated, and to understand how important such violations can be. Indeed,
the next several chapters will show what happens if the world is not perfect.
Let™s see now how the Modigliani and Miller proof works. For simplicity, take it as given that
The proof is simpler if
we assume a ¬xed the ¬rm has already decided on what projects to take. M&M believed that this was a necessary
investment policy for
assumption, but it turns out not to matter in their perfect market. (We will discuss this at the
the moment.
end of this section.)
The ¬rm wishes to consider how to ¬nance its projects. Because we all agree on all current
The Modigliani-Miller
proposition states that and future projects™ expected cash ¬‚ows and proper discount rates, we agree on the present
capital structure does
value of these projects today. Call the value of the projects under a hypothetical best capital
not matter in this ideal
structure “PV.” (This is [almost by de¬nition] the present value that the ¬rm™s projects can fetch
in our perfect capital market, of course.) The M&M claim is that the present value of the ¬rms™
projects must equal the present value of the ¬rms™ issued claims today. In other words, if the
¬rm has no liabilities and issues 100% equity, the equity must sell for the PV of the projects.
If the ¬rm instead were ¬nanced by 50% debt (or by other liabilities) and 50% equity, the two
together must sell for the same PV. If the ¬rm issues x% debt and (1-x%) equity, the two together
must sell for PV. According to theory, the value of the ¬rm should be determined by the net
present value of its projects, regardless of capital structure. So, why does the capital structure
not matter?

The Full Repurchase (Takeover) Argument Assume that the managers could ¬nd”and ac-
tually did choose”a capital structure that makes the ¬rm worth $1 less than PV. For
example, assume that the ¬rm is worth PV=$100 under the optimal capital structure of
80% equity and 20% debt; and assume further that the ¬rm is worth only $99 under the
capital structure of 50% equity and 50% debt that the ¬rm has actually chosen. Then, all
you need to do to get rich is to purchase all old equity and all old debt, i.e., the entire
¬rm, for $99. Now issue claims duplicating the optimal capital structure (assumed to be
80% equity/20% debt). These claims will sell for $100, and you pocket an instant arbitrage
pro¬t of $1.
¬le=capideal.tex: RP
Section 21·3. Modigliani and Miller (M&M), The Formal Way In Perfect Markets.

Unfortunately, you would not be the only one to notice this opportunity. After all, in- Competition: Others
would want to arbitrage,
formation is universally shared. So, the old managers would simply ask for bids from
too”until the M&M
other investors. The only price at which no one will overbid you for the right to purchase proposition works (¬rm
the ¬rm™s current securities is $100. But notice that this means that the value of the old value is as if it was
¬nanced optimally).
securities is equal to the price that the ¬rm is worth under the optimal capital structure.
So, regardless of the ¬nancial structure that managers choose, they can sell their claims
for $100, i.e., the present value of their projects.

The Partial Repurchase (Homemade Leverage) Argument This argument even works if you
do not buy 100% of the ¬rm, but only 1% of the ¬rm. That is, if you buy 1% of all the ¬rm™s
securities, you will receive 1% of the projects™ payo¬s. You can then sell your securities
repackaged to imitate the payo¬s under the presumably better capital structure for 1% of
the ¬rm™s higher value, and receive an arbitrage pro¬t of 1%·$1.
You will see the gory details in two paragraphs, but there is an important caveat to this The Full Repurchase
method is more general.
argument: it allows you only to obtain the cash ¬‚ows of securities under any arbitrary
and presumably better capital structure, but not the control rights! If the better capital
structure has more value because it allows you to ¬re management and change what the
¬rm is doing, you have to rely on the full repurchase argument.

Actually, the M&M argument should not come as a surprise to you. In Section 5·3.B (Page 96), You had already seen
this argument!
without calling it the M&M argument, we had already made use of it in the context of ¬nancing
a house. We found that neither the house value nor the cost of capital were in¬‚uenced by your
debt vs. equity choice: the house was worth what it was worth. This was M&M precisely. It is
the same argument. So, let us put our general but verbal-only proof into the framework of a
more concrete scenario analysis for our $100 ¬rm. To accomplish this as simply as possible,
we assume the world is risk-neutral and all securities have to o¬er an expected rate of return
of 10%. (We will work an example in a risk-averse world in Section 21·5.A.)
Linking M&M under
uncertainty to
state-contingent payoffs
The Full Scenario Analysis Table 21.1 shows the value of a ¬rm if the projects will be worth (under risk-neutral
either $60 or $160. The expected future value is $110, the present value is $100. Under
hypothetical capital structure LD (“little debt”), the ¬rm issues debt with face value $55.
Consequently, bondholders face no uncertainty, and will pay $55/(1 + 10%) = $50. Equity
holders will receive either $5 or $105, and are thus prepared to pay $55/(1 + 10%) =
$50. Simply adding the value of the ¬rm™s claims adds up to the same $100. Under
hypothetical capital structure MD (“much debt”), the ¬rm issues debt with face value $94.
Consequently, bondholders will now receive either $60 or $94, and are willing to pay $70
today. Equity holders will receive $0 or $66, and are willing to pay $30 for this privilege.
Again, the value of all claims adds to $100.
Arbitraging does not
require purchase and
sale of the entire ¬rm; it
The Partial Scenario Analysis The more surprising fact is that investors can relever securities
also works with a
themselves”they do not need the ¬rm to do it for them. In our example, assume that fraction of the ¬rm.
the ¬rm has chosen capital structure LD, but you really, really like capital structure MD,
because you would really, really like to own a security that pays $0.60 in the bad state and
$0.94 in the good state. This would cost you 1% of the bond™s $70 price, or $0.70. How
can you purchase the existing LD securities to give you the MD security that you prefer
without any cooperation by the LD ¬rm?
What happens if you purchase d bonds and e stocks in our LD ¬rm? You will receive
payo¬s of d · $55 + e · $5 in the bad scenario, and d · $55 + e · $105 in the good scenario.
You want to end up with $0.60 in the bad scenario, and $0.94 in the good scenario”two
equations, two unknowns:

d · $55 + e · $5 = $0.60 d ≈ 0.0034
Bad Luck
’ (21.1)
d · $55 + e · $105 = $0.94 e ≈ 0.0106
Good Luck
¬le=capideal.tex: LP
532 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

Table 21.1. Illustration of the M&M Proposition with Risk-Neutral Investors

Bad Luck Good Luck Future Ex- Today™s
Prob: 1/2 1/2 pected Value Present Value

Project FM $60 $160 $110 $100

Capital Structure LD: Bond with Face Value FV=$55
Bond(FV=$55) DT $55 $55 $55 $50
Equity EQ $5 $105 $55 $50

Capital Structure MD: Bond with Face Value FV=$94
Bond(FV=$94) DT $60 $94 $77 $70
Equity EQ $0 $66 $33 $30

The cost of capital in this example is 10% for all securities, which is equivalent to assuming risk-neutrality. Later in
this chapter, we will work an example in which the cost of capital is higher for riskier projects.

So, if you purchase 0.0034 bonds and 0.0106 of the equity, you will end up with $0.60
in the bad state, $0.94 in the good state”exactly the same that an MD ¬rm would have
given you!. How much would you have to pay to get these payo¬s? The cost today would
be d · $50 + e · $50 = 0.0034 · $50 + 0.0106 · $50 = $0.70, exactly the same that your
desired payo¬s would have cost you if the ¬rm itself had chosen an MD capital structure.
In e¬ect, you have manufactured the capital structure payo¬s that you like without the
cooperation of the ¬rm itself. By repeating this exercise, you can replicate the payo¬s of
any claims in any kind of capital structure.
From here, it is an easy step to the M&M argument. If the value of the ¬rm were higher
under the MD capital structure than it is under the LD capital structure, you could man-
ufacture for yourself at lower cost from the current capital structure the securities that
would end up with a higher market value, sell them, and earn an arbitrage pro¬t.

Important: In Modigliani and Miller™s perfect world:

• Arbitrage restrictions force the value of the ¬rm to be the same, regardless
of the ¬rm™s mix of debt (liabilities) and equity.

• Because ¬nancing and investing are two entirely separate issues, managers
can ignore ¬nancing issues when they make investment decisions.

If the world is not perfect, neither need be the case.

This is good news and bad news. It is good news that you now know where to focus your e¬orts.
Know what to care
about and what not to You should try to increase the value of the underlying projects”by increasing their expected
care about!
cash ¬‚ows, or by reducing their cost of capital, or both. It is bad news that you now know that
to the extent that the world is perfect, you cannot add value by ¬ddling around with how you
¬nance your projects.
¬le=capideal.tex: RP
Section 21·4. Dividends.

The above proof of the Modigliani-Miller proposition followed the M&M 1957 paper in assuming In a perfect world, ¬rms
always undertake the
that the ¬rm™s real investment decisions had already been decided upon. It turns out that this
best projects.
is not necessary as long as investment decisions are reversible, because the perfect markets
assumption also implies that management™s project policy should not matter. If the ¬rm were
not to undertake all positive NPV projects and reject all negative NPV projects, you could buy
all the shares, ¬re the old management, institute the best underlying ¬rm investment project
policy, resell all the shares, and earn the di¬erence in ¬rm value as an immediate arbitrage.
Alas, you would again not be the only one: everyone could do this. Therefore, in this perfect
world, ¬rms always do the right thing. They maximize the ¬rm™s NPV, and they are worth
exactly what they should be worth under the optimal operating policy.
Solve Now!
Q 21.3 Under what assumptions does capital structure not matter?

Q 21.4 Explain the M&M argument to your 10-year old brother.

Q 21.5 What does risk-neutrality “buy” us in the M&M argument?

Q 21.6 In the example from Table 21.1, how would you purchase the equivalent of 5% of the
equity of a hypothetical MD ¬rm, if all that was traded were the securities of the LD ¬rm?

21·4. Dividends

The second Modigliani-Miller proposition is even simpler than the ¬rst: it states that corporate In a perfect world,
dividends do not create
dividend policy should not matter in a perfect market setting, either. From the corporate
perspective, if managers pay $1 in dividends, this money has to come from somewhere. As
Merton Miller noted, dividends do not fall like manna from heaven, so no value is created or
destroyed when ¬rms pay dividends. Money that was previously owned by investors but held
inside the corporate shell is just being moved to the same investors, so that it is now outside the
corporate shell. The owners do not have any more or any less wealth because of the dividend
payment. From the M&M arbitrage perspective, if managers undertook a dividend policy that
destroyed value, then any investor could step in to purchase the ¬rm, ¬re the management,
institute the better dividend policy, and resell the ¬rm for the di¬erence. With many investors
vying to do this if even just a penny can be earned, the only ¬rm value and dividend policy
that do not allow anyone to arbitrage (get rich without e¬ort) is the value of the ¬rm under an
optimal dividend policy.
Like the M&M capital structure proposition, the point of the M&M dividend proposition is not The M&M logic helps us
think in our imperfect
to argue that dividends do not matter. It is to point out what perfect market violations must be
the case for dividend policy to matter, and how much these violations can matter. For example,
if it costs a roundtrip premium of $10 million to purchase and then resell a ¬rm, then it cannot
be that the wrong dividend policy destroys more than $10 million. If it did, you could make
money even in our imperfect world.
As of 2005, the average dividend yield of large ¬rms is around 1% of ¬rm value per year. This The situation today, and
a preview.
is probably so low that the real-world market frictions are larger than what you could earn by
correcting this policy if it were incorrect. That is, if the optimal payout were 0% or 2% instead of
1%, the maximum 1% value increase is too little to overcome the transaction costs that would
allow someone to step in and correct it. Interestingly, there is some good evidence that the
M&M assumptions are indeed violated: when ¬rms announce dividend increases, their values
usually go up, and when they announce dividend decreases, their values usually go down. Can
you speculate which M&M assumption is most likely violated? Most ¬nance professors believe
that paying dividends sends a credible signal from management about future ¬rm prospects
and good managerial behavior (managers will not waste the money on themselves). This violates
the M&M assumption that everyone has the same information: in the real world, managers have
inside information that investors do not have.
¬le=capideal.tex: LP
534 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

21·5. The Weighted Average Cost of Capital (WACC) in a Per-
fect M&M World

You now understand why the value of the ¬rm does not depend on the ¬nancing in a perfect
Let us do contingent
claims under market. This is equivalent to stating that the overall cost of capital to the ¬rm does not depend
risk-aversion, i.e., usually
on the debt/equity ratio of the ¬rm. To show this, we want to repeat the house example from
higher expected rates of
Section 5·3.B. It is important that you realize that our argument also works in the context
return to equity than to
of a risk-averse world, just as long as the world is still perfect. Therefore, we shall work our
earlier examples again but allow riskier securities to have higher expected rates of returns. Our
example will draw on your knowledge of net present value, the capital asset pricing model, and
capital structure concepts. The reason why this is important is that it introduces the concept of
the “weighted average cost of capital” (or WACC) in the corporate context. (In the next chapter,
we will generalize WACC to a world in which corporations pay income tax.)

21·5.A. The Numerical Example In a Risk-Averse World Where Riskier Equity Must O¬er
Higher A Expected Rate of Return

Our example will again be our house with the mortgage from Section 5·3.A (Page 93), so make
All tools learned in
Section 5·3.B still apply sure you remember the concepts from Chapter 5. Brie¬‚y, in that Chapter, the expected rate of
under risk aversion.
return was the same for projects of all risk classes. In contrast, in this chapter, we take into
account the fact that investors are risk-averse, so that riskier securities have to o¬er higher
expected rates of return. The basic tools will be exactly the same as those in Section 5·3.A:
payo¬ tables, promised rates of return, and expected rates of return.
The weighted average cost of capital (or WACC) is the value-weighted average cost of capital
Our goal: show that the
weighted cost of capital of the ¬rm™s debt and equity. Not surprisingly, the expected rate of return on the ¬rm is
from debt and equity
determined by the risk of the assets, the expected rate of return on the bonds is determined by
must be the cost of
the risk of the bonds, and the expected rate of return on the equity is determined by the risk
capital of the ¬rm, and
be independent of how
of the shares. We want to show that if the perfect markets arbitrage condition holds”that is,
the ¬rm is ¬nanced.
bonds and stocks together cost the same as the ¬rm”then you can compute the cost of capital
for the ¬rm as the weighted cost of capital of stocks and bonds. Put di¬erently, if you know
any two costs of capital, you can compute the third one. For example, if you know the cost of
capital for the ¬rm and the cost of capital for the bonds, then you can determine the cost of
capital for the equity. Or, if you know the cost of capital for the debt and equity, you know the
cost of capital for the ¬rm”the WACC. And regardless of the ¬rm™s debt ratio, the WACC in
our perfect world is always constant and determined by the risk of the underlying assets.
From Chapter 27, we know that debt and equity are contingent claims on the underlying project.
This example applies to
more than just houses. Although we continue calling this project a house (as we did in Section 5·3.A), you can consider
the corporation to be the same as an unlevered house, the mortgage the same as corporate lia-
bilities, the levered house equity ownership the same as corporate equity, and the possibilities
of sunshine and tornadoes the same as future scenarios that the ¬rm might face. There is no
conceptual di¬erence.
The probability of sunshine is 80%, and the probability of a tornado is 20%. If the sun shines,
Recap the example
parameters. the project is worth $100,000; if the tornado strikes, the project is worth only $20,000. The
appropriate cost of capital (at which investors are willing to borrow or save) is 10% for the
overall project. We will retain this cost of capital for the project. We had also computed earlier
that the house must then be worth $76,363.64.
The novelty is that we now assume that Treasury bonds pay a lower expected rate of return,
Here we introduce
different costs of capital: which is equivalent to assuming that investors are risk-averse. The debt on the house is not
Risk aversion causes
exactly risk-free, though. We assume that a particular risky bond that promises to pay $28,125
expected interest rates
does require a 6% expected rate of return. (This 6% expected rate of return must be higher
on debt to be lower than
the expected rates of than the true risk-free rate of return [e.g., 5%], and lower than the 10% required expected rate
return on the project.
of return for projects that are of the riskiness of “unlevered house” ownership.) Table 21.2
summarizes our model inputs. Our goal is to determine now what the appropriate cost of
capital for the levered equity is.
¬le=capideal.tex: RP
Section 21·5. The Weighted Average Cost of Capital (WACC) in a Perfect M&M World.

Table 21.2. All Provided Inputs For Valuing The House

Financing Financing
Scheme 1 Scheme 2
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)= 80% $100,000.00
prob(Tornado)= 20% $20,000.00
E Future Payo¬
E Rate of Return (E (˜))
r 10% 6%
Price P0 Today

Step 1: Determine how much the house owners receive if they own the entire house (Scheme AE Compute the
state-contingent payoffs.
for “all equity”) vs. if they promise $28,125 to bond holders and retain only the levered equity
(Scheme DE for “debt and equity”). Naturally, in each state, the bond and the levered equity
together must own the entire house, so:

Financing Financing
Scheme AE Scheme DE
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)= 80% $100,000.00 $100,000.00 $28,125.00 $71,875.00
prob(Tornado)= 20% $20,000.00 $20,000.00 $20,000.00 $0.00
E Future Payo¬
E Rate of Return (E (˜))
r 10% 6%
Price P0 Today

Step 2: Compute the expected value of each security, using the probabilities of sunshine vs. Compute the expected
tornado. Note that the expected payo¬s of the bond and the levered stock together must add
up to the expected payo¬ on the house (i.e., as if the house were 100% equity ¬nanced).

Financing Financing
Scheme AE Scheme DE
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)= 80% $100,000.00 $100,000.00 $28,125.00 $71,875.00
prob(Tornado)= 20% $20,000.00 $20,000.00 $20,000.00 $0.00
E Future Payo¬ $84,000.00 $26,500.00 $57,500.00
E Rate of Return (E (˜))
r 10% 6%
Price P0 Today

Step 3: Discount the expected cash ¬‚ows by the appropriate cost of capital demanded by the Discount the expected
payoffs on the overall
capital providers:
project and on the debt.

Financing Financing
Scheme AE Scheme DE
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)=80% $100,000.00 $100,000.00 $28,125.00 $71,875.00
prob(Tornado)=20% $20,000.00 $20,000.00 $20,000.00 $0.00
E Future Payo¬ $84,000.00 $26,500.00 $57,500.00
E Rate of Return (E (˜))
r 10% 6%
Price P0 Today $76,363.64 $25,000.00
¬le=capideal.tex: LP
536 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

Step 4: Invoke our “perfect market” assumptions. Everyone can buy or sell without transaction
Determine the value of
the levered equity. costs, taxes, or any other impediments. By “absence of arbitrage,” the value of the house if
¬nanced by a bond plus levered equity must be the same as the value of the house if it is
100% equity ¬nanced. Put di¬erently, if you own all of the bond and all of the levered equity
ownership, you own the same thing as the house”and vice-versa. Now use the arbitrage
condition that the value of the levered equity plus the value of the bond should equal the total
house value.

Financing Financing
Scheme AE Scheme DE
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)=80% $100,000.00 $100,000.00 $28,125.00 $71,875.00
prob(Tornado)=20% $20,000.00 $20,000.00 $20,000.00 $0.00
E Future Payo¬ $84,000.00 $26,500.00 $57,500.00
E Rate of Return (E (˜))
r 10% 6%
Price P0 Today $76,363.64 $25,000.00 $51,363.64

Step 5: Levered equity ownership, which sells for $51,363.64 and expects to pay o¬ $57,500.00,
Compute the
appropriate expected o¬ers an expected rate of return of $57, 500.00/$51, 363.64 ’ 1 = +11.95%.
rate of return on the
levered equity.
Scheme AE Scheme DE
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)=80% $100,000.00 $100,000.00 $28,125.00 $71,875.00
prob(Tornado)=20% $20,000.00 $20,000.00 $20,000.00 $0.00
E Future Payo¬ $84,000.00 $26,500.00 $57,500.00
E Rate of Return (E (˜))
r 10% 6% 11.95%
Price P0 Today $76,363.64 $25,000.00 $51,363.64

Now summarize the rates of return in the two possible states on each ¬nancial claim:
Small Detour: Compute
the riskiness of a dollar
investment in each
Contingent Expected
¬nancial instrument.

Tornado Sunshine Appropriate
$20, 000 $100, 000 $84, 000
Unlevered Ownership ’1= ’1= ’1=
$76, 364 $76, 364 $76, 364

’73.81% +30.95% +10.00%
$20, 000 $28, 125 $26, 500
Loan Ownership ’1= ’1= ’1=
$25, 000 $25, 000 $25, 000

’20.00% +12.50% +6.00%
$0 $71, 875 $57, 500
Levered (post-Loan) Ownership ’1= ’1= ’1=
$51, 364 $51, 364 $51, 364

’100.00% +39.93% +11.95%

We started knowing only the cost of capital on our bond (6%) and on our ¬rm (10%), and we
were able to determine the cost of capital on our levered equity (11.95%).
As was also the case in the example with risk-neutral investors (Figure 5.2, Page 100), the rates
Debt is less risky than
unlevered ownership of return to levered equity are more risky than those to unlevered ownership, which in turn are
which is less risky than
more risky than those to the corporate loan. But, whereas these risk di¬erences did not a¬ect
levered ownership.
the expected rates of return in our risk-neutral world, they do in our risk-averse world. The
cost of capital (the expected rate of return at which you, the owner, can obtain ¬nancing) is now
higher for levered equity ownership than it is for unlevered ownership, which in turn is higher
than it is for loan ownership. We had worked out exactly how high this expected rate of return
¬le=capideal.tex: RP
Section 21·5. The Weighted Average Cost of Capital (WACC) in a Perfect M&M World.

on levered equity ownership must be by resorting to the “absence of arbitrage” argument in
the perfect M&M world: Given the expected rate of return on the house and on the bond, we
could determine the expected rate of return on levered equity ownership. (Alternatively, if we
had known the appropriate expected rate of return on levered equity ownership and the rate
of return on the bond, we could have worked out the appropriate expected rate of return on
unlevered ownership.)
In the real world, house owners would naturally like to pay the lowest rate of return possible How the CAPM ¬ts in!
on loans. Similarly, anyone selling a ¬rm would like to receive the highest price (lowest cost of
capital) possible. What the issuer of the loan can get away with in the end will depend on the
demand and supply of capital. Here it was 6% for the debt and 11.95% for the levered equity.
However, this does not tell us why demand and supply met at 6% for the debt and 11.95% for
the levered equity in our example (although one implies the other). So, what model can tell
us the appropriate expected rate of return for a risky security? But, of course: the CAPM from
Chapter 17! To show that everything ¬ts together, we will put the CAPM into the WACC formula
in Sections 21·7 and 21·A, below.

21·5.B. The WACC Formula (Without Taxes)

We can now translate the numerical example into a formula for the “weighted average cost of A Line-by-line derivation
of the WACC formula.
capital,” or WACC. No matter which state will come about, the debt (all liabilities) and equity
(all stock) together own the ¬rm, which we shall call FM:

$28, 125 + $71, 875 = $100, 000

$20, 000 + = $20, 000
Tornado: $0

+ = .
Either: DTt=1 EQt=1 FMt=1

Therefore, the expected value of debt and equity together must be equal to the expected value
of the ¬rm:
$57, 500 + $26, 500 = $84, 000
E (EQt=1 ) + E (DTt=1 ) = E (FMt=1 ) .

You can rewrite this in terms of today™s values and expected rates of return (E(˜)) from time
t = 0 to t = 1:

+ =
$57, 500 $26, 500 $84, 000

= $51, 363.64 · (1 + 11.95%) + $25, 000 · (1 + 6%) = $76, 363.64 · (1 + 10%)
E (EQt=1 ) + E (DTt=1 ) = E (FMt=1 )

= EQt=0 · 1 + E (˜EQ,t=0,1 ) + DTt=0 · 1 + E (˜DT,t=0,1 ) = FMt=0 · 1 + E (˜FM,t=0,1 )
r r r .

Omit the time subscripts on the expected rates of return to reduce clutter. (There is no risk of
confusion because we only consider two time periods anyway.) Divide all three terms by FMt=0
to express this formula in terms of percentages of ¬rm value:
$51, 363.64 $25, 000.00
· (1 + 11.95%) + · (1 + 6%) = 1 + 10%
$76, 363.64 $76, 363.64
EQt=0 DTt=0
· 1 + E (˜EQ ) + · [1 + E (˜DT )] = [1 + E (˜FM )] .
r r r
FMt=0 FMt=0
¬le=capideal.tex: LP
538 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

EQt=0 /FMt=0 is the weight of equity in the ¬rm™s value, so we can call it wEQ,t=0 . Similarly,
DTt=0 /FMt=0 is wDT,t=0 . It is common to just omit the time-subscript if the time is now, so we
can write our formula as

67.26% · (1 + 11.95%) + 32.74% · (1 + 6%) = 1 + 10%
wEQ · 1 + E (˜EQ ) + wDT · [1 + E (˜DT )] = [1 + E (˜FM )] .
r r r

Multiply the weight percentages into the brackets,

67.26% + 67.26% · 11.95% + 32.74% + 32.74% · 6% = 1 + 10%
+ wEQ · E (˜EQ ) + + wDT · E (˜DT ) = 1 + E (˜FM ) .
wEQ r wDT r r

Because debt and equity own the ¬rm, wDT + wEQ = 1, and the “+1” terms on both sides cancel.
We have arrived at the weighted average cost of capital (WACC) Formula:

WACC = 67.26% · 11.95% + 32.74% · 6% = 10%
WACC = wEQ · E (˜EQ ) + wDT · E (˜DT ) = E (˜FM ) .
r r r

Almost always, the term WACC is used to describe the expected weighted average rates of return,
not the actual weighted average rates of return; no one calls it the expected WACC. Chapter 22
will explain how WACC must be modi¬ed in the presence of corporate income taxes.
One important reminder: the cost of debt in the WACC formula is not the interest rate that
The promised debt
interest rate is not used the bank is charging. The bank™s quoted interest rate is the promised rate of return to debt,
in the WACC formula;
and therefore higher than the expected interest rate that is in the WACC formula (higher by
the expected rate is.
the default premium). In addition, non-¬nancial liabilities may carry higher or lower expected
rates of return. So, how do you ¬nd the expected rate of return on debt? Pretty much the same
way as you ¬nd the expected rate of return on equity or anything else: use the CAPM (if you
believe it). Conveniently, the CAPM provides the expected rates of return, which is the sum
of the time-premium and the systematic risk premium, and appropriately ignores the debt™s
idiosyncratic risk and default premium. So you can try to estimate the beta from the debt™s
historical monthly rates of return, and then substitute it into the CAPM formula. Fortunately,
debt betas are often fairly small, especially for short-term and low-risk debt, in which case you
will end up with an E(˜DT ) reasonably close to the risk-free rate.

21·5.C. A Graphical Illustration

You can gain some more intuition by extending our numerical example into a graph. Recall:
Consider different
capital structures now.

Scheme AE Scheme DE
Levered Equity
100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)=80% $100,000.00 $100,000.00 $28,125.00 $71,875.00
prob(Tornado)=20% $20,000.00 $20,000.00 $20,000.00 $0.00
E Future Payo¬ $84,000.00 $26,500.00 $57,500.00
E Rate of Return (E (˜))
r 10% 6% 11.95%
Price P0 Today $76,363.64 $25,000.00 $51,363.64
Capital Structure Weight (Security Price/Firm Value) 32.74% 67.26%
Promised Rate of Return (Bond Promise/Bond Price - 1) 12.5%

How would the promised rate of return, the expected rate of return, and the debt-equity ratio
change if the ¬rm changed the amount it borrowed? For illustration™s sake, allow me to set the
risk-free rate at 5.55%, and to linearly increase the cost of debt capital, so that debt is the entire
¬rm, its cost of capital is 10%. (In our perfect world, this is also the ¬rm™s WACC, no matter
how it ¬nances itself.) This allows me to set the cost of capital for debt that promises to pay
¬le=capideal.tex: RP
Section 21·5. The Weighted Average Cost of Capital (WACC) in a Perfect M&M World.

$28,125 at the 6% that we used in our example. Figure 21.1 shows that we can identify three

A Risk-Free Debt Domain Until the bond promises to payo¬ $20,000, the debt is risk-free. The
debt therefore has a constant cost of capital of exactly 5.55%. But the expected rate of
return to equity is not constant. Equity enjoys the 10% cost of capital equal to that of
the ¬rm, only if there is exactly zero debt. As soon as the ¬rm takes on any debt, the
expected rate of return to the equity has to increase. For example, if the ¬rm takes on
$15,000 in debt, the equity requires an expected rate of return of 11.0% to participate.
(An exercise below asks you to compute the equity™s required rate of return if the ¬rm
takes on $10,000 in debt.)
The Risky Debt Domain in Which Debt and Equity Coown the Firm At a debt value today of
$18,948.37 today, the debt becomes risky. Once the debt becomes risky, the debt™s
promised rate of return must exceed its expected rate of return. (If you look carefully,
you might also note that the debt ratio changes its slope. This is because a little bit of
extra promised payo¬ does not translate into the same amount of cash raised”even debt
investors require both default and risk compensation.) Our example has already worked
out some of these numbers. If the debt promises to pay o¬ $28,125, we know that the
debt ratio is 32.74%, the debt promised rate of return is 12.5%, the expected rate of return
is only 6%, and the expected rate of return on equity is 11.95%. (This again illustrates
the mistake of comparing an equity expected rate of return [e.g., from the CAPM] to the
interest quoted by the bank. The debt has a lower cost of capital, not a higher cost of
The Debt Owns the Firm If the ¬rm promises $100,000 or more to creditors, equity owners
never receive anything. Thus, they are unwilling to provide any capital, which is why the
green line ends at $100,000. The bond now assumes the ¬rm™s cost of capital of 10%, and
any promise to pay more than $100,000 is entirely irrelevant. (This is why the promised
bond rate of return increases diagonally.)

In the real world, it would be rare to rely on exactly this ¬gure. Instead, everyone is accustomed
to place the ¬rm™s debt ratio on the x-axis. The lower graph in Figure 21.1 does this. The vertical
rise in the promised rate of return once the debt owns 100% of the ¬rm is because the ¬rm can
only o¬er to pay an expected rate of return 10%, regardless of what rate of return it promises
to its creditors.
Few ¬rms have binomial payo¬s. A more common situation is one in which the ¬rm has Switch from the
binomial to a more
normally distributed payo¬s”anything between $0 and in¬nity. Figure 21.2 illustrates such
realistic ¬rm example.
a ¬rm. I chose it to have expected payo¬s of $84,000, with a standard deviation of $32,000,
which matches the mean and standard deviation of the binomial example. The calculations
proceed very similarly to those in the binomial case, except there are now many more cases
than just payo¬s of $20,000 and $100,000 to work out. (Better trust me on this one!) Again, the
risk-free rate is 5.55%, the ¬rm™s overall cost of capital is 10%, and the debt™s required interest
rate rises smoothly with the weight of debt in the ¬rm™s capital structure. (However, it is not
linear anymore, but relates to the covariance of the debt payo¬ with respect to the ¬rm payo¬.)
What remains the same and what changes?
The higher the payo¬s that the bond promises, the higher is the debt ratio, the higher are the The WACC remains
constant. The expected
expected rates of return on both the bond and the stock, and the higher is the promised rate
rates of return on debt
of return on the bond. Most importantly, the ¬rm™s WACC remains at the ¬rm™s cost of capital and equity both increase
of 10%! There are some di¬erences, but they are minor. There is now no region in which the as the ¬rm takes on
debt”and steeply if the
bond is risk-free, and equity owns part of the ¬rm even if the bond promises to pay more than
debt ratio is high.
$100,000. You might also notice how steeply the expected rate of return on the stock (and the
bond™s promised [but not expected] rate of return) rises when the debt™s weight in the capital
structure increases. At a debt ratio of 90%, the equity must o¬er an expected rate of return of
close to 25%. Moreover, even though the expected rate of return on the bond remains below
the ¬rm™s WACC of 10%, the ¬rm must promise its creditors an interest rate as high as 30% as
compensation. Of course, this is not the ¬rm™s cost of debt! It is the expected rate of return to
creditors paid by the ¬rm that is the ¬rm™s cost of debt capital.
¬le=capideal.tex: LP
540 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

Figure 21.1. Expected and Promised Rates of Return in the House Example

Rates of Return (and Debt Ratio), in %
debt is debt
riskfree risky shares owns
with firm
equity outright

worked out

in the text

Bo urn
ed et
De is f R
om o
Pr ate

Expected Stock Rate of Re
of 10% rf urn
Expected Bond Rate of Ret

0 20 40 60 80 100 120

Bond Promised Payoff, in Thousand Dollars

debt is
riskfree risky
Rates of Return, in %

worked out
in the text

Promised Bo

Rate of Ret

Expected Stock Rate of Return
of 10% rf
Expected Bond Rate of Return

0 20 40 60 80 100

Debt Ratio, in %

This ¬gure illustrates the binomial house valuation example in a perfect world. The ¬rm™s WACC is always 10%. The
risk-free rate is 5.55%. (It increases linearly with the fraction of debt in the ¬rm™s capital structure.) Below $18,957,
the debt is risk-free and pays 5.55%. The equity already requires a higher expected rate of return, however, starting
at 10%. The text worked out the case in which the ¬rm has promised to pay $28,125, and requires an expected rate
of return of 6%. Therefore, it has to promise a rate of return of 12.5%, and owns 32.74% of the ¬rm. (Equity would
have to receive a rate of return of 11.95%.) The lower graph changes the x-axis to the debt ratio, which is also plotted
in the upper graph.
¬le=capideal.tex: RP
Section 21·5. The Weighted Average Cost of Capital (WACC) in a Perfect M&M World.

Figure 21.2. Expected and Promised Rates of Return in a Normally Distributed Example
Rates of Return (and Debt Ratio), in %


tio nd
tR Bo rn

De d tu
ise Re
om of
Pr ate


. 27
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