ñòð. 27 |

Â

Â

Â

Â

Â

Â

Â

Â

Â

Â ion

t

Â Op

all

Â C

he

Â T

ant

rr

Â Wa

e

Th

Â E

Firm Value,

$50,000

Time 1

ï¬le=corpclaims.tex: RP

521

Section 20Â·3. More About Stock.

A number of companies have sold such put options on their own stock. For example, in 1995, Some speciï¬c corporate

examples [via Ted

Microsoft generated cash by selling puts that speculated that the share price of Microsoft would

Azarmi].

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 ï¬ve years. Moreover, because Microsoftâ€™s

share price did not end up dropping, it did not have to pay anything to put purchasers. How-

ever, put options can also backï¬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. Maytagâ€™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 oneâ€™s own stock.)

The Time Dimension

Again, the payoï¬€ 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!)â€”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-ï¬fth of the ï¬rmâ€™s equity.

Write down the payoï¬€ tables (and draw the payoï¬€ diagrams). Reï¬‚ect on how they diï¬€er.

Q 20.9 Write down the equity payoï¬€ diagram if the ï¬rm has the following capital structure:

â€¢ 1,000 senior bonds with promised payoï¬€s of $100 million, convertible into 50 million new

equity shares.

â€¢ 500 junior bonds with promised payoï¬€s of $50 million, convertible (at the bond holderâ€™s

discretion) into 15 million new shares.

â€¢ 100 million equity shares for the rest of the ï¬rm.

It is easiest to work with aggregate ï¬gures, i.e., consider the ï¬rm value in increments of $50

million. Hint: At what equity value and at what ï¬rm value (the two are not the same) would the

senior convert?

ï¬le=corpclaims.tex: LP

522 Chapter 20. Corporate Financial Claims.

20Â·4. Summary

In the real world, ï¬rms are ï¬nanced by a whole set of diï¬€erent ï¬nancial claims. The same ï¬rm

Large ï¬rms usually have

multiple claims may have senior debt, junior debt (perhaps with a conversion feature), equity, and a warrant.

outstanding.

The right way to think about all these claims still often involves the â€œmagicâ€ payoï¬€ table (and

payoï¬€ diagram): if the ï¬rm ends up worth very little, only the senior debt is paid. If the ï¬rm

is worth a little more, both the senior and the junior debt are paid. If the ï¬rm is worth even

more, the equity will become worth something; and ï¬nally, the warrant and/or the conversion

feature will be valuable.

The chapter covered the following major points:

â€¢ The two most basic building blocks of capital structure are debt and equity. These diï¬€er

in their cash ï¬‚ow rights and in their control rights:

â€“ Debt has ï¬rst rights to the distribution of cash ï¬‚ows. It is â€œsenior.â€ It can force the

ï¬rm into bankruptcy if payments are not made.

â€“ Equity gets only what is left over after debt has been satisï¬ed. It is â€œjunior.â€ It is in

control of the ï¬rm, unless the ï¬rm ï¬nds itself in ï¬nancial distress.

â€¢ Payoï¬€ tables and payoï¬€ diagrams are often a good way to describe debt and equity, be-

cause they are primarily state-contingent claims, where the ï¬rm value is the state. The

paradigm is not always perfect, because the value of claims can also depend on factors

other than ï¬rm value.

â€¢ 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.

â€¢ 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.

â€¢ Convertible bonds allow their owners to convert their bonds into shares. They can there-

fore be considered as part-debt, part-equity.

â€¢ Corporate borrowing comes in thousands of diï¬€erent varieties. It can be plain, convertible,

callable, or convertible and callable. It can be ï¬xed-rate or ï¬‚oating-rate. It can be short-

term or long-term. It can have detailed covenants of many kinds. And so on.

â€¢ The lines between diï¬€erent ï¬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.

ï¬le=corpclaims.tex: RP

523

Section 20Â·4. Summary.

Solutions and Exercises

1. A control right is the right to inï¬‚uence decisions, speciï¬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 stuï¬€.

3.

Medical Costs Insurance Payout

$0 $0

$250 $0

$500 $0

$750 $225

$1,000 $450

$2,000 $1,350

â€¦ â€¦

$11,500 $9,900

$11,600 $9,990

$11,611 $10,000

$11,700 $10,000

$12,000 $10,000

$13,000 $10,000

â€¦ â€¦

The â€œslopeâ€ is zero until $500 is reached, then 90% until $11,611.11 is reached, then zero again.

4. See text.

5.

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

â€¦ â€¦ â€¦ â€¦

6. If the ï¬rm is worth less than $2 million, the bondholders own the entire ï¬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

indiï¬€erent between converting and not converting. The payoï¬€ 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 ï¬rm into bankruptcy if their dividends are not paid.

8. All numbers are quoted in thousands.

ï¬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 ï¬rm value, Firm Aâ€™s equity shares are worth $800, which translates into $8/share. At

$8 share, warrant holders are indiï¬€erent between exercising and not exercising. b At $1,100 ï¬rm value, Firm

Aâ€™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 ï¬rmâ€™s equity value. c Convertible bond holders are

indiï¬€erent between converting their bond into equity if the ï¬rm value is $1,000. After all, they would receive

20% of the ï¬rmâ€™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 diï¬ƒcult, but it does become easy once you realize the following:

â€¢ If the junior does not convert, then the seniorâ€™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 ï¬rm value reaches $350 million, because the junior creditors

still would have their â€œ$50 million ï¬rstâ€ claim.

â€¢ If the senior has converted, then the juniorâ€™s 15 million in new equity shares would represent 15/165

of the ï¬rm. This is about 9.1% of the ï¬rm value. Therefore, at a ï¬rm value of $550 million (solve

xÂ·15/165 = $50), the junior would be indiï¬€erent between exercising and not exercising.

These two insights make it easy to write down the payoï¬€ table (note my odd stepping when convenient for

explanation), all in million dollars:

Junior

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 indiï¬€erent to converting)

$350 $100 $50 $200

(at $353, senior has converted â€” 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 indiï¬€erent to converting)

$550 $167 $50 $333

(at $561, junior has converted â€” 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.)

CHAPTER 21

Capital Structure and Capital Budgeting in a

Perfect Market

Should A Company Issue Stocks or Bonds?

last ï¬le change: Feb 23, 2006 (15:02h)

last major edit: Feb 2005

This chapter explains how to look at the basics of ï¬nancing: in a â€œperfectâ€ world (no transaction

costs, no taxes, etc.) the value of the ï¬rm is determined by the value of the projects (the NPV),

not by whether the ï¬rm is ï¬nanced with equity or leverage. This chapter also explains the

basics of the weighted average cost of capital formula (WACC). Subsequent chapters will explain

ï¬nancing in the real world, not in our perfect world.

525

ï¬le=capideal.tex: LP

526 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

21Â·1. Conceptual Basics

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

The best way to think about an optimal ï¬rm structure is as follows: the current owners of a

The ï¬rmâ€™s charter

deï¬nes the ï¬rm ï¬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 ï¬rm todayâ€”that is, the price that new

agreement of who

investors are willing to pay to acquire the ï¬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 diï¬€erent owners and

stakeholders (such as customers, workers, and suppliers). This agreement on how to split up

future earnings is the ï¬rmâ€™s ï¬nancial structure or capital structure: it is the rules that specify

who receives the proceeds of (usually uncertain) future cash ï¬‚ows. The capital structure is

rarely fully explicit or set in stone. Instead, it also encompasses the rules on how the ï¬rm may

be governed in the future, which in turn may inï¬‚uence the capital structure in the future, which

in turn may inï¬‚uence the ï¬nancing arrangement today.

21Â·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

value?

or as representatives of the ï¬rm. The common view is that it is the task of management to

maximize shareholder wealth. But equity is only one part of the ï¬rmâ€™s capital structure. There

are also liabilities that have claims on the ï¬rmâ€™s assets. There are pension obligations, accounts

payable, and ï¬nancial debtâ€”and many ï¬nancial instruments with both debt and equity features.

(For illustration, we will use ï¬nancial debt as a standin for all liabilities.) Does it even make a

diï¬€erence as to whom management is representing?

The legal situation in the United States is that management has a legal ï¬duciary duty to

When is there a

potential problem? shareholdersâ€”except if decisions made by the ï¬rm can threaten its insolvency. In this case,

managementâ€™s legal ï¬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 beneï¬t from a managerâ€™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 ï¬rmâ€™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 ï¬nancial debt

by $3. (You will later learn how easy it is to do exactly this.) This â€œexpropriativeâ€ transaction

would destroy $2 in the net value of the ï¬rm. But even in our perfect world, this is the type of

situation that creates a dilemma for management: should management maximize ï¬rm value

or shareholder value? Recall that it is shareholders who vote managers into oï¬ƒce and allow

them to stay there. Whether this transaction destroys ï¬rm value or not, when the time comes,

managers will ï¬nd it in their interest to execute this transaction because doing so raises equity

value and with it managementâ€™s chances of being reappointed.

However, there is a ï¬‚y in the ointment. Put yourself into the shoes of the old ï¬rm owners

Bond buyers understand

future conï¬‚icts of today, who are trying to set up the corporate charter that maximizes the value of their ï¬rm,

interest. If managers

i.e., for sale to new investors at the highest price. You want to ï¬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 ï¬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 aï¬€airs is also appointed (qualiï¬ed co-determination).

ï¬le=capideal.tex: RP

527

Section 21Â·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 ï¬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 ï¬rm that will not destroy $2 of value in the future. There-

the needs of

fore, management today has a choice: it can either ï¬nd a way to commit today not to exploit bondholdersâ€”even

bondholders in the future; or it can sell the ï¬rm today for a lower net present value, which though they are voted

into power only by

takes into account value destruction tomorrowâ€”because everyone realizes that managers will

equity holders and are

be trapped tomorrow into destroying $2 of ï¬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 ï¬rm in the future, which in turn

maximizes the value of the ï¬rm today.

Important:

â€¢ In deciding on an appropriate price to pay, the buyers of ï¬nancial securities

take into account what the ï¬rm is likely to do in the future.

â€¢ The theoretical notion of the optimal capital structure is the structure that

maximizes the value of the ï¬rm, not the value of the equity.

In this theoretical world, management should maximize ï¬rm owner value, not shareholder Conï¬‚icts of interest

arise between

value. Practically, the two objectives diï¬€er only rarely (and usually only when ï¬rms are close to

shareholders and

ï¬nancial distress), so the popular mantra of â€œshareholder value maximizationâ€ is fortunately bondholders, but they

usually synonymous with â€œtotal value maximization.â€ But strictly speaking, the best capital are dwarfed by the

conï¬‚ict between

structure does not have as its goal maximizing shareholder value but ï¬rm value. In any case, in

managers and owners.

the real world, managers are far more conï¬‚icted with respect to their own welfare (the agency

conï¬‚icts we ï¬rst discussed in Section 7Â·6) than they are with respect to favoring shareholders

at the expense of bondholders.

The advantage of a ï¬rm that is committed to maximizing ï¬rm value in the future is that it Committing to

maximizing ï¬rm value

can obtain a better price for its ï¬nancial securities (a lower interest rate for its bonds) today.

gives a better interest

Therefore, the ï¬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 ï¬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 (â€œbuyer bewareâ€) 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 managementâ€™s (ownersâ€™) own

structure to maximize

interest today to commit not to exploit future owners and bondholders tomorrowâ€”especially ï¬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 ï¬rms to

ability to commit to restrain itself would value the ï¬rm more highly than the current manage- improve Capital

ment team. It could purchase the ï¬rm and make an immediate proï¬t. Therefore, competition Structures.

among management teams can push ï¬rms towards the best capital structure. Again, the gen-

eral thread emphasized throughout this chapter is that ï¬rms that can commit to do â€œthe right

thingâ€ tomorrow (ex-post ) are worth more today (ex-ante). It is a direct consequence that ï¬rms

that maximize ï¬rm value are worth more than ï¬rms that maximize just shareholder value.

ï¬le=capideal.tex: LP

528 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

Solve Now!

Q 21.1 Explain the diï¬€erence between ex-ante and ex-post, especially in the capital structure

context. Give an example where the two diï¬€er.

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

21Â·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 ï¬rmsâ€™ capital structure decisions. Although the M&M theory involves

some complex algebraic calculations, it is actually based on some surprisingly simple ideasâ€”

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 ï¬rst

Nobel Prize; Merton Miller would receive his own Nobel Prize a few years later.)

Anecdote: M&M: Milk, Cream, and Pizza

How diï¬ƒcult it is to summarize brieï¬‚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 partâ€”but, of course, only in partâ€”

for the work in ï¬nance. The television camera crews from our local stations in Chicago immediately descended

upon me. â€œWe understand,â€ they said, â€œthat you worked with Modigliani some years back in developing these

M&M theorems and we wonder if you could explain them brieï¬‚y to our television viewers.â€ â€œHow brieï¬‚y?â€, I

asked. â€œOh, take 10 seconds,â€ 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: â€œYou donâ€™t have to go into details. Just give us the main points in simple,

common sense terms.â€

The main point of the ï¬rst or cost-of-capital article was, in principle at least, simple enough to make. It said that

in an economistâ€™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 ï¬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 ï¬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 oï¬€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. â€œThink of the ï¬rm,â€ I said, â€œas 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 ï¬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

milk.â€

(continues.)

ï¬le=capideal.tex: RP

529

Section 21Â·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. â€œHave you anything simpler?â€, 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 â€œpartitioningâ€ a ï¬rmâ€™s payoï¬€s in each

possible state of the world among the group of its capital suppliers. â€œThink of the ï¬rm,â€ I said, â€œas

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.â€

Again there was a whispered conference among the camera crew and the director came back and said:

â€œProfessor, we understand from the press release that there were two M&M propositions. Maybe we

should try the other one.â€

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 â€œdividendâ€ 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 somewhereâ€”either

from cutting back on real investment or from further sales (or reduced purchases) of ï¬nancial in-

struments. The M&M dividend proposition oï¬€ered no advice as to which source or how much to

tap. It claimed, rather, that once the ï¬rm had made its real operating and investment decisions, its

dividend policy would have no eï¬€ect on shareholder value. Any seeming gain in wealth from raising

the dividend and giving the shareholders more cash would be oï¬€set by the subtraction of that part

of their interest in the ï¬rm sold oï¬€ to provide the necessary funds. To convey that notion within

my allotted 10 seconds I said: â€œThe 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 oï¬€.â€

Once again whispered conversation. This time, they shut the lights oï¬€. 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 donâ€™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 ï¬rst. One writerâ€”David Durand, the same critic who had so strongly attacked the Markowitz

modelâ€”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 didnâ€™t hold exactly; but, of course, empirical

relations never do.

ï¬le=capideal.tex: LP

530 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

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

Markets

To begin with, Modigliani and Miller argued that under perfect conditions, the total value of

Consider a perfect

market world. all ï¬nancial securities is the same, regardless of whether the ï¬rm is ï¬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 ï¬rm depended on how it is ï¬nanced. Because there should

be no arbitrage in real life, it follows that ï¬rms should be able to choose any mix of securities

without impact on the ï¬rmsâ€™ values. This perfect world that M&M describe relies on the familiar

perfect world assumptions (e.g., in Section 6Â·1).

â€¢ 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 ï¬nancial distress before bankruptcy occurs, either.)

â€¢ Capital markets are perfectly competitive, with a large number of investors competing for

many securities.

â€¢ There are no taxes.

â€¢ There are no diï¬€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.

Letâ€™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 ï¬xed the ï¬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 ï¬rm wishes to consider how to ï¬nance its projects. Because we all agree on all current

The Modigliani-Miller

proposition states that and future projectsâ€™ expected cash ï¬‚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 â€œPV.â€ (This is [almost by deï¬nition] the present value that the ï¬rmâ€™s projects can fetch

world.

in our perfect capital market, of course.) The M&M claim is that the present value of the ï¬rmsâ€™

projects must equal the present value of the ï¬rmsâ€™ issued claims today. In other words, if the

ï¬rm has no liabilities and issues 100% equity, the equity must sell for the PV of the projects.

If the ï¬rm instead were ï¬nanced by 50% debt (or by other liabilities) and 50% equity, the two

together must sell for the same PV. If the ï¬rm issues x% debt and (1-x%) equity, the two together

must sell for PV. According to theory, the value of the ï¬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 ï¬ndâ€”and ac-

Arbitrage!

tually did chooseâ€”a capital structure that makes the ï¬rm worth $1 less than PV. For

example, assume that the ï¬rm is worth PV=$100 under the optimal capital structure of

80% equity and 20% debt; and assume further that the ï¬rm is worth only $99 under the

capital structure of 50% equity and 50% debt that the ï¬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

ï¬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

proï¬t of $1.

ï¬le=capideal.tex: RP

531

Section 21Â·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

tooâ€”until the M&M

other investors. The only price at which no one will overbid you for the right to purchase proposition works (ï¬rm

the ï¬rmâ€™s current securities is $100. But notice that this means that the value of the old value is as if it was

ï¬nanced optimally).

securities is equal to the price that the ï¬rm is worth under the optimal capital structure.

So, regardless of the ï¬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 ï¬rm, but only 1% of the ï¬rm. That is, if you buy 1% of all the ï¬rmâ€™s

securities, you will receive 1% of the projectsâ€™ payoï¬€s. You can then sell your securities

repackaged to imitate the payoï¬€s under the presumably better capital structure for 1% of

the ï¬rmâ€™s higher value, and receive an arbitrage proï¬t of 1%Â·$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 ï¬‚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 ï¬re management and change what the

ï¬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 5Â·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 ï¬nancing

a house. We found that neither the house value nor the cost of capital were inï¬‚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 ï¬rm. To accomplish this as simply as possible,

we assume the world is risk-neutral and all securities have to oï¬€er an expected rate of return

of 10%. (We will work an example in a risk-averse world in Section 21Â·5.A.)

Linking M&M under

uncertainty to

state-contingent payoffs

The Full Scenario Analysis Table 21.1 shows the value of a ï¬rm if the projects will be worth (under risk-neutral

pricing).

either $60 or $160. The expected future value is $110, the present value is $100. Under

hypothetical capital structure LD (â€œlittle debtâ€), the ï¬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 ï¬rmâ€™s claims adds up to the same $100. Under

hypothetical capital structure MD (â€œmuch debtâ€), the ï¬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 ï¬rm; it

The Partial Scenario Analysis The more surprising fact is that investors can relever securities

also works with a

themselvesâ€”they do not need the ï¬rm to do it for them. In our example, assume that fraction of the ï¬rm.

the ï¬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 bondâ€™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 ï¬rm?

What happens if you purchase d bonds and e stocks in our LD ï¬rm? You will receive

payoï¬€s of d Â· $55 + e Â· $5 in the bad scenario, and d Â· $55 + e Â· $105 in the good scenario.

You want to end up with $0.60 in the bad scenario, and $0.94 in the good scenarioâ€”two

equations, two unknowns:

d Â· $55 + e Â· $5 = $0.60 d â‰ˆ 0.0034

Bad Luck

â‡’ (21.1)

d Â· $55 + e Â· $105 = $0.94 e â‰ˆ 0.0106

Good Luck

ï¬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- Todayâ€™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 stateâ€”exactly the same that an MD ï¬rm would have

given you!. How much would you have to pay to get these payoï¬€s? The cost today would

be d Â· $50 + e Â· $50 = 0.0034 Â· $50 + 0.0106 Â· $50 = $0.70, exactly the same that your

desired payoï¬€s would have cost you if the ï¬rm itself had chosen an MD capital structure.

In eï¬€ect, you have manufactured the capital structure payoï¬€s that you like without the

cooperation of the ï¬rm itself. By repeating this exercise, you can replicate the payoï¬€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 ï¬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 proï¬t.

Important: In Modigliani and Millerâ€™s perfect world:

â€¢ Arbitrage restrictions force the value of the ï¬rm to be the same, regardless

of the ï¬rmâ€™s mix of debt (liabilities) and equity.

â€¢ Because ï¬nancing and investing are two entirely separate issues, managers

can ignore ï¬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 eï¬€orts.

Know what to care

about and what not to You should try to increase the value of the underlying projectsâ€”by increasing their expected

care about!

cash ï¬‚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 ï¬ddling around with how you

ï¬nance your projects.

ï¬le=capideal.tex: RP

533

Section 21Â·4. Dividends.

The above proof of the Modigliani-Miller proposition followed the M&M 1957 paper in assuming In a perfect world, ï¬rms

always undertake the

that the ï¬rmâ€™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 managementâ€™s project policy should not matter. If the ï¬rm were

not to undertake all positive NPV projects and reject all negative NPV projects, you could buy

all the shares, ï¬re the old management, institute the best underlying ï¬rm investment project

policy, resell all the shares, and earn the diï¬€erence in ï¬rm value as an immediate arbitrage.

Alas, you would again not be the only one: everyone could do this. Therefore, in this perfect

world, ï¬rms always do the right thing. They maximize the ï¬rmâ€™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 â€œbuyâ€ 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 ï¬rm, if all that was traded were the securities of the LD ï¬rm?

21Â·4. Dividends

The second Modigliani-Miller proposition is even simpler than the ï¬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

value.

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 ï¬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 ï¬rm, ï¬re the management,

institute the better dividend policy, and resell the ï¬rm for the diï¬€erence. With many investors

vying to do this if even just a penny can be earned, the only ï¬rm value and dividend policy

that do not allow anyone to arbitrage (get rich without eï¬€ort) is the value of the ï¬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

world.

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 ï¬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 ï¬rms is around 1% of ï¬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 ï¬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 ï¬nance professors believe

that paying dividends sends a credible signal from management about future ï¬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.

ï¬le=capideal.tex: LP

534 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

21Â·5. The Weighted Average Cost of Capital (WACC) in a Per-

fect M&M World

You now understand why the value of the ï¬rm does not depend on the ï¬nancing in a perfect

Let us do contingent

claims under market. This is equivalent to stating that the overall cost of capital to the ï¬rm does not depend

risk-aversion, i.e., usually

on the debt/equity ratio of the ï¬rm. To show this, we want to repeat the house example from

higher expected rates of

Section 5Â·3.B. It is important that you realize that our argument also works in the context

return to equity than to

debt.

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 â€œweighted average cost of capitalâ€ (or WACC) in the corporate context. (In the next chapter,

we will generalize WACC to a world in which corporations pay income tax.)

21Â·5.A. The Numerical Example In a Risk-Averse World Where Riskier Equity Must Oï¬€er

Higher A Expected Rate of Return

Our example will again be our house with the mortgage from Section 5Â·3.A (Page 93), so make

All tools learned in

Section 5Â·3.B still apply sure you remember the concepts from Chapter 5. Brieï¬‚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 oï¬€er higher

expected rates of return. The basic tools will be exactly the same as those in Section 5Â·3.A:

payoï¬€ 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 ï¬rmâ€™s debt and equity. Not surprisingly, the expected rate of return on the ï¬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 ï¬rm, and

be independent of how

of the shares. We want to show that if the perfect markets arbitrage condition holdsâ€”that is,

the ï¬rm is ï¬nanced.

bonds and stocks together cost the same as the ï¬rmâ€”then you can compute the cost of capital

for the ï¬rm as the weighted cost of capital of stocks and bonds. Put diï¬€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 ï¬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 ï¬rmâ€”the WACC. And regardless of the ï¬rmâ€™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 5Â·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 ï¬rm might face. There is no

conceptual diï¬€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 â€œunlevered houseâ€ 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.

ï¬le=capideal.tex: RP

535

Section 21Â·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

Bond

100% Equity promises $28,125 after $28,125 obligation

prob(Sunshine)= 80% $100,000.00

prob(Tornado)= 20% $20,000.00

E Future Payoï¬€

E Rate of Return (E (Ëœ))

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 â€œall equityâ€) vs. if they promise $28,125 to bond holders and retain only the levered equity

(Scheme DE for â€œdebt and equityâ€). 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

Bond

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 Payoï¬€

E Rate of Return (E (Ëœ))

r 10% 6%

Price P0 Today

Step 2: Compute the expected value of each security, using the probabilities of sunshine vs. Compute the expected

payoffs.

tornado. Note that the expected payoï¬€s of the bond and the levered stock together must add

up to the expected payoï¬€ on the house (i.e., as if the house were 100% equity ï¬nanced).

Financing Financing

Scheme AE Scheme DE

Levered Equity

Bond

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 Payoï¬€ $84,000.00 $26,500.00 $57,500.00

E Rate of Return (E (Ëœ))

r 10% 6%

Price P0 Today

Step 3: Discount the expected cash ï¬‚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

Bond

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 Payoï¬€ $84,000.00 $26,500.00 $57,500.00

E Rate of Return (E (Ëœ))

r 10% 6%

Price P0 Today $76,363.64 $25,000.00

ï¬le=capideal.tex: LP

536 Chapter 21. Capital Structure and Capital Budgeting in a Perfect Market.

Step 4: Invoke our â€œperfect marketâ€ assumptions. Everyone can buy or sell without transaction

Determine the value of

the levered equity. costs, taxes, or any other impediments. By â€œabsence of arbitrage,â€ the value of the house if

ï¬nanced by a bond plus levered equity must be the same as the value of the house if it is

100% equity ï¬nanced. Put diï¬€erently, if you own all of the bond and all of the levered equity

ownership, you own the same thing as the houseâ€”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

Bond

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 Payoï¬€ $84,000.00 $26,500.00 $57,500.00

E Rate of Return (E (Ëœ))

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 oï¬€ $57,500.00,

Compute the

appropriate expected oï¬€ers an expected rate of return of $57, 500.00/$51, 363.64 âˆ’ 1 = +11.95%.

rate of return on the

levered equity.

Scheme AE Scheme DE

Levered Equity

Bond

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 Payoï¬€ $84,000.00 $26,500.00 $57,500.00

E Rate of Return (E (Ëœ))

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 ï¬nancial claim:

Small Detour: Compute

the riskiness of a dollar

investment in each

Contingent Expected

ï¬nancial instrument.

Tornado Sunshine Appropriate

$20, 000 $100, 000 $84, 000

Unlevered Ownership âˆ’1= âˆ’1= âˆ’1=

$76, 364 $76, 364 $76, 364

âˆ’73.81% +30.95% +10.00%

$20, 000 $28, 125 $26, 500

Loan Ownership âˆ’1= âˆ’1= âˆ’1=

$25, 000 $25, 000 $25, 000

âˆ’20.00% +12.50% +6.00%

$0 $71, 875 $57, 500

Levered (post-Loan) Ownership âˆ’1= âˆ’1= âˆ’1=

$51, 364 $51, 364 $51, 364

âˆ’100.00% +39.93% +11.95%

We started knowing only the cost of capital on our bond (6%) and on our ï¬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 diï¬€erences did not aï¬€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 ï¬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

ï¬le=capideal.tex: RP

537

Section 21Â·5. The Weighted Average Cost of Capital (WACC) in a Perfect M&M World.

on levered equity ownership must be by resorting to the â€œabsence of arbitrageâ€ 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 ï¬ts in!

on loans. Similarly, anyone selling a ï¬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 ï¬ts together, we will put the CAPM into the WACC formula

in Sections 21Â·7 and 21Â·A, below.

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

We can now translate the numerical example into a formula for the â€œweighted average cost of A Line-by-line derivation

of the WACC formula.

capital,â€ or WACC. No matter which state will come about, the debt (all liabilities) and equity

(all stock) together own the ï¬rm, which we shall call FM:

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

Sunshine:

(21.2)

$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 ï¬rm:

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

(21.3)

E (EQt=1 ) + E (DTt=1 ) = E (FMt=1 ) .

You can rewrite this in terms of todayâ€™s values and expected rates of return (E(Ëœ)) from time

r

t = 0 to t = 1:

+ =

$57, 500 $26, 500 $84, 000

= $51, 363.64 Â· (1 + 11.95%) + $25, 000 Â· (1 + 6%) = $76, 363.64 Â· (1 + 10%)

(21.4)

E (EQt=1 ) + E (DTt=1 ) = E (FMt=1 )

= EQt=0 Â· 1 + E (ËœEQ,t=0,1 ) + DTt=0 Â· 1 + E (ËœDT,t=0,1 ) = FMt=0 Â· 1 + E (ËœFM,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 ï¬rm value:

$51, 363.64 $25, 000.00

Â· (1 + 11.95%) + Â· (1 + 6%) = 1 + 10%

$76, 363.64 $76, 363.64

(21.5)

EQt=0 DTt=0

Â· 1 + E (ËœEQ ) + Â· [1 + E (ËœDT )] = [1 + E (ËœFM )] .

r r r

FMt=0 FMt=0

ï¬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 ï¬rmâ€™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% Â· (1 + 11.95%) + 32.74% Â· (1 + 6%) = 1 + 10%

(21.6)

wEQ Â· 1 + E (ËœEQ ) + wDT Â· [1 + E (ËœDT )] = [1 + E (ËœFM )] .

r r r

Multiply the weight percentages into the brackets,

67.26% + 67.26% Â· 11.95% + 32.74% + 32.74% Â· 6% = 1 + 10%

(21.7)

+ wEQ Â· E (ËœEQ ) + + wDT Â· E (ËœDT ) = 1 + E (ËœFM ) .

wEQ r wDT r r

Because debt and equity own the ï¬rm, wDT + wEQ = 1, and the â€œ+1â€ terms on both sides cancel.

We have arrived at the weighted average cost of capital (WACC) Formula:

WACC = 67.26% Â· 11.95% + 32.74% Â· 6% = 10%

(21.8)

WACC = wEQ Â· E (ËœEQ ) + wDT Â· E (ËœDT ) = E (ËœFM ) .

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 modiï¬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 bankâ€™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-ï¬nancial liabilities may carry higher or lower expected

rates of return. So, how do you ï¬nd the expected rate of return on debt? Pretty much the same

way as you ï¬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 debtâ€™s

idiosyncratic risk and default premium. So you can try to estimate the beta from the debtâ€™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(ËœDT ) reasonably close to the risk-free rate.

r

21Â·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

Bond

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 Payoï¬€ $84,000.00 $26,500.00 $57,500.00

E Rate of Return (E (Ëœ))

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 ï¬rm changed the amount it borrowed? For illustrationâ€™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

ï¬rm, its cost of capital is 10%. (In our perfect world, this is also the ï¬rmâ€™s WACC, no matter

how it ï¬nances itself.) This allows me to set the cost of capital for debt that promises to pay

ï¬le=capideal.tex: RP

539

Section 21Â·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

regions:

A Risk-Free Debt Domain Until the bond promises to payoï¬€ $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 ï¬rm, only if there is exactly zero debt. As soon as the ï¬rm takes on any debt, the

expected rate of return to the equity has to increase. For example, if the ï¬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 equityâ€™s required rate of return if the ï¬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 debtâ€™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 payoï¬€ does not translate into the same amount of cash raisedâ€”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 oï¬€ $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

capital!)

The Debt Owns the Firm If the ï¬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 ï¬rmâ€™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 ï¬gure. Instead, everyone is accustomed

to place the ï¬rmâ€™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 ï¬rm is because the ï¬rm can

only oï¬€er to pay an expected rate of return 10%, regardless of what rate of return it promises

to its creditors.

Few ï¬rms have binomial payoï¬€s. A more common situation is one in which the ï¬rm has Switch from the

binomial to a more

normally distributed payoï¬€sâ€”anything between $0 and inï¬nity. Figure 21.2 illustrates such

realistic ï¬rm example.

a ï¬rm. I chose it to have expected payoï¬€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 payoï¬€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 ï¬rmâ€™s overall cost of capital is 10%, and the debtâ€™s required interest

rate rises smoothly with the weight of debt in the ï¬rmâ€™s capital structure. (However, it is not

linear anymore, but relates to the covariance of the debt payoï¬€ with respect to the ï¬rm payoï¬€.)

What remains the same and what changes?

The higher the payoï¬€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 ï¬rmâ€™s WACC remains at the ï¬rmâ€™s cost of capital and equity both increase

of 10%! There are some diï¬€erences, but they are minor. There is now no region in which the as the ï¬rm takes on

debtâ€”and steeply if the

bond is risk-free, and equity owns part of the ï¬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

bondâ€™s promised [but not expected] rate of return) rises when the debtâ€™s weight in the capital

structure increases. At a debt ratio of 90%, the equity must oï¬€er an expected rate of return of

close to 25%. Moreover, even though the expected rate of return on the bond remains below

the ï¬rmâ€™s WACC of 10%, the ï¬rm must promise its creditors an interest rate as high as 30% as

compensation. Of course, this is not the ï¬rmâ€™s cost of debt! It is the expected rate of return to

creditors paid by the ï¬rm that is the ï¬rmâ€™s cost of debt capital.

ï¬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

100

Rates of Return (and Debt Ratio), in %

debt is debt

riskfree risky shares owns

with firm

equity outright

80

worked out

tio

in the text

nd

60

a

Bo urn

tR

b

ed et

De is f R

om o

Pr ate

R

40

20

turn

Expected Stock Rate of Re

WACC

of 10% rf urn

Expected Bond Rate of Ret

0

0 20 40 60 80 100 120

Bond Promised Payoff, in Thousand Dollars

100

debt is

riskfree risky

80

Rates of Return, in %

worked out

in the text

60

nd

Promised Bo

40

urn

Rate of Ret

20

Expected Stock Rate of Return

WACC

of 10% rf

Expected Bond Rate of Return

0

0 20 40 60 80 100

Debt Ratio, in %

This ï¬gure illustrates the binomial house valuation example in a perfect world. The ï¬rmâ€™s WACC is always 10%. The

risk-free rate is 5.55%. (It increases linearly with the fraction of debt in the ï¬rmâ€™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 ï¬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 ï¬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.

ï¬le=capideal.tex: RP

541

Section 21Â·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

100

Rates of Return (and Debt Ratio), in %

80

tio nd

a

tR Bo rn

60

b

De d tu

ise Re

om of

Pr ate

R

40

ñòð. 27 |