tock

Expected S

20

turn

Rate of Re

WACC

of 10% rf

Expected Bond Rate of Return

0

0 20 40 60 80 100 120

Bond Promised Payoff, in Thousand Dollars

100

80

Rates of Return, in %

60

nd

Bo urn

ed et

40

is f R

om o

Pr ate

R

20

te of Return

Expected Stock Ra

WACC

of 10% rf

Expected Bond Rate of Return

0

0 20 40 60 80 100

Debt Ratio, in %

This ¬gure illustrates a ¬rm that has normally distributed payo¬s. This ¬rm has expected payo¬s of $84,000, with

a standard deviation of $32,000. (This matches the mean and standard deviation of the binomial example.) The

risk-free rate is 5.55%, the ¬rm™s cost of capital (WACC) is 10%.

¬le=capideal.tex: LP

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

21·5.D. A Major Blunder: If all securities are more risky, is the ¬rm more risky?

Many practitioners commit a serious logical mistake. They argue as follows:

More debt does not

increase the ¬rm™s cost

of capital, because it

1. If the ¬rm takes on more debt, the debt becomes riskier and the cost of capital for the

increases both debt and

equity cost of capital. debt (rDT ) rises.

2. If the ¬rm takes on more debt, the equity becomes riskier and the cost of capital for the

equity (rEQ ) rises.

3. Because the ¬rm consists of only debt and equity, the ¬rm also becomes riskier when the

¬rm takes on more debt.

The ¬rst two statements are correct. With higher leverage, the cost of capital on debt increases,

The fact that both debt

and equity become because the debt becomes riskier: in default, the debt is less likely to receive what it was

riskier as the ¬rm takes

promised. The equity also becomes riskier: the cost of capital on equity rises, because in

on more leverage does

¬nancial default, which is now also more likely to occur, more cash goes to the creditors before

not mean that the ¬rm

becomes riskier.

equity holders receive anything. It is only the ¬nal conclusion””therefore, the ¬rm becomes

riskier””that is wrong. The reason is that when the ¬rm takes on more debt, the weight on

the debt (wDT ) increases and the weight on the equity (wEQ = 1 ’ wDT ) decreases. Because the

cost of capital for debt (rDT ) is lower than the cost of capital for equity (rEQ ), the weighted sum

remains the same. Here is an example, in which I have made up appropriate costs of capital to

illustrate the point:

30% · 5.0% + 70% · 12.2% =

Low Debt 10%

(21.9)

60% · 6.0% + 40% · 16.0% =

High Debt 10%

wDT · E (˜DT ) + wEQ · E (˜EQ ) = E (˜FM ) .

r r r

This example shows that statements 1 and 2 are correct and statement 3 is incorrect: the costs

of capital for both debt and equity are higher when the ¬rm has more leverage, but the overall

cost of capital for the ¬rm has not changed. In the perfect M&M world, the overall cost of

capital is independent of the mix between debt and equity.

21·5.E. The E¬ect of Leverage Price-Earnings Ratios (Again)

We have actually already discussed the e¬ect of leverage on price-earnings ratios in Section 10·3.E,

How earnings are split

between debt and equity but you had to trust me blindly that debt o¬ered a lower expected rate of return than equity.

Let us repeat it quickly with our two capital structures.

30% · 5.0% + 70% · 12.2% =

Low Debt 10%

(21.10)

60% · 6.0% + 40% · 16.0% =

High Debt 10%

wDT · E (˜DT ) + wEQ · E (˜EQ ) = E (˜FM ) .

r r r

Say your ¬rm requires a capital investment of $10 million today, and expects to generate $11

million in return next year. It exists for only one year, so you have no accruals and the $1

million extra cash ¬‚ows is also net income. In the low-debt capital structure, your ¬rm would

raise $3 million from creditors and expect to pay them interest of $150,000”although its

promised interest payments would likely be a little higher to compensate creditors for default.

Shareholders would ¬nance $7 million and expect a return of $7.84 million. In the high-debt

capital structure, creditors would provide $6 million and expect $360,000 in interest payments,

shareholders would provide $4 million and expect a net return of $640,000.

Equity

Share P/E

Low Debt 70% 8.33

High Debt 40% 6.25

¬le=capideal.tex: RP

543

Section 21·6. The Big Picture: How to Think of Debt and Equity.

Side Note: Some other textbooks may talk about the e¬ect of leverage on earnings per share (EPS). However,

this is meaningless. The same capital structure can exist under di¬erent numbers of shares. Equity can provides

its $7 million with 1 million shares valued at $7/share (and expected EPS of $0.70/share) or with 100,000 shares

valued at $70/share (and expected EPS of $7/share). Any EPS ¬gure is possible.

Solve Now!

Q 21.7 In Figure 21.1, compute the equity™s expected rate of return if the ¬rm borrows $9,474.18.

Q 21.8 In a perfect market, if the cost of equity for a company is 15% and the cost of debt is 10%,

and if the company is ¬nanced by 80% debt and by 20% equity, what would be the company™s

cost of equity if it reduced its debt from 80% to 50%, so that it could qualify as a AAA rated ¬rm

with a debt interest rate of 8%?

Q 21.9 Compared to Firm B, Firm A has both a higher cost of capital for its debt and a higher

cost of capital for its equity. Does this necessarily imply that Firm A has a higher cost of capital

for the overall ¬rm than Firm B?

21·6. The Big Picture: How to Think of Debt and Equity

Important: Think of debt (all liabilities) and equity (all stocks), contingent claims,

and contingent claims valuation, in a perfect M&M world, as follows:

The Value of Claims

• An “absence of arbitrage” relationship ensures that the total value of a pro-

ject equals the sum-total of the values of all its ¬nancing instruments.

• Debt and equity claims “partition” the ¬rm™s payo¬s in future states of the

world. These payo¬s are contractually speci¬ed.

The Risk of Claims

• Levered equity (i.e., the residual claim after the liabilities are satis¬ed) is

riskier than full ownership, which in turn is riskier than the liabilities.

The Cost of Capital

• In a world of uncertainty, riskier investments have to o¬er higher expected

rates of return. This implies that levered equity has to o¬er a higher expected

rate of return than outright ownership, which in turn has to o¬er a higher

expected rate of return than the liabilities.

• The absence of arbitrage implies that the capitalization-weighted average

expected rate of return is

(21.11)

WACC = wEquity · E (˜Equity ) + wLiabilities · E (˜Liabilities ) = E (˜Firm ) ,

r r r

where the weights wEquity and wLiabilities are the values of equity and liabilities

when quoted as a fraction of the overall ¬rm value today.

• The project™s WACC remains the same, no matter how the ¬rm is ¬nanced.

It is determined by the underlying payo¬s of the project.

¬le=capideal.tex: LP

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

If the ¬rm™s debt ratio is very high, it is also not uncommon to see very high expected rates

Some more factoids.

of return on the equity”multiple times that of the ¬rm™s WACC. You should also realize that

the quoted rate of return on debt can be higher than the expected rate of return on equity. It is

only the expected rate of return on debt that should not exceed the expected rate of return on

equity. (The exceedingly rare exception are companies that have very negative betas.)

21·7. Using the CAPM and WACC Cost of Capital in the NPV

Formula

We have yet to combine NPV, WACC, and the CAPM. This actually turns out to be easy: we

NPV, WACC, and CAPM

are often all used use the CAPM to provide appropriate expected rates of return on debt and equity, compute

together.

the weighted average to obtain a WACC, and then use this WACC as the denominator in the

NPV formula. For example, consider a project that can be ¬nanced with low-risk debt with a

market-beta of 0.1, worth $400 today; and high-risk equity with a market-beta of 2.5, worth

$250 today. The risk-free rate of return is 4%; the equity premium is 3%. What is the cost of

capital of this project?

The standard method is to ¬rst compute the appropriate expected rates of return for the debt

and the equity,

E (˜debt,M ) = 4% + ·0.1 = 4.3%

r ,

3%

(21.12)

E (˜equity,M ) = 4% + ·2.5 = 11.5% ,

r 3%

E (˜i,M ) = rF + E (˜M ) ’ rF ·βi,M

r r .

Second, compute the WACC:

$400 $250

WACC = · 4.3% + · 11.5%

$400 + $250 $400 + $250

(21.13)

= 61.5% · 4.3% + 38.5% · 11.5%

= .

7.1%

An alternative method relies on the weighted average project beta,

$400 $250

β¬rm,M = · 0.1 + · 2.5 = 1.025 .

$400 + $250 $400 + $250 (21.14)

β¬rm,M = wdebt · βdebt,M + wequity · βequity,M .

This means that the project™s cost of capital is

E (˜¬rm,M ) = 4% + ·1.025 = 7.1%

r 3%

(21.15)

E (˜i,M ) = rF + E (˜M ) ’ rF ·βi,M

r r ,

which is the same as the cost of capital estimate in Formula 21.13. We can now use this cost of

capital estimate to discount the project™s expected cash ¬‚ows to obtain a present value estimate.

For example, if the project earns $800 with probability 48% and $600 with probability 52%, then

48%·$800 + 52%·$600

(21.16)

PV = ≈ $650.

1 + 7.1%

¬le=capideal.tex: RP

545

Section 21·8. Summary.

Solve Now!

Q 21.10 Assume the risk-free rate of return is 3% and the equity premium is 4%. A ¬rm worth

$100 (all numbers in millions of dollars) has a market beta of 3. A new project costing $10

appears, which is expected to pay o¬ $11 next year. The beta of this new project is 0.5. However,

the ¬rm evaluates all projects by its overall cost of capital. Would this ¬rm take the project? How

do the beta and the value of the ¬rm change if it takes the project vs. if it does not take it?

21·8. Summary

The chapter covered the following major points:

• Entrepreneurs have an incentive to set up a capital structure that maximizes ¬rm value,

not equity value, even if”or because”managers later would want to behave opportunis-

tically.

• In the perfect market scenario of M&M, the value of all securities is equal to the value

of the ¬rm™s underlying projects, and thus unrelated both to the ¬nancing split between

debt and equity and to the ¬rm™s dividend policy.

• Arbitrage in this perfect world forces the ¬rm™s cost of capital to be invariant to the split

between debt (liabilities) and equity. It is equal to the weighted average cost of capital

(WACC) of debt (liabilities) and equity.

• Higher leverage does not imply that the overall cost of capital increases, even though both

debt (liabilities) and equity become riskier.

• The CAPM is compatible with this perfect-world point of view.

¬le=capideal.tex: LP

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

Appendix

A. Advanced Appendix: Compatibility of Beta, the WACC, and

the CAPM Formulas in a Perfect World.

For the nerd, we now show that the “leverage adjustment for beta” Formula 17.16 (Page 437), the

Reconciling Beta, WACC,

and the CAPM. WACC Formula 21.11 (Page 543), and the CAPM Formula 17.2 (Page 423) are mutually compatible,

at least in the perfect markets scenario.

In this chapter, we developed the basic WACC formula (the cost of capital for the overall ¬rm

Recap WACC.

FM”not to be confused with F, the subscript for the risk-free security):

E (˜FM ) = wDT ·E (˜DT ) + wEQ ·E (˜EQ ) .

r r r

where FM is the ¬rm, DT is the total debt, and EQ is the total equity of the ¬rm.

Substitute the CAPM Formula 17.2 into the three expected rates of return in the WACC Formula:

Recap CAPM.

rF + E (˜M ) ’ rF · βFM,M = wDT · rF + E (˜M ) ’ rF · βDT,M

r r

+ wEQ · rF + E (˜M ) ’ rF · βEQ,M

r .

Now pull out the risk-free rates of return,

And show the

compatibility.

rF + E (˜M ) ’ rF · βFM,M

r

= wDT ·rF + wEQ ·rF + wDT · E (˜M ) ’ rF · βDT,M + wEQ · E (˜M ) ’ rF · βEQ,M

r r .

Recognize that (wEQ + wDT ) = 1, so (wEQ + wDT )·rF = rF , so

E (˜M ) ’ rF · βFM,M = wDT · E (˜M ) ’ rF · βDT,M + wEQ · E (˜M ) ’ rF · βEQ,M .

r r r

Divide by E(˜M ) ’ rF to obtain

r

βFM,M = wDT ·βDT,M + wEQ ·βEQ,M ,

which is exactly our relationship in Formula 17.16, which relates betas to one another! Indeed,

all three formulas share the same intuition: ¬rms and securities with higher betas are riskier

and thus have to o¬er higher expected rates of return.

Solve Now!

Q 21.11 A ¬rm consists of 60% equity with a market-beta of 2, and 40% debt with a market-beta

of 0.25. The equity premium is 5%, the risk-free rate is 3%. Compute the ¬rm™s cost of capital

from the overall-¬rm beta. Then compute the equity™s cost of equity capital and debt™s cost of

debt capital, and compute the ¬rm™s cost of capital from these costs of capital.

¬le=capideal.tex: RP

547

Section A. Advanced Appendix: Compatibility of Beta, the WACC, and the CAPM Formulas in a Perfect World..

Solutions and Exercises

1. Ex-ante means “before the fact”; Ex-post means “after the fact.” To the extent that owners can set up a

situation (charter) that encourages best (i.e., from a perspective of the ¬rm) ex-post behavior, the ex-ante

value (for which the ¬rm can be sold right now) is maximized. However, if the situation (charter) is such that

owners will later try to expropriate others or such that managers make bad decisions in the future, the ex-ante

value today is less.

2. Yes. See the example of the $3 for $1 transaction in the text.

3. A perfect market: no transaction costs, perfect competition, no taxes and bankruptcy costs, and no di¬erences

in information.

4. The idea is to explain it really simply. Milk, cream, and pizza are handy metaphors.

5. Nothing really. We do not need it. We only use it because it makes the Tables simpler to compute.

6.

d · $55 + e · $5 = $0 · 5% d ≈ ’0.003

Bad Luck

’ (21.17)

d · $55 + e · $105 = $66 · 5% e ≈ +0.033

Good Luck

So you would purchase 3.3% of the LD equity and sell (issue) 0.3% of the equivalent of the LD debt. The equity

would cost you e · $50 = $1.65, the debt issue would give you $0.15 in proceeds. Your net cost would thus be

$1.50”as it should be, because purchasing 5% of the MD equity would have cost you 5% of $30, which also

comes to $1.50.

7. The ¬rm must repay $10,000 at the 5.55% risk-free cost of capital. Therefore, equity expects to receive

$74,000. This requires an investment of $66,889.46, which is the ¬rm value $76,363.64 minus the $9,474.18

that the creditors provide. Therefore, the equity™s expected rate of return is 10.63%.

8. First, compute the WACC without taxes: 0.8 · 10% + 0.2 · 15% = 11%. We now want to compute the cost of

debt that satis¬es 0.5 · 8% + 0.5 · x = 11%, so x = 14%.

9. No. The example in the “Important Error” Section illustrates this fallacy.

10. The solution proceeds the same way as in the text on Page 441. The project should have an appropriate rate

of return of

(21.18)

E ( r ) = 3% + 4%·0.5 = 5% .

˜

With a 10% expected rate of return, the project would not be taken if the ¬rm used a beta of 3 (implying a

cost of capital of 15%, which is higher than the expected rate of return of 10%). Nevertheless, the project has

a net present value of

$10·(1 + 10%)

(21.19)

NPV = ’$10 + = $0.48 ,

1 + 5%

which is lost if the project is not taken. (Please con¬rm that this is also the outcome if the combined ¬rm

value is computed.) The value is destroyed because the ¬rm managers are making the mistake of not taking

the positive NPV project, which is because they do not understand that projects should be evaluated by the

projects™ own costs of capital, not the ¬rm™s cost of capital.

Using the ¬rst method, the overall beta is 60%·2 + 40%·0.25 = 1.3. Using the CAPM, the cost of capital is

11.

E (˜i ) = rF + [E (˜M ) ’ rF ] · βi,M = 3% + 5% · 1.3 = 9.5%. Using the second method, the equity™s cost of

r r

capital is E (˜EQ ) = rF + [E (˜M ) ’ rF ] · βEQ,M = 3% + 5% · 2 = 13%, the debt™s cost of capital is E (˜DT ) =

r r r

rF + [E (˜M ) ’ rF ] · βDT,M = 3% + 5% · 0.25 = 4.25. Therefore, the ¬rm™s cost of capital can also be computed

r

as E (˜i ) = 60%·13% + 40%·4.25% = 9.5%.

r

(All answers should be treated as suspect. They have only been sketched, and not been checked.)

¬le=capideal.tex: LP

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

CHAPTER 22

Corporate Taxes and A Tax Advantage of Debt

The Tax-Adjusted Weighted Average Cost of Capital (APV, WACC)

last ¬le change: Feb 19, 2006 (13:46h)

last major edit: Mar 2005

The presence of taxes may be the most glaring violation of the M&M assumptions in the real

world. If the ¬rm has to pay corporate income taxes, managers can use capital structure policy

to create value. To do so, they rely either on the APV formula or on the tax-adjusted WACC

formula. This chapter explains it all.

Note that the relevant tax that this chapter is concerned with is only the corporate income

tax, not the personal income tax. (You can assume our shareholders”or yourself”to be a

tax-exempt pension fund, if this helps you.) The next chapter will consider both corporate and

personal income taxes.

549

¬le=capctaxes.tex: LP

550 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

22·1. Capital Budgeting If Equity and Debt Were Equally Taxed

Table 22.1. Hypothetical Firm

Investment Cost in Year 0 $200

Before-Tax Return in Year 1 $280

Before-Tax Net Return From Year 0 to Year 1 $80

Corporate Income Tax Rate („) 30%

Appropriate Cost of Capital From 0 to 1 12%

In this chapter, we will follow a simple hypothetical ¬rm that is described in Table 22.1. As

A Basic Corporate

Example in Section 6·5.B, we follow the standard practice to denote the tax rate with the Greek letter „

(tau). We start with a brief discussion of an entirely unrealistic case, in which the ¬rm faces

the same tax rate, regardless of how it is ¬nanced (i.e., unlike as in the real world, interest

payments are not tax-deductible). This helps to illustrate that investors care about after-tax

returns, not before-tax returns.

Consider ¬nancing our ¬rm entirely with equity. With $280 in before-tax earnings on the $200

Taxes mean that the

after-tax rate of return investment, our ¬rm has a “before corporate income tax” internal rate of return of ($280 ’

is lower than the

$200)/$200 = 40%. But, with taxes to the tune of 30% on its net return of $80, Uncle Sam

before-tax rate of return.

collects $24. The ¬rm™s “post corporate income tax” net rate of return is therefore only ($256’

$200)/$200 = 28%.

Now assume that ¬nancial markets are such that investors are willing to provide capital for

Investors demand a

certain rate of return, projects with this riskiness at a rate of E(˜FM ) = 12%. (In this chapter, we will again omit

r

regardless of how the

time subscripts if there is little risk of confusion.) Naturally, investor-owners do not care what

¬rm gets there.

happens inside the ¬rm, only what the ¬rm will pay them in the end. It is all the same to them

if the ¬rm earns 12% before-tax and manages to avoid all corporate income taxes; or if the ¬rm

earns 24% and pays half of it in corporate income taxes; or if the ¬rm earns 600%, of which 98%

is con¬scated by the government (1 + 600% · (1 ’ 98%) = 1 + 12%). From the perspective of our

small ¬rm, we are “price-takers” when it comes to raising capital”we have to accept the rate

of return asked for by our investors, which is determined by the competition among capital

providers and capital consumers. The investors care only about the cash that will ultimately

return to them”and this (after-corporate income tax) rate of return has to be 12%. Now, if our

example ¬rm faces a 30% corporate tax rate, it must earn 17.14% in before-tax rate of return

in order to be able to o¬er investors 12% in actual rate of return. Check this: an investment of

$100 that turns into $117.14 has to pay Uncle Sam 30% in taxes on income of $17.14 for a total

income tax of $5.14, which leaves the ¬rm $112 to return to its investors after the corporate

income tax is paid. Returning to our “$280 before-corporate-income-tax” ¬rm, with its 12%

required after-corporate-income tax cost of capital, the ¬rm™s actual PV is

E (CFafter-tax ) $256

PV = = = $228.57 . (22.1)

1 + E (˜after-tax ) 1 + 12%

r

Again, note that the present value is calculated with both the after-tax expected cash ¬‚ows and

the after-tax cost of the capital.

¬le=capctaxes.tex: RP

551

Section 22·2. Di¬erential Debt and Equity Taxation in The U.S. Tax Code.

22·2. Differential Debt and Equity Taxation in The U.S. Tax

Code

We now move on to a more realistic tax code. In many countries”the United States included” Tax codes worldwide

violate the M&M no-tax

individuals and corporations face similar tax treatments, tax schedules, and tax rates. Although

assumption.

tax code details vary from year to year, country to country, state to state, county to county, and

even city to city, most tax codes are pretty similar in spirit. Thus, the tax concepts in this book

apply relatively universally.

Recall from Chapter 6·4 that when corporations earn money, the form of payout matters. Firms Tax codes subsidize

borrowing: ¬rms can

pay taxes on their earnings net of interest payments. That is, unlike dividend distributions or

pay interest from

money used to repurchase shares or money reinvested, the IRS considers interest payments to before-tax income, but

be a cost of production. Therefore, it allows the payment of interest to be treated as a before- pay dividends from

after-tax income.

tax expense rather than as an after-tax distribution of earnings. The result is that a corporation

saves on taxes when it distributes its earnings in the form of interest payments. For example,

if PepsiCo™s operations really produced $100, and if $100 in interest was owed, then PepsiCo

could pay the full $100 to its creditors and Uncle Sam would get nothing. However, if not

paid out in interest, Uncle Sam would ¬rst collect corporate income taxes, say, 30%. PepsiCo

could only keep (or distribute) the $70 that would be left over. The point of this chapter is

to show how to exploit this di¬erence in the relative tax treatment between payments to debt

and all other uses of money. It allows the astute CFO to add value by choosing a clever capital

structure.

At this point, you are probably wondering why you would not always ¬nance your ¬rm with Preview: With too much

debt, other forces may

as much debt as possible. The short answer is that if you were in a world in which corporate

increase the cost of

incomes taxes are the only distortion, then having as much corporate leverage as possible would capital.

indeed be ideal. However, there is more going on. If you take on too much corporate debt,

eventually other forces will raise the ¬rm™s cost of capital to the point that further increases

in debt are no longer value-increasing. These forces will be the subject of the next chapters.

But let us ¬rst understand how managers should do capital budgeting when there are only

corporate income taxes, and no other taxes or perfect market distortions.

Solve Now!

Q 22.1 A debt-equity hybrid security is making a payout of $500 to its holders. If the ¬rm is in

the 33% tax bracket, how much does the ¬rm have to earn if the IRS designates the payment an

interest payment? How much does the ¬rm have to earn if the IRS designates the payment a

dividend distribution?

Anecdote: Special Tax Breaks and Corporate Welfare

“Special Income Tax Provisions” are tax breaks enacted by Congress for speci¬c activities, often on behalf of a

single corporation. The special income tax provisions amounts are commonly estimated to be about $1 trillion

a year”more than the total amount of federal discretionary spending! These provisions are a main reason

why corporations”large corporations, really”have paid less and less in income taxes relative to the rest of the

population and relative to other OECD countries. In 1965, corporate income taxes were 4.1% of U.S. GDP; in

2000, about 2.5%; in 2002, about 1.5%. And, for comparison to the 1.5% U.S. tax rate, in 2000, Germany™s rate

was 1.8%, Canada™s rate 4.0%.

It would be wonderful if the low U.S. corporate income tax rate would attract businesses to locate into the United

States and to create jobs. Alas, because the low e¬ective corporate income tax rates come about through strange

corporate tax shelters (often through relocation of headquarters into foreign countries), the United States often

ends up with the worst of both worlds: both incentives for companies to move out of the United States and low

corporate income tax receipts. The only president in recent history to buck the trend may have been Ronald

Reagan, who slashed both the corporate income tax and the ability of companies to circumvent it.

Source: “Testimony of Robert S. McIntyre” (www.ctj.org), Director of “Citizens for Tax Justice.”

¬le=capctaxes.tex: LP

552 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

22·3. Firm Value Under Different Capital Structures

You know that ¬rms are indi¬erent between debt and equity in a perfect world. You also know

Introducing an interest

tax subsidy leads to a that Uncle Sam subsidizes ¬rms that pay interest, relative to ¬rms that either retain earnings

corporate preference for

or pay dividends or repurchase shares. Therefore, you know that, on tax grounds, ¬rms should

debt.

have a preference for debt. Your goal now is to determine the exact value of the ¬rm in the

presence of this tax subsidy for debt interest payments.

The next chapter describes why too much debt can create other costs, which raise the

¬rm™s overall cost of capital. Fortunately, the formulas in this chapter will still hold,

because these other costs will be re¬‚ected only in the cost of capital (E (˜)) components.

r

Table 22.2. Two Financing Scenarios for a Safe 1-Year Firm

Scenario Assumptions:

Investment Cost in Year 0 $200.00

Before-Tax Return in Year 1 $280.00

Before-Tax Net Return From Year 0 to Year 1 $80.00

Corporate Income Tax Rate („) 30%

Appropriate Average Cost of Capital From 0 to 1 12%

Scenario EF: 100% Equity Financing.

Taxable Pro¬ts Next Year $80.00

Corporate Income Taxes Next Year (30% of $80) $24.00

Owners Will Keep Next Year $56.00

Scenario DF: $200 Debt Financing at 11%. Rest is Levered Equity.

Interest Payments $22.00

Taxable Pro¬ts Next Year $58.00

Corporate Income Taxes Next Year $17.40

Equity Owners Will Keep Next Year $40.60

$22.00 + $40.60 = $62.60

Equity+Debt Owners Will Keep Next Year

¬le=capctaxes.tex: RP

553

Section 22·3. Firm Value Under Di¬erent Capital Structures.

22·3.A. Future Corporate Income Taxes and Owner Returns

We begin with Table 22.2, which works out the value of a speci¬c ¬rm under two ¬nancing If the ¬rm is debt

¬nanced, then there is

scenarios.

more money that can be

paid to owners-total.

This is the money that

Equity Financing (EF) Scenario In the all-equity scenario, the ¬rm does not exploit the help of

the IRS does not get.

the IRS. It earns $280 on an investment of $200. At a 30% corporate income tax rate, it

will pay corporate income taxes of 30% · $80 = $24. It can then pay out the remaining

$56 in dividends.

Debt Financing (DF) Scenario In the debt ¬nancing scenario, the ¬rm borrows $200 today at

an interest rate of 11% for interest payments next year of $22. Therefore, its corporate

pro¬ts will be $80 ’ $22 = $58, on which it would have to pay Uncle Sam $17.40. This

permits owners (creditors and shareholders”and you may be both) to receive $62.60, the

sum of $22 for its creditors and $40.60 for its equity holders.

Relative to the 100% equity ¬nanced case (in which owners keep $56.00), the debt ¬nanced case

(in which owners keep $62.60) increases the ¬rm™s after-tax cash ¬‚ow by $6.60. A quicker way

to compute the tax savings is to multiply the tax rate by the interest payments: If the IRS allows

the ¬rm to deduct $22 in interest payments, the ¬rm will save $22·30% = $6.60 in corporate

income taxes. These $6.60 in tax savings will occur next year, and will therefore have to be

discounted back. It is common but not unique to use the ¬rm™s cost of capital to discount the

tax shelter for a growing ¬rm. The appendix explains the appropriate discount rate in greater

detail, but just realize that whether you discount the much smaller tax shelter of $6.60 by the

debt cost of capital (11%) or a higher, say, 15% (the ¬rm™s cost of capital), would only make a

di¬erence between $5.95 and $5.74. In our context, 21 cents on a $280 expected cash ¬‚ow is

a third-order issue.

Solve Now!

Q 22.2 A $1 million construction project is expected to return $1.2 million. You are in a 45%

combined federal and state marginal income tax bracket. Your annual income is $200,000 per

year. If you ¬nance the project with an $800,000 mortgage at an interest rate of 5%, how much

will Uncle Sam receive? If you ¬nance the project with cash, how much will Uncle Sam receive?

Q 22.3 Continue. If the appropriate project interest rate is 8%, what is the PV of the tax savings

from ¬nancing the project with a mortgage?

Anecdote: The RJR Buyout Tax Loophole

In a Leveraged Buyout (L.B.O. for short) leverage can increase dramatically”and this can signi¬cantly reduce

corporate income taxes. In 1988, First Boston™s plan to take over R.J.R. Nabisco relied on an esoteric tax loophole

just about to be closed. By “monetizing” its food operations, a fancy way to increase leverage, the deferring

of taxes would have saved an estimated $3-$4 billion dollars of R.J.R.™s corporate income taxes”which would

have increased the annual federal U.S. de¬cit by 2 percent!

¬le=capctaxes.tex: LP

554 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

22·4. Formulaic Valuation Methods: APV and WACC

You now know how to compute the tax cash ¬‚ows and the tax shelter. You have also thought

But we need a formula

that works for about reasonable discount rates. Your next goal is to ¬nd a valuation formula that allows you

intermediate leverage.

to compute the ¬rm value today not only for the current ¬nancing arrangement, but also for

The choices are WACC,

other ¬nancing arrangements that you might contemplate for your ¬rm. You can essentially

APV, or ¬nancials.

use three methods to value the ¬rms under di¬erent ¬nancing scenarios:

1. You can construct the pro formas for the ¬rm under the new hypothetical capital structure,

and then value the cash ¬‚ows directly. Without describing it as such, you have actually

already done this in Chapter 9, but we shall work out another example in the next section.

2. You can compute an adjusted present value (or APV for short), which adds back the tax

subsidy. (This is basically what you have been doing above.)

3. You can generalize the weighted average cost of capital (WACC) formula to re¬‚ect the

preferential treatment of debt by suitably lowering the debt™s cost of capital.

This section explains the APV and WACC in some detail. The next section will work out a multi-

year example that shows how to use them. Properly applied, all methods should provide very

similar answers.

Before we start, it is important that you realize that our model will be just that”a model. We

Be aware of the many

simpli¬cations that we will determine the tax savings of a company that come from the presence of debt if the ¬rm

shall make.

faces a ¬xed marginal income tax rate. This is a ¬rst-order ¬nancial issue, but our model will

ignore many other possibly important tax issues, such as delayed income tax payments, tax-

loss carryforwards, recapture of past tax payments, di¬erent marginal corporate income tax

rates at di¬erent income levels, the possibility to default on income tax payments, state taxes,

foreign taxes, special tax incentives, or outright tax evasion.

22·4.A. Adjusted Present Value (APV): Theory

APV decomposes the value of the ¬rm into two components: the value of the ¬rm if it were

The main idea of APV:

value the ¬rm, then add all equity-¬nanced and fully taxed, plus a tax subsidy for each dollar that can be declared as

the tax subsidy.

“interest” rather than as “dividend.” In our example from Table 22.2, $256 ($280 pro¬t minus

$24 in corporate taxes) is the expected cash ¬‚ow of our ¬rm if it is 100% equity ¬nanced. The

APV method then adds the tax subsidy. For example,

Zero interest payments If the ¬rm is all equity ¬nanced, the tax subsidy is zero.

High interest payments If the ¬rm could have the interest payments of, say a utopian $80, the

IRS would believe that the ¬rm had not earned a penny. Therefore, the owners could keep

an extra $24 above the $256 all-taxed scenario next year.

Normal interest payments If the ¬rm will have interest payments of, say $19, the IRS would

see $280 ’ $19 = $261 in return minus $200 investment cost for a net return of $61. The

IRS would therefore collect 30%·$61 = $18.30, which is $5.70 less than the $24 that the IRS

would have collected if the ¬rm had been 100% equity ¬nanced. Alternatively, you could

have directly calculated the expected tax savings as „ ·(E(˜DT )·DT) = 30%·($19) = $5.70.

r

Let™s make a formula out of this method. Your ¬rst step to a more general valuation formula

Computing tax savings,

given the debt level. in the presence of corporate income taxes is to relate the amount of debt today to the interest

payments next year. Let™s return to our example, in which you borrow $200 at an interest rate

of 11%. The expected interest payments are now

Expected Interest Payment = 11% · $200 = $22

(22.2)

Expected Interest Payment = E (˜DT ) · DT .

r

¬le=capctaxes.tex: RP

555

Section 22·4. Formulaic Valuation Methods: APV and WACC.

One important error to avoid is that you must use the expected debt interest (of 11%), not the

quoted bank interest rate, which could be considerably higher than 11%. (This would not matter

for large ¬rms that are not too highly levered, but it would matter elsewhere.) Continuing on,

the future tax savings relative to an all-equity ¬nanced ¬rm is the amount of corporate income

tax that the ¬rm will not have to pay on the interest.

Expected Tax Saving = 30% · [11% · $200] = $6.60

(22.3)

Expected Tax Saving = · [E (˜DT ) · DT] .

„ r

In words, Uncle Sam will expect to receive $6.60 less from the owners of the project, because

$22 in pro¬t repatriation was designated as “interest.”

The $6.60 in tax savings have to be discounted, because they will occur next year. So the APV APV discounts tax

savings and adds them

Formula computes the discounted value of an all-equity ¬nanced ¬rm (with after-tax cash ¬‚ows

to an all-equity type

of $256 next year), and then adds back the discounted tax savings: hypothetical ¬rm.

$200 debt at 11% interest,

30% · $22

$256

i.e.,$22 interest payment APV = + = $234.52

1 + 12% 1 + 11%

discounted at 11%:

(22.4)

E (CF) „ · [E (˜DT ) · DT]

r

APV = +

1 + E (˜FM ) 1 + E (˜DT )

r r

Value as Tax Subsidy

if 100% equity +

APV = from Interest .

¬nanced Payments

As described at length in Section a, you could also reasonably use the ¬rm™s cost of capital to

discount the tax savings,

$200 debt at 11% interest,

30% · $22

$256

i.e.,$22 interest payment APV = + = $234.46

1 + 12% 1 + 12%

discounted at 12%:

(22.5)

E (CF) „ · [E (˜DT ) · DT]

r

APV = +

1 + E (˜FM ) 1 + E (˜FM )

r r

Value as Tax Subsidy

if 100% equity +

APV = from Interest .

¬nanced Payments

The di¬erence of 6 cents is trivial. APV generalizes easily to multiple years: just compute the

tax savings for each year and add them up, the same way that you would add up present values.

We will work such a multi-period example in the next section.

Important: The Adjusted Present Value (APV) Formula computes an “as if all

equity ¬nanced” PV (i.e., after corporate income tax) and then adds back the tax

subsidy: Value as if Firm is 100% Tax Subsidies

(22.6)

equity ¬nanced and fully + from Interest

=

APV0

taxed. Payments

If the project lasts for only one period, this translates into

interest

payment

(22.7)

E (CF1 ) E („ · rDT ·DT)

˜

APV0 = + .

1 + E (˜FM ) 1 + E (˜DT )

r r

¬le=capctaxes.tex: LP

556 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

22·4.B. APV: Application to a 60/40 Debt Financing Case

In our example, the ¬rm with $200 debt is worth $234.46 today, which comes to a debt ratio

An example of how to

value a ¬rm ¬nanced of $200/$234.46 ≈ 85%. Now assume that the ¬rm instead considers a new capital structure,

with 60% debt.

in which it would borrow only $139.16. The ¬rm has determined that this lower-debt capital

structure would reduce its debt cost of capital to 9% per annum”after all, at such low levels, it

is risk-free, so risk-averse investors would be willing to accept a lower expected rate of return.

What would the ¬rm™s value become?

According to the APV Formula, you begin with the value of a 100%-equity ¬rm, which is

Problem solved.

$256/(1 + 12%), and add back the tax subsidy. Interest payments on $139.16 of debt will

be 9% · $139.16 = $12.52 next year. Taxes saved will be 30% · $12.52 = $3.76 next year. This

is worth $3.45 today. Therefore

30% · 9% · $139.16

$256.00

APV = +

1 + 12% 1 + 9%

= + = $232.02

$228.57 $3.45

(22.8)

E (CF) „ · E (˜DT ) · DT

r

APV = +

1 + E (˜FM ) 1 + E (˜DT )

r r

= “as if all equity ¬nanced” ¬rm + .

tax subsidy

If you prefer discounting the expected tax shelter with the ¬rm™s cost of capital, use

30% · 9% · $139.16

$256.00

APV = +

1 + 12% 1 + 12%

= + = $231.93

$228.57 $3.36

(22.9)

E (CF) „ · E (˜DT ) · DT

r

APV = +

1 + E (˜FM ) 1 + E (˜FM )

r r

= “as if all equity ¬nanced” ¬rm + .

tax subsidy

(Again, the cost of capital on the tax shelter makes little di¬erence, here only $3.45 ’ $3.36 =

$0.09.) This is the APV answer: In the presence of corporate income taxes, a ¬rm ¬nanced with

$139.16 in debt would be worth about $232.

22·4.C. Tax-Adjusted Weighted Average Cost of Capital (WACC) Valuation: Theory

The second method to compute the value of the ¬rm uses a tax-adjusted weighted average cost

To show that WACC and

APV are compatible, we of capital formula. If you start with the APV Formula and manipulate it, it will be most apparent

derive the tax-adjusted

that the two methods can yield the same value, at least if you start from Formula 22.9. We will

WACC Formula from the

stick with the same parameters: our 60/40 debt/equity ¬nancing, a 30% corporate income tax

APV Formula.

rate, a 9% cost of debt capital, and $280 in before-tax return ($256 after-tax return in the all-

equity case). As before, the ¬rm borrows $139.16 at a 9% interest rate for net interest payments

of $12.52. The corporate income tax shield is 30% of $12.52 or $3.76. The APV Formula 22.9

values the ¬rm at

=$3.76

$12.52

30% · (9% · $139.16)

$256

(22.10)

PV = + = $231.93

1 + 12% 1 + 12%

E (CF) „ · [E (˜DT ) · DT]

r

PV = + .

1 + E (˜FM ) 1 + E (˜FM )

r r

¬le=capctaxes.tex: RP

557

Section 22·4. Formulaic Valuation Methods: APV and WACC.

The main di¬erence between APV and WACC is that the WACC method expresses the debt as a

ratio of ¬rm value,

60% = $139.16/$231.93 $139.16 = 60% · $231.93

(22.11)

wDT = ’ = wDT · PV .

DT/PV DT

Substitute this DT into the APV formula,

30% · [9% · (60% · $231.93)]

$256

PV = + = $231.93

1 + 12% 1 + 12%

(22.12)

E (CF) „ · [E (˜DT ) · (wDT · PV)]

r

PV = + .

1 + E (˜FM ) 1 + E (˜FM )

r r

We now have PV on both sides of the equation, so you want to solve for PV. This requires a

couple of algebraic steps.

1. Multiply both sides by [1 + E(˜FM )] = (1 + 12%) to make the denominator disappear,

r

(1 + 12%) · $231.93 = $256 + 30% · [9% · (60%·$231.93)]

(22.13)

[1 + E (˜FM )] · PV = E (CF) + „ · [E (˜DT ) · (wDT ·PV)]

r r .

2. Move the second term to the left side,

(1 + 12%) · $231.93 ’ 30% · [9% · (60%·$231.93)] = $256

(22.14)

[1 + E (˜FM )] · PV ’ „ · [E (˜DT ) · (wDT ·PV)] = E (CF) .

r r

3. Pull out the PV,

$231.93 · 1 + 12% ’ 30% · 9% · 60% = $256

(22.15)

PV · 1 + E (˜FM ) ’ „ · E (˜DT ) · wDT = E (CF) .

r r

4. Divide both sides by the PV multiplier,

$256 $256

$231.93 = =

1 + 12% ’ 30% · 9%·60% 1 + 10.38%

(22.16)

E (CF) E (CF)

= = .

PV

1 + E (˜FM ) ’ „ · [E (˜DT ) · wDT ] 1 + WACC

r r

This is the “real” WACC valuation formula, i.e., the WACC formula that also works when ¬rms

pay corporate income tax. The big idea of tax-adjusted WACC is to discount the “as-if-100%-

equity ¬nanced and fully taxed” cash ¬‚ows (of E(CF) = $256), not with ¬rm™s cost of capital

E(˜FM ) = 12%, but with a reduced interest rate that comes from the corporate income tax

r

subsidy on interest payments. The term which does this”relative to our earlier no-tax WACC

Formula 21.11”is „ · wDT · E(˜DT ) = 30% · 60%·9% ≈ 1.62%. Therefore, your revised discount

r

rate is 1 + 12% ’ 30%·9%·60% ≈ 1 + 10.38%. 10.38% is the (tax-adjusted) WACC.

¬le=capctaxes.tex: LP

558 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

The WACC formula is often slightly rearranged. Split E(˜FM ) into its cost of equity and

r

The more common form

of WACC also breaks out cost of debt components, E(˜FM ) = wDT ·E(˜DT ) + wDT ·E(˜DT ). In our example, to keep

r r r

equity cost of capital.

the weighted average ¬rm cost of capital at the constant E(˜FM ) = 12%, solve E(˜FM ) =

r r

wDT ·E(˜DT ) + wEQ ·E(˜EQ ) = 60%·9% + 40%·E(˜EQ ) = 12%, and ¬nd E(˜EQ ) = 16.5%. Sub-

r r r r

stitute this into Formula 22.16, and you get the more common version of the WACC formula,

$256 $256

PV = =

1 + 10.38% 1 + 40%·16.5% + (1 ’ 30%) · 60% · 9%

(22.17)

E (CF) E (CF)

PV = = .

1 + WACC 1 + wEQ ·E (˜EQ ) + (1 ’ „) · wDT · E (˜DT )

r r

Your new WACC formula generalizes your old M&M WACC formula, because when the corporate

The tax-adjusted WACC

generalizes ordinary tax rate „ is zero, the tax subsidy is useless, „ is zero, and the tax-adjusted WACC formula sim-

WACC.

pli¬es to your older and simpler WACC formula. But your new WACC formula can also handle

¬rms with positive corporate income tax rates. (Incidentally, about half of all publicly traded

¬rms in the United States have a marginal tax rate of zero, e.g., due to tax-loss carryforwards

or due to clever tax shelters. For these companies, the corporate tax rate may indeed be close

to zero, which means that they cannot use debt to reduce their cost of capital”and they could

therefore use the simpli¬ed M&M version of the WACC formula which ignores the tax subsidy

of interest.)

Important:

• The (tax-adjusted) weighted average cost of capital (WACC) formula dis-

counts the future cash ¬‚ows with a lower cost of capital that re¬‚ects the

corporate income tax shelter,

E (CF)

PV = ,

1 + WACC

(22.18)

where WACC = E (˜FM ) ’ „ · E (˜DT ) · wDT

r r

(22.19)

= wEQ · E (˜EQ ) + wDT · E (˜DT ) · (1 ’ „) .

r r

The expected cash ¬‚ows must be the cash ¬‚ows “as if the ¬rm were all equity

¬nanced and therefore fully taxed.”

• This formula is a generalization of the WACC formula from the perfect M&M

world. It is therefore this formula that is usually called the WACC formula.

• It is not clear how to use the WACC formula in a multi-period setting.

The WACC formula is so common that it is worth memorizing.

Unfortunately, you can only use the WACC formula in a multi-period setting if the cost of

capital, the ¬rm™s debt ratio, and the ¬rm™s tax rate stay constant. In this case, a present value

formula would look something like

∞

E (CFt )

(22.20)

PV0 = .

t

1 + [ wEQ · rEQ,0,t + wDT · E (˜DT,0,t ) · (1 ’ „) ]

r

t=1

If these quantities are not all constant, no one knows how to compute a proper WACC. It is

not unusual for ¬rms to plan on high debt ¬nancing upfront that they lateron pay back, which

is a situation that the WACC formula can handle. Therefore, the WACC formula can often serve

just as a rough approximation. The APV method is generally more ¬‚exible than WACC method.

¬le=capctaxes.tex: RP

559

Section 22·4. Formulaic Valuation Methods: APV and WACC.

The Hamada Equation gives a formula to adjust a ¬rm™s beta for both leverage and

Digging Deeper:

corporate income taxes:

(22.21)

βLevered = βUnlevered · [1 + (1 ’ „) · (DT/EQ)] .

Solve Now!

Q 22.4 Consider a 25/75 debt/equity ¬nancing case for your ¬rm: the before-tax return is $280,

the tax rate is 30%, the overall cost of capital is 12%, and debt when the ¬rm is 25% debt ¬nanced

must o¬er an expected rate of return of 8%. First compute the WACC, then compute the debt as

25% the WACC value, and show how the APV yields the same result.

Q 22.5 Consider ¬nancing your ¬rm with $100 debt: the before-tax return is $280, the tax rate

is 30%, the overall cost of capital is 12%, and this debt must o¬er an expected rate of return of

8.7%. First compute the APV, then compute the capital structure in ratios, and ¬nally show that

the WACC yields the same result.

Q 22.6 If you are thinking of debt in terms of a (constant) fraction of ¬rm value, would you

prefer WACC or APV? If you are thinking of debt in terms of a (constant) dollar amount, would

you prefer WACC or APV?

22·4.D. A Major Blunder: Applying APV and WACC to the Current Cash Flows

Unfortunately, both WACC and APV are often used incorrectly. Analysts frequently forget that Another Common

Mistake: Make sure you

the correct expected cash ¬‚ow in the present value numerator is the “as if fully equity ¬nanced

use the correct Cash

and fully after-tax” cash ¬‚ows”$256 in our example. It is neither the before-tax project cash Flow to Discount.

¬‚ows ($280 in our example), nor the after-tax cash ¬‚ows under the current ¬nancing scheme

(e.g., $280 ’ 9% · $139.16 = $267.48). If you have worked through the examples in this chapter,

you should understand why this would provide the wrong answer. Unlike errors in the discount

rate applied to the tax shelter”which is a modest error”using the wrong cash ¬‚ows would be

a big error.

Important: WACC and APV operate with expected “as if 100% equity-¬nanced

and after corporate income tax” cash ¬‚ows, not the ¬rm™s current cash ¬‚ows (which

depend on the current debt/equity ¬nancing).

Solve Now!

Q 22.7 A ¬rm in the 20% marginal tax bracket is currently ¬nanced with $500 debt and $1,000

equity. The debt carries an interest rate of 6%, the equity™s cost of capital is 12%. The risk-

free rate is 4%, the equity premium is 3%. What is the ¬rm™s beta? The ¬rm is pondering a

recapitalization to $1,000 debt, which would increase the debt™s interest rate to 8%. The ¬rm

exists only for 1 more year. What would the new equity be worth?

Solve Now!

Q 22.8 A ¬rm in the 40% income tax bracket has an investment that costs $300 in year 0, and

o¬ers a pretax return in year 1 of $500. Assume that the ¬rm™s cost of capital, as provided by the

external capital markets, is approximately E(˜DT ) = 15% + wDT · 5%. Compute the APV, WACC,

r

and a WACC-based value if the ¬rm borrows $50 to ¬nance it. Repeat if the ¬rm borrows $100.

¬le=capctaxes.tex: LP

560 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

22·5. A Sample Application of Tax-Adjusting Valuation Tech-

niques

Table 22.3. Income Statement of Hypothetical Machine

Year 1 2 3 4 5 6

Gross Sales (Revenues) $70 $70 $70 $70 $70 $70

“ Cost of Goods Sold (COGS) $5 $5 $5 $5 $5 $5

“ Selling, General & Administra- $5 $5 $5 $5 $5 $5

tive Expenses (SG&A)

= EBITDA (Net Sales) $60 $60 $60 $60 $60 $60

“ Depreciation $25 $50 $50 $25 $0 $0

= EBIT (Operating Income) $35 $10 $10 $35 $60 $60

“ Interest Expense $0 $5 $5 $5 $5 $5

= EAIBT (or EBT) $35 $5 $5 $30 $55 $55

“ Corporate Income Tax (at 40%) $14 $2 $2 $12 $22 $22

Net Income

= $21 $3 $3 $18 $33 $33

Excerpts From the Cash Flow Statement

Year 1 2 3 4 5 6

Net Income $21 $3 $3 $18 $33 $33

+ Depreciation $25 $50 $50 $25 $0 $0

Total Operating Activity

= $46 $53 $53 $43 $33 $33

Capital Expenditures “$75 “$75 - - - -

Total Investing Activity

= “$75 “$75 - - - -

Financing Cash Flow - - - - - -

Net Equity Issue +$26 - - - - -

Dividends - - “$53 “$43 “$33 “$8

Net Debt Issue +$25 - - - - “$25

Total Financing Activity

= +$51 - “$53 “$43 “$33 “$33

Net Change In Cash

= +$22 “$22 $0 $0 $0 $0

We now apply our corporate income-tax related valuation techniques on a multi-year, pro forma

Let™s value a pro forma

¬rm. ¬rm. (Pro formas will be discussed in detail in Chapter 29. Think of a pro forma as a forward

projection of the ¬nancial statements.) In Table 22.3, we return to our basic hypothetical

machine from Chapter 9, Table 9.6 on Page 209. To make the example more useful, assume an

appropriate debt interest rate of 20%, so a loan of $25 must o¬er an expected $5 in interest

per annum. The appropriate overall cost of capital for our ¬rm is 30%. The corporate income

tax rate is 40%. Table 22.3 shows that shareholders pay in $26 and receive a total of $137

in dividends. Debtholders invest $25 and receive $25 in total interest payments. (Your ¬rm

follows an odd capital distribution policy, but so be it.) What is your ¬rm worth?

¬le=capctaxes.tex: RP

561

Section 22·5. A Sample Application of Tax-Adjusting Valuation Techniques.

22·5.A. The Flow-To-Equity Direct Valuation from the Pro Forma Financials

Our ¬rst method to value the ¬rm uses the pro forma cash ¬‚ows directly. This is sometimes also Method 1: Direct Flows

from the Financials.

called the ¬‚ow-to-equity method. First, use the project cash ¬‚ow formula 9.18 from Page 221,

which tells you that the project cash ¬‚ows that you need for your NPV valuation are

Computing Project Cash Flows, $25 Debt Financing

Year 1 2 3 4 5 6

Total Operating Activity $46 $53 $53 $43 $33 $33

+ Total Investing Activity “$75 “$75 - - - -

+ Interest Expense - $5 $5 $5 $5 $5

Project Cash Flows

= “$29 “$17 +$58 +$48 +$38 +$38

Then discount these cash ¬‚ows, using the assumed 30% cost of capital on the overall ¬rm:

’$29 ’$17 +$58 +$48

NPV = + + +

1 + 30% (1 + 30%) (1 + 30%) (1 + 30%)4

2 3

(22.22)

+$38 +$38

+ + = .

$28.95

(1 + 30%) (1 + 30%)6

5

So, you would be willing to pay $28.95 today for the right to buy (and ¬nance) the ¬rm, which

will initiate next year with this exact capital structure. But wait: did you not forget about

the tax-shelter that came with the debt? No, you did not! The pro forma itself had already

incorporated the correct interest expense, which had reduced the corporate income tax, and

thus increased our project™s cash ¬‚ows.

Digging Deeper: I am using 30% as the post-tax cash ¬‚ow discount rate here, too. This ignores the fact that

the tax shelter cash ¬‚ows that we are adding to the ¬rm value have a lower discount rate. Fortunately, this is

usually a very small discrepancy in terms of the ¬rm™s overall cost of capital.

22·5.B. APV

Our second method to value our ¬rm is APV. But be careful: the cash ¬‚ows in Formula (22.22) Method 2, APV demands

a detour: we must

are not the cash ¬‚ows that you need for the APV analysis, because these are not the cash ¬‚ows

construct

as if 100% equity ¬nanced. APV states that you want to use the as if 100% equity ¬nanced cash as-if-100%-equity

¬‚ows, and then add back the tax shield. If you used the cash ¬‚ows in Formula 22.22 and then ¬nancials.

added the tax-shield (due to the interest payment designation), you would mistakenly count

the tax-shield twice. We must therefore start over to ¬nd the correct expected cash ¬‚ows. You

already know that if the ¬rm were fully equity ¬nanced, the tax obligation would go up. By

how much? You can intuit this even before you write down the full ¬nancials. In years 2“6,

the taxable net income would be $5 more, so at your 40% corporate income tax rate, you would

have to pay not $2, but $4 in taxes. This means that you would have to pay an extra $2 in

taxes each year. To make sure this intuition is right, construct the ¬nancials of a 100% equity

¬nanced ¬rm:

Abbreviated Income Statement, As If 100% Equity Financed

Year 1 2 3 4 5 6

= EBIT (Operating Income) $35 $10 $10 $35 $60 $60

“ Interest Expense $0 $0 $0 $0 $0 $0

= EAIBT (or EBT) $35 $10 $10 $35 $60 $60

“ Corporate Income Tax (at 40%) $14 $4 $4 $14 $24 $24

Net Income

= $21 $6 $6 $21 $36 $36

¬le=capctaxes.tex: LP

562 Chapter 22. Corporate Taxes and A Tax Advantage of Debt.

Abbreviated Cash Flow Statement, 100% Equity Financed

Year 1 2 3 4 5 6

Net Income $21 $6 $6 $21 $36 $36

+ Depreciation $25 $50 $50 $25 $0 $0

Total Operating Activity

= $46 $56 $56 $46 $36 $36

Capital Expenditures “$75 “$75 - - - -

Total Investing Activity

= “$75 “$75 - - - -

We can now reuse our present value cash ¬‚ow formula on the 100% equity ¬nanced version of

our ¬rm:

Computing Project Cash Flows, 100% Equity Financed

Year 1 2 3 4 5 6

Total Operating Activity $46 $56 $56 $46 $36 $36

+ Total Investing Activity “$75 “$75 - - - -

+ Interest Expense $0 $0 $0 $0 $0 $0

Project Cash Flows “$29 “$19 +$56 +$46 +$36 +$36

Comparing this to the equivalent table on Page 561, you can see that the project cash ¬‚ows in

your 100% equity ¬nanced ¬rm has indeed lost the tax shelter of $2 in each of years 2“6. The

intuition was correct!

Now discount these “as if 100% equity ¬nanced” cash ¬‚ows with the ¬rm™s appropriate cost of

Return to the main task:

APV valuation capital, which is assumed to be 30%. Standing at time 0, this gives you

’$29 ’$19 +$56 +$46

NPVProject, 100% Equity Financed = + + +

(1 + 30%) (1 + 30%)2 (1 + 30%)3 (1 + 30%)4

(22.23)

+$36 +$36

+ + = .

$25.20

(1 + 30%)5 (1 + 30%)6

The APV formula tells you that you now need to add back the expected tax shield from the debt.

The interest tax shields in years 2“6 are the interest payments ($5 per year) multiplied by the

corporate tax rate (40%), or $2 per year. What is the value of this tax shelter?

+$2 +$2 +$2

$0

NPVTax Shelter = + + +

(1 + 30%) (1 + 30%) (1 + 30%) (1 + 30%)4

2 3

(22.24)

+$2 +$2

+ + = .

$3.75

(1 + 30%) (1 + 30%)6

5

Therefore, the APV method tells you that the ¬rm value is

(22.25)

APVt=0 = $25.20 + $3.75 = $28.95 .

This is the same answer that you found in Formula 22.22.

¬le=capctaxes.tex: RP

563

Section 22·5. A Sample Application of Tax-Adjusting Valuation Techniques.

22·5.C. WACC

Our third method to value the ¬rm is WACC. Start again with the ¬rm™s cash ¬‚ows, as if 100% Method 3: WACC

equity ¬nanced.

Computing Project Cash Flows, 100% Equity Financed

Year 1 2 3 4 5 6

Project Cash Flows “$29 “$19 +$56 +$46 +$36 +$36