2. the fact that expenses that can be paid from before-tax income are better

than expenses that must be paid from after-tax income;

3. how to compute the average tax rate;

4. how to obtain the marginal tax rate;

5. the fact that capital gains enjoy preferential tax treatment;

6. why the average and marginal tax rates di¬er, and why the marginal tax

rate is usually higher than the average tax rate.

As already noted, you will later have to pay special attention to three facts: that corporations What you will learn later

about taxes.

and individuals can deduct certain interest expenses; that capital gains are taxed at lower tax

rates; and that some retirement account investment returns are tax-exempt. These features

of the tax code o¬er individuals and corporations opportunities to legally reduce their tax

obligations.

Solve Now!

For all questions here, assume that this investor has $1,000 in valid interest deductions.

Q 6.17 What are the average and marginal federal tax rates for a single individual earning

$5,000? Repeat for a corporation.

Q 6.18 What are the average and marginal federal tax rates for a single individual earning

$50,000? Repeat for a corporation.

Q 6.19 What are the average and marginal federal tax rates for a single individual earning

$50,000,000? Repeat for a corporation.

¬le=frictions.tex: LP

134 Chapter 6. Dealing With Imperfect Markets.

6·5. Working With Taxes

In one sense, taxes are very similar to transaction costs”they take a “cut,” making investments

less pro¬table. However, taxes are often orders of magnitude bigger and thus more important

than ordinary transaction costs and”except for illustrative examples”you should not simply

assume them away, which is quite di¬erent from what you can sometimes do with transaction

costs. (Ignoring taxes may be a good assumption for the tax-exempt Red Cross, but probably

not for you or for the ordinary corporation!) Another di¬erence between taxes and transaction

costs is that taxes are higher on pro¬table transactions, whereas plain transaction costs do not

care whether you made money or lost money. In addition, taxes often have many more nuances.

We now try to understand better how to work with income taxes.

6·5.A. Taxes in Rates of Returns

In the end, all you probably care about are your after-tax returns, not your pre-tax returns.

Taxable Investors

(unlike tax-exempt It should not matter whether you receive $100 that has to be taxed at 50% or whether you

investors) care about

receive $50 that does not have to be taxed. This leads to a recommendation analogous to that

post-tax in¬‚ows and

for transaction costs”work only in after-tax money. For example, say you invest $100,000 in

out¬‚ows.

after-tax money to earn a return of $160,000. Your marginal tax rate is 25%. Taxes are on the

net return of $60,000, so your after-tax net return is

75% · $60, 000 = $45, 000

(6.9)

(1 ’ „) · Before-Tax Net Return = After-Tax Net Return .

(The tax rate is often abbreviated with the Greek letter „, tau.) In addition, you will receive your

original investment back, so your after-tax rate of return is

$145, 000 ’ $100, 000

rAfter Tax = = 45% . (6.10)

$100, 000

6·5.B. Tax-Exempt Bonds and the Marginal Investor

In the United States, there are bonds that are issued by governmental entities, whose interest

Municipal bonds™

interest payments are payments are legally tax-exempt”the reasoning of the federal government being that it does

legally exempt from

not want to burden states™ or local governments™ e¬orts to raise money. If you own one of these

income taxes.

bonds, you do not need to declare the interest on your federal income tax forms, and sometimes

not even on your state™s income tax form, either. (The arrangement di¬ers from bond to bond.)

The most prominent tax-exempt bonds are called municipal bonds or muni bonds or even

munis for short. As their name suggests, they are usually issued by municipalities such as the

City of Los Angeles (CA) or the City of Canton (OH).

On May 31, 2002, the Wall Street Journal reported on Page C12 that tax-exempt municipal 7-12

The May 2002 Situation.

year highly rated bonds (AA) o¬ered an annualized interest rate of 5.24%. Bonds of similar risk

issued by corporations o¬ered an interest rate of about 6.76%. Which one would be a better

investment for you? Well, it depends.

If you invested $1,000 into munis at a 5.24% interest rate, you would receive $52.40 at year™s

Comparing After-Tax

Returns of Tax-Exempt end. You would get to keep all of it, because these bonds are tax-exempt. If you invested

and Taxable Bonds.

$1,000 in taxable bonds at a 6.76% interest rate, you would receive $67.60 at year™s end. If

your income tax rate is 0%, you would clearly prefer the $67.60 to the $52.40. However, if your

marginal income tax rate is 30%, Uncle Sam would collect $20.28 and leave you with $47.32.

Your after-tax rate of return is

rpost-tax = (1 ’ 30%) · 6.76% = 70% · 6.76% ≈ 4.73%

(6.11)

rpost-tax = (1 ’ „) · rpre-tax .

¬le=frictions.tex: RP

135

Section 6·5. Working With Taxes.

With a 30% tax rate, you would prefer the tax-exempt bond that pays $52.40.

Economists sometimes like to talk about a hypothetical marginal investor. This is an investor High-income tax bracket

individuals should prefer

whose marginal income tax rate is such that she would be exactly indi¬erent between buying

tax-exempt bonds;

the tax-exempt and the taxable bond. Using Formula 6.11, the marginal investor has a tax rate low-income tax bracket

of individuals should prefer

5.24% taxable bonds.

5.24% = (1 ’ „marginal ) · 6.76% „marginal = 1 ’ ≈ 22.5%

”

6.76%

(6.12)

rpost-tax

rpost-tax = (1 ’ „marginal ) · rpre-tax = 1’

„marginal .

”

rpre-tax

Any investor with a marginal income tax rate above 22.5% (such as a high-income retail investor)

should prefer the tax-exempt bond. Any investor with a marginal income tax rate below this

income tax rate (such as a tax-exempt pension fund investor) should prefer the taxable bond.

Unfortunately, unlike the U.S. Treasury, municipalities can and have gone bankrupt, so that Munis do have default

(credit) risk. See next

they may not fully repay. (The most prominent recent default was the Orange County (CA)

chapter.

default in December 1994.) Municipal bonds are not an entirely risk-free investment.

Solve Now!

Q 6.20 On May 31, 2002, for short-term bonds, the Bond Market Data Bank Section in the Wall

Street Journal (Page C15) indicates the ratio between the equivalent yields of AAA municipal and

Treasury securities to be around 74.6%. What is the marginal investor™s tax rate?

Q 6.21 On May 31, 2002, for long-term bonds, the Bond Market Data Bank Section in the Wall

Street Journal (Page C15) computes the ratio between the AAA municipal and Treasury securities

to be around 92% (for short-term municipal bonds). What is the marginal investor™s tax rate?

6·5.C. Taxes in NPV

Again, as with transaction costs, you should take care to work only with cash in the same units” Compute everything in

After-Tax Dollars!

here, this means cash that you can use for consumption. Again, it should not matter whether

you receive $100 that has to be taxed at 50% or whether you receive $50 that does not have to

be taxed. As far as NPV is concerned, everything should be computed in after-tax dollars. This

includes all cash ¬‚ows, whether today or tomorrow, whether cash in¬‚ows or out¬‚ows.

Important: Do all NPV calculations in after-tax money.

Unfortunately, you cannot simply discount pre-tax cash ¬‚ows with the pre-tax cost of capital You must compute the

after-tax opportunity

(wrong!) and expect to come up with the same result as when you discount after-tax cash ¬‚ows

cost of capital.

with after-tax costs of capital (right!).

For example, consider a project that costs $10,000 and returns $13,000 next year. Your tax An example”how to pick

your opportunity cost of

rate is 40%, and 1-year equivalently risky bonds return 25% if their income is taxable, and 10%

capital.

if their income is not taxable. First, you must decide what your opportunity cost of capital is.

Section 6·5.B tells you that if you put $100 into taxables, you will own $125, but the IRS will

con¬scate ($125’$100)·40% = $10. You will thus own $115 in after-tax income. Tax-exempts

grow only to $110, so you prefer the taxable bond”it is the taxable bond that determines your

opportunity cost of capital. Your equivalent after-tax rate of return is therefore 15%. This 15%

is your after-tax “opportunity” cost of capital”it is your best use of capital elsewhere.

¬le=frictions.tex: LP

136 Chapter 6. Dealing With Imperfect Markets.

Return to your $10,000 project now. You know that your taxable project returns 30% taxable

You must apply it to the

after-tax expected cash ($3,000), while taxable bonds return 25% ($2,500), so NPV should tell you to take this project.

¬‚ows.

Uncle Sam will con¬scate 40%·$3, 000 = $1, 200, leaving you with $11,800. Therefore, the NPV

of your project is

$11, 800

NPV = ’$10, 000 + = $260.87

1 + 15%

(6.13)

E (CF1 )

NPV = + .

CF0

1 + E (r0,1 )

It makes intuitive sense: if you had invested money into the bonds, you would have ended up

with $11,500. Instead, you will end up with $11,800, the $300 di¬erence occurring next year.

Discounted, the $261 seems intuitively correct. Of course, there is an in¬nite number of ways

of getting incorrect solutions, but recognize that none of the following calculations that use

the pre-tax expected cash ¬‚ows (and try di¬erent discount rates) give the same correct result:

$13, 000

NPV ≠ ’$10, 000 + = $400

1 + 25%

$13, 000 (6.14)

NPV ≠ ’$10, 000 + = $1, 304.35

1 + 15%

$13, 000

NPV ≠ ’$10, 000 + = $1, 818.18

1 + 10%

You have no choice: you cannot work with pre-tax expected cash ¬‚ows. Instead, you need to

go through the exercise of carefully computing after-tax cash ¬‚ows and discounting with your

after-tax opportunity cost of capital.

You know that computing after-tax cash ¬‚ows is a pain. Can you at least compare two equally

Can you compare two

projects based on taxable projects in terms of their pre-tax NPV? If one project is better than the other in pre-

pre-tax NPV?

tax terms, is it also better in after-tax terms? If yes, then you could at least do relative capital

budgeting with pre-tax project cash ¬‚ows. This may or may not work, and here is why. Compare

project SAFE that costs $1,000 and will provide $1,500 this evening; and project UNSAFE that

costs $1,000 and will provide either $500 or $2,500 this evening with equal probability. The

expected payout is the same, and the cost of capital is practically 0% for 1 day. If you are in the

20% marginal tax bracket, project SAFE will leave the IRS with 20% · ($1, 500 ’ $1, 000) = $100,

and you with +$400 in after-tax net return. Project UNSAFE will either give you $1,500 or “$500

in taxable earnings.

• If you can use the losses to o¬set other gains elsewhere, then you would either send

$1, 500 · 20% = $300 extra to the IRS; or you would send $100 less to the IRS (because

your taxable pro¬ts elsewhere would be reduced). In this case, project SAFE and UNSAFE

would have the same expected tax costs and after-tax cash ¬‚ows.

• If you drop into a di¬erent tax bracket beyond an additional net income of $1000, say 25%,

then project UNSAFE becomes less desirable than project SAFE. For the $1,500 income,

the ¬rst $500 would still cost you $100 in tax, but the remaining $1,000 would cost you

$250. Thus, your project™s marginal tax obligation would be either $350 or ’$100, for an

expected tax burden of $125. (The same logic applies if your losses would make you fall

into a lower tax bracket”the UNSAFE project would become less desirable.)

• If you have no gains elsewhere to use your project tax loss against, then the UNSAFE

project would again be worth less. Corporations can ask for a tax refund on old gains, so

this factor is less binding than it is for individuals, who may have to carry the capital loss

forward until they have su¬cient income again to use it”if ever.

Thus, whether you can compare projects on a pre-tax basis depends on whether you have

perfect symmetry in the applicable marginal tax rates across projects. If you do, then the

project that is more pro¬table in after-tax terms is also more pro¬table in pre-tax terms. This

would allow you to simply compare projects by their pre-tax NPVs. If gains and losses face

di¬erent taxation”either because of tax bracket changes or because of your inability to use

the tax losses elsewhere”then you cannot simply choose the project with the higher pre-tax

NPV. You will have to go through the entire after-tax NPV calculations and compare these.

¬le=frictions.tex: RP

137

Section 6·5. Working With Taxes.

Important: You can only compare projects on a before-tax NPV basis if the tax

treatment is absolutely symmetric. This requires consideration of your overall tax

situation.

You now know how to discount projects in the presence of income taxes. However, you do not WACC and APV

unfortunately have to

yet know how to compute the proper discount rate for projects that are ¬nanced by debt and

wait.

equity, because debt and equity face di¬erent tax consequences. Unfortunately, you will have

to wait until Chapter 22 before we can do a good job discussing the two suitable methods”

called APV and WACC”to handle di¬erential taxation by ¬nancing. Until we will have covered

investments in Part III, you just do not have all the necessary pieces, and your goal must be to

understand formulas, rather than just eat them.

Solve Now!

Q 6.22 You have a project that costs $50,000 and will return $80,000 in three years. Your

marginal tax rate is 37.5%. Treasuries pay a rate of return of 8% per year, munis pay a rate of

return of 3% per year. What is the NPV of your project?

6·5.D. Tax Timing

In many situations, the IRS does not allow reinvestment of funds generated by a project without Do not forget that even

when in¬‚ows require

an interim tax penalty. This can be important when you compare one long-term investment to

after-tax dollars,

multiple short-term investments that are otherwise identical. For example, consider a farmer sometimes out¬‚ows are

in the 40% tax bracket who purchases grain that costs $300, and that triples its value every taxed again.

year.

• If the IRS considers this farm to be one long-term two-year project, the farmer can use the

¬rst harvest to reseed, so $300 seed turns into $900 in one year and then into a $2,700

harvest in two years. Uncle Sam considers the pro¬t to be $2,400 and so collects taxes of

$960. The farmer is left with post-tax pro¬ts of $1,440.

• If the IRS considers this production to be two consecutive one-year projects, then the

farmer ends up with $900 at the end of the ¬rst year. Uncle Sam collects 40%·$600 =

$240, leaving the farmer with $660. Replanted, the $660 grows to $1,980, of which

the IRS collects another 40%·$1, 980 = $792. The farmer is left with post-tax pro¬ts

of 60%·$1, 980 = $1, 188.

The discrepancy between $1,440 and $1,188 is due to the fact that the long-term project can

avoid the interim taxation. Similar issues arise whenever an expense can be reclassi¬ed from

“reinvested pro¬ts” (taxed, if not with some credit at reinvestment time) into “necessary main-

tenance.”

Although you should always get taxes right”and really know the details of the tax situation

that applies to you”be aware that you must particularly pay attention to getting taxes right if

you are planning to undertake real estate transactions. These have special tax exemptions and

tax depreciation writeo¬s that are essential to getting the project valuation right.

Solve Now!

Q 6.23 It is not uncommon for individuals to forget about taxes, especially when investments

are small and payo¬s are large but rare. Presume you are in the 30% tax bracket. Is the NPV of

a $1 lottery ticket that pays o¬ taxable winnings of $10 million with a chance of 1 in 9 million

positive or negative? How would it change if you could purchase the lottery ticket with pre-tax

money?

¬le=frictions.tex: LP

138 Chapter 6. Dealing With Imperfect Markets.

6·6. In¬‚ation

We have now discussed all violations from the assumptions necessary for our perfect market

Back to our perfect

markets assumptions. Utopia. So, what are we doing now? If you return to our perfect markets assumptions, you will

see that “no in¬‚ation” was not among them. In¬‚ation is the process by which goods cost more

in the future than they cost today”in which the price level is rising and money is losing its

value.

So, in¬‚ation is actually not a market imperfection per se. If today we quoted everything in

Known in¬‚ation

applicable everywhere is dollars, and tomorrow we quote everything in cents”so that an apple that cost 1 currency

irrelevant.

unit today will cost 100 currency units tomorrow, an in¬‚ation of 10,000%”would it make

any di¬erence? Not really. The apple would still cost the same in terms of foregone other

opportunities, whether it is 1 dollar or 100 cents.

However, we have made a big assumption here”in¬‚ation applied equally to everything, and

...but in¬‚ation is often

not applicable especially applied equally to all contracts across time. See, if you had contracted to deliver

everywhere.

apples at 1 currency unit tomorrow, whatever currency units may be, you could be in big

trouble”you would have promised to sell your apples at 1 cent (1 currency unit) instead of $1.

Most ¬nancial contracts are denominated in such “nominal” terms”that is, in plain currency

units”so in¬‚ation would matter. Of course, in¬‚ation would not be much of a concern for a

¬nancial contract that would be “in¬‚ation-indexed.”

What e¬ect does in¬‚ation have on returns? On (net) present values? This is the subject of this

Our agenda.

section. As before, we start with interest rates and then proceed to net present values.

6·6.A. De¬ning the In¬‚ation Rate

The ¬rst important question is how you should de¬ne in¬‚ation. Is the rate of change of the

The CPI is the most

common in¬‚ation price of apples the best measure of in¬‚ation? What if apples (the fruit) become more expensive,

measure.

but Apples (the computers) become less expensive? De¬ning in¬‚ation is somewhat tricky. To

solve this problem, economists have invented baskets or bundles of goods that are deemed

to be representative, for which they can then measure an average price change. The o¬cial

source of most in¬‚ation measures is the Bureau of Labor Statistics (B.L.S.), which determines

the compositions of a number of prominent bundles (indexes), and publishes the average total

price of these bundles on a monthly basis. The most prominent such in¬‚ation measure is a

hypothetical bundle of average household consumption, called the Consumer Price Index (or

CPI). (The CPI components are roughly: housing 40%, food 20%, transportation 15%, medical

care 10%, clothing 5%, entertainment 5%, others 5%.) The Wall Street Journal prints the percent

change in the CPI at the end of its column Money Rates. (On May 31, 2002, the Consumer Price

Index was increasing at a rate of 1.6%/year.) A number of other indexes are also in common

use as in¬‚ation measures, such as the Producer Price Index (PPI) or the broader GDP De¬‚ator.

They typically move fairly similarly to the CPI. There are also more specialized bundles, such

as computer in¬‚ation indexes (the price of equivalent computer power does not in¬‚ate, but

de¬‚ate, so the rate is usually negative), or indexes for prices of goods purchased in a particular

region.

Anecdote: The German Hyperin¬‚ation of 1922

The most famous episode of hyperin¬‚ation occurred in Germany from August 1922 to November 1923. Prices

more than quadrupled every month. The price for goods was higher in the evening than in the morning! Stamps

had to be overprinted by the day, and shoppers went out with bags of money”that were worthless at the end

of the day. By the time Germany printed 1,000 billion Mark Bank Notes, no one trusted the currency anymore.

This hyperin¬‚ation was stopped only by a drastic currency and ¬nancial system reform. But high in¬‚ation is

not just a historic artifact. For example, many Latin American countries experienced annual doubling of prices

in the early 1980s.

The opposite of in¬‚ation is de¬‚ation (negative in¬‚ation)”a process in which the price level falls. Though much

rarer, it happens. In fact, in November 2002, Business Week reported that an ongoing recession and low demand

continue to force an ongoing decline in Japanese prices.

Many economists now believe that a modest in¬‚ation rate between 1% and 3% per year is a healthy number.

¬le=frictions.tex: RP

139

Section 6·6. In¬‚ation.

The o¬cial in¬‚ation rate is not just a number”it is important in itself, because many contracts The CPI matters”even if

it is wrong.

are rate-indexed. For example, even if actual in¬‚ation is zero, if the o¬cially reported CPI

rate is positive, the government must pay out more to social security recipients. The lower the

o¬cial in¬‚ation rate, the less the government has to pay. You would therefore think that the

government has the incentive to understate in¬‚ation. But strangely, this has not been the case.

On the contrary, there are strong political interest groups that hinder the B.L.S. from even just

improving on mistakes in the CPI because it would result in lower o¬cial in¬‚ation numbers.

In 1996, the Boskin Commission, consisting of a number of eminent economists, found that

the CPI overstates in¬‚ation by about 74 basis points per annum”a huge di¬erence. The main

reasons are that the B.L.S. has been tardy in recognizing the growing importance of such factors

as computer and telecommunication e¬ective price declines, and the role of superstores such

as Wal-Mart.

One ¬nal warning:

Important: The common statement “in today™s dollars” is ambiguous. Some

people mean “in¬‚ation adjusted.” Other people mean present values (i.e., “com-

pared to an investment in risk-free bonds”). When in doubt, ask!

Solve Now!

Q 6.24 Using information from a current newspaper or the WWW, ¬nd out what the current

in¬‚ation rate is.

6·6.B. Real and Nominal Interest Rates

To work around in¬‚ation, you ¬rst need to learn the di¬erence between a nominal return and Nominal is what is

normally quoted. Real is

a real return. The nominal return is what everyone usually quotes”a return that has not been

what you want to know.

adjusted for in¬‚ation. In contrast, the real return somehow “takes out” in¬‚ation from the

nominal return in order to calculate a return “as if” there had been no price in¬‚ation to begin

with. It is the real return which re¬‚ects the fact that, in the presence of in¬‚ation, a dollar in

the future will have less purchasing power than a dollar today. It is the real rate of return

that measures your trade-o¬ between present and future consumption, taking into account the

change in prices.

Consider a simple no-uncertainty scenario: assume that the in¬‚ation rate is 100% per year, An Extreme 100%

In¬‚ation Rate Example:

and you can buy a bond that promises a nominal interest rate of 700% (the bond payout is

Prices Double Every

quadruple your pay-in). What is your real rate of return? To ¬nd out, assume that $1 buys one Year.

apple today. With an in¬‚ation rate of 100%, you need $2 next year to buy the same apple. Your

investment return will be $1 · (1 + 700%) = $8 for today™s $1 of investment. But this $8 now

applies to apples costing $2 each. So, your $8 will buy 4 apples, and not 8 apples. Your real

rate of return is

4 Apples ’ 1 Apples

rreal = = 300% (6.15)

.

1 Apples

For each dollar invested today, you will be able to purchase only 300 percent more apples next

year (not 700% more apples) than you could purchase today. This is because the purchasing

power of your dollar next year will be reduced by half.

The correct formula to adjust for in¬‚ation is again a “one-plus” type formula. In our example, The Conversion Formula

from Nominal to Real

it is

Rates.

(1 + 700%) = (1 + 100%) · (1 + 300%)

(6.16)

(1 + rnominal ) = (1 + In¬‚ation Rate) · (1 + rreal ) .

¬le=frictions.tex: LP

140 Chapter 6. Dealing With Imperfect Markets.

Turning this formula around solves for real rates of return,

1 + 700%

(1 + rreal ) = = 1 + 300%

1 + 100%

(6.17)

(1 + rnominal )

(1 + rreal ) = .

(1 + In¬‚ation Rate)

Important: The relation between nominal rates of return (rnominal ), real rates

of returns (rreal ), and in¬‚ation (π ) is

(6.18)

(1 + rnominal ) = (1 + rreal ) · (1 + π ) .

As with compounding, if both in¬‚ation and the nominal interest rate are small, the mistake of

For small rates,

adding/subtracting is ok. just subtracting the in¬‚ation rate from the nominal interest rate to obtain the real interest rate

is not too grave. The di¬erence is a cross-term (see Page 21),

rreal = rnominal ’ π ’ rreal ·π . (6.19)

cross-term

For example, as of mid-2004, the o¬cial CPI in¬‚ation rate ¬‚uctuated month-to-month from

about 2.5% to 3% per annum. The 10-year Treasury bond paid 4% per annum on October 31,

2004. Therefore, if you believe that the in¬‚ation rate will remain at 2.5% per annum, you would

presume a real rate of return of about 4% ’ 2.5% ≈ 1.5%”though this would ignore the cross-

term. The more accurate computation would be (1+4%)/(1+2.5%)’1 ≈ 1.46%. The cross-term

di¬erence of 4 basis points is swamped by your uncertainty about the future in¬‚ation rate”at

least as of 2004. However, when in¬‚ation and interest rates are high”as they were, e.g., in the

late nineteen-seventies”then the cross-term can make quite a meaningful di¬erence.

A positive time-value of money”the fact that money tomorrow is worth more than money

Real interest rates can

be negative. today”is only true for nominal quantities, not for real quantities. Only nominal interest rates

are never negative. In the presence of in¬‚ation, real interest rates not only can be negative, but

often have been negative. In such situations, by saving money, you would have ended up with

more money”but with less purchasing power, not more purchasing power. Of course, if there

are goods or project that appreciate with in¬‚ation (in¬‚ation hedges, such as real estate or gold),

and to the extent that these goods are both storable and traded in a perfect market, you would

not expect to see negative real rates of return. After all, you could buy these projects today

and sell them next year, and thereby earn a real rate of return that is positive.

Solve Now!

Q 6.25 Using information from a current newspaper or the WWW, ¬nd out what the annualized

current 30-day nominal interest rate is.

Q 6.26 Using the information from the previous two questions, determine the annualized current

real interest rate?

Q 6.27 From memory, write down the relationship between nominal rates of return (rnominal ),

real rates of return (rr ), and the in¬‚ation rate (π ).

Q 6.28 The nominal interest rate is 20%. In¬‚ation is 5%. What is the real interest rate?

¬le=frictions.tex: RP

141

Section 6·6. In¬‚ation.

Q 6.29 The in¬‚ation rate is 1.5% per year. The real rate of return is 2% per year. A perpetuity

project that payed $100 this year will provide income that grows by the in¬‚ation rate. Show

what this project is truly worth. Do this in both nominal and real terms. (Be clear on what never

to do.)

6·6.C. Handling In¬‚ation in Net Present Value

When it comes to in¬‚ation and net present value, there is a simple rule: never mix apples and The most fundamental

rule is to never mix

oranges. The beauty of NPV is that every project, every action is translated into the same units:

apples and oranges.

today™s dollars. Keep everything in the same units in the presence of in¬‚ation, so that this NPV Nominal cash ¬‚ows must

advantage is not lost. When you use the NPV formula, always discount nominal cash ¬‚ows with be discounted with

nominal interest rates.

nominal costs of capital, and real (in¬‚ation-adjusted) cash ¬‚ows with real (in¬‚ation-adjusted)

costs of capital.

Let™s show this. Return to our “apple” example. With 700% nominal interest rates and 100% A Previous Example

Revisited.

in¬‚ation, the real interest rate is (1 + 700%)/(1 + 100%) ’ 1 = 300%. What is the value of a

project that gives 12 apples next year, given that apples cost $1 each today and $2 each next

year? There are two methods you can use.

1. Discount the nominal value of 12 apples next year ($2·12 = $24) with the nominal interest

rate. Thus, the 12 future apples are worth

Nominal Cash Flow1 $24

= = $3 . (6.20)

1 + nominal rate0,1 1 + 700%

2. Discount real cash ¬‚ows (i.e., 12A) with the real interest rate. Thus, the 12 future apples

are worth

Real Cash Flow1 12A

= = 3A , (6.21)

1 + real rate0,1 1 + 300%

in today™s apples. Because an apple costs $1 today, the eight apples are worth $3.

Both methods arrive at the same result. The opportunity cost of capital is that if you invest

one apple today, you can quadruple your apple holdings by next year. Thus, a 12 apple harvest

next year is worth 3 apples to you today. The higher nominal interest rates already re¬‚ect the

fact that nominal cash ¬‚ows next year are worth less than they are this year.

Important:

• Discount nominal cash ¬‚ows with nominal interest rates.

• Discount real cash ¬‚ows with real interest rates.

Either works. Never discount nominal cash ¬‚ows with real interest rates, or vice-

versa.

If you want to see this in algebra, the reason that the two methods come to the same result is Usually, use nominal

interest rates.

that the in¬‚ation rate cancels out,

12A · (1 + 100%)

$24 12A

PV = = =

1 + 700% 1 + 300% (1 + 300%) · (1 + 100%)

(6.22)

R · (1 + π )

N R

= = = ,

1+n 1+r (1 + r ) · (1 + π )

where N is the nominal cash ¬‚ow, n the nominal interest rate, R the real cash ¬‚ow, r the real

interest rate, and π the in¬‚ation rate. Most of the time, it is easier to work in nominal quantities.

¬le=frictions.tex: LP

142 Chapter 6. Dealing With Imperfect Markets.

Nominal interest rates are far more common than real interest rates, and you can simply use

published in¬‚ation rates to adjust the future price of goods to obtain future expected nominal

cash ¬‚ows.

Solve Now!

Q 6.30 If the real interest is 3% per annum, the in¬‚ation rate is 8% per annum, then what is the

value of a $500,000 payment next year?

Q 6.31 If the real interest is 3% per annum, the in¬‚ation rate is 8% per annum, then what is the

value of a $500,000 payment every year forever?

Q 6.32 In¬‚ation is 2% per year, the interest rate is 8% per year. Our perpetuity project has cash

¬‚ows that grow at 1% faster than in¬‚ation forever, starting with $20 next year.

(a) What is the real interest rate?

(b) What is the project PV?

(c) What would you get if you grew a perpetuity project of $20 by the real growth rate of 1%,

and then discounted at the nominal cost of capital?

(d) What would you get if you grew a perpetuity project of $20 by the nominal growth rate of

3%, and then discounted at the real cost of capital?

Doing either of the latter two calculation is not an uncommon mistake.

Q 6.33 You must value a perpetual lease. It will cost $100,000 each year in real terms”that

is, its proceeds will not grow in real terms, but just contractually keep pace with in¬‚ation. The

prevailing interest rate is 8% per year, the in¬‚ation rate is 2% per year forever. The ¬rst cash

¬‚ow of your project next year is $100,000 quoted in today™s real dollars. What is the PV of the

project? (Warning: watch the timing and amount of your ¬rst payment.)

6·6.D. Interest Rates and In¬‚ation Expectations

Nominal Interest Rate Levels

Should you take in¬‚ation into account? Absolutely. As an investor, like the market overall, you

In¬‚ation affects the level

of the nominal interest probably care more about real returns than nominal rates. Therefore, when purchasing ¬nancial

rate.

investments, you must form an expectation of how this investment will a¬ect your purchasing

power. For example, if the nominal interest rate is 5%, you may prefer spending more money

today if you believe the in¬‚ation rate to be 10% than if you believe it to be only 6%. Of course,

if you have no better alternatives, you might still want to save money even if your real rate of

return is negative. Be this as it may, you would expect nominal interest rates in the economy

to be higher when in¬‚ation is higher. This also means that you would expect nominal rates

to go up when in¬‚ation rate expectations are going up. Similarly, you would expect nominal

rates to go down when in¬‚ation rate expectations are going down. Now, many investors also

believe that stocks are good in¬‚ation hedges, in that they appreciate automatically in value when

the in¬‚ation rate increases”after all, they are just claims on real projects, which presumably

similarly experience a price increase. In the end, the exact real interest rates in the economy

are determined by the demand and supply for capital, which is determined by these kinds of

considerations.

¬le=frictions.tex: RP

143

Section 6·6. In¬‚ation.

TIPS and Short-Term Bonds as “In¬‚ation Hedges”

But what if you wanted to purchase a bond that is truly risk-free, i.e., a bond that promises a In¬‚ation is uncertainty

speci¬ed amount of purchasing power (a real amount, not a nominal amount)? The problem

is that you do not yet know fully what future in¬‚ation will be. In¬‚ation is a random variable,

because you do not yet know what in¬‚ation will be over the bond™s holding period. You can

estimate it, but you do not really know.

What you want is a bond that pays out 1% more in interest if in¬‚ation were to turn out 1% In¬‚ation-Adjusted

Treasury Bonds [TIPS].

higher. In 1997, the U.S. Treasury reintroduced such in¬‚ation-adjusted bonds. They are called

Treasury In¬‚ation Protected Securities (or TIPS, or sometimes just CPI Bonds).

In late October 2004, the 10-year T-bond o¬ered 4.02% per annum, while the 10-year TIPS An example: the October

2004 situation.

o¬ered 1.6% per annum. If in¬‚ation turns out to be above 2.38% per annum over the 10-year

interval, then the TIPS will have been the better purchase. If in¬‚ation turns out to be lower

than 2.56% per annum, then the plain T-bond will have been the better purchase. A volatile

oil price in 2004 had caused the in¬‚ation rate to ¬‚uctuate dramatically”it troughed at 1.69%

in March 2004 and peaked at 3.27% in June 2004. Ladies and Gentlemen”place your bet.

TIPS are not the only way you can reduce your worry about future in¬‚ation. Short-term bonds Short-term securities

also help you “hedge”

are another way to reduce the e¬ect of future in¬‚ation. In¬‚ation increases are associated with

against in¬‚ation.

higher interest rates. Thus, an in¬‚ation increase would allow a short-term bond investor to

earn a higher interest rate upon reinvestment.

Does Future In¬‚ation Drive the Yield Curve Slope?

Now let us return to our question about what determines the slope of the ordinary Treasury It is harder to see why

the expectation of

yield curve. Recall from Page 65 that you might demand a higher long-term interest rate if

in¬‚ation would affect

you believed that future in¬‚ation will increase. For example, if you believe that in¬‚ation will the slope of the yield

be much higher from year 5 to year 10, you would be less inclined to accept the same 5% per curve.

annum for the 10-year Treasury bond that you might accept for the 5-year Treasury bond. After

all, what you will end up getting back from your 10-year bond will be worth much less to you!

You could also demand extra compensation if you were less certain about in¬‚ation from 5-years

out to 10-years out than about in¬‚ation from now to 5-years out. Fortunately, you can now put

this to the test using TIPS. In October 30, 2004, the yield curves was as follows”with implied

in¬‚ation rates computed for you:

Ordinary Implied

T-Bonds TIPS In¬‚ation

3-month 1.90% (n/a)

5-year 3.29% 0.90% 2.4%

10-year 4.02% 1.60% 2.4%

30-year 4.79% 2.06% 2.7%

Remember that the TIPS returns are una¬ected by in¬‚ation, so neither your expectation nor

your uncertainty about future in¬‚ation should in¬‚uence the TIPS yield curve”and yet it is

almost as steep as the ordinary yield curve. The 5-year and 10-year T-bond vs. TIPS interest

spread even embody the same in¬‚ation expectation of 2.4% per annum. The yield di¬erence

between the 5-year and the 30-year T-bond is about 1.5%, similar to the 1.2% di¬erence between

Anecdote: In¬‚ation-Adjusting Bonds

As it turns out, in¬‚ation-adjusted bonds had already been invented once before! The world™s ¬rst known

in¬‚ation-indexed bonds were issued by the Commonwealth of Massachusetts in 1780 during the Revolutionary

War. These bonds were invented to deal with severe wartime in¬‚ation and discontent among soldiers in the

U.S. Army with the decline in purchasing power of their pay. Although the bonds were successful, the concept

of indexed bonds was abandoned after the immediate extreme in¬‚ationary environment passed, and largely

forgotten. In 1780, the bonds were viewed as at best only an irregular expedient, since there was no formulated

economic theory to justify indexation.

Source: Robert Shiller.

¬le=frictions.tex: LP

144 Chapter 6. Dealing With Imperfect Markets.

the 5-year and the 30-year TIPS. So, in¬‚ation uncertainty can account for only a small fraction

of the steepness of this yield curve. There must be something other than in¬‚ation that makes

investors prefer shorter-term T-bonds to longer-term T-bonds and borrowers prefer longer-

term T-bonds to shorter-term T-bonds by so much that they are willing to agree on several

hundred basis points less compensation per annum on the short-term rate. Of course, it may

be that the horizon-dependent expectations or uncertainties about in¬‚ation will play a more

important role in the future”but in October 2004, they just did not.

Solve Now!

Q 6.34 On May 31, 2002, the Wall Street Journal reported on Page C10 that a 30-year CPI bond

o¬ered a real yield of about 3.375%/year. The current in¬‚ation rate was only 1.6%/year, and

a normal 30-year Treasury bond o¬ered a nominal yield of 5.6%/year. Under what scenario

would you be better o¬ buying one or the other?

6·7. Multiple Effects

Of course, in the messy real world, you can su¬er in¬‚ation, transaction costs, imperfect markets,

and taxes all at once, not just in isolation. In fact, there are so many possible real-world

problems that no one can possibly give you a formula for each one. Thus, it is more important

that you realize you must approach the real world thinking about two issues.

1. To what extent is the assumption of a perfect market appropriate? For example, in the

case of large and possibly tax-exempt companies, you may consider it reasonable to get

away with assuming a perfect market, thinking about the direction in which market im-

perfections would push you, and judging the magnitude thereof. This can often give a

reasonable answer without enormous complications that a perfect answer would require.

2. How can you handle a new situation in which you face particular sets of market imper-

fections? To answer such new thorny questions, you should internalize the method of

“thinking by numerical example.” You really need to become able to work out formulas

for yourself when you need them.

6·7.A. How to Work Problems You Have Not Encountered

For example, let™s see how you could approach a situation with both taxes and in¬‚ation. Always

Taxes and In¬‚ation:

Interactions? start by making up some numbers you ¬nd easy to work with. Let™s say you are considering an

investment of $100. Further, assume you will earn a 10% rate of return on your $100 investment

and Uncle Sam will take „ = 40% (or $4 on your $10). Therefore, you get $110 before taxes but

end up with only $106 in nominal terms. What you have just calculated is

(6.23)

$100 · [1 + 10% · (1 ’ 40%)] = $106 .

Translate this into an algebraic formula,

$100 · [1 + 10% · (1 ’ 40%)] = $106 .

(6.24)

CF0 · 1 + rnominal,pre-tax · (1 ’ „) = .

CF1

Now you need to determine what your $106 is really worth, so you must introduce in¬‚ation.

Pick some round number, say, a rate of π = 5% per annum. Consequently, $106 is worth in

purchasing power

$106

= $100.95

1 + 5%

(6.25)

CF1

= V0 .

1+π

¬le=frictions.tex: RP

145

Section 6·7. Multiple E¬ects.

So, your post-tax post-in¬‚ation real rate of return is $100.95/$100’1 ≈ 0.95%. Again, knowing

the numerical result, you need to translate your numbers into a formula. You computed

$100·[1+10%·(1’40%)]

’ $100

$100.95 ’ $100 1+5%

rpost-tax, real = =

$100 $100

10% · (1 ’ 40%) ’ 5%

= = 0.95%

1 + 5%

(6.26)

CF0 ·[1+rnominal,pre-tax ·(1’„)]

’ CF0

V0 ’ CF0 1+π

rpost-tax, real = =

CF0 CF0

rnominal,pre-tax · (1 ’ „) ’ π

= .

1+π

This is, of course, not a formula that anyone remembers. However, it is both useful and a nice

illustration of how you should approach and simplify complex questions”numerical example

¬rst, formula second.

6·7.B. Taxes on Nominal Returns?

Here is an interesting question: if the real rate remains constant, does it help or hurt an investor If the real rate stays

constant, does in¬‚ation

if in¬‚ation goes up? Let™s assume that the real rate of return is a constant 20%. If in¬‚ation is 50%,

hurt an investor? Yes, if

then the nominal rate of return is 80% (because (1 + 50%) · (1 + 20%) = 1 + 80%): you get $180 there are taxes!

for a $100 investment. Now add income taxes to the tune of 40%. The IRS sees $80 in interest,

taxes $32, and leaves you with $48. Your $148 will thus be worth $148/(1 + 50%) = $98.67 in

real value. Instead of a 20% increase in real purchasing power when you save money, you now

su¬er a $98.67/$100 ’ 1 ≈ 1.3% decrease in real purchasing power. Despite a high real interest

rate, Uncle Sam ended up with more, and you ended up with less purchasing power than you

started with. The reason is that although Uncle Sam claims to tax only interest gains, because

the interest tax is on nominal interest payments, you can actually lose in real terms. Contrast

this with the same scenario without in¬‚ation. In this case, if the real rate of return were still

20%, you would have been promised $20, Uncle Sam would have taxed you $8, and you could

have kept $112 in real value.

Important: Higher in¬‚ation rates hurt taxable investors who earn interest in-

come, even if real interest rates seem to remain constant. This is because the IRS

taxes nominal returns, not real returns.

For much of the post-war U.S. history, real rates of return on short-term government bonds

have indeed been negative for taxed investors.

In¬‚ation and taxes have an interesting indirect e¬ect on equilibrium interest rates. You know When in¬‚ation increases,

even real interest rates

that holding the agreed-upon interest ¬xed, in¬‚ation bene¬ts borrowers and hurts lenders,

must also increase.

because lenders who receive interest must pay taxes on the nominal amount of interest, not

the real amount of interest. The reverse holds for borrowers. For example, assume interest

rates are 3% and there is no in¬‚ation. A savings account holder with $100 in the 33% tax bracket

has to pay 1% to Uncle Sam ($1), and gets to keep 2% ($2). Now assume that interest rates are

12% and in¬‚ation is 9%. The savings account holder would now have to pay 4% ($4) in taxes,

and own $108 the coming year. However, because money has lost 9% of its value, the $108

is worth less than $100 the following year. In e¬ect, although real rates are identical in the

no-in¬‚ation and in¬‚ation scenarios, a lender who pays taxes on nominal interest receipts gets

to keep less in real terms if there is in¬‚ation. (It is straightforward to check that the opposite

is true for borrowers.) The implication of this argument is simple: to compensate lenders for

their additional tax burdens (on nominal interest), real interest rates must rise with in¬‚ation.

¬le=frictions.tex: LP

146 Chapter 6. Dealing With Imperfect Markets.

Solve Now!

Q 6.35 If your tax rate is 20%, what interest rate do you earn in after-tax terms if the pre-tax

interest rate is 6%?

Q 6.36 If your tax rate is 40%, what interest rate do you earn in after-tax terms if the pre-tax

interest rate is 6%?

Q 6.37 If the private sector is a net saver, e.g., leaving the public sector as a net borrower, does

Uncle Sam have an incentive to reduce or increase in¬‚ation?

Q 6.38 You are in the 33.33% tax bracket. A project will return $14,000 for a $12,000 investment”

a $2,000 net return. The equivalent tax-exempt bond yields 15%, and the equivalent taxable bond

yields 20%. What is the NPV of this project?

Q 6.39 Compare a 10-year zero bond and a 10% coupon bond, both paying 10%, with an appro-

priate (economy-wide) interest rate of 10%. If the IRS does not collect interim interest on the zero

bond, and the marginal tax rate is 25%, then what is the relative NPV of the two bonds?

Q 6.40 Assume you have both taxes and in¬‚ation. You are in the 20% tax bracket, and the

in¬‚ation rate is 5%/year. A 1-year project o¬ers you $3,000 return for a $20,000 investment.

Taxable bonds o¬er a rate of return of 10%/year. What is the NPV of this project? Extra-credit if

you can derive the formula yourself!

Q 6.41 Advanced question: Return to the apples example from Section 6·6, in which the in¬‚ation

rate was 100% and the nominal rate of interest was 700%. Now, assume that there is also a 25%

default rate. That is, 25% of all apples are returned with worms inside, and will therefore not be

sellable (and be worth $0). What is your real rate of return? What is the formula?

Q 6.42 Really advanced question: Return to the taxes-and-in¬‚ation example from Section 6·7.

A 10% nominal rate of return, a tax rate of 40%, and an in¬‚ation rate of 5%. (We worked out that

the post-in¬‚ation, post-tax rate of return was 0.95%.) Now, add a default rate, d, of 2%, where

all money is lost (’100% return). What is the real, post-in¬‚ation, post-tax, post-default rate of

return? (Hint: Losses are tax-deductible, too. Assume that the default rate reduces the nominal

rate of return (on which taxes are charged), because you do not just take 1 such loan, but 1

million, which practically assures you of the exact default rate without any sampling variation.)

¬le=frictions.tex: RP

147

Section 6·8. Summary.

6·8. Summary

The chapter covered the following major points:

• If markets are not perfect, even expected borrowing and lending rates can be di¬erent.

This is di¬erent from the fact that even in perfect markets, promised borrowing and

lending rates can be di¬erent.

• If markets are not perfect, capital budgeting decisions can then depend on the cash posi-

tion of the project owner.

NPV and interest rate computations can still be used, although it then requires special

care in working with correct and meaningful inputs (especially for the cost of capital).

This is usually best done by thinking in terms of concrete examples ¬rst, and translating

them into formula later.

• Transaction costs and taxes are market imperfections that reduce earned rates of return.

• Transaction costs can be direct (such as commissions) or indirect (such as search or wait-

ing costs). It is often useful to think of round-trip transaction costs.

• Financial assets™ transaction costs tend to be very low, so that it is reasonable in many

(but not all) circumstances to just ignore them.

• In the real world, buyers often prefer more liquid investments. To induce them to pur-

chase a less liquid investment may require o¬ering them some additional expected rate

of return.

• Many ¬nancial markets have such low transaction costs and are often so liquid that they

are believed to be fairly e¬cient”there are so many buyers and so many sellers, that it

is unlikely that you would pay too much or too little for an asset. Such assets are likely

be worth what you pay for them.

• The tax code is complex. For the most part, individuals and corporations are taxed simi-

larly. You must understand

1. how income taxes are computed (the principles, not the details);

2. that expenses that can be paid from before-tax income are better than expenses that

must be paid from after-tax income;

3. how to compute the average tax rate;

4. how to obtain the marginal tax rate;

5. that capital gains enjoy preferential tax treatment;

6. why the average and marginal tax rates di¬er, and why the marginal tax rate is usually

higher than the average tax rate.

• Taxable interest rates can be converted into equivalent tax-exempt interest rates, given

the appropriate marginal tax-rate.

• Tax-exempt bonds are usually advantageous for investors in high-income tax brackets.

You can compute the critical tax rate investor who is indi¬erent between the two.

• Long-term projects often su¬er less interim taxation than short-term projects.

• You should do all transaction cost and tax net present value calculations with after-

transaction cash ¬‚ows and after-tax costs of capital.

• Like taxes and transaction costs, in¬‚ation can also cut into returns. However, in a perfect

market, it can be contracted around and therefore neutralized.

¬le=frictions.tex: LP

148 Chapter 6. Dealing With Imperfect Markets.

• The relationship between nominal interest rates, real interest rates and in¬‚ation rates is

(6.27)

(1 + rnominal ) = (1 + rreal ) · (1 + π ) .

Unlike nominal interest rates, real interest rates can and have been negative.

• In NPV, you can either discount real cash ¬‚ows with real interest rates, or discount nominal

cash ¬‚ows with nominal interest rates. The latter is usually more convenient.

• TIPS are bonds whose payments are indexed to the future in¬‚ation rate, and which there-

fore o¬er protection against future in¬‚ation. Short-term bond buyers are also less ex-

posed to in¬‚ation rate changes than long-term bond buyers.

• Empirically, in¬‚ation seems to be able to explain the level of the yield curve, but not its

slope.

• The IRS taxes nominal returns, not real returns. This means that higher in¬‚ation rates

disadvantage savers and advantage borrowers.

¬le=frictions.tex: RP

149

Section 6·8. Summary.

Solutions and Exercises

1. No di¬erences in information, no market power, no transaction costs, no taxes.

2. It means that borrowing and lending rates are identical, and that there is a unique price at which stu¬ is

selling for (i.e., its value).

3. In most neighborhoods, there are plenty of supermarkets, ¬ercely competing for business. Losing one addi-

tional or gaining one additional supermarket probably makes little di¬erence. There are also plenty of buyers.

In many ways, supermarkets are a fairly competitive business, and their products are usually priced not far

away from their closest competitors. Believe it or not: supermarkets typically earn a gross spread on goods of

only about 2%! This has to pay for space and personnel. However, in some senses, this “supermarket market”

is not perfectly competitive and frictionless: there is sales tax, so third parties cannot easily “arbitrage” prod-

uct, i.e., sell a product that is less expensive in one supermarket to the other, more expensive supermarket.

Plus, once you are at the supermarket, it is often cheaper just to buy the goods there, than it is to drive to

another supermarket.

4. Yes! Surplus?

5. It helps you evaluate what violations really mean.

6. An e¬cient market is one in which the market uses all available information. In a perfect market, market

pressures will make this come true, so a perfect market should be e¬cient. However, an e¬cient market need

not be perfect.

7. You would have to borrow $100 at an interest rate of 10% in order to take the project. If you take the project,

you will therefore have $1, 000 · 1.08 ’ $110 = $970 next period. If instead you invest $900 at the 4% savings

rate, you will receive only $936. So, de¬nitely take the project.

8. Say you invest I. If you put it into the bank, you receive I · (1 + 4%). If you put I into the project, you receive

$1, 000 · (1 + 8%) from the project, borrow ($1, 000 ’ I) at an interest rate of (1 + 10%). Therefore, you must

solve

(6.28)

I · (1 + 4%) = $1, 000 · (1 + 8%) ’ ($1, 000 ’ I) · (1 + 10%)

The solution is I = $333.33, which means that if you want to consume more than $1,666.66, you should not

take the project. Check: [1] If you consume $1,700, you have a remaining $300 to invest. The bank would pay

$312 next year. The project would pay o¬ $1,080, but you would have to borrow $700 and pay back $770,

for a net of $310. You should not take the project [2] If you consume $1,600, you have a remaining $400 to

invest. The bank would pay $416 next year. The project would pay o¬ $1,080, but you would have to borrow

$600 and pay back $660, for a net of $420. You should take the project.

9. Yes! Stated rates include a default premium. A perfect market is about equality of expected rates, not about

equality of promised rates?

10. First, default rates are high. (This is not necessarily a di¬erence in expected rates of returns.) Second, infor-

mation di¬erences about default probabilities are high. Banks cannot easily determine which entrepreneurs

are for real, and which ones will die and take the bank™s money to their graces. The entrepreneurs may or

may not be better at knowing whether their inventions will work. (This can be a market imperfection.)

11. Do it! This information can be found in the Yield Comparisons exhibit in the Credit Markets section in the

WSJ.

12. The appropriate 7-year interest rate would now be about (1 + 8%)7 ’ 1 ≈ 71%. Therefore, a $1 million house

that you would resell in 7 years for $1 million would cost you a direct $60, 000/(1+8%)7 ≈ $35, 009 in present

value of commissions. If you were paying all future real estate commissions for this house, the present value

of this cost would be $84,507. Therefore, the capitalized value of all future brokerage commissions would be

lower (only about 8.5% of house value) than the 15% that we found in the text for lower interest rates.

13. DELL is an even larger stock than PepsiCo. Therefore, a round-trip transaction would probably cost a bid-ask

spread of between 0.1% and 0.3%. On a $10,000, the bid-ask cost would be around $20, and broker fees would

probably be around $10 to $30 with a discount broker. Thus, $50 is a reasonable estimate.

14. Direct: Broker Costs. Market-Maker or Exchange Costs (Bid-Ask Spread). Indirect: Research Costs; Search

(for Buyer/Seller) Costs; Anxiety.

15. You need to assume a proper discount rate for the $4,000. A reasonable assumption is an annuity. At a 7%

interest rate, this value is around $46,281 today. Therefore

x · (1 ’ 8%)

(6.29)

’($1, 000, 000 + $5, 000) + $46, 281 + =0.

1 + 7%

Therefore, x ∼ $1.115, 032 million, so the capital appreciation must be 11.5% per annum. Note how the

$5,000 must be added to the upfront cost, not subtracted!

¬le=frictions.tex: LP

150 Chapter 6. Dealing With Imperfect Markets.

16. A liquidity premium is an upfront lower price to compensate you for transaction costs later on.

17. Taxable income is $4,000. Individual: Tax Rate of 10%, so taxes are $400. Average and marginal tax rates

are 10%. Corporation: Tax Rate of 15%, so taxes are $600. Average and marginal tax rates are 15%.

18. The taxable income is $49,000. Individual: Taxes are $715 + $3, 285 + $19, 950 · 25% = $8, 987.50. Average

Tax Rate (relative to taxable income) is 18.3%. The marginal tax rate is 25%. Corporation: Taxes are $7,500.

The marginal and average tax rate are both 15%.

19. Taxable Income: $49,999,000. Individual: 35% · ($49, 999, 000 ’ $319, 100) + $92, 592 = $17, 480, 557. The

average tax rate is 34.96%. Marginal tax rate is 35%. At very high-income levels, the marginal and average tax

rates are close. Corporation: 35% · ($49, 999, 000 ’ $18, 300, 000) + $6, 381, 900 = $17, 476, 550. The average

tax rate is 34.95%. The marginal tax rate is 35%.

20. 25.4%.

21. 8%. The ratio is very high by historical standards, which means that the marginal investor™s income tax rate

of 8% is quite low.

22. The T-bond will pay $108 before tax. You will therefore earn $105 after taxes. The muni will pay only $103.

So, your opportunity cost of capital is 5%. The project itself will have to pay taxes on $30,000, so you will

have $18,750 net return left after taxes, which comes to an amount of $68,750. Your project NPV is therefore

$68, 750

’$50, 000 + ≈ +$9, 389 . (6.30)

(1 + 5%)3

This is a great project!

23. The $1 is paid from after-tax income, so leave it as is. The $10 million is taxed, so you will only receive

$7 million. With a 1 in 9 million chance of winning, the expected payo¬ is 78 cents. Therefore, the NPV is

negative for any cost of capital. If you could pay with pre-tax money, the ticket would cost you only 70 cents

in terms of after-tax money, so for interest rates of below 10% or so, the lottery would be a positive NPV

investment.

24. Do it! (As of 2002, it should be between 1% and 2% per year.) This rate can be found at the end of the Money

Rates box in the WSJ.

25. Do it! (This changes too often to give a useful ¬gure here.) It can also found in the Money Rates box.

26. Do it!

27. (1 + rnominal ) = (1 + rreal ) · (1 + π ).

28. (1 + 20%)/(1 + 5%) = (1 + 14.29%). The real interest rate is 14.29%.

29. In nominal terms, the rate of return is n0,1 = (1 + 2%) · (1 + 1.5%) ’ 1 = 3.53%, the cash ¬‚ow will be $101.50.

Therefore, PV = $101.50/(3.53% ’ 1.5%) = $5, 000. In real, in¬‚ation-adjusted terms, the rate of return is 2%,

the $101.50 next year are still worth $100 in today™s dollars, so PV = $100/2% = $5, 000. Never discount

$100 by 3.53%, or $101.50 by 2%.

30. The nominal interest rate is (1 + 3%) · (1 + 8%) ’ 1 = 11.24%. Therefore, the cash ¬‚ow is worth about $449,478.

31. $4.448 million.

32.

(a) 5.88%.

(b) The correct PV is

$20 · (1 + 3%)2

$20 · (1 + 3%)

$20

PV = + + + ···

1 + 8% (1 + 8%)2 (1 + 8%)3

(6.31)

$20

= = .

$400

8% ’ 3%

(c) Project value is not $20/(8% ’ 1%) ≈ $285.71.

(d) Project value is not $20/(5.88% ’ 3%) ≈ $694.44.

33. The ¬rst nominal cash ¬‚ow next period is $102,000. Now, you can switch to nominal quantities throughout

(the nominal cash ¬‚ow next year, the nominal interest rate, and as nominal growth rate the in¬‚ation rate).

You would therefore use next year™s nominal cash ¬‚ow”a CF of $102,000”in the formula,

$102, 000 (6.32)

PV = = $1, 700, 000 .

8% ’ 2%

It is a¬rmatively not $100, 000/6% ≈ $1, 666, 666.

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151

Section 6·8. Summary.

34. If in¬‚ation were to remain at 1.6%/year, the plain Treasury bond would o¬er a higher real rate of return

because (1 + 5.6%)/(1 + 1.6%) ’ 1 ≈ 3.9%/year. But if in¬‚ation were to rise in the future, the TIPS could end

up o¬ering the higher rates of return.

35. For every $100, you receive $6. Uncle Sam takes 20% of $6, or $1.20. So, your after tax rate of return is

$4.80/$100 = 4.8%. You could have also computed (1 ’ 20%) · 6% = 4.8% directly.

36. For every $100, you receive $6. Uncle Sam takes 40% of $6, or $2.40. So, your after tax rate of return is

$3.60/$100 = 3.6%. You could have also computed (1 ’ 40%) · 6% = 3.6% directly.

37. Increase. In the real world, interest rates may have to rise to compensate private savers for this extra “tax”

on money.

Your opportunity cost of capital is determined by the tax-exempt bond, because 66.7% · 20% < 15%. Your

38.

project™s $2,000 will turn into 66.7%·$2, 000 = $1, 334 after-tax earnings, or $13,334 after-tax cash ¬‚ow.

Therefore, your NPV is ’$12, 000 + $13, 334/(1 + 15%) = ’$405.22. Check: The after-tax rate of return of

the project™s cash ¬‚ow are $13, 334/$12, 000 ’ 1 ≈ 11%. This is less than 15%. You are better o¬ investing in

tax-exempt bonds.

39. The coupon bond has an after-tax rate of return of 7.5%. Start with $1,000 of money. Reinvestment yields

an after-tax rate of return of 7.5% ($75 in the ¬rst year on $1,000). So, after 10 years, you are left with

$1, 000·1.07510 = $2, 061. In contrast, the zero bond has a single pre-tax payout of $1, 000 · (1 + 10%)10 =

$2, 593.74, for which the IRS would collect $1, 593.74 · 25% = $398.43 in year 10, for a post-tax zero-bond

payout of $2,195. The tax savings on the zero bond are therefore $134 in 10 years, or $52 in present value.

What is your after-tax rate of return on taxable bonds? $100 will grow to $110 ((1 + 10%) · $100 = $110)

40.

pre-tax, minus the 20% what Uncle Sam collects. Uncle Sam takes (1 + 10%) · $100 = $110, subtracts $100,

and then leaves you with only 80% thereof:

80% · ($110 ’ $100)

rafter-tax = = 8%

$100

(6.33)

(1 ’ „) · (CF1 ’ CF0 )

= ,

CF0

where „ is your tax-rate of 20%. (CF1 ’ CF0 )/CF0 is the pre-tax rate of return, so this is just

(6.34)

rafter-tax = 80% · 10% = (1 ’ „) · rpre-tax .

Now, in pre-tax terms, your project o¬ers a 15% rate of return. In after-tax terms, the project o¬ers

80%·$3, 000 = $2, 400 net return, which on your investment of $20,000 is a 12% after-tax rate of return.

(On the same $20,000, the taxable bond would o¬er only 80%·($22, 000 ’ $20, 000) = $1, 600 net return (8%).

So, you know that the NPV should be positive.) Therefore, the project NPV is

$20, 000 + 80%·($22, 400 ’ $20, 000)

NPV = ’$20, 000 + ≈ $740.74

1 + 8%

(6.35)

CF0 + (1 ’ „)·(CF1 ’ CF0 )

= + .

CF0

1 + rafter-tax

You can now easily substitute any other cash ¬‚ows or interest rates into these formulas to obtain the NPV.

Note how everything is computed in nominal dollars, so we do not need the information about the in¬‚ation

rate!

Your numeraire is one apple (1a) that costs $1. You will get $8 in nominal terms, next year (a · (1 +

41.

rnominal,pre-tax ) = a·(1+700%) = 8·a). This will purchase apples that cost $2 each ((1+π ) = (1+100%) = $2),

i.e., 4 apples (a · (1 + rnominal,pre-tax )/(1 + π ) = 1 · (1 + 700%)/(1 + 100%) = 4). However, one of the ap-

ples (d = 25%) is bad, so you will get only 3 apples (a1 = a0 · (1 + rnominal,pre-tax )/(1 + π ) · (1 ’ d) =

1a0 · (1 + 700%)/(1 + 100%) · 75% = 3·a0 , where d is the 25% default rate) . Therefore, the real rate of return

is (a1 ’ a0 )/a0 or

1+700%

(1a · · 75%) ’ 1a

1+100%

rreal,post-tax,post-default = = 300% ’ 1 = 200%

1a

(6.36)

1+rnominal,pre-tax

[1a · · (1 ’ d)] ’ 1a

1+π

rreal,post-tax,post-default = .

1a

The “1a” of course cancels, because the formula applies to any number of apples or other goods.

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152 Chapter 6. Dealing With Imperfect Markets.

42. WARNING: THIS ANSWER HAS NOT BEEN CHECKED, AND MIGHT BE WRONG: Instead of 10%, you earn

only 98%·10% + 2%·(’100%) = 7.8%. Translated into a formula, this is (1 ’ d) · rnominal,pre-tax + d · (’100%) =

rnominal,pre-tax ’ d·(1 + rnominal,pre-tax ) = 10% ’ 2%·(1 + 10%) = 7.8%. Now, in Formula 6.26,

CF0 ·[1+rnominal,pre-tax ·(1’„)]

’ CF0

V0 ’ CF0 1+π

= =

rpost-tax, real, post-default

CF0 CF0 (6.37)

rnominal,pre-tax · (1 ’ „) ’ π

= ,

1+π

replace the nominal interest rate rnominal,pre-tax with the default reduced nominal rate rnominal,pre-tax ’ d·(1 +

rnominal,pre-tax ), so the new formula is

V0 ’ CF0

rpost-default, post-tax, real =

CF0

CF0 ·[1+(rnominal,pre-tax ’d·(1+rnominal,pre-tax ))·(1’„)]

’ CF0

1+π

=

CF0

(6.38)

(rnominal,pre-tax ’ d·(1 + rnominal,pre-tax )) · (1 ’ „) ’ π

=

1+π

7.8% · (1 ’ 40%) ’ 5%

= = ’0.3% .

1 + 5%

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

CHAPTER 7

Capital Budgeting (NPV) Applications and Advice

Tips and Tricks!

last ¬le change: Feb 24, 2006 (16:55h)

last major edit: Apr 2004, Dec 2004

The previous chapters have developed all necessary concepts in capital budgeting. This means

that almost everything in the rest of the book will “just” help you estimate and understand the

NPV inputs better, help you in particular applications, or elaborate on trade-o¬s that you”the

decision maker”face. But this does not mean that you are done. Applying the relatively simple

concept of NPV in the real world can be very di¬cult.

In this current chapter, we cover a collection of topics in which the application of NPV is often

challenging. You will almost surely encounter these complications in your future corporate

practice. In fact, I will try to help you avoid the kind of common mistakes that companies

commit almost every day”mistakes that cost them value. In later chapters, we will look at

other re¬nements. In Part II, we will bring in ¬nancial accounting information to work with

the numbers that corporations use and report. In Part III, we will work on understanding what

determines the cost of capital. In Part IV, we will work on how debt and equity ¬nancing

in¬‚uences the net present value. And, our crowning achievement will be Chapter 29, in which

we will develop a pro forma that will have to bring everything together.

153

¬le=npvadvice.tex: LP

154 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

7·1. The Economics of Project Interactions

So far, we have considered projects in isolation. We computed the costs and bene¬ts necessary

An example of

interacting projects. to make our decision whether to accept or reject. Unfortunately, in the real world, projects are

not always isolated. For example, an aquarium may add a large shark to its exhibition tank at

a cost of $50,000 for projected additional ticket receipts of $120,000; or it may add a large

octopus at a cost of $75,000 for projected additional ticket receipts of $200,000.

Shark Octopus

Ticket Receipts + $120,000 + $200,000

Creature Cost “ $50,000 “ $75,000

Net = $70,000 = $125,000

Regrettably, adding both the shark and the octopus would not increase project value by

$195,000, because octopuses are known to have negative e¬ects on similarly sized sharks”

they eat them. Thus, the best achievable project value is only $125,000 (skip the shark!). On

the other hand, stocking the aquarium with an octopus plus some lobsters would cost only

$75,000 plus a couple of dollars for the lobsters”which allows the octopus to remain alive. If

you do not add the lobsters, you would end up with a starved and expiring octopus and thus

not much audience. So, you either want to add the octopus and the lobsters together, or neither.

In general, the question we are considering in this section is how you should deal with projects

that have mutual interactions. In other words, how should you stock the aquarium?

7·1.A. The Ultimate Project Selection Rule

Important: The Ultimate Project Selection Rule: Consider all possible project

combinations, and select the combination of projects that gives the highest overall

NPV.

Optimal project selection is easier said than done. It is easier for two projects at a time, as it

There are too many

possible action choices was in our aquarium example, because there are only four options to consider: take neither,

in the real world to

take one, take the other, or take both. But the complexity quickly explodes when there are

evaluate (to compute

more projects. For three projects, there are eight options. For four projects, there are sixteen

NPV for). You need rules

and heuristics!

options. For ten projects, there are about a thousand options. For twenty projects, there are

over a million options. (The formula for the number of choices is 2N , where N is the number

of projects.) Even the simplest corporate projects can easily involve hundreds of decisions

that have to be made. For our little aquarium, there are about 54,000 di¬erent ¬sh species to

consider”and each may interact with many others. These choices do not even consider the fact

that some projects may allow other projects to be added in the future, and that many projects

are not just “accept” or “reject,” but “how much project to take.”

To help us determine which projects to take, we need to ¬nd suitable heuristics, i.e., rules that

A Greedy Algorithm?

simplify decisions even if they are not always correct. One common heuristic algorithm is to

consider project combinations, one at a time. Start with the project combination that, if you

were only allowed to take two projects (one pair from a set of many di¬erent projects), would

give you the highest NPV. Then take this pair as ¬xed, i.e., treat it as a single project. Now see

which project adds the most value to your existing pair. Continue until adding the best remain-

ing project no longer increases value. Computer scientists call this the greedy algorithm. It is a

good heuristic, because it drastically cuts down the possible project combinations to consider,

and usually gives a pretty good set of projects. There are many possible enhancements to this

algorithm, such as forward and backward iterations, in which one considers replacing one pro-

ject at a time with every other option. Full-¬‚edged algorithms and combinatorial enhancements

¬le=npvadvice.tex: RP

155

Section 7·1. The Economics of Project Interactions.

that guarantee optimal choice are really the domain of computer science and operations re-

search, not of ¬nance. Yet many of these algorithms have been shown to require more time

than the duration of the universe, unless you make simpli¬cations that distort the business

problem so much that the results seem no longer trustworthy. Fortunately, economics is in

our ¬nance domain, and it can also help us simplify our project selection problem.

7·1.B. Project Pairs and Externalities

We just mentioned considering projects in pairs. This is not only common practice, but also Project combinations

can be classi¬ed into

clari¬es the economic issues. With two projects, we can decompose the total net present value

positive, zero, and

into three terms: negative interaction

combinations.

(7.1)

Overall NPV = NPV Project One + NPV Project Two + NPV Interactions .

If you were to stock both the shark and the octopus, you would get ticket receipts of $200,000

[octopus] but pay $125,000 [octopus and shark], for a net of $75,000. Therefore,

= + + (’$120, 000)

$75, 000 $70, 000 $125, 000

NPV Octopus eats Shark,

NPV Aquarium With Both = NPV Shark + NPV Octopus + .

so no more Shark ticket receipts

(7.2)

The ¬nal term suggests that we can classify project combinations into one of three di¬erent

categories:

1. Projects with zero interactions.

2. Projects with positive interactions.

3. Projects with negative interactions.

Interactions are also sometimes called externalities in economics, because one project has

external in¬‚uences on other projects”sometimes imposing external costs and sometimes pro-

viding external bene¬ts. We now discuss these three cases.

Zero Project Interactions

Most projects in this world are independent”they have no mutual interactions. For example, a Project independence is

the most common case,

mall in Maine probably has no e¬ect on a mall in Oregon. It neither steals customers from Ore-

and allows simple

gon nor attracts extra customers. Independent project payo¬s permit the separate evaluation decision making.

of each project. This makes decision-making very easy:

• Taking each Positive NPV project increases ¬rm value.

• Taking each Zero NPV project leaves ¬rm value unchanged.