10.1Cash Flows

The most important inputs for evaluating a real investment are the incremental cash

ows that can be attributed to the investment. Chapter 9 discussed these in detail. The

critical lesson learned is to use incremental cash ows: the rm’s cash ows with the

investment project less its cash ows without the project. Such a view of a project’s

cash ows necessarily excludes sunk costs. Sunk costs are incurred whether or not the

project is adopted. It also allows for synergies and other interactions between a new

project and the rm’s existing projects. These additional cash ows need to be

accounted for when one values a project.⁵

10.2Net Present Value

The net present value (NPV)of an investment project is the difference between the

project’s present value (PV), the value of a portfolio of nancial instruments that track

the project’s future cash ows, and the cost of implementing the project. Projects that

create value are those whose present values exceed their costs, and thus represent sit-

uations where a future cash ow pattern can be produced more cheaply, internally,

within the rm, than externally, by investing in nancial assets. These are called pos-

itive net present valueinvestments.

In some cases, such as those associated with riskless projects discussed in this chap-

ter, the tracking portfolio will perfectly track the future cash ows of the project. When

perfect tracking is possible, the value created by real asset investment is a pure arbitrage

gain achievable by taking the project along with an associated short position in the track-

ing portfolio. Since shorting the project’s tracking portfolio from the nancial markets

offsets the project’s future cash ows, a comparison between the date 0 cash ows of

the project and the tracking portfolio is the only determinant of arbitrage. Value is cre-

ated if there is arbitrage, as indicated by a positive NPV,and value is destroyed if there

is a negative NPV.⁶

The nancial assets in the tracking portfolio are thus the zero point

on the NPVmeasuring stick, as the real assets are always measured in relation to them.

The perspective we provide here on the valuation of real assets is one that allows

corporations to create value for their shareholders by generating arbitrage opportuni-

ties between the markets for real assetsand nancial assets.In other words, by mak-

ing all nancial assets zero-NPVinvestments we are implicity assuming that it is

impossible to make money by buying IBM stock and selling short Microsoft stock.

However, Microsoft may be able to make money by investing in a new operating

system and financing it by selling its own stock. In an informationally efficient

4

See Chapter 18.

⁵One cannot ignore project nancing when considering the possibility that the rm can create value

from its nancing decisions by altering the rm’s tax liabilities. For this reason Chapter 13, which

focuses on taxes and valuation, considers both cash ows and the tax effect of the nancing cash ows.

⁶This is true even if nancial assets are not fairly valued; that is, even if the nancial markets are not

informationally efcient.

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Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

332Part IIIValuing Real Assets

financial market (defined in Chapter 3),financial securities are always fairly

priced—but bargains can and do exist in the market for real assets even when the

nancial markets are informationally efcient. Because of its special abilities or cir-

cumstances, Microsoft can develop and market a new operating system better than

its competitors and, as a result, can create value for its shareholders.

Discounted Cash Flow and Net Present Value

When the cash ows of a project are riskless, they can be tracked perfectly with a com-

bination of default-free bonds. For convenience, this chapter uses zero-coupon bonds,

which are bonds that pay cash only at their maturity dates (see Chapter 2) as the secu-

rities in the tracking portfolio. This subsection shows that the net present value is the

same as the discounted value of the project’s cash ows.⁷The discount rates are the

yields to maturity of these zero-coupon bonds.

Yield-to-Maturity of a Zero-Coupon Bond: The Discount Rate.⁸The per-period

yield-to-maturityof a zero-coupon bond is the discount rate (compounded once per

period) that makes the discounted value of its face amount (the bond’s payment on its

maturity date) equal to the current market price of the bond; that is, the rthat makes

t

$1

P

(1r)^t

t

where

P current bond price per $1 payment at maturity

t number of periods to the maturity date of the bond

When markets are frictionless, a concept dened in Chapter 4, and if there is no arbi-

trage, the yields to maturity of all zero-coupon bonds of a given maturity are the same.

AFormula forthe Discounted Cash Flow.We now formally dene the Discounted

Cash Flow (DCF) of a riskless project. Aproject has riskless cash ows

C, C, C,. . ., C

012T

where

C the (positive or negative) cash ow at date t.Positive numbers represent

t

cash inows(for example, when a sale is made) and negative numbers

represent cash outows(for example, when labor is paid), which are

positive costs.

The discounted cash owof the project is

CCC
12T
DCF C. . .	(10.1)
01r(1r)2(1r)T
12T

⁷Although this is also true for risky cash ows, in many of these cases, we would not discount cash

ows to compute the NPVof a risky cash ow stream. See Chapter 12 for further detail.

⁸See Chapter 2 for more on yield-to-maturity.

Grinblatt679Titman: Financial	III. Valuing Real Assets	10. Investing in Risk679Free	© The McGraw679Hill
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Strategy, Second Edition

Chapter 10

Investing in Risk-Free Projects

333

where

r the per period yield-to-maturity of a default-free zero-coupon bond maturing

t

at date t

Aproject’s discounted cash flow is the sum of all of the discounted future cash

flows plus today’s cash ow, which is usually negative, since it represents the initial

expenditure needed to start the project. The “discounted futurecash ow stream,” equa-

tion (10.1) with Comitted on the right-hand side, is often used interchangeably with

0

the term “present value of the project’s future cash ows” or simply “project present

value.” Similarly, the sum of the present value of the future cash ows plus today’s

cash ow, referred to as “the net present value of the project,” is then used inter-

changeably with the term “discounted cash ow stream.”

Using Different Discount Rates at Different Maturities.Many formulas compute

present values (and net present values) using the same discount rate for cash ows that

occur at different times. This simplication makes many formulas appear elegant and

simple. However, if default-free bond yields vary as the maturity of the bond changes,

the correct approach must use discount rates that vary depending on the timing of the

cash ows. These discount rates, often referred to as the costs of capital(or costs of

nancing), are the yields-to-maturity of default-free zero-coupon bonds.⁹

Project Evaluation with the Net Present Value Rule

This subsection shows why the NPVrule, “adopt the project when NPVis positive,” is

sensible. Below, we show that the NPVrule is consistent with the creation of wealth

through arbitrage.

Arbitrage and NPV.Adopting a project at a cost less than the present value of its

future cash ows (that is, positive NPV) means that nancing the project by selling

short this tracking portfolio leaves surplus cash in the rm today. Since the future cash

that needs to be paid out on the shorted tracking portfolio matches the cash ows com-

ing in from the project, the rm that adopts the positive NPVproject creates wealth

risklessly. In short, adopting a riskless project with a positive NPVand nancing it in

this manner is an arbitrage opportunity for the rm.

Hence, when there are no project selection constraints, net present value offers a

simple and correct procedure for evaluating real investments:

Result 10.1

The wealth maximizing net present value criterion is that:

•	All projects with positive net present values should be accepted.
•	All projects with negative net present values should be rejected.

The Relation between Arbitrage, NPV,and DCF.Below, we use a riskless project

tracked by a portfolio of zero-coupon bonds to illustrate the relation between NPVand

arbitrage. This illustration points out—at least for riskless projects—that net present value

and discounted cash ow are the same. Begin by looking at a project with cash ows

at two dates, 0 (today) and 1 (one period from now). The algebraic representation of the

cash ows of such a project is given in the rst row below the following time line.

⁹When cash ows are risky, the appropriate discount rate may reect a risk premium, as Chapter 11

discusses.

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Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

334Part IIIValuing Real Assets

Cash Flows at Date

01

Algebra	C	C
	0	1
Numbers	$10 million	$12 million

The second row of the time line is a numerical example of the cash ows at the same

two dates. Inspection of the time line suggests that default-free zero-coupon bonds,

maturing at date 1, with aggregated promised date 1 payments of C($12 million), per-

1

fectly track the futurecash ows of this project. Let

CP the current market value of these tracking bonds C/(1r), where

11

r yield-to-maturity of these tracking bonds (an equivalent way of

expressing each bond’s price) 1/P1

For example, if each zero-coupon bond is selling for $0.93 per $1.00 of face value,

then P $0.93 and r 7.5 percent (approximately). Since, by the denition of the

yield-to-maturity, P 1/(1 r), the issuance of these tracking bonds, in a face amount

of C, results in a date 0 cash ow of

1

CP C/(1 r), or numerically, $12 million.93 $12 million/1.075

11

and a cash ow of C, or $12 million, at date 1.

1

Hence, a rm that adopts the project and issues C($12 million) in face value of

1

these bonds has cash ows at dates 0 and 1 summarized by the time line below.

Cash Flows at Date

01

	C
	1
Algebra	C	0
	01r

	$12 million
Numbers	$10 million	0
	1.075

Since the date 1 cash ow from the combination of project adoption and zero-coupon

bond nancing is zero, the rm achieves arbitrage if CC/(1 r), the value under

01

date 0, is positive, which is the case here.

The algebraic symbols or number under date 0 above, the project’s NPV,is the sum

of the discounted cash ows of the project, including the (undiscounted) cash ow at

date 0. It also represents the difference between the cost of buying the tracking invest-

ment for the project’s futurecash ows, C/(1 r) (or $12,000,000/1.075), and the

1

cost of initiating the project, C($10 million). If this difference, CC/(1 r)

001

(or $10 million $12 million/1.075), is positive, the future cash ows of the proj-

ect can be generated more cheaply by adopting the project (at a cost of C) than by

0

investing in the project’s tracking investment at a cost of C/(1 r).

1

Example 10.1 extends this idea to multiple periods.

Grinblatt683Titman: Financial	III. Valuing Real Assets	10. Investing in Risk683Free	© The McGraw683Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 10

Investing in Risk-Free Projects

335

Example 10.1: The Relation between Arbitrage and NPV

Consider the cash ows below:

Cash Flows (in $ millions) at Date

0123

20401030

Explain how to nance the project so that the combined future cash ows from the project

and its nancing are zero.What determines whether this is a good or a bad project?

Answer:The cash ows from the project at dates 1, 2, and 3 can be offset by

1.	issuing—that is, selling—short, zero-coupon bonds maturing at date 1 with a facevalue of $40 million,

2.	purchasing zero-coupon bonds maturing at date 2 with a face value of $10 million,and

3.	selling short zero-coupon bonds maturing at date 3 with a face value of $30 million.

The cash ows from this bond portfolio in combination with the project are

Cash Flows at Date

0123

V000

where Vrepresents the cost of the tracking portfolio less the $20 million initial cost

of the project, and thus is the project’s NPV.Clearly, if Vis positive, the project is

good because it represents an arbitrage opportunity.If Vis negative, it represents a bad

project.

All riskless projects can have their future cash ows tracked with a portfolio of

zero-coupon bonds. Shorting the tracking portfolio thus offsets the future cash ows of

the project. To see this, let C, C, . . . , Cdenote the project’s cash ows at dates

12T

1, 2, . . . , T,respectively. Ashort position in a zero-coupon bond maturing at date 1

with a face value of Coffsets the rst cash ow; shorting a zero-coupon bond with a

1

face value of Cpaid at date 2 offsets the second cash ow; and so forth. Aportfolio

2

that is short by these amounts creates a cash ow pattern similar to that in Example

10.1, with zero cash ows at future dates and possibly a nonzero cash ow at date 0.

The logic of the net present value criterion for riskless projects and the equivalence

between net present value and discounted cash ow follows immediately. Result 10.2

summarizes our discussion of this issue as follows:

Result 10.2

For a project with riskless cash ows, the NPV—that is, the market value of the project’stracking portfolio less the cost of initiating the project—is the same as the discounted valueof all present and future cash ows of the project.

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Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

336Part IIIValuing Real Assets

Financing Versus Tracking the Real Asset’s Cash Flows.With frictionless markets,

a rm can in fact realize arbitrage prots if it nances a positive NPVreal asset by issu-

ing the securities that track its cash ow. However, our analysis does not require the

rm to nance the project in this way. In chapter 14 we show that with frictionless mar-

kets, if the cash ow pattern is unaffected, the nancing choice does not affect values.

Present Values and Net Present Values Have the Value Additivity Property

Aconsequence of no arbitrage is that two future cash ow streams, when combined,

have a value that is the sum of the present values of the separate cash ow streams.

An arbitrage opportunity exists when an investor can purchase two cash ow streams

separately, put them together, and sell the combined cash ow stream for more than

the sum of the purchase prices of each of them. Arbitrage is also achieved if it is

possible to purchase a cash ow stream and break it up into two or more cash ow

streams (as in equity carve-outs), and to sell them for more than the original purchase

price. Chapter 9 noted that the present value from combining two (or more) cash ow

streams is the sum of the present values of each cash ow stream. In this section, we

are learning that this value additivityproperty is closely linked to the principle of no

arbitrage.

The net present value, which, as seen earlier, is also so closely linked to the prin-

ciple of no arbitrage in nancial markets, possesses the value additivity property. NPV

value additivity is apparent from the DCF formula, equation (10.1).If project Ahas

cash ows C, C, . . . Cand project B has cash owsC, C, . . . , C,then,

A0A1ATB0B1BT

because cash ows only appear in the numerator terms of equation (10.1), NPV(C

A0

C, CC, . . . , CC)is the sum of NPV(C, C, . . . , C) and

B0A1B1ATBTA0A1AT

NPV(C, C, . . . , C).

B0B1BT

Implications of Value Additivity forProject Adoption and Cancellation.Value

additivity implies that the value of a rm after project adoption is the present value of

the rm’s cash ows from existing projects, plus cash for future investment, plus the

net present value of the adopted project’s cash ows. (If one is careful to dene the

cash ows of the project as incremental cash ows, this is true even when synergies

exist between the rm’s current projects and the new project.) Value additivity also

works in reverse. Thus, if the net present value of a project is negative, a rm that has

just adopted the project would nd that its incremental cash ow pattern for canceling

the project and acquiring the tracking bonds is the stream given in the table below,

which has a positive value under date 0:

Cash Flows at Date

01

C
1
C	0
01r

This table demonstrates that a negative net present value for adoption of the project implies

a positive net present value for canceling the project, once adopted, and vice versa.