Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8

7/25/2019 Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8

1/8

Uses, Abuses, and Alternatives to the Net-Present-Value Rule

Author(s): Stephen A. RossSource: Financial Management, Vol. 24, No. 3 (Autumn, 1995), pp. 96-102Published by: Wileyon behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665561.

Accessed: 08/09/2013 23:33

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Wileyand Financial Management Association Internationalare collaborating with JSTOR to digitize, preserve

and extend access to Financial Management.

http://www.jstor.org
http://www.jstor.org/action/showPublisher?publisherCode=blackhttp://www.jstor.org/action/showPublisher?publisherCode=fmahttp://www.jstor.org/stable/3665561?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/3665561?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=fmahttp://www.jstor.org/action/showPublisher?publisherCode=black


2/8

ontemporary

s s u e s

U s e s

Abuses

n d

lternatives t

t h e

Net Present Valueu l e

Stephen

A.

Ross

Stephen

A.

Ross

is

SterlingProfessor

of

Economics

and

Finance at YaleSchool

of Management,

Yale

University,

New

Haven,

CT.

M

No

student

can

leave

the

introductory

inance course

without having masteredthe net-present-valuerule. It is

the meat of

most textbooks and lies at

the

core of what

financialacademics hink

hey

have to

offer

CFOs,

corporate

treasurers,

investment

bankers,

and

practitioners

of all

stripes.

In

fact,

it is

not

uncommon o

spend

a

considerable

amount

of

time

in

class

making

sure that the

student

understands

all the

wrong ways

of

thinking

about

investment decision

making-from

the IRR rule to the

payback period. Wrong,

of

course,

because

they

don't

coincide

with the NPV rule.

The

simplest

statement

f

the

NPV

rule

s that

you

should

discard

projects

with

negative

NPVs

and

undertake all

projects

with

positive

NPVs. If

we

are

being

careful,

we add

the

caveat that

a

positive

recommendation o

take a

project

should

only

be made

if

taking

on the

project

doesn't

prevent

us

from

undertaking

ome other

project.

Like all

good

rules,

this

one

containsmuch

truth,

butI am

going

to

take

the

contrary

view.

I

have become convinced

that it is

time to revisit the

usefulness

of NPV

and to

reconsider

just

how much stock

we want to

place

in it.

Perhaps

most

surprising,

will

argue

he merits

of alternative

rules-modified

versions

of

NPV-that seem

to endure n

practice

despite

theirconflictwiththe NPV

rule. In

general,

though,I believe that we now have superiorways to make

investment

decisions.

Section

I

describes

the

problems

with the

way

we

currently

use the

NPV

rule.

This

section-and much

of this

paper--draws

heavily

on

my

work with Jon

Ingersoll

as

reported

n

Ingersoll

and Ross

(1992).

Section

II

offers a

simple

example

that

shows the

ubiquitous

need

for

alternatives

or modifications

to

the

NPV

rule.

Section

III

examines some of the ad hoc rules

used

n

practice

and

makes

a case that they may not be as undesirable as accepted

wisdom holds. Section IV outlines

a

general

approach

o

making

wise investment

decisions

as an

alternative o

the

NPV

rule and

provides

a

practical

ule-of-thumb

djustment

to the NPV

rule. Section

V

briefly

summarizeswhat the

paper

has said on these

matters.

I.

The

Good,

the

Bad,

and

the

Ugly

of the

NPV

A

firm is

considering

a

major

investment. The

project

involves the

completion

of a

majorpower

source,

and

t will

cost

$100

million

upfront.

One

year

after

making

the

investment,

he

firm will be

able

to

liquidate

ts

stake in the

project

for

$110

million.

The

project

s

big enough

so that

the

top management

will

make the

decision.

The

management

are all

graduates

of the

top

management programs,

and

they

have a

firm

allegiance

to the utilizationof the best available inancial

heory

in their

decision-making. They

are

certain that the

project

is

absolutely

riskless.

The

market

agrees

with

their assessment

and will finance the

project

at riskless

interest

rates.

A

quick

look at the latest results from the bond

market

reveals that he current ieldcurve s flatat 10.3%, .e., short

rates,

long

rates,

and all

intermediaterates are 10.3%.

A

back-of-the-envelope

alculation eveals that

he

project

has

an NPV of

approximatelynegative

$300,000.

Armed

with

this

information,

management ejects

the investment.

Dejected

at

having

o turndown the

project,

butconvinced

that

they

have made the

right

decision,

management

s

soon

to

get

some

good

news.

An

independent

nvestor

approaches

them and offers

to

buy

the

rights

to

the

project.

When asked

why

she

would

pay anything

for a worthless

project,

the

investor

answers hatshe subscribes

o the

greater

ool

theory

This

paper

was the FMA

Keynote

Address at the 1994 FMA Annual

Meeting,

October 12, 1994. The

author s

grateful

o

Jon

Ingersoll

and the

Editors or

their

helpful

comments.

All errorsare his own.

Financial

Management,

Vol.

24,

No.

3,

Autumn

1995,

pages

96-102.

This content downloaded from 142.103.160.110 on Sun, 8 Sep 2013 23:33:17 PMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/8

ROSS

/

USES,

ABUSES,

AND

ALTERNATIVESO

THE

NET-PRESENT-VALUE

ULE

97

and believes that the

project

can

be

flipped

at

a

profit.

Management

s

unconvinced

but is

more

than

happy

to sell

the

rights

to

the

project

for

$10,000

and move on to

more

profitableopportunities.

In

fact,

the

investor,

true

to her

word,

just

a few

days

later is

able

to

flip

the

project

for

$30,000,

netting

a

quick

$20,000

profit.

The

purchaser

is a

sharp

real

estate

developer

who weatheredthe 1980s and with

few

opportunities

in

development

is

casting

about

for

other

activities.

As luck would have

it,

one

month

after

purchasing

the

project

he

one-year

nterestrate

suddenly

drops

50

basis

points,

from

10.3%

o 9.8%.

Computing

he

NPV at

the

new

lowered

interest rate results in

approximately

$200,000.

Without

a

second

thought,

the

ex-developer

finances

the

investment

collateralizing

he

loan

with

the

proceeds

from

the

project

and

pockets

about

$200,000.

Deducting

the

$30,000

he

paid

the

dealmaker,

the

developer

nets

a

$170,000

profit

from a one-month

holding.

A.

The Good:

Rejecting

an InvestmentWhen It

Should

Be

Rejected

The

$100

million

project generates

$110

million one

year

later. With interest rates

at

10.3%,

the same

$100

million

grows

to

$110.3

million when invested

n a

one-year

discount

bond.

Investing

in

the

project

s

dominated

by

the

opportunities

available in the

capital

markets,

and the

managers' finance training hasn't let them down-yet.

The

key

point

to

keep

in mind is that

capital

market

alternatives

are

always freely

available

and

undertaking

themdoes not alter he set of alternatives hatare

open

to an

investor.

Hence,

if

a

project

s dominated

by

a

capital

market

alternative,

hen

there

are no other

financing

considerations

that would

justify taking

on

the

dominated

project.

B.

The

Bad:

Rejecting

an InvestmentWhenIt

Should

Be

Accepted

Selling

the

project

for

$10,000

was

tantamount to

rejecting

the investment.

Even

though

the NPV

of this

project s negative,$10,000 seems like a bargainbasement

price

for a

$100

million

project.

Suppose

that interestrates

were

exactly

10%.

Simply

because the NPV

calculation

yields

a zero for

NPV,

does

anyonereally

believe the

project

is worthless?What s

wrong

with the NPV

analysis?

This

project

is

more

than

just

a one-time investment.It

also

includes

the

rights

to the investment.

Simply

because

current nterest rates don't

justify making

the investment

doesn't

mean

that

this will

always

be the

case. Nor does the

fact that the

yield

curve is flat

preclude

the

possibility

that

interestrates could fall

below 10%

and

bring

the NPV into

the

positive range.

The

rights

to the

project

are the

rights

to

the interest rate

option

inherent in the

project.

Any

such

project

has such

rights.

When

the investment s

undertaken,

it will

have

positive

NPV.

Since

nothing

orces

the

holder o

take the

project

when the NPV is

negative,

it will only be

undertaken

t

positive

NPVs.

Thus,

the

holder

profits

from

declines in the

one-year

rate and has limited

liability

if

interestrates rise. This

project

s

equivalent

o a call

option

on a

one-year

bond.

Simply

because the

option

isn't

in-the-money oday

doesn't mean

that

t is

worthless.

C. The

Ugly:

Accepting

an InvestmentWhen

It

ShouldBe

Rejected

The subtlesterrorof all in the

application

f

the NPV rule

is

accepting

he

project,

.e.,

making

the

investment,

simply

because the

NPV

is

positive.

What

if the

interest

rate were

9.999%?

Do we

undertake

he

investment to

realize the

gain

of

$1,000?

What

is

wrong

with this

application

of

the NPV rule?

Actually, nothing.

What

s

wrong

is

the

general

ailure o

seriously

consider the caveat to the NPV

rule,

namely,

undertake he

project

as

long

as

doing

so

doesn't

interfere

with

the

ability

to

take

on a

competing

project.

Undertaking

the

project implies

that

we

are not

taking

on the

project

tomorrowor

next week

or

at

any

otherfuture

ime.

Every project

competes

with

itself

delayed

in

time.

This

is the

essence of the

problem

with

applying

the

NPV

rule,

and it is anotherway to understand ptionality.In a capital

budgeting

context with

a

budget

constraint,

undertaking

a

project

means

taking

on that feasible

combination of

projects

that

maximizes the

NPV.

Clearly

with

interestrate

uncertainty,

we

trade off the

value

of

taking

on

the

project

today against

the lost

opportunity

cost of

foregoing

the

option

to undertake he

project

at some later

date when

interestrates are more

favorable.

This same

reasoning

can

also resolve the

problem

of

rejecting

the

project

when

it

should be

accepted,

i.e.,

of

selling

the

rights

o the

project

oo low.

Selling

the

project

o

the dealmaker

is

not

only

selling today's

project,

it is

also the sale of all the potentialfutureprojects.Weighing

each

such

project

by

the

probability

suitably

risk

adjusted,

i.e.,

the

martingale probability)

and

taking

the

expected

valuewe

get

the value of those

future

projects.

D.

Taking

Optionality

nto

Account

Ingersoll

and Ross

(1992)

use a

specific process

for

the

dynamics

of

interestrate

movements

to

develop

an

exact

formula or the value of

the

project

when

viewed with all of

its

optionality

ntact,

and

they

determine

he interestrate

at

which

it is

optimal

to

exercisethe

option

and

undertake he

investment.There s no

reasonto

display

the

formulas rom



4/8

98

FINANCIAL ANAGEMENTAUTUMN 995

that

paper;

uffice

it to

say

thatthereare at least two

lessons

to be learned rom the exact

analysis.

First,

not

surprisingly,

or

our

project

he

interestrate at

which it is optimal to exercise is a rate below 10% at

which the

project

is well

in-the-money

and

generates

adequate

NPV to

compensate

for

the loss of

the

option.

Second,

naturally,

the

volatility

of interest rates is a

significant

determinant f the value of

the

project

and

of

the

cutoff

rate at which the

project

is undertaken.

Generally

speaking,

the

higher

the

volatility,

the more valuable is the

optionality

and,

therefore,

he more

valuable is the

project

and

the lower the

optimal

nterestrate at which it should be

exercised.

It is

fascinating

hat

his,

the

simplest

nvestment

xample

possible,

contains so much

in the

way

of

optionality

and

carries such a potentialfor misleadinganalysis.Of course,

projects

are never this

straightforward.

An investment

that is

not undertaken

today

cannot be warehoused

forever.Over

ime

it

changes-often

in

uncertain

ways-and

aftera certain

period,

t

may

no

longer

be available.But

what

is the more

realistic

polar

case? That of a

project

that

disappears

altogether

if

it's

not undertaken

at this exact

moment,

or one

that can be

undertakenwithout

alteration t

any

time

in the future?

Cast

in

this

stark

ight,

both

appear

o be extremes.

But

given

a

choice,

I

believe the

infinitely

lived

project

s much

the

preferred

anonical

example.

As shown

in

Ingersoll

and

Ross (1992), most of the

option

value is developed in a

relatively

short

period

of time. Since

most

projects simply

don't

go away

if

they

are not undertaken

right

now,

the

perpetuity

xample

is a

good approximation

o

reality.

There

is a

long

literature

on

investment

projects

with

embedded

options.

Almost all actual

projects

have

optionality

inherent

in

the cash

flows

themselves. Often

investments

open

up

the

possibility

of

profitable

future

opportunities

to invest.

These are

important

ssues,

but

they

are

relatively

clear

conceptually,

and

they

are

separate

from

the issues

raised here.

The value

of

any project

comes

from

three

sources.

First,

from

its

in-the-money-value,

which is

simply

its NPV if the

option

were to be exercised

today.

Second,

from

the valueof the embedded

options

built

into

the

project

itself.

We are

talking

aboutthe third and

ubiquitous

source

of value.

Every project,

whetheror not it

explicitly

contains

options,

always

is an

option

on the

movement

of

capital

costs

and

prices.

In

assessing

investmentvalue

andin

making

nvestment

decisions,

we must now

recognize

the

impact

of this

sourceof value

on eventhe

most

straightforward

f

projects.

As a

practical

matter,

ngersoll

and Ross

(1992)

show that

this source of value

is

generally

large

enough

to have a

significant mpact.

II.

Another

Example

The above

reasoning

extends to

nearly

all

investment

decision-making; t certainly oses none of its force when

uncertainty

s

introduced.

When cash flows are

random,

we

have

learned

to value

them

by

applying

a

modified

version

of

the

NPV rule. We take the

expected

cash

flows,

apply

a risk

adjustment,

and discount the

resulting

certainty-equivalent

lows. A not

entirely

naccurate

ariant

discounts the

expected

cash flows at

risk-adjusted

osts

of

capital,

but

it is

preferable

o

apply

the risk

adjustment

o the

cash flows

using

the

martingale

r

risk-adjusted xpectation.

Not all valuation

problems

with random cash

flows,

though, require

he full

artillery

of derivative

pricing.

Ross

(1978)

displayed

a

rich

class of

problems

hat

had mmediate

and simple solutions. For example, supposethatT periods

from the initial

nvestment-say

$

1-the

payoff

on

a

project

will be

proportional

to the value of some

marketed

asset,

S

(where

S

is inclusive of reinvested

dividends).

Typically,

S could be

a

stock

or a market

ndex,

and

P

will

be the

proportionality

onstant.

Thus

if

$1

is investedat time

t,

the

payoff

will be

St

+

T

at time t

+

T.

Since the

asset itself is

equivalent

o a claim on the value

St + T

at time t +

T,

it follows

without resort to

any fancy

derivative

pricing

analysis hat

he current alue of the

payoff

PSt

+T

is

simply

PSt.

But,it is

certainly

not the case

thatthe

rightto undertake hisprojecthas a current alueof PSt- 1.

Just as

in

the

risk-free

example

of Section

I,

insofar

as it is

possible

to

delay making

the initial

investment,

he

project

has an

option

value.

In

fact,

clearly

the current

value of this

project

s

simply

the

value of a call

option

with an exercise

price

of

1

and

a

maturityequal

to the

length

of time that the

project

can

be

delayed.

Even

though

the

cash flows involve no

optionality

in and of

themselves,

the

ability

to

delay

confers

optionality

on the

project.

As

before,

the

project

value

is

enhanced

by

the value of the

option,

which,

in

turn,

derives

from the

opportunity

o

profit

from

changing

uture

valuations

of the

project'spayoff.

It

is worth

noting

that to some extent this

problem

of

neglected

optionality

is

mitigated

by

the flow

through

of

inflationary

xpectations

o cash flows.

In a

simple

Fisherian

world,

the

nominal discountrate is the real rate

of interest

plus

the

expected

inflation

rate. If cash flows in the

futureare

expected

to increasewith

inflation,

then

changes

in the inflation

ratewill leave the

NPV

unaltered.

Thus,

the

optionality

associated with

financing

will

only

be with

respect

to

uncertainty

n the real rateof interest.

Of

course,

to theextentthat

he cash flows on a

project

arenot

perfectly

proportional

othe

price

evel that s embodied n the interest



5/8

ROSS

/

USES,

ABUSES,

ANDALTERNATIVES

OTHE

NET-PRESENT-VALUE

ULE 99

rate,

the

impact

of

uncertainty

n

the

interestrate

will not be

limited

solely

to

uncertainty

n

real

rates of interest.

Ill. HurdleRates and Other Rules

Finance scholars

have

always

been

puzzled

by

the

durability

of a host of

investmentrules that

seem to survive

and even thrive

despite

their

obvious

shortcomings.

Ross,

Westerfield,

and Jaffe

(1993)

report

on a numberof

these,

including

the

payback

period,

the

IRR,

and the hurdle

rate rule.

The latter

requires

an investment to not

merely

have a

positive

NPV but to

have a

sufficiently

positive

NPV.

Often,

this is

put

in

the form of

requiring

hat

the

IRR exceed the current

market rate of

interest

by

an

additional

amount,

say

3%.

Such

rules are

interesting

for a number of

reasons,

and not

surprisingly,

they

have attractedmuch attention.

Asymmetric

information

arguments

or their existence are

probably

the most

prevalent

rationalizations.

For

example,

Antle

and

Eppen

(1985)

and Antle and

Fellingham

(1990)

have

argued

orcefully

hat

he incentivesof

managers

within

hierarchiesare such as

to rewardthem for

amassing

more

control over

corporate

resources.

These studies

argue

that

firms

requireprojects

o

satisfy

hurdle

rates

in

excess of the

interestcosts

in the

capital

markets o serve

as a brakeon this

tendency

to overinvest.

Alternatively,high

hurdlerates

may

simply

be a

practical

way

to deal with

uncertainty.

These theoriesall

have

merit,

butour ook at

the NPV rule

suggests

that

another

explanation

s at work.

Figure

1

plots

the NPV of a

projectalong

with the

option

adjusted

NPV,

or

OANPV,

which is the

simple

NPV

plus

the value of the

interest

rate

options

inherent

in the

right

to

delay

the

project.

Since

the

project

can be

delayed

indefinitely,

the

OANPV

is never zero no matter how

large

is the

interestrate

and

negative

is the NPV. At an interest rate

equal

to the

OAIRR,

he value of these

rights

s zero because

the

option

is

sufficiently in-the-money

that the

gain

from

delaying

to

preserve

the

option

is

just

offset

by

the loss

in value from not

immediately xercising

t and

realizing

the

positive NPV. As can be seen, the OAIRR lies below

the

IRR

so as to insure

that currentexercise is

sufficiently

valuable.The difference

between the IRR andthe OAIRR s

the extra amountthat must be added to the current nterest

rate to find the

hurdlerate.The

project

will be

undertaken

f

the IRR exceeds this hurdle rate. The exact hurdle rate

adjustment

depends

on interest rate

volatility

(or

cost of

capitalvolatility)

and on the exact cash flow structure f the

project,

but as a

practical

matter within wide families

of

potential projects,

some such

adjustment

s sensible and

improves

on the

simple

use of the NPV or the IRRrules. I

will

explore

this matter

n

greater

detail

in

the next section.

It would be difficult

to make a

similar defense of

the

payback period

rule.

In the

simplest

form of the

payback

rule,

an investment

is

accepted

or

rejected

dependingon whetherthe cash flows addup to the initial

investment

by

some

specified

time

period, e.g.,

three

years.

Typically

the

projects

o which it is

applied

nvolve a

single

upfront

nvestmentand

a

subsequent

treamof

positive

cash

flows.

Typically,

too,

the rule manifests

tself in industries

like entertainment where

projects

can be

delayed

for

meaningfulperiods

of time.

In

such

cases,

often

the

firm

has made infrastructure

investments that realize their

return

through

a stream of

projects

with excess

returns

for all rates within broad

historical

ranges.

The

limitationon

the firm is not so much

capital

as it is the

capacity

o

apply

imitedreservesof human

and

managerial apital.

While this is

quite

different

rom the

sort of

optionality

we

have

analyzed,

nonetheless it does

embody

the same

principle

that

undertaking

n investment

at

any point

in

time is at the cost

of

foregoing

the

option

to

undertake other

projects

that

may

become available. In

effect,

while

projects

are

not

being delayed

o take

advantage

of

lower

interest

rates,

they may

be

put

on hold to take

advantage

of

superior

alternative

projects.

IV.

A

General Alternative

o the

NPV

Rule and a

Simple

Rule

of Thumb

The lesson from this

analysis

of the NPV rule is

that we

must treat all investment

problems

as

option

valuation

problems.

Consider

a

project

with

cash flows of

c(t)

over

time.Forease of

exposition,

we will assume hat hese flows

are

deterministic

and

leave the extension to random

cash

flows for later.

If

P(t)

denotes

the currentvalue of a

pure

discountbond

paying

$1

t

years

from

now,

then the NPV of a

project

f

it

is undertaken

oday

is

given by

NPV = p(t)c(t)dt

Ignoring

the

possibility

that the

projectmay

change

if

it

is

delayed,

the above NPV formula tells us

what

will

be

realized at

any

time that the

project

is undertaken.

The

decision to undertakehe

project,

hen,

s

a

decision torealize

this NPV

and to

forego

the

opportunity

o

realize a

higher

NPV if interestrates

should

happen

o fall.

Evaluating

his trade-offbetween

the current ealization

and the

potential

future value

is a

complex

mathematical

problem.

It is dealt with in

detail

in

Ingersoll

and Ross

(1992).

In

essence,

though,

f we are

willing

to make some



6/8

100

FINANCIAL

ANAGEMENTAUTUMN

995

Figure

1. NPV and OANPV

OANPV

OAIRR

IRR

Valuef

ighto delay

Interest

ate

NPV

reasonable

assumptions

aboutthe behaviorof

interest

rates,

thenthe solution to the problemcan be characterizedn an

intuitively

appealing

ashion.

Assume, first,

that there is some

rate,

perhaps

he short

rateof

interest

or,

maybe,

the

five-year

rate,

hatcan be used

to characterizehe movement

of

interest ates.As an

intuitive

matter,

clearly

the solution

will

be to undertake he

project

whenever this relevant interest rate

falls below some

appropriate

urdle

rate,

r*.

Letting,

IRR

denote the internal

rate of return or the

project,

then it's clear that

an

optimal

choice

of r*

will

be less thanIRR. In

effect,

we

will

demand

that he

project

be

in-the-money

o

give

us

enough

additional

value from

undertaking

t

to offset the lost

opportunity

of

waitingfora possible drop n interestrates.

For

any

choice

of

r*,

the formula for the value of the

project, including

the

option

to

delay choosing

it,

will be

given by

the

expected

discounted

value over all

possible

future interest

rate

paths.

If we let

NPV(r*)

denote the net

present

value of the

project

when the interestrate is

r*,

then

we

can

compute

the value

of

the

project ncluding

he value

of the

option

to

delay

by following

a simulation

procedure.

First,

we

simulate

a future nterestrate

scenario.

Second,

we

markthe

first time

in

this scenariothat

the

interestrate falls

to r*.

Third,

we

compute

the discountedvalue

of

NPV(r*)

along

the interestrate

path

for

this

scenario

up

to the marked

time. In

other

words,

if

the marked time is

t,

then we

computethepathcontribution,

fr(s)ds

e-

S

NPV(r*)

If the

path

never crosses

r*,

then

its contribution s zero.

We

repeat

his

process

n times

and

average

the

resulting

contributions,

ontribution(j):

(1)I

contribution

j)

This

sum is

an

approximation

to the exact

expected

discounted

net

present

value of

the

project

with

the exercise

rule, r*,

r(s)ds r(s)ds

VALUE

=

E(e1-4

NPV(r*))=

NPV(r*)E(e1-f)

where

I

is the random ime

at

which

the interest ate

path

irst

hits r*.

While the above

procedure

s

appropriate

or

large

and

complex

projects,

t is useful to

have

an

approximation

or

simpler

decisions. For

simple point input

and

point output

projects,

Ingersoll

and Ross

(1992)

developed

an

analytic

formula

or

the value

including

he

option

o wait.

Using

this

formula,

an

approximation

s

available for

determining

he

cutoff hurdlerate at which a

more

general

project

should be



7/8

ROSS

USES, ABUSES,

AND

ALTERNATIVESO

THE

NET-PRESENT-VALUEULE

101

undertaken.Let T

denote the durationof the

project,

and

let

rT

denote the

yield

on

a

zero

coupon

bond

with

maturity

T.

The

appendix

shows that the

optimal

yield

at which

to

undertake project s approximated y

rT

=

IRR

-

a/-

where a?J- is the standarddeviation of

the interest rate

at level

r.

An interest

ateof 6%andan annual

proportional

tandard

deviation of 20%

(i.e.,

oY(dr/r)

0.2 and

o(dr)

=

0.2

x

6%

=

120

bp) produces

a

= 0.2 x

0.06/4=0.06

=

0.049

This would result

n

a differencebetween the

optimal

cutoff

andtheIRR of

r*(T)

IRR

=

-o/2-

=

-0.035

or

3.5%.

Notice that the

optimal

cutoff is lower

than the IRR

which

captures

the sense that the

option

to

undertake he

project

has to be

in-the-money

o be

exercised. As an IRR

rule,

this would meanthatwe

undertake he

project

when

the

IRR

exceeds themarket ateof interest

by

3.5%.

Notice, too,

thatthis is a

significant

difference n

that

t

is

on

the

orderof

half of the

interestrate tself.

V. Conclusion

For most

investments,

the usefulness of the NPV

rule

is severely limited. As a formal matter, it applies only

in those cases where

the investment

opportunity

nstantly

disappears

f it is not

immediately

undertaken.n

fact,

the

vast

majority

of investments have

a not

insignificant

time

period

over which

they may

be

undertaken,

nd this

implies

that

they

have an embedded

optionality

on their

own

valuation

that is exercised when

the initial investment

is

made. We must

take

very seriously

the caveat

to the

NPV

that it

applies only

in cases where

an investment

does

not

preclude

some

alternative

nvestment,

because

every

investment

ompetes

with

itself

delayed

n

time. It is

not that

the NPV rule is

wrong,

rather t is

somewhat

rrelevant,

nd

atbest, it must

generally

be modified to be useful.

Because

nearly

all

investments involve the

option

to

undertake them when

financing

alternatives are

more

favorable,

n

general,

the

preferredway

to

deal with

such

investment

decisions

is

to treat

them as

serious

options

on

the

financing

environment.

As we have

shown,

when

evaluating

investments,

optionality

is

ubiquitous

and

unavoidable.

If

modem

finance is to have

a

practical

and

salutary

mpact

on

investment-decision

making,

it

is

now

obliged

to treat

all

major

investment

decisions as

option

pricing problems.

U

References

Antle,

R.

and G.

Eppen,

1985,

CapitalRationing

and

Organizational

lack

in

CapitalBudgeting, Management

Science

(February),

163-174.

Antle,

R. and J.

Fellingham,

1990,

Capital

Rationing

and

Organizational

Slack in

a Two Period

Model,

Journal

of

Accounting

Research

(Spring),

1-24.

Ingersoll,

J. and

S.

Ross,

1992,

Waiting

to

Invest,

Journal

of

Business

(January),

1-30.

Ross, S.,

R.W.

Westerfield,

and J.F.

Jaffe,

1993,

Corporate

Finance,

3rd

ed.,

Homewood,IL,

Irwin.

Ross,

S.,

1978,

A

Simple

Approach

o the

Valuationof

Risky

Assets,

Journal

of

Business

(July),

453-475.



8/8

102

FINANCIAL

ANAGEMENT

AUTUMN

995

Appendix

This

appendix

uses

the results

of

Ingersoll

and

Ross

(1992) toderiveanapproximationor theoptimalhurdle ate

at which an

investmentshould

be undertaken.

We will

startwith

point

input-point

outputprojects

and

let

I

denote

the

ratio of

the

point

investment to the

point

output

at time T. The

interestrate

dynamics

are

given

by

the

Ito

equation

or the

short

rate,r,

dr

=

Xrdt+

a

Trdz

where

z

is a

Brownian

process.

This can

be

thought

of as

either the

actual

dynamics

of the interest rate

or the risk

neutral

dynamics

with risk

adjustment

oefficient,

k.

Under hisassumption,he term structure f interest ates

can be

solved and the

price

of a zero

coupon

bond is

given

by:

p(r,T)

=

e-b(T)r

where

b(T)=

2(er

-

1)

(y

-

X)(eY

-1)

+

2y

and

7 = XA2+ 202

The

Ingersoll

andRoss

formula or the

optimal

short-term

interestrate at which to

exercise

the

option

to

undertake he

project

s

given by:

r*

=r?

+?

l

(v

-

b(T))r

where

ro is the

instantaneous

ate

at which the

project

has

a

zeroNPV

ro 1nI

b(T)

and

V

(2

We can recast this

solutionin

terms

of

the cutoff

hurdle

rateon a

T-period

bondand the IRRfor

the

project.

The

IRR

is

given

by

IRR

=- nI

Hence,

thehurdle

rate

expressed

n

terms

of a

T-period

bond

yield

is:

Sb(T)IRR

Il(v

-

b(T)

rT=

T

=

IRR

Tn v

To

simplify

the

computations,

we assume

that

the

local

expectations hypothesis

holds and

that there is

no

interest

rate

risk

premiumper

se,

i.e.,

we set X = 0.

Approximating

the hurdlerate for

small choices of

a,

through

a

Taylor

expansion,produces

rT

=

IRR-

(J

In

general,

this

approximation

s valid for

well-behaved

projects,

.e.,

projects

witha

single

IRR.

Interestingly,

ince

the

approximation

s

independent

f

time, T,

there

s

no

need

to estimate heduration

of the

project.

This content downloaded from 142.103.160.110 on Sun, 8 Sep 2013 23:33:17 PM
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

Documents

Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8