ross, 1995, uses, abuses, and alternatives to the net-present-value rule 8
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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.
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7/25/2019 Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8
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.
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7/25/2019 Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8
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
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7/25/2019 Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8
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
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7/25/2019 Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8
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
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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
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7/25/2019 Ross, 1995, Uses, Abuses, And Alternatives to the Net-Present-Value Rule 8
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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.
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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.
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