CALCULATING ABANDONMENT VALUE
USING OPTION PRICING THEORY
Stewart C. Myers
Saman Majd
#1462-83
Revised:
May 1983
August 1983
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Calculating Abandonment Value Using Option Pricing Theory
Stewart C. Myers*
and
Saman Majd**
Sloan School of Management, M.I.T.
May 1983
(revised: August 1983)
-
11^1_-_1___1_1__ __
.
Calculating Abandonment Value Using Option Pricing Theory
ABSTRACT
Conventional capital budgeting procedure values investment
projects
as
if they will
be
undertaken
for
a given
economic
life, and assigns a prespecified salvage value to the assets at
the end of the life.
This ignores the value of the option
to
abandon the project early.
This paper models the abandonment option as an
put
option on a dividend paying
yield and exercise price.
the abandonment
value
stock, with varying
A general procedure for
is presented, along with
examples illustrating its practical importance.
some
American
dividend
calculating
numerical
1
Calculating Abandonment Value Using Option Pricing Theory
Modern finance advocates the use of net present value
evaluating capital investments.
a
capital
expected
investment
after-tax
budgeting
has
The net present value (NPV) of
project
cash
is
the
flows.
concentrated
in
Most
on
present
of
the
value
the work
on
difficult
specifying the appropriate discount rate.
of
its
capital
problem
Modelling the
of
cash
flows to the project is equally important, however.
Consider the choice between two production
technologies.
Technology A employs standard machine tools which have an active
second
hand
market.
Technology
B
uses
custom
designed,
specialized equipment for which there is no second hand market.
The two technologies produce an identical product and
revenues,
but
technology
operating costs.
is more
efficient
has
lower
are
worn out and scrapped.
If the two alternatives' cash flows
are
projected
assumptions and
under
then
Production continues
and
until the machines
rates,
these
known.
"early"
But
discounted
the NPV of B is greater than
calculations would
11111_
B
identical
it
is
at
that of A.
the
same
These
presume that the duration of production
not known:
if
production may
be
is
halted
(before the machines are worn out) then technology A's
.1
.__
greater
salvage
value
makes
it
relatively
more
attractive.
Technology A's value in the second-hand market increases the net
present value of using it.
Standard capital budgeting procedure assigns an
salvage value
to
project life.
However, salvage value also affects the value of
an
because
investment
early.
is
assets at
of
the end
the option
of their
expected
to
(pre-determined)
abandon
the
project
The project will be abandoned if the value of continuing
less
than
the
salvage
value
at
that
time.
Conventional
capital budgeting fails to take this option into account.
The
true
value
of
a
project
includes
its
abandonment
value, which depends on the salvage value and the optimal time
to abandon.
The optimal time to abandon is of course not
when the project
is undertaken,
but will depend on
known
subsequent
performance.
Robichek and Van Horne [16] provide an early analysis of
abandonment value.
They recognise that the option to abandon a
project
be
early
can
practical importance.
valuable,
and
illustrate
the
option's
Their examples, however, assume that the
project will be abandoned as soon as the salvage value exceeds
the present value of the remaining expected cash flows.
Dyl and
Long [4] emphasise that the optimal time to abandon the project
3
will not, in general, be the first instance where salvage value
exceeds the present value of the remaining expected cash flows.
Rather,
the
recognize
abandonment
that,
if
the
decision
at
each
project
is
not
point
in
time
must
abandoned,
the
firm
retains the option to abandon in the future.
Unfortunately, Robichek and Van Horne's solution procedure
(as corrected by Dyl and Long) is not practical when applied
realistic situations.
to
The subsequent finance literature has not
provided a practical approach to solving for the value of the
abandonment option and the optimal time for abandonment.
The option to abandon a project is formally equivalent to
an
American
put
option
and
can
be
valued
techniques developed to value options on stocks
is not a simple put option:
flows
and
has
an
by
applying
1/
.
However, it
the project yields uncertain
uncertain
salvage
value.
the
These
cash
factors
significantly complicate the solution procedure. 2
This paper presents a general procedure for estimating the
abandonment value
of a capital
investment
project.
The
next
section specifies the abandonment option as a contingent claim,
and
discusses
value.
valuation
___l___l_______rs__l__I______
some
Section
of the
important
II describes our
procedure, and
section
factors
simplifying
III
1__1_1_
presents
that
affect
its
assumptions
and
some
numerical
examples
of
uncertainty
analysis.
the
in
calculations.
the salvage value
Section
can be
IV
discusses
incorporated
Section V offers some concluding comments.
in
how
the
I.
Problem Specification
The option to abandon a project is formally equivalent to
an American put option on a dividend-paying stock:
the exercise
price of the put is the salvage value of the project;
the
flows from the project are equivalent to the dividend
on the stock.
Also, the
cash
payments
project can be abandoned at any
time
during its life.
Each of these factors affects the abandonment value of the
project, and to
solve for the abandonment value, each must be
explicitly modelled.
This section discusses how these
affect the abandonment value.
factors
The specific assumptions we make
to implement the solution technique are described in section II.
A.
Cash Flows and Payout Ratios
In the classical model of stock valuation, the value of a
stock
is
the
present
value
of
its
expected
dividends.
Similarly, the value of a project is determined by the present
value
of
future
its
cash
arrives, the
---------
------I-'x
-------^--
expected cash
flows
are
value
of the
---------
flows.
revised
as
Since expectations
new
information
project varies
---------P-
about
randomly
randomly about
its
---
6
expected value.
therefore
The uncertainty in the value of the project
related
to
that
of
the
cash
flows,
although
is
the
relationship is generally complex.
To apply the contingent claim valuation techniques to our
problem we must specify the
on the
process generating
flow, rather than project value,
the cash
flows.
Project cash
is the natural state variable.
However, we can express cash flow as a function of asset
t
=
Ct/Vt
.
the
However, in capital budgeting we normally
value of the project.
focus
stochastic process generating
These "payout ratios"
value:
( y ) can be functions of
time and project value.
The
following
simple
example
shows
how we
restate
forecasts of cash flows in terms of payout ratios.
the
Suppose that
cash flows are forecasted to be constant over the life of
project,
so
that
it
resembles a
simple
annuity
the
la).
(Figure
Since the value of the project at any time is the present
value
of the remaining cash flows, we can derive the expected path
the project value over time from the forecasted cash flows.
project value
will
decline over
time as
these two sets of forecasts we can
in
Figure
lb.
of
The
From
forecast the payout ratios
implied by the expected path of project value:
in this
the payout ratios will increase over time (Figure
c).
example
7
Suppose there is a forecast error in the cash flow.
happens
simple
is,
t
to
the
conditional
forecast
of
the
payout
What
ratio?
assumption is that the payout ratio is constant.
the
percent
forecast
error
=[c(-Etl(Ct)] /Et 1(Ct)
t
t-1 ](
,
in
causes
the
the
Vt = Et-1(Vt) [1+Et]
change in project value:
That
cash
same
A
flow,
percentage
.
Hence
the
forecasted payout ratio is unchanged.
If the assumption that payout
project
value
necessary
future
for
cash
budgeting
seems
using
flows,
unduly
restrictive,
a single
which
ratios are independent
is
note
risk-adjusted
standard
that
it
is
rate
to
discount
procedure
in
capital
3/ .
Other assumptions about the effect of forecast errors
the
cash
of
flows
on
forecasted
payout
ratios
are
possible.
mean-reverting cash flow, for example, causes forecasted
ratios to be
functions of both time and project value.
in
A
payout
More
complex specifications of payout ratios as functions of time and
project value are also possible.
The payout ratios used in the numerical valuations below
are constant or functions of time only.
___
__11_1_1______1_11_1-----__11
---_1___
I
111
B. Asset Life and Salvage Value
Conventional capital budgeting treats salvage by assigning
a prespecified salvage value to the asset at the end of
predetermined
life.
But
what
determines
the
some
'life'
of
a
project?
The physical life of an asset depends on the time at which
it will wear out and must be replaced.
of the asset.
This is the maximum life
Physical life could be infinite or very
For example, think of land or a hydroelectric facility
The
economic life of the asset is
during which the asset is being used.
is
terminated,
if
the
asset
is
4/
.
the length of time
Thus even if the
still
long.
being employed
project
in an
alternative use, its economic life continues.
The project life is not fixed, but is determined by the
decision to abandon.
It is solved for simultaneously with the
value
to
of
the
abandonment
option
value
will
abandon.
therefore
life.
I,-'---
·--- --------~I'~`~'- ~~~c~~~--~-~-~--~~~'~~'"
The
also
determinants
determine
the
of
the
project
9
A
project may
never
be
abandoned.
In
that
case
the
project life, the economic life and the physical life will all
be equal.
Generally, however, the project life is less than the
economic life, which is less than the physical life.
If the decision to abandon determines project life,
determines the life of the abandonment option?
the
project can
be
abandoned
at
In
any time during
what
principle,
the
asset's
physical life, therefore this is the appropriate maturity
for
the abandonment option.
The salvage value of an asset can vary over time and
not be known in
is
an
active
technological
predictable
advance.
Standardized assets, for which there
second-hand
change,
salvage
may
market
may
have
schedule.
and
a
which
experience
relatively
Assets
little
stable
subject
and
to
rapid
technological change may have unpredictable salvage values.
The salvage value at any time is the market value of
asset in its next most productive use.
It is net of any
costs
of converting from one use to the other, and
incorporates the
value of any subsequent options to abandon.
Some assets with
several
possible
uses may
their physical life.
be
abandoned
several
ill
times
during
Most land, for example, has many different
productive uses and infinite physical life.
1·_1_11
the
._
In such cases, the
10
is
option to abandon (i.e., to switch from one use to another)
like
an
'option on
if
an option':
the project
is
abandoned
before its economic life is over, its salvage value includes the
value of terminating its next tour of duty, and the next
gets
another
abandonment
specification of the
option.
stochastic
Therefore
properties of
a
user
complete
salvage
value,
including its relation to project value, is not an easy thing to
write
down
or
analyze.
However,
we
can
cope
with
uncertain
salvage values under the assumptions described in section IV.
11
II.
Solution Procedure
Valuing an option requires solving a partial
equation
whose
5/
claim
contingent
conditions
boundary
.
Sometimes
define
the
the
equation
differential
nature
and
of
the
boundary
conditions allow for a closed-form solution, as in the
classic
Black-Scholes equation for the value of a European call
option.
More generally, however, a closed-form solution does not exist,
forcing us to seek a numerical approximation.
partial
The
½ o2App
value is:
where
P is
the
equation
differential
+
value
[rP-y(P,t)] A
of
the
for
the
abandonment
-rA + A = 0,
underlying
project,
o
is
the
standard deviation of the rate of return on the project, r
is
the riskless interest rate, Y(P,t) is the payout ratio, and A is
the abandonment value.
Two
boundary
abandonment
the
specifying
conditions
(1) if the value
option are:
nature
of
the
of the project
is
zero, the value of the option is the salvage value at that time:
A(P=O,t)
= S(t);
infinitely
lim
P-+oo
large,
A(P,t) = 0.
(2) as
the
the
value
of
value
the
of
the
option
project
tends
to
becomes
zero:
12
A third condition
is
applies when the option
exercised:
the value of the project is the greater of the salvage value
the value of continuing
or
abandonment).
(but with optimal future
This condition is invoked at each point in time, and thus
the
optimal abandonment schedule is solved for implicitly.
Finally, a terminal boundary must be assigned, where
the
This boundary is
the
value of the project is taken as zero.
physical
life
of
the
asset,
maturity of the option is:
and
the
boundary
condition
A(P,t=T) = max[S(T)-P(T),O].
The most general specification of the abandonment
does not allow a closed-form solution because:
salvage
values are
value in a
uncertain
and
are
(generally) complex manner;
related
depend on time and project value in a
manner;
6/
option
(1) the future
to
the
project
(2) the future payout
ratios, as defined in the previous section,
can
at
are uncertain
and
(generally) complex
(3) the abandonment option is an American option for
which early exercise will generally be optimal, and the timing
of
early
exercise
must
be
jointly
determined
with
the
abandonment value.
We focus
flows
value.
and
first on
the uncertainty regarding future
project values by assuming a deterministic
cash
salvage
Specifically, we assume that the initial salvage value
13
is known and subsequently declines at a known constant rate.
We
also assume that the payout ratios are constant over the entire
physical life of the asset.
uncertain
but
couples
This allows the cash flows to
their
uncertainty
to
that
of
be
the
7/
project 7/
Since there is no closed-form solution to even this simple
formulation of the abandonment problem, we must use a numerical
approximation
technique.
Several
techniques
for
such
approximations are available.
We employ an explicit form of the
finite difference technique
.
This
numerical
approximation
results
in
a
relationship
between the value of the abandonment option in any period and
its values in the next period.
We can employ this
relationship
recursively, starting with the values at the terminal boundary,
and working back to the abandonment values at the start of
project.
In
addition,
abandonment decision.
sufficient
optimal
the
procedure
dictates
the
optimal
Since the project present value (PV) is a
statistic for the value of the future cash
flows,
abandonment is expressed as a schedule of project
over time.
Should the project PV fall below this schedule
any time the project will be abandoned.
be equal to
the
This schedule will
the present value of remaining cash flows
(as
PVs
at
not
in
Robichek and Van Horne) because our recursive procedure includes
14
the value of optimal future abandonment at each point.
current
abandonment
future abandonment.
decision
implicitly
accounts
for
Thus the
optimal
15
III.
Numerical Examples for the Simplified Abandonment Problem
For the base case calculations we assume a constant payout
ratio (constant Y ), and also that:
1.
The initial project present value (PV) is 100.
2.
Project cash flow and PV are forecasted to decline by 8
percent per year.
Hence in the absence of salvage, the
project would be perpetual (i.e., economic life = physical
life = m ).
However, we arbitrarily terminate
the
physical/life of the asset in the distant future (after 70
years)
.
This is the terminal boundary where the
project value is zero.
3.
The initial salvage value is 50, declining
by 5 percent per year.
4.
The standard deviation of forecast error of project PV is
20 percent per year.
5.
The real risk-free rate of interest is 2 percent per year.
Figure
exponentially
2 shows how the forecasted project PV and salvage value
change over time.
Note that the initial PV of 100 is based
on
the assumption that the project will never be abandoned prior to
the
end of the asset's physical
life,
and
therefore does not
include any allowance for salvage value, even at the
date.
terminal
Therefore the abandonment value we calculate is the extra
value due to the option to abandon in favor of the salvage value
at any time during the project life.
-·-------
---
III
16
Table I shows the results of the base case calculations.
Each entry in the table is the abandonment value given project
start
first column gives the abandonment value at the
The
value.
of
the
project
for
initial
different
project
values.
abandonment
Thus, since the initial forecasted PV is 100, the
value is 5.63, or approximately 6 percent of project value.
If
this project required an initial investment of 100, making
its
NPV (without any salvage) zero, the abandonment value would make
the project worthwhile.
The
Table I:
optimal
abandonment
decision
is also
indicated
in
if at any time the project value falls below the PV
corresponding to the line drawn in the table, the project should
In our base case, for example, if the
be abandoned.
value
in year 2 is less than 24.53,
abandoned.
the
project
project
should
be
Similarly, if the project value is less than 20.09
in year 9, the project should be abandoned.
The direction of the change in abandonment value can
predicted from standard option pricing theory:
be
(1) an increase
in the salvage value (exercise price) will increase the value of
the abandonment option (put option);
(2) an increase in the
volatility of the value of the project (the underlying asset)
will increase the value of the option;
forecasted
_X·____1_____1·_
lll^-__·_._tlll-l_
·)·.._.1-·---174--11--1--_·_--1--1---111-
project
.._...
PV
(the
current
(3) an increase in the
value
of
the
underlying
17
asset) will decrease the value of the option;
(4) a decrease in
the physical life (the maturity of the option) will decrease the
value of the option.
These results were verified
numerically
and are presented graphically in Figure 3.
The abandonment value calculated by our procedure is added
to
the
present
value of cash flows
obtain the total project value.
the
present value
of cash
(without any salvage)
In our base case, for
flows
without
salvage
example,
is
100,
abandonment value is approximately 6, making the total
value
106.
How
does
this
differ
from
the
to
the
project
conventional
calculation of project value?
The conventional approach assumes that the project will be
terminated when the salvage value equals the present value
the remaining cash flows
(i.e.,
when the salvage value
the forecasted project value without savlage).
equals
The conventional
project value is therefore the present value of the
forecasted
cash flows up to the termination date plus the present value
the
forecasted
salvage
value
at
that
of
date.
Since
of
the
termination date is the time when the salvage value equals the
present value of remaining cash flows, the conventional
project
value will not change when salvage value is substituted for the
value of remaining cash flows.
In our base case the
project life is 23.1 years (see Figure 2).
forecasted
The present value of
_
18
and the
present
4, making the total
project
cash flows up to this termination date is 96,
value of salvage at this date is
present
value
100.
is
This
also
the
project
present
value
without any salvage.
The forecasted project life can be changed by varying the
initial salvage value.
the longer
value
of
The smaller the initial salvage value,
the forecasted project life, the larger the present
cash
flows
up
to
present value of salvage.
salvage,
termination,
The total
remains constant at
100,
as
and
the
smaller
project value,
described
the
including
above.
results are shown in Table II for a range of forecasted
These
project
lives.
Also
value
shown in Table II
The
calculation.
abandonment or
salvage
extra value due
is
abandonment
is the corresponding
project
100.
present
The
value
abandonment value
to the option to terminate the
early
without
is
project early,
and this is added to the present value of cash flows to get
total
project value.
with
conventional
from
the
option.
project
The results show that the
allowance for
value
with
for
the
the
project value
salvage can be very
allowance
the
different
abandonment
19
We also performed
varying
payout
ratios.
an
illustrative calculation
Specifically,
we
assumed
assuming
that
the
project payout ratio is zero in the first five years, 2 percent
in the next ten years, and 20 percent per year thereafter.
This
implies the pattern for the forecasted project value shown
Figure 4.
in
In this example, if the initial forecasted PV is 100,
the abandonment value is 3.54,
or approximately 4 percent
of
project value.
__________________ 1___11_1______1_____ ____
_____
20
IV.
Modelling Uncertain Salvage Values
Abandonment
physical
life:
can
the
occur more than
asset
another several times.
may be
once during
switched
option
Unfortunately,
is
this
one
asset's
use
to
Included in the salvage value each time
abandonment occurs is the option to abandon
abandonment
from
an
an
option
makes
on
a
complete
again.
sequence
Hence the
of
specification
options.
of
the
stochastic properties of the salvage value extremely difficult.
As
a first
step towards
introducing
uncertainty
salvage value we make two simplifying assumptions.
in
the
The first is
that identical assets are being used elsewhere, and that these
assets are traded.
Thus the salvage value is a market
price.
Second, we assume that the stochastic processes for the project
value and the salvage value are:
dP/P = (a - Yp)
p
p
dt+
a
P
dZ ,
p
and,
dS/S = (
S
- Y) dt =
s dZs
S
S
S
Here P is the project value, a the expected rate of change
P
P, Y
the project payout ratio,
rate
of
change
of
P,
and
of
the standard deviation of the
dz
the
standard
Weiner
process
P
generating the unexpected changes in P
the salvage value,
S, are
11/
.
similarly defined.
The parameters for
The project and
21
salvage
values are
coefficient
for
similar
instantaneous correlation
[9] has valued an option to
another.
to
his,
interpreted
analysis
with
P .
Margrabe
asset
correlated,
as
The assumptions
where
the
above make our
stochastic
salvage
one of his risky assets.
assumes
no
payouts
from
the
strictly apply to the abandonment option
However,
exchange one
we
can
use
the
assets,
framework
value
However,
it
can
be
since
his
would
option,
valuation
procedure
developed
and
shows
that
after
suitable
in the case of deterministic salvage.
(1) Redefine
the
state
project to salvage value (X=P/S).
2
this new state variable is
exercise
riskless
salvage
The appendix derives the differential equation for this
transformation,
option's value obeys the same differential equation as
follows:
not
12/
earlier to value the abandonment option with uncertain
value.
risky
price
equal
interest
to
rate.
abandonment with uncertain
the
applies
This transformation is as
variable
as
the
ratio
of
(2) The standard deviation of
2
2
a = a -2po a +o
X
P
ps s
unity.
With
(3) Set
the
(4) Substitute
Y
the
these
adjustments,
four
salvage is reduced to an
option with known and constant exercise price.
XII__
for
equivalent
22
Table
III
presents
when salvage value
abandonment
value
results
is uncertain.
is
sensitive
of
abandonment
calculations
As the results indicate,
to
changes
in
both
the
the
correlation between salvage and project values and the standard
deviation of salvage value.
23
V.
Conclusions
It
is
easy to
think
American put option, and
insight
to
practical
application in
of
the
abandonment
somewhat more
use.
This
option
difficult
as
to put
paper discusses
an
that
problems
of
some detail and presents numerical estimates
of
abandonment value for halfway realistic examples.
The
obvious
next step is to expand the numerical valuation program to allow
project
payout
ratios
(i.e.,
cash
flow to
value
ratios)
to
depend on project value as well as time.
Such
problems.
a
program
could
help
Here is one example.
solve
a
variety
of
other
Suppose you expect to need
new plant ready to produce turbo-encabulators in 36 months.
design A is chosen construction must begin immediately.
B is more expensive, but you can wait 12 months before
ground.
Figure
construction
deadline.
5
costs
shows
for
the
the
two
cumulative
designs
present
up
to
the
a
If
Design
breaking
value
of
36-month
Assume the designs, once built, are equally efficient
and have equal production capacity.
A standard discounted cash flow analysis would rank design
A ahead of B.
falls
and the
But
suppose the demand for turbo-encabulators
new factory
_
is not needed:
__
then, as
Figure
5
III
24
shows, the firm would be better off with design B provided
the
project is abandoned before month 24 13/
This is also an abandonment value problem.
asset
is
assuming
Think
the
present
the
firm
of putting
value
must
the
of the
complete
present
expenditure in an escrow account.
be larger
for
turbo-encabulator
construction
value
of
design B than design
balance in
the
of
required
the
project
plant.
construction
The account would of course
A.
abandoned before month 36, however, its
unspent
The underlying
If either
design is
'salvage value' is the
escrow account.
This
is the exercise
price at which the firm can 'put' the turbo-encabulator project
between month zero and month 36.
We are back to a standard abandonment option, where
exercise
price
is
investment
in
value
equal
is
commitment to
the
determined
design being
to
(1)
by
the
pattern
valued.
project
complete construction
cumulative
Project net
present
14/
of
value
, less
present
assuming
(2) the
is completed.
(3)
construction
The value of (3) reflects the option to
part of the construction costs comprising (2).
a
present
value of construction costs assuming this commitment, plus
the present value of the option to abandon before
the
recover
25
Design B could be more valuable than A, if B's abandonment
value outweighed its higher cost.
It would be interesting to analyse the choice between gas
turbines,
coal,
and
nuclear
power
plants
pricing framework.
__
·______I
_I_____
using
this
option
III
26
APPENDIX
This
(PDE)
appendix
for
the
stochastic,
abandonment
and
option value
derives the
shows
is
partial differential
option
that with
when
the
salvage
equation
value
suitable transformation,
also a solution to
the
is
the
PDE for deterministic
salvage.
The derivation follows the methodology of Merton [12] and
the
reader
is
referred to
his
references
for
the
supporting
literature.
For simplicity, the derivation that follows assumes
a
type
European
abandonment
option;
American type option is straightforward.
reported
in
section
the
extension
to
an
(The numerical results
IV of the text refer,
of course, to
the
American abandonment option.)
The
stochastic
processes
describing
the
project
value
without salvage, P, and the salvage value, S, are:
dP/P = (
- yp)dt + a dZ ,
dS/S = (
- Ys)dt +
and,
The
rate
of
deviation ac
change
of
dZs
P
has
, the project payout
expectation
Ua
P
and
ratio is yp, and dz
standard
is
standard Weiner process generating the unexpected changes in
-1-
____....
____
___X-~-lll~~-
...--
the
P.
27
The
parameters for
salvage
values
are
S are similarly defined.
correlated
with
The
project and
instantaneous
correlation
coefficient P.
Let F(P,S,t) be the solution to the PDE:
2
P2 F pp+papa PSFp+2S2 F SS+(r-p)PFp+(r-
P
PP
PSS
)
=
SFrF+Ft
PP
S
O
subject to the boundary conditions:
F(P=O,S,t) = S(t),
lim F(P,S,t) = 0,
Pi*W
and
F(P,S,t=T) = max[S(T)-P(T), 0].
From Ito's lemma,
2
2
dF:[F + (a -Yp)PFp+(as-Y)SFs+½ o P F + pa a PSF
t
p p
p
S
S
p
pp
ps
S2 F
+½ o
]
] dt+ [pPFp] dz + [SFs
dz S
5
5
P
P P
s
Substituting from the PDE above, we have,
dF = FpdP + FdS + [rF-(r-yp)PFp-(r-YS)SFs]dt.
Consider the portfolio formed by investing the fractions x in P,
y
in
S, and
the
remainder
in
riskless Treasury bills.
The
dynamics for the portfolio are,
dY = xY (dP + ypPdt)/P +yY (dS + ysSdt)/S + (-x-y)Yrdt.
Choose
the
investment
x=FpP/Y and y=FsS/Y.
dY = F
according
to
the
rules:
Then,
(dP + ypPdt) + F (dS + YSdt) + (Y-F -F S)rdt
= dF + (Y-F)rdt.
__II__
proportions
II
28
If the
amount
initially
invested
in the
portfolio
is
Y(t=O) = F(P,S,t=O), then it is clear that Y(t) = F(P,S,t)
at
all subsequent times.
Further, the value of the portfolio, Y,
is equal to the function F(P,S,t) at the boundaries given above,
which by construction are identical to the boundaries for the
abandonment option.
Since the portfolio has the same payoffs as
the abandonment option, then to avoid dominance, the value of
the
abandonment
option
must be
given
by A(P,S,t)
= Y(t)
=
F(P,S,t).
The PDE for the abandonment option above can be simplified
by the transformation:
G(X,t) = A(P,S,t)/S, where X=P/S.
This
leads to the PDE:
2
½2x XC2G
where
x
=
+ (
Up -2p2p
- Yp) XG
+
Y
-
The
G + G
= 0,
new boundary conditions
are:
G(X=O,t) = 1,
lim G(X,t) = 0,
X-tCO
and
G(X,t=T) = max[1-X(T), 0].
This is identical to a formulation of the abandonment
option
with deterministic exercise price (equal to unity), and with the
riskless rate replaced by Y s .
The intuition behind this transformation is clear:
of the salvage value as the numeraire.
In these units,
project
its
value
is
P/S
(=X),
----
and
variance
think
the
is
-11-1-1-1-1-1- 1--------- -----------"- -1- - . --I-..---1-------- --.I---.-
111~~~~~~"
-
29
2
2
2
ax = ap -2Paps +o s.
exercise
The
price
is
now
known
and
consider
the
When
the
constant.
To
see why Ys
that
portfolio
is
replaces the riskless rate,
to
used
replicate
the
option.
salvage value is uncertain and is represented by a traded asset,
is used
this asset
to
hedge against
salvage asset
price.
The
which
includes
the
cash
changes
earns a fair
flows
to
the
in
the exercise
total rate of
asset.
return
However
the
exercise price changes only as the price of the salvage asset.
The difference between the total return to the salvage asset and
the rate of change of exercise price is the opportunity cost
holding an option
directly.
on the salvage asset rather than holding it
This difference is the payout ratio Y
the PDE instead of the riskless rate.
r^s
·
I
-- -
1
111
-----
of
which enters
30
FOOTNOTES
*
**
Professor of Finance, Sloan School of Management, MIT.
Doctoral candidate, Sloan School of Management, MIT.
1. For the seminal works on option pricing, see Black and
Scholes
[1]
and
Merton
[11].
For
a
comprehensive
review
article, see Smith [18].
2.
option.
Kensinger
[8]
analyses project abandonment as
a
put
However, his analysis assumes that the option is of the
'European' type with a non-stochastic exercise price.
This
is
equivalent to assuming that the project can only be abandoned at
one time, and that the salvage value
is known with certainty.
This misses important features of the option.
·----
·-- ··------ ·-- -·--·----··-·----- ·-- ------·-· -·----·---·-·--·-··--·------·--·-·----·----
·---·-·-·-- ··-,,----------- ---- -----------·--------------- ·------
31
3.
using
a
For
a
single
discussion
of
risk-adjusted
the
assumptions
discount
rate,
necessary
see
for
Myers
and
asset with very
long
Turnbull [14].
4.
physical
Here
life.
vintages.
Consider
There is
which maintains
could
is another example of an
live
the
a
fleet
of
trucks
a program of maintenance
fleet's
indefinitely.
productive
However,
it
of
different
and
replacement
capacity.
has
The
fleet
abandonment
value:
its owner may decide to get out of the trucking business.
5.
to
Black and Scholes [1] and Merton [11] were the
derive
such
options.
partial
differential
equations
for
first
financial
Much of the subsequent literature on option
pricing
has followed their methodology and assumptions.
6.
be
Options on stocks with uncertain dividend payouts can
valued
given
distribution of
specific
assumptions
stock price and
about
dividend payout
Geske [6].
____III__I_____^L___I___·
___
the
ratio.
joint
See
32
7.
Note that this is not a causal
relationship
the value of the project and the cash flows.
between
The causality runs
the other way, from cash flows to value.
8.
For
a
discussion
of
numerical
methods
for
solving
partial differential equations and examples of their application
to problems in financial economics, see Brennan and Schwartz [2]
and
3], Mason [10],
Parkinson
[15], and Schwartz [17].
Geske
and Shastri [7] provide a useful summary of the major numerical
methods.
9.
Because our
solution procedure
starts
at
a terminal
boundary and works back recursively, we need
a finite
for
has
our
calculations.
Since
the
base
case
an
horizon
infinite
horizon, we approximate this by setting the boundary far in the
future, at 70 years.
Sensitivity analysis shows that this does
not induce a significant error in the initial abandonment value.
10. This implicitly assumes that there is no
option
for the salvage asset.
If there
abandonment
is, then the
market
value of the salvage asset includes the value of this option,
and
the
simple
dynamics
posited
for
S
will
no
longer
be
33
appropriate.
In particular,
as will no longer be constant and
may depend on S in a complex manner.
11. For
a
detailed
discussion
of
the
assumptions
underlying the use of such processes in financial economics see
Merton
131.
The
use
of these processes is
option pricing literature
standard
in
the
(see Black and Scholes [1] and Merton
[11]).
12. Fischer
5] also values a European call
stochastic exercise price.
Asset
Pricing
expected
Model
return
salvage asset
if
on
can
He also describes how the
be
used
a security
this
option
to
infer
perfectly
security is
not
the
Capital
equilibrium
correlated
traded.
with
with
Stultz
the
[19]
extends Margrabe's results to more general European options
on
the maximum or minimum of two risky assets.
13. We assume for simplicity that construction outlays are
totally lost if the project is abandoned before construction is
complete.
Our story is easily adapted if some of the
outlays
can be recovered.
~~~~~ ~~~~_1_11~~~~~~~~~____~~~~~~~~~~~_~~~~--
34
14. This present value would include the present value of
abandoning after month 36.
35
REFERENCES
1.
Black, F., and M. Scholes, "The Pricing of Options and
Corporate
Liabilities",
Journal of Political Economy,
May/June
1973, 637-659.
2.
Brennan, M.,
and
E.Schwartz,
"Finite
Difference
Methods and Jump Processes Arising in the Pricing of Contingent
Claims:
A
Synthesis",
Journal of Financial and Quantitative
Analysis, September 1978, 461-474.
3.
Brennan, M.,
Retractable
Bonds
and
and
E.Schwartz,
Callable
Bonds",
"Savings
Bonds,
Journal of Financial
Economics, August 1977, 67-88.
4.
Budgeting:
5.
Price
Dyl, E., and H. Long, "Abandonment Value and
Capital
Comment", Journal of Finance, March 1969, 88-95.
Fischer, S.,
"Call
Option
Pricing When
The
Exercise
Is Uncertain, And The Valuation of Index Bonds", Journal
36
of Finance, March 1978, 169-176.
6.
Geske, R.,
"The
Pricing
of
Stochastic
Options With
Dividend Yield", Journal of Finance, May 1978, 617-625.
7.
and
Geske, R.,
A
Approximation:
Shastri,
K.
Comparison
of
"Valuation
By
Valuation
Alternative Option
Techniques", Working Paper 13-82, Graduate School of Management,
U.C.L.A.,
8.
1982.
Kensinger, J.,
"Project Abandonment as a Put
Option:
Dealing With the Capital Investment Decision and Operating Risk
Using
Option
Pricing
Theory",
Working Paper 80-121,
Edwin
L.
Cox School of Business, Southern Methodist University, 1980.
9.
Margrabe, W., "The Valuation of an Option to Exchange
One Asset for Another", Journal of Finance, March 1978, 177-186.
10. Mason, S.,
"The Theory of the
Numerical
Certain Free Boundary Problems Arising in Financial
Analysis
of
Economics",
37
unpublished
Ph.D.
Management, M.I.T.,
dissertation,
chapter
3, Sloan
School
of
1979.
11. Merton, R.C.,
"Theory
of
Rational
Option
Bell Journal of Economics and Management Science,
Pricing",
Spring
1973,
141-183.
12. Merton, R.C., "On the Pricing of Contingent Claims and
the Modigliani-Miller Theorem",
November 1977,
13.
Journal of Financial Economics,
2 4 1-249 .
Merton, R.C.,
"On
The
Mathematics
and
Economics
Assumptions of Continuous Time Models", in "Financial Economics:
Essays in Honor of Paul Cootner", edited by W.
Sharpe and
C.
Cootner, Prentice-Hall, 1982.
14. Myers, S.C.,
the
Capital
Asset
and
Pricing
S.Turnbull,
Model:
"Capital
Good
News
Budgeting
and
Bad
and
News",
Journal of Finance, May 1977, 321-333.
^-~-"Y~"~~~""""~"I""
------------------
III
38
15. Parkinson, M.,
"Option
Pricing:
The
American
Put",
The Journal of Business, January 1977, 21-36.
16. Robichek, A., and J. Van Horne, "Abandonment Value and
Capital Budgeting", Journal of Finance, December 1967, 577-590.
17. Schwartz, E.S.,
Implementing
a New
"The
Approach",
Valuation
of
Warrants:
Journal of Financial Economics,
January 1977, 79-93.
18. Smith, C.,
"Option
Pricing:
A
Review",
Journal of
Financial Economics, January/March 1976, 1-51.
19. Stulz, R.,
Risky Assets:
"Options on the Minimum or Maximum of
Analysis and Applications",
Economics, July 1982, 161-185.
Two
Journal of Financial
-
Ct
___
t
Figure la:
Forecasted cash flows.
-·-_ _~~~~~~~ ~~_~~~____
~~DR~~~r~~~lu(~~~~~~rrsll~~~~~
~
~
~
~
----
V
t
F
Figure
t
b:
Forecasted project value.
~i
___~^
------__~__
__~____~______1~__~I__
,a··-···-------·
---- ·- ·--·-------------
0
Figure
_a
t
c:
Forecasted payout ratios.
I
___
III
100
50
0
t
23.1
Forecasted
project life
Figure 2:
Forecasted project value and salvage value
(base case example, constant payout ratio).
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...
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Figure 4:
t
15
Forecasted project value
(varying payout ratio).
~~~~~`-"-~~~~~~~"~~~~~~~~~~~'~·
T-1 1-
, ,-
-1
- - --
-- -, -
---
-- ..
-·
-
-----
-
-
Present value
of cumulative
construction cost
Total cost of design
B
ne (months)
0
12
24
36
t
Start A
Figure
__
Start B
Cumulative construction cost of
two plant designs.
I
---^----
__11___
Table III.
Abandoment value with stochastic salvage value a
b
correlation coefficient (0)
+0.9
+0.5
0.00
-0.5
-0.9
variance
0.00
5.55c
5.55
5.55
5.55
5.55
of salvage
0.04
2.18
5.55
8.83
11.45
13.13
value ( 2):
0.08
3.75
7.80
11.45
14.63
16.18
0.12
3.88
9.85
13.50
16.53
19.63
S
NOTES:
a.
For these calculations, we assume Ys = 0.07.
Otherwise the assumptions are those of the base
case with deterministic salvage.
b.
The variance of the rate of change of project value in
units of salvage value,
x=P/S,
c.
is
02
x
2
P
-
2
+
ps
2
s
When the uncertainty in salvage value is zero, the problem
reduces to our base case with deterministic salvage.
difference between the abandonment value when
The
= 0 in
s
this table and our base case solution is due to round-off
error.