Real Options - Part 2 (Ch. 22)
THE TIMING OPTION
·
A
project does not have to be taken right away.
With any project, you always have the option to wait, and see how factors
relevant to the NPV of the project change.
It could be demand for the product, or production technology, or
something else.
·
It is
potentially useful to wait before you decide about a project only if there will
be some new information that will affect the NPV of the project.
In other words, the timing option is valuable only when new information
is involved. The value of the
timing option reflects the value of the new information you expect to receive by
waiting.
·
Waiting
to take a project always implies some costs too.
Waiting allows competitors to get an advantage over you; this will
adversely affect your future cashflows. Also,
even if nothing about the project changed (no new info, no adverse impact on
cashflows), by waiting you would get the same NPV at a later date.
This loss in PV is always one component of the cost of waiting. The value of the timing option reflects the costs of waiting
as well.
·
Combining
both issues, it makes sense to postpone the project only when the expected value
of the new information is greater than the cost of waiting.
·
Like
before, we view the project as a call option on the PV of the project’s future
cashflows, and the exercise price is the investment required.
The difference is that this time we are viewing it as an American option.
We can exercise the option (i.e. invest in the project) either today or
tomorrow.
·
Quick
review of relevant facts about American call options:
o
If you
have an American call option on a stock that doesn’t pay dividends, you would
never exercise the option early.
§
The
option is worth more alive than dead
§
If you
sell the option today, you get C (the value of the call)
§
If you
exercise it today, you get the intrinsic value, P0 - X
§
The
value of the call is always more than the intrinsic value (the difference being
the time value of the option)
§
So you
would always sell it rather than exercise early; by exercising early you lose
the time value of the option
o
However,
an American call option on a dividend-paying stock is another story.
Depending on how large the dividend is, it
can make sense to exercise the option just before the ex-dividend date.
§
If you
wait till the maturity date, you lose the dividend.
§
If you
exercise just before the ex-dividend date, you get the dividend.
§
You
will exercise early if the dividend is more than the time value of the option
§
The
dividend constitutes a benefit of exercising early; the time value constitutes a
cost of exercising early
§
Or,
flipping it around, the dividend is the cost of waiting while the time value is
the benefit of waiting
·
The
timing option is really like an American call option on a dividend-paying stock.
In both cases there is a cost of waiting and a benefit of waiting.
·
Example:
o
Investment
in a project is $350 million whether the project is taken today or tomorrow (one
year from today).
o
Required
return on the project is 10%; the riskfree rate is 6%.
o
If you
take the project today, you invest $350 million at time 0, and expected
cashflows are $36 million per year forever, starting at time 1.
o
If you
wait till tomorrow, you will find out whether demand for the product increases
or decreases.
§
With a
probability of .45 demand will fall. By
investing $350 million at time 1 you would receive expected cashflows of $25
million per year forever, starting at time 2.
§
With a
probability of .55 demand will rise. By
investing $350 million at time 1 you would receive expected cashflows of $45
million per year forever, starting at time 2.
(Note:
This is why at time 0 your future expected cashflows are $36 million.
With a probability of .45 you expect 25 million per year, and with a
probability of .55 you expect 45 million per year, for an expected cashflow of
.45*25 + .55*45 = 36 million)
Decision
Tree Approach:
 |
If
you take the project today, NPV0 = 36/.1 – 350 = 10 million |
|
If
you wait: |
 |
With
a probability of .45, NPV1 = 25/.1 – 350 = -100 million =>
you would not take the project |
|
With
a probability of .55, NPV1 = 45/.1 – 350 = 100 million =>
you would take the project |
|
The
time 0 value resulting from waiting = .55*100/1.1 = $50 million |
Therefore,
it’s better to wait
Option
Pricing Approach
o
The
timing option is like an American call option on a dividend-paying stock.
And normally we cannot apply the Black-Scholes formula to an American
call option on a dividend-paying stock without complicated adjustments for
dividends.
o
But ultimately we need to do the same thing we did using the decision
tree approach: compare the time 0 value from exercising the option today with
the time 0 value from waiting and exercising tomorrow.
o
The value from exercising today is the same as before, the NPV0
of 10 million
o
And if the option is not exercised today, it becomes a European call
option – can only be exercised tomorrow (at maturity)
o
So the value of the project if you wait is just the value today of a
European call option on the time 1 PV of the project’s future cashflows.
o
The exercise price is $350 million.
o
The maturity is 1 year.
o
The expected time 1 value of the project’s cashflows = 36/.1 = 360
million. (or, if you like you can
go .45*(25/.1) + .55*(45/.1) = 360 million)
o
P0 is the PV today of this $360 million = 360/1.1 = 327.27
million
o
Unlike the previous example, this time we’ll compute the standard
deviation of the project’s annual return.
§
The value today of the project’s cashflows = 327.27.
§
With a probability of .45, the value tomorrow will be 25/.1 = 250
million. This represents a return
of 250/327.27 – 1 = -23.61%
§
With a probability of .55, the value tomorrow will be 45/.1 = 450
million. This represents a return
of 450/327.27 – 1 = 37.5%
§
The expected return over the year must equal the required return of 10%.
We verify this. E(R) = .45 *
(-.2361) + .55 * .375 = 10%
§
The variance of the annual return = .45*(-.2361 - .10)^2 + .55*(.375 -
.10)^2 = .0924
The standard deviation of the
annual return = .0924^0.5 = 30.40%
o
The value of the call option is then:
d1
= { ln (P0/X)
+ (R + 0.5 s2)*
T}/{s Ö
T }
= { ln (327.27/350) + (.06 + .5*.0924)}/{0.304} =
0.1285 = .13
d2
= d1 - s Ö
T =
.1285 - 0.304 = -0.1755 = -0.18
N(d1)
= 0.55173, N(d2) = 0.42857
C
= P0
N(d1) - X e-RT N(d2) = 327.27*0. 55173 -
(350 *
(e-.06) * 0.
42857)
= 180.566 – 141.264 = 39.30
o
So the
bottom line is:
o
If we
exercise the option today, we get an NPV of $10 million today
o
If we
keep the option alive and wait till tomorrow, we have an investment opportunity
worth $39.30 million today
o
Therefore,
it’s better to wait
o
We
make the same decision under both approaches.
However:
§
This time the approximation error using decision trees is much larger
than the first example
§
Also this time the decision tree approach led us to overestimate the
value of the option
Example
#2 (in-class exercise)
If
a project is taken today, investment is $520 million and expected future
cashflows will be $64 million forever. If
we wait till next year, investment will still be $520 million.
However, new information about demand will arrive.
With a probability of 0.4, demand will turn out to be higher than
expected, and expected future cashflows will go up to $70 million forever.
With a probability of 0.6, demand will turn out to be lower than expected
and expected future cashflows will be $60 million forever.
The project’s required return is 12%, and the riskfree rate is 6%.
a)
Using the decision tree approach, should you take the project today, or wait
till tomorrow?
b)
Using the option pricing approach, should you take the project today, or wait
till tomorrow?
THE
ABANDONMENT OPTION
·
This
is really the opposite of the first example.
Follow-on investments represent a firm exercising the option to expand
when things improve. Now we focus on the option to abandon a project if things go
badly (demand falls, or production costs increase sharply or selling prices
decline).
·
A
project requires a time 0 investment of $320 million.
·
The
required return for the project is 15%, while the riskfree rate is 8%.
·
The
project is expected to generate future cashflows with a time 1 PV of $350
million. However, there is
considerable uncertainty about the actual time 1 PV; it is equally likely to be
$320 million or $330 million or $400 million.
(The probability of each outcome is 1/3.)
·
At
time 1, the project’s assets will have an after tax salvage value of $340
million.
·
Ignoring
the option to abandon, the project has a NPV of 350/1.15 – 320 = 304.35 –310
= -15.65 million
·
If the
time 1 PV turns out to be $400 million, the company will continue the project.
However, if the time 1 PV turns out to be lower, then instead of
continuing the project (and getting cashflows worth $320 million or $330
million), the company will abandon the project (and take the after-tax salvage
value of $340 million).
·
Using
the decision tree approach to account for the value of the abandonment option:
o
With a
probability of 1/3 we get a time 1 PV of 400 million
o
With a
probability of 2/3 we get a time 1 PV of 340 million
o
The
value of the project with the abandonment option = (.333*400 + .667*340)/1.15
– 320 = -6.96 million
o
This
implies the value of the abandonment option = -6.96 – (-15.65) = 8.70 million
·
The
option to abandon the project is really a put option.
The company has the right to “sell” the project at time 1 for $340
million. It is a put option on the
PV of the project’s future cashflows, with an exercise price of $340 million.
If the PV of the cashflows is less than the exercise price, the company
will “sell” the project.
·
The
value of a put option is given by the put-call parity formula:
Value
of put = C - P0 + X e-RT
·
So to
value the put option, we first use the Black Scholes formula to value a call
with the same exercise price
o
Maturity
= 1 year
o
Rf = 8% per year
o
X = $340 million
o
P0 = PV today of the project’s cashflows = $304.35 million
o
Once again we compute the standard deviation of the project’s annual
return.
The value of the
cashflows today is $304.35 million. The
possible returns are:
Prob
Time 1 value Return
1/3
320
320/304.35 = 5.143%
1/3
330 330/304.35
= 8.429%
1/3
400 400/304.35
= 31.429%
(A good idea to verify that the E(R)
is 15%:
E(R)
= .333*.05143 + .333*.08429 + .333*.31429 = 15%)
The variance of the return =
.333*(.05143 - .15)^2 + .333*(.08429 - .15)^2 +.333*(.31429 - .15)^2 = .0137
The standard deviation of the
one-year return = .0137^0.5 =
11.69%
o
The value of the call option is then:
d1
= { ln (P0/X)
+ (R + 0.5 s2)*
T}/{s Ö
T }
= { ln (304.35/340) + (.08 + .5*.0137)}/{0.1169} = -0.2047
d2
= d1 - s Ö
T = -0.2047 - 0.1169 = -0.3216
N(d1)
= 0.42073
N(d2)
= 0.37449
C
= P0
N(d1) - X e-RT N(d2) = 304.35*0.42073 -
(340 * (e-.08) *
0.37449)
= 128.048 – 117.537 = 10.51
o
Value of put = 10.51 – 304.35 + 340 * (e-.08) = 10.51
– 304.35 + 313.86 = 20.02
o
The NPV of the project without the put option to abandon was -15.65
million.
The NPV with the put option =
-15.65 + 20.02 = 4.37 million
o
Note that the decision tree approach grossly undervalued the put option,
and therefore the project. It would
have led us to erroneously reject the project.
BONUS:
THE INTEREST RATE OPTION
·
We
won’t actually go ahead and value this option, because it is a lot more
complicated than the examples we have looked at.
But it is important to recognize that such an option exists.
·
Suppose
you have a project with no other real option, and when you compute the NPV, it
turns out to be negative. So
obviously, you don’t take the project.
·
Now
ask yourself: what is the value of this investment opportunity?
If you were to try and sell the rights to the project, how much should
you get?
·
At
first blush, we would say zero. It’s not a project worth taking, so we don’t take it.
No one else will take it either, so it has no value.
·
If we
look deeper, though, all our NPV computation is saying is “At today’s
interest rates, the PV of future cashflows < investment, so we don’t want
to take the project today.”
·
Interest
rates can either increase or decrease. If
the interest rate drops enough, the NPV of the project can become positive.
·
We
have identified an investment opportunity.
Today the NPV may be negative, but tomorrow it could turn positive.
In effect, we hold an option to invest in the project in the future if
interest rates drop enough. Like
before, it’s a call option on the future PV of the project.
But this time, whether the option is exercised or not depends on interest
rate movements rather than anything to do with the project per se.
·
Valuing
such an option is a lot more complicated than valuing say the timing option.
For example, the standard deviation of interest rate changes and the
standard deviation of the project’s returns will both have to figure in the
valuation formula.
·
But
the important point is: just because a project has negative NPV today and no
other real options, that does not mean it has no value.
You don’t junk these projects and forget about them.
You mothball them, and consider them again if and when interest rates
drop.