Practice
Problems – Chapter 22
1)
Out Of The World Inc. is well established in the space tourism business.
They are considering whether to enter the emerging market for
weddings-in-orbit. Entering
involves investing $475 billion today. The
required return on the investment would be 18% per year.
Cashflows will start next year with $40 billion but will then grow at 9%
per year forever. The riskfree rate
is 11% per year. If they enter the
market today, they will have the opportunity to invest in a follow-up project
after three years. This second
project will require a time 3 investment of $1250 billion, and the same required
return as the first one. Cashflows
will start at time 4 with an expected cashflow of $100 billion, and the
cashflows are then expected to grow at 9% per year forever.
However, there is considerable uncertainty about the time 3 PV of the
second project’s cashflows. The
PV is equally likely to be $586 billion more than expected or $586 billion less
than expected. These numbers
correspond to an annual standard deviation of 50%.
a)
What is the NPV of the first project?
b)
What is the expected NPV at time 3 of the second project?
c)
Using the decision tree approach, what is the total value of both projects
together?
d)
Using the option pricing approach, what is the total value of both projects
together?
2)
A company is considering entry into a new product market.
Entering today entails a project with an investment of $100,000 and a
required return of 12% per year. Cashflows
will start next year with $6,500 and then grow at 5% per year forever.
The riskfree rate is 9% per year. If
they enter the market today, they will have the opportunity to invest in a
follow-up project after four years. This
second project will require a time 4 investment of $300,000, and the same
required return as the first one. Cashflows
will start at time 6. The first
cashflow is expected to be $22,000, and the cashflows will then grow at 5% per
year forever. However, there is
considerable uncertainty about the time 5 PV of the second project’s cashflows.
With a probability of 0.4 the PV could be $60,000 more than expected, and
with a probability of 0.6 the PV could be $40,000 less than expected.
These numbers correspond to an annual standard deviation of 12.2853%.
a)
What is the NPV of the first project?
b)
What is the expected NPV at time 4 of the second project?
c)
Using the decision tree approach, what is the total value of both projects
together?
d)
Using the option pricing approach, what is the total value of both projects
together?
3) Your company is evaluating whether to enter a new market today. The project requires an investment of $15 million and has a required return of 12% per year. It will generate cashflows that will start next year with $1.25 million and grow at 4% per year forever. The riskfree rate is 6% per year. Launching this project today will allow you to introduce a new, improved version of the product after three years. This second project will require a time 3 investment of $25 million. It will have the same required return as the first. Cashflows are expected to start at time 4 with $1.8 million, and grow at an expected rate of 4% per year forever. However, there is considerable uncertainty about the time 3 PV of the cashflows from the second project. With a probability of 0.5 the PV could be $11 million more than expected, and with a probability of 0.5 the PV could be $11 million less than expected. These numbers correspond to an annual standard deviation of 40%. Using the option pricing approach, what is the project’s NPV taking the value of the real option into account?
4)
A project can be taken either today or tomorrow (one year from today).
Investment will be $100 million, whenever you take the project.
The project’s required return is 15%, and the riskfree rate is 7.5%.
If the project is taken today, expected future cashflows are $16 million
forever. If we wait till tomorrow,
we will learn whether production costs increase or decrease.
With a probability of 0.5, costs will fall and expected future cashflows
will increase to $18 million forever. With
a probability of 0.5, costs will go up and expected future cashflows will be $14
million forever.
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?
5)
If a project is taken today, investment is $750 million and expected
future cashflows will be $98.6 million forever.
If we wait till next year, investment will still be $750 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 $110 million forever.
With a probability of 0.6, demand will turn out to be lower than expected
and expected future cashflows will be $91 million forever.
The project’s required return is 13%, and the riskfree rate is 4%.
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?
6)
A project can be taken either today or tomorrow (one year from today).
Investment will be $365,000, whenever you take the project.
The project’s required return is 11%, and the riskfree rate is 4%.
If the project is taken today, expected future cashflows are $41,000
forever. If we wait till tomorrow,
we will learn whether selling prices increase or decrease.
With a probability of 0.6, the selling price goes up and expected future
cashflows increase to $45,000 forever. With
a probability of 0.4, the selling price goes down and expected future cashflows
are $35,000 forever.
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?
7)
A project requires investment of $400,000 today.
The probability distribution of the time 1 PV of the future cashflows is
as follows:
Prob.
PV1
0.25
$390,000
0.25
$410,000
0.25
$440,000
0.25
$500,000
If
the project is abandoned at time 1, the after-tax salvage value of the assets is
$425,000. The project’s required
return is 12.5%, and the riskfree rate is 5%.
a)
Using the decision tree approach, what is the NPV of this project today, after
taking the value of the abandonment option into account?
b)
Using the option pricing approach, what is the NPV of this project today, after
taking the value of the abandonment option into account?
8)
If $750 million is invested in a project today, the time 1 PV of the
resulting cashflows will have the following probability distribution:
Prob.
PV1
0.1
$700 million
0.2
$750 million
0.3
$800 million
0.4
$900 million
If
the project is abandoned at time 1, the after-tax salvage value of the assets is
$780 million. The project’s
required return is 10%, and the riskfree rate is 6%.
a)
Using the decision tree approach, what is the NPV of this project today, after
taking the value of the abandonment option into account?
b)
Using the option pricing approach, what is the NPV of this project today, after
taking the value of the abandonment option into account?
9)
Investing $240 million in a project today generates the following
probability distribution for the time 1 PV of the resulting cashflows:
Prob.
PV1
0.25
$222 million
0.3
$250 million
0.45
$333 million
If
the project is abandoned at time 1, the after-tax salvage value of the assets
will be $295 million. The
project’s required return is 14%, and the riskfree rate is 6%.
a)
Using the decision tree approach, what is the NPV of this project today, after
taking the value of the abandonment option into account?
b)
Using the option pricing approach, what is the NPV of this project today, after
taking the value of the abandonment option into account?
10)
Mark the following statements with a T or an F to indicate
whether they are true or false (no explanations required or considered):
a)
In the Black-Scholes formula, N(d1) is related to the
probability that the option will be exercised on the maturity date.
b)
When we use the Black-Scholes formula to value the option to make
follow-on investments, P0 is the PV today of the future NPV of the
follow-on (or second stage) project.
c)
The option to make follow-on investments is a call option on the future
PV of the first stage project.
d)
When the second stage project occurs several years after the first stage
project, we use the standard deviation of annual returns in the Black-Scholes
formula; if the second stage project occurred several months after the first
stage project, we would use the standard deviation of monthly returns.
e)
The timing option is an American call option; it can be exercised before
maturity.
f)
The timing option is valuable only because waiting may lead to some new
information being revealed before you decide whether to accept the project or
not.
g)
The only time an American call option on a dividend-paying stock might be
exercised early (before maturity) is just before an ex-dividend date; the higher
the dividend, the more likely you are to exercise early.
h)
The abandonment option is a put option on the PV of the project’s
cashflows; the exercise price is the investment amount.