Answers and Solutions: 26 – 1
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Chapter 26
Real Options
26-1 a. Real options occur when managers can influence the size and risk of a project’s cash
flows by taking different actions during the project’s life. They are referred to as real
because they are a part of another project.
changing market conditions. This includes the option to contract or expand
production. Growth options allow a company to expand if market demand is higher
c. Decision trees are a form of scenario analysis in which different actions are taken in
different scenarios.
26-2 Postponing the project means that cash flows come later rather than sooner; however,
26-3 Timing options make it less likely that a project will be accepted today. Often, if a firm
26-4 Having the option to abandon a project makes it more likely that the project will be
accepted today.
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
26-1 a. 0 1 2 20
├─────┼─────┼────── ────┤
PV @
0 1 2 3 21 Yr. 1
50% Prob. 0 –20 3.8 3.8 3.8 26.69
Tax imposed: NPV @ Yr. 1 = (-20 + 15.45)/(1.13) = -4.027
would not be undertaken. The value of this option of waiting one year is evaluated as
0.5($0) + (0.5)($ 5.920) = $2.96 million.
r= 13%
Answers and Solutions: 26 – 3
26-2 a. 0 1 2 3 4
├─────┼─────┼─────┼─────┤
-8 4 4 4 4
NPV = $4.6795 million.
| | | | | | |
90% Prob. 0 0 -9 4.2 4.2 4.2 4.2 $13.313
Low CF scenario: NPV = (-9 + 6.974)/(1.1)2 = -$1.674
Since the NPV of waiting two years is less than going ahead and proceeding with the
project today, it makes sense to drill today.
10%
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26-3 a. 0 1 2 20
├─────┼─────┼────── ────┤
-300 40 40 40
NPV = –$19.0099 million. Don’t purchase.
50% Prob. 0 –300 30 30 30 30 -$78.9889
| | | | | |
50% Prob. 0 –300 50 50 50 50 45.3430
| | | | | |
50% Prob. 0 –300 30 30 + 280 0 0 -$27.1468
| | | | | |
13%
26-4 a. 0 1 14 15
| | | |
-6,200,000 600,000 600,000 600,000
Using a financial calculator, input the following data: CF0 = -6,200,000;
CF1-15 = 600,000; I/YR = 12; and then solve for NPV = -$2,113,481.31.
c. If they proceed with the project today, the project’s expected NPV = (0.5 –
$2,113,481.31) + (0.5 $1,973,037.39) = −$70,221.96. So, Hart Enterprises would not
do it.
Taxes -6,200,000 6,000,000 0 0 -$ 842,857.14
No Taxes | | | |
50% Prob. -6,200,000 1,200,000 1,200,000 1,200,000 1,973,037.39
12%
Answers and Solutions: 26 – 6
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e. NPV @
0 1 Yr. 0
50% Prob. | |
Taxes NPV = ? -1,500,000 $ 0.00
+300,000 = NPV @ t = 1
No Taxes | |
r = 12%
wouldn’t do
26-5
a.
0 1 2
40% Prob. | | |
ENPV = 23,388 (0.40) – 11,322 (0.60) = $2,562
b. 0 1 2 3 4
40% Prob. | | | | |
The PV of the 20,000 payment in year 2 at the risk free rate is 20,000/(1.06)2 = 17,800.
The PV of the 4 cash flows of 25,000 in years 1 through 4 at the cost of capital of 10%
is 79,247.
E[NPV] = 23,388 (0.40) – 11,322 (0.60) = $2,562. This is the expected NPV of the
project including the growth option. The value of the growth option itself is the
r = 10%
r = 10%
Answers and Solutions: 26 – 8
26-6 P = PV of all expected future cash flows if project is delayed. From Problem 14-1 we
know that PV @ Year 1 of Tax Imposed scenario is $15.45 and PV @ Year 1 of Tax Not
Imposed Scenario is $26.69. So the PV is:
d1 = ln[18.646/20] + [0.08 + .5(.0687)](1) = 0.1688
(.0687)0.5 (1)0.5
d2 = 0.1688 – (.0687)0.5 (1)0.5 = -0.0933
V = P[N(d1)] –
trRF
Xe
[N(d2)]
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26-7 P = PV of all expected future cash flows if project is delayed. From Problem 26-1 we
know that PV @ Year 2 of Low CF Scenario is $6.974 and PV @ Year 2 of High CF
t = 2.
rRF = 0.06.
d2 = 1.9010 – (.0111)0.5 (2)0.5 = 1.7520
Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:
trRF
= $10.479(0.9713) – $9e(-0.06)(2)(0.9601)
= $10.178 – $7.664
26-8 P = PV as of time zero of all expected future cash flows if the project is repeated starting
in year 2. Note it includes both the good cash flows and the bad cash flows since as of
now, we don’t know which outcome will result, and P excludes the $20,000 investment
in the franchise.
60% Prob. 5,000 5,000
EPV of cash flows (as of time 0) = 35,858(0.40) + 7,172(0.60) = $18,646 = P.
The strike price, X, is the cost to extend the franchise at the end of year 2, and is $20,000.
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The time to expiration is the time you decide whether or not to extend the franchise, and
is at the end of year 2.
Although the problem stated to assume the variance of the project’s rate of return was
0.2025, we’ll also calculate it using the direct method. First calculate the rates of return
using the decision tree. To do this, calculate the present values of the two branches as of
the exercise date, year 2, and the rates of return assuming the initial value of the
Bad | | | | |
60% Prob. 8,678 = PV of two 5,000 cash flows
Annual rate of return in good state = [43,388/18,646]1/2 -1 = 52.54%
twice: 0.21 = (1.10)2 – 1.]
The variance is
σ2 = (0.5254 – 0.0195)2(0.40) + (-0.3178 – 0.0195)2 (0.60) = 0.1706