Chapter 12
Real Options
ANSWERS TO END-OF-CHAPTER QUESTIONS

12-1 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 options because they deal with real as opposed to financial assets. They are also called managerial options because they give opportunities to managers to respond to changing market conditions. Sometimes they are called strategic options because they often deal with strategic issues. Finally, they are also called embedded options because they are a part of another project.

b. Investment timing options give companies the option to delay a project rather than implement it immediately. This option to wait allows a company to reduce the uncertainty of market conditions before it decides to implement the project. Capacity options allow a company to change the capacity of their output in response to changing market conditions. This includes the option to contract or expand production. Growth options allow a company to expand if market demand is higher than expected. This includes the opportunity to expand into different geographic markets and the opportunity to introduce complementary or second-generation products. It also includes the option to abandon a project if market conditions deteriorate too much.

c. Decision trees are a form of scenario analysis in which different actions are taken in different scenarios.


12-2 Postponing the project means that cash flows come later rather than sooner; however, waiting may allow you to take advantage of changing conditions. It might make sense, however, to proceed today if there are important advantages to being the first competitor to enter a market.

12-3 Timing options make it less likely that a project will be accepted today. Often, if a firm can delay a decision, it can increase the expected NPV of a project.

12-4 Having the option to abandon a project makes it more likely that the project will be accepted today.











SOLUTIONS TO END-OF-CHAPTER PROBLEMS

12-1 a. 0 1 2 20
+-----+-----+------ · · · ----+
-20 3 3 3

NPV = $1.074 million.

b. Wait 1 year:

PV @
0 1 2 3 21 Yr. 1
Tax imposed | | | | · · · |
50% Prob. 0 -20 2.2 2.2 2.2 15.45

Tax not imposed | | | | · · · |
50% Prob. 0 -20 3.8 3.8 3.8 26.69

Tax imposed: NPV @ Yr. 1 = (-20 + 15.45)/(1.13) = -4.027
Tax not imposed: NPV @ Yr 1 = (-20 + 26.69)/ (1.13) = 5.920
Expected NPV = .5(-4.027) + .5(5.920) = 0.947

Note though, that if the tax is imposed, the NPV of the project is negative and therefore 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.
Since the NPV of waiting one year is greater than going ahead and proceeding with the project today, it makes sense to wait.


12-2 a. 0 1 2 3 4
+-----+-----+-----+-----+
-8 4 4 4 4

NPV = $4.6795 million.

b. Wait 2 years:

PV @
0 1 2 3 4 5 6 Yr. 2
| | | | | | |
10% Prob. 0 0 -9 2.2 2.2 2.2 2.2 $6.974

| | | | | | |
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
High CF scenario: NPV = (-9 + 13.313)/(1.1)2 = $3.564
Expected NPV = .1(-1.674) + .9(3.564) = 3.040

If the cash flows are only $2.2 million, the NPV of the project is negative and, thus, would not be undertaken. The value of the option of waiting two years is evaluated as 0.10($0) + 0.90($3.564) = $3.208 million.
Since the NPV of waiting two years is less than going ahead and proceeding with the project today, it makes sense to drill today.


12-3 a. 0 1 2 20
+-----+-----+------ · · · ----+
-300 40 40 40

NPV = -$19.0099 million. Don’t purchase.

b. Wait 1 year:

NPV @
0 1 2 3 4 21 Yr. 0
| | | | | · · · |
50% Prob. 0 -300 30 30 30 30 -$78.9889

| | | | | · · · |
50% Prob. 0 -300 50 50 50 50 45.3430

If the cash flows are only $30 million per year, the NPV of the project is negative. However, we’ve not considered the fact that the company could then be sold for $280 million. The decision tree would then look like this:

NPV @
0 1 2 3 4 21 Yr. 0
| | | | | · · · |
50% Prob. 0 -300 30 30 + 280 0 0 -$27.1468

| | | | | · · · |
50% Prob. 0 -300 50 50 50 50 45.3430

The expected NPV of waiting 1 year is 0.5(-$27.1468) + 0.5($45.3430) = $9.0981 million.
Given the option to sell, it makes sense to wait 1 year before deciding whether to make the acquisition.


12-4 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 = 12; and then solve for NPV = -$2,113,481.31.

b. 0 1 14 15
| | · · · | |
-6,200,000 1,200,000 1,200,000 1,200,000

Using a financial calculator, input the following data: CF0 = -6,200,000;
CF1-15 = 1,200,000; I = 12; and then solve for NPV = $1,973,037.39.

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.

d. Since the project’s NPV with the tax is negative, if the tax were imposed the firm would abandon the project. Thus, the decision tree looks like this:
NPV @
0 1 2 15 Yr. 0
50% Prob. | | | · · · |
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
Expected NPV $ 565,090.13

Yes, the existence of the abandonment option changes the expected NPV of the project from negative to positive. Given this option the firm would take on the project because its expected NPV is $565,090.13.

e. NPV @
0 1 Yr. 0
50% Prob. | |
Taxes NPV = ? -1,500,000 $ 0.00
+300,000 = NPV @ t = 1

No Taxes | |
50% Prob. NPV = ? -1,500,000 2,232,142.86
+4,000,000 = NPV @ t = 1 Expected NPV $1,116,071.43

If the firm pays $1,116,071.43 for the option to purchase the land, then the NPV of the project is exactly equal to zero. So the firm would not pay any more than this for the option.



12-5 P = PV of all expected future cash flows if project is delayed. From Problem 15-3 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:

P = [0.5(15.45)+ 0.5(26.690] / 1.13 = $18.646.
X = $20.
t = 1.
rRF = 0.08.
s2 = 0.0687.

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

From Excel function NORMSDIST, or approximated from Table 13E-1 in Extension to Chapter 13:
N(d1) = 0.5670
N(d2) = 0.4628

Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:

V = P[N(d1)] - [N(d2)]
= $18.646(0.5670) - $20e(-0.08)(1)(0.4628)
= $10.572 - $8.544
= $2.028 million.

12-6 P = PV of all expected future cash flows if project is delayed. From Problem 13-4 we know that PV @ Year 2 of Low CF Scenario is $6.974 and PV @ Year 2 of High CF Scenario is $13.313. So the PV is:

P = [0.1(6.974)+ 0.9(13.313] / 1.102 = $10.479.
X = $9.
t = 2.
rRF = 0.06.
s2 = 0.0111.

d1 = ln[10.479/9] + [0.06 + .5(.0111)](2) = 1.9010
(.0111)0.5 (2)0.5

d2 = 1.9010 - (.0111)0.5 (2)0.5 = 1.7520

From Excel function NORMSDIST, or approximated from Table 12E-1 in Extension to Chapter 12:
N(d1) = 0.9713
N(d2) = 0.9601

Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:

V = P[N(d1)] - [N(d2)]
= $10.479(0.9713) - $9e(-0.06)(2)(0.9601)
= $10.178 - $7.664
= $2.514 million.



SOLUTION TO SPREADSHEET PROBLEMS

12-7 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for FM11 Ch 12 P7 Build a Model.xls) and on the instructor’s side of the textbook’s web site, http://brigham.swcollege.com.


MINI CASE


Assume that you have just been hired as a financial analyst by Tropical Sweets Inc., a mid-sized California company that specializes in creating exotic candies from tropical fruits such as mangoes, papayas, and dates. The firm's CEO, George Yamaguchi, recently returned from an industry corporate executive conference in San Francisco, and one of the sessions he attended was on real options. Since no one at Tropical Sweets is familiar with the basics of real options, Yamaguchi has asked you to prepare a brief report that the firm's executives could use to gain at least a cursory understanding of the topics.
To begin, you gathered some outside materials the subject and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a question-and-answer format. Now that the questions have been drafted, you have to develop the answers.


a. What are some types of real options?

Answer: 1. Investment timing options
2. Growth options
a. Expansion of existing product line
b. New products
c. New geographic markets
3. Abandonment options
a. Contraction
b. Temporary suspension
c. Complete abandonment
4. Flexibility options.


b. What are five possible procedures for analyzing a real option?

Answer: 1. DCF analysis of expected cash flows, ignoring option.
2. Qualitatively assess the value of the real option.
3. Decision tree analysis.
4. Use a model for a corresponding financial option, if possible.
5. Use financial engineering techniques if a corresponding financial option is not available.


c. Tropical Sweets is considering a project that will cost $70 million and will generate expected cash flows of $30 per year for three years. The cost of capital for this type of project is 10 percent and the risk-free rate is 6 percent. After discussions with the marketing department, you learn that there is a 30 percent chance of high demand, with future cash flows of $45 million per year. There is a 40 percent chance of average demand, with cash flows of $30 million per year. If demand is low (a 30 percent chance), cash flows will be only $15 million per year. What is the expected NPV?

Answer: Initial Cost = $70 Million
Expected Cash Flows = $30 Million Per Year For Three Years
Cost Of Capital = 10%
PV Of Expected CFs = $74.61 Million
Expected NPV = $74.61 - $70
= $4.61 Million

Alternatively, one could calculate the NPV of each scenario:
Demand Probability Annual Cash Flow
High 30% $45
Average 40% $30
Low 30% $15

Find NPV of each scenario:
PV High: N=3 I=10 PV=? PMT=-45 FV=0
PV= 111.91
NPV High = $111.91 - $70 = $41.91 Million.
PV Average: N=3 I=10 PV=? PMT=-30 FV=0
PV= 74.61
NPV Average = $74.61 - $70 = $4.71 Million.
PV Low: N=3 I=10 PV=? PMT=-15 FV=0
PV= 37.30
NPV Low = $37.30 - $70 = -$32.70 Million.

Find Expected NPV:
E(NPV)=.3($41.91)+.4($4.61)+.3(-$32.70)
E(PV)= $4.61.









d. Now suppose this project has an investment timing option, since it can be delayed for a year. The cost will still be $70 million at the end of the year, and the cash flows for the scenarios will still last three years. However, Tropical Sweets will know the level of demand, and will implement the project only if it adds value to the company. Perform a qualitative assessment of the investment timing option’s value.

Answer: If we immediately proceed with the project, its expected NPV is $4.61 million. However, the project is very risky. If demand is high, NPV will be $41.91 million. If demand is average, NPV will be $4.61 million. If demand is low, NPV will be -$32.70 million. However, if we wait one year, we will find out additional information regarding demand. If demand is low, we won’t implement project. If we wait, the up-front cost and cash flows will stay the same, except they will be shifted ahead by a year.

The value of any real option increases if the underlying project is very risky or if there is a long time before you must exercise the option.

This project is risky and has one year before we must decide, so the option to wait is probably valuable.

e. Use decision tree analysis to calculate the NPV of the project with the investment timing option.

Answer: The project will be implemented only if demand is average or high.
Here is the time line:
0 1 2 3 4
High $0 -$70 $45 $45 $45
Average $0 -$70 $30 $30 $30
Low $0 $0 $0 $0 $0

To find the NPVC, discount the cost at the risk-free rate of 6 percent since it is known for certain, and discount the other risky cash flows at the 10 percent cost of capital.

High: NPV = -$70/1.06 + $45/1.102 + $45/1.103 +$45/1.104 = $35.70
Average: NPV = -$70/1.06 + $30/1.102 + $30/1.103 +$30/1.104 = $1.79
Low: NPV = $0.

Expected NPV = 0.3($35.70) + 0.4($1.79) + 0.3($0) = $11.42.

Since this is much greater than the NPV of immediate implementation (which is $4.61 million) we should wait. In other words, implementing immediately gives an expected NPV of $4.61 million, but implementing immediately means we give up the option to wait, which is worth $11.42 million.

f. Use a financial option pricing model to estimate the value of the investment timing option.

Answer: The option to wait resembles a financial call option-- we get to “buy” the project for $70 million in one year if value of project in one year is greater than $70 million. This is like a call option with an exercise price of $70 million and an expiration date of one year.
X = Exercise Price = Cost Of Implement Project = $70 Million.
RRF = Risk-Free Rate = 6%.
T = Time To Maturity = 1 year.
P = Current Price Of Stock = Current Value Of The Project’s Future Cash Flows.
? 2 = Variance Of Stock Return = Variance Of Project’s Rate Of Return.

We explain how to calculate P and ?2 below.
Just as the price of a stock is the present value of all the stock’s future cash flows, the “price” of the real option is the present value of all the project’s cash flows that occur beyond the exercise date. Notice that the exercise cost of an option does not affect the stock price. Similarly, the cost to implement the real option does not affect the current value of the underlying asset (which is the PV of the project’s cash flows). It will be helpful in later steps if we break the calculation into two parts. First, we find the value of all cash flows beyond the exercise date discounted back to the exercise date. Then we find the expected present value of those values.

Step 1: Find the value of all cash flows beyond the exercise date discounted back to the exercise date. Here is the time line. The exercise date is year 1, so we discount all future cash flows back to year 1.
0 1 2 3 4
High $45 $45 $45
Average $30 $30 $30
Low $15 $15 $15

High: PV1 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91
Average: PV1 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61
Low: PV1 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30

The current expected present value, P, is:
P = 0.3[$111.91/1.1] + 0.4[$74.61/1.1] + 0.3[$37.30/1.1] = $67.82.

For a stock option, ?2 is the variance of the stock return, not the variance of the stock price. Therefore, for a real option we need the variance of the project’s rate of return. There are three ways to estimate this variance. First, we can use subjective judgment. Second, we can calculate the project’s return in each scenario and then calculate the return’s variance. This is the direct approach. Third, we know the projects value at each scenario at the expiration date, and we know the current value of the project. Thus, we can find a variance of project return that gives the range of project values that can occur at expiration. This is the indirect approach.

Following is an explanation of each approach.

Subjective estimate:
The typical stock has ?2 of about 12%. Most projects will be somewhat riskier than the firm, since the risk of the firm reflects the diversification that comes from having many projects. Subjectively scale the variance of the company’s stock return up or down to reflect the risk of the project. The company in our example has a stock with a variance of 10%, so we might expect the project to have a variance in the range of 12% to 19%.

Direct approach:
From our previous analysis, we know the current value of the project and the value for each scenario at the time the option expires (year 1). Here is the time line:

Current Value Value At Expiration
Year 0 Year 1
High $67.82 $111.91
Average $67.82 $74.61
Low $67.82 $37.30

The annual rate of return is:
High: Return = ($111.91/$67.82) – 1 = 65%.
High: Average = ($74.61/$67.82) – 1 = 10%.
High: Return = ($37.30/$67.82) – 1 = -45%.

Expected Return = 0.3(0.65) + 0.4(0.10) + 0.3(-0.45)
= 10%.

s2 = 0.3(0.65-0.10)2 + 0.4(0.10-0.10)2 + 0.3(-0.45-0.10)2
= 0.182 = 18.2%.

The direct approach gives an estimate of 18.2% for the variance of the project’s return.

The indirect approach:
Given a current stock price and an anticipated range of possible stock prices at some point in the future, we can use our knowledge of the distribution of stock returns (which is lognormal) to relate the variance of the stock’s rate of return to the range of possible outcomes for stock price. To use this formula, we need the coefficient of variation of stock price at the time the option expires. To calculate the coefficient of variation, we need the expected stock price and the standard deviation of the stock price (both of these are measured at the time the option expires). For the real option, we need the expected value of the project’s cash flows at the date the real option expires, and the standard deviation of the project’s value at the date the real option expires.
We previously calculated the value of the project at the time the option expires, and we can use this to calculate the expected value and the standard deviation.

Value At Expiration
Year 1
High $111.91
Average $74.61
Low $37.30

Expected Value =.3($111.91)+.4($74.61)+.3($37.3)
= $74.61.
svalue = [.3($111.91-$74.61)2 + .4($74.61-$74.61)2
+ .3($37.30-$74.61)2]1/2
= $28.90.

Coefficient Of Variation = CV = Expected Value / svalue
CV = $74.61 / $28.90 = 0.39.

Here is a formula for the variance of a stock’s return, if you know the coefficient of variation of the expected stock price at some point in the future. The CV should be for the entire project, including all scenarios:
?2 = LN[CV2 + 1]/T = LN[0.392 + 1]/1 = 14.2%.















Now, we proceed to use the OPM:

V = $67.83[N(d1)] - $70e-(0.06)(1)[N(d2)].

d1 =
= 0.2641.

d2 = d1 - (0.142)0.5(1)0.5 = 0.2641 - 0.3768
= -0.1127.

N(d1) = N(0.2641) = 0.6041.

N(d2) = N(-0.1127) = 0.4551.


therefore,

V = $67.83(0.6041) - $70e-0.06(0.4551)
= $10.98.

g. Now suppose the cost of the project is $75 million and the project cannot be delayed. But if Tropical Sweets implements the project, then Tropical Sweets will have a growth option. It will have the opportunity to replicate the original project at the end of its life. What is the total expected NPV of the two projects if both are implemented?

Answer: Suppose the cost of the project is $75 million instead of $70 million, and there is no option to wait.
NPV = PV of future cash flows - cost
= $74.61 - $75 = -$0.39 million.
The project now looks like a loser. Using NPV analysis:
NPV = NPV Of Original Project + NPV Of Replication Project
= -$0.39 + -$0.39/(1+0.10)3
= -$0.39 + -$0.30 = -$0.69.
Still looks like a loser, but you will only implement project 2 if demand is high. We might have chosen to discount the cost of the replication project at the risk-free rate, and this would have made the NPV even lower.


h. Tropical Sweets will replicate the original project only if demand is high. Using decision tree analysis, estimate the value of the project with the growth option.

Answer: The future cash flows of the optimal decisions are shown below. The cash flow in year 3 for the high demand scenario is the cash flow from the original project and the cost of the replication project.

0 1 2 3 4 5 6
High -$75 $45 $45 $45 -$70 $45 $45 $45
Average -$75 $30 $30 $30 $0 $0 $0
Low -$75 $15 $15 $15 $0 $0 $0

To find the NPV, we discount the risky cash flows at the 10 percent cost of capital, and the non-risky cost to replicate (i.e., the $75 million) at the risk-free rate.

NPV high = -$75 + $45/1.10 + $45/1.102 + $45/1.103 + $45/1.104
+ $45/1.105 + $45/1.106 - $75/1.063
= $58.02
NPV average = -$75 + $30/1.10 + $30/1.102 + $30/1.103 = -$0.39
NPV average = -$75 + $15/1.10 + $15/1.102 + $15/1.103 = -$37.70

Expected NPV = 0.3($58.02) + 0.4(-$0.39) + 0.3(-$37.70) = $5.94.

Thus, the option to replicate adds enough value that the project now has a positive NPV.

i. Use a financial option model to estimate the value of the growth option.

Answer: X = Exercise Price = Cost Of Implement Project = $75 million.
RRF = Risk-Free Rate = 6%.
T = Time To Maturity = 3 years.
P = Current Price Of Stock = Current Value Of The Project’s Future Cash Flows.
?2 = Variance Of Stock Return = Variance Of Project’s Rate Of Return.

We explain how to calculate P and ?2 below.

Step 1: Find the value of all cash flows beyond the exercise date discounted back to the exercise date. Here is the time line. The exercise date is year 1, so we discount all future cash flows back to year 3.

0 1 2 3 4 5 6
High $45 $45 $45
Average $30 $30 $30
Low $15 $15 $15

High: PV3 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91
Average: PV3 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61
Low: PV3 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30

The current expected present value, P, is:
P = 0.3[$111.91/1.13] + 0.4[$74.61/1.13] + 0.3[$37.30/1.13] = $56.05.

Direct approach for estimating ?2:

From our previous analysis, we know the current value of the project and the value for each scenario at the time the option expires (year 3). Here is the time line:

Current Value Value At Expiration
Year 0 Year 3
High $56.02 $111.91
Average $56.02 $74.61
Low $56.02 $37.30

The annual rate of return is:
High: Return = ($111.91/$56.02)(1/3) – 1 = 25.9%.
High: Average = ($74.61/$56.02)(1/3) – 1 = 10%.
High: Return = ($37.30/$56.02)(1/3) – 1 = -12.7%.

Expected Return = 0.3(0.259) + 0.4(0.10) + 0.3(-0.127)
= 8.0%.

s2 = 0.3(0..259-0.08)2 + 0.4(0.10-0.08)2 + 0.3(-0.127-0.08)2
= 0.182 = 2.3%.

This is lower than the variance found for the previous option because the dispersion of cash flows for the replication project is the same as for the original, even though the replication occurs much later. Therefore, the rate of return for the replication is less volatile. We do sensitivity analysis later.


The indirect approach:
First, find the coefficient of variation for the value of the project at the time the option expires (year 3).







We previously calculated the value of the project at the time the option expires, and we can use this to calculate the expected value and the standard deviation.

Value At Expiration
Year 3
High $111.91
Average $74.61
Low $37.30

Expected Value =.3($111.91)+.4($74.61)+.3($37.3)
= $74.61.
svalue = [.3($111.91-$74.61)2 + .4($74.61-$74.61)2
+ .3($37.30-$74.61)2]1/2
= $28.90.

Coefficient Of Variation = CV = Expected Value / svalue
CV = $74.61 / $28.90 = 0.39.

To find the variance of the project’s rate or return, we use the formula below:
?2 = LN[CV2 + 1]/T = LN[0.392 + 1]/3 = 4.7%.

Now, we proceed to use the OPM:

V = $56.06[N(d1)] - $75e-(0.06)(3)[N(d2)].

d1 =
= -0.1085.

d2 = d1 - (0.047)0.5(3)0.5 = -.1085 - 0.3755
= -0.4840.

N(d1) = N(-0.1080) = 0.4568.

N(d2) = N(-0.4835) = 0.3142.
Therefore,

V = $56.06(0.4568) - $75e-(0.06)(3)(0.3142)
= $5.92.

Total Value = NPV Of Project 1 + Value Of Growth Option
=-$0.39 + $5.92
= $5.5 million

j. What happens to the value of the growth option if the variance of the project’s return is 14.2 percent? What if it is 50 percent? How might this explain the high valuations of many dot.com companies?

Answer: If risk, defined by ?2, goes up, then value of growth option goes up (see the file ch 12 mini case.xls for calculations):
?2 = 4.7%, option value = $5.92
?2 = 14.2%, option value = $12.10
?2 = 50%, option value = $24.09

If the future profitability of dot.com companies is very volatile (i.e., there is the potential for very high profits), then a company with a real option on those profits might have a very high value for its growth option.