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Chapter 12Real Options

ANSWERS TO END-OF-CHAPTER QUESTIONS

12-1 a. Real options occur when managers can influence the size and risk of a projects cashflows by taking different actions during the projects life. They are referred to as realoptions because they deal with real as opposed to financial assets. They are alsocalled managerial options because they give opportunities to managers to respond tochanging market conditions. Sometimes they are called strategic options becausethey 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 thanimplement it immediately. This option to wait allows a company to reduce theuncertainty of market conditions before it decides to implement the project. Capacityoptions allow a company to change the capacity of their output in response tochanging 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 geographicmarkets and the opportunity to introduce complementary or second-generation

products. It also includes the option to abandon a project if market conditionsdeteriorate too much.

c. Decision trees are a form of scenario analysis in which different actions are taken indifferent 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 firmcan 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 beaccepted today.

Answers and Solutions: 12 - 1

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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) = -4.027Tax not imposed: NPV @ Yr 1 = (-20 + 26.69)/ (1.13) = 5.920Expected 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 thereforewould not be undertaken. The value of this option of waiting one year is evaluated as0.5($0) + (0.5)($ 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.


r= 13%

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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 = -$1.674High CF scenario: NPV = (-9 + 13.313)/(1.1) 2 = $3.564Expected 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 as0.10($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.


10%

r = 10%

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12-3 a. 0 1 2 20

-300 40 40 40

NPV = -$19.0099 million. Dont 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, weve not considered the fact that the company could then be sold for $280million. 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) = $9.0981million.

Given the option to sell, it makes sense to wait 1 year before deciding whether tomake the acquisition.


13%

r = 13%

r = 13%

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12-4 a. 0 1 14 15| | | |

-6,200,000 600,000 600,000 600,000

Using a financial calculator, input the following data: CF 0 = -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: CF 0 = -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 projects expected NPV = (0.5 -$2,113,481.31) + (0.5 $1,973,037.39) = -$70,221.96. So, Hart Enterprises would notdo it.

d. Since the projects NPV with the tax is negative, if the tax were imposed the firmwould 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 projectfrom negative to positive. Given this option the firm would take on the project becauseits 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 theoption.


12%

12%

r= 12%

r = 12%

}wouldnt do

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12-5 P = PV of all expected future cash flows if project is delayed. From Problem 15-3 weknow that PV @ Year 1 of Tax Imposed scenario is $15.45 and PV @ Year 1 of Tax NotImposed 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.r RF = 0.08. 2 = 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 toChapter 13:

N(d 1) = 0.5670 N(d 2) = 0.4628

Using the Black-Scholes Option Pricing Model, you calculate the options value as:

V = P[N(d 1)] - t r RF Xe [N(d 2)]

= $18.646(0.5670) - $20e (-0.08)(1) (0.4628)= $10.572 - $8.544= $2.028 million.


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

P = [0.1(6.974)+ 0.9(13.313] / 1.10 2 = $10.479.X = $9.t = 2.r RF = 0.06. 2 = 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 toChapter 12:

N(d 1) = 0.9713 N(d 2) = 0.9601

Using the Black-Scholes Option Pricing Model, you calculate the options value as:

V = P[N(d 1)] - t r RF Xe [N(d 2)]

= $10.479(0.9713) - $9e (-0.06)(2) (0.9601)= $10.178 - $7.664= $2.514 million.


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SOLUTION TO SPREADSHEET PROBLEMS

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


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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 fruitssuch as mangoes, papayas, and dates. The firm's CEO, George Yamaguchi, recentlyreturned from an industry corporate executive conference in San Francisco, and one of thesessions he attended was on real options. Since no one at Tropical Sweets is familiar withthe basics of real options, Yamaguchi has asked you to prepare a brief report that thefirm'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 todraft a list of pertinent questions that need to be answered. In fact, one possible approachto the paper is to use a question-and-answer format. Now that the questions have beendrafted, you have to develop the answers.

a. What are some types of real options?

Answer: 1. Investment timing options2. Growth options

a. Expansion of existing product line b. New productsc. New geographic markets

3. Abandonment optionsa. Contraction

b. Temporary suspension

c. Complete abandonment4. 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.

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c. Tropical Sweets is considering a project that will cost $70 million and willgenerate expected cash flows of $30 per year for three years. The cost of capitalfor this type of project is 10 percent and the risk-free rate is 6 percent. Afterdiscussions with the marketing department, you learn that there is a 30 percentchance 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 peryear. What is the expected NPV?

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

= $4.61 Million

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

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d. Now suppose this project has an investment t iming option, since it can bedelayed for a year. The cost will still be $70 million at the end of the year, andthe cash flows for the scenarios will still last three years. However, TropicalSweets will know the level of demand, and will implement the project only if itadds value to the company. Perform a qualitative assessment of the investment

timing options 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 $41.91million. If demand is average, NPV will be $4.61 million. If demand is low, NPVwill be -$32.70 million. However, if we wait one year, we will find out additionalinformation regarding demand. If demand is low, we wont implement project. If we wait, the up-front cost and cash flows will stay the same, except they will beshifted 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 waitis probably valuable.

e. Use decision tree analysis to calculate the NPV of the project with the investmenttiming 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 $45Average $0 -$70 $30 $30 $30Low $0 $0 $0 $0 $0

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

High: NPV = -$70/1.06 + $45/1.10 2 + $45/1.10 3 +$45/1.10 4 = $35.70Average: NPV = -$70/1.06 + $30/1.10 2 + $30/1.10 3 +$30/1.10 4 = $1.79Low: 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 anexpected NPV of $4.61 million, but implementing immediately means we give up theoption to wait, which is worth $11.42 million.

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f. Use a financial option pricing model to estimate the value of the investmenttiming 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 dateof one year.X = Exercise Price = Cost Of Implement Project = $70 Million.R RF = Risk-Free Rate = 6%.T = Time To Maturity = 1 year.P = Current Price Of Stock = Current Value Of The Projects Future Cash Flows. 2 = Variance Of Stock Return = Variance Of Projects 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 stocks future cash flows, the

price of the real option is the present value of all the projects cash flows that occur beyond the exercise date. Notice that the exercise cost of an option does not affectthe stock price. Similarly, the cost to implement the real option does not affect thecurrent value of the underlying asset (which is the PV of the projects cash flows). Itwill be helpful in later steps if we break the calculation into two parts. First, we findthe value of all cash flows beyond the exercise date discounted back to the exercisedate. 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 tothe exercise date. Here is the time line. The exercise date is year 1, so we discountall future cash flows back to year 1.

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

High: PV 1 = $45/1.10 + $45/1.10 2 + $45/1.10 3 = $111.91Average: PV 1 = $30/1.10 + $30/1.10 2 + $30/1.10 3 = $74.61Low: PV 1 = $15/1.10 + $15/1.10 2 + $15/1.10 3 = $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 projects rate of return.There are three ways to estimate this variance. First, we can use subjective judgment.Second, we can calculate the projects return in each scenario and then calculate thereturns variance. This is the direct approach. Third, we know the projects value ateach scenario at the expiration date, and we know the current value of the project.

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Thus, we can find a variance of project return that gives the range of project valuesthat 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 thanthe firm, since the risk of the firm reflects the diversification that comes from havingmany projects. Subjectively scale the variance of the companys stock return up or down to reflect the risk of the project. The company in our example has a stock witha 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 valuefor each scenario at the time the option expires (year 1). Here is the time line:

Current Value Value At ExpirationYear 0 Year 1

High $67.82 $111.91Average $67.82 $74.61Low $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%.

2 = 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 projectsreturn.

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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 stocks 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 projects cash flows at the date the real optionexpires, and the standard deviation of the projects value at the date the real optionexpires.We previously calculated the value of the project at the time the option expires, andwe can use this to calculate the expected value and the standard deviation.

Value At ExpirationYear 1

High $111.91Average $74.61Low $37.30

Expected Value =.3($111.91)+.4($74.61)+.3($37.3)= $74.61.

value = [.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 / valueCV= $74.61 / $28.90 = 0.39.

Here is a formula for the variance of a stocks return, if you know the coefficient of variation of the expected stock price at some point in the future. The CV should befor the entire project, including all scenarios:2 = LN[CV 2 + 1]/T = LN[0.39 2 + 1]/1 = 14.2%.

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Now, we proceed to use the OPM:

V = $67.83[N(d 1)] - $70e -(0.06)(1)[N(d 2)].

d1 = 5.0)1(

5.0)142(.

)15)](0.142/206.0[()$67.83/$70ln( ++

= 0.2641.

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

N(d 1) = N(0.2641) = 0.6041.

N(d 2) = 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 bedelayed. But if Tropical Sweets implements the project, then Tropical Sweetswill have a growth option. It will have the opportunity to replicate the originalproject at the end of its life. What is the total expected NPV of the two projectsif both are implemented?

Answer: Suppose the cost of the project is $75 million instead of $70 million, and there is nooption 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. Wemight have chosen to discount the cost of the replication project at the risk-free rate,and this would have made the NPV even lower.

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h. Tropical Sweets will replicate the original project only if demand is high. Usingdecision 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 inyear 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 6High -$75 $45 $45 $45 -$70 $45 $45 $45Average -$75 $30 $30 $30 $0 $0 $0Low -$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 + $45/1.10 2 + $45/1.10 3 + $45/1.10 4 + $45/1.10 5 + $45/1.10 6 - $75/1.06 3

= $58.02 NPV average = -$75 + $30/1.10 + $30/1.10 2 + $30/1.10 3 = -$0.39 NPV average = -$75 + $15/1.10 + $15/1.10 2 + $15/1.10 3 = -$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.R RF = Risk-Free Rate = 6%.T = Time To Maturity = 3 years.P = Current Price Of Stock = Current Value Of The Projects Future Cash Flows.2 = Variance Of Stock Return = Variance Of Projects 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 tothe exercise date. Here is the time line. The exercise date is year 1, so we discountall future cash flows back to year 3.

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

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High: PV 3 = $45/1.10 + $45/1.10 2 + $45/1.10 3 = $111.91Average: PV 3 = $30/1.10 + $30/1.10 2 + $30/1.10 3 = $74.61Low: PV 3 = $15/1.10 + $15/1.10 2 + $15/1.10 3 = $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 valuefor each scenario at the time the option expires (year 3). Here is the time line:

Current Value Value At ExpirationYear 0 Year 3

High $56.02 $111.91Average $56.02 $74.61Low $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%.

2 = 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 dispersionof cash flows for the replication project is the same as for the original, even thoughthe replication occurs much later. Therefore, the rate of return for the replication isless 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 optionexpires (year 3).

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

Value At ExpirationYear 3

High $111.91Average $74.61Low $37.30

Expected Value =.3($111.91)+.4($74.61)+.3($37.3)= $74.61.

value = [.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 / valueCV= $74.61 / $28.90 = 0.39.

To find the variance of the projects rate or return, we use the formula below:2 = LN[CV 2 + 1]/T = LN[0.39 2 + 1]/3 = 4.7%.

Now, we proceed to use the OPM:

V = $56.06[N(d 1)] - $75e -(0.06)(3)[N(d 2)].

d1 = 5.0)3(5.0)047.0()3)](0.047/206.0[()$56.06/$75ln( ++

= -0.1085.

d2 = d 1 - (0.047) 0.5(3)0.5 = -.1085 - 0.3755= -0.4840.

N(d 1) = N(-0.1080) = 0.4568.

N(d 2) = 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

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j. What happens to the value of the growth option if the variance of the projectsreturn is 14.2 percent? What if it is 50 percent? How might this explain thehigh 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 = $5.922 = 14.2%, option value = $12.102 = 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 profitsmight have a very high value for its growth option.

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