6/25/2003

Chapter 12. Real Options

REAL OPTIONS: THE INVESTMENT TIMING OPTION

Cost= ($70)WACC= 10%Risk-free rate= 6%

Demand Prob.High 0.3 $45 $13.50 Average 0.4 $30 $12.00 Low 0.3 $15 $4.50

Expected CF= $30.00

Procedure 1: DCF OnlyYear 1 2 3Expected CF $30.00 $30.00 $30.00

NPV= $4.61

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

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.

a. What are some types of real options? Answer: See Chapter 12 Mini Case Show

b. What are the five steps for analyzing a real option? Answer: See Chapter 12 Mini Case Show

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 per year. What is the expected NPV?

Annual Cash Flow

Prob. x (CF)

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: See Chapter 12 Mini Case Show

Procedure 3: Decision Tree Analysis

a. Scenario Analysis: Proceed with Project TodayCost Future Cash Flows NPV this Prob. Data for

Year 0 Prob. 1 2 3 Scenario x NPV Std Deviation

$45 $45 $45 $41.91 $12.57 417

30%-$70 40% $30 $30 $30 $4.61 $1.84 0

30%$15 $15 $15 -$32.70 -$9.81 417

Expected NPV of Future CFs = $4.61 835

Standard Deviation= $28.89

Coefficient of Variation = 6.27

b. Decision Tree Analysis: Implement in One Year Only if OptimalCost Future Cash Flows NPV this Prob. Data for

Year 0 Prob. 1 2 3 4 x NPV Std Deviation

-$70 $45 $45 $45 $35.70 $10.71 177

30%$0 40% -$70 $30 $30 $30 $1.79 $0.71 37

30%$0 $0 $0 $0 $0.00 $0.00 39

Expected NPV of Future CFs = $11.42 253

Standard Deviation= $15.91

Coefficient of Variation = 1.39

Notes:

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

Procedure 4: Analysis with a Financial Option

=Variance of PV

Scenarioa

a Discount the cost of the project at the risk-free rate, since the cost is known. Discount the operating cash flows at the WACC.

The option to defer the project is like a call option. The company has until Year 1 to decide whether or not to implement the project, so the time to maturity of the option is one year. If the company exercises the option, it must pay an exercise price equal to the cost of implementing the project. If the company does implement the project, it gains the value of the project. If you exercise a call option, you will own a stock that is worth whatever its price is. If the company implements the project, it will gain a project, whose value is equal to the present value of its cash flows. Therefore, the present value of a project's future cash flows is analogous to the current value of a stock. The rate of return on the project is equal to its cost of capital. To find the value of a call option, we need the standard deviation of its rate of return; to find the value of this real option, we need the standard deviation of the projects expected rate of return.

Find the Year 1 Value and Risk of Future Cash Flows If Project is Deferred

Future Cash Flows PV at Prob. Data forYear 0 Prob. 1 2 3 4 Year 1 x Value Std Deviation

$45 $45 $45 $111.91 $33.57 417

30%40% $30 $30 $30 $74.61 $29.84 0

30%$15 $15 $15 $37.30 $11.19 417

Expected Year 1 Value of Future CFs = $74.61 835

Standard Deviation of value at Year 1= $28.89

Coefficient of Variation at Year 1 = 0.39

Find the current value of future cash flows if project is deferred (note: this is the estimate of P).

Current Value = ear 1 Value = $74.61 = $67.82 (1+WACC) 1.10

P = $67.82

Use the direct approach to estimate the variance of the project's rate of return.

Probability Data for

Probability Std Deviation

$111.91 65.00% 0.30 19.5% 9.1%

High

$67.82 Average $74.61 10.0% 0.40 4.0% 0.0%

Low

$37.30 -45.0% 0.30 -13.5% 9.1%

1.00

Expected return = 10.0% 18.1%

42.6%

The option to defer the project is like a call option. The company has until Year 1 to decide whether or not to implement the project, so the time to maturity of the option is one year. If the company exercises the option, it must pay an exercise price equal to the cost of implementing the project. If the company does implement the project, it gains the value of the project. If you exercise a call option, you will own a stock that is worth whatever its price is. If the company implements the project, it will gain a project, whose value is equal to the present value of its cash flows. Therefore, the present value of a project's future cash flows is analogous to the current value of a stock. The rate of return on the project is equal to its cost of capital. To find the value of a call option, we need the standard deviation of its rate of return; to find the value of this real option, we need the standard deviation of the projects expected rate of return.

The first step is to find the value of the project's future cash flows, as of the time the option must be exercised. We also need the standard deviation of the project's value as of the date it must be exercised. Finally, we need the present value of the project's future cash flows.

PVYear 0 PVYear 1 Return x ReturnYear 1

Standard deviation of return =

18.1%

CV =Coefficient of Variation = 0.39

Now use the following formula to estimate the variance of the project's rate of return.

t = time until the option expires = 1

14.2%

Find the Value of a Call Option Using the Black-Scholes Model

Financial Option Real Option

Risk-free interest rate = Risk-free interest ratet = Time until the option expires = Time until the option expiresX = Exercise price = Cost to implement the projectP = Current price of the underlying stock = Current value of the project

Variance of the stock's rate of return = Variance of the project's rate of return

rRF = 6%t = 1X = $70.00P = $67.82

14.2%

= 0.2637

= -0.1131N(d1)= = 0.6040N(d2)= = 0.4550

V = = $ 10.97

REAL OPTIONS: THE GROWTH OPTION

Direct estimate of s2 = Variance of return =

Use the indirect approach to estimate the variance of the project's rate of return. Start by estimating the coefficient of variation, CV, of the project's value at the time the option expires. This was done in an earlier step.

Indirect estimate of s2 =

rRF =

s2 =

s2 =

d1 = { ln (P/X) + rRF + s2 /2) ] t } / (s t1/2 )

d2 = d1 - s (t 1 / 2)

P[ N (d1) ] - Xe-rRF t [ N (d2) ]

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 total expected NPV of the two projects if both are implemented?

σ 2= ln [CV2 + 1 ]t

Cost= $75 WACC= 10%

Risk-free rate = 6%

Original ProjectCost Future Cash Flows NPV this Prob. Data for

Year 0 Prob. 1 2 3 Scenario x NPV Std Deviation

$45 $45 $45 $36.91 $11.07 417.45

30%-$75 40% $30 $30 $30 -$0.39 -$0.16 0.00

30%$15 $15 $15 -$37.70 -$11.31 417.45

Expected NPV = -$0.39 834.90

Standard Deviation= $28.89

Coefficient of Variation = (73.25)

NPV without growth option:NPV = -$0.39

Expected NPV is you simply repeat project at time 3:NPV = NPV of project 1 + PV of repeated project

NPV = +NPV = -$0.39 + -$0.30NPV = -$0.69

Decision Tree: Implement the repeated project only if demand is high.Cost Future Cash Flows NPV this

Year 0 Prob. 1 2 3 4 5 6 Scenario

$45 $45 -$30 $45 $45 $45 $58.0230%

-$75 40% $30 $30 $30 $0 $0 $0 -$0.3930%

$15 $15 $15 $0 $0 $0 -$37.70

Expected NPV =

Standard Deviation=

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 total expected NPV of the two projects if both are implemented?

NPV1 NPV1 / (1+WACC)3

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.

Coefficient of Variation =

Notes: 1. The CF in Year 3 includes the cost to implement the second project if it is optimal to do so.

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

Financial Option Approach

Find the value and risk of the future cash flows as of the time the option expires.

Cost Future Cash Flows PV at Year 0 Prob. 1 2 3 4 5 6 Year 3

$45 $45 $45 $111.9130%40% $30 $30 $30 $74.6130%

$15 $15 $15 $37.30

Expected value at Year 3 =

Standard Deviation of value at Year 3=

Coefficient of Variation at Year 3=

Find the current value of future cash flows if project is deferred (note: this is the estimate of P).

Current Value = ear 3 Value = $74.61 = $56.05

1.33

P = $56.05

Use the direct approach to estimate the variance of the project's rate of return.

Annual Data for

1 2 Return Probability

$111.91 25.9% 0.30 7.8%High

$56.05 Average $74.61 10.0% 0.40 4.0%Low

$37.30 -12.7% 0.30 -3.8%

1.00

Expected return = 8.0%

2. When finding the NPV, the cost to implement the second project is discounted at the risk-free rate; other cash flows are discounted at the cost of capital.

(1+WACC)3

PVYear 0 PVYear 3 x Return2005

15.0%

2.3%

CV =Coefficient of Variation = 0.39

Now use the following formula to estimate the variance of the project's rate of return.

t = time until the option expires = 3

4.7%

Find the Value of a Call Option Using the Black-Scholes ModelSensitivity Analysis

Base Case Case 1 Case 2

6% 6% 6%t = 3 3 3X = $75.00 $75.00 $75.00P = $56.05 $56.05 $56.05

4.70% 14.20% 50.00%

= -0.1085 0.1559 0.5215

= -0.4840 -0.4968 -0.7032N(d1)= = 0.4568 0.5619 0.6990N(d2)= = 0.3142 0.3097 0.2410

V = = $ 5.92 $ 12.10 $ 24.08

Total Value = Value of Project 1 + Value of growth optionTotal Value = -$0.39 + $5.92Total Value = $ 5.53

Standard deviation of return =

Direct estimate of s2 = Variance of return =

Use the indirect approach to estimate the variance of the project's rate of return. Start by estimating the coefficient of variation, CV, of the project's value at the time the option expires. This was done in an earlier step.

Indirect estimate of s2 =

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?

rRF =

s2 =

d1 ={ ln (P/X) + [r RF + s 2 /2) ] t } (s t1/2 )

d2 = d1 - s (t 1 / 2)

P[ N (d1) ] - Xe-rRF t [ N (d2) ]

σ 2= ln [CV2 + 1 ]t

Chapter 12. Real Options

Std Deviation

=Variance of PV

Std Deviation

=Variance of PV

Std Deviation

=Variance of PV

Note: use the NORMSDIST function.

Use the indirect approach to estimate the variance of the project's rate of return. Start by estimating the coefficient of variation, CV, of the project's value at the time the option expires. This was done in an earlier step.

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 total expected NPV of the two projects if both are implemented?

Std Deviation

=Variance of PV

Data forProb. Std Deviationx NPV

$17.40 1,010

-$0.16 0

-$11.31 -

$5.94 1,010 =Variance of PV

$31.78

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 total expected NPV of the two projects if both are implemented?

5.35

Data forProb. Std Deviationx NPV

$33.57 417

$29.84 -

$11.19 417

$74.61 835 =Variance of PV

$28.89

0.39

Std Deviation

1.0%

0.0%

1.3%

2.3% =Variance of PV

2. When finding the NPV, the cost to implement the second project is discounted at the risk-free rate; other cash

Note: we used the NORMSDIST function.

Use the indirect approach to estimate the variance of the project's rate of return. Start by estimating the coefficient of variation, CV, of the project's value at the time the option expires. This was done in an earlier step.