There

are four types of real options :

Investment

timing option Growth

option ( 3 types )Expansion of existing product lineNew productsNew geographic marketsAbandonment

option ( 3 types)ContractionTemporary suspensionComplete abandonmentFlexibility

option

The

five possible procedures for analyzing a real option are:

Use

discounted cash flow (DCF) valuation and ignores any real options by assuming

their values are zero.Use

DCF valuation and include a qualitative recognition of any real option’s value.Use

decision tree analysis.Use

standard model for financial option.Develop

a unique, project specific model using financial engineering technique.

Cost

of project is $70mCash flow $30m per year for 3 years WACC is 10%Prior to the discussion where probabilities were assigned:NPV= -70m + 30* (1-1.1^-3)/0.1 = $4.61m After assessing the probabilities, NPV is:NPV (chance of high demand) = -70+45(1-1.1^-3)/0.1 = $41.91 mNPV (chance of average demand) = -70+30(1-1.1^-3)/0.1= $4.61 mNPV (chance of low demand) = -70+15(1-1.1^-3)/0.1= $-32.67 mExpected NPV= 41.91*0.3 + 4.61*0.4 + (-32.67)*0.3 = $ 4.62 mRelying

on NPV, we should proceed with the project because NPV is positive. However, the project seems very risky. In case of high demand, NPV

will be very high compared to the other two scenarios. In case of moderate

demand, NPV will be positive yet much small compared to high demand. In case of

low demand, the project will incur massive loss (high negative NPV). In such case, since the project is very risky , the timing option

seems of high value and it is better to wait and assess more the market before deciding

whether to execute the project and not .Relying

on timing option, company will go with the project that adds value to the company,

that is, the one with high demand and moderate demand. It will not take the project

in case of low demand.

0

1

2

3

4

high

0

-70

45

45

45

average

0

-70

30

30

30

low

0

0

0

0

0

The cost of project will be discounted at risk free and remaining

cash stream will be discounted at WACC.

When demand is high NPV is:

NPV= -70*1.06^-1 + 45*((1-1.1^-3)/0.1)*1.1^-1 = $35.7m

When demand is average, NPV is:

NPV= -70*1.06^-1 + 30*((1-1.1^-3)/0.1)*1.1^-1 = $1.79m

In case of low demand, project will not be take and NPV=$0

Expected NPV= 0.3*35.7 + 0.4*1.79 + 0.3*0= $11.43 m

Since the expected NPV of the timing option is higher than the NPV

of the immediate performing, it is better to wait one year then execute the

project.

We

need 5 inputsX = Exercise

Price = Cost Of Implement Project = $70 Million.RRF = Risk-Free Rate = 6%.T = Time

to Maturity = 1 year.P = Current

Value of the Project’s Future Cash Flows.? 2 = Variance of Project’s Rate of

Return.The current price of stock is the present value of cash

flows beyond the exercise price discounted back to exercise date.

0

1

2

3

4

high

45

45

45

average

30

30

30

low

15

15

15

NPV of high demand = 45*(1-1.1^-3)/0.1= $111.91m

NPV of average demand= 30*(1-1.1^-3)/0.1= $74.61m

NPV of low demand = 15*(1-1.1^-3)/0.1= $37.3m

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.

?2 is the variance of the stock return and there are 3

methods to calculate it:

The subjective method: the variance of average company’s stock is

about 12%. Projects are riskier

than the firm, so the variance will be higher. 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%.

The

direct method:

We

calculated the current value of the project and the value for each scenario at

the time the option expires.

current

value

value

at expiration

high

67.82

111.91

average

67.82

74.61

low

67.82

37.3

The

annual rate of return is:

High

return = ($111.91/$67.82) – 1 = 65%.

Average

return = ($74.61/$67.82) – 1 = 10%.

Low

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

variance based on this approach is 18.2%.

The

third method is the indirect method.

We

need to calculate the coefficient of variation. To calculate the CV, we need

the expected value of project’s cash flows at date the real option expires and

the standard deviation at that date.

The

value of the project at the time the option expires has been calculated and it

can be used to calculate the expected value and the standard deviation.

value

at expiration

high

111.91

average

74.61

low

37.3

Expected

Value =0.3($111.91) +0.4($74.61) +0.3($37.3)

= $74.61.

sValue = .3($111.91-$74.61)2

+ .4($74.61-$74.61)2

+ .3($37.30-$74.61)21/2

= $28.90.

Coefficient

of Variation = CV = Expected

Value / svalue

CV =

$74.61 / $28.90 = 0.39.

?2 = LN CV2 + 1/T = LN 0.392

+ 1/1 = 14.2%.

Now, we

proceed to use the OPM:

V =

$67.83N (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.

Under

new cost, NPV will be (ignoring demand):

year

0

1

2

3

4

5

6

cash

stream

-75

30

30

30-75

30

30

30

NPV= -75+-75*1.1^-3 + 30* (1-1.1^-6) = $-0.69m

Taking into consideration that cash flow may vary depending on

demand as in question c,

The expected NPVs will be:

Under high demand:

year

0

1

2

3

4

5

6

cash

stream

-75

45

45

45-75

45

45

45

NPV= -75-75*1.1^-3 + 45*(1-1.1^-6)/0.1= $64.64m

Under average demand:

year

0

1

2

3

4

5

6

cash

stream

-75

30

30

30-75

30

30

30

NPV= -75+-75*1.1^-3 + 30* (1-1.1^-6) = $-0.69m

Under low demand:

year

0

1

2

3

4

5

6

cash

stream

-75

15

15

15-75

15

15

15

NPV= -75+-75*1.1^-3 + 15* (1-1.1^-6) = $-66.02m

Expected NPV= 64.64*0.3 + (-0.69)*0.4+ (-66.02)*0.3= $-0.69m

The company will take the growth option only in case of high demand.

The

company will take growth option only in case of high demand.The expected future cash flow is:

0

1

2

3

4

5

6

high

-75

45

45

-75+45

45

45

45

average

-75

30

30

30

low

-75

15

15

15

To calculate NPV, we discount future cash flow at WACC and cost of reinvestment

at risk free rate.

NPV of high demand=

-75 + 45*(1-1.1^-6)/0.1 -75*1.06^-3 =$58.02m

NPV of average demand:

-75 + 30*(1-1.1^0-3)/0.1= $-0.39

NPV of low demand= -75+15*(1-1.1^-3)/0-1=$-37.7

Expected NPV= 0.3($58.02) + 0.4(-$0.39) + 0.3(-$37.7) = $5.94m

X = Exercise Price = Cost of Implement Project = $75 million.

rRF = Risk-Free Rate =

6%.

t = Time to Maturity = 3 years.

P

= Current Value of the Project’s Future Cash Flows.

?2 = Variance of Project’s Rate of Return.

To find P, we need to find the PV of cash flows discounted back to

exercise date.

0

1

2

3

4

5

6

high

45

45

45

average

30

30

30

low

15

15

15

High: PV3

= 45*(1-1.1^-3)/0.1= $111.91m

Average: PV3

= 30*(1-1.1^-3)/0.1 = $74.61m

Low: PV3

= $15*(1-1.1^-3)/0.1 = $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.

To

estimate ?2, we use

the following approaches,

Direct

approach:

The

current value of project and value of project at expiration of each scenario

was calculated previously.

Current

Value ( year 0)

Value

At Expiration (year 3 )

high

56.02

111.91

average

56.02

74.61

low

56.02

37.3

The

annual rate of return is:

High

Return = ($111.91/$56.02) (1/3) – 1 = 25.9%.

Average

return = ($74.61/$56.02) (1/3) – 1 = 10%.

Low

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%.

The indirect approach:

We need to find the coefficient of variation of the project at the

time the option expires.

The value of the project at the time the option expires was

calculated previously 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

37.3

Expected Value =0.3($111.91)

+0.4($74.61) +0.3($37.3)

= $74.61.

sValue = .3($111.91-$74.61)2

+ .4($74.61-$74.61)2

+

.3($37.30-$74.61)21/2

=

$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%.

V = $56.06N (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.

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) Value of growth option goes up when ?2

If the future profitability of dot.com companies is highly

unstable and have chance of high earnings, then a company with a real option on

those profits might have a very high value for its growth option.