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There Expected NPV= 0.3*35.7 + 0.4*1.79 + 0.3*0=

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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

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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.

 

x

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