Solutions to Practice Problems – Chapter 22

1 a) NPV of First Stage Project = 40/(.18 - .09) – 475 = -30.56 billion

b) Expected time 3 NPV of second stage project = 100/(.18 - .09) – 1250 = 1111.11 – 1250 = -138.89 billion

c) For the decision tree approach, we consider the actual time 3 outcomes for the second stage project:

So with a probability of .5, we would end up with a time 3 NPV of 447.11 billion.  The value today of this investment opportunity is then .5*447.11/(1.18^3) = 136.06 billion

The total NPV of both projects together = -30.56 + 136.06 = 105.51 billion => we would accept the project today

d) The second stage project is a call option on the time 3 PV of the project’s future cashflows, with an exercise price of $1250 million. 

Maturity = 3 years

Annual standard deviation = 50%

R_f = 11% per year

X = $1250 billion

P₀ = the value today (three years before maturity) of the project’s exp. cashflows

     =1111.11/(1.18^3) = $676.26 billion

d₁ = { ln (676.26/1250) + (.11 + .5*(.5^2))*3}/{0.5*(3^0.5)} = 0.1047

d₂ = 0.1047 - 0.5*(3^0.5) = -0.7613

N(d₁) = 0.53984

N(d₂) = 0.24197

C =  676.26*0.54381 -  (1250 * e^-.11*3 *0.22663) = 367.7551 – 203.6621 = 147.62

Total NPV of project = -30.56 + 147.62 = 117.07 billion  => we would accept the project today

2 a) NPV of First Stage Project = 6,500/(.12 - .05) – 100,000 = -7,142.86

b) Need to be careful with the PVs since cashflows start 2 years after the investment.

Expected time 5 PV of second stage project’s cashflows = 20,000/(.12 - .05) = 314,285.71

Expected time 4 NPV of second stage project = 314,285.71/1.12 – 300,000 = -19,387.76

c) For the decision tree approach, we consider the actual outcomes for the second stage project:  

So with a probability of .4, we would end up with a time 4 NPV of 34,183.67.  The value today of this investment opportunity is then .4*34,183.67/(1.12^4) = 8,689.74

The total NPV of both projects together = -7,142.86 + 8,689.74 = 1,546.88 => we would accept the project today

d) The second stage project is a call option on the time 4 PV of the project’s future cashflows (since the second stage project involves investing at time 4), with an exercise price of $300,000. 

Maturity = 4 years

Annual standard deviation = 12.2853%

R_f = 9% per year

X = $300,000

P₀ = the value today (three years before maturity) of the project’s exp. cashflows

     =314,285.71/(1.12^5) = $178,334.15

d₁ = { ln (178,334.15/300,000) + (.09 + .5*(0.122853^2))*4}/{0.122853*(4^0.5)} = -0.5288

d₂ = -0.5288 - 0.122853*(4^0.5) = -0.7745

N(d₁) = 0.29807

N(d₂) = 0.22065

C = 178,334.15*0.29807 -  (300,000 * e^-.09*4 *0.22065) = 53,156.06 – 46,182.68 = 6,973.38

Total NPV of project = -7,142.86 + 6,973.38 = -169.48 => we would reject the project today

3) NPV of First Stage Project = 1.25/(.12 - .04) – 15 = 0.625 million

Second Stage Project:

The second stage project is a call option on the time 3 PV of the project’s future cashflows, with an exercise price of $25 million. 

Maturity = 3 years

Annual standard deviation = 40%

R_f = 6% per year

X = $25m

P₀ = the value today (three years before maturity) of the project’s exp. cashflows

     = [1.8/(.12 - .04)]/(1.12^3) = $16.015 million

d₁ = { ln (P₀/X) + (R + 0.5 s²)* T}/{s Ö T } 

          = { ln (16.015/25) + (.06 + .5*(.4^2))*3}/{0.4*(3^0.5)} = -0.037

d₂ = d₁ - s Ö T  = -0.037 - 0.4*(3^0.5) = -0.729

N(d₁) = 0.48404

N(d₂) = 0.23270

C =  P₀ N(d₁) - X e^-RT N(d₂)  = 16.015*0.48404 -  (25 * e^-.06*3 *0.2327)

   = 7.752 – 4.859 = 2.893

Total NPV of project = 0.625 + 2.893 = 3.518 million => we would accept the project today

4 a) If you take the project today, NPV = 16/.15 – 100 = 6.67 million

The decision tree approach would compute the NPV if you wait as follows:

So with a probability of .5, we would end up with a time 1 NPV of 20.  The value today of this investment opportunity is then .5*20/1.15 = 8.696 million.  Therefore, it is better to wait rather than take the project today.

b) Under the option pricing approach, if we wait, the project is a call option on the time 1 PV of the project’s future cashflows. 

Maturity = 1 year

R_f = 7.5% per year

X = $100 million

P₀ = the value today of the project’s exp. cashflows = (.5*120 + .5*93.33)/1.15 = 92.754 million

Annual standard deviation is computed as follows:

Variance of annual return = Prob of R₁*[ R₁ - E(R)]² + Prob of R₂*[ R₂ - E(R)]²

= .5*(.29375-.15)² + .5*(.00625-.15)² = .0207

Annual standard deviation = 14.38%

d₁ = { ln (92.754/100) + .075 + .5*(0.1438^2)}/{0.1438} = 0.0703

d₂ = 0.0703 - 0.1438 = -.0734

N(d₁) = 0.52791

N(d₂) = 0.47209

C = 92.754*0.52791 -  (100 * e^-.075 *0.47209) = 48.966 – 43.798 = 5.168 million.

This time it is better to take the project today.

5 a) If you take the project today, NPV = 98.6/.13 – 750 = 8.462 million

The decision tree approach would compute the NPV if you wait as follows:

So with a probability of .4, we would end up with a time 1 NPV of 96.15.  The value today of this investment opportunity is then .4*96.15/1.13 = 34.037 million.  Therefore, it is better to wait rather than take the project today.

b) Under the option pricing approach, if we wait, the project is a call option on the time 1 PV of the project’s future cashflows. 

Maturity = 1 year

R_f = 4% per year

X = $750 million

P₀ = the value today of the project’s exp. cashflows = (.4*846.154 + .6*700)/1.13 = 671.205 million

Annual standard deviation is computed as follows:

Variance of annual return = Prob of R₁*[ R₁ - E(R)]² + Prob of R₂*[ R₂ - E(R)]²

= .4*(.26065-.13)² + .6*(.0429-.13)² = .0114

Annual standard deviation = 10.6675%

d₁ = { ln (671.205/750) + .04 + .5*.0114}/{0.106675} = -0.6122

d₂ = -0.612 - 0.106675 = -0.7189

N(d₁) = 0.27094

N(d₂) = 0.23577

C = 671.205*0.27094 -  (750 * e^-.04 *0.23577) = 181.856 – 169.894 = 11.962 million.

Once again, it is better to wait than to take the project today.

6 a) If you take the project today, NPV = 41,000/.11 – 365,000 = 7,727.27

The decision tree approach would compute the NPV if you wait as follows:

So with a probability of .6, we would end up with a time 1 NPV of 44,090.91.  The value today of this investment opportunity is then .6*44,090.91/1.11 = 23,832.92. 

Therefore, it is better to wait rather than take the project today.

b) Under the option pricing approach, if we wait, the project is a call option on the time 1 PV of the project’s future cashflows. 

Maturity = 1 year

R_f = 4% per year

X = $365,000

P₀ = the value today of the project’s exp. cashflows = (.6*409,090.91 + .4*318,181.82)/1.11 = 335,790

Annual standard deviation is computed as follows:

Variance of annual return = Prob of R₁*[ R₁ - E(R)]² + Prob of R₂*[ R₂ - E(R)]²

= .6*(.21829-.11)² + .4*(-.05244-.11)² = .0176

Annual standard deviation = 13.263%

d₁ = { ln (335,790/365,000) + .04 + .5*.0176)}/{0.13263} = -0.2610

d₂ = -0.261 - 0.13263 = -0.3936

N(d₁) = 0.39743

N(d₂) = 0.34828

C = 335,790*0.39743 -  (365,000 * e^-.04 *0.34828) = 133,453.15 – 122,137.67 = 11,315.48.

Once again, it is better to wait than to take the project today.

7 a) Under the decision tree approach, the firm will abandon the project at time 1 whenever the PV is less than the abandonment value of $425,000.

So the expected future cashflows are .25*425,000 + .25*425,000 + .25*440,000 + .25*500,000 = 447,500

The value of the project after accounting for the abandonment option = 447,500/.125 – 400,000 = -2,222.22.  So, using the decision tree approach, we wouldn’t accept this project.

b) Ignoring the option to abandon, the NPV of the project is (.25*390,000 + .25*410,000 + .25*440,000 + .25*500,00)/1.125 – 400,000 = -13,333.33

The abandonment option is a one-year put option on the time 1 PV of the project’s cashflows, with an exercise price equal to the abandonment value of $425,000.

Maturity = 1 year

R_f = 5% per year

X = $425,000

P₀ = the value today of the project’s exp. cashflows = 386,666.67

Annual standard deviation is computed as follows:

Variance of annual return = .25*(.00862 -.125)² + .25*(.06034-.125)² + .25*(.13793 -.125)² + .25*(.29310-.125)² =  .0115

Annual standard deviation = 10.74%

First we compute the value of a call option:

d₁ = { ln (386,666.67/425,000) + .05 + .5*(0.1074^2)}/{0.1074} = -0.3608

d₂ = -0.3608 - 0.1074 = -0.4682

N(d₁) = 0.35943

N(d₂) = 0.31919

C = 386,666.67*0.35943 - (425,000 * e^-.05 *0.31919) = 138,979.60 – 129,039.74 = 9,939.86.

The value of the put is then C - P₀ + X e^-RT = 9,939.86 – 386,666.67 + 425,000* e^-.05 = 27,545.70.

The total NPV of the project after taking into account the abandonment option = NPV without abandonment option + value of abandonment option = -13,333.33 + 27,545.70 = 14,212.36

8 a) Under the decision tree approach, the firm will abandon the project at time 1 whenever the PV is less than the abandonment value of $780 million.

So the expected future cashflows are .1*780 + .2*780 + .3*800 + .4*900 = 834 million

The value of the project after accounting for the abandonment option = 834/1.1 – 750 = 8.18 million. 

b) Ignoring the option to abandon, the NPV of the project is (.1*700 + .2*750 + .3*800 + .4*900)/1.1 – 750 = 745.45 - 750 = -4.55 million

The abandonment option is a one-year put option on the time 1 PV of the project’s cashflows, with an exercise price equal to the abandonment value of $780 million.

Maturity = 1 year

R_f = 6% per year

X = $780 million

P₀ = the value today of the project’s exp. cashflows = 745.45 million

Annual standard deviation is computed as follows:

The expected return is just the required return of 10%.

Variance of annual return = .1*(-.06098 -.1)² + .2*(.0061-.1)² + .3*(.07317 -.1)² + .4*(.20732-.1)² =  .0092

Annual standard deviation = 9.58%

First we compute the value of a call option:

d₁ = { ln (745.45/780) + .06 + .5*(0.0958^2)}/{0.0958} = 0.2013

d₂ = 0.2013 - 0.0958 = 0.1055

N(d₁) = 0.49601

N(d₂) = 0.49016

C = 745.45*0.49601 - (780 * e^-.06 *0.49016) = 369.75 – 338.02 = 31.73 million.

The value of the put is then C - P₀ + X e^-RT = 31.73 – 745.45 + 780* e^-.06 = 20.85 million.

The total NPV of the project after taking into account the abandonment option = NPV without abandonment option + value of abandonment option = -4.55 + 20.85 = 16.31 million.

9 a) Under the decision tree approach, the firm will abandon the project at time 1 whenever the PV is less than the abandonment value of $295 million.

So the expected future cashflows are  .25*295 + .3*295 + .45*333 = 312.10 million

The value of the project after accounting for the abandonment option = 312.1/1.14 – 240 = 33.77 million. 

b) Ignoring the option to abandon, the NPV of the project is (.25*222 + .3*250 + .45*333)/1.14 – 240 = 245.92 - 240 = 5.92 million

The abandonment option is a one-year put option on the time 1 PV of the project’s cashflows, with an exercise price equal to the abandonment value of $295 million.

Maturity = 1 year

R_f = 6% per year

X = $295 million

P₀ = the value today of the project’s exp. cashflows = 245.92 million

Annual standard deviation is computed as follows:

The expected return is just the required return of 14%.

Variance of annual return = .25*(-0.09727-.14)² + .3*(.01659 -.14)² + .45*(.35409-.14)² =  .0393

Annual standard deviation = 19.82%

First we compute the value of a call option:

d₁ = { ln (245.92/295) + .06 + .5*(0.1982^2)}/{0.1982} = -0.5164

d₂ = -0.5164 - 0.1982 = -0.7146

N(d₁) = 0.30154

N(d₂) = 0.23886

C = 245.92*0.30154 - (295 * e^-.06 *0.23886) = 74.155 – 66.360 = 7.795 million.

The value of the put is then C - P₀ + X e^-RT = 7.795 – 245.92 + 295* e^-.06 = 36.694 million.

The total NPV of the project after taking into account the abandonment option = NPV without abandonment option + value of abandonment option = 5.92 + 36.694 = 45.62 million.

10 a) In the Black-Scholes formula, N(d₁) is related to the probability that the option will be exercised on the maturity date. T That’s what it represents at an intuitive level.  It’s not mathematically equal to the probability the option will be exercised, but it roughly corresponds to it.  And higher values of N(d₁) represent a higher likelihood of exercise.

b)   When we use the Black-Scholes formula to value the option to make follow-on investments, P₀ is the PV today of the future NPV of the follow-on (or second stage) project. F It’s the PV today of the future PV of the follow-on (or second stage) project.

c)   The option to make follow-on investments is a call option on the future PV of the first stage project. F It’s a call option on the future PV of the second stage project.

d)   When the second stage project occurs several years after the first stage project, we use the standard deviation of annual returns in the Black-Scholes formula; if the second stage project occurred several months after the first stage project, we would use the standard deviation of monthly returns. F In the Black-Scholes formula we always use the standard deviation of annual returns.  (All variables are measured annually: time to maturity, R_f and σ.)

e)   The timing option is an American call option; it can be exercised before maturity. T That’s exactly what it is – we can exercise it either today or tomorrow, not just at maturity.

f)    The timing option is valuable only because waiting may lead to some new information being revealed before you decide whether to accept the project or not. T Without new information there would only be a cost of waiting and no benefit.  You would never wait, and the option to wait would have no value.

g)   The only time an American call option on a dividend-paying stock might be exercised early (before maturity) is just before an ex-dividend date; the higher the dividend, the more likely you are to exercise early. T Any time you exercise before maturity, you lose the time value of the option, so there is always a cost of exercising early.  There is a benefit to exercising early only just before an ex-dividend date.  By not exercising you lose the dividend; by exercising you get the dividend.  So you will exercise early only on ex-dividend dates.  And you exercise only when the dividend is greater than the time value of the option, so the higher the dividend the more likely you are to exercise early.

h)   The abandonment option is a put option on the PV of the project’s cashflows; the exercise price is the investment amount. F It is a put option but the exercise price is the abandonment value – the after-tax salvage value you get by abandoning the project.

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