1. A call option confers the right, without the obligation, to buy an asset at a given price on or before a given date. A put option confers the right, without the obligation, to sell an asset at a given price on or before a given date. You would buy a call option if you expect the price of the asset to increase. You would buy a put option if you expect the price of the asset to decrease. A call option has unlimited potential profit, while a put option has limited potential profit; the underlying asset’s price cannot be less than zero.
2. a. The buyer of a call option pays money for the right to buy….
b. The buyer of a put option pays money for the right to sell….
c. The seller of a call option receives money for the obligation to sell….
d. The seller of a put option receives money for the obligation to buy….
3. The intrinsic value of a call option is Max [S – E,0]. It is the value of the option at expiration.
4. The value of a put option at expiration is Max[E – S,0]. By definition, the intrinsic value of an option is its value at expiration, so Max[E – S,0] is the intrinsic value of a put option.
5. The call is selling for less than its intrinsic value; an arbitrage opportunity exists. Buy the call for $10, exercise the call by paying $35 in return for a share of stock, and sell the stock for $50. You’ve made a riskless $5 profit.
6. The prices of both the call and the put option should increase. The higher level of downside risk still results in an option price of zero, but the upside potential is greater since there is a higher probability that the asset will finish in the money.
7. False. The value of a call option depends on the total variance of the underlying asset, not just the systematic variance.
8. The call option will sell for more since it provides an unlimited profit opportunity, while the potential profit from the put is limited (the stock price cannot fall below zero).
9. The value of a call option will increase, and the value of a put option will decrease.
10. The reason they don’t show up is that the U.S. government uses cash accounting; i.e., only actual cash inflows and outflows are counted, not contingent cash flows. From a political perspective, they would make the deficit larger, so that is another reason not to count them! Whether they should be included depends on whether we feel cash accounting is appropriate or not, but these contingent liabilities should be measured and reported. They currently are not, at least not in a systematic fashion.
11. The option to abandon reflects our ability to shut down a project if it is losing money. Since this option acts to limit losses, we will underestimate NPV if we ignore it.
12. The option to expand reflects our ability to increase production if the new product sells more than we initially expected. Since this option increases the potential future cash flows beyond our initial estimate, we will underestimate NPV if we ignore it.
13. This is a good example of the option to expand.
14. With oil, for example, we can simply stop pumping if prices drop too far, and we can do so quickly. The oil itself is not affected; it just sits in the ground until prices rise to a point where pumping is profitable. Given the volatility of natural resource prices, the option to suspend output is very valuable.
15. There are two possible benefits. First, awarding employee stock options may better align the interests of the employees with the interests of the stockholders, lowering agency costs. Secondly, if the company has little cash available to pay top employees, employee stock options may help attract qualified employees for less pay.
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.
Basic
1. a. The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $55 – [$45/1.055] = $12.35
The intrinsic value is the amount by which the stock price exceeds the exercise price of the call, so the intrinsic value is $10.
b. The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $55 – [$35/1.055] = $21.82
The intrinsic value is the amount by which the stock price exceeds the exercise price of the call, so the intrinsic value is $20.
c. The value of the put option is $0 since there is no possibility that the put will finish in the money. The intrinsic value is also $0.
2. a. The calls are in the money. The intrinsic value of the calls is $3.
b. The puts are out of the money. The intrinsic value of the puts is $0.
c. The Mar call and the Oct put are mispriced. The call is mispriced because it is selling for less than its intrinsic value. If the option expired today, the arbitrage strategy would be to buy the call for $2.80, exercise it and pay $80 for a share of stock, and sell the stock for $83. A riskless profit of $0.20 results. The October put is mispriced because it sells for less than the July put. To take advantage of this, sell the July put for $3.90 and buy the October put for $3.65, for a cash inflow of $0.25. The exposure of the short position is completely covered by the long position in the October put, with a positive cash inflow today.
3. a. Each contract is for 100 shares, so the total cost is:
Cost = 10(100 shares/contract)($7.60)
Cost = $7,600
b. If the stock price at expiration is $140, the payoff is:
Payoff = 10(100)($140 – 110)
Payoff = $30,000
If the stock price at expiration is $125, the payoff is:
Payoff = 10(100)($125 – 110)
Payoff = $15,000
c. Remembering that each contract is for 100 shares of stock, the cost is:
Cost = 10(100)($4.70)
Cost = $4,700
The maximum gain on the put option would occur if the stock price goes to $0. We also need to subtract the initial cost, so:
Maximum gain = 10(100)($110) – $4,700
Maximum gain = $105,300
If the stock price at expiration is $104, the position will have a profit of:
Profit = 10(100)($110 – 104) – $4,700
Profit = $6,000
d. At a stock price of $103 the put is in the money. As the writer you will make:
Net loss = $4,700 – 10(100)($110 – 103)
Net loss = –$2,300
At a stock price of $132 the put is out of the money, so the writer will make the initial cost:
Net gain = $4,700
At the breakeven, you would recover the initial cost of $4,700, so:
$4,700 = 10(100)($110 – ST)
ST = $105.30
For terminal stock prices above $105.30, the writer of the put option makes a net profit (ignoring transaction costs and the effects of the time value of money).
4. a. The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $80 – 70/1.06
C0 = $13.96
b. Using the equation presented in the text to prevent arbitrage, we find the value of the call is:
$80 = [($95 – 75)/($95 – 90)]C0 + $75/1.06
C0 = $2.31
5. a. The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $70 – $45/1.05
C0 = $27.14
b. Using the equation presented in the text to prevent arbitrage, we find the value of the call is:
$70 = 2C0 + $60/1.05
C0 = $6.43
6. Each option contract is for 100 shares of stock, so the price of a call on one share is:
C0 = $1,200/100 shares per contract
C0 = $12
Using the no arbitrage model, we find that the price of the stock is:
S0 = $12[($65 – 45)/($65 – 60)] + $45/1.05
S0 = $90.86
7. a. The equity can be valued as a call option on the firm with an exercise price equal to the value of the debt, so:
E0 = $1,050 – [$1,000/1.05]
E0 = $97.62
b. The current value of debt is the value of the firm’s assets minus the value of the equity, so:
D0 = $1,050 – 97.62
D0 = $952.38
We can use the face value of the debt and the current market value of the debt to find the interest rate, so:
Interest rate = [$1,000/$952.38] – 1
Interest rate = .05 or 5%
c. The value of the equity will increase. The debt then requires a higher return; therefore the present value of the debt is less while the value of the firm does not change.
8. a. Using the no arbitrage valuation model, we can use the current market value of the firm as the stock price, and the par value of the bond as the strike price to value the equity. Doing so, we get:
$1,200 = [($1,400 – 800)/($1,400 – 1,000)]E0 + [$800/1.04]
E0 = $287.18
The current value of the debt is the value of the firm’s assets minus the value of the equity, so:
D0 = $1,200 – 287.18
D0 = $912.82
b. Using the no arbitrage model as in part a, we get:
$1,200 = [($1,700 – 500)/($1,700 – 1,000)]E0 + [$500/1.04]
E0 = $419.55
The stockholders will prefer the new asset structure because their potential gain increases while their maximum potential loss remains unchanged.
9. The conversion ratio is the par value divided by the conversion price, so:
Conversion ratio = $1,000/$80
Conversion ratio = 12.50
The conversion value is the conversion ratio times the stock price, so:
Conversion value = 12.5($90)
Conversion value = $1,125.00
10. a. The minimum bond price is the greater of the straight bond value or the conversion price. The straight bond value is:
Straight bond value = $37.50(PVIFA4.5%,40) + $1,000/1.04540
Straight bond value = $861.99
The conversion ratio is the par value divided by the conversion price, so:
Conversion ratio = $1,000/$40
Conversion ratio = 25
The conversion value is the conversion ratio times the stock price, so:
Conversion value = 25($38)
Conversion value = $950.00
The minimum value for this bond is the convertible floor value of $950.00.
b. The option embedded in the bond adds the extra value.
11. a. The minimum bond price is the greater of the straight bond value or the conversion value. The straight bond value is:
Straight bond value = $70(PVIFA9%,30) + $1,000/1.0930
Straight bond value = $794.53
The conversion ratio is the par value divided by the conversion price, so:
Conversion ratio = $1,000/$65
Conversion ratio = 15.38
The conversion price is the conversion ratio times the stock price, so:
Conversion value = 15.38($50)
Conversion value = $769.23
The minimum value for this bond is the straight bond floor value of $794.53.
b. The conversion premium is the difference between the current stock price and conversion price, divided by the conversion price, so:
Conversion premium = ($65 – 50)/$50 = 30%
12. The value of the bond without warrants is:
Straight bond value = $105(PVIFA12%,15) + $1,000/1.1215
Straight bond value = $897.84
The value of the warrants is the selling price of the bond minus the value of the bond without warrants, so:
Total warrant value = $1,000 – 897.84
Total warrant value = $102.16
Since the bond has 25 warrants attached, the price of each warrant is:
Price of one warrant = $102.16/25
Price of one warrant = $4.09
13. If we purchase the machine today, the NPV is the cost plus the present value of the increased cash flows, so:
NPV0 = –$1,500,000 + $280,000(PVIFA12%,10)
NPV0 = $82,062.45
We should not purchase the machine today. We would want to purchase the machine when the NPV is the highest. So, we need to calculate the NPV each year. The NPV each year will be the cost plus the present value of the increased cash savings. We must be careful however. In order to make the correct decision, the NPV for each year must be taken to a common date. We will discount all of the NPVs to today. Doing so, we get:
Year 1: NPV1 = [–$1,375,000 + $280,000(PVIFA12%,9)] / 1.12
NPV1 = $104,383.88
Year 2: NPV2 = [–$1,250,000 + $280,000(PVIFA12%,8)] / 1.122
NPV2 = $112,355.82
Year 3: NPV3 = [–$1,125,000 + $280,000(PVIFA12%,7)] / 1.123
NPV3 = $108,796.91
Year 4: NPV4 = [–$1,000,000 + $280,000(PVIFA12%,6)] / 1.124
NPV4 = $96,086.55
Year 5: NPV5 = [–$1,000,000 + $280,000(PVIFA12%,5)] / 1.125
NPV5 = $5,298.26
Year 6: NPV6 = [–$1,000,000 + $280,000(PVIFA12%,4)] / 1.126
NPV6 = –$75,762.72
The company should purchase the machine two years from now when the NPV is the highest.
Intermediate
14. a. The base-case NPV is:
NPV = –$1,800,000 + $420,000(PVIFA16%,10)
NPV = $229,955.54
b. We would abandon the project if the cash flow from selling the equipment is greater than the present value of the future cash flows. We need to find the sale quantity where the two are equal, so:
$1,400,000 = ($60)Q(PVIFA16%,9)
Q = $1,400,000/[$60(4.6065)]
Q = 5,065
Abandon the project if Q < 5,065 units, because the NPV of abandoning the project is greater than the NPV of the future cash flows.
c. The $1,400,000 is the market value of the project. If you continue with the project in one year, you forego the $1,400,000 that could have been used for something else.
15. a. If the project is a success, present value of the future cash flows will be:
PV future CFs = $60(9,000)(PVIFA16%,9)
PV future CFs = $2,487,533.69
From the previous question, if the quantity sold is 4,000, we would abandon the project, and the cash flow would be $1,400,000. Since the project has an equal likelihood of success or failure in one year, the expected value of the project in one year is the average of the success and failure cash flows, plus the cash flow in one year, so:
Expected value of project at year 1 = [($2,487,533.69 + $1,400,000)/2] + $420,000
Expected value of project at year 1 = $2,363,766.85
The NPV is the present value of the expected value in one year plus the cost of the equipment, so:
NPV = –$1,800,000 + ($2,363,766.85)/1.16
NPV = $237,730.045
b. If we couldn’t abandon the project, the present value of the future cash flows when the quantity is 4,000 will be:
PV future CFs = $60(4,000)(PVIFA16%,9)
PV future CFs = $1,105,570.53
The gain from the option to abandon is the abandonment value minus the present value of the cash flows if we cannot abandon the project, so:
Gain from option to abandon = $1,400,000 – 1,105,570.53
Gain from option to abandon = $294,429.47
We need to find the value of the option to abandon times the likelihood of abandonment. So, the value of the option to abandon today is:
Option value = (.50)($294,429.47)/1.16
Option value = $126,909.25
16. If the project is a success, present value of the future cash flows will be:
PV future CFs = $60(18,000)(PVIFA16%,9)
PV future CFs = $4,975,067.39
If the sales are only 4,000 units, from Problem #14, we know we will abandon the project, with a value of $1,400,000. Since the project has an equal likelihood of success or failure in one year, the expected value of the project in one year is the average of the success and failure cash flows, plus the cash flow in one year, so:
Expected value of project at year 1 = [($4,975,067.39 + $1,400,000)/2] + $420,000
Expected value of project at year 1 = $3,607,533.69
The NPV is the present value of the expected value in one year plus the cost of the equipment, so:
NPV = –$1,800,000 + $3,607,533.69/1.16
NPV = $1,309,942.84
The gain from the option to expand is the present value of the cash flows from the additional units sold, so:
Gain from option to expand = $60(9,000)(PVIFA16%,9)
Gain from option to expand = $2,487,533.69
We need to find the value of the option to expand times the likelihood of expansion. We also need to find the value of the option to expand today, so:
Option value = (.50)($2,487,533.69)/1.16
Option value = $1,072,212.80
17. a. The value of the call is the maximum of the stock price minus the present value of the exercise price, or zero, so:
C0 = Max[$65 – ($75/1.05),0]
C0 = $0
The option isn’t worth anything.
b. The stock price is too low for the option to finish in the money. The minimum return on the stock required to get the option in the money is:
Minimum stock return = ($75 – 65)/$65
Minimum stock return = .1538 or 15.38%
which is much higher than the risk-free rate of interest.
18. B is the more typical case; A presents an arbitrage opportunity. You could buy the bond for $800 and immediately convert it into stock that can be sold for $1,000. A riskless $200 profit results.
19. a. The conversion ratio is given at 20. The conversion price is the par value divided by the conversion ratio, so:
Conversion price = $1,000/20
Conversion price = $50
The conversion premium is the percent increase in stock price that results in no profit when the bond is converted, so:
Conversion premium = ($50 – 46)/$46
Conversion premium = .0870 or 8.70%
b. The straight bond value is:
Straight bond value = $40(PVIFA5%,20) + $1,000/1.0520
Straight bond value = $875.38
And the conversion value is the conversion ratio times the stock price, so:
Conversion value = 20($46)
Conversion value = $920.00
c. We simply need to set the straight bond value equal to the conversion ratio times the stock price, and solve for the stock price, so:
$875.38 = 20S
S = $43.77
d. There are actually two option values to consider with a convertible bond. The conversion option value, defined as the market value less the floor value, and the speculative option value, defined as the floor value less the straight bond value. When the conversion value is less than the straight-bond value, the speculative option is worth zero.
Conversion option value = $950 – 920 = $30
Speculative option value = $920 – 875.38 = $44.62
Total option value = $30.00 + 44.62 = $74.62
Challenge
20. The straight bond value today is:
Straight bond value = $68(PVIFA10%,25) + $1,000/1.1025
Straight bond value = $709.53
And the conversion value of the bond today is:
Conversion value = $44.75($1,000/$150)
Conversion value = $298.33
We expect the bond to be called when the conversion value increases to $1,300, so we need to find the number of periods it will take for the current conversion value to reach the expected value at which the bond will be converted. Doing so, we find:
$298.33(1.12)t = $1,300
t = 12.99 years.
The bond will be called in 12.99 years.
The bond value is the present value of the expected cash flows. The cash flows will be the annual coupon payments plus the conversion price. The present value of these cash flows is:
Bond value = $68(PVIFA10%,12.99) + $1,300/1.1012.99 = $859.80
21. We will use the bottom up approach to calculate the operating cash flow. Assuming we operate the project for all four years, the cash flows are:
| Year | 0 | 1 | 2 | 3 | 4 | |
| Sales | $7,000,000 | $7,000,000 | $7,000,000 | $7,000,000 | ||
| Operating costs | 3,000,000 | 3,000,000 | 3,000,000 | 3,000,000 | ||
| Depreciation | 2,000,000 | 2,000,000 | 2,000,000 | 2,000,000 | ||
| EBT | $2,000,000 | $2,000,000 | $2,000,000 | $2,000,000 | ||
| Tax | 760,000 | 760,000 | 760,000 | 760,000 | ||
| Net income | $1,240,000 | $1,240,000 | $1,240,000 | $1,240,000 | ||
| +Depreciation | 2,000,000 | 2,000,000 | 2,000,000 | 2,000,000 | ||
| Operating CF | $3,240,000 | $3,240,000 | $3,240,000 | $3,240,000 | ||
| Change in NWC | –$2,000,000 | 0 | 0 | 0 | $2,000,000 | |
| Capital spending | –8,000,000 | 0 | 0 | 0 | 0 | |
| Total cash flow | –$10,000,000 | $3,240,000 | $3,240,000 | $3,240,000 | $5,240,000 |
There is no salvage value for the equipment. The NPV is:
NPV = –$10,000,000 + $3,240,000(PVIFA16%,3) + $5,2400,000/1.164
NPV = $170,687.46
The cash flows if we abandon the project after one year are:
| Year | 0 | 1 | ||||
| Sales | $7,000,000 | |||||
| Operating costs | 3,000,000 | |||||
| Depreciation | 2,000,000 | |||||
| EBT | $2,000,000 | |||||
| Tax | 760,000 | |||||
| Net income | $1,240,000 | |||||
| +Depreciation | 2,000,000 | |||||
| Operating CF | $3,240,000 | |||||
| Change in NWC | –$2,000,000 | $2,000,000 | ||||
| Capital spending | –8,000,000 | 6,310,000 | ||||
| Total cash flow | –$10,000,000 | $11,550,000 |
The book value of the equipment is:
Book value = $8,000,000 – (1)($8,000,000/4)
Book value = $6,000,000
So the taxes on the salvage value will be:
Taxes = ($6,000,000 – 6,500,000)(.38)
Taxes = –$190,000
This makes the aftertax salvage value:
Aftertax salvage value = $6,500,000 – 190,000
Aftertax salvage value = $6,310,000
The NPV if we abandon the project after one year is:
NPV = –$10,000,000 + $11,550,000/1.16
NPV = –$43,103.45
If we abandon the project after two years, the cash flows are:
| Year | 0 | 1 | 2 | |||
| Sales | $7,000,000 | $7,000,000 | ||||
| Operating costs | 3,000,000 | 3,000,000 | ||||
| Depreciation | 2,000,000 | 2,000,000 | ||||
| EBT | $2,000,000 | $2,000,000 | ||||
| Tax | 760,000 | 760,000 | ||||
| Net income | $1,240,000 | $1,240,000 | ||||
| +Depreciation | 2,000,000 | 2,000,000 | ||||
| Operating CF | $3,240,000 | $3,240,000 | ||||
| Change in NWC | –$2,000,000 | 0 | $2,000,000 | |||
| Capital spending | –8,000,000 | 0 | 5,240,000 | |||
| Total cash flow | –$10,000,000 | $3,240,000 | $10,480,000 |
The book value of the equipment is:
Book value = $8,000,000 – (2)($8,000,000/4)
Book value = $4,000,000
So the taxes on the salvage value will be:
Taxes = ($4,000,000 – 6,000,000)(.38)
Taxes = –$760,000
This makes the aftertax salvage value:
Aftertax salvage value = $6,000,000 – 760,000
Aftertax salvage value = $5,240,000
The NPV if we abandon the project after two years is:
NPV = –$10,000,000 + $3,240,000/1.16 + $10,480,000/1.162
NPV = $581,450.65
If we abandon the project after three years, the cash flows are:
| Year | 0 | 1 | 2 | 3 | ||
| Sales | $7,000,000 | $7,000,000 | $7,000,000 | |||
| Operating costs | 3,000,000 | 3,000,000 | 3,000,000 | |||
| Depreciation | 2,000,000 | 2,000,000 | 2,000,000 | |||
| EBT | $2,000,000 | $2,000,000 | $2,000,000 | |||
| Tax | 760,000 | 760,000 | 760,000 | |||
| Net income | $1,240,000 | $1,240,000 | $1,240,000 | |||
| +Depreciation | 2,000,000 | 2,000,000 | 2,000,000 | |||
| Operating CF | $3,240,000 | $3,240,000 | $3,240,000 | |||
| Change in NWC | –$2,000,000 | 0 | 0 | $2,000,000 | ||
| Capital spending | –8,000,000 | 0 | 0 | 2,620,000 | ||
| Total cash flow | –$10,000,000 | $3,240,000 | $3,240,000 | $7,860,000 |
The book value of the equipment is:
Book value = $8,000,000 – (3)($8,000,000/4)
Book value = $2,000,000
So the taxes on the salvage value will be:
Taxes = ($2,000,000 – 3,000,000)(.38)
Taxes = –$380,000
This makes the aftertax salvage value:
Aftertax salvage value = $3,000,000 – 380,000
Aftertax salvage value = $2,620,000
The NPV if we abandon the project after two years is:
NPV = –$10,000,000 + $3,240,000(PVIFA16%,2) + $7,860,000/1.163
NPV = $236,520.56
We should abandon the equipment after two years since the NPV of abandoning the project after two years has the highest NPV.
