CHAPTER 25
OPTIONS AND CORPORATE SECURITIES
Learning Objectives
LO1 The basics of call and put options and how to calculate their payoffs and profits.
LO2 The factors that affect option values and how to price call and put options using
no arbitrage conditions.
LO3 The basics of employee stock options and their benefits and disadvantages.
LO4 How to value a firm’s equity as an option on the firm’s assets and use of option
valuation to evaluate capital budgeting projects.
LO5 The basics of convertible bonds and warrants and how to value them.
Answers to Concepts Review and Critical Thinking Questions
1.
(LO1) 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 zer .
2.
(LO1)
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.
(LO1) The intrinsic value of a call option is Max [S – E,0]. It is the value of the option at expiration.
4.
(LO1) 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.
(LO2) 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.
(LO2) 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.
(LO2) False. The value of a call option depends on the total variance of the underlying asset, not just the
systematic variance.
8.
(LO1) 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.
(LO2) The value of a call option will increase, and the value of a put option will decrease.
10. (LO1) The reason they don’t show up is that the 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.
25-1
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.
(LO2)
a.
The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $67 – [$45/1.043] = $23.85
The intrinsic value is the amount by which the stock price exceeds the exercise price of the call, so the
intrinsic value is $22.
b.
The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $67 – [$35/1.043] = $33.44
The intrinsic value is the amount by which the stock price exceeds the exercise price of the call, so the
intrinsic value is $32.
c.
2.
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.
(LO1)
a.
The calls are not in the money. The intrinsic value of the calls is $0.
b.
The puts are in the money. The intrinsic value of the puts is $85 – 84 = 1.
c.
The Mar call is out of the money.
The October put is mispriced because it sells for less than the July put. To take advantage of this, sell the
July put for $10.85 and buy the October put for $10.45, for a cash inflow of $0.40. The exposure of the
short position is completely covered by the long position in the October put, with a positive cash inflow
today.
3.
(LO1)
a.
Each contract is for 100 shares, so the total cost is:
Cost = 10(100 shares/contract)($0.23)
Cost = $230
b.
If the stock price at expiration is $30, the payoff is:
Payoff = 10(100)($30 – 28)
Payoff = $2000
If the stock price at expiration is $29, the payoff is:
Payoff = 10(100)($29 – 28)
Payoff = $1,000
c.
Remembering that each contract is for 100 shares of stock, the cost is:
25-2
Cost = 10(100)($2.10)
Cost = $2,100
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)($28) – $2,100
Maximum gain = $25,900
If the stock price at expiration is $23, the position will have a profit of:
Profit = 10(100)($28 – 23) – $2,100
Profit = $2,900
d.
At a stock price of $25 the put is in the money. As the writer you will make:
Net loss = $2,100 – 10(100)($28 – 25)
Net loss = –$900
At a stock price of $31 the put is out of the money, so the writer will make the initial cost:
Net gain = $2,100
At the breakeven, you would recover the initial cost of $2,900, so:
$2,100 = 10(100)($28 – ST)
ST = $25.90
For terminal stock prices above $25.90, the writer of the put option makes a net profit (ignoring
transaction costs and the effects of the time value of money).
4.
(LO2)
a.
The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $85 – 75/1.06
C0 = $14.25
b.
Using the equation presented in the text to prevent arbitrage, we find the value of the call is:
$85 = [($95 – 53)/($95 – 90)]C0 + $53/1.06
C0 = $4.167
5.
(LO2)
a.
The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $75 – $60/1.06
C0 = $18.396
b.
Using the equation presented in the text to prevent arbitrage, we find the value of the call is:
$75 = 1.38C0 + $64/1.06
C0 = $10.635
25-3
6.
(LO2)
Each option contract is for 100 shares of stock, so the price of a call on one share is:
C0 = $1,150/100 shares per contract
C0 = $11.50
Using the no arbitrage model, we find that the price of the stock is:
S0 = $11.50[($68 – 47)/($68 – 60)] + $47/1.03
S0 = $75.82
7.
(LO4)
a.
The equity can be valued as a call option on the firm with an exercise price equal to the value of the debt:
Method 1:
The two possible future asset values are equal or greater to the future value of debt, which implies that
the debt is risk-free. For this special case where the debt is risk-free, we can calculate equity value as:
E0 = $104 – [$100/1.05]
E0 = $8.76
Method 2:
The solution for the general case, which works for both risk-free and risky debt, is shown below:
The lower possible future asset value is equal to:
$100 = min[$100, $129]
Replicate this lower future asset value by investing today in the risk-free asset, at a present value of:
$100 / 1.05 = $95.238
The exercise price of the option is the future value of the debt, equal to its face value of $100. So if the
asset turns out to be worth the upper value of $129 in one year, the option will then be worth:
$129 - $100 = $29
The required number of call options is then the upper future value minus the lower future value, divided
by the value of the option:
($129 - $100) / $29 = 1.00 call options
With the firm is presently valued at $104, we can replicate the value of the firm as:
$104 = 1.00 C0 + $95.238
C0 = $8.762
b.
The current value of debt is the value of the firm’s assets minus the value of the equity, so:
D0 = $104 – 8.762
D0 = $95.238
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 = [$100/$95.238] – 1
Interest rate = .05 or 5%
The calculated interest rate is equal to the risk free rate because the value of the firm’s assets is always
equal to or greater than the face value of debt, and therefore the debt is risk-free.
25-4
Note that if the lower possible future value was below the face value of debt, for example the two
possible future values were $100 and $129, then the debt would be risky and the calculated interest rate
would have turned out to be greater than the risk free rate.
c.
8.
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.
(LO4)
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:
$109 = [($138 – 92)/($138 – 100)]E0 + [$92/1.06]
E0 = $18.345
The current value of the debt is the value of the firm’s assets minus the value of the equity, so:
D0 = $109 – 18.345
D0 = $90.65
b.
Using the no arbitrage model as in part a, we get:
$109 = [($160 – 80)/($160 – 100)]E0 + [$80/1.06]
E0 = 25.146
The stockholders will prefer the new asset structure because their potential gain increases.
9.
(LO5) The conversion ratio is the par value divided by the conversion price, so:
Conversion ratio = $100/$28
Conversion ratio = 3.571428571
The conversion value is the conversion ratio times the stock price, so:
Conversion value = 3.571428571 ($37)
Conversion value = $132.143
10. (LO5)
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 = $2.70(PVIFA3.50%,60) + $100/1.03560
Straight bond value = $80.04
The conversion ratio is the par value divided by the conversion price, so:
Conversion ratio = $100/$35
Conversion ratio = 2.857142857
The conversion value is the conversion ratio times the stock price, so:
Conversion value = 2.857142857 ($34)
Conversion value = $97.14285714
The minimum value for this bond is the convertible floor value of $97.14.
25-5
b.
The option embedded in the bond adds the extra value.
11. (LO5)
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 = $7.5(PVIFA9%,30) + $100/1.0930
Straight bond value = $84.59
The conversion ratio is the par value divided by the conversion price, so:
Conversion ratio = $100/$5.5
Conversion ratio = 18.182
The conversion price is the conversion ratio times the stock price, so:
Conversion value = 18.182($4.2)
Conversion value = $76.3636
The minimum value for this bond is the straight value of $84.59.
b.
The conversion premium is 0
12. (LO5) The value of the bond without warrants is:
Straight bond value = $5.5(PVIFA7%,25) + $100/1.0725
Straight bond value = $82.52
The value of the warrants is the selling price of the bond minus the value of the bond without warrants, so:
Total warrant value = $100 – 82.52
Total warrant value = $17.48
Since the bond has 20 warrants attached, the price of each warrant is:
Price of one warrant = $17.48/20
Price of one warrant = $0.874
13. (LO4) If we purchase the machine today, the NPV is the cost plus the present value of the increased cash flows,
so:
NPV0 = –$1,600,000 + $310,000(PVIFA14%,10)
NPV0 = $16,995.85
We should purchase the machine today because the NPV is positive. We would want to purchase the machine
when the NPV is the highest positive value. 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:
25-6
Year 1: NPV1 = [–$1,505,000 + $310,000(PVIFA14%,9)] / 1.14
NPV1 = 24,890.59
Year 2: NPV2 = [–$1,410,000 + $310,000(PVIFA14%,8)] / 1.142
NPV2 = 21,581.88
Year 3: NPV3 = [–$1,315,000 + $310,000(PVIFA14%,7)] / 1.143
NPV3 = $9,702.37
Year 4: NPV4 = [–$1,220,000 + $310,000(PVIFA14%,6)] / 1.144
NPV4 = –$8,592.90
Year 5: NPV5 = [–$1,125,000 + $310,000(PVIFA14%,5)] / 1.145
NPV5 = –$31,548.99
Year 6: NPV6 = [–$1,125,000 + $310,000(PVIFA14%,4)] / 1.146
NPV6 = –$101,025.95
The NPV is the highest one year from now.
Intermediate
14. (LO4)
a.
The base-case NPV is:
NPV = –$1,600,000 + $409,500(PVIFA14%,10)
NPV = $535,999.35
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:
$1,200,000 = ($63)Q(PVIFA14%,9)
Q = $1,200,000 / [$63(4.946371837)]
Q = 3850.83
Abandon the project if Q < 3850.56 units, because the NPV of abandoning the project is greater than the
NPV of the future cash flows.
c.
The $1,200,000 is the market value of the project. If you continue with the project in one year, you
forego the $1,200,000 that could have been used for something else.
15. (LO4)
a.
If the project is a success, present value of the future cash flows will be:
PV future CFs = $63(9,000)(PVIFA14%,9)
PV future CFs = $2,804,592.83
From the previous question, if the quantity sold is3,500, we would abandon the project, and the cash flow
would be $1,200,000. 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 = [($2,804,592.83+ $1,200,000)/2] + $409,500
Expected value of project at year 1 = $2,411,796.42
25-7
The NPV is the present value of the expected value in one year plus the cost of the equipment, so:
NPV = –$1,600,000 + ($2,411,796.42)/1.14
NPV = $515,610.89
b.
If we couldn’t abandon the project, the present value of the future cash flows when the quantity is 3,500
will be:
PV future CFs = $63(3,500)(PVIFA14%,9)
PV future CFs = $1,090,674.99
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 = $1,200,000 – 1,090,674.99
Gain from option to abandon = $109,325.01
We need to find the value of the option to abandon times the likelihood of abandonment. So, the value of
the option to abandon today is:
Option value = (.50)($ 109,325.01)/1.14
Option value = $47,949.57
16. (LO4) If the project is a success, present value of the future cash flows will be:
PV future CFs = $63(18,000)(PVIFA14%,9)
PV future CFs = $5,609,185.66
If the sales are only 3,500 units, from Problem #14 & 15, we know we will abandon the project, with a value
of $1,200,000. 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 = [($5,609,185.66 + $1,200,000)/2] + $409,500
Expected value of project at year 1 = $3,814,092.83
The NPV is the present value of the expected value in one year plus the cost of the equipment, so:
NPV = –$1,600,000 + $3,814,092.83/1.14
NPV = $1,745,695.47
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 = $63(9,000)(PVIFA14%,9)
Gain from option to expand = $2,804,592.83
We need to find the value of the option to expand times the likelihood of expansion. We also need to find the
value of the option to expand today, so:
Option value = (.50)($ 2,804,592.83)/1.14
Option value = $1,230,084.57
17. (LO2)
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[$54 – ($60/1.05),0]
C0 = $0
25-8
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 = ($60 – 54)/$54
Minimum stock return = .1111 or 11.11%
which is much higher than the risk-free rate of interest.
18. (LO5) B is the more typical case; A presents an arbitrage opportunity. You could buy the bond for $80 and
immediately convert it into stock that can be sold for $100. A riskless $20 profit results.
19. (LO5)
a.
The conversion ratio is given at 20. The conversion price is the par value divided by the conversion ratio:
Conversion price = $100/20
Conversion price = $5
The conversion premium is the percent increase in stock price that results in no profit when the bond is
converted, so:
Conversion premium = ($5 – 4.5)/$4.5
Conversion premium = .1111 or 11.11%
b.
The straight bond value is:
Straight bond value = $3.25(PVIFA4.5%,40) + $100/1.04540
Straight bond value = $77.00
And the conversion value is the conversion ratio times the stock price, so:
Conversion value = 20($4.5)
Conversion value = $90.00
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:
$77 = 20S
S = $3.85
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 = $96 – 90 = $6
Speculative option value = $90 – 77 = $13
Total option value = $6.00 + 13 = $19.00
25-9
20.(LO4) a. The NPV of the project is the sum of the present value of the cash flows generated by the
project. The cash flows from this project are an annuity, so the NPV is:
NPV = –$38,000,000 + $8,000,000(PVIFA11%,8)
NPV = $3,168,982.09
b. The company should abandon the project if the PV of the revised cash flows for the next seven years is
less than the project’s aftertax salvage value. Since the option to abandon the project occurs in
Year 1, discount the revised cash flows to Year 1 as well. To determine the level of expected
cash flows below which the company should abandon the project, calculate the equivalent
annual cash flows the project must earn to equal the aftertax salvage value. We will solve for
C2, the revised cash flow beginning in Year 2. So, the revised annual cash flow below which it
makes sense to abandon the project is:
Aftertax salvage value = C2(PVIFA11%,7)
$25,000,000 = C2(PVIFA11%,9)
C2 = $25,000,000 / PVIFA11%,7
C2 = $5,305,381.74
Challenge
21. (LO5) The straight bond value today is:
Straight bond value = $4.8(PVIFA8%,25) + $100/1.0825
Straight bond value = $65.84
And the conversion value of the bond today is:
Conversion value = $3.21($100/$9)
Conversion value = $35.67
We expect the bond to be called when the conversion value increases to $130, 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:
$35.67(1.11)t = $130
t = ln(130/35.67) / ln(1.11) = 12.39 years.
The bond will be called in 12.39 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 = $4.8(PVIFA8%,12.39) + $130/1.0812.39 = $75.41
22. (LO4) 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
Sales
0
1
2
3
4
$10,900,000
$10,900,000
10,900,000
10,900,000
Operating costs
4,100,000
4,100,000
4,100,000
4,100,000
Depreciation
3,750,000
3,750,000
3,750,000
3,750,000
25-10
EBT
3,050,000
3,050,000
3,050,000
3,050,000
Tax
1,159,000
1,159,000
1,159,000
1,159,000
Net income
1,891,000
1,891,000
1,891,000
1,891,000
+Depreciation
3,750,000
3,750,000
3,750,000
3,750,000
Operating CF
5,641,000
5,641,000
5,641,000
5,641,000
Change in NWC
–$900,000
0
0
0
900,000
Capital spending
–15,000,000
0
0
0
0
Total cash flow
–$15,900,000
$5,641,000
$5,641,000
$5,641,000
$6,541,000
There is no salvage value for the equipment. The NPV is:
NPV = –$15,900,000 + $5,641,000 (PVIFA13%,3) + $6,541,000/1.134
NPV = $1,430,979.60
The cash flows if we abandon the project after one year are:
Year
0
Sales
1
$10,900,000
Operating costs
4,100,000
Depreciation
3,750,000
EBT
$3,050,000
Tax
1,159,000
Net income
$1,891,000
+Depreciation
3,750,000
Operating CF
$5,641,000
Change in NWC
–$900,000
$900,000
Capital spending
–15,000,000
12,335,000
Total cash flow
–$15,900,000
$18,876,000
The book value of the equipment is:
Book value = $15,000,000 – (1)($15,000,000/4)
Book value = $11,250,000
So the taxes on the salvage value will be:
Taxes = ($11,250,000 – 13,000,000)(.38)
Taxes = –$665,000
This makes the aftertax salvage value:
Aftertax salvage value = $13,000,000 – 665,000
25-11
Aftertax salvage value = $12,335,000
The NPV if we abandon the project after one year is:
NPV = –$15,900,000 + $18,876,000/1.14
NPV = 804,424.78
If we abandon the project after two years, the cash flows are:
Year
0
1
2
$10,900,000
10,900,000
Operating costs
4,100,000
4,100,000
Depreciation
3,750,000
3,570,000
EBT
3,050,000
3,050,000
Tax
1,159,000
1,159,000
Net income
1,189,000
1,189,000
+Depreciation
3,750,000
3,375,000
Operating CF
5,641,000
5,641,000
Sales
Change in NWC
–$900,000
0
900,000
Capital spending
–15,000,000
0
9,050,000
Total cash flow
–$15,000,000
$5,641,000
$15,591,000
The book value of the equipment is:
Book value = $15,000,000 – (2)($15,000,000/4)
Book value = $7,500,000
So the taxes on the salvage value will be:
Taxes = ($7,500,000 – 10,000,000)(.38)
25-12
Taxes = –$950,000
This makes the aftertax salvage value:
Aftertax salvage value = $10,000,000 –950,000
Aftertax salvage value = $9,050,000
The NPV if we abandon the project after two years is:
NPV = –$15,900,000 + $5,641,000/1.14 + $15,591,000/1.142
NPV = $1,302,075.34
If we abandon the project after three years, the cash flows are:
Year
0
Sales
1
2
3
$10,900,000
10,900,000
10,900,000
Operating costs
4,100,000
4,100,000
4,100,000
Depreciation
3,750,000
3,750,000
3,750,000
EBT
3,050,000
3,050,000
3,050,000
Tax
1,159,000
1,159,000
1,159,000
Net income
1,891,000
1,891,000
1,891,000
+Depreciation
3,750,000
3,750,000
3,750,000
Operating CF
5,641,000
5,641,000
5,641,000
Change in NWC
–$900,000
0
0
900,000
Capital spending
–15,000,000
0
0
6,075,000
Total cash flow
–$15,900,000
5,641,000
5,641,000
12,616,000
The book value of the equipment is:
Book value = $15,000,000 – (3)($15,000,000/4)
Book value = $3,750,000
So the taxes on the salvage value will be:
Taxes = ($3,750,000 – 7,500,000)(.38)
25-13
Taxes = –$1,425,000
This makes the aftertax salvage value:
Aftertax salvage value = $7,500,000 – 1,425,000
Aftertax salvage value = $6,075,000
The NPV if we abandon the project after two years is:
NPV = –$15,900,000 + $5,461,000(PVIFA14%,2) + $12,616,000/1.163
NPV = $2,253,286.69
We should abandon the equipment after three years since the NPV of abandoning the project after three years
has the highest NPV.
APPENDIX 25A
A1. Accurate values for the standard normal distribution are used here based on Excel’s NORMSDIST function. If
standard normal values are taken from Table 25A.1 instead, the final value for the option will differ slightly.
a.
d1 = –1.593726 ; d2 = –1.698726 ; N(d1) = .05549; N(d2) = .04468
C0 = $31.00(.05549) – [37.50/1.071/4](.04468) = $0.073
b.
d1 = 3.2264; d2 = 3.1203; N(d1) = .9994; N(d2) = .9991
C0 = $40(.9994) – [29/1.031/2](.9991) = $11.43
c.
d1 = 2.1992; d2 = 1.9914; N(d1) = .9861; N(d2) = .9768
C0 = $89(.9861) – [63/1.123/4](.9768) = $31.24
d.
d1 = .3486; d2 = .0486; N(d1) = .6363; N(d2) = .5194
C0 = $97(.6363) – [99/1.08](.5194) = $14.11
e.
S0 = 0, so C0 = 0
f.
T = ∞, so C0 = S0 = $125
g.
E = 0, so C0 = S0 = $129
h.
 = 0, so d1 and d2 go to +∞, so N(d1) and N(d2) go to 1. This is the no risk call option formula given in
the text. C0 = S0 – E/(1+R)t ; C0 = $121 – [113/1.061/2] = $11.24
i.
for  = ∞, d1 goes to +∞ so N(d1) goes to 1, and d2 goes to –∞ so N(d2) goes to 0; C0= S0 = $50
25-14
A2. Using S = $3,400, E = $2,950, t = 1, R = .045,  = .29 = 0.5385:
d1 = .6165; d2 = .0779; N(d1) = .7312; N(d2) = .5311
Value of equity = $3,400(.7312) – [2,950/1.045](.5311) = $986.92
Value of debt = $3,400 – 986.92 = $2,413.08
A3. a.
Project A:
using S = $3,400 + $135 = $3,535, E = $2,950, t = 1, R = .045,  = .39 = 0.6245:
d1 = .6940; d2 = .0495; N(d1) = .7498; N(d2) = .5197
Value of equityA = $3,535(.7498) – [2,950/1.045](.5197) = $1,183.49
Value of debtA = $3,535 – 1,183.49 = $2,351.51
Project B:
using S = $3,400 + $215 = $3,615, E = $2,950, t = 1, R = .045,  = .22 = 0.4690:
d1 = .7639; d2 = .2948; N(d1) = .7775; N(d2) = .6159
Value of equityB = $3,615(.7775) – [2,950/1.045](.6159) = $1,071.99
Value of debtB = $3,615 – 1,071.99 = $2,543.01
b.
Although the NPV of project B is higher, the equity value with project A is higher. While NPV
represents the increase in the value of the assets of the firm, in this case, the increase in the value of the
firm’s assets resulting from project B is mostly allocated to the debtholders resulting in a smaller increase
in the value of the equity. Stockholders would therefore prefer project A even though it has a lower NPV.
c.
Yes. If the same group of investors has equal stakes in the firm as bondholders and stockholders, then
total firm value matters and project B should be chosen, since it increases the value of the firm to $3,615
instead of $3,535.
d.
Stockholders may have an incentive to take on more risky, less profitable projects if the firm is
leveraged; all else the same, the higher the firm’s debt load the greater is this incentive.
25-15