CHAPTER 17 B – 1
26. a. The combined value of equity and debt of the two firms is:
Debt = $13,035.46 + 18,834.35 = $31,869.81
b. For the new firm, the combined market value of assets is $39,300, and the combined face value
of debt is $37,000. Using Black-Scholes to find the value of equity for the new firm, we find:
d1 = [ln($39,300/$37,000) + (.06 + .162/2) 1]/(.16
1
) = .8319
d2 = .8319 – (.16
1
) = .6719
N(d1) = .7973
N(d2) = .7492
Putting these values into the Black-Scholes model, we find the equity value is:
c. The change in the value of the firm’s equity is:
d. In a purely financial merger, when the standard deviation of the assets declines, the value of the
equity declines as well. The shareholders will lose exactly the amount the bondholders gain. The
27. a. Using the equation for the PV of a continuously compounded lump sum, we get:
b. Using the Black-Scholes model to value the equity, we get:
CHAPTER 17 B – 2
d2 = .2392 – (.60
2
) = –.6093
N(d1) = .5945
N(d2) = .2711
Putting these values into Black-Scholes:
c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the
firm’s equity, so:
Value of risky bond = $67,862.81 – 25,944.01
Value of risky bond = $41,918.80
Using the equation for the PV of a continuously compounded lump sum to find the return on
debt, we get:
d. The value of the debt with five years to maturity at the risk-free rate is:
PV = $75,000 e–.05(5) = $58,410.06
Using the Black-Scholes model to value the equity, we get:
d1 = [ln($58,000/$75,000) + (.05 + .602/2) 5]/(.60
5
) = .6656
d2 = .6656 – (.60
5
) = –.6761
N(d1) = .7472
Putting these values into Black-Scholes:
Equity = $58,000(.7472) – ($75,000e–.05(5))(.2495)
Equity = $28,761.92
CHAPTER 17 B – 3
Using the equation for the PV of a continuously compounded lump sum to find the return on
debt, we get:
value of assets to meet or exceed the face value of debt is higher than if the company only operates
for two more years.
28. a. Using the equation for the PV of a continuously compounded lump sum, we get:
PV = $75,000 e–.06(5) = $55,561.37
b. Using Black-Scholes model to value the equity, we get:
d1 = [ln($61,000/$75,000) + (.06 + .502/2) 5]/(.50
5
) = .6425
d2 = .6425 – (.50
5
) = –.4755
N(d1) = .7397
N(d2) = .3172
Putting these values into Black-Scholes:
Equity = $61,000(.7397) – ($75,000e–.06(5))(.3172)
CHAPTER 17 B – 4
c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the
firm’s equity, so:
Value of risky bond = $55,561.37 – 22,060.39
Value of risky bond = $33,500.98
Using the equation for the PV of a continuously compounded lump sum to find the return on
debt, we get:
Return on debt:
$33,500.98 = $75,000e–R(5)
d. Using the equation for the PV of a continuously compounded lump sum, we get:
PV = $75,000 e–.06(5) = $55,561.37
Using the Black-Scholes model to value the equity, we get:
d1 = [ln($61,000/$75,000) + (.06 + .602/2) 5]/(.60
5
) = .7404
d2 = .7404 – (.60
5
) = –.6012
N(d1) = .7705
N(d2) = .2738
Putting these values into Black-Scholes:
Equity = $61,000(.7705) – ($75,000e–.06(5))(.2738)
CHAPTER 17 B – 5
Using the equation for the PV of a continuously compounded lump sum to find the return on
e. From c and d, bondholders lose: $29,216.16 – 33,500.98 = –$4,284.81
From c and d, stockholders gain: $31,783.84 – 27,499.02 = $4,284.81
This is an agency problem for bondholders. Management, acting to increase shareholder wealth
29. a. Since the equityholders of a firm financed partially with debt can be thought of as holding a call
option on the assets of the firm with a strike price equal to the debt’s face value and a time to
expiration equal to the debt’s time to maturity, the value of the company’s equity equals a call
option with a strike price of $95 million and 1 year until expiration.
In order to value this option using the two-state option model, first draw a tree containing both
the current value of the firm and the firm’s possible values at the time of the option’s expiration.
Next, draw a similar tree for the option, designating what its value will be at expiration given
Value of company (in millions)
Equityholders’ call option price with a strike of $95
(in millions)
Today
1 year
Today
1 year
$125
$30
=Max(0, $125 – 95)
$103
?
$85
$0
=Max(0, $85 – 95)
CHAPTER 17 B – 6
value falls, the percentage decrease in value over the period is –17.48 percent [= ($85/$103) –
1]. We can determine the risk-neutral probability of an increase in the value of the company as:
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
And the risk-neutral probability of a decline in the company value is:
ProbabilityFall = 1 – ProbabilityRise
Using these risk-neutral probabilities, we can determine the expected payoff to the equityholders’
call option at expiration, which will be:
Expected payoff at expiration = (.6303)($30,000,000) + (.3698)($0)
Expected payoff at expiration = $18,907,500.00
Since this payoff occurs 1 year from now, we must discount it at the risk-free rate in order to find
its present value. So:
PV(Expected payoff at expiration) = $18,907,500.00/1.07
outstanding. So, the price per share is:
Price per share = Total equity value/Shares outstanding
Price per share = $17,670,560.75/300,000
Price per share = $58.90
of riskless debt is $88,785,046.73 (= $95,000,000/1.07). The firm’s debt is worth less than the
present value of riskless debt since there is a risk that it will not be repaid in full. In other words,
the market value of the debt takes into account the risk of default. Since there is a chance that the
company might not repay its debtholders in full, the debt is worth less than $88,785,046.73.
million, the equityholders will exercise their call option, and they will receive a payoff of $40
million at expiration. However, if the firm’s value decreases to $70 million, the equityholders
will not exercise their call option, and they will receive no payoff at expiration.
Value of company (in millions)
Equityholders’ call option price with a strike of $95
(in millions)
Today
1 year
Today
1 year
$135
$40
=Max(0, $135 – 95)
$103
?
$70
$0
=Max(0, $70 – 95)
If the project is successful and the company’s value rises, the increase in the value of the company
over the period is 31.07 percent [= ($135/$103) – 1]. If the project is unsuccessful and the
company’s value falls, the decrease in the value of the company over the period is –38.14 percent
[= ($70/$103) – 1]. We can use the following expression to determine the risk-neutral probability
of an increase in the value of the company:
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
ProbabilityFall = 1 – ProbabilityRise
ProbabilityFall = 1 – .6186
Expected payoff at expiration = .6186($40,000,000) + .3814($0)
Expected payoff at expiration = $24,744,615.38
its present value. So:
PV(Expected payoff at expiration) = ($24,744,615.38/1.07)
PV(Expected payoff at expiration) = $23,125,808.77
CHAPTER 17 B – 8
The current value of the company is equal to the value of its equity plus the value of its debt. In
order to find the value of the company’s debt, we can subtract the value of the company’s equity
from the total value of the company, which yields:
VL = Debt + Equity
And the new share price is:
Share price = Total equity/Shares
The riskier project increases the value of the company’s equity and decreases the value of the
30. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the
dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a
dividend is paid, reducing the upside potential of the call by the amount of the dividend. The
b. Using the Black-Scholes model with dividends, we get:
d1 = [ln($87/$80) + (.05 – .02 + .502/2) (4/12)]/(.50
12/4
) = .4696
d2 = .4696 – (.50
12/4
) = .1809
N(d1) = .6807
N(d2) = .5718
C = $87e–(.02)(4/12)(.6807) – ($80–.05(4/12))(.5718)
31. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the
dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a
dividend is paid. The price of a put option will increase when the dividend yield increases.
b. Using put-call parity to find the price of the put option, we get:
$87e–.02(4/12) + P = $80e–.05(4/12) + 13.84
32. N(d1) is the probability that “z” is less than or equal to N(d1), so 1 – N(d1) is the probability that “z” is
greater than N(d1). Because of the symmetry of the normal distribution, this is the same thing as the
33. From put-call parity:
P = E × e–Rt + C – S
Substituting the Black-Scholes call option formula for C and using the result in the previous question
produces the put option formula:
P = E × e–Rt + C – S
34. Based on Black-Scholes, the call option is worth $50! The reason is that present value of the exercise
price is zero, so the second term disappears. Also, d1 is infinite, so N(d1) is equal to one. The problem
is that the call option is European with an infinite expiration, so why would you pay anything for it
since you can never exercise it? The paradox can be resolved by examining the price of the stock.
35. The delta of the call option is N(d1) and the delta of the put option is N(d1) – 1. Since you are selling
a put option, the delta of the portfolio is N(d1) – [N(d1) – 1]. This leaves the overall delta of your
position as 1. This position will change dollar for dollar in value with the underlying asset. This