# Zero Coupon Bonds: Value of Option to Default on the Bonds

2. Assume your firm has zero coupon bonds maturing in 10 years with a face value of $183.75 million.

a) If the assets are worth $150 million, what is the value of the stocks and the bonds?

b) If the assets are worth $400 million, what is the value of the stocks and the bonds?

c) If the assets are worth $400 million, the standard deviation is 0.60 and the risk-free rate is the T-bond rate, what is the value of the option to default on the bonds? What is the value of the stockholders' call option? Use Black and Scholes to get the answer to part c.

Note: For parts a and b, assume the options would be exercised immediately. Be sure to label your answer.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

See the attached file.

2. Assume your firm has zero coupon bonds maturing in 10 years with a face value of $183.75 million.

a) If the assets are worth $150 million, what is the value of the stocks and the bonds?

Equity is a call option. If this option is exercised today, shareholders will receive their share only when after satisfying the financial claims of the debt holders.

The payoff to equity investors on liquidation are

= V - D if V > D

= 0 if V <D

where,

V = Value of the firm

D = Face Value of the outstanding debt

Here, V= $150 million D=$183.75 million and V<D

Hence,

S = Value of the stocks = 0

D = Value of bonds = Min(V,D) = Min( 150 million, 183.75 million) = $150 million

b) If the assets are worth $400 million, what is the value of the stocks and the bonds?

Here, V= $400 million D=$183.75 million and V>D

Hence,

S = Value of the stocks = V-D = 400 million -183.75 million =216.25 million

D = Value of bonds = Min(V, D) = Min( 400 million, 183.75 million) = $183.75

c) If the assets are worth $400 million, the standard deviation is 0.60 and the risk-free rate is the T-bond rate, what is the value of the option to default on the bonds? What is the value of the stockholder's call option? Use Black and Scholes to get the answer to part c.

Note: For parts a and b, assume the options would be exercised immediately. Be sure to label your answer.

r=Assume that 10 year T-Bond rate is 4%

T=10 years

V=Current value of assets = $400 million

X = face value of outstanding debts = $183.75 million

Sigma = 0.60

C=V*N(d1) - Xe-rTN(d2)

d1=(ln(So/X)+(r+sigma^2/2)T)/sigma*square root of T

d2=(ln(So/X)+(r-sigma^2/2)T)/sigma*square root of T

d1=(ln(400/183.75)+(4%+0.60^2/2)*10)/(0.60*(10)^0.5)

=1.5695

d2=(ln(400/183.75)+(4%-0.60^2/2)*10)/(0.60*(10)^0.5)

=-0.3279

Look at the value in standard normal distribution

N(d1)=0.9417

N(d2)=0.3715

C=400*0.9417-183.75*exp(-4%*10)*0.3715

=330.92

Value of equity = $330.92 million

Value of debt = $400 million - $330.92 million = $69.08 million

Value of option to default on the bonds = Face value of bond - current value of bond = 183.75 million - 69.08 million = 114.67 million

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