Question

A company has a single zero coupon bond outstanding that matures in five years with a face value of $39 million. The current value of the company’s assets is $29 million and the standard deviation of the return on the firm’s assets is 39 percent per year. The risk-free rate is 3 percent per year, compounded continuously.

  

a.	What is the current market value of the company’s equity? (Do not round intermediate calculations and enter your answer in dollars, not millions of dollars, rounded to 2 decimal places, e.g., 1,234,567.89.)
b.	What is the current market value of the company’s debt? (Do not round intermediate calculations and enter your answer in dollars, not millions of dollars, rounded to 2 decimal places, e.g., 1,234,567.89.)
c.	What is the company’s continuously compounded cost of debt? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
d.	The company has a new project available. The project has an NPV of $2,800,000. If the company undertakes the project, what will be the new market value of equity? Assume volatility is unchanged. (Do not round intermediate calculations and enter your answer in dollars, not millions of dollars, rounded to 2 decimal places, e.g., 1,234,567.89.)
e.	Assuming the company undertakes the new project and does not borrow any additional funds, what is the new continuously compounded cost of debt? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Answer 1

a). Equity will be valued as a call option, using Black-Scholes model:

Inputs:
Current stock price (S)	Current asset value	29,000,000
Strike price (K)	Face value of zero coupon bond	39,000,000
Time until expiration(in years) (t)	Maturity of zero coupon bond	5.000
volatility (s)	Standard deviation of return on assets	39.0%
risk-free rate (r)		3.00%

d1 = {ln(S/K) + (r +s^2/2)t}/(s(t^0.5))

= [ln(29/39) + (3% + (39%^2)/2)*5]/(39%*(5^0.5)) = 0.2683

d2 = d1 - (s*(t^0.5)) = 0.2683 - (39%*(5^0.5)) = -0.6028

Using normal distribution table,

N(d1) = 0.6058 and N(d2) = 0.2730

Call option value = S*N(d1) - N(d2)*K*(e^(-rt))

= (29,000,000*0.6058) - (0.2730*39,000,000*e^(-3%*5)) = 8,403,268.96

Equity is valued at $8,403,268.96

b). Market value of debt = V - E = 29,000,000 - 8,403,268.96 = 20,596,731.04

c). If the continuously compounded cost of debt is rd then

Market value of debt = Face value of debt*e^(-rd^5) (5 years being the time to maturity)

20,596,731.04 = 39,000,000*e^(-rd^5)

e^(rd^5) = 39,000,000/20,596,731.04

rd^5 = ln(1.8935)

rd = 1/5*0.6384 = 0.1277 or 12.77%

d). If the project is undertaken the new asset value of the company is 29,000,000 + 2,800,000 = 31,800,000

Again, using Black-Scholes model, equity is valued as a call option:

Inputs:
Current stock price (S)	3,18,00,000.00
Strike price (K)	3,90,00,000.00
Time until expiration(in years) (t)	5.000
volatility (s)	39.0%
risk-free rate (r)	3.00%

d1 = {ln(S/K) + (r +s^2/2)t}/(s(t^0.5))

= [ln(31.8/39) + (3% + (39%^2)/2)*5]/(39%*(5^0.5)) = 0.3740

d2 = d1 - (s*(t^0.5)) = 0.2683 - (39%*(5^0.5)) = -0.4981

Using normal distribution table,

N(d1) = 0.6458 and N(d2) = 0.3092

Call option value = S*N(d1) - N(d2)*K*(e^(-rt))

= (31,800,000*0.6458) - (0.3092*39,000,000*e^(-3%*5)) = 10,156,642.28

Equity is valued at $10,156,642.28

e). New market value of debt = V - E = 31,800,000 - 10,156,642.28 = 21,643,357.72

Market value of debt = Face value of debt*e^(-rd^5)

e^(rd^5) = Face value of debt/Market value of debt

rd = 1/5*[ln(Face value of debt/Market value of debt)]

=1/5*ln(39,000,000/21,643,357.72)] = 0.1178 or 11.78%