Question

A company has a single zero coupon bond outstanding that matures in five years with a face value of $38 million. The current value of the company’s assets is $28 million and the standard deviation of the return on the firm’s assets is 40 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 $2,700,000. 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.)

	a. Market value of equity b. Market value of equity c. Cost of debt % d. Market value of equity e. Cost of debt %

Answer 1

(a) The firm has a single zero-coupon bond as debt with a maturity of 5 years. Debtholders of the firm will have to be repaid their entire face value due worth $ 38 million in case of liquidation and any remaining assets will then be paid put to the firm's equity holders. Now at the time of liquidation, the equity holders will have a positive payoff only when the firm's asset value is greater than the debt outstanding (zero-coupon bond's face value to be repaid), otherwise, the payoff to equity holders is zero. This implies that the payoff to equity holders is similar to the payoff of a call option. By the law of one price, if two financial assets have similar payoffs then they should also have equal prices and can be priced using the same method.

Therefore, Equity Value = Value of a Call Option with strike price = Debt Face Value, Underlying asset's price = Total Asset value of the firm, Option Tenure = Debt Maturity, Risk-Free Rate = 3%, Standard Deviation (SD) = SD of Return on Firm's Assets.

Underlying Asset Price = $ 28 million, Face Value = $ 38 million, Risk-Free Rate = 3% and Tenure = 5 years, Standard Deviation = 40%

Using an online Black Scholes Calculator to solve the above call structure we get:

Equity Value = Call Option Price = $ 8.27 million

(b) Firm's Asset Value = $ 28 million, Equity Value = $ 8.27 million

Current Debt Value = 28 - 8.27 = $ 19.73 million

(c) Let the continuously compounded cost of debt be r1 %

Therefore, 19.73 = 38 / e^(r1 x 5)

e^(r1 x 5) = 38/19.73 = 1.926

r1 x 5 = = 0.65545

r1 = 0.13109 or 13.109 % ~ 13.11 %

(d) If the firm undertakes a project with NPV of $ 2.7 million, the firm's asset value goes up by the NPV of the project, other conditions remaining constant.

Therefore, New Asset Value = 28 + 2.7 = $ 30.7 million

Using the same online option calculator, we get:

Equity Value = Call Price = $ 9.97 million

(e) New Debt Value = New Firm Value - Equity Value = 30.7 - 9.97 = $ 20.73 million

Let the continuously compounded cost of debt be r2

Therefore, 20.73 = 38 / e^(r2 x 5)

e^(r2 x 5) = 38 / 20.73 = 1.83309

r2 x 5 = = 0.606

r2 = 0.1212 or 12.12 %