Question

Sunburn Sunscreen has a zero coupon bond issue outstanding with a $10000 face value that matures in one year. The current market value of the firm’s assets is $10600. The standard deviation of the return on the firm’s assets is 32 percent per year, and the annual risk-free rate is 7 percent per year, compounded continuously. what are the market values of the firm’s equity and debt based on the Black-Scholes model? Use “risk-free bond -put option” to find out the value of debt.

Answer 1

It is a well-known fact that in the case of liquidation, the firm's debtholders have the first right over liquidation proceeds. For example, if the liquidation proceeds (liquidation value) of a firm is say V and the firm's outstanding debt is say D, then an amount D is repaid (from the liquidation proceeds) first to the firm's debt holders, followed by a payoff worth V - D to the firm's equity holders. In fact, if the debt value of D is more than the liquidation proceeds V, then the firm's debtholders get the entire liquidation proceeds with payoffs to equity holders being zero.

The payoff to Equityholders:

Payoff = V - D if V>D and Payoff = 0 if V<D

As is observable, the equity holders have a payoff structure similar to a call option. thereby implying that a firm's equity can be valued in a manner similar to a call option.

Face Value of Debt = $ 100000 which is Equivalent to the call option's strike price, Current Value of Asset = $ 10600 which is Equivalent to the current spot price of the asset, Standard Deviation of Return on Assets = 32 % is equivalent to volatility of returns of the underlying asset and annual risk-free rate is 7 % continuously compounded.

Using an online Black-Scholes model based option pricing calculator, we get:

Market Value of Firm's Equity = Call Price = $ 2003.76

Market Value of Debt = Current Value of Firm's Assets = 10600 -2003.76 = $ 8596.24

The value of the corresponding put option (put option with same underlying, same strike price and equal maturity) can be calculated using the put-call parity as shown below:

Call Price + PV of Strike Price = Current Price of Underlying Asset + Put Price

Put Price = 2003.76 + 10000 / e^(0.07 x 1) - 10600 = $ 727.698 ~ $ 727.7