Question

FSA is a privately held firm. As an analyst trying to determine the value of FSA’s common stock and bonds, you have estimated the market value of the firm’s assets to be $1 million and the standard deviation of the asset return to be .3. The debt of FSA, which consists of zero-coupon bank loans, will come due one year from now at its face value of $1 million. Assuming that the risk-free rate is 5 percent, use the Black-Scholes Model to estimate the value of the firm’s equity and debt.

Answer 1

Let's use the Black Scholes model to value equity as call option.

$Value of Call = S\times N(d{1}) - K\times e^{(-R{f}\times T)}\times N(d2))$

Where

$d1 = \frac{ln(\frac{S}{K}) +(R{f} +\sigma ^{2} / 2) \times T}{\sigma \sqrt{T}}$

d2 = d1 - $\sigma \sqrt{T}$

S = Market Value of the firm's asset = $ 1 million

K = Strike price = Face Value of debt outstanding = $ 1 million

T = Life of the option = Due time of zero-coupon debt = 1 year

$\sigma$= standard deviation of the asset return = 0.3

r = risk free rate = 5%

$d1 = \frac{ln(\frac{1}{1}) +(0.05 + 0.3^{2} / 2) \times 1}{0.3 \sqrt{1}} = 0.32$

d2 = 0.32 - 0.3 x $\sqrt{1}$ = 0.02

N(d1) = 0.62

N(d2) = 0.51

Value of equity as call option

$= S\times N(d{1}) - K\times e^{(-R{f}\times T)}\times N(d2))$

$= 1\times 0.32 - 1\times e^{(-0.05\times 1)}\times 0.02$

=$ 0.1423 mn

Value of debt = Value of asset - Value of equity = S - E = 1 - 0.1423 = $ 0.8577 mn