$21-28.$ You would like to know the unlevered beta of Schwartz Industries $($SI$).$ SI's value of outstanding

equity is $$400$ million, and you have estimated its beta to be $1\mathrm{.}2.$ SI has four$-$year zero$-$coupon

debt outstanding with a face value of $$100$ million that currently trades for $$75$ million. SI pays

no dividends and reinvests all of its earnings. The four$-$year risk$-$free rate of interest is currently

$5\mathrm{.}13\%.$ Use the Black$-$Scholes formula to estimate the unlevered beta of the firm.

The equity can be interpreted as a four$-$year call option on the firm's assets with a strike price of $$100$

million. The current market value of assets is $400+75=$$$475$ million.

Using the Black$-$Scholes formula, we can get the implied volatility of assets: Call option value $=400,$

$K=100,S=475,T=4,r=0\mathrm{.}0513=>=60\mathrm{.}6\%.$

Using the volatility:

${}_U=\frac{{}_E}{(1+\frac{D}{E})}=\frac{1\mathrm{.}2}{0\mathrm{.}980(1+\frac{75}{400})}=1\mathrm{.}03$