XYZ Corporation has $$500$ million in zero coupon debt outstanding, due in five years. The firm had earnings before interest and taxes of $$40$ million in the most recent year $($the tax rate is $40\%).$ These earnings are expected to grow $5\%$ a year in perpetuity, and the firm paid no dividends. The firm had a return on capital of $12\%$ and a cost of capital of $10\%.$ The annualized standard deviation in firm values of comparable firms is $12\mathrm{.}5\%.$ The five$-$year bond rate is $5\%.$

a$.$ Estimate the value of the firm.

b$.$ Estimate the value of equity, using an option pricing model.

c$.$ Estimate the market value of debt and the appropriate interest rate on the debt.