Question $3$

SkyBus, a low$-$cost airline company, owns and operates $10$ airplanes, each estimated to be worth $$100$ million in one year. The firm has also estimated EBIT per airplane for next year as follows:

$\backslash$table$[[\backslash$table$[[$Possible Business State$],[$in One Year$]],$Probability,$\backslash$table$[[$EBIT$],[$per airplane$]]],[$High travel demand,$0\mathrm{.}75,$$$10$ million$],[$Low travel demand,$0\mathrm{.}25,0]]$

Assume that airplanes are the only assets that the firm owns and that the value of the firm's total cash flows thereafter $("$terminal value"$)$ equals zero. ${?}^1$ The marketplace for aircraft is known to be very slow. A normal transaction, with no transaction costs, takes a year to complete. However, if an urgent transaction is required, the transaction can be completed immediately at a cost of $50\%$ of the fleet value $("$fire sales"$).$ For simplicity, assume that the business state $($and thus the firm's cash flow$)$ is uncorrelated with the market portfolio and that the effective tax rate equals zero. The risk$-$free rate is $10\%.$

a$)$ Suppose the firm is $100\%$ equity$-$financed. Find the unlevered firm value.

b$)$ Suppose instead that the firm has issued one$-$year zero$-$coupon bond with a face value of $$100$ million $($i$.$e$.,$ due in one year$).$ If the firm's EBIT falls below the bond face value, it will have to invoke fire sales of its airplanes. Determine the value of the zero$-$coupon bond and the value of levered equity.