$1\mathrm{.}6$Monsters Incorporated $($MI$)$is ready to launch a new product. Depending upon the success of this product, MI will have a value of either $$100$million$,$ $$150$million$,$ or $$191$million$,$ with each outcome being equally likely. The cash flows are unrelated to the state of the economy $($i$.$e$.$risk from the project is diversifiable$)$so that the project has a beta of $0$and a cost of capital equal to the risk$-$free rate, which is currently $5\%.$Assume that the capital markets are perfect.

Assuming that in the event of default, $20\%$of the value of MI's assets will be lost in bankruptcy costs, what is the initial value of MI's equity without leverage? why do you take the pv value?