You are establishing a new company. You have no money of yours and you can only raise debt to finance your activity. There are $2$ mutually exclusive projects $($Projects A and B$)$ you can undertake, both require and initial investment of $50.$ Project A$$s assets will produce in $5$ years a cash flow of $60$ with probability $0\mathrm{.}5$; and $50$ with probability $0\mathrm{.}5.$ Project B$$s assets will produce in $3$ years a payoff of $40$ with probability $0\mathrm{.}8$; and a payoff of $100$ with probability $0\mathrm{.}2.$ Assume that agents are risk neutral and that the market interest rate is $0.$ Also, debt holders are competitive, that is$,$ they will only require, in expectation, the market interest rate. Finally assume that the company cannot commit ex$-$ante $($i$.$e$.$ before issuing debt$)$ to choose one or the other project.

a$)$ Which project will be undertaken? Why? $[7$ Points$]$

b$)$ What is the face value of the debt issued by the company? $[7$ Points$]$

c$)$ If the firm could commit to choose one project, how richer would the shareholder be compared to the case of no possibility of commitment? Why? $[5$ Points$]$