QUESTION $5$a $[12$ Marks$]$

You are evaluating a loan request of $$13\mathrm{.}0$ million from Precise Corp $($The bank will generously only require an expected return of the principal$).$ The firm has an existing debt repayment obligation of $$5\mathrm{.}0$ million. It has $$4\mathrm{.}0$ million in equity. The firm has two projects, A and B $.$ An investment in A will yield a payoff of $$11\mathrm{.}5$ million with a probability of $0\mathrm{.}8$ and $$6\mathrm{.}0$ million with a probability of $0\mathrm{.}2.$ An investment in $B$ will yield a payoff of $$14\mathrm{.}0$ million with a probability of $0\mathrm{.}4$ and $2\mathrm{.}5$ million with a probability of $0\mathrm{.}6.$ The firm has assets$-$in$-$place that generates $$10$ million with a probability of $0\mathrm{.}75$ and $$2\mathrm{.}5$ million with a probability of $0\mathrm{.}25.$ Assume that the distributions of payoffs for A and $B$ are common knowledge, and the payoff from $A$ is statistically independent of the payoff for B$.$ However, as a bank officer, you cannot observe the firm's project choice.

What rate should the bank charge in order to breakeven? Given this rate will a Nash equilibrium result?

QUESTION $5$b $[8$ Marks$]$

Consider the case if an investment in B had a revised payoff of $$18\mathrm{.}0$ million with a probability of $0\mathrm{.}4$ and $2\mathrm{.}5$ million with a probability of $0\mathrm{.}6.$ All other information was the same as in Question $5$a$.$