I. $(40$ points$)$ Hearthstone Communities, a real estate developer, would like to fund a new affordable housing project for seniors. To do this, they will need funding to cover various costs. Assume the following: The probability that the project will be successful is $80\%,$ and the probability of failure is $20\%.$ The company needs to raise $$1$M $($one million dollars$),$ and in a successful state, the project will be worth $$1\mathrm{.}5$M$,$ while it will be worth its initial $$1$M if it fails. NexaCore Technologies, a software developer, wants to fund a new project in a new digital money transfer technology. They also need funding to cover various costs. Assume the following: The probability that the project will be successful is $80\%,$ and the probability of failure is $20\%.$ The company needs to raise $$1$M$,$ and in a successful state, the project will be worth $$1\mathrm{.}5$M$,$ while it will be worth its initial $$1$M if it fails. The bankruptcy costs, which are the same to all lenders, for the two projects are as follows: Hearthstone's project has a high recovered value under liquidation of gamma$=90\%.$ That is$,$ for every $$100$ worth of assets liquidated, lenders recover $$90.$ NexaCore's project has a low recovered value under liquidation of gamma$=10\%.$ That is$,$ for every $$100$ worth of assets liquidated, lenders recover $$10.$ Assumptions for both projects: Potential lenders require a rate of return of $5\%.$ Monitoring Costs are $$0\mathrm{.}005$ per each $$1$ that the bank funds. Assume that when banks monitor a borrower, they observe the return on the borrower's project. That is$,$ the borrower cannot lie to the bank if the bank monitors it$.$ The projects are independent of one another Suppose a bank were to finance only Nexacore. To what should the bank set the face value of the loan to Nexacore if it monitors the borrower?