Consider a borrower that can choose between two projects, S and R, each of which will pay off a random amount one period hence. Project S will yield £250 with probability 0.9 and zero with probability 0.1 one period hence. Project R will yield £350 with probability 0.4 and nothing with probability 0.6 one period hence. The bank’s cost of funds is equal to the riskless interest rate of 10%.
As a banker, you cannot control your borrower’s project choice directly because you cannot observe this choice. You are restricted to making unsecured loans. Assume universal risk neutrality and that the discount rate is zero. Moreover, you can charge this borrower 300 basis points above your break-even interest rate before the borrower switches to another bank.
a. Compute the expected payoffs of the borrower and the bank under the following two scenarios: (1) the bank and the borrower can contract with each other over only one period, and (2) the bank and the borrower can contract over two time periods. In case (1), the borrower will request a single loan of £150, and in case (2), the borrower will need a sequence of two £150 loans, with the ability to choose between S and R in each period.
b. Analyze what should be the choice of contracting horizon, both from the bank’s and the borrower’s perspective.