A customer has approached a large bank asking for a $100,000 one-year loan at a 12% interest rate (principal and interest repayable after one year). The bank has to decide whether or not to grant the loan. If the bank does not approve this loan application, the $100,000 will be invested in government bonds that earn a 6% annual return.

Historical data shows that the bank approved one thousand similar loans in the past and that only 50 customers defaulted on their loan (5%). If this customer defaults on the loan, the bank will lose the entire $100,000.

1. Assuming the bank is risk-neutral, what should the bank do? Draw and include a decision tree.

2. Generate and include a risk profile to your decision tree in Q1. Taking risk into consideration, what would be your recommendation to the bank?

3. Draw and analyze a decision tree with 6% customer default rate. Also, generate a one-way sensitivity graph for the effect of customer default rate on the expected value of approving the loan. How does this affect your analysis and recommendation?

4. Suppose the credit rating agency would be able to predict defaults with perfect accuracy. What is the maximum that the bank should be willing to pay in order to obtain this information, i.e. to know for sure whether or not the customer will default before approving or rejecting the loan? Draw and include an appropriate decision tree to answer this question.

5. Calculate the conditional probabilities of not defaulting (i) given a favorable recommendation (ii) a negative recommendation. Do you agree with the bank staffs reasoning?

6. Should the bank hire the credit rating agency to provide a recommendation for this customer? Draw and include an appropriate decision tree and its risk profile to answer this question.