FIN 204: Money and Banking Homework #2: Solutions. Due: 8 Apr. 2008, 14:30, Instructor: Petar Stankov

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1 I strongly encourage you to work in groups and discuss your solutions, and only then write them down for me. This will save you time AND will improve your understanding of the subject. I do NOT advise you however to work on separate problems and then copy the rest from the other members of the group. This may spoil your exam preparation, as the questions you get on the homeworks will be similar to those you will solve during the exams. Although the deadline for this particular homework is well after the midterm, I encourage you to solve it before the exam because some of the exam problems may be similar to the ones appearing here. You can find more solutions and class materials at the class web-site. This homework covers the material covered up to and including lecture 6. Emphasis is given to the new material. Problem 1. (45 pts.) You are a bank credit manager. A client asks you for a 1-year credit of e , and she does not have a collateral. She files an application, and based on her answers your credit score calculations show that her probability of default is 2%. You offer her an interest rate of 10%. She accepts, signs the contract, and drives away with the money. Inflation is 0% and is expected to remain 0%. a) (5pts.) What is the expected profit for the bank from this loan, if the bank does not pay an interest to those and has rather taken them from its excess reserves? In case the client does not default, the bank s profit is In case the client defaults, the bank loses both the interest payments, and the principal. Therefore, in case the client defaults, the bank loss is Thus: eπ = 0.98 (10000) ( ) = = b) (5 pts.) After several months, you receive additional information about the client that tells you her investment project is not going as expected. After receiving this information, you think her probability of default is now 7%. What is your expected profit from this loan? eπ = 0.93 (10000) ( ) = = c) (5 pts.) You decide that this client brings too much risk to the bank under 7% risk of default, and decide you want to sell this loan to another financial institution. You design a contract in which selling the loan means that you are willing to accept 1

2 certain amount of money, and in exchange, you grant the buyer of the loan the exclusive right to receive the interest payments, as well as the principal from this client s loan. How much are you willing to accept for the loan, provided the risk of default is 7% but the other financial institution thinks everything is going fine with her project, and that there is no risk of default? You are willing to accept the principal + your expected profit which in this case is Therefore, you are willing to accept d) (5 pts.) Suppose that, after all, the client has not defaulted but you sold the loan to another financial institution under the conditions set up in c). What is the actual profit that the other financial institution made, provided that it did not change the interest payments for the client? π = = e) (5 pts.) Now suppose the client s probability of default is still 7% but she has a collateral worth , and in case she defaults, you can take this collateral any moment you want, and without any costs. What is the bank s expected profit from this loan now? In case the client defaults, the bank still loses but the collateral is , so the actual loss in case of default is only Therefore: eπ = 0.93 (10000) ( 20000) = = f) (5 pts.) For how much are you willing to sell the loan now? What is your profit? You are willing to accept the principal + your expected profit which in this case is Therefore, you are willing to sell the loan for Your profit is g) (5 pts.) Suppose that the client did not default. How much profit did the buyer of the loan make? = h) (10 pts.) Suppose the other financial institution has an alternative investment opportunity, and compares the returns from both opportunities before engaging into one of them. That is, the financial institution would invest the same amount of money the bank asks for the loan (in f)) into the second option. Suppose that the second option is to buy municipal bonds that promise 3% interest rate. Which ivestment option will the second financial institution choose? The return from the first option is (that is, the profit), while the return 2

3 from the second option is ( ) = = Therefore, it does not make sence to buy the loan because its expected return is lower than the return on the municipal bonds. An alternative way to solve is to compare the rates of return from the two investment options: in the first case the financial institution makes = 1.95%, while in the second option it makes 3% rate of return. Therefore, the financial institution chooses to invest in municipal bonds. Problem 2. (12 pts.) Suppose that you are the manager of a bank that has e15 million of fixed-rate assets, e30 million of rate-sensitive assets, e25 million of fixed-rate liabilities, and e20 million of rate-sensitive liabilities. a) (6 pts.) Conduct a gap analysis for the bank, and show what will happen to bank profits if interest rates rise by 5 percentage points. The gap is +10,000,000. This means that if interest rates rise, tha bank starts making more income out of its assets than payment on its liabilities. Both increase but the higher exposure to rate-sensitive assets brings more money to the bank. Mathematically: +10, 000, 000 (+0.05) = +500, 000 which are added to the bank profits. b) (3 pts.) What would happen if the interest rates fall by 5% points? The bank will start paying less on its sources of finance (liabilities) but it will make even less than that on its assets because its assets are more than the liabilities. Mathematically: +10, 000, 000 ( 0.05) = 500, 000 which are subtracted from the bank s profits. c) (3 pts.) Assume you do not know about duration analysis. What actions could you take to reduce the bank s interest-rate risk if the interest rates are expected to fall in the next few years? If the interest rates are expected to fall, I will make less and less money out from my rate-sensitive assets. Therefore, I would like to have less rate-sensitive assets, and more rate-sensitive liabilities. This will reduce my gap, and I will start losing less money. If the gap goes negative, I will start making profit from a fall in the interest rates. Problem 3. (8pts.) Suppose that you are the manager of a bank whose $100 billion of assets have an average duration of four years and whose $90 billion of liabilities have an average duration of six years. 3

4 a) (6 pts.) Conduct a duration analysis for the bank, and show what will happen to the net worth of the bank if interest rates falls by 2 percentage points. (Assets) = Assets [ (Interest rate) (Duration)] = 100 [ ( 0.02) (4)] = +8 mln. (Liabilities) = Liabilities [ (Interest rate) (Duration)] = 90 [ ( 0.02) (6)] = mln. Therefore, the net worth will change by (Assets) (Liabilities) = 2.8 mln. b) (2 pts.) Suppose you do not know gap analysis. What actions could you take to reduce the bank s interest-rate risk? You can either increase the duration of the assets, or decrease the duration of the liabilities. Problem 4. (10pts.) The bank you own has the following balance sheet: a) (5pts.) Suppose the bank suffers a deposit outflow of $50 million with a required reserve ratio on deposits of 10%. State at least 2 measures you may want to take to keep your bank in a stable financial situation. There is a 50-mln. deposit outflow, so now the deposits are 450 mln. It corresponds to holding 45 mln. required reserves. The bank has 75 mln. reserves which means that it can cover 30 mln. of the outflow with its excess reserves. However, the other 20 mln. are more problematic. The bank has several options to balance its sheet. One: call in loans for 20 mln.; Two: raise capital for 20 mln.; Three: borrow from other banks. b) (5 pts.) What are the disadvantages of each of these methods of coping with such a large outflow of deposits? Calling in loans damages confidence in the bank. Raising capital reduces the ROE and makes the owners less rich. Borrowing from other banks comes at a cost. 4

5 Problem 5. (5 pts.) Why has the development of overnight loan markets made it more likely that banks will hold fewer excess reserves? State at least 2 reasons. Because the bank can meet a large withdrawal activity by borrowing from other banks. Furthermore, holding excess reserves comes at a cost: they do not allow the bank to make money on them. Knowing the fact that it can borrow overnight from another bank if it needs to, the bank loans a higher proportion of its own excess reserves. Problem 6. (5 pts.) Suppose the income tax rate is τ = 15%, and returns on bonds are taxed as income. A bond promises 10% interest, and the inflation is 5%. What is the real interest rate on the bond? Taxes reduce income, in general (unless they are negative). Since the 10% you are making on the bonds are taxed with 15%, what remains after taxation is (1 τ) 10% = 8.5%. Inflation is 5%. Then, the real interest rate you are getting on your bond is not 5 but 3.5%. Problem 7. (15 pts.) Find the Bank data and statistics section. Click on Historical statistics on banking, and then on Commercial Bank reports. Select the state of Wisconsin. a) (3 pts.) Find by how many has the number of commercial banks declined from 1995 to = 122. b) (3 pts.) Look at the trend in bank branches. Does the public appear to have more or less access to banking facilities? c) (9 pts.) Present a printed graph 1 made in a statistical application of your choice 2 depicting the following: 1) The trend in the number of institutions from 1966 till 2005; 2) The trend in the number of bank offices. (Hint: First, paste data into a text application. A small briefcase will appear at the down-right angle of the data with paste options. Choose Keep text only option. It will align the data in columns. Only then copy and paste into a spreadsheet. Otherwise, you will have to type the numbers one by one.) 1 You do not have to print it in color 2 Open-source applications are also fun;-) 5

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