Minneapolis Community and Technical College Money and Banking Economics
ECON 4721 – Money and Banking Homework 1: Banking Fundamentals Due Date: 02/08/2024 Spring 2024 Exercise 1: Adverse Selection in the Market for Funds Consider a very similar scenario to the one we discussed in class in which there is a representative household that has $1 available for investment. There are two candidate firms, each of them requiring $1 to finance an investment project. One of these firms is ”safe”, in the sense that the investment will produce a gross return of $1.15 with absolute certainty. The other firm is ”risky”, in the sense that with some probability p the investment project will be successful and produce a gross return of $1.6 but with probability (1 − p), the project will fail and produce a return of $0. Let’s assume that if a firm gets financing through the household and the project succeeds, then the firm must repay the $1 dollar plus an interest rate r. As we did in class, we are assuming limited liability, so if the project fails, then the firm pays nothing to the household. For the following three questions, assume that the household can perfectly distinguish between the safe and the risky firm (there is no private information here). a) (5 points) For which values of r would the safe firm and the household be both willing to engage in business (i.e., the firm would be willing to borrow and the household would be willing to lend)? (derive both the lower and upper bound for r). b) (5 points) If p = 0.4, would the household be willing to lend to the risky firm? c) (5 points) What is the threshold value for p, such that if the household were to lend to the risky firm, then both parts would be getting zero expected profit. For the remaining questions, assume now that there is private information, in which the household cannot perfectly distinguish between types of firms (but each firm knows if it’s either safe or risky). In particular, let’s assume that the household assigns a probability q that a firm is risky and 1 − q that the firm is safe. 1 Econ 4721-Money and Banking HW 1 d) (5 points) Write down the household’s expected profit equation, as a function of both p and q e) (5 points) Show that the expected profit equation derived above is an increasing function of p and a decreasing function of q f) (5 points) Assume now that q = 0.5 and p = 0.6. What is the minimum rate of return r that the household would be demanding in order to invest in a firm? g) (5 points) Based on your last answer, will there be any firm able to borrow in this setup? Briefly explain Exercise 2: Moral Hazard in the Market for Funds Suppose there is only one firm seeking a loan of $1. The firm has two options: invest this loan into a safe project that would generate a gross return of $1.15 with certainty, or invest it into a risky project that generates a gross return of $1.5 with probability p and zero with probability 1 − p. Suppose there is a lender willing to make the loan at a return rate of r = 0.08. a) (10 points) What is the value of p such that the firm is indifferent between investing the loan at the risky project or the safe one? b) (5 points) Assume now that p is exactly the value you derived above. Continue to assume that r = 0.08. What is the expected return for the lender if the firm invests in the risky project? Exercise 3: Credit Risk Suppose a representative bank like the one we studied in class. On the asset side of their T-account, the bank has loans for $100, government issued bonds (completely risk-free) for $20 and some cash reserves for $15. To finance these assets, on the liability side, the bank is considering three options: • Option 1: Deposits for $100 and borrowing for $0 (leaving an equity of $35) • Option 2: Deposits for $80, borrowings for $40 (leaving an equity of $15) • Option 3: Deposits for $80, borrowing for $20 (leaving an equity of $35) Suppose further that the interest rate on loans is r loans = 0.12, the interest rate on the government issued bonds is r f = 0.03. The deposits in turn, pay an interest of r dep = 0.02 and finally the bank borrowed (from a larger financial institution) at an interest rate of r debt = 0.1. However, there is risk associated with the bank’s assets. In particular, there is a probability p that 10% of the loans the bank made, will default. 2 Econ 4721-Money and Banking HW 1 a) (15 points) Incorporating the risk of default, write down the bank’s expected profit as a function of p, for each of the three possible options. b) (10 points) Assume now that p = 0.5. What is the bank’s expected ROE for each of the three cases? If the bank’s objective were to maximize expected ROE, which of the three options would the bank choose? Exercise 4: Liquidity Risk Suppose a representative bank like the one we studied in class. On the liability side of their T-account, the bank has $100 in deposits and $0 in borrowing. On the asset side, the bank is analyzing three options: • Option 1: Loans for $100, government issued bonds for $20 and cash reserves for $20 (leaving an equity of $40) • Option 2: Loans for $80, government issued bonds for $40 and cash reserves for $20 (leaving an equity of $40) • Option 3: Loans for $90, government issued bonds for $20 and cash reserves for $10 (leaving an equity of $20) Suppose further that the interest rate on loans is r loans = 0.12, the interest rate on the government issued bonds is r f = 0.03. The deposits in turn, pay an interest of r dep = 0.02. However, there is risk associated with the bank’s liabilities. In particular, there is a probability p that 40% of the deposits are suddenly withdrawn by the depositors. If such an event where to happen, the bank would first use its cash reserves to meet depositors demand. If reserves prove insufficient, then the bank would sell government issued debt at face value (i.e., without any discount on the sale price). If, in turn, this is not enough to meet the depositors’ withdrawals, then the bank would be forced to sell some of its loans accepting a discount of 50% (this means, for example, that if the bank wanted to raise $20 in cash selling loans, it would have to sell $40 worth of loans). a) (15 points) Incorporating the risk of withdrawal (liquidity risk), write down the bank’s expected profit as a function of p, for each of the three possible options. b) (10 points) Assume now that p = 0.5. What is the bank’s expected ROE for each of the three cases? If the bank’s objective were to maximize expected ROE, which of the three options would the bank choose? 3
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