I've been reading my textbook and looking through my class notes and I just can't seem to make sense out of this problem. Any help would be appreciated!! I'm totally lost! :/
Suppose a bank has $100 million dollars of assets to invest. It can either invest in risky or safe loans. Safe loans will be worth $105 million in one year with certainty. Risky loans will be worth either $70 M or $130 M in one year, each with equal probability. Notice that risky loans have an expected value next year of 0.5 * 70 + 0.5 * 130= $100 M, so that risky loans are socially inefficient relative to safe loans: safe loans have both a higher average return and lower uncertainty.
a) Suppose the bank has $80 million in one-year time deposits. For simplicity, assume that they pay no interest, so that the bank's liability will still be $80 M in one year. Assume that the deposits are insured by the government, and for simplicity assume that the bank does not have to pay a premium for this insurance. If the bank's assets are worth less than $80 M in one year, the government will shut the bank down and pay the difference between $80 million and the value of assets. [HINT: SET UP THE BANK’S BALANCE SHEET]
Compute the following if the bank invests its assets in safe loans: (a) the probability that the bank will fail; (b) the expected value of the bank's net worth; and (c) the expected size of the government's bailout in one year. Do the same assuming the bank invests in risky loans. Which investment strategy would the government prefer the bank to undertake? Which strategy will the bank choose, assuming that the bank's primary objective is to ensure its survival, and its secondary objective is to maximize its expected net worth.