Homework 3 for Banking & Regulation PDF

Homework 3 for Banking & Regulation Prof. Tanju Yorulmazer Question 1 We have a risk-neutral bank. Bank has a pool of loans that has a return of 10 with probability 0.5 and 3 with probability 0.5. If the bank holds the loans on its balance sheet it incurs a...

Homework 3 for Banking & Regulation Prof. Tanju Yorulmazer Question 1 We have a risk-neutral bank. Bank has a pool of loans that has a return of 10 with probability 0.5 and 3 with probability 0.5. If the bank holds the loans on its balance sheet it incurs a cost of 0.5 due to capital requirements. Bank can also securitize the loans to overcome capital requirements. a) What is the maximum price a risk-averse buyer with u(x) = x1/3 would be willing to pay for the loan? Would the bank be willing to sell the loans in that case? b) Suppose, in addition to the risk-averse investors as in part a), there are also risk-neutral investors who are willing to buy the loans. However, risk-neutral investors have limited funds of 3.5. Would the bank be willing to sell the loans? How? Question 2 Bank issues two loans. Each loan yields a return R with probability p and 0 otherwise. Probabilities of success of individual loans are independently and identically distributed (i.i.d.). The bank finances the loan with equity (k) and debt (d). Debt has a net cost of 0 (bank debtors break- even in expectation). Equity is costly, where 1 unit of bank equity costs c > 1. Suppose the bank keeps the loans on its balance sheet. Expected return from each loan is pR. For each loan, bank needs to hold equity k and d=1-k, where the cost of equity is ck. Hence, the expected profit for the bank from holding the loans on its balance sheet is given as: E(Profit) = 2pR – 2(1-k) – 2ck = 2pR – 2 – 2k(c-1). For the rest, suppose that R = 4, p = 0.5, k = 0.5 and c = 2.5. a) What is the expected profit of the bank in this case from holding the loans on its balance sheet? b) Suppose the bank wants to avoid capital costs and securitizes the two loans. Bank sells the loans individually to risk-averse investors with utility functions u(x) = x1/2. What is the revenue that the bank would get if it sold 2 loans to two risk-averse investors? c) Bank can pool the two loans and create two securities by splitting equally the claims on the cash flows. What is the maximum price the risk-averse investors would be willing to pay for such a security and would the bank be willing to pool and sell the loans? d) What happens when the bank issues n loans with n→ and each investor acquires a share of 1/n in the loan portfolio? Question 3 Consider a bank that holds government bonds and consumer loans as assets. In particular, the bank holds a zero-coupon government bond with maturity of 10 years that promises to pay 15m in year 10. The bank’s loan portfolio matures in 2 years: it pays 2m in the first year and 12m at the end of the 2nd year. The bank has 20m (present value) worth of liabilities with a duration of 4 years. The risk-free interest rate is 5%. a) What is the (present) value of bank’s assets? What is the value of bank’s equity? b) What is the duration of the bank’s assets? c) What change in the interest rate would make the bank insolvent? d) How can the bank eliminate interest rate risk? Explain the possibilities in this case. Keeping everything else the same, what should the duration of the bank’s liabilities be to make the bank’s value insensitive to changes in the interest rate? 2