Bank issues two loans. Each loan yields a return with probability p and 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. 

a) Suppose the keeps the loans on its balance sheet. What is the expected profit that the bank makes? 

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. 

Expected 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= 25 

b) What is the expected profit of the bank in this case from holding the two loans on its balance sheet? 

c) 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(R) = R^1/2. What is the revenue that the bank would get if it sold 2 loans to two risk-averse investors? 

d) 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? 

e) What happens when the bank issues N loans with N→∞ and each investor acquires a share of 1/N in the loan portfolio?