Consider a bank that invests an amount y in risky assets (loans) at t=0. At t=2, each unit invested in the risky asset will yield R>1 and so investment y will return yR.

Banks assets are funded with short-term debt s, long-term debt l and equity e so that: y = s + l + e. The gross interest rate that short-term creditors receive from t=0 to t=1 is 1, the gross interest rate

they receive from t=1 to t=2 is rs > 1, and the long-term creditors receive rl at t=2.

A fraction of short-term creditors do not roll over their debt at t=1 (i.e. s withdraw), where has (approximation) a Normal distribution N(mean,std). The bank can repay such creditors by liquidating the risky asset. When one unit of the risky asset is liquidated at t=1, it yields R.

Suppose y=100, e = 10, s=70, l=20, rl = 1.2, rs = 1.1, mean = 0.6, = 0.5 andstd = 0.1.

Assets Liabilities

Short-term debt s=70 y=100 Long-term debt l=20

Equity e=10

1. a) For what values of R is the bank fundamentally insolvent? For what values of R is the bank fundamentally solvent?

2. b) Suppose R =1.5 (so an investment of 100 would return 150 at t=2). What is the probability that the bank will be solvent at t=2?

3. c) Suppose that at t=1, we learn that R=1.2 and alfa=0.9. The central bank decides to introduce an asset purchase program. What is the minimum price the central bank needs to buy the risky asset at to prevent the bank from failing? How much cash does the central need to conduct the asset purchase program at that price?