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.
Bank’s 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 α of short-term creditors do not rollover their debt at t=1 (i.e. s withdraw), where α has
(approximation) a Normal distribution N(μ,σ). 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 τR.
Suppose y=100, e = 10, s=70, l=20, rl = 1.2, rs = 1.1, μ = 0.6, τ = 0.5 and σ = 0.1.
Table 1: Bank’s balance sheet
Assets Liabilities
Short-term debt s=70
y=100 Long-term debt l=20
Equity e=10
a) 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?
b) Suppose that at t=1, we learn that R=1.2 and =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 assets at to prevent the bank from failing? How much cash does the central bank need
to conduct the asset purchase program at that price?