1. The arrows represent exposure. For instance, bank 2 is exposed to bank 1 (there is an arrow from bank 2 to bank 1), and that means bank 2s assets contain securities (debt or equity) issued by bank 1.

Suppose each banks balance sheet is of the following form:

Assets	Liability
20% in loans	(1 L1) fraction in debt
	Equity
80% in bank-issued securities	fraction in equity

For instance, 80% of bank 2s assets are securities issued by bank 1; 80% of bank 3s assets are securities issued by bank 2, and etc. (follow the arrows in Figure 1). The loans held by different banks are all separate from each other.

Throughout this question, you dont need to think in dollar values, but only need to think about values in proportional terms (e.g. loans are 20% of total asset value in the balance sheet above).

Recall how investors are paid in bankruptcy: if a bank goes bankrupt, then its equity becomes worthless, and its debt holders get all remaining asset value with a haircut. For instance, suppose L = 10 (so equity is 10% of total assets), and suppose the asset value of a bank declines by 30%. Then, it goes bankrupt, and its equity becomes worthless. Its debt is now worth 1-30%/90%=7/9 as much as before. So the holders of its equity suffer a loss of 100% (worthless now), and the holders of its debt suffer a proportional loss of 1-7/9=2/922.2%

In all sub-questions below, we want to figure out what happens if bank 1 suffers a negative shock: the value of its loans suddenly becomes zero. (The bank-issued securities on bank 1 balance sheet is not affected, and the loans held by other banks are also not affected.)

For simplicity, suppose all four banks have the same leverage ratio of L before the negative shocked happened. Please answer the questions below.

1. How high can the leverage ratio L be for there to be no bankruptcy among all four banks in response to this negative shock?^[1]
2. Now suppose banks hold debt securities issued by other banks.^[2]Suppose all banks have a leverage ratio of L = 15. How many banks will go bankrupt? Please be specific in your answer which banks are bankrupt and why. Support your answer with calculations.

Hint: there is a chain reaction. When bank 1 gets the negative shock, the value of its equity and its ability to pay back debt may be affected. That will impact the asset values of bank 2, because bank 2 holds securities issued by bank 1. And then that impacts bank 3, and so on...

3. Still assume that L = 15 for all banks. Now suppose banks hold equity securities issued by other banks. How many banks will go bankrupt?
4. What do you learn from the previous three questions? Please focus onhow inter-bank relationships impact the likelihood of wide-spread bank failures.

[1] That is, even bank 1 does not go bankrupt.

[2] Thus, 80% of bank 2s assets are debt issued by bank 1; 80% of bank 3s assets are debt issued by bank 2, etc.

Bank 1 Bank 2 Bank 4 Bank 3 Bank 1 Bank 2 Bank 4 Bank 3