# Model

We are using the Eisenberg and Noe article to give us a framework to formulate our model as well as Matyasasovska’s Set-valued measures of systemic risk and central clearing counterparty article to help with the extensions.  As an extension to the Eisenberg and Noe paper, we make an addition of a society and a CPP node to the network.  The society node is important because it is the way that we will be measuring and comparing the effectiveness of the network with and the network without a CCP.

We used MATLAB to model the interactions that the banks have with one another in the event of one or many defaults.  Initially we developed the code from the Eisenberg-Noe article.  In Figure 1, we have a network without a CCP.  This is an example of the interconnectivity between four banks. Each letter (or node) represents a bank in our system.  The arrows represent the liabilities that one bank has to another bank. In other words, each line is a cash flow between two banks.

Figure 1: Directed network of banks without a CCP

Defining our code:

• We define each bank as a node.
• We defined the innitial starting cash from each bank as the operating cash flow.
• We make an equity vector of each of the nodes, which is each bank’s current amount of cash.  This amount of money is what each bank will use to pay off each liability.
• The nominal liabilities matrix is defined by the liability of one node to another. We assume that all claims are nonnegative and that no bank or node has a claim with itself.
• The total payments made by the nodes is represented by the another vector which represents the total dollar payment from one node to the other nodes in the network. In terms of Figure 1, this vector would be a representation of how much one bank pays another bank.
• The obligation vector represents the total dollar amount the bank owes to all other nodes.
• The relative liabilities matrix represents the nominal liability of one node to another. This number is a proportion of the node’s total liabilities and represents what one bank owes to another bank. We assume that all debt claims have equal priority.
• The total payments received by one bank are equal to the sum of all of the payments times the relative liabilities matrix.
• The clearing payment vector is defined as the minimum between the obligation vector and the operating cash flow added to the total payments received by a specific bank. This represents banks paying the smaller value between the amount they owe to the network and the amount they are capable of paying.
• Our iterative process begins when one or more banks default, and we add these banks to a default matrix.  This matrix is an iteratively growing binary matrix where each column corresponds to information relevant to the current iteration.
• The clearing vector is updated with this information by determining how much of a bank’s liabilities can be paid in the next iteration.
• This process continues unless the new column of defaulting banks is identical to the column that was just being cleared. In other words, if nothing changes after an iteration, then the code stops.

Our next step is to add the society node, which is shown below. A new row and column is added to the liabilities matrix to allow for this addition. There are only liabilities from banks to society and no liabilities from society to banks.  The way society is visually added to a network is shown in Figure 2.  This network operates the same way as in Figure 1.

Figure 2: Directed network of banks with Society node implementation

Next, we create the CCP network by adding the CCP node to the previous model in Figure 2.  This was done by breaking each connection the bank has to one another, and having each liability or cash flow go to the CCP node first. The intercepting of these liabilities is shown in Figure 3.  Instead of each bank having liabilities owed directly to other banks, the liabilities going to and from a certain bank will be netted through CCP.  Thus, the CCP acts as the regulator of the system.

Figure 3: Directed network of banks with Society and CCP node implementation

After the code for the CCP network was completed, both networks were applied to a real-world application of 10 banks.  The banks that we included in our model are JPMorgan Chase, Bank of America, Wells Fargo, Citigroup, Goldman Sachs, Morgan Stanley, U.S. Bancorp, Bank of New York Mellon, PNC Bank, and Capital One. The data from the 2015 10-K reports from these banks were used to analyze the systemic risk associated with a network with and without a CCP.

Measuring our results:

The following inequality was used to assess the systemic risk associated with different scenarios for the network with and without the CCP:

R + C ≥ S,

where S represents the amount society would receive if there are no defaults, R represents the amount society did receive and C represents the amount needed to ensure that society receives what it is owed. It is impossible for us to find a number for the amount that one individual bank owes to another bank. Therefore, we looked up the total liabilities that each of the banks owe, and we created a range that encompasses the variance of the values that we thought would be appropriate as individual liabilities between all of the banks.  The liabilities were determined by a set of uniformly distributed random variables ranging from 100 to 1000 million dollars. From this, the expected values for R and S are found after running 10,000 trials to represent our systemic risk inequality more accurately.  Thus:

E[R] + C ≥ E[S].

The inequality can then be solved for C to determine the amount needed to give to society to make sure it receives 100% of what it should.

To be certain that there will be a default in the networks, we chose to expose them to four different forms of shock:

1. All banks only have access to 50% of their operating cash flow.
2. All banks only have access to 10% of their operating cash flow.
3. The largest bank only has access 10% of its operating cash flow.
4. The three largest banks only have access to 10% of their operating cash flow.

We graphically analyzed the effect of the CCP reducing systemic risk with the value of the initial investment into the CCP in our model, as the independent variable and C as the dependent variable. This parameter ranged from zero to one billion dollars.