The Three Algorithms

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The optimization algorithm, fixed point algorithm, and fictitious default algorithms yielded the same results in both the static and dynamic implementations. Banks' payments and wealths may differ on a decimal level across algorithms.  

 

Despite the comparable performance of the algorithms, each algorithm has different advantages. The main advantage of the fictitious default algorithm is providing insight into the banks’ systemic risk, based on how many “waves” (or rounds) of defaults are required to induce a given bank in the system to fail. Disadvantages of the fictitious default algorithm include complexity and difficulty of implementation. The optimization and fixed point algorithms are both easier to implement, but do not provide as much insight into bank defaults at individual iterations or time periods. The main advantage of the optimization algorithm is that, since it implements a linear program, a sensitivity analysis is fairly easy to conduct, which provides interesting insights to the given banking system being investigated.  

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Static Implementation vs. Dynamic Implementation

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Incorporating a discrete time component had no effect on the static model’s results when uniform payments are made over uniform time intervals (proportional breakdown). For example, when we worked with three time periods and used liabilities matrices and cash flow vectors equal to ⅓ of the total liabilities and cash flow, the results were the same as the results in the static model. As long as the liabilities matrix is split up proportionally over the time periods, the results will be the same as in the static implementation, regardless of the breakdown of the cash flow vector.

 

When payments are not uniform across all time periods, it is possible to set up the liabilities and cash flows in the dynamic model to save a bank from default or to trigger a bank to default. A drastic change is frequently needed from the static model inputs to see this type of result. For example, a drastic change in the breakdown of the incremental liabilities matrices (compared to the proportional breakdown yielding the same results as the static model) is required to yield significant deviation from the static model’s results.

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European Bank Data

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One explanation for no banks defaulting in the static implementation of the 2011 European bank data is that there was no major crisis in 2011. By changing the liabilities matrix and/or cash flow vector and passing the modified data through our algorithms, we were able to interpret the results of simulated economic events, which we referred to as stress tests.  Depending on the nature and severity of the changes we make, we found that certain shocks to the system caused an

ensuing wave of bank defaults.

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Implications and Extensions

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There are many different directions our analysis can be taken for additional detailed conclusions. One direction could be experimenting with different breakdowns of liabilities and cash flows using the dynamic model to try to save some banks from defaulting. We did not consider such an example with the 2011 European bank data, so it might be interesting to see if the dynamic model prevents banks from defaulting in say, the 3% cash flow reduction example. Another area of exploration could be conducting additional stress tests on the 2011 European bank data by changing the liabilities matrix and cash flow vector. We only considered a small subset of economic and financial events that could occur, and simulating additional scenarios may be helpful to those who seek to understand the nature of the banks’ reliances on each other. An additional direction one could take is conducting a sensitivity analysis on the 2011 European bank data to gain further insight into the banks’ financial health. Finally, our dynamic model could be implemented with a continuous time component, which would be even more robust than the discrete time component. However, a discrete time implementation is still quite realistic considering payments are often made daily before end of day.