1. Mengyang Wei, Miguel Leon-Ledesma, Gianluca Marcelli, Sarah Spurgeon
A dynamic network model of banking system stability
Mengyang Wei, Miguel Leon-Ledesma, Gianluca Marcelli, Sarah Spurgeon
Good afternoon everyone, thanks for having me here. Today I am going to introduce a dynamic network model
to study the stability of the banking system.

2. The University of Kent Canterbury Campus
First I would like to give a short introduction of where I am from. I am currently a PhD student in University of Kent, Canterbury campus. Canterbury is a nice and historical town located in the southeast of the UK. There are three faculties at Kent, with 18 academic schools and 728 academics.
There are three faculties at Kent, comprising 18 academic schools
Number of academics: 728

3. The group Prof Miguel León-Ledesma Mengyang Wei, PhD Student
Professor of Economics
This the team. I am the PhD student. As this is an interdisciplinary project, I have three supervisors from different research areas. Prof Miguel León-Ledesma from the School of Economics, he advises on the economics side such as macro economic modeling and financial regulations and helps with the model validation. Prof Sarah Spurgeon advises me on control analysis and control design. Dr Gianluca Marcelli is a modeler who helped building the model which I am going to introduce today.
Dr Gianluca Marcelli
Lecturer in Engineering
Prof Sarah Spurgeon OBE, FREng
Professor of Control Engineering

4. Outline of the presentation
Introduction
Overview of the Model
Preliminary Results
Conclusions and Future Work
My presentation today has four parts. First part is the introduction of the project including the motivation, background as well as the aim of the project. Then I will present some details of the dynamic network model. Then some preliminary results. Last part will be the conclusions and future work.
As this is still a working in progress project, I will present only some preliminary results and I will appreciate your feedback.

5. Introduction ------Motivation
The financial meltdown demonstrated the failure of bank regulation on an individual basis and showed the need to give special emphasis to systemic risk.
Financial regulators have now adopted a macro-prudential to mitigate the risk of the financial system as a whole.
More attention has been given to links between banks (interbank borrowing and lending) which can facilitate the spread of bankruptcy.
The motivation of this project is the financial crisis, which showed that banks should be regulated at system level rather that individual level.
The financial crisis exposed the risk of contagion caused by the links between the banks.
A failure in a small number of banks can spread to other banks through inter-bank loans, then causing the paralysis of the whole banking system.
There is a need, then, to give special emphasis to systemic risk.

6. Introduction ------Background
Bank1
Bank2
Bank3
Bank4
Bank5
Bank6
Introduction Background
Several network models have been
proposed to study the stability of the banking system.
Iori et al., 2006, showed that size and connectivity of banks affect contagion.
Nier et al., 2007, studied how contagious defaults is affected by capitalization, connectivity and the size of interbank exposure.
May and Arinaminpathy, 2010, explored the interplay between complexity and stability in simplified models of the banking system.
Acemoglu et al., 2015, studied how network interactions can function as a mechanism for propagation and amplification of microeconomic shocks.
Several network models have been proposed to study the stability of the banking system.
Here, I mentioned 4 papers that used models of the banking system to study how bank size, connectivity and other relevant parameters affect contagion and interbank exposure.
Later, I will point out what elements I have taken from these two papers (Iori et al, and May et al.)

7. Introduction -------Aim and objectives of the project
Bank1
Bank2
Bank3
Bank4
Bank5
Bank6
Introduction Aim and objectives of the project
In this project a network model has
been developed to study the stability of the banking system.
The banking system is represented as a network where the nodes are the individual banks.
A dynamic model (DM) based on a system of differential equations has been developed to describe the network.
Engineering Control Theory will be applied, for the first time, to analyse and minimizing systemic risk within a network of banks.
Following the control analysis, recommendations for financial regulators and operators will be proposed to help preserving financial stability.
The model we developed aims at studying a the stability of the banking system. In this model the banking system is represented by a network where the nodes are the individual banks. Differential equations has been developed to describe the dynamics of network.
We are currently applying Control Theory on this model to analyse and minimise systemic risk. We expect to propose some recommendations following the control analysis for the financial regulators to help preserving financial stability.
Today I am going to show you this dynamic model and preliminary results.
For the control analysis, we are still working.

8. Outline of the presentation
Introduction
Overview of the Model
Preliminary Results
Conclusion and Future Work
Now I will going to some details of the model

9. Overview of the Dynamic Model -------Structure of each bank
Each bank is modelled as made of six banking components/activities:
Net-worth (N)
Investment (I)
Deposits (D)
Cash (C)
Interbank Borrowing (B)
Interbank Lending (L)
The basic structure of each bank is described, following a paper by May et al This figure shows the 6 components/activities of a bank.
Deposits (D) and interbank borrowings (B). Investment(I), cash (C) and interbank lending (L). The net-worth of the bank is the different between the asset and liability.
We introduced the Cash as new component compared et al. The cash is an important component in our model.
Liability
Asset
Page 9

10. Overview of the Dynamic Model -------Structure of each bank
The links between any two banks, i and j, are made by interbank lending and borrowings, 𝐿ij and 𝐵ij.
The link rate, lr (0<lr<1), represents the average proportions of links of any bank.
The behaviour of Di , Ci, Ii, 𝐿ij and 𝐵ij are described by time derivatives.
The differential equations containing Di , Ci, Ii, 𝐿ij and 𝐵ij prescribe the dynamics of the system.
Links between banks forming the network: the links between banks are made by the loans/borrowings.
The links between any two banks i and j are made of the interbank loans and borrowings, Li,j and Bi,j.
A parameter called link rate, lr (0<lr<1), has been developed to represent the average proportions of links of any bank. The link rate take values between 0 to 1, the higher link rate the more links in the network.
The behaviour of different bank components will be described by the differential equations prescribing the dynamics of the system. Next I will explain the Time evolution of each bank component.

11. Overview of the Dynamic Model -------Time evolution of banks components
Cash:
𝑑𝐶 𝑖 𝑑𝑡 = 𝑑𝐷 𝑖 𝑑𝑡 − 𝑑𝐼 𝑖 𝑑𝑡 − 𝑔 𝑖 𝐷 𝑖 + 𝑝 𝑖 𝐼 𝑖 + 𝑖≠𝑗 𝑏 𝑖𝑗 − 𝑖≠𝑗 ℎ 𝑖𝑗 𝐵 𝑖𝑗 − 𝑖≠𝑗 𝑙 𝑖𝑗 + 𝑖≠𝑗 𝑘 𝑖𝑗 𝐿 𝑖𝑗
Differential equation governing the change in time of the cash.
Importantly, when the cash becomes ‘negative’, the bank fails and is removed from the system.
𝑔 𝑖 , 𝑝 𝑖 , ℎ 𝑖𝑗 and 𝑘 𝑖𝑗 are interest rates.
This is the diff. eq. describing how cask changes with time.
This is the change in cash do to deposits…n it is positive because…
This is the change in cash due to investments. It is negative because…
The third term is the interests paid by the bank toi depositors (it is negative)..

12. Overview of the Dynamic Model -------Time evolution of banks components
Deposits:
𝐷 𝑖 = 𝐷 + 𝐷 𝜎 𝐷 𝜀 𝑡
𝜀t ~ N(0,1)
The deposits are assumed to be assigned by an exogenous signal.
𝐷 represents an average size of the deposits.
𝜎 𝐷 represents the amplitude of the shock.
Fluctuations (shocks) in the deposit are caused by random payments/withdrawals of deposits.
The deposits are assumed to be assigned by an exogenous signal which introduces shocks in the system.
Di is the total deposit of bank it is equal to the dbar which is the average size of the deposit plus a random shock. sigmad is the amplitude of the shock and epsilon_t is random variable normally distributed.

13. Overview of the Dynamic Model -------Time evolution of banks components
Investments:
𝑑𝐼 𝑖 𝑑𝑡 = min 𝐶 𝑖 −𝑟 𝐷 𝑖 + , 𝑜𝑝𝑝 𝑖 −𝑤 𝐼 𝑖 −𝑣 𝐼 𝑖
𝑜𝑝𝑝 𝑖 = 𝑜𝑝𝑝 + 𝜎 𝑜𝑝𝑝 𝜂 𝑡
𝜂𝑡 ~ N(0,1).
A bank invests depending on cash availability and investment opportunity, 𝑜𝑝𝑝 𝑖 .
r is the reserve ratio.
−𝑤 𝐼 𝑖 is the proportion of total investment that has matured.
−𝑣 𝐼 𝑖 is the proportion of total investments that has been lost due to defaults.
This the dif. Eq. for the inv.
A bank invest if … and if …
The other two term are … matured inv. and inv lost due to defaults

14. Overview of the Dynamic Model -------Time evolution of banks components
Interbank borrowing and lending:
𝑏 𝑖𝑗 = 𝑙 𝑗𝑖 = 𝑚𝑖𝑛 [ 𝑟 𝐷 𝑖 − 𝐶 𝑖 + , 𝐶 𝑗 −𝑟 𝐷 𝑗 + ]
𝑑𝐵 𝑖𝑗 𝑑𝑡 = 𝑏 𝑖𝑗 + 𝐵 𝑖𝑗 𝛼 𝑖𝑗
Fluctuations in deposits may force banks to borrow money to meet the reserve ratio requirements.
The first equation shows the current change of borrowing/lending between two banks.
The second equation shows the total borrowing/lending.
𝛼 𝑖𝑗 is proportion of the total borrowing repaid at a given time.
Fluctuations in deposits may force banks to borrow money to meet the reserve ratio requirements.
The first equation shows the current change of borrowing/lending between two banks. Bij is the money bank I borrows from bank j, it equals the money bank j lend to bank I and it is the minimum value of these two. First is the money bank I needs to borrow, which is equals to , since bank I need to borrow money to reach its reserve requirement. And the other one is the money that bank j can provided.
The second equation shows the total borrowing/lending. Which equals to the current change of borrowing plus the repayment of the borrowing form previous period.

15. Overview of the Dynamic Model -------Time evolution of banks components
Interbank interest rates:
ℎ 𝑖𝑗 = 𝑘 𝑗𝑖 = ℎ 0 + 𝑎 𝑒 𝑦− 𝐵 𝑖𝑗 𝐶 𝑗 𝑧 +1
ℎ 0 is the basic interest rate applied for lending and borrowing.
The interest rate changes according to the debt of the borrowing bank and the cash of the lending bank: ….
The interest rate is a dynamic parameter, which depends on th estate of the borrowing and lending banks. This parameter, as others, makes the system non-linear.

16. Overview of the Dynamic Model -------Simulation of the model
Banks’ cash changes due to interest payments to depositors and changes of deposits due to stochastic shocks.
Banks with extra cash lend and invest.
If the cash of a bank falls below the value required by the reserve ratio that bank has to borrow from other banks.
Those banks that are left with ‘negative’ cash, are removed from the system. Their remaining assets are distributed to depositors and lending banks.
After any default liquidation a new simulation step starts.
At the end of the simulation, the banks that survived are counted and other relevant quantities are calculated.

17. Overview of the Dynamic Model ------- Implementation using Simulink (Matlab)
Before the results, let’s have a quick look at the model implementation. The model is implemented using Matlab Simulink, this figure shows the top level of the model in the Simulink. The blocks are subsystems for different bank components.
An over view of the model in Simulink

18. Outline of the presentation
Introduction
Overview of the Model
Preliminary Results
Conclusions and Future Work
Now I am going to show you some preliminary results generated from the Simulink simulation

19. Preliminary Results -------Number of surviving banks for different reserve ratios, r.
Link rate = 0
Link rate = 0.3
Link rate = 0.8
Link rate = 1
We look into how the number of surviving banks changes according to the reserve ratios. The testing period is 1000 days and at the beginning there are 50 banks in total. The amplitude of the shock is 0.3. The line in different colour represent different reserve ratios. And the figures are for different link rate cases. In the first figure when link rate is 0, as the reserve ratio increase more bank survive to the end of the testing period. The similar trend can be captured as the link rate changes. We can also see when the link rate increases, the failure of banks get slower. So from these results we can see both the reserve ratio and link rate contribute to the stability of the system.
r = r = r = r =0.7
𝜎 𝐷 =0.3

20. Preliminary Results -------Number of surviving banks for different reserve ratios, r.
Link rate = 0
Link rate = 0.3
Link rate = 0.8
Link rate = 1
Now let s move to another situation where the amplitude of the shock increases to in the first figure we can still find out more bank survive as the reserve ratio increases, but because of the high amplitude of the shock all banks failed quickly around day 60. when link rate increases, the reserve ratio no longer contribute to the stability, as it increasing, fewer bank survived.
r = r = r = r =0.7
𝜎 𝐷 =0.5

21. Preliminary Results -------Number of surviving banks for different reserve ratios and link rate.
At day 1000
𝜎 𝐷 =0.3
At day 100
𝜎 𝐷 =0.5
Waterfall plot showing the number of surviving banks as function of both link rate and reserve ratio.

22. Preliminary Results -------Contagion for different reserve ratios and link rate.
We also quantify the contagion using this model. To quantify the contagion, at the end of each simulation, we calculate the proportion of failed banks with significant unpaid loans. If the unpaid loan is larger the negative cash of the bank when it failed, which means the bank could survive if other bank repaid their borrowing to this failed bank. we recogenaise this failed bank has a signigicant unpaid loans.
Proportion (in different colours) of failed banks with significant unpaid loans as function of link rate and reserve ratios, at day 1000.
𝜎 𝐷 =0.3

23. Outline of the presentation
Introduction
Overview of the Model
Preliminary Results
Conclusions and Future Work

24. Conclusions
A model based on differential equations has been developed to study the banking system stability as a function of link rate, reserve ratio and deposit shocks.
Some behaviours of our model are similar to previous models.
Reserve ratio and link rate have a positive effect on the stability of the systems.
For high values of the shocks, high reserve ratios may have a detrimental effect on the survival of banks.
The model also allows the quantification of contagion, which shows some non-linear behaviour as function of reserve ratio and link rate.

25. Future Work
The model will be further expanded by the application of control-theory analysis.
Feedback mechanisms will be implemented to allow the preservation of stability.
New measures of contagion will be implemented.

26. Thank you