Dear Sir/Madam
I have an urgent programming assignment. The deadline is in max 6 days from today.
Please find the attachements and two supporting websites:
http://people.brandeis.edu/~blebaron/classes/agentfin/GaiKapadia.html?fbclid=IwAR0KYiShhpxEkGh14tF0Jh_I9UPoBDMh9EgOBsGHXeel521mIYnWyO7HrNA#id2
https://stackoverflow.com/questions/56969062/financial-contagion-epidemic-spread-model-meets-problem
Please tell me your thoughts,
Best,
WP383 Electronic copy available at: http://ssrn.com/abstract=1577043 Working Paper No. 383 Contagion in financial networks Prasanna Gai and Sujit Kapadia March 2010 Electronic copy available at: http://ssrn.com/abstract=1577043 Working Paper No. 383 Contagion in financial networks Prasanna Gai(1) and Sujit Kapadia(2) Abstract This paper develops an analytical model of contagion in financial networks with arbitrary structure. We explore how the probability and potential impact of contagion is influenced by aggregate and idiosyncratic shocks, changes in network structure, and asset market liquidity. Our findings suggest that financial systems exhibit a robust-yet-fragile tendency: while the probability of contagion may be low, the effects can be extremely widespread when problems occur. And we suggest why the resilience of the system in withstanding fairly large shocks prior to 2007 should not have been taken as a reliable guide to its future robustness. Key words: Contagion, network models, systemic risk, liquidity risk, financial crises. JEL classification: D85, G01, G21. (1) Australian National University and Bank of England. Email:
[email protected] (2) Bank of England. Email:
[email protected] The views expressed in this paper are those of the authors, and not necessarily those of the Bank of England. The paper is forthcoming in Proceedings of the Royal Society A. We thank Emma Mattingley, Nick Moore, Barry Willis and, particularly, Jason Dowson for excellent research assistance. We are also grateful to Kartik Anand, Fabio Castiglionesi, Geoff Coppins, Avinash Dixit, John Driffill, Sanjeev Goyal, Andy Haldane, Simon Hall, Matteo Marsili, Robert May, Marcus Miller, Emma Murphy, Filipa Sa, Nancy Stokey, Merxe Tudela, Jing Yang, three anonymous referees and seminar participants at the Bank of England, the University of Oxford, the University of Warwick research workshop and conference on ‘World Economy and Global Finance’ (Warwick, 11–15 July 2007), the UniCredit Group Conference on ‘Banking and Finance: Span and Scope of Banks, Stability and Regulation’ (Naples, 17–18 December 2007), the 2008 Royal Economic Society Annual Conference (Warwick, 17–19 March 2008), and the 2008 Southern Workshop in Macroeconomics (Auckland, 28–30 March 2008) for helpful comments and suggestions. This paper was finalised on 8 October 2009. The Bank of England’s working paper series is externally refereed. Information on the Bank’s working paper series can be found at www.bankofengland.co.uk/publications/workingpapers/index.htm Publications Group, Bank of England, Threadneedle Street, London, EC2R 8AH Telephone +44 (0)20 7601 4030 Fax +44 (0)20 7601 3298 email
[email protected] © Bank of England 2010 ISSN 1749-9135 (on-line) Electronic copy available at: http://ssrn.com/abstract=1577043 Contents Summary 3 1 Introduction 5 2 The model 10 3 Numerical simulations 20 4 Liquidity risk 26 5 Relationship to the empirical literature 28 6 Conclusion 29 Appendix: Generating functions 30 References 32 Working Paper No. 383 March 2010 2 Summary In modern �nancial systems, an intricate web of claims and obligations links the balance sheets of a wide variety of intermediaries, such as banks and hedge funds, into a network structure. The advent of sophisticated �nancial products, such as credit default swaps and collateralised debt obligations, has heightened the complexity of these balance sheet connections still further. As demonstrated by the �nancial crisis, especially in relation to the failure of Lehman Brothers and the rescue of American International Group (AIG), these interdependencies have created an environment for feedback elements to generate ampli�ed responses to shocks to the �nancial system. They have also made it dif�cult to assess the potential for contagion arising from the behaviour of �nancial institutions under distress or from outright default. This paper models two key channels of contagion in �nancial systems. The primary focus is on how losses may potentially spread via the complex network of direct counterparty exposures following an initial default. But the knock-on effects of distress at some �nancial institutions on asset prices can force other �nancial entities to write down the value of their assets, and we also model the potential for this effect to trigger further rounds of default. Contagion due to the direct interlinkages of interbank claims and obligations may thus be reinforced by indirect contagion on the asset side of the balance sheet particularly when the market for key �nancial system assets is illiquid. Our modelling approach applies statistical techniques from complex network theory. In contrast to most existing theoretical work on interbank contagion, which considers small, stylised networks, we demonstrate that analytical results on the relationship between �nancial system connectivity and contagion can be obtained for structures which re�ect the complexities of observed �nancial networks. And we provide a framework for isolating the probability and spread of contagion when claims and obligations are interlinked. The model we develop explicitly accounts for the nature and scale of macroeconomic and bank-speci�c shocks, and the complexity of network structure, while allowing asset prices to interact with balance sheets. The interactions between �nancial intermediaries following shocks make for non-linear system dynamics, whereby contagion risk can be highly sensitive to small changes in parameters. Working Paper No. 383 March 2010 3 Our results suggest that �nancial systems may exhibit a robust-yet-fragile tendency: while the probability of contagion may be low, the effects can be extremely widespread when problems occur. The model also highlights how seemingly indistinguishable shocks can have very different consequences for the �nancial system depending on whether or not the shock hits at a particular pressure point in the network structure. This helps explain why the evidence of the resilience of the system to fairly large shocks prior to 2007 was not a reliable guide to its future robustness. The intuition underpinning these results is as follows. In a highly connected system, the counterparty losses of a failing institution can be more widely dispersed to, and absorbed by, other entities. So increased connectivity and risk sharing may lower the probability of contagious default. But, conditional on the failure of one institution triggering contagious defaults, a high number of �nancial linkages also increases the potential for contagion to spread more widely. In particular, high connectivity increases the chances that institutions which survive the effects of the initial default will be exposed to more than one defaulting counterparty after the �rst round of contagion, thus making them vulnerable to a second-round default. The effects of any crises that do occur can, therefore, be extremely widespread. Working Paper No. 383 March 2010 4 1 Introduction In modern �nancial systems, an intricate web of claims and obligations links the balance sheets of a wide variety of intermediaries, such as banks and hedge funds, into a network structure. The advent of sophisticated �nancial products, such as credit default swaps and collateralised debt obligations, has heightened the complexity of these balance sheet connections still further. As demonstrated by the �nancial crisis, especially in relation to the failure of Lehman Brothers and the rescue of American International Group (AIG), these interdependencies have created an environment for feedback elements to generate ampli�ed responses to shocks to the �nancial system. They have also made it dif�cult to assess the potential for contagion arising from the behaviour of �nancial institutions under distress or from outright default.1 This paper models two key channels of contagion in �nancial systems by which default may spread from one institution to another. The primary focus is on how losses can potentially spread via the complex network of direct counterparty exposures following an initial default. But, as Cifuentes et al (2005) and Shin (2008) stress, the knock-on effects of distress at some �nancial institutions on asset prices can force other �nancial entities to write down the value of their assets, and we also model the potential for this effect to trigger further rounds of default. Contagion due to the direct interlinkages of interbank claims and obligations may thus be reinforced by indirect contagion on the asset side of the balance sheet particularly when the market for key �nancial system assets is illiquid. The most well-known contribution to the analysis of contagion through direct linkages in �nancial systems is that of Allen and Gale (2000).2 Using a network structure involving four banks, they demonstrate that the spread of contagion depends crucially on the pattern of interconnectedness between banks. When the network is complete, with all banks having exposures to each other such that the amount of interbank deposits held by any bank is evenly spread over all other banks, the impact of a shock is readily attenuated. Every bank takes a small `hit' and there is no contagion. By contrast, when the network is `incomplete', with banks only having exposures to a few counterparties, the system is more fragile. The initial impact of a 1See Rajan (2005) for a policymaker's view of the recent trends in �nancial development and Haldane (2009) for a discussion of the role that the structure and complexities of the �nancial network have played in the �nancial turmoil of 2007-09. 2Other strands of the literature on �nancial contagion have focused on the role of liquidity constraints (Kodres and Pritsker (2002)), information asymmetries (Calvo and Mendoza (2000)), and wealth constraints (Kyle and Xiong (2001)). As such, their focus is less on the nexus between network structure and �nancial stability. Network perspectives have also been applied to other topics in �nance: for a comprehensive survey of the use of network models in �nance, see Allen and Babus (2009). Working Paper No. 383 March 2010 5 shock is concentrated among neighbouring banks. Once these succumb, the premature liquidation of long-term assets and the associated loss of value bring previously unaffected banks into the front line of contagion. In a similar vein, Freixas et al (2000) show that tiered systems with money-centre banks, where banks on the periphery are linked to the centre but not to each other, may also be susceptible to contagion.3 The generality of insights based on simple networks with rigid structures to real-world contagion is clearly open to debate. Moreover, while not being so stylised, models with endogenous network formation (eg Leitner (2005) and Castiglionesi and Navarro (2007)) impose strong assumptions which lead to stark predictions on the implied network structure that do not re�ect the complexities of real-world �nancial networks. And, by and large, the existing literature fails to distinguish the probability of contagious default from its potential spread. However, even prior to the current �nancial crisis, the identi�cation of the probability and impact of shocks to the �nancial system was assuming centre-stage in policy debate. Some policy institutions, for example, attempted to articulate the probability and impact of key risks to the �nancial system in their Financial Stability Reports.4 Moreover, the complexity of �nancial systems means that policymakers have only partial information about the true linkages between �nancial intermediaries. Given the speed with which shocks propagate, there is, therefore, a need to develop tools that facilitate analysis of the transmission of shocks through a given, but arbitrary, network structure. Recent events in the global �nancial system have only served