Integrating Systemic Risk and Risk Analysis Using Copulas
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摘要
Systemic risk research is gaining traction across diverse disciplinary research communities, but has as yet not been strongly linked to traditional, well-established risk analysis research. This is due in part to the fact that systemic risk research focuses on the connection of elements within a system, while risk analysis research focuses more on individual risk to single elements. We therefore investigate how current systemic risk research can be related to traditional risk analysis approaches from a conceptual as well as an empirical point of view. Based on Sklar's Theorem, which provides a one-to-one relationship between multivariate distributions and copulas, we suggest a reframing of the concept of copulas based on a network perspective. This provides a promising way forward for integrating individual risk(in the form of probability distributions) and systemic risk(in the form of copulasdescribing the dependencies among such distributions)across research domains. Copulas can link continuous node states, characterizing individual risks, with a gradual dependency of the coupling strength between nodes on their states, characterizing systemic risk. When copulas are used for describing such refined coupling between nodes,they can provide a more accurate quantification of a system's network structure. This enables more realistic systemic risk assessments, and is especially useful when extreme events(that occur at low probabilities, but have high impacts) affect a system's nodes. In this way, copulas can be informative in measuring and quantifying changes in systemic risk and therefore be helpful in its management. We discuss the advantages and limitations of copulas for integrative risk analyses from the perspectives of modeling, measurement, and management.
Systemic risk research is gaining traction across diverse disciplinary research communities, but has as yet not been strongly linked to traditional, well-established risk analysis research. This is due in part to the fact that systemic risk research focuses on the connection of elements within a system, while risk analysis research focuses more on individual risk to single elements. We therefore investigate how current systemic risk research can be related to traditional risk analysis approaches from a conceptual as well as an empirical point of view. Based on Sklar's Theorem, which provides a one-to-one relationship between multivariate distributions and copulas, we suggest a reframing of the concept of copulas based on a network perspective. This provides a promising way forward for integrating individual risk(in the form of probability distributions) and systemic risk(in the form of copulasdescribing the dependencies among such distributions)across research domains. Copulas can link continuous node states, characterizing individual risks, with a gradual dependency of the coupling strength between nodes on their states, characterizing systemic risk. When copulas are used for describing such refined coupling between nodes,they can provide a more accurate quantification of a system's network structure. This enables more realistic systemic risk assessments, and is especially useful when extreme events(that occur at low probabilities, but have high impacts) affect a system's nodes. In this way, copulas can be informative in measuring and quantifying changes in systemic risk and therefore be helpful in its management. We discuss the advantages and limitations of copulas for integrative risk analyses from the perspectives of modeling, measurement, and management.
Systemic risk research is gaining traction across diverse disciplinary research communities, but has as yet not been strongly linked to traditional, well-established risk analysis research. This is due in part to the fact that systemic risk research focuses on the connection of elements within a system, while risk analysis research focuses more on individual risk to single elements. We therefore investigate how current systemic risk research can be related to traditional risk analysis approaches from a conceptual as well as an empirical point of view. Based on Sklar's Theorem, which provides a one-to-one relationship between multivariate distributions and copulas, we suggest a reframing of the concept of copulas based on a network perspective. This provides a promising way forward for integrating individual risk(in the form of probability distributions) and systemic risk(in the form of copulasdescribing the dependencies among such distributions)across research domains. Copulas can link continuous node states, characterizing individual risks, with a gradual dependency of the coupling strength between nodes on their states, characterizing systemic risk. When copulas are used for describing such refined coupling between nodes,they can provide a more accurate quantification of a system's network structure. This enables more realistic systemic risk assessments, and is especially useful when extreme events(that occur at low probabilities, but have high impacts) affect a system's nodes. In this way, copulas can be informative in measuring and quantifying changes in systemic risk and therefore be helpful in its management. We discuss the advantages and limitations of copulas for integrative risk analyses from the perspectives of modeling, measurement, and management.
引文
Allen,F.,and D.Gale.2000.Financial contagion.Journal of Political Economy 108(1):1-33.
Allen,F.,A.Babus,and E.Carletti.2012.Asset commonality,debt maturity and systemic risk.Journal of Financial Economics104(3):519-534.
Aven,T.2016.What defines us as professionals in the field of risk analysis?Risk Analysis 37(5):854-860.
Bak,P.,and M.Paczuski.1995.Complexity,contingency,and criticality.Proceedings of the National Academy of Sciences of the USA 92(15):6689-6696.
Battiston,S.,D.D.Gatti,M.Gallegati,B.Greenwald,and J.E.Stiglitz.2012.Liaisons dangereuses:Increasing connectivity,risk sharing,and systemic risk.Journal of Economic Dynamics and Control 36(8):1121-1141.
Battiston,S.,M.Puliga,R.Kaushik,P.Tasca,and G.Caldarelli.2012.Debtrank:Too central to fail?Financial networks,the FEDand systemic risk.Scientific Reports 2(8):Article 541.
Bedford,T.,and R.Cooke.2002.Vines:A new graphical model for dependent random variables.Annals of Statistics 30(4):1031-1068.
Borgomeo,E.,G.Pflug,J.W.Hall,and S.Hochrainer-Stigler.2015.Assessing water resource system vulnerability to unprecedented hydrological drought using copulas to characterize drought duration and deficit.Water Resources Research 51(11):8927-8948.
Borrvall,C.,B.Ebenman,and T.Jonsson.2000.Biodiversity lessens the risk of cascading extinction in model food webs.Ecology Letters 3(2):131-136.
Boss,M.,M.Summer,and S.Thurner.2004.Contagion flow through banking networks.In Computational Science-ICCS 2004,Proceedings,Part III,ed.M.Bubak,G.D.van Albada,P.M.A.Sloot,and J.Dongarra,1070-1077.Berlin:Springer.
Centeno,M.A.,M.Nag,T.S.Patterson,A.Shaver,and A.J.Windawi.2015.The emergence of global systemic risk.Annual Review of Sociology 41(1):65-85.
Folke,C.,S.Carpenter,B.Walker,M.Scheffer,T.Elmqvist,L.Gunderson,and C.S.Holling.2004.Regime shifts,resilience,and biodiversity in ecosystem management.Annual Review of Ecology,Evolution,and Systematics 35(1):557-581.
Gai,P.,and S.Kapadia.2010.Contagion in financial networks.Proceedings of the Royal Society A 466:2401-2423.
Gardner,M.R.,and W.R.Ashby.1970.Connectance of large dynamic(cybernetic)systems:Critical values for stability.Nature228(5273):784.
Grossi,P.,and H.Kunreuther(eds.).2005.Catastrophe modeling:Anew approach to managing risk.Boston,MA:Springer.
Haefeli,D.,and P.M.Liedtke(eds.).2012.Insurance and resolution in light of the systemic risk debate.Geneva,Switzerland:The Geneva Association.
Haldane,A.G.,and R.M.May.2011.Systemic risk in banking ecosystems.Nature 469(7330):351-355.
Hanel,R.,S.A.Kauffman,and S.Thurner.2007.Towards a physics of evolution:Critical diversity dynamics at the edges of collapse and bursts of diversification.Physical Review E 76:Article030101.
Helbing,D.2013.Globally networked risks and how to respond.Nature 497(7447):51-59.
Hochrainer-Stigler,S.,G.Boza,C.Colon,S.Poledna,E.Rovenskaya,and U.Dieckmann.2018.Resilience of systems to individual risk and systemic risk.In Resource guide on resilience,Volume2,ed.M.-V.Florin,and I.Linkov.Lausanne,Switzerland:EPFLInternational Risk Governance Center(IRGC).
Hochrainer-Stigler,S.,J.Linnerooth-Bayer,and A.Lorant.2017.The European Union Solidarity Fund:An assessment of its recent reforms.Mitigation and Adaptation Strategies for Global Change 22(4):547-563.
Hofert,M.,and M.Ma¨chler.2011.Nested Archimedean copulas meet R:The nacopula package.Journal of Statistical Software 39(9):1-20.
Jain,S.,and S.Krishna.2001.A model for the emergence of cooperation,interdependence,and structure in evolving networks.Proceedings of the National Academy of Sciences of the USA 98(2):543-547.
Joe,H.2014.Dependence modeling with copulas.Chapman and Hall/CRC.
Jeong,B.,W.Lee,D.S.Kim,and H.Shin.2016.Copula-based approach to synthetic population generation.PLoS ONE 11(8):Article e0159496.
Jongman,B.,S.Hochrainer-Stigler,L.Feyen,J.C.J.H.Aerts,R.Mechler,W.J.W.Botzen,and L.M.Bouwer,et al.2014.Increasing stress on disaster-risk finance due to large floods.Nature Climate Change 4(4):264-268.
Kessler,D.2014.Why(re)insurance is not systemic.Journal of Risk and Insurance 81(3):477-488.
Kovacevic,R.M.,G.C.Pflug,and A.Pichler.2014.Measuring and managing risk.In Investment risk management,ed.H.K.Baker,and G.Filbeck,17-41.Oxford:Oxford University Press.
Kurowicka,D.,and R.Cooke.2006.Uncertainty analysis with high dimensional dependence modelling.West Sussex,UK:Wiley.
Kurowicka,D.,and H.Joe.2010.Dependence modeling:Vine copula handbook.Singapore:World Scientific Publishing Co.
Little,R.G.2002.Controlling cascading failure:Understanding the vulnerabilities of interconnected infrastructures.Journal of Urban Technology 9(1):109-123.
May,R.M.1972.Will a large complex system be stable?Nature238(5364):413-414.
May,R.M.,and N.Arinaminpathy.2010.Systemic risk:The dynamics of model banking systems.Journal of the Royal Society Interface 7(46):823-838.
McCann,K.S.2000.The diversity-stability debate.Nature 405(6783):228-233.
McNeil,A.J.,R.Frey,and P.Embrechts(eds.).2015.Quantitative risk management:Concepts,techniques and tools.Princeton,NJ:Princeton University Press.
Nelsen,R.B.2006.An introduction to copulas.New York:Springer.
Page,S.E.2015.What sociologists should know about complexity.Annual Review of Sociology 41(1):21-41.
Pimm,S.L.,and J.H.Lawton.1978.On feeding on more than one trophic level.Nature 275(5680):542-544.
Pflug,G.,and A.Pichler.2018.Systemic risk and copula models.Central European Journal of Operations Research 26(2):465-483.
Pflug,G.C.,and W.Ro¨misch.2007.Modeling,measuring and managing risk.Singapore:World Scientific Publishing Co.
Poledna,S.,and S.Thurner.2016.Elimination of systemic risk in financial networks by means of a systemic risk transaction tax.Quantitative Finance 16(10):1599-1613.
Scheffer,M.,and S.R.Carpenter.2003.Catastrophic regime shifts in ecosystems:Linking theory to observation.Trends in Ecology&Evolution 18(12):648-656.
Sklar,A.1959.Distribution functions with n dimensions and their margins(Fonctions de re′partition a`n dimensions et leurs marges).Publications de l’Institut Statistique de l’Universite′de Paris 8:229-231(in French).
Sklar,A.1973.Random variables,bivariate distribution functions and copulas.Kybernetika 9:449-460.
Thurner,S.,and S.Poledna.2013.DebtRank-transparency:Controlling systemic risk in financial networks.Scientific Reports 3(5):Article 1888.
Tilman,D.,and J.A.Downing.1996.Biodiversity and stability in grasslands.In Ecosystem management,ed.F.B.Samson and F.L.Knopf,3-7.New York:Springer.
Timonina,A.,S.Hochrainer-Stigler,G.Pflug,B.Jongman,and R.Rojas.2015.Structured coupling of probability loss distributions:Assessing joint flood risk in multiple river basins.Risk Analysis 35(11):2102-2119.
WEF(World Economic Forum).2018.The global risk report.Geneva,Switzerland:WEF.
1For the definition of the bivariate and the multivariate Joe copula,see Joe(2014).
Allen,F.,A.Babus,and E.Carletti.2012.Asset commonality,debt maturity and systemic risk.Journal of Financial Economics104(3):519-534.
Aven,T.2016.What defines us as professionals in the field of risk analysis?Risk Analysis 37(5):854-860.
Bak,P.,and M.Paczuski.1995.Complexity,contingency,and criticality.Proceedings of the National Academy of Sciences of the USA 92(15):6689-6696.
Battiston,S.,D.D.Gatti,M.Gallegati,B.Greenwald,and J.E.Stiglitz.2012.Liaisons dangereuses:Increasing connectivity,risk sharing,and systemic risk.Journal of Economic Dynamics and Control 36(8):1121-1141.
Battiston,S.,M.Puliga,R.Kaushik,P.Tasca,and G.Caldarelli.2012.Debtrank:Too central to fail?Financial networks,the FEDand systemic risk.Scientific Reports 2(8):Article 541.
Bedford,T.,and R.Cooke.2002.Vines:A new graphical model for dependent random variables.Annals of Statistics 30(4):1031-1068.
Borgomeo,E.,G.Pflug,J.W.Hall,and S.Hochrainer-Stigler.2015.Assessing water resource system vulnerability to unprecedented hydrological drought using copulas to characterize drought duration and deficit.Water Resources Research 51(11):8927-8948.
Borrvall,C.,B.Ebenman,and T.Jonsson.2000.Biodiversity lessens the risk of cascading extinction in model food webs.Ecology Letters 3(2):131-136.
Boss,M.,M.Summer,and S.Thurner.2004.Contagion flow through banking networks.In Computational Science-ICCS 2004,Proceedings,Part III,ed.M.Bubak,G.D.van Albada,P.M.A.Sloot,and J.Dongarra,1070-1077.Berlin:Springer.
Centeno,M.A.,M.Nag,T.S.Patterson,A.Shaver,and A.J.Windawi.2015.The emergence of global systemic risk.Annual Review of Sociology 41(1):65-85.
Folke,C.,S.Carpenter,B.Walker,M.Scheffer,T.Elmqvist,L.Gunderson,and C.S.Holling.2004.Regime shifts,resilience,and biodiversity in ecosystem management.Annual Review of Ecology,Evolution,and Systematics 35(1):557-581.
Gai,P.,and S.Kapadia.2010.Contagion in financial networks.Proceedings of the Royal Society A 466:2401-2423.
Gardner,M.R.,and W.R.Ashby.1970.Connectance of large dynamic(cybernetic)systems:Critical values for stability.Nature228(5273):784.
Grossi,P.,and H.Kunreuther(eds.).2005.Catastrophe modeling:Anew approach to managing risk.Boston,MA:Springer.
Haefeli,D.,and P.M.Liedtke(eds.).2012.Insurance and resolution in light of the systemic risk debate.Geneva,Switzerland:The Geneva Association.
Haldane,A.G.,and R.M.May.2011.Systemic risk in banking ecosystems.Nature 469(7330):351-355.
Hanel,R.,S.A.Kauffman,and S.Thurner.2007.Towards a physics of evolution:Critical diversity dynamics at the edges of collapse and bursts of diversification.Physical Review E 76:Article030101.
Helbing,D.2013.Globally networked risks and how to respond.Nature 497(7447):51-59.
Hochrainer-Stigler,S.,G.Boza,C.Colon,S.Poledna,E.Rovenskaya,and U.Dieckmann.2018.Resilience of systems to individual risk and systemic risk.In Resource guide on resilience,Volume2,ed.M.-V.Florin,and I.Linkov.Lausanne,Switzerland:EPFLInternational Risk Governance Center(IRGC).
Hochrainer-Stigler,S.,J.Linnerooth-Bayer,and A.Lorant.2017.The European Union Solidarity Fund:An assessment of its recent reforms.Mitigation and Adaptation Strategies for Global Change 22(4):547-563.
Hofert,M.,and M.Ma¨chler.2011.Nested Archimedean copulas meet R:The nacopula package.Journal of Statistical Software 39(9):1-20.
Jain,S.,and S.Krishna.2001.A model for the emergence of cooperation,interdependence,and structure in evolving networks.Proceedings of the National Academy of Sciences of the USA 98(2):543-547.
Joe,H.2014.Dependence modeling with copulas.Chapman and Hall/CRC.
Jeong,B.,W.Lee,D.S.Kim,and H.Shin.2016.Copula-based approach to synthetic population generation.PLoS ONE 11(8):Article e0159496.
Jongman,B.,S.Hochrainer-Stigler,L.Feyen,J.C.J.H.Aerts,R.Mechler,W.J.W.Botzen,and L.M.Bouwer,et al.2014.Increasing stress on disaster-risk finance due to large floods.Nature Climate Change 4(4):264-268.
Kessler,D.2014.Why(re)insurance is not systemic.Journal of Risk and Insurance 81(3):477-488.
Kovacevic,R.M.,G.C.Pflug,and A.Pichler.2014.Measuring and managing risk.In Investment risk management,ed.H.K.Baker,and G.Filbeck,17-41.Oxford:Oxford University Press.
Kurowicka,D.,and R.Cooke.2006.Uncertainty analysis with high dimensional dependence modelling.West Sussex,UK:Wiley.
Kurowicka,D.,and H.Joe.2010.Dependence modeling:Vine copula handbook.Singapore:World Scientific Publishing Co.
Little,R.G.2002.Controlling cascading failure:Understanding the vulnerabilities of interconnected infrastructures.Journal of Urban Technology 9(1):109-123.
May,R.M.1972.Will a large complex system be stable?Nature238(5364):413-414.
May,R.M.,and N.Arinaminpathy.2010.Systemic risk:The dynamics of model banking systems.Journal of the Royal Society Interface 7(46):823-838.
McCann,K.S.2000.The diversity-stability debate.Nature 405(6783):228-233.
McNeil,A.J.,R.Frey,and P.Embrechts(eds.).2015.Quantitative risk management:Concepts,techniques and tools.Princeton,NJ:Princeton University Press.
Nelsen,R.B.2006.An introduction to copulas.New York:Springer.
Page,S.E.2015.What sociologists should know about complexity.Annual Review of Sociology 41(1):21-41.
Pimm,S.L.,and J.H.Lawton.1978.On feeding on more than one trophic level.Nature 275(5680):542-544.
Pflug,G.,and A.Pichler.2018.Systemic risk and copula models.Central European Journal of Operations Research 26(2):465-483.
Pflug,G.C.,and W.Ro¨misch.2007.Modeling,measuring and managing risk.Singapore:World Scientific Publishing Co.
Poledna,S.,and S.Thurner.2016.Elimination of systemic risk in financial networks by means of a systemic risk transaction tax.Quantitative Finance 16(10):1599-1613.
Scheffer,M.,and S.R.Carpenter.2003.Catastrophic regime shifts in ecosystems:Linking theory to observation.Trends in Ecology&Evolution 18(12):648-656.
Sklar,A.1959.Distribution functions with n dimensions and their margins(Fonctions de re′partition a`n dimensions et leurs marges).Publications de l’Institut Statistique de l’Universite′de Paris 8:229-231(in French).
Sklar,A.1973.Random variables,bivariate distribution functions and copulas.Kybernetika 9:449-460.
Thurner,S.,and S.Poledna.2013.DebtRank-transparency:Controlling systemic risk in financial networks.Scientific Reports 3(5):Article 1888.
Tilman,D.,and J.A.Downing.1996.Biodiversity and stability in grasslands.In Ecosystem management,ed.F.B.Samson and F.L.Knopf,3-7.New York:Springer.
Timonina,A.,S.Hochrainer-Stigler,G.Pflug,B.Jongman,and R.Rojas.2015.Structured coupling of probability loss distributions:Assessing joint flood risk in multiple river basins.Risk Analysis 35(11):2102-2119.
WEF(World Economic Forum).2018.The global risk report.Geneva,Switzerland:WEF.
1For the definition of the bivariate and the multivariate Joe copula,see Joe(2014).