Modeling Contagion and Systemic Risk Daniele Bianchi 1 Monica Billio 2 Roberto Casarin 2 Massimo Guidolin 3 1 University of Warwick, Warwick Business School 2 University C´ a Foscari of Venice, Department of Economics 3 Bocconi University, Department of Finance and IGIER
Motivation ◮ Group of Ten (2001,p.126): “ Systemic financial risk is the risk that an event will trigger a loss of economic value or confidence in [...] a substantial portion of the financial system that [...] have significant adverse effects on the real economy.” ◮ Propagation of a single shock to the economy through cross-firm linkages or chain reactions ◮ World Bank, very restrictive definition: “ Contagion occurs when cross-country correlations increase during crisis times relative to correlations during tranquil times.” ◮ Structural breaks in correlations ◮ Punchline: Understand the structure/dynamics of financial networks is critical
Motivation Question: ◮ How we can make inference on unobservable time-varying cross-firm financial linkages? Issues: ◮ Identification of networks in dynamic time-series contexts ◮ System-wide inference in large dimensions ◮ Structural uncertainty ◮ Interaction with sources of systematic risk
This Paper ◮ Builds on network analysis ◮ Undirected graphical model to make system-wide inference on financial networks ◮ Time-varying network structure ◮ Different “regimes” of network connectivity ◮ Contagion as a shift concept ◮ Factor pricing approach ◮ Systemic and systematic risks are not mutually exclusive ◮ Exposures to systematic risks change across regimes of aggregate network connectivity ◮ Bayesian inference on the network structure and parameters jointly ◮ MCMC, robust finite-sample approach
Findings and Overview We focus on 100 Blue Chips from the S&P100 Index, daily: ◮ Network centrality ◮ Financial (Energy) firms are central when systemic risk is high (low) ◮ Network centrality does not depend on market values ◮ Connectivity is not constant over time ◮ Two “regimes” of systemic risk; high systemic risk from late 90s to early 2000 (i.e. dot.com bubble, Gramm-Leach-Bliley Act, financial scandals, etc.) and across the great financial crisis. ◮ Firm-level financial fragility and aggregate early warning ◮ High centrality of a firm in the network = Financial fragility ◮ High systemic risk anticipates market-wide financial distress conditions ◮ Systematic risks exposures change across regimes of systemic risk
Reference Literature ◮ Gaussian graphical models ◮ Giudici and Green (1999), Carvalho and West (2007), Carvalho, West and Hassam (2007), Wang and West (2009), Nakajima and West (2013),... ◮ Growing empirical research on financial networks for systemic risk measurement purposes ◮ Pairwise correlations and Granger causality (e.g. Billio, Getmansky, Lo and Pellizon 2012) ◮ Partial correlations (e.g. Barigozzi and Brownlees 2014, Brownlees, Nualart, and Sun 2015) ◮ System-wide inference based on VARs (e.g. Diebold and Yilmaz 2014,2015) ◮ ...
Background Gaussian graphical model (undirected): ◮ Characterize the conditional independence structure of a set of random variables by an undirected graph G t = ( V , E t ) , with | V | = p nodes and E t the set of edges at time t Markov Property: G t implies that, for all 1 � i < j � N , e ij , t = 0 ⇐ ⇒ X i , t ⊥ X j , t | X V \{ i , j } , Network characterization: From G t = ( V , E t ) we can characterize the network as a sequence of p × p “weighted” adjacency matrices A t with entries � σ ij , t if i � = j are connected at time t a ij , t = 0 otherwise
Background 1 2 4 5 3 6 7 8
Background Graphical model (undirected): ◮ Characterize the conditional independence structure of a set of random variables by an undirected graph G t = ( V , E t ) , with | V | = p nodes and E t the set of edges at time t Markov Property: ◮ G t implies that, for all 1 � i < j � p , e ij , t = 0 ⇐ ⇒ X i , t ⊥ X j , t | X V \{ i , j } , Network characterization: From G t = ( V , E t ) we can characterize the network as a sequence of p × p “weighted” adjacency matrices A t with entries � σ ij , t if i � = j are connected at time t a ij , t = 0 otherwise
Background 1 2 4 5 3 6 7 8
Background Graphical model (undirected): ◮ Characterize the conditional independence structure of a set of random variables by an undirected graph G t = ( V , E t ) , with | V | = p nodes and E t the set of edges at time t Markov Property: ◮ G t implies that, for all 1 � i < j � p , e ij , t = 0 X i , t ⊥ X j , t | X V \{ i , j } , ⇐ ⇒ Network characterization: ◮ From G t = ( V , E t ) we can characterize the network as a sequence of p × p adjacency matrices A t with entries � 1 if i � = j are connected at time t a ij , t = 0 otherwise
Background 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 A t = 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
Our Model ◮ Seemingly Unrelated Regression with Graph restrictions y t = X ′ t β t + ε t , ε t ∼ N p ( 0 , Σ t ( G t )) , ◮ y t = ( y 1 t , . . . , y pt ) ′ : vector of returns in excess of the risk-free rate; ◮ X t = diag { ( x 1 t , . . . , x pt ) } : matrix of systematic risk factors; ◮ ε t = ( ε 1 t , . . . , ε pt ) ′ : normal random errors; ◮ β t = ( β 1 t , . . . , β pt ) ′ : time-varying exposures to systematic risks; ◮ Σ t ( G t ) ∈ M ( G t ) : residual covariance matrix ◮ G t : time-varying (state-dependent) graph;
Our Model ◮ Seemingly Unrelated Regression with Graph restrictions y t = X ′ t β t + ε t , ε t ∼ N p ( 0 , Σ t ( G t )) , ◮ y t = ( y 1 t , . . . , y pt ) ′ : vector of returns in excess of the risk-free rate; ◮ X t = diag { ( x 1 t , . . . , x pt ) } : matrix of systematic risk factors; ◮ ε t = ( ε 1 t , . . . , ε pt ) ′ : normal random errors; ◮ β t = ( β 1 t , . . . , β pt ) ′ : time-varying exposures to systematic risks; ◮ Σ t ( G t ) ∈ M ( G t ) : residual covariance matrix ◮ G t : time-varying (state-dependent) graph; ◮ Markov regime-switching dynamics K K K � � � β t = β k I { k } ( s t ) , Σ t = Σ k ( G k ) I { k } ( s t ) , G t = G k I { k } ( s t ) , k = 1 k = 1 k = 1 ◮ s t represents the state of system-wide connectedness, and evolves as a Markov chain process with transition probability P ( s t = i | s t − 1 = j ) = π ij , i , j = 1, . . . , K .
Regimes Identification ◮ Systemic risk is identified through a connectivity measure h ( G k ) ⊂ R ◮ Several connectivity measures are used in the literature: ◮ Average degree Does not discriminate the “quality” of linkages ◮ Closeness and betweenness Assume a predetermined path Weighted eigenvector centrality measure n x i , k = 1 a ij , k x j , k = 1 � � σ ij , k x j , k λ k λ k j = 1 j ∈ N ( i , k ) A firm with a small number of relevant connections may outrank one with a large number of mediocre linkages
Regimes Identification ◮ Systemic risk is identified through a connectivity measure h ( G k ) ⊂ R ◮ Several connectivity measures are used in the literature: ◮ Average degree Does not discriminate the “quality” of linkages ◮ Closeness and betweenness Assume a predetermined path Weighted eigenvector centrality measure n x i , k = 1 a ij , k x j , k = 1 � � σ ij , k x j , k λ k λ k j = 1 j ∈ N ( i , k ) A firm with a small number of relevant connections may outrank one with a large number of mediocre linkages
Regimes Identification ◮ Systemic risk is identified through a connectivity measure h ( G k ) ⊂ R ◮ Several connectivity measures are used in the literature: ◮ Average degree Does not discriminate the “quality” of linkages ◮ Closeness and betweenness Assume a predetermined path ◮ Weighted eigenvector centrality measure n x i , k = 1 a ij , k σ ij , k x j , k = 1 � � σ ij , k x j , k λ k λ k j = 1 j ∈ N ( i , k ) ◮ A firm with a small number of relevant connections may outrank one with a large number of mediocre linkages
Regimes Identification ◮ Systemic risk is identified through a connectivity measure h ( G k ) ⊂ R ◮ Several connectivity measures are used in the literature: ◮ Average degree Does not discriminate the “quality” of linkages ◮ Closeness and betweenness Assume a predetermined path ◮ Weighted eigenvector centrality measure n x i , k = 1 a ij , k σ ij , k x j , k = 1 � � σ ij , k x j , k λ k λ k j = 1 j ∈ N ( i , k ) ◮ A firm with a small number of relevant connections may outrank one with a large number of mediocre linkages
Regimes Identification ◮ “Regimes” of systemic risk are identified by imposing the constraint p p 1 x i ,1 < . . . < 1 � � h ( G 1 ) < . . . < h ( G K ) ⇒ x i , K , = p p i = 1 i = 1 ◮ Increasing (average) network connectivity corresponds to an increasing aggregate systemic risk.
Recommend
More recommend
