Modeling Contagion and Systemic Risk Daniele Bianchi 1 Monica Billio - - PowerPoint PPT Presentation

Modeling Contagion and Systemic Risk

Daniele Bianchi1 Monica Billio2 Roberto Casarin2 Massimo Guidolin3

1University of Warwick,

Warwick Business School

2University C´

a Foscari of Venice, Department of Economics

3Bocconi 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 Gt = (V , Et), with |V | = p nodes and Et the set of edges at time t Markov Property: Gt implies that, for all 1 i < j N, eij,t = 0 ⇐ ⇒ Xi,t⊥Xj,t|XV \{i,j}, Network characterization: From Gt = (V , Et) we can characterize the network as a sequence of p × p “weighted” adjacency matrices At with entries aij,t =

• σij,t

if i = j are connected at time t

• therwise

Background

1 2 3 4 5 6 7 8

Background

Graphical model (undirected):

◮ Characterize the conditional independence structure of a set of

random variables by an undirected graph Gt = (V , Et), with |V | = p nodes and Et the set of edges at time t Markov Property:

◮ Gt implies that, for all 1 i < j p,

eij,t = 0 ⇐ ⇒ Xi,t⊥Xj,t|XV \{i,j}, Network characterization: From Gt = (V , Et) we can characterize the network as a sequence of p × p “weighted” adjacency matrices At with entries aij,t =

• σij,t

if i = j are connected at time t

• therwise

Background

1 2 3 4 5 6 7 8

Background

Graphical model (undirected):

◮ Characterize the conditional independence structure of a set of

random variables by an undirected graph Gt = (V , Et), with |V | = p nodes and Et the set of edges at time t Markov Property:

◮ Gt implies that, for all 1 i < j p,

eij,t = 0 ⇐ ⇒ Xi,t⊥Xj,t|XV \{i,j}, Network characterization:

◮ From Gt = (V , Et) we can characterize the network as a sequence of

p × p adjacency matrices At with entries aij,t =

• 1

if i = j are connected at time t

• therwise

Background

At =             1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1            

Our Model

◮ Seemingly Unrelated Regression with Graph restrictions

yt = X ′

tβt + εt,

εt ∼ Np (0, Σt (Gt)) ,

◮ yt = (y1t, . . . , ypt)′: vector of returns in excess of the risk-free rate; ◮ Xt = diag{(x1t, . . . , xpt)}: matrix of systematic risk factors; ◮ εt = (ε1t, . . . , εpt)′: normal random errors; ◮ βt = (β1t, . . . , βpt)′: time-varying exposures to systematic risks; ◮ Σt (Gt) ∈ M(Gt): residual covariance matrix ◮ Gt: time-varying (state-dependent) graph;

Our Model

◮ Seemingly Unrelated Regression with Graph restrictions

yt = X ′

tβt + εt,

εt ∼ Np (0, Σt (Gt)) ,

◮ yt = (y1t, . . . , ypt)′: vector of returns in excess of the risk-free rate; ◮ Xt = diag{(x1t, . . . , xpt)}: matrix of systematic risk factors; ◮ εt = (ε1t, . . . , εpt)′: normal random errors; ◮ βt = (β1t, . . . , βpt)′: time-varying exposures to systematic risks; ◮ Σt (Gt) ∈ M(Gt): residual covariance matrix ◮ Gt: time-varying (state-dependent) graph;

◮ Markov regime-switching dynamics

βt =

K

• k=1

βkI{k}(st), Σt =

K

• k=1

Σk (Gk) I{k}(st), Gt =

K

• k=1

GkI{k}(st),

◮ st represents the state of system-wide connectedness, and evolves as

a Markov chain process with transition probability P(st = i|st−1 = j) = πij, i, j = 1, . . . , K.

Regimes Identification

◮ Systemic risk is identified through a connectivity measure h(Gk) ⊂ 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 xi,k= 1 λk

n

• j=1

aij,kxj,k = 1 λk

• j∈N(i,k)

σij,kxj,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(Gk) ⊂ 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 xi,k= 1 λk

n

• j=1

aij,kxj,k = 1 λk

• j∈N(i,k)

σij,kxj,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(Gk) ⊂ 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

xi,k = 1 λk

n

• j=1

aij,kσij,kxj,k = 1 λk

• j∈N(i,k)

σij,kxj,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(Gk) ⊂ 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

xi,k = 1 λk

n

• j=1

aij,kσij,kxj,k = 1 λk

• j∈N(i,k)

σij,kxj,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

h(G1) < . . . < h(GK) = ⇒ 1 p

p

• i=1

xi,1 < . . . < 1 p

p

• i=1

xi,K,

◮ Increasing (average) network connectivity corresponds to an

increasing aggregate systemic risk.

Bayesian Estimation: Priors

◮ Dirichlet conjugate prior for transition probs

(πk1, . . . , πkK) ∼ Dir (δk1, . . . , δkL) ,

◮ Bernoulli prior for the graph

p(Gk) ∝

• i,j

ψeij (1 − ψ)(1−eij) = ψ|Ek| (1 − ψ)T−|Ek| ,

◮ Conjugate Hyper-Inverse Wishart prior for covariances

Σk ∼ HIWGk (dk, Dk) ,

◮ Conjugate normal prior

βk ∼ Np (mk, Mk) ,

Bayesian Estimation: Gibbs Sampler

• 1. Draw s1:T: Forward-filtering Backward-Sampling procedure (see

Fr¨ uhwirth-Schnatter 1994, and Carter and Kohn 1994).

• 2. Draw Σk conditional on y1:T, s1:T, Gk and βk.
• 3. Draw βk conditional on y1:T, s1:T and Σk.
• 4. Draw πk conditional on y1:T, s1:T.
• 5. Draw Gk: Local-move Metropolis-Hastings based on the conditional

posterior pk(G|y1:T, s1:T). A candidate G

′ is sampled from a

proposal distribution q(G

′

k|Gk) and accepted with probability

α = min

• 1, pk(G

′

k|y1:T, s1:T)q(Gk|G

′

k)

pk(Gk|y1:T, s1:T)q(G

′

k|Gk)

• ,

This add/delete edge move proposal is accurate despite entails a substantial computational burden (see, e.g. Jones et al. 2005, Carvalho, West and Massam 2007).

Empirical Results: Data and Factor Models

Factor Pricing Models:

◮ CAPM, Three-factor Fama-French, I-CAPM

Data:

◮ We take 100 Blue Chip companies that compose the S&P100 Index ◮ Sample 05/10/1996-10/31/2014, 18 years of daily data (more than

400,000 firm-day observations)

◮ SMB, HML and Market factor from Kenneth French website ◮ Macro factors (I-CAPM): Aggregate dividend-yield, Default-spread

(Baa- long-term gov bond) and Term-spread (long-short term gov bonds). Data from FredII St. Louis, Ibbotson and CRSP.

Empirical Results: Systemic Risk Probability

1996 1998 2000 2002 2004 2006 2008 2010 2012 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Aggregate Systemic Risk Probability

Model−Implied Systemic Risk Probability and NBER Recessions − CAPM

1996 1998 2000 2002 2004 2006 2008 2010 2012 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Model−Implied Systemic Risk Probability and NBER Recessions − I−CAPM Aggregate Systemic Risk Probability

Empirical Results: Changes in Market Betas Low vs High Connectivity, I-CAPM

• Cons. Disc.
• Cons. Stap.

Energy Financials Health Care Industrials Materials Tech TelecUtils −0.6 −0.4 −0.2 0.2 0.4 0.6

Empirical Results: Network, Low Systemic Risk, I-CAPM

MMM DD EMR GE UTX T VZ ABT BAX BMY MKR PFE ALL BAC BRKB MO CL AXP BK USB WFC AIG AMGN BIIB FDX GILD APC APA CVX COP HAL OXY SLB AAPL C JPM TXN JNJ MDT BA GD HON LMT CVS WAG COF CAT FCX EXC FDX QCOM CSCO INTC ORCL KO MCD PEP SO PG CMCSA TWX DIS COST TGT WMT DVN XOM DOW EMC NSC RTN HPQ HD LOW IBM MSFT LLY MS NKE UNP SPG SBUX UNH

Empirical Results: Network, High Systemic Risk, I-CAPM

MMM DD EMR GE UTX T VZ ABT BAX BMY MKR PFE HAL FCX EXC MO CL AXP OXY USB SLB COP AMGN BIIB FDX GILD JPM C CVX AIG ALL BK WFC AAPL APA APC TXN JNJ MDT BA GD HON LMT CVS WAG DVN CAT XOM BRKB FDX QCOM CSCO INTC ORCL KO MCD PEP SO PG CMCSA TWX DIS COST TGT WMT COF BAC DOW EMC NSC RTN HPQ HD LOW IBM MSFT LLY MS NKE UNP SPG SBUX UNH

Empirical Results: Firm-Level Network Centrality, I-CAPM

2 4 6 8 10 12 14 16 18 20 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

C AIG BAC JPM BK WFC APC CSCO USB PG INTC UTX BMY OXY SLB CVX ALL JNJ DVN XOM JNJ COP CAT UTX APC XOM INTC WFC WMT TGT OXY BAC LLY ABT APA CSCO UNP BA HON AXP

Median Weighted Eigenvector Centrality Ranking

I−CAPM

High Systemic Risk Low Systemic Risk

Empirical Results: Network Centrality and Market Values

w i,k = γ0 + γ1xi,k + ǫi,k, for i = 1, ..., N,

Panel A: Weighted Eigenvector Centrality CAPM Fama-French I-CAPM Coeff t-stat R2 τ Coeff t-stat R2 τ Coeff t-stat R2 τ High 0.012 0.908 0.002 0.054 0.001 0.649 0.001 0.061 0.011 0.654 0.003 0.045 Low 0.015 1.072 0.016 0.084 0.041 0.871 0.006 0.085 0.032 1.032 0.004 0.055 Panel B: Standard Eigenvector Centrality CAPM Fama-French I-CAPM Coeff t-stat R2 τ Coeff t-stat R2 τ Coeff t-stat R2 τ High 0.099 0.861 0.011 0.062 0.003 0.123 0.003 0.059 0.021 0.782 0.012 0.048 Low 0.021 1.592 0.021 0.091 0.002 0.231 0.008 0.085 0.042 1.321 0.023 0.052

Empirical Results: Network Centrality and Value Losses

AM%Li,k = γ0 + γ1xi,k + ǫi,k, for i = 1, ..., N,

Panel A: Weighted Eigenvector Centrality CAPM Fama-French I-CAPM Coeff t-stat R2 τ Coeff t-stat R2 τ Coeff t-stat R2 τ High 0.551 2.061 0.091 0.211 0.671 2.194 0.101 0.205 0.891 1.981 0.112 0.198 Low 0.213 1.651 0.045 0.181 0.391 1.691 0.062 0.171 0.691 1.759 0.061 0.169 Panel B: Standard Eigenvector Centrality CAPM Fama-French I-CAPM Coeff t-stat R2 τ Coeff t-stat R2 τ Coeff t-stat R2 τ High 0.421 1.951 0.078 0.191 0.671 1.981 0.083 0.198 0.521 1.931 0.08 0.185 Low 0.172 1.761 0.055 0.188 0.401 1.641 0.051 0.185 0.301 1.761 0.054 0.160

Empirical Results: Network Connectivity and Macro-Financial Variables

I{st} = 1 if p (st = 1|y1:T) 0.5

• therwise

for t = 1, ..., T,

Probit Regression (Dep: Systemic Risk Indicator, Indep: Changes in Macro-Financial Variables) M1 M2 M3 M4 M5 M6 M7 M8 Term 0.656∗∗∗ Credit 2.321∗∗∗ Default 1.998∗∗∗ DY 0.306 PE

• 0.021

VIX

• 0.002

Distress 0.424∗∗∗ Mkt Unc 0.001 Pseudo R2 0.01 0.03 0.12 0.01 0.01 0.02 0.09 0.01

Empirical Results: Systemic Risk and Financial Distress

1996 1998 2000 2002 2004 2006 2008 2010 2012 0.2 0.4 0.6 0.8 1 1996 1998 2000 2002 2004 2006 2008 2010 2012 −2 2 4 6 8

St.Louis Fed Financial Distress Index

Conclusions

◮ Unifying framework to measure network connectivity for systemic

risk management purposes

◮ Dynamic, state-dependent, network structure ◮ Robust Bayesian inference

◮ Financial firms are central for systemic risk management, above and

beyond their market valuations

◮ Systemic and systematic risks should not be considered

independently

◮ Aggregate systemic risk partly correlates with the business cycle and

aggregate financial distress