Network Connectivity and Systematic Risk
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1 Network Connectivity and Systematic Risk Monica Billio 1 Massimiliano Caporin 2 Roberto Panzica 3 Loriana Pelizzon 1,3 1 University Ca Foscari Venezia (Italy) 2 University of Padova (Italy) 3 Goethe University Frankfurt (Germany) March, 2015 Billio, Caporin, Panzica and Pelizzon N&S March, / 47
2 Research questions Introduction There is a general agreement on the traditional decomposition of an asset (portfolio) total risk into systematic and idiosyncratic components following the CAPM model Systematic risks comes from the dependence of returns on common factors Idiosyncratic risks are asset-specific But... There is also a recent consensus on the existence of network interconnections/systemic risks that affect asset pricing Billio, Caporin, Panzica and Pelizzon N&S March, / 47
3 Research questions (Challenging) Research questions What is the impact of network interconnections on the asset return loading to common factors? How does network interconncetions affect asset pricing in a factor model? Does network interconnections endanger the power of diversification? Billio, Caporin, Panzica and Pelizzon N&S March, / 47
4 Literature review - Part I Literature review An increasing literature investigates the role of interconnections between different firms and sectors, functioning as a potential propagation mechanism of idiosyncratic shocks throughout the economy. Canonical idea: microeconomic shocks average out and thus, only have negligible aggregate effects (Lucas, 1977, among others). Similarly, these shocks have little impact on asset prices. However: Acemoglou et al. (2011) use network structure to show the possibility that aggregate fluctuations may originate from microeconomic shocks to firms. Such a possibility is discarded in standard macroeconomics models due to a diversification argument. Shock propagation in static networks: Horvath (1998, 2000), Dupor (1999), Shea (2002), and Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2011). Billio, Caporin, Panzica and Pelizzon N&S March, / 47
5 Literature review - Part I Measuring systemic links Role of idiosyncratic risk in asset pricing: The CAPM predicts that only systematic risk is priced and expected excess returns satisfy two-fund separation. This prediction is contradicted by: Ozsoylev and Walden (2011): study a static model of asset price formation in large information networks, mainly centred on the relation between price volatility and network connectedness. Ang, Hodrick, Xing, and Zhang (2006): who show that idiosyncratic volatility risk is priced in the cross-section of expected stock returns, a regularity which is not subsumed by size, book-to-market, momentum, or liquidity effects. Buraschi and Porchia (2014) and Branger et al. (2014): dynamic network connectivity has implications on diversification and asset pricing Billio, Caporin, Panzica and Pelizzon N&S March, / 47
6 Literature review - Part I Measuring systematic risk Traditional models consider several risk factors, one of them being the market Expected returns depend on risk premiums of common factors and betas (loadings) to the factors Standard decomposition of assets (and portfolios) total risk into a systematic component (driven by the risk factors) and an idiosyncratic component (driven by asset-specific shocks) Diversification acts on the idiosyncratic part and aims at reducing its impact in a portfolio Billio, Caporin, Panzica and Pelizzon N&S March, / 47
7 Literature review - Part I Pricing and diversification in multifactor models General multi-factor model for a k dimensional vector of risky assets (m risk factors) R t = E [R t ] + BF t + ɛ t (1) Expected returns depend on the factor risk premiums Λ E [R t ] = r f + BΛ (2) Standard total risk decomposition VAR [R t ] = BVAR [F t ] B + VAR [ɛ t ] (3) Σ R = BΣ F B + Ω (4) Billio, Caporin, Panzica and Pelizzon N&S March, / 47
8 Literature review - Part I Pricing and diversification in multifactor models Portfolio risk decomposition (ω being a k dimensional vector of portfolio weights) VAR [ω R t ] = ω BVAR [F t ] B ω + ω VAR [ɛ t ] ω (5) ω Σ R ω = ω BΣ F B ω + ω Ωω (6) Diversification benefit lim k ω Ωω = ν (7) Special case: uncorrelated idiosyncratic shocks with average variance σ 2 lim k ω Ωω = 1 k σ2 = 0 (8) Billio, Caporin, Panzica and Pelizzon N&S March, / 47
9 Systemic and systematic risks Adding network connections in multifactor models Network connections represent contemporaneous relations across assets that co-exists with the dependence on common risk factors Network connections (A) are contemporaneous relations across endogenous variables A (R t E [R t ]) = BF t + ɛ t (9) The simultaneous equation system above is not identified unless we impose some restriction; k number of assets is much larger than m the number of factors Assumption 1: the idiosyncratic shocks are uncorrelated, that is Ω is a diagonal matrix Assumption already taken into account in multi-factor models Billio, Caporin, Panzica and Pelizzon N&S March, / 47
10 Systemic and systematic risks Adding Network connections in multifactor models Assumption 2: In networks, nodes are connected (in general) to a subset of the network total number of nodes, where connections represent links across assets; shocks are transmitted through the network Intuition: connected assets are neighbours assets Spatial Econometrics tools introduced to capture the spatial dependence across subjects Billio, Caporin, Panzica and Pelizzon N&S March, / 47
11 Systemic and systematic risks Adding Network connections in multifactor models Networks can be used to define spatial matrices An example of a spatial matrix (10) Spatial matrices can be row normalized Spatial matrices might be generalized in such a way values are associated with the intensity of the link across assets Billio, Caporin, Panzica and Pelizzon N&S March, / 47
12 Systemic and systematic risks Adding Network connections in multifactor models An example of a network Networks provide information on the existence of links across assets and on the strength/intensity of the link Networks can be represented as spatial matrices Billio, Caporin, Panzica and Pelizzon N&S March, / 47
13 Systemic and systematic risks A general framework with Network connections and systematic risks By means of spatial matrices we can impose a structure on matrix A and rewrite the simultaneous equation system A = I ρw (11) (R t E [R t ]) = ρw (R t E [R t ]) + BF t + ɛ t (12) The coefficient ρ represents the impact coming from neighbours and by now we assume it is a scalar This simultaneous equation system corresponds to a Spatial Auto Regression Panel model where the covariates (risk factors) are common across all subjects (at least in a simplified representation) Billio, Caporin, Panzica and Pelizzon N&S March, / 47
14 Systemic and systematic risks Literature review - Part II Few works with similar intuition Fernandez (2011) adds a spatial term to the CAPM for Value-at-Risk estimation and recovers the W from correlations Fernandez-Aviles et al. (2012) show that asset proximity should be measured by financial quantities and not geographical locations Arnold et al. (2013) combine different financial closeness measures within a risk management framework Asgharian et al. (2013) use countries economic relations to analyse stock markets co-movements Wied (2013) risk management and parameter stability testing in spatial models for stock market data Keiler and Eder (2013) attach a systemic risk interpretation to the ρ with W coming from correlations Blasques et al. (2014) make the ρ time-varying with W coming from bank s cross-border exposures Billio, Caporin, Panzica and Pelizzon N&S March, / 47
15 Systemic and systematic risks Our contributions Consider a multifactor model and focus on both pricing and diversification effects Measure links starting from a network that capture causality relations Show that network connections act as an inflating factor on the asset loadings to the common factors Show that network elements impact on both the systematic and idiosyncratic risk components Generalize the model in two directions First by adding heterogeneity to the asset reaction to the network links A = I RW (13) R = diag (ρ 1, ρ 2,..., ρ N ) (14) Billio, Caporin, Panzica and Pelizzon N&S March, / 47
16 Systemic and systematic risks Our contributions Then, by allowing for a time-variation in the W matrix A A t = I RW t (15) In this way the matrices W t capture the dynamic in the links across assets (the evolution of the newtork), while the spatial coefficients included in R represent the impact on each asset of the neighbours, and are assumed to be time-independent The W t matrices are also assumed to be known at time t and thus we condition the model estimation and the analysis to their availability We assume the evolution of W t is smooth or acts at a time scale much lower than that affecting the evolution of asset returns (i.e. for monthly returns the W change with, say, a yearly frequency) Billio, Caporin, Panzica and Pelizzon N&S March, / 47
17 Systemic and systematic risks Network links impact on model structure From the simultaneous equation system A t (R t E [R t ]) = BF t + ɛ t (16) The parameters in B are the structural betas; recast the model into the reduced form R t = E [R t ] + (A 1 t B)F t + A 1 t ɛ t (17) = E [R t ] + B t F t + η t (18) Two relevant consequences: the presence of Network links lead to a reduced form model (standard linear factor model) with dynamic betas and whose idiosyncratic shocks are contemporaneously correlated due to the interaction between the W t matrices and the spatial coefficients in R mildly heteroskedastic due to the smooth evolution of W t Billio, Caporin, Panzica and Pelizzon N&S March, / 47
18 Systemic and systematic risks Network links impact on expected returns For simplicity focus on the scalar case ρ with time invariant W (I ρw ) (R t E [R t ]) = BF t + ɛ t (19) It holds that (I ρw ) 1 = I + ρw + ρ 2 W 2 + ρ 3 W 3... (20) Therefore the model corresponds to R t = E [R t ] + BF t + ρ j W j BF t + η t (21) j=1 Billio, Caporin, Panzica and Pelizzon N&S March, / 47
19 Systemic and systematic risks Network links impact on expected returns The expected returns equal E [R t ] = r f + BΛ + ρ j W j BΛ (22) Therefore, as ρ > 0 and elements in W are positive, the presence of systemic links inflates the loading to the factors with an impact on the asset expected returns In addition, if W (or even ρ) change over time we have a dynamic in the expected returns driven by the network links Expected returns increase as a consequence to an increase in ρ or a change in W with subsequent effects on prices The use of asset-specific ρ allows for an heterogeneous reaction of the assets to changes in W j=1 Billio, Caporin, Panzica and Pelizzon N&S March, / 47
20 Systemic and systematic risks Network links impact on risk A risk decomposition applies also in our case, starting from the reduced form model VAR [R t ] = A 1 BVAR [F t ] B ( A 1) + A 1 VAR [ɛ t ] ( A 1) Σ R = A 1 BΣ F B ( A 1) + A 1 Ω ( A 1) (23) (24) In this case, the systematic and idiosyncratic contributions to total risk are also influenced by the existence of Network connections Obviously, if network links are absent, that is ρ = 0 we get back to the traditional risk decomposition Billio, Caporin, Panzica and Pelizzon N&S March, / 47
21 Systemic and systematic risks Network links impact on risk We can play around this decomposition to recover a more insightful one Σ R = A 1 BΣ F B ( A 1) + A 1 Ω ( A 1) (25) = BΣ F B + AΩA (26) = BΣ B F + AΩA ± BΣ F B ± Ω (27) = BΣ F B }{{} + }{{} Ω + ( BΣ F B BΣ F B ) + (AΩA Ω) (28) }{{}}{{} i ii iii iv We have thus four terms in the risk decomposition i The systematic component ii The idiosyncratic component iii The network impact on the systematic component iv The network impact on the idiosyncratic component Billio, Caporin, Panzica and Pelizzon N&S March, / 47
22 Systemic and systematic risks Network links impact on risk This has effects on the diversification benefits which can be analytically derived in a special case Consider K uncorrelated idiosyncratic shocks with average variance σ 2 and a W matrix where all assets are linked to each other, we have lim K ω AΩ η A ω = K + ρ 2 (K + ρ) 2 (ρ 1) 2 σ2 = 0 (29) Diversification benefits still present but the decrease of the idiosyncratic component of the portfolio variance is much slower Billio, Caporin, Panzica and Pelizzon N&S March, / 47
23 Systemic and systematic risks Network links impact on risk Portfolio idiosyncratic risk across different ρ levels and increasing number of assets. The case ρ = 0 corresponds to the absence of spatial links and is the standard result for diversification benefits. Billio, Caporin, Panzica and Pelizzon N&S March, / 47
24 Systemic and systematic risks Simulated examples Question: is the proposed framework providing insightful features? Simulations: 100 assets, common factor volatility 15% yearly, betas to the common factor U (0.8, 1.2), idiosyncratic volatilities U (20%, 40%) Spatial matrices W : market matrix, all asset cross-dependent, 1 k 1 k I k; two-neighbours, tri-diagonal matrix; random, W elements follows w i Bern (0.3) Evaluate the effects of different ρ values Billio, Caporin, Panzica and Pelizzon N&S March, / 47
25 Systemic and systematic risks Simulated examples Beta values across assets: structural betas (in blue) and augmented betas (in red) across different spatial matrices W. Billio, Caporin, Panzica and Pelizzon N&S March, / 47
26 Systemic and systematic risks Simulated examples Beta values across assets: structural betas (in blue) and augmented betas (in red) across different values for ρ with the random matrix W. Billio, Caporin, Panzica and Pelizzon N&S March, / 47
27 Systemic and systematic risks Simulated examples Relative variance decomposition without network links (ρ = 0) across the 100 simulated assets Billio, Caporin, Panzica and Pelizzon N&S March, / 47
28 Systemic and systematic risks Simulated examples Relative variance decomposition with network links (ρ = 0.25) and market matrix W across the 100 simulated assets Billio, Caporin, Panzica and Pelizzon N&S March, / 47
29 Systemic and systematic risks Simulated examples Relative variance decomposition with systemic links (ρ = 0.5) and random matrix W across the 100 simulated assets Billio, Caporin, Panzica and Pelizzon N&S March, / 47
30 Systemic and systematic risks Simulated examples 1/N portfolio variance decomposition across different values of ρ with the same random matrix W (relative decomposition) Billio, Caporin, Panzica and Pelizzon N&S March, / 47
31 Systemic and systematic risks Simulated examples 1/N portfolio idiosyncratic component variance - relative contraction with increasing N and across ρ values Billio, Caporin, Panzica and Pelizzon N&S March, / 47
32 Systemic and systematic risks Model estimation We do not impose a method for estimating the network links, those can come from any approach or measurement framework; we do impose that the W is known before estimating the other model parameters In this work we estimate W by means of Granger causality, see Billio et al. (2012) Estimation of the factor loadings, spatial coefficients, and idiosyncratic shocks by Full Information Maximum Likelihood; computational advantages come from the possibility of concentrating out the factor loadings (and the error variances) If needed impose technical assumptions guaranteeing the non singularity of A t and the identification of the parameters in R Billio, Caporin, Panzica and Pelizzon N&S March, / 47
33 Empirical example Data description We consider the industrial sector indexes available from the Kenneth French website over the range , thus allowing for pre-crisis ( ) and crisis + post-crisis ( ) samples of the same length (7 years) We take the decomposition of the market into 48 economic sectors/industries We recover the common factors from the same source: the market index (composite index of NYSE-AMEX-NASDAQ); the three additional factors of Fama-French and Carhart, SMB, HML and MOM We make use of daily data to estimate the networks and monthly data to estimate the full model Billio, Caporin, Panzica and Pelizzon N&S March, / 47
34 Empirical example Estimating networks We estimate networks, monitoring connections across economic sectors, by means of Granger Causality tests on daily data depurated by Garch(1,1) Estimation of the networks is based on a daily date over yearly samples We thus have a sequence of matrices W t with time index evolving over years We stress we thus have a reduced form model with heteroskedastic innovations Billio, Caporin, Panzica and Pelizzon N&S March, / 47
35 Empirical example Estimating networks Granger causality model used to define the Spatial Matrix Consider a set M of series on which we want to identify systemic links Focus on two series of interest X t and Y t taken from M X t = m j=1 a jx t j + m j=1 b jy t j + m j=1 c jz t j + m j=1 d jf t j + ɛ i Y t = m j=1 e jx t j + m j=1 r jy t j + m j=1 g jz t j + m j=1 h jf t j + η i In addition to the causing series the equations include the effect of a common factor F t and two additional background series included in Z t taken from M Billio, Caporin, Panzica and Pelizzon N&S March, / 47
36 Empirical example Estimating networks For each pair of causing-caused series X t and Y t we run a Granger Causality test for each of possible pair of background series Z t Y G X if b j is different from 0 X G Y if e j is different from 0 If both b j and e j are different from 0, feedback relation We pick then the worst case, the higher p-value on causality tests, and consider it as our reference quantity obtaining a P-Value Matrix Billio, Caporin, Panzica and Pelizzon N&S March, / 47
37 Empirical example Estimating networks The link between the Granger Causality and the Network analysis is the Adjacency Matrix A In Network analysis, assuming that we have N series, the adjacency matrix is N N with values determining the existence of links and their strength The P-value matrix is recast into an Adjacency matrix In out case, if the series i causes, in the sense of Granger, the series j, at a given confidence level, then in A, the element a ij = 1 otherwise a ij = 0 Starting from adjacency matrix we are able to compute Network measures, but Adjacency matrices are also Spatial matrices W Billio, Caporin, Panzica and Pelizzon N&S March, / 47
38 Empirical example Estimated networks Estimated network for the year 2002 Billio, Caporin, Panzica and Pelizzon N&S March, / 47
39 Empirical example Estimated networks Estimated network for the year 2008 Billio, Caporin, Panzica and Pelizzon N&S March, / 47
40 Empirical example Model output Estimated ρ for selected sectors and over the two samples Billio, Caporin, Panzica and Pelizzon N&S March, / 47
41 Empirical example Model output The estimated reduced form beta parameters B t = (I RW t ) 1 B are time-varying given that the W matrix is time-varying To summarize findings, we take averages of the reduced form betas over years and recover the average impact of network links on the reduced form betas Focus mostly on market impact Billio, Caporin, Panzica and Pelizzon N&S March, / 47
42 Empirical example Model output Estimated β for selected sectors and over different periods Billio, Caporin, Panzica and Pelizzon N&S March, / 47
43 Empirical example Model output Estimated Sector volatility decomposition Billio, Caporin, Panzica and Pelizzon N&S March, / 47
44 Empirical example Model output Estimated Sector volatility decomposition Billio, Caporin, Panzica and Pelizzon N&S March, / 47
45 Empirical example Model output Estimated equally weighted portfolio variance Billio, Caporin, Panzica and Pelizzon N&S March, / 47
46 Empirical example Model output Main findings Network impact coefficients ρ i increase during the crisis period leading to a larger impact of connections on stock returns Total exposure to factor shocks increases for most sectors Billio, Caporin, Panzica and Pelizzon N&S March, / 47
47 Empirical example Future developments Deeper evaluation of risk factor exposure and pricing implications; analysis of diversification impact of network exposure Time-variation in the ρ coefficients Different design of the W matrices taking into account the strength of the relation across subjects Comparison between US and Europe Billio, Caporin, Panzica and Pelizzon N&S March, / 47
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