systemic and systematic risk - monica billio, massimiliano caporin, roberto panzica, loriana...

Systemic and

systematic risks

SYstemic Risk TOmography:

Signals, Measurements, Transmission Channels, and Policy Interventions

Monica Billio, University Ca' Foscari Venezia (Italy) Massimiliano Caporin, University of Padova (Italy) Roberto Panzica, Goethe University Frankfurt (Germany) Loriana Pelizzon, University Ca' Foscari Venezia (Italy) and Goethe University Frankfurt (Germany) SYRTO Code Workshop Workshop on Systemic Risk Policy Issues for SYRTO July, 2 2014 - Frankfurt (Bundesbank-ECB-ESRB)

Research questions

Research questions

There is a general agreement on the traditional decomposition of an asset(portfolio) total risk into systematic and idiosyncratic components followingthe 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 systemic risks

Billio, Caporin, Panzica and Pelizon S&S July, 2014 2 / 37

Research questions

Research questions

Systemic risk definition: any set of circumstances that threatens the stabilityof, or public confidence in, the financial system

Systemic risk is a function of a system

Systemic risk arises endogenously from a system

Systemic risk is a function of connections between and the structure offinancial institutions or, more generally, between the companies and/oreconomic sectors


Research questions

Research questions

Do systemic and systematic risks overlap (market risk is bothsystematic and systemic), or are we able to separate those two risksources?

How idiosyncratic risks generates systemic risk and reduces the powerof diversification?


Literature review

Measuring systemic links

An increasing literature in economics investigates the role of interconnectionsbetween different firms and sectors, functioning as a potential propagationmechanism of idiosyncratic shocks throughout the economy.

Canonical idea: Lucas (1977), among others, that states that suchmicroeconomic shocks would average out and thus, would only havenegligible aggregate effects. Similarly, these shocks would have little impacton asset prices.

However:

Acemoglou et al. (2011) use network structure to show the possibility thataggregate 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).


Literature review

Measuring systemic links

Role of idiosyncratic risk in asset pricing: The CAPM predicts that onlysystematic risk is priced and expected excess returns satisfy two-fundseparation. This prediction is contradicted by:

Ozsoylev and Walden (2009) study a static model of asset price formation inlarge information networks, mainly centred on the relation between pricevolatility and network connectedness.

Ang, Hodrick, Xing, and Zhang (2006), who show that idiosyncratic volatilityrisk is priced in the cross-section of expected stock returns, a regularity whichis not subsumed by size, book-to-market, momentum, or liquidity effects.

Buraschi and Porchia (2014): dynamic network connectivity has implicationson diversification and asset pricing


Literature review

Measuring systematic risk

Traditional models consider several risk factors, one of them being the market

Standard decomposition of assets (and portfolios) total risk into a systematiccomponent (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 impactin a portfolio

Recent literature on diversification and systemic risk:

Ozsoylev and Walden (2009), Ang, Hodrick, Xing, and Zhang (2006),Buraschi and Porchia (2014), Wagner (2011) and Battiston and Tasca(2013)

They suggest it is not any more true!


Literature review


General multi-factor model for a k−dimensional vector of risky assets (m riskfactors) [avoiding for simplicity the introduction of the risk-free and assumingfactors have zero-mean]

Rt = α + BFt + εt (1)

Standard total risk decomposition

VAR [Rt ] = BVAR [Ft ]B′ + VAR [εt ] (2)

ΣR = BΣFB′ + Ω (3)


Literature review


Portfolio risk decomposition (ω being a k−dimensional vector of portfolioweights)

VAR [ω′Rt ] = ω′BVAR [Ft ]B′ω + ω′VAR [εt ]ω (4)

ω′ΣRω = ω′BΣFB′ω + ω′Ωω (5)

Diversification benefitlimk→∞ω

′Ωω = ν (6)

where ν is a small (possibly zero?) quantity


Systemic and systematic risks

A general framework with systemic and systematic risks

Systemic links represents relations across assets that co-exists with thedependence on common risk factors

Systemic links are relations across endogenous variables

ARt = BFt + εt (7)

The simultaneous equation system above is not identified unless we imposesome restriction; k number of assets is much larger than m the number offactors

Assumption 1: the idiosyncratic shocks are uncorrelated, that is Ω is adiagonal matrix

Assumption already taken into account in multi-factor models




Assumption 2: the matrix of contemporaneous relations has a structuredriven by systemic links

Systemic links are summarized by networks;

In networks, nodes are connected (in general) to a subset of the network totalnumber of nodes;

The connections represents links across assets,

Shocks on one asset are transmitted to the connected ones

Intuition: connected assets are neighbours assets ⇒ Spatial Econometricstools introduced to capture the spatial dependence across subjects

Networks can be used to define spatial matrices




An example of a spatial matrix0 1 0 0 1 01 0 1 0 0 11 0 0 1 0 10 1 1 0 0 01 0 0 0 0 10 1 0 1 0 0

(8)

Spatial matrices can be row normalized

Spatial matrices might be generalized in such a way values are associatedwith the intensity of the link across assets




An example of a network

Networks provide information on the existence of links across assets and onthe strength/intensity of the link

Networks can be represented as spatial matrices




By means of spatial matrices we can impose a structure on matrix A andrewrite the simultaneous equation system

A = I − ρW (9)

Rt = ρWRt + BFt + εt (10)

The coefficient ρ represents the impact coming from neighbours and by nowwe assume it is a scalar

This simultaneous equation system corresponds to a Spatial Auto RegressionPanel model where the covariates (risk factors) are common across allsubjects (at least in a simplified representation)




The model is similar to Spatial Econometric approaches, in particular theSpatial Auto Regression, see the books by Anselin (1988) and LeSage andKelley (2009), among others, and therein cited references

Being (potentially) applied on a large cross-section with repeatedobservations over time, the model is also related to the Spatial Panelliterature, see Elhorst (2003) and Anselin (2006)

The model might be also generalized to the spatio-temporal approacheswhere dynamic behaviours are introduced in the model structure




If the reference (the true) model becomes

ARt = BFt + εt (11)

The parameters in B are the structural betas while standard linear factormodels would estimate reduced-form betas from the system

Rt = (A−1B)Ft + A−1εt (12)

= BFt + ηt (13)

Therefore, the model we propose might capture some of the observedcorrelation across idiosyncratic residuals observed in reduced form linearfactor models




A risk decomposition applies also in our case, starting from the reduced formmodel

VAR [Rt ] = A−1BVAR [Ft ]B′ (A−1)′ + A−1VAR [εt ]

(A−1

)′(14)

ΣR = A−1BΣFB′ (A−1)′ + A−1Ω

(A−1

)′(15)

In this case, the systematic and idiosyncratic contributions to total risk arealso influenced by the existence of systemic links

Obviously, if systemic links are absent, that is ρ = 0 we get back to thetraditional risk decomposition




We can play around this decomposition to recover a more insightful one

ΣR = A−1BΣFB′ (A−1)′ + A−1Ω

(A−1

)′(16)

= BΣF B′ +AΩA′ (17)

= BΣF B′ +AΩA′ ± BΣFB

′ ± Ω (18)

= BΣFB′︸︷︷︸

i

+ Ω︸︷︷︸ii

+(BΣF B

′ − BΣFB′)︸︷︷︸

iii

+ (AΩA′ − Ω)︸︷︷︸iv

(19)

We have thus four terms in the risk decomposition

i The structural systematic componentii The structural idiosyncratic componentiii The systemic impact on the structural systematic componentiv The systemic impact on the idiosyncratic component




Question: is the proposed framework providing insightful features?

Simulations: 100 assets, common factor volatility 15% yearly, betas to thecommon factor U ∼ (0.8, 1.2), idiosyncratic volatilities U ∼ (20%, 40%)

Spatial matrices W : ”market matrix”, all asset cross-dependent, 1k1′k − Ik ;”two-neighbours”, tri-diagonal matrix; ”random”, W elements followswi ∼ Bern (0.3)

Evaluate the effects of different ρ values




Relative variance decomposition without systemic links (ρ = 0) across the100 simulated assets




Relative variance decomposition with systemic links (ρ = 0.25) and ”market”matrix W across the 100 simulated assets




Relative variance decomposition with systemic links (ρ = 0.5) and ”random”matrix W across the 100 simulated assets




1/N portfolio variance decomposition across different values of ρ with thesame ”random” matrix W (relative decomposition)




1/N portfolio idiosyncratic component variance - relative contraction withincreasing N and across ρ values




Relevant generalizations of the proposed framework

Evaluate alternative designs for common factorsSystemic links are time-dependent: deal with dynamics into parametermatricesSystemic impacts might be asset specific: make the ρ asset-specific

A more general form for the matrix A

A⇒ At = I −RWt (20)

R = diag (ρ1, ρ2, . . . , ρN) (21)




Estimation of the spatial (or weight) matrices might be performed on arelatively low frequency or when extraordinary events take place

Estimates of the matrices W might be valid for sub-samples, or, say,before/during/after crises

The matrices Wt capture the dynamic in the links across assets, while thespatial coefficients R represents the impact on each asset of the neighbors,and are assumed to be time-independent

When the model is applied to high frequency data (weekly/daily)heteroskedasticity in the idiosyncratic component (and factors) must betaken into account

Note that the time-change in the Wt induces a form of heteroskedasticty onthe reduced-form model even when the underlying data have a low frequency(monthly/quarterly)



Estimating systemic links

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 Xt and Yt taken from M

Xt =∑m

j=1 ajXt−j +∑m

j=1 bjYt−j +∑m

j=1 cjZt−j +∑m

j=1 djFt−j + εi

Yt =∑m

j=1 ejXt−j +∑m

j=1 rjYt−j +∑m

j=1 gjZt−j +∑m

j=1 hjFt−j + ηi

In addition to the causing series the equations include the effect of a commonfactor Ft and two additional background series included in Zt taken from M




For each pair of causing-caused series Xt and Yt we run a Granger Causalitytest for each of possible pair of background series Zt

Y ⇒G X if bj is different from 0

X ⇒G Y if ej is different from 0

If both bj and ej are different from 0, feedback relation

We pick then the worst case, the higher p-value on causality tests, andconsider it as our reference quantity obtaining a P-Value Matrix

Causality test makes use of autocorrelation and heteroskedasticity consistentstandard errors

Alternatively, series might be also pre-filtered to remove GARCH-type effects




The link between the Granger Causality and the Network analysis is theAdjacency Matrix A

In Network analysis, assuming that we have N series, the adjacency matrix isN × 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 agiven confidence level, then in A, the element aij = 1 otherwise aij = 0

Starting from adjacency matrix we are able to compute Network measures,but Adjacency matrices are also Spatial matrices W


Empirical example

Data description

We consider the industrial sector indices available from the Kenneth Frenchwebsite

We take the decomposition of the market into 48 economic sectors/industries

The market proxy (the common factor) is also recovered from the samesource (composite index of NYSE-AMEX-NASDAQ) and is used as theunique common factor

Data are considered at the daily and monthly frequencies


Empirical example

Systemic links

Networks, monitoring connections across economic sectors, have beenestimated by means of Granger Causality tests on daily data depurated byGarch(1,1)

Estimation of the networks is based on a daily date over yearly samples

We thus have a sequence of matrices Wt with time index evolving over years

On spatial returns models estimated on frequencies higher than the year thisinduces a time variation in the model coefficients and mild heteroskedasticityover reduced-form innovations


Empirical example

Systemic links

Estimated network for the year 2002


Empirical example

Systemic links

Estimated network for the year 2008


Empirical example

Spatial Model output

The model has been estimated using monthly data and for different samples

A sample starting in 1992 and ending at December 2006 thus before thesubprime and EU sovereign crises

A sample starting in January 2007 and ending at December 2013 includingthus both crises


Empirical example

Model output

Estimated ρ for selected sectors and over different periods


Empirical example

Model output

Estimated β for selected sectors and over different periods


Empirical example

Future developments

Effects on diversification of systemic links

Time-variation in the ρ coefficients

Shorter term estimation of the spatial matrices W

Generalize the approach to induce locally explosive patterns in (diversified)portfolio risk due to the presence of systemic links


This project has received funding from the European Union’s

Seventh Framework Programme for research, technological

development and demonstration under grant agreement n° 320270

www.syrtoproject.eu

This document reflects only the author’s views.

The European Union is not liable for any use that may be made of the information contained therein.

systemic and systematic risk - monica billio, massimiliano caporin, roberto panzica, loriana...

Economy & Finance