Detecting Correlation StruDetecting Correlation StruDetecting Correlation Strugb kby Network Cby Network Cby Network Cy

Takashi Isogai (Bank of Japan JAIST) ETakashi Isogai (Bank of Japan, JAIST) Eg ( p , )

•Motivation: The correlation of returns is a key conce•Motivation: The correlation of returns is a key conceyapproach to detect the high dimensional correlation strapproach to detect the high dimensional correlation strpp gclustering for reducing portfolio risk (VaR Expected Shclustering for reducing portfolio risk (VaR, Expected Shg g p ( , p

•Method: GARCH filtering and a hierarchical network•Method: GARCH filtering and a hierarchical network deal with the fat tailness of return distribution and the hdeal with the fat-tailness of return distribution and the h

Fi di•Findings: Hierarchical groups are detected The cuFindings: Hierarchical groups are detected. The cusome sectors are grouped almost together The groupsome sectors are grouped almost together. The group

1 Correlation of fat tail stock returns1. Correlation of fat-tail stock returnsP tf li i k V i ( ) C i ( )Portfolio risk = Variance (r) + Covariance (r)Portfolio risk Variance (r) Covariance (r)

Stock return feature: P bl f li l tiStock return feature: F t t il d l tilit l t i

Problem of linear correlationFat tail and volatility clustering

2

3

Corr = 0y g

2

Corr = 0

Volatility of Nikkei by GARCH(1,1)

1

1y y ( , )

0

z1

0

z2

zr -1

1

ttt zr

-2

-

White noisezt ～IID(0,1) -3

-2White noise0 50 100 150 200 250 300

Index

0 50 100 150 200 250 300

Index

z is assumed asred: Normalbl Sk t

0 0

blue: Skew-t

-5 -5

dd 10 h k

10x1 -10

x2

add 10σ shockWhite noise

-1x

-15

x

-15

0-

Corr = really?

-20

-2Corr , really?

0 50 100 150 200 250 300 0 50 100 150 200 250 300

-25

Index Index

•Focus on covariance of z rather than r•Focus on covariance of zi,t rather than ri,t

•GARCH filtering to separate volatilities σi t andr

GARCH filtering to separate volatilities σi,t and i i d i ti ith l ti t i P ri.i.d innovations zi t with correlation matrix Pti,t t

Multivariate GARCH for stock returns rMultivariate GARCH for stock returns ri,t

for over 1400 Stocks listed at Tokyo Stock Exchange 1st section- for over 1400 Stocks, listed at Tokyo Stock Exchange, 1st sectionwith complete time series

0with complete time series

ttt zr 111 0

ttt ,1,1,1

0=> Zi t has correlation Pt

ttt zr 222 0 i,t t

ttt ,2,2,2

•Hard to estimate parameters due to high dimensionality•Hard to estimate parameters due to high dimensionality - Exclude cross effects (no volatility spillovers, diagonal specifications)( y p , g p )

•Vector GARCH(1 1) volatility equations:•Vector GARCH(1,1) volatility equations:22222 21,22,1

21,11,1

21,22,1

21,11,10,1

2,1 ttttt rr

22222

,,,,,,,,,,

21,22,2

21,11,2

21,22,2

21,11,20,2

2,2 ttttt rr 1,22,21,11,21,22,21,11,20,2,2 ttttt

•Simplify time varying P as constant over time P•Simplify time varying Pt as constant over time P- CCC-GARCH (Bollerslev[1990]) ccc: constant conditional correlationCCC GARCH (Bollerslev[1990]) ccc: constant conditional correlation

2 Clustering stock returns2. Clustering stock returnsg•Current 33 sector classification: the best grouping?Current 33 sector classification: the best grouping?

Heat map of correlation matrixHeat map of correlation matrixof stock returns: corr(r r )of stock returns: corr(rit, rjt)

•Find more data oriented grouping•Find more data-oriented grouping using correlation matrix P = {Pij}using correlation matrix P {Pij}

C l ti t i {P } ⇒ dj tf

• Correlation matrix {Pij} ⇒ adjacent Not so informative

{ ij} jmatrix {A } ⇒ Network clusteringmatrix {Aij} ⇒ Network clusteringj

- Divisive hierarchical- Divisive, hierarchicalM d l it i i ti + t l l t i 2- Modularity maximization + spectral clustering 2(Newman[2006])( [ ])

R l ti li it bl k d b- Resolution limit problem… ; work around by recursive clustering; simple but it works. g; p

Sorted in the sector order

ReferencesReferences[1] B ll l T [1990] “M d li th C h i Sh t N i l E h R t A M lti i[1] Bollerslev, T., [1990]. “Modeling the Coherence in Short-run Nominal Exchange Rates: A Multivaria[2] Lancichinetti, A. and Fortunato, S., [2011]. “Limits of modularity maximization in community detect[3] Newman, M.E.J., [2006]. “Modularity and community structure in networks.” Proceedings of the Na[ ] , , [ ] y y g

Views expressed here are those of the author and do not necessarilyViews expressed here are those of the author and do not necessarily

ucture of Stock Returnsucture of Stock Returnsucture of Stock ReturnslClusteringClusteringClusteringg

E mail: takashi isogai@boj or jpE‐mail: takashi.isogai@boj.or.jpg @ j jp

ept for portfolio risk management We propose a newept for portfolio risk management. We propose a new p p g p pructure of market wide stock returns by networkructure of market-wide stock returns by network yhortfall) with effective diversification of investmenthortfall) with effective diversification of investment. )

clustering with modularity maximization is combined toclustering with modularity maximization is combined to high dimensional correlation structurehigh dimensional correlation structure. g

rrent sector classification is partially effective; stocks inrrent sector classification is partially effective; stocks in p properties are identified by classification tree analysisp properties are identified by classification tree analysis.

3 Building hierarchical group structure3. Building hierarchical group structure

Macro view of clusteringMacro view of clusteringM d l it i i ti

gModularity maximization

1 ji CCww

AQ1

maxarg

jiji

ijC

CCW

AW

Q ,22

maxarg

jiC WWji 22 ,,

• wi, wj is the sum of weights of i, j gvertex (stock) i, jvertex (stock) i, j

• W is the sum of wi over all stocksW is the sum of wi over all stocks. • δ( ) is 1 if both stocks are in the

•Bx: intermediate layers• δ( ) is 1 if both stocks are in the

same class otherwise 0y

•Gx: stock groupssame class, otherwise 0.

G stoc g oups- Bx and Gx are all ‘communities’ - For modularity limit;For modularity limit;

What about modifying wij -> wij+rδWhat about modifying wij -> wij+rδijas in Q ? <currently r=0>as in QAFG? <currently r=0>

•Recursive clustering: B1: all stocksg

When indivisible by modularityB1: all stocks

When indivisible by modularity maximization, reset the subnet ,as the new whole group and g pstart again.

rr

St i l t i t h 4 i ld• Stop recursive clustering at h=4 yields 24 groups smaller than 33 sectors24 groups, smaller than 33 sectors.

- with many automobile companies Understanding group propertiesand related industries

Understanding group propertieshow stocks are divided into groups at each level; which factor- how stocks are divided into groups at each level; which factor l k l i d t i i th bdi i i t lplays a key role in determining the subdivision at every layer.

- shared properties of a group reflect investors’ views of those p p g pstocks.stocks.

la er3layer3 The most relevantrelevant propertyproperty

layer4layer4Classification tree at every hierarchy

l 2

Classification tree at every hierarchyModel: Group = f(continuous and categoricallayer2 Model: Group = f(continuous and categorical

properties of stocks)layer1 properties of stocks)$• TOPIX beta, size, PBR (Fama and French), $/¥ rate beta,

l toverseas sales, sectors, …

C diti l ti l t d l f t k l ifi tiConditional sequential tree model for stock classificatione.g., B1 G27: f(L1 Tree * L2 Tree * L3 tree * L4 tree)e g , G ( _ ee _ ee 3_t ee _t ee)

it f t ID f l li t d t k t k ith li it d i d t t- merits: forecast group ID of newly listed stocks, stocks with limited price data, etc.

4 F th t i4. Further topicsBipartite graph 24 groups vs 33 sectors pBipartite graph – 24 groups vs. 33 sectors

•Quantify risk (VaR ES)Quantify risk (VaR, ES) d ti ff treduction effects

b d tf li i l ti-by random portfolio simulation

Multivariate GARCH with•Multivariate GARCH with non-diagonal specificationnon-diagonal specification

- Market-wide analysis by reduced y ysize of GARCH models33 sectors24 groups size of GARCH models,33 sectors24 groups

- volatility spillovers and dynamic •Group integrities are significantly different y p yconditional correlation (DCC

Group integrities are significantly different depending on sectors conditional correlation (DCC

GARCH)depending on sectors.

i hi di ifi d h GARCH)…- e.g., service, machinery,… are diversified, whereas t t ti b ki t t dtransportation, banking, energy… are concentrated.

t G li d ARCH M d l ” R i f E i d St ti ti V l 72 N 3 498 505ate Generalized ARCH Model.” Review of Economics and Statistics, Vol.72, No.3, pp.498–505.ion.” Physical Review E, Vol.84, No.6, p.066122.ational Academy of Sciences of the United States of America, Vol.103, No.23, pp.8577–82.y , , , pp

y reflect the official views of the Bank of Japany reflect the official views of the Bank of Japan.