Detecting Correlation Stru Detecting Correlation Stru Detecting Correlation Stru b k by Network C by Network C by Network C Takashi Isogai (Bank of Japan JAIST) E Takashi Isogai (Bank of Japan, JAIST) E • Motivation: The correlation of returns is a key conce • Motivation : The correlation of returns is a key conce approach to detect the high dimensional correlation str approach to detect the high dimensional correlation str clustering for reducing portfolio risk (VaR Expected Sh clustering for reducing portfolio risk (VaR, Expected Sh • Method: GARCH filtering and a hierarchical network • Method : GARCH filtering and a hierarchical network deal with the fat tailness of return distribution and the h deal with the fat-tailness of return distribution and the h Fi di • Findings: Hierarchical groups are detected The cu Findings : Hierarchical groups are detected. The cu some sectors are grouped almost together The group some sectors are grouped almost together. The group 1 Correlation of fat tail stock returns 1. Correlation of fat-tail stock returns P tf li ik V i ( ) C i ( ) Portfolio risk = Variance (r)+ Covariance (r) Portfolio risk Variance (r) Covariance (r) Stock return feature: P bl f li l ti Stock return feature: F tt il d l tilit l t i Problem of linear correlation Fat tail and volatility clustering 2 3 Corr = 0 2 Corr = 0 Volatility of Nikkei by GARCH(1,1) 1 1 0 z1 0 z2 z r -1 1 t t t z r -2 - White noise z t ~IID(0,1) -3 -2 White noise 0 50 100 150 200 250 300 Index 0 50 100 150 200 250 300 Index z is assumed as red: Normal bl Sk t 0 0 blue: Skew-t -5 -5 dd 10 h k 10 x1 -10 x2 add 10σ shock White noise -1 x -15 x -15 0 - Corr = really? -20 -2 Corr , 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 z i,t rather than r i,t • GARCH filtering to separate volatilities σ it and r GARCH filtering to separate volatilities σ i,t and iidi ti ith l ti ti P r i.i.d innovations z it with correlation matrix P t i,t t Multivariate GARCH for stock returns r Multivariate GARCH for stock returns r i,t for over 1400 Stocks listed at Tokyo Stock Exchange 1st section - for over 1400 Stocks, listed at Tokyo Stock Exchange, 1st section with complete time series 0 with complete time series t t t z r 1 1 1 0 t t t , 1 , 1 , 1 0 => Z it has correlation P t t t t z r 2 2 2 0 i,t t t t t , 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) •Vector GARCH(1 1) volatility equations: •Vector GARCH(1,1) volatility equations: 2 2 2 2 2 2 1 , 2 2 , 1 2 1 , 1 1 , 1 2 1 , 2 2 , 1 2 1 , 1 1 , 1 0 , 1 2 , 1 t t t t t r r 2 2 2 2 2 2 1 , 2 2 , 2 2 1 , 1 1 , 2 2 1 , 2 2 , 2 2 1 , 1 1 , 2 0 , 2 2 , 2 t t t t t r r 1 , 2 2 , 2 1 , 1 1 , 2 1 , 2 2 , 2 1 , 1 1 , 2 0 , 2 , 2 t t t t t •Simplify time varying P as constant over time P •Simplify time varying P t as constant over time P - CCC-GARCH (Bollerslev[1990]) ccc: constant conditional correlation CCC GARCH (Bollerslev[1990]) ccc: constant conditional correlation 2 Clustering stock returns 2. Clustering stock returns •Current 33 sector classification: the best grouping? Current 33 sector classification: the best grouping? Heat map of correlation matrix Heat map of correlation matrix of stock returns: corr(r r ) of stock returns: corr(r it, r jt ) • Find more data oriented grouping • Find more data-oriented grouping using correlation matrix P= {P ij } using correlation matrix P {P ij } C l ti ti {P } ⇒ dj t f • Correlation matrix {P ij } ⇒ adjacent Not so informative ij matrix {A } ⇒ Network clustering matrix {A ij } ⇒ Network clustering - Divisive hierarchical - Divisive, hierarchical Mdl it i i ti + t l l t i 2 - Modularity maximization + spectral clustering 2 (Newman[2006]) R l ti li it bl k db - Resolution limit problem… ; work around by recursive clustering; simple but it works. Sorted in the sector order References References [1] B ll l T [1990] “M d li th Ch i Sh t N i lE h Rt 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 Views expressed here are those of the author and do not necessarily Views expressed here are those of the author and do not necessarily ucture of Stock Returns ucture of Stock Returns ucture of Stock Returns l Clustering Clustering Clustering E mail: takashi isogai@boj or jp E‐mail: [email protected] ept for portfolio risk management We propose a new ept for portfolio risk management. We propose a new ructure of market wide stock returns by network ructure of market-wide stock returns by network hortfall) with effective diversification of investment hortfall) with effective diversification of investment. clustering with modularity maximization is combined to clustering with modularity maximization is combined to high dimensional correlation structure high dimensional correlation structure. rrent sector classification is partially effective; stocks in rrent sector classification is partially effective; stocks in p properties are identified by classification tree analysis p properties are identified by classification tree analysis. 3 Building hierarchical group structure 3. Building hierarchical group structure Macro view of clustering Macro view of clustering Mdl it i i ti Modularity maximization 1 j i C C w w A Q 1 max arg j i j i ij C C C W A W Q , 2 2 max arg j i C W W j i 2 2 , , • w i , w j is the sum of weights of i j vertex (stock) i, j vertex (stock) i, j • W is the sum of w i over all stocks W is the sum of w i 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 0 •Gx: stock groups same class, otherwise 0. - Bx and Gx are all ‘communities’ - For modularity limit; For modularity limit; What about modifying w ij ->w ij +rδ What about modifying w ij -> w ij +rδ ij as in Q ? <currently r=0> as in Q AFG ? <currently r=0> •Recursive clustering: B1: all stocks When indivisible by modularity B1: all stocks When indivisible by modularity maximization, reset the subnet as the new whole group and start again. r r St i l t i th 4 i ld • Stop recursive clustering at h=4 yields 24 groups smaller than 33 sectors 24 groups, smaller than 33 sectors. - with many automobile companies Understanding group properties and related industries Understanding group properties how 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 dt ii th bdi i i t l plays a key role in determining the subdivision at every layer. - shared properties of a group reflect investors’ views of those stocks. stocks. la er3 layer3 The most relevant relevant property property layer4 layer4 Classification tree at every hierarchy l 2 Classification tree at every hierarchy Model: Group = f(continuous and categorical layer2 Model: Group = f(continuous and categorical properties of stocks) layer1 properties of stocks) $ • TOPIX beta, size, PBR (Fama and French), $/¥ rate beta, l t overseas sales, sectors, … C diti l ti l t dlf t k l ifi ti Conditional sequential tree model for stock classification e.g., B1 G27: f(L1 Tree * L2 Tree * L3 tree * L4 tree) it f t ID f l li t d t k t k ith li it d i dt t - merits: forecast group ID of newly listed stocks, stocks with limited price data, etc. 4F th t i 4. Further topics Bipartite graph 24 groups vs 33 sectors Bipartite graph – 24 groups vs. 33 sectors • Quantify risk (VaR ES) Quantify risk (VaR, ES) d ti ff t reduction effects b d tf li i l ti -by random portfolio simulation Multivariate GARCH with • Multivariate GARCH with non-diagonal specification non-diagonal specification - Market-wide analysis by reduced size of GARCH models 33 sectors 24 groups size of GARCH models, 33 sectors 24 groups - volatility spillovers and dynamic • Group integrities are significantly different conditional 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 td transportation, banking, energy… are concentrated. t G li d ARCH M d l ” R i fE i d St ti ti V l72 N 3 498 505 ate 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 reflect the official views of the Bank of Japan y reflect the official views of the Bank of Japan.