Group Correlation Structures Embedded in Financial Markets Comparative Study Between Japan and U.S.

Transcription

1 /Review 1 2 Group Correlation Structures Embedded in Financial Takeo YOSHIKAWA 1 and Hiroshi IYETOMI 2 Abstract We study group correlation structures of financial markets in Japan and U.S. from a network-theoretic point of view. The correlation matrix of stock price changes, purified by the random matrix theory, is regarded as an adjacency matrix for a network. The weighted links in the networks thus constructed can have negative sign corresponding to anticorrelation between stocks. To identify groups in such a network, we search for the optimum decomposition of nodes which maximizes the total sum of weights of links inside groups. We then find that the network of Tokyo Stock Exchange is decomposed into four groups. The stock prices comove almost perfectly inside the groups and move oppositely across the groups. Also we apply the same analysis to the S&P 500 stocks. The U.S. stock market shows frustrated behavior similar to that embedded in the Japanese market. Keywords principal component analysis, financial market, RMT: Random Matrix Theory, group correlation, network, frustration 1. T N N 1 N T Graduate School of Science and Technology, Niigata University, 8050, ikarashi 2-no-cho, Nishi-ku, Niigata 2 Faculty of Science, Niigata University, 8050, ikarashi 2-no-cho, Nishi-ku, Niigata Received: 28 July 2013, 6 September RMT [1] N T Q = T /N 1 RMT [2, 3] S&P 500 RMT RMT [4 6]

2 Group Correlation Structures Embedded in Financial RMT [4,7,8] [9] [10] [11] [12] [13 15] [16] 2. TSE S&P n t S n,t (n = 1,2,...,N; t = 1,2,...,T + 1) {S n,t } R n,t lns n,t+1 lns n,t 0 1 : G n,t R n,t R n (1) σ n {R n,t } σ n : R n 1 T σ n 1 T T t=1 T t=1 R n,t, (2) (R n,t R n ) 2. (3) G n,t (n,t) N T G G 2. SICC 33 S&P 500 Global Industry Classification Standard (GICS) GICS 10 C C = 1 T GGT (4) G T G 3. RMT : C = N l=1 λ l u l u T l (5) λ 1 >λ 2 >...>λ N (5) 1 80% 0 1 N T H 1 T HHT H 1 T HHT N RMT Q = T /N N,T [17, 18]: ρ(λ )= Q (λ+ λ )(λ λ ). (6) 2π λ ( λ ± = 1 ± 2 1/Q) [λ,λ + ] S&P 500 Q = T /N Fig. 1 Fig. 2 (6) λ + RMT λ 13 ( = 2.17)>λ + ( = 2.11)>λ 14 ( = 2.07) 13 S&P 500 Oukan Vol.7, No.2 93

3 Yoshikawa, T. and Iyetomi, H. Fig. 1: Probability density distribution of the eigenvalues of the correlation matrix in the Tokyo Stock Exchange and the corresponding distribution ρ (λ), Eq. (6), predicted by the RMT. Fig. 3: Polarization of each sector defined by Eq. (7) in the statistically meaningful eigenvectors for the Tokyo Stock Exchange. Fig. 2: Same as Fig. 1, but in the S&P 500. λ 9 ( = 3.07)>λ + ( = 2.86)>λ 10 ( = 2.71) 9 Fig. 3 Fig. 4 S&P 500 u l λ l >λ + u 1 SICC S&P 500 GICS S : Ps j s u l, j j s u l, j. (7) 1 [4,7,9] S&P 500 u 2 u 6 u 7 S&P 500 S&P 500 GICS Fig. 4: Same as Fig. 3, but for the S&P

4 Group Correlation Structures Embedded in Financial C group (m) m = l=2 λ l u l u T l (m = 2,...,N c ) (10) m = N c C group 5. Fig. 5: Distribution of the matrix elements of C group and C market + C group in the Tokyo Stock Excange. Fig. 6: Same as Fig. 5, but in the S&P 500. C [3, 4, 8] C C market + C group + C random (8) = λ 1 u 1 u T 1 + N c l=2 λ l u l u T l + N l=n c +1 λ l u l u T l. (9) N N c RMT TSE N c = 13 S&P 500 N c = 9 C random 3 1 λ 1 2 TSE λ 1 = 133 λ2 = 15 S&P 500 λ 1 = 241 λ2 = 17 u 1 C market Fig. 5 Fig. 6 S&P 500 C group C market + C group C group C market + C group C group [19] i σ i {σ} = {σ 1,σ 2,...,σ N } {σ} [19,20] Q({σ}) : Q({σ}) 1 m i, j [A i, j p i, j ]δ (σ i,σ j ), p i, j = k ik j m. (11) A i, j (i, j) k i = j A i, j i m = i j A i, j δ() i j p i, j i j i j Zachary [21] Zachary Oukan Vol.7, No.2 95

5 Yoshikawa, T. and Iyetomi, H. Table 1: Decomposition of stocks into communities at varied m in the Tokyo Stock Exchange. C group and C group + C random have almost the same structures. Fig. 7: Four communities obtained through the modularity maximization for the Karate Club network[21], and the dashed line depicts its actual split into two groups. 2 Fig. 7 Community m I II III IV (C group ) C group + C random Kim and Jeong[8] Pan and Sinha[9] C group C group 3 [14]: F ({σ})= A i, j δ (σ i,σ j ). (13) i, j 3. 5 [13, 14] : H ({σ}) i, j [ A i, j ( )] γ + p + i, j γ p i, j δ ( ) σ i,σ j, (12) p ± i, j i j γ ± F ({σ}) γ ± = 0 Table 2: Decomposition of stocks into communities at varied m in the S&P 500. C group and C group + C random have the same structures. Community m I II III IV (C group ) C group + C random C group C group (m) C group + C random Table 1 Table 2 simulated annealing [22] S&P (l = 2 3 4)

6 Group Correlation Structures Embedded in Financial Fig. 8: Fragmentation of the business sectors into communities in the Tokyo Stock Exchange. The dotted line represents the fraction level of 2/3. Fig. 10: Frustrated stock group structure in the Tokyo Stock Exchange. The arrows designate the anticorrelation relationship between communities with the corresponding values of P corr, Eq. (14), represented by their width. Fig. 9: Same as Fig. 8, but in the S&P 500. S&P (l = 2 3) l = 9 3 C random 64% S&P % RMT Fig. 8 Fig. 9 II 29 III S&P I Consumer Discretionary Financials II Industrials Consumer Staples Telecommunication Services Utilities III Energy Materials IV Information Technology Fig. 11: Same as Fig. 10, but in the S&P : P corr [ ] within/between Cgroup i, j [. (14) within/between Cgroup ]i, j within/between Fig. 10 Fig. 11 S&P % 4 Oukan Vol.7, No.2 97

7 Yoshikawa, T. and Iyetomi, H S&P IT 3 j d j : [ λ2 ] d T j = u 2, j,..., λ Nc u Nc, j. (15) 13 S&P i j : [ ] Cgroup i, j = dt i d j (16) 4 3 Fig. 12 Fig Fig. 12: Three-dimensional correlation state vectors of the stocks in the Tokyo Stock Exchange. The stocks belonging to different communities are distinguished by varied degrees of shading. The arrows point to the positions of the center of gravity in individual communities with their designations. 8. RMT S&P unbalanced Fig. 13: Same as Fig. 12, but in the S&P

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