Quang Nguyen - PhD Co-authors: Dinh Nguyen, Thu Hoang, Phat Huynh

Transcription

1 FINANCIAL MARKET RISK ANALYSIS THROUGH CROSS-CORRELATION S EIGENVECTOR COMPONENTS DISTRIBUTION Quang Nguyen - PhD Co-authors: Dinh Nguyen, Thu Hoang, Phat Huynh John von Neumann Math. Finance Chair Vietnam National University in Ho Chi Minh city FNET-KYOTO-07/2013

2 Objective Random matrix theory (a statistical physics method) application to the cross-correlation matrix of the stock market Study of eigenvalue spectrum and eigenvector can reveal useful information about the inter-dependency of the stocks Market risk the un-diversifiable risk can be determined through this analysis 2

3 Outline 1. Introduction 2. Empirical results 3. One-factor model of correlation matrix 4. Market risk and the comp. of the 1 st eigenvector 5. Future work and Conclusion 3

4 Introduction Stock price & daily price change Pt Pr t, t ln P Randomness r t t t 1 The stock market: An interdependent system of hundreds to thousand stocks t 4

5 Introduction Portfolio 1 2 N (,,..., ) Portfolio risk Risk r r r P Diversification i 2 i i j i j ( ).var( )..cov(, ) i i j Risk Market _ risk P Diversifiable _ risk C Port. risk ,2 1, N 2,1 2, N N,1 N, Specific risk Market risk Nb of stock 5

6 Introduction Portfolio 1 2 N (,,..., ) Portfolio risk Risk r r r P Diversification i 2 i i j i j ( ).var( )..cov(, ) i i j Risk Market _ risk P Diversifiable _ risk C Port. risk ,2 1, N 2,1 2, N N,1 N, Specific risk Market risk Nb of stock 6

7 Market risk Market risk (systematic risk) is a notion of the (complex) system, not of individual stock Market risk depends on the relation/correlation between stocks (components in the system) No correlation = no market risk = no risk at all (as N ) Stock price is often embedded by noise Random matrix method is therefore a suitable tool 7

8 Random Matrix Theory (RMT) Random entries Eigenvalue spectrum Q=T/N γ (finite) The limiting spectra is given by Marcenko-Pastur Law C RMT ,2 1, N 2,1 2, N N,1 N,2 ~ i. i. d. N(0,1) C. u. u RMT i i i... 1 Q ( )( ) g( ), 2 density, Q Q λ 8

9 Empirical data US market - 1: market mode - > +: sector modes - < +: noise V. Plerou et al. Phy. Rev. E

10 Empirical data Vietnamese market Q. Nguyen, Physica A

11 Largest eigenvalue Different group reported different value of 1 in the range of , about This mode is the collective behaviour of all stock represent by their correlation, which is not null The value of 1 must be derived from non-null value of collective correlation 11

12 Largest eigenvalue We found a phenomenological relation 1 N (an analytic derivation of this formula is expected in the future work) How can this average correlation explain the eigenvalue spectra? 12

13 One-factor model Constant correlation model C t r N(0, C) Experimental RMT Shifted RMT

14 Empirical data - IPR Empirical data Simulation data

15 Eigenvectors 15

16 1 st eigenvector Positive Relatively uniform Some groups found relation to the stock market capital - We do not find such dependence 0.15 Component of U max Market capital

17 1 st eigenvector However, it is natural to think about the correlation of each stock to all others stocks Influence factor: IF i j ij 17

18 1 st eigenvector However, it is natural to think about the correlation of each stock to all others stocks 0.12 Influence factor: 0.1 IF i j ij Component of U max Sum of correlation coefficient Q. Nguyen et al., in preparation 18

19 Component of 1 st eigenvector 0.05 Volume Market capital Component of U max Average daily volume 19

20 1 st eigenvector Minimum Spanning Tree and U 1 s components 20

21 1 st eigenvector s components The distribution of IF is not random High u 1 i= high IF: stock that correlate the most to all other stocks DEVELOPED MARKET - Institutional investors - Fundamental driven - Market leading: high market cap. stock EMERGING MARKET - Retail investors - Herding driven - Market leading: popular stock (no def.) (the case of Vietnam: securities stocks) 21

22 U 1 portfolio Is the highest coherent mode Is not the commonly defined market portfolio (derived from the Efficient Market Theory) Its risk is higher than the EMT s It has high market risk but still eliminate specific risk 22

23 Nature of stock correlation Why stock correlation is almost positive? 23

24 Why stock corr. is almost positive? - Firm i: i i i i F O F ( S, M ) 0 M O F ( S, O ) j i i j t t 1 j O F ( S, O O... O ) j i i 1 2 N t t 1 t 1 t 1 Therefore, ij is very likely to be positive unless there is a strict negative correlation between S i and S j 24

25 Source of instability? The market: O F ( S, O O... O ) N t t 1 t 1 t 1 O F ( S, O O... O ) N t t 1 t 1 t 1 O F ( S, O O... O ) N N N 1 2 N t t 1 t 1 t 1 Positive feedback Non-linearity Agent-based simulation could be the efficient tool 25

26 Conclusions One factor model: Magnitude of the maximum eigenvalue Shift of the bulk 1 st eigenvector analysis 1 st eigenvector s component ~ Influence factor Discuss the market mode 26

27 Selected References - V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, H.E. Stanley, Phys. Rev. Lett. 83 (1999) V. Plerou, P. Gopikrishnan, B. Rosenow, Luis A. Nunes Amaral, T. Guhr, and H.E. Stanley, Phys. Rev. E65, (2002). - J. Kwapien, S. Drozdz, P. Oswiecimka, Physica A 359 (2006) E.P. Wigner, Ann. Math. 53,(1951) V.A. Marchenko, L.A. Pastur, Math. USSR-Sbornik 1 (1967) A. Utsugi, K. Ino, and M. Oshikawa, Phys. Rev. E70, (2004). - S. Cukur, M. Eryigit, R. Eryigit, Physica A 376 (2007) Q. Nguyen, Physica A, 392 (2013) Q. Nguyen, D. Nguyen, T. Hoang, P. Huynh, in preparation 27

28 THANK YOU FOR YOUR ATTENTION! 28