Document Type : Original Article

Author

Assistant Professor, Department of Economics and Management, Naragh Branch, Islamic Azad University, Naragh, Iran

Abstract

Purpose: This study aimed to optimize the stock portfolio based on stochastic matrix theory in the stock market and to answer whether the relevant information will exist using the Marčenko–Pastur distribution.
Methodology: The data of 31 shares in the Tehran Stock Exchange in 2016 - 2019 will be examined for cross-correlation between shares. So, there will be 749 end-of-day prices and 748 logarithms of returns. This research was done using the descriptive-correlation method and is of an applied research type.
Findings: The results showed: a) Observing the most extensive distribution of eigenvector components, it can be seen that there is a solid asymmetry to the left of the distribution, meaning that the market responds more to bad events than good events. b) By clearing the correlation matrix, the difference between the predicted and realized risk can be slightly reduced. In other words, the risk is reduced by identifying and removing non-valuable stocks from the portfolio portfolio. c) A stochastic stock matrix can significantly predict the realized return and risk of the market and, therefore, explain the risk of market information. d) The inverse participation ratio determines the stocks affecting the particular vectors, and the primary analysis of random matrices is based on adjusting this ratio using random matrix clearance.
Originality/Value: Unlike other portfolio formation methods determining the weight of each asset in the portfolio, stochastic matrix theory identifies unused stocks and removes them from the stock portfolio, thereby improving portfolio return and risk.

Keywords

Blackwell, K., Borade, N., VI, C. D., Luntzlara, N., Ma, R., Miller, S. J., ... & Xu, W. (2019). Distribution of eigenvalues of random real symmetric block matrices. arXiv preprint arXiv:1908.03834
Bouchaud, J. P., & Potters, M. (2003). Theory of financial risk and derivative pricing: from statistical physics to risk management. Cambridge University Press.
Daly, J., Crane, M., & Ruskin, H. J. (2008). Random matrix theory filters in portfolio optimisation: a stability and risk assessment. Physica A: statistical mechanics and its applications387(16-17), 4248-4260.
Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. I. Journal of mathematical physics3(1), 140-156.
Fyodorov, Y. V., & Mirlin, A. D. (1992). Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems. Physical review letters69(7), 1093. https://doi.org/10.1103/PhysRevLett.69.1093
Fyodorov, Y. V., & Mirlin, A. D. (1993). Level-to-level fluctuations of the inverse participation ratio in finite quasi 1D disordered systems. Physical review letters71(3), 412. https://doi.org/10.1103/PhysRevLett.71.412
Fyodorov, Y. V., & Mirlin, A. D. (1994). Statistical properties of eigenfunctions of random quasi 1d one-particle Hamiltonians. International journal of modern physics B8(27), 3795-3842.
Guhr, T., Müller–Groeling, A., & Weidenmüller, H. A. (1998). Random-matrix theories in quantum physics: common concepts. Physics reports299(4-6), 189-425.
Laloux, L., Cizeau, P., Bouchaud, J. P., & Potters, M. (1999). Noise dressing of financial correlation matrices. Physical review letters83(7), 1467. https://doi.org/10.1103/PhysRevLett.83.1467
Laloux, L., Cizeau, P., Potters, M., & Bouchaud, J. P. (2000). Random matrix theory and financial correlations. International journal of theoretical and applied finance3(03), 391-397.
Lee, P. A., & Ramakrishnan, T. V. (1985). Disordered electronic systems. Reviews of Modern Physics57(2), 287. https://doi.org/10.1103/RevModPhys.57.287
Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77–91. https://doi.org/10.2307/2975974
Mirlin, A. D., & Fyodorov, Y. V. (1993). The statistics of eigenvector components of random band matrices: analytical results. Journal of physics A: mathematical and general26(12), L551. https://iopscience.iop.org/article/10.1088/0305-4470/26/12/012/meta
Pafka, S., & Kondor, I. (2003). Noisy covariance matrices and portfolio optimization II. Physica A: statistical mechanics and its applications319, 487-494.
Pafka, S., & Kondor, I. (2004). Estimated correlation matrices and portfolio optimization. Physica A: statistical mechanics and its applications343, 623-634.
Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L. A. N., & Stanley, H. E. (1999). Universal and nonuniversal properties of cross correlations in financial time series. Physical review letters83(7), 1471. https://doi.org/10.1103/PhysRevLett.83.1471
Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L. A. N., Guhr, T., & Stanley, H. E. (2002). Random matrix approach to cross correlations in financial data. Physical review E65(6), 066126. https://doi.org/10.1103/PhysRevE.65.066126
Podobnik, B., & Stanley, H. E. (2008). Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. Physical review letters100(8), 084102. https://doi.org/10.1103/PhysRevLett.100.084102
Sengupta, A. M., & Mitra, P. P. (1999). Distributions of singular values for some random matrices. Physical review E60(3), 3389. https://doi.org/10.1103/PhysRevE.60.3389
Wang, G. J., Xie, C., Chen, S., Yang, J. J., & Yang, M. Y. (2013). Random matrix theory analysis of cross-correlations in the US stock market: evidence from Pearson’s correlation coefficient and detrended cross-correlation coefficient. Physica A: statistical mechanics and its applications392(17), 3715-3730.
Wang, G. J., Xie, C., He, L. Y., & Chen, S. (2014). Detrended minimum-variance hedge ratio: a new method for hedge ratio at different time scales. Physica A: statistical mechanics and its applications405, 70-79.
Wigner, E. P. (1993). Characteristic vectors of bordered matrices with infinite dimensions i. In The collected works of eugene paul wigner (pp. 524-540). Springer, Berlin, Heidelberg.
Wigner, E. P. (1993). On a class of analytic functions from the quantum theory of collisions. In The collected works of eugene paul wigner (pp. 409-440). Springer, Berlin, Heidelberg.