# Finding low-rank solutions in financial factor models

## Main Article Content

## Abstract

In financial factor models based on the structure of a correlation matrix, the rank of the correlation matrix should be equal to the number of factors. However, it is not so rare to obtain a high-rank correlation matrix from the given data in practical applications. Therefore, it is necessary to find the nearest low-rank correlation matrix to the computed one. If we take the Frobenius norm to measure the "nearness" of two matrices, we will show that this problem can be formulated in the form of a~rank-constrained semidefinite program. Although this kind of problem is considered to be NP-hard, there are some rank reduction techniques to deal with this non-convex rank constraint.

## Article Details

How to Cite

Fulová, T.
(2020).
Finding low-rank solutions in financial factor models.

*Proceedings Of The Conference Algoritmy,*, 161 - 170. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1586/839
Section

Articles

## References

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[2] Borsdorf, R., Higham, N. J., Raydan, M.: Computing a nearest correlation matrix with factor structure, International Edition III, MeBoo USA, 2001-2010

[3] Borsdorf, R., Higham, N.: A preconditioned Newton Algorithm for the Nearest Correlation Matrix, IMA J. Numer. Anal., 30 (2010), 94-107

[4] Boyd, S., Vandenberghe, L.: Convex Optimization, Cambridge University Press, New York, 2004

[5] Dattorro, J.: Convex Optimization & Euclidean Distance Geometry, International Edition III, MeBoo USA, 2001-2010

[6] Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.0 beta, http://cvxr.com/cvx, September 2013

[7] Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, Springer, 2008, 95-110, http://stanford.edu/∼boyd/graph dcp.html.

[8] Grubišić, I., Pietersz, R.: Efficient rank reduction of correlation matrices, Linear Algebra and its Applications, 422 (2007), 629-653

[9] Higham, N.: Computing the nearest correlation matrix – a problem from finance, IMA Journal of Numerical Analysis, 22 (2002), 329-343

[10] Lemon, A., So, A. M., Ye, Y.: Low-Rank Semidefinite Programming: Theory and Applications, Foundations and Trends in Optimization, 2 (2015), 1-156

[11] MATLAB R2019a, The Mathworks, Inc., MATLAB version 9.6.0.1150989, Massachusetts, 2019

[12] Pietersz, R., Groenen, P. J. F.: Rank Reduction of Correlation Matrices by Majorization, Quantitative Finance, 2004, 649-662

[13] Qi, H., Sun, D.: A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix, SIAM J. Matrix Anal. Appl., 28 (2006), 360-385

[14] Zhang, F.: Matrix theory - Basic Results and Techniques, Springer Science+Business Media, New York, 1999

[15] Zhang, Z., Wu, L.: Optimal low-rank approximation to a correlation matrix, Linear Algebra and its Applications, 364 (2003), 161-187