Report

Financial Applications of RMT Max Timmons May 13, 2013 Main Application: Improving Estimates from Empirical Covariance Matricies • Overview of optimized portfolios in finance • Brief overview of relevant RMT facts • Statistical evidence from financial markets Overview: Modern Portfolio Theory • Classic goal is to maximize return and minimize return • Consider a portfolio P of N assets where pi is the amount of capital invested in asset i and Ri is the expected return of asset i • Expected return is RP=∑Ni=1piRi • σ2P=∑Ni,j=1piCijpj where C is the covariance matrix • Optimal portfolio minimizes σ2P for a given RP and involves inverting C. This places a large weight on the eigenvectors of C with the smallest eigenvalues. • Want to distinguish the true covariance matrix from statistical due in the empirical covariance matrix as sample size is not large compared to size of matrix • Empirical covariance matrix Cij=1/T* ∑Tt=1δxi(t)δxj(t) where δxi(t) is the price changes Relevant RMT Facts • If δxi(t) are independent, identical distributed, random variables then we have a Wishart matrix or Laguerre ensemble (i.e. all assets have uncorrelated returns) • If Q=T/N≥1 is fixed than as N∞, T∞ the Marcenko-Pastur law gives the exact distribution of eigenvalues • In particular λmaxmin=σ2(1+1/Q±2sqrt(1/Q)) Evidence from Financial Markets • Is the independence of all assets a good assumption? • The Marcenko-Pastur law predicts the distribution of small eigenvalues pretty well but there are much larger eigenvalues than predicted (from NY and Tokyo stock markets) • The largest eigenvalue roughly corresponds to the overall performance of the market • Other large eigenvalues correspond to specific industries • Looking at the discrepancies between the MarcenkoPastur prediction and the data provides actual information on covariance that is not due to noise References • Laloux, L. Cizeau, P. Potters, M. Bouchaud, J. Random Matrix Theory and Financial Correlations. Int. J. Theor. Appl. Finan. 03, 391 (2000) • Utsugi, A. Ino, K. Oshikawa, M. Random matrix theory analysis of cross correlations in financial markets. Physical Review E 70, 026110 (2004). • Plerou, V et al. Random matrix approach to cross correlations in financial data. Phys. Rev. E 65, 066126 (2002)