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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

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Overview: Modern Portfolio Theory Classic goal is to maximize return and minimize return Consider a portfolio P of N assets where p i is the amount of capital invested in asset i and R i is the expected return of asset i Expected return is R P =∑ N i=1 p i R i σ 2 P =∑ N i,j=1 p i C ij p j where C is the covariance matrix Optimal portfolio minimizes σ 2 P for a given R P 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 C ij =1/T* ∑ T t=1 δx i (t)δx j (t) where δx i (t) is the price changes

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Relevant RMT Facts If δx i (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 λ max min =σ 2 (1+1/Q±2sqrt(1/Q))

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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 Marcenko- Pastur prediction and the data provides actual information on covariance that is not due to noise

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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)

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