mt_slides

```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)
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