PORTFOLIO SELECTION USING TIKHONOV FILTERING TO ESTIMATE THE COVARIANCE MATRIX2012.04.06
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Markowitz’s Mean-Variance portfolio selection problem (MV problem)chooses weights for stocks in a portfolio based on an estimatedcovariance matrix of stock returns. Our study proposes to reduce noisein the estimation using a Tikhonov filter function. The new methoddecreases the contribution of the smaller eigenvalues of a correlationmatrix gradually by using a Tikhonov filtering function. To derive theTikhonov filtering, we construct a linear model based on principalcomponent analysis and formulate an optimization problem that findsappropriately noise-filtered factors. Using the filtered factor data,we estimate a Tikhonov covariance matrix. In addition, we propose amethod for effectively choosing the Tikhonov parameter, whichdetermines the filtering intensity. We put previous estimators into acommon framework and compare their filtering functions for eigenvaluesof the correlation matrix.We performed empirical experiments to evaluatecovariance estimators. We evaluate the covariance estimators usingreturn data from the NYSE, AMEX, and NASDAQ. We collected the monthlydata from January 1958 to December 2007 from the CRSP database (theCenter for Research in Security Prices). For the MV portfolioselection problem, the Tikhonov choice was among the most efficientportfolios and the best estimates of risk. Moreover, the Tikhonovestimator performs relatively well in the circumstance of insufficienthistorical data.