Application of PCA and Random Matrix Theory to Passive Fund Management

Summary

We use principal component analysis (PCA) for extracting principal components having larger-power in cross correlation from risky assets (Elton and Gruber 1973), and random matrix theory (RMT) for removing noise in the correlation and for choosing statistically significant components (Laloux et al 1999, Plerou et al 1999) in order to estimate expected correlation in portfolio optimization problem. In addition to correlation between every pairs of asset returns, the standard mean-variance model of optimal asset allocation requires estimation of expected return and risk for each assets. Asset allocation is, in practice, quite sensitive to how to estimate the expected return. We applied estimation based on “beta” (following the idea of Black and Litterman 1992) to portfolio optimization for 658 stocks in Tokyo Stock Exchange (TSE). By using daily returns in TSE and verifying that TSE has qualitatively similar principal components as NYSE (Plerou et al 1999), we show (i) that the error in estimation of correlation matrix via RMT is more stable and smaller than either historical, single-index model or constant-correlation model, (ii) that the realized risk-return in TSE based on our method outperforms that of index-fund with respect to Sharpe ratio, and (iii) that the optimization gives a practically reasonable asset allocation.