Essays on Applications of the Factor Model

Abstract

Estimating the volatilities and correlations of asset returns plays an important role in portfolio and risk management. As of late, interest in the estimation of the covariance matrix of large dimensional portfolios has increased. Estimating large dimensional covariance poses a challenge in that the cross-sectional dimension is often similar to or bigger than the number of observations available. Simple estimators are often poorly conditioned with some small eigenvalues, and so are unsuitable for many real world applications, including portfolio optimization and tracking error minimization. The first chapter introduces our two large dimensional covariance matrix estimators. We estimate the large dimensional realized covariance matrix by using the methods of asymptotic principal components analysis based factor modeling and singular value decomposition. In the second chapter, we show though simulation that our proposed estimators are closer to the true covariance matrix than the current popular shrinkage estimator. We also simulate conducting the out sample portfolio performance tests and find that the portfolios constructed based on our proposed estimators have lower risk than portfolios constructed using the shrinkage matrix. Using S&P 500 stocks from 1926 to 2011, we back test our proposed covariance matrix. In addition, the portfolios constructed based on our proposed estimators exhibit lower risk than portfolios constructed using the shrinkage matrix. The third chapter proposes a new volatility index--a cross-sectional volatility index of residuals using factor model. The cross-sectional volatility index moves closely with the VIX for the S&P 500 stock universe. It is a non-parametric, model-free volatility index, which could be estimated at any frequency for any region, sector, and style of world equity market and also does not depend on any option pricing. We provide some interpretation of the cross-sectional volatility index of residuals as a proxy for aggregate economic uncertainty, and show a high correlation between the VIX index and the corresponding cross-sectional volatility index of residuals based on the S&P 500 universe. Our results show that the portfolio hedged based on the cross-sectional volatility index of residuals has a much higher Sharpe ratio than the portfolio without hedge. Overall, these findings suggest that the cross-sectional volatility index of residuals is intimately related to other volatility measures where and when such measures are available, and that it can be used as a reliable proxy for volatility when such measures are not available.