| This thesis aims to develop techniques for improving portfolio optimization. The second chapter presents an improved covariance matrix estimator in the mean-variance optimization setting. Sample covariance matrix can be singular when the number of observations is less than the number of assets, and nearly singular when the number of observations exceeds the number of assets. Since the sample covariance matrix is not well-conditioned, using it as an input in mean-variance optimization can result in unreasonable "optimal" portfolios and badly biased estimates of Sharpe ratios. We address this problem by imposing constraints on the Sharpe ratio, asset return variances, and the variance of the global minimum variance portfolio. Our simulations show that the Constrained Maximum Likelihood Estimator (CMLE) performs better than the sample covariance matrix. Moreover, when the shrinkage approach is applied to the CMLE and single index covariance matrix, it performs better than the shrinkage of the sample covariance matrix and the single index covariance matrix of Ledoit and Wolf (2004). During the last two decades Value-at-Risk (VaR) has become the most commonly used measure of market risk due to its ease of calculation and simple interpretation. However, VaR has some undesirable mathematical characteristics such as lack of subadditivity and convexity. Conditional Value-at-Risk (CVaR), defined as the expected loss conditional on a loss larger than the VaR is an intuitively appealing coherent risk measure (Artzner et al. (1999)). However, tractable methods to optimize portfolios based on CVaR are not readily available. In the third chapter, we use the volatility dispersion trading strategy to illustrate that the quantile regression approach developed by Bassett et al. (2004) to risk management with CVaR allows for the easy solution of this otherwise difficult hedging and optimization problem. Credit risk is more difficult to model than market risk because the loss distribution is asymmetric and "fat-tailed" relative to the normal distribution. In the fourth chapter, we use a standard bond portfolio to demonstrate that credit risk optimization can be carried out using the quantile regression approach to compute CVaR developed by Bassett et al. (2004). |
