Markowitz minimum variance portfolio optimization using new machine learning methods

The use of improved covariance matrix estimators as an alternative to the sample covariance is considered an important approach for enhancing portfolio optimization. In this thesis, we propose the use of sparse inverse covariance estimation for Markowitz minimum variance portfolio optimization, using existing methodology known as Graphical Lasso [16], which is an algorithm used to estimate the inverse covariance matrix from observations from a multivariate Gaussian distribution. We begin by benchmarking Graphical Lasso, showing the importance of regularization to control sparsity. Experimental results show that Graphical Lasso has a tendency to overestimate the diagonal elements of the estimated inverse covariance matrix as the regularization increases. To remedy this, we introduce a new method of setting the optimal regularization which shows performance that is at least as good as the original method by [16]. Next, we show the application of Graphical Lasso in a bioinformatics gene microarray tissue classification problem where we have a large number of genes relative to the number of samples. We perform dimensionality reduction by estimating graphical Gaussian models using Graphical Lasso, and using gene group average expression levels as opposed to individual expression levels to classify samples. We compare classification performance with the sample covariance, and show that the sample covariance performs better. Finally, we use Graphical Lasso in combination with validation techniques that optimize portfolio criteria (risk, return etc.) and Gaussian likelihood to generate new portfolio strategies to be used for portfolio optimization with and without short selling constraints. We compare performance on synthetic and real stock market data with existing covariance estimators in literature, and show that the newly developed portfolio strategies perform well, although performance of all methods depend on the ratio between the estimation period and number of stocks, and on the presence or absence of short selling constraints.