A Statistical Response to Challenges in Vast Portfolio Selection

Abstract

The thesis is written in response to emerging issues brought about by an increasing number of assets allocated in a portfolio and seeks answers to puzzling empirical findings in the portfolio management area. Over the years, researchers and practitioners working in the portfolio optimization area have been concerned with estimation errors in the first two moments of asset returns. The thesis comprises several related chapters on our statistical inquiry into this subject. Chapter 1 of the thesis contains an introduction to what will be reported in the remaining chapters.

A few well-known covariance matrix estimation methods in the literature involve adjustment of sample eigenvalues. Chapter 2 of the thesis examines the effects of sample eigenvalue adjustment on the out-of-sample performance of a portfolio constructed from the sample covariance matrix. We identify a few sample eigenvalue adjustment patterns that lead to a definite improvement in the out-of-sample portfolio Sharpe ratio when the true covariance matrix admits a high-dimensional factor model.

Chapter 3 shows that even when the covariance matrix is poorly estimated, it is still possible to obtain a robust maximum Sharpe ratio (MSR) portfolio by exploiting the uneven distribution of estimation errors across principal components. This is accomplished by approximating the vector of expected future asset returns using a few relatively accurate sample principal components. We discuss two approximation methods. The first method leads to a subtle connection to existing approaches in the literature, while the second one named the ``spectral selection method" is novel and able to address main shortcomings of existing methods in the literature.

A few academic studies report an unsatisfactory performance of the optimized portfolios relative to that of the 1/N portfolio. Chapter 4 of the thesis reports an in-depth investigation into the reasons behind the reported superior performance of the 1/N portfolio. It is supported by both theoretical and empirical evidence that the success of the 1/N portfolio is by no means due to the failure of the portfolio optimization theory. Instead, a major reason behind the superiority of the 1/N portfolio is its adjacency to the mean-variance optimal portfolio.

Chapter 5 examines the performance of randomized 1/N stock portfolios over time. During the last four decades these portfolios outperformed the market. The construction of these portfolios implies that their constituent stocks are in general older than those in the market as a whole. We show that the differential performance can be explained by the relation between stock returns and firm age. We document a significant relation between age and returns in the US stock market. Since 1977 stock returns have been an increasing function of age apart from the oldest ages. For this period the age effect completely dominates the size effect.