Mitigating Low-Sample Issues in Portfolio Analysis: Applications of Shrinkage and Gaussian Graphical Models

Abstract

In manyfinancial applications, such as portfolio allocation, factor modeling, and volat ility estimation, the number of assets often exceeds the number of historical return observations. Estimating the covariance matrix of returns, and at times its inverse (the precision matrix), is essential for these applications. However, when observations are fewer than the number of assets, traditional covariance and inverse covariance (pre cision) matrix estimation become unstable. This challenge is not unique to finance; fields such as genetics face similar issues, where the number of markers exceeds the sample size. This thesis examines the challenges of portfolio optimization, factor construction, and volatility modeling in high-dimensional, low-sample settings, where the imbal ance between the number of assets and observations renders standard covariance matrix methods unreliable. In such cases, stabilizing both covariance and precision matrix estimation is necessary to avoid noise and inaccuracies. This work explores shrinkage techniques for covariance estimation alongside Gaussian Graphical Models (GGMs) for precision matrix estimation, offering a comprehensive toolkit for financial practitioners working with limited data. A key contribution of this thesis is the exploration of shrinkage and thresholding methods for covariance matrix estimation in the context of expected utility portfolios, demonstrating how these techniques improve out-of-sample portfolio performance. Additionally, precision matrix estimators are applied to minimum variance portfolio construction, mitigating the noise from covariance matrix inversion and enhancing portfolio optimization outcomes. We find that direct estimators of the precision matrix offer a reliable approach to solving the minimum variance portfolio problem across daily, weekly, and monthly rebalancing horizons. The impact of noisy covariance and precision matrices on other financial applica tions is also explored. Shrinkage-based Principal Component Analysis (PCA) methods have been used to address the issue of low samples in PCA. Building on the ideas of shrinkage-based PCA, we formulate a GGM-based approach that utilizes the preci sion matrix for eigenvalue decomposition. The thesis also examines PCA from the iv perspective of factor construction, finding that shrinkage and GGM-based estimation provide better estimates of statistical factors in low-sample regimes. Furthermore, we study the asset pricing implications of using GGM and shrinkage-based PCA methods, employing performance metrics such as asset mispricing. Portfolio management often requires dynamic estimates of the relationships between different asset returns, necessitating the use of multivariate GARCH models. This thesis develops the Dynamic Conditional Precision Matrix (DCPM)-GARCH model, which extends the Dynamic Conditional Correlation (DCC)-GARCH framework by introducing a new approach to estimate the dynamic conditional precision matrix. While DCC-GARCH remains a widely used model for estimating dynamic conditional covariances, DCPM-GARCH provides a complementary perspective by focusing on the precision matrix, which captures the underlying conditional dependence structure more directly. Both approaches offer valuable insights into dynamic relationships in f inancial markets, with DCC estimating time-varying covariances and DCPM focusing on time-varying precision matrices. These methods are tested for dynamic portfolio allocation for the weekly rebalancing horizons. The GGM-based approach introduced in DCPM is also adaptable to other multivariate models, such as the BEKK-GARCH framework. By combining shrinkage-based and precision-based methods, this thesis offers a comprehensive framework for addressing the challenges of high-dimensional, low sample environments. While shrinkage methods stabilize covariance estimation, precision matrix estimation enhances portfolio construction, factor modeling, and volatility estimation. Together, these approaches provide financial practitioners with effective tools for decision-making in data-constrained environments.