Congrats to Dr. Alex Bernstein for completing his PhD!

Title:  Long-Only Minimum Variance Portfolio Composition for Factor Models

Committee: Shkolnik (Chair), Fouque, Franks

Abstract: The Markowitz (Mean-Variance) framework has greatly influenced portfolio construction. Historically, constraints, namely the Long-Only constraint have necessitated the use of numerical optimizers. In our work, we develop a closed-form solution to the Long-Only minimum variance portfolio under a risk-factor model assumption, up to the solution of a fixed-point equation. This approach allows us to develop a highly efficient optimization method, referred to as the fast fixed-point (FFP) algorithm, and to directly derive the sensitivity of portfolio weights with respect to model parameters. This fixed-point solution also enables us to directly examine the number of active (i.e., strictly positive) weights of a long-only minimum-variance portfolio. We prove limit theorems that determine the fraction of assets held as the number of assets in the portfolio grows to infinity. We derive these results in the setting of a single-index model, and illustrate how the distribution of stock betas affect the number of active weights. This has potential practical impacts on investing, e.g. how the inclusion of negative beta stocks leads to more diversified long-only portfolios. We confirm our theoretical findings on an empirical dataset using a principal component based risk model.