On the long-only minimum variance portfolio under single factor model

Abstract

The minimum-variance portfolio (MVP) has become an essential part of modern portfolio theory, largely due to the availability of its analytical formula and its good out-of-sample performance. When extra constraints such as the long-only constraints are added, MVP in general does not admit an analytical formula anymore. An exceptional case is when the single-factor model holds among the securities considered. It is known that there exists a semi-closed form formula, which relies on an unknown quantity called the long-only beta threshold (βL). This note conducts a detailed study of βL and establishes its optimality property that it is the smallest among all possible thresholds. We also develop a simple bisection method for computing βL. Finally we include a mathematical proof for this semi-closed form formula. The results reported provide a deep understanding how the long-only MVP are constructed.