Understanding the Long-Only Minimum Variance Portfolio

TL;DR

This paper explores the structure of long-only minimum variance portfolios using factor models, providing explicit solutions for one-factor models and geometric insights for multiple factors, supported by empirical US stock data.

Contribution

It offers a rigorous explicit description for one-factor models and a geometric framework for multi-factor models of long-only minimum variance portfolios.

Findings

Explicit solution for 1-factor models

Geometric description for multi-factor models

Empirical validation with US stock data

Abstract

For a covariance matrix coming from a factor model of returns, we investigate the relationship between the long-only global minimum variance portfolio and the asset exposures to the factors. In the case of a 1-factor model, we provide a rigorous and explicit description of the long-only solution in terms of the parameters of the covariance matrix. For $q>1$ factors, we provide a description of the long-only portfolio in geometric terms. The results are illustrated with empirical daily returns of US stocks.