Minimum Variance Portfolio Composition

Abstract

Empirical studies document that equity portfolios constructed to have the lowest possible risk have surprisingly high average returns. We derive an analytic solution for the long-only minimum variance portfolio under the assumption of a single-factor covariance matrix. The equation for optimal security weights has a simple and intuitive form that provides several insights on minimum variance portfolio composition. While high idiosyncratic risk can lead to a low security weight, high systematic risk takes the large majority of investable securities out of long-only solutions. The relatively small set of securities that remain have market betas below an analytically specified threshold beta. The math also shows that the ratio of portfolio beta to the threshold beta dictates the portion of ex-ante portfolio variance that is market-factor related. We verify and illustrate the portfolio mathematics using historical data on the U.S. equity market and explore how the single-factor analytic results compare to numerical optimization under a generalized covariance matrix. The analytic and empirical results of this study suggest that minimum variance portfolio performance is largely a function of the long-standing empirical critique of the traditional CAPM that low beta stocks have relatively high average returns.

Keywords: Portfolio Theory, Mean Variance Optimization, Market Model, Idiosyncratic Risk

JEL Classification: G11