Are minimum variance portfolios in multi-factor models long in low-beta assets?

Abstract

Within the one-factor capital asset pricing model (CAPM), the minimum-variance portfolio (MVP) is known to have long positions in those assets of the underlying investment universe whose betas are less than a well-defined long-short threshold beta. We study the structure of MVPs in more general multi-factor asset pricing models and clarify the low-beta puzzle for multi-factor models: For multi-factor models we derive a similar criterion in terms of the betas with explicit closed-form formulas. But the structural relationship is now more involved and the long-short threshold turns out to be asset-specific. The results rely on recursive inverse-free formulas for the precision matrix, which hold for multi-factor models and allow quick computation of that inverse matrix without the need to invert matrices going beyond diagonal ones. We illustrate our findings by analyzing S &P 500 asset returns. Our empirical results of the S &P 500 constituents between 2019 and 2022 confirm the theoretical findings and shows that the minimum variance portfolio is long in low-beta assets when applying estimates of the established asset-specific thresholds.