Portfolio selection with active strategies: how long only constraints shape convictions

In the recent years, an intense effort has been dedicated to the research on equity factors. If the academic literature questions their rewarding nature, the financial industry competes for the best portfolio implementation, either in long-short or in long only vehicles. While both solutions try to exploit the same factors while controlling risk, two kinds of managers co-exist in the meantime: fundamental, discretionary stock-pickers, and quantitative, systematic managers. The contribution of this paper is twofold. The core message is that a quantitative long only portfolio, built with any factor as a future returns’ proxy and a risk control, ends up to be high conviction portfolios: the long only constraint polarizes naturally the portfolio on a concentrated set of nonzero positions. In this respect, actively managed, quantitative long only portfolios share some similarity with discretionary stock pickers. Beyond this message, and backed by numerical experiments, we derive theoretical results and closed-form formulas to show in addition that: (i) selected stocks are those that realize a trade-off between a low β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} and a high loading on the factor; (ii) the thresholds driving this selection are endogenous, leading to a recursive procedure to select the stocks. In other words: the portfolio selection problem may be solved linearly instead of using an optimizer. We highlight in particular the crucial role played by low β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} stocks and by the co-linearity of the risk model with the factor used to forecast returns.