Portfolio Selection With Active Strategies: How Long Only Constraints Shape Convictions

Abstract

We explore in this paper the drivers of equity portfolio selection with an active strategy, to be understood as the combination of the use of a rewarded factor as an expected return, and a risk management made thanks to a risk model. The main message of the paper is that quantitative long only portfolios (built in a Markowitz-driven way) are high conviction portfolios, with few strong bets, and hence few effective (or non-zero) positions. In this respect, they share some similarity with discretionary stock pickers. This conclusion is valid either the objective of the fund is to follow the rewarded factor, either it is to target a risk strongly different from the risk of the market. We derive theoretical results and show that: (i) the long only constraint induce naturally a high concentration of the portfolio which becomes naturally parsimonious; (ii) closed-form formulas may be derived for the weights of the portfolio either for a Minimum Variance portfolio, either for a Managed Volatility portfolio with a rewarding factor; (iii) in the case of the Managed Volatility portfolio with a rewarded factor, the stocks that are selected are those that realize a trade-off between a low β and a high factor loading, both relatively to (respectively) a risk threshold and a factor threshold; (iv) those thresholds are endogenous in the sense that they depend on the risk structure and implicitly of the stocks finally kept in the portfolio, leading to a recursive procedure to select the stocks and the weights of the final portfolio. In particular, this means that the portfolio selection problem may be solved linearly instead of using an optimizer. A strong message following our results is also the essential role played by low β stocks, and by the interaction of the factor with the risk model, as the selectivity effect is higher for factors with a lower co-linearity with the risk model.

Keywords: Portfolio Selection, Long Only, Long-Short, Factors, Sparsity, Optimization, Number of positions, Benchmark

JEL Classification: G11