Expected mean return—standard deviation efficient frontier approximation with low‐cardinality portfolios in the presence of the risk‐free asset

Abstract

AbstractIn the expected mean return, standard deviation portfolio selection problem, the first step is usually to derive the set of efficient portfolios, which in the space of objective function values is represented by the efficient frontier. With modern methods and software, it is an easy task even for thousands of assets provided that the problem is continuous. However, investors often introduce the requirement to limit the number of assets in portfolios (portfolio cardinality). The resulting mixed‐integer quadratic formulations are computationally much more complex. In this work, we assume that besides risky assets, the risk‐free asset is available to the investor, and short selling is not allowed. Since in this case the efficient frontier cannot be directly derived by a quadratic solver, we propose a naive but intuitive heuristic to approximate the efficient frontier in the presence of the risk‐free asset. In contrast to the general‐purpose evolutionary heuristics, we exploit the underlying mechanism of portfolio composition. We show by numerical experiments that in large‐scale instances it works well, even compared to a state‐of‐the‐art evolutionary multiobjective optimization algorithm. Moreover, the heuristic produces portfolios of remarkably limited cardinalities.