Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure

Abstract

Portfolio selection is recognised as the birth-place of modern finance; portfolio optimisation has become a developed tool. However, efficient frontiers are piece-wisely made up by connected parabolic segments; such structure can be rendered only by parametric quadratic programming. Overlooking the structure can both get incomplete results and cause difficulties in e-constraint approaches or weighted-sums approaches. There has been no research to systematically parametrically compute efficient frontiers and report and analyse the structure up until now; in such an area, this article contributes to the literature. I utilise the software of parametric quadratic programming, set up practical portfolio selection models, build batches of 5-stock problems up to 1800-stock problems, analyse the structure, and report the findings. For example, the numbers of parabolic segments can quadratically increase with problem sizes, so fixed numbers of points are insufficient approximations of efficient frontiers. Contrary to common assumptions, an efficient frontier is not smooth in the presence of kinks. Moreover, I utilise the structure for rebalancing portfolios, propose two models to minimise rebalancing cost, transform them into linear programming or integer programming, and solve them. This article can help scholars and practitioners obtain a comprehensive picture of efficient frontiers and perceive the structure.

Keywords:

• Multi-objective
• quadratic programming
• parametric quadratic programming
• portfolio selection
• efficient set
• nondominated set

Acknowledgments

The author wishes to thank two anonymous referees for their highly constructive comments.

Notes

1 Also acknowledged is the research for general parametric quadratic programming (e.g., the studies of Benveniste 1981; Bank, Guddat, Klatte, Kummer, & Tammer, 1983; Ferreau, Kirches, Potschka, Bock, & Diehl, 2014); “general” means that the research is not specifically designed for portfolio optimization and that the parabolic-segment structure is not explicitly demonstrated.

2 Data source: 2018 Investment Company Fact Book, pp.13–14, Investment Company Institute, www.icifactbook.org, 2 September 2018.

3 Data source: Wharton Research Data Services, <https://wrds-web.wharton.upenn.edu/wrds/>, 15 February 2017.