Parametrically computing efficient frontiers and reanalyzing efficiency-diversification discrepancies and naive diversification

Abstract

Portfolio selection is recognized as the birth-place of modern finance. Weighted-sums methods or e-constraint methods are normally utilized for portfolio optimization, but the results are only approximations of efficient frontiers. One concern of portfolio selection is efficiency-diversification discrepancies that efficient frontiers lack diversification. Scholars typically analyze the discrepancies by using weighted-sums methods or e-constraint methods, studying only a specific portfolio, and utilizing small-scale portfolio selection. Some scholars find that portfolio selection is not consistently better than naive diversification. We utilize parametric quadratic programming, exhaustively sample US stocks, build batches of 5-stock problems up to 1800-stock problems, obtain the structure of (whole) efficient frontiers, and propose new diversification measures on the basis of the structure. We find that (1) setting upper bounds can be more effective in changing diversification status than setting right-hand sides or setting the numbers of constraints can, (2) portfolio selection can substantially outperform naive diversification at least in sample so the cost of naive diversification can be prohibitive, and (3) efficiency-diversification discrepancies can arise due to efficient frontiers’ nature of having relatively small numbers of stocks and cannot be easily reconciled.

Keywords:

• Multiple-objective optimization
• portfolio optimization
• parametric quadratic programming
• efficient set
• diversification

Notes

1 For notations, we use normal symbols (e.g., k) to denote scalars and use bold-face symbols (e.g., $\mathbf{x}$ or $\mathbf{\Sigma }$) to denote vectors or matrices with normal symbols (e.g., x_i) for the elements. We define $\mathbf{x}\ge \mathbf{y}$ in the normal style (i.e., $x_i\ge y_i,i=1\dots n$, and $\mathbf{x}\ne \mathbf{y}$).

2 With $\mathbf{\Sigma }$ of (2) as positive semidefinite (as documented by Brockwell and Davis 1987, pp. 33–35), $z_1^n={(\frac{1}{\mathbf{n}})}^T\mathbf{\Sigma }\frac{1}{\mathbf{n}}\ge 0$. If $z_1^n=0$ (very rare), ${\mathbf{z}}^n$ is nondominated of (2) because $z_1^n\ge 0$, and $\frac{1}{\mathbf{n}}$ of (3) is a perfect combination of efficiency and diversification.

Additional information

Funding

This work was supported by the National Social Science Fund of China 2018 under Grant 18BGL063.