AN OBTAINABLE AND EFFICIENT SET IN THE STANDARD MEAN-VARIANCE SMALL PORTFOLIO SELECTION MODEL: A NON-MARKOWITZ APPROACH

Abstract

In this study, we performed an innovative approach to analyzing an obtainable and efficient set on a small portfolio using 3-month historical data on the daily movements of stock prices of industrial companies, represented by the Dow Jones Industrial Average Index (DJIA). Under a small portfolio, we mean a two-member, three-member, and a four-member portfolio. In the analysis of the two-member portfolio, the explicit form of the variance function concerning the portfolio return was determined. The analytical function of the sixth-degree polynomial was obtained, and the diagram of this function was a parabola of the same order. An efficient portfolio set is defined as the part of a smooth curve where the first derivative of the function is greater than or equal to zero. Alternatively, the explicit form of the variance function concerning the portions of portfolio securities was determined, which has a quadratic form whose diagram is a quadratic parabola. The efficient set, in this case, determined by the implicit equation of variables, which represents the portions of the securities, will be part of the straight line that forms the constraint of the portfolio. The identical procedure was conducted in the analysis of the three-member and four-member portfolios. The partitive portfolio model was defined as every subset of the general portfolio with a growing number of members. This study of the partitive portfolios significantly simplifies the procedure for obtaining the efficient set of multi-member portfolios.

Keywords

• mean-variance
• obtainable set
• efficient set
• partitive portfolio