Markowitz Random Set and Its Application to the Paris Stock Market Prices

Abstract

In this paper, we will combine random set theory and portfolio theory, through the estimation of the lower bound of the Markowitz random set based on the Mean-Variance Analysis of Asset Portfolios Approach, which represents the efficient frontier of a portfolio. There are several Markowitz optimization approaches, of which we denote the most known and used in the modern theory of portfolio, namely, the Markowitz's approach, the Markowitz Sharpe's approach and the Markowitz and Perold's approach, generally these methods are based on the minimization of the variance of the return of a portfolio. On the other hand, the method used in this paper is completely different from those denoted above, because it is based on the theory of random sets, which allowed us to have the mathematical structure and the graphic of the Markowitz set. The graphical representation of the Markowitz set gives us an idea of the investment region. This region, called the investment zone, contains the stocks in which the rational investor can choose to invest. Mathematical and statistical estimation techniques are used in this paper to find the explicit form of the Markowitz random set, and to study its elements in function of the signs of the estimated parameters. Finally, we will apply the results found to the case of the returns of a portfolio composed of 200 assets from the Paris Stock Market Prices. The results obtained by this simulation allow us to have an idea on the stocks to recommend to the investors. In order to optimize their choices, these stocks are those which will be located above the curve of the hyperbola which represents the Markowitz set.