Portfolios Optimization under Constraints

Abstract

This article selects 10 companies in the financial sector, energy sector and consumption sector, as well as SPX500 index. This paper uses two models, not only the Markowitz model but also the index model, to calculate the correlation coefficient matrix, minimum variance, maximum Sharpe ratio, capital allocation line and so on to analyze the return rate and volatility of 10 specific companies. Four limitations were calculated for Markowitz model and Index model respectively and the two models were compared under the same constraints. Because common financial constraints and specific industries are rarely noticed in reality, the results of this paper reflect the following three aspects: First, in order to strike a balance between risk and return, SPX is an investment worth considering due to its high correlation coefficient; the second is that for certain investors with added constraints, the capital allocation line performs relatively poorly, as does the minimum variance boundary. Thirdly, because the Markowitz model uses stock's covariance while the beta and alpha of stocks are components that the index model uses to construct a portfolio, the results show that under certain risk conditions, Markowitz model is inferior to index model in pursuing maximum return and minimum risk under certain return conditions.