Application of Quadratic Programming Using The Beale Method

Abstract

Portfolio stock is an alternative for investors in making investment decisions. Portfolio stock can be modeled into a quadratic programming model using mean variance Markowitz which can be solved using the Beale method. This research aims to find the optimal results of portfolio stock. The data used in this research are stock’s data on 10 companies that pay out the largest dividends in the IDX High Dividend 20 category for the 2022. There are two selected stock portfolios. The first stock portfolio is a negative covariance stock portfolio with a much different expected return level difference, namely the stock portfolio of PT Astra International Tbk and PT Indo Tambangraya Megah Tbk. The second stock portfolio is a negative covariance stock portfolio with a not much different expected return level difference, namely PT Bank Rakyat Indonesia (Persero) Tbk and PT Mitra Pinasthika Mustika Tbk. Based on Markowitz mean variance, the objective function for the first stock portfolio is Z=-1,305x_1-9,832x_2-0,137x_1x_2+0,681x_1^2+2,358x_2^2 with constraints x_1+x_2<=100  and x_1,x_2>=0 . After calculating using the Beale method, the expected profit rate is 3,432%  and the risk is 0,505% with the optimal proportion for PT Astra International Tbk is 75,063% and the optimal proportion for PT Indo Tambangraya Megah Tbk is 24,937%.  In addition, the objective function for the second stock portfolio is Z=-2,426x_1-2,528x_2-0,292x_1x_2+0,366x_1^2+0,508x_2^2 with constraints x_1+x_2<=100 and x_1,x_2>=0. With the Beale method, the expected profit rate is  2,471% and the risk is 0,141% with the optimal proportion for PT Bank Rakyat Indonesia (Persero) Tbk is 56,077% and the optimal proportion for PT Mitra Pinasthika Mustika Tbk is 43,923%.