The first group includes DIS, KORS, OCN, YELP, FORD, X, ETP, EMG, BBBY and AAPL. [7], we can have the following equivalent definition of variance:where . In optimization, condition (19) is known as the complementary condition. (iii)“” stands for the i-th row vector of corresponding to , if . (iv)“RCMVP” stands for the no-short-sales-constrained minimum-variance portfolio model with a multivariate regression form. An optimal set of weights is one in which the portfolio achieves an acceptable baseline expected rate of return with minimal volatility. ( Log Out /  And then we assign initial weights of stocks as equal. cons = ({'type': 'eq', 'fun': lambda x: np.sum(x)-1.0}) (vi)“-RMVP” stands for the -regularization minimum-variance portfolio model. There are several points I want to mention. In this section, we will investigate the difference and relation between the -regularization minimum-variance portfolio model and the -norm-constrained minimum-variance portfolio. (3)For , the optimal portfolios by -NCMVP are the same as those by MVP. Make sure you enter all the required information, indicated by an asterisk (*). Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. Jagannathan and Ma [5] imposed a no-short-sale constraint to the minimum-variance portfolio (MVP) model and gave some insightful explanations and demonstrations why the “wrong” constraint helps find a solution with better out-of-sample performance. That is, the upper bound of in -RMVP is . Moreover, it is obvious that(2)Suppose that the two weight vectors and are minimizers for the objective function in -RMVP, corresponding to the values and , respectively, and both satisfy the constraint . Minimum variance portfolio is an attractive theoretical op- ... not enough to build a real-life minimum vari-ance portfolio. Then, we select any . The authors would like to thank Professors Grant and Boyd for providing the optimization package CVX for numerical experiments. (2)Considering , , , and , conditions (15)–(19) are satisfied. In our project, we are going to use the mean variance method to do the portfolio optimization. , we add an -regularization term to the objective function in to obtain the following -regularization minimum-variance portfolio model (-RMVP) as follows: where and τ is a regularization parameter that allows us to adjust the relative importance of the penalization in our optimization. If all the are nonnegative, but some of the are negative, then we have . And then we allocate weights of assets two hours after the market opens. M. Andrecut, “Portfolio optimization in R,” 2013, V. DeMiguel, L. Garlappi, and R. Uppal, “Optimal versus naive diversification: how inefficient is the 1/, Z. Dai, H. Zhu, and F. Wen, “Two nonparametric approaches to mean absolute deviation portfolio selection model,”, A. Ling, J. Sun, and X. Yang, “Robust tracking error portfolio selection with worst-case downside risk measures,”, K. Natarajan, D. Pachamanova, and M. Sim, “Incorporating asymmetric distributional information in robust value-at-risk optimization,”, L. Yan, F. Xu, J. Liu, K. L. Teo, and M. Lai, “Stability strategies of demand-driven supply networks with transportation delay,”, X. J. Yu, S. Y. Xie, and W. J. Xu, “Optimal portfolio strategy under rolling economic maximum drawdown constraints,”, L. Zhang and Z. Li, “Multi-period mean-variance portfolio selection with uncertain time horizon when returns are serially correlated,”, S. Zhu and M. Fukushima, “Worst-case conditional value-at-risk with application to robust portfolio management,”, V. K. Chopra and W. T. Ziemba, “The effect of errors in means, variances, and covariances on optimal portfolio choice,”, L. E. Ghaoui, M. Oks, and F. Oustry, “Worst-case value-at-risk and robust portfolio optimization: a conic programming approach,”, R. C. Green and B. Hollifield, “When will mean-variance efficient portfolios be well diversified?”. In our numerical experiments, we select 10 assets from S&P 500 with 500 historical daily stock return data for the illustration purpose. (2)For , the optimal portfolios by -NCMVP are the same as those by MVP. In our numerical experiments, the tested portfolio models have the following meanings:(i)“MVP” stands for the minimum-variance portfolio model. The test results of the -NCMVP are given in Table 3. The reason for this is . ( Log Out /  In our project, we are going to use the mean variance method to do the portfolio optimization. And then we calculate the return data basing on the original data. But as for the difference between the return of group1 and return of benchmark is a little larger than that in group 2. The answers to these questions obviously have important bearings on the debate about portfolio selection problems. Ledoit and Wolf [3, 4] proposed replacing the sample covariance matrix with a weighted average of the sample covariance matrix and a low-variance target estimator matrix, . This implies -NCMVP with is equivalent to the no-short-sale-constrained minimum-variance minimization problem since -NCMVP with has the sparsest solution. However, in this paper, by analyzing the KKT conditions (necessary and sufficient ones) of Lagrangian functions, we investigate the relation between the portfolio weight -norm-constrained method and the objective function -regularization method in minimum-variance portfolio selection problems and give the range of parameters for the two models and the corresponding relationship of parameters. [6] depends on solving the traditional minimum-variance model with the additional norm constraint on the portfolio-weight vector. Copyright © 2019 Zhifeng Dai. We apply these models to construct optimal portfolios and test the proposed propositions by employing real market data. # min var optimization We put the 4 stocks in each group. Note that because of the convexity of the norm , solving the above models is a easy task for which the standard software solution exists. All the codes were run on Matlab 2015a. The rest of this paper is organized as follows: In Section 2, we introduce some existing minimum-variance portfolio models. (2) What is the range of parameters for the two models? The second one is that the stocks we pick up are not suitable. We define R as the matrix, the row of which equals , where . ». R = np.matrix('14; 12; 15; 7')/100 “CMVP” stands for the no-short-sales-constrained minimum-variance portfolio model (Jagannathan and Ma, 2003).

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