Minimum variance portfolio models mentioned by traditional literature were all brought out by exerting different penalties, and these models always ignored interactions between high-dimensional assets. When variables highly correlated, Lasso could not get suitable variables. So we referred to Li Caiyan and Li Hongzhe's way that combined graph structure penalty with MVP model with L1 penalty. And the correlations between assets by graph structure are described in order to get assets more accurately. The purpose of the paper is to provide theoretical and decisive inferences. The MVP model with graph structure is showed below: Minimize ωT∑ω+λ1‖ω‖1+λ2ωTLω Subject to ωT1p=1 For concise computation, we made λ=λ1+λ2 and θ=λ1/λ1+λ2, so the problem could be transmitted to this one: Minimize ωT∑ω+λθ‖ω‖1+λ(1-θ)ωTLω Subject to ωT1p=1 When ∑、L、λ and θ are given, it is the model that we fully focused on. Some properties of solves are given under the structure of regulated methods. With combinations between an coordinate algorithm and improvement of Yen, an efficient algorithm is also brought about.A-shares data of Shanghai Stock Market is used to conduct empirical analysis. Time varied from Jan.4th to Mar.29th in 2013. The data contained 56 observations and 818 stocks. The empirical analysis showed that: (1) The model with graph structure is better than other models in returns. (2) The model with graph structure is feasible in returns. (3) The sharp ratio of Graph1-MVP is positive, and the sharp ratio of Graph3-MVP is the highest among all models with negative ratio. (4) The probability of short is comparable small and this result could be due to the equation constriction that sum of weights is 1. (5)L1-MVP and Graph1-MVP could get the least number of assets. In summary, It is concluded that MVP model with graph structure can be advantageous in choose of asset. In the future, we could do further on the penalty of MVP model, or different penalties in other models could be considered to get better portfolios.
SU Zhi, QIN Lei, FANG Tong
. Merger Efficiency Evaluation of Two-Stage Production System Based on Non-Cooperative Game Theory[J]. Chinese Journal of Management Science, 2015
, 23(9)
: 65
-70
.
DOI: 10.16381/j.cnki.issn1003-207x.2015.09.008
[1] Markowitz H. Portfolio selection[J]. The Journal of Finance, 1952, 7(1): 77-91.
[2] Jagannathan R, Ma Tongshu. Risk reduction in large portfolios: Why imposing the wrong constraints helps[J]. The Journal of Finance, 2003, 58(4): 1651-1684.
[3] Tibshirani R. Regression shrinkage and selection via the lasso[J]. Journal of the Royal Statistical Society. Series B (Methodological), 1996,58(11): 267-288.
[4] Brodie J, Daubechies I, De Mol C, et al. Sparse and stable Markowitz portfolios[J]. Proceedings of the National Academy of Sciences, 2009, 106(30): 12267-12272.
[5] DeMiguel V, Garlappi L, Nogales F J, et al. A generalized approach to portfolio optimization: improving performance by constraining portfolio norms[J]. Management Science, 2009, 55(5): 798-812.
[6] Hoerl A E, Kennard R W. Ridge regression: Biased estimation for nonorthogonal problems[J]. Technometrics, 1970, 12(1): 55-67.
[7] Yen Y M. A note on sparse minimum variance portfolios and coordinate-wise descent algorithms. Working Paper, Available at SSRN 1604093, 2010.
[8] Zou Hui, Hastie T. Regularization and variable selection via the elastic net[J]. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2005, 67(2): 301-320.
[9] Carrasco M,Noumon N. Optimal portfolio selection using regularization. Working Paper, Université de Montréal, 2011.
[10] Fan Jianqing, Zhang Jingjin, Yu Ke. Vast portfolio selection with gross-exposure constraints[J]. Journal of the American Statistical Association, 2012, 107(498): 592-606.
[11] Fernandes M, Rocha G, Souza T. Regularized minimum variance portfolios using asset group information[J]. Working Paper, Bradford University,2011.
[12] Li Caiyan, Li Hongzhe. Variable selection and regression analysis for graph-structured covariates with an application to genomics[J]. The Annals of Applied Statistics, 2010, 4(3): 1498-1516.
[13] Huang Jian, Ma Shuangge, Li Hongzhe, et al. The sparse laplacian shrinkage estimator for high-dimensional regression[J]. Annals of Statistics, 2011, 39(4): 2021-2046.
[14] Friedman J, Hastie T,Höfling H, et al. Pathwise coordinate optimization[J]. The Annals of Applied Statistics, 2007, 1(2): 302-332.
[15] 丁志国,苏治,赵晶.资产系统性风险跨期时变的内生性:由理论证明到实证检验[J].中国社会科学, 2012, (4):87-108.
