一类稀疏投资组合双层参数估计模型及其应用

doi:10.16381/j.cnki.issn1003-207x.2019.09.002

摘要/Abstract

摘要： 带基约束的投资组合问题是近年来投资组合领域的热点问题，但是参数不确定性直接影响了模型的效果。带基约束的投资组合问题所涉及的参数不仅包括以往研究认为非常重要的预期收益率，还包括控制投资组合规模的稀疏度，尤其是最优稀疏度估计方面的专门研究还十分匮乏。为了使带基约束的投资组合模型更好地为投资决策服务，本文从投资者效用出发，用双层规划的思想构建了带基约束的投资组合双层参数估计模型。然后根据模型的特点，设计了无导数优化算法框架，并基于ADMM对算法子问题进行求解。本文实验针对真实的市场数据给出了预期收益率和最优稀疏度的估计，接着通过与等权重策略和含上下界约束的均值-方差模型进行比较，说明了模型及算法的有效性和实用性。最后，将本文提出的双层参数估计模型推广到了更一般的形式。

关键词: 基数约束, 参数估计, 双层模型, 无导数优化, 最优稀疏度

Abstract: The portfolio selection problem with cardinality constraint is a hot issue in recent years. Much attention is paid to solve this problem because managing a portfolio with many assets often leads to high transaction costs and is a rather time-and energy-consuming experience in practice. However, the parameter uncertainty directly affects the effect of the model and makes it difficult to achive best performance of the portfolio. The parameters of the portfolio selection problem with cardinality constraint include not only the expected rate of return which was considered to be very important in previous studies, but also the sparsity controlling the size of the portfolio. Especially, there hasn't been much special research on estimation of portfolio's sparsity. In order to select optimal parameters better for investment decision, a sparse bi-level parameter estimation model is constructed for portfolio with cardinality constraint. The outer layer of the model is designed to maximize the utility of the portfolio which is measured by Sharpe ratio, while the inner layer of the model is designed to minimize the risk of a portfolio under a given expected return.The outer layer function is non-convex and non-smooth, so it is difficult to solve by the traditional gradient method. What's more, A framework of derivative-free optimization algorithm is built based on the model and we use ADMM to solve the sub problems. In particular, ADMM can get the closed-form solution of the sub-problem, which shows that the algorithm is effective.In this paper, numerical experiments are conducted by real-life data from OR-library and some Chinese stock markets(SSE 50、CSI 100)to estimate the expected rate of return and sparsity. The estimated sparsity is much smaller than the number of risky assets contained in each index,which will greatly ease the difficulty of portfolio management and decrease the transaction cost. The effectiveness of the model and algorithm is illustrated through comparison with the classical MV model with bounded constraints and the equal weight strategy(naive strategy).Specifically,our method improves the Sharpe ratio by at least 18.99% compared with 1/N strategy and by at least 39.03% compared with MV model which has upper and lower bounds out of sample. Finally, the proposed sparse bi-level parameter estimation model is extended to a general form, which can help estimate more accurate upper and lower bounds as well as other parameters. This is the direction of our future research.

Key words: cardinality constraints, parameter estimation, bi-level model, derivative-free optimization, optimal sparsity

中图分类号:

C931
F830

徐凤敏, 景奎, 梁循. 一类稀疏投资组合双层参数估计模型及其应用[J]. 中国管理科学, 2019, 27(9): 15-25.

XU Feng-min, JING Kui, LIANG Xun. A Class of Bi-level Parameter Estimation Models for Sparse Portfolios and Its Application[J]. Chinese Journal of Management Science, 2019, 27(9): 15-25.

参考文献

[1] Kourtis A, Dotsis G, Markellos R N. Parameter uncertainty in portfolio selection:Shrinking the inverse covariance matrix[J]. Journal of Banking & Finance, 2012, 36(9):2522-2531.
[2] Kan R, Zhou G. Optimal portfolio choice with parameter uncertainty[J]. Journal of Financial & Quantitative Analysis, 2007, 42(3):621-656.
[3] Dai Zhifeng, Wen Fenghua. Some improved sparse and stable portfolio optimization problems[J]. Finance Research Letters, 2018,DOI:10.1016/j.frl.2018.02.026
[4] Markowitz H. Portfolio selection[J]. Journal of Finance, 1952, 7(1):77-91.
[5] Woodside-Oriakhi M, Lucas C, Beasley J E. Heuristic algorithms for the cardinality constrained efficient frontier[J]. European Journal of Operational Research, 2011, 213(3):538-550.
[6] Branda M. Diversification-consistent data envelopment analysis with general deviation measures[J]. European Journal of OperationalResearch,2013,226(3):626-635.
[7] Chang T J, Meade N, Beasley J E, et al. Heuristics for cardinality constrained portfolio optimisation[J]. Computers & OperationsResearch,2000,27(13):1271-130.
[8] Li Duan, Sun Xiaoling, Wang Jun. Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection[J]. Mathematical Finance, 2010, 16(1):83-101.
[9] Best M J, Grauer R R. On the sensitivity of mean-variance-efficient portfolios to changes in asset means:Some analytical and computational results[J]. Review of Financial Studies, 1991, 4(2):315-342.
[10] Broadie M. Computing efficient frontiers using estimated parameters[J]. Annals of Operations Research, 1993, 45(1):21-58.
[11] Sharpe W. Capital asset prices:A theory of market equilibrium under conditions of risk[J]. Journal of Finance, 1964, 19(3):425-442.
[12] Fama E, French K. Common risk factors in the returns on stocks and bonds[J]. Journal of Financial Economics, 1993, 33(1):3-56.
[13] Qian E, Hua R, Sorensen E. Quantitative equity portfolio management:Modern technique and applications[M]. London:Chapman & Hall/crc,2007.
[14] Ross S. The arbitrage theory of capital asset pricing[J]. Journal of Economic Theory, 2015, 13(3):341-360.
[15] Black F, Litterman R. Global asset allocation with equities, bonds, and currencies[J].The Journal of Fixed Income, 1991(9):7-18.
[16] Black F, Litterman R. Global portfolio optimization[J].Financial Analysts Journal,1992, 48(5):28-43.
[17] Lejeune M. A var black-litterman model for the construction of absolute return fund-of-funds[J]. Quantitative Finance, 2011, 11(10):1489-1501.
[18] Ledoit O, Wolf M. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection[J]. Journal of Empirical Finance, 2003,10(5):603-621.
[19] Ledoit O, Wolf M. A well-conditioned estimator for large-dimensional covariance matrices[J]. Journal of Multivariate Analysis, 2004, 88(2):365-411.
[20] Rudd A. Optimal selection of passive portfolios[J]. Financial Management, 1980, 9(1):57-66.
[21] Montfort K, Draat L, Visser E. Index tracking by means of optimized sampling[J]. Journal of Portfolio Management, 2008, 34(2):143-151.
[22] Focardi S, Fabozzi F. A methodology for index tracking based on time-series clustering[J]. Quantitative Finance, 2004, 4(4):417-425.
[23] Dose C, Cincotti S. Clustering of financial time series with application to index and enhanced index tracking portfolio[J]. Physica A Statistical Mechanics & Its Applications, 2005, 355(1):145-151.
[24] Wang Meihua, Xu Fengmin, Dai Yuhong. An index tracking model with stratified sampling and optimal allocation[J]. Applied Stochastic Models in Business & Industry, 2017, 34(1):144-157.
[25] Ben-Tal A, Nemirovski A. Robust convex optimization[J]. Mathematics of Operations Research, 1998, 23(4):769-805.
[26] El-Ghaoui L, Lebret H. Robust solutions to least-square problems to uncertain data matrices[J]. SIAM Journal of Matrix Analysis&Applications,1996,51(4):543-55.
[27] Goldfarb D, Iyengar G. Robust portfolio selection problems[J]. Mathematics of Operations Research, 2003, 28(1):1-38.
[28] Lobo M, Boyd S.The worst-case risk of a portfolio[R]. Working Paper, Stanford University,2000.
[29] Tutuncu R, Koenig M. Robust asset allocation[J]. Annals of Operation Research, 2004, 132(1):157-187.
[30] Garlappi L, Uppal R, Wang T. Portfolio selection with parameter and model uncertainty:A multi-prior approach[J]. Review of Financial Studies, 2007,20(1):41-81.
[31] Gürkan G, Özge A, Robinson S. Sample-path solution of stochastic variational inequalities[J]. Mathematical Programming, 1999, 84(2):313-333.
[32] Lin Guihua, Xu Mengwei, Ye J. On solving simple bilevel programs with a nonconvex lower level program[J]. Mathematical Programming, 2014, 144(1-2):277-305.
[33] Zhang Dali, Lin Guihua. Bilevel direct search method for leader-follower problems and application in health insurance[J]. Computers & Operations Research, 2014, 41(1):359-373.
[34] Chen Xiaojun, Kelley C, Xu Fengmin,et al. A smoothing direct search method for monte carlo-based constrained nonsmooth nonconvex optimization[J]. SIAM Journal on Scientific Computing, 2018, 40(4):2174-2199.
[35] Merton R C. On estimating the expected return on the market:An exploratory investigation[J]. Journal of Financial Economics, 1980, 8(4):323-361.
[36] Chopra VK, Ziemba WT. The effect of errors in means, variances, and covariances on optimal portfolio choice[J]. Journal of Portfolio Management, 1993, 19(2):6-11.
[37] 许启发, 周莹莹, 蒋翠侠. 带有范数约束的CVaR高维组合投资决策[J]. 中国管理科学, 2017,(2):40-49.
[38] Zhao Zhihua, Xu Fengmin, Wang Meihua, et al. A sparse enhanced indexation model with l_1/2 norm and its alternating quadratic penalty method[J]. Journal of the Operational Research Society, 2018,(3):1-14.
[39] Beasley J E, Meade N, Chang, et al. An evolutionary heuristic for the index tracking problem[J]. European Journal of Operational Research,2003, 148(3), 621-643.
[40] Brodie J, Daubechies I, De M C, et al. Sparse and stable Markowitz portfolios[J]. Proceedings of the National Academy of Sciences of the United States of America, 2009,106(30):12267-12272.
[41] 楼振凯. 应急物流系统LRP的双层规划模型及算法[J]. 中国管理科学, 2017,(11):151-157.
[42] 李镜儒. 一种用于求解二次双层规划问题和双层证券投资组合优化模型的基于神经网络的混合算法[D].成都:电子科技大学, 2015.
[43] Sharp W F.Mutual fund performance[J]. Journal ofBusiness,1966,39(7):119-138.
[44] Kolda T G, Lewis R M, Torczon V. Optimization by direct search:New perspectives on some classical and modern methods[J]. SIAM Review, 2003, 45(3):385-482.
[45] Boyd S, Parikh N, Chu E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers[J]. Foundations & Trends in Machine Learning, 2010, 3(1):1-122.
[46] Blumensath T, Davies M E. Iterative hard thresholding for compressed sensing[J]. Applied & Computational Harmonic Analysis, 2009, 27(3):265-274.
[47] Xu Fengmin, Lu Zhaosong, Xu Zongben. An efficient optimization approach for a cardinality-constrained index tracking problem[J]. Optimization Methods & Software, 2016, 31(2):258-271.