基于CVaR投资组合优化问题的非光滑优化方法

doi:10.16381/j.cnki.issn1003-207x.2017.10.002

中国管理科学 ›› 2017, Vol. 25 ›› Issue (10): 11-19.doi: 10.16381/j.cnki.issn1003-207x.2017.10.002

基于CVaR投资组合优化问题的非光滑优化方法

张清叶^1,2, 高岩¹

1. 上海理工大学管理学院, 上海 200093;
2. 河南工学院, 河南新乡 453003

收稿日期:2015-06-10 修回日期:2017-01-04 出版日期:2017-10-20 发布日期:2017-12-15
通讯作者: 张清叶(1978-),女(汉族),河南新乡人,河南工学院讲师,博士,研究方向:投资组合优化、非光滑优化,E-mail:zhangqingye123@163.com. E-mail:zhangqingye123@163.com
基金资助:
国家自然科学基金资助项目（11171221）；高等学校博士学科点专项科研基金项目（20123120110004）；上海市一流学科项目（XTKX2012）

A Nonsmooth Optimization Method for Portfolio Optimization Based on CVaR

ZHANG Qing-ye^1,2, GAO Yan¹

1. School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. Henan Institute of Techonology, Xinxiang, 453003, China

Received:2015-06-10 Revised:2017-01-04 Online:2017-10-20 Published:2017-12-15

摘要/Abstract

摘要： 对选定的风险资产进行组合投资，以条件风险价值（CVaR）作为度量风险的工具，建立单期投资组合优化问题的CVaR模型。目标函数中含有多重积分与极大值函数，首先利用蒙特卡洛模拟产生情景矩阵将多重积分计算转化成求和运算，之后目标函数为分片光滑（非光滑）函数，设计相应的非光滑优化方法并给出其收敛性分析。初步的数值试验表明了本文算法的有效性。

关键词: 投资组合, 条件风险价值, 非光滑优化, 束方法

Abstract: Portfolio selection is an important issue in finance. It aims to determine how to allocate one's wealth among a given asset pool to maximize the return and minimize the risk. Different from the accepted return, there are many risk measures. Nevertheless, among all risk measures, conditional value-at-risk (CVaR) is widely accepted, and in this paper it is adopted. As there is a nonsmooth term in the expression of CVaR, an optimization problem containing CVaR cannot be solved by classical algorithms based on gradient. Though there is an extensive literature on tackling optimization problem containing CVaR, such as linear programming method, intelligent optimization algorithms and nonsmooth optimization methods, etc, literatures on solving this problem by bundle method are scarce. And the literature on this aspect in this paper is enriched. That is, a bundle method is investigated for portfolio selection problem based on CVaR. Specifically, a single-period portfolio optimization model, which takes CVaR as the objective function coupled with a prescribed minimal level of the expected return, is formulated at first. By exploring the structure of the model, a proximal bundle method is proposed. At the same time, the convergence analysis of the method is given as well. Finally, an illustrative numerical example is presented, where assets' returns are assumed to be normally distributed and their mean and the covariance matrix known. By Monte Carlo sampling method, several scenario matrices are generated. Then, not only the bundle method, but linear programming method, subgradient algorithm, genetic algorithm and smoothing method are adopted to solve the model as well. By comparing the results of the different methods, conclusions are drawn:linear programming method and subgradient algorithm are inefficient, genetic algorithm, smoothing method and bundle method are feasible. Further, among three feasible algorithms, bundle method takes the least amount of CPU time. So, the proximal bundle method is efficient and can be regarded as a new solution method for not only portfolio optimization problem but other problems containing CVaR.

Key words: portfolio, Conditional Valu-at-Risk (CVaR), nonsmooth optimization, bundle method

中图分类号:

F830
O221

张清叶, 高岩. 基于CVaR投资组合优化问题的非光滑优化方法[J]. 中国管理科学, 2017, 25(10): 11-19.

ZHANG Qing-ye, GAO Yan. A Nonsmooth Optimization Method for Portfolio Optimization Based on CVaR[J]. Chinese Journal of Management Science, 2017, 25(10): 11-19.

参考文献

[1] Markowitz H M. Portfolio selection[J]. Journal of Finance, 1952, 7(1):77-91.

[2] 张鹏,张卫国,张逸菲.具有最小交易量限制的多阶段均值_半方差投资组合优化[J].中国管理科学,2016,24(7):11-17.

[3] Liu Mingming, Gao Yan. An algorithm for portfolio selection in a frictional market[J]. Applied Mathematics and Computation, 2006,182(2):1629-1638.

[4] 刘彬蔚,房勇.引入股指期货的两阶段投资组合选择模型[J].中国管理科学,2015,23(S1):419-423.

[5] Jorion P. Value at risk: The new benchmark for controlling market risk [M]. BurrRidge.Illinois:Irwin Professional Publication,1997.

[6] Artzner P, Delbaen F, Eber J M, et al. Coherent measures of risk[J]. Mathematical Finance, 1999, 9(3): 203-228.

[7] Rockafellar R T, Uryasev S. Optimization of conditional value-at-risk [J]. Journal of risk, 2000, 2: 21-42.

[8] 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,2017,25(2):40-49.

[9] Zhang Qingye, Gao Yan. Optimal consumption-portfolio problem with CVaR constraints[J]. Chaos, Solitons and Fractals, 2016, 91:516-521.

[10] Palmquist J, Uryasev S, Krokhmal P.Portfolio optimization with conditional value-at-risk objective and constraints [J]. Journal of risk, 2002, 4:43-68.

[11] 陈剑利,李胜宏. CVaR风险度量模型在投资组合中的运用[J].运筹与管理, 2004, 13(1):96-99.

[12] 朱文娟,高岩.考虑分段交易费用的CVaR投资组合优化[J].统计与决策,2008,(24):36-38.

[13] Tong Xiaojiao, Qi Liqun, Wu F, et al. A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset [J].Applied Mathematics and Computation, 2010, 216(6):1723-1740.

[14] Meng Fanwen, Sun Jie, Goh M. A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure[J].Journal of Optimization Theory and Applications, 2010, 146(2): 399-418.

[15] Lim C, Sherali H D, Uryasev S. Portfolio optimization by minimizing conditional value-at -risk via nondifferentiable optimization[J]. Computational Optimization and Applications, 2010, 46(3):391-415.

[16] Takano Y, Nanjo K, Sukegawa N, et al. Cutting plane algorithms for mean-CVaR portfolio optimization with nonconvex transaction costs[J].Computational Management Science,2015,12(2):319-340.

[17] Beliakov G, Bgirov A. Nonsmooth optimization methods for computation of the Conditional Value-at-Risk and portfolio optimization [J]. Optimization, 2006, 55(5-6):459-479.

[18] Pang Liping,Chen Shuang, Wang Jinhe. Risk management in portfolio applications of non-convex stochastic programming[J]. Applied Mathematics and Computation, 2015, 258:565-575.

[19] 李国成,肖庆宪. CVaR投资组合问题求解的一种混合元启发式搜索算法[J].运筹与管理,2014,23(6):229-235.

[20] Qin Quande, Li Li, Cheng Shi. A novel hybrid algorithm for mean-CVaR portfolio selection with real-world constraints[M]//Tan Ying,Shi Yuhui,Coello CAC.Advances in swarm intelligence. Springer International Publishing, 2014.

[21] Mäkelä M M. Survey of bundle method for nonsmooth optimization[J]. Optimization methods and software, 2002,17(1):1-29.

[22] 高岩.非光滑优化[M].北京:科学出版社,2008.

[23] Kiwiel K C. Method of descent for nondifferentiable optimization[M].Berlin:Springer Verlag, 1985.

[24] Belloni A. Introduction to bundle methods[J]. Lecture Notes for IAP, 2005.

基于CVaR投资组合优化问题的非光滑优化方法

A Nonsmooth Optimization Method for Portfolio Optimization Based on CVaR

PDF (PC)

可视化

被引次数

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	金博轶, 闫庆悦, 于文广. 养老金统筹账户与个人账户最优组合策略研究[J]. 中国管理科学, 2020, 28(2): 48-57.
[2]	刘维奇, 张燕. 资产定价与劳动成本占比[J]. 中国管理科学, 2020, 28(12): 1-11.
[3]	费为银, 张繁红, 杨晓光. 通胀对高管的股权激励和工作努力策略的影响[J]. 中国管理科学, 2020, 28(11): 1-11.
[4]	金秀, 尘娜, 王佳. 安全投资转移视角下的跨行业投资组合模型及实证[J]. 中国管理科学, 2020, 28(11): 12-22.
[5]	朱鹏飞, 唐勇, 钟莉. 基于小波-高阶矩模型的投资组合策略——以国际原油市场为例[J]. 中国管理科学, 2020, 28(10): 24-35.
[6]	郭范勇, 潘和平. 基于β系数优化的动态投资组合策略研究[J]. 中国管理科学, 2019, 27(7): 1-10.
[7]	姚鸿, 王超, 何建敏, 李亮. 银行投资组合多元化与系统性风险的关系研究[J]. 中国管理科学, 2019, 27(2): 9-18.
[8]	李爱忠, 汪寿阳, 彭月兰. 考虑随机利率和通货膨胀的连续时间资产组合选择[J]. 中国管理科学, 2019, 27(2): 61-70.
[9]	陶莉, 高岩, 朱红波, 曹磊. 有可再生能源和电力存储设施并网的智能电网优化用电策略[J]. 中国管理科学, 2019, 27(2): 150-157.
[10]	周忠宝, 任甜甜, 肖和录, 吴士健, LIU Wenbin. 基于相对财富效用的多阶段投资组合博弈模型[J]. 中国管理科学, 2019, 27(1): 34-43.
[11]	彭子衿, 徐维军. 基于股价预测的泛证券投资组合策略[J]. 中国管理科学, 2018, 26(9): 1-10.
[12]	王建建, 何枫, 吴子轩, 陈丽莉. 改进区间可接受度的证券投资组合区间二次规划模型[J]. 中国管理科学, 2018, 26(9): 11-18.
[13]	谭浩, 吴卫星. 股东为什么在解禁后会减持股票?——基于背景风险分析的股票解禁效应研究[J]. 中国管理科学, 2018, 26(7): 9-17.
[14]	凌爱凡, 陈骁阳. 嵌入GARCH波动率估计的B1ack-Litterman投资组合模型[J]. 中国管理科学, 2018, 26(6): 17-25.
[15]	刘炳越, 姬强, 范英. 黄金是否为原油的“避险天堂”?——基于组合收益及其波动视角[J]. 中国管理科学, 2018, 26(11): 1-10.