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。
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.
[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.
