摘要： 考虑到方差、下半方差和绝对偏差等度量投资组合风险的局限性以及单阶段投资决策不符合投资者的实际投资行为等因素，本文将风险价值（Value-at-Risk，简称VaR）作为风险度量标准应用到多阶段投资组合优化中．由于中国股票市场不允许卖空，因此本文在不允许卖空的情况下，在约束条件中同时考虑了交易费用和投资比例，建立了一个均值--VaR多阶段投资组合优化模型．考虑到粒子群算法具有收敛速度快，结构简单以及需要调控的参数比较少等优点，运用带有罚函数处理机制的粒子群算法对新建立的多阶段投资组合优化模型进行求解．求解得到了不同路径下各阶段资产的最优投资策略，从运算结果可以看出，在不同的投资路径下投资者的投资行为基本一致，在第一阶段对自己看好的股票买入，经过第一阶段股市的波动，在第二阶段对自己看好的股票继续买入，对不看好的股票不买入或者直接卖出，这种投资行为符合投资者的实际投资行为，说明本文所提出的模型具有合理性．

Abstract: Taking into account the limitations of risk measures such as variance, lower semi-variance and absolute deviation, and the fact that single-stage investment decision does not accord with investors' actual investment behavior, Value-at-Risk (VaR) is applied in this paper as a measure of risk for multi-stage portfolio optimization. Because short selling is not allowed in Chinese stock market, we consider both transaction costs and investment proportions in constraints and establish a multi-stage mean-VaR portfolio optimization model. Considering that PSO has the advantages of fast convergence, simple structure and less parameters to be regulated, the proposed multi-stage portfolio optimization model is solved by using PSO with penalty function processing mechanism. The optimal investment strategy at each stage under different paths is obtained. It can be seen from the results that the investor's investment behavior is consistent under different investment paths, buying the stocks they are optimistic about in the first stage. After the fluctuation of the stock market in the first stage, the investor continues to buy the stocks which are optimistic in the second stage. Meanwhile, the investor does not buy or sell stocks that are not good. This kind of investment behavior accords with the actual investment behavior of investors. Thus the proposed model is reasonable.

Key words: multi-stage, particle swarm optimization, Value-at-Risk, transaction costs 陈 涛, 程希骏, 马利军, 符永健. R-藤Pair Copula模型下的投资组合最优套期保值比例研究 [J]. 工程数学学报, 2018, 35(6): 611-621. 董 艳. 基于摄动方法的关卡期权定价及其误差分析 [J]. 工程数学学报, 2018, 35(4): 375-384. 林建伟, 李慧敏. 跳扩散模型下具有违约资产重组公司债券的定价 [J]. 工程数学学报, 2017, 34(4): 367-374. 高岳林, 余雅萍. 基于混合量子粒子群优化的投资组合模型及实证分析 [J]. 工程数学学报, 2017, 34(1): 21-30. 地址：西安市咸宁西路28号西安交通大学数学与统计学院
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