带有基数约束的指数跟踪问题及其粒子群算法求解

Particle swarm optimization approach to index tracking problem with a cardinality constraint

随着指数衍生产品日益受到重视,指数化投资组合常被传统的消极基金管理者或机构所采用,而用有限的资金按指数构成比例进行投资显然是不现实的,所以指数的最优误差追踪就显得更加重要。将追踪误差定义为证券投资组合收益率与所追踪的指数基准收益率之差的均值平方和的平方根,建立了基数约束(即总资产数不超过某个特定整数K)下的跟踪误差最小化模型。由于引入显示的基数约束使得该模型是一个非线性混合整数规划问题,传统算法难以有效求解,为此设计了一个粒子群算法求解基数约束下的指数跟踪模型,实际算例表明,算法是有效的。

With the index derivative production are paid attention to day and day, index portfolio are often used by investor or financial setup, but it is not real that investors invest index according to the proportion of index fund-construction with limited fund in number. So the minimum tracking error of index is becoming increasingly important. The tracking error is defined as the standard deviations of the returns＇ differences （i. e. the root-mean-squared de- viation） between the portfolio and the benchmark of index, and portfolio optimization model of the minimum tracking error is established with cardinality constraints. This portfolio optimization model, with a restriction on the cardinality constraints, is a nonlinear and mixed integer programming problem, for which efficient algorithms do not exist. A particle swarm optimization （PSO） algorithm is proposed to solve the tracking error minimization problem. The numerical results show that PSO approach is successful in portfolio optimization.