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

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 agiven 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,aproximal 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.