带二阶随机占优约束的投资组合优化问题的松弛割平面法
|
摘要
结合割平面法和风险价值(Value at Rick,VaR)近似方法,提出了一种松弛的割平面法,用来求解带二阶随机占优(Second order dominance,SSD)约束的投资组合优化问题,该松弛算法的最优值和解是带SSD约束的投资组合优化问题的近似最优值和近似解。随着VaR置信度β趋近于1,该近似解的收敛性被证明。两个市场数据的实证研究表明,当置信度β小于但接近于1时,松弛算法求得的投资组合的表现要优于带SSD约束的优化问题求得的投资组合的表现,优于相应的市场指数。
By combining the cutting plane method with Value at Risk( VaR) approximation,a relaxation cutting plane( RCP) method for portfolio optimization with second order dominance( SSD) constraints is proposed. The optimal value and optimal solutions of the relaxation method are approximation of the optimal value and optimal solutions of portfolio optimization with SSD constraints. The convergence analysis of the approximated solutions has been investigated when VaR confidence level β close to 1. Moreover,it is shown by testing empirically on two sets of market data that when VaR confidence level β less but close to 1,these portfolios constructed by the relaxation method outperform both the portfolios constructed by solving portfolio optimization problems with SSD constraints and index portfolios.
By combining the cutting plane method with Value at Risk( VaR) approximation,a relaxation cutting plane( RCP) method for portfolio optimization with second order dominance( SSD) constraints is proposed. The optimal value and optimal solutions of the relaxation method are approximation of the optimal value and optimal solutions of portfolio optimization with SSD constraints. The convergence analysis of the approximated solutions has been investigated when VaR confidence level β close to 1. Moreover,it is shown by testing empirically on two sets of market data that when VaR confidence level β less but close to 1,these portfolios constructed by the relaxation method outperform both the portfolios constructed by solving portfolio optimization problems with SSD constraints and index portfolios.
引文
[1]MOSLER K C,SCARSINI M.Stochastic orders and decision under risk[M].Hayward,California:Institute of Mathematical Statistics,1991.
[2]WHITMORE G A,FINDLAY M C.Stochastic dominance:an approach to decision-making under risk[M].Lexington,Massachusetts:Daniel Collamore Heath and Company,1978.
[3]DENTCHEVA D,RUSZCZYN'SKI A.Optimization with stochastic dominance constraints[J].SIAM Journal on Optimization,2003,14(2):548-566.
[4]DENTCHEVA D,RUSZCZYN'SKI A.Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints[J].Mathematical Programming,2004,99(2):329-350.
[5]DENTCHEVA D,RUSZCZYN'SKI A.Portfolio optimization with stochastic dominance constraints[J].Journal of Banking&Finance,2006,30(2):433-451.
[6]HOMEM-DE-MELLO T,MEHROTRA S.A cutting-surface method for uncertain linear programs with polyhedral stochastic dominance constraints[J].SIAM Journal on Optimization,2010,20(3):1250-1273.
[7]SUN H,XU H,WANG Y.A smoothing penalized sample average approximation method for stochastic programs with second-order stochastic dominance constraints[J].Asia-Pacific Journal of Operational Research,2013,30(3):1-25.
[8]RUDOLF G,RUSZCZYN'SKI A.Optimization problems with second order stochastic dominance constraints:duality,compact formulations,and cut generation methods[J].SIAM Journal on Optimization,2008,19(3):1326-1343.
[9]FBIN C I,MITRA G,ROMAN D.Processing second-order stochastic dominance models using cutting-plane representations[J].Mathematical Programming,2011,130(1):33-57.
[10]SUN H,XU H,MESKARIAN R,et al.Exact penalization,level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints[J].SIAM Journal on Optimization,2013,23(1):602-631.
[11]KELLEY J E.The cutting-plane method for solving convex programs[J].Journal of the Society for Industrial and Applied Mathematics,1960,8(4):703-712.
[12]王伟,邓春林.随机占优理论及其在证券投资组合风险模型中的应用[J].湘潭大学学报(哲学社会科学版),2014,38(6):74-78.
[13]罗晓琴,成央金,杨柳,等.二阶随机占优约束保险资金资产组合优化[J].湖南工业大学学报,2013,27(2):99-104.
[14]吴敏,胡支军,陈璟.二阶随机占优约束下考虑偏度的投资组合模型[J].数学的实践与认识,2014,44(19):90-98.
[15]LESHNO M,LEVY H.Preferred by“all”and preferred by“most”decision makers:almost stochastic dominance[J].Management Science,2002,48(8):1074-1085.
[16]MARKOWITZ H.Portfolio selection[J].The Journal of Finance,1952,7(1):77-91.
[17]ROCKAFELLAR R T,URYASEV S.Optimization of conditional value-at-risk[J].Journal of Risk,2000,2(3):21-41.
[18]OGRYCZAK W,RUSZCZYN'SKI A.From stochastic dominance to mean-risk models:semideviations as risk measures[J].European Journal of Operational Research,1999,116(1):33-50.
[19]SUN H L,XU H F.Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints[J].Mathematical Programming,2014,143(1-2):31-59.
[20]ANDERSON E,XU H,ZHANG D.Confidence levels for CVaR risk measures and minimax limits[R].Sydney:The University of Sydney,2014.
[21]ARTZNER P,DELBAEN F,EBER J M,et al.Thinking coherently[J].Risk,1997,10(11):68-71.
[22]SHARPE W F.The sharpe ratio[J].Journal of Portfolio Management,1994,21(1):49-58.
[23]SORTINO F A,PRICE L N.Performance measurement in a downside risk framework[J].The Journal of Investing,1994,3(3):59-64.
[2]WHITMORE G A,FINDLAY M C.Stochastic dominance:an approach to decision-making under risk[M].Lexington,Massachusetts:Daniel Collamore Heath and Company,1978.
[3]DENTCHEVA D,RUSZCZYN'SKI A.Optimization with stochastic dominance constraints[J].SIAM Journal on Optimization,2003,14(2):548-566.
[4]DENTCHEVA D,RUSZCZYN'SKI A.Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints[J].Mathematical Programming,2004,99(2):329-350.
[5]DENTCHEVA D,RUSZCZYN'SKI A.Portfolio optimization with stochastic dominance constraints[J].Journal of Banking&Finance,2006,30(2):433-451.
[6]HOMEM-DE-MELLO T,MEHROTRA S.A cutting-surface method for uncertain linear programs with polyhedral stochastic dominance constraints[J].SIAM Journal on Optimization,2010,20(3):1250-1273.
[7]SUN H,XU H,WANG Y.A smoothing penalized sample average approximation method for stochastic programs with second-order stochastic dominance constraints[J].Asia-Pacific Journal of Operational Research,2013,30(3):1-25.
[8]RUDOLF G,RUSZCZYN'SKI A.Optimization problems with second order stochastic dominance constraints:duality,compact formulations,and cut generation methods[J].SIAM Journal on Optimization,2008,19(3):1326-1343.
[9]FBIN C I,MITRA G,ROMAN D.Processing second-order stochastic dominance models using cutting-plane representations[J].Mathematical Programming,2011,130(1):33-57.
[10]SUN H,XU H,MESKARIAN R,et al.Exact penalization,level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints[J].SIAM Journal on Optimization,2013,23(1):602-631.
[11]KELLEY J E.The cutting-plane method for solving convex programs[J].Journal of the Society for Industrial and Applied Mathematics,1960,8(4):703-712.
[12]王伟,邓春林.随机占优理论及其在证券投资组合风险模型中的应用[J].湘潭大学学报(哲学社会科学版),2014,38(6):74-78.
[13]罗晓琴,成央金,杨柳,等.二阶随机占优约束保险资金资产组合优化[J].湖南工业大学学报,2013,27(2):99-104.
[14]吴敏,胡支军,陈璟.二阶随机占优约束下考虑偏度的投资组合模型[J].数学的实践与认识,2014,44(19):90-98.
[15]LESHNO M,LEVY H.Preferred by“all”and preferred by“most”decision makers:almost stochastic dominance[J].Management Science,2002,48(8):1074-1085.
[16]MARKOWITZ H.Portfolio selection[J].The Journal of Finance,1952,7(1):77-91.
[17]ROCKAFELLAR R T,URYASEV S.Optimization of conditional value-at-risk[J].Journal of Risk,2000,2(3):21-41.
[18]OGRYCZAK W,RUSZCZYN'SKI A.From stochastic dominance to mean-risk models:semideviations as risk measures[J].European Journal of Operational Research,1999,116(1):33-50.
[19]SUN H L,XU H F.Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints[J].Mathematical Programming,2014,143(1-2):31-59.
[20]ANDERSON E,XU H,ZHANG D.Confidence levels for CVaR risk measures and minimax limits[R].Sydney:The University of Sydney,2014.
[21]ARTZNER P,DELBAEN F,EBER J M,et al.Thinking coherently[J].Risk,1997,10(11):68-71.
[22]SHARPE W F.The sharpe ratio[J].Journal of Portfolio Management,1994,21(1):49-58.
[23]SORTINO F A,PRICE L N.Performance measurement in a downside risk framework[J].The Journal of Investing,1994,3(3):59-64.