基于相对财富效用的多阶段投资组合博弈模型
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摘要
现有投资组合优化模型普遍假设投资者之间是相互独立的,然而在实际投资过程中投资者往往是相互影响的,尤其是对于机构投资者而言,考虑这种相互影响显得尤为重要。本文基于多阶段投资组合理论和纳什均衡思想,考虑投资者决策之间的影响关系,以每个投资者自身终端相对财富的期望效用水平最大化为目标,构建了多阶段投资组合博弈模型,并给出了纳什均衡投资策略和相应值函数的解析解。最后,采用累计经验分布函数和确定性等价等指标,对纳什均衡投资策略与传统策略进行了比较,结果表明纳什均衡策略更具优越性。
Most existing portfolio optimization models assume that investors are independent.However,the investors especially institutional investors,usually affect each other in actual investment.Based on the portfolio optimization and Nash game theories,the competition relationship is considered into investors' decision-making,and a multi-period portfolio game model is constructed by maximizing the expected utility of the relative terminal wealth of each competitor.The close forms of the Nash equilibrium investment strategy and the corresponding value function are obtained.By using the empirical cumulative distribution function and the certainty equivalent,the Nash equilibrium investment strategy is compared with the traditional strategy in our simulations.The results show that the Nash equilibrium strategy is more effective for investors.
Most existing portfolio optimization models assume that investors are independent.However,the investors especially institutional investors,usually affect each other in actual investment.Based on the portfolio optimization and Nash game theories,the competition relationship is considered into investors' decision-making,and a multi-period portfolio game model is constructed by maximizing the expected utility of the relative terminal wealth of each competitor.The close forms of the Nash equilibrium investment strategy and the corresponding value function are obtained.By using the empirical cumulative distribution function and the certainty equivalent,the Nash equilibrium investment strategy is compared with the traditional strategy in our simulations.The results show that the Nash equilibrium strategy is more effective for investors.
引文
[1]Markowitz H.Portfolio selection[J].Journal of Finance,1952,7(1):77-91.
[2]Smith K V.A transition model for portfolio revision[J].Journal of Finance,1952,22(3):425-439.
[3]Li Duan,Ng W L.Optimal dynamic portfolio selection:Multiperiod mean-variance formulation[J].Mathematical Finance,2000,10(3):387-406.
[4]Zhou X Y,Li Duan.Continuous-time mean-variance portfolio selection:A stochastic LQ framework[J].Applied Mathematics&Optimization,2000,42(1):19-33.
[5]Yin G,Zhou Xunyu.Markowitz’s mean-variance portfolio selection with regime switching:From discretetime models to their continuous-time limits[J].IEEETransactions on Automatic Control,2004,49(3):349-360.
[6]Zhu Shushang,Li Duan,Wang Shouyang.Risk control over bankruptcy in dynamic portfolio selection:A generalized mean-variance formulation[J].IEEE Transactions on Automatic Control,2004,49(3):447-457.
[7]郭文旌,胡奇英.不确定终止时间的多阶段最优投资组合[J].管理科学学报,2005,8(2):13-19.
[8]Wu Huiling,Li Zhongfei.Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow[J].Insurance:Mathematics and Economics,2012,50(3):371-384.
[9]Shen Yang,Zhang Xin,Siu T K.Mean-variance portfolio selection under a constant elasticity of variance model[J].Operations Research Letters,2014,42(5):337-342.
[10]周忠宝,刘佩,喻怀宁,等.考虑交易成本的多阶段投资组合评价方法研究[J].中国管理科学,2015,23(5):1-6.
[11]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.
[12]周忠宝,肖和录,任甜甜,等.全链接分散化多阶段投资组合评价研究[J].管理科学学报,2017,20(6):89-100.
[13]Mossin J.Optimal multiperiod portfolio policies[J].Journal of Business,1968,41(2):215-229.
[14]Samuelson P A.Lifetime portfolio selection by dynamic stochastic programming[J].The Review of Economics and Statistics,1969,50(3):239-246.
[15]Winkler R L,Barry C B.A Bayesian model for portfolio selection and revision[J].Journal of Finance,1975,30(1):179-192.
[16]Dumas B,Luciano E.An exact solution to a dynamic portfolio choice problem under transactions costs[J].Journal of Finance,1991,46(2):577-595.
[17]刘海龙,樊治平,潘德惠.带有交易费用的证券投资最优策略[J].管理科学学报,1999,20(4):44-47.
[18]anako gˇlu E,zekici S.Portfolio selection in stochastic markets with exponential utility functions[J].Annals of Operations Research,2009,166(1):281-297.
[19]常浩,荣喜民.借贷利率限制下的效用投资组合[J].系统工程学报,2012,27(1):26-34.
[20]姚海祥,李仲飞.基于非参数估计框架的期望效用最大化最优投资组合[J].中国管理科学,2014,22(1):1-9.
[21]Bodnar T,Parolya N,Schmid W.On the exact solution of the multi-period portfolio choice problem for an exponential utility under return predictability[J].European Journal of Operational Research,2015,246(2):528-542.
[22]Browne S.Stochastic differential portfolio games[J].Journal of Applied Probability.2000,37(1):126-147.
[23]Bensoussan A,Frehse J.Stochastic games for N players[J].Journal of Optimization Theroy and Applications.2000,105(3):543-565.
[24]张卫国,罗军,吴丙山.风险投资中的可转换证券与双重道德风险研究[J].管理科学,2005,18(2):27-32.
[25]Basu A,Ghosh M K.Stochastic differential games with modes and applications to portfolio optimization[J].Stochastic Analysis and Applications,2007,25(4):845-867.
[26]曹国华,耿朝刚.竞争机制下风险投资机构间决策的博弈分析[J].系统工程学报,2009,24(5):596-601.
[27]Leong C K,Huang W.Stochastic differential game of capitalism[J].Journal of Mathematical Economics,2010,46(4):552-561.
[28]Espinosa G E,Touzi N.Optimal investment under relative performance concerns[J].Mathematical Finance,2015,25(2):221-257.
[29]Bensoussan A,Chi C S,Yam S P C,et al.A class of non-zero-sum stochastic differential investment and reinsurance games[J].Automatica,2014,50(8):2025-2037.
[30]Meng Hui,Li Shuanming,Jin Zhuo.A reinsurance game between two insurance companies with nonlinear risk processes[J].Insurance:Mathematics and Economics,2015,62:91-97.
[31]Chi S P,Chi C S,Wong H Y.Non-zero-sum reinsurance games subject to ambiguous correlations[J].Operations Research Letters,2016,44(5):578-586.
[32]Guan Guohui,Liang Zongxia.A stochastic Nash equilibrium portfolio game between two DC pension funds[J].Insurance:Mathematics and Economics,2016,70:237-244.
[33]Yan Ming,Peng Fanyi,Zhang Shuhua.A reinsurance and investment game between two insurance companies with the different opinions about some extra information[J].Insurance:Mathematics and Economics,2017,75:58-70.
[34]Deng Chao,Zeng Xudong,Zhu Huiming.Non-zero-sum stochastic differential reinsurance and investment games with default risk[J].European Journal of Operational Research.2018,264(3):1144-1158.
[35]Balduzzi P,Lynch A W.Transaction costs and predictability:some utility cost calculations[J].Journal of Financial Economics,1999,52(1):47-78.
[36]Nash J F.Equilibrium points in n-person games[J].Proceedings of the National Academy of Sciences of the United States of America,1950,36(1):48-49.
[37]Merton R C.Optimum consumption and portfolio rules in a continuous-time model[J].Journal of Economic Theory,1970,3(4):373-413.
[38]Tehranchi M.Explicit solutions of some utility maximization problems in incomplete markets[J].Stochastic Processes and their Applications,2004,114(1):109-125.
[39]Soyer R,Tanyeri K.Bayesian portfolio selection with multi-variate random variance models[J].European Journal of Operational Research,2006,171(3):977-990.
[2]Smith K V.A transition model for portfolio revision[J].Journal of Finance,1952,22(3):425-439.
[3]Li Duan,Ng W L.Optimal dynamic portfolio selection:Multiperiod mean-variance formulation[J].Mathematical Finance,2000,10(3):387-406.
[4]Zhou X Y,Li Duan.Continuous-time mean-variance portfolio selection:A stochastic LQ framework[J].Applied Mathematics&Optimization,2000,42(1):19-33.
[5]Yin G,Zhou Xunyu.Markowitz’s mean-variance portfolio selection with regime switching:From discretetime models to their continuous-time limits[J].IEEETransactions on Automatic Control,2004,49(3):349-360.
[6]Zhu Shushang,Li Duan,Wang Shouyang.Risk control over bankruptcy in dynamic portfolio selection:A generalized mean-variance formulation[J].IEEE Transactions on Automatic Control,2004,49(3):447-457.
[7]郭文旌,胡奇英.不确定终止时间的多阶段最优投资组合[J].管理科学学报,2005,8(2):13-19.
[8]Wu Huiling,Li Zhongfei.Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow[J].Insurance:Mathematics and Economics,2012,50(3):371-384.
[9]Shen Yang,Zhang Xin,Siu T K.Mean-variance portfolio selection under a constant elasticity of variance model[J].Operations Research Letters,2014,42(5):337-342.
[10]周忠宝,刘佩,喻怀宁,等.考虑交易成本的多阶段投资组合评价方法研究[J].中国管理科学,2015,23(5):1-6.
[11]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.
[12]周忠宝,肖和录,任甜甜,等.全链接分散化多阶段投资组合评价研究[J].管理科学学报,2017,20(6):89-100.
[13]Mossin J.Optimal multiperiod portfolio policies[J].Journal of Business,1968,41(2):215-229.
[14]Samuelson P A.Lifetime portfolio selection by dynamic stochastic programming[J].The Review of Economics and Statistics,1969,50(3):239-246.
[15]Winkler R L,Barry C B.A Bayesian model for portfolio selection and revision[J].Journal of Finance,1975,30(1):179-192.
[16]Dumas B,Luciano E.An exact solution to a dynamic portfolio choice problem under transactions costs[J].Journal of Finance,1991,46(2):577-595.
[17]刘海龙,樊治平,潘德惠.带有交易费用的证券投资最优策略[J].管理科学学报,1999,20(4):44-47.
[18]anako gˇlu E,zekici S.Portfolio selection in stochastic markets with exponential utility functions[J].Annals of Operations Research,2009,166(1):281-297.
[19]常浩,荣喜民.借贷利率限制下的效用投资组合[J].系统工程学报,2012,27(1):26-34.
[20]姚海祥,李仲飞.基于非参数估计框架的期望效用最大化最优投资组合[J].中国管理科学,2014,22(1):1-9.
[21]Bodnar T,Parolya N,Schmid W.On the exact solution of the multi-period portfolio choice problem for an exponential utility under return predictability[J].European Journal of Operational Research,2015,246(2):528-542.
[22]Browne S.Stochastic differential portfolio games[J].Journal of Applied Probability.2000,37(1):126-147.
[23]Bensoussan A,Frehse J.Stochastic games for N players[J].Journal of Optimization Theroy and Applications.2000,105(3):543-565.
[24]张卫国,罗军,吴丙山.风险投资中的可转换证券与双重道德风险研究[J].管理科学,2005,18(2):27-32.
[25]Basu A,Ghosh M K.Stochastic differential games with modes and applications to portfolio optimization[J].Stochastic Analysis and Applications,2007,25(4):845-867.
[26]曹国华,耿朝刚.竞争机制下风险投资机构间决策的博弈分析[J].系统工程学报,2009,24(5):596-601.
[27]Leong C K,Huang W.Stochastic differential game of capitalism[J].Journal of Mathematical Economics,2010,46(4):552-561.
[28]Espinosa G E,Touzi N.Optimal investment under relative performance concerns[J].Mathematical Finance,2015,25(2):221-257.
[29]Bensoussan A,Chi C S,Yam S P C,et al.A class of non-zero-sum stochastic differential investment and reinsurance games[J].Automatica,2014,50(8):2025-2037.
[30]Meng Hui,Li Shuanming,Jin Zhuo.A reinsurance game between two insurance companies with nonlinear risk processes[J].Insurance:Mathematics and Economics,2015,62:91-97.
[31]Chi S P,Chi C S,Wong H Y.Non-zero-sum reinsurance games subject to ambiguous correlations[J].Operations Research Letters,2016,44(5):578-586.
[32]Guan Guohui,Liang Zongxia.A stochastic Nash equilibrium portfolio game between two DC pension funds[J].Insurance:Mathematics and Economics,2016,70:237-244.
[33]Yan Ming,Peng Fanyi,Zhang Shuhua.A reinsurance and investment game between two insurance companies with the different opinions about some extra information[J].Insurance:Mathematics and Economics,2017,75:58-70.
[34]Deng Chao,Zeng Xudong,Zhu Huiming.Non-zero-sum stochastic differential reinsurance and investment games with default risk[J].European Journal of Operational Research.2018,264(3):1144-1158.
[35]Balduzzi P,Lynch A W.Transaction costs and predictability:some utility cost calculations[J].Journal of Financial Economics,1999,52(1):47-78.
[36]Nash J F.Equilibrium points in n-person games[J].Proceedings of the National Academy of Sciences of the United States of America,1950,36(1):48-49.
[37]Merton R C.Optimum consumption and portfolio rules in a continuous-time model[J].Journal of Economic Theory,1970,3(4):373-413.
[38]Tehranchi M.Explicit solutions of some utility maximization problems in incomplete markets[J].Stochastic Processes and their Applications,2004,114(1):109-125.
[39]Soyer R,Tanyeri K.Bayesian portfolio selection with multi-variate random variance models[J].European Journal of Operational Research,2006,171(3):977-990.