CONTENTS CONTENTS 1 Introduction 1.1 Research Background 1.2 Multi-period Risk Measures 1. 3 Multi-stage Portfolio Selection 11124 1.3.1 Scenario Tree Technique 1.3.2 Statistical Approaches 3.3 Distributionally robust Optimization Approaches 1. 4 Contribution and Outline of the dissertation 2 Introduction to Multi-period Risk Measures 2.1 Risk Measures and Multi-stage Portfolio Selection 2.2 Properties of Multi-period Risk Measures 2.3 Time Consistency… 2.3. 1 Dynamic Time Consistency 2.3.2 Weakly Time Consistency 23.3 Time Consistency of Optimal Policies…… 2.4 Classification of Multi-period Risk Measures 2.4.1 Terminal Wealth risk measures……… 2.4.2 Additive Risk measures 2.4.3 Nested Risk measures 3 Nested CVaR risk Measure with Joint Information Framework and its Appli- cation in Multi-stage Port folio Selection 3.1 Joint Information Framework 3.1.1 Joint Information Process 312 Return Factors and Their Dynamics…… 3.2 Regime-dependent Nested Risk Measure 3.3 Regime-dependent Multi-stage Portfolio Selection 3.4 Empirical Analy 36 3.4.1 Data Set and Parameter Estimation 3.4.2 Optimal Portfolio Selection and Evaluation 3.4.3 Out-of-sample Performances 3.4.4 Superiority of Joint Information Framework
CONTENTS CONTENTS 1 Introduction ·············································································································· 1 1.1 Research Background ························································································· 1 1.2 Multi-period Risk Measures ··············································································· 1 1.3 Multi-stage Portfolio Selection ·········································································· 2 1.3.1 Scenario Tree Technique ·············································································· 4 1.3.2 Statistical Approaches ················································································· 4 1.3.3 Distributionally Robust Optimization Approaches ··································· 6 1.4 Contribution and Outline of the Dissertation ··················································· 7 2 Introduction to Multi-period Risk Measures ·························································· 10 2.1 Risk Measures and Multi-stage Portfolio Selection ·········································· 10 2.2 Properties of Multi-period Risk Measures ························································ 12 2.3 Time Consistency ······························································································· 14 2.3.1 Dynamic Time Consistency ········································································ 14 2.3.2 Weakly Time Consistency ··········································································· 15 2.3.3 Time Consistency of Optimal Policies ························································ 15 2.4 Classification of Multi-period Risk Measures ··················································· 16 2.4.1 Terminal Wealth Risk Measures ································································· 16 2.4.2 Additive Risk Measures ··············································································· 18 2.4.3 Nested Risk Measures ·················································································· 20 3 Nested CVaR Risk Measure with Joint Information Framework and its Application in Multi-stage Portfolio Selection ····································································· 24 3.1 Joint Information Framework ············································································ 24 3.1.1 Joint Information Process ··········································································· 24 3.1.2 Return Factors and Their Dynamics ·························································· 25 3.2 Regime-dependent Nested Risk Measure ·························································· 27 3.3 Regime-dependent Multi-stage Portfolio Selection ··········································· 28 3.4 Empirical Analysis ······························································································ 36 3.4.1 Data Set and Parameter Estimation ·························································· 36 3.4.2 Optimal Portfolio Selection and Evaluation ·············································· 38 3.4.3 Out-of-sample Performances ······································································· 43 3.4.4 Superiority of Joint Information Framework ············································· 45 XI
西安交通大学博士学位论文 3.5 Conclusion 4 Multi-period Worst-case Risk Measure and its Application in Multi-stage Port folio Selection 4. 1 Definition to Multi-period Worst-case Risk Measure 4.2 Multi-stage robust Portfolio Selection with w CVaR 4.3 Empirical results 4.4 Conclusion… 5 Additive robust risk Measure with Moments Uncertainty and its Application in Multi-stage Investment Policy Selection 5. 1 Single-stage Robust Investment Policy Selection with Moments Uncertainty 57 5.2 Multi-stage Robust Investment Policy Selection with Additive Risk Measure and Moments Uncertainty 53 Empirical Results…… 5.3.1 Data Sets 5.3.2 Estimating Parameters in Dynamic Uncertainty Set 3.3 Out-of-sample Performance 5.4 Conclusion 6 Nested Robust Risk Measure with Moments Uncertainty and its Application in Multi-stage Portfolio Selecti 6.1 Single-stage robust Mean-CVaR Portfolio Selection Model with the New Uncertainty Set 6.2 Multi-stage Robust Portfolio Selection Model with Nested Risk Measure and Moments Uncertainty… 6.3 Empirical results 6.4 Conclusio 7 Regime-dependent Multi-period Robust Risk Measures and Their Applications in multi- stage investment problems………… 7.1 Multi-period Worst-regime Risk Measure and Multi-period Mixed Worst- Risk me 7.2 Multi-stage Robust Investment Models with wrCVaR and mw CVaR 7.3 Empirical Results 7. 4 Conclusion 8 Target Reaching Risk Measure and its Application in Multi-stage Investment Policy Selection 8.1 Probabilistic Target Reaching Time and New Dynamic Risk Measure 8.2 Properties of Target Reaching Risk Measure
‹SœåÆƨƆÿ© 3.5 Conclusion ··········································································································· 47 4 Multi-period Worst-case Risk Measure and its Application in Multi-stage Portfolio Selection ················································································································ 48 4.1 Definition to Multi-period Worst-case Risk Measure ······································· 48 4.2 Multi-stage Robust Portfolio Selection with wCVaR ······································· 50 4.3 Empirical Results ······························································································· 55 4.4 Conclusion ··········································································································· 56 5 Additive Robust Risk Measure with Moments Uncertainty and its Application in Multi-stage Investment Policy Selection ································································· 57 5.1 Single-stage Robust Investment Policy Selection with Moments Uncertainty 57 5.2 Multi-stage Robust Investment Policy Selection with Additive Risk Measure and Moments Uncertainty ························································································ 60 5.3 Empirical Results ······························································································· 64 5.3.1 Data Sets ······································································································ 65 5.3.2 Estimating Parameters in Dynamic Uncertainty Sets ······························· 65 5.3.3 Out-of-sample Performance ········································································· 66 5.4 Conclusion ··········································································································· 67 6 Nested Robust Risk Measure with Moments Uncertainty and its Application in Multi-stage Portfolio Selection ···················································································· 69 6.1 Single-stage Robust Mean-CVaR Portfolio Selection Model with the New Uncertainty Set ········································································································· 69 6.2 Multi-stage Robust Portfolio Selection Model with Nested Risk Measure and Moments Uncertainty ······························································································· 74 6.3 Empirical Results ······························································································· 77 6.4 Conclusion ··········································································································· 81 7 Regime-dependent Multi-period Robust Risk Measures and Their Applications in Multi-stage Investment Problems ··········································································· 82 7.1 Multi-period Worst-regime Risk Measure and Multi-period Mixed Worstcase Risk Measure ···································································································· 82 7.2 Multi-stage Robust Investment Models with wrCVaR and mwCVaR ············ 85 7.3 Empirical Results ······························································································· 89 7.4 Conclusion ··········································································································· 94 8 Target Reaching Risk Measure and its Application in Multi-stage Investment Policy Selection ············································································································· 95 8.1 Probabilistic Target Reaching Time and New Dynamic Risk Measure ·········· 95 8.2 Properties of Target Reaching Risk Measure ···················································· 97 XII
CONTENTS 8.3 Relationships with Variance and VaR 8.4 Multi-stage Investment Policy Selection with Target Reaching Risk Measure 106 8.4.1 Sub-problem with Fixed Probabilistic Target Reaching Time 107 4.2 Analytical Solution of Auxiliary Problem ……110 4.3 Optimal Multiplier. 8.4.4 Optimal Probabilistic Target Reaching Time .. 114 8.5 Empirical Results …114 8.5.1 Application to US Security Market 114 8.2 Comparison with Dynamic MV Model………… ……116 8. 6 Conclusion 9 Conclusions and Prospect 118 9.1 Conclusion 9.2 Prospect 120 Acknowledgements 121 References Achievements129
CONTENTS 8.3 Relationships with Variance and VaR ······························································· 102 8.4 Multi-stage Investment Policy Selection with Target Reaching Risk Measure 106 8.4.1 Sub-problem with Fixed Probabilistic Target Reaching Time ·················· 107 8.4.2 Analytical Solution of Auxiliary Problem ·················································· 110 8.4.3 Optimal Multiplier ······················································································· 113 8.4.4 Optimal Probabilistic Target Reaching Time ············································ 114 8.5 Empirical Results ······························································································· 114 8.5.1 Application to US Security Markets ··························································· 114 8.5.2 Comparison with Dynamic MV Model ······················································· 116 8.6 Conclusion ··········································································································· 117 9 Conclusions and Prospects ······················································································ 118 9.1 Conclusions ········································································································· 118 9.2 Prospects ············································································································· 120 Acknowledgements ······································································································· 121 References ····················································································································· 122 Achievements129 XIII
西安交通大学博士学位论文 主要符号表 实数域 RRT n维实向量空间 投资区间长度 风险资产数目 全为1的向量 全体事件集 Ft 截止到第t期所有已知信息的σ域 概率度量 F条件下的概率度量,P=P|F C F可测的随机变量所处的概率空间 Ltt 从第t期到T期的乘积空间,CT=Ct×…×C 第t期的随机损失 X 从第t期到第T期的随机损失过程,Xr=(X,X+1,…,Xr) 第t期的财富 第t期无风险资产的收益 第讠个风险资产在第t期的随机收益 n个风险资产在第t期的随机收益向量,rt=(r2,n2,…,r) Rt 第t期风险资产的超额收益向量,Rt=r1-re 第t期投资在第i个风险资产上的财富量 第t期的投资决策向量,u=(ul,…,a) 从第t期到第T期的投资策略,r=(u,u+1,…,ur-1 期望算子 方差 协方差 单期风险度量
‹SœåÆƨƆÿ© ÃጓL R ¢Íç R n n ë¢ï˛òm T ›]´m› n ºx]Í8 e è 1 ï˛ Ω NØá8 Ft é1 t œ§kÆ&E σ ç P V«›˛ Pt Ft ^áeV«›˛, Pt = P|Ft Lt Ft åˇëÅC˛§?V«òm Lt,T l1 t œ T œ¶»òm, Lt,T = Lt × · · · × LT Xt 1 t œëÅõî Xt,T l1t œ1T œëÅõîLß, Xt,T = (Xt , Xt+1, · · · , XT ) xt 1 t œ„L r 0 t 1 t œÃºx]¬Ã r i t 1 i áºx]31 t œëŬà rt n áºx]31 t œëŬÃï˛, rt = (r 1 t , r2 t , · · · , rn t ) > Rt 1 t œºx]á¬Ãï˛, Rt = rt − r 0 t e u i t 1 t œ›]31 i áºx]˛„L˛ ut 1 t œ›]˚¸ï˛, ut = (u 1 t , · · · , un t ) > ut,T l1 t œ1 T œ›]¸—, ut,T = (ut , ut+1, · · · , uT −1) E œ"éf σ 2 ê Cov ê ρ ¸œºx›˛ XIV
主要符号表 pt 第t期的单期风险映射 Pt 从第T期到第t期的条件风险映射 多期风险度量 机制数 M 全体机制集 第t期的机制 第j种机制,j=1,…,J 机制转换概率矩阵 PM(l1-1,L)从机制l-1转换到机制L的概率 截止到第t期所有已知机制信息的σ域 分布P的不确定集 第t期条件分布P的不确定集 XV
ÃጓL ρt 1 t œ¸œºxN ρt,T l1 T œ1 t œ^áºxN {ρt,T } T t=0 ıœºx›˛ J ÅõÍ M NÅõ8 It 1 t œÅõ I j 1 j ´Åõ, j = 1, · · · , J PM Åõ=ÜV«› PM(It−1, It) lÅõ It−1 =ÜÅõ It V« M é1 t œ§kÆÅõ&E σ ç P ©Ÿ P ÿ(½8 Pt 1 t œ^á©Ÿ Pt ÿ(½8 XV