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postgraduate thesis: Mean variance portfolio management : time consistent approach
Title  Mean variance portfolio management : time consistent approach 

Authors  
Advisors  Advisor(s):Yung, SP 
Issue Date  2013 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Wong, K. [黃國全]. (2013). Mean variance portfolio management : time consistent approach. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5153743 
Abstract  In this thesis, two problems of time consistent meanvariance portfolio selection have been studied: meanvariance assetliability management with regime switchings and meanvariance optimization with statedependent risk aversion under shortselling prohibition.
Due to the nonlinear expectation term in the meanvariance utility, the usual Tower Property fails to hold, and the corresponding optimal portfolio selection problem becomes timeinconsistent in the sense that it does not admit the Bellman Optimality Principle. Because of this, in this thesis, timeconsistent equilibrium solution of two meanvariance optimization problems is established via a game theoretic approach.
In the first part of this thesis, the time consistent solution of the meanvariance assetliability management is sought for. By using the extended HamiltonJacobi Bellman equation for equilibrium solution, equilibrium feedback control of this MVALM and the corresponding equilibrium value function can be obtained. The equilibrium control is found to be affine in liability. Hence, the time consistent equilibrium control of this problem is state dependent in the sense that it depends on the uncontrollable liability process, which is in substantial contrast with the time consistent solution of the simple classical meanvariance problem in Björk and Murgoci (2010), in which it was independent of the state.
In the second part of this thesis, the time consistent equilibrium strategies for the meanvariance portfolio selection with state dependent risk aversion under shortselling prohibition is studied in both a discrete and a continuous time set tings. The motivation that urges us to study this problem is the recent work in Björk et al. (2012) that considered the meanvariance problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth. There is no shortselling restriction in their problem and the corresponding time consistent control was shown to be linear in wealth. However, we discovered that the counterpart of their continuous time equilibrium control in the discrete time framework behaves unsatisfactory, in the sense that the corresponding “optimal” wealth process can take negative values. This negativity in wealth will change the investor into a risk seeker which results in an unbounded value function that is economically unsound. Therefore, the discretized version of the problem in Bjork et al. (2012) might yield solutions with bankruptcy possibility. Furthermore, such “bankruptcy” solution can converge to the solution in continuous counterpart as Björk et al. (2012). This means that the negative risk aversion drawback could appear in implementing the solution in Björk et al. (2012) discretely in practice. This drawback urges us to prohibit shortselling in order to eliminate the chance of getting nonpositive wealth. Using backward induction, the equilibrium control in discrete time setting is explicit solvable and is shown to be linear in wealth. An application of the extended HamiltonJacobiBellman equation leads us to conclude that the continuous time equilibrium control is also linear in wealth. Also, the investment to wealth ratio would satisfy an integral equation which is uniquely solvable. The discrete time equilibrium controls are shown to converge to that in continuous time setting. 
Degree  Master of Philosophy 
Subject  Portfolio management  Mathematical models 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/196026 
HKU Library Item ID  b5153743 
DC Field  Value  Language 

dc.contributor.advisor  Yung, SP   
dc.contributor.author  Wong, Kwokchuen   
dc.contributor.author  黃國全   
dc.date.accessioned  20140321T03:50:06Z   
dc.date.available  20140321T03:50:06Z   
dc.date.issued  2013   
dc.identifier.citation  Wong, K. [黃國全]. (2013). Mean variance portfolio management : time consistent approach. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5153743   
dc.identifier.uri  http://hdl.handle.net/10722/196026   
dc.description.abstract  In this thesis, two problems of time consistent meanvariance portfolio selection have been studied: meanvariance assetliability management with regime switchings and meanvariance optimization with statedependent risk aversion under shortselling prohibition. Due to the nonlinear expectation term in the meanvariance utility, the usual Tower Property fails to hold, and the corresponding optimal portfolio selection problem becomes timeinconsistent in the sense that it does not admit the Bellman Optimality Principle. Because of this, in this thesis, timeconsistent equilibrium solution of two meanvariance optimization problems is established via a game theoretic approach. In the first part of this thesis, the time consistent solution of the meanvariance assetliability management is sought for. By using the extended HamiltonJacobi Bellman equation for equilibrium solution, equilibrium feedback control of this MVALM and the corresponding equilibrium value function can be obtained. The equilibrium control is found to be affine in liability. Hence, the time consistent equilibrium control of this problem is state dependent in the sense that it depends on the uncontrollable liability process, which is in substantial contrast with the time consistent solution of the simple classical meanvariance problem in Björk and Murgoci (2010), in which it was independent of the state. In the second part of this thesis, the time consistent equilibrium strategies for the meanvariance portfolio selection with state dependent risk aversion under shortselling prohibition is studied in both a discrete and a continuous time set tings. The motivation that urges us to study this problem is the recent work in Björk et al. (2012) that considered the meanvariance problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth. There is no shortselling restriction in their problem and the corresponding time consistent control was shown to be linear in wealth. However, we discovered that the counterpart of their continuous time equilibrium control in the discrete time framework behaves unsatisfactory, in the sense that the corresponding “optimal” wealth process can take negative values. This negativity in wealth will change the investor into a risk seeker which results in an unbounded value function that is economically unsound. Therefore, the discretized version of the problem in Bjork et al. (2012) might yield solutions with bankruptcy possibility. Furthermore, such “bankruptcy” solution can converge to the solution in continuous counterpart as Björk et al. (2012). This means that the negative risk aversion drawback could appear in implementing the solution in Björk et al. (2012) discretely in practice. This drawback urges us to prohibit shortselling in order to eliminate the chance of getting nonpositive wealth. Using backward induction, the equilibrium control in discrete time setting is explicit solvable and is shown to be linear in wealth. An application of the extended HamiltonJacobiBellman equation leads us to conclude that the continuous time equilibrium control is also linear in wealth. Also, the investment to wealth ratio would satisfy an integral equation which is uniquely solvable. The discrete time equilibrium controls are shown to converge to that in continuous time setting.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.subject.lcsh  Portfolio management  Mathematical models   
dc.title  Mean variance portfolio management : time consistent approach   
dc.type  PG_Thesis   
dc.identifier.hkul  b5153743   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5153743   