Research On Optimal Strategies Of Dynamic Asset Allocation Problems Under Multi-period Mean-variance Model

Asset allocation is one of the most important links in the investment process.In recent years,dynamic asset allocation problems,including portfolio selection problems,asset liability management problems,and pension investment management problems,are all hot topics in the field of mathematical and financial research.This thesis pays more attention on linking reality and focuses on the optimal investment decision problem based on the influence of background risk(such as interest rate risk)and the impact of market state on the return of risky assets.First,we study the time-consistent strategy of portfolio selection problem based on stochastic interest rate risk and uncontrollable liability.Second,we discuss the time-consistent strategy for a DC pension plan based on stochastic interest rate risk and regime switching.Then,the pre-commitment strategy and equilibrium strategy of DC pension plans with regime switching and a return of premiums clause are considered.Finally,the time-consistent asset-liability management with stochastic cash flows and imperfect information is studied.In chapter 1,we first introduce the research background of dynamic asset allocation.Then,the main work of this thesis is introduced.Finally,the Bellman optimality principle,the methods to deal with the problem of time inconsistency,and some assumptions and knowledges of matrix theory are presented.Chapter 2 discusses the time-consistent strategy for a multi-period mean-variance portfolio selection problem.We first consider the general portfolio selection problem based on stochastic interest rate risk.The investor invests in a financial market consisting of one risk-free asset and n risky assets.We use discrete-time Vasicek model to characterize the dynamics of interest rate proposed by Yao et al.(2016d)[96].We regard this problem as a non-cooperative game whose equilibrium strategy is the desired time-consistent strategy.Using the extended Bellman equation,the analytical expressions of the equilibrium strategy and the equilibrium value function are deduced and the corresponding equilibrium effective frontier is obtained.Then,we extend our model to the situation with a liability and obtain the corresponding equilibrium strategy and efficient frontier.Next,some properties of our equilibrium strategy,including a two-fund separation theorem,are proposed.Finally,a numerical example with real data from the China market is given to illustrate the effects of stochastic interest rates and uncontrollable liability on the equilibrium strategy and the efficient frontier.Chapter 3 studies the time-consistent strategy for the defined contribution(DC)pension plan.The pension investor can invest the pension funds in a financial market consisting of one risk-free asset and n risky assets.Unlike chapter 2,the interest rate in this chapter is characterized by discrete-time Ho-Lee model,and the interest rate as well as the returns of the risky assets depends on the market states.The evolution of the market states is described by a Markov chain,where the transition matrixes are time-varying.Using the game theory,the extended Bellman equations,and the matrix representation techniques,we derive the analytical expressions for the equilibrium strategy and equilibrium efficient frontier.Finally,using the real data from the UK market,we analyze the equilibrium strategy and the equilibrium effective frontier by the numerical method.Chapter 4 considers the pre-commitment and equilibrium strategies for a DC pension plan during the accumulation phase under a multi-period mean-variance framework,where a pension member contributes a predetermined amount of money as a premium and then the manager of the pension fund invests the premium in a financial market to increase the value of the accumulation.To protect the rights of pension members who die before retirement,a return of premiums clause is introduced,under which a member who dies before retirement can withdraw all the premiums she has contributed.We assume that the financial market also consists of one risk-free asset and n risky assets,and the returns of the risky assets depend on the market states.The evolution of the market states is described by a Markov chain,where the transition matrixes are time-varying.Using the embedding technique and the dynamic programming method,we obtain the pre-commitment strategy and the corresponding efficient frontier in closed form.Applying the game theory and the extended Bellman equation,we derive the analytical expressions of the equilibrium strategy and the corresponding efficient frontier.For the two obtained investment strategies and their corresponding efficient frontiers,as well as the impact of regime switching and the return of premiums clause on them,some interesting theoretical and numerical results are found.Chapter 5 studies the time-consistent strategy for a asset-liability management(ALM)problem with stochastic cash flows under imperfect information.Imperfect information means that both observable and unobservable states exist in the financial market,where the dynamics of the unobservable market state process are formulated by a discrete-time finite-state hidden Markov chain with time-varying transition probability matrices.In our model,not only the returns of the risky assets and liability,but also the cash flows depend on both the observable and unobservable market states.By adopting sufficient statistics method,the ALM problem with imperfect information is transformed into one with complete information.Then,we derive the analytical expressions of the equilibrium strategy,the equilibrium value function and the equilibrium efficient frontier by the extended Bellman equation.Finally,a numerical example based on real data from the China market is used to analyze the impact of imperfect information on the equilibrium strategy and the corresponding efficient frontier.