Investment Opportunities And Portfolio Var Constraint Mean - Variance Model

Modern portfolio selection theory is concerned with the allocation of wealth among a basket of securities in an uncertain economy so as to achieve a reasonable balance between the return of the portfolio and the associated risk. The model was first proposed and solved in the single period Mean-Variance portfolio selection model by Harry Markowitz. This work has become the foundation of modern finance and the framework of using expected return as a measure of return rate of a portfolio and using variance as a measure of risk is founded in the contemporary financial theories. The chance constraint and the VaR constraint are constructed respectively according to the prespecified expected rates of return and confidence level. When securities' rates of return obey normal distribution, a kind of chance-constrained and VaR-constrained portfolio selection with short selling are put forward, and then the 'mean-variance efficient set is subdivided. This paper differs from previous work in several respects. First, risk-free security is introduced to the chance-constrained mean-variance model. Second, a portfolio selection model under constraints of both investment chance and VaR is established. Existence and uniqueness of the optimal solutions to these models are discussed. Moreover, the explicit expressions of the optimal solutions are obtained.This paper contains four chapters:Chapter 1 introduces the standard Mean-Variance portfolio selection model and the properties of the 'Mean-Variance efficient set'. The definitions of the chance constraints and VaR constraints are given in this chapter.Chapter 2 discusses the chance-constrained portfolio selection model when a risk-free security exists in the economy. This chapter begins by the assuming that risk-free lending is allowed, but risk-free borrowing is not. Then existence and uniqueness of the optimal solution to chance-constrained Mean-Variance model are discussed. Moreover, the explicit expressions of the optimal portfolio are obtained. The similar results are obtained if risk-free borrowing is allowed but at a rate higher than the lending rate.Chapter 3 adds VaR constraint to the chance-constrained mean-variance modeland constructs a portfolio selection model under constraints of both investment chance and VaR. According to the assumption that whether a risk-free security exists or not, the paper discusses respectively the optimal solutions to these models and the explicit expressions of the optimal portfolios.Chapter 4 introduces the dynamic continuous time model on the basis of the continuous time Mean-Variance model and investigates the cruxes of the problem.