M-V Portfolio Selection And Portfolio Consumption Decisions

The emergence of portfolio theory really makes mathematical methods enter the investment field. Thus, mathematical finance as an independent branch of the finance theory develops quickly. But there still exist shortcomings in previous study of portfolio theory, such as the mean-variance portfolio theory is only involved with the deterministic time horizon and its market is very simple; The investment consumption theory is only involved with a single consumption good and the investment objects are only a bond and some risky stocks. However, the consumption good and investment object are becoming more various. So the previous some conclusions of investment theory can not satisfy the requirement of real situations. The aim of this paper is to study mean variance portfolio theory and portfolio consumption theory further to fit better real situations. The main results are listed as follows.(1) M-V portfolio selection of deterministic time horizon. The mean variance models are formulated respectively in three market cases: (i) The stock prices follow jump diffusion process; (ii) Fixed consumption is considered; (iii) The market coefficients are stochastic processes. By using stochastic dynamic programming principle and martingale approach to solve these models, the optimal investment strategies and the efficient frontier are presented explicitly. By comparing them with conclusions of the classical continuous time model and analyzing the influence of the fixed consumption on investment, the main results are derived as follows: (i) The models discussed here extend the classical model discussed by Zhou and Li[208] and can be better applied to the real situations; (ii) The selection of optimal investment strategies is affected by the fixed consumption. When the final expected return is fixed, investment on stocks grows with the consumption. The efficient frontier moves downward (upward) if consumption increase (decrease) . Thus, we characterize the internal relations between investment and consumption.(2) M-V portfolio selection of random time horizon. The random time horizon mean variance models corresponding to three market situations :discrete time, continuous time and jump diffusion process are formulated respectively and the relevant efficient frontier is defined. In the former two situations, the exit time is assumed to be a random variable and in the last situation the exit time is assumed to be a stochastic process. By solving the three models respectively, the explicit expressions of the investment strategies and the efficient frontiers are presented. By number example and by comparing the constant time horizon with the random time horizon we find that the portfolio decisions are affected by distribution of the exit time and the random time horizon model extends the constant horizon model as a special case.(3) portfolio consumption decision with special consumption. Being different from classical portfolio consumption problem, here two special portfolio consumption problems whose consumption are fixed consumption style and a combination of a perishable with a durable consumption good respectively are considered. The modelsmaximizing utility of consumption or final wealth corresponding to the above two cases are formulated. For the HARA utility function and separable isoelastic utility function, the optimal portfolio and consumption rules are derived explicitly. By analyzing the influences of fixed consumption style and durable consumption good on investment decisions, the following results are presented, (i) The intuition that the portfolio selection decision is not affected by the fixed consumption style is not true. As a fact, investment varies with consumption contrarily. The degree of such influence is decided by market risk price and riskless rate, (ii) The influence of durable consumption good on investment consumption decisions is different from that of perishable consumption good. So it is necessary to consider them separately when one invests and consumes.(4) portfolio consumption decision concl...