Optimal Portfolio Selection Problem Based On Stochastic Benchmark

In the financial market,portfolio as an effective risk diversification tool is very important for investors.The core of its theory is to effectively allocate assets in an uncertain environment.The existing literature on the choice of investors’ portfolio strategy mostly consider the utility of absolute wealth or the mean-variance problem,or only pays attention to the objective function corresponding to the relative performance of investors.The absolute wealth and relative performance of investors are not taken into account at the same time.While relative performance and absolute wealth are equally important in investment practice,and both of them need to be considered simultaneously.Therefore,in this paper we investigate the portfolio selection problem based on a stochastic benchmark.Investor can invest her wealth in a financial market consisting of a risk-free asset and a risky stock.The benchmark is driven by a stochastic process and is correlated with the risky stock.Then we try to solve the optimal portfolio selection by maximizing her expected terminal wealth utility and using mean-variance criterion.Firstly,we studied the optimal portfolio selection problem based on stochastic benchmark under the utility maximization goal.Under the investor’s power utility function and logarithm utility function,the investor’s optimal portfolio selection strategy and value function are obtained respectively.The results show:The investor’s optimal investment strategy depends on the investor’s relative risk aversion coefficient,the various parameters of the financial market,and the volatility of the common risk factor of the benchmark process and the risky stock.However,it is independent of the instantaneous rate of return of the benchmark process and the volatility of the independent factor.The relationship between the investor’s optimal portfolio strategy and the various parameters depends on whether the investor is risk-tolerant or risk-intolerant.Relative performance concerns will change the investor’s inherent risk-taking trend,and the relative performance concerns is not the same for risk-tolerant investors and risk-intolerant investors.As the relative wealth sensitivity increases,the risk-tolerant investor will reduce the proportion of capital investment in risky stocks,while the risk-tolerant investors will increase the proportion of capital investment in risky stocks.As investor’s risk aversion increases,investors who only consider absolute wealth will reduce the proportion of capital invested in risky stocks,but the proportion of investors who invest in risky stocks will increase or decrease depending on the Sharpe ratio of risky stocks.By using numerical calculation,we found the relationship between the utility profit or loss of investors considering relative performance and the main parameters of the model.The relationship between investor utility profit and relative wealth sensitivity depends on the investor’s risk tolerance.For risk-tolerant investors,utility profit or loss is an increasing function of relative wealth sensitivity,while for risk-intolerant investors,utility profit or loss is increasing firstly and then decreasing.The relationship between the investor’s utility profit and loss and the investor’s relative risk aversion coefficient depends on the investor’s relative wealth sensitivity.Then,we considered the mean-variance optimal portfolio selection problem based on stochastic benchmark,the investor’s optimal portfolio selection strategy and expressions of the effective frontier of the portfolio are derived.The results show:The optimal portfolio strategy and effective frontier of the mean-variance portfolio problem based on stochastic benchmarks are very different from the optimal portfolio strategy and effective frontier of the mean-variance portfolio problem considering only absolute wealth.The introduction of benchmark assets will change the optimal portfolio strategy and effective frontier.The investor’s optimal portfolio strategy is an increasing function of relative wealth sensitivity.The increase of the volatility of the benchmark asset and the risk stock common factor will increase investment in the risky stock.The optimal portfolio strategy is a decreasing function of the volatility of the benchmark asset specific factor.The greater the relative wealth sensitivity of investors,the greater the risk they bear under the same expected relative returns.As the sensitivity to relative wealth increases,the effective frontier figure becomes flatter,and the risk is more sensitive to changes in returns.It means that the increase in the same return for investors,the greater the relative wealth sensitivity,the more the risk of investors will take.