Research On Optimal Investment Strategy Selection Considering Benchmark Under Model Uncertainty

In financial markets,it is very important for investors to construct optimal investment portfolios because investment portfolios can effectively diversify investment risks.Optimal investment portfolio selection is to explore how rational investors make intertemporal optimal asset allocation for a certain initial wealth by selecting asset types and quantities in the financial market under uncertain circumstances,so that investors can maximize the utility of future consumption or final wealth,thus achieve a balance between the maximization of asset portfolio returns and the minimization of risk.On the one hand,investors hope to maximize the absolute wealth of investment,because greater absolute wealth can bring greater wealth utility;on the other hand,investors also expect investment effects to exceed a certain benchmark,that is,to have higher relative returns.From the perspective of fund managers,if the investment returns can exceed those of their peers,they can attract more customers and manage more funds.In addition,it is also very difficult to accurately estimate the return and volatility of risky assets,and investors face model uncertainty in portfolio selection.Therefore,construct a portfolio selection model that considers both benchmark and model uncertainty,and explore the impact of benchmark process and model uncertainty on investment selection strategies and value functions.It is of positive significance to expand portfolio theory and guide investment practice.First,this paper studies the optimal investment strategy choice for both absolute wealth and relative wealth investors under model uncertainty.Assuming that the investor has power law utility function,the Hamilton-Jacobi-Bellman(HJB)equation satisfied by the optimal investment strategy and the value function of the investor under model uncertainty is obtained by using dynamic programming theory.By solving the HJB equation,the investor’s optimal investment strategy and the explicit solution of the maximum expected terminal wealth utility are obtained.Through sensitivity analysis,the influence of model uncertainty on the optimal investment strategy and the maximum expected terminal wealth utility is discussed,and the influence of the main parameters of the model on the optimal investment strategy and the maximum expected terminal wealth utility is given respectively through numerical simulation.The research results show that: when the benchmark process is independent of the stock price process,compared with investors who do not consider model uncertainty,investors who consider model uncertainty will invest a smaller proportion of funds in risky stocks;when the volatility of the unique risk factor of the stock price process is completely uncertain,investors will not invest any proportion of funds in risky stocks;when the investor is completely uncertain about the volatility of the common risk factor of the risky stock price process and the benchmark process,and when the risky stock price process and the benchmark process are independent,investors will not invest in risky stocks;but if the risky stock price process and the benchmark process are related,investors will invest a certain proportion of funds in risky stocks;the relationship between optimal investment strategy and relative wealth sensitivity depends on investor’s relative risk aversion coefficient and model aversion coefficient;model uncertainty will lead to investors’ loss of utility,and relative wealth and model uncertainty will change investors’ risk-taking.Secondly,we study the optimal investment strategy for investors who consider both absolute return and relative return when facing model uncertainty.Under the condition that investors have exponential utility function,the HJB equation is obtained by using dynamic programming principle,which maximizes the weight and utility of investors’ expected absolute return and relative return and satisfies the maximum expected terminal return utility.By solving the HJB equation,an explicit solution of the investor’s optimal investment strategy and the utility of the maximum expected terminal income is obtained.The influence of model uncertainty on the optimal investment strategy and the utility of the maximum expected terminal income is studied through sensitivity analysis,and the influence of the main parameters of the model on the optimal investment strategy and the utility of the maximum expected terminal income are given respectively through numerical simulation.The research results show that the investment ratio of investors in risky stocks is a fixed constant,and investors will not short risky stocks.If investors are completely uncertain about the volatility of the risky stock price process’ specific risk factor,investors will not invest any proportion of their wealth in risky stocks;if the price process of risky stock is related to the benchmark process,investors will still invest a certain proportion of funds in risky stock even if they are completely uncertain about the volatility of the common risk factors of risky stock price process and benchmark process;the optimal investment strategy of investors increases with the increase of their relative return sensitivity;the optimal investment strategy is a decreasing function of the risk aversion coefficient specific to the risky stocks that investors face;the uncertainty of the model will lead to the loss of investors’ utility.