Optimal Portfolio Selection Game Problem With Interacting Agents Under Relative Performance Concerns

As an effective risk diversification tool,portfolio is very important for investors.The core theory of portfolio is to allocate assets effectively in an uncertain environment,so as to achieve the balance between maximization of returns and minimization of risks.For individual investors,investment decisions are related to their own risk appetite.For institutional investors,they are eager to attract new clients and manage more money.And they worry about their careers,so they always try to get better returns than their competitors.In order to weigh the benefits and risks,institutional investors must consider the investment behavior of their competitors when deciding on their investment strategies.Compared with the investment behavior of individual investor,competition makes investment decisions more complicated.Therefore,it is of great significance in both theory and practice to study the optimal portfolio selection with interacting agents under relative performance concerns,analyze the impact of competition among institutional investors on risk taking and the effect on the terminal wealth utility.Firstly,we study the discrete time optimal portfolio selection game problem between two interacting agents when each agent takes into account his relative performance by comparison to his competitor under the goal of utility maximization.Based on different investment opportunity sets,the Nash equilibrium portfolio strategy and the value function of each agent are obtained in closed forms for the case of each agent with an exponential utility function.The results show that the Nash equilibrium portfolio strategy and the value function of each agent not only depend on the institutional investor's own risk aversion coefficient and sensitivity to relative wealth,but also on the competitor's risk aversion coefficient and their sensitivity to relative wealth,as well as the parameters of risky stocks and risk-free interest rate.Competition can change the agent risk taking.When the correlation between risky stocks is positive or has a strong negative correlation,competition will increase the investment in risky stocks.When risk stocks are independent from each other,the value function with different investment opportunity sets is smaller than that of with the same investment opportunity sets.Using certainty equivalent as an index to measure investment performance,it was found that the certainty equivalent of institutional investors that consider relative performance does not exceed the certainty equivalent of institutional investors that do not consider relative performance.Then we study the discrete time optimal portfolio selection game problem between two interacting agents when each agent takes into account his relative performance by comparison to his competitor under the mean-variance criteria.The results show that the Nash equilibrium portfolio strategy of institutional investors based on the mean-variance criteria is similar to the Nash equilibrium portfolio strategy based on utility maximization.Va R value is used as an index to measure investment performance.If institutional investors only consider absolute wealth,the Va R value will be greater than they consider the relative wealth.Under the same expected relative return level,the more institutional investors consider relative wealth,the more the risk they will take.As competitors' relative wealth sensitivity increases,the effective frontier graph becomes flatter.Finally,we study the optimal portfolio selection game problem between two interacting agents over multiple periods,comparing the effects of asset specialization and asset diversification on the investment strategy and terminal wealth utility.The results show that the Nash equilibrium investment strategy and the value function of institutional investors under asset specialization is quite different from that under asset diversification,and the correlation coefficient between the return rate of risky stocks has different influence on the optimal investment strategy under the two conditions.Under asset diversification,the value function of institutional investors is related to their absolute risk aversion coefficient and the parameters of risky stocks,but it is independent of relative wealth sensitivity and competitors' absolute risk aversion coefficient.When the returns of risky stocks are independent,institutional investors will adopt asset diversification strategies.The utility gain and losses of institutional investors are related to the Sharpe ratio of risky stocks.And the utility gain and loss of institutional investors is a decreasing function of relative wealth conscern.With the increase of market competition,institutional investors will prefer the diversification strategy of assets.