Research On Optimal Portfolio Selection Of The Investor Under Relative Performance Concern

As an important investment tool in the financial market,portfolio can spread risk and allocate assets effectively in an uncertain environment.On the one hand,the investor hopes to maximize the absolute return of the investment,because it will increase his final wealth.On the other hand,it is reasonable for the investor to care about the relative return relative to a certain benchmark,because the higher relative return reflects the better investment results.For institutional investors,higher relative return means it can attract more funds.Therefore,the research on the optimal investment strategy that considers both the absolute return of investment and the relative return of investment has important theoretical and practical significance.First,we study the optimal portfolio problem which considering both absolute wealth and relative performance in continuous time.Under the general objective function,we use the principle of dynamic programming to obtain the optimal investment strategy and the Hamilton-Jacobi-Bellman equation satisfied by the value function.Under goals of maximizing the probability of exceeding a certain percentage of the benchmark,minimizing the expected time when investment performance exceeds a certain percentage of the benchmark,or maximizing the expected time when investment performance is below a certain percentage of the benchmark,minimizing the expected discount loss or maximizing the expected discount reward,we obtain three kinds of explicit expressions of optimal investment strategy and value function by solving the corresponding HJB equations,then discuss the influence of investors' relative performance concern on optimal investment strategies.We also consider the relationship between the optimal investment strategy and value function and the main parameters of the model by numerical examples.The results show that:Under the goal of maximizing the probability of exceeding a certain percentage of the benchmark,the increase in investor's relative performance sensitivity will make investors increase investment in risky assets,take more risks,and reduce the probability of exceeding a certain percentage of the benchmark.Under the goal of minimizing expected time which investment performance exceed a certain percentage of the benchmark,the relationship between expected time and investor's relative performance sensitivity depends on relationship between the rate of return and volatility of benchmark.As far as the goal of maximizing discounted returns is concerned,we can get different forms of optimal investment strategies and value functions by calculating different parameter values.We investigate a continuous-time optimal portfolio selection problem for a riskaverse investor based on a relative log-return.The objective of the investor is to exceed the performance of a stochastic benchmark that is not perfectly correlated with the risky asset.Investor chooses a dynamic portfolio strategy in order to maximize her expected terminal utility of the weight sum of absolute log-return and relative log-return.By using the dynamic programming principle,the corresponding Hamilton-JacobiBellman equation of the optimal portfolio strategy and the value function is established.Furthermore,closed-form expressions of the optimal portfolio strategy and the value function under the investor with a exponential utility function are derived.The effect of the relative return on the optimal portfolio strategy is also analyzed.The result shows that the relative return works against a investor's intrinsic risk-taking tendency.Finally,numerical examples are provided to illustrate how the optimal portfolio strategy and the value function change when some model parameters vary.The results show that:If the investor not only considers the absolute return but also considers the relative returns,then the investor's optimal investment strategy will not depend on the expected return rate of the benchmark process,but the investor's value function is not only related to its own risk aversion coefficient and financial market parameters,but also depends on the various parameters of the benchmark process.The optimal investment strategy can be explained in two parts.The first part only considers the optimal investment strategy that maximizes the absolute return utility,and the second part takes hedging to deal with the benchmark process risk strategy.At the same time,due to the different degree of influence of absolute return risk and relative return risk on investors,the degree and direction of investors' utility changes caused by changes in different model parameters are also quite different.