Equilibrium strategies for Mean-variance problem

This research is devoted to study equilibrium strategies in a game theoretical framework for the mean-variance problem. The thesis explores the investment behaviour and interlinks between different types of equilibrium strategies. In order to find the open-loop strategy in discrete time, we incorporate the idea based on Hu et al. (2012) and the concept of open-loop strategies in engineering study. In engineering study, there are two types of strategies: open-loop and closed-loop control strategies. We find the interpretations for both strategies in a Nash equilibrium context from a financial perspective. This thesis extends the literature by providing the existence and uniqueness of the solution of open-loop equilibrium strategy in discrete time. Our findings point to the causes of different equilibrium strategies in the existing literature. We show the common issue of equilibrium strategies, i.e. that the amount of money invested in risky assets decays to 0 as time moves away from the maturity. Furthermore, the closed-loop strategy tends to a negative limit depending on the assets’ Sharpe ratio. We call this phenomenon as Mean-variance puzzle. The reason is that the variance term penalises the wealth changes quadratically as well as the expectation only increases linearly. By drawing in the concepts from behavioural economics, we solve this puzzle by using the present-biased preference. The advantage of the present-biased preference is that equilibrium investors have the flexibility to adjust their risk attitude based on their anticipated future. We simulate three types of control strategies existing in the literature and compare the investment performance. Furthermore, we evaluate the performance with respect to different rebalancing periods.