Optimal Investment Selection Game Problems for Institutional Investors under Log Returns

In this paper we study a continuous-time optimal portfolio selection game problems between two risk-averse institutional investors when each investor takes into account his relative return by comparison to her competitor. Both institutional investors can invest freely in the risk-free asset and only one of the two correlated risky stocks is available to each investor, reflecting asset specialization. Each 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. We first characterize explicitly the unique Nash equilibrium portfolio strategies. Secondly, the Nash equilibrium portfolio strategy and the value function of each investor are obtained in closed forms for the case of each investor with an exponential utility function. The effects of the relative return on the Nash equilibrium portfolio strategies and the value function are also analyzed. Numerical examples are also provided to illustrate how the Nash equilibrium portfolio strategies change when some model parameters vary. The investment performance between the Nash equilibrium portfolio strategy and the traditional strategy is also compared by using the certainty equivalent. The results reveal that the relative return can change the investment performance of the institutional investors.