Continuous-time stochastic mutual fund management game between active and passive funds

This paper investigates a Stackelberg game between a mutual fund manager and an investor. Throughout the paper, we follow the literature (see, e.g. Li and Sethi. A review of dynamic stackelberg game models. Discrete Contin. Dyn. Syst. B, 2016, 22(1), 125) on Stackelberg games, i.e. leader-follower games, and call the leader and follower, respectively, her and him. In our model, the leader (she) refers to the mutual fund manager, while the follower (he) is the investor. The individual investor (he), manages an active mutual fund and can only allocate his wealth among a risk-free asset, the active mutual fund, and a passive index fund. The passive index fund is composed of an exogenously given portfolio of all stocks in the market. The investor aims to maximize the expected constant relative risk aversion utility of his terminal wealth, while the mutual fund manager's objective is to maximize the expected value of the accumulative discounted management fees from the investor. Assume that the mutual fund manager faces portfolio constraints, which reflects the investment objective and style of the fund. We first focus on a special case in which the mutual fund is a sector fund, investing only in a small subset of available stocks in the market, namely a specific sector or industry. Then, this special case is extended to the general case in which the portfolio constraint is given by a nonempty, closed convex set. By applying the dynamic programming principle approach, we solve two associated Hamilton-Jacobi-Bellman (HJB) equations and obtain Stackelberg equilibrium strategies for both the mutual fund manager and the investor in closed-form expressions. Finally, in the special case we provide numerical examples to analyze the effects of some parameters on the equilibrium strategies.