Asset management, High Water Mark and flow of funds

We analyze the asset manager's portfolio problem when he is remunerated through a High Water Mark incentive fee and a management fee, and the assets under management are characterized by in/outflow of funds as a function of the performance of the fund with respect to a benchmark. Once we solve numerically the investment problem, we show that the presence of a flow fund induces risk in excess in case of a High Water Mark defined on the pure performance of the fund. Instead a High Water Mark defined on the assets under management leads to a more prudent investment strategy.