  We analyze a family of portfolio management problems under relative
performance criteria, for fund managers having CARA or CRRA utilities and
trading in a common investment horizon in log-normal markets. We construct
explicit constant equilibrium strategies for both the finite population games
and the corresponding mean field games, which we show are unique in the class
of constant equilibria. In the CARA case, competition drives agents to invest
more in the risky asset than they would otherwise, while in the CRRA case
competitive agents may over- or under-invest, depending on their levels of risk
tolerance.
