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 time horizon in log-normal markets. We construct explicit time-independent equilibrium strategies for both the finite population games and the corresponding mean field games, which we show are unique in the class of time-independent 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.
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