Optimal Portfolio with Ratio Type Periodic Evaluation under Short-Selling Prohibition

This paper studies some unconventional utility maximization problems when the ratio type relative portfolio performance is periodically evaluated over an infinite horizon. Meanwhile, the agent is prohibited from short-selling stocks. Our goal is to understand impacts of the short-selling constraint and the periodic reward structure on the long-run optimal portfolio strategy. Under logarithmic and power utilities, we first reformulate the original problem into an auxiliary one-period optimization problem. To cope with the auxiliary problem with no short-selling, the dual control problem is introduced and studied, which gives the characterization of the candidate optimal portfolio within one period. With the help of the obtained results from the auxiliary problem, the value function and the optimal constrained portfolio for the original problem with periodic evaluation over an infinite horizon can be further derived and verified, allowing us to discuss some interesting financial implications under the new performance paradigm.