Periodic Evaluation with Non-Concave Utility (with Cong Qin and Harry Zheng).
+ submitted. [Abstract|SSRN]
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+ A fund manager's performance is often evaluated annually and compared with a benchmark, such as a market index. In addition, the manager may be subject to trading constraints, such as limited use of leverage, no short-selling, and a forced liquidation clause. We formulate this as a periodic evaluation problem with a non-concave utility, a stochastic reference point, and trading constraints. The value function is characterized as the unique solution to a Hamilton-Jacobi-Bellman equation with periodic terminal and boundary conditions, which must be imposed carefully due to possible discontinuities at the terminal time and/or on the liquidation boundary. We find that, at the evaluation time, future investment opportunities induce a discontinuity in the value function on the liquidation boundary, leading to a substantial change in local risk-aversion. More importantly, this local concavity/convexity weakens and shifts inward from the liquidation boundary to the interior region as the evaluation horizon increases. As a result, the joint effect of periodic evaluation and forced liquidation can generate highly nonlinear investment strategies, which is helpful in understanding the complexity of trading strategies in the loss region.
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