  In hedge funds, convex compensation schemes are popular to stimulate a
high-profit performance for portfolio managers. In economics, non-monotone risk
aversion is proposed to argue that individuals may not be risk-averse when the
wealth level is low. Combining these two ingredients, we study the optimal
control strategy of the manager in incomplete markets. Generally, we propose a
wide class of utility functions, the Piecewise Symmetric Asymptotic Hyperbolic
Absolute Risk Aversion (PSAHARA) utility, to model the two ingredients,
containing both non-concavity and non-differentiability as some abnormalities.
Significantly, we derive an explicit optimal control for the family of PSAHARA
utilities. This control is expressed into a unified four-term structure,
featuring the asymptotic Merton term and the risk adjustment term. Furthermore,
we provide a detailed asymptotic analysis and numerical illustration of the
optimal portfolio. We obtain the following key insights: (i) A manager with the
PSAHARA utility becomes extremely risk-seeking when his/her wealth level tends
to zero; (ii) The optimal investment ratio tends to the Merton constant as the
wealth level approaches infinity and the negative Merton constant when the
wealth falls to negative infinity, implying that such a manager takes a
risk-seeking investment as the wealth falls negatively low; (iii) The convex
compensation still induces a great risk-taking behavior in the case that the
preference is modeled by SAHARA utility. Finally, we conduct a real-data
analysis of the U.S. stock market under the above model and conclude that the
PSAHARA portfolio is very risk-seeking and leads to a high return and a high
volatility (two-peak Sharpe ratio).
