Paper: arXiv 2406.00435

Authors: Yang Liu, Zhenyu Shen

Abstract

In hedge funds, convex compensation schemes are adopted 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 family 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. Technically, we propose an additional assumption and prove concavification techniques of non–concave utility functions with a left unbounded domain in incomplete markets. Next, 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. Furthermore, we provide a detailed asymptotic analysis and numerical illustration of the optimal portfolio. We obtain several key insights, including that 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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically rigorous framework introducing the PSAHARA utility family and concavification techniques for optimal control in incomplete markets. While the empirical validation uses real U.S. stock market data to demonstrate manager risk-taking behaviors, the primary strength lies in its theoretical proofs, asymptotic derivations, and analytical closed-form solutions.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Control Strategy for Portfolio Managers with PSAHARA Utility in Incomplete Markets] --> B[Model Formulation: Combine Non-monotone Risk Aversion & Convex Compensation]
    B --> C[Methodology: Prove Concavification Techniques for Left-Unbounded Non-Concave Utility]
    D[Inputs: U.S. Stock Market Real Data & Economic Parameters] --> E[Computational Analysis: Derive Explicit 4-Term Control Strategy & Asymptotic Analysis]
    C --> E
    E --> F[Outcomes: PSAHARA Portfolio Induces Risk-Seeking Behavior with High Returns & High Volatility]