PSAHARA Utility Family: Modeling Non-monotone Risk Aversion and Convex Compensation in Incomplete Markets
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910681107791872 |
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| author | Liu, Yang Shen, Zhenyu |
| author_facet | Liu, Yang Shen, Zhenyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00435 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | PSAHARA Utility Family: Modeling Non-monotone Risk Aversion and Convex Compensation in Incomplete Markets Liu, Yang Shen, Zhenyu Optimization and Control Mathematical Finance 91B16, 91G10 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. |
| title | PSAHARA Utility Family: Modeling Non-monotone Risk Aversion and Convex Compensation in Incomplete Markets |
| topic | Optimization and Control Mathematical Finance 91B16, 91G10 |
| url | https://arxiv.org/abs/2406.00435 |