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Portfolio Optimization under Convex Incentive Schemes

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  • Maxim Bichuch
  • Stephan Sturm
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    Abstract

    We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function $g$ of the terminal wealth. The manager's own utility function $U$ is assumed to be smooth and strictly concave, however the resulting utility function $U \circ g$ fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor.

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    File URL: http://arxiv.org/pdf/1109.2945
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1109.2945.

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    Date of creation: Sep 2011
    Date of revision: Oct 2013
    Handle: RePEc:arx:papers:1109.2945

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    Web page: http://arxiv.org/

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    1. Frey, RĂ¼diger, 1997. "Derivative Asset Analysis in Models with Level-Dependent and Stochastic Volatility," Discussion Paper Serie B 401, University of Bonn, Germany.
    2. Zhegal, Amina & Touzi, Nizar & Bouchard, Bruno, 2004. "Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility," Economics Papers from University Paris Dauphine 123456789/1531, Paris Dauphine University.
    3. Jennifer N. Carpenter, 2000. "Does Option Compensation Increase Managerial Risk Appetite?," Journal of Finance, American Finance Association, vol. 55(5), pages 2311-2331, October.
    4. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
    5. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
    6. Stavros Panageas & Mark M. Westerfield, 2009. "High-Water Marks: High Risk Appetites? Convex Compensation, Long Horizons, and Portfolio Choice," Journal of Finance, American Finance Association, vol. 64(1), pages 1-36, 02.
    7. B. Bouchard & N. Touzi & A. Zeghal, 2004. "Dual formulation of the utility maximization problem: the case of nonsmooth utility," Papers math/0405290, arXiv.org.
    8. Stephen A. Ross, 2004. "Compensation, Incentives, and the Duality of Risk Aversion and Riskiness," Journal of Finance, American Finance Association, vol. 59(1), pages 207-225, 02.
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    Cited by:
    1. Paolo Guasoni & Johannes Muhle-Karbe & Hao Xing, 2013. "Robust Portfolios and Weak Incentives in Long-Run Investments," Papers 1306.2751, arXiv.org, revised Aug 2014.

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