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Dynamic quasi-concave performance measures

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  • Sara Biagini
  • Jocelyne Bion-Nadal
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    Abstract

    We define Conditional quasi concave Performance Measures (CPMs), on random variables bounded from below, to accommodate for additional information. Our notion encompasses a wide variety of cases, from conditional expected utility and certainty equivalent to conditional acceptability indexes. We provide the characterization of a CPM in terms of an induced family of conditional convex risk measures. In the case of indexes these risk measures are coherent. Then, Dynamic Performance Measures (DPMs) are introduced and the problem of time consistency is addressed. The definition of time consistency chosen here ensures that the positions which are considered good tomorrow are already considered good today. We prove the equivalence between time consistency for a DPM and weak acceptance consistency for the induced families of risk measures. Finally, we extend CPMs and DPMs to dividend processes.

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    File URL: http://arxiv.org/pdf/1212.3958
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    Paper provided by arXiv.org in its series Papers with number 1212.3958.

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    Date of creation: Dec 2012
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    Handle: RePEc:arx:papers:1212.3958

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    Web page: http://arxiv.org/

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    1. Alexander Cherny & Dilip Madan, 2009. "New Measures for Performance Evaluation," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 22(7), pages 2371-2406, July.
    2. Jocelyne Bion-Nadal, 2008. "Dynamic risk measures: Time consistency and risk measures from BMO martingales," Finance and Stochastics, Springer, Springer, vol. 12(2), pages 219-244, April.
    3. Johannes Leitner, 2008. "Optimal Portfolios With Lower Partial Moment Constraints And Lpm-Risk-Optimal Martingale Measures," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 18(2), pages 317-331.
    4. Marco Frittelli & Marco Maggis, 2011. "Conditional Certainty Equivalent," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 41-59.
    5. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, Springer, vol. 9(4), pages 539-561, October.
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