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Dynamic quasi-concave performance measures

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  • Sara Biagini
  • Jocelyne Bion-Nadal

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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  • Sara Biagini & Jocelyne Bion-Nadal, 2012. "Dynamic quasi-concave performance measures," Papers 1212.3958, arXiv.org.
    • Handle: RePEc:arx:papers:1212.3958
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    References listed on IDEAS

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    1. Bion-Nadal, Jocelyne, 2009. "Time consistent dynamic risk processes," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 633-654, February.
    2. Jocelyne Bion-Nadal, 2008. "Dynamic risk measures: Time consistency and risk measures from BMO martingales," Finance and Stochastics, Springer, vol. 12(2), pages 219-244, April.
    3. Sara Biagini & Mustafa Pinar, 2012. "The best gain-loss ratio is a poor performance measure," Papers 1209.6439, arXiv.org, revised Dec 2012.
    4. Detlefsen, Kai & Scandolo, Giacomo, 2005. "Conditional and dynamic convex risk measures," SFB 649 Discussion Papers 2005-006, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    6. Johannes Leitner, 2008. "Optimal Portfolios With Lower Partial Moment Constraints And Lpm‐Risk‐Optimal Martingale Measures," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 317-331, April.
    7. Antonio E. Bernardo & Olivier Ledoit, 2000. "Gain, Loss, and Asset Pricing," Journal of Political Economy, University of Chicago Press, vol. 108(1), pages 144-172, February.
    8. Alexander Cherny & Dilip Madan, 2009. "New Measures for Performance Evaluation," The Review of Financial Studies, Society for Financial Studies, vol. 22(7), pages 2371-2406, July.
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