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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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    1. 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.
    2. 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.
    3. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    4. Alexander Cherny & Dilip Madan, 2009. "New Measures for Performance Evaluation," Review of Financial Studies, Society for Financial Studies, vol. 22(7), pages 2371-2406, July.
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