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Distribution-Invariant Dynamic Risk Measures

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  • Weber, Stefan

Abstract

The paper provides an axiomatic characterization of dynamic risk measures for multi-period financial positions. For the special case of a terminal cash flow, we require that risk depends on its conditional distribution only. We prove a representation theorem for dynamic risk measures and investigate their relation to static risk measures. Two notions of dynamic consistency are proposed. A key insight of the paper is that dynamic consistency and the notion of 'measure convex sets of probability measures' are intimately related. Measure convexity can be interpreted using the concept of compound lotteries. We characterize the class of static risk measures that represent consistent dynamic risk measures. It turns out that these are closely connected to shortfall risk. Under weak additional assumptions, static convex risk measures coincide with shortfall risk, if compound lotteries of acceptable respectively rejected positions are again acceptable respectively rejected. This result implies a characterization of dynamically consistent convex risk measures.

Suggested Citation

  • Weber, Stefan, 2003. "Distribution-Invariant Dynamic Risk Measures," SFB 373 Discussion Papers 2003,53, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:200353
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/5446 is not listed on IDEAS
    2. Stefan Jaschke & Uwe Küchler, 2001. "Coherent risk measures and good-deal bounds," Finance and Stochastics, Springer, vol. 5(2), pages 181-200.
    3. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    4. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    5. Carlier, G. & Dana, R. A., 2003. "Core of convex distortions of a probability," Journal of Economic Theory, Elsevier, vol. 113(2), pages 199-222, December.
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    Cited by:

    1. Geman, Hélyette & Ohana, Steve, 2008. "Time-consistency in managing a commodity portfolio: A dynamic risk measure approach," Journal of Banking & Finance, Elsevier, vol. 32(10), pages 1991-2005, October.
    2. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2004. "Dynamic monetary risk measures for bounded discrete-time processes," Papers math/0410453, arXiv.org.
    3. Berend Roorda & J. M. Schumacher & Jacob Engwerda, 2005. "Coherent Acceptability Measures In Multiperiod Models," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 589-612, October.

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    More about this item

    Keywords

    Dynamic risk measure; capital requirement; measure of risk; dynamic consistency; measure convexity; shortfall risk;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G28 - Financial Economics - - Financial Institutions and Services - - - Government Policy and Regulation
    • G18 - Financial Economics - - General Financial Markets - - - Government Policy and Regulation

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