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Dynamic coherent risk measures

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  • Riedel, Frank

Abstract

Monetary measures of risk like Value at Risk or Worst Conditional Expectation assess the risk of financial positions. The existing risk measures are of a static, one period nature. In this paper, I define dynamic monetary risk measures and I present an axiomatic approach that extends the class of coherent risk measures to the dynamic framework. The axiom of translation invariance has to be recast as predictable translation invariance to account for the release of new information. In addition to the coherency axioms, I introduce the axiom of dynamic consistency. Consistency requires that judgements based on the risk measure are not contradictory over time. I show that consistent dynamic coherent risk measures can be represented as the worst conditional expectation of discounted future losses where the expectations are being taken over a set of probability measures that satisfies a consistency condition.

Suggested Citation

  • Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
  • Handle: RePEc:eee:spapps:v:112:y:2004:i:2:p:185-200
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    References listed on IDEAS

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    1. Ioannis Karatzas & Jaksa Cvitanic, 1999. "On dynamic measures of risk," Finance and Stochastics, Springer, vol. 3(4), pages 451-482.
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    4. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    5. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    6. Sarin, Rakesh & Wakker, Peter P, 1998. "Dynamic Choice and NonExpected Utility," Journal of Risk and Uncertainty, Springer, vol. 17(2), pages 87-119, November.
    7. Berend Roorda & J. M. Schumacher & Jacob Engwerda, 2005. "Coherent Acceptability Measures In Multiperiod Models," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 589-612.
    8. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
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