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Stochastic Root Finding and Efficient Estimation of Convex Risk Measures


  • Jörn Dunkel

    () (Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom)

  • Stefan Weber

    () (Institut für Mathematische Stochastik, Insurance and Financial Mathematics, Gottfried Wilhelm Leibniz Universität Hannover, 30167 Hannover, Germany)


Reliable risk measurement is a key problem for financial institutions and regulatory authorities. The current industry standard Value-at-Risk has several deficiencies. Improved risk measures have been suggested and analyzed in the recent literature, but their computational implementation has largely been neglected so far. We propose and investigate stochastic approximation algorithms for the convex risk measure Utility-Based Shortfall Risk. Our approach combines stochastic root-finding schemes with importance sampling. We prove that the resulting Shortfall Risk estimators are consistent and asymptotically normal, and provide formulas for confidence intervals. The performance of the proposed algorithms is tested numerically. We finally apply our techniques to the Normal Copula Model, which is also known as the industry model CreditMetrics. This provides guidance for future implementations in practice.

Suggested Citation

  • Jörn Dunkel & Stefan Weber, 2010. "Stochastic Root Finding and Efficient Estimation of Convex Risk Measures," Operations Research, INFORMS, vol. 58(5), pages 1505-1521, October.
  • Handle: RePEc:inm:oropre:v:58:y:2010:i:5:p:1505-1521
    DOI: 10.1287/opre.1090.0784

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    References listed on IDEAS

    1. Jeff Linderoth & Alexander Shapiro & Stephen Wright, 2006. "The empirical behavior of sampling methods for stochastic programming," Annals of Operations Research, Springer, vol. 142(1), pages 215-241, February.
    2. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    3. Stefan Weber, 2006. "Distribution‐Invariant Risk Measures, Information, And Dynamic Consistency," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 419-441, April.
    4. Paul Glasserman & Wanmo Kang & Perwez Shahabuddin, 2008. "Fast Simulation of Multifactor Portfolio Credit Risk," Operations Research, INFORMS, vol. 56(5), pages 1200-1217, October.
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    Cited by:

    1. Zhaolin Hu & Dali Zhang, 2018. "Utility‐based shortfall risk: Efficient computations via Monte Carlo," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(5), pages 378-392, August.

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