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On stability of operational risk estimates by LDA: From causes to approaches


  • Zhou, Xiaoping
  • Durfee, Antonina V.
  • Fabozzi, Frank J.


The stability of estimates is critical when applying advanced measurement approaches (AMA) such as loss distribution approach (LDA) for operational risk capital modeling. Recent studies have identified issues associated with capital estimates by applying the maximum likelihood estimation (MLE) method for truncated distributions: significant upward mean-bias, considerable uncertainty about the estimates, and non-robustness to both small and large losses. Although alternative estimation approaches have been proposed, there has not been any comprehensive study of how alternative approaches perform compared to the MLE method. This paper is the first comprehensive study on the performance of various potentially promising alternative approaches (including minimum distance approach, quantile distance approach, scaling-based bias correction, upward scaling of lower quantiles, and right-truncated distributions) as compared to MLE with regards to accuracy, precision and robustness. More importantly, based on the properties of each estimator, we propose a right-truncation with probability weighted least squares method, by combining the right-truncated distribution and minimizing a probability weighted distance (i.e., the quadratic upper-tail Anderson–Darling distance), and we find it significantly reduces the bias and volatility of capital estimates and improves the robustness of capital estimates to small losses near the threshold or moving the threshold, demonstrated by both simulation results and real data application.

Suggested Citation

  • Zhou, Xiaoping & Durfee, Antonina V. & Fabozzi, Frank J., 2016. "On stability of operational risk estimates by LDA: From causes to approaches," Journal of Banking & Finance, Elsevier, vol. 68(C), pages 266-278.
  • Handle: RePEc:eee:jbfina:v:68:y:2016:i:c:p:266-278
    DOI: 10.1016/j.jbankfin.2016.01.014

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    References listed on IDEAS

    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Sonja Huber, 2010. "(Non-)robustness of maximum likelihood estimators for operational risk severity distributions," Quantitative Finance, Taylor & Francis Journals, vol. 10(8), pages 871-882.
    3. Xiaoping Zhou & Rosella Giacometti & Frank J. Fabozzi & Ann H. Tucker, 2014. "Bayesian estimation of truncated data with applications to operational risk measurement," Quantitative Finance, Taylor & Francis Journals, vol. 14(5), pages 863-888, May.
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    5. Dahen, Hela & Dionne, Georges, 2010. "Scaling models for the severity and frequency of external operational loss data," Journal of Banking & Finance, Elsevier, vol. 34(7), pages 1484-1496, July.
    6. Ames, Mark & Schuermann, Til & Scott, Hal S., 2014. "Bank Capital for Operational Risk: A Tale of Fragility and Instability," Working Papers 14-02, University of Pennsylvania, Wharton School, Weiss Center.
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    8. Chavez-Demoulin, V. & Embrechts, P. & Neslehova, J., 2006. "Quantitative models for operational risk: Extremes, dependence and aggregation," Journal of Banking & Finance, Elsevier, vol. 30(10), pages 2635-2658, October.
    9. Cope, Eric W. & Piche, Mark T. & Walter, John S., 2012. "Macroenvironmental determinants of operational loss severity," Journal of Banking & Finance, Elsevier, vol. 36(5), pages 1362-1380.
    10. Christian Menn & Svetlozar Rachev, 2009. "Smoothly truncated stable distributions, GARCH-models, and option pricing," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 411-438, July.
    11. Chernobai, Anna & Jorion, Philippe & Yu, Fan, 2012. "The Determinants of Operational Risk in U.S. Financial Institutions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 46(06), pages 1683-1725, February.
    12. Paul Embrechts & Marco Frei, 2009. "Panjer recursion versus FFT for compound distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 497-508, July.
    13. Allen, Linda & Bali, Turan G., 2007. "Cyclicality in catastrophic and operational risk measurements," Journal of Banking & Finance, Elsevier, vol. 31(4), pages 1191-1235, April.
    14. Kim, Joseph Hyun Tae & Hardy, Mary R., 2007. "Quantifying and Correcting the Bias in Estimated Risk Measures," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 37(02), pages 365-386, November.
    15. J. D. Opdyke, 2014. "Estimating Operational Risk Capital with Greater Accuracy, Precision, and Robustness," Papers 1406.0389,, revised Nov 2014.
    16. Xiaolin Luo & Pavel V. Shevchenko & John B. Donnelly, 2009. "Addressing the Impact of Data Truncation and Parameter Uncertainty on Operational Risk Estimates," Papers 0904.2910,
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    More about this item


    Operational risk modeling; Maximum likelihood estimation; Bias correction; Robust estimation; Right-truncated distribution; Probability weighted least squares method;

    JEL classification:

    • G21 - Financial Economics - - Financial Institutions and Services - - - Banks; Other Depository Institutions; Micro Finance Institutions; Mortgages
    • G28 - Financial Economics - - Financial Institutions and Services - - - Government Policy and Regulation


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