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Bayesian estimation of probabilities of default for low default portfolios

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  • Dirk Tasche

Abstract

The estimation of probabilities of default (PDs) for low default portfolios by means of upper confidence bounds is a well established procedure in many financial institutions. However, there are often discussions within the institutions or between institutions and supervisors about which confidence level to use for the estimation. The Bayesian estimator for the PD based on the uninformed, uniform prior distribution is an obvious alternative that avoids the choice of a confidence level. In this paper, we demonstrate that in the case of independent default events the upper confidence bounds can be represented as quantiles of a Bayesian posterior distribution based on a prior that is slightly more conservative than the uninformed prior. We then describe how to implement the uninformed and conservative Bayesian estimators in the dependent one- and multi-period default data cases and compare their estimates to the upper confidence bound estimates. The comparison leads us to suggest a constrained version of the uninformed (neutral) Bayesian estimator as an alternative to the upper confidence bound estimators.

Suggested Citation

  • Dirk Tasche, 2011. "Bayesian estimation of probabilities of default for low default portfolios," Papers 1112.5550, arXiv.org, revised Aug 2013.
    • Handle: RePEc:arx:papers:1112.5550
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    References listed on IDEAS

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    1. Dirk Tasche, 2009. "Estimating discriminatory power and PD curves when the number of defaults is small," Papers 0905.3928, arXiv.org, revised Mar 2010.
    2. Kiefer, Nicholas M., 2009. "Default estimation for low-default portfolios," Journal of Empirical Finance, Elsevier, vol. 16(1), pages 164-173, January.
    3. Orth, Walter, 2011. "Default probability estimation in small samples: With an application to sovereign bonds," Discussion Papers in Econometrics and Statistics 5/11, University of Cologne, Institute of Econometrics and Statistics.
    4. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    5. Nicholas M. Kiefer, 2011. "Default estimation, correlated defaults, and expert information," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 26(2), pages 173-192, March.
    6. Dirk Tasche, 2002. "Expected Shortfall and Beyond," Papers cond-mat/0203558, arXiv.org, revised Oct 2002.
    7. Katja Pluto & Dirk Tasche, 2006. "Estimating Probabilities of Default for Low Default Portfolios," Springer Books, in: Bernd Engelmann & Robert Rauhmeier (ed.), The Basel II Risk Parameters, chapter 0, pages 79-103, Springer.
    8. Bernd Engelmann & Robert Rauhmeier (ed.), 2006. "The Basel II Risk Parameters," Springer Books, Springer, number 978-3-540-33087-5, December.
    9. Orth, Walter, 2011. "Default probability estimation in small samples - with an application to sovereign bonds," MPRA Paper 33778, University Library of Munich, Germany.
    10. Bernd Engelmann & Robert Rauhmeier (ed.), 2011. "The Basel II Risk Parameters," Springer Books, Springer, number 978-3-642-16114-8, December.
    11. Tasche, Dirk, 2002. "Expected shortfall and beyond," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1519-1533, July.
    12. Alexander Shapiro & Jos Berge, 2002. "Statistical inference of minimum rank factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 67(1), pages 79-94, March.
    13. Kiefer, Nicholas M., 2010. "Default Estimation and Expert Information," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(2), pages 320-328.
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    Cited by:

    1. Andrius Grigutis, 2023. "Probabilistic Overview of Probabilities of Default for Low Default Portfolios by K. Pluto and D. Tasche," Papers 2303.06148, arXiv.org.
    2. Oliver Blümke, 2020. "Estimating the probability of default for no‐default and low‐default portfolios," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 69(1), pages 89-107, January.
    3. Oliver Blümke, 2022. "Multiperiod default probability forecasting," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 41(4), pages 677-696, July.
    4. Blümke, Oliver, 2018. "On the cyclicality of default rates of banks: A comparative study of the asset correlation and diversification effects," Journal of Empirical Finance, Elsevier, vol. 47(C), pages 65-77.
    5. Yi-Ping Chang & Chih-Tun Yu, 2014. "Bayesian confidence intervals for probability of default and asset correlation of portfolio credit risk," Computational Statistics, Springer, vol. 29(1), pages 331-361, February.
    6. Nendel, Max & Streicher, Jan, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Journal of Mathematical Economics, Elsevier, vol. 109(C).
    7. Denis Surzhko, 2017. "Bayesian Approach to PD Calibration and Stress-testing in Low Default Portfolios," Journal of Applied Finance & Banking, SCIENPRESS Ltd, vol. 7(2), pages 1-6.

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    JEL classification:

    • G2 - Financial Economics - - Financial Institutions and Services
    • E5 - Macroeconomics and Monetary Economics - - Monetary Policy, Central Banking, and the Supply of Money and Credit

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