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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.

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Paper provided by arXiv.org in its series Quantitative Finance Papers with number 1112.5550.

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Date of creation: Dec 2011
Date of revision: Apr 2012
Handle: RePEc:arx:papers:1112.5550

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  1. 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.
  2. Kiefer, Nicholas M., 2006. "Default Estimation for Low-Default Portfolios," Working Papers 06-08, Cornell University, Center for Analytic Economics.
  3. Dirk Tasche, 2009. "Estimating discriminatory power and PD curves when the number of defaults is small," Quantitative Finance Papers 0905.3928, arXiv.org, revised Mar 2010.
  4. Dirk Tasche, 2002. "Expected Shortfall and Beyond," Quantitative Finance Papers cond-mat/0203558, arXiv.org, revised Oct 2002.
  5. 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.
  6. Alexander Shapiro & Jos Berge, 2002. "Statistical inference of minimum rank factor analysis," Psychometrika, Springer, vol. 67(1), pages 79-94, March.
  7. Kiefer, Nicholas M., 2010. "Default Estimation and Expert Information," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(2), pages 320-328.
  8. Katja Pluto & Dirk Tasche, 2004. "Estimating Probabilities of Default for Low Default Portfolios," Quantitative Finance Papers cond-mat/0411699, arXiv.org, revised Apr 2005.
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