The robustness of simulation-based Markovian transition probabilities for ultra-small samples of non-performing credit

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The analysis of systemic credit risk by financial regulators is largely affected by the paucity of data. Supervisors receive reports on proportions of performing and non-performing aggregate loan classes of individual banks-corporate, mortgage and consumer-often with very short history. The transition to different proportions of aggregate credit qualities can be seen as a Markov process and Christodoulakis [J. Credit Risk, 2007, 3(3), 25-39] proposed a simulation-based method for the estimation of transition probabilities under non-negativity constraints, based on the work of Kloek and van Dijk [Econometrica, 1978, 46(1), 1-19] and van Dijk and Kloek [J. Econometrics, 1980, 14, 307-328]. This paper provides Monte Carlo robustness checks for the performance of this estimator in comparison to Least Squares and Restricted Least Squares (OLS-R) in the presence of ultra-small samples. When true transition probabilities are very large or very small, the least squares estimators severely violate nonnegativity restrictions, leading to a spuriously small bias as compared to Monte Carlo integration (MCI). This result is intensified for low volatility regimes and small samples. The bias of MCI diminishes in higher volatility regimes and larger samples, irrespective of the number of Bayesian replications. Regarding the accuracy statistics, we observe that the variance of every estimator increases in smaller samples and lower volatility regimes. MCI is revealed as more accurate or at least equivalent to OLS-R in all cases. Our empirical applications with real US and European data on aggregate credit portfolios revealed significant violations of non-negativity constraints by least squares methods, which contribute to favourable conclusions for MCI. © 2011 Copyright Taylor and Francis Group, LLC.
Original languageEnglish
Pages (from-to)1847-1864
Number of pages17
JournalQuantitative Finance
Issue number12
Publication statusPublished - Dec 2011


  • Credit risk
  • Markov process
  • Non-performing credit
  • Probability
  • Transition


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