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Estimation of Regulatory Credit Risk Models


This article estimates a general credit risk model with both macroeconomic and latent credit factors for Spanish banks during the period 2004–2010. The proposed framework allows to estimate with bank level data both a credit risk model in line with the standard of Basel II and generalized models. I find evidence of persistence in the credit latent factor and of a significant effect of GDP growth and interbank rates on loan default rates. The estimated default correlation is low across specifications, indicating a positive relation between bank concentration and financial stability. The model is also used to calculate the impact on the probabilities of default of stressed economic scenarios.

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  1. 1.

    I refer throughout the document to the regulatory structure for International Convergence of Capital Measurement and Capital Standards of the Basel Committee of Banking Supervision (2004) as Basel II.

  2. 2.

    See also Gordy (2003) for an analysis of the application of Vasicek (2002) to internal ratings models.

  3. 3.

    The relation between competition and bank risk taking has been also studied in Chiappori et al. (1995), Hellmann et al. (2000), Matutes and Vives (2000) and Repullo (2004). Acharya (2009) evaluates bank closure policy and capital requirements in a duopoly model in which banks choose credit correlation.

  4. 4.

    An account of the restructuring process of the Spanish banking sector can be found on the Bank of Spain webpage at:

  5. 5.

    The defaults of corporate U.S bonds have been thoroughly analyzed. Recently, Duffie et al. (2009) find that the addition of a second factor in a continuous-time latent factor model is necessary to control for correlation in corporate bond defaults.

  6. 6.

    The probability Pr(d i j r t < 0|ι t ) in Eq. 1 is the base for the application of the Kalman filter and estimation of all the models in the article. The probabilities Pr(d i j r t < 0|ι t − 1) in Eqs. 4 and 5 are discussed because they serve to form expected default rates at time t for out-of-sample forecasting or modelization of expectations in theoretical models.

  7. 7.

    The unconditional (on ξ t ) correlation across defaults of borrowers i and i′ is defined by \(Corr(d_{ijrt},d_{i^{\prime }jrt})= 1/[1+(1-\rho )(1-F^{2})/\rho ]\). I focus on conditional correlation throughout the article, assuming that ι t − 1 is the relevant information set for banks at t.

  8. 8.

    The study does not consider credit cooperatives, a form of mutual banking, and financial institutions with narrow operations in terms of geography or credit products. In particular, I do not include specialist banks and small subsidiaries of foreign banks with a focus on a representative function or wholesale operations.

  9. 9.

    The information requirements on interest rate reporting can be found in the Bank of Spain order 1/2010 published in BOE (2010), in accordance with the European Central Bank Regulations (CE) 290/2009. Banks with assets in excess of 1,500 million euros and euro denominated deposits in excess of 500 million euros are required to report interest rates. The Bank of Spain can also require interest rate information to banks that do not satisfy this threshold.

  10. 10.

    The interest rate on other loans to households serves as a proxy for the rates on personal loans, as it weights the interest rates on consumer loans and other loans to households with no mortgage guarantees.

  11. 11.

    The application of maximum likelihood estimation with Kalman filtering follows the presentation of Chapter 13 of Hamilton (1994).

  12. 12.

    See Watson (1989) and Hamilton (1994) for asymptotic results. I compute the derivatives in Eq. 14 with the numerical differentiation suite for Matlab® of John D’Errico, which is available at

  13. 13.

    The classic references in the field include Geweke (1977), Sargent and Sims (1977), and Stock and Watson (1989). Hamilton (1994) provides a general exposition of this literature.

  14. 14.

    Approximate factor models allow for cross sectional correlation of idiosyncratic components whereas an exact factor model uses an i.i.d. assumption. Doz et al. (2012) use an exact factor model as proxy for an approximate factor model and show that the bias introduced by approximation disappears asymptotically. The results in Doz et al. (2012) apply to exact factor models as an special case.

  15. 15.

    I have used the test in Harris and Tzavalis (1999) to check for non-stationarity in the panel of transformed default rates. The test rejects the presence of a unit root in the panel data set. This is a regression-based test with panel fixed effects and it controls for contemporaneous correlation through common time effects, but it does not allow to test whether the common time effects are themselves persistent.

  16. 16.

    The bootstrap procedure across the time dimension t is analogous to the cross-sectional bootstrap described in Section 4. In step (1), I draw time periods with replacement from the original sample, with each draw containing the information of all risk classes at period t.

  17. 17.

    The number of banks that minimizes the probability of bank failure with ρ < 0.1 is 1 for most of the risk-shifting possibilities in the calibration considered in Martinez-Miera and Repullo (2010). See Figs. 3 and 6 in Martinez-Miera and Repullo (2010) for results with Cournot and circular city models.

  18. 18.

    These adjustments are not considered for 2004-2010, as the Spanish financial sector was very stable in this period. There are no failures or mergers between banks in the sample, and the relative weight of acquisition targets in total credit in the Spanish financial sector is on average below 2 %. Estimation results are unaffected by this low level of corporate acquisitions.

  19. 19.

    The Statistical Bulletin of the Bank of Spainreports in Section 4.14 an aggregate figure of 612,000 million euros for the total household mortgages of credit institutions at year end 2010.

  20. 20.

    The forecasting accuracy of the models is also linked to its capacity to rank banks based on default rates. The average ranking differences between data and forecasts are small, but observations with larger forecasting errors can be ranked imprecisely.


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I thank Javier Mencía, Jesús Saurina, Carlos Trucharte, the editor and two anonymous referees for their useful comments on this project. This article is my sole responsibility and, in particular, it does not necessarily represent the views of the Bank of Spain or the Eurosystem.

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Correspondence to Carlos Perez Montes.

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Perez Montes, C. Estimation of Regulatory Credit Risk Models. J Financ Serv Res 48, 161–191 (2015).

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  • Credit risk
  • Default correlation
  • Stress test
  • State space model
  • Bootstrap
  • MLE

JEL Classification

  • E0
  • G21