The Annals of Applied Statistics

Disentangling and assessing uncertainties in multiperiod corporate default risk predictions

Miao Yuan, Cheng Yong Tang, Yili Hong, and Jian Yang

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Measuring the corporate default risk is broadly important in economics and finance. Quantitative methods have been developed to predictively assess future corporate default probabilities. However, as a more difficult yet crucial problem, evaluating the uncertainties associated with the default predictions remains little explored. In this paper, we attempt to fill this blank by developing a procedure for quantifying the level of associated uncertainties upon carefully disentangling multiple contributing sources. Our framework effectively incorporates broad information from historical default data, corporates’ financial records, and macroeconomic conditions by (a) characterizing the default mechanism, and (b) capturing the future dynamics of various features contributing to the default mechanism. Our procedure overcomes the major challenges in this large scale statistical inference problem and makes it practically feasible by using parsimonious models, innovative methods, and modern computational facilities. By predicting the marketwide total number of defaults and assessing the associated uncertainties, our method can also be applied for evaluating the aggregated market credit risk level. Upon analyzing a US market data set, we demonstrate that the level of uncertainties associated with default risk assessments is indeed substantial. More informatively, we also find that the level of uncertainties associated with the default risk predictions is correlated with the level of default risks, indicating potential for new scopes in practical applications including improving the accuracy of default risk assessments.

Article information

Ann. Appl. Stat., Volume 12, Number 4 (2018), 2587-2617.

Received: June 2016
Revised: April 2018
First available in Project Euclid: 13 November 2018

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Mathematical Reviews number (MathSciNet)

Competing risks corporate default probability EM algorithm dynamic factor model high-dimensional time series prediction interval


Yuan, Miao; Tang, Cheng Yong; Hong, Yili; Yang, Jian. Disentangling and assessing uncertainties in multiperiod corporate default risk predictions. Ann. Appl. Stat. 12 (2018), no. 4, 2587--2617. doi:10.1214/18-AOAS1170.

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  • Altman, E. I. (1968). Financial ratios, discriminant analysis, and the prediction of corporate bankruptcy. J. Finance 23 589–609.
  • Altman, E., Resti, A. and Sironi, A. (2004). Default recovery rates in credit risk modelling: A review of the literature and empirical evidence. Economic Notes 33 183–208.
  • Bai, J. and Ng, S. (2008). Large dimensional factor analysis. Foundations and Trends in Econometrics 3 89–163.
  • Bai, J. and Wang, P. (2015). Identification and Bayesian estimation of dynamic factor models. J. Bus. Econom. Statist. 33 221–240.
  • Bańbura, M. and Modugno, M. (2014). Maximum likelihood estimation of factor models on datasets with arbitrary pattern of missing data. J. Appl. Econometrics 29 133–160.
  • Beaver, W. H. (1966). Financial ratios as predictors of failure. J. Acc. Res. 4 71.
  • Beaver, W. H. (1968). Market prices, financial ratios, and the prediction of failure. J. Acc. Res. 6 179.
  • Beaver, W. H., Correia, M. and McNichols, M. F. (2012). Do differences in financial reporting attributes impair the predictive ability of financial ratios for bankruptcy? Rev. Acc. Stud. 17 969–1010.
  • Bharath, S. T. and Shumway, T. (2008). Forecasting default with the merton distance to default model. Rev. Financ. Stud. 21 1339–1369.
  • Böhning, D., Dietz, E., Schaub, R., Schlattmann, P. and Lindsay, B. G. (1994). The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family. Ann. Inst. Statist. Math. 46 373–388.
  • Bräuning, F. and Koopman, S. J. (2014). Forecasting macroeconomic variables using collapsed dynamic factor analysis. Int. J. Forecast. 30 572–584.
  • Campbell, J. Y., Hilscher, J. and Szilagyi, J. (2008). In search of distress risk. J. Finance 63 2899–2939.
  • Chava, S. and Jarrow, R. A. (2004). Bankruptcy prediction with industry effects. Rev. Finance 8 537–569.
  • Diebold, F. X. and Mariano, R. S. (1995). Comparing predictive accuracy. J. Bus. Econom. Statist. 13 253–263.
  • Ding, A. A., Tian, S., Yu, Y. and Guo, H. (2012). A class of discrete transformation survival models with application to default probability prediction. J. Amer. Statist. Assoc. 107 990–1003.
  • Duan, J.-C., Sun, J. and Wang, T. (2012). Multiperiod corporate default prediction—A forward intensity approach. J. Econometrics 170 191–209.
  • Duffie, D. (2011). Measuring Corporate Default Risk. Oxford Univ. Press, Oxford.
  • Duffie, D. and Lando, D. (2001). Term structures of credit spreads with incomplete accounting information. Econometrica 69 633–664.
  • Duffie, D., Saita, L. and Wang, K. (2007). Multi-period corporate default prediction with stochastic covariates. J. Financ. Econ. 83 635–665.
  • Duffie, D., Eckner, A., Horel, G. and Saita, L. (2009). Frailty correlated default. J. Finance 64 2089–2123.
  • Durbin, J. and Koopman, S. J. (2012). Time Series Analysis by State Space Methods, 2nd ed. Oxford Statistical Science Series 38. Oxford Univ. Press, Oxford.
  • Giesecke, K. and Kim, B. (2011). Systemic risk: What defaults are telling us. Manage. Sci. 57 1387–1405.
  • Giesecke, K., Longstaff, F., Schafer, S. and Strebulaev, I. (2011). Corporate bond default risk: A 150-year perspective. J. Financ. Econ. 102 233–250.
  • Hillegeist, S. A., Keating, E. K., Cram, D. P. and Lundstedt, K. G. (2004). Assessing the probability of bankruptcy. Rev. Acc. Stud. 9 5–34.
  • Hong, Y. (2013). On computing the distribution function for the Poisson binomial distribution. Comput. Statist. Data Anal. 59 41–51.
  • Hong, Y., Meeker, W. Q. and McCalley, J. D. (2009). Prediction of remaining life of power transformers based on left truncated and right censored lifetime data. Ann. Appl. Stat. 3 857–879.
  • Hosmer, D. W., Lemeshow, S. and Sturdivant, R. X. (2013). Applied Logistic Regression. John Wiley and Sons Ltd.
  • Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data, 2nd ed. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ.
  • Koopman, S. J. and Lucas, A. (2005). Business and default cycles for credit risk. J. Appl. Econometrics 20 311–323.
  • Koopman, S. J. and Lucas, A. (2008). A non-Gaussian panel time series model for estimating and decomposing default risk. J. Bus. Econom. Statist. 26 510–525.
  • Koopman, S. J., Lucas, A. and Monteiro, A. (2008). The multi-state latent factor intensity model for credit rating transitions. J. Econometrics 142 399–424.
  • Koopman, S. J., Lucas, A. and Schwaab, B. (2011). Modeling frailty-correlated defaults using many macroeconomic covariates. J. Econometrics 162 312–325.
  • Koopman, S. J., Lucas, A. and Schwaab, B. (2012). Dynamic factor models with macro, frailty, and industry effects for U.S. default counts: The credit crisis of 2008. J. Bus. Econom. Statist. 30 521–532.
  • Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: Inference for the number of factors. Ann. Statist. 40 694–726.
  • Lawless, J. F. and Fredette, M. (2005). Frequentist prediction intervals and predictive distributions. Biometrika 92 529–542.
  • Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. Wiley, New York.
  • Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. J. Finance 29 449–470.
  • Ohlson, J. A. (1980). Financial ratios and the probabilistic prediction of bankruptcy. J. Acc. Res. 18 109.
  • Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors. Biometrika 95 365–379.
  • Peng, X. and Kou, S. (2009). Default clustering and valuation of collateralized debt obligations. Working paper.
  • Schwaab, B., Koopman, S. J. and Lucas, A. (2017). Global credit risk: World, country and industry factors. J. Appl. Econometrics 32 296–317.
  • Shumway, T. (2001). Forecasting bankruptcy more accurately: A simple hazard model. J. Bus. 74 101–124.
  • Stock, J. H. and Watson, M. W. (2002). Forecasting using principal components from a large number of predictors. J. Amer. Statist. Assoc. 97 1167–1179.
  • Tsay, R. S. (2010). Analysis of Financial Time Series, 3rd ed. Wiley, Hoboken, NJ.
  • Tsay, R. S. (2014). Multivariate Time Series Analysis: With R and Financial Applications. Wiley, Hoboken, NJ.
  • Volkova, A. Y. (1996). A refinement of the central limit theorem for sums of independent random indicators. Theory Probab. Appl. 40 791–794.
  • Yuan, M., Tang, C. Y., Hong, Y. and Yang, J. (2018). Supplement to “Disentangling and assessing uncertainties in multiperiod corporate default risk predictions.” DOI:10.1214/18-AOAS1170SUPP.
  • Zmijewski, M. E. (1984). Methodological issues related to the estimation of financial distress prediction models. J. Acc. Res. 22 59.

Supplemental materials

  • Supplement to “Disentangling and assessing uncertainties in multiperiod corporate default risk predictions”. Detail of the EM algorithm.