Table of Contents
ISRN Probability and Statistics
Volume 2012, Article ID 832175, 42 pages
http://dx.doi.org/10.5402/2012/832175
Review Article

Credit Portfolios, Credibility Theory, and Dynamic Empirical Bayes

Department of Statistics, Stanford University, Stanford, CA 94305, USA

Received 21 October 2012; Accepted 11 November 2012

Academic Editors: I. Beg and M. Scotto

Copyright © 2012 Tze Leung Lai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Robbins, “An empirical Bayes approach to statistics,” in Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. I, pp. 157–163, University of California Press, Los Angeles, Calif, USA, 1956. View at Zentralblatt MATH
  2. C. Stein, “Inadmissibility of the usual estimator for the mean of a multivariate normal distribution,” in Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. I, pp. 197–206, University of California Press, Los Angeles, Calif, USA, 1956. View at Zentralblatt MATH
  3. T. L. Lai, Y. Su, and K. H. Sun, “Dynamic empirical Bayes models and their applications to longitudinal data analysis and prediction,” Statistica Sinica. In press.
  4. N. E. Breslow and D. G. Clayton, “Approximate inference in generalized linear mixed models,” Journal of the American Statistical Association, vol. 88, no. 421, pp. 9–25, 1993. View at Publisher · View at Google Scholar
  5. T. R. Fleming and D. P. Harrington, Counting Processes and Survival Analysis, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1991.
  6. P. K. Andersen, Ø. Borgan, R. D. Gill, and N. Keiding, Statistical Models based on Counting Processes, Springer Series in Statistics, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar
  7. O. Aalen, “Nonparametric inference in connection with multiple decrement models,” Scandinavian Journal of Statistics, vol. 3, no. 1, pp. 15–27, 1976. View at Google Scholar · View at Zentralblatt MATH
  8. W. Nelson, “Theory and applications of hazard plotting for censored failure data,” Technometrics, vol. 14, no. 4, pp. 945–966, 1972. View at Publisher · View at Google Scholar
  9. E. I. Altman, “Measuring corporate bond mortality and performance,” Journal of Finance, vol. 44, no. 4, pp. 909–922, 1989. View at Publisher · View at Google Scholar
  10. E. L. Kaplan and P. Meier, “Nonparametric estimation from incomplete observations,” Journal of the American Statistical Association, vol. 53, pp. 457–481, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. D. R. Cox, “Regression models and life-tables,” Journal of the Royal Statistical Society Series B, vol. 34, pp. 187–220, 1972. View at Google Scholar · View at Zentralblatt MATH
  12. W. R. Lane, S. W. Looney, and J. W. Wansley, “An application of the Cox proportional hazards model to bank failure,” Journal of Banking & Finance, vol. 10, no. 4, pp. 511–531, 1986. View at Publisher · View at Google Scholar
  13. J. D. Kalbeisch and R. L. Prentice, The Statistical Analysis of Failure Time Data-Chichester-Brisbane, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1980.
  14. S. H. Lee and J. L. Urrutia, “Analysis and prediction of insolvency in the propertyliability insurance industry: a comparison of logit and hazard models,” Journal of Risk and Insurance, vol. 63, no. 1, pp. 121–130, 1996. View at Publisher · View at Google Scholar
  15. C. G. McDonald and L. M. Van de Gucht, “High-yield bond default and call risks,” The Review of Economics and Statistics, vol. 81, no. 3, pp. 409–419, 1999. View at Publisher · View at Google Scholar
  16. W. H. Beaver, “Market prices, financial ratios, and the prediction of failure,” Journal of Accounting Research, vol. 6, no. 2, pp. 179–192, 1968. View at Publisher · View at Google Scholar
  17. T. Shumway, “Forecasting bankruptcy more accurately: a simple hazard model,” The Journal of Business, vol. 74, no. 1, pp. 101–124, 2001. View at Publisher · View at Google Scholar
  18. S. Chava and R. A. Jarrow, “Bankruptcy prediction with industry effects,” Review of Finance, vol. 8, pp. 537–569, 2004. View at Google Scholar
  19. S. A. Hillegeist, E. K. Keating, D. P. Cram, and K. G. Lundstedt, “Assessing the probability of bankruptcy,” Review of Accounting Studies, vol. 9, pp. 5–34, 2004. View at Publisher · View at Google Scholar
  20. D. R. Cox, “Partial likelihood,” Biometrika, vol. 62, no. 2, pp. 269–276, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. N. E. Breslow, “Covariance analysis of censored survival data,” Biometrics, vol. 30, no. 1, pp. 89–99, 1974. View at Publisher · View at Google Scholar
  22. T. L. Lai and H. Xing, Statistical Models and Methods for Financial Markets, Springer Texts in Statistics, Springer, New York, NY, USA, 2008. View at Publisher · View at Google Scholar
  23. D. Duffe and K. J. Singleton, Credit Risk: Pricing, Measurement, and Management, Princeton Series in Finance, Princeton University Press, 2003.
  24. D. Lando, Credit Risk Modeling: Theory and Applications, Princeton Series in Finance, Princeton University Press, 2004.
  25. R. C. Merton, “On the pricing of corporate debt: the risk structure of interest rates,” Journal of Finance, vol. 29, no. 2, pp. 449–470, 1974. View at Google Scholar
  26. F. Black and J. C. Cox, “Valuing corporate securities: some effects of bond indenture provisions,” Journal of Finance, vol. 31, no. 2, pp. 351–367, 1976. View at Google Scholar
  27. O. Aalen, “Nonparametric estimation of partial transition probabilities in multiple decrement models,” The Annals of Statistics, vol. 6, no. 3, pp. 534–545, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. O. O. Aalen and S. Johansen, “An empirical transition matrix for non-homogeneous Markov chains based on censored observations,” Scandinavian Journal of Statistics, vol. 5, no. 3, pp. 141–150, 1978. View at Google Scholar · View at Zentralblatt MATH
  29. T. R. Fleming, “Nonparametric estimation for nonhomogeneous Markov processes in the problem of competing risks,” The Annals of Statistics, vol. 6, no. 5, pp. 1057–1070, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. O. A. Vasicek, “Probability of loss on a loan portfolio,” Working Paper, KMV Corporation, 1987. View at Google Scholar
  31. O. A. Vasicek, “The distribution of loan portfolio value,” RISK, vol. 15, no. 12, pp. 160–162, 2002. View at Google Scholar
  32. P. Schonbucher, “Factor models for portfolio credit risk,” Working Paper, Department of Statistics, Bonn University, 2000. View at Google Scholar
  33. D. X. Li, “On default correlation: a copula function approach,” The Journal of Fixed Income, vol. 9, no. 4, pp. 43–54, 2000. View at Publisher · View at Google Scholar
  34. O. A. Vasicek, “Limiting loan loss probability distribution,” Working Paper, KMV Corporation, 1991. View at Google Scholar
  35. M. B. Gordy, “A comparative anatomy of credit risk models,” Journal of Banking & Finance, vol. 24, no. 1-2, pp. 119–149, 2000. View at Publisher · View at Google Scholar
  36. R. Frey and A. J. McNeil, “Dependent defaults in models of portfolio credit risk,” Journal of Risk, vol. 6, no. 1, pp. 59–92, 2003. View at Google Scholar
  37. F. Salmon, “The formula that killed Wall Street,” Significance, vol. 9, no. 1, pp. 16–20, 2012. View at Publisher · View at Google Scholar
  38. J. W. Vaupel, K. G. Manton, and E. Stallard, “The impact of heterogeneity in individual frailty on the dynamics of mortality,” Demography, vol. 16, no. 3, pp. 439–454, 1979. View at Google Scholar · View at Scopus
  39. K. G. Manton, E. Stallard, and J. W. Vaupel, “Methods for comparing the mortality experience of heterogeneous populations,” Demography, vol. 18, no. 3, pp. 389–410, 1981. View at Google Scholar · View at Scopus
  40. K. G. Manton, E. Stallard, and J. W. Vaupel, “Alternative models for the heterogeneity of mortality risks among the aged,” Journal of the American Statistical Association, vol. 81, no. 395, pp. 635–644, 1986. View at Google Scholar · View at Scopus
  41. J. J. Heckman and B. Singer, “Econometric duration analysis,” Journal of Econometrics, vol. 24, no. 1-2, pp. 63–132, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. J. Heckman and B. Singer, “A method for minimizing the impact of distributional assumptions in econometric models for duration data,” Econometrica, vol. 52, no. 2, pp. 271–320, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. P. Hougaard, “Life table methods for heterogeneous populations: distributions describing the heterogeneity,” Biometrika, vol. 71, no. 1, pp. 75–83, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. P. Hougaard, “Survival models for heterogeneous populations derived from stable distributions,” Biometrika, vol. 73, no. 2, pp. 387–396, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. D. Clayton and J. Cuzick, “Multivariate generalizations of the proportional hazards model,” Journal of the Royal Statistical Society Series A, vol. 148, no. 2, pp. 82–117, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  46. D. Duffie, A. Eckner, G. Horel, and L. Saita, “Frailty correlated default,” Journal of Finance, vol. 64, no. 5, pp. 2089–2123, 2009. View at Publisher · View at Google Scholar · View at Scopus
  47. O. Cappé, E. Moulines, and T. Rydén, Inference in Hidden Markov Models, Springer Series in Statistics, Springer, New York, NY, USA, 2005.
  48. G. Celeux and J. Diebolt, “The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem,” Computational Statistics Quarterly, vol. 2, pp. 73–82, 1985. View at Google Scholar
  49. G. C. G. Wei and M. A. Tanner, “A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithms,” Journal of the American Statistical Association, vol. 85, no. 411, pp. 699–704, 1990. View at Google Scholar
  50. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 6, no. 6, pp. 721–741, 1984. View at Google Scholar · View at Scopus
  51. C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer Texts in Statistics, Springer, New York, NY, USA, 2nd edition, 2004.
  52. D. Duffie, L. Saita, and K. Wang, “Multi-period corporate default prediction with stochastic covariates,” Journal of Financial Economics, vol. 83, no. 3, pp. 635–665, 2007. View at Publisher · View at Google Scholar · View at Scopus
  53. M. Davis and V. Lo, “Infectious defaults,” Quantitative Finance, vol. 1, no. 4, pp. 382–387, 2001. View at Publisher · View at Google Scholar
  54. S. Deng, K. Giesecke, and T. L. Lai, “Sequential importance sampling and resampling for dynamic portfolio credit risk,” Operations Research, vol. 60, no. 1, pp. 78–91, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  55. O. Aalen, “A model for nonparametric regression analysis of counting processes,” in Mathematical Statistics and Probability Theory, vol. 2 of Lecture Notes in Statistics, pp. 1–25, Springer, New York, NY, USA, 1980, Proceedings of the 6th International Conference, Wisła, Poland, 1978. View at Google Scholar · View at Zentralblatt MATH
  56. F. W. Huffer and I. W. McKeague, “Weighted least squares estimation for Aalen's additive risk model,” Journal of the American Statistical Association, vol. 86, no. 413, pp. 114–129, 1991. View at Google Scholar
  57. A. T. Payandeh Najafabadi, “A new approach to the credibility formula,” Insurance: Mathematics & Economics, vol. 46, no. 2, pp. 334–338, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  58. H. Bühlmann, “Experience rating and credibility,” The Astin Bulletin, vol. 4, no. 3, pp. 199–207, 1967. View at Google Scholar
  59. H. Bühlmann, “Experience rating and credibility,” The Astin Bulletin, vol. 5, no. 2, pp. 157–169, 1969. View at Google Scholar
  60. W. James and C. Stein, “Estimation with quadratic loss,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 361–379, University of California Press, Berkeley, Calif, USA, 1961.
  61. H. Robbins, “Some thoughts on empirical Bayes estimation,” The Annals of Statistics, vol. 11, no. 3, pp. 713–723, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  62. H. Bühlmann and A. Gisler, A Course in Credibility Theory and Its Applications, Universitext, Springer, Berlin, Germany, 2005.
  63. C. A. Hachemeister, “Credibility for regression models with application to trend,” in Credibility: Theory and Applications (Proceedings of the Berkeley Actuarial Research Conference on Credibility, University of California, Berkeley, Calif, USA, 1974; dedicated to E. A. Lew), P. M. Kahn, Ed., pp. 129–169, Academic Press, New York, 1975, With a discussion by Al Quirin. View at Google Scholar · View at Zentralblatt MATH
  64. E. W. Frees, V. R. Young, and Y. Luo, “A longitudinal data analysis interpretation of credibility models,” Insurance: Mathematics & Economics, vol. 24, no. 3, pp. 229–247, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  65. E. W. Frees, V. R. Young, and Y. Luo, “Case studies using panel data models,” North American Actuarial Journal, vol. 5, no. 4, pp. 24–42, 2001. View at Google Scholar · View at Zentralblatt MATH
  66. W. S. Jewell, “Credible means are exact Bayesian for exponential families,” The Astin Bulletin, vol. 8, pp. 77–90, 1974. View at Google Scholar
  67. A. E. Renshaw, “Actuarial graduation practice and generalised linear and non-linear models,” Journal of the Institute of Actuaries, vol. 118, no. 2, pp. 295–312, 1991. View at Publisher · View at Google Scholar
  68. S. Haberman and A. E. Renshaw, “Generalized linear models and actuarial science,” Journal of the Royal Statistical Society Series D, vol. 45, no. 4, pp. 407–436, 1996. View at Google Scholar · View at Scopus
  69. J. A. Nelder and R. J. Verrall, “Credibility theory and generalized linear models,” The Astin Bulletin, vol. 27, no. 1, pp. 71–82, 1997. View at Google Scholar
  70. Z. Landsman and U. E. Makov, “Credibility evaluation for the exponential dispersion family,” Insurance: Mathematics & Economics, vol. 24, no. 1-2, pp. 23–29, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  71. Z. Landsman and U. E. Makov, “On credibility evaluation and the tail area of the exponential dispersion family,” Insurance: Mathematics & Economics, vol. 27, no. 3, pp. 277–283, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  72. K. Antonio and J. Beirlant, “Actuarial statistics with generalized linear mixed models,” Insurance: Mathematics & Economics, vol. 40, no. 1, pp. 58–76, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  73. T. L. Lai and K. H. Sun, “Evolutionary credibility theory: a generalized linear mixed modeling approach,” The North American Actuarial Journal. In press.
  74. S. L. Zeger and B. Qaqish, “Markov regression models for time series: a quasi-likelihood approach,” Biometrics, vol. 44, no. 4, pp. 1019–1031, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  75. T. L. Lai and M.-C. Shih, “Nonparametric estimation in nonlinear mixed effects models,” Biometrika, vol. 90, no. 1, pp. 1–13, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  76. R. Wolfinger and M. O'Connell, “Generalized linear mixed models a pseudo-likelihood approach,” Journal of Statistical Computation and Simulation, vol. 48, no. 3-4, pp. 233–243, 1993. View at Publisher · View at Google Scholar
  77. Q. Liu and D. A. Pierce, “A note on Gauss-Hermite quadrature,” Biometrika, vol. 81, no. 3, pp. 624–629, 1994. View at Google Scholar · View at Zentralblatt MATH
  78. A. Yafune, M. Takebe, and H. Ogata, “A use of Monte Carlo integration for population pharmacokinetics with multivariate population distribution,” Journal of Pharmacokinetics and Biopharmaceutics, vol. 26, pp. 103–123, 1998. View at Google Scholar
  79. T. L. Lai and M.-C. Shih, “A hybrid estimator in nonlinear and generalised linear mixed effects models,” Biometrika, vol. 90, no. 4, pp. 859–879, 2003. View at Publisher · View at Google Scholar
  80. T. L. Lai, M. -C. Shih, and S. P. Wong, “A new approach to modeling covariate effects and individualization in population pharmacokinetics-pharmacodynamics,” Journal of Pharmacokinetics and Pharmacodynamics, vol. 33, no. 1, pp. 49–74, 2006. View at Publisher · View at Google Scholar · View at Scopus
  81. T. L. Lai, M.-C. Shih, and S. P.-S. Wong, “Flexible modeling via a hybrid estimation scheme in generalized mixed models for longitudinal data,” Biometrics, vol. 62, no. 1, pp. 159–318, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  82. T. L. Lai and S. P.-S. Wong, “Statistical models for the Basel II internal ratings-based approach to measuring credit risk of retail products,” Statistics and its Interface, vol. 1, no. 2, pp. 229–241, 2008. View at Google Scholar · View at Zentralblatt MATH
  83. K. Y. Liang and S. L. Zeger, “Longitudinal data analysis using generalized linear models,” Biometrika, vol. 73, no. 1, pp. 13–22, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  84. M. S. Pepe and G. L. Anderson, “A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data,” Communications in Statistics B, vol. 23, no. 4, pp. 939–951, 1994. View at Google Scholar
  85. M. S. Pepe and D. Couper, “Modeling partly conditional means with longitudinal data,” Journal of the American Statistical Association, vol. 92, no. 439, pp. 991–998, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  86. B. Efron and C. Morris, “Data analysis using Stein's estimator and its generalizations,” Journal of the American Statistical Association, vol. 70, no. 350, pp. 311–319, 1975. View at Publisher · View at Google Scholar
  87. B. Efron and C. Morris, “Stein's paradox in statistics,” Scientific American, vol. 236, no. 5, pp. 119–127, 1977. View at Publisher · View at Google Scholar
  88. L. D. Brown, “In-season prediction of batting averages: a field test of empirical Bayes and Bayes methodologies,” The Annals of Applied Statistics, vol. 2, no. 1, pp. 113–152, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  89. C. Hsiao, C. Kim, and G. Taylor, “A statistical perspective on insurance rate-making,” Journal of Econometrics, vol. 44, no. 1-2, pp. 5–24, 1990. View at Publisher · View at Google Scholar
  90. M. West, P. J. Harrison, and H. S. Migon, “Dynamic generalized linear models and Bayesian forecasting,” Journal of the American Statistical Association, vol. 80, no. 389, pp. 73–83, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  91. A. T. Payandeh Najafabadi, “A new approach to the credibility formula,” Insurance: Mathematics & Economics, vol. 46, no. 2, pp. 334–338, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  92. Basel Committee on Banking Supervision, “An explanatory note on the Basel II IRB risk weight functions,” 2005, http://www.bis.org/bcbs/irbriskweight.pdf.
  93. C. A. Calhoun and Y. Deng, “A dynamic analysis of fixed- and adjustable-rate mortgage terminations,” Journal of Real Estate Finance and Economics, vol. 24, no. 1-2, pp. 9–33, 2002. View at Publisher · View at Google Scholar · View at Scopus
  94. J. P. Fine and R. J. Gray, “A proportional hazards model for the subdistribution of a competing risk,” Journal of the American Statistical Association, vol. 94, no. 446, pp. 496–509, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  95. R. J. Gray, “A class of K-sample tests for comparing the cumulative incidence of a competing risk,” The Annals of Statistics, vol. 16, no. 3, pp. 1141–1154, 1988. View at Publisher · View at Google Scholar
  96. J. P. Klein and Y. Shu, “Multi-state models for bone marrow transplantation studies,” Statistical Methods in Medical Research, vol. 11, no. 2, pp. 117–139, 2002. View at Publisher · View at Google Scholar
  97. T. H. Scheike, M.-J. Zhang, and T. A. Gerds, “Predicting cumulative incidence probability by direct binomial regression,” Biometrika, vol. 95, no. 1, pp. 205–220, 2008. View at Publisher · View at Google Scholar
  98. J. P. Klein and P. K. Andersen, “Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function,” Biometrics, vol. 61, no. 1, pp. 223–229, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  99. F. Graw, T. A. Gerds, and M. Schumacher, “On pseudo-values for regression analysis in competing risks models,” Lifetime Data Analysis, vol. 15, no. 2, pp. 241–255, 2009. View at Publisher · View at Google Scholar
  100. T. L. Lai, Y. Su, and K. H. Sun, “A new approach to dynamic panel data and its applications to default modeling,” Working Paper, Department of Statistics, Stanford Unviersity, 2012. View at Google Scholar
  101. M. Gu and T. L. Lai, “Repeated significance testing with censored rank statistics in interim analysis of clinical trials,” Statistica Sinica, vol. 8, no. 2, pp. 411–428, 1998. View at Google Scholar · View at Zentralblatt MATH
  102. Y. Jin, T. L. Lai, and S. Yong, “Evaluating econometric forecasts, with applications to mortgage default prediction and forecasting insurance claims,” Working Paper, Department of Statistics, Stanford Unviersity, 2012. View at Google Scholar
  103. T. L. Lai and S. P.-S. Wong, “Statistical models for the Basel II internal ratings-based approach to measuring credit risk of retail products,” Statistics and its Interface, vol. 1, no. 2, pp. 229–241, 2008. View at Google Scholar · View at Zentralblatt MATH
  104. S. Deng, K. Giesecke, and T. L. Lai, “Sequential importance sampling and resampling for dynamic portfolio credit risk,” Operations Research, vol. 60, no. 1, pp. 78–91, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH