Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models

Misile Kunene; Joseph Kyalo Mung’atu; Euna Nyarige

doi:doi:10.11648/j.ajtas.20211003.12

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Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models

Misile Kunene, Joseph Kyalo Mung’atu, Euna Nyarige

Published in American Journal of Theoretical and Applied Statistics (Volume 10, Issue 3)

Received: 6 May 2021 Accepted: 24 May 2021 Published: 31 May 2021

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Abstract

Generalized Linear Mixed Models (GLMMs) can be used to model the occurrence of defaults in a loan or bond portfolio. In this paper, we used a Bernoulli mixture model, a type of GLMMs, to model the dependency of default events. We discussed how Bernoulli mixture models can be used to model portfolio credit default risk, with the probit normal distribution as the link function. The general mathematical framework of the GLMMs was examined, with a particular focus on using Bernoulli mixture models to model credit default risk measures. We showed how GLMMs can be mapped into Bernoulli mixture models. An important aspect in portfolio credit default modelling is the dependence among default events, and in the GLMM setting, this may be captured using the so called random effects. Both fixed and random effects influence default probabilities of firms and these are taken as the systemic risk of the portfolio. After describing the model, we also conducted an empirical study for the applicability of our model using Standard and Poor’s data incorporating rating category (fixed effect) and time (random effect) as components of the model that constitute to the systemic risk of the portfolio. We were able to find the estimates of the model parameters using the Maximum Likelihood (ML) estimation method.

Published in	American Journal of Theoretical and Applied Statistics (Volume 10, Issue 3)
DOI	10.11648/j.ajtas.20211003.12
Page(s)	146-151
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Portfolio Credit Risk, Generalized Linear Mixed Models, Bernoulli Mixture Models, Dependency, Risk Measures

References

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[2]	Egloff, D., Leippold, M. and Vanini, P., "A simple model of credit contagion," Journal of Banking & Finance, vol. 31, p. 2475–2492., 2007.
[3]	Giesecke, K. and Weber, S., "Cyclical correlations, credit contagion, and portfolio losses," Journal of Banking & Finance, vol. 28, p. 3009–3036, 2004.
[4]	Giesecke, K. and Weber, S., "Credit contagion and aggregate losses.," Journal of Economic Dynamics and Control, vol. 30, p. 741–767., 2006.
[5]	Bangia, A., Diebold, F. X., Kronimus, A., Schagen, C. and Schuermann, T., "Ratings migration and the business cycle, with application to credit portfolio stress testing," Journal of Banking & Finance, vol. 26, p. 445–474, 2002.
[6]	C. S. F. Boston, "CreditRisk+: A credit risk management framework Technical Report, Technical report, Credit Suisse First Boston," 1997.
[7]	Molins, J. and VIives, E, "Model risk and credit risk," Risk and Decision Analysis, vol. 6, p. 65–78, 2016.
[8]	H. Joe, Multivariate models and multivariate dependence concepts, CRC Press, 1997.
[9]	Frey R., McNeil A. J., "Modelling dependent defaults," ETH, Zurich, 2001.
[10]	C. Martin Robert, "On the pricing of corporate debt: the risk structure of interest rates," The Journal of Finance, vol. 29, no. 2, pp. 449--470, 1974.
[11]	Boston, Credit Suisse First, "CreditRisk+: A credit risk management framework," 1997.
[12]	J. K. Lindsey, Applying generalized linear models, Springler Science & Business Media, 2000.
[13]	McNeil, Alexander J and Frey, Rüdiger and Embrechts, Paul, Quantitative risk management: concepts, techniques and tools-revised edition, Princeton university press, 2015.
[14]	Breslow, Norman E and Clayton, David G, "Approximate inference for generalized linear mixed models," Journal of the American Statistical Association, vol. 88, no. 421, pp. 2-25, 1993.
[15]	Gourieroux, Monfort and Gourieroux, Christian and Monfort, Alain and Monfort, Director Alain and others, Simulation based econometric methods, Oxford university press, 1996.
[16]	Frey, R. and McNeil, A. J., "Dependent defaults in models of portfolio credit risk," Journal of Risk, vol. 6, pp. 59-92, 2003.

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APA Style

Misile Kunene, Joseph Kyalo Mung’atu, Euna Nyarige. (2021). Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models. American Journal of Theoretical and Applied Statistics, 10(3), 146-151. https://doi.org/10.11648/j.ajtas.20211003.12

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ACS Style

Misile Kunene; Joseph Kyalo Mung’atu; Euna Nyarige. Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models. Am. J. Theor. Appl. Stat. 2021, 10(3), 146-151. doi: 10.11648/j.ajtas.20211003.12

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AMA Style

Misile Kunene, Joseph Kyalo Mung’atu, Euna Nyarige. Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models. Am J Theor Appl Stat. 2021;10(3):146-151. doi: 10.11648/j.ajtas.20211003.12

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@article{10.11648/j.ajtas.20211003.12,
  author = {Misile Kunene and Joseph Kyalo Mung’atu and Euna Nyarige},
  title = {Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {3},
  pages = {146-151},
  doi = {10.11648/j.ajtas.20211003.12},
  url = {https://doi.org/10.11648/j.ajtas.20211003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211003.12},
  abstract = {Generalized Linear Mixed Models (GLMMs) can be used to model the occurrence of defaults in a loan or bond portfolio. In this paper, we used a Bernoulli mixture model, a type of GLMMs, to model the dependency of default events. We discussed how Bernoulli mixture models can be used to model portfolio credit default risk, with the probit normal distribution as the link function. The general mathematical framework of the GLMMs was examined, with a particular focus on using Bernoulli mixture models to model credit default risk measures. We showed how GLMMs can be mapped into Bernoulli mixture models. An important aspect in portfolio credit default modelling is the dependence among default events, and in the GLMM setting, this may be captured using the so called random effects. Both fixed and random effects influence default probabilities of firms and these are taken as the systemic risk of the portfolio. After describing the model, we also conducted an empirical study for the applicability of our model using Standard and Poor’s data incorporating rating category (fixed effect) and time (random effect) as components of the model that constitute to the systemic risk of the portfolio. We were able to find the estimates of the model parameters using the Maximum Likelihood (ML) estimation method.},
 year = {2021}
}

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TY  - JOUR
T1  - Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models
AU  - Misile Kunene
AU  - Joseph Kyalo Mung’atu
AU  - Euna Nyarige
Y1  - 2021/05/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211003.12
DO  - 10.11648/j.ajtas.20211003.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 146
EP  - 151
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211003.12
AB  - Generalized Linear Mixed Models (GLMMs) can be used to model the occurrence of defaults in a loan or bond portfolio. In this paper, we used a Bernoulli mixture model, a type of GLMMs, to model the dependency of default events. We discussed how Bernoulli mixture models can be used to model portfolio credit default risk, with the probit normal distribution as the link function. The general mathematical framework of the GLMMs was examined, with a particular focus on using Bernoulli mixture models to model credit default risk measures. We showed how GLMMs can be mapped into Bernoulli mixture models. An important aspect in portfolio credit default modelling is the dependence among default events, and in the GLMM setting, this may be captured using the so called random effects. Both fixed and random effects influence default probabilities of firms and these are taken as the systemic risk of the portfolio. After describing the model, we also conducted an empirical study for the applicability of our model using Standard and Poor’s data incorporating rating category (fixed effect) and time (random effect) as components of the model that constitute to the systemic risk of the portfolio. We were able to find the estimates of the model parameters using the Maximum Likelihood (ML) estimation method.
VL  - 10
IS  - 3
ER  -

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Author Information

Misile Kunene

Financial Mathematics, Department of Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Joseph Kyalo Mung’atu

Statistics, Department of Statistics and Actuarial Science, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Euna Nyarige

Statistics, Department of Mathematics and Statistics, Machakos University, Machakos, Kenya

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Plain Text BibTeX RIS

APA Style

Misile Kunene, Joseph Kyalo Mung’atu, Euna Nyarige. (2021). Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models. American Journal of Theoretical and Applied Statistics, 10(3), 146-151. https://doi.org/10.11648/j.ajtas.20211003.12

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ACS Style

Misile Kunene; Joseph Kyalo Mung’atu; Euna Nyarige. Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models. Am. J. Theor. Appl. Stat. 2021, 10(3), 146-151. doi: 10.11648/j.ajtas.20211003.12

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AMA Style

Misile Kunene, Joseph Kyalo Mung’atu, Euna Nyarige. Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models. Am J Theor Appl Stat. 2021;10(3):146-151. doi: 10.11648/j.ajtas.20211003.12

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@article{10.11648/j.ajtas.20211003.12,
  author = {Misile Kunene and Joseph Kyalo Mung’atu and Euna Nyarige},
  title = {Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {3},
  pages = {146-151},
  doi = {10.11648/j.ajtas.20211003.12},
  url = {https://doi.org/10.11648/j.ajtas.20211003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211003.12},
  abstract = {Generalized Linear Mixed Models (GLMMs) can be used to model the occurrence of defaults in a loan or bond portfolio. In this paper, we used a Bernoulli mixture model, a type of GLMMs, to model the dependency of default events. We discussed how Bernoulli mixture models can be used to model portfolio credit default risk, with the probit normal distribution as the link function. The general mathematical framework of the GLMMs was examined, with a particular focus on using Bernoulli mixture models to model credit default risk measures. We showed how GLMMs can be mapped into Bernoulli mixture models. An important aspect in portfolio credit default modelling is the dependence among default events, and in the GLMM setting, this may be captured using the so called random effects. Both fixed and random effects influence default probabilities of firms and these are taken as the systemic risk of the portfolio. After describing the model, we also conducted an empirical study for the applicability of our model using Standard and Poor’s data incorporating rating category (fixed effect) and time (random effect) as components of the model that constitute to the systemic risk of the portfolio. We were able to find the estimates of the model parameters using the Maximum Likelihood (ML) estimation method.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Dependent Credit Default Risk Modelling Using Bernoulli Mixture Models
AU  - Misile Kunene
AU  - Joseph Kyalo Mung’atu
AU  - Euna Nyarige
Y1  - 2021/05/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211003.12
DO  - 10.11648/j.ajtas.20211003.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 146
EP  - 151
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211003.12
AB  - Generalized Linear Mixed Models (GLMMs) can be used to model the occurrence of defaults in a loan or bond portfolio. In this paper, we used a Bernoulli mixture model, a type of GLMMs, to model the dependency of default events. We discussed how Bernoulli mixture models can be used to model portfolio credit default risk, with the probit normal distribution as the link function. The general mathematical framework of the GLMMs was examined, with a particular focus on using Bernoulli mixture models to model credit default risk measures. We showed how GLMMs can be mapped into Bernoulli mixture models. An important aspect in portfolio credit default modelling is the dependence among default events, and in the GLMM setting, this may be captured using the so called random effects. Both fixed and random effects influence default probabilities of firms and these are taken as the systemic risk of the portfolio. After describing the model, we also conducted an empirical study for the applicability of our model using Standard and Poor’s data incorporating rating category (fixed effect) and time (random effect) as components of the model that constitute to the systemic risk of the portfolio. We were able to find the estimates of the model parameters using the Maximum Likelihood (ML) estimation method.
VL  - 10
IS  - 3
ER  -

Copy | Download