Exploring Bayesian Hierarchical Models for Multi-Level Credit Risk Assessment: Detailed Insights

International Journal of Computer Science & Information Technology (IJCSIT) Vol 16, No 3, June 2024
DOI: 10.5121/ijcsit.2024.16306 67
EXPLORING BAYESIAN HIERARCHICAL MODELS
FOR MULTI-LEVEL CREDIT RISK
ASSESSMENT: DETAILED INSIGHTS
Sanjay Moolchandani
Vice President, Citibank N.A., New Jersey, USA
ABSTRACT
In this paper, we examine the use of Bayesian Hierarchical Models (BHMs) for multi-level credit risk
assessment while focusing on their advantages compared to conventional valuation approaches of single-
level models. Unlike most traditional methodologies, which consider events either separately or condition
on an aggregate measure, each of the BHMs systematically incorporates data from different levels — loan
or obligor level and institution level — to provide a more holistic view of credit risk under numerous
uncertainties and dependencies. The paper reviews basic theoretical underpinnings of BHMs, such as
Bayesian inference and hierarchical Modeling, while giving examples on how these mechanisms work in
practice within the context of estimating default risk. In addition, the paper outlines computational
challenges, highlights the role of prior distributions, and explains that BHMs could potentially be
combined with machine learning for dynamic risk assessments. The paper highlights a real-world
application, and provides detailed insights into how BHMs can help improve both the accuracy and
interpretability of credit risk assessments.
KEYWORDS
Bayesian Hierarchical Models, Credit Risk Assessment, Financial Risk Management, Multi-level
Modeling, Bayesian Inference, Default Risk, Machine Learning Integration.
1. INTRODUCTION
Credit risk assessment is one of the primary tools in financial risk management. It requires the
evaluation of the default risk, which is a critical part of lending and can be used by financial
institutions for credit decisions aswell as risk-management strategies[1]. Classical credit risk
models, e.g. logistic regression or decision trees, usually work on a single level of data: either one
individual loan or one counterparty. These models do not account for the complex hierarchical
structure of credit risk data—loans are nested within borrowers, and borrowers are in-turn nested
within institutions or companies[2].
Bayesian hierarchical models (BHMs) provide a robust framework for multi-level credit risk
assessment, offering nuanced insights by incorporating various levels of data and
uncertainties[3].
This paper delves into the intricacies of Bayesian hierarchical models, their application in credit
risk assessment, and the benefits they offer over traditional methods[4]. It will provide a holistic
view of the advanced statistical approach through its theoretical underpinnings, practical
implementation and real-world applications.

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2. THEORETICAL FOUNDATIONS OF BAYESIAN HIERARCHICAL MODELS
Following section gives the theoretical foundations of Bayesian hierarchical models:
2.1. Bayesian Inference
Bayesian Inference works on the principle of updating the probability of the hypothesis as new
evidencesor information is added. Bayesian approach quantify uncertainty using prior beliefs
which are updated in proportion to the strength of the evidence from new data[5]. At the heart of
Bayesian inference is Bayes' theorem.
𝑃(𝜃|𝑑𝑎𝑡𝑎) =
𝑃(𝑑𝑎𝑡𝑎|𝜃)𝑃(𝜃)
𝑃(𝑑𝑎𝑡𝑎)
(1)
Where:
 𝑃(𝜃|𝑑𝑎𝑡𝑎) 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝜃 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎
 𝑃(𝑑𝑎𝑡𝑎|𝜃) 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝜃
 𝑃(𝜃) 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑖𝑜𝑟 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝜃
 𝑃(𝑑𝑎𝑡𝑎) 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎
Bayesian inference allows for the incorporation of prior knowledge and the updating of this
knowledge with new data, providing a flexible and dynamic approach to statistical modelling [6].
2.2. Hierarchical Modelling
Hierarchical models are known as multi-level models, which uses data that have structure at more
than one level. For credit risk, this could be disaggregated into borrower-level data, loan-level
data, and institution-level data. A Hierarchical modelsaccounts for the dependency: it allows
analysis of data at different levels (i.e., within and between the variability) simultaneously [7].
A hierarchical model typically consists of:
• Level 1 (Individual level): The basic observational unit (e.g., individual loans).
• Level 2 (Group level): Groups of observational units (e.g., borrowers).
• Level 3 (Higher group level): Larger groups (e.g., financial institutions).
These levels are modeled with varying parameters, which can be correlated or independent,
providing a rich structure to capture complex relationships [8].
3. BAYESIAN HIERARCHICAL MODELS IN CREDIT RISK ASSESSMENT
3.1. Model Structure
In a Bayesian hierarchical model for credit risk assessment, the data might be structured as
follows:
• Level 1 (Loan level): Variables like, loan amount, interest rate, duration, and default status.
• Level 2 (Borrower level): Variables like, credit score, income, employment status, and
other demographic information.
• Level 3 (Institution level): Institution type, market conditions, and regulatory environment.

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The model can be expressed with the following notation:
Level 1 model (Loan Level)
𝑦𝑖𝑗 = 𝛽0𝑗 + 𝛽1𝑗𝑥𝑖𝑗 + 𝜖𝑖𝑗 (2)
Where:
• 𝑦𝑖𝑗is the is the default status of the loan i for the borrower j,
• 𝑥𝑖𝑗are the loan-level predictors
• 𝛽0𝑗 and 𝛽𝑖𝑗 are the borrower specific coefficients,and
• 𝜖𝑖𝑗 is the error term.
Level 2 model (Borrower Level)
𝛽0𝑗 = 𝛾00 + 𝛾01𝜔𝑗 + 𝑢0𝑗 (3)
𝛽1𝑗 = 𝛾10 + 𝛾11𝜔𝑗 + 𝑢1𝑗 (4)
Where:
• 𝜔𝑗are the borrower-level predictors,
• 𝛾00 , 𝛾10are the intercepts
• 𝛾01 , 𝛾11are the slopes, and
• 𝑢0𝑗, 𝑢1𝑗are the random effects.
Level 3 model (Institution Level)
𝛾00 = 𝛿000 + 𝛿001𝑧𝑘 + 𝑣00𝑘 (5)
𝛾10 = 𝛿100 + 𝛿101𝑧𝑘 + 𝑣10𝑘 (6)
Where:
• 𝑧𝑘are the institution-level predictors,
• 𝛿000 , 𝛿100are the intercepts
• 𝛿001 , 𝛿101are the slopes, and
• 𝑣00𝑘, 𝑣10𝑘are the random effects.

70
Figure1: Bayesian hierarchical model structure for credit risk assessment
3.2. Prior Distributions
Prior distributions are one of the essential ingredients in Bayesian hierarchical models. Priors can
be informative or non-informative:
• Informative: It incorporatesprior knowledge or expert opinions into the model. For
example, historical default rates can inform the prior distribution of default probabilities
[11].
• Non-informative: It is used when there is limited or no prior knowledge, allowing the data
to speak for itself [12].
The choice of priors is crucial as it propagates through to the posterior distribution and, hence,
the inferences from the model [13].
4. PRACTICAL IMPLEMENTATION
4.1. Data Preparation
Implementing a Bayesian hierarchical model requires meticulous data preparation. The following
steps outline a typical process:
1. Gather Data: Collect data at all relevant levels (loan, borrower, institution).
2. Preprocess Data: Deal with missing values, outliers, and inconsistencies.
3. Transform Data: Transform data as applicable, Normalize or standardize.
4. Variable Selection: Use domain knowledge and statistical tests to derive suitable
predictors.
4.2. Software and Tools
Several software tools and libraries facilitate the implementation of Bayesian hierarchical models
[14][15][16][17]:

71
• R: Packages such as brms, rstan, and lme4 offer robust functionalities for Bayesian
modeling.
• Python: Libraries like PyMC3, Stan, and TensorFlow Probability provide powerful tools
for Bayesian inference.
• Stan: A probabilistic programming language that integrates with R and Python, ideal for
specifying and fitting complex Bayesian models.
4.3. Model Fitting and Evaluation
Process of fitting a Bayesian hierarchical model
1. Specification of the model: Hierarchical Structure and Prior Distribution
2. Parameter Estimation: Markov Chain Monte Carlo (MCMC) methodto sample from the
posterior distribution.
3. Convergence Diagnostics Assess whether the MCMC chains have converged to a steady
state (Diagnostics like trace plots or Gelman-Rubin used)
4. Checking: Do posterior predictive checks to assess how well your model fits and where it
may diverge.
5. Compare models: Compare different models using criteria such as the Deviance
Information Criterion (DIC) or Widely Applicable Information Criterion (WAIC)[18]
5. REAL-WORLD APPLICATION
5.1. Mortgage Default Risk Preparation
A practical application of Bayesian hierarchical models in credit risk assessment is the evaluation
of mortgage default risk. This involves assessing the likelihood of a borrower defaulting on their
mortgage based on loan-level, borrower-level, and institution-level data.
5.1.1. Data Description
• Loan-level data: Loan amount, interest rate, loan-to-value ratio, payment history.
• Borrower-level data: Credit score, income, employment status, age [19].
• Institution-level data: Bank type, regulatory environment, economic indicators.
5.1.2. Model Implementation
1. Model Specification:
• Loan-level model: Default status as a function of loan amount, interest rate, and loan-
to-value ratio.
• Borrower-level model: Loan-level coefficients as functions of credit score, income, and
employment status.
• Institution-level model: Borrower-level coefficients as functions of bank type and
economic indicators.
2. Parameter Estimation:
• Use MCMC sampling to estimate the posterior distributions of the parameters[20].

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3. Model Checking and Validation:
• Perform posterior predictive checks to ensure the model accurately captures the default
risk.
• Validate the model using out-of-sample data.
5.1.3. Results and Insight
The Bayesian hierarchical model provides several advantages:
• Granular Insights: By incorporating data at multiple levels, the model captures the
nuanced factors influencing default risk.
• Uncertainty Quantification: The posterior distributions offer a measure of uncertainty
for each parameter estimate, aiding in risk management.
• Flexible Prior Incorporation: The ability to include prior knowledge enhances the
model's robustness, especially in the presence of limited data.
6. ADVANTAGES AND CHALLENGES
6.1. Advantages
1. Improved Accuracy: BHMs account for multi-level data structures, leading to more
accurate risk assessments [21].
2. Robust Uncertainty Estimates: The Bayesian framework provides comprehensive
uncertainty estimates for model parameters [22].
3. Flexibility: BHMs can incorporate various types of data and prior information, making
them adaptable to different contexts [23].
4. Enhanced Interpretability: The hierarchical structure allows for the decomposition of
effects at different levels, facilitating a better understanding of the factors driving credit
risk [24].
6.2. Challenges
1. Computational Complexity: Fitting Bayesian hierarchical models, especially with large
datasets, can be computationally intensive [25].
2. Model Specification: Defining the appropriate hierarchical structure and priors requires
domain expertise and careful consideration [26].
3. Convergence Issues: Ensuring the convergence of MCMC chains can be challenging,
necessitating the use of diagnostics and potentially more advanced sampling techniques
[27].
7. FUTURE DIRECTIONS
7.1. Integration with Machine Learning
Probabilistic Machine Learning offers a lot of potential when combined with Bayesian
hierarchical models in application for credit risk assessment. Hybrid models can leverage the best
of both worlds—using BHMs to bring in domain knowledge and uncertainty quantification, while
using machine learning models for dealing with big data and complex interactions.

73
7.2. Real-time Risk Assessment
The use of real-time data and Bayesian hierarchical models seems likely to improve the speed
and precision of credit risk assessments. The big innovation here will be in developing algorithms
and systems that can actually update a risk assessment dynamically as new data comes in.
7.3. Advanced Priors and Hierarchical Structures
Employing more intricate priors and hierarchical structures would be an important aspect to
continue refining Bayesian hierarchical models. Including non-linear relationships, interactions,
and more sophisticated prior distributions will improve the model's ability to capture the nuances
of credit risk.
8. CONCLUSION
Bayesian hierarchical models combine data across multiple levels and incorporate prior
understanding to articulate better insight into credit risk. While challenges remain, the potential
benefits in terms of accuracy, uncertainty quantification, and interpretability make BHMs a
valuable tool for financial risk management.
Given emerging technologies and data sources, further advances in the development and
integration of Bayesian hierarchical models will continue to expand their use case and improve
their utility for credit risk assessment. During these ever-evolving times, BHMs become essential
to formulating a sound credit risk assessment and enriching the strength of well-functioned
financial systems as they evolve.
REFERENCES
[1] Altman, E. I., & Saunders, A. (1998). Credit risk measurement: Developments over the last 20
years. Journal of Banking & Finance, 21(11-12), 1721-1742.
[2] McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts,
techniques, and tools. Princeton University Press.
[3] Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian
data analysis. CRC press.
[4] Ghosh, J. K., & Ramamoorthi, R. V. (2003). Bayesian nonparametrics. Springer.
[5] Bernardo, J. M., & Smith, A. F. M. (2000). Bayesian Theory. John Wiley & Sons.
[6] Kruschke, J. K. (2015). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (2nd
ed.). Academic Press.
[7] Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models.
Cambridge University Press.
[8] Goldstein, H. (2011). Multilevel Statistical Models (4th ed.). Wiley.
[9] Smith, J., & Allen, D. (2020). "Loan-level data in Bayesian hierarchical models for credit risk
assessment." Journal of Financial Risk Management.
[10] Brown, T., & Harris, M. (2018). "Borrower characteristics in credit risk modeling." Journal of
Banking and Finance.
[11] O'Hagan, A., & Forster, J. (2004). Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian
Inference (2nd ed.).
[12] Kass, R. E., & Wasserman, L. (1996). "The Selection of Prior Distributions by Formal Rules."
Journal of the American Statistical Association, 91(435), 1343-1370.
[13] Gelman, A. (2006). "Prior Distributions for Variance Parameters in Hierarchical Models." Bayesian
Analysis, 1(3), 515-533.
[14] Carpenter, B., et al. (2017). Stan: A Probabilistic Programming Language. Journal of Statistical
Software.

74
[15] Salvatier, J., Wiecki, T. V., Fonnesbeck, C. (2016). Probabilistic Programming in Python using
PyMC3. PeerJ Computer Science.
[16] Vehtari, A., Gelman, A., Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out
cross-validation and WAIC. Statistics and Computing.
[17] Bürkner, P.-C. (2017). brms: An R Package for Bayesian Multilevel Models using Stan. Journal of
Statistical Software.
[18] Gelman, A., Vehtari, A., Simpson, D., Margossian, C. C., Carpenter, B., Yao, Y., Kennedy, L.
(2020). Bayesian Workflow. arXiv preprint arXiv:2011.01808
[19] Thomas, L. C., Crook, J. N., & Edelman, D. B. (2017). Credit Scoring and Its Applications (2nd
ed.). SIAM.
[20] Gilks, W. R., Richardson, S., & Spiegelhalter, D. (1995). Markov Chain Monte Carlo in Practice.
Chapman and Hall/CRC.
[21] Lee, P. M. (2012). Bayesian Statistics: An Introduction (4th ed.). Wiley.
[22] Banerjee, S., Carlin, B. P., & Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial
Data (2nd ed.). CRC Press.
[23] Congdon, P. (2005). Bayesian Models for Categorical Data. Wiley.
[24] Greenberg, E. (2012). Introduction to Bayesian Econometrics (2nd ed.). Cambridge University
Press.
[25] Brooks, S., Gelman, A., Jones, G., & Meng, X. L. (Eds.). (2011). Handbook of Markov Chain
Monte Carlo. CRC Press.
[26] Geman, S., &Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian
restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, (6), 721-
741.
[27] Betancourt, M. (2017). A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint
arXiv:1701.02434.
AUTHOR
Sanjay Moolchandani has over 20 years of experience in Banking, Risk, and
Financial technology. He is a seasoned expert in developing and managing large-
scale IT projects and sophisticated risk management solutions. In addition to his
strategic vision and analytical capabilities, Sanjay is widely recognized for delivering
innovative solutions for Banking and Risk Technology using next-generation
technology.
His extensive expertise spans Credit & Market Risk, Investment Banking processes,
Forecasting and Pricing models, and Risk Governance & Compliance. He has successfully led numerous
high-impact projects across global financial institutions in Japan, EU, UK and US. Sanjay's comprehensive
understanding of risk calculations and methodologies, coupled with his deep knowledge of industry
regulations such as Basel II, Basel III, Basel IV, FRTB, and CCAR has enabled him to develop and
implement innovative technology solutions for top-tier investment banks.
Sanjay's academic background includes an MSc in Mathematical Trading and Finance from Bayes
Business School, London (UK), a Post Graduate Diploma in Finance from Pune (India), and a Bachelor of
Engineering from Bhila Institute of Technology, India. He holds the Financial Risk Manager® (FRM)
designation from the Global Association of Risk Professionals (GARP).

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