A Functional Representation of Credit Behavior for Probability of Default Modeling

Paper: arXiv 2610.07888

Authors: Jonas Brunholm, Bjarne Højgaard, Thomas D. Nielsen, Orimar Sauri

Abstract

This paper proposes a framework for modeling probability of default via functional data analysis. By representing a series of credit variables as functions, we investigate whether intra-monthly information improves default predictability in linear models. We further show how a range of widely used variables, among them available funds, utilization rate, and overdraft, can all be derived from three quantities observed over time. Namely, 1) the type of each account, 2) the balance of the account, and 3) the size of the credit limit on the account. Retaining these quantities as continuous-time processes, rather than reducing them to monthly aggregated values, yields a continuous faithful representation of the borrower. When analyzing these three processes, we discovered that recurring events associated with the ordinal position among banking days created strong cyclical patterns. We therefore develop a relative time framework that aligns the recurring events across borrowers, ensuring that borrowers possess the same cyclical pattern, regardless of real time. We assess the framework using functional logistic regression. This approach accommodates the continuous representation while remaining closely related to a logistic regression model commonly used in credit risk practice, due to strict regulatory constraints. We show that, when equipped with an effective functional representation of transactional trajectories, the proposed model attains predictive performance on par with XGBoost while consistently outperforming logistic regression. Importantly, the model balances predictive performance of a machine learning model with the interpretability of linear default models, potentially enabling financial institutions to use the model, even under strict regulation.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a novel approach to PD modeling using functional data analysis, demonstrating strong empirical results. The methodology is well-explained and addresses a practical problem in credit risk with an interpretable solution. The combination of advanced mathematical techniques with robust empirical validation places it in the ‘Holy Grail’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Improve PD Modeling with Intra-Monthly Info & Functional Data] --> B(Methodology: Functional Data Analysis)
    B --> C{Inputs: Credit Account Type, Balance, Credit Limit (Continuous Time)}
    C --> D[Computational Process: Functional Logistic Regression & Relative Time Framework]
    D --> E{Outcomes: Improved Predictability, Cyclical Pattern Alignment}
    E --> F[Key Findings: Performance on par with XGBoost, Outperforms Logistic Regression, Interpretable]

Cited References (36)

1. Guidelines on the application of the definition of default under Article 178 of Regulation (EU) No 575/2013 (2016) — link
2. Final report on guidelines on PD estimation, LGD estimation and the treatment of defaulted exposures (2017) — link
3. Regulation (EU) No 575/2013 on prudential requirements for credit institutions and investment firms ( Capital Requirements Regulation ) (2013) — link
4. Prediction of default probability by using statistical models for rare events (2019) — link
5. Statistical and machine learning models in credit scoring: A systematic literature survey (2020) — link
6. Benchmarking state-of-the-art classification algorithms for credit scoring: A n update of research (2015) — link
7. Consumer credit-risk models via machine-learning algorithms (2010) — link
8. A boosted decision tree approach using B ayesian hyper-parameter optimization for credit scoring (2017) — link

0 of 36 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.