Probability of Default Over the Contract Life under Resolution Cmn 4.966/2021

Ariadne Bitencourt Ferreira Domene Schneider; Aline Araujo Perini

DOI: 10.22167/2675-6528-202602685

Article derived from a Course Conclusion Work (TCC), with content based on the student’s original work and adapted to the editorial format of the E&S Magazine with the support of the ResumeAI tool, an artificial intelligence solution developed by Instituto Pecege for textual synthesis and organization.

Summary

The adoption of Resolution CMN No. 4.966/2021 increased the importance of estimating the probability of default over different time horizons, especially for the measurement of expected credit loss. The study evaluated whether discrete-time survival analysis constituted a methodologically adequate approach to estimate the probability of default over the lifetime of a personal credit portfolio. For this purpose, an anonymized internal database, structured in a monthly panel by contract, with observations from January 2021 to November 2025, was used. The model was estimated by logistic regression with time effects, incorporating registration, contractual, behavioral, and macroeconomic variables. The comparison between specifications prioritized probabilistic error metrics, discrimination, and adherence to the temporal structure of risk, with evaluation on training, validation, test, and out-of-time test sets. The results indicated that the approach produced informative estimates at different horizons and maintained adequate performance outside the development period. Variables related to the history of delays showed the greatest contribution to the model, while the discretization of age into broad bands did not imply a relevant loss of fit. It was concluded that discrete-time survival analysis constituted a viable methodological alternative for the analyzed product, although its use in a complete regulatory environment still demands further developments, especially regarding the prospective treatment of covariates.

Keywords: Personal credit; Term structure; IFRS 9; Expected credit loss; Logistic Regression.

1. Introduction

The 2008 global financial crisis highlighted relevant limitations of the incurred loss accounting model, which recognized credit risk only after its materialization. In response, IFRS 9 replaced this logic with a forward-looking expected loss approach, requiring consideration of historical information, current conditions, and reasonable and supportable estimates about the future (Dong; Oberson, 2022; IASB, 2014). This standard became mandatory internationally from 2018 and, in Brazil, was incorporated by CPC 48 and, for financial institutions authorized to operate by the Central Bank of Brazil, by Resolution CMN No. 4.966/2021 (Picker et al., 2016; CPC, 2016; CMN, 2021).

In this new regulatory framework, the estimation of expected loss directly depends on the measurement of the probability of default over different time horizons. For Stage 1 assets, the provision reflects default events in the subsequent twelve months. For Stage 2 and 3 assets, the measurement considers the remaining term of the operation, making the estimation of the probability of default over the lifetime a central component of the standard’s implementation (IASB, 2014; CMN, 2021). Furthermore, lifetime probability of default models influence the accounting classification of contracts, with the deterioration of the probability of default being a quantitative signal to identify a significant increase in credit risk (EBA, 2023; Botha et al., 2025).

The requirement to incorporate forward-looking information makes the development of these models even more challenging. The measurement of expected loss under IFRS 9 cannot be based solely on observed history, but must reflect reasonable expectations about the future evolution of the economic environment. This implies considering macroeconomic variables or scenarios in the estimation to anticipate risk deteriorations. Reports monitoring the implementation of the standard indicate that the incorporation of forward-looking information and multiple scenarios is a recurring practice (EBA, 2023). Despite its relevance, estimating the temporal structure of Nonpayment requires simultaneously dealing with censoring, different contract maturity times, covariates that vary over time, and the need for consistency between monthly conditional probabilities, 12-month Nonpayment probability, and cumulative probability to maturity (Saavedra et al., 2024).

The estimation of the probability of default over the lifetime can be approached by several methodological strategies. Among the relevant alternatives, state transition models, such as Markov chains, and continuous-time survival models, such as the Cox model (Saavedra et al., 2024; Botha; Verster, 2026), stand out. Transition models, although flexible for multiple outcomes, require greater formulation complexity and can hinder governance in retail portfolios with large data volumes. Continuous-time models, despite their statistical robustness, are less natural for data observed at regular monthly intervals, such as credit data, and demand additional treatment for multiple occurrences in the same period (Botha; Verster, 2026).

Given these considerations, discrete-time survival analysis, estimated by logistic regression, emerges as the most suitable alternative for this study. This approach preserves the advantages of survival models, such as the treatment of censoring and the explicit modeling of the temporal evolution of risk, by directly adapting to the monthly structure of the analytical base. It allows for the incorporation of fixed and time-varying covariates, including macroeconomic factors, which are essential for prospective estimates of expected loss (Saavedra et al., 2024; Botha; Verster, 2026). An additional advantage is the ability to directly model the conditional risk of default in the period, which enables the coherent derivation of the probability of default within 12 months and the cumulative probability over the remaining life of the operation, crucial for Resolution CMN No. 4.966/2021.

Despite the relevance of these models, their development remains challenging, requiring approaches that deal with the complexity of the temporal structure of credit risk and the incorporation of prospective information. Thus, this work aimed to evaluate whether discrete-time survival analysis constituted a methodologically adequate approach to estimate the probability of nonpayment over the lifetime of a personal credit portfolio.

2. Material and Methods

The study was characterized as applied research, quantitative, with descriptive and analytical objectives, using secondary data. The methodological design was based on the analysis of historical corporate databases, without direct intervention. The central approach consisted of discrete-time survival analysis, estimated by logistic regression, to model the probability of nonpayment over the lifetime in a personal credit portfolio.

The data originated from a financial institution, with formal authorization for academic use. The databases comprised historical credit operations information, used exclusively for modeling and statistical analysis. All data were rigorously anonymized, without information that allowed for direct or indirect identification of clients, and the results were disclosed in an aggregated manner.

The research followed the ethical principles of CNS Resolution nº 510/2016, which regulates research in Human and Social Sciences. Because it used secondary, anonymized, and authorized access data, the study did not constitute research with direct intervention in human beings, implying no risks. The analyses were performed in a secure computational environment, ensuring data integrity and confidentiality.

The study population included personal credit contracts for individuals, without payroll deduction, with installments due between January 2021 and December 2025. Contracts already in initial default and those renegotiated, which represented 3.17% of eligible contracts, were excluded. The unit of analysis was the contract-month, structured in a monthly panel.

The analytical base comprised monthly observations between January 2021 and November 2025. Each panel row recorded the contract age, time at risk, status on the reference date, and the occurrence of the event in the subsequent month. The month of December 2025 was not used as a reference, as the response variable requires observation of the subsequent outcome.

The estimation of the probability of default over the lifetime was performed by discrete-time survival analysis, using logistic regression with time effects. This approach handled censoring and modeled the temporal evolution of risk, adjusting to the monthly structure of the data (Saavedra et al., 2024; Botha; Verster, 2026). The model incorporated fixed and time-varying covariates, including macroeconomic factors.

The explanatory variables were grouped into customer characteristics, contract characteristics, external credit information (bureau and SCR), and macroeconomic variables, as detailed in Appendix I of the original work. From the monthly conditional default probabilities, the 12-month default probability and the cumulative probability over the remaining life of the operation were derived, which is essential for CMN Resolution No. 4.966/2021.

The event of interest was defined as default, characterized by a delay exceeding ninety days. The monthly response variable assumed a value of 1 for default in the subsequent month and 0 in other eligible cases. Contracts were monitored until default, settlement, or censoring, without reintroduction into the panel after a default event, in alignment with regulatory treatment.

For the sample partitioning, a hybrid strategy was adopted. Contracts with the first observation between January 2021 and December 2023 comprised the development sample (70% training, 15% validation, 15% test), separated at the identifier level. Contracts with the first observation between January 2024 and November 2025 were reserved for the out-of-time test set.

The comparison between the model specifications prioritized metrics that evaluate the probabilistic quality of predictions, such as the Brier score, which measures the mean squared errors between the predicted probability and the observed outcome at fixed horizons. Complementarily, the Brier skill score was used for relative performance and the area under the ROC curve (AUC) for discrimination.

To evaluate the temporal evolution of default, observed rates and average conditional probabilities predicted by age group were compared. The weighted monthly default Brier score and the weighted mean absolute error, which consider the distribution of observations by contract age, were calculated.

3. Results and Discussion

The results obtained from the analytical base constructed for the study allowed a comprehensive evaluation of the adequacy of discrete-time survival analysis for estimating the probability of default in a personal credit portfolio. Initially, the characterization of the database revealed a substantial volume of information, essential for the robustness of the modeling. Subsequently, the exploratory analysis of the temporal behavior of default and the comparison between different model specifications provided crucial insights into the contribution of explanatory variables and the predictive approach’s predictive capacity.

The database used comprised monthly observations of personal credit contracts between January 2021 and November 2025, totaling 59 months of monitoring. During this period, 64,844 unique contracts were analyzed, generating 782,541 monthly observations. The average default rate observed in the portfolio was 0.66%, with significant variations, fluctuating between a minimum of 0.22% and a maximum of 1.1% over time. This volume of data is considered sufficient to support the estimation of survival models, both in terms of individual contracts and observations over time.

The evolution of the monthly default rate in the analyzed portfolio showed an upward trend between 2021 and 2025, albeit with notable volatility. In 2021, default rates remained at relatively low levels, but from 2022 onwards, an increase and greater fluctuation were observed, with more pronounced peaks in 2023, 2024, and especially in 2025. For example, the default rate reached 0.45% in July 2021, 0.64% in May 2022, 0.84% in May 2023, and 0.99% in May 2025. This dynamic behavior of nonpayment, which reflects both the frequency of default and the composition of the portfolio at risk, highlighted the importance of evaluating the model on test sets outside the development period to verify its generalization capability in different economic and operational environments.

The representation of contract age in the model was carried out through age bands, which proved to be an effective approach. The distribution of observations and the default rates associated with each age grouping revealed distinct patterns. Contracts with ages between three and four months showed a default rate of 0.72%, while those between five and six months registered 0.71%. The bands from nine to twelve months and above 39 months exhibited rates of 0.75%, the highest, suggesting that the risk of nonpayment may be higher in the initial and very advanced stages of the contract’s life. Intermediate bands, such as 13 to 26 months and 27 to 38 months, presented rates of 0.62% and 0.53%, respectively, indicating a lower probability of default in this period. This discretization of age into broad bands did not imply a relevant loss of fit compared to more granular alternatives, favoring modeling parsimony.

The choice of the final model was guided by the comparison between different specifications, prioritizing metrics that evaluate the probabilistic quality of predictions, such as the Weighted Brier Score (WBS), and discrimination ability, such as the Area Under the ROC Curve (AUC), used as an auxiliary criterion. The objective was not only the best fit within the development sample but also the ability to represent the temporal dynamics of default, maintain satisfactory performance outside the training period, and avoid excessive complexity. The final decision was based on the aggregated metrics by contract age in the validation, test, and out-of-time (OOT) test sets.

The comparison between the reference model and the reduced model, which used 21 variables in contrast to the 32 in the reference model, demonstrated very close performances. The Weighted Brier Score (WBS) for the reference model was 0.000930 in validation, 0.001226 in testing, and 0.001789 in out-of-time testing. For the reduced model, the values were 0.000935, 0.001235, and 0.001778, respectively. This proximity in results, also reflected in the Weighted Brier Skill (WBSK), which was 0.819964 for the reference model and 0.819022 for the reduced model in validation, indicated the possibility of simplifying the model specification without a material loss of fit, promoting greater parsimony.

The removal of the SCR (Central Bank Credit Information System) and credit card information block resulted in a limited impact on performance metrics. The specification without this information, with 24 variables, presented a WBS of 0.000932 in validation, 0.001225 in testing, and 0.001788 in out-of-time testing, values very close to those of the reference model. The WBSK for this specification was 0.819570 in validation. This finding suggests that, for the analyzed portfolio, the group of variables related to SCR and credit card offered a marginal contribution to the model’s overall predictive performance, which may be relevant for simplification decisions in contexts where obtaining or processing this information is more complex.

In contrast, the specification that excluded the delay history from the contract showed a sharp deterioration in performance across all evaluation sets. The WBS for this specification was 0.004960 in validation, 0.005557 in testing, and 0.008486 in out-of-time testing, values significantly higher than those of the other specifications. The Weighted Brier Skill (WBSK) was drastically lower, reaching 0.040115 in validation. This substantial loss of predictive capability compared to the naive benchmark showed that the variables related to delay history concentrate a crucial portion of the model’s explanatory power, reinforcing their importance for the quality of fit and the accuracy of nonpayment predictions (Botha and Verster, 2026).

The results also highlighted an important methodological point in the selection and treatment of covariates for lifetime default probability models. Although lag and other behavioral variables are predictively relevant, the model performance evaluation was based on the observed values of these covariates over time, rather than on an explicit prospective projection exercise. This distinction is fundamental, as, in the practical application of lifetime PD models, not all covariates can be treated in the same way, especially in a regulatory environment that requires prospective estimates of expected loss (Saavedra et al., 2024).

Some characteristics, such as contract age, are naturally projectable or can be maintained without great difficulty. Macroeconomic variables can be incorporated through external scenarios, a common practice required in many financial institutions. However, behavioral variables such as the number of outstanding installments, outstanding balance, days in arrears, or recent history of arrears depend on the borrower’s future behavior. Freezing these covariates in their current state implies an assumption that the contract’s behavior will remain unchanged, while projecting them requires additional hypotheses about the future evolution of payments, which introduces an extra layer of uncertainty and complexity.

Thus, the performance gain observed with realized values of these covariates does not, by itself, guarantee equivalent performance in a prospective application environment. For a proper evaluation in a complete regulatory context, it would be necessary to measure model error under scenarios with frozen or explicitly projected covariates, which was not the focus of this study. This limitation points to the need for caution in interpreting the results for regulatory implementation purposes, where the projection capability of covariates is as important as their historical predictive power.

The work, therefore, demonstrated that discrete-time survival analysis constitutes a methodologically viable approach to estimate the probability of lifetime default in personal credit portfolios, producing informative estimates and maintaining adequate performance outside the development period. The ability to build a leaner specification, with limited loss of fit, and the critical relevance of the delay history variables were central findings. However, application in a complete regulatory environment still demands further developments, especially regarding the prospective treatment of behavioral covariates, which represent a significant challenge for the calibration of the probability of default over longer horizons.

4. Conclusion

This work aimed to evaluate whether discrete-time survival analysis constituted a methodologically adequate approach to estimate the lifetime probability of default in a personal credit portfolio, in the context of CMN Resolution No. 4.966/2021. It was found that the approach produced informative estimates across different horizons and maintained adequate performance outside the model development period. The comparison between specifications indicated that discrete-time survival analysis represented a viable methodological alternative for the analyzed product, allowing the construction of a leaner specification with limited loss of fit. It was observed that representing contract age in broad bands did not imply a relevant loss of performance, favoring modeling parsimony. Additionally, the results showed that variables related to the contract’s own delinquency history concentrated a substantial portion of the model’s predictive capacity, reinforcing its usefulness in estimating the temporal dynamics of nonpayment.

However, the study highlighted an important limitation for prospective use in a full regulatory environment. The performance gain observed with realized values of behavioral covariates, such as delay and debtor balance, does not guarantee equivalent performance in a prospective application environment, as the projection of these variables depends on the borrower’s future behavior. In this regard, the work did not fully contemplate all the developments required for complete regulatory application, such as the evaluation of model error under frozen or explicitly projected covariates. As future developments, we suggest the formal evaluation of the model under different prospective treatment strategies for covariates, the incorporation of explicit cure criteria for eventual re-entry of contracts after default, and the investigation of the impact of using macroeconomic scenarios and distinct projection rules for behavioral variables on the calibration of the probability of default over longer horizons.

Bibliographic References

Botha, A.; Oberholzer, E.; Larney, J.; de Jongh, R. Defining and comparing SICR-events for classifying impaired loans under IFRS 9. Annals of Operations Research, 2025, p. 1–38. DOI: 10.1007/s10479-025-06546-3.

Botha, A.; Verster, T. Approaches for modelling the term-structure of default risk under IFRS 9: a tutorial using discrete-time survival analysis. International Journal of Data Science and Analytics, 2026. DOI: 10.1007/s41060-026-01032-w

Comitê De Pronunciamentos Contábeis [CPC]. Pronunciamento Técnico CPC 48 – Instrumentos Financeiros. São Paulo: CPC, 2016.

Conselho Monetário Nacional [CMN]. Resolução nº 4.966, de 25 de novembro de 2021. Brasília: CMN, 2021.

Dong, M.; Oberson, R. Moving toward the expected credit loss model under IFRS 9: capital transitional arrangement and bank systematic risk. Accounting and Business Research, v. 52, n. 6, p. 641–679, 2022.

European Banking Authority [EBA]. Report on the implementation of IFRS 9 by EU institutions. Luxembourg: Publications Office of the European Union, 2023.

International Accounting Standards Board [IASB]. IFRS 9 – Financial Instruments. London: IFRS Foundation, 2014.

Picker, R.; Clark, K.; Dunn, J.; Kolitz, D. L.; Livne, G.; Loftus, J.; Tas, L. Van der. Applying IFRS Standards. Hoboken: John Wiley & Sons, 2016.

SAAVEDRA, C. A. P. B.; et al. Probability of default for lifetime credit loss for IFRS 9 using competing risks survival analysis. Expert Systems with Applications, 2024.

Article originating from the Final Course Work of the Specialization in Finance and Controllership of the MBA USP/Esalq

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