# New Definition of Default—Recalibration of Credit Risk Models Using Bayesian Approach

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## Abstract

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## 1. Introduction

- One-step approach—the introduction of the new definition of default (DoD) and recalibration of all relevant models in one step;
- Two-step approach—first the introduction of the new DoD and then recalibration of relevant models.

## 2. Implications of the New Definition of Default—Literature Review

- Data sources for modeling acquisition;
- Simulation of data according to the new definition of default;
- Back-test of all A-IRB models: PD, LGD, EAD;
- Recalibration of all models that showed material change during back-test;
- Recalibration of all IFRS 9 models as a result of A-IRB models recalibration and redevelopment;
- New Margin of Conservatism (MoC) calculation for existing models and the new DoD;
- Assessment of Risk Weighted Assets (RWA) impact of this change;
- Additional validation of rating systems.

## 3. Proposed Recalibration and Re-Development Methods for Credit Risk Parameters

- Simulation of data concerning the new definition of default for different portfolios.
- Construction of Analytical Based Tables (ABTs) for model’s re-calibration process.
- Construction of codes for recalibration process automatization—model development.
- Construction of codes for validation process automatization—model validation.
- Re-calibration of LGD/CR, EAD/CCF parameters.
- ○
- Scaling factors.
- ○
- Regression models adjusting the old risk parameter to the new one.

- Re-development of models in case of negative back-test after re-calibration of models.
- ○
- Models including a-priori distributions (Bayesian approach).
- ○
- Parametric models.
- ○
- Non-parametric models.

- Validation of the model’s re-calibration/re-development process.

#### 3.1. Probability of Default PD—Model Recalibration

#### 3.2. Loss Given Default LGD—Model Recalibration

_{cure},

_{cure}—the economic loss associated with cured cases expressed as a percentage of EAD.

- For the cure rate parameter (CR):
- ○
- scaling factor (including no-loss),
- ○
- logistic regression model,
- ○
- full recalibration using the Bayesian approach,
- ○
- recalibration using survival methods (with censored observations).

- For the secured and unsecured recovery rate (SRR, URR):
- ○
- calibration using the scaling factor,
- ○
- linear or non-linear regression model,
- ○
- full calibration using regression models including simultaneous calibration of both parameters.

#### 3.3. Exposure at Default EAD—Model Recalibration

Scenario | Parameters Description |

1 | Exposure below limit |

2 | Exposure equal limit or above (>0) |

3 | Exposure positive, limit 0 |

4 | Exposure and limit equal 0 |

- credit conversion factor (CCF) in the first case;
- the ratio of limit in the observation date for a facility to all limits for all facilities in a sample in the second case;
- the ratio of exposure at observation date for a facility to all exposures for all facilities in a sample in the third case;
- constant value estimated on the sample in the fourth case.

#### 3.4. The Bayesian Approach in PD Recalibration

- PD
_{sim}—PD estimated on simulated data; - PD
_{emp}—PD estimated on empirical data, at start this population is much smaller than the simulated one; - prior information comes from simulated data;
- likelihood comes from empirical data;
- posterior distribution is a combination of prior distribution and likelihood;
- final PD is based on posterior distribution characteristics;

_{sim}does not pass the alignment on empirical data, which means that there is no match between simulated PD

_{sim}and empirical PD

_{emp}.

_{sim}on simulated data with additional information from empirical data:

- $\pi (P{D}_{sim}|P{D}_{emp})$—is the posterior distribution under condition of PD
_{emp}, - $\pi \left(P{D}_{sim}\right)$—is the prior distribution,
- $f(P{D}_{emp}|P{D}_{sim})$—likelihood—conditional distribution of $P{D}_{emp}$ parameter under the condition of $P{D}_{sim}$.

- The Bayesian approach fits estimate parameters very well with the use of knowledge from different data sources, both internal and external. However, before applying the Bayesian technique the following problems should be solved. The choice of prior distribution for PD simulated. The prior should be based on portfolio characteristics and the uncertainty in data;
- The choice of MCMC technique, such as Gibbs sampling, Hastings–Metropolis or other when explicit posterior cannot be found;
- The possibility of including other prior information that is not derived from the data such as future macroeconomic conditions in some selected industries;
- The possibility of calculating confidence intervals for PD and using a more conservative estimator.

## 4. PD Recalibration—Application on Real Data

- Building the new PD model on the joint population for simulated and empirical defaults, i.e., a mixed population. The model is built with use of logistic regression;
- Calculation of Long Run Average (LRA) on the simulated data. LRA is the average of default rates calculated within the given period of time;
- Adjusting LRA through the Bayesian methodology, which combines both the simulated and empirical data. The role of empirical data is to adjust the LRA calculated on the simulated data;
- Final recalibration of the PD parameter estimated in the 1st step at the facility level according to the posterior mean calculated at the step 3. The final step is also performed with Bayesian approach.

- -
- Absolute breach in the past;
- -
- Relative breach in the past;
- -
- Maximum DpD in the past;
- -
- Amount of maximum arrears in the past;
- -
- Total obligations;
- -
- The age of the customer;
- -
- Account balance.

_{sim}is a random variable normally distributed around E(LRA

_{sim}).

_{sim}) and std(LRA

_{sim}) are the expected value of LRA and standard deviation of LRA, respectively, calculated on the simulated data.

_{sim}is directly derived from the historical data. The distribution of the variable PD

_{sim}is considered as prior distribution. This is the case where prior distribution is more objective as it is based on historical data. We used many different prior distributions which are allowed by SAS MCMC procedure and the final posterior results were very similar which further proves robustness of the posterior estimators.

_{emp}parameter on the empirical data is the binomial distribution with the number of trials that equals the number of observations n

_{emp}in the empirical data and the probability of success equals the PD

_{sim}calculated on the simulated data. Hence, the likelihood can be viewed as a conditional distribution of the PD parameter on empirical data given the PD

_{sim}calculated on the simulated data is defined as follows:

_{emp}is a random variable binomially distributed (Joseph 2021) with parameters n

_{emp}and PD

_{sim}.

_{cal}) depending on the simulated PD (PD

_{sim}) and two additional parameters a and b as follows:

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**LGD recalibration—differences between the old and new default period. Source: own elaboration.

**Figure 6.**PD distribution: (

**a**) Raw PD; (

**b**) PD calibrated with and without the Bayesian approach. Source: own elaboration using SAS Enterprise Guide.

Model | Re-calibration of IRB models (end of year 2021) | Re-development of models using new data (years 2022/2023) |

PD | Default data—simulated Additional data—empirical | Simulated data (option of immediate redevelopment) Empirical data (redevelopment possible after 2 years of data collection) |

LGD | Default data—simulated Collection data—real empirical data after default date | Empirical/mixed data, No simulated data |

Year | Count | Default Rate | Arrears Max [Thousand EUR] | Arrears Average [Thousand EUR] |
---|---|---|---|---|

2008 | 6939 | 0.72% | 10.59 | 6.36 |

2009 | 8973 | 0.57% | 15.93 | 3.32 |

2010 | 7502 | 0.72% | 14.29 | 2.64 |

2011 | 6889 | 1.32% | 19.29 | 4.82 |

2012 | 7860 | 2.01% | 19.21 | 4.19 |

2013 | 8993 | 2.62% | 26.60 | 7.72 |

2014 | 9774 | 2.34% | 32.68 | 7.00 |

2015 | 10,725 | 2.45% | 28.64 | 8.52 |

2016 | 12,545 | 2.18% | 22.42 | 6.40 |

2017 | 16,832 | 1.73% | 36.84 | 20.93 |

2018 | 3290 | 1.52% | 19.81 | 6.21 |

Simulated Data (Prior) | Development Data | Empirical Data | Posterior |
---|---|---|---|

0.017499 | 0.017424 | 0.015198 | 0.0155 |

Parameter | Mean | STD | 95% HPD Interval | |
---|---|---|---|---|

a | 0.88 | 0.0896 | 0.7229 | 1.0791 |

b | −5.4135 | 0.2399 | −5.8676 | −4.9243 |

sigma_a | 0.4013 | 0.298 | 0.00168 | 0.9242 |

sigma_b | 0.9423 | 0.0535 | 0.8326 | 1 |

Parameter | Mean | STD | 95% HPD Interval | |
---|---|---|---|---|

a | 1.1087 | 0.0192 | 1.0736 | 1.1498 |

b | −5.9081 | 0.0512 | −6.0129 | −5.8132 |

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**MDPI and ACS Style**

Ptak-Chmielewska, A.; Kopciuszewski, P.
New Definition of Default—Recalibration of Credit Risk Models Using Bayesian Approach. *Risks* **2022**, *10*, 16.
https://doi.org/10.3390/risks10010016

**AMA Style**

Ptak-Chmielewska A, Kopciuszewski P.
New Definition of Default—Recalibration of Credit Risk Models Using Bayesian Approach. *Risks*. 2022; 10(1):16.
https://doi.org/10.3390/risks10010016

**Chicago/Turabian Style**

Ptak-Chmielewska, Aneta, and Paweł Kopciuszewski.
2022. "New Definition of Default—Recalibration of Credit Risk Models Using Bayesian Approach" *Risks* 10, no. 1: 16.
https://doi.org/10.3390/risks10010016