BCR Approach for Regulatory Reporting (PD in Low-Default Portfolios)

Attached: Excel workbook with full BCR implementation (macro-driven, quarterly defaults, confidence intervals, and model diagnostics)
https://docs.google.com/spreadsheets/d/1ltVYX4vhGeTllOQkEWKrED2DV33eb55M/edit?usp=sharing&ouid=115594792889982302405&rtpof=true&sd=true

Excel Implementation Details (see link above)

• Historical GDP YoY data as a macro factor

• Z-score standardization of macro series

• Conditional PD computed using Vasicek model

• Expected defaults calculated for each period

• Goal Seek used to backsolve for portfolio PD

• Bayesian posterior estimate and 95% confidence bound from Beta distribution

  
  
  

 BCR Approach for Regulatory Reporting (PD in Low-Default Portfolios)

To calculate PD for low-default portfolios like sovereigns, large corporates, or prime mortgages, the Benjamin, Cathcart, and Ryan (BCR) methodology (2006) is a robust statistical approach used in both regulatory modeling and internal risk management:

It leverages:

- Vasicek (asymptotic single risk factor) model

-Observed macroeconomic conditions (e.g., GDP YoY growth)

-Asset correlation

-Observed default count over a given period

Excel Implementation Steps (Included in Attached File)

Inputs:

• #Obligors (e.g., 54,000)

• Observed defaults (e.g., 150)

• Correlation (e.g., 0.20)

• Historical GDP YoY (%) path as macro risk driver

VIF (Variation Inflation Factor) vs. GVIF (Generalized VIF):

 VIF (Variation Inflation Factor) vs. GVIF (Generalized VIF):

Handling Multicollinearity with Factor Variables
When building regression models, checking multicollinearity is crucial.

For continuous predictors, VIF helps identify multicollinearity. But categorical variables with multiple levels (factors) need special treatment

A factor with k levels is represented by d=k−1 dummy variables. These dummies are inherently correlated because they encode the same categorical feature.
In this case we use the Generalized Variance Inflation Factor (GVIF) — which measures multicollinearity jointly for all dummy variables representing a factor.

When calculating GVIF for a factor variable, we regress all its dummy variables simultaneously on the other predictors in the model
GVIFj​=(1/ (1−Rj^2​​)^d)

Adjusted GVIF=GVIF^(1/(2⋅d))

d = number of dummy variables (degrees of freedom for the factor).
This adjustment scales GVIF to be comparable to standard VIF values.


Key reasons to consider GVIF:
Treating the factor’s dummies jointly prevents misleading interpretations of multicollinearity.
It reflects the true inflation of variance caused by correlation between the factor and other predictors.
Helps in making informed decisions about feature selection and model stability.

Incorporating IPCC Climate Projections into Probability of Default:

 For a detailed description od theory and excel worbook example, refer to the attached link

https://drive.google.com/drive/folders/1vAcH8ge2KEFBPxxJt06cPgIr5XVKhyOr?usp=sharing

Huber M-estimation (CCF for EAD):

 Huber M-estimation is a robust regression technique used to address the influence of outliers on model parameters. It is used to calculate CCF (Credit Conversion Factor) models for EAD (Exposure at Default).

Huber M-estimation ensures robust parameter estimation by minimizing the impact of extreme observations (e.g., outliers in utilization rates, credit line drawdowns, or other key drivers).

Huber M-estimation uses a loss function (1) that transitions from squared error to absolute error depending on a threshold 𝛿:
δ based on the expected distribution of residuals.

e.g. δ=m * Stdev of errors
- If ∣ri∣≤m * Stdev, weight wi=1

- If ∣ri∣>m * Stdev the weight wi= m * Stdev / ∣ri∣


Steps:
Y (CCF) =β0​+ β1​X+ ϵ,
fit the Ordinary Least Squares (OLS) regression:
β^​=(X^TX)^−1 * X^TY
residuals ri=yi−y^i

Define the Huber loss function (1) to calculate weights.

Using weights, modify the regression:
β^​=(X^T*W*X)^−1X^T*W*Y

Change Point Detection Time Series

 

Change Point Detection Methods

Kernel Change Point Detection:

Kernel change point detection method detects changes in the distribution of the data, not just changes in the mean or variance.

Kernel Method is utilized to map the data into a high-dimensional feature space, where changes are more easily detectable. This approach uses the Maximum Mean Discrepancy (MMD) to measure the difference between the distributions of segments of the time series.

Steps:

1-      Data and Kernel Function: Consider a univariate time series {x1,x2,…,xn} We start by choosing a kernel function k(x,y) to measure similarity between points.

2-      Construction of Kernel Matrix: kernel matrix K is constructed, where each element K_ij=k(xi,xj)

For the linear kernel, this is: Kij=xi⋅xj​ (X^TX)

 

3-      Maximum Mean Discrepancy (MMD):

 

MMD measures how different two groups of data are by comparing the average of all pairwise similarities within each group and between the groups or compares two distributions to see if they are different.

MMD is used to measure the difference between the distributions before and after a candidate change point t.

For each candidate change point t

In the above equation:

-          The first term measures the similarity within the first segment.

-          The second term measures the similarity within the second segment.

-          The third term measures the similarity between the two segments.

 

4-      To detect the change point, we compute the MMD values are computed for all possible change points t and choose the one that maximizes the MMD value:

Excel Example :https://docs.google.com/spreadsheets/d/1IdC-ss1VjaL2QVQdABNwuIPfRphDtlZi/edit?usp=sharing&ouid=115594792889982302405&rtpof=true&sd=true

Climate Risk (DICE)

 

https://docs.google.com/document/d/1OGZzZ6tsBvYRXfyeB95PLPTGG-6dFSg8/edit?usp=sharing&ouid=115594792889982302405&rtpof=true&sd=true

Modeling Low Default Portfolio

The BCR Approach (Modeling Low Default Portfolio):

Benjamin, Cathcart and Ryan proposed adjustments to the Pluto & Tasche or called Confidence Based Approach that is widely applied by banks.

Pluto& Tasche propose a methodology for generating grade level PDs by the most prudent estimation, the BCR approach concentrates on estimating a portfolio-wide PD that then apply to grade level estimates and result in capital requirements.

- Independent case: The conservative portfolio estimate as in the BCR setting is therefore given by the solution of (1).

- Dependent Case: Assumed that there is a single normally distributed risk-factor Y to which all assets are correlated and that each asset has the correlation √ρ with the risk factor Y.

For a given value of the common factory=Y the conditional probability of default given a realization of the systematic factor Y is given by (2). The probability of default is equivalent to finding p such that the above is true.

- Multi-Period: Multi-period case:

The, the conditional probability of default given a realization of the systematic factor for t years as in the Vasicek model (3)

Estimation Method:

Steps of Execution:

1-     Draw N samples from N(λ,Σ) where λ is a zero vector with the same length as the time period and Σ is the correlation matrix as in the Vasicek model.

2-     Equation (4)

3-     Find p such that f(p) is close to zero using the following iteration:

-         Set the number of iterations:

n = log2((phigh−plow)/δ) where [plow, phigh]is the interval p is believed to be in and δ is the accepted error.

-         For n number of iterations, the midpoint, pmid, of the interval is calculated. It is then checked if f(pmid)>0 or<0.

If it’s the first case the lower bound is set equal to the midpoint, in the second case the higher bound is set equal to the midpoint.

-         When the n iterations are done the estimated probability of default is set to the final midpoint