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

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

 

Python Generic Code (Probability of Default Model Development, Validation and Testing):

Python Generic Code (Probability of Default Model Development, Validation and Testing):
In the link below:

1- PD_Factors.csv: CSV with factors and required data
2- PD_Model_Generic_Python (Doc and PDF): Python generic code (as in steps)
3- PD_Estimate_Steps_Python: Steps in word as in Screenshot below

Asymptotic Single Risk Factor Model (ASRF):

 Key assumptions: asymptoticity, a single risk factor, and normality.

PD assumptions and violations:

Probability of observing D defaults over N (total number of exposures in the credit portfolio) independent random draws follows a binomial distribution.
PD ASRF model Binomial Distribution assumptions:
  i.  Each asset in the rating grade has default probability P.
 ii.  Each pair of assets has default correlation ρ
iii.  The conditional correlation between any two assets is constant even if the number of defaults increases.
iv.   Normal distribution assumption for the systematic factor
 
ASRF model assumptions may get violated:

Assumptions (i) and (ii):
Let x1, ..,xn be random indicator variables representing the default behavior of the assets where xj =1 indicates the default of asset j. Define as the probability of default of asset j given that assets 1 to j-1 are known to have defaulted.
Assumptions (i) and (ii) imply that:
P1 = P and P2 = P + (1 - P )*ρ
When assets are independent ρ = 0 than these assumptions lead to the Binomial distribution with Pj = P.
However, If ρ > 0, then x1, ..,xn are not independent than the assumption of Binomial distribution is violated.
Assumption (iii):
If the conditional correlation between any two assets increase as the number of defaults increases will lead to increase in default probability.
The increasing default probability given other defaults results in fatter tails of the Correlated Binomial distribution. Contrast assumption (3) with the Binomial distribution where the independence assumption implies that ρj = ρ for all j=1,..,n assets
 
Assumption (iv):
Systematic factor may follows an autoregressive process.

Modeling Low Default Portfolio Dependent Case:

 Modeling Low Default Portfolio Dependent Case:

VASCIEK MODEL: Dependence between the default is explained by by Vasicek model.

By using conditional probability from the Vasicek model in the case where there are no defaults, the probability of default is the solution of below equations:

Low Default Portfolio (PD)

 Modeling Low Default Portfolio (Independent Default Events):

Pluto and Tasche method for calculating probability of default for portfolios with none or very few observations of defaults.
One-sided upper confidence bound as an estimator of PD.

Assumptions:
- n >0 borrowers in the portfolio.
- At the end of the observation period 0≤ d < n defaults are observed among the n borrowers.
- Default events are independent, hence the number of defaults in a portfolio is binomially distributed:
nCr * p^r * ((1-p)^(n-r)
n is the total number of borrowers, r is the total number of defaults and p is the probability of default.
PD to be logical, it should have the following characteristic:
p1 <= p2 <=p3 <=p4..........
It also means that p1=p2=p3=p4=p5...... In this scenario, all the 500 borrowers belong to the same risk characteristic, i.e. homogenous borrowers.

E.g: 

https://drive.google.com/file/d/1OmGmQV-AsYPsfdRYowSy1bkZArgEFMvN/view?usp=sharing