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