Basel II Latent Factor Credit Capital Model
The BAsel II committee used the Vasicek Credit Model as the framework for providing a minimum capital formula for banks to use.Derivations
The Basel model calculates the credit loss in the 99.9% quantile of the systemic factor, labeled Z in the Vasicek model. Then, it caluclates the probability of default for each loan given the realization of the systemic factor.
{% PD_i(Z) = Prob(A_i \leq \Phi^{-1}(PD_i)|Z) %}
{% = Prob(w_i Z + \sqrt{1-w_i^2} e_i \leq \Phi^{-1}(PD_i)) %}
{% =Prob(e_i \leq \frac{\Phi^{-1}(PD_i) - w_iZ}{\sqrt{1-w_i^2}}) %}
{% = \Phi[\frac{\Phi^{-1}(PD_i) - w_i Z}{\sqrt{1 - w_i^2}}] %}
Once the probabilitiy of a single loan defaulting given Z has been calculated, the
expected
loss can be computed.
{% \mathbb{E}[Loss|Stress] = EAD \times LGD \times \Phi[\frac{\Phi^{-1}(PD) - w_i \Phi^{-1}(\alpha)}{\sqrt{1 - w_i^2}}] %}
Note, this assumes that the LGD is given, or non random. Typically, an average value is used here.
Next, the committee assumes that banks provision for the expected value of credit loss (CECL) so they subtract that value from the model.
{% EAD \times LGD \times \Phi[\frac{\Phi^{-1}(PD) - w_i \Phi^{-1}(\alpha)}{\sqrt{1 - w_i^2}}] - EAD \times LGD \times PD %}
In addition to the capital calculated from the above model, the committee added an additional term for some loan types to account for
losses due to deteriating credit quality, as opposed to default. This is the so called maturity adjustment, used primarily for long dated loans.