# ASRF model

## Definition

ASFR Model (asymptotic single factor risk model) is a simplified Credit Portfolio risk model that underpins the Basel II capital requirements

## Assumptions

The following is an indicative list of assumptions / design choices in constructing the ASFR model

• One year risk horizon
• Single factor model
• Normal distributions
• Homogeneous and large pool of exposures

## Formulae

Below are the explicit formulae for some banks’ major products:

• Corporate exposures,
• Small-medium enterprise (SME) exposures
• Residential mortgages and
• Qualifying revolving retail exposure. (S being Min(Max(Sales Turnover),5),50 )

In the formulas below:

### Corporate Exposures

The exposure for corporate loans is calculated as follows[1]

#### Correlation Parameter

${\displaystyle R=0.12{\frac {1-e^{-50PD}}{1-e^{-50}}}+0.24\left(1-{\frac {1-e^{-50PD}}{1-e^{-50}}}\right)}$

The correlation parameter offers a simple approach to capture an element of Default Correlation. The peculiar presence of exponentials is for normalization purposes

${\displaystyle b=(0.11852-0.05478\ln(PD))^{2}}$

The maturity adjustment offers a simple approach to capture residual credit risks that may not materialise within the one-year risk horizon implied by the ASRF model.

#### Capital requirement

${\displaystyle K=\left[{\mbox{LGD}}\,N\left({\sqrt {\frac {1}{1-R}}}G({\mbox{PD}})+{\sqrt {\frac {R}{1-R}}}G(0.999)\right)-({\mbox{LGD}}\,{\mbox{PD}})\right]{\frac {1+(M-2.5)b}{1-1.5b}}}$

#### Risk-weighted assets

${\displaystyle RWA=K12.5EAD}$

### Correlation adjustment for Large Financial Institutions

A multiplier of 1.25 is applied to the corporate correlation parameter of all exposures to financial institutions meeting the following criteria[2]

• Regulated financial institutions whose total assets are greater than or equal to US \$100 billion. Regulated financial institutions include, but are not limited to, prudentially regulated Insurance Companies, Broker/Dealers, Banks, Thrifts and Futures Commission Merchants
• Unregulated financial institutions, regardless of size.

#### Correlation Parameter

${\displaystyle R=1.25*[0.12{\frac {1-e^{-50PD}}{1-e^{-50}}}+0.24\left(1-{\frac {1-e^{-50PD}}{1-e^{-50}}}\right)]}$

For small and medium enterprises with annual Sales Turnover below 50 million euro, the correlation may be adjusted as follows:[3]

#### SME Correlation

${\displaystyle R=0.12{\frac {1-e^{-50*PD}}{1-e^{-50}}}+0.24\left(1-{\frac {1-e^{-50PD}}{1-e^{-50}}}\right)-0.04(1-{\frac {\max(S-5,0)}{45}})}$

In the above formula, S is the enterprise's annual sales turnover in millions of euro.

### Residential mortgage exposures

The exposure related to residential mortgages can be calculated as this[4]

#### Correlation

${\displaystyle R=0.15}$

#### Capital Requirement

${\displaystyle K=LGD\left[N\left({\sqrt {\frac {1}{1-R}}}G(PD)+{\sqrt {\frac {R}{1-R}}}G(0.999)\right)-PD\right]}$

#### Risk-weighted assets

${\displaystyle RWA=K12.5EAD}$

### Qualifying revolving retail exposures (credit card products)

The exposure related to unsecured retail credit products can be calculated as follows:[5]

#### QRRE Correlation Assumption

${\displaystyle R=0.04}$

#### Capital Requirement

${\displaystyle K=LGD\left[N\left({\sqrt {\frac {1}{1-R}}}G(PD)+{\sqrt {\frac {R}{1-R}}}G(0.999)\right)-PD\right]}$

#### Risk-weighted assets

${\displaystyle {\mbox{RWA}}=K12.5{\mbox{EAD}}}$

## References

1. Basel III: A global regulatory framework for more resilient banks and banking systems