IRB formula

IRB formulas are in banking Applied formulas with which the minimum capital requirements for credit risks are determined. These formulas belong to the internal ratings-based approach (IRB for short, internal ratings based approach ).

General

The Solvency Regulation (SolvV) of January 2007 was based on the capital requirements of Basel II and contained provisions for determining the capital requirements of lending by banks . The regulatory tasks of the SolvV have been performed by the Capital Adequacy Regulation ( CRR; Regulation (EU) No. 575/2013 (PDF) ) since January 2014 . The IRB formulas contained in this - like the entire CRR - aim to use these presence indicators to provide banking supervisors with an opportunity to ensure a functioning banking system and a stable financial market and thus to identify bank insolvency risks in good time using standardized formulas.

The IRB formulas are only to be used for the two internal rating procedures. They do not apply to the IRB Basis Approach (IRBB), in which - with the exception of the probability of default - all risk parameters are fixed by the banking supervisory authority. In addition to the uniform conversion factor of 0.75% to determine the default loan amount (EaD) and the fixed default loss rate of 45% for senior secured receivables , the remaining term in the IRB basis approach is uniformly set at 2.5 years (Art. 162 (1) CRR) .

content

The internal ratings-based approach (IRB approach) enables banks to determine individual risk parameters, which are included in the regulatory IRB formulas, based on their own data history or external data sources. These include the default loan amount , default probability and default loss rate . These are hypothetical quantities that are based on stochastic probabilities . This is based on the following assumptions:

• The company returns depend on the development of a systematic and an unsystematic factor from,
• insolvency occurs when the company returns below a threshold and
• the loan portfolio consists of a large number of loans; it is completely granular .

The IRB formulas are based on an asymptotic “one-factor asset value credit risk model” (ASRF model) with a time horizon of one year. It builds on the work of Oldřich Vašíček with his stochastic interest rate model and Michael B. Gordy, which was designed for non-banks in general . With these asset value models (company value models), the company value is compared to a default threshold. Specifically, the market value of the assets is offset by the book value of a company's liabilities :

${\displaystyle {\text{Ausfallschwelle}}={\frac {\text{Marktwert der Aktiva}}{\text{Buchwert der Verbindlichkeiten}}}}$.

The default threshold is reached when the market value of the assets corresponds to the book value of the liabilities . However, if the market value falls below these liabilities, the company becomes insolvent.

The IRB formulas are based on a model based on this, which compares the standardized returns on a borrower's assets with a default threshold. The following applies to the probability of default , with which the return assumed as standard normally distributed falls below a critical value ${\displaystyle PD}$ ${\displaystyle Y_{i}}$ ${\displaystyle D}$

${\displaystyle PD=P(Y_{i}<D)}$

So the failure threshold is where the inverse of the distribution function is the standard normal distribution . The following is assumed for the return : ${\displaystyle D=G(PD)}$ ${\displaystyle G}$ ${\displaystyle Y_{i}}$

${\displaystyle Y_{i}={\sqrt {R}}\cdot X+{\sqrt {1-R}}\cdot e_{i}}$

This shows the systematic risk that affects all companies (economic situation) and the company-specific risk. One can show that if and the standard normal distribution is, that then the returns are standard normally distributed and the correlation of the returns is. ${\displaystyle X}$ ${\displaystyle e_{i}}$ ${\displaystyle X}$ ${\displaystyle e_{i}}$ ${\displaystyle Y_{i}}$ ${\displaystyle R}$

If one assumes a certain realization of the systematic risk as a condition, then the returns and thus the default events are stochastically independent. Because of the law of large numbers , the relative frequency of default events in a sufficiently large loan portfolio will then almost coincide with the corresponding conditional probability of default. ${\displaystyle X}$

In the IRB procedure, it is assumed that the systematic risk will be realized with a probability of only 0.1%. The corresponding conditional probability of default indicates the frequency of loan defaults, which is only exceeded rarely. If you put this value of the systematic risk in the above formulas, you get: ${\displaystyle X=G(0{,}001)=-G(0{,}999)}$

${\displaystyle P(-{\sqrt {R}}\cdot G(0{,}999)+{\sqrt {1-R}}\cdot e_{i}<G(PD))}$

${\displaystyle =P\left(e_{i}<{\frac {1}{\sqrt {1-R}}}\cdot G(PD)+{\sqrt {\frac {R}{1-R}}}\cdot G(0{,}999)\right)}$

${\displaystyle =N\left({\frac {1}{\sqrt {1-R}}}\cdot G(PD)+{\sqrt {\frac {R}{1-R}}}\cdot G(0{,}999)\right)}$

Here the distribution function is the standard normal distribution. ${\displaystyle N}$

This is the core of the formula used in Articles 153 and 154 of the Capital Adequacy Regulation (CRR). The expected loss covered by the interest margin must be deducted . The factor 12.5 is neutralized with a core capital ratio of 8%. The factor 1.06 should ensure that the capital adequacy requirements do not decrease compared to Basel I. In Article 153 CRR it is also multiplied by a maturity adjustment factor.

literature

• Christian Cech: The IRB formula. Version 1.01, March 2004, University of Applied Sciences of bfi Vienna

Individual evidence

1. ^ Jochen Klement: Credit Risk Trading: Basel II and Internal Markets in Banks. 2007, p. 217 FN 622
2. ↑ Oldřich Vašíček: An Equilibrium Characterization of the Term Structure. In: Journal of Financial Economics 5 (2), 1977, pp. 177-188
3. ^ Michael B. Gordy: A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules. In: Journal of Financial Intermediation 12 (3), July 2003, pp. 199-232
4. ↑ Peter J Crosbie, Jeffrey R. Bohn: Modeling Default Risk. KMV, 2002, Figure 1
5. ↑ Christian Cech: The IRB formula - for calculating the minimum capital required for credit risk. March 2004, p. 7 ff.