History of the Basel internal-ratings-based (IRB) credit risk regulation

Abstract

In December 2019 the Basel Committee has launched the consolidated Basel framework. The framework inherits the Basel II internal ratings-based (IRB) approach for the credit risk with mostly no changes. The absence of the material methodological changes is unexpected given the fact that the key shortcomings of the IRB approach stay unresolved. The paper objective is therefore to list its earlier discussed shortcomings and to address the new ones. Latter include the unbalanced treatment of PD and LGD parameters, as well as methodological inconsistency in expected and unexpected loss treatment.

Keywords

Basel Committee IRB credit risk regulation structural model

1. Introduction

2018 was a remarkable year for the credit risk-management. From one side, it was a fifty-year anniversary of Z-score family of models as described in detail by Altman (2018). Those models form the foundation for the probability of default (PD) estimation. From another side, just on the eve of the 2018 year the Basel Committee on Banking Supervision (BCBS) adopted the finalized version of Basel III. Here and below we will use interchangeably the terms such as BCBS, the regulator, the supervisor. Discussing specifics between the regulator and the supervisor falls out of the current paper scope. Basel III has endowed the Basel II allowance for the use of internal (internal-ratings-based, IRB) models, including the PD ones, for the regulatory purposes of calculating capital adequacy ratio. Though Basel II models – particularly the underlying (Vasicek, 1987) one – were much explained (e.g., BCBS, 2005a; Ozdemir & Miu, 2009) and criticized (for instance, Gordy & Howells, 2004, Illing & Paulin, 2005; Witzany, 2013; Zimper, 2014), to the best of author’s knowledge, there are still four issues left unattended. Those include the possibility of negative default correlation, the necessity to recalibrate the supervisory risk-weighting formula, and unbalanced treatment of two input and output parameter sets. Input parameters include the default probabilities (PDs) and the losses given default (LGDs). Output parameters include the expected losses (EL) and the unexpected ones (UL). These four issues lead to the inconsistent estimate of the credit risk as we show it later.

That is why the objective of the paper is to trace the evolution of regulatory treatment of internal models from the very first consultative paper in 1999 (BCBS, 1999) to its recent one 20 years later (BCBS, 2019a). Such an investigation allows getting how the regulatory model shortcomings were tackled and which ones still remain unresolved. The banking risk regulation evolution was generally discussed by Goodhart (2011) and Penikas (2015) before Basel II introduction and afterwards, respectively. Nevertheless, the credit risk regulation evolution and, particularly, the internal (IRB) models were not discussed in depth. All the known works deal only with the final, not consultative versions of the regulation (e.g., Gordy, 2004; Illing & Paulin, 2005; Ozdemir & Miu, 2009; Gordy et al. 2014). Investigation of such an evolution may provide one with the hints of how to validate the credit scoring models (e.g. the Basel IRB maturity adjustment component may be benchmarked to international financial reporting standards (IFRS) 9 life-time expected loss computation concept; the granularity adjustment might be used for internal capital adequacy assessment process (ICAAP) purposes).

The paper is thus organized as follows. Section 2 presents the literature review. It explains in detail the logic of Vasicek (1987) model being of interest for the beginners. At each stage of transformation – where applicable – the known shortcomings are presented with references to other research papers. Such shortcomings include the choice of the confidence level, factor distribution assumptions (including marginal and joint normality), PD-LGD correlation, exposure concentration (or portfolio non-granularity). Two additional shortcomings are also discussed in the section, i.e. allowing for negative default correlation and the need to recalibrate the supervisory risk-weighting formula. Section 3 discusses another two previously non-discussed shortcomings, i.e. unbalanced treatment of PDs and LGDs, and that of EL and UL. This section is intended for the advanced risk-managers and regulators. If Section 2 is a rehearsal of previous findings by others, Section 3 comports the novel material developed by the author himself. Section 4 concludes by presenting policy implications and paper contribution to the field.

2. Literature review (Vasicek model)

The founding brick of the regulatory treatment of the internal models is the Vasicek (1987) model. That is why we would first present it with transformations and derivations that original paper lacks. Then we would present the following regulatory enhancements to the baseline model, namely:

1.
Adjustment for recovery efficiency;
2.
Maturity adjustment;
3.
Granularity adjustment;
4.
Double default treatment.

2.1 Baseline model

The starting point for Vasicek (1987) model is the Merton (1974) model. The default occurs when the company assets become less in value than its debt amount. Let us introduce the process for the asset value dynamics because such a dynamics is driven by a single systemic factor. That is why the Vasicek model is also called a single risk-factor (ASRF) one. However, Pykhtin (2004) shows the shortcomings of capturing only one systemic factor and proposed model extension to several systemic factors. Tarashev (2005) reviews six models that may drive the asset value as alternative to the approaches of Vasicek (1987) and Pykhtin (2004). However, none of those six gets anchored into the Basel IRB regulation. The following one of Vasicek (1987) does.

$\displaystyle A_{i}=Y\sqrt{R_{i}}+{\varepsilon}_{i}\sqrt{1-R_{i}}$

where $A_{i}\sim N(0;1)$ – the $i^{th}$ asset value belonging to $i^{th}$ company ( $i^{th}$ bank borrower);

$N(0;1)$ – the cumulative distribution function for the standard normal (Gaussian) distribution with mean zero and variance of one; and $N^{-1}(∼{})$ is it’s inverse;

$Y\sim N(0;1)$ – the systemic factor that co-depends with the asset value, i.e.,

$\displaystyle\textit{Cor}\left(A_{i};Y\right)=R_{i}$

where $R_{i}\in[0;1]$ is the value of ‘asset correlation’. (Foulcher et al., 2005) also call it a ‘latent correlation’.

${\varepsilon}_{i}\sim N(0;1)$ – the idiosyncratic factor that co-depends with asset value, i.e.,

$\displaystyle\textit{Cor}\left(A_{i};{\varepsilon}_{i}\right)=1-R_{i}$

${\varepsilon}_{i}$ is independent both to the systemic factor, i.e., $\textit{Cor}\left(Y;{\varepsilon}_{i}\right)=$ 0, and to the other idiosyncratic factors, i.e., $\textit{Cor}\left({\varepsilon}_{i};{\varepsilon}_{j}\right)=$ 0 given $\forall j:i\neq j$ .

The Basel Committee suggests using the following formula for asset correlation ( $R_{i}$ ).

$\displaystyle R_{i}=R_{0}+R_{\textit{MIN}}\left(\frac{1-e^{-w{PD}_{i}}}{1-e^{-% w}}\right)+R_{\textit{MAX}}\left(1-\frac{1-e^{-w{\textit{PD}}_{i}}}{1-e^{-w}}\right)$

where $\textit{PD}_{i}$ is the probability of default of the $i^{th}$ borrower;

$w$ is the calibrated weighting coefficient;

$R_{0}$ – the fixed asset correlation component independent of the PD;

$R_{\textit{MIN}}$ , $R_{\textit{MAX}}$ stand for the minimum and maximum asset correlation values applicable for the given exposure asset class (see Annex 1 for particular values).

The idea underlying the asset correlation formula is the following. The higher PD is, the lower the R value is. This means, the less creditworthy the borrower is, the less its credit risk is augmented when crisis comes in. The reason for this is that the borrower’s credit risk assessment is already relatively high.

The above formula can still be simplified. For instance, Lopez (2002, p. 18) offers the following for the general case of corporate exposures:

$\displaystyle R_{i}=R_{\textit{MAX}}-R_{\textit{MIN}}\left(\frac{1-e^{-w{% \textit{PD}}_{i}}}{1-e^{-w}}\right)$

Repullo (2013) suggests the even simpler formula that may yield the same output, i.e.

$\displaystyle R_{i}=R_{\textit{MIN}}\left(1+e^{-w\textit{PD}_{i}}\right)$

For the last case consider the largest probability of default of $\textit{PD}_{i}=$ 100%. For both formulas $R_{i}(\textit{PD}_{i}=100\%)=$ 0.24, whereas for the smallest one of $\textit{PD}_{i}=$ 0% both equal to $R_{i}\left(\textit{PD}_{i}=0\%\right)=$ 0.12.

The key implication from the regulatory asset correlation formula is that there is a non-positive dependence of the asset correlation and the default probability, i.e.:

$\displaystyle\textit{Cor}\left(R_{i};\textit{PD}_{i}\right)\leqslant 0$

or equivalently there is a non-negative dependence of the asset correlation and the idiosyncratic factor realization, i.e.:

$\displaystyle\textit{Cor}\left(R_{i};{\varepsilon}_{i}\right)\geqslant 0$

Remember that Vasicek (1987) assumes the weights for the factors ( $\sqrt{R_{i}}$ and $\sqrt{1-R_{i}}$ ), or factor loading, to be constant. When one considers that the factor loading are variable as the regulator imposed, the ultimate loss distribution may have different properties than Vasicek (1987) expected.

Note that a negative neither asset, nor default correlation ( $r_{i}$ such that ${r_{i}}^{2}=R_{i}$ ) was neither assumed, nor tried to be implemented since (Vasicek, 1987) because the asset value dynamics equation uses $\sqrt{R_{i}}$ . This is the first shortcoming. There is an alternative, i.e. to use the following equation in a way like most researchers write it after (Vasicek, 1987) (see, for example, Gordy, 2000, p. 141, eq. (C.1); BCBS, 2005c, p. 21):

$\displaystyle A_{i}=Y\cdot r_{i}+{\varepsilon}_{i}\sqrt{1-\left({r_{i}}^{2}% \right)}$

For instance, Nagpal and Bahar, (2001, p. 97) empirically justify the presence of negative default correlation ( $r_{i}$ ), e.g. for the investment grade insurance companies and most times non-investment grade hi-tech companies.

We may suspect two reasons why the negative default correlation was skipped when IRB framework was developed. First, it was difficult to simulate one. The known approaches to generating negatively correlated binary outcomes refer to dates much later than the ones when Basel II (BCBS, 2006) was adopted, e.g. for the Bernoulli distribution (see Witt, 2014; Kruppa et al., 2018) and for the Poisson one (see Chiu et al., 2017). Second, for the infinite number of borrowers negative correlation is not feasible as follows from (Preisser & Qaqish, 2014). Discussing asset correlation when the number of borrowers is finite falls out of the current paper scope.

Let us introduce the probability that the idiosyncratic factor ${\varepsilon}_{i}$ hits its quantile value $x_{{\varepsilon}_{i}}$ .

$\displaystyle P\left({\varepsilon}_{i}\leqslant x_{{\varepsilon}_{i}}\right)=N% \left(x_{{\varepsilon}_{i}}\right)=p_{i}$

where

$\displaystyle p_{i}\in\left[0;1\right]$

Then

$\displaystyle x_{{\varepsilon}_{i}}=N^{-1}\left(p_{i}\right)$

where $p_{i}\rightarrow 0$ . Such a requirement for $p_{i}$ is introduced more from the convenience perspective. Given the symmetry of the normal distribution, it follows from the above that

$\displaystyle-x_{{\varepsilon}_{i}}=-N^{-1}\left(p_{i}\right)=N^{-1}\left({1-p% }_{i}\right)$

$\displaystyle x_{{\varepsilon}_{i}}={-N}^{-1}\left({1-p}_{i}\right)$

Similarly, for the systemic factor it would be

$\displaystyle x_{Y}={-N}^{-1}\left(1-p_{Y}\right)$

where $p_{Y}$ may be regarded as the probability of crisis occurrence.

The weighted sum of the normally distributed random variables $Y$ and ${\varepsilon}_{i}$ is a normal one ( $A_{i}$ ). Thus we may define the default condition and the probability of default ( $\textit{PD}_{i}$ ) when the value of the company (borrower) assets ( $A_{i}$ ) becomes less than the amount of its total outstanding debt ( $D_{i}$ ).

Li (2000) has shown that copula model can be applied to reflect non-linear risk dependencies within credit portfolio. He uses the simplest Gaussian copula to illustrate his idea. Industry seems to have evaluated copulas of Li (2000) to be simpler than the multi-factor risk models of Pykhtin (2004). That is why copulas seem to have become the basis for the credit risk evaluation for the securitized pools of loans in the collateralized debt obligations (CDOs). Later copula as a tool was blamed as a ‘formula that killed Wall-Street’. Salmon (2009) argues that copulas did not capture true credit risk profile. In fact Salmon (2009) was wrong as copula might have captured it if more complex families (e.g. Archimedean) were used. But because of the computational complexities, investment banks might have chosen to use the Gaussian copula where one parameter is used to model risk dependencies of the millions of loans in a pool. By construction, it is obvious that such a model materially underestimates risks. This is why Kupiec (2006) criticized the use of exactly Gaussian copula for portfolio credit risk.

Getting back to the baseline model, we want to define the marginal default probability ( $\textit{PD}_{i}$ ) per single exposure (borrower or facility where the facility level is used in the retail IRB model; and a borrower one – in the non-retail; discussing the methodological challenges of such a difference falls out of the scope of the current paper) as if it were a part of an infinitely granular portfolio with an infinite number of tiny in size loans.

$\displaystyle A_{i}\leqslant D_{i}$

$\displaystyle P\left(A_{i}\leqslant D_{i}\right)=N\left(D_{i}\right)=\textit{% PD}_{i}$

where

$\textit{PD}_{i}$ – the probability of default for $i^{th}$ company (borrower); and

$D_{i}$ is the debt amount held by $i^{th}$ company and as $D_{i}=N^{-1}\left(\textit{PD}_{i}\right)$ then:

$\displaystyle A_{i}\leqslant N^{-1}\left(\textit{PD}_{i}\right)$

$\displaystyle Y\sqrt{R}+{\varepsilon}_{i}\sqrt{1-R}\leqslant N^{-1}\left(% \textit{PD}_{i}\right)$

Move systemic factor to the right of the inequality

$\displaystyle{\varepsilon}_{i}\sqrt{1-R}\leqslant N^{-1}\left(\textit{PD}_{i}% \right)-Y\sqrt{R}$

Divide both parts by $\sqrt{1-R}$ to have the expression ready to apply the Normal probability density function definition.

$\displaystyle{\varepsilon}_{i}\leqslant\frac{N^{-1}\left(\textit{PD}_{i}\right% )-Y\sqrt{R}}{\sqrt{1-R}}$

Remembering that

$\displaystyle x_{Y}={-N}^{-1}\left(1-p_{Y}\right)$

we may substitute $Y$ and rewrite as

$\displaystyle{\varepsilon}_{i}\leqslant\frac{N^{-1}\left(\textit{PD}_{i}\right% )+N^{-1}\left(1-p_{Y}\right)\sqrt{R}}{\sqrt{1-R}}$

Given

$\displaystyle P\left({\varepsilon}_{i}\leqslant x_{{\varepsilon}_{i}}\right)=N% \left(x_{{\varepsilon}_{i}}\right)=p_{i}$

and setting $x_{{\varepsilon}_{i}}$ to the above expression, i.e.

$\displaystyle x_{{\varepsilon}_{i}}=\frac{N^{-1}\left(\textit{PD}_{i}\right)+N% ^{-1}\left(1-p_{Y}\right)\sqrt{R}}{\sqrt{1-R}}$

we may write the probability of $i^{th}$ company (bank borrower) default subject to realization of both the idiosyncratic and systematic factors as follows:

$\displaystyle P\left({\varepsilon}_{i}\leqslant\frac{N^{-1}\left(\textit{PD}_{% i}\right)+N^{-1}\left(1-p_{Y}\right)\sqrt{R}}{\sqrt{1-R}}\right)=N\left(\frac{% N^{-1}\left(\textit{PD}_{i}\right)+N^{-1}\left(1-p_{Y}\right)\sqrt{R}}{\sqrt{1% -R}}\right)=p_{i}\left(\textit{PD}_{i};p_{Y}\right)$

The Basel Committee assumed in 2001 that the approach should tolerate default only once in 200 years originally, i.e. $p_{Y}=$ 0.5% (BCBS, 2001b, p. 36). Later it made the requirement more conservative, i.e. once in 1000 years (BCBS, 2005a, p. 11), i.e. $p_{Y}=$ 0.1%. Though this may look reasonable as the regulator wants the crisis to hit the system as rarely as possible. However, such an over-safety may be too expensive for the banks. Thus the second shortcoming of the IRB model seems to be its unrealistic confidence level. The material financial distress tends to occur every 5 to 10 years and not once in 1000 years. For instance, during the last 20 years one may at least recall Asian crisis of 1997; dotcom bubble burst in 2001; the Great Recession in 2007–09, i.e. the confidence level parameter ( $p_{Y}$ ) should be readjusted 100-times upwards to $p_{Y}=\frac{3}{20}\approx$ 15%. Zimper (2014) also shows that the confidence level, i.e., bank’s own (not the borrower’s) default probability should be much more than 0.1%.

Thus, Basel Committee approach suggests the following formula for capital requirements to start with (later on it would be extended by considering other loan features as recovery, maturity etc.):

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)=N\left(\frac{N^{-1}\left% (\textit{PD}_{i}\right)+N^{-1}\left(0.999\right)\sqrt{R}}{\sqrt{1-R}}\right)$

The third shortcoming of the IRB model is the assumption of the Normal distribution for the risk components. For instance, (Witzany, 2013) shows that one should consider the log-normal distribution. It results in up to 100% higher risk estimate and henceforth higher capital requirements.

The structural (Merton-type) default model does not account for the several loan features that in practice significantly differentiate the risk per exposure. Let us describe the required adjustments according to the following list by paying special attention to the RWA value calibration that was never shown before in such a breakdown:

(1)
Recovery efficiency;
(2)
Loan maturity;
(3)
Exposure amount;
(4)
Non-infinite-granularity of the loan portfolio;
(5)
Double-default;

2.2 Recovery efficiency

An actual loss per the credit exposure depends not solely on the fact of the default occurrence, but also on how much the bank is able to work out in case of the non-retail loans or to recover (collect) in case of the retail ones, or inversely how much it ultimately losses subject to (given) default. That is why the regulator distinguishes the probability of default (PD) and the loss given default (LGD) as presented at the Fig. 1.

Figure 1.

General scheme of credit default process.

Thus the capital requirement becomes equal to the following

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)\cdot\textit{LGD}$

Basel II (BCBS, 2006) offers two options when estimating capital requirements for corporate loans: foundation (F-IRB) and advanced (A-IRB) ones. As for the foundation one, the bank has to develop only PD models. LGD and exposure at default (EAD) values are chosen from the predefined list by the regulator (an equivalent to the fixed risk-weights of Basel I (BCBS, 1988)). Foundation LGD was first 50% in 2001; 45% in 2006; and is set to 40% from 2022.

An A-IRB bank should develop all types of models, i.e. PD, LGD, and EAD. As for the retail loans only advanced approach is allowed. No maturity adjustment is applicable to retail loans.

Multiplication sign in front of LGD means the parameters’ independence. However, Ozdemir and Miu (2009) and Meng et al. (2010) discuss that this is not the case often in practice. Typically the riskier the borrower is (the higher the PD is), the less amount recovered by the bank is (the higher LGD is), i.e. there is a positive PD-LGD correlation (PLC). Same time obtaining reliable PLC estimates in general and LGD ones in particular is challenging because of the recovery data insufficiency. As of today, one of the longest LGD time series is available since 1998 from the not-for-profit banking association Global Credit Data (URL: https://www.globalcreditdata.org/).

As PLC is difficult to be robustly assessed (one needs rich default and collection data), the regulator required LGD with downturn adjustment ( $\textit{LGD}_{\textit{DT}}$ ) to be used.

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)\cdot\textit{LGD}_{% \textit{DT}}$

The downturn adjustment was thought to offset rise in risk estimate by preserving formula with multiplication sign. Altman (2011) refers to the US regulation and presents the following formula for the downturn adjustment of LGD.

$\displaystyle\textit{LGD}_{\textit{DT}}=8\%+92\%\cdot\textit{LGD}$

However, The US Federal Reserve (2006, p. 340) states that the above-mentioned LGD should already reflect the downturn, i.e. the above mentioned formula is not a downturn adjustment. All it offers is a more conservative capital treatment of bank estimates of LGD (the higher the bank LGD estimate is, the less is the regulatory add-on, see Fig. 2).

Figure 2.

The US approach to regulatory LGD value. Note: LGD_in – LGD (input); LGD_out – LGD (output; suggested by (Altman, 2011) as $\textit{LGD}_{\textit{DT}}$ ).

2.3 Maturity adjustment

The longer the original contract duration (maturity) is, the higher the risk of not paying back is all else being equal. This is how maturity adjustment (MA) comes in. The Basel Committee previewed maturity adjustment in the form of a multiplier to the overall capital requirements.

Then the regulatory capital requirements are as follows.

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)\cdot\textit{LGD}_{% \textit{DT}}\cdot\textit{MA}$

In 2001 MA was suggested as the following one (version 01; mult_01)

$\displaystyle\textit{MA}=1+\left(M-3\right)b_{i}$

where

$M$ – credit facility original (not residual) maturity, measured in years;

$\displaystyle b_{i}=\frac{0.0235\left(1-\textit{PD}_{i}\right)}{{\left(\textit% {PD}_{i}\right)}^{0.44}+0.047\left(1-\textit{PD}_{i}\right)}$

In 2003 it was readjusted to be (version 02; Mult_03)

$\displaystyle b_{i}=\left(0.08451-0.05898\log\left(\textit{PD}_{i}\right)% \right)^{2}$

2006 version is as follows (version 03; Mult_04).

$\displaystyle\textit{MA}=\frac{1+\left(M-2.5\right)b_{i}}{1-1.5b_{i}}$

where

$\displaystyle b_{i}={\left(0.11852-0.05478\ln\left(\textit{PD}_{i}\right)% \right)}^{2}$

Note that the type of logarithm evolved since 2003 to 2006 from the logarithm to the base of two ( $\log$ ) to the natural one ( $\ln$ ). Consider important to also underscore the precision used in the formula, i.e., five digits after the comma. In case upper rounding is applied, there would be no change in coefficient since 2003, i.e., it stays as 0.1 and 0.06 as follows requiring no additional quantitative survey to be undertaken and enabling to implement the IRB regulation earlier:

$\displaystyle b_{i}={\left(0.1-0.06\ln\left(\textit{PD}_{i}\right)\right)}^{2}$

The impact of maturity multiplier recalibration can be seen from the Fig. 3.

Figure 3.

Maturity adjustment evolution calibration.

2.4 Granularity adjustment (GA)

Loans may differ in size. Then one has to account for the exposure at default (EAD).

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)\cdot\textit{LGD}_{% \textit{DT}}\cdot\textit{MA}\cdot\textit{EAD}$

In case when the EAD distribution is not uniform, there is a concentration risk. It means that the very same number of defaults may correspond to the materially larger amount of loss in money terms. That is why granularity adjustment (GA) was suggested to deal with it

The version of the capital requirements (or more precisely risk-weighted assets, RWA) for the regulatory purposes looks as follows (BCBS, 2001a, par. 508–515). Further it was extended by Gordy (2004) and Gordy and Lütkebohmert (2013).

$\displaystyle\textit{RWA}_{\textit{ADJ}}=\textit{RWA}+\textit{GA}$

where

$\textit{RWA}_{\textit{ADJ}}$ – the amount of risk-weighted assets adjusted for granularity;

RWA – baseline level of non-retail risk-weighted assets;

$\displaystyle\textit{GA}=\textit{EAD}\cdot\frac{\textit{GSF}}{\hat{n}}-0.04% \cdot\textit{RWA}$

where

EAD – total non-retail exposure;

GSF – granularity scaling factor;

$\hat{n}$ – number of the effective loans.

$\displaystyle\textit{GSF}=\left(0.6+1.8\cdot\textit{LGD}_{\textit{AG}}\right)% \cdot\left(9.5+13.75\cdot\frac{\textit{PD}_{\textit{AG}}}{F_{\textit{AG}}}\right)$ $\displaystyle F_{\textit{AG}}=\sum_{b}{s_{b}\cdot F_{b}}$

where AG – aggregate (index), see below.

$b$ – internal grade in each portfolio $t$ .

$s_{b}$ – exposure share of risk grade $b$ in total exposure.

$\displaystyle F_{b}=N\left(1.118\cdot N^{-1}\left(\textit{PD}_{b}\right)+1.288\right)$

GA was supposed to be applicable to corporate, bank, sovereign exposures.

$\displaystyle\hat{n}=\frac{1}{\sum_{b}{A_{b}\cdot H_{b}\cdot s^{2}_{b}}}$

We may deem such a non-straightforward computation of $\widehat{n}$ as a way to demonstrate risk-and exposure-weighted equivalent for the number of borrowers. There is a parallel to such approach in the quality of life studies. When one wishes to define the effective number of consumers within the family, one conventionally assigns 1 to an adult and a less than unity value to children and retired people (Gardes, et al., 1999). Here the same logic applies. The more risky the borrowers with the greater share in total loan exposure are, the higher the effective number corresponding to them is.

$\displaystyle A_{b}=\frac{\textit{LGD}^{2}_{b}\cdot\left(\textit{PD}_{b}\cdot% \left(1-\textit{PD}_{b}\right)-0.033\cdot F^{2}_{b}\right)+0.25\cdot\textit{PD% }_{b}\cdot\textit{LGD}_{b}\cdot\left(1-\textit{LGD}_{b}\right)}{\textit{LGD}^{% 2}_{\textit{AG}}\cdot\left(\textit{PD}_{\textit{AG}}\cdot\left(1-\textit{PD}_{% \textit{AG}}\right)-0.033\cdot F^{2}_{\textit{AG}}\right)+0.25\cdot\textit{PD}% _{\textit{AG}}\cdot\textit{LGD}_{\textit{AG}}\cdot\left(1-\textit{LGD}_{% \textit{AG}}\right)}$

$\displaystyle\textit{PD}_{\textit{AG}}=\sum_{b}{s_{b}\cdot\textit{PD}_{b}}$

$\displaystyle\textit{LGD}_{\textit{AG}}=\frac{\sum_{b}{s_{b}\cdot\textit{PD}_{% b}\cdot\textit{LGD}_{b}}}{\sum_{b}{s_{b}\cdot\textit{PD}_{b}}}$

$\textit{LGD}_{b}$ – exposure-weighted LGD in grade $b$ .

$\displaystyle H_{b}=\frac{\sum_{b}{\textit{EAD}^{2}_{i}}}{{\left(\sum_{b}{% \textit{EAD}_{i}}\right)}^{2}}$

Above we provide an illustration for the GA impact on the RWA amount. HHI stands for Herfindal-Hirshman Index (a measure of concentration that sums up squares of portions). It is measured in percentage points where the value of 0 reflects the absence of concentration and 100 stands for the highest level of concentration. As one can see from the Fig. 4, the impact of the GA on RWA is linear with respect to the HHI value. The more variable the risk estimates are (see upper solid line for the uniformly distributed PDs), the higher the RWA add-on is compared to the case where all borrowers are assigned with the same (single) PD value.

Figure 4.

Granularity adjustment calibration.

Thus the capital requirements are as follows.

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)\cdot\textit{LGD}_{% \textit{DT}}\cdot\textit{MA}\cdot\textit{EAD}\cdot\textit{GA}$

Though the logic to adjust the credit risk estimate and the capital requirements for the concentration is intuitive and necessary, it was abandoned in the upcoming versions of Basel II starting from (BCBS, 2004). The most probable reason is its computational complexity. Same time we continue presenting GA for the sake of generality for the IRB framework.

Similar to PLC, Ozdemir and Miu (2009) note that PD-EAD correlation (PEC) is not nill in practice, though regulator assumes those as independent ones because the multiplication sign is used. Thus EAD estimate is also recommended by the regulator to be downturn-adjusted ( $\textit{EAD}_{\textit{DT}}$ ). RWA calibration when both issues are present (non-granularity and PLC) is provided in (Ermolova & Penikas, 2019).

Thus the capital requirements are as follows.

$\displaystyle p_{i}\left(\textit{PD}_{i};0.1\%\right)\cdot\textit{LGD}_{% \textit{DT}}\cdot\textit{MA}\cdot\textit{EAD}_{\textit{DT}}\cdot\textit{GA}$

2.5 Total capital requirements calibration

There are other two adjustments that are intended to make the total capital requirement more conservative: alpha multiplier ( $\alpha$ ) and an output floor ( $\phi$ ). The former is an overall capital buffer that was introduced to smooth the transition from the Basel I or Basel II standardized approach to the Basel II IRB one. Initially it was put as 1.50 in 2003, then decreased to 1.06 in 2006, and ultimately abandoned from 2022, i.e. set to 1.00.

The final amount of the credit risk per credit exposure should be estimated as follows:

$\displaystyle\textit{CR}_{i}=\alpha\cdot p_{i}\left(\textit{PD}_{i};0.1\%% \right)\cdot\textit{LGD}_{i}\cdot\textit{EAD}_{i}\cdot{\hat{M}}_{i}\left(% \textit{PD}_{i}\right)$

where the credit risk capital requirement ( $\textit{CR}_{i}$ ) is decomposed into expected loss component ( $\textit{EL}_{i}$ ) and unexpected one ( $\textit{UL}_{i}$ ), i.e.,

$\displaystyle\textit{CR}_{i}=\textit{EL}_{i}+\textit{UL}_{i}$

where

$\displaystyle\textit{EL}_{i}=\textit{PD}_{i}\cdot\textit{LGD}_{\textit{DT}}% \cdot\textit{EAD}_{\textit{DT}}$

$\textit{EL}_{i}$ may sometimes be referred to as provisions, though from the accounting perspective there might be differences in details (e.g. whether it is the incurred losses under international accounting standard (IAS) 39 or life-time expected losses under IFRS 9).

Rationale for the EL – UL decomposition is the following. When there is no risk (PD $=$ 0), there is neither EL, nor UL. When the risk is a sure thing (PD $=$ 100%), then the loss is fully expected. Hence UL $=$ 0. The extra capital requirements over EL appear at the interim stages when 0 $<$ PD $<$ 100%. Visual representation is given at Fig. 5. The sum of the UL and EL equals to the Value-at-Risk estimate for the loan book total losses’ estimate (BCBS, 2005a).

Figure 5.

EL and UL Decomposition. Note: PD ranges from 0 to 100%.

The Basel Committee distinguishes two broad exposure (asset) classes: corporate (non-retail) and retail claims. BCBS (2019a) introduced for consistency the following asset classes as separate ones: subordinated debt, covered bonds, non-regulatory equity capital instruments. The capital requirements calibration for the UL part of existing asset classes is available at the Fig. 6.

Figure 6.

Capital requirements calibration per various loan classes.

As for the corporate asset class, lending to small and medium enterprises (SMEs) has preferential treatment (line corresponding to the capital requirements for the corporate SME loans is below the one for the general corporate ones at the Fig. 6), i.e. all other things being equal the UL part for SME loan is regarded to be smaller for the regulatory purposes. In the same way retail revolving (credit card overdrafts) loans have preferential treatment than the mortgage ones.

Output floor means benchmarking the internal model risk (capital requirement per facility) estimate to a standardized (fixed, predetermined) risk-weight ( $\textit{RW}_{\textit{ST}}$ ) taken with an adjusted coefficient, i.e.

$\displaystyle\textit{RW}=\textit{max}\left(\textit{UL}_{\textit{IRB}};\phi% \cdot\textit{RW}_{\textit{ST}}\right)$

2.6 Double-default framework

Another modification of the IRB model is the treatment of double-default (DD). This is a low risk situation when the payback on the credit exposure is additionally guaranteed by the third party (guarantor, G), not solely by a borrower (obligor, B, O). It is considered to be low risky as the probability of the joint default of the obligor and the guarantor is less than the PD of a single obligor or a guarantor taken separately. There were two variations how to account for the double-default in the IRB framework.

Figure 7.

Double default calibration. Note: LGD $g=$ 5%; LGD $o=$ 45%; $w=$ 0%.

BCBS (2001a, pp. 38, par. 183) offers the following. Covered part of the exposure is assigned with the adjusted default probability $\widetilde{\textit{PD}}$ :

$\displaystyle\widetilde{\textit{PD}}=w\cdot\textit{PD}_{B}+\left(1-w\right)% \cdot\textit{PD}_{G}$

Where the weight $w=\left\{0;0.15\right\}$ is chosen by the bank. Uncovered part of the exposure is assigned to the PD of the underlying obligor. Thus mostly full substitution of PD takes place.

The final version of the capital requirements (unexpected loss, UL) has the following modification (BCBS, 2006, p. 66, par. 284).

$\displaystyle\textit{UL}_{\textit{DD}}=\textit{UL}_{o}\cdot\left(0.15+160\cdot% \textit{PD}_{g}\right)$

where

$\displaystyle\textit{UL}_{o}=\left(p_{o}\left(\textit{PD}_{o};0.1\%\right)-% \textit{PD}_{o}\right)\cdot\textit{LGD}_{g}\cdot\textit{EAD}_{o}\cdot\widehat{% \textit{MA}}\left(\textit{min}\left(\textit{PD}_{o};\textit{PD}_{g}\right)\right)$

Thus, in 2001 the double default framework was reflected only in the PD, whereas in 2006 it required the substitution of both PD for maturity adjustment and LGD. In 2017 the Basel Committee decided to exclude the possibility of the DD treatment.

2.7 Capital adequacy ratio

The capital requirements ( $\textit{CR}_{i}$ ) decomposition into the EL and UL parts is used to define regulatory treatment of the IRB risk estimates, i.e. to set capital adequacy ratio (CAR). When introducing one, the regulator wants a bank to sufficiently cover its risks with its own funds (capital). Risk is defined as the Value-at-Risk (VaR) measure comprising the expected and unexpected parts. We will omit here the discussion of risk-measures, though in market risk regulation the Basel Committee made a shift from the Value-at-Risk (VaR) to the Expected Shortfall (ES) one (BCBS, 2013).

$\displaystyle K\geqslant\textit{Risk}$

where $K$ is the bank capital, the most pressing ratio refers to the common equity tier one (CET1) capital (it includes initial stock, paid-in capital and retained earnings).

$\displaystyle K\geqslant\textit{EL}+\textit{UL}$ $\displaystyle\textit{K-EL}\geqslant\textit{UL}$ $\displaystyle\frac{\textit{K-EL}}{\textit{UL}}\geqslant 1$

Though it was possible to use the above general comparison of the residual capital amount (K-EL) versus the unexpected loss amount (UL), the regulator decided to introduce a $\theta$ multiplier as follows.

$\displaystyle\textit{CAR}=\frac{\textit{K-EL}}{\theta\cdot\textit{UL}}% \geqslant\frac{1}{\theta}$

For instance, for the Basel I purpose the multiplier was set equal to $\theta=$ 12.5, so that the CAR minimum was 8% $=\frac{1}{\theta}=\frac{1}{12.5}$ . As the reader may understand, such an operation does not make difference in essence. There is only one rationale coming to the author’s mind why such a multiplier came in. When observing the regulatory constraint in the form of minimum CAR level, the value of 8% is psychologically more appealing than 100%, or 1, as former is smaller than the latter (however, given the former implies increase of risk-measure by $\theta$ , both approaches yield the same).

In order to assure the financial stability system-wide the minimum capital requirements (CAR) are being constantly raised according to the following schedule from the start of Basel II in 2000s to the perspective treatment of Basel III in 2022, see Fig. 8.

Figure 8.

Evolution of Minimum CAR requirement.

CAR for tier 1 capital in Basel I was 4% and for the sum of the tier 1 and 2 capitals equaled to already mentioned 8%. CAR for the CET1 was de facto 2 percent during the Basel II era of 2004–06. Basel III required banks to raise it to at least 4.5 percent and to 12 percent with all three capital buffers fully phased in. For the details, please, refer to the Annex of (Caruana, 2010). Capital buffers include conservation, countercyclical, for the systemically important financial institutions (SIFIs) ones. Each costs 2.5 percent of RWA, see Basel III phase-in arrangements (URL: http://www.bis.org/bcbs/basel3/basel3_phase_in_arrangements.pdf). The stock markets’ recovery by 2013 might have relieved regulators’ concerns about the capital base. As a result, new capital requirements were introduced, known as total loss absorbing capacity (TLAC). In essence, the requirement was to raise the CET1 CAR to 18 percent of RWA (BCBS, 2016a, p. 10).

The last shortcoming of the IRB approach is the absence of regular calibration of the above described parameters R and M. The calibration is preserved disregarding the change in economic environment that occurred since the last parameter values calibration in 2004. Since then the list of the Basel Committee members presenting survey data to calibrate parameters increased from a dozen close to three dozens. Same time inputs to the Vasicek formula are required to be regularly validated and at least annually updated in case of the models’ forecasting ability deterioration. This means that given no changes in Basel III version of 2017, the overall risk assessment for regulatory purposes might be biased even though the inputs such as PD, LGD, EAD might be sufficiently correct and accurate.

3. Other additional shortcomings of the IRB approach

The two additional shortcomings of the Vasicek model are the following:

1.
Unbalanced treatment of PDs and LGDs; and
2.
Unbalanced treatment of EL and UL.

When we say unbalanced treatment, we mean that the capital requirements and the indication of the financial standing of the financial institution may vary because of the parameter manipulation whereas the actual change in the risk profile does not occur. This is important to understand for the regulators and the financial analysts covering the financial services’ sector and especially the banking one.
3.1 Unbalanced treatment of the PDs and LGDs

Let us consider the following example of a portfolio with 200 loans granted initially. Thus for the default definition of 90 days (standard one set in (BCBS, 2006) and (BCBS, 2017a)) past due the probability of default is 28%, whereas the one for 180 days (was used by some EU regulators having exercised its right for national discretion) equals to 23%. However, the expected loss is the same for any default definition, i.e., 22% or 45 non-performing loans out of 200 total ones (see Table 1 and Fig. 9a).

Table 1
Schedule for the hypothetical loan portfolio redemption

(1)	(2)	(3)	(4)	(5)	(6)
dpd	# fully redeemed loans	Cumul. Redeemed # loans	Lost $=$ 200 – (3)	PD $=$ (3)/200	LGD $=$ 45/(4)
1	100	100	100	50%	45%
15	10	110	90	45%	50%
30	9	119	81	41%	56%
45	8	127	73	37%	62%
60	7	134	66	33%	68%
75	6	140	60	30%	75%
90	5	145	55	28%	82%
105	4	149	51	26%	88%
120	3	152	48	24%	94%
135	2	154	46	23%	98%
150	1	155	45	23%	100%
165	0	155	45	23%	100%
180	0	155	45	23%	100%

Note: # – number of; PD $=$ number of cumulatively redeemed claims/200 loans; LGD $=$ 45%/number of lost loans.

Upper solid line at the Fig. 9b shows that the longer the default definition threshold is, the larger the minimum capital requirement is given the same expected loss amount (we assume this is an A-IRB bank). It comes from the fact that variation in LGD contributes more to the model than variation in PD.

Figure 9.

Capital requirements rise the longer the default definition is for varying LGD.

Lower dashed line at the Fig. 9b shows that within the F-IRB framework the longer default definition would result in smaller capital requirements. That could have been the reason why the regulator introduced ‘unlikely to pay’ (UTP) criteria (BCBS, 2006, p. 100. par. 453) to account for defaults earlier. However, at the A-IRB world UTP criteria triggering would result in less capital requirements all else being equal.

As we see from the Fig. 9, depending on the IRB approach in place – foundation or advanced – the UTP criteria may result in both higher and lower capital requirements. That is why we recommend to exclude UTP to base the credit risk capital requirements on the quantitative parameters such as 90 days past due only.

3.2 Unbalanced treatment of EL and UL

The breakdown into expected and unexpected losses is based on the idea that expected losses stand for the first moment of the loss distribution (BCBS, 2005a). The residual of VaR and EL is UL. Nevertheless, the accounting treatment does not impose any requirement on what should be deducted from capital in the numerator of CAR (currently it is EL) and what should be benchmarked against the residual amount of capital as the denominator of CAR (today it is $\theta\cdot\textit{UL}$ ). In practice it is impossible to differentiate which loss type (expected or unexpected one) did occur. All the bank or the regulator can do is to run a back-test of the actual losses against the total risk estimate (i.e. sum of EL and UL). Back-testing losses separately against either EL or UL only is methodologically inconsistent. This is why the bank might be prone to manipulation when choosing the breakdown into EL and UL. Here are two cases.

First, the Fig. 10 shows that in case the bank has capital in excess of the total risk (consider upper line), then for the bank it is more beneficial to allocate most risk to EL. It implies increase in CAR or granting more loans.

On the opposite, if the bank’s capital is less than its total risk (consider lower line at the Fig. 10), the bank may be incentivized to allocate most risk to the UL in order to demonstrate higher CAR. Policy implications here are to impose the credit risk regulation by merely comparing capital to the sum of the EL and UL to avoid possibilities for capital arbitrage.

Second, when one raises the minimum capital requirement (CAR), i.e. adjusts upwards right hand-side of the inequality ( $\frac{1}{\theta}$ ) by $\beta>1$ times, this is equivalent (if inversely engineered) to requiring the following

$\displaystyle K\geqslant\textit{EL}+\beta\cdot\textit{UL}$

This means that regulator does not require more capital against expected losses. Take the following example in Table 2. Let there be two banks. The total risk amount for the two banks is the same (Risk $=$ EL $+$ UL $=$ 100). Bank 1 is less risky. Most part of its total risk forms unexpected loss (90 out of 100). Bank 2 has larger portion of the total risk attributed to the expected loss (50 out of 100) and is thus considered a riskier one.

Table 2
Computational example for changes in CAR regulatory requirements impact

#	Category	Bank 1	Bank 2
1.	Total risk amount	100	100
2.	EL	10	50
3.	UL	90	50
4.	Risk Taken (qualitative judgment)	Low	High
5.	K	200	200
6.	CAR $=$ (K – EL)/(12.5 * UL)	17%	24%
7.	CAR min	OK	OK
8.	T0	8%	8%
9.	T1 $\left(\beta\cdot\textit{CAR\_min}\left(\textit{T0}\right)\right)$ where $\beta=$ 1.25	10%	10%
10.	CAR under Option 1: $EL+\beta\cdot\textit{UL}$ where $\beta=$ 1.25	13.5%	19.2%
11.	CAR under Option 2: $\beta\cdot\left(\textit{EL}+\textit{UL}\right)$ where $\beta=$ 1.25	13.3%	17.6%
12.	Drop in CAR from its current value in line (6), pp
13.	Option 1	$-$ 3.5	$-$ 4.8
14.	Option 2	$-$ 3.7	$-$ 6.4

Note: pp – percentage points.

Figure 10.

Capital Arbitrage Illustration when CAR varies because of EL-UL redistribution (total risk estimate being constant).

Suppose both banks have the same initial capital ( $K=$ 200). Then the capital adequacy ratio for bank 1 is 17% whereas for bank 2 it is 24%. Let the initial minimum CAR be 8% as set by the regulator. Both banks comply with the capital requirement. Suppose the regulator wishes to increase the level of the financial soundness. For this purpose it raises minimum CAR level from 8% to 10%, i.e. adjusts 1.25 times higher ( $\beta=$ 1.25; see line 9 in the Table 2). But such a regulatory change implies no changes for two banks from the stakeholder (e.g. financial analyst or the regulator) perspective as CAR does not change. What may seem to stakeholders is that bank 2 is more sound as its CAR remains higher (24% of bank 2 versus 17% of bank 1).

Nevertheless, if the regulator wishes to give a signal to the industry upon its regulatory reforms, it is recommended to use another approach, namely, option 1 (non-proportionate treatment of UL and EL, see line 10 in the Table 2) and option 2 (proportionate treatment of the EL-UL components, see line 11 in the Table 2). Then changes in regulation would signal to a market different financial soundness of the two banks. When CAR for bank 2 decreases more from such a reform than for bank 1, stakeholders may see that bank 2 takes on more risks, than the bank 1 (by $-$ 4.8 or $-$ 6.4 percentage points for bank 2 versus $-$ 3.5 or $-$ 3.7 for bank 1 for option 1 or 2, respectively).

4. Conclusion (policy implications)

To sum up the paper contribution to the field we may list three points:

We collected in one place the evolution of regulatory parameters used to govern the capital treatment for the credit risk within the internal-ratings-based (IRB) models (for concise summary, please, consult Annexes 1 and 2);

We collected the existing criticism of the IRB models in a single place;

We raised the shortcomings of the IRB approach that were not discussed before. It includes the following:

(a)

The current capital requirements can be manipulated by the choice of default definition. Particularly, allowing for the various unlikely to pay criteria may effectively decrease capital requirements. That is why we suggest only a single quantitative measure of 90 days past due to be used to assure level playing field and model output comparability between the banks.

(b)

The current capital requirements can be manipulated by the decomposition of total risk into the expected and unexpected parts. As a result, the bank may choose the composition of its loan book (i.e. the proportions of high and low risky borrowers) so that to either augment its CAR or to have the opportunity to offer more loans. Here are two suggestions. First, the capital regulation should target comparing total capital to total loss estimate without its decomposition into the expected and unexpected parts via CAR computation. Second, in order to produce information signals about the bank’s soundness the regulator should prefer adjusting exposure-specific risk-weights instead of adjusting overall CAR threshold. Thus a sectoral countercyclical capital buffer (BCBS, 2019b) should be preferred to the general one (BCBS, 2017b).

Footnotes

Acknowledgments

Author acknowledges the anonymous reviewer for the useful and detailed comments. The paper was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and supported within the framework of a subsidy by the Russian Academic Excellence Project ‘5-100’.

Annex 1. Asset Correlation function parameters evolution

#	Exposure type name	Year	$R_{0}$ , %	$R_{\textit{MIN}}$ , %	$R_{\textit{MAX}}$ , %	Coef. ( $w$ )	Ref.
1.	General (fixed)	2001	20	n/a	n/a	n/a	(BCBS, 2001b, pp. 36, par. 172)
2.	General (variable)	2002	n/a	10	20	$-$ 50	(Lopez, 2002, p. 18)
3.	Corporate	2003	n/a	12	24	$-$ 50	(BCBS, 2005c)
4.	Small and medium enterprises (SME)	2003	$-$ 4	12	24	$-$ 50	(BCBS, 2004)
5.	Highly-volatile commercial real estate (HVCRE)	2003	n/a	12	30	$-$ 50	(BCBS, 2004)
6.	Plastic cards (qualifying revolving retail)	2003	n/a	2	17	$-$ 35	(BCBS, 2004)
7.	Other retail	2003	n/a	2	11	$-$ 50	(BCBS, 2004)
8.	Plastic cards (qualifying revolving retail)	2004	4	n/a	n/a	n/a	(BCBS, 2004)
9.	Residential Mortgage (retail)	2004	15	n/a	n/a	n/a	(BCBS, 2004)
10.	Other retail	2004	n/a	3	16	$-$ 35	(BCBS, 2004)
11.	Systemically important financial institutions (SIFI)	2009	n/a	15 $=$ 1.25 * 12	30 $=$ 1.25 * 24	$-$ 50	(BCBS, 2009, p. 30)

Note: n/a – not applicable.

Annex 2. Summary of IRB credit risk regulation evolution

		Basel II						Basel III
	BCBS paper No.	50	ca02	cp3	107	118	128	d424
No.	Year	(BCBS, 1999)	(BCBS, 2001a)	(BCBS, 2003)	(BCBS, 2004)	(BCBS, 2005b)	(BCBS, 2006)	(BCBS, 2017a)
1.	IRB general	NO	$+$	$+$	$+$	$+$	$+$	$+$
2.	$\alpha$ multiplier (scaling factor)	1	1	1.5	1.06	1.06	1.06	1
3.	Granularity Adj.	- -	$+$	–	–	–	–	–
4.	LGD (F-IRB), %	–	50	50	45	45	45	40
5.	Double default	–	v01	v01	v01	v01	v02	NO
6.	Maturity Adj.	–	v01	v02	v03	v03	v03	v03
7.	Output floor (period)	–	Temporary for # of years (Y)					Constant after 3 years
			2Y	2Y	3Y	3Y	3Y
8.	Output floor ( $\phi$ ), % of RW_ST	–	–	90-80	95-90-80	95-90-80	95-90-80	95-90-80 – 72.5
9.	Confidence level, $1-p_{Y}$ , %	–	99.5	99.9

Note: v – version; # – number; Y – year.

Though the internal models (the IRB approach) were not yet allowed in (BCBS, 1999), a discussion paper (BCBS, 2000) soon followed to describe the banking practice and to lay the ground for its further regulatory approval.

(BCBS, 2017a) is effective from 2022. (BCBS, 2016b) discussed the option to forbid developing probability of default models for the low-default portfolios (LDP), but eventually the committee did not preserve such a restriction. Discussing challenges of the LDP modeling falls out of the scope.

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History of the Basel internal-ratings-based (IRB) credit risk regulation

Abstract

Keywords

1. Introduction

2. Literature review (Vasicek model)

1. Adjustment for recovery efficiency; 2. Maturity adjustment; 3. Granularity adjustment; 4. Double default treatment. 2.1 Baseline model

(1) Recovery efficiency; (2) Loan maturity; (3) Exposure amount; (4) Non-infinite-granularity of the loan portfolio; (5) Double-default; 2.2 Recovery efficiency

Table 1 Schedule for the hypothetical loan portfolio redemption

Table 2 Computational example for changes in CAR regulatory requirements impact

Footnotes

Acknowledgments

Annex 1. Asset Correlation function parameters evolution

Annex 2. Summary of IRB credit risk regulation evolution

References

1.
Adjustment for recovery efficiency;
2.
Maturity adjustment;
3.
Granularity adjustment;
4.
Double default treatment.

2.1 Baseline model

(1)
Recovery efficiency;
(2)
Loan maturity;
(3)
Exposure amount;
(4)
Non-infinite-granularity of the loan portfolio;
(5)
Double-default;

2.2 Recovery efficiency

Table 1
Schedule for the hypothetical loan portfolio redemption

Table 2
Computational example for changes in CAR regulatory requirements impact