The calibration of the IRB supervisory formula : a case study.
Date: 2023-11-10
Institutions: European Banking Authority
Subjects: Italy, bank, banking, case study, financial regulation, financial requirements, financial risk

EBA STAFF PAPER SERIES 

N. 17 – 09/2023 

THE CALIBRATION OF THE IRB 
SUPERVISORY FORMULA – A 
CASE STUDY 

by Simone Casellina, Fabio Salis, Giovanni Tessiore, Roberto Ugoccioni 
and Franco Varetto 

The EBA Staff Papers Series describe research in progress by the author(s) and are published to elicit 
comments and to further debate. Any views expressed are solely those of the author(s) and so cannot be 
taken to represent those of the European Banking Authority (EBA) or to state EBA policy. Neither the EBA nor 
any person acting on its behalf may be held responsible for the use which may be made of the information 
contained in this publication, or for any errors which, despite careful preparation and checking, may appear. 

 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA 

ABSTRACT 

The  level  of  capital  requirement  generated  by  the  IRB  approach 
depends crucially on the asset correlation, a parameter that enters the 
regulatory risk weight formula and is  determined by the Regulators. 
Several studies have estimated the asset correlations and found that 
the  empirical  values  are  materially 
lower  than  the  regulatory 
calibration  included  in  the  Basel  framework.  However,  the  simple 
comparison between different estimates of this parameter does not 
easily translates into a clear economic interpretation. In this paper, we 
use detailed data from Italian banks to show how to extract from the 
regulatory  risk  measures  easily  interpretable  figures  i.e.  the  Worst-
Case  Default  Rate  (WCDR)  and  the  Worst-Case  Loss  (WCL)  and  we 
show how the asset correlation influences these measures. We then 
provide  a  rationale  for  the  regulatory  calibration  in  terms  of 
corrections  to  well-known  limits  of  the  underlying  models  like  the 
assumption  of  perfect  granularity.  We  claim  that  our  approach  can 
provide a better understating of the IRB risk measures fostering their 
transparency and reliability but also simplifying the comparison among 
different  banks.  We  apply  the  proposed  approach  exploiting  some 
data sources (publicly available and proprietary). As the data used is 
mainly  referred  to  the  Italian  system  and,  in  particular,  to  only  two 
banks,  the  empirical  results  obtained  are  meant  just  to  provide  a 
practical example. 

KEYWORDS 

Bank Capital; Regulation; Basel 2; Credit Risk; Asset Correlation; Value-at-Risk 

JEL CODES 

C15; G21; G32

Page 2 

EBA STAFF PAPER SERIES 

THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

1.  Introduction1 

Motivation 

About 20 years ago, the Basel Committee on Banking Supervision (BCBS) introduced into the system of prudential 
regulation  for  banks  a  risk-based  framework  (named  Basel  II),  allowing  financial  institutions  to  use  internal 
models to calculate minimum capital requirements for major risk types.  For credit risk, the BCBS proposed a 
measure of regulatory capital based on risk measures internally estimated by the banks, and presented the first 
version of the Supervisory Formula (SF) i.e. a “closed” formula aimed at replicating the results of the portfolio 
models developed by major investment banks and consulting firms (mainly from the US). The BCBS opted for a 
structural model and, in particular for the gaussian2 Single Risk Factor Model derived from the Merton-Vasicek 
(MV) model. The banking regulation on credit risk issued by the BCBS has undergone a progressive evolution 
starting from the initial amendment proposal submitted for consultation in 19993, which proposed the use of 
systems based on internal rating models (IRB) for the purpose of calculating the capital requirement. Detailed 
clarifications were provided by the consultation paper published in January 20014. 

The basic idea of the regulation consists in establishing the minimum amount of capital that the bank must hold 
to protect depositors (and the entire economic system)  from the insolvency of the bank itself. The minimum 
capital has been quantified in different ways depending on the versions of the proposed regulation, but all share 
the principle of protecting the bank from adverse credit events (peak losses) that may affect the loan portfolio 
up to a certain confidence interval over the time horizon of one year. The worst-case credit loss of the portfolio 
obtainable from the MV model is based on the probability of default conditional on the realisation of an extreme 
event, which has a small but not zero probability of occurrence. 

The level of capital requirement generated by the IRB approach depends crucially on the asset correlation, a 
parameter that enters the regulatory risk weight formula and is calibrated by the Regulator. Estimating the asset 
correlation  parameter  is  challenging,  however,  according  to  Chernih  (2006),  there  appears  to  be  a  growing 
consensus in the literature on the range for asset correlations and the current regulatory correlations are larger 
than the empirical results reported in this literature. However, differences in the estimating approaches and in 
the type of data, may produce quite different results. The simple comparison of different estimated values of 
this parameter does not provide an easy interpretation. For example, what it does entail in practice to assume 
an asset correlation equal to 1% or 20% is not self-evident. 

In this paper, we offer a different perspective that allows us to evaluate the adequacy of the asset correlation 
estimate for a given portfolio without the need to make comparisons with other estimates, perhaps based on 
data  that  is  not  representative  of  the  portfolio  in  question.  We  show  that  the  IRB  credit  risk  measure  is 
substantially  obtained  by  rescaling  the  distance  between  the  expected  probability  of  default  (PD)  and  the 
stressed PD. The regulatory IRB Supervisory Formula5 (SF hereinafter) can be seen as an algorithm whose main 
purpose is to provide the stressed PD given the expected PD (estimated by the banks) and the asset correlation. 
The stressed PD can then be compared with the observed default rates. This comparison provides a simple way 
to evaluate the realism (and the level of conservatism) implied by a given level of asset correlation. We then turn 
our attention to the minimum regulatory capital requirements. We show that this quantity can be seen as the 

1 The opinions expressed are those of the Authors and do not involve responsibility of the institutions. 
2 In practice, the external/systemic factor is assumed to be a standard normal random variable 
3 BCBS (1999a) 
4 BCBS (2001a). in preparation for this regulatory proposal, a survey of the state of the credit risk models available was carried 
out by the Basel Committee, BCBS (1999b). 
5 Article 153 and 154 of the Regulation EU 2013/575 (the CRR) 

Page 3 

EBA STAFF PAPER SERIES 

 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

estimated worst-case loss (WCL) given the confidence level and how this measure is influenced by the stressed 
PD.  This  enables  us  to  make  comparisons  with  different  risk  measures  obtained  by  relaxing  some  of  the 
hypothesis underlying the IRB approach and this in turn permits to quantify the role of such hypothesis. 

Structure of the paper 

The  rest  of  the  paper  is  organised  as  follows:  Section  2  provides  a  review  of  the  literature  regarding  the 
estimation of the asset correlation. In Section 3, four different sources of data (two publicly available and two 
propriétaire data) are used to estimate the asset correlation. The data is mainly referred to the Italian system 
and in particular to two Italian banks. We find values for the asset correlations in line with the existing literature. 
In Section 4 we show the effect of asset correlation on the determination of the stressed PD. The latter is then 
compared with the worst-case level of default rates actually observed in a time frame which includes the Great 
Financial Crisis of 2007-2008. We argue that this comparison enables to appreciate, in a more natural way, the 
level  of  conservatism  implicit  in  the  regulatory  calibration  of  the  asset  correlation.  In  Section  5  we  turn  the 
attention to the regulatory minimum capital requirements, and we show that the level of conservatism of the 
stressed PD can be seen as covering risk components that the Basel Committee avoided formalising for the sake 
of simplicity and due to lack of data. We empirically verify this hypothesis by processing the data of real portfolios 
of  two Italian banks.  In detail,  we  have  analysed  the difference between  the  measure  of  the  worst-case  loss 
provided by the SF that uses the regulatory asset correlation and the measure that would be obtained using the 
asset correlation estimated based on real default data but relaxing some of the assumptions underlying the IRB 
model. The results of this empirical exercise cannot be easily extended given the limited sample (two banks) at 
our  disposal.  However,  our  findings  suggest  that  by  expressing  the  regulatory  risk  measure  in  terms  of  WCL 
(which  should  be  covered  by  provisions  and  capital),  instead  of  unexpected  loss  multiplied  by  12.56  and 
disentangling its implicit components, could greatly simplify the comprehension of these metrics by the market 
participants. Transparency and the possibility of making comparisons among banks would benefit. 

2.  Literature Review 

The default correlation 

In his book on credit-risk modelling, Bolder (2018) states that the dependency on default correlation is at the 
heart of credit-risk modelling. The imposition of even quite modest amounts of default dependence has dramatic 
effects on risk estimates. Loosely speaking, without the hypothesis that defaults are somehow correlated, the 
number of defaults generated by a portfolio of 𝑁 borrowers could be represented with a binomial distribution7 
of  parameters  𝑁  and  𝑝  where  𝑝  is  the  probability  that  a  single  borrower  incurs  a  default.  In  that  case,  the 
expected value of the default rate (number of defaults divided by 𝑁) would be 𝑝 and the variance 𝑝(1 − 𝑝)/𝑁.  

The fact that the variance decreases with 𝑁 implies that, for 𝑁 sufficiently large, the variance – and consequently 
the unexpected loss – would go to zero. In other words, assuming default independence, for sufficiently large 
portfolios, all idiosyncratic risks can be diversified completely. On the other hand, with even a limited amount of 
correlation  among  the  defaults,  a  systematic,  non-diversifiable  source  of  risk  is  introduced.  From  a  risk 
management perspective,  a significant  underestimation  of  the credit  risk  exposure can  derive  from  omission 

6 This would also align the prudential treatment of the credit risk with that of the market risk. 
7  Let  𝐷𝑖  represent  the  dichotomous  variable  assuming  value  1  if  the  𝑖 − 𝑒𝑠𝑖𝑚  borrower  defaults  and  0  otherwise.  Then 
𝔼(𝐷𝑖) = 𝑝  is  the  expected  probability  of  default.  Assume  that  in  a  portfolio  there  are  𝑁  borrowers  having  the  same 
probability of default. Then, under the hypothesis that the defaults are independent, the total number of defaults is the sum 
of 𝑁 independent Bernoulli variables so ∑ 𝐷𝑖

 is a binomial random variable with parameters 𝑁 and 𝑝. 

𝑁
𝑖=1

Page 4 

EBA STAFF PAPER SERIES 

 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

(Das, 2006) or underestimation (Jorion, 2006) of the default correlation. The Annex presents a simple example 
built with real data showing the difference between the actually observed variance of a series of default rates 
and the theoretical variance obtained under the hypothesis of independence. 

In credit risk modelling, the dependency on the default correlation is typically introduced in two different ways 
(Bluhm,  2016):  reduced  form  and  structural  models.  The  aim  of  these  models  is  to  obtain  the  theoretical 
probability  distribution  of  the  defaults  and  to  derive  from  it  the  risk  measures.  In  reduced-form  models,  the 
defaults of the borrowers are seen as exogenous events occurring with a given probability and a given default 
correlation. In structural models, the defaults are made endogenous, in the sense that the model explains their 
occurrence through a mechanism and the risk measures are derived from the mechanism that generates the 
defaults. It is under this second type of models that the asset correlation plays a role. 

A third approach, quite different from the reduced and structural models, is the econometric approach where 
we try to link the historical series of default rates to macroeconomic variables, such as the GDP. In these models, 
an effort is made to identify the external factors that determine the dynamics of the default rates, while the risk 
measure is obtained by stressing these external factors, for example by assuming a strong reduction in GDP. 

The reduced form and the structural models and the econometric models share the goal of providing a measure 
of stressed default rate, but they arrive at this result in quite different ways. The reduced form and the structural 
models  allow  to  construct  the  entire  probability  distribution  of  the  default  rates  and  to  derive  from  this  the 
stressed default rate (the WCDR) given the desired confidence level. With econometric models, on the other 
hand, the stressed default rate is obtained by providing a stressed scenario.  

Under the structural framework, the default of a borrower is an event connected with an external factor, which 
is in common with all the borrowers of the portfolio. It is through this common dependence on the external 
factor that the defaults are correlated, while the strength of the dependence between the common factor and 
the probability of default is named asset correlation. In other words, the hypothesis of default independence is 
substituted  by  the  hypothesis  of  correlated  defaults  and  in  turn  this  correlation  stems  from  the  connection 
between the probability of default and a factor in common with all the borrowers that can be thought of as the 
state of the economy. 

Stock prices or default rates 

The asset correlation can be estimated either directly, by exploiting equity prices, or indirectly (passing through 
the  default  correlation),  by  exploiting  data  on  default  rates.  Using  default  data,  one  can  estimate  default 
correlations directly and then retrieve the asset correlation. Using asset value data, one can directly estimate 
asset correlations and then obtain the default correlations. 

Using equity prices appears more appealing for several reasons: the original derivation of the Merton model  
refers to companies whose equities are traded on the financial markets; data about equity prices is more easily 
found while banks’ default data is not public, and the time series is usually longer8. However, it must also be 
considered  that  equity  prices  are  usually  referred  only  to  large  corporates  while  banks’  portfolios  include  a 
substantial number of SMEs9 and exposures towards households. In general, market-based data can be rationally 
used whenever there is a liquid and reasonably efficient secondary market. 

8 The Basel 2 default definition has been implemented by the banks since 2006, which implies that the longest time series 
available with a homogeneous definition of default spans 15 years. 
9 From Piccone (2019) pag 78, the share of corporates with more than 250 employees between the non-financial firms is 
equal to 0.47% in Germany, 0.14% in France and 0.09% in Italy while the share of enterprises with less than 10 employees is 
82% in Germany, 95% in France and in Italy. 

Page 5 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

The  main  difference  between  estimates  obtained  through  default  data  and  market  data  is  the  sample 
composition. Market data is available only for publicly traded enterprises, and these are usually large corporates. 
With default data it is possible to estimate the default correlation (and derive the asset correlation) for any sector 
including SMEs and households. Asset correlation estimates based on market data tend to be larger than those 
based on default data. The higher estimated asset correlation stemming from market data could be led by the 
fact that large corporates are more subject to the financial markets where the pressures of market participants, 
herd behaviour and the tendency to anticipate the effects of shocks, may lead to a higher dependency from 
common factors. These aspects may lead to the question of whether it is fair to derive from market data-based 
estimates the measure of a crucial parameter, such as the asset correlation for sectors where the idiosyncratic 
components can be expected to be structurally (at least, in the short term) more important than the external 
conditions. 

Duellmann (2010) investigates why estimates of asset correlations based on equity prices tend to be considerably 
higher than those based on default rates. It is shown that differences between these estimates are caused by a 
substantial downward bias of the estimates based on default rates. More specifically, by means of a simulation 
study, it is shown that the direct estimation of asset correlations from equity returns is superior to an estimation 
from default rates, both in terms of bias and efficiency. The problem of the downward bias in the estimation of 
the asset correlation based on default rates, has been discussed also by Gordy (2002) and Resti (2008). 

From  a  prudential  perspective,  the  possibility  that  the  asset  correlation  may  not  be  stable  over  different 
economic scenarios is crucial. In Lee (2011) it is found that the  asset correlations are asymmetric and have a 
procyclical dynamic: they tend to rise during economic downturns but decline during economic upturns. Working 
with  stock  prices  could  enable  to  better  explore  these  dynamics.  However,  the  relationship  between  the 
probabilities of default (implicit in the CDS) and the actual default rates must be further investigated. 

Asset correlations: estimates vs regulatory 

For Italian banks, Sironi (2003) obtained estimated asset correlations consistently lower than the value proposed 
by the BCBS; exploiting publicly available data stemming from the Italian Credit Register, Resti (2008) estimated 
for  the  non-financial  sector  (Corporates  and  SMEs)  a  value  equal  to  2.4%;  Curcio  (2011)  estimated  the  asset 
correlations for firms classified in different buckets of size and area obtaining values that are in the range of 0.5% 
and 5%. For German banks Duellmann (2013), using a dataset including about 250 thousand of borrowers per 
year (SMEs and Large Corporates), obtained asset correlations not higher than 2%. Henneke (2006), by examining 
a typical German SME portfolio of Small and Mid-Sized Enterprises, also found asset correlations to be lower 
than  those  assumed  in  the  capital  accord.  In  a  study  of  the  Bank  of  Japan,  Hashimoto  (2009),  the  estimates 
referred  to  large  and  medium-sized  corporates,  SMEs  and  households  and  to  different  industrial  sectors, 
showcased that asset correlations never exceeded the value of 4.5%. In his survey, Frye (2008) reports the results 
in Table 1 below. It can be noticed that the asset correlations obtained from default data, as in our  case, are 
systematically lower than the ones obtained from asset value data.  

Page 6 

EBA STAFF PAPER SERIES 

 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Table 1: Estimated asset correlations reported in Frye (2008) 

As regards, it is also interesting to mention the results obtained by Moody’s/KMW researchers. In detail, in Zhang 
(2008) the estimated values are in line with the Bcbs proposals, the data used are referred to publicly traded U.S. 
non-financial firms,  in  practice  U.S.  Large Corporates.  Chernih (2010),  by summarizing different  studies  using 
rating and default data from rating agencies suggested the asset correlation parameter for the corporate sector 
to be around 10%. Siarka (2014) exploiting data obtained from financial institutions operating in Poland obtains 
for Retail loans estimates of the asset correlation in the range 1% – 3%. The author concludes that: << […] it 
appears that the correlation between retail borrowers’ assets is very low, much lower than that recommended 
under the IRB approach>>. Geidosch (2014)  uses several estimation approaches to obtain an estimate  of the 
asset correlation for US residential mortgage-backed security transactions, and the result is about 6%. Duellmann 
(2014) highlights that a possible underestimation of the asset correlation could result from the limited length of 
the time series. However, in that paper, the estimates are obtained with data covering 7 years and not including 
a complete business cycle. Dietsch (2016), by using a unique and comprehensive data set related to France and 
Germany capturing a significant part of lending towards SME and large corporates, obtained estimates of the 
asset correlation in the range of 0.5% - 2%. In a recent paper, Di Clemente (2020), by exploiting one of the data 
sources that we used for this paper (i.e. the historical time series of default rates  for Italy that can be freely 
downloaded from the website of the Bank of Italy) estimated asset correlations in the range of 1%-3%.  

In Resti (2008) it is studied the effect on the estimation of the asset correlation of working with default rates 
stemming  from  samples  of  borrowers  that  are  not  homogeneous  in  terms  of  probability  of  default  and  he 
concludes that the effect is a downward bias. These considerations are in line with those in Gordy (2002) about 
the possible sources of bias in the estimation of the asset correlation. For example, Blumke (2017), while stressing 
the importance of working with homogeneous portfolios, analyzed Standard and Poor’s rating and default data 
concluding that asset correlation parameters for banks and several industry sectors, are close to the regulatory 
value or even exceed it. 

However, it must be noticed that data exploit in this paper spans at least 16 years (from 2007 to 2022), including 
the effect of the Great Financial Crisis. Moreover, we estimated the asset correlation at rating grade level and 
this should entail a good level of homogeneity.  The item of the possible underestimation of the asset correlations 
is resumed further in this paper where a simple, heuristic solution is also suggested. 

Lastly, it is worth mentioning the possibility that the asset correlation is not stable over the economic cycle. One 
of  the  Great  Financial  Crisis  aftermath  is  that  correlations  between  assets  in  periods  of  stress  can  differ 
substantially from those seen in normal circumstances. The problem of “correlation breakdown” during periods 

Page 7 

EBA STAFF PAPER SERIES 

 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

of greater volatility is well known. The subprime CDSs crisis was induced largely by badly calibrated mortgage 
default  correlations.  Also,  with  reference  to  1998,  when  the  market  conditions  were  heavily  stressed,  a 
comprehensive study (BIS (1999)) found that the average correlation between yield spreads rose significantly 
during the crisis.  

3.  Estimating asset correlations 

Data 

In the following sections, we exploit four data sources. For all of them, the length of the time series ensures the 
full coverage of different economic cycles, including the effect of the 2008 financial shock and the consequent 
2011-2012 European sovereign debt crisis. The magnitude of the 2008 crisis is clearly observable in the default 
rates (see Figure 2). All the data sources are referred to non-financial corporations. 

The first data source is the time series of the default rates published by Moody’s. The observation period dates 
to the 80s (so more than 40 years are covered) it is also available the distinction between Investment and not-
Investment grades. The main con of this data source is that it refers mainly to US large enterprises. 

The second data source is made available by the Bank of Italy, and it is based on the Italian Credit Register10. The 
period covered is 2006 – 2022 (17 years). The data refers to practically all Italian non-financial corporations. The 
default definition is based on the Regulatory definition of default (Article 178 Regulation (EU) 575/2013) that in 
turn is in line with the Basel 2/3 rules. This definition is wider than the insolvency legal criterion. 

The  third  and  fourth  sources  are  not  publicly  available.  Cerved  Group  S.p.A.  is  a  group  that  operates  as  a 
commercial  information  agency:  it  assesses  the  solvency  and  creditworthiness  of  companies,  monitors  and 
manages credit risk and defines marketing strategies. Cerved operates as a rating agency through Cerved Rating 
Agency S.p.A. We exploit Cerved data referred to non-financial corporations segmented in rating grades. The 
main con, in this case is the definition of the default that, on the contrary for the other data sources, is not aligned 
with the Basel definition. In the Cerved database, the judiciary definition of default is used. 

Lastly, we exploit the data referred to the regulatory asset class Corporates of two Italian multiregional banks 
whose internal credit risk models have been authorized for their use for regulatory purposes and are currently 
in place. This is the most granular and complete data that we use in this paper. Clearly, it is referred to just two 
particular banks and even if these two are quite differentiated by size and regional areas of main interest, they 
still do not constitute a representative sample of the underlying banking system. In detail, two different data sets 
were provided. The first data set (DS1) provides the time series of the yearly default rates observed for each 
rating grade between 01/01/2007-01/01/2022 of the banks’ master scale and it is used to estimate the asset 
correlation. The second data set (DS2) contains data at the borrower level at a given (not specified) reference 
date.  For  each  borrower,  this  data  set  contains  the  exposure  at  default,  the  estimated  LGD  (inclusive  of  the 
downturn effect), the rating grade and the associated probability of default and the RWA computed by the bank 
along with the SF. The DS2 is used to compute the risk measures exploiting the SF both with the regulatory and 
the estimated asset correlation. It is also used to obtain the risk measures through Monte Carlo simulation.  

10 https://www.bancaditalia.it/statistiche/raccolta-dati/centrale-rischi/index.html?com.dotmarketing.htmlpage.language=1 

Page 8 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Table 2: Data Sources Used11 

Estimates 

There exist several methods that can be exploited for the estimation of the asset correlation (R), such as the 
likelihood  method  which  permits  a  direct  estimation  of  the  R  parameter,  or  the  method  of  moments,  which 
provides an estimation of the default correlation, from which in turn the R is derived. We used eight different 
estimation approaches to estimate the asset correlation parameter. The estimation was done at the aggregate 
level for the first and second data sources, while for the third and fourth data sources, the estimation was done 
for  each  rating  grade.  The  chapter  9  of  Baesens  (2016)  provides  details  about  all  but  one  these  estimation 
approaches together with SAS codes implementing them. The estimator ML2 is the same used in Dietsch (2016). 

The results referred to one of the four data sources are shown in the table below. The probability of default (PD) 
associated with each rating grade was computed as the simple average of the annual default rates. In the second 
column the value of the R parameter obtained with the regulatory expression (7) is reported. The difference 
between the regulatory and the estimated value of the asset correlation is quite wide with all the estimators. In 
particular, for lower levels of the PD the difference is higher. For example, for the IG rating grade of Mood’y data, 
the regulatory asset correlation is about 23.5% against 12.7% obtained with one of the estimators. 

Indeed, the Regulatory calibration is obtained under the assumption that lower PD levels are connected with 
higher levels of asset correlation. The effect of this assumption is evident in Figure 1 which represents the series 
of the regulatory and estimated asset correlation for the specific case of Bank2. The decreasing trend of R with 
respect to the PD is observed also in the estimated values but only up to a certain PD threshold beyond which R 
starts to grow again. 

The values we obtain range between 0.3% and 12% in line with Gordy (2002) and Dietsch (2004). However, we 
claim  that  the  simple  comparison  between  different  levels  of  asset  correlations  does  not  provide  sufficient 
interpretative elements. Consider the case of Bankit data. The Regulatory asset correlation is equal to  13.9% 
while the higher estimated value is 3.7% (ML1). The difference is remarkable, but it is difficult to associate these 
numbers with any concrete interpretation, so it is not easy to understand which of the two is more realistic. The 
next Section aims at providing an interpretative scheme in which to insert these numbers. 

11 The Cerved model has 10 rating grades. We aggregated some of them for this analysis. 

Page 9 

EBA STAFF PAPER SERIES 

rangeTMoody's1981-2022422 gradesMarketTraded CorporatesBankit2006-202217NoBasel 2/3Non Fin. CorpsCerved2008-2022158 gradesJudiciary Non Fin. CorpsBank12007-2022169 gradesBasel 2/3CorporatesBank22009-20221415 gradesBasel 2/3CorporatesDefinition of defaulttime seriesRatingsSourcePortfolio 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Table 3: Regulatory and estimated asset correlations for each rating grade 

      Source: own elaboration 

Page 10 

EBA STAFF PAPER SERIES 

MM1MM2MM3MM4MM5ML1ML2Cerved30.05%23.68%0.3%0.7%0.2%0.7%0.2%0.9%0.4%40.18%22.97%0.6%0.8%0.5%0.8%0.5%0.7%0.4%50.52%21.25%1.3%1.4%1.2%1.3%1.2%1.2%0.9%61.36%18.07%2.5%2.6%2.4%2.3%2.4%2.4%1.5%73.19%14.43%2.9%2.9%2.7%2.6%2.7%2.6%1.5%86.14%12.56%2.6%2.7%2.5%2.2%2.5%2.6%1.3%912.79%12.02%2.1%2.1%1.9%1.7%1.9%2.0%1.2%1042.23%12.00%2.4%2.5%2.2%2.0%2.2%2.5%1.9%Moody'sIG0.09%23.47%11.1%12.4%10.7%12.7%10.5%7.3%7.2%SG4.26%13.42%7.2%7.4%6.9%7.3%6.9%7.5%5.8%Bankit3.70%13.89%3.4%3.4%3.2%3.1%3.2%3.70%1.0%Bank110.06%23.64%3.4%12.2%1.4%11.0%1.2%1.4%0.4%20.17%23.02%4.1%6.9%3.3%6.3%3.5%5.0%5.1%30.34%22.14%0.7%2.3%0.5%2.1%0.5%3.6%4.1%40.69%20.48%1.6%2.4%1.5%2.3%1.5%2.8%2.3%51.30%18.26%1.0%1.5%0.9%1.4%0.9%1.8%1.2%62.54%15.37%1.5%1.9%1.4%1.6%1.4%2.0%1.6%75.09%12.94%4.3%4.6%4.0%3.5%4.0%5.0%3.8%810.85%12.05%5.3%5.4%4.9%4.2%4.9%5.7%4.1%930.13%12.00%4.4%4.6%4.0%3.8%4.0%4.6%3.1%Bank210.21%22.83%5.0%6.6%4.5%6.3%4.5%7.0%7.1%20.35%22.06%1.5%2.3%1.2%1.8%1.2%3.0%3.0%30.37%21.99%3.3%3.9%3.0%3.7%3.1%4.5%3.1%40.43%21.69%3.6%3.8%3.3%3.6%3.3%4.0%3.5%50.70%20.47%4.0%4.2%3.8%4.0%3.8%3.7%1.7%60.99%19.33%3.5%3.6%3.2%3.5%3.2%3.5%1.7%71.46%17.78%2.2%2.3%2.0%2.1%2.0%2.3%2.1%82.21%15.98%2.0%2.1%1.8%2.0%1.8%1.9%1.5%93.23%14.39%2.2%2.3%2.0%2.2%2.0%2.0%1.2%106.01%12.60%2.9%3.0%2.7%2.9%2.7%2.6%2.0%1110.20%12.07%2.9%2.9%2.6%3.0%2.6%2.6%2.2%1215.69%12.00%2.7%2.8%2.4%2.8%2.4%2.5%2.1%1320.45%12.00%4.0%4.1%3.7%3.6%3.7%3.9%2.5%1424.84%12.00%4.4%4.5%4.0%3.6%4.0%4.3%2.8%1531.90%12.00%4.2%4.6%3.8%3.4%3.8%4.6%2.6%Estimated RMethod of MomentsMaximum LikelihoodRegulatory RPDRatingData Source 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Figure 1: Regulatory vs Estimated (ML1) values of R, Bank 2 

      Source: own elaboration 

4.  The IRB and the stressed PD 

Credit risk models: WCL and WCDR 

In general terms, the risk measure, also known as Unexpected Loss (UL), is obtained from the difference between 
the Worst-Case Loss (WCL) and the expected loss (EL) that a portfolio can generate. The expected loss is obtained 
by  multiplying  the  expected  values  of  the  probability  of  default  (PD),  the  loss-given  default  (LGD)  and  the 
exposure at default (EAD12). The WCL represents the highest loss that is expected to be observed with a given 
level of confidence and its computation would require to know highest levels that are expected for the PD, LGD 
and EAD. A simplification is obtained by keeping constant the LGD and EAD and considering only the difference 
between  the  highest  expected  probability  of  default  (the  Worst-Case  Default  Rate,  WCDR)  and  the  average 
probability of default. 

𝑈𝐿 = 𝑊𝐶𝐿 − 𝐸𝐿 = 

12 In general, the three terms PD, LGD and EAD could be not independent. In this case the average of the product does not 
necessarily  coincide  with  the  product  of  the  averages.  However,  it  is  possible  to  assume  that  the  three  variables  are 
conditionally independent. In other terms, their reciprocal dependency is shaped by a common factor and once this factor is 
fixed (realised) the variables are independent. 

Page 11 

EBA STAFF PAPER SERIES 

 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

= 𝑊𝐶𝐷𝑅 ∗ 𝐿𝐺𝐷 ∗ 𝐸𝐴𝐷 − 𝑃𝐷 ∗ 𝐿𝐺𝐷 ∗ 𝐸𝐴𝐷 = 

    = [𝑊𝐶𝐷𝑅 − 𝑃𝐷] ∗ 𝐿𝐺𝐷 ∗ 𝐸𝐴𝐷                                                                          (1) 

Consider the simple introduction to credit risk modelling provided by the Chapter 15 - Section 8 of Hull (2015) 
where it is stated that under the gaussian Merton-Vasicek (MV) model, the Worst-Case Default Rate (WCDR), 
defined as the (1 − 𝛼) − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 of the distribution of the default rates, can be computed as: 

𝑊𝐶𝐷𝑅 = 𝛷 [

𝛷−1(𝑃𝐷)+√𝑅𝛷−1(𝛼)
√1−𝑅

]                  

                                  (2) 

where 𝑃𝐷 is the expected probability of default and 𝑅 is the asset correlation. In practice, the WCDR is a given 
percentile  of  the  default  rate  distribution:  ℙ(𝐷𝑟 > 𝑊𝐶𝐷𝑅) = 𝛼.  This  expression  is  obtained  under  the 
hypothesis that the default of a borrower is triggered by another variable that can be either observable (the asset 
value of a firm) or latent (i.e. not observable variable and generically referred to as the creditworthiness). 

The IRB model 

The bulk of the BCBS proposal was the definition of the maximum or worst-case loss attainable given the level of 
confidence. If the LGD and EAD are defined as constants, the only variable is the probability of default (PD) and 
the 𝑊𝐶𝐿 is equal to the product EAD*LGD*stressed PD where the stressed PD is provided by WCDR formula (1). 
Around  this  general  framework,  various  proposals  have  been  articulated  in  the  context  of  the  consultation 
process.  In  the  2001  proposal,  in  the  risk  weighting  formula  appeared  the  following  expression  (see  BCBS 
(2001a)): 

                                                        𝛷[1.118 ∗ 𝛷−1(𝑃𝐷) + 1.288]                                                                         (3) 

Notice that the (2) is a version of the (1) i.e. the MV formula for obtaining the stressed PD. Notice also that 1.118 
= 1/√1 − 0.2 and 1.288 = √0.2/(1 − 0.2)Φ−1(0.995). This implies that the asset correlation was set at 20% and 
the confidence level at 99.5%.  

Despite its centrality, the BCBS decided not to allow banks to use their own estimate of the asset correlation, 
mostly  because  the  estimation  of  correlation  parameters  for  defaults  was,  at  the  time,  an  area  of  evolving 
empirical and theoretical research, but also because the possibility to set the correlation values was seen as a 
policy instrument to be used for compensating for known limits of the models. It is also relevant to know that in 
this first proposal, the weighting formula was defined without subtracting the expected loss. In practice it was 
required to cover with capital the entire WCL. 

Later  and  in  response  to  feedback  from  practitioners,  the  Bcbs  proposed  an  alternative  formula  for  capital 
calculation, where the asset correlation was a decreasing function of the borrowers’ probability of default and, 
for  the  firms,  an  increasing  function  of  their  dimension.  For  the  calibration  of  the  asset  correlation  for  the 
Corporates portfolio, the BCBS applied two approaches – one direct and another survey-based or indirect. 

Under the indirect approach, the BCBS collected from major banks around the world, information regarding their 
internal economic capital allocations against large corporate loans. For each institution, the data was used to 
estimate  the  implied  risk  weights  (i.e.  relative  economic  capital  requirements)  that  each  bank  attributed  to 
corporate loans having particular PD, LGD and maturity configurations. The BCBS also undertook a number of 
studies  to  independently  estimate  appropriate  risk  weights  for  large  corporate  loans  using  formal  credit  risk 
models (the direct approach)13. It is worth point out that the BCBS mentioned specifically that these studies (both 
the direct and the indirect approach) has been conducted only on data referred to large corporates.  

13 See the paragraphs 166-169 of BCBS 2001 b 

Page 12 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

The final proposed expression for the computation of the risk measure, also known as SF, can be expressed as 
follows14 (for example as expressed in the Article 153 (1) of the CRR): 

                                 𝑅𝑊 = {𝐿𝐺𝐷 ∗ 𝛷 (

(𝑃𝐷)

𝛷−1
√1−𝑅

+ √𝑅𝛷−1

(0.999)

√1−𝑅

) − 𝐿𝐺𝐷 ∗ 𝑃𝐷} ∗ 1+(𝑀−2.5)∗𝑏

1−1.5∗𝑏

∗ 1.06 ∗ 12.5             (4) 

Having in mind the expression (1) it can be easily verified that the SF can be expressed as:                          

                                                     𝑅𝑊 = {𝐿𝐺𝐷 ∗ 𝑊𝐶𝐷𝑅
⏟        
𝑤𝑜𝑟𝑠𝑡 𝑐𝑎𝑠𝑒 𝑙𝑜𝑠𝑠

− 𝐿𝐺𝐷 ∗ 𝑃𝐷
⏟      
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑜𝑠𝑠

} ∗ 𝜓                                                             (5) 

Once 𝛼 has been set to 0.999 and ψ =

1+(𝑀−2.5)∗𝑏

1−1.5∗𝑏

∗ 1.06 ∗ 12.5. 

It can be also noticed that the LGD parameter is the same whether associated with either the average probability 
of default or the (1 − 𝛼) − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 of the distribution of the default rates. This observation is confirmed by 
Article 158 of the CRR which states that the calculation of the expected loss shall be based on the same input 
figures of PD, LGD and exposure value which are used for the computation of the risk weights. This implies that 
the  expected  loss  is  computed  with  the  downturn  LGD.  This  peculiarity  is  also  highlighted  in  a note  by  BCBS 
(2005b) where it is said that in theory the expected loss should be associated with the expected LGD and not 
with the stressed LGD. It can be imagined that this approach depends on the difficulty encountered by banks, at 
least  in  the  early  years,  in  distinguishing  the  downturn  LGD  from  the  expected  LGD.  With  this  in  mind,  the 
expression (4), i.e. the SF, can be further simplified as follows: 

                                                                   𝑅𝑊 = {𝑊𝐶𝐷𝑅 − 𝑃𝐷} ∗ 𝐿𝐺𝐷 ∗ 𝜓     

     (6) 

This expression showcases the risk weight is essentially obtained as the difference between the WCDR, which is 
the (1 − 𝛼) − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 of the distribution of the default rates, and the expected value of the default rate 
scaled by the factor 𝐿𝐺𝐷 ∗ ψ. 

The proposed weighting formula is based on models of the MV type with a single systemic factor. It was Gordy 
(2003) who clarified the conceptual basis behind this formula. One of the intentions was to make the capital 
requirement depend solely on the characteristics of the individual debtor  whilst being independent from the 
portfolio composition. Gordy has shown that this result is obtainable only if the systemic factor is only one and 
if the portfolio is granular i.e. there is no concentration. 

Since actual banking portfolios do not have the granularity characteristics required by the theoretical model, 
Gordy  (2004)  proposed  a  granularity  adjustment  as  the  final  step  in  calculating  the  portfolio  credit  VaR. 
Granularity has to do with the degree of concentration of credit exposures: the finer the granularity of a portfolio, 
the greater the diversification and elimination of the idiosyncratic risk component of credit positions. Consistent 
with  the  Gordy  scheme,  the  2001  proposal  of  the  Committee  provides  for  an  important  adjustment  for 
granularity  (Granularity  Adjustment)  to  take  into  account  the  fact  that  bank  portfolios  very  rarely  have  the 
characteristics of perfect granularity. 

With the proposal of April 2003 some changes are introduced with respect to the first version (BCBS, 2003a; 
BCBS, 2003b): 

• 

The fixed correlation coefficient, R, of 20% is abandoned and a function, whose coefficient assumes a 
value  between  12%  and  24%  depending  on  the  value  of  the  PD,  is  adopted.  The  function  of  the 

14 The maturity adjustment [1 + (𝑀 − 2.5)]/[1 − 1.5 ∗ 𝑏] reflects the assumption that mid-long term loans are more risky 
than short terms loans. b = (0.11852 – 0.05478*ln(PD))2 

Page 13 

EBA STAFF PAPER SERIES 

 
 
 
     
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

correlation  coefficient  is  modified  in  the  case  of  small  and  medium-sized  enterprises,  to  take  into 
account the fact that SMEs are on average riskier than large companies. 

• 

• 

• 

The increase in the confidence interval to 99.9% was chosen by the Committee to take into account the 
estimation errors that inevitably occur in the calculations of PD, LGD and EAD and model errors. 

The most important difference with respect to the previous proposal concerns the cancellation of the 
granularity  adjustment,  probably  due  to  the  considerable  additional  complexity  that  it  would  have 
imposed on banks, if adopted, in the context of an already very complex regulation. There had been 
numerous negative comments on this adjustment and several requests for simplification. 

The capital requirement is referred exclusively to the unexpected loss (UL), while it was recognized that 
the expected loss is absorbed by the provisions. 

Obtaining the WCL from the risk weights is straightforward. The risk-weighted assets (RWAs) are obtained by 
multiplying the exposures at default by the risk weights. 𝑅𝑊𝐴 = 𝐸𝐴𝐷 ∗ 𝑅𝑊 where 𝑅𝑊 is the expression (4) or 
equivalently the expression (6). The amount 8%*RWA is the minimum required capital (MRC), and it coincides 
with the measure of the unexpected loss which is the difference between the WCL with a confidence level 99.9% 
and the expected loss. To obtain the WCL is then sufficient to sum to 8%*RWA the expected loss amount (ELA). 
This  amount  can  also  be  named  Total  Loss15,  since  it  is  the  sum  of  unexpected  and  expected  losses.  This 
computation is not formally correct, because the risk weights formula also includes also the 1.06% factor and the 
maturity adjustment factor. However, 8%*RWA is actually the minimum required capital, while the expected loss 
must be covered by provisions or, in case the level of provisions is lower than the expected loss, by additional 
capital. Therefore, 8%*RWA+ELA is in practice the amount of capital and provisions that the Regulation requires 
to hold. 

Consider the following example where an exposure of 1 euro million is associated with PD equal to 1% and LGD 
equal to 25%; the expected loss then amounts to simply 1%*25% = 0.25% multiplied by the EAD, which implies 
that, in normal conditions, it can be expected to loss 2,5 thousand of euro (0.25%*1 euro million). The WCDR 
with a confidence level 99.9% is equal to 14%; this implies that the expected probability that the default rate will 
exceed this level is 0.01% and that the WCL is computed by assuming that the default rate is equal to this value. 
The unexpected loss (8%RWA) is equal to 34,5 euro thousand and it represents the maximum deviation from the 
expected loss that is expected. Finally, the WCL in monetary terms is equal to 37 euro thousand (3.7% of the 
EAD) obtained by summing the expected and unexpected loss. 

It should be noticed that what does practically mean that the risk weight is 43.2% is not so immediate, while 
knowing that the WCL is 3.7% (or 37 € thousand) gives a more natural point of view. Further, by supposing that 
this exposure is backed by 50 € thousand of capital, the capital on RWA ratio would be 11.6%, which is higher 
than the regulatory minimum (8%) but it should be recognized that it is not so easy to understand what it does 
represent in concrete that the difference between the ratio and the minimum is 3.6%? Instead by comparing the 
capital  with  the  WCL,  it  is  revealed  that  the  capital  amount  would  be  sufficient  to  cover  the  estimated  WCL 
(generated by the occurrence of a default rate of 14%) plus additional 13 € thousand of losses.    

15 See also Cannata (2020) 

Page 14 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
 
 
 
 
 
  
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Table 4: Example of computation of the WCL from regulatory figures 

EAD 
LGD 
M 
PD 
R = Eq. (7) 
WCDR = Eq. (2) 
maturity adjustment 
ELA = EAD*PD*LGD 
RW = Eq. (4) 
RWA = EAD*RW 
MRC = 8%RWA 
WCL = MRC + ELA 
WCL/EAD 

€ 1,000,000.0 
25.00% 
1 
1.00% 
19.28% 
14.03% 
1 
€ 2,500.0 
43.20% 
€ 431,528.2 
€ 34,522.3 
€ 37,022.3 
3.70% 

The asset correlation and the WCDR (or stressed PD) 

For illustrative purposes, in this section we show in practice how it is possible to obtain the stressed PD i.e. the 
estimate of the WCDR using the MV model. This is done both by estimating the asset correlation from real data, 
then exploiting the asset correlation provided by the BCBS for the SF. This enables to verify how the value of the 
asset correlation influences the estimated WCDR. 

In this section we use the publicly available data provided by Bankit. It is worth mentioning that we are working 
with aggregate data while the regulatory formula applies at a more granular level. However, here we are focusing 
the attention on the computation of the estimated WCDR only. In Section 5, where we extend the analysis to the 
computation of the estimated portfolio WCL, we work with counterparty level data.  

Observed WCDR 

The  following chart  represents  the  time  series  of  the  annual  default  rates  of  non-financial  firms  in  Italy.  The 
average of these 17 points is 3.70% and this represents the estimated average probability of default (long run 
PD). The observed WCDR, as of 2013, is equal to 6.20%. We could use this number as the estimate of the stressed 
PD thus the measure of risk i.e. the unexpected loss would be proportional to the quantity 6.20% - 3.70% = 2.51% 
(i.e. observed WCDR minus the average of the default rates).  

Say for example that the LGD is equal to 50%; leaving aside the factor 𝜓 for simplicity, we have (see Eq. (5)): 
Worst Case Loss = 6.20%*50% = 3.10% and Expected Loss = 3.70%*50% = 1.85%. The unexpected loss would be 
(6.20% - 3.70%)*LGD = 2.51%*50% = 1.25%. Said in other terms, for a portfolio of 1 mln of euro, the Worst-Case 
Loss would be 31,022 € of which 18,494 € is the expected loss component, while the unexpected loss (that should 
be covered by the capital) would amount to 12,528 €. 

Page 15 

EBA STAFF PAPER SERIES 

 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Figure 2: annual default rates for non-financial firms in Italy 

 Source: Banca d’Italia 

The observed WCDR could be considered as not sufficiently prudent. Even if the observation period includes the 
worst financial crisis of the last few decades, it is possible that the time series was too short to give the possibility 
to observe the 99.9-esim percentile of the distribution of the default rates. For this reason, we need a model. 

Estimated WCDR 

Both the reduced form and structural models provide the entire probability distribution of the default rates from 
which we can derive any desired quantile. Both approaches rely on two parameters i.e. the expected value of 
the default rates 𝔼(𝐷𝑟), usually named unconditional or long run 𝑃𝐷 and estimated as the simple average of the 
default  rates,  and  the  default  correlation  (for  the  reduced  form  models)  or  the  asset  correlation  (for  the 
structural models). Under the MV model, the expression (2) provides the 𝛼 percentile given the parameters 𝑃𝐷 
and 𝑅. 𝑃𝐷 is estimated as the average of the observed default rates which we know (see Figure 2) being equal 
to 3.70%. 

With the data underlying Figure 2 and using the Maximum Likelihood estimator proposed by Duellman (2004), it 
is possible to obtain an estimate of 𝑅 equal to 0.03697 (see Table 3, ML1). Setting 𝛼 = 99.9% the expression (2) 
produces 11.21%, this being the estimate of the WCDR. In other terms, we expect that the probability of the 
default rates exceeding the value of 11.21% is only 0.1%. As it can be appreciated from Figure 3 - a, the estimated 
WCDR is well higher than the observed WCDR. With an LGD equal to 50%, the Worst-Case Loss is 11.21%*50% = 
5.61% which is 56,068 € with a portfolio of 1 mln euros. The unexpected loss (and then the capital requirements) 
is (11.21% - 3.70%)*50% = 3.76% i.e. 37,573 €. This amount must be compared with 12,528 €, which was the 
unexpected loss obtained referring to the observed WCDR.  

Page 16 

EBA STAFF PAPER SERIES 

 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Regulatory WCDR 

Now  we  use  the  regulatory16  value  of  𝑅.  Under  the  IRB  framework,  for  the  Corporates  portfolio,  the  asset 
correlation is a function of the 𝑃𝐷: 

                                               𝑅 = 0.12 ∗

1−𝑒−50∗𝑃𝐷
1−𝑒−50 + 0.24 ∗ (1 −

1−𝑒−50∗𝑃𝐷
1−𝑒−50 )                                               (7) 

With 𝑃𝐷 = 3.70% the expression (7) provides for 𝑅 a value equal to 0.1389. This is the regulatory estimate of the 
asset correlation. Feeding the expression (2) with this parameter, the estimated WCDR is 22.60%. With an LGD 
equal to 50%, the worst-case loss for a portfolio of 1 mln euros is now 113,022 € and the unexpected loss is 
94,528 €.  

The  following figure provides  the  comparison  between  the  observed  default  rates  and  the  estimated  WCDR, 
obtained with the same model but with different values of the R parameter. It is also shown the difference in 
terms  of  risk  measures  expressed  in  monetary  terms.  It  is  important  to  understand  that  with  fixed  LGD  and 
exposures  (1 mln  in  this  case)  there  is  a  strict  connection  between  the  risk  measure (from  which the  capital 
requirements are derived) and the level of the stressed PD (or estimated WCDR) provided by the model. 

Figure 3: Observed vs estimated WCDR for non-financial firms in Italy and implied loss for a 1 mln portfolio 

(a)  Estimated WCDR vs Observed Default rates 

(b)  Estimated WCL 

Source, own elaboration 

The following table synthetises the results obtained. The observed WCDR is the highest default rate observed 
in the time series used for the estimation of the PD. For example, for the time series of the default rates of the 
Moody’s class Investment Grades (IG), the observed WCDR in 42 years was 0.62% against an average equal 
about 0.09%. The estimated WCDR is obtained with the expression (2) that is the core of IRB SF using the 
regulatory asset correlation (i.e. expression (7)) or using the estimated asset correlation. Considering again the 
Moody’s class IG, it can be seen that the estimated WCDR obtained with the regulatory asset correlation is 

16 Cfr Article 153 (1) of the CRR 

Page 17 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

2.49% (4 times the observed WCDR) while the estimated WCDR obtained with the estimated asset correlation 
is 1.54% (2.5 times the observed WCDR). 

Table 5: WCDR Observed vs Estimated 

Source, own estimation 

Page 18 

EBA STAFF PAPER SERIES 

Regulatory REstimated RCerved30.05%0.10%1.28%0.14%40.18%0.30%3.35%0.41%50.52%0.94%7.31%1.34%61.36%2.64%13.38%4.14%73.19%6.01%20.84%8.97%86.14%10.21%29.46%14.67%912.79%19.71%44.89%24.44%1042.23%51.84%80.93%61.70%Moody'sIG0.09%0.62%2.49%1.54%SG4.26%12.09%25.06%18.10%Bankit3.70%6.20%22.60%11.21%Bank110.06%0.34%1.79%1.07%20.17%0.55%3.57%1.40%30.34%0.70%5.64%1.66%40.69%1.39%9.00%2.42%51.30%1.92%12.93%3.38%62.54%3.91%18.43%6.28%75.09%9.42%26.99%16.55%810.85%17.95%41.62%30.43%930.13%40.96%70.57%55.78%Bank210.21%0.66%4.14%1.69%20.35%0.61%5.74%1.43%30.37%0.81%6.01%1.89%40.43%1.02%6.64%1.99%50.70%1.54%9.19%3.12%60.99%2.03%11.28%3.79%71.46%2.65%13.87%4.20%82.21%3.83%17.18%5.70%93.23%5.49%20.85%8.10%106.01%10.79%29.15%15.00%1110.20%17.13%39.57%22.65%1215.69%24.97%50.49%30.91%1320.45%32.30%58.44%42.03%1424.84%36.41%64.47%49.11%1531.90%42.97%72.39%57.78%EstimatedObservedWCDRPDRatingData Source 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

5.  Estimated vs regulatory WCL 

Estimated vs Regulatory WCL 

In this section we compare the WCL computed as total loss (expected and unexpected loss) obtained with the SF  
and the calibration of the asset correlation provided by the equation (7) i.e. the regulatory calibration of the SF, 
with the total loss obtained with the same SF but with the estimated value of the asset correlation. Using the 
borrower level data stemming from the two banks’ Corporate portfolios, we computed the RWA at single name 
level with the expression (4) and obtained the Total Loss contribution of each borrower as 8%*RWA + Expected 
Loss Amount, then we summed the results. All the borrowers classified in a given rating grade shared the same 
probability of default and consequently the same value of R was assigned both when using the regulatory and 
the  estimated values  for  the asset  correlation.  The  LGD  and  exposures at  default  were instead  different  and 
specific for each borrower. 

We obtained the sum of the Total Loss computed at borrower level, then we scaled this amount by the sum of 
the exposures at default. The results are represented in the next figure for the two banks. The Total Loss (i.e. the 
estimated WCL observable with 99.9% of confidence level) is 4.71% and 6.31% of the exposure value. When the 
regulatory value of the parameter R is substituted with the estimated value, the Total Loss decreases to 1.54% 
and 2.47% of the exposure value. In relative terms, the regulatory figures (obtained with the value of R provided 
by the expression (7)) are about three times (3.1 and 2.6 respectively) the values obtained with the same model 
(the SF) using instead the estimated asset correlation17. 

Figure 4: Total Loss for the Corporates portfolio with the SF: regulatory vs estimated asset correlation 

Bank 1 

Bank 2 

Source: own estimation on banks internal data 

The difference we obtained in terms of Total Loss is considerable and deserves some additional thought. First, 
we have seen in section 2 that derivation of the SF relays on the hypothesis that the portfolio is not concentrated 

17 Also in Dietsch (2013) it can be found, with reference to the French banking system, that the regulatory minimum capital 
requirements are about two times the level of the capital that can be obtained by fully calibrating a portfolio credit risk model. 

Page 19 

EBA STAFF PAPER SERIES 

 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

(perfectly granular). This hypothesis is quite unrealistic given that data refer to a Corporates portfolio. It is thanks 
to the infinite granularity hypothesis that the measure of risk can be obtained with a closed formula like the SF.  

The concentration effect 

Abandoning the hypothesis of perfect granularity implies either correcting the SF with an adjustment like the 
one  proposed  by  Gordy  or  switching  to  simulation-based  methods.  In  order  to  verify  the  impact  of  the 
concentration on the risk measures, we used a Monte Carlo (MC) approach. In practice, the results obtained with 
the MC approach should approximately be equal to the results obtained with the SF in case the concentration of 
the portfolios (the single rating grades) was low. In case the concentration is high, the MC approach should return 
a higher measure of risk. 

Here we briefly depict how the MC approach works. The SAS codes that have been used are included in the 
Annex.  The  values  y  and  𝑧  of  the  common  factor  𝑌𝑡  and  of  the  idiosyncratic  factor  𝑍𝑖,𝑡  are  generated  from 
independent standard normal variables. The creditworthiness of each borrower of the DS2 is computed  with the 
parameter R estimated for each rating class (see Table 3). Each borrower is then considered as defaulted if his 
credit  worthiness  is  less  than  𝛷−1(𝑃𝐷)  with  the  PD  depending  on  the  rating  class.  Summing  the  loss 
(exposure*LGD)  of  all  the  defaulted  borrowers  produces  the  total  losses  obtained  given  the  scenario  𝑦.  This 
process is repeated many times. In this way it is possible to obtain the entire probability distribution of the total 
losses, thus letting us derive empirically the 𝛼 − 𝑒𝑠𝑖𝑚  percentile of this distribution. We repeated this procedure 
for each rating grade and then summed the result for each bank18. To stabilise the results, we exploit a variance 
reduction technique consisting in sampling for 𝑌𝑡 more heavily from the left tail of the normal distribution and 
then reweighting the results to obtain correct measures19. The results we obtained were indeed quite stable. 

The  impact  of  the  concentration  effect  appears  wide  for  both  banks  (see  the  figure  below).  In  detail,  the 
regulatory Total Loss is now only 1.2 and 1.8 times (respectively for the two banks) higher than the measure 
obtained exploiting the MC approach with the estimated value of R. These results can lead to think that the 
regulatory measures (SF with regulatory R) already include a correction for the concentration effect and this 
correction is embedded in the quantification of the asset correlation.  

18 Summing the 𝛼 −VaR obtained for each rating class it is equivalent to assume that the risks in the rating grades are perfectly 
correlated that is, there are not beneficial effects due to portfolio diversification. This is in line with the general setting of the 
Basel  II  that  for  example  does  not  recognize  the  possible  diversification  effect  due  to  segmentation  of  the  portfolio  by 
countries or sectors of the counterparties. This setting, in turn coherent with the single risk factor hypothesis, allows to obtain 
a considerable simplification. The scenarios (the y values) where re-generated for each rating class. We also tried to keep the 
same scenarios for all rating classes and the results were substantially the same. 
19 See Bolder (2018) chapter 8 

Page 20 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Figure 5: Total Loss obtained with the SF (regulatory and estimated asset correlation) and WCL 
obtained with Monte Carlo simulations and estimated asset correlation 

Bank 1 

Bank 2 

  Concentration Effect 

A better understanding is provided by comparing the two measures of risk at rating grade level. The next figure 
shows the level of the Total Loss obtained with the SF using the regulatory calibration of R for each rating grade 
of the Bank1 portfolio and the WCL obtained with the MC approach but with the estimated R. It is also shown 
the Herfindahl index (a measure of the concentration) computed at rating level. It can be seen that for most of 
the rating grades the MC measure of risk is lower than the SF measure. This implies that the concentration effect 
is not sufficient to compensate for the reduction of R (see Table 3). It can also be noticed that the Herfindahl 
index for these rating grades is quite low. However, for the first and the last rating grade the MC measure is 
higher than the SF measure and this occurs in correspondence with higher values of the Herfindahl index. What 
happens in these cases is that the concentration effect is able to more than compensate for the reduction of the 
asset correlation.  

It  is  also  notable  to  see  how  the  risk  measures  may  change  dramatically  when  the  concentration  effect  is 
considered. The rating grade 7 has PD which is about 50 times higher than the PD of the rating grade 1. However, 
the measure of risk obtained with the MC approach (i.e. considering the concentration effect) is higher for the 
rating grade 1 (8.7% vs 6.7%). Moreover, for the first and last rating grade, where the concentration appears high 
(Herfindahl  >  10%),  the  risk  measure  obtained  with  the  MC  approach  and  the  estimated  asset  correlation  is 
higher  than  the  regulatory  measure.  This  implies  that  in  some  cases,  the  regulatory  calibration  of  the  asset 
correlation may not adequately compensate for the lack of consideration of the concentration effect.  

Page 21 

EBA STAFF PAPER SERIES 

 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Figure 6: Total Loss with the SF (regulatory asset correlation) vs WCL with the MC simulations 
(estimated asset correlation) at rating grade level, Bank 1 

Other corrections 

We now briefly consider other shortcomings of the MV model and the related corrective measures. It is known 
that the estimation of the asset correlation encounters some limitations (see for example Gordy (2002)). One of 
these limitations is represented by the hypothesis that all the borrowers classified in a given cluster or rating 
class have the same level of probability of default. If this hypothesis is not satisfied, i.e. if the borrowers have 
different probability of defaults, then the resulting estimated asset correlation will be biased. In particular, it is 
known  that  the  estimates  are  downward  biased.  Resti  (2008)  suggested  to  multiply  the  estimated  asset 
correlation  by  1.5.  Based  on  this  analysis,  this  factor  should  ensure  robust  estimates  even  in  the  case  the 
hypothesis of homogeneous probability of default is violated. The other limitation is the length of the time series 
of default rates. With short ranges of years of observation (less than 20, according to Gordy) of the default rates, 
the estimates are known to be downward biased. 

In our case, we are working with borrowers classified in rating grades through a scoring system that includes 
several risk indicators and expert evaluations. We can then expect that the borrowers in each rating grade are 
quite homogenous in terms of probability of default, but still some sources of heterogeneity may exist. More 
serious is the problem of the length of the observation period. The number of years of observation of the default 
rates at our disposal is only 14. For these reasons we decided to apply a corrective factor as the one suggested 
by Resti and, namely, we adopted a factor equal to 220.  Notice for example that with this correction, the asset 
correlation for the first rating grade becomes 11.68%*2 = 23.36%, which is not so far from the regulatory value 
equal to 23.41% (see Table 3). However, for the other rating grades, the difference is still wide. 

Another  subtle  shortcoming  of  the  gaussian  MV  model  relates  to  a  quite  technical  aspect  known  as  tail 
correlation.  The  main  feature  of  the  MV  model  is  to  provide  a  method  to  generate  a  predefined  level  of 
correlation among the default events. However, it can be shown that as we increase the confidence level (𝛼), the 
correlation  between  default  events  tends  to  decrease  and  for  high  𝛼  levels  this  reduction  is  important.  This 
problem is strictly related to the hypothesis that the common factor 𝑍𝑡 has normal distribution. An approach to 
overcome this problem consists in abandoning the normal distribution and adopt another one, such as the T 
distribution.  

20 A similar expedient has been used also in Tola (2010) 

Page 22 

EBA STAFF PAPER SERIES 

 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

The departure from the Gaussian hypothesis for the common factor deserves some attention21. The MV is based 
on  the  normal  distribution  because  the  log-yields  of  the  activities  are  by  assumption  described  with  the 
geometric Brownian motion (which in essence is a transformation of the normal standard); if we abandon the 
normal distribution, to have standardized returns linked by a t-copula (and not by a Gaussian copula that has 
zero correlations in the upper and lower tails), for consistency’s sake, it is also necessary that the calculation of 
the distance-to-default is adjusted and the same is true for the estimate of the conditional PD which must be 
obtained  referring  to  a  T-distribution.  It  follows  that  the  regulatory  formula  should  be  updated  to  insert  the 
inverse of the t-distribution for the PD and the common factor. For example, the 99.9% quantile of a standard 
normal variable is -3.09 while the same quantile of the Student t with 3 of freedom is about -10.21. 

With the aim of controlling also for this problem, in our MC exercise, we multiplied the firms’ asset value by the 
random  variable  √𝑣/𝑊𝑖,𝑡  where  𝑊𝑖,𝑡  is  a  𝜒2(𝑣)  random  variable  with  𝑣  degrees  of  generated  freedom.  In 
practice,  √𝑣/𝑊𝑖,𝑡ri,t  is  a  standard  T-distributed  random  variable  with  𝑣  degrees  of  freedom.  The  degrees  of 
freedom have been set so as to reproduce a default correlation equal to two times the estimates obtained with  
one of estimators in Table 3 and a level of tail dependency equal to 5%. With the aim to better exploit the fat-
tail characteristic of the T distribution, we also computed the Expected Shortfall (ES) or tail VaR as the average 
of the losses exceeding the VaR. This further step can be considered as a heuristic way to account for other not-
easily quantifiable shortcomings of the underlying model as the model risk.     

Lastly,  besides  the  volatilities  and  the  correlations,  an  additional  feature  of  default  rates  that  is  commonly 
empirically observed but not accounted for by the class of models backing the SF is the persistence of the shocks. 
The persistence, measured by the serial correlation or autocorrelation, is the ability of shocks to reflect their 
effects  even  in  periods  subsequent  to  their  occurrence.  Adopting  a  one-year  perspective  brings  to  implicitly 
assume that the effects of a shock (adverse scenario) observed in a given period, do not spread over the next 
periods. Default data referred to SMEs and Households could reveal a lower-than-expected volatility but a quite 
high persistence that the SF is not suited to account for. We have not implemented this correction because a 
different class of models, which makes it possible to take into account the dynamics of default rates, should be 
considered. 

Figure 7 shows the results for the two banks. Now it is possible to retrieve the level of the TL obtained with SF 
and the regulatory value for the asset correlation. The correction to the estimated asset correlation obtained by 
multiplying the estimates with a factor of 2, produces an increase of the risk measure by some percentage points. 
The  impact  of substituting  the  normal  distribution  with  the  T distribution  for  the  generation  of  the common 
factor has a more material impact. 

21 See Bolder (2018) section 4.3, 4.4 

Page 23 

EBA STAFF PAPER SERIES 

 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Figure 7: Total Loss with the SF (regulatory and estimated asset correlation) and WCL and ES with 
Monte Carlo simulations with estimated asset correlation multiplied by 2 and imposing a 5% of 
tail dependency 

  Concentration Effect ; 

 bias correction ; 

 tail correlation ; 

 expected shortfall 

 difference from the regulatory measure 

Page 24 

EBA STAFF PAPER SERIES 

 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

6.  Conclusions 

The level of capital requirement generated by the IRB approach depends crucially on the asset correlation – a 
parameter that enters the regulatory risk weight formula and is determined by the Regulators. Several studies 
have estimated the asset correlation and found values lower than the regulatory calibration.  Exploiting different 
data sources, we confirm the findings from existing literature. However, we believe that the simple comparison 
between the estimated and Regulatory values of this parameter does not offer an easy interpretation. We then 
propose a different point of view.  

The IRB Regulatory Supervisory Formula (SF) can be seen as an algorithm taking the long-run PD as input and 
providing  the  stressed  PD  as  output.  Looking  at  the  stressed  PD  provides  a  natural  interpretation  of  the  risk 
measure produced by the SF. In fact, it is possible to compare the obtained stressed PD with the historical series 
of the default rates. From this comparison, it is possible to appreciate the level of prudence inherent in the SF. It 
is also possible to translate the Risk Weighted Assets into estimated Worst-Case Loss (the highest economic loss 
that can be expected with a given level of confidence) and this provides an easier economic interpretation of the 
obtained risk measures.  

We  compare  the  regulatory  risk  measure  with  the  risk  measures  obtained  by  substituting,  in  the  SF,  the 
regulatory asset  correlation with the one that can be estimated from observed default rates. At least for the 
data at our disposal, justifying the observed difference in the risk measure, requires retrieving the concentration 
effect plus other margins of prudence. In other terms, we have verified that the current regulatory calibration of 
the SF seems to already account for a series of known limits of the reference model that the Basel Committee 
avoided formalizing for sake of simplicity such as the hypothesis of perfect granularity and the uncertainty related 
to  the  structure  of  the  default  correlation.  This  implies  that  while  the  asset  correlation  input  seems 
overestimated and the effect of concentration seems not considered, the aggregate risk measure does not seem 
mis-calibrated if both these inputs are correctly accounted for. 

While the empirical results presented cannot easily be extended outside the analyzed sample that it is limited to 
two banks, they demonstrate the advantages of the proposed method that, by focusing the attention on the 
stressed PD and Worst-Case Loss instead of on unexpected loss multiplied by 12.5 (i.e. the Risk Weighted Assets), 
enables to disentangle the risk components (like the concentration effect). This kind of analysis could be included 
in the Pillars 3 of the IRB validated banks for transparency, benchmarking and better comprehension of these 
metrics by the market participants. 

Page 25 

EBA STAFF PAPER SERIES 

 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

7.  Annex 

The variance of the default rates: a concrete example 

Consider the following simple example built with real data22 coming from the Corporates portfolio of an Italian 
bank. The chart below represents the number of borrowers (performing at the beginning of each year - 𝑁𝑡) and 
the number of new defaults (𝐷𝑡) observed during the year 𝑡 for a given rating class or grade of the portfolio. 
Thanks  to  this  data,  it  is  possible  to  compute  15  annual  default  rates  (𝐷𝑟𝑡 = 𝐷𝑡/𝑁𝑡 ; 𝑡 = 1,2 … 15),  and  the 
(simple23) average of these default rates is an estimate of the long run probability of default for this rating grade. 
The average of the default rates is equal to 1.273%. The number of borrowers is quite stable across the years 
and the average number of borrowers is equal to 2235. We then have an estimate of the parameters 𝑝 and 𝑁 of 
a binomial distribution and we can derive the expected variance as 1.273%(1-1.273%)/2235 = 0.00056%. 

We can also compute the variance empirically and the result is 0.00166%. The empirically observed variance is 3 
times the variance of a binomial random variable with parameters 𝑁 = 2235 and 𝑝 = 1.273% and this result 
(i.e.  the  observed  variance  of  the  default  rates  being  higher  than  the  variance  implied  by  the  hypothesis  of 
independence) is commonly found when dealing with credit portfolios. This evidence, in turn, casts doubts on 
the hypothesis that the defaults are independent and consequently on the opportunity to employ the binomial 
model in this context. 

Figure 8:  New defaults and Number of borrowers for a 
rating grade of a Corporates portfolio 

 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑁 = 2235 

 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐷𝑒𝑓𝑎𝑢𝑙𝑡 𝑟𝑎𝑡𝑒 =

∑

2020
𝑡=2007
15

𝐷𝑟𝑡

= 1.273% 

 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑎 𝐵𝑖(𝑁 = 2235, 𝑝 = 1.273%) = 0.00056% 

 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 =

Source: bank internal data 

∑

2020
𝑡=2007

(𝐷𝑟𝑡−1.273%)2

15

= 0.00166% 

22 The details about the data used are in the Section 4. 
23 Article 180 (1) (a) of the Regulation (EU) No 575/2013 (CRR) states that << institutions shall estimate PDs by obligor grade 
from long-run averages of one-year default rates >> but it is not clear whether the average should be simple or weighted. 
However,  in  the  EBA  (2017)  guidelines  paragraph  81  it  specified  that  institutions  should  calculate  the  observed  average 
default rates as the arithmetic (i.e. simple) average of all one year default rates. 

Page 26 

EBA STAFF PAPER SERIES 

 
 
 
 
 
 
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Monte Carlo simulation of the MV model 

What follows is an example of Sas code that can be used to obtain a Monte Carlo evaluation of the VaR at given 
confidence level under the MV model. The code exploits a variance-reduction technique, in practice the scenarios 
𝑌𝑡 are extracted from a normal distribution shifted toward negative numbers. 

The data set <<rating>> contains the data at borrower level referred to as a given rating grade (the PD is the 
same for all the borrowers). In particular, the variable severity is built as EAD*LGD. The program generates a 1xM 
vector of scenarios (extractions from a normal distribution with a central parameter equal to &med) and the 
NxM matrix of the idiosyncratic effects (extracted from a standard normal distribution) where N is number of 
clients in the rating grade. Z_E is the matrix containing the creditworthiness computed for each borrower and 
under each scenario. D is a (0,1) NxM matrix where 1 is assigned when the creditworthiness is less than Φ−1(𝑃𝐷). 
LZ is a Mx2 matrix in which the first column contains the loss rate observed under each scenario and the second 
column contains the realized scenarios.  

In the data step, after the iml procedure, it is computed the weighting factor (wgt) of each scenario. This passage 
is  needed  to  correct  for  the  fact  the  scenarios  were  not  extracted  from  a  standard  normal  distribution.  The 
univariate procedure is then used to compute the percentile of the distribution at the desired level of confidence. 

%let med = -1.4; 
data _null_; 
    set rating end=eof; 

if eof then threshold= probit(PD); 
if eof then call symput (‘Thr’,threshold); 

end; 
proc iml; 
    use rating; 

read all var {severity} into A[colname = varNames]; 
call randseed(-1); 
Z = j(1,&M); 
E = j(&N,&M); 
call randgen(Z,"Normal",&med,1); 
call randgen(E,"Normal",0,1); 
Z_E = sqrt(&ro)*Z+sqrt(1-&ro)*E; 
D = (Z_E<&Thr); 
LZ = (A`*D)`/&S_ead ||Z`; 
create Ldistr from LZ[colname={"L" "Z"}]; 

append from LZ; 

close Ldistr; 

quit; 

data Ldistr(drop=Z f g); 

set accum end=eof; 
retain S_wgt 0; 
f=PDF('NORMAL',Z,0,1); 
g=PDF('NORMAL',Z,&med.,1); 
wgt = f/g; 

run; 

proc univariate data=Ldistr noprint; 

var L; 
weight wgt; 
output out=Pctl_TL(keep= P_95 p_98 P_99 P_99_5 P_99_7 P_99_8 P_99_9)  

pctlpre=P_ pctlpts= 95 to 100 by 0.1; 

run; 

Page 27 

EBA STAFF PAPER SERIES 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

References 

Baesens B., Rosch D,. Scheule H., (2016), Credit Risk Analytics. Wiley 

Bank for International Settlements (BIS), Committee on the Global Financial System (1999): A Review of Financial 
Market Events in Autumn 1998. Basel, October. 

Bcbs, (1999a), A new capital adequacy framework, [bcbs50] 

Bcbs (1999b), Credit risk modelling: current practices and applications [bcbs49] 

Bcbs, (2001a), Overview of the new Basel Capital Accord, [bcbsca02] 

Bcbs (2001b), The Internal Rating-Based Approach – Consultative Document, [bcbsca05] 

Bcbs (2003a), Overview of the new Basel Capital Accord, [cp3ov] 

Bcbs (2003b), The new Basel Capital Accord – Consultative Document, [bcbscp3] 

Bcbs (2004), International Convergence of Capital Measurement and Capital Standard, [bcbs107] 

Bcbs (2005a), International Convergence of Capital Measurement and Capital Standard – A Revised Framework, 
[bcbs118] 

Bcbs (2005b), An explanatory note on the Basel II IRB risk weight functions, [irbriskweight] 

Bcbs (2006), International Convergence of Capital Measurement and Capital Standard – A Revised Framework, 
Comprehensive Version, [Bcbs128] 

Bcbs (2017), Basel III: Finalising post-crisis reforms, [d424] 

Bluhm C., Overbeck L., Wagner C., (2016), Introduction to Credit Risk Modelling. (Second Edition). Chapman & 
Hall/CRC. London. 

Bofondi M., Ropele T., (2011), Macroeconomic Determinants of Bad Loans: Evidence from Italian Banks. 
Questioni di Economia e Finanza (Occasional Papers)  

Bolder D., (2018), Credit-Risk Modelling: Theoretical Foundations, Diagnostic Tools, Practical Examples, and 
Numerical Recipes in Python. Springer 

Bonaccorsi di Patti E., Cascarino G., (2020), Modelling the dynamics of nonperforming loans in Italy. Notes on 
Financial Stability and Supervision, No. 19 

Blümke, O., (2017) Asset Correlation, Diversification and the Basel Accord: A Comparative Study. Available at 
SSRN: https://ssrn.com/abstract=2970534 or http://dx.doi.org/10.2139/ssrn.2970534 

Cannata F., Casellina S., Chionsini G., (2020), Time to Go Beyond RWA Variability for IRB Banks: An Empirical 
Analysis. EBA staff papers 

Chernih A., Vanduffel S., Henrard L., Asset correlations: A literature review and analysis of the impact of 
dependent loss given defaults, WP 

Chernih A., Vanduffel S., Henrard L., (2010) Reconciling credit correlations. Journal of Risk Model Validation 4 
(2), 47–64. 

Curcio D., Gianfrancesco I., Malinconico A., (2011), Investigating Implied Asset Correlation and Capital 
Requirements: Empirical Evidence from the Italian Banking System. Banks and Bank Systems 

Das S. R., Freed L., Geng G., Kapadia N., (2006), Correlated Default Risk. The Journal of Fixed Income  

Di Clemente A., (2020), Modeling Portfolio Credit Risk Taking into Accountthe Default Correlations Using a 
Copula Approach:Implementation to an Italian Loan Portfolio. Journal of risk and financial management. 

Page 28 

EBA STAFF PAPER SERIES 

THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Dietsch M., Fraisse H., (2013), How different is the regulatory capital from the economic capital: the case of 
business loans portfolios held by the major banking groups in France. Débats économiques et financiers, n°1, 
ACP – Banque de France 

Dietsch M., Duellmann K., Fraisse H., Koziol P., Ott C. (2016), Support for the SME Supporting Factor: Multi-
Country Empirical Evidence on Systematic Risk Factor for SME Loans. Bundesbank Discussion Paper No. 45/2016 

Duellmann K., Koziol P., (2014), Are SME Loans Less Risky than Regulatory Capital Requirements Suggest? 
Journal of Fixed Income 

Duellmann K., Kull J., Kunisch M., (2010), Estimating asset correlations from stock prices or default rates—
Which method is superior?  Journal of Economic Dynamics & Control 

Duellmann K., Gehde-Trapp M., (2004), Systematic Risk in Recovery Rates: An Empirical Analysis of Us 
Corporate Credit Exposures. Bundesbank Series 2 Discussion Paper No. 2004,02 

EBA, (2017), Guidelines on PD estimation, LGD estimation and the treatment of defaulted exposures 

EU (2013), European Parliament and Council, Regulation (EU) No 575/2013, Official Journal of the European 
Union 

Frye J., (2008), Correlation and asset correlation in the structural portfolio model, Journal of Credit Risk 

Geidosch, M. (2014) Asset Correlation in Residential Mortgage-Backed Security Reference Portfolios. Journal of 
Credit Risk, Vol. 10, No. 2, 2014 

Gordy M., Heitfield E., (2002), Estimating Default Correlations from Short Panels of Credit Rating Performance 
Data. Working Paper, Federal Reserve Board 

Gordy  M.,  (2003),  A  risk-factor  model  foundation  for  ratings-based  capital  rules,  Journal  of  Financial 
Intermediation 

Gordy M, (2004), Granularity Adjustment in Portfolio Credit Risk Measurement. Risk Management and Financial 
Institutions. Wiley 

Hashimoto T., (2009), Asset correlation for credit risk analysis.  Empirical study of default data for Japanese 
companies. Bank of Japan Working Paper Series 

Henneke J., Trück S., (2006), Asset correlations and capital requirements for SME in the revised Basel II 
framework. Banks and Bank systems 

Hull J., (2015), Risk Management and Financial Institutions, (fourth ed), Wiley 

Jorion P., (2006), Value at Risk: the New Benchmark for managing Financial Risk. (Second Edition). McGraw Hill  

Lucas D., (1995), Default correlation and credit analysis, Journal of Fixed Income 

Lee S., Lin C., Yang C., (2011), The asymmetric behavior and procyclical impact of asset correlations, Journal of 
Banking & Finance 

Merton R., (1974), On the pricing of corporate debt: the risk structure of interest rates, Journal of Finance 

Piccone B., (2019), L’Italia: molti capitali, pochi capitalisti, Il Sole 24 Ore 

Quagliariello  M.,  Marcucci  J.,  (2009),  Asymmetric  effects  of  the  business  cycle  on  bank  credit  risk.  Journal  of 
Banking & Finance 

Quagliariello M., (2019), Are Stress Tests Beauty Contests?, EBA staff papers 

Resti A., (2008), Pillar II in the New Basel Accord, Risk Books 

Schonbucher P., (2001), Factor models: portfolio credit risks when defaults are correlated. Journal of Risk Finance 

Siarka, P. (2014) Asset Correlation of Retail Loans in the Context of the New Basel Capital Accord. Journal of Credit 
Risk, Vol. 10, No. 2 

Page 29 

EBA STAFF PAPER SERIES 

THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

Sironi  A.,  Zazzara  C,  (2003),  The  Basel  Committee  proposals  for  a  new  capital  accord:  implications  for  Italian 
banks. Review of Financial Economics 

Tola V., (2010), Measuring Sectoral/Geographic Concentration Risk. Bank of Italy Occasional Papers 

Vasicek O., (1987), Probability of loss on loan portfolio, w.p. KMV 

Vasicek O., (1991), Limiting loan loss probability distribution, w.p. KMV 

Vasicek O., (2002), Loan portfolio value, Risk 

Zhang J., Zhu F., Lee J., (2008), Asset Correlation, Realized Default Correlation, and Portfolio Credit Risk. Moody’s 
KMV technical working paper 

Page 30 

EBA STAFF PAPER SERIES 

 
 
 
THE CALIBRATION OF THE IRB SUPERVISORY FORMULA  

ACKNOWLEDGEMENTS 

We are grateful to V. Bonten, F. Cannata, G. Cascarino, S. Cuneo, L. Giaccheirni, 
A. Ilari, R. Kopp, A. Leardi, D. Mokas, M. Quagliariello, M. Reggi, A. Resti, G. Rinna, 
R. Russo and E. Sabatini for useful comments and suggestions on a preliminary 
draft of the paper 

Simone Casellina:  
Bank sector analyst, European Banking Authority   

Fabio Salis:  
Chief Risk Officer, Creval 

Giovanni Tessiore:  
Head of Risk Advanced Analytics & Innovation, Cerved 

Roberto Ugoccioni:  
Senior Model Expert, Intesa San Paolo 

Franco Varetto:  
Past Contract Professor at Politecnico of Turin 

EUROPEAN BANKING AUTHORITY 

20 avenue André Prothin CS 30154 
92927 Paris La Défense CEDEX, France 

Tel.  +33 1 86 52 70 00 
E-mail: info@eba.europa.eu 

https://eba.europa.eu/ 

ISBN 978-92-9245-923-9 
ISSN 2599-7831 

DOI 10.2853/080166 
DZ-AH-23-003-EN-N 

© European Banking Authority, 2023. 

Any reproduction, publication and reprint in the form of a different publication, whether printed or 
produced electronically, in whole or in part, is permitted, provided the source is acknowledged. 
Where copyright vests in a third party, permission for reproduction must be sought directly from the 
copyright holder. 

This paper exists in English only and it can be downloaded without charge from 
www.eba.europa.eu/staffpapers, where information on the EBA Staff Paper Series can also be 
found. 

Page 31 

EBA STAFF PAPER SERIES