Sector concentration risk: A model for estimating capital requirements

Mathematical and Computer Modelling 54 (2011) 1765–1772

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Sector concentration risk: A model for estimating capital requirements J. David Cabedo Semper ∗ , Jose Miguel Tirado Beltrán Departamento de Finanzas y Contabilidad, Universitat Jaume I, Av. de Vicent Sos Baynat, s/n, E12071 Castellón de la Plana, Spain

article

info

Article history: Received 13 October 2010 Received in revised form 25 November 2010 Accepted 30 November 2010 Keywords: Concentration risk Credit risk Concentration index Capital requirements

∗

abstract The 2004 Basel Committee on Banking Supervision Accord (known as Basel II) provides a common framework for banks to determine their minimum capital requirements for solvency purposes. For credit risk (the most important one for banking) Basel II uses an asymptotic single risk factor (ASRF) model and, as we demonstrate in the paper, assumes two fundamental hypotheses: Firstly, that there is only one risk factor common to all banks; and secondly, that the number of debtors in bank portfolios is high enough to ensure that no single debtor’s behaviour can have a significant impact on the portfolio value as a whole. This allows capital requirements to be estimated by using a model based on the percentage of defaulting borrowers (x). The model only requires values for two variables: the probability of default and loss if default occurs. Using a 99% likelihood and assuming that all sectors are equally correlated, the model estimates x through the cumulative distribution function for the Gaussian distribution. But many bank portfolios do not fit these hypotheses, and therefore the ASRF model underestimates actual capital requirements. Thus, a surcharge for concentration risk is required. There are two kinds of concentration risk (sector and name concentration risk), each one corresponding to the violation of one of the above mentioned hypotheses. Supervisory authorities are currently developing models to incorporate this surcharge into banking solvency rules. In Spain, the Spanish Central Bank bases its surcharge proposal for sectorial concentration on the Herfindahl–Hirschman Index (HHI). In this paper we show that HHI treats all sectors as equally risky and propose an alternative index (CI) in which sectors are weighted according to risk. Moreover, our index also incorporates the relations between each pair of sectors (in the HHI framework no sectorial relationship is considered). Our proposal is based on an adjusted variance–covariance matrix, in which negative covariances have been equalled to 0. We demonstrate the HHI is a particular case of our proposed index, by means of simplifying hypotheses. As we will show, the proposed index has two fundamental properties: it is lower and upper bounded; and it decreases as concentration and/or risk decreases. These properties allow the index to be incorporated into bank risk management models. In this way bank estimations can improve upon those based on the supervisory model and, according to banking rules, can also be used for determining the capital surcharge for sectorial concentration. © 2010 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +34 964387150; fax: +34 964728565. E-mail addresses: [email protected] (J.D. Cabedo Semper), [email protected] (J.M. Tirado Beltrán).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.086

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1. Introduction The Basel Committee on Banking Supervision (from here on the Committee) is a forum for regular banking supervision cooperation. The Committee regularly issues papers and surveys on various aspects of banking supervision and other related topics. The most important of the documents it has produced are the capital adequacy rules issued1 in 2004, also known as the Basel II Accord. These rules are not compulsory for banks but all the members of the Committee2 have committed to incorporating them into their national legislation. The Basel II Accord is divided into three parts: Pillar I establishes rules for calculating capital requirements in view of the most important risks for banks: credit, market and operational risks. Pillar II establishes a set of rules for banking supervision; and Pillar III deals with market discipline. Within Pillar I, the Committee offers a double possibility for calculating the credit risk capital requirements: the standardised approach and the internal ratings based (IRB) approach. In the standardised approach the credit portfolio is divided into several categories, with assets classified according to their rating. In the IRB approach the Committee provides a function based on the fraction of defaulted securities. Banks must apply this function entering the values of a number of internally calculated variables. The function provided by the Committee is based on the Asymptotic Single Risk Factor (ASRF) model. Neither the standardised nor the IRB approaches consider the degree of concentration of the credit portfolio. That is why, in accordance with Pillar II rules, the supervisory authorities can fix a concentration surcharge for the credit risk capital requirements estimated in Pillar I. So far, the literature on concentration risk has been limited. Following Kalkbrener [2] there are two kinds of portfolio concentration: sectorial concentration and name concentration, both stemming from the hypotheses assumed by the ARSF model. Therefore, some research has focused on introducing modifications into this model in order to estimate the concentration risk. Garcia et al. [3] proposed an adjustment to the ASRF model based on the Herfindahl–Hirschman concentration index. Düllmann [4] replicates the actual credit portfolio with a simulated one, as the Moody’s Binomial Expansion Technique (BET) does. But contrary to this technique, Düllmann [4] assumes a simple dependence scheme that allows economic capital to be estimated in a simple way. The starting point for Pykhtin’s [5] paper is also a simulated portfolio. He generates a one-factor dependent portfolio with a loss distribution similar to the actual portfolio one. Pykhtin [5] estimates Value at Risk (VaR) by introducing a multifactor adjustment into the simulated portfolio VaR. Düllmann and Masschelein [6] simulate several portfolios with different degrees of concentration, and, by using this multifactor adjustment, estimate the economic capital. Düllmann and Masschelein [7] highlight the importance of sectorial concentration for calculating the economic capital needed to cover credit risk. For this, it is necessary not only to consider the degree of concentration in specific economic (or geographical) sectors but also the relationships between them. Bonti et al. [8] also stress the importance of these relationships: they propose using a multifactor model for estimating economic capital, which should take into account the relationships between economic (or geographical) sectors. Stemming from a different research path, Uberti and Figine [9] propose an index for quantifying name concentration in a credit portfolio. This index may be used within Pillar II for determining the capital surcharge required for this kind of concentration. Our paper is part of this same line of research. Firstly we demonstrate why the function provided by the Committee in the IRB approach does not take into account the degree of concentration of the credit portfolio. And, secondly, we develop a sector concentration index that can be used for estimating the concentration surcharge for credit risk capital requirements. The remainder of the paper has been structured as follows: in Section 2 we analyse the Asymptotic Single Risk Factor (ASRF) model. In Section 3 we focus on concentration risk, differentiating between name concentration risk and sector concentration risk. In Section 4 we set out our proposal for measuring sector concentration risk. And in the last section we provide a summary with our main conclusions. 2. The ASRF model In accordance with the IRB method of Pillar I, the minimum capital requirements for credit risk must be calculated as a function of a series of variables, of which the proportion of securities defaulted, is the key one. It must be estimated as follows3 :

[ √ ] ρ 1 −1 −1 · N (0.999) + √ · N (PD) x=N √ 1−ρ 1−ρ

(1)

where ρ denotes the constant coefficient of correlation between every pair of assets belonging to the credit portfolio; PD is the probability of default; and N represents the cumulative distribution function of a standard normal distribution.

1 See Basel Committee on Banking Supervision [1]. 2 The central banks of the European Union countries and the United States of America are members of the Committee. 3 We focus our analysis on corporate exposures. These are, generally, the largest within the credit portfolio of a bank. Anyway, all the conclusions of this section can be extended to the other components (exposures) of the portfolio.

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Expression (1) can be derived from the ARSF model, which is based on the Merton model. Merton4 [11] stated that a company attends its payment obligations when the value of its assets is higher than the value of its liabilities. If this is not the case, companies prefer to default. Therefore, for the company i, the probability of defaulting can be modelled as the probability of the value of its assets (AV i ) falling below that of its liabilities (LV i ). PD = P (AV i < LVi ).

(2)

Factor models use expression (2) as the starting point, modelling the value of the assets through a series of factors. One of these models is the ASRF model, which assumes the following hypothesis: Hypothesis 1. The value of a company’s assets depends on a unique risk factor, which is the same for all firms. The relationship between this factor and the value of the company’s assets can be modelled as follows: AV i =

 √ ρ · Z + 1 − ρ · εi

(3)

where Z denotes the common risk factor and εi the idiosyncratic risk component. Both variables (Z and εi ) are assumed to be independent and identically distributed (i.i.d.) according to a standard normal distribution function. Furthermore, the model assumes that the correlation coefficient for every pair of exposures (ρ ) remains constant. If the risk factor has a particular value, say z, then we can estimate the conditional default probability as stated below: PD(z ) = P (AV i < LVi |Z = z ).

(4)

Hypothesis 2. The loss threshold is the same for all borrowers5 :

∀i.

LVi = LV

(5)

Under this hypothesis, considering (3) and the distribution of εi , the conditional default probability can be reformulated as follows: PD(z ) = N

[

] √ ρ·z . 1−ρ

LV −

(6)

√

That is, the conditional default probability is the same for all debtors. If we take into account the independence between exposures, the number of defaults (D) follows a binomial distribution. Therefore, the probability that the number of defaults equals a given value (δ ) can be expressed as follows: P (D = δ|Z = z ) =

  C

δ

· PD(z )δ · [1 − PD(z )]C −δ .

(7)

Here C represents the number of debtors in the portfolio. By applying the law of iterated expectations we can transform a conditional probability into an unconditional one and calculate it in the following manner: P (D ≤ m) =

∫ m − δ=0

∞

 

−∞

δ

C

[ ·N

√

ρ·z √ 1−ρ

LV −

 ]δ [ [ ]]1−δ √ LV − ρ · z · 1−N · n(z )dz √ 1−ρ

(8)

n(z ) denotes the standard normal density function. Despite expression (8) being a tractable one, it is common to work with the fraction of defaulted borrowers instead of its absolute value. At this point a third hypothesis is required: Hypothesis 3. Granularity. The number of debtors included in the bank portfolio is large enough for no single debtor to be able to affect significantly the value of the portfolio: C → ∞. Under this hypothesis the law of large number ensures, in conditional terms, that the fraction of defaulting debtors (D/C ) equals the probability of default6 : And therefore, similarly to [8], we can set an expression for the unconditional probability:

 P

D C

 ≤x

[  ] ∫ D =E P ≤ x|Z = z = C

∫

∞

 P

−∞

D C

 ≤ x|Z = z · n (z ) dz =

∞

P (PD(z ) ≤ x|Z = z ) · n(z )dz .

=

(9)

−∞

By fixing z ∗ and writing the threshold for the liability value as a function of the probability of default:

  1 z∗ = √ · 1 − ρ · N −1 (x) − LV ; ρ

4 See [10]. 5 This hypothesis is consistent with the third one. 6 See [12].

LV = N −1 (PD).

(10)

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We can solve (9) as follows:

 P

D C

 ∫ ≤x =

∞ −z ∗

n(z )dz = N (z ∗ ).

(11)

Therefore, the cumulative distribution function of the fraction of defaulted debtors can be written as follows: F (x) = P



D C



[

1

≤x =N √ · ρ



1−ρ·N

−1

(x) − N

−1

(PD)

]

.

(12)

If we predetermine a level for the statistical likelihood (SL), we may write:

 P

D C



[

1

≤ x = SL → N √ · ρ



1−ρ·N

−1

(x) − N

−1

(PD)

]

= SL.

(13)

From (13) we can calculate x as:

[ √ ] ρ 1 −1 −1 x=N √ · N (SL) + √ · N (PD) . 1−ρ 1−ρ

(14)

For a 99.9% statistical likelihood level, it becomes obvious that (1) and (14) are the same expression. Summarizing: Basel II uses the ARSF model for determining minimum capital requirements for credit risk, which involves two basic hypotheses:

• The probability of default depends on a unique risk factor, which is the same for all debtors. • The portfolio is conformed by a large enough number of debtors to ensure that no single borrower can affect significantly the value of the portfolio (granularity). 3. Concentration risk under a supervisory approach Concentration risk has traditionally been covered by limiting the amount of exposure to any single debtor. Spanish banking regulation (see [13]) limits this exposure in two ways:

• By fixing a cap (25% of the equity) for the total exposure to a single business group. • By fixing another cap (800% of the equity) for the sum of large exposures. Large exposures are those over 10% of the total value of equity. However the Committee has not followed the path of limiting exposures to guarantee bank solvency. It has preferred to do so by requiring a minimum amount of capital for this aim. However Basel II has not envisaged any minimum capital requirement for concentration risk. Why? Because of the hypotheses assumed by the model used for quantifying credit risk capital requirements (which were highlighted in the preceding section). These hypotheses allow the use of the model by a wide range of entities. This is an advantage, but it also involves a drawback: the exclusion of concentration risk. That is why the Committee has envisaged additional capital requirements within Pillar II, and supervisory authorities are currently carrying out a series of analyses to determine the method for calculating the capital requirements for risk concentration. In 2008 the Spanish authorities issued a set of guidelines7 for banks’ self-assessment of capital requirements. In accordance with these, banks have been sending supervisory authorities specific data and information about their risk exposure. The Spanish guidelines8 specify that, regarding concentration risk, banks are to choose between the following two possibilities:

• A general approach, with banks using their own methods for quantifying the capital surcharge for their portfolio concentration.

• A simplified approach, with surcharge calculated by means of two predetermined concentration indexes: one for name concentration and the other for the sectorial concentration. Name concentration derives from large exposures to the same debtor. This was actually the only kind of concentration contemplated by the traditional approach when seeking to limit exposures. The need for a capital surcharge for this kind of concentration arises from the violation of one of the hypotheses of the ASRF model: the granularity one. The non-fulfilment of the other hypothesis (default depending on a unique risk factor) brings about the need for an additional surcharge to deal with sectorial concentration.9 In this area and within the simplified approach, the Spanish

7 See [14]. 8 No capital surcharge for concentration risk is now required from Spanish banks. However, the self-evaluation of capital necessities requires an estimation of concentration risk requirements. Currently this is done for purely informative purposes. In the future, however, this information will certainly be used by the supervisory authority for determining the capital surcharge for assumed risk. 9 For simplicity we will refer to this kind of concentration as sectorial. We must, however, be aware that it also comprises geographical concentration.

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supervisory authorities require the application of the Herfindahl–Hirschman Index (HHI), a generic concentration measure defined as follows: HHI =

n −

sh2i

shi =

i =1

si n ∑

(15) si

i=1

where si denotes the portfolio exposure (in monetary units) to sector i. This index measures suitably the level of concentration, but there is an important drawback: exposures to all sectors are given equal value. This is not valid for a credit portfolio, as each sector has its own risk which must be considered when analysing portfolio concentration. In the following section we will propose an alternative calculation, which could be used by banks within the general approach. 4. Proposal of an index for sector concentration risk The HHI (15) can be re-expressed as a matrix product: HHI = SH T · I · SH

(16)

where I is the identity matrix and SH represents the vector of the fractions of exposure to each sector. SH T = sh1



shn .



···

sh2

(17)

‘‘I’’ summarizes the weights given by the index to the exposure fractions of the different sectors. As it is the identity matrix, all the sectors are equally weighted, which is not valid for the credit portfolio of a bank. Think in terms of economic sectors; some industries are riskier than others, and concentration in the riskiest sectors does not have the same implications as concentration in low risk sectors. The former, being worse, require a higher capital reserve for solvency purposes. We, therefore, need an index that weights exposures relative to each sector’s risk. Furthermore, this index must also consider the relationship between every pair of sectors: the value of the index must be higher when portfolio concentration is greater in highly correlated sectors. Our proposal for a concentration index (CI ) meets both requirements. It can be expressed as follows: CI = SH T · VCM · SH

(18)

where VCM is the following matrix:

σ12 ∗ σ12 VCM =   ··· ∗ σ1n

σ12 ∗ 2 σ2 ··· ∗ σ2,

∗

∗

σi2 =

∗

σi2 n

max(σ ) 2 i

i=1

··· ··· ··· ···

 σij = max 0;

(19)

∀i = 1, . . . , n; σi2−1 ≥ σi2

 ∗

 σ1 ∗ σ2n   ···  ∗ 2 σn ∗

(20)

 σij n

max(σi2 )

 

∀i = 1, . . . , n.

(21)

i=1

We treat the negative covariance as null because we are not seeking a full quantification of portfolio risk, but rather, a quantification of risk weighted factors. To this end, positive correlations must not be compensated by negative ones. It is evident that HHI is a particular case of CI, in which all sectors have the same variance and no correlation between sectors exists. CI has some properties that make it appropriate for portfolio concentration risk. Property 1. The maximum CI is reached when all the portfolio investments are located in the maximum variance sector. Proof. We can write (18) as follows:

   − 1 − 2 2 ∗∗   CI = n · shi · σi + 2 · shi · shj · σij    i=1 max(σi2 ) i=1  i̸=j i=1 

n

n

∗∗

σij = max(0; σij ).

(22)

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From (22) it becomes obvious that the CI value will be equal to 1 when all exposures correspond to sector one.10 Let us bear in mind that in accordance with (20) sector one has the highest variance. Therefore: sh1 = 1 ∧ shj = 0 ∀j = 2, . . . , n → CI = 1.

(23)

Now, suppose that you have another portfolio with the same amount of exposure but shared between sector 1 and sector 2. CI value will be:

  CI (2) = sh21 · ∗ σ12 + sh22 · ∗ σ22 + 2 · sh1 · sh1 · ∗ σ12 .

(24)

If we take into account11 :

(A) sh2 = (1 − sh1 )

(B) ∗ σ22 = α(2) · ∗ σ12 |α(2) < 1

(C) ∗ σ12 = β(12) · ∗ σ12 |β(12) < 1.

(25)

We may write (22) as: CI (2) = sh21 + (1 − sh1 )2 · α(2) + 2 · sh1 · (1 − sh1 ) · β(12) · ∗ σ12





(26)

where the factor for the adjusted variance of sector 1 is a linear combination of:

• 1, with a coefficient equal to sh21 • α(2) , with a coefficient equal to (1 − sh1 )2 • And β(12) with a coefficient equal to 2 · sh1 (1 − sh1 ). The sum of the above mentioned coefficients equals 1. Taking this into account and considering that both α(2) and β(12) have positive values lower than 1, we can conclude:

[sh21 + (1 − sh1 )2 · α(2) + 2 · sh1 · (1 − sh1 ) · ∗ β(12) ] < 1 → CI (2) < CI (1) .

(27)

If we have another portfolio with exposures in the three highest variance sectors, the concentration index can be estimated by using an expression similar to (26): sh21 + sh21 · α(2) + (1 − sh1 − sh2 )2 · α(3) + 2 · sh1 · sh2 · β(12) CI (3) = · ∗ σ12 + 2 · sh1 · (1 − sh1 − sh3 ) · β(13) + 2 · sh2 · (1 − sh1 − sh3 ) · β(23)

[

]

(28)

where the factor for the adjusted variance of sector 1 is a linear combination of:

• • • • • •

1, with a coefficient equal to sh21 α(2) , with a coefficient equal to sh22 α(3) , with a coefficient equal to (1 − sh1 − sh2 )2 β(12) with a coefficient equal to 2 · sh1 sh2 β(13) with a coefficient equal to 2· sh1 (1 − sh1 − sh2 ) And β(23) with a coefficient equal to 2 · sh2 (1 − sh1 − sh2 ).

The sum of these coefficients equals 1. In view of this, and considering that both αs and βs have positive values lower than 1, we may conclude: CI (3) < CI (1) .

(29)

This argumentation can be extended to n sectors.



Property 2. When introducing exposures in a new sector into a portfolio, the index will decrease only when the risk (variance) of the new sector is lower than the portfolio’s original variance. Proof. We take as a starting point a portfolio with exposures in two sectors, A and B. Its index being: CI (AB) =

 sh2A · σA2 + sh2B · σB2 + 2 · shA · shB · ∗∗ σAB  ∗∗  σAB = max(0; σAB ).  max(σA2 ; σB2 )

(30)

The numerator of (30) can be split into two factors: sh2A · σA2 + sh2B · σB2 + 2 · shA · shB · ∗∗ σAB =

1 I(AB)

  · IA · σA2 + IB · σB2 + 2 · IA · IB · ∗∗ σAB

(31)

where I(j) expresses investment in sector j. According to the CI definition the following condition must be met:

 2 IA · σA2 + IB · σB2 + 2 · IA · IB · ∗∗ σAB ≤ σAB



where σ

2 AB

is the variance value for the initial portfolio.

10 We have expressed this index as CI . (1) 11 Let’s not forget that ∗ σ 2 < ∗ σ 2 → ∗ σ 2

1

12

< ∗ σ 21 .

(32)

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If, in this portfolio we introduce investments in a third sector with the condition, 2 σC2 < σAB .

(33)

Following the arguments used from (26) we may conclude: CI (ABC ) < CI (AB) . This argumentation can also be used when we have a portfolio with 3 or more sectors.

(34) 

Property 3. CI has a lower boundary: Its minimum value occurs when sectors are uncorrelated and every change in the fraction invested in a sector involves a decrease in the index. For investments in a large enough number of sectors CI will tend to 0. Proof. As we have seen, the adjusted covariance must be positive or zero. Therefore, the minimum CI will correspond to the set of portfolios with the least correlated sectors. Moreover, within this set, CI will decrease as investments in high variance sectors are replaced by investments in low variance sectors. When the relative weight of high variance sectors decreases, two opposing movements take place: 1. CI drops in relation to two factors: the decrease of the relative weight of the high variance sector; and the level of the variance. 2. CI rises as a consequence of the increase in relative weight of low variance sectors. It also depends on the relative weight and the variance of the sectors. When movement 1 is stronger than movement 2, CI drops. Otherwise CI grows or remains stable. Finally, when the number of sectors is high enough and the portfolio tends to be equally weighted, shi → 0 ∀i, the value of the index will also tend to 0. In accordance with this third property, when all the sectors have the same variance, the minimum CI value corresponds to an equally weighted portfolio. Finally we must emphasize that we have defined and analysed CI in the context of concentration on economic sectors. But it can easily be extended to geographical concentration. We only need to introduce specific risk factors (sectors) for these areas and proceed with them as with the other sectors.  Concluding remarks Basel II has generalised the approach of minimum capital requirements for guaranteeing the solvency of banks. Furthermore, it has established the methods and models to be used to determine these requirements for the main risks facing banks: market, operational and credit risk. The requirements for the latter are based on the ASRF model, which assumes two fundamental hypotheses: The probability of default depends on a unique risk factor, which is the same for all the debtors. And the portfolio is made up by a number of debtors large enough to ensure that no single borrower can affect significantly the value of the portfolio (granularity hypothesis). The violation of these hypotheses involves an additional risk (concentration risk) not covered by the ARSF model, and therefore, a re-evaluation of the capital requirements. In accordance with Pillar II of the Basel Accord, national supervisory authorities can require a capital surcharge for risks that are not included in the Pillar I (concentration risk being one of them). For sector concentration risk the Spanish supervisory authorities estimate this surcharge in two ways: a simplified approach based on the Herfindahl–Hirschman Index; and a general approach, with banks using their internally developed methods. In this paper we propose a concentration index that overcomes the drawbacks of HHI when used in the concentration risk environment: HHI considers all exposures equal regardless of the sector they belong to; and fails to contemplate linkages between sectors. The proposed index is based on an adjusted variance–covariance matrix in which no negative covariance is contemplated. As has been demonstrated the index has upper and lower boundaries, and drops as the level of concentration decreases (if, simultaneously, variance does not rise). Therefore, it may be used by banks to determine their capital surcharges. One of the greatest problems for the implementation of this index is likely to be the estimation of the variance and covariance for the various sectors. These are not directly observable variables and, therefore, proxies are needed. The variance and covariance of the sectorial indexes of the stock exchange may be suitable proxies. References [1] Basel Committee on Banking Supervision, Basel II: international convergence of capital measurement and capital standards: a revised framework, Bank for International Settlements, Basel, November, 2005. [2] M. Kalkbrener, An axiomatic approach to capital allocation, Mathematical Finance 15 (2005) 425–437. [3] J.C. Garcia, J.A. de Juan, A. Kreinin, D. Rosen, A simple multi-factor factor adjustment for the treatment of diversification in credit capital rules, in: Workshop Concentration Risk in Credit Portfolios, Frankfurt/Eltville, 2005. [4] K. Düllmann, Measuring business sector concentration by an infection model, Deutsche Bundesbank Discussion Paper Series 2, Banking and Financial Studies, No. 3, 2006. [5] M. Pykhtin, Multi-factor adjustment, Risk Magazine (March) (2004) 85–90. [6] K. Düllmann, N. Masschelein, A tractable model to measure sector concentration risk in credit portfolios, Journal of Financial Services Research 32 (2007) 55–79. [7] K. Düllmann, N. Masschelein, The impact of sector concentration risk in loan portfolios on economic capital, Financial Stability Review, National Bank of Belgium, 2006, pp. 175–186.

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[8] G. Bonti, M. Kalkbrener, C. Lotz, G. Stahl, Credit risk concentrations under stress, in: Workshop Concentration Risk in Credit Portfolios, Frankfurt/Eltville, 18–19, November, 2005. [9] P. Uberti, S. Figini, How to measure single-name credit risk concentrations, European Journal of Operational Research 202 (2010) 232–238. [10] M. Gordy, A comparative anatomy of credit risk models, Journal of Banking and Finance 24 (1–2) (2000) 19–149. [11] R. Merton, On the pricing of corporate debt: the risk structure of interest rates, Journal of Finance 34 (1974) 449–470. [12] P.J. Schönbucher, Factor models for portfolio credit risk, Journal of Risk Finance 3 (1) (2001) 45–56. [13] Banco de España, Circular 3/2008, de 22 de mayo, del Banco de España, a entidades de crédito, sobre determinación y control de los recursos propios mínimos, Boletín Oficial del Estado de 10 de Junio, 2008. [14] Banco de España, Guía del proceso de autoevaluación del capital de las entidades de crédito, Boletín Oficial del Estado de 10 de Junio, 2008.