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How to measure single-name credit risk concentrations
European Journal of Operational Research 202 (2010) 232–238
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
How to measure single-name credit risk concentrations Pierpaolo Uberti a, Silvia Figini b,* a b
Department of Quantitative Methods for Business and Economic Sciences, University of Milano Bicocca, Italy Department of Statistics and Applied Economics, University of Pavia, Italy
a r t i c l e
i n f o
Article history: Received 26 November 2008 Accepted 6 May 2009 Available online 12 May 2009 Keywords: Credit risk Concentration Gini index Herfindahl–Hirschman index Granularity adjustment
a b s t r a c t Credit risk concentration is one of the leading topics in modern finance, as the bank regulation has made increasing use of external and internal credit ratings. Concentration risk in credit portfolios comes into being through an uneven distribution of bank loans to individual borrowers (single-name concentration) or in a hierarchical dimension such as in industry and services sectors and geographical regions (sectorial concentration). To measure single-name concentration risk the literature proposes specific concentration indexes such as the Herfindahl–Hirschman index, the Gini index or more general approaches to calculate the appropriate economic capital needed to cover the risk arising from the potential default of large borrowers. However, in our opinion, the Gini index and the Herfindahl–Hirschman index can be improved taking into account methodological and theoretical issues which are explained in this paper. We propose a new index to measure single-name credit concentration risk and we prove the properties of our contribution. Furthermore, considering the guidelines of Basel II, we describe how our index works on real financial data. Finally, we compare our index with the common procedures proposed in the literature on the basis of simulated and real data. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Credit risk forecasting and credit risk concentration are two of the leading topics in modern finance, as the bank regulation has made increasing use of external and internal credit ratings (Basel committee on banking supervision, 2005). Considering credit risk forecasting process, banks need to discriminate good customers from bad customers in terms of their creditworthiness. The second point of interest of credit risk is concentration of exposures in credit portfolios. It may arise from two types of imperfect diversification. The first type, name concentration, relates to imperfect diversification of idiosyncratic risk in the portfolio either because of its small size or because of large exposures to specific individual obligors. The second type, sector concentration, relates to imperfect diversification across systematic components of risk, namely sectoral factors. The existence of concentration risk violates one or both of two key assumptions of the asymptotic single-risk-factor (ASRF) model that underpins the capital calculations of the Internal Ratings Based (IRB) approaches of the Basel framework.
* Corresponding author. Tel.: +39 0382984660. E-mail addresses: [email protected] (P. Uberti), silvia.fi[email protected] (S. Figini). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.05.001
The ASRF model in the new Basel capital framework does not allow for the explicit measurement of concentration risk. In the riskfactor model frameworks that underpin the IRB risk weights of Basel II, credit risk of a portfolio is caused by two main sources, systematic and idiosyncratic risks. Systematic risk represents the effect of unexpected changes in macroeconomic and financial market conditions on the performance of borrowers. Borrowers may differ in their degree of sensitivity to systematic risk, but few firms are completely indifferent to the wider economic conditions in which they operate. Therefore, the systematic component of portfolio risk is unavoidable and only partly diversifiable. Meanwhile idiosyncratic risk represents the effects of risks that are particular to individual borrowers. As a portfolio becomes more fine-grained, in the sense that the largest individual exposures account for a smaller share of total portfolio exposure, idiosyncratic risk is diversified away at the portfolio level. This risk is totally eliminated in an infinitely granular portfolio (one with a very large number of exposures) as unsystematic risk vanishes in Capital Asset Pricing Model (CAPM). The ASRF model framework underlying the IRB approach is based on two key assumptions. The first one is that bank portfolios are perfectly fine-grained and the second one is that there is only one source of systematic risk. When these two assumptions hold, one can easily calculate required capital level depending on only one systematic risk. In case of well-diversified portfolio, the capital required for a loan does not depend on the portfolio it is added to.
P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238
This simplicity makes the new IRB framework applicable to a wider range of countries and institutions. However, if any of two assumptions is violated, there is no guarantee that the IRB approach and ASRF model will be accurate. Starting from an idealized portfolio model, Kalkbrener (2005) has discussed the concepts of name and sector concentration and has demonstrated that a multifactor model is needed as the basis for stressing sector concentration. Seminal contribution to measure credit risk concentrations under stress are given by Egloff et al. (2004), Embrechts (2007), Glasserman and Li (2003), Gordy (2004). Statistical models play an important role in studying concentration risk in credit portfolios. Despite this strong motivation, the literature on the topic is quite rare. The Basel committee on banking supervision (2004), Gordy (2003) and Wilde (2001) consider the asymptotic single-risk-factor model foundation. Looking at the Value at Risk (VaR) adjustment for sector concentration, a simple multi-factor adjustment for the treatment of diversification in credit capital rules is proposed by Cespedes et al. (2004) and Pykhtin (2004). Solutions to extend this model by means of a granularity adjustment are described in Dembo et al. (2004), Tasche et al. (2003), Gordy (2004), Gouriroux et al. (2000), Martin and Wilde (2002) and Wilde (2001). On the basis of previous evidences and references, we propose in this paper a novel methodological approach to measure singlename credit concentration risk. To reach this objective, we describe a new concentration credit index and we compare our proposal with the common approach described by the Basel committee on banking supervision. The paper is organized as follows. Section 2 describes heuristic methods available for measuring concentration risk. Section 3 shows our proposal to measure concentration risk. Section 4 reports two applications and finally, Section 5 the conclusions. 2. Measuring single-name concentration risk The approaches available for measuring single-name concentration can be broken down into model-free (heuristic) and modelbased methods. The first approach is to adapt indices of concentration such as Herfindahl–Hirschman index (Kwoka, 1977) and Gini coefficient (Gini, 1921). While these indices could be good measures for concentration itself, they do not seem to serve well for concentration risk because they do not take distribution of different quality obligors into account. The second approach is granularity adjustment suggested by Gordy (2003) and Gordy (2004): its difficulties in implementation and huge data requirement make it hard to be performed in practice. Usually practitioners use both approaches to measure the concentration risk of their portfolio. However the concentration measurement index such as Herfindahl–Hirschman index could not measure the actual risk accurately, granularity adjustment sometimes overestimates the actual concentration risk of a portfolio. Before to describe our concentration index, in this section we review classical concentration measures proposed by Basel II for single-name credit concentration risk measurement. 2.1. Herfindahl–Hirschman index (HHI) The HHI is in general a statistical measure of concentration. The HHI is defined as the sum of the squares of the relative portfolio shares of all borrowers.
HHI ¼
n X i¼1
x2i ;
233
where xi is the portfolio share of borrower i, and n is the number of borrowers under observation. The HHI ranges from 1=n to 1, so that the normalized HHI index can be written as:
HHI ¼
H 1n : 1 1n
Well-diversified portfolios with a large number of small credits have an HHI value close to zero, whereas heavily concentrated portfolios can have a considerably higher HHI value. In the extreme case where we observe only one credit, the HHI takes the value of 1. In the context of the measurement of concentration risk, the HHI formula is included as a main component of a number of approaches. However, HHI itself has some drawbacks to be used for measuring concentration risk. At first, it does not consider distribution of exposures across credit ratings: portfolios with the same HHI values can have different sizes of concentration risks. Secondly, it does not allow concentration risk to be expressed directly as economic capital, so it needs additional functions to calculate economic capital for concentration risk. We remark that HHI does not take quality of a portfolio into consideration. It implies that the portfolios which are differently distributed across the credit ratings can have the same HHI. 2.2. Gini index ðGÞ The G coefficient provides a further method of measuring single-name concentration. This ratio can be interpreted as a concentration index, i.e. a measure of the deviation of a distribution of exposure amounts from an even distribution. In general G is a summary statistic of the Lorenz curve (Lorenz, 1905) and a measure of inequality in a population. More formally, G could be expressed as:
G ¼ 1 þ 1=n ð2=n2 xÞðx1 þ 2x2 þ 3x3 þ þ nxn Þ;
ð1Þ
where xi ; i ¼ 1; . . . ; n is the credit amount by each borrower in a specific portfolio in decreasing order of size and x is the mean of P x ¼ 1n ni¼1 xi . G is thus a weighted sum of xi ; i ¼ 1; . . . ; n, equal to the shares, with the weights determined by rank order position. A coefficient close to zero signifies a homogeneous portfolio in which all of the exposure amounts are distributed equally; a coefficient close to one means a highly concentrated portfolio. A fundamental disadvantage of using G to measure concentration, however, is the fact that the size of the portfolio is not taken into consideration. For example, a portfolio with a few equal sized loans has a lower coefficient than a better diversified, larger credit portfolio containing loans of different amounts. Moreover, the G index may rise if a relatively small loan to another borrower is added to the portfolio despite the fact that this diminishes the concentration. For these reasons, G is only of limited suitability for measuring single-name concentration risk. 2.3. Granularity adjustment ðGAÞ The GA for the ASRF (see e.g. Wilde, 2001) model constitutes an approximation formula for calculating the appropriate economic capital needed to cover the risk arising from the potential default of large borrowers. GA is an extension of the ASRF model which forms the theoretical basis of the IRB approaches. Through this adjustment, originally omitted single-name concentration is integrated into the ASRF model. The GA can be calculated as the difference between unexpected loss in the real portfolio and in an infinitely granular portfolio with the same risk characteristics. To measure the GA, the guidelines of the Basel committee on banking supervision, proposes the following equation:
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GA ¼ C HHI
n X
xi ;
ð2Þ
i¼1
where C is a proportional constant which is a function of the ðq; PD; LGDÞ. In particular, q is the correlation coefficient, PD, the probability of default, estimated by statistical predictive models and LGD is the loss given default. We remark that C is a well calibrated parameter and it is fixed on the basis of the regulation. Basel II proposes to use GA on the basis of a fixed value for LGD ¼ 45%, coefficient of correlation q ¼ 18% and the constant C. The constant C is derived on the basis of particular value of PD, (see e.g. Basel committee on banking supervision, 2004). In particular, fixed q ¼ 18%; LGD ¼ 45% and the PD, C is derived as shown in Table 1. This is an important point because fixed choices for q and LGD, PD and C leads to have biased estimation for concentration risk. Furthermore it is recognized that each subject presents different risk in terms of solvency expressed by the PD. Considering this point, we remark that HHI and G are risk independent. The GA consider the specific PD, only through the constant C. We also remark that, as we know from Basel II, C is derived only for specific values of PD. Empirical evidence shows that it was possible to establish an approximately linear relationship between the granularity adjustment and the HHI for the portfolios. This indicates that the HHI is suitable as a measure of single-name concentration, in particular in view of its relatively simple calculation method. However, in the case of small portfolios, which usually have an higher HHI value, different borrower specific probabilities of default play a greater role than in the case of large portfolios with low HHI values. Thus, for such small portfolios, the GA leads to a wider dispersion than in the case of more diversified portfolios with low HHI values, in which the effects of the different probabilities of default tend to be evened out. This finding shows that, at least for relatively small portfolios for which idiosyncratic risk plays a greater role, a GA holds out more promise for providing information than the HHI. The accurately and transparently measurement of credit concentration risk motivates the present paper, which supports a novel and general way to measure concentration risk. In particular, we propose a new index which consider both the measures: risk and credit size. We point out that as measures of risk we consider the PD, variances or more in general credit scores measurements. We describe our proposal in Section 3.
exposures to a single credit, or asset class, but also from linkages between asset classes and risk. Let us consider the classical portfolio selection scheme (Markowitz, 1950), as in the mean-variance analysis. Consider the variance–covariance matrix ðn nÞ; R ¼ frij g, for i ¼ 1; . . . ; n and j ¼ 1; . . . ; n, associated with the chosen portfolio. As usual R is a definite positive square real matrix. If we assume that the covariance matrix R describes the risk of the portfolio r2P , we can write it in the classical way as:
r2P ¼ x0 R x ¼
n X
x2i rii þ
i¼1
n n X X
xi xj rij
ð3Þ
i¼1 j¼1;i–j
where x 2 Rn is the vector that contains the relative shares of the assets in portfolio, while n is the number of assets. The vector x is such P that xi P 0 for i ¼ 1; . . . ; n and ni¼1 xi ¼ 1. In order to simplify the calculation assume for the time being that xi ¼ 1n, for i ¼ 1; . . . ; n. Therefore it is possible to rewrite the risk r2P of the portfolio as:
r2P ¼
n n n X 1X 1 n1 X rij rii þ n i¼1 n n i¼1 j¼1;i–j nðn 1Þ
ð4Þ
where the second term is multiplied and divided by n 1. The first term in Eq. (4) is the idiosyncratic risk component while the second one is the market risk component. In other words, the first term represents the risk of the portfolio due to the variance of every single asset while the second term represents the risk for an investor who decided to invest on that specific market. When the dimension of the portfolio increases, with n ! 1, the first term in Eq. (4) vanishes and:
lim r2P ¼ lim
n!1
n!1
n n X n1 X rij ij : ¼r n i¼1 j¼1;i–j nðn 1Þ
ð5Þ
ij is the mean of the covariances and it represents the The value r minimum value of risk that an investor has to take working on that specific market. Let us focus the attention on the first term specific risk component. The idea is to use the first term in Eq. (4) in order to measure inequality and concentration at the same time. In particular, we are interested not only in inequality but also in diversification. We can define a new inequality index in the following way:
IV ¼
n X
x2i rii ;
i ¼ 1; . . . ; n
ð6Þ
i¼1
Let us now rewrite Eq. (6) in a more compact and convenient way:
IV ¼ x0 V x
3. Our proposal As we have described in the previous sections, there are indexes useful to measure concentration that in our opinion must be improved. Historical experience shows that concentration of credit risk in asset portfolios has been one of the major causes of bank distress. This is true both for individual institutions as well as banking systems at large. The failures of large borrowers like Enron, Worldcom and Parmalat were the source of sizeable losses in a number of banks. These examples illustrate the importance of measuring concentration risk in credit portfolios of banks that arises not only from
ð7Þ
where V is the diagonal positive definite square matrix of the variances. The matrix V contains the risk information available about the activities in the portfolio and is equivalent to define a relation of order within the activities. In other words we can order the activities from the more risky one to the lower one using the rii coefficients, i.e. the variances. According with previous assumptions we note that IV P 0. In order to obtain a normalized index, IV 2 ð0; 1, we simply observe that IV 6 maxfrii ; i ¼ 1; . . . ; ng. In fact, the extreme value of the index is obtained when the portfolio contains the most risky activity only. Therefore we can define a normalized index IV as:
Table 1 Granularity adjustment approach. PD
0.5%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
C
0.773
0.784
0.848
0.885
0.909
0.927
0.939
0.948
0.955
0.959
0.963
P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238
Pn IV ¼
2 i¼1 xi
rii
i ¼ 1; . . . ; n:
;
maxfrii g
ð8Þ
If no risk information is available on the activities in the portfolio, i.e. V ¼ k I; k 2 Rþ , where I is the identity matrix, we obtain:
IV ¼
k x0 Vx ¼ maxfrii g
Pn
2 i¼1 xi
k
¼
n X
x2i ¼ HHI:
ð9Þ
i¼1
In other words, when we are not able to distinguish more risky credits from lower risky ones and we associate them the same variance, we obtain the HHI index as a special case of the IV index. Furthermore, it is possible to interpret the HHI index as a particular measure of idiosyncratic risk of a portfolio that considers only the relative weights: more precisely, we remark that the HHI index assumes constant variance across credit. 3.1. Properties of our proposed index In this section we prove that the index IV satisfies a number of useful properties: Property 3.1. In the case of absence of inequality and absence of concentration the index IV ! 0. Proof. Absence of inequality means that xi ¼ 1n ; i ¼ 1; . . . ; n while absence of concentration means n ! 1; then:
lim IV ¼ lim
n!1
n!1
1 n
Pn
1 i¼1 n
rii
maxfrii g
¼ 0;
i ¼ 1; . . . ; n:
Property 3.2. In the case of maximum inequality and maximum concentration the value of the index must be equal to 1. Proof. Maximum inequality means that we have only one activity in the portfolio with weight equal to 1 and r ¼ maxfrii g, then:
IV ¼
x2 r ¼ 1: maxfrii g
235
Hardy et al. (1934), mathematical economists, have demonstrated the equivalence that prevails between inequality measures, dominance between income distributions and transfer principles. Accordingly, a wide literature has been devoted to such transfers aiming at proposing some well suited measures of inequality as well as deriving, by means of decision theory, the behavior of decision maximizers under risk and uncertainty (see e.g. Chateauneuf et al., 2002). In the spirit of Dalton’s transfer (1920), Kolm (1976) proposed the diminishing transfer principle, which postulates that an income transfer, valued to be d > 0 from a higher-income individual to a lower-income one, yields a better impact on social welfare indices insofar as incomes ðxÞ are the lowest possible, given that individuals’ ranking remains unchanged after such transfers. However, as mentioned by Mehran (1976) and Kakwani (1980), these transfers do not involve the difference between individuals’ ranks. Hence, they propose the transfer sensitivity property. Other principles have been proposed, developing some new transfer concepts such as Fleurbaey and Michel’s (2001) proportional transfer principle, exhibiting a small loss in the transfer between the donor and the recipient, or Gajdos (2004) a-spread simulating global changes in a distribution with simultaneous progressive transfers (from one higher-income agent to all lower-income ones) and simultaneous regressive transfers (from the same agent to all higher-income ones) in order to obtain the implications on the decision maker’s behavior. Without exploring new transfer principles, let us now suppose that transfers may occur between borrowers that are under observation following Property 3.4. Before introducing Property 3.5, let us define homogeneity of a statistical variable. A statistical variable shows maximum homogeneity when: xk ¼ 1 for some k between 1 and n, and xi ¼ 0 for all i – k; A statistical variable shows minimum homogeneity when: xi ¼ 1n 8 i ¼ 1; . . . ; n.
Property 3.3. If Y ¼ a X, where X is a diagonal covariance matrix, with a 2 ð0; 1Þ, then IV ðYÞ ¼ IV ðXÞ.
A statistical measure is an index of homogeneity if it is maximum under maximum homogeneity and minimum under minimum homogeneity.
Proof. By assumptions we have:
Property 3.5. The IV is definitely a homogeneity index, IV 2 ½0; 1.
2
a r11
6 Y ¼6 4 0 0
0 .. .
0
0
a rnn
3
Proof. In the case of minimum homogeneity the value of the index IV is defenetely 0 as shown in 3.1. In the case of maximum homogeneity the value of the index IV is equal to 1 as shown in 3.2.
7 7 0 5:
Note that the new index IV satisfies the properties of a homogeneity index only when n ! 1. This depends from the fact that the proposed index gives a measure of homogeneity and concentration at the same time.
Therefore,
Pn IV ðYÞ ¼
P
rii Þ a ð ni¼1 x2i rii Þ ¼ ¼ IV ðXÞ; i ¼ 1; . . . ; n: maxfa rii g a ðmaxfrii gÞ 2 i¼1 xi ða
Property 3.4. Let 0 6 r11 6 6 rnn and 0 < h 6 xj . If we move the weight h from xj to xj1 the value of the index IV decreases.
4. Application
and this implies IV decreases. h
In order to explain the performance of our index we refer to the guidelines of Basel II and described also in the documentation of Basel committee on banking supervision (2005). As described in Section 2 and referring to Table 1, Basel II proposes to use the GA on the basis of a fixed value for LGD; q and the constant C. As suggested by Basel committee on banking supervision (2005), we consider two portfolios, A and B, which are composed in the following way:
Let’s observe that Property 3.4 is very similar to the principle of transfers (see e.g. Dalton, 1920) but it is more meaningful in this case respect to the principle of transfers.
– A is composed by 10 statistical units ðx1 ; . . . ; x10 Þ each one with relative frequency equal to 10% (same size);
Proof. By hypothesis we have
rðj1Þðj1Þ 6 rjj ; then
ðxj þ hÞ2 rðj1Þðj1Þ þ ðxj hÞ2 rjj < x2j1 rðj1Þðj1Þ þ x2j rjj
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P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238
– B is composed by 100 statistical units ðx1 ; . . . ; x100 Þ each one with relative frequency equal to 1%. Basel committee on banking supervision supposes that the PD for the statistical units in portfolio A is equal to the PD for the statistical units in portfolio B and it is equal to 0.01. On the basis of Table 1 we consider C ¼ 0:784. Simple calculation show that the GA for portfolio A is equal to GAðAÞ ¼ 7:84 and for portfolio B is equal to GAðBÞ ¼ 0:784. We remark that, the GA when PD is constant in a specific portfolio, and consequently C is constant and LGD and q fixed, go to zero when the granularity increases. Considering the data at hand we remark that our index proposed in Section 3 shows for portfolio A a specific value of IA ¼ 0:1 and for portfolio B, IB ¼ 0:01. We present a new example. Considering a portfolio C composed by 100 statistical units with xi ¼ 0:01 for i ¼ 1; . . . ; 100 and PD equal to 10%: in this situation IC ¼ 0:01 which is equal to the index for portfolio B. Therefore, the value of the index is strictly decreasing with the increasing size of the portfolio. This empirical evidence leads us to propose the following remark. P Let ni¼1 x2i ! 0 and rii ¼ k 2 R, i.e. max rii ¼ k, then IV ðnÞ is a strictly decreasing function of n. In fact, we have:
Pn IV ðnÞ ¼
2 i¼1 xi
k
rii
Pn ¼
2 i¼1 xi k
k
! 0:
We obtain a strictly decreasing index with the increasing size of the portfolio if we assume that all the risks associated with the single asset classes are equal. Furthermore, we remark that it is very easy to derive, on the basis of our proposal, a measure for the credit concentration that is independent by C, LGD and q. More precisely, in order to compute our index we only use xi (the portfolio share of the borrower i, with ð0 < xi < 1Þ) and rii , the PD for the ith statistical unit. Our approach derives as result a specific value bounded between 0 and 1 ð0 6 PD 6 1Þ, independently from fixed value for LGD and q. This is an important point because specific choice for q and LGD, PD and C leads to have unbiased estimation for concentration risk. As we can understand from the above report, suppose that for a specific statistical unit in portfolio A we observe PD ¼ 4:735%: in this case we have not available C to compute the GA. But, in this case our index propose a specific value of 0.01. Application results show that in general the value of the index IV is not strictly decreasing with the growing size of the portfolio, like, for example for HHI index. This fact seems coherent to us, because the reduction of the risk of the portfolio due to the diversification effect could not be enough when we add to the portfolio an high risk component. Nevertheless, we can prove that it is possible to obtain a fixed value of the index if we diversify enough the portfolio. In order to explain how our index work, we consider an empirical analysis based on annual 1996–2004 financial real data for 1003 companies from Creditreform, which is one of the major rating agencies for SMEs in Germany. In order to meet the requirements of Basel II, many SMEs in Germany face the challenge of adjusting their financing structure and putting it on a more orderly footing. Alternative forms of financing play an increasingly important role. Still, bank loans play an extremely important role in the external financing of corporate activities of German SMEs. To measure credit concentration we use the total amount of loan and the corresponding PD estimated on the basis of the longitudinal predictive model described in Figini and Fantazzini (2007), that will now be briefly described.
On the basis of our real dataset, we present the following specification for longitudinal models: for observation k; ðk ¼ 1; . . . ; KÞ, time t; ðt ¼ 1; . . . ; TÞ and sector j; j ¼ 1; . . . ; J, let Y ktj denote the response solvency variable, let X ktj denote a p 1 vector of candidate predictors (fixed effects), and let Z ktj denote a q 1 vector of candidate predictors (random effects), where 1k Nð0; RÞ are random effects for SMEk and R is the covariance matrix. The elements of Y ktj ¼ ðy1tj ; . . . ; yktj Þ0 are modelled as conditionally independent random variables from a simple exponential family:
Y ktj hktj bðhktj Þ þ cðY ktj ; /Þ ; aktj ð/Þ
pðY ktj jX ktj ; Z ktj ; 1k Þ / exp
ð10Þ
where hktj is the canonical parameter related to the linear predictor gktj ¼ X 0ktj b þ Z 0ktj 1k with a p 1 vector of fixed effects regression coefficient b and a q 1 vector of subject specific random effects 1k Nq ð0; RÞ; / is a scalar dispersion parameter and aktj ; b; c are known functions with aktj ð/Þ ¼ x/ , where xktj is a known weight ktj (see e.g. Dunson and Cai, 2006). We are interested in predicting the expectation of the response as a function of the covariates. The expectation of a simple binary response is just the probability that the response is 1:
EðY ktj jX ktj ; Z ktj ; 1k Þ ¼ pðY ktj ¼ 1jX ktj Þ:
ð11Þ
In linear regression, this expectation is modelled as a linear function b0 X ktj of the covariates. For binary responses, as in our case, this approach may be problematic because the probability must lie between 0 and 1, whereas regression lines increase (or decrease) indefinitely as the covariate increases (or decreases). Instead, a nonlinear function is specified in one of two ways:
pðY ktj ¼ 1jX ktj Þ ¼ hðb0 X ktj Þ;
ð12Þ
or
gfpðY ktj ¼ 1jX ktj Þg ¼ b0 X ktj ¼ mk ;
ð13Þ
where mk is referred to as the linear predictor. These two formulations are equivalent if the function hð:Þ is the inverse of the link function gðÞ. We have introduced two components of a generalized linear model: the linear predictor and the link function. The third component is the distribution of the response given the covariates. For binary response, this is always specified as Bernoulli ðpk Þ. Typical choice of link function g is the logit link. The logit link is appealing because it produces a linear model for n o pðY ¼1jx Þ the log of the odds, ln 1pðYktj ¼1jxk Þ , implying a multiplicative modktj
k
el for the odds themselves (see e.g. Dobson, 2002). To relax the assumption of conditional independence among the firms given the covariates, we can include a subject-specific random intercept 1k Nð0; wÞ in the linear predictor:
gfpðY ktj ¼ 1jX ktj ; 1k Þg ¼ b0 X ktj þ 1k :
ð14Þ
This is a simple example of a generalized linear mixed model because it is a generalized linear model with both fixed effects X ktj and a random effects 1k . The available data was made up of about 1000 SMEs. Among the corresponding 1000 records, 280 were incompleted or economically illogical and therefore have been discarded. The resulting filtered data set is composed of 702 SMEs and for each statistical unit we know the estimated PD and the total loan. For the loan variable, the data at hand shows a minimum value of 37,878.36 and a maximum value of 1,66,831.4. The median is equal to 99,962.950 and the coefficient of variation is equal to 0.19. The kurtosis and the skewness are respectively equal to 0.187 and 0.127. Considering the loan variable, Fig. 1 depicts the relative histogram. As we observe from Fig. 1, the loan variable can be approx-
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P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238 Table 2 HHI, G and IV .
Fig. 1. Histogram for the loan.
imated with a normal density probability function. Statistical tests, such us the Kolmogorov Smirnov and the Shapiro Wilk, confirm this empirical evidence. More precisely, considering the loan variable, the first quartile (25% of the SMEs) is equal to 86,966.65, the median (50% of the SMEs) is equal to 99,962.95, and the third quartile (75% of the SMEs) is equal to 1,13,439.6. For the variable PD, estimated on the basis of the longitudinal model, the minimum value is 0 and the maximum value is 0.867. The median is equal to 0.076 and the coefficient of variation is equal to 1.012. The kurtosis and the skewness are respectively equal to 3.448 and 1.523. Sorting the PD, Fig. 2 reports the histogram. The distibution is clearly not normal. More precisely, considering PD, the first quartile (25% of the SMEs) is equal to 0.0248, the median (50% of the SMEs) is equal to 0.0763, and the third quartile (75% of the SMEs) is equal to 0.20. Bringing toghether the PD derived from the longitudinal model and the variable loan, the HHI is equal to 0.00148, the G index is equal to 0.110 and IV is equal to 0.000176. We divide the data
Segment
HHI
G
IV
A B
0.08 0.079
0.074 0.0572
0.00037 0.00033
set in two segments. The first segment A is composed of 351 loans with a total amount less than 100,000 euros. For segment A the maximum value for PD is equal to 0.709 with a coefficient of variation of 0.970. The second segment B is composed by 351 loans with a total amount greater than 1,00,000 euros. For segment B the maximum value for PD is equal to 0.876 with a coefficient of variation of 1.058. We computed for segment A and B the HHI index, the G index and the IV index. The results are in Table 2. To better compare the G, the HHI index and the index IV we propose an example based on the available data. We consider 8 different loans, with different probabilities of default and consider four different portfolios P1 ; P2 ; P3 ; P4 , as reported in Table 3. For simplicity of calculation we assume the same value of 50,000 euros for every loan. Let now evaluate the values of the concentration indexes associated with the four portfolios of Table 4. In particular we compute G, HHI and our proposed index, IV . We are working with the probabilities of default, so that we fix the value of the normalization constant at 1 in order to calculate the value of the index IV . As we can see in Table 4 the value of G is equal to 0 for portfolio P1 ; P2 ; P3 ; P4 . This evidence shows that G does not take into account the size of the portfolio and the risk of the loans. If we consider the HHI index we can see that the value of the index depends only from the share of every single loan. Also in this case the risk of the loans is not considered. Considering the data at hand, the value of HHI is strictly decreasing when the size of the portfolio increases; this means that differentiation automatically decreases the risk. We underline that the value of HHI is strictly positive for every portfolio in the example: this fact shows that it is impossible to achieve the best theoretical value, i.e. HHI ¼ 0, in a real portfolio. We finally focus the attention on the behaviour of IV . First of all, we see that this index takes into account the risk of the loans. In fact, if we consider portfolios P 1 and P 2 we observe that the portfolio with the more risky loans (in this case case P 2 ) has a greater va-
Table 3 Example: portfolios of loans. Loans
Probability of default
P1
P2
P3
P4
a b c d e f g h
0.08 0.1 0.11 0.4 0.41 0.47 0.5 0.2
50,000 50,000 50,000 – – – – –
– – – 50,000 50,000 50,000 – –
50,000 50,000 50,000 – – – 50,000 –
50,000 50,000 50,000 – – – 50,000
Table 4 Values of concentration.
Fig. 2. Histogram for the PD.
Portfolios
G
HHI
IV
P1 P2 P3 P4
0 0 0 0
0:3 0:3
0:032 0:142 0.049375 0.030625
0.25 0.25
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lue for the index. Furthermore, the value of IV depends on the size of the portfolio as shown on portfolios P 1 ; P 2 and P3 . The more interesting property is that the value of the index is not strictly decreasing but it depends from the added risk loan in a portfolio. In particular, if we add loan g to the portfolio P 1 the value of the index IV increases, while adding the loan h the value decrease. In the first case the differentiation effect is not enough to counterbalance the increase of the risk.
5. Conclusions In this paper we propose a new index to measure single-name concentration for credit risk. We think that our contribution could be interesting from a theoretical and an empirical point of view. In particular, banks and financial institutions, based on our proposal, are able to improve the single-name credit risk concentration measurement. Acknowledgements The authors acknowledge financial support from the MIUR-FIRB 2006–2009. We thank the referees and Prof. Paolo Giudici for useful comments and suggestions. References Basel committee on banking supervision, 2004. International convergence of capital measurement and capital standards: a revised framework. Working Paper available at. Basel committee on banking supervision, 2005. Amendment to the Capital Accord to Incorporate Market Risks, Basel. Cespedes, G., Keinin, G.C., Rosen, D., 2004. A simple multi-factor factor adjustment for the treatment of diversification in credit capital rules. Algorithmics Working Paper. Chateauneuf, A., Gajdos, T., Wilthien, P.H., 2002. The principle of strong diminishing transfer. Journal of Economic Theory 103, 311–333.
Dalton, H., 1920. The measurement of inequality of incomes. Economic Journal 30, 348–361. Dembo, A., Deuschel, J.D., Duffie, D., 2004. Large portfolio losses. Finance and Stochastics 8 (1). Dobson, A.J., 2002. Introduction to Generalised Linear Model. Chapman & Hall. Dunson, D.B., Cai, B., 2006. Bayesian covariance selection in generalized linear mixed models. Biometrics 62, 446–457. Egloff, D., Leippold, M., Vanini, P., 2004. A simple model of credit contagion. Working Paper, University of Zurich. Embrechts, P., 2007. Aggregating Risk Capital: Lessons Learned From Basel II, available at
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
How to measure single-name credit risk concentrations Pierpaolo Uberti a, Silvia Figini b,* a b
Department of Quantitative Methods for Business and Economic Sciences, University of Milano Bicocca, Italy Department of Statistics and Applied Economics, University of Pavia, Italy
a r t i c l e
i n f o
Article history: Received 26 November 2008 Accepted 6 May 2009 Available online 12 May 2009 Keywords: Credit risk Concentration Gini index Herfindahl–Hirschman index Granularity adjustment
a b s t r a c t Credit risk concentration is one of the leading topics in modern finance, as the bank regulation has made increasing use of external and internal credit ratings. Concentration risk in credit portfolios comes into being through an uneven distribution of bank loans to individual borrowers (single-name concentration) or in a hierarchical dimension such as in industry and services sectors and geographical regions (sectorial concentration). To measure single-name concentration risk the literature proposes specific concentration indexes such as the Herfindahl–Hirschman index, the Gini index or more general approaches to calculate the appropriate economic capital needed to cover the risk arising from the potential default of large borrowers. However, in our opinion, the Gini index and the Herfindahl–Hirschman index can be improved taking into account methodological and theoretical issues which are explained in this paper. We propose a new index to measure single-name credit concentration risk and we prove the properties of our contribution. Furthermore, considering the guidelines of Basel II, we describe how our index works on real financial data. Finally, we compare our index with the common procedures proposed in the literature on the basis of simulated and real data. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Credit risk forecasting and credit risk concentration are two of the leading topics in modern finance, as the bank regulation has made increasing use of external and internal credit ratings (Basel committee on banking supervision, 2005). Considering credit risk forecasting process, banks need to discriminate good customers from bad customers in terms of their creditworthiness. The second point of interest of credit risk is concentration of exposures in credit portfolios. It may arise from two types of imperfect diversification. The first type, name concentration, relates to imperfect diversification of idiosyncratic risk in the portfolio either because of its small size or because of large exposures to specific individual obligors. The second type, sector concentration, relates to imperfect diversification across systematic components of risk, namely sectoral factors. The existence of concentration risk violates one or both of two key assumptions of the asymptotic single-risk-factor (ASRF) model that underpins the capital calculations of the Internal Ratings Based (IRB) approaches of the Basel framework.
* Corresponding author. Tel.: +39 0382984660. E-mail addresses: [email protected] (P. Uberti), silvia.fi[email protected] (S. Figini). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.05.001
The ASRF model in the new Basel capital framework does not allow for the explicit measurement of concentration risk. In the riskfactor model frameworks that underpin the IRB risk weights of Basel II, credit risk of a portfolio is caused by two main sources, systematic and idiosyncratic risks. Systematic risk represents the effect of unexpected changes in macroeconomic and financial market conditions on the performance of borrowers. Borrowers may differ in their degree of sensitivity to systematic risk, but few firms are completely indifferent to the wider economic conditions in which they operate. Therefore, the systematic component of portfolio risk is unavoidable and only partly diversifiable. Meanwhile idiosyncratic risk represents the effects of risks that are particular to individual borrowers. As a portfolio becomes more fine-grained, in the sense that the largest individual exposures account for a smaller share of total portfolio exposure, idiosyncratic risk is diversified away at the portfolio level. This risk is totally eliminated in an infinitely granular portfolio (one with a very large number of exposures) as unsystematic risk vanishes in Capital Asset Pricing Model (CAPM). The ASRF model framework underlying the IRB approach is based on two key assumptions. The first one is that bank portfolios are perfectly fine-grained and the second one is that there is only one source of systematic risk. When these two assumptions hold, one can easily calculate required capital level depending on only one systematic risk. In case of well-diversified portfolio, the capital required for a loan does not depend on the portfolio it is added to.
P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238
This simplicity makes the new IRB framework applicable to a wider range of countries and institutions. However, if any of two assumptions is violated, there is no guarantee that the IRB approach and ASRF model will be accurate. Starting from an idealized portfolio model, Kalkbrener (2005) has discussed the concepts of name and sector concentration and has demonstrated that a multifactor model is needed as the basis for stressing sector concentration. Seminal contribution to measure credit risk concentrations under stress are given by Egloff et al. (2004), Embrechts (2007), Glasserman and Li (2003), Gordy (2004). Statistical models play an important role in studying concentration risk in credit portfolios. Despite this strong motivation, the literature on the topic is quite rare. The Basel committee on banking supervision (2004), Gordy (2003) and Wilde (2001) consider the asymptotic single-risk-factor model foundation. Looking at the Value at Risk (VaR) adjustment for sector concentration, a simple multi-factor adjustment for the treatment of diversification in credit capital rules is proposed by Cespedes et al. (2004) and Pykhtin (2004). Solutions to extend this model by means of a granularity adjustment are described in Dembo et al. (2004), Tasche et al. (2003), Gordy (2004), Gouriroux et al. (2000), Martin and Wilde (2002) and Wilde (2001). On the basis of previous evidences and references, we propose in this paper a novel methodological approach to measure singlename credit concentration risk. To reach this objective, we describe a new concentration credit index and we compare our proposal with the common approach described by the Basel committee on banking supervision. The paper is organized as follows. Section 2 describes heuristic methods available for measuring concentration risk. Section 3 shows our proposal to measure concentration risk. Section 4 reports two applications and finally, Section 5 the conclusions. 2. Measuring single-name concentration risk The approaches available for measuring single-name concentration can be broken down into model-free (heuristic) and modelbased methods. The first approach is to adapt indices of concentration such as Herfindahl–Hirschman index (Kwoka, 1977) and Gini coefficient (Gini, 1921). While these indices could be good measures for concentration itself, they do not seem to serve well for concentration risk because they do not take distribution of different quality obligors into account. The second approach is granularity adjustment suggested by Gordy (2003) and Gordy (2004): its difficulties in implementation and huge data requirement make it hard to be performed in practice. Usually practitioners use both approaches to measure the concentration risk of their portfolio. However the concentration measurement index such as Herfindahl–Hirschman index could not measure the actual risk accurately, granularity adjustment sometimes overestimates the actual concentration risk of a portfolio. Before to describe our concentration index, in this section we review classical concentration measures proposed by Basel II for single-name credit concentration risk measurement. 2.1. Herfindahl–Hirschman index (HHI) The HHI is in general a statistical measure of concentration. The HHI is defined as the sum of the squares of the relative portfolio shares of all borrowers.
HHI ¼
n X i¼1
x2i ;
233
where xi is the portfolio share of borrower i, and n is the number of borrowers under observation. The HHI ranges from 1=n to 1, so that the normalized HHI index can be written as:
HHI ¼
H 1n : 1 1n
Well-diversified portfolios with a large number of small credits have an HHI value close to zero, whereas heavily concentrated portfolios can have a considerably higher HHI value. In the extreme case where we observe only one credit, the HHI takes the value of 1. In the context of the measurement of concentration risk, the HHI formula is included as a main component of a number of approaches. However, HHI itself has some drawbacks to be used for measuring concentration risk. At first, it does not consider distribution of exposures across credit ratings: portfolios with the same HHI values can have different sizes of concentration risks. Secondly, it does not allow concentration risk to be expressed directly as economic capital, so it needs additional functions to calculate economic capital for concentration risk. We remark that HHI does not take quality of a portfolio into consideration. It implies that the portfolios which are differently distributed across the credit ratings can have the same HHI. 2.2. Gini index ðGÞ The G coefficient provides a further method of measuring single-name concentration. This ratio can be interpreted as a concentration index, i.e. a measure of the deviation of a distribution of exposure amounts from an even distribution. In general G is a summary statistic of the Lorenz curve (Lorenz, 1905) and a measure of inequality in a population. More formally, G could be expressed as:
G ¼ 1 þ 1=n ð2=n2 xÞðx1 þ 2x2 þ 3x3 þ þ nxn Þ;
ð1Þ
where xi ; i ¼ 1; . . . ; n is the credit amount by each borrower in a specific portfolio in decreasing order of size and x is the mean of P x ¼ 1n ni¼1 xi . G is thus a weighted sum of xi ; i ¼ 1; . . . ; n, equal to the shares, with the weights determined by rank order position. A coefficient close to zero signifies a homogeneous portfolio in which all of the exposure amounts are distributed equally; a coefficient close to one means a highly concentrated portfolio. A fundamental disadvantage of using G to measure concentration, however, is the fact that the size of the portfolio is not taken into consideration. For example, a portfolio with a few equal sized loans has a lower coefficient than a better diversified, larger credit portfolio containing loans of different amounts. Moreover, the G index may rise if a relatively small loan to another borrower is added to the portfolio despite the fact that this diminishes the concentration. For these reasons, G is only of limited suitability for measuring single-name concentration risk. 2.3. Granularity adjustment ðGAÞ The GA for the ASRF (see e.g. Wilde, 2001) model constitutes an approximation formula for calculating the appropriate economic capital needed to cover the risk arising from the potential default of large borrowers. GA is an extension of the ASRF model which forms the theoretical basis of the IRB approaches. Through this adjustment, originally omitted single-name concentration is integrated into the ASRF model. The GA can be calculated as the difference between unexpected loss in the real portfolio and in an infinitely granular portfolio with the same risk characteristics. To measure the GA, the guidelines of the Basel committee on banking supervision, proposes the following equation:
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GA ¼ C HHI
n X
xi ;
ð2Þ
i¼1
where C is a proportional constant which is a function of the ðq; PD; LGDÞ. In particular, q is the correlation coefficient, PD, the probability of default, estimated by statistical predictive models and LGD is the loss given default. We remark that C is a well calibrated parameter and it is fixed on the basis of the regulation. Basel II proposes to use GA on the basis of a fixed value for LGD ¼ 45%, coefficient of correlation q ¼ 18% and the constant C. The constant C is derived on the basis of particular value of PD, (see e.g. Basel committee on banking supervision, 2004). In particular, fixed q ¼ 18%; LGD ¼ 45% and the PD, C is derived as shown in Table 1. This is an important point because fixed choices for q and LGD, PD and C leads to have biased estimation for concentration risk. Furthermore it is recognized that each subject presents different risk in terms of solvency expressed by the PD. Considering this point, we remark that HHI and G are risk independent. The GA consider the specific PD, only through the constant C. We also remark that, as we know from Basel II, C is derived only for specific values of PD. Empirical evidence shows that it was possible to establish an approximately linear relationship between the granularity adjustment and the HHI for the portfolios. This indicates that the HHI is suitable as a measure of single-name concentration, in particular in view of its relatively simple calculation method. However, in the case of small portfolios, which usually have an higher HHI value, different borrower specific probabilities of default play a greater role than in the case of large portfolios with low HHI values. Thus, for such small portfolios, the GA leads to a wider dispersion than in the case of more diversified portfolios with low HHI values, in which the effects of the different probabilities of default tend to be evened out. This finding shows that, at least for relatively small portfolios for which idiosyncratic risk plays a greater role, a GA holds out more promise for providing information than the HHI. The accurately and transparently measurement of credit concentration risk motivates the present paper, which supports a novel and general way to measure concentration risk. In particular, we propose a new index which consider both the measures: risk and credit size. We point out that as measures of risk we consider the PD, variances or more in general credit scores measurements. We describe our proposal in Section 3.
exposures to a single credit, or asset class, but also from linkages between asset classes and risk. Let us consider the classical portfolio selection scheme (Markowitz, 1950), as in the mean-variance analysis. Consider the variance–covariance matrix ðn nÞ; R ¼ frij g, for i ¼ 1; . . . ; n and j ¼ 1; . . . ; n, associated with the chosen portfolio. As usual R is a definite positive square real matrix. If we assume that the covariance matrix R describes the risk of the portfolio r2P , we can write it in the classical way as:
r2P ¼ x0 R x ¼
n X
x2i rii þ
i¼1
n n X X
xi xj rij
ð3Þ
i¼1 j¼1;i–j
where x 2 Rn is the vector that contains the relative shares of the assets in portfolio, while n is the number of assets. The vector x is such P that xi P 0 for i ¼ 1; . . . ; n and ni¼1 xi ¼ 1. In order to simplify the calculation assume for the time being that xi ¼ 1n, for i ¼ 1; . . . ; n. Therefore it is possible to rewrite the risk r2P of the portfolio as:
r2P ¼
n n n X 1X 1 n1 X rij rii þ n i¼1 n n i¼1 j¼1;i–j nðn 1Þ
ð4Þ
where the second term is multiplied and divided by n 1. The first term in Eq. (4) is the idiosyncratic risk component while the second one is the market risk component. In other words, the first term represents the risk of the portfolio due to the variance of every single asset while the second term represents the risk for an investor who decided to invest on that specific market. When the dimension of the portfolio increases, with n ! 1, the first term in Eq. (4) vanishes and:
lim r2P ¼ lim
n!1
n!1
n n X n1 X rij ij : ¼r n i¼1 j¼1;i–j nðn 1Þ
ð5Þ
ij is the mean of the covariances and it represents the The value r minimum value of risk that an investor has to take working on that specific market. Let us focus the attention on the first term specific risk component. The idea is to use the first term in Eq. (4) in order to measure inequality and concentration at the same time. In particular, we are interested not only in inequality but also in diversification. We can define a new inequality index in the following way:
IV ¼
n X
x2i rii ;
i ¼ 1; . . . ; n
ð6Þ
i¼1
Let us now rewrite Eq. (6) in a more compact and convenient way:
IV ¼ x0 V x
3. Our proposal As we have described in the previous sections, there are indexes useful to measure concentration that in our opinion must be improved. Historical experience shows that concentration of credit risk in asset portfolios has been one of the major causes of bank distress. This is true both for individual institutions as well as banking systems at large. The failures of large borrowers like Enron, Worldcom and Parmalat were the source of sizeable losses in a number of banks. These examples illustrate the importance of measuring concentration risk in credit portfolios of banks that arises not only from
ð7Þ
where V is the diagonal positive definite square matrix of the variances. The matrix V contains the risk information available about the activities in the portfolio and is equivalent to define a relation of order within the activities. In other words we can order the activities from the more risky one to the lower one using the rii coefficients, i.e. the variances. According with previous assumptions we note that IV P 0. In order to obtain a normalized index, IV 2 ð0; 1, we simply observe that IV 6 maxfrii ; i ¼ 1; . . . ; ng. In fact, the extreme value of the index is obtained when the portfolio contains the most risky activity only. Therefore we can define a normalized index IV as:
Table 1 Granularity adjustment approach. PD
0.5%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
C
0.773
0.784
0.848
0.885
0.909
0.927
0.939
0.948
0.955
0.959
0.963
P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238
Pn IV ¼
2 i¼1 xi
rii
i ¼ 1; . . . ; n:
;
maxfrii g
ð8Þ
If no risk information is available on the activities in the portfolio, i.e. V ¼ k I; k 2 Rþ , where I is the identity matrix, we obtain:
IV ¼
k x0 Vx ¼ maxfrii g
Pn
2 i¼1 xi
k
¼
n X
x2i ¼ HHI:
ð9Þ
i¼1
In other words, when we are not able to distinguish more risky credits from lower risky ones and we associate them the same variance, we obtain the HHI index as a special case of the IV index. Furthermore, it is possible to interpret the HHI index as a particular measure of idiosyncratic risk of a portfolio that considers only the relative weights: more precisely, we remark that the HHI index assumes constant variance across credit. 3.1. Properties of our proposed index In this section we prove that the index IV satisfies a number of useful properties: Property 3.1. In the case of absence of inequality and absence of concentration the index IV ! 0. Proof. Absence of inequality means that xi ¼ 1n ; i ¼ 1; . . . ; n while absence of concentration means n ! 1; then:
lim IV ¼ lim
n!1
n!1
1 n
Pn
1 i¼1 n
rii
maxfrii g
¼ 0;
i ¼ 1; . . . ; n:
Property 3.2. In the case of maximum inequality and maximum concentration the value of the index must be equal to 1. Proof. Maximum inequality means that we have only one activity in the portfolio with weight equal to 1 and r ¼ maxfrii g, then:
IV ¼
x2 r ¼ 1: maxfrii g
235
Hardy et al. (1934), mathematical economists, have demonstrated the equivalence that prevails between inequality measures, dominance between income distributions and transfer principles. Accordingly, a wide literature has been devoted to such transfers aiming at proposing some well suited measures of inequality as well as deriving, by means of decision theory, the behavior of decision maximizers under risk and uncertainty (see e.g. Chateauneuf et al., 2002). In the spirit of Dalton’s transfer (1920), Kolm (1976) proposed the diminishing transfer principle, which postulates that an income transfer, valued to be d > 0 from a higher-income individual to a lower-income one, yields a better impact on social welfare indices insofar as incomes ðxÞ are the lowest possible, given that individuals’ ranking remains unchanged after such transfers. However, as mentioned by Mehran (1976) and Kakwani (1980), these transfers do not involve the difference between individuals’ ranks. Hence, they propose the transfer sensitivity property. Other principles have been proposed, developing some new transfer concepts such as Fleurbaey and Michel’s (2001) proportional transfer principle, exhibiting a small loss in the transfer between the donor and the recipient, or Gajdos (2004) a-spread simulating global changes in a distribution with simultaneous progressive transfers (from one higher-income agent to all lower-income ones) and simultaneous regressive transfers (from the same agent to all higher-income ones) in order to obtain the implications on the decision maker’s behavior. Without exploring new transfer principles, let us now suppose that transfers may occur between borrowers that are under observation following Property 3.4. Before introducing Property 3.5, let us define homogeneity of a statistical variable. A statistical variable shows maximum homogeneity when: xk ¼ 1 for some k between 1 and n, and xi ¼ 0 for all i – k; A statistical variable shows minimum homogeneity when: xi ¼ 1n 8 i ¼ 1; . . . ; n.
Property 3.3. If Y ¼ a X, where X is a diagonal covariance matrix, with a 2 ð0; 1Þ, then IV ðYÞ ¼ IV ðXÞ.
A statistical measure is an index of homogeneity if it is maximum under maximum homogeneity and minimum under minimum homogeneity.
Proof. By assumptions we have:
Property 3.5. The IV is definitely a homogeneity index, IV 2 ½0; 1.
2
a r11
6 Y ¼6 4 0 0
0 .. .
0
0
a rnn
3
Proof. In the case of minimum homogeneity the value of the index IV is defenetely 0 as shown in 3.1. In the case of maximum homogeneity the value of the index IV is equal to 1 as shown in 3.2.
7 7 0 5:
Note that the new index IV satisfies the properties of a homogeneity index only when n ! 1. This depends from the fact that the proposed index gives a measure of homogeneity and concentration at the same time.
Therefore,
Pn IV ðYÞ ¼
P
rii Þ a ð ni¼1 x2i rii Þ ¼ ¼ IV ðXÞ; i ¼ 1; . . . ; n: maxfa rii g a ðmaxfrii gÞ 2 i¼1 xi ða
Property 3.4. Let 0 6 r11 6 6 rnn and 0 < h 6 xj . If we move the weight h from xj to xj1 the value of the index IV decreases.
4. Application
and this implies IV decreases. h
In order to explain the performance of our index we refer to the guidelines of Basel II and described also in the documentation of Basel committee on banking supervision (2005). As described in Section 2 and referring to Table 1, Basel II proposes to use the GA on the basis of a fixed value for LGD; q and the constant C. As suggested by Basel committee on banking supervision (2005), we consider two portfolios, A and B, which are composed in the following way:
Let’s observe that Property 3.4 is very similar to the principle of transfers (see e.g. Dalton, 1920) but it is more meaningful in this case respect to the principle of transfers.
– A is composed by 10 statistical units ðx1 ; . . . ; x10 Þ each one with relative frequency equal to 10% (same size);
Proof. By hypothesis we have
rðj1Þðj1Þ 6 rjj ; then
ðxj þ hÞ2 rðj1Þðj1Þ þ ðxj hÞ2 rjj < x2j1 rðj1Þðj1Þ þ x2j rjj
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P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238
– B is composed by 100 statistical units ðx1 ; . . . ; x100 Þ each one with relative frequency equal to 1%. Basel committee on banking supervision supposes that the PD for the statistical units in portfolio A is equal to the PD for the statistical units in portfolio B and it is equal to 0.01. On the basis of Table 1 we consider C ¼ 0:784. Simple calculation show that the GA for portfolio A is equal to GAðAÞ ¼ 7:84 and for portfolio B is equal to GAðBÞ ¼ 0:784. We remark that, the GA when PD is constant in a specific portfolio, and consequently C is constant and LGD and q fixed, go to zero when the granularity increases. Considering the data at hand we remark that our index proposed in Section 3 shows for portfolio A a specific value of IA ¼ 0:1 and for portfolio B, IB ¼ 0:01. We present a new example. Considering a portfolio C composed by 100 statistical units with xi ¼ 0:01 for i ¼ 1; . . . ; 100 and PD equal to 10%: in this situation IC ¼ 0:01 which is equal to the index for portfolio B. Therefore, the value of the index is strictly decreasing with the increasing size of the portfolio. This empirical evidence leads us to propose the following remark. P Let ni¼1 x2i ! 0 and rii ¼ k 2 R, i.e. max rii ¼ k, then IV ðnÞ is a strictly decreasing function of n. In fact, we have:
Pn IV ðnÞ ¼
2 i¼1 xi
k
rii
Pn ¼
2 i¼1 xi k
k
! 0:
We obtain a strictly decreasing index with the increasing size of the portfolio if we assume that all the risks associated with the single asset classes are equal. Furthermore, we remark that it is very easy to derive, on the basis of our proposal, a measure for the credit concentration that is independent by C, LGD and q. More precisely, in order to compute our index we only use xi (the portfolio share of the borrower i, with ð0 < xi < 1Þ) and rii , the PD for the ith statistical unit. Our approach derives as result a specific value bounded between 0 and 1 ð0 6 PD 6 1Þ, independently from fixed value for LGD and q. This is an important point because specific choice for q and LGD, PD and C leads to have unbiased estimation for concentration risk. As we can understand from the above report, suppose that for a specific statistical unit in portfolio A we observe PD ¼ 4:735%: in this case we have not available C to compute the GA. But, in this case our index propose a specific value of 0.01. Application results show that in general the value of the index IV is not strictly decreasing with the growing size of the portfolio, like, for example for HHI index. This fact seems coherent to us, because the reduction of the risk of the portfolio due to the diversification effect could not be enough when we add to the portfolio an high risk component. Nevertheless, we can prove that it is possible to obtain a fixed value of the index if we diversify enough the portfolio. In order to explain how our index work, we consider an empirical analysis based on annual 1996–2004 financial real data for 1003 companies from Creditreform, which is one of the major rating agencies for SMEs in Germany. In order to meet the requirements of Basel II, many SMEs in Germany face the challenge of adjusting their financing structure and putting it on a more orderly footing. Alternative forms of financing play an increasingly important role. Still, bank loans play an extremely important role in the external financing of corporate activities of German SMEs. To measure credit concentration we use the total amount of loan and the corresponding PD estimated on the basis of the longitudinal predictive model described in Figini and Fantazzini (2007), that will now be briefly described.
On the basis of our real dataset, we present the following specification for longitudinal models: for observation k; ðk ¼ 1; . . . ; KÞ, time t; ðt ¼ 1; . . . ; TÞ and sector j; j ¼ 1; . . . ; J, let Y ktj denote the response solvency variable, let X ktj denote a p 1 vector of candidate predictors (fixed effects), and let Z ktj denote a q 1 vector of candidate predictors (random effects), where 1k Nð0; RÞ are random effects for SMEk and R is the covariance matrix. The elements of Y ktj ¼ ðy1tj ; . . . ; yktj Þ0 are modelled as conditionally independent random variables from a simple exponential family:
Y ktj hktj bðhktj Þ þ cðY ktj ; /Þ ; aktj ð/Þ
pðY ktj jX ktj ; Z ktj ; 1k Þ / exp
ð10Þ
where hktj is the canonical parameter related to the linear predictor gktj ¼ X 0ktj b þ Z 0ktj 1k with a p 1 vector of fixed effects regression coefficient b and a q 1 vector of subject specific random effects 1k Nq ð0; RÞ; / is a scalar dispersion parameter and aktj ; b; c are known functions with aktj ð/Þ ¼ x/ , where xktj is a known weight ktj (see e.g. Dunson and Cai, 2006). We are interested in predicting the expectation of the response as a function of the covariates. The expectation of a simple binary response is just the probability that the response is 1:
EðY ktj jX ktj ; Z ktj ; 1k Þ ¼ pðY ktj ¼ 1jX ktj Þ:
ð11Þ
In linear regression, this expectation is modelled as a linear function b0 X ktj of the covariates. For binary responses, as in our case, this approach may be problematic because the probability must lie between 0 and 1, whereas regression lines increase (or decrease) indefinitely as the covariate increases (or decreases). Instead, a nonlinear function is specified in one of two ways:
pðY ktj ¼ 1jX ktj Þ ¼ hðb0 X ktj Þ;
ð12Þ
or
gfpðY ktj ¼ 1jX ktj Þg ¼ b0 X ktj ¼ mk ;
ð13Þ
where mk is referred to as the linear predictor. These two formulations are equivalent if the function hð:Þ is the inverse of the link function gðÞ. We have introduced two components of a generalized linear model: the linear predictor and the link function. The third component is the distribution of the response given the covariates. For binary response, this is always specified as Bernoulli ðpk Þ. Typical choice of link function g is the logit link. The logit link is appealing because it produces a linear model for n o pðY ¼1jx Þ the log of the odds, ln 1pðYktj ¼1jxk Þ , implying a multiplicative modktj
k
el for the odds themselves (see e.g. Dobson, 2002). To relax the assumption of conditional independence among the firms given the covariates, we can include a subject-specific random intercept 1k Nð0; wÞ in the linear predictor:
gfpðY ktj ¼ 1jX ktj ; 1k Þg ¼ b0 X ktj þ 1k :
ð14Þ
This is a simple example of a generalized linear mixed model because it is a generalized linear model with both fixed effects X ktj and a random effects 1k . The available data was made up of about 1000 SMEs. Among the corresponding 1000 records, 280 were incompleted or economically illogical and therefore have been discarded. The resulting filtered data set is composed of 702 SMEs and for each statistical unit we know the estimated PD and the total loan. For the loan variable, the data at hand shows a minimum value of 37,878.36 and a maximum value of 1,66,831.4. The median is equal to 99,962.950 and the coefficient of variation is equal to 0.19. The kurtosis and the skewness are respectively equal to 0.187 and 0.127. Considering the loan variable, Fig. 1 depicts the relative histogram. As we observe from Fig. 1, the loan variable can be approx-
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P. Uberti, S. Figini / European Journal of Operational Research 202 (2010) 232–238 Table 2 HHI, G and IV .
Fig. 1. Histogram for the loan.
imated with a normal density probability function. Statistical tests, such us the Kolmogorov Smirnov and the Shapiro Wilk, confirm this empirical evidence. More precisely, considering the loan variable, the first quartile (25% of the SMEs) is equal to 86,966.65, the median (50% of the SMEs) is equal to 99,962.95, and the third quartile (75% of the SMEs) is equal to 1,13,439.6. For the variable PD, estimated on the basis of the longitudinal model, the minimum value is 0 and the maximum value is 0.867. The median is equal to 0.076 and the coefficient of variation is equal to 1.012. The kurtosis and the skewness are respectively equal to 3.448 and 1.523. Sorting the PD, Fig. 2 reports the histogram. The distibution is clearly not normal. More precisely, considering PD, the first quartile (25% of the SMEs) is equal to 0.0248, the median (50% of the SMEs) is equal to 0.0763, and the third quartile (75% of the SMEs) is equal to 0.20. Bringing toghether the PD derived from the longitudinal model and the variable loan, the HHI is equal to 0.00148, the G index is equal to 0.110 and IV is equal to 0.000176. We divide the data
Segment
HHI
G
IV
A B
0.08 0.079
0.074 0.0572
0.00037 0.00033
set in two segments. The first segment A is composed of 351 loans with a total amount less than 100,000 euros. For segment A the maximum value for PD is equal to 0.709 with a coefficient of variation of 0.970. The second segment B is composed by 351 loans with a total amount greater than 1,00,000 euros. For segment B the maximum value for PD is equal to 0.876 with a coefficient of variation of 1.058. We computed for segment A and B the HHI index, the G index and the IV index. The results are in Table 2. To better compare the G, the HHI index and the index IV we propose an example based on the available data. We consider 8 different loans, with different probabilities of default and consider four different portfolios P1 ; P2 ; P3 ; P4 , as reported in Table 3. For simplicity of calculation we assume the same value of 50,000 euros for every loan. Let now evaluate the values of the concentration indexes associated with the four portfolios of Table 4. In particular we compute G, HHI and our proposed index, IV . We are working with the probabilities of default, so that we fix the value of the normalization constant at 1 in order to calculate the value of the index IV . As we can see in Table 4 the value of G is equal to 0 for portfolio P1 ; P2 ; P3 ; P4 . This evidence shows that G does not take into account the size of the portfolio and the risk of the loans. If we consider the HHI index we can see that the value of the index depends only from the share of every single loan. Also in this case the risk of the loans is not considered. Considering the data at hand, the value of HHI is strictly decreasing when the size of the portfolio increases; this means that differentiation automatically decreases the risk. We underline that the value of HHI is strictly positive for every portfolio in the example: this fact shows that it is impossible to achieve the best theoretical value, i.e. HHI ¼ 0, in a real portfolio. We finally focus the attention on the behaviour of IV . First of all, we see that this index takes into account the risk of the loans. In fact, if we consider portfolios P 1 and P 2 we observe that the portfolio with the more risky loans (in this case case P 2 ) has a greater va-
Table 3 Example: portfolios of loans. Loans
Probability of default
P1
P2
P3
P4
a b c d e f g h
0.08 0.1 0.11 0.4 0.41 0.47 0.5 0.2
50,000 50,000 50,000 – – – – –
– – – 50,000 50,000 50,000 – –
50,000 50,000 50,000 – – – 50,000 –
50,000 50,000 50,000 – – – 50,000
Table 4 Values of concentration.
Fig. 2. Histogram for the PD.
Portfolios
G
HHI
IV
P1 P2 P3 P4
0 0 0 0
0:3 0:3
0:032 0:142 0.049375 0.030625
0.25 0.25
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lue for the index. Furthermore, the value of IV depends on the size of the portfolio as shown on portfolios P 1 ; P 2 and P3 . The more interesting property is that the value of the index is not strictly decreasing but it depends from the added risk loan in a portfolio. In particular, if we add loan g to the portfolio P 1 the value of the index IV increases, while adding the loan h the value decrease. In the first case the differentiation effect is not enough to counterbalance the increase of the risk.
5. Conclusions In this paper we propose a new index to measure single-name concentration for credit risk. We think that our contribution could be interesting from a theoretical and an empirical point of view. In particular, banks and financial institutions, based on our proposal, are able to improve the single-name credit risk concentration measurement. Acknowledgements The authors acknowledge financial support from the MIUR-FIRB 2006–2009. We thank the referees and Prof. Paolo Giudici for useful comments and suggestions. References Basel committee on banking supervision, 2004. International convergence of capital measurement and capital standards: a revised framework. Working Paper available at
Dalton, H., 1920. The measurement of inequality of incomes. Economic Journal 30, 348–361. Dembo, A., Deuschel, J.D., Duffie, D., 2004. Large portfolio losses. Finance and Stochastics 8 (1). Dobson, A.J., 2002. Introduction to Generalised Linear Model. Chapman & Hall. Dunson, D.B., Cai, B., 2006. Bayesian covariance selection in generalized linear mixed models. Biometrics 62, 446–457. Egloff, D., Leippold, M., Vanini, P., 2004. A simple model of credit contagion. Working Paper, University of Zurich. Embrechts, P., 2007. Aggregating Risk Capital: Lessons Learned From Basel II, available at