Single-name concentration risk in credit portfolios: a comparison of concentration indices

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1 UCD GEARY INSTITUTE DISCUSSION PAPER SERIES Sigle-ame cocetratio risk i credit portfolios: a compariso of cocetratio idices Raffaella Calabrese Uiversity College Dubli Fracesco Porro Uiversit`a degli Studi di Milao-Bicocca Geary WP2012/14 May 2012 UCD Geary Istitute Discussio Papers ofte represet prelimiary work ad are circulated to ecourage discussio. Citatio of such a paper should accout for its provisioal character. A revised versio may be available directly from the author. Ay opiios expressed here are those of the author(s) ad ot those of UCD Geary Istitute. Research published i this series may iclude views o policy, but the istitute itself takes o istitutioal policy positios.

2 Sigle-ame cocetratio risk i credit portfolios: a compariso of cocetratio idices Raffaella Calabrese, Fracesco Porro Abstract For assessig the effect of udiversified idiosycratic risk, Basel II has established that baks should measure ad cotrol their credit cocetratio risk. Cocetratio risk i credit portfolios comes ito beig through a ueve distributio of bak loas to idividual borrowers (sigle-ame cocetratio) or through a ubalaced allocatio of loas i productive sectors ad geographical regios (sectoral cocetratio). To evaluate sigle-ame cocetratio risk i the literature cocetratio idices proposed i welfare (Gii Idex) ad moopoly theory (Herfidahl- Hirschma idex, Theil etropy idex, Haah-Kay idex, Hall-Tidema idex) have bee used. I this paper such cocetratio idices are compared by usig as bechmark six properties that esure a cosistet measuremet of sigle-ame cocetratio. Fially, the idices are compared o some portfolios of loas. 1 Itroductio The Asymptotic Sigle-Risk Factor (ASRF) model (Gordy, 2003) that uderpis the Iteral Ratig Based (IRB) approach i the Basel II Accord (Basel Committee o Bakig Supervisio, BCBS 2004) assumes that idiosycratic risk has bee diversified away fully i the portfolio, so that ecoomic capital depeds oly o systematic risk cotributios. Systematic risk represets the effect of uexpected chages i macroecoomic ad fiacial market coditios o the performace of borrowers. O the other had, idiosycratic risk represets the effects of risks that are particular to idividual borrowers. I order to iclude idiosycratic risk i ecoomic capital, Basel II (BCBS, 2004) requires that baks estimate cocetratio risk. Cocetratio risks i credit portfolios arise from a uequal distributio of loas to Raffaella Calabrese Uiversity College Dubli, raffaella.calabrese@ucd.ie Fracesco Porro Uiversità degli Studi di Milao-Bicocca, fracesco.porro1@uimib.it 1

3 2 Raffaella Calabrese, Fracesco Porro sigle borrowers (sigle-ame cocetratio) or idustrial or regioal sectors (sector cocetratio). This paper is focused oly o the sigle-ame cocetratio, i particular i the cotext of loa portfolios. Five cocetratio measures, which have bee proposed i welfare ad moopoly theory, are compared as regards six desirable properties for measuremets of sigleame cocetratio risk. The first idex is the Gii coefficiet, a widely applied iequality measure. To uderstad the differece betwee iequality ad cocetratio, a portfolio that cotais few large exposures is cosidered. By icludig i such portfolio may small exposures so that eve i aggregate their share of the portfolio exposure is very low. Therefore, cocetratio has ot bee sigificatly affected, but the degree of iequality i borrowers exposures has greatly icreased. It follows that iequality measures are very sesitive to the umber of small exposures. Some other idices (the Haah-Kay idex, the Herfidahl-Hirschma idex, the Hall-Tidema idex) have bee proposed i moopoly theory, ad oe (the Theil idex) arises from the iformatio theory. Sigle-ame cocetratio risk, aalogously to idustrial cocetratio, is due to both a small umber of loas ad high credit exposures i a portfolio. It follows that these idices, ulike the Gii idex, deped o the umber of loas i portfolio. By ormalizig these idices, the importat iformatio of the umber of loas i the portfolio would be lost, so the ormalizatio is ot applied to them i this paper. A iterestig theoretical result of this work is that the Haah-Kay idex, the Herfidahl-Hirschma idex ad the Hall-Tidema idex satisfy all the six desirable properties for measuremets of sigle-ame cocetratio risk. I order to compare the features of the cocetratio idices set up by the six properties, six portfolios with differet levels of sigle-ame cocetratio risk are aalyzed. The portfolio with the highest cocetratio risk is cosidered compliat to the regulatio of the Bak of Italy (Baca d Italia, 2006). Both the total exposure of a portfolio ad the umber of loas are chaged i order to aalyse their impact o the idices of sigle-ame cocetratio risk. The mai results of this umerical applicatio are that the Reciprocal of Haah-Kay idex with α = 3, the Herfidahl-Hirschma idex ad the Hall-Tidema idex stress the impact of lager exposures. O the cotrary, the Reciprocal of Haah-Kay idex with α = 0.5 ad the Theil idex uderlie the importace of smaller exposures. The paper is orgaized as follows. Sectio 2 defies the six properties of a sigleame cocetratio idex ad the relatioships amog them. Sectios 3, 4 ad 5 ivestigate which properties are satisfied respectively by the Gii idex, the idustrial cocetratio idices ad the Theil etropy idex. I sectio 6 the five idices are used to measure the sigle-ame cocetratio risk of six portfolios. Sectio 7 is devoted to coclusios.

4 Sigle-ame cocetratio risk 3 2 Properties of a sigle-ame cocetratio idex Cosider a portfolio of loas. The exposure of the loa i is represeted by x i 0 ad the total exposure of the portfolio is x i = T. I the followig, a portfolio is deoted by the vector of the shares of the amouts of the loas s = (s 1,s 2,...,s ): the share s i 0 of i-th loa is defied as s i = x i /T. It follows that s i = 1. Wheever the shares of the portfolio s eed to be ordered, the correspodig portfolio obtaied by the icreasig rakig of the shares will be deoted by s (.) = (s (1),...,s () ). It is clear that ay reasoable cocetratio measure C must satisfy C(s) = C(s (.) ). Wheever it is ecessary, i order to remark the umber of the loas i the portfolio, the sigle-ame cocetratio measure will be deoted with C. The followig six properties are desirable oes that a sigle-ame cocetratio measure C should satisfy. Ideed they were bor i a differet framework, evertheless their traslatio to credit aalysis ca be cosidered successful (cfr [3], [7] ad [15]). 1. (Trasfer priciple) The reductio of a loa exposure ad a equal icrease of a bigger loa that preserve the order must ot decrease the cocetratio measure. Let s = (s 1,s 2,...,s ) ad s = (s 1,s 2,...,s ) be two portfolios such that s ( j) h k = j s (k) = s ( j+1) + h k = j + 1 (1) otherwise, where s (k) s j < s j+1, 0 < h < s ( j+1) s ( j), h < s ( j+2) s ( j+1). (2) The C(s) C(s ) 2. (Uiform distributio priciple) The measure of cocetratio attais its miimum value, whe all loas are of equal size. Let s = (s 1,s 2,,s ) be a portfolio of loas. The C(s) C(s e ), where s e is the portfolio with equal-size loas, that is s e = (1/,...,1/). 3. (Lorez-criterio) If two portfolios, which are composed of the same umber of loas, satisfy that the aggregate size of the k biggest loas of the first portfolio is greater or equal to the size of the k biggest loas i the secod portfolio for 1 k, the the same iequality must hold betwee the measures of cocetratio i the two portfolios. Let s = (s 1,s 2,...,s ) ad s = (s 1,s 2,...,s ) be two portfolios with loas. If i=ks (i) s (i) for all k = 1,...,, the C(s) C(s ). i=k 4. (Superadditivity) If two or more loas are merged, the measure of cocetratio must ot decrease. Let s = (s 1,...,s i,...,s j,...,s ) be a portfolio of loas, ad s = (s 1,...,s i 1,s i+1,...,s j 1,s m,s j+1,...,s ) a portfolio of 1 loas such that s m = s i + s j. The C (s) C 1 (s ).

5 4 Raffaella Calabrese, Fracesco Porro 5. (Idepedece of loa quatity) Cosider a portfolio cosistig of loas of equal size. The measure of cocetratio must ot icrease with a icrease i the umber of loas. Let s e, = (1/,...,1/) ad s e,m = (1/m,...,1/m) be two portfolios with equalsize loas ad m, the C (s e, ) C m (s e,m ). 6. (Irrelevace of small exposures) Gratig a additioal loa of a relatively low amout must ot icrease the cocetratio measure. More formally, if s deotes a share of a loa ad a ew loa with a share s s is grated, the the cocetratio measure must ot icrease. Let s = (s 1,s 2...,s ) be a portfolio of loas with total exposure T. The, there exists a share s such that for all s = x/(t + x) s the portfolio of + 1 loas s = (s 1,s 2,...,s +1 ) with shares { s xi /(T + x) i = 1,2,..., i = x/(t + x) i = + 1 is cosidered. It holds that C(s) C(s ). A few remarks o the aforemetioed properties ca be useful. The first three properties have bee proposed for the cocetratio of icome distributio. I the first three properties the umber of loas of the portfolio is fixed, while i the others chages. This meas that the properties 4, 5 ad 6 poit out the ifluece of the umber of the loas o the cocetratio measure. The priciple of trasfers ad the Lorez-criterio have bee proposed at the begiig of the last cetury:the former has bee itroduced by Pigou i 1912 (see [16]) ad Dalto i 1920 (see [5]), the latter is related to the Lorez curve proposed i 1905 by Lorez (see [14]). The property 4 ca be applied more tha oe time by settig up the merge of three or more loas. Fially, the properties 4 ad 5 have bee suggested i the field of the idustrial cocetratio where the issue of moopoly is very importat. Theorem 1 (Lik amog the properties). If a cocetratio measure satisfies the properties 1 ad 6, the it satisfies all the aforemetioed six properties. Proof. The outlie of the proof is the followig. It ca be proved that a cocetratio idex satisfyig property 1 fulfills also properties 2 ad 3. Further, if a cocetratio measure satisfies the properties 1 ad 6, the it meets the property 4. Fially, properties 2 ad 4 imply the property Property 1 property 3 Let s = (s 1,s 2,...,s ) ad s = (s 1,s 2,...,s ) be two portfolios of loas each. Let the shares of the two portfolios be such that k s (i) k s (i) k = 1,..., 1. Let deote the differece vector s (.) s (.), which is = ( 1, 2,..., ) = (s (1) s (1),s (2) s (2),...,s () s ()). By costructio k i 0 for k = 1,..., 1 ad i = 0. The idea is to costruct, by iductio, a sequece of t vector of shares m 1,...,m t such that

6 Sigle-ame cocetratio risk 5 - m 1 = s (.) ad m t = s (.) ; - m h is obtaied from m h 1 by a trasfer of part of share of a certai loa to a bigger oe. Suppose that m h 1 = (m h 1 1,...,m h 1 ) is already defied ad satisfies the three restrictios: 1) m h 1 i 0 i = 1,...,; 2) mh 1 i = 1; 3) the m h 1 i are decreasigly raked. Itroduce the vector h 1 = s (.) mh 1, with i h 1 = 0. Let α be the first value of the idex i i {1,...,} such that h 1 < 0, which is α = mi i {1,...,} h 1 {i : i < 0}. By costructio, m h 1 α > s (α) ad mh 1 α = s (α) h 1 α. Cosider ow the loa β, where β is the first value of the idex i i {1,...,α} that realizes the miimum of the strictly positive values of i h 1 β = h 1 mi { i i {1,...,α} : h 1 i > 0}. Such a value β always exists because the first o-ull compoet of vector h 1 caot be egative. By costructio, m h 1 β < s (β) ad mh 1 β = s (β) h 1 β. Sice β < α ad sice the compoets of vector m h 1 are decreasigly ordered, it results m h 1 β m h 1 α. By a trasfer of the positive quatity mi( α h 1, h 1 β ) from the loa α to the loa β, the vector m h ca be obtaied with shares m h 1 m h β + mi( α h 1, h 1 β ) i = β i = m h 1 α mi( α h 1, h 1 β ) i = α otherwise m h 1 i The shares m h i with i = 1,2,..., is decreasigly ordered ad this vector is deoted by m h. The vector m h is obtaied from m h 1 by a trasfer of a positive amout from the share α to a bigger share β. If the property 1 holds, the C (m h ) C (m h 1 ). Followig this procedure it is possible to costruct by iductio a sequece of vector of shares: this sequece starts from the portfolio s ad eds with the portfolio s by icreasig the cocetratio at each step. Fially, it should be showed that the umber of iteratio of the procedure is fiite. At each step, a compoet of m h is replaced by the respective compoet value of s. The umber of the compoets of s is fiite, therefore the algorithm coverges after a fiite umber of iteratios. Fially, it holds that C (s) = C (m 1 ) C (m h 1 ) C (m h )...C (m t ) = C (s )

7 6 Raffaella Calabrese, Fracesco Porro 2. Property 1 property 2 The proof is similar to the previous oe. The idea is to costruct a sequece of trasfers which trasforms ay portfolio of loas i the portfolio with equalamout loas by decreasig the cocetratio at each step. Thus, the miimum value of the sigle-ame cocetratio idex is obtaied whe all the shares have the same value. 3. Properties 1 ad 6 property 4 Let s = (s 1,...,s i,...,s j,...,s ) be a portfolio of loas, ad s = (s 1,...,s i 1,s i+1,...,s j 1,s m,s j+1,...,s ) a portfolio obtaied by the merge s m = s i + s j. If s i s j cosider the portfolio s 1 = (s 1,...,s i 1,0,s i+1,...,s j 1,s m,s j+1,...,s ) is obtaied by a trasfer from a smaller loa (s i ) to a bigger oe (s j ). Sice the property 1 holds, C (s) C (s 1 ). Now, if the property 6 is satisfied, the loa with ull amout ca be removed from s 1 with o cocetratio icrease. The result is the portfolio s ad it holds that C (s) C (s 1 ) C 1 (s ), therefore the property 4 is true. 4. Properties 2 ad 4 property 5 Sice the property 2 is satisfied, C +1 (1/,...,1/,0) C +1 (1/(+1),...,1/(+ 1)). After a merge, by property 4, it follows that C (1/,...,1/) C +1 (1/,...,1/,0), ad therefore C (1/,...,1/) C +1 (1/( + 1),...,1/( + 1)). This meas that C (s e, ) C +1 (s e,+1 ). By iteratio, the property 5 holds true. 3 A iequality idex: Gii coefficiet The Gii coefficiet (G) was itroduced by Gii at the begiig of the last cetury [8]. It is a measure of the distace betwee the equalitaria ad the cosidered situatio. I the case of a portfolio of loas s = (s 1,...,s ) it is defied as G = ( i + 1)s (i). (3) It is a ormalized idex, sice its rage is the iterval [0,1]: it assumes value 0 if the loas have the same amout, while it equals 1 if the total amout correspods to a uique loa. The major drawback is that the G idex does ot take ito accout the portfolio size, therefore it ca be cosidered more a iequality idex tha a cocetratio idex. Theorem 2. The G idex satisfies the properties 1, 2, 3 ad 5. Proof. Properties 1, 2 ad 3 Let s = (s 1,...,s ) a portfolio to which the trasfer (2) is applied ad so the portfolio s = (s 1,...,s ) is obtaied. The differece betwee the G idex of these two portfolios is

8 Sigle-ame cocetratio risk 7 G(s ) G(s) = 2 [ ( ) ( )] ( j + 1) s 1 ( j) s ( j) + ( j) s ( j+1) s ( j+1) = 2 [( j + 1)h + ( j)( h)] 1 = 2h 1 > 0. The G(s ) > G(s). As the Theorem 1 states, the property 1 implies the properties 2 ad 3, which are therefore satisfied by the G idex. Property 5 Cosider two portfolios with equal-amout loas: s e, = (1/,...,1/) ad s e,m = (1/m,...,1/m). From the defiitio of the G idex (3), it follows that G(s e, ) = G(s e,m ) = 0. Hece, it holds that G (s e, ) G m (s e,m ). 4 Idustrial cocetratio idices 4.1 Haah-Kay idex For the idustrial cocetratio Haah ad Kay [10] have proposed the followig idex (HK) HK = ( s α i ) 1 1 α with α > 0 ad α 1. The HK idex is iversely proportioal to the level of cocetratio: by icreasig cocetratio the HK idex decreases. For this reaso i this paper, as i Becker, Dullma ad Pisarek [3], the Reciprocal of Haah-Kay (RHK) idex is cosidered: RHK = ( s α i ) 1 α 1 α > 0 ad α 1, (4) so that the RHK idex is proportioal to the level of cocetratio. For a portfolio with equal-size loas the RHK idex is RHK = [ ( ) ] 1 1 α α 1 = 1.

9 8 Raffaella Calabrese, Fracesco Porro If the portfolio cosists of oly oe o-ull share, the RHK idex is equal to 1. The role of the elasticity parameter α is to decide how much weight to attach to the upper portio of the distributio relative to the lower. High α gives greater weight to the role of the highest credit exposures i the distributio ad low α emphasizes the presece or the absece of the small exposures. Theorem 3. The RHK idex satisfies all the six properties cosidered i Sectio 2. Proof. From theorem 1 if the 1 ad 6 properties are satisfied, all the six properties of a cocetratio measure are satisfied. Property 1 1 Let s ad s two portfolios that satisfy the coditio (2). The followig differece is computed ( ) 1 f (h) = RHK(s ) RHK(s) = s α k + (s j + h) α + (s j h) α k i, j α 1 ( s α ) 1 α 1 k. The fuctio f (h) is cotiuous for h > 0 ad lim h 0 f (h) = 0. The derivative of f (h) is ( ) 2 α f (h) h = α α 1 α 1 s α k + (s j + h) α + (s j h) α [ (s j + h) α 1 (s i h) α 1]. k i, j (5) I order to determie the sig of this derivative, two cases are cosidered 1. 0 < α < 1 I the equatio (5) the first ad the third factors of the product are egative ad the secod factor is positive, hece the derivative is positive. 2. α 1 I the equatio (5) all the factors are positive, hece the derivative is positive. Property 6 Let s ad s two portfolios that satisfy the coditios give i the property 6. The followig differece is computed g( x) = RHK(s ) RHK(s) = [ ( ) α ( ) ] 1 [ xi x α α 1 ] 1 α 1 + s α i. T + x T + x The fuctio g( x) is cotiuous for x > 0 ad lim x 0 g( x) = 0, so the etry of a ew loa with isigificat exposure x i the portfolio has isigificat impact o the RHK idex. 1 The proof is similar to the oe suggested by Becker, Dullma ad Pisarek [3].

10 Sigle-ame cocetratio risk 9 The derivative of g( x) is computed by obtaiig g( x) x [ = α ( ) ] α xi + x α 2 α α 1 T + x T + x α 1 x ( i x α 1 xi α 1 ) (T + x) α+1. (6) I order to determie the sig of this derivative for s < s, two cases are cosidered: 1. 0 < α < 1 I the equatio (6) the first factor of the product is egative ad the secod ad the third factors are positive, hece the derivative is egative. 2. α 1 I the equatio (5) the first ad the secod factors are positive ad the third factor is egative, hece the derivative is egative. This meas that eve if the itroductio of a ew loa with exposure x causes a egligible chage i the RHK idex, it slightly decreases. Set the equatio (6) equal to zero, the superior limit s for s is obtaied s = [ s α i ] 1 α 1 = RHK. It follows that if a ew loa has a share s higher tha the RHK idex, the effect of the ew loa i reducig the share of the existig large exposures is offset to some extet by the fact that its exposure is large. The ext idex represets a particular case of the RHK idex for a give value of the elasticity parameter α Herfidahl-Hirschma idex By cosiderig α = 2, the RHK idex (4) becomes HH = s 2 i the Herfidahl-Hirschma idex (HH) proposed by Herfidahl [11] as a idustrial cocetratio idex, whose root has bee proposed by Hirschma [12]. For this reaso, this idex is kow as Herfidahl-Hirschma idex. It is defied as the sum of squared portfolio shares of all borrowers. By cosiderig the square of the portfolio share s i i the HH idex, small exposures affect the level of cocetratio less tha a proportioal relatioship. The mai advatage of the HH idex is that it satisfies all the six properties of a idex of credit cocetratio, because it is a particular case of the RHK idex.

11 10 Raffaella Calabrese, Fracesco Porro 4.2 Hall-Tidema idex The last idustrial cocetratio idex here aalysed has bee proposed by Hall ad Tidema (HT) ad defied as HT = 1 2 ( i + 1)s (i) 1. This idex weights each loa with a value depedig o its rak: this feature gives more importace to big loas ad to the total umber of the loas. If the amouts of all the loas are equal, the it results HT = i=0 ( i + 1) 1 = 1. If there is oly oe loa i the portfolio, the value of the idex is oe. A importat result i the literature is the lik betwee the HT ad the G idices HT = 1 ( 1)G. Theorem 4. The HT satisfies all the six properties cosidered i sectio 2. Proof. Property 1 Let s = (s 1,s 2,...,s ) ad s = (s 1,s 2,...,s ) be two portfolios of loas each oe, as i property 1. The, sice the Gii coefficiet G satisfies the property 1, it holds G(s) G(s ). Hece G(s) G(s ) ( 1)G(s) ( 1)G(s ) HT (s) HT (s ). Property 6 Let s = (s 1,s 2,...,s ) ad s = (s 1,s 2,...,s,s +1 ) be two portfolios of ad + 1 loas, respectively, as stated i property 6. Cosider the differece HT (s) HT (s ):

12 Sigle-ame cocetratio risk 11 HT (s) HT (s ) = 1 = 2[s (1) + ( 1)s (2) + + s () ] 1 1 2[( + 1) s + s (1) + ( 1)s (2) + + s () ] 1 T = 2[x (1) + ( 1)x (2) + + x () ] T T + x 2[x (1) + ( 1)x (2) + + x () ] T + x(2 + 1) = c [T (2 + 1) 2[x (1) + ( 1)x (2) + + x () ] + T ] = c [2T + 2T 2[x (1) + ( 1)x (2) + + x () ]] > c [2T + 2T 2T ] > 0. where c is a costat greater tha 0. It follows that HT (s) > HT (s ) ad therefore property 4 holds true. 5 A idex from iformatio theory: Theil etropy idex The RHK idex is udefied for α = 1 but its behaviour ca be aalysed whe α is close to 1. Let α = 1 + h, the limit of the RHK for h 0 is computed. By applyig the Taylor expasio, for h 0 we obtai s h+1 i (s i + hs i logs i ) = 1 + h s i logs i. (7) By computig the logarithm of the RHK idex ad by cosiderig the result (7), it is obtaied 2 lim logrhk = lim log( s h+1 i α 1 h 0 ) 1 h = lim h 0 By cosiderig the Theil (TH) etropy idex [17] T H = ( ) 1 h h s i logs i = s i logs i. (8) s i log 1 s i, (9) the result (8) ca be writte as a trasformatio of the TH idex, so obtaiig the followig relatioship lim RHK = exp( T H). α 1 Aalogously to the HK idex, the TH idex is iversely proportioal to the level of cocetratio. To obtai a directly proportioal measure of cocetratio from 2 if for ay i, s i = 0, the the quatity s i logs i is covetioally defied as 0

13 12 Raffaella Calabrese, Fracesco Porro the expressio (9), it is preferred to cosider DT H = max{t H} T H = log T H = Whe all the loas have the same exposure, the DTH idex is DT H = log log = 0. s i logs i + log (10) If a portfolio with oly oe o-ull share is cosidered, the DTH idex is equal to log. By cosiderig the logarithm of the portfolio share s i, DTH gives relatively more weight to smaller loas. Theorem 5. The DTH idex satisfies the properties 1,2,3,4, ad 5. Proof. Property 1 Cosider the differece betwee the TH idices of the two portfolios s ad s that satisfy the assumptios of the property 1: DT H(s ) DT H(s ) = (s j +h)log(s j +h)+(s i h)log(s i h) s j logs j s i logs i. (11) Because the secod derivative of the fuctio xlogx is o-egative o the iterval (0,1), xlogx is a covex fuctio. Let s j = α(s j + h) + (1 α)(s i h) s i = (1 α)(s j + h) + α(s i h) where h > 0, s i < s j ad α = s j s i + h s j s i + 2h (0,1). Because xlogx is a covex fuctio, the followig iequalities are satisfied It follows that s j logs j α(s j + h)log(s j + h) + (1 α)(s i h)log(s i h)) s i logs i (1 α)(s j + h)log(s j + h) + α(s i h)log(s i h)). s i logs i + s j logs j (s j + h)log(s j + h) + (s i h)log(s i h)). This meas that the differece (11) is positive ad so the property 1 is satisfied. Property 4 Cosider two portfolios of loas s ad s as i property 4. The:

14 Sigle-ame cocetratio risk 13 DT H 1 (s ) DT H (s) = s m logs m + log( + 1) [s i logs i + s j logs j + log] = s m logs m s i logs i s j logs j + log( + 1) log = log ss m + log + 1 s s i i ss j j = log (s i + s j ) s i+s j s s + log + 1 i i ss j j ( ) si + s si ( ) j si + s s j j = log + log + 1 > 0, (12) sice all the argumets of the logarithms i (12) are greater tha 1. s i s j 6 Numerical applicatios I this sectio six portfolios of loas are cosidered ad the idices preseted i the previous sectios are calculated o them. For the costructio of the most cocetrated portfolio, the large exposure limits of the Bak of Italy (Baca d Italia, 2006) is cosidered. I this aalysis the exposure of the portfolio T is 1,000 euros. Therefore, the miimum regulatory capital charge of 8% is 80 euros ad this is cosidered the capital requiremet of the bak. The Bak of Italy establishes that a exposure is defied as large if it amouts to 10% or more of the bak s regulatory capital, i this case a exposure is large if it is greater tha or equal to 8 euros. Accordig to the Bak of Italy s regulatio, a large exposure must ot exceed 25% of the regulatory capital, i this case 20 euros. The sum of all large exposures is limited to eight times the regulatory capital, which correspods to 640 euros i this case. By cosiderig this regulatio, the portfolio with the highest cocetratio risk P1 cosists of 32 exposures equal to 20 euros, 51 equal to 7 euros ad oe equal to 3 euros. Hece, the total exposure of the portfolio P1 is 1,000 euros ad its umber of loas is 84. I order to obtai the portfolio P2, each exposure of 20 euros i the portfolio P2 is divided ito two exposures of 10 euros. It follows that the total exposure of the portfolio P2 remais costat (T = 1,000) ad the umber of the loas of the portfolio icreases ( = 116). Moreover, the portfolio P3 is obtaied from the portfolio P2 by mergig two exposures of 10 euros i oe of 20 euros. From the portfolio P3, by eglectig the exposure of 20 euros the portfolio P4 is defied. Fially, the last two portfolios are obtaied by itroducig i P4 a medium exposure of 7 euros for the portfolio P5 ad a low exposure of 3 euros for the portfolio P6. It is importat to highlight that both the total exposure T ad the umber of loas ca chage i these six portfolios.

15 14 Raffaella Calabrese, Fracesco Porro } {{}} {{} 3 T = 1,000 = 84 P } {{}} {{} 3 T = 1,000 = 116 P } {{}} {{} 3 T = 1,000 = 115 P } {{}} {{} 3 T = 980 = 114 P4 } {{}} {{} 3 T = 987 = 115 P } {{}} {{} 3 3 T = 983 = 115 P Table 1 summarizes the values of the sigle-ame cocetratio idices cosidered for the six portfolios. The portfolio P3 shows a higher sigle-ame cocetra- G HH HT DTH RHK(α = 3) RHK(α = 0.5) P P P P P P Table 1 The values of the cocetratio idices are computed for the six portfolios ad multiplied by tio of the portfolios P4 ad P5, this orderig is satisfied by all the idices except the RHK idex with α = 0.5. This result is maily due to the characteristic of the RHK with α = 0.5 to stress the importace of smaller exposures. It is iterestig the compariso of the portfolios P4 ad P6, eve if it falls i the property 6. For the RHK idex with α = 0.5, α = 3 ad the HH idex, the results i Table 1 are coheret with the proof i Sectio 4 that these idices satisfy the property 6 of the irrelevace of small exposures. Moreover, the results i Table 1 show that the G ad the DTH do ot satisfied the property 6, accordig to what showed i the previous sectio. Fially, for the HT idex the result ca be icoheret with the proof i the Subsectio 4.2. I particular, the HT idex satisfies the property 6 but its superior limit s of small exposures is lower tha 3 (the exposure s added to the portfolio). Ideed, by addig a exposure equal to oe to the portfolio, the HT idex decreases. The compariso betwee the portfolios P3 ad P2 falls i the subadditivity property 4. Eve if the G idex does ot satisfy this property, i this particular case G satisfies the orderig established by the property 4. The portfolio P5 is obtaied

16 Sigle-ame cocetratio risk 15 by addig a medium exposure to the portfolio P4, for this reaso the orderig of the cocetratio risks of these portfolios is ambiguous. From the portfolio P4 to P5, the G ad the DTH idices show a icrease of the cocetratio risk, o the cotrary the other idices a decrease. All the idices agree that the cocetratio risk decreases from P4 to P2. It follows that the impact of the icrease of the umber of loas i the portfolio is higher tha the impact of larger loas. Fially, the portfolio P5 shows a higher cocetratio risk tha that of the portfolio P6. It is importat to highlight that oly the HH idex satisfies this orderig. Refereces 1. Baca d Italia (2006). New regulatios for the prudetial supervisio of baks. Circular 263 of 27 December Basel Committee o Bakig Supervisio (2004). Iteratioal Covergece of Capital Measuremet ad Capital Stadards: A Revised Framework. Jue, Basel, BIS. 3. Becker, S.Dullma, K., Pisarek, V. (2004). Measuremet of cocetratio risk - A theoretical compariso of selected cocetratio idices. upublished Workig Paper, Deutsche Budesbak. 4. Cowell, F.A. (1977) Measurig Iequality: Techiques For The Social Scieces. Joh Wiley, New York. 5. Dalto, H. (1920). The measuremet of the iequality of icomes. Ecoomics Joural, 30, pp Gordy, M. B., A risk-factor model foudatio for ratigs-based bak capital rules. Joural of Fiacial Itermediatio, 12(3), Ecaoua, D., Jaquemi, A. (1980). Degree of moopoly, idices of cocetratio ad threat of etry. Iteratioal ecoomic review, Gii, C. (1921). Measuremet of Iequality of Icomes. Ecoomic joural 31, pp Hall, M., Tidema (1967). Measures of cocetratio. Joural of America statistical society, 62, 17, pp Haah L., Kay, J. A. (1977). Cocetratio i moder idustry. Mac Milla Press, Lodo. 11. Herfidahl, O. (1950) Cocetratio i the U.S. Steel Idustry. Dissertio, Columbia Uiversity. 12. Hirschma, A. (1945) Natioal power ad the structure of foreig trade. Uiversity of Califoria Press, Berkeley. 13. Hirschma, A. (1964). The paterity of a idex. America Ecoomic Review, 54, 5, pp Lorez, M. O. (1905). Methods of measurig the cocetratio of wealth. Publicatios of the America Statistical Associatio, 9, 70, pp Lutkebohmert, E. (2009). Cocetratio risk i credit portfolios. Spriger-Verlag. 16. Pigou, A.C. (1912). Wealth ad Welfare. Macmilla Co., Lodo. 17. Theil, H. (1967). Ecoomics ad Iformatio Theory. North Hollad, Amsterdam.