Bounds of Herfindahl-Hirschman index of banks in the European Union

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1 MPRA Muich Persoal RePEc Archive Bouds of Herfidahl-Hirschma idex of bas i the Europea Uio József Tóth Kig Sigismud Busiess School, Hugary 4 February 2016 Olie at MPRA Paper No , posted 8 August :43 UTC

2 JÓZSEF TÓTH Bouds of Herfidahl-Hirschma idex of bas i the Europea Uio Abstract The level of cocetratio ca be measured by differet idicators. The article overviews the calculatio methods of the so called CR3 idex, Gii-idex ad Herfidahl-Hirschma idex. The paper maes evidece for existece of such upper ad lower bouds of Herfidahl-Hirschma idex that could be determied by samplig if the total maret value ad the total umber of maret players are ow. Furthermore, it proves that the differece of the upper ad lower bouds decreases if the sample is supplemeted by ew uits of the aalyzed populatio. I order to apply the fidigs, the bouds of the Herfidahl-Hirschma idex of the ba maret of the Europea Uio are determied i case of total assets, ow fuds, et iterest icome ad et fee icome of Europea bas. It ca be doe because these aggregated values ad the umber of Europea bas are ow. Joural of Ecoomic Literature (JEL): C01, C51, G21 Key Words: Herfidahl-Hirschma idex, cocetratio, ba 1. Itroductio Measurig of the cocetratio level of the ba maret is importat for more reasos. O the oe had, it is importat because evaluatio of the processes effectig o the ris level of the baig activity supports mitigatio of the riss hidde i the maret cocetratio. I 90 s, the Europea baig maret wet through i a sigificat cocetratio process. As a result of it, huge bas have bee created. For example, the balace sheet total of the three largest bas i the Uited Kigdom is close to two ad half times higher tha the GDP of the UK i The situatio of other Europea Uio member states is the same. It bears extreme high ris. Baruptcy of a large

3 ba might cause umaageable cosequeces. Therefore, measurig the cocetratio level ad maig decisio o this result are importat i the moder baig idustry. Geerally, the fiacial govermets have tools to measure cocetratio via atioal reportig systems. O the other had, determiatio the stregth of the competitio supports the decisios of the maret players. If a ba wats to eter a ew maret, measurig of the competitio level is a uavoidable tas. Also, a ba operatig withi a maret should aalyse the possibilities of gaiig ew maret share. Sice, the competitio level depeds o the cocetratio level ad vice versa, the ew ad old maret players should measure the cocetratio of the maret. Eve more, whe maig decisios, ivestors of baig idustry also have to have iformatio o baig maret cocetratio level. Sice survey of the etire maret is extremely expesive ad caot be doe i umerous cases, samplig the maret is the acceptable way to get iformatio about the maret cocetratio. Though differet methods have already bee elaborated (Naueberg, Basu ad Cha, Naldi ad Flamii, Micheliia ad Picforda), their applicatio is ot always easily feasible i case of fiacial idustry. However, sice the baig is a special maret, umerous data are reported to the Europea ad atioal cetral bas. If these data could be used, it would mae the calculatio of the cocetratio level easier. The mai goal of this paper is to itroduce such tool that applicable for maret players ad decisio maers for determiatio of the baig maret cocetratio i a simple way based o data o baig activities disclosed by the Europea Cetral Ba ad based o samplig. The applied idicator is the Herfidahl-Hirschma idex (HHI).

4 2. Calculatio of Herfidahl-Hirschma idex The cocetratio level ca be measured based o balace sheet total, et iterest icome, umber of cliets etc. This paper uses balace sheet ad profit or loss statemet data. The formula of HHI is calculated as follows: HHI = HHI = ( x i i=1, where is the umber of the maret players, x i is the value of the ith maret player, ( x i i=1 T 100)2 or T = i=1 x i, ad 0 < i, i Z +, Z + (whe calculatig the cocrete values bellow, figures are give i percetage form but i other cases i ormal form). This idex is easily ca be calculated if all of the measured value of the maret players is ow. For example, if the maret cosists of four maret players, the aalysed maret cocetratio value is the total assets of the maret players ad their values are {120;200;80;500}. I that case the sum of the total assets is 900 ad the Herfidahl-Hirschma idex is the followig: HHI = ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 = The HHI values are betwee zero ad The higher level of the cocetratio is expressed by higher level of the idex. If the maret share of the maret players is ot ow ad umerous maret players operate i the maret, determiatio of the HHI value is ot so easy.

5 If there is o possibility to get iformatio about all of the maret shares the Herfidahl-Hirschma idex is to be estimated. This paper aims to mae prove for the followig two allegatios: a. If the umber of maret players of the whole baig maret ad aggregated value of their balace sheet or profit or loss statemet items are ow, bouds of Herfidahl-Hirschma idex are determiable based o samplig i coectio with the cocetratio of the balace sheet or profit or loss statemet items i questio. b. By icreasig the sample size, the iterval determied by bouds specified above decreases. I other words the estimatio of the Herfidahl-Hirschma idex is more accurate if the sample size icreases. Though the total assets related HHI idex is also disclosed by the Europea Cetral Ba, but whe calculatig, isurace istitutios ad pesio fuds are also icluded. Usig the elaborated method itroduced bellow, ay balace sheet data or ay profit or loss statemet data cocetratio i relatio with fiacial istitutios ca be estimated. 3. Former Herfidahl-Hirschma idex estimatio methods Whe estimatig the idex, the base coditio of Naueberg, Basu ad Cha (1997) was that the maret share of the largest maret players is ow but other s maret share is uow. They aalysed the liely distributio of the uow maret participats maret share. I their approach the maret share of the uow maret participats is give i a positive iteger value which is expressed i percetage. They origiated the issue from the wellow combiatorial problem where m pieces ball should be distributed i q boxes ad all of the balls must be ordered to oe of the boxes as well as each box should be filled up at least oe ball. I this problem the m meas the

6 maret share give as a iteger withi a maret which must be distributed amog q maret players. Accordig to the example of the authors, if the maret has maret players ad the maret share of the three smallest firms (q=3) is 8 % (q=8%) tha the followig distributio ad probability ca be expressed by permutatio: Table 1: Example for Neueberg et al. s forecast of the Herfidahl- Hirschma idex Number Permutatios Sum HHI value Probability Expected value X -2 X -1 X m /21=0,14 5, /21=0,29 8, /21=0,29 7, /21=0,14 3, /21=0,14 3,14 Sum 28 Source: Based o data of Neueberg et al., ow editig

7 I the example above, the expected HHI value of the uow maret shares is 28. Therefore, the HHI of the whole maret is the HHI of the ow shares of the maret players elarged by 28. Also, Naldi ad Flamii (2014) made iterval forecast for the Herfidahl- Hirschma idex. I their approach, the maret share of the M largest maret 2 players is also ow but other s maret share is uow. HHI = i=1 s i as well as 1 = i=1 s i, if the s i expresses the maret share of the ith maret player i the maret where there are maret participats ad the maret share of the M largest maret players is ow. I that case Naldi ad Flamii determied the miimal value of the HHI as follows: HHI mi = i=1 M M 2 M s 2 i + (1 i=1 s i). Whe determiig the maximal value of the Herfidahl-Hirschma idex, they differetiated two cases. If the aggregated value of the uow maret share is less tha the least ow maret share M M 2 i=1 i=1. tha HHI max = s i 2 + (1 s i ) If the aggregated value of the uow maret share is higher tha the least ow maret share tha HHI max = M s 2 i + s 2 M i=1 M + (1 s i s M Q, where Q = 1 i=1 s i i=1 ) 2 M. s M Furthermore, Micheliia ad Picforda (1985) applied the cocetratio ratios i order to determie the lower ad upper bouds of Herfidahl- Hirschma idex, where the icome of the eterprises was the aalysed maret value. 4. The ew model Durig the model compilatio it is supposed that the aggregated value aalysed ad the umber of uits of the populatio is ow. Furthermore, it

8 is supposed that result of a radom sample from the give populatio is also available. Propositio 1 Let 0 < x i deote the value of the uits of a populatio which has Z + uits where i ad i Z +. Let the sample size origiated from the populatio is (1 <, Z + ) ad x = G, where G = i=1 x i deotes the mea of the uow uits as well as let at least oe uow uit value which i ot equal to at least oe other uow uit value. I that case, by substitutig each uow uit by their mea, the HHI value of the populatio composed by ow uits ad pieces x is lower tha the HHI of the origial populatio. If the propositio is true, the lower boud level of the HHI is determiable by samplig i case where the aggregated value ad the umber of uits of the populatio are ow. It meas that owig the aggregated value of a populatio (for example the aggregated value of the total assets of bas fuded o the area of Europea Uio) ad the umber of uits of a populatio (the umber of those bas), tha a lower boud determiable (by calculatio of the mea of the total assets of the uow bas ad substitutig each uow total asset value by their mea). I this case the HHI value of the populatio is higher tha the lower boud. For example let the A the set of the followig values: A= {5;10;20;25;40;50;60;80;100}. A has 9 uits ad sum of the elemets is 390. The HHI value of the populatio is HHI = ( ) 2 + ( ) 2 + ( ) 2 + ( ) ( )2 + ( )2 + ( )2 + ( )2 + ( )2 = =

9 If the sample has 4 uits tha 5 uits is uow. For example let the sample is the followig: {25;50;80;100}. Therefore {5;10;20;40;60} are uow. The essece of the method is that the uow uits are substituted by their mea durig the HHI calculatio. Though, {5;10;20;40;60} are uow, their mea is determiable because accordig to the origial assumptio, sum of the value of the uits ad umber of the uits is ow. Sice sum of the uits is 390, sum of the ow elemets is =255 ad the umber of the uow uits is 5, therefore, the mea of the uow uits is = 27. Let A the populatio of the ow values ad 5 pieces 27: A = {27;27;27;25;27;50;27;80;100}. modified populatio is the followig: I that case the HHI value of the HHI = ( ) + ( ) + ( ) + ( ) + + ( )2 + ( )2 + ( )2 + ( )2 + + ( )2 = By substitutig the 5 uow elemets by their mea, such value was determiable which is lower tha the HHI of the origial populatio. I this case the HHI value is higher tha Beside the deotatios i the Propositio 1, let sum of the value of the uits of the populatio T = i=1 x i. Let deote HHI the Herfidahl-Hirschma idex of such populatio that composed by ow uits ad pieces x. The order of the uits does ot have effect o HHI value, therefore, let the uow elemets are the first uits (x 1, x 2, x 3,, x ), ad x +1, x +2, x +3,, x the ow elemets of the modified populatio. I that case, HHI < HHI statemet must be prove.

10 Sice 1 = x 1 = x 2 = x 3 = = x G G G G is ot possible (there is at least oe uow uit value which i ot equal to at least oe other uow uit value), therefore, the value of the expressio ( 1 x 1 G )2 + ( 1 x 2 G )2 + ( 1 x 3 G )2 + + ( 1 x G )2 1 2 x i + 2 G (x i G )2. is higher tha zero. After leavig the bracets 0 < i=1 Therefore, 0 < 1 2 x i i=1 ( x i i=1 Sice i=1 x i G = 1, thus, G + G )2. 0 < (x i i=1 As 1 2 = 1. Accordigly, 1 < x i 2. It G )2. i=1 G 2 meas ( G )2 < i=1 x 2 i. Multiplyig both side of the iequality by ( 1, G 2 ( ) T i=1 < ( x i, where G the mea of the first elemets. Icreasig both side of the iequality by ( x i i=+1 the iequality will be the G 2 followig: ( ) T G 2 ( ) T + ( x i i=+1 < ( x i i=1 ( x i i=+1. Therefore, + ( x i i=+1 < ( x i i=1 statemet give i the propositio. +. Thus, HHI HHI. It is exactly the Whe calculatig the cocetratio of the ba maret of the Europea Uio, the mea of the values of uow uits is differet i case of differet member states. Sice i case of ay uits (where 1< ), substitutig the values of uits by their mea the value of HHI decreases, therefore, the Propositio 1 maes possible that the mea of the uow values are to be calculated as per member states. By usig mea of uow values as per member states, the value of HHI decreases comparig with HHI value of the origial populatio.

11 Propositio 2 Let 0 < x i deote the value of the uits of a populatio which has Z + uits where i ad i Z +. Let the sample size origiated from the populatio is (1 <, Z + ) where G = i=1 x i deotes the sum of the uow uits. By excludig the uow uits from the populatio ad supplemetig it by G, the Herfidahl-Hirschma idex value of the populatio created o this way (it has + 1 uits) is higher tha the Herfidahl-Hirschma idex value of the origial populatio (which has uits). Keepig the above example, A= {5;10;20;25;40;50;60;80;100}. It has ie uits, the sum of the value of the uits is 390, the value of the HHI is The sample is {25;50;80;100}. The essece of the method is that the uow uits are excluded ad the populatio is supplemeted by sum of their values. The HHI of the populatio created o this way is higher tha the HHI value of the origial populatio. Sice sum of the uits of A is 390 ad sum of the value of the uits i the sample =255, therefore, G= = 135. Accordig to the propositio, the HHI of the A populatio is less tha the HHI of the populatio composed by {25;50;80;100;135} uits (HHI ). It is less actually, sice HHI = ( )2 + ( )2 + ( )2 + + ( )2 + ( )2 = By excludig the 5 uow elemets ad supplemetig the populatio by their aggregated value, such value was determiable which is higher tha the HHI of the origial populatio. I this case the HHI value is less tha

12 Beside the deotatios i the Propositio 2, let sum of the value of the uits of the populatio T = i=1 x i. Let deote HHI the Herfidahl-Hirschma idex of such populatio that composed by + 1 ({G, x +1, x +2, x +3,, x }) uits. I that case, HHI < HHI statemet must be prove. Sice 0 < x i i ay case, 0 < 2 (x 1 x 2 + x 1 x x 1 x + x 2 x 3 + x 2 x x 2 x + + x 1 x ). 2 2 Icreasig the right side of iequality by 0 = i=1 x i x i i=1, 2 0 < (x 1 + x 2 +x x )(x 1 + x 2 +x x ) x i. i=1 Therefore, 0 < G i=1 x i ad x i iequality by ( 1, i=1 ( x i by ( x i i=+1 the iequality will be the followig: ( x 2 i T ) + i=1 i=1 < G 2. Multiplyig both side of the < ( G. Icreasig both side of the iequality ( x 2 i T ) < ( G 2 T ) + ( x 2 i T ). i=+1 i=+1 Therefore, HHI < HHI. It is exactly the statemet give i the propositio. Whe calculatig the cocetratio of the ba maret of the Europea Uio,, the sum of the values of uow uits is differet i case of differet member states. Sice i case of ay uits (where 1< ) the HHI value of the origial populatio is less tha the HHI, the Propositio 2 maes possible that the sum of the uow values are to be calculated as per member states. By usig sum of the uow values as per member

13 states, the value of HHI icreases comparig with HHI value of the origial populatio. Propositio 3 Supplemetig the sample size by a additioal data collectio, the value of HHI of the modified populatio determied i Propositio 1 will icrease. I the example used at Propositio 1 the sample size was 4. Let the sample size ow 5 where the ow values are {25;40;50;80;100}. Sum of their values is = 295. The mea of the uow values is = Therefore, the value of HHI : HHI = ( 23, )2 + ( 23, )2 + ( 23, )2 + ( )2 + + ( )2 + ( )2 + ( 23, )2 + ( )2 + ( )2 = Whe four uits were ow was the value of the HHI (which was less tha the HHI of the origial populatio). By addig ew data to the sample the HHI value (which was also less tha the origial HHI) icreased ( ). Therefore, the accuracy of the forecast is better. Let HHI x = ( x i i=1 ad HHI y = ( y i i=1 the Herfidahl-Hirschma idexes of the populatios where is the umber of uits i both cases. Let T = i=1 x i = i=1 y i where i,i Z +, Z + ad let HHI x < HHI y. Let supplemeted both populatio by uits z +1, z +2, z +3, z +l. I that case HHI x = ( x i T )2 +l + ( z i i=1 i=+1 T )2 ad

14 +l HHI y = ( y i T )2 + ( z i i=1 i=+1 T )2 where T is sum of the values of the i=1 < populatios. Multiplyig both side of the iequality ( x i ( y i i=1 by ( T T )2 the ew expressio is ( x i i=1 < ( y i Icreasig both side of the iequality by +l ( x i T )2 + ( z i i=1 i=+1 T )2 < ( y i T )2 + T )2 +l ( z i i=+1 T )2, +l ( z i i=1 i=+1 T )2. i=1 T )2. Therefore, HHI x < HHI y. I other words if umber of uits ad sum of the uits of two populatios is the same ad the HHI value of the first populatio is less tha the HHI value of the secod oe as well as both of the populatios are supplemeted with the same ew uits, relatio of the HHI values does ot chage (the less remais the less). Usig the deotatios above, let the first uits of the populatio the uow uits (x 1, x 2, x 3,, x ), ad x +1, x +2, x +3,, x the ow uits where x = G. By additioal data collectio let x 1, x 2, x 3,, x l (l < ) are ow. Accordig to the Propositio 1, the Herfidahl-Hirschma idex of the subpopulatio (HHI ) which has uits ad composed by x 1, x 2, x 3,, x l ow uits ad l uow uits (their value is the mea of the uow uits) is higher tha Herfidahl-Hirschma idex of such subpopulatio (HHI l ) that has uits (their value is x ): HHI < HHI l. That is, the HHI value is lower if fewer uits are ow. Sice the umber of uits is, sum of the uits is G i both of the subpopulatios as well as the HHI value is lower i case of subpopulatio where fewer uits are ow, supplemetig the subpopulatios by x +1, x +2, x +3,, x ow uits, the HHI value will be lower where fewer uits are ow.

15 Propositio 4 Supplemetig the sample size by a additioal data collectio, the value of HHI of the modified populatio determied i Propositio 2 will decrease. I the example used at Propositio 2 the sample size was 4. Let the sample size ow 5 where the ow values are {25;40;50;80;100}. Sum of their values is = 295. The sum of the uow values is = 95. Therefore the value of the HHI : HHI = ( )2 + ( ) + ( )2 + ( )2 + + ( )2 + ( )2 = Whe four uits were ow was the value of the HHI (which was higher tha the HHI of the origial populatio). By addig ew data to the sample the value (which is also higher tha the origial HHI) decreased ( ). Therefore, the accuracy of the forecast is better. Usig the deotatios above, let the first uow uits of the populatio deoted by x 1, x 2, x 3,, x, ad the ow uits deoted by x +1, x +2, x +3,, x where G = i=1 x i. By additioal data collectio let x 1, x 2, x 3,, x l (l < ) are ow ad let deote J the sum of such uits that are ot ow after the additioal (secod) data collectio (J = i=l+1 x i ). Therefore, x 1 + x x l + J = G. Accordig to the Propositio 4 the Herfidahl-Hirschma idex of the populatio (HHI l ) composed by uits x 1, x 2, x 3,, x l, J, x +1, x +2, x +3,, x is less tha Herfidahl-Hirschma idex (HHI ) of the populatio composed by uits G, x +1, x +2, x +3,, x : HHI l < HHI. x x x l 2 + J 2 < (x 1 + x x l + J) 2, sice 0 < x i i ay case. It meas that x x x l 2 + J 2 < G 2. Therefore,

16 ( x 1 T ) 2 + ( x 2 T ) ( x l T ) 2 + ( J T ) 2 + ( x +1 T ) 2 + ( x +2 T ) ( x T ) 2 I other words: HHI l < HHI. < ( G T ) 2 + ( x +1 T ) 2 + ( x +2 T ) ( x T ) 2 5. Practical applicatio of method I order to use the theoretical result above, a sample has bee selected from the ba maret of Europea Uio. Durig the selectio the followig aspects were tae ito accout: - at least five bas as per member states of Europea Uio had to be covered by samplig, - at least oe to third of the total assets had to be covered by samplig as per member states. Based o these aspects 164 bas were selected (the average value is 5,86 as per member states). The sample covers 72.40% of the whole ba maret of the Europea Uio (as per assets). The lower ad upper bouds were determied i case of total assets, ow fuds, et iterest icome ad et fee icome. The followig chart depicts the lower ad upper bouds of the aalysed balace sheet or icome statemet data (data of et iterest icome are icomplete for 2014, these data are ot idicated). Chart 1: HHI bouds of total assets, ow fuds, et iterest icome ad et fee icome i the baig maret of Europea Uio

17 Source: Ow calculatio The cocetratio level of the total assets, ow fuds, et iterest icome ad et fee icome was determiable by samplig. By usig this method the cost of cocetratio measuremet could be cut. However, it ca be used oly if the aalysed aggregated value of the populatio as well as the umber of maret players is ow. I case of Europea bas, the method is applicable because the Europea Cetral Ba discloses these data. The lower ad upper values of HHIs mae possible to determie the cocetratio level by samplig. If the size of the sample icreases the accuracy is better. It is observable if the lower ad upper values are ordered to the sample size. For example, the followig chart depicts how differece of upper ad lower HHI values approaches each other whe the sample size grows. At first, 28 bas are i the sample (from each member state). After that, the sample icreases by 28 ew bas (from each member state) ad the differece betwee the upper ad lower HHIs decreases. Chart 2 : Chage i the lower ad upper HHI values by icreasig the sample size.

18 Total assets 2014 Total assets 2013 Ow fud 2014 Ow fud 2013 Net iterest icome 2014 Net iterest icome 2013 Net fee icome 2014 Net fee icome 2013 Source: Ow calculatio If the calculatio relates to the separated maret of member states, the cocetratio level icreases. Thought, the sample size is ot too high (the average is 5.86 per member state) the differece betwee the upper ad lower HHI bouds is uder 500 poits i umerous cases. The followig table summarizes the differeces measured (data of et iterest icome are icomplete as for year 2014). Table 2: Differeces betwee upper ad lower HHI bouds i case of 28 member states of Europea Uio (umber of member states, pieces) Differeces betwee the upper ad lower HHIs (poit) , , , , ,

19 Total assets 2014 Total assets 2013 Ow fud 2014 Ow fud 2013 Net iterest icome 2014 Net iterest icome 2013 Net fee icome 2014 Net fee icome 2013 Total assets 2014 Total assets 2013 Ow fud 2014 Ow fud 2013 Net iterest icome 2014 Net iterest icome 2013 Net fee icome 2014 Net fee icome 2013 Differeces betwee the upper ad lower HHIs (poit) 600, , , , , Sum Source: ow calculatio The most accurate measuremet was i case of et iterest icome regardig year 2014 where close to half of the cases less tha 500 poits was the differece. Whe aalysig the applicability of the propositios it ca be determied that how may cases was the cocetratio level uambiguously classified. The followig table summarises the result. Table 3: Classificatio based o the upper ad lower HHI bouds Cases The category ca be uambiguously determied The upper HHI boud is i oe category higher tha the lower oe

20 Total assets 2014 Total assets 2013 Ow fud 2014 Ow fud 2013 Net iterest icome 2014 Net iterest icome 2013 Net fee icome 2014 Net fee icome 2013 Cases The upper HHI boud is higher tha 1800 ad the lower is less tha Sum Source: ow calculatio I more tha half of the cases the upper ad lower HHI boud either i the same cocetratio category or the upper HHI boud is i oe category higher tha the lower oe. 6. Coclusio Differet methods are elaborated for measurig the cocetratio level of the maret. The Herfidahl-Hirschma idex is oe of the most applied maret cocetratio measuremet idexes. Sice the actual maret shares of the maret players are ot ow i the practice, the Herfidahl-Hirchma idex is to be estimated. Beside other previously elaborated methods, upper ad lower bouds of Herfidahl-Hirschma idex could be determied based o a sample origiated from the etire populatio if the aggregated value of the aalysed maret is ow. It is the case whe aalysig the Europea ba maret cocetratio. Sice, the Europea Cetral Ba publishes the aggregated balace sheet ad profit or loss statemet items ad the umber of the maret participats, the method itroduced above is applicable. Usig these data, lower ad upper bouds ca be determied based o samplig.

21 The differece of the upper ad lower bouds decreases if the sample is supplemeted by ew uits of the aalysed populatio. The practical applicatio of the method shows also the same result. The method itroduce above maes possible for the maret players to determie lower ad upper bouds of the HHI idex i a ew way i coectio with maret of a member state of the Europea Uio. Selectig 6-8 bigger maret players fiacial statemets operatig i the territory of the member state of the Europea Uio ad usig their data as well as usig data of the Europea Cetral Ba, the coutry specific cocetratio level ca be estimated. Refereces: 1. Europea Cetral Ba (2015). Report o fiacial structures. Europea Cetral Ba 2. Europea Cetral Ba (2016). Statistical wirehouse 3. Flemig, J., Nellis, G. (2000). Priciples of Applied Statistics. Thomso Learig, 2d editio, p Herfidahl, O. C. (1950). Cocetratio i the U.S. steel idustry. Upublished doctoral dissertatio, Columbia Uiversity 5. Micheliia, C.,Picford, M. (1985). Estimatig the Herfidahl Idex from Cocetratio Ratio Data. Joural of the America Statistical Associatio, vol. 80 issue 390, pp Naldi, M., Flamii, M. (2014). Iterval Estimatio of the Herfidahl-Hirschma Idex Uder Icomplete Maret Iformatio. IEEE Computer Society, pp Naueberg, E., Basu, K., Chad, H. (1997). Hirschma-Herfidahl idex determiatio uder icomplete iformatio. Applied Ecoomics Letters, vol. 4 issue 10, pp