Applying Stress-Testing On Value at Risk (VaR) Methodologies

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1 62 Invesmen Managemen and Fnancal Innovaons, 4/2004 Applyng Sress-Tesng On Value a Rsk (VaR) Mehodologes José Manuel Fera Domínguez 1, María Dolores Olver Alfonso 2 Absrac In recen years, Value a Rsk (VaR) mehodologes,. e., aramerc VaR, Hsorcal Smulaon and he Mone Carlo Smulaon have experenced specacular growh whn he new regulaory framework whch s Basle II. Moreover, complemenary analyses such a Sress-esng and Back-esng have also demonsraed her usefulness for fnancal rsk managers. In hs paper, we develop an emprcal Sress-Tesng exercse by usng wo hsorcal scenaros of crss. In parcular, we analyze he mpac of he 11-S aacks (2001) and he Lan Amerca crss (2002) on he level of rsk, prevously calculaed by dfferen sascal mehods. Consequenly, we have seleced a Spansh sock porfolo n order o focus on marke rsk. Key words: Sress-Tesng, Value a Rsk, Marke Rsk Managemen. I. Inroducon From a concepual pon of vew, Value a Rsk (VaR) needs o be defned prevously n erms of ceran parameers (me horzon, level of confdence and currency n reference), as well as some heorecal hypoheses. One of hem has o do wh sably whch supposes ha he VaR esmae s obaned for normal marke condons. Ths prncple mples he excluson of exreme scenaros characerzed by hgh volaly levels ha are defned by Joron (1997) as Even Rsk. Sress-Tesng s a useful ool for fnancal rsk managers because gves us a clear dea of he vulnerably of a defned porfolo. By applyng Sress-esng echnques we measure he poenal loss we could suffer n a hypohecal scenaro of crss. In he words of Wllam McDonough, he presden of he New York Federal Commsson Bank, One of he mos mporan funcons of Sress-esng s o denfy hdden vulnerables, ofen he resul of hdden assumpons, and make clear o radng managers and senor managemen he consequences of beng wrong n her assumpons. II. Scenaro Analyss Broadly speakng, here are dfferen ways o develop he Sress-Tesng exercse. Dowd (1998) dsngushes hree man approaches: Hsorcal Scenaros of Crss: Scenaros are chosen from hsorcal dsasers such as he US sock marke crash of Ocober 1987, he bond prce falls of 1994, he Mexcan crss of 1994, he Asan crss of 1997, he Argennean crss of 2001, ec. Sylzed Scenaros: Smulaons of he effecs of some marke movemens n neres raes, exchange raes, sock prces and commody prces on he porfolo. These movemens are expressed n erms of boh absolue and relave changes. As he Dervaves olcy Group (1995) suggess: o arallel yeld curve n ±100 bass pons. o Yeld curve shfs of ±25 bass pons. o Sock ndex changes of ±10%. o Currency changes of ±6%. o Volaly changes of ±20%. 1 rofesor Asocado, Fnance Deparmen, ablo de Olavde Unversy, Span. 2 rofesora Tular, Fnance Deparmen, Unversy of Sevlle, Span.

2 Invesmen Managemen and Fnancal Innovaons, 4/ Hypohecal Evens: A reflecon process n whch we have o hnk abou he poenal consequences of ceran hypohecal suaons such as an earhquake, an nernaonal war, a errors aack, ec. Hsorcal Scenaros of Crss Scenaro Analyss Sylzed Scenaros Hpohecal Evens III. Mehodologcal Issues Fg. 1. Types of Scenaro Analyss (Dowd, 1998) Man Assumpons In hs paper, we wan o evaluae he response of Value a Rsk mehodologes o he Sress-esng exercse based on hsorcal scenaros of crss. The frs sep s o calculae VaR esmaes by hree alernave mehods: aramerc VaR, Hsorcal Smulaon and he Mone Carlo Smulaon. In a second par, we pu press on hose esmaes by nroducng boh he sressed volaly and he correlaon observed n wo scenaros of crss; n parcular, he mpac of he 11-S aack n New York (2001) and he Lan Amercan Crss of July orfolo The seleced porfolo consss of fve common Spansh socks, such as: TELEFÓNICA (TEF), BBVA (BBVA), BSCH (SAN), ENDESA (ELE), RESOL (RE). Those shares are he blue chps of he Spansh Marke and hey represen more han 50% of he IBEX-35. I s also mporan o defne he nal value of he poson (porfolo), as well as he parcular weghs of each sock. In ha sense, we are gong o nves equally dvded among he shares (Table 1). Moreover, he dae used o calculae VaR has been se on 30 Augus If we wan o asses he global poson, we only have o mulply respecve prces and number of shares. In ha parcular case, we have chosen he same wegh for each sock,. e., 20%. Inal poson (euros) Table 1 Fecha VeR 30/08/2002 TEF ELE BBVA SAN RE TOTAL N o deulos Cozacón Valor eso 20% 20% 20% 20% 20% 100% Tme Horzon In hs paper, we have seleced a me wndow from 28 January 2000 o 30 Augus 2002 and consss of 651 days of radng. For hs perod, we have ransformed daly prce seres no logarhmc reurn seres by usng he followng formula:

3 64 Invesmen Managemen and Fnancal Innovaons, 4/2004 R ln. (1) 1 In oher words, our sample daa s composed of 650 hsorcal daly reurns. Secondly, we have calculaed he hsorcal volaly for each reurn seres as he followng equaon llusraes: T 2 ( R ) 1 T 1 =1, , (2) where sample sandard devaon, T oal number of observaons, medum reurn of he seres, R reurn of ndvdual asse. Fnally, n order o buld up he sress-esng exercse, we have chosen wo hsorcal scenaros whch are characerzed for her respecve hgh level of volaly: 11-S errors aacks n New York (2001) Brazlan crss (July 2002). The daly volales for each parcular common sock n our porfolo have been calculaed by usng a moble monhly wndow (20 days of marke radng) as Fgure 1 llusraes. We also plo (Fgure 3) he daly volaly observed for he Spansh Sock Marke Index (IBEX-35). Boh chars reflec how rsk, n erms of volaly, ncreases afer hese nernaonal evens occur. 7,00% 6,00% Scenaro I Scenaro II 5,00% volaldad dara 4,00% 3,00% 2,00% 1,00% 0,00% 25/02/ /03/ /04/ /05/ /06/ /07/ /08/ /09/ /10/ /11/ /12/ /01/ /02/ /03/ /04/ /05/ /06/ /07/ /08/ /09/ /10/ /11/ /12/ /01/ /02/ /03/ /04/ /05/ /06/ /07/ /08/2002 horzone emporal TEF ELE BBVA SAN RE Fg. 2. Daly volaly for ndvdual socks

4 Invesmen Managemen and Fnancal Innovaons, 4/ ,00% 3,50% Scenaro I Scenaro II 3,00% 2,50% 2,00% 1,50% 1,00% 0,50% 0,00% 25/02/ /03/ /04/ /05/ /06/ /07/ /08/ /09/ /10/ /11/ /12/ /01/ /02/ /03/ /04/ /05/ /06/ /07/ /08/ /09/ /10/ /11/ /12/ /01/ /02/ /03/ /04/ /05/ /06/ /07/ /08/2002 bex Fg. 3. Daly volaly for IBEX-35 VaR arameers Value a Rsk (VaR) ndcaes he maxmum loss whch we can ncur on a parcular me horzon wh a defned level of confdence. In oher words, VaR, as a sascal esmae, requres he followng parameers: The me horzon wll be one day,.e., we wll esmae daly VaR, or DeaR (Daly Earnngs a Rsk). The level of confdence has been se a 95%. The currency used for reporng VaR fgures s he Euro. robably 95% pobably of &L < VaR 5% probably of loss beyond VaR Loss rof -C VaR 0 Value (poson) Fg. 4. VaR Concep IV. Sress-Tesng On VaR Mehodologes In general, he bass of he Sress-Tesng exercse s o recalculae he Value a Rsk esmae by usng a hgher volaly han he observed one for he hsorcal wndow seleced,.e., 651 radng days. For hs purpose, we have compued he daly volales for each scenaro of crss. These are presened n Table 2.

5 66 Invesmen Managemen and Fnancal Innovaons, 4/2004 Daly volaly for boh scenaro of crss Table 2 Escenaro I Escenaro II Fecha VeR TEF ELE BBVA SAN RE 30/08/ % 1.81% 2.35% 2.52% 2.13% Fecha TEF ELE BBVA SAN RE 11/10/ % 1.96% 4.73% 4.90% 3.48% Fecha TEF ELE BBVA SAN RE 09/08/ % 4.51% 4.99% 5.63% 3.66% From an operaonal pon of vew, he man problem wh Sress-Tesng appears when ncorporang correlaon. Emprcal evdence 1 demonsraes ha correlaon s no consan over me; moreover, flucuaes n perods of crss. As Aragonés and Blanco (2000) pon ou, f we pu pressure on correlaon coeffcens n an arbrary way, probably, he newly calculaed correlaon marx wll no be posve defned and, as a consequence, s elemens wll no have nernal conssency. For hs reason, s srongly recommended no only pressng volales up, bu also he correlaon marx. In pracce, once we have calculaed he correlaon coeffcens beween pars of socks usng a monhly moble wndow, we can selec he correlaon observed for hose days of maxmum volaly levels, whch corresponds o 11/10/2001 and 09/08/2002, respecvely. From here, we have desgned boh sressed correlaon marces (Tables 3 and 4) whose deermnans are posve: 0. (3) Correlaon marx scenaro I Table 3 Marz de correlacón: Escenaro I TEF ELE BBVA SAN RE TEF 100% 56.82% 66.24% 74.49% 45.88% ELE 56.82% 100% 74.07% 73.49% 65.36% BBVA 66.24% 74.07% 100% 93.68% 80.05% SAN 74.49% 73.49% 93.68% 100% 76.14% RE 45.88% 65.36% 80.05% 76.14% 100% Correlaon marx scenaro II Table 4 Marz de correlacón: Escenaro I TEF ELE BBVA SAN RE TEF 100% 74.87% 74.39% 76.97% 54.05% ELE 74.87% 100% 85.19% 84.50% 69.38% BBVA 74.39% 85.19% 100% 89.75% 76.99% SAN 76.97% 84.50% 89.75% 100% 59.24% RE 54.05% 69.38% 76.99% 59.24% 100% 1 Jackson (1996) and Mor, Ohsawa and Shmzu (1996) analysed such phenomena.

6 Invesmen Managemen and Fnancal Innovaons, 4/ Accordng o Alexander y Legh (1997), o ensure ha he correlaon marx s posve defned, mus comply wh he Cholesky mahemacal propery, ha s: where, Correlaon marx, A Cholesky marx, T A Transposed Cholesky marx. T A A, (4) We have also verfed ha sressed correlaon marces can be decomposed no Cholesky facors as Tables 5 and 6 llusrae. Cholesky marx scenaro I Table 5 Marz de Cholesky: Escenaro I TEF ELE BBVA SAN RE TEF 100% 0.00% 0.00% 0.00% 0.00% ELE 56.82% 82.29% 0.00% 0.00% 0.00% BBVA 66.24% 44.27% 60.43% 0.00% 0.00% SAN 74.49% 37.87% 45.63% 30.57% 0.00% RE 45.88% 47.75% 47.19% 7.67% 57.70% Cholesky marx scenaro II Table 6 Marz de Cholesky: Escenaro II TEF ELE BBVA SAN RE TEF 100% 0.00% 0.00% 0.00% 0.00% ELE 74.87% 66.29% 0.00% 0.00% 0.00% BBVA 74.39% 44.50% 49.86% 0.00% 0.00% SAN 76.97% 40.54% 28.98% 39.90% 0.00% RE 54.05% 43.61% 34.85% % 57.58% Sress-Tesng and aramerc VaR Sress-Tesng s very easy o apply when dealng wh he paramerc mehodology because we only have o esmae on 30/08/2002 he sressed VaR for each scenaro of crss as formula 5 ndcaes: * VaR( sressed) 1,6449, (5), daly where nal value of he poson mananed n sock ( Euros), *, daly daly volaly of he sock assocaed o a sressed scenaro, * Z depends on he level of confdence; a 95% confdence s value s equal o -1,6449. * Z

7 68 Invesmen Managemen and Fnancal Innovaons, 4/2004 In Tables 7 and 8 we presen he ndvdual VaR esmaes assocaed wh boh scenaros of crss. We can defne a new magnude whch s raw VaR, wh he aggregaon of ndvdual VaR s, so gve us a global measure of rsk whou sandng dversfcaon benefs. If we wan o have a more realsc dea of he rsk exposure, s necessary o nroduce anoher esmae, whch s dversfed VaR or ne VaR. For ncorporang dversfcaon effecs, we apply he followng formula: VaR porfolo VaR VaR V VaR 1, 2, n, T *, V V (6) : Column vecor of dmenson (nx1) whch represens non dversfed nd- * * vdual VaR s. I s calculaed from he produc of V Z T VaR VaR VaR : The ransposed vecor of V s calculaed as V V 1, 2, n, Z T T * *. Indvdual VaR scenaro I Table 7 Escenaro I TEF ELE BBVA SAN RE Valor ncal Volaldad dara 3,31% 1,96% 4,73% 4,90% 3,48% Z (95%) 1,6449 1,6449 1,6449 1,6449 1,6449 VeR ndvdual 1.090, , , ,17 Indvdual VaR scenaro II Table 8 Escenaro II TEF ELE BBVA SAN RE Valor ncal Volaldad dara 5.11% 4.51% 4.99% 5.63% 3,66% Z (95%) 1,6449 1,6449 1,6449 1,6449 1,6449 VeR ndvdual 1.681, , , , ,43 In Tables 9 and 10 we have compued he dversfed VaR for our porfolo n boh sressed scenaros. Moreover, we have also calculaed anoher neresng esmae, whch s EaR (Earnng a Rsk). I s he maxmum gan we can expec wh a ceran confdence level whn a seleced me perod. In parcular, we have esmaed a 95% percenle. We noce ha boh fgures, VaR and EaR, concde because of he underlyng assumpon of normal dsrbuon. Correlaed VaR scenaro I Table 9 Escenaro I Nvel de confanza Horzone emporal VeR correlaconado 5.391,11 95% 1 da EaR correlaconado 5.391,11 Rao VeR/EaR 100% Rao VeR/Valor de la carera 5.39% Table 10

8 Invesmen Managemen and Fnancal Innovaons, 4/ Correlaed VaR scenaro II Escenaro II Nvel de confanza Horzone emporal VeR correlaconado 7.061,43 95% 1 da EaR correlaconado 7.061,43 Rao VeR/EaR 100% Rao VeR/Valor de la carera 7.06 % Sress-Tesng and he Mone Carlo Smulaon The Mone Carlo Smulaon s based on he generaon of random prces as follows: 1 e, (7) where s he smulaed prce, 1 s he curren prce of he sock, s a random varable whch s dsrbued as a normal sandardzed,.e., wh =0 and =1, s he daly volaly of he sock, s an adjused facor whch ransforms daly volaly no wder me horzons. In hs paper, as VaR s esmaed one day hence, s value s equal o one. In he case of a porfolo, composed by mulple asses, he prevous formula canno be appled because s only vald for a sngle asse. Therefore, he process of generang random numbers s more complex; n oher words, he hsorcal correlaon beween shares should be ncorporaed n such a process. For hs reason, and from a mehodologcal pon of vew, he normal random numbers,, should be ransformed no correlaed random numbers, Z, by usng he Cholesky Marx: ZTEF Z ELE Z BBVA ZSAN Z RE 5x1 Correlaed random numbers * A 5x5 Choleskymarx x1 random numbers, (8) where Z s a vecor of ransformed normal varables whch embodes he hsorcal correlaon, s a vecor of normal sandardzed varables, * A s he sressed Cholesky Marx for each scenaro of crss as Tables 5 and 6 show, respecvely. For smulang correlaed and sressed prces from curren prces (see Table 1) we should generae Z vecors, as he sub- ndex ndcaes n he followng equaon: TEF, ELE, BBVA, SAN, RE, 9,17 e 12,10 e 10,01 e 6,81 e 13,30 e * TEF ZTEF * ELE Z ELE * BBVA Z BBVA * SAN Z SAN * RE Z RE =1, (9)

9 70 Invesmen Managemen and Fnancal Innovaons, 4/2004 Once we have compued he random pahs for ndvdual sock prces, we can oban he smulaed value for he porfolo by mulplyng number of shares and smulaed prces. We can also calculae he smulaed prof and loss dsrbuon as: s & Ls W W, (10) where W s he smulaed value for he scenaro, W s he curren porfolo value on 30 Augus 2002, whch s Euros. If we pu n order each smulaed resul for he porfolo from low o hgh, we can drecly nfer boh VaR and EaR esmaes as 5% and 95% percenles of ha dsrbuon as well as oher parameers such as sandard devaon and meda (Tables 11 and 12). Mone Carlo Smulaon scenaro I Table 11 érdda máxma ,52 Gananca máxma ,29 romedo 118,62 Desvacón esándar 3.330,85 Nvel de confanza Horzone emporal VeR 5.179,94 95% 1 da EaR 5.623,93 Rao VeR/EaR 92,11% Rao VeR/Valor poscón 5.18% Mone Carlo Smulaon scenaro II Table 12 érdda máxma ,87 Gananca máxma ,28 romedo -28,46 Desvacón esándar 4.340,78 Nvel de confanza Horzone emporal VeR 7.032,16 95% 1 da EaR 7.261,98 Rao VeR/EaR 96,84% Rao VeR/Valor poscón 7,03% Sress-Tesng and Hsorcal Smulaon To some exen, Sress-Tesng appears o be a mechancal process based on ncreasng he volaly and correlaon followng a ceran mahemacal formulaon. In conras, when applyng Sress-Tesng on a Hsorcal Smulaon, hs exercse presens a clear dfference. In ha sense, correlaon can no be sressed drecly because s ncorporaed n he hsorcal smulaed prce seres. So, he praccal mplemenaon goes hrough he followng seps: Selecon of wo hsorcal wndows assocaed o boh scenaros of crss. In parcular, we have compued he prevous 20 days of radng from 11/10/2001 for he frs scenaro, and 20 days of radng from 9/08/2002 for he second one. Compuaon of hsorcal sock reurns for each me wndow. Generaon of hsorcal smulaed prces by usng he followng formula: R, (11) e

10 Invesmen Managemen and Fnancal Innovaons, 4/ where s he smulaed prce for he scenaro, s he curren prce of he sock, R s he hsorcal reurn 1,2, From hs pon, he process s dencal o ha descrbed for he Mone Carlo Smulaon. In Tables 13 and 14 we sum up all he calculaons for each scenaro analyzed. Hsorcal Smulaon scenaro I Table 13 VeR correlaconado 4.456,76 Nvel de confanza Horzone emporal EaR correlaconado 4.540,95 95% 1 da Rao VeR/EaR 98,15% Rao VeR/Valor poscón 4,46% Escenaro I Hsorcal Smulaon scenaro II Table 14 VeR correlaconado 5.769,37 Nvel de confanza Horzone emporal EaR correlaconado 6.858,42 95% 1 da Rao VeR/EaR 84,12% Rao VeR/Valor poscón 5,77% Escenaro II Fnally, we conclude wh a comparson among he resuls of he Sress-Tesng as Table 15 llusraes. Summary Table 15 aramérco Normal Escenaro I Escenaro II VeR 2.978, , ,43 EaR 2.978, , ,43 Rao VeR/EaR 100,00% 100,00% 100,00% Rao VeR/Valor poscón 2,98% 5,39% 7,06% Mone Carlo Normal Escenaro I Escenaro II VeR 2.810, , ,16 EaR 3.155, , ,98 Rao VeR/EaR 89,04% 92,11% 96,84% Rao VeR/Valor poscón 2,81% 5,18% 7,03% Smulacón Hsórca Normal Escenaro I Escenaro II VeR 2.817, , ,37 EaR 2.727, , ,42 Rao VeR/EaR 103,29% 98,15% 84,12% Rao VeR/Valor poscón 2,82% 4,46% 5,77%

11 72 Invesmen Managemen and Fnancal Innovaons, 4/2004 V. Conclusons Afer applyng Sress-esng on VaR mehodologes, he man conclusons obaned from Table 15 are as follows: 1. In general, he Sress-Tesng exercse always mples a hgher level of rsk measured n erms of VaR. As Table 15 reflecs, VaR fgures ncrease for boh sressed scenaros. 2. The mpac of Brazlan crss (scenaro II) n our porfolo s greaer han ha of he 11-S errors aacks. Tha s due o he narrow relaonshp beween he Spansh frms (BSCH, RESOL, TELEFÓNICA, BBVA AND ENDESA, whose shares are ncluded n he porfolo) and he Lan Amercan counres such as Argenna, Brazl, ec. 3. The response of VaR mehodologes o he Sress-Tesng exercse s no he same. For boh scenaros of crss, aramerc VaR s he mos reacve. In conras, n erms of EaR, he Mone Carlo Smulaon demonsraes more sensvy. 4. From he mehodologcal pon of vew, we should ensure he nernal conssency of he Sress-esng exercse. For ha reason, we mus verfy ha he Correlaon marx s posve defned and, hus can be decomposed no s Cholesky facors. References 1. Alexander, C. y Legh, C. (1997), On he Covarance Marces Used n Value a Rsk Models, The Journal of Dervaves, volumen 4, nº3. 2. Aragonés, J. y Blanco, C. (2000), Valor en Resgo: Aplcacón a la Gesón Empresaral, rámde. 3. Aragonés, J., Blanco, C. y Dowd, K. (2001), Incorporang Sress Tess no Marke Rsk Modellng, Dervaves Quarerly, Insuonal Invesor, prmavera. 4. Arzner,., Delbaen, F., Eber J. y Heah, D. (1997), Thnkng Coherenly, Rsk, Volumen 10, nº 11, novembre. 5. (1999), Coheren Measures of Rsk, Mahemacal Fnance, volumen 9, nº 3, julo. 6. Basle Commee of Bankng Supervson (1996), Supervsory Framework for he Use of Back-esng n Conjuncons wh he Inernal Model Approach o Marke Rsk Capal Requremens, enero. 7. Beder, T. (1995), VaR: Seducve bu Dangerous, Fnancal Analys Journal 51, sepembre-ocubre. 8. -(1996), Repor Card on VaR: Hgh oenal bu Slow Sarer, Bank, Accounng and Fnance, volumen Bes,. (1998), Implemenng Value a Rsk, John Wley & Sons, Reno Undo. 10. Blanco, C. e Ihle, G. (1999), Makng Sense of Backesng, Fnancal Engneerng News, agoso. 11. Boudoukh, J., Rchardson, M. y Whelaw, R. (1995), Expec he Wors, Rsk nº 8, sepembre. 12. Brer, G. (1950), Verfcaon of Forecass Expressed n Terms of robably, Monhly Weaher Revew, Carrllo, S. y Lamohe,. (2001), Nuevos Reos en la Medcón del Resgo de Mercado, erspecvas del Ssema Fnancero, nº Chrsoffersen,. (1996), Evaluang Inernal Forecass, Mmeo, 15. Research Deparmen Inernaonal Moneary Fund, Forhcomng n he Inernaonal Economc Rewew. 16. Cohen, R. (1998), Caraceríscas y Lmacones del Valor en Resgo como Medda del Resgo de Mercado, ponenca ncluda en La Gesón del Resgo de Mercado y de Crédo. Nuevas Técncas de Valoracón, Fundacón BBV, Blbao. 17. Comsón Naconal del Mercado de Valores (1998), Crcular 3/1998 de 22 de sepembre sobre Operacones en Insrumenos Dervados de las Insucones de Inversón Colecva, Madrd.

12 Invesmen Managemen and Fnancal Innovaons, 4/ Crnkovc, C. y Drachman, J. (1995), A Unversal Tool o Dscrmnae Among Rsk Measuremen Technques, Mmeo, Corporae Rsk Managemen Group, J.. Morgan. 19. Danelsson, J. y de Vres, C. (1997), Exreme Reurns, Tal Esmaon and Value a Rsk, Mmeo, Unversy of Iceland, Tnbergen Insue and Erasmus Unversy. 20. De la Cruz, J. (1998), Una Evaluacón Críca de las Nuevas Meddas de Gesón Y Conrol del Resgo de Mercado desde una erspecva de Supervsón, ponenca ncluda en La Gesón del Resgo de Mercado y de Crédo. Nuevas Técncas de Valoracón, Fundacón BBV, Blbao. 21. Dervaves olcy Group (1995), A Framework for Volunary Oversgh, New York. 22. Dowd, K. (1998), Beyond Value a Rsk. The New Scence of Rsk Managemen, John Wley & Sons, Reno Undo. 23. Group of Thry, Global Dervaves Sudy Group (1993a), Dervaves: racces and rncples. Survey of Indusry racce, Washngon, D.C, julo. 24. Fnger, C. (1997), A Mehodology o Sress Correlaons, Rskmercs Monor, Fourh Quarer. 25. Fran, J. y Meegan, C. (1996), Marke Rsk: An Inroducon o he Concep and Analycs of Value a Rsk, Mmeo, Economc Analyss Research and ublcaons Deparmen, Cenral Bank of Ireland. 26. González, M. (2000), Errores y posbles solucones en la aplcacón del Value a Rsk, Fundacón de las Cajas de Ahorros Confederadas para la Invesgacón Económca y Socal, documeno de rabajo nº Greenspan, A. (2000), Remarks on Bankng Evoluon, 36 h Annual Conference on Bank Srucure and Compeon of he Federal Reserve Bank of Chcago, Federal Reserve Bank of Chcago, Chcago, Illnos, mayo. 28. Hll, B. (1975), A Smple General Approach o Inference Abou he Tal of a Dsrbuon, Annals of Sascs Joron,. (1997), Value a Rsk: he New Benchmark for Conrollng Dervaves Rsk, The McGraw-Hll companes. 30. Jackson,. (1996), Rsk Measuremen and Capal Requremens for Banks, Bank of England Quarer Bullen, mayo. 31. Jackson,., Maude, D. y erraudn, W. (1997), Bank Capal and Value a Rsk, Journal of Dervaves 4, prmavera. 32. Kupec,. (1995), Technques for Verfyng he Accuracy of Rsk Measuremen Models, The Journal of Dervaves, Lerman, R. (1996), Ho Spos and Hedges, Journal of orfolo Managemen Specal Issue. 34. López, J. (1996), Regulaory Evaluaon of Value a Rsk Models, Mmeo, Research and Marke Analyss Group, Federal Reserve Bank of New York. 35. Mahoney, J. (1996), Forecas bases n Value a Rsk Esmaons: Evdence from Foregn Exchange and Global Equy orfolos. Mmeo, Federal Reserve Bank of New York. 36. Mor, A., Ohsawa, M. y Shmzu, T. (1996), A Framework for More Effecve Sress Tesng, Bank of Japan Insue for Moneary and Economc Sudes, Dscusson paper nº 96-E age, M. y Cosa, D. (1996), The Value a Rsk of a orfolo of Currency Dervaves Under Wors-Case Dsrbuonal Assumpons, Mmeo, Susquehanna Invesmen Group and Deparmen of Mahemacs, Unversy of Vrgna. 38. Robnson, G. (1996), More Hase, Less recson, Rsk, volumen 9, nº 9, sepembre. 39. Rockafellar, R. T. (1970), Convex Analyss, rnceon Mahemacs, volumen 28, rnceon Unversy ress. 40. Rockafellar, R. T. y Uryasev, S. (1999), Opmzaon of Condonal Value a Rsk, sepembre, hp://www.se.ufl.edu/uryasev

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