THE USE IN BANKS OF VALUE AT RISK METHOD IN MARKET RISK MANAGEMENT. Ioan TRENCA *

Transcription

1 ANALELE ŞTIINłIFICE ALE UNIVERSITĂłII ALEXANDRU IOAN CUZA DIN IAŞI Tomul LVI ŞnŃe Economce 009 THE USE IN BANKS OF VALUE AT RISK METHOD IN MARKET RISK MANAGEMENT Ioan TRENCA * Absrac In sophscaed marke envronmens, banks wh suffcen ludy can normally hedge agans marke volaly. The resulng ne effecve open poson deermnes he amoun of he porfolo ha remans exposed o marke rsk, whch Value a Rsk can measure. In conras wh radonal rsk measures, VaR provdes an aggregae vew of a porfolo s rsk ha accouns for advanage, correlaons, and curren posons. As a resul, s ruly a forward-lookng rsk measure ha apples no only o dervaves bu also o all fnancal nsrumens. Furhermore, he mehodology can also be broadened from marke rsk o oher ypes of fnancal rsk, usng Dela-Normal Mehod, Hsorcal Smulaon, or Mone Carlo Smulaon. Key words: Value a Rsk mehod, marke rsk managemen, marke volaly, fnancal rsk, porfolo s rsk JEL classfcaon: G1, G3 Value a Rsk s he mehodology used o esmae he marke rsk o whch a bank s exposed, and also for deermnng, he banks mnmum capal reured o cover hs rsk. I measures he maxmum loss lkely o be los n a porfolo n a gven perod, and for a gven confdence nerval. VaR mehodology was born n 1994, when he Presden of J.P. Morgan Invesmen Bank, Denns Weahersone, asked employees o subm a repor every day abou he bank s degree of rsk ogeher wh a correspondng rsk measure. Thus n Ocober 1994, he well-known deparmen of Rsk Mercs was esablshed whn he bank, specalzed n he rsk sudy and analyss. The rsk measure used has become known under he name of VaR. I s currenly used worldwde by many banks, nvesmen funds, brokerage frms, and nonfnancal companes. Value a Rsk s he fnal sep n he evoluon of rsk managemen nsrumens, combnng he relaonshp beween prce and performance wh he probably of unancpaed marke movemens. I akes no accoun he correlaons beween fnancal asses of he porfolo and he advanage effec. Ths has a dual role, boh for measurng marke rsk on an negraed bass, and deermnng he mnmum capal reured o cover he banks' marke rsk. In a model based on VaR, daa on bank posons, prces, volaly, and rsk facors are nroduced. Rsks covered by he model mus nclude all ems of ner- * Ioan TRENCA (oan.renca@econ.ubbcluj.ro), PhD, Professor, Babes-Bolya Unversy of Cluj- Napoca., Faculy of Economcs and Busness Managemen.

2 The Use n Banks of Value a Rsk Mehod n Marke Rsk Managemen 187 es, shares, commodes, opons and foregn exchange posons, balance shee and offbalance shees. Provsons of he Basel II Agreemen In recognon of he ncreasngly large banks exposure o marke rsk, and n order o ake advanage of he dscplne mposed by capal reuremens, he Basel Commee on Bankng Supervson Issues promoed he 1988 Capal Agreemen, addng n January 1996 capal expendure specfc o marke rsk. I ncludes a se of uanave and ualave sandards for he process of rsk managemen ha are appled o banks usng nernal models. Thus, models of nernal rsk evaluaon, used by banks, ener no a common concepual framework, assessng aggregae exposure o marke rsk of he enre porfolo. The 1996 Amendmen o he Basel II has brough an added capal margn for he marke rsk, ncludng for he bank s radng porfolo (radng book - composed of fnancal nsrumens owned for a shor erm o be sold, whch are marke marked) and for oher porfolos conssng of fnancal nsrumens, manly creds (bankng book). To esmae he marke rsk, banks may use he sandard mehod, ogeher wh he nernal models for deermnng he VaR. Inernal models are more advanageous for large banks, as hey ake no accoun he correlaons beween asses, and reure a much lower capal cos. The assessmen of marke rsk hrough he VaR mehod, respecng he Basel II provsons, mus mee ceran condons: he daly evaluaon of he marke rsk relaed o he neres raes and capal nsrumens of he ransacons porfolo; he daly assessmen of he currency rsk rae of he Bank's porfolo; use a rus level of 99% for he VaR calculaon; use an nsan prce shock euvalen o a prce assocaed movemen of 10 days; use a hsorcal observaon perod of a leas one year; updang of daa ses a leas once every 3 monhs and her reassessmen whenever marke prces change; recognon of emprcal correlaons beween major rsk caegores such as neres raes, exchange raes, prces of capal nsrumens and goods, ncludng he volaly of opons n each caegory of rsk facors he possbly of carryng ou operaons of sress-esng and back-esng; esablshng and monorng VaR lms; esablshng a separae capal margn o cover he rsk of specfc neres raes and capal nsrumens. Sandard capal marke rsk of he Basel Commee reures ha VaR be calculaed daly and capal reuremens relaed o marke rsk are me daly. Capal reuremens are expressed as he maxmum value of he prevous day's VaR and he average of daly VaR ndcaors for he las 60 workng days. Ths s hen mulpled by an addonal mulpler k (whose mnmum value s 3) desgnaed by he naonal supervsory auhores accordng o he ualy of rsk managemen relaed o he banks owned porfolo. Thus, he margn of marke rsk relaed o momen s: 60 1 MRP = max( k VaR, VaR ) + MRS, where:

3 188 Ioan TRENCA MRS- represens he specfc margn rsk o each le n he porfolo, varyng accordng o he sensvy of each le o he marke changes. K was creaed n order o provde addonal proecon o banks ha are no very srong and banks operang n an unsable marke. Buldng he Value a Rsk Model Value a rsk measures he probable maxmum loss regsered on a ceran poson or a posons porfolo n a gven perod and for a gven confdence nerval. The bgges advanage of usng hs mehodology s ha a sngle number summarzes he bank's exposure o several rsk varables. To deermne VaR he followng parameers should be se: Tme horzon for he rsk (): depends on he rsk facors and he maury of he porfolo posons. For more accurae rsk measuremen of he esablshed benchmark, s recommended he calculaon of VaR on a shor me horzon; for he bank capal adeuacy n relaon o marke rsk exposure, s recommended o use a longer nerval. Confdence probably (α) or percenage of rsk olerance (1 - α) should reflec he bank's averson owards he capal cos ha wll exceed VaR. Greaer averson o rsk, and he cos of hgher capal adeuacy wll lead o he esablshmen of a hgh level of confdence. I s recommended ha manan whn he margn of 95% -99%, f no, VaR accuracy wll suffer. Correspondng VaR confdence level α s gven by he smalles number l such ha he probably of loss L o exceed l s no greaer han 1 - α, as follows: VaRα = nf { l R : P( L > l) 1 α} = nf { l R : FL ( l) α} VaR s deermned as he dfference beween he expeced value of he porfolo n he chosen me horzon wh a ceran probably p and he lowes value of he porfolo (gven by he level), a he same me horzon and wh he same probably, as follows: VaR = V V = V 1+ R ) V (1 + R ) = V ( R R ), where: m 0 ( m 0 0 m V0 - curren marke value of porfolo; V - expeced value of he porfolo on he me horzon m V - he lowes value ha a porfolo can record (level) on he chosen me horzon, correspondng wh he confdence nerval; R - he porfolo s average yeld on he me horzon ; m R - yeld correspondng level Value a rsk graph can be represened as follows:

4 The Use n Banks of Value a Rsk Mehod n Marke Rsk Managemen 189 The lowes value of he porfolo reurn a he chosen me horzon wh a ceran probably α s deermned from he dsrbuon of reurn: R 1 α = P ( x < R ) = f ( x) dx. If he cumulave dsrbuon funcon s unknown, and especally f s he cumulave normal dsrbuon, hen he correspondng yeld level can be deermned by he relaonshp: R = α σ + R m Sarng from he prevous relaonshp, VaR formula s obaned assumng normal dsrbuon: VaR = Vm V = V0 ( Rm R ) = V0 [ Rm ( α σ + Rm )] = V0 α σ, where, V 0 - Curren marke value of porfolo α - level of rus; σ- porfolo volaly I s noed ha an ncrease n porfolo volaly wll lead o he flaenng of he yeld dsrbuon curve, whch wll ncrease VaR. If he perod of deenon s shor, changng he average yeld wll no have a sgnfcan mpac on he calculaon of VaR, bu f he perod of deenon s hgh, he changes of average profably wll be sgnfcan. VaR wll also ncrease along wh he confdence levels. VaR s a mehod ofen used, bu mperfec, beng ue dffcul o esmae. Somemes hs can lead o false conclusons, because underesmaes he freuency of small reurn. In addon, he ndcaor s very dffcul o deermne for a bank wh a much-dversfed porfolo of les. Therefore, banks should always es he accuracy of VaR mehodology

5 190 Ioan TRENCA hrough sress-esng and back-esng ess, especally f VaR s used by banks for capal adeuacy o marke rsks. Alernaves o VaR Due o lmaons of VaR, varous alernaves have developed. One of hem s Condonal VaR, whch can cause an expeced porfolo loss when VaR s exceeded, as follows: P ( x < ) = xf ( x) / f ( x) Condonal VaR s a superor VaR measure for uanfyng marke rsk ha can be used successfully o opmze marke porfolos of banks, regardless of wheher follows a normal dsrbuon law, or no. The man dsadvanage of he mehod s ha does no allow he effecve mplemenaon of he back esng mehods. Anoher alernave o VaR s he Margnal VaR, whch may be a facor n he decson o ncorporae or no a new le n he porfolo, assessng he margnal conrbuon of he new le relang o he whole porfolo VaR. Ths value s deermned by calculang VaR sensvy value nvesed n he porfolo s componen. Incremenal VaR represens he VaR effec on a new ransacon. If a porfolo componen s suffcenly small n relaon o he porfolo value, mgh be consdered ha margnal VaR remans consan as he value of x ends o 0. I s deermned as he dfference beween VaR relaed o he nal porfolo poson ( VaR ) and he VaR relaed o he new porfolo posons ( VaR + ): p a VaR ncremena l = VaR p+ a VaR If he VaR decreases, hen he new ransacon wll reduce he rsk of he porfolo, or, on he conrary, wll enhance. The a coeffcen represens a change n one or more componens of he porfolo, n whch mus be aken no accoun her aggregae effec Dfference wh margnal VaR s ha can uanfy a larger modfcaon n he porfolo composon. Alhough s dffcul o mplemen because nvolves a oal revaluaon of he porfolo, he mehod successfully apples where a new ransacon nvolves porfolo exposure o new rsks. If a s he new value ha wll be nvesed n asse, and W s he presen value of he porfolo, hen he rsk of he new porfolo wll be gven by he followng relaonshp: σ pf 1 = W σ pf 0 + a σ + Waσ pf 0, To deermne he sze of new ransacons leadng o mnmze he porfolo rsk, he derved of order I of he laer expresson n repor wh a wll be deermned: σ pf 1 = aσ + Wσ pf 0, a σ Eualng o zero hs expresson, we wll oban: pf 0, σ pf 0 a = W = Wβ σ σ p p

6 The Use n Banks of Value a Rsk Mehod n Marke Rsk Managemen 191 Value a Rsk Models VaR based models combne he poenal modfcaon of each poson, resulng from specfc rsk facors varaons, wh he probably of such varaons. Value changes are aggregaed o segmens level from he ransacon regsry, and o he level of radng markes. VaR value can be calculaed usng he followng mehodologes: he Analycal mehod (co-varaon- varaon mehod or dela-normal); hsorcal smulaon mehod; Mone Carlo smulaon mehod. Analycal mehod nvolves assumng a normal dsrbuon of porfolo secures values yelds. Profably s consdered gradually ndependen, no beng nfluenced by he prevous day's profably. To calculae he poenal modfcaon of curren porfolo value, s calculaed he average and sandard devaon of porfolo les reurn n order o acheve a combnaon of ndvdual posons sensvy o rsk facors from he co-varaon marx, represenng he rsk facors volaly and he correlaons beween asses. I s one of he mos easly mplemened mehodologes; also presens some dsadvanages, because he normal assumpon on whch s based s raher rarely me n pracce. Mos dsrbuons acually have oblong als (fa als); characerzed by a large number of unforeseen evens, n whch case VaR can no esmae well he large losses. On he oher hand, volaly and correlaon coeffcens are varable n me, havng a sgnfcan mpac especally f porfolos conan opons Hsorcal Smulaon Mehod calculaes he hypohecal value of a change n he curren porfolo dependng on hsorcal varaons of he rsk facors. The grea advanage of he mehod s ha makes no assumpon regardng he dsrbuon of profably, usng he emprcal dsrbuon obaned from analyss of pas daa, whle beng a relavely smple calculaon. Because s no dependen on assumpons regardng he parameers of he markes evoluon, hs mehodology can be adaped o lepokurc, asymmerc and oher abnormal dsrbuons. The dsadvanage of he mehod les n he fac ha predcs he fuure developmen based on pas daa, whch could lead o naccurae forecass f he rend of he pas no longer comples, or f he porfolo changes. In he case of Smulaon Mone Carlo Mehod, he dsrbuon of porfolo reurn s obaned by generang dfferen scenaros for he consdered rsk facors, and calculang he porfolo value n hese crcumsances. The mehod s flexble and can be appled o all ypes of porfolos, bu reures a larger power of calculaon and he careful choce of evaluaon models for porfolo s fnancal asses. If a large enough number of possble prof or loss values recorded by he porfolo s smulaed, hen can buld a probably densy, generang he VaR based on he lowes percenle of he dsrbuon. The frs sep s o choose a sochasc model for he behavor of prces, one of he mos freuenly used beng he geomerc Brownan moon. Ths mples ha prces of fnancal asses are no correlaed over me, he varan decreasng as he me ncreases. The change of he value of porfolo asses may be descrbed as follows: ds = S d S dw, where µ + σ S s he value of fnancal asses; µ s he expeced yeld per un of me; σ s he fnancal asse volaly;

7 19 Ioan TRENCA 1 DW s a Wener process, whch can be wren as dw = ϕ(d), where φ s a random varable wh a sandard normal dsrbuon. Insan reurn of fnancal asses vares dependng on he rend µd and he random varable µ, n pracce he dscree model beng used. Thus, f s he freuency wh whch he asse reurn s measured, and S s change n prce n he me nerval, hen we have: S = µ + σ ϕ S Thus, he fnancal asses yeld s consdered o have a normal dsrbuon wh average µ and sandard devaon. Value a rsk of he porfolo les wll hen deermne he dsrbuon of acon prce a he me T (S (T)). Tesng model accuracy The Basel Commee recommends ha banks use regularly rgorous sress-esng programs o denfy evens ha could have a negave mpac on he bank s capal poson. Sress ess should have boh a ualave and a uanave naure. Quanave crera should denfy plausble sress scenaros ha mgh arse n he marke. Qualy crera am assessng he Bank's ably o absorb bg losses, and he measures he bank can ake o reduce rsk. Sress-esng mehodology nvolves several seps: revew nformaon on he acual hghes losses recorded durng a gven perod, compared wh he esmaed losses from he bank's nernal rsk assessmen; smulaon of exreme sress scenaros, by ncorporang boh he large prce varaons, and large reducons n he level of ludy ha are assocaed wh hese evens; evaluae he degree of banks exposure sensvy o marke rsk from changng assumpons on volaly and correlaons; resumpon of bank s specfc sress scenaros, whch surprse he characerscs of he radng porfolo of a bank n he mos adverse condons. In addon, banks are recommended also o use back esng, whch s based on esng a sample of daa from 50 days ago unl he day on whch VaR s calculaed. Through, hey are ryng o deermne how ofen and by wha amoun he VaR lm has been exceeded. Banks can use wo approaches: he bnary loss funcon approach-whch helps deermne he facor k for capal adeuacy. The es can be descrbed as follows: 1, perderea < VaR T = 0, perderea VaR The es resul s: he maxmum number of exceedances of he VaR on a horzon of 50 days suppored by he Basel Commee s 4, f no, he model used n he calculaon of VaR s no suable. he uadrac loss funcon approach s used o compare dfferen VaR models, and consss n he followng es, where P s he porfolo loss : 1 + ( P VaR ), P < VaR T = 0, P VaR

8 The Use n Banks of Value a Rsk Mehod n Marke Rsk Managemen 193 The es resul st = T. Applyng he mehodology of VaR on he BCR porfolo rae usng he analycal mehod In he analyss of exposure o currency rsk, BCR uses VaR mehodology, beng denfed long and shor posons held by each bank on s porfolo of foregn currency. On , he Bank had he followng currency srucure of he porfolo, conssng of en currences: Table no. 1. Currences Porfolo held by he bank Currency Curren poson Exchange rae ( ) Long Poson Shor Poson AUD 11361,13, ,450 CAD ,175, ,300 CHF 18754,944, ,750 DKK ,069 0, ,150 EUR ,873 3, ,000 GBP ,189 4, ,600 HUF 5736,140 1, ,500 JPY ,94, ,500 NOK 708,634 0, ,600 USD ,505, ,000 In he nex sep, has been aken no accoun he exchange raes of en of he porfolo currences over a perod of 5 days (from o ), calculang connuously her daly reurn, hrough he logarhmc mehod, and he volaly, hrough average suare devaon. The daa obaned are summarzed n he followng able: Table no.. Daly Reurn and Volaly Currency AUD CAD CHF DKK EUR Average 0,0005 0, , , ,00007 Devaon 0, ,0069 0,0050 0, ,00376 Currency GBP HUF JPY NOK USD Average -0, , , , ,00040 Devaon 0, , , , ,00571 Daly evoluon of he porfolo reurn rae s graphcally represened as follows:

9 194 Ioan TRENCA The nex sep s o deermne correlaon coeffcens beween porfolo currences and he consrucon of he correlaon coeffcens marx correspondng o all currences of he porfolo. Calculaon of correlaon coeffcens s represened n he followng formula: σ j ρ = ; σ σ T σ, = ( R, R ) ( R, R ) / T 1; = 1, n, j = 1, n j j j = 1 Table no. 3. The marx of correlaon coeffcens j AUD CAD CHF DKK EUR GBP HUF JPY NOK USD AUD 1, ,4564 0,093 0,5063 0,5310 0, ,3471-0,1081 0, ,1883 CAD 0,4564 1, , , , ,5051 0, , , ,6768 CHF 0,093 0, , , ,8854 0, , , ,5683 0,7090 DKK 0,5063 0, , , , ,733 0,5584 0, ,6985 0,81467 EUR 0,5310 0, ,8854 0, , ,7144 0,589 0, , ,8183 GBP 0, ,5051 0, ,733 0,7144 1, ,6505 0,5995 0, ,7138 HUF 0,3471 0, , ,5584 0,589 0,6505 1, ,0940 0,3901 0,1136 JPY -0,1081 0, , , , ,5995-0,0940 1, , ,7737 NOK 0, , ,5683 0,6985 0, , ,3901 0, , ,50857 USD 0,1883 0,6768 0,7090 0, ,8183 0,7138 0,1136 0,7737 0, ,00000 In deermnng he probably wh whch o calculae he maxmum loss relaed o he bank s porfolo, s used a confdence coeffcen α =.33 correspondng o a probably of 99%, whch s also he recommendaon and BNR To calculae he daly VaR, correspondng o each currency n he porfolo, he followng relaonshp wll be used: VaR = V, 0 α σ, where: V,0 - represens he curren marke value of he bank's exposure n each currency porfolo on ; σ - volaly of each currency n he porfolo

10 The Use n Banks of Value a Rsk Mehod n Marke Rsk Managemen 195 The prevous formula wll deermne he maxmum possble losses relaed o foregn currences n each porfolo. Daly maxmum possble losses: Currency Volaly Ne Poson Daly VaR (99%) AUD 0, , ,1354 CAD 0, , ,675 CHF 0, , ,9318 DKK 0, , ,3786 EUR 0, , ,0301 GBP 0, , ,148 HUF 0, , ,036 JPY 0, , ,4445 NOK 0, , ,048 USD 0, , ,070 I s noed from he able ha he larges possble daly loss ( bllon) may be caused by holdng a shor poson n EUR, whch was followed by holdng a shor poson on USD; he maxmum possble loss ha could record n hs case s The maxmum possble loss f he currency would be uncorrelaed would be he amoun of losses for each ndvdual currency, recordng a value of 586,74.58 le. Because currences are correlaed, s necessary o calculae he daly VaR ndcaor for he currency porfolo of he bank, akng no accoun he correlaon beween he currences presened n he marx of correlaon coeffcens, as follows: VaR n n pf = = 1 j= 1 VaR VaR ρ11 ρ1... ρ1 n VaR1 ρ 1 ρ... ρ n VaR VaR pf = ( VaR1 VaR... VaRn ) ρ n1 ρ n... ρ nn VaRn I s obaned a value of VaR ndcaor correspondng o he currency porfolo, for a day, of le. To deermne he maxmum possble loss ha can be recorded on a horzon of 10 days (h), he followng formula wll be appled: VaR = VaR h pf, h achevng a value of le. I s noed ha calculaed VaR, akng accoun of correlaons beween currences, s lower han he VaR calculaed by ndvdual aggregang VaR ndcaors for each currency separaely. The daa obaned can be summarzed as follows: VaR for 1 day ,7089 le Probably 99% Level of confdence,33 Tme horzon 10 days VaR for 10 days ,4571 le. pf j ρ j

11 196 Ioan TRENCA Conclusons Value a rsk s he mos used mehod of uanfyng he marke rsk, beng also a measure for deermnng mnmum capal lm reured for banks o cover he exposed marke rsk. Ths lm s prescrbed by he Basel II hrough a se of uanave and ualave reuremens. Esmang he maxmum loss of a fnancal nsrumens porfolo, hs mehod nvolves he arbrary choce of wo parameers: me horzon and rae of rsk olerance. Because measures wh a ceran error, he rsk exposure, as of he confdence percenage and used smplfcaons, varous alernaves o VaR have developed: Condonal VaR, Margnal VaR and VaR Incremenal. In pracce, here are used several mehods for deermnng he ndcaor, he bes-known are: paramerc mehod, he hsorcal smulaon and Mone Carlo smulaon mehod. In choosng one of he mehods, mus be aken no accoun he accuracy and speed of each model. Paramerc mehod s smple, bu s based on he assumpon of normaly. Hsorcal mehod s easly mplemened, bu does no accuraely capure he rsk of fuure evens. The mos powerful of hem s he Mone Carlo smulaon, whch reures a power calculaon measure. References Greunng, H., Braanovc S., Analyzng Bankng Rsk, A Framework for Assesng Corporae Governance and Fnancal Rsk, The World Bank, Edura Irecson, Bucureş, 004 Governance and Fnancal Rsk, The World Bank, Edura Irecson, Bucureş Hull J., Opons, Fuures and Oher Dervaves, Edura Prence Hall, USA, 1999 Jackel P., Mone Carlo mehods n fnance, Edura John Wley&Sons, USA, 00 Joron P. Value a Rsk: he new benchmark for managng fnancal rsk, Edura McGraw-Hll, USA, 001 Joron P., Fnancal Rsk Manager Handbook, Edura Wley&Sons, England, 003 Angelds T., Mau R., Shenoy C., A robus VaR model under dfferen me perods and weghng schemes, Revew of Quanave Fnance and Accounng, Boson, vol. 18, pg. 197 Basel Commee on Bankng Supervson, The New Basel Accord, 003, hp:// Chpalka N., Daar V., The relevance of value-a-rsk dsclosures: evdence from he LTCM crss, Journal of Fnancal Regulaon and Complance, London, vol. 14, pg. 174 Shjn S., Kumar A., Bhaacharyya S., The relaonshp beween sze, value, and marke rsk: some evdence, Invesmen Managemen & Fnancal Innovaons, vol. 4, pg. 15 Yakov Ben-HaM, Value-a-rsk wh nfo-gap uncerany, The Journal of rsk fnance, 005;6;5, pg. 388