Reaching extreme events with conditional and unconditional models

Transcription

1 Reaching exreme evens wih condiional and uncondiional models LAMPROS KALYVAS Deparmen for he Supervision of Financial and Credi Insiuions Bank of Greece 3 Amerikis S., 1050, Ahens GREECE lkalyvas@bankofgreece.gr COSTAS SIRIOPOULOS Deparmen of Business Adminisraion Universiy of Paras Universiy Campus, Rion, 6500, Paras GREECE siriopoulos@eap.gr NIKOLAOS DRITSAKIS Deparmen of Applied Informaics Universiy of Macedonia 156, Egnaia Sr., 54006, Thessaloniki GREECE dris@uom.gr Absrac: - Classical models ofen fail o predic values arising from crises evens, parly, because hey are based on he assumpion ha financial reurn series follow normal disribuion. On he oher hand, recen empirical sudies have shown ha financial daa exhibi non-normaliy and heavy-ails. Hisorical Simulaion is able o overcome obsacles arising from assumpions abou he shape of he risk facors disribuion. Moreover, Exreme Value Theory (EVT) is becoming a key elemen for modern risk managers and regulaors, in he analysis of heavy-ailed daa and he esimaion of exreme quaniles as i derives he laws governing rare evens wihou implying any saisical disribuion assumpions for he daase. Choosing from he resriced specrum of hisorical simulaion and EVT he key issue is wheher o choose a condiional or uncondiional model. Key-Words: - Exreme Value Theory, GARCH models, condiionaliy, risk managemen, capial requiremens 1 Inroducion The recen hisory exhibis a broad se of examples on financial disasers bu heir consequences on marke paricipans varied from defaul evens o sharp decrease in profiabiliy. Learning from hese lessons, regulaors are rying o formulae a sound and sable financial sysem. In order o achieve heir objecives hey imposed conservaive capial requiremens by he use of he sandard approach (Basle, 1996). On he oher hand, financial insiuions, being complian wih rules imposed by regulaory agencies, ry o saisfy shareholders appeie for coninuously increasing profis, wihou underaking exreme risks. Boh marke paricipans and regulaors are under a coninuous discussion, sriving o find a bilaerally acceped heoreical common ground. Wihin his consensus, quaniaive probabilisic risk managemen echniques (Value-a-Risk ype approaches) were implemened as inernal models, widely acceped by boh as he mos suiable way for measuring individual marke risk exposures. VaR radiionally involves hree mehods developed hroughou he las decade. Variance - Covariance mehod is based on he assumpion ha he join disribuion of risk facors can be approximaed by an a priori heoreical disribuion (usually Normal) which ends o underesimae he ails of he reurn disribuion. Hisorical Simulaion is subjec o even risk because of he lack of sufficien daa se (Longin, 000). Finally, Mone Carlo simulaion is subjec o model error 1

2 risk because i is ofen based on models seleced from a limied specrum of models. Empirical lieraure (Raaikainen, 00) is moving beyond he aforemenioned VaR models owards Exreme Value Theory (EVT) models. EVT refers o wo alernaive classes of models for exreme values: he Block Maxima approach (BM) and he Peaks-Over-Threshold (POT) approach. In our paper we will focus on he semiparameric POT, srucured around he Hill esimaor (Hill, 1975) in an aemp o examine Theoreical Background.1 Hisorical Simulaion Hisorical simulaion-based mehods have become increasingly popular because of heir easiness o implemen, heir abiliy o capure second-order risks hrough various means, and heir independence of he covariance marix of he risk facors. Bu he mos imporan feaures are ha differen disribuion assumpions can be adaped for differen risk facors (e.g. fa ails, lepokurosis, sochasic volailiy, mean-reversion, ec.) and ha valuaion of complex insrumens can be carried ou..1.1 Uncondiional approach Uncondiional hisorical based esimaion of VaR or widely known uncondiional hisorical simulaion VaR (UHS) is he simples way o compue, in erms of minimal compuaional power and parameric assumpions, he marke risk of financial insrumens. These are, mainly, he moivaions ha force many financial insiuions o use he aforemenioned mehod. The sric saionariy assumpion underlying UHS implies ha he disribuion of he risk facors realized in he pas can be used o simulae he disribuion of changes in value of he presen porfolio. Therefore, he lengh of daa available has a direc effec on he abiliy o predic exreme economic recessions in he fuure. The VaR esimae for he +1 day, given a probabiliy level q, is given by he empirical quanile Q q of a window of n independenly and idenically disribued (iid) observaions up o dae, ha is ( r, r r ) VaR ˆ 1 Q,..., (1) + = q 1 n + 1 For example, examining a window of n=5000 observaions, he 1% VaR esimae is he 50 h negaive value of he sample ascending order represenaive European Union marke daa in order o selec he bes VaR model from (un)condiional EVT and hisorical simulaion approaches. Finally, we examine he effec of each mehod on he calculaion of capial requiremens for banking secor s rading book, under he new Basel Accord and, possible dilemmas ha regulaors will face rying o impose he mos efficien mehod. saisic. In his case 49 exreme evens are oping ou he esimaion procedure. The iid assumpion implies ha he model assigns an equal probabiliy weigh of 1/n o each observaion (Prisker, 001). However, volailiy of asse reurns ends o appear in clusers, ha is, high volailiy observaions and low volailiy observaions are grouped ogeher (Bollerslev, 1986). Boudoukh, Richardson, and Whielaw (1998), deviaed he problem by assigning higher weighs o he recen pas observaions han he disan pas observaions, formulaing a generalised version of he hisorical simulaion..1. B-A.G.V. Filer specificaion Adaping he assumpion of normaliy, variancecovariance mehods (no examined here) aemp o capure he condiional heeroskedasiciy of he asse reurns by esimaing variance-covariance marices a every poin in ime. By conras, uncondiional hisorical simulaion mehod does no assume any reurn disribuion bu, ypically, i is unable o capure condiional heeroskedasiciy. In his conex, Barone-Adesi, Giannopoulos, and Vosper (1999), inroduced a varian of he hisorical simulaion mehodology referred as filered hisorical simulaion (FHS). This approach inroduces an innovaion ha capures boh condiional heeroskedasiciy and non-normaliy of asse reurns wihin a unique model, improving he performance of classical variance-covariance and hisorical simulaion mehods ha are currenly in use. Assuming lognormal asse reurns as X r = ln () X 1

3 where X is he index value a ime and X -1 is he lagged index value. B-A.G.V. mehodology proposes he esimaion of an ARMA(1,1)-GARCH(1,1) wihou excluding alernaive models r = a0 + a1r 1 + aε 1 + ε (3) = + + β 0 β1ε 1 β σ 1 σ (4) wih ε ~ N 0, σ ( ) Alhough forecasing for muliple days ahead is feasible we focused on nex day predicions. The VaR is esimaed from he following sequence of equaions afer obaining he fuure reurn disribuion from 1000 simulaions. ε e = (5) σ 1 = β 0 + β1ε β σ σ + + (6) z σ (7) = + 1 e = a0 + a1r + a z + z+ 1 r (8) For he remaining σ +s, ε erm in he firs equaion of he sequence is subsiued by z +s-1. Evenually, a quanile is derived form he forecased disribuion of reurns. The s-day ahead VaR is calculaed by subracing he aforemenioned quanile from -day acual value. A criicism on his mehod is ha when a VaR model is examined using real daa, he abiliy o undersand he properies of he VaR model is obfuscaed by he simulaneous occurrence of oher ypes of model errors including errors in pricing, errors in GARCH models, and oher poenial flaws in he VaR mehodology (Prisker, 001).. Exreme Value Theory Exreme Value Theorem or Fisher-Tippe Theorem and is varians (Fisher and Tippe, 198), originally applied o model rare phenomena came from hydrology and climaology. Albei being among he oldes heories in saisical engineering, only recenly have sysemaically being employed o explain exreme, possibly ou of sample, behaviors came from he field of insurance and finance (McNeil, 1999; Kellezi and Gilli, 000; Blum and Dacorogna, 003)...1 Uncondiional approach Semi-parameric POT: The financial risk managemen, in general, is focused on heavy-ailed disribuions in he maximum domain of aracion of GEV, ha is, ξ>0, for which α ( x) = x L( x) 1 F (9) Excep for saisical properies, he erm L(x) can be eliminaed from he saisical esimaes (Blum and Dacorogna, 003), wihou imposing any significan bias. In such cases, Hill (1975) proposed an esimaor of ξ ha was mean o be used in cases ha ξ>0 (Freche class models). Firs original X i daa, ha are assumed o be independenly idenically disribued (i.i.d.), are arranged in a descending order, as, Χ (1) Χ () Χ (n). In urn, he ail index ξ is ˆ 1 ξ (10) m 1 Η n, m = ln X ( i) ln X ( m) m 1 i= 1 Index m denoes he m h hreshold elemen or, oherwise saed, he cu-off poin of he descending ordered sample, over which exreme values are realized. The difficuly in hreshold deerminaion is ha a high hreshold provides us wih lile informaion abou he behavior of exremes, while a low hreshold inroduces some of he average observaions from he daase disribuion, increasing he bias incorporaed in he esimaion. Dacorogna e al (001) and Blum and Dacorogna (003) defined m as he square roo of he number (n) of observaions and found his a fairly good approximaion of he rue value, reaining he rade off beween bias and availabiliy of daa. In order o esimae he exreme quanile Q q, wihou bias, he formula (11) proposed by Dacorogna e al (001) should be applied on large samples. For a specified probabiliy level q exreme quanile is given by ξ m Q ˆ q = X ( m) nq (11).. Condiional approach McNeil and Frey (000) inroduced a wo-sep esimaion procedure called condiional Exreme Value Theory Sep 1: They fi a GARCH-ype model o he logreurn daa by quasi-maximum likelihood. Tha is, 3

4 maximize he log-likelihood funcion of he sample assuming normal innovaions. Sep : I is assumed ha he sandardized residuals compued in Sep 1 are realizaions of a whie noise. In urn, in order o esimae he ails of hese innovaions, EVT is used. Finally, he quanile of he innovaions are calculaed for a given level of q. Le he following equaion represen he behavior of log-negaive reurns r = a0 + a1r 1 + ε (1) where α 0 and α 1 are parameers o be esimaed, r -1 is he lagged log-reurn and ε indicaes he residua series. If we suppose ha he condiional variance σ of ε follows a GARCH(1,1) process, his is generaed by equaion (4) The equaion (4) is esimaed by maximizing he log-likelihood funcion of a sample of n observaions. The sep 1 ends by calculaing esimaes of he condiional mean and variance for 1-day ahead forecass ˆ µ aˆ 1 0 aˆ + = + 1r (13) ˆ σ ˆ ˆ ˆ ˆ ˆ + 1 = β 0 + β1ε + β σ (14) where ˆ ε = r ˆ µ In he presen paper he one day ahead forecas for VaR(e) q is given by applying equaion (11) on he negaive sandardized residuals. Consequenly, one-day ahead forecas of he VaR q is given by ^ VaR q aˆ + aˆ r ˆ () e q σ VaR (15) ^ 3 Daa - Mehodological Issues In his paper we examined he behavior of four (4) European Union sae counries sock exchange indices, namely ASE-G (Greece), CAC-40 (France), DAX (Germany) and FTSE-100 (UK). The analysis was based on daa provided by Bloomberg Professional online conribuor. The daa we have aken ino consideraion are daily observaions ranging from 03/01/1984 o 06/11/003 or 500 observaions for he FTSE-100 Index, from 09/07/1987 o 06/11/003 or 4096 observaions for he CAC-40 Index, from 1/06/1976 o 06/11/003 or 6874 observaions for he DAX Index and, from 04/01/1987 o 06/11/003 or 4195 observaions for he ASE-G Index. The saring poin of our mehodology is o find he economeric model ha bes describes he behavior for he enire daase for each of four sock index reurns. The parameer esimaes were obained by he mehod of quasi-maximum likelihood. Tha is, he log-likelihood funcion of he daa was consruced by assuming ha innovaions are condiionally normally disribued. The esimaes of he parameers will be used in he forecasing of one-day condiional VaR for boh Hisorical Simulaion and Exreme Value approach. In urn, we esimaed he condiional and uncondiional EVT quanile by using esimaion procedure of Dacorogna e. al. (1995). The uncondiional VaR was esimaed by he pure reurn daa. Alernaively, a varian of condiional VaR, inroduced by McNeil and Frey (000), was esimaed by, iniially, esimaing he uncondiional VaR of he sandardized residuals and hen subsiuing in he AR(1) GARCH(1,1) equaion (16) which was proved o be he bes specificaion for all reurn series. 4 Resuls and Discussion Alhough according o uncondiional mehods ASE-G appears o be he riskies index, i is he hird riskies among examined indices according o condiional mehods. This proves ha he curren condiions are indeed posing Greece among he developed counries in marke risk erms. Exacly he opposie view is demonsraed by DAX index, as Germany is currenly affeced by economic recession. Insead, FTSE-100 is he less risky index under any approximaion. All coefficiens esimaed for AR(1)- GARCH(1,1) are saisically significan a 99% level, excep for he AR(1) erm concerning CAC- 40 Index ha is saisically significan a 95% level and he consan erm in he AR(1) regression for ASE-G Index ha is saisically insignifican. However, in general, all AR(1)-GARCH(1,1) resuls are overwhelmingly acceped and were used for boh EVT and HS condiional approaches. I can be observed ha he MA(1) erm was dropped ou from he original B-A.G.V. filer specificaion. 4

5 Table 1: Coefficiens of AR(1)-GARCH(1,1)* AR(1) GARCH(1,1) Index α 0 α 1 β 1 β DAX ASE CAC FTSE [0.7] [0.014] [0.0000] [0.06] [0.001] *all β 0 esimaors significan bu very close o zero [ ]: probabiliies of coefficiens A naural indicaor or he expeced value of he VaR, given a cerain confidence level, is expeced o be he value of he order saisic provided by he muliplicaion of he confidence level wih he number of observaions included in he populaion of he asse under discussion. According o ha view, uncondiional hisorical simulaion VaR is considered o be he naural or he expeced VaR. Table : Daily Value a Risk (%) Hisorical Prob. Index Simulaion EVT U C U C ASE % CAC DAX FTSE ASE % CAC DAX FTSE ASE % CAC DAX FTSE U: uncondiional C: condiional Comparing he resuls, condiional hisorical simulaion, for 95% and 99% probabiliy level, provides higher values han he uncondiional hisorical simulaion only for DAX Index. For he highes probabiliy level examined (99.9%) uncondiional forecass are higher for all sock exchanges. Being in line wih he above resuls, condiional Exreme Values Theory forecass yields lower values for all probabiliy levels for all indices, excep for DAX Index for 95% level. In general, condiional models yield lower esimaes for VaR. Tha was happened because condiional models ake ino accoun he curren condiions in he economy expressed by he curren clusered volailiy. Insead, uncondiional models capure exreme evens ha have appeared cerain imes in he price hisory. In he above conex, i is apparen ha uncondiional models are suiable o carry ou sress esing experimens, while condiional models are properly suied for on-going daily VaR esimaions. In oher words, he former class of models is direced for use by regulaors for sress esing he enire credi sysem; while, he laer class of models seems o be suiable for praciioners. An alernaive uncondiional hisorical simulaion approach, ha diminishes he influence of far disan pas evens, would be he use of a smaller ime-moving sample of fixed lengh involved in he esimaion of VaR. Once his approach akes ino consideraion he curren and near pas evens, conceivably, i is a beer on-going VaR esimaor compared wih large sample based hisorical simulaion. In conras, we canno use ime-moving sample echnique in order o esimae uncondiional EVT models because as he sample size decreases he bias of he ail index esimaor increases. Thus, uncondiional semi-parameric EVT model can be considered as a genuine sress-esing model. Looking for links beween hisorical simulaion and EVT approaches i can be seen ha HS produces sysemaically lower values for one-day VaR forecass, apar from he uncondiional class of models under which he image was invered. The Basle Commiee (1996) allows banks o consider price shocks equivalen o a shor holding period such as a day, bu i recommends a holding period of en (10) days. Thus, banks are implicily encouraged o conver one-day VaR o muliple-days VaR. The mos popular mehodology is o muliply he original one-day esimae by he roo of en ( roo of en law ) ha obviously will yield very high values. In conras o ha view, Danielsson and De Vries (000) sugges ha should be done by he scaling facor T 1/a (T is he horizon lengh expressed in days) for original esimaes made using EVT. This evenually leads o lower muliday VaRs han would be obained from he normal rule (Danielsson e al, 1998). 5

6 5 Conclusion The objecive of he paper is o idenify he mehod of VaR esimaion ha well balances he endency of regulaors for conservaism and he need of financial insiuions of having heir shareholders saisfied. Our resuls show ha varians of VaR can be manipulaed in various ways. Firs, regulaors are inclined o use full sample-based uncondiional hisorical simulaion and EVT approaches, as hey yield he highes VaR values, in order o es he vulnerabiliy of he credi sysem. Second, here is no unique recipe for boh regulaors and supervisors in applying a paricular model. Third, we canno make a clear saemen recognizing he beer model because differen models are consruced for differen purposes. However, we can explicily observe ha uncondiional specificaions provide saver predicions, albei we are no sure abou heir accuracy. In order o es heir accuracy, he fuure researcher should be focus on dynamic back-esing echniques. Dynamic back-esing should be carried ou in order o see he rue effeciveness of VaR sysems in economic erms, insead of applying hose sysems from a subjecive aspec of view. Anoher issue for furher research is o es he accuracy of he scaling facor proposed by Basel Commiee, by comparing he scaled one-day ahead VaRs (by he square roo of en) wih he original en-day ahead VaRs based on en-day period reurns. References: [1] Barone-Adesi, G., Giannopoulos, K. and Vosper, L., VaR wihou Correlaions for Porfolios of Derivaive Securiies, Journal of Fuures Markes, 19, 1999, pp [] Basle Commiee, Overview of he Amendmen of he Capial Accord o Incorporae Marke Risk, Basle Commiee on Banking Supervision, [3] Blum P. and M. Dacorogna, Exreme Forex Moves, RISK, February 003, pp [4] Bollerslev, T., Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 31, 1986, pp [5] Boudoukh, J., Richardson, M., and Whielaw, R., The Bes of Boh Worlds, RISK, 11, May 1998, pp [6] Dacorogna, M., Gencay, R., Muller, U., Olsen, R. and Pice, O., An inroducion o high frequency finance, 001, Academic Press. [7] Dacorogna, M., Muller, U., Pice, O. and De Vries, C., The Disribuion of Exremal Foreign Exchange Rae Reurns in Exremely Large Daa Ses, June 1995, Preprin, Olsen & Associaes, [8] Danielsson, Jon, de Vries, Casper G. and Jorgensen, Bjorn N., The value of Value-a- Risk: Saisical, Financial, and Regulaory Consideraions, Economic Policy Review, Federal Reserve Bank of New York, Ocober 1998, pp [9] Danielsson, Jon and de Vries, Casper G., Value-a-Risk and Exreme Reurns, Annals D Economie e de Saisique, 60, 000, pp [10]Embrechs, P., Kluppelberg, C. and Mikosch, T., Modeling exremal evens, Applicaions of Mahemaics Sochasic Modeling and Applied Probabiliy, 33, 1997, Springer. [11]Gaussel, N., Legras, J., Longin, F. and Rabemananjara, R., Beyond he VaR horizon, Quans, CCF, 37, June 000. [1]Gunher, Jeffrey W., Levonian, Mark E. and Moore, Rober R., Can he Sock Marke ell bank supervisors anyhing hey don already know?, Economic and Financial Review, Federal Reserve Bank of Dallas, Second Quarer 001, pp. -9. [13]Hill, B., A simple general approach o inference abou he ail of he disribuion, Annals of Saisics, 3(5), 1975, pp [14]Kellezi, E. and Gilli, M., Exreme Value Theory for ail-relaed risk measures, Preprin, Deparmen of Economerics and FAME, 000 Universiy of Geneva. [15]Longin, Francois M., From Value-a-Risk o sress esing: an exreme value approach, Journal of Banking and Finance, 4, 000, pp [16]McNeil, Alexander J., Inernal Modeling and CAD II, RISK Books, 1999, pp [17]McNeil, Alexander J. and Frey, R., Esimaion of ail-relaed risk measures for heeroskedasic financial ime series: an Exreme Value Approach, Journal of Empirical Finance, 7, 000, pp [18]Prisker, M., The hidden dangers of hisorical simulaion, Federal Reserve Board, Financial and Economics Discussion Series, June 001. [19]Raaikainen, J., Inroducion o modern risk managemen, Survey, Universiy of Jyväskylä, 00 6