Forecasting Value at Risk with Historical and Filtered Historical Simulation Methods

Transcription

1 U.U.D.M. Projec Repor 2009:15 Forecasing Value a Risk wih Hisorical and Filered Hisorical Simulaion Mehods Ghashang Piroozfar Examensarbee i maemaik, 30 hp Handledare och examinaor: Maciej Klimek Sepember 2009 Deparmen of Mahemaics Uppsala Universiy

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3 Absrac The dissaisfacion wih he previous parameric VaR models in esimaing he marke values during pas few years has pu heir reliabiliy in quesion. As a subsiue, non-parameric and semi-parameric echniques were creaed, which are he subjecs of his hesis. We sudy he Hisorical Simulaion and Filered Hisorical Simulaion as wo powerful alernaives o primary models in VaR measuremen. In addiion, we apply hese mehods o en years daa of he OMX index, o show how well hey work.

4 Acknowledgemens Here I begin by acknowledging my deb of graiude o my ousanding supervisor, Professor Maciej Klimek, for his paience, valuable suggesions ha subsanially shaped his hesis. Thanks a lo for inroducing his subjec and his consan guidance and encouragemen hrough i. I also wan o hank all he saff of he deparmen of mahemaics a Uppsala Universiy, and my eachers during pas wo years, especially Professor Johan Tysk and Erik Eksröm for heir deep consideraion o financial mahemaics sudens. I should definiely hank my parens and my brohers, Dr Poorang Piroozfar and Dr Arjang Piroozfar, for heir supporive role and endless generosiy in giving me advices, whenever I ask for. And finally, Wih he special hanks o my mom, as wihou her exreme invaluable forbearance, I could never ever sar and finish his projec.

5 Conens 1 Inroducion..1 2 Value a Risk Risk Definiion from he Financial Poin of View VaR VaR Formula..5 3 Hisorical Simulaion and Filered Hisorical Simulaion Conceps HS and FHS HS HS Shorcomings BRW FHS The Mos Suiable Time Series Model for our Mehod: ARMA-GARCH Volailiy Heeroskedasiciy ARCH, GARCH and ARMA-GARCH Time Series Models ARMA-GARCH Preference in Modelling Marke Volailiy Mehodologies of Hisorical Simulaion and Filered Hisorical Simulaion HS FHS Theoreical Mehod of Obaining Fuure Reurns Fuure Reurns Esimaion Simulaion of he Fuure Reurns Compuaion of VaR (Same Mehod in HS and FHS) Empirical Sudies Daily Reurns Plos Empirical Sudy of HS and VaR Compuaion by HS Mehod Empirical Disribuions, Forecasing Prices and 1%VaR for 5-Day Horizon Acual Prices Comparison...59

6 5.3 Empirical Sudy of FHS and VaR Compuaion by FHS Mehod Empirical Disribuions, Forecasing Prices and 1%VaR for 5-Day Horizon Comparison Running Anoher Empirical Sudy HS FHS Conclusion..80 Appendix A (Malab Source Codes).. 81 References..85

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8 1 Inroducion Value a Risk (VaR) as a branch of risk managemen has been a he cenre of aenion of financial managers during pas few years, especially afer he financial crises in 90 s. And now, afer he marke failure in 2008, he demand for a precise risk measuremen is even higher han before. Risk managers ry o review he previous mehods, as hey hink one of he mos imporan causes of he recen crisis was mismanagemen of risk. In addiion o VaR s fundamenal applicaion which is measuring he risk, i also has oher usages relaed o risk, such as conrolling and managing i. VaR as a widespread mehod is applicable in any kind of insiuions which are somehow involved wih financial risk, like financial insiuions, regulaors, nonfinancial corporaions or asse managers. There are differen approaches o VaR models for esimaing he probable losses of a porfolio, which differ in calculaing he densiy funcion of hose losses. The primary VaR mehods were based on parameric approaches and some imposed assumpions, which in real cases did no work. One of he mos imporan assumpions could be menioned as he normal disribuion of he densiy funcion of he daily reurns. Empirical evidence shows he prediced loss or profi by his disribuion underesimaes he ones in real world. So a non-parameric mehod, based on hisorical reurns of marke, called hisorical simulaion (HS) has been inroduced as a subsiue. Bu some of he disadvanages of his mehod (especially is inabiliy o model he mos recen volailiy of marke) make i inefficien. Therefore Barone-Adesi e al, inroduced a number of effecive refinemens of his mehod, by mixing i wih some parameric echniques i.e. GARCH ime series models (as his kind of models are able o reveal volailiy clusers), which leads o a new mehod called filered hisorical simulaion (FHS). Invesigaing how well each of hese mehods (HS and FHS) works in VaR measuremen field is he main purpose of his hesis. In his hesis, which is based on paper [4], secion 2 is allocaed o he explanaion of he VaR. In secion 3 we will explain conceps of HS and FHS as a new generaion of VaR measuremen mehods. In secion 4 we will go hrough he heory behind hem, in secion 5 we will examine heir 1

9 applicaion for measuring VaR in pas few years OMX 1 index daily prices. Finally secion 6 is devoed o conclusions. 2 Value a Risk This secion conains inerpreaion of Value a Risk (VaR), is formula and an inroducion of differen approaches relaed o is measuremen. 2.1 Risk Definiion from he Financial Poin of View When we are confroned wih he word Risk, he firs definiion ha comes o our mind is loss disaser, bu he financial heory ake a more comprehensive look a his word. Alhough here is no any unique definiion for his word in finance, we could menion risk as a possibiliy of losses due o unexpeced oucomes caused by he financial marke movemens. And he probabiliy of he loss occurrence more han expeced amoun, in a specific ime period is he VaR measuremen. The sandard deviaion of he unexpeced oucomes (σ ) which is called volailiy, is he mos common risk measuremen ool. There are four ypes of financial risks: - Ineres rae risk - Exchange rae risk - Equiy risk - Commodiy risk Volailiy changes do no have any rend. There will be a higher probabiliy o increase or decrease in value for a more volaile insrumen. Increased volailiy could occur wih any posiive or negaive unexpeced changes in price. The volailiy of financial markes is a source of risk, which should be conrolled as precise as possible. When we encouner a volailiy diagram and we see a lo of flucuaion over ime, an imporan quesion arises i.e. wheher he risk is unsable or hese flucuaions relaed o our esimaion mehod and hey only reflecs he noise in daa. 1 The OMX Index is a marke value-weighed index, ha racks he sock price performance of he mos liquid issues raded on he Sockholm Sock Exchange (SSE).[31] 2

10 2.2 VaR VaR could be defined as an easy mehod for measuring marke volailiy of unexpeced oucomes (risk) wih he help of saisical echniques. On he oher hand he purpose of VaR is measuremen of wors expeced loss a a special ime period (holding period) and he special given probabiliy under assumpion of he normal marke condiion e.g. if a bank announces ha he daily VaR of is rading porfolio is $50 million a he 1% confidence level i means ha here is only 1 chance ou of 100 for a loss greaer han $50 million over a one day period (when he ime horizon is 1 day) i.e. he VaR measure is an esimae of more han $50 million decline in he porfolio value ha could occur wih 1% probabiliy over he nex rading day. The wo imporan facors in defining VaR of a porfolio, is he lengh of ime and he confidence level ha he marke risk is measured. The choice of hese wo facors compleely changes he essence of he VaR model. The choice of ime horizon could differ from few hours o one day, one monh, one year ec. For insance for a bank, holding period of 1-day could be effecive enough, as banks have highly liquid currencies. This amoun could change o 1 monh for a pension organizaion. Abou he deerminaion of he confidence level when a company encouners an exernal regulaory 2, his number should be very small, such as 1% of confidence level or less for banks, bu for inernal risk measuremen modelling in companies i could increase o around 5%. [18] VaR models are based on he assumpion ha he componens of he porfolio do no change over he holding period. Bu his assumpion is only accurae for he shor holding periods, so mos of he ime he discussion of he VaR rounds abou he one-day holding period. When he prediced VaR hreshold is conradiced by he observed asse reurn, his is called VaR break, which could be a good VaR approach accuracy crierion. I s good o menion ha alhough VaR is a necessary ool for conrolling risk, i is no sufficien, because i should be accompanied by he limiaions and conrols plus an independen risk managemen funcions. 3 The early VaR mehods including Variance-Covariance approach and Simulaion, which are also called parameric mehods, were based on he linear muliplicaion of he variance-covariance risk facors esimaes. 2 Regulaory agency: An independen governmenal commission esablished by legislaive ac in order o se sandards in a specific field of aciviy, or operaions, in he privae secor of he economy and o hen enforce hose sandards. [35]. 3 Independen risk managemen sysem as a key componen of risk managemen in an organizaion, including a srong inernal conrol environmen and an inegraed, insiuionwide sysem for measuring and limiing risk, is an imporan sraegic and acical suppor o managemen and board of direcors. [13], [17]. 3

11 Some of he imporan shorcomings of hese mehods, moivaed risk managers o look for beer esimaion of VaR, despie heir worldwide repuaion. Number of heoreical assumpions is pu on daa properies by hese mehods. One of hese assumpions is abou he densiy funcion of he risk facors which should be adjused o a one or higher-dimensional Gaussian disribuion i.e. mulivariae normal disribuion (normal disribuion is menioned here because hey could be defined by only heir firs and second momens) and has consan mean and variance. Bu empirical resuls show somehing differen which is emphasis on he nonnormaliy of he daily asse price changes. They indicae significan and more common occurrences of he losses higher han VaR, caused by excess kurosis 4 (volailiy of volailiy) in comparison o he ones prediced by normal disribuion. Anoher imporan disadvanage of hese mehods was he large number of inpus hey required o be able o work well. Because, as a maer of fac all daa covariance s should be menioned 5. And finally lack of good provision of he VaR esimaes during he financial crises lead risk managers o search for beer ones. So hisorical simulaion (HS) models based on non-parameric mehods and filered hisorical simulaion (FHS), as a mixure of parameric and non-parameric mehods appeared. 4 An imporan benchmark for he fuure reurns of a sock or porfolio, i.e. he probabiliy ha he fuure reurns will be significanly large or small depends on he difference of he kurosis coefficien and number 3, which is he kurosis of normal disribuion (higher kurosis coefficien from normal disribuion kurosis leads o a more probable oo large or oo small fuure reurns). Kurosis explains he flaness degree of a disribuion. The Kurosis of a normal disribuion is 3. This measure is a crierion o check wheher he sample disribuion is close o normal disribuion. Lef ail of he empirical disribuion, which is used o compue VaR, will no fi a normal disribuion for he large values of he kurosis. 5 By he normal disribuion assumpion of he porfolio reurns of such mehods, hey need o esimae he expeced value and sandard deviaion of reurns of all he asses. 4

12 2.3 VaR Formula If we menion confidence level in VaR measuremen as below: Confidence. Level = (1 α)100% VaR formula will be like his: VaR α =inf{l:prob[loss > L] 1 α }, where L is he lower hreshold of loss, which means ha he probabiliy of losing more han L in a special ime horizon (e.g. 1 day) is a mos (1 α). In he conex of compuer simulaion, given α, if we make he probabiliy below equal o1 α (by varying L), we will find he value of L as VaR a he specific ime horizon. 6 No. of. Simulaions. wihvalue. < P L Prob [Loss > L] =, N where N (muliple of 1000) is a number of simulaions we have done wih HS or FHS mehod, and P is he iniial amoun of invesmen. 3 Hisorical Simulaion and Filered Hisorical Simulaion Conceps In his secion we explain fundamenal concep of HS and FHS, and he reason behind using hese wo mehods which are accompanied by explanaion of oher significan definiions and ime series models relaed o he risk managemen field. 3.1 HS and FHS As a shor and sraighforward definiion of hese wo mehods we could caegorize hem by he following srucure: When volailiy and correlaion are consan, we use classical HS, bu as we confron he ime-varying volailiy hen we should change our sraegy o he FHS. 6 α is a number beween zero and one ( 0< α < 1), which usually is chosen equal o 0.99 or

13 For specifying a model o a porfolio he simples assumpion could be sandard normal disribuion of he porfolio s shock reurns, because his disribuion has no parameers and he model will soon be ready for forecasing risks. Bu we know ha his assumpion is no rue for mos asses of a porfolio. So an imporan quesion arises here abou selecing he bes subsiue disribuion. Insead of looking for a special disribuion, risk managers rely on resampling mehods for modelling such as HS and FHS HS HS which is also known as boosrapping simulaion, gaher marke raw values of risk in a special pas period of ime, and calculae heir changes over ha period o be used in he VaR measuremen. HS could be menioned as a good resampling alernaive mehod because of is simpliciy and lack of disribuional assumpion abou underlying process of reurns (finding a disribuion, fi o all he asses of a porfolio is no a simple approach). The main assumpions of HS are: - Chosen sample period could describe he properies of asses very well, - There is a possibiliy of repeaing he pas in he fuure i.e. he replicaion of he paerns appeared in he volailiies and correlaions of he reurns in hisorical sample, in he fuure. On he oher hand pas could be a good crierion of he fuure forecas HS Shorcomings HS named as a simple explained concep mehod which is based on he imporan fac in risk managemen i.e. nonlineariies 8 and non-normal disribuion. Bu is simpliciy ends o below disadvanages: 7 HS is resriced in markes which are developing fas e.g elecriciy, because of he conradicion of he pas repea in he fuure assumpion. 8 Nonlineariy means ime dependency of he price of a financial insrumen such as derivaives. 6

14 - The imporance of he pas reurns is as much as he recen reurns as observaions are equally weighed, which causes lack of good weigh allocaion o he sudden increase in volailiy of he recen pas which has significan influence on he near fuure. 9 I means ha HS has negleced he more effecive role of he recen pas, by equalizing is relaed weigh o weigh of he furher away pas. - HS needs a long enough series of raw daa (found by Vlaar [21], [34] hrough esing he accuracy of he differen VaR models on Duch ineres rae porfolio), and his is he mos difficuly of HS echnique in esimaing VaR of new risks and asses because here is no hisorical daa available in hese wo cases. [2] - If he number of observaions is oo large hen almos he las one which migh describe he fuure beer, has he same affec of he firs observaion as hey have equal weighs (shown by Brooks and Persand [9]: VaR models could end o inaccurae esimaes when he lengh of hisorical daa sample is no chosen correcly, by esing sensiiviy of he VaR models wih respec o he sample size and weighing mehods). [2] - The assumpion of consan volailiy and covariance of he raw reurns, leads o he fac ha he sensiiviy of he VaR canno be checked. This means ha he volailiy updae 10 does no happen in his model, and so he marke changes won be refleced. - In HS, ofen, resuls are based on one of he mos recen crises and he oher facors won be esed. So HS causes VaR o be compleely backward-looking, i.e. when a company or organizaion ries o proec is resources by he VaR measuremen based on he HS srucure, in principal i organizes o proec iself from he passed crisis no he nex one. - As HS mos commonly choose marke daa from he las 250 days, he window effec problem could also happen in his process. This means, when 250 days passes from a special crisis, ha will be omied from he window and he VaR will dramaically drop from day o he oher. I could cause doub for raders abou he inegriy of his mehod as hey know ha nohing special has happened during hose wo days where he drop happened. 9 As an evidence of he VaR models failure, we could menion he bank experimens of he unusual number of VaR excess days in Augus and Sepember 1998, which followed by he lack of aenion o he reurns join disribuion a ha ime inerval. 10 Volailiy updae could be obained by dividing hisorical reurns by he corresponding hisorical volailiies (which is called normalizing he hisorical reurns), and muliplying resul by he curren volailiy. 7

15 3.1.3 BRW By he poins we menioned above as HS deficiencies, differen researchers such as Boudoukh e al. or Barone-Adesi e al. aemped o decrease hem as much as possible. As closer pas reurn o he presen ime could forecas fuure beer han he far pas one, an exponenial weighing echnique which is known as BRW, was suggesed by Boudoukh, Richardson and Whielaw [12], [6] as a ool o decrease he effec of chosen sample s size on VaR esimaion. In his modificaion insead of equally weighed reurns, heir weigh is based on heir prioriy of happening by using facor called decay facor, e.g. if we assume 0.99 for he decay facor and P for he probabiliy weigh of he las observaion, he one observaion before ha will receive he 0.99P weigh, P will goes o he weigh of one before and so on. So we could say ha he original mehod of HS is a special case of BRW mehod wih decay equal o 1. Boudoukh e al. examined he accuracy of heir mehod by compuing he VaR of sock porfolio, before and afer he 19/10/1987(19 h of Ocober) when a marke crash happened, wih se of reurn for 250 days. They show ha he VaR compued by HS mehod didn show any changes he day afer he crash because all he days had same weighs bu by BRW, VaR showed he effec of he crash. Cabado and Moya [12] showed ha he beer forecasing of VaR could be reached by using he parameers of a ime series model which has been fied o he hisorical daa. They showed he improvemen in he VaR resul by fiing an Auoregressive Moving Average (ARMA) model o he oil price daa from 1992 o 1998 and use his model o forecas reurns wih 99% confidence inerval for he holdou period of 1999 [12]. One of he reasons of such an improvemen could be inerpreed as, he measuremen of risk has higher sensiiviy o he oil prices variance changes wih ime series model han HS. Hull and Whie [33] suggesed anoher mehod of adjusing he hisorical daa o he volailiy changes. They used GARCH models for obaining daily esimaion of he variance changes over he menioned period in he pas. 8

16 3.1.4 FHS As we noified before he main deficiency of HS as a non-parameric mehod is is disabiliy in modelling he volailiy dynamics of he reurns. Also in he parameric mehod (which is no he subjec of his paper and we do no go hrough i in deail), he choice of righ disribuion (if i could be found) is criical, so Hull and Whie [33] and Barone-Adesi e al [4], [33], combined he wo previous mehods o receive one called FHS, which is a semi-parameric echnique 11 and has he below imporan prioriies o he HS: - Wihou any aenion o he disribuion of he observaions you could use volailiy model, - Conforming he hisorical reurns (by filering hem) o show curren informaion abou securiy risk. Filered expression relaed o he fac ha we do no use raw reurns in his mehod, bu we use series of shocks ( z ), which are GARCH filered reurns. Also Barone-Adesi and Giannopoulos [2], [33] discussed, as he curren siuaion of he marke is embedded in risk forecas by FHS, i works beer han HS. FHS is a Mone Carlo approach which is he combinaion of parameric modelling of risk facor volailiy and non-parameric modelling of innovaions, which has he bes usage in 10-day, 1% VaR. An imporan assumpion of he FHS is ha he reurn vecors is i.i.d, which means ha is correlaion marix 12 is consan 13 (his assumpion could be unrealisic for he long ime series). For making reurns i.i.d we should remove serial correlaions and volailiy clusered from he daa. Serial correlaion could be removed by adding MA erm in condiional mean equaion and o remove he volailiy clusers we should model he reurns as GARCH processes. GARCH models are based on he normal disribuion of he residual asse reurns assumpion. Non-normaliy of he residual reurns will conradic he efficiency of GARCH esimaes, alhough hey migh sill be consisen. So we could say ha every ime series which generaes i.i.d residuals from reurns is good enough for our modelling. [4] In he simulaion we only use hisorical disribuion of he reurn series and we do no use any heoreical disribuion. As we menioned in he previous paragraph any ime series model which generaes i.i.d residuals 11 The parameric par of his echnique is he GARCH esimaion of residuals. 12 Correlaion marix is a marix which is members are correlaion coefficiens. 13 In a Mean-Variance porfolio when he correlaion of reurns beween each securiy pairs is he same, his is called consan correlaion model [20]. 9

17 from our reurns is suiable for us, so we inroduce ARMA-GARCH which removes he serial correlaions 14 by MA and volailiy clusers by GARCH. To apply he FHS, he esimaion of he parameers of GARCH (1, 1) is by quasi-maximum likelihood esimaion (QMLE). QMLE behaves as he reurns are normally disribued, bu even if hey are no normally disribued, i esimaes he parameers and condiional volailiies. 3.2 The Mos Suiable Time Series Model for our Mehod: ARMA- GARCH Volailiy Volailiy which is almos accepable forecasing is used o measure risk in credi insiues is an imporan facor a he cenre of aenion of risk managemen echniques. Volailiy, measure he size of occurred errors in differen variables of financial marke modelling such as reurn. For a large number of models, volailiy is no consan and is ime varying. Volailiy as an unobservable fac should be esimaed from daa. The predicabiliy of financial marke volailiy is a considerable propery, wih vas usages in risk managemen. VaR will increase as volailiy increase. So invesors will ry o change he diversificaion of heir porfolio o decrease he number of hose asses which heir volailiy has been prediced o be increased. By predicable volailiy consrucion, opions value changes 15 (opion are kind of asses, srongly dependen o volailiy) will have predicable srucure. Also changes in volailiy, influence he equilibrium asse prices. Thus whoever could forecas volailiy changes more precisely, will have higher abiliy in conrolling he risk of marke. So each echnique could saisfy his forecasing is really valuable in he financial world. 14 When he assumpion of corr( εε 1) = 0is conradiced, his is called serial correlaion, which means error erms do no follow an independen disribuion and are no sricly random. 15 Volailiy is an imporan benchmark for specifying which kind of opions should be bough or sold. Ending opions in-he-money is in he direc relaion wih he underlying conrac price flucuaions. So opion s value goes up and down wih respec o he value of he conrac. As volailiy soars, he possibiliy of receiving higher oucomes ou of conracs will be higher, so opion s value will increase, and vice versa. 10

18 3.2.2 Heeroskedasiciy The oher facor is heeroskedasiciy which represens he non-consan variance of error erms, ( i.e. violaion of he following condiion 2 2 var( ε ) = var( y ) = σ, σ is consan.)[30] i i On he oher hand dependence of he residual variance on he independen variables is ermed heeroskedosiciy (we could also define i as variable variance of residuals ) [26]. This facor mos of he ime happens by he cross secional daa 16. As a mos imporan consequence of heeroskedasiciy, we could menion is influence on he efficiency of he OLS esimaors. Alhough he OLS lineariy and unbiasedness 17 won be affeced by heeroskedasiciy, i violaes heir minimum variance propery and by his way hey won be efficien, consisen and herefore bes esimaors anymore. There are hree main approaches o deal wih his problem of heeroskedasiciy: - Changing he model, - Transforming daa for receiving more sable variaion, - Consider he variance as a funcion of predicor and modelling i wih respec o his propery. (For more informaion see [28]) In finance his facor usually could be found in sock prices. The volailiy level of hese equiies is no predicable during differen ime inervals. These wo facors in and canno be covered by any linear or nonlinear AR or ARMA processes. So we inroduce very well-known ype of volailiy models which saisfies all he above specializaions: auoregressive condiional heeroskedasiciy (ARCH) and generalized auoregressive condiional heeroskedasiciy (GARCH) models ARCH, GARCH and ARMA-GARCH Time Series Models In economic and financial modelling, he mos undeermined par of any even is fuure. By gahering more new informaion as ime passes, we could modify he effecs of fuure uncerainy in forecasing our models. 16 In Cross Secional Daa we look a differen areas during he same year, in spie of he ime series which we look a one area during so many years. 17 An unbiased esimaor of a parameer is he one wih he expecaion value equal o he value of he esimaed parameer. 11

19 In finance, asse prices are he bes forecas of fuure benefi of marke, so hey are oo sensiive o each news. ARCH/GARCH models can be named as a measuremen ool of he news process inensiy ( News Clusering as an inerpreaion of he Volailiy Clusering ). There are many differen facors ha influence he spreading seps of he news and heir affecs on he prices. Alhough macroeconomics models could emperae he affecs of such news, ARCH/GARCH modelling are more well-known relaed o heir abiliy in improving he volailiies which has been made during hese processes. For simpliciy we define he ARCH (1) process here and hen show he complee exension of such processes in inroducing GARCH process. ARCH (1) The processε, Z is ARCH (1), if E[ ε F ] = 0, 1 Wihω > 0, α 0 and σ = ω+ αε Var ( ε F ) σ ε = and z = is i.i.d (srong ARCH) 18 σ Var ( ε F ) σ 1 = (semi-srong ARCH) - ( 1, 1, 2,..., 1, 2,... ) P ε ε ε ε ε = σ (weak ARCH). [15] In ARCH model he volailiy is a funcion of squared lagged shocks 2 ( ε 1 ). In he generalized forma of his model i.e. GARCH model, volailiy also depends on he pas squared volailiies. As a general definiion we can call GARCH model as an unpredicable ime series wih sochasic volailiy. In srong GARCH, here is ε = σ z where σ is F 1 -measurable, i.e. σ (volailiy) only depends on he informaion available ill ime -1 and he i.i.d innovaions z wih E[ z ] = 0 E ε F = 0, and ( ) 1 2 Var ( ε F ) σ Var z =. For his ime series we also have [ ] 1 =, which means ha ε is unpredicable and excep he cases where σ is consan, i is condiionally heeroskedasic. [15] 1 18 This means in ARCH models he condiional variance of ε is a linear funcion of he lagged squared error erms. 12

20 GARCH (p,q) The processε, Z is GARCH (p,q), if E[ ε F ] = 0, 1 q p = + i i + j j i= 1 j= 1 σ ω αε β σ and 2 - Var ( ε F ) σ ε = and z = is i.i.d (srong GARCH) σ Var ( ε F ) σ = (semi-srong GARCH) 1 - ( 1, 1, 2,..., 1, 2,... ) P ε ε ε ε ε = σ (weak GARCH) 2 2 The sufficien bu no necessary condiion forσ > 0 a.s. ( P σ > 0 =1) are ω > 0, α 0, i=1,,q and β 0, j=1,,p.[15] i j ARMA-GARCH Model ARMA-GARCH model is: [21] p q y = ϕiy i + ψiε i + ε i= 1 i= 1 r s 2 ε = η h, h = α0 + αε + β h i= 1 i= 1 i i i i where ε is he random residual and equal o η h r s 2 = α0 + αε i i + βih i i= 1 i= 1 ), η is i.i.d random variable wih mean zero and variance 1, and α 0 is consan. h, ( η is modelled by If s=0: ARMA-GARCH changes o ARMA-ARCH, If q=0, s=0 and r=1: ARMA-GARCH changes o AR-ARCH (1). 13

21 There are wo ypes of parameers in he above equaions: - Se of parameers of condiional mean denoed by m, - Se of parameers of he condiional variance h denoed byδ. [21] In pracice, firs of all we esimae m, hen he residuals from he esimaed condiional mean will be calculaed, afer ha δ can be esimaed (he mehod of calculaion δ will be menioned below) and h o receive more efficien esimaor of m. finally we use he esimaed [21] Leas square Esimaor (LSE) of m 0 (rue value of m ) :[21] Assume y,..., 1 y n are he given observaions. Then he LSE of m 0, i.e. m, r 1 could be defined as he value inθ, a compac subse of R +, which minimize s n n 2 = ε. = 1 Weiss [21], showed ha m is consisen for m0 and nm ( m) N(0, A) wih 0 ε ε ε ε ε ε A E E E = m m ε m m m m m= m 0 Panula [21] obained he asympoic disribuion of he LSE for he AR model wih ARCH (1) errors, and gave an explici form for A. [21] 14

22 Bu he resuls in Weiss and Panula [21] needs he finie fourh momen 19 condiion for he y. By now no one have menioned he LSE of m0 for he ARMA-GARCH model. However Weiss resul for LSE could be exended o ARMA-GARCH model. The LSE is equivalen o he MLE of m 0, when GARCH reduces o an i.i.d whie noise process. [21] If he fourh momen is finie, he LSE is consisen and asympoically normal 20, bu i is inefficien for ARMA-ARCH/GARCH models. So we should use MLE in such a case. The log-likelihood funcion is: Lm ( ) = l, n = ε l = ln h 2 2 h where h is a funcion of m and y, and will be calculaed by he below recursion: h r s 2 = α0 + αε i i + βih i i= 1 i= 1, h0 = a posiive consan If we define m = max m θ L( m), as we didn assume ha he η is normal, hen m called he QMLE of m. Weiss showed for he ARMA- GARCH, QMLE is consisen and asympoically normal under finie fourh momen condiion. Ling and Li showed if he finie fourh momen condiion be rue hen a locally consisen and asympoically normal QMLE exis for he ARMA-GARCH. [21] 19 Fourh sandardized momen is μ4 4 σ where μ4 = E ( X μ)4 he fourh momen around he mean is and σ is he sandard deviaion. 20 An asympoically normal esimaor is an esimaor which is consisen and is disribuion around he rue parameer (in our example rue parameer is m 0 ) is a normal 1 disribuion wih he sandard deviaion decreasing proporionally o n. m is asympoically normal for some A(A is an asympoic variance of he esimaor). 15

23 Esimaion of δ : [21] Considering he following ARCH(r) model: ε η h 1 2 =, h = α + αε + + αε (1) r r where α 0 > 0, αi 0 (i=1,,r) are adequae for h > 0 and η are i.i.d random variables wih mean zero and variance 1. For esimaing he parameers of he model (1) he easies echnique is LSE. So we wrie he model as below: ε α α ε α ε ξ = r r + where difference ξ = ε h and ξ can be menioned as a maringale δ = ( α, α,..., α ) and 2 2 ε = (1, ε,..., ε + ). Then LSE of δ Le 0 1 is equal o 1 n n δ = ε 1ε 1 ε 1ε = 2 = 2 r r which Weiss and Panula [21] showed ha δ is consisen and asympoically normal. (They assume ha he 8 h momen of ε exiss). In general, for esimaing he parameer δ, maximum likelihood esimaion (MLE) will be used. Condiional log-likelihood wih respec o heε as observaions, =1,,n can be wrien as below: n ε L( δ ) = l, l = lnh, 2 2 h = A sochasic series { x i } is a maringale difference sequence wih respec o he { y i } if : Ex ( i+ 1 yi, yi 1,...) = 0, i 16

24 where h is a funcion of ε. Assumeδ θ, and δ0 is a rue value ofδ. If we define δ arg max L( δ ) =, δ θ sinceη (condiional error) is no assumed o be normal, δ will be he QMLE. 3.3 ARMA-GARCH Preference in Modelling Marke Volailiy During pas years researches abou he ARCH/GARCH model, researchers always were ineresed o use his model o analyze he volailiy of financial marke daa wihou any respec o he esimaion of condiional mean. Bu i was no reachable because if he condiional mean is no esimaed sufficienly, hen consrucion of consisen esimaes of he ARCH process won happen and ha ends o he failure in he saisical inference and empirical analysis wih respec o he ARCH elemens. Thus we should no ignore he imporance of he esimaion of he condiional mean alhough he mos ineresing par for us is invesigaing he volailiy of daa. The condiional mean is given by AR or ARMA model. So why we don use ARMA model and why should we look for he oher models such as ARMA-GARCH? Because he condiional variances of whie noise are no consan, so we should generae a new AR or ARMA model compleely differen from he radiional one which assumes he errors are i.i.d or maringales differences wih a consan condiional variance. So as he saisical properies of he radiional AR or ARMA model could no cover he properies of our case, we inroduce anoher model called ARMA-GARCH model. 4 Mehodologies of Hisorical Simulaion and Filered Hisorical Simulaion In he following secion we will explain he srucure of heory behind hese wo echniques, and VaR compuaion wih he help of hem in deail. 17

25 4.1 HS In his echnique afer gahering a leas one year of recen daily hisorical reurns (esimaed reurns), simulaion of reurns will be he nex sep. Thus we choose a small number (compare o he lengh of our observed daily hisorical reurns) for upper hreshold of period we wan o forecas, called T, and hen we selec T-random reurns wih replacemen from our observaion daa se (simulaed reurns). By puing hese simulaed reurns in below formula we form a simulaed price series, which is recursively updaed up o he las day (T): [2] P = P (1 + r )(1 + r )...(1 + r ) s + T s s+ 1 s+ 2 s+ T where P s is our iniial price (oucome), r s + is he simulaed reurn of he -h day of our horizon, which has been chosen (randomly and wih replacemen) from se of hisorical reurns and P s+ is he simulaed price of ha day. This simulaion should be repeaed for N imes (N is a muliple of 1000 for receiving more accurae resul), which ends o he [ 1 ], [ 2 ],..., [ ] P P P N as our simulaed prices in period [,+1]. By aking average of hese simulaed prices relaed o each day of our horizon and compare i wih he acual price, which was received from he exac reurn of he corresponding day, we could examine accuracy degree of our mehod. 4.2 FHS As we explained before for removing shorcomings of HS mehod, FHS was suggesed by Barone-Adesi e al. as a refinemen of HS echnique. In his mehod we fi an ARMA ime series model o our daa se and hen use he parameers of his model for VaR forecasing plus GARCH ime series for esimaing he ime-varying volailiy of he model. So we could wrie an ARMA-GARCH (1, 1) model as below: r = μr + θε + ε (1) 1 1 h = ω+ α( ε + γ) + βh (2)

26 where he equaion (1) is ARMA(1,1) modelling of r reurns wih μ as an AR(1) erm and θ as MA erm, and equaion (2) is GARCH (1,1) modelling of random residualsε (noises), which defines volailiy of he random residuals ε as a funcion of las period volailiy h 1, closes residualε 1 and consan ω wihω > 0, α 0, β 0 o guaranee ha each soluion of he equaion (2) is posiive. Random residuals are assumed o be unpredicable and condiionally heeroskedasic (excep when σ is consan). We could formulae wo above properies of he ε as below: E[ ε F 1] = 0 (Unpredicabiliy) (3) Var ε F = h (Condiional heeroskedasiciy) (4) [ ] 1 In (3) F 1 is called informaion se a ime -1. Also he sandardized residual reurns e = ε h are i.i.d wih mean 0 and variance 1. (Menion ha he only observed daa we have are r 0, r 1,..., r s ) Our aim is o predic he empirical disribuion of r, which will be obained by he process we will explain, bu firs of all, we ake a quick look a he heoreical echnique: Theoreical Mehod of Obaining Fuure Reurns If we hink of λ = ( μθωα,,,, βγ, ) as an iniial choice of λ we could use below algorihm o find he se of corresponding residuals, { e1, e2,..., e s} : - Assuming iniial values of residual and volailiy of he residual is equal 2 ω + αγ o 0 and respecively: 1 α β 19

27 ε 0 = 0, h 0 2 ω + αγ = (Uncondiional variance of GARCH (1, 1) formula) 1 α β - The iniial sandardized residual reurn is calculaed easily by he above amouns: - For = 1,2,..., s we findε, h and e ε 0 0 = = h0 e 0 ε = via following process: h - Pu ε 1 in equaion (1) and receive ε - Pu ε 1 and h 1 in equaion (2) and receive h - Finding e hrough e ε = formula h - Repeaing hese hree seps for = 1,2,..., s we receive { } e1, e2,..., es Fuure Reurns Esimaion Now, as an iniial sep in simulaing fuure reurns, we need he esimaed ones, so: 1. Firs of all we find he following parameers λ = ( μθωα,,,, βγ, ) i.e. he esimaion of our ARMA-GARCH (1,1) model parameers λ = ( μθωα,,,, βγ, ). For such esimaion we use quasi maximum likelihood esimaion (QMLE) mehod as i behaves reurns are normally disribued and i could also esimae he parameers when his assumpion is conradiced. Thus λ = arg max λ L( λ) where L is likelihood funcion and λ is QMLE of λ. 22 These residuals differ wih respec o he iniial choice of λ changes. 20

28 2. When we found λ, we repea he algorihm was explained in a heoreical mehod, for obaining he esimaed residuals and hen use hem in forming he simulaed values: - Assuming ε 0, h 0 as iniial values of he esimaed residual and esimaed volailiy of residual respecively. - So he iniial value of he esimaed sandardized residual reurn will be obained by: e = 0 ε 0 h - For = 1,2,..., s we could find ε, h and ε e = via following h process: 0 - Pu ε 1 in equaion (1) and receive ε - Pu ε 1 and h 1 in equaion (2) and receive h - Finding e hrough ε e = formula h - Repeaing hese hree seps for = 1,2,..., s we receive { e1, e2,..., e s} 3. The final process is simulaion (which should be repeaed housands of imes for reaching he accepable resul, as he number of replicaions increases, he esimae converges o he rue value [18]) and i is based on he following algorihm: Simulaion of he Fuure Reurns - As he mos recen esimaed daa could forecas fuure beer han he ohers, so we describe he iniial values of he simulaed residual and volailiy of he residual as below: h = h, ε = ε s s s s 21

29 - Selec a se {,..., es+ 1 es+ T} which has T elemens 23, and is consruced randomly bu wih replacemen from he se e1, e2,..., e s. { } - For = s+ 1,..., s+ T, we will find: h 2 = ω + αε ( + γ) + βh, 1 1 ε e h =, r = μr 1+ θε 1+ βh 1, = s+ 1 r = μr 1+ θε 1+ βh 1, = s+ 2, s+ 3,..., s+ T, P = P 1(1 + r ), = s+ 1 P = P 1(1 + r ), = s+ 2,... s+ T - If we hink N as a number of simulaions, we obain N simulaed reurns and prices for each period such as[ +, 1], i.e.: [ 1 ], [ 2 ],..., [ ] r r r N, [ 1 ], [ 2 ],..., [ ] P P P N - We use he above simulaed P s for finding he prediced empirical probabiliy disribuion of P, e.g. 1 * N P N n = 1 [ n] which is he prediced expeced value of P. 23 T as a number of forecasing days is considerably smaller han s, which means ha wih a large number of hisorical daa we could only forecas shor period of ime in fuure. 22

30 4.3 Compuaion of VaR (Same Mehod in HS and FHS) By obaining he disribuion of he values [ 1 ], [ 2 ],..., [ ] P P P N, we could repor he VaR a he specified level of confidence e.g. (1 α)100% and ime horizon, by making he probabiliy below equal o 1 α (by varying L): No. of. Simulaions. wihvalue. < P L Prob [Loss > L] = N P: Iniial invesmen N: Toal number of simulaions Thus L will be he (1 α)100% VaR. 23

31 5 Empirical Sudies 5.1 Daily Reurns Plos As an empirical sudy we colleced 10 years of OMX Index daily closing prices from1999 o 2009 and calculaed: - Daily simple reurns: R P = 1 P 1 - Daily logarihmic reurns: P r = ln( ) = ln(1 + R ) P 1 - Some of heir saisical properies such as Min., Max., Average, Sandard deviaion, Skewness, Excess Kurosis, Also ploed he daily simple and log reurns and heir empirical disribuion for he whole period and some of subinervals such as , , , and

32 : Table 1: Simple Reurns Log Reurns Min Max Mean S.Dev Skewness Excess Kurosis As we know, skewness is a measure of symmery, which is equal o zero for normal disribuion. This number for each symmeric disribuion is also zero. Negaive skewness (lef-skewed disribuion) means ha he lef ail of he disribuion is longer and he disribuion disposed o he righ and vice versa for posiive skewness. Excess kurosis is a measure of peakedness or flaness of daa in comparison o normal disribuion. This number for normal disribuion is equal o zero. A disribuion wih negaive excess kurosis is called playkuric, which has lower peak around mean (han normal disribuion) and fla disribuion (hin ails). A lepokuric (disribuion wih posiive excess kurosis), is a peaked disribuion wih faer ails. By he number we received in able 1, we could say ha he disribuion of he daa beween , when we use simple reurns, is a peaked disribuion wih heavier ails han a normal disribuion which is also no symmeric. The heavy ail means ha if we assume normal disribuion for hese daa, we will underesimae all evens in he ails, which could end o a no precise simulaion of he fuure reurns. By ake a look a he second column of he able 1, we could conclude ha disribuion, using log reurns, again ends o a peaked disribuion wih faer ails han normal disribuion bu raher symmeric. When we compare hese resuls wih he hisograms in figures 3 and 4, we will find ha hey conform. 25

33 Figure 1 Figure 2 26

34 Figure 3 Figure 4 27

35 Below we will go hrough he yearly daa ( one year means 252 days in our samples): : Table 2: Simple Reurns Log Reurns Min Max Mean E S.Dev Skewness Excess Kurosis Again by ake a look a he Skewness line in able 2, we could conclude ha he disribuion of boh simple and log reurns should be almos symmeric, and boh of hem have high peaked, heavy ailed disribuions. 28

36 Figure 5 Figure 6 29

37 Figure 7 Figure 8 30

38 : Table 3: Simple Reurns Log Reurns Min Max Mean S.Dev Skewness Excess Kurosis

39 Figure 9 Figure 10 32

40 Figure 11 Figure 12 33

41 : Table 4: Simple Reurns Log Reurns Min Max Mean S.Dev Skewness Excess Kurosis

42 Figure 13 Figure 14 35

43 Figure 15 Figure 16 36

44 : Table 5: Simple Reurns Log Reurns Min Max Mean S.Dev Skewness Excess Kurosis

45 Figure 17 Figure 18 38

46 Figure 19 Figure 20 39

47 : Because of lack of complee daa in 2004 we chose 252 days from 2003 and 2005: Table 6: Simple Reurns Log Reurns Min Max Mean S.Dev Skewness Excess Kurosis

48 Figure 21 Figure 22 41

49 Figure 23 Figure 24 42

50 By comparing he above simple and log reurns plos in each period separaely, i is clear ha hey are prey similar o each oher, which we expeced by he similariy of heir formulas. 5.2 Empirical Sudy of HS and VaR Compuaion by HS Mehod As an empirical sudy of HS, we simulaed 24 he price (oucome) of 12 weeks (here week means 5 business days) i.e. 12 series of 5-day horizon in 2009 and In 2008 we sared by daa from o (including financial crisis), simulaing price of one week afer , hen go one week furher in our daa i.e. ' o and again simulaing one week afer he las dae of our daa and so on, for 5000 imes, wih he iniial price equal o In 2007 we sared by o and coninue by repeaing he above process, for simulaing prices in 12 weeks. From hese simulaed horizons we chose he 5 h day simulaed prices, ook he average of hem, o receive he forecasing price. Also we compued he acual prices of hese days and 1%VaR for each week (5-day horizon). Here are he resuls: 24 By MATLAB coding of HS mehod, available a appendix A. 25 MM-DD-YYYY. 43

51 5.2.1 Empirical Disribuions, Forecasing Prices and 1%VaR for 5-Day Horizon : Figure 25 Forecasing price of he 5 h day (1 s week): % VaR: As he received prices by using simple reurns and log reurns are really close o each oher, we will only use simple reurns in our calculaions. 44

52 Figure 26 Forecasing Price of he 5 h day (2 nd week): % VaR: 32.7 Figure 27 Forecasing Price of he 5 h day (3 rd week): % VaR:

53 Figure 28 Forecasing price of he 5 h day (4 h week): % VaR: Figure 29 Forecasing price of he 5 h day (5 h week): % VaR:

54 Figure 30 Forecasing price of he 5 h day (6 h week): % VaR: Figure 31 Forecasing price of he 5 h day (7 h week): % VaR:

55 Figure 32 Forecasing price of he 5 h day (8 h week): % VaR: Figure 33 Forecasing price of he 5 h day (9 h week): % VaR:

56 Figure 34 Forecasing price of he 5 h day (10 h week): % VaR: Figure 35 Forecasing price of he 5 h day (11 h week): % VaR: 35 49

57 Figure 36 Forecasing price of he 5 h day (12 h week): % VaR:

58 2008: Figure 37 Forecasing price of he 5 h day (1 s week): % VaR:

59 Figure 38 Forecasing price of he 5 h day (2 nd week): % VaR: Figure 39 Forecasing price of he 5 h day (3 rd week): % VaR:

60 Figure 40 Forecasing price of he 5 h day (4 h week): % VaR: Figure 41 Forecasing price of he 5 h day (5 h week): % VaR:

61 Figure 42 Forecasing price of he 5 h day (6 h week): % VaR: Figure 43 Forecasing price of he 5 h day (7 h week): % VaR:

62 Figure 44 Forecasing price of he 5 h day (8 h week): % VaR: Figure 45 Forecasing price of he 5 h day (9 h week): % VaR:

63 Figure 46 Forecasing price of he 5 h day (10 h week): % VaR: Figure 47 Forecasing price of he 5 h day (11 h week): % VaR:

64 Figure 48 Forecasing price of he 5 h day (12 h week): % VaR:

65 5.2.2 Acual Prices For hese forecasing prices, we also calculaed he acual prices, by he below formula: [2] P = P(1 + r )(1 + r )...(1 + r ) s+ T s s+ 1 s+ 2 s+ T where P s is our iniial price(oucome), r s+ is he acual reurn (no he simulaed one ) of he -h day of our horizon and P s + is our acual price of ha day. Here are he resuls: - Acual prices for he 5 h day of he firs 12 weeks in 2009, respecively:

66 - Acual price for he 5 h day of he firs 12 weeks in 2008, respecively: Comparison In he below able, we gahered he acual, forecasing prices and he VaR corresponding o each, o make clear how well our simulaion has worked: Week Acual Prices Forecasing Prices VaR 1 s nd rd h h h h h h h h h Table 8: 1% VaR Esimaes for 5-Day Horizon (HS Mehod, 2008) 59

67 Week Acual Prices Forecasing Prices VaR 1 s nd rd h h h h h h h h h Table 7: 1% VaR Esimaes for 5-Day Horizon (HS Mehod, 2009) Above ables make clear ha he HS mehod for measuring VaR, in his case i.e. 5-day horizon and 1% confidence level, was approximaely successful, excep in one iem i.e. he 9 h week of he 2009, where he break has occurred. (Menion ha he break has happened in 2009, which he simulaion has made based on he crisis period of he marke) 60

68 5.3 Empirical Sudy of FHS and VaR Compuaion by FHS Mehod As we explained in subsecion 4.2, in his mehod we model our reurns by an ARMA (1,1), and he random residuals by GARCH (1,1) processes. Wih he help of hese wo ime series modelling, we will receive he esimaed se of parameers menioned in i.e. ˆλ ( ˆ γ is assumed o be equal o 0). 27 In he nex sep, we followed he explained processes in 4.2.2, 4.2.3, 4.3 and obain he prediced empirical probabiliy disribuion of prices and 1% VaR for 5-day horizon. 28 (All hese seps has been done for he same daes in 2008 and 2009, we did for HS) Here are he resuls: Empirical Disribuions, Forecasing Prices and 1%VaR for 5-Day Horizon 2009: Figure 49 Forecasing price of he 5 h day (1 s week): % VaR: Using ITSM sofware (presened by [8]) 28 By MATLAB coding of FHS mehod, available a appendix A. 61

69 Figure 50 Forecasing price of he 5 h day (2 nd week): % VaR: 53.8 Figure 51 Forecasing price of he 5 h day (3 rd week): % VaR:

70 Figure 52 Forecasing price of he 5 h day (4 h week): % VaR: Figure 53 Forecasing price of he 5 h day (5 h week): % VaR:

71 Figure 54 Forecasing price of he 5 h day (6 h week): % VaR: 33.4 Figure 55 Forecasing price of he 5 h day (7 h week): % VaR:

72 Figure 56 Forecasing price of he 5 h day (8 h week): % VaR: Figure 57 Forecasing price of he 5 h day (9 h week): % VaR: