Spectral Risk Measures with an Application to Futures Clearinghouse Variation Margin Requirements By John Cotter and Kevin Dowd

Transcription

1 Specral Risk Measures wih an Applicaion o Fuures Clearinghouse Variaion Margin Requiremens By John Coer and Kevin Dowd Absrac This paper applies an AR(1)-GARCH (1, 1) process o deail he condiional disribuions of he reurn disribuions for he S&P5, FT1, DAX, Hang Seng, and Nikkei225 fuures conracs. I hen uses he condiional disribuion for hese conracs o esimae specral risk measures, which are coheren risk measures ha reflec a user s risk-aversion funcion. I compares hese o more familiar VaR and Expeced Shorfall (ES) measures of risk, and also compares he precision and discusses he relaive usefulness of each of hese risk measures in seing variaion margins ha incorporae ime-varying marke condiions. The goodness of fi of he model is confirmed by a variey of backess. Keywords: Specral risk measures, Expeced Shorfall, Value a Risk, GARCH, clearinghouse. JEL Classificaion: G15 Ocober John Coer is a he Cenre for Financial Markes, School of Business, Universiy College Dublin, Carysfor Avenue, Blackrock, Co. Dublin, Ireland; john.coer@ucd.ie. Kevin Dowd is a he Cenre for Risk and Insurance Sudies, Noingham Universiy Business School, Jubilee Campus, Noingham NG8 1BB, UK; Kevin.Dowd@noingham.ac.uk. Coer s conribuion o he sudy has been suppored by a Universiy College Dublin School of Business. We hank seminar paricipans a CASS Business School, UCD Cenre for Financial Markes, Tor Vergaa Annual Conference 25, he join conference of he European Cenral Bank and he Federal Reserve Bank of Chicago 25 on Issues Relaed o Cenral Counerpary Clearing and he Deusche Bundesbank Conference 25 in celebraion of he 8h birhday of Professor Benoî B. Mandelbro. 1

2 1 INTRODUCTION Fuures clearinghouses se margins requiring raders o pay a deposi o minimise defaul risk and acs as counerpary o all rades ha ake place wihin is exchanges. This ensures ha individual raders do no have o concern hemselves wih credi risk exposures o oher raders, because he clearinghouse assumes all such risk iself. However, i also means ha he clearinghouse has o manage his risk, and one way i does so by imposing margin requiremens. These consis of an iniial margin (deposi) and a variaion or daily margin. The iniial margin represens he deposi a fuures rader mus give o a Clearinghouse o iniiae a rade whereas in conras he variaion margin is he exra deposi required of he rader once a margin call is made. 1 In modelling iniial margins he focus is ypically on exreme confidence levels for exraordinary marke evens so as o minimise he probabiliy ha he associaed quanile is exceeded (Longin (1999) and Coer (21)). This is equivalen o sress esing for very low probabiliy evens. I can also be hough of as requiring uncondiional risk modelling over a long forecas horizon, and mos previous lieraure has focused on iniial margins modelled uncondiionally. 2 The variaion margin can hough of as supporing he iniial margin afer i has been breached o help avoid rader defaul. 3 Bu whereas he iniial margin is inended o reflec longrun condiions, he variaion margin is inended o reflec curren marke condiions. 1 See Hull (23) for deails of margin requiremens for fuures. Essenially here are four elemens o a margin accoun: As well as he iniial margin; here is he mainenance margin ha represens he minimum balance of he margin accoun ha mus no be breached; he margin call where an invesor is informed ha hey have o op up heir margin accoun and he variaion margin represening he amoun ha he invesor mus add o heir margin accoun as a resul of he margin call o bring i up o he value of he iniial margin. If he rader defauls on paying he variaion margin he broker closes ou he posiion by selling he conrac. 2 Many approaches have been followed in seing iniial margins. For insance Figlewski (1984), Edwards and Nefci (1988), Warshawsky (1989), Booh e al (1997), and Longin (1999) use differen uncondiional saisical disribuions (Gaussian, hisorical or exreme value disribuion). In conras, Brennan (1986) proposes an economic model for broker cos minimizaion where margins are endogenously deermined, and Craine (1992) and Day and Lewis (24) model he disribuions of he payoffs o fuures raders and he poenial losses o he fuures clearinghouse in erms of he payoffs o barrier opions. 3 For illusraive purposes we assume ha he iniial margin is zero and ha he variaion margin represens he full margin requiremen deposi of raders. In realiy he variaion margin represens he exra funds required o bring he raders deposi back o he value of he iniial margin bu would vary relaive o he benchmark iniial margin by he exen of he price dynamics for he condiional disribuion of fuures. 2

3 This means ha variaion margins should be modelled condiionally (so ha hey can ake accoun of curren marke dynamics) and a more convenional (i.e., nonexreme) confidence levels. The need o ake accoun of curren marke condiions suggess ha we should use some sor of GARCH process (e.g., as in Barone-Adesi e al. (1999), McNeil and Frey (2), Giannopoulos and Tunaru (25) and Coer (26)). We wan convenional confidence levels because he clearinghouse is concerned abou he prospec of possible defaul in he near fuure, bu he confidence levels should no be oo low because ha would involve very frequen changes in variaion margins, and his would be difficul o implemen in pracice. In he lieraure margins have been ypically modelled as a quanile or VaR. 4 The clearinghouse hen selecs a paricular confidence level, and ses he margin as he VaR a his confidence level. 5 However, he VaR has been heavily criicised as a risk measure as i does no saisfy he properies of coherence and, mos paricularly, because he VaR is no subaddiive (Arzner e al. (1999), Acerbi (24)). 6 This paper examines how one may obain variaion margins using a condiional modelling framework, bu using risk measures ha are superior o he VaR. The model used is a AR(1)-GARCH(1,1), and he oher risk measures considered are he Expeced Shorfall (ES), which is he average of he wors p losses, where p is he ail probabiliy or 1 minus he confidence level, and Specral Risk Measures (SRMs), which are risk measures ha ake accoun of he user s (in his case, he clearinghouse s) degrees of risk aversion. Boh hese ypes of risk measure are coheren and, from a heoreical poin of view, demonsrably superior o he VaR. 4 An excepion is Baes and Craine (1999) who esimae a measure for he expeced addiional funds required assuming he fuures price move has exhaused he iniial margin. These addiional funds fund however would no jus include he variaion margins bu would come from a variey of sources including he remaining asses of losing fuures raders, he remaining asses of he Clearinghouse and possibly fund arising from cenral bank inervenion. 5 The Clearinghouse has a difficul balancing ac for he margin sysem inroduced beween minimising counerpary risk and remaining compeiive for rades: oo high a margin implies low counerpary risk bu also an uncompeiive environmen for fuures raders and vice versa. 6 The failure of VaR o be subaddiive can hen lead o srange and undesirable oucomes: in he presen case, he use of he VaR o se margin requiremens akes no accoun of he magniude of possible losses exceeding VaR, and can herefore leave he clearinghouse heavily exposed o very high losses exceeding he VaR. For insance, because VaR is no-subaddive, using he VaR o se margin requiremens migh encourage invesors o break up heir accouns o reduce overall margin requiremens, and in so doing leave he clearinghouse exposed o a hidden residual risk agains which he clearinghouse has no effecive collaeral from is invesors. 3

4 This paper provides esimaes of condiional VaR, ES and specral risk measures for long posiions in 6 index fuures conracs: S&P5, he FTSE1, he DAX, he Hang Seng, and he Nikkei225 indexes. The adequacy of he fis is confirmed by he resuls of a variey of backess. The esimaed risk measures illusrae he prevailing price dynamics of he condiional disribuions of he fuures. The paper also evaluaes he precision of hese esimaes using a number of differen ways of esimaing heir precision. This paper is organised as follows. Secion 2 reviews he risk measures o be examined and explores heir properies. Secion 3 looks a condiional risk modelling where reurns follow an AR(1)-GARCH(1,1) process. Secion 4 discusses daa issues and model esimaion, and secion 5 evaluaes he fied model. Secions 6, 7 and 8 presen resuls for he VaR, ES and SRM risk measures in urn. Secion 9 discusses our findings and secion 1 concludes. The paper is hen followed by wo appendices, one discussing he parameric boosrap used in he simulaions and he oher discussing how numerical inegraion mehods may be used o esimae Specral Risk Measures. 2. MEASURES OF RISK We are ineresed in risk measures ha are weighed-averages of he quaniles of he reurn disribuion. If specify our risk measure q p is he 1p% quanile of he reurn disribuion, hen we can M φ as: 1 Mφ = φ( p) q pdp (1) for some arbirary weighing funcion φ ( p). The specificaion of φ ( p) hen deermines he risk measure iself. The bes-known risk measure in his class is he Value-a-Risk (VaR). The VaR a he 1α% confidence level is equal o q 1 α, so he risk measure (1) is equivalen o he VaR when φ ( p) akes he form of a Dirac dela funcion ha gives 4

5 he q quanile an infinie weigh (such ha φ ( p) dp = 1) and gives every oher 1 α quanile a weigh of zero. This means ha ha he VaR weighing funcion places no weigh on ail quaniles, and implies ha a VaR user is (in some sense) held o care more abou he prospec of a loss equal o VaR han abou he prospec of a loss greaer han he VaR. This is of course a raher srange propery and one ha leads he VaR o be non-subaddiive 7 and herefore no coheren (see Acerbi (24, p. 174); see also Arzner e alia (1999)). 8 A second risk measure is he Expeced Shorfall (ES), which is he average of he wors ( 1 α )1% of losses, i.e.: ESα = 1 α q p dp (2) α The ES is herefore based on a weighing funcion ha gives each ail quanile he same weigh, and gives every oher quanile a weigh of zero: 1 φ( p ) = 1 α if p α p = (3) p < α 7 Le A and B represen any wo porfolios, and le ρ (.) be a measure of risk over a given horizon. The risk measure ρ (.) is subaddiive if i saisfies ρ ( A + B) ρ( A) + ρ( B). Subaddiiviy reflecs risk diversificaion and is he mos imporan a priori crierion we would expec a respecable risk measure o saisfy. However, i can be demonsraed ha VaR is no subaddiive unless we impose he empirically implausible requiremen ha reurns are ellipically disribued. Is non-subaddiiviy makes i very difficul o regard he VaR as a respecable measure of risk. 8 There are also oher reasons o believe ha VaR does no si well wih well-behaved uiliy or risk aversion funcions. For example, he VaR is no consisen wih expeced uiliy maximisaion excep in he very unusual case where risk preferences are lexicographic (Grooveld and Hallerbach, 24, p. 33). Anoher example emerges from he downside risk lieraure (see, e.g., Bawa (1975) and Fishburn (1977)), which suggess ha we can hink of downside risk in erms of lower-parial momens (LPMs), where he LPM of order k around a below-arge reurn r * is equal o E[max(, r * r) ]. The parameer k reflecs he degree of risk aversion, and he user is risk-averse if k > 1, risk-neural if k = 1, and risk-loving if < k < 1. From he LPM perspecive, we would choose he VaR as our preferred risk measure only if k = (Grooveld and Hallerbach, 24, p. 35), and his suggess ha he use of he VaR as a preferred risk measure indicaes a srong degree of negaive risk aversion. k 5

6 Unlike he VaR, he ES has he aracion of being a coheren risk measure. However, since he ES weighing funcion gives he same weigh o all ail quaniles, he choice of he ES as a risk measure would imply ha he user is risk-neural a he margin beween beer and worse ail oucomes, and his is inconsisen wih risk-aversion. 9 Furhermore, like he VaR, he ES is condiioned on a parameer, he confidence level, whose value is ypically difficul o esablish and is ofen seleced arbirarily. If we wish o have a risk measure ha akes accoun of user risk-aversion, we can use a specral risk measure in which φ ( p) is obained from he user s riskaversion funcion. In pracice, his requires ha we specify he form his funcion akes, bu a plausible choice is an exponenial risk-aversion funcion which implies he following weighing funcion: kp ke φ ( p) = (4) 1 k e where k (, ) is he user s coefficien of absolue risk-aversion (see Acerbi (24, p. 178)) or Coer and Dowd (26)). This funcion depends on a single condiioning parameer, he coefficien of absolue risk aversion, he value of which reflecs he risk aversion of he user. A specral risk-aversion funcion is illusraed in Figure 1. This shows how he weighs rise as we encouner he prospec of higher losses and he rae of increase depends on k: he more risk-averse he user, he more rapidly he weighs rise. Inser Figure 1 here Unlike he case wih he VaR or ES confidence level, he value of he riskaversion parameer is in principle known or a leas ascerainable, and his means ha specral risk measures avoid he condiioning parameer arbirariness implici in using 9 Again from he downside risk lieraure he ES is he ideal risk measure if k=1, implying ha he user is risk-neural (Grooveld and Hallerbach, 24, p. 36). 6

7 he oher wo risk measures. Given he risk-aversion funcion, seing he value of he user s risk-aversion parameer ensures ha he specral risk measure akes a unique value: 1 k 1 kp Mφ = φ( p) q pdp = e q pdp k 1 e (5) which would ypically be esimaed using some form of numerical inegraion or quadraure mehod (e.g., a rapezoidal rule, Simpson s rule, pseudo or quasi Mone Carlo, ec.). 1 The fac ha an SRM akes accoun of he user s aiude o risk also means ha an SRM is a subjecive risk measure in a way ha he VaR or ES are no: so wo users wih he same porfolio bu differing degrees of risk-aversion would face SRMs of differen values, bu sill face he same VaR or ES. As well as reflecing user risk-aversion, a specral risk measure is also coheren provided φ ( p) is nonnegaive for all p [,1], provided φ ( p) dp = 1 and provided φ ( p) saisfies he weakly increasing propery of giving higher losses weighs ha are no smaller han hose of lower losses. In erms of Figure 1, his laer propery means ha φ ( p) mus never fall as we go from lef o righ along he x-axis (Acerbi (24, proposiion 3.4) This propery indicaes ha he key o coherence is ha a risk measure mus give higher losses a leas he same weigh in (1) as lower losses. This also helps explain why he ES is coheren and why he VaR is no, and ells us ha he VaR s mos prominen inadequacies are closely relaed o is failure o saisfy he weakly increasing propery (see also Acerbi (24, p. 173)) MODELLING CONDITIONAL RISK Following McNeil and Frey (2) and Coer (26), we model he daily reurn process by a normal AR(1)-GARCH(1,1) process. This process supposes ha daily reurns r are condiionally normal, i.e., 1 More deails on such mehods can be found in sandard references (e.g., Kreyszig (1999, pp ) or Miranda and Fackler (22, chaper 5). 7

8 2 r ~ N( µ, σ ) (6) where he mean µ obeys an AR(1) process: µ E[ r ] ρr + ε ; ρ < 1 (7) = = 1 and process: ε is iid zero-mean normal, and where he variance 2 σ obeys a GARCH(1,1) σ ; ω, α, β, α + β < 1 (8) 2 = ω αr 1 βσ 1 The AR(1)-GARCH(1,1) is a popular and parsimonious model which ofen provides a reasonable fi o daily reurn daa. This model allows daily reurns o have some degree of persisence, o have a volailiy ha exhibis persisence bu also alernaes beween periods of low and high volailiy, and o have moderae degrees of skewness and excess kurosis. Now le Mφ, ( µ, σ ) be our ime-varying forecas of a risk measure for day. Noe ha his is wrien as a funcion of he forecass of he day- mean and volailiy, µ and σ, and hese wo parameers are sufficien o calibrae he forecas of he risk measure forecas because we have assumed ha reurns are condiionally normal. For any of he hree ypes of risk measure VaR, ES or SRM i is easy o show ha he following relaionship holds: M µ, σ ) = µ + σ M (,1) (9) φ, ( φ, This relaionship is handy compuaionally, because i reduces he ask of risk forecasing o he wo very simple smaller asks calculaing he sandard normal risk measure M φ, (,1), and forecasing he parameers µ and σ - and his can 8

9 someimes lead o major gains in compuaional efficiency. 11 Once we specify he relevan condiioning parameer, hen he firs ask becomes sraighforward and we can focus on he second. To illusrae, suppose ha for he VaR and he ES we se he confidence level α o be 95%. And if he risk measure is he VaR, hen he sandard normal risk measure M φ (,1) becomes:, M φ, (,1) = z α = z. 95 = (1) M, ( µ, σ ) µ + σ φ = If he risk measure is he ES, hen φ( z ) (. 95), (,1) = α φ z φ = = (11) 1 α.5 M M, ( µ, σ ) µ + σ φ = where φ (.) is he value of he sandard normal densiy. On he oher hand, if he risk measure is an SRM and we ake he ARA coefficien o be, say, 5, hen (,1) becomes: M φ, 1 k, (,1) = kp Mφ e z dp k p (12) e φ M, ( µ, σ ) µ + σ = In each case risk forecasing now boils down o he ask of forecasing µ and σ. Table 1 gives a se of alernaive values of M φ (,1) for differen risk measures and condiioning parameers. So, for insance, if one was ineresed in he ES a he 99% confidence level, one would use (11) wih (,1) = and he, M φ, only informaion hen needed o esimae he ES would be forecass of he parameers 11 This issue is discussed furher in Appendix 1. 9

10 µ and σ. Similarly, if one was ineresed in he SRM wih ARA=1, one would use (12) wih (,1) =2.4916, and so forh. M φ, Inser Table 1 here 4. DATA AND MODEL ESTIMATION Our daa se consiss of daily geomeric reurns (aken as he difference beween he naural logarihms of consecuive end-of-day prices) for he mos heavily raded index fuures ha is, he S&P5, FTSE1, DAX, Hang Seng and Nikkei 225 fuures beween January 1 2 and December Daily price daa was obained from Daasream represening he full calendar period excluding weekends giving 782 close-of-day reurns. This sample period is hen spli ino wo subsamples: an iniial esimaion period (covering all of 2 and 21) o provide an iniial GARCH fi; and a rolling esimaion period (encompassing all of 22) over which he model is updaed on a daily basis. As a preliminary, Table 2 provides some descripive saisics for he full sample period are oulined in Table 2. The mean reurns are near zero bu negaive, and he sandard deviaions of reurns are in excess of 1% per day for all indexes. Mos reurns have a small negaive skew, and all reurns exhibi moderae degrees of excess kurosis. The Ljung-Box saisics applied o he reurn series give a mixed picure wo of hem appear o have a significan dependence srucure, and he oher hree do no. These resuls sugges ha we migh wish o ake accoun of possible dependence in he reurn process, and are he reason why we chose o model reurns using an AR(1) process. For heir par, he Ljung-Box saisics applied o squared reurns indicae ha hese have very significan dependence srucures, and his finding is reinforced by he significan ARCH effecs indicaed by he resuls of Engle s (1981) LM es. These laer es resuls suppor he exisence of ime-varying volailiy dynamics and sugges ha some form of GARCH-ype process is called for. Inser Table 2 here 1

11 The AR(1)-GARCH(1,1) was firs applied o each fuures index daily reurns for Esimaion was by Maximum Likelihood. We hen obain forecass for each week day in 22 by updaing parameers on a daily rolling window basis giving 259 ses of ime-varying condiional parameers. Table 3 repors he average values of he parameers and of heir associaed diagnosics. Our resuls are very much as expeced: GARCH effecs are significan and parameer values across he differen indices are similar o each oher and o hose repored by many oher sudies. 12 The residual diagnosic resuls also sugges ha he residuals are independen and ha he model is well-specified. Inser Table 3 here Figure 3 shows plos of he esimaed GARCH daily volailiies in each of our five fuures indices over The volailiy of he S&P is a lile more han 1% for mos of he firs half of 22, bu hen i rises aferwards and wice peaks a around 2.5%. The FTSE volailiy is similar, bu peaks a greaer values and is noiceably higher in he second half of he year. The DAX volailiies show a similar paern bu are a lile higher han he FTSE ones. The Asian markes are much more ranquil: he Hang Seng volailiy is generally a lile under 1.5% in early 22 and hen rises a lile quie differen for mos of he second half of he year, bu is always well under 2%; he Nikkei is mosly beween 1% and 2% and peaks a lile beyond 2% in early March, bu hen falls back again is sable for he res of he year. Thus, Figure 3 shows ha condiions varied somewha across differen markes, and perhaps he mos noiceable difference is ha wesern markes exhibied much less ranquilliy in laer 22 han did heir eas Asian counerpars. Inser Figure 2 here 12 We presen average values only for breviy, bu here was relaively lile flucuaion in parameer esimaes on a day-o-day basis. Furher summary saisics deails (deviaions ec) are available on reques. 13 We only show resuls for long posiions because here is lile asymmery in he index reurns: explicily addressing he VaRs of shor posiions would herefore provide lile subsanial addiional informaion. 11

12 5. MODEL EVALUATION In order o provide a more formal evaluaion of he model, we mus firs se ou is main predicions. One useful predicion relaes o he model s probabiliy inegral ransform (PIT) values. The PITs are he values of he realised reurns afer hey have been pu hrough he following ransformaion: p = F r ) (13) ( where F (.) is he forecased cumulaive densiy funcion made he previous day. A sample of PIT values is prediced o be disribued as sandard uniform under he null hypohesis of model adequacy. 14 Accordingly, Diebold e alia (1998, p. 869) sugges ha a useful diagnosic of model adequacy is o plo he PITs and check visually if hey are close o he prediced uniform disribuion. The PIT values from our model applied o he fuures reurns are presened in Figure 3. The fied lines are very close o he 45 line prediced under he null, and his srongly suggess ha he fis are good ones. Inser Figure 3 here Anoher predicion of he model is ha he frequency of exceedances ha is, he frequency of observed losses exceeding VaR should be compaible wih he predicions of he model. This lead o he Kupiec es: for he 95% VaR, he model predics ha 5% of observaions should be exceedances. Wih 259 observaions, a prediced 5% exceedance probabiliy means ha here are prediced exceedances, so we would es he hypohesis ha he number of exceedances is accepably close o 13. We can es his predicion using a binomial es. 14 A formal proof of his predicion is provided by Diebold e alia (1998, pp. 865, ). 12

13 A hird predicion relaes o he sandardized residuals. Combining and rearranging (6) and (7) leads o he predicion ha he sandardized residuals should be iid N(,1), viz.: ε / σ ε / σ ~iid N(,1) (14) Thus, ε / σ are prediced o be boh iid and sandard normal. However, he resuls of Table 3 have already esablished ha ε are independen, so we can work on he basis of ha he iid predicion is saisfied. We herefore focus on exbook ess of sandard normaliy aking iid for graned. These include: a z-es of he predicion ha ε / σ has a mean of zero; a -es of he predicion ha ε / σ has a mean of zero; a variance-raio es of he predicion ha ε / σ has a uni variance; and a Jarque-Bera es of he predicion ha ε / σ has zero skewness and a kurosis of 3. The resuls of hese ess are presened in Table 4. For each fuures index, his Table presens he number of exceedances, he mean, sandard deviaion, skewness and kurosis of ε / σ, and he prob-values of each of he above ess. Going hrough hese resuls one index a a ime: The S&P does well by all ess excep perhaps he Kupiec one, which yield a prob-value of 3.7%. This laer resul suggess ha he number of exceedances (19) migh be raher high, which migh indicae ha he model under-esimaes he 95% VaR. The FTSE resuls are similar, excep ha he prob-value of he Kupiec es is now.3. This presens srong evidence ha he number of exceedances (23) is oo high, and ha he model under-esimaes he 95% VaR. The DAX does well wih he possible excep of he z-es and -es, which are boh significan a under he 5% level. This migh indicae a problem wih he predicion ha sandardised residuals should have a zero mean. 13

14 The Hang Seng does well for all ess excep he Jarque-Bera. The Jarque-Bera is significan a under he 1% level, and his suggess ha he sandardized residuals are no normal. The Nikkei passes all ess easily. In sum, of 25 es resuls, 2 are no significan, 3 are significan a he 5% level and 2 are significan a he 1% level. Overall, we would sugges ha his is a reasonably good performance which suggess ha he model s forecass are fairly accurae. Inser Table 4 here 6. RESULTS FOR VALUE-AT-RISK Figure 4 shows plos of he esimaed 95% VaRs and he esimaed bounds of heir 9% confidence inervals for long posiions in each of our five fuures indices over 22. Broadly speaking, he esimaed VaRs show much he same paerns as he GARCH volailiies and pain a similar picure abou marke condiions. For heir par, he confidence bounds in Figure 4 sugges ha he uncerainy in VaR forecass had a endency o move wih he VaR forecass hemselves. This endency is paricularly pronounced wih he FSTE, where he confidence bounds are iniially quie narrow bu become much wider when he forecased VaR peaks in Augus and hen again in Ocober. We see comparable increases in he widhs of he S&P and DAX confidence bounds when hey also peak a much he same imes. Inser Figure 4 here Table 5 gives VaR resuls compued for he 22 average daily values of he inpu parameers. The Table repors esimaed VaRs and a variey of precision merics for our VaR esimaes: hese are he VaR sandard error (SE); he sandardized VaR SE, which is he SE divided by he esimaed VaR; he 9% confidence inerval for he VaR; and he sandardized 9% confidence inerval for he VaR, which is same confidence inerval divided by he esimaed VaR. The firs and he hird of hese give 14

15 esimaes of precision in absolue erms, whereas he second and fourh give esimaes of precision relaive o he size of he esimaed VaR. These precision merics are based on a fully parameric boosrap applied o he AR(1)-GARCH(1,1) process. This boosrap gives esimaes of precision merics based on simulaed VaRs (or oher risk measures) based on informaion available he previous day. Since he volailiy follows a GARCH process, his means ha he curren-day volailiy is already known (from (8)). Hence i is imporan o appreciae ha he only noise in he boosrap process is he sampling variaion of he daily mean, and his means ha he noisiness of he simulaed VaRs is driven enirely off he noise in he simulaed means. Furher deails of he boosrap are given in Appendix 1. Before examining he resuls, i is also imporan o appreciae ha if σ is given in any boosrap simulaions (as is he case here), hen (9) ells us ha he sandard error of any of he risk measures considered here mus be equal o he sandard deviaion of he µ. This also means ha, for any given ses of parameers, he VaR, ES and SRM mus all have he same SE. We would emphasise ha hese predicions are no a produc of any srangeness in our algorihm, because he algorihm fully reflecs he srucure of he AR(1)-GARC(1,1) process and our assumpions abou he informaion available a any poin in ime. Insead, hese predicions are generaed by he underlying srucure of he model iself. The Table shows ha he VaRs are generally quie close. The S&P, FTSE, Hang Seng and Nikkei are quie close in he region 2.3% o 2.6%. The Hang Seng lowes a 2.338%, and he DAX is an oulier a 3.557%. The oher hree are close o he Hang Seng, so he DAX is an oulier. The Table repors ha he VaR SEs are proporional o he VaRs, and he sandardized SEs are all he same a.33%. The Table also shows ha he bounds of he confidence inerval reflec he sizes of he esimaed VaRs, bu he sandardized confidence inervals he confidence inervals divided by he esimaed VaRs are he same: he sandardized 9% confidence inerval is always [.51,1.499], i.e., plus or minus 5% of he esimaed VaR. The precision sory is herefore ha in absolue erms, he level of precision varies inversely wih he size of he VaR, bu in relaive or sandardized erms, he level of precision is always he same. 15

16 Inser Table 5 here 7. RESULTS FOR EXPECTED SHORTFALL The corresponding ES resuls are presened in Table 6. Perhaps he mos sriking feaure in his Table is ha he esimaed ES is always approximaely 25.5% greaer han he comparable earlier VaR. This suggess ha whaever he values of µ and σ migh be, he normal ES esimaed wih hese parameer values will be close o imes he normal VaR esimaed wih hese same parameer values. 15 I is herefore no surprising o discover ha he ES plos in Figure 5 have he same shape as he early VaR ones, and herefore have he same inerpreaions. However, we would emphasise ha his relaionship is approximae raher han exac: if we plo he raio of he ES o he VaR over ime, we do no ge a sraigh line, bu noisy process ha oscillaes around a sraigh line and is close o i. Inser Table 6 here Inser Figure 5 here The precision merics in Table 6 are also as expeced: he ES has he same SE as he VaR, and herefore has a small sandardized SE. The bounds of he ES confidence level are hen pushed ou by he exen of he difference beween he ES and he VaR, bu he ES confidence inerval has he same widh as he VaR one. And because he ES exceeds he ES, he ES mus have a narrower sandardised SE han he VaR: he sandardized ES confidence inerval is now plus or minus 4%. Thus, in absolue erms, he ES is esimaed wih he same precision as he VaR, bu in relaive erms, i is esimaed more precisely han he VaR. And, if we compare he ES resuls o each oher, resuls show ha in absolue erms precision varies inversely wih he 15 For any given confidence level and given empirically plausible parameer values, he normal ES is a slighly varying greaer-han-one muliple of he normal VaR. The muliple iself can be seen in he raio of he sandard normal ES o he sandard normal VaR. So, for example, wih a confidence level of 95% he raio of he sandard normal ES o he sandard normal VaR can be seen from Table 1 o be 2.627/1.6449= as in Table 6. 16

17 size of he esimaed ES, bu in relaive erms, he level of precision is always he same. 8. RESULTS FOR SPECTRAL RISK MEASURES The corresponding SRM resuls are presened in Table These show ha he SRM calibraed on an ARA coefficien of 5 is now in he region of imes he value of is 95% VaR counerpar, 17 and his implies (and Figure 6 confirms) ha he SRM plos have much he same shapes as he VaR ones in Figure 3. Of course, we should keep in mind ha whereas he normal ES always exceeds he normal VaR predicaed on a given confidence level, he normal SRM predicaed on a chosen ARA does no always exceed he normal VaR predicaed on a given confidence level: if he ARA is relaively low, and he confidence level high, hen he SRM can be lower han he VaR (as is eviden from Table 1). Inser Table 7 here Inser Figure 6 here The precision merics in Table 7 are also as expeced: he SRM has he same SE as he VaR, and herefore has a smaller sandardized SE. The bounds of he SRM confidence level are pushed ou by he difference beween he SRM and he VaR, and he SRM confidence inerval has he same widh as he VaR one; and he sandardized SRM confidence inerval is now plus or minus 36%. In absolue erms, he SRM is esimaed wih he same precision as he VaR, bu in relaive erms, i is esimaed more precisely. (However, for reasons ha will be apparen from he previous paragraph, we would expec he SRM o be relaively less precisely esimaed han he VaR in cases where he ES is smaller raher han larger han he VaR.) Comparing he 16 More deails on he esimaion of he SRMs are provided in Appendix Reminiscen of he las noe bu one, he SRM predicaed on a paricular ARA coefficien is a slighly varying bu no-necessarily-greaer-han-one muliple of he α VaR, where he muliple can again be inferred from he resuls of Table 1. In his case, Table 1 ells us ha he sandard normal SRM predicaed on an ARA of 5 is equal o , and he 95% sandard normal VaR is The raio /1.6449= , which is he value repored in Table 7. 17

18 SRM resuls o each oher, we find (as wih he VAR and ES) ha absolue precision varies inversely wih he size of he esimaed risk measure, bu in relaive erms, he level of precision is always he same. 9. DISCUSSION For all hree risk measures, we herefore ge he same precision sory : in absolue erms precision varies inversely wih he size of he esimaed risk measure, bu in relaive erms, he level of precision is always he same. Second, all risk measures indicae he relaive riskiness of differen conracs and how he risk changes over ime. For insance, each of he condiional risk measures show he ime varying naure of volailiy for he respecive indices during 22. All indices exhibi specific dynamics reflecing high condiional volailiy a a cerain ime and hereby having large associaed risk esimaes followed by decreasing condiional volailiy owards he end of he sample as indicaed by he decreasing risk esimaes. However as all he risk esimaes are driven by he same condiional process he same paern emerges for each risk measure. Clearly hey all sugges ha he Hang Seng conrac is he leas risky index and ha he DAX is he mos risky. Thus, he use of any of hese measures for seing variaion margins would herefore lead o he former ones having he lowes margins and he DAX he highes. Third, boh esimaes of ES and VaR are direcly comparable o each oher as heir key parameer is he confidence level, unlike he SRM, whose key parameer is he degree of risk aversion. Moreover here are also disincions in erms of he inerpreaion and usefulness of he risk measures. Firs, he use of VaR o esimae variaion margins only allows he Clearinghouse o esimae a quanile and he associaed defaul probabiliy. Thus, he condiional VaR is limied as i gives he Clearinghouse no idea of he size of heir exposure beyond he probabiliy level chosen. However, esimaing a condiional VaR (and he oher measures) allows he Clearinghouse o esimae heir variaion margins based on he ime-varying dynamics prevalen a any ime. 18

19 Second, in principle he ES is more useful o he clearinghouse han he VaR because i akes accoun of he sizes of losses higher han he VaR iself. I also has he helpful inerpreaion ha i ells he clearinghouse he loss an invesor can expec o make on i experiencing a loss ha exceeds a chosen VaR hreshold. So if he clearinghouse ses a VaR-based variaion margin, hen he ES ells he clearinghouse he expeced defaul loss for he invesor experiencing a loss ha exceeds is margin. Also, by esimaing a condiional ES he clearinghouse has an indicaion of he expeced losses assuming a quanile has been breached based on he prevailing price dynamics. Third, in seing variaion margins, he specral risk measures are in principle he mos useful, because hey alone ake accoun of he user s degree of risk aversion. 1. SUMMARY AND CONCLUSIONS Clearinghouses manage counerpary risk hrough a margin accoun. In addiion o he iniial margin, a rader mus add a variaion margin if a margin call akes place. The modelling of variaion margins encompasses he price dynamics of he condiional disribuion during he lifeime of he fuures conrac. This paper esimaes variaion margins using an AR(1)-GARCH (1, 1) process ha models boh he condiional mean and volailiy using hree differen risk measures: he VaR, he Expeced Shorfall (ES), and specral-coheren risk measures. Alhough he imevarying approach allows for a daily updae of variaion margins based on each risk measure, he Clearinghouse may have variaion margins ha reflec he prevailing dynamics bu updaed less frequenly, hereby exploiing he volailiy persisence in fuures. Previous sudies have esimaed margins for fuures reurns using a VaR measure where margins are associaed wih quaniles measured a confidence levels. In conras, he use of ES allows he Clearinghouse o ge an esimae of he expeced losses assuming he margin level is breached. Moreover, he use of SRMs allows he Clearinghouse o incorporae risk aversion ino he margin esimaes. This paper illusraes he properies and he associaed relaive meris of he measures underpinned by he condiional disribuion of fuures. 19

20 REFERENCES Acerbi, C., 22. Specral measures of risk: a coheren represenaion of subjecive risk aversion. Journal of Banking and Finance 26: Acerbi, C., 24. Coheren represenaions of subjecive risk-aversion, in G. Szego (Ed), Risk Measures for he 21 s Cenury, Wiley, New York, pp Arzner, P., F. Delbaen, J.-M. Eber, D. Heah, Coheren measures of risk. Mahemaical Finance 9, Barone-Adesi, G., K. Giannopoulos, and L. Vosper, VaR wihou correlaions for porfolios of derivaives securiies, Journal of Fuures Markes, 19, Baes, D., and R. Craine, Valuing he fuures marke clearinghouse's defaul exposure during he 1987 crash. Journal of Money, Credi, and Banking, 31, Bawa, V. S., Opimal rules for ordering uncerain prospecs. Journal of Financial Economics 2: Bollerslev, T., Chou, R., and K. Kroner, ARCH modelling in finance: a review of he heory and empirical evidence, Journal of Economerics, 52, Booh, G. G., Brousssard, J.P., Marikainen, T., and Puonen, V., Pruden margin levels in he Finnish sock index fuures marke. Managemen Science 43, Brennan, M.J., A heory of price limis in fuures markes, Journal of Financial Economics, 16, Coer, J. 21. Margin exceedences for European sock index fuures using exreme value heory. Journal of Banking and Finance 25, Coer, J., 26, Varying he VaR for uncondiional and condiional environmens, Journal of Inernaional Money and Finance, Forhcoming. Coer, J., Dowd, K., 26, Exreme specral risk measures: an applicaion o fuures, Elecronically available on Science Direc: hp:// LisID= &_sor=d&view=c&_acc=C5221&_version=1&_urlVersio n=&_userid=1&md5=e64db1df9db6e25e4c d9aebd 2

21 Craine R., Are fuures margins adequae? Working Paper, Universiy of California Berkley. Day, T.E., and C.M. Lewis, 24, Margin adequacy and sandards: an analysis of he crude oil fuures markes. Journal of Business, 77, Diebold, FX, TA Gunher, and AS Tay, Evaluaing densiy forecass wih applicaions o financial risk managemen. Inernaional Economic Review 39: Edwards, F. R., and S. N. Nefci, Exreme price movemens and margin levels in fuures markes. Journal of Fuures Markes 8, Engle, R.F., Auoregressive condiional heeroskedasiciy wih esimaes of he variance of UK Inflaion, Economerica, 5, Figlewski, S., Margins and marke inegriy: margin seing for sock index fuures and opions. The Journal of Fuures Markes, 4, Fishburn, P. C., Mean-risk analysis wih risk associaed wih below-arge reurns. American Economic Review 67: Giannopoulos, K., and R. Tunaru, 25. Coheren risk measures under filered hisorical simulaion. Journal of Banking and Finance, 29, Grooveld, H., Hallerbach, W. G., 24. Upgrading value-a-risk from diagnosic meric o decision variable: a wise hing o do?, in G. Szegö (Ed.) Risk Measures for he 21 s Cenury. Wiley, New York, pp Hsieh, D.A., Implicaions of nonlinear dynamics for financial risk managemen, Journal of Financial and Quaniaive Analysis, 28, Hull, J. C., 23. Opions, Fuures, and oher Derivaives, 5 h ediion, Prenice Hall, New Jersey. Jarque, CM, and AK Bera, A es for normaliy of observaions and regression residuals, Inernaional Saisical Review 55: Kreyszig, E Advanced Engineering Mahemaics. 8 h ediion. New York: Wiley. Kupiec, P., Techniques for verifying he accuracy of risk managemen models. Journal of Derivaives 3:

22 Kusuoka, S., 21. On law invarian coheren risk measures. Advances in Mahemaical Economics 3: Longin, F., Opimal margin levels in fuures markes: exreme price movemens. Journal of Fuures Markes, 19, McNeil, A. J., and R. Frey, 2. Esimaion of ail-relaed risk for heeroscedasic financial ime series: an exreme value approach. Journal of Empirical Finance, 7, Miranda, M. J., and P. L. Fackler, 22. Applied Compuaional Economics and Finance. MIT Press, Cambridge MA and London. Warshawsky, M. J., The adequacy and consisency of margin requiremens: he cash, fuures and opions segmens of he equiy markes. The Review of Fuures Markes 8,

23 APPENDIX 1: AN EFFICIENT PARAMETRIC BOOTSTRAP The precision merics (i.e., sandard errors and confidence inervals) repored in he paper were esimaed using a parameric boosrap: his boosrap is moivaed by he idea ha our precision merics we should make full use of he srucure of he model. Now imagine he following problem. We would like o esimae precision merics for a risk measure Mφ, ( µ, σ ) for day, using informaion available for day -1. To do so, we need a boosrap procedure ha gives us a se of, say, m randomly chosen values of M ( µ, σ ), where hese simulaed values make use of he informaion we φ, have abou day -1 and are generaed in a way ha reflecs he AR(1)-GARCH(1,1) model, i.e., we wan a fully parameric boosrap. The logic behind his boosrap is as follows: we firs noe from (9) ha we can consruc a value of Mφ, ( µ, σ ) if we have values of he parameers µ and σ. We also know from he GARCH(1,1) process (8) ha σ is deermined by our informaion from he previous day, i.e., σ is already given. Any randomly simulaed value of Mφ, ( µ, σ ) mus hen be driven by a randomly simulaed value of µ. This means ha we need o simulae values of µ and (7) ells us ha µ is a zero-mean normal wih an unknown variance. Le us suppose for he momen ha we know his variance. We now simulae a value of µ from his normal disribuion and inpu his value and our given value of σ ino (9) o simulae a random value of he risk measure M ( µ, σ ). We hen do his a large number of imes m o give us a large φ, sample of simulaed Mφ, ( µ, σ ) values. Our precision merics hen follow naurally: if we wan he sandard error of Mφ, ( µ, σ ), we esimae his as he sandard deviaion of Mφ, ( µ, σ ) ; and if we wan a confidence inerval, we can use a convenional exbook formula. I remains o show how we obain he variance of µ. The easies way o obain his numerically is o firs assume an arbirary variance for µ and use his in (6) o simulae a se of reurns. (We can acually use he same se of simulaed random numbers as before, bu his is jus an issue of compuaional efficiency.) We hen 23

24 esimae he variance of hese reurns and find his o be, say, * σ. However, he reurns should have a variance of σ. To ge he correc variance for boh reurns and µ, we herefore have o muliply he original assumed variance by value of he variance of variance. µ is hen equal o * / σ σ * / σ. The rue σ imes he original assumed We should noe wo oher poins abou his boosrap. Firs, he random noise in our simulaed Mφ, ( µ, σ ) values is driven off he random noise in he µ process. Since (9) also ells us ha Mφ, ( µ, σ ) moves pari passu wih µ, his implies ha he sandard error of Mφ, ( µ, σ ) mus be equal o he volailiy of µ. This laer predicion also means ha for any given se of parameers, he VaR, ES and SRM all have he same sandard error, and he resuls in Tables 5 o 7 reflec hese implicaions. The final poin o noe is ha we use (9) in our algorihm in order o ensure ha we only ever direcly esimae he sandard normal risk measure (,1 ), and M φ, ha we do his only once in each boosrap exercise; all esimaes of Mφ, ( µ, σ ) are hen obained indrecly by insering he sandard normal esimae M φ (,1 ) and he parameers, µ and σ ino (9). This is much more efficien compuaionally han esimaing Mφ, ( µ, σ ) direcly (i.e., by inpuing parameer values ino he relevan version of (1)). So, for example, if we are dealing wih an SRM, he former approach requires us o invoke a numerical inegraion rouine only once, bu he laer requires us o invoke i a large number of imes. The former approach is much faser compuaionally. Thus, our boosrap algorihm is compuaionally efficien as well as fully parameric. 24

25 APPENDIX 2 : ESTIMATING SPECTRAL RISK MEASURES USING NUMERICAL INTEGRATION Unlike he case wih he esimaion of VaR or ES, he esimaion of specral risk measures ypically requires us o compue he value of an inegral. Where reurns are normal wih mean µ and sandard deviaion σ, hen equaions (9) and (12) ell us ha he SRM predicaed on a coefficien of absolue risk aversion equal o k is given by: where M φ, ( φ, µ, σ ) = µ + σ M (,1) (A2.1) 1 k kp M e z pdp k e φ, (,1) = (A2.2) 1 where z p is he 1p% quanile of he sandard normal disribuion. We herefore have o calculae he inegral in (A2.2). We can do so using a numerical inegraion or quadraure mehod, in which we approximae he coninuous inegral by a discree equivalen: we discreise he coninuous variable p ino a number N of discree slices, where he approximaion ges beer as N ges larger. To apply numerical inergraion, we have o selec a value of N and choose a suiable numerical inegraion mehod, and our choices include rapezoidal and Simpson s rules, quasi-mone Carlo mehods (e.g., such as hose using Niederreier or Weyl algorihms) and pseudo-mone Carlo mehods. To evaluae he accuracy of hese mehods, Figure A2.1 shows plos of esimaed sandard normal SRMs agains N, where he SRM is predicaed on an ARA coefficien of 5, using he rapezoidal rule, Simpson s rule, Niederreier and Weyl quasi-mone Carlo, and pseudo-mone Carlo numerical inegraion mehods. We consider N values from 1 o 5. The firs four converge upwards owards he rue value of and are all more or less accurae when N reaches 5. By conras, he pseudo-mone Carlo gives very volaile esimaes, and converges much more slowly. We can herefore eliminae he pseudo-mone Carlo mehod as very 25

26 unreliable compared o he ohers. Of he remaining four, he rapezoidal and Simpson s rule esimaes are smooher and converge somewha faser han he quasi- Mone Carlo mehods. There is very lile difference beween he rapezoidal and Simpson s rule esimaes, and boh are very accurae for N values of 2 or more. However, he rapezoidal rule is easier compuaionally. Taking on board issues of boh accuracy and compuaional efficiency, we hen seleced he rapezoidal rule wih N=3 as he combinaion of N value and inegraion mehod o be used o produce he SRM esimaes in he paper: his combinaion produces highly accurae resuls fairly quickly. Inser Figure A2.1 here 26

27 TABLES Table 1: Values of Sandard Normal Risk Measures α VaR ES ARA SRM Noes: VaR, ES and SRM are Value-a-Risk, Expeced Shorfall and Specral Risk Measures for sandard normally disribued reurns, α is he confidence level and ARA is he coefficien of absolue risk aversion. The SRM measures are esimaed using he rapezoidal numerical inegraion mehod wih N=3. Table 2: Summary Saisics for Daily Reurn Daa S&P FTSE DAX HANG SENG NIKKEI Mean Sd Dev Skewness Kurosis Minimum Maximum Q(12) ** * Q(12) ** ** ** ** ** ARCH(12) ** ** ** 6.997** 6.726** Noes: Based on he 782 close-of-day % reurns for each of he saed indexes over he period January 1 2 o December Mean, sandard deviaion, minimum and maximum are in percenage form. Q(12) is he 12-lag Ljung-Box es saisic applied o he reurns series, Q 2 (12) is he same es saisic applied o he squared reurns series, and ARCH(12) is he Engle (1981) LM es for up o welve-order ARCH effecs. * represen significan a 5% level and ** represens significance a 1% level. 27

28 Table 3: Mean GARCH Parameer Esimaes and Diagnosics for Fuures Indexes S&P5 FTSE1 DAX Hang Seng Nikkei 225 ρ (.592) (.362) (.411) (.383) (.179) ω (.97) (.25) (.98) (.15) (.66) α (.1) (.) (.) (.1) (.11) β (.) (.) (.) (.) (.) Q(12) R (.339) (.4) (.61) (.134) (.452) Q(12) 2 R (.) (.) (.) (.3) (.1) ARCH(12) - R (.16) (.) (.) (.18) (.6) AIC BIC JB-Z (45.39) (.16) (.6) (.13) (.1) (.4) Q(12) Z (.345) (.349) (.471) (.266) (.883) Q(12) 2 Z (.299) (.927) (.562) (.97) (.19) ARCH(12) - Z (.35) (.91) (.579) (.968) (.173) Noes: The Table presens average GARCH model coefficiens and average diagnosics. Marginal significance levels using Bollerslev-Wooldridge sandard errors are displayed in parenheses. Pre-model diagnosics are applied o he reurns series R and pos model diagnosics are applied o he sandardised filered reurn series Z. Q(12) is he Ljung-Box es applied o he indicaed series, and Q 2 (12) is he Ljung-Box es applied o he squared indicaed series. ARCH(12) is he Engle (1981) LM es for up o welfh order ARCH effecs. Marginal significance levels for he model diagnosics are given in parenheses. 28

29 Table 4: Model Evaluaion Resuls S&P FTSE DAX Hang Seng Nikkei Numbers of exceedances Summary saisics for sandardized residuals Mean Sd Skewness Kurosis Tes Prob-values of es saisics Kupiec.373*.3** z-es * es * Variance raio Jarque-Bera **.1633 Noes: The resuls apply o 259 daily observaions over 22. The number of exceedances refers o he number of loss reurns exceeding forecased VaR. The sandardized residuals are he residuals of (6) divided by forecased σ. The ess are wo-sided version of: he Kupiec binomial es ha he frequency of exceedances equals 1 minus he confidence level of 95%, z- and -ess of he predicion ha sandardized residuals have a mean equal o, a variance-raio es of he predicion ha sandardized residuals have a variance equal o 1, and a Jarque-Bera es ha he sandardized residuals are normal. * indicaes significance a 5% level and ** indicaes significance a 1% level. Table 5: Value-a-Risk Resuls VaR SE S. SE 9% CI S. 9% CI LB UB LB UB S&P FTSE DAX HS Nikkei Noes: Esimaes of VaR are in daily % reurn erms and VaR is predicaed on a 95% confidence level. SE is he VaR sandard error, S. SE is he sandard error of he sandardized VaR (i.e., VaR SE divided by he mean VaR), 9% CI is he 9% confidence inerval for he VaR, S. 9% CI is he 9% confidence inerval for he sandardized VaR, and LB and UB refer o he lower and upper bounds of confidence inervals. Sandard errors and confidence inervals are esimaed using a parameric boosrap applied o he 22-average values of he 259 ses of daily parameers of he AR(1)-GARCH(1,1) model. To faciliae comparabiliy of resuls across Tables 5 o 7, simulaions are carried ou using he same ses of seed numbers for he pseudo-random number generaor. 29