Beyond the Sample: Extreme Quantile and Probability Estimation æ

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1 Beyond he Samle: Exreme Quanle and Probably Esmaon Jon Danelsson London School of Economcs and Insue of Economc Sudes a Unversy of Iceland Caser G. de Vres Erasmus Unversy Roerdam and Tnbergen Insue December 997 Absrac Economc roblems such as large clams analyss n nsurance and value-a-rsk n fnance, requre assessmen of he robably P of exreme realzaons Q: Ths aer rovdes a sem-aramerc mehod for esmaon of exreme P;Q combnaons for daa wh heavy als. We solve he long sandng roblem of esmang he samle hreshold of where he al of he dsrbuon sars. Ths s accomlshed by he combnaon of a conrol varae ye devce and a subsamle boosra echnque. The subsamle boosra aans convergence n robably, whereas he full samle boosra would only rovde convergence n dsrbuon. Ths erms a comlee and comrehensve reamen of exreme P;Q esmaon. Keywords: Exreme value heory, al esmaon, rsk analyss Corresondng auhor: C. G. de Vres, Tnbergen Insue, P.O. Box 738, 3000 DR Roerdam, The Neherlands, e-mal cdevres@few.eur.nl. Danelsson s e-mal s jond@h.s. Ths and relaed aers can be downloaded from h:// jond/research. Danelsson benefed from an HCM fellowsh of he EU and he Research Conrbuon of he Icelandc banks. We are graeful o Holger Drees, Laurens de Haan, Marc Henry, and Lang Peng for many useful suggesons. Some daa suded n he aer was obaned from Olsen and Assocaes.

2 INTRODUCTION Inroducon Economc analyss ofen deends on assessmen of he robably P of exreme quanles Q : For examle, nsurance comanes focus on he robably of run and commercal banks use he value-a-rsk mehodology o calculae he loss ha can be ncurred wh a gven low robably on her radng orfolo. In addon, value-arsk s used as he bass for deermnaon of he caal adequacy of fnancal nsuons. Accurae esmaon of he borderlne n-samle and he ou-of-samle P; Q combnaons s essenal for hese roblems. The al characerscs are also moran for economerc ssues such as he convergence rae of regresson esmaors and he selecon of arorae es sascs. In hs aer we develo a sem-aramerc esmaor for he als of he dsrbuon. From sascal exreme value heory, see e.g. Leadbeer, Lndgren and Roozen (983), we know ha he lm law for he exreme order sascs s one of hree yes whch are deermned by wheher he dsrbuon has a fne endon or no, and by wheher he als of he denses are declnng exonenally fas or by a ower. The dsrbuon s sad o be heavy aled n he case of ower declne so ha no all momens are bounded; oherwse he dsrbuon s sad o be hn aled. Hence, f one s only neresed n he exreme P; Q combnaons, one can rely on he asymoc form of he al of dsrbuon nsead of havng o model he whole dsrbuon. Ths gves he al focused sem-aramerc mehod an advanage over oher mehods n al alcaons. Ths ales boh o non-aramerc mehods n general, and aramerc mehods where he ye of dsrbuon s unknown. Esmang he arameers of he wrong dsrbuon ycally mles ncorrec exreme P; Q esmaes, boh because of mssecfcaon, and because he daa n he cener of he emrcal dsrbuon have oo much nfluence over he arameer esmaes of he wrong model; whle f only he als are modelled, hs nfluence s absen. The sem aramerc mehod may also be sueror o a non aramerc aroach because he laer s dffcul o use for consrucng ou-of-samle P; Q esmaes. If he daa are generaed by a heavy aled dsrbuon, hen s dsrbuon has, o a frs order aroxmaon, a Pareo ye al: P fx éxgax, ; aé0; é0; as x!:by usng he conce of regular varaon, defned below, he Pareo naure of he als of such dsrbuons as he non-normal sable, Suden-, and he Fréche, can easly be verfed. The exonen equals he number of bounded momens, and a s a scalng consan. Esmaes of are needed n order o consruc exreme P; Q esmaes. Suose here exss a hgh hreshold s above whch ax, s a good aroxmaon of P fx éxg,andlex denoe he samle realzaons such ha X é s: Then he maxmum lkelhood esmaor for = of he lef runcaed Pareo dsrbu- 2

3 INTRODUCTION on s he average of he log X =s : Ths esmaor s known as he Hll (975) esmaor. I has been shown by Hall (982) and Golde and Smh (987) ha here exss a unque sequence of hresholds s n as a funcon of he samle sze n such ha he bas squared and varance of he Hll esmaor vansh a he same rae. Moreover, hs sequence mnmzes he asymoc mean squared error (AMSE) of =^. As we show laer, gven hs sequence and he Hll esmae, he consrucon of exreme P; Q esmaes s sraghforward. The roblem s herefore fndng he omal hreshold s n : Unl now, has no been known how o esmae s n ; exce under very resrcve assumons, see e.g. he recen survey by Embrechs, Kuelberg and Mkosch (997). Ths roblem has hamered he raccal mlemenaon and adoon of exreme value mehods, because a key ar of he sascal rocedure remaned arbrary. Mos emrcal aers roceed by long esmaes of = agans dfferen choces for s n : Subsequenly, by eyeballng such a lo one res o locae s n where he bas squared and varance have he aearance of beng n balance. In hs aer we solve he long sandng roblem of esmang s n hrough a boosra of he MSE of d =, and by mnmzng he boosra MSE hrough he choce of sn : I s, however, no sraghforward o consruc such a boosra, because he heorecal benchmark value = s unknown. To solve hs roblem we use he dea behnd conrol varaes n Mone Carlo esmaon, see e.g. Hendry (984). We subrac from he Hll esmaor an alernave esmaor whch converges n he MSE sense a he same rae, albe wh a dfferen mullcave consan. Hence, hs dfference sasc converges a he same rae and has a known heorecal benchmark whch equals zero n he lm. The square of hs dfference sasc roduces a vable esmae of he MSE ë=ë ha can be mnmzed wh resec o he choce of he hreshold s n : Unforunaely, can be shown ha he convenonal boosra rocedure of hs dfference sasc from he enre samle only generaes s n levels whch relave o he omal level converge n dsrbuon. To aan he desred convergence n robably, we show ha one needs o creae resamles of smaller sze han he orgnal samle wh a subsamle boosra echnque. The reason why he full samle boosra echnque fals s due o he lneary of he esmaors n log X =s : By boosrang on he enre samle, one n essence recreaes he full samle esmaes, bu he use of smaller resamles roduces a weak law of large numbers effec. We resen he maeral n a comrehensve self conaned manner, wh known roven resuls eher n he Aendx or referenced, wh he benef ha sascal exreme value heory becomes accessble o economss. There s a hos of neresng alcaons n economcs and economercs, some whch have been underexloed. Lorean and Phlls (994) use esmaes of heavness of he als o deermne wheher he fourh momen s bounded or no, n order o decde uon he roer asymoc dsrbuon for he CUSUM sasc. Kearns and Pagan (997) dscuss he ssue of al 3

4 2 THEORY ndex esmaon wh deenden fnancal reurns daa. Akgray, Booh and Sefer (988), Koedjk, Schafgans and de Vres (990) and Longn (996) use al esmaes o consruc a nesed es n order o dscrmnae beween such non-nesed models lke he sum sable and Suden- dsrbuons. Informaon abou he heavness of he dsrbuon s also useful for obanng he convergence seed of OLS esmaors n regresson analyss, snce he seed of convergence deeroraes f he nnovaons have fne varance bu are heavy aled nsead of beng normally dsrbued. Booh, Broussaard, Markanen and Paonen (997) analyze he deermnaon of margn calls n fuures markes, and Jansen and de Vres (99) sudes he redcon of boom and crashes. Large clams analyss n nsurance economcs s suded by Berlan, Teugels and Vyncker (994), and Embrechs, Kuelberg and Mkosch (997). We look a one alcaon n some deal, Value-a-Rsk. In addon, we rovde an exensve amoun of Mone Carlo exermens o es our esmaor. We smulae from a number of..d. and deenden heavy aled dsrbuons and sochasc rocesses, and esmae back known heorecal quanes. 2 Theory We defne he dea of heavy als rgorously by means of he conce of regular varaon n subsecon 2. and hen develo a whole class of al esmaors. The roeres of hese esmaors and he omal choce of s n for a gven dsrbuon are dscussed n subsecon 2.3. Subsecon 2.4 shows how s n can be esmaed from he daa f he dsrbuon s unknown. The las subsecon rovdes he exreme robably quanle P; Q esmaors. 2. Regular Varaon A dsrbuon funcon F x s sad o vary regularly a nfny wh al ndex f, F x lm!, F = ; é0; xé0: () x, The roery of regular varaon mles ha he uncondonal momens of X larger han are unbounded. In hs sense he class of regular varyng dsrbuons s heavy aled. Ths assumon s essenally he only assumon whch s needed for he analyss of he al behavor of X. For exosory reasons we only focus on he uer al of he dsrbuon; he analyss of he lower al s analogous. A more dealed aramerc form for he uer al of F x can be obaned by akng a second order exanson of F x as x!. Whle here are several exansons 4

5 2 THEORY 2.2 k-momen Rao Tal Index Esmaors ossble, de Haan and Sadmuller (996) show ha here are only wo non-rval exansons. The frs exanson s F x =,ax, +bx, + o, x, ; é 0; as x!: (2) Under a mld exra condon, he exanson (2) mles he followng exanson for he densy f x =ax,, + ab + x,,, + o, x,,, : (3) Here, he heory wll be develoed on he bass of he exanson for he densy (3). The densy exanson faclaes he unfed, comrehensve and sreamlned reamen of sascal exreme value heory gven below. The exanson (3) ales o he well known cases of non-normal sum sable, Suden-, Fréche, and oher fa aled dsrbuons. The oher non-rval second order exanson s: F x =,ax, ë + b log x + o log xë : The second order erm n hs exanson decays more slowly han he algebrac rae of he second order erm n (2) and (3). 2.2 k-momen Rao Tal Index Esmaors Consder he condonal k,h order log emrcal momen from a samle X ; :::; X n of n..d. draws from F x : u k s n M MX = log X s n k X és n ; (4) where s n s a hreshold ha deends on n, andm s he random number of excesses. An alernave defnon s u k m n m n Xmn = log X s n k ; s n = X mn+ ; where X are he descendng order sascs, and s n s a random hreshold. Noe ha u k s n and u k m n are funcons of he hghes realzaons of X. These wo defnons yeld dencal resuls and are used nerchangeably deendng on he exedency of he roofs. Danelsson, Jansen and de Vres (996) nroduced he followng class of esmaors for he nverse of he frs order al ndex, =:. For hs class he esmaor we develo for = s conssen. Bu he slow decay of he second order erm makes hs class suffcenly dfferen from he oher class such ha does no easly f whn he sreamlned resenaon of he curren aer,.e. would make he resen aer overly long. 5

6 2 THEORY 2.2 k-momen Rao Tal Index Esmaors Defnon The k-momen rao esmaor, denoed as w k s n, for he nverse al ndex s w k s n = d u k s n = ku k, s n ; (5) where k =;2; ::: are neger valued, and u 0 s n =: The secfc case where k =, andw s n = u s n, s known as he Hll esmaor roosed by Hll (975). The heorecal roeres of he Hll esmaor are well documened by e.g. Hall (982) and Golde and Smh (987), who develo he heory resecvely on he bass of u k m n and u k s n. The Hll esmaor s consdered here as ar of he class of momen rao esmaors. There are several moves for a consderaon of he whole class of k-momens rao esmaors. Below we show ha he varous members of he k,class have under ceran condons beer bas and mean squared error roeres han oher members. Secondly, we show ha a leas wo elemens from hs class are needed o n down he omal hreshold s n. Hall (982) and Golde and Smh (987) rovde roofs of he momen roeres of he Hll sasc w. We exend he roofs o he general case of w k. The roofs are conaned n Aendx A. Theorem For he class of random varables ha sasfy (3), he asymoc bas of he k momen rao esmaor w k s n s E w k s n, =, bk,2 + k s, + o, s, n n : (6) Theorem 2 Suose he hreshold s n s chosen such ha Ms =an! n robably n as n!. Then f (3) ales Var w k s n, = k M + o ; (7) 2 M where k = 2k! 2k, 2! 2k,! +, k! k,! k!k,! : (8) The frs few values of he k funcon are gven n Table. Noe he rad ncrease as k ncreases. 6

7 2 THEORY 2.3 Omal Choce of s n k k Table : Values of he k funcon Togeher hese wo heorems mly ha for ceran sequences of s n he w k are conssen esmaors f he X are..d. and sasfy he densy resrcon (3). The w k esmaors are also conssen under varous forms of deendency. Leadbeer, Lndgren and Roozen (983) conans an exensve reamen of ARMA ye deendence, whch reservers he regular varaon roery, and de Haan, Resnck, Roozen, and de Vres (989) subsequenly roved ha he uncondonal dsrbuon of ARCH rocesses sasfy he regular varaon roery. Hsng (99) and Resnck and Sărcă (996) show ha he Hll esmaor s a conssen esmaor under resecvely ARMA and ARCH ye deenden rocesses. 2.3 Omal Choce of s n The asymoc mean squared error (AMSE) of w k s n follows from Theorems and 2: AMSE w k s n k s n a 2 n + b2 2 2k,4 s,2: (9) 2k n + Whch of he wo erms on he rgh hand sde of (9) asymocally domnaes he oher, s deermned by he rae by whch s n! as n! : Moreover, here s a unque sequence s n whch asymocally balances he wo erms. We derve hs sequence from he frs order AMSE =@s n = 0; can be verfed ha he second order condon s sasfed as well. Theorem 3 Suose ha Ms =an! n robably and ha (3) ales. The unque n AMSE mnmzng asymoc hreshold s n s " 2ab 2 3 2k,3 +2 s n w k = n +2 ; (0) + 2k k and he assocaed asymocally mnmal MSE of w k s n equals " MSE ëw k s n ë = k a + 2ab 2 3 2k, k k 2+ n, 2 2+ : () 7

8 2 THEORY 2.3 Omal Choce of s n The asymoc number of exceedances m n where he bas and varance ars are balanced s comued by combnng (0) wh he suoson Ms =an! : n m n w k =a " 2ab 2 3 2k,3 + 2k k, +2 n 2 +2 : (2) From (9-) s sraghforward o show ha f s n ends o nfny a a rae below n =2+ ; he bas ar n he MSE wll domnae, whle conversely he varance ar domnaes f s n ends o nfny more radly han n =2+ : For he class w k s n we show ha on he bass of he AMSE creron he only wo elemens of neres are w and w 2 : Theorem 4 The w and w 2 sascs are he only wo esmaors n he class w k ;k = ;2;3:::; whch are no domnaed, n he sense of he AMSE creron, for all = 2 R + combnaons. Proof. From () we have ha, for a gven n, 2k MSE = c k 2=2+ 2+ ; + and where cé0:comarng we fnd ha hs s equvalen wh MSE w k, R MSE w k ; + R k k, : Now noe ha ë k = k, ë = domnaes += for all values of = é 0 f k = k, ée2:7: From Table s evden ha hs holds for k =3;4; ::: Because + R2 as R ; we fnd: Corollary For é 0, when he Hll and he w 2 sascs are each evaluaed a her own asymoc MSE mnmzng hresholds: MSE w R MSE w 2 as R : 8

9 2 THEORY 2.4 Esmaon of s n From he revous formulas for he asymoc bas and varance can be seen ha boh he (omal) varance and bas squared dffer only wh resec o he mullcave consans = and =2: Hence = k BIAS 2 w k, R BIAS 2 w k, += R : k, Ths mles ha f w 2 has a lower MSE han w,henw s asymocally more based han he w 2 : Therefore, w 2 domnaes w n boh he MSE and bas sense for he cases where he frs order erm x, n he asymoc exanson of F x converges more radly han he second order erm x,. The asymoc dsrbuon of w k s gven n Theorem 5. The analyss follows Golde and Smh (987) who dscuss he secal case of he Hll sasc. Theorem 5 Suose we choose s n such ha s n n, 2+! h2ab 2 3 2k,3 +,2k k, 2+ n robably as n!:then r M wk s n, k! N, 2 sgn b ; n dsrbuon. The usefulness of Theorem (5) deends on a number of facors. Frs, he on esmae w k s n s condonal on he choce of s n : If he asymocally omal hreshold s n can be esmaed by ^s n such ha ^s n =s n converges n robably o, hen he asymoc normaly of w k s n also ales o he case where s n s relaced by s esmaed value. Hall and Welsh (985) showed ha hs holds whou resrcons on he convergence rae; s an mlcaon of he regular varaon roery of he MSE n (). Second, he lmng normal dsrbuon has a mean whch deends on he unknown facor =2 sgn b : If hs laer facor can be esmaed conssenly, hen follows from Slusky s heorem ha he clam of he asymoc normaly of w k s n also ales o he case where he bas facor mus be esmaed. Ths ssue wll be aken u a he end of he nex secon. 2.4 Esmaon of s n The esmaon of s n s a non-rval roblem, and u o now he ublshed work on hs roblem s by Hall (990). Bu Hall s aer only gves a very aral soluon because 9

10 2 THEORY 2.4 Esmaon of s n assumes ha he frs and second order al ndces are equal,.e. =. Noneheless Hall s aer conans he moran suggeson o emloy a boosra rocedure o esmae s n.snces n asymocally mnmzes he MSE, he dea s o solve hs mnmzaon roblem by he boosra. Suose ha for k =we oban he sandard boosra equvalen of he execaon MSEëw ë=e " w s n, 2 by RX w;r s n, ew ~s n 2 ; (3) R r= where w ;r s calculaed on a boosra resamle of he orgnal samle, R s he number of boosra resamles, and ew ~s n s some conssen nal esmae. Then one could mnmze hs sasc by choce of s n : There are, unforunaely, hree roblems wh hs aroach. Frs, hs rocedure fals o ck u he bas. The log-lneary of he esmaor mles ha he boosra execaon E R of w ;r gven he emrcal dsrbuon funcon F n equals E R ëw ;r s n j F n ë=w s n : Hence, he bas s calculaed o be zero. Bu, as was shown n he revous secon, he omal s n mus be such ha he bas squared and varance are of he same order of magnude. Hall (990) ncely solved hs roblem by roosng a subsamle boosra rocedure. Suose he boosra resamles are no of sze n; bu of smaller sze n é n; such ha n =n! 0 as n ;n! : Relace w ;r s n n (3) by w ;r s.then n he average bas from he subsamle esmaes does no cancel agans he bas of he full samle nal esmae ew s n because he laer s of smaller order, a he corresondng omal values of s n. Second, he rocedure only works when he frs and second order al ndexes are resrced o be equal. The reason for hs resrcon can be undersood as follows. Once he omal hreshold s n has been esmaed for he subsamle sze n ; has o be nflaed o fnd he corresondng full samle equvalen. From (0) follows ha he relaonsh beween he omal hreshold values for he dfferen samle szes s: s n =s n n=n 2+ : (4) The corresondng formula for he number of excesses s from (2) m n = m n n=n 2 2+ : (5) 0

11 2 THEORY 2.4 Esmaon of s n Golde and Smh (987) show ha (4) and (5) are equvalen soluons o he choce of a hreshold. Hall assumes = and focuses on (5), so ha he exonen equals 2=3. Ths resrcon ales o ceran classes of fa aled dsrbuons lke he ye II exreme value dsrbuon and he sable dsrbuons. Bu for oher dsrbuons, lke he Suden- class where = 2 and equals he degrees of freedom, he resrcon does no aly. For a sasfacory general reamen of he class of heavy aled denses he exonen n (4) or (5) has o be esmaed. The hrd roblem s ha he mnmzaon of he subsamle boosra MSE mn sn R RX w;r s, ew n ~s n 2 r= s sll condonal on an nal full samle esmae ew ~s n : Bu for he enre rocedure o work hs nal esmae has o be such ha ~s n =s n! n robably, and hence requres an arorae choce of es n : If = hen one mgh use w ~m n where ~m n = n 2=3. Ths would sll gnore he mullcave consan n (2). Bu for he general case hs s of no aval anyway, and hence ~m n or ~s n have o be esmaed. However, we se ou o fnd s n n he frs lace. Hence, hs roblem undermnes he enre rocedure skeched hus far. Boh he second and he hrd roblem are solved below. We roose relacng w ;r s n by a sasc for whch he rue value s known,.e. s ndeenden of ; bu wh an AMSE ha has he same convergence rae, albe wh a dfferen mullcave consan, as he AMSE of he w k sasc. When he rue value s known, he boosra AMSE s easly mlemened, because we do no need an nal esmae lke ew es n n (6), and he omal s n or m n can be esmaed. The sasc we roose o use s: (6) z s n w 2 s n, w s n : (7) We showed earler he conssency of all w k sascs as esmaors of = for s n cn 2+.Thezsn converges o 0 as n!:hence, MSEëzë =Eëz 2 ë:we now show ha he AMSE ëzë has he same order of magnude as he AMSE ëw k ë : Theorem 6 If s n s such ha Ms =an! n robably, hen n and E ëz s n ë= b s, + o, s, n n Var ëz s n ë = 2 M + o : M ;

12 2 THEORY 2.4 Esmaon of s n Corollary 2 Suose ha Ms =an! n robably, hen he AMSE ëzë mnmzng asymoc hreshold level s n z n reads s n z = 2ab n 2+ : (8) By comarng s n w k from (0) wh he s n z from (8) we see ha! s n z s n w k = 2 + 2k,4 2+ k : 2k,2 Hence he wo hreshold values only dffer wh resec o her mullcave consans, bu ncrease a he same rae wh resec o he samle sze n: From Corollary 2 we know ha he asymoc MSE ëzë s mnmzed by s n z from (8). However, he raccal roblem of fndng hs value remans. One ossbly s a boosra of z 2 s ;.e. calculae he boosra average =RP R n r z2 s, and mnmze hs average wh resec o s: Unforunaely, for he followng reason hs does n;r no roduce an esmae whch s asymoc o s n z. In he roof o Theorem (5) we showed ha Mu k s asymocally normally dsrbued. By he Taylor exanson from he roof o Theorem (6) hen readly follows ha Mz s also asymocally normally dsrbued. Hence Mz 2 s asymoc o a 2 dsrbued random varable. The mean of Mz 2 s easly shown o be asymoc o M mes he value of he AMSE ëzë as saed n (40) n Aendx A. Bu because Mz 2 only converges n dsrbuon, he average of he full samle boosra values Mz 2 has he same dsrbuonal roeres as Mz 2 : To show hs, use he log-lneary of he u and u 2 n he n;r daa and consder he Taylor exanson of z as gven n he roof o Theorem 6. Insead of he convergence n dsrbuon, for sascal uroses one would lke o have convergence n robably. We wll now show ha he desred convergence n robably can be obaned hrough he subsamle boosra rocedure. Noe ha he argumen for usng he subsamle boosra rocedure s dfferen from Hall s (990) argumen concernng he bas. Before we urn o he roof, we wll rovde an nuve accoun of wha he subsamle boosra rocedure acheves. Consder he Hll esmaor w s n = M MX Y s n ; = MP M Y s n s where Y s n log X =s n gven ha X é s n and where X are descendng order sascs. From Theorem 5 we know ha for s n = s n ; 2

13 2 THEORY 2.4 Esmaon of s n asymocally normally dsrbued. Now suose ha s n s no of he order n 2+ ; cf. (0). Then follows from (9) ha eher he bas domnaes asymocally, f s n = o n 2+ ; or ha he varance domnaes, f n =2+ = o s n : Ths resul for he Hll sasc frs aeared n Hall (982). Now consder akng subsamle resamles of sze n such ha n = O n," ; where 0 é"é:le s n be he AMSE mnmzng hreshold level for he samle sze n : Because n n,", s = o n n 2+ : The subsamle boosra average of he Hll sasc s: R RX r M r XMr Y ;r s n = R RX r w ;r s n : In he orgnal samle, snce s n é s n ; he observaons are ordered as follows: X ::: X M é s n X M + ::: X T é s n X T + ::: The boosraed sasc w ;r s n evaluaed a he subsamle omal hreshold value s n s herefore an average from he se Y s ; :::; Y n M s ;Y n M+ s ; :::; Y n T s n : I follows ha as he number of subsamles R ncreases R RX r M r XMr Y ;r s n! T TX Y s n (9) n robably. Ths resul can be undersood as follows. For a gven n he number of resamles R,say,forwhchM r has he secfc sze m, m 2f;2; :::; n g ; becomes more numerous as R ncreases. Hence, evenually he law of large numbers kcks n such ha he sum of he averages P R P m Y;r s n =m dvded by he number of P resamles R for whch M r equals m T converges o Y =T: The weghed average, Pwh weghs R =R, of hese averages for secfc m values hen also converges o T Y =T: By he heorem from Hall (982) we hen have ha as n! Tn TnP Y s, n! (20) E w s n ;n, n robably. And hence hs s also ales o he lef hand sde of (9). Noe ha w s n ;nsands for he Hll sasc calculaed from he full samle bu condonal on he smaller, subsamle omal, hreshold value s n : The dea s ha subsamle 3

14 2 THEORY 2.4 Esmaon of s n boosra averages condonal on he subsamle omal hreshold value are comarable o he corresondng full samle sasc evaluaed a a smaller hreshold han s n : Bu condonal on hs smaller hreshold value, he full samle sasc converges n robably raher han n dsrbuon. Ths embeddng dea s he essence of he roof o our man resul. Theorem 7 Suose model (3) ales. Le n = O n," for some 0 é"ébe he boosra resamle sze. For gven n le R!and deermne ^s n such ha s mnmal. Then, as n! R RX r ëz r ^s n ;n ë 2 ^s n z=s n z! n robably. Noe, s n z was gven n (8). k Proof. Use he above shorhand noaon Y k s = log X ;r ;r s ; where X;r s: By he Taylor exanson of z from he roof o Theorem (6) we can wre (usng he shorhand s for s n ) R = R RX ëz r s ;n ë 2 (2) 8 é! Mr X 2! Mr Y ;r s + 2 X 2 Y 2 ;r 4 s r RX : +4 2 r XMr M r, 4 Y ;r s + M r M r 2 XMr,2 Y ;r s M r XMr XMr Y 2 ;r s Y 2 ;r s For each of he erms whn he curled brackes we can use argumens smlar o he ones ha were used o fnd he asymoc values n (9) and (20) by frs drvng R! and subsequenly akng n!. To hs end suose ha s = o s n : Hence, for he second erm on he RHS of (2) asymocally R RX r M 2 r Mr X Y ;r! o: M r

15 2 THEORY 2.4 Esmaon of s n = R RX r M 2 r XMr Y ;r 2 + R 2 m s + +bs, 2 RX r M 2 r XMr Mr 6=j + bs, + X j! 2 ; Y ;r Y j;r where m s an s n : The las se, when n!whle ms, =n! a; follows n from he argumens n he roof o Theorem 2. To see how he frs se can be obaned consder e.g. he frs erm. By he reasonng ha was aled o arrve a (9), we fnd ha as R! R RX r M 2 r XMr ëy ;r s ë 2! m s T TX Y 2 s n robably. Where n addon o he argumens behnd (9) we used lm R! Xn = R R = m s : By smlar reasonng one fnds he asymoc values of he oher erms as: R RX r M 2 r Mr X Y 2 ;r! 2 20 m s bs, R RX r M r +bs, R RX r 2 M r +bs, R RX r M 2 r XMr Y ;r + bs, + XMr Y 2 ;r 2 + XMr Mr Y ;r 2! ;! bs, + 2 X Y 2 ;r! 2 + bs, ; ; 5

16 2 THEORY 2.4 Esmaon of s n 4 m s bs, 2! + bs, bs, + Subsue hese exressons no he arorae laces whn he curled brackes n (2). Afer some rearrangemen, one arrves a R RX r ëz r s ;n ë 2 s b a 2 n s 2! ; (22) for any s = o s n : By Corollary 2 hs boosra MSE ëzë value s mnmzed a s = s ; where s n s gven n (8). Moreover, s sraghforward o show ha on n he one hand for s 2 0; s n he rgh hand sde of (22) s monoonc and declnng n s. On he oher hand for s = o s n and s é s n ; he rgh hand sde of (22) s monoonc and ncreasng. Thus s n can be locaed asymocally by searchng for he mnmum o RX ëz r s ;n ë 2 R as s s ncreased from zero. r Remark An analogous rocedure, and roof ales o he nerreaon of he Hll sasc wh a fxed number of excesses and a random hreshold. In ha case one decreases he number of excesses from he maxmal number o he value where booms ou. R RX r ëz r s ;n ë 2 : Remark 2 A roof of hs clam by means of bounds on he uer class sequences for he emrcal dsrbuon funcon of he unform dsrbuon s avalable from Danelsson, de Haan, Peng and de Vres (997). Ths roof ales o he more general class (2), bu s also more nvolved. An alogeher dfferen aroach s aken n a recen manuscr by Drees and Kaufmann (997). They esablsh a law of eraed logarhm for mw. The resul s hen used o consruc a sequence ^m n whch s asymoc o m n : The above yelds an esmae ^s z of he omal hreshold s n z such ha n ^s z =s n z! n robably. A smlar saemen ales o he esmae for n 6

17 2 THEORY 2.4 Esmaon of s n ^m n z for he omal number of hghes order sascs m n z : Noe ha m n z can be calculaed n he same way as ha m n w k n (2) was obaned: m n z =a 2ab , 2+ n 2 2+ : (23) Whle s more exeden o resen he heorecal dervaons n erms of he hreshold nerreaon, however, n racce he mnmzaon of he boosraed MSE ëzë s done n erms of he ndex m: In he end we are no neresed n he omal m n z from (23), bu raher we need he omal m n w k as n (2). These wo quanes are relaed as follows: " 2 m n z m n w k = + 2k,4 k, + 2 : (24) Hence, a converson from ^m n z o ^m n w k requres a conssen esmae of he rao of he frs and second order al arameers =: Thefollowngresul exloshefac ha m n z vares regularly, cf.() and (23). Theorem 8 A conssen esmaor for = s d= = log ^m z n (25) 2 log n, 2 log ^m n z: Theorem (8) n combnaon wh (24) mles ha 2 ^m w n 2= ^m z log ^m 2 log n,2 log ^mn z log n n n 2 log n, 2 log ^m z (26) n s a conssen esmaor for m n w 2 : Smlar exressons can be obaned for ^m n w : Bu hese esmaors do no exlo all he nformaon whch s avalable n he full samle, because hese are resrced o he subsamle sze n : The second converson we need s o go from ^m n w 2 o ^m n w 2 : Corollary 3 Under he condons of Theorems (3) and (7) n robably. ^m n w k m n w k 2 n 2+=! (27) n 7 z

18 2 THEORY 2.4 Esmaon of s n Proof. Combne he resuls from boh roosons. One mgh conemlae usng relaon (27) as an equaly and o relace = n he exonen by d = from (25). Unforunaely, even hough he d = esmaes n (25) s conssen, s rae of convergence s unknown. Ths frusraes usng d = n (27) because = aears n heexonen (and hence sconvergence rae maybe ooslow,.e. less han " log n). A soluon s o do a second boosra on an even furher reduced subsamle sze n 2 ; and o choose n 2 handly such ha he mullcave facor n (27) can be relaced by a known value. Theorem 9 Le n = O n," for some 0 é " é =2 and choose n 2 = n 2=n: Suose ^m z s he conssen esmaor of m n z from he subsamle boosra 2 n2 rocedure on subsamle resamles of sze n 2 : Then n robably. Proof. Smlar o Corollary 3 we have ha and ^m n z2! (28) m n z ^m n z 2 ^m n z m n z 2 n 2+= n ^m z 2 n 2 n 2+= ^m z n n 2 P,! P,! : Dvson combned wh he fac ha we choose nn 2 =n 2 =yelds he clam. Combne resul (28) wh (26) o arrve a he conssen esmaor ^m n w 2 = ^m n 2 z2 log m 2 log n,2 log ^mn z z log n n ^m z n 2 log n 2, 2 log ^m z : (29) n The oher varans lke ^m n w follow easly. We have shown how s n ; or m n,and= can be esmaed on he bass of a double subsamle boosra rocedure. Ths rocedure ress on a choce for he subsamle szes n = n," ; where é"é0;and n 2 2 = n 2=n: Asymocally any n such ha é"é0yelds a conssen esmae of. Hence, asymoc argumens rovde lle 2 gudance n choosng beween any of he n, whch s desred for raccal uroses. We roose he followng creron. 8

19 2 THEORY 2.5 Predcon of Exremes The bass for our esmaor of s he mnmzaon of her AMSE : The subsamle boosra yelds esmaes of he AMSE z and AMSE z n : By he same argumens as were used n he roof o Theorem 9, one can show n2 ha h 2 AMSE ëz n = AMSE ë z (30) n 2 s asymoc o AMSE z m n : The dea s hen o choose n by arg mn n h 2 AMSE ëz n = ë Choosng n n hs way kees he esmaed MSE o a mnmum. AMSE, z n 2n : (3) Remark 3 Noe ha n conras o he revous leraure, no arbrary choce of arameers, n arcular m n or s n, has o be made n our rocedure. Only he unng arameers concernng he grd sze over whch n s vared and he number of boosra resamles has o be chosen. These are dcaed by he avalable comung me. Fnally, we need o address how sgn b can be esmaed conssenly. Esmaon of sgn b s needed n Theorem 5 for he urose of hyohess esng. Ths can be acheved as follows. Recall he mean of z s n from Theorem 6: E ëz s n ë = cs, sgn b+o, s, n n ; where cé0:ths suggess he followng conssen esmaor sgn ëb = sgn z s n ; wh s n é s n : (32) Noe ha we choose s n é s n, or alernavely m n é m n ; o guaranee ha he bas asymocally domnaes he varance. We also exermened wh he followng esmaor for sgn b sgn ëw 2, w ë, ëw 4, w 3 ë : I s sraghforward o check ha hs esmaor s also conssen. 2.5 Predcon of Exremes The rmary objecve of he aer s o develo beer esmaors for borderlne n samle and ou of samle quanle and robably P; Q combnaons. The roeres of he quanle and al robably esmaors follow from he roeres of d =: Ou of samle P; Q esmaes are relaed n he same fashon as he n samle P; Q esmaes,.e. we esablsh an ou of samle Bahadur-Kefer resul. 9

20 2 THEORY 2.5 Predcon of Exremes Consder wo excess robables and wh é =n é ; where n s he samle sze. Assocaed wh and are large quanles x and x, where x :,Fx =, and x :, F x =. Snce é =n, s lkely ha x é max fx ;:::;X n g: The quanle x can be esmaed by exraolang he emrcal dsrbuon funcon F n x by means of s regular varaon roeres. Usng he exanson of F x n (2) wh é 0 we have = x x +bx, +bx, + o + o x, x, ; so ha x = x = +bx, +bx,, + o x, + o x, A = : (33) Ths suggess he followng esmaor. Ignore he hgher order erms n he exanson, relace by m=n and x by he m +-h descendng order sasc, and subsue for = an w k esmaor. Ths yelds: ^x = X m+ m n wk : (34) Alernavely, emloy he hreshold nerreaon of he w k ;.e. he robably s relaced by he random varable M=n wh x fxed a s n.thsgves ^x = x M n wk : (35) Theorem 0 Suose ha he condons of Theorem 5 do hold. In addon ake x = ^s n ;m=ënë, decreasng bu n!. Suose ha n n converges o a consan whch may be zero. Then he quanle esmaor ^x s asymocally normally dsrbued: m ^x sgn b, k N, ; : log m=n x 2 2 The roof for he fxed number of order sascs s smlar and omed. 20

21 3 ESTIMATION AND SIMULATION An esmaor for he reverse roblem can be develoed as well. Rewre (33 ) and use = x, +bx, + o x, ; (36) x +bx, + o x, ^ = M n x x ^ : (37) Theorem Under he same condons as n Theorem0, he excess robably esmaor ^ s asymocally normally dsrbued, ha s m n dsrbuon. log x =x ^, k! N 2 sgn b ; 2 2 Noe ha he asymoc dsrbuons of he normed quanles and robables dffer by a mullcave facor of, 2 : Ths s a Bahadur-Kefer ye resul for ou of samle P; Qcombnaons, cf. Serflng (980). In words, does no maer from whch axs one looks a he dsance beween he emrcal dsrbuon funcon and he dsrbuon funcon, even f ou of samle he emrcal dsrbuon funcon s relaced by he ; ^x or ^; x curves. Remark 4 The algorhm for comung ; ^x or ^; x and w 2 m n s as follows. For Pa gven choce of n é n draw R boosra resamles of sze n. Calculae R ëz R r r m ;n n ë 2,.e. he boosra MSE of he dfference sasc z a each m n ; and fnd he ^m n whch mnmzes hs boosra MSE. Reea hs rocedure for an even smaller resamle sze n 2, where n 2 = n 2 =n. Ths yelds ^m n 2. Subsequenly calculae ^m n from (29). Fnally, esmae = by w 2 ^m n. The choce for n s made from (3). By usng hs rocedure wo unng arameers have o be chosen, he number of boosra resamles and he search grd sze. Las, one esmaes he desred P; Q combnaons from eher (34) or (37). 3 Esmaon and Smulaon We nvesgae he erformance of our esmaors, boh wh Mone Carlo exermens and alcaon o real world roblems. In he frs exermen, we evaluae he al 2

22 3 ESTIMATION AND SIMULATION 3. Mone Carlo Exermens ndex and quanle esmaors for a number of heavy aled dsrbuons and sochasc rocesses whle n he second exermen, we generae daa from one model and esmae back anoher fully aramerc model. In he alcaons we frs evaluae he al shae of several fnancal reurns, and hen nvesgae he deermnaon of caal requremens by Value-a-Rsk mehods. 3. Mone Carlo Exermens We generae seudo random numbers from several known dsrbuons and sochasc rocesses. These are lsed n Table 2. The al ndex esmaor w 2 and he quanle esmaor ^x are aled o smulaed daa, and he resuls comared wh her heorecal values. For he Suden- he al ndex equals he degrees of freedom; for he exreme value of he Fréche ye and he log Pareo dsrbuons he hyerbolc coeffcen equals he al ndex; and for he non normal symmerc sable he characersc exonen equals he al ndex. The log Pareo has a dsrbuon for whch he second order erm decays slower han he ower decay of (2), cf. Foonoe (): F x =,x, ë + log xë : I can be obaned as he dsrbuon of he roduc of wo..d. Pareo random varaes. The Suden(3) SV 0:;0:9 model denoes he followng rocess Y = U W H ; ; Pr ëu =,ë=0:5; Pr ëu =ë=0:5 H = Q + H,; =0:; =0:9; Q N0; s, 2 3 W = ; Z 3 : Z 2 Ths rocess generaes volaly clusers bu Y sll reflecs he far game roery of fnancal reurns, and s herefore relaed o he ARCH class of models. I was desgned n hs secfc way because follows ha he margnal dsrbuon funcon of Y has a Suden-(3) dsrbuon, for whch we know all heorecal arameers of he exanson a nfny. The MA(,) Suden (3) refers o he MA rocess, Y = X +X + where he X are..d. Suden-(3) dsrbued. Therefore Y has a convolued Suden- margnal dsrbuon, for whch we can comue all he relevan heorecal values. The las rocess o be smulaed s he GARCH(,) rocess wh normal nnovaons. We use wo rocesses, GARCH(2.0) 0:05;0:8;0:2 and GARCH(4.0) 0:05;0:6;0:2 where he number n he brackes s he heorecal al ndex, and he subscred values are he mean, MA, and AR arameers of he volaly rocess. Kearns and Pagan (997) sugges ha al ndex esmaon may no be sraghforward for fnancal reurns daa 22

23 3 ESTIMATION AND SIMULATION 3. Mone Carlo Exermens Table 2: Smulaon Resuls: Parameers Dsrbuon Parameer Mean s.e. RMSE True Suden = :02 0:075 0:075 :000 = :398 0:305 0:675 2:000 m=m :702 0:69 0:984 :000 Suden 4 = 0:286 0:054 0:064 0:250 = 0:600 0:65 0:93 0:500 m=m 2:702 :982 2:60 :000 Sable:4 = 0:670 0:047 0:065 0:74 = :497 0:268 0:565 :000 m=m 7:425 :993 6:726 :000 Sable:8 = 0:392 0:040 0:68 0:556 = :226 0:204 0:304 :000 m=m 2:708 6:492 2:698 :000 Tye II Exreme () = :026 0:062 0:067 :000 = 2:042 0:623 :23 :000 m=m 2:46 :27 :844 :000 Tye II Exreme (4) = 0:257 0:06 0:07 0:250 = 2:043 0:623 :24 :000 m=m 2:462 :27 :844 :000 Log Pareo (4) = 0:302 0:020 0:055 0:250 = 2:068 0:702 2:83 0:000 m=m Suden 3 SV 0:;0:9 = 0:360 0:060 0:066 0:333 = 0:705 0:8 0:85 0:667 m=m 2:484 :627 2:200 :000 MA(,) Suden (3) = 0:33 0:077 0:079 0:333 = 0:664 0:239 0:239 0:667 m=m 5:994 5:06 7:03 :000 GARCH(2.0) 0:05;0:8;0:2 = 0:485 0:2 0:3 0:500 = 0:905 0:37 m=m GARCH(4:0 0:05;0:6;0:2 = 0:326 0:073 0:05 0:250 = 0:695 0:232 m=m 23

24 3 ESTIMATION AND SIMULATION 3. Mone Carlo Exermens due o he resence of volaly clusers. By ncludng he GARCH and SV rocesses n he smulaon, where he arameers have been chosen wh an eye owards he Kearns and Pagan (997) aer, we can nvesgae hs ssue. We conclude ha our mehod sll erforms well n hese cases. The smulaons conss of 250 relcaons wh samle sze 5,000. The mnmum samle sze for ruden alcaon of exreme value mehods les around,500. Gven he currenly avalable szes of fnancal daa ses, 5000 s a reasonable number. The smulaon esmaon roceeds as oulned n Remark 4. Esmaon was erformed by searchng over he mnmum MSE n by varyng n n ses of 300 from 800 u o 4; 200; as suggesed n Remark. We noe ha n an alcaon o a arcular daa se, a much fner grd can easly be mlemened. For each choce of n and n 2 we drew 500 subsamles n he boosra rocedure. In Table 2 we reor he esmae w 2 of =, he esmaes of he rao =, andhe rao of he omal number of hghes order sascs m n o he heorecal value m n where he laer value s known. For each value we reor he mean, sandard error (s.e.), roo mean squared error (RMSE), and he heorecal value, where hese values are known. Table 3 reors resuls from he quanle esmaon, wh robables =n and =3n: We show he heorecal values, he mean, he coeffcen of varaon over he smulaons, and he average of he samle maxma. From Table 2 we see ha he al ndex esmaor works reasonably well, w 2 s mosly whn wo sandard errors of he rue =; and s ofen whn one sandard error of =: Ths holds also for he deenden daa. The sable dsrbuon wh a characersc exonen close o 2 s, however, more heavly based. Ths s due o he fac ha when he characersc exonen equals 2, he sable law swches from beng fa aled o he normal dsrbuon whch has hn als. Thus whle = jums a he lef end from he oen nerval 0:5; o 0, he esmaor smoohly nerolaes beween 0:5 and 0; see e.g. Gelens, Sraemans and de Vres (996) for furher deals. The esmae of = s less recse han he esmae of =: The reason s ha he measuremens of second order al arameers s more dffcul, because hese are second order arameers of he Taylor exanson of he dsrbuon a nfny. The exreme realzaons are less nformave abou hs second order behavor han he frs order behavor. Neverheless, he = esmaes are ofen whn wo sandard errors of he rue values. The omal number of order sascs m n s a funcon of he second order arameers, and hence s no surrsng o observe a smlar behavor as for =: For mos economc uroses he usefulness of our rocedure resdes n he esmaon of ou-of-samle P; Q combnaons, raher han n he recse value of he al ndex =: Table 3 reors quanle esmaes ^x on he bass of he esmaor n (34). We reor he quanle esmae for he borderlne n samle robably =n and he ou- 24

25 3 ESTIMATION AND SIMULATION 3. Mone Carlo Exermens Table 3: Smulaon Resuls: Quanle Esmaon Dsrbuon Predced Samle n True mean c.v. mean c.v. Suden :6 653:6 : : : : 47 Suden :95 :54 : 8 3:68 : :450 5:97 : 23 Sable(.40) :8 33:4 :47 435:2 : :57 282:8 : 32 Sable(.80) :398 2:0 : 2 60:68 : :028 32:66 : 26 Tye II Exreme () : : : 39 Tye II Exreme (4) :409 8:547 :08 9:875 : :067 :35 :0 Log Pareo (4) :65 7:02 : 9:35 : :09 23:76 : 3 Suden 3 SV 0:;0: :598 8:63 : 2 24:9 : :432 28:07 : 26 MA Suden :452 22:3 : 26 25:52 : :243 32:7 : 34 GARCH(2.0) :05;0:8;0: :06 : 57 8:46 : :0 : 7 GARCH(4.0) :05;:6;: :737 : 30 5:548 : :94 : 37 25

26 3 ESTIMATION AND SIMULATION 3. Mone Carlo Exermens Table 4: Performance of GARCH and Wors Case Analyss Wors Case GARCH Exreme True x =5000 7:60 7:60 7:60 mean of mn,24:32,9:06,7:54 s.e. of mn 20:99 8:60 4:8 max of mn,0:,6:48,9:60 mn of mn,3:07,58:48,33:77 of-samle robably =3n n =5;000 : To esmae x =n, he fnancal ndusry ofen uses eher he so called wors case analyss or hsorcal smulaon. In he former case, one uses he maxmum or mnmum samle realzaons, and n he laer case one uses he average of he exreme realzaons n boosraed relcaons of he samle daa. Table 3 reors he average of he maxmum n he 250 smulaons,.e. he average wors case analyss. As can be seen from Table 3, hs rocedure nvarably carres wh more uncerany and bas han he sem-aramerc mehod. Our sem-aramerc echnque s n essence a mehod whch exraolaes he al shae of he emrcal dsrbuon funcon. Hence, reles on he conrbuon of more han a sngle order sasc, and hereby reduces varance and bas. In fac, one can show ha he average wors case analyss overredcs he quanles, and ha f our sem aramerc mehod overredcs as well, he former mehod s necessarly more uward based. From Table (3) we see he quanle esmaor ^x erforms decenly when judged by he mean and he coeffcen of varaon. The coeffcen of varaon (c.v.),.e. s.e./mean, s more convenen han he sandard error for he urose of comarson, and hence we only reor he c.v. Snce he rue mean s unknown n he GARCH case, we use he samle mean. The c.v. does no change much when movng from = =n o = =3n n he exreme value echnque case. Moreover, he erformance of he quanle esmaor s farly homogeneous across dsrbuons and sochasc rocesses n erms of he c.v. The wors case analyss shows c.v. s whch are conssenly larger, a leas wce as large, han c.v. s for he sem aramerc mehod. A comarson beween our sem-aramerc and fully aramerc aroaches s also of neres. We se u an exermen where he researcher s gven he knowledge ha he daa s fa aled, bu no he correc class of aramerc models. We erform one exermen wh samle szes 5,000 and 250 relcaons where daa s generaed from he Suden- SV 0:3;0:68 model, whle he GARCH(,) model s esmaed back. The esmaed GARCH(,) model s subsequenly used o generae 500 seres of 5,000 observaons where each smulaed seres s used o evaluae one smulaed esmae of he ^x =n quanle. Agan hs aroach s comared wh he sem-aramerc and he wors case analyss. 26

27 3 ESTIMATION AND SIMULATION 3.2 Asse Reurns and Value-a-Rsk From Table 4 we see ha GARCH ouerforms he wors case analyss, bu he semaramerc rocedure s sll beer and has much lower varance. The reason s ha he al rocedure s less rone o model mssecfcaon because does no rely on secfc dsrbuonal assumons. I only uses he lm exanson for heavy aled dsrbuons. The sem-aramerc al esmaes herefore do no have o serve wo masers by machng he arameers o sasfy boh al and cener characerscs of he model. 3.2 Asse Reurns and Value-a-Rsk The leraure already conans al ndex and quanle esmaes for fnancal seres. Tycally hose esmaes are obaned from alyng a grahcal rocedure for locang he sar of he al, see e.g. Embrechs, Kuelberg and Mkosch (997). We used a se of he hghes frequency daa from he Olsen comany on foregn exchange rae quoes. The daa se conans.4 mllon quoes on he USD-DM so conrac from Ocober 992 o Seember 993. These quoes are aggregaed no equally saced 0 mnue reurns, and we use he frs and las 5,000 observaons. The daa are as n Danelsson and de Vres (997), and a comlee descron of he daa can be found here. Smlarly, we use he frs and las 5,000 daly reurns from he daly S&P 500 ndex over he erod 928 o 997. For hese four daases we comue a number of sandard sascs. The mean and sandard error are annualzed by usng a facor of 250 and for he sock ndex and FOREX daases resecvely. In addon he skewness, kuross, and he mnmum log reurn are reored. Subsequenly, we aled our esmaon rocedure o he lower al of he daa, and reor esmaes of =; =; ^x =n ; and ^x =3n : Beween brackes we gve he 95% confdence band. From Table 5 we noe ha he FOREX daa are farly symmerc, whle he S&P daa clearly reflecs he abysmal erod of he 930 s, and he fa years of he resen decade. Neverheless, he black Monday of 987 s resen n he las 5,000 S&P reurns samle as can be seen from he mnmum. The al ndex esmaes hover around 3. The second order arameer aears o be larger han : From an economc on of vew he neresng esmaes are he quanle esmaes. Table 5 reveals ha he rsk of nvesng n socks has come down over me, and ha he rsk n he FOREX marke s farly sable over he erod of one year. The oher alcaon concerns he Value-a-Rsk (VaR) on a orfolo of asses. Fnancal nsuons have o regularly assess her caal adequacy o cover adverse marke movemens, and have o reor a number whch reflecs he (mnmum) loss of her orfolo f a negave reurn n he lowes quanle maeralzes; hs loss s called VaR. For examle, he lowes quanle may be he 0.5% quanle, and he VaR s he loss ha maeralzes f exacly hs reurn maeralzes. Several mehods are used n racce, e.g. hsorcal smulaon (HS), he J. P. Morgan RskMercs, and normal movng 27

28 0:276 0:346 0:30 0:374 = 0:24; 0:3 :3;:36 0:27; 0:32 0:33; 0:40 ; :97 :05 :60 :760 = =n,3:3,8:68,0:690,0:668 ^x =3n,8:0,2:60,0:958,:0 ^x,6:0;,23:,: 4;,5: 9,: 86;,: 7,: 88;,: 4 ; 28 Table 5: Daly S&P 500 and Olsen DM/US 0 Mnue forex. Lower Tal Daase S&P 500 S&P mnue DM/US 0 mnue DM/US Observaons Frs 5,000 Las 5,000 Frs 5,000 Las 5,000 Annualzed,: 75 0:0 2:9, 23:0 mean 26:4 4:7 9:2 2:8 s.e. 0:089,3:26 0:464,0:084 skewness 8:43 77:96 7:886 7:90 Kuross,3:66,22:8,0:693,0:655 Mnmum ;,:8;,7:,7:82;,0:9,0:62;,0:85,0:58;,0:9 3 ESTIMATION AND SIMULATION 3.2 Asse Reurns and Value-a-Rsk

29 3 ESTIMATION AND SIMULATION 3.2 Asse Reurns and Value-a-Rsk Table 6: Esmaon Resuls: Average Number of Realzed Porfolos ha were Larger han VaR Predcons Tal Percenage 5 0:5 0: 0:005 Execed Number of Volaons :05 Execed Frequency of Volaons 20 days 200 days 3:8 years 77 years RskMercs 52:457:39 0:652:73 4:852:06 :58:29 Hsorcal Smulaon 43:240:75 3:692:39 0:95:03 Tal Esmaor 43:4:0 4:232:55 :06:3 0:060:23 Daly observaons n esng = 000 over erod 9305 o Wndow sze n HS and TK = 500, nal sarng dae for wndow Random orfolos = 500. Sandard errors n arenhess. Probables exressed n recenages wh sum=00% average. HS s non-aramerc and uses he lowes quanles of a hsorcal samle. RskMercs combnes he normal nnovaons wh an IGARCH model. For 500 randomly weghed orfolos of seven US socks,.e. J.P. Morgan, 3M, McDonalds, Inel, IBM, Xerox, and Exxon, we calculae he VaR on he bass of he frs wo ndusry sandard mehods, and our sem-aramerc aroach. To mlemen our rocedure on he orfolo, we creaed vecors of orfolo reurns by smulang from he orgnal reurn er ndvdual sock and hen combnng hese on he bass of nal orfolo weghs o arrve a a vecor of orfolo reurns (hs s he same rocedure as followed by hsorcal smulaon.) The lengh of he vecor was se a,500. To hs vecor we aled our al esmaor rocedure. We hen reserve,000 days a he end of he samle, do sequenal one day VaR redcon, and coun he number of days where he realzed reurn exceeded he VaR. The resuls of hese dfferen echnques owards he VaR roblem are n Table (6). We can see ha he IGARCH normal based RskMercs erforms well a he 5% level, bu s unable o coe wh lower robables. Usng he emrcal dsrbuon funcon (hsorcal smulaon), gves resuls whch are smlar o he sem aramerc aroach n samle. I can no rovde esmaes for he 0.005% level, or an even whch haens once every 77 years, snce we only have wndows wh 500 days. The sem aramerc al esmaor reforms well a all robably levels, exce hose whch are far nsde he samle. 29

30 A MATHEMATICAL DERIVATIONS 4 Concluson In hs aer we develo a comlee sem aramerc mehod for esmang large,.e. borderlne n samle and ou-of-samle, robably-quanle P; Q combnaons for heavy aled dsrbuons. The key arameer o be esmaed s he al ndex whch deermnes he al shae and domnaes he large P; Q esmaors. For boh al ndex esmaon and large P; Q esmaon, s essenal o know he number of exreme order sascs ha have o be aken no accoun. In hs aer we solve he hhero unknown deermnaon of he omal number of exreme order sascs by means of a wo se subsamle rocedure n combnaon wh a conrol varae ye mehod. The subsamle boosra rocedure s essenal for convergence n robably; a full samle boosra would delver convergence n dsrbuon. The heory s resened n a comrehensve and self conaned manner. In addon o esablshng he heorecal roeres of our esmaors, we subjec hem o a number of Mone Carlo exermens, where we smulae from a varey of heavy aled dsrbuons and sochasc rocesses. These exermens are desgned so ha he fne samle roeres of he esmaors can be esablshed for mos DGP s. For he mos frequenly used fa aled dsrbuons, and sochasc rocesses such as GARCH, we demonsrae ha he esmaors have good erformance. There are a number of roblems n economcs where exreme value analyss lays a key role. We focus on wo alcaons n fnance. Frs we nvesgae he change n rsk over me wh of he daly SP-500 ndex, and 0 mnue foregn exchange quoes, hen we aly he mehod o he roblem of deermnng caal requremens n radng orfolos wh he Value-a-Rsk mehod. We demonsrae ha here are large gans o be made n recson f one uses he sem aramerc mehod advocaed here, raher han a fully aramerc or a non aramerc aroach. We are currenly workng on several oher alcaons of our al esmaor, e.g. he roblem of caal deermnaon for fnancal nsuons, omal hedgng, oons rcng, and nra day rsk managemen. Furher heorecal work s focused on he rae of convergence roblems n regresson esmaors. A Mahemacal Dervaons Before we can roof he frs wo heorems we need he followng calculus resul. The argumen s used reeaedly. Lemma Gven he model (3) and condonal on M!, E ëu k s n ë = k! + bs, n + o s, k + k n ; (38) 30

31 A MATHEMATICAL DERIVATIONS for s n!as n!. Proof. From calculus we have he followng resul: Z log x k Z x,, dx = s, log y k y,, dy s s Z = ks, log y k, y,, dy = k! k, s, Z = k! k s, : y,, dy Now aly hs resul wce o comue he condonal execed value of log x=s k when he densy adheres o (3). Hence, he condonal execaon n (38) follows from as s!: E ëu k së = = Z log x s k f x dx, F s s " k! +bs, + bs, k + k + o, s, : To oban he mean of w k one frs akes a frs order exanson of w k n u k and u k,. Subsequenly, usng he fxed hreshold nerreaon of u k s n, one needs o comue he condonal execed value of log x=s k : Ths s faclaed by means of he calculus resul from Lemma () above. Proof of Theorem. Develo w k s n no a frs order Taylor exanson, of he rao of he wo argumens u k =k! and u k,= k,! around he on = k ; = k, and wh remander o: w k = + uk k, k!,, k,2 uk, k k,!, + o: (39) k, By alcaon of Lemma () we ge asymocally E w k s n, bs, = k, From hs resul he clam n (6) s easly esablshed. + k, k,2 bs, + k, + o, s, : 3