USatnadbleCrMSeUndniUvvedodelingofMarkettitValueatRiskaniversityofCalolizfaorrnRiaa,chSeavntaBarbaraenrisviteyErsdituyarodfoKSacrhlswruarhtez,GermanyColoIrOraoicdnftoaoCKalhifionrdnaian,oLvaosAngelesAbSbecrshto1o5l,o2f00M1ines ienvgalTmuhaaterikopenatspasenhrdoewxcartmehdaiinttesVstaathlbueleuasteRofisskta(rbValaceRtP)-issatmribpulet-ioannsdinfomreocadsetl--ammlseaotl"hdoemdvoefdlooreplsisnimagunfloeawrtihnteiggchhpnoviraqtlefuuoemleisfooaorrkfeesttthiVemaVaRtaiRnmgoccdooenrl irnedgleantoicuoetnp,\rtnniocerlwe-1Iidoenupstencrdaoseendst,ouafnccdrtesidkoietwiendstdruepmeenndtesvn',ibauntdioansss:essinesdeppoerntfdoelniot,VsayRmimnevtarric-Oneofthemostimportnanttasksof -taorenetdpdaoitteqtnuietaisal,iltcyloomssusmecshotdahistaiatesrm,aetigixnhcgthamoncigcgeurarrtabitoeencsa,',eoronprmo atrs-Croisfosisskk(VaR).Inthetextbesleodwm,tehtehomdoalrokgeytVfoarResitmimpaliteisonthoefrisksistheValueatKn ,,,oetshpaencikalKlyr,-vIaotoofvUanoifvBerrsaivtyo
aomwtfcielaaalAFsnegossxeivrcmtVteeshaeanaRdilstlcy,$ormnwee $icoghr2haeemnxsgtaielmlpsioopslesi,bilfetlhoessdaoivleyrVaaRcerftoarinagpievreiondpoofrttfiomlioeinnathtethmea9rk5etpecrocnentcon dencelevel,it,2amVailRlion=5VpaeRrtc;,one-daylosseswohne rdeenceinterval:Pr isde nedastheupperboundoftheone-sidedwcisthecon dencel[e vPel( an)d< PVa( R)]==1 Pct;(1)(wrhiettehurn)intheport( )istherelativechangeTrehTeP foliova luPeoverthetimehorizon .pcpe(stt)se==n l,oganSd(tt);isSt(ht)eicstu(tr hr)een=ptoPtritm(ftoe+ v)a luPe(att);t,thetimeperiodis[t;T],,,hioteMensc, erentwaysofthTteehCevaatrrrailaodnistciiemonualaltmioent,hoadnsdasrteretshse-iHonsqbuoostbaweaniremsveedr,o nnatnhceianlodramtaaloaftsesnumvipotliaotnetfohrenthoermdaislittryibaustsiuomnp-otcfioov ,.ltvialpetsoiosoenfsd eixsPthriibbwiutitt\hiofanata",lLopgarynoddoouf,ccatonnod-fjmouinrTntbhmueollve(xe1memoapirniertiducraranlws .ibs9nate9aGicn7nkit)gvsaiesmonnfdien ttahahrsedmosedunsqs'mautaadpustotrseieeontans(,acetrxiesdcfeaitscstqoukrauylitrteyovsacilhsua,)hoeliimationofdistributionalformsofportfo-fr1Ineftee2rrSSrenekonaeepclweGrCsni,uareepsstdesteeoitdGRinsucitispraktioblMnud,otaita,thestableParetiandistributionsseemtobethemostins,mFoinstgeprr,odFanoeinonlnsduga(neBl1cr9he,m9daa8ton),dBd(B1ias9ht9ar7itb)(i1o9n9s7o)f,)ry;hoaMweonscg(kC1ia9una9lndl6od)cR;haMRc(a1hitc9eth9vne6iv(k)1;,9(MR1995ai9t)ct;4hn,Ceivkh1,o9aab9nn5addan,RoPsvlaaea,cnoMhCdoleealv1mlt9em(e91v59i,b9tt1M)e;,1eit9Mot9nna3inkBd,aeanlnbkdrionRtga(Sc1hu9ep6ve2r,(v1i19s99io66n3);a(,F1a91m9996a)3.(b1,96a5n)d;2a(1998).aand1993b);Mittnik,Rachev,and
mepretaaeprrrToaskrm:hetteee dtaepntahrasdpeirwlecreirlceleoddxindneietaxm p,Vrimc ,eilesustssa,eb loefstableParetiandistributionsinmodelinglodistributionsaredescribedbyfourparam-isnadnededs,kescwtaatnbieolsens,moafondddieslti -snaamncpilael-raentudrnfos,reacnads,t-tehvearleufaotrieo,\inbourtmioanls"pmososdeeslsinthgefoarddhiitgihvitvyalpuredospoefrttyh:eaVlainR-Isut\niisanteibdtdvhieilteifpoytyercnpaeirsnsoetdpiemoexfrat itynindpigseropapvegoinadienrdtefsonaaltinosiatnrlaybtlitecafbroalrenmd(ouormlajsovifanortrilaypbaslrteaamwblieett)herrtashnedosmamvear -isstkruwmitehnotsu,",\symmetry"and\skewness":fmioonifrnedtdVephreooaprRodteffnofoutdhlsireieosnnpgtrim,,ruwemeaepcnoptnlsy-'tsphyeemrvmisaeroitarrgnicacned-iezcpeoedvnaadrsieanfnotcl, ivmnniddaaoersmaal itnyanascseu-omripetniotendfdoerscthrieptdioisntroibfusttiaobnleodfei stmnraiebntuhcitoaidoln,-fernatmpeowrtof8roklpioroefisnetshntetr,uormenseepn-fetacsc't:tivoserylmym,meveatlruicatainodnp2tioorntf1o\0lioNdicsroceurdsimstesriasckrle"-Fro4mSeetahede nitionofVaR=VaRt; ineqVuaatiRtainedfromthepron(1),theVaRvaluesareob-lsoGamrowsokbiaanbdiliRtyachdeisvt(r1ib99u6t):3
wp(bwpohdretreF P1 c=F(x)=Pr( P6x )P( VaR)= Z1VaRfisthecumulatived P(x)dx;istributionfunction(cdf)ofIfff)otolhifoe ,andfinthep P(x)istheprobabilitydensityfunctiontrueibtriuoetnvi,oienVwaaR\sncoarnmbael",. tThraditionoarrltlfyoa,liinoScitonhnigsslisseettstAionfsgas,estihtnegldeisatsrsiebtu,twiohnicohfdaespseetnrdesonlyononeriskrze1d uc:ced,teaonu dnnitdvhiaenrgsiattathenedna(o1rrd mdacel)vtidhaitspitorenirb,cue ,heoidfpetrnhotebi lseetmadnodbfyacradtlwcnuoolarptmtuainrarnaglmsdViieassttRaerrsiibsss:uutmthtioheenndewntTrohirbemure1tailo c=nd(izsf)utrniiscb ZXtu1 g(x)dx=iotinon zZ1 cthefsotfraunnrdecttauirordnn1, (z)dz=N(z1 c);withX =z1 c + ;snXormisatlhdeepnosirttyfofluioncretitounr,n,Ng((zx))iisstthheencourmmuallatdiivse-isthelowestrwithmean andstandarddeviation ,andX tciaahnlilIsybnaesmsmsuaanmlylepratipeotphtnulairincsnabttahiaotesneasmdgiaonivgnvenentishttuceoodrcnseo and,therefore,wanhde,retherefore,VaR= X =z1 c :Y0istdhoeY0X = Y0z1esinnoittiaexlist, c ;5Iff P(x) P.
conoPlintioseoristntfVoslaoiofRmasasnetysaasrseetds,ectohmepcoomsepduitnattoion\boufilVdainRgibslpocekrfso"r,, PXXpistheportfolioreturninoneperWedenote: Nisthenumberofassetsintheporitofodl,io,ii(0is),twheheir-tehPaisissetthreetluorgn-sipnotonperperiod( =1),Xi= P(1)=Pi(1) Thawelilypi,osXicanbetrhneriskfactorthaticeenotferasslsienteai,rliy=6i1n;t:h::e;-rtthfoeliio-trhetausset'sweightintheportfolio,i=1;:::;=tri(xwnotation,XXp=XNi=1wiXi:1;w2;:::;wNX)T=,(Xp=wTX;theportfoliovar:Vianceis1;X2;:::;XN)Twhere (Xp)=wT w=X=Ni1wi2 ii+XNi=1XjNi=1 i j;6=jwiwj ijoafndretthuernijis-itoshnthateshseveatis-rt,iah naceofreturnsonthei-thasset, iisthestandarddeviationapsoratIffolialniloelapVroarcRtofmoblbaioisneradettiouonirsnsnsetth,e cijovisartihaencorrelationbetweenthereturnsonthei-thtsohfearnnVeoorjmoinaltlyvacnreioamrbmaletasrl,ilxy,d isaislt=sroib[n uoitjer]dm,,1at6lhi6N,,arRm=al distributionassumptionisThe6IftherisY0z1 c (XP);(a5sinacaseofanoption),thenalinearapprox-
wheTreh (Xp)istheportfoliostandarddeviation(thepocrmeoatvulalryriaduniss,= eiesd,kebfsaytcimtaoacrtsoimoVrtfoliovolatility),(Xp)nb(iXonp)(intfharetiissokinmroepdfleulsicnteecsaatrsoee,exvposurestonor-inadluivaitdiounaloafstsheettatiTonnhsed) -hagotesresaitcnaetaibonllynesdwisetlraidbnauatltyaiozonefsttehaneresvtmiaobloalreteematdhoedeqenulioanrtgmeodafliismttyrairbakusesttuiomannpad-ltsiesdwitDhperosvcidrinipgat nance-orientedpropertiesofeinretimesetohodsofestimatingparametersofcAonrastnadnotmscv>ari0abalnedRdissaidtobestable7iffofraSntyaab>le0Danidstbr>,issRttra2ibbaulerteahRthat1idonistfruinbcuttiioonnsst+dcbooRpni2oe=dscoRf+Rdindependenthavecand=ddenotestheequalityind losedformexpressionsforthew ehRscribedbytheircharacteresticfunctions:( )=(R)arecommonlyR6( )=E((eexxpp((iiRR ))))==eexxppn j j 1 i sign( )tan 2 +i o;if 6=1;j j 1+i 2sign( )ln +i ;if 7eOr1eft;e n=1;iRisstisthhceealilsnecddae lexpoafrsatmab-stableoreitPleitaryr,e,t o0>s<ta0b l,e6an2d; iisstthheelsokceawtinoensspapraarmameteetre,r, -L evy-stable(for <2).
dmwTioesotdirwneibrldiiunitctegaiotRoenftr heedSud cee(ps e;nt od;e tnh) lee=mRp2oi,rnictihatlesnsptaturhdaeimesse,ttaetbrhlsee,htbhaeecvoisnmtgreos1ngh<eigrh t ehnr<(itlthhaenspdtaetabhkleeosdfmitshatrelliebdruetn isoinitsys,) us>,t1h, ra=me0t,erth eIf >0,the dRricandthecharacteristicfunctionisis(t r)ib=utEio(nexisp(sikRe w)e)d=toexpf j j +i g:i s= s<tro0n,(ees Rdsasm)t .oeTb=he0,scthaleenptahreamsteatberle(rtahnedvoomlatvialirtiya)bl eaRiscalledsymmetric -inedeexxporefssstedasR= Ro,whereRohasllaowusnaitnyscastleabplaerraamnedtoemr,:ael ri t enyi=t ionanodfsskteawndnaersdspdaervaiamteiotner. set (otnilglooanwt-iiGnngagutpshrseoiapnbe)erhtdiaeivsst:iorirbuotfiothnesand !lim+1 P(R> )=k 1+2 wheTre !lim+1 P(R< )=k 1 2 ;hepk =-thabsol u(t2e 1) co s( =2);if 6=1;k = 2;if =19: nitemoment,EjRjp=R10P(jRjp>x)dx;isifp< or =2,andanc8e -sta7blelawsexhibit nitemeansbutin nitevari-
TrandSthaubsl,etdhiestsreicbountdiomnsopmoesnstesosftahneynon-Gaussianstabledistributionisin :daexlin eairscaogmbinationofvstaarbiEoailbximtaleymvwipanilrtdeiha:ebxIlfeRw1it;hRt2h;:e:s:a;mRne e,saRmie Sa n(d ip;a ri; i),thenR=iP=n1wiRiisastablereasnwdoitmh(a)if ameters:6= 1 ==,(s(ijgwn1(jw 1i)) 1+((j w 1 j+jw1j 11()j) w n++j n ) +) +s1(ijgw;nn(jw nn)) n(jwnj n) ;(b) =w1 1+ +wn n;if =1 ===,wjswig1nj (1w+1) 1 j+w1jjw n1j+ n ; +sig1 1+ +jww1nj 1n+ +jwnnj( wnn) njwnj n;SincetheParet 2(w1lnjw1j 1 1+ +wnlnjwnj n n):timateriskbyvarioa-nstcaebalenddisdterpibeuntdioenncsehabvyecionr rneiltevariances,onecannotes-eovxanirsitwAtahnehcrcea>>0andb>0there1andR2areindepaRen1d+enbtRc2o=dpcRh+tDhatt(heIvfeactroarnadroemstvaebclteowriitshsttahbelesawith >1,,portfolioreturns)-adnidmsetanbslieonlaawls,vaencdtoirstihsegmivaeinnabdyv:antageofthefoliobyassumingthat(R1;:::;Rd)isaunct (R1;:::;Rd)ofaport-8
(a)if R6=( )1,===E R( 1; 2exepx8<:p( i T;R::):; d)SZd ( Ts 1 isign( Ts)tan 2 (ds)+i T 9=;;(b)if R=1,where ( )=exp8<: SZd ( Ts 1+i 2sign( Ts)isaboundednonnegatiln Ts (ds)+i T 9=;;incnhatamergaerdcatneadrissputeinccittfruavnlecmcttieooarnsu(s2Sd)avnedm ,susreis thies(a)if .Then,the6=1 R( )=exp( j j Z0 Z0 Z02 jcos( ; )j (1 sgn(cos( ; )))tan 2 dH( )+i T )(b);if R=1,( )=exp( j j Z0 Z0 Z02 jcos( ; )j where(1 sgn(cos( ; ))) 2ln( jcos( ; )j)dH( )+i T );sin d 2focros dg iv1e;n sbiny i1ts::p:soilnar dc o3ocrodsin adt e2s;,:: :( ; sicno s1 :1:):,swine dde n1o;t esin 1:::cos( ; )= dY 1i=1sin isin i!+ dY 2i=1sin isin i!cos d 9++1ccooss 1dc o1 s 1:
lineIafr co>mb1in,athen isthemeanvector, = tionofthecomponentsofastablevectorRsatis estherelation:(wTwR)= SfpolioViwewithingweRigh=(w1R1+ +wdRd)=SZd wTs (ds):ts(wR11;;::::::w;Rd,d ) a(swtThRe)vwecitllorbeofthindividualreturnsinaport-s eiemndsdidleeain srcntlyered,ibtoounatbetiohocnevasent:,rua ds eit=tiohne a cloiivnsatrtehiraeptrivoeantarittaiootineosnot,[R1;R2] = 1@ (w1R1+@w1R2) w1=0;w2=1=SZns1s2< 1> (ds);re)p.(EeRnTs1dht;eieRnmm2c)eaasittstririoxuancotsfu orcsefovavPamercaiotarontarigomn(t1hsee[t<Rienird; Rsivji6od] )fua1aSln6trdeatibux6r<lnekds>,D=jxjksgn(x)(signedin1i6tshtejrip6borudt,,ewshtiemreattihnegptrhiemsatrayblceonpcaerranmiesttehrseatnaidlbtheehiar2ap-btieynurdtieoxns(.thIethinadsebxeeonfsptraobpiolisteyd)tinhsatteaitdisofm otrteinugsetfhueltwoheovlaeludaitedirevctiolyritmseuthseodofis\ccelanitmere" -udeiss:triitbuistidonesimgnoeddelifnogr. Wttiengsutghgeesotaomthetahpopdr,oawchhiecsh: arebasedontheasymptoticParethWefeousrhpaallracmoentseirdseorfsthefollowingestimtao-pC19h9o5bba)n,oKvl,ebMaantoeve,vM,Meliatmtneidk,,aannddRRaacchheevv((11996),GamrowskiandRtaacbhleevu(n1i9v9a4ri,a1te99la5wa,s,asnedee(1st9i9m6a),tiMonitotnf:ikthandRachev(1991),Rachev9a9n4d),SKeonzGuubpotwask(1i9a9n3d).RFaochretvhe(1m99u4lt)i,vMarciaCtuelcloacshe((11e991n399dS43ee,bne)1c,9eofstableedspisetcrtibraultemderaestuurren,s,thseeeinCdheexngofanstdabRialicthy,evth(e19c9o5v)a,rGiaatimonroawnsdtestsforde-sRe9c5ataci,,aXnidn(11999963)),.Heathcote,Cheng,andRachev(1995),MittnikkiaannddRRaacchheevv10
:theHill,thePickands,andthemodi edunconditional ^orwH1i5isdescribedbyll=k11jP=kktbhpeorien1ln(Xn+1 j:tXn) lnXn k:n;sj:wnhdeerneotthesetthaeilja-trheaor\dstearrsttsa",k::i:s,cXonm1p6;theintegertrIthuaetdioeno ;:buittmifuitstisbteooadsemqaulal,tetlhyesemstaimllastootrhiastnXotna ckc:-ttbisepstriomvaedtotrhiastt,hfeorcosntadbitlieonPatoreetixapnlidciitsltyridbeutteiromnsi,nethteheorderstastaisdtvicanXtna gek:,vastinrtPeei ^racnskoanteiaohvrdbreeie, ^kH,iPlladoloeelslan,oatndHRillaechsteivm(a1t9o9r8ics)cfoonusnisd-inenteosttohimfreodarHetrnielstl>oeasor1teb0imt;pa0ari0not0oduru(nsacebeeriedaFtsoheingdeluyPerset1im17a) necersessaermybtloehitasveaseynmorpmtootuicsifcokrannd>sa1n0d0;t,0f0o0r(esxeaemFpilgeu,rwei2th18 )=.-,PTaihncked=s\oersitgiminaatl "+1ic:nka ndXsne stihemodi edunconditional2km+1a:tnor19takestheformln(Xn)l n2ln(Xn 2k+1:n XnensatmiTmehadete\Ptohifcek amndsestimatorrequireschoiceoftheoptimalk ,4wk+h1i:cn);4k<n:thetrueunknown .MittnikandRachev(1996)proposedanewthaidleepsendsonisoodbi teadineudncboynadpitpiolynianlgPthicekannodnslin(MeaUrlPe)asetsstqimuaarteosr"m,e ^:Mittnik,Pke2=X2;X1 1k1+";X14eFroerdetaaoilleslloantheHill,Pickands,andthemodi edunconditionalPickandsestimators,see11v1:56nHG6iilvle:(n:1:9a76s5a),lReaocfhoebvse(1rv9a9t8ico)nsanXd1r,e:f:e:re,nXcens,!ifmi1gku(rfkerX(on1mn,:nt1,htetohte)=n)!nrnu0/=vSea5ojl0, ,saotttifhcttehhiseeseaeeqmssuttpaiimmlletaaosttioozXrrejrf: ^i=re1=s0,v ^0e0(rk0y;(nstm)h;eanl),airfsgneu ^doer(rsekath;ns)msaulglgveasltusevaolfuke,otfh ^,twhheereesatism a=(21a,9tt7,h5feo).rtrvueery ,otfhek,sa mplesizeisn=500,000,k=1,:::,n/2=250, .
F i=gu1re:91:Hillestimatorfor10,000standardstableobservationswithindex Fi=gu1re:92:Hillestimatorfor500,000standardstableobservationswithindex12
XaMk11it==tni k,X2XkPkn n 2kk++11::nnXXn n 22 2 kk++11::nn ;X2= Xn 3k+1:nXn 2 3k+a ol1Xn 4k+1:nXn 2 4k+11::nn ;e1lla ,;andk2= 34kk 11 :fSatruldeisessddeepmenodnestnrtaotend atnhtdhaatRn ^aiMcnhUtePhve(isc1a9as9pe8pocr)foexfoiitmuhnedrtthheatHtihlleoorpPtiicmkaaalnkdsfoerst ^ 2[,):nnqgMumaonedttiheleolidanspgporfoeascthimesa,tcihnagrasctatebrliestpicarfaumncettieorns(wCiFth)spteeecsartrivhaFmenmadaimqetpuotareerorssasp,eneoaxdrntdfhsiiRebysommitloelamfxd(se1iatmt9ar7usibcm1mle)(tiMuhchteeLbioi )(fs1si9tga8nb6e)idliotf yoer reev> ss0ett%iidmm,qaa7utt5ooa%rrnss,tiaoalenfredatel9dlc5ehf%roniuviqqreuudseatu,ansbwtiilnhleegispchfaurnpacrmtoievotindesresdo,ffcoo rnv seiss2taemn[0tp:6ale;n2dq:0ua]asanyntmidlep st:o2ttihc[a ell15y;%1n,]%hdae-l,inthetails(belowthe5% im95tta%quantile),(1972aand1972b)proposedseveraglCthFemsaemthpoldesCFtotheiKamannrouedduMmctWoradistance,theminimumr-thmeandistance,andthemethodof:((h11e9989e08s,)ti1m9o8d1i) (r1aa9nb7sle1fo)premavraatlmouae\ttecedernstthderei ",sRvaiachaevfa,satnFdouPraieorletlrlaan(s1f9o9r8ms)RNa22oc10lhHFaeonevra,(taa1hdn9cddo9itt8Peio,aan)Coarlchlieolerslenamtfgiec(,pr1efau9nun9tcnd8eec)dsRti;,PatsohecnaeheuaeAlvdsroneadnsaittyinutseirnmgenduiamteerpicoainltasp-pbroyxliimneaatrioinntoefrpinotleagtiroanls.(n1d,9H(915o9))m;bF,euaenrdveLregietrchan(d19M75c)(1981);Mittnik,13
():(caXaRrlaet6cfatieristicfiunngcttihoenFinouarnieerigThrbaonrshfooormdo(fFtThe)omriegtihnotd= setttinhgeVfx le(ta1s))tgi6maisPteth(ej:Xreja>lpa1a)rt6ofKatR0hae(1ch arRaecftefrXis(tti)cgf)udntc2t3io,nfofrx(atl)laan>dt0, =oesse1a=c(c1u thartastient1a)il<es1t=,iswehesicthimlaeatidosntoofathneacdheaqruaacteristicfuhnecctoionn- >=1r25thedistributionofreturnsrissymmetrteevaluationofis:thecharacteristicfunctionofrisgivenbyfr(t)ic- -stable24,thattionsR=,thn1ei=r=ni1, rgii:vreFanopropblrasoeragrcvehaevtsaiolfunRessr1o;f:n::(t)=,e; trhjnce,wtj =eti cEfbueyinrctthti=eonseaoimtf p oljbecstmje re:vaaIn-fteristicfunciPntionofthecenteredobservatonsidertheempiricalcharac-tf^hRe;nt(ht)eo=reRtieca ln1chkP=anr1aecitReristicfunction,fR(ito)n,sis:rf^eRa;lna(tn)d=pons1iktP=inv1ee,iwRket:haBveecathuasetc^suNcohwthtahteRM0pr of^bRl;enm ofkfRte (stt=;i ^m;n1ac^tkP)i=n n1g=co sRM(0aR nn1kdt)=n1icsosreducedtodetermining ^andRwheTSrtehi=StreeeippM r21eisasuÆcientlylargevalue.(Rkt) e (c^t) ^ dtisminimal,.aliGzaitionoftheFTmethodisperformedinthefollowingsteps:.r,Cio=nv1se,tnr:u:tc:ht,enassetreturnsr1,:::,rn,computethecenteredreturnst,hewhsaemreprle=chn1aiPr==sitioojnn (0,,j= 1],f^26::.:, , istnheXmaximalfunction(tj)=1k=n1cos(Rktvj)a;lueoft, isthenumberof2243ETmhepliarisctailneevqaiularleifteyreisncbeys,LseememMait3tnoikn,,SDhoigryaagyneovgl(u1,9a8n4)dChenyao(1997).in22 65FA>osr1w,ceowmhhapivcuehtaaimltdrieoepannldiceyepsusoerubxpgsiosgetsreeevsnste,csde,wthioenafthathal lve ed nocraehsnstocsnmiaoltrpeltauyrnadsaigtan,i ctatnintgroalne en= =
^andc^suchthatjP= 1 n1kP=n1cos(Rktj) e (c^tj) ^ is -\Ia\rnennsttoatuhaWrlrbyminlzseeaesde"icl"tcsoVhttnVaerioiirbnupwtrieoopncoefrnotslilidaRuRcmtmeaaenasausluyrersemeodissmweeraspaapslstytaiabnblgeleVaRmodel,\stable" theYen/BritishPound(BP)exchangerasets: tthheeDBPeu/tUscSh$eexchangerate,ate, theS&P500Minadrekx(DM)/BPexchangerate, tthheDAX30index,, AttheeNCAikCk40index,shheorDtodweseJciro2inp2et5sioiCnnodomefxm,thoedditaietsPriceIndex(DJCPI). nitionofVaRinequation(1)R,,VaRt; ,areao1nftheFTmethodweselecPterd[ thPetf(o ll)ow<in gVgraidRstt;e p]s ht1: if0c;6(2)t:>0d;1hn;tah=mt=e2d02 t0=h 1a=0t1000a00p0;>ohr1dF;eThrt-tTo=aei2ml0p (Fh=Ta1s0Ti0z)0e:.tiThfeh0eta6nilutbme6ehrai0cv:ai1ol;rrh,etwsu=eltrse2t a0n6r ee=d11;re0tph0ho0ter0=tm0e;de2si0 =50000;ififhn0ns:e1eacrt6itont=
SYBDeSD&ePAMrn/iX/P/eUsBB3S50PP0$:suFem666ri7212vnb3858aae2575tnriocoinafslda71t111......7r77e74ie440---p-1s11e1......(nDcy)CNDAiJkCCkPe4i0I225DDDDaaaattttaaaassssttttrrrreeeaaammm824677351068171...010402...688570---111...333000... edhenocriezolenve l,i. ,t( )istherelativechangeintheportfoliohootrimaRniFmzoeoernntthhteopveurrpthoseetoimfteehstoirnigzoVna Ramn dodPtetli(ss )thn=eanccuRrtr; ,itsloy,witeitake =,seifttohcehtoiomseet e1s,tiRmta t2e,V::a:R,,ethsteiTWmsahaemetoepdblteeamsinizpe\ sytHainebgnVlwa:Rt=VaR(Rt 1;Rt 2;:::;Rt lw):(3)lwparametleecrt"ehi,,VaRis(e\weentdiorensoatmppolientofooubtstehrveaptrioensesn,,lewt=.N,whereNisinnormal")VaRmeasurementsatthecon dencelevelc(tiw)o steps:(ii)cattlcinuglempiricaldatabyastable(normal)distribution,(pMarLa\)mS,(teFnatoobeurlremrs"iaeator li)nftTdtdgreiinansangstirsVaRasthenegativeofthe(1-c)thquantileofa ttedstabletiifisboerusimmtiopnan()oe,rnatneddFuosuinrigerthTrreaensmfoertmho-Tdas:ilm(FaTxTim) mbcn,yp>.,FftoThreaosnukdrewFdnaTetTass spetrarteiirensaspgm,owntehtedeeiransgsucmone tdheantcediisnttreibrvuatlisonasreofprreetsuenrntesdarinel ociastieoqnuaplartaomzeetreor. atdeesnwceerinetceormvaplsut(eCdI)apfoprlytihnegFthTeaSnTABLEd2287TEvhaeluSaTtAioBnLoEfpparroagmraemterissdofessctraibbleeddiinstNriboutionsisprovidedFTT1la6n(1997)..
%esdhCouwI;sei(ndiig)thabootstrapmethodwith1000repli-inastuÆacsieetntoffo1r0o0b0tarienplicationsis:(i) nlywovuharorsyeexinpbgeorbuimonudensndtcsso;,i(nsetsof1000replicationsgenerated:(ii)ng95r%eliaCbIlefo9r9% ici)idveadryuinpgt9o9t%woCId,ewciimthalinpsoiginnts;95%CIfor :Parametersoi cantvariationofleftSeriesMeNormalfstableandnoSren/BPtmaablledensities -tMhoed- FMFTTLT11..6647[[,,11..6668]]--00..023[,[,
..823[[,]..002270[,,311..8799][[,]][--00..001550,,],00..005671][[0..56902[00..,,00..6602]][..20042,]][[00..55665,,700..5589]][..177,9[,611..8857]][[,][.69,][--00..0157042,,]00..67908[--00..035934,,000..100931][][][00..06687,,00..7734]]]FTTT11..4544[[00..066..7656,,29400..7727]DJCPI[,][,][[00..,,900..662][[11..,,011..4677]]4002..0120,2[[--00..127,00..110320]][[--01..44540182,,00..381660]]][[,,]]FTT11..5569[[11..15532,,811..6667]][[,][[00..0033..4333,,500..3379]][,,00..120][[11..,,911..5659]][,]][[,,] ]rightbelowtsthweeesrteimcaatlecsualrethe95%CIs,thenextCIsarethe99%%%%(95%)VaRwasdateetdermatinceodn adsetnhceenleevgealtsivce=of9t9h%e1a%nd(5c%=)iFnotregcraallcurelaptriensgensttaatbiolenqfouramntiolefstweusedourprogram,builtontheccsatonamdbpl9ea5na%yndoVunaroRrcmoemsatlipmVuataatRetisomanersaewsruepohrteedcuimnuTlaatbilveesd3isatnrdibu4t,.. dailypriceliecvaell,s,sof:30 Biad essatettsiielmyadrraeetecmetuodmprippnruast,readmbneytoserurmbst,arla,ctainngdstabledensitieswiththeML,FT,andFTthee1m8piricalVARfromthemodelVARestimatesT.
SYeerni/esBPTablEe3:Empirical,Normal,and9Stable99%VaRestimates TheCIsrightbelowtheestimatesarethe95%CIs,the[,2I2..442734]][[22..328828,,23..807305]]dta ^hniYsedep AnlFVaTdailyempirical,normal,andstableVAR*sarethe99% %and95%,errettourinllsu,stdreantseittiehsa,tandVaRestimationarecV;o=,evsuÆcientlynarrow,wceonsh odwenscteabinletedrveanlssitfioesrtahleuVesaRofcon denceintervalsfor ^Yen;FTandsR9t5a%abtlesleeavenemdlctn=oo9rbm9e%alolnimsgortedoaelltlsyhseienparatehraaetbtealeyitlwfaoerereesntitmhaete"staoibl",cwencedodmtahpteacr"oecnpe nedrteefonrr"%eRr*Valan,uRtmhgbereerssattaaerberletthhemannoedgteahtleiivneegmv(apMlui1reL,FT,andFT=T9)5p%roavniddecd=e9v9a%%VAVRaeRsti(mseaetesF,iVgAurRe*s=3--1V0A,
SYeerni/esBPTablEem4:pEirmicapliricaNl,oNrmoarmlal,anMd9L5S%taVblaeR95%FSVTtaaRbleestimatesBDPM//UBSP$1F [[,,11..014876]][0[00..,,511..133229]].978712[0..,4[0898,11..057929]][,61[[[,1..155888]]SD&[,,00..784994]][0[00..,,801..849148]][[,]CNAACX430011..58018911..69243311..474596[[,,,111..411][[,,911..452033]].550303]][[11..4309555,,211..562510]]2D2iJ5kCkPeiI11..80566611..92279411..70331[[,,11..893474]][,[[,]][,,12..823890]][1[11..,,02..501222]][ TheCIsr[,,11..024070]][[,,111..168185]]3mttsheaetnhDdoAd5Xs):.3i0FghTtbelowtheestimatesarethe95%CIs,thenextCIsarethe99%%,S&P500,andDAXin30thiendaipcepsl,icaantidonMsLof-twfoorfu9ortuarbe7leVdaaaRptaprtsohexatsinm(tasheteieonTn,aobtrhmeledea5lFd)(spstearoebvleTidaVebdalRem5eo)%daellVreerasRttihmeasanttieumdnadttheeers:ethmmepeanirnoicaabls9o9l%,lreto(hvFeidTne)odmrmmeoatrhleomadcodelingicsu4ra2t%ebia32sLeeqtubamls;:bm;s=VaRm;s-VaREmpirical;-FTTmeAthBomds,=an(dsP=8s1{jbam;ssejr)i=e8s,.wheremdenotesnormal,stable-ML,stable-FT,and20
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25
26
27
28
29
30
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SYBDeriesTable5:NBoiarsmesalof9N9%ormVaMalRLamn-d99S%tabVlaeSRt9aE9bm%lepiVriacRalestimatesSDCNDM&ePAAiMnJkP//CXk/UBBSPP$---000...423544180000...243643870000F...140T323361000F...T585190T547e435i000202---000...123650288-000...211620670---000...010984399000...610987126AeCanPIa5bs--00...362394819100...434091836000...421139426200...55785550Vpilutebiaso-00arRic,atle9xV5ca%eRptcfootrnh eadlYleednnac/etaBlePsveeteslx,.ctThhahenesgtneaobrralmeteaVlsaVeRraieResstm(ismeeaeastTuersaebwmleeern6et)-rmaotereseerxiaecst(YVeanR/BesPt,imBaPt/eUS$,andDM/BP),thenormalmethodreesxucltheadngineaitnonddtihcneeost,&VPaR5,0t0h,oDugAhXt3h0e,eCstAimC40,:inBgiaasreesooffc9No5m%orpmVaaraalRbamlnemagnitudes.-d95Stable95%VaRestimatesBDeSD&ePrAMnP/iX//eUsBB35S00PP$N0-0000o0....% ...102270438-0-0-0...000375459---000....01009175496---000...000556285-000...010623355---000...010897750---000...000581656 Mmedaennotaessnormal,-8M7L,,-00..007505lutebiabso-In-VsaamRpelesteimxaamteisn,atwiohnichofVaRmodelse-FTTmethods. thestablemodelinggeniserparlelyferrresultsshionwecd:3e3dby naonncsiearlvaintsivtietuatniodnsacacnudratreeg9u9l%a-
t tohres3n3o,rmalapproachleadstooverlyoptimisticf9r9o%mVaacRVVonesstimation,forecastsoflossesinthe tthhee9st5a%blVeamRerovedasettliisvmeuanptidooeinnr,etsotifmview,thenormalmodelingisacceptableforaaRRmmeeaassuurementsarecloseratoteththeeem95p%,tthheensotramblael9955%%-VInaRtWhmiesotsdeesectltiintoghnebwyneucliolnmvvhaepyspalturoiigatnahgtteeispoirstnehtdehiocaftfoterdVeeqcVaausaaRtRtiinoewgnsiptt(hri1om)(w h=e1re) holdsatanytimetPtistherelativeP:chra[ ngPet(<re tuVrna)Ronof1dayitn]=the1 pocr;tfol(4)iVsatRhecopno rtdfeonlicoerleetvuerlnatmomentt,VaRtistheVaRmeiaosuvraeluaet, t;Pcti=,-tcte,siwsthtisherepeecrVufroarrRemnetdtitisbmthye,chte2ck[i1n;gTw],haentdheTrPisrt[Rhet<len gtVhaoRftt](4)Lheotldbst,=1 RfRt< VaRtg;< VaRt = 10;;pprroobbaabbiilliittyy==1c c:tbenotebyEthenumberofexceedings(Rt< VaRt)35overtheaetis3nt3loeinmgiailndteisrtvrailbu[1t,iTon]..IWfeeqcuaantifoon(4)isvalid,thenthevariableE=tP=T1bthasaInvermulateatestingrule:rejectthenullhypothesissta3b4Sleeemtlhoeodfe9l9sin%iggnVwia aRsca4en2s%ctiemsxmataiiolflnerfothradnatuandseerrietshefrnoomrmTaalbmleo1d,(3e)ls.,anexceedingimpliesacasewhenactuallossesexceededthepredicted34
bfourtFtiP=Eo0r Tlartge T(1a ndc)stucÆT citen6tlxy2horigthP=E0V aRTt con( 1d encc)etcT t>1 xlevels,theb2:inomialdistri-loanrgceaTnbise: Hcence,thetestingrulewheEre<zpT(is1t hecancexif)p %zs1t axn=d2pardT(n1o rmca)clqouraEnt>ileT.(T1h ecb)o+unzd1 sxo=f2padTm(i1ss ibcle)cV;5ex%ceaenddinT1g%sEandexceedingsfrequencies,ET,fortestingatleveaRablaere7providedinTable7.:AdmissibleVlofsigi cancedcVoeannR ce-LeonfgathaRexceedingsandexceedingfrequenciestestingAdemxciseseEidbilnegVs,aRAdfrmeqisEusiebnlceieVsa,Rl9e5vc%el,in1t5e5T0r0v00al,Sig5n%i cancelevel,xSigni can/cTelevel,x99W%[11%5%1%[671,3,839]][1[546,3,964]][[%,%]eexam155i0n000ed[2[9,8,2]1][0[6,1,203]][%,%][[23..8703%%,,76..2207%%]][00..4600%%,,11..6400%%]][[00..0400%,%]paraforecastingpropertiesofstableandVaRmode%,%]- E mAvweitnedrso:wlengthslw=260observasideredthefollowingolaeblnusgeatrthvsatoiofntess(tdinagtainotveerrva6lsyeTa=rs)5,00tidonayss(danatdaTo=ve1r5010yedaary)=,ptwrwonehdoicurcmahcneaalrsneemuemooutdbhteesalriitdsnegnooofsrfimegaxncacicl %tcecsVepaVtRaaRbcloe(mlorpsasunetgsae)-,.,tinisogtnasblintoeenrmlvyaolfdooerflitnhgeisBPno/tUsSa$,mbeDydoAnttXshhtea3rat0st,tneaausbnmle-FTmethodareoutsideo500observations,exceed-tdhbaeCtrAsCinocfr4\e0astsinainbdgliect".=eTS1a5&b0lP0e3o5wlengthfrom260observationsto1560
SYeerni/esBPLaetngthofTable8:99%VaRexceedingsinteersvtianlg,TWindowle99%VaRexceedings15500001ENormEa/lTngthE=2F6TE0/
ioemnvbgoesdnetrehvleiealnetgwvioainnitnesdstorhweedu9c9e%,,erinforsmomsenocarmseas,lingT26h0eo9b5%VaRnormalestimmaattieosns(.excepttheDAX30series),obtainedus-lprearntogevAtihdegdenseusreÆarlvclayietnwiotonr9ss5,e%narseVthaweRitnheoirnmtahleVpaeRrmmiessaisbulreermaenngtes..ITnhceresatsaibnlge-tFhTemwientdhoowd erofVaRmodelssuggeststhat:5 9thte5er%%imVataeRs9es9t%imVaato1560observtaeRts,i,-taalliolcwainlsldfteohxra,ht e-tashvkeewstnaebslse, d-islotrciabtuiotino,nasnadre :te rs-sctoanb lremmsotdhealtin,giyncdtaaepeildtsu:ar(neisd)-ltaanilas;ly(siii)sgcp1droisevtdTeririthbnerumat\eteinadnts:gRsseatnfsrd"omcuos\repBdoBrin"atttethohebe\soAesnAtfdeuAsadt"-indicesisproivid edSi ni(T ia;b lei; 1i),wherei=1;:::;\", yc=at2egaonrdizaatsioynmmofesttraicbldeisdtrisibtruibtiuotniohnass,aafsnkroeertwmunranelsssdbipsytarrinabomurtemitoanlttthhoisee= <2indicatethickertailsthanthetailsofthenormearlstiskioIehnnwigngisheeesnsrske..erwaTIfleh, deast< issmaller,thetailsareheavierandthepeakofsota0tb, p>oin0t,,whichrevealsheavy-tailedness,andtdheinsdkieces,thetailindex iasnbdenloowrmzaelrod, ttedwnemespsipriacraal,msettaebrle ,37layedinFigure16andinAppendixA.
IdnGGe-x TabRtlsesu10er:NoMrrmalandStableParameterEstsIinag-orimatesofBondIndice(yiteayatur) aeixlSnkeS setswab-leLtioocna-G1G230400222ggUUoovvSS--tt13--3500.. Sca ..22074511..773891--00..11334400..00336200........01810200..00523900..002979CCCCC3C4C1C23A4A1A2AA2A2A3A33A3A4A4BBAAAAA73-5--1350000...000322289000....-2-0-0..11156500..00331100..114950-00..11015100..00227900........--00....00323700..1095861---1730000...000332239000...231531532111...7706452533-3--00..11232300..00239300..110541-00..11617300..00323900..109594 ......-6--00..01422500....114800iancdheixndseerxiesse:t,MexecreriplltLHy0nAc1h,,..S1o0u4rce38
Figure15:H0A1DailyReturns39
mcoen(stanSc ntlisdunmscpieoroncvrsei3dd6ei:tmreotruernasdheqauveatsekeinwdedandheavy-taileddistributions,VaRmesure-tahreddeviationor,incaseofisctaatbiolendoifstrriisbkutthioanns,syscmalmeeptarircammeetaesru),awblheicmhoisdpelrienfgerprerdodbuyc esnacnocnisaelrivnasttiivtuetiaonndsaancdcurreagtuela99%VaResti-liovses"esVwaRereesgtrimeaatteerstehxacneeodbseemrvpeidricloalVaR,\foCroencasestrs- T httVtehhhaeeeRnstoarbmlealmmooddeleilninggugnidveerseostviemrlayteospttihmessi9sets5i,%cfoVraeRca,stsoflossesinthestnaeobsrltmeimmalaotmdioeonldi,negling99%forishiagchcevpatlaubelseofofrthteheV9a5R%\iens-csarmibpelse",athnedcosnkcelowunsni :StableandNorm4a0lFittingoftheHOA1Index
o6emfksptiinarbeicslaeslomafnosadinleygFlislnieisgure17:,theportfolioretehnebiy-thRpas=set,turnisiP=n1iw=iR1i,whereRiisthereturnonthei-thasset,w;::::;n,iP=n1wi=,eaR:bil(ei )riaSnn dd(eo pmRein,vda erRniaite;b 0l)em-3ss7top,adrbeiol=leviniad1gni;e:dsn:a:(tnih:i;a)>ol1io( Ri= Ri6=0,we\cenptteiro"ntihs4ea1lRwiayosbssuerpvpaotriotends:bRy ith=eRemip ir )and
ITnGad1be0xle211:%cicaalVl,,%........0....0....1218931243150000.....57039400..23464900..23578800..2335633108212000...925380155000...412830200..41934400....57520200..68899000..34578432000...235680101000...2345450...247367900..23097000..25836000..12365000..12364700......34786700..35816400.....123246431000...12565800..121298maerHCa4as0muAAree41tmeresn0a0..85652700....,the\independent",aandlionmeavracroiambblinationofin-wRhPe=,reiP =n1wiRifollowsastableisthetailindRPla wS: ( Rp; Rp;0);rameter,ex, RpisthXescaleparameter Rpistheskewnesspa- RP="ni=1(jwij Ri) # 1;42
rpaamraTemtheuertsse,:rtth aeidistribu tRioPn=iP=n1[signPn(wi) R iR(jwij Ri) ]oftheporit=f1ol(ijowrijeturin) s:ischaracterizedbythreepilsinexdoegxen()o f, Rskpewanndes sR pRcpa,andscale -1threSe;::S:ttees;ppTte;21pi:s:Findestimatesof Riand Rnbeimplementedin:=E1va;:lu:a:t; ttingsetsofRit,t= ^liRoparameters Rpand Rp:P="Xni=1(jwij ^Ri) # 1(5)Havin ^RP=iP=n1hsignn(wi) ^Ri(jwij ^Ri) iiP=1(jwij ^Ri) (6)pwoeritgAfhostlieoadnVgraeeiltRstiimluurssntrstaahttoeeinosnnoefgpaatirvaemoefttehrseoafptphreopproirattfeolqiouacnretdilietorfistk, ^RPand ^RP,theinodfictehseofmtehtehoindv,eportfolioriskisesthimeaR^ f21SAAe14c,,tiCCon23AA541.,,BCCy34aAAs41s,u,maCnp1dtAiC2,,DCe3sAcsrt2imp,teCinot4nAgo2rfa,dCe1cAo3rp,Co2raAt3e,bCo3nAds3:,CC14AindicesisgiveninTable10iqnhuRdeiirex andA13,,(iion,theindicesare(i)characterizedbythesametail,bin ogRnidt,hraeenstduamr n)( ),,ltchuelaateveortahgeerosftatbhlee pavraalmuesforinwdeexst -:rR^fVmreaoesRmtarsiuectsrthtieevimmep:earntentevseiwo((dg(iixrveeBpno)riatnreedTcailnobdslToeeaet1sbo2lne)oth1dte0oapnpoetardit oebremvuecrhyi)n;ittihaelsnteawblestVabaRletpiavPreaT omhefSett1h e:e7r0ses(81f(to0i smll(oaT:6c5wtaeibnsl)9th; geqa0urf1eo1ra):;le0a)(ll1h.)e-F(up2or)trhatfreoerlio ^onUc,P%w=eVsa0hR:6a5llassume =,c ^aUlcPul=ate d0a:-,5ray3ti8enaAgr:sd,1i3g{it{\afArftoAemrAl"5e,ttt2oer{7\\yCAe"aArd"se,,n34o-t\{eAsffr"tot,hhm4ee{7mR^\taPoBtuB-rditiystbriabnud:ti1on{.frFoomr1thteo3anyaealyrsz,e2d{pforrotmfo3lio,4130B";adigitafterletter\A"denotescredtiot
t9Vmttlpoyhepenidc9ea9nl%%,,sottnhim-edanitvoeensr.-sdiT ivheedersuVi paepRde,rsttbhaobeulesnud9m9o%fotfhtehpeosrttafonldio-ta,.%a87%-bbdllieeveVVrasaRiR eeedsvtasimtluaabattleeios9ns5l%%ectViVoanaRR,-divle9rs5i% eVda9R9.%TVhearRefaonred,aabsleed1d2o:wSntawbalredFsmeasurementoftheportfolionon-diversi ed95%BonCCittingoftheBondIndicCCdC1C234i1A2AAAndA1A111icesM(ya212735eta-u3rrsit)yS-0ta. aw2t7i th= 54CCC1C3---57--00..11111600..00239100..01949-4-130--00..11406700..00321700..108587C3C4C1C23A4A1A2AA2A2A3A3337541473--5--157--00..10095800..00239300..110433850--00..11248400..00322700..10954713----17350---00..11220500..00330200..-d-170elingo--0f00...011P4245830ortfolio00...000R333355isk000...011948810fof ,wedevelop4n4dividualasset.
paTRvdR ionretPstWrfwodibielimohsutertivierboatinhsulukotoiadaoftoneslsote,=iP=n1wiRivalurems;in(iini)g pnodritnfgoliaocVeratRa:in(iq),itmheo1f%siqmuualnattiiloen,swihsicthocporrorejescptonpdossstiobltehepo9r9t%folVioaRretcuornn-timee:sRRi1P;Rait2;t:i:m:e;RTiT+;i1=g1i;v:e:n::;n(;i)(iio)bwseerivgahttisonofspoofrtifnodliioviadsusaeltsrweturnsoveri;eis=im1u;l:a:t:io;;t:r:e:t;, (association)struc-r0a;n0d).omAresyscmamlinegtrticra ns-sfotable(S S)ranmdommetrviacrsiatrbilcetlycasntabbele:inRteir prSet eRdi(a sRia;efbaoxberpllmoePlwsaari) aStivoanrioafbalesnoisrmobatlarinaneddobmyavpaprilaybinleg(,nothrmeadlepvaernidaebnlecse,tahmeodnegpeSn (i)Gisanormalrandomhavta:riablY SS2 ( G;0;0)=Newithazeromean:(ii)Yisas(ymmetric -st(a0b;l2e rG2a)n,domvariable, <2:(iii)Sisapo sYit;iv0e;0 2),-strandTohSmen ,S 2 Y2ablerandomvariable:iv)SantdheGG2 cos 4 2;1;0 (,sayrmeminedrescalingtrantserfpiocernm :s(tih)egRennsoftheportfolioretYdomvariableG:Simulatiourn=vSa21luGe:sRPi'~iwithctahnebseamdieviddeepdeinndteontcweostfrraugc--(19409P4) u45
Sowf Rain( hereoRrmi;a0l;0ra).ndBoamsedvaorinabPlero:pertRy1,Ricanbeexpressedasatransformationi=Si12GiSGi SN;(7)2( Gi;0;0)=(0;2 G2i);(8)Si S R2i GR22ii cos 4 R2i;1;0 ;(9)tbhyieisindependentofGidgeenptReehdarneenadptdedeeennopRt~dmeennrdceeesnccsatelrianumgctoutr;nraiegn.=GsHf1o;rm::a:t;;enic=e,1t;h:e::d;,bweeexgepnlainedin=ormS~ia21lG~via,rwiahblesG~i,maintainingtheinitialdependence41,thene,rawteeT(iih)ecsoimmpuuassuSSSmttteelattiniognsR~aPre=peereS~iisasimulatedvalueofSi;iP=nr1fowrimR~ 123R::E\Csteinmtaert"estthaebRleipoabrsaemrveatteirosofRi: Ri, Rlil,ow ::iA=s0suandconsiderRi asRin:s:RRi i =S RRii( R iR;i0.;0F)u;rith=er1;o:n:,:w;(t 9rTu);+n(c1iai=)tetd hce(cTo+va1rmiaen:ce(im)aRtirixcaonf(bGei)d1e6cio6mnpiosseeqduaalcctoortdingtoexpressions(7)-iRj;i)(hGeic)1o6vai6rinanactejT ;i=1;:::;n;j=1;:::;n,usingexponentialweighting:c2i;T+1jT=(1 )XkKtimmaetTrix+o1f,=0 kRi2;T k;(10)ds\41VariablesGi,whicch2ije;Tn+1jT=(1 )Xterformulas(1)kKa=n0d kRi;T kRj;T k;(11)(8),"coeafrsettrrvuuencrctyuatrieemdpofvoarGtluaaneustssofioafrnsttvhaaberliedaebvslacerrsiipa(bGleis)16(Ri6i)n16isi6\"sefrwomebtehleievdeeptehnadtentthhceee44e32nRTTcaahhtcieishoaenslvegvcoetartiloiutanehlmaf(ors1si9us9Rm9imi) ÆeRmciiee=,(VaR).46
wtEcolxhuptesitromeenreeTinnTg+ti;a1( GljTwAiseRdaigCehdnHteoictneaeg ysea(fc6atf)cso-t)(ro47e4rc),a0st<fo r<tim1e;KT+(c6countforvolatilityandcoornrse'laltaigosn.(GARCH-type)form45c:)-(7)canbeexpressedinrecursive2i;T+1jT= c2i;TjT 1+(1 )Ri2c2;T(G1SSij;T+1jT= c2ij;TjT 1+(1 )Ri;TRj;ttGee2pp;:54::Generateavalueofthemultivariatenormal;rTa:ndomvaria::S;iGmnu)withthecovariancematrix T+=SStie p6S R2i 2l ca2tevalues4o f stablerandomvari:CompuiR2tie cR~ois= S R2ii12Gablesi;i;=1;10 ;::;:i;=;:::;::aRCeapploceruatlftaoStleitoeR~pVPsa4R=-7miPa=n1lawrgiRe~~,mR~CeePvn4a-Atdlu2gis,artatCredib1epAucot3oiro,,reiCsbk3eoaAnfso3dur,sr:eCemqC4ueA1naA3tll,1yaC,sw1CtAeh2ieA4g,h1ntC,eeg2CdaA3tr4iAev,t1euC,or3Cnfs4aAospn1,eci edquantileiCnd1iAce2s,Cof2Ath2e,9tsfohc5lra%liotpwtrVieionatnuRgrotnmhfseienoaandsbiutcorheveseemsiedsenigntidvseiicnnesitnwarToeapsbyrlomec1me0deuotrrfeicSs-:e c(-tisi),pueaotanerntdtfohaClesi4so9uA9rm4%-pausloathenegativeofthee1sc%rib(5e%d)alqguoarintthilme.;(Iini)Sctaelpcu3laotfiocalculatetnheopfothrtefo9l9io%re(t9u5r%n)sVsiamR-itnitosnasn,(reKdmeDra)ttoarinxeds tAiTmn+ad1te,erwsaoecncu-uDsreaadcrlydinio gfe(srAiemnDtu)ltarstutianotcniast,iwoneADKD=suxpFstics:je(x) Fs(x)j;wsmimho4eurleatFeed(xc)distheempirica=lcsuuxpm(ulpatjFiFveee((xdx)e)(n 1si tFysF(xfue(n)xjc)t))ion;(cdf)andFs(x)'(a1r9i9(61)).lyweighted
faD0cetcoTarya ble1pT3or:uPinntcosa,rtt(if%oonl)ioV9a9R%PofVortrafRoSlyiom9Vm5a%eRtricDepeKnDodliemsntoatgnCocreroedvitRAeDtnudarernlrissnogn-0............--9o95088..10124855...420353426553...624515022000...110217877The1-9988..14666955....82075400..)addsertio%vehdVigaihnRerSeevscattimatesinTable13arewithinthe99%VaRrange(,,increasingthedecayfac-Tt9i9o%%,eacsatyhfeadcteocra,yinfacgteonrergarlo,wrse,dtuhce-lCeaeottnesridoebrsaetriovnatioofnabylarpgoesritinvuemcboerrreolaftitoanilhiinogbhtseaerirlvVsaa(),hiendgeecnaeyrafla, riesrkenisceabreistkweaedndetdhebpyoharnotfdoalsiaos eetrcisttkoseaxahmrailnsodknT""malyyar,znetfesdhaudcel\=\,nhthigehleorwreartKinDgsacnodnAtrDib>usttaeSttimastnoidcres-ffaocrtcoor=,awsees;wtohueldnor-etcroumncmateinodnmtheethcohdoicbeetotferthaeccdoeucnatysavMmCersIin cTaatibolnee1 5ecwtesraetpoto-htrretudnmecacaratgyioinnfaamclt9eo9trh%,asntadndth-aelonnoe-tr9u9n%caVtiaoRn,,,ifnogrinaldnVeuxamRmbeoarfks0e, %ndVtahRat:tthheeWedrosneiin etfTcadpaanstobdai-lallootnheerrismkas,,riinskdseefodr,rloen15,wenoticethattheC4A1andC3A44tf8oeli
DecaTyabFlaect1o4r:MaTrrguinncaaltRioinsk,(f%or)SyVmaCRmase>etsr:iVScMtaaDRanerdpg-eiannloadnleentCRrCeadatsiitensRg:--99o959000300022T5BoBnod1---99500004ableN9o92nCCId1n512dI:AAnicMd11eiscaer(sgDinecaalMyVfaaaRrcg,Stand-alone99%Va3R,4andDiversi catio4nE =,)VaRDiveEr0s. i0 ...579033941000...1127611280...245497545000...256808559000.....714890803000...8259830600..10094260...59192820500..7914099000...110222770900...3583208230000....2579914601100000....{skeweddependent49
uW8eesqanuDSdatneatipbleModelingofPortfolioRiskforSkeweddefryievpinnogrdtfaeolpniootrrtifsCokliorRePVdabRiytReturnsfgreonmertahteincgoansdtrisutributionofitspossibleval-tapcdhreeecenndacdRsieteePnoarcefnevtduap(lrtaounhsresestsfoRoscslhioassetswithskeweddependentcrcetdeidtdreistturrinbsu,
ectthe\cumulative"skewnessandmaintaintihoensdoe-fitnnitodon{)twafoomrionsnkdgeewpthneneemsdse..nTtInhpeaonrrt,dsew:retthooebdt aorisntthtpahate,,oPy(niiea)nsgtaemnseuarimanttiaooifnnthtionhrfgeetehttwhepeooprodgtreieotpnfnoeesnrl:iadote(esnidk),lpaainonnraddttifiv(ooiilindioi)s-tionA)isnttaoblerandomavraerpiarbovleidRed beSlo w( .; ;0)canbedecomposed(indistribu-wheretwoindependentstablerandomvariablesR(1)andRR=dRR(2):(1)+R(2);R((12 ) S ( 1; 1;0); ) ==S( 11 ( 2; 2;0);1 ++ 2 ) 1;(12) .STuhpepno,sefotrhmaut:la(si)(1R2()1)anisda(1sy3m) cma=entrb 1 + 22 2 :(13)2ices 1 rteadbulceevdatroiabthlee:f ol1lo=wi0;(ii) 1= 2= ngexpressions:Fromequations(14)and(15), =12 2:;((1145)) w e2==h2a ve 1 ;520 :
dwanishdtIrnsibereksueuwtimeodn,)Rai(ns2tt)ablerandomvariableR S ( ; ;0)canbedecomposed(in:otwoindepeRndentstablerandomvariables:symmetricR(1)R=dR(1)+R(2);(16)RweipU asrintSig tiRomiR((12)) SS 22 11 ;;02; 0; 0 ;;((1178))n(e RtRhiodology(16){(18),wecandivideindividualcreditreturnsi;i ntRoi;t0)intothe\dependence"and\skewness",wherehe\symmetry"and\skewness"fragmentsRR=R:idi(1)+Ri(2);i(1) S Ri 2 pRai(r1t);siR=(1)1R(2) R1i Ri;0;0 ;i S Ri 2 R1i Ri;2 Ri;0 ;RCiwor;a:n:d:;Rni(,2a)raerdeeipnednedpeenntdeanntd,(iii=1;:::;,wesuppose:(i)i'instBstaeeanqsedudetneotrnlmy,PssrRyompi(2em)rtaeyctrc1iocu(ntseetremfoSrsecsRtkiie(o1wn)ne7ex)sp,s)lRoaRfini(i(1R2)d) ;i=1;:::;n,' dRein c2e ( aR1sis oRciiation)among;0;0 canbewherenasatraRnsformationofanormalrandomvariable:Gi(1)=Si12Gi;i S2( Gi;dtheeteSRdramenpidennoedmtehnreecsednS0;0 )G2=;afToimhr=moern1aeg;tfio:oR:rniNi( 0c;o2s G2i )4 ;i S R2i 2 R2i R2 eR2i;1;0!;iisindepe:s;(,1)fromthedependenceaimntoonRgi(G1)i'msawientcaainn5,1orthedependenceamongRi.
theAtwdodicnogmspeponareantteslyoftthheedpeopretnfodleioncreetaunrndssRkePwnesstermsofRi's,weobtainwRhP(2e)re=RP(1)TheW=iP=n1wiRi(1)=iP=nR1P=RP1)+RP2):((;(19)iP=n1wwiSi12Giisthe\dependence"componentandSsteeimpsui1mluliaRtei(2t)hiesRthPev\asklueewsnbeasss"(19):R~P=R~:atEiosntismaarteeesxteacbuletepdaaraccmoertdeirnsgotfoRtih:e nReix,t aRlgi,o riRthi,m 4R6i:P(1)+R~P(2).awatsseituSgSimmtteheteepipnT 2g3R:+::iE\1=Cve0natenrd"=ctoh necsRidierobRsi eravsatRioin:sR:iR i S= RRii( :ron,weshall,a luTa+t1ethecioj;vTa+ri1ajTn ce;mi=atr1i;x:o::fn;nor;mja=l r1Rain;d oRmi;v0a);riia=ble1s;(:G::i);1n;:::;n,usingexponentialc2i;T+1jT=(1 )Xc2kK=0 kRi2;T k;wij;T+1jTtohteirmeeTT+;1=(1 )XkK=0 kRi;T kRj;T k; jTisdaenotesaforecastfortimeT+1conditionaloninformationup(G1SSS;ttGe2p;:4:::G;Genne)rdawetceayithatfvhaaecltucoeorv,oa0frit<ahne cem<mu1lat;tiKvriaxriis ataTe+'lelaGgs=.ie p5S: RSiimulatevaluesofstablerandomvariablesSStteepp76:2 21 cR22ii R2i cos 4 R2i;1;0 ;i=1;:::;n.:CGoemnepruatteeRR~~i(1)=Si12Gi;i=1;:::;~PSDStteepp98::RCeaplceualtaSteteR~i(P2)= S Ri 2 R1i Ri;2 Ri;0 ;i=1;:::;=n1wiR~i(1)+iP=n1wiR~i(2).raer4teW-deeirsitveaportfolioVpsaR4-8esatilmaragteenausmtbheernoefgtaimtivesetooffaorcmhoasnenR~(Step1-Step9)fortheriskas-6uTprhnriesssoealnngtoiernditdhiimnceTssaamreedpeopretnfodleinotosfkeinwdeidc-e
ablepT1or6u1:PortfolioVaRforSkewedDependentCreditReturnsin5n0t-cs-99a,50t(i%on)9945P%..o93rV38t90afoRlioV9a5R23%..--9o950555...41409133..203069556...645433000.............)eof9S9e%,..347300..1145falalRwithinthe99%VaRrange(,..%h u-is9n,9ct% tdom=;i4tr(usilied)edgtaodpescertlnooindermeuascloelwryteh:hec(enoi) incchraenagseesslnosweervstatKivDest1 ahus6=:hi9oe4aenRmcoaomfnamdrsegkbaietisnnhwuaaeerlted1iVmo%daneeR-ssnp9eoe9dfn%,nWwtVheeeawcRoudledtormtse,-irltisokn-frteweoppa-orfltastcoautnolafdrtiPtmohnoeosd:cerrlt(eid)-uftatsecadtnormreomdieulm,i;n(tiih),weassumethataportfolioincludesnthereturnon,ththeei-pthortafsosleiot,rweituirsnthisegwieviegnhtbyofRthpe=i-thiP=n1aswsieRt,i,iw=he1r;e::R::i;nis,iP=1wi=-risk-
wfrewcrheietedhSrfiueactorYi:traRiirpeptsuokarsnneasdntdRhberas,iU=aiisit+hebrieYsiid+uUali;representingcompensation(2f0o)iareconstantrfrRreeefactorYi:i.(iin)-Ttdhhoempreonraotrfioseli4o7,inis=tr1u;m::e:nt:; tr-ity ;ri(siki)-awniittu hurnanadise croolnbuyiniYnatige rrfep-ay
,thespreadUi=,trehtausrnasloowncreditrating,sayBBB,owlieoacassnetchisoaossew,aapsrcirsekdiotfrtahteinBgBABA-sAw:=aj+bjYja+sUimj,iltahreswtearpmwUitjhaacccooumntpsanfoyrtthhaetchraesdiat((iiWi)e)CDriemefadpuitoltsr-eifsrtkehesepffaroceltaloodrwssiUnYgassumptionsonthecomponentsofmodel(20):iiaarThen,reessttrriiccttllyyssttaable,Ui S Ui( Ui; Ui;0)48, >1.(iii)UianthdeYpioarrtefoilnioderpeteunrdnenRtble,Yi S Yi( Yi; Yi;0)49,
>,i=1;::::;=A+YPe+coUmPp;osedintothreecomponents:foliocreditriskA,=esXnaggregatee ectofunderlyingfactors,UPrepresentsport-i=1wiai;YP=Xni=1wibiYi;UP=Xni=1wiUi:risoie=;Sfe:eee:cav:tca:iholu;;np(oi1ri0)tfeodsliitsoicmucasrseteidnsigtcrrUeisPdkitaUsrPiaskicnuevtmwauloulaastttiievopens:rfeo(sriu)sltqinuogaflneintpidfoyivrintidfgoulcaiorleUadsiit-,prf447inWdeepinetnerdpernett,ethsxyeamymimelidnetesrspicrpeoadrdetpfaoeslniotdheecnrcter,,,,liquidity,andotherthenwieistathkeeYceitnt=erZeditr etZu ri,=If1t;h:e::re;,Zi,arenon-centered,54
oA1bp0tparinoxedCimuarstieinodgnsimtoofdRtehleis(2crkediEtrvisaklpureamtiuiomnvalfuoesrUPortfolioAssetsiforportfolioassetscanbewhU^0):erea^iandb^it=Rti a^i b^iYti;(21)iaretheOLSestimates,a^i=tP=T1YiT2ttP=T1Rit tP=T1YittP=T1RitYittP=T1Yi2t tP=T1Yit 2;(22)b^i=TtP=TT1PRTitYit WtP=T1YittP=T1Ritt=1Yi2t AtP=T1Yit 2;(2byssruemteuermantnahsatalooytnrzs3)Estireaea^tccurrreenddsiiti=1;:::;n;t=1;::::Tiandto-b^rnriii,ssakkgn-iofvenbyexpressions(22)and(23),,theYebiUo,aSnndcdsoraappcoprrelaydtiengboonndes-,faRcti,orarmeoddeeslcr(i2b0e);iandbiareconstantsRitspread,Ui:,i=ai+biYi+Ui3nAAd24i,cesasinSection5(seeTia=bl1e;1::0:)::;,inReCr2eAt1u,rnRsCo3nA1t,hReCsa4Am1e,tfRuRrGn1sOo,R2nanCd2AR2C,,WReCc4hAo2o,sRe,Ca1sA3c,orRreCs2pAo3n,dRinCg3Ac3r,edRiC4A3,RC1A4,RC2A4,,RthGe2iOn2d,icReGs3oOf2U,Sgovernmentbondsinthesamet-mriasktu-frrieteyfbaacntodr:sY,,eiafrws,eRcCon1Asi1d,etrhreentutrhnesronthettOahge2e,isnreed(,i,Wfoelloawppinrogxaimppartoeatchhe(p2er-diÆ)criuenntOsLofStrheegrOesLsSiornesgorfesmsioodnesla(r2e0g),(ii)computetheresiduals'seriesU^)-:(50ForanalysisofasymptoticpropertiesofivOeLnSinestAimpapteonrsd(i2x2)Ba,nTda(b2l3e)),ti3gioit{naafssumptionforthedisturbanceterm,seeGotzenberger,Rachev,andSchswtaabrtlezfrtoemrl5etttoer7\yGea"rsd,en4o{:1{from1to3years,2{from3to
^fkitU^hsaieprraereplsohtotewdniinnFFiigguurree1189aannddiinnFAipgpuresofienaddivsiedruiaeslrU^eituerxnhsibsitvolatilitycluesntdeirxs by sStotshymatm0tcaa, diosdk GelA0inR(^etscanbecapturedspreadsofthec7o)'ionfdpiic,aeresanmatareeiltefeadrte-vtsaatliiumleedsatoaefnsd ina<d1:6,lmicaotset1 f=the1bTable17presentsthefollowing and valuesofthecreditrisks:3ondindiceswithamaturitybandfromonetothreeyears:AAAbonds:se ta=bl1e:3a9n33d3ananondrdm a =l = tt00in::002111;;BABABbboonnddss:: ==11:4:31729anandd ==0:000:;loAtsboofntdhse:^^lmitioieadsreelaisnnhgdowwnfaetllontcaaFiplsitg,uurbreyeswhereRUanceasanexplicitfunctionofpastinformation:i;t=ai+biYi;t+Ui; "ii;;tt= Ni;(t0";i;1t);t;(;((222456)))i2;t=ci+Xj=p1
iitisWi=1;j i2;t j+Xj=q1 i;jUi2;t j;(27);:::;n;t=1;:::;T:wbeassehdalolnnatmheemodel(24){(27)asaGARCH(p,q)-normalmodelbecausecadouerttreoeccltoorgGrreAalmaRtsCiosHnug-fdgueennpsctoetsrniomvdnaealsnulicoetiye ndthatp=q=1isadeqsfeusotaa,^,RRpAuCC42CAAo14e,Æcientsofm ofvolatilities:i2;t=ci+
i i2;t 1+ iUi2;t 1:(28),RRCC13AA24,,aRnCd2oARd2e,lR(2C43)A{2,(R26C)4aAn2d,R(2C81)Aw3,itRhCR2iA23,fRRCC31AA31,,RRCC42AA31,,RRCC31AA14,,paoepalrpstkeesUdni;itn=Apqpcein+dix
iBC,,tRhGe2GO2A,RRCGH3O(12,,1R) di2e;tm 1on+st riaUtei2;tt h1at "i;taredisplayedinFigures20andin5t6heGARCHcreditriskserieshavelower
Figure18:OLSCreditRiskPremiumoftheC1A1BondIndexpbteavxeloetsprittIonomifngnooaaeltidlhtnolyeeetilecaixpncalpogosweamrtnfpdolslei5xosoeifogfihnt2dicvoindtueaxlt,,waeneuvmraiblsukearstewofiptophratarfacocmliooeutncertreisndgritafproiirdsklGyUAiPnRcCbreaHasseeed ,,sokfeUwePd,wdeepseenpdaernatteclryedinit-cRI1n1ri55es2t3kdFAhMiotinserrtarsPriepsiecfkpcoesrtr'aoeiorntwfosacuemlfoiulolloawtCivtehreee -tvmearGoyviAinnRggCvaHol,atsieleitiEensgbleyaenxdpoKroner(1995).(o1w9s95t)
TInarOdibsiLlkceCCeSCos11fc7Ar:b1eoSdtitableandNormalFittingoftheOLSCreditRiskPremiumsofBondindicesndM(ya1et-au3rrsit)yMeaNnormdSetaavlniadtaiordn Stable CC2341AAAA1112735-1--5-17003000..0....0..00475511..353238-00..00181C9-00..00000100....549506--00..00223600..00000000........1..-3-000..00370400..00001000..001259-00...000062501000..00000100........-2--0000....000080714437000..0....0012590...000441288dC234AAA444735---1570000...000000...1014912634111...554258720---000...000275475000...000000101000... o,-ecdrsiehsdtoriwtib-ruiimstkiopnplesrm:emeinnimtdaeutpimoensndforefonmtth,-prhieseknn-,dpbreeynmttihueCmasrdeadrdiet:iivt(iit)Rychipsarkorapscetretryizoefdsbtyabtlheevsaarmiaebtlaesil(insedex ;(ii)w3)h,etrheeportfoliocreditriskUUPP =SiP=neSection 1wiUiisstablydistributed: isthetailindex, Upisthe( sUcap;le Up;0);rameter,parameter, Upistheskewnesspa-58
Figure19:StableandNormalFittingofC1A1OLS-Credit-Risks UP="Xni=1(jwij Ui) # 1 ;(29)ConsideraportfUP=iP=n1[signn(wi) Ui(jwij Ui) ]iP=1(jwij Ui) :(30)Staeicltiinodne1x0 ..AWsseutmakeoecli roed=oift1-erquallyweightedOLS-credit-riskpremimumsfrom:4is7k2-,ptrhemeaiuvmersagareindependentandhavethesameb prsdeooUreniimne,dosaiuidnnmoddei ssceUensrio,(seceesowmenedetsiTottiaiobmbnelaevtoee1fsr7tya)h,rreeeassnrtaerdmpaioreneritctneaogdim:lipeninnuoedwtTfeeaxtpbohal teeh :t-r ikUsi-,(eTrtepTaudbriltnesr1ios8nk)thoherutSnsfio,ntelgU^cidoeeaPtpovivab iaertataSeominmf1:ee4ttdu7hec2ereh(s0s1tafi:%0prm1op5am(l5;yt %eitnsh0)goef:0qfop3u r8rameav;n0urieouvserpyasrmamalelt,ewreesatsismuamtees (T=))(h2fe9t)h9e{9%(U^3P0()-9:5d ^%isU)tPrcib=reud0tii:to0n1V:5a,Rt ^UiPsd=er iv0e:()%te(r9s,5%we)Icnomoprduetredtoinastsweossspteoprtsf:ol(iopcei)sriemndduitlearntiisntkg,CawredeidosbtirttiabiRuntiiposonkrtsofoflitohecrUedPit==1wiUi
OTLbaSobncleCCCrCdeC11d8iAnit:d1irScitseaksbolefFMittaintugrio1tfy-3t(hyeeaOrLs)SCrSetdaib tleRpisakraPmreemteiru smastw = ed C2C3412AAAAA11122735-1-57--000...000091009-000...000000001000...000134825CCC3---1350--000...0002173920-00...000000111000...000512259C3C4C1C2C3AC4A1A2A3A24A2A3A3A3A3754444713--5---17-1350--0-00......----13570---0000....000011751013-0000......0..00400139459365Figure20:CreditRisks:OLSandGARCH60
STvaas hlUuiWsie(s se;eUsc(iitimi;i)o0unin;l0a)efte;xeriarUm=inPign1a;epp:sor:ptt:lhyf;:uUtiion .t(hi)egUein'~miawtiigotyhnftorhfoemasnaSomercmetiaodnep7e:ndencestructurewhereGUSi S2(Glrandomvariable:i=Si12 Gi;i;0;0)=N(0;2 G2i);Gi S Udeip;T(einh=dee1ndSiisind2eip en dUG22eini tcoofsG i;i4 = 1 U2i;:::;;1n;:0 ;;ce:ep:.:eNn;,ceweefaomgremonnedgreapUteeinU~dcanbeexplainedbythedependenceamongie=ntSn~io21rG~mi,alwvhaerrieabS~liesisG~ai,sipmreusleartveidngvatlhueeionfitSiai;lAAiis)paocnartlcifloululliasottrecarUte~idPoint=VofiPa=ntR1hwecaiaUn~,swureeedstfrimomatethcereU~(osstreetneeartneSpdeoocrifttnifdooTenlpiao1ebn0ole)-,ttsitrohunnensoctcasteioenmmteothao decatpthepofratcfotoliro,creredduicttiVonaRofinthaeptraurnticcautleadrinredfaitrriskresidupalrsoaocfhthleedintovetshtemsemntalglersatdVeainRdimceesashuarvee-nreooegt-tasritmnui
shtci,,nwdeAcDonsstidaetirsttihcesnp0ou-ttartuionncaotifthemtahrogdinaalndVaaRrb,isttraanridly-alsoenleectVathRe,ddievcearysi faccattoiornofe ,4oiaAmn4d,,p0Tar,nohhadecicgChrh4eaedAnsitd2tctrbiohsonkentdprdirebcuatyienmdiiicouefnmassc:tottofohr=%,of5b4oTnhde isncydamitcimeosentarreye (isteioenTiasbplleau16si):theskewnessparametersofcreditriskspremiums61
Deca0y. 8f5acTtaobrlepT1or9u1:PortfolioCreditVaRforSymmetricCreditRisksin5n0t-cs-9a,0t(i%on)993%%--99o95503333....7356199406431111....8899956663877...866998820000................61675400..215647cohplioeornfisdcSkeecfontritotnhCe8rs:kee(diw)ietsdpRdlietipsienkndsdievnidtucarledcirterdiistksr,iswkespUriopinotsoeaskrneeddwWisntkeeersiwdssikncvsoei;dsmse(pifiroi)neevnatslukbaeytwecnotemhsesbippnoainrrttgfso;tlh(ioeii)dcer )e:nce"and\skewness"partUs,applyingm etSh oUdio(l oUgyi; (1U6i);0{)(in1t8o)(tsheee\Sdeecptieonn-whereUUi(1)i= dUSi( 1U)i+ 2U i(2 )U1;i Ui;0;0 (2);1pi S Ui 2 U1i Ui;2 Ui;sy;am:r:tm:s;eUnt(ric0 ;i,1)araceonmddepUpoie(n2ne)dneatnrsetUiandependent,i=1;:::;:(i)Ui(1);i=i(n1)de(xipi)laUini(2d);epie=nde1n;c:e::;n,,pcoremtBepydonaPesrnoatpsteUrrati(ny2s)f1dore(mspeiaecttiSosekncetowiofnnaes7ns)o,orfmUUia(1il)'sr. (association)amongUi'sandandSo mUiv a2r ia bU1lie :Ui;0;0 canbeinter-62
TCarebBdloeitn2CCR0:isMkParrgeimnaiulmVasRof,SBtoanndd-Ianldoincees99(D%ecVaayRf,acatnodrD=, NcaotitornE ectsforCd123IAAAnd111icesMarg00inalVaRStand-aloneVaRDiversi cautniocation)0...121706532000...123950115E000.. ..101425543000........120891063000...241954039000.....235420457000...112421672A420...001794912000...123644836000...(S1)Si =SSi221(G Gii;0;0 )UG2i=Randomi U2Ni( 0;c2o s G2i )4 ;i S U2i 2 2 U2i;1;0!;theCdoempebnidneinnrgecsecalingitirsaninsdfoerpmenatdieonntsooffGnoi;rmi=al1v;a:r:i:;n:,hHfereondmceep,tehwnededecneapcneendadenredinvsceketablesGiintoUi(1)maintainehamedoenpgeGndi'(1),orthetwocomponentsoftheporUtfPol=ioUcreditriskUwP:nesstermsofUi's,weobtainUP(1)+UP(2);P(1)=Xni=1wiUi(1)=Xni=1wiSi2163Gi
(e1g)frpgaeoi);isgsrmtttehfeodenlimt\"saP(2)nb=Xncomepeio=vna1eforportflonwutaiUtaen(2ddaUsPa(2)suismthofet\hsekedwepneenssd"enccoemapnod-s1tlratedtheirapplicationsonanalysisoliocreditriskassessmentanddemon-Taovgeiefsatc-atnailbeedeamndp/looyresdkfeowrerdisdkisetvraibluuaftiroentuorfnasnoyn -poh2seurVealCtuoeo-manta-rRckilesutkas(nVidoaRnc),orlnetthsasse--{nytoeturshmtneidnaegdlreme{-ar,Rnoohvmtiisddeteaoesrssuicacrrtaieiblsmefsaeiwcmntteouslrllaytieovna,luMatoinotneoChfeasrltoosVimauRlactoiomn-,i ..pusmtrlraiogetxThsiiosthm-ntibsaeedtswitoanieon gsrgeklcnoVtwotpeadrRqpoubrpayeoonsdsmtetiuismlocedeswroibthusatsVmaaRllnesutmimbaetreosfsoibnsceervitatiisonnsotinrteihlsitaobrliecailnsiamp--tahetelesmaapisrpselpiesucib jceacttiiovne.. avtetyimoa-tplalioirlwciscasatolioantfomndaaoelrrfykisvseitestavabaeclnercdiu -trhmeasltmaboldeemlinogdefolirnghigghenvenarlasulhelyoswopsfrotthhdeautcVesastRacboclneosnVe radvRenmceoldeevleinl:goutperforms VthaeRneosrtmimalamteest,hwohdiclheaidssptroefoevrerreldyboypt imnainstciicafloirnescataittsiuvtsetiooandaccurate99%fnsandregulators, BaVtsheaedRnoeonsrtmimalatmioond,elinglossesinthe99%thepropertiesisofacsctaepbtleabdliestfroirbuthtieon9s5,%:-fiancdtoepremndoednetl,,--taigfygrcergeadtieter ,rwisekqausaann-
tdiimsteTrihbsueuptsietoranibollreea PdainregtitaonthmeoCdeLlT,w(hCielentsrhaalrLinimgitthTehmeaoirnempr)o, ,taiodndiotfiomnaoldreels-tvhaircykintgaivlos,laotinlietyc,,,MiralelAnbad-el ls-phosrobtoarsvteb-irldveae,endddAplohnegn-ormanegneadienp :nnhaale-laysvtsayisbtlaeilGs,AtRimCeH-vamryoidneglinRMaAsu-bstsaebqluee-nGtApRaCpeHr.,andfractionaolf-sVtaaRbleestGimAaRtCioHnR(M1a555Forsomepreliminaryresultss59itc679thTF8neo)hiv,kre,s(Md1ePi9sitamc9tou5onles),Paolella,andRaeceheLviu(1a9n9d7,Br1o9r9s8ean,(a1n9d95)1,99M8ibt)t,niPk,anRoarcshkaev,,ManitdtnPika,olaenlldalleiaols,naaonrfedsnRtaaambclheede-GvaA(s1RA9C9R7HM).Am-o ,MittnikandRachev(1995)and
::StableandNormalFittingoftheG302Index66
::C3A2DailyReturns67
::VaREstimationfortheC3A2Index68
::StableandNormalFittingoftheC4A2Index69
::C3A3DailyReturns70
::VaREstimationfortheC3A3Index71
:%%%.%-1-1-00..01732000..00508800..-00..--00..001237CC-C-C-000...0116370910000....0000474462780000....00000101149---000...000101086C1C2C341A2A3ACAA2A2A223-3-3-0-00...001361613000...000462826000..........000246091-000...-00....00154600..001207--00..001052--0-0..00381600..00238300..-000...111665604000...000391083000...0001115499---0-000....00001111586372
:DSi %tV aVRaRFEixsteidm0. a2t5= s%wFitihxeFdix e=d1 ...47934200..23135963000...123135631CCC5000...925771937000...412624754000.......3468053246581000...1232464705000..........:CCooee Æ.cientDsveoapfreinOadbLelSntRegressions10200..080842742234RC1A3eVaRrGiCablesCoe .10200..0903887RCC23AA1C4A1RGC20200..070760118332RC2A3RGC20200..08401656072207591RR1RGCC302000...080030553567543005RRC3A3RGC30200..080553025915RCRC1C2AA2RGC40202RRGGC10..80407323256720022000...905015176353RRCC4C1A2AA34RGCC402000...080070374RGC10200..8070648904006303609123C34AA22RRGGCC3400202000.....7060081269264022000...80800047337012915173
:oGe ÆriacbVielanerstisanceCeqoeu ...090033759891600..000004582955GAARRCCCHH(1(1))200...518018E6563-860175200...000200E3254-480166RC3A1RGC20200..80030484991449849000...000003948616GAARRCCCHH(1(1))700...618737E0010-100945500...700600E5477-210046RC4A1RGC30200..80903490419400..000000131365353109GAARRCCCCHH(1(1))000...018036006147574296100...600600E3297-710065RC1A2RGC40200..80807257843600..000030543181GAARRCCCHH(1(1))4000....70187053E030379-5440541300...200200E8874-550625RC2A2RGC10200..90406402126900..000030883805GAARRCCCHH(1(1))100...309491E6447-23087657002....90037008E22E55--980467RRCRC3A2R4A2RGRGGCC20200..88990418263100..((11((11))))......-0409011665RC40200..-3055C12AA3R3RRCRGGCC1200201..000023207719000...00000030324242101GAARRCCCHH(1(1))700...018681E0714-640846009...00900297E10-380567200..09005326084300..0000026264547GAAARRRCCCCHHH((11(1))200...018138E7708-410525003...00200132E40-960416C34AA3R3RRCRGGCC3400200..09015584400800..000013101549GAARRCCCHH(1(1)))0000....1902200545032226882573002...00200042E42-170715200..09044201706200..000012370288GAARRCCCHH(1(1))000...704700701979739405250030....0010006174E514-980905C12AA4R4RRCRGGCC1200201..00032648560100..00003547608GGAAARRRCCCCHH(1(1))-003...61230791E99-710486050...0500807E1689-13902077200..08077906117800..000003813999GAARRCCC3CHHH(1((11)))5000....7928743E5125-060935030...0900700E660-6106364AA44RRGGCC3400200..08097346542500..000013267465GAARRCCCHH(1(1))300...9199400E117033-810066450700....0100702E371-730026200..08087741002400..000012389932GAARRCCCHH(1(1))000...018074090108933408100...070002002E53278-9100569574
::StableandNormalFittingofC1A2OLS-Credit-Risks75
::StableandNormalFittingofC1A3OLS-Credit-Risks76
::StableandNormalFittingofC3A3OLS-Credit-Risks77
::StableandNormalFittingofC1A4OLS-Credit-Risks78
:C2A1CreditRisks::C3A1CreditRisks:OLSandGARCH79
:C4A1CreditRisks::C1A2CreditRisks:OLSandGARCH80
R[1e]fCAetiorruansrldsee"nR,,srn1ita9tt8ei0eo,noa\nPaBraanmkeintegrSEusptiemrvaitsiioonn,fo1r99S9y,m\mCreetdriictSRtableDistribu-[2]BalEconomicReview,21,:[3]Chernegn,,an1tci9oe9n,5s5,",\.MultivariateStableFuturesPrices",[4]CinhogbotbinhasenooDnvia,,,thtS,,ilt\atTntiii1lmke3,."gT,-rpRartaienchgTeeivrm,Ve1er9Sl9ea6rgi,,es1\"3M,oindePl-.[5]LDiukMeliohuocohdelE,.,te19w73h,en\OSnamthpeliAngsyfmropmtotaicSNtaobrlmeaDliitsytroibfuthtieonM"a,,.,,D1e9p7a1r,tm\SetnatbolefSDtiastributionsinStatisticalInferencea"ls[6]Du,1,948-957.,[7]EGnAgRleC,HR".F,.,eo1r9y95(,1\9M96u)t,lits1itv2iac2rs-i,[8]DFnaeissmtsrai3,b8EuE(t..1i,oa91nn69sd56")5,R,J.\oTRurhonella,Bl1eo9hf7aA1v,mio\",JournalofBusi-[9]Fama,,[10]FederalReserveSystemTaskForceonIncalAssociation,66,\:isCkuMrroendtelSs,ta1t9e9o8f,[11]GCmFeeuterricveSrtgaebrl,"ghin,,s\CoOragnpoiEteaÆtlcaAiednetqIunafceyre"-aRPtealmRatiresokdwTskoip,,1996,\Testingtlh,eedVs,al\iSdtitaytisotficVsaalnud[12]cs",Amsterdam:,oSsnpurrAesp"p,liiendHPeryodbea,bPilirtoyhoarnodvTPiymkee,SaenridesR,aVcohleuvmeedIs:.,AAptphleiends[13]GstaambilsretoiwCcssAkaPin,MdB".A,,ohbf3ei0lvCi7,ta-y13li92,,an\,iAcSaTalneRsttaabBlaerVbearrsai,onDoefptahretmPeanrtetoo-f[14]GILnaadwmussr"to,rwiaPslkroMi,,1995b,\,(aetdioitnosr,),S2u,
[15]PsGeatmPrroiwKcisnkgi",,,cPhervo,ba1b9il9it4y,\aSntdabRleelaMteoddeFlsieilnds,TeNsteawblYeoArks-:[16]GsuloretniacarenzllmusermDneunbotPesrr:geeTsrsh,,eG22S.,..PRaarcehteivan,",arWtzo,r1k9in9g9,P\aPpeerrf,orUmnaivnecresiMtyeao-f[17]Guptonh,ceGu,.,tmC"a,.CnNy..Finger,,1997,\CreditMetricsTM-Tech-[18]HSyematmhceotrtey,",.,,t:,,51995,\TestingMultivariate[19]oHfilal,,b1u9t7io5n,"\,AASninmaplsleofGSetnaetriastliAcsp,p3r(o5a)c,h11t4o6,[20]JthaerrTower,.,,"ll,,R19ev9i7e,w\,[[2212]]1JK0Pl(e2bM)a,on4rog8timationva1o,n-5fL,,5lJe,.\,t"e,arnTsd"h,,avtJ,iPo1n9M,94Po,rr\-lBatedFields,NewYork:Pley,andRe-[23],:tailoBnGirukoihfd:ioSsn,taP1ta9isr9ta8imc,ae\ltCTerhesca,,naetBidosan,s\se"Ad,[24]Koutrouvelis,.,r,1938111,-3\",CommunicationsinfoSrtathtiestEicsstim{aStiimonuloafttiohne[25]KDoutrouvelis,,.B,1109,801,7-\-typeEstimationoftheParameters[26]Kozubos,JournalofAmericanStatisticalAssociation,75,,,li1n9g94F,i\nTanhceiaTlhDeoartya"o,fEGueroompeeatnricJoSutarnbalel[27]LGiAu,:oFrsienna,n1c9ia9l5M,\Modaexliinmgu,m74L,[28]Lmoenngte"r,sFtaoeuyr,",.JZoaunrgnaarli,of1A99p6p,lie\dREisckoMnoemtriectsrTicMs,T1e0ch,-[29]MJoaunrdneallborfotB,.,ss194n60,7(,N\1eT9w6h7eY)V,o3ark9lu:..",
[30]oMJfRetoauPnrodnleaitllibcroaofltB,.,m1y199676313ab,(\,1\N9Te6wheMVeatrhioadtisoninoSftaCtiesrttiacianlEScpoencoumlaitcivs"e,PJoriucrensa"l[31]Mandelbrot,.,3),421-440.,[32]MeanMndoedl eblreotB,aBs ,s12le96s62P(1,r\9oS6cu3er)s, eicfusldateifPs:auFlaLit esvEy,mCpoirmiqputeess[33]McCduus254(1962),,.,a1n9d96C,\.,aeldAs,ppHlaicnadtbioonoskooffSSttaabtilsetiDcsis-trSibtuattiisotnics"a,l[34]dttMaiotcntiChonPuoildloscihn,.,nc1e9,8A6,m\SsitmerpdlaemC:,f1S4t,,,i,."P,aCoolemllma,-SimulationandCompu--[35]Mittn9-1136.,ri5butionalModelsforth,1998a,\UnconditionalandConn--[36]Mitt,'sIndex",Asia-Paci cFinaoDfoSwtnna-itkSiidSe.,,"v,,T1e9c9h8nbic,\aTlRheepPorret,diIcntsitointutoef[37],seMaSntadbElecPonaoremtieatnricDs,UniversityofKiel,,",,1C9o9m8cm,\uAniTcaatiiloEnsstiimnaSttoartifsotrictshe-[38],,lnei2tldli7ae,E,-esvt,a1b9le97P,\roMceosdseelsi"n,gMthaenPusecrrsiipstte,nInce-[39]M "i,ttAnpikp,,m1e9t9r6ic,s\,TUainlivEesrtsiimtyatoifonKioefl,[40]MnmaittitvneikS,"v,,tEe1rc9so9,n39oa,m,5\-[41]MAsistetntiRk,,t1a9b9le3bL,\aRwes'palyndtoSoCm,o1me2Em(xe3tn)e,ts26o1n-3'-loi1.,,\shS,et1ia2rbTAnsions",Econo-[42]Mittnik,.()PR,prl3aio4cccha7eet-3siv8so,,F\RAinellataentrcenidaaltTeMoMpoicudslet"lii,vnBagr"oia,sttioennS:abiusastneriris-,83
[43]MRaitttens:ikA,SG.,,,d1e9l9"6,,P\rDoicseterdibinugtsioonfothfeESxecmhangeLonikeDliahtoa,,,RSaecphte,v1,,o1ganoglu,,1997,\Maximinuamr[44]Mittnik996,Sozopol,",Manuscript,Institute[45]Mittnik,Sic.,,UniversityofKiel,,iendFs,i\nAanPcer:acRStaioccmaheevE,,r1et9i9c8a,l\ASstpaebcletsP",-SSttpaapbtillsiectaicDtsiiosntrsi"b,uBtioosntso",lWGuidetoHeavyTails:StatisticalTechniquesand[46]Nolan,.,1998a,\Mn:axBiiomrrkukhmianugLseiPkrae,pl7ihe9ro-,[47]Nolan,J.,.,1998b,\ParameterizationsofStableDistributions",Statistics[48]1Nr3oi(bl4auP)nt,i,,,t\stN"er,usmC,3oe8mr,-DSteoncshitaisetsicanMdodDeilss-,[49]Panorska,,,,1995,\StableGARCHModels[50].,"o,mAbp,,s1L9e7tt5e,rsT,h8e15E,[51]PickandsJ.,1975,\SetaSttiastbleLaws",Biometrica,62,"S,tJaotiusrtincasl,o3(icalInferenceUsingExtremeOrderStatistics",[52].,1972a,\Est1im),-[53].,1972b,\AppliedMultivariateAntion,67(340),842-846.(eVicrlinaatgRl,)PDMroalysis",NewYork:Holt,Rine-[54]Racheavn,,,"nt,oMvfaaM-nJaeogrtecomuvraey,nB919..30H,,r\LSisaotpfotlwva,,uM1re9as9r9k,e\tTRecishk-[55]Rachev,,ib1utedFinancialReturns".forMod-[56]Rachev,,1993,\TetsStcoiennAces,s1993,,ct1i4o(n1)o,"a,nPdroombabVilairtyiabalneds84
[57]PS&armoHcaoerodnitsky,,1994,\:StochasticModelswithIn niteVariance",NewYork:Chapman85
DAUSANNEPDIVAAUERRPTSPMILTEIYENDTOPFORFCOASBTLAAIFBTOIILSRITNTIYCSIANAPDNGNSNDDTTIMATUABTTAHEREOBMAFARSTATIA,-cMheAMvI@LAl:@-HNE0MSPEIWVAAAAEIRENLNRTS:,eLOSANGELESaESSOO,DNCAPSCL9AH0Z0O9AO5,LBOX951481E-OIOMVLLIAODSIIRELONA:@,DkChOiOnSdC8a0nH4@
