Risk Management with the Multivariate Generalized Hyperbolic Distribution

Transcription

1 Economtrcs Rsk Managmnt wth th Multvarat Gnralzd Hyprbolc Dstrbuton Calbratd by th Mult-Cycl EM Algorthm by: Marcl Holtslag h quston, mntond by Pagan (996), s whthr th mor compl modls to captur th natur of th condtonal dnsty of rturns ar bttr sutd compard to smplr but asr to us modls. Gradually, th tradtonal Gaussan dstrbuton to modl fnancal rturns has bn rplacd by svral othr vabl dstrbutons sutabl to captur th mprcally obsrvd havy tal bhavor, kurtoss and pakdnss. hs study contrbuts towards th furthr dvlopmnt of th ffctvnss of th multvarat gnralzd hyprbolc dstrbuton (MGHyp) whn t s usd to forcast th possbl nt day portfolo loss. It dffrntats by ntroducng an asst portfolo modl va DCC-MGARCH and trs to rduc th ovral calbraton tm. By rvwng th subclasss: normal nvrs gaussan (NIG), multvarat hyprbolc (Hyp) and th 0-dmnsonal multvarat hyprbolc dstrbuton (KHyp) followng from th MGHyp class, a rcommndaton s mad about th ovrall prformanc. DCC-Multvarat GARCH h undrlyng portfolo modl s basd on th DCC(,)-MGARCH(,) of Engl (999). It covrs th tm dpndnt corrlatons among th assts and as Laplant t al. (008) mntons, statstcal vdnc dos not support an vn mor complcatd modl f dalng wth fnancal rturns. h modl assums log rturn srs from k assts that ar condtonally multvarat normal dstrbutd wth constant man μ and covaranc matr H t. h nformaton st I t conssts of all known nformaton untl tm t. r t I t I ~ N( μ, H t ) wth Ht = DRD t t t standard dvatons h t, spcfd by k unvarat GARCH(,) spcfcatons gvn by: h = ω + α ( r μ ) + β h t, t, t, for =... k wth suffcnt rstrctons on paramtrs α + β < and non ngatv varancs. h k k dynamc corrlaton matr, R t, wth ons on th th dagonal s gvn by: R = dag{ Q } Q dag{ Q } t t t t wth Q t th dagonal transformaton matr of th dynamc corrlaton matr. whr D s a k k dagonal matr of tm varyng t Marcl Holtslag Marcl Holtslag obtand hs Mastr s dgr n Fnancal Economtrcs at th Unvrsty of Amstrdam n May 0 and durng hs study h has contrbutd to svral VSAE commtts. hs artcl s a brf summary basd on hs Mastr s thss wrttn undr th suprvson of Dr. Smon Broda and Prof. Dr. Ptr Boswjk. Qt = ( α β) Q+ αε ε' + βqt t t Lastly, lt th standardzd rsduals ε t b dnotd by ε = D ( r μ) t t t h uncondtonal covaranc matr Q s a k k matr stmatd from th standardzd rsduals ε t. A dsgn fatur of th DCC modl maks t possbl to stmat t as a two stp optmzaton problm usng th (quas) mamum log-lklhood mthod. As long as th frst stp paramtr stmats ar consstnt, th scond stp paramtr stmats ar consstnt as wll assumng rasonabl rgularty condtons and a contnuous functon 8 AENORM vol. 9 (73) Dcmbr 0

2 Economtrcs n th nghborhood of th tru paramtr valus. Aftr th stmaton procss, th standardzd rsduals ar usd to calbrat th multvarat gnralzd hyprbolc dstrbuton. Multvarat gnralzd hyprbolc dstrbuton Although a vast ltratur has bn wrttn dscrbng all sorts of dffrnt havy tald asymmtrcal dstrbutons, t s th dnsty of Barndorff-Nlsn (977) that s qut ntrstng. Whl th orgnal papr concntrats ts rsarch to modl mass-sz dstrbutons of aolan sand dposts, th ndpndnt calbraton of th thrd and fourth momnts showd potntal to modl fnancal rturns. Snc 99 thr srous attmpts wr mad to mplmnt th MGHyp dnsty for fnancal problms and ths papr uss th latst, most gnral, vrson covrd by Protassov (004) and McNl t al. (005) snc t handls th unwllng tractablty ssus. h drvaton s not that complcatd f on starts wth th assumpton of th Normal-Man-Varanc-Mtur d X = μ+ Wγ + WA () d k wth Z ~ N k (0, I k ), A wth AA ' = of dmnson k k and μ and γ ar paramtr vctors n k. h rmanng stp to drv th MGHyp dstrbuton s to assum that th mtur wght W follows a Gnralzd Invrs Gaussan or N (, χψ, ) dstrbuton. hs rsults n th followng MGHyp dnsty prsson: f ( ) = p{( μ)' γ} ψ ( χψ ) k ( π) K ( χψ ) k K k χψ k ψ ( χψ ) wth χ and ψ dfnd by ( )' ( ) = ' + ( ) χ = μ μ + χ ψ γ γ ψ and K k th Bssl functon of th thrd knd. Each paramtr dfns on partcular part or shap of th dnsty; controls th shap, χ th pakdnss, ψ th dffrnc btwn th statstcal skwnss and kurtoss stmats, μ s th locaton vctor, s th dsprson matr and γ controls th skwnss vctor. All s functon argumnts (, χψ,, μ,, γ) ar n gnral unknown, hnc som stmaton procdur s ncssary. s consdrd to b prdfnd snc t taks qut som tm to stmat ts valu whl th dffrnc btwn a prdfnd or stmatd s nglctabl. h othr fv paramtrs ar stmatd by mans of th Epctaton- Mamzaton (EM) algorthm. Calbraton by EM h am of th Epctaton-Mamzaton algorthm s to mamz th condtonal pctaton of th full modl log-lklhood functon such that f th datast s ncomplt, consstnt paramtrs could stll b stmatd. Each traton of th EM algorthm conssts of two stps, calld th Epctaton stp that dals wth th mssng valus and th Mamzaton stp to stmat th paramtrs. Dfnng th E-stp Lt, strctly non-ngatv, dnot th currnt EM cycl and lt Θ dnot th collcton of paramtrs, μ,, χ, ψ and γ at cycl (p) such that th E-stp s dfnd as Q( Θ Θ ) =I E[ log f ( Θ), Θ complt obsrvd ] Unfortunatly, th complt data spcfcaton dpnds not only on th obsrvatons, but also on th mssng varabls w snc th obsrvatonal data for th GIG dstrbuton s unavalabl. Estmatng th jont dnsty f ( w, ) s thrfor qut dffcult n ts prsnt form but f somhow t s known that f ( w Θ) has bn ralzd, ths knowldg can provd nformaton whthr f ( w; Θ) has also bn ralzd. Assum f ( w Θ ) > 0such that f( Θ ) = f( w; Θ) f( w Θ) complt Lt b a vctor of dmnson k contanng standardzd rsduals of th DCC(,)-MGARCH(,) modl of k assts at som tm whr [... ], assum that all obsrvaton vctors ar capturd n a k vctor (,...,..., ) and lt th latnt varabls w = (w,...,w,..., w ) b drawn by N (, χψ, ) gvn as χ ( χψ ) f ( w ; χψ,, ) = w K ( χψ ) ( χ + ψ w w ) wth K ( χψ ) as th modfd Bssl functon of th thrd knd wth nd. h prsson for th condtonal dnsty f( w Θ) s drvd usng th Normal-Man- Varanc-Mtur (). f( w μ,, γ) = ( π ) k k w ( μ)' ( μ) w ( μ )' γ w γ ' γ By smply substtutng th found dnsty prssons, on vntually fnds th compltly dfnd E-stp as Q Q Q ( ΘΘ ) = (, μ,, γ) + (, χ, ψ) AENORM vol. 9 (73) Dcmbr 0 9

3 Economtrcs k Q () = log I E[ log w Θ ] + ( μ)' I E [ w Θ ] ( μ)' ( μ) ' γ γ E k I [ w Θ ] log( π ) Q () = ( ) I E [ χ log w Θ ] I E [ ψ I E [ w Θ ] log χ + logψ log [ K ( ) χψ ] w Θ ] It can b shown that all thr condtonal pctatons followng from Q () and Q () ar actually dfnd by th frst momnt of th Gnralzd Invrs Gaussan dstrbuton f on utlzs Bays rul. Dfnng th M-stp ) h updatd paramtrs Θ ar found va th scond ( ) ( ) ( ) stp by sparatly mamzng (, p, p, p Q ) μ γ and (, Q, ) χ ψ. St th drvatv of Q () wth rspct to μ, γ and qual to zro and smply solv th systm of unknowns. ( ) δ p+ ( ) γ = δ η ) ( ) p δ γ + = μ = δ ( p ) ( p ) ( p ) + = δ ( μ + )( μ + )' ) ) + η γ ( γ ') Mamzng th quas log-lklhood functon Q () wth rspct to χ and ψ s prformd by a numrcal mamzaton mthod. χ ψ ma ( ) ξ δ η χψ, log( χ) + log( ψ) log[ K ( χψ )] Snc only Q () s optmzd, nstad of th complt loglklhood, t s rqurd to rcalculat th wghts δ, η and ξ usng th updatd paramtrs μ, and γ of th currnt cycl (p). Furthrmor, to addrss th nar sngularty problm for, th dsprson matr s scald usng th dtrmnant of th sampl covaranc matr. ) = cov( ) k ) k ) h E- and M-stp kps updatng th paramtrs untl th dffrnc btwn th cycls (p) and (p-) s nglctabl. Rsk assssmnt h on day ahad forcast portfolo rturn s stmatd by th undrlyng DCC-MGARCH portfolo modl wth th rsduals followng th MGHyp dstrbuton, calbratd by th prvous 500 obsrvatons usng th condtonal VaR approach wth th nomnal covrag lvls 95% and 99%. A rollng wndow usng th prvous 000 obsrvatons has bn tstd, but no sgnfcant dffrncs compard wth th rollng wndow of 500 wr notcabl. Lt b th asst wghts, sum up to on and lt H t+ b th forcast DCC-MGARCH covaranc matr such that th forcast portfolo rturn s dnotd by ' = t ' μ + + ' t + r H wth ε as th MGHyp dstrbuton. o stmat th condtonal VaR usng a multvarat dnsty modl s rathr complcatd du to th ntgrand, but by ntroducng wghts th problm translats to an ordnary unvarat cas by whch w all know how to solv. h proof s asy. Assum a multvarat lnar functon BX + b, assum that th ntrcpt vctor b = 0 and that k matr B s actually a wghtng vctor such that th sum of th wghts quals on. Fnally, lt X dnot th Normal-Man-Varanc-Mtur and W dstrbutd by th GIG dstrbuton such that th multvarat cas translats to th unvarat on. Applcaton and rmarks h qually wghtd portfolo s constructd by th S&P 500 top tn consttunts by markt cap; Appl Inc (AAPL), Chvron Corp (CVX), Gnral Elctrc (GE), Intl Busnss Machns Corp (IBM), JP Morgan Chas & Co (JPM), Mcrosoft Corp (MSF), Proctr and Gambl (PG), A& (), Wlls Fargo & Co (WFC) and Eon Mobl Corp (XOM). h fnt sampl =,766, for th prod 0/0/000 to 0/0/0, s formd by takng daly ngatv log rturns of th adjustd daly clos prc. It s apparnt from dscrptv statstcs that ght out of th tn assts ndurd a loss ovr th covrd prod du to th lqudty crss. Furthrmor, all tn asst rturns hbt havy tal and asymmtrcal proprts. Fnally, Marda s tst to tst whthr th portfolo s gaussan dstrbutd s rjctd at a % sgnfcanc lvl wth p-valu Rsults By rvwng th subclasss: normal nvrs gaussan (NIG), multvarat hyprbolc (Hyp) and th 0- dmnsonal multvarat hyprbolc (KHyp) a comparson s mad whch of th hyprbolc subclasss prforms th bst to handl th obsrvd havy tal and asymmtry. Fgur shows two of th twlv outcoms of th tm srs analyss llustratng th rsk volatons by + markngs whl tabl and prsnts th statstcal ε 0 AENORM vol. 9 (73) Dcmbr 0

4 Economtrcs data. It s apparnt from tabl and that Chrstoffrsn uncondtonal covrag tst ndcats strong vdnc to rjct th null of corrct covrag for all thr symmtrc dstrbutons usng th nomnal 95% covrag lvl. Of ths thr dstrbutons, only th NIG s found to b statstcally sgnfcant for th Chrstoffrsn condtonal tst. Whl th lattr dstrbutons all undrstmat rsk, no sgnfcant problms ar found whn stmatng th rsk usng th asymmtrcal dstrbutons. For th nomnal 99% covrag lvl no strong statstcal sgnfcanc s found of undr or ovrstmatd rsk for th symmtrcal as wll as th asymmtrcal dstrbutons. It sms suspcous that all thr symmtrcal dstrbutons outprform th asymmtrcal dstrbuton basd on th MSE valu. Frstly, th portfolo s havly skwd accordng to Marda s tst. Scondly, th asymmtrcal dstrbuton nsts th symmtrcal dstrbuton such that on should pct that at last on asymmtrcal subclass could outprform a symmtrcal Fgur. h on day ahad condtonal Valu at Rsk usng th Hyprbolc dstrbuton for th nomnal 95% (nd ln from top) and 99% (3rd ln from top) covrag lvl. Condtonal VaR volatons ar ndcatd by + markngs wth 95% blow th 99% covrag lvl. abl. Backtstng tst statstcs basd on th 95% covrag lvl. h p-valus ar dnotd btwn th parnthss and () ndcats a rjcton at % (5%) sgnfcanc lvl. All thr symmtrc subclasss ar rjctd by th Kupc tst at % sgnfcanc lvl. ˆβ Covrag Indpndnc Condtonal MSE asym NIG (0.0707) (0.7696) (0.870) asym HYP (0.0880) (0.7668) (0.34) asym 0dm Hyp (0.98) (0.736) (0.4004) sym NIG (0.00) (0.4890) (0.0038) sym HYP (0.003) (0.5685) (0.00) sym 0dm Hyp (0.0084) (0.6466) (0.08) abl. Backtstng tst statstcs basd on th 99% covrag lvl. h p-valus ar dnotd btwn th parnthss and () ndcats a rjcton at % (5%) sgnfcanc lvl. ˆβ Covrag Indpndnc Condtonal MSE asym NIG (0.3085) (0.58) (0.53) asym HYP (0.94) (0.488) (0.7796) asym 0dm Hyp (0.759) (0.394) (0.384) sym NIG (0.987) (0.3776) (0.967) sym HYP (0.655) (0.446) (0.664) sym 0dm Hyp (0.0949) (0.3459) (0.59) AENORM vol. 9 (73) Dcmbr 0

5 Economtrcs subclass. Du to th parsmonous bhavor of th MSE, mor obsrvatons ar pland by th smplr and lss compl symmtrcal dstrbuton. hs rsults n lowr MSE valus and falsly ndcats th bttr sutd modl. It probably could plan th odd rankng, and thrfor ts advcd not to rank th modls basd on th MSE valu. If on only compars th stups f th sam assumptons ar usd,.g. symmtry and nomnal covrag lvl, ts follows from tabls and that n ths four cass th NIG dstrbuton outprforms. hs rsult s not surprsng and has bn documntd n othr paprs, for nstanc by Protassov (004) and McNl t al. (005). Protassov, R. (004). Em-basd mamum lklhood paramtr stmaton for multvarat gnralzd hyprbolc dstrbutons wth fd. Statstcs and Computng, 4: Calbraton tm mprovmnt h calbraton of th MGHyp dnsty by EM s consdrd to b n gnral a slow optmzaton procss. Lt th tm to optmz on cycl of th backtstng analyss b gvn as fv mnuts, such that stmatng th full backtstng sampl (,66 cycls) taks a shockng ght days to complt. It would smply tak too much tm for an mprcal study wth twlv dffrnt dstrbutons, namly 96 days.hs study proposs th us of paralll procssng usng mult cor dsktops. h ffctvnss of th paralll procssng unt s notcabl snc t rducs th runnng tm by 600%. Usng on of th symmtrc dstrbutons rsults a runnng tm of mrly s hours whl th asymmtrc dstrbutons taks svn hours to complt. hs s prfctly planabl bcaus th symmtrc cas assums γ = 0 such that t sn t stmatd by th EM algorthm. Rfrncs Barndorff-Nlsn, O. (977). Normal nvrs gaussan dstrbutons and stochastc volatlty modllng. Scandnavan Journal of Statstcs, 4:0-3. Engl, R. (999). Dynamc condtonal corrlaton - a smpl class of multvarat garch modls. Economtrca, 50: Laplant, J., Dsrochrs, J., and J.Prfontan (008). h garch(,) modl as a rsk prdctor for ntrnatonal portfolos. Intrnatonal Busnss and Economc Rsarch Journal, 7:3-34. McNl, A., Fry, R., and Embrchts, P. (005). Quanttatv Rsk Managmnt: Concpts, chnqus and ools. Prncton Unvrsty Prss, Prncton, Nw Jrsy. Pagan, A. (996). h conomtrcs of fnancal markts. Journal of Emprcal Fnanc, 3:5-0. Full sampl rducd by th frst 500 calbraton obsrvatons. AENORM vol. 9 (73) Dcmbr 0