PORTFOLIO CHOICE WITH HEAVY TAILED DISTRIBUTIONS 1. Svetlozar Rachev 2 Isabella Huber 3 Sergio Ortobelli 4

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1 PORTFOLIO CHOIC WITH HAVY TAILD DISTRIBUTIONS Sveloar Rachev Isabella Huber 3 Sergo Orobell 4 We are graeful o Boryaa Racheva-Joova Soya Soyaov ad Almra Bglova for he comuaoal aalyss ad helful commes. S. Rachevs research was suored by gras from Dvso of Mahemacal Lfe ad Physcal Sceces College of Leers ad Scece Uversy of Calfora Saa Barbara ad he Deusche Forschugsgemeschaf. Orobells research has bee arally suored uder Murs 4% 6% 3 ad CNR-MIUR-Legge 95/95. Uversy of Karlsruhe Germay ad Uversy of Calfora Saa Barbara 3 Uversy of Karlsruhe Germay 4 Uversy of Bergamo Ialy. Coac auhor: Dearme MSIA Va de Caaa 47 Bergamo Ialy

2 Absrac: Ths aer aalyes orfolo seleco models wh heavy aled reur dsrbuos. Frsly we eame vesor s omal choces whe we assume resecvely eher Gaussa or sable o-gaussa ucodoal dsrbued de reurs. The we aromae dscree me omal allocaos assumg reurs followg a ARMA rocess. Secodly we eame dffere erformace measures alerave o Share Rao. I arcular we aalye several allocao roblems whch cosder orfolo seleco models based o dffere erformace raos. For each allocao roblem we dscuss a e-os mul-erod orfolo seleco aalyss order o descrbe ad comare he samle ah of fal wealh rocesses. Fally we descrbe furher auoregressve orfolo choce models. Key words: Sable dsrbuos orfolo seleco ARMA models. AMS Subec Classfcao: 9B8 67 6G

3 . Iroduco Over he las ffy years he roblem of omal orfolo seleco has los oe of s allure or morace for he facal commuy. Jus cosder he followg reaso whch wll aeal mmedaely o every vesor. A a me whe more ha ffy erce of all facal asses Norh Amerca are corolled by eso or muual fuds a lo of eole aarely emloy someoe else o maage her moey. Also hey are obvously wllg o ay hgh fees or eeses for hese servces ad are aurally very eresed how well her fuds are erformg. The mea-varace aalyss develoed by Marow ad Tob geeraled o a eulbrum heory by Share Ler ad Moss ad o a er-emoral heory by Samuelso ad Mero was he frs heory o gve rgorous resuls o he orfolo seleco roblem erms of he mea ad he varace. However may crcsms ad emrcal reecos have uderled he rsc lms of he mea-varace aromao. Probably Roll [4] [5] [6] was he frs o clearly udersad he weaesses of he heory ad he emrcal defceces. O he oher had he fudameal wor of Madelbro [5] [6] ad Fama [] has sared cosderable eres sudyg he emrcal dsrbuo of facal asses. The ecess uross foud Madelbro s ad Fama s vesgaos led hem o reec he ormal assumo geerally used o usfy he mea-varace aroach ad o roose he sable Parea dsrbuo as a sascal model for asse reurs. The Fama ad Madelbro s coecure was suored by umerous emrcal vesgaos he subseue years see amog ohers M ad Rachev [3] Orobell Rachev ad Schwar [9]. Ths aer reses ad dscusses codoal ad ucodoal orfolo seleco models for reurs wh heavy als. Frsly we eame he case of sub Gaussa K-sable dsrbued reurs. Ths assumo erms a mea rs aalyss rey smlar o he Marow-Tob mea varace oe. As a maer of fac hs model adms he same aalycal form for he effce froer bu he arameers he wo models have a dffere meag. Therefore he mos mora dfferece s gve by he way of esmag he arameers. I order o comare he erformace of Gaussa ad sable models we aalye a vesme allocao roblem. I cosss of he mamao of he mea mus a measure of orfolo rs. The comarso made bewee he sable sub-gaussa ad he ormal aroach erms of he allocao roblem has dcaed ha he sable sub-gaussa allocao s more rs reservg ha he ormal oe ad ca gve more oorues of earg. Precsely he sable aroach dfferely from he ormal oe cosders he comoe of rs due o he fa als. Secodly we eame dyamc orfolo choces whe reurs follow a ARMA model. Thus he mulsage orfolo allocao roblem we aalye he vesor s choces cosderg a ARMA model for he fuure scearos of orfolo reurs. The we comare vesor s omal allocaos obaed whe he resduals are eher K-sable dsrbued or Gaussa dsrbued. Thus order o value he mac of hese dsrbuoal assumos we roose ad eame a vesme allocao roblem. Fally we roose a e-os dyamc comarso amog orfolo seleco models based o he dffere erformace measures. I arcular we use Germa mare daa ad we comare he eeced logarhmc uly of vesors whe he mare orfolo s comued mamg a gve erformace measure. I Seco we roduce orfolo heory whe reurs are ucodoally sable dsrbued. I Seco 3 we comare he sable sub-gaussa mulvarae aroach wh he

4 ormal mulvarae oe. Seco 4 rooses a comarso amog mul-sage codoal orfolo choce models. Seco 5 assesses he ably o forecas omal choces of dffere erformace measures. Fally we brefly summare he resuls.. The sub-gaussa -sable model I hs seco we aalye he roblem of omal allocao amog asses: of hose asses are sable dsrbued rsy asses wh reurs [... ] ad he h asse s rs-free wh reur. Assume he vecor of rsy reurs [... ] s sub-gaussa K-sable dsrbued wh < K <. The he characersc fuco of has he followg form e e Q µ where Q s a osve defe -mar P s he mea vecor. The erm s defed by [ ~ ~ ] ~ where ~ µ s he cered reur he covarao [ ~ ~ ] bewee wo oly symmerc sable radom varables ~ ad ~ s gve by [ ~ ~ ] s s sg s ds arcular S ~ s ds [ ~ ~ ]. Here Rds s he secral measure ad has S suor o he u crcle S. Ths model ca be cosdered as a secal case of Owe-Rabovch s ellcal model see Owe ad Rabovch []. However o esmae rocedure of he model arameers s gve he ellcal models wh fe varace. I our aroach we use ad o rovde a sascal esmaor of he sable effce froer. To esmae he effce froer for reurs gve by we eed o cosder a esmaor for he mea vecor P ad a esmaor for he dserso mar Q. The esmaor of P s gve by he vecor µˆ of samle averages. Usg lemma.7.6 Samorodsy Tau [8] we ca wre for every [ ~ ~ ~ ~ ] 3 ~ ~ where ~ sg ~ ~ ad he scale arameer ca be wre ~. The ca be aromaed by he mome mehod suggesed by Samorodsy Tau [8] Proery..7 he case U for every

5 ~ u s udu ~ Moreover he followg lemma holds. Lemma For ay sub-gaussa _-sable dsrbued vecor ~ [ ~... ~ ] wh ull mea ad <! < follows ha ~ for ay [ ] ad ~ ~ for ay [. I arcular whe he vecor ~ [ ~... ~ ] s mulvarae ormal dsrbued wh ull mea ad varace covarace mar V [v ] follows ha v ~ for ay ad ~ ~ v v 3 4 for ay. Proof Suose we have a seuece of radom varables X L µ { X : X dµ < } ha coverges dsrbuo o a radom varable X L µ he he momes curves g X ha are aalyc fucos as Z [ coverge uformly o g X he erval [ ]. Cosder a seuece X S where ad he he seuece X coverges dsrbuo o a Gaussa radom varable X wh ull mea ad varace eual o. Thus X lm u s lm u X udu s. udu 4

6 ad s udu u. Because ~ ~ ~ ~ ~ ad for ay... he elemes of dserso mar ] [ Q ca be defed ~ ~ f A for every 5 [ where A ad f ~ ~ ~ ~ ~. I addo as he V Q where V s he varace-covarace mar ad cosderg ha ~ ~ ~ f f s a aalyc fuco he hess holds. The above suggess he followg esmaor ] ˆ [ ˆ Q for he eres of he uow covarao mar Q 6 ~ ~ ˆ ˆ N N where he ] s esmaed as follows 7 ~ ˆ ˆ N N I s mora o observe ha he bes deeds o K ad o he umber of observaos we have. The rae of covergece of he emrcal mar ] ˆ [ ˆ Q o he uow mar Q o be esmaed wll be faser for a large samle f s as small as ossble see Rachev [] Lamaa Orobell ad Rachev [3]. Uder hese assumos he wealh e W assocaed o he orfolo s gve by W S d e W W W

7 ad W he w where K s he de of sably W Q s he scale dserso arameer W s he sewess arameer ad W e. Recall ha whe W < W W W ad W W he W secod order sochascally domaes W ad every rs averse vesor refers W o W see Orobell [8]. Thus whe he reurs [... ] are oly sub-gaussa K-sable dsrbued ad ulmed shor sales are allowed every rs averse vesor wll choose a omal orfolo amog he orfolo soluos of he followg omao roblem: m Q subec o ; 8 µ e mw for some gve mea m W. Therefore every omal orfolo ha mames a gve cocave uly fuco u belogs o he mea-dserso froer m f m µ e Q µ e 9 m f m < µ e Q µ e where µ ; m µ e ; e [...] ; ad Q. Besdes he omal orfolo weghs sasfy he followg relao: m Q µ e. µ e Q µ e Noe ha 9 ad have he same form as he mea-varace froer. However eve f Q s a symmerc mar s osve defe he esmaor roosed he sub-gaussa case see formulas 6 ad 7 geerally s o symmerc. Therefore we could oba a cosse suao whch has sable dsrbuo wh egave suared scale arameer. However mos of he cases ˆ Q ˆ Qˆ Q > for every vecor R because geerally s a osve defe mar 4. Suose [... ] s K -sable sub-gaussa dsrbued wh mea µ ad dserso mar Q. If we aromae he vecor dsrbuo wh a K -sable sub-gaussa law wh 4 Observe ha for every we ca verfy ha R we ge Q > Q ˆ Qˆ ˆ f ad oly f Q ˆ Qˆ s osve defe order o avod sable orfolos arameer esmaors. Moreover we observe ha he symmerc mar he dserso mar Q whose sascal roeres have o be roved. Q ˆ Qˆ s a osve defe mar. Thus wh egave scale s a alerave esmaor of

8 < K < K mea µ ad dserso mar Q he ˆ A Q ˆ Q ad here are o A coseueces of hs aromao error because he resuls of he orfolo seleco roblem 8 do o chage. Whle f > > we cao guaraee he same resuls of he orfolo choce roblem. Ths frs dfferece s oe of he reasos for cosderg ad sudyg he covergece roeres of he esmaor see Rachev [] ad he suably of he model. Moreover ehbs he wo fud searao roery for boh he sable ad he ormal case see Ross [7] bu he mar Q ad he arameer ] have dffere meags. I he ormal case Q s he varace-covarace mar ad ] s he sadard devao whle he sable case Q s a dserso mar ad Q s he scale dserso arameer. Accordg o he wo-fud searao roery of he sub-gaussa K-sable aroach we ca assume ha he mare orfolo s eual o he rsy age orfolo uder he eulbrum codos as he classc mea-varace Caal Asse Prcg Model CAPM. Therefore every omal orfolo ca be see as he lear combao bewee he mare orfolo Q µ e e Q µ e Q e ad he rsless asse reur. Followg he same argumes as Share Ler Moss s mea-varace eulbrum model he reur of asse s gve by: m Qe where m wh e [...] he vecor wh he h comoe ad ero Q all he oher comoes. 3. A comarso bewee he ormal mulvarae dsrbuoal assumo ad he sable sub-gaussa oe I hs seco we eame ad comare he sable sub-gaussa assumo wh he ormal dsrbuoal oe. Thus we mlcly assume ha reurs are uuely deermed by he mea m ad 5 ha s eher he scale arameer of sable dsrbuos or he sadard devao of ormal dsrbuos. I a rece wor Orobell Rachev ad Schwar [9] comare he sable o-gaussa assumo ad he ormal oe by aalyg omal allocaos bewee a rsless reur ad a bechmar de. Three dffere dees have bee ae o cosderao: CAC4 DAX3 ad S&P5. Ne we eed Orobell Rachev ad Schwar s comarso o he mulvarae case. Ths comarso s formally ad heorecally dffere from he revous oe because here he bechmar de s gve by he mare orfolo whch geerally wll chage f he dsrbuoal assumos chage oo. Thus as a coseuece of Roll [4] [5] [6] Dybvg ad Ross [8] [9] aalyss we observe ha: a a vesor who fs he reur dsrbuos wh a o! -sable sub-gaussa dsrbuo wll cosder as effce he choce of aoher vesor who fs he reur dsrbuos wh a o! -sable sub-gaussa dsrbuo wh ; ad b he sable CAPM s sll subec o some of he crcsm already addressed o he classcal oe.

9 Neverheless seems ha he sable case beer elas he emrcal daa. Ths s he ma reaso why we erre ad aalye he dffere behavor here bewee he vesor who fs he daa wh o sable sub-gaussa dsrbuo ad he vesor who fs he daa wh he o ormal dsrbuo. 3. A omal allocao roblem Frs we cosder he omal allocao amog 4 asses: 3 of hose asses are rsy asses wh reurs [... 3 ] ad he 4h s rsfree wh a aual rae of 6%. We aalye he orfolo choce roblems whe shor sales are allowed ad whe shor sales are o allowed. I vew of hs comarso we dscuss ad sudy he dffereces orfolo choce roblems whou eamg hem so as o choose oe of he wo assumos Gaussa or sub-gaussa. I our comarso we use daly daa ae from 3 eraoal rsy dees valued USD ad uoed from Jauary 995 o Jauary 998. I he aalyss roosed we frs cosder he mamum lelhood esmao of he sable arameers ad of he Gaussa oes for every rsy asse. Thus Table I assembles he aromag arameers obaed from usg Cogy Sysem 5. I order o comare he dffere sable sub-gaussa o dsrbuos ad he o ormal dsrbuos for he asse reurs we assume ha he vecor s sub-gaussa K-sable dsrbued wh K K where K.7488 rereses he average of he dees of sably ad K.8856 rereses he mamum of he dees of sably see Table I 6. Moreover whe he followg ables we cosder he de of sably K we mlcly assume ha he reurs are oly ormal dsrbued. Thus every orfolo of rsy asses s sable dsrbued he followg way: d S m where! s oe of he cosdered dees of sably Q s he resecve scale arameer Q [ ] s he dserso mar wh s he sewess arameer ad m rereses he mea of. Observe ha he mar Q s esmaed wh he mehod defed he revous seco ad hus deeds o he de of sably K for. As observed revously he rae of covergece of he emrcal mar Qˆ o he uow mar Q wll be faser for a large samle f s as small as ossble. However sudyg sable smulaed daa we obaed very good aromaos eve usg o oo small we refer o Lamaa Orobell ad Rachev [3] for furher sudes o hs roblem. I our esmaos we use.6 relave o!.7488 ad.7 relave o! We assume he vesors wsh o mame he followg uly fucoal: U W W c W W 3 5 Ths sofware s develoed by FAalyca Ic. 6 We cosder dffere dees of sably order o value he effecs of heavy-aledess o he orfolo seleco roblems.

10 where c ad are osve real umbers W s he reur of he orfolo s he rs-free asse reur ad s he age orfolo reur gve by euao. Wh referece o he allocao roblem 3 we observe ha rs averse vesors should choose a orfolo W ha mames he uly fucoal 3 for some real a ad some [. We ow ha for orfolo reur W s dsrbued accordg o a sable law S ; m ad W whe a. Now order o solve he asse allocao roblem ma W c W W oce frs ha for all [ ad < < we ge where U W W c W W m c H H a A see he above lemma ad Samorodsy ad Tau [8]. The above relao aalyes he sable o-gaussa case. Whe he vecor adms a o ormal dsrbuo.e.! he for all > U W W c W W m c Hece he real omal soluo of he roblem he mora case s gve by " sg m sg! c V ad 5 where s gve by ad H he sable V he ormal case case. 4 < < Aga oe would eec ha he omal allocao s dffere because he cosa V ad he mar Q are dffere he sable sub-gaussa ad he ormal case..

11 3. Sable versus ormal omal allocao: a frs comarso We aalye he dffereces omal allocaos wh referece o roblem 3 whe he vesor chooses:. o ormal dsrbuo or. o! sable sub-gaussa dsrbuo where!.7488;!.8856 as a model for he asse reurs hs/her orfolo. Uder hese dscve assumos he vesors wh uly fucoal 3 have dffere formao abou he dsrbuoal behavor of daa. I arcular we eame he dffere mare orfolo comoso ad he dffere vesor s wealh allocao he rsless asse. Frs whe shor sales are allowed ad whe shor sales are o allowed we eame omal allocao amog he rsless reur ad 3 de-daly reurs: DAX 3 DAX Performace CAC 4 FTS all share FTS FTS acuares 35 Reuers Commodes Ne 5 Smle average Ne 3 weghed soc average Ne 3 smle soc average Ne 5 Ne 5 soc average Ne 3 Bre Crude Physcal Bre curre moh Cor No Yellow ces Coffee Brala Dow Joes Fuures Dow Joes Commodes Dow Joes Idusrals Fuel Ol No Goldma Sachs Commody S&P 5. We use he rsless reur 6%.a.. Usg he esmaed daly de arameers we ca comue he dserso marces ad he aromag mare orfolos. The dserso mar Q s gve by eher he varacecovarace mar he ormal case or he mar Q he sable cases whch deeds o he de of sably! for!.7488 ad! Therefore as show by Tables II III he mare orfolo weghs Q µ e e Q µ e Q e chage uder he dffere dsrbuoal assumos. We observe ha he mare orfolo comoso does o chage ecessvely whe we use eher he asymmerc esmaor 6 ad Q ˆ ˆ Q 7 of mar Q or he symmerc oe. However usg daly daa he elemes of he dserso marces are of order -6. Thus he aromao he daa used could be deerma for he elemes of he marces. I arcular Table II reses he mare orfolo weghs whe we cosder all 3 asse reurs ad shor sales are allowed. Table III gves he mare orfolo weghs whe o shor sales are allowed. Uder hs cosra we value he mare orfolo weghs erms of he rsy orfolo comosos whch mame he eeded Share rao.e. he mare orfolo weghs are he soluo of he followg omao roblem ma e.... I hs case he omal allocao s reduced oly amog he four rsy asses: DAX

12 Performace FTS all Share Ne 3 weghed soc average Dow Joes Idusrals ad he rsless oe. As argued by Roll [4] [5] Dybvg ad Ross [8] dffere mare orfolos mly a comleely dffere secury mare le aalyss. Thus he aroach whch aes o accou shor sales reses more oorues of earg ha he aroach wh o shor sales cosra. Therefore domaes he oher aroaches. Besdes f he reurs are oly _ sable sub-gaussa dsrbued for some deermed he he Gaussa aroach s effce. Sce geeral effce ad effce orfolos ca lo above ad below he real secury mare le. The aalyss of Tables II ad III os ou ha he comoso of he mare orfolo s srcly led o he de of sably. I fac we see ha he allocao of he mare orfolo each asse comoe s geerally moooe wh resec o he sably de. The he uo suggess ha he sable sub-gaussa aroaches ae more o cosderao he comoe of rs because of he heavy als. Recall ha he al behavor of every sable o-gaussa d dsrbuo X µ wh <! < s gve by S ± lm P ± X > C 6 where C. Therefore he fa als of smaller sably dees uderle he cos rs of he loss comoe of every orfolo. I arcular uder he dverse dsrbuoal assumo we dsgush he dffere erceo of rs he mare orfolo comoes. Ths ssue ca be easly aalyed he mare orfolo weghs wh referece o he 3 reurs whe o shor sales are allowed. I fac Table I shows ha he de of sably of FTS all Share s greaer ha he oher dees of sably of he asses DAX Performace Ne 3 weghed soc average Dow Joes Idusrals. Observe ha Table III he comoe of he FTS all Share he mare orfolo creases wh he de of sably! of he sub- Gaussa aroach ad he comoe of he oher asses DAX Performace Ne 3 weghed soc average Dow Joes Idusrals decreases wh he de of sably. Thus he mare orfolos obaed uder Gaussa ad sub-gaussa dsrbuoal hyoheses cosder he rss due o heavy als dfferely. O he oher had he mea of mare orfolos decreases wh he de of sably. However f we acce he dea ha he mare orfolos rerese some sese he mare behavor he accordg o he classc mea-rs erreao a omal orfolo ha has a greaer mea has also a greaer rs. Ths fac aears clear eough whe we cosder ad comare he dserso measures Q every mea-rs lae for Q µ e every mare orfolo weghs for every ad. Observe ha e Q µ e Q e ~ Q s he dserso measure of mare orfolo cosderg he! -sable Parea aroach. Therefore for every fed mea-rs lae.e. for every fed! sable dsrbuoal aroach we ca comare he mare orfolo rs osos cosderg her rs oso ~ varyg. Accordg o a mea-rs erreao we could observe ha he mare orfolo wh a greaer mea adms also a greaer dserso measure ~ ay mears lae see ables IV ad V. However we oba dffere resuls f we use he Value a Rs

13 VaR R as he rs measure of he mare orfolos whch s mlcly defed by he followg eualy: VaR su{ y : P $ y > }. I fac VaR measures he rs of loss whch s rereseed by he lef al of he mare orfolo dsrbuo. The as reored by Table VI we could observe ha he VaR. ad VaR.5 of he Gaussa mare orfolo are greaer ha he aalogous VaR of he Sub-Gaussa mare orfolos. I hs sese he Gaussa mare orfolo s rser ha he sable mare orfolos because does o cosder he rs due o he heavy als. Moreover comarg he dffere VaR umbers we ca defy he mare orfolo wh de of sably!.7488 as he leas rsy. As a coseuece of relao 6 follows ha every sable o-gaussa dsrbuo d X µ wh <! < has he roery ha S X X < or f < 7 X X or f 8 Hece he wegh of he rs measure X - X omao roblem 3 s geerally greaer for he vesors who use he sable laws for asse reurs whe s ue close o he de of sably K. I Tables VII ad VIII we ls he omal allocao for he ormal ad he sable f. Recall ha s he omal rooro of fuds vesed he rs free asse whch mames W - c W - W where W. We have chose.45 Table VII ad.55 Table VIII so ha s srcly less ha all dees of sably! he daa se. O he oher had we wa o value ad comare he dffere effecs of beg more dsa or closer o he sably arameers!. For ay gve allocao roblem we remar bold characer ad alcs resecvely he greaes ad he smalles allocao he rsless asse. Boh ables show ha he greaes dversy s amog he omal allocaos corresodg o small rs averso coeffce c. I coras he very rs averse vesors would ae a less rsy oso for every dsrbuoal hyohess ad hece he allocaos he rsless asse do o chage very much. As we see from hese ables whe.45 ad.55 he vesors who f he daa wh he Gaussa aroach geerally assume a less rsy oso ha he vesors who f he daa wh he sub-gaussa aroach. Thus f he sable sub-gaussa aromao reses greaer erformace oorues ha he Gaussa oe as observed by may emrcal aalyses he sable vesors have more oorues of earg ha he Gaussa vesors. I arcular he vesors wh!.7488 sable sub-gaussa aroach ves less he rsless asse ha he vesors who f he daa wh he oher aroaches. However f we cosder much closer o! he omao roblem 3 he as a coseuece of 7 ad 8 he vesors who f he daa wh! sable sub-gaussa aroach assume a less rsy oso ha he vesors who f he daa wh he Gaussa aroach. I hs case he sable vesor has a very rs reservg behavor because he does o refer allocag oo much wealh he rsy asse. I hs sese uo suggess ha he sable aroach wh lower de of sably geerally s more rs reservg ha wh hgher de of sably because he comoe of rs due o he fa als of asse reurs s ae o accou. Therefore he sably de lays a sraegc role he sable omal orfolo seleco. Coversely he above

14 omao roblem ca be regarded as a oorue measure of he magude o be gve o he comoe of rs due o he heavy als. The morace gve o s uvely led o he codos of he mare whch he vesor oeraes. 4. Porfolo seleco wh ARMA models The sudy ad he aalyss of he emrcal behavor of daa see amog ohers Agray [] Cambell [5] Fama ad Frech [] Frech Schwer ad Sambaugh [] have reored evdece ha he codoal frs ad secod momes of soc reurs are me varyg ad oeally ersse esecally whe reurs are measured over log horos. However he evdece observed o emoral deedece for daly soc reurs s o srog see amog ohers Fama ad Frech [] ad s sll a source of coroversy wheher soc mare rce models should clude log memory or o see amog ohers Lo [4]. I hs sese he ucodoal models of revous secos rerese vald ad dcave orfolo choce models whe reurs are measured over shor horos. If facal modelg volves formao o as mare movemes s o he ucodoal reur dsrbuo whch s of eres bu he codoal dsrbuo whch s codoed o he formao coaed as reur daa or a more geeral formao se. The class of auo-regressve movg average ARMA models s a aural caddae for codog o he as of a reur seres. These models esecally whe he seres are saoary are wdely used o redc fuure movemes of rces coss ad reurs. The reaso for hs oulary les her smlcy ad because hey gve a good aromao of a very wde class of saoary seueces. ARMA models have he roery ha he codoal dsrbuo s homosedasc.e. cosa codoal volaly. I arcular f we model asse reurs wh a ARMA model of auo-regressve order ad a movg average order reurs assume he followg form: a a % b % where {% } s a whe ose rocess. To secfy he orders ad we ca follow he sadard Bo-Jes defcao echues see amog ohers Bo ad Jes [3] Browell ad Davs [4] ad we sec he samle auocorrelao fucos SACFs ad samle aral auocorrelao fucos SPACFs of he reur seres. I arcular we ca assume ha he seuece of ovaos {% } s a fe varace rocess cossg of..d. symmerc radom varables he doma of ormal araco of a symmerc sable dsrbuo wh de of sably ad scale arameer >. Tha s N N / % d Y as N uy where Y has characersc fuco Y u e e. Uder hese assumos Mosh Gadrch Kluelberg ad Adler [7] deermed esmaors for hs rocess based o he samle erodogramm of ad suded her asymoc roeres. Oher esmaors a Gaussa- Newo ye ad a M-esmaor for he arameers of he ARMA rocess wh fe varace ovaos were roosed ad suded by Davs [7]. I s eresg o observe ha coras o ARMA rocesses wh fe varace he sable case we geerally oba a esmaor he rae u

15 of covergece of whch s cosderably faser. Ne we cosder a orfolo choce ARMA model ad we wa o comare he mac of sable resduals ad Gaussa oes. So we roose a dyamc orfolo choce amog hree rsy dees Dow Joes Idusral DAX ad FTS all share. We cosder 5 orfolos of hese asses ad we assume ha orfolo reurs adm he form a a % b% where we suose he seuece of ovaos {% } are eher Sable or Gaussa dsrbued. We cosder 83 daly observaos of de reurs from /4/995 ll /3/98 ad for each orfolo we verfy saoary ad we esmae he arameers of he model. By frs comarso bewee he Gaussa ad he sable o-gaussa hyohess aears clear ha sable dsrbuos aromae much beer he resduals ha he Gaussa oe. As a maer of fac wh he Kolmogorov-Smrov es we ca comare he emrcal cumulave dsrbuo fuco cdf F of he resduals corresodg o several orfolos wh eher a smulaed Guassa or a smulaed Sable dsrbuo fed advace. By hs frs aalyss we ca geerally reec a 5% cofdece level he hyohess of ormaly because we oba ha he robably ha he emrcal dsrbuo of he resduals s Gaussa s o average amog dffere orfolos eual o. -6 << 5%. I addo we could observe ha o average amog dffere orfolos su F F. 875 where F s he fed cdf of he Gaussa law. Boh fucos F ad F are show o Fgure. I coras geerally we cao reec a 5% cofdece level he hyohess ha he resduals follow a sable law because he robably ha he emrcal dsrbuo of he resduals s sable s o average amog dffere orfolos eual o.%. I addo we could observe ha o average amog dffere orfolos su F F. 75 whe F s he cdf of a sable law. Boh fucos F ad F are show o Fgure. I order o value he mac of dffere dsrbuoal aromaos o vesor s orfolo choce we cosder a dyamc asse allocao aroach very smlar o hose roosed by Boeder [] ad Orobell Rachev ad Schwar [9]. So we geerae abou 5 al asse allocaos. These allocaos are he smulaed o he fuure by usg he ecoomc scearos whch are geeraed uder he Gaussa ad sable assumos for he ovaos of he me seres models. Fuure ecoomc scearos are smulaed a daly ervals. Oe se of scearo s geeraed by assumg ha resduals of he varables are..d. ormal ad aoher se of scearo s geeraed by assumg ha resduals are..d. sable. The horo of eres s days ad wo scearos are geeraed for each day so 4 ossble ecoomc scearos are cosdered for each al orfolo. The -day scearo ree s reeaed mes. We assume ha vesors wsh o mame he followg fucoal of fal wealh:.5 U WT WT c WT WT where c s a coeffce of vesor s rs averso. Therefore he vesor wll choose amog he al orfolos he orfolo wegh vecor ] whch [ 3 s 3 S Ws T S s mames he uly fucoal U W T cosderg ha WT s he mea of fal S wealh ha we oba wh orfolo ; WT Ws T WT s he measure of rs S

16 T & assocaed o ; W s R s he fal wealh ha we oba wh orfolo uder T s R s s s 33 s scearo s {... S} ; s he reur of orfolo uder scearo s {... S} me erod ad s s he rae of reur of -h asse uder scearo s {... S} me erod. The calculaos for hs emrcal aalyss were erformed eher Malab evrome or rogrammed Delh. Table IX summares he resuls of hs comarso. I arcular we observe ha for each rs averso coeffce we oba greaer eeced uly usg sable dsrbued resduals. Thus s mlcly cofrmed ha we have resumed a beer dsrbuoal aromao. Jus as he case of he revous emrcal comarso we observe ha here es subsaal dffereces bewee he orfolo allocaos uder he dffere dsrbuoal aroaches. 5. A comarso amog dffere erformace raos Ths seco descrbes he dffere erformace raos eamed he orfolo seleco roblems. I arcular we aalye he roblem of omal allocao amog asses: 9 of hose asses are rsy wh reurs couously comouded [... 9 ] ad he h s s rs-free wh reur. No shor sellg s allowed.e. he orfolo rsy weghs [] for every... ad he rsless wegh are eual or greaer ha ero. Assume ha all orfolos are uuely deermed by he mea ad a rs measure cosse wh some sochasc domace order. The vesor wll choose oe omal orfolo - a lear combao bewee he rsless asse ad a omal rsy orfolo. The omal rsy orfolo s gve by he orfolo ha mame he erformace rao. Thus for ay erformace measure d we comue a "mare orfolo" M ha s he soluo of he followg omao roblem: ma s.. e ;... For dffere erformace measures we oba dffere omal orfolos. Therefore he mare orfolo comoso M M... 9 M foud for each erformace measure d s based o a dffere rs erceo. I arcular we cosder he followg erformace measures roosed leraure see Orobell Rachev Bglova ad Soyaov [] : Share rao: STD where s he rsfree asse ad STD s he sadard devao of orfolo Mma rao: MM

17 where m T MM $ $ ad s he vecor of reurs a me 3 Sable rao: where Q ad ] [ Q ha ca be esmaed wh formulas 6 ad 7. 4 MAD rao: 4 where N N ad os ou he -h observao of vecor r 5 G Rao: where > T T T T 6 Farell-Tble Rao:! "! " where T T /! "! " ma ad T T /! "! " ma. We use f r / f r ad we use / where s he de of sably of vecor. 7 Soro-Sachell Rao: where T T ad we suose /. 8 VaR99% RATIO: 99% VaR where % 99 VaR s he 99% value a rs of orfolo ha s % 99 VaR s he oose of % uale ha s. % 99 $ VaR P

18 9 CVaR99% rao: CVaR where CVaR. 99% [.* ] r $VaR99% I arcular assumg ha o shor sales are allowed we eame omal allocao amog he rsless reur LIBOR ad 9 reurs o he Germa mare Addas_salomo AG Basf AG Bayershe Moore Were AG Coeal AG Bayer AG Hoechs AG Freseus Medcal Care AG MAN AG Heel KGAA. We cosder daly daa he erod Le us suose ha a vesor wh logarhmc uly fuco vess hs wealh he mare assumg ha mare orfolo s deermed mamg oe of he revous erformace measures. The we wa o comare he samle ah ad fal wealh ha we oba uder he dffere aroaches. Thus every day ad for each erformace measure d we have o solve wo omao roblems: he frs order o deerme mare orfolo ad he secod o deerme he omal eeced uly o he effce froer. For each erod we use he las 5 observaos. Therefore whou cosderg rasaco coss ad aes we frs deerme he mare orfolo as soluo of omao roblem. Thus afer erods we ge he mare M orfolo comoso eeced uly gve by: where ma 99% ad he vesor wll choose he orfolo ha mames hs/her 5 s.. $ $ log. " /! s he corresodg -h observao of he LIBOR. I hs way we ge he omal vesme a [] he rsless ad he vecor comoso of rs asses M - M afer erods. Therefore he fal wealh a he -h se s gve by " 5 5. W W rf M r /! - Observg ha Farell-Tble rao ad Mma rao ae o cosderao he heavy als of orfolo reurs he hs e-os mul-erod aalyss geerally cofrms ha heavy als have a fudameal mac orfolo choces. As a maer of fac we observe ha Farell- Tble rao ad Mma rao rese he bes erformaces durg all erod cosdered as we ca see by Fgure 3 ad 4. However a more geeral heorecal ad emrcal aalyss wh furher dscusso sudes ad comarsos of hese erformace raos does o eer he obecve of hs aer ad wll be he subec of fuure research. 6. Coclusos I hs aer we have show ha he classcal orfolo choce models ca be geeraled assumg sable dsrbuos for he uderlyg radom varables ad ha he geeraled models o oly are heorecally usfable ad emrcally esable bu hey geerally have beer erformace ha he resecve Gaussa models whe asse reurs ehb heavy als.

19 By comarg he o ormal dsrbuo wh he o sable sub-guassa oe has occurred ha he resuls receved from he eamed omal allocao roblems are subsaally dffere. I arcular he sable mare orfolo s geerally less rsy ha he Gaussa mare orfolo. Ths uve resul s cofrmed by comarso of he omal allocaos whe he dffere dsrbuoal hyoheses are assumed. Therefore he vesors who f he daa wh he sable dsrbuos are geerally more rs reservg ha he vesors who f he daa wh he Gaussa law because sable laws ae o accou he comoe of rs due o heavy als. Thus we fd ha he al behavor of sub-gaussa ad Gaussa aroaches could mly subsaal dffereces he asse allocao. These resuls are emrcally cofrmed f we comare orfolo choces obaed whe cosderg orfolo reurs ha follow a ARMA model wh sable or Gaussa dsrbued resduals. As a maer of fac we observed ha he dsrbuo of resduals s asymmerc ad leourc ad he hyohess of ormaly s usually reeced uder sascal esg. I addo we show ha he aromao gve by he sable resduals mly beer erformace for rs averse vesors. Fally Aleravely o he revous aalyss we roduce ad comare several erformace measures. The emrcal comarso cofrms ha he classc Share Rao reses worse forecas ables ha oher erformace measures roosed leraure. Amog he alerave models roosed he Farell-Tble rao ad he Mma rao rese beer erformace ad hs resul emhase he mac of heavy als orfolo choces. Refereces [] V. Agray Codoal heeroscedascy me seres of soc reurs: evdece ad forecas Joural of Busess [] G. C.. Boeder Modellg ad maageme of asses ad lables of eso las he Neherlads Worldwde Asse ad Lably Modelg W. T. Zemba ad J. M. Mulvey eds. Cambrdge Uversy Press Cambrdge [3] G.. P. Bo ad G.M. Jes Tme seres aalyss: forecasg ad corol d ed. Holde-Day Sa Fracsco976. [4] P. J. Browell ad R.A. Davs Tme seres: heory ad mehods d ed. SrgerNew Yor99. [5] J. Cambell Soc reurs ad he erm srucure Joural of Facal coomcs [6] J. Chambers S.J. Mallows ad B. Suc A mehod for smulag sable radom varables Joural of he Amerca Sascal Assocao [7] R. Davs Gauss-Newo ad M-esmao for ARMA rocesseswh fe varace Sochasc Processes ad Alcaos [8] P. Dybvg ad S. Ross The aalyss of erformace measureme usg a secury mare le Joural of Face a. [9] P. Dybvg ad S. Ross D_ereal formao ad erformace measureme usg a secury mare le Joural of Face b. []. Fama The behavor of soc mare rces Joural of Busess []. Fama ad K. Frech Permae ad emorary comoes of soc rces Joural of olcal coomy [] K. Frech G. Schwer ad R. Sambaugh eced soc reurs ad volaly Joural of Facal coomcs

20 [3] F. Lamaa S. Orobell ad S. Rachev Value a rs wh sable dsrbued reurs o aear Aals of Oerao Research3. [4] A. Lo Log erm memory soc mare rces coomerca [5] B. Madelbro The varao of cera seculave rces Joural of Busess [6] B. Madelbro The varao of some oher seculave rces Joural of Busess [7] T. Mosh T. Gadrch C. Kluelberg ad R.J. Adler 995: Parameer esmao for ARMA models wh fe varace ovaos The Aals of Sascs [8] S. Orobell The classfcao of aramerc choces uder uceray: aalyss of he orfolo choce roblem Theory ad Decso [9] S. Orobell S. T. Rachev ad. Schwar The roblem of omal asse allocao wh sable dsrbued reurs Techcal Reor Uversy of Karlsruhe o aear he Volume Sochasc Processes ad Fucoal Aalyss Marcel Deer dor 3. [] S. Orobell S. Rachev A. Bglova ad S. Soyaov A comarso amog erformace measures orfolo heory Techcal reor se o SM 4. [] J. Owe ad R. Rabovch O he class of ellcal dsrbuos ad her alcaos o he heory of orfolo choce Joural of Face [] S. Rachev Probably mercs ad he sably of sochasc models Wley Chcheser 99. [3] S. Rachev ad S. M Sable area models face Wley Chcheser. [4] R. Roll A crue of he asse rcg heory s ess Joural of Facal coomcs [5] R. Roll Ambguy whe erformace s measured by he secury mare le Joural of Face [6] R. Roll Tesg a orfolo for e ae mea-varace effcecy Sudes he Maageme Sceces. lo ad M. Gruber eds. Norh- Hollad Amserdam 979. [7] S. Ross Muual fud searao facal heory-he searag dsrbuos Joural of coomc Theory [8] G. Samorodsy ad M. S. Tau Sable o Gaussa radom rocesses: sochasc models wh fe varace Chama ad Hall New Yor 994.

21 Gaussa Parameers Sable Parameers ASSTS Mea µ Sadard devao 6 Ide of sably Sable sewess 9 Sable mea µ Sable scale arameer 6 DAX DAX CAC FTS all share FTS FTS a uares RUTRS Commodes NIKKI 5 Smle Average NIKKI 3 Wegh. Soc Av NIKKI 3 Smle Soc Av NIKKI NIKKI 5 Soc Average NIKKI BRNT Crude BRNT Curre Moh CORN. Yellow Res COFF BRAZILIAN DOW JONS FUTURS DOW JONS Commodes DOW JONS INDUSTRIALS FUL OIL N GOLDMAN SACHS Comm S&P Table I Mamum lelhood esmaos of he Gaussa ad Sable asse reur arameers cosderg daly daa from /3/95 o /3/98

22 ASSTS Gaussa Sable Weghs Weghs for K.8856 Weghs for K.7488 DAX DAX CAC FTS all share FTS FTS a uares RUTRS Commodes NIKKI 5 Smle Average NIKKI 3 Wegh. Soc Av NIKKI 3 Smle Soc Av NIKKI NIKKI 5 Soc Average NIKKI BRNT Crude BRNT Curre Moh CORN. Yellow Res COFF BRAZILIAN DOW JONS FUTURS DOW JONS Commodes DOW JONS INDUSTRIALS FUL OIL N GOLDMAN SACHS Comm S&P Table II Sable sub-gaussa ad Gaussa mare orfolo weghs whe shor sales are allowed

23 ASSTS Gaussa Sable Weghs Weghs for K.8856 Weghs for K.7488 DAX FTS all share NIKKI 3 Wegh. Soc Av DOW JONS INDUSTRIALS OTHRS Table III Sable sub-gaussa ad Gaussa mare orfolo weghs whe o shor sales are allowed MAN PARAMTRS Gaussa Mare Porfolo Mare Porfolo K.8856 Sable Mare Porfolo K STANDARD DVIATION DISPRSION IF K DISPRSION IF K Table IV Sable sub-gaussa ad Gaussa mare orfolo arameers for every mea-dserso lae whe shor sales are allowed MAN PARAMTRS Gaussa Mare Porfolo Mare Porfolo K.8856 Sable Mare Porfolo K STANDARD DVIATION DISPRSION IF K DISPRSION IF K Table V Sable sub-gaussa ad Gaussa mare orfolo arameers for every mea-dserso lae whe shor sales are allowed

24 PARAMTRS Gaussa Mare Porfolo Mare Porfolo K.8856 Sable Mare Porfolo K.7488 VaR % WITH SHORT SALS VaR 5% WITH SHORT SALS VaR % NO SHORT SALS VaR 5% NO SHORT SALS Table VI Sable sub-gaussa ad Gaussa mare orfolo Values a Rs

25 Allocao _ he rsless asse cosderg he mare orfolo o 3 asses whe ulmed shor sales are allowed Coeffce c of omal allocao Sable _ Omal allocao Omal allocao The omao whe roblem Gaussa whe whe Allocao _ he rsless asse cosderg he mare orfolo o 3 asses whe ulmed shor sales are o allowed Coeffce c of omal allocao Sable _ Omal allocao Omal allocao The omao whe roblem Gaussa whe whe Table VII Ths able comues he allocao _ he rsless 6% aual rae daly for dffere rs averso coeffce of he omao roblem ma W.45 W W where W r ad r s eher he Gaussa Mare orfolo for or he sub-gaussa mare orfolo for or

26 Allocao _ he rsless asse cosderg he mare orfolo o 3 asses whe ulmed shor sales are allowed Coeffce c of omal allocao Sable _ Omal allocao Omal allocao The omao whe roblem Gaussa whe whe Allocao _ he rsless asse cosderg he mare orfolo o 3 asses whe Table VIII ulmed shor sales are o allowed Coeffce c of omal allocao Sable _ Omal allocao Omal allocao The omao whe roblem Gaussa whe whe Ths able comues he allocao _ he rsless 6% aual rae daly for dffere rs averso coeffce of he omao roblem ma W.55 W W where W r ad r s eher he Gaussa Mare orfolo for or he sub-gaussa mare orfolo for or

27 Fgure Kolmogorov-Smrov es for Gaussa dsrbued resduals

28 Fgure Kolmogorov-Smrov es for Sable dsrbued resduals

29 Rs averso arameer c eced Uly Sable ovaos Omal Porfolo comoso Dow DAX Joes Idusral FTS all share Rs averso arameer c Gaussa ovaos eced Uly Omal Porfolo comoso DAX Dow Joes Idusral FTS all share Table IX Mamum W W W ad orfolo comoso cosderg reur scearo geeraed wh a ARMA model uder he Gaussa ad sable assumos for he ovaos of he me seres models

30 Fgure 3

31 Fgure 4