Optimal portfolio allocation with Asian hedge funds and Asian REITs

Transcription

1 Omal orfolo allocaon wh Asan hedge funds and Asan ehan Höch HVB-Insue for Mahemacal Fnance echnsche Unversä München German E-mal: ah Hwa Ng Drecor Rsk Managemen Insue Naonal Unvers of ngaore E-mal: ürgen Wolf echnsche Unversä München German and Naonal Unvers of ngaore E-mal: Rud Zags Drecor HVB-Insue for Mahemacal Fnance echnsche Unversä München German E-mal: Absrac: Durng he as ears he nsuonal neres n nvesmens no hedge funds and real esae nvesmen russ has grown consderabl. In hs aer he benefs of nvesng n hese asse classes are analzed b alng models ha recognze hgher-order momens or he whole reurn dsrbuon lke he ower-ul Omega and core-value model. rng o oban more general resuls han hose we can fnd from hsorcal daa onl we modelled he asse reurns b Markov swchng rocesses and dd a Mone Carlo sud. Whn hs desgn we analzed he omal allocaons o hedge funds and sacall and wh monhl reallocaons based on daa from Asan markes. Our man fndngs are ha n he sac case he ul model and he core model are domnan whereas he meanvarance model aears o be he model of frs choce n he dnamc case. In boh sengs hedge funds are he mos domnan asse of he omal orfolos. are manl used for dversfcaon and added a comarabl lower raes. ewords: Alernave Invesmens Asse Allocaon Hgher-Order Momens Markov-wchng Auoregressve Model INRODUCION he omal allocaon of nsuonal nvesors caal o varous asse classes s ofen referred o as ner-asse or mxed-asse dversfcaon. Invesors have been searchng for ossbles o enhance orfolo erformance besde he radonal asse classes bonds and socks. I s obvous ha nvesmens whch deend on dfferen facors drvng he reurns are lkel o rovde for he bes dversfcaon oenal. hus man nvesors seekng dversfcaon looked for socks and bonds n oher markes.e. nernaonalzed he orfolo. Unforunael n mes of general downurn on bond and sock markes or bg crss as he Asan crss 997 he Russan deb crss 998 he burs of he nerne bubble and he aack on he World rade Cener n New York dversfcaon s mos needed b he nvesor bu ver dffcul o oban whn hese radonal asse classes even f he orfolo s dversfed nernaonall. Berero and Maer (99) ng and Wadhwan (99) and ng e al. (99) analzed he erod surroundng he crash of 987 and found greaer negraon of he world s sock markes. Wha s mos needed b nsuonal nvesors a hese mes are asses wh low correlaon wh resec o bond and sock markes. hs exlans he growng nsuonal neres n nvesmens no hedge funds and real esae durng he las ears. If rsk s defned as sandard devaon has been demonsraed ha he ncluson of hedge funds romses rsk reducon whou loss of execed reurn. Bu as co and Horvah (98) have shown raonal nvesors have also clear references for hgher order momens.

2 Unforunael ncludng hedge funds and real esae nvesmen russ () no orfolos ver ofen leads o lower skewness and hgher excess kuross (see e.g. Brooks and a () or Brunel ()) whch s he exac oose of wha nvesors refer. herefore relng on he mean and he sandard devaon onl s dangerous. Mos of he sudes nvesgang gans n orfolo erformance when alernave nvesmens are ncluded carr ou he analss b nroducng he alernave asses no a orfolo of radonal bonds and equ. Zobrowsk and Zobrowsk (997) for examle added U real esae o bonds and socks and analzed he rsk reducon n he omal mean-varance orfolos. he found ha real esae rsk had been grossl underesmaed and ha for nvesors wh a low-rsk olerance he hgher levels of real esae rsk can elmnae all real esae dversfcaon benefs. Lee and evenson (5) used real esae nvesmen russ o add o he orfolo of socks and bonds. her analss shows ha aracveness as a dversfcaon asse ncreases wh he lengh of he holdng erod. In addon rovde for reurn enhancemen roeres a he lower end of he effcen froner and for rsk reducon a he o end of he effcen froner. Conover e al. () confrm hs b showng ha foregn real esae has a sgnfcan wegh n effcen nernaonal orfolos. udes concernng hedge funds n he mxed-asse orfolo are numerous as well: Marelln and Vasse (6) e.g. analzed he reurn enhancemen benefs and rsk reducon benefs of addng dfferen hedge fund sles no exsng orfolos. McFall Lamm () as well as Brunel () and Brunner and Hafner (6) fnd ha hedge funds have he oenal o mrove he mean-varance roeres of a orfolo bu creae ver unaracve hgher momens. he major of emrcal sudes we found n he leraure concenrae on analzng he effecs of ncludng jus one alernave asse class n a mxed-asses orfolo. In hs sud we wan o concenrae on he Asan markes and fnd he secal mlcaons of addng Asan alernave asses o he orfolo of socks and bonds. herefore we choose an Asan hedge funds ndex and an Asan ndex whch are added o he orfolo a he same me. Furhermore we analze he dfferences beween a sac omzaon lke n Brunner and Hafner (6) and a dnamc orfolo omzaon as erformed b e.g. Grauer and Hakansson (987). he goal of hs sud s o examne he omal orfolo fracons of alernave nvesmens (hedge funds and ). We look a hree dfferen nvesor es wh each havng hree dfferen nvesmen horzons: ear ears and 5 ears. We dsngush one ver conservave nvesor A one moderael rsk averse nvesor B and one more aggressve nvesor C. As menoned above we sar wh a sac omzaon lke n Brunner and Hafner (6).e. we look a he nvesmen horzon as one sngle erod. Moreover we also run a dnamc omzaon as nroduced b e.g. Grauer and Hakansson (987). In hs ar he nvesors are allowed o rearrange he orfolo weghs a he end of each monh o resond o marke develomens. We drecl comare he allocaon resuls wh he resuls of he sacall omzed allocaons. Evenuall we analze he sensv of he orfolo weghs o bases of he hedge funds daa. We accomlsh hs b correcng for he survvorsh and backfll bases and rerunnng he orfolo omzaon. he remander of hs aer s organzed as follows: In econ we examne he samle of Asan asse reurns and esablsh her man characerscs. econ covers he model used for he reurn rocesses and econ nroduces he echnque aled for he smulaon of he reurn ahs. he nvesor es we look a are defned n econ 5. econ 6 descrbes he resuls of he sac omzaon econ 7 he resuls of he dnamc omzaon. We rovde for a sensv analss wh resec o he nu arameers n econ 8 and conclude n econ 9. HE AIAN DAA E We use ndces o rox for log reurns of socks and bonds from Asan counres. he MCI All Counr Asa Prce Index s used for nvesmens no socks (n he followng referred o as ocks) whle Asan bond reurns are reresened hrough he.p. Morgan Emergng Markes Bond Prce Index (from now on referred o as ). As currenc rsk beween dfferen Asan counres can be elmnaed (for examle hrough currenc swas) he orfolo srucurng s looked a searael and for convenence boh ndces are used n her U.. Dollar reresenaon. he samle daa of monhl log reurns covers he me erod from anuar unl ul 6.e. 79 reurns (hree reurns were aken ou due o he fac ha he ver small samle showed exremel uncal behavour. hese are he monhs Arl ul and eember ). Obvousl hs samle has ver uncommon and herefore unexeced sascal roeres: socks show a ver low mean of.% er monh (.%.a.) comared o a mean of whch s more han four mes hgher (.85% er monh or.%.a.). In hs sud hedge fund reurns are reresened b he Eurekahedge Asan Hedge Fund Index whch s comosed of hedge funds of all sles and across all Asan counres. reurns are roxed b he EPRA/NAREI Asa oal Reurn Index whch also comrses dfferen sles and regons. We observe ha he EPRA/NAREI Asa oal Reurn Index (from now on referred o as ) has a much hgher sandard devaon as he Eurekahedge Asan Hedge Fund Index (from now on jus ). Esecall show more han normal hghl negave reurns and more reurns near he mean. Boh roeres ndcae osve excess kuross. urrsngl Hedge Fund reurns have no exreme negave values a all bu end o show more han normal osve values. We wll analze hese mressons n more deal b scrunzng he samle daa se: he summar of he sascs of all reurn dsrbuons s shown n able.

3 able : ascs of asse reurns Monhl Log Reurns ocks HFs Mean Mean (.a.) andard devaon andard devaon (.a.) kewness Excess kuross Mnmum Maxmum.... arque-bera--value arque-bera-asc Auocorrelaon (lag) Auocorrelaon (lag) Auocorrelaon (lag) % crcal value ±.9 ±.9 ±.9 ±.9 Ljung-Box--value Ljung-Box-asc Q() value for squared reurns Q() for squared reurns Hedge funds daa of world ndces s suosed o have hgher mean reurns han equ (see e.g. Brunel ()). he negave skewness of reurns s n lne wh our execaons even f s ver small n absolue erms (-.). On he oher hand he hgh negave value of excess kuross (-.68) s ver surrsng. Brooks and a () amongs ohers showed ha n general he low sandard devaon and hgh mean reurn of hedge funds come a he rce of unfavorable low skewness and hgh excess kuross. Conrar he observed n-samle excess kuross of -.68 s ver welcomed b nvesors. he above menoned cal osve auocorrelaon found n hedge fund ndces s confrmed n hs samle b alng he Ljung-Box-es (see e.g. Box and Ljung (978)). Hedge Funds auocorrelaon coeffcen of lag s.7 and herefore hgh enough o exceed he crcal value of he 5% sgnfcance level. hs confrms he resuls of oher sudes e.g. Brooks and a () who found evdence of hghl sgnfcan osve frs-order auocorrelaon n hedge fund daa. Probable exlanaons for he auocorrelaon n hedge fund reurns are llqud exosure.e. he roblem of unavalable marke rces of nvesmens and erformance smoohng.e. nenonall reored low volale reurns. We also esed for effecs of volal cluserng b comung he Ljung-Box-sasc Q() for squared reurns. Volal cluserng descrbes he henomenon ha large changes n rces end o follow large changes whle small changes end o follow small changes. We can rejec he null hohess for no auocorrelaon n squared resduals for onl. ha gves hn for volal clusers n he daa. Alogeher he daa n he hsorcal samle shows unexecedl low mean reurn and surrsngl aracve excess kuross. In comarson o he radonal asse classes he levels of sandard devaon and skewness confrm he fndngs of oher sudes abou world ndces of hedge funds. he sgnfcan osve auocorrelaon roeres also are n lne wh oher sudes and ndcae he above exlaned llqud exosure or nenonal erformance smoohng (see e.g. Brunner and Hafner (6)). Lee and evenson (5) and McFall Lamm () have analzed he benefs of addng no a orfolo of radonal asses. he found ha reurns call have a mean and a sandard devaon beween ha of bonds and socks. In he hsorcal samle of hs sud he reurns of ocks are ver low and have a ver hgh mean whch acuall les even above ha of. he negave skewness (-.7) and osve excess kuross (.86) are conssen wh he resuls of Zobrowsk and Zobrowsk (997). Alng he arque-bera- (see e.g. Bera and arque (98)) and Ljung-Box-es resecvel shows sgnfcan non-normal and auocorrelaon resen n he daa. mlar o he reurns he auocorrelaon for lag s ver hgh and osve (.) ndcang some sor of llqud exosure. Boh auocorrelaons of and wll be aken accoun for wh he e of Markov swchng models we aled for he smulaon of he reurn seres. If we comare he reurns of he alernave nvesmens n he samle wh hose of radonal he do no look ver sueror. hs s due o he ver uncal samle dsrbuon of. For laer analss we correc he mean reurn levels of all asses o weaken he roblems caused b hese raher unusual secfcs. hose raher srange roeres mgh be caused b he crss on he sock marke whch negavel affecs man hedge fund sraeges as well. herefore he hedge fund ndex used as rox for nvesmens n Asan hedge funds mgh also have suffered more han usual. Inroduced no a orfolo of and ocks he alernave asses sll have he oenal o rovde for dversfcaon even f he are no adjused a all. hs s shown b seng u equall weghed orfolos. able conans he descrve sascs of he orfolos wh dfferen fracons of alernave nvesmens (% n he frs column means ha % are nvesed n alernave nvesmens.e. he orfolo conans of 5% and 5% ocks. % means ha % are nvesed n alernave nvesmens (% n and % n ) he remanng 8% are nvesed n radonal asse classes (% n and % n ocks) ec.). Inroducng alernave nvesmens no he radonal orfolo seems ver romsng n a mean-varance world bu ends o render he values for he hrd and fourh momen n an unfavourable manner. he reurn seres of alernave nvesmens are nenonall creaed o have low or even negave correlaon o radonal asse classes o rovde hgher dversfcaon. herefore he erformance measured b he hare Rao can be mroved b addng and o he sarng orfolo.

4 able : Descrve sascs of equall weghed orfolos Mean andard kewness Excess hare (.a.) devaon kuross rao Porfolo fracon of AIs Adjused hare Rao (.a.) % % % % % % he radonal mean-varance omzaon does no realze he negave arbues assocaed wh skewness and excess kuross of alernave nvesmens reurn dsrbuons. When he adjused hare Rao s aled hs becomes obvous: he unaracve feaures of alernave asses negavel nfluence he erformance. As was shown n Brunel () alernave asses hgh weghs n he omal mean-varance orfolos mgh jus be a wa of ang for undesred negave skewness and osve excess kuross. PORFOLIO OPIMIZAION MODEL We wll comare four dfferen omzaon models. he frs one s he radonal mean-varance orfolo omzaon whch maxmzes he mean-varance radeoff of he nvesor. I s defned b λ max w w Σw () w where λ descrbes he rsk averson of he nvesor Σ denoes he covarance-marx and he vecor of execed asse reurns. We also al a full nvesmen consran and rohb shor-sellng. nce we oned ou he morance of hgher momens for he orfolo omzaon we also emlo a ower-ul omzaon and orfolo omzaons whch maxmze he erformance measures Ω and core-value. he ower ul omzaon roblem s gven b max E U w [ ( R( w) )] γ max E ( R( w) ) γ < γ () w γ max E[ ln( R( w) )] γ w where γ s he rsk averson arameer and R(w) he orfolo reurn for orfolo weghs w. Ω s defned as he rao of robabl weghed gans o losses n resec of he reurn hreshold τ.e. he usde oenal dvded b he downsde oenal (see eang and hadwck ()). he accordng Ω-omzaon roblem s gven b E maxωτ ( R( w) [ R( w) τ ] ) max () w w E[ τ R( w) ] where τ s he loss hreshold. he core-value defnes he rsk averson of he nvesor as λ c. he usde and downsde oenals are gven as n he formula of Ω wh he rsk-free rae r f used as hreshold. he accordng core-value omzaon s gven b maxcoreλ c ( R( w) ) w [ R( w) r ] λ E r R [ ] max E f c f w () w where λ c s he rsk averson arameer. MODELING AND IMULAION OF AE REURN he smulaon of he reurns has o allow for he fac ha he eculares of alernave asses reurn dsrbuons negavel affec he nvesor. Amongs ohers Clark (97) has shown ha he mxure of normal dsrbuons gves he oorun o generae dfferen values for skewness and excess kuross and herefore les us model a wder range of dsrbuons han s obanable hrough he use of a sngle dsrbuon (see mmermann ()). A serous drawback of hese me-ndeenden mxure models s her nabl o caure he henomena of volal cluserng resen n man reurn daa seres. he general dea of Markov swchng models s o assume a me-deendence of he dsrbuon arameers bu resrc he ossble number of dfferen realzaons o a fne number. A he same me exends he dea of ARCH models n ha no jus he volal s me-deenden bu also he mean and oher arameers. nce he oneerng work of Hamlon hese models have been broadl acceed and gven rse o a vas research leraure (see e.g. Hansen (99) m (99) Debold e al. (99) Psaradaks and ola (998) and Clarda e al. () and man oher aers). In our sud we consder orfolos made u of four asses: bonds socks hedge funds and. he daa n he hsorcal samle s ver lle. Fng a mulvarae Markov swchng model s herefore ver unlkel o resul n sable arameers. Hence we f unvarae Markov swchng models for he asses and ake no accoun he correlaon beween he asse reurns hrough he resduals used for he smulaon. We al he Markov swchng model whch allows for sae-ndeenden (frs lag) auoregressve dnamcs as defned n mmermann () o model he reurn rocesses of he asses: Φ ( ) (5) where denoes he reurn a me he sae a me he mean reurn n sae he sandard devaon n sae he nnovaon a me and Φ he auocorrelaon arameer. he changes of he saes are modeled b a Markov chan. here are wo ossble saes: f he reurn rocess s n sae hen f s n sae hen. he ranson robables of changng from sae a me - no sae j a me are gven b P (6) wh P( j - ) j. he nnovaons are assumed o be ndeendenl dsrbued wh resec o all as and fuure realzaons of he sae varable and..d N().

5 Of course we exec Φ <. If he Markov chan s ergodc.e. < < and > hen here exss a unque saonar dsrbuon wh uncondonal robables of (7) for he rocess beng n sae and - for he rocess beng n sae. nce we wan he momens of he smulaed daa o mach hose of he hsorcal samle we now esmae he unknown arameer vecor Θ b equang he samle momens and he heorecal dsrbuon momens (for brev we om he formulas here and refer he reader o mmermann ()). he acual esmaon of he values of Θ s done b numercall mnmzng he squared devaon beween he rocess momens and he accordng samle momens. Hsorcal reurns do no necessarl have o be a good redcor for fuure reurns. Moreover snce our hsorcal daa se s somewha uncal we wan o nclude marke exers reurn forecass for all asse classes. Alogeher hs nformaon s aken no accoun b underakng wo ses: frs we adjus he overall mean of each asse class wh he exer forecass b usng he Black-Lerman model wh a confdence arameer τ se equal o 95% (see Black and Lerman (99)). able : Black-Lerman-adjused mean reurns ocks HFs Hsorcal mean reurn (er.85%.%.98%.76% monh) Forecas (.a.) 5.% 9.75%.% 6.75% Forecas (er monh).%.8%.8%.56% BL-adjused mean reurn 5.8% 9.6%.96% 6.8% (.a.) BL-adjused mean reurn (er monh).%.78%.8%.57% ubsequenl he BL-adjused level of he mean reurn ogeher wh he sascal cenral momens of he hsorcal samle are used o oban he arameer esmaes of he Markov swchng rocesses of all asses. he resulng arameer vecors for unvarae Markov swchng rocesses are dslaed n able. able : Parameer fs for unvarae Markov swchng rocesses Asse Φ ocks E HFs For each ar of asses he heorecal covarance of reurns s comued deenden on he covarance of her resduals. hs s accomlshed b exendng he roof of Prooson n mmermann () for he case of wo asses. Because we need o ake no accoun ha he resduals sem from dfferen regmes (namel he regmes he corresondng rocesses are n ha ver me) we roceed as follows: frs we derve he equaon for he heorecal covarance of he asses deenden on he known arameer values of he asse rocesses and he unknown sae-deenden correlaons. he corresondng formula and a comrehensve dervaon can be found n Aendx A. hen we lug all known arameer values no he derved equaon o deermne he values of he sae-deenden correlaons beween he asses. We al a echnque smlar o he mehod of momens as descrbed above. hs s how we overcome he roblem of daa scarc and use mulvarae behavour noneheless. For convenence able 5 shows he covarance n he hsorcal samle and he covarance whch s heorecall mled b he saedeenden correlaons we derved above. We fnd he values o le close o he emrcal covarance marx. he sae-deenden correlaons of he resduals are used o draw he resduals for all asses smulaneousl deenden on he acual saes of he asses. Each smulaed ah consss of 6 reurns.e. 5 ears for each asse. Alogeher we smulae ahs wh a Mone Carlo aroach. able 5: heorecal (mled) covarance beween asses Covarance n (%) ocks Hedge Funds.7 (.) ocks..6 (.5) (.5) (.) (.68) (.8) (-.) (.7) (.8) (.75) 5 DEFINIION OF INVEOR YPE o comare he erformance of omal orfolos for dfferen models we se he rsk averson arameers n all models o values whch reresen he same nvesor n erms of rsk averson. herefore we use a characersc benchmark orfolo for each nvesor consdered. We look a hree dfferen nvesors defned b her dsncve benchmark orfolos of and ocks (see able 6). able 6: Characersc benchmark weghs Benchmark weghs Invesor A Invesor B Invesor C w 75% 5% 5% BM BM wocks 5% 5% 75% hese orfolos are used o conssenl deermne he rsk averson arameers and he loss hresholds for all omzaon models. We do hs b seng each arameer o he value whch makes he omal weghs whn he corresondng framework mach he weghs n he benchmark orfolo.e. we choose hose values for λ γ τ and λ c whch le he weghs of he benchmark BM orfolo w aear omal n he resecve framework (see able 7).

6 able 7: Rsk averson arameers Rsk averson λ γ τ (.a.) λ c ear %. Invesor A ears % 7. 5 ears % REUL OF AIC OPIMIZAION We sar wh he sac omzaon and al all models as nroduced n econ for all hree nvesor es and all nvesmen horzons. he omal orfolos for hs case.e. wh no reallocaons durng he nvesmen erod are reored below n able 8. All weghs n % Invesor A Invesor B Invesor B Invesor C ear %.8 ears %.85 5 ears % 6.9 ear %.7 ears %.6 5 ears % 5.6 able 8: Omal orfolo weghs n % ear ears 5 ears MV U Om c MV U Om c MV U Om c 7 ocks HFs ocks HFs Porfolo Allocaons From able 8 we fnd ha across all nvesors models and nvesmen horzons seem o be he asse of frs choce. he allocaon o s a leas 57% (Invesor A ear Ω-model) bu n man cases Hedge Funds consue he enre orfolo (e.g. Invesor C ears core-model). hs confrms he resuls of man auhors who analzed omal orfolos wh nvesmens no hedge funds (see e.g. a () or Brunner and Hafner (6)). Furher we can see ha are hardl allocaed a all. Whle he Ω-model us fracons from 5% o 7% no and allocaes all asse classes for all nvesors and all me horzons do no aear noabl n an oher omal orfolo whn all oher frameworks. Brunner and Hafner (6) who suded he omal allocaons for dfferen hedge fund sles found smlar resuls for equ hedge sle. he cure of s no ha dfferen: f are allocaed a all we fnd raher lle fracons of eher % or 9-% across he frameworks. If we look for examle a Invesor A for he ear me horzon hs Ω-omal orfolo conans of 6% Hedge Funds % 7% ocks and 7%. he omal orfolos for he oher models have an nvesmen beween 9% and 96% n beween % and 8% n ocks and onl he ul omal orfolo has % of. 6. Consraned Omzaon In real allocaons regularl are consraned. herefore we al maxmum consrans (8% for and ocks resecvel % for and 5% for ) and rerun all orfolo omzaons (see able 9). able 9: Omal orfolo weghs n % (wh consrans aled) ear ears 5 ears All weghs n % MV U Om c MV U Om c MV U Om c Invesor C 5 ocks HFs Invesor A ocks Hedge Funds he dfferences n he omal orfolos amongs he nvesor es do no aear o be ver bg. hs of course s a leas n ar a resul of he values of he rsk averson arameers. More dsnc resuls ma be obaned b usng exernall defned benchmark reurns and rsk aversons whch dffer o a greaer exen. Hageln and Pramborg (5) dd so and aledγ -values n he range of [-6; ] n he ul framework whch resuled n more dsnc allocaons. In such a case he comarson of dfferen models s ver dffcul as s hardl ossble o conssenl arameerze hem. he aroach of usng benchmark orfolos as aled here guaranees hs conssenc. Invesor B Invesor C ocks Hedge Funds ocks Hedge Funds Wh he alcaon of he consrans he allocaons change essenall. All omal orfolos u % no (he maxmum admssble fracon) leavng a far fracon o redsrbue as all unconsraned orfolos

7 assgned hgher weghs o before. Accordng o he ndvdual rsk aversons.e. deenden on he nvesor e he omzaon models refer dsnc asses: smlar o he unconsraned omzaon he Ω-framework allocaes dfferenl han he oher hree frameworks. I ncreases he fracon of jus slghl comared o he unconsraned weghs ( are beween % for Invesor A and a me horzon of 5 ears and % for Invesor C on he ear me horzon). In all orfolos a fracon smlar o ha of s nvesed no ocks. For lower levels of rsk averson hese fracons ncrease and he Ω-model nvess more no he rsker asses and ocks whch romse a hgher mean reurn. Clearl he mos referred asse class s. In he Ω-omzaon we are lookng for reurns ha exceed he loss hresholdτ. Invesor A moses he lowes value of.6%.a. (for nvesmens of 5 ears) for he loss hreshold and Invesor C he hghes of 5.7%.a. (for ear). Alhough are he asse class whch romse he lowes mean reurn for each nvesmen horzon of arox. 5.%.a. we see ha hs reurn s ver close o he loss hresholds. hs mean reurn comes ogeher wh he lowes volal of all asses. herefore a orfolo wh a hgh fracon of earns he requred reurn a a ver low level of rsk causng he downsde o be ver small and he Ω-measure o be ver hgh. Wh an nvesmen of % no he model bes on hgh reurns. he omal weghs n he oher omzaon frameworks behave slghl dfferen: agan are assgned he maxmum ossble fracon of %. Afer allocang he admssble 5% no (jus nvesor A sas slghl below 5%) he remander s sl u amongs and ocks (n fac hese hree models (mean-varance- uland core model) seek o allocae hgher fracons of he orfolo no when he maxmum consran of 5% s loosened). Whle he reurn-seekng nvesor e C (lowes values of rsk averson) does no allocae a all nvesors A and B are more araced b he less rsk (n erms of sandard devaon). he mos conservave nvesor A us even more mone no he less rsk (%-9%) han no ocks (%-5%). hs s also shown b long he omal orfolo weghs along he mean-varance effcen froner for he unconsraned and he consraned case (see Fgure ). Almos he enre fracon of has o be allocaed no he oher asses due o he maxmum bound. For orfolos wh a sandard devaon of less han 5% receve he hghes ar. hen for hgher levels of sandard devaon and mean ocks become more aracve o make sure ha he hgher levels of mean reurn are acheved. From he x-axs we see ha he rsk secrum wh he consraned orfolos has been exended. hs s rmarl caused b and he hgh weghs of ocks as hese wo asses exhb he hghes sandard devaons. A more dealed reor on he ndvdual dfferences beween he frameworks for each nvesor e can be found n Aendces B C and D. Inv. A:.9 Inv. B:.95 Inv. C:.7 omal weghs ocks sandard devaon Inv. A:.79 Inv. B:.6 Inv. C:.9 omal weghs ocks sandard devaon Fgure : Mean-varance omal weghs (unconsraned vs. consraned) wh suermosed effcen froner of ear reurns o sum u we can sa ha for all nvesors models and me horzons he maxmum admssble fracon of % s nvesed no. Invesors B and C almos alwas allocae he maxmum of 5% n Invesor A sas slghl below. he remander s allocaed o and ocks accordng o he nvesor s rsk averson. he fracon of decreases wh he rsk averson and les beween % and 9% for he mean-varance ul- and core-model and beween % and 66% for he Ω-model. he reverse s rue for socks wh omal allocaons beween % and 67% for mean-varance ul and core and beween % and 6% for he Ω-model. Our examle nvesor from econ 6. (Invesor A ears) now has an Ω-omal orfolo ha consss of % Hedge Funds 65% % ocks and %. All oher omal orfolos u less mone no (%-9%) and nves more n ocks (%-9%) and (7%-%). hs means ha he Ω-model us mos of he remander (afer nvesng % n ) n he rskless whereas he oher models are more dversfed. o comare he resuls of he dfferen omzaon models we also calculaed he robabl of ouerformance.e. he ercenage of scenaros n whch a model ouerformed he oher. I urned ou ha for he one-ear me horzon here s no bg dfference beween he dfferen models (maxmum ercenage of ouerformance of 55%) wh he ul model beng he bes. For he and 5 ear me horzon he coremodel clearl ouerforms he oher models (ercenage of ouerformance beween 6% and 8%) mean reurn mean reurn

8 ear able summarzes all erformance measures and comares hem o hose of he sac omzaon. In general he resuls confrm wha we execed: he erformance of he orfolos wh monhl reallocaons exceeds he erformance of he sacall omzed ones. here are jus a few cases where he erformance measures ndcae a oorer erformance of he dnamcall omzed orfolos. o shed lgh on hese observaons we roduce scaer los of he annualzed reurns of he sacall and dnamcall omzed orfolos (see Fgure ).6.. Resuls of dnamc omzaon For each of he hree nvesmen horzons we comue he cumulave orfolo reurns and calculae he same erformance measures as n he revous econ 6 (see able )....6 reurns of sacall omzed orfolo Framework: Omega reurns of sacall omzed orfolo...6 reurns of sacall omzed orfolo.8 Framework: core.8 7. Framework: ul reurns of dnamcall omzed orfolo Framework: mean-varance.8 reurns of dnamcall omzed orfolo nce we wan o comare he resuls of hs dnamc omzaon wh hose obaned above hrough he sac omzaon we use he dencal nvesmen horzons of and 5 ears for he same nvesor e A. o smlf he analss we do no consder an ransacon coss. hus he nvesor can freel reallocae a he end of each monh. A gven seres of orfolo reurns should be evaluaed conssenl as good or bad ndeenden of he model or e of omzaon sems from. herefore nvesor A ales he same values for he rsk averson arameers (of course he hresholds τ and he rsk free rae are rescaled for he new erod lengh of one monh). hs seng ensures he comarabl beween he aroaches of sac and dnamc omzaon. Invesor A (dnamc) hare Omega core We make he followng observaons for he erformances: snce we comue Ω for he cumulave reurns a he end of he consdered nvesmen horzon he Ω-values do no necessarl have o be he hghes for he Ω-framework even hough hs framework maxmzes he Ω-value on he monhl bass. Of course hs s also rue for he corevalue. On he consdered daa he ul-omzed orfolo creaes he bes Ω-erformance for 5 ears (7). hs s manl due o a ver low downsde (.6). When nerreng he resuls of he sac omzaon we found hgh smlares beween he allocaons and erformances of he mean-varance model and he ul model. For he orfolos wh monhl reallocaons hs cure changes slghl. he dfferences become obvous esecall when erformance s measured b Ω and corevalue alhough he means and sandard devaons (and herefore he hare Rao) are ver smlar. reurns of dnamcall omzed orfolo he aroach wh reallocaon was emloed b Grauer and Hakansson (986) Grauer and Hakansson (987) Grauer and Hakansson (995) as well as Hageln and Pramborg (5) who emloed he ower-ul model as descrbed above. he rocedure aled b hem uses a rollng wndow of nu daa whle movng hrough me.e. along he reurn ah. We use he same aroach here. However whle he menoned sudes use hsorcal daa onl we amlf he resuls b erformng he omzaon on all he smulaed ahs from above. he lengh of he rollng wndow s se o reresenng a ears me horzon (Hageln and Pramborg (5) also reor he resuls of her omzaon wh a wndow lengh of.e. ears. her resuls wh a ears wndow are smlar o he ones wh he ears wndow.). Invesor A (sac) hare Omega core MV Ul Omega core MV Ul Omega core MV Ul Omega core reurns of dnamcall omzed orfolo In hs secon we ck nvesor A o analze a dfferen seng: we allow he nvesor o redsrbue he enre orfolo a he end of each monh. he seu we used n econ 6 was a sac one: even f he marke envronmen develos unfavourabl no reallocaons wll ake lace a all. Now we dvde he me from he resen unl he end of he nvesmen horzon n dsnc erods each erod havng he lengh of one monh. hus we ge erods for he ear 6 erods for he ears and 6 erods for he 5 ears horzon. A he end of each of hese erods he nvesor s allowed o make a new allocaon decson. ears 7 DYNAMIC OPIMIZAION WIH MONHLY REALLOCAION able : Performance of omal orfolos (sacall vs. dnamcall omzed) 5 ears If we wan o comare he omzaon models b means of he erformance measures hare-rao Ω and core-value we can see ha he Ω-model erforms bes n erms of hare-rao and of course Ω bu wors n erms of corevalue. he hghes core-value s obaned wh meanvarance ul- and core-model reurns of sacall omzed orfolo.8 Fgure : caer lo of ears reurns (.a.): sac vs. dnamc omzaon (Inv. A)

9 In hese los we fnd he reasons for he observaons made above: for he mean-varance he ul and he Ω-model we fnd effecvel onl reurns above he lne ndcang ha he resul of he dnamcall omzed orfolo s almos never below ha of he sacall omzed orfolo. Bu for he vas major of ahs he aaned reurn exceeds ha of he sacall omzed counerar causng hgher levels of mean reurns for he dnamcall omzed orfolos. In addon we fnd ha he clouds of dos sread over a larger range for he dnamcall omzed orfolos n all case. o he dnamcall omzed orfolos above showed a hgher sandard devaon. Neher effec domnaes he oher n all cases whch causes he hare Rao o show beer erformance of he dnamcall omzed orfolos no n all cases (when erformance s measured b Ω and core-value.e. b ncororang hgher momens all orfolos erform beer n he dnamcallomzed case). We fnd ha all hree models roduce raher smlar los for all hree me horzon: he ercenage of scenaros whch he sacall omzed orfolos succeed n do no ncrease for longer me horzons. In fac he fracon decreases slghl (see able below). If we look a he core-omal orfolo we see ha for he reurns of ear me horzon behaves lke he oher models bu when he nvesmen horzon s exended we fnd ha he sacall omzed orfolo erforms beer han he dnamcall omzed orfolo for a growng number of scenaros (more han 5% for he 5 ear me horzon). We also fnd ha he range of orfolo reurns ends o be larger for he sacall omzed orfolo (whch s conrar o he oher hree models). able : Percenage of scenaros n whch he sacall omzed orfolo earns a hgher reurn han he dnamcall omzed orfolo Mean-var. model Ul model Omega model core model r 5.96 % 5. %.8 %.78 % rs 5.8 %.87 %.9 % 9.9 % 5 rs 5. %.8 %.7 % 5.7 % 7. Omal allocaons Le us now have a look a he weghs of he dnamcall omzed orfolos. For hs urose we comue he mean of he omal wegh of each asse for each on n me across all scenaros (see Fgure ). For all models s save o sa ha some me o adjus s needed n whch he average of he omal weghs changes heavl. For he longer horzons of and 5 ears resecvel we observe ha he average orfolo weghs swng no a relavel sable allocaon afer arox. ears (.e. erods). For reasons of comarson we add he omal allocaon of he sac omzaons for each case (see Fgure ). nce he sacall omzed orfolo weghs of he meanvarance ul and core models clearl emhaszed and nvesed heavl no hs asse and he dnamcall omzed weghs on average allocae n all asses we fnd he fracons of o be a lo smaller n he dnamc case (e.g. Invesor A nvess beween % afer ear and 6% afer 5 ears). he fracons of are found o be a lo hgher snce on average all dnamcall omzed orfolos conan and he sacall omzed allocaons hardl assgn (e.g. Invesor A nvess beween % (core-model) afer ear and 5% afer ears). us he Ω-omal orfolos led o more balanced allocaons n he sac omzaon. herefore he comarson of hese weghs wh he average weghs of he dnamcall omzed orfolos exhbs ha he almos mach esecall for he horzons of and 5 ears. dnamc weghs dnamc weghs dnamc weghs dnamc weghs Model: Mean-varance; nvesmen horzon ears ocks me Model: Ul; nvesmen horzon ears ocks me Model: Omega; nvesmen horzon ears ocks me Model: core; nvesmen horzon ears ocks me Fgure : Omal (mean) weghs over me for ears (Inv. A) sac weghs sac weghs sac weghs sac weghs

10 akng all he asecs no consderaon can be sad ha and la a major role n he omal orfolos. On average all frameworks allocae subsanall no boh asse classes. he Ω-omal and he coreomal orfolos are more domnaed b wh fracons of 6% for he horzons of and 5 ears and 5% and more for he ear horzon whle he mean-varanceomal and he ul-omal orfolos assgn -5% o and 5-5% o (deendng on he nvesmen horzon). Due o he more balanced allocaons he dnamcall omzed orfolos creae lower downsdes han her sacall omzed counerars. hs s nuve snce wh monhl reallocaons he orfolo s able o face bad develomens n sngle asses wh lower allocaons. Agan we ake a closer look a he ear me horzon for Invesor A and hs omal orfolos: for he mean-varance and he ul-model he omal orfolos are que smlar (% ocks5% HFs5% %) whereas he Ω- and core-omal nvesors u more mone no (6%) and (%) and less n ocks (%) and (%). If we comare hs o he resuls obaned above n he sac case we can see ha besdes he Ω-model where here are no bg changes n he dnamc case he nvesor allocaes less n and more no ocks and. All he resuls are n lne wh resuls from revous sudes (see e.g. Grauer and Hakansson (995) or Hageln and Pramborg (5)) ha boh and are allocaed n he omal orfolo and ha hese allocaons can be remarkabl hgh. Hageln and Pramborg (5) found ha a rsk avers nvesor heavl nvess no hedge funds (for a orfolo of bonds socks and hedge funds). he also showed ha for a less rsk averse nvesor even undversfed orfolos aear o be omal whch allocae all caal no for subsanal erods of mes. As n he sac case we also calculaed he robabl of ouerformance for he dfferen models n he dnamc case. Here he mean-varance model clearl ouerformed all oher models and he core-model s ouerformed b all oher models. If we comare he omzaon models b means of he erformance measures we can no decde whch model s bes. Accordng o he hare-rao he Ω-model erforms bes he hghes Ω-values are aaned b he ul and core-model and he hghes core-value s reached b mean-varance and ul model. 8 DYNAMIC OPIMIZAION UNDER MARE CONDIION he resuls from econ 7 showed he omal allocaon for an nvesor n a dnamc seng for dfferen omzaon models. In real such an nvesor would have o face furher consrans and eculares ha would ossbl affec hs decsons and hence he allocaons shown above. hree of such eculares are dscussed n hs secon namel daa bases n hedge funds daa lock-u erods and ransacon coss. 8. Daa bases Due o he eculares of he hedge fund markes a coule of bases can be found n hedge funds daa. he mos common and robabl bes-analzed bases are selecon survvorsh and backfll bas. he frs one (selecon bas) s caused b he volunar reors of reurns. Wh no oblgaon o reor bad resuls oorl erformng funds ma decde no o reor he reurns o daa collecors. hs resuls n daabases whch conan jus he well-erformng funds. In addon funds ha do no wan o grow anmore.e. are no lookng for new nvesors ofen do no reor her reurns o daabases an longer. hs ma cause an oose effec. Alogeher he selecon bas descrbes he roblem ha reurn daabases ma no adequael reresen he rue hedge funds unverse. he second bas survvorsh bas occurs because man daa vendors jus reor reurns of funds ha are sll n oeraon. Funds whch are closed down durng ha erod ofen are excluded. A major reason for ceasng oeraons of a fund s bad erformance (roughl % of newl esablshed hedge funds do no survve he frs hree ears (see.8 of Brooks and a ()). herefore daa avalable from daabases end o overesmae he ossble reurns (see a ()). Lang () found ha hs overesmaon exceeds %.a. whle a () reors %-%.a. Backfll bases are caused b fllng n hsorcal reurns of newl added hedge funds. When a hedge fund s added o an ndex or ncluded n a daabase can choose he enr dae. Of course he fund managemen chooses a dae whch makes he erformance aear ver well. hus he backfll bas s an uward bas whch s esmaed b Fung and Hseh () o be arox.. ercenage ons.a. akng hese bases no accoun we correc he Hedge Funds mean accordngl b subracng %.a. from he mean reurn and rerun all he ses descrbed above.e. correcon of mean level wh Black-Lerman model f and arameer esmaon comuaon of sae-deenden correlaons smulaon and evenuall orfolo omzaon. For brev we do no reor all he ses bu focus on he changes n he omal orfolos: Fgure dslas he allocaons smlarl o Fgure of he unadjused omzaon for he se whch correcs he Hedge Fund mean b %.a. We fnd he allocaons o decreased sgnfcanl and ha of ocks ncreased. Bu smlar o above he allocaon fnds a sable balance afer ears - of course wh less n he omal orfolos. For he Ω-model hs sable orfolo almos maches he omal weghs of he sac omzaon. E.g. for he ear me horzon of Invesor A we fnd he followng omal orfolos: for he mean-varance consss of % % ocks % HFs and % for he ul-model of % % ocks

11 5% HFs and 5% for he Ω-model of 5% 5% ocks 5% HFs and 5% and for he coremodel of % % ocks 5% HFs and %. All n all we can sa ha he sensv of he orfolo allocaon owards correcons for bases s raher small. Alhough here are some changes n he omal orfolos he srucure s que smlar. dnamc weghs dnamc weghs dnamc weghs Model: Mean-Varance; nvesmen horzon ears ocks me Model: Ul; nvesmen horzon ears ocks me Model: Omega; nvesmen horzon ears me Model: core; nvesmen horzon ears ocks sac weghs sac weghs sac weghs Lock u erods o far we have no consdered ha man hedge fund managers esablsh so called lock u erods. hese lock u erods descrbe a me nerval he nvesor s requred o hold he once bough shares a mnmum. nce he managers are more or less free n choosng hese nervals we use an average of ear. he model s adjused n a wa ha he fracon of he orfolo nvesed no hedge funds s no ncluded n he monhl omzaon whn he lock u erod. us afer one ear he weghs of hedge funds wll be adjused o reresen he develomens of he asses. All oher asses are reallocaed as before on a monhl bass. We found ha he allocaons do no change fundamenall. Agan afer wo ears he weghs swng no a sable allocaon whch s almos exacl he same as n he case whou lock u erods. herefore we conclude ha he lock u erods do no nfluence he allocaon rocess o a grea exen. 8. ransacon coss nce we excluded ransacon coss from our analss so far an nvesor could no earn he reored reurns n real. When ransacon coss.e. coss for bung and sellng asses are consdered he reurns of he dnamcall omzed orfolos wll be reduced. Assumng a realsc se of ransacon coss (ocks: 5bs : 5bs : bs and HFs: 5bs) led o he resul ha n a smle seng of ransacon coss rooronal o he amoun reallocaed he reurns of he dnamcall omzed orfolos are onl reduced b a small amoun and are sll clearl above he sac ones. Furhermore we found ha he average ransacon coss dffer beween he omzaon models wh he lowes ransacon cos for he meanvarance model. E.g. he average dscoun on reurns er monh.e. he value b whch he monhl reurns decrease when consderng ransacon cos for Invesor A on a ear me horzon s.7.7% for he mean-varance model. for he ul-model.7 for he Ω-model and.7 for he core-model. dnamc weghs ocks me Fgure : Omal (mean) weghs over me for ears (Inv. A) HF mean mnus % sac weghs CONCLUDING REMAR hs sud has examned he aracveness of hedge funds and real esae nvesmen russ () wh Asan background n a mxed-asse orfolo of Asan bonds and Asan socks. A lo of ror sudes have shown ha reurns of alernave asses esecall hedge funds end o exhb non-normal behavour (see e.g. Brooks and a () Amn and a () and McFall Lamm ()) and ha hese dearures from normal are decsve o he nvesor

12 (see e.g. co and Horvah (98)). B usng unvarae Markov swchng models we accoun for man secfcs of alernave nvesmen reurns or fnancal asse reurns n general: auocorrelaon smoohed daa non-normal daa bases ec. Earler sudes on he afermahs of he hgher momen characerscs of alernave nvesmens as e.g. Brunner and Hafner (6) showed ha ncororang hese feaures leads he hgh orfolo fracons of hedge funds n he mean-varance framework decrease due o unfavourabl negave skewness and hgh excess kuross. In hs arcle we found n a sac one-erod world all hree reresenave nvesor es o heavl allocae no hedge funds across all orfolo omzaon models. For lower degrees of rsk averson no seldom he enre orfolo s nvesed no hedge funds. Even he models ncludng hgher momens (he ul Ω and core frameworks) refer hedge funds. do no la a major role n he omal orfolos. he Ω-framework s he onl model whch allocaes n all omal orfolos. he conradcon o he resuls of oher sudes bascall grounds on he fac ha n our sud on Asan alernave nvesmens he samle daa of hedge funds exhbs ver uncal feaures whch aear o be ver aracve o he nvesor: he skewness s almos and he excess kuross acall low and herefore aealng. he arcle also encomasses some knd of dnamc orfolo omzaon. Here we allowed for monhl reallocaons of he enre orfolo. We found ha for he mean-varance he ul and he Ω-model he orfolo reurn a he end of he nvesmen horzon vruall alwas exceeded ha of he sac omzaon whou monhl reallocaons when ransacon coss were excluded. For hese hree models he robabl ha he sacall omzed orfolo earns a hgher reurn as he dnamcall omzed orfolo decreased when we ncreased he nvesmen horzon. he core-model roduced resuls conrar o hose of he oher models. For nvesmen horzons of and 5 ears we observed consderable numbers of scenaros n whch he dnamcall omzed orfolo led o a reurn worse han ha of he sac counerar. Furhermore he ercenage of hese scenaros ncreased wh ncreasng horzon lengh. For a 5 ears horzon we obaned more scenaros n whch he sacall omzed orfolo erformed beer han scenaros hs orfolo erformed worse n. One reason mgh be found n he hghes average allocaons o. nce romse he lowes level of mean reurn amongs all asse classes consdered n he sud he resecve orfolo reurn wll be raher low. Analzng he orfolo weghs across all scenaros revealed ha and la a major role n all omal orfolos. All n all we can sa ha on average all frameworks allocae subsanall no boh asse classes. he Ω-omal and he core-omal orfolos are more domnaed b wh fracons of 6% for he horzons of and 5 ears and 5% and more for he ear horzon whle he mean-varance-omal and he ulomal orfolos assgn -5% o and 5-5% o (deendng on he nvesmen horzon). Comarng he dfferen omzaon models b means of ouerformance robables led o he resul ha n he sac case he ul model ( ear me horzon) and he core model ( and 5 ear me horzon) are domnan whereas he mean-varance model aears o be he model of frs choce n he dnamc case. We also found ha he omal allocaons n he dnamc models swng no a relavel sable allocaon afer arox. ears. I was also shown ha hese omal allocaons n he dnamc seng are relavel sable owards bases ha call aear n daa. o sum u an nvesor ha does no reallocae durng he lfeme of hs nvesmen obans he bes erformance wh a core-value model whch nvess almos everhng no and nohng no and. A more dversfed orfolo can be obaned whn he Ω- framework (6% % 7% ocks and 7% for Invesor A over ears) whch leads o he hghes hare-raos and Ω-values. he dnamcall omzed orfolos are all que well dversfed and sable owards bases n HF daa. An nvesor who has he oorun o reallocae hs orfolo should use he mean-varance framework ha leads o an allocaon of arox. % % ocks % HFs and %. We wan o add one las ssue whch we hnk s of neres for fuure research: for boh omzaons wh and whou reallocaons we saed he assumon of no consumon a all durng he nvesmen me. In real nvesors do need mone a ceran mes. A oular suggeson o cure hs roblem s he use of one orfolo for each of he consumon mes. hen each orfolo s omzed ndeendenl of he ohers. hs ma no be omal from an overall ersecve of all orfolos. hs ssue s ackled n he feld of asse-labl-managemen whch ncororaes no jus he asses n he orfolo consumon rocess bu also he lables. REFERENCE Amn G.. and a H. M. (). Welcome o he Dark de: Hedge Fund Aron and urvvorsh Bas Over he Perod 99-. he ournal of Alernave Invesmens 6() Bera A.. and arque C. M. (98). Effcen ess for normal homoscedasc and seral ndeendence of regresson resduals. Economcs Leers 6() Berero E. and Maer C. (99). rucure and Performance Global Inerdeendence of ock Markes around he Crash of Ocober 987. Euroean Economc Revew (6) Black F. and Lerman R. (99). Global Porfolo Omzaon Fnancal Analss ournal 8(5) 8-. Box G. E. P. and Ljung G. M. (978). On a measure of a lack of f n me seres models. Bomerka Brooks C. and a H. M. (). he ascal Proeres of Hedge Fund Index Reurns and her Imlcaons for Invesors. he ournal of Alernave Invesmens 5() 6-. Brunel. (). Revsng he role of n dversfed orfolos. he ournal of Wealh Managemen 7() 5-8. Brunner B. and Hafner R. (6). Modelng Hedge Fund Reurns: An Asse Allocaon Persecve. In and

13 Managed Fuures A Handbook for Insuonal Invesors. Rsk Books. Clarda R.H. arno L. alor M.P. and Valene G. (). he Ou-of-amle uccess of erm rucure Models as Exchange Rae Predcors: A e Beond. ournal of Inernaonal Economcs Clark P.C. (97). A subordnaed sochasc rocess model wh fne varance for seculave rces Economerca Conover C.M. Frda H.. and rmans G.. (): Dversfcaon Benefs from Foregn Real Esae Invesmens. ournal of Real Esae Porfolo Managemen 8() 7-5. Debold F.X. Lee.H. and Wenbach G.C. (99). Regme wchng wh me-varng ranson Probables. In C. Hargreaves ed. me eres Analss and Conegraon Oxford Unvers Press 8-. Fung W. and Hseh D.A. (). Performance characerscs of hedge funds and commod funds: Naural vs. surous bases. ournal of Fnancal and Quanave Analss 5() 9 7. Grauer R. and Hakansson N. (986). A half cenur of Reurns on Levered and unlevered orfolos of socks bonds and blls ournal of Busness Grauer R. and Hakansson N. (987). Gans from Inernaonal Dversfcaon: Reurns on Porfolos of ocks and. he ournal of Fnance () Grauer R. and Hakansson N. (995): Gans from Dversfng no Real Esae: hree Decades of Porfolo Reurns Based on he Dnamc Invesmen Model. Real Esae Economcs Hageln N. and Pramborg B. (5). Evaluang Gans from Dversfng no Usng Dnamc Invesmen raeges. In B. chacher ed. Inellgen Hedge Fund Invesng -5. Rsk Books. Hansen B.E. (99). he Lkelhood Rao es under Nonsandard Condons: esng he Markov wchng Model of GNP. ournal of Aled Economercs a H. M. (). hngs ha nvesors should know abou. he ournal of Wealh Mangemen 5() 6-7. eang C. and hadwck W. (). A unversal erformance measure. Workng aer. m C.. (99). Dnamc Lnear Models wh Markov swchng. ournal of Economercs 6 -. ng M. A. and Wadhwan. (99). ransmsson of Volal beween ock Markes. Revew of Fnancal udes 5-. ng M. A. enana E. and Wadhwan. (99). Volal and Lnks beween Naonal ock Markes Economerca 6() 9-9. Lee. and evenson. (5). he Case for n he mxed-asse orfolo n he shor and long run. ournal of Real Esae Porfolo Managemen () Lang B. (). Hedge funds: he lvng and he dead. ournal of Fnancal and Quanave Analss 5() 9 6. Marelln L. and Vassé M. (6). Omal allocaon o hedge funds. Rsk March McFall Lamm R. (). Asmmerc Reurns and omal Hedge Fund orfolos. he ournal of Alernave Invesmens 6() 9-. Psaradaks Z. and ola M. (998). Fne-amle Proeres of he Maxmum Lkelhood Esmaor n Auoregressve Models wh Markov wchng. ournal of Economercs co R. and Horvarh P. (98): On he drecon of reference for momens of hgher order han he varance. he ournal of Fnance (5) mmermann A. (). Momens of Markov swchng models. ournal of Economercs 96() 75-. Zobrowsk B.. Zobrowsk A.. (997). Hgher Real Esae Rsk and Mxed-Asse Porfolo Performance. ournal of Real Esae Porfolo Managemen () 7-5. A PROOF OF COVARIANCE FORMULA uose we wan o comue he sae-deenden resduals correlaons of asse and asse. As defned above he reurn rocesses of boh asses are gven b Φ and Φ wh. Alng he dervaons n mmermann () he covarance of boh rocesses can be exressed as follows: [ ] [ ] [ ] [ ] Cov Ε Ε Ε Ε ) ( ) ( ) ( ) ( cons cons D C B A P Φ Φ Φ Φ Φ Φ Ε Φ Φ Ε cons cons Cov wh ae combnaons: and where and denoe sae of asse and asse and and denoe sae of asse and asse resecvel;

14 ranson robabl marx: P wh e.g. Uncondonal sae robables where denoes he uncondonal robabl for he rocess beng n sae { } and { }. (A): ubsung backwards no Φ leads drecl o Φ (B): Assumons o..d. N() o j Cov j Cov (C): Defnons of and and wh { }. (D): Proer of sae-sead robables P Noaons: o elemen-b-elemen mullcaon oeraor o sae-ndeenden overall mean of asse o sae-deenden resduals correlaons of asse and asse. B AIC REUL FOR INVEOR A As we saed above he Ω-model shows rofoundl dfferen omal allocaons han he oher frameworks due o he fac ha Invesor A us abou 5% of he orfolo value no for all nvesmen horzons f he or she nvess accordng o he Ω model. he oher models refer and ocks esecall for longer erods. her allocaons are more alke. We have a more horough look a he dfferences beween he erformance of he Ω framework and he oher frameworks. Fgure 5 shows he scaer-lo of he reurn seres of he ul-omal and Ω-omal orfolos for ears. For each of he ahs one on s loed. he black lne marks he locaons for whch he reurn of boh orfolos would be he same. cenaros n whch he ul framework earned a hgher reurn as he Ω framework le below ha lne and vce versa. Frs of all we fnd ha he cloud of dos sreches ou along he black lne. herefore eher boh models come off raher well or raher badl n all scenaros. In he lo we can see ha he Ω- orfolo leads o beer resuls for low levels of reurn and ha good scenaros end o be more favourable for he ul-orfolo. he cloud aears o be ver dense wh jus a few scenaros wh exreme reurns. Whle a he frs glance boh reurn seres have almos equal means for each of he me horzons he varance seems o dffer rofoundl. he range of (annualzed) reurns of he ulorfolo s [-.7;.] whle he reurns of he Ω-orfolo le n he nerval [-.;.5] reurn of Ul-omal orfolo reurn of Omega-omal orfolo Invesor A wh ears nvesmen horzon Fgure 5: caer-lo of he reurns of he ul- and Ω-omal orfolos

15 Even hough he ul-orfolo exhbs a hgher mean han he resecve Ω-orfolo s hare Rao s worse. hs s due o he hgher sandard devaon whch dmnshes he mean advanage. We can conclude ha he Ω-model looks que romsng for nvesors of e A. Due o he raher low loss hresholds of hs nvesor e (.6% o.%.a.) he model res o use o generae less rsk reurns and nvess n all asses raher han concenrang all mone n one sngle asse. As he Ω-measure s comued b dvdng he usde hrough he downsde oenal hs aears o be a good sraeg. If he nvesor was neresed on hgher reurns would be beer o follow anoher sraeg referabl he core model. We found ha he Ω-model s beaen b all oher models n he bes scenaros. hs comes a he rce of an unaracve erformance of he oher models n worse scenaros. C AIC REUL FOR INVEOR B mlar o nvesor A nvesor B s mean-varance model and he ul model lead o (almos) he same omal orfolos for all nvesmen horzons. he core model ends o allocae more aggressvel and he Ω model nvess agan n all asses avalable. All models allocae no jus whn he ear me horzon. he reason s he low correlaon o ocks and and herefore a low sandard devaon of he whole orfolo. In all unconsraned orfolos he bgges ar s assgned o whch rovde for he hghes mean reurn: he usde creaed b s bg enough no o care oo much abou he downsde. Bu wh he alcaon of he consrans he orfolos canno creae enough usde: he maxmum consran lms he nvesmen no. he Omega- and ore-models r o lm he downsde n order o maxmze he orfolo erformance. ubsequenl and are allocaed. he have he lowes ndvdual sandard devaons low correlaons o ocks and and are even negavel correlaed wh each oher. he also seem romsng for measures ncororang hgher momens: Boh asses show osve skewness and negave excess kuross for and 5 ears. herefore he orfolo downsdes can be lmed b ncludng hem no he omal allocaon. Recall ha Ω s comued b dvdng usde hrough downsde oenal. Boh oenals are deermned b alng he ndvdual loss hreshold τ. Invesor B ales hgher hresholds han nvesor A for all me horzons: 5.7%.a. for ear 5.%.a. for ears and 5.7%.a. for 5 ears. herefore nvesor B needs o allocae hgher fracons of he orfolo o asses ha exceed hs reurn. In addon we need o consder he downsde. hs s equall moran as he usde whn he comuaon of Ω. We exec orfolos exhbng reasonabl hgh mean reurns above he hreshold wh a he same me havng he smalles downsde o erform bes whn he Ω-framework. For hs reason we comare he omal Ω-orfolo wh he more aggressve allocaon of he core model (boh for ears nvesmen erod). We analze n deal he Ω and core-value of he orfolos and he resecve usdes and downsdes: nce he omal core-orfolo s enrel made u of no dversfcaon effecs ake lace and he sandard devaon exceeds ha of he Ω-omal orfolo. As execed he usde s ver hgh snce romse he hghes mean reurn of.%.a. As above we fnd ha he Ω-orfolo erforms beer for lower levels of reurns whle he core-orfolo clearl ouerforms he Ω- orfolo n good scenaros. he values of he usdes and downsdes confrm hs: For boh measures Ω and corevalue he resecve usde of he Ω-orfolo (. and.) s below ha of he core-orfolo (.7 and.). he Ω-omal mnmzes he downsde beer han he core-omal orfolo. hs leads o a beer erformance accordng o Ω where he comuaon us equal weghs o usde and downsde. Whn he core framework he rsk averson arameer deermnes o wha exen he usde s reduced b he downsde. he less rsk averse an nvesor s he lower wll be he λ c -value. ha sofens he morance of he downsde and consequenl favours sraeges wh hgher usdes. Invesor B s λ c -value for ears s aarenl low enough (.85) o le he Ω-omal orfolo aear worseerformng n se of a lower core-downsde. he downsde s jus no as decsve as s when erformance s measured b Ω. able : ascs of ears reurns of unconsraned Ω- and core-orfolos Invesor B ears nvesmen horzon of orfolo ascs reurns Performance measures core-model Unconsraned Omega-model Unconsraned Mean.8.98 Mean (.a.).6.6 andard devaon..6 andard dev. (.a.).6.97 kewness.6.5 Excess kuross hare rao Omega Usde.7.7 Omega Downsde.6.9 Omega core Usde core Downsde.. core-value..96

16 D AIC REUL FOR INVEOR C We now urn o he leas rsk averse nvesor. hs nvesor ales he lowes (absolue) values for he rsk averson arameers. Wh wha has been sad above we exec he downsde oenals of he orfolos o be of less neres and he man goal o be he maxmzaon of he usdes. In he unconsraned omzaon all Ω-omal orfolos sll allocae all asses even f make u jus lle fracons and clearl domnae he orfolo. he oher frameworks have he mos exreme allocaons amongs all nvesors. Agan ocks are ncluded n he ear orfolos due o he hgh mean whn ha erod. Bu for and 5 ears he orfolos are almos comleel nvesed no he asse wh he hghes mean reurn. When consrans are aled he orfolo allocaons o and are se o he maxmum lm. he remander s u no ocks comleel. us he Ω model sll nvess no all four asses. Whle nvesor B reans a lle fracon of n he oher frameworks for reasons of dversfcaon nvesor C looks more for hgh reurns and he maxmzaon of he usde. hus ocks aear o be he mos romsng asse and wh 5% of he orfolo nvesed no here are alogeher hree asses for dversfcaon. are no requred for furher dversfcaon snce he rsk averson s consderabl low. We menoned above ha a less rsk averse nvesor ends o u more effor no he maxmzaon of he usde han no he mnmzaon of he downsde when weghs are subjec o consrans. he more rsk averse an nvesor s he hgher wll be he mac of he downsde o reduce he erformance. hs exlans wh he fracon of (as he asse wh he wors mean) s lower n he omal orfolo of nvesor C. Also whn he core model nvesor C emhaszes he maxmzaon of he usde (Ωusde as well as core-usde are hgher for nvesor C s orfolo) even f he downsde rses consderabl. When he core-value s comued for nvesor B we fnd ha he erformance of nvesor C s more exreme orfolo s worse han hs own more balanced orfolo. hs s because he downsde s weghed hgher due o he hgher value of λ c 6.9. he cure changes comleel when he rsk averson of nvesor C s aled: he lower value of λ c (5.6) reduces he mac of he downsde. herefore he orfolo wh he hgher usde s referred and ranked hgher.