PRACTICAL PORTFOLIO OPTIMIZATION

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1 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD PRACICAL PORFOLIO OPIMIZAION Isaac Swale echcal Repor RD Apex Research Lmed Lodo Eglad e-mal: Absrac hs paper s o he porfolo opmzao problem for whch wo geerc models are preseed he coex of a propreary solver called GENO: 1 he frs s a psedo-dyamc model wh a sgle sae varable ha s mea for he sgle holdg-perod case; he secod s a rly dyamc model wh several sae varables ha apples o boh sgle ad ml-perod scearos. Boh models ca hadle praccal resrcos sch as exposre, cardaly ad rod-lo cosras; boh ca accommodae o-radoal rs measres; ad he sal wo-eleme crero se comprsg porfolo reward ad porfolo rs may be agmeed by ay mber of sbobecves deemed ecessary. he paper also preses a robs opmzao model of he porfolo problem ha explcly accos for daa ceray; he robs model ca also accommodae sophscaed rs measres, ad oe sch fco s preseed Appedx B. he compromse solo cocep s sed o compe porfolos ha are Pareo-effce; several mercal examples show ha he porfolos hs fod are o oly opmal he compromse sese, b hey also have a compeve (ad ofe he hghes) reward-rs rao a proxy for he Sharpe rao. A lmed emprcal aalyss shows ha he robs model s speror o s o-robs coerpar erms of boh omal performace, ad he poeal for avodg oppory coss de daa ceray. Key Words: Porfolo Seleco; Robs Opmzao; Ml-obecve Programmg; Evoloary Algorhms; Compromse Solo; Cohere Rs Measres; Ierval Programmg. 1 Irodco he porfolo problem frs formlaed by Harry M. Marowz [42] s cocered wh how vesors ogh o deploy her wealh facal asses, ad hs recommedao s a sraegy he called he E-V Rle : he vesor shold selec oe of hose porfolos wh mmm varace [V] for a gve expeced rer [E] or more, ad maxmm expeced rer for gve varace or less [Paraphrased from 42, p.82; emphass added] Oe may formally sae Marowz ormave decso model erms of ly heory as follows: 2 assme a vesor has a (sbecve) ly fco U, ad here are secres wh radom sgle-perod rers r ad covarace marx s [c ] whch he wshes o ves; he he bes wealh deployme sraegy may be fod by solvg he followg mahemacal program, 3 vz.: MP1: Max E Uw, r, 0 w 0; 0 Where: E s he mahemacal expecao operaor; w 0 s he vesor's al wealh; s he proporo of wealh allocaed o he -h asse; r s he sgle-perod rae of rer of he -h asse. 1 hs s a acroym for: Geeral Evoloary Nmercal Opmzer. 2 he semal ex for ly heory s [72], b a very readable acco may also be fod [39]. 3 he abbrevao MP or he coraco M-program shall ofe be sed place of mahemacal program hrogho hs paper. 1 Copyrgh : Apex Research Ld

2 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD MP1 s a sochasc opmzao problem, ad oe cold se a sochasc algorhm o solve. B, apar from he perls assocaed wh calbrag ly fcos (see Remars Appedx A), sochasc algorhms per se are compaoally demadg ad raher cmbersome. Ad addo, her solo ca be msleadg whe here s ambgy he choce of a dsrbo for he radom parameers. Ad so, alhogh moder egraed sochasc programmg evromes sch as SPIE [71] do allevae some of he dffcles, he fac sll remas ha sch a drec approach wold be effce. Ad was hs recogo of sch compaoal dffcles a ha me, coco wh hs qes o recocle he E-V Rle wh ly heory ha promped Marowz o sgges a dffere approach by posg he followg qeso: If a vesor wh a parclar sgle perod ly fco aced oly o he bass of expeced rer ad varace, cold he vesor acheve almos maxmm expeced ly? Aleravely, f oe ew he expeced vale ad varace of a probably dsrbo of rer o a porfolo ca oe gess farly closely s expeced ly? [Paraphrased from 41, p.471] Marowz oo a pragmac approach o hs qeso ad derved a deermsc fco ha oe cold se place of he ly fco. As show Appedx A, hs mea-varace (MV) crero s, essece, a aylor seres approxmao of a geeral ly fco. Emprcal evdece shows ha he MV crero s qe represeave of a varey of sadard ly fcos [1, 23, 35, 38, 53]; frhermore, alhogh he MV crero s oly a secod-order approxmao, sffces mos cases,.e. o advaage s gaed by reag hgher order erms he aylor seres [29]. For mos praccal prposes herefore, he sochasc obecve fco MP1 s ormally replaced by he MV crero ad he reslg decso model s a woerm, -obecve opmzao problem ha cldes a measre of he model-ser s olerace for rs, vz.: MP2: Max 1 1 r c 1; Where: r s he expeced rer for he -h asse; c s he covarace of rers bewee he -h ad -h asses; s he proporo of wealh vesed he -h asse; s he vesor s rs olerace. Ad accordace wh he EV rle, he MV model s sally saed ad sded oe of wo ways: () as MP3a where he porfolo rer s reqred o eqal or srpass a arbrarly fxed arge rer, r ; () as MP3b where he porfolo varace s reqred o be wh a arbrarly fxed arge varace,. Research s focsed more o MP3a (somemes who he mmm-rer cosra) becase solos comped by hs mmm varace model are more sable wh respec o perrbaos he p daa [12]. MP3a: M c 1 1 r r ; 1; 0 MP3b: Max 1 r c ; 1; hs paper corbes a solo mehod whose ma arbes are smplcy, praccaly, ad he fac ha he ml-obecve essece of he porfolo opmzao s o sppressed. he paper s orgazed as follows: 2 dscsses praccal sses porfolo seleco; 3 preses wo basc decso models for he problem; 4 dscsses specal cosras; 5 preses a mehod for hadlg ceray; he solo o he models s explaed 6; mercal examples are preseed 7 ad dscssed 8; 9 smmarses ad draws some coclsos. he exposo occasoally meos a solver called GENO hs s descrbed deal [61]. 2 Copyrgh : Apex Research Ld

3 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Praccal Isses Preamble. he MV model s crrely he sadard ormave heory for vesor behavor; s dversfcao prcples are he fodao of Moder Porfolo heory, ad s he bass for may oher mpora advaces facal ecoomcs, cldg he celebraed Capal Asse Prcg Model [59]. However, Rchard Mchad oes ha: here s some evdece ha a mber of expereced vesme professoals have expermeed wh MV opmzers oly o abado he effor whe hey fod her MV porfolos o be ve ad who obvos vesme vale. Gve he sccess of he MV model as a cocepal framewor, ad he avalably for early 30 years of [he Crcal Le algorhm 4 ] for compg effce froers, remas oe of he osadg pzzles of moder face ha MV opmzao has ye o mee wh wdespread accepace by he vesme commy [...] Does hs Marowz opmzao egma reflec a cosdered dgeme by he vesme commy ha sch mehods are o worhwhle? [Paraphrased from 46, p.31] he evdece meoed by Mchad may be aecdoal, b s oeheless real ad he egma sll exss more ha 20 years o; Kolm [34, p.7] qoes a moder seor porfolo maager a a large hedge fd hs: We do se MVO, Blac-Lerma or ay oher mehodology. We always combe sgals sg home grow echqes ha we develop o s or eeds. he same holds for porfolo cosrco: here are a lo of compeg obecves play ha mae hard o cas as a classcal opmzao problem. ha s why we eded p wh a lo of echqes ha loo ad-hoc, b are robs ad wor well for s. hey are probably o opmal a mahemacal sese, b persoally I cosder opmaly o be ll-defed a osy sysem le he soc mare. We ever sped a lo of me ryg o mae hgs opmal. Robsess s far more mpora o s. Dagoss. he relcace o he par of some pracoers o se he MV model s rgg. Mchad [46] has examed hs Marowz opmzao egma ad hs dagoss s ml-faceed he ma sses are: Error Maxmzao. he ve characer of may opmzed MV-porfolos ca be raced o he fac ha MV-opmzers are fdameally esmao-error maxmzers. he daa reqred for MV opmzao are ecessarly esmaes ha are sbec o esmao errors ad he opmzao process sgfcaly over-weghs (der-weghs) hose secres ha have large (small) esmaed rers, egave (posve) correlaos ad small (large) varaces. hese secres are he oes mos lely o have large esmao errors [9, 46]. Isably of Solos. MV opmzaos ca be very sable whereby small chages p daa assmpos leads o large chages he solo. he ma reaso for hs s les he drasc loss of sascal effcecy (see foooe 8) ha some esmaors sffer whe he Gassa error assmpo s volaed [12, 46, 70]. No-Ivesably of Allocaos. Sadard MV Opmzers ofe prodce porfolos ha are o vesable. Becase mos Opmzers assme coos varables, hey ca somemes prodce porfolo allocaos ha are o covee s, or are o sgfca eogh qaes, or boh. Ad ye s ow ha soal asses maagers ypcally mae vesmes large dollar cremes, or rod los of socs [46]. Specal Cosras. Sadard MV Opmzers gore some mpora cosras ha apply pracce. For example, der Germa Ivesme Law, here exss law called he rle ha cosras he composo of mal fd porfolos. hs rle saes: p o 5% of he vale of he mal fd may be vesed secres ad moey mare srmes of he same sser. hs lm may be exeded o 10% f he coracal erms ad codos of he vesme fd so provde ad, a he same me, he oal vesme he asses of sch ssers does o exceed 40% of he e asse vale of he fd [22, p.11]. he MV model depeds o forecass of fre eves ad as sch herely eals oppory coss ; hese are he poeal coss of holdg a spposedly opmal porfolo based as s o a parclar forecas of asse rers ad her correlao whe he sad forecass do o acally pa o as expeced. Sce hs s sally he case pracce becase we ca ever ow he fre wh absole ceray hs paper proposes ha a porfolo s oppory cos shold be par of s evalao. Ad so, whls acowledgg Mchad s Isably dagoss, porfolo robsess shall be reaed hs sese. 4 For a rece revew ad ypcal applcao of he Crcal Le algorhm, see [51]. 3 Copyrgh : Apex Research Ld

4 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD here s also he qeso of wheher he MV model realscally ecapslaes vesor behavor. he model s ormave, ad as sch, oe may expec ha cles wold oly have cofdece s prescrpos f hese reflec commo sese. Roy [57] qesoed he fdameal premse of he MV model, amely ha vesors shold maxmze her expeced ly: he dd o beleve ha a accrae ly fco cold be derved for ay vesor, ad herefore vesors wold fd praccally mpossble o maxmse her expeced ly. Isead, Roy posed ha vesors ormally decde sg a sraegy he calls Safey of Prcpal Frs hey se some mmm accepable rer (dabbed he dsaser level ), ad her preferred porfolo s ha whose rer s leas lely o fall below he dsaser level,.e. ha whch exhbs he leas dowsde rs. Nawroc observes ha Marowz hmself was a early propoe of a smlar dea: [Marowz] realzed ha (a) oly dowsde rs s releva o vesors, ad (b) secry dsrbos may o be Gassa dsrbed, whch case he se of varace s correc. I 1959, he sggesed wo dowsde rs measres a sem-varace comped from he mea rer, ad a sem-varace comped from a arge rer. B Marowz sayed wh varace becase was smpler o compe [Paraphrased from 50, p.3] Roy s posve heory has sce spawed a spae of research o dowsde rs ad s varos he measres: a bref ole of he maor srads of hs research s as follows: Lower Paral Mome he Lower Paral Mome of he radom varable Z s defed as follows: LPM( ) E Max Z; 0 LPM sbsmes a sgfca mber of he ow Vo Nema-Morgaser ly fcos, as well as he whole gam of hma behavor from rs-seeg o rs-eral o rs-averse [50]. he rage parameer s se eqal o he hreshold from whch he devaos are calclaed. he (sbecve) parameer represes he vesor s rs ade rs-averse vesors are expeced o choose a coeffce greaer ha 1 (gvg more wegh o large devaos); rs-eral vesors are expeced o choose a eqal o 1; ad he rsseeers wold choose a ha s less ha 1; for more deals, see e.g. [33]. Vale-a-Rs I 1995 he Ivesme Ba, JP Morga, released daa o varaces ad covaraces across varos secry ad asse classes ha had sed erally o maage rs; called he servce RsMercs ad sed he erm Vale a Rs o descrbe he rs measre ha emerged from hese daa. hs erm (sally abbrevaed as VaR ) refers o a rs oo ha addresses he qeso: Wha s he mos I cold lose o hs vesme? VaR s formally defed as follows: sppose he radom varable Z has a ellpcal dsrbo (see foooe 5) deoed by F Z ; he he Vale-a-Rs of Z a probably level α s gve by: 1 VaR (Z) F Z ( ) f{z R : FZ (Z) } I oher words, VaR s smply a alerave oao for he qale fco of he dsrbo, F Z, evalaed a probably level α. Ad p l he developme of he heory of cohere rs measres (see below) where VaR was show o o coform o hs mpora dea geeral, VaR was he de faco sadard measre of rs facal servces frms ad was begg o fd accepace he o-facal secor as well. For a hsorcal revew of VaR, cldg he assocaed echcal deals, see [47]. Coherecy hs erm apples o rs measres ha sasfy a se of axoms frs arclaed by Arzer, Delbae, Eber & Heah [3, 4]; a very readable acco of coherecy ha also exeds her formlao may be fod [56]. he heory of Cohere Rs Measres developed coslao wh rs-maageme pracoers s a aemp o provde obecve crera for evalag dffere qaave measres of rs. A cohere rs measre s a real-valed fco o he space of real-valed radom varables ha exhbs: A1. Mooocy: For ay wo radom varables, f Z 1 Z 2, he (Z 2 ) (Z 1 ) A2. Posve Homogeey: For λ 0 we have ha (λz) = λ (Z) A3. raslao Ivarace: For ay α R we have ha (Z + α) = (Z) - α A4. Sb-addvy: For ay wo radom varables Z 1 ad Z 2 we have (Z 1 + Z 2 ) (Z 1 ) + (Z 2 ) 4 Copyrgh : Apex Research Ld

5 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD hese properes are cosse wh commo sese: A1 mples ha f porfolo Z 2 s preferred o porfolo Z 1, he Z 1 s more rsy ha Z 2 ; A2 reflecs he fac ha rs grows drec proporo o he sze of he porfolo; A3 meas ha f a compoe wh deermsc ocome s added o he crre holdg, he rs of he ew porfolo s redced by he same amo; ad A4 esseally saes ha he rs of a porfolo cao exceed he sm of rss of s dvdal compoes, oher words, a merger does o creae exra rs [4]. I he porfolo seleco coex, A4 s he mos mpora propery s wha maes dversfcao beefcal. he case for cohere dowsde rs measres porfolo opmzao s coesable. I a deal world where porfolo rers are ellpcally dsrbed, hs reqreme wold o be so crcal becase, ha scearo, mos rs measres wold sasfy wha Embrechs, McNel & Srama [16, 17] call he Fdameal heorem of Iegraed Rs Maageme, whch s esseally a saeme o he sably of rs measres der varos probablsc assmpos ha may formally be smmarsed hs: 1. For ellpcally dsrbed 5 porfolos, VaR s a cohere rs measre: s moooc, posve homogeeos ad raslao vara, flfls he sb-addvy propery regardless of he probably level α sed 2. I a ellpcal world, amog all porfolos wh he same expeced rer, he porfolo mmzg VaR s he MV-mmzg porfolo. hs saeme apples o oher rs measres ha may or may o be cohere; parclar, apples o he Lower Paral Mome eve hogh hs measre voles he posve homogeey ad raslao varace axoms. 3. For o-ellpcally dsrbed porfolos, mos (f o all) of he sadard resls from he prevos saemes do o hold, parclar, VaR s more ha qesoable he secod saeme above appears o mgae he commo se of porfolo varace a o-cohere, o-dowsde measre of rs. B foraely, here s o garaee ha real porfolo rers are always gog o be ellpcally dsrbed. Ad vew of he compellg heory of coherecy, oe mplcao ha hs eals s clear: he real-world, all ormave models for porfolo opmzao ms, a he very leas, have he poeal for accommodag a cohere rs measre. Closg Remars. I s worh og a he ose ha o oly does GENO have a real capacy for accommodag ay rs measre, also allows clso of exra crera he MV decso model ha may be sed o address oher sses meoed above; a bref smmary of s capably profle s as follows: O No-Ivesably of Allocaos. he fac ha GENO ca hadle dscree varables ha may be bary, eger or real meas ha he o-vesably problem does o arse hs s dscssed 4. O Ad hoc Cosras. he GENO framewor allows clso o he MV model of ad hoc cosras (sch as he Germa Ivesme Law) va bary varables or logc proposos hs s dscssed 4. O Isably of Solos. Dealg wh porfolo sables arsg from esmao errors s wh GENO s capacy. However, he emphass hs paper shall be o a relaed ad more praccal sse, amely dealg wh he oppory coss of holdg a parclar porfolo he face of esmao errors. o ha ed, a robs MV model s preseed 5 ad s qales are esed Example 4. O Error Maxmsao. GENO allows agmeao of he sadard MV crero se o clde fcos ha may be desged o mmse he adverse effecs of esmao errors hs s dscssed 5. O Alerave Rs Measres. Evoloary algorhms are very versale: hey ca accommodae ay obecve fco ha ca be mapped oo a ordal evalao fco, ad sad obecve fco does o eve have o be a closed-form expresso [43, p.37]. hs, alhogh he mercal example preseed hs paper adops porfolo varace as he rs measre, he M-programs ha may sccessflly be addressed by GENO may clde varos oher ypes of measres ha cold be of he dowsde, o-dowsde ad / or cohere or ocohere varees, ad cldg eve hose ha ca oly be defed algorhmcally. 5 A ellpcal dsrbo s oe whose probably desy s cosa o ellpsods he bvarae case, he coors les of he desy srface are ellpses. Ellpc dsrbos are symmerc ad -modal, ad hey possess a sefl propery o porfolo heory, amely ha a lear sm of ellpc dsrbos s also ellpc. Examples of ellpc dsrbos clde he Gassa, Laplace, Sde s, Cachy ad Logsc dsrbos. 5 Copyrgh : Apex Research Ld

6 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Porfolo Opmzao: Ml-obecve Dyamc Models Preamble. hs seco preses wo alerave models of he porfolo allocao problem, boh of whch are ml-obecve opmzao problems defed over a dscree doma. he frs s a sgle-perod psedodyamc model he psedo-dyamcs arsg from he fac ha vesme allocao decsos are made serally, b all a he sar of he holdg perod; he secod s a rly dyamc ml-perod model whch s assmed ha vesme decsos are made smlaeosly a each decso sage. he secod model may also be operaed sgle-perod mode by merely declarg he opmzao horzo parameer as = 1. he formlaos assme ha all asse rers are draw from ellpcal probably dsrbos, ad ha sad dsrbos are also saoary ; see e.g., [27, p.23]. he followg oao apples: Noao. he erm Op deoes a operaor whose operad s a dscree se of crero fcos; s a commad o opmze he operad ; may be dsrbed oo he elemes of he crero se, or facored o from sch a colleco; whe he operad s a sgleo, he Op meas mmze or maxmze depedg o he coex. he assocaed erm arg op meas he argme ha opmzes he operad ; whe he operad s a sgleo, meas arg m or arg max depedg o he coex. he Seqeal Decsos Model. Gve a asse verse of sze, le he prme deoe marx rasposo; le,,, ) deoe he fracoal seqeally appled vesme allocaos he asses; le ( 1 2 x ( x1,x2,,x 1) represe he sae of a vesor s wealh, he compoe x deog he balace ye o be vesed as a me ; le he vecor r r,r,,r ) deoe he expeced vales for he asse rers ( 1 2 over a sgle holdg perod; ad le symbol Op deoe a operaor ha separaely opmzes each eleme of a crero se, say {f 1, f 2,, f m }. If asse rers are radom varables draw from a o probably dsrbo wh covarace marx [c ], he he porfolo problem may be modelled as a b-obecve dyamc opmzao problem o he dscree doma N = {1, 2,, } as follows: MP4: Op g ( ), h( ) Sbec o: x 1 x x 1 1 x [0,1] {1, 2,..., } Where: g ( ) r, r s he porfolo rer; 1 h () c s he porfolo varace. 1 1 Remars. he M-program MP4 s a wo-po bodary vale problem, ad he examples [61] aes o GENO s effcacy o hs class of problems. I s easy o mpleme whe oly sadard cosras apply, as s he case above. B he real-world, cosras perag o asse exposre, rasaco coss ad vesably are ofe ecessary, ad hese reder MP4 raher awward o mpleme. Ad was hs awwardess ha promped he formlao of a secod model explaed ex. 6 Copyrgh : Apex Research Ld

7 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD he Parallel Decsos Model. he parallel-decsos model s a rly dyamc M-program whch he basc premse s ha vesme allocaos are made adem a each decso sage; he formlao proceeds as follows. Gve a asse verse of sze, le he marx ] deoe he fracoal vesme [ 0 1 decsos made over me he colm vecor beg appled a he -h sace; le x x x x ] [ 1 2 deoe a sae marx ha descrbes he vesor s asse holdgs over me; le he colm vecors of he marx ] deoe he expeced sgle-perod asse rers a me = 1, 2,, ; assme ha [ 1 2 each vecor comprses radom rers r draw from a o probably dsrbo wh covarace marx [ c ], ad ha he sad dsrbos s ellpcal, depede ad decal over me; le he prme deoe marx rasposo; le h deoe a ml-perod measre of porfolo rs ad g deoe he porfolo s oal rer over he perod [0, ]; 6 he he ml-perod porfolo problem may be modelled as a b-obecve dyamc opmzao problem o he dscree me doma {0, 1, 2,, } as: MP5: Op g ), h( ) ( Sbec o: x 1 x ; {1, 2,, }; {0, 1, 2,, } 1 1 ; {1, 2,, }; {0, 1, 2,, } [0,1] ; {1, 2,, }; {0, 1, 2,, } x 0 0 Where: g ) r [ ] s he oal porfolo rer over [0, ] ( 1 ) H (L, ) 1 h ( s a cohere measre of he porfolo rs over [0, ]. Remars. he M-program MP5 s a ml-perod model, ad as sch oe eeds o specfy a rs crero ha s me-cosse,.e. oe ha esres compably of cosecve decsos mpled by he measre [37]. o ha ed, a approprae measre s oe formlaed by Hardy & Wrch [26] called he Ieraed Codoal al Expecao (ICE). he ICE sasfes he coherecy axoms of Arzer, e al. [3, 4], as well as he reqremes of dyamc cossecy ad relevacy as arclaed by Redel [55]. he followg s a descrpo of he formla for he recommeded rs measre h: assme he plag horzo [0, ] s dvded o sb-ervals, each of legh years; le elemes of he se N {1,2,,} deoe he mber of erao sages ad defe a sage locaor varable by m ; le deoe he cmlave dsrbo fco of he sadardzed Gassa radom varable z; ad le ad be parameers of a log-ormal asse prce process ha characerses a radom seqece of loss varables L ha are sed o assess he porfolo s rs; he, a ay decso sage [ 0, ] m : m N cofdece level s (see Appedx B, eqao B11): H, 1 z 2 L, L expm [ 0.5 ] 1 m, we have ha = m, ad he ICE a he dervao of eqao (1) ad how cold be mplemeed pracce s explaed flly Appedx B. (1) 6 Noe ha g r where r s he marx race operaor; he formla for h saed below s explaed flly Appedx B Copyrgh : Apex Research Ld

8 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Hadlg Specal Cosras Preamble. Icorporag specal cosras o he M-V model s he sbec of mch crre research, b GENO affords some smple mehods hs regard. Example 3 below whch volves exposre ad rodlo cosras s a case po: he GENO framewor, sch cosras are, he ma, mplemeed by merely eerg he reqred mbers o he p daa marx. Oherwse, oe ca always clde sable ad hoc mahemacal expressos for sch specal cosras as follows. Modellg Mehods. here are wo oher ypes of cosras ha have garered aeo he lerare: () cardaly cosras specfy pper ad lower bods o he sze of he porfolo o be cosrced from a gve asse verse, here deoed by he se E sch cosras may be reqred for porfolo dversy ad effce moorg; () by- hresholds specfy he mmm amo ha may vesed a parclar asse sch cosras pera o he eed o mmse rasaco coss. Boh ypes are easly accommodaed he GENO framewor: by- hresholds are mplemeed by smply declarg a fgre for he lowerbod varables, UCB ad LSB (cf. Example 3) ad cardaly cosras are deal wh as follows. Frs oe ha, he mere ac of declarg he sze of he sae vecor esablshes he sze of he asse verse E ; f he asse composo of he proposed porfolo cocdes wh E, he cardaly cosras are rreleva; oherwse, cardaly cosras may be mplemeed drecly va bary varables, or sg Boolea varables ogeher wh some logc proposos as follows. he Drec Approach. o cosrc a porfolo of sze N [N L, N U ], from he asse verse E, where N L s mmm ad N U s he maxmm porfolo sze, oe approach sggesed by Se, Brae & Schmec [65] s o rodce o MP5 he followg se of cosras whch respecvely for : L ad U are he lower ad pper bods [bl, bu ] (2a) E 1 b N [ N L, N U ] (2b) b {0, 1} (2c) Smlarly, f D s he sbse of he asse verse ha s sbec o he rle der Germay Ivesme Law [22], he he cosra may be modelled as follows. 0.05b 0.05 D (3a) D b 0.4 (3b) D b D (3c) he GDP Approach. A mch more expressve approach wold be o adop a modellg echqe devsed by Chemcal Egeers a Carege-Mello Uversy called Geeralsed Dscve Programmg (GDP) [25]. he mehod employs Boolea varables ad logc proposos o express cosras of he eher-or ype. For he sae of llsrao, cosder he se of relaos: 8 Copyrgh : Apex Research Ld

9 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD b [L, U ] (4a) {0} (4b) b he Boolea varable b 1 2 E b b re (4c) f ( b) re (4d) b {re, False} (4e) b dcaes wheher he -h asse s ( b = re), or o ( b = False) of a parclar porfolo; eqao (4c) esres ha he proposed porfolo coas a leas oe asse; ad he fco f (b) s a proposo ha splaes frher codos; for example, f mgh coa erms sch as b b ( whch s he exclsve-or operaor) o mpleme a dversy reqreme ha saes: asses m ad cao smlaeosly be par of ay parclar porfolo ; ad of corse, several sch proposos are allowed. Closg Remars. Noe ha boh mehods oled above lmaely resl a Mxed-Bary Program, ad he mercal examples [61] show ha GENO s well capable of solvg hs class of problems. m 5 Accommodag Uceray Preamble. he solo o a porfolo problem s o compable less or l oe ows he model parameers,.e. he mea vecor ad he covarace marx of he asse rers. I pracce herefore, porfolo opmzao s sally a wo-sep procedre ha proceeds as follows: gve some me seres daa o asse rers coverg a hsorcal perod, () sascal esmaes of he mea vecor ad covarace marx of he asse rers are comped based o he observed daa; () he esmaes comped a Sep 1 are he regarded as f hey were he re poplao parameers for he ow probably dsrbo ha geeraed he observed daa; hey are plgged o he MV model, ad he laer s solved o prodce he (spposedly) opmal asse allocaos. A poeal sorce of weaess hs plg- solo sraegy s a Sep 1, he sascal esmao sage. I pracce, less here are srog a pror reasos o sgges oherwse,... s sally assmed ha he observaos are [Gassa] dsrbed. hs assmpos s parly sfed by a appeal o he Ceral Lm heorem. However, [he Gassa dsrbo] ca rarely be ae for graed, ad he famos remar of Pocaré remas ap:... everyoe beleves he (Gassa) laws of errors, he expermeers becase hey h s a mahemacal heorem, he mahemacas becase hey h s a emprcal fac [Paraphrased from 27, p.116] B oce he Gassa assmpo s cas o dob, he robsess of he esmaors employed becomes a ey sse: rs o ha some esmaors sffer a drasc loss of sascal effcecy 7 f he daa sppled hem s o exacly Gassa. hs s maly wha les a he roo of he sably-of-opmzed-porfolos problem defed by Mchad [46] a re-calbrao of a MV solver sg a o-robs esmaor ad ew daa ha s o as Gassa as he orgal sample resls sgfcaly dffere esmaes for he model parameers, ad hece solver geeraes asse allocaos ha are qe removed from he old oes. 7 A good sascal esmaor s oe ha s based ad has a small mea sqare error; f a esmaor s based, he s mea sqare error cocdes wh s varace, ad so gve wo esmaors ha are based, a choce of he oe wh he lower varace s cosse wh mmsg he mea sqare error. hs goodess crera s measred by relave effcecy : a esmaor e 1 of a parameer p s sad o be relavely more effce ha aoher esmaor e 2 f boh e 1 ad e 2 are based esmaors of p, b he varace of e 1 s less ha ha of e 2. 9 Copyrgh : Apex Research Ld

10 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD here are several ways of aclg hs problem, each wh s ow mers ad demers; however a dealed dscsso of hese mehods s beyod he scope of hs paper; eresed readers may wsh o cosl a rece srvey [20]. he mos obvos approach s o se robs esmaors a Sep 1 of he plg- sraegy; a secod approach s o formlae robs verso of he MV problem assmg cera characerscs for he asse rer dsrbos; a hrd approach esseally combes he wo sages of he plg- sraegy porfolos are geeraed sg opmal esmao oe sgle sep sch as [12]; a forh approach s o combe he vesor s pror belefs o asse rers wh evdece from hsorcal observaos ad he se esmaors based o Bayes heorem. Modellg Mehod. he mehod proposed hs paper s esseally a plg- sraegy he ve of üücü & Koeg [70]: s basc raoale s ha, sce sascal po esmaes are always accompaed by some degree of mprecso o maer how sophscaed he esmaor sed may be, s prde o se ervals or ceray ses wh whch he parameers qeso may be assmed or show o le wh a cera degree of cofdece. he mehod s a adapao of a olear erval programmg (NLIP) echqe for dealg wh daa ceray ha s preseed by Jag, Ha, L & L [32]. he formlao s as follows. Cosder a geerc M-program whch he cosra ad obecve fcos are deermed by a decso vecor U R codoal o a daa vecor d R m, vz.: MP6a: f ( d, ) ( d, ) 0; d, U Max 0 f1 I pracce, here s always some degree of ceray as o he re vale of he vecor d. hs gorace may be modelled by seg he elemes of d as closed ervals d q IR, q = {1, 2,, m} o R raher ha sgle pos. Ad he rodco of he ervals d q o MP6a mples ha he fcos f ad c also assme erval-le qales, ad oe may characerse hem by he md-pos ad rad, or by he ed-pos o her respecve erval rages. Exacly how he ervals d q propagae hrogh he fcos f 0 ad f 1 may be wored o aalycally or emprcally sg he calcls of (real) erval mbers [36, 48]. Jag, e al. [32] base her NLIP formlao o a preferece relao for closed ervals IR ha was frs sggesed by Ishbch & aaa [31]; a smplfed defo of he sad relao s as follows: DEFINIION 1 [Ierval Preferece Relao] Cosder wo closed ervals A = [a L, a R ] ad B = [b L, b R ] o he real le ha pera o he vales aaed by a crero fco of a opmzao problem. he ervals may also be characersed by a ordered par comprsg he cere or md-po m a 0.5(a L + a R ), ad he rads or half-wdh w a 0.5(a R a L ), vz., A = (m a, w a ) ad B = (m b, w b ); a sefl dey s: [a L, a R ] m a + w a [ 1, 1]. If he assocaed opmzao problem s oe of maxmzao, he he erval preferece relao A P B (read: A s beer ha B ) mples he followg: A P B (m a m b ) (w a w b ) (5a) Smlarly, f he assocaed opmzao problem s oe of mmzao, he A A P B mples he followg: P B (m a m b ) (w a w b ) (5b) REMARKS 1 hs s a smplfed verso of he preferece relao [31] ha reas oly he esseal elemes. Le f 0L ad f 0R deoe he lef ad rgh ed-pos, respecvely, of he obecve fco erval dced by he rodco of he daa ervals d q IR, q = {1, 2,, m}; le f 1L ad f 1R be smlarly defed for he cosra fco f 1 ; ad le m 0 ad w 0 deoe he md-po ad rads fcos wh respec o he ervals of f 0 (); he clearly: 10 Copyrgh : Apex Research Ld

11 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD f 0L () If 0( d, ) d f 0R () Sp 0 ( d, ) d f (6a) f (6b) Noe ha f 0L () ad f 0R () are deermsc becase he ceray assocaed wh he daa vecor d s effecvely elmaed by he If ad Sp operaos. Ad by defo: m 0 () 0.5(f 0L () + f 0R ()) w 0 () 0.5(f 0R () f 0L ()) (7a) (7b) he preferece relaos DEFINIION 1 mplcly comprses wo choce measres: () a cardal preferece ecapslaed by he order relao o he erval md-pos; () a averso for ceray ecapslaed by he order relao o he erval wdhs. Accordgly, f oe apples he preferece relao DEFINIION 1 o he fcos f 0 ad f 1 MP6a, he oe obas a robs verso of he orgal M-program, amely he erval-dced, b-obecve opmzao problem ha may be saed hs: m MP6b: Op m ( ), w ( ) f ( ) 0; di ; U 0 0 1R R he md-po fco m 0 () s correlaed o he expeced vale of he paral fco f 0 (), ad he rads fco w 0 () s correlaed o he varace V[ f 0 ]. For example, f oe assmes, prely for he sae of argme, ha f 0 s formly dsrbed o [f 0L, f 0R ], he: m 0 () E[ f 0 () ] w 0 () 3 f ( 0 ) (8a) V (8b) I he case of MP6b herefore, he approprae erpreao of he operaor Op whe dsrbed over he crero se { m 0, w 0 } MP6b s ha assmes he meags maxmze ad mmze, respecvely becase maxmzg m 0 s cosse wh he orgal program MP6a, ad mmzg w 0 s cosse wh redcg he adverse effecs of he ceray d. hs, MP6b s effec a reward-rs opmzao formlao maxmzg he md-po fco m 0 also maxmzes he expeced vale of f 0,.e. he reward ; ad sofar varace measres rs, he mmzg he rads fco w 0 blds o he solo a cera degree of robsess o he esmao rs emaag from varaos he daa vecor d. hs fac may be sed o formlae a robs verso of he porfolo problem. For example, cosder a sgle-perod verso of MP5: he cera daa are he asse rers r ad each eleme of he covarace marx [c ]; a applcao of DEFINIION 1 separaely o he rer fco g ad he rs fco h wold resl a crero se comprsg for elemes; he robs verso of MP5 may be saed hs: MP7: Op m ( ), w ( ), m ( ), w ( ) g g h h Sbec o: x x 1 ; {1, 2,, }; {0, 1, 2,, } 1 1 ; {1, 2,, }; {0, 1, 2,, } [0,1] ; {1, 2,, }; {0, 1, 2,, } x Copyrgh : Apex Research Ld

12 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Where a each decso sage : m g () 0.5(g L () + g R ()); w g () 0.5(g R () g L ()); m h () 0.5(h L () + h R ()); w h () 0.5(h R () h L ()). MP7 may seem raher complcaed o mpleme de o he fac ha he reqse operads, amely he edpo fcos g L (), g R (), h L () ad h R (), are defed va opmzao processes (accordg o eqao 6). B hs problem s more appare ha real: he daa compoes feare learly boh g ad h, ad so s easy o derve explc expressos for he ed-po fcos sg erval aalyses of he fcos qeso. Noe ha for he sgle-holdg perod verso of MP5, he porfolo rer fco degeeraes o he er prodc g =, r whch s blear. Ad so f he vecors r L ad r R deoe collecos of he lef ad rgh-ed pos of he asse rers vecor r, he from he aral erval exeso of g wh respec o r, follows mmedaely ha; see [48, Chap. 5]: 8 g L (), r L g R (), r R (9a) (9b) Oe does o eed o explcly deerme he ed-po perag o h becase he fcos lmaely reqred by MP7.e. m h ad w h may be saed drecly sg he defos of marx md-po ad marx wdh ; he former s exacly he defo (7a) appled o each erval eleme of a marx; he laer s a scalar defed as he spremm orm of a marx; see [48, 7.1]. o ha ed, le c L ad c R deoe he lef ad rgh ed-pos of he erval perag o he covarace marx eleme, c ad le he w c deoe he wdh of he covarace marx [c ]; he s easy o show ha: m h () [0.5(c L + c R )] w h () w c, (10a) (10b) he wdh fco w c, whch as meoed earler s defed as he spremm orm of a marx, s: w c Max0.5(c c ), R (10c) Closg Remars. Noe ha f he erval vecor d s degeerae.e. whe all s elemes are specfc cosas ad here s o ceray abo her acal vales he eqaos (9a) ad (9b) coalesce o he ormal porfolo rer g =, r; all wdh fcos vash; eqao (10a) becomes he ormal porfolo varace, ad hs MP7 revers o he ormal, o-robs formlao of MP5. Noe also ha m h ad w h are s as easly derved for he cohere rs fco recommeded for MP5 becase he daa elemes, ad, are argmes of moooc fcos; see [48, 5.2]. Ad all cases, he correc erpreao of Op whe dsrbed over he crero se { m g, w g, m h, w h } s max, m, m ad m respecvely. L 8 A alerave approach wold be o se a calcls-of-varaos argme as follows: g =, r g =, r g R g L =, (r R r L) g R g L =, r R, r L (g R(), r R) (g L(), r L) 12 Copyrgh : Apex Research Ld

13 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Solo of he Models Preamble. he oo of opmzao s ambgos he -obecve coex: he verb opmze s a commad ha s versally dersood o mea compe he exrema of some crero fco. B he same cao be sad of ml-obecve problems becase he varos crero fcos volved evalae caddae solos dsparae ways: ae par-wse, some may compee he sese ha a mproveme wh respec o oe degrades he solo as assessed he oher; ohers may collde he sese ha a mproveme oe eals he same he oher; ad ohers may be oally depede. I s o srprsg herefore ha here exss dffere ypes of opma for sch problems ad frhermore, some of he opmzao oos yeld o-qe solos. A commo approach o he qadary posed by he oqeess problem s o rodce a model-ser or decso-maer.e. he perso for whom he comped solos are eded ad reqre sch a perso o arclae prefereces a some sage drg he solo process. B hs paper shall oly be cocered wh oe he compromse solo. he raoale for he compromse solo s preseed [62] ad shall o be repeaed here. he Compromse Solo ad s Compao. he compromse solo cocep ha was frs rodced by Salvadze [58] ad laer depedely preseed by Y [76] ad Zeley [77]. I s based o he commosese ad compellg oo ha he bes opo s a feasble po ha yelds vales ha are closes o a deal ocome he deal beg ha po a whch each crero s opmzed o he flles exe possble. he raoale for he compromse solo s bes explaed erms of he wo-dmesoal ocome space depced Fgre 1 below whch f 1 (x) ad f 2 (x) are fe-valed crero fcos of a decso vecor x, ad he colleco of all sch feasble ocomes coses he ocome se. Assocaed wh each ocome vecor are for bodary ocome vecors 1, 2, 3 ad 4 ; hese are pos where les ha are parallel o he axes f 1 ad f 2 ersec he bodary of he se, whch shall hereafer be deoed by. he verces of he smalles recagle eclosg comprse he opa se, ad for ay gve verex vecor z, each of s dmesos represes he bes possble ocome,.e. he global solo, ha cold be aaed by maxmzg or mmzg a parclar crero depedely. However, oly oe verex wold be releva ay gve scearo ad sch a verex s coveoally called he deal po. I Fgre 1 below, z 1 s he deal po whe boh crera are reqred o be mmzed; whereas z 4 s for he case where crero 1 s o be mmzed, ad crero 2 maxmzed. Fgre 1: Ocome Se of a b-obecve Opmzao Problem f 2 z 4 4 z f 1 z 1 3 z 3 13 Copyrgh : Apex Research Ld

14 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Oe may defe he compromse solo wo sages as follows: DEFINIION 2 [he Ideal Po]: Le R deoe a ocome se; le L ad U deoe he lower ad pper bods respecvely for he crero f a assmg all oher ocomes rema cosa; le z, deoe he deal po for he problem a had, he he coordaes of z are gve by scalars z defed as: z Sp U ( ) If L ( ),, f he f he h crero reqresmaxmzg h crero reqresmmzg REMARKS 2: Becase deal ocomes are ormally o oly aaable, a compromse s reqred. DEFINIION 3 [he Compromse Solo]: Le he po z be he deal po a gve ml-obecve opmzao problem; he he compromse solo s a member of hose feasble corols whose ocomes are closes o he deal ocome as measred by some dsace fco sch as he Ecldea or chebycheff merc; hs, erms of he chebycheff merc, he compromse solo s a feasble vecor x* whose correspodg ocome vecor * belogs o a se of ocomes ha s defed as: (11a) z ) : arg m z (11b) ( REMARKS 3: he defo of (z ) eals wo processes: () he obvos mmzao process deoed by he arg m operaor; () he less obvos search process ha s spposed o deleae he bodary se. he raoale derlyg hs solo cocep may be explaed as follows: (a) here s o qeso ha, f were achevable, he deal ocome vecor z wold cose he opmal solo o he ml-obecve opmzao problem der sdy; (b) b sce hs s sally o he case, oe has o compromse dowwards from he deal ocome z o a less-ha-deal ocome ω* ha correspods o a feasble vecor x* ad obvosly, he exe of he dowward compromse,.e. he qay of crero vale ha ms be gve p alog each dmeso, has o be mmal, hece he dsace-mmzg operao he defo of he solo se, ad he splao ha ω*. Closg Remars. he compromse solo s logcally sod ad compellg; dspeses wh he eed o calbrae ly fcos as s he case oher sgle-po MV solo echqes sch as [15]; he modelser s o eve reqred o express prefereces a ay po he solo process; raher, mplc o he mehod s a assmpo of vesor behavor ha effecvely assers he followg: Assmpo: he model-ser s raoal ad wold arally prefer he deal solo; b sce sch a solo s sally o aaable, he model-ser wll accep he compromse solo as beg he bes ha ca be doe he gve crcmsaces. I addo, le may ser-medaed ml-obecve solo mehods ha ed o become cmbersome ad eve seless as we crease he mber of obecves [10, p.4] he sze of he problem, as measred by he dmeso of, s largely rreleva. I order o mpleme he compromse solo, oe ms frs evalae (11a) ad hs ca be a seros hddle pracce. If s possble o reformlae he crero fcos f sch ha z 0, he ha s wha shold be doe; falg ha, he a echqe volvg a moooc fco called he Geeralzed Loss rasform (or g-loss rasform) may be sed sead he g-loss rasform s explaed deal [62]. B assmg he deal vecor z s ow, he he arg m operao (11b) may be vewed as a commad o fd a solo o a olear eqao sysem, ad so he mehods explaed [63] are applcable; he wo obecve mehods preseed here, amely he compose-merc mehod (code ame: EDR) ad he NCP mehod (code ame: ECR), are sed o Example 4, prmarly for he sae of effcecy. 14 Copyrgh : Apex Research Ld

15 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Nmercal Examples Preamble. hs seco preses some mercal examples whose prpose s o assess he deas ad echqes preseed hs paper. he reqse p daa s from varos sorces: he frs example s from [40] ad provdes a basc comparso of he solo mehod employed here wh GENO; he secod example s from [44] ad serves o compare GENO o he Blac-Lerma model as well as he re-sampled effcecy mehod of Rchard ad Rober Mchad; 9 he hrd example s from [19] llsraes he hadlg of oradoal ad hoc cosras ad assesses GENO s solo per se; he las example employs daa from [70] ad serves o assess he robs opmzao model formlaed ad preseed 5 above. Example 1: A sgle-perod MV opmzao problem 10 MP8: Op g ( ), h( ) Sbec o: x 1 x ; {1, 2,..., 5} x 1 1 x [0,1] Where: 5 g ( ) r, r s he porfolo rer; 1 5 h () c s he porfolo varace ables 1A: Expeced Aal Rers (%) [40, p.163] Asse Name SECURIY 1 SECURIY 2 SECURIY 3 SECURIY 4 SECURIY 5 MEAN REURN ables 1B: Covarace Marx [40, p.163] Covarace Marx SECURIY 1 SECURIY 2 SECURIY 3 SECURIY 4 SECURIY 5 SECURIY SECURIY SECURIY SECURIY SECURIY US Pae #6,003,018, December A bref descrpo of he re-sampled effcecy mehod may be fod Becer, Gürler & Hbbel [6] who prese a exesve smlao sdy ha compares o he radoal Marowz approach. 10 MP6 s a seqeal-decsos model; hs was sed o geerae he frs se of resl repored below; he secod se of resls was geeraed by a parallel-decsos model whch, for he sae brevy, s o re-saed here. 15 Copyrgh : Apex Research Ld

16 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD GENO Op [he Seqeal Decsos Model] Geerao me Nmber (sec) Porfolo Loss 11 Porfolo Rs Asse Name: SECURIY 1 SECURIY 2 SECURIY 3 SECURIY 4 SECURIY 5 Allocao: Porfolo Rer (%): Porfolo Rs: Rer-Rs Rao: GENO Op [he Parallel Decsos Model] Geerao me Nmber (sec) Porfolo Loss Porfolo Rs Asse Name: SECURIY 1 SECURIY 2 SECURIY 3 SECURIY 4 SECURIY 5 Allocao: Porfolo Rer (%): Porfolo Rs: Rer-Rs Rao: Remars o Example 1. Leberger ally solves hs problem by qadrac programmg,.e. sg MP3a, he mmm varace verso of he MV problem; he he geeraes a secod effce porfolo sg he wo-fd heorem [40, p.163]. As ca be see above, boh GENO models geerae seqeces ha coverge, b o wo dffere porfolos; he for porfolos are aalysed ad compared able 9 of he Porfolo Loss s here defed as he recprocal of he oal porfolo rer. hs s so ha he org of he ocome space becomes he deal po whch s reqred he compromse solo scheme (see 6). hs s permssble whe porfolo rer are o-egave (as s he case for all he examples hs paper); b for a more geeral approach, see he geeralzed Loss rasform preseed [62]. 16 Copyrgh : Apex Research Ld

17 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Example 2: A sgle-perod MV opmzao problem MP9: Op g ( ), h( ) Sbec o: x 1 x x 1 1 x [0,1] {1, 2,..., 8} Where: 8 g ( ) r, r s he porfolo rer; 1 8 h () c s he porfolo varace ables 2A: Expeced Rers ad Sadard Devaos (%) [44] Asse Class Ero Bods US Bods Caada Frace Germay Japa UK US Mea Rer Sd Devao ables 2B: Correlao Marx [44] Correlao Ero Bods US Bods Caada Frace Germay Japa UK US Ero Bods US Bods Caada Frace Germay Japa UK US Noes o Example 2. Mchad, e al. [44] se hs example o assess her re-sampled effcecy mehod of porfolo opmzao agas he Blac-Leerma model; able 2C below, he GENO resls are appeded o hose from [44] for a hree-way comparso; frher commes may be fod der able Copyrgh : Apex Research Ld

18 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD GENO Op Geerao me Nmber (sec) Porfolo Loss Porfolo Rs Asse Name: Ero Bods US Bods Caada Frace Germay Japa UK US Allocao: Porfolo Rer (%): Porfolo Rs: ables 2C: Comparave performace of porfolos: GENO verss ohers from [44, able 3] Mare (%) Blac-Lerma (%) Mchad MSR (%) GENO (%) Ero Bods US Bods Caada Frace Germay Japa UK US PORFOLIO REURN (%) PORFOLIO RISK (%) REURN-RISK RAIO Remars o Example 2. I her evalao of her re-samplg mehod agas he Blac-Lerma model, oe coclso ha Mchad, Escher & Mchad draw s ha der he same codos, he Mchad MSR porfolo s beer dversfed, less bechmar cerc, ad less sbec o large rsy allocaos [Paraphrased from 44, p.12] Admedly, he GENO porfolo s also less dversfed ha he Mchad MSR. B hs appare weaess s more ha compesaed for by he speror rer-rs rao; 12 ad ay case, he GENO framewor allows oe o easly dce more dversy va exposre cosras as llsraed by Example 3 below. 12 Roy [57] was he frs o sgges a vara of he reward-rs rao called he Safey-Frs Rao as a measre for evalag a vesme sraegy s vale; Sharpe laer appled Roy s deas o he mea-varace framewor of Marowz ad developed a merc ow commoly ow as he Sharpe Rao, oe of he bes ow performace evalao measres porfolo maageme [60]; he reward-rs rao s eqvale o he Sharpe rao whe he bechmar rer assocaed wh he laer s zero, sch as s he case whe he bechmar asse s Cash [14]; a porfolo wh a relavely hgh reward-rs rao awards larger rers for each addoal of rs [19]. 18 Copyrgh : Apex Research Ld

19 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Example 3: A sgle-perod MV opmzao problem wh exposre ad rod-lo cosras MP10: Op g ), h( ) ( -1-1 Sbec o: x 1 x ; {1, 2, 3, 4}; {0} 1 1; {1, 2, 3, 4}; {0} [0, 1] ; {1, 2, 3, 4}; {0} [0, 1] { : 0.13, Z} ; {3}; {0} x0 0 ; = 1; = 4. Where: g ) r [ ] s he porfolo rer; ( h ( ) c s he porfolo varace ables 3A: Expeced Rers ad Sadard Devaos [19] Asse Class US Bods US Large-caps Eqy US Small-caps Eqy EAFE Ieraoal Eqy Mea Rae of Rer (%) Sadard Devao ables 3B: Correlao Marx [19] Correlao Coeffces US Bods US Large-caps Eqy US Small-caps Eqy EAFE Ieraoal Eqy US Bods US Large-caps Eqy US Small-caps Eqy EAFE Ieraoal Eqy Noes o Example 3. hs example s a sgle-perod mplemeao of he parallel-decsos model ha serves o llsrae, er ala, he ease wh whch oe may mpleme exposre ad rod-lo cosras he GENO framewor. Exposre cosras ypcally places a sbecve pper lm o he vesme proporo allocaed o specfc asses, ad rod-lo cosras are aemps o comply wh dscreeess reqremes for vesmes cera asses. I hs parclar case, here s a pper bod or exposre cosra o oe asse deoed by 1 ad a rod-lo cosra o a secod asse deoed by 3. he frs cosra s mplemeed va daa sppled o GENO (see able 3C & 3D), whereas he secod s mplemeed by a combao of a explc dscree se-cosra (cosra #4 MP10) as well as he daa sppled. 19 Copyrgh : Apex Research Ld

20 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD GENO Op Geerao me Nmber (sec) Porfolo Loss Porfolo Rs Asse Class: US Bods US Large-caps Eqy US Small-caps Eqy EAFE Ieraoal Eqy Ivesme Allocao: Porfolo Rer: Porfolo Rs: able 3C: GENO Ip Daa Marx for he MV opmzao problem wh exposre ad rod-lo cosras VARIABLE NAME X1 X2 X3 X4 UCB LCB 1e-6 1e-6 1e-6 1e-6 USB LSB 1e-6 1e-6 1e-6 1e-6 Ial Sae Vecor Fal Sae Vecor Dscree Vales able 3D: Leged o he p daa for he MV opmzao problem wh exposre ad rod-lo cosras Corol varables (): UCB Upper Corol Bod; LCB Lower Corol Bod; Sae varables (x): USB Upper Sae Bod; LSB Lower Sae Bod he eres UCB = 0.3 ad Dscree Vales = 0.13 mpleme he followg cosras respecvely: a) Exposre: Fds allocaed o US Bods (varable x 1) shold o exceed 30% of he oal fd; b) Rod-lo: Ivesme US Small-Caps Eqy (varable x 3) shold be eger mlples of 0.13, say. 13 Remars o Example 3. hs example shows how oe may easly deal wh mpora praccal maers (sch as he Germa rle meoed 2) sg GENO; he framewor acally accommodaes eve more complex real-world scearos ha oe may formlae, e.g. va dscve programmg (see 5). 13 Sch a dscree vesme proporo wold have bee derved as follows: le p ad deoe he crre mare prce ad he rod-lo qay respecvely for he asse qeso. he c = p s he cos crrecy s of oe rod-lo, ad based o a oal vesme bdge b, he dscree rod-lo vesme qay o a per--of-vesme bass s smply c/b. 20 Copyrgh : Apex Research Ld

21 Praccal Porfolo Opmzao [rev. 5] echcal Repor RD Example 4: A sgle-perod MV opmzao problem: he robs approach MP11: Op m ), w ( ), m ( ), w ( ) g ( g h h Sbec o: x 1 x 1 1 [0,1] x 0 0 {1, 2,, 5} {0} Where: m g () 0.5(g L () + g R ()) w g () 0.5(g R () g L ()) m h () 0.5(h L () + h R ()) w h () 0.5(h R () h L ()) ables 4A: Expeced Rers (%) [70] Asse Name 14 R1000 Growh R1000 Vale R2000 Growh R2000 Vale LBGC Bod Mea Rer (%) ables 4B: Covarace Marx [70] Covarace Marx R1000 Growh R1000 Vale R2000 Growh R2000 Vale LBGC Bod R1000 Growh R1000 Vale R2000 Growh R2000 Vale LBGC Bod Noes o Example 4. hs example affords a assessme of he robs opmzao model preseed 5 sg daa prepared by üücü & Koeg [70, p.13]. B sead of solvg drecly as a ml-obecve problem, a ad hoc adapao of he compose-merc mehod for solvg olear eqao sysems ha s descrbed [63] was sed for he sae of effcecy. A o-robs model was also r assmg omal vales for he expeced rer ad he covarace marx for comparso boh resls are depced below. 14 Leged: R1000 Growh Rssell 1000 dex for large cap growh socs; R1000 Vale Rssell 1000 dex for large cap vale socs; R2000 Growh Rssell 2000 dex for small cap growh socs; R2000 Vale Rssell 1000 dex for small cap vale socs; LBGC Bod Lehma Brohers US Iermedae Goverme/Cred Bod dex for fxed come secres 21 Copyrgh : Apex Research Ld