The UPM/LPM Framework on Portfolio Performance Measurement and Optimization

Transcription

1 he UPM/LPM Framewor on Porfolo Performane Measuremen and Opmzaon Lngwen Kong U.U.D.M. Proje Repor 2006:0 Examensaree maema, 20 poäng Handledare oh examnaor: Johan ys Deemer 2006 Deparmen of Mahemas Uppsala Unversy

2

3 Asra In real lfe he nvesmen reurn s no normally dsrued and he nvesors audes o upsde poenal and downsde rs wh respe o a enhmar are dfferen. he upper paral momen (UPM) and he lower paral momen (LPM) n plae of mean and varane as reward and rs measure have een suggesed o solve hs asymmer prolem (Farnell and le (2002)/Moreno, Cumova and Nawro (2004)). In hs maser hess, we dsuss he empral properes of he UPM/LPM framewor as performane measure and also applaons o porfolo opmzaon. In order o evaluae he enef of he performane measuremen, we onras he ranng resuls from he UPM/LPM performane measuremen and onvenonal ses. he model sensvy o parameer hange and he esmaon rs are also esed. Regardng porfolo opmzaon prolems, when employng he sewness suden- dsrued sample, we show ha he UPM/LPM porfolo opmzaon model provdes dfferen weghs from onvenonal models, and exhs a more effen froner. We fnd ha hgher momens an e of grea sgnfane for performane ranng and porfolo seleon ased on UPM/LPM framewor. And y hoosng approprae enhmar and upper or lower order, he UPM/LPM model s ale o refle nvesors varous asymmer preferenes.

4 Anowledgemens I would le han my supervsor, Professor Johan ys, and oher eahers n he Fnanal Mahemas Programme, Uppsala Unversy, for nrodung he suje of Fnanal Mahemas and provdng a sold foundaon for my fuure researh wor; I am also hanful o Professor Nawro, for explanng hs prevous wor on UPM/LPM model; my graude also elongs o my parens and frends, Zhnan Ln, Bo Pan, Chao Cheng, Yu Meng, for supporng and enouragng me o pursue my areer n Fnane.

5 Conen:. Inroduon Convenonal Framewor n Invesmen heory Mean-Varane Framewor Mean-Below arge semvarane Framewor Mean-Lower Paral Momen Framewor Requremen for a new framewor.. 3. A New framewor: Upper Paral Momen/ Lower Paral Momen Defnon Congruene wh Uly Funon Porfolo Performane Measuremen Based on UPM/LPM Framewor In-sample Comparson Sensvy o Benhmar Shf Esmaon Rs Porfolo Opmzaon ased on UPM/LPM model Opmzaon mehod In-sample omparson Effen Froner Influene of Benhmar Shf Conluson 42 Referene...43

6 . Inroduon Marowz (952) proposed ha nvesors expe o e ompensaed for ang addonal rs and provded a framewor for measurng rs and reward. Based on hs onep, Sharpe (966) nrodued he reward-o-volaly rao, nown as he Sharpe rao, he frs major aemp o reae a measure for omparson of porfolos on a rs-adjused ass. hs mean-varane analyss requres assumpons on he nvesor s uly funon, namely a quadra uly funon, or on he normaly of he reurns dsruon. I s well nown however ha a quadra uly funon s nonssen wh raonal human ehavor. Moreover, he fnanal nsrumens reurns are oserved of non-normal dsruon,.e. wh posve/negave sewness or/and fa-al. he rs free rae of reurn as enhmar s also quesonale. In an asymmeral world, he good (aove he enhmar) volaly and ad volaly (elow he enhmar) may e srongly dfferen. wo ypes of asymmer should e noed, asymmery n preferene o good and ad volaly from he enhmar, and asymmery n preferene o small and large devaons from he enhmar. F Sorno, K Bernardo (998) proposed a fund performane measure, he U-P rao, whh aes no aoun he nvesors asymmer preferene eween upsde poenal and downsde rs aordng o a spefed enhmar. Moreover S Farnell and L le (2002) proposed a generalzed rs-reward rao and D Moreno, D Cumova and D Nawro (2004) provded proedure o apply UPM/LPM model n porfolo opmzaon. In hs hess we apply generalzed UPM/LPM model o asses wh sew suden- dsrued reurn, whh maye more lose o hedge fund reurn dsruon. Our wor nludes wo sdes, one s o he he enef of UPM/LPM performane measuremen for ranng porfolos ompared o onvenonal models and o es he model sensvy o parameer hange and he esmaon rs; he oher wor s o analyze how UPM/LPM porfolo opmzaon model provde asse alloaon weghs when he asses are sew suden- dsrued and nvesors do are aou hgher momens of porfolo reurn. - -

7 In Seon 2, some onvenonal reward-rs framewors are presened and her weanesses are dsussed. In Seon 3, we nrodue he defnon of UPM and LPM, as well as s orrespondng porfolo performane measuremen and s represened uly funons. In Seon 4 we analyze how he UPM/LPM performane measuremen onsders varous nvesors preferenes, and also he he sensvy o enhmar shf and s esmaon rs. In Seon 5, he UPM/LPM model wll e appled o porfolo opmzaon and he haraerss n porfolo seleon are analyzed. Fnally, we presen he onlusons n Seon

8 2. Convenonal Framewor n Invesmen heory 2. Mean-Varane Framewor he modern porfolo heory along wh he onep of rs/reward sared wh he pulaon of paper Porfolo Seleon y Marowz (952). He denfed ha wo faors should e onsdered n porfolo seleon, he reward and he rs. Reward s defned y expeed reurn whle he rs s defned y varane. he esmaon formulaon s as elow, μ = 2 2 σ = ( μ ), = X X = X X where 2 μ X s reward, σ s rs, s he numer of oservaons, and X s he rae of reurn durng me. he nvesor has o mae a radeoff eween rs and reurn. Asse alloaon s performed y solvng an opmzaon prolem. he opmal porfolos are hose ha maxmze he expeed reurn for a gven varane or mnmze he varane for a gven expeed reurn. An effen froner urve onssng of all suh opmal porfolos ould e onsrued. hs mean-varane framewor (MV) n porfolo opmzaon prolem an e saed as elow, suje o MnmzeV = ω C ω, ω μ ω = E, A ω =, ω 0, where C s he ovarane marx, μ = μ Lμ ) s he expeed rae of reurn for asse ( n o asse n, ω = ω Lω ) s he wegh veor, and A s lnear equales onsrans ( n - 3 -

9 marx. For example, one of hese equaly onsrans saes ha he sum of wegh ω s one, u oher speal ondons on spefed asses an also e added. he las nequaly onsran for shor-sellng fordden ould e added or no. he Lagrange funon for hs prolem s, L = ω C ω λ ( A ω ) λu ( μ ω E), 2 where λ ( λ Lλ m ) and λ denoe Lagrangan mulplers for he onsrans. hen we = u an use sequenal quadra programmng o solve hs quadra opmzaon prolem. he sequenal quadra programmng algorhm s emedded n he Mala opmzaon ool ox. Based on he MV framewor, Sharpe (966) nrodued a reward-o-varaly porfolo performane measuremen. hs so-alled Sharpe Rao s defned y, μ X rf SR =, σ Here devaon. μ X s he expeed reurn, rf s he rs free rae of reurn, and σ s he sandard MV analyss reles on he resrve assumpons ha eher he nvesor s uly funon s quadra or he reurns are normally dsrued. he orrespondng uly funon s saed elow, U ( r) r 2 = r, where s nvesor's margnal rae of susuon of expeed reurn for varane, r s he wealh he nvesor possesses

10 Fgure 2. quadra uly funon (Soure: marowz (959)) Fgure 2. shows he relaonshp eween uly and wealh r. Quadra uly funons are unappealng, eause hey mply nreasng asolue rs averson. ha s, nvesors wh hs ype of uly funon requre hgher rs premums for a gven nvesmen as her wealh nreases. hs s oserved o e onrary o oh nuon and oserved nvesor ehavor. Marowz (979) demonsraes ha he MV approah ould e used o maxmze he expeed Bernoull s uly funon, u s lmed y he assumpon ha all people are rs-averse and prefer erany, whh s no rue for some nvesors enefed y aepng small amouns and agreeng o ae losses,.e. opon wrers. When he reurn s non-normally dsrued, he reurn and varane may nauraely desre reward and rs for no apurng hgher momens of reurn dsruon suh as sewness and uross (fa als). he reason for he wde aepane of MV analyss s s ompuaonal smply. he desrale propery of varane or sandard devaon s ha apures he reurns for he whole dsruon. As a rs measure, varane or sandard devaon somemes hs our goal, onsderng ha our am s prng a rsy asse, where we fous on apurng he saly around a enral endeny. However, some nvesors or nvesmen sraegy may wegh dfferenly eween upsde poenal and downsde rs relave o some enhmar

11 2.2 Mean-Below arge semvarane Framewor Varane as a rs measure nspres argumens sne defnes all devaon from mean as rs. An alernave formulaon uses he elow-arge semvarane (SV) (Marowz 959) as a measure of rs. he SV s an asymmer rs measure fousng on he reurns elow a spefed reurn arge. he esmaon formula s saed as follows, 2 SV( X, τ ) = { MAX (0,( τ X ))}. = Here s he numer of oservaons, X s he rae of reurn durng me, τ s he reurn arge and MAX s he maxmzaon funon. he opmzaon prolem an e formulaed as elow (Marowz 993), suje o Mnmze = z 2, z n = τ ω X + y, = z 0, 0, y μ ω = E, A ω =, ω 0, where s he numer of oservaons, n denoes he numer of asse, τ s he enhmar durng me, μ = μ Lμ ) s he expeed rae of reurn for asse o ( n asse n, ω = ω Lω ) s he wegh veor, A s lnear equales onsrans marx, ( n and X s he reurn of asse durng me. he Lagrange for hs prolem s, - 6 -

12 * L = f ( ω, z) λ ( z ( τ X ω + y)) λ ( A ω ) λu ( μ ω E), 2 where f ω, 2 (, z) = z = * * * and λ = ( λ Lλ ), λ λ Lλ ), = ( m u onsrans. hs s also a quadra opmzaon prolem. λ denoe he Lagrangan mulplers for he he orrespondng performane measure, nown as Sorno Rao (994), s defned y, rf RSV X = μ x ( ), SV where he rs par n denomnaor denoed as sandard devaon n Sharpe Rao s replaed y elow arge sandard devaon. Compared o he MV framewor, he rs n he erm of SV onsders he nvesor s asymmer aude, so he nformaon onaned n he upsde of he dsruon does no onrue o he rs u s apured n he mean of he dsruon. And he reurn arge s se aordng o he nvesor s averson. herefore hs framewor s more algned wh nvesors perepon ha loss weghs more han reurns. hs framewor represens he followng uly funon (Mao970), U ( r) = r for all r τ, 2 τ ( ) for all < τ U ( r) = τ r r and > 0, where r s he nvesor s wealh, τ s he enhmar or reurn arge and s nvesor's margnal rae of susuon of expeed reurn for varane. he relaonshp eween uly and wealh s deped n Fgure 2.3 a a =

13 hus, he mean-sv framewor relaxes he assumpon on he asse reurn dsruon. Moreover, he porfolo seleon prolem s also a quadra opmzaon prolem. 2.3 Mean-Lower Paral Momen Framewor Movng from he mean-sv framewor o mean-lower paral momen (mean-lpm) framewor s o lerae he nvesor from a onsran of havng only one uly funon o a sgnfan numer of uly funons. In he mean-lpm framewor, expeed reurn s sll used as he reward par, u he rs par s expressed y a general measuremen, he lower paral momen (Fshurn 977). I s defned y, LPM [{ MAX ( X )} ] a = E 0,, where E means expeed value, s he enhmar, a s he order of he lower paral momen, and MAX s he maxmzaon funon. I ould e esmaed y, a LPM ( a, τ ) = { MAX (0, τ X )}, = where s he numer of oservaons, X s he rae of reurn durng me, τ s he enhmar or reurn arge durng me. LPM wh he order 0 < a < an express rs seeng, a = rs neuraly, and a > rs averson ehavor for a group of he nvesors. Rs averson means he furher reurns fall elow he enhmar, he more he nvesor dsle hem, whle rs seeng represens he aude of advenure lovers. For a =, LPM eomes o he expeed devaon of reurns elow a arge, whle for a = 2, LPM s analogous o he varane elow he arge reurn, or SV. he opmzaon prolem s, Mnmze LPM ω a = = { } a MAX (0, τ X ω), - 8 -

14 suje o μ ω = E, A ω =, ω 0, where X = x Lx ) s he reurn veor for n asses durng me, ω = ω Lω ) s ( n ( n he wegh veor and oher noaons are he same as he defnon n LPM. For a s a onvex opmzaon prolem holdng ha eah loal mnmzer s a gloal mnmzer. Nawro (99, 992) proposed a reonsrued LPM formula for a ha he opmzaon prolem urns o e a quadra opmzaon prolem. In hs approah, he downsde rs par s saed as elow, n n = j= 2 E( LPM ) = ω ω CLPM = ω LPM + ω ω CLPM, for a, P j j n j j j where LPM = = [ { }] a MAX 0,( x ) τ, for a, CLPM j = = a [ MAX { 0,( x )}] ( τ x ) τ, for a >, j CLPM j = I = τ, for a =, { MAX [ 0.( x )]} ( τ x j ) LPM = CLPM j, for = j. Noe ha n hese formulas, E LPM ) s ounded a a. And I {} = for x > 0, oherwse, I {} x = 0 ( p. he opmzaon prolem s, x Mnmze LPM ω = ω L ω p, - 9 -

15 suje o where CLPM L = M CLPM μ ω = E, n A ω =, ω 0, L O L CLPM M CLPM n nn. In hs ase, he Lagrangan funon s L = ω L ω λ ( A ω ) λu ( μ ω E), 2 where λ = ( λ Lλ m ) and λu denoe Lagrangan mulplers for onsrans. So he porfolo opmzaon prolem an e solved n he same way as mean-varane model. he porfolo performane measuremen ould e defned y, RLPM a τ μ x rf ( X ) =. a LPM ( a, τ ) Insead of squarng he elow arge devaon and ang square roo n performane measuremen alulaons, he devaon n mean-lpm performane measuremen ould e adjused y order a. Reallng MV and mean-sv framewor, hey only provde us wh one ype of uly funon, hene only one ype of nvesor s preferene. he uly funons mpled n mean-lpm framewor are saed as elow (Fshurn 977), U ( r) = r for all X τ, a τ ( for all < τ U ( r) = τ r) X and >

16 he uly funon for dfferen rs order a s llusraed n Fgure 2.3. Fgure 2.3 uly funon n M-LPM framewor ( a = 0/ 0.5// 2 / 4 ) (soure: Fshurn (977)) hs uly funon s paral suppored y he ongruene wh von Neumann- Morgensern uly funon for elow-enhmar par, where apures nvesors varous rs preferenes, suh as rs averson for a>, rs neuraly for a=, and rs seeng for 0<a<. Bu he lnear uly funon aove he enhmar par mples only one neuraly aude oward aove-enhmar reurn. hs s he mos ommon rsm of mean-lower paral momen model. 2.4 Requremen for a new framewor wo ypes of asymmery should e onsdered and ul no he new model, he asymmery n preferene eween he upsde and downsde volaly from he enhmar. he asymmery n preferene eween he small and large devaon from he enhmar. he former asymmery desres he nvesors dfferen audes o he upsde gan and downsde loss. he laer asymmery refles he nvesor s nlnaon (n he ase of - -

17 expeed gans) or dsle (n he ase of expeed losses) for he exreme evens. So he new porfolo model s requred o expresse nvesors arrary preferenes y managng non-normal dsrued asses, suh as ang no aoun sewness and uross or oher hgher momens affe nvesmen deson. In he followng seons we wll nrodue a UPM/LPM framewor ha fulflls hese requremens

18 3. A New framewor: Upper Paral Momen/ Lower Paral Momen 3. Defnon Consder an asse wh random oal reurn X over a eran perod. Is performane s measured n omparson wh a enhmar reurn. he rs s represened y LPM, defned he same as n mean-lpm framewor, LPM [{ MAX ( X )} ] a = E 0,. Here he reward s replaed y UPM, also nown as upsde poenal, defned y (Farnell and le 2002), UPM [{ MAX ( X ) } ] = E 0,, where s he orders of upper paral momen, s he enhmar, E denoes expeed value, and MAX s he maxmzaon funon. he man dfferene o mean-lpm model s he replaemen of he expeed porfolo reurn wh he UPM, whh apures he haraers of upper reurns devang from he enhmar. he UPM onans mporan nformaon aou how ofen and how far nvesor wshes o exeed he enhmar. As n he LPM alulaon, n UPM he dfferen orders represen dfferen nvesor ehavors, > for poenal seeng, = for poenal neuraly or < for poenal averson. Poenal seeng means he hgher he reurns aove he enhmar, he happer he nvesor. he poenal averson desres a raher onservave sraegy on he upsde, for example, a sraeg ulzng a shor all or a shor pu and her dynam replaon wh so and onds. hus dfferen orders an e used o solve varous preferenes. If moderae devaons from he enhmar are relavely harmless when ompared o large devaons, hen a hgh order for he lef-sded momen s suale. Ve versa, f small suessful ouomes over he enhmar are relavely appreaed wh respe o exeponal large saes, hen a low order for he rgh-sded momen well fs he purpose. Hene, he ofen rzed uly - 3 -

19 neuraly aove he enhmar ha s nheren n he mean-lpm framewor s elmnaed now. he UPM and LPM an e esmaed y, LPM UPM = = { MAX (0, X )} = a τ, { MAX (0, X )} = τ, where he s he numer of oservaons, τ s he enhmar durng me, a and are he orders of lower paral momen and upper paral momen, X s he reurn for he asse durng me. he performane rao ased on UPM/LPM model for an asse wh oal reurn X and enhmar s defned for any a, > 0 y, [{ MAX (0, X ) } ] { MAX (0, X )} a / a, E Φ ( X ) =. / a E [ ] Is analyss an e aomplshed usng hsoral daa as proxes for ex-ane asse ehavor. he formula s saed elow, Φ a, ( X ) = a = = { MAX (0, X τ )} { MAX (0, τ X )} a, Φ a ( X ) an e seen as he reward-o-varaly, or, n eonom erms, as he shadow pre for un of rs for he exess reurn. When he enhmar τ s fxed, he hgher he ndex a, Φ ( X ), he more preferale he rsy asse X s. hs s a seleon rule provdng - 4 -

20 preferene ranng for a se of omparale asses a he same enhmar. And as menoned efore, he hgher he orders a and are, he more emphass s gven o he exreme evens on he dsruon als. he orders n he UPM and LPM are no requred o e equal. On he onrary, an asymmeral preferene on he exreme favorale and unfavorale evens s que normal n real lfe desons. For example, for =, a = 2, we ge, [{ MAX (0, X MAR} ] { MAX (0, MAR X )} R( X ) = E, E [ ] 2 where MAR s he mnmum aepale reurn, as enhmar n our general formaon. I s defned y Sorno (999) as a penson fund performane measuremen. 3.2 Congruene wh Uly Funon In he mean-lpm model, rs s measured y he LPM; s parally ompale wh he expeed von Neumann-Morgensern uly heory. Exendng smlar houghs o he UPM/LPM model, s possle o denfy analyally a uly funon dependng only on he lower and he upper paral momen up o order a and. Le e a fxed enhmar, A he se of all random varales and X A has fne n n paral momens UPM (, j) and LPM (, j) for all posve negers j < n. If he ojeve funon of an expeed uly maxmzer on A n as only on he ass of he LPM and UPM up o order a and, hen he assoae uly funon has he followng form (Farnell and le 2002), (X ) U = = a j= j= j j ( X ) j h ( X ), j, for for X X > - 5 -

21 where and h are non-negave parameers. he expeed uly hen has he followng form, j j a [ X )] = j LPM (, j) + E U ( h UPM (, j), j= j= j where a and e he lower and upper paral momen order. Le j = 0, 0, for for j a j = a h j = 0, 0, for for j j = hen he expeed uly funon has he followng formaon, a [ ( X )] = LPM (, a) + h UPM (, ) = h UPM (, ) LPM (, a) E U a h. Sne he uly funon s defned as a lnear ransformaon, an e expressed y a lnear omnaon of UPM and LPM. In hs ase, he uly funon ould e expressed n he followng, U ( X ) ) U ( X ) ) X = ( X for all, a = h ( X for all X < and h > 0. Boh pars of uly funon are properly shaped y he orders a and respevely, In Fgure 3.2., we an see he dfferen shape of uly funon orrespondng o dfferen orders a and

22 u(r) a=4 =4 a=3 =3 a= = a=.5 =.5 a=.2 = x fgure3.2.uly funon (=0 h=) he reverse S-shaped uly funon desred n general for a > and > s onssen wh nsurane agans losses and ang es for gans (Fgure3.2., a = = 3 or 4), he upper par of uly funon s onvex and he lower par s onave. he larger he values of order a and are, he seeper he reverse S-shape urve s. he order 0 < a < and 0 < < (Fgure3.2., a = = 0. 5 or 0.2) orrespond o he S-shaped uly funons, where he upper par of uly funon s onave and he lower par s onvex. he smaller he value of order a and are, he flaer he S-shape urve s. hey apure he nvesors endeny o mae rs-averse hoes relave o UPM and rsseeng hoes relave o LPM. In hs ase nvesors may e very rs-averse o small losses u wll ae on nvesmens wh a small proaly of very large losses. Also, for ndvduals wh he poenal reurn and rs seeng ehavor, an e apured y 0 < a < and >, whh mples a onvex uly funon. In addon, rs neuraly ( a = ) n omnaon wh poenal averson or poenal seeng or poenal neural, - 7 -

23 .e. lnear loss funon and onave or onvex upper poenal funon, an e expressed. Smlarly, he upper poenal neuraly ( = ) n omnaon wh downsde rsaverson or rs-seeng, mplyng a lnear uly funon aove he enhmar reurn and onave or onvex elow he enhmar reurn, s allowed. Lnear gan and loss funon an e also assumed y a = and = (Fgure 3.2.), whh means ha he gans and losses are evaluaed proporonally o her exenson

24 4. Performane Measuremen Based on UPM/LPM Framewor 4.. In-Sample Comparson In order o he how a, Φ ( X ) ndex onsders he hgher momens and nvesors a, asymmer preferene, we ompare he value Φ of wo nvesmens wh he same reurn and varane u dfferen values of sewness. ale 4.. desres he sas properes of hese wo nvesmens. nvesmen A nvesmen B Reurn/Proaly (0 oservaons) Mean Varane Sewness Kuross ale 4.. asses reurns ls (Narow 999) Sne he frs momen (mean) and he seond momen (varane) of wo asses reurns are same, he Sharpe Rao gves he same ran, n oher words, ould no ell he dfferene eween hese nvesmens. And he measuremen ased on MSV s nluded n he mean-lpm framewor for a = 2, so we only ompare he resuls ased on mean- LPM and UPM/LPM. I s lsed n ale 4..2, UPM() LPM(a) a, Φ MLPM A B A B A B A B a=0.2 = a=0.5 = a= = a=2 = a=3 = a=2 = ale 4..2 nvesmen performane rao value ased on UPM/LPM and MLPM (=5) - 9 -

25 In hs expermen, he asses reurns are no normally dsrued and he nvesors do are aou he hgher momens refleed y he order a and. As he order a nreases, represenng he nvesor from rs-seeng o rs-averse, he value of LPM nreases. For a < he Invesmen A s onsdered o e more rsy han B, alhough he value of sewness ndaes ha Invesmen B has possly n gger loss. hs resul s onssen wh s uly funon, reallng ha a advenure-seeng ehavor. For a < uly funon desres a > nvesmen B s more rsy han A o nvesors eause of negave sewness. On he oher sde, as he order nreases, represenng ha he nvesors move o poenal seeng, he value of UPM nreases. For < nvesmen A s onsdered o e less rewarded, and hs suaon s hanged for., Aove all, he value of Φ a ( X ) for nvesmen B s larger han A a a, <, whh means nvesmen B s preferred o nvesmen A for nvesors who sasfy wh small aove-enhmar reurn and agree o ae g loss; he value for nvesmen A s larger a order a, >, whh means nvesmen A s preferred for nvesors who are poenal seeng and rs averson. Noe ha he rans of mean-lpm and UPM/LPM performane measuremen are dfferen for a = 2 and = 0. 5, where he nvesors show averson o all nds of volales. In hs ase, ells ha nvesmen B s more suale eause he larger devaed reurns happen less frequenly. I s eer han mean/lpm rao n ha an e modfed aordng o he nvesors varous audes o up-enhmar volaly. 4.2 Sensvy o Benhmar Shf Φ a, Noe ha a, Φ ( X ) s a funon of enhmar. he hoe of he enhmar s an exogenous queson wh respe o analyss of he performane. For example, he enhmar n Sharpe Rao s he rs free rae of reurn; whle n Sorno s ndex, s he mnmum aeped reurn. In UPM/LPM model, he enhmar s a sujeve hoe o nvesors. Reallng he uly funon, enhmar s a n pon separang nvesors ehavors aordng o dfferen audes owards upsde poenal and downsde rs. In hs seon, we nvesgae he sensvy of a, Φ ( X ) ndex o enhmar shf

26 Inuvely he hgher he enhmar s, he lower he possly o ea, herefore he hgher he enhmar s, he lower he performane ndex should e. Sne [{ MAX ( X ) } ] UPM = E 0, = ( X ) df( X ) s a dereasng funon of, and [{ MAX ( X )} ] a a LPM = E 0, = ( X ) df( X ) s an nreasng funon of ; he a, rao ( X ) urns ou o e a dereasng funon of he enhmar for a gven order a Φ and. heoreally, we an derve he frs paral dervave of ndex wh respeve o enhmar, +, a a ( ) ( ) ( ) ( ) ( ) Φ x df x x df x X a, = + Φ + a ( ) ( ) ( ) ( ) x df x x df x ( X ) < 0. hs formula shows ha he exen of sensvy o enhmar shf s relaed o he order a and as well as he dsruon of underlyng asse. In order o analyze how a, Φ ( X ) from dfferen samples rea o he hange of enhmar, we use he SN-paage n R o generae 200 random varales for eah of 4 sewness suden- dsruons. he mean and sandard devaon (SDev) are all around. he frs dsruon s wh negave sewness and fa-al. he seond one s wh negave sewness and smaller exess uross. he hrd one s wh small sewness u larger exess uross. he las one s generaed from normal dsruon. ale 4.2. desres he as sas properes of hese daa. X X2 X3 X4 Mean SDev Sewness Exess Kuross ale 4.2. sample desrpon - 2 -

27 , hen we alulae he Φ a ( X ) value when enhmar hanges from = 0 o =. Gven a se of order ( a, ), we oserve ha he preferene ranng for hese four nvesmens may hange as he hange of enhmar. When nvesors who are rs averse and poenal seeng, Φ dsruon he hghes ran whn he range from a, a = = 3 (Fgure 4.2.), for (X ) gves asse X4 wh normal = 0 o =. he ran reversal happens eween asse X2 and X3 a = When a = 2, = 0. 5 (Fgure 4.2.2), for nvesors who are averse o oh nds of volaly, here s no domnan asse and ran reversal happens eween asse X and X4 a = he reversal ould happen eween dfferen asses a dfferen enhmar pon, so an onlude ha he sensvy for enhmar shf s relaed o he asse reurn dsruon as well as he nvesor s asymmer preferene denoed y he LPM and UPM orders a,. UPM/LPM X X2 X3 X4 UPM/LPM X X2 X3 X Fgure 4.2. a,, Φ ( X ) vs. ( a = = 3) Fgure Φ a ( X ) vs. ( a = 2 = 0. 5 ) Nex we apply oher performane measuremens o he same samples n order o ompare her sensvy o enhmar shf. Beause of he same mean and varane for all asses, Sharpe rao s almos equal for oh rades, so here s no ran hange. We only ompare performane measuremens ased on mean-lpm and UPM/LPM framewor. Here we se a = 2, = 0.5 n UPM/LPM model and a = 2 n mean-lpm model

28 UPM/LPM M/SD , Fgure Φ a ( X ) vs. ( a = 2 / = 0. 5 ) Fgure mean/arge semdevaon vs. I s oserved n Fgure and ha he ran reversal happens eween X and X4 when usng mean-lpm measuremen, whle n a, Φ ( X ) ndex, happens o same asses u a = 0.6. Addonally, he slope n Fgure s no as seep as he one n Fgure when s small, demonsrang less sensvy o enhmar shf. he reason s ha UPM s a funon of enhmar, hus he effe of enhmar shf ould e magnfed y order. 4.3 Esmaon Rs In order o he esmaon rs, we generae random varales wh normal dsruon, where mean = and sandard devaon = 2. By varyng he sample sze from 60 o 600 daa pons, we smulae 00 reurns me seres for eah sample sze and ompue her a, Φ ( X ) values a sx suaons where dfferen enhmars and orders ( = 0 / 0.5/, ( a, = 2) / ( a = 2, = 0.5) ) are used. hen, we alulae he average value of he esmaed a, Φ ( X ) for eah sample sze a eah suaon. We also ompue he sandard error of esmaed a, Φ ( X ) around s rue value, whh s alulaed y usng he parameer n he rue dsruon. hs leads o Fgure 4.3., he sensvy of esmaed a, Φ ( X ) o sample sze under dfferen enhmars and he orders s shown n

29 he lef sde; n he rgh sde, he orrespondng sandard error of esmaed a, Φ ( X ) wh respeve o sample sze s also deped. We an see he mpa of larger sample szes on he reduon of he esmaon error. average value when =0,a==2 average sandard error when =0,a==2 avg.hea average value: heoreal value: avg.sandard error average value when =0.5,a==2 average sandard error when =0.5,a==2 avg.hea avg.sandard error

30 average value when =,a==2 average sandard error when =,a==2 avg.hea avg.sandard error average value when =0,a=2,=0.5 average sandard error when =0,a=2,=0.5 avg.hea avg.sandard error

31 average value when =0.5,a=2,=0.5 average sandard error when =0.5,a=2,=0.5 avg.hea avg.sandard error average value when =,a=2,=0.5 average sandard error when =,a=2,=0.5 avg.hea avg.sandard error Fgure 4.3. average value of esmaed a, Φ ( X ) o sample sze hange We fnd ha when sample sze elow 20, he esmaed value of a, Φ ( X ) shows sgnfan devaon from s rue value and he error erm redues rapdly when more daa ould e used, whereas when sample sze s larger han 240, larger sample sze do no provde sgnfan mprovemen. he unerany lned o he esmaon of a, Φ ( X ) ends o reah a floor afer 240 daa pons. We an herefore argue ha a me seres should nlude a leas 20 oservaons o gve onssen resuls wh a, Φ ( X ). In real lfe, f monhly reurn s used, a leas en years daa are requesed. However, more aurae resuls ould e oaned wh more daa used eyond 240, whh are ovously

32 shown n Fgures 4.3., he rgh par; he sandard error onnues o derease when sample sze s eyond 240. Moreover, we ompare he relave sandard error, whh s ompued as dvdng he sandard error y he average esmaed a, Φ ( X ) value. Fgure shows he hange of relave error as sample sze nreases under spefed enhmar and he orders. We an see he enhmar and he UPM/LPM orders do affe he onvergene,.e., relave error a = (Fgure lef) or a = 2, = 2 (Fgure rgh) domnae all he expermens. Bu n oh suaons, he a, Φ ( X ) value dd no onverge o he heoreal one. Mos me n our rals, hs value s over esmaed. I may omes from he asymmer defnon of he rs and he reward, for example, dfferen UPM and LPM orders provde dfferen power o error or enhmar far away from mean dvdes UPM and LPM wh unequal numer of oservaons. So when he enhmar omes o mean or he UPM and LPM orders are se equal, he esmaed value more lely onverges o he heoreal one. sandard error/hea when a==2 sandard error/hea when =0.5 sandard error/hea =0 =0.5 = sandard error/hea a=2/=2 a=2/= Fgure relave sandard error o daa sze a dfferen enhmar or orders

33 5. Porfolo Opmzaon ased on UPM/LPM framewor Gven he poenal usefulness of UPM/LPM model, we expe ould e used n porfolo seleon. he opmzaon prolem s, suje o or Mnmze E( LPM P ) = τ ω = { MAX (0, ω ( X ))} = E( UPM P ) = { MAX (0, ω ( X τ ))} = UPM p, ω =, a, suje o Maxmze E( UPM P ) = τ ω = { MAX (0, ω ( X ))} = a { MAX (0, ( X ))} LPM p E ( LPM P ) = ω τ =,, ω =. Where ω ( n = ω Lω ) s he wegh veor, X = ( x Lx n ) s he rae of reurn veor durng me, and τ s he enhmar durng me. We onsder o do he porfolo seleon from n asses and smply ge he expe value y averagng he values durng me perod. We are loong for a legmae wegh ω ( ω Lω ) = wh mnmal n downsde rs gven porfolo Upper Paral Momen UPM p or maxmal upper poenal gven downsde rs LPM p. he wegh veor ω s legmae whenever fulflls he onsrans. If we wan o he he effen froner of UPM / LPM varyng λ and oanng all he soluons o he prolem as elow, regon, an e found y

34 suje o Mnmze E ω ( LPM P p ω =. ) λe( UPM ), In he followng seon, we wll nrodue an opmzaon mehod ased on UPM/LPM framewor developed y Nawro, Moreno and Cumova (2004). 5. Opmzaon mehod he E ( UPM ) P and E LPM ) are reformulaed as elow, ( P where n n 2 E( LPM ) = ω ω CLPM = ω LPM + ω ω CLPM, P = j= n n = j= j j 2 E( ULPM ) = ω ω CUPM = ω UPM + ω ω CUPM, P j j n j n j j j j j LPM = = [ { }] a MAX 0,( x ) τ, for a, CLPM j = = a [ MAX{ 0,( x )}] ( τ x ) τ, for a >, j CLPM j = I = τ, for a =, { MAX [ 0.( x )]} ( τ x j ) UPM = [ { }] MAX 0,( ) x = τ, for, CUPM j = [ MAX{ 0,( x τ )}] ( x j τ ), for > = CUPM j = I = { MAX [ 0.( x τ )]} ( x j τ ), for =,

35 LPM = CLPM j ; UPM = CUPM j, for any = j. Noe ha under hs formulaon E LPM ) and E UPM ) are ounded for a,. ( p ( p And I = for x > 0, oherwse, I {} = 0. Unforunaely, he approprae opmzaon {} x x mehod for a, < are sll unnown. he general opmzaon prolem s saed as elow, suje o Mnmze LPM ω UPM = ω L ω p, = ω U ω UPM, p = A ω =, ω 0, where CLPM L = M CLPM n L O L CLPM M CLPM n nn, U CUPM = M CUPM n L O L CUPM M CUPM n nn, a A = M am L O L a a n M mn, = ( L m ). Here A s he equaly onsran marx wh m rows for m lnear equales onsrans. In hs ase, Lagrangan funon s,

36 where λ L = ω L ω λ ( A ω ) λu ( ω U ω UPM ), 2 2 ( λ Lλ m ) and λ denoe Lagrangan mulplers for onsrans. = u In order o oan he effen froner n UPM/LPM regon, we solve he followng opmzaon prolem, suje o Mnmze LPM p λ UPM P, ω A ω =, ω 0. By varyng λ from 0 o, all ses of soluons ompose he effen froner. For λ = 0, he gloal mnmal porfolo s found, whle λ = he maxmum E(UPM ) porfolo s found. he relaonshp s desred n Fgure 5.. Fgure 5.. he relaonshp eween λ and feasle se ( UPM, LPM ) (soure: Nawro, Moreno and Cumova (2004)) - 3 -

37 hs reonsruon of E LPM ) and E UPM ) smplfy he porfolo seleon o a ( p ( p quadra opmzaon prolem. o es f he weghs oaned here are vald o onsru he effen porfolo n he UPM/LPM regon, we ompare hs froner o he one oaned from Marowz MV opmzaon model n UPM/LPM regon. Fgure 5..2 shows a onave UPM/LPM froner ras alongsde he MV froner and domnaes he MV froner almos all he me. hus hs mehod emodes he properes of UPM/LPM model meanwhle eeps he ompuaon smply. effen froner effen froner upm(=2) upm(=) Marowz: UPM/LPM: lpm(a=2) lpm(a=2) Fgure 5..2 effen froner y UPM/LPM model and MV model 5.2 In-sample omparson When applyng UPM/LPM model o porfolo opmzaon, we expe ould help o manage hgher momens of he porfolo reurn aordng o a spefed nvesmen preferene requremen. We generae en groups of samples, 20 random varales of eah, wh mulvarae sewness suden- dsruon. he sas propery of sample se s lsed n ale We apply hree dfferen porfolo opmzaon models, ased on UPM/LPM framewor, MV framewor and mean-lpm framewor, o hese samples and ompare he alloaon resuls as well as he sass properes of he opmal porfolos

38 Mean SDev. Sem Dev Sewness Kuross Y Y Y Y Y Y Y Y Y Y ale 5.2. sample desrpon We fous on he gloal mnmal rs porfolo, whh s loaed n he exreme lef of effen froner. We oan hs porfolo y mnmzng he rs whou onsderaon he reward level. Due o s ndependene of reward, we an solae he nfluene of rs defnon on he porfolo alloaon. Consderng ha he rs defnons under UPM/LPM and mean-lpm framewor are denal, we only ompare he resuls from UPM/LPM and MV model. In UPM/LPM approah, we oan 4 porfolos y nreasng he order a from o 4. ale and Char 5.2. desre he alloaon resuls. I learly shows ha MV porfolo prefers asse Y3, whh has he smalles varane among he en asses, whle as he order a nreases, he UPM/LPM porfolo weghs more on asse Y4, whh has posve sewness

39 MV LPM LPM2 LPM3 LPM4 Y Y Y Y Y Y Y Y Y Y ale asse alloaon asse proporon Asse Alloaon MV LPM LPM2 LPM3 LPM4 model seleon Y Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y0 Char 5.2. asse alloaon he sas properes of porfolo reurns are presened n ale I shows ha he MV gloal mnmal rs porfolo has he lower sandard devaon (SDev), whle all UPM/LPM gloal mnmal rs porfolos exep he one wh a = have lower elow arge sandard devaon (SD). I should no e a surprse sne he varane s defned as rs o e mnmzed n MV framewor whle he downsde par s defned as rs n UPM/LPM framewor u wh dfferen orders. Among UPM/LPM porfolo, he LPM2-34 -

40 porfolo ( a = 2 ) has he lowes SD sne n hs ase rs eomes he SV; he LPM porfolo ( a = ) has a hgher SDev and SD onssen wh he rs neural aude. he major ssue onernng he UPM/LPM framewor s s apaly o manage hgher momens. As he order a nrease, he porfolo s sewness nreases. Moreover, he porfolo sewness n LPM2, LPM3 and LPM4 are larger han he weghed sum of ndvdual asse sewness, whh shows he aly of UPM/LPM model n managng sewness. MV LPM LPM2 LPM3 LPM4 Mean Sandard Dev Below arge SD Sewness Kuross Effen Froner ale porfolo desrpon In order o analyze he relaonshp eween rs and reward defned y MV, mean-lpm and UPM/LPM framewor, we ompare her effen froners n he followng hree rs-reward oordnaes, Mean-Sandard devaon UPM/LPM( a = 2, = 3 ): orrespondng o rs averson and poenal see UPM/LPM( a = 2, = ): orrespondng o rs averson and poenal neural In he followng expermens, we fx he enhmar =

41 effen froner mean MV: ULPM(a=2,=3): ULPM(a=2,=): ULPM(a=,=2): MLPM(a=2): sandard dev. Fgure 5.3. effen froner n mean-varane oordnae In he mean-sandard devaon oordnae (Fgure 5.3.), he MV effen froner domnaes all he oher froners. he loses froner o he MV effen froner are he ones from he UPM/LPM ( a = 2, = ) and he mean-lpm ( a = 2 ) model, whose ojeve funon are loses o he MV framewor, so are he downsde par n her uly funons. However, he UPM/LPM ( a = 2, = ) and he mean-lpm ( a = 2 ) opmal porfolos sarfe some effeny n mean-sandard devaon regon y jus mnmzng he SV oher han all he volaly devang from he mean. he furhes froner from he MV effen froner s he one from he UPM/LPM ( a =, = 2 ) model, for nvesors who are rs neural and poenal see

42 effen froner upm(=3) MV: ULPM(a=2,=3): ULPM(a=2,=): ULPM(a=,=2): MLPM(a=2): lpm(a=2) Fgure effen froner n UPM/LPM(a=2,=3) oordnae In he UPM/LPM ( a = 2, = 3 ) oordnae, s as expeed ha he UPM/LPM( a = 2, = 3 ) opmal porfolos domnaes all he oher opmal porfolos (Fgure 5.3.2). he UPM/LPM ( a = 2, = 3 ), he mean-lpm ( a = 2 ), and he UPM/LPM ( a = 2, = ) model provdes he same gloal mnmal rs porfolo, eause oh of hem have he same rs defnon, so does he uly funon n he elow enhmar par. he furhes froner from UPM/LPM ( a = 2, = 3 ) effen froner s he one from UPM/LPM( a =, = 2 ) model for nvesors who are rs neural

43 effen froner upm(=) MV: ULPM(a=2,=3): ULPM(a=2,=): ULPM(a=,=2): MLPM(a=2): lpm(a=2) Fgure effen froner n UPM/LPM(a=2,=) oordnae When omes o he UPM/LPM ( a = 2, = ) regon, he froner from UPM/LPM ( a = 2, = ), mean-lpm ( a = 2 ) are smlar as hey do n he oher ases, eause hey have he smlar opmzaon funon for he smlar uly funons. 5.4 Influene of Benhmar Shf Noe ha n he UPM/LPM framewor, a rse of enhmar nreases he rs perepon of an asse and redues he reward perepon. I s expeed ha he use of dfferen enhmars should have an mpa on porfolo seleon. In order o nvesgae he nfluene of enhmar shf on he effen froner, we onras he effen froners oaned y nreasng he enhmar from 0, 0.2 o 0.4 and leavng he oher parameers unhanged ( a = 2 and = 3/ a = 2 and = ). Reallng he ongruen uly funons n UPM/LPM framewor, enhmar plays as he n pon separang he audes owards lower par, downsde rs, and upper

44 par, poenal reurn. As nreases, he lower par o e mnmzed n porfolo opmzaon nreases and upper par dereases, so he effen froner would shf o he down rgh dreon n he orrespondng UPM/LPM oordnae u o deferen exen aordng o he order, whh s shown n Fgure effen froner effen froner upm(=3) upm(=) =0: =0.2: =0.4: lpm(a=2) lpm(a=2) Fgure 5.4. effen froner hange y he enhmar shf Nex, we he he nfluene of enhmar shf on gloal mnmal rs porfolo. Here we hoose wo dfferen order se, a = 2 and = 3 / a = 2 and =. he asse alloaon of hese wo gloal mnmal rs porfolos are desred n ale 5.4./Char 5.4. and ale 5.4.2/ Char In oh ases ould e oserved ha as enhmar arses, asse Y2 and Y4 wh relavely small sem-devaon are weghed more, whle asse Y7 wh he larges sem-devaon among he seleed asses s redued. Moreover he asse Y4 wh posve sewness s weghed mos n oh porfolos

45 GMP =0 GMP =0.2 GMP =0.4 Y Y Y Y Y Y Y Y Y Y ale 5.4. Gloal Mnmal Rs Porfolo when =0, 0.2, 0.4 (a=2, =3) Asse Proporon Gloal Mnmal Rs Porfolo Alloaon GMP =0 GMP =0.2 GMP =0.4 Y Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y0 Char 5.4. Gloal Mnmal Rs Porfolo when =0, 0.2, 0.4 (a=2, =3)

46 GMP =0 GMP =0.2 GMP =0.4 Y Y Y Y Y Y Y Y Y Y ale Gloal Mnmal Rs Porfolo when =0, 0.2, 0.4 (a=2, =) Asse Proporon Gloal Mnmal Rs Porfolo Alloaon GMP =0 GMP =0.2 GMP =0.4 Y Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y0 ale Gloal Mnmal Rs Porfolo when =0, 0.2, 0.4 (a=2, =) - 4 -

47 6. Conluson UPM/LPM framewor provdes a new defnon of rs and reward y lower paral momen and upper paral momen. In hs maser hess, we emphasze he mporane of UPM/LPM model o refle nvesors asymmer preferenes n he nvesmen deson proess and also show ha he onvenonal framewors are nsuffen o adequaely assess he nvesmen. In our empral expermens for analyzng he properes of UPM/LPM performane measuremen, we fnd ha s ale o norporae hgher momens of porfolo reurns n order o provde a usom ranng; y alerng UPM/LPM orders as well as enhmar he UPM/LPM performane ndex an ran porfolos dfferenly aordng o nvesors varous preferenes. Regardng o he esmaon rs, n he ase of a normally dsrued sample, a leas 20 daa s requesed o oan a relale performane analyss. When s appled o porfolo opmzaon, we have shown ha ompared o mean-varane and mean-lower paral momen models he UPM/LPM model gves dfferen alloaon weghs and a more effen froner n s rs/reward regon. In parularly, we apply he UPM/LPM model o mulvarae sewness suden- sample, and fnd ha as nvesors eome more rs averse wll wegh more on he asses wh less elow-arge varane or larger posve sewness

48 Referenes. Fshurn, Peer C., "Mean-Rs Analyss Wh Rs Assoaed Wh Below- arge Reurns", Ameran Eonom Revew, 977, v67(2), Harlow, W. V., "Asse Alloaon In A Downsde-Rs Framewor", Fnanal Analys Journal, 99, v47(5), Haugen, R., Modern Invesmen heory, Prene Hall, Hogan, Wllam W. and James, M. Warren., "Compuaon of he Effen Boundary In he E-S Porfolo Seleon Model", Journal of Fnanal and Quanave Analyss, 972, v7(4), Levy, H. and Marowz, H. M., Approxmang Expeed Uly y a Funon of Mean and Varane, Ameran Eonom Revew, Jun, 979, Vol. 69 Issue 3, p308, 0p. 6. Marowz, H. M., "Porfolo Seleon", Journal of Fnane, 952, v7(), Marowz, H. M., Porfolo Seleon: Effen Dversfaon of Invesmens, John Wley & Sons, In., Marowz, H. M.; odd, P.; Xu, G.; Yamane, Y., Compuaon of meansemvarane effen ses y he Cral Lne Algorhm, Annals of operaons researh, 993, vol.45, No., Mao, J. C.., Models of Capal Budeng, E-V Vs E-S, he Journal of Fnanal and Quanave Analyss, 970, Vol. 4, No

49 0. Moreno, D.; Cumova, D.; Nawro, D., A Gene Algorhm for UPM/LPM porfolo, Compung n Eonoms and Fnane, 2006, No Nash, S. G.; Sofer, A., Lnear And Nonlnear Programmng, he MGraw-Hll, Nawro, D. N., "Opmal Algorhms And Lower Paral Momen: Ex Pos Resuls", Appled Eonoms, 99, v23(3), Nawro, D. N., "he Charaerss of Porfolos Seleed By n-degree Lower Paral Momen", Inernaonal Revew of Fnanal Analyss, 992, v(3), Nawro, D. N., "A Bref Hsory Of Downsde Rs Measures", Journal of Invesng, 999, v8 (3,Fall), Womersley, R.S. and Lau, K., Porfolo Opmzaon Prolems'', n A. Eason and R. L. May eds., Compuaonal ehnques and Applaons, 996, Sharpe, W. F., "Muual Fund Performane", Journal of Busness, 966, v39(), Par II, Sorno, F. A. and Pre, Lee N. "Performane Measuremen n Downsde Rs Framewor," Journal of Invesng, 994. (PRI wese). 8. Sorno, F. A. and Forsey, H. J., "On he Use And Msuse Of Downsde Rs", Journal of Porfolo Managemen, 996, v22(2,wner), Sorno, F. A. and Kuan, B., he U-P Sraegy: A Paradgm Shf n Performane Measuremen, Penson Researh Insue,

50 20. Soyanov, S.; Rahev, S.; Faozz, F., Opmal Fnanal Porfolo, Worng paper, Deparmen of Proaly and Appled Sass, Unversy of Calforna, Sana Barara, USA, le, L. and Farnell, S., "Sharpe hnng wh Asymmeral Preferenes", Soal Sene Researh Newor, O