RISK AND INVESTMENT OPPORTUNITIES IN PORTFOLIO OPTIMIZATION. Ali Argun Karacabey. Ankara University, Turkey.

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1 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey RISK AND INVESMEN OPPORUNIIES IN PORFOLIO OPIMIZAION Ali Argu Krcbey Akr Uiversiy, urkey. ABSRAC Mrkowiz legedry work bou porfolio opimizio is cceped o be he pioeer of he moder porfolio heory. Corry o is heoreicl repuio, i hs o bee used eesively. Koo d Ymzki (99) proposed ew porfolio opimizio model s lerive o Mrkowiz s me-vrice model. Mrkowiz s me vrice model d he me bsolue deviio models regrd risk i erms of deviios h my be eiher posiive (upwrd) or egive (dowwrd) i relio o he epeced reur. I oher words boh of he models pelize o oly he egive (dowwrd) deviios bu lso he posiive (upwrd) deviios. I his pper, ew model h kes io cosiderio boh risk d beer ivesme opporuiy is proposed. he differece bewee he proposed model d he oher porfolio opimizio models is heir obecives. he proposed model ssumes h ivesor ws o choose porfolio wih higher upside deviios d lower dowside deviios. Key word: porfolio opimizio, MAD, dowside risk JEL Codes: G, G2, G32 Fculy of Poliicl Scieces, 659, Akr, urkey.

2 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey I. INRODUCION Mrkowiz s legedry sudy of porfolio opimizio is regrded s he pioeerig work of moder porfolio heory. I he Mrkowiz model, risk is sed i erms of he prediced vrice of porfolio reur, fucio h is qudric i he decisio vribles. All oher fucios d cosris re ssumed o be lier (Shrpe, 97). he obecive of he model is o form he efficie porfolios. Corry o is heoreicl repuio, i hs o bee used eesively. he wo impor resos why Mrkowiz s model hs o bee implemeed c be summrized s follows (Elo, Gruber, Pdberg, 976; Koo d Ymzki, 99): (i) he difficuly i esimig he correlio mrices, (ii) he compuiol difficuly of he qudric progrmmig model. Shrpe (97) climed h if he essece of porfolio lysis problem could be dequely cpured i form suible for lier progrmmig mehods, he prospec for prcicl pplicio would be grely ehced. Shrpe (97) d Soe (973) ried o cover he porfolio problem io lier progrmmig model. Koo d Ymzki (99) proposed ew porfolio opimizio model s lerive o Mrkowiz s me-vrice model. hey employed L me bsolue deviio s risk mesure ised of vrice, i order o overcome he problem of compuiol difficuly. he MAD model is sid o be vible lerive becuse (i) i does o require he covrice mri of he reurs, d (ii) MAD porfolios hve fewer sses (Sim, 997). I is lso rgued h s he umber of sses decreses, he rscio coss of he porfolio will lso decrese. he MAD porfolio opimizio model hs 2+2 rows where is he umber of periods. Feisei d hp (993) reformuled he MAD porfolio opimizio model so h he umber of rows decresed o +2, which implies h he mimum umber of socks ivesed i decreses from 2+2 o +2. Chg (25) modified Feisei d hp s model so h his model hs fewer vribles d he sme umber of cosris. 2

3 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey All of hese opimizio models cosider risk s he deviio of reurs from he epeced or me reur. However, hese models ssume h here is o differece bewee posiive d egive deviio. O he oher hd, for ivesor posiive deviio is desirble, while egive deviio is o. So, i his pper ew model h differeies posiive d egive deviios is proposed. Accordig o his proposed model ivesor simuleously wishes o mimize posiive deviios d miimize egive deviios. II. REVIEW OF HE MEAN VARIANCE AND MAD PORFOLIO OPIMIZAION MODELS Mrkowiz (952) cosiders wo rules while formulig he porfolio opimizio model. Firs, he ivesor does (or should) mimize epeced reurs, d secodly, he ivesor does (or should) cosider epeced reur desirble hig d vrice of reur udesirble hig. he cocep of efficie porfolio hs emerged i ccordce wih hese wo rules. he Mrkowiz porfolio opimizio model employs vrice s he mesure of risk, d he obecive of he model is o fid ou he weighigs of sses h miimizes he vrice of porfolio d esure reur equl o or bigger h he epeced reur. Accordigly he mhemicl model for sses is s follows: Miimize subec o i= = = = r ρm =... Where: σ i = covrice bewee sses i d, = he mou ivesed i sse, r = he epeced reur of sse per period, σ i i () 3

4 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey ρ = prmeer represeig he miiml re of reur required by ivesor, M = ol mou of he fud, d u = mimum mou of moey which c be ivesed i sse. Koo d Ymzki (99) iroduced he L risk fucio (me bsolue deviio-mad) w () = ER E R ised of = he L 2 risk (vrice) fucio where R is rdom vrible represeig he re of reur per period of he sse. hey proved h hese wo mesures re he sme if (R R ) re mulivrie ormlly disribued. So he Koo-Ymzki MAD porfolio opimizio model becomes s follows: Miimize subec o w() = E = = E[R ] R ρm E = R =... Koo d Ymzki ssumed h he epeced vlue of he rdom vrible c be pproimed by he verge from he d. So: r = E[R ] = r = Where r is he relizio of rdom vrible R durig period (where = ). hus, w() is pproimed by = = (r r ). Deoig =r -r (= d = ), model (2) c be epressed s follows. (2) 4

5 Europe Jourl of Fice d Bkig Reserch Vol.. No Miimize subec o = = = r = ρm =... Ali Argu Krcbey Koo d Ymzki replced he model wih model (4) which is equivle o model (3). Miimize subec o = y y y = - = + r = = ρm =... =... =... Accordig o Koo d Ymzki he MAD porfolio opimizio model s dvges over he Mrkowiz s model re (i) his model does o use he covrice mri which herefore does o eed o be clculed, (ii) solvig his lier model is much esier h solvig qudric model, (iii) he mimum umber of sses h re ivesed i is 2+2 (if u = ) while Mrkowiz s model my coi s my s sses, d (iv) c be used s corol vrible o resric he umber of sses. Feisei d hp (993) modified he MAD porfolio opimizio model d proposed ew model h is equivle o Koo d Ymzki s bu hs limi of +2 o he umber of ozero sses i he opiml porfolio o he ssumpio h here is o upper limi o he ivesme u = - i sse. (3) (4) 5

6 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey hey subrc o-egive surplus vribles 2v d 2w from ech of he cosris i problem 4 i order o replce he iequliies wih equliies: y y + = = 2v - 2w = = I order o elimie y, hey subrc (6) from (5) d divide by 2. hus he porfolio opimizio model becomes: (5) (6) Miimize subec o = v (v = = r + w ) - w ρ M (7) = = =... =... v, w, =... Compred wih Koo d Ymzki s opimizio model uder he u = ssumpio, model (7) llows ivesme i mos +2 sses i he opiml porfolio. Accordig o Koo d Ymzki his mes h he opiml porfolio produced by (7) should hve lower rscio coss compred wih he opiml porfolio obied by model (4). Chg (25) reformule Feisei d hp s model by iroducig coiuous vrible d ; 6

7 Europe Jourl of Fice d Bkig Reserch Vol.. No Miimize subec o d d (2d = = - = = r ρm ) =... =... Ali Argu Krcbey (8) = =... Chg (25) proved h model (8) is equivle o model (7) while he umbers of ddiiol coiuous vribles d uiliry sig cosris re hlf of he Feisei d hp s model. So he CPU ime d he umber of ierios eeded o fid he opiml soluio re decresed. III. PROPOSED OPIMIZAION MODEL Mrkowiz s me vrice model d he me bsolue deviio models regrd risk i erms of deviios h my be eiher posiive (upwrd) or egive (dowwrd) i relio o he epeced reur. I oher words boh of he models pelize o oly he egive (dowwrd) deviios bu lso he posiive (upwrd) deviios. However, he differece bewee egive d posiive deviios is crucil. Negive deviio is regrded s udesirble for mos ivesors while, posiive deviio is desirble. Followig he work of Mrkowiz (959), i which he proposed h semivrice replce vrice s he mesure of risk, dowside risk hs bee he subec of umerous sudies [Grooveld d Hllebch (999), Michlowski d Ogryczk (2)]. he geerl coclusio cocerig dowside risk is h he lef-hd side of reur disribuio ivolves risk while he righ-hd side cois he beer ivesme opporuiies Grooveld d Hllebch (999). 7

8 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey Some of hese sudies proposed porfolio opimizio models which employ dowside risk s he mesure of risk. A porfolio opimizio model which icorpores dowside risk s he mesure of risk oly pelizes dowside deviios bu does o ke upside deviios io cosiderio. hese models re similr o he me vrice or MAD models s ll of hem shre he obecive of miimizig risk, he lef-hd side of reur disribuio. Perhps he mos impor deficiecy of hese models is h hey do o ke io cosiderio he beer ivesme opporuiies, he righ hd side of reur disribuio. I his pper ew model h kes io cosiderio boh risk d beer ivesme opporuiies re proposed. he differece bewee he proposed model d he oher porfolio opimizio models is heir obecives. he proposed model ssumes h ivesor ws o choose porfolio wih higher upside deviios d lower dowside deviios. I oher words he/she hs wo simuleous obecives. Firs he/she ws o mimize upside deviios d secod ws o miimize dowside deviios. hese wo obecives c be merged d resed s he sigle obecive of mimizig he differece bewee he upside d dowside deviios. For securiies (= ) durig periods (= ) he dowside risk (NMAD) d beer ivesme opporuiies (PMAD) c be show i his wy: NMAD = N = mi [, r -r ] (9) PMAD = P = m [, r -r ] () Accordig o hese defiiios, he obecive of he proposed model c be sed s follows: Mimize subec o = = = r = P. ρm - = = N. =... () 8

9 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey his model is similr o model (2) if we ssume h = P. / = Y d = Mimize = subec o Y = = = P = _ N r N. / = Z. = Z _ Y _ Z ρm = = =... =... =... (2) his ew model hs 2+ vribles d 2+2 cosris. As cosequece of developmes i compuer echology, he umber of vribles d cosris re o loger crucil problem. Bu his ew model differs from oher porfolio opimizio models sice i cosiders risk i erms of egive deviios from he me d lso kes he beer ivesme opporuiies io cosiderio. Accordig o oher opimizio models, he securiy wih he lowes deviio mog securiies wih he sme reur is regrded s he mos desirble wheres his model regrds securiy which hs higher posiive d lower egive deviios s more desirble. I oher words, his ew model ries o fid equilibrium poi bewee risk d beer ivesme opporuiies. III. COMPARISON OF HE OPIMIZAION MODELS I his secio, i order o evlue he performce of he porfolio opimizio models, porfolios developed ccordig o he proposed model, me vrice d me bsolue deviio models re compred for differe ivesme horizos. he dbse cosiss of socks icluded i he ISE-, he well-kow ide of he Isbul Sock Echge o he srig de of he lysis, Jury 2. 9 of he socks re ecluded becuse of missig d. 9

10 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey I he firs sge of he lysis, 48 differe d ses A,A2 A48- which iclude d for 9 socks over 2 mohs re prepred. A icludes d from Jury 2 o December 2 (- 2 mohs); A2 icludes d from Februry 2 o Jury 2 (2-3 mohs) d so o. For every d se, he me-vrice, MAD d proposed opiml porfolios re pu ogeher usig models, 7 d 2. I he secod sge, i order o ke io cosiderio he effec of he porfolio horizo, he ssumpio ivolvig modificio of he porfolios every moh is replced wih modificio of he porfolios i every 3 mohs, 6 mohs, yer d 2 yers. For ech ssumpio, he reurs of he 3 porfolios developed ccordig o he models, 7 d 2, re clculed, d he hese reurs re compred. Porfolios modified more h oce yer show h he proposed model s reurs defiiely eceed he reurs of he oher wo models (Figure ). Whe he ivesme horizo eceeds yer, he me bsolue deviio models produce higher reurs. Figure : Reurs of he models for differe ivesme horizos,35,3,25,2,5,,5 moh 3 mohs 6 mohs yer 2 yer MV MAD PRO Eve hough ordiry ivesor geerlly evlues his/her ivesme performce oly ccordig o he reurs of he porfolio, for vlid evluio, clculig he mou of he reur

11 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey for every ui of risk bore would be more pproprie. o clcule he rewrd per risk equio 3 is employed: Porfolio Performce = me reur/risk (3) I is obvious h porfolio performce clculed usig equio (3) will differ ccordig o he risk mesure chose. I his pper 3 differe risk mesures re used. hese re sdrd deviio, dowside risk d me bsolue deviio. For every porfolio d for every ivesme horizo, 3 performce mesures re clculed usig hese 3 differe risk mesures. Besides, well kow porfolio performce mesure, Shrpe rio is clculed. he performce of he porfolios which were modified every moh is show i Figure 2. Compred wih he me reurs i Figure, he differece bewee he performces of differe porfolios is smller. his idices h, idepede of he risk mesure seleced, porfolios composed ccordig o he proposed model re riskier h he oher porfolios. Figure 2: Differe Risk Mesures of he Porfolios MV MAD PRO,45,4,35,3,25,2,5,,5 Sdrd Deviio Dowside Risk MAD Usig he proposed model, he icresed risk for porfolios is see o oly i hose porfolios revised mohly bu lso i ll ivesme horizos. ble summrizes some sisics of he porfolios. Aoher coclusio of he ble is h porfolios usig he proposed model ouperform he oher porfolios for ivesme horizos shorer h yer.

12 Europe Jourl of Fice d Bkig Reserch Vol.. No ble : Summry Sisics of he Porfolios Ali Argu Krcbey Moh 3 Mohs 6 Mohs Opimizio Model Me Reur Sdrd Deviio Dowside Risk MAD MV,2,26,64,93 MAD,5,9,52,84 PRO,33,82,82,4 MV,6,32,67,99 MAD,2,32,73,92 PRO,8,78,82,32 MV,3,27,67,96 MAD,6,4,75,3 PRO,27,73,79,33 Whe compred o me vrice d me bsolue deviio models, he ew proposed model geeres more risky porfolios. Bu for idividul ivesors who geerlly mke ivesme decisios bsed o reur proposed model cosiues more rcive porfolios. ble 2 summrizes he performce mesures of he porfolios. For d 6 moh periods, proposed model ouperforms he oher porfolios. For 3 moh period, me vrice porfolio shows beer performce lhough proposed model s me reur is he highes. 2

13 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey ble 2: Performce Mesures of he Porfolios Me Reur / MAD Me Reur / Dowside Risk Shrpe Rio Opimizio Model Moh 3 Mohs 6 Mohs MV,96,88,29 MAD,26,288,79 PRO,82,42,234 MV,8,239,62 MAD,9,64,3 PRO,4,22,36 MV,,94,35 MAD,6,23,55 PRO,59,342,23 Whe compred o me vrice d me bsolue deviio models, he ew proposed model geeres more risky porfolios. Bu for idividul ivesors who geerlly mke ivesme decisios bsed o reur proposed model cosiues more rcive porfolios. IV. CONCLUDING REMARKS Mos of he reserch bou porfolio opimizio fer Mrkowiz focused o overcomig he compuiol burde of he me vrice model. Koo d Ymzki sugges ew model which cceps MAD s risk mesure ised of vrice. hey lso proved h miimizig MAD is similr o miimizig vrice if he reurs of he socks re mulivrie ormlly disribued. Mrkowiz s me vrice model d he me bsolue deviio models ccep risk s he deviio h c be posiive (upside) or egive (dowside) from he epeced reur. I oher words boh of he models puish o oly he egive (dowside) deviios bu lso he posiive (upside) deviios. However, he differece bewee egive d posiive deviios is crucil. 3

14 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey Negive deviio is cceped o be udesirble for mos of he ivesor while posiive deviio is desirble. I his pper ew model h kes io cosiderio boh risk dowside deviios d beer ivesme opporuiies upside deviios-, is proposed. he differece bewee he proposed model d he oher porfolio opimizio models is heir obecives. he proposed model ssumes h ivesor ws o choose porfolio wih higher upside deviios d lower dowside deviios. I his pper his heoreiclly righ issue is esed wih rel d of emergig mrke. he resuls of he lysis showed h he porfolios of he proposed model re riskier bu geere higher reurs. So his model c be useful for ivesors whose ivesme evluio mosly depeds o reur. REFERENCES Chg, C. (25). A Modified Gol Progrmmig Approch for he Me Absolue Deviio Porfolio Opimizio Model. Applied Mhemics d Compuio, (Aricle I Press). Elo, E. J., Gruber, M.J. d Pdberg, W. (976). Simple Crieri for Opiml Porfolio Selecio. he Jourl of Fice, Vol. 3, No. 5, pp Feisei, C.D. d hp, M.N. (993). Noes: A Reformulio of Me- Absolue Deviio Porfolio Opimizio Model. Mgeme Sciece, Vol. 39, No. 2, pp Grooveld, H. d Hllerbch, W. (999). Vrice vs Dowside Risk: Is here Relly h Much Differece? Europe Jourl of Operiol Reserch, Vol. 4, Issue 2, pp Koo, H. d Ymzki, H. (99). Me-Absolue Deviio Porfolio Opimizio Model d IS Applicios o okyo Sock Mrke. Mgeme Sciece, Vol. 37, No. 5, pp Mrkowiz, H. (952). Porfolio Selecio. he Jourl of Fice, Vol. 7, No., pp Michlowski, W. d Ogryczk, W. (2). Eedig he MAD Porfolio Opimizio Model o Icorpore Dowside Risk Aversio. Nvl Reserch Logisics, Vol. 48, İssue 3, pp Shrpe, W. (97). A Lier Progrmmig Approimio for he Geerl Porfolio Alysis Problem. Jourl of Ficil d Quiive Alysis, December, pp Sim, Y. (997). Esimio Risk i Porfolio Selecio: he Me Vrice Model Versus he Me Absolue Deviio Model. Mgeme Sciece, 4

15 Europe Jourl of Fice d Bkig Reserch Vol.. No Ali Argu Krcbey Vol. 43, No., pp Soe, B. (973). A lier Progrmmig Formulio of he Geerl Porfolio Selecio Problem. Jourl of Ficil d Quiive Alysis, Vol. 8, pp