Efficient Frontier - Comparing Different Volatility Estimators

Transcription

1 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 Efficie Froier - Comparig Differe Volailiy Esimaors Tea Poklepović, Zdravka Aljiović, Mario Maković Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ Absrac Moder Porfolio Theory (MPT) accordig o Markowiz saes ha ivesors form mea-variace efficie porfolios which maximizes heir uiliy. Markowiz proposed he sadard deviaio as a simple measure for porfolio risk ad he lower semi-variace as he oly risk measure of ieres o raioal ivesors. This paper uses a hird volailiy esimaor based o iraday daa ad compares hree efficie froiers o he Croaia Sock Marke. The resuls show ha rage-based volailiy esimaor ouperforms boh mea-variace ad lower semi-variace model. Keywords Variace, lower semi-variace, rage-based volailiy. I. INTRODUCTION ODERN Porfolio Theory was iroduced by Nobel MLaureae Harry Markowiz [15] i his semial paper which chaged he way porfolios were maaged uil he. This heory focuses o porfolio diversificaio ad risk corol. Ivesors form porfolios accordig o he mea variace efficiecy crieria. This meas ha ivesors maximize heir reur across all possible porfolios ad accep he risk accordig o heir risk aversio. Markowiz describes a risk averse ivesor as a subjec who prefers a higher reur versus a lower reur ad who a he same ime is prepared o accep more risk if such ivesme icreases he expeced reur. Such a ivesor opimizes he expeced porfolio reur give he porfolio risk. Moder porfolio heory is based o he efficie froier (EF) of ivesmes, i.e. he spie of porfolios wih maximum expeced reur across all possible porfolios give a cerai amou of porfolio risk. The porfolio risk is of crucial iformaio o he ivesor ad herefore eeds o be quaified. The volailiy of he porfolio reur is ofe cosidered as he risk of cocer. Sice volailiy is o observable i eeds o be esimaed. Markowiz proposed o quaify porfolio risk by meas of he volailiy of fiacial asses. He used he sadard deviaio of fiacial asses as a simple measure of risk ad he lower semi-variace as he more complex esimaor. The lower semi-variace is accordig o Markowiz he oly volailiy esimaor i which a raioal ivesor migh be ieresed i. O he oher side he sadard deviaio has become oe of he mos popular risk esimaors i pracice due o he simpliciy i usig ad T. Poklepović is wih he Faculy of Ecoomics, Uiversiy of Spli, Spli, Croaia (phoe: ; fax: ; poklepo@efs.hr). Z. Aljiović is wih he Faculy of Ecoomics, Uiversiy of Spli, Spli, Croaia ( zdravka.aljiovic@efs.hr). M. Maković is wih he Bick Bak, Barbara Srozzilaa 310 Amserdam, 1083 HN Neherlads ( mmakic@homail.com). udersadig his measure i porfolio maageme. However, boh esimaors use oly oe sigle daily observable price chage o deermie he volailiy of fiacial reurs. A aleraive volailiy esimaor used i his paper is based o high frequecy daa. Volailiy esimaed by meas of high frequecy daa is also called realized volailiy ad ca be cosidered ubiased. I pracice, however, he implemeaio of high frequecy daa is limied by several reasos. Firs, high frequecy daa is o available for all securiies. This is especially rue for securiies raded i emergig markes where he radig volume is ofe isufficie as he daa frequecy becomes smaller. Secodly, as he frequecy becomes smaller microsrucure effecs emerge which iduce a upward bias i he esimaed volailiy. Thirdly, here is a serious calculaio complexiy due o he exesive amou of daa ha is required for esimaig he daily volailiy or he variacecovariace marix. For example o calculae he volailiy of 250 radig days based o 5-miue ierval observaios aroud iraday price observaios are required. Moreover, for esimaig he EF of a porfolio cosisig of 20 asses more ha a millio observaios will be required. A more pracical mehodology o esimae he iraday volailiy is by meas of ope, high, low ad closig prices (OHLC). This paper uses he Parkiso [17] rage-based volailiy esimaor for exreme price jumps, which are characerisic for emergig markes like Croaia. Accordig o empirical research performed by [4] he rage-based volailiy esimaor is he leas biased volailiy esimaor usig OHLC daa whe measurig he volailiy of he Croaia sock marke. A sigifica shorcomig however, of he rage-based volailiy esimaor is ha o mulivariae aalogue of he iraday rage exiss, which meas ha he esimaio of he variacecovariace marix is o sraighforward. A simple esimaor of he codiioal variace-covariace marix of reurs was proposed by [12]. This mehodology is used o cosruc he EF based o he rage-based volailiy esimaor. This paper compares EF based o 3 differe volailiy esimaors usig a porfolio of socks from he Croaia Sock Marke. The oulie of he remaider of he paper is as follows. Secio II reviews he lieraure o moder porfolio heory wih focus o he lieraure o differe approaches o esimaig he volailiy. Secio III describes he moder porfolio heory, which is he basis of his research. Secio IV preses he lower semi-variace approach i esimaig he efficie froier ad Secio V he iraday volailiy approach. The sock price daa is described i Secio VI. The resuls of he empirical research are preseed i Secio VII. Secio VIII cocludes. Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

2 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ II. REVIEW OF PREVIOUS RESEARCH I his semial paper, [15] describes Moder Porfolio Theory i a quaiaive model ha solves he complex problem of capial allocaio across asses i such a way ha i miimizes he variace of he porfolio give a expeced reur. Markowiz proposed a simple square roo opimisaio ha resuls i he mea-variace efficie porfolio ad suggesed o use he sadard deviaio for esimaig he porfolio volailiy. The sadard deviaio is a saisically correc esimaor of he volailiy of reurs if he observed ime series are derived from a ormal disribuio. This, however, is o always he case. Oe of he sylized facs of fiacial reurs as described i [7] is he o-ormaliy of fiacial reurs. The disribuio ofe suffers from posiive skewess ad lepokurosis. Oher sylized facs iclude amogs ohers heeroscedasiciy ad ime varyig correlaios of fiacial reurs. Therefore he sadard deviaio, which assumes ormaliy by defaul, is expeced o uderesimae he rue volailiy of he disribuio. Moivaed by he defiiio of risk, as a fiacial loss or dowside risk, [16] proposes a ew defiiio of risk cosiderig oly he egaive resuls. The lower semi-variace measures he dispersio of he reurs below a give arge reur. Markowiz explais ha he usage of he lower semi-variace is jusified by wo reasos. Firsly, raioal ivesors are oly ieresed i limiig he volailiy ha ca cause a egaive resul. Secodly, if he fiacial ime series are o ormally disribued he he sadard deviaio will uderesimae he rue risk of he porfolio. I hese cases he lower semivariace, as a measure of dowside risk, should be used isead. I is show i [14] ha here is a grea suppor i he marke for usig he lower semi-variace as a risk measure. Ivesors are more sesiive o losses below a cerai hreshold he o gais beyod a cerai hreshold. I [5] he formula for lower semi-variace is geeralized ad defied as he lower parial mome (LPM). Four differe LPM volailiy esimaors are compared i [13] ad i is show ha he LPM proposed by Markowiz is suiable for corollig risk whe he disribuio of he asses is o ormal. Boh esimaors ha were proposed by Markowiz use oly oe sigle daily price observaio o deermie he variacecovariace marix. This meas ha all oher price observaios ha are available whe high frequecy daa a used are igored. Oe of he rece heories focusig o volailiy esimaors are described i he lieraure of high frequecy daa. The realized volailiy esimaor is proposed i [8], which is he squared sum of iraday reurs. Accordig o [3] his volailiy esimaor is heoreically ubiased whe he frequecy sample goes o zero, bu will i ur iduce microsrucure effecs. The realized volailiy is esimaed by usig all marke available iraday iformaio. Aoher pracical disadvaage of his mehod is ha i requires a exesive amou of iraday price observaios for esimaig a EF. A reasoable aleraive o usig high frequecy daa is o use volailiy esimaors ha require oly 4 sadard available iraday price observaios, i.e. he OHLC esimaors. OHLC esimaors, geerally, assume ha asse prices follow a Geomeric Browia Moio (GBM) i.e. he price of he asse o day is idepede of he price of he same asse o day -1 ad ha he price of he asses are sochasic hrough ime. GBM wihou drif is assumed i [17] ad i proposes a rage based volailiy esimaor. This esimaor uses he maximum differece bewee he maximum ad he miimum iraday price for esimaig he volailiy. The ope ad closig prices are icluded i [11] ad hey propose a esimaor, which uses all four OHLC iraday price observaios. A esimaor ha follows a GBM wih drif is proposed i [18]. This esimaor is useful whe he drif is o-zero. Sigifica differeces bewee OHLC esimaors ha are popular i he lieraure are foud i [9] ad hey coclude ha he choice of he OHLC volailiy esimaor is impora. OHLC volailiy esimaors ha are popular i he lieraure are compared i [4], agais he ubiased high frequecy based volailiy esimaor. They show ha he Parkiso rage-based volailiy esimaor is he leas biased esimaor for esimaig he volailiy of he Croaia Sock marke compared o oher OHLC volailiy esimaors. The compariso is performed agais he high frequecy based realized volailiy which is he heoreically ubiased volailiy esimaor. The daa used i heir research spas a period of 5 years ad icludes he rece credi ad bak crisis of 2007 ad They cofirm he fidigs of [9] by meas of loss fucios ad ime varyig codiioal correlaios ad coclude ha he OHLC volailiy esimaors are sigificaly differe from each oher. This paper follows he resuls of [4] ad uses he Parkiso rage-based volailiy esimaor i esimaig he iraday volailiy. The codiioal variacecovariace marix proposed i [12] is used o cosruc he EF by meas of mea-variace. EF are compared i [10], [19] ad [20] based o he sadard deviaio ad he lower semi-variace ad coclude ha i is possible o cosruc a EF based o he lower semivariace ha lies o he lef side of he EF based o he simple sadard deviaio. They coclude ha his EF is sochasically domia compared o he sadard deviaio proposed by Markowiz. I is possible o reduce he risk of a porfolio by usig he lower semi-variace as a measure of he porfolio volailiy [10]. Accordig o [6], i is o possible o compare EF based o differe volailiy esimaors sice he risk esimaors are o ideical, i.e. he x-axis o he meavariace coordiae sysem is differe. They coclude ha he oly meaigful way of comparig EF is by ex-pos aalysis ad ha he locaio of he EF o he mea-variace coordiae sysem does o add valuable iformaio. This research compares EF based o differe volailiy esimaors: sadard deviaio, lower semi-variace ad he iraday volailiy esimaor. The quesios of ieres are wheher he volailiy esimaor iflueces he locaio of he efficie froier o he mea-variace coordiae sysem ad wheher he locaio of he efficie froier o he meavariace coordiae sysem deermies he performace of he efficie porfolios by he ex-pos aalysis. Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

3 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ III. MARKOWITZ MODERN PORTFOLIO THEORY Accordig o Moder Porfolio Theory (MPT) ivesors use mea-variace opimizaio o cosruc a efficie porfolio. MPT relies o he followig assumpios: he ivesme horizo is oe period (oe moh, oe year, ec.); ivesors opimize heir expeced reur across all possible porfolios; he expeced porfolio reur depeds o he expeced reur ad he risk of he ivesme; ivesors are raioal ad prefer a higher reur compared o a lower reur, ad also have aversio o risk; here is o ax, o iflaio ad here is o rasacio or oher coss ivolved; all ivesors have free ad ulimied access o releva iformaio a he same ime; all socks are ifiiely divisible. This se of assumpios creaes a heoreical world i which ivesors operae accordig o MPT. This world is differe from he real world, bu icorporaes almos all elemes average ivesors ake io accou whe makig ivesme decisios. Accordig o MPT ivesors will spread heir porfolio o divers or corol he risk ad a he same ime hey wa o maximize heir expeced reur. Opimizaio is based o mea-variace efficiecy, which meas ha, give a predeermied porfolio risk, ivesors will choose he porfolio ha maximizes heir reur. The sadard deviaio is a popular volailiy esimaor ha requires oly oe price observaio per day. This esimaor is symmerical ad assumes ha he reurs follow a ormal or mulivariae ormal disribuio. Cosiderig ha every efficie porfolio has he highes reveue alog wih defied rae of risk c, mahemaically we may defie efficie porfolio as follows [1]: Subjec o: max ER ( ) (1) c (2) i 1, i 0, i1, 2,..., (3) i1 The expeced porfolio reur is defied as ad he porfolio risk as ER ( ) ier ( i) ' ER ( ) ER ( )' (4) i1 ' S. (5) i1 j1 i j ij ERdeoes ( ) he vecor colum of expeced reurs, E( R i ) he expeced reur of he sock i, he vecor colum of weighs of a sock i a porfolio, i he weigh of a sock i i porfolio, S he variace-covariace marix, ij he covariace of reurs of socks i ad j, ad he umber of socks. IV. LOWER SEMI-VARIANCE APPROACH Whe he reurs do o follow a ormal disribuio, he sadard deviaio ofe uderesimaes he rue volailiy [7]. The lower semi-variace measures he variace below a cerai hreshold. Oly reurs ha are uderperformig his hreshold are ake io accou i deermiig he porfolio risk. Porfolios opimized wih his volailiy esimaor are expeced o assig less weigh o socks ha are uderperformig ad more weigh o socks whose reurs exceed he hreshold. Whe he risk is measured by lower semi-variace, he porfolio opimizaio problem becomes he problem of quadraic programmig of he followig form give i [2]: wih cosrais: mi T 2 pz (6) 1,, 1,2,..., z r E R T j j j j 1 j1 j1 z 0, 1, 2,..., T 1 ER ( ) j 0, j 1, 2,...,. j r is he reur of he share j i period, T is he umber of j, p, 1,2,..., T, is probabiliy ha he vecor of radig days. reurs of porfolio π, R ( R1, R2,..., R ) akes value r ( r1,, r2,,..., r, ). Usually, 1 p. Model s variables are T j, j 1, 2,..., ad z, 1, 2,..., T. V. INTRADAY VOLATILITY APPROACH The sadard deviaio ad he lower semi-variace use oe sigle daily price observaio i deermiig he volailiy of he porfolio. All oher releva iformaio available o he ivesor is igored. Iraday volailiy esimaors use more daily price observaios i compuig he volailiy. These esimaors do o rely o he ormal disribuio. I is show i [8] ha he ubiased volailiy esimaor could be cosruced by meas of high frequecy daa. The radig volume i emergig markes is ofe isufficie o esure high frequecy daa for all required socks. Due o limiaios, we follow [4] which showed ha he Parkiso rage-based volailiy esimaor is he leas biased OHLC esimaor whe esimaig he volailiy of he Croaia Sock Marke. The Parkiso rage based volailiy esimaor uses wo iraday price observaios o deermie he spread: he highes ad he lowes iraday observaios. The rage-based volailiy esimaor is give by: Rage ii, j j 1 H i l 4l2 Li 2 (7) (8) Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

4 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ I (8) H deoes he highes ad L deoes he lowes observed iraday price. Usig he highes ad he lowes price observaios Parkiso proposes a volailiy esimaor for high volaile markes. This esimaor follows a GBM wihou drif ad uses oly exreme price movemes o calculae he volailiy. The porfolio risk as defied i (5) requires he variacecovariace marix for ipu. The off-diagoal elemes of he variace-covariace marix of he rage-based volailiy esimaor are o direcly observable. A simple model, ha is based o he expoeially weighed movig average (EWMA), developed o esimae he off-diagoal elemes of he variace-covariace marix whe usig he rage-based volailiy esimaor, is proposed i [12]. This model combies he rage-based ad he reur-based approaches. The reurbased volailiy esimaor is give by: l p R 2 i, ii, p i, 1 where p is he price of sock i o day. i, The esimaor is based o he mulivariae EWMA model of he codiioal variace-covariace marix give by: (9). (10) ˆ ˆ ij ij 1 ij ; i, j 1,..., R R R,, 1, 1 where λ is he sigle decay facor, which is ypically se o 0.94, esimaed by JP Morga as he average value of decay facor ha miimizes he mea square error of daily ou-ofsample codiioal volailiy forecass for a wide rage of asses. Covariace is give by: r l r l. (11) R ij, i, j, The diagoal ad off-diagoal elemes of he rage-based esimaor of he codiioal variace-covariace marix are give by where ˆ Rage Rage Rage ii, 1 ii, 1 ij, R Rage Rage ij ii, jj, ˆ 1 ; i j; i, j 1,..., ˆ ˆ ; i j; i, j 1,..., R R ii, jj, (12) R R ij, ij ; i, j 1,..., (13) ˆ ˆ Fially, he elemes of he variace-covariace marix of he rage-based volailiy model are calculaed by ˆ ˆ. (14) Rage Rage R ij ii, jj, ij Now, whe he rage-based variace-covariace marix is kow, we proceed wih seps (1) o (5) o calculae he mea variace porfolio. VI. DATA AND DESCRIPTIVE STATISTICS The porfolios cosruced i his research cosis of a ivesme i 10 socks from he CROBEX idex. The daa spas from 12 h March 2013 o 13 h December 2013 ad cous 191 price observaios. The followig socks are icluded: AD Plasik d.d. (ADPL), Alaska Plovidba d.d. (ATPL), Belje d.d. (BLJE), Djuro Djakovic Holdig d.d. (DDJH), Dalekovod d.d. (DLKV), Valamar Adria Holdig d.d. (DOMF), Ericsso Nikola Tesla d.d. (ERNT), Hrvaski Telekom d.d. (HT), Igra d.d. (INGR) ad Vupik d.d. (VPIK). Table I shows he descripive saisics icludig he sample size, miimum, maximum, expeced reurs, he volailiy a he ed of he period for each volailiy esimaor, skeweess, kurosis ad he Jarque-Berra es for ormaliy of reurs. The descripive saisics shows ha all asses show asymmeric behaviour ad lepokurosis, i.e. deviaio from he ormal disribuio. The Jarque-Berra es shows ha oe of he socks follows a ormal disribuio. Fig. 1 shows values of he hree observed volailiy esimaors for each sock. I ca be cocluded ha, o average, he highes volailiy is esimaed by sadard deviaio ad ha he lowes volailiy is esimaed by he lower semisadard deviaio. However, he rage-based volailiy esimaor yields mixed resuls, i.e. for cerai socks i esimaes volailiy higher ha he sadard deviaio does, ad someimes i provides volailiy of a sock lower ha he lower semi-sadard deviaio does. Fig. 1 Volailiy esimaors VII. EMPIRICAL RESULTS I his paper he locaio of he EF o he mea-variace coordiae sysem ad he performace of hree differe volailiy esimaors i he ex-pos or ou-of sample aalysis are compared. I he firs par of he aalysis, he EF is compued a 20 differe risk levels for he porfolios usig mea-variace, lower semi-variace ad iraday rage-based volailiy approach. The appropriae weighs, reurs ad sadard deviaios are preseed i Tables II-IV. Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

5 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 TABLE I DESCRIPTIVE STATISTICS ADPL ATPL BLJE DDJH DLKV DOMF ERNT HT INGR VPIK N Mi -0,0284-0,0648-0,0520-0,0573-0,3075-0,0341-0,1278-0,1133-0,0750-0,0735 Max 0,0412 0,0731 0,1010 0,0825 0,2597 0,0289 0,0422 0,0329 0,1443 0,0724 Expeced reur 0, , , , , , , , , ,00152 Variace 0, , , , , , , , , ,00040 Sadard deviaio 0, , , , , , , , , ,02002 Lower semi-variace 0, , , , , , , , , ,00019 Lower semi-sadard deviaio 0, , , , , , , , , ,01387 Rage-based volailiy 0, , , , , , , , , ,01737 Skeweess 0,24 0,49 1,04 0,69-0,18-0,19-3,44-4,51 1,21 0,10 Kurosis 3,66 0,16 5,04 1,37 7,20 0,39 31,17 43,18 4,37 1,65 Jarque-Berra 5,18 71,51 66,93 36,02 140,47 55, , ,71 61,05 14,83 Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ TABLE II EFFICIENT PORTFOLIOS USING MEAN-VARIANCE MODEL ADPL ATPL BLJE DDJH DLKV DOMF ERNT HT INGR VPIK S. dev. (%) Reur (%) 0,3453 0,0590 0,0347 0,0402 0,0020 0,1867 0,1208 0,1765 0,0000 0,0348 0,5680-0,0392 0,5818 0,1680 0,0000 0,0000 0,0000 0,1363 0,1139 0,0000 0,0000 0,0000 0,7000 0,0153 0,6344 0,2279 0,0000 0,0000 0,0000 0,0516 0,0862 0,0000 0,0000 0,0000 0,8000 0,0266 0,6615 0,2772 0,0000 0,0000 0,0000 0,0000 0,0613 0,0000 0,0000 0,0000 0,9000 0,0354 0,6547 0,3257 0,0000 0,0000 0,0000 0,0000 0,0196 0,0000 0,0000 0,0000 1,0000 0,0429 0,6277 0,3723 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,1000 0,0496 0,5819 0,4181 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,2000 0,0557 0,5389 0,4611 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,3000 0,0614 0,4979 0,5021 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,4000 0,0669 0,4583 0,5417 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,5000 0,0722 0,4196 0,5804 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,6000 0,0774 0,3817 0,6183 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,7000 0,0824 0,3445 0,6555 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,8000 0,0874 0,3077 0,6923 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,9000 0,0923 0,2713 0,7287 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 2,0000 0,0971 0,2353 0,7647 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 2,1000 0,1019 0,1995 0,8005 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 2,2000 0,1067 0,1640 0,8360 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 2,3000 0,1114 0,0935 0,9065 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 2,5000 0,1208 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 2,7682 0,1333 TABLE III EFFICIENT PORTFOLIOS USING LOWER SEMI-VARIANCE MODEL ADPL ATPL BLJE DDJH DLKV DOMF ERNT HT INGR VPIK S. dev. (%) Reur (%) 0,5268 0,1216 0,0000 0,0030 0,0000 0,2010 0,0802 0,0675 0,0000 0,0000 0,4500 0,0000 0,5271 0,1218 0,0000 0,0028 0,0000 0,2010 0,0802 0,0672 0,0000 0,0000 0,4502 0,0001 0,5307 0,1238 0,0000 0,0007 0,0000 0,2007 0,0797 0,0643 0,0000 0,0000 0,4518 0,0010 0,5711 0,1484 0,0000 0,0000 0,0000 0,1900 0,0724 0,0181 0,0000 0,0000 0,4724 0,0100 0,6018 0,1950 0,0000 0,0000 0,0000 0,1423 0,0610 0,0000 0,0000 0,0000 0,5074 0,0200 0,6273 0,2513 0,0000 0,0000 0,0000 0,0716 0,0499 0,0000 0,0000 0,0000 0,5636 0,0300 0,6537 0,3074 0,0000 0,0000 0,0000 0,0000 0,0389 0,0000 0,0000 0,0000 0,6374 0,0400 0,6362 0,3417 0,0000 0,0000 0,0000 0,0000 0,0221 0,0000 0,0000 0,0000 0,6805 0,0450 0,6181 0,3762 0,0000 0,0000 0,0000 0,0000 0,0057 0,0000 0,0000 0,0000 0,7282 0,0500 0,5498 0,4502 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,8346 0,0600 0,4748 0,5252 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,9530 0,0700 0,3998 0,6002 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,0788 0,0800 0,3247 0,6753 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,2096 0,0900 0,2872 0,7128 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,2764 0,0950 0,2497 0,7503 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,3441 0,1000 0,1747 0,8253 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,4816 0,1100 0,0996 0,9004 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,6214 0,1200 0,0246 0,9754 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,7629 0,1300 0,0021 0,9979 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,8056 0,1330 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,8085 0,1333 Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

6 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ TABLE IV EFFICIENT PORTFOLIOS USING RANGE-BASED VOLATILITY MODEL ADPL ATPL BLJE DDJH DLKV DOMF ERNT HT INGR VPIK S. Dev. (%) Reur (%) 0,1772 0,0618 0,0443 0,0376 0,0242 0,1955 0,1552 0,2002 0,0410 0,0631 0,3085-0,0599 0,1990 0,0727 0,0302 0,0218 0,0169 0,1979 0,1724 0,2013 0,0357 0,0522 0,3102-0,0497 0,2208 0,0836 0,0162 0,0060 0,0095 0,2002 0,1896 0,2025 0,0304 0,0412 0,3152-0,0396 0,2448 0,0936 0,0023 0,0000 0,0028 0,2023 0,2078 0,1992 0,0217 0,0255 0,3238-0,0294 0,2742 0,1079 0,0000 0,0000 0,0000 0,2028 0,2233 0,1807 0,0099 0,0012 0,3380-0,0192 0,3002 0,1341 0,0000 0,0000 0,0000 0,1957 0,2349 0,1344 0,0007 0,0000 0,3611-0,0091 0,3237 0,1639 0,0000 0,0000 0,0000 0,1882 0,2451 0,0791 0,0000 0,0000 0,3942 0,0011 0,3470 0,1939 0,0000 0,0000 0,0000 0,1807 0,2553 0,0232 0,0000 0,0000 0,4355 0,0113 0,3426 0,2457 0,0000 0,0000 0,0000 0,1647 0,2471 0,0000 0,0000 0,0000 0,4849 0,0214 0,3185 0,3129 0,0000 0,0000 0,0000 0,1428 0,2259 0,0000 0,0000 0,0000 0,5482 0,0316 0,2944 0,3800 0,0000 0,0000 0,0000 0,1209 0,2047 0,0000 0,0000 0,0000 0,6219 0,0418 0,2703 0,4472 0,0000 0,0000 0,0000 0,0989 0,1836 0,0000 0,0000 0,0000 0,7027 0,0519 0,2462 0,5144 0,0000 0,0000 0,0000 0,0770 0,1624 0,0000 0,0000 0,0000 0,7885 0,0621 0,2221 0,5816 0,0000 0,0000 0,0000 0,0551 0,1413 0,0000 0,0000 0,0000 0,8779 0,0723 0,1980 0,6487 0,0000 0,0000 0,0000 0,0332 0,1201 0,0000 0,0000 0,0000 0,9697 0,0824 0,1739 0,7159 0,0000 0,0000 0,0000 0,0113 0,0989 0,0000 0,0000 0,0000 1,0634 0,0926 0,1438 0,7845 0,0000 0,0000 0,0000 0,0000 0,0717 0,0000 0,0000 0,0000 1,1587 0,1028 0,1074 0,8545 0,0000 0,0000 0,0000 0,0000 0,0381 0,0000 0,0000 0,0000 1,2554 0,1129 0,0710 0,9245 0,0000 0,0000 0,0000 0,0000 0,0044 0,0000 0,0000 0,0000 1,3535 0,1231 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,4530 0,1333 The resuls of he hree differe volailiy esimaors show ha usig differe volailiy esimaor yields wih differe weighs of socks i a porfolio, which resuls i differe rages of reur ad risk. However, he highes risk is foud wih mea-variace model. I rages from 0,57%, whe diversifyig porfolio ad ivesig i all bu oe sock (INGR) yieldig a reur of -0,04%, o 2,77% whe ivesig i oly oe share (ATPL), yieldig a reur of 0,13%. Whe cosiderig lower semi-variace, he risk rages from 0,45%, whe diversifyig porfolio ad ivesig i ADPL, ATPL, DDJH, DOMF, ERNT ad HT yieldig 0,00% reur, o 1,81% whe ivesig i oly oe sock (ATPL) yieldig reur of 0,13%. The lowes risk is measured wih rage-based esimaor ragig from 0,31%, whe diversifyig risk ad ivesig i all he socks wih differe weighs yieldig he egaive reur of -0,06%, o 1,45% whe ivesig i ATPL yieldig he reur of 0,13%. The highes reur for all esimaors is 0,13% sice hey all have he same sock i he las porfolio. I ca be cocluded ha perhaps iraday rage-based volailiy esimaor uderesimaes he risk compared o he wo oher volailiy esimaors. However, he resuls of he expos aalysis es he performaces of he models. The compued efficie froiers are ploed o he meavariace coordiae sysem for he mea-variace model, lower semi-variace ad iraday rage-based volailiy model ad are preseed i Figs I he secod sep, a ex-pos aalysis is performed by ivesig i a porfolio of socks usig he calculaed porfolio weighs. The sock reurs o he ex radig day ad he calculaed weighs are used o calculae for each model he porfolio reurs. The resuls are preseed i Table V. Fig. 2 The efficie froier based o he mea-variace model Fig. 3 The efficie froier based o he lower semi-variace model Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

7 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 performace of he 1-day ex-pos aalysis, we coclude ha he rage-based volailiy model ouperforms boh he meavariace ad he lower semi-variace models whe cosrucig EF. Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ Fig. 4 The efficie froier based o he rage-based esimaor model TABLE V THE RETURNS (IN %) FOR THREE OBSERVED ESTIMATORS ON 16 TH DECEMBER 2013 mea-variace lower semi-variace rage-based volailiy model 1 0,5039 0,3518 0, ,0791 0,3513 0, ,2381 0,3466 0, ,4154 0,3030 0, ,3587 0,1332 0, ,3007-0,1211 0, ,2405-0,3796 0, ,1840-0,3372 0, ,1301-0,2947 0, ,0779-0,1983 0, ,0270-0,0996 0, ,0228-0,0009 0, ,0718 0,0978 0, ,1202 0,1471 0, ,1681 0,1965 0, ,2155 0,2952 0, ,2625 0,3939 0, ,3092 0,4926 0, ,4019 0,5222 0, ,5249 0,5242 0,5249 Assumig ha 20 ivesors wih differe risk aversios ives a equal amou of moey, he highes reur would be obaied if he iraday rage-based volailiy esimaor approach were used. This ivesme sraegy would yield i a posiive reur for all risk aversios, while for porfolios from 1 o 15 i would yield i he highes porfolio reur amogs all risk measures. Porfolio reur for rage-based volailiy esimaor rages from 0,86% whe diversifyig risk o 0,52% whe ivesig i oly oe sock, i.e. ADPL. Whe he ivesme is based o he mea-variace or lower semivariace approach, i yields boh posiive ad egaive reurs, depedig o he risk aversio. Moreover, i yields lower posiive reurs for more diversified porfolios ha he iraday rage-based volailiy approach. Noice ha he las porfolio cosiders a 100% ivesme i a sigle sock, i.e. ATPL, ad hereby deoes he porfolio wih he highes risk. This porfolio will ear he same amou regardless of he chose volailiy esimaors. Give he mea-variace coordiaes of he EF ad he VIII. CONCLUSION Accordig o Markowiz, raioal ivesors are oly ieresed i he lower semi-variace because his esimaor measures he risk of losses below a cerai hreshold, i.e. losses of ieres o he ivesor. Raioal ivesors are cocered abou losses, because hey wa o corol heir porfolio risk a every poi i ime. Whe fiacial reurs do o follow a ormal disribuio he sadard deviaio ca be replaced by he lower semi-variace. Sice boh esimaors use a sigle daily price observaio i esimaig he volailiy, iraday volailiy esimaors ca be cosidered as a aleraive. Accordig o [8] iraday volailiy esimaors are assumed o be ubiased. However, high frequecy daa iduce microsrucure effecs ad some pracical limiaio sice hey do o exis for all asses. The rage based iraday volailiy esimaor has gaied ieres i rece lieraure. I is a more efficie esimaor ha he daily squared close-o-close reur ad i is relaively robus o microsrucure effecs. However, sice here is o mulivariae aalogue of he rage-based volailiy esimaor, he codiioal variace-covariace marix is esimaed by a EWMA-based model, which forms he basis for he mea-variace porfolio esimaio. The EFs are cosruced based o all hree volailiy esimaors. Their performaces are compared i he ex ou-of-sample radig day. The EF based o hese hree ypes of volailiy measures show differe levels of expeced reurs, porfolio risk ad porfolio diversificaio. Thus, he EF differs i locaio o he mea-variace coordiaes. The resuls of he hree differe volailiy esimaors show ha he highes risk is foud wih mea-variace model ad he lowes risk is measured wih rage-based esimaor. The efficie porfolios based o he iraday rage-based volailiy esimaor ouperforms he aleraive volailiy esimaors for mos risk levels whe cosiderig he ivesme i hese porfolios ad he reurs o he ex radig day. This research shows ha he choice of he risk esimaor is impora i cosrucig he EF sice he porfolio weighs differ ad hus he choice of he ivesme. For furher research, we sugges o exed his heoreical research by icludig a loger period ad o iclude more volaile periods like he rece credi crisis of 2007 ad Iraday volailiy has he ieresig propery of usig muliple iraday observaios o deermie he daily volailiy. Accordig o Markowiz, raioal ivesors are oly ieresed i he risk of a egaive reur. Therefore i would be ieresig o ivesigae he performace of a semivariace versio of he rage-based volailiy model o a se of fiacial asses. Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/

8 World Academy of Sciece, Egieerig ad Techology Ieraioal Joural of Mahemaical, Compuaioal, Physical, Elecrical ad Compuer Egieerig Vol:9, No:4, 2015 ACKNOWLEDGMENT This work has bee fully suppored by Croaia Sciece Foudaio uder he projec Volailiy measureme, modellig ad forecasig (5199). Ieraioal Sciece Idex, Mahemaical ad Compuaioal Scieces Vol:9, No:4, 2015 wase.org/publicaio/ REFERENCES [1] Aljiović, Z., Marasović, B., Šego, B., Fiacijsko modeliraje, II izmijejeo i dopujeo izdaje, Ekoomski fakule Spli, 2011, pp [2] Aljiović, Z., Marasović, B., Vidović, J., The aleraive risk measures i Excel, MIPRO 2010, Proceedigs of he 33 rd Ieraioal Coveio: Compuers i Educaio, Opaija, Croaia, 2010, pp [3] Aderse, T.G., Bollerslev, T., Diebold, F.X., ad Labys, P., The disribuio of realized exchage rae volailiy, Joural of he America Saisical Associaio 96, 2001, pp [4] Arerić, J. ad Maković, M., Rakig Ope, High, Low, Closig Volailiy Esimaors, Spli Faculy of Ecoomics - Workig paper, [5] Bawa, S. ad Vijay S., "Opimal Rules For Orderig Ucerai Prospecs", Joural of Fiacial Ecoomics, v2(1), 1975, pp [6] Cheg, P. ad Wovero, M., "MPT ad he Dowside Risk Framework: A Comme o Two Rece Sudies", Joural of Real Esae Porfolio Maageme, vol 07, o.2, [7] Co, R., " Empirical properies of asse reurs: sylized facs ad saisical issues", Joural of Quaiaive Fiace, vol. 01, 2001, pp [8] Dacaroga, M.M., Muller, U.A., Olse, R.B, ad Pice, O.V., Modellig shor erm volailiy wih GARCH ad HARCH models, published i Noliear Modellig of High frequecy Fiacial Time Series., edied by Chrisia Duis ad Bi Zhou, Joh Wiley, Chicheser, 1998, pp [9] Duque, J., ad Paxso, D., Empirical Evidece o Volailiy Esimaors, Workig Paper, Uiversidade Tecica de Lisboa ad Uiversiy of Macheser, [10] Foo, T., Eg, S., Asse Allocaio i a Dowside Risk Framework, Joural or Real Esae Porfolio Maageme, Vol.06, No.3, [11] Garma, M.B., ad Klass, M.J., O he Esimaio of Securiy Price Volailiies from Hisorical Daa, Joural of Busiess, 53, 1980, pp [12] Harris, R.D.F., Yilmaz, F. Esimaio of he Codiioal Variace- Covariace Marix of Reurs usig he Iraday Rage, Uiversiy of Exerer, XFi Cere for Fiace ad Ivesme, Workig paper: 07/11, Ocober, 2007 [13] Koo, H., Waki, H., Yuuki, A., Porfolio Opimizaio uder Lower Parial Risk Measures, Asia-Pacific Fiacial Markes, vol.9,, 2002, pp , Kluwer. [14] Mao, James C. T., "Models Of Capial Budgeig, E-V Vs. E-S" Joural of Fiacial ad Quaiaive Aalysis, v5(5), 1970, pp [15] Markowiz, H., Porfolio Selecio: Efficie Diversificaio of Ivesmes, Joural of Fiace, Vol.7, No.1, 1952, pp [16] Markowiz, H., Porfolio Selecio: Efficie Diversificaio of Ivesmes, Joh Wiley ad Sos, [17] Parkiso, M., The Exreme Value Mehod for Esimaig he Variace of he Rae of Reur, Joural of Busiess, 53, 1980, pp [18] Rogers, L.C.G., ad Sachell, S.E., Esimaig Variace from High, Low, ad Closig Prices, Aals of Applied Probabiliy 1, 1991, pp [19] Sig, T. F., Og, S. E., Asse Allocaio i a Dowside Risk Framework, Joural of Real Esae Porfolio Maageme, Vol.6, o.3, 2000, pp [20] Siviaides, P. S., A Dowside-risk Approach o Real Esae Porfolio Srucurig, Joural of Real Esae Porfolio Maageme, Vol.4, o.2, 1998, pp Ieraioal Scholarly ad Scieific Research & Iovaio 9(4) scholar.wase.org/1999.7/