The APT with Lagged, Value-at-Risk and Asset Allocations by Using Econometric Approach

Transcription

1 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 he AP wih Lagged, Value-a-Risk and Asse Allocaions by Using Economeric Approach Sukono Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences Padadaran Universiy Bandung, Indonesia Abdul alib Bin Bon Deparmen of Producion and Operaions Universii un Hussein Onn Malaysia Malaysia Musafa Mama Faculy of Informaics and Compuing Univesii Sulan Zainal Abidin Kuala erengganu, Malaysia Sudrada Supian Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences Padadaran Universiy Bandung, Indonesia Absrac In his paper analyzed he AP wih lagged, Value-a-Risk and asse allocaions by using economic approach. I is assumed ha socks were analyzed following he model of Arbirage Pricing heory (AP) wih lagged. Where he facor risk premium in he pas affecs he presen changes in sock reurn. he reurn of facor is assumed o have non consan volailiy and here is effec of long memory. Long memory effecs are esimaed using he rescaled range mehod (R/S) or Geweke and Porer- Hudak (GPH) mehod. he mean and non consan volailiy is esimaed using ARFIMA-GARCH models. he porfolio risk level is measured by he Value-a-Risk (VaR). Asse allocaion problem solving carried ou using he Lagrangian muliplier echnique and he Kuhn-ucker mehod. he purpose of his research forms he efficien porfolio and deermines opimal porfolio weighs. As a numerical illusraion, we analyze some socks raded on he capial marke in Indonesia. Keywords AP, ARFIMA, GARCH, Value-a-Risk, Asse Allocaion. I. INRODUCION Formulaion of Arbirage Pricing heory (AP) has imporan implicaions in deermining sock prices [3]. I s saed ha he reurn of a sock (or porfolio) will be affeced by one or several explanaory variables (facor index). However, AP does no menion (explicily) wha variables affec he sock reurn. o deermine he facors ha influence he degree of sensiiviy and magniude of asse reurns o each facor, is o se a number of facors ha allegedly had an influence on sock reurn [7; 11]. hese facors include he indusrial variables (egg marke index, alernaive producs, ec.) and economic variables (egg inflaion, ineres raes, ec.) [9; 17]. AP applies he law of one price, in equilibrium, he relaionship beween risk and sock reurn occurs in one area (if here is more han one facor). his siuaion can be achieved hrough a process of arbirage. Arbiraors will cause all he porfolio is locaed in one and he same area [19; 11]. he locaion of each porfolio will be deermined by he proporion (weigh) of funds invesed in he esablishmen of a porfolio [9; 3]. Deerminaion of he proporion (weigh) porfolio is a problem ha should be sough he soluion. o deermine he proporion (weigh) can be conduced using a porfolio opimizaion [1; 13]. he 53

2 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 composiion obained proporions will affec he reurn expecaions and risk porfolio [17; 15]. Highly popular porfolio risk is measured using a Value-a-Risk (VaR) [1; 4; 5]. In his paper, he formulaion of AP as a means of deermining he sock price will be expanded by considering he facors of lagged. Where he facors of he las period is assumed influence he formulaion of AP. he reurn of facors in AP is assumed o have non consan volailiy and here is an effec of long memory. Non consan volailiy and long memory effec will be analyzed using ARFIMA-GARCH models []. he mean and variance of reurns of socks esimaed based on he AP wih he lagged, which has non consan volailiy and long memory effecs. Using he mean esimaor and he variance will be arranged he problem of Asse allocaion. Asse allocaion is based on he mean-var approach [18]. he selemen of asse allocaion problem is based on Lagrangian muliplier echnique and he Kuhn-ucker mehods [17]. hus he analysis could be performed, because many socks have characerisics such as he discussion here. he aim is o esablish an efficien porfolio and deermine he proporion (weigh) porfolio opimally. he empirical research carried ou on a few socks ha are raded a he capial markes in Indonesia. II. MEHODOLOGIES Deermining socks reurn. Le P i and r i denoe he prices and he reurns of sock i ( i 1,..., socks ha are analyzed), respecively, a he ime ( 1,..., N and N is he number of, denoes he period of daa observaion). Socks reurn r i is calculaed using he formula ri ln ( Pi / Pi 1 ). Le F and r respecively denoe he price and he reurn of facor index ( 1,..., M and M is he number of facors index in he AP), a he ime, 1,..., he facors index reurn r are calculaed by r ln ( F / F 1 ) [; 8]. A. Mean Modeling. In he same way o calculae r i, In he nex sage we idenify he exisence of long memory effec in he daa reurn of facor index using he rescale range mehod (R/S) or Geweke and Porer-Hudak (GPH) mehod. he parameer esimaion of fracional difference index d, 1,..., M, is performed using he maximum likelihood mehod [; 1]. he confidence inerval (1 c )1 % for d is d zc /. d d d zc /. d where d denoes esimaor of d d d, and z c denoes he percenile of sandard normal disribuion a he significance level c. Le and respecively denoe he mean and sandard deviaion of d. We can es he null hypohesis H : d agains H1 : d using saisic zd ( d d ) / d. We reec H if he value zd zc/ or zd z 1 c / [14; 16]. Fracional difference process is defined as: d (1 B ) r a,.5 d.5 ; (1) where { a } is he error componen which is he whie noise process, and B denoes he backshif operaor? If he sequence d of fracional difference (1 B) r is following he model of ARMA( p, q ), hen we call r auoregressive fracionally inegraed moving average degree p, d and q process, or ARFIMA( p, d, q ) [; 16]. he ARMA( p, q ) follows he following form r p q g 1 h 1 g r g a h a h, () wih consan and g ( g 1,..., p) and h ( h 1,..., q) he parameer coefficiens of mean model of facors index reurn, 1,..., M. We assume ha { a } is he error sequence of whie noise process wih mean zero and variance a [; ]. Sages of mean modeling process include: (i) Idenificaion of he model, (ii) parameers esimaion, (iii) diagnosic ess, and (iv) Predicion []. 533

3 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 B. Non Consan Volailiy Modeling he non consan volailiy of he reurns of facor index is modeled using generalized auoregressive condiional heeroscedasic (GARCH) models. Suppose and respecively denoe he mean and non consan volailiy of reurn of facor index ( 1,..., M and M denoes he number of facors index in he AP), a he ime ( 1,..., and is he period of daa observaion). he error a can be calculaed as a r [; 3]. he non consan volailiy will follow he GARCH model of degree m and n or GARCH( m, n ), if m n a, 1 k a k k l 1 l l. (3) where is a consan and k ( k 1,..., m) and l ( l 1,..., n) denoe he parameer coefficiens of non consan volailiy model of facor index reurn ( 1,..., M ). Here we assume { } is he sequence independen and idenically disribuion (iid) random variable wih mean zero and variance 1, m ax ( m, n) ( ) 1 k 1 k k [; ]., k, l, and he sages of non consan volailiy modeling include: (i) he esimaion of mean model, (ii) esing he effec of ARCH, (iii) Model idenificaion, (iv) Non consan volailiy model esimaion, (v) Diagnosic es, and (vi) Predicion. We furher use he mean model () and he non consan volailiy model (3), o calculae r (1), i.e. he 1-sep ahead predicion afer ime period of he mean and he variance []. (1) C. Modeling of Sock Reurn under AP wih Lagged In his secion expand he AP o AP model wih lagged. I is known ha r i he reurn of sock i a he ime, and r reurns he index facor a he ime. Suppose r is he risk free asse reurn a he ime ( 1,..., and he period of daa observaion). AP regression model wih lagged expressed as equaion or i can be wrien ino r ) ( i r i i1 ( r1 r ) i11 ( r1 1 r 1 i1 r r )... i1l ( r1 L r L )... ) ( i ( r r ) i 1 ( r 1 r 1 i r r )... i L ( r L r L )... ) ( im ( rm r ) im 1 ( rm 1 r 1 im rm r )... iml ( rm L r L ) u i, M L. (4) ri r i il ( r r ) u i 1 Assumed ha { u i } is he whie noise of regressions residual [7; 11; 19]. Where i and i ( i 1,..., N ; 1,..., M ;,..., L and L is lengh of lagged), respecively declare consans and parameer coefficiens of regression for he AP wih lagged of sock reurn i a he ime. o esimae he consan i and parameer coefficiens regression of equaion (4) can be performed using he leas squares mehod. Lengh of lagged il esimaed based on he Ad-Hoc mehod, namely by looking a he consisency changes of parameer coefficiens sign, posiive (+) coninue or he negaive (-) coninued, when lagged exended. Referring Blume (1971), he abiliy of he index facors explain changes in individual sock reurns ranging beween 5%-51% rae coefficien of deerminaion [9]. 534

4 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 As previously described, and successively saes he mean and variance of he index reurn facor a he ime. Suppose ha and, in succession saes mean and variance of reurn risk-free asse. Based on he equaion (4), he mean sock reurn i a he ime, which i can be esimaed using he following equaion: M i E ( ri ) i i ( ) L. (5) 1 I is assumed ha E [( r r ), ( r ' ' r ' )], where, ' 1,..., M, ' and, ',..., L, '. Sock reurn variance i a he ime, ha u i i can be esimaed using he equaion M L i V a r ( ri ) ( ) i u 1. (6) Where V a r ( u ) he regressions residual variance of AP wih is lagged of sock reurn i a he ime. Based on he i assumpions in equaion (6), covariance beween sock i wih sock i ', which are saed o ii ' be esimaed wih equaion M L ii ' C o v ( ri, ri ' ) i i ' ( ) 1 ; i i'. (7) Esimaor mean, variance and covariance of sock reurn i ( i 1,..., N and N he number of sock ha were analyzed), a he ime ( 1,..., and he period of daa observaion), hen used for he following porfolio formaion. i D. Asse Allocaion Based on Mean-VaR Le r denoe he reurn of porfolio a he ime, and be deermined using he equaion [9; 17]: w i ( i 1,..., N ) weigh of sock i. Reurn of porfolio r can N r w i 1 i r i N ; erms wi 1 and w i 1 i 1 ( i 1,..., N ). (9) Suppose μ ( 1... i ), i 1,..., N is he mean vecor, and w ( w1... w N ) he weigh vecor of porfolio. From equaion (9), he weigh vecor w follows he propery e w 1, where e (1... 1). he mean of porfolio reurn can be esimaed using he equaion: he variance of porfolio reurn N w i 1 i i can be esimaed using he equaion: μ w. (1) N N N w 1 i i w 1 ' i wi ' ii ' ; i i ' i i i w Σw. (11) where ii ' C o v ( ri, ri ' ) denoes he covariance beween sock i and sock i ' [17]. Value-a-Risk (VaR) of an invesmen porfolio based on sandard normal disribuion approach is calculaed using he equaion [18; 4]: where W 1/ V a R W ( zc ) W { zc ( ) } w μ w Σ w. (1) he number of fund is allocaed in he porfolio and z c is he percenile of sandard normal disribuion a he significance level c. When i is assumed W 1 uni, he equaion (1) becomes: 535

5 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 1/ V a R ( zc ) { zc ( ) } w μ w Σ w. (13) A porfolio w * is called (mean-var) efficien if here is no oher porfolio w wih and V a R V a R [17]. o obain he efficien porfolio, we used he obecive funcion, o maximize { V a R }, where denoes he invesor risk olerance facor. For he invesor wih he risk olerance herefore we mus solve an opimizaion problem [17; 1]: 1 / M a x im iz e { μ w μ w z c ( w Σ w ) } such ha e w 1 Equaion (14) is a quadraic concave opimizaion problem. Is Lagrangian funcion can be wrien as 1/ L(, ) ( 1) z ( ) ( c 1). (14) w μ w w Σ w e w. Using he Kuhn-ucker heorem, he opimal soluion can be obained using he firs derivaives, as follows [18; 6]: 1/ L / ( 1) z c / ( ) w μ Σ w w Σ w e and L / e w 1. (15) Solving he equaion (15) as he funcion of, we obain a quadraic funcion A B C. where A e Σ e, B ( 1)( μ Σ e e Σ μ ) and C ( 1) ( ) z c of he quadraic funcion [18]: μ Σ μ. he soluion can be obained direcly from he soluion 1 For, we obain he weigh vecor w as 1/ { B ( B 4 A C ) } / A ;. (16) 1 1 ( 1) Σ μ Σ e w 1 1 ( 1) e Σ μ e Σ e. (17) By subsiuing he vecor w ino he equaion (1) we obain he mean value of porfolio reurn. When vecors w are subsiued ino he equaion (13), we obain he value of he invesmen porfolio risk levelv a R. he ses of poin pairs (, V a R ) form a graph of efficien fronier. Among he efficien fronier, here are opimum porfolios, which have he larges raio / V a R [18]. III. RESULS AND ANALYSIS A. he Daa For empirical sudy, we analyze he following socks: ASII, RUB, BBCA, BBRI, HMSP, and LKM, and are denoed as S 1 unil S 5. As he facors index, we use Composie Sock Price Index (IHSG), he rae of inflaion, exchange rae of he rupiah agains he euro, he rupiah agains he U.S. dollar, and he rupiah agains he yen, and are denoed as F1 unil F 5. For he risk-free asse daa, we use a governmen bond price. he daa are obained from hp:// he period of observaion is January, 11 unil December 3, 15. he empirical analysis is done using he sofware s of: MS Excel 7, Eviews 6, Maple 9.5 and R. B. Empirical Resuls In his sudy, he facors index used are F1 unil F 5, as described above. We firs calculae he reurns of each facor index, hen idenify he exisence of he effecs of long memory in he reurns, and finally esimaed he mean and volailiy models of he reurns. 536

6 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 Idenificaion of long memory effecs. o idenify he effec of long memory, we esimae he parameers of fracional d difference ( 1,..., 5 ) as in equaion (1). he esimaion is performed using he rescale range mehod (R/S) or Geweke and Porer-Hudak (GPH) mehod. he resuls are summarized in able-1. ABLE I. Sock d Inervals d Confidence F < 1 HE IDENIFICAION OF HE LONG MEMORY EFFECS Effec of Long z Memory d < Significan F < d <.5.57 Significan F < d 3 < No Significan F < d 4 < Significan F < d 5 < No Significan o ensure he exisence of long memory paerns, we es he hypohesis H : d agains, H1 : d, 1,..., 5. he values of saisic z is given in able above, while for he level of significance c.95, from he sandard normal disribuion able values, we obain Z.9 5 / Because of he values z 1, z and z 4 are larger han he value Z.9 5 /, i is concluded ha he es resuls are significan, he reurns of facor index daa, F 1, F and F 4 have long memory effecs. However, in F3 and F5 here are no long memory effecs. In he nex sep, we idenify and esimae he bes mean and volailiy models o difference fracional daa F 1, F and F 4, where for F 3 and F 5, he analysis is applied direcly o he reurns daa. d of he reurns Idenificaion and esimaion of mean models. Idenificaion of he mean models is done using he sample auocorrelaion funcion (ACF) and parial auocorrelaion funcion (PACF). Based on he paerns of ACF and PACF of each facor index reurns (or he fracional differenced daa), we obain he bes models for F 1 unil F 5, which also passed he sandard diagnosic check. he resuls are summarized in able. Idenificaion and esimaion of volailiy models. We furher idenify and esimae of volailiy model using generalized auoregressive condiional heerscedasiciy (GARCH) models. Based on he correlogram of quadraic residual, we selec he plausible volailiy model for he daa. Esimaion of volailiy models of each facors index reurn is done simulaneously wih he mean models. he resuls, obained for he bes model which is also passed he diagnosic checks, are given in abel-. a ABLE II. Facor ( F ) HE ESIMAION RESULS OF MEAN AND VOLAILIY MODEL OF FACOR INDEX REURNS Model Mean and Volailiy Equaions F 1 ARFIMA(1, 1 d,)- GARCH(1,1) F ARFIMA(1, d,1)- GARCH(1,1) F AR(1)- 3 GARCH(1,) F 4 ARFIMA(1, 4 d,)- GARCH(,1) r r1 1 a a r r a 1 a a r r3 1 a a r r4 1 a a a

7 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 F AR(1)- 5 GARCH(,1) r r5 1 a a a Esimaed regression model of AP wih lagged. In his secion esimaion of AP model wih lagged, conduced by esimaing regression models of each of he five sock reurn daa, agains he daa of reurn of he five-facor index. Esimaion made refers o he equaion (4), helped by Eviews 4 sofware. he reurn of risk-free asse daa (bond) is relaively consan, herefore, aken he mean size and variance. o simplify he wriing, for example i ri , i 1,..., 5 and he risk premium of facor index wih he lagged I r ( 1,..., 5 and,1,..., L, where L lengh of is lagged). he Esimaion resuls wrien covering he regression equaion wih -saisic values, wrien in parenheses below each parameer coefficiens. Similarly, given coefficien of deerminaion R i of, and Durbin-Wason saisic (D-W) from each of he regression. he regression equaion of AP wih lagged hose given in able-3. ABLE III. REGRESSION MODEL ESIMAION RESULS OF AP WIH LAGGED As presened by Blume (1971), he abiliy of he index facors o explain changes in individual sock reurns ranging beween 5% -51% rae coefficien of deerminaion. Looking a he resuls in able-3, i appears ha he coefficien of deerminaion R i of regression models of each sock worh nearly 51%. Means he reurn of five shares S 1 unil S 5 a relaively srong correlaion wih he reurn of five risk premium of he index facor I ( 1,..., 5 and,1,..., L, where L lengh of is lagged). Based D-W saisic whose value is also relaively small, showing he five regression models in able-3 is significan. I can also be shown he residuals u i of each regression model are normally disribued, wih zero mean and variance u i. Esimaor value u i esimae he mean and variance of sock reurn. of each regression equaions are given in able-4. Regression model is hen used o he esimaed mean and variance values of socks reurn. he values of consans, coefficien of parameers and u i regression of residual variance in able-3, and he mean value esimaor r (1) and variance (1), hen used o esimae he mean and variance values of sock reurn S 1 unil S 5. he mean value esimaed using equaion (6), while he variance value is esimaed based on equaion (7). he esimaion resuls are given in able

8 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 ABLE IV. Socks ( S i ) HE ESIMAION RESULS OF MEAN AND VARIANCE OF SOCKS REURN u i i i S S S S S In his sudy assumed ha he covariance beween sock reurn i and i ' a he ime, ii ' ( i, i' 1,..., 5 and i i' ), because is values are also very small close o zero. his means ha beween sock reurn i and i ' a he ime where he cross correlaion does no occur. he values of he mean esimaor porfolio opimizaion process. i and he variance i will be used for he following Porfolio opimizaion. Porfolio opimizaion in his case is based on he mean-var model, using he Maple 9.5. sofware. Firs, he values of he mean esimaor in able-4 column is formed ino a vecor mean of μ i. Second, referring o he number of shares o be analyzed as much as five, mean vecor defined ideniy is e hird, he values of he variance esimaor in able-4 a column of i, is formed ino he covariance marix and deermined inverse, as follows: Σ.113 and Σ Risk olerance in his sudy are deermined by simulaion. olerance values ogeher and vecors and marix, when subsiued ino equaion (15) will be obained muliplier value as. Where hen subsiued ino equaion (16) will be obained by porfolio weigh vecor w. Weigh vecor w, hen used o calculae he mean value of porfolios using equaion (1), and calculae he risk level of VaR using equaion (13). For he values of risk olerance some calculaion resuls are given in able-5. μ e 1 Σ ABLE V. RISK OLERANCE, WEIGH, MEAN AND PORFOLIO RISK Weigh w 1 w w 3 w 4 w 5 V a R V a R 539

9 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, For values of risk olerance is no longer feasible, because he porfolio weigh does no qualify. A collecion of poins (, V a R ) o form he surface efficienly (efficien fronier), as shown in Figure-1. Raio beween oward V a R he values given in las column of able-5 while he graph is given in Figure-. Fig. 1. Efficien Fronier of Porfolio Fig.. Mean and VaR Raio of Porfolio C. Discussion Based on he calculaion resuls given in able-5, can be seen ha wih he risk olerance of. obained VaR porfolio composiion ha produces a minimum, ha is equal o.434 wih he expeced reurn of porfolio amouned o he amoun of risk olerance can sill be improved bu wih he condiion ha he resuling values of weighed porfolio of real 5 w i 1 ( i 1,..., 5 ) and qualify w 1 i 1 i. In his case he value will be a mos risk olerance as Where he resuling composiion of he porfolio wih he highes expeced reurn of porfolio ha is equal o wih VaR a Any increase in he value of risk olerance will cause he increase in expeced reurn porfolio which is also accompanied by an increase in he Value-a-Risk porfolio. Efficien porfolios lie along he line wih risk olerance of 7.377, as given by he graph of he surface efficienly (efficien fronier) in Figure-1. Along ha line of invesors eligible o inves in a porfolio consising of socks S 1 unil S. Of course, each invesor should choose one based on preference or risk olerance level ha is believed. For he avoidance of risk invesors will usually ses he risk olerance is small, while for invesors challenger will risk aking a big risk olerance. Large-size specified risk olerance, of course, will affec he large-size of porfolio expeced reurn obained. Having obained a series of efficien porfolios, he nex sep is deermining he opimum porfolio composiion. Every invesor wans a porfolio of invesmen ha can yield big reurns, bu accompanied by a small degree of risk. If i is assumed ha invesor s preferences based only on expeced reurn and risk of he porfolio. Selecion of he opimum porfolio can be deermined based on he composiion of he efficien porfolios ha generae expeced reurn and he Value-a-Risk of porfolio wih he larges raio. Based on he calculaion resuls given in he las column of able-5, shows ha he raio of expeced reurn and VaR he larges porfolio is.7877, or obained when risk olerance reaches 644. he raio of expeced reurn and VaR coninues o increase a inervals of risk olerance 6.44 and decreased in he inerval Up-and-downs of hese raios can be seen graph given in Figure-. Based on he values in able-5, obained he resuls ha based on he model of asse allocaion mean-var, he opimum porfolio composiion prepared from he sock S 1 unil S 5 he porfolio composiion wih he weigh vecor w ( ). Where is he opimum porfolio composiion o generae he expeced reurn of.185 wih Value-a-Risk of

10 Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 IV. CONCLUSION Mahemaical of Arbirage Pricing heory (AP) model can be expanded ino AP wih lagged. Based on i he esimaors of mean, variance and covariance of sock reurn are formulaed. he empirical research conduced on five sock S1 unil S 5, and five index facor of F1 unil F 5. Where reurn of he index facors in he AP wih lagged analyzed using ARFIMA-GARCH model approach. Based on he analysis ha he hree index reurn facor, namely F 1, F and F 4 here are significan effecs of long memory. Parameers esimaor of mean, variance and covariance, hen used for he analysis of asse allocaion problem based on he mean-var model. Based on he resuls of opimizaion, efficien porfolios are formed for he values of risk olerance An opimum occurs in he value of porfolio risk olerance Opimum porfolio has composiion allocaion weigh of w , wih a mean porfolio reurn of.185 and Value-a-Risk of ACKNOWLEDGEMENS We would like o hank he financial suppor from Deparmen of Higher Educaion, Minisry of Higher Educaion Malaysia hrough he Fundamenal Research Gran Scheme (FGRS) Vo hanks also submied o he Insiue for Research and Communiy Services, Padadaran Universiy, which has been providing faciliies o carry ou his research. REFERENCES [1] Alexader, C. (Edior). (1999). Risk Managemen and Analysis. Volume 1 : Measuring and Modelling Financial Risk. New York: John Wiley & Sons Inc. [] Blum, M. (1971). On he Assessmen of Risk. Journal of Finance, XXVI (March, 1971), pp poral.acm.org/ciaion.cfm?id=8351 (Rerieved on November 7, 8). [3] Bodie, Z., Kane, A. & Marcus, A.J. (1999). Invesmens. New York: Irwin / McGraw-Hill. [4] Benninga, S. & Wiener, Z. (1998). Value a Risk (VaR). Journal of Mahemaica in Iducaion and Research. Vol. 7, No. 4, [5] Cheng, S., Liu, Y., & Wang, S. (4). Progress in Risk Measuremen. AMO-Advanced Modelling and Opimizaion, Vol. 6, No. 1, 4, pp. 1-. [6] Cole, F., De Moor, L., De Ryck, P. & Smeds, J. (7). he Performance Evaluaion of Hedge Funds: Are Invesors Mislead by Sandard Mean-Variance Saisics?. HUB Research Paper 7/4. November 7. [7] Connor, G. & Koraczyk, R.A. (1993). he Arbirage Pricing heory and Mulifacor Models of Asse Reurns. Working Paper #139. USA: Kellogg Graduae School of Managemen, Norhwesern Universiy. hp:// wp139.pdf. (Rerieved on December 13, 1). [8] Dowd, K. (). An Inroducion o Marke Risk Measuremen, New Delhi, India: John Wiley & Sons, Inc. [9] Elon, E.J. & Gruber, M.J., (1991). Modern Porfolio heory and Invesmen Analysis. Fourh Ediion. New York: John Wiley & Sons, Inc. [1] Gaivoronski, A. & Pflug, G. (5). Value-a-Risk in Porfolio Opimizaion: Properies and Compuaional Approach. Journal of Risk, Vol. 7, No., pp. 1-31, Winer 4-5. [11] Huberman, G. & Wang, Z. (5). Arbirage Pricing heory. Saff Repor No. 16. Federal Reserve Bank of New York. Augus 5. hp:// /.../AP-Huberman-Wang.pdf. (Rerieved on December 3, 1). [1] Jinwen Wu. (7). he Sudy of he Opimal Mean-VaR Porfolio Selecion, Inernaional Journal of Business and Managemen. Vol, No 5 7, pp hp:// ournal/index.php/ibm/ aricle/view/51. (Rerieved on December 1, 9). [13] Huyen PHAM. (7). Opimizaion Mehods in Porfolio Managemen and Opion Hedging. Working Paper. Laboraoire de Probabilies e Modeles Aleaoires, CNRS, UMR Universie Paris 7. [14] Kang, S.H. & Yoon, S-M. (5). Value-a-Risk Analysis of he Long Memory Volailiy Process: he Case of Individual Sock Reurn. Working Paper. School of Commerce. Universiy of Souh Ausralia. hp://www. korfin.org/ daa/ournal/1-1-4.pdf. (Rerieved on February 1, 8). [15] Korn, O. & Koziol, C. (4). Bond Porfolio Opimizaion: A Risk-Reurn Approach. Working Paper. WHU-Oo Beisheim School of Managemen. Burglaz, D Vallendar, Germany. [16] Korkmaz,., Cevic, E.I. & Ozaac, N. 9. esing for Long Memory in ISE Using ARFIMA-FIGARCH Model and Srucural Break es, Inernaional Research Journal of Finance and Economics, ISSN Issue 6 9. hp://www. Euroournals. com/finance.hm. (Rerieved on April 9, 1). [17] Paner, H.H., Boyle, D.D., Cox, S.H., Dufresne, D., Gerber, H.U., Mueller, H.H., Pedersen, H.W., & Pliska, S.R. (1998). Financial Economics. Wih Applicaions o Invesmens, Insurance and Pensions. Schaumberg, Illinois: he Acuarial Foundaion. [18] Sukono, Subanar & Rosadi, D., (9). Mean-VaR Porfolio Opimizaion under CAPM by Non Consan Volailiy in Marke Reurn. Working Paper. Presened in 5 h Inernaional Conference on Mahemaics, Saisics and heir Applicaion a Andalah Universiy, Padang, Wes Sumaera-Indonesia, June 9-11, 9. [19] Shanken, J. (199). he Curren Sae of he Arbirage Pricing heory. he Journal of Finance, Vol. 47, No. 4. (Sep., 199), pp hp:// (Rerieved on July 8, 8). 541

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