ROBUST OPTIMIZATION IN PORTFOLIO SELECTION BY m- MAD MODEL APPROACH

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1 Ecoomc Computato a Ecoomc Cyberetcs Stues a Research, Issue 1/218; Vol. 52 Alreza GHAHARANI, M.Sc. E-mal: alghahtara@gmal.com Departmet of Iustral Egeerg K.N. oos Uversty of echology ehra, Ira Assocate Professor Amr Abbas NAJAFI, PhD E-mal: aaaaf@ktu.ac.r Departmet of Iustral Egeerg K.N. oos Uversty of echology ehra, Ira ROBUS OPIMIZAION IN PORFOLIO SELECION BY m- MAD MODEL APPROACH Abstract: he portfolo selecto problem s oe of the ma vestmet maagemet problems. I the portfolo selecto problem, robustess s sought agast ucertaty or varablty the value of the parameters of the problem. I ths paper, a extee mea absolute evato moel ame the m-mad moel s apple to costruct a ew robust portfolo selecto moel that s solvable to real-worl problems. he m-mad moel s a lear programmg moel a allows us to measure rsk usg owse evatos wth the ablty to pealze larger owse evatos. It also has a better performace of rsk-averse prortes. he results of the performace aalyss of the moel show that the solutos of the m-mad moel are compatble wth respect to seco-egree stochastc omace. Keywors: Portfolo Optmzato, Lear Programmg, Dowse Rsk, Stochastc Domace, Robust Optmzato. JEL Classfcato: C61, G11 1. Itroucto Portfolo optmzato s the process of aalyzg a portfolo a maagg the assets wth t. Markowtz (1952) has presete the Moer Portfolo heory (MP) a tre to maxmze the retur a cotrol rsk through the mmzato of the varace of the portfolo retur as a rsk measure. I spte of some avatages, the Markowtz moel has two ffcultes: (a) quaratc programmg problems are more ffcult to solve, a (b), for practcal markets, the sze of the covarace matrx for solvg the moel s very large a ffcult to estmate. o DOI: 14818/ /

2 Alreza Ghahtara, Amr Abbas Naaf overcome these ffcultes, may researchers have tre to preset lear programmg for the portfolo selecto problem. Koo a Yamazak (1991) suggeste the mea absolute evato (MAD) for rsk measure as a lear moel. Absolute evato the MAD moel s sestve agast ay upwar or owwar movemet of the mea. A vestor who uses the MAD moel s assume to have a costat trae-off: a ut evato from the mea rate of retur. hs assumpto oes ot allow for the stcto of rsk assocate wth larger losses. It ca be argue that the varablty of the rate of retur above the mea shoul ot be pealze. Sce the vestors are cocere wth the uerperformace of a portfolo rather tha the over performace of a portfolo, Markowtz (1959) propose owse rsk measures. Subsequetly, Mchalowsk a Ogryczak (21) suggeste the m-mad moel. he m-mad moel s a true owse rsk measure that ca stgush larger losses. Researchers such as (Kellerer et al 2, Mas et al 23, Cho et al 23, Papahrstooulou a Dotzauer 24, a Rockafellar a Uryasev 2) extee the moels by presetg smlar eas o the rsk measure for a lear programmg formulato. he parameters o the metoe moels are efe wth ther omal value a t s assume that all parameters are costat. However the real worl, we eal wth the problems where robustess s sought agast ucertaty or varablty the value of parameters of the problem. I the recet years, a boy of the lterature s evelopg uer the ame of robust optmzato to coser ucertaty the value of parameters of the moel. Soyster (1973) propose a lear optmzato moel to costruct a soluto that s feasble for all ata that belog a covex set. he solutos of the Soyster moel are too coservatve the sese a t causes to gve up too much of optmalty for the omal problem orer to esure robustess. he seco step forwar for evelopg a theory for robust optmzato was take epeetly by Be-al a Nemrovsk (2) a El-Ghaou et al (1998). hey use ellpsoal ucertaty set. hs moel ca aust the coservatsm. However, ths moel s ot lear whch ca be problematc the real worl problems. Aother evelopmet o robust optmzato has bee oe by Bertsmas a Sm (24). hs moel s lear, applcable a exteable to screte optmzato a ca flexbly aust the level of coservatsm of the robust solutos terms of probablstc bous of costrat volatos. I ths paper, we use Bertsmas a Sm methoology for evelopmet of our moel. here are some practcal moels of robust optmzato face. El-Ghaou et al (23) propose a robust portfolo moel uer a ucertaty of covarace matrx wtch s evelope by sem-efte programmg (SDP) a cosers worst case value-at-rsk. utucu a Koeg (24) evelope a robust portfolo optmzato problem formulate a quaratc program (QP). Kawas a hele (28) evelope a log robust portfolo moel to coser the heavy tale property of stock prces. Moo a Yao (211) evelope a robust mea absolute evato moel for portfolo optmzato. Quarata a Zaffaro (28) evelope a robust optmzato of cotoal value at rsk. Che a a (29) evelope 28 DOI: 14818/ /

3 Robust Optmzato Portfolo Selecto by m-mad Moel Approach robust portfolo selecto base o asymmetrc measures of varablty of stock returs. I ths paper we evelop a robust moel for the m-mad moel. he m- MAD moel s a lear a owse rsk measure. he results of the m-mad moel are seco orer stochastc omace (SSD). he rest of the paper s orgaze as follows. I Secto 2, we expla the MAD a m-mad moels. We propose robust optmzato of m-mad moel Secto 3. he computato results of emprcal stuy base o hstorcal ata are scusse Secto 4. Fally, the cocluso comes Secto m-mad moel J eotes a set of securtes cosere for vestmet. he rate of retur for each securty J s represete by a raom varable R wth Let 1,2,..., a gve mea E( R ). Further, let X ( x1,x2,...,x ) eotes a vector of securtes weghts (ecso varables) efg a portfolo. he weghts must satsfy a set of costrats that form a feasble set Q. he weghts must sum to oe a there s ot short sellg: X ( x1,x2,...,x ) : x 1,x ; 1,..., (1) 1 Each portfolo X efes a correspog raom varable R R x that represets a portfolo rate of retur. he mea rate of retur for portfolo X s gve as: ( X ) E( RX ) x 1 X MAD moel Koo a Yamazak (1991) tre to represet a lear rsk measure. hey efe the mea absolute evato from a mea as follows. R P ( ) ( x ) E x ( x ) ( x ) x Where P X eotes a probablty measure uce by the raom varable R X. May authors (Koo, 1999; Feste a hapa, 1993; Zeos a Kag, 1993) have pote out that the MAD moel opes up opportutes for more specfc moelg of the owse rsk, because absolute evato may be cosere as a measure of owse rsk. Namely, the mea absolute evato (X) equals twce the (owses) absolute sem evato. )2( DOI: 14818/ /

4 Alreza Ghahtara, Amr Abbas Naaf ( x ) E(max ( x ) Rx,) )3( Accorg to (Koo, 1999), the followg parametrc optmzato problem s calle the MAD moel; max ( x ) ( x ) : X Q )4( he propose exteso to the MAD moel allows oe to fferetate betwee owse a upse rsk, a to pealze larger owse evatos. It thus proves for better moelg of rsk averse prefereces. Note that such a exteso s some ways equvalet to replacg (X) wth u (x) efe as: u( x ) E( u(max{ ( x ) Rx, })) )5( Where, u s some covex pealty fucto. It s assume that r t s the realzato of the raom varables R urg the pero t (where t=1,,) that s avalable from hstorcal ata. It s also assume that the expecte value of R ca be approxmate by: 1 r t t 1 herefore, moel (4) for a screte set of realzatos r t ca be rewrtte as the followg LP moel (Feste a hapa, 1993): max 1 Subect to X Q t x ( r 1 t 1 t )x t ;t 1,..., t ;t 1,..., )9( If the rate of retur s ormally strbute multvarate, the the MAD moel s equvalet to the Markowtz moel (Koo, 1999). Recetly, the MAD moel was further valate by Ogryczak a Ruszcysk (1999). hey emostrate that f the trae-off coeffcet s boue by 1, the the MAD moel s partally cosstet wth seco egree stochastc omace (Whtmore a Flay, 1978) Extee MAD Moel he extee MAD moel s to fferetate betwee the varous levels of owse evatos, a to pealze the larger oes. Koo (1999) has alreay propose such a exteso of the MAD moel for portfolo optmzato. He )6( )7( )8( 282 DOI: 14818/ /

5 Robust Optmzato Portfolo Selecto by m-mad Moel Approach cosere atoal mea evatos from some target rate of retur preefe as beg proportoal to the meas rate of retur: x ) E(max k( x ) R, ) for k )1( k ( x If k=1 the 1( x ) ( x ) 1 a ths moel s lke absolute sem evato use the orgal MAD moel. Oe may attempt to augmet the owse rsk measure by pealzg atoal evatos for several k<1. I terms of pealty fucto (5), ths approach s equvalet to troucg a covex pecewse lear fucto wth break pots proportoal to the mea of R x. Koo (1999) evelope MAD moel wth atoal owse evato as follow: max ( x ) ( x ) ( x ) : X Q )11( k k Where, s the basc trae-off parameter a k s a atoal parameter. We refer to ths moel as k-mad. Assumg that has value 1. Mea surplus evato Emax ( x ) ( x ) Rx, ees to be pealze by a value, let's say 2, of a trae-off betwee the surplus evato a a mea evato, whch leas to maxmzato of the followg equato: ( x ) 1( ( x ) 2 E(max ( x ) ( x ) Rx,)) )12( Oe mght wsh to pealze seco level surplus evato exceeg that mea. m max( x ) ( k ) ( x ) : X Q )13( 1 k 1 Where, 1,...,m are the assume kow trae-off coeffcets. rae-off coeffcets are measure as follow: k 1 k ; 1,...,m he moel formulate as follow: m max( x ) ( x ) : X Q )15( 1 We wll refer to moel (15) as the recursve m-level MAD moel (Mchalowsk a Ogryczak, 21). If trae-off coeffcets are postve a ot greater tha oe a satsfyg: m )16( he the results of the m-mad moel wll be SSD (Mchalowsk a Ogryczak, 21). I ato, the m-mad moel wth m=2 s formulate as a LP problem. max J 1 x 1 t 1 t1 2 t 1 t2 )14( )17( DOI: 14818/ /

6 Alreza Ghahtara, Amr Abbas Naaf Subect to X Q )18( t 2 t1 ( r 1 1 ( r t t )x )x 1 L1 ;t 1,..., L1 ;t 1,..., t1,t 2 ;t 1,..., A geeral m-mad moel ca be formulate as a LP moel. m maxw w Subect to X Q w 1 x 1 w t t 1 1 k t w t k r t 1 ; 1,...,m x ;t 1,...,, ;t 1,...,, 1,...,m I the above formulato (x) a (x) atoal varables wa. w 1,...,m )19( )2( )21( )22( )23( )24( )25( )26( )27( are explctly represete usg 3. Robust m-mad moel hs secto evelops a robust reformulato of the m-mad moel. here are three ks of robust optmzato base o ucertaty sets. Soyster (1973), Be- al a Nemrovsk (2), a Bertsmas a Sm (24) evelope Robust Optmzato. he result of Soyster moel prouces solutos that are too coservatve. Be-al a Nemrovsk (2) assume that the ucertaty sets are ellpso. Wth ths ucertaty set the robust couterparts are olear; the last moel of robust was evelope by Bertsmas a Sm (24). he robust couterparts of Bertsmas a Sm are lear.i m-mad moel, the expecte 1 retur, of asset s approxmate by r t t 1, whch meas that a actual 284 DOI: 14818/ /

7 Robust Optmzato Portfolo Selecto by m-mad Moel Approach retur caot be exactly obtae a has ucertaty. Let J the set of coeffcets, J that are subect to parameter ucertaty. ~, J take values accorg to symmetrc strbuto wth mea equal to the omal value the terval [ ˆ, ˆ ]. We trouce a parameter, ot ecessary teger that takes values the terval,. he role of the parameter s to aust the robustess of the propose moel agast the level of coservatsm of the soluto. For the chose, coser a subset S satsfyg the cotos S J a S. For the gve set S a a coeffcet rˆ, where, S \ J we lke to allow a certa level of evatos costrats. It s clear by the costructo of robust formulato that f up to of the J coeffcets chage wth ther bous, a up to oe coeffcet ˆ chages by( ) ˆ, a the the soluto of problem wll rema feasble a flexble. I other wors, we stpulate that ature wll be restrcte ts behavor, that oly a subset of the coeffcets wll chage orer to aversely effect o soluto (Bertsmas a Sm, 24). here are two places the m-mad moel that exsts. At frst we shoul chage obectve fucto to costrat. he form of costrats has to be lke ax b so the m-mad moel chages to: max k )28( Subect to X Q )29( 1 x w 1 t t 1 x t m 1 1 k 1 w k w k ; 1,...,m r t 1 ;t 1,...,, x t 1,...,m ;t 1,...,, 1,...,m We use the followg formulato for (32). hs formulato s exactly use for (3). )3( )31( )32( )33( max k Subect to )34( DOI: 14818/ /

8 Alreza Ghahtara, Amr Abbas Naaf X Q )35( 1 x ˆ y s w 1 t 1 x s t ˆ y Note that m 1 ( t 1 k 1 w s w k ( max s, s, \s )ˆ y k ; 1,...,m 1 r ;t 1,...,, t x s max s, s )ˆ y t ;t 1,...,, 1,..., m 1,...,m, \s )36( )37( )38( )39( (36) a (38) are same a we ca use ust oe for both of them. If s chose as a teger: max ˆ s s J, s B ( x, ) x )4( We ee the follow proposto that reformulate (34) to (39) as a lear x : costrat. For gve vector B ( x, ) max ˆ x ( )ˆ s s J, s, J \s s he above formulato s equal to followg lear optmzato problem: x B ( x, ) max ˆ x z J subect to: J Z Z 1 J We coser ual formulato of above lear optmzato problem as follow: )41( )42( )43( )44( 286 DOI: 14818/ /

9 Robust Optmzato Portfolo Selecto by m-mad Moel Approach m P z )45( J Subect to: Z P P ˆ x J J Z )48( By strog ualty, sce problems (42) to (44) are feasble a boue for all, J, the ual problems (45) to (48) are also feasble a boue a ther obectve values coce. I ato, B ( x, ) s equal to (36) a (38). herefore we ca reformulate the robust m-mad moel base o Bertsmas a Sm (24) as follow: max k )49( Subect to X Q )5( m x w z P w 1 t t 1 t 1 ; 1,...,m k x wk rtx z P t ; t 1,...,, 1,..., m z k1 P y 1 1 Rˆ ;t 1,...,, 1,...,m t P J, Z y x ;t 1,...,; 1,...,m y, y 1,..., N 4. Computatoal result I ths secto, we show how robust optmzato approach ca be mplemete to the m-mad moel. We show that the robust optmzato of the m-mad moel ca measure ow se mea absolute evato wth ucertaty coeffcets. We use real ata from New York facal market. he ata comes from New York stock exchage betwee Aprl, 212 a Aprl 1, 213 for 1 stocks. he stocks that we use ths case stuy s as follow: )46( (47) )51( )52( )53) )54( )55) )56( )57( )58( DOI: 14818/ /

10 Alreza Ghahtara, Amr Abbas Naaf Amazo, bak of Amerca, bak of Motreal, Exxo Mobl, face book, FeEx, for, geeral electrc, geeral motors a yahoo X1 to X1 respectvely refer to above stocks. Summary of ata s table 1: 1 able1: Summary of Data from New York stock exchage X1 X2 X3 X4 X5 X6 X7 X8 X X We wat to share vestmet betwee these 1 stocks. he pealty parameters are calculate as follow: 2 1 a 2 We coser. 5a the we have 1. 5 a We coe the robust m-mad moel by lgo software as a lear programmg the we solve that wth fferet robust prce parameter. For ucertaty parameter we coser 2 percet volatlty for each. We summarze the results of obectve fucto able 2. I ths table we show the value of the obectve fucto a the prce of robustess a the value of ecso varables. 288 DOI: 14818/ /

11 Robust Optmzato Portfolo Selecto by m-mad Moel Approach able 2: he value of the obectve fucto for varous prce of robustess X1 X2 X3 X4 X5 X6 X7 X8 X9 X1 Obectve Fucto I ths table we show fferet soluto base o fferet prce of robustess. As we show table by crease of C the obectve fucto reuce. he frst row s about the stuato that It meas there s t ay ucertaty parameters. =. Whe = the result s equal to the stuato that we o t use robust approach. hs row show fferet betwee the use of robust approach a orgal m-mad. X4 has the best rate of retur ths set of stocks a x5 refer to face book rate of retur that has the worst rate of retur ths set. As show table the moel try to maxmze x4 because t has the best effect o portfolo a all stuato of x4 has the most share portfolo a x5 has the less share portfolo. From, = to =1 there s a lttle fferet betwee soluto because we use small.vestors ca use ths strategy f they atcpate small volatlty market. From row =1 to =1 the ucertaty of parameters goes up. A the solutos have bg chage. But from =5 to =1 the soluto o t chage ay more. here s a mathematcal expla for ths pheomea. Robust optmzato coser the optmum soluto worst case of ucertaty a from =5 to =1 the best soluto that rema feasble are acheve. DOI: 14818/ /

12 Alreza Ghahtara, Amr Abbas Naaf Base o formato the table, by crease of level of robustess the level of coservatsm creases a obectve fucto s reuce. As we show the robust metho of m-mad moel that we represet, coser ucertaty coeffcet the goo way. Expermetal results show that by creasg the level of robustess how the moel s reactg. Whe, the prce of robustess creases, the coservatsm of soluto has crease. 5. Coclusos hs paper evelope a ew robust moel portfolo optmzato by usg the m-mad approach. hs moel has avatages o computatoal complexty a proves robust solutos uer parameter ucertaty.i ato, we show coservatsm of obectve fucto agast the level of robustess ths moel. For future evelopmet, we offer use of robust optmzato goal programmg for portfolo selecto problem. REFERENCES [1] Be-al, A., Nemrovsk, A. (2), Robust Solutos of Lear Programmg Problems Cotamate wth Ucerta Data; Mathematcal Programmg, 88,pp ; [2] Bertsmas, D., Sm, M. (24), he prce of Robustess; Operatos Research, 52,pp.35 53; [3] Che, W., a, S. (29), Robust Portfolo Selecto Base o Asymmetrc Measures of Varablty of Stock Returs; Joural of Computatoal a Apple Mathematcs, 232,pp ; [4] Cho, L., Mas, R., Speraza, M.G. (23), Sem-absolute Devato Rule for Mutual Fus Portfolo Selecto; Aals of Operatos Research, 124,pp ; [5] El Ghaou, L., Oks, M., Oustry, F. (23), Worst-case Value-at-rsk a Robust Portfolo Optmzato; A Coc Programmg Approach; Operatos Research, 51,pp ; [6] El-Ghaou, L., Oustry, F., Lebret, H. (1998), Robust Solutos to Ucerta Sem Defte Programs; SIAM Joural of Optmzato, 9,pp.33-52; [7] Feste, C.D., hapa, M.N. (1993), A Reformulato of a Mea-Absolute Devato Portfolo Optmzato Moel; Maagemet Scece, 39,pp ; [8] Kawas, B., hele, A. (28), A Log-robust Optmzato Approach to Portfolo Maagemet; Workg Paper; [9] Kellerer, H., Mas, R., Speraza, M.G. (2), Selectg Portfolos wth Fxe Costs a Mmum rasacto Lots; Aals of Operatos Research, 99,pp ; 29 DOI: 14818/ /

13 Robust Optmzato Portfolo Selecto by m-mad Moel Approach [1] Koo, H., Yamazak, H. (1991), Mea-absolute Devato Portfolo Optmzato Moel a Its Applcatos to okyo Stock Market; Maagemet Scece, 37,pp,519 31; [11] Koo, H. (1999), Pecewse Lear Rsk Fucto a Portfolo Optmzato; Joural of the operatos research socety of Japa, 33,pp, ; [12] Markowtz, H. (1952), Portfolo Selecto; he Joural of Face, 7,pp, 77 91; [13] Markowtz, H. (1959), Portfolo Selecto: Effcet Dversfcato of Ivestmets; Joh Wley & Sos, New York; [14] Mas, R., Ogryczak, W., Graza Speraza, M. (23), LP Solvable Moels for Portfolo Optmzato: A Classfcato a Computatoal Comparso; IMA Joural of Maagemet Mathematcs, 14,pp,187 22; [15] Mchalowsk, W., Ogryczak, W. (21), Exteg the MAD Portfolo Optmzato Moel to Icorporate Dowse Rsk Averso; Naval research logstcs 48(3),pp, 185-2; [16] Moo, Y., Yao,.A. (211), Robust Mea Absolute Devato Moel for Portfolo Optmzato; Computers & Operatos Research, 38,pp, ; [17] Ogryczak, W., Ruszczysk, A. (1999), From Stochastc Domace to Mea-Rsk Moels: Sem Devatos as Rsk Measures; Europea oural of operatoal research, 116,pp,33-5; [18] Papahrstooulou, C., Dotzauer. E. (24), Optmal Portfolos Usg Lear Programmg Moels; Joural of the Operatoal Research Socety 55.pp ; [19] Quarata, A.G., Zaffaro, A. (28), Robust Optmzato of Cotoal Value at Rsk a Portfolo Selecto; Joural of Bakg & Face, 32,pp, ; [2] Rockafellar, R.., Uryasev, S. (2), Optmzato of Cotoal Value at Rsk; Joural of Rsk, 3,pp,21 41; [21] Soyster, A.L. (1973), Covex Programmg wth Set-clusve Costrats a Applcatos to Iexact Lear Programmg; Operatos Research, 21,pp,1154 7; [22] utucu, R., Koeg, M. (24), Robust Asset Allocato; Aals of Operatos Research, 132,pp,157 87; [23] Whtmore, G.A., Flay, M.C. (1978), Stochastc Domace; A approach to Decso-makg uer Rsk, Lexgto, MA; [24] Zeos, S.A., Kag, P. (1993), Mea-absolute Devato Portfolo Optmzato for Mortgage-backe Securtes; Aals of operatoal research, 45,pp, DOI: 14818/ /

LINEARLY CONSTRAINED MINIMIZATION BY USING NEWTON S METHOD

LINEARLY CONSTRAINED MINIMIZATION BY USING NEWTON S METHOD Jural Karya Asl Loreka Ahl Matematk Vol 8 o 205 Page 084-088 Jural Karya Asl Loreka Ahl Matematk LIEARLY COSTRAIED MIIMIZATIO BY USIG EWTO S METHOD Yosza B Dasrl, a Ismal B Moh 2 Faculty Electrocs a Computer