Revista Economică 69:5 (2017) PORTFOLIO OPTIMIZATION - APPLICATION OF SHARPE MODEL USING LAGRANGE. Vasile BRĂTIAN 1

Size: px

Start display at page:

Download "Revista Economică 69:5 (2017) PORTFOLIO OPTIMIZATION - APPLICATION OF SHARPE MODEL USING LAGRANGE. Vasile BRĂTIAN 1"

Sybil Cole
5 years ago
Views:

1 PORTFOLIO OPTIMIZATION - APPLICATION OF SHARPE MODEL USING LAGRANGE Vasle BRĂTIAN Luca Blaga Uversty of Sbu, Romaa Abstract: Ths aer resets the model develoed by Wllam Share regardg the determato of the structure of the effectve securtes ortfolo ad the alcato of ths model o the Romaa catal market. I ths resect, the ortfolo of shares used our aalyss s a ortfolo of shares of the facal vestmet comaes (SIF), lsted o the Bucharest Stock Echage (BVB), ad for determg the structure of the effcet ortfolo, there s bult ad mmzed a fucto of tye Lagrage. Also, to suort racttoers, the aer also resets a seres of mathematcal demostratos of varables used modelg. Keywords: moder ortfolo theory, Share model, lagraga JEL classfcato: C, G, G7. Itroducto Ths aer resets theoretcally ad alcably the model develoed by Wllam Share regardg the determato of the structure of the effcet securtes ortfolo. The stock ortfolo used our aalyss s a ortfolo of shares of facal vestmet comaes (SIF) lsted o the Bucharest Stock Echage (BVB), ad for determg the structure of the effcet ortfolo, there s bult ad mmzed a fucto of tye Lagrage. Also, to suort racttoers, the aer also resets a seres of mathematcal demostratos of varables used modelg. The fudametals of the moder theory of the ortfolo were ut forward by Harry Markowtz ad Wllam Share. I the sese of Markowtz, The rocess of selectg a ortfolu may be dvded to two stages. The frs Assoc. Prof. PhD,Faculty of Ecoomcs, Deartmet of Face ad Accoutg, Luca Blaga Uversty Sbu, Sbu, Romaa, e-mal: vasle.brata@ulbsbu.ro 8

2 stage starts wth belefs about the future erformaces of avalable securtes. The secod stage starts wth the relevat belefs about future erformaces ad wth the choce of ortfolu. (Markowtz, 95,.77). Share, PhD studet of Markowtz, s workg out a smlfed model of the relatoshs amog securtes, dcates the maer whch t allows the ortfolu aalyss roblem to be smlfed, ad rovdes evdece o the costs as well as the desrablty of usg the model for ractcal alcatos of the Markowtz techque. (Share, 963,.77). The structure of the effcet ortfolo, accordg to the two researchers metoed above, s characterzed by the hghest roftablty for a gve level of rsk or equvalet, the lowest rsk for a gve level of roftablty. I other words, accordg to the retur-rsk crtero, vestors cosder themselves ratoal ad ursue the lowest ortfolo rsk for a gve level of eected retur. Markowtz ostulated that a vesttor should mamze eected ortfolo retur (μ ) whle mmzg ortfolo varace of retur (σ ) Rubste (,.4). The most mortat hyothess of the ortfolo's moder theory s that the roftablty of each securtes the ortfolo s the form of a radomly dstrbuted varable, characterzed by average (eectato) ad varace (rsk). As a result, the eected retur of the ortfolo ad ortfolo rsk are all of ths kd.. Lterature revew The frst studes o the otmzato of the securtes ortfolo were carred out by Markowtz (95) ad Roy (95). Subsequetly, the works of Tob (958); Treyor (96); Share (963, 964); Lter (965); Moss (966), made substatal cotrbutos to the moder theory of the ortfolo (see Holto, 3,. 5-6). A successful work, whch the cotrbutos of may authors to the moder theory of the ortfolo are reseted, s the work of Eltoa ad Gruber (997), ad a ffty-year retrosectve of ths theory s made by Rubste (). The terest o ths toc s qute hgh amog researchers, ad the alcatos of theory, dfferet markets, are may. Here are some of these works, as follows: Koo, Yamazak (99); Ledot, Wolf (3); Huag, Qao(); Blbao, Areas, Jmeez, Gladsh, Rodrguez(6). We also meto here some artcles wrtte by Romaa authors o ths ssue, amely: Turcas, Dumter, Brezeau, Farcas, Corou (7); Aghelache, Aghel 9

3 (4); Badea, Petrescu, Stegarou, Ștefa (); Balteș, Dragoe (5); Dma, Crstrea (9); Aghelache G, Aghelache C (4); Paat, Dacoescu (); Stacu, Predescu (); Zavera (7). It should be metoed that most of the works wrtte by Romaa authors we ca see the use of the Markowtz model or other models derved from the moder theory of the ortfolo ad almost o Share model. That beg sad, we are also makg a very mortat statemet, amely that oe of the works that we cosder referece the mortace ad use of lagragea otmzato roblems s that of Fsher (4). 3. Methodology The Share model rovdes, accordg to the vestor's rsk averso degree, the structure of the effcet ortfolo (PE), e, the structure of the mmum rsk ortfolos for a eected retur to be hgher tha the eected retur of the mmum rsk ortfolo (PR m) whch s located o the so-called effcecy border. I ths model, the retur ad the rsk of the facal asset (eressed by varace) are gve by the followg formal eressos: R = α + β R m + ε, resectvely: σ = β σ m + σ ε, where: ε ~N(, σ ε ); β = cov(r,r m ). Portfolo rsk σ (R m ) (eressed by varace) s gve by the followg formal eresso: σ ε + β = σ m. To determe the structure of the effcet ortfolo, the objectve fucto (mmzg ortfolo varace value ad mlctly mmzg ortfolo volatlty) has three restrctos, resectvely: = β = β ; = μ = μ ; = =. As a result, Lagrage fucto (L) has the followg formal eresso (Brăta; Bucur; Oreaa, 6, ): L m 3 ()

4 To mmze L, the otmal codtos are: L 3 L m ( ) L ( ) L L 3 () The two above systems are equvalet because: + L = [ + ( σ ε + β σ m ) + λ ( β β ) = = ]+ = = )]= ( μ μ )+λ 3 ( = [ = σ ε ]+ ( = β β )]+ ( = μ μ )]+ 3 ( = )]= σ ε + ( = β ) ] + ( = μ ) ]+ 3 ( = )] = σ ε + λ β + λ μ + +λ 3 ; (3) + L β = β [ ( σ ε + β σ m ) + λ ( β β ) = = ]+ = ) β ( = μ μ )+λ 3 ( ]= + β σ m + λ ( )+ + = β σ m + λ ( ); (4)

5 L = [ λ λ ( σ ε + β σ m ) + λ ( = β β ) + = ]+ = = )]= λ ( μ μ )+λ 3 ( = [ λ = σ ε ]+ λ ( = β β )]+ λ ( = μ μ )]+ λ 3 ( = )] = +( = β β ) ++= = β + +β ( ) ; (5) L = [ λ λ ( σ ε + β σ m ) + λ ( = β β ) + = ]+ = = )]= λ ( μ μ )+λ 3 ( = [ λ = σ ε ]+ λ ( = β β )]+ λ ( = μ μ )]+ λ 3 ( = )] = + + ( = μ μ ) = = μ μ ; (6) L = [ λ 3 λ 3 ( σ ε + β σ m ) + λ ( = β β ) + = ]+ = = )]= λ 3 ( μ μ )+λ 3 ( = [ λ 3 = σ ε ]+ λ ( = β β )]+ 3 λ ( 3 = μ μ )]+ λ 3 ( = )] = =. (7) 3

6 3 I matr form, the last system above () s wrtte as follows: 3 m (8) 3 m (9)

7 4 The volatlty (as a measure of rsk) of effcet ortfolo retur (σ PE ) s determed as follows: m PE () Note: The effcecy border has as a etreme ot the mmum rsk ortfolo (PR m) ad reresets the total ortfolos betwee PR m ad the oe wth μ σ =. These ortfolos betwee PR m ad μ σ = are called effcet ortfolos (PE). Lagrage fucto for the mmum rsk ortfolo (PR m) has the followg formal eresso (Dragotă, (coordator),. 8-83): j j j L () To mmze L, the otmal codtos are: j j j L L ()

8 5 I the matr eresso, the above system () s wrtte as follows: (3) (4) The eected retur (µ PRm) s gve by the followg formal eresso: μ PRm = ( ) ( μ μ μ ) (5) Volatlty of retur for the mmum rsk ortfolo (σ PE ) s determed as follows: PR... m (6)

9 4. Alcato of the Share ortfolo otmzato model to the Romaa catal market Net we aly the Share ortfolo otmzato model usg the methodology descrbed above. I ths resect, the data used our aalyss are the daly closg rces of the shares of the facal vestmet comaes (SIF), lsted o the Bucharest Stock Echage ( The erod of aalyss s from 8..6 to 3..7, oe year. Also, the BET-FI de s also aalyzed the above-metoed erod ( Whereas the structure of the effcet ortfolo s lked to a gve eected retur, greater tha the eected retur of the ortfolo wth the mmum rsk, we wll frst aroach the ortfolo wth mmum rsk. After determg the roftablty of the mmum rsk ortfolo, we wll offer a hgher eected retur tha ths ad determe the structure of the effcet ortfolo usg the Share model. That beg sad, wth regard to the mmum rsk ortfolo (PR m) the followg values were obtaed: 3 4 = 5 ( λ ) E E E E E E E E E E E E 5.546E 5.96E E E 5.546E E E E 5.96E E E 5 ( ) ( ) 3 = 4 5 ( λ ) 6

10 (.469, , E 5) ( ).46% 7.6% 3 4.4% = % 5.74 ( λ ) ( 3,56649E 5) As a result, the structure of the ortfolo wth mmum rsk s the followg: SIF shares =,46%; SIF shares = 7,6%; SIF3 shares = 4,4%; SIF4 shares = 7,78%; SIF5 shares =,74%. For ths structure, the eected retur ad the mmum rsk are: μ PRm = (.46% 7.6% 4.4% 7.78%.74%),498%,763%,9% =,7346%,389% (,% ) Note: To the above eected retur corresods a yearly eected retur of 8,5%. σ PRm = (.46% 7.6% 4.4% 7.78%.74%) E E E E E 5,46% E E E E 5 7,6%.6933E E E 5.546E 5.96E 5 4,4% = ( 3.683E E E E 5.96E 5.546E E E 5 ) 3.544E 5 (,74%) 7,78% =, σ PRm = σ PRm =.597% Note: To the above volatlty corresods a yearly volatlty of 9,48%. 7

11 Regardg the structure of the effcet ortfolo (PE), usg µ PE=,9% (to ths eected retur corresods a yearly eected retur of,68%), the followg values were obtaed: = β λ λ ( λ 3 ) =,434,63, ,95 (,734, E , ,7 ) (,9% ) β λ λ ( λ 3 ) = = ( ) 8

12 (,9% ) 4.7%.4% % 4 4.5% 5 = 4.8% β,97733 λ,3 λ,454 ( λ 3 ) (,3 ) As a result, the structure of the ortfolo wth mmum rsk s the followg: SIF shares =4,7%; SIF shares =,4%; SIF3 shares = 33,6%; SIF4 shares = 4,5%; SIF5 shares = 4,8%. For ths structure, volatlty s: σ PRm = (4.7%.4% 33.6% 4.5% 4.8%,97733) (,434,63,583 4,7%,4% 33,6%,734 4,5% =,583 4,8% 3.436E 5) (,97733) =,3894 σ PRm = σ PRm =.68% Remarks:. To the above volatlty corresods a yearly volatlty of: 9,8%. 9

13 . coeffcets β from the regresso fucto are: β SIF =,855534; β SIF =,7578; β SIF3 =,573595; β SIF4 =,57497; β SIF5 =, Varaces of resdues (σ ε ) are: σ εsf =,434; σ εsf =,63; σ εsf3 =,583; σ εsf4 =,734; σ εsf5 =,583; 4. The calculatos were made Ecel. 5. Coclusos The results obtaed by us, followg the alcato of the Share model o the Romaa catal market usg Lagrage, ca be sytheszed as follows: a) µ (yearly)prm = 8,5% σ (yearly)prm = 9,48% b) µ (yearly)pe =,68% σ (yearly)pe = 9,8% Sf =,46% Sf = 7,6% Sf3 = 4,4% Sf4 = 7,78% ( Sf5 =,74%) Sf = 4,7% Sf =,4% Sf3 = 33,6% Sf4 = 4,5% ( Sf5 = 4,8%) Refereces. Aghelache, C.; Aghel, M.C, (4), The model W.F. Share model of the global regresso utlzed for the ortfolo selecto, Revsta Româă de Statstcă, Sulmet r.7;. Aghelache, G., Aghelache, C., (4), Dversfyfg the rsk through ortfolo vestmet, Theoretcal ad Aled Ecoomcs, Vol. XXI, No. 9; 3. Balteș, N.; Dragoe, A.; (5), Study Regardg The Markowtz Model of Portfolo Selecto, Revsta Ecoomcă, Vol 67, Sulmet; 4. Brăta, V., (coordator), (6); Bucur, A.; Oreaa, C., Fațe cattatve evaluarea valorlor moblare ș gestuea ortofolulu, Edtura ULBS, Sbu; 5. Blbao, A.; Areas, M.; Jmeez, M.; Perez Gladsh, B.; Rodrguez, M., (6), A eteso of Share`s sgle de model; ortfolo selecto wth eert betas, Oeratoal Research Socety, Vol. 57, No.; 6. Bodea, L.; Petrescu, M.; Stegarou, I.; Ștefa, C., (), Alcato of Markowtz Model o the Stock Market from Romaa wth the Global Facal Crss, Recet Advaces Busess

14 Admstrato, Proccedgs of the 4th WSEAS Iteratoal Coferece a Busess Admstrato, Uversty of Cambrdge, UK, avalable at htt:// 7. Dma, B.; Crstea, Ș., (9), Modelul RskMetrcs de evaluare a rsculu de ortofolu, Audt Facar, Aul VII, Nr. 5; 8. Dragotă, V., (coordator), (9); Dragotă, M.; Dăma, O.; Stoa, A.; Mtrcă, E.; Lăcătuș, C.; Maațe, D.; Țâțu, L.; Hâdoreau, C., Gestuea ortofolulu de valor moblare, ed. a -a, Edtura Ecoomcă, Bucureșt; 9. Eltoa, E.; Gruber, M., (997), Moder ortfolu theory, 95 to Date, Joural of Bakg & Face, Vol., No. -;. Fsher, M., (4), The Lagraga Relaato Method for Solvg Iteger Programmg Problems, Maagemet Scece, Vol. 5, No.;. Holto, G., (3),Value-at-rsk. Theory ad Practce, Elsever Scece, USA;. Huag, X.; Qao, L., (), A rsk de model for mult-erod ucerta ortfolo selecto, Iformato Sceces, Vol 7; 3. Koo, H.; Yamazaky, H., (99), Mea Absolute Devato Portfolo Otmzato Model ad Its Alcatos to Tokyo Stock Market, Maagemet Scece, Vol. 37, No.5; 4. Ledot, O.; Wolf, M., (3), Imroved estmato of the covarace matr of stock returs wth a alcato to ortfolo selecto, Joural of Face, Vol., No.5; 5. Lter, J., (965), The valuato of ssk assets ad the selecto of rscky vestemets stock ortofolos ad catal budgets, Rewew of Ecoomcs ad Statstcs, Vol. 47, No.; 6. Markowtz, H., (95), Portfolu Selecto, The Joural of Face, Vol. 7, No. ; 7. Moss, J., (966), Equlbrum a catal assets market, Ecoometrca, Vol. 34, No. 4; 8. Paat, I.; Dacoescu, T., (), Partculartăț ale alcăr teore modere a ortofolulu î cazul acțulor lstate la Bursa de Valor Bucureșt, MPRA Paer No.4, avalable at htts://mra.ub.u-mueche.de/4448/; 9. Roy, A., (95), Safety frst ad the holdg of assets, Ecoometrca, Vol., No.3;. Rubste, M., (), Markowts`s Portfolu Selecto : A Ffty Year Retrosectve, The Joural of Face, Vol. LVII, No.3;. Share, W., (963), A Smlfed Model for Portfolu Aalyss, Maagemet Scece, Vol.9, No.;. Share, W., (964), Catal assets rces: A theory of market equlbrum uder codtos of rsk, Joural of Face, Vol. 9, No. 3; 3. Stacu, S.; Predescu, M., (), Procesul de selecțe a uu ortofolu otm î codț de certtude e ața de catal d Româa, Stud ș Cercetăr de calcul Ecoomc ș Cberetcă Ecoomcă, Vol. 44, Nr. 3/4; 4. Tob, J.,(958), Lqudty referece as behavor towards rsk, The Revew of Ecoomc Studes, Vol. 5, No. ; 5. Treyor, J., (96), Towards a Theory of market value of rsky assets, uulshed mauscrt; 6. Turcas, F.; Dumter, F.; Brezeau, P.; Farcaș, P.; Corou, S., (7), Practcal asects of ortfolu selecto ad otmsato o the catal market, Ecoomc Research Ekoomska Istražvaja, Vol. 3, No.; 7. Zavera, I., (7),Alcato of Markowtz Model o Romaa Stock Market, Holstca, Vol. 8, No.; 8.

Multi Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions.

Multi Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions. It. Joural of Math. Aalyss, Vol. 8, 204, o. 4, 87-93 HIKARI Ltd, www.m-hkar.com http://dx.do.org/0.2988/jma.204.30252 Mult Objectve Fuzzy Ivetory Model wth Demad Depedet Ut Cost ad Lead Tme Costrats A