MATHEMATICAL MODELLING OF RISK IN PORTFOLIO OPTIMIZATION WITH MEAN- EXTENDED GINI APPROACH

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1 SCIREA Joural of Mathematcs December 21, 2016 Volume 1, Issue 2, December 2016 MATHEMATICAL MODELLING OF RISK IN PORTFOLIO OPTIMIZATION WITH MEAN- EXTENDED GINI APPROACH Lam Weg Hoe 1,2,*, Lam Weg Sew 1,2 1 Departmet of Physcal ad Mathematcal Scece, Faculty of Scece, Uverst Tuku Abdul Rahma, Kampar, Perak, Malaysa 2 Cetre for Mathematcal Sceces, Cetre for Busess ad Maagemet, Uverst Tuku Abdul Rahma, Kampar, Perak, Malaysa *Correspodg E-mal: whlam@utar.edu.my Abstract Ivestors wsh to mmze the rsk ad acheve the target rate of retur ther vestmet. The mea-exteded G model has bee proposed portfolo optmzato to mmze the portfolo rsk. The mea s used as the expected retur of the vestors ad exteded G s used as rsk measure. The objectve fucto of ths model s to mmze the portfolo exteded G. The objectve of ths paper s to costruct the optmal portfolo by employg the mea-exteded G model Malaysa stock market. The data of ths study cossts of weekly retur of 20 stocks that lsted Malaysa stock market. The mea-exteded G model s solved wth LINGO software ths study. The results of ths study show that the composto of stocks the optmal portfolo s dfferet. Furthermore, the mea-exteded G optmal portfolo wll gve the mea retur at ad portfolo rsk at Ths study s sgfcat because the vestors ca mmze 190

2 the portfolo rsk ad acheve the target rate of retur Malaysa stock market wth the mea-exteded G model. Keywords: Mea Retur, Rsk, Optmal Portfolo, Portfolo Composto, LINGO Software 1. Itroducto Ivestors wsh to fd the trade-off betwee the rsk ad retur ther vestmet of assets. The vestors desre to mmze the vestmet rsk ad maxmze ther retur. Shalt ad Ytzhak [1] has troduced the exteded G as a measure of rsk. The meaexteded G (MEG) approach has bee studed by the past researchers [2-5]. The meaexteded G model s a optmzato model that used to costruct the optmal portfolo. The objectve of ths paper s to costruct the optmal portfolo by employg the meaexteded G model Malaysa stock market. The rest of the paper s structured as follows. The ext secto descrbes the materals ad methods appled ths study. Secto 3 dscusses about the emprcal results of ths study. Secto 4 cocludes the paper. 2. Materal ad Methods 2.1 Data The data of ths study comprses weekly retur of 20 stocks that lsted Malaysa stock market. The perod of ths study covers from July 2011 utl Jue Table 1 shows the ame lst of 20 stocks ths study wth abbrevato. Table 1: Name Lst of 20 Stocks wth Abbrevato Abbrevato Name of Stocks ASUPREM AZRB BENALEC BPURI Astral Supreme Berhad Ahmad Zak Resources Berhad Bealec Holdgs Berhad Ba Pur Holdgs Bhd 191

3 CRESBLD EKOVEST FAJAR GADANG GAMUDA HSL IJM JAKS KEURO KIMLUN MITRA MUDAJYA MUHIBAH PRTASCO PUNCAK WCT Crest Bulder Holdgs Berhad Ekovest Berhad Fajarbaru Bulder Group Bhd Gadag Holdgs Bhd Gamuda Berhad Hock Seg Lee Berhad IJM Corporato Berhad Jaks Resources Berhad Kumpula Europlus Berhad Kmlu Corporato Berhad Mtrajaya Holdgs Berhad Mudajaya Group Berhad Muhbbah Egeerg (M) Bhd Protasco Berhad Pucak Naga Holdgs Berhad WCT Holdgs Berhad 2.2 Mea-Exteded G Model Fgure 1 dsplays the costructo process of the optmal portfolo wth mea-exteded G model. Fgure 1. Costructo Process of the Optmal Portfolo wth Mea-Exteded G Model 192

4 The mathematcal model of mea-exteded G (MEG) model s formulated as follows: 1 Mmze z w cov{ x,[1 Fp ( p)] } (1) Subject to 1 1 E( p) w E( ) (2) 1 x w 1 (3) w 0, 1,..., (4) where s a parameter determg the relatve weght attrbuted to varous portos of the probablty dstrbuto, F p ( p) s the cumulatve probablty dstrbuto of the portfolo returs p, w s the weght vested asset, x s the retur of asset. Objectve fucto (1) mmzes the portfolo exteded G. Costrat (2) mples that the vestors ca acheve the expected rate of retur. Costrat (3) mples that the sum of weghts of the assets equals to oe. Costrat (4) mples that the weghts of all the assets are postve. The optmal portfolo s costructed by employg the mea-exteded G model wth the parameter, s set as 4 ths study. Besdes that, the optmal portfolo composto for each stock wll be geerated ths study. The mea retur ad rsk of the optmal portfolo wll also be preseted ths paper. The mea-exteded G model s solved wth LINGO software ths study. 3. Results ad Dscussos Table 2 presets the stock selecto of the mea-exteded G (MEG) model. Table 2: Stock Selecto wth Mea-Exteded G Model Stocks Weghts (%) ASUPREM 0.96 AZRB 0.00 BENALEC

5 BPURI 0.00 CRESBLD 6.29 EKOVEST FAJAR 0.00 GADANG 6.63 GAMUDA HSL IJM 9.73 JAKS 0.00 KEURO 6.55 KIMLUN 0.00 MITRA 0.00 MUDAJYA 0.00 MUHIBAH 0.00 PRTASCO PUNCAK 0.00 WCT 0.00 As show Table 2, those stocks wth postve values dcate that they are selected by the MEG model. The compoets of the optmal portfolo are ASUPREM, BENALEC, CRESBLD, EKOVEST, GADANG, GAMUDA, HSL, IJM, KEURO ad PRTASCO. AZRB, BPURI, FAJAR, JAKS, KIMLUN, MITRA, MUDAJYA, MUHIBAH, PUNCAK ad WCT are ot vested the optmal portfolo of the mea-exteded G model because these stocks gve the value 0.00%. Fgure 2 dsplays the optmal portfolo composto of MEG model. 194

6 Fgure 2: Optmal Portfolo Composto of MEG Model As show Fgure 2, based o the vestmet fud, the optmal portfolo cossts of GAMUDA (30.52%), EKOVEST (13.88%), HSL (13.14%), PRTASCO (11.44%), IJM (9.73%), GADANG (6.63%), KEURO (6.55%), CRESBLD (6.29%), ASUPREM (0.96%) ad BENALEC (0.87%). GAMUDA s the most domat stock the optmal portfolo whereas BENALEC s the smallest compoet the optmal portfolo. Table 3 presets the portfolo mea retur ad rsk of the mea-exteded G model. Table 3: Optmal Portfolo Mea Retur ad Rsk of the Mea-Exteded G Model Optmal Portfolo MEG Portfolo Mea Retur Portfolo Rsk As reported Table 3, the optmal portfolo of the mea-exteded G model gves the portfolo mea retur at ad portfolo rsk at It dcates that the vestors ca acheve the target retur at wth the mmum rsk at

7 4. Coclusos Ths paper dscusses the mathematcal modellg of rsk for the vestmet Malaysa stock market wth the mea-exteded G model. The optmal portfolo s costructed by employg the mea-exteded G model ths study. The results of ths study dcate that the mea-exteded G model geerates dfferet composto of stocks the optmal portfolo. Furthermore, the vestors ca get the target rate of retur wth mmum rsk Malaysa stock market. The future research of ths study should be exteded to the stocks other coutres besdes Malaysa. Refereces [1] Shalt, H. ad Ytzhak, S., Mea-G, portfolo theory, ad the prcg of rsky assets. Joural of Face, 39: [2] Butterworth, D. ad Holmes, P., The hedgg effectveess of U.K. stock dex futures cotracts usg a Exteded Mea G approach: Evdece for the FTSE 100 ad FTSE md250 cotracts. Multatoal Face Joural, 9(3-4): [3] Shalt, H. ad Greeberg, D., Hedgg wth stock dex optos: A Mea- Exteded G approach. Joural of Mathematcal Face, 3: [4] Shalt, H. ad Ytzhak, S., Evaluatg the Mea-G approach to portfolo selecto. The Iteratoal Joural of Face, 1(2): [5] Shalt, H. ad Ytzhak, S., The Mea-G effcet froter. The Joural of Facal Research, 28: