Financial Risk Management in Portfolio Optimization with Lower Partial Moment

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1 Amercan Journal of Busness and Socety Vol., o., 26, pp Fnancal Rsk Management n Portfolo Optmzaton wth Lower Partal Moment Lam Weng Sew, 2, *, Lam Weng Hoe, 2 Department of Physcal and Mathematcal Scence, Faculty of Scence, Unverst unku Abdul Rahman, Kampar Campus, Kampar, Perak, Malaysa 2 Centre for Mathematcal Scences, Centre for Busness and Management, Unverst unku Abdul Rahman, Kampar Campus, Kampar, Perak, Malaysa Abstract he mean-lower partal moment model has been proposed n portfolo optmzaton to mnmze the portfolo rsk. hs model employs mean as the return measure and lower partal moment as the rsk measure. Lower partal moment s a downsde rsk measure n portfolo optmzaton. he nvestors wll be able to mnmze the portfolo rsk n ther nvestment by usng the mean-lower partal moment model. he objectve of ths paper s to construct the optmal portfolo usng the mean-lower partal moment model wth the degree set as, 2 and 3. he data of ths study comprses weekly return of 2 component stocks of FSE Bursa Malaysa Kuala Lumpur Composte Index (FBMKLCI) n Malaysa stock market. In partcular, the results of ths study show that MAXIS s the largest component stock n the optmal portfolo of the mean-lower partal moment model wth degree, 2 and 3. Besdes that, the optmal portfolo of the mean-lower partal moment model wth degree 2 gves the portfolo mean return at. and portfolo rsk at.. hs study s sgnfcant because the nvestor can acheve the target rate of return wth mnmum portfolo rsk. Keywords Mean Return, Rsk, Sem Varance, Optmal Portfolo, Portfolo Performance Receved: July 7, 26 / Accepted: August, 26 / Publshed onlne: September 3, 26 he Authors. Publshed by Amercan Insttute of Scence. hs Open Access artcle s under the CC BY lcense. Introducton Portfolo optmzaton s an nvestment strategy n asset selecton and allocaton []. Portfolo optmzaton s a mathematcal approach n makng nvestment decsons of varous assets to be ncluded n the portfolo. Portfolo s a groupng of fnancal assets such as stocks that are held by the nvestors [2]. Investors desre to acheve the target rate of return at the mnmum portfolo rsk. Fshburn [3] has ntroduced the mean-lower partal moment model n portfolo optmzaton. he mean-lower partal moment model s a mathematcal model n portfolo optmzaton that used to mnmze the portfolo rsk and can acheve the nvestors target rate of return. he portfolo rsk s represented by the lower partal moment. he lower partal moment s the downsde rsk that defnes rsk n terms of probabltyweghted functons of devatons below the target return. he mean-lower partal moment model has been studed by the past researchers n portfolo optmzaton [-9]. he lower partal moment also has been used as rsk measure n handlng system desgn []. he objectve of ths paper s to construct the optmal portfolo usng the mean-lower partal moment model based on the components from FSE Bursa Malaysa Kuala Lumpur Composte Index (FBMKLCI) n Malaysa stock market. he rest of the paper s organzed as follow. he next secton descrbes the materals and methods employed n ths study. Secton 3 dscusses about the emprcal results of ths study. Secton concludes the paper. * Correspondng author E-mal address: lamws@utar.edu.my (L. W. Sew), whlam@utar.edu.my (L. W. Hoe)

2 2 Lam Weng Sew and Lam Weng Hoe: Fnancal Rsk Management n Portfolo Optmzaton wth Lower Partal Moment 2. Materals and Methods 2.. Data Fgure shows the constructon process of the optmal portfolo wth mean lower partal moment model. he data of ths study comprses weekly return of 2 component stocks of FSE Bursa Malaysa Kuala Lumpur Composte Index (FBMKLCI) n Malaysa stock market. FBMKLCI s the man ndcator of economy n Malaysa. he perod of ths study covers from ovember 29 untl December 2. able shows the component stocks of FBMKLCI n Malaysa stock market. able. Component Stocks of FBMKLCI n Malaysa Stock Market. Stocks AMMB AXIAA BA CIMB DIGI GEIG BHD GEIG MALAYSIA HLB BHD IOI KLK MALAYA BAK MAXIS PEROAS DAGAGA BHD PEROAS GAS BHD PUBLIC BAK BHD PPB GROUP BHD RHB ELEKOM EAGA UMW An optmal portfolo s constructed by usng the mean-lower partal moment model. he optmal portfolo composton for each stock wll be generated. he summary statstcs of the optmal portfolo wll also be calculated. he return of the stocks s determned as below []. R t P, t P, t = () P, t Rt s the return of stock at tme t, P, t s the closng prce of stock at tme t P, t s the closng prce of stock at tme t-. he mean return of the stock s calculated as below [2]. r = R (2) t = r s the mean return of stock, Rt s the return of stock at tme t, s the number of observatons t Fgure. Constructon Process of the Optmal Portfolo wth Mean Lower Partal Moment Model. Based on the determnaton of the optmal portfolo composton usng the lower partal moment model, the mean return of the optmal portfolo s formulated as follow [3]. r p n = rx (3) r p s the mean return of the optmal portfolo, r s the mean return of stock, s the amount of the fund nvested n stock, n s the number of stocks Mean-Lower Partal Moment Model Fshburn [3] has defned the lower partal moment of order α around τ as follows: τ α LPMα( τ ; R) ( τ R) df( R) = E{(max[, τ R]) α } () F( R) s the cumulatve dstrbuton functon of the nvestment return R, τ s the benchmark return, α s the order or degree of the lower partal moment he mathematcal model of mean-lower partal moment

3 Amercan Journal of Busness and Socety Vol., o., 26, pp model [3] s formulated as follows: Subject to Mnmze α (max[, τ Rpt]) (5) t= E( R ) = µ (6) = (7) s the number of perods, x () R Pt s the return of the optmal portfolo at tme t, R s the return of stock, s the weght of each stock n the optmal portfolo, s the total number of stocks, µ s the target rate of return. Equaton (5) s the objectve functon of the model whch mnmzes the portfolo lower partal moment. Constrant (6) ndcates that the portfolo mean return equals to the nvestors expected rate of return. Constrant (7) ndcates that the total weghts of all stocks n the optmal portfolo are one. Constrant () ndcates that the weght of each stock n the optmal portfolo s postve. τ and α wll be set as the expected return and 2 respectvely n ths study. Besdes that, the optmal portfolo of mean-lower partal moment wll also be generated n ths study wth the degree α set as and 3. Settng α as 2 and τ as the expected return wll produce the mean-sem varance model. he mean-sem varance model s formulated as follows: Subject to 2 Mnmze (max[, E( R) Rpt]) (9) t = E( R ) = µ () = () s the number of perods, x (2) E( R) s the expected return, R Pt s the return of the optmal portfolo at tme t, R s the return of stock, s the weght of each stock n the optmal portfolo, s the total number of stocks, µ s the target rate of return. Equaton (9) s the objectve functon of the model whch mnmzes the portfolo sem varance subject to the constrants ()-(2). he mean-lower partal moment model has been used to overcome the dsadvantage of Markowtz [] mean-varance model. Markowtz [] was the poneer of portfolo optmzaton by ntroducng the mean-varance model. Varance s used as rsk measure whle the mean return s used as the expected return n the mean-varance model. he objectve functon of the mean-varance model s to mnmze the portfolo varance whch s the portfolo rsk. he lower partal moment s the more approprate rsk measure because the mean-varance model wll not only penalze the downsde devaton but also the upsde devaton whch s desrable for the nvestors. 3. Emprcal Results able 2 presents the optmal portfolo composton of the mean-lower partal moment model wth the degree α set as (MLPM), 2 (MLPM2) and 3 (MLPM3) n percentage. Stocks able 2. Optmal Portfolo Composton (%). MLPM MLPM2 MLPM3 AMMB AXIAA... BA CIMB DIGI..5.6 GEIG BHD... GEIG MALAYSIA... HLB BHD... IOI KLK.7.. MALAYA BAK.92.. MAXIS PEROAS DAGAGA BHD... PEROAS GAS BHD... PUBLIC BAK BHD PPB GROUP BHD RHB... ELEKOM... EAGA... UMW...

4 23 Lam Weng Sew and Lam Weng Hoe: Fnancal Rsk Management n Portfolo Optmzaton wth Lower Partal Moment Based on able 2, the optmal portfolo of the mean-lower partal moment model wth degree conssts of AMMB (.77%), BA (3.95%), CIMB (.6%), IOI (6.35%), KLK (.7%), MALAYA BAK (.92%), MAXIS (6.5%), PUBLIC BAK BHD (2.67%) and PPB GROUP BHD (6.5%). AXIAA, DIGI, GEIG BHD, GEIG MALAYSIA, HLB BHD, PEROAS DAGAGA BHD, PEROAS GAS BHD, RHB, ELEKOM, EAGA and UMW are not selected to be ncluded n the optmal portfolo of the mean-lower partal moment model wth degree because these stocks gve the value.%. MAXIS s the largest component stock because t comprses 6.5% n the optmal portfolo. In constrast, CIMB s the smallest component stock n the optmal portfolo of mean-lower partal moment model wth the smallest percentage at.6%. Moreover, the optmal portfolo of the mean-lower partal moment model wth degree 2 conssts of AMMB (5.9%), BA (.%), CIMB (6.%), DIGI (.5%), IOI (2.55%), MAXIS (53.5%), PUBLIC BAK BHD (3.%) and PPB GROUP BHD (.99%). AXIAA, GEIG BHD, GEIG MALAYSIA, HLB BHD, KLK, MALAYA BAK, PEROAS DAGAGA BHD, PEROAS GAS BHD, RHB, ELEKOM, EAGA and UMW are not selected to be ncluded n the optmal portfolo of the meanlower partal moment model wth degree 2 because these stocks gve the value.%. MAXIS s the largest component stock because t comprses 53.5% n the optmal portfolo. On the other hand, DIGI s the smallest component stock n the optmal portfolo of mean-lower partal moment model wth the smallest percentage at.5%. In addton, the optmal portfolo of the mean-lower partal moment model wth degree 3 conssts of AMMB (.6%), BA (9.3%), CIMB (.6%), DIGI (.6%), IOI (.%), MAXIS (55.26%) and PUBLIC BAK BHD (9.25%). AXIAA, GEIG BHD, GEIG MALAYSIA, HLB BHD, KLK, MALAYA BAK, PEROAS DAGAGA BHD, PEROAS GAS BHD, PPB GROUP BHD, RHB, ELEKOM, EAGA and UMW are not selected to be ncluded n the optmal portfolo of the meanlower partal moment model wth degree 3 because these stocks gve the value.%. MAXIS s the largest component stock because t comprses 55.26% n the optmal portfolo. In constrast, DIGI s the smallest component stock n the optmal portfolo of mean-lower partal moment model wth the smallest percentage at.6%. hese results ndcate that dfferent degree of the mean-lower partal moment model wll gve dfferent optmal portfolo composton. able 3 presents the summary statstcs for the optmal portfolo of the mean-lower partal moment model wth the degree α set as, 2 and 3. able 3. Summary Statstcs for the Optmal Portfolo Performance of the Mean-Lower Partal Moment Model. Optmal Portfolo MLPM MLPM2 MLPM3 Portfolo Mean Return... Portfolo Rsk...2 Portfolo Skewness Portfolo Kurtoss As reported n able 3, the optmal portfolo of the meanlower partal moment model wth degree gves the portfolo mean return at. and portfolo rsk at.. he optmal portfolo skewness as well as kurtoss value s -. and.972 respectvely. Besdes that, the optmal portfolo of the mean-lower partal moment model wth degree 2 gves the portfolo mean return at. and portfolo rsk at.. hs mples that the nvestors can acheve the portfolo mean return at. wth the mnmum rsk of.. he optmal portfolo skewness as well as kurtoss value s -.3 and.79 respectvely. Furthermore, the optmal portfolo of the mean-lower partal moment model wth degree 3 gves the portfolo mean return at. and portfolo rsk at.2. he optmal portfolo skewness as well as kurtoss value s -.97 and.792 respectvely. hese results ndcate that dfferent degree of the mean-lower partal moment model wll also gve dfferent optmal portfolo rsk, portfolo skewness and portfolo kurtoss. Fgure 2, 3 and dsplay the hstogram for the optmal portfolo return dstrbuton of the mean-lower partal moment model Fgure 2. Hstogram for the Optmal Dstrbuton of the Mean-Lower Partal Moment Model wth Degree.

5 Amercan Journal of Busness and Socety Vol., o., 26, pp Fgure 3. Hstogram for the Optmal Dstrbuton of the Mean-Lower Partal Moment Model wth Degree Fgure. Hstogram for the Optmal Dstrbuton of the Mean-Lower Partal Moment Model wth Degree 3. As dsplayed n Fgure 2, 3 and, the dstrbuton of the portfolo return of the mean-lower partal moment model shows postve and negatve return. Besdes that, the dstrbuton for the optmal portfolo return of the meanlower partal moment model exhbts skewness.. Concluson hs paper dscusses the applcaton of the mean-lower partal moment model n Malaysa stock market. he meanlower partal moment model s the portfolo optmzaton model that appled to mnmze the portfolo rsk. he portfolo rsk s represented by the lower partal moment. he optmal portfolo s constructed by usng the mean-lower partal moment model wth the degree set as, 2 and 3 n ths study. he results of ths study show that dfferent degree of the mean-lower partal moment model wll gve dfferent optmal portfolo composton. Furthermore, dfferent degree of the mean-lower partal moment model wll also gve dfferent optmal portfolo rsk, portfolo skewness and portfolo kurtoss. hs study s sgnfcant because the nvestors can acheve the target rate of return at mnmum level of rsk n ther nvestment. he future research of ths study should be extended to the assets n other countres. References [] Gtman, L. J., Joehnk, M. D. and Smart, L. J. (2). Fundamentals of Investng. th Edton, Pearson. [2] Relly, F. K. and Brown, K. C. (22). Investment Analyss and Portfolo Management. th Edton, Mason, South Western Cengage Learnng. [3] Fshburn, P. C. (977). Mean-rsk analyss wth rsk assocated wth below-target returns. he Amercan Economc Revew, 67: [] Grootveld, H. and Hallerbach, W. (999). Varance vs downsde rsk: Is there really that much dfference?. European Journal of Operatonal Research, : [5] Harlow, W. V. (99). Asset allocaton n a downsde rsk framework. Fnancal Analysts Journal, 7: 2-. [6] awrock, D.. (99). Optmal algorthms and lower partal moment: ex post results. Appled Economcs, 23: [7] Samet, G. (25). Measurng the fnancal rsk level n emergng and developed markets: tradtonal and alternatve methods. Asan Socal Scence, : [] Sng,. F. and Ong, S. E. (2). Asset allocaton n a downsde rsk framework. Journal of real estate portfolo management, 6: [9] Usman, A., Syed, Z. A. S. and Qasar, A. (25). Robust analyss for downsde rsk n portfolo management for a volatle stock market. Economc Modellng, : [] Pratk, M., Marc, G. and Edward, H. (25). Robust materal handlng system desgn wth standard devaton, varance and downsde rsk as rsk measures. Int. J. Producton Economcs, 7: 5-2. [] Lam, W. S., and Lam, W. H. (2). Selecton of moble telecommuncatons companes n portfolo optmzaton wth mean-varance model. Amercan Journal of Moble Systems, Applcatons and Servces, :9-23. [2] Bode, Z., Kane, A. and Marcus, A. J. (2). Investments. 7th Edton, ew York, McGraw-Hll. [3] Jones, C. P. (2). Investments Prncples and Concepts. th Edton, John Wley & Sons. [] Markowtz, H. (952). Portfolo selecton. Journal of Fnance, 7: 77-9.