Leyla M. Mamedova Ph.D., Associate Professor, Baku State University/Department of Mathematical Economics / Azerbaijan, Baku city

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1 Iteratioal Joural of Latest Egieerig ad Maageet Research (IJLEMR) Volue 03 - Issue 04 April 2018 PP Coparative Aalyses of the Ivestet Portfolio o the Basis of a Multi-Criterial Optiizatio Model i the Stock Market with Liear Covolutio Method Leyla M. Maedova Ph.D., Associate Professor, Baku State Uiversity/Departet of Matheatical Ecooics / Azerbaija, Baku city Abstract: I this paper, a ethod for reducig the uber of criteria for the ulti objective optiizatio proble is proposed.the result is two really coflictig criteria i which the iproveet of ay of the ievitably leads to the deterioratio of others.the Markowitz odel, odified by the additio of two criteria, reductio of their oe-criterio optiizatio proble by eas of a liear covolutio of the criteria is cosidered.the article aalyzes the stock arket papers of the Aerica stock arket i real period. Key words: liear covolutio, ulti-objective optiizatio Markowitz odel, efficiet frotier, Lagrage ethod, covariace, portfolio yield, portfolio of securities Figure 1. Efficiet frotier The fud arket fors a echais for attractig ivestets to the ecooy, buildig relatioships betwee those who eed additioal fiacial resources ad those who wat to ivest surplus icoe.portfolio ivestet allows you to pla, evaluate, ad oitor the fial outcoes of all ivestet activities i various sectors of the stock arket. Optiizatio of the structure of the securities portfolio is oe of the ost iportat tasks of akig decisios i ivestig i the stock arket.the purpose of securities portfolio optiizatio is the foratio of a portfolio of securities that would satisfy the requireets of the ivestor, the eterprise, both i ters of profitability ad possible risk, which is achieved through the distributio of securities i the portfolio.i geeral, portfolio optiizatio cocers ot oly the foratio of a portfolio of ivestet projects, a loa portfolio, etc.the core of portfolio optiizatio is to select fro a set of alterative objects the subset that, withi a give period, will brig the optial portfolio to the portfolio ower, that is, the best outcoe.criteria of optiizatio ca be several; tedecies of their iproveet ca cotradict each other.the optial result i differet issues is uderstood as either the axiu profit or the specified profit level uder the iiu risk, possibly takig ito accout additioal outer costraits ad the prefereces of the decisio-aker. Each ivestor seeks to create such a portfolio of securities, which would provide the axiu possible icoe with iial risk.there are two probles: how to forecast reveue based o statistical data ad how to easure risk. I the classical forulatio of Markowitz, the proble of choosig the optial portfolio is reduced to the theory of a effective set of portfolios, or the socalled effective boudary.the essece of the theory is that if there are securities available to the ivestor, each with its expected retur E( r i ), where,2,.,, the there is oe cobiatio of securities i the portfolio that iiizes the portfolio risk at each set value expected retur o the portfolio.fig. 1 shows that whatever the value of the expected retur is deteried by the ivestor (for exaple, E(r )), always by scalig the securities we ca fid a portfolio i which the risk level reaches a iiu value (i Figure 1 - poit B) [1]. The expected yield of a security i the Markowitz odel is calculated as the atheatical expectatio of its returs for the previous period of tie, the risk is the stadard deviatio of these yields, ad the covariace 87 Page

2 Iteratioal Joural of Latest Egieerig ad Maageet Research (IJLEMR) Volue 03 - Issue 04 April 2018 PP is give by the forula σ=v ij σ i σ j, where V ij is the coefficiet of the pairwise liear correlatio betwee the yields of the two assets [2]. The ivestor's task i Markowitz's odel boils dow to the followig: fro a set of portfolios with the expected rate of retur E(r p ), oe ust fid oe that would esure a iiu level of risk. I other words, the ivestor's task ca be reduced to solvig the followig syste: j =1 θ i θ j b ij i, b ij = cov R i, R j, θ i = 1 i θ i = p, θ 1 0,., θ 0 (1) where p the value of the portfolio efficiecy selected by the ivestor; θ i share of the i -th security i the portfolio; i ea of effectiveess of R i i-th security We pass fro the oo objective odel of Markowitz to the odel of ulticriteria optiizatio, that is, o our cases to the odel two-criterio optiizatio[3] [4]: j =1 θ i θ j b ij i, b ij = cov R i, R j, i θ i ax θ = θ 1,, θ, θ i = 1 (2) Here we will apply the ethod of liear covolutio for ulti-objective portfolio optiizatio.fro the odel with two criteria (2) by usig the ethod of liear covolutio, oe ca pass to a odel with oe criterio.the siplest ad ost frequetly used ethod for reducig the ulticriteria proble to siglecriterio is liear covolutio.weighted oegative coefficiets α i are desigated, deotig the iportace of each criterio, ad the liear cobiatio of objective fuctios [5] [6] is axiized, i.e. the proble is solved: g x = x X i f i x i 0, i = 1,,, i = 1 This task ivolves cobiig the criteria fro the above proble by costructig a liear cobiatio f i x, i = 1,2,, (costructig a weighted su of partial criteria) ad passig to a sigle-objective proble: i f i x i x X i = cost > 0, i = 1,2,.,, i = 1 Where i are deteried by experts. However, this approach of deteriig i, based o the subjective opiio of experts, ultiately leads to the fact that the solutio of proble (2), (3) will be largely subjective.i (3) 88 Page

3 Iteratioal Joural of Latest Egieerig ad Maageet Research (IJLEMR) Volue 03 - Issue 04 April 2018 PP this paragraph, aother way of deteriigof i, i = 1,2,,. First we assue that all the criteria i (1) are ot raked.i this case, the followig ethod of covolutio of the criteriaf i x fro (1). Let there be give poitsx (1), x (2),, x r X.Let's calculate the values y i (k) = f i x k, i = 1,2,,, k = 1,, r, (4) We costruct a liear cobiatio: y 1,,, x k = 1 f 1 x k + 2 f 2 x k + + f x k, k = 1,.., r, (5) Here it is proposed to choose oliear prograig probles: (y 1, 2,,, x 1 y i 1 ) 2 + (y 1, 2,,, x 2 y i 2 ) (y 1, 2,,, x r y i r ) 2 i = 1 i 0, i = 1,,. i (6) 1, 2,, For its uerical solutio, you ca use various tools, for exaple, a office applicatio of Excel spreadsheets. Now let all the criteriaf i x, i = 1,2,,,raked as follows: f 1 x = f 2 x = = f x (7) где f p x = f p+1 x, p = 1,, 1, (7) eas that the criteriof p x is ot less preferable tha the criteriof p+1 x. However, the degree of preferece of f p x forf p+1 x is t arked. I such case, obviously i, i = 1,,,ust satisfy the additioal coditio 1 2 (8). The the proble of approxiate calculatio of i, i = 1,2,,,i the case of their rakig accordig to (7) with the solutio [7] [8], reduces to the solutio of the optiizatio proble of(4)-(6), (8). For solvig the proble of optiizatio of the ivestet portfolio, a holistic review of all idicators of the portfolio should be ade.i a holistic view, it ust be take ito accout that axiizig the values of soe idicators ca be accopaied by iiizig the values of others.particular criteria for ulti-criteria optiizatio of the ivestet portfolio are: Maxiizatio of the predicted retur o the securities portfolio; Miiizig the risk of the fored portfolio; Takig ito accout the above two goals, our task is reduced to two-criterio optiizatio. After deteriig the approxiate values of 1, 2, the quadratic prograig proble is solved [9]: 1 ( j =1 θ i θ j b ij ) 2 ( i θ i ) i i, j = 1,.,, b ij = cov(r i, R J ) θ i = 1 θ 1 0,, θ 0 1 0, 2 0, = 1 Let us give a exaple of the portfolio proble of Markowitz with shares of the Aerica stock arket at the begiig of 2016: Chevro, Walt Disey, Caterpillar, AT&TиAdobe Syste. I calculatig the (9) 89 Page

4 Iteratioal Joural of Latest Egieerig ad Maageet Research (IJLEMR) Volue 03 - Issue 04 April 2018 PP expected retur of the portfolio we will use real data reflectig the value of the idices for the period 01/01/ /01/2018 (106 tradig weeks) [10]: The yield of each security ca be calculated accordig to the rules: R t = P t P t 1 100% (yield as a percetage of the ivested aout). P t 1 Here P t is the price of the security i period t. The average yield E(r i ) is defied as the arithetic ea of historical returs for 106 weeks.next, we fid the variaces ad stadard deviatios of these idices. As a result, we obtai 5-diesioal vectors: r= {0.449, 0.141, 0.956, 0.153, 0.765} σ 2 = {4.463, 5.316, 9.896, 5.495, 9.078} σ= {2.112, 2.306, 3.146, 2.344, 3.013} Let's ake the covariace atrix of these shares: First Solar Walt Disey Caterpillar AT&T Adobe Syst First Solar Walt Disey Caterpillar AT&T Adobe Syste Table 1. Covariace Usig the ethod of deteriig ofα 1, α 2, described aboveθ 1 = 2,23, θ 2 = 0,60, θ 3 = 0,67, θ 4 = 0,31, θ 1 = 0,35we fidα 1 = 0.5; α 2 = 0.5. We defie statioary poits. Let us fid the extree of the fuctio [14]: F(θ) = 0.5 (4.463 θ θ θ θ θ θ 1 θ θ 1 θ θ 1 θ θ 1 θ θ 2 θ θ 2 θ θ 2 θ θ 3 θ 4 + ( 0.106) θ 3 θ 5 + ( 1.085) θ 4 θ 5 ) 0.5 (0.449 θ θ θ θ θ 5 ) We rewrite the restrictio of the proble i a iplicit for: φ 1 θ = 1 θ 1 + θ 2 + θ 3 + θ 4 + θ 5 = 0 Let us copose the auxiliary Lagrage fuctio: L θ, λ, μ = 0.5 (4.463 θ θ θ θ θ θ 1 θ θ 1 θ θ 1 θ θ 1 θ θ 2 θ θ 2 θ θ 2 θ θ 3 θ 4 + ( 0.106) θ 3 θ 5 + ( 1.085) θ 4 θ 5 ) 0.5 (0.449 θ θ θ θ θ 5 ) + λ 1 1 θ 1 + θ 2 + θ 3 + θ 4 + θ 5 Differetiatig the fuctio, we forulate the syste of equatios: 4,463 θ 1 + 0,929 θ 2 + 1,094 θ 3 + 1,036 θ 4 0,425 θ 5 λ 1 0,225 = 0 0,929 θ 1 + 5,316 θ 2 + 2,435 θ 3 + 1,516 θ 4 + 1,991 θ 5 λ 1 0,071 = 0 1,094 θ 1 + 2,435 θ 2 + 9,896 θ 3 + 1,317 θ θ 5 λ 1 0,478 = 0 1,036 θ 1 + 1,516 θ 2 + 1,317 θ 3 + 5,495 θ 4 0,542 θ 5 λ 1 0,077 = 0 0,425 θ 1 + 1,991 θ θ 3 0,542 θ 4 + 9,078 θ 5 λ 1 0,383 = 0 1 θ 1 + θ 2 + θ 3 + θ 4 + θ 5 = 0 Solvig the syste of equatios by the iverse atrix ethod, we fially obtai: θ (1) =(0.3475, , , , ), λ 1 = This poit satisfies all coditios.the fuctio value:f(θ)= Page

5 Iteratioal Joural of Latest Egieerig ad Maageet Research (IJLEMR) Volue 03 - Issue 04 April 2018 PP θ i r i = I the portfolio, the decisio aker will obtai the followig cobiatio of shares: θ 1 = 34,75 θ 2 = 4,31% θ 3 = 13,68% θ 4 = 22,78% θ 4 = 24,48% Ad the profitability of the etire portfolior p = 51.54% Copared with the previous decisio, give the subjective decisios of the decisio aker, you ca specify other cobiatios of the criteria ad, solvig the probles, we get the followig results: If 1 = 0,75 2 = 0,25the: θ (1) =(0.3424, , , , ), λ 1 = This poit satisfies all coditios. The fuctio value:f(θ)= θ i r i = I the portfolio, the decisio aker will obtai the followig cobiatio of shares: θ 1 = 34.24% θ 2 = 7.95% θ 3 = 10.98% θ 4 = 24.21% θ 4 = 22.62% Ad the profitability of the etire portfolior p = 48.03% Let s exaie the situatio with 1 = 0,4 2 = 0,6. The we will obtai the followig result: θ (1) =(0.3512, , 0.157, 0.217, ), λ 1 = This poit satisfies all coditios. The fuctio value:f(θ)=0, θ i r i = I the portfolio, the decisio aker will obtai the followig cobiatio of shares: θ 1 = 35,12% θ 2 = 1,60% θ 3 = 15.70% θ 4 = 21,70% θ 4 = 25,88% Ad the profitability of the etire portfolior p = 54.16% 5 Suary As ca be see fro the three cases whe the decisio aker akes a ratioal decisio (i.e., gives ore preferrig to risk criteria), the portfolio returs less. With aggressive choice (preferece for high yield), the profitability of the etire portfolio is icreased. Refereces: [1]. Fabozzi J. Frak, Ivestet Maageet: Tras. with Eglish. - Moscow: INFRA, (Series "Uiversity textbook"). p [2]. Burei A. N: Securities portfolio aageet. M., Scietific ad Techical Society aed after acadeicia S.I. Vavilov, 2008 [3]. Seechi EA, Deiseko A.O. About a ethod of covolutio of criteria i ulticriteria probles ad its applicatio i solvig probles of optiizatio of securities portfolios., Scietific article., Scietific joural "Fudaetal Research" pages: [4]. Seechi E. A, Deiseko AO, Multicriteria atheatical odels of decisio-akig o the securities arket i coditios of ucertaity., Scietific joural / Kuba State Uiversity, 64 (10), Page

6 Iteratioal Joural of Latest Egieerig ad Maageet Research (IJLEMR) Volue 03 - Issue 04 April 2018 PP [5]. Schwarz DT Iteractive ethods for solvig the ulticriteria optiizatio proble. Overview. Scietific article: "Sciece ad Educatio", electroic scietific ad techical joural of the MSTU. N.E. Baua., Pp [6]. Shapki A. S, Shapki V. A: Matheatical ethods ad odels of operatios research. Textbook. 6th editio., Moscow 2016., p: [7]. Orujov E.G., Matheatical Ecooics. Baku State Uiversity. Matheatical-Ecooics departet. Textbook. Baku, "Khazar Uiversity" publishig house. 2016, pp [8]. Maita L.A., Usability of optiality i fiite - diesioal oliear task optiizatio. M., p Tutorial., Moscow State Istitute of Electroics ad Matheatics. M., p [9]. Lotov, AV Pospelova, I. I, Multicriteria proble of decisio akig., Textbook., Moscow State Uiversity. Moscow State Uiversity, Faculty of Coputatioal Matheatics ad Cyberetics.Moscow 2008., p [10]. = &dt=31&t=11&yt=2016&to= &p=10&f=MCD_160101_161231&e=.csv&c =MCD&dtf=4&tf=1&MSOR=1&stie=o&stiever=1&sep=1&sep2=4&datf=2&at=1&fsp=1 ( ) [11]. Fedoseev AA, Modificatio of the Markowitz odel by takig ito accout additioal characteristics of securities.,izvestiya Tula State Uiversity; Natural scieces P [12]. Zicheko A.S. Bolkvadze I.R. Vuchkov YA, Applicatio of the ethod of liear covolutio of criteria for optiizatio of fiacial support of the orgaizatio's activities., Scietific article: "Fiacial Maageet" joural. p: [13]. Pisaruk NN Ivestigatio of operatios. BSU. Misk , p [14]. Maedova L.M., KaziovSh.E. About a proble of optial ivestet of the stock arket., Caspia Joural of Applied Matheatics, Ecology ad Ecooics. V.5. No Page

ELEMENTARY PORTFOLIO MATHEMATICS

QRMC06 9/7/0 4:44 PM Page 03 CHAPTER SIX ELEMENTARY PORTFOLIO MATHEMATICS 6. AN INTRODUCTION TO PORTFOLIO ANALYSIS (Backgroud readig: sectios 5. ad 5.5) A ivestor s portfolio is the set of all her ivestets.