Portfolio Optimization using Higher Order Moments of the Stocks Returns Distribution: The Case of Bucharest Stock Exchange

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1 Internatonal Journal of Academc Research n Economcs and Management Scences Portfolo Optmzaton usng Hgher Order Moments of the Stocks Returns Dstrbuton: he Case of Bucharest Stock Exchange Mrcea BAHNA Bucharest Unversty of Economc Studes - Faculty of Fnance, Insurance, Bankng and Stock Exchange, Bucharest, Romana Emal: mrceabahna@yahoo.com DOI: /IJAREMS/v5-4/2272 URL: Abstract he Modern Portfolo heory, based on Markowtz s (1952) work, propose a portfolo selecton that consder only the frst two moments from a tme seres of returns. In spte of the popularty of Markowtz s portfolo selecton, many crtques have been emergng throughout the years. All ths crtcs are about the hypothess that Modern Portfolo heory uses n order to get the equlbrum on captal markets constran lke the absence of transactons cost and assets fnancal effcency. he am of ths paper s to use hgher return moments such as skewness and kurtoss for portfolo selecton. A seres of theoretcal papers ponted out that portfolo wth excess skewness and smaller kurtoss are preferred by ndvdual nvestors. Usng polynomal goal programmng we make a comparson of two dfferent strateges of portfolo selecton bases on Bucharest Stock Exchange quotes. Intrnsc reusable preference parameters for hgher order moments have resulted wth respect to BSE shares. Shares havng returns wth low senstvty to the market evoluton get to be the most selected ones. Keywords : Portfolo Selecton,Optmzaton, Hgher Moments Polynomal Goal Programmng. JEL: C44, C61, C63, G11 1. Introducton For a long perod of tme, the problem of selecton and portfolo management has been a very attractve one for the nvestors. Followng the Markowtz s work, the assets return s n generally assocated wth the average return and the rsk s descrbed by the varance of returns. (Markowtz, 1952) model for assets s based on some assumptons very restrctve, and all the factors ncluded n ths model are not suffcent for establshment of all the crtera that 20

2 Internatonal Journal of Academc Research n Economcs and Management Scences nfluence the nvestment decson. One hypothess of the model, the normal dstrbuton of returns s reected by most of emprcal studes on ths area, many researchers suggestng that s not possble to make a model tractable wthout consderng hgher moments of assets returns lke skewness or kurtoss. here are several reasons to beleve that nvestors take nto account hgher order of assets return dstrbuton lke skewness and kurtoss. Harry Markowtz, n hs book about portfolo theory based on the mean and varance recognzes the need to ncorporate skewness alongsde mean and varance of assets returns n order to make more accurate nvestment decsons. In a statc context, Markowtz defnes the effcent portfolo lne n whch expected return can be mproved only by exposure to a hgher rsk. hs approach, n whch the utlty functon depends only on the frst two moments of returns dstrbuton s vable only n the context Neumann-Morgenstern axom or, n other words, f returns are normally dstrbuted. Gven the falure of ths condton by certan securtes wthn a portfolo, a lot of studes come to contradct the model, showng the need to redefne the model and the man assumptons. In our paper, based on ths last suggeston, usng polynomal goal programmng, we establsh how the presence of a dstrbuton of assets returns whch s dfferent from normal dstrbuton, wll nfluence the portfolo selecton, and more exactly the weghts that an nvestor wll use n hs portfolo constructon. We wll use for ths purpose the most lqud assets from Bucharest Stock Exchange, and to optmze our portfolo we wll use ths model. In the frst secton we brefly present Markowtz portfolo and ts crtques. In the next secton we present the methodology for portfolo constructon usng hgher moments and the data base. In the fourth secton we present the emprcal results whle the last secton concludes our work n ths area. 2. Lterature Revew he concern for usng hgher moments of returns dstrbuton n fnance can be dentfed snce (Kendall&Hll, 1953), (Mandelbrot, 1963), (Cootner, 1964) and (Fama, 1965) who found a sgnfcant presence of the skewness (asymmetry) and excess kurtoss n assets returns dstrbuton. Emprcal studes on nvestors preferences for skewness can be dentfed snce the (Ardtt,1967) and (Kraus&Ltzenberger, 1976) who have dentfed nvestors preference for postve skewness. hese emprcal fndngs have led to new areas of research dedcated to the ntroducton of hgher moments n the study of the portfolo theory and asset prcng models, promoters of ths drecton beng consdered (Samuelson, 1970) and (Rubnsten, 1973). In terms of portfolo theory(samuelson, 1970), based on the work of (Marschak, 1938) regardng the decson based on trple condtonng and (Levy, 1969) research regardng cubc utlty functon was the frst on who had taken nto account the mportance of hgher moments of returns dstrbuton n order to study portfolo management. In the area of asset valuaton Rubnsten (1973) s the frst one to propose a valuaton model based on of hgher moments of returns dstrbuton. hus, extended the tradtonal CAPM model of (Sharpe,1964), (Lntner,1965) and (Mossn,1966) wth a control measure to take nto account the effects of systematc co-skewness n asset valuaton. A confrmaton of hs deas 21

3 Internatonal Journal of Academc Research n Economcs and Management Scences was made through the work of (Krau&Ltzenberger, 1976 ) whch reformulated the orgnal dea and made the frst emprcal study of tr-factors CAPM model n the US market. Lately (Ang&Chua, 1979) developed a measure of absolute rsk adusted performance based on three moments of returns dstrbuton. Durng the last 30 years a new approach regardng portfolo selecton usng hgher moments, namely Polnomal Goal Programmng (PGP) was ntroduced by (ay &Leonard, 1988). Lately, (La, 1991) used PGP n order to explore ncorporaton of nvestor s preferences n the constructon of a portfolo wth skewness. (Leung et al.,2001) used PGPn order to solve meanvarance-skewness model. In the three moments framework, (Chunhachnda et al.,1997), (Wang & Xa,2002), (Sun&Yan,2003), (Prakash et al.,2003) used PGP to construct optmal portfolos also. Despte the volume of research paper n the feld of portfolo optmzaton the usage of models takng nto account hgh order moments have only recently become popular among researchers as shown n Azm s revew paper on the subect (Azm, 2010). hus the need to focus on BSE becomes nherent, also takng nto account prevous studes regardng the effcency of Romanan captal markets, hstorcal volatlty and tradng costs from an ntra-day perspectve, performed by Dragotă et al.(2009), Cepo (2014a, 2014b), Cepo and Radu(2014) and Radu and Cepo(2015). hese studes concluded that Bucharest Stock Exchange has a week form of effcency but n comparson wth other markets tradng cost are much hgher. 3. he Modern Portfolo heory In ths secton we dscuss the model proposed by Henry Markowtz n Let s assume that we have N assets (=1,2,, N), and the perod avalable for the hstorcal prces s. he prce for the asset at the moment t wll be represented as P(t). he return of asset at the moment t s R(t) and s gven n equaton (1): ( P ( t) R ) ln ( 1) t P t 1 he expected return for the same asset s gven n equaton (2): 1 R t 2 E R t1 he varance of asset s the measure of rsk, and s represented by the equaton (3). he covarance between two assets s calculated usng equaton (4). 22

4 Internatonal Journal of Academc Research n Economcs and Management Scences cov E R ( t) E( R ) R ( t) E( R ) 3 1 R, R ER t ER R t ER R t ER R t ER 4 In ths context we can defne the expected return and rsk of a portfolo usng equaton (5) and (6), where w represents the relatve weght of asset n portfolo. I 1 E RP w ER 5 I 1 1 I, 1 t1 t P w 2w w cov R R 6 Followng Markowtz s dea, the optmzaton model s gven by the equaton (7). In ths context, an nvestor wll choose a portfolo wth the hghest expected return on a gven level of rsk, or vce versa. max E w 2 P w 0, R w ER N P 1 1 N 1 N 1 w w c w 1 I 1,2,, N 7 he optmzaton problem gven n equaton (7) s conducted consderng normal dstrbuton of asset returns. Among ths dea whch s reected by almost all the authors that have make research on t (Mandelbrot 1963, Fama-1965), other assumptons are made, n order to fnd an elegant soluton for ths model, lke the absence of transactons cost and fnancal effcency of captal markets. Many emprcal studes, most of them n exchanges from emergent markets, reect those assumptons, so, n order to fnd tractable model n portfolo selecton management, other realstc hypothess must be made. 4. Hgher Moments n Portfolo Selecton. Methodology and data base descrpton In ths study, our purpose s to establsh how the presence of a dstrbuton of assets returns whch s dfferent from normal dstrbuton, wll nfluence the portfolo selecton, and more exactly the weghts that an nvestor wll use n hs portfolo constructon. One of the man hypotheses s that nvestors on captal market are wllng to nvest small amounts of money 23

5 Internatonal Journal of Academc Research n Economcs and Management Scences compared to daly traded volume, or n other words they consder hgher moments n nvestment decson. In addton to ths, t s worthy to menton that the dea behnd ths model starts from the assumpton that we don t want to get as hgh as possble returns but we want to get as consstent as possble returns. Our optmzaton program, polynomal goal programmng, s a routne that allow fndng a soluton usng hgher moments lke skewness and kurtoss n our analyss. Our obectve functons are presented n system (7): Mean M Varance V Skewness S Kurtoss K x x X X M VX 3 x EX M M 4 x EX M M 7 In system (7), M s return dstrbuton, M s ther mean, X=(x1, x2,..., xn ) s the vector of weghts that each asset has n portfolo, whle V, S and K are varance-covarance, skewnesscoskewness and kurtoss-cokurtoss matrx of M. Our goal s to obtan a maxmum value for expected return and skewness of our portfolo whle varance and kurtoss have mnmum values. For ths purpose we wll follow two steps. he frst one s to optmze every moment gven a certan restrcton (S1), and the second one s to obtan a vector of weghts usng result from the frst step (S2). he frst step s summarzed n system (8), where I s a vector of ones: MaxMean M x X M MnVarance V x X VX 3 MaxSkewness Sx EX M M 8 4 MnKurtoss Kx EX M M Wth restrctons : X I 1 X 0 When we run each equaton n system (8) accordng to ths two restrcton we wll fnd optmal values for mean, varance, skewness and kurtoss (M*, V*, S*, K*). o combne those four optmzaton problems nto one, accordng to polynomal goal programmng, we need to defne the varables d1, d2, d3, and d4; those varables are gong to quantfy the devatons of mean, varance, skewness and kurtoss (M, V, S and K) from the optmal values (M*, V*, S*, K*). 24

6 Internatonal Journal of Academc Research n Economcs and Management Scences he fnal step can be summarzed n system (9) and wll return the optmal vector of weghts for or optmzaton model: d d d d Mn Z * * * * M V S K R1: X M d1 M * * R2: X VX d2 V 3 * R3: EX M M S 9 4 * R4: EX M M V R5: X I 1 R6: X 0 R7 : d 0, 1,,4 We ve extracted closng prces for all shares lsted at BSE n standard and premum categores and ran multple selecton crtera: market captalsaton, lqudty, return/rsk, stocks dversfcaton wth respect to each ndustry. In terms of software package we ve used FrontlneSolvers optmzaton software. In the next secton we wll present the results. 5. Results As an emprcal evdence for the proposed effcent selecton model we analyzed the Romanan Stock Market between 2009 and We plan here to compare the results of 4 moment optmal space wth 2 moment asset allocaton and to try to explan each result. he perod we have covered ranges between March 2nd 2009 and June 11th 2014, takng nto account daly observatons gven by stock prces. We only take nto account stock market shares daly closng prces, we restrct the possblty of short sellng and don t gve the nvestors the possblty to use the rskless asset. Usng lqudty, market captalzaton, mean return, ndustry dversty we settle for 20 shares from the standard and premum categores of the Bucharest Stock Exchange: LV, BRD, SNP, GN, EL, BIO, BRK, DAFR, SIF1, SIF2, SIF3, SIF4, SIF5, BM, AL, ALU, RRC, SCD, EBS and CMP. As stated before, we frst determne each ndvdual goal, n terms of optmum value for each of the 4 moments of the return (M*, V*, S* and K* respectvely), then we use these values n determnng the cumulate goal by determnng the value of the aggregate functon that takes nto account all 4 moments followed by the computaton of each range of values for preference parameters that wll yeld dfferent results n accordance wth nvestors preferences. Computng the optmal values for each of the 4 moments of the return, as expressed n equaton system (8) we get the followng set of results each determned by a portfolo of weghts for each of the 20 selected shares. 25

7 Internatonal Journal of Academc Research n Economcs and Management Scences M* V* S* K* able-optmal-results-each-of-the-4-momentsof-the-return We then follow the second set of optmzaton, frst determnng Z when the (lambda) parameters don t have any nfluence (.e. equal to1) and we get Z equal to Another very mportant step n determnng and analyzng nvestors preferences s solvng the optmzaton for extreme values of the parameters. Keepng 3 parameters equal to 1 (no preference) we get the values for hgh, medum and low preferences for each of the 4 moments: we compute the value for whch each momentum gets equal or very close to ts optmum (M*, V*, S*, K*), we compute the value for each lambda when the mprovement versus the pont where Z=5.85 s vsble (.e. >= 0.01) and set the medum preference parameters at equal dstance between these 2 values. For more consstent preference parameters we ve added nsde (9) an extra constrant makng the d's not beng too spread 1 apart: Varance(d1;d2;d3;d4) less than Below we aggregate the results for (lambda) M V S K Low Medum Hgh able-moments-preferences-(lambda) Unlke prevous studes of (La, 2006), (Aracoglu, 2010), (Geseckw, 2010), (Kemalbay, 2011), (Škrnarć, 2013), ust to name a few, where MVSK parameters only take values of 1, 2 and 3 for low, medum and hgh respectvely, we ve tred to lnk preference parameters to ntrnsc characterstcs 2 of the Romanan market, thus resultng n the above values 3. In a smlar computaton for returns for the same shares, we ve reached to smlar results/values 1 In a smlar effort on a 13 shares portfolo (AL, AZO, BIO, BRD, BRK, COMI, DAFR, SIF2, SIF5, SNP, BM, EL and GN) for march2009-december2010 we ntally got huge dscrepances between the lambda values for lower order vs hgher order moments of the dstrbutons:.e. a hgh preference parameter of 2000 for mean versus a medum lambda of 2 for skewness. 2 Hedorn,.; Kaser, D.G.; Muschol, A. (2007) Portfolooptmerung mt Hedgefonds unter Berückschtgung höherer Momente der Vertelung (No. 77). Workng paper seres//frankfurt School of Fnance & Management. 26

8 Internatonal Journal of Academc Research n Economcs and Management Scences for the preference parameters wth moments of order 3 and 4 more senstve to the exponental lambda used for expressng nvestors preferences. Our results are consstent wth Daves and Kat (2004, 2009) work, preference parameters for the frst two moments havng much bgger values then the ones fore skewness and kurtoss. An absolute method to elmnate the possblty of selectng local optmal ponts hasn't been ntroduced nsde ths methodology only partally guaranteeng the global optmzaton, however, smlar to (Brec, 2011), relyng on multple startng postons and mproved software optmsaton routne our results guarantee an ncreased accuracy. Usng the low-mean-hgh values for the preferences parameters (lambda) we then follow up on the weghts that each share should have n the portfolo and we propose a possble nterpretaton of the results. 27

9 Internatonal Journal of Academc Research n Economcs and Management Scences M Medum Hgh Low Hgh Low * Hgh Hgh V Medum Hgh Low Low Hgh * Medum Hgh S Medum Low Hgh Hgh Low * Low Hgh K Medum Low Hgh Low Hgh * Low Hgh LV BRD SNP 28% 25% 3% 2% 15% 1% 6% 13% GN 6% 5% 8% EL 4% 14% BIO 4% 10% 91% BRK 6% DAFR SIF1 SIF2 5% SIF3 SIF4 1% SIF5 11% 2% 9% BM 3% 4% 10% AL 18% 2% 42% 39% 20% 39% 3% 30% ALU 5% RRC 12% 6% 53% 23% 19% 47% 22% 16% 32% SCD 29% 21% 40% 8% 21% 2% 11% 19% 6% EBS 5% 9% 12% 6% 1% 12% 18% CMP 1% 17% 2% 1% 2% 100% 22% M V S K able-optmzaton-portfolos-results As long as one obectve ncreases n preference sze at least one of the other three drops n the nvestors nterest, and as the computaton shows ts d ncreases. LV, BRD, BRK, DAFR, SIF3 28

10 Internatonal Journal of Academc Research n Economcs and Management Scences and SIF4 are beng left out; we can conclude these shares have medocre value when t comes to all four dstrbuton parameters; t can be observed that these shares dsplay the hghest values for the market senstvty (Beta >=1.2). CMP although preferred for ts superor return wth a total portfolo allocaton for the case where mean n the absolute crtera fals to mpress when superor moments are also taken nto account, the share proporton not exceedng 2%. On the opposte sde of the selecton demand, SNP, AL, RRC, SCD and EBS are the most preferred shares by the nvestors; RRC and SCD are selected n a sgnfcant proporton even on opposte preference portfolos (hgh mean-skewness vs hgh varance-kurtoss; cases 2-3 and 4-5); these shares whch appear to be the most preferred seem to have nelastc market senstvty wth beta to as low as 0.51 for SCD. CMP, SIF5 and SIF2, although preferred n the modern portfolo selecton (Markowtz, Sharpe, reynor), fal to be selected when hgher order parameters are taken nto account for the portfolo optmsaton: CMP s only present for ts superor return whle the other two barely make t nsde the presented portfolos. A hgh competton between optmsed values of skewness and kurtoss has been exhbted n our resultng portfolos, manly when the preference parameters are at least medum. 6. Conclusons Growng fears of extreme rsks on the markets and the request to have consstent returns vs speculatve ones make the need to use hgher orders of return nevtable. Capturng harmonzed n sze preference parameters usng dsperson condton for dstanceto-optmum parameters we manage to better explan the way preferences for the dfferent moments mpact the portfolo selecton. We manage to compute reusable preference weghts for all 4 moments of the shares returns makng nvestors on BSE more easly dynamcally manage ther portfolos usng skewness and kurtoss among ther crtera of optmzaton. Shares whch at frst sght capture nvestor s attenton when only usng modern portfolo optmzaton models fal to perform, not beng selected when skewness and kurtoss preferences are taken nto account: n our proposed tme nterval selecton, shares whch are less senstve to market fluctuatons tend to be preferred even n opposte ends portfolos (hgh vs low preference for certan parameters). Further efforts need to be made to address the problem of substtutablty computaton for hgh order moments of the stocks returns n order to further generalse the provded results. References 1) Ang, J.; J. Chua (1979), Composte Measures for the Evaluaton of Investment Performance, Journal of Fnancal Quanttatve Analyss 14, pp ; 2) Aracoglu, B.; Demrcan, F. (2010) Mean Varance Skewness Kurtoss Approach to Portfolo Optmzaton: An Applcaton n Istanbul Stock Exchange, Ege Academc Revew 11.Specal Edton:

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