Portfolio selection models based on characteristics of return distributions

Transcription

1 Workig Papers No. 14/2013 (99) PAWEŁ WNUK LIPINSKI Portfolio selectio models based o characteristics of retur distributios Warsaw 2013

2 Portfolio selectio models based o characteristics of retur distributios PAWEŁ WNUK LIPINSKI Faculty of Ecoomic Scieces, Uiversity of Warsaw wuklipiski.pawel@gmail.com [eabstract This article cocers the problem of optimal portfolio selectio. The objective of this paper is to idicate the best method ad criteria for optimal portfolio selectio. I order to achieve the objective six models icludig such optimizatio criteria as mea, variace, skewess, kurtosis ad trasactio costs are aalyzed. The method of fuzzy multi-objective programmig is used to trasform multiple coflictig criteria ito a sigle objective problem ad to fid optimal portfolios. I order to idicate the best portfolio selectio model a simulatio based o five years data from Jauary 1, 2007 to December 31, 2011 was coducted. The portfolios were costructed from WIG20 stocks ad WIBID 3M as risk-free asset. Keywords: optimal portfolio, portfolio selectio, fuzzy multi-objective programmig, skewess, kurtosis JEL: G11, C61 Ackowledgtmets: The origial versio of the paper was prepared as a master thesis. The author would like to thak his supervisors, Dr Paweł Sakowski ad Dr Robert Ślepaczuk, for their valuable support ad assistace durig the research. Isightful commets provided by the participats of the semiar Modellig ad Forecastig Returs ad Volatility o Capital Markets are also gratefully ackowledged. Workig Papers cotai prelimiary research results. Please cosider this whe citig the paper. Please cotact the authors to give commets or to obtai revised versio. Ay mistakes ad the views expressed herei are solely those of the authors.

3 1. Itroductio The appropriate choice of securities to the portfolio is a importat issue for the asset maagemet o fiacial markets. Sice the foudatio of the Warsaw Stock Exchage i 1991 we ca observe a dyamic developmet of the Polish fiacial market 1. Durig the last te years savigs of Polish citizes more tha doubled 2 ad the capitalizatio of the WSE icreased fourfold 3. Thus, the society is becomig richer ad people face the problem of choosig the best way to ivest their moey. The choice of securities to the portfolio ca be doe with the applicatio of portfolio selectio models. The use of these models ca help to aswer the questio i which ad how may securities of each kid to ivest. I this paper several portfolio selectio models are aalysed, which iclude such criteria as average rate of retur, the risk quatified as variace ad kurtosis, trasactio costs ad skewess, which is a measure of distributio asymmetry. I this study the followig two research hypotheses are set up: Portfolio optimizatio models allow obtaiig better tha average results. Additio of higher momets to the stadard mea-variace model improves the results. Moreover, attempts are made to aswer the followig research questios: Does the additio of risk-free rate to portfolio of stocks improve performace of the models? Does costraiig trasactio costs i the optimizatio process traslate ito better results? I order to verify the hypotheses ad aswer the research questios a empirical study was coducted. Six models icludig criteria based o portfolio distributio of returs are compared. The models are compared o the basis of a simulatio that was performed for a five years period from Jauary 1, 2007 to December 31, The portfolios are composed of stocks from WIG20 idex ad WIBID3M ad are recostructed each quarter. The method of fuzzy multi-objective programmig is used to trasform multiple coflictig criteria ito a sigle-objective problem ad to fid the optimal portfolio. The remaider of the paper is structured as follows. After itroductio we come to the literature review i the secod sectio. I the third sectio the methodology ad data of the research are described, as well as the assumptios of the study are formulated. The fourth sectio is devoted to the mai results of the simulatio. I this sectio the problems of icorporatig trasactio costs as optimizatio costrait ad of addig a risk-free asset to the portfolio are aalysed. I the fifth sectio the results of a broad sesitivity aalysis are preseted. The ifluece of boudary costraits to a positio i oe security ad of chage i the momet ad frequecy of portfolio recostructio o the results is aalysed. Subsequetly, the impact of trasactio costs o models performace is preseted. The paper eds with the idicatio of the optimal models characteristics ad coclusios. 1 access July 20, W_bakach_mamy_prawie.html, access July 20, access July 20,

4 2. Literature review The questio of how to optimally select assets to a portfolio has its log history. It started i 1952 whe Harry Markowitz published his groud-breakig paper etitled Portfolio Selectio. This paper was a milestoe i the developmet of moder portfolio theory ad its practical applicatios. Markowitz assumes that the ivestor cosiders expected retur as a desirable thig ad risk as a udesirable thig. Retur is quatified as the mea ad risk as the variace of the rates of returs of securities. The ivestors are assumed to look for a balace betwee the maximizatio of the rate of retur ad miimizatio of risk of their ivestmet decisios (Markowitz, 1952). The itroductio of mathematical represetatio of risk ad retur made possible the applicatio of optimizatio tools i the portfolio maagemet problems. Sice the ivestors dislike risk, they will choose a portfolio that maximizes retur, give a fixed level of risk. Otherwise, the ivestors will choose a portfolio with a miimum risk, give a fixed level of retur. This relatioship is called the mea-variace efficiecy ad a portfolio that fulfils this coditio is called a efficiet portfolio. I other words, if a portfolio is iefficiet the there exists aother portfolio with either smaller variace ad o smaller mea, or with larger mea ad o larger variace. The set of all efficiet portfolios forms the efficiet frotier. Sice the ivestors may have differet prefereces toward required risk ad retur, their optimal portfolios might be differet, but their choices must be o the efficiet frotier (Wag ad Xia, 2002). Sice the semial paper of Markowitz (1952), i which the mea-variace model was proposed, umerous studies cocerig portfolio theory were published. The Markowitz model gaied widespread acceptace as a tool for portfolio selectio ad revolutioized the way people thik about the choice of assets to portfolio. However, there is still a debate amog researchers whether higher momets tha variace should be icluded i the process of portfolio selectio. May researchers (e.g. Arditti, 1971; Samuelso, 1970; Koo ad Suzuki, 1995; Jea, 1971; Scott ad Horvath, 1980) claim that portfolio aalysis should be exteded to higher momets uless the ivestors utility fuctio is quadratic ad the assets returs are ormally distributed. Moreover, Scott ad Horvath (1980) show that for ivestor that is cosistet i the directio of preferece of momets, the preferece directio is positive for positive values of each odd cetral momet ad egative for each eve cetral momet. It meas that ivestors should prefer high skewess ad low kurtosis. Cotrary to the assumptios of the mea-variace model, there is ample evidece (Arditti, 1971; Tag ad Shum, 2003; Chuhachida et al., 1997; Prakash et al., 2003) showig that securities ad portfolio returs are ot ormally distributed. The asymmetry of the distributio is measured by the third momet skewess. Whe skewess is positive it idicates that the right tail of the distributio is log i compariso to the left tail. For the ivestmet portfolio it traslates ito high but rare gais ad small but more frequet losses, assumig that mea is ear zero. Geerally, ivestors would prefer a portfolio with higher third momet, whe the first ad the secod momets are the same. Furthermore, they ca eve prefer a portfolio with a larger skewess at the expese of larger variace ad smaller mea (Koo et al., 1993). Research papers cocerig the applicatio of higher momets i portfolio selectio (e. g. Koo ad Suzuki, 1995; Lai, 1991; Chuhachida, 1997; Ryoo, 2007; Prakash et al., 2003) cocetrate maily o the first three momets i.e. the mea-variace-skewess framework. Empirical studies suggest that the icorporatio of skewess ito a ivestor s portfolio might result i a improved optimal portfolio (Ryoo, 2007; Joro ad Na, 2006). The fourth momet, kurtosis, is ofte omitted by researchers, though it is also importat for portfolio selectio, especially whe the retur distributio is o-ormal. Kurtosis reflects the probability of extreme evets. The 2

5 higher is the kurtosis, the larger is the probability of extreme evets. Therefore it seems to be reasoable to take ito accout kurtosis i the portfolio selectio process, which was applied i e. g. Lai et al. (2006) or Yu ad Lee (2011). As ivestors might cosider differet factors tha mea ad variace, accordig to their prefereces, the choice of a flexible ad comprehesive portfolio selectio method is very importat. Polyomial goal programmig (Lai et al., 2006; Chuhachida et al., 1997; Lai, 1991; Prakash et al., 2003) ad fuzzy multi-objective programmig (Zimmerma, 1978; Lee ad Li, 1993; Yu ad Lee, 2011) are methods that allow selectig optimal portfolio i a multi-objective framework. These methods allow the decisio maker to choose portfolio selectio criteria ad the to fid optimal portfolio that best suits their prefereces. I this paper the applicatio of the fuzzy multi-objective programmig i portfolio selectio is aalysed. Although may studies were coducted i the field of moder portfolio theory, the questio which model ad which criteria should be icluded i the portfolio selectio is still ope to debate. The appearace of more ad more powerful computers ad more efficiet algorithms for solvig complex optimizatio problems suggests that portfolio theory will develop i the directio of more advaced multi-objective models. This paper aims to add value to this strad of literature ad aalyse the ratioale of applyig multi-objective models i portfolio selectio. 3. Methodology ad data 3.1 Data descriptio Data used i the study cover the period from Jue 30, 2006 to February 14, All the aalysed time series are daily closig prices. All the data were dowloaded from the website of Polish data provider Stocks of compaies icluded i the WIG20 idex are aalysed i the research. For a give poit of time oly the stocks of compaies that were at this time i WIG20 idex could have bee chose. This meas that whe the WIG20 costituets chaged also the stocks that were aalysed chaged. Therefore for a give poit of time always 20 stocks were aalysed. Such a approach allowed avoidig survivorship bias. Additioally, the stock prices were adjusted by splits ad divideds to better reflect real ivestmet value. Table 1. Costituets of WIG20 idex o Jue 30, 2006 PKN ORLEN TP S.A. PEKAO S.A. KGHM PKO BP BANK BPH AGORA BZ WBK PGNIG PROKOM NETIA TVN LOTOS BRE MOL GTC KĘTY BORYSZEW BIOTON ORBIS * Table 1 presets costituets of WIG20 idex o Jue 30, The iformatio was obtaied from the website of the Warsaw Stock Exchage, I Table 1 the costituets of WIG20 idex at the begiig of the aalysed period are preseted, whereas Table 2 shows the compositio of the WIG20 idex at the last day of the aalysed period. Durig the time of more tha 5 years 50% of the idex was subject to chage, which is a sigificat value. 3

6 Table 2. Costituets of WIG20 idex o February 14, 2012 PKN ORLEN TP S.A. PEKAO S.A. KGHM PKO BP GETIN PGE BANK HANDLOWY PGNIG ASSECOPOL PBG TVN LOTOS BRE BOGDANKA GTC PZU KERNEL TAURON CEZ * Table 2 presets costituets of WIG20 idex o February 14, The iformatio was obtaied from the website of the Warsaw Stock Exchage, Additioally, the data of WIG idex, Alliaz Akcji FIO mutual fud ad Alliaz OFE pesio fud for the same period are used. They serve as bechmarks to the models results. These particular fuds are chose, because they were amog the best fuds i the aalysed period i their categories Alliaz Akcji FIO amog mutual fuds ad Alliaz OFE amog pesio fuds. As a approximatio of risk-free rate WIBID3M 3 moth Warsaw Iterbak Bid Rate is applied. 3.2 Model specificatio I the research six multi-objective models, which iclude criteria of retur, variace, skewess, kurtosis ad trasactio costs, are aalysed. The aforemetioed criteria are based maily o the characteristics of the distributios of the portfolio compoets returs. Apart from that trasactio costs are subject to aalysis sice it is supposed that the icorporatio of trasactio costs i the optimizatio process might improve the results. The aalysed multiobjective models are the M (mea), the MV (mea, variace), the MS (mea, skewess), the MVS (mea, variace, skewess), the MVK (mea, variace, kurtosis) ad the MVSK (mea, variace, skewess, kurtosis). The mathematical represetatio of the models, which is preseted i the remaider of this subsectio, is similar to the oe show i paper by Yu ad Lee (2011) with some ecessary modificatios. The M model cosists of two objectives, the maximizatio of portfolio retur ad the miimizatio of trasactio costs, as it is show by equatios (3.1) ad (3.2). The detailed represetatio of the M model is as follows: Max i=1 r i w i, (3.1) Mi i=1 p(l i + s i ), (3.2) s.t. i=1 (w i + pl i + ps i ) = 1, (3.3) 0 w i 0,2, (3.4) for i=1,,, where is the umber of available securities; ri is the retur o security i; wi is the weight of the security i i the portfolio; p is the trasactio cost expressed i percetages; li is the ratio of the value of i securities bought by the ivestor to the portfolio value; si is the ratio of the value of i securities sold by the ivestor to the portfolio value. Costrait (3.2) represets the trasactio costs that are paid by each portfolio recostructio. The costs are a liear fuctio of the ratio of bought ad sold securities value to 4

7 the portfolio value. Thus, the miimizatio of trasactio costs is equivalet to miimizatio of portfolio turover. Costrait (3.3) shows the available budget allocated to ivestmet i securities ad trasactio costs of buyig ad sellig. Obviously, the product of li ad si (lisi) is equal to zero sice a give security is either bought or sold. Costrait (3.4) represets the lower ad the upper bouds of the total positio i each security. This costrait idicates that short sellig is ot allowed i the model. Such a assumptio is set up because o the Warsaw Stock Exchage short sellig is allowed oly from July 1, Therefore the research would be urealistic whe i the aalysed period the possibility of sellig securities short would exist. Additioally the maximum share of a sigle security i the portfolio is assumed to be 20% so as the model does ot cocetrate too much o few securities. This assumptio is a subject to chage i the sesitivity aalysis (subsectio 5.1). The MV is a triple objective model cotaiig all the criteria of the M model ad additioally a objective of variace miimizatio (3.5). The specificatio of the model is as follows: Max i=1 r i w i, (3.1) Mi i=1 p(l i + s i ), (3.2) Mi i=1 j=1 w i w j σ ij, (3.5) s. t. Costraits (3.3) ad (3.4), where σ ij is the covariace betwee security i ad security j. Whe i=j the σ ij represets the variace of the security. The MS is a triple objective model icludig the criteria of the M model additioally with the objective of skewess maximizatio (3.6). The model is represeted as follows: Max i=1 r i w i, (3.1) Mi i=1 p(l i + s i ), (3.2) Max E ( wt 3 (r r ) ), σ s. t. Costraits (3.3) ad (3.4), (3.6) where E is the expected value operator; superscript T is the trasposig operator; w = (w 1,, w ) is the vector of portfolio weights; r = (r 1,, r ) T is the vector of securities returs; r = (r 1,, r ) T is the vector of expected returs; σ is the stadard deviatio of the portfolio returs. The MVS is a model with four objectives, which icludes all the criteria of MV model additioally with the objective of skewess maximizatio. The specificatio of the MVS is as follows: (3.1) Max i=1 r i w i, Mi i=1 p(l i + s i ), (3.2) 4 access May 26,

8 Mi i=1 j=1 w i w j σ ij, (3.5) Max E ( wt 3 (r r ) ), σ s. t. Costraits (3.3) ad (3.4), (3.6) The MVK is a model with four objectives, icludig all the criteria of MV model ad additioally a criterio of kurtosis miimizatio (3.7). The MVK is represeted as follows: Max i=1 r i w i, (3.1) Mi i=1 p(l i + s i ), (3.2) Mi i=1 j=1 w i w j σ ij, (3.5) Mi E ( wt 4 (r r ) ), σ s. t. Costraits (3.3) ad (3.4), The MVSK is the most complex model with five objectives of retur maximizatio (3.1), costs miimizatio (3.2), variace miimizatio (3.5), skewess maximizatio (3.6) ad kurtosis miimizatio (3.7). The MVSK is represeted as follows: Max i=1 r i w i, (3.1) Mi i=1 p(l i + s i ), (3.2) Mi i=1 j=1 w i w j σ ij, (3.5) (3.7) Max E ( wt (r r ) σ 3 ) Mi E ( wt (r r ) ), σ s. t. Costraits (3.3) ad (3.4), 4, (3.6) (3.7) The above described multi-criteria models are solved by the method of fuzzy multiobjective programmig. This approach preseted i papers by Zimmerma (1978), Lee ad Li (1993) ad Yu ad Lee (2011) allows to trasfer a multi-objective model ito a sigle-objective model. The fuzzy multi-objective programmig uses the cocept of fuzzy sets ad the aspiratio level λ. At first the program forces each objective to achieve its aspiratio level, ad the it provides a trade-off betwee coflictig criteria. The method requires ideal ad ati-ideal values of variables to be provided i advace. The reformulatio of a multi-objective model ito a sigle-objective model by the use of fuzzy multi-objective programmig is preseted usig the MV model as a example: Max λ s. t. λ r r a r i r a, (3.8) 6

9 λ σ σ a σ i σ a, λ c c a r =, c i c a i=1 r i w i, (3.9) (3.10) (3.11) i=1 j=1, σ = w i w j σ ij (3.12) c = i=1 p(l i + s i ), (3.13) i=1 (w i + pl i + ps i ) = 1, (3.3) 0 w i 0,2, for i=1,,, where r* is the retur of the portfolio; ra is the ati-ideal retur of the portfolio; ri is the ideal retur of the portfolio; σ* is the risk of the portfolio measured as stadard deviatio; σa is the ati-ideal risk of the portfolio; σi is the ideal risk of the portfolio; c* is the trasactio cost of the portfolio; ca is the ati-ideal trasactio cost; ci is the ideal trasactio cost. Costraits (3.8)-(3.10) respectively preset the goals of simultaeously maximizig retur, miimizig risk ad miimizig trasactio costs of the portfolio. Here the ideal ad ati-ideal values of the criteria must be provided. I this research it is assumed that the ideal ad ati-ideal values for mea, variace, skewess ad kurtosis of the portfolio are calculated as the best ad worst historical observatios take from all the aalysed securities at every rebalacig. The ideal value of trasactio costs is 0 ad the ati-ideal is 0,8%, which is 0,4% multiplied by 2, whereas 0,4% is the most commo value of trasactio costs charged by Polish brokerage houses i case of idividual ivestors 5. Thus i the research the 0,4% trasactio cost was assumed as the most appropriate oe. I this case the ati-ideal value of 0,8% meas that at rebalacig the whole compositio of the portfolio is chaged. The other five models are trasformed ito sigleobjective models through fuzzy multi-objective programmig i a similar way that was show for the MV model. 3.3 Research assumptios The mai part of the research is a simulatio of a ivestmet which is based o the models preseted i subsectio 3.2. The ivestmet period is assumed to be 5 years, from Jauary 1, 2007 to December 31, The iitial ivestmet value is 1 millio Polish Zloty ad the portfolio ca cosist of 20 biggest Polish stocks ad a risk-free rate. I the mai part of the research the holdig period of the portfolio is 3 moths ad the historical data used for estimatio of models parameters is 6 moths. The maximum share of oe compay s stocks i the portfolio is set to 20%, but there is o such costrait for the risk-free rate, which meas that the maximum share of risk-free rate i the portfolio is 100%. I the sesitivity aalysis the maximum share is chaged to 10% ad 40%. Thus, o Jauary 1, 2007 a portfolio of stocks is costructed based o parameters estimated o data from July 1, 2006 to December 31, Subsequetly the (3.4) 5 The iformatio is based o the summary of Polish brokerage houses charges, foud o the website access Jue 1,

10 portfolio is held 3 moths without rebalacig. O the April 1, 2007 the portfolio is recostructed based o parameters estimated o data from October 1, 2006 to March 31, The this portfolio is held without chages till July 1, This process is cotiued till the ed of Such a simulatio is coducted for each of the six models. Additioally, differet versios of the models are compared. I the mai research: Models icludig trasactio costs i the optimizatio process are compared with models without this costrait; Models cosistig oly of WIG20 stocks are compared with models cosistig of WIG20 stocks ad risk-free rate. Whereas i the sesitivity aalysis: The maximum share of oe compay i the portfolio is chaged from 20% to 10% ad 40% for each model; The startig poit of the rebalacig is moved from the begiig of the quarter to the middle of the quarter for all the models. The frequecy of portfolio recostructio is icreased from 3 moths to 1 moth. I order to better compare the models the followig statistics were applied: Aual Retur Compouded (ARC), calculated accordig to the followig formula (Feibel, 2003): N ARC = N R t where: 252 is the assumed umber of trasactio days i a year; R t is a daily logarithmic log-retur. Aual Stadard Deviatio (ASD), expressed by the formula (Feibel, 2003): t=1 ASD = N (R t R ) 2 Retur to Risk ratio, a simple measure of retur per uit of risk (Culp, 2001): N t=1 RR = ARC ASD Average trasactio cost, calculated i the followig way: average trasactio cost = 1 NR trasactio cost t t=1 where: trasactio costt is trasactio cost at time t expressed as % of ivested capital, NR is the umber of times the portfolio is recostructed durig the simulatio period. NR 8

11 ml PLN Trasactio costs are icluded i all the models aalysed i this study. The results obtaied for each model are compared with the performace of the three bechmarks: WIG20 idex, Alliaz Akcji FIO mutual fud ad Alliaz OFE pesio fud i the same period. Additioally a equally weighted portfolio of WIG20 stocks is costructed (WIG20eq), which serves as the fourth bechmark showig a alterative of passive ivestmet i WIG20 stocks with equal weights. The equally weighted portfolio of WIG20 stocks is used istead of WIG20 idex, sice the data of stocks applied i the study is corrected by divideds which is ot the case for WIG20 idex. Thus, compariso of the models with equally weighted portfolio of WIG20 is more reliable tha with WIG20 idex. Table 3. Basic statistics for the bechmarks i the period from Jauary 1, 2007 to December 31, 2011 WIG20eq WIG Alliaz OFE Alliaz Akcji FIO ARC -9,13% -5,89% 2,34% -4,47% ASD 27,47% 24,67% 7,15% 20,96% RR -0,33-0,24 0,33-0,21 * Table 3 presets basic statistics for four bechmarks: WIG20eq, WIG, Alliaz OFE ad Alliaz Akcji FIO i the period from Jauary 1, 2007 to December 31, Figure 1. The four bechmarks i the period from Jauary 1, 2007 to December 31, ,4 1,2 1 0,8 0,6 0,4 0, WIG20eq WIG Alliaz OFE Alliaz Akcji FIO * Figure 1 presets the equity lies for four bechmarks: WIG20eq, WIG, Alliaz OFE ad Alliaz Akcji FIO i the period from Jauary 1, 2007 to December 31, The amout ivested at the begiig of the period is 1 millio Polish Zloty. I Table 3 basic statistics for all the bechmarks are preseted. I the studied period Alliaz OFE pesio fud was the best both i case of retur ad risk. It was the oly bechmark with 9

12 positive retur ad positive value of retur to risk ratio. For the other three bechmarks the retur is egative ad therefore the iterpretatio of retur to risk ratio is ambiguous. Defiitely the worst performace shows the WIG20eq with aual retur of less tha -9% ad the highest aual stadard deviatio of more tha 27%. WIG ad Alliaz Akcji FIO performed similar to each other with a slight advatage of the mutual fud. Figure 1 presets the performace of the four bechmarks i the studied period. It is clearly visible that Alliaz OFE pesio fud is characterized by differet behaviour tha the other three bechmarks durig the crisis i 2008 is suffers small losses, whereas WIG, WIG20eq ad Alliaz Akcji FIO lose about half of their value i the worst momet of the crisis. Moreover, these three bechmarks behave similarly to each other. All the calculatios preseted i the study were performed i the Microsoft Excel, versio Results I this sectio the results for all six models are described. Tables with aforemetioed statistics as well as figures are preseted i order to provide detailed view of models performace. The models icludig trasactio costs i the optimizatio process are compared with models without this costrait. Also models buildig portfolio from stocks are compared to models choosig from stocks ad risk-free rate. 4.1 Models with ad without trasactio costs as the optimizatio costrait I the model specificatio, preseted i sectio 3.2, the costrait (3.10) cocers trasactio costs. Yu ad Lee (2011) proposed the implemetatio of trasactio costs i this way as a importat improvemet to the model. I this sectio the models icludig the costrait (3.10) are compared with models for which trasactio costs are ot cosidered as oe of the optimizatio costraits. It is worth otig that i all models trasactio costs are take ito accout, but for some they are also cosidered as a optimizatio costrait. Table 4. Basic statistics for models with trasactio costs as the optimizatio costrait M MV MS MVK MVS MVSK ARC -11,61% -9,31% -14,78% -7,29% -13,01% -12,83% ASD 29,03% 28,79% 29,57% 28,08% 29,31% 29,05% average t.c. 0,21% 0,21% 0,27% 0,21% 0,27% 0,27% skewess 6-0,19-0,18-0,08-0,28-0,10-0,11 kurtosis 7 2,09 2,11 1,95 2,26 1,99 1,78 * Table 4 presets basic statistics for all the models i the period from Jauary 1, 2007 to December 31, The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value. For these models trasactio costs are icluded i the optimizatio process as a costrait. 6 Skewess is calculated accordig to the formula ( 1)( 2) (x j x s deviatio, x j are the rates of returs ad x is the average rate of returs. 7 Kurtosis is calculated accordig to the formula (+1) ( 1)( 2)( 3) (x j x s ) 3, where is the sample size, s stadard ) 4 3( 1)2 ( 2)( 3). 10

13 ml PLN Table 4 presets basic statistics for all six models icludig trasactio costs as a optimizatio costrait. Each model suffered sigificat losses i the studied period. The highest losses occurred for MS, whereas the lowest for MVK. It is worth otig that additio of skewess deteriorated results i case of every model the MS, MVS ad MVSK performed worse tha M, MV ad MVK respectively. The level of risk is similar for every model with about 29% aual stadard deviatio. The average trasactio cost was above 0,2%, which meas that o average more tha 25% of the portfolio was replaced by each rebalacig. As far as skewess is cocered all the values are egative, which is probably a cosequece of the large drawdow o the market durig the world-wide crisis i The models icludig skewess as optimizatio costrait MS, MVS, MVSK obtaied higher values of skewess tha their couterparts without this costrait M, MV, MVK. It shows that the additio of skewess allows ifluecig skewess of the portfolio distributio of returs, but the improvemet is relatively small. The value of kurtosis higher tha 0 iforms that the distributio is leptokurtic with fatter tails tha ormal distributio ad the pheomeo of leptokurtosis is preset i distributios of returs of all the models. Comparig models with kurtosis as optimizatio costrait MVK, MVSK with their couterparts without this costrait MV ad MVS it caot be stated that additio of kurtosis decreases kurtosis of portfolio returs. I case of MV model the kurtosis icreased, whereas for MVS the kurtosis decreased whe the additioal costrait for kurtosis was imposed. O Figure 2 the MV, MVK, MVS models ad WIG20eq are preseted. The equity lies of the portfolios are very similar to each other. The MVK model with the highest ARC from models i Table 4, outperforms WIG20eq, while the MVS, the model with the lowest ARC, performs worse tha WIG20eq. Based o the abovemetioed results it caot be stated that the models obtai good results or eve outperform the bechmark. I the ext part of this sectio it is checked whether the removal of trasactio costs optimizatio costrait iflueces the results. Figure 2. WIG20eq ad the MV, MVK, MVS with trasactio costs as a optimizatio costrait 1,6 1,4 1,2 1 0,8 0,6 0,4 0, MV MVK MVS WIG20eq * Figure 2 presets equity lies for MV, MVK, MVS models ad bechmark WIG20eq i the period from Jauary 1, 2007 to December 31, The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value. The amout ivested at the begiig of the period is 1 millio Polish Zloty. For these models trasactio costs are icluded i the optimizatio process as a costrait. 11

14 ml PLN Table 5 presets basic statistics for models ot icludig trasactio costs i the optimizatio process. The aual retur is egative but the results are much better tha i case of models preseted i Table 4. The volatility is about 29%, a very similar value i compariso to models from Table 4. This meas that the iclusio of trasactio costs costrait i the optimizatio process does ot have ay ifluece o the volatility of the models results. The average trasactio cost amouts to about 0,48% which is more tha two times higher as i case of models from Table 4. This is cosistet with expectatios, sice models that costrai trasactio costs obtai much lower values of costs. Similarly to the results preseted i Table 4 also for models from Table 5 the additio of skewess as a optimizatio costrait improves skewess of the portfolio rates of returs, which is visible by comparig M, MV, ad MVK with MS, MVS ad MVSK models. Also the additio of kurtosis allows decreasig its value i the portfolios distributios of returs MV ad MVS models have fatter tails tha MVK ad MVSK. Table 5. Basic statistics for models without trasactio costs as the optimizatio costrait M MV MS MVK MVS MVSK ARC -3,75% -6,50% -8,90% -4,78% -9,94% -7,66% ASD 29,07% 28,78% 28,95% 28,21% 28,78% 28,11% average t. c. 0,46% 0,48% 0,49% 0,48% 0,48% 0,49% skewess -0,13-0,14-0,04-0,14-0,04 0,00 kurtosis 2,06 2,26 2,18 2,08 2,17 2,00 * Table 5 presets basic statistics for all the models i the period from Jauary 1, 2007 to December 31, The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value. For these models trasactio costs are ot icluded i the optimizatio process as a costrait. Figure 3. WIG20eq ad the M, MVS without trasactio costs as optimizatio costrait 1,6 1,4 1,2 1 0,8 0,6 0,4 0, M MVS WIG20eq * Figure 3 presets equity lies for M, MVS models ad bechmark WIG20eq i the period from Jauary 1, 2007 to December 31, The portfolios of the models are recostructed at the begiig of every quarter, based o halfyear historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value. The amout ivested at the begiig of the period is 1 millio Polish Zloty. For these models trasactio costs are ot icluded i the optimizatio process as a costrait. 12

15 Figure 3 shows performace of two models M, MVS ad WIG20eq. Durig the first four years the models outperform the bechmark. At the ed of the studied period MVS slightly uderperforms i compariso to WIG20eq, whereas M obtais better results. Figure 4. Compariso of ARC 8 for models with ad without trasactio costs as a optimizatio costrait 0% -2% -4% -6% -8% -10% -12% -14% -16% M MV MS MVK MVS MVSK ARC for models without trasactio costs as optimizatio costrait ARC for models with trasactio costs as optimizatio costrait * Figure 4 presets Aual Retur Compouded for all the models i the period from Jauary 1, 2007 to December 31, Models with ad without trasactio costs as a optimizatio costrait are preseted. The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value. Figure 4 presets ARC for models with ad without trasactio costs as a optimizatio costrait. The Figure cofirms the coclusios draw from compariso of statistics from Table 4 ad 5. All the models ot havig the trasactio costs costrait achieve higher returs tha their couterparts, which have this costrait. These results are surprisig, because it was expected that models, for which trasactio costs are costraied, should achieve higher returs due to lower costs. It tured out that ideed the models cotrollig trasactio costs i optimizatio process bear lower costs, but it does ot traslate ito higher returs. The cause of this situatio may be the fact that the costrait (3.10) may heavily ifluece the choice of the optimal portfolio allowig oly small chages i its compositio. Thus, i situatios where market coditios chaged ad the held portfolio was o loger optimal, the trasactio costs costrait may allow i the recostructio to obtai oly suboptimal portfolio as far as the shape of the returs distributio is cocered. The results idicate that savigs obtaied by lower trasactio costs i models cotrollig costs do ot compesate the lost retur that occurred due to additioal trasactio costs costrait. Therefore i the portfolio choice, the models without the costrait of trasactio costs should be used, which is cotrary to what Yu ad Lee (2011) propose. Probably the iclusio of trasactio costs might improve the results, but it should be doe i differet way, so that the other optimizatio costraits would ot be so strogly affected. 8 The ARC is preseted o Figures i this paper ad ot Retur to Risk ratio, because for may models the rate of returs are egative, which makes the iterpretatio of RR ambiguous RR ca be properly iterpreted oly for positive values. 13

16 4.2 Additio of risk-free asset to the models I this subsectio the additio of risk-free asset (WIBID3M) to the portfolio is aalysed. It is supposed that risk-free asset, as a portfolio compoet of differet characteristics from stocks, ca add value to the portfolio performace ad ca help reduce losses durig periods of dowturs. Here oly models without trasactio costs as a optimizatio costrait are preseted, sice i previous sectio it was show that they perform better from models cotrollig trasactio costs. Table 6. Basic statistics for all the models with risk-free asset ad bechmarks M MV MS MVK MVS MVSK WIG20eq WIG Alliaz OFE Alliaz Akcji FIO ARC 8,41% 4,59% 3,01% 4,92% 3,10% 0,69% -9,13% -5,89% 2,34% -4,47% ASD 22,21% 14,92% 21,98% 14,94% 16,16% 16,34% 27,47% 24,67% 7,15% 20,96% RR 0,38 0,31 0,14 0,33 0,19 0,04-0,33-0,24 0,33-0,21 skewess 0,15 0,35 0,16 0,34 0,43 0,37-0,25-0,37-0,46-0,69 kurtosis 2,54 3,70 3,06 3,68 3,17 3,17 2,99 2,65 1,85 8,89 * Table 6 presets basic statistics for all the models ad bechmarks i the period from Jauary 1, 2007 to December 31, The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value ad i risk-free rate is 100% of the portfolio value. I Table 6 the statistics for all the models ad bechmarks are show. The aual retur for all models is positive, which is a great result takig ito accout that the studied period icludes the fiacial crisis. The models also perform well i compariso to the bechmarks. All of them beat WIG20eq, WIG ad Alliaz Akcji FIO sigificatly. Five models have higher rate of retur tha the best bechmark Alliaz OFE. As far as both risk ad retur are cocered M, MV ad MVK obtaied similar results to Alliaz OFE. Apart from much higher rate of retur of the models with risk free rate i compariso to the models preseted i the previous sectio, the additio of risk free rate allowed also to reduce sigificatly the volatility of the models portfolio returs. The distributios of returs for all the bechmarks are egatively skewed ad o the cotrary the distributios of returs of each models portfolios are characterized by positive skewess. Comparig it with the results from Table 5 it ca be stated that the additio of risk-free rate improved skewess of the portfolios sigificatly. As far as kurtosis is cocered all models ad bechmarks have positive kurtosis showig that leptokurtosis is preset i each preseted ivestmet strategy. Figure 5 presets the frequecy of the portfolios daily returs for two variats of MV model: oe composed oly of stocks ad the secod cosistig of stocks ad risk-free rate. The aalysis of the Figure allows to state that the model with additio of risk-free rate is characterized by higher mea, much lower dispersio aroud the mea ad higher skewess, resultig probably from much shorter left tail of the distributio, whe it is compared to the model for which the ivestmet i risk-free rate is ot allowed. This Figure cofirms the coclusios draw from aalysis of Tables 5 ad 6 ad visually shows the beefits of icorporatio of risk-free asset ito portfolio selectio process. 14

17 ml PLN Figure 5. The frequecy of the daily rates of returs for two cases of the MV model with ad without risk-free rate % -8% -7% -6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% MV model without risk-free rate MV model with risk-free rate * Figure 5 presets a histogram for rates of returs for MV models i the period from Jauary 1, 2007 to December 31, Two variats of MV model are preseted: oe that allowed ivestmet oly i WIG20 stocks, ad secod oe that eabled ivestmet i WIG20 stocks ad risk-free rate (WIBID 3M). The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value ad i risk-free rate is 100% of the portfolio value. Figure 6. M, MVSK models with risk-free rate compared with Alliaz OFE ad WIG20eq 2 1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0, M MVSK WIG20eq Alliaz OFE * Figure 6 presets equity lies for M, MVSK models ad two bechmarks: WIG20eq, Alliaz OFE i the period from Jauary 1, 2007 to December 31, The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value ad i risk-free rate is 100% of the portfolio value. The amout ivested at the begiig of the period is 1 millio Polish Zloty. 15

18 Figure 6 presets the equity lies of bechmarks ad models with the highest ad lowest aual rate of retur Alliaz OFE, WIG20eq, M ad MVSK models respectively. The models outperform the WIG20eq sigificatly. The worst of the models i case of retur MVSK performs very similar to the best bechmark Alliaz OFE. Summig up the iformatio from Table 6 ad Figure 6 it ca be cocluded that the additio of risk-free asset improves the results sigificatly. Not oly volatility is dimiished but also the retur is improved. It is worth otig that the models with risk free asset protect the ivested capital, which is a importat quality especially i periods of large dowturs like the oe that occurred i Figure 7. M model, WIG ad share of risk-free rate i M model portfolio * Figure 7 presets equity lies for M model ad WIG i the period from Jauary 1, 2007 to December 31, The grey colums represet the share of risk-free rate i M model portfolio i a give quarter. The share is measured o the left vertical axis. The portfolio of the model is recostructed at the begiig of every quarter, based o halfyear historical data. The trasactio cost is 0,4% of the trasactio value. The maximum ivestmet i stocks of oe compay is 20% of portfolio value ad i risk-free rate is 100% of the portfolio value. The amout ivested at the begiig of the period is 1 millio Polish Zloty. Figure 7 shows the performace of the M model i compariso to WIG ad additioally the share of risk-free asset i the M model s portfolio i the studied period. The model did ot choose the risk-free asset i 2007 at all. But i the 2008, whe the situatio o the fiacial market was gettig worse, the share of risk-free asset was gradually icreasig. Thus, i 2008 ad i the first half of 2009 the model ivested large part of the portfolio value i the risk free asset, which proved to be a great strategy agaist losses that occurred o markets i that time. Whe the situatio o the stock exchage was becomig better the model stopped ivestig i risk-free asset, which allowed makig a high profit. The model ivested also i the risk-free rate i the last quarter of 2011, which was a reactio to large losses o the market i third quarter of But this time the model did ot foresee the dowtur of August The reaso for this may be the fact that the model is rebalaced oly quarterly ad therefore caot react sufficietly fast to 16

19 sudde chages o the market. It is also possible that the model did ot react strog eough to the losses, because its parameters are estimated o 6 moth historical data. With shorter periods of historical data, e.g. 3 moths, the model would react stroger to recet evets o the market. 5. Sesitivity aalysis I this sectio the ifluece of chages i the research assumptios o the results is preseted. Firstly, the chage of the upper boud of a positio i oe security from 20% to 10% ad 40% is examied. The the shift of the startig poit of portfolio recostructio from the begiig of the quarter to the middle of the quarter is ivestigated. Subsequetly, mothly portfolio rebalacig is compared to quarterly rebalacig. The the ifluece of trasactio costs o the results is studied ad at the ed the optimal models are described. The sesitivity aalysis is coducted for all the six models without trasactio costs as a optimizatio costrait ad with risk-free asset. Therefore the models that performed the best accordig to sectio 4 are aalysed. 5.1 Chage of upper boud of a positio i oe security Table 7 presets basic statistics for all the models with upper boud of a positio i oe security shifted from 20% to 10% ad 40%. This shift cocers oly stocks, sice i case of risk free rate always the ivestmet up to 100% is possible. Geerally the decrease i the upper boud of a positio improves the results for all models, except from M model for which the highest RR is for upper boud of 20%. Table 7. Basic statistics for models with upper boud of a positio i oe security of 10%, 20% ad 40% upper boud of a positio of 0,1 upper boud of a positio of 0,2 upper boud of a positio of 0,4 M MV MS MVK MVS MVSK ARC 5,03% 7,53% 2,44% 7,03% 5,69% 1,57% ASD 19,17% 14,81% 18,06% 14,82% 14,99% 15,81% RR 0,26 0,51 0,14 0,47 0,38 0,10 ARC 8,41% 4,59% 3,01% 4,92% 3,10% 0,69% ASD 22,21% 14,92% 21,98% 14,94% 16,16% 16,34% RR 0,38 0,31 0,14 0,33 0,19 0,04 ARC 0,46% 2,83% -2,40% 2,87% 0,30% -1,19% ASD 25,45% 14,74% 24,65% 14,74% 15,82% 16,06% RR 0,02 0,19-0,10 0,19 0,02-0,07 * Table 7 presets basic statistics for all the models i the period from Jauary 1, 2007 to December 31, Three variats of the models are distiguished for which maximum ivestmet i stocks of oe compay is 40%, 20% ad 10%. The maximum ivestmet i risk-free rate is 100% of the portfolio value. The portfolios of the models are recostructed at the begiig of every quarter, based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. The aalysis of Table 7 allows to state that i case of rate of retur the models with upper boud of a positio i oe security of 20% outperform models with higher upper boud of 40% ad models with upper boud of 10% outperform models with upper boud of 20% i 4 out of 6 17

20 cases. This shows that it is worth to impose a restrictio of a maximal share of oe security i the portfolio ad that the results are highly affected by the level of the imposed bouds. Too high value of the boudary deteriorated the results. It allowed a higher cocetratio i a smaller umber of securities, which proved ot to be a good ivestmet strategy. Additioal coclusio, which ca be draw from the aalysis of Table 7, is that models with skewess have lower rates of retur tha their couterparts without skewess. Models M, MV, MVK achieve better results tha MS, MVS ad MVSK respectively. 5.2 Chage of the portfolio recostructio momet to the middle of the quarter I this sectio the ifluece of the momet of portfolio recostructio o the stability of the results is examied. So far all the results were preseted for models with portfolio recostructio occurrig at the begiig of each quarter durig the studied period. Now the momet of portfolio recostructio is shifted to the middle of the quarter. Table 8. Basic statistics for models with differet momets of portfolio recostructio Begiig of the quarter rebalacig M MV MS MVK MVS MVSK ARC 8,41% 4,59% 3,01% 4,92% 3,10% 0,69% ASD 22,21% 14,92% 21,98% 14,94% 16,16% 16,34% RR 0,38 0,31 0,14 0,33 0,19 0,04 ARC 4,08% 9,12% -0,29% 9,33% 0,68% 2,45% Middle of the quarter ASD 22,44% 14,89% 20,71% 15,01% 16,43% 16,77% rebalacig RR 0,18 0,61-0,01 0,62 0,04 0,15 * Table 8 presets basic statistics for all the models i the period from February 14, 2007 to February 14, Two variats of the models are distiguished for which the momet of portfolio recostructio is at the begiig of the quarter ad i the middle of the quarter. The maximum ivestmet i stocks of oe compay is 20% of portfolio value ad i risk-free rate is 100% of the portfolio value. The portfolios of the models are recostructed based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. Table 8 shows the results both for models with the portfolio recostructio momet at the begiig of the quarter ad i the middle of the quarter. As far as the volatility is cocered, the momet of portfolio recostructio almost does ot ifluece it. The chages i volatility are egligible. But the rate of retur ad likewise retur to risk ratio chage sigificatly. Nevertheless for both variats of the portfolio recostructio momets the three best models are the same M, MV ad MVK. But the hierarchy chages: with begiig of the quarter rebalacig the M model is the best oe, whereas with middle of the quarter rebalacig the MVK is the best oe. This part of sesitivity aalysis proves that there is ay model that outperforms other oes irrespective of assumptios made. The shift of the portfolio recostructio momet chages the hierarchy of the models. The aalysis also shows how importat the assumptios are ad how sesitive the models ca be for slight chages i them. 18

21 5.3 Mothly portfolio recostructio I this sectio the frequecy of portfolio recostructio is chaged. So far for all the preseted models it was assumed that compositio of the portfolio is subject to chage oce a quarter. O the oe had this approach allowed to bear quite a low trasactio costs, whe the portfolio is recostructed oly four times a year, but o the other had a quarter is quite a log time ad whe sudde chages occur o the market, the model reacts with a large delay. I this sectio it is examied whether mothly rebalacig of the portfolio, istead of quarterly, ca improve the results. Table 9. Basic statistics for models with differet frequecy of portfolio recostructio quarterly portfolio recostructio M MV MS MVK MVS MVSK ARC 8,41% 4,59% 3,01% 4,92% 3,10% 0,69% ASD 22,21% 14,92% 21,98% 14,94% 16,16% 16,34% RR 0,38 0,31 0,14 0,33 0,19 0,04 mothly ARC 7,48% 8,32% 3,75% 7,70% 4,83% 2,22% portfolio ASD 22,73% 14,94% 23,05% 14,99% 16,40% 16,43% recostructio RR 0,33 0,56 0,16 0,51 0,29 0,14 * Table 9 presets basic statistics for all the models i the period from Jauary 1, 2007 to December 31, Two variats of models are distiguished for which the portfolios are rebalaced mothly ad quarterly. The maximum ivestmet i stocks of oe compay is 20% of portfolio value ad i risk-free rate is 100% of the portfolio value. The portfolios of the models are recostructed based o half-year historical data. The trasactio cost is 0,4% of the trasactio value. Table 9 shows basic statistics for models with both mothly ad quarterly portfolio rebalacig. For five out of six models the rate of returs ad retur to risk ratios are higher whe the compositio of the portfolio is chaged oce a moth. The largest improvemet is for MV ad MVK models, for which the aual rate of retur icreases from 4,59% ad 4,92% to 8,32% ad 7,7% respectively. The volatility for most of the models does ot chage sigificatly. Although mothly rebalacig does ot outperform quarterly rebalacig for all the models, it should be stressed that i case of mothly portfolio recostructio the total trasactio costs should be much higher. The ifluece of trasactio costs o the results is examied i the ext sectio. 5.4 The ifluece of trasactio costs o models performace I this sectio the ifluece of trasactio costs o the models performace is aalysed. I the whole research the assumed level of trasactio costs is 0,4% of the trasactio value. This size of costs is close to reality i case of idividual ivestors. But istitutioal ivestors executig large deals ca pays much lower costs i proportio to the trasactio value. Thus, i this sectio the sesitivity of the results to the level of trasactio costs is checked. Three levels of costs are compared: 0,1%, 0,2% ad 0,4%. The results are preseted for the best model with quarterly ad mothly recostructio whe it comes to value of retur to risk ratio, i.e. M ad MV models respectively. The compariso of models with differet frequecy of portfolio recostructio 19