Portfolio Rebalancing under Uncertainty Using Meta-heuristic. Algorithm

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1 Portfolio Rebalacig uder Ucertaity Usig Meta-heuristic Algorithm Mostafa Zadieh a,*, Seyed Omid Mohaddesi b a Departmet of Idustrial Maagemet, Maagemet ad Accoutig Faculty, Shahid Beheshti Uiversity, Tehra, Ira Abstract b Departmet of Fiacial Egieerig, Raja Uiversity, Qazvi, Ira I this paper, we solve portfolio rebalacig problem whe security returs are represeted by ucertai variables cosiderig trasactio costs. The performace of the proposed model is studied usig costat-proportio portfolio isurace (CPPI) as rebalacig strategy. Numerical results showed that ucertai parameters ad differet belief degrees will produce differet efficiet frotiers, ad affect the performace of the proposed model. Moreover, CPPI strategy performs as a isurace mechaism ad limits dowside risk i bear markets while it allows potetial beefit i bull markets. Fially, usig a globally optimizatio solver ad geetic algorithm (GA) for solvig the model, we cocluded that the problem size is a importat factor i solvig portfolio rebalacig problem with ucertai parameters ad to gai better results, it is recommeded to use a meta-heuristic algorithm rather tha a global solver. Keywords: Portfolio Rebalacig; Trasactio Costs; Costat-Proportio Portfolio Isurace (CPPI); Ucertaity Theory; Meta-heuristic Algorithm. 1. Itroductio Portfolio rebalacig is oe of the major compoets of portfolio maagemet process [1]. Geerally, asset allocatio is performed by ivestors i accordace with their level of risk seekig or risk aversio as well as their expected portfolio retur. Meawhile, the objective of portfolio rebalacig is to restore a portfolio to its origial state ad primarily optimal allocatio. Portfolio rebalacig is cosidered to be oe of the risk cotrol techiques; sice as time passes, chages i asset prices lead to gais ad losses o each asset i portfolio which will cause a icrease or decrease i the weight ivested i that asset, ad as a result the risk of ivestmet will be icreased. This will make the portfolio to deviate beyod a certai threshold from the ivestor s expected risk [2]. Portfolio rebalacig is a dyamic process of buyig ad sellig of assets i portfolio so that the optimal weights ivested i each asset is maitaied over time. I this situatio cosiderig trasactio cost associated with buyig ad/or sellig of a asset is importat. I fact, trasactio cost is oe of the mai costraits for modelig portfolio rebalacig that help i creatig more realistic models [3]. May researchers studied portfolio rebalacig problem cosiderig trasactio costs icludig Su et al. [4], 2006, Fag et al. [5], 2006, Fadaei-Nezad ad Baaeia [6], 2010, Yu ad Lee [7], 2011, Woodside-Oriakhi et al. [8], 2013, Gupta et al. [9], 2014, Wag et al. [10], 2014, Che et al. [11], 2014, Qi et al. [12], 2014, Rabbai [13], 2014 ad Kumar et al. [14], I order to measure portfolio performace differet scholars practiced various models cosiderig differet parameters icludig risk, retur, liquidity, ivestmet horizo ad so forth. To the best of our kowledge, however, all solved their models assumig buy-ad-hold (B&H) as rebalacig strategy. * Correspodig Author. Phoe: +98 (21) ; Mobile: ; address: m_zadieh@sbu.ac.ir 1

2 Other (Iovative) CPPI Buy & Hold Ucertai Radom Fuzzy Trasactio Costs Ivestmet Horizo Liquidity Retur Risk Other Markowitz Multi-period Markowitz (Mea-Variace) Multi-objective I additio, the most otable characteristic of security returs is ucertaity [15]. I classical portfolio theory, security returs were described by radom variables, ad back the probability theory was the mai mathematical tool for hadlig ucertaity. However, ucertaity is varied i real ad complex world, especially whe huma factors are ivolved, ad radomess is ot the oly type of ucertaity i reality. Cosequetly, security market as oe of the most complex markets i the world, cotais almost all kids of ucertaity. I particular, the security returs are sesitive to various factors icludig ecoomic, social, ad political, ad very importatly, people s psychological factors. Due to this complex ature of fiacial markets, historical data may be isufficiet to reflect the future returs of securities i real situatios [12]. Aother feasible approach for estimatig probability distributio of security returs is usig belief degrees evaluated by experts. To deal with this belief degrees, Liu [16] fouded the cocept of ucertai measure ad ucertaity theory. Liu also proposed ucertai programmig for solvig optimizatio problems ivolvig ucertai variables. I this area, there have bee may studies amog which we ca refer to vehicle routig ad project schedulig problems, shortest path problem ad stock model [17]. I particular, Ya ad Huag [15, 18] applied Liu s ucertaity theory to portfolio selectio problem usig ucertai variables whe securities returs are either radom or fuzzy. Moreover, i order to solve ucertai portfolio optimizatio model, Zhag et al. ad Che proposed meta-heuristic algorithms [19, 20]. Yet there are t too may studies i the literature o portfolio rebalacig uder ucertaity usig experts subjective evaluatios. Oe of the few studies o ucertai portfolio rebalacig model is the oe doe by Qi et al. [12] i They used buy-ad-hold strategy ad solved their proposed model by umerical examples for small size problems. Table 1 represets the gap aalysis i the literature of the portfolio rebalacig ad the status of the preset study i the cotext. Accordig to the type of mathematical programmig model, cosidered parameters ad mai costraits of the model, types of variables ad the rebalacig strategy, we categorized the aforemetioed works which clearly idicates the gap i the cotext. Table 1. Aalysis of gap i the literature Model Parameters ad Costraits Type of Variables Rebalacig Strategy Ref. [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [21] [22] [23] Curret Study 2

3 Accordigly, the cotributio of our study is first to itegrate costat-proportio portfolio isurace (CPPI) strategy with ucertai variables i portfolio rebalacig problem. Secod, the results of solvig our itegrated model is compared with a correspodig ucertai model with buy-ad-hold (B&H) strategy. Third, the portfolio rebalacig model with CPPI strategy is solved usig a meta-heuristic algorithm which has bee ever ivestigated i the cotext. The objective of this paper is to study portfolio rebalacig problem usig ucertai variables whe security returs are estimated by experts evaluated belief degrees. Here, we attempt to solve the model usig costat-proportio portfolio isurace (CPPI) as rebalacig strategy for differet size problems. The performace of CPPI strategy ad B&H strategy is compared by solvig the model usig real data from Tehra Stock Exchage (TSE). I additio, a geetic algorithm is proposed to solve the ucertai portfolio rebalacig model ad its effectiveess is measured ad illustrated by solvig real examples for differet size problems. The rest of the paper is orgaized as follows. I Sectio 22, we review some basic defiitios ad fudametals of ucertaity theory ad portfolio rebalacig. Sectio 3 formulates the portfolio rebalacig model whe security returs are described by ucertai variables. I Sectio 4, we preset details of the solutio algorithm for solvig the model. The model is tested i Sectio 5 usig 11 problems i three differet sizes cosiderig 100 compaies selected from Tehra Stock Exchage. This sectio also discusses the obtaied results. Fially, we coclude the paper ad provide some suggestios for future researches i Sectio Prelimiaries Ucertaity theory was first itroduced by Liu [16] i 2007 ad further developed by other researchers. Today ucertaity theory is cosidered as a brach of axiomatic mathematics for modelig belief degrees. Here we review some basic cocepts, defiitios ad properties of ucertaity theory icludig ucertai measure, ucertai variable ad ucertaity distributio, which will be used i the whole paper. A ucertai measure M o the σ-algebra L is defied as a umber M{Λ} which is assiged to each evet Λ to idicate the belief degree with which we believe Λ will happe. Obviously the assigmet of this umber is ot arbitrary, ad the ucertai measure M must have certai mathematical properties. To isure this ad i order to ratioally deal with belief degrees, Liu [16] proposed the followig three axioms: Axiom 1. M{Γ} = 1. Axiom 2. M{Λ} + M{Λ c } = 1 for ay evet Λ. Axiom 3. For every coutable sequece of evets Λ 1, Λ 2, we have M { Λ i } M{Λ i }. (1) Product ucertai measure was defied by Liu [24] i 2009 ad produced the forth axiom of ucertaity theory. Axiom 4. Let (Γ k, L k, M k ) be ucertaity spaces for k = 1,2, ; The product ucertai measure M is a ucertai measure satisfyig M { Λ k } = M k {Λ k } k=1 k=1 (2) where Λ k are arbitrarily chose evets from L k for k = 1,2,, respectively. Ucertai variable is a fudametal cocept i ucertaity theory [17], ad is defied as a fuctio ξ from a ucertaity space (Γ, L, M) to the set of real umbers such that {ξ B} is a evet for ay Borel set B. I order to describe ucertai variable ξ Liu itroduced ucertaity distributio Φ which is defied by 3

4 for ay real umber x. Φ(x) = M{ξ x} For example, a liear ucertai variable is a ucertai variable ξ i which has a liear ucertaity distributio 0, if x a, x a Φ(x) = { b a, if a x b, (4) 1, if x b, deoted by L(a, b) where a ad b are real umbers with a < b. A ormal ucertai variable is a ucertai variable ξ i which has a ormal ucertaity distributio (3) π(e x) Φ(x) = (1 + exp ( 3σ )), x R deoted by N(e, σ) where e ad σ are real umbers with σ > 0. 1 (5) I additio to ucertaity distributio, a fuctio Φ 1 is called a iverse ucertaity distributio of a ucertai variable ξ if ad oly if M{ξ Φ 1 (α)} = α, for all α [0,1]. Liu [16] also stated that ucertai variables ξ 1, ξ 2,, ξ are cosidered idepedet if for ay Borel sets B 1, B 2,, B we have M { (ξ i B i )} = M{ξ i B i } I order to represet the size of ucertai variable, Liu [16] proposed the expected value of ξ as + E[ξ] = M{ξ x} dx provided that at least oe of the two itegrals is fiite. 0 0 M{ξ x} dx (6) (7) For example, the liear ucertai variable ξ ~ L(a, b) has a expected value E[ξ] = (a + b)/2. The ormal ucertai variable ξ ~ N(e, σ) has a expected value e, which meas E[ξ] = e. Lemma 1 [24] Let a ad b be two real umbers, ad ξ ad η two ucertai variables. The we have E[aξ + b] = ae[ξ] + b. Further, if ξ ad η are idepedet, the E[aξ + bη] = ae[ξ] + be[η ]. A degree of the spread of the distributio aroud its expected value is how Liu defied the variace of ucertai variable [17]. Let ξ be a ucertai variable with fiite expected value e. The the variace of ξ is V[ξ] = E[(ξ e) 2 ]. This meas that the variace is the expected value of (ξ e) 2, ad owig to the fact that (ξ e) 2 is a oegative ucertai variable, we also have + V[ξ] = M{(ξ e) 2 x}dx 0 For example, the liear ucertai variable ξ ~ L(a, b) has the variace V[ξ] = (b a) 2 12, ad the ormal ucertai variable ξ ~ N(e, σ) has the variace σ 2. Lemma 2 [24] Let a ad b be real umbers, ad ξ a ucertai variable with fiite expected value, the V[aξ + b] = a 2 V[ξ]. Further, let e be the expected value of ucertai variable ξ. the V[ξ] = 0 if ad oly if M{ξ = e} = 1. This meas the ucertai variable ξ is basically the costat e. Lemma 3 [17] Let ξ be a ucertai variable with regular ucertaity distributio Φ ad fiite expected value e. The (8) V[ξ] = (Φ 1 (α) e) 2 dα 1 0 (9) 4

5 Lemma 4 [17] Assume ξ 1, ξ 2,, ξ are idepedet ucertai variables with regular ucertaity distributios Φ 1, Φ 2,, Φ, respectively. If f(x 1, x 2,, x ) is strictly icreasig for x 1, x 2,, x m ad strictly decreasig for x m+1, x m+2,, x, the expected value ad variace of the ucertai variable ξ = f(ξ 1, ξ 2,, ξ ) are as follows 1 E[ξ] = f(φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1 (1 α))dα 0 1 V[ξ] = (f(φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1 (1 α)) e) 2 dα 0 (10) (11) 2.1. Rebalacig strategy Idividual ad istitutioal ivestors use various strategies for rebalacig their portfolio i order to optimize their ivestmet process. Perold ad Sharpe [25] itroduced four dyamic strategies for portfolio rebalacig which iclude: 1) buy-ad-hold; 2) costat mix; 3) costat-proportio portfolio isurace (CPPI); ad 4) optio-based portfolio isurace (OBPI). As metioed earlier, o scholar has solved the portfolio rebalacig problem usig ucertai variables ad CPPI strategy to date. I this paper, we use CPPI strategy to solve portfolio rebalacig problem while ucertai variables of the model are estimated by experts evaluated belief degrees Costat-proportio portfolio isurace The CPPI strategy is a self-fiacig dyamic strategy i which by ivestig a portio of the wealth i risky asset equal to a costat multiple of cushio, the ivestor limits the dowside risk while gaiig some upside potetial [26]. The cushio is equal to the differece betwee value of the portfolio at time t (W t ) ad the floor which the ivestor refuses the value of the portfolio to go below (F t ), ad is defied as C t = W t F t I this case ad at ay time t, if W t > F t, the exposure to the risky asset (the amout of wealth ivested i risky asset) is obtaied by mc t m(w t F t ), where m > 1 is a costat multiplier. If W t F t, the etire portfolio is ivested i risk-free bods. I fact, CPPI always keeps a costat multiple of the cushio as exposure to risky asset like stocks, ad the rest of the wealth is ivested i risk-free govermet savigs or treasury bods [27]. Thus, if E t represets the exposure to the risky asset, ad E t W t a fractio of the total wealth to be ivested i risky assets at ay time t, the E t W t = mi {m (1 F t W t ), 1} where m > 1 is a costat multiplier. The value of the multiplier m is derived based o the ivestor s risk seekig or risk aversio, ad typically by aswerig what is the probability of the maximum oe-day loss from ivestig i risky asset. The multiplier will be the iverse of this percetage of the loss. For istace, if a ivestor estimates the maximum probable loss of 20%, the multiplier value will be equal to 5. Values betwee 3 ad 6 are more frequetly used as multiplier [26]. 3. Ucertai portfolio rebalacig model I this sectio, the bi-objective rebalacig problem is formulated usig ucertai variables ad cosiderig trasactio costs. As previously metioed, i this paper we use CPPI strategy for modelig portfolio rebalacig problem. For this purpose, we assume that the ivestors allocate their wealth i accordace with CPPI strategy assumptios, ad by dividig it betwee risky assets (stocks) ad 1 riskfree asset (participatio bods i this paper). The parameters ad variables used to formulate the mathematical model are described as follows: ξ i : the ucertai retur of the i-th asset, u i : the maximal fractio of the wealth allocated to the i-th asset, l i : the miimal fractio of the wealth allocated to the i-th asset, 5 (12) (13)

6 x 0 i : the iitial proportio of the total fuds ivested i the i-th asset, i.e., before rebalacig, x + i : the proportio of the i-th asset to be purchased durig rebalacig, x i : the proportio of the i-th asset to be sold durig rebalacig, x i : the fial proportio of the total fuds ivested i the i-th asset, i.e., after rebalacig, y i : a biary variable idicatig whether the i-th asset is cotaied i the portfolio, where y i = { 1, if the i th asset is cotaied i the portfolio 0, otherwise, z i : a biary variable idicatig whether the i-th asset is purchased or sold, where z i = { 1, if the i th asset is purchased, 0, if the i th asset is sold, h: the maximum umber of assets i the portfolio, b i : the trasactio cost of buyig a proportio of the i-th asset, s i : the trasactio cost of sellig a proportio of the i-th asset, C(x): the total trasactio costs icurred by rebalacig the portfolio, W t : the total value of the portfolio (etire wealth) at time t, F t : the miimum value accepted by the ivestor, which they refuse the value of the portfolio to go below, m: the costat multiplier of CPPI strategy. Agreemet The zeroth asset (i.e., x 0 ) will idicate the risk-free asset (i.e., participatio bods) i rest of the paper, thus we also have the followig otatios x 0 0 : the iitial proportio of the total fuds ivested i the risk-free asset, i.e., before rebalacig, x 0 + : the proportio of the risk-free asset to be purchased durig rebalacig, x 0 : the proportio of the risk-free asset to be sold durig rebalacig, x 0 : the fial proportio of the total fuds ivested i the risk-free asset, i.e., after rebalacig Trasactio costs Trasactios cost is oe the pricipal costraits of portfolio rebalacig problem which helps i creatig more realistic models by itegratig market frictios [3]. I fact, trasactio costs make a decrease i portfolio retur; hece, cosiderig these costs is sesible for portfolio rebalacig as well as portfolio optimizatio problems. We assume that x 0 stads for the iitial portfolio ad our goal is to achieve optimal portfolio x by adjustig the weights of each asset i portfolio. I this case, trasactio costs o purchases are measured by the amout added to the portfolio x 0, ad trasactio costs o sales are measured by the amout deducted from the iitial portfolio x 0. If the ivestor pays trasactio costs proportioal to b i for every added amout to the i-th asset (x i + ), ad proportioal to s i for every deducted amout from the i-th asset (x i ), the total trasactio costs of rebalacig the portfolio will be derived by C(x) = (b i x i + + s i x i ) i=0 where for i = 1,2,,, s i ad b i idicate trasactio costs of stocks (risky assets), while s 0 ad b 0 idicate trasactio costs of participatio bods (risk-free asset); ad are obtaied by Table 2 ad Table 3, respectively Objectives The proposed model is a bi-objective optimizatio model i which the returs o the assets are cosidered as ucertai variables. The objectives of the model will be described below. Objective 1: Portfolio retur The mai goal of ivestmet is to produce retur. Accordig to Markowitz mea-variace portfolio theory [28], the ivestmet retur ca be modeled by meas of expected retur. Thus, the first objective of our proposed model, is to maximize portfolio retur calculated by ucertai expected value. I additio, (14) 6

7 the trasactio costs eed to be cosidered i objective fuctio as a factor makig a decrease i portfolio retur. Assume the returs of assets are represeted by ucertai variables ξ i for i = 0,1,2,,. The objective fuctio to maximize portfolio retur after adjustig trasactio costs is expressed as max E [ x i ξ i ] C(x) i=0 where x i idicates the proportio of total fuds ivested i the i-th asset ad C(x) is calculated by Equatio (14). Table 2. Trasactio commissio & fees for stocks i Tehra Stock Exchage Objective 2: Portfolio risk Fee Descriptio The Buyer The Seller Fees (%) Fees (%) Brokerage Fees 0.4% 0.4% TSE Commissio 0.032% 0.048% SEO Commissio 0.032% 0.048% Clearig Fees 0.022% 0.033% Taxes - 0.5% Total Commissio & Fees 0.486% 1.029% Table 3. Trasactio commissio & fees for participatio bods Fee Descriptio The Buyer The Seller Fees (%) Fees (%) Brokerage Fees 0.063% 0.063% TSE Commissio % % Total Commissio & Fees % % Huag [15] describes the optimal portfolio as the oe miimizig the risk while maximizig the retur. To be more precise, whe higher levels of returs wo t be attaied uless we take more risk, or takig less risk wo t be possible uless we udertake lower levels of retur, we will achieve the optimal portfolio. Accordig to Markowitz model [28], the risk of the portfolio is measured by meas of variace. Let ξ i, i = 0,1,2,, be the ucertai returs of assets. The the objective fuctio to miimize the risk of the portfolio is described as (15) mi V [ x i ξ i ] where x i idicates the proportio of total fuds ivested i the i-th asset. i=0 (16) 3.3. Costraits Rebalacig costrait: The fial proportio of the total fuds ivested i the i-th asset after rebalacig is calculated by x i = x i 0 + x i + x i, i = 0,1,2,,. Complemetarity costrait o buyig ad sellig a asset: I ay timeframe, the ivestor must either buy a proportio of the i-th asset or sell it. I other words, it is impossible to buy ad sell some proportio of the same asset simultaeously [3]. Thus, x i + ad x i are complemetary ad the followig costrait must hold (17) x i +. x i = 0, i = 0,1,2,,. (18) 7

8 I order to better cotrol the above costrait i our model, it is trasformed to the liear form by replacig with the followig two costraits where z i {0,1}. x i + z i, x i (1 z i ), i = 0,1,2,,, i = 0,1,2,,, (19) (20) Capital budget costrait: I the proposed model, we assume that the portfolio rebalacig process is fiaced by curret assets i the portfolio (i.e., the portfolio is self-fiacig), ad the ivestor does ot add ay additioal capital to the portfolio durig rebalacig. The trasactio costs are also paid by the capital i the portfolio. Moreover, sice we use CPPI strategy for rebalacig, the exposure is derived by Equatio (13). Thus, the followig costrait must hold x i + C(x) = mi {m (1 F t W t ), 1}. Risk-free asset costrait: The amout of ivestmet i risk-free asset (participatio bods) is equal to the subtractio of total wealth ad the exposure to the risky asset. Thus this amout is calculated by (21) x 0 = 1 mi {m (1 F t W t ), 1}. (22) Maximal ad miimal fractio of the wealth allocated to oe asset: The maximal ad miimal fractio of the wealth that ca be allocated to a specific asset i the portfolio may deped o several factors [3]. For example, the ivestor may cosider the price or value of the asset to the average price or value of all assets i the portfolio, the miimum volume that ca be ordered ad traded i the market, the past behavior of the price or traded volume of the asset, the available iformatio about the issuer of the asset, or the treds i a particular idustry. Geerally, the costraits correspodig to lower bouds (l i ) ad upper bouds (u i ) o ivestmet i a particular asset, are added to the portfolio rebalacig model i order to avoid a large umber of low volume ivestmets ad also to esure adequate diversificatio i portfolio. Thus, the followig costraits must hold where (0 l i u i 1, i). x i u i y i, x i l i y i, i = 0,1,2,, i = 0,1,2,, (23) (24) Number of assets: I order to effectively maage risky assets i the portfolio, ad to avoid large umber of low volume ivestmets, the followig costrait is defied for maagig the umber of risky assets. where y i {0,1}. y i h. No short sellig costrait: I order to avoid short sellig of assets the followig three costraits must hold (25) x i 0, x i + 0, x i 0, i = 0,1,2,,. (26) 3.4. The decisio problem The bi-objective ucertai portfolio rebalacig model is formulated as follows 8

9 { max f 1 (x) = E [ x i ξ i ] C(x) i=0 mi f 2 (x) = V [ x i ξ i ] i=0 Subject to: x i = x 0 i + x + i x i, x + i z i, x i (1 z i ), x i i = 0,1,2,,, i = 0,1,2,,, i = 0,1,2,,, + C(x) = mi {m (1 F t W t ), 1}, x 0 = 1 mi {m (1 F t ), 1}, W t y i h, x i u i y i, i = 0,1,2,,, x i l i y i, i = 0,1,2,,, x i 0, i = 0,1,2,,, x + i 0 i = 0,1,2,,, x i 0, i = 0,1,2,,, y i, z i {0,1}, i = 0,1,2,,. (27) 3.5. Solutio methodologies Huag [29] itroduced so-called Method i order to calculate expected value ad variace of the portfolio i mea-variace model whe securities returs are described by ucertai variables with differet ucertaity distributios Method Assume ξ i is a ucertai variable with ucertaity distributio Φ i, ad k i a positive umber for i = 1,2,,, respectively. Let Ψ i represet the ucertaity distributios of k i ξ i, i = 1,2,,, respectively. The, we have Ψ i 1 (α) = k i Φ i 1 (α). Now cosiderig Ψ to be the ucertaity distributio of k 1 ξ 1 +k 2 ξ k ξ, we have (28) Ψ 1 (α) = Ψ 1 i (α) = k i Φ 1 i (α). This meas the ucertaity distributio Ψ of k 1 ξ 1 +k 2 ξ k ξ ca be represeted o a computer as show i Table 4. Suppose ξ = k 1 ξ 1 +k 2 ξ k ξ is a ucertai variable. Accordig to Lemma 4 ad Method, the expected value ad variace of ucertai variable ξ are as follows (29) E[ξ] = j=1 k it i/j, (( k i t i/j ) E[ξ]) 2 j=1 V[ξ] =. (30) (31) 9

10 Table 4. Presetatio of Method o computer α i Φ 1 1 (α i ) t 1/1 t 1/2 t 1/3 t 1/ Φ 1 2 (α i ) t 2/1 t 2/2 t 2/3 t 2/ Φ 1 3 (α i ) t 3/1 t 3/2 t 3/3 t 3/ Φ 1 (α i ) t /1 t /2 t /3 t / Ψ 1 (α i ) k i t i/1 k i t i/2 k i t i/3 k i t i/ As a result, the objectives of the proposed portfolio rebalacig model ca be replaced by followigs max f 1 (x) = j=1 i=0 x it i/j C(x), (( i=0 x i t i/j ) e) 2 j=1 mi f 2 (x) =, (32) (33) where e = j=1 i=0 x i t i/j is the expected value of portfolio, ad t i/j, i = 0,1,2,,, ad j = 1,2,,, are obtaied from Method by iverse ucertaity distributio Φ 1 i accordig to Table ε-costrait method There are various approaches for solvig a multi-objective mathematical programmig (MOMP) problem. Miettie [30] classified them ito four categories: 1) o-prefereces methods; 2) a priori methods; 3) a posteriori methods; ad 4) iteractive methods. While i o-prefereces methods the decisio maker (DM) has o participatio i the solutio process, a priori methods ask for the DM prefereces ad opiios before the solutio process. I a posteriori methods which are also called geeratio methods, first the Pareto optimal set (or a represetatio of it) is geerated ad the the DM selects the most preferred solutio. I iteractive methods, the DM gets ivolved i the solutio process by correctig his/her prefereces i each iteratio ad after of beig preseted oly part of the Pareto optimal poits. Cosiderig portfolio selectio problems we geerally search for every possible combiatio of assets that geerates differet efficiet portfolios with differet combiatios of risk expected retur accordig to the ivestors prefereces. These differet efficiet portfolios will form the efficiet frotier [29]. Afterwards, each ivestor ca fid his/her ow optimal portfolio from the efficiet frotier accordig to their risk prefereces. Thus, solvig the bi-objective portfolio rebalacig problem falls ito the a posteriori methods category i which the Pareto optimal set is represeted by the efficiet frotier, ad the the DM (here the ivestor) will choose the optimal solutio accordig to his/her prefereces. Miettie [30] itroduced two basic a posteriori methods; the weightig method ad the ε-costrait method. While the ε- costrait method ca fid every Pareto optimal solutio of ay MOMP regardless of covexity of the problem, the weightig method fails to fid all of the Pareto optimal solutios whe the problem is ocovex. Accordigly, we utilize the ε-costrait method i order to solve the bi-objective ucertai portfolio rebalacig problem. I this method, oe of the objective fuctios is optimized by formulatig other objective fuctios as costraits, ad trasferrig them to the costrait part of the model [31]. Assume the followig MOMP problem 10

11 max (f 1 (x), f 2 (x),, f p (x)) s. t. x S. The usig ε-costrait method we will have the followig sigle-objective problem max f 1 (x) s. t. f 2 (x) ε 2, f 3 (x) ε 3, f p (x) ε p, x S. (34) (35) The Pareto optimal solutios of the problem are obtaied by iitializatio ad parametric variatio i the RHS of the costraied objective fuctios (i.e., ε i ) ad the solvig the model for these differet parameters. Cosequetly, usig ε-costrait method, the objective fuctio correspodig to portfolio retur i our proposed model is formulated as j=1 x i t i/j (36) C(x) λ, where λ represets the miimum expected retur required by the ivestor. By solvig the model for differet values of λ, the solutios obtaied will form a efficiet frotier. I additio, the miimum expected retur required by the ivestor is obviously greater tha the retur o risk-free asset. Thus λ r f. Fially, the ucertai portfolio rebalacig model is formulated as follows (37) { (( i=0 x i t i/j ) e) 2 j=1 mi f 2 (x) =, Subject to: j=1 x i t i/j C(x) λ, x i = x 0 i + x + i x i, i = 0,1,2,,, x + i z i, i = 0,1,2,,, x i (1 z i ), i = 0,1,2,,, x i + C(x) = mi {m (1 F t W t ), 1}, x 0 = 1 mi {m (1 F t ), 1}, W t y i h, λ r f, x i u i y i, x i l i y i, x i 0, x + i 0 x i 0, y i, z i {0,1}, i = 0,1,2,,, i = 0,1,2,,, i = 0,1,2,,, i = 0,1,2,,, i = 0,1,2,,, i = 0,1,2,,. (38) 4. Solutio Algorithm Due to the complexity of the oliear ucertai portfolio rebalacig problem, it is hard to solve the model usig exact methods, ad thus, a meta-heuristic algorithm is utilized i this sectio to solve the proposed model. There are various classificatios o the types of meta-heuristics icludig the type of the 11

12 search strategy (i.e. local or global), sigle solutio agaist populatio-based searches, ature-ispired agaist o-ature ispired, etc. [32]. For istace, Shahvari ad Logedra [33, 34], 2016, ad Shahvari et al. [35], 2012, utilized local search-based tabu search (TS) algorithms ad tabu search/path relikig (TS/PR) algorithm for their researches. Other researchers employed global search methods that are usually populatio-based meta-heuristics, such as o-domiated sortig geetic algorithm-ii (NSGA-II) ad odomiated rakig geetic algorithms (NRGA) i the work of Sadeghi ad Niaki [36], 2015, or particle swarm optimizatio (PSO) ad geetic algorithm (GA) i the works of Mousavi et al. [37, 38], 2014, ad Pasadideh et al. [39], 2013, or hybrid algorithm based o GA i the work of Mousavi ad Niaki [40], Furthermore, Mousavi et al. [41-43], 2014 to 2016, used ature-ispired meta-heuristics icludig fruit fly optimizatio algorithms (FFOA) ad harmoy search algorithm. Selectig the appropriate meta-heuristic algorithm highly depeds o the problem itself, ad geetic algorithm (GA) has bee commoly ivestigated i the cotext of portfolio optimizatio ad ucertai programmig [3, 19, 44, 45]. As a result, we utilize GA i our study to solve the ucertai portfolio rebalacig problem. Besides, we will use a globally optimizatio solver based o brach-ad-boud cocept i order to validate the results obtaied ad to verify the performace of the proposed GA. Geetic Algorithm (GA) first became popular through the work of Joh Hollad [46] i early 1970s ad has bee further developed by others the. The algorithm provides a efficiet search method for large spaces that evetually leads to fidig the optimal solutio. I geeral, a GA cosists of followig elemets [47]: a ecodig mechaism for represetig each solutio i form of a chromosome, a populatio of chromosomes, a fitess fuctio assigig scores to each chromosome, geetic operators icludig selectio accordig to fitess, crossover to produce ew offsprig ad radom mutatio of ew offsprig to produce ew populatio. Fially, the evolutio process is stopped accordig to a predetermied termiatio coditio Chromosome ecodig I this paper, we cosider each chromosome as a array with + 1 elemets. Assume x = (x 0, x 1, x 2,, x ) is a possible solutio of the problem, the for ay umber k = 1,2,, + 1 correspodig to a gee, the gee value represets the fial proportio of the wealth ivested i (k 1)-th asset. Agreemet The first gee of each chromosome represets the fial proportio of the wealth ivested i risk-free asset (i.e., x 0 ) Iitializatio Iitializatio of each gee except the first oe is take place by producig uiformly oegative umbers betwee zero ad the exposure (E t W t ). The value of the first gee is obtaied by Equatio (22). The iitializatio esures that costraits correspodig to Equatio (26) will hold. I additio, chromosome x 0 represets the iitial portfolio before rebalacig, ad accordig to Equatio (17) we have x i x i 0 = x i + x i. Accordig to the above equatio, subtractig the gees of chromosome x 0 from correspodig gees of each chromosome x i gives us the tradable amout. Thus, Equatio (17) will be satisfied. After subtractio, a positive value remaiig i each gee idicates addig some proportio to the i-th asset, while a egative value idicates sellig some proportio from the i-th asset. Obviously, the result of the subtractio of chromosomes gees is either positive or egative, thus the Equatio (18) will also be satisfied Repairig mechaism I order to satisfy other costraits ad to prevet GA operators from producig ifeasible solutios, we eed to desig some repairig mechaisms. For satisfyig costrait of Equatio (21), the value of each gee oly gees correspodig to risky assets eed to be adjusted. Thus 12

13 x i = x i E t W t (39) x i where x i is the repaired (adjusted) value of each gee. Moreover, if the maximum umber of risky assets (h) does ot meet i a chromosome after iitializatio, we will use the followig mechaism: 1. Geerate a radom iteger h betwee 1 ad h; 2. Sort gees umber 2 to + 1 i ascedig order based o their values; 3. Replace the value of first h gees with zero. Thus, i each chromosome, the maximum umber of h gees (assets) have values ad the costrait i Equatio (25) will be satisfied. We cosider the miimal fractio of the wealth allocated to the i-th asset equal to zero i this paper (l i = 0). I order to satisfy the costrait correspodig to maximal fractio of wealth i each asset, the adjusted value of each gee obtaied by Equatio (39) is compared with predetermied maximal fractio for its correspodig asset (u i ), ad it will be reduced to u i if it is greater tha its predetermied value. The, the excess amout of the gee will be added to other ozero gees. If all gees have their maximum predetermied value, they all will be reduced to u i. Sice λ is oe of the iputs of the problem, before ruig the algorithm the costrait of Equatio (37) will be cotrolled already Fitess fuctio The fitess of each chromosome is evaluated by a fitess fuctio. Sice all costraits have bee satisfied so far leavig the costrait of Equatio (36), we cosider it while formulatig the fitess fuctio. First, we defie a pealty fuctio to esure that Equatio (36) is satisfied ad to make the actual retur of the portfolio exceed miimum expected retur. Thus λ (e C(x)), if λ > (e C(x)) p(x) = { 0, otherwise, where e = j=1 i=0 x i t i/j. Icorporatig the objective fuctio (33) ad the pealty fuctio (40), the fitess fuctio of the algorithm ca be defied as follows fitess = exp ( k(f 2 (x) + M. p(x))), where k is a positive costat ad M a large positive umber. The egative expoet trasforms the miimizatio problem ito its equivalet maximizatio problem for GA to solve. I additio, the expoetial fuctio with costat k cofies the fitess rage ad thus alleviates the selectio pressure of chromosomes with higher fitess, to prevet the GA from premature covergece [48]. O the other had, The large positive umber M forces the solutio to meet costrait (36) before miimizig the portfolio risk Geetic operators Crossover operator We use roulette wheel method i this paper for selectig chromosomes out of their populatio i order to perform crossover. For this purpose, we have to select a umber of chromosomes from populatio relative to crossover rate (p c ). The probability of the i-th chromosome of beig selected is equal to p i = (42) N f i where f i is the fitess value of chromosome i ad N is the umber of idividual chromosomes i populatio. Parets are radomly ad pairwise selected from chose chromosomes ad the the crossover operator is performed to produce ew offsprig. Here we use oe-poit crossover. As metioed earlier, the value of the first gee is equal to the proportio of wealth ivested i risk-free asset which is obtaied from Equatio (22), ad therefore it should t be cosidered durig crossover process. Thus, a radom umber is geerated betwee 2 ad the legth of the chromosome, the all gees of paret chromosomes are swapped beyod that poit. f i (40) (41) 13

14 Mutatio operator Chromosomes selectio for mutatio operator is totally radom ad their quatity is determied based o mutatio rate (p m ). We use two approaches i this paper for radom mutatio; radom swap of gees ad radom replacemet of them. The former icludes radom selectio of two oegative gees ad swappig them with each other. The latter icludes radom selectio of oe gee ad chagig its value by iitializatio mechaism metioed i sectio It must be oted that the mutatio does ot apply to the first gee (i.e., risk-free asset). Elitism To maitai ad use best solutios i previous geeratios, we use a elitism operator i which it trasfers the best solutios of each iteratio to the ext geeratio without ay chage. The elitism rate is calculated by the followig formula p r = 1 p c p m. (43) 5. Computatioal results I this sectio we preset umerical examples to test proposed ucertai portfolio rebalacig model ad to illustrate correspodig computatioal results. To do so, we cosider three cotexts i which we study the performace of the model. First, we study the proposed GA i compare with a globally optimizatio solver. The, the performace of CPPI strategy is studied uder the proposed ucertai portfolio rebalacig model. Fially, the impact of belief degrees ad cosiderig ucertai parameters i the model is ivestigated. GA performace The mathematical model is solved usig BARON Solver i GAMS [49]. Moreover, the proposed geetic algorithm is coded ad implemeted i MATLAB, ad o a persoal computer with a 2.2 GHz Itel Core i7 CPU ad 4 GB of RAM. To geerate rebalacig problems, 100 securities are selected from Tehra Stock Exchage (TSE) ad usig experts evaluatios, ormal ucertaity distributios are estimated for the returs o each asset. Evetually, 11 sample problems are divided ito 3 differet sizes of small, medium ad large i order to evaluate the performace of the proposed model. Table 5 represets differet sizes of sample problems ad correspodig umber of assets i each oe. Table 5. Problem size ad umber of assets Problem Size Small Medium Large Quatity of Risk- Quatity of Risky Free Asset(s) Asset(s) To determie the feasible regio of each problem we first code the ucertai portfolio rebalacig model i GAMS, ad by ruig BARON computatioal system, a feasible set is specified for λ. The by ruig GA i the search space ad for differet values of λ, the correspodig results of 5 times implemetatio of the algorithm are recorded for each problem. Besides, Figure 1 illustrates a example of GA covergece diagram for the proposed ucertai portfolio rebalacig model. To compare the results, we calculate the ratio of performace deviatio (RPD) for BARON ad the proposed GA. RPD represets the superiority of GA results over BARON i which it meas the more 14

15 egative the RPD is, the lower is the risk obtaied by GA i compare with BARON. RPD is calculated by the followig formula RPD = Risk GA Risk BARON Risk BARON 100 (44) Figure 1. Covergece of the GA to maximize the fitess fuctio Solvig the portfolio rebalacig problem with ucertai parameters, we observed that BARON does ot guaratee globally optimal solutio, ot eve after fidig 20 feasible solutios for each sample problem, ad the solver is termiated by reachig its time limit. Table 6 represets the results of solvig the model by GA ad BARON icludig average processig time (CPU time) ad the correspodig RPD calculated for each sample problem. I additio, for better uderstadig of results, correspodig efficiet frotiers for each sample problem are illustrated i Figure 2, Figure 3 ad Figure 4, for small, medium ad large size problems, respectively. It ca be see that icreasig the problem size, makes the gap betwee BARON ad GA results greater while GA provides better solutios. Moreover, by icreasig the problem size, BARON processig time sharply icreases while GA reaches better solutios i more reasoable times. Table 6. Results correspodig to GAMS & GA i differet problem sizes Quatity of Risky Asset(s) Average Processig Time (Sec) BARON GA RPD% % % % % % % % % % % % CPPI performace 15

16 To study the performace of CPPI strategy i our proposed model, we used daily prices of 50 securities i Tehra Stock Exchage from late March 2015 to early July Cosiderig iitial wealth of 100,000 USD, ad the floor value of 70,000 USD, the ucertai portfolio rebalacig problem is solved cotiuously ad the best portfolio is selected i each period. I additio, for better uderstadig of CPPI performace, we also solve the model by buy-ad-hold (B&H) strategy cosiderig mothly rebalacig ad the results correspodig to the fial wealth after rebalacig are compared. Moreover, to better realize the effects of each strategy o model, we simulate hypothetical bull market ad bear market situatios ad the portfolio rebalacig model is also solved cosiderig these situatios. Figure 2. Efficiet frotiers obtaied by BARON solver vs. GA, for small problem with 10 risky assets (Left), with 15 risky assets (Middle), ad with 20 risky assets (Right) Figure 3. Efficiet frotiers obtaied by BARON solver vs. GA, for medium problem with 30 risky assets (Left), with 40 risky assets (Middle), ad with 50 risky assets (Right) Figure 5 (Left) illustrates CPPI performace versus B&H i flat market. Sice CPPI allocates more moey to risky assets as the total wealth icreases, there will be a upside potetial for CPPI strategy by icreasig stock prices. I meatime, if there is a drop i stock prices (i which CPPI has allocated more moey to), there will be more reductio o wealth level for CPPI tha for B&H. This ca be observed i Figure 5 (Left) as the wealth level o CPPI lowers ito the B&H wealth level. Figure 5 (Middle) illustrate similar situatio for the total wealth level i a bear market. It ca be see that CPPI better cotrols risk tha B&H as the prices fall. Although allocatig more moey to stocks at first has caused CPPI wealth levels to declie faster i compare with B&H, by gradually trasferrig moey to risk-free bods, CPPI isures that the wealth will ever fall below the floor of 70,000. Figure 5 (Right) compares the two strategies i a bull market. As the prices rise, it ca be observed that CPPI has more potetial i gaiig profit ad icreasig the portfolio retur, ad cosequetly icreasig the total wealth. 16

17 Figure 4. Efficiet frotiers obtaied by BARON solver vs. GA, for large problem with 60 risky assets (Up-Left), with 70 risky assets (Up-Middle), with 80 risky assets (Up-Right), with 90 risky assets (dow-left), ad with 100 risky assets (Dow-Right) Figure 5. Performace of CPPI vs. B&H i chagig portfolio value (total wealth) i Flat market (Left), Bear market (Middle), ad Bull market (Right) Belief degrees effect Belief degrees are results of experts subjective evaluatios, ad therefore differet people might produce differet belief degrees. That is why it is importat to ivestigate the impact of chages i belief degrees i proposed ucertai portfolio rebalacig problem. For this purpose, we cosider six differet levels for belief degrees o 10 risky assets where from level 1 to 6, the belief degrees become more coservative ad cotai wider rages. Normal ucertaity distributios estimated for each asset i each level are show i Table 7. The zeroth asset i each level represets the risk-free participatio bods. The ucertai portfolio rebalacig problem is solved for each level. Figure 6 illustrates the efficiet frotiers correspodig to each level. Accordig to the efficiet frotiers it ca be observed that i a costat risk 17

18 level we will gai lower returs as the belief degrees become wider (from level 1 to level 6), which meas that experts evaluatios or their prefereces have bee more coservative. 6. Coclusio ad future research I this paper, the portfolio rebalacig problem was modeled cosiderig ucertai variables, trasactio costs ad costat-proportio portfolio isurace (CPPI) as rebalacig strategy. Our proposed model was solved usig BARON Solver i GAMS ad Geetic Algorithm i MATLAB. BARON is a computatioal system for solvig o-covex optimizatio problems to global optimality. Solvig umerical examples with real data showed that BARON does ot guaratee global optimality, ot eve after fidig 20 feasible solutios for each example, ad the solver is termiated by reachig its time limit. Therefore, it is recommeded that a meta-heuristic algorithm such as GA be utilized i order to reduce processig time ad to obtai better solutios especially i large size problems. The results showed that the proposed algorithm i this paper performs well. I compare with BARON solver, it reduced the processig time by 90% o average, ad improved the portfolio risk by 2%, 6% ad 18% for small, medium ad large problems, respectively. Moreover, the efficiet frotiers (Pareto solutios) obtaied by GA are more preferable, especially for problems with large sizes. O the other had, cosiderig differet levels of belief degrees cofirmed chagig i model solutios ad efficiet frotier diagrams. Therefore, cosiderig ucertai variables affects portfolio rebalacig model ad wider belief degree rages make more coservative results. Furthermore, CPPI strategy performs as a isurace mechaism ad limits dowside risk i bear markets while it allows potetial beefit i bull markets. Therefore, CPPI strategy has better performace tha buy-ad-hold strategy i portfolio rebalacig problem especially i bearish ad bullish markets. Future researches ca iclude liquidity, price volatility, jumps i asset prices, the possibility of loas ad short sellig i their model i order to better illustrate real market situatios. I additio, i order to improve the performace of the algorithm, future researches ca cosider other meta-heuristics such as artificial bee coloy algorithm. Furthermore, it is recommeded to study the performace of the model i other markets such real states. Fially, cosiderig dyamic values for multiplier m i CPPI strategy, discrete-time CPPI, ad applyig other strategies icludig OBPI i ucertai portfolio rebalacig problem is suggested for future studies. Table 7. Estimated ucertaity distributios of returs o each asset for differet belief degree levels Asset i Level 1 Level 2 Level 3 0 Costat Costat Costat N( , ) N( , ) N( , ) 2 N( , ) N( , ) N( , ) 3 N( , ) N( , ) N( , ) 4 N( , ) N( , ) N( , ) 5 N( , ) N( , ) N( , ) 6 N( , ) N( , ) N( , ) 7 N( , ) N( , ) N( , ) 8 N( , ) N( , ) N( , ) 9 N( , ) N( , ) N( , ) 10 N( , ) N( , ) N( , ) Asset i Level 4 Level 5 Level 6 0 Costat Costat Costat N( , ) N( , ) N( , ) 2 N( , ) N( , ) N( , ) 3 N( , ) N( , ) N( , 0.031) 4 N( , ) N( , ) N( , ) 5 N( , ) N( , 0.023) N( , ) 6 N( , ) N( , ) N( , ) 7 N( , ) N( , ) N( , ) 8 N( , ) N( , ) N( , ) 9 N( , ) N( , ) N( , ) 10 N( , ) N( , ) N( , ) 18

19 Figure 6. Efficiet frotiers correspodig to differet levels of belief degrees. The more coservative the belief degrees are evaluated, the lower the efficiet frotier curve will be. Refereces [1] C. Xu, K. Kijima, J. Wag, ad A. Ioue, "Portfolio rebalacig with VaR as risk measure," Iteratioal Joural of Iovative Computig, Iformatio ad Cotrol, vol. 4, pp , [2] E. G. Lim, Do Reserve Portfolios Respod to Exchage Rate Chages Usig a Portfolio Rebalacig Strategy? a Ecoometric Study Usig Cofer Data: Iteratioal Moetary Fud, [3] P. Gupta, G. Mittal, ad M. K. Mehlawat, "Expected value multiobjective portfolio rebalacig model with fuzzy parameters," Isurace: Mathematics ad Ecoomics, vol. 52, pp , [4] W. Su, A. Fa, L.-W. Che, T. Schouweaars, ad M. A. Albota, "Optimal rebalacig for istitutioal portfolios," The Joural of Portfolio Maagemet, vol. 32, pp , [5] Y. Fag, K. K. Lai, ad S.-Y. Wag, "Portfolio rebalacig model with trasactio costs based o fuzzy decisio theory," Europea Joural of Operatioal Research, vol. 175, pp , [6] M. E. Fadaei-Nezad ad H. Baaeia, "Portfolio rebalacig model with trasactio costs based o fuzzy decisio theory," preseted at the 8th Iteratioal Maagemet Coferece, Tehra, [7] J.-R. Yu ad W.-Y. Lee, "Portfolio rebalacig model usig multiple criteria," Europea Joural of Operatioal Research, vol. 209, pp , [8] M. Woodside-Oriakhi, C. Lucas, ad J. E. Beasley, "Portfolio rebalacig with a ivestmet horizo ad trasactio costs," Omega, vol. 41, pp , [9] P. Gupta, G. Mittal, ad M. K. Mehlawat, "A multicriteria optimizatio model of portfolio rebalacig with trasactio costs i fuzzy eviromet," Memetic Computig, vol. 6, pp , [10] M. Wag, C. Li, H. Xue, ad F. Xu, "A New Portfolio Rebalacig Model with Trasactio Costs," Joural of Applied Mathematics, vol. 2014, [11] W. Che, H. Ma, Y. Yag, ad M. Su, "Applicatio of artificial bee coloy algorithm to portfolio adjustmet problem with trasactio costs," Joural of Applied Mathematics, vol. Volume 2014, p. 12, [12] Z. Qi, S. Kar, ad H. Zheg, "Ucertai portfolio adjustig model usig semiabsolute deviatio," Soft Computig, pp. 1-9, [13] M. Rabbai, "The effect of multi period model Portfolio Ivestmet Rebalacig o Retur of portfolio, by Metaheuristic Algorithm," Master of Sciece, Departmet of Fiacial Egieerig, Raja Uiversity, Qazvi, [14] P. Kumar, G. Pada, ad U. Gupta, "Portfolio rebalacig model with trasactio costs usig iterval optimizatio," OPSEARCH, pp. 1-34, [15] X. Huag, "Mea-risk model for ucertai portfolio selectio," Fuzzy Optimizatio ad Decisio Makig, vol. 10, pp ,

20 [16] B. Liu, Ucertaity Theory. Berli: Spriger, [17] B. Liu, Ucertaity Theory, Fifth ed. Beijig , Chia: Departmet of Mathematical Scieces, Tsighua Uiversity, [18] L. Ya, "Optimal portfolio selectio models with ucertai returs," Moder Applied Sciece, vol. 3, p. p76, [19] B. Zhag, J. Peg, ad S. Li, "Ucertai programmig models for portfolio selectio with ucertai returs," Iteratioal Joural of Systems Sciece, pp. 1-10, [20] W. Che, "A Artificial Bee Coloy Algorithm for Ucertai Portfolio Selectio," The Scietific World Joural, vol. 2014, [21] G. Mittal ad M. K. Mehlawat, "A multiobjective portfolio rebalacig model icorporatig trasactio costs based o icremetal discouts," Optimizatio, vol. 63, pp , [22] R. Masii, W. Ogryczak, ad W. G. Speraza, "Rebalacig ad Idex Trackig," i Liear ad Mixed Iteger Programmig for Portfolio Optimizatio, ed: Spriger, 2015, pp [23] P. A. Krokhmal, S. P. Uryasev, ad G. M. Zrazhevsky, "Comparative Aalysis of Liear Portfolio rebalacig strategies: a applicatio to hedge fuds," U of Florida ISE Research Report, [24] B. Liu, Ucertaity theory: a brach of mathematics for modelig huma ucertaity, 3rd ed. Berli: Spriger, [25] A. F. Perold ad W. F. Sharpe, "Dyamic strategies for asset allocatio," Fiacial Aalysts Joural, pp , [26] R. Cot ad P. Takov, "Costat proportio portfolio isurace i the presece of jumps i asset prices," Mathematical Fiace, vol. 19, pp , [27] F. Black ad A. Perold, "Theory of costat proportio portfolio isurace," Joural of Ecoomic Dyamics ad Cotrol, vol. 16, pp , [28] H. Markowitz, "Portfolio selectio*," The joural of fiace, vol. 7, pp , [29] X. Huag, Portfolio aalysis: from probabilistic to credibilistic ad ucertai approaches vol. 250: Spriger, [30] K. Miettie, Noliear Multiobjective Optimizatio: Spriger US, [31] G. Mavrotas, "Effective implemetatio of the ε-costrait method i multi-objective mathematical programmig problems," Applied mathematics ad computatio, vol. 213, pp , [32] Z. Beheshti ad S. Shamsuddig, "A review of populatio-based meta-heuristic algorithms," It. J. Adv. Soft Comput. Appl, vol. 5, pp. 1-35, [33] O. Shahvari ad R. Logedra, "A Ehaced tabu search algorithm to miimize a bi-criteria objective i batchig ad schedulig problems o urelated-parallel machies with desired lower bouds o batch sizes," Computers & Operatios Research, [34] O. Shahvari ad R. Logedra, "Hybrid flow shop batchig ad schedulig with a bi-criteria objective," Iteratioal Joural of Productio Ecoomics, vol. 179, pp , [35] O. Shahvari, N. Salmasi, R. Logedra, ad B. Abbasi, "A efficiet tabu search algorithm for flexible flow shop sequece-depedet group schedulig problems," Iteratioal Joural of Productio Research, vol. 50, pp , [36] J. Sadeghi ad S. T. A. Niaki, "Two parameter tued multi-objective evolutioary algorithms for a bi-objective vedor maaged ivetory model with trapezoidal fuzzy demad," Applied Soft Computig, vol. 30, pp , [37] S. M. Mousavi, V. Hajipour, S. T. A. Niaki, ad N. Aalikar, "A multi-product multi-period ivetory cotrol problem uder iflatio ad discout: a parameter-tued particle swarm optimizatio algorithm," The Iteratioal Joural of Advaced Maufacturig Techology, vol. 70, pp , [38] S. M. Mousavi, A. Bahreiiejad, S. N. Musa, ad F. Yusof, "A modified particle swarm optimizatio for solvig the itegrated locatio ad ivetory cotrol problems i a two-echelo supply chai etwork," Joural of Itelliget Maufacturig, pp. 1-16, [39] S. H. R. Pasadideh, S. T. A. Niaki, ad S. M. Mousavi, "Two metaheuristics to solve a multi-item multiperiod ivetory cotrol problem uder storage costrait ad discouts," The Iteratioal Joural of Advaced Maufacturig Techology, vol. 69, pp , [40] S. M. Mousavi ad S. T. A. Niaki, "Capacitated locatio allocatio problem with stochastic locatio ad fuzzy demad: a hybrid algorithm," Applied Mathematical Modellig, vol. 37, pp ,