research article a new portfolio rebalancing model with...

Research ArticleA New Portfolio Rebalancing Model with Transaction Costs

Meihua Wang1 Cheng Li1 Honggang Xue1 and Fengmin Xu2

1 School of Finance and Economics Xirsquoan Jiaotong University Xirsquoan 710061 China2 School of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Fengmin Xu fengminxumailxjtueducn

Received 4 December 2013 Revised 1 March 2014 Accepted 4 March 2014 Published 28 April 2014

Academic Editor Farhad Hosseinzadeh Lotfi

Copyright copy 2014 Meihua Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A portfolio rebalancing model with self-finance strategy and consideration of V-shaped transaction cost is presented in this paperOurmain contribution is that a new constraint is introduced to confirm that the rebalance necessity of the existing portfolio needs tobe adjustedThe constraint is constructed by considering both the transaction amount and transaction cost without any additionalsupply to the investment amount The V-shaped transaction cost function is used to calculate the transaction cost of the portfolioand conditional value at risk (CVaR) is used to measure the risk of the portfolios Computational tests on practical financial datashow that the proposed model is effective and the rebalanced portfolio increases the expected return of the portfolio and reducesthe CVaR risk of the portfolio

1 Introduction

During the past six decades many papers have been writtenon the theory and practice of portfolio selection with mostresearchers concentrating on the initial investment Howeverwith the elapse of time the initial portfolio may becomenonoptimal If an investor hopes to hold the investment onthe portfolio in the following periods it is necessary to adjustthe portfolio based on either maximizing expected return ofthe portfolio or minimizing the risk of the portfolio Thisis the so-called portfolio rebalancing (revision adjusting)problem [1]

Our main concern is self-finance portfolio rebalancingproblem which means that the investor will not supply anyadditional investment amount Most researchers focus on themodeling and algorithm design based on various transactioncost functions Jouini and Kallal [2] establish the theorywith consideration of transaction costs based on martin-gale measures Arnott and Wagner [3] find that ignoringtransaction costs often results in an inefficient portfolio inpractice Yoshimoto [4] also reaches the same conclusionthrough empirical analysis Konno and Yamamoto [5] aimto minimize the transaction costs associated with rebalancein the framework of mean absolute deviation with concave

transaction cost function Then they obtain a concave mini-mization problem and propose a branch-and-bound methodto get optimal rebalance amount Lobo et al [6 7] Bestand Hlouskova [8] propose a self-finance strategy with lineartransaction costs A self-finance constraint is included inthe portfolio rebalancing model The investor aims eitherto maximize the expected return of the resulting portfolioafter paying transaction costs under a given tolerated level ofrisk and other constraints or to minimize the total rebalancetransaction costs subject to a specified requirement on theexpected return of the portfolio risk self-finance and otherconstraints Best and Hlouskova [9] consider the problem ofmaximizing as an expected utility function of 119899 assets withlinear transaction costs and present a method for solvingthe 3n-dimensional problem based on optimality conditionsfor the higher-dimensional problem The new method iscompared to the barrier method implemented in Cplex in aseries of numerical experiments and outperforms the barriermethod by a larger and larger factor as the size of thetransaction cost increases Mitchell and Braun [10] extendthe standard portfolio selection problem to consider convextransaction costs incurred when rebalancing an investmentportfolio They suggest rescaling by the funds availableafter paying transaction costs and then obtain a fractional

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 942374 7 pageshttpdxdoiorg1011552014942374

2 Journal of Applied Mathematics

programming problem which can be reformulated as anequivalent convex program of size comparable to the modelwithout transaction costs However existing models do notconsider whether the initial portfolio is worthwhile to adjustor not

In the existing literature on portfolio rebalancing prob-lem variance [6 8 11] absolute deviation [5] and entropicrisk measure [12] are usually used to measure the risk of theportfolio In recent years shortfall measure has become sopopular and practical in the risk management area Value atrisk (VaR) is an important shortfall risk measurement butit has some undesirable mathematical characteristics such aslack of subadditivity and convexity [13] VaR is not coherentexcept the case when the return rate of portfolios is ofnormal distribution To overcome these shortcomings recentresearch focuses on coherent riskmeasurement especially onCVaR Tyrrell Rockafellar andUryasex [14] propose the use ofCVaR in portfolio selection problemsThemain advantage ofCVaR is thatwhetherCVaR is used to be an objective functionor a constraint in portfolio selection problems the resultingmodel can be converted into a linear programming problembased on discrete samples [15] Furthermore there is also anequivalent linear programming if one deals with continuousproblems [16]

Our main contribution is adding a constraint to con-firm the necessity of rebalance The portfolio rebalancingproblemwith self-finance strategy is modified by introducinga constraint of expected excess return requirement in thispaper CVaR is used to measure the risk of portfoliosTransaction cost is one of the main factors for an investorto take into account in adjusting the existing portfolio Itis commonly assumed in literature that the transaction costis a V-shaped function of the transaction amount and thenwe accept this However the proposed model can be easilyextended to handle more complex transaction cost functionsThe expected excess return on portfolios is specified by theinvestor and when the expected excess return is satisfied therebalancing on the initial portfolio will be performed One ofthemain contributions of this paper is to construct the returnrequirement constraint to confirm the necessity of rebalance

The rest of the paper is organized as follows The newmodel with self-finance strategy is presented in Section 2where the constraint for expected excess return of portfolioswill be given Computational tests are stated in Section 3Conclusion and future researches are given in Section 4

Throughout the paper prime (1015840) denotes transposition ofvectors without special declaration The notation 119904

119894or (119904)119894is

used to denote the 119894th component of the vector 119904

2 A New Portfolio Rebalancing Model

The portfolio rebalancing model with self-finance strategyand an excess return requirement will be presented in thissection It is assumed that the investor holds a portfolio whichis optimal at the beginning of a previous period Howeverwith the elapse of time the initial portfolio needs to beevaluated whether it is still optimal or not due to changesof conditions in financial markets or disclosure of much

more information If it becomes nonoptimal rebalancingprocedure will be carried out on the portfolio withoutproviding any additional supply to pay for the transactioncosts That means that paying for transaction costs will comefrom the capital of selling some assets The rebalancing goalin [7] is to maximize the expected return of the portfoliosubject to some practical constraints Different from [7]the rebalancing goal of this paper is to minimize the CVaRrisk of the portfolio subject to some practical constraintsincluding specified excess return when the initial portfolioneeds to be adjusted The problem whether the portfolioneeds to be adjusted will be controlled by the constraint theexpected excess return should be satisfied Only when this istrue the portfolio rebalancing procedure is worthwhile to beperformed

21 The Requirement of Excess Return Assume that aninvestor holds a portfolio consisting of 119899 assets and 119904 =

(1199041 1199042 119904

119899)1015840 is used to denote the initial portfolio where 119904

119894

denotes the amount (such as dollars) invested on the 119894th assetLet 119903 = (119903

1 1199032 119903

119899)1015840 be the return rates of the assets with

119903 = (1199031 1199032 119903

119899)1015840 being the expected return rate of the given

period Let 119909 = (1199091 1199092 119909

119899)1015840 be the amount transacted

in all assets with 119909119894gt 0 for buying and 119909

119894lt 0 for selling

After transacting the resulting portfolio is 119904 + 119909The expectedreturn from the adjusting is

1199031015840119909 (1)

The transaction cost associated with transaction amount 119909 isgiven by

119862 (119909) =

119899

sum

119894=1

119862119894(119909119894) (2)

where 119862119894(119909119894) is the transaction cost on the 119894th asset when 119909

119894

amount is traded Hence the expected excess return from theadjusting is

1199031015840119909 minus 119862 (119909) (3)

It can be seen that the transaction costs associated withpurchasing or selling have a significant effect in portfoliorebalancing problem

Let 120572 gt 0 be an additional excess return requirementgiven by the investor Then the constraint controlling theportfolio rebalancing can be expressed as

1199031015840119909 minus 119862 (119909) ge 120572 (4)

It can be seen from (4) that if there exists no feasiblesolution119909 to satisfy (4) the rebalancing on the portfolio is notnecessary and the investor can still hold the portfolio Hencethe new portfolio rebalancing model is expressed as follows

min 119891 (119904 + 119909)

st 1199031015840119909 minus 119862 (119909) ge 120572

119909 isin M

(119875)

Journal of Applied Mathematics 3

where 119891(119904 + 119909) is a risk measure of the portfolio 119904 + 119909

and M is the set of feasible portfolios restricted by somepractical constraints which will be discussed in the followingsubsections

22 The CVaR Risk of Portfolios It is known that CVaR is acoherent risk measure and becomes more and more popularin practical risk management area Moreover no matter thatthe CVaR is used as an objective function or a constraint theresulting portfolio rebalancing model can be converted intoa linear programming problem

Let 119897(119909 119903) be the loss function associatedwith the decisionvector 119909 and the return vector 119903 of portfolios Let 119901(119903) be thedensity function of the return vector 119903 Then the probabilityof 119897(119909 119903) not exceeding a threshold 119908 is given by

120595 (119909 119908) = int

119897(119909119903)le119908

119901 (119903) 119889119903 (5)

Definition 1 ((VaR) [14]) The VaR risk of the loss associatedwith a decision vector 119909 and a specified probability level 120579 isin

(0 1) is the value

VaR120579(119909) = min 119908 isin R 120595 (119909 119908) ge 120579 (6)

Definition 2 ((CVaR) [14]) The CVaR risk of the loss associ-ated with a decision vector 119909 and a specified probability level120579 isin (0 1) is given by

CVaR120579(119909) =

1

1 minus 120579

int

119897(119909119903)geVaR120579(119909)119897 (119909 119903) 119901 (119903) 119889119903 (7)

Substituting (7) into the problem (119875) generates

min119909

1

1 minus 120579

int

119897(119909119903)geVaR120579(119909)119897 (119909 119903) 119901 (119903) 119889119903

st 1199031015840119909 minus 119862 (119909) ge 120572

119909 isin M

(8)

Since the value of VaR120579(119909) in the problem is unknown it is

difficult to solve the problem Tyrrell Rockafellar andUryasexin [14 15] propose the following auxiliary function to replacethe CVaR risk function

119865120579(119909 119908) = 119908 +

1

1 minus 120579

E [119897 (119909 119903) minus 119908]+

= 119908 +

1

1 minus 120579

int [119897 (119909 119903) minus 119908]+119901 (119903) 119889119903

(9)

Here E120585 denotes the expectation of the random variable 120585and [119897(119909 119903) minus 119908]

+= max 119897(119909 119903) minus 119908 0

Theorem 3 (see [14]) Minimizing CVaR of the loss associatedwith 119909 and 120579 over all 119909 isin X is equivalent to minimizing119865120579(119909 119908) over all (119909 119908) isin X times R in the sense that

min119909isinX

CVaR120579(119909) lArrrArr min

(119909119908)isinXtimesR119865120579(119909 119908) (10)

The theorem indicates that the solutions of both opti-mization problems are the same When rebalancing is con-sidered for a portfolio 119904 the loss function 119897(119909 119903) = minus(119904 + 119909)

1015840119903

Then the objective function associated with a given probabil-ity 120579 isin (0 1) can be expressed as

min(119909119908)isin(XtimesR)

119865120579(119909 119908)

= min(119909119908)isin(XtimesR)

119908 +

1

1 minus 120579

int [minus(119904 + 119909)1015840119903 minus 119908]

+

119901 (119903) 119889119903

(11)

The function 119865120579(119909 119908) can further be approximated when

the values of the return vector 119903 are given by a collectionof discrete vectors 119903

1 1199032 119903

119879 where 119903

119905 is the returnrate of assets at the subperiod 119905 then a correspondingapproximation to the function 119865

120579(119909 119908) is

119865120579(119909 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905minus 119908]

+

(12)

Although the function 119865120579(119909 119908) is not differentiable the

resulting problem can be readily solved either by line searchtechniques or by converting it into a linear programmingproblem

23 The Self-Finance Constraint The new model is estab-lished under the self-finance strategy The self-finance con-straint can be expressed by the total value of the initialportfolio equal to the sum of the value of the final portfolioand the transaction costs

11015840 (119904 + 119909) + 119862 (119909) = 11015840119904 (13)

where 1 is an 119899-dimensional row vector with all entries equalto one Equation (13) can be simplified as

11015840119909 + 119862 (119909) = 0 (14)

This constraint says that the total transaction costs areobtained by selling assets more than buying assets

24 Short Sale Constraints Due to market regulations shortsale constraints should be considered Short sale limitation ineach asset is represented by

119904119894+ 119909119894ge minus119901119894 119894 = 1 2 119899 (15)

where 119901119894is the maximum amount of short sale permitted for

the 119894th asset If short sale is not permitted 119901119894is set to 0

25 Constraints for Share Restrictions Due to investorrsquos lim-ited investment capability and diversification requirementthe investment amount or proportion on each asset has alower bound and an upper bound Consider

119897119894le 119904119894+ 119909119894le 119906119894

(16)


or

119871119894le

119904119894+ 119909119894

sum119899

119894=1119904119894

le 119880119894 (17)

where 119897119894(or 119871119894) and 119906

119894(or 119880119894) are the lower bound and the

upper bound of the investment amount (or proportion) onthe 119894th asset after transacting If short sale is not permittedthen 119897

119894ge 0 and short sale constraint (15) can be omitted

With all of the above constraints the new rebalancemodel is given as follows

min119909119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905 minus 119908]

+

st 1199031015840119909 minus 119862 (119909) ge 120572

11015840119909 + 119862 (119909) = 0

119897119894le 119904119894+ 119909119894le 119906119894 119894 = 1 2 119899

(18)

Let 119889119861

119894(119889119878119894) be the transaction cost ratio for buying or

selling one unit amount of the 119894th asset 119894 = 1 2 119899 Then

119862119894(119909119894) =

119889119861

119894119909119894 if 119909

119894gt 0

minus119889119878

119894119909119894 if 119909

119894lt 0

(19)

Variables 119909+119894and 119909

minus

119894 119894 = 1 2 119899 are introduced for

convenience of calculation with 119909+

119894= max0 119909

119894 and 119909

minus

119894=

minusmin0 119909119894 Then 119909+

119894ge 0 119909minus

119894ge 0 for 119894 = 1 2 119899 and the

transaction costs function 119862119894(119909119894) could be represented as

119862119894(119909119894) = 119889119861

119894119909+

119894+ 119889119878

119894119909minus

119894 (20)

where we have 119909119894= 119909+

119894minus 119909minus

119894

Since there are [sdot]+ representations in (18) the problem isnot differentiable Let

119902119905= [minus(119904 + 119909)

1015840119903119905minus 119908]

+

119905 = 1 2 119879 (21)

Then

119902119905ge minus(119904 + 119909)

1015840119903119905minus 119908 119905 = 1 2 119879 (22)

and the problem (18) can be converted into the following

min119909+119909minus119902119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

119902119905

st119899

sum

119894=1

119903119894(119909+


119894)

minus

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) ge 120572

119902119905ge minus(119904 + 119909

+minus 119909minus)

1015840

119903119905minus 119908

119899

sum

119894=1

(119909+


119894) +

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) = 0

119897119894le 119904119894+ 119909+


119894le 119906119894

119902119905ge 0 119909

+

119894ge 0 119909

minus

119894ge 0

119894 = 1 2 119899 119905 = 1 2 119879

(23)

Problem (23) is a linear programming and algorithms such asthe simplex algorithm and interior point algorithms [17 18]can be employed to find its solution

3 Computational Results

In this section wewill test the proposed portfolio rebalancingmodel with practical data Assume that the investor holdsan existing portfolio which is an optimal portfolio investedone year ago As time goes by the return and the risk of theexisting portfolio have changed and whether the portfolio isstill optimal is unknown The proposed rebalancing modelwill be used to help the investor to decide whether to adjustthe portfolio or not

31 Test Data Sets and Steps Themodel will be tested on thedata sets described in [19] Beasley has built an OR-Librarywhich is a publicly available collection of test data sets for avariety of operations research problems A series of weeklyclosing prices from 1992 to 1994 for 15 component stocks ofSampP index are selected

The data sets include 105 history weekly prices of the 15selected component stocks by which we could obtain 104weekly return rates for each stock (it is assumed that there are52 weeks per year) The time period for each week is denotedas 119905 = 1 2 104 An initial portfolio is constructed basedon the first 52 return rates of the 15 stocks at time 119905 = 52After one year later (at time 104) it is necessary to evaluate theoptimality of the existing portfolio based on the last 52 returnrates of these stocks The test experiments will be performedin the following two stages

Stage 1 The data sets are divided into two parts The first52 return rates (one year) are applied to construct the initialportfolio The investor follows the initial portfolio to makeinvestments on the selected stocks at time 119905 = 52

Stage 2 After one year has passed (at time 119905 = 104) theproposedmodelwill be used to evaluatewhether the portfoliois still optimal or not by using the last 52 return rates If it isnot optimal adjusting to the portfolio is necessary and theoptimal adjusting strategywill be given by using the proposedrebalancing model

All the computational tests are run on a personal com-puter with Pentium Pro 1794MHZ and 512MB memory

32 The Initial Portfolio Assume that the investor has theinitial portfolio with a total wealth of119882 dollar at time 119905 = 52


Transaction costs are not considered in constructing theinitial portfolio so that the total amount of the portfolio is119882 Yearly risk of CVaR is also used to measure the risk of theportfolio The initial portfolio is constructed by minimizingthe CVaR risk of the portfolio subjecting to specified returnrate That is the initial portfolio is constructed based onthe first 52 weekly return rates of the selected 15 componentstocks and it is the solution of the following portfolio selectionmodel

min119904

CVaR120579(119904)

st119899

sum

119894=1

119903119894119904119894ge 120582119882

119899

sum

119894=1

119904119894= 119882

120576119894119882 le 119904

119894le 120575119894119882

119894 = 1 2 119899

(24)

where 119904 = (1199041 1199042 119904

119899)1015840 (119899 = 15) is the initial portfolio 119904

119894

is the investment amount on the 119894th stock at time 119905 = 52 119903119894

is yearly expected return rate of the 119894th stock estimated fromthe data 119903119905

119894over time period 119905 = 1 to 119905 = 52 CVaR

120579(119904) is the

CVaR risk of the portfolio 119904 with a specified confidence level120579 isin (0 1)119882 is the total amount of the portfolio 120582 gt 0 is therequirement of the expected return specified by the investorand 120576119894and 120575119894are the investment lower and upper bound scale

limitations on the 119894th stock Note that the transaction cost tothe initial portfolio is not considered that is the transactioncost will be supplied by the investor so that the total amountof the initial portfolio is119882

Problem (24) could be converted into a linear program-ming by replacing CVaR

120579(119904) with the function

119865120579(119904 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus1199041015840119903119905minus 119908]

+

(25)

where 119879 = 52 The ldquolinprogrdquo function in MATLAB is used tosolve the resulting linear programming

The following parameter values are used in the model togenerate the initial optimal portfolio

(1) confidence level 120579 = 95(2) initial wealth119882 = $100000000(3) 120582 = 10 (the maximal yearly return rate among the

15 stocks is 2766 and the minimal one is minus1425)(4) 119889119894= 0002 for 119894 = 1 2 119899 (transaction cost per unit

amount investment)(5) lower bound 120576

119894= 0 and upper bound 120575

119894= 020 for

investment on each stock

The generated initial portfolio is given in the secondcolumn of Table 1

When model (23) is used to test and to adjust the initialportfolio at time 119905 = 104 the expected return rate vector

119903 is estimated from 119903119905 119905 = 53 54 104 and the other

parameter values such as 120579 119889119894 120576119894 and 120575

119894in the model are kept

unchanged Since problem (23) is a linear programming it isalso solved using the ldquolinprogrdquo function in MATLAB

33 Results and Analysis Test results obtained by model (23)show that the initial portfolio is not optimal at time 119905 =

104 and it is necessary to adjust the portfolio Table 1 givesadjusted portfolio for different levels of requirement specifiedby the value of 120572The second column gives the initial optimalportfolio at time 119905 = 52 The values are the investmentamounts ($) on each stock and the total wealth of the initialportfolio is $100000000The expected wealth of the portfolioafter one year is $113811773 and the CVaR risk of the portfoliois $12807285

The initial portfolio will be held for one year (until at119905 = 104) The third column gives the value of the portfolioon each stock before rebalance at 119905 = 104 It can be seen fromthe results in this column that the total wealth is $100425613before rebalance If the investor holds the portfolio withoutany change then the expected wealth of the portfolio is$104321125 and its CVaR risk is $17050812

The fourth column to sixth column list final portfolios119904 + 119909 generated using model (23) with different values of 120572It can be seen from column 4 that after rebalancing the totalwealth is reduced to $100206042 after rebalance because ofthe transaction cost but the expected wealth of the portfoliois increased and the CVaR risk of the portfolio is greatlyreduced

Figure 1 gives the 120572-CVaR curves for some values of 119889119861119894

(or 119889119878119894) 119894 = 1 2 119899 where the vertical axis denotes the

requirements of expected returns of the portfolio specifiedby the value of 120572 in (23) the horizontal axis denotes thecorresponding CVaR risk of the portfolio and (119904 + 119909) is finalportfolio after rebalancing It can be observed from the figurethat the 120572-(CVaR) is similar to efficient frontiers in portfolioselection and that the value of CVaR increases along withthe requirement raising in expected return Moreover thelarger the 119889

119861 is the higher the curve is When 119889119861 become

larger the investor will endure larger risk (CVaR) for thesame requirement of the expected excess return This can beunderstood from another point of view larger transactioncosts will lead to smaller excess return

4 Conclusion

A new portfolio rebalancing model is proposed The modelis different from the existing models due to the constraint forevaluating rebalance feasibility The constraint is constructedby considering the amount in transaction and the transactioncost Computational tests based on an initial portfolio aremade for different requirements of expected returns Testsshow that the introduced constraint effectively controls therebalance of the portfolio and when the rebalance is neces-sary the proposed model gives an optimal portfolio whichsatisfy the investorrsquos requirement and give some guidancefor the investment behavior in financial market Since theproposed model is based on the condition of single portfolio


Table 1 Results for initial portfolio and final portfolio (unit $)

Stock number list Initial portfolioat 119905 = 52

Portfolio beforerebalanceat t = 104

Final portfolio(119904 + 119909) for

120572 = 001

119899

sum

119894=1

119904119894


120572 = 002

119899

sum

119894=1

119904119894


120572 = 003

119899

sum

119894=1

119904119894

1 0 0 5820477 5845410 58552152 0 0 6151855 7628376 91476493 0 0 5657514 4682076 40624934 1907243 1803596 0 0 05 667971 713507 7866669 8157215 84208336 1702319 1524101 14291614 14119415 139673517 10222692 9770488 20085123 19774512 193714978 13732542 10152372 0 0 09 18972734 19675658 20085123 20085123 2008512310 0 0 2967647 1938595 102095211 0 0 0 0 012 20000000 18535111 0 0 013 20000000 21298069 5005473 3337153 150713814 0 0 3540840 4543087 531419215 12794498 16952710 8733708 10093850 11450478Total wealth 100000000 100425613 100206042 100204812 100202922Expected wealth 113811773 104321125 105325381 106329637 107333893CVaR 12807285 17050812 7433953 7809457 8188147Total transaction costs mdash mdash 219572 220802 222692

06 08 1 12 14 16 18 2 220

2

4

6

8

10

12

14

16

18

Expe

cted

exce

ss re

turn

requ

irem

ent120572

($)

times106

times107

Without TC

CVaR120579(s + x) ($)

With dB = 0002

With dB = 0005

With dB = 001

Figure 1 Return requirement 120572-CVaR curve with and withouttransaction costs

rebalancing application of the new research into dynamicportfolio rebalancing problem would have significant prac-tical value

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China Postdoctoral Science Foun-dation 2013M530418 National Natural Science Foundationsof China 71171158 and 11101325

References

[1] H Konno and A Wijayanayake ldquoMean-absolute deviationportfolio optimization model under transaction costsrdquo Journalof the Operations Research Society of Japan vol 42 no 4 pp422ndash435 1999

[2] E Jouini and H Kallal ldquoMartingales and arbitrage in securitiesmarketswith transaction costsrdquo Journal of EconomicTheory vol66 no 1 pp 178ndash197 1995

[3] R D Arnott andW HWagner ldquoThemeasurement and controlof trading costsrdquo Financial Analysts Journal vol 46 pp 73ndash801990

[4] A Yoshimoto ldquoThe mean-variance approach to portfolio opti-mization subject to transaction costsrdquo Journal of the OperationsResearch Society of Japan vol 39 no 1 pp 99ndash117 1996


[5] H Konno and R Yamamoto ldquoMinimal concave cost rebalanceof a portfolio to the efficient frontierrdquo Mathematical Program-ming B vol 97 no 3 pp 571ndash585 2003

[6] M S Lobo M Fazel and S Boyd ldquoPortfolio optimizationwith linear and fixed transaction costsrdquo Technical ReportDepartment of Electrical Engineering Stanford UniversityStanford Calif USA 2002

[7] M S Lobo M Fazel and S Boyd ldquoPortfolio optimizationwith linear and fixed transaction costsrdquo Annals of OperationsResearch vol 152 pp 341ndash365 2007

[8] M J Best and J Hlouskova ldquoPortfolio selection and transac-tions costsrdquo Computational Optimization and Applications vol24 no 1 pp 95ndash116 2003

[9] M J Best and J Hlouskova ldquoAn algorithm for portfoliooptimization with transaction costsrdquoManagement Science vol51 no 11 pp 1676ndash1688 2005

[10] J E Mitchell and S Braun ldquoRebalancing an investment portfo-lio in the presence of convex transaction costs includingmarketimpact costsrdquo Optimization Methods amp Software vol 28 no 3pp 523ndash542 2013

[11] M J Best and J Hlouskova ldquoQuadratic programming withtransaction costsrdquoComputersampOperations Research vol 35 no1 pp 18ndash33 2008

[12] W Zhong ldquoPortfolio optimization under entropic riskmanage-mentrdquo Acta Mathematica Sinica vol 25 no 7 pp 1113ndash11302009

[13] P Artzner F Delbaen J-M Eber and D Heath ldquoCoherentmeasures of riskrdquo Mathematical Finance An InternationalJournal of Mathematics Statistics and Financial Economics vol9 no 3 pp 203ndash228 1999

[14] R Tyrrell Rockafellar and S Uryasex ldquoOptimization of condi-tional value-at- riskrdquo Journal of Risk vol 2 pp 21ndash41 2000

[15] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[16] A Balbas B Balbas and R Balbas ldquoCAPM and APT-likemodels with riskmeasuresrdquo Journal of Banking and Finance vol34 no 6 pp 1166ndash1174 2010

[17] C Roos T Terlaky and J-Ph Vial Theory and Algorithm forLinear Optimization An Interior Point Approach John Wiley ampSons New York NY USA 1986

[18] Y Y Ye Interior Point Algorithms Theory and Analysis JohnWiley amp Sons New York NY USA 1997

[19] J E Beasley N Meade and T-J Chang ldquoAn evolutionaryheuristic for the index tracking problemrdquo European Journal ofOperational Research vol 148 no 3 pp 621ndash643 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


programming problem which can be reformulated as anequivalent convex program of size comparable to the modelwithout transaction costs However existing models do notconsider whether the initial portfolio is worthwhile to adjustor not

In the existing literature on portfolio rebalancing prob-lem variance [6 8 11] absolute deviation [5] and entropicrisk measure [12] are usually used to measure the risk of theportfolio In recent years shortfall measure has become sopopular and practical in the risk management area Value atrisk (VaR) is an important shortfall risk measurement butit has some undesirable mathematical characteristics such aslack of subadditivity and convexity [13] VaR is not coherentexcept the case when the return rate of portfolios is ofnormal distribution To overcome these shortcomings recentresearch focuses on coherent riskmeasurement especially onCVaR Tyrrell Rockafellar andUryasex [14] propose the use ofCVaR in portfolio selection problemsThemain advantage ofCVaR is thatwhetherCVaR is used to be an objective functionor a constraint in portfolio selection problems the resultingmodel can be converted into a linear programming problembased on discrete samples [15] Furthermore there is also anequivalent linear programming if one deals with continuousproblems [16]

Our main contribution is adding a constraint to con-firm the necessity of rebalance The portfolio rebalancingproblemwith self-finance strategy is modified by introducinga constraint of expected excess return requirement in thispaper CVaR is used to measure the risk of portfoliosTransaction cost is one of the main factors for an investorto take into account in adjusting the existing portfolio Itis commonly assumed in literature that the transaction costis a V-shaped function of the transaction amount and thenwe accept this However the proposed model can be easilyextended to handle more complex transaction cost functionsThe expected excess return on portfolios is specified by theinvestor and when the expected excess return is satisfied therebalancing on the initial portfolio will be performed One ofthemain contributions of this paper is to construct the returnrequirement constraint to confirm the necessity of rebalance

The rest of the paper is organized as follows The newmodel with self-finance strategy is presented in Section 2where the constraint for expected excess return of portfolioswill be given Computational tests are stated in Section 3Conclusion and future researches are given in Section 4

Throughout the paper prime (1015840) denotes transposition ofvectors without special declaration The notation 119904

119894or (119904)119894is

used to denote the 119894th component of the vector 119904

2 A New Portfolio Rebalancing Model

The portfolio rebalancing model with self-finance strategyand an excess return requirement will be presented in thissection It is assumed that the investor holds a portfolio whichis optimal at the beginning of a previous period Howeverwith the elapse of time the initial portfolio needs to beevaluated whether it is still optimal or not due to changesof conditions in financial markets or disclosure of much

more information If it becomes nonoptimal rebalancingprocedure will be carried out on the portfolio withoutproviding any additional supply to pay for the transactioncosts That means that paying for transaction costs will comefrom the capital of selling some assets The rebalancing goalin [7] is to maximize the expected return of the portfoliosubject to some practical constraints Different from [7]the rebalancing goal of this paper is to minimize the CVaRrisk of the portfolio subject to some practical constraintsincluding specified excess return when the initial portfolioneeds to be adjusted The problem whether the portfolioneeds to be adjusted will be controlled by the constraint theexpected excess return should be satisfied Only when this istrue the portfolio rebalancing procedure is worthwhile to beperformed

21 The Requirement of Excess Return Assume that aninvestor holds a portfolio consisting of 119899 assets and 119904 =

(1199041 1199042 119904

119899)1015840 is used to denote the initial portfolio where 119904

119894

denotes the amount (such as dollars) invested on the 119894th assetLet 119903 = (119903

1 1199032 119903

119899)1015840 be the return rates of the assets with

119903 = (1199031 1199032 119903

119899)1015840 being the expected return rate of the given

period Let 119909 = (1199091 1199092 119909

119899)1015840 be the amount transacted

in all assets with 119909119894gt 0 for buying and 119909

119894lt 0 for selling

After transacting the resulting portfolio is 119904 + 119909The expectedreturn from the adjusting is

1199031015840119909 (1)

The transaction cost associated with transaction amount 119909 isgiven by

119862 (119909) =

119899

sum

119894=1

119862119894(119909119894) (2)

where 119862119894(119909119894) is the transaction cost on the 119894th asset when 119909

119894

amount is traded Hence the expected excess return from theadjusting is

1199031015840119909 minus 119862 (119909) (3)

It can be seen that the transaction costs associated withpurchasing or selling have a significant effect in portfoliorebalancing problem

Let 120572 gt 0 be an additional excess return requirementgiven by the investor Then the constraint controlling theportfolio rebalancing can be expressed as

1199031015840119909 minus 119862 (119909) ge 120572 (4)

It can be seen from (4) that if there exists no feasiblesolution119909 to satisfy (4) the rebalancing on the portfolio is notnecessary and the investor can still hold the portfolio Hencethe new portfolio rebalancing model is expressed as follows

min 119891 (119904 + 119909)

st 1199031015840119909 minus 119862 (119909) ge 120572

119909 isin M

(119875)






120595 (119909 119908) = int

119897(119909119903)le119908

119901 (119903) 119889119903 (5)


(0 1) is the value

VaR120579(119909) = min 119908 isin R 120595 (119909 119908) ge 120579 (6)


CVaR120579(119909) =

1

1 minus 120579

int

119897(119909119903)geVaR120579(119909)119897 (119909 119903) 119901 (119903) 119889119903 (7)


min119909

1

1 minus 120579

int

119897(119909119903)geVaR120579(119909)119897 (119909 119903) 119901 (119903) 119889119903

st 1199031015840119909 minus 119862 (119909) ge 120572

119909 isin M

(8)



119865120579(119909 119908) = 119908 +

1

1 minus 120579

E [119897 (119909 119903) minus 119908]+

= 119908 +

1

1 minus 120579

int [119897 (119909 119903) minus 119908]+119901 (119903) 119889119903

(9)


+= max 119897(119909 119903) minus 119908 0


min119909isinX


(119909119908)isinXtimesR119865120579(119909 119908) (10)


1015840119903



119865120579(119909 119908)


119908 +

1

1 minus 120579

int [minus(119904 + 119909)1015840119903 minus 119908]

+

119901 (119903) 119889119903

(11)



1 1199032 119903

119879 where 119903


120579(119909 119908) is

119865120579(119909 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905minus 119908]

+

(12)




11015840 (119904 + 119909) + 119862 (119909) = 11015840119904 (13)


11015840119909 + 119862 (119909) = 0 (14)



119904119894+ 119909119894ge minus119901119894 119894 = 1 2 119899 (15)




119897119894le 119904119894+ 119909119894le 119906119894

(16)


or

119871119894le

119904119894+ 119909119894

sum119899

119894=1119904119894

le 119880119894 (17)

where 119897119894(or 119871119894) and 119906





min119909119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905 minus 119908]

+

st 1199031015840119909 minus 119862 (119909) ge 120572

11015840119909 + 119862 (119909) = 0

119897119894le 119904119894+ 119909119894le 119906119894 119894 = 1 2 119899

(18)

Let 119889119861



119862119894(119909119894) =

119889119861

119894119909119894 if 119909

119894gt 0

minus119889119878

119894119909119894 if 119909

119894lt 0

(19)

Variables 119909+119894and 119909

minus



119894= max0 119909

119894 and 119909

minus

119894=

minusmin0 119909119894 Then 119909+

119894ge 0 119909minus

119894ge 0 for 119894 = 1 2 119899 and the


119862119894(119909119894) = 119889119861

119894119909+

119894+ 119889119878

119894119909minus

119894 (20)

where we have 119909119894= 119909+


119894


119902119905= [minus(119904 + 119909)

1015840119903119905minus 119908]

+

119905 = 1 2 119879 (21)

Then

119902119905ge minus(119904 + 119909)

1015840119903119905minus 119908 119905 = 1 2 119879 (22)


min119909+119909minus119902119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

119902119905

st119899

sum

119894=1

119903119894(119909+


119894)

minus

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) ge 120572

119902119905ge minus(119904 + 119909

+minus 119909minus)

1015840

119903119905minus 119908

119899

sum

119894=1

(119909+


119894) +

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) = 0

119897119894le 119904119894+ 119909+


119894le 119906119894

119902119905ge 0 119909

+

119894ge 0 119909

minus

119894ge 0

119894 = 1 2 119899 119905 = 1 2 119879

(23)












min119904

CVaR120579(119904)

st119899

sum

119894=1

119903119894119904119894ge 120582119882

119899

sum

119894=1

119904119894= 119882

120576119894119882 le 119904

119894le 120575119894119882

119894 = 1 2 119899

(24)

where 119904 = (1199041 1199042 119904


119894




120579(119904) is the





119865120579(119904 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus1199041015840119903119905minus 119908]

+

(25)







119894= 020 for

















4 Conclusion







120572 = 001

119899

sum

119894=1

119904119894


120572 = 002

119899

sum

119894=1

119904119894


120572 = 003

119899

sum

119894=1

119904119894


06 08 1 12 14 16 18 2 220

2

4

6

8

10

12

14

16

18

Expe

cted

exce

ss re

turn

requ

irem

ent120572

($)

times106

times107

Without TC

CVaR120579(s + x) ($)

With dB = 0002

With dB = 0005

With dB = 001





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120595 (119909 119908) = int

119897(119909119903)le119908

119901 (119903) 119889119903 (5)


(0 1) is the value

VaR120579(119909) = min 119908 isin R 120595 (119909 119908) ge 120579 (6)


CVaR120579(119909) =

1

1 minus 120579

int

119897(119909119903)geVaR120579(119909)119897 (119909 119903) 119901 (119903) 119889119903 (7)


min119909

1

1 minus 120579

int

119897(119909119903)geVaR120579(119909)119897 (119909 119903) 119901 (119903) 119889119903

st 1199031015840119909 minus 119862 (119909) ge 120572

119909 isin M

(8)



119865120579(119909 119908) = 119908 +

1

1 minus 120579

E [119897 (119909 119903) minus 119908]+

= 119908 +

1

1 minus 120579

int [119897 (119909 119903) minus 119908]+119901 (119903) 119889119903

(9)


+= max 119897(119909 119903) minus 119908 0


min119909isinX


(119909119908)isinXtimesR119865120579(119909 119908) (10)


1015840119903



119865120579(119909 119908)


119908 +

1

1 minus 120579

int [minus(119904 + 119909)1015840119903 minus 119908]

+

119901 (119903) 119889119903

(11)



1 1199032 119903

119879 where 119903


120579(119909 119908) is

119865120579(119909 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905minus 119908]

+

(12)




11015840 (119904 + 119909) + 119862 (119909) = 11015840119904 (13)


11015840119909 + 119862 (119909) = 0 (14)



119904119894+ 119909119894ge minus119901119894 119894 = 1 2 119899 (15)




119897119894le 119904119894+ 119909119894le 119906119894

(16)


or

119871119894le

119904119894+ 119909119894

sum119899

119894=1119904119894

le 119880119894 (17)

where 119897119894(or 119871119894) and 119906





min119909119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905 minus 119908]

+

st 1199031015840119909 minus 119862 (119909) ge 120572

11015840119909 + 119862 (119909) = 0

119897119894le 119904119894+ 119909119894le 119906119894 119894 = 1 2 119899

(18)

Let 119889119861



119862119894(119909119894) =

119889119861

119894119909119894 if 119909

119894gt 0

minus119889119878

119894119909119894 if 119909

119894lt 0

(19)

Variables 119909+119894and 119909

minus



119894= max0 119909

119894 and 119909

minus

119894=

minusmin0 119909119894 Then 119909+

119894ge 0 119909minus

119894ge 0 for 119894 = 1 2 119899 and the


119862119894(119909119894) = 119889119861

119894119909+

119894+ 119889119878

119894119909minus

119894 (20)

where we have 119909119894= 119909+


119894


119902119905= [minus(119904 + 119909)

1015840119903119905minus 119908]

+

119905 = 1 2 119879 (21)

Then

119902119905ge minus(119904 + 119909)

1015840119903119905minus 119908 119905 = 1 2 119879 (22)


min119909+119909minus119902119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

119902119905

st119899

sum

119894=1

119903119894(119909+


119894)

minus

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) ge 120572

119902119905ge minus(119904 + 119909

+minus 119909minus)

1015840

119903119905minus 119908

119899

sum

119894=1

(119909+


119894) +

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) = 0

119897119894le 119904119894+ 119909+


119894le 119906119894

119902119905ge 0 119909

+

119894ge 0 119909

minus

119894ge 0

119894 = 1 2 119899 119905 = 1 2 119879

(23)












min119904

CVaR120579(119904)

st119899

sum

119894=1

119903119894119904119894ge 120582119882

119899

sum

119894=1

119904119894= 119882

120576119894119882 le 119904

119894le 120575119894119882

119894 = 1 2 119899

(24)

where 119904 = (1199041 1199042 119904


119894




120579(119904) is the





119865120579(119904 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus1199041015840119903119905minus 119908]

+

(25)







119894= 020 for

















4 Conclusion







120572 = 001

119899

sum

119894=1

119904119894


120572 = 002

119899

sum

119894=1

119904119894


120572 = 003

119899

sum

119894=1

119904119894


06 08 1 12 14 16 18 2 220

2

4

6

8

10

12

14

16

18

Expe

cted

exce

ss re

turn

requ

irem

ent120572

($)

times106

times107

Without TC

CVaR120579(s + x) ($)

With dB = 0002

With dB = 0005

With dB = 001





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










or

119871119894le

119904119894+ 119909119894

sum119899

119894=1119904119894

le 119880119894 (17)

where 119897119894(or 119871119894) and 119906





min119909119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus(119904 + 119909)1015840119903119905 minus 119908]

+

st 1199031015840119909 minus 119862 (119909) ge 120572

11015840119909 + 119862 (119909) = 0

119897119894le 119904119894+ 119909119894le 119906119894 119894 = 1 2 119899

(18)

Let 119889119861



119862119894(119909119894) =

119889119861

119894119909119894 if 119909

119894gt 0

minus119889119878

119894119909119894 if 119909

119894lt 0

(19)

Variables 119909+119894and 119909

minus



119894= max0 119909

119894 and 119909

minus

119894=

minusmin0 119909119894 Then 119909+

119894ge 0 119909minus

119894ge 0 for 119894 = 1 2 119899 and the


119862119894(119909119894) = 119889119861

119894119909+

119894+ 119889119878

119894119909minus

119894 (20)

where we have 119909119894= 119909+


119894


119902119905= [minus(119904 + 119909)

1015840119903119905minus 119908]

+

119905 = 1 2 119879 (21)

Then

119902119905ge minus(119904 + 119909)

1015840119903119905minus 119908 119905 = 1 2 119879 (22)


min119909+119909minus119902119908

119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

119902119905

st119899

sum

119894=1

119903119894(119909+


119894)

minus

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) ge 120572

119902119905ge minus(119904 + 119909

+minus 119909minus)

1015840

119903119905minus 119908

119899

sum

119894=1

(119909+


119894) +

119899

sum

119894=1

(119889119861

119894119909+

119894+ 119889119861

119894119909minus

119894) = 0

119897119894le 119904119894+ 119909+


119894le 119906119894

119902119905ge 0 119909

+

119894ge 0 119909

minus

119894ge 0

119894 = 1 2 119899 119905 = 1 2 119879

(23)












min119904

CVaR120579(119904)

st119899

sum

119894=1

119903119894119904119894ge 120582119882

119899

sum

119894=1

119904119894= 119882

120576119894119882 le 119904

119894le 120575119894119882

119894 = 1 2 119899

(24)

where 119904 = (1199041 1199042 119904


119894




120579(119904) is the





119865120579(119904 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus1199041015840119903119905minus 119908]

+

(25)







119894= 020 for

















4 Conclusion







120572 = 001

119899

sum

119894=1

119904119894


120572 = 002

119899

sum

119894=1

119904119894


120572 = 003

119899

sum

119894=1

119904119894


06 08 1 12 14 16 18 2 220

2

4

6

8

10

12

14

16

18

Expe

cted

exce

ss re

turn

requ

irem

ent120572

($)

times106

times107

Without TC

CVaR120579(s + x) ($)

With dB = 0002

With dB = 0005

With dB = 001





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min119904

CVaR120579(119904)

st119899

sum

119894=1

119903119894119904119894ge 120582119882

119899

sum

119894=1

119904119894= 119882

120576119894119882 le 119904

119894le 120575119894119882

119894 = 1 2 119899

(24)

where 119904 = (1199041 1199042 119904


119894




120579(119904) is the





119865120579(119904 119908) asymp 119908 +

1

(1 minus 120579) 119879

119879

sum

119905=1

[minus1199041015840119903119905minus 119908]

+

(25)







119894= 020 for

















4 Conclusion







120572 = 001

119899

sum

119894=1

119904119894


120572 = 002

119899

sum

119894=1

119904119894


120572 = 003

119899

sum

119894=1

119904119894


06 08 1 12 14 16 18 2 220

2

4

6

8

10

12

14

16

18

Expe

cted

exce

ss re

turn

requ

irem

ent120572

($)

times106

times107

Without TC

CVaR120579(s + x) ($)

With dB = 0002

With dB = 0005

With dB = 001





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120572 = 001

119899

sum

119894=1

119904119894


120572 = 002

119899

sum

119894=1

119904119894


120572 = 003

119899

sum

119894=1

119904119894


06 08 1 12 14 16 18 2 220

2

4

6

8

10

12

14

16

18

Expe

cted

exce

ss re

turn

requ

irem

ent120572

($)

times106

times107

Without TC

CVaR120579(s + x) ($)

With dB = 0002

With dB = 0005

With dB = 001





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a new portfolio rebalancing model with...

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