Research ArticleA Numerical Study for Robust Active Portfolio Managementwith Worst-Case Downside Risk Measure

Aifan Ling1 and Le Tang2

1 School of Finance Jiangxi University of Finance amp Economics Nanchang 330013 China2 Jiangxi University of Technology Nanchang 330098 China

Correspondence should be addressed to Aifan Ling aiffling163com

Received 9 December 2013 Accepted 22 January 2014 Published 1 April 2014

Academic Editor Chuangxia Huang

Copyright copy 2014 A Ling and L TangThis 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

Recently active portfolio management problems are paid close attention by many researchers due to the explosion of fundindustries We consider a numerical study of a robust active portfolio selection model with downside risk and multiple weightsconstraints in this paper We compare the numerical performance of solutions with the classical mean-variance tracking errormodel and the naive 1119873 portfolio strategy by real market data from China market and other markets We find from the numericalresults that the tested active models are more attractive and robust than the compared models

1 Introduction

The choice of an optimal portfolio of assets has becomea major research topic in financial economics The mean-variance model proposed by Markowitz (1952 1956) [12] provided a fundamental basis of portfolio selection forthe theoretical and practical applications today Analyticalexpression of the mean-variance efficient frontier couldbe derived by solving convex quadratic programs and theoptimal portfolio can be found when the expected returnsand covariance matrix of risk assets are exactly estimatedBased on Markowitzrsquos mean-variance model Roll (1992) [3]proposed an active portfolio management model which iscalled tracking error portfoliomodel in the literature Roll [3]used the variance of tracking error to measure how closelya portfolio follows the index to which it is benchmarkedMotivated by Rollrsquos seminal paper many researches pay closeattention to the active portfolio selection problems see [4ndash9]and recent papers [10ndash18]

Let r = (1199031 119903

119899)119879

isin R119899 denote the return vectorof the 119899 risk assets where 119903

119894is the gross rate of return for

the 119894th risky asset The investorrsquos position is described byvector w = (119908

1 119908

119899)119879

isin R119899 where the 119894th component119908119894represents the proportion invested to the 119894th risky asset

Let w119887

isin R119899 denote a fixed portfolio that investor seeks

to outperform w119887also is called benchmark portfolio in the

literature Define the tracking error of portfolio w relative tobenchmark portfolio w

119887as Δ119908 = (w minus w

119887)119879r In order to

obtain a good tracking error portfolio w Roll [3] consideredthe solution of the following variance tracking error (VTE)problem

VTEmax120583119879w (w minus w119887)119879

Σ (w minus w119887)le120590vte

119899

sum

119894=1

119908119894=1

(1)

where 120583 and Σ are the expectation and covariance matrix ofreturn vector r respectively and 120590mv is the preset trackingerror level When 120583 and Σ are known exactly similar toMarkowitzrsquos mean-variance portfolio problem (1) can besolved as a convex quadratic programming

Since the gain and the loss are symmetric on the meanvalue in the variance Markowitz (1959) [19] proposed to usethe semivariance of portfolio to control risk Bawa (1975) [20]Bawa and Lindenberg (1977) [21] and Fishburn (1977) [22]later introduced a class of downside risk measure known asthe lower partial moment (LPM) to better suit different riskprofits of the investors Because LPM mainly control the lossof portfolio it becomes a popular risk controlling tool intheory and practice see [23ndash28]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 912389 13 pageshttpdxdoiorg1011552014912389

2 Mathematical Problems in Engineering

Generally LPM can be expressed as

LPM119898(120588) = E [((120588 minus 119883)

+)119898

] (2)

where119883 is the random variable for example the asset returnof risky asset 120588may be a target that investors want to outper-form119898 is a parameter which can take any nonnegative valueto model the risk attitude of an investor (119886)

+= max(119886 0)

E[sdot] andPsdot below expresses the expectation and probabilityof random variable For the case of119898 = 0 1 2 we have

LPM119898(120588) = E [((120588 minus 119883)

+)119898

]

=

P 119883 le 120588 119898 = 0

E [(120588 minus 119883)+] 119898 = 1

E [((120588 minus 119883)+)2

] 119898 = 2

(3)

That is to say that LPM0is nothing but the probability of

the asset return falling below the benchmark index LPM1

is the expected shortfall of the investment falling below thebenchmark index and LPM

2is an analog of the semivariance

here however the deviation is in reference to a preset targetor benchmark return instead of the mean

In the last five to ten years robust portfolio optimiza-tion problems based on the robust optimization techniquedeveloped by Ben-Tal and Nemirovski (1998) [29] are thefocus of many financial and economic researchers Basedon robust portfolio optimization theory one can deal withmodels with uncertainty parameters or uncertainty distri-bution For instance Costa and Paiva (2002) [30] consid-ered a robust framework of model (1) with 120583 and Σ ina polytopic uncertainty set described by its vertices ElGhaoui et al (2003) [31] proposed a worst-case Value-at-Risk (VaR) robust optimizationmodel inwhich they assumedthat the distribution of returns is partially known Onlybounds on the mean and covariance matrix are availablein El Ghaoui et alrsquos model and the proposed model canbe solved by a second order cone programming approachGoldfarb and Iyengar (2003) [32] proposed a robust factormodel and later their models were developed and extendedby Erdogan et al (2008) [33] to a robust index tracking andactive portfolio management problem see also Ling and Xu(2012) [34] for a similar research Zhu et al (2009) [28]proposed a robust framework based on LPM constraintswith uncertainty discrete distribution Glabadanidis (2010)[11] considered a robust and efficient strategy to track andoutperform a benchmark in which he proposed a sequentialstepwise regression and relative method based on factormodels of security returns Chen et al (2011) [23] developedsome tight bounds on the expected values of LPM underthe framework of robust optimization models arising fromportfolio selection problems More robust portfolio modelsbased on different parameters uncertainty or distributionuncertainty can be found in the recent survey given byFabozzi et al (2010) [35]

As we know some policies are commonly found inthe contracts between investors and portfolio managers andrequire that the weights of certain types of assets shouldbe smaller higher or equal to a given percentage in some

funds We call this type of restriction in this paper weightsconstraint(s) of a portfolio For instance weights constraintappears frequently in some index funds (ETF) stock stylefunds (the weights invested in stocks is not less than a presetpercentage of the market value of fund) bond style funds(the weights invested in stocks is not larger than a presetvalue) QFII (Qualified foreign institutional investors) fundsand QDII (Qualified domestic institutional investor) fundsrequire that the weights invested in foreign markets not belarger than a preset value However it is to be regretted thatweights constraint is rarely used in the current tracking error(robust) portfolio literature Recently Bajeux-Besnainou et al(2011) [36] first introduce this kind of constraint into amean-variance index tracking portfoliomodel In their paperBajeux-Besnainou et al described a comparison with andwithout weights constraint and showed that the influence ofweights constraint is very remarkable and cannot be ignoredfor the returns of portfolio

Motivated by the topics of active portfolio managementthat are paid close attention by many researchers and thefact that there exists the rare model with explicit solutionsin the robust active portfolio management literature in thispaper we further consider a numerical study of robust activeportfolio selection model that is proposed by Ling et al in[37] In the current framework we test the models in [37]using the real market data which includes ten stock indexesfrom Shanghai Stock Exchange (SHH) and Shenzhen StockExchange (SHZ) which are called domestic assets and fourstock indexes from other stock exchanges which are calledforeign assets Our numerical studies give the comparisonswith the classical (VTE) and the naive 1119873 portfolio strategyconsidered by DeMiguel et al (2009) [38] Numerical resultsindicate that the proposed active portfolio selection modelcan obtain better numerical performance for real marketdata Most of proofs of theorems in the current paper canbe found in [37] but in order to keep the completeness ofreading we still give these main proofs in Appendix

This paper is arranged as follows In Section 2 we givethe robust active portfolio problem with multiple weightsconstraints and establish the robust active portfolio modelsWe explore the explicit solutions of the proposed modelsin Section 3 In Section 4 we do some numerical tests andcomparisons based on real market data

2 Robust Active Portfolio Problems

LetI be a subset of 1 2 119899 If assets inI are restrictedto a limited weight we call I the set of restricted assets Forgiven constant 119902 weights constraint requires

e119879Iw = (le) 119902 (4)

where eI = (1198901 119890

119899)119879

isin R119899 is an index vector with119890119894= 1when 119894 isin I and 119890

119894= 0when 119894 notin IThe notation ldquo= (le

)rdquo means the equality only (or inequality only) weights con-straint Under the estimation of 120583 and Σ is exactly obtained

Mathematical Problems in Engineering 3

and selling short is allowed Bajeux-Besnainou et al [36] con-sidered the following active portfolio optimization problemwith single weight constraint

max E [w119879r] (w minus w119887)119879

Σ (w minus w119887) le 120590

e119879Iw = (le) 119902 e119879w = 1

(5)

and get a closed solution of this problem As pointed inSection 1 the classical mean-variance framework may not beappropriate since it controls not only the loss of portfolio butalso the gain of portfolio

In our framework we use LPM to control the risk ofportfolio bywhichwe in fact only control the loss of portfolioAdditionally we make no assumption on the distribution ofreturns r Instead we assume that we only partially knowsome knowledge about the statistical properties of returnsr and the underlying distribution function is only knownto belong to a certain set D of distribution functions Fof the returns vector r Under the uncertainty set D of theunderlying distribution we have the worst-case LPM

119898

Definition 1 For any 119898 ge 0 and real number 120588 the worst-case lower-partial moment (WCLPM for short) of randomvariable119883 with respect to 120588 underF isin D is defined by

WCLPM119898(120588) = sup

FisinD

E [((120588 minus 119883)+)119898

] (6)

In tracking error portfolio selection problems investorswant in fact to find a portfolio w such that the return ofportfolio w can be close to or outperform the return ofbenchmark w119879

119887120583 For this we call w119879

119887120583 minus w119879120583 the loss of

portfoliow relative tow119879119887120583 and control the risk of portfolio by

restricting the loss Based on this idea we maximize on onehand the return w119879120583 and on the other hand control the lossby usingworst-case LPM

119898Thus themean-WCLPM

119898robust

tracking error problem with multiple weights constraints canbe written as

maxw

w119879120583

st supFisinD

E [((w119879119887r minus w119879r)

+)119898

] le 120590119898

e119879I119894w = (le) 119902119894 119894 = 1 119901

e119879w = 1

(7)

where 120590119898(119898 = 0 1 2) are the preset constants The second

class of inequalities constraints 119901weights constraints expressthat several classes of assets are restricted to the limitedweights where I

119894sube 1 2 119899 and 119902

119894(119894 = 1 119901)

are constants We call these constraints multiple weightsconstraints When 119901 = 1 it is the single weight constraintIn this paper we only consider the cases of 119898 = 0 1 2

andmultiple equality weights constraintsThe correspondingresults with multiple inequality weights constraints can beobtained by the similar methods

Because we are interested in developing robust trackingerror portfolio policies and exploring the explicit solutions ofproblems (7) in the rest of this paper we assume that D is

the set of allowable distributions of returns r with the knownmean and covariance that is

D = r | E [r] = 120583Cov (r) = Σ ≻ 0 (8)

For convenience we sometimes denote r isin D by r sim (120583 Σ)Our model can be extended to the case of parametersuncertainty for which parameters are not estimated exactly

Similar to Erdogan et al [33] and Bajeux-Besnainou et al[36] we normalize the benchmark portfolio w

119887that is

e119879w119887= 1 and introduce a new variable y = w minus w

119887 Notice

the w119879120583 = y119879120583 + w119879119887120583 then optimization problem (7) with

multiple equality weights constraints can be written into thefollowing form

(RSm) maxy

y119879120583

st suprisinD

E [((minusy119879r)+)119898

] le 120590119898

e119879I119894y = 119902119894minus e119879I119894w119887 119894 = 1 119901

e119879y = 0

(9)

We call y a self-financing portfolio and w a fully investingportfolio We mention that 119902

119894minus e119879I119894w119887 119894 = 1 119901 can be

positive or negative most likely between minus1 and 1 since 119902119894and

e119879I119894w119887 are in most cases between 0 and 1 In the rest of thepaper we suppose that 119901 + 1 vectors e and eI119894 (119894 = 1 119901)

are linearly independent and 119899 gt 119901 + 1 always holds Ifthere exists certain eI119896 such that eI119896 can be expressed as thelinear combination of eI119894 (119894 = 1 119901 119894 = 119896) then e119879I119896y =

119902119896minus e119879I119896w119887 becomes a redundant constraint Additionally

the linear equality constraints will lead to a unique feasiblesolution y if 119899 = 119901 + 1

The following lemmas indicate that a tight upper boundof the worst-case LPM

119898can be obtained explicitly which will

be helpful for our analysis later We extend the proof in [23]for WCLPM

0to a general case

Lemma2 (see [23 39]) Let 120585 be a random variable withmean120583 and variance 120590 but have unknown distribution and 120588 is anarbitrary real number Then we have the following

(a) For the case of119898 = 0

sup120585sim(120583120590)

P 120585 le 120588 =

1

1 + (120588 minus 120583)2

1205902

if 120588 le 120583

1 if 120588 ge 120583

(10)

(b) For the case of119898 = 1

sup120585sim(120583120590)

E [(120588 minus 120585)+] =

120588 minus 120583 + radic1205902+ (120588 minus 120583)

2

2 (11)

(c) For the case of119898 = 2

sup120585sim(120583120590)

E [((120588 minus 120585)+)2

] = [(120588 minus 120583)+]2

+ 1205902

= 1205902 120588 le 120583

(120588 minus 120585)2

+ 1205902 120588 ge 120583

(12)

4 Mathematical Problems in Engineering

Lemma 3 (see [40]) For any a isin R119899 denote

1198781= a119879r | r isin D

1198782= 120578 | E (120578) = a119879120583Var (120578) = a119879Σa

(13)

Then 1198781= 1198782

Lemma 4 (see [37]) For any r isin D 120578 isin 1198782 and any real

vector a isin R119899 the following equality

suprisinD

E [((120588 minus a119879r)+)119898

] = sup120578sim(a119879120583a119879Σa)

E [((120588 minus 120578)+)119898

] (14)

holds

Lemma 4 establishes the relationship of single variable120578 and multivariable r which is useful for our analysis laterWe end this section by introducing some notations that willbe used frequently Unless otherwise specified in the restof paper bold letter expresses a vector uppercase letter is amatrix and lowercase is a real number 0 is a zero matrix orzero vector by context For simplicity we denote eI119896 by e

119896

119896 = 1 119901 and 119902119894minus e119879I119894w119887 by 119889119894 and thus 119889

119894isin [minus1 1] Let

119872 = [e1 e

119901] isin R

119899times119901 d = (119889

1 119889

119901)119879

isin R119901

(15)

Then from the assumption that e1 e

119901and e are linearly

independent matrix 119872 has column full rank Let 119860 =

119872119879Σminus1119872 isin R119901times119901 Then matrix 119860 is positive definite via

Σminus1

≻ 0 where the notation 119885 ⪰ 119884 (or 119885 ≻ 119884) means matrix119885 minus 119884 is positive semidefinite (or positive definite) Furtherwe denote

119861 = 119868 minus119872119860minus1119872119879Σminus1

isin R119899times119899

119862 = Σminus1119872119860minus1

isin R119899times119901

(16)

where 119868 is a unit matrix with reasonable dimensions Thenfor these matrices we have the following results

Lemma 5 (a) 119861119872 = 0(b)119872T

Σminus1119861 = 0

(c) 119861119879119862 = 0(d) 119861119879Σminus1119861 = Σ

minus1119861

(e) 119862119879Σ119862 = 119860minus1

( f) Matrix 119861 is positive semidefinite

The conclusions are straightforward for (a)ndash(e) For (f)we have from (d) Σminus1119861 = 119861

119879Σminus1119861 ⪰ 0 since Σminus1 ≻ 0 Hence

119861 is a positive semidefinite matrixBased on matrices 119861 119862 and Σ

minus1 we denote additionally

119886 = e119879Σminus1e 119887 = e119879Σminus1120583 119888 = 120583119879Σminus1120583

1198860= e119879119862d 119887

0= 120583119879119862d 119888

0= d119879119860minus1d

1198861= e119879Σminus1119861e 119887

1= e119879Σminus1119861120583 119888

1= 120583119879Σminus1119861120583

1198862= 11988611198881minus 1198872

1 119887

2= 11988611198870minus 11988601198871 119888

2= 11988611198880+ 1198862

0

(17)

Without loss the generality we assumeΣminus1119861 = 0 thenwe havethat

119886119888 gt 1198872 119886

1gt 0 119888

1gt 0

1198862gt 0 119888

2ge 0

(18)

and 1198882= 0 if and only if d = 0 via 119860minus1 ≻ 0

3 The Solutions

In this sectionwewill explore the explicit solution of problem(RSm) Notice that y = wminusw

119887 then we can rewrite separably

problem (RSm) into the following three problems

maxy

y119879120583 suprisinD

P y119879r le 0 le 1205900 119872119879y = d e119879y = 0

(19)

maxy

y119879120583 suprisinD

E [(minusy119879r)+] le 1205901 119872119879y = d e119879y = 0

(20)

maxy

y119879120583 suprisinD

E [((minusy119879r)+)2

] le 1205902 119872119879y = d e119879y = 0

(21)

which is corresponding to the case of 120588 = 0 in the worst-caseLPM119898

31 Maximization of Information Ratio A Variation of119882119862119871119875119872

0 Let us first consider problem (19) The inequality

Py119879r le 0 le 1205900is usually called chance constraint or Royrsquos

safety-first rule in the literature Generally speaking 1205900is

taken far less than 12 which leads to (11205900) minus 1 gt 0 Hence

from Lemma 2(a) and Lemma 4 we have that when y119879120583 ge 0

suprisinD

P y119879r le 0 le 1205900lArrrArr

1

1 + (minusy119879120583)2y119879Σyle 1205900

lArrrArr

(minusy119879120583)2

y119879Σyge (

1

1205900

minus 1)

lArrrArry119879120583

radicy119879Σyge radic(

1

1205900

minus 1)

(22)

The last inequality means that information ratio (or sharperatio) of the portfolio is not less than a preset constant Bythis characteristic we consider a variation version of problem(19) in which we minimize the WCLPM

0 that is

miny suprisinD

P y119879r le 0 | st 119872119879y = d e119879y = 0 (23)

which results from (22) the maximization of informationratio (IR for short)

MAXIR = maxy

y119879120583

radicy119879Σy 119872119879y = d e119879y = 0

(24)

Mathematical Problems in Engineering 5

Problem (24) is related to the mean-variance with multipleweights constraints (MVMWC for short) for a given param-eter 1205901015840

0

(MVMWC) maxy

y119879120583 y119879Σy le 1205901015840

0 119872119879y = d e119879y = 0

(25)

If 119901 = 0 that is there is not weights constraint then problem(25) is nothing but Rollrsquos VTE problem If 119901 = 1 that is119872 = e

1 then problem (25) becomes the case of single weight

constraint problem which has been considered by Bajeux-Besnainou et al [36]

The explicit solution of problem (24) without weightsconstraints has been obtained in the literature (eg see [1623] for detail) But the methods in the literature cannotbe used directly for problem (24) because of the existenceof multiple weights constraints Thus we need to find thedifferent method from the literature to obtain the explicitsolution To this end we deal with this problem in two stagesto get the explicit solution of (24) We introduce a parameterat first stage and parameterize the solution of (24) and thenmaximize IR of the solution with respect to the parameter inthe second stage

Theorem 6 There exists a real number say 1205910 such that the

explicit solution of problem (24) can be expressed as

ylowast1205910=

1

1205910

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(26)

Proof See Theorem 1 in [37] see also Appendix for detail

To obtain the portfolio with the maximum informationratio that is the optimal solution of problem (24) what isthen the value of parameter 120591

0 As the second stage we

replace ylowast1205910into the objective function (24) Then we get the

information ratio of portfolio ylowast1205910 denoted by MAX

1205910 which

is a function of parameter 1205910 Thus we can get the maximum

information ratio by maximizing MAX1205910with respect to 120591

0

That is we have

MAXIR = max1205910gt0

MAX1205910= max1205910gt0

(ylowast1205910)119879

120583

radic(ylowast1205910)119879

Σylowast1205910

(27)

Let 120591lowast0be the optimal solution of (27) then portfolio ylowast

120591lowast

0

isthe optimal solution of (24) for which it has the maximuminformation ratio Hence we have the results below and itsproof is straightforward

Theorem 7 The maximizer of problem (27) is given by

120591lowast

0=11988611198870minus 11988601198871

11988611198880+ 1198862

0

(28)

when 11988611198870minus 11988601198871

gt 0 and the optimal solution of (24) isobtained by

ylowastIR =

(11988611198880+ 1198862

0)

11988611198870minus 11988601198871

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(29)

Theorem 8 (see [37]) If ylowast1205910

in (26) is obtained for certainpositive number 120591

0 then there exists a 1205901015840

0gt 0 such that both

problems (24) and (25) have the same optimal solution

Let ylowast0be the optimal solution of (24) in view of the proof

of Theorem 6 in Appendix if 12059010158400is chosen and satisfies

radic1198862(11988611205901015840

0minus 1198882)

1198862

=(ylowast0)119879

Σylowast0

(120583119879ylowast0) (30)

then the optimal portfolio for problem (25) is also the optimalsolution of (24) The relationship between problems (24)and (25) can be presented as follows Solving (24) will geta portfolio with the maximum IR However the optimalportfolio of problem (25) is not necessarily the one with themaximum IR If we solve (25) with 120590

1015840

0given by (30) then the

optimal portfolio of (25) will have the maximum IR too

32 The Mean-1198821198621198711198751198721 TheWCLPM

1constraint in prob-

lem (20) from Lemmas 2 and 4 can be written as

suprisinD

E [(minusy119879r)+] le 1205901lArrrArr

minusy119879120583 + radicy119879Σy + (minusy119879120583)2

2le 1205901

lArrrArr y119879Σy minus 4120590

1y119879120583 le 4120590

2

1

y119879120583 ge minus21205901

(31)

In fact it can verify that

miny y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1 gt minus2120590

1 (32)

Hence inequality constraint y119879120583 ge minus21205901is redundant Thus

for any preset 1205901gt 0 we can rewrite problem (20) as

WCLPM1maxy

y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1

119872119879y = d e119879y = 0

(33)

Here we call 21205901the upper bound of tracking error under

WCLPM1constraint and denote the upper bound of tracking

error by TE1= 21205901 The following lemma gives the feasible

condition of problem (33)

Lemma 9 If 1205901is chosen to satisfy

1205901gt 120590lowast

1 (34)

6 Mathematical Problems in Engineering

then problem (33) is feasible where

120590lowast

1

=

radic(11988611198870minus11988601198871)2

+(11988611198881minus1198872

1+1198861) (11988611198880+1198862

0)minus(11988611198870minus 11988601198871)

2 (11988611198881minus 1198872

1+ 1198861)

(35)

Proof Let

119891 (y) = y119879Σy minus 41205901y119879120583 (36)

and S11

= y 119891(y) le 41205902

1 S12

= y isin R119899 119872119879y = d e119879y =

0 Then this lemma is to verify that the set S1= S11⋂S12

is nonempty for any 1205901gt 120590lowast

1 Now for any 120590

1gt 0 consider

the following subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 119872119879y = d e119879y = 0 (37)

and denote its optimal value by 1198911205901

which is a quadraticfunction of parameter 120590

1 Thus (for the computation of 119891

1205901

and 120590lowast

1 see Appendix for detail)

1198911205901le 41205902

1(38)

gives a quadratic inequality with respect to 1205901 Solving (38)

we have that the equality in (38) holds when 1205901= 120590lowast

1gt 0

where 120590lowast1is given by (35) Hence set S

1to be nonempty and

has at least an element when 1205901gt 120590lowast

1 This implies problem

(33) is feasible The proof is completed

Lemma 9 indicates that problemWCLPM1is well defined

and meaningful The following results give the explicit solu-tion of problemWCLPM

1

Theorem 10 Let the condition of Lemma 9 be satisfied Thenthe optimal solution of problem (33) can be expressed explicitlyas

ylowast1= (1205821+ 21205901) (Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(39)

where

1205821

=

radic4 (11988611198881minus 1198872

1+ 1198861) 1205902

1+ 4 (119886

11198870minus 11988601198871) 1205901minus (11988611198880+ 1198862

0)

radic11988611198881minus 1198872

1

(40)

Proof See Theorem 4 in [37] see also Appendix for detail

33 The Mean-1198821198621198711198751198722 In this subsection we consider

mean-WCLPM2portfolio problem In view of Lemma 2(c)

and Lemma 4 the WCLPM2constraint can be formulated as

((minusy119879120583)+)2

+ y119879Σy le 1205902 (41)

Then we can write problem (7) with119898 = 2 as

WCLPM2maxy

y119879120583 (41) 119872119879y = d e119879y = 0 (42)

If we add constraint y119879120583 ge 0 then problem (42) is nothingbut the classical M-Vmodel with weights constraints On theother hand if we add constraint y119879120583 lt 0 then constraint (41)is

(y119879120583)2

+ y119879Σy = y119879 (Σ + 120583120583119879) y le 120590

2 (43)

which is more conservative than the variance of portfolio forthe same preset value 120590

2since (Σ + 120583120583

119879) minus Σ = 120583120583

119879 is apositive semidefinite matrix We call radic120590

2 instead of 120590

2 the

upper bound of tracking error under WCLPM2constraint

and denote the upper bound by TE2

= radic1205902 Similar to

Lemma 9 we can also give the feasible condition of problem(42)

Lemma 11 Let

1205902gt11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (44)

Then problem (42) is feasible

Proof Let

120590lowast

2= miny ((minusy119879120583)

+)2

+ y119879Σy 119872119879y = d e119879y = 0

(45)

It follows from solving straightforwardly the problem that itsoptimal solution is

ylowast1205902=

[1198861V minus (119886

11198870minus 11988601198871)]

11988611198881minus 1198872

1

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(46)

where

V =(11988611198881minus 1198872

1) (11988601198871minus 11988611198870)+

1198861(1198861+ 11988611198881minus 1198872

1)

+11988611198870minus 11988601198871

1198861

(47)

Then we get that

120590lowast

2=11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (48)

Hence the problem (42) is feasible when 1205902satisfies 120590

2gt 120590lowast

2

This completes the proof

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Mathematical PhysicsAdvances in

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Generally LPM can be expressed as

LPM119898(120588) = E [((120588 minus 119883)

+)119898

] (2)

where119883 is the random variable for example the asset returnof risky asset 120588may be a target that investors want to outper-form119898 is a parameter which can take any nonnegative valueto model the risk attitude of an investor (119886)

+= max(119886 0)

E[sdot] andPsdot below expresses the expectation and probabilityof random variable For the case of119898 = 0 1 2 we have

LPM119898(120588) = E [((120588 minus 119883)

+)119898

]

=

P 119883 le 120588 119898 = 0

E [(120588 minus 119883)+] 119898 = 1

E [((120588 minus 119883)+)2

] 119898 = 2

(3)

That is to say that LPM0is nothing but the probability of

the asset return falling below the benchmark index LPM1

is the expected shortfall of the investment falling below thebenchmark index and LPM

2is an analog of the semivariance

here however the deviation is in reference to a preset targetor benchmark return instead of the mean

In the last five to ten years robust portfolio optimiza-tion problems based on the robust optimization techniquedeveloped by Ben-Tal and Nemirovski (1998) [29] are thefocus of many financial and economic researchers Basedon robust portfolio optimization theory one can deal withmodels with uncertainty parameters or uncertainty distri-bution For instance Costa and Paiva (2002) [30] consid-ered a robust framework of model (1) with 120583 and Σ ina polytopic uncertainty set described by its vertices ElGhaoui et al (2003) [31] proposed a worst-case Value-at-Risk (VaR) robust optimizationmodel inwhich they assumedthat the distribution of returns is partially known Onlybounds on the mean and covariance matrix are availablein El Ghaoui et alrsquos model and the proposed model canbe solved by a second order cone programming approachGoldfarb and Iyengar (2003) [32] proposed a robust factormodel and later their models were developed and extendedby Erdogan et al (2008) [33] to a robust index tracking andactive portfolio management problem see also Ling and Xu(2012) [34] for a similar research Zhu et al (2009) [28]proposed a robust framework based on LPM constraintswith uncertainty discrete distribution Glabadanidis (2010)[11] considered a robust and efficient strategy to track andoutperform a benchmark in which he proposed a sequentialstepwise regression and relative method based on factormodels of security returns Chen et al (2011) [23] developedsome tight bounds on the expected values of LPM underthe framework of robust optimization models arising fromportfolio selection problems More robust portfolio modelsbased on different parameters uncertainty or distributionuncertainty can be found in the recent survey given byFabozzi et al (2010) [35]

As we know some policies are commonly found inthe contracts between investors and portfolio managers andrequire that the weights of certain types of assets shouldbe smaller higher or equal to a given percentage in some

funds We call this type of restriction in this paper weightsconstraint(s) of a portfolio For instance weights constraintappears frequently in some index funds (ETF) stock stylefunds (the weights invested in stocks is not less than a presetpercentage of the market value of fund) bond style funds(the weights invested in stocks is not larger than a presetvalue) QFII (Qualified foreign institutional investors) fundsand QDII (Qualified domestic institutional investor) fundsrequire that the weights invested in foreign markets not belarger than a preset value However it is to be regretted thatweights constraint is rarely used in the current tracking error(robust) portfolio literature Recently Bajeux-Besnainou et al(2011) [36] first introduce this kind of constraint into amean-variance index tracking portfoliomodel In their paperBajeux-Besnainou et al described a comparison with andwithout weights constraint and showed that the influence ofweights constraint is very remarkable and cannot be ignoredfor the returns of portfolio

Motivated by the topics of active portfolio managementthat are paid close attention by many researchers and thefact that there exists the rare model with explicit solutionsin the robust active portfolio management literature in thispaper we further consider a numerical study of robust activeportfolio selection model that is proposed by Ling et al in[37] In the current framework we test the models in [37]using the real market data which includes ten stock indexesfrom Shanghai Stock Exchange (SHH) and Shenzhen StockExchange (SHZ) which are called domestic assets and fourstock indexes from other stock exchanges which are calledforeign assets Our numerical studies give the comparisonswith the classical (VTE) and the naive 1119873 portfolio strategyconsidered by DeMiguel et al (2009) [38] Numerical resultsindicate that the proposed active portfolio selection modelcan obtain better numerical performance for real marketdata Most of proofs of theorems in the current paper canbe found in [37] but in order to keep the completeness ofreading we still give these main proofs in Appendix

This paper is arranged as follows In Section 2 we givethe robust active portfolio problem with multiple weightsconstraints and establish the robust active portfolio modelsWe explore the explicit solutions of the proposed modelsin Section 3 In Section 4 we do some numerical tests andcomparisons based on real market data

2 Robust Active Portfolio Problems

LetI be a subset of 1 2 119899 If assets inI are restrictedto a limited weight we call I the set of restricted assets Forgiven constant 119902 weights constraint requires

e119879Iw = (le) 119902 (4)

where eI = (1198901 119890

119899)119879

isin R119899 is an index vector with119890119894= 1when 119894 isin I and 119890

119894= 0when 119894 notin IThe notation ldquo= (le

)rdquo means the equality only (or inequality only) weights con-straint Under the estimation of 120583 and Σ is exactly obtained

Mathematical Problems in Engineering 3

and selling short is allowed Bajeux-Besnainou et al [36] con-sidered the following active portfolio optimization problemwith single weight constraint

max E [w119879r] (w minus w119887)119879

Σ (w minus w119887) le 120590

e119879Iw = (le) 119902 e119879w = 1

(5)

and get a closed solution of this problem As pointed inSection 1 the classical mean-variance framework may not beappropriate since it controls not only the loss of portfolio butalso the gain of portfolio

In our framework we use LPM to control the risk ofportfolio bywhichwe in fact only control the loss of portfolioAdditionally we make no assumption on the distribution ofreturns r Instead we assume that we only partially knowsome knowledge about the statistical properties of returnsr and the underlying distribution function is only knownto belong to a certain set D of distribution functions Fof the returns vector r Under the uncertainty set D of theunderlying distribution we have the worst-case LPM

119898

Definition 1 For any 119898 ge 0 and real number 120588 the worst-case lower-partial moment (WCLPM for short) of randomvariable119883 with respect to 120588 underF isin D is defined by

WCLPM119898(120588) = sup

FisinD

E [((120588 minus 119883)+)119898

] (6)

In tracking error portfolio selection problems investorswant in fact to find a portfolio w such that the return ofportfolio w can be close to or outperform the return ofbenchmark w119879

119887120583 For this we call w119879

119887120583 minus w119879120583 the loss of

portfoliow relative tow119879119887120583 and control the risk of portfolio by

restricting the loss Based on this idea we maximize on onehand the return w119879120583 and on the other hand control the lossby usingworst-case LPM

119898Thus themean-WCLPM

119898robust

tracking error problem with multiple weights constraints canbe written as

maxw

w119879120583

st supFisinD

E [((w119879119887r minus w119879r)

+)119898

] le 120590119898

e119879I119894w = (le) 119902119894 119894 = 1 119901

e119879w = 1

(7)

where 120590119898(119898 = 0 1 2) are the preset constants The second

class of inequalities constraints 119901weights constraints expressthat several classes of assets are restricted to the limitedweights where I

119894sube 1 2 119899 and 119902

119894(119894 = 1 119901)

are constants We call these constraints multiple weightsconstraints When 119901 = 1 it is the single weight constraintIn this paper we only consider the cases of 119898 = 0 1 2

andmultiple equality weights constraintsThe correspondingresults with multiple inequality weights constraints can beobtained by the similar methods

Because we are interested in developing robust trackingerror portfolio policies and exploring the explicit solutions ofproblems (7) in the rest of this paper we assume that D is

the set of allowable distributions of returns r with the knownmean and covariance that is

D = r | E [r] = 120583Cov (r) = Σ ≻ 0 (8)

For convenience we sometimes denote r isin D by r sim (120583 Σ)Our model can be extended to the case of parametersuncertainty for which parameters are not estimated exactly

Similar to Erdogan et al [33] and Bajeux-Besnainou et al[36] we normalize the benchmark portfolio w

119887that is

e119879w119887= 1 and introduce a new variable y = w minus w

119887 Notice

the w119879120583 = y119879120583 + w119879119887120583 then optimization problem (7) with

multiple equality weights constraints can be written into thefollowing form

(RSm) maxy

y119879120583

st suprisinD

E [((minusy119879r)+)119898

] le 120590119898

e119879I119894y = 119902119894minus e119879I119894w119887 119894 = 1 119901

e119879y = 0

(9)

We call y a self-financing portfolio and w a fully investingportfolio We mention that 119902

119894minus e119879I119894w119887 119894 = 1 119901 can be

positive or negative most likely between minus1 and 1 since 119902119894and

e119879I119894w119887 are in most cases between 0 and 1 In the rest of thepaper we suppose that 119901 + 1 vectors e and eI119894 (119894 = 1 119901)

are linearly independent and 119899 gt 119901 + 1 always holds Ifthere exists certain eI119896 such that eI119896 can be expressed as thelinear combination of eI119894 (119894 = 1 119901 119894 = 119896) then e119879I119896y =

119902119896minus e119879I119896w119887 becomes a redundant constraint Additionally

the linear equality constraints will lead to a unique feasiblesolution y if 119899 = 119901 + 1

The following lemmas indicate that a tight upper boundof the worst-case LPM

119898can be obtained explicitly which will

be helpful for our analysis later We extend the proof in [23]for WCLPM

0to a general case

Lemma2 (see [23 39]) Let 120585 be a random variable withmean120583 and variance 120590 but have unknown distribution and 120588 is anarbitrary real number Then we have the following

(a) For the case of119898 = 0

sup120585sim(120583120590)

P 120585 le 120588 =

1

1 + (120588 minus 120583)2

1205902

if 120588 le 120583

1 if 120588 ge 120583

(10)

(b) For the case of119898 = 1

sup120585sim(120583120590)

E [(120588 minus 120585)+] =

120588 minus 120583 + radic1205902+ (120588 minus 120583)

2

2 (11)

(c) For the case of119898 = 2

sup120585sim(120583120590)

E [((120588 minus 120585)+)2

] = [(120588 minus 120583)+]2

+ 1205902

= 1205902 120588 le 120583

(120588 minus 120585)2

+ 1205902 120588 ge 120583

(12)

4 Mathematical Problems in Engineering

Lemma 3 (see [40]) For any a isin R119899 denote

1198781= a119879r | r isin D

1198782= 120578 | E (120578) = a119879120583Var (120578) = a119879Σa

(13)

Then 1198781= 1198782

Lemma 4 (see [37]) For any r isin D 120578 isin 1198782 and any real

vector a isin R119899 the following equality

suprisinD

E [((120588 minus a119879r)+)119898

] = sup120578sim(a119879120583a119879Σa)

E [((120588 minus 120578)+)119898

] (14)

holds

Lemma 4 establishes the relationship of single variable120578 and multivariable r which is useful for our analysis laterWe end this section by introducing some notations that willbe used frequently Unless otherwise specified in the restof paper bold letter expresses a vector uppercase letter is amatrix and lowercase is a real number 0 is a zero matrix orzero vector by context For simplicity we denote eI119896 by e

119896

119896 = 1 119901 and 119902119894minus e119879I119894w119887 by 119889119894 and thus 119889

119894isin [minus1 1] Let

119872 = [e1 e

119901] isin R

119899times119901 d = (119889

1 119889

119901)119879

isin R119901

(15)

Then from the assumption that e1 e

119901and e are linearly

independent matrix 119872 has column full rank Let 119860 =

119872119879Σminus1119872 isin R119901times119901 Then matrix 119860 is positive definite via

Σminus1

≻ 0 where the notation 119885 ⪰ 119884 (or 119885 ≻ 119884) means matrix119885 minus 119884 is positive semidefinite (or positive definite) Furtherwe denote

119861 = 119868 minus119872119860minus1119872119879Σminus1

isin R119899times119899

119862 = Σminus1119872119860minus1

isin R119899times119901

(16)

where 119868 is a unit matrix with reasonable dimensions Thenfor these matrices we have the following results

Lemma 5 (a) 119861119872 = 0(b)119872T

Σminus1119861 = 0

(c) 119861119879119862 = 0(d) 119861119879Σminus1119861 = Σ

minus1119861

(e) 119862119879Σ119862 = 119860minus1

( f) Matrix 119861 is positive semidefinite

The conclusions are straightforward for (a)ndash(e) For (f)we have from (d) Σminus1119861 = 119861

119879Σminus1119861 ⪰ 0 since Σminus1 ≻ 0 Hence

119861 is a positive semidefinite matrixBased on matrices 119861 119862 and Σ

minus1 we denote additionally

119886 = e119879Σminus1e 119887 = e119879Σminus1120583 119888 = 120583119879Σminus1120583

1198860= e119879119862d 119887

0= 120583119879119862d 119888

0= d119879119860minus1d

1198861= e119879Σminus1119861e 119887

1= e119879Σminus1119861120583 119888

1= 120583119879Σminus1119861120583

1198862= 11988611198881minus 1198872

1 119887

2= 11988611198870minus 11988601198871 119888

2= 11988611198880+ 1198862

0

(17)

Without loss the generality we assumeΣminus1119861 = 0 thenwe havethat

119886119888 gt 1198872 119886

1gt 0 119888

1gt 0

1198862gt 0 119888

2ge 0

(18)

and 1198882= 0 if and only if d = 0 via 119860minus1 ≻ 0

3 The Solutions

In this sectionwewill explore the explicit solution of problem(RSm) Notice that y = wminusw

119887 then we can rewrite separably

problem (RSm) into the following three problems

maxy

y119879120583 suprisinD

P y119879r le 0 le 1205900 119872119879y = d e119879y = 0

(19)

maxy

y119879120583 suprisinD

E [(minusy119879r)+] le 1205901 119872119879y = d e119879y = 0

(20)

maxy

y119879120583 suprisinD

E [((minusy119879r)+)2

] le 1205902 119872119879y = d e119879y = 0

(21)

which is corresponding to the case of 120588 = 0 in the worst-caseLPM119898

31 Maximization of Information Ratio A Variation of119882119862119871119875119872

0 Let us first consider problem (19) The inequality

Py119879r le 0 le 1205900is usually called chance constraint or Royrsquos

safety-first rule in the literature Generally speaking 1205900is

taken far less than 12 which leads to (11205900) minus 1 gt 0 Hence

from Lemma 2(a) and Lemma 4 we have that when y119879120583 ge 0

suprisinD

P y119879r le 0 le 1205900lArrrArr

1

1 + (minusy119879120583)2y119879Σyle 1205900

lArrrArr

(minusy119879120583)2

y119879Σyge (

1

1205900

minus 1)

lArrrArry119879120583

radicy119879Σyge radic(

1

1205900

minus 1)

(22)

The last inequality means that information ratio (or sharperatio) of the portfolio is not less than a preset constant Bythis characteristic we consider a variation version of problem(19) in which we minimize the WCLPM

0 that is

miny suprisinD

P y119879r le 0 | st 119872119879y = d e119879y = 0 (23)

which results from (22) the maximization of informationratio (IR for short)

MAXIR = maxy

y119879120583

radicy119879Σy 119872119879y = d e119879y = 0

(24)

Mathematical Problems in Engineering 5

Problem (24) is related to the mean-variance with multipleweights constraints (MVMWC for short) for a given param-eter 1205901015840

0

(MVMWC) maxy

y119879120583 y119879Σy le 1205901015840

0 119872119879y = d e119879y = 0

(25)

If 119901 = 0 that is there is not weights constraint then problem(25) is nothing but Rollrsquos VTE problem If 119901 = 1 that is119872 = e

1 then problem (25) becomes the case of single weight

constraint problem which has been considered by Bajeux-Besnainou et al [36]

The explicit solution of problem (24) without weightsconstraints has been obtained in the literature (eg see [1623] for detail) But the methods in the literature cannotbe used directly for problem (24) because of the existenceof multiple weights constraints Thus we need to find thedifferent method from the literature to obtain the explicitsolution To this end we deal with this problem in two stagesto get the explicit solution of (24) We introduce a parameterat first stage and parameterize the solution of (24) and thenmaximize IR of the solution with respect to the parameter inthe second stage

Theorem 6 There exists a real number say 1205910 such that the

explicit solution of problem (24) can be expressed as

ylowast1205910=

1

1205910

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(26)

Proof See Theorem 1 in [37] see also Appendix for detail

To obtain the portfolio with the maximum informationratio that is the optimal solution of problem (24) what isthen the value of parameter 120591

0 As the second stage we

replace ylowast1205910into the objective function (24) Then we get the

information ratio of portfolio ylowast1205910 denoted by MAX

1205910 which

is a function of parameter 1205910 Thus we can get the maximum

information ratio by maximizing MAX1205910with respect to 120591

0

That is we have

MAXIR = max1205910gt0

MAX1205910= max1205910gt0

(ylowast1205910)119879

120583

radic(ylowast1205910)119879

Σylowast1205910

(27)

Let 120591lowast0be the optimal solution of (27) then portfolio ylowast

120591lowast

0

isthe optimal solution of (24) for which it has the maximuminformation ratio Hence we have the results below and itsproof is straightforward

Theorem 7 The maximizer of problem (27) is given by

120591lowast

0=11988611198870minus 11988601198871

11988611198880+ 1198862

0

(28)

when 11988611198870minus 11988601198871

gt 0 and the optimal solution of (24) isobtained by

ylowastIR =

(11988611198880+ 1198862

0)

11988611198870minus 11988601198871

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(29)

Theorem 8 (see [37]) If ylowast1205910

in (26) is obtained for certainpositive number 120591

0 then there exists a 1205901015840

0gt 0 such that both

problems (24) and (25) have the same optimal solution

Let ylowast0be the optimal solution of (24) in view of the proof

of Theorem 6 in Appendix if 12059010158400is chosen and satisfies

radic1198862(11988611205901015840

0minus 1198882)

1198862

=(ylowast0)119879

Σylowast0

(120583119879ylowast0) (30)

then the optimal portfolio for problem (25) is also the optimalsolution of (24) The relationship between problems (24)and (25) can be presented as follows Solving (24) will geta portfolio with the maximum IR However the optimalportfolio of problem (25) is not necessarily the one with themaximum IR If we solve (25) with 120590

1015840

0given by (30) then the

optimal portfolio of (25) will have the maximum IR too

32 The Mean-1198821198621198711198751198721 TheWCLPM

1constraint in prob-

lem (20) from Lemmas 2 and 4 can be written as

suprisinD

E [(minusy119879r)+] le 1205901lArrrArr

minusy119879120583 + radicy119879Σy + (minusy119879120583)2

2le 1205901

lArrrArr y119879Σy minus 4120590

1y119879120583 le 4120590

2

1

y119879120583 ge minus21205901

(31)

In fact it can verify that

miny y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1 gt minus2120590

1 (32)

Hence inequality constraint y119879120583 ge minus21205901is redundant Thus

for any preset 1205901gt 0 we can rewrite problem (20) as

WCLPM1maxy

y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1

119872119879y = d e119879y = 0

(33)

Here we call 21205901the upper bound of tracking error under

WCLPM1constraint and denote the upper bound of tracking

error by TE1= 21205901 The following lemma gives the feasible

condition of problem (33)

Lemma 9 If 1205901is chosen to satisfy

1205901gt 120590lowast

1 (34)

6 Mathematical Problems in Engineering

then problem (33) is feasible where

120590lowast

1

=

radic(11988611198870minus11988601198871)2

+(11988611198881minus1198872

1+1198861) (11988611198880+1198862

0)minus(11988611198870minus 11988601198871)

2 (11988611198881minus 1198872

1+ 1198861)

(35)

Proof Let

119891 (y) = y119879Σy minus 41205901y119879120583 (36)

and S11

= y 119891(y) le 41205902

1 S12

= y isin R119899 119872119879y = d e119879y =

0 Then this lemma is to verify that the set S1= S11⋂S12

is nonempty for any 1205901gt 120590lowast

1 Now for any 120590

1gt 0 consider

the following subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 119872119879y = d e119879y = 0 (37)

and denote its optimal value by 1198911205901

which is a quadraticfunction of parameter 120590

1 Thus (for the computation of 119891

1205901

and 120590lowast

1 see Appendix for detail)

1198911205901le 41205902

1(38)

gives a quadratic inequality with respect to 1205901 Solving (38)

we have that the equality in (38) holds when 1205901= 120590lowast

1gt 0

where 120590lowast1is given by (35) Hence set S

1to be nonempty and

has at least an element when 1205901gt 120590lowast

1 This implies problem

(33) is feasible The proof is completed

Lemma 9 indicates that problemWCLPM1is well defined

and meaningful The following results give the explicit solu-tion of problemWCLPM

1

Theorem 10 Let the condition of Lemma 9 be satisfied Thenthe optimal solution of problem (33) can be expressed explicitlyas

ylowast1= (1205821+ 21205901) (Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(39)

where

1205821

=

radic4 (11988611198881minus 1198872

1+ 1198861) 1205902

1+ 4 (119886

11198870minus 11988601198871) 1205901minus (11988611198880+ 1198862

0)

radic11988611198881minus 1198872

1

(40)

Proof See Theorem 4 in [37] see also Appendix for detail

33 The Mean-1198821198621198711198751198722 In this subsection we consider

mean-WCLPM2portfolio problem In view of Lemma 2(c)

and Lemma 4 the WCLPM2constraint can be formulated as

((minusy119879120583)+)2

+ y119879Σy le 1205902 (41)

Then we can write problem (7) with119898 = 2 as

WCLPM2maxy

y119879120583 (41) 119872119879y = d e119879y = 0 (42)

If we add constraint y119879120583 ge 0 then problem (42) is nothingbut the classical M-Vmodel with weights constraints On theother hand if we add constraint y119879120583 lt 0 then constraint (41)is

(y119879120583)2

+ y119879Σy = y119879 (Σ + 120583120583119879) y le 120590

2 (43)

which is more conservative than the variance of portfolio forthe same preset value 120590

2since (Σ + 120583120583

119879) minus Σ = 120583120583

119879 is apositive semidefinite matrix We call radic120590

2 instead of 120590

2 the

upper bound of tracking error under WCLPM2constraint

and denote the upper bound by TE2

= radic1205902 Similar to

Lemma 9 we can also give the feasible condition of problem(42)

Lemma 11 Let

1205902gt11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (44)

Then problem (42) is feasible

Proof Let

120590lowast

2= miny ((minusy119879120583)

+)2

+ y119879Σy 119872119879y = d e119879y = 0

(45)

It follows from solving straightforwardly the problem that itsoptimal solution is

ylowast1205902=

[1198861V minus (119886

11198870minus 11988601198871)]

11988611198881minus 1198872

1

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(46)

where

V =(11988611198881minus 1198872

1) (11988601198871minus 11988611198870)+

1198861(1198861+ 11988611198881minus 1198872

1)

+11988611198870minus 11988601198871

1198861

(47)

Then we get that

120590lowast

2=11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (48)

Hence the problem (42) is feasible when 1205902satisfies 120590

2gt 120590lowast

2

This completes the proof

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

and selling short is allowed Bajeux-Besnainou et al [36] con-sidered the following active portfolio optimization problemwith single weight constraint

max E [w119879r] (w minus w119887)119879

Σ (w minus w119887) le 120590

e119879Iw = (le) 119902 e119879w = 1

(5)

and get a closed solution of this problem As pointed inSection 1 the classical mean-variance framework may not beappropriate since it controls not only the loss of portfolio butalso the gain of portfolio

In our framework we use LPM to control the risk ofportfolio bywhichwe in fact only control the loss of portfolioAdditionally we make no assumption on the distribution ofreturns r Instead we assume that we only partially knowsome knowledge about the statistical properties of returnsr and the underlying distribution function is only knownto belong to a certain set D of distribution functions Fof the returns vector r Under the uncertainty set D of theunderlying distribution we have the worst-case LPM

119898

Definition 1 For any 119898 ge 0 and real number 120588 the worst-case lower-partial moment (WCLPM for short) of randomvariable119883 with respect to 120588 underF isin D is defined by

WCLPM119898(120588) = sup

FisinD

E [((120588 minus 119883)+)119898

] (6)

In tracking error portfolio selection problems investorswant in fact to find a portfolio w such that the return ofportfolio w can be close to or outperform the return ofbenchmark w119879

119887120583 For this we call w119879

119887120583 minus w119879120583 the loss of

portfoliow relative tow119879119887120583 and control the risk of portfolio by

restricting the loss Based on this idea we maximize on onehand the return w119879120583 and on the other hand control the lossby usingworst-case LPM

119898Thus themean-WCLPM

119898robust

tracking error problem with multiple weights constraints canbe written as

maxw

w119879120583

st supFisinD

E [((w119879119887r minus w119879r)

+)119898

] le 120590119898

e119879I119894w = (le) 119902119894 119894 = 1 119901

e119879w = 1

(7)

where 120590119898(119898 = 0 1 2) are the preset constants The second

class of inequalities constraints 119901weights constraints expressthat several classes of assets are restricted to the limitedweights where I

119894sube 1 2 119899 and 119902

119894(119894 = 1 119901)

are constants We call these constraints multiple weightsconstraints When 119901 = 1 it is the single weight constraintIn this paper we only consider the cases of 119898 = 0 1 2

andmultiple equality weights constraintsThe correspondingresults with multiple inequality weights constraints can beobtained by the similar methods

Because we are interested in developing robust trackingerror portfolio policies and exploring the explicit solutions ofproblems (7) in the rest of this paper we assume that D is

the set of allowable distributions of returns r with the knownmean and covariance that is

D = r | E [r] = 120583Cov (r) = Σ ≻ 0 (8)

For convenience we sometimes denote r isin D by r sim (120583 Σ)Our model can be extended to the case of parametersuncertainty for which parameters are not estimated exactly

Similar to Erdogan et al [33] and Bajeux-Besnainou et al[36] we normalize the benchmark portfolio w

119887that is

e119879w119887= 1 and introduce a new variable y = w minus w

119887 Notice

the w119879120583 = y119879120583 + w119879119887120583 then optimization problem (7) with

multiple equality weights constraints can be written into thefollowing form

(RSm) maxy

y119879120583

st suprisinD

E [((minusy119879r)+)119898

] le 120590119898

e119879I119894y = 119902119894minus e119879I119894w119887 119894 = 1 119901

e119879y = 0

(9)

We call y a self-financing portfolio and w a fully investingportfolio We mention that 119902

119894minus e119879I119894w119887 119894 = 1 119901 can be

positive or negative most likely between minus1 and 1 since 119902119894and

e119879I119894w119887 are in most cases between 0 and 1 In the rest of thepaper we suppose that 119901 + 1 vectors e and eI119894 (119894 = 1 119901)

are linearly independent and 119899 gt 119901 + 1 always holds Ifthere exists certain eI119896 such that eI119896 can be expressed as thelinear combination of eI119894 (119894 = 1 119901 119894 = 119896) then e119879I119896y =

119902119896minus e119879I119896w119887 becomes a redundant constraint Additionally

the linear equality constraints will lead to a unique feasiblesolution y if 119899 = 119901 + 1

The following lemmas indicate that a tight upper boundof the worst-case LPM

119898can be obtained explicitly which will

be helpful for our analysis later We extend the proof in [23]for WCLPM

0to a general case

Lemma2 (see [23 39]) Let 120585 be a random variable withmean120583 and variance 120590 but have unknown distribution and 120588 is anarbitrary real number Then we have the following

(a) For the case of119898 = 0

sup120585sim(120583120590)

P 120585 le 120588 =

1

1 + (120588 minus 120583)2

1205902

if 120588 le 120583

1 if 120588 ge 120583

(10)

(b) For the case of119898 = 1

sup120585sim(120583120590)

E [(120588 minus 120585)+] =

120588 minus 120583 + radic1205902+ (120588 minus 120583)

2

2 (11)

(c) For the case of119898 = 2

sup120585sim(120583120590)

E [((120588 minus 120585)+)2

] = [(120588 minus 120583)+]2

+ 1205902

= 1205902 120588 le 120583

(120588 minus 120585)2

+ 1205902 120588 ge 120583

(12)

4 Mathematical Problems in Engineering

Lemma 3 (see [40]) For any a isin R119899 denote

1198781= a119879r | r isin D

1198782= 120578 | E (120578) = a119879120583Var (120578) = a119879Σa

(13)

Then 1198781= 1198782

Lemma 4 (see [37]) For any r isin D 120578 isin 1198782 and any real

vector a isin R119899 the following equality

suprisinD

E [((120588 minus a119879r)+)119898

] = sup120578sim(a119879120583a119879Σa)

E [((120588 minus 120578)+)119898

] (14)

holds

Lemma 4 establishes the relationship of single variable120578 and multivariable r which is useful for our analysis laterWe end this section by introducing some notations that willbe used frequently Unless otherwise specified in the restof paper bold letter expresses a vector uppercase letter is amatrix and lowercase is a real number 0 is a zero matrix orzero vector by context For simplicity we denote eI119896 by e

119896

119896 = 1 119901 and 119902119894minus e119879I119894w119887 by 119889119894 and thus 119889

119894isin [minus1 1] Let

119872 = [e1 e

119901] isin R

119899times119901 d = (119889

1 119889

119901)119879

isin R119901

(15)

Then from the assumption that e1 e

119901and e are linearly

independent matrix 119872 has column full rank Let 119860 =

119872119879Σminus1119872 isin R119901times119901 Then matrix 119860 is positive definite via

Σminus1

≻ 0 where the notation 119885 ⪰ 119884 (or 119885 ≻ 119884) means matrix119885 minus 119884 is positive semidefinite (or positive definite) Furtherwe denote

119861 = 119868 minus119872119860minus1119872119879Σminus1

isin R119899times119899

119862 = Σminus1119872119860minus1

isin R119899times119901

(16)

where 119868 is a unit matrix with reasonable dimensions Thenfor these matrices we have the following results

Lemma 5 (a) 119861119872 = 0(b)119872T

Σminus1119861 = 0

(c) 119861119879119862 = 0(d) 119861119879Σminus1119861 = Σ

minus1119861

(e) 119862119879Σ119862 = 119860minus1

( f) Matrix 119861 is positive semidefinite

The conclusions are straightforward for (a)ndash(e) For (f)we have from (d) Σminus1119861 = 119861

119879Σminus1119861 ⪰ 0 since Σminus1 ≻ 0 Hence

119861 is a positive semidefinite matrixBased on matrices 119861 119862 and Σ

minus1 we denote additionally

119886 = e119879Σminus1e 119887 = e119879Σminus1120583 119888 = 120583119879Σminus1120583

1198860= e119879119862d 119887

0= 120583119879119862d 119888

0= d119879119860minus1d

1198861= e119879Σminus1119861e 119887

1= e119879Σminus1119861120583 119888

1= 120583119879Σminus1119861120583

1198862= 11988611198881minus 1198872

1 119887

2= 11988611198870minus 11988601198871 119888

2= 11988611198880+ 1198862

0

(17)

Without loss the generality we assumeΣminus1119861 = 0 thenwe havethat

119886119888 gt 1198872 119886

1gt 0 119888

1gt 0

1198862gt 0 119888

2ge 0

(18)

and 1198882= 0 if and only if d = 0 via 119860minus1 ≻ 0

3 The Solutions

In this sectionwewill explore the explicit solution of problem(RSm) Notice that y = wminusw

119887 then we can rewrite separably

problem (RSm) into the following three problems

maxy

y119879120583 suprisinD

P y119879r le 0 le 1205900 119872119879y = d e119879y = 0

(19)

maxy

y119879120583 suprisinD

E [(minusy119879r)+] le 1205901 119872119879y = d e119879y = 0

(20)

maxy

y119879120583 suprisinD

E [((minusy119879r)+)2

] le 1205902 119872119879y = d e119879y = 0

(21)

which is corresponding to the case of 120588 = 0 in the worst-caseLPM119898

31 Maximization of Information Ratio A Variation of119882119862119871119875119872

0 Let us first consider problem (19) The inequality

Py119879r le 0 le 1205900is usually called chance constraint or Royrsquos

safety-first rule in the literature Generally speaking 1205900is

taken far less than 12 which leads to (11205900) minus 1 gt 0 Hence

from Lemma 2(a) and Lemma 4 we have that when y119879120583 ge 0

suprisinD

P y119879r le 0 le 1205900lArrrArr

1

1 + (minusy119879120583)2y119879Σyle 1205900

lArrrArr

(minusy119879120583)2

y119879Σyge (

1

1205900

minus 1)

lArrrArry119879120583

radicy119879Σyge radic(

1

1205900

minus 1)

(22)

The last inequality means that information ratio (or sharperatio) of the portfolio is not less than a preset constant Bythis characteristic we consider a variation version of problem(19) in which we minimize the WCLPM

0 that is

miny suprisinD

P y119879r le 0 | st 119872119879y = d e119879y = 0 (23)

which results from (22) the maximization of informationratio (IR for short)

MAXIR = maxy

y119879120583

radicy119879Σy 119872119879y = d e119879y = 0

(24)

Mathematical Problems in Engineering 5

Problem (24) is related to the mean-variance with multipleweights constraints (MVMWC for short) for a given param-eter 1205901015840

0

(MVMWC) maxy

y119879120583 y119879Σy le 1205901015840

0 119872119879y = d e119879y = 0

(25)

If 119901 = 0 that is there is not weights constraint then problem(25) is nothing but Rollrsquos VTE problem If 119901 = 1 that is119872 = e

1 then problem (25) becomes the case of single weight

constraint problem which has been considered by Bajeux-Besnainou et al [36]

The explicit solution of problem (24) without weightsconstraints has been obtained in the literature (eg see [1623] for detail) But the methods in the literature cannotbe used directly for problem (24) because of the existenceof multiple weights constraints Thus we need to find thedifferent method from the literature to obtain the explicitsolution To this end we deal with this problem in two stagesto get the explicit solution of (24) We introduce a parameterat first stage and parameterize the solution of (24) and thenmaximize IR of the solution with respect to the parameter inthe second stage

Theorem 6 There exists a real number say 1205910 such that the

explicit solution of problem (24) can be expressed as

ylowast1205910=

1

1205910

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(26)

Proof See Theorem 1 in [37] see also Appendix for detail

To obtain the portfolio with the maximum informationratio that is the optimal solution of problem (24) what isthen the value of parameter 120591

0 As the second stage we

replace ylowast1205910into the objective function (24) Then we get the

information ratio of portfolio ylowast1205910 denoted by MAX

1205910 which

is a function of parameter 1205910 Thus we can get the maximum

information ratio by maximizing MAX1205910with respect to 120591

0

That is we have

MAXIR = max1205910gt0

MAX1205910= max1205910gt0

(ylowast1205910)119879

120583

radic(ylowast1205910)119879

Σylowast1205910

(27)

Let 120591lowast0be the optimal solution of (27) then portfolio ylowast

120591lowast

0

isthe optimal solution of (24) for which it has the maximuminformation ratio Hence we have the results below and itsproof is straightforward

Theorem 7 The maximizer of problem (27) is given by

120591lowast

0=11988611198870minus 11988601198871

11988611198880+ 1198862

0

(28)

when 11988611198870minus 11988601198871

gt 0 and the optimal solution of (24) isobtained by

ylowastIR =

(11988611198880+ 1198862

0)

11988611198870minus 11988601198871

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(29)

Theorem 8 (see [37]) If ylowast1205910

in (26) is obtained for certainpositive number 120591

0 then there exists a 1205901015840

0gt 0 such that both

problems (24) and (25) have the same optimal solution

Let ylowast0be the optimal solution of (24) in view of the proof

of Theorem 6 in Appendix if 12059010158400is chosen and satisfies

radic1198862(11988611205901015840

0minus 1198882)

1198862

=(ylowast0)119879

Σylowast0

(120583119879ylowast0) (30)

then the optimal portfolio for problem (25) is also the optimalsolution of (24) The relationship between problems (24)and (25) can be presented as follows Solving (24) will geta portfolio with the maximum IR However the optimalportfolio of problem (25) is not necessarily the one with themaximum IR If we solve (25) with 120590

1015840

0given by (30) then the

optimal portfolio of (25) will have the maximum IR too

32 The Mean-1198821198621198711198751198721 TheWCLPM

1constraint in prob-

lem (20) from Lemmas 2 and 4 can be written as

suprisinD

E [(minusy119879r)+] le 1205901lArrrArr

minusy119879120583 + radicy119879Σy + (minusy119879120583)2

2le 1205901

lArrrArr y119879Σy minus 4120590

1y119879120583 le 4120590

2

1

y119879120583 ge minus21205901

(31)

In fact it can verify that

miny y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1 gt minus2120590

1 (32)

Hence inequality constraint y119879120583 ge minus21205901is redundant Thus

for any preset 1205901gt 0 we can rewrite problem (20) as

WCLPM1maxy

y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1

119872119879y = d e119879y = 0

(33)

Here we call 21205901the upper bound of tracking error under

WCLPM1constraint and denote the upper bound of tracking

error by TE1= 21205901 The following lemma gives the feasible

condition of problem (33)

Lemma 9 If 1205901is chosen to satisfy

1205901gt 120590lowast

1 (34)

6 Mathematical Problems in Engineering

then problem (33) is feasible where

120590lowast

1

=

radic(11988611198870minus11988601198871)2

+(11988611198881minus1198872

1+1198861) (11988611198880+1198862

0)minus(11988611198870minus 11988601198871)

2 (11988611198881minus 1198872

1+ 1198861)

(35)

Proof Let

119891 (y) = y119879Σy minus 41205901y119879120583 (36)

and S11

= y 119891(y) le 41205902

1 S12

= y isin R119899 119872119879y = d e119879y =

0 Then this lemma is to verify that the set S1= S11⋂S12

is nonempty for any 1205901gt 120590lowast

1 Now for any 120590

1gt 0 consider

the following subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 119872119879y = d e119879y = 0 (37)

and denote its optimal value by 1198911205901

which is a quadraticfunction of parameter 120590

1 Thus (for the computation of 119891

1205901

and 120590lowast

1 see Appendix for detail)

1198911205901le 41205902

1(38)

gives a quadratic inequality with respect to 1205901 Solving (38)

we have that the equality in (38) holds when 1205901= 120590lowast

1gt 0

where 120590lowast1is given by (35) Hence set S

1to be nonempty and

has at least an element when 1205901gt 120590lowast

1 This implies problem

(33) is feasible The proof is completed

Lemma 9 indicates that problemWCLPM1is well defined

and meaningful The following results give the explicit solu-tion of problemWCLPM

1

Theorem 10 Let the condition of Lemma 9 be satisfied Thenthe optimal solution of problem (33) can be expressed explicitlyas

ylowast1= (1205821+ 21205901) (Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(39)

where

1205821

=

radic4 (11988611198881minus 1198872

1+ 1198861) 1205902

1+ 4 (119886

11198870minus 11988601198871) 1205901minus (11988611198880+ 1198862

0)

radic11988611198881minus 1198872

1

(40)

Proof See Theorem 4 in [37] see also Appendix for detail

33 The Mean-1198821198621198711198751198722 In this subsection we consider

mean-WCLPM2portfolio problem In view of Lemma 2(c)

and Lemma 4 the WCLPM2constraint can be formulated as

((minusy119879120583)+)2

+ y119879Σy le 1205902 (41)

Then we can write problem (7) with119898 = 2 as

WCLPM2maxy

y119879120583 (41) 119872119879y = d e119879y = 0 (42)

If we add constraint y119879120583 ge 0 then problem (42) is nothingbut the classical M-Vmodel with weights constraints On theother hand if we add constraint y119879120583 lt 0 then constraint (41)is

(y119879120583)2

+ y119879Σy = y119879 (Σ + 120583120583119879) y le 120590

2 (43)

which is more conservative than the variance of portfolio forthe same preset value 120590

2since (Σ + 120583120583

119879) minus Σ = 120583120583

119879 is apositive semidefinite matrix We call radic120590

2 instead of 120590

2 the

upper bound of tracking error under WCLPM2constraint

and denote the upper bound by TE2

= radic1205902 Similar to

Lemma 9 we can also give the feasible condition of problem(42)

Lemma 11 Let

1205902gt11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (44)

Then problem (42) is feasible

Proof Let

120590lowast

2= miny ((minusy119879120583)

+)2

+ y119879Σy 119872119879y = d e119879y = 0

(45)

It follows from solving straightforwardly the problem that itsoptimal solution is

ylowast1205902=

[1198861V minus (119886

11198870minus 11988601198871)]

11988611198881minus 1198872

1

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(46)

where

V =(11988611198881minus 1198872

1) (11988601198871minus 11988611198870)+

1198861(1198861+ 11988611198881minus 1198872

1)

+11988611198870minus 11988601198871

1198861

(47)

Then we get that

120590lowast

2=11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (48)

Hence the problem (42) is feasible when 1205902satisfies 120590

2gt 120590lowast

2

This completes the proof

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Lemma 3 (see [40]) For any a isin R119899 denote

1198781= a119879r | r isin D

1198782= 120578 | E (120578) = a119879120583Var (120578) = a119879Σa

(13)

Then 1198781= 1198782

Lemma 4 (see [37]) For any r isin D 120578 isin 1198782 and any real

vector a isin R119899 the following equality

suprisinD

E [((120588 minus a119879r)+)119898

] = sup120578sim(a119879120583a119879Σa)

E [((120588 minus 120578)+)119898

] (14)

holds

Lemma 4 establishes the relationship of single variable120578 and multivariable r which is useful for our analysis laterWe end this section by introducing some notations that willbe used frequently Unless otherwise specified in the restof paper bold letter expresses a vector uppercase letter is amatrix and lowercase is a real number 0 is a zero matrix orzero vector by context For simplicity we denote eI119896 by e

119896

119896 = 1 119901 and 119902119894minus e119879I119894w119887 by 119889119894 and thus 119889

119894isin [minus1 1] Let

119872 = [e1 e

119901] isin R

119899times119901 d = (119889

1 119889

119901)119879

isin R119901

(15)

Then from the assumption that e1 e

119901and e are linearly

independent matrix 119872 has column full rank Let 119860 =

119872119879Σminus1119872 isin R119901times119901 Then matrix 119860 is positive definite via

Σminus1

≻ 0 where the notation 119885 ⪰ 119884 (or 119885 ≻ 119884) means matrix119885 minus 119884 is positive semidefinite (or positive definite) Furtherwe denote

119861 = 119868 minus119872119860minus1119872119879Σminus1

isin R119899times119899

119862 = Σminus1119872119860minus1

isin R119899times119901

(16)

where 119868 is a unit matrix with reasonable dimensions Thenfor these matrices we have the following results

Lemma 5 (a) 119861119872 = 0(b)119872T

Σminus1119861 = 0

(c) 119861119879119862 = 0(d) 119861119879Σminus1119861 = Σ

minus1119861

(e) 119862119879Σ119862 = 119860minus1

( f) Matrix 119861 is positive semidefinite

The conclusions are straightforward for (a)ndash(e) For (f)we have from (d) Σminus1119861 = 119861

119879Σminus1119861 ⪰ 0 since Σminus1 ≻ 0 Hence

119861 is a positive semidefinite matrixBased on matrices 119861 119862 and Σ

minus1 we denote additionally

119886 = e119879Σminus1e 119887 = e119879Σminus1120583 119888 = 120583119879Σminus1120583

1198860= e119879119862d 119887

0= 120583119879119862d 119888

0= d119879119860minus1d

1198861= e119879Σminus1119861e 119887

1= e119879Σminus1119861120583 119888

1= 120583119879Σminus1119861120583

1198862= 11988611198881minus 1198872

1 119887

2= 11988611198870minus 11988601198871 119888

2= 11988611198880+ 1198862

0

(17)

Without loss the generality we assumeΣminus1119861 = 0 thenwe havethat

119886119888 gt 1198872 119886

1gt 0 119888

1gt 0

1198862gt 0 119888

2ge 0

(18)

and 1198882= 0 if and only if d = 0 via 119860minus1 ≻ 0

3 The Solutions

In this sectionwewill explore the explicit solution of problem(RSm) Notice that y = wminusw

119887 then we can rewrite separably

problem (RSm) into the following three problems

maxy

y119879120583 suprisinD

P y119879r le 0 le 1205900 119872119879y = d e119879y = 0

(19)

maxy

y119879120583 suprisinD

E [(minusy119879r)+] le 1205901 119872119879y = d e119879y = 0

(20)

maxy

y119879120583 suprisinD

E [((minusy119879r)+)2

] le 1205902 119872119879y = d e119879y = 0

(21)

which is corresponding to the case of 120588 = 0 in the worst-caseLPM119898

31 Maximization of Information Ratio A Variation of119882119862119871119875119872

0 Let us first consider problem (19) The inequality

Py119879r le 0 le 1205900is usually called chance constraint or Royrsquos

safety-first rule in the literature Generally speaking 1205900is

taken far less than 12 which leads to (11205900) minus 1 gt 0 Hence

from Lemma 2(a) and Lemma 4 we have that when y119879120583 ge 0

suprisinD

P y119879r le 0 le 1205900lArrrArr

1

1 + (minusy119879120583)2y119879Σyle 1205900

lArrrArr

(minusy119879120583)2

y119879Σyge (

1

1205900

minus 1)

lArrrArry119879120583

radicy119879Σyge radic(

1

1205900

minus 1)

(22)

The last inequality means that information ratio (or sharperatio) of the portfolio is not less than a preset constant Bythis characteristic we consider a variation version of problem(19) in which we minimize the WCLPM

0 that is

miny suprisinD

P y119879r le 0 | st 119872119879y = d e119879y = 0 (23)

which results from (22) the maximization of informationratio (IR for short)

MAXIR = maxy

y119879120583

radicy119879Σy 119872119879y = d e119879y = 0

(24)

Mathematical Problems in Engineering 5

Problem (24) is related to the mean-variance with multipleweights constraints (MVMWC for short) for a given param-eter 1205901015840

0

(MVMWC) maxy

y119879120583 y119879Σy le 1205901015840

0 119872119879y = d e119879y = 0

(25)

If 119901 = 0 that is there is not weights constraint then problem(25) is nothing but Rollrsquos VTE problem If 119901 = 1 that is119872 = e

1 then problem (25) becomes the case of single weight

constraint problem which has been considered by Bajeux-Besnainou et al [36]

The explicit solution of problem (24) without weightsconstraints has been obtained in the literature (eg see [1623] for detail) But the methods in the literature cannotbe used directly for problem (24) because of the existenceof multiple weights constraints Thus we need to find thedifferent method from the literature to obtain the explicitsolution To this end we deal with this problem in two stagesto get the explicit solution of (24) We introduce a parameterat first stage and parameterize the solution of (24) and thenmaximize IR of the solution with respect to the parameter inthe second stage

Theorem 6 There exists a real number say 1205910 such that the

explicit solution of problem (24) can be expressed as

ylowast1205910=

1

1205910

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(26)

Proof See Theorem 1 in [37] see also Appendix for detail

To obtain the portfolio with the maximum informationratio that is the optimal solution of problem (24) what isthen the value of parameter 120591

0 As the second stage we

replace ylowast1205910into the objective function (24) Then we get the

information ratio of portfolio ylowast1205910 denoted by MAX

1205910 which

is a function of parameter 1205910 Thus we can get the maximum

information ratio by maximizing MAX1205910with respect to 120591

0

That is we have

MAXIR = max1205910gt0

MAX1205910= max1205910gt0

(ylowast1205910)119879

120583

radic(ylowast1205910)119879

Σylowast1205910

(27)

Let 120591lowast0be the optimal solution of (27) then portfolio ylowast

120591lowast

0

isthe optimal solution of (24) for which it has the maximuminformation ratio Hence we have the results below and itsproof is straightforward

Theorem 7 The maximizer of problem (27) is given by

120591lowast

0=11988611198870minus 11988601198871

11988611198880+ 1198862

0

(28)

when 11988611198870minus 11988601198871

gt 0 and the optimal solution of (24) isobtained by

ylowastIR =

(11988611198880+ 1198862

0)

11988611198870minus 11988601198871

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(29)

Theorem 8 (see [37]) If ylowast1205910

in (26) is obtained for certainpositive number 120591

0 then there exists a 1205901015840

0gt 0 such that both

problems (24) and (25) have the same optimal solution

Let ylowast0be the optimal solution of (24) in view of the proof

of Theorem 6 in Appendix if 12059010158400is chosen and satisfies

radic1198862(11988611205901015840

0minus 1198882)

1198862

=(ylowast0)119879

Σylowast0

(120583119879ylowast0) (30)

then the optimal portfolio for problem (25) is also the optimalsolution of (24) The relationship between problems (24)and (25) can be presented as follows Solving (24) will geta portfolio with the maximum IR However the optimalportfolio of problem (25) is not necessarily the one with themaximum IR If we solve (25) with 120590

1015840

0given by (30) then the

optimal portfolio of (25) will have the maximum IR too

32 The Mean-1198821198621198711198751198721 TheWCLPM

1constraint in prob-

lem (20) from Lemmas 2 and 4 can be written as

suprisinD

E [(minusy119879r)+] le 1205901lArrrArr

minusy119879120583 + radicy119879Σy + (minusy119879120583)2

2le 1205901

lArrrArr y119879Σy minus 4120590

1y119879120583 le 4120590

2

1

y119879120583 ge minus21205901

(31)

In fact it can verify that

miny y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1 gt minus2120590

1 (32)

Hence inequality constraint y119879120583 ge minus21205901is redundant Thus

for any preset 1205901gt 0 we can rewrite problem (20) as

WCLPM1maxy

y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1

119872119879y = d e119879y = 0

(33)

Here we call 21205901the upper bound of tracking error under

WCLPM1constraint and denote the upper bound of tracking

error by TE1= 21205901 The following lemma gives the feasible

condition of problem (33)

Lemma 9 If 1205901is chosen to satisfy

1205901gt 120590lowast

1 (34)

6 Mathematical Problems in Engineering

then problem (33) is feasible where

120590lowast

1

=

radic(11988611198870minus11988601198871)2

+(11988611198881minus1198872

1+1198861) (11988611198880+1198862

0)minus(11988611198870minus 11988601198871)

2 (11988611198881minus 1198872

1+ 1198861)

(35)

Proof Let

119891 (y) = y119879Σy minus 41205901y119879120583 (36)

and S11

= y 119891(y) le 41205902

1 S12

= y isin R119899 119872119879y = d e119879y =

0 Then this lemma is to verify that the set S1= S11⋂S12

is nonempty for any 1205901gt 120590lowast

1 Now for any 120590

1gt 0 consider

the following subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 119872119879y = d e119879y = 0 (37)

and denote its optimal value by 1198911205901

which is a quadraticfunction of parameter 120590

1 Thus (for the computation of 119891

1205901

and 120590lowast

1 see Appendix for detail)

1198911205901le 41205902

1(38)

gives a quadratic inequality with respect to 1205901 Solving (38)

we have that the equality in (38) holds when 1205901= 120590lowast

1gt 0

where 120590lowast1is given by (35) Hence set S

1to be nonempty and

has at least an element when 1205901gt 120590lowast

1 This implies problem

(33) is feasible The proof is completed

Lemma 9 indicates that problemWCLPM1is well defined

and meaningful The following results give the explicit solu-tion of problemWCLPM

1

Theorem 10 Let the condition of Lemma 9 be satisfied Thenthe optimal solution of problem (33) can be expressed explicitlyas

ylowast1= (1205821+ 21205901) (Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(39)

where

1205821

=

radic4 (11988611198881minus 1198872

1+ 1198861) 1205902

1+ 4 (119886

11198870minus 11988601198871) 1205901minus (11988611198880+ 1198862

0)

radic11988611198881minus 1198872

1

(40)

Proof See Theorem 4 in [37] see also Appendix for detail

33 The Mean-1198821198621198711198751198722 In this subsection we consider

mean-WCLPM2portfolio problem In view of Lemma 2(c)

and Lemma 4 the WCLPM2constraint can be formulated as

((minusy119879120583)+)2

+ y119879Σy le 1205902 (41)

Then we can write problem (7) with119898 = 2 as

WCLPM2maxy

y119879120583 (41) 119872119879y = d e119879y = 0 (42)

If we add constraint y119879120583 ge 0 then problem (42) is nothingbut the classical M-Vmodel with weights constraints On theother hand if we add constraint y119879120583 lt 0 then constraint (41)is

(y119879120583)2

+ y119879Σy = y119879 (Σ + 120583120583119879) y le 120590

2 (43)

which is more conservative than the variance of portfolio forthe same preset value 120590

2since (Σ + 120583120583

119879) minus Σ = 120583120583

119879 is apositive semidefinite matrix We call radic120590

2 instead of 120590

2 the

upper bound of tracking error under WCLPM2constraint

and denote the upper bound by TE2

= radic1205902 Similar to

Lemma 9 we can also give the feasible condition of problem(42)

Lemma 11 Let

1205902gt11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (44)

Then problem (42) is feasible

Proof Let

120590lowast

2= miny ((minusy119879120583)

+)2

+ y119879Σy 119872119879y = d e119879y = 0

(45)

It follows from solving straightforwardly the problem that itsoptimal solution is

ylowast1205902=

[1198861V minus (119886

11198870minus 11988601198871)]

11988611198881minus 1198872

1

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(46)

where

V =(11988611198881minus 1198872

1) (11988601198871minus 11988611198870)+

1198861(1198861+ 11988611198881minus 1198872

1)

+11988611198870minus 11988601198871

1198861

(47)

Then we get that

120590lowast

2=11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (48)

Hence the problem (42) is feasible when 1205902satisfies 120590

2gt 120590lowast

2

This completes the proof

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Problem (24) is related to the mean-variance with multipleweights constraints (MVMWC for short) for a given param-eter 1205901015840

0

(MVMWC) maxy

y119879120583 y119879Σy le 1205901015840

0 119872119879y = d e119879y = 0

(25)

If 119901 = 0 that is there is not weights constraint then problem(25) is nothing but Rollrsquos VTE problem If 119901 = 1 that is119872 = e

1 then problem (25) becomes the case of single weight

constraint problem which has been considered by Bajeux-Besnainou et al [36]

The explicit solution of problem (24) without weightsconstraints has been obtained in the literature (eg see [1623] for detail) But the methods in the literature cannotbe used directly for problem (24) because of the existenceof multiple weights constraints Thus we need to find thedifferent method from the literature to obtain the explicitsolution To this end we deal with this problem in two stagesto get the explicit solution of (24) We introduce a parameterat first stage and parameterize the solution of (24) and thenmaximize IR of the solution with respect to the parameter inthe second stage

Theorem 6 There exists a real number say 1205910 such that the

explicit solution of problem (24) can be expressed as

ylowast1205910=

1

1205910

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(26)

Proof See Theorem 1 in [37] see also Appendix for detail

To obtain the portfolio with the maximum informationratio that is the optimal solution of problem (24) what isthen the value of parameter 120591

0 As the second stage we

replace ylowast1205910into the objective function (24) Then we get the

information ratio of portfolio ylowast1205910 denoted by MAX

1205910 which

is a function of parameter 1205910 Thus we can get the maximum

information ratio by maximizing MAX1205910with respect to 120591

0

That is we have

MAXIR = max1205910gt0

MAX1205910= max1205910gt0

(ylowast1205910)119879

120583

radic(ylowast1205910)119879

Σylowast1205910

(27)

Let 120591lowast0be the optimal solution of (27) then portfolio ylowast

120591lowast

0

isthe optimal solution of (24) for which it has the maximuminformation ratio Hence we have the results below and itsproof is straightforward

Theorem 7 The maximizer of problem (27) is given by

120591lowast

0=11988611198870minus 11988601198871

11988611198880+ 1198862

0

(28)

when 11988611198870minus 11988601198871

gt 0 and the optimal solution of (24) isobtained by

ylowastIR =

(11988611198880+ 1198862

0)

11988611198870minus 11988601198871

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(29)

Theorem 8 (see [37]) If ylowast1205910

in (26) is obtained for certainpositive number 120591

0 then there exists a 1205901015840

0gt 0 such that both

problems (24) and (25) have the same optimal solution

Let ylowast0be the optimal solution of (24) in view of the proof

of Theorem 6 in Appendix if 12059010158400is chosen and satisfies

radic1198862(11988611205901015840

0minus 1198882)

1198862

=(ylowast0)119879

Σylowast0

(120583119879ylowast0) (30)

then the optimal portfolio for problem (25) is also the optimalsolution of (24) The relationship between problems (24)and (25) can be presented as follows Solving (24) will geta portfolio with the maximum IR However the optimalportfolio of problem (25) is not necessarily the one with themaximum IR If we solve (25) with 120590

1015840

0given by (30) then the

optimal portfolio of (25) will have the maximum IR too

32 The Mean-1198821198621198711198751198721 TheWCLPM

1constraint in prob-

lem (20) from Lemmas 2 and 4 can be written as

suprisinD

E [(minusy119879r)+] le 1205901lArrrArr

minusy119879120583 + radicy119879Σy + (minusy119879120583)2

2le 1205901

lArrrArr y119879Σy minus 4120590

1y119879120583 le 4120590

2

1

y119879120583 ge minus21205901

(31)

In fact it can verify that

miny y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1 gt minus2120590

1 (32)

Hence inequality constraint y119879120583 ge minus21205901is redundant Thus

for any preset 1205901gt 0 we can rewrite problem (20) as

WCLPM1maxy

y119879120583 y119879Σy minus 41205901y119879120583 le 4120590

2

1

119872119879y = d e119879y = 0

(33)

Here we call 21205901the upper bound of tracking error under

WCLPM1constraint and denote the upper bound of tracking

error by TE1= 21205901 The following lemma gives the feasible

condition of problem (33)

Lemma 9 If 1205901is chosen to satisfy

1205901gt 120590lowast

1 (34)

6 Mathematical Problems in Engineering

then problem (33) is feasible where

120590lowast

1

=

radic(11988611198870minus11988601198871)2

+(11988611198881minus1198872

1+1198861) (11988611198880+1198862

0)minus(11988611198870minus 11988601198871)

2 (11988611198881minus 1198872

1+ 1198861)

(35)

Proof Let

119891 (y) = y119879Σy minus 41205901y119879120583 (36)

and S11

= y 119891(y) le 41205902

1 S12

= y isin R119899 119872119879y = d e119879y =

0 Then this lemma is to verify that the set S1= S11⋂S12

is nonempty for any 1205901gt 120590lowast

1 Now for any 120590

1gt 0 consider

the following subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 119872119879y = d e119879y = 0 (37)

and denote its optimal value by 1198911205901

which is a quadraticfunction of parameter 120590

1 Thus (for the computation of 119891

1205901

and 120590lowast

1 see Appendix for detail)

1198911205901le 41205902

1(38)

gives a quadratic inequality with respect to 1205901 Solving (38)

we have that the equality in (38) holds when 1205901= 120590lowast

1gt 0

where 120590lowast1is given by (35) Hence set S

1to be nonempty and

has at least an element when 1205901gt 120590lowast

1 This implies problem

(33) is feasible The proof is completed

Lemma 9 indicates that problemWCLPM1is well defined

and meaningful The following results give the explicit solu-tion of problemWCLPM

1

Theorem 10 Let the condition of Lemma 9 be satisfied Thenthe optimal solution of problem (33) can be expressed explicitlyas

ylowast1= (1205821+ 21205901) (Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(39)

where

1205821

=

radic4 (11988611198881minus 1198872

1+ 1198861) 1205902

1+ 4 (119886

11198870minus 11988601198871) 1205901minus (11988611198880+ 1198862

0)

radic11988611198881minus 1198872

1

(40)

Proof See Theorem 4 in [37] see also Appendix for detail

33 The Mean-1198821198621198711198751198722 In this subsection we consider

mean-WCLPM2portfolio problem In view of Lemma 2(c)

and Lemma 4 the WCLPM2constraint can be formulated as

((minusy119879120583)+)2

+ y119879Σy le 1205902 (41)

Then we can write problem (7) with119898 = 2 as

WCLPM2maxy

y119879120583 (41) 119872119879y = d e119879y = 0 (42)

If we add constraint y119879120583 ge 0 then problem (42) is nothingbut the classical M-Vmodel with weights constraints On theother hand if we add constraint y119879120583 lt 0 then constraint (41)is

(y119879120583)2

+ y119879Σy = y119879 (Σ + 120583120583119879) y le 120590

2 (43)

which is more conservative than the variance of portfolio forthe same preset value 120590

2since (Σ + 120583120583

119879) minus Σ = 120583120583

119879 is apositive semidefinite matrix We call radic120590

2 instead of 120590

2 the

upper bound of tracking error under WCLPM2constraint

and denote the upper bound by TE2

= radic1205902 Similar to

Lemma 9 we can also give the feasible condition of problem(42)

Lemma 11 Let

1205902gt11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (44)

Then problem (42) is feasible

Proof Let

120590lowast

2= miny ((minusy119879120583)

+)2

+ y119879Σy 119872119879y = d e119879y = 0

(45)

It follows from solving straightforwardly the problem that itsoptimal solution is

ylowast1205902=

[1198861V minus (119886

11198870minus 11988601198871)]

11988611198881minus 1198872

1

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(46)

where

V =(11988611198881minus 1198872

1) (11988601198871minus 11988611198870)+

1198861(1198861+ 11988611198881minus 1198872

1)

+11988611198870minus 11988601198871

1198861

(47)

Then we get that

120590lowast

2=11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (48)

Hence the problem (42) is feasible when 1205902satisfies 120590

2gt 120590lowast

2

This completes the proof

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

then problem (33) is feasible where

120590lowast

1

=

radic(11988611198870minus11988601198871)2

+(11988611198881minus1198872

1+1198861) (11988611198880+1198862

0)minus(11988611198870minus 11988601198871)

2 (11988611198881minus 1198872

1+ 1198861)

(35)

Proof Let

119891 (y) = y119879Σy minus 41205901y119879120583 (36)

and S11

= y 119891(y) le 41205902

1 S12

= y isin R119899 119872119879y = d e119879y =

0 Then this lemma is to verify that the set S1= S11⋂S12

is nonempty for any 1205901gt 120590lowast

1 Now for any 120590

1gt 0 consider

the following subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 119872119879y = d e119879y = 0 (37)

and denote its optimal value by 1198911205901

which is a quadraticfunction of parameter 120590

1 Thus (for the computation of 119891

1205901

and 120590lowast

1 see Appendix for detail)

1198911205901le 41205902

1(38)

gives a quadratic inequality with respect to 1205901 Solving (38)

we have that the equality in (38) holds when 1205901= 120590lowast

1gt 0

where 120590lowast1is given by (35) Hence set S

1to be nonempty and

has at least an element when 1205901gt 120590lowast

1 This implies problem

(33) is feasible The proof is completed

Lemma 9 indicates that problemWCLPM1is well defined

and meaningful The following results give the explicit solu-tion of problemWCLPM

1

Theorem 10 Let the condition of Lemma 9 be satisfied Thenthe optimal solution of problem (33) can be expressed explicitlyas

ylowast1= (1205821+ 21205901) (Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(39)

where

1205821

=

radic4 (11988611198881minus 1198872

1+ 1198861) 1205902

1+ 4 (119886

11198870minus 11988601198871) 1205901minus (11988611198880+ 1198862

0)

radic11988611198881minus 1198872

1

(40)

Proof See Theorem 4 in [37] see also Appendix for detail

33 The Mean-1198821198621198711198751198722 In this subsection we consider

mean-WCLPM2portfolio problem In view of Lemma 2(c)

and Lemma 4 the WCLPM2constraint can be formulated as

((minusy119879120583)+)2

+ y119879Σy le 1205902 (41)

Then we can write problem (7) with119898 = 2 as

WCLPM2maxy

y119879120583 (41) 119872119879y = d e119879y = 0 (42)

If we add constraint y119879120583 ge 0 then problem (42) is nothingbut the classical M-Vmodel with weights constraints On theother hand if we add constraint y119879120583 lt 0 then constraint (41)is

(y119879120583)2

+ y119879Σy = y119879 (Σ + 120583120583119879) y le 120590

2 (43)

which is more conservative than the variance of portfolio forthe same preset value 120590

2since (Σ + 120583120583

119879) minus Σ = 120583120583

119879 is apositive semidefinite matrix We call radic120590

2 instead of 120590

2 the

upper bound of tracking error under WCLPM2constraint

and denote the upper bound by TE2

= radic1205902 Similar to

Lemma 9 we can also give the feasible condition of problem(42)

Lemma 11 Let

1205902gt11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (44)

Then problem (42) is feasible

Proof Let

120590lowast

2= miny ((minusy119879120583)

+)2

+ y119879Σy 119872119879y = d e119879y = 0

(45)

It follows from solving straightforwardly the problem that itsoptimal solution is

ylowast1205902=

[1198861V minus (119886

11198870minus 11988601198871)]

11988611198881minus 1198872

1

(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e)

+ 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(46)

where

V =(11988611198881minus 1198872

1) (11988601198871minus 11988611198870)+

1198861(1198861+ 11988611198881minus 1198872

1)

+11988611198870minus 11988601198871

1198861

(47)

Then we get that

120590lowast

2=11988611198880+ 1198862

0

1198861

+((11988601198871minus 11988611198870)+)2

1198862

1+ 1198861(11988611198881minus 1198872

1) (48)

Hence the problem (42) is feasible when 1205902satisfies 120590

2gt 120590lowast

2

This completes the proof

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

By Lemma 11 problem (42) is well-defined and mean-ingful The following results give the explicit expression ofsolution of problem (42)

Theorem 12 Let the condition of Lemma 11 be satisfied Thenproblem (42) has the explicit solution as follows

ylowast2=

1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

[Σminus1119861120583 minus

1198871

1198861

Σminus1119861e]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(49)

where

Vlowast =

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus ((minus119887

2)+)2

1198861(1198861+ 1198862)

+

radic1198862(11988611205902minus 1198882)

1198861

(1 minus radic1198861

1198861+ 1198862

)(1198872)+

1198872

+(1198861+ 1198862) 1198872+ 1198862(minus1198872)+

1198861(1198861+ 1198862)

120590lowast

2le 1205902le

1198872

2

11988611198862

+1198882

a1

1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0 1205902gt

1198872

2

11988611198862

+1198882

1198861

(50)

Proof See Theorem 5 in [37] see also Appendix for detail

Wemention that the optimal solution ylowast2 of problem (42)

is consistent with problem (25) when 1198872ge 0 or 120590

2gt 1198872

211988611198862+

11988821198861 This is not surprising Because the term ((minusy119879120583)

+)2

vanishes in problem (42) in these cases

4 Numerical Results

In this section we consider the performance of modelsWCLPM

1 WCLPM

2 and MV using real market data We

choose ten stock indexes from Shanghai Stock Exchange(SHH) and Shenzhen Stock Exchange (SHZ) which arecalled domestic assets and four stock indexes from otherstock exchanges which are called foreign assets All fourteenrisky assets are listed in Table 1 Our data covers the periodfrom January 1 2000 to March 18 2011 and includes 2663samples for each asset We group the whole period into twosubperiods that is the first subperiod is from January 1 2000to April 21 2006 and the second subperiod is from April24 2006 to March 18 2011 The data in the first subperiodincludes 1663 samples of each asset and is used as the in-sample test set Other 1000 samples are used as the out-of-sample test setThe statistic properties of samples of all assetsare reported in Table 2 We take the benchmark as the naive1119873 portfolio [38] with119873 = 14 We fix TE = 005 in the nextnumerical experiment

Table 1 Chosen 14 indexes as the risky assets

Number Asset name SymbolsDomestic assets

1 SSE Composite Index 001SS2 SSE A-share Index 002SS3 SSE B-share Index 003SS4 SSE Industrial Index 004SS5 SSE Commercial Index 005SS6 SSE Properties Index 006SS7 SSE Utilities Index 007SS8 SSE Component Index 001SZ9 SSE Component A 002SZ10 SSE Composite Index 106SZ

Foreign Assets11 BSE SENSEX BSESN12 FTSE Bursa Malaysia KLCI KLSE13 Hang Seng Index HSI14 NIKKEI 225 N225

Here we compare the performance of three modelsWCLPM

1 MVMWC and IRMWC (information ratio prob-

lem with multiple weights constraints ie problem (24)) Weconsider the following three cases of weights constraints

119901 = 1 11990811+ 11990812+ 11990813+ 11990814

le1

20

119901 = 2

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

119901 = 4

11990811+ 11990812+ 11990813+ 11990814

le1

20

1199081+ 1199083+ 1199085+ 11990811+ 11990814

le1

14

1199087+ 11990812+ 11990814

le1

14

1199088+ 1199082+ 11990813+ 11990814

le1

20

(51)

The wealth invested to foreign assets is restricted by theweights constraint Table 3 gives the return variance and IRof optimal portfolios obtained by these models Figures 1 2and 3 give the comparisons of portfolio value obtained inthesemodelsThe portfolio value in these figures is computedby

PV119905=

14

sum

119894=1

119908119894times Asset

119894119905 (52)

where w = (1199081 119908

14)119879 is the obtained optimal portfolios

and Asset119894is the value of the 119894th market index at the 119905th

exchange day Some interesting results from our numericalcomparisons can be found as follows

(1) The tracking error of in-sample data from Figures 1ndash3is less than that of out-of-sample data and the tracking errorof model IRMWC is the least among these models whatever

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 Asset market daily returns from January 1 2000 to March 18 2011

Asset Mean (10minus3) Standard deviation Minimum MaximumPanel A in-sample data from January 01 2000 to April 21 2006

001SS 04263 02019 minus00654 01096002SS 04194 01970 minus00651 01098003SS 08951 04687 minus01029 01840004SS 04381 01997 minus00631 01137005SS 03720 02390 minus00753 01039006SS 02563 03519 minus00974 01054007SS 05023 01960 minus00634 01089001SZ 04986 02282 minus00693 01163002SZ 04399 02284 minus00680 01208106SZ 02936 02172 minus00682 01056BSESN 05748 02476 minus01181 01046KLSE 02094 00862 minus00634 00649HSI 01150 01749 minus00929 00601N225 minus00510 02112 minus00901 00722

Panel B out-of-sample data from April 24 2006 to March 18 2011001SS 00172 04691 minus01130 00903002SS 00138 04692 minus01128 00903003SS 06245 06482 minus01572 00937004SS 01737 04982 minus01186 00895005SS 05640 05404 minus01261 00918006SS 01804 09067 minus01817 00951007SS 01058 05745 minus01365 00938001SZ 04622 05780 minus01210 00916002SZ 04951 05776 minus01206 00916106SZ 06488 05608 minus01339 00852BSESN 02246 04266 minus01268 01599KLSE 02428 02111 minus01925 01986HSI 00759 04654 minus01358 01341N225 minus06399 03917 minus01272 01323

the in-sample and out-of-sample data is ButmodelWCLPM1

from Table 3 has the best expectation return among theseconsidered models for all 119901 = 1 2 4 and in-sample and out-of-sample dataThis is understandable sincemodelWCLPM

1

has an excess MV region where it also exceeds the returnof MVMWC Moreover from the view of portfolio valueWCLPM

1can not only exceed benchmark but can also get

greater terminal wealth than that of models MVMWC andIRMWC

(2) It seems to be not understandable that the trackingerrors of all models from Figures 4 5 and 6 increases as thenumber of restricted assets 119901 increases But it is in fact notsurprising because the feasible sets of all models reduce as 119901increases this clearly leads to a larger error relative to the caseof without restricted assets Additionally from Figures 4ndash6we also find that models WCLPM

1and MVMWC are more

sensible than model IRMWC with respect to the number ofrestricted assets

5 Conclusions

We considered a numerical extension of three active portfolioselection problems with the worst-case 119898 = 0- 1- and2-order lower partial moment risk and multiple weightsconstraints which are proposed in [37] Using ten stocksfrom China market and four stocks from other market wecompared the numerical performance with VTE and ldquo1Nrdquostrategy Clearly WCLPM

1from the numerical results can

obtain the better expectation excess return and informationratio than when the tracking error taken is small ModelMVMWC will be another good choice when a large trackingerror is required and sell-shorting is forbidden

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 3 The comparisons of return variance and information ratio of optimal solutions

Model Mean return (10minus3) Standard deviation Information ratio

Panel A in-sample dataBenchmark 03850 11993119890 minus 04 mdashVTE 04304 00527 00782

119901 = 1

WCLPM1 04522 00548 00818MVMWC 04256 00510 00759IRMWC 01300 00170 00781

119901 = 2

WCLPM1 04412 00548 00813MVMWC 04204 00510 00755IRMWC 01365 00173 00786

119901 = 4

WCLPM1 04526 00543 00666MVMWC 04278 00512 00626IRMWC 01325 00139 00607

Panel B out-of-sample dataBenchmark 02278 33696119890 minus 004 mdashMV 03925 00631 00102

119901 = 1

WCLPM1 04282 00642 00104MVMWC 04033 00598 00099IRMWC 01237 00235 00103

119901 = 2

WCLPM1 03851 00637 00102MVMWC 03542 00593 00101IRMWC 01032 00237 00102

119901 = 4

WCLPM1 04268 00649 00101MVMWC 04006 00611 00101IRMWC 01260 00208 00101

Appendix

Technical Proofs

Proof of Theorem 8 Let ylowast0be an optimal solution of problem

(24) Then ylowast0must satisfy the first order optimal condition

of problem (24) Thus we have

120583

radic(ylowast0)119879

Σylowast0

minus

(120583119879ylowast0)

radic(ylowast0)119879

Σylowast0

Σylowast0

(ylowast0)119879

Σylowast0

minus 12058201e minus119872120582

02= 0

(A1)

where 12058201

isin R 12058202

isin R119901 are the Lagrange multipliers Let01

= 12058201radic(ylowast0)119879Σylowast0and

02= 12058202radic(ylowast0)119879Σ119910lowast

0 then equality

(A1) can be reduced as

120583 minus 1205910Σylowast0minus 01e minus119872

02= 0 (A2)

where 1205910= (120583119879ylowast0)(ylowast0)119879Σylowast0is a constant Thus combining

(A2) and the equality constraints 119872119879ylowast0= d and e119879ylowast

0= 0

we get that

ylowast1205910=

1198871

1205910

(Σminus1119861120583

1198871

minusΣminus1119861e

1198861

) + 1198860(119862d1198860

minusΣminus1119861e

1198861

)

(A3)This completes the proof

The Computation of 1198911205901

and 120590lowast

1 Consider the following

subproblem

miny 119891 (y) = y119879Σy minus 41205901y119879120583 | st 119872119879y = d e119879y = 0

(A4)

and denote its optimal value by 1198911205901 Let ylowast

1205901be its optimal

solution Then by the KKT optimal condition we have2Σylowast1205901minus 41205901120583 + 12058211e +119872120582

12= 0

119872119879ylowast1205901= d e119879ylowast

1205901= 0

(A5)

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of samples

Port

folio

val

ue

MVMWCBenchmarkIRMWC

In-sampleperformance

Out-of-sampleperformance

WCLPM1

times104

p = 1 single weights constrain

Figure 1 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there is single weight constraint

0 500 1000 1500 2000 25000

2

4

6

8

10

12

The number of Samples

Port

folio

val

ue

times104

p = 2 two weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 2 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are two weights constraints

where 12058211

isin R and 12058212

isin R119901 are the Lagrange multiplierSolving the equality system of KKT conditions we get

ylowast1205901= 21205901(Σminus1119861120583 minus

1198871

1198861

Σminus1119861e) + 119886

0(119862d1198860

minusΣminus1119861e

1198861

)

(A6)

Substituting it into the objective function and using someresults in Lemma 5 it follows that

1198911205901= minus

4 (11988611198881minus 1198872

1)

1198861

1205902

1minus4 (11988611198870minus 11988601198871)

1198861

1205901

+11988611198880+ 1198862

0

1198861

(A7)

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

ue

times104

p = 4 four weights constraints

MVMWCBenchmarkIRMWC

WCLPM1

In-sampleperformance

Out-of -sampleperformance

Figure 3 The comparison of portfolio values of models WCLPM1

MVMWC IRMWC and benchmark for in-sample and out-of-sample data when there are four weights constraints

0 500 1000 1500 2000 2500The number of samples

0

2

4

6

8

10

12

Port

folio

val

uetimes10

4

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

with different weights constraintsThe comparisons of model WCLPM1

Figure 4 The comparison of portfolio values of model WCLPM1

for in-sample and out-of-sample data and different values of 119901

and it is a quadratic function of parameter 1205901 Solving directly

the quadratic equation in 1205901

1198911205901= 41205902

1 (A8)

we get a positive root of this equation as

120590lowast

1= (radic(119886

11198870minus 11988601198871)2

+ (11988611198881minus 1198872

1+ 1198861) (11988611198880+ 1198862

0)

minus (11988611198870minus 11988601198871) ) times (2 (119886

11198881minus 1198872

1+ 1198861))minus1

(A9)

This obtains equality (35) Thus inequality 1198911205901

lt 41205902

1holds

when 1205901gt 120590lowast

1 This is the conclusion of Lemma 9

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 500 1000 1500 2000 2500The number of samples

The comparisons of model MVMWCwith different weights constraints

0

2

4

6

8

10

12

Port

folio

val

ue

times104

In-sampleperformance

Out-of-sampleperformance

p = 1

p = 2

p = 4

Figure 5 The comparison of portfolio values of model MVMWCfor in-sample and out-of-sample data and different values of 119901

0 500 1000 1500 2000 2500The number of samples

Port

folio

val

ue

The comparisons of IRMWC withdifferent weights constraints

05

1

15

2

25

3In-sample

performanceOut-of-sample

performance

p = 1

p = 2

p = 4

times104

Figure 6The comparison of portfolio values of model IRMWC forin-sample and out-of-sample data and different values of 119901

Proof of Theorem 10 Let ylowast1be the optimal solution of prob-

lem (33) Then from KKT condition we have

(1 + 412059011205821)120583 minus 2120582

11Σylowast1minus 12058212e minus119872120582

13= 0

(ylowast1)119879

Σylowast1minus 41205901ylowast1120583 = 4120590

2

1

119872ylowast1= d

e119879ylowast1= 0

(A10)

where 12058211

ge 0 12058212

isin R and 12058213

isin R119901 are the Lagrangemultipliers Solving directly KKT system (A10) it gets theconclusions of Theorem 10 The proof is finished

Proof of Theorem 12 Let V = y119879120583 Then we can rewriteproblem (42) as

maxyV120583119879y

st ((minusV)+)2

+ y119879Σy le 1205902

119872119879y = d

e119879y = 0

120583119879y = V

(A11)

here we view V as an auxiliary variable The Lagrangefunction of this problem is

119871 (y V 1205822112058222 12058223)

= 120583119879y minus 120582

21e119879y minus 120582119879

22(119872119879y minus d)

minus 12058223[((minusV)

+)2

+ y119879Σy minus 1205902] minus 12058224(120583119879y minus V)

(A12)

where 12058221

isin R 12058222

isin R119901 12058223

ge 0 and 12058224

isin R arethe Lagrange multipliers Let the pair (ylowast

2 Vlowast) be the optimal

solution of (A11) Then the pair (ylowast2 Vlowast) satisfies the first

optimal condition that is

120597119871

120597y= minus2120582

23Σylowast2minus 12058221e minus119872120582

22+ (1 minus 120582

24)120583 = 0

120597119871

120597V= minus2120582

23(minusVlowast)

++ 12058224

= 0

((minusVlowast)+)2

+ (ylowast2)119879

Σylowast2= 1205902

119872119879ylowast2= d e119879ylowast

2= 0 120583

119879ylowast2= Vlowast

(A13)

By the first equation of (A13) it gets that

ylowast2= minus

1

212058223

[12058221Σminus1e + Σ

minus111987212058222minus (1 minus 120582

24) Σminus1120583]

(A14)

Substituting ylowast2into equation119872

119879ylowast2= d we obtain that

12058222

= minus [12058221119860minus1119872119879Σminus1e + 2120582

23119860minus1d minus (1 minus 120582

24) 119860minus1119872119879Σminus1120583]

(A15)

where 119860 = 119872119879Σminus1119872 Substituting 120582

22into ylowast

2and using

Lemma 5 we further get

ylowast2=

1

212058223

[(1 minus 12058224) Σminus1119861120583 minus 120582

21Σminus1119861e + 2120582

23119862d]

(A16)

Combining equation e119879ylowast2= 0 it gets that

12058221

= (1 minus 12058224)1198871

1198861

+21198860

1198861

12058223 (A17)

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Notice that 1minus12058224

= 1minus212058223(minusVlowast)+Thus ylowast

2can be reduced

as

ylowast2= (

1

212058223

minus (minusVlowast)+)[Σminus1119861120583 minus

1198871

1198861

Σminus1119861120583]

+ 1198860[119862d1198860

minusΣminus1119861e

1198861

]

(A18)

Substituting ylowast2into the last equation 120583119879ylowast

2= Vlowast of (A13) we

get

1

212058223

=1198861Vlowast minus (119886

11198870minus 11988601198871)

11988611198881minus 1198872

1

+ (minusVlowast)+ (A19)

On the other hand substituting ylowast2into the third equation of

(A13) we can obtain another expression of 1212058223as

1

212058223

=

radic(11988611198881minus 1198872

1) [11988611205902minus (11988611198880+ 1198862

0) minus 1198861((minusVlowast)

+)2

]

11988611198881minus 1198872

1

+ (minusVlowast)+

(A20)

Hence combining (A19) and (A20) and noticing the nota-tions in Lemma 5 we have that Vlowast satisfies the quadraticequation

1198862

1(Vlowast)2 + 119886

11198862((minusVlowast)

+)2

minus 211988611198872Vlowast + 119887

2

2

minus 1198862(11988611205902minus 1198882) = 0

(A21)

Next we discuss the solutions of (A21) by the different signof 1198872(1) 1198872ge 0 Notice that Vlowast = 120583119879ylowast

2is the expectation excess

return of self-finance portfolio ylowast2 Thus generally speaking

Vlowast is increasing as the preset tracking error 1205902increases

Thus (A21) has only an efficient positive solution as follows(Equation (A21) has in this case another solution

Vlowast =1198872

1198861+ 1198862

minus

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882)

1198861(1198861+ 1198862)

lt 0 (A22)

when Vlowast lt 0 and 1205902is taken to satisfy 120590

2gt (1198872

211988611198862)+(11988821198861)

but it is not an efficient solution since Vlowast is decreasing as 1205902

increases)

Vlowast =1198872

1198861

+

radic1198862(11988611205902minus 1198882)

1198861

ge 0(A23)

(2) 1198872lt 0 If Vlowast le 0 then (A21) has the solution

Vlowast =1198872

1198861+ 1198862

+

radic11988611198862radic(1198861+ 1198862) (11988611205902minus 1198882) minus 1198872

2

1198861(1198861+ 1198862)

le 0

(A24)

when 1205902is chosen to satisfy 120590lowast

2le 1205902le (1198872

211988611198862) + (11988821198861)

If 1205902gt (1198872

211988611198862) + (119888

21198861) then the side of right hand of

equality (A24) is positive and therefore equality (A24) is nota solution of (A21) In the case of 120590

2gt (1198872

211988611198862) + (11988821198861)

the solution equation (A21) has still the form of (A23)Summarizing the two cases above we get that the solution

of (A21) can be expressed as (50) This completes the proof

Conflict of Interests

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

Acknowledgments

The authors are in debt to two anonymous referees andLead Guest Editor Professor Chuangxia Huang for theirconstructive comments and suggestions to improve thequality of this paper This work is supported by NationalNatural Science Foundations of China (nos 71001045 and71371090) Science Foundation of Ministry of Education ofChina (13YJCZH160) Natural Science Foundation of JiangxiProvince of China (20114BAB211008) and Jiangxi Universityof Finance and Economics Support Program Funds forOutstanding Youths

References

[1] H M Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol7 pp 77ndash91 1952

[2] H Markowitz ldquoThe optimization of a quadratic functionsubject to linear constraintsrdquoNaval Research LogisticsQuarterlyvol 3 pp 111ndash133 1956

[3] R Roll ldquoAmean-variance analysis of the tracking errorrdquo Journalof Portfolio Management vol 18 no 1 pp 13ndash22 1992

[4] G J Alexander and A M Baptista ldquoActive portfolio manage-ment with benchmarking adding a value-at-risk constraintrdquoJournal of Economic Dynamics amp Control vol 32 no 3 pp 779ndash820 2008

[5] P Jorion ldquoPortfolio optimization with tracking-error con-straintsrdquo Financial Analysts Journal vol 59 no 5 pp 70ndash822003

[6] M Rudolf H J Wolter and H Zimmermann ldquoA linearmodel for tracking error minimizationrdquo Journal of Banking andFinance vol 23 no 1 pp 85ndash103 1999

[7] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] Y Zhao ldquoAdynamicmodel of active portfoliomanagementwithbenchmark orientationrdquo Journal of Banking and Finance vol 31no 11 pp 3336ndash3356 2007

[10] H H Chena H T Tsaib and D K J Linc ldquoOptimal mean-variance portfolio selection using Cauchy-Schwarz maximiza-tionrdquo Applied economics vol 43 pp 2795ndash2801 2011

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[11] P Glabadanidis ldquoRobust and efficient strategies to trackand outperform a benchmarkrdquo working paper 2010httppapersssrncomsol3paperscfmabstract-id=1571250

[12] I Guedj and J Huang ldquoAre ETFs replacing index mutualfundsrdquo Manuscript 2009 httppapersssrncomsol3paperscfmabstract-id=1108728

[13] V Glode ldquoWhy mutual funds lsquoUnderperformrsquordquo Manuscript2010 httppapersssrncomsol3paperscfmabstract id=1121436

[14] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[15] C X Huang X Gong X H Chen and F H Wen ldquoMeasuringand forecasting volatility in Chinese stock market using HARCJM modelrdquo Abstract and Applied Analysis vol 2013 ArticleID 143194 13 pages 2013

[16] S Liu and R Xu ldquoThe effects of risk aversionon optimizationrdquoMSCI Barra Research 2010 httpssrncomabstract=1601412

[17] M Wang C Xu F Xu and H Xue ldquoA mixed 0-1 LP forindex tracking problem with CVaR risk constraintsrdquo Annals ofOperations Research vol 196 pp 591ndash609 2012

[18] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[19] H M Markowitz Porfolio Selection Ecient Diversification ofInvestment John Willey New York NY USA 1959

[20] V S Bawa ldquoOptimal rules for ordering uncertain prospectsrdquoJournal of Financial Economics vol 2 no 1 pp 95ndash121 1975

[21] V S Bawa and E B Lindenberg ldquoCapital market equilibrium ina mean-lower partial moment frameworkrdquo Journal of FinancialEconomics vol 5 no 2 pp 189ndash200 1977

[22] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo American Economics Review vol 67 no2 pp 116ndash126 1977

[23] L Chen S He and S Zhang ldquoTight bounds for some riskmeasures with applications to robust portfolio selectionrdquoOper-ations Research vol 59 no 4 pp 847ndash865 2011

[24] H Grootveld and W Hallerbach ldquoVariance vs downside riskis there really that much differencerdquo European Journal ofOperational Research vol 114 no 2 pp 304ndash319 1999

[25] W V Harlow ldquoAsset allocation in a downside-risk frameworkrdquoFinancial Analysis Journal vol 475 pp 28ndash40 1991

[26] FWen L Zhong c Xie and S David ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals vol20 no 2 pp 133ndash140 2012

[27] L Yu S Zhang and X Y Zhou ldquoA downside risk analysis basedon financial index tracking modelsrdquo in Stochastic Finance part1 pp 213ndash236 2006

[28] S Zhu D Li and S Wang ldquoRobust portfolio selection underdownside risk measuresrdquo Quantitative Finance vol 9 no 7 pp869ndash885 2009

[29] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[30] O L V Costa and A C Paiva ldquoRobust portfolio selection usinglinear-matrix inequalitiesrdquo Journal of Economic Dynamics ampControl vol 26 no 6 pp 889ndash909 2002

[31] L El Ghaoui M Oks and F Oustry ldquoWorst-case value-at-risk and robust portfolio optimization a conic programmingapproachrdquoOperations Research vol 51 no 4 pp 543ndash556 2003

[32] D Goldfarb and G Iyengar ldquoRobust portfolio selection prob-lemsrdquoMathematics of Operations Research vol 28 no 1 pp 1ndash38 2003

[33] E Erdogan D Goldfarb and G Iyengar ldquoRobust active portfo-lio managementrdquoThe Journal of Computational Finance vol 11no 4 pp 71ndash98 2008

[34] A F Ling and C X Xu ldquoRobust portfolio selection involvingoptions under a ldquomarginal + jointrdquo ellipsoidal uncertainty setrdquoJournal of Computational andAppliedMathematics vol 236 no14 pp 3373ndash3393 2012

[35] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annals ofOperations Research vol 176 pp 191ndash220 2010

[36] I Bajeux-Besnainou R Belhaj D Maillard and R PortaitldquoPortfolio optimization under tracking error and weights con-straintsrdquo Journal of Financial Research vol 34 no 2 pp 295ndash330 2011

[37] A Ling X Yang and L Tang ldquoRobust LPM active portfolioselection problems with multiple weights constraintsrdquo Journalof Management Science in China vol 16 no 8 pp 32ndash46 2013

[38] V DeMiguel L Garlappi and R Uppal ldquoOptimal versus naivediversification how inefficient is the 1119873 portfolio strategyrdquoReview of Financial Studies vol 22 no 5 pp 1915ndash1953 2009

[39] D Bertsimas and I Popescu ldquoOn the relation between optionand stock prices a convex optimization approachrdquo OperationsResearch vol 50 no 2 pp 358ndash374 2002

[40] I Popescu ldquoRobust mean-covariance solutions for stochasticoptimizationrdquo Operations Research vol 55 no 1 pp 98ndash1122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of