Applying Least Squares Support Vector Machines to Mean ...elie.korea.ac.kr/~cfdkim/papers/SupportVectorMachines.pdf · Wolf optimizer algorithm. ... introduced by Markowitz []. By

Research ArticleApplying Least Squares Support Vector Machines toMean-Variance Portfolio Analysis

JianWang and Junseok Kim

Department of Mathematics Korea University Seoul 02841 Republic of Korea

Correspondence should be addressed to Junseok Kim cfdkimkoreaackr

Received 22 March 2019 Revised 3 June 2019 Accepted 17 June 2019 Published 27 June 2019

Academic Editor Georgios Dounias

Copyright copy 2019 JianWang and Junseok KimThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Portfolio selection problem introduced byMarkowitz has been one of the most important research fields in modern finance In thispaper we propose a model (least squares support vector machines (LSSVM)-mean-variance) for the portfolio management basedon LSSVM To verify the reliability of LSSVM-mean-variance model we conduct an empirical research and design an algorithmto illustrate the performance of the model by using the historical data from Shanghai stock exchange The numerical results showthat the proposedmodel is useful when compared with the traditional Markowitz model Comparing the efficient frontier and totalwealth of both models our model can provide a more measurable standard of judgment when investors do their investment

1 Introduction

Portfolio is the combination of securities such as foreignexchange stocks and othermarket instruments Stock invest-ment has become very common for household investors toinvolve in the stock market Investors used many technicalmethods to minimize risk and optimize return Among themethods Markowitz model developed by Harry Markowitzin 1952 had serious practical limitations due to complexitiesinvolved in compiling the variance covariance expectationstandard deviation of each asset to other assets in theportfolio In recent years many works have been done byscholars to make the portfolio theory more efficient In[1] the authors presented the different variants of the goalprogramming model that has been applied to the financialportfolio selection problem In [2] the authors optimizedthe portfolio based on entropy and higher moments byusing a polynomial goal programmingmodel and the resultsindicated that the proposed method is suited for portfoliomodels which have higher moments Portfolio optimizationtechniques also significantly improved the return-risk trade-off performances using multiobjective evolutionary modelproposed in [3] The authors in [4] used the multiperiodmean-variance model to investigate a defined contributionpension plan investment problem during the accumulation

phase To incorporate social responsibility a modificationof the Markowitz model was proposed in [5] Konno [6]proposed a mean-absolute deviation portfolio optimizationmodel and applied it to Tokyo stock market The results ofnumerical experiments showed that the model generated aportfolio quite similar to that of the Markowitz model withina fraction of time required to solve the latter In [7] theauthors used the independently estimated possibilistic returnrates to deal with a portfolio selection problem

However there are few scholars who used the methodsof machine learning to modify the Markowitz model As weknow the return rate in the mean-variance model refers tothe historical return rate which can also be called historicalvolatility Historical volatility refers to the standard deviationof the underlying asset price changes over the past periodof time which represents the past volatility law The actualvolatility in the trading point cannot be determined but canonly be predictedwith historical volatility and currentmarketinformation In this paper we predict the actual volatilitywith historical volatility by using machine learning As oneof important applications of the machine learning LSSVMhas been used to deal with various financial problems suchas stock price prediction [8 9] and regression [10] Mustaffa[11] optimized LSSVM for nonvolatile financial predictionIn [12] the authors proposed a time series forecasting

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 4189683 10 pageshttpsdoiorg10115520194189683

2 Mathematical Problems in Engineering

model which used LSSVM optimization based on GreyWolf optimizer algorithm For time series prediction [13ndash15] Van Gestel [16] used the LSSVM regression within theevidence framework to infer nonlinear models for predictingtime series and volatility By regularizing least squares fuzzysupport vector regression Khemchandani [17] handled thefinancial time series forecasting In [18] the authors opti-mized LSSVMmodel by weight particle swarm optimizationand they forecasted stock return accurately

In our study we apply LSSVM regression model to tradi-tional Markowitz model and an efficient result is achieved byour proposed model when we compare the efficient frontierand total wealth of both models

The contents of this paper are as follows in Section 2 weprovidemodel description aboutmean-variance LSSVMandLSSVM-mean-variance model In Section 3 we describe thedata set and software which will be taken to do the empiricalresearch in this paper In Section 4 we will show and describethe empirical results In Section 5 conclusions are given

2 Model Description

21 Mean-Variance Model Modern portfolio theory was firstintroduced by Markowitz [19] By the model Markowitzproposed the formulation of an efficient frontier shown in atwo-dimension graphic from it investors can choose theirfinancial portfolio to maximize return for a given level ofrisk as measured by the variance of returns Suppose thatthe investorrsquos wealth is 1198820 the weights on the 119899 assets are1205961 1205962 sdot sdot sdot 120596119899 and the return rate in the future is 119877119901 thenthe investorrsquos wealth in the future will be W = 1198820(1 + 119877119901)Investors usually determine the proportion of investmentin each asset at the initial stage to maximize the expectedinvestment value Then the process can be formulated as

max119908119894

119864 [119880 (119877119901)] = 119864[119880( 119899sum119894=1

119908119894119877119894)]

119904119905 119899sum119894=1

119908119894 = 1(1)

By Taylor expansion

119864 [119880 (119877119901)] = 119880 (119864 (119877119901))+ 119864 [119877119901 minus 119864 (119877119901)]1198801015840 (119864 (119877119901))+ 12119864 ((119877119901 minus 119864 (119877119901))2)11988010158401015840 (119864 (119877119901))+ sdot sdot sdot+ 1119899119864 ((119877119901 minus 119864 (119877119901))119899)119880(119899) (119864 (119877119901))+ sdot sdot sdot

(2)

Assuming that the series 1198771 1198772 sdot sdot sdot 119877119899 follow normal distri-bution then the above function depends on themean and the

variance of 119877119901 Suppose 119880(∙) is a concave function then wecan simplify (1) to

min119908119894

1205902 (119877119901) = 119899sum119894=1

1205962119894 1205902 (119877119894) +sum119894 =119895

120596119894120596119895120590 (119877119894 119877119895)

119904119905 119877119901 = 119899sum119894=1

119908119894119864 (119877119894)

and119899sum119894=1

119908119894 = 1

(3)

where 119877119901 represents the portfoliorsquos expected return whichthe investor expected The mathematical form of (3) is thequadratic programming problem that can be solved by aLagrangian method We give the first-order condition

[[[[

Σ 119890 1198771198901015840 0 01198771015840 0 0

]]]][[[120596lowast12058211205822]]]= [[[[

01119877119901]]]] (4)

where 119877 is the return mean vector composed by119873 assets 119890 isthe119873times 1 unit column vector and Σ is the119873times119873 covariancematrix of return Then the final investment proportion isoptimally satisfied by

120596lowast = 119886 + 119887119877119901119886 = 120573Σminus1119890 minus 120572Σminus1119877120573120575 minus 1205722 119887 = 120575Σminus1119877 minus 120572Σminus1119890120573120575 minus 1205722

(5)

120572 = 1198771015840Σminus1119890120573 = 1198771015840Σminus1119877120575 = 1198901015840Σminus1119890

(6)

22 Least Square Support Vector Machine Support vectormachine (SVM) has been successfully applied for financialproblems especially in time series forecasting LSSVM isthe least squares formulation of SVM and was developed byPelckmans [20] LSSVM is the combination of structural riskminimization and VC dimension theory [21] and usually isused for classification as well as regression such as patternrecognition fitting functions [22 23] and data analysis Thealgorithm of LSSVM is introduced as follows The followingregression model is constructed by using a nonlinear map-ping function 120593() 119877119899 997888rarr 119877119899ℎ which maps the input datato a higher dimensional feature space

119910 (119909) = 120596119879120593 (119909) + 119887 (7)

where 119909 isin 119877119899 119910 isin 119877120596119879 is the weight vector and 119887 is the biasAssume a training set as

119878 = (1199091 1199101) sdot sdot sdot (119909119894 119910119894) | 119909119894 isin 119877119899 119910119894 isin 119877 (8)

Mathematical Problems in Engineering 3

The original optimization problem is

min 12 1205962 (9)

and the LSSVM can be formulated as

min 119869 (120596 119890) = 12 1205962 + 12120574119871sum119894=1

1198902119897 (10)

subject to the constraints

119910 (119909) = 120596119879120593 (119909119897) + 119887 + 119890119897 119897 = 1 2 sdot sdot sdot 119871 (11)

where 120574 is the regularization parameter and 119890119897 is the randomerrors Using a Lagrange multiplier method we have

119871 (120596 119887 119890 120572) = 119869 (120596 119890)minus 119871sum119894=1

120572119897 [120596119879120593 (119909119897) + 119887 + 119890119897 minus 119910119897] (12)

where 120572119897are the Lagrange multipliers from the optimizationconditions by partially differentiating with respect to eachparameter yielding

120597119871120597120596 = 0 997888rarr 120596 = 119871sum119894=1

120572119897120593 (119909119897) 120597119871120597119861 = 0 997888rarr

119871sum119894=1

120572119897 = 0120597119871120597119890119897 = 0 997888rarr 120572119897 = 120574119890119897120597119871120597120572119897 = 0 997888rarr 120596119879120593 (119909119897) + 119887 + 119890119897 minus 119910119897 = 0

(13)

After elimination of parameters 120596 and 119890 we obtain thefollowing matrix solution

[[[0 11198791198711119871 Ω + 119868120574

]]][119887120572] = [

0119910] (14)

where the composition of thematrixΩ isΩ119894119895 = 120593(119909119894)119879120593(119909119895) =119870(119909119894 119909119895) 119894 119895 = 1 sdot sdot sdot 119871 Here 119870(119909119894 119909119895) is the radial basisfunction (RBF) kernel function For regression models theRBF kernel is often applied because of its influence and speedin training process [24]

119870(119909 119909119897) = exp(minus1003817100381710038171003817119909 minus 1199091198971003817100381710038171003817221205902 ) (15)

Then we can obtain the regression function as

119910 (119909) = 119871sum119897=1

120572119897119870(119909 119909119897) + 119887 (16)

1205902 in the above function is the kernel width and we applyit to adjust the degree of generalization To make theLSSVM model we should optimize the parameters 120574 and1205902 In this paper to do the comparison test between thetraditional mean-variance model and the proposed LSSVM-mean-variance model we take 1205902 = 006 and 120574 = 5 unlessotherwise specified

23 LSSVM forMean-VarianceModel In this section we givea description of applying LSSVM to mean-variance modelWe first select a portfolio and then calculate the returnsof the assets in the portfolio As mentioned above in theLSSVM model we take the matrix of assetsrsquo returns as thetraining sets by the process of Section 22 we get a regressionmatrix of assetsrsquo returns Then we use returns and regressionreturns to do the test and follow the steps described inSection 21 Finally we compare the efficient frontier and finalwealth return for the two methods which will be shown inSection 4

3 Data Set and Software

We select a portfolio consistimg of three assets which arechosen fromShanghai stockmarket To do a buy-and-sell testwe use the historical data for the stock ldquo-zgyh-rdquo ldquo-nyyh-rdquo andldquo-jtyh-rdquo from August 09 2018 to October 26 2018 We takethe data fromAugust 09 2018 toOctober 25 2018 intomean-variancemodel and LSSVM-mean-variancemodelThere are50 data in total In the LSSVMmodel we divide the data into atraining set with 39 data and a test set with 10 data Then wecompare the performance with the two models on October26 2018 To do a buy-and-hold test we use the historical datafor the stock ldquo-zgyh-rdquo ldquo-nyyh-rdquo and ldquo-jtyh-rdquo fromMarch 102017 toMarch 12 2018We take the data of closing price every5 days then there are 50 data in total In the LSSVM modelwe also divide the data into a training set with 39 data and atest set with 10 dataThen we compare the performance withthe twomodels onMarch 19 2018 For the calculation processMATLAB R2016a will be used

4 Empirical Results

41 Empirical Results of Buy-and-Sell Strategy From thestocks data chosen in Section 3 we can see the stocksrsquo pricetrend as shown in Figure 1

The body in the candlestick usually consists of an openingprice and a closing price the price excursions below or abovethe body are called the wicks For a stock during the timeinterval represented the wick contains the lowest and highestprices as well as the body contains the opening and closingprices The red body of a candlestick indicates the securityhas a higher closed price than it opened the opening price atthe bottom and the closing price at the top The green bodyof a candlestick indicates the security has a lower closed pricethan it opened the opening price at the top and the closingprice at the bottom

Nowwe select the real historical data of the stocks ldquo-zgyh-rdquo ldquo-nyyh-rdquo and ldquo-jtyh-rdquo from August 09 2018 to October25 2018 Taking the closing data to the calculation of return


Table 1 Proportion of each asset for traditional mean-variance model

Investment proportion combination Proportion of ldquo-zgyh-rdquo Proportion of ldquo-nyyh-rdquo Proportion of ldquo-jtyh-rdquo1 07994 00000 020062 06483 00000 035173 04972 00000 050284 03461 00000 065395 01950 00000 080506 00635 00349 090167 00000 00349 090168 00000 04608 053929 00000 07304 0269610 00000 10000 00000

date20180809 20181025 20181026

Stoc

k pr

ice

33335

34345

35355

36365

37375

38

zgyh

(a)

20180809 20181025 20181026date

Stoc

k pr

ice

32

33

34

35

36

37

38

39

4

nyyh

(b)

20180809 20181025 20181026date

Stoc

k pr

ice

53

54

55

56

57

58

59

6

jtyh

(c)

Figure 1 Candlestick chart for ldquo-zgyh-rdquo ldquo-nyyh-rdquo and ldquo-jtyh-rdquo from August 09 2018 to October 26 2018 (For interpretation of thereferences to color in this figure the reader is referred to the web version of this article The data sources were downloaded from the web siteldquohttpquotesmoney163comstockrdquo)

rate for every day the total number of data is 50 Then weget 49 return rate data for each asset The return rate ofldquo-jtyh-rdquo is shown in Figure 2 and reflected by ldquolowastrdquo Thenwe take return rate to the calculation process of Markowitzmodel As a result Table 1 shows the proportion of each assetfor traditional mean-variance model we set 10 investmentproportion combinations

As a comparison we calculate the proportion of eachasset for LSSVM-mean-variance model by using the LSSVM

regression We take the return rate mentioned above to theLSSVM model described in Section 22 we divide the datainto a training set with 39 data and a test set with 10 dataThen we get the regression data which is shown in Figure 2and reflected by ldquoordquo

Then we take regression return rate toMarkowitz modelTable 2 shows the proportion of each asset for LSSVM-mean-variance model Here we also set 10 investment proportioncombinations


Table 2 Proportion of each asset for LSSVM-mean-variance model


0 5 10 15 20 25 30 35 40 45 50t

Retu

rn ra

te

Mean-Variance modelLSSVM-Mean-Variance model

minus003

minus002

minus001

0

001

002

003

004

(a)

0 5 10 15 20 25 30 35 40 45 50t

0

002

004

006

008

Retu

rn ra

te


minus002

minus004

(b)

0 5 10 15 20 25 30 35 40 45 50t

0

001

002

003

Retu

rn ra

te


minus004

minus003

minus002

minus001

minus005

(c)

Figure 2 Return rate of mean-variance model and LSSVMmean-variance model for (a) ldquo-zgyh-rdquo (b) ldquo-nyyh-rdquo and (c) ldquo-jtyh-rdquo

Each investment proportion combination in the tableresponds to a maximum return for a given level of risk asmeasured by the standard varianceThepoints are constitutedby mean and standard variance forming an efficient frontier

As seen in Figure 3 the efficient portfolio frontier forLSSVM-mean-variancemodel has a better performance thanthe traditional model It is possible to check that portfolios

corresponding to the new proposed method can improve thereturn at the same risk For the same expectation of bothmodels the LSSVM-mean-variance model can reduce therisk for investors

According to the investment proportion combinationsshown in Tables 1 and 2 we perform a simulation test Weassume that the initial total wealth is 100 million and we buy


Table 3 Proportion of each asset for mean-variance model for buy-and-hold for 5-day strategy


Efficient frontier of Mean-Variance modelEfficient frontier of LSSVM-Mean-Variance model

0011 0012 0013 0014 0015 0016 0017001Portfolios standard deviation

06

08

1

12

14

16

18

Port

folio

s exp

ecta

tions

times10minus3

Figure 3 Efficient portfolio frontier of mean-variance model andLSSVMmean-variance model

the asset portfolio depending on the proportion combinationon October 25 2018 and sell the portfolio on October 262018 Under the two models the total wealth on October 262018 is shown in Figure 4 the point in the figure representsthe performance of each combination By the simulation thewealth invested by using the new model helps investors earnmore than the traditional model when taking the buy-and-sell strategy

42 Empirical Results of Buy-and-Hold for 5-Day StrategyAs the proposed model conducted by using buy-and-sellstrategy we get a satisfied result However the data set weselected is small and between summer and autumn whichmakes people think that the above results have specificseasonality To address this concern we select a long data setcovering all seasons of the year fromMarch 10 2017 toMarch12 2018 The candlestick charts for three stocks are shown inFigure 5 In addition in order to show that our model does


2 3 4 5 6 7 8 9 101Investment proportion combination

994

996

998

100

1002

1004

1006

1008

Tota

l wea

lth

Figure 4 Total wealth for each investment proportion combinationof two models

not only work for buy-and-sell strategy we take the data ofclosing price every 5 days for buy-and-hold strategy

The return rate of ldquo-jtyh-rdquo for buy-and-hold for 5-daystrategy is shown in Figure 6 and is reflected by ldquolowastrdquoThen we take return rate to the calculation process ofMarkowitz model Table 3 shows the proportion of eachasset for traditional mean-variance model and we set 10investment proportion combinations Similar to buy-and-sellstrategy each investment proportion combination in the tableresponds to a maximum return for a given level of risk asmeasured by the standard varianceThepoints are constitutedby mean and standard variance forming an efficient frontier

As seen in Figure 7 in the buy-and-hold strategy theefficient portfolio frontier for LSSVM-mean-variance modelhas a better performance than the traditional model It ispossible to check that portfolios corresponding to the newproposed method can improve the return at the same riskFor the same expectation of both models the LSSVM-mean-variance model can reduce the risk for investors


Table 4 Proportion of each asset for LSSVM-mean-variance model for buy-and-hold for 5-day strategy


20170310 20180311 20180312date

zgyh

34

36

38

4

42

44

46

48

5

Stoc

k pr

ice

(a)

20170310 20180311 20180312date

nyyh

32343638

442444648

5

Stoc

k pr

ice

(b)

20170310 20180311 20180312

date

Stoc

k pr

ice

55

6

65

7

75

jtyh

(c)

Figure 5 Candlestick chart for ldquo-zgyh-rdquo ldquo-nyyh-rdquo and ldquo-jtyh-rdquo fromMarch 10 2017 to March 12 2018 (For interpretation of the referencesto color in this figure the reader is referred to the web version of this article The data sources were downloaded from the web siteldquohttpquotesmoney163comstockrdquo)


According to the investment proportion combinationsshown in Tables 3 and 4 we perform a simulation test forbuy-and-hold for 5-day strategy We assume that the initialtotal wealth is 100 million and we buy the asset portfoliodepending on the proportion combination onMarch 12 2018

and sell the portfolio on March 19 2018 Under the twomodels the total wealth on March 19 2018 is shown inFigure 8 the point in the figure represents the performanceof each combination By the simulation the newmodel helpsinvestors earn more than the traditional model when takingthe buy-and-hold strategy

To illustrate that this result is not caused by the specificperiod we selected we calculate the total wealth of each dayin 15 days from August 30 2018 to September 19 2018 for the



minus012

minus01

minus008

minus006

minus004

minus002

0002004006

Retu

rn ra

te

5 10 15 20 25 30 35 40 45 500t

(a)


minus015

minus01

minus005

0

005

01

Retu

rn ra

te

5 10 15 20 25 30 35 40 45 500t

(b)


5 10 15 20 25 30 35 40 45 500t

minus01

minus008

minus006

minus004

minus002

0002004006008

Retu

rn ra

te

(c)

Figure 6 Return rate of mean-variance model and LSSVM mean-variance model for buy-and-hold for 5-day strategy of (a) ldquo-zgyh-rdquo (b)ldquo-nyyh-rdquo and (c) ldquo-jtyh-rdquo


times10minus3

002 0022 0024 0026 0028 003 0032 0034 00360018Portfolios standard deviation

1

2

3

4

5

6

7

8

Port

folio

s exp

ecta

tions

Figure 7 Efficient portfolio frontier of mean-variance model andLSSVMmean-variance model for buy-and-hold for 5-day strategy



985

99

995

100

1005

101

1015

Tota

l wea

lth

Figure 8 Total wealth for each investment proportion combinationof two models for buy-and-hold for 5-day strategy


0

2

40

Combination

0

65

date810

02

1015

Diff

04

06

Figure 9 Total wealth difference of two models

two models according to the former 15 days The calculationsteps are taken as same as the above process We set the totalwealth of mean-variance model as 1198791198821 and 1198791198822 for theLSSVM-mean-variance model the difference of two modelsis defined by

119863119894119891119891 = 1198791198822 minus 1198791198821 (17)

As shown in Figure 9 almost all the difference values aregreater than 0 which indicates that the optimized model hasa higher yield of each day in 15 days

5 Conclusion

Machine learning over the last few years has resulted in apotential opportunity for investors to invest in the finan-cial market with a smarter and profitable way Combiningmachine learning technologywith financial investment it canentirely change the way we make investment decisions Thispaper gives an overview of how the two technologies can becombined into a powerful tool and proposes the LSSVM-mean-variance algorithm with the aim of maximizing returnfor a given level of risk as measured by the variance ofreturns The efficiency of the proposed method is measuredby empirical data namely efficient frontier and total wealthComparing the efficient frontier and total wealth of bothmodels the curve of mean-variance model is always belowthe proposed model This shows that our model has a higheryield under the same risk and has more total wealth undereach combination our model performs a more measurablestandard of judgment when investors do their investmentWe confirm the efficiency through the strategy both buy-and-sell and buy-and-hold The encouraging performance showsthat our proposed method may become a promising modelfor the context of study and the results indicate a positiveopportunity to be explored in the future

Data Availability

Data and source program codes in this paper are availableupon request from the corresponding author

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026)

References

[1] B Aouni C Colapinto and D La Torre ldquoFinancial portfoliomanagement through the goal programming model currentstate-of-the-artrdquo European Journal of Operational Research vol234 no 2 pp 536ndash545 2014

[2] M Aksaraylı and O Pala ldquoA polynomial goal programmingmodel for portfolio optimization based on entropy and highermomentsrdquo Expert Systems with Applications vol 94 pp 185ndash192 2018

[3] B Y Qu Q Zhou J M Xiao J J Liang and P N SuganthanldquoLarge-scale portfolio optimization using multiobjective evo-lutionary algorithms and preselection methodsrdquo MathematicalProblems in Engineering vol 2017 14 pages 2017

[4] L Wang and Z Chen ldquoNash equilibrium strategy for a DCpension plan with state-dependent risk aversion a multiperiodmean-variance frameworkrdquo Discrete Dynamics in Nature andSociety vol 2018 17 pages 2018

[5] S M Gasser M Rammerstorfer and K WeinmayerldquoMarkowitz revisited Social portfolio engineeringrdquo EuropeanJournal of Operational Research vol 258 no 3 pp 1181ndash11902017


[6] H Konno and H Yamazaki ldquoMean absolute deviation port-folio optimization model and its application to Tokyo stockexchangerdquoManagement Science vol 37 pp 519ndash531 1991

[7] M Inuiguchi andT Tanino ldquoPortfolio selection under indepen-dent possibilistic informationrdquo Fuzzy Sets and Systems vol 115no 1 pp 83ndash92 2000

[8] X Zhou Z Pan G Hu S Tang and C Zhao ldquoStock marketprediction on high-frequency data using generative adversarialnetsrdquoMathematical Problems in Engineering vol 2018 11 pages2018

[9] Z Wang J Hu and Y Wu ldquoA bimodel algorithm with data-divider to predict stock indexrdquo Mathematical Problems inEngineering vol 2018 14 pages 2018

[10] D Prayogo and Y T T Susanto ldquoOptimizing the predictionaccuracy of friction capacity of driven piles in cohesive soilusing a novel self-tuning least squares support vector machinerdquoAdvances in Civil Engineering vol 2018 Article ID 6490169 9pages 2018

[11] Z Mustaffa and Y Yusof ldquoOptimizing LSSVM using ABC fornon-volatile financial predictionrdquo Australian Journal of Basicand Applied Sciences vol 5 no 11 pp 549ndash556 2011

[12] Z Mustaffa M H Sulaiman and M N M Kahar ldquoLS-SVMhyper-parameters optimization based on GWO algorithm fortime series forecastingrdquo in Proceedings of the 4th InternationalConference on Soware Engineering and Computer SystemsICSECS 2015 pp 183ndash188 Malaysia August 2015

[13] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in BitcoinmarketrdquoChaos Solitons amp Fractals vol 106pp 28ndash34 2018

[14] S Lahmiri ldquoAsymmetric and persistent responses in pricevolatility of fertilizers through stable and unstable periodsrdquoPhysica A Statistical Mechanics and its Applications vol 466pp 405ndash414 2017

[15] S Lahmiri and S Bekiros ldquoTime-varying self-similarity inalternative investmentsrdquo Chaos Solitons amp Fractals vol 111 pp1ndash5 2018

[16] T Van Gestel J A K Suykens D-E Baestaens et al ldquoFinan-cial time series prediction using least squares support vectormachines within the evidence frameworkrdquo IEEE TransactionsonNeural Networks and Learning Systems vol 12 no 4 pp 809ndash821 2001

[17] R Khemchandani Jayadeva and S Chandra ldquoRegularizedleast squares fuzzy support vector regression for financial timeseries forecastingrdquo Expert Systems with Applications vol 36 no1 pp 132ndash138 2009

[18] W Shen Y Zhang and X Ma ldquoStock return forecast withLS-SVM and particle swarm optimizationrdquo in Proceedings ofthe 2009 International Conference on Business Intelligence andFinancial Engineering BIFE 2009 pp 143ndash147 China 2009

[19] H Markowitz ldquoPortfolio selectionrdquoe Journal of Finance vol7 no 1 pp 77ndash91 1952

[20] K Pelckmans J A Suykens T Van Gestel et al ldquoLS-SVMlaba matlabc toolbox for least squares support vector machinesrdquoTutorial KULeuven-ESAT vol 142 pp 1-2 2002

[21] V N Vapnik and A Y Chervonenkis ldquoOn the uniform conver-gence of relative frequencies of events to their probabilitiesrdquo inMeasures of Complexity pp 11ndash30 Springer Cham Switzerland2015

[22] X Chen J Yang J Liang and Q Ye ldquoRecursive robustleast squares support vector regression based on maximumcorrentropy criterionrdquoNeurocomputing vol 97 pp 63ndash73 2012

[23] H Nie G Liu X Liu and Y Wang ldquoHybrid of ARIMA andSVMs for short-term load forecastingrdquo Energy Procedia vol 16pp 1455ndash1460 2012

[24] NXin XGuHWuYHu andZ Yang ldquoApplication of geneticalgorithm-support vector regression (GA-SVR) for quantitativeanalysis of herbal medicinesrdquo Journal of Chemometrics vol 26pp 353ndash360 2012

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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Hindawiwwwhindawicom Volume 2018Volume 2018

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Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

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model which used LSSVM optimization based on GreyWolf optimizer algorithm For time series prediction [13ndash15] Van Gestel [16] used the LSSVM regression within theevidence framework to infer nonlinear models for predictingtime series and volatility By regularizing least squares fuzzysupport vector regression Khemchandani [17] handled thefinancial time series forecasting In [18] the authors opti-mized LSSVMmodel by weight particle swarm optimizationand they forecasted stock return accurately

In our study we apply LSSVM regression model to tradi-tional Markowitz model and an efficient result is achieved byour proposed model when we compare the efficient frontierand total wealth of both models

The contents of this paper are as follows in Section 2 weprovidemodel description aboutmean-variance LSSVMandLSSVM-mean-variance model In Section 3 we describe thedata set and software which will be taken to do the empiricalresearch in this paper In Section 4 we will show and describethe empirical results In Section 5 conclusions are given

2 Model Description

21 Mean-Variance Model Modern portfolio theory was firstintroduced by Markowitz [19] By the model Markowitzproposed the formulation of an efficient frontier shown in atwo-dimension graphic from it investors can choose theirfinancial portfolio to maximize return for a given level ofrisk as measured by the variance of returns Suppose thatthe investorrsquos wealth is 1198820 the weights on the 119899 assets are1205961 1205962 sdot sdot sdot 120596119899 and the return rate in the future is 119877119901 thenthe investorrsquos wealth in the future will be W = 1198820(1 + 119877119901)Investors usually determine the proportion of investmentin each asset at the initial stage to maximize the expectedinvestment value Then the process can be formulated as

max119908119894

119864 [119880 (119877119901)] = 119864[119880( 119899sum119894=1

119908119894119877119894)]

119904119905 119899sum119894=1

119908119894 = 1(1)

By Taylor expansion

119864 [119880 (119877119901)] = 119880 (119864 (119877119901))+ 119864 [119877119901 minus 119864 (119877119901)]1198801015840 (119864 (119877119901))+ 12119864 ((119877119901 minus 119864 (119877119901))2)11988010158401015840 (119864 (119877119901))+ sdot sdot sdot+ 1119899119864 ((119877119901 minus 119864 (119877119901))119899)119880(119899) (119864 (119877119901))+ sdot sdot sdot

(2)

Assuming that the series 1198771 1198772 sdot sdot sdot 119877119899 follow normal distri-bution then the above function depends on themean and the

variance of 119877119901 Suppose 119880(∙) is a concave function then wecan simplify (1) to

min119908119894

1205902 (119877119901) = 119899sum119894=1

1205962119894 1205902 (119877119894) +sum119894 =119895

120596119894120596119895120590 (119877119894 119877119895)

119904119905 119877119901 = 119899sum119894=1

119908119894119864 (119877119894)

and119899sum119894=1

119908119894 = 1

(3)

where 119877119901 represents the portfoliorsquos expected return whichthe investor expected The mathematical form of (3) is thequadratic programming problem that can be solved by aLagrangian method We give the first-order condition

[[[[

Σ 119890 1198771198901015840 0 01198771015840 0 0

]]]][[[120596lowast12058211205822]]]= [[[[

01119877119901]]]] (4)

where 119877 is the return mean vector composed by119873 assets 119890 isthe119873times 1 unit column vector and Σ is the119873times119873 covariancematrix of return Then the final investment proportion isoptimally satisfied by

120596lowast = 119886 + 119887119877119901119886 = 120573Σminus1119890 minus 120572Σminus1119877120573120575 minus 1205722 119887 = 120575Σminus1119877 minus 120572Σminus1119890120573120575 minus 1205722

(5)

120572 = 1198771015840Σminus1119890120573 = 1198771015840Σminus1119877120575 = 1198901015840Σminus1119890

(6)

22 Least Square Support Vector Machine Support vectormachine (SVM) has been successfully applied for financialproblems especially in time series forecasting LSSVM isthe least squares formulation of SVM and was developed byPelckmans [20] LSSVM is the combination of structural riskminimization and VC dimension theory [21] and usually isused for classification as well as regression such as patternrecognition fitting functions [22 23] and data analysis Thealgorithm of LSSVM is introduced as follows The followingregression model is constructed by using a nonlinear map-ping function 120593() 119877119899 997888rarr 119877119899ℎ which maps the input datato a higher dimensional feature space

119910 (119909) = 120596119879120593 (119909) + 119887 (7)

where 119909 isin 119877119899 119910 isin 119877120596119879 is the weight vector and 119887 is the biasAssume a training set as

119878 = (1199091 1199101) sdot sdot sdot (119909119894 119910119894) | 119909119894 isin 119877119899 119910119894 isin 119877 (8)



min 12 1205962 (9)


min 119869 (120596 119890) = 12 1205962 + 12120574119871sum119894=1

1198902119897 (10)


119910 (119909) = 120596119879120593 (119909119897) + 119887 + 119890119897 119897 = 1 2 sdot sdot sdot 119871 (11)


119871 (120596 119887 119890 120572) = 119869 (120596 119890)minus 119871sum119894=1

120572119897 [120596119879120593 (119909119897) + 119887 + 119890119897 minus 119910119897] (12)


120597119871120597120596 = 0 997888rarr 120596 = 119871sum119894=1

120572119897120593 (119909119897) 120597119871120597119861 = 0 997888rarr

119871sum119894=1

120572119897 = 0120597119871120597119890119897 = 0 997888rarr 120572119897 = 120574119890119897120597119871120597120572119897 = 0 997888rarr 120596119879120593 (119909119897) + 119887 + 119890119897 minus 119910119897 = 0

(13)


[[[0 11198791198711119871 Ω + 119868120574

]]][119887120572] = [

0119910] (14)


119870(119909 119909119897) = exp(minus1003817100381710038171003817119909 minus 1199091198971003817100381710038171003817221205902 ) (15)


119910 (119909) = 119871sum119897=1

120572119897119870(119909 119909119897) + 119887 (16)





4 Empirical Results







date20180809 20181025 20181026

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20180809 20181025 20181026date

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(c)









0 5 10 15 20 25 30 35 40 45 50t

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minus003

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0

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004

(a)

0 5 10 15 20 25 30 35 40 45 50t

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006

008

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minus002

minus004

(b)

0 5 10 15 20 25 30 35 40 45 50t

0

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003

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minus004

minus003

minus002

minus001

minus005

(c)











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994

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20170310 20180311 20180312date

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32343638

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20170310 20180311 20180312

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minus012

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0002004006

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5 10 15 20 25 30 35 40 45 500t

(a)


minus015

minus01

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0

005

01

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te

5 10 15 20 25 30 35 40 45 500t

(b)


5 10 15 20 25 30 35 40 45 500t

minus01

minus008

minus006

minus004

minus002

0002004006008

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(c)



times10minus3


1

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985

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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018










min 12 1205962 (9)


min 119869 (120596 119890) = 12 1205962 + 12120574119871sum119894=1

1198902119897 (10)


119910 (119909) = 120596119879120593 (119909119897) + 119887 + 119890119897 119897 = 1 2 sdot sdot sdot 119871 (11)


119871 (120596 119887 119890 120572) = 119869 (120596 119890)minus 119871sum119894=1

120572119897 [120596119879120593 (119909119897) + 119887 + 119890119897 minus 119910119897] (12)


120597119871120597120596 = 0 997888rarr 120596 = 119871sum119894=1

120572119897120593 (119909119897) 120597119871120597119861 = 0 997888rarr

119871sum119894=1

120572119897 = 0120597119871120597119890119897 = 0 997888rarr 120572119897 = 120574119890119897120597119871120597120572119897 = 0 997888rarr 120596119879120593 (119909119897) + 119887 + 119890119897 minus 119910119897 = 0

(13)


[[[0 11198791198711119871 Ω + 119868120574

]]][119887120572] = [

0119910] (14)


119870(119909 119909119897) = exp(minus1003817100381710038171003817119909 minus 1199091198971003817100381710038171003817221205902 ) (15)


119910 (119909) = 119871sum119897=1

120572119897119870(119909 119909119897) + 119887 (16)





4 Empirical Results







date20180809 20181025 20181026

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20180809 20181025 20181026date

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(c)









0 5 10 15 20 25 30 35 40 45 50t

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0

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(a)

0 5 10 15 20 25 30 35 40 45 50t

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(b)

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(c)











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(b)

20170310 20180311 20180312

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minus012

minus01

minus008

minus006

minus004

minus002

0002004006

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5 10 15 20 25 30 35 40 45 500t

(a)


minus015

minus01

minus005

0

005

01

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te

5 10 15 20 25 30 35 40 45 500t

(b)


5 10 15 20 25 30 35 40 45 500t

minus01

minus008

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minus002

0002004006008

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times10minus3


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tions




985

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0

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0

65

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04

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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018











date20180809 20181025 20181026

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(c)









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minus015

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(b)


5 10 15 20 25 30 35 40 45 500t

minus01

minus008

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0002004006008

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times10minus3


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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018











0 5 10 15 20 25 30 35 40 45 50t

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20170310 20180311 20180312date

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20170310 20180311 20180312

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minus012

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(a)


minus015

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5 10 15 20 25 30 35 40 45 500t

(b)


5 10 15 20 25 30 35 40 45 500t

minus01

minus008

minus006

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minus002

0002004006008

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(c)



times10minus3


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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018













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minus012

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5 10 15 20 25 30 35 40 45 500t

(a)


minus015

minus01

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(b)


5 10 15 20 25 30 35 40 45 500t

minus01

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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018











20170310 20180311 20180312date

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(a)


minus015

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minus01

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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018










minus012

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(a)


minus015

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04

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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018









0

2

40

Combination

0

65

date810

02

1015

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04

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119863119894119891119891 = 1198791198822 minus 1198791198821 (17)


5 Conclusion


Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018



































Journal of












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Journal of












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Documents

Applying Least Squares Support Vector Machines to Mean ...elie.korea.ac.kr/~cfdkim/papers/SupportVectorMachines.pdf · Wolf optimizer algorithm. ... introduced by Markowitz []. By