Using investment satisfaction capability index based particle swarm optimization to construct a stock portfolio

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Using investment satisfaction capability index based particle swarm

optimization to construct a stock portfolio

Jui-Fang Chang

a,*

, Peng Shi

b,c

a

Department of International Business, National Kaohsiung University of Applied Sciences, Taiwan b

Department of Computing and Mathematical Sciences, University of Glamorgan, UK c

Division of Information Technology, Engineering and the Environment, University of South Australia, Australia

a r t i c l e

i n f o

Article history:

Received 14 August 2009

Received in revised form 30 March 2010 Accepted 10 May 2010

Available online 16 May 2010 Keywords:

Artiﬁcial intelligence Particle swarm algorithms

Investment satisfaction capability index Particle swarm optimization

a b s t r a c t

The goal of this study is to construct an enhanced process based on the investment satisfied capability index (ISCI). The process is divided into two stages. The first stage is to apply the Process Capability Indices (PCI) for quality management so as to develop a new perfor-mance appreciation method. Investors can utilize the ISCI index to rapidly evaluate individ-ual stock performance and then select those stocks which can lead to achieve investment satisfaction. In the second stage, a particle swarm optimization (PSO) algorithm with mov-ing interval windows is applied to find the optimal investment allocation of the stocks in this portfolio. Based on those algorithms we can ensure investment risk control and obtain a more profitable stock investment portfolio.

1. Introduction

In the recent years, the field of artificial intelligence (AI) has developed at lightening speed[37]. The usage of AI in the financial field has also reached maturity. In this type of research, the characteristics of particle swarm optimization (PSO) can be used to help the capital allocation in a stock portfolio[2,8,9,19,26,32].

Modern portfolio selection theory originated with Markowitz[29]. He introduced portfolio selection theory and pub-lished the Mean–Variance model (MV model) in 1952. Investors spend value time and resources in the hope of gaining the maximum expected returns but without take market risk into account. The Markowitz’s Mean–Variance model suggests that investors can obtain the highest expected returns when the risk level is ﬁxed, or they can have the lowest risk when the expected return level is ﬁxed.

Although the Mean–Variance model provides principles for portfolio selection, various factors must be considered before investing, especially in the security market. In the face of this dilemma, the investment satisfaction capability index (ISCI) applies the Process Capability Indices (PCI) of quality management in the process of selecting the stocks which can lead to investment satisfaction. During the purchasing of stock portfolios, the investor always refers to the portfolio performance evaluation. Traditional performance indices can be inferred from the Capital Assets Pricing Model (CAPM). Treynor[35] developed the Treynor Index in 1965. It can be used by investors to obtain the expected return given systematic risk. In fol-lowing year, Sharpe[33]investigated 34 funds and developed the Sharpe Index. Jensen[23]developed the Jensen Index to compare the investment of the mutual funds with buy and hold market portfolios. It can be used to determine which invest-ment can lead to the gaining of higher expected returns. In general, the investinvest-ment satisfaction degree (ISD) is not taken into account in these indices.

*Corresponding author. Tel.: +886 3814526x3800. E-mail address:rose@cc.kuas.edu.tw(J.-F. Chang).

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The Process Capability Index CPLhas been widely used in the microelectronics manufacturing industry as a capability

measure of processes with one-sided speciﬁcation limits. It can be applied in the proposed setting to judge whether its pro-cesses meet the preset capability requirement and run under the desired quality conditions. The method has been utilized by Kane[24]and Pearn et al.[30]and is also widely adopted by the academic professionals[4,5,7,11–13,15,16,28,31].

Particle swarm optimization (PSO) is a kind of swarm intelligence that is based on social-psychological principles and provides insights into social behavior, as well as contributing to engineering applications. PSO algorithm was first described in 1995 by Eberhart et al.[20]. PSO is similar to the genetic algorithm since both methods maintain a population of potential solutions. However, PSO provides potential solutions through particles flying across the problem hyperspace while in the genetic algorithm the solution is provided through selection and reproduction. Shi and Eberhart[34]advanced the theory of ‘‘Inertia Weight” to change the velocity of particles, and to find the balance between a global search and a local search. They probed into the inertia weight factor and provided a better computation of the inertia weight factor and study factor: wmax: 0.9; wmin: 0.4; and c1= c2= 2.0. Eberhart et al.[20]arranged the advance version of the PSO algorithm. The

improve-ment which is improve-mentioned above can be divided into three main parts: inertia weight, constriction factor and tracking and optimizing dynamic systems[1,3,19–22,27,36].

PSO has been applied to several optimization problems as a multi-factor and optimal model for solution of security investment optimizing combination problems in security market[6]. A particle swarm solver is developed and tested on var-ious restricted and unrestricted risky investment portfolios[25]. PSO is combined with the cardinality constrained Mean– Variance model to present a heuristic approach to portfolio optimization problem[17]. PSO is faster than GA, both in terms of number of iterations and in terms of total running time[10,18,38,39]. The proposed approach is evaluated on applications from the ﬁeld of computational ﬁnance, namely portfolio optimization and time series forecasting[14].

A number of stocks were evaluated by the ISCI. The ISD is set at 70%. The ratio of return and standard deviation are cal-culated using the stocks’ historical ratio of return, provided by the Taiwan Economic Journal (TEJ). PSO was utilized to allo-cate the capital. From this empirical study, it was found that: (i) our portfolio algorithm ISCI_PSO performs well to obtain a more proﬁtable stock investment portfolio; (ii) the ISCI_PSO algorithm tends to offer outstanding performance for short-term applications.

This paper is divided into ﬁve sections. A brief survey of the ISCI and PSO has been presented in Section1. In Section2, we discuss the detailed procedure of the proposed algorithm and its theoretical aspects while the experimental design is de-scribed in Section3. The empirical experiments are described in Section4. Finally, Section5it devoted to the conclusions.

2. Methodology

In this section, we focus primarily on stock. Three parts are included in this section. Firstly, ISCI utilizes the PCI of quality management for the selection of stock. Secondly, ISCI is used to evaluate the overall level of performance of that stock. Then PSO is utilized to allocate the capital. Finally, the experimental data are explained.

2.1. Capability index CPL

The capability index CPLhas been widely used in the manufacturing sector as well as the service industry to provide

com-mon quantitative measures of process potential and performance[1,2]. The index is deﬁned as follows:

CPL¼

l

LSL

3

r

ð1Þ

where LSL is the lower speciﬁcation limit;

l

is the process mean; and

r

is the standard deviation of the process. CPLmeasures

the capability of a larger-the-better process with a lower speciﬁcation limit LSL. For a normal distribution process with a one-sided speciﬁcation limit LSL, the process yield is

PðX > LSLÞ ¼ P

l

X 3

r

<

l

LSL 3

r

¼ P 1 3Z < CPL ¼ PðZ > 3CPLÞ ¼ 1

U

ð3CPLÞ ¼

U

ð3CPLÞ ð2Þ

where Z follows the standard normal distribution N (0, 1) with the cumulated distribution functionU(). 2.2. Investment satisfaction capability index (ISCI)

Investment satisfaction degree (ISD) is always the core element considered by investors while making investment choices. Our research aims to make use of the investment satisfaction capability index (ISCI) to analyze the overall level of performance of the stock. In addition, we consider the mathematical advantage of the one-to-one radio for the indicators of ISCI and ISD. Investors will be able to obtain rapid information about the individual performance of the stock by using the ISCI index to check whether the level of satisfaction of the return on investment (ROI) is achievable for the particular stock. Finally, we derive the formulas for the ISCI, probability density function (PDF) and the uniformly minimum-variance unbi-ased estimator (UMVUE). We also introduce a complete set of procedures, calculus-bunbi-ased examination and the rules of

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anal-ysis for judging the overall performance of the stock. It is hoped that this method will aid investors in making better stock investment choices and to increase the proﬁt for the investment portfolio.

We analyzed the level of performance of all the listed stock in order to judge the overall performance of an individual stock. Since investors have different levels of satisfaction for the ROI, they also have their own lowest reference limit (LRL) for the desired ROI. Despite this fact, investors would at least want their ROI set above the risk free rate. Hence, we make the following assumption for the LRL of the ROI: the risk free rate with a risk premium for a 210 day treasury rate is 1.75%, with the annual risk premium being set at 10% in this study. The LRL of the annual return is 11.75%, which will become 0.2260% for the weekly ROI (11.75%/52, 52 weeks per year).

In order to choose those stocks showing better performance within the various listed companies, it is necessary to in-crease their ROI. To do this, we make LRL the lower speciﬁcation reference limit of the annual ROI and weekly stock ROI. Furthermore, because the movement of the price of the stock is random, we can treat the stock return rate X as a random variable. When the random variable is lower than the LRL of the daily return rate (X < LRL), it indicates that the performance of the particular stock does not meet the expected ROI. Due to the above, we deﬁne the weekly stock return lower reward limit of ISCI as follows:

CSL¼

l

LRL

3S LRL ¼ Rfþ premium ð3Þ

where Rfis the 210 day treasury rate, and premium is the risk free premium. S is the sample standard deviation in the weekly

ROI for individual stock.

In order to judge ISCI, this study makes use of statistics sampling to estimate the ROI of the non-parameter (

l

,

r

) for the individual stock. Let Xibe i samples of stock chosen from the weekly ROI under the assumption of Xi N (

l

i,

r

i). Now

calcu-late the sample mean Xiand the sample deviation Si, from the chosen samples, to estimate the

l

iand

r

iin the matrix. We are

able to use the natural estimator in the index CSL, as shown below: b CSL¼ X ðRfþ premiumÞ 3S ð4Þ X ¼1 n Xn i¼1 Xi S ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn i¼1 ðXi XÞ 2 =ðn 1Þ v u u t

where X is the average weekly return rate for individual stock, S is the sample standard deviation in the weekly ROI for indi-vidual stock.

Under the normal distribution assumption of Chou and Owen[16], the estimation of bCSLcan be written as Cntn1(d), with

Cn¼ ð3

ffiffiffi n p

Þ1and tn1(d) being the non-central t distribution with n 1 degree of freedom and the non-centrality parameter

d¼ 3pffiffiffinCSL.

However, since bCSLis an estimation, as in Pearn et al.[31], an exact factor bn1= [2/(n 1)]1/2C[(n 1)/2]/C[(n 2)/2]

can be added to bCSLand the unbiased estimator bn1CbSL, simpliﬁed as eCSL, can be obtained. Hence, we have EðeCSLÞ ¼ CSL, from

bn1< 1, varðeCSLÞ < varðbCSLÞ. Here eCSLis the UMVUE of CSL, so it would be more suitable to make use of UMVUE eCSLfor our

calculation. The probability density function (PDF) of eCSLis f ðxÞ ¼3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n=ðn 1Þ p 2n=2 bn1 ffiffiffin p

C

½ðn 1Þ=2 Z 1 0 yðn2Þ=2_exp1 2 y þ 3x ffiffiffiffiffiffipny bn1 ffiffiffiffiffiffiffiffiffiffiffiffi n 1 p d 2 " # ( ) dy ð5Þ where bn1¼ ½2=ðn 1Þ1=2C½ðn 1Þ=2=C½ðn 2Þ=2; d ¼ 3 ffiffiffin p CSL.

As the level of satisfaction with investment rises, the risk also increases. To reﬂect this reality when judging the perfor-mance of the individual stock, we set the ISD to be 70%. Next, we select those stocks that pass the investment perforperfor-mance testing to be components of the stock portfolio. Some error exists during the sampling of the statistics and when analyzing the performance of investment in different categories. Hence, there is a need to make use of testing statistics to more objec-tively evaluate the whole performance. Assuming that the level of satisfaction of the investor is C, the testing of the perfor-mance of the individual stock can be expressed as follows:

H0: CSL6C indicates the investment performance does not reach the level of satisfaction of the investor;

H1: CSL> C indicates the investment performance reaches the level of satisfaction of the investor;

C is a constant setted by the investor.

If CSL> C, the investment performance reaches the level of satisfaction of the investor; whereas if CSL6C, the investment

performance does not reach the level of satisfaction of the investor. We treat the minimum unbiased estimator eCSLas the

testing statistics for the performance of individual stock at signiﬁcance level

a

. The rejection zone will be deﬁned as feCSLPc0g and c0as the critical value of the rejecting zone. The value of signiﬁcance level

a

is determined by the following

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2006 and 16.5631% in 2007. However, the monthly accumulated ROI of the stock portfolios making use of PSO is better than the average allocation, being 54.2964% in 2006 and 41.0297% in 2007. The results of Experiment II further confirm the ben-efits brought about by PSO. Its application to stock selection improves the performance of the investors. The preceding anal-ysis confirms the viability of the application of PSO in the finance sector.

5. Conclusions

Data adopted from the years 2005 to 2007 were used in this study for training and testing. We constructed a decision-making system, formed by the investment satisfaction capability index (ISCI) CSLand PSO, to apply to the structure of a stock

investment portfolio.

In the first stage, we provided investors with a new strategy of using the ISCI to optimize their choices of stocks. In the second stage, we made use of the PSO to make investment decisions regarding the allocation of capital investment for each specific stock in the portfolio. Depending on the choice of components for our stock, we created two different portfolios. In Experiment I, the only variation is the position weighting of the stock in the portfolio. After further analysis, the results show that the accumulated ROI of these portfolios were all better than the ROI of the TWSI portfolios. Experiment II was conducted based on the concern that investors must be flexible when making decision. Thus the components of their portfolios vary each month thereby reflecting the need for investment flexibility. The results of Experiment II show a comparison of the cap-ital allocation while making use of the PSO and average allocation. The yearly accumulated ROI is better when making use of PSO than the average allocation; likewise, the monthly accumulated ROI is better when utilizing PSO instead of average allo-cation. The results of Experiment II further confirm the benefits brought about by application of PSO to the portfolio. It has proven to improve on the performance of the investors.

The system should help to set up a highly profitable stock investment portfolio which can also be changed to meet the demand of a volatile stock market environment. Investors can make use of our suggested procedure to make better decisions on their investments. Based on this system, investors can obtain better investment risk control and acquire a more profitable stock investment portfolio, while maintaining flexibility responsive to the stock market environment. This system can be used as suggested or changed to meet individual requirements. Otherwise, the LRL and ISD set up in our research are ex-pected to vary due to personal preference. In this paper, we have set the ISD at 70%, thus we suggest use Fuzzy, Grey, or other algorithms to evaluate ISD in the future research.

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