A Hybrid Model for Portfolio Selection based on Grey Relational Analysis and RS Theories

Kuang Yu Huang1, Chuen-Jiuan Jane2, Ting-Cheng Chang3 Department of Management Informationon1,3, Department of Finance and Risk Management2

Ling Tung University Taichung ity, Taiwan

kyhuang@teamail.ltu.edu.tw1, ltc869@teamail.ltu.edu.tw2, tcchang@teamail.ltu.edu.tw3

Abstract—In this study, the Grey Relational Analysis (GRA) model is combined with Fuzzy C-Means (FCM) clustering scheme and Rough Set (RS) theory to create an automatic portfolio selection mechanism. In the proposed approach, 53 financial indices are collected automatically for each stock item every quarter and a GRA model is used to consolidate these indices into six predetermined financial ratios (Grey Relational Grades (GRGs)). The GRGs of the stock items are then clustered using a FCM scheme and the resulting cluster indices are processed using RS theory to identify the lower approximate set within the stock system. The stock items within the lower approximate set are filtered in accordance with established investment principles and the six GRGs of each surviving stock item are then consolidated to a single GRG indicating the overall merit of the corresponding stock item in terms of its ability to maximize the rate of return on the investment portfolio. It is shown that the rate of return on the investment portfolio selected using the proposed GRA/FCM/RS system is higher than the average rate of return predicted by the variation in the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) over the same period.

Keywords-Grey Relational Analysis; Rough Set; Stock Portfolio; ROE; TAIEX

I. INTRODUCTION In the past, the decision as to which stocks to include

within an investment portfolio was generally made on a qualitative or intuitive basis. However, in recent decades, many applications have been proposed for predicting future stock market trends and selecting suitable stock portfolios. Typically such applications include the use of genetic algorithms (GAs) [1], artificial neural networks (ANNs) [2], fuzzy logic schemes [3], statistical forecasting mechanisms [4], or Rough Set (RS) models [5,6]. Of these various techniques, RS models are particularly advantageous for classifying multiple stock items in terms of their financial performance. RS theory was first introduced more than twenty years ago [7] and has developed into a powerful automatic classification technique with potential applications in a diverse range of fields, including machine learning, forecasting, knowledge acquisition, decision analysis, knowledge discovery, pattern recognition, and data mining. Real-world information systems are commonly

characterized by a large number of objects and attributes. Thus, in applying RS theory to the classification of such systems, some form of attribute reduction scheme is often used to eliminate the conditional attributes which have little or no affect on the classification decision, thereby simplifying the decision table and enabling the decision rules to be more easily identified. For example, Shen [8] demonstrated the use of Rough Set Attribute Reduction (RSAR) techniques in supervised and unsupervised learning applications, and in a later study [9], proposed an attribute selection technique based on a fuzzy RS approach for complex systems monitoring purposes. Bazan [10] compared and contrasted the performance of various dynamic and non-dynamic RS methods for extracting decision rules from decision tables. Jensen [11] presented a fuzzy RS approach for the attribute reduction of complex systems with minimal loss of information.

In general, the results presented in [8-11] confirm the desirability of reducing the dimensions of the target data system before applying RS theory for classification purposes. In attempting to classify multiple stock items with the aim of identifying which items to include within a stock portfolio and which items to exclude, a number of problems arise: (1) the information system is complex and comprises both a large number of data objects and a large number of data attributes (i.e. financial indices); (2) of the many financial indices (conditional attributes), some have a more critical effect on the investment decision (dependent attribute) than others; (3) some of the financial indices are interrelated, i.e. they not only exert a direct effect on the ultimate investment decision, but also affect the value(s) of one or more of the other data attributes; and (4) the stock market system is characterized by inherent vagueness and uncertainty, and thus definitive predictions of the future stock market performance are virtually impossible to obtain. Consequently, in developing an automatic stock portfolio selection system, a method is required for not only reducing the dimensionality of the stock market system in order to simplify the classification and selection processes, but also for dealing with its inherent vagueness and uncertainty. In practice, both issues can be resolved using the Grey Relational Analysis (GRA) model prescribed in Grey System theory.

Grey System theory, proposed by Deng [12] in 1982, provides a powerful technique for dealing with systems

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characterized by poor, incomplete and uncertain information. One of the most fundamental components of Grey System theory is that of GRA [13], in which information from the grey system is used to quantify the respective effects of the various factors within the system in terms of Grey Relational Grades (GRGs). In practice, GRA provides the means to “weight” the various factors within an uncertain system in terms of their effect on the system outcome and therefore provides an ideal basis for classification systems. As a result, GRA provides an ideal tool for analyzing the complex inter-relationships amongst the individual parameters in systems with multiple performance characteristics and has therefore been widely applied in a variety of optimization, decision-making and classification problems in the finance [14], business, economics, design, manufacturing and production fields .

The current study combines RS theory and GRA to construct an automated stock portfolio selection system. The feasibility of the proposed approach is demonstrated using stock data extracted from the financial database maintained by the Taiwan Economic Journal (TEJ) [5].

The remainder of this paper is organized as follows. Section 2 discusses the various methodologies used in the present study, including Grey Relational Analysis (GRA), Fuzzy C-Means (FCM) clustering, and Rough Set (RS) theory. Section 3 describes the detailed processing steps in the proposed automatic stock selection system and verifies its performance by comparing the rate of return on an investment portfolio comprising stocks in Taiwan with the average rate of return predicted by the variation in the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) over the equivalent period. Finally, Section 4 provides some brief concluding remarks.

II. REVIEW OF RELATED METHODOLOGIES

A. Grey Relational Analysis Grey Relational Analysis (GRA) methods provide an

effective means of solving multiple-criteria decision problems by ranking the potential solutions in terms of their so-called Grey Relational Grades (GRGs) such that the optimal solution can be easily determined [15]. In the stock portfolio selection system proposed in the present study, the GRA method is used to simplify the stock classification and selection processes by consolidating the values of the multiple attributes of each data object into a single attribute value representing one particular subsystem of the total stock system. In the present study, the GRA model is used to reduce the number of inputs to the stock classification and selection processes in each financial quarter, k. Let )()(),( 00 jxjxjk ikki −=Δ , where ni ,...,2,1= ,

Nj ,...,2,1= , in which N is the number of attributes observed. Furthermore, let )(0 jx k and )( jxik denote the j th attribute values of the reference object and the

comparative object, respectively. Choosing the GRA model proposed by Deng [12] for illustration purposes, the grey relational coefficient has the form

)(),()()())(),((

max0

maxmin0 kjk

kkjxjxi

ikk Δ+ΔΔ+Δ=

ζζγ ,

where ),(minmin(k)min jkoiji

Δ=Δ∀∀

,

),(maxmax)( 0max jkk iji

Δ=Δ∀∀

, and ζ is the distinguishing

coefficient and has a value between 0 and 1. (Note that ζ is generally assigned a value of 0.5 to minimize the error.) The Grey Relational Grade (GRG) represents the degree of correlation between two sequences and is generally defined as the average of the respective grey relational coefficients,

i.e. ==Γ=

N

jikkikki jxjx

Nxxk

1000 ))(),((1),()( γγ .Having

calculated the GRG for each sequence, the sequences are ranked using a Grey relational ranking procedure. For example, if ),(),( 00 jkkikk xxxx Γ>Γ , the corresponding

ranking is denoted as jkik xx .

B. Fuzzy C-Means (FCM) Clustering Method [16] In the stock portfolio selection system proposed in this

study, a GRA model is used to reduce the number of attributes of each data object within the stock market system from 53 financial indices to 6 consolidated financial ratios. The GRGs (i.e. financial ratios) of the stock items are then clustered using the Fuzzy C-Means (FCM) method developed by Dunn in 1973 [17] and refined by Bezdek in 1981 [18].

C. Rough Set Theory The FCM method described above produces a set of

indices with a value between 0 and 1 which indicate the degree of belonging of each data object to each cluster within the dataset. In the stock selection system proposed in this study, these indices are then processed using Rough Set (RS) theory in order to identify the lower and upper approximate sets within the stock market system such that an initial assessment of the potential stocks for inclusion within the portfolio can be made.

RS theory was originally introduced by Pawlak in 1982 [19] as a means of handling the vagueness and uncertainty inherent in many decision-making processes. RS theory is based on the assumption that every object in the universe of discourse is associated with a particular set of information (i.e. attributes). Objects characterized by the same informa-tion are regarded as indiscernible. The indiscernibility relationships amongst all the objects in the universe of discourse provide the basic mathematical basis for RS theory. Typical problems amenable to RS processing include classifying sets of objects in terms of their attribute values, checking the dependencies (either full or partial) between attributes, reducing the number of attributes required to classify the system of interest, analyzing the significance of individual attributes, generating decision rules, and so on.

In RS theory, knowledge about the universe of discourse is represented using so-called information systems. A

typical information system has the form ),,,( qq fVAUS = ,

where U is a non-empty finite set of objects and A is a non-empty finite set of attributes which describe these objects. Generally speaking, the attributes in set A can be partitioned into a set of conditional attributes φ≠C and a set of decision attributes φ≠D , i.e. A = DC and φ=DC . For each attribute, qVAq ,∈ represents the

domain of q , i.e. qVV = . Finally, VAUf q →×: is an

information function defined such that qVqxf ∈),( for Aq ∈∀ and Ux ∈∀ .

For an attribute subset P , CP ⊆ and φ≠P , the equivalence relation is denoted by

}),(),(:),{( PqqyfqxfUUyxI P ∈∀=×∈= . In other words, the equivalence relation PI partitions

U into a family of disjoint subsets PIU / known as quotient sets of U : }:]{[/ UxxIU pP ∈= , where

px][ denotes the equivalence class determined by x with

respect to P , i.e. }),(:{][ Pp IyxUyx ∈∈= . In a decision table, in which the values of all the

attributes (both conditional and decision) are listed for each object in the dataset, some objects are likely to be characterized by the same set of attribute values. As a result, it is impossible to discern between them on the basis of their attribute values alone. The objects which are indistinguishable from one another when evaluated using a particular subset of all the attributes define an equivalence or indiscernibility relationship. In RS theory, this indiscernability of the data objects is handled using the concept of approximate sets.

Assume that the information system ),,,( qq fVAUS = is represented in the form of a decision table in which

UX ⊆ and AR ⊆ . The upper and lower approximate sets of X are denoted as )(XR and )(XR , respectively, and are defined as

}][|{)( φ≠∈= XxUxXR p ,

}][|{)( XxUxXR p ⊆∈= ,

where px][ denotes the equivalence class determined by x with respect to P , i.e. }),(:{][ Pp IyxUyx ∈∈= . The lower approximate set )(XR contains all the elements ( x ) which have the same rank when evaluated in terms of the X th decision attribute, while the upper approximate set

)(XR contains all the elements ( x ) which may have the same rank when processed in accordance with the X th decision attribute. Furthermore, the boundary set of X is defined as the difference between the two approximate sets, i.e. )()()( XRXRXBN R −= .

In the present context, the lower approximate set is comprised of stock items which are known unambiguously

to have an adequate financial performance (as indicated by the corresponding values of the consolidated financial ratios). By contrast, the upper approximate set contains both high-performing and low-performing stock items. Thus, in considering which stock items to include within the investment portfolio, only those within the lower approximate set are considered.

III. AUTOMATIC STOCK MARKET PORTFOLIO SELECTION SYSTEM

In this section, the GRA, FCM and RS methodologies are integrated to construct an automatic stock portfolio selection system. In the proposed approach, the stock items are collected automatically every quarter and the 53 financial indices of each item are consolidated into 6 normalized financial ratios using a GRA model. The stock items are then clustered in accordance with their financial ratio values using a FCM scheme. Following the clustering process, RS theory is applied to identify the stocks within the lower approximate set. Finally, the GRA model is re-applied to rank the filtered stocks in terms of their likely contribution in maximizing the rate of return on the stock portfolio such that a definitive choice of stocks for inclusion within the portfolio can be made.

A. GRA Attribute Reduction Mechanism In the proposed stock selection scheme, a GRA model is

used initially to consolidate the attributes of the extracted stock items in order to simplify the FCM and RS procedures, and is then used once again to consolidate the attributes of the stocks filtered in accordance with Buffet’s principles in order to general a single performance measure for each stock item.

The GRA model is used to reduce the 53 financial indices for each stock item to 6 predetermined financial ratios, namely (1) the rate per share, (2) the growth, (3) the credit capacity, (4) the operating capacity, (5) the statutory ratio, and (6) the profitability, where ratios (1)~(5) are taken as the conditional attributes of the stock item and index (6) is taken as the decision attribute. Note that the contents of each financial ratio, i.e. the financial indices which constitute each ratio, are pre-defined, and thus the purpose of the GRA model is simply to compute the normalized value of each financial ratio. In other words, for each stock item, the GRA model outputs six normalized financial ratios, i.e. Grey Relational Grades (GRGs).

In this study, the information system is assumed to have the form ),,,( qq fVAUS = , where U is a non-empty finite set of objects (stock items) and A is a finite set of attributes (financial indices) describing these objects. Following the application of the GRA model, a modified information system with the form ),,ˆ,( qq fVAUS = is obtained, where

A is a set of six consolidated attributes (financial ratios) describing the same set of objects. The financial ratios are clustered using the FCM method and the resulting cluster indices are then processed using RS theory in order to

identify the corresponding lower approximate set. Assuming that U is the domain of discourse and R is

the set of equivalences of U , then the RS problem can be formulated as

UX ⊆ is )(,))(,)(( XBNXRXR R , where X is the set of elements; }:]{[/ UxxIU pP ∈= ,

)(/ RINDU are the quotient sets of U ; px][ denotes the equivalence class determined by x with respect to P , i.e.

}),(:{][ Pp IyxUyx ∈∈= ; φ is the zero set; R is the

attribute set of X which includes the conditional attribute set C and the decision-making attribute set ( D ); )(XR is the lower approximate set of X ; )(XR is the upper approximate set of X ; and )(XBNR is the boundary set of X . Every element in the domain of discourse U ( UX ⊆ ) has an attribute set R , which describes the particular value of X . In the model developed in this study, the attributes are the financial ratios, and every X ( UX ⊆ ) of U is assigned an appropriate set of

conditional attribute and decision-making attribute values ),,...,,( 1521 DCCCR = .

As described above, the selected GRA model is also used to reduce the six financial ratios of each stock item remaining after the stocks within the lower approximate set have been filtered using the general investment principles prescribed by Buffet. In this case, the GRA model takes the six consolidated financial ratios (GRGs) of each stock item as the input and generates a single GRG which indicates the overall performance of the corresponding stock item. The GRGs are ranked in descending order such that the stock items with a better financial performance are placed above those with a poorer performance and the ranked sequence is then taken as the input to the final stock selection decision.

B. Data Extraction In the present study, the performance of the proposed

stock selection mechanism was evaluated using electronic stock data extracted from the TEJ database [12] over the period extending from the first quarter of 1998 to 2/27/2009. In general, financial statements for a particular accounting period are subject to a certain delay before publication. For example, annual reports are published after four months, half-yearly reports after two months, and first and third quarterly reports (without notarization) after a minimum of one month. As a result, the stock selection system can only be executed three times in every 12 month period, namely 5/31~09/22, 9/22~11/15 and 11/15~05/31 the following year.

C. Detailed Processing Steps in Stock Selection System Figure 1 illustrates the overall framework of the

proposed stock selection system. The detailed processing steps within each stage of the system are summarized in the following paragraphs. Step 1: Data collection and attribute determination

In each quarter, the 53 attributes of each specified stock item within the TEJ database are collected automatically, and the user is given the opportunity to modify the choice of financial ratios used for attribute reduction in the initial GRA process, to select a new GRA model for attribute reduction purposes, and to modify the decision-making attributes used to filter the stocks in the lower approximate set. Step 2: Data preprocessing

Having collected the relevant financial data for each

Data collection and attribute determination

Start

Data preprocessing preprocessing

Attribute reduction using GRA model

Fuzzy C-means clustering

Selection and filtering of feasible stocks

Fund allocation

Renew the GRA Method

No

Yes

Yes

No

Continue investment

End

Proceed to next quarter investment

Modeling applicable

Figure 1 Flow chart of proposed stock selection model.

quarterly period, a basic pre-processing operation is performed to improve the efficiency of the GRA attribute reduction process. Specifically, the data records containing missing fields (i.e. missing financial indices) are immediately rejected, and the Box Plots method is applied to resolve the data outlier problem by establishing an inter-quartile range such that any data points falling outside this range can be automatically assigned a default value depending on the interval within which they fall. Step 3: Attribute reduction using GRA model

For the stock records which remain after the pre-processing operation, the GRA model normalizes the values of each of the 53 financial indices and then computes the corresponding financial ratios. Step 4 Fuzzy C-Means clustering

The financial ratios of the stock items (i.e. five conditional attributes 51 ~ CC and one decision attribute 1D ) are clustered into three groups (i.e. three groups per financial ratio, giving a total of 18 clusters for the entire dataset) using a FCM clustering algorithm. Step 5: Selection and filtering of feasible stocks

The cluster indices generated by the FCM clustering scheme are processed using RS theory to identify the stock items within the lower approximate set. These stock items are then filtered in accordance with the general investment guidelines proposed by Buffett in order to identify the final

set of stocks for possible inclusion within the investment portfolio. Step 6: Fund allocation

The six financial ratios of each stock item remaining after the filtering operation are consolidated to a single GRG (i.e. an overall performance indicator) by the selected GRA model. The GRGs of all the surviving stock items are then arranged in descending order and the first five stock items are chosen for investment purposes. Step 7: Check validity of modeling

The rate of return on the stock portfolio constructed at the end of quarter k is compared at the end of quarter k+1 with the average rate of return implied by the variation in the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) over the equivalent financial period. If the rate of return is acceptable, a decision is made as to whether or not the model should be run for a further quarter using the existing GRA model. However, if the rate of return is deemed unacceptable, the suitability of the GRA model is reviewed and a new GRA model is adopted as appropriate.

D. Verification and Evaluation of Proposed Stock Selection System In this section, the validity of the proposed stock

selection system is evaluated by considering two illustrative examples using electronic stock data extracted from the TEJ database. The first example considers the financial data relating to Q1 1998 to Q1 2003 and is intended to confirm the basic validity of the proposed approach, while the second example considers the financial data relating to Q1 2003 to 2/27/2009 and is intended to demonstrate the

performance of the proposed stock selection system. Note that in both cases, the stock selection system is implemented using the GRA model proposed by Wen [14].

Table 1 compares the rate of return on the selected stock portfolio in each of the 17 investment periods between 1998 and 2003 with the variation in the TAIEX index over the equivalent periods. It can be seen that the rate of return obtained using the proposed stock selection system is invariably higher than the variation of the TAIEX index in each investment period. Moreover, the accumulated rate of return obtained from the proposed system (70.01) over the 17 investment periods is significantly higher than the TAIEX return (7.58), and thus the basic validity of the proposed approach is confirmed. Table 2 compares the rate of return on the selected stock portfolio with the average variation in the TAIEX index over each of the 17 investment periods between 2003 and 2009. As shown, the accumulated rate of return on the stocks selected using the proposed system (65.96) is significantly higher than the accumulated rate of return on all the stocks within the TAIEX system over the equivalent period (-9.09%). Consequently, the superior performance of the proposed stock selection system is confirmed.

IV. CONCLUSIONS AND DISCUSSIONS This study has utilized a Grey Relational Analysis (GRA)

attribute reduction mechanism, a Fuzzy C-Means (FCM) clustering scheme and a Rough Set (RS) classification

Table 1 Rate of return on selected stock portfolio and average variation of TAIEX index over 17 quartersbetween 1998 and 2003.

Investment period TAIEX GRA-based

98/06/01~98/09/21 -7.58 -10.41

98/09/21~98/11/16 -0.36 7.63 98/11/16~99/05/31 4.46 33.20

99/05/31~99/09/27 6.06 14.16 99/09/27~99/11/15 -2.77 1.03

99/11/15~00/05/31 18.48 6.02 00/05/31~00/09/21 -22.58 -11.09

00/09/21~00/11/15 -17.11 -23.31 00/11/15~01/05/31 -12 13.89

01/05/31~01/09/21 -28.86 -30.22 01/09/21~01/11/15 22.60 15.95

01/11/15~02/05/31 28.89 21.33 02/05/31~02/09/23 -23.74 -12.87

02/09/23~02/11/15 11.21 14.38 02/11/15~03/06/02 -2.51 -5.50

03/06/02~03/09/22 20.94 24.76 03/09/22~03/11/17 4.87 11.06

Accumulated rate of return 7.58 70.01

Table 2 Rate of return on selected stock portfolio and average variation of TAIEX index over 17 quarters between2003 and 2009.

Investment period TAIEX GRA-based

03/09/22~03/11/17 4.87 11.06

03/11/17~04/05/31 0.43 -6.27 04/05/31~04/09/21 -0.48 -0.19

04/09/21~04/11/15 -0.72 -4.33 04/11/15~05/05/31 1.78 -0.47

05/05/31~05/09/21 0.93 8.60 05/09/21~05/11/15 -0.60 -1.7.3

05/11/15~06/06/01 13.96 14.56 06/06/01~06/09/21 0.25 -11.21

06/09/21~06/11/15 5.04 -7.86 06/11/15~07/05/31 12.55 82.01

07/05/31~07/09/21 11.79 -1.57 07/09/21~07/11/15 -2.20 2.90

07/11/15~08/06/02 -2.03 2.29 08/06/02~08/09/22 -29.96 -22.36

08/09/22~08/11/17 -27.34 -11.62 08/11/17~09/02/27 2.64 12.14

Accumulated rate of return -9.09 65.96 scheme to construct an automatic stock market portfolio

selection system. The general validity of the GRA attribute reduction mechanism has been confirmed by constructing an automatic stock price forecasting system comprising an artificial neural network (ANN) and a GRA model. Meanwhile, the validity and performance of the stock selection system have been demonstrated by comparing the rate of return on the selected stock portfolio over 33 quarters between 1998 and 2009 with the average rate of return predicted by the TAIEX index over an equivalent period. The experimental results presented in this study support the following conclusions:

The accumulated rate of return on the portfolio constructed using the proposed automatic stock selection system exceeds the accumulated rate of return predicted by the variation in the TAIEX index over the corresponding period. Thus, the overall validity and performance of the proposed stock selection system are confirmed. As a result, the GRA/FCM/RS model proposed in this study provides an effective and computationally efficient solution for the classification not only of stock market systems, but also of any data system characterized by a large number of attributes and inherent uncertainty.

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