Combined DEMATEL technique with a novel MCDM model for exploring portfolio selection based on CAPM

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Combined DEMATEL technique with a novel MCDM model for exploring

portfolio selection based on CAPM

Wen-Rong Jerry Ho

a

_{, Chih-Lung Tsai}

b

_{, Gwo-Hshiung Tzeng}

c,d,*

_{, Sheng-Kai Fang}

b a

Department of Finance, Chinese Culture University, Yang-Ming-Shan, Taipei 111, Taiwan

b

Department of Banking and Finance, Kainan University, No. 1, Kainan Road, Luzhu Shiang, Taoyuan 33857, Taiwan

c

Institute of Management of Technology, National Chiao Tung University, 1001, Ta-Hsueh Road, Hsin-Chu 300, Taiwan

d

Institute of Project Management, Department of Business and Entrepreneurial Administration, Kainan University, No. 1, Kainan Road, Luzhu Shiang, Taoyuan 33857, Taiwan

a r t i c l e

i n f o

Keywords:

MCDM (multiple criteria decision making) DEMATEL (decision making trial and evaluation laboratory)

ANP (analytical network process) VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje)

CAPM (capital asset pricing model) Portfolio

a b s t r a c t

This research proposes a novel MCDM model, including DEMATEL, ANP, and VIKOR for exploring portfolio selection based on CAPM. We probe into the influential factors and relative weights of risk-free rate, expected market return, and beta of the security. The purpose of this research is to establish an invest-ment decision model and provides investors with a reference of portfolio selection most suitable for investing effects to achieve the greatest returns. Taking full consideration of the interrelation effects among criteria/variables of the decision model, this paper examined leading semiconductor companies spanning the hottest sectors of integrated circuit (IC) design, wafer foundry, and IC packaging by experts. Empirical findings revealed that risk-free rate was affected by budget deficit, discount rate, and exchange rate; expected market return was affected by country risk, industrial structure, and macroeconomic fac-tors; and beta of the security was affected by firm-specific risk and financial risk. Also, the factors of the CAPM possessed a self-effect relationship according to the DEMATEL technique. In the eight evaluation criteria, macroeconomic criterion was the most important factor affecting investment decisions, followed by exchange rate and firm-specific risk. In portfolio selection, leading companies in the wafer foundry industry outperformed those in IC design and IC packaging, becoming the optimal portfolio of investors during the time that this study was conducted.

1. Introduction

Markowitz (1952)introduced Mean–Variance Portfolio Model; moreover, Sharpe (1964), Lintner (1965), and Mossin (1966)

subsequently referenced his model to propose the CAPM1_noting

that expected return on a security is impacted by risk-free rate, expected market return, and beta of the security. Coupled with the model’s ability to predict expected stock return and formally link to-gether the notions of risk and return, it is widely applied to help investors make investment decisions. However, investors care about how much return that they expect to earn. In other words, they are more interested in determining what factors inﬂuence the CAPM’s three ﬁxed variables, and the level of importance of each individual factor. The CAPM only explained that three important factors impact expected stock return but a description of more detailed factors was not available. Therefore, through extensive literature review, this

study identiﬁed the factors that were to be inﬂuenced by risk-free rate, expected market return, and beta of the security, and examined the level of importance of these factors in an effort to make up for the inadequacy of the CAPM.

Preceding studies on expected stock return were mostly focused on exploring the relationship between expected stock return and macroeconomic factors such as money supply (Bilson, Brailsford, & Hooper, 2001; Kwon & Shin, 1999; Mandelker & Tandon, 1985; Robi-chek & Cohn, 1974; Rogalski & Vinso, 1977), inflation (Balduzzi, 1995; Fama, 1981; Gultekin, 1983; Kim & In, 2005; Park & Ratti, 2000), and interest rates (Abugri, 2008; Domian, Gilster, & Louton, 1996; Geske & Roll, 1983; Kim & Wu, 1987). Yet, the results of these researches were not consistent in that the studies were conducted on unidirectional relationships. For investors, the message conveyed was simply what factors influence expected stock return and whether the influence was positive or negative. Consequently, these findings contribute little to the goal of constructing a complete set of expected stock return pricing model, and particularly a comprehen-sive analysis of factors and interactive relationships. In addition, studies on the relative weights among variables were insufficient, and MCDM was seldom used by researches for portfolio selection such asEhrgott, Klamroth, and Schwehm (2004)andLee, Tzeng, Guan, Chien, and Huang (2009).

*Corresponding author. Address: Department of Business and Entrepreneurial Administration, Kainan University, No. 1, Kainan Road, Luzhu Shiang, Taoyuan 33857, Taiwan.

E-mail address:ghtzeng@mail.knu.edu.tw(G.-H. Tzeng).

1

The CAPM refers to EðRiÞ ¼ Rfþ ½EðRmÞ Rf b, where E(Ri) denotes expected

return on a security; Rfdenotes risk-free rate; E(Rm) denotes expected market return;

bdenotes beta of the security.

Contents lists available atScienceDirect

Expert Systems with Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a

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Hence, the purpose of this study is to supplement the CAPM and that of previous findings on expected stock return in establishing an investment decision model by experts to provide investors with a reference of portfolio selection most suitable for investing effects to achieve the greatest returns. This research adopted the CAPM and a novel hybrid MCDM model consisting of combined DEMATEL with ANP and VIKOR. By reviewing literatures, we identified the sub-factors of risk-free rate, expected market return, and beta of the security in order to establish the investment decision model. Through a survey of experts, we employed DEMATEL technique to analyze the causal relationships between complex factors and then to build a network relation map (NRM) among criteria for portfolio evaluation. The weights of each factor of MCDM problem for selecting the best portfolio will then be derived by utilizing the Analytic Network Process (ANP) based on the NRM. Afterward, we ranked the data to recognize cardinal factors. Evaluation objects were taken from leadership companies of the hottest stocks in the semiconductor sectors: IC design, wafer foundry, and IC pack-aging. We then identified the most suitable investment by VIKOR and offered a complete depiction and testing of the decision model for the reference of investors.

The rest of this paper is organized as follows: in Section2, we identify the sub-factors inﬂuencing risk-free rate, expected market return, and beta of the security on expected stock return pricing model in order to construct the evaluation criteria based on litera-ture review. In Section3, the depiction and application of the novel MCDM are included. Section4shows an empirical study of select-ing the optimal portfolio by usselect-ing the proposed evaluation model, and the results are discussed. The conclusions and remarks are pro-vided in the ﬁnal section.

2. Expected stock return pricing model

The purpose of this section is to identify the inﬂuential factors of expected stock return based on past literatures and discuss the regions of scarcity in these studies. To make up for such a gap, this study conducted a literature review of the CAPM’s three main fac-tors – risk-free rate, expected market return, and beta of the secu-rity – for the sake of more accurately identifying the evaluation criteria affecting expected stock return.

2.1. Related literature on factors influencing expected stock return Expected stock return is the basis of comparison of required rate of return of investors in the whole stock market. Investors will not invest in a portfolio unless the required rate of return is higher than expected stock return. Therefore, what investors emphasize would be to estimate the expected stock return to provide them with a reference of portfolio selection for decision making and to make profits in the stock market. The factors influencing expected stock return are diverse, and can be classified into financial and macroeconomic factors. According toChen, Roll, and Ross (1986), macroeconomic variables were found to be significant in explain-ing expected stock returns. Moreover, because macroeconomic variables can be quantifiably analyzed, much empirical research, in recent years, has been done by using macroeconomic status’s proxy variables to investigate the relationship between macroeco-nomic variables and expected stock return.

Macroeconomic factors that were detected to be inﬂuential of expected stock return contained money supply (Bilson et al., 2001; Kwon & Shin, 1999; Mandelker & Tandon, 1985; Robichek & Cohn, 1974; Rogalski & Vinso, 1977), inﬂation (Balduzzi, 1995; Fama, 1981; Gultekin, 1983; Kim & In, 2005; Park & Ratti, 2000), interest rates (Abugri, 2008; Domian et al., 1996; Geske & Roll, 1983; Kim & Wu, 1987), and industrial production (Chen et al.,

1986; Fama, 1990; Ferson & Harvey, 1998). These literatures are thoroughly discussed as follows.

In the related studies of money supply, Robichek and Cohn (1974) analyzed the data, current money supply and inflation, between January 1963 and October 1970 revealing that the rela-tionship between contemporaneous amount of money supply and stock price was negligible. However, Rogalski and Vinso (1977)concluded that changes in money supply may have influ-ences on real economic activity, thereby having a lagged influence on stock returns, which implied a positive relationship between changes in money supply and stock returns. Whereas,Mandelker and Tandon (1985)found that under situations where monetary policy is not credible, money supply innovations may affect stock returns negatively through its effects on inflation uncertainty. In recent studies,Kwon and Shin (1999)investigated whether macro-economic variables such as foreign exchange rate, money supply and oil price are significant explanatory factors of stock market re-turns. Monthly data from Korea between 1980 and 1992 were used to carry out the empirical study. Results indicated that stock price indices are correlated with the production index, exchange rate, trade balance, and money supply which provide a direct long-run equilibrium relation with each stock price index. Bilson et al. (2001)used a least squares procedure to test whether macroeco-nomic variables have explanatory power over stock returns in twenty emerging markets. The results indicated that the money supply variable is positively significant in six markets.

On the topic of inflation, Fama (1981) employed a simple rational expectations version of the quantity theory of money find-ing that the results had statistically reliable negative relations between real stock returns and measures of expected and unex-pected inflation.Gultekin (1983), on the contrary, conducted an empirical research to investigate the relation between stock returns and inflation in 26 countries observing that countries with bigger rates of inflation generally have bigger nominal stock returns, thus leading to a positive correlation between inflation and stock returns.Balduzzi (1995)reexamined the proxy hypothe-sis ofFama (1981)as the main explanation for the negative corre-lation between stock returns and inflation. This paper used quarterly data on industrial-production growth, monetary-base growth, CPI inflation, three-month Treasury-bill rates, and returns on the equally-weighted NYSE portfolio, for the 1954–1976 and 1977–1990 periods. Using time-series techniques, he found that production growth induced only a weak negative correlation be-tween inflation and stock returns, and explained less of the covari-ance between the two series than inflation and interest-rate innovations. However,Park and Ratti (2000)found that monetary policy tightens shocks generated statistically significant move-ments in inflation and expected real stock returns, and that these movements go in opposite directions, even though Kim and In (2005) in their empirical results showed that there is a positive relationship between stock returns and inflation at the shortest scale (1-month period) and at the longest scale (128-month peri-od), while a negative relationship is shown at the intermediate scales.

In the related literatures of interest rate,Geske and Roll (1983), according to their empirical research, declared that stock market returns are negatively associated with nominal interest rates.

Kim and Wu (1987)discovered that a factor characterized by inter-est rate and money supply was one of three very significant factors to explain stock returns. Furthermore, based on the empirical find-ings ofDomian et al. (1996), drops in interest rates are followed by twelve months of excessive stock returns, while increases in inter-est rates have little effect. Abugri (2008) observed that interest rates and exchange rates are significant in three out of the four markets examined to explain market returns. As for the literatures of industrial production, Chen et al. (1986) found that several

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macro-variables were signiﬁcant in explaining expected stock re-turns, which included industrial production. Besides, growth in ex-pected real activity, such as industrial productivity has been found byFama (1990)as well asFerson and Harvey (1998)to be posi-tively related to stock returns.

By reviewing the above literatures, we observe that previous lit-eratures mostly discussed changes in stock returns through macro-economic factors such as money supply, inﬂation, interest rate, and industrial production. Besides, the conclusions are inconsistent in that the direction of these researches generally concentrates on the exploration between the correlation of variables and stock re-turns as well as unidirectional explanatory power. Furthermore, in terms of the above macroeconomic variables, the message to investors is vague, since we only comprehend the inﬂuence of a macroeconomic variable on stock returns as positive or negative, but not the impact on separate stock returns and relative weight, which is very important for investors to make the optimal decision. 2.2. Criteria of expected stock return for portfolio selection

Sharpe (1964), Lintner (1965), andMossin (1966)proposed the CAPM revealing that risk-free rate, expected market return, and beta of the security have an inﬂuence on expected stock return, and a reasonable expectation of stock return could be obtained by the summation of risk-free rate and the return of bearing sys-tematic risk.Fama and MacBeth (1973)showed that the average return on a portfolio of stocks was positively related to the beta of the portfolio; namely, expected stock return was affected by risk-free rate, expected market return, and beta of the security, a ﬁnding consistent with the CAPM. However, the model only noted that three critical factors impacting expected stock return but not the detailed ingredients. Consequently, this research tried to point out the sub-factors of the three factors and set up a pricing scale for portfolio selection.

The so-called risk-free rate is the assumption that all investors can have unlimited lending and borrowing under the risk-free rate of interest in a financial market. We, therefore, identify the sub-factors of risk-free rates by exploring the related literature of inter-est rate.Cebula, Hung, and Manage (1992)investigated the impact of federal budget deficits on nominal long-term interest rates in the United States. It was found that the federal deficits had positive and significant impact on the long-term rates of interest during the period 1955–1985. Hence, budget deficit is one of the important factors of risk-free rate (Cebula, 1998; Knot & de Haan, 1999). In addition,Thornton (1986)intended to clarify the relationship be-tween the Federal Reserve’s discount rate and market interest rates. The evidence showed that a statistically significant effect of a change in the discount rate on both the federal funds and Trea-sury bill rates immediately following the discount rate change (Rai, Seth, & Mohanty, 2007; Thornton, 1998). Besides,Pi-Anguita (1998)measured capital mobility in France by analyzing the direc-tion of causality between the real exchange rate and the interest rate. Cointegration and Granger causality tests showed that the direction of causality between the two variables reversed in 1987, the date at which capital controls started to be lifted in France (Chow & Kim, 2006; Nakagawa, 2002).

On the other hand,Madura, Tucker, and Wiley (1997)reported the results of an investigation of factors hypothesized to explain dif-ferences in mean returns across entire national stock markets. Un-like most studies focused solely on US stocks, the study did not ﬁnd a beta or size effect in the assessment of national stock market movements. The most relevant factor for explaining disparate re-turns across markets is country risk (Rouwenhorst, 1999; Serra, 2000). Moreover, Stock Price Indices, according toRoll (1992), were compared across countries in an attempt to explain why they exhib-ited such disparate behavior. Each country’s industrial structure was

empirically documented as one of the explanatory influences and played a major role in explaining market return (Hong, Torous, & Valkanov, 2007; Serra, 2000). And then,Hooker (2004)investigated the predictive power of several candidate macroeconomic factors for emerging market equity returns. The results provided strong sup-port for several financial factors as significant predictors of excess re-turns (Abugri, 2008; Nikkinen, Omran, Sahlström, & Äijö, 2006).

Beta of a security is the instrument of measurement of system-atic risk. Therefore, by reviewing the associated literatures of sys-tematic risk, we point out the sub-factors of beta of a security.

Rosenberg and McKibben (1973)tried to combine both accounting data for the firm and the previous history of stock prices to provide efficient predictions of the probability distribution of returns. They predicted two parameters of the distribution of returns for each security in each year: the response to the overall market return (b), and the variance of the part of risk, specific to the security, that was uncorrelated with the market return. Results showed that the method proposed for the analysis of specific risk seemed to have been highly successful. The estimated relationships were signifi-cant, and the signs of all significant coefficients corresponded with a prior intuition. That is to say, beta of a security is influenced by firm-specific risk (Cai, Faff, Hillier, & Mohamed, 2007; Lee & Jang, 2007). Besides,Hamada (1972)attempted to tie together some of the notions associated with the field of corporation finance with those associated with security and portfolio analyses. The outcome presented that approximately 21–24% of the observed systematic risk of common stocks can be explained merely by the added finan-cial risk taken on by the underlying firm with its use of debt and preferred stock. Corporate leverage did count considerably. Conse-quently, financial risk is one of the sub-factors of beta of a security (Faff, Brooks, & Kee, 2002; Patel & Olsen, 1984).

Based on the CAPM, three factors (dimensions) impact on ex-pected stock return: (1) risk-free rate, (2) exex-pected market return, and (3) beta of the security. In addition, reviewing literature shows that risk-free rate is affected by three criteria: budget deficit, dis-count rate, and exchange rate; expected market return is affected by three criteria: country risk, industrial structure, and macroeco-nomic factors; and beta of the security is affected by two criteria: firm-specific risk and financial risk which are interpreted inTable 1. 3. A novel MCDM model with DEMATEL technique

As any criterion may impact each other, this study used the DEMATEL technique to acquire the structure of the MCDM prob-lems. The weights of each criterion from the structure are obtained by utilizing the ANP. The VIKOR technique will be leveraged for calculating compromise ranking and gap of the alternatives. In short, the framework of evaluation contains three main phases: (1) constructing the network relation map (NRM) among criteria by the DEMATEL technique, (2) calculating the weights of each cri-terion by the ANP based on the NRM, and (3) ranking or improving the priorities of alternatives of portfolios through the VIKOR. 3.1. The DEMATEL for constructing a NRM

The DEMATEL method (Gabus & Fontela, 1972; Ou Yang, Shieh, Leu, & Tzeng, 2008) was utilized to investigate the interrelations among criteria to build a NRM. The technique has been successfully applied in many situations, such as vehicle telematics system, development strategies, management systems, e-learning evalua-tions, and knowledge management (Lin, Hsieh, & Tzeng, (2010); Lin & Tzeng, 2009; Tsai & Chou, 2009; Tzeng, Chiang, & Li, 2007; Wu, 2008). The method can be arranged as follows:

Step 1: Obtain the direct-inﬂuence matrix by scores. Respondents are required to point out the degree of direct inﬂuence among

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each criterion. We suppose that the comparison scales, 0, 1, 2, 3 and 4, stand for the levels from ‘‘no inﬂuence” to ‘‘very high inﬂuence”. Then, the graph which can describe the interrela-tionships between the criteria of the system is shown in

Fig. 1. For instance, an arrow from w to y symbolizes that w impacts on y, and the score of inﬂuence is 1. The direct-inﬂu-ence matrix, A, can be derived by indicated one criterion i impact on another criterion j as aij.

Step 2: Calculate the normalized direct-inﬂuence matrix S. S can be calculated by normalizing A through Eqs.(1) and (2).

S ¼ m A ð1Þ m ¼ min 1 max i Pn j¼1jaijj ; 1 max j Pn i¼1jaijj 2 4 3 5 ð2Þ

Step 3: Derive the total direct-inﬂuence matrix T. T of NRM can be derived by using a formula(3), where I denotes the identity matrix; i.e., a continuous decrease of the indirect effects of problems along the powers of S, e.g., S2_{, S}3_{, . . . , S}q _and limq!1Sq¼ ½0nn, where S ¼ ½sijnn, 0 6 sij< 1 and 0 6Pisij or P

jsij< 1 only one column or one row sum equals 1, but not all. The total-inﬂuence matrix is listed as follows:

T ¼ S þ S2þ þ Sq

¼ SðI þ S þ S2þ þ Sk1ÞðI SÞðI SÞ1 ¼ SðI SqÞðI SÞ1

when q ? 1, Sq_{= ½0} nn, then

T ¼ SðI SÞ1 ð3Þ

where T = ½tijnn, i, j = 1, 2, . . . , n.

Step 4: Construct the NRM based on the vectors r and c. The vectors r and c of matrix T represent the sums of rows and col-umns respectively, which are shown as Eqs.(4) and (5).

r ¼ ½rin1¼ Xn j¼1 tij " # n1 ð4Þ d ¼ ½dj_n1¼ Xn i¼1 tij " #0 1n ð5Þ where ridenotes the sum of the ith row of matrix T and displays the sum of direct and indirect effects of criterion i on another cri-teria. Also, djdenotes the sum of the jth column of matrix T and represents the sum of direct and indirect effects that criterion j has received from another criteria. Moreover, when i = j(ri+ di), it presents the index of the degree of inﬂuences given and re-ceived; i.e. (ri+ di) reveals the strength of the central role that factor i plays in the problem. If (ri di) is positive representing that other factors are impacted by factor i. On the contrary, if (ri di) is negative, other factors has inﬂuences on factor i and thus the NRM can be constructed (Huang, Shyu, & Tzeng, 2007; Liou, Tzeng, & Chang, 2007;Tzeng et al., 2007).

3.2. The ANP for calculating weights of criteria based on the NRM The analytic hierarchy process (AHP) supposes independence among criteria, which is not reasonable in the real world.Saaty (1996)thus extended AHP to ANP to resolve problems with depen-dence or feedback between criteria, which primarily divides prob-lems into numerous different clusters and every cluster includes multiple criteria. Moreover, there is outer dependence among clus-ters and inner dependence within the criteria of clusclus-ters as illus-trated in Fig. 2. In addition, we ﬁgured the relative weights of Table 1

Explanation of criteria.

Dimensions Evaluation criteria

Descriptions Proposed scholars

Risk-free rate (D1) Budget deﬁcit

(C1)

It occurs when a government intends to spend more money than it takes in Cebula et al. (1992), Cebula (1998), and Knot and Haan (1999)

Discount rate (C2) An interest rate that a central bank charges depository institutions that

borrow reserves from it

Thornton (1986, 1998), andRai et al. (2007)

Exchange rate (C3)

Between two currencies indicates how much one currency is worth in terms of the other

Pi-Anguita (1998), Nakagawa (2002), andChow and Kim (2006)

Expected market return (D2)

Country risk (C4) The probability that changes in the business environment adversely affect

the value of assets in a speciﬁc country

Madura et al. (1997), Rouwenhorst (1999), and

Serra (2000)

Industrial structure (C5)

Different ﬁnancial markets have various industrial structures Roll (1992), Serra (2000), andHong et al. (2007)

Macroeconomic factors (C6)

It deals with the performance, structure, and behavior of a national economy as a whole

Hooker (2004), Nikkinen et al. (2006), and

Abugri (2008)

Beta of the security (D3)

Firm-speciﬁc risk (C7)

A risk that affects a very small number of assets Rosenberg and McKibben (1973), Cai et al. (2007), andLee and Jang (2007)

Financial risk (C8) Any risk associated with any form of ﬁnancing Hamada (1972), Patel and Olsen (1984), and Faff et al. (2002)

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criteria of respective matrices by pair-wise comparison and modi-fying the weights as eigenvectors. Then we integrated multiple matrices into a supermatrix, because the capacity to examine the inner and outer dependence of clusters is the largest beneﬁt of a supermatrix as Eq.(6).

ð6Þ

There are three steps for the decision process of ANP. First, the decision problem and the structure of problem were built to offer an evident depiction of the problem and separate it into a relation network structure as shown inFig. 1. Second, not only is pair-wise comparison matrix established, but also eigenvalue and eigenvec-tor were ﬁgured. Pair-wise comparison is composed of clusters and criteria. Furthermore, the pair-wise comparison of clusters was separated into comparison of criteria within and between clusters. We utilize ratio scale (1–9) to determine the level of importance of the comparison. In addition, the data deriving from the survey of ANP were combined and transferred into pair-wise comparison matrix by geometric average. After building the matrix, we re-ceived the eigenvector Wii through an equation: Aw ¼ kmaxw, where A is pair-wise comparison matrix, w ¼ ðw1; . . . ;wi; . . . ;wnÞ0 is the eigenvector, wiis the eigenvalue, then

kmax¼ 1 n Xn i¼1 ðAwÞi wi where ðAwÞi¼ Pn

j¼1aijwjand n equals the number of comparative criteria. Third, the supermatrix, tagged W (as shown in Eq.(6)),

was formed. It was constructed by the dependence table obtained from the interrelations among criteria, and the eigenvectors re-ceived from the pair-wise comparison matrix served as the weights of it. No inner dependence among criteria or clusters was shown by a blank or zero. ByWu and Lee (2007), the usage of power matrix by Wh_{(multiplication) and lim}

h?1Whis a ﬁxed convergence value; therefore, we can acquire weights in every criterion.

3.3. The VIKOR for ranking and improving the alternatives

Opricovic (1998) proposed the compromise ranking method

(VIKOR) as one applicable technique to implement within MCDM. Suppose the feasible alternatives are represented by A1, A2, . . . , Ak, . . . , Am. The performance score of alternative Akand the jth cri-terion is denoted by fkj; wjis the weight (relative importance) of the jth criterion, where j = 1, 2, . . ., n, and n is the number of crite-ria. Development of the VIKOR method began with the following form of Lp-metric: Lpk¼ Xn j¼1 ½wjðjfj fkjjÞ=ðjfj f j jÞ p ( )1=p

where 1 6 p 6 1; k = 1, 2, . . ., m; weight wjis derived from the ANP. To formulate the ranking and gap measure Lp¼1_k (as Sk) and Lp¼1k (as Qk) are used by VIKOR (Opricovic, 1998; Opricovic & Tzeng, 2002,

2004, 2007; Tzeng, Lin, & Opricovic, 2005; Tzeng, Teng, Chen, & Opricovic, 2002). Sk¼ Lp¼1k ¼ Xn j¼1 ½wjðjfj fkjjÞ=ðjfj fjjÞ Qk¼ Lp¼1k ¼ max_j fðjf j fkjjÞ=ðjfj fjjÞjj ¼ 1; 2; . . . ; ng

The compromise solution minkLpkshows the synthesized gap to be the minimum and will be selected for its value to be the closest to the aspired level. Besides, the group utility is emphasized when p is small (such as p = 1); on the contrary, if p tends to become inﬁ-nite, the individual maximal regrets/gaps obtain more importance in prior improvement (Freimer & Yu, 1976; Yu, 1973) in each dimension/criterion. Consequently, minkSkstresses the maximum group utility; however, minkQkaccents on the selecting the mini-mum from the maximini-mum individual regrets/gaps. The compromise ranking algorithm VIKOR has four steps according to the above-mentioned ideas.

Step 1: Obtain an aspired or tolerable level. We calculate the best f

j values (aspired level) and the worst fjvalues (tolerable level) of all criterion functions, j = 1, 2, . . . , n. Suppose the jth function denotes beneﬁts: f

j ¼ maxkfkj and fj¼ minkfkj or these values can be set by decision makers, i.e., f

j is the aspired level and f

j is the worst value. Further, an original rating matrix can be converted into a normalized weight-rating matrix by using the equation: rkj¼ ðjfj fkjjÞ=ðjfj fjjÞ.

Step 2: Calculate mean of group utility and maximal regret. The values can be computed respectively by Sk¼Pnj¼1wjrkj(the syn-thesized gap for all criteria) and Qk¼ maxjfrkjjj ¼ 1; 2; . . . ; ng (the maximal gap in k criterion for prior improvement). Step 3: Calculate the index value. The value can be counted by Ri¼

v

ðSk SÞ=ðS SÞ þ ð1

v

ÞðQk QÞ=ðQ QÞ, where k = 1, 2, . . . , m. S

¼ miniSi or setting S*= 0 and S= maxiSior setting S_{= 1; Q}*_{= min}

iQior setting Q*= 0 and Q= maxiQior setting Q¼ 1; and

v

is presented as the weight of the strategy of the maximum group utility.

Step 4: Rank or improve the alternatives for a compromise solu-tion. Order them decreasingly by the value of Sk, Qkand Rk. Pro-pose as a compromise solution the alternative (A(1)_{) which is} arranged by the measure min{Rk|k = 1, 2, . . . , m} when the two

Cluster 1

Element 1 Element 2 Element 3

Cluster 3

Cluster 2

Alternative

Case 1 Case 2 Case 3

Cluster 4

Innerdependence

Outerdependence

Feedback

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conditions are satisﬁed: C1. Acceptable advantage: R(A(2)₎ R(A(1)_{) P 1/(m 1), where A}(2) _{is the second position in the} alternatives ranked by R. C2. Acceptable stability in decision mak-ing: Alternative A(1)_{must also be the best ranked by S}

kor/and Qk. When one of the conditions is not satisfied, a set of compro-mise solutions is selected. The comprocompro-mise solutions are com-posed of: (1) Alternatives A(1)_{and A}(2)_{if only condition C2 is} not satisfied, or (2) Alternatives A(1)_{, A}(2)_{, . . . , A}(M)_{if condition} C1 is not satisfied. A(M) _{is calculated by the relation} R(A(M)_{) R(A}(1)_{) < 1/(m 1) for maximum M (the positions of} these alternatives are close).

The compromise-ranking method (VIKOR) is applied to deter-mine the compromise solution and the solution is adoptable for decision-makers in that it offers a maximum group utility of the majority (shown by min S), and a maximal regret of minimum indi-viduals of the opponent (shown by min Q). This model utilizes the DEMATEL and ANP processes in Sections3.1 and 3.2 to get the weights of criteria with dependence and feedback and employs the VIKOR method to acquire the compromise solution.

4. Empirical case – using semiconductor portfolio as an example

In this section, an empirical study is displayed to illustrate the application of the proposed model for evaluating and selecting the best portfolio.

4.1. Background and problem descriptions

Stock markets have signiﬁcant impact on most of the people in the world. Taiwan especially, there are around eight million ac-counts of security meaning that one invests in the stock market among three people; investment of stocks thus becomes an impor-tant tool when managing one’s capital (Taiwan Stock Exchange Corporation). However, choosing portfolios with stable growth and booming prospects in the gradually larger stock market is like ﬁshing for a needle in the ocean. Moreover, there are many factors that investors concern the most affecting stock returns; conse-quently, it is a tough problem for investors to evaluate and select

Table 2

The initial inﬂuence matrix A for criteria.

Criteria C1 C2 C3 C4 C5 C6 C7 C8 C1 0.000 2.267 2.133 2.933 2.000 3.800 1.800 1.533 C2 1.733 0.000 2.867 1.933 2.467 2.733 1.067 2.867 C3 2.000 2.333 0.000 2.200 2.467 2.933 2.467 2.467 C4 3.067 2.400 2.800 0.000 2.467 2.600 1.667 1.933 C5 1.267 1.467 1.933 1.867 0.000 3.067 2.267 2.200 C6 2.933 3.267 3.267 2.667 3.000 0.000 2.400 3.067 C7 1.000 1.067 1.067 1.067 1.267 1.133 0.000 2.867 C8 1.067 1.467 1.200 1.200 1.333 1.533 2.533 0.000 Table 3

The normalized direct-inﬂuence matrix S for criteria.

The total-inﬂuence matrix T for criteria.

The total-inﬂuence matrix T for dimensions.

Dimensions D1 D2 D3

D1 3.022 3.631 2.304

D2 3.480 3.385 2.418

D3 1.258 1.360 0.936

Table 6

The sum of inﬂuences given and received on dimensions.

ri di ri+ di ri di D1 8.956 7.760 16.716 1.197 D2 9.282 8.376 17.658 0.907 D3 3.554 5.658 9.212 2.103 Risk-free rate Expected market return Beta of the security Budget deficit Discount rate Exchange rate Country risk Industrial structure Macroeconomic Firm-specific risk Financial risk Portfolio of IC design Portfolio of wafer foundry Portfolio of IC packaging

Fig. 3. The impact NRM of investment decision.

Table 7

The novel unweighted supermatrix.

Criteria C1 C2 C3 C4 C5 C6 C7 C8 C1 0.252 0.324 0.338 0.482 0.216 0.235 0.212 0.147 C2 0.368 0.267 0.374 0.195 0.193 0.229 0.209 0.317 C3 0.38 0.408 0.288 0.323 0.59 0.536 0.579 0.536 C4 0.201 0.158 0.162 0.243 0.321 0.333 0.117 0.138 C5 0.368 0.182 0.137 0.355 0.254 0.362 0.316 0.219 C6 0.431 0.661 0.701 0.402 0.426 0.305 0.567 0.643 C7 0.61 0.268 0.592 0.468 0.514 0.471 0.338 0.584 C8 0.39 0.732 0.408 0.532 0.486 0.529 0.662 0.416

Note: the novel ANP has the relationship of feedback, so the diagonal matrix was derived by normalizing the diagonal matrix of the total-inﬂuence matrix from DEMATEL.

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portfolios to maximize their returns. In order to make investors know what the criteria of portfolio selection are, this study thus explores the criteria in the experts’ point of view and constructs an investment decision model. In addition, because Taiwan is a leading country in the industry of semiconductor in the world, this research picked the most suitable portfolio from semiconductor companies spanning the hottest sectors of (IC) design, wafer foun-dry, and IC packaging to offer investors with a reference of portfo-lio selection.

4.2. Data collection

The experts with professional knowledge of finance and spe-cialty of investment were the objects of this research, comprising consultants of investment, scholars of finance, and managers of mutual funds. Furthermore, the background of experts are de-scribed as follows: consultants of investment are good at security analysis in financial holding companies and institutions of invest-ment, scholars of finance are those who have the specialty of man-agement of investment and the experience of teaching financial courses in a university, and managers of mutual funds are in charge

of managing and investing the capital of customers in a mutual fund management company. Experts’ perspectives on the diverse criteria and the performance of every portfolio within the criteria were received by personal interviews and filling out the question-naires. A total of 15 objects were divided into five consultants of investment, five scholars of finance, and five managers of mutual funds. This investigation was carried out in April 2009, and it took 40–80 min for every expert to fill out the questionnaire and to be interviewed.

4.3. Constructing the NRM by DEMATEL

To analyze the interrelationships between the eight determi-nants summarized through literatures, the DEMATEL method introduced in Section3.1will be utilized in the decision problem structure. First, the direct-influence matrix A for criteria was pre-sented (seeTable 2). Then, the normalized direct-influence matrix S for criteria can be calculated by Eq.(1)(seeTable 3). Third, the total direct-influence matrix T for criteria/dimensions was derived based on Eq.(3)(seeTables 4 and 5). Finally, the NRM was con-structed by the r and d in the total direct-influence matrix T (see

Table 6) as shown inFig. 3.

4.4. Calculating weights of each criterion by ANP

The primary survey targets included investment consultants, financial scholars, and managers of mutual funds. The level of importance (global weights) of eight criteria can be calculated by ANP shown asTables 7–10. Results showed that experts were most concerned with macroeconomic factors and exchange rate, and least concerned with budget deficit and country risk. Research find-ings indicated that the level of importance was higher in macroeco-nomic factors, exchange rate, and firm-specific risk. Specifically, macroeconomic factors scored the highest at 0.200, followed by ex-change rate at 0.165, and firm-specific risk at 0.127. The level of importance assessed for budget deficit and country risk was rela-tively lower, and the two criteria averaged 0.088. From the stand-point of dimensions, experts considered exchange rate relatively most important among the three criteria of ‘‘risk-free rate”. Among the three criteria of the ‘‘expected market return” dimension, ex-perts found ‘‘macroeconomic factors” as important. As for the ‘‘beta of the security” dimension, experts considered ‘‘financial risk” as important. In contrast, experts considered ‘‘firm-specific risk” as less important. This finding revealed that the experts believed mac-roeconomic factors could not be overlooked by investors when Table 8

Weighting the unweighted supermatrix based on total-inﬂuence normalized matrix.

The stable matrix of ANP when power limit h ? 1 (ANP).

The weights of criteria for evaluating portfolio and total performance (SAW method).

Dimensions/criteria Local weight Global weight (by ANP) Portfolio of IC design (A1) Portfolio of wafer foundry (A2) Portfolio of IC packaging (A3)

Risk-free rate (D1) 0.355 7.437 8.457 8.443

Budget deﬁcit 0.265 0.094(7) 6.933 6.467 6.667

Discount rate 0.270 0.096(6) 6.867 7.867 7.733

Exchange rate 0.465 0.165(2) 8.133 9.933 9.867

Expected market return (D2) 0.384 6.270 7.197 6.714

Country risk 0.211 0.081(8) 5.733 5.933 5.867

Industrial structure 0.268 0.103(5) 7.867 7.667 7.733

Macroeconomic factors 0.521 0.200(1) 5.667 7.467 6.533

Beta of the security (D3) 0.261 7.555 7.188 7.569

Firm-speciﬁc risk 0.487 0.127(4) 9.333 7.667 7.467

Financial risk 0.513 0.134(3) 5.867 6.733 7.667

Total performance 7.032(3) 7.642(1) 7.551(2)

Example:

Calculating total performance by global weights:

0:094 6:933 þ 0:096 6:867 þ 0:165 8:133 þ 0:081 5:733 þ 0:103 7:867 þ 0:200 5:667 þ 0:127 9:333 þ 0:134 5:867 ¼ 7:032. Calculating total performance by local weights:

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picking portfolios. Also, relative to the dimension of beta of the security, experts were less concerned because the mean of the dimension was substantially lower than others. Besides, the syn-thesized scores were then calculated to derive the total perfor-mance as illustrated in Table 10. Results showed that the total performance was highest in portfolio of wafer foundries, followed by portfolio of IC packaging and IC design. Therefore, the invest-ment decision model provided by this study indicated that inves-tors are suggested to invest in a portfolio of wafer foundries. 4.5. Compromise ranking by VIKOR

The VIKOR technique was applied for compromise ranking after the weights of determinants was calculated by ANP in Section4.4.

Calculation results (Table 11) demonstrated that the total gaps were highest in portfolio of IC design, followed by portfolio of IC packaging and IC design. Therefore, both VIKOR and ANP came to the same conclusions that the investment decision model provided by this study indicated that investors are suggested to invest in a portfolio of wafer foundries.

4.6. Implications and discussion

The empirical results were discussed as follows. In the ﬁrst place, the most important criterion calculated by ANP when mak-ing investment decisions was macroeconomic factors weightmak-ing 0.200. If the macroeconomic condition is bad, the demand of most industries will decrease, not to mention the semiconductor Table 11

The weights of criteria for evaluating portfolio and total performance (VIKOR method).

Dimensions/criteria Local weight Global weight (by ANP) Portfolio of IC design (A1) Portfolio of wafer foundry (A2) Portfolio of IC packaging (A3)

Risk-free rate (D1) 0.355 0.253 0.154 0.156

Budget deﬁcit 0.265 0.094(7) 0.307 0.353 0.333

Discount rate 0.270 0.096(6) 0.313 0.213 0.227

Exchange rate 0.465 0.165(2) 0.187 0.007 0.013

Expected market return (D2) 0.384 0.373 0.280 0.329

Country risk 0.211 0.081(8) 0.427 0.407 0.413

Industrial structure 0.268 0.103(5) 0.213 0.233 0.227

Macroeconomic factors 0.521 0.200(1) 0.433 0.253 0.347

Beta of the security (D3) 0.261 0.245 0.281 0.243

Firm-speciﬁc risk 0.487 0.127(4) 0.067 0.233 0.253

Financial risk 0.513 0.134(3) 0.413 0.327 0.233

SA1 Total gaps 0.297(3) 0.236(1) 0.245(2)

QA1 Maximal gaps 0.433(3) 0.407(1) 0.413(2)

Example:

Calculating dimension gap by dimensions of local weights: SD1¼ d p¼1 D1 ¼ P3 j¼1wDj1 fD1 j fkjD1 fD1 j fjD1 ¼ 0:265 106:933 100 þ 0:270 106:867 100 þ 0:465 108:133 100 ¼ 0:253. QD1¼ d p¼1 D1 ¼ 0:313.

Calculating total gap by criteria of global weights: SA1¼ d p¼1 A1 ¼ P8 j¼1wj f jfA1 j f jfj ¼ 0:094 106:933 100 þ 0:096 106:867 100 þ 0:165 108:133 100 þ 0:081 105:733 100 þ 0:103 107:867 100 þ 0:200 105:667 100 þ 0:127 109:333 100 þ 0:134 105:867 100 ¼ 0:297. QA1¼ d p¼1 A1 ¼ max f jfA1 j f jfjjj ¼ 1; . . . ; n ¼ 0:433. 1 2 10 15 25 i i r+d D1 (16.716, 1.197),

gaps:

D =0.253, A₁ D =0.154, A2 DA3=0.156 Risk-free rate D2 (17.658, 0.907), gaps:D =0.373,A₁ D =0.280,A2 D =0.329 A3

Expected market return i i

r−d

0

20

D3 (9.212, -2.103), gaps:D =0.245,A₁ D =0.281,A2 D =0.243 A3

Beta of the security -1

-2

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industry. Therefore, an economic recession will lead to the returns of portfolios of semiconductor to be lower than during prosperous macroeconomic conditions. On the contrary, the returns of portfo-lios of semiconductor will be higher when the macroeconomic con-dition thrives. The macroeconomic criterion was the most signiﬁcant factor when considering portfolio selection. Secondly, an exchange rate weighting 0.165 was ranked number 2 of the eight criteria. It is a critical criterion of returns of portfolio espe-cially in an island country, such as Taiwan, because exports play a very crucial role in the Gross National Product (GNP) in this kind of country. The returns of portfolio, whose products are for export, will become higher when the currency of the country depreciates. However, the returns of portfolio of exports will go down when the money of the country appreciates. Thirdly, the compromise rank-ing by VIKOR showed that the portfolio of wafer foundries is the best investment target among the industry of semiconductor fol-lowed by IC packaging and design. It is very sensible in that the output value of wafer foundries and IC packaging of Taiwan is one of the largest in the whole world. The wafer foundry industry in particular has been anchoring the economy of Taiwan. Fourthly,

Fig. 4showed that the dimension of risk-free rate was the primary factor that people should improve ﬁrst when the returns of portfo-lios became bad. That was why the central banks of most countries adjusted their risk-free rate immediately when a ﬁnancial crisis broke out.

The portfolio selection model provided by this study can be used in most of the countries of the world. There are some differ-ences that investors should keep in mind when applying this model: the level of importance of the eight criteria could be varied according to the situations of the country and investors can select the portfolios that they want to invest in and compare them and then make the optimal investment decision.

5. Conclusions and remarks

The CAPM is used all over in the financial field as an important reference in evaluating stock returns. Mathematical models have demonstrated that risk-free rate, expected market return, and beta of the security are influential of stock returns. However, it is uncer-tain how the sub-factors impact these three factors. Moreover, the level of importance of these three factors for estimating stock re-turns is also not mentioned, even though the understanding of the importance of these factors and sub-factors can be beneficial for investors when selecting their portfolios.

Because the factors possessed interrelations and self-feedback relationships proven by the DEMATEL technique, ANP was then utilized to calculate each weight of the eight criteria. Empirical re-sults presented that macroeconomic factors were rated number 1, followed by exchange rate, firm-specific risk, financial risk, indus-trial structure, discount rate, budget deficit, and country risk. Though investors have to take into account the effect of all factors when making decisions of portfolio selection, experts noted that macroeconomic factors should be given the most important weight. Therefore, when considering portfolios selection, investors can look more closely in this direction. In the standpoint of portfo-lio selection, the consolidated score of the eight criteria for evalu-ation was the highest for portfolio of wafer foundries, followed by portfolio of IC packaging as well as IC design, and the result was consistent with the VIKOR method. As a consequence, experts found that a portfolio of wafer foundries in the semiconductor industry was the most suitable for investment, and this result was coherent with the real-world situations of the samples in this study.

Previous models that studied stock returns were mainly focused on macroeconomic variables and the identiﬁcation of factors that

influenced stock returns. However, few theoretical models were derived from mathematical equations and a description of more detailed factors was not accessible. This study identified the factors and sub-factors that were found to be influential in stock returns by reviewing literature to construct a theoretical model based on the CAPM, using the novel MCDM to explore the relationships between factors and sub-factors, and surveying the views of experts for optimal portfolio selection. Combining the practical experience of experts with the mathematic model assists the CAPM becoming a more helpful model for investors to make investment decisions. Furthermore, these features are not provided by preced-ing studies. To sum up, this study has integrated the CAPM and no-vel MCDM to discuss the issues of stock returns and portfolio selection, and further studies can utilize the framework to solve the problems with multiple criteria.

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