Int. J. Production Economics 134 (2011) 58–66
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Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
Two-step multi-criteria model for selecting optimal portfolio
Branka Marasovic´ n, Zoran Babic´
Faculty of Economics, University of Split, Matice hrvatske 31, 21 000 Split, Croatia article info abstract
Article history: In spite of a large number of multi-criteria models applied to solve the problem of optimal portfolio Received 31 December 2009 selection and a large number of market criteria and accounting criteria proposed for these models, the Accepted 28 April 2011 problem of portfolio containing securities from different industries has not yet been adequately solved. Available online 8 May 2011 Namely, neither can stocks of companies from different industries be compared using the same criteria Keywords: nor can the weight of a particular criteria be equal for them all. Therefore this paper develops a new Financial risk management two-step model that will overcome the shortcomings of the previously used models. The model is Portfolio selection divided into two different but related pillars: the choice of different industries to form the overall Multi-criteria portfolio and the choice of portfolio for each industry. The multi-criteria model used in this paper is a PROMETHEE modified multi-criteria programming model based on the PROMETHEE II approach. The selected model has been applied at the Zagreb Stock Exchange (ZSE) as a real case. & 2011 Elsevier B.V. All rights reserved.
1. Introduction problem of quadratic programming which consists of minimizing risk while keeping in mind an expected return which should be Several traumatic events (terroristic attacks, natural disasters, guaranteed. business scandals, bank and corporate failures, etc.) over the past The importance of Markowitz’s work is affirmed by the Nobel decade have prompted corporate, government, and non-profit Prize for Economics he won in 1990. However, parallel to organizations to embrace enterprise risk management (ERM). Last introducing the Markowitz model in the common usage its few years, that trend has been recognized by the investment limitations and drawbacks were being noticed. The assumptions industry and now it takes into account a broader, enterprise risk of the Markowitz model for portfolio optimization are the perspective. The development and current status of ERM is following: presented in papers (Wu and Olson, 2010a, b). Although today’s investment industry leaders have accepted the necessity of a utility function which presents the investor’s preferences is a managing different types of risk, financial risk remained the most quadratic function and influential risk in operations in the financial industry. Portfolio the returns have normal distribution. selection models are usually a must in the process of diagnosing risk exposures (Wu and Olson, 2010c). So, that is the reason why These assumptions were the starting point for many critics of we are focused (in this paper) on this important financial risk this model. The majority of the empirical tests on the capital management tool. markets resulted in asymmetrical and (or) leptokurtic distribu- The first model for portfolio optimization has been developed tion (Cloquette et al., 1995). In such distributions, variance is not in 1952 by H.M. Markowitz and with that model he laid the an adequate risk measure. Having recognized the drawbacks of foundation of the modern portfolio theory. His model is based variance as risk measure, new models for the selection of optimal upon only two criteria: return and risk (Markowitz, 1959). The portfolio which use alternative measures, like lover partial risk risk is measured by the variance of returns’ distribution. Marko- measures, Value-at-Risk, and Conditional Value-at-Risk, have witz shows how to calculate portfolio which has the highest been developed (Konno et al., 2002; Rockafaller and Uryasev, expected return for a given level of risk, or the lowest risk for a 2000; Yau et al., in press). Generally it is obvious that over the given level of expected return (the so-called efficient portfolio). past 10 years the field of financial risk management and enter- The problem of portfolio selection, according to this theory, is a prise risk management have experienced a fast and advanced growth at an incredible speed (Wu and Olson, 2008, 2009a, b). However, contrary to the expectations of the modern portfolio
n theory, the tests carried out on a number of financial markets Corresponding author. Tel.: þ385 21 430 600; fax: þ385 21 430 701. E-mail addresses: [email protected] (B. Marasovic´), have revealed the existence of other indicators, besides return [email protected] (Z. Babic´). and risk, important in portfolio selection. Considering the
0925-5273/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2011.04.026 B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66 59 importance of variables other than return and risk, selection of the PROMETHEE II method, which has been done in the Section 3. the optimal portfolio becomes a multi-criteria problem which The modification is made for the all preference functions from the should be solved by using the appropriate techniques. The multi- PROMETHEE method and positive and negative flow function. criteria nature of the portfolio selection was well presented in the The selected model has been applied at the Zagreb Stock paper by Khoury et al. (1993), and at present an arsenal of multi- Exchange (ZSE) as a real case. The Zagreb Stock Exchange (ZSE) dimensional and multi-criteria methods such as factor analysis, is a major stock market in Croatia and its market value is more goal programming, AHP, ELECTRE, MINORA, ADELAIS, etc. is than sixty billion dollars. already being applied in portfolio selection (Bouri et al., 2002; This paper is organized as follows: following this introduction, Ogryczak, 2000; Zopounidis, 1999; Shing and Nagasawa, 1999). A in Section 2, we describe the two-step model. Section 3 presents review of MCDA methods applied in portfolio selection and the multi-criteria method which is a modification of the PRO- management is well presented in a paper by Zopoundis and METHEE II method. Section 4 presents its application to the Doumpos (2002). Croatian capital market. Section 5 summarizes the paper and There are a vast number of criteria that can be taken into indicates the possible directions for further research. consideration in portfolio selection and that are usually classified into two groups: accounting criteria and those based on market values. The accounting criteria are obtained analyzing audit 2. Two-step model reports, income statements, quarterly balance sheets, dividend records, sales records, etc. There are a large number of them such In portfolio selection, investors, both individual and institu- as profitability indicators, liquidity and solvency indicators, and tional ones, are governed by a number of criteria that provide an indicators of financial structure of the company. They are used by insight into the current value and an estimation of the future the analysts (or managers) to give a synthesized and clear idea value of stocks making up the portfolio, and thus also the about the firm’s financial situation. The other criteria are market portfolio itself. Such an approach to portfolio selection in which criteria which contain all the information used by the stock we are faced with a number of conflicting criteria requires the use analysts to appreciate a stock’s performance. The criteria used of adequate multi-criteria decision making methods. As we have at this level are the mean return, total risk (variance), systematic already stated in the introduction, a large number of models been risk (beta), the size measured by the stock capitalization, the PER applied in solving this problem (price earning ratio), stock liquidity, and others. The use of one In this paper, the authors present a new multi-criteria model criteria or the other depends on the manager’s attitude and that, unlike the previous models, allows for the specific features of objectives (Albadvi et al., 2007; Bouri et al., 2002). industries (e.g. characteristic multi-annual cycles in shipping, Nevertheless, in spite of a large number of both proposed seasonality in tourism, effect of weather on food industry, etc.) multi-criteria models for optimal portfolio selection and market and the market’s different perceptions of prospects for different and accounting criteria proposed in these models, the problem of industries to be taken into account when choosing the optimal selection of optimal portfolio containing stocks of companies portfolio. These facts result in deviations in the mean value of from different industries has not yet been adequately solved. some criteria in different industries. Also, the criteria which are The problem arises because the evaluation of stocks from very important in one industry can have a significantly lower different industries generally requires different criteria, and even weight in another, and there are some criteria appearing only in a when the same criteria are used they need not have the same certain industry, while in others they will be non-existent (i.e. weight for them all. their weight is zero) (e.g. criterion credits/deposits in banking). Therefore this paper sets out to develop a new two-step model To overcome the stated drawbacks of the previous models we that will overcome the drawbacks of the previously used models. have developed a new model consisting of two steps. We first The model is divided into two different but related pillars: the select a set of stocks to form an optimal portfolio and then choice of different industries to form the overall portfolio and the determine the industries to which the selected stocks belong. choice of portfolio for each industry. In the first step, we After determining the set of corresponding industries (I1, I2, ..., In) determine a set of criteria relevant in the evaluation of industries, we take the first step in which we determine the share of each and applying the multi-criteria model, we select a portfolio industry in the portfolio (y1, y2, ..., yn). To determine the share of consisting of industries, i.e. we determine the share of each each industry in the portfolio we use a multi-criteria model industry in the portfolio. In the second step, we select an optimal shown in the next section. The criteria which are basis for portfolio for each industry. Consequently, in this step each selection optimal portfolio we will call effective criteria. The industry is observed separately. This approach allows for the review of criteria most frequently used in industry evaluation is criteria to be taken into consideration in every industry not to be given in Albadvi et al. (2007). Selection of the set of effective equally weighted. criteria and determination of the weight of those criteria should A set of criteria is determined to evaluate the stocks from the be carry out in cooperation with both macroeconomic and same industry and the optimal portfolio is selected for each financial experts. The weight of effective criteria could be esti- industry applying the multi-criteria model. Finally, we calculate mated with any known method for determining the weights of weights of stocks in the optimal portfolio using the previously criteria and in this paper we propose the Saaty’s AHP method, or calculated weights of industries and weights of stocks in the more precisely, its eigenvalue procedure. portfolio of each industry. The multi-criteria model by which we Type of each criterion is Max/Min. Anyone who wants to select determine the weights of industries in the portfolio and select the the right stocks for investment has to determine investment portfolio of each industry is a modified model proposed by Bouri strategies such as the maximum investment per industry or (in Bouri et al., 2002) which is based on the PROMETHEE II company in proportion to his/her capital. These strategies are approach. Namely, since the number of portfolios that can be determined based on investor’s expected rate of return and made up from a set of pre-selected stocks is infinite we cannot degree of risk taking. Also, the type of some criteria could depend use the original version of PROMETHEE II. The PROMETHEE II is a on investment strategies of decision-maker.
MCDA method which makes a selection among a finite number of In the second step, we evaluate the stocks (Si1, Si2, ..., Siki) activities whereas in our case a portfolio is designed for an infinite within an industry Ii (i¼1, 2, ..., n). Furthermore, the effective number of possibilities. For that reason, it is necessary to modify criteria in company evaluation are chosen and by multi-criteria 60 B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66
Existing Industries in Stock Exchange
Effective Criteria in Industry Evaluation Determining share of each and weights of those industry in portfolio Investment Strategies criteria (estimated Tool: multi-criteria in Industry using the AHP programming model presented methot) in next section
Share of industries in optimal portfolio
Effective Criteria in Company Evaluation Determining share of each stock in and weights of those industry portfolio within each Investment Strategies criteria (estimated industry in Company using the AHP Tool: multi-criteria programming method) model presented in next section
Share of stocks in optimal industry portfolio
Share of stocks in optimal portfolio
Fig. 1. The two-step model for selecting optimal portfolio.
model shares of particular stocks (zi1, zi2, ..., ziki) in an optimal so the higher the net flow F(P) is, the better the alternative. industry portfolio Ii (i¼1, 2, ..., n) are determined. The shares of Positive and negative flows are calculated by pairwise compar- stocks in the industry portfolio are also obtained by the multi- isons of all the alternatives and for every criterion. criteria model shown in the next section. In this step, depending Since, in our case, the number of possible portfolios that can be on the investment strategy of the decision maker, the criteria to made up from a set of alternatives (set of industries in the first be used are also selected from the set of market and accounting step of model and set of stocks in the second step of the model) is criteria to evaluate stocks, and then their weights are determined. infinite, it is impossible to compare all pairs of portfolios. There- It is important to note that we are now allowed to attach fore, Khoury and Martel (1990), as well as Zmitri et al. (1998) different weights to the same criteria in different industries. Also, suggest a different procedure which evaluates each alternative (or in this way we can use the criteria important for the stocks of only rather its positive and negative flow) by comparison with two one industry or only for a number of similar industries (like the fiction portfolios: one ideal (P) and the other anti-ideal (P). The aforementioned criterion credits/deposits important only in the positive flow F þ(P) is then obtained by comparison with anti- evaluation of stocks in the banking industry). ideal, where the higher the Fþ (P) the better the alternative
Finally, with xij we denote the share of stock Sij i¼1, 2, ..., n; (it could be said the more distant from the anti-ideal it is). j¼1, 2, ..., ki in the optimal portfolio and it is given as Accordingly, the closer to the ideal the better the alternative, or the lesser the F (P) the better the alternative. According to the x ¼ y z ð1Þ ij i ij PROMETHEE II method, the portfolio is better if it has a higher net Fig. 1 shows the flow diagram of the two-step model. flow F. For criteria Cj, which have to be maximized, we will have
CjðPÞ¼maxCjðAiÞð3Þ 3. The multi-criteria model i
where A¼{A1, A2, ..., AN} is the set of N alternatives. In the same The multi-criteria model applied in this paper is based on the way, for the same criteria we will have PROMETHEE II approach (Brans et al., 1984). In accordance with CjðPÞ¼minCjðAiÞð4Þ the PROMETHEE II method each alternative P is evaluated with i two flows. The positive flow Fþ (P) indicates how much an alternative is better than the others (in all criteria). Accordingly, Without loss of generality we can suppose that all criteria are to the higher the Fþ (P) is, the better the alternative. The negative be maximized. flow F (P) indicates how much better than P the other alter- The set of possible solutions is the set of portfolios which can natives are, i.e. how much P is dominated by the others. Accord- be formed from observed alternatives. It is important to bear in ingly, the lesser the F (P) is, the better the alternative. mind fact that set of alternatives in the first step of model is PROMETHEE II calculates the net flow F as the difference consisted of industries and in the second step it is consisted of between these two flows, i.e. stocks within an industry. The evaluation of the portfolio P according to criterion j is obtained by multiplying the share of þ FðPÞ¼F ðPÞ F ðPÞð2Þ each alternative Ai in the portfolio with the evaluation of B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66 61 alternative i according to criterion j: Φ XN 1 CjðPÞ¼ aiCjðAiÞð5Þ i ¼ 1 where ai is the share invested in Ai in the portfolio P. Naturally,
XN () ()+ C ()P C ()P ai ¼ 1 ð6Þ CP CP p i ¼ 1 Fig. 2. Positive flow of linear preference function. For each criterion Cj (j¼1,2, ..., n) the preference functions are defined as in the PROMETHEE method where indifference (q) and preference (p) thresholds are certain numbers from the interval Φ ½0, CjðPÞ CjðPÞ , i.e. the following is applicable: 1 0rq, prCjðPÞ CjðPÞð7Þ
By analogy, if Gauss criterion is used, relation (7) is also applicable to the parameter s. Naturally, qrp is always true. Let us assume, which is in line with the economic significance of such thresholds, ( ) − ( ) ( ) C (P) C P p C P C P that pA½q,ðCjðPÞ CjðPÞÞ=2 , i.e. the highest value of preference threshold cannot exceed half the span between the lowest and Fig. 3. Negative flow of linear preference function. the highest value according to that criterion. Bouri et al. (2002) used only linear preference function from following form: the PROMETHEE and in this paper we will, in addition to that, 8 take into consideration the linear preference function with < Cj ðPÞ CjðPÞ , CjðPÞ4CjðPÞ p ð Þ¼ p ð Þ indifference threshold (type V) and Gaussian criteria (type VI), Fj P : 12 1, C ðPÞ C ðPÞ p as they most generally display the relations between pairs of j r j alternatives. The graph of that function can be seen in Fig. 3. The linear preference function from the PROMETHEE method Consequently, the better the portfolio P by j-criterion, or the (type III) has the following form: higher the Cj(P) (assuming that the criterion is maximized) the ( d lower the Fj ðPÞ is. p , 0rdop CðdÞ¼ ð8Þ Let us note that positive and negative flows defined in this 1, dZp way, unlike of PROMETHEE, are calculated separately for each where d is the difference in evaluation of the two alternatives by criterion. Finally, the net flow of the portfolio P (for every the same criterion. criterion) is calculated as the difference, i.e. In our case, when the portfolio P is compared to the anti-ideal þ FjðPÞ¼Fj ðPÞ Fj ðPÞ, ð13Þ þ ðPÞ, i.e. when we calculate Fj ðPÞ, the difference d presents the ‘‘distance’’ from the anti-ideal (by j-criterion), or the higher the d, so that we obtain the preference is closer to 1. Consequently, F ðPÞ¼ 8j C ðPÞ C ðPÞ p XN > j j j > p , CjðPÞrCjðPÞþpj > j djðPÞ¼CjðPÞ CjðPÞ¼ aiCjðAiÞ CjðPÞð9Þ <> þ i ¼ 1 0, CjðPÞþpj oCjðPÞrCjðPÞ pj ð14Þ > > p þ þ C ðPÞ C ðPÞ As for each P it is always CjðPÞZCjðPÞ, this means that it is always :> j j j þ p þ , CjðPÞ4CjðPÞ pj j dj(P)Z0. Therefore 8 C ðPÞ C ðPÞ < j j As each portfolio is compared separately to the ideal when Fj ðPÞ , 0rCjðPÞ CjðPÞop F þ ðPÞ¼ ¼ p ð10Þ is calculated, and separately to the anti-ideal when F þ ðPÞ is j : ð Þ ð ÞZ j 1, Cj P Cj P p calculated, we generally need not have the same preference thresholds in the functions of positive and negative flow. If we The higher the difference d ðPÞ¼C ðPÞ C ðPÞ the higher will be the j j j mark the preference thresholds for the positive flow with p , and value of the positive flow, or when it exceeds p it will be 1. If we j for the negative flow with p þ , the net flow by j-criterion can be want to present the positive flow as the function of the portfolio j graphically presented as in Fig. 4. P (i.e. of Cj(P)) we have 8 The linear preference function with indifference threshold < CjðPÞ CjðPÞ from the PROMETHEE method (type V) has the following form: , CjðPÞoCjðPÞþp 8 þ ðPÞ¼ p ð11Þ Fj : > 0, d q 1, CjðPÞZCjðPÞþp <> r d q , q d p CðdÞ¼> p q o r ð15Þ þ :> The graph of Fj ðPÞ is presented in Fig. 2. 1, d4p It is obvious from the figure that the positive flow is higher if þ the Cj(P) is higher (assuming that all criteria are maximized). Consequently, taking again that djðPÞ¼CjðPÞ CjðPÞ for Fj ðPÞ we Analogously, the negative flow Fj ðPÞ is considered for the obtain 8 differences (‘‘distance’’) of the portfolio P from the ideal ðPÞ. The > 0, CjðPÞrCjðPÞþqj smaller the distance dj(P) the better the portfolio is. As <> C ðPÞ C ðPÞ q þ j j j ð ÞZ ð Þ , C ðPÞþq C ðPÞ C ðPÞþp Cj P Cj P for each possible portfolio P (and for every criterion Fj ðPÞ¼ p q j j o j r j j ð16Þ > j j that has to be maximized) the difference dj(P) is defined in that :> 1, CjðPÞ4CjðPÞþpj case as: djðPÞ¼CjðPÞ CjðPÞ, so that the negative flow has the 62 B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66
Φ ()P
1
0 C ()P CP() CP()+ p CP()− p C ()P
−1
Fig. 4. Net flow function for linear preference function.
Φ ()P
1
()+ CP p C ()P CP()− pCP()− q () CP()CP()+ q C P
-1
Fig. 5. Net flow function for linear preference function with indifference threshold.
or taking that djðPÞ¼CjðPÞ CjðPÞ for Fj ðPÞ we obtain Φ ()P 8 > 0, C ðPÞZC ðPÞ q þ > j j j <> C ðPÞ C ðPÞ q þ 1 j j j þ þ ð Þ¼ þ þ , C ðPÞ p C ðPÞ C ðPÞ q ð Þ Fj P > p q j j r j o j j 17 > j j C ()P :> þ 1, CjðPÞoCjðPÞ pj C ()P C ()P Finally, for Fj(P) we obtain -1 C ()P + CP() 8 > 1, CjðPÞrCjðPÞþqj > > C ðPÞ C ðPÞ p Fig. 6. Net flow function for Gauss’s preference function. > j j j > p q , CjðPÞþqj oCjðPÞrCjðPÞþpj > j j <> þ 0, CjðPÞþp oCjðPÞrCjðPÞ p FjðPÞ¼> j j ð18Þ > þ > p CjðPÞþCjðPÞ > j ð Þ þ ð Þ ð Þ þ > p þ q þ , Cj P pj rCj P oCj P qj > j j The net flow is given graphically in Fig. 6 (the proof for the zero of > : Z þ the net flow is given in Appendix A). 1, CjðPÞ CjðPÞ qj Positive, negative and net flows are calculated separately for or graphically as in Fig. 5. each particular criterion, i.e. for each criterion Cj (j¼1,2, ..., n)we þ The Gaussian preference function with indifference threshold have to calculate Fj ðPÞi Fj ðPÞ, and the net flow is obtained as a from the PROMETHEE method (type VI) has the following form: weighted sum of the difference between those flows, i.e. d2 Xn CðdÞ¼ exp ð19Þ FðPÞ¼ w ðF þ ðPÞ F ðPÞÞ ð23Þ 2s2 j j j j ¼ 1 þ Thus for Fj ðPÞ we obtain ! For any possible portfolio P naturally we will have ½C ðPÞ C ðPÞ 2 þ ðPÞ¼1 exp j j ð20Þ Fj 2 FðPÞrFðPÞrFðPÞ: ð24Þ 2sj
The weights wj of the criteria are obtained in agreement with the while for Fj ðPÞ we obtain ! decision maker (and/or by some of the methods for determining ½C ðPÞ C ðPÞ 2 the weights of criteria). For obtaining the weights in our case F ðPÞ¼1 exp j j ð21Þ j 2 we use the AHP method and its eigenvalue procedure with 2sj þ pairwise comparisons obtained by a group of experts, which will The net flow for the Gaussian preference function has the form be explained in more detail in the case study. ! ! Finally, the optimal portfolio (a ,a , ..., a ) is one that solves: ½C ðPÞ C ðP Þ 2 ½C ðPÞ C ðPÞ 2 1 2 N ð Þ¼ j j þ j j ð Þ Fj P exp 2 exp 2 22 2sj 2sj þ MaxFðPÞð25Þ B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66 63 subject to: share of each industry in the optimal portfolio are of course
XN selected with the aim of ensuring the best investment yielding ai ¼ 1 ð26Þ the highest eventual profit. For criteria selection and determina- i ¼ 1 tion of their weights we have surveyed ten experts: 5 scientists from the Faculty of Economics in Split and 5 financial managers 0rai raMi ð27Þ from the capital market. The selected effective criteria are the following: the expected increasing demand for the industry where ai is the share invested in alternative Ai in the portfolio; aMi products, the stability of products’ sale, pricing of the products is the maximum proportion to invest in alternative Ai in portfolio P; N is the number of alternatives which can be included in by the government, industry investments’ share of the GNP, the portfolio P. quarterly growth rate of gross added value and gross domestic In the first step of two-step model when we use this multi- product, the industry’s P/E average, the industry’s current value, criteria model to determine the share of each industry in portfolio the average transaction volume in the industry. we have (in accordance with notation from previous section) The importance (weight) of the chosen criteria (for our purpose) has been estimated using the Saaty’s AHP method, or N¼n, ai¼yi, i¼1, 2, ..., n. Similarly, in the second step of two-step model when we determinate shares of stocks in an optimal more precisely, by its eigenvalue procedure. Using the Saaty 1–9 industry portfolio we have (in accordance with notation from scale (see Babic´ and Tomic´-Plazibat, 1998; Saaty, 2001; Yurdakul and Tansel, 2004) pairwise comparisons are carried out for every previous section) N¼ki, aj¼zij, j¼1, 2, ..., ki, i¼1, 2, ..., n. pair of criteria. The applied criteria are compared in pairs to determine which one is more important (which one contributes 4. Application to Croatian capital market more to the given aim, i.e. which one better determines which industry is more suitable for investment) on the 1–9 scale. The two-step model presented above will be used for selection The matrix A is obtained by surveying the aforementioned 10 of an optimal portfolio in the Croatian capital market which is the experts and the final values in the matrix are obtained by the largest capital market in the South-East Europe among non-EU geometrical mean of their estimates. After obtaining the complete member countries (the Western Balkan Region). The stocks to be Saaty matrix by the eigenvalue vector method (using Expert included in this portfolio are the most liquid stocks from the Choice) we have the weights for each of the chosen criteria and Croatian capital market from food, construction, hotels & motels, they are shown in Fig. 7. maritime transport, and commercial banking industries (here we The next task is the selection of individual preference func- use industry classification given by the analysts of Hypo Group tions for the PROMETHEE, or more precisely, for our modified Alpe Adria and FIMA Securities Limited Liability Company for form of this method. trading securities with minor modification). These stocks are: In PROMETHEE II, the preference function of each criterion is Podravka, Kras,ˇ Franck, IPK Kandit, Viro, Tehnika, Viadukt, IGH, determined mostly by the nature of the criterion and the Hidroelektra niskograd, Istraturist, Riviera Holding, Plava Laguna, decision-maker’s viewpoint. There are six types of predefined Suncˇani Hvar, HUP Zagreb, Atlantska plovidba, Tankerska plo- preference functions which cover most of the applications (Brans vidba, Jadroplov, Uljanik plovidba, Zagrebacˇka banka, Privredna et al., 1984) which are as follows: usual criterion, quasi-criterion, banka, Jadranska banka, Karlovacˇka banka, and Centar banka. The criterion with linear preference, level criterion, criterion with selected companies and their corresponding industries are shown linear preference and indifference area, and Gaussian criterion. In in Table 1. We therefore start with 23 companies in which we this case, the preference function and the values of the thresholds could possibly invest. for those functions for each criterion have been identified with In the first step, we determine the share of each of the 5 chosen the help of the previously mentioned financial managers. In this industries in the optimal portfolio. The criteria determining the paper, we use the criterion with linear preference (type III), the
Table 1 Selected industries and companies.
Industry Food Construction Hotels & motels Maritime transport Commercial banking
Company Podravka Tehnika Istraturist Atlantska plovidba Zagrebacˇka banka Krasˇ Viadukt Riviera Tankerska plovidba Privredna banka Franck IGH Holding Jadroplov Jadranska banka IPK Kandit Hidroelektra niskograd. Plava Laguna Uljanik plovidba Karlovacˇka banka Viro Suncˇani Hvar Centar banka HUP Zagreb
Fig. 7. Estimation of criteria weights using Expert Choice. 64 B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66 criterion with linear preference and indifference area (type V), experts. The weights of effective criteria have been estimated as and the Gaussian criterion (type VI) as they most generally show before by the eigenvalue method. Table 4 shows the chosen the relations between pairs of alternatives. Table 2 presents the effective criteria for evaluation of stocks within each industry chosen criteria for industry evaluation with the corresponding with the corresponding weights. criteria weights obtained by the eigenvalue method, and criteria It can be seen from the tables that the experts have chosen values for each of the industries. different criteria for the selection of portfolios of different The values for the first three criteria, qualitative scale from industries. Thus for instance, the criterion loans/deposit ratio is very low to very high, are obtained by the estimate of financial important for the commercial banking industry, but not for other experts and scientists. The values for other criteria are taken from industries, which means that for other industries this criterion the Statistical Yearbook (The Statistical Year book, 2007), financial could be considered as having weight 0. It is important to statements of companies and the web pages of the Zagreb Stock emphasize that criteria which are immanent in all industries Exchange (The Zagreb Stock Exchange, History Trading, 2007). (e.g. stock monthly return) do not necessarily have the same Programming the net flow function and solving problem weight in each industry. Furthermore, we use return lower semi- (25)–(27) in MATLAB we obtain the solution given in Table 3. variance as a measure of risk because monthly returns of To avoid portfolios that are not diversified, we introduce the considered stocks do not have normal distribution and for that constraint of the maximal share of an industry in the portfolio reason variance does not represent an adequate measure of risk. ymax¼0.3. The values of criteria for the stocks quoted on the Zagreb Stock In the second stage, we determine the share of each stock in Exchange are taken from the Reuters data base. the optimal industry portfolio. The criteria for stock evaluation Again, programming the net flow function and solving pro- within an industry are also chosen in collaboration with financial blem (25)–(27) in MATLAB we obtain the optimal portfolios for
Table 2 Decision matrix for industry evaluation.
Expected The stability of Pricing of the Industry Quarterly growth Industry’s Industry’s Average increasing products’ sale products by the investments’ rate of gross P/E current value transactions demand for the government share of GNP added value and average volume in the industry gross domestic industry products product
Type of criterion Max Max Min Max Max Min Max Max Weight 0.157 0.024 0.033 0.048 0.106 0.331 0.231 0.071 Unit Scalea Scale Scale Percent Percent Number Millions of HRK Number Preference function III III III V V VI VI V Food 3 4 5 3.3 2.19 38.88 13819,4 21131,1 Construction 2 2 2 5.7 3.16 58.8 9680,7 18788,4 Hotels & motels 4 3 3 3.1 7.74 47.35 6881,5 6526,8 Maritime transport 5 1 1 6.1 3.25 10.8 17678,9 21726,29 Commercial banks 1 5 4 5.1 2.88 16.6 86271,3 4281
a Scale: 1: very low, 2: low, 3: average, 4: high, 5: very high.
Table 3 Shares of industries in optimal portfolio.
Food Construction Hotels & motels Maritime transport Commercial banks
0.1342 0.0000 0.2658 0.3000 0.3000
Table 4 Effective criteria list in company evaluation.
Food, hotels Effective criteria Stock Return Beta P/E ratio Total debt to total Long-term P/B ratio Competitive situation Liquidity & motels in company monthly lower semi- coefficient assets ratio debt to in the industry (market ratio evaluation return variance equity ratio share) Type of criterion Max Min Min Min Min Min Min Max Max Weight 0.110322 0.120533 0.10677 0.122309 0.097447 0.096781 0.101665 0.112542 0.131632 Unit Percent Percent Number Number Percent Percent Number Percent Percent
Transport- Effective criteria in Stock Return Beta P/E ratio Total debt to total Long-term P/B ratio Net asset value to Liquidity marine company monthly lower semi- coefficient assets ratio debt to enterprise value ratio ratio evaluation return variance equity ratio Type of criterion Max Min Min Min Min Min Min Max Max Weight 0.110322 0.120533 0.10677 0.122309 0.097447 0.096781 0.101665 0.112542 0.131632 Unit Percent Percent Number Number Percent Percent Number Number Percent
Commercial Effective criteria in Stock Return Beta P/E ratio Total long-term Loans to P/B ratio banks company monthly lower semi- coefficient debt to enterprise deposit ratio evaluation return variance value ratio Type of criterion Max Min Min Min Min Max Min Weight 0.136501 0.162867 0.151332 0.149135 0.129909 0.138149 0.132107 Unit Percent Percent Number Number Percent Percent Number B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66 65
Table 5 Shares of companies in industry portfolio.
Industry Food Company Podravka Krasˇ Franck IPK Kandit Viro Weight 0.3000 0.3000 0.3000 0.0000 0.1000
Industry Hotels & motels Company Istraturist Riviera Holding Plava Laguna Suncˇani Hvar HUP Zagreb Weight 0.1000 0.0000 0.3000 0.3000 0.3000
Industry Maritime transport Company Atlantska plovidba Tankerska plovidba Jadroplov Uljanik plovidba Weight 0.1000 0.3000 0.3000 0.3000
Industry Commercial banks Company Zagrebacˇka banka Privredna banka Jadranska banka Karlovacˇka banka Centar banka Weight 0.0000 0.3000 0.1000 0.3000 0.3000
Table 6 industries is left to the decision maker (i.e. investor) but it has to Optimal portfolio for Croatian capital market. be in compliance with the directions provided by the professional literature. Besides the choice of criteria, the decision maker is also Industry Company Weight allowed to determine the importance of each criterion, which is Food Podravka 0.040260 the basis for determination of criteria weights in the model. Krasˇ 0.040260 In the second step, we select the optimal portfolio for each Franck 0.040260 individual industry. Assuming that the principles stated in pro- IPK Kandit 0.000000 fessional literature are followed, the effective criteria in company Viro 0.013420 evaluation and criteria weights are again left to the decision Hotels & motels Istraturist 0.026580 maker (investor). Let us note that the choice of criteria, which Riviera Holding 0.000000 Plava Laguna 0.079740 may be different for different industries, and their weights, which Suncˇani Hvar 0.079740 may also be different for different industries, will depend on the HUP Zagreb 0.079740 investment strategy of the decision maker. Transport-marine Atlantska plovidba 0.030000 Finally, based on the shares of industries in the portfolio Tankerska plovidba 0.090000 calculated in the first step and the shares of company stock in Jadroplov 0.090000 the single industry portfolio calculated in the second step, we Uljanik plovidba 0.090000 determine the share of each company stock in the total portfolio. Commercial banks Zagrebacˇka banka 0.000000 The multi-criteria model used to determine the share of each Privredna banka 0.090000 single industry in the portfolio and to select the portfolio of each Jadranska banka 0.030000 single industry is a modified multi-criteria programming model Karlovacˇka banka 0.090000 Centar banka 0.090000 based on the PROMETHEE II approach. The advantage of this model is that by preference functions it can accurately present the relation of each portfolio to the ideal and anti-ideal. Nevertheless, a future each single industry (Table 5). To avoid non-diversified portfolios research might also consider some other multi-criteria models for we again introduce constraints of the maximal share of a portfolio selection, e.g. extension of the TOPSIS method which also particular stock in the industry portfolio zmax¼0.3. compares alternatives with the ideal and anti-ideal. Finally, the optimal portfolio of stocks quoted on the Zagreb The contribution of the two-step model is in the possibility to Stock Exchange according to relation (1) is presented in Table 6. select the criteria characteristic for a particular industry, which are It is to be noted that this optimal portfolio is obtained on non-existent in other industries and are unavoidable for evaluation the basis of commercial, financial and macroeconomic data for of company stocks in that particular industry. In addition, the the period of 01/01/2007–30/06/2008. Each change of data (e.g. weights of particular criteria may be different in different indus- the price of a particular stock, the publication of financial tries. Consequently, due to these characteristics of the here pre- statements, the change in GDP, etc.) affects the optimal portfolio, sented multi-criteria model with the two-step structure, we can therefore to form an optimal portfolio at the present moment assume that the obtained results or the optimal portfolios calcu- these data should be updated. Besides the input data, we would lated by its application are more representative than the results of also have to change the criteria and some criteria weights in most multi-criteria models for stock portfolio optimization. accordance with the investor’s preferences. Namely, most approaches described in scientific and professional literature do not allow a selective choice of criteria for particular industries and perform stock comparison based on uniform criteria 5. Conclusion with uniform weights for all stocks while avoiding the criteria that are characteristic only for a particular industry. This paper presents a two-step model for selection of the optimal portfolio. Portfolio optimization in this model is per- formed first by defining the focused set of stocks. Then the stocks Appendix A. Zero of Gaussian preference function are classified by the corresponding industries. In the first step of portfolio selecting, we determine the share of each industry in the For zero point Cj(P) holds ! ! portfolio. Evaluation of industries is carried out by a multi-criteria ½C ðPÞ C ðPÞ 2 ½C ðPÞ C ðPÞ 2 model and the obtained result is the share of each industry in the F ðPÞ¼ exp j j þexp j j ¼ 0 j 2s2 2s2 optimal portfolio. The choice of the criteria set for comparison of j j þ 66 B. Marasovic´, Z. Babic´ / Int. J. Production Economics 134 (2011) 58–66
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