A hybrid approach for constructing suitable and optimal portfolios

Expert Systems with Applications 38 (2011) 5620–5632

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A hybrid approach for constructing suitable and optimal portfolios Pankaj Gupta a,⇑, Masahiro Inuiguchi b, Mukesh K. Mehlawat a a b

Department of Operational Research, University of Delhi, Delhi, India Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Japan

a r t i c l e

i n f o

Keywords: Portfolio selection Fuzzy linear programming Analytical hierarchy process Risk Cluster analysis

Dedicated to my beloved father Sh. Govind Ram

a b s t r a c t The purpose of the paper is to propose a hybrid approach for asset allocation with simultaneous consideration of suitability and optimality. In this approach, recourse is taken to multiple methodologies to simultaneously capture the impersonal characteristics of the financial assets as well as the personal preferences of the investors. Accordingly, the paper draws on Investors’ survey to capture their preferences, cluster analysis to categorize the financial assets, the analytical hierarchy process for obtaining local weights (performance scores) of the financial assets corresponding to the four key asset allocation criteria and fuzzy multi-objective linear programming model for portfolio selection.  2010 Elsevier Ltd. All rights reserved.

1. Introduction The main objective in a portfolio selection problem is to obtain optimal proportions of the assets for creating a portfolio which respects investors’ preferences regarding the return/risk of the portfolio. Modern portfolio analysis started from pioneering research work of Markowitz (1952). The portfolio selection model formulated by him is called mean–variance model in which return is quantified as the mean and risk as the variance. Konno and Yamazaki (1991) used the absolute deviation risk function to formulate mean-absolute deviation portfolio selection model. Furthermore, Speranza (1993) employed semi-absolute deviation to measure risk in portfolio selection model. The above mentioned approaches assumed ‘‘perfect information’’ whereas investor experience is invariably characterized by vagueness. Hence, decisions must be made under conditions of uncertainty. Though, probability theory is one of the main techniques used for analyzing uncertainty in finance, the financial market is also affected by several non-probabilistic factors such as vagueness and ambiguity. Investors are commonly provided with information based upon linguistic descriptions such as high risk, low profit, high interest rate, etc. With the introduction of fuzzy set theory (FST) by Zadeh (1965), it was realized that imperfect knowledge of the returns on the assets and the uncertainty involved in the behavior of financial markets may be captured by means of fuzzy quantities and/or fuzzy constraints. A review of literature on application of FST in portfolio selection suggests a variety of approaches in doing so. Ammar (2008) discussed fuzzy portfolio optimization problem in a convex quadratic programming ⇑ Corresponding author. Address: Flat No. 01, Kamayani Kunj, Plot No. 69, Indraprastha Extension, Delhi 110 092, India. E-mail addresses: [email protected], [email protected] (P. Gupta). 0957-4174/$ - see front matter  2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.10.073

framework and provided an acceptable solution for it. Bilbao-Terol, Pérez-Gladish, Arenas-Parra, and Rodríguez-Uría (2006) applied fuzzy compromise programming for portfolio selection. Chen and Huang (2009) presented portfolio selection model with fuzzy returns and risks. In their paper, cluster analysis was performed to categorize equity mutual funds into several clusters on the basis of certain evaluation indices. Fei (2007) studied the optimal consumption and portfolio choice with ambiguity and anticipation. In Huang (2007a, 2007b) portfolio selection models have been studied in which the return rates are stochastic variables with fuzzy information and the return rates contain both randomness and fuzziness. A new definition of risk has been used in Huang (2008) for portfolio selection in fuzzy environment. Inuiguchi and Tanino (2000) discussed portfolio selection problem with independently estimated possibilistic return rates and proposed a new possibilistic programming approach based on the worst regret. Li and Xu (2009) proposed a new portfolio selection model in hybrid uncertain environment considering the returns of each security to be fuzzy random variable. Lin and Liu (2008) proposed three possible models for portfolio selection with minimum transaction lots and presented corresponding genetic algorithms to solve them. Qin, Li, and Ji (2009) presented mathematical models for portfolio selection using cross-entropy to minimize the divergence of the fuzzy investment return from a priori one and designed hybrid intelligent algorithm to solve these models under fuzzy environment. Tiryaki and Ahlatcioglu (2009) used fuzzy analytic hierarchy process in portfolio selection to provide both ranking and weighting information to the investors. Vercher, Bermúdez, and Segura (2007) proposed two fuzzy portfolio selection models where they minimize the downside risk constrained by a given expected return. Vercher (2008) presented models for portfolio selection in which the returns on securities are considered fuzzy numbers and optimal portfolio is obtained using

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semi-infinite programming in a soft framework. Zhang, Wang, Chen, and Nie (2007) proposed portfolio selection models based on lower and upper possibilistic means and possibilistic variances, respectively. Zhang (2007) discussed the portfolio selection problem for bounded assets based on upper and lower possibilistic means and variances considering uncertain returns of the risky assets in capital markets as fuzzy numbers. The foregoing studies of portfolio selection consider expected return and risk as the two fundamental factors that govern investors’ choice. However, it is often found that not all the relevant information for an investment decision can be captured in terms of return and risk only. The other considerations/criteria might be of equal, if not greater, importance to investors. By considering them in the model, it may be possible to construct optimal portfolios such that a deficit on account of the traditional criteria, i.e. return and risk may be more than compensated by portfolio performance on the other criteria, resulting into greater overall satisfaction for investors. There are studies that consider more criteria than return and risk, e.g. Arenas-Parra, Bilbao-Terol, and Rodríguez-Uría (2001), Ehrgott, Klamroth, and Schwehm (2004), Fang, Lai, and Wang (2006), Gupta, Mehlawat, and Saxena (2008, 2010). However, a common limitation of the works reviewed thus far, has been the neglect of suitability issue in portfolio selection. Suitability is a behavioral concept that refers to the propriety of the match between investor-preferences and portfolio characteristics. Financial advisors and investment companies use various techniques to profile investors and then recommend a suitable asset allocation. It may be justifiably posited that a substantial improvement may be brought about in the models for investment decision-making if one could incorporate investor-characteristics/ preferences. Bolster and Warrick (2008) developed a model of suitability for individual investors based on their personal attributes. Other than this, to the best of our knowledge, there is not much research on the incorporation of suitability criterion in portfolio selection. In this paper, the focus of the research is to attain the convergence of suitability and optimality in portfolio selection using analytical hierarchy process (AHP) and fuzzy multi-objective linear programming (FMOLP) frameworks. Towards this, firstly a typology of investors is evolved using the inputs from a primary survey of investor preferences (Gupta et al., 2005). We, then implement cluster analysis (CA) to categorize the chosen sample of financial assets into various clusters based on three evaluation indices. Next, the local weights (performance scores) of each asset within a cluster with respect to the four key asset allocation criteria, namely, return, risk, liquidity and suitability are calculated using AHP. These weights are then utilized as coefficients of the objective functions corresponding to the four criteria in the FMOLP model. The FMOLP model is solved by considering a weighted-additive model using the weights (relative importance) of the four key criteria that directly influence the asset allocation decision. These criteria weights are also calculated using AHP. In order to improve portfolio performance on individual objective(s) as per investors’ preferences, we use an interactive approach that focuses on maximizing the worst lower bound to obtain an efficient solution close to the best upper bound of each objective function. The solution procedure controls the search direction via updating both the membership function (MF) values and the aspiration levels with the interaction of the investors until they are satisfied with the portfolio construction. This paper is organized as follows. In Section 2, we describe the research methodology and present portfolio selection model based on fuzzy decision theory. The proposed model is test-run in Section 3 drawing on a 36-month data series in respect of assets listed on the National Stock Exchange (NSE), Mumbai, India. This section also pertains to a discussion of the results obtained. In Section 4, we furnish our concluding remarks.

2. Methodology The paper rests on the following four-stage research design. 2.1. Investor typology Investor diversity is characterized by variations in their demographic, socio-cultural, economic and psychographic factors, each factor comprising in turn, a host of interrelated variables. It may be noted that there are many variables that impinge upon investment decision-making and any list would at best be illustrative. Moreover, rather than the impact of an individual variable, the investor behavior is influenced by the constellation or configuration of the various variables at a point of time. Since these variables are not static, their relative influence undergoes a change with the passage of time. Therefore, it is usual to capture the dynamic nature of these variables through life-stage analysis whereby the time element is broken into discrete stages of life to reflect the combined effect of the various variables. Investment advisors map the investors on these variables with a view to assessing their investment preferences. Such an assessment facilitates recommendation of the appropriate investment alternatives that have at least a prima facie suitability for the investors. The individual investor, then, may pick and choose from among these alternatives. This paper captures the investor diversity in terms of a typology developed from a primary survey (Gupta et al., 2005). The survey relied on structured questionnaires covering a variety of interrelated aspects, such as investors’ economic and financial position, including income and types of investment held, past experiences, future investment intentions, etc. Analysis of the survey data showed variation in investor behavior across these variables. From a behavioral perspective such a variation is understandable as even though they would prefer more of return and liquidity over less and less of risk over more, different investors would order these investment objectives differently. From a careful scrutiny of the survey data, we have culled a triadic typology of investor behavior: return seekers, safety seekers and liquidity seekers, for details see Gupta et al. (2010). 2.2. Clustering of assets Just as different investor types may be showing distinct ordering of return, risk and liquidity criteria, likewise different assets may also be showing distinct characteristics vis-à-vis these criteria. Given that not all the assets in the market would be appropriate for any one given investor type, it would be desirable to see if these assets could be stratified into different clusters on the basis of some pre-defined characteristics. We consider three evaluation indices to perform cluster analysis, namely, asset returns, standard deviation denoting risk and liquidity. Since the measurement unit of each index will be different, we need to perform normalization for ensuring that the result of clustering is on the same basis. We apply z-score transformation for the purpose. In the present study, K-means method (Kaufman & Rousseeuw, 2005) is used for clustering of the assets. In order to find the most suitable number of clusters (k) for the input data set, we rely on the silhouette coefficients (Kaufman & Rousseeuw, 2005). The silhouette coefficient s(i) is computed as per the following steps: 1. For the ith object, calculate its average distance to all other objects in its cluster; call this value ai. 2. For the ith object and any cluster not containing the object, calculate the object’s average distance to all the objects in the given cluster. Find the minimum such value with respect to all the clusters; call this value bi. bi ai 3. For the ith object, the silhouette coefficient is sðiÞ ¼ maxða . ;b Þ i

i

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The value of the silhouette coefficient of an object can vary between 1 and 1, which indicates how much that object belongs to the cluster in which it is classified. The closer the value is to 1, the higher the degree that the object belongs to its cluster. In the present study, the silhouette coefficients are used to quantify the quality of assignment of an asset to a particular cluster. The silhouette value of a cluster is the average of the silhouette coefficients of all data items belonging to the cluster. Kaufman and Rousseeuw (2005) proposed the following interpretation of the silhouette value of a cluster: 0.71 6 cluster silhouette 6 1 means it is a strong cluster; 0.51 6 cluster silhouette 6 0.7 means it is a reasonable cluster; 0.26 6 cluster silhouette 6 0.5 means it is a weak or artificial cluster; cluster silhouette < 0.25 means no cluster is found. The silhouette value for k is the average silhouette values of the k clusters. The most suitable k is the one with the highest average silhouette value.

2.3. Modeling with the AHP Here, we calculate the local weights (performance scores) of each asset within a cluster with respect to key asset allocation criteria and the weights (relative importance) of the key criteria when making the asset allocation decision. The AHP model used here comprises five levels of hierarchy. Level 1 represents the overall goal, i.e. asset allocation. Level 2 represents the key criteria (return, risk, liquidity and suitability) that directly influence the goal. At level 3, return criterion is broken into short term return, long term return and forecasted return; risk criterion is broken into standard deviation, risk tolerance and microeconomic risk; suitability criterion is broken into income and savings, investment objectives and investing experience. At level 4, suitability sub-criteria are further broken into 11 sub-criteria that may affect the choice for a particular asset. At the bottom level of the hierarchy, the alternatives (i.e. assets) are listed (please refer to Fig. 1 for complete structural hierarchy). In AHP, the elements of each level of the decision hierarchy are rated using pairwise comparison. After all the elements have been compared pair by pair, a paired comparison matrix is formed. The order of the matrix depends on the number of elements at each level. The number of such matrices at each level depends on the number of elements at the immediate upper level that it links to. After developing all the paired comparison matrices, the eigenvector or the relative weights (i.e. the degree of the relative importance amongst the elements) and the maximum eigenvalue (kmax) are calculated for each matrix. The kmax value is an important validating parameter in AHP. It is used as a reference index to screen information by calculating the consistency ratio of the estimated vector in order to validate whether the paired comparison matrix provides a completely consistent evaluation. The consistency ratio is calculated as per the following steps: 1. Calculate the eigenvector or the relative weights and kmax for each matrix of order n. 2. Compute the consistency index (CI) for each matrix of order n as:

CI ¼ ðkmax  nÞ=ðn  1Þ: 3. The consistency ratio (CR) is calculated as:

CR ¼ CI=RI;

Fig. 1. Structural hierarchy for suitability of assets.

where RI is a known random consistency index obtained from a large number of simulation runs and varies according to the order of matrix. The acceptable CR value for a matrix at each level is 0.1. If CI is sufficiently small then pairwise comparisons are probably consistent enough to give useful estimates of the weights. If CI/ RI 6 0.10, the degree of consistency is satisfactory. However, if CI/RI > 0.10, serious inconsistencies may exist and hence AHP may not yield meaningful results. The evaluation process should therefore be reviewed and improved. The eigenvectors are used to calculate global weights if there is an acceptable degree of consistency for the selection criteria. In the present study to determine the local weights of the assets with respect to the quantitative measures of performance, we have relied on actual data, that is, the past performance of the assets. The question for pairwise comparison of quantitative criteria is considered as: ‘‘Of two elements i and j, how many times i is preferred to j’’. If the values for the alternatives i and j are, respectively, wi and wj, the preference of the alternative i to j is equal to wi/wj. Therefore, the pairwise comparison matrix is

0

1

w1 =w1 B w =w B 2 1 B @

w1 =w2

. . . w1 =wn

w2 =w2

. . . w2 =wn C C C: A

wn =w1

wn =w2

...

. . . wn =wn

As this matrix is consistent (Saaty, 2000), the weight of each element is its relative normalized amount, i.e. weight of the ith element = Pwn i . i¼1

wi

The priority of the alternative i to j for negative criteria, such as risk, is equal to wj/wi. The pairwise comparison matrix is therefore

0

1

w1 =w1

w2 =w1

. . . wn =w1

B w =w B 1 2 B @

w2 =w2

. . . wn =w2 C C C: A

w1 =wn

w2 =wn

...

. . . wn =wn

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The above matrix is consistent (Saaty, 2000) and the weight of the i ith element (for negative criteria) = P1=w . n i¼1

1=wi

2.4. FMOLP model for portfolio selection Operationally, formulating an asset portfolio requires an estimate of distributions of future returns/risk for the various assets. Distributed randomly as they are over the chosen time horizon, such estimates at best represent the investor’s subjective/intuitive interpretation of the information set available at the time of constructing the portfolio. The same information set may be interpreted differently by different investors. Under such circumstances, the issue of constructing a portfolio of assets becomes the one of a choice from a ‘‘fuzzy’’ set of subjective/intuitive interpretations, the term ‘‘fuzzy’’ being suggestive of the diversity of both the investor’s objective functions as well as that of the constraints. Here, we formulate the general FMOLP model for asset allocation. We assume that investors allocate their wealth among n assets. We introduce some notations as follows: ri: the AHP-local weight of the ith asset with respect to return. bi: the AHP-local weight of the ith asset with respect to risk. li: the AHP-local weight of the ith asset with respect to liquidity. si: the AHP-local weight of the ith asset with respect to suitability. h1: the AHP-weight of the return criterion. h2: the AHP-weight of the risk criterion. h3: the AHP-weight of the liquidity criterion. h4: the AHP-weight of the suitability criterion. xi: the proportion of total funds invested in the ith asset, yi: the binary variable indicating whether the ith asset is contained in the portfolio or not,

yi ¼



The sign () indicates the fuzzy environment and symbol J denotes the fuzzified version of P and has linguistic interpretation ‘‘essentially greater than or equal to’’. Z 01 , Z 02 , Z 03 and Z 04 are the aspiration levels. 2.4.2. Constraints  Capital budget constraint on the assets: n X

xi ¼ 1:

i¼1

 Maximal fraction of the capital that can be invested in a single asset:

xi 6 ui yi ;

i ¼ 1; 2; . . . ; n:

 Minimal fraction of the capital that can be invested in a single asset:

xi P l i y i ;

i ¼ 1; 2; . . . ; n:

The maximal and minimal fractions of the capital budget being allocated to various assets in the portfolio would depend upon a consideration of a number of factors. For example, one may consider price/value relative of the asset vis-à-vis the average of the price/value of all the assets in the chosen portfolio, minimal lot size that can be traded at the market, the past behavior of the price/volume of the asset, information available about the issuer of the asset, trends in the industry of which it is a part, etc. In other words, investors would refer to various fundamental factors affecting the company and the industry. Since investors would differ in their interpretation of the available information, they may allocate the same overall capital budget differently. The constraints corresponding to lower bounds li

1; if ith asset is contained in the portfolio; 0;

otherwise:

2.4.1. Objectives

1

Z 1 ðxÞ ¼

n X

r i xi J Z 01 :

i¼1

Cluster

 Return criterion The objective function of the return criterion is expressed as:

2

 Risk criterion The objective function of the risk criterion is expressed as:

Z 2 ðxÞ ¼

n X

bi xi J Z 02 :

i¼1

3

 Liquidity criterion In respect of an asset, liquidity may be measured with respect to the proportion, called the turnover rate, between the average stock traded at the market and the tradable stock (shares held by public) of that asset. Generally, investors prefer greater liquidity. The objective function of the liquidity criterion is expressed as:

Z 3 ðxÞ ¼

n X

li xi J Z 03 :

i¼1

 Suitability criterion The objective function of the suitability criterion is expressed as:

Z 4 ðxÞ ¼

n X i¼1

si xi J Z 04 :

Silhouette Value Fig. 2. Cluster analysis (average silhouette = 0.6841).

Table 1 Results of cluster analysis.

Return rate Std. deviation Liquidity rate Category

Cluster 1 (56 assets)

Cluster 2 (51 assets)

Cluster 3 (43 assets)

0.14084 0.44474 0.00513 Liquid assets

0.27078 0.52338 0.00152 High-yield assets

0.18015 0.32999 0.00340 Less risky assets

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and upper bounds ui on the investment in individual assets (0 6 li, ui 6 1, li 6 ui, "i) are incorporated to avoid a large number of very small investments (lower bounds) and at the same time to ensure sufficient diversification of the investment (upper bounds). It may be noted that the lower and upper bounds have to be chosen carefully so that the problem has feasible solution.  Number of assets held in a portfolio: n X

ðFMOLPÞ max Z 1 ðxÞ ¼

n X

ri xi J Z 01 ;

ð1Þ

bi xi J Z 02 ;

ð2Þ

li xi J Z 03 ;

ð3Þ

si xi J Z 04 ;

ð4Þ

i¼1

max Z 2 ðxÞ ¼

n X i¼1

max Z 3 ðxÞ ¼

n X i¼1

yi ¼ h;

max Z 4 ðxÞ ¼

i¼1

n X i¼1

where h is the number of assets that the investor desires in the portfolio. Of all the assets in a given set, investors would like to pick up the ones that are likely to yield the desired satisfaction of their preferences. It is not necessary that all the assets from the given set may configure in portfolio as well. Investors would differ with respect to the number of assets they can effectively handle in a portfolio.  No short selling of assets:

xi P 0;

subject to

n X

n X

yi ¼ h;

ð6Þ

xi 6 ui yi ; xi P l i y i ; xi P 0;

i ¼ 1; 2; . . . ; n; i ¼ 1; 2; . . . ; n;

ð7Þ ð8Þ

i ¼ 1; 2; . . . ; n;

yi 2 f0; 1g;

ð9Þ

i ¼ 1; 2; . . . ; n:

ð10Þ

Using fuzzified aspiration levels with respect to the linguistic term ‘‘essentially greater than or equal to’’, the fuzzy linear MFs (Zimmermann, 1978) are defined for each objective function, Zp(x), p = 1, 2, 3, 4 of the FMOLP model as follows:

2.4.3. The portfolio selection problem The FMOLP model of portfolio selection using assets belonging to a particular cluster is formulated as: Table 2 Fundamental scale for pairwise comparisons. Numerical values

Equally important, likely or preferred Moderately more important, likely or preferred Strongly more important, likely or preferred Very strongly more important, likely or preferred Extremely more important, likely or preferred Intermediate values to reflect compromise Reciprocals for inverse comparison

ð5Þ

i¼1

i ¼ 1; 2; . . . ; n:

Verbal scale

xi ¼ 1;

i¼1

lZp ðxÞ ¼

1 3 5 7 9 2, 4, 6, 8 Reciprocals

8 1; > > > < > > > :

Z p ðxÞZ Lp L ZU p Z p

if Z p ðxÞ P Z Up ; ; if Z Lp 6 Z p ðxÞ 6 Z Up ; p ¼ 1; 2; 3; 4;

ð11Þ

if Z p ðxÞ 6 Z Lp ;

0;

where Z Up is the best upper bound, Z Lp is the worst lower bound of the pth objective function and are calculated as follows:

Table 3 Data characteristics of assets for quantitative criteria from cluster 1, cluster 2 and cluster 3. Cluster

Criteria

Assets A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

Cluster 1

Expected short term return Expected long term return Risk (standard deviation) Liquidity

0.02996 0.11213 0.44464 0.00611

0.00329 0.06742 0.39196 0.00027

0.17208 0.15058 0.43562 0.00359

0.20376 0.15292 0.44955 0.00478

0.21379 0.28213 0.66807 0.01378

0.06883 0.10498 0.43017 0.00570

0.08069 0.11671 0.46513 0.00937

0.03861 0.06898 0.40143 0.00410

0.19037 0.20361 0.53714 0.01149

0.07952 0.17279 0.48292 0.00634

Cluster 2

Expected short term return Expected long term return Risk (standard deviation) Liquidity

0.17377 0.19278 0.37430 0.00032

0.16111 0.21366 0.41053 0.00126

0.20249 0.21711 0.45425 0.00124

0.09481 0.19819 0.52204 0.00134

0.35012 0.40086 0.65059 0.00352

0.28332 0.30831 0.61920 0.00241

0.14264 0.27858 0.60419 0.00631

0.17333 0.27477 0.58206 0.00415

0.08511 0.17133 0.45553 0.00180

0.19311 0.17985 0.36399 0.00060

Cluster 3

Expected short term return Expected long term return Risk (standard deviation) Liquidity

0.10119 0.13587 0.29466 0.00088

0.02790 0.16622 0.38806 0.00504

0.13992 0.14868 0.31859 0.00201

0.09751 0.14382 0.31315 0.00251

0.22892 0.17354 0.28442 0.00164

0.26715 0.26879 0.55536 0.01234

0.14870 0.10492 0.27518 0.00059

0.16130 0.11692 0.28486 0.00221

0.12905 0.10255 0.25876 0.00045

0.12591 0.12328 0.24466 0.00092

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

Cluster 1

Expected short term return Expected long term return Risk (standard deviation) Liquidity

0.01838 0.09370 0.44688 0.00185

0.17869 0.05809 0.32967 0.00164

0.15004 0.10051 0.45308 0.00570

0.06406 0.11579 0.38205 0.00532

0.05924 0.09073 0.32435 0.00076

0.16050 0.11533 0.39966 0.00111

0.15501 0.06188 0.32673 0.00168

0.04218 0.10039 0.69076 0.00989

0.03700 0.04261 0.49083 0.00083

0.14039 0.11500 0.39374 0.00832

Cluster 2

Expected short term return Expected long term return Risk (standard deviation) Liquidity

0.24057 0.29892 0.47987 0.00028

0.15152 0.29428 0.50224 0.00104

0.12064 0.26969 0.50553 0.00153

0.32033 0.34734 0.64998 0.00104

0.08501 0.23776 0.58731 0.00155

0.06110 0.21098 0.45066 0.00376

0.27495 0.30107 0.43539 0.00069

0.18855 0.29633 0.48685 0.00084

0.18030 0.36700 0.54553 0.00042

0.39583 0.30120 0.45199 0.00032

Cluster 3

Expected short term return Expected long term return Risk (standard deviation) Liquidity

0.15810 0.19492 0.40147 0.00548

0.11333 0.15927 0.33135 0.00267

0.10104 0.12072 0.24734 0.00132

0.11453 0.14921 0.27544 0.00413

0.12426 0.12386 0.28463 0.00086

0.08967 0.15267 0.25163 0.00165

0.24725 0.23114 0.34483 0.00473

0.10843 0.16003 0.34627 0.00218

0.26765 0.16188 0.28071 0.00672

0.13041 0.15436 0.25564 0.00972

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P. Gupta et al. / Expert Systems with Applications 38 (2011) 5620–5632 Table 4 Input data for asset allocation using AHP for cluster 1, cluster 2 and cluster 3. Cluster

Criteria

Local weights (performance scores) of the assets A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

Cluster 1

Return Risk Liquidity Suitability

0.03058 0.04655 0.05953 0.05753

0.01874 0.06382 0.00262 0.07834

0.07692 0.06214 0.03500 0.07044

0.07963 0.04988 0.04654 0.04364

0.09918 0.02667 0.13428 0.03385

0.03666 0.04589 0.05550 0.03651

0.04199 0.04012 0.09128 0.03358

0.02717 0.06318 0.03996 0.04954

0.08183 0.03045 0.11199 0.03450

0.05522 0.05047 0.06176 0.04793

0.19281 0.13521 0.57190 0.10008

Cluster 2

Return Risk Liquidity Suitability

0.03771 0.04371 0.00920 0.02433

0.04015 0.04531 0.03657 0.01869

0.04760 0.05014 0.03593 0.06251

0.03292 0.05140 0.03890 0.04326

0.07999 0.05491 0.10227 0.06531

0.06217 0.05062 0.06992 0.06759

0.04831 0.05155 0.18340 0.05801

0.05080 0.05093 0.12050 0.04196

0.02864 0.04627 0.05225 0.05576

0.03973 0.04670 0.01742 0.02370

0.47625 0.35620 0.07040 0.09715

Cluster 3

Return Risk Liquidity Suitability

0.03816 0.04739 0.01294 0.03223

0.03480 0.03149 0.07401 0.06366

0.04719 0.05281 0.02956 0.06602

0.03960 0.04591 0.03694 0.04995

0.06791 0.05050 0.02411 0.02217

0.09143 0.02281 0.18142 0.07537

0.04006 0.06697 0.00867 0.02012

0.04424 0.05223 0.03244 0.05021

0.03745 0.06927 0.00656 0.08286

0.04081 0.07812 0.01359 0.05344

0.35557 0.48111 0.09056 0.07276

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

Cluster 1

Return Risk Liquidity Suitability

0.02346 0.04601 0.01800 0.04549

0.06057 0.06158 0.01597 0.06094

0.05781 0.04177 0.05556 0.03647

0.03853 0.05902 0.05181 0.06080

0.03182 0.06035 0.00740 0.06212

0.06946 0.06886 0.01085 0.06538

0.05573 0.05675 0.01635 0.06693

0.03581 0.04815 0.09637 0.04484

0.02169 0.03916 0.00812 0.02685

0.05718 0.03920 0.08110 0.04432

0.19281 0.13521 0.57190 0.10008

Cluster 2

Return Risk Liquidity Suitability

0.05822 0.04498 0.00818 0.06477

0.04877 0.04908 0.03021 0.04867

0.04273 0.04766 0.04454 0.04379

0.07131 0.04879 0.03035 0.05158

0.03535 0.04618 0.04510 0.05751

0.03058 0.05606 0.10917 0.05776

0.06097 0.05653 0.02013 0.05584

0.05482 0.05857 0.02453 0.06293

0.06088 0.05421 0.01225 0.04449

0.06836 0.04638 0.00919 0.05153

0.47625 0.35620 0.07040 0.09715

Cluster 3

Return Risk Liquidity Suitability

0.06113 0.02996 0.08053 0.06354

0.04595 0.03693 0.03922 0.06235

0.03613 0.06867 0.01940 0.03426

0.04633 0.04914 0.06063 0.04225

0.04024 0.04795 0.01260 0.02198

0.04112 0.06883 0.02431 0.02319

0.08061 0.03558 0.06946 0.07660

0.04685 0.03834 0.03200 0.05426

0.07335 0.04992 0.09877 0.03498

0.04665 0.05716 0.14283 0.07057

0.35557 0.48111 0.09056 0.07276

Table 5 Pay-off table.

Table 8 Attainment values of the quantitative criteria. Objective functions Return (Z1)

Return (Z1) Risk (Z2) Liquidity (Z3) Suitability (Z4) a b

a

0.08553 0.03629 0.09612 0.03874

Risk (Z2) 0.05366 0.06597a 0.01156b 0.06751

Liquidity (Z3) 0.07311 0.03485b 0.11127a 0.03736b

Suitability (Z4) b

0.04760 0.06154 0.01883 0.07129a

Table 6 Results of the W-FMOLP model.

(Return) (Risk) (Liquidity) (Suitability)

Criteria

Value

Short term return (ST) Long term return (LT) Risk (RI) Liquidity (LI)

0.13794 0.16136 0.56233 0.00876

Rao (1987). The weights of the key criteria of asset allocation calculated via AHP are used to reflect relative importance of these criteria in portfolio selection. The weighted additive model proposed for assets allocation within a particular cluster is formulated as follows:

Upper bound. Lower bound.

Z1 Z2 Z3 Z4

Criteria weight (relative importance)

Objective function value

Membership function values

0.06655 0.04508 0.08539 0.04773

0.49961 0.32862 0.74044 0.30566

ðW-FMOLPÞ max

4 X

hp kp

p¼1

kp 6 lZp ðxÞ; p ¼ 1; 2; 3; 4;

subject to

0 6 kp 6 1; p ¼ 1; 2; 3; 4; and Constraints ð5Þ—ð10Þ; where hp is the weight of the pth objective among the fuzzy goals P such that hp P 0 and 4p¼1 hp ¼ 1.

Z Up ¼ max Z p ðxÞ; p ¼ 1; 2; 3; 4; x2X

and Z Lp ¼ min Z p ðxÞ; p ¼ 1; 2; 3; 4; x2X

3. Numerical illustrations

where X is the solution space. In the present study, we solve the FMOLP model using ‘‘weighted additive approach’’ proposed in Tiwari, Dharmar, and

Presented hereunder are the results of an empirical study for which we have relied on a data set of daily closing prices in respect

Table 7 The proportions of the assets in the portfolio. Assets Proportions

1 0.00000

2 0.00000

3 0.26400

4 0.02500

5 0.10000

6 0.00000

7 0.02000

8 0.00000

9 0.25000

10 0.02600

Assets Proportions

11 0.00000

12 0.00000

13 0.00000

14 0.00000

15 0.00000

16 0.00000

17 0.00000

18 0.30000

19 0.00000

20 0.01500

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Table 9 Iterative results. Iteration number 1

2

3

4

5

Z1

0.07930 (0.06655 6 Z1 6 0.08553)

0.07009 (0.04760 6 Z1 6 0.08553)

0.05745 (0.04760 6 Z1 6 0.08553)

0.07050 (0.04760 6 Z1 6 0.08553)

0.06627 (0.04760 6 Z1 6 0.08553)

Z2

0.04695 (0.03485 6 Z2 6 0.06597)

0.04628 (0.03485 6 Z2 6 0.06597)

0.03950 (0.03485 6 Z2 6 0.06597)

0.04677 (0.04508 6 Z2 6 0.06597)

0.04662 (0.04508 6 Z2 6 0.06597)

Z3

0.06888 (0.01156 6 Z3 6 0.11127)

0.08011 (0.01156 6 Z3 6 0.11127)

0.09942 (0.08539 6 Z3 6 0.11127)

0.07884 (0.01156 6 Z3 6 0.11127)

0.08163 (0.01156 6 Z3 6 0.11127)

Z4

0.05003 (0.03736 6 Z4 6 0.07129)

0.04993 (0.04773 6 Z4 6 0.07129)

0.03855 (0.03736 6 Z4 6 0.07129)

0.05049 (0.04773 6 Z4 6 0.07129)

0.04948 (0.03736 6 Z4 6 0.07129)

Table 10 The proportions of assets in the portfolio corresponding to investors’ preferences. Proportions

Iteration Iteration Iteration Iteration Iteration

Iteration Iteration Iteration Iteration Iteration

1 2 3 4 5

ST

LT

0.18299 0.14911 0.11415 0.15023 0.13696

RI

0.17632 0.16568 0.15182 0.16559 0.15864

0.48797 0.54038 0.56964 0.54024 0.55717

LI

Assets

0.00707 0.00822 0.01020 0.00809 0.00838

1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

0.35000 0.35000 0.02000 0.35000 0.31200

0.22300 0.02500 0.00000 0.02500 0.02500

0.10000 0.10000 0.10000 0.10000 0.10000

0.00000 0.00000 0.00000 0.00000 0.00000

0.02000 0.02000 0.26600 0.00000 0.02000

0.00000 0.00000 0.00000 0.00000 0.00000

0.25000 0.25000 0.25000 0.25000 0.20000

0.02600 0.02600 0.02600 0.02600 0.02600

Assets 11

12

13

14

15

16

17

18

19

20

0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.02300 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

0.01600 0.00000 0.00000 0.01600 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.21400 0.30000 0.21800 0.30000

0.00000 0.00000 0.00000 0.00000 0.00000

0.01500 0.01500 0.01500 0.01500 0.01700

of 150 assets listed on NSE, Mumbai, India, the premier market for financial assets.

Table 11 Pay-off table. Objective functions

Return (Z1) Risk (Z2) Liquidity (Z3) Suitability (Z4) a b

Return (Z1)

Risk (Z2)

Liquidity (Z3)

Suitability (Z4)

0.07270a 0.05104b 0.05759 0.05747

0.05655 0.05713a 0.02928b 0.05934

0.04704b 0.05207 0.13116a 0.05123b

0.06661 0.05158 0.07202 0.06554a

Upper bound. Lower bound.

3.1. Clustering First, we normalize the data to ensure that all indices are in the same unit. Next, for cluster analysis we use the K-means method tool of the MATLAB 7.0 software. To determine the appropriate number of clusters (k), we experiment with a range of values for k. The average silhouette value 0.6841 for k = 3 is the highest in comparison with the other values of k (see Fig. 2). To overcome local minima, we used the optional ‘replicates’ parameter. The corresponding results are summarized in Table 1, where the mean value of each variable (index) is provided. On the basis of computational results, we propose the following three clusters of assets.

Table 12 Results of the W-FMOLP model.

Z1 Z2 Z3 Z4

(Return) (Risk) (Liquidity) (Suitability)

Table 14 Attainment values of the quantitative criteria.

Objective function value

Membership function value

Criteria

Value

0.06313 0.05479 0.04993 0.05925

0.62692 0.61616 0.20272 0.56042

Short term return (ST) Long term return (LT) Risk (RI) Liquidity (LI)

0.25215 0.33203 0.54833 0.00172

Table 13 The proportions of the assets in the portfolio. Assets Proportions

1 0.00000

2 0.00000

3 0.00000

4 0.00000

5 0.20000

6 0.19000

7 0.02000

8 0.00000

9 0.00000

10 0.00000

Assets Proportions

11 0.00000

12 0.00000

13 0.00000

14 0.00000

15 0.00000

16 0.01600

17 0.17000

18 0.20000

19 0.18900

20 0.01500

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P. Gupta et al. / Expert Systems with Applications 38 (2011) 5620–5632

(i) Cluster 1: liquid assets Assets in cluster 1 are categorized as liquid assets, as mean value for liquidity is the highest in this cluster. This cluster is typified by low but widely varying returns. (ii) Cluster 2: high-yield assets Assets in cluster 2 are categorized as high-yielding ones, since they have rather high returns. On the expected lines of return/risk relationship, these assets also show high standard deviation. Although, investors may profit from the high returns, they also have to endure the high risk. However, these assets have low liquidity amongst all the clusters indicating that high-yielding investment involves a longer time horizon. (iii) Cluster 3: less risky assets Assets in cluster 3 are categorized as less risky assets, since compared to other clusters, these assets manifest the lowest standard deviation for the cluster. The return is not high but medium. The liquidity is medium too.

Table 17 Pay-off table. Objective functions

Return (Z1) Risk (Z2) Liquidity (Z3) Suitability (Z4) a

Z1

Z2

Z3

Z4

Z1

Z2

Z3

Z4

3

0.06946 (0.06313 6 Z1 6 0.07270) 0.05204 (0.05104 6 Z2 6 0.05713) 0.04620 (0.02928 6 Z3 6 0.13116) 0.05619 (0.05123 6 Z4 6 0.06554)

0.06254 (0.04704 6 Z1 6 0.07270) 0.05542 (0.05479 6 Z2 6 0.05713) 0.04094 (0.02928 6 Z3 6 0.13116) 0.05569 (0.05123 6 Z4 6 0.06554)

0.06406 (0.04704 6 Z1 6 0.07270) 0.05437 (0.05104 6 Z2 6 0.05713) 0.05179 (0.02928 6 Z3 6 0.13116) 0.05985 (0.05925 6 Z4 6 0.06554)

4

5

6

0.06504 (0.04704 6 Z1 6 0.07270) 0.05386 (0.05104 6 Z2 6 0.05713) 0.05349 (0.04993 6 Z3 6 0.13116) 0.06052 (0.05925 6 Z4 6 0.06554)

0.06313 (0.06313 6 Z1 6 0.07270) 0.05527 (0.05479 6 Z2 6 0.05713) 0.03934 (0.02928 6 Z3 6 0.13116) 0.05559 (0.05123 6 Z4 6 0.06554)

0.05441 (0.04704 6 Z1 6 0.07270) 0.05592 (0.05479 6 Z2 6 0.05713) 0.06692 (0.02928 6 Z3 6 0.13116) 0.05925 (0.05925 6 Z4 6 0.06554)

Liquidity (Z3)

Suitability (Z4)

0.04030b 0.07149a 0.01744b 0.05356b

0.07111 0.04049 0.13965a 0.06410

0.06486 0.04600 0.05352 0.07678a

Table 18 Results of the W-FMOLP model.

Z1 Z2 Z3 Z4

(Return) (Risk) (Liquidity) (Suitability)

Objective function value

Membership function value

0.06209 0.05368 0.07432 0.06230

0.52207 0.55048 0.46541 0.37651

The asset clusters discussed above have a prima facie suitability for the investor types identified in Section 2.1. Thus, the cluster of liquid assets is apparently suitable for liquidity seekers; the cluster of high-yield assets is apparently suitable for return seekers; and the cluster of less risky assets is more appropriate for the safety seekers.

Iteration number 2

Risk (Z2)

0.08203a 0.03188b 0.11370 0.07298

Upper bound. Lower bound.

b

Table 15 Iterative results.

1

Return (Z1)

3.2. AHP weighted scores The AHP procedure relies on pairwise comparisons to evaluate the importance of the criteria, sub-criteria, and alternatives; refer to Fig. 1 for complete hierarchy. It may be noted that for the empirical testing of the AHP model, we have randomly chosen 20 assets from each cluster to ensure consistency in the application of the research methodology. For the data in respect of pairwise comparisons involving qualitative measures (criteria and sub-criteria), we have relied on inputs from investors that are based on the verbal scale (Saaty, 2000) listed in Table 2. For the data in respect of pairwise comparisons involving quantitative measures, the real quantitative data listed in Table 3 is used for the three clusters. It may be noted that the average returns listed in Table 3 are the average of the averages, i.e. the average monthly returns (average 12-month performance and average 36-month performance). The monthly returns in turn are based on the daily returns in respect of each of the assets. Liquidity of the assets is measured with respect to the respective turnover rates (average 36-month performance).

Table 16 The proportions of the assets in the portfolio corresponding to investors’ preferences. Proportions

ST

LT

RI

LI

Iteration Iteration Iteration Iteration Iteration Iteration

1 2 3 4 5 6

0.30591 0.23236 0.26292 0.27608 0.23750 0.20040

0.34427 0.33970 0.33105 0.32920 0.34121 0.29787

0.59813 0.53575 0.56018 0.57000 0.53587 0.51032

0.00159 0.00141 0.00178 0.00184 0.00135 0.00230

Iteration Iteration Iteration Iteration Iteration Iteration

1 2 3 4 5 6

Assets 1

2

3

4

5

6

7

8

9

10

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.02000 0.00000 0.00000 0.02000 0.02000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.20000 0.20000 0.20000 0.20000 0.20000 0.20000

0.02500 0.02500 0.19000 0.19000 0.02500 0.02500

0.02000 0.02000 0.02000 0.02000 0.02000 0.02000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

11

12

13

14

15

16

17

18

19

20

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.50000 0.00000 0.10000 0.19400 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.01600 0.01600 0.01600 0.00000 0.28411

0.17000 0.17000 0.17000 0.17000 0.17000 0.17000

0.06000 0.20000 0.20000 0.20000 0.20000 0.20000

0.01000 0.34900 0.10400 0.01000 0.35000 0.08089

0.01500 0.00000 0.00000 0.00000 0.01500 0.00000

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Table 19 The proportions of the assets in the portfolio. Assets Proportions

1 0.00000

2 0.00000

3 0.00000

4 0.00000

5 0.01900

6 0.25000

7 0.00000

8 0.00000

9 0.15600

10 0.30000

Assets Proportions

11 0.00000

12 0.00000

13 0.00000

14 0.00000

15 0.00000

16 0.02000

17 0.10000

18 0.00000

19 0.14000

20 0.01500

compared. Finally, pairwise comparisons of asset 1 through asset 20 are made with respect to various sub-criteria and liquidity criterion. The local weights (performance scores) assigned to an asset with respect to the four key criteria are found by tracing the paths that lead from the respective key criterion down to the asset, multiplying the weights of the branches in the path to determine the weight of the path, and adding these path weights together. The weights (relative importance) of the four key criteria with respect to the overall goal are obtained from their pairwise comparisons with respect to the goal. Table 4 presents the local weights of the 20 assets for cluster 1, cluster 2 and cluster 3 and also the criteria weights. For the sake of brevity, we have not included all the pairwise comparison details for calculating the AHP weights of the criteria, sub-criteria, and alternatives. However, weights used in calculation of the performance scores of the assets for all the three clusters are provided in Appendix (A1–A3).

Table 20 Attainment values of the quantitative criteria. Criteria

Value

Short term return (ST) Long term return (LT) Risk (RI) Liquidity (LI)

0.19499 0.17462 0.34066 0.00506

The pairwise comparison process used in this study moves from the top of the hierarchy down. In Fig. 1, the four key criteria are first compared with respect to the overall goal. The three subcriteria beneath the return criterion are then pairwise compared. Similarly, the sub-criteria beneath the risk criterion and the suitability criterion are pairwise compared. The various sub-criteria beneath each of the suitability sub-criteria are also pairwise

Table 21 Iterative results. Iteration number

Z1 Z2 Z3 Z4

Z1 Z2 Z3 Z4

1

2

3

0.07905 (0.06209 6 Z1 6 0.08203) 0.03712 (0.03188 6 Z2 6 0.07149) 0.09864 (0.01744 6 Z3 6 0.13965) 0.06746 (0.05356 6 Z4 6 0.07678)

0.04165 (0.04030 6 Z1 6 0.08203) 0.06909 (0.05368 6 Z2 6 0.07149) 0.03639 (0.01744 6 Z3 6 0.13965) 0.05356 (0.05356 6 Z4 6 0.07678)

0.07856 (0.06209 6 Z1 6 0.08203) 0.03629 (0.03188 6 Z2 6 0.07149) 0.09234 (0.01744 6 Z3 6 0.13965) 0.07351 (0.06230 6 Z4 6 0.07678)

4

5

6

0.07228 (0.04030 6 Z1 6 0.08203) 0.04028 (0.03188 6 Z2 6 0.07149) 0.10591 (0.07432 6 Z3 6 0.13965) 0.07239 (0.06230 6 Z4 6 0.07678)

0.07926 (0.06209 6 Z1 6 0.08203) 0.03574 (0.03188 6 Z2 6 0.07149) 0.09332 (0.07432 6 Z3 6 0.13965) 0.07216 (0.06230 6 Z4 6 0.07678)

0.05209 (0.04030 6 Z1 6 0.08203) 0.05921 (0.05368 6 Z2 6 0.07149) 0.07432 (0.07432 6 Z3 6 0.13965) 0.06608 (0.06230 6 Z4 6 0.07678)

Table 22 The proportions of assets in the portfolio corresponding to investors’ preferences. Proportions

ST

LT

RI

LI

Iteration Iteration Iteration Iteration Iteration Iteration

1 2 3 4 5 6

0.24453 0.12056 0.23753 0.21592 0.23970 0.15698

0.22259 0.13223 0.22779 0.21358 0.22976 0.15693

0.38101 0.25403 0.38697 0.37047 0.38859 0.31820

0.00671 0.00248 0.00628 0.00721 0.00635 0.00506

Iteration Iteration Iteration Iteration Iteration Iteration

1 2 3 4 5 6

Assets 1

2

3

4

5

6

7

8

9

10

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.02000 0.02000 0.02000 0.02000 0.02000 0.02000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.01900 0.01900 0.01900 0.01900 0.01900 0.00000

0.25000 0.00000 0.25000 0.25000 0.25000 0.21300

0.00000 0.02000 0.00000 0.00000 0.02000 0.02000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.21000 0.03500 0.03500 0.00000 0.20400

0.02600 0.30000 0.02600 0.02600 0.02600 0.30000

11

12

13

14

15

16

17

18

19

20

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.02300 0.00000 0.00000 0.00000 0.02300

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.02000 0.24477 0.02000 0.02000 0.02000 0.02000

0.51000 0.00000 0.61500 0.43000 0.63000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.14000 0.00000 0.00000 0.00000 0.00000 0.00000

0.01500 0.16323 0.01500 0.20000 0.01500 0.20000

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P. Gupta et al. / Expert Systems with Applications 38 (2011) 5620–5632

3.3. Asset allocation The 20 financial assets of each cluster comprise the population from which we attempt to construct a portfolio comprising 8 assets with the corresponding upper and lower bounds of capital budget allocation. The reason to construct a portfolio comprising 8 assets is based on an understanding of investor behavior as evident from the survey (Gupta et al., 2005). The survey shows that portfolio diversification by investors lies in the narrow range of 3–10 assets. In what follows, we present computational results corresponding to three types of investor behavior.  Cluster 1 for liquidity seekers. Here, we use the local weights of the assets with respect to the four key asset allocation criteria listed in Table 4. In order to define the linear MFs for the four objective functions of the FMOLP model, we solve the multi-objective problem as a single objective problem to obtain the data set for the values of the best upper bound ðZ Up Þ and the worst lower bound ðZ Lp Þ, given in the pay-off table (Table 5). For illustration purpose, we take the return criterion to show the construction of the MF according to Eq. (11):

lZ1 ðxÞ ¼

8 > < 1;

if Z 1 ðxÞ P 0:08553;

Z 1 ðxÞ0:04760 ; > 0:03793

:

0;

ð12Þ

if 0:04760 6 Z 1 ðxÞ 6 0:08553; if Z 1 ðxÞ 6 0:04760;

Next, we develop the weighted model, i.e. W-FMOLP model. The weights (hp) of the four key criteria calculated using AHP are provided in Table 4, under the heading ‘‘Criteria weight’’. These weights can also be termed as ‘‘investors’ preferences’’. Finally, we obtain portfolio selection strategy by solving W-FMOLP model using LINDO (Schrage, 1997) software. The corresponding computational results are summarized in Table 6. Table 7 presents proportion of the assets in the obtained portfolio. The achievement levels of the various objectives are consistent with the investor’s preferences. The values of the quantitative measures of asset performance corresponding to the obtained solution are given in Table 8. It is possible to further improve portfolio performance on individual objective(s) as per investors’ preferences. However, it must be noted that because of the multi-objective nature of the problem there may be a compensatory variation on the other performance

Table A1 Weight calculations of 20 assets for cluster 1 using AHP. Attribute

Subattribute

Weight

Asset weights A2

A3

A4

A5

A6

A7

A8

A9

A10

Return

ST LT FR

0.55714 0.32024 0.12262

0.01436 0.04820 0.05829

0.00158 0.02898 0.06996

0.08248 0.06473 0.08353

0.09766 0.06574 0.03398

0.10247 0.12128 0.02655

0.03299 0.04513 0.03123

0.03867 0.05017 0.03573

0.01851 0.02965 0.06009

0.09124 0.08753 0.02416

0.03811 0.07428 0.08318

Risk

SD RT MR –

0.52468 0.33377 0.14156 –

0.04838 0.04720 0.03819 0.05953

0.05489 0.07037 0.08147 0.00262

0.04939 0.08209 0.06234 0.03500

0.04786 0.04933 0.05866 0.04654

0.03220 0.02198 0.01718 0.13428

0.05001 0.03264 0.06185 0.05550

0.04625 0.03702 0.02470 0.09128

0.05359 0.06959 0.08360 0.03996

0.04005 0.02186 0.01509 0.11199

0.04455 0.07084 0.02436 0.06176

IN SO SA SR

0.22230 0.07692 0.55357 0.14721

0.09075 0.08566 0.03980 0.04534

0.08079 0.08056 0.08187 0.02540

0.08220 0.04084 0.06551 0.06181

0.03047 0.04409 0.04343 0.06535

0.01716 0.02005 0.01219 0.14857

0.03057 0.02774 0.04233 0.02406

0.03490 0.02942 0.02075 0.04403

0.05910 0.03113 0.06926 0.02362

0.02240 0.02192 0.01450 0.11641

0.08297 0.08566 0.02135 0.08662

AG DE TH R/L

0.23595 0.54536 0.10153 0.11716

0.03921 0.08604 0.04611 0.03947

0.09272 0.14502 0.01974 0.07557

0.04156 0.13788 0.02029 0.04008

0.03862 0.03502 0.02482 0.03868

0.01590 0.02003 0.11504 0.01109

0.04232 0.02903 0.04329 0.03966

0.03920 0.03172 0.09031 0.02063

0.02319 0.03275 0.02299 0.06711

0.01594 0.02283 0.11884 0.01587

0.03877 0.04365 0.04748 0.01751

HL EH ED

0.62322 0.23949 0.13729

0.04987 0.04599 0.05985

0.02586 0.02664 0.03081

0.02616 0.08116 0.03081

0.04472 0.08490 0.10865

0.01476 0.15088 0.02143

0.04987 0.02464 0.05235

0.04943 0.04440 0.10425

0.04463 0.02622 0.05912

0.02777 0.12132 0.10785

0.04914 0.08156 0.05632

Liquidity Suitability

IS 0.63335

IO 0.26050

IE 0.10616

A1

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

Return

ST LT FR

0.55714 0.32024 0.12262

0.00881 0.04028 0.04608

0.08565 0.02497 0.03958

0.07191 0.04321 0.03185

0.03070 0.04977 0.04473

0.02839 0.03900 0.02864

0.07693 0.04958 0.08749

0.07430 0.02660 0.04748

0.02022 0.04315 0.08749

0.01773 0.01832 0.04848

0.06729 0.04944 0.03148

Risk

SD RT MR –

0.52468 0.33377 0.14156 –

0.04814 0.04531 0.03976 0.01800

0.06526 0.04225 0.09351 0.01597

0.04748 0.03371 0.03960 0.05556

0.05631 0.05137 0.08707 0.05181

0.06633 0.04200 0.08145 0.00740

0.05383 0.08931 0.07637 0.01085

0.06585 0.04580 0.04889 0.01635

0.03115 0.08081 0.03416 0.09637

0.04383 0.04308 0.01260 0.00812

0.05464 0.02343 0.01914 0.08110

IN SO SA SR

0.22230 0.07692 0.55357 0.14721

0.04575 0.04203 0.03891 0.05023

0.03081 0.02667 0.10000 0.01561

0.02834 0.03048 0.04134 0.02624

0.04268 0.04969 0.06415 0.05126

0.02857 0.03700 0.09697 0.03846

0.08696 0.08162 0.05957 0.04515

0.04694 0.04598 0.09686 0.02447

0.08696 0.14440 0.01254 0.04309

0.04811 0.04335 0.01610 0.01519

0.02357 0.03170 0.06258 0.04909

AG DE TH R/L

0.23595 0.54536 0.10153 0.11716

0.04254 0.04388 0.01336 0.03872

0.10470 0.02807 0.01387 0.11279

0.04058 0.02974 0.04434 0.03606

0.07270 0.04733 0.04561 0.05967

0.09907 0.03080 0.01236 0.09989

0.07645 0.04683 0.10277 0.07234

0.09164 0.04254 0.09036 0.11329

0.01049 0.08381 0.03409 0.01125

0.01477 0.04134 0.04367 0.01762

0.05962 0.02168 0.05065 0.07269

HL EH ED

0.62322 0.23949 0.13729

0.10095 0.04560 0.03120

0.05108 0.01484 0.05488

0.05289 0.02691 0.06038

0.14258 0.04429 0.02213

0.05290 0.02455 0.05106

0.09624 0.04188 0.02447

0.05119 0.01694 0.03430

0.02590 0.04180 0.03919

0.02588 0.01597 0.02128

0.01819 0.03951 0.02966

Liquidity Suitability

IS 0.63335

IO 0.26050

IE 0.10616

5630

P. Gupta et al. / Expert Systems with Applications 38 (2011) 5620–5632

measures. After getting solution, suppose the investor is not satisfied with the first objective, i.e. Z1 (return criterion). The lower bound of the return criterion is revised with the value achieved for Z1 listed in Table 6, i.e. 0.06655. The problem W-FMOLP is resolved with the new parameters. The procedure is continued until the investor is satisfied with the obtained portfolio. The solutions of all the iterations performed are given in Table 9. Table 9 also shows the revised lower bounds of the various objective functions at each iteration. Table 10 presents attainment values of the quantitative measures of asset performance and the proportion of the assets in the obtained portfolios.  Cluster 2 for return seekers. The data set for the values of the best upper bound ðZ Up Þ and the worst lower bound ðZ Lp Þ are given in the pay-off table (Table 11). To develop the W-FMOLP model, we use the weights (hp) of the key asset allocation criteria calculated using AHP listed in Table 4. We obtain portfolio selection strategy by solving W-FMOLP model. The corresponding computational results are summarized in Table 12. Table 13 presents proportion of the assets in the obtained

portfolios. The achievement levels of the various objectives are consistent with the investor’s preferences. The values of the quantitative measures of asset performance corresponding to the obtained solution are given in Table 14. The results of the various iterations performed in order to further improve portfolio performance on individual objective(s) as per investor’s preferences are given in Table 15. Table 15 also shows the revised lower bounds of the various objective functions at each iteration. Table 16 presents attainment values of the quantitative measures of asset performance and the proportion of the assets in the obtained portfolios corresponding to the iterations performed.  Cluster 3 for safety seekers. As performed above, the data set for the values of the best upper bound ðZ Up Þ and the worst lower bound ðZ Lp Þ are given in the pay-off table (Table 17). To develop the W-FMOLP model, we use the weights (hp) of the key asset allocation criteria calculated using AHP listed in Table 4. We obtain portfolio selection strategy by solving W-FMOLP model.

Table A2 Weight calculations of 20 assets for cluster 2 using AHP. Attribute

Subattribute

Weight

A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

Return

ST LT FR

0.32024 0.55714 0.12262

0.04480 0.03597 0.02709

0.04154 0.03986 0.03781

0.05221 0.04051 0.06781

0.02445 0.03698 0.03664

0.09027 0.07479 0.07678

0.07305 0.05752 0.05489

0.03678 0.05197 0.06178

0.04469 0.05126 0.06462

0.02194 0.03196 0.03103

0.04979 0.03355 0.04149

Risk

SD RT MR –

0.54375 0.34595 0.11030 –

0.06580 0.01729 0.01768 0.00920

0.05999 0.02517 0.03606 0.03657

0.05422 0.04316 0.05191 0.03593

0.04718 0.05567 0.05883 0.03890

0.03786 0.07547 0.07454 0.10227

0.03978 0.06169 0.06934 0.06992

0.04076 0.06357 0.06703 0.18340

0.04231 0.06328 0.05471 0.12050

0.05407 0.03775 0.03457 0.05225

0.06767 0.02210 0.02053 0.01742

IN SO SA SR

0.27763 0.06346 0.55526 0.10365

0.01863 0.01858 0.01930 0.01855

0.00825 0.00914 0.00824 0.04598

0.06460 0.08790 0.06191 0.04630

0.03279 0.06855 0.05470 0.02163

0.05938 0.07138 0.05895 0.09002

0.07297 0.08549 0.06069 0.08075

0.06123 0.08092 0.05836 0.05645

0.04195 0.04289 0.04472 0.01062

0.05818 0.04453 0.07558 0.01056

0.00891 0.00822 0.00887 0.05791

AG DE TH R/L

0.24956 0.54777 0.12761 0.07506

0.01751 0.01742 0.07678 0.07210

0.04188 0.00872 0.04061 0.06964

0.09339 0.07173 0.03972 0.03573

0.04359 0.05017 0.02744 0.02326

0.08589 0.06975 0.08107 0.01836

0.08311 0.08347 0.01969 0.02518

0.06524 0.06262 0.01411 0.02865

0.08284 0.05346 0.01703 0.02603

0.03795 0.06684 0.03613 0.03463

0.00910 0.00835 0.03789 0.17002

HL EH ED

0.55714 0.12262 0.32024

0.01835 0.01845 0.09969

0.03546 0.03905 0.04921

0.03842 0.03869 0.05030

0.01980 0.02084 0.03764

0.08177 0.08465 0.02452

0.07404 0.07614 0.02497

0.06340 0.06229 0.02937

0.01161 0.05731 0.02704

0.01637 0.01033 0.04343

0.04163 0.01181 0.17371

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

Liquidity Suitability

IS 0.63335

IO 0.26050

IE 0.10616

Asset weights

Return

ST LT FR

0.32024 0.55714 0.12262

0.06203 0.05577 0.05938

0.03906 0.05490 0.04625

0.03110 0.05032 0.03862

0.08259 0.06480 0.07142

0.02192 0.04436 0.02949

0.01575 0.03936 0.02945

0.07089 0.05617 0.05687

0.04861 0.05529 0.06895

0.04649 0.06847 0.06395

0.10205 0.05619 0.03567

Risk

SD RT MR –

0.54375 0.34595 0.11030 –

0.05133 0.03786 0.03605 0.00818

0.04904 0.04837 0.05154 0.03021

0.04872 0.04438 0.05274 0.04454

0.03789 0.06359 0.05611 0.03035

0.04194 0.05134 0.05093 0.04510

0.05465 0.05882 0.05437 0.10917

0.05657 0.05892 0.04884 0.02013

0.05059 0.06758 0.06967 0.02453

0.04515 0.06566 0.06292 0.01225

0.05449 0.03833 0.03164 0.00919

IN SO SA SR

0.27763 0.06346 0.55526 0.10365

0.05261 0.06388 0.07282 0.05425

0.04307 0.05584 0.05488 0.03758

0.03897 0.03875 0.03795 0.07251

0.05759 0.03531 0.05027 0.06821

0.06802 0.04083 0.06671 0.04052

0.08862 0.05352 0.04741 0.05280

0.06213 0.05462 0.06688 0.05557

0.08267 0.06550 0.05758 0.05751

0.03520 0.03272 0.04027 0.06141

0.04423 0.04143 0.05392 0.06088

AG DE TH R/L

0.24956 0.54777 0.12761 0.07506

0.05169 0.06519 0.14825 0.03696

0.05081 0.04560 0.03797 0.03981

0.06249 0.03647 0.03807 0.03733

0.05773 0.04548 0.03586 0.02396

0.03202 0.06943 0.04244 0.02945

0.03420 0.06014 0.07790 0.03834

0.01256 0.04344 0.03727 0.07007

0.04316 0.07114 0.03965 0.07027

0.06261 0.03201 0.07341 0.07040

0.03224 0.03858 0.07869 0.07980

HL EH ED

0.55714 0.12262 0.32024

0.04435 0.07560 0.05479

0.04168 0.07705 0.05204

0.06870 0.06184 0.03353

0.07019 0.07117 0.02634

0.03553 0.03905 0.03555

0.06041 0.03245 0.05147

0.05682 0.06523 0.05038

0.06141 0.04623 0.05454

0.07531 0.06392 0.03102

0.08472 0.04790 0.05045

Liquidity Suitability

IS 0.63335

IO 0.26050

IE 0.10616

5631

P. Gupta et al. / Expert Systems with Applications 38 (2011) 5620–5632

The corresponding computational results are summarized in Table 18. Table 19 presents proportion of the assets in the obtained portfolio. The achievement levels of the various objectives are consistent with the investor’s preferences. The values of the quantitative measures of the asset performance corresponding to the obtained solution are given in Table 20. The results of the various iterations performed in order to further improve portfolio performance on individual objective(s) as per investor’s preferences are given in Table 21. Table 21 also shows the revised lower bounds of the various objective functions at each iteration. Table 22 presents attainment values of the quantitative measures of asset performance and the proportion of the assets in the obtained portfolios corresponding to the iterations performed. A comparison of the solutions listed in Tables 8, 14 and 20 highlights that if investors are liquidity seekers they will obtain a higher level of liquidity in comparison to return seekers and

safety seekers, albeit they may have to settle for similar variability of return/risk. If the investors are return seekers they will obtain a higher level of expected returns in comparison to liquidity seekers and safety seekers, but that supposes assuming a higher risk level. If the investors are safety seekers, they will obtain a lower level of risk in comparison to liquidity seekers and return seekers, but that supposes accepting medium level of expected returns. Our results indicate that the model developed here is capable of yielding optimal portfolios not only for each category of investors but can also accommodate individual preferences within each category following an interactive procedure. It may be noted that by revising lower bounds of the linear MFs, we can obtain different portfolio constructions by solving the W-FMOLP model. It is important to point out that for some choices of the MF there may be no improvement in solution with the revised lower bound(s). In such instances, we will have to modify the lower bound(s) for the various scenarios to find a satisfactory solution.

Table A3 Weight calculations of 20 assets for cluster 3 using AHP. Attribute

Subattribute

Weight

Asset weights A2

A3

A4

A5

A6

A7

A8

A9

A10

Return

ST LT FR

0.45767 0.41601 0.12632

0.03511 0.04393 0.03017

0.00968 0.05375 0.06344

0.04855 0.04808 0.03931

0.03383 0.04650 0.03772

0.07942 0.05611 0.06506

0.09269 0.08691 0.10173

0.05159 0.03392 0.01845

0.05596 0.03781 0.02292

0.04478 0.03316 0.02505

0.04369 0.03986 0.03353

Risk

SD RT MR –

0.58889 0.25185 0.15926 –

0.05092 0.04269 0.04178 0.01294

0.03866 0.02388 0.01702 0.07401

0.04710 0.04247 0.09032 0.02956

0.04791 0.04080 0.04657 0.03694

0.05275 0.05156 0.04048 0.02411

0.02702 0.01441 0.02055 0.18142

0.05452 0.08888 0.07835 0.00867

0.05267 0.04233 0.06626 0.03244

0.05798 0.09604 0.06863 0.00656

0.06133 0.09935 0.10661 0.01359

IN SO SA SR

0.22596 0.10462 0.50693 0.16250

0.02969 0.03008 0.02785 0.02733

0.09665 0.05851 0.08182 0.08436

0.09665 0.05851 0.08033 0.10046

0.05531 0.05896 0.05279 0.05201

0.00837 0.00977 0.00857 0.00926

0.09158 0.10572 0.09656 0.08267

0.01544 0.01095 0.00925 0.00896

0.05531 0.06740 0.05378 0.05201

0.09075 0.10581 0.09678 0.09323

0.02969 0.03008 0.02835 0.02733

AG DE TH R/L

0.23734 0.54536 0.10784 0.10946

0.04819 0.04656 0.02284 0.03996

0.01811 0.01226 0.04280 0.01853

0.02810 0.01652 0.04161 0.02086

0.04637 0.02501 0.04333 0.02108

0.04595 0.04396 0.07148 0.04066

0.00975 0.00874 0.11671 0.00859

0.04494 0.04671 0.01181 0.07204

0.04753 0.04436 0.01169 0.03856

0.09683 0.08611 0.01225 0.07521

0.10348 0.14811 0.01224 0.12467

HL EH ED

0.54848 0.21061 0.24091

0.02315 0.02625 0.03630

0.08239 0.04979 0.01945

0.04372 0.09076 0.06864

0.04372 0.10464 0.10560

0.02874 0.10644 0.01215

0.13703 0.08425 0.02052

0.01231 0.01253 0.01924

0.04511 0.01354 0.06290

0.01223 0.01304 0.01888

0.02385 0.01319 0.08862

Liquidity Suitability

IS 0.63335

IO 0.26050

IE 0.10616

A1

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

Return

ST LT FR

0.45767 0.41601 0.12632

0.05485 0.06303 0.07766

0.03932 0.05150 0.05173

0.03506 0.03903 0.03047

0.03974 0.04825 0.06388

0.04311 0.04005 0.03048

0.03111 0.04936 0.05023

0.08578 0.07474 0.08119

0.03762 0.05174 0.06415

0.09286 0.05234 0.07185

0.04525 0.04991 0.04099

Risk

SD RT MR –

0.58889 0.25185 0.15926 –

0.03737 0.01924 0.01953 0.08053

0.04528 0.02403 0.02644 0.03922

0.06066 0.08609 0.07075 0.01940

0.05447 0.04108 0.04218 0.06063

0.05272 0.03928 0.04403 0.01260

0.05963 0.08683 0.07441 0.02431

0.04351 0.02500 0.02301 0.06946

0.04333 0.02472 0.04139 0.03200

0.05345 0.04780 0.04024 0.09877

0.05869 0.06351 0.04146 0.14283

IN SO SA SR

0.22596 0.10462 0.50693 0.16250

0.05531 0.10497 0.09579 0.08267

0.09158 0.05953 0.07688 0.09278

0.01306 0.01513 0.01101 0.00892

0.02969 0.02981 0.02782 0.02733

0.01069 0.00786 0.00884 0.00960

0.01502 0.01565 0.01463 0.01424

0.09158 0.09627 0.10417 0.09156

0.05531 0.06036 0.05895 0.07454

0.01302 0.01425 0.00876 0.00872

0.05531 0.06036 0.05708 0.05201

AG DE TH R/L

0.23734 0.54536 0.10784 0.10946

0.01564 0.01255 0.09509 0.01247

0.02486 0.01749 0.04819 0.02065

0.08290 0.08526 0.02343 0.07591

0.04547 0.08355 0.02308 0.07591

0.04865 0.04521 0.04176 0.04038

0.02836 0.01776 0.07304 0.07521

0.02536 0.01778 0.08806 0.02065

0.04725 0.04497 0.07483 0.02272

0.09254 0.08355 0.07052 0.07788

0.09973 0.11354 0.07524 0.11807

HL EH ED

0.54848 0.21061 0.24091

0.01213 0.02747 0.09013

0.04453 0.04990 0.05461

0.04449 0.08095 0.09669

0.07867 0.02595 0.06392

0.02385 0.02773 0.09142

0.04511 0.04956 0.06388

0.07797 0.09481 0.01672

0.04511 0.02692 0.01783

0.08025 0.05179 0.03281

0.09566 0.05049 0.01968

Liquidity Suitability

IS 0.63335

IO 0.26050

IE 0.10616

5632

P. Gupta et al. / Expert Systems with Applications 38 (2011) 5620–5632

4. Conclusions The focus of the research work undertaken has been to attain the convergence of suitability and optimality in portfolio selection using AHP and FMOLP frameworks. Based on a survey data, investor behavior has been categorized as: return seekers, safety seekers and liquidity seekers. Cluster analysis has been performed to categorize the chosen sample of financial assets into three clusters corresponding to rate of return, standard deviation and liquidity. One to one mapping of investor behavior and asset clusters has been performed. AHP has been used to calculate the local weights (performance scores) of each asset with respect to the four key asset allocation criteria. These weights have been used as objective function coefficients in the FMOLP model. Using the weights (relative importance) of the key allocation criteria calculated via AHP, a weighted-additive model has been developed to attain convergence of the dual goals of suitability and optimality in investment decision-making. Numerical illustrations based on 20-asset universe from each of the three clusters have been presented to illustrate the effectiveness of the proposed portfolio selection model. Acknowledgements The first author acknowledges the research grant received under a scheme for strengthening R & D Doctoral Research Programme of University of Delhi, Delhi, India. Appendix A See Tables A1, A2, A3. References Ammar, E. E. (2008). On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem. Information Sciences, 178, 468–484. Arenas-Parra, M., Bilbao-Terol, A., & Rodríguez-Uría, M. V. (2001). A fuzzy goal programming approach to portfolio selection. European Journal of Operational Research, 133, 287–297. Bilbao-Terol, A., Pérez-Gladish, B., Arenas-Parra, M., & Rodríguez-Uría, M. V. (2006). Fuzzy compromise programming for portfolio selection. Applied Mathematics and Computation, 173, 251–264. Bolster, P. J., & Warrick, S. (2008). Matching investors with suitable optimal and investable portfolios. The Journal of Wealth Management, 10(4). doi:10.3905/ jwm.2008.701851. spring. Chen, L.-H., & Huang, L. (2009). Portfolio optimization of equity mutual funds with fuzzy return rates and risks. Expert Systems with Applications, 36, 3720–3727. Ehrgott, M., Klamroth, K., & Schwehm, C. (2004). An MCDM approach to portfolio optimization. European Journal of Operational Research, 155, 752–770.

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