Decision Making: Applications in Management and Engineering
Vol. 2, Issue 2, 2019, pp. 138-158.
ISSN: 2560-6018
eISSN: 2620-0104
DOI: https://doi.org/10.31181/dmame2003079b
* Corresponding author.
E-mail addresses: sanjibb@acm.org (S. Biswas), gautam.bandyopadhyay@dms.nitdgp.ac.in (G.
Bandyopadhyay), banhi.guha@gmail.com (B. Guha), malay.bhattacharjee100@gmail.com (M.
Bhattacharjee),
AN ENSEMBLE APPROACH FOR PORTFOLIO SELECTION
IN A MULTI-CRITERIA DECISION MAKING FRAMEWORK
Sanjib Biswas 1*, Gautam Bandyopadhyay 2, Banhi Guha 3 and Malay
Bhattacharjee 2*
1 Calcutta Business School, Diamond Harbour Road, Bishnupur, West Bengal, India
2 Department of Management Studies, National Institute of Technology, Durgapur, West
Bengal, India
3 Amity University, Kolkata, West Bengal, India
Received: 2 February 2019;
Accepted: 10 August 2019;
Available online: 23 August 2019.
Original scientific paper
Abstract: Investment in Mutual Funds (MF) has generated increasing interest
among the investors over last few decades as it provides an opportunity for
flexible and transparent choice of funds to diversify risk while having return
potential. MF are essentially a portfolio wherein investors’ funds are invested
in the securities traded in the capital market while sharing a common
objective. However, selection and management of different asset classes
pertaining to a particular MF are done by an active fund manager under
regulatory supervision. Hence, for an individual investor, it is important to
assess the performances of the MF before investment. Performances of MF
depend on several criteria based on risk-return measures. Hence, selection of
MF is subject to satisfying multiple criteria. In this paper, we have adopted an
ensemble approach based on a two-stage framework. Our sample consists of
the open ended equity large cap funds (direct plan) in India. In the first stage,
the efficiencies of the funds are analyzed using DEA for primary selection of
the funds. In order to rank the funds based on risk and return parameters for
investment portfolio formulation, we have used MABAC approach in the
second stage wherein criteria weights have been calculated using the Entropy
method.
Keywords: Mutual Fund, Portfolio Selection, Multi-Criteria Decision Making,
Data Envelopment Analysis (DEA), Entropy, Multi-Attribute Border
Approximation Area Comparisons (MABAC).
mailto:sanjibb@acm.org
mailto:gautam.bandyopadhyay@dms.nitdgp.ac.in
mailto:banhi.guha@gmail.com
mailto:malay.bhattacharjee100@gmail.com
An ensemble approach for portfolio selection in a multi-criteria decision making framework
139
1. Introduction
The aftermath of the economic liberalization, Indian Capital Market (ICM) has
witnessed significant changes in investment pattern. With the development of
Information and Communication Technology (ICT), there is no dearth of information
regarding available investment opportunities. Moreover, with increasing efforts from
the Govt. of India (GOI), MF have emerged as a preferred choice for common investors
because of many reasons. First, it provides reportedly high return as compared to
other popular investment options like Fixed Deposits (FD), National Savings
Certificates (NSC), Public Provident Funds (PPF), and other postal savings. Alongside,
with increasing awareness and available information, risk can also be brought down
to an affordable level by carefully evaluating the performances of the MF for the
prudent selection of funds to invest. The concept of MF resembles the selection of
portfolio wherein the active fund managers invest the total amount invested in other
asset classes such as stocks. The objective is to generate sizeable return out of the
investment made in the portfolio while minimizing the risk through diversification i.e.
by appropriate selection of the securities and allocating optimum weights dynamically
(Gupta et al., 2018). MF in India have a long stint with ICM since the inception of Unit
Trust of India (UTI) in 1963. Recent reports(AMFI, India, 2018)indicate a significant
growth in the asset base of the Indian MF Industry (IMI) from INR 5.05 trillion (31st
March, 2008) to INR 22.20 trillion (February, 2018)despite the event of bankruptcy of
the renowned US bank Lehman Brothers in September, 2008. Therefore, a large
number of investors have been attracted by the promising nature of IMI. However, in
order to ensure the possibility of considerable return at an affordable risk, one has to
select the funds apt to his/her risk appetite and financial goal.
With this preamble, in this study, we have focused on open ended equity MF(direct
plans) in India belonging to the large cap segment. The equity MF segment accounts
for around 50.7% share of the total asset base of the industry in February, 2018 as
reported by AMFI. Unlike the close ended funds (CF), open ended funds (OF) allow the
investors to buy and sell the units of the funds on a continuous basis, which means the
new investors are allowed to enter at convenience and so as the existing investors can
exit whenever needed. Moreover, for CF the unit capital is fixed and also there is a limit
on sales. In other words, OF allow greater flexibility for the investors than CF. Large
cap funds show relatively stable movements in the return as compared to the mid-cap
and small cap funds. We have considered direct plans only since, unlike regular plans,
they impose less pressure on expense ratio as no intermediate commission is involved.
In essence, we like to perceive the performance of MFs from a common investors’ point
of view without imposing significant burden on Net Assets Value (NAV). In our
approach, we have used Non-Parametric Methods (NPM). Literatures manifest
comparatively less evidence of such methods than their traditional parametric
counterparts (Babalos et al., 2011). Selection of MF needs to satisfy the objectives
pertaining to several criteria based on risk and return and time horizon, etc. Therefore,
among NPM, DEA has been a popular method, though, moderate evidences of the use
of MCDM techniques have been found in the state-of-the art. The reason lies in the
fundamental use of DEA to assess and differentiate between the efficient and non-
efficient Decision Making Units (DMU). MCDM techniques allow to rank the DMU
based on a number of criteria. Hence, for identifying efficient DMU or MF while ranking
them based on performance parameters, we propose a two-stage assessment
framework. The rest of the paper proceeds as follows. Section 2 highlights some of the
related work while in the section 3 we discuss the research methodology. Section 4
Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
140
summarizes the results and put forward necessary discussions on the same. Finally,
section 5 concludes the paper and posits some future scope of research.
2. Related Work
A plethora of research has been conducted on MF for understanding the nature and
setting performance measurement framework with an objective to inflate the
expected utility while reducing the risk level. In one of the seminal works in the stated
field, Markowitz postulated the mean-variance model related to efficiently diversified
portfolio (Markowitz, 1952). In the following works, the researchers (Tobin, 1958;
Markowitz, 1959; Sharpe, 1966; Jensen, 1968; Treynor, 1965) further explained and
extended the framework by introducing new risk measures such as semi-variance and
risk adjusted performance metrics such as reward to volatility ratio, alpha and reward
to variability ratio based on the Capital Assets Pricing Model (CAPM). The objective
was to assess portfolio performance with respect to the benchmark with an objective
to minimize the systematic risk which is represented by beta. The authors exhibited
that unsystematic risk can be reduced through diversification of the portfolio. Based
on these measures, several researchers and practitioners worked on evaluating the
performances of the MF (Kacperczyk et al., 2005; Pedersen &RudholmAlfvin, 2003;
Eling & Schuhmacher, 2007; Plantinga & de Groot, 2002; Redman et al., 2000). In the
Indian context also, across different periods, many researchers (Barua&Verma, 1991;
Jaydev, 1996; Gupta, 2000; Sehgal&Jhanwar, 2008; Tripathy, 2004; Anand &
Murugaiah, 2006; Anitha et al., 2011; Arora, 2015; Kundu, 2009) worked on selection
of funds based the criteria like Sharpe ratio, Jensen ratio, and Sortino ratio, alpha, beta,
NAV, timing to market, and selectivity skill following traditional statistical approaches.
Over the years apart from the traditional parametric approaches, applied
operations research techniques have also been adopted by the researchers. The
authors (Pendaraki et al., 2004; Sharma & Sharma, 2006) have applied goal
programming to evaluate the performances of MF for formulating the portfolio. DEA
has been a widely accepted method by the researchers and practitioners (Murthi&
Choi, 2001; Murthi et al., 1997; Anderson et al., 2004; Sengupta, 2003; Daraio & Simar,
2006; Babalos et al., 2015; McMullen & Strong, 1998; Wilkens & Zhu, 2001; Tarim&
Karan, 2001; Galagedera & Silvapulle, 2002; Chang, 2004; Carlos Matallín et al., 2014;
Nguyen-Thi-Thanh, 2006; Chu et al., 2010; Tsolas, 2011; Morey & Morey, 1999; Basso
& Funari, 2001; Briec et al., 2004; Zhao et al., 2011; Jaro & Na, 2006; Kooli et al., 2005;
Haslem & Scheraga, 2003)among the non-parametric applied operations research
techniques. The researchers have considered the variables like standard deviation,
expense ratio, loads, turnover, beta, costs, fund size, variance, percentage of periods
with negative return, lower semi-variance, sales charges, operating expenses, cash
percentage, P/E ratio, P/B ratio, total assets, lower mean, lower semi-skewness, and
excess kurtosis as input while considering the variables like return, deviations from
median return, capital flow, skewness, Sharpe ratio, upper semi-variance, upper semi-
skewness, and Jensen’s α as output in assessing the performances of the funds under
study. There has been another string of the literature in which MCDM techniques are
applied for selection of MF (Pendaraki et al., 2005; Lin et al., 2007; Gladish et al., 2007;
Chang et al., 2010; Babalos et al., 2011; Alptekin, 2009; Karmakar et al., 2018;
Pendaraki & Zopounidis, 2003; Sielska, 2010). Attribute based classification
approaches like UTADIS (UTilités Additives DIScriminantes) (Pendaraki et al., 2005;
Lin et al., 2007), fuzzy MCDM techniques (Gladish et al., 2007), distance based MCDM
methods like TOPSIS (Lin et al., 2007; Chang et al., 2010; Alptekin, 2009; Karmakar
An ensemble approach for portfolio selection in a multi-criteria decision making framework
141
et al., 2018; Sielska, 2010) and EDAS (Karmakar et al., 2018), outranking methods
such as PROMETHEE (Sielska, 2010) and PROMETHEE II (Pendaraki&Zopounidis,
2003) have been selected for portfolio selection issue based on the parameters like
Sharpe ratio, Sortinoratio, Treynorratio, Jensen’s α, AUM, beta, standard deviation,
NAV, annualized return, average return, Information ratio, and R-squared. In this
context, in paper (Babalos et al., 2011) the authors used Stochastic Multi-criteria
Acceptability Analysis (SMAA-2) framework for assessing performances of MF.
Predominantly, objective weight method using Euclidean Distance has been applied
for calculating criteria weight. However, some authors like Chang et al. (2010)
experimented with different distance measures for examining performances of the
MF.
3. Data and Methodology
3.1. Development of the framework
In this study, we have followed a non-parametric two stage framework wherein we
have ranked the funds under study. In the first stage, we have applied DEA to appraise
the efficiencies of the funds for a primary level selection. For a refined selection in a
common setting for investment choice among the relatively efficient portfolios, at
stage two the MCDM technique like MABAC has been used. The entropy method has
been employed for calculation of criteria weight in this regard. Figure 1 depicts the
framework followed in this study. We have started our analysis with total 48 number
of open ended equity large cap funds under direct plan.
In the seminal works of Markowitz (1952, 1959), the underline assumption
considered first two central moments of the utility function of the return which is
treated as a normal distribution. However, the studies (Lau et al., 1990; Cambell &
Hentsche, 1992) arguably reported that portfolio returns are always not normally
distributed in practice. Hence, it is imperative to consider higher moments. Average
investors prefer lower value of Kurtosis (Scott & Horvath, 1980) as it entails a higher
degree of sensitivity of the funds with respect to non-favorable market condition. In
view of this, for filling the gap in the literature in Indian context, this study considers
Kurtosis as one of the inputs. Expense ratio puts load on the profitability as it covers
the management fees and operating expense.
The third quartile return has a typical significance in the sense that it indicates
relative closeness to the highest value than the average return. The ratio PSV to NSV
signifies inclination of the deviation of the return from the mean towards the higher
side which essentially acts as a favorable proposition for the investors. We have used
standard Variable Returns to Scale (VRS) model as it resembles real life situation
compared to Constant Returns to Scale (CRS) which reflects the proportionate change
in the output with respect to the input (Ali & Seiford, 1990). Further, we have
calculated super efficiency in order to discriminate the funds to a considerable extent.
Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
142
Figure 1. Research Framework
The higher value of net assets of a fund indicates the greater possibility of a good
return. The Sharpe ratio, α, and Sortino ratio specify the excess return with respect to
risk, i.e. risk-adjusted performance while beta captures the systematic risk. To what
extent, the portfolio return with respect to the benchmark, i.e. index, adjusts the
volatility of the return is reflected in the value of Information ratio which stands
beneficial for the investors. The value of R-squared manifests the nature of
diversification of the portfolio, which leads to reduction of unsystematic risk.
3.2. Sample
In our study, we focus on open ended and large cap equity mutual funds under
direct plan. In this context, we have excluded the fixed maturity plan, and plans
suspended for sales which otherwise leaves lesser chance to gain more at calculated
risk. For spotting the funds, we have referred the Value research online data base and
subsequently for collecting information on the performance criteria. Appendix 1
shows the descriptive statistics of the total 48 funds selected initially for the study.
Returns of the last 12 quarters (i.e. Sep 2015 to Jun 2018) have been considered for
calculating distribution based parameters used in DEA.
3.3. Data Envelopment Analysis (DEA)
DEA evaluates a set of peer entities (homogenous) i.e. Decision Making Units
(DMUs) having multiple inputs or outputs. As introduced by Charnes et al. (1978), this
technique has gained extensive importance by the researchers in measuring
performance efficiency in terms of the frontiers or envelop rather than the central
tendency as in the case of fitting a regression model. In DEA, the efficiency of the
homogenous entities is calculated by using the linear programming method.
Calculations for input oriented, constant return to scale (CRS) are as follows:
min 𝜃
Subject to:
∑ xij
n
j=1 λj ≤ θxit i = 1,2, … , m; Input Constraint (1)
∑ yrj
n
j=1 λj ≥ yrt r = 1,2, … , s; Output Constraint (2)
Where, 𝜆𝑗 ≥ 0 ∀ 𝑖, 𝑗, 𝑟
For variable return to scale (VRS) the sets of equations are:
min 𝜃
Stage 1
DEA
Input/ Output Variables
Input: Kurtosis; Expense Ratio
Output: Third quartile return; ratio
between positive semi-variance (PSV) to
negative semi-variance (NSV)
Stage 2
MABAC
Efficiency based selection
Final
Ranking
Portfolio
Selection
Criteria
Net Assets, 5-Year Annualized
Return, Sharpe Ratio, Sortino
Ratio, Information Ratio,
Alpha, R-Squared, and Beta.
Entropy
Method for
Criteria
Weight
Funds under study
An ensemble approach for portfolio selection in a multi-criteria decision making framework
143
Subject to:
∑ xij
n
j=1 λj ≤ θxit i = 1,2, … , m; Input Constraint (3)
∑ yrj
n
j=1 λj ≥ yrt r = 1,2, … , s; Output Constraint (4)
Where, ∑ λj
n
j=1 = 1 ; λj ≥ 0 ∀ i, j, r
If two or more DMUs are found to be efficient (i.e. θ = 1 or 100%) then the Super-
efficiency value is calculated to discriminate them. For VRS the super efficiency is
calculated as below:
min 𝜃
Subject to
∑ xij
n
j=1 λj ≤ θxit i = 1,2, … , m; Input Constraint (5)
∑ yrj
n
j=1 λj ≥ yrt r = 1,2, … , s; Output Constraint (6)
Where, ∑ λj
n
j=1 = 1; λj ≥ 0 ∀ i, j, r ; j ≠ t.
3.4. Entropy Method
It is an objective method for calculating criteria weights based on relative
information content wherein, higher value of the entropy (degree of disorder)
indicates more information content for the respective criterion (Shannon, 1948). The
steps are as given below where,
𝑎𝑖 : i
th alternative where i = 1,2,3,…..m;
𝑐𝑗 : j
th criterion where j = 1,2,3,…..n;
𝑥𝑖𝑗 : j
th criterion value for the ith alternative;
Step1. Normalization of the criteria.
For this purpose, we have used the enhanced accuracy method of normalization
Zeng et al. (2013) as mentioned in (Jahan & Edwards, 2015). Accordingly, the
normalized matrix R = [𝑟𝑖𝑗 ]𝑚 ×𝑛
is given by:
𝑟𝑖𝑗 = 1-
𝑥𝑗
𝑚𝑎𝑥
−𝑥𝑖𝑗
∑ (𝑥𝑗
𝑚𝑎𝑥−𝑥𝑖𝑗)
𝑚
𝑖=1
(Beneficial Criteria) (7)
𝑟𝑖𝑗 = 1-
𝑥𝑖𝑗−𝑥𝑗
𝑚𝑖𝑛
∑ (𝑥𝑖𝑗−𝑥𝑗
𝑚𝑖𝑛)𝑚𝑖=1
(Non-beneficial Criteria) (8)
Step 2. Entropy calculation for the criterion
Entropy of the jth criterion is given by:
𝐻𝑗 = -
∑ 𝑓𝑖𝑗 ln 𝑓𝑖𝑗
𝑚
𝑖=1
ln 𝑚
, 𝑖 = 1,2, … 𝑚; 𝑗 = 1,2, . . 𝑛 (9)
Where,
𝑓𝑖𝑗 =
𝑟𝑖𝑗
∑ 𝑟𝑖𝑗
𝑚
𝑖
, 𝑖 = 1, 2, … 𝑚; 𝑗 = 1,2, … (10)
Step 3. Calculation of the entropy weight for the criterion
Entropy weight of the jth criterion is determined by:
𝑤𝑗 =
1−𝐻𝑗
𝑛−∑ 𝐻𝑗
𝑛
𝑗=1
, where ∑ 𝑤𝑗
𝑛
𝑗=1 = 1 (11)
Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
144
3.5. Multi-Attribute Border Approximation Area Comparisons (MABAC)
Since its proposal (Pamučar & Ćirović, 2015), this method has drawn significant
attention from the researchers for its inherent computational ease and stability. Unlike
TOPSIS, this method classifies the performances of the criteria into two areas such as
Upper Approximation Area (UAA) for ideal solutions and Lower Approximation Area
(LAA) for non-ideal solutions instead of calculating the distance of any solution from
the ideal and non-ideal solutions. In a sense, this method examines the relative
strength and weakness of each alternative with respect to the others pertaining to
each criterion (Roy et al., 2016). This method has been widely applied by the
researchers for solving multi-criteria decision making problems such as railway
management (Sharma et al., 2018; Vesković et al., 2018)medical tourism site selection
(Roy et al., 2018) and selection of hotels (Yu et al., 2017).
Let, D is the initial decision matrix represented by
𝑎𝑖 : i
th alternative where i = 1,2,3,…..m;
𝑐𝑗 : j
th criterion where j = 1,2,3,…..n;
𝑥𝑖𝑗 : j
th criterion value for the ith alternative;
The broad steps under this method are given below.
Step 1. Normalization of the criteria values
𝑟𝑖𝑗 =
(𝑥𝑖𝑗− 𝑥𝑖
−)
(𝑥𝑖
+− 𝑥𝑖
− )
; For beneficial criteria (12)
𝑟𝑖𝑗 =
(𝑥𝑖𝑗− 𝑥𝑖
+
)
(𝑥𝑖
−− 𝑥𝑖
+ )
; For non-beneficial criteria (13)
Where, 𝑥𝑖
+ and 𝑥𝑖
− are the maximum and minimum criteria values respectively.
Step 2. Construction of weighted normalization matrix (Y)
Elements of Y are given by:
𝑦𝑖𝑗 = 𝑤𝑗 (𝑟𝑖𝑗 + 1) ; Where, 𝑤𝑗 are the criteria weight. (14)
Step 3. Determination of the Border Approximation Area (BAA)
The elements of the Border Approximation Area (BAA) T is given by:
𝑡𝑗 = (∏ 𝑦𝑖𝑗
𝑚
𝑖=1 )
1/𝑚
(15)
Where, m is the total number of alternatives &𝑡𝑗 corresponds to each criterion.
Step 4. Calculation of the matrix Q related to the separation of the alternatives from
BAA
Q = Y-T (16)
A particular alternative 𝑎𝑖 is said to be belonging to the Upper Approximation Area
(UAA) i.e. 𝑇+ if 𝑞𝑖𝑗 > 0 or Lower Approximation Area (LAA) i.e. 𝑇
− if 𝑞𝑖𝑗 < 0or BAA i.e.
T if 𝑞𝑖𝑗 = 0.
The alternative 𝑎𝑖 is considered to be the best among the others if more numbers
of criteria pertaining to it possibly belong to 𝑇+.
Step 5. Ranking of the alternatives
It is done according to the final values of the criterion functions as given by
Si = ∑ qij
n
j=1 for j = 1,2, … n and i = 1,2, … m (17)
Higher the value, the better is the rank.
In this study for carrying out DEA, we have used Lingo (version 11) software while
for MCDM related calculations, Microsoft Office excel (version 2010) is utilized.
An ensemble approach for portfolio selection in a multi-criteria decision making framework
145
4. Results and discussions
We have considered total 48 funds initially. However, before analyzing them using
DEA, we have checked whether the condition of the required number of DMUs is
satisfied or not. In this regard, though there are several studies, we have followed one
of the widely accepted study conducted by Banker et al. (1989). According to the study,
the rule of the thumb is n >= max {p × q, 3(p + q)} where, p be the number of inputs
and q is the number of outputs used in the analysis, and n is the number of DMUs to be
considered. Our model has two inputs and two outputs. Hence, it satisfies the
condition. The top 20 funds (i.e. DMUs) based on the result of DEA considering VRS
model is given in the table 1 while the details is included in the appendix 2.
The performance criteria values of the above funds (Table 1) are given in the
appendix 1. We then have used MCDM model for ranking the above mentioned funds.
For calculating criteria weights, we have used a modified Entropy method. The results
are listed in tables 2-3. After calculating the criteria weights, we then proceed to the
stage 2 i.e. ranking of the funds (primary selection through DEA) using the MABAC
technique. The results are given in tables 4-6.
From the result, it is evident that the top five funds (i.e. A27, A19, A38, A22 and
A25; the names are given in appendix 1) are rated 4-star and 5-star by Value research
and except one of them, and their risk grades are above average or more. On the other
hand, bottom five funds (i.e. A40, A42, A21, A3 and A5; the names are given in
appendix 1) are rated 2-star or below and having an overall average risk grade. Hence,
this study also conforms to the market based rating of the funds by Value research.
In order to check the dependability of the result obtained from MABAC, we have
also ranked the funds selected from DEA result using TOPSIS technique. In line with
the method suggested by Hwang and Yoon (1981), we have obtained the rankings as
given in the table 7. For checking consistency with MABAC based ranking, we have
performed Spearman’s Rank Correlation test using IBM SPSS 22, which is 0.952
(significant at 0.01 level). Hence, the result obtained from MABAC is acceptable.
Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
146
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An ensemble approach for portfolio selection in a multi-criteria decision making framework
147
Table 2. Normalization Table
Alt. C1 C2 C3 C4 C5 C6 C7 C8
A3 0.9450 0.9338 0.9315 0.9315 0.8427 0.9350 1.0000 0.9600
A5 0.9477 0.9561 0.8910 0.8883 0.9688 0.9165 0.8951 0.9850
A10 0.9450 0.9446 0.9688 0.9694 0.9866 0.9611 0.9161 1.0000
A13 0.9453 0.9381 0.9564 0.9514 0.9839 0.9499 0.9930 0.9600
A14 0.9460 0.9406 0.9470 0.9477 0.9304 0.9453 1.0000 0.9600
A15 1.0000 0.9627 0.9315 0.9333 0.9831 0.9350 0.9510 0.8650
A19 0.9454 0.9834 0.9844 1.0000 0.9914 0.9815 0.7832 0.9800
A21 0.9453 0.9359 0.9377 0.9369 0.8585 0.9389 1.0000 0.9550
A22 0.9446 0.9809 0.9782 0.9928 0.9902 0.9784 0.7762 0.9850
A24 0.9449 0.9400 0.9470 0.9459 0.9346 0.9446 1.0000 0.9550
A25 0.9459 0.9509 0.9751 0.9712 0.9917 0.9655 0.9231 0.9550
A27 0.9446 1.0000 1.0000 0.9946 1.0000 1.0000 0.8531 0.7850
A32 0.9445 0.9314 0.9315 0.9279 0.9634 0.9363 0.9930 0.9600
A36 0.9449 0.9383 0.9377 0.9405 0.9144 0.9399 1.0000 0.9550
A37 0.9445 0.9333 0.9470 0.9441 0.9756 0.9452 0.9930 0.9700
A38 0.9841 0.9808 0.9502 0.9423 0.9848 0.9491 0.9301 0.9400
A40 0.9456 0.9371 0.9439 0.9459 0.8882 0.9435 1.0000 0.9550
A42 0.9445 0.9375 0.9439 0.9441 0.8936 0.9426 1.0000 0.9550
A43 0.9445 0.9345 0.9470 0.9441 0.9774 0.9457 0.9930 0.9600
A48 0.9478 0.9401 0.9502 0.9477 0.9405 0.9459 1.0000 0.9600
Table 3. Hj values and criteria weights
Hj 0.748 0.748 0.748 0.748 0.747 0.748 0.747 0.747
wj 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
Table 4. Normalization Result
Alt. C1 C2 C3 C4 C5 C6 C7 C8
A3 0.009 0.035 0.371 0.387 0.000 0.221 1.000 0.814
A5 0.058 0.359 0.000 0.000 0.802 0.000 0.531 0.930
A10 0.009 0.193 0.714 0.726 0.915 0.534 0.625 1.000
A13 0.016 0.098 0.600 0.565 0.898 0.399 0.969 0.814
A14 0.028 0.134 0.514 0.532 0.558 0.345 1.000 0.814
A15 1.000 0.456 0.371 0.403 0.892 0.221 0.781 0.372
A19 0.017 0.759 0.857 1.000 0.945 0.779 0.031 0.907
A21 0.014 0.065 0.429 0.435 0.100 0.268 1.000 0.791
A22 0.003 0.722 0.800 0.935 0.938 0.741 0.000 0.930
A24 0.007 0.125 0.514 0.516 0.584 0.336 1.000 0.791
A25 0.026 0.284 0.771 0.742 0.947 0.587 0.656 0.791
A27 0.002 1.000 1.000 0.952 1.000 1.000 0.344 0.000
A32 0.001 0.000 0.371 0.355 0.767 0.237 0.969 0.814
A36 0.008 0.100 0.429 0.468 0.456 0.280 1.000 0.791
A37 0.000 0.028 0.514 0.500 0.845 0.344 0.969 0.860
A38 0.714 0.720 0.543 0.484 0.904 0.390 0.688 0.721
Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
148
Alt. C1 C2 C3 C4 C5 C6 C7 C8
A40 0.021 0.083 0.486 0.516 0.289 0.324 1.000 0.791
A42 0.000 0.088 0.486 0.500 0.323 0.313 1.000 0.791
A43 0.000 0.045 0.514 0.500 0.856 0.350 0.969 0.814
A48 0.061 0.126 0.543 0.532 0.622 0.351 1.000 0.814
Table 5. Border Approximation Area Matrix
BAA 0.1347 0.1552 0.1907 0.1919 0.2066 0.1730 0.2181 0.2189
Table 6. Ranking of the Funds
Fund A3 A5 A10 A13 A14 A15 A19 A21 A22 A24
Sum (Si) -0.134 -0.154 0.101 0.056 0.002 0.073 0.173 -0.101 0.144 -0.005
Rank 19 20 6 8 12 7 2 18 4 13
Fund A25 A27 A32 A36 A37 A38 A40 A42 A43 A48
Sum (Si) 0.111 0.173 -0.050 -0.048 0.019 0.156 -0.050 -0.051 0.017 0.017
Rank 5 1 15 14 9 3 16 17 11 10
Table 7. TOPSIS Ranking
Fund A3 A5 A10 A13 A14 A15 A19 A21 A22 A24
TOPSIS Rank 20 15 7 8 12 3 4 19 5 13
Fund A25 A27 A32 A36 A37 A38 A40 A42 A43 A48
TOPSIS Rank 6 1 14 16 10 2 17 18 9 11
5. Conclusion
We have made an attempt to assess the funds from two perspectives such as
efficiency and performance. Accordingly, we have filtrated the funds through a two
stage process using DEA at stage 1 and MABAC at stage 2. The rationale behind this
study lies in the selection of the funds to form an investment portfolio based on their
return distribution and performance parameters encompassing risk-return tradeoff.
In effect, this study not only has adjudged the funds on efficiency dimension, but also
sets out to establish a ranking based on risk-return criteria. Our results conform to the
market based rating of the funds. A combination of DEA-Entropy-MABAC turns this
study considerably different from the existing contributions in the Indian context as
far as the approach is concerned. The results of this study provide the investors a
broader perspective for selection of the portfolio. However, future research shall be
required to focus more clinical approach by considering fundamental parameters and
stock level analysis. It is important to analyze the stocks on which the fund managers
invest the amount invested by the investors both on fundamental dimension and
organizational dimensions to ascertain the decision and establish a causal relationship
among the stock performance and MF performance. Further, the efficiencies of the
fund houses need to be examined. There are requirements to investigate the
relationship between investors’ sentiments and market performance of the MF. Also,
a consistency between the performances of the funds belonging to different categories
can be thought of. Finally, the framework used in this study can be further explored
for different other applications.
An ensemble approach for portfolio selection in a multi-criteria decision making framework
149
Acknowledgement: The authors would like to express sincere gratitude to all
reviewers for providing valuable comments for further improvement. The first draft
of this paper has been presented during the First International Conference on
“Frontiers of Operations Research & Business Studies (FORBS 2018)” organized by
Calcutta Business School in collaboration with Operational Research Society of India
(ORSI), Kolkata Chapter (Oct 11-13, 2018) and subsequently some modifications have
been incorporated in response to the reviewers’ comments.
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Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
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An ensemble approach for portfolio selection in a multi-criteria decision making framework
157
Appendix 2. Calculation of efficiency using DEA
Funds
under
study
Input Output VRS
result
Rank
VRS Kurtosis Kurtosis
(normalized)
Expense
Ratio
PSV/
NSV
Q3
DMU1 -0.4904 0.5013 1.2600 1.0549 6.6575 16.07% 31
DMU2 -0.7490 0.3612 1.3100 1.0393 7.1600 13.37% 36
DMU3 -0.9942 0.2284 0.5100 0.6925 6.3250 27.70% 20
DMU4 -0.5445 0.4720 0.9400 0.5358 7.4000 16.62% 30
DMU5 0.4301 1.0000 1.0500 1.1908 7.5125 41.96% 17
DMU6 0.2381 0.8960 1.4800 0.7074 5.9725 9.31% 43
DMU7 -0.0014 0.7662 1.5100 0.7634 5.5975 9.27% 44
DMU8 -0.9583 0.2478 1.4700 0.6868 6.0125 10.06% 41
DMU9 -0.5300 0.4798 0.5800 0.6661 6.8850 22.95% 25
DMU10 0.0687 0.8043 2.0900 1.7130 6.0650 100% 7
DMU11 -0.5220 0.4842 1.1600 0.7371 4.7950 12.24% 38
DMU12 -1.0308 0.2085 0.6400 0.7048 6.3925 22.51% 26
DMU13 -1.3921 0.0128 0.1500 1.0249 7.3475 152.72% 2
DMU14 -1.0697 0.1874 0.1500 0.6978 6.6575 83.42% 11
DMU15 -0.9863 0.2327 1.2600 1.6584 8.5375 100% 7
DMU16 -0.5595 0.4639 1.5100 0.7604 6.7950 9.57% 42
DMU17 -1.2878 0.0693 1.1700 0.9866 6.7800 15.06% 32
DMU18 -1.0226 0.2130 0.5700 0.7052 6.5200 25.08% 22
DMU19 -0.8677 0.2969 0.4400 1.1672 9.8025 146.00% 3
DMU20 -0.0496 0.7401 1.0600 1.0686 6.1950 21.37% 27
DMU21 -1.0472 0.1996 0.4300 0.7086 6.5225 32.76% 18
DMU22 -0.9146 0.2715 0.5600 1.1773 9.7425 102.00% 6
DMU23 -1.0924 0.1751 1.8300 0.5602 6.7950 8.18% 46
DMU24 -1.0202 0.2143 0.1700 0.7021 6.5925 73.49% 13
DMU25 -0.7569 0.3569 0.7500 0.7040 8.0900 31.69% 19
DMU26 -0.7229 0.3754 0.9600 0.6534 6.3100 14.85% 33
DMU27 -1.3407 0.0406 2.9400 0.7030 12.0225 100% 7
DMU28 -0.6121 0.4354 1.6100 0.7645 3.8550 9.03% 45
DMU29 -0.5916 0.4465 1.1500 1.0974 5.9675 24.09% 24
DMU30 -0.3986 0.5510 2.0400 0.7255 6.5800 7.13% 47
DMU31 -1.0855 0.1789 0.6900 0.6874 6.5025 21.11% 28
DMU32 -1.4157 0.0000 1.1500 1.0384 7.1800 100% 7
DMU33 -1.3228 0.0504 1.5200 0.5927 5.8700 14.05% 35
DMU34 -0.8733 0.2939 1.1500 0.6444 7.5100 14.71% 34
DMU35 -0.7201 0.3768 0.5000 0.7821 4.9900 26.93% 21
DMU36 -0.9688 0.2421 0.2900 0.7066 6.6100 45.86% 16
DMU37 -1.4146 0.0006 0.2900 1.0276 7.0450 303.43% 1
DMU38 -0.9501 0.2523 1.3200 1.5284 7.0700 78.19% 12
DMU39 -0.6039 0.4398 1.1800 0.8370 7.2125 12.26% 37
DMU40 -1.0468 0.1999 0.2900 0.7034 6.6075 46.89% 15
DMU41 -0.1160 0.7042 0.5500 0.8305 5.9050 24.10% 23
DMU42 -1.0639 0.1906 0.2100 0.7021 6.6850 62.64% 14
Biwas et al/Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 138-158
158
Funds
under
study
Input Output VRS
result
Rank
VRS Kurtosis Kurtosis
(normalized)
Expense
Ratio
PSV/
NSV
Q3
DMU43 -1.4037 0.0065 0.2200 1.0335 7.2400 141.03% 4
DMU44 -0.8221 0.3216 0.7400 0.6315 6.6900 19.13% 29
DMU45 -1.0498 0.1983 2.1500 0.4023 6.5025 6.97% 48
DMU46 -0.6233 0.4293 1.2200 0.7896 6.4950 11.76% 39
DMU47 -0.8883 0.2857 1.4300 0.7594 5.8050 10.29% 40
DMU48 -1.0725 0.1860 0.1200 0.6911 6.5875 125.00% 5