Advances in Fuzzy Systems
Volume 2018 (2018), Article ID 4279236, 9 pages
https://doi.org/10.1155/2018/4279236

Research Article

Optimization of Risk and Return Using Fuzzy Multiobjective Linear Programming

Darsha Panwar, Manoj Jha, and Namita Srivastava

Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal (MP), India

Correspondence should be addressed to Darsha Panwar; panwar.darsha@gmail.com

Received 11 May 2018; Revised 25 July 2018; Accepted 14 August 2018; Published 3 September 2018

Academic Editor: Zeki Ayag

Copyright © 2018 Darsha Panwar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Stock selection poses a challenge for both the investor and the finance researcher. In this paper, a hybrid approach is proposed for asset allocation, offering a combination of several methodologies for portfolio selection, such as investor topology, cluster analysis, and the analytical hierarchy process (AHP) to facilitate ranking the assets and fuzzy multiobjective linear programming (FMOLP). This paper considers some important factors of stock, like relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), and price to earnings growth ratio (PEG ratio), apart from the risk and return and stocks which are included within these same factors. Employing fuzzy multiobjective linear programming, optimization is performed using seven objective functions viz., return, risk, relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), price to earnings growth ratio (PEG ratio), and AHP weighted score. The FMOLP transforms the multiobjective problem to a single objective problem using the “weighted adaptive approach” in which the weights are calculated by AHP or choices by the investors. The FMOLP model permits choices in solution.

1. Introduction

Due to the uncertainty of return it is not easy to select the stocks. The main aim of portfolio selection is to obtain an accurate ratio of the assets to ensure that the investor gets the maximum return with minimum risk.

Professor Markowitz initially presented the problem of portfolio selection [1]. He proposed the Markowitz model or mean-variance model (MV) for portfolio selection reiterating the fact that investing in more than one stock is less risky than investing in a single stock. Konno and Yamazaki [2] introduced an improved version of the Markowitz model in which the risk is calculated by the mean absolute deviation (MAD). Speranza [3] advanced a linear programming model, in which the risk is calculated by the semiabsolute deviation method. Gupta et al. [4] projected the hybrid approach for portfolio selection using a combination of multiple methodologies like investor’s behavioral survey, cluster analysis, analytical hierarchy process, and fuzzy mathematical programming. Ganasekaran and Ramaswami [5] proffered a portfolio optimization model applying the neurofuzzy framework. Gupta et al. [6] obtained ethical stock performance using the AHP technique and portfolio selection done by the FMCDM technique. Mehlawat [7] presented a detailed computation procedure of the AHP and applied FMCDM technique. Sanokolaei [8] proposed the fuzzy method for portfolio optimization based on the mean absolute deviation risk function. Sadati and Doniavi [9] advocated their portfolio selection model based on the possibility model with the fuzzy random variable parameter and applied the harmony search algorithm. Konak and Bagei [10] applied fuzzy linear programming for portfolio optimization. Wang et al. [11] introduced a new risk index variable called equilibrium risk value (ERV) of the random fuzzy expected value (EV) and the EV-ERV model was used for portfolio selection.

A literature survey revealed several drawbacks in the K-means algorithm used for clustering and improper scaling because it involves identification of the number of clusters. In AHP, the stocks are ranked based the criteria of return, risk, liquidity, dividend, alpha, beta and stock prices, etc.

This study presents a hybrid approach for portfolio selection with multiple methodology. First, the X-means algorithm needs to be performed for cluster analysis, which is an extended version of the K-means clustering. The drawbacks have been improved in X-means. In X-means, the number of clusters does not need to be specified. Then by applying the AHP, the stocks for all three clusters must be ranked. In this paper, some new features for stock selection have been included, such as relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), and price to earnings growth ratio (PEG ratio), which have not been used earlier in the AHP. Optimization is done using fuzzy multiobjective linear programming with seven objective functions viz., return, risk, relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), price to earnings growth ratio (PEG ratio), and AHP weighted score. The daily closing price, number of shares, turnover rate, earning per share, price to earnings ratio, price to earnings growth ratio, and market cap for all the 15 stocks selected are taken from the BSE, Bombay Stock Exchange, Mumbai, India (https://www.bseindia.com), from February ’15 to January ’16.

This paper is organized in four sections as follows: Section 2 includes an account of the research methodology, the FMOLP algorithm, and its working process with reference to each of the seven objectives, viz., return, risk, relative strength index, coefficient of variation, earning yield, price to earnings growth ratio, and AHP weighted score. Section 3 presents the numerical illustrations, while Section 4 contains the concluding remarks.

2. Methodology

To solve the multiobjective linear programming problem, the following step-by-step strategy is used.

2.1. Investor Behavior Pattern

Investor behavior plays an important role in the selection of stocks as each individual stock-holder will have a specific decision-making style. Three main categories of investors can be identified, viz., money makers, liquidity lovers, and risk averse investors, according to their investment topology [12]. The survey done above is based only on the characteristics of return, risk, and liquidity. Return, risk, and liquidity are the basic factors used in stock selection; however, some more important features, as listed below, need to be considered prior to selecting the stocks:

(i)J. Welles Wilder introduced the relative strength index in 1978. This evaluates the current and historical performance of a stock based on today’s closing prices. RSI normally falls within the 30-70 range.(ii)Coefficient of variation enables the evaluation of the value of instability relative to the return rate.(iii)Earning yield is the percentage of each amount invested in the stock which the company has received.(iv)A comparative calculation or relation between the stock price, EPS, and the growth of the companies is defined by the price to earnings growth ratio.(v)Market cap is used to classify the company size, which is of greater importance than the stock price.

2.2. Cluster

For every investor, the approaches employed in stock selection are different. Generally, however, the investors predominantly observe all the three aspects of return, risk, and liquidity. Therefore, based on these three points, stocks can be better categorized under three groups, with qualities like high return, minimum risk, and liquid stocks. Cluster analysis is a technique used to divide data into groups by which similar objects are placed within the same cluster which is different from the other cluster objects. To formulate the clusters, the X-means [13] clustering algorithm is used. It is an extended version of the K-means which attempts to automatically determine the number of clusters. It starts with just one centroid and then iteratively increases the centroid, as required. If a cluster is divided into two subclusters, then the data distribution is done using the Bayesian Information Criteria (BIC) which is a statistical model.

The proposed research includes investor topology, clustering, the AHP, and optimization technique for portfolio selection. Different investors employ different approaches for investing in the stock market. Based on the preferences, the investors are divided into three different clusters:

(a)Investors who are willing to take only higher returns(b)Investors who are not interested in taking more risks, even if the returns are less(c)Investors who are neither in favor of greater risk nor favor low returns and who only desire secure investment (liquidity lovers)

Therefore, based on these three points, stocks are divided into three groups, with qualities like high return, minimum risk, and liquid stocks.

2.3. AHP

AHP technique developed by Thomas L. Saaty [14] is a multicriteria decision-making (MCDM) tool. It has a particular application in group decision-making. Hierarchy structure design, weight analysis, and consistency proof are the three main steps of AHP for ranking the object. Figure 1 shows the 4-level hierarchy structure of AHP. Firstly, form a pair wise comparison matrix for each criterion with respect to its parent criteria.

Figure 1: Hierarchy structure.

Each entry of the judgmental matrix A is formed by the following rules:

To compare two things, we have a well-defined 1-9 scale which is given by Satty.

For the matrix A of order “n” the normalized Eigenvector is called Priority vector.

where “w” is known as weight of the objects. is the highest Eigen value of the matrix A. The consistency index (CI) for each nth order matrix is calculated as

The consistency ratio (CR) is calculated as

where RI the random index is determined by on the order of the matrix.

The matrix is consistent if CR ≤ 0.10. However, if CR> 0.10, inconsistencies exist and pairwise comparisons need revision.

2.4. Portfolio Selection Model

The fuzzy multiobjective linear programming (FMOLP) [15] technique is commonly used for optimization. The MOLP can be changed to a single objective utilizing the membership functions.

2.4.1. Portfolio Selection Problem

The multiobjective portfolio selection problem with seven objective functions such as return, risk, relative strength index, coefficient of variation, earning yield, price to earnings growth ratio, and AHP weight and some notations are introduced as follows:

 : return of the stock, : the proportion of the total fund invested in the stock, : the binary variable indicating whether the stock is contained in the portfolio or not, i.e., ,  : risk of the stock, : relative strength index of the stock, : coefficient of variation of the stock, : the AHP weight of the stock, : earning yield of the stock, : p/e growth ratio of the stock, : the maximum fraction of the stock, : the minimum fraction of the stock, : total number of stocks in each cluster , : number of stock in a selected portfolio .

2.4.2. Parameters Used

(i) Return. The return of the portfolio is written as

Where .

(ii) Risk. The semiabsolute deviation of return of the portfolio below the expected return over the past period t, , can be written as

Consequently, the expected semiabsolute deviation of return of the portfolio ) below the expected return becomes

where represents portfolio risk.

Above risk function converted into linear function as optimization technique is for linear problem

(iii) Relative Strength Index (RSI). The RSI of the portfolio is written as

where and .

(iv) Coefficient of Variation (CV). The CV of the portfolio is written as

where of the stock.

(v) Earning Yield (EY). The EY of the portfolio is written as

where of the stock.

(vi) Price to Earnings Growth Ratio (PEG Ratio). The PEG ratio of the portfolio is written as

where of the stock.

(vii) AHP Weight. The AHP weight of the portfolio is written as

where is weight of stock.

2.4.3. Constraints

Investment economical restriction on the stocks:

(i) Sum of proportion of stocks should be 1

(ii) Number of stocks held in a portfolio is

(iii) The maximum percentage of the investment which can be invested in a stock:

(iv) The minimum percentage of the investment which can be invested in a stock is

The upper and lower bounds have been taken to avoid too many large investments and in the same manner too many small investments.

2.4.4. The Decision Problem

Assuming that, after solving (18) with the constraints ((25)–(31)), the solution is , then the other objective functions are also similarly calculated at When this process is repeated for (19) through to (24) you will get seven solutions with respect to each objective.

Next, identify the best upper bound (ub) and worst lower bound (lb) for all the objectives.

The membership function for , , , , and is defined by

where is the satisfaction degree of the objective function for a given solution X.

Convert the multiobjective problem into a single objective using “weighted adaptive approach” based on AHP-criteria weight in respect of each objective.

Model I

The solution obtained on solving Model I is the first iteration. The old lower bound will be replaced by the first iteration, only when improvement is required. This process must be repeated until the investors are satisfied with the solution.

3. Numerical Illustration

The results of an experimental study built on a data set of 147 assets registered in the BSE, Mumbai, India (from February-’15 to January-’16), are as follows.

3.1. Cluster Analysis

For performing cluster analysis, X-means tool of the Rapid Miner version 5.2 software is used. And the initial distribution of first centroid is performed by K-means clustering. The result of the X-means algorithm is shown in Table 1.

Table 1: Cluster result.

Parameters
Cluster 1 (46 stocks)
Cluster 2 (78 stocks)
Cluster 3 (23stocks)

average return
0.0441
0.0154
0.0731

average risk
0.0547
0.0344
0.0744

turnover rate
0.0010
0.0005
0.0010

Category
Liquid
less risky
high return

As per the topology of investors discussed in Section 2.1,

(i)Cluster 1 includes liquid stocks as the liquidity is highest when compared with the other clusters, and risk is medium. This cluster is suitable for those investors who are interested in liquid stocks and medium risk.(ii)Cluster 2 includes high return stocks, as the average value of return is higher in comparison to the other clusters. This cluster is meant for those investors who are focused only on maximum returns.(iii)Cluster 3 contains less risky stocks, as the average risk value is low when compared with the other clusters. This cluster is good for those investors who are risk averse.

Symbolic representations of stocks from each cluster are shown in Table 2.

Table 2: Stocks for each cluster.

Symbol
Cluster 1
Cluster 2
Cluster 3

S1
Whbrady
Blue Star
Kinetic Eng.

S2
Nelco Ltd.
Great Estate
Tokyo Plast

S3
Nocil Ltd
Swaraj Engine
Force Motor

S4
Ceat Limited
Bajfinance
Kg Denim

S5
Nucleus S/w Exports Ltd.
Finolex Ind.
Zenith Fiber

S6
Sauras.Cem.
Bharat Pet.
Jenson Nicolson

S7
Fedder.Llyod
Lakshmi Mill
NIIT Ltd.

S8
Dcw Ltd.
Jsw steel
Tata Elxsi

S9
Eveready Ind. India Ltd.
Pel
Century Ext

S10
Himachal Fertilizer
Swan Eng
Jasch Indust

S11
Timex Group
Pfizer Ltd.
Medi-caps

S12
Camph.& All
Sri Adhikari Brothers Tel. Net. Ltd.
Pas.Acrylon

S13
Andhra Petro
Kajaria Cer.
Modi Rubber

S14
Sha Eng Pla
Asian Paints
Mafatlal Ind

S15
Majestic Aut
Lic Housing Finance
Panyam Cement

3.2. Numerical Calculation of AHP Weights

In this segment under the criteria and subcriteria in AHP, stocks are ranked according to the investor preference. The weights are given in Table 3.

Table 3: Weight of criteria and subcriteria.

Criteria
weight
sub-criteria
Weight

Basic factor
0.3529
Risk
0.1569

Return
0.1961

Growth factor
0.2353
PEG Ratio
0.1176

Earning Yield
0.1176

Variation factor
0.2353
Relative Strength Index
0.0642

Coefficient of Variation
0.0642

Liquidity
0.1070

Market cap
0.1765

Tables 4–6 represent the input data for all three clusters.

Table 4: Input data for Cluster 1.

Stocks
Return
Risk
RSI
CV
EY
PEG
AHP weight

S1
0.0628
0.0402
56.897
1.5707
5.61
3.24
0.1126

S2
0.0296
0.0638
50.689
5.2433
5.29
0.51
0.0466

S3
0.0369
0.0471
52.540
3.0668
6.88
0.75
0.0592

S4
0.0314
0.0544
50.626
4.8066
4.757
1.72
0.1101

S5
0.0326
0.0534
51.306
3.7950
6.04
2.5
0.0553

S6
0.0512
0.0620
51.430
3.8625
5.9
-0.97
0.0501

S7
0.0323
0.0661
50.823
4.9407
13.41
-1.23
0.0534

S8
0.0417
0.0366
52.184
2.2204
2.73
0
0.0486

S9
0.0371
0.0534
54.159
3.4756
3.86
1.3
0.0655

S10
0.0301
0.0522
50.700
3.8591
5.05
1.68
0.0713

S11
0.0673
0.0483
55.167
1.9584
0.94
39.64
0.1287

S12
0.0653
0.0486
53.923
1.7890
6.86
0.51
0.0532

S13
0.0308
0.0592
49.333
4.6172
6.6
3.12
0.0475

S14
0.0628
0.0402
56.897
1.5707
3.99
1.43
0.0552

S15
0.0556
0.0669
49.732
3.1074
5.04
0
0.0428

Table 5: Input data for Cluster 2.

Stocks
Return
Risk
RSI
CV
EY
PEG
AHP weight

S1
0.0110
0.0151
51.256
3.701
2.8
1.53
0.0443

S2
0.0008
0.0161
50.669
55.590
10.91
-4.49
0.0617

S3
0.0111
0.0172
51.076
3.810
4.99
4.94
0.0609

S4
0.0342
0.0174
44.017
1.320
5.27
1.3
0.0880

S5
0.0048
0.0188
49.703
10.388
5.76
0.86
0.0416

S6
0.0192
0.0190
52.799
2.506
7.91
0.88
0.0904

S7
0.0017
0.0197
49.607
31.856
5.33
7.06
0.0712

S8
0.0058
0.0203
50.001
9.464
8.16
1.49
0.0739

S9
0.0102
0.0213
51.640
5.128
5.87
0.51
0.0489

S10
0.0186
0.0215
52.378
2.956
0.47
-20.5
0.0739

S11
0.0128
0.0218
50.822
4.196
6.37
-3.28
0.0447

S12
0.0349
0.0220
56.296
1.718
5.08
0
0.0702

S13
0.0207
0.0229
53.777
2.986
3.47
2.36
0.0590

S14
0.0073
0.0232
51.026
7.824
2.69
4.74
0.1038

S15
0.0052
0.0235
50.436
12.596
9.08
1.03
0.0674

Table 6: Input data for Cluster 3.

Stocks
Return
Risk
RSI
CV
EY
PEG
AHP weight

S1
0.0924
0.0668
53.803
1.796
-5.03
0.00
0.0379

S2
0.0915
0.0721
53.639
1.891
6.19
0.84
0.0717

S3
0.0907
0.0873
55.482
2.482
4.68
-0.99
0.1235

S4
0.0892
0.0650
53.335
1.778
14.13
-6.02
0.0592

S5
0.0883
0.0478
59.405
1.269
15.15
-13.41
0.0569

S6
0.0860
0.0669
51.985
2.230
-8.41
0.00
0.0385

S7
0.0807
0.0797
54.253
2.446
1.27
0.00
0.0578

S8
0.0805
0.0647
56.497
1.887
5.34
0.56
0.1658

S9
0.0779
0.1038
49.735
3.147
12.92
-2.13
0.0446

S10
0.1027
0.0758
54.414
1.989
10.03
1.70
0.1056

S11
0.0740
0.0602
52.276
2.414
3.79
-3.25
0.0417

S12
0.0734
0.0689
50.528
4.414
17.19
0.41
0.0703

S13
0.0704
0.0790
52.559
2.946
1.16
-1.34
0.0328

S14
0.0701
0.0537
55.462
1.839
-1.57
0.00
0.0397

S15
0.0698
0.0794
52.630
3.182
15.67
-217.43
0.0540

3.3. FMOLP Calculation

Upper and lower bound for each cluster are given by Table 7.

Table 7: Upper bound and lower bound.

cluster 1
cluster 2
cluster 3

Objective
Ub
Lb
Ub
Lb
Ub
Lb

Return
0.0661
0.0311
0.0333
0.0014
0.0980
0.0725

Risk
0.0643
0.0385
0.0222
0.0156
0.0743
0.0511

RSI
56.7298
50.7244
55.0748
50.2265
58.0193
51.4598

CV
5.0742
1.6272
43.6201
1.7508
3.8445
1.5350

EY
10.4818
2.9446
9.9886
4.4508
16.4061
4.2379

PEG ratio
23.4067
-0.2626
5.9784
-2.1260
1.2829
-82.2285

AHP weight
0.1195
0.0507
0.0971
0.0522
0.1442
0.0526

The iterations for each cluster are given by Tables 8–10.

Table 8: Iterations for Cluster 1.

Cluster 1
f1
f2
f3
f4
f5
f6
f7

iteration 1
0.0661
0.0643
56.7298
5.0742
10.4818
23.4067
0.1195

iteration 2
0.0311
0.0385
50.7244
1.6272
2.9446
-0.2626
0.0507

iteration 3
0.0647
0.0455
55.7323
1.8668
3.1830
23.2575
0.1176

iteration 4
0.0643
0.0451
55.2060
1.7007
6.1264
2.4610
0.0774

iteration 5
0.0643
0.0451
55.2060
1.7007
6.1264
2.4610
0.0774

Table 9: Iterations for Cluster 2.

Cluster 2
f1
f2
f3
f4
f5
f6
f7

iteration 1
0.0333
0.0222
55.0748
43.6201
9.9886
5.9784
0.0971

iteration 2
0.0014
0.0156
50.2265
1.7508
4.4508
-2.1260
0.0522

iteration 3
0.0332
0.0202
51.3810
2.2922
5.1847
0.7219
0.0772

iteration 4
0.0329
0.0202
51.3191
2.4010
5.1672
0.7754
0.0782

iteration 5
0.0329
0.0202
51.3191
2.4010
5.1672
0.7754
0.0782

Table 10: Iterations for Cluster 3.

Cluster 3
f1
f2
f3
f4
f5
f6
f7

iteration 1
0.0980
0.0743
58.0193
3.8445
16.4061
1.2829
0.1442

iteration 2
0.0725
0.0511
51.4598
1.5350
4.2379
-82.2285
0.0526

iteration 3
0.0888
0.0562
56.8504
1.5181
14.1940
-9.6902
0.0628

iteration 4
0.0849
0.0737
55.8689
2.1708
5.4825
0.0042
0.1442

iteration 5
0.0901
0.0734
54.3120
2.0392
10.3194
-3.9619
0.0847

4. Assets Allocation

The numerical results for each cluster are shown in Table 11.

Table 11: Results for each cluster.

Stock
cluster 1
cluster 2
cluster 3

S1
0.0225
0
0

S2
0
0
0.3435

S3
0
0
0.0560

S4
0
0.0225
0.0225

S5
0
0
0.0225

S6
0
0.5555
0

S7
0
0.0225
0

S8
0
0
0

S9
0
0
0

S10
0
0
0.5555

S11
0.5555
0
0

S12
0.3770
0.3770
0

S13
0
0
0

S14
0.0225
0.0225
0

S15
0.0225
0
0

Thus, from the results it is clear that Cluster 1 contains the high liquidity stocks, Custer 2 includes the low risk stocks, and Cluster 3 has the high return stocks, although the main objective of minimization of risk and maximization of return is to be preserved.

4.1. Comparison

Risk/return ratio (CV) is very helpful to choosing the stocks. Investors are risk averse, as they want to consider stocks with a low risk and a high degree return.

Proposed approach gives better results as compared to the approach presented in Gupta et al. [4] as the CV (risk/return) is 0.81, 0.70, and 0.61 for Cluster 1, Cluster 2, and Cluster 3, respectively, in this model while Dr. Gupta’s [4] methodology results are 0.77, 1.12, and 0.65 for the same clusters. Based on the above results, the investor would like to invest with lower CV, since the lower value of risk/return ratio indicate a better risk-return trade-off.

5. Conclusion

This paper presented a hybrid approach that was adopted while investigating the problem of portfolio selection. The hybrid approach involved important components such as Behavior Survey, Cluster Analysis, AHP, and FMOLP. Cluster analysis is done using the X-means algorithm, which gives a better fit to the data in the clusters as the number of clusters was decided by itself. In this paper, a few new and important criteria like RSI, CV, EY, and the PEG ratio have been considered, which are very helpful for beginners and a good start for stock selection. The FMOLP transforms a multiobjective problem to a single objective one using the “weighted adaptive approach” in which the weights are calculated by the AHP or chosen by the investors. The FMOLP model permits choices in solution. The main advantage of the model proposed is—if the investor is not satisfied with the portfolio he/she can change the weights of objective functions or recalculate the AHP model—based on the preferences of the decision-maker and thus achieves improved results. This approach gives better results as risk/return ratio is lower which indicates better risk-return trade-off.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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