Advances in Fuzzy Systems

Volume 2018, Article ID 4279236, 9 pages

https://doi.org/10.1155/2018/4279236

## Optimization of Risk and Return Using Fuzzy Multiobjective Linear Programming

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

Correspondence should be addressed to Darsha Panwar; moc.liamg@ahsrad.rawnap

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.