Optimization of Risk and Return Using Fuzzy Multiobjective Linear Programming
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.
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 . 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  introduced an improved version of the Markowitz model in which the risk is calculated by the mean absolute deviation (MAD). Speranza  advanced a linear programming model, in which the risk is calculated by the semiabsolute deviation method. Gupta et al.  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  proffered a portfolio optimization model applying the neurofuzzy framework. Gupta et al.  obtained ethical stock performance using the AHP technique and portfolio selection done by the FMCDM technique. Mehlawat  presented a detailed computation procedure of the AHP and applied FMCDM technique. Sanokolaei  proposed the fuzzy method for portfolio optimization based on the mean absolute deviation risk function. Sadati and Doniavi  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  applied fuzzy linear programming for portfolio optimization. Wang et al.  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.
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 . 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.
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  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.
AHP technique developed by Thomas L. Saaty  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.
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 asThe consistency ratio (CR) is calculated aswhere 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)  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 asWhere .
(ii) Risk. The semiabsolute deviation of return of the portfolio below the expected return over the past period t, , can be written asConsequently, the expected semiabsolute deviation of return of the portfolio ) below the expected return becomeswhere 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 aswhere and .
(iv) Coefficient of Variation (CV). The CV of the portfolio is written aswhere of the stock.
(v) Earning Yield (EY). The EY of the portfolio is written aswhere of the stock.
(vi) Price to Earnings Growth Ratio (PEG Ratio). The PEG ratio of the portfolio is written aswhere of the stock.
(vii) AHP Weight. The AHP weight of the portfolio is written aswhere is weight of stock.
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 isThe 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 bywhere 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 IThe 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.
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.
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.
3.3. FMOLP Calculation
Upper and lower bound for each cluster are given by Table 7.
4. Assets Allocation
The numerical results for each cluster are shown in Table 11.
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.
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.  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  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.
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.
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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