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 [
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 (
This paper is organized in four sections as follows: Section
To solve the multiobjective linear programming problem, the following step-by-step strategy is used.
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 [ 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. Coefficient of variation enables the evaluation of the value of instability relative to the return rate. Earning yield is the percentage of each amount invested in the stock which the company has received. 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. 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 [
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: Investors who are willing to take only higher returns Investors who are not interested in taking more risks, even if the returns are less 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 [
Hierarchy structure.
Each entry of the judgmental matrix A is formed by the following rules:
For the matrix A of order “n” the normalized Eigenvector is called Priority vector.
The matrix is consistent if CR ≤ 0.10. However, if CR> 0.10, inconsistencies exist and pairwise comparisons need revision.
The fuzzy multiobjective linear programming (FMOLP) [
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:
Above risk function converted into linear function as optimization technique is for linear problem
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
Next, identify the best upper bound (ub) and worst lower bound (lb) for all the objectives.
The membership function for
Convert the multiobjective problem into a single objective using “weighted adaptive approach” based on AHP-criteria weight in respect of each objective.
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.
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
Cluster result.
| | | |
---|---|---|---|
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
Symbolic representations of stocks from each cluster are shown in Table
Stocks for each cluster.
| | | |
---|---|---|---|
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 |
In this segment under the criteria and subcriteria in AHP, stocks are ranked according to the investor preference. The weights are given in Table
Weight of criteria and subcriteria.
| | | |
---|---|---|---|
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
Input data for Cluster 1.
| | | | | | | |
---|---|---|---|---|---|---|---|
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 |
Input data for Cluster 2.
| | | | | | | |
---|---|---|---|---|---|---|---|
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 |
Input data for Cluster 3.
| | | | | | | |
---|---|---|---|---|---|---|---|
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 |
Upper and lower bound for each cluster are given by Table
Upper bound and lower bound.
| | | ||||
---|---|---|---|---|---|---|
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
Iterations for Cluster 1.
| | | | | | | |
---|---|---|---|---|---|---|---|
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 |
Iterations for Cluster 2.
| | | | | | | |
---|---|---|---|---|---|---|---|
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 |
Iterations for Cluster 3.
| | | | | | | |
---|---|---|---|---|---|---|---|
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 |
The numerical results for each cluster are shown in Table
Results for each cluster.
| | | |
---|---|---|---|
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.
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. [
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.
The authors declare that they have no conflicts of interest.