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An investor would like to build a balanced portfolio with stocks representing different sectors. Several researchers have attempted the portfolio selection problem by different methods. Many of these methods consider companies of different sectors together. However, it can be argued that the attributes affecting the company’s growth vary for different sectors. Therefore, it is advisable to compare a company with the companies of the same sector. There are many options for the selection of a stock from a particular sector. A stock ranking method is proposed by using MADM methods based on overall performance under a stochastic environment. Of many MADM methods, SAW, AHP, TOPSIS, and VIKOR are applied. Usually, Euclidean distances (2-norm) are considered in the implementation of TOPSIS and VIKOR methods. In this work, this norm is generalized to p-norm, where p > 1. The model is tested for 13 companies in the field of Information Technology sector (IT) listed on National Stock Exchange in India and 13 criteria as performance indicators of a company. A MATLAB GUI system is developed and the results are obtained for several values of p in case of TOPSIS and VIKOR methods besides other methods. As the result indicates, the ordering is not much affected by different values of p in certain range. Moreover, higher values of p have adverse effect on the ordering. The proposed model is able to provide better information on the overall performance of a particular stock in comparison with its peers. The results obtained by various methods clearly separate good companies from inferior companies though the exact ordering slightly differs.

Nowadays, due to the complexity and diversity involved in investments, the evaluation and ranking of companies is an important issue. It is necessary to promote a method for identifying efficient and superior firms. Ranking of stocks differentiates efficient companies from non-efficient ones. In order to optimize the return, effective selection process of stocks for portfolio investments is one of the most important decision making processes in competitive capital markets. Several researchers have investigated the problem with different perspectives.

Quah (2008) presented methodologies to select equities based on soft-computing models which focus on applying fundamental analysis for equities screening and compared the performance of three soft-computing models [

Fatma Tiryaki and Beyza Ahlatcioglu (2009) used the fuzzy AHP for the portfolio selection problem [

Sevastjanov Pavel and Dymova Ludmila (2009) suggested a new method for stock screening with the use of multiple criteria decision making and optimization [

Ricardo Giglio and Sergio Da Silva (2009) have proposed ranking of the stocks listed on Bovespa Stock Exchange according to their relative efficiency based on the algorithmic complexity theory [

The efficient portfolio can only be built when the stocks in the portfolio constitute a right mix. Therefore, selection of the best stock among the existing ones in the same industry becomes crucial. Accordingly, there is a need for a study to provide a mechanism for performance evaluation. This paper intends to develop a decision making method to assign ranks to the companies (alternatives) on the basis of several financial and non-financial (quantitative and qualitative) parameters (attributes) which affect the performance of the company. Owing to the regulator SEBI (Securities and Exchange Board of India) investors can access all the information and data of the listed companies. All the data considered in this paper is available at www.nseindia.com.

Three kinds of analyses, namely, technical analysis, bottom-up analysis, and social analysis help us understand where the company ranks among its peers. Stock price is a good indicator of a company’s financial health, if not driven by rumors or speculation. Speculation in the prices of a stock is measured by the value of beta. It is an indicator of a stock’s standard deviation, or volatility. For a technical analyst, this volatility is of worth as traders prefer stocks with high beta. But for investment purposes, fundamentals of a company, i.e. bottom-up and social analysis, should be taken into account. The present work deals with the selection of stocks for investment purpose. So, in this work, higher values of beta are discouraged by giving it smaller weightage. Basically, earnings drive stock prices. Several financial ratios, such as total income, net profit, operating profit margin, net profit margin, are useful to measure the earnings. Investors can compare these financial ratios of a company with those of other companies in the same sector. Dividends and bonus are the rewards by the companies to their shareholders for holding their stocks. Warren Buffett prefers to buy or sell a company’s stock based on its intrinsic value. Net worth and return on net worth are the indicators of the intrinsic value of a company. Promoter holding and FII + DII holding reflect the confidence of promoters and of high net worth investors respectively. Reliability is a qualitative attribute, to some extent, subjective also, in the selection process. It is based on the brand value of the products and/or services offered by the company as well as quality of the management. The effects of some of the attributes such as book value of a stock, earnings per share are reflected in the attributes, namely, net worth and net profit respectively. So, they need not to be considered separately as attributes.

The proposed method is validated by considering the data for 13 Indian IT companies for last five years [NSE]. The companies considered are TCS, HCL Tech, Wipro, Persistent Systems, Mphasis, Hexaware, Vakrangee, Infosys, KPIT Tech, Zensar Tech, NIIT Tech, Sonata Software, Mastek, in this order. The results obtained are matched with the general opinions of informed investors. As the fluctuations of currency affect every company of the IT sector and its effect will appear in net profit numbers, it is not considered as a separate attribute. The data used for promoter holding and FII + DII holding as well as bonus must be recent. However, the fluctuations in rest of the data are smoothed by taking average of five years of these data. Reliability is assigned a value in the scale of 1 to 9, the smaller the number, the lesser the reliability.

MADM is an approach employed to solve problems involving selection from among a finite number of alternatives. There are numerous applications of these methods in management science, economics, psychometrics, marketing research, applied statistics, decision theory, and many more to name a few. Of the many MADM methods, the following four commonly used methods are considered in this work.

1) Simple Additive Weight method (SAW);

2) Analytic Hierarchy Process method (AHP);

3) Technique for Order Preference by Similarity to Ideal Solution (TOPSIS);

4) Compromise Ranking method (or VIsekriterijumskoKOmpromisnoRangiranje―VIKOR).

In TOPSIS method, the distances are generalized as p-norms. In VIKOR method also, generally researchers fix the value of p as 1. In both these cases, several values of p are tested in the range [1, ∞] and their effects are examined. Critic method is used to find weight vector in TOPSIS method and eigenvalue method is used in AHP method. The GUI system developed incorporates all the above methods and assigns ranks to the companies as per the user’s choice of method.

To the best of our knowledge the present is the first report on employing MADM methods for ranking of Indian stocks.

In 1947, a book “Theory of Games and Economic Bahavior” published by von Neumann and Morgenstern opened the door to Multiple Attribute Decision-making Methods (MADM) [

The Simple Additive Weighting (SAW) method is probably the best known and widely used method for MADM. Here each attribute is given a weight, and the sum of all weights must be equal to 1. Each alternative is assessed with regard to every attribute. The composite performance score which determines the ranking is given by

Alternatives | Attributes | ||||
---|---|---|---|---|---|

B_{1} (w_{1}) | B_{2} (w_{2}) | B_{3} (w_{3}) | B_{n} (w_{n}) | ||

A_{1 } | a_{11 } | a_{12 } | a_{13} | a_{1n} | |

A_{2 } | a_{21 } | a_{22 } | a_{23} | a_{2n} | |

A_{n } | a_{n}_{1 } | a_{n}_{2 } | a_{n}_{3} | a_{nn} |

where

The analytic hierarchy process (AHP) is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It was developed by Thomas L. Saaty in the 1970s [

The n elements

The matrix

genvalue of

erage CI calculated from a large number of randomly generated reciprocal matrices. The consistency ratio (CR)

of a reciprocal matrix

acceptable, and it reflects an informed judgment attributable to the knowledge of the analyst regarding the problem under study. J. Alonso and M. Lamata derived the equation

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision analysis method, which was originally developed by Hwang and Yoon in 1981 with further developments by Yoon in 1987, and Hwang, Lai and Liu in 1993 [

As an illustration,

1) Let

2) Construct the weighted normalized decision matrix

3) Determine ideal and negative-ideal solutions.

Let

Determine the worst alternative

4) The TOPSIS method evaluates the separation measures by considering l^{2}-norm. One of the contributions in this paper is to define generalized separation measures using l^{p}-norm with real

Also, for_{¥}-norm is defined as

5) The relative closeness to the ideal solution, which will be used for the ranking of options, is calculated as in formula

Clearly,

An attempt has also been made to modify this method by using weighted Euclidean distances, rather than creating a weighted decision matrix, referred to as modified TOPSIS method, by Deng et al. (2000) [

VIKOR method was developed for multicriteria optimization of complex systems. It was originally developed by Serafim Opricovic to solve decision problems with conflicting and non-commensurable (different units) criteria in his Ph.D. dissertation in 1979 [

1) The first step is to determine the objective, and to identify the pertinent evaluation attributes. Determine the best

2) Compute the values

3) Compute the value

where

4) Rank the alternatives. Sort the values of S, R and Q in decreasing order. The results are three ranking lists. The compromise ranking list for a given v is obtained by ranking with

5) For given attribute weights, propose a compromise solution, alternative

a) Acceptable advantage:

b) Acceptable stability in decision making: Alternative

This compromise solution is stable within a decision making process, which could be: voting by majority rule (when

Usually, researchers take the value of p as 1. In this paper, other values of p have also been tried.

If the criteria are assigned weights as per the choice of decision maker, the subjectivity enters into the picture. This may result into misleading conclusions. One of the ways to remove this subjectivity is to assign data-de- pendent weights. In the present implementation, a method, called critic method, suggested by H. Kazan has been used in the TOPSIS method. Similarly, one of the several methods discussed for assigning objective weights, namely eigenvalue method, is used in AHP method by S. Gao [

In critic method [

The weight

where

The GUI software, incorporating some of the known methods and modifications suggested in this paper, makes it possible to select a small group of “good” stocks (and to reject a small group of “inferior” stocks) on the basis of different types of meaningful financial ratios considered. The results are presented in the following ^{nd} company is selected by all the methods. Similarly, 6^{th} company is not selected by only one method, namely, AHP. 4^{th} company is selected as an exception by VIKOR method that can be neglected. Similar comments apply to the rejection of inferior companies. For example, 13^{th }and 5^{th} companies are declared as inferior companies by all the methods.

As far as effects of varying p are considered, the p-TOPSIS method exhibits the uniform ranking for p in the range [

Selected screen shots as in Figures 2-4 show the actual execution process. The upper table titled “Company Information” contains the collected data which remains fixed. The column titles indicate the attributes. The attributes considered along with their abbreviations are: Total Income (TI), Net Profit (NP), Net Worth (NW), Return on Net worth (RON), Stock Price (SP), Promoter Holding (PH), FII + DII Holding (FII), Operating Profit Margin (OPM), Net Profit Margin (NPM), Dividend Payout Ratio (DPR). The user has to just select the

Name of Method | Ranking | |||||||
---|---|---|---|---|---|---|---|---|

Good | Inferior | |||||||

SAW | 2 | 6 | 1 | 7 | 13 | 5 | 12 | 10 |

TOPSIS | 2 | 6 | 7 | 1 | 13 | 5 | 3 | 12 |

p-TOPSIS (p = 4.5) | 6 | 7 | 2 | 1 | 13 | 5 | 3 | 8 |

p-TOPSIS (p = 16) | 6 | 1 | 7 | 2 | 13 | 5 | 9 | 3 |

Modified TOPSIS | 6 | 7 | 2 | 1 | 13 | 5 | 3 | 10 |

AHP | 1 | 7 | 2 | 8 | 13 | 5 | 12 | 11 |

VIKOR (p = 1) | 6 | 2 | 8 | 1 | 13 | 12 | 5 | 10 |

VIKOR (p = 2) | 6 | 8 | 2 | 4 | 13 | 12 | 7 | 5 |

VIKOR (p = 1.5) | 6 | 8 | 2 | 1 | 13 | 12 | 5 | 7 |

method among the methods listed in the table “Select the Method”. If the method requires the value of p, then a separate box will be displayed asking the user to enter the value. After this, when the user clicks the button “Decision Maker”, the results are displayed in the table titled “Output” and the four good and four inferior companies are displayed at proper place.

Collective performance assessment of n entities in the context of chosen n attributes is an important aspect in decision making in finance, engineering, social sciences, management, etc. There are several methods, called multiple attribute decision making methods, developed for this purpose. In the present note by replacing Euclidean norm by the p-norms, 1< p < ∞ in R^{n}, we have modified two of these methods: TOPSIS method and VIKOR method into p-TOPSIS and p-VIKOR methods respectively. We analyze performance of 13 companies of IT sector listed on Indian Stock Exchanges considering 13 financial and non-financial attributes using each of the MADM methods, namely, SAW, AHP, p-TOPSIS, p-VIKOR (with different values of p).

We concentrate on ranking of the first four best performing companies and the last four worst performing companies to assess stability of these methods for different values of p. It is found that for values of p in the interval [

As a way forward, other MADM methods such as ELECTRE, PROMETHEE, COPRAS will be tried for. Similarly, to delve into the ranking of entities of other asset classes is also an interesting idea. To devise a completely new method for the purpose may be considered as an excellent contribution to the theory of MADM methods.

The authors are thankful to Prof. S. J. Bhatt for suggesting p-TOPSIS method and for reading the manuscript. The work is supported by UGC-SAP-DRS-F.510/1/DRS-III/2015 (SAP), to the Department of Mathematics, S P University.

Haresh V.Dedania,Vipul R.Shah,RajeshC. Sanghvi, (2015) Portfolio Management: Stock Ranking by Multiple Attribute Decision Making Methods. Technology and Investment,06,141-150. doi: 10.4236/ti.2015.64016