“Ranking of firms by performance using I-distance method”

The objective of this article is to rank firms by their financial performance using statisti- cal I-distance method, which has the ability to determine both ranking and important factors. For this purpose, the method was first applied to 110 Turkish industrial firms without any sectorial separation and then to 7 different sectors, and various findings about firms, sectors and variables were obtained. The I-distance method is used to get rid of the high correlation between variables during the analysis. The reason for choosing the I-distance method is that it allows you to sort the variables by importance and eliminate insignificant variables, as well as take into account correlations between variables. The authors believe that the method is superior to other alternative methods thanks to these qualities. Through a number of analyses, it was possible to see positions of firms both within the whole sample and their own sectors. Furthermore, this method provided valuable information on which factors were important in assessing firms’ financial per- formance. It has been observed in the analyses that the most effective factors in ranking firms and separating them from each other were profitability ratios, and the fact that liquidity and financial leverage ratios are not effective at all. When examined from a sec-toral perspective, the nonmetal mining sector and the chemical, petroleum and plastic sectors seem to be better than other sectors in the performance rankings.


INTRODUCTION
One of the most important objectives of enterprises is to meet the expectations of their shareholders at the maximum level by increasing their market value. It is of vital importance for firms to achieve this objective in today's increasingly competitive environment. In particular, firms operating in the same sector need to make the right decisions so as to increase their competitiveness in the national and international area. While these decisions are made based on the experience of the management, the past financial information of the enterprise is also of great significance. This is because the performance of any firm can be evaluated with the help of the financial ratios that are calculated using the firm's financial information.
There are various indicators that can be used for financial analyses with the aim of examining the current status of firms and making comparisons. By using these indicators, which are mostly proportional, it is possible to acquire information about the liquidity status of firms, the usage status of their assets, their financial structures, profitability statuses and market values. The analysis results provide detailed information about the performance of the firm in terms of managers, shareholders, credit institutions, customers and investors.
The performance evaluation of the firms can be based on a single variable or multiple variables. Some examples of evaluation based on When we look at the studies in the literature made for the purpose of performance evaluation, no full consistence can be observed regarding the variable selection. Although the selection of variables varies relatively per sector, it seems that arbitrary choices are made in general. However, the first and the most important stage of a quantitative research is correct selection of the variables. As we mentioned above, many studies that have been conducted up to this point have used multivariate mathematical methods. The main reasons for such extensive use of these methods are: not requiring any assumption, having a simple theoretical structure and arbitrary selection of variables. Since selection of the analysis variables is based on subjective value judgments in such mathematical methods, it is possible to encounter very different variables in different studies on the same subject. In addition, these methods cannot take the relationships between variables into account sufficiently.
Excessive number of variables used in any research can complicate the analysis and increase the workload. On the other hand, if some important variables are excluded from the research, a considerable amount of information would be ignored, which prevents the analysis findings from being reliable.
High correlations between explanatory variables in statistical analyses are certainly undesirable, because only one of the highly correlated variables already contains the information necessary for the research and the duplication of the same information leads to erroneous results.
Following the above explanations, when selecting variables in any research, it is very important to include as much information as possible, on the one hand, and to avoid unnecessary and repetitive knowledge, on the other hand.
In this study, we have used a number of indicators to take into account as much information as possible while assessing Turkish industrial enterprises according to their financial performance. We select the I-distance method to get rid of the high correlation between variables during our analysis. The reason for choosing the I-distance method is that it allows you to sort the variables by importance and eliminate insignificant variables, as well as take into account correlations between variables. We believe that the method is superior to other alternative methods thanks to these qualities. Different methods sometimes form similar and sometimes different sequences. But based on the findings, it is not logical to judge which of these is better. These methods can only be compared to one another according to their theoretical bases or easiness of calculation. For this reason, instead of making a choice between sequences generated by different methods in such studies, they are seen to be combined into a single sequence using methods such as arithmetic mean or Borda rule (Borda, 1781). 1 The list is compiled and published annually by Istanbul Chamber of Industry. 2 The list is compiled and published annually by Fortune magazine.
In the next part of our study, we are primarily concerned with major and new works performed on the evaluation of firm performance. Then, we describe the I-distance method and the variables of the research in detail. We sort the firms included in the Turkish Industry Index in their entirety and in their sectors, respectively, and simultaneously determine important variables, per I-distance method. In this study, although I-distance ranking is accepted as the basis, firms are ranked with some other methods as well. For this purpose, weights of the financial ratios used in the study are calculated by ENTROPY method. The firms are then ranked according to their financial performance with TOPSIS and VIKOR methods. We finalize our study by interpreting the findings of the analysis and making sector and method comparisons.

METHODS
Generally, financial ratios are used to evaluate the financial performance of an enterprise. These ratios provide important information about the liquidity, operating efficiency, financial structure, profitability and market value of the enterprise. The ratios we use in this study are provided in Table 1 and grouped as follows (Brigham & Houston, 2007;Brealey et al., 2001).
1. Liquidity ratios: measures the ability of firms to meet their short-term debt obligations. The main ratios in this group are the current ratio, acid test ratio and cash ratio.
• Current ratio: it shows the ability of firms to cover short-term debts of their current assets so as to continue their activities. It is assumed that the ideal value is generally around 2, although it varies depending on the state of the sector.
• Acid test (liquidity) ratio: represents the ability of the remaining current assets to cover the enterprise's short-term debts after stocks are excluded from the current assets. It is desirable that this ratio is around 1.
• Cash ratio: shows the capacity of the enterprise to pay short-term debts when the entity's receivables cannot be collected and sales decrease, and it is desirable that it be around 0.20.
2. Activity ratios: are useful to determine whether enterprises are actively using their assets. The main ratios in this group are the total assets turnover ratio, receivable turnover ratio, inventory turnover rate and equity turnover ratio.
• Total assets turnover ratio: is used to measure the efficiency of the enterprise's assets. A low rate indicates that an enterprise cannot effectively use its assets.
• Receivable turnover rate: shows the ability of the enterprise to collect its receivables. A high ratio means that the collection of receivables is not problematic and that it has an efficient collection policy.
• Inventory turnover ratio: shows how effectively an enterprise turns its inventory into sales. A high rate indicates that the enterprise has an effective inventory policy and that it has increased its ability to compete with its competitors.
• Equity turnover ratio: this is used to determine the effectiveness of the investments financed by the shareholders of the enterprise. A high ratio indicates that the funds that shareholders have allocated to the firm are used efficiently.
3. Financial leverage ratios: this group of ratios shows the firm's capital structure, i.e. its ability to meet its long-term debt obligations. The main ratios in this group are the leverage ratio, the debt-equity ratio and long-term debt ratio.
• Leverage ratio: shows how much of the firm assets are covered by debt. A higher rate indicates greater financial risk for the firm.
• Debt-equity ratio: shows the relationship between the firm's foreign assets and its own equity capital.
• Long-term debt ratio: shows how much the enterprise has long-term debts in its total debts.
4. Profitability ratios: shows the firm's ability to make profits over its sales, assets and equity. The main ratios in this group are the return on assets, net profit margin, gross profit margin, return on equity, operating margin, and earnings before interests and taxes margin. It is desirable that these ratios are as high as possible.
• Return on assets (ROA): shows how effectively the firm's assets are being used.
• Net profit margin: shows how much profit the firm has in proportion to net sales after taxes.
• Gross profit margin: shows how much gross profit the firm has in proportion to net sales.
• Return on equity (ROE): this shows the return of the shareholders on their capital invested in the firm.
• Operating margin: shows how profitable business operations are.
• Earnings before interests and taxes margin (EBIT): shows the profitability of the enterprises before interest, depreciation and tax compared to net sales.
5. Market value ratios: these rates, which are generally applicable to public firms, show the effect of the stocks of the enterprises on the market value. They are considered to be reflecting the investors' thoughts about past and future performance of the enterprise. For this reason, other market value ratios except for the earning per share ratio are somewhat subjective. Hence, we use only the earning per share ratio of the market value ratios in this research.
• Earnings per share (EPS): this ratio is one of the most important factors determining the market value of a firm. Using this ratio, an investor can see how much profit one makes per each share of the firm owned.
We have only used earnings per share as a market value ratio, as can be seen in Table 1. However, there are also other indicators under this main group, such as market value/book value and price to earnings ratio. But since they are of interest only to stock investors and, based on subjective evaluations, we did not include them in this study.
In the I-distance method, the distance of each unit to a reference unit is calculated. This reference unit may be a hypothetical unit with the best value, the worst value or the average value for each variable.
The I-distance value depends on the calculation sequence of the variables. That is to say, the discriminant value of the most important variable (which provides the most information) should be calculated first, and then other variables are to be added according to the significance order.
The partial correlation coefficients shown in equation (1) are intended to prevent duplication (being used repeatedly) of the information contained in variables. However, the presence of negative partial correlations necessarily leads to duplication.
To avoid this, I-squared distance given in equation (2) below should be used instead of I-distance in practice:

ANALYSIS AND FINDINGS
In this study, 110 companies from 144 manufacturing companies in BIST Industry Index were ranked in terms of their performance based on 2015 data using a multi-variable method. Due to the fact that the data are not available or bad, we have not included 34 firms in the analysis. We used various financial ratios as variables in the ranking. We have selected seventeen variables from the main groups of liquidity, activity, financial leverage, profitability and market value ratios. First, we calculated the I 2 -distance values for each firm and ranked the firms accordingly with the assumption that there are no correlations between variables. We assumed that all variables are of equal importance at this stage. I 2 -distance shows the square of the distance of firms from a reference point. In our study, we accepted a hypothetical firm with the best value for each variable as a reference point. The best values are the maximum values for 12 of the 17 variables and the fixed values as described in Table 1 for the remaining 5. They are 2 for current ratio, 1 for liquidity ratio, 0.20 for cash ratio, 0.50 for leverage ratio and 1 for debt-equity ratio. According to the results provided in Table 2, the top 5 firms with the best performance at this stage are ranked as EGEEN, TUPRS, POLTK, BAGFAS and ADEL.
We looked at the simple linear correlation coefficients between the rankings and the variables in Table 2 in order to exclude the insignificant variables from the analysis by specifying the significance levels of the variables. Table 3 shows  the results of this review. As you can see, 11 out of 17 variables were significant at the level of 0.01. Liquidity ratio, equity turnover ratio, long-term debt ratio, leverage ratio, debt-equity ratio and cash ratio are permanently excluded from the analysis, because they are insignificant, i.e. they do not have any distinctive effect on firm performance ranking. The magnitude of the correlation coefficients between variables and ranking also informs us of the order in which the variables should be introduced into the distance calculation in the next step. Accordingly, the most important variable in the second phase is return on equity, and it should be included first in the calculation.
The current ratio with the correlation coefficient of 0.03929 in the second phase of our analysis, and the total assets turnover ratio with the correlation coefficient of 0.11887 in the third phase were insignificant. Table 4 shows the I 2 -distance values calculated based on the remaining 9 variables and the final firm ranking. The top 5 firms were ranked as EGEEN, DGZTE, ADEL, BAGFS and FMIZP. While the three companies were still in the top, DGZTE, which was the 8 th in the initial ranking, rose to the second position and FMISP, which was the 51 st , rose to the fifth place. These changes reveal the misleading results of the presence of the variables that are gradually excluded from the analysis.
The final variables and correlation coefficients provided in Table 5 indicate that all the remaining variables are significant. It is clear that the most important variable here is return on assets, and the least important is the inventory turnover ratio. Although there is a significant difference between the correlation coefficients for these two variables, both are significant at the level of 0.01 due to the size of our sample volume. We have conducted performance evaluations of firms in the BIST Industry Index without introducing any subsector divisions, and then applied  the same analyses separately to each sector and obtained the findings shown in Table 6. The subsectors to which the firms belong are provided in the appendix A. When Table 6 is examined, it could be seen that some variables are not significant in any sector. These are long term debt ratio, debtequity ratio and equity turnover ratio. According to Table 5, these variables were not found to be significant in the total rankings of BIST industry index firms as well. Another noteworthy point in Table 6 is that ROA, ROE and net profit margin variables were significant with high correlation values in all the sectors. In addition, variables that are significant in six of the 7 sectors are operating margin and EBIT. According to these findings, we can say that the profitability ratios of the firms are determinant in the in-sector performance evaluations of firms.
When the sectors are examined separately, it could be seen that in the metal products and machine sector, firm rankings were calculated according to 11 of the 17 variables, whereas in the textile, leathver sector only 5 variables were used. This can be explained by the fact that the firms in the textile sector have more similar values than the firms in the other sectors in terms of many financial ratios.
For comparison purposes, the firms are ranked with TOPSIS (Hwang & Yoon, 1981) and VIKOR (Opricovic & Tzeng, 2004), in addition to I 2distance method. The variable weights required for these methods were obtained by ENTROPY (Wang & Lee, 2009). When the weights of the variables in Table 7 are compared with Table 5, there are remarkable differences. For example, liquidity ratio, cash ratio and debt-equity ratio, which are not included in Table 5, since they are excluded from analysis, are the most important variables by ENTROPY meth- od. Reason for this is that, in I-distance method, the probable effects of these variables are already made by other variables in the analysis. To put it more precisely, since the ENTROPY method does not eliminate the variables, the weights determined by it do show only the absolute importance of each variable. On the contrary, since the I-distance method eliminates variables by considering partial correlations between them, the remaining variables include discrimination effects of some variables that are eliminated, as well as their own effects.
TOPSIS and VIKOR results obtained by using ENTROPY weights are presented in Tables 8 and  9, respectively. The TOPSIS method determined EGEEN, DGZTE, CMBTN, TUPRS and ATKAR as the top 5 firms, while VIKOR method placed ADEL, TTRAK and EGEEN in the top row. As shown in Table 9, the VIKOR method places them in the same order of superiority, as it can not find sufficient evidence to distinguish some firms from each other. As a result, the resulting image resembles a sort of clustering rather than ranking.
In the examinations made, the greatest difference between I 2 -distance and TOPSIS was seen for FMZIP firm. FMZIP was ranked 109 th by TOPSIS while taking the 5 th place by I 2 -distance method. The VIKOR method placed this firm into the 10 th group.
As another example, BAGFS firm is in the 4 th place in I 2 -distance, is in the 104 th place in TOPSIS and in the 2 nd group in VIKOR. The reason for these different findings can be explained by the variables considered and their significance ratings, but it may be useful to evaluate these firms in detail for each variable to determine which the correct rank is. Even though there are individual differences, matrix of serial correlation is established in Table 10 to see if there is any truly meaningful difference between the sequences of the methods. According to this matrix, it is seen that the sequence formed by I 2 -distance has a meaningful and positive relation with ones formed by both TOPSIS and VIKOR.

CONCLUSION
In this study, the I 2 -distance method was used to evaluate the financial performance of Turkish industrial firms based on multiple variables. Through a number of analyses, the positions of firms both within the whole sample and within their own sectors could be seen. Furthermore, this method provided valuable information on which factors were important in assessing firms' financial performance.
The I 2 -distance method was compared with TOPSIS and VIKOR methods which are among many methods that can be used in evaluation and ranking problems. The ENTROPY method was used to determine the significance levels of the variables in this comparative study, and it was found that there  was no significant difference between the general rankings, even though there were small differences between the firms' ranks. Despite this similarity between the findings of the methods, the I 2 -distance method is concluded to be superior to the other methods, because it takes into account correlations between variables, eliminates unnecessary variables, and thus reduces the number of variables the researcher has to deal with.
According to the I 2 -distance method, considered to be a powerful, reliable and useful method, 16 of the top 20 firms in the total ranking are from 3 sectors, which are the chemical, petroleum, plastic, metal products, machine and nonmetal mining product. Very few of the firms in the other sectors entered the top 20. In terms of the average proximity to the reference points of the sectors, the chemical, petroleum, plastic sector and the nonmetal mining product sector are the closest, while wood, paper, printing, and basic metal are the most distant sectors. Profitability ratios were highly decisive in the total ranking, whereas liquidity and financial leverage ratios had no influence whatsoever.
In separate evaluations of sectors, the variables that are significant differ according to the sectors. The number of variables here ranges from 5 to 11. In these analyses, ROA, ROE and net profit margin proved to be the most important variables.
Firms, which at the bottom in the financial performance ranking, may consider this as a danger signal and they can make new decisions to improve their situation after determining which financial ratios cause this problem. More detailed financial interpretations of the above findings are beyond the objective and scope of this study. However, an improvement that can be made in the implementation part of our study appears in the specification of the reference point in the I 2 -distance method. While we used the maximum values in determining the reference points for most of the variables in our study, we used fixed values for some, i.e. for current ratio, acid test (liquidity) ratio, cash ratio, leverage ratio and debtequity ratio. This is because these values are required to be around a specific, fixed value rather than high or low values. The word "around" here is in fact very appropriate for the concept of fuzziness. For this reason, a fuzzy I 2 -distance method can be developed by taking such fixed values as fuzzy.   -SASA  -TMSN  -KUTPO  ---SODA  -TOASO  -MRDIN  ---TMPOL  -TTRAK  -NUHCM  ---TUPRS  -ULUSE  -TRKCM  -----VESBE  -UNYEC  -----VESTL  -USAK  --